### Nuprl Lemma : inverse-rpower

`∀[x:ℝ]. ∀[n:ℕ]. ((r1/x^n) = (r1/x)^n) supposing x ≠ r0`

Proof

Definitions occuring in Statement :  rdiv: `(x/y)` rneq: `x ≠ y` rnexp: `x^k1` req: `x = y` int-to-real: `r(n)` real: `ℝ` nat: `ℕ` uimplies: `b supposing a` uall: `∀[x:A]. B[x]` natural_number: `\$n`
Definitions unfolded in proof :  uall: `∀[x:A]. B[x]` member: `t ∈ T` uimplies: `b supposing a` all: `∀x:A. B[x]` implies: `P `` Q` prop: `ℙ` nat: `ℕ` false: `False` ge: `i ≥ j ` not: `¬A` satisfiable_int_formula: `satisfiable_int_formula(fmla)` exists: `∃x:A. B[x]` top: `Top` and: `P ∧ Q` rat_term_to_real: `rat_term_to_real(f;t)` rtermConstant: `"const"` rat_term_ind: rat_term_ind pi1: `fst(t)` true: `True` rtermDivide: `num "/" denom` rneq: `x ≠ y` guard: `{T}` or: `P ∨ Q` iff: `P `⇐⇒` Q` rev_implies: `P `` Q` less_than: `a < b` squash: `↓T` less_than': `less_than'(a;b)` pi2: `snd(t)` decidable: `Dec(P)` bool: `𝔹` unit: `Unit` it: `⋅` btrue: `tt` uiff: `uiff(P;Q)` bfalse: `ff` sq_type: `SQType(T)` bnot: `¬bb` ifthenelse: `if b then t else f fi ` assert: `↑b` nequal: `a ≠ b ∈ T ` rev_uimplies: `rev_uimplies(P;Q)` rtermMultiply: `left "*" right` rtermVar: `rtermVar(var)`
Lemmas referenced :  rpower-nonzero req_witness rdiv_wf int-to-real_wf rnexp_wf istype-nat rneq_wf real_wf nat_properties full-omega-unsat intformand_wf intformle_wf itermConstant_wf itermVar_wf intformless_wf istype-int int_formula_prop_and_lemma istype-void int_formula_prop_le_lemma int_term_value_constant_lemma int_term_value_var_lemma int_formula_prop_less_lemma int_formula_prop_wf ge_wf istype-less_than rnexp_zero_lemma assert-rat-term-eq2 rtermDivide_wf rtermConstant_wf rless-int rless_wf decidable__le intformnot_wf int_formula_prop_not_lemma istype-le subtract-1-ge-0 ifthenelse_wf eq_int_wf rmul_wf subtract_wf itermSubtract_wf int_term_value_subtract_lemma eqtt_to_assert assert_of_eq_int intformeq_wf int_formula_prop_eq_lemma eqff_to_assert bool_cases_sqequal subtype_base_sq bool_wf bool_subtype_base assert-bnot neg_assert_of_eq_int rneq_functionality rnexp-req req_weakening req_functionality rdiv_functionality rtermMultiply_wf rtermVar_wf rmul_functionality req_inversion
Rules used in proof :  sqequalSubstitution sqequalTransitivity computationStep sqequalReflexivity isect_memberFormation_alt introduction cut lambdaFormation_alt extract_by_obid sqequalHypSubstitution dependent_functionElimination thin hypothesisEquality independent_functionElimination hypothesis inhabitedIsType isectElimination closedConclusion natural_numberEquality independent_isectElimination sqequalRule isect_memberEquality_alt because_Cache isectIsTypeImplies universeIsType setElimination rename intWeakElimination approximateComputation dependent_pairFormation_alt lambdaEquality_alt int_eqEquality voidElimination independent_pairFormation functionIsTypeImplies inrFormation_alt productElimination imageMemberEquality baseClosed dependent_set_memberEquality_alt unionElimination equalityElimination equalityTransitivity equalitySymmetry equalityIstype promote_hyp instantiate cumulativity

Latex:
\mforall{}[x:\mBbbR{}].  \mforall{}[n:\mBbbN{}].  ((r1/x\^{}n)  =  (r1/x)\^{}n)  supposing  x  \mneq{}  r0

Date html generated: 2019_10_29-AM-09_58_04
Last ObjectModification: 2019_04_01-PM-11_19_47

Theory : reals

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