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Theorem ablomul 25558
Description: Nonzero complex number multiplication is an Abelian group operation. (Contributed by Steve Rodriguez, 12-Feb-2007.) (New usage is discouraged.)
Assertion
Ref Expression
ablomul  |-  (  x.  |`  ( ( CC  \  { 0 } )  X.  ( CC  \  { 0 } ) ) )  e.  AbelOp

Proof of Theorem ablomul
Dummy variables  x  y  z are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 cnex 9562 . . . 4  |-  CC  e.  _V
2 difexg 4585 . . . 4  |-  ( CC  e.  _V  ->  ( CC  \  { 0 } )  e.  _V )
31, 2ax-mp 5 . . 3  |-  ( CC 
\  { 0 } )  e.  _V
4 mulnzcnopr 10191 . . 3  |-  (  x.  |`  ( ( CC  \  { 0 } )  X.  ( CC  \  { 0 } ) ) ) : ( ( CC  \  {
0 } )  X.  ( CC  \  {
0 } ) ) --> ( CC  \  {
0 } )
5 ovres 6415 . . . . . 6  |-  ( ( x  e.  ( CC 
\  { 0 } )  /\  y  e.  ( CC  \  {
0 } ) )  ->  ( x (  x.  |`  ( ( CC  \  { 0 } )  X.  ( CC 
\  { 0 } ) ) ) y )  =  ( x  x.  y ) )
6 eldifsn 4141 . . . . . . . 8  |-  ( x  e.  ( CC  \  { 0 } )  <-> 
( x  e.  CC  /\  x  =/=  0 ) )
7 eldifsn 4141 . . . . . . . 8  |-  ( y  e.  ( CC  \  { 0 } )  <-> 
( y  e.  CC  /\  y  =/=  0 ) )
8 mulcl 9565 . . . . . . . . . 10  |-  ( ( x  e.  CC  /\  y  e.  CC )  ->  ( x  x.  y
)  e.  CC )
98ad2ant2r 744 . . . . . . . . 9  |-  ( ( ( x  e.  CC  /\  x  =/=  0 )  /\  ( y  e.  CC  /\  y  =/=  0 ) )  -> 
( x  x.  y
)  e.  CC )
10 mulne0 10187 . . . . . . . . 9  |-  ( ( ( x  e.  CC  /\  x  =/=  0 )  /\  ( y  e.  CC  /\  y  =/=  0 ) )  -> 
( x  x.  y
)  =/=  0 )
119, 10jca 530 . . . . . . . 8  |-  ( ( ( x  e.  CC  /\  x  =/=  0 )  /\  ( y  e.  CC  /\  y  =/=  0 ) )  -> 
( ( x  x.  y )  e.  CC  /\  ( x  x.  y
)  =/=  0 ) )
126, 7, 11syl2anb 477 . . . . . . 7  |-  ( ( x  e.  ( CC 
\  { 0 } )  /\  y  e.  ( CC  \  {
0 } ) )  ->  ( ( x  x.  y )  e.  CC  /\  ( x  x.  y )  =/=  0 ) )
13 eldifsn 4141 . . . . . . 7  |-  ( ( x  x.  y )  e.  ( CC  \  { 0 } )  <-> 
( ( x  x.  y )  e.  CC  /\  ( x  x.  y
)  =/=  0 ) )
1412, 13sylibr 212 . . . . . 6  |-  ( ( x  e.  ( CC 
\  { 0 } )  /\  y  e.  ( CC  \  {
0 } ) )  ->  ( x  x.  y )  e.  ( CC  \  { 0 } ) )
155, 14eqeltrd 2542 . . . . 5  |-  ( ( x  e.  ( CC 
\  { 0 } )  /\  y  e.  ( CC  \  {
0 } ) )  ->  ( x (  x.  |`  ( ( CC  \  { 0 } )  X.  ( CC 
\  { 0 } ) ) ) y )  e.  ( CC 
\  { 0 } ) )
16 ovres 6415 . . . . 5  |-  ( ( ( x (  x.  |`  ( ( CC  \  { 0 } )  X.  ( CC  \  { 0 } ) ) ) y )  e.  ( CC  \  { 0 } )  /\  z  e.  ( CC  \  { 0 } ) )  -> 
( ( x (  x.  |`  ( ( CC  \  { 0 } )  X.  ( CC 
\  { 0 } ) ) ) y ) (  x.  |`  (
( CC  \  {
