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Theorem xdivmul 28232
Description: Relationship between division and multiplication. (Contributed by Thierry Arnoux, 24-Dec-2016.)
Assertion
Ref Expression
xdivmul  |-  ( ( A  e.  RR*  /\  B  e.  RR*  /\  ( C  e.  RR  /\  C  =/=  0 ) )  -> 
( ( A /𝑒  C )  =  B  <->  ( C xe B )  =  A ) )

Proof of Theorem xdivmul
Dummy variable  x is distinct from all other variables.
StepHypRef Expression
1 xdivval 28226 . . . . 5  |-  ( ( A  e.  RR*  /\  C  e.  RR  /\  C  =/=  0 )  ->  ( A /𝑒  C )  =  (
iota_ x  e.  RR*  ( C xe x )  =  A ) )
213expb 1206 . . . 4  |-  ( ( A  e.  RR*  /\  ( C  e.  RR  /\  C  =/=  0 ) )  -> 
( A /𝑒  C )  =  (
iota_ x  e.  RR*  ( C xe x )  =  A ) )
323adant2 1024 . . 3  |-  ( ( A  e.  RR*  /\  B  e.  RR*  /\  ( C  e.  RR  /\  C  =/=  0 ) )  -> 
( A /𝑒  C )  =  (
iota_ x  e.  RR*  ( C xe x )  =  A ) )
43eqeq1d 2431 . 2  |-  ( ( A  e.  RR*  /\  B  e.  RR*  /\  ( C  e.  RR  /\  C  =/=  0 ) )  -> 
( ( A /𝑒  C )  =  B  <->  ( iota_ x  e.  RR*  ( C xe x )  =  A )  =  B ) )
5 simp2 1006 . . 3  |-  ( ( A  e.  RR*  /\  B  e.  RR*  /\  ( C  e.  RR  /\  C  =/=  0 ) )  ->  B  e.  RR* )
6 xreceu 28229 . . . . 5  |-  ( ( A  e.  RR*  /\  C  e.  RR  /\  C  =/=  0 )  ->  E! x  e.  RR*  ( C xe x )  =  A )
763expb 1206 . . . 4  |-  ( ( A  e.  RR*  /\  ( C  e.  RR  /\  C  =/=  0 ) )  ->  E! x  e.  RR*  ( C xe x )  =  A )
873adant2 1024 . . 3  |-  ( ( A  e.  RR*  /\  B  e.  RR*  /\  ( C  e.  RR  /\  C  =/=  0 ) )  ->  E! x  e.  RR*  ( C xe x )  =  A )
9 oveq2 6313 . . . . 5  |-  ( x  =  B  ->  ( C xe x )  =  ( C xe B ) )
109eqeq1d 2431 . . . 4  |-  ( x  =  B  ->  (
( C xe x )  =  A  <-> 
( C xe B )  =  A ) )
1110riota2 6289 . . 3  |-  ( ( B  e.  RR*  /\  E! x  e.  RR*  ( C xe x )  =  A )  -> 
( ( C xe B )  =  A  <->  ( iota_ x  e. 
RR*  ( C xe x )  =  A )  =  B ) )
125, 8, 11syl2anc 665 . 2  |-  ( ( A  e.  RR*  /\  B  e.  RR*  /\  ( C  e.  RR  /\  C  =/=  0 ) )  -> 
( ( C xe B )  =  A  <->  ( iota_ x  e. 
RR*  ( C xe x )  =  A )  =  B ) )
134, 12bitr4d 259 1  |-  ( ( A  e.  RR*  /\  B  e.  RR*  /\  ( C  e.  RR  /\  C  =/=  0 ) )  -> 
( ( A /𝑒  C )  =  B  <->  ( C xe B )  =  A ) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    <-> wb 187    /\ wa 370    /\ w3a 982    = wceq 1437    e. wcel 1870    =/= wne 2625   E!wreu 2784   iota_crio 6266  (class class class)co 6305   RRcr 9537   0cc0 9538   RR*cxr 9673   xecxmu 11408   /𝑒 cxdiv 28224
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1665  ax-4 1678  ax-5 1751  ax-6 1797  ax-7 1841  ax-8 1872  ax-9 1874  ax-10 1889  ax-11 1894  ax-12 1907  ax-13 2055  ax-ext 2407  ax-sep 4548  ax-nul 4556  ax-pow 4603  ax-pr 4661  ax-un 6597  ax-cnex 9594  ax-resscn 9595  ax-1cn 9596  ax-icn 9597  ax-addcl 9598  ax-addrcl 9599  ax-mulcl 9600  ax-mulrcl 9601  ax-mulcom 9602  ax-addass 9603  ax-mulass 9604  ax-distr 9605  ax-i2m1 9606  ax-1ne0 9607  ax-1rid 9608  ax-rnegex 9609  ax-rrecex 9610  ax-cnre 9611  ax-pre-lttri 9612  ax-pre-lttrn 9613  ax-pre-ltadd 9614  ax-pre-mulgt0 9615
This theorem depends on definitions:  df-bi 188  df-or 371  df-an 372  df-3or 983  df-3an 984  df-tru 1440  df-ex 1660  df-nf 1664  df-sb 1790  df-eu 2270  df-mo 2271  df-clab 2415  df-cleq 2421  df-clel 2424  df-nfc 2579  df-ne 2627  df-nel 2628  df-ral 2787  df-rex 2788  df-reu 2789  df-rmo 2790  df-rab 2791  df-v 3089  df-sbc 3306  df-csb 3402  df-dif 3445  df-un 3447  df-in 3449  df-ss 3456  df-nul 3768  df-if 3916  df-pw 3987  df-sn 4003  df-pr 4005  df-op 4009  df-uni 4223  df-iun 4304  df-br 4427  df-opab 4485  df-mpt 4486  df-id 4769  df-po 4775  df-so 4776  df-xp 4860  df-rel 4861  df-cnv 4862  df-co 4863  df-dm 4864  df-rn 4865  df-res 4866  df-ima 4867  df-iota 5565  df-fun 5603  df-fn 5604  df-f 5605  df-f1 5606  df-fo 5607  df-f1o 5608  df-fv 5609  df-riota 6267  df-ov 6308  df-oprab 6309  df-mpt2 6310  df-1st 6807  df-2nd 6808  df-er 7371  df-en 7578  df-dom 7579  df-sdom 7580  df-pnf 9676  df-mnf 9677  df-xr 9678  df-ltxr 9679  df-le 9680  df-sub 9861  df-neg 9862  df-xneg 11409  df-xmul 11411  df-xdiv 28225
This theorem is referenced by:  xdivrec  28234
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