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Theorem mulcan1g 25181
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 9066 . . . 4  |-  ( ( A  e.  CC  /\  B  e.  CC )  ->  ( A  x.  B
)  e.  CC )
213adant3 977 . . 3  |-  ( ( A  e.  CC  /\  B  e.  CC  /\  C  e.  CC )  ->  ( A  x.  B )  e.  CC )
3 mulcl 9066 . . . 4  |-  ( ( A  e.  CC  /\  C  e.  CC )  ->  ( A  x.  C
)  e.  CC )
433adant2 976 . . 3  |-  ( ( A  e.  CC  /\  B  e.  CC  /\  C  e.  CC )  ->  ( A  x.  C )  e.  CC )
52, 4subeq0ad 9413 . 2  |-  ( ( A  e.  CC  /\  B  e.  CC  /\  C  e.  CC )  ->  (
( ( A  x.  B )  -  ( A  x.  C )
)  =  0  <->  ( A  x.  B )  =  ( A  x.  C ) ) )
6 simp1 957 . . . 4  |-  ( ( A  e.  CC  /\  B  e.  CC  /\  C  e.  CC )  ->  A  e.  CC )
7 subcl 9297 . . . . 5  |-  ( ( B  e.  CC  /\  C  e.  CC )  ->  ( B  -  C
)  e.  CC )
873adant1 975 . . . 4  |-  ( ( A  e.  CC  /\  B  e.  CC  /\  C  e.  CC )  ->  ( B  -  C )  e.  CC )
96, 8mul0ord 9664 . . 3  |-  ( ( A  e.  CC  /\  B  e.  CC  /\  C  e.  CC )  ->  (
( A  x.  ( B  -  C )
)  =  0  <->  ( A  =  0  \/  ( B  -  C
)  =  0 ) ) )
10 subdi 9459 . . . 4  |-  ( ( A  e.  CC  /\  B  e.  CC  /\  C  e.  CC )  ->  ( A  x.  ( B  -  C ) )  =  ( ( A  x.  B )  -  ( A  x.  C )
) )
1110eqeq1d 2443 . . 3  |-  ( ( A  e.  CC  /\  B  e.  CC  /\  C  e.  CC )  ->  (
( A  x.  ( B  -  C )
)  =  0  <->  (
( A  x.  B
)  -  ( A  x.  C ) )  =  0 ) )
12 subeq0 9319 . . . . 5  |-  ( ( B  e.  CC  /\  C  e.  CC )  ->  ( ( B  -  C )  =  0  <-> 
B  =  C ) )
13123adant1 975 . . . 4  |-  ( ( A  e.  CC  /\  B  e.  CC  /\  C  e.  CC )  ->  (
( B  -  C
)  =  0  <->  B  =  C ) )
1413orbi2d 683 . . 3  |-  ( ( A  e.  CC  /\  B  e.  CC  /\  C  e.  CC )  ->  (
( A  =  0  \/  ( B  -  C )  =  0 )  <->  ( A  =  0  \/  B  =  C ) ) )
159, 11, 143bitr3d 275 . 2  |-  ( ( A  e.  CC  /\  B  e.  CC  /\  C  e.  CC )  ->  (
( ( A  x.  B )  -  ( A  x.  C )
)  =  0  <->  ( A  =  0  \/  B  =  C )
) )
165, 15bitr3d 247 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 177    \/ wo 358    /\ w3a 936    = wceq 1652    e. wcel 1725  (class class class)co 6073   CCcc 8980   0cc0 8982    x. cmul 8987    - cmin 9283
This theorem is referenced by:  mulcan2g  25182  axcontlem2  25896  axcontlem7  25901
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1555  ax-5 1566  ax-17 1626  ax-9 1666  ax-8 1687  ax-13 1727  ax-14 1729  ax-6 1744  ax-7 1749  ax-11 1761  ax-12 1950  ax-ext 2416  ax-sep 4322  ax-nul 4330  ax-pow 4369  ax-pr 4395  ax-un 4693  ax-resscn 9039  ax-1cn 9040  ax-icn 9041  ax-addcl 9042  ax-addrcl 9043  ax-mulcl 9044  ax-mulrcl 9045  ax-mulcom 9046  ax-addass 9047  ax-mulass 9048  ax-distr 9049  ax-i2m1 9050  ax-1ne0 9051  ax-1rid 9052  ax-rnegex 9053  ax-rrecex 9054  ax-cnre 9055  ax-pre-lttri 9056  ax-pre-lttrn 9057  ax-pre-ltadd 9058  ax-pre-mulgt0 9059
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3or 937  df-3an 938  df-tru 1328  df-ex 1551  df-nf 1554  df-sb 1659  df-eu 2284  df-mo 2285  df-clab 2422  df-cleq 2428  df-clel 2431  df-nfc 2560  df-ne 2600  df-nel 2601  df-ral 2702  df-rex 2703  df-reu 2704  df-rab 2706  df-v 2950  df-sbc 3154  df-csb 3244  df-dif 3315  df-un 3317  df-in 3319  df-ss 3326  df-nul 3621  df-if 3732  df-pw 3793  df-sn 3812  df-pr 3813  df-op 3815  df-uni 4008  df-br 4205  df-opab 4259  df-mpt 4260  df-id 4490  df-po 4495  df-so 4496  df-xp 4876  df-rel 4877  df-cnv 4878  df-co 4879  df-dm 4880  df-rn 4881  df-res 4882  df-ima 4883  df-iota 5410  df-fun 5448  df-fn 5449  df-f 5450  df-f1 5451  df-fo 5452  df-f1o 5453  df-fv 5454  df-ov 6076  df-oprab 6077  df-mpt2 6078  df-riota 6541  df-er 6897  df-en 7102  df-dom 7103  df-sdom 7104  df-pnf 9114  df-mnf 9115  df-xr 9116  df-ltxr 9117  df-le 9118  df-sub 9285  df-neg 9286
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