0 } )  X.  ( CC  \  {
0 } ) ) ) z )  =  ( ( x (  x.  |`  ( ( CC  \  { 0 } )  X.  ( CC 
\  { 0 } ) ) ) y )  x.  z ) )
1715, 16stoic3 1614 . . . 4  |-  ( ( x  e.  ( CC 
\  { 0 } )  /\  y  e.  ( CC  \  {
0 } )  /\  z  e.  ( CC  \  { 0 } ) )  ->  ( (
x (  x.  |`  (
( CC  \  {
0 } )  X.  ( CC  \  {
0 } ) ) ) y ) (  x.  |`  ( ( CC  \  { 0 } )  X.  ( CC 
\  { 0 } ) ) ) z )  =  ( ( x (  x.  |`  (
( CC  \  {
0 } )  X.  ( CC  \  {
0 } ) ) ) y )  x.  z ) )
1853adant3 1014 . . . . 5  |-  ( ( x  e.  ( CC 
\  { 0 } )  /\  y  e.  ( CC  \  {
0 } )  /\  z  e.  ( CC  \  { 0 } ) )  ->  ( x
(  x.  |`  (
( CC  \  {
0 } )  X.  ( CC  \  {
0 } ) ) ) y )  =  ( x  x.  y
) )
1918oveq1d 6285 . . . 4  |-  ( ( x  e.  ( CC 
\  { 0 } )  /\  y  e.  ( CC  \  {
0 } )  /\  z  e.  ( CC  \  { 0 } ) )  ->  ( (
x (  x.  |`  (
( CC  \  {
0 } )  X.  ( CC  \  {
0 } ) ) ) y )  x.  z )  =  ( ( x  x.  y
)  x.  z ) )
20 eldifi 3612 . . . . . 6  |-  ( x  e.  ( CC  \  { 0 } )  ->  x  e.  CC )
21 eldifi 3612 . . . . . 6  |-  ( y  e.  ( CC  \  { 0 } )  ->  y  e.  CC )
22 eldifi 3612 . . . . . 6  |-  ( z  e.  ( CC  \  { 0 } )  ->  z  e.  CC )
23 mulass 9569 . . . . . 6  |-  ( ( x  e.  CC  /\  y  e.  CC  /\  z  e.  CC )  ->  (
( x  x.  y
)  x.  z )  =  ( x  x.  ( y  x.  z
) ) )
2420, 21, 22, 23syl3an 1268 . . . . 5  |-  ( ( x  e.  ( CC 
\  { 0 } )  /\  y  e.  ( CC  \  {
0 } )  /\  z  e.  ( CC  \  { 0 } ) )  ->  ( (
x  x.  y )  x.  z )  =  ( x  x.  (
y  x.  z ) ) )
25 ovres 6415 . . . . . . . 8  |-  ( ( y  e.  ( CC 
\  { 0 } )  /\  z  e.  ( CC  \  {
0 } ) )  ->  ( y (  x.  |`  ( ( CC  \  { 0 } )  X.  ( CC 
\  { 0 } ) ) ) z )  =  ( y  x.  z ) )
2625eqcomd 2462 . . . . . . 7  |-  ( ( y  e.  ( CC 
\  { 0 } )  /\  z  e.  ( CC  \  {
0 } ) )  ->  ( y  x.  z )  =  ( y (  x.  |`  (
( CC  \  {
0 } )  X.  ( CC  \  {
0 } ) ) ) z ) )
27263adant1 1012 . . . . . 6  |-  ( ( x  e.  ( CC 
\  { 0 } )  /\  y  e.  ( CC  \  {
0 } )  /\  z  e.  ( CC  \  { 0 } ) )  ->  ( y  x.  z )  =  ( y (  x.  |`  (
( CC  \  {
0 } )  X.  ( CC  \  {
0 } ) ) ) z ) )
2827oveq2d 6286 . . . . 5  |-  ( ( x  e.  ( CC 
\  { 0 } )  /\  y  e.  ( CC  \  {
0 } )  /\  z  e.  ( CC  \  { 0 } ) )  ->  ( x  x.  ( y  x.  z
) )  =  ( x  x.  ( y (  x.  |`  (
( CC  \  {
0 } )  X.  ( CC  \  {
0 } ) ) ) z ) ) )
294fovcl 6380 . . . . . . . . 9  |-  ( ( y  e.  ( CC 
\  { 0 } )  /\  z  e.  ( CC  \  {
0 } ) )  ->  ( y (  x.  |`  ( ( CC  \  { 0 } )  X.  ( CC 
\  { 0 } ) ) ) z )  e.  ( CC 
\  { 0 } ) )
3025, 29eqeltrrd 2543 . . . . . . . 8  |-  ( ( y  e.  ( CC 
\  { 0 } )  /\  z  e.  ( CC  \  {
