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Theorem mulid 23858
Description: The group identity element of nonzero complex number multiplication is one. (Contributed by Steve Rodriguez, 23-Feb-2007.) (Revised by Mario Carneiro, 17-Dec-2013.) (New usage is discouraged.)
Assertion
Ref Expression
mulid  |-  (GId `  (  x.  |`  ( ( CC  \  { 0 } )  X.  ( CC  \  { 0 } ) ) ) )  =  1

Proof of Theorem mulid
Dummy variables  x  y are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 ablomul 23857 . . . 4  |-  (  x.  |`  ( ( CC  \  { 0 } )  X.  ( CC  \  { 0 } ) ) )  e.  AbelOp
2 ablogrpo 23786 . . . 4  |-  ( (  x.  |`  ( ( CC  \  { 0 } )  X.  ( CC 
\  { 0 } ) ) )  e. 
AbelOp  ->  (  x.  |`  (
( CC  \  {
0 } )  X.  ( CC  \  {
0 } ) ) )  e.  GrpOp )
31, 2ax-mp 5 . . 3  |-  (  x.  |`  ( ( CC  \  { 0 } )  X.  ( CC  \  { 0 } ) ) )  e.  GrpOp
4 mulnzcnopr 9997 . . . . . 6  |-  (  x.  |`  ( ( CC  \  { 0 } )  X.  ( CC  \  { 0 } ) ) ) : ( ( CC  \  {
0 } )  X.  ( CC  \  {
0 } ) ) --> ( CC  \  {
0 } )
54fdmi 5579 . . . . 5  |-  dom  (  x.  |`  ( ( CC 
\  { 0 } )  X.  ( CC 
\  { 0 } ) ) )  =  ( ( CC  \  { 0 } )  X.  ( CC  \  { 0 } ) )
63, 5grporn 23714 . . . 4  |-  ( CC 
\  { 0 } )  =  ran  (  x.  |`  ( ( CC 
\  { 0 } )  X.  ( CC 
\  { 0 } ) ) )
7 eqid 2443 . . . 4  |-  (GId `  (  x.  |`  ( ( CC  \  { 0 } )  X.  ( CC  \  { 0 } ) ) ) )  =  (GId `  (  x.  |`  ( ( CC 
\  { 0 } )  X.  ( CC 
\  { 0 } ) ) ) )
86, 7grpoidval 23718 . . 3  |-  ( (  x.  |`  ( ( CC  \  { 0 } )  X.  ( CC 
\  { 0 } ) ) )  e. 
GrpOp  ->  (GId `  (  x.  |`  ( ( CC 
\  { 0 } )  X.  ( CC 
\  { 0 } ) ) ) )  =  ( iota_ y  e.  ( CC  \  {
0 } ) A. x  e.  ( CC  \  { 0 } ) ( y (  x.  |`  ( ( CC  \  { 0 } )  X.  ( CC  \  { 0 } ) ) ) x )  =  x ) )
93, 8ax-mp 5 . 2  |-  (GId `  (  x.  |`  ( ( CC  \  { 0 } )  X.  ( CC  \  { 0 } ) ) ) )  =  ( iota_ y  e.  ( CC  \  {
0 } ) A. x  e.  ( CC  \  { 0 } ) ( y (  x.  |`  ( ( CC  \  { 0 } )  X.  ( CC  \  { 0 } ) ) ) x )  =  x )
10 ax-1cn 9355 . . . . . . 7  |-  1  e.  CC
11 ax-1ne0 9366 . . . . . . 7  |-  1  =/=  0
12 eldifsn 4015 . . . . . . 7  |-  ( 1  e.  ( CC  \  { 0 } )  <-> 
( 1  e.  CC  /\  1  =/=  0 ) )
1310, 11, 12mpbir2an 911 . . . . . 6  |-  1  e.  ( CC  \  {
0 } )
14 ovres 6245 . . . . . 6  |-  ( ( 1  e.  ( CC 
\  { 0 } )  /\  x  e.  ( CC  \  {
0 } ) )  ->  ( 1 (  x.  |`  ( ( CC  \  { 0 } )  X.  ( CC 
