### Nuprl Lemma : rmul-rdiv

`∀[x,y,a,b:ℝ].  (((x/a) * (y/b)) = (x * y/a * b)) supposing (b ≠ r0 and a ≠ r0)`

Proof

Definitions occuring in Statement :  rdiv: `(x/y)` rneq: `x ≠ y` req: `x = y` rmul: `a * b` int-to-real: `r(n)` real: `ℝ` 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` false: `False` implies: `P `` Q` not: `¬A` rat_term_to_real: `rat_term_to_real(f;t)` rtermDivide: `num "/" denom` rat_term_ind: rat_term_ind rtermMultiply: `left "*" right` rtermVar: `rtermVar(var)` pi1: `fst(t)` and: `P ∧ Q` true: `True` all: `∀x:A. B[x]` pi2: `snd(t)` prop: `ℙ`
Lemmas referenced :  assert-rat-term-eq2 rtermMultiply_wf rtermDivide_wf rtermVar_wf int-to-real_wf istype-int rmul-neq-zero req_witness rmul_wf rdiv_wf rneq_wf real_wf
Rules used in proof :  sqequalSubstitution sqequalTransitivity computationStep sqequalReflexivity isect_memberFormation_alt introduction cut extract_by_obid sqequalHypSubstitution isectElimination thin natural_numberEquality hypothesis lambdaEquality_alt int_eqEquality hypothesisEquality independent_isectElimination approximateComputation sqequalRule independent_pairFormation dependent_functionElimination independent_functionElimination because_Cache universeIsType isect_memberEquality_alt isectIsTypeImplies inhabitedIsType

Latex:
\mforall{}[x,y,a,b:\mBbbR{}].    (((x/a)  *  (y/b))  =  (x  *  y/a  *  b))  supposing  (b  \mneq{}  r0  and  a  \mneq{}  r0)

Date html generated: 2019_10_29-AM-09_56_49
Last ObjectModification: 2019_04_01-PM-07_09_13

Theory : reals

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