### Nuprl Lemma : qinv_wf

`∀[r:ℚ]. 1/r ∈ ℚ supposing ¬↑qeq(r;0)`

Proof

Definitions occuring in Statement :  qinv: `1/r` rationals: `ℚ` qeq: `qeq(r;s)` assert: `↑b` uimplies: `b supposing a` uall: `∀[x:A]. B[x]` not: `¬A` member: `t ∈ T` natural_number: `\$n`
Definitions unfolded in proof :  uall: `∀[x:A]. B[x]` uimplies: `b supposing a` member: `t ∈ T` rationals: `ℚ` all: `∀x:A. B[x]` quotient: `x,y:A//B[x; y]` and: `P ∧ Q` subtype_rel: `A ⊆r B` squash: `↓T` prop: `ℙ` so_lambda: `λ2x y.t[x; y]` so_apply: `x[s1;s2]` sq_type: `SQType(T)` implies: `P `` Q` guard: `{T}` true: `True` b-union: `A ⋃ B` tunion: `⋃x:A.B[x]` bool: `𝔹` unit: `Unit` ifthenelse: `if b then t else f fi ` pi2: `snd(t)` not: `¬A` false: `False` qeq: `qeq(r;s)` callbyvalueall: callbyvalueall evalall: `evalall(t)` btrue: `tt` qinv: `1/r` has-value: `(a)↓` has-valueall: `has-valueall(a)` uiff: `uiff(P;Q)` so_lambda: `λ2x.t[x]` so_apply: `x[s]` bfalse: `ff` rev_uimplies: `rev_uimplies(P;Q)` int_nzero: `ℤ-o` nequal: `a ≠ b ∈ T ` decidable: `Dec(P)` or: `P ∨ Q` satisfiable_int_formula: `satisfiable_int_formula(fmla)` exists: `∃x:A. B[x]` top: `Top`
Lemmas referenced :  rationals_wf b-union_wf int_nzero_wf bool_wf qeq_wf qeq_refl qeq-functionality subtype_rel_b-union-left member_wf squash_wf true_wf istype-universe quotient-member-eq qeq-equiv qinv-wf subtype_base_sq bool_subtype_base equal-wf-T-base assert_wf istype-void valueall-type-has-valueall int-valueall-type evalall-reduce eqtt_to_assert assert_of_eq_int int_subtype_base product-valueall-type eq_int_wf set-valueall-type nequal_wf int_nzero_properties decidable__equal_int full-omega-unsat intformnot_wf intformeq_wf itermMultiply_wf itermVar_wf itermConstant_wf istype-int int_formula_prop_not_lemma int_formula_prop_eq_lemma int_term_value_mul_lemma int_term_value_var_lemma int_term_value_constant_lemma int_formula_prop_wf intformand_wf int_formula_prop_and_lemma equal_wf not_wf int-subtype-rationals qeq-wf
Rules used in proof :  sqequalSubstitution sqequalTransitivity computationStep sqequalReflexivity isect_memberFormation_alt introduction cut sqequalRule sqequalHypSubstitution axiomEquality equalityTransitivity hypothesis equalitySymmetry universeIsType extract_by_obid isectElimination thin intEquality productEquality promote_hyp lambdaFormation_alt equalityIsType3 hypothesisEquality baseClosed inhabitedIsType pointwiseFunctionality pertypeElimination productElimination natural_numberEquality applyEquality independent_isectElimination productIsType equalityIsType4 dependent_functionElimination lambdaEquality_alt imageElimination universeEquality because_Cache instantiate cumulativity independent_functionElimination imageMemberEquality unionElimination equalityElimination functionIsType equalityIsType1 callbyvalueReduce sqleReflexivity isintReduceTrue independent_pairEquality multiplyEquality setElimination rename approximateComputation dependent_pairFormation_alt int_eqEquality isect_memberEquality_alt voidElimination independent_pairFormation lambdaFormation

Latex:
\mforall{}[r:\mBbbQ{}].  1/r  \mmember{}  \mBbbQ{}  supposing  \mneg{}\muparrow{}qeq(r;0)

Date html generated: 2019_10_16-AM-11_47_12
Last ObjectModification: 2018_10_11-PM-01_25_28

Theory : rationals

Home Index