\  { 0 } ) ) ) x )  =  ( 1  x.  x ) )
1513, 14mpan 670 . . . . 5  |-  ( x  e.  ( CC  \  { 0 } )  ->  ( 1 (  x.  |`  ( ( CC  \  { 0 } )  X.  ( CC 
\  { 0 } ) ) ) x )  =  ( 1  x.  x ) )
16 eldifi 3493 . . . . . 6  |-  ( x  e.  ( CC  \  { 0 } )  ->  x  e.  CC )
1716mulid2d 9419 . . . . 5  |-  ( x  e.  ( CC  \  { 0 } )  ->  ( 1  x.  x )  =  x )
1815, 17eqtrd 2475 . . . 4  |-  ( x  e.  ( CC  \  { 0 } )  ->  ( 1 (  x.  |`  ( ( CC  \  { 0 } )  X.  ( CC 
\  { 0 } ) ) ) x )  =  x )
1918rgen 2796 . . 3  |-  A. x  e.  ( CC  \  {
0 } ) ( 1 (  x.  |`  (
( CC  \  {
0 } )  X.  ( CC  \  {
0 } ) ) ) x )  =  x
206grpoideu 23711 . . . . 5  |-  ( (  x.  |`  ( ( CC  \  { 0 } )  X.  ( CC 
\  { 0 } ) ) )  e. 
GrpOp  ->  E! y  e.  ( CC  \  {
0 } ) A. x  e.  ( CC  \  { 0 } ) ( y (  x.  |`  ( ( CC  \  { 0 } )  X.  ( CC  \  { 0 } ) ) ) x )  =  x )
213, 20ax-mp 5 . . . 4  |-  E! y  e.  ( CC  \  { 0 } ) A. x  e.  ( CC  \  { 0 } ) ( y (  x.  |`  (
( CC  \  {
0 } )  X.  ( CC  \  {
0 } ) ) ) x )  =  x
22 oveq1 6113 . . . . . . 7  |-  ( y  =  1  ->  (
y (  x.  |`  (
( CC  \  {
0 } )  X.  ( CC  \  {
0 } ) ) ) x )  =  ( 1 (  x.  |`  ( ( CC  \  { 0 } )  X.  ( CC  \  { 0 } ) ) ) x ) )
2322eqeq1d 2451 . . . . . 6  |-  ( y  =  1  ->  (
( y (  x.  |`  ( ( CC  \  { 0 } )  X.  ( CC  \  { 0 } ) ) ) x )  =  x  <->  ( 1 (  x.  |`  (
( CC  \  {
0 } )  X.  ( CC  \  {
0 } ) ) ) x )  =  x ) )
2423ralbidv 2750 . . . . 5  |-  ( y  =  1  ->  ( A. x  e.  ( CC  \  { 0 } ) ( y (  x.  |`  ( ( CC  \  { 0 } )  X.  ( CC 
\  { 0 } ) ) ) x )  =  x  <->  A. x  e.  ( CC  \  {
0 } ) ( 1 (  x.  |`  (
( CC  \  {
0 } )  X.  ( CC  \  {
0 } ) ) ) x )  =  x ) )
2524riota2 6090 . . . 4  |-  ( ( 1  e.  ( CC 
\  { 0 } )  /\  E! y  e.  ( CC  \  { 0 } ) A. x  e.  ( CC  \  { 0 } ) ( y (  x.  |`  (
( CC  \  {
0 } )  X.  ( CC  \  {
0 } ) ) ) x )  =  x )  ->  ( A. x  e.  ( CC  \  { 0 } ) ( 1 (  x.  |`  ( ( CC  \  { 0 } )  X.  ( CC 
\  { 0 } ) ) ) x )  =  x  <->  ( iota_ y  e.  ( CC  \  { 0 } ) A. x  e.  ( CC  \  { 0 } ) ( y (  x.  |`  (
( CC  \  {
0 } )  X.  ( CC  \  {
0 } ) ) ) x )  =  x )  =  1 ) )
2613, 21, 25mp2an 672 . . 3  |-  ( A. x  e.  ( CC  \  { 0 } ) ( 1 (  x.  |`  ( ( CC  \  { 0 } )  X.  ( CC  \  { 0 } ) ) ) x )  =  x  <->  ( iota_ y  e.  ( CC  \  { 0 } ) A. x  e.  ( CC  \  { 0 } ) ( y (  x.  |`  (
( CC  \  {
0 } )  X.  ( CC  \  {
0 } ) ) ) x )  =  x )  =  1 )