0 } ) )  ->  ( y  x.  z )  e.  ( CC  \  { 0 } ) )
31 ovres 6415 . . . . . . . . 9  |-  ( ( x  e.  ( CC 
\  { 0 } )  /\  ( y  x.  z )  e.  ( CC  \  {
0 } ) )  ->  ( x (  x.  |`  ( ( CC  \  { 0 } )  X.  ( CC 
\  { 0 } ) ) ) ( y  x.  z ) )  =  ( x  x.  ( y  x.  z ) ) )
3231eqcomd 2462 . . . . . . . 8  |-  ( ( x  e.  ( CC 
\  { 0 } )  /\  ( y  x.  z )  e.  ( CC  \  {
0 } ) )  ->  ( x  x.  ( y  x.  z
) )  =  ( x (  x.  |`  (
( CC  \  {
0 } )  X.  ( CC  \  {
0 } ) ) ) ( y  x.  z ) ) )
3330, 32sylan2 472 . . . . . . 7  |-  ( ( x  e.  ( CC 
\  { 0 } )  /\  ( y  e.  ( CC  \  { 0 } )  /\  z  e.  ( CC  \  { 0 } ) ) )  ->  ( x  x.  ( y  x.  z
) )  =  ( x (  x.  |`  (
( CC  \  {
0 } )  X.  ( CC  \  {
0 } ) ) ) ( y  x.  z ) ) )
34333impb 1190 . . . . . 6  |-  ( ( x  e.  ( CC 
\  { 0 } )  /\  y  e.  ( CC  \  {
0 } )  /\  z  e.  ( CC  \  { 0 } ) )  ->  ( x  x.  ( y  x.  z
) )  =  ( x (  x.  |`  (
( CC  \  {
0 } )  X.  ( CC  \  {
0 } ) ) ) ( y  x.  z ) ) )
3527oveq2d 6286 . . . . . 6  |-  ( ( x  e.  ( CC 
\  { 0 } )  /\  y  e.  ( CC  \  {
0 } )  /\  z  e.  ( CC  \  { 0 } ) )  ->  ( x
(  x.  |`  (
( CC  \  {
0 } )  X.  ( CC  \  {
0 } ) ) ) ( y  x.  z ) )  =  ( x (  x.  |`  ( ( CC  \  { 0 } )  X.  ( CC  \  { 0 } ) ) ) ( y (  x.  |`  (
( CC  \  {
0 } )  X.  ( CC  \  {
0 } ) ) ) z ) ) )
3634, 28, 353eqtr3d 2503 . . . . 5  |-  ( ( x  e.  ( CC 
\  { 0 } )  /\  y  e.  ( CC  \  {
0 } )  /\  z  e.  ( CC  \  { 0 } ) )  ->  ( x  x.  ( y (  x.  |`  ( ( CC  \  { 0 } )  X.  ( CC  \  { 0 } ) ) ) z ) )  =  ( x (  x.  |`  (
( CC  \  {
0 } )  X.  ( CC  \  {
0 } ) ) ) ( y (  x.  |`  ( ( CC  \  { 0 } )  X.  ( CC 
\  { 0 } ) ) ) z ) ) )
3724, 28, 363eqtrd 2499 . . . 4  |-  ( ( x  e.  ( CC 
\  { 0 } )  /\  y  e.  ( CC  \  {
0 } )  /\  z  e.  ( CC  \  { 0 } ) )  ->  ( (
x  x.  y )  x.  z )  =  ( x (  x.  |`  ( ( CC  \  { 0 } )  X.  ( CC  \  { 0 } ) ) ) ( y (  x.  |`  (
( CC  \  {
0 } )  X.  ( CC  \  {
0 } ) ) ) z ) ) )
3817, 19, 373eqtrd 2499 . . 3  |-  ( ( x  e.  ( CC 
\  { 0 } )  /\  y  e.  ( CC  \  {
0 } )  /\  z  e.  ( CC  \  { 0 } ) )  ->  ( (
x (  x.  |`  (
( CC  \  {
0 } )  X.  ( CC  \  {
0 } ) ) ) y ) (  x.  |`  ( ( CC  \  { 0 } )  X.  ( CC 
\  { 0 } ) ) ) z )  =  ( x (  x.  |`  (
( CC  \  {
0 } )  X.  ( CC  \  {
0 } ) ) ) ( y (  x.  |`  ( ( CC  \  { 0 } )  X.  ( CC 
\  { 0 } ) ) ) z ) ) )
39 ax-1cn 9539 . . . 4  |-  1  e.  CC
40 ax-1ne0 9550 . . . 4  |-  1  =/=  0
41 eldifsn 4141 . . . 4  |-  ( 1  e.  ( CC  \  { 0 } )  <-> 
( 1  e.  CC  /\  1  =/=  0 ) )
4239, 40, 41mpbir2an 918 . . 3  |-  1  e.  ( CC  \  {
0 } )
