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Theorem mulcan1g 24084
Description: A generalized form of the cancellation law for multiplication. (Contributed by Scott Fenton, 17-Jun-2013.)
Assertion
Ref Expression
mulcan1g  |-  ( ( A  e.  CC  /\  B  e.  CC  /\  C  e.  CC )  ->  (
( A  x.  B
)  =  ( A  x.  C )  <->  ( A  =  0  \/  B  =  C ) ) )

Proof of Theorem mulcan1g
StepHypRef Expression
1 mulcl 8821 . . . 4  |-  ( ( A  e.  CC  /\  B  e.  CC )  ->  ( A  x.  B
)  e.  CC )
213adant3 975 . . 3  |-  ( ( A  e.  CC  /\  B  e.  CC  /\  C  e.  CC )  ->  ( A  x.  B )  e.  CC )
3 mulcl 8821 . . . 4  |-  ( ( A  e.  CC  /\  C  e.  CC )  ->  ( A  x.  C
)  e.  CC )
433adant2 974 . . 3  |-  ( ( A  e.  CC  /\  B  e.  CC  /\  C  e.  CC )  ->  ( A  x.  C )  e.  CC )
5 subeq0 9073 . . 3  |-  ( ( ( A  x.  B
)  e.  CC  /\  ( A  x.  C
)  e.  CC )  ->  ( ( ( A  x.  B )  -  ( A  x.  C ) )  =  0  <->  ( A  x.  B )  =  ( A  x.  C ) ) )
62, 4, 5syl2anc 642 . 2  |-  ( ( A  e.  CC  /\  B  e.  CC  /\  C  e.  CC )  ->  (
( ( A  x.  B )  -  ( A  x.  C )
)  =  0  <->  ( A  x.  B )  =  ( A  x.  C ) ) )
7 simp1 955 . . . 4  |-  ( ( A  e.  CC  /\  B  e.  CC  /\  C  e.  CC )  ->  A  e.  CC )
8 subcl 9051 . . . . 5  |-  ( ( B  e.  CC  /\  C  e.  CC )  ->  ( B  -  C
)  e.  CC )
983adant1 973 . . . 4  |-  ( ( A  e.  CC  /\  B  e.  CC  /\  C  e.  CC )  ->  ( B  -  C )  e.  CC )
107, 9mul0ord 9418 . . 3  |-  ( ( A  e.  CC  /\  B  e.  CC  /\  C  e.  CC )  ->  (
( A  x.  ( B  -  C )
)  =  0  <->  ( A  =  0  \/  ( B  -  C
)  =  0 ) ) )
11 subdi 9213 . . . 4  |-  ( ( A  e.  CC  /\  B  e.  CC  /\  C  e.  CC )  ->  ( A  x.  ( B  -  C ) )  =  ( ( A  x.  B )  -  ( A  x.  C )
) )
1211eqeq1d 2291 . . 3  |-  ( ( A  e.  CC  /\  B  e.  CC  /\  C  e.  CC )  ->  (
( A  x.  ( B  -  C )
)  =  0  <->  (
( A  x.  B
)  -  ( A  x.  C ) )  =  0 ) )
13 subeq0 9073 . . . . 5  |-  ( ( B  e.  CC  /\  C  e.  CC )  ->  ( ( B  -  C )  =  0  <-> 
B  =  C ) )
14133adant1 973 . . . 4  |-  ( ( A  e.  CC  /\  B  e.  CC  /\  C  e.  CC )  ->  (
( B  -  C
)  =  0  <->  B  =  C ) )
1514orbi2d 682 . . 3  |-  ( ( A  e.  CC  /\  B  e.  CC  /\  C  e.  CC )  ->  (
( A  =  0  \/  ( B  -  C )  =  0 )  <->  ( A  =  0  \/  B  =  C ) ) )
1610, 12, 153bitr3d 274 . 2  |-  ( ( A  e.  CC  /\  B  e.  CC  /\  C  e.  CC )  ->  (
( ( A  x.  B )  -  ( A  x.  C )
)  =  0  <->  ( A  =  0  \/  B  =  C )
) )
176, 16bitr3d 246 1  |-  ( ( A  e.  CC  /\  B  e.  CC  /\  C  e.  CC )  ->  (
( A  x.  B
)  =  ( A  x.  C )  <->  ( A  =  0  \/  B  =  C ) ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 176    \/ wo 357    /\ w3a 934    = wceq 1623    e. wcel 1684  (class class class)co 5858   CCcc 8735   0cc0 8737    x. cmul 8742    - cmin 9037
This theorem is referenced by:  mulcan2g  24085  axcontlem2  24593  axcontlem7  24598
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1533  ax-5 1544  ax-17 1603  ax-9 1635  ax-8 1643  ax-13 1686  ax-14 1688  ax-6 1703  ax-7 1708  ax-11 1715  ax-12 1866  ax-ext 2264  ax-sep 4141  ax-nul 4149  ax-pow 4188  ax-pr 4214  ax-un 4512  ax-resscn 8794  ax-1cn 8795  ax-icn 8796  ax-addcl 8797  ax-addrcl 8798  ax-mulcl 8799  ax-mulrcl 8800  ax-mulcom 8801  ax-addass 8802  ax-mulass 8803  ax-distr 8804  ax-i2m1 8805  ax-1ne0 8806  ax-1rid 8807  ax-rnegex 8808  ax-rrecex 8809  ax-cnre 8810  ax-pre-lttri 8811  ax-pre-lttrn 8812  ax-pre-ltadd 8813  ax-pre-mulgt0 8814
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3or 935  df-3an 936  df-tru 1310  df-ex 1529  df-nf 1532  df-sb 1630  df-eu 2147  df-mo 2148  df-clab 2270  df-cleq 2276  df-clel 2279  df-nfc 2408  df-ne 2448  df-nel 2449  df-ral 2548  df-rex 2549  df-reu 2550  df-rab 2552  df-v 2790  df-sbc 2992  df-csb 3082  df-dif 3155  df-un 3157  df-in 3159  df-ss 3166  df-nul 3456  df-if 3566  df-pw 3627  df-sn 3646  df-pr 3647  df-op 3649  df-uni 3828  df-br 4024  df-opab 4078  df-mpt 4079  df-id 4309  df-po 4314  df-so 4315  df-xp 4695  df-rel 4696  df-cnv 4697  df-co 4698  df-dm 4699  df-rn 4700  df-res 4701  df-ima 4702  df-iota 5219  df-fun 5257  df-fn 5258  df-f 5259  df-f1 5260  df-fo 5261  df-f1o 5262  df-fv 5263  df-ov 5861  df-oprab 5862  df-mpt2 5863  df-riota 6304  df-er 6660  df-en 6864  df-dom 6865  df-sdom 6866  df-pnf 8869  df-mnf 8870  df-xr 8871  df-ltxr 8872  df-le 8873  df-sub 9039  df-neg 9040
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