2719, 26mpbi 208 . 2  |-  ( iota_ y  e.  ( CC  \  { 0 } ) A. x  e.  ( CC  \  { 0 } ) ( y (  x.  |`  (
( CC  \  {
0 } )  X.  ( CC  \  {
0 } ) ) ) x )  =  x )  =  1
289, 27eqtri 2463 1  |-  (GId `  (  x.  |`  ( ( CC  \  { 0 } )  X.  ( CC  \  { 0 } ) ) ) )  =  1
Colors of variables: wff setvar class
Syntax hints:    <-> wb 184    = wceq 1369    e. wcel 1756    =/= wne 2620   A.wral 2730   E!wreu 2732    \ cdif 3340   {csn 3892    X. cxp 4853    |` cres 4857   ` cfv 5433   iota_crio 6066  (class class class)co 6106   CCcc 9295   0cc0 9297   1c1 9298    x. cmul 9302   GrpOpcgr 23688  GIdcgi 23689   AbelOpcablo 23783
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1591  ax-4 1602  ax-5 1670  ax-6 1708  ax-7 1728  ax-8 1758  ax-9 1760  ax-10 1775  ax-11 1780  ax-12 1792  ax-13 1943  ax-ext 2423  ax-rep 4418  ax-sep 4428  ax-nul 4436  ax-pow 4485  ax-pr 4546  ax-un 6387  ax-cnex 9353  ax-resscn 9354  ax-1cn 9355  ax-icn 9356  ax-addcl 9357  ax-addrcl 9358  ax-mulcl 9359  ax-mulrcl 9360  ax-mulcom 9361  ax-addass 9362  ax-mulass 9363  ax-distr 9364  ax-i2m1 9365  ax-1ne0 9366  ax-1rid 9367  ax-rnegex 9368  ax-rrecex 9369  ax-cnre 9370  ax-pre-lttri 9371  ax-pre-lttrn 9372  ax-pre-ltadd 9373  ax-pre-mulgt0 9374  ax-mulf 9377
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 966  df-3an 967  df-tru 1372  df-ex 1587  df-nf 1590  df-sb 1701  df-eu 2257  df-mo 2258  df-clab 2430  df-cleq 2436  df-clel 2439  df-nfc 2577  df-ne 2622  df-nel 2623  df-ral 2735  df-rex 2736  df-reu 2737  df-rmo 2738  df-rab 2739  df-v 2989  df-sbc 3202  df-csb 3304  df-dif 3346  df-un 3348  df-in 3350  df-ss 3357  df-nul 3653  df-if 3807  df-pw 3877  df-sn 3893  df-pr 3895  df-op 3899  df-uni 4107  df-iun 4188  df-br 4308  df-opab 4366  df-mpt 4367  df-id 4651  df-po 4656  df-so 4657  df-xp 4861  df-rel 4862  df-cnv 4863  df-co 4864  df-dm 4865  df-rn 4866  df-res 4867  df-ima 4868  df-iota 5396  df-fun 5435  df-fn 5436  df-f 5437  df-f1 5438  df-fo 5439  df-f1o 5440  df-fv 5441  df-riota 6067  df-ov 6109  df-oprab 6110  df-mpt2 6111  df-er 7116  df-en 7326  df-dom 7327  df-sdom 7328  df-pnf 9435  df-mnf 9436  df-xr 9437  df-ltxr 9438  df-le 9439  df-sub 9612  df-neg 9613  df-div 10009  df-grpo 23693  df-gid 23694  df-ablo 23784
This theorem is referenced by: (None)
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