43 ovres 6415 . . . . 5  |-  ( ( 1  e.  ( CC 
\  { 0 } )  /\  x  e.  ( CC  \  {
0 } ) )  ->  ( 1 (  x.  |`  ( ( CC  \  { 0 } )  X.  ( CC 
\  { 0 } ) ) ) x )  =  ( 1  x.  x ) )
4442, 43mpan 668 . . . 4  |-  ( x  e.  ( CC  \  { 0 } )  ->  ( 1 (  x.  |`  ( ( CC  \  { 0 } )  X.  ( CC 
\  { 0 } ) ) ) x )  =  ( 1  x.  x ) )
4520mulid2d 9603 . . . 4  |-  ( x  e.  ( CC  \  { 0 } )  ->  ( 1  x.  x )  =  x )
4644, 45eqtrd 2495 . . 3  |-  ( x  e.  ( CC  \  { 0 } )  ->  ( 1 (  x.  |`  ( ( CC  \  { 0 } )  X.  ( CC 
\  { 0 } ) ) ) x )  =  x )
47 reccl 10210 . . . . . 6  |-  ( ( x  e.  CC  /\  x  =/=  0 )  -> 
( 1  /  x
)  e.  CC )
48 recne0 10216 . . . . . 6  |-  ( ( x  e.  CC  /\  x  =/=  0 )  -> 
( 1  /  x
)  =/=  0 )
4947, 48jca 530 . . . . 5  |-  ( ( x  e.  CC  /\  x  =/=  0 )  -> 
( ( 1  /  x )  e.  CC  /\  ( 1  /  x
)  =/=  0 ) )
506, 49sylbi 195 . . . 4  |-  ( x  e.  ( CC  \  { 0 } )  ->  ( ( 1  /  x )  e.  CC  /\  ( 1  /  x )  =/=  0 ) )
51 eldifsn 4141 . . . 4  |-  ( ( 1  /  x )  e.  ( CC  \  { 0 } )  <-> 
( ( 1  /  x )  e.  CC  /\  ( 1  /  x
)  =/=  0 ) )
5250, 51sylibr 212 . . 3  |-  ( x  e.  ( CC  \  { 0 } )  ->  ( 1  /  x )  e.  ( CC  \  { 0 } ) )
53 ovres 6415 . . . . 5  |-  ( ( ( 1  /  x
)  e.  ( CC 
\  { 0 } )  /\  x  e.  ( CC  \  {
0 } ) )  ->  ( ( 1  /  x ) (  x.  |`  ( ( CC  \  { 0 } )  X.  ( CC 
\  { 0 } ) ) ) x )  =  ( ( 1  /  x )  x.  x ) )
5452, 53mpancom 667 . . . 4  |-  ( x  e.  ( CC  \  { 0 } )  ->  ( ( 1  /  x ) (  x.  |`  ( ( CC  \  { 0 } )  X.  ( CC 
\  { 0 } ) ) ) x )  =  ( ( 1  /  x )  x.  x ) )
55 recid2 10218 . . . . 5  |-  ( ( x  e.  CC  /\  x  =/=  0 )  -> 
( ( 1  /  x )  x.  x
)  =  1 )
566, 55sylbi 195 . . . 4  |-  ( x  e.  ( CC  \  { 0 } )  ->  ( ( 1  /  x )  x.  x )  =  1 )
5754, 56eqtrd 2495 . . 3  |-  ( x  e.  ( CC  \  { 0 } )  ->  ( ( 1  /  x ) (  x.  |`  ( ( CC  \  { 0 } )  X.  ( CC 
\  { 0 } ) ) ) x )  =  1 )
583, 4, 38, 42, 46, 52, 57isgrpoi 25401 . 2  |-  (  x.  |`  ( ( CC  \  { 0 } )  X.  ( CC  \  { 0 } ) ) )  e.  GrpOp
594fdmi 5718 . 2  |-  dom  (  x.  |`  ( ( CC 
\  { 0 } )  X.  ( CC 
\  { 0 } ) ) )  =  ( ( CC  \  { 0 } )  X.  ( CC  \  { 0 } ) )
60 mulcom 9567 . . . 4  |-  ( ( x  e.  CC  /\  y  e.  CC )  ->  ( x  x.  y
)  =  ( y  x.  x ) )
6120, 21, 60syl2an 475 . . 3  |-  ( ( x  e.  ( CC 
\  { 0 } )  /\  y  e.  ( CC  \  {
0 } ) )  ->  ( x  x.  y )  =  ( y  x.  x ) )
62 ovres 6415 . . . 4  |-  ( ( y  e.  ( CC 
\  { 0 } )  /\  x  e.  ( CC  \  {
0 } ) )  ->  ( y (  x.  |`  ( ( CC  \  { 0 } )  X.  ( CC 
\  { 0 } ) ) ) x )  =  ( y  x.  x ) )
