### Nuprl Lemma : qmul_com

`∀[r,s:ℚ].  ((r * s) = (s * r) ∈ ℚ)`

Proof

Definitions occuring in Statement :  qmul: `r * s` rationals: `ℚ` uall: `∀[x:A]. B[x]` equal: `s = t ∈ T`
Definitions unfolded in proof :  uall: `∀[x:A]. B[x]` member: `t ∈ T` uiff: `uiff(P;Q)` and: `P ∧ Q` rev_uimplies: `rev_uimplies(P;Q)` uimplies: `b supposing a` all: `∀x:A. B[x]` exists: `∃x:A. B[x]` nat_plus: `ℕ+` cand: `A c∧ B` not: `¬A` subtype_rel: `A ⊆r B` prop: `ℙ` qdiv: `(r/s)` top: `Top` ifthenelse: `if b then t else f fi ` btrue: `tt` mk-rational: `mk-rational(a;b)` int_nzero: `ℤ-o` nequal: `a ≠ b ∈ T ` implies: `P `` Q` satisfiable_int_formula: `satisfiable_int_formula(fmla)` false: `False` bfalse: `ff` decidable: `Dec(P)` or: `P ∨ Q`
Lemmas referenced :  assert-qeq qmul_wf q-elim nat_plus_properties int-subtype-rationals assert_wf qeq_wf2 not_wf equal-wf-base rationals_wf int_subtype_base qinv-elim qmul-elim isint-int mk-rational_wf satisfiable-full-omega-tt intformand_wf intformeq_wf itermVar_wf itermConstant_wf intformless_wf int_formula_prop_and_lemma int_formula_prop_eq_lemma int_term_value_var_lemma int_term_value_constant_lemma int_formula_prop_less_lemma int_formula_prop_wf nequal_wf qeq-elim mul_nzero assert_of_eq_int decidable__equal_int intformnot_wf itermMultiply_wf int_formula_prop_not_lemma int_term_value_mul_lemma qdiv_wf
Rules used in proof :  sqequalSubstitution sqequalTransitivity computationStep sqequalReflexivity isect_memberFormation introduction cut extract_by_obid sqequalHypSubstitution isectElimination thin hypothesisEquality hypothesis productElimination independent_pairFormation independent_isectElimination dependent_functionElimination setElimination rename addLevel impliesFunctionality applyEquality sqequalRule natural_numberEquality because_Cache baseClosed isect_memberEquality voidElimination voidEquality dependent_set_memberEquality lambdaFormation dependent_pairFormation lambdaEquality int_eqEquality intEquality computeAll multiplyEquality isintReduceTrue unionElimination hyp_replacement equalitySymmetry Error :applyLambdaEquality,  axiomEquality

Latex:
\mforall{}[r,s:\mBbbQ{}].    ((r  *  s)  =  (s  *  r))

Date html generated: 2016_10_25-AM-11_50_51
Last ObjectModification: 2016_07_12-AM-07_47_33

Theory : rationals

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