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Theorem divmulap 8442
Description: Relationship between division and multiplication. (Contributed by Jim Kingdon, 22-Feb-2020.)
Assertion
Ref Expression
divmulap  |-  ( ( A  e.  CC  /\  B  e.  CC  /\  ( C  e.  CC  /\  C #  0 ) )  -> 
( ( A  /  C )  =  B  <-> 
( C  x.  B
)  =  A ) )

Proof of Theorem divmulap
Dummy variable  x is distinct from all other variables.
StepHypRef Expression
1 divvalap 8441 . . . . 5  |-  ( ( A  e.  CC  /\  C  e.  CC  /\  C #  0 )  ->  ( A  /  C )  =  ( iota_ x  e.  CC  ( C  x.  x
)  =  A ) )
213expb 1182 . . . 4  |-  ( ( A  e.  CC  /\  ( C  e.  CC  /\  C #  0 ) )  ->  ( A  /  C )  =  (
iota_ x  e.  CC  ( C  x.  x
)  =  A ) )
323adant2 1000 . . 3  |-  ( ( A  e.  CC  /\  B  e.  CC  /\  ( C  e.  CC  /\  C #  0 ) )  -> 
( A  /  C
)  =  ( iota_ x  e.  CC  ( C  x.  x )  =  A ) )
43eqeq1d 2148 . 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 982 . . 3  |-  ( ( A  e.  CC  /\  B  e.  CC  /\  ( C  e.  CC  /\  C #  0 ) )  ->  B  e.  CC )
6 receuap 8437 . . . . 5  |-  ( ( A  e.  CC  /\  C  e.  CC  /\  C #  0 )  ->  E! x  e.  CC  ( C  x.  x )  =  A )
763expb 1182 . . . 4  |-  ( ( A  e.  CC  /\  ( C  e.  CC  /\  C #  0 ) )  ->  E! x  e.  CC  ( C  x.  x )  =  A )
873adant2 1000 . . 3  |-  ( ( A  e.  CC  /\  B  e.  CC  /\  ( C  e.  CC  /\  C #  0 ) )  ->  E! x  e.  CC  ( C  x.  x
)  =  A )
9 oveq2 5782 . . . . 5  |-  ( x  =  B  ->  ( C  x.  x )  =  ( C  x.  B ) )
109eqeq1d 2148 . . . 4  |-  ( x  =  B  ->  (
( C  x.  x
)  =  A  <->  ( C  x.  B )  =  A ) )
1110riota2 5752 . . 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 408 . 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 190 1  |-  ( ( A  e.  CC  /\  B  e.  CC  /\  ( C  e.  CC  /\  C #  0 ) )  -> 
( ( A  /  C )  =  B  <-> 
( C  x.  B
)  =  A ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 103    <-> wb 104    /\ w3a 962    = wceq 1331    e. wcel 1480   E!wreu 2418   class class class wbr 3929   iota_crio 5729  (class class class)co 5774   CCcc 7625   0cc0 7627    x. cmul 7632   # cap 8350    / cdiv 8439
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-in1 603  ax-in2 604  ax-io 698  ax-5 1423  ax-7 1424  ax-gen 1425  ax-ie1 1469  ax-ie2 1470  ax-8 1482  ax-10 1483  ax-11 1484  ax-i12 1485  ax-bndl 1486  ax-4 1487  ax-13 1491  ax-14 1492  ax-17 1506  ax-i9 1510  ax-ial 1514  ax-i5r 1515  ax-ext 2121  ax-sep 4046  ax-pow 4098  ax-pr 4131  ax-un 4355  ax-setind 4452  ax-cnex 7718  ax-resscn 7719  ax-1cn 7720  ax-1re 7721  ax-icn 7722  ax-addcl 7723  ax-addrcl 7724  ax-mulcl 7725  ax-mulrcl 7726  ax-addcom 7727  ax-mulcom 7728  ax-addass 7729  ax-mulass 7730  ax-distr 7731  ax-i2m1 7732  ax-0lt1 7733  ax-1rid 7734  ax-0id 7735  ax-rnegex 7736  ax-precex 7737  ax-cnre 7738  ax-pre-ltirr 7739  ax-pre-ltwlin 7740  ax-pre-lttrn 7741  ax-pre-apti 7742  ax-pre-ltadd 7743  ax-pre-mulgt0 7744  ax-pre-mulext 7745
This theorem depends on definitions:  df-bi 116  df-3an 964  df-tru 1334  df-fal 1337  df-nf 1437  df-sb 1736  df-eu 2002  df-mo 2003  df-clab 2126  df-cleq 2132  df-clel 2135  df-nfc 2270  df-ne 2309  df-nel 2404  df-ral 2421  df-rex 2422  df-reu 2423  df-rmo 2424  df-rab 2425  df-v 2688  df-sbc 2910  df-dif 3073  df-un 3075  df-in 3077  df-ss 3084  df-pw 3512  df-sn 3533  df-pr 3534  df-op 3536  df-uni 3737  df-br 3930  df-opab 3990  df-id 4215  df-po 4218  df-iso 4219  df-xp 4545  df-rel 4546  df-cnv 4547  df-co 4548  df-dm 4549  df-iota 5088  df-fun 5125  df-fv 5131  df-riota 5730  df-ov 5777  df-oprab 5778  df-mpo 5779  df-pnf 7809  df-mnf 7810  df-xr 7811  df-ltxr 7812  df-le 7813  df-sub 7942  df-neg 7943  df-reap 8344  df-ap 8351  df-div 8440
This theorem is referenced by:  divmulap2  8443  divcanap2  8447  divrecap  8455  divcanap3  8465  div0ap  8469  div1  8470  recrecap  8476  rec11ap  8477  divdivdivap  8480  ddcanap  8493  rerecclap  8497  div2negap  8502  divmulapzi  8530  divmulapd  8579  caucvgrelemrec  10758  odd2np1  11576  sqgcd  11723
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