6362ancoms 451 . . 3  |-  ( ( x  e.  ( CC 
\  { 0 } )  /\  y  e.  ( CC  \  {
0 } ) )  ->  ( y (  x.  |`  ( ( CC  \  { 0 } )  X.  ( CC 
\  { 0 } ) ) ) x )  =  ( y  x.  x ) )
6461, 5, 633eqtr4d 2505 . 2  |-  ( ( x  e.  ( CC 
\  { 0 } )  /\  y  e.  ( CC  \  {
0 } ) )  ->  ( x (  x.  |`  ( ( CC  \  { 0 } )  X.  ( CC 
\  { 0 } ) ) ) y )  =  ( y (  x.  |`  (
( CC  \  {
0 } )  X.  ( CC  \  {
0 } ) ) ) x ) )
6558, 59, 64isabloi 25491 1  |-  (  x.  |`  ( ( CC  \  { 0 } )  X.  ( CC  \  { 0 } ) ) )  e.  AbelOp
Colors of variables: wff setvar class
Syntax hints:    /\ wa 367    /\ w3a 971    = wceq 1398    e. wcel 1823    =/= wne 2649   _Vcvv 3106    \ cdif 3458   {csn 4016    X. cxp 4986    |` cres 4990  (class class class)co 6270   CCcc 9479   0cc0 9481   1c1 9482    x. cmul 9486    / cdiv 10202   AbelOpcablo 25484
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1623  ax-4 1636  ax-5 1709  ax-6 1752  ax-7 1795  ax-8 1825  ax-9 1827  ax-10 1842  ax-11 1847  ax-12 1859  ax-13 2004  ax-ext 2432  ax-rep 4550  ax-sep 4560  ax-nul 4568  ax-pow 4615  ax-pr 4676  ax-un 6565  ax-cnex 9537  ax-resscn 9538  ax-1cn 9539  ax-icn 9540  ax-addcl 9541  ax-addrcl 9542  ax-mulcl 9543  ax-mulrcl 9544  ax-mulcom 9545  ax-addass 9546  ax-mulass 9547  ax-distr 9548  ax-i2m1 9549  ax-1ne0 9550  ax-1rid 9551  ax-rnegex 9552  ax-rrecex 9553  ax-cnre 9554  ax-pre-lttri 9555  ax-pre-lttrn 9556  ax-pre-ltadd 9557  ax-pre-mulgt0 9558  ax-mulf 9561
This theorem depends on definitions:  df-bi 185  df-or 368  df-an 369  df-3or 972  df-3an 973  df-tru 1401  df-ex 1618  df-nf 1622  df-sb 1745  df-eu 2288  df-mo 2289  df-clab 2440  df-cleq 2446  df-clel 2449  df-nfc 2604  df-ne 2651  df-nel 2652  df-ral 2809  df-rex 2810  df-reu 2811  df-rmo 2812  df-rab 2813  df-v 3108  df-sbc 3325  df-csb 3421  df-dif 3464  df-un 3466  df-in 3468  df-ss 3475  df-nul 3784  df-if 3930  df-pw 4001  df-sn 4017  df-pr 4019  df-op 4023  df-uni 4236  df-iun 4317  df-br 4440  df-opab 4498  df-mpt 4499  df-id 4784  df-po 4789  df-so 4790  df-xp 4994  df-rel 4995  df-cnv 4996  df-co 4997  df-dm 4998  df-rn 4999  df-res 5000  df-ima 5001  df-iota 5534  df-fun 5572  df-fn 5573  df-f 5574  df-f1 5575  df-fo 5576  df-f1o 5577  df-fv 5578  df-riota 6232  df-ov 6273  df-oprab 6274  df-mpt2 6275  df-er 7303  df-en 7510  df-dom 7511  df-sdom 7512  df-pnf 9619  df-mnf 9620  df-xr 9621  df-ltxr 9622  df-le 9623  df-sub 9798  df-neg 9799  df-div 10203  df-grpo 25394  df-ablo 25485
This theorem is referenced by:  mulid  25559
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