### Nuprl Lemma : rmul_preserves_req

`∀[x,y,z:ℝ].  uiff(x = z;(x * y) = (z * y)) supposing y ≠ r0`

Proof

Definitions occuring in Statement :  rneq: `x ≠ y` req: `x = y` rmul: `a * b` int-to-real: `r(n)` real: `ℝ` uiff: `uiff(P;Q)` 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` uiff: `uiff(P;Q)` and: `P ∧ Q` rev_uimplies: `rev_uimplies(P;Q)` implies: `P `` Q` prop: `ℙ` rdiv: `(x/y)` all: `∀x:A. B[x]` req_int_terms: `t1 ≡ t2` false: `False` not: `¬A` top: `Top`
Lemmas referenced :  req_functionality rmul_wf rmul_functionality req_weakening req_witness req_wf rneq_wf int-to-real_wf real_wf rdiv_wf rinv_wf2 itermSubtract_wf itermMultiply_wf itermVar_wf itermConstant_wf req_transitivity rmul-rinv rmul-rinv3 req-iff-rsub-is-0 real_polynomial_null istype-int real_term_value_sub_lemma istype-void real_term_value_mul_lemma real_term_value_var_lemma real_term_value_const_lemma
Rules used in proof :  sqequalSubstitution sqequalTransitivity computationStep sqequalReflexivity isect_memberFormation_alt introduction cut independent_pairFormation extract_by_obid sqequalHypSubstitution isectElimination thin hypothesisEquality hypothesis because_Cache independent_isectElimination productElimination independent_functionElimination universeIsType sqequalRule independent_pairEquality isect_memberEquality_alt isectIsTypeImplies inhabitedIsType natural_numberEquality dependent_functionElimination approximateComputation lambdaEquality_alt int_eqEquality voidElimination

Latex:
\mforall{}[x,y,z:\mBbbR{}].    uiff(x  =  z;(x  *  y)  =  (z  *  y))  supposing  y  \mneq{}  r0

Date html generated: 2019_10_29-AM-09_40_11
Last ObjectModification: 2019_04_01-PM-07_01_17

Theory : reals

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