Nuprl Lemma : qmul_assoc

[r,s,t:ℚ].  (((r s) t) (r t) ∈ 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 implies:  Q subtype_rel: A ⊆B iff: ⇐⇒ Q int_nzero: -o nequal: a ≠ b ∈  satisfiable_int_formula: satisfiable_int_formula(fmla) false: False prop: qmul: s qeq: qeq(r;s) so_lambda: λ2x.t[x] so_apply: x[s] callbyvalueall: callbyvalueall has-value: (a)↓ has-valueall: has-valueall(a) ifthenelse: if then else fi  bfalse: ff decidable: Dec(P) or: P ∨ Q
Lemmas referenced :  assert-qeq qmul_wf q-elim nat_plus_properties iff_weakening_uiff assert_wf qeq_wf2 int-subtype-rationals equal-wf-base rationals_wf int_subtype_base istype-assert qdiv-int-elim full-omega-unsat intformand_wf intformeq_wf itermVar_wf itermConstant_wf intformless_wf istype-int 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 valueall-type-has-valueall product-valueall-type int-valueall-type evalall-reduce 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_alt cut introduction extract_by_obid sqequalHypSubstitution isectElimination thin hypothesisEquality hypothesis productElimination independent_pairFormation independent_isectElimination dependent_functionElimination setElimination rename lambdaFormation_alt independent_functionElimination applyEquality sqequalRule closedConclusion natural_numberEquality baseClosed because_Cache dependent_set_memberEquality_alt approximateComputation dependent_pairFormation_alt lambdaEquality_alt int_eqEquality Error :memTop,  universeIsType voidElimination equalityIstype inhabitedIsType sqequalBase equalitySymmetry intEquality productEquality independent_pairEquality callbyvalueReduce multiplyEquality unionElimination hyp_replacement applyLambdaEquality

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



Date html generated: 2020_05_20-AM-09_13_22
Last ObjectModification: 2020_01_24-PM-03_56_21

Theory : rationals


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