Nuprl Lemma : qmul_com

[r,s:ℚ].  ((r s) (s r) ∈ ℚ)


Proof




Definitions occuring in Statement :  qmul: s rationals: uall: [x:A]. B[x] equal: 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: supposing a all: x:A. B[x] exists: x:A. B[x] nat_plus: + cand: c∧ B not: ¬A subtype_rel: A ⊆B prop: qdiv: (r/s) top: Top ifthenelse: if then else fi  btrue: tt mk-rational: mk-rational(a;b) int_nzero: -o nequal: a ≠ b ∈  implies:  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


Home Index