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Theorem divmul 10217
Description: Relationship between division and multiplication. (Contributed by NM, 2-Aug-2004.) (Revised by Mario Carneiro, 17-Feb-2014.)
Assertion
Ref Expression
divmul  |-  ( ( A  e.  CC  /\  B  e.  CC  /\  ( C  e.  CC  /\  C  =/=  0 ) )  -> 
( ( A  /  C )  =  B  <-> 
( C  x.  B
)  =  A ) )

Proof of Theorem divmul
Dummy variable  x is distinct from all other variables.
StepHypRef Expression
1 divval 10216 . . . . 5  |-  ( ( A  e.  CC  /\  C  e.  CC  /\  C  =/=  0 )  ->  ( A  /  C )  =  ( iota_ x  e.  CC  ( C  x.  x
)  =  A ) )
213expb 1198 . . . 4  |-  ( ( A  e.  CC  /\  ( C  e.  CC  /\  C  =/=  0 ) )  ->  ( A  /  C )  =  (
iota_ x  e.  CC  ( C  x.  x
)  =  A ) )
323adant2 1016 . . 3  |-  ( ( A  e.  CC  /\  B  e.  CC  /\  ( C  e.  CC  /\  C  =/=  0 ) )  -> 
( A  /  C
)  =  ( iota_ x  e.  CC  ( C  x.  x )  =  A ) )
43eqeq1d 2445 . 2  |-  ( ( A  e.  CC  /\  B  e.  CC  /\  ( C  e.  CC  /\  C  =/=  0 ) )  -> 
( ( A  /  C )  =  B  <-> 
( iota_ x  e.  CC  ( C  x.  x
)  =  A )  =  B ) )
5 simp2 998 . . 3  |-  ( ( A  e.  CC  /\  B  e.  CC  /\  ( C  e.  CC  /\  C  =/=  0 ) )  ->  B  e.  CC )
6 receu 10201 . . . . 5  |-  ( ( A  e.  CC  /\  C  e.  CC  /\  C  =/=  0 )  ->  E! x  e.  CC  ( C  x.  x )  =  A )
763expb 1198 . . . 4  |-  ( ( A  e.  CC  /\  ( C  e.  CC  /\  C  =/=  0 ) )  ->  E! x  e.  CC  ( C  x.  x )  =  A )
873adant2 1016 . . 3  |-  ( ( A  e.  CC  /\  B  e.  CC  /\  ( C  e.  CC  /\  C  =/=  0 ) )  ->  E! x  e.  CC  ( C  x.  x
)  =  A )
9 oveq2 6289 . . . . 5  |-  ( x  =  B  ->  ( C  x.  x )  =  ( C  x.  B ) )
109eqeq1d 2445 . . . 4  |-  ( x  =  B  ->  (
( C  x.  x
)  =  A  <->  ( C  x.  B )  =  A ) )
1110riota2 6265 . . 3  |-  ( ( B  e.  CC  /\  E! x  e.  CC  ( C  x.  x
)  =  A )  ->  ( ( C  x.  B )  =  A  <->  ( iota_ x  e.  CC  ( C  x.  x )  =  A )  =  B ) )
125, 8, 11syl2anc 661 . 2  |-  ( ( A  e.  CC  /\  B  e.  CC  /\  ( C  e.  CC  /\  C  =/=  0 ) )  -> 
( ( C  x.  B )  =  A  <-> 
( iota_ x  e.  CC  ( C  x.  x
)  =  A )  =  B ) )
134, 12bitr4d 256 1  |-  ( ( A  e.  CC  /\  B  e.  CC  /\  ( C  e.  CC  /\  C  =/=  0 ) )  -> 
( ( A  /  C )  =  B  <-> 
( C  x.  B
)  =  A ) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    <-> wb 184    /\ wa 369    /\ w3a 974    = wceq 1383    e. wcel 1804    =/= wne 2638   E!wreu 2795   iota_crio 6241  (class class class)co 6281   CCcc 9493   0cc0 9495    x. cmul 9500    / cdiv 10213
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1605  ax-4 1618  ax-5 1691  ax-6 1734  ax-7 1776  ax-8 1806  ax-9 1808  ax-10 1823  ax-11 1828  ax-12 1840  ax-13 1985  ax-ext 2421  ax-sep 4558  ax-nul 4566  ax-pow 4615  ax-pr 4676  ax-un 6577  ax-resscn 9552  ax-1cn 9553  ax-icn 9554  ax-addcl 9555  ax-addrcl 9556  ax-mulcl 9557  ax-mulrcl 9558  ax-mulcom 9559  ax-addass 9560  ax-mulass 9561  ax-distr 9562  ax-i2m1 9563  ax-1ne0 9564  ax-1rid 9565  ax-rnegex 9566  ax-rrecex 9567  ax-cnre 9568  ax-pre-lttri 9569  ax-pre-lttrn 9570  ax-pre-ltadd 9571  ax-pre-mulgt0 9572
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 975  df-3an 976  df-tru 1386  df-ex 1600  df-nf 1604  df-sb 1727  df-eu 2272  df-mo 2273  df-clab 2429  df-cleq 2435  df-clel 2438  df-nfc 2593  df-ne 2640  df-nel 2641  df-ral 2798  df-rex 2799  df-reu 2800  df-rmo 2801  df-rab 2802  df-v 3097  df-sbc 3314  df-csb 3421  df-dif 3464  df-un 3466  df-in 3468  df-ss 3475  df-nul 3771  df-if 3927  df-pw 3999  df-sn 4015  df-pr 4017  df-op 4021  df-uni 4235  df-br 4438  df-opab 4496  df-mpt 4497  df-id 4785  df-po 4790  df-so 4791  df-xp 4995  df-rel 4996  df-cnv 4997  df-co 4998  df-dm 4999  df-rn 5000  df-res 5001  df-ima 5002  df-iota 5541  df-fun 5580  df-fn 5581  df-f 5582  df-f1 5583  df-fo 5584  df-f1o 5585  df-fv 5586  df-riota 6242  df-ov 6284  df-oprab 6285  df-mpt2 6286  df-er 7313  df-en 7519  df-dom 7520  df-sdom 7521  df-pnf 9633  df-mnf 9634  df-xr 9635  df-ltxr 9636  df-le 9637  df-sub 9812  df-neg 9813  df-div 10214
This theorem is referenced by:  divmul2  10218  divcan2  10222  divrec  10230  divcan3  10238  div0  10242  div1  10243  recrec  10248  rec11  10249  divdivdiv  10252  ddcan  10265  rereccl  10269  div2neg  10274  divmulzi  10302  divmuld  10349  crreczi  12273  odd2np1  14028  sqgcd  14178
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