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Theorem mulid 21039
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 21038 . . . 4  |-  (  x.  |`  ( ( CC  \  { 0 } )  X.  ( CC  \  { 0 } ) ) )  e.  AbelOp
2 ablogrpo 20967 . . . 4  |-  ( (  x.  |`  ( ( CC  \  { 0 } )  X.  ( CC 
\  { 0 } ) ) )  e. 
AbelOp  ->  (  x.  |`  (
( CC  \  {
0 } )  X.  ( CC  \  {
0 } ) ) )  e.  GrpOp )
31, 2ax-mp 8 . . 3  |-  (  x.  |`  ( ( CC  \  { 0 } )  X.  ( CC  \  { 0 } ) ) )  e.  GrpOp
4 mulnzcnopr 9430 . . . . . 6  |-  (  x.  |`  ( ( CC  \  { 0 } )  X.  ( CC  \  { 0 } ) ) ) : ( ( CC  \  {
0 } )  X.  ( CC  \  {
0 } ) ) --> ( CC  \  {
0 } )
54fdmi 5410 . . . . 5  |-  dom  (  x.  |`  ( ( CC 
\  { 0 } )  X.  ( CC 
\  { 0 } ) ) )  =  ( ( CC  \  { 0 } )  X.  ( CC  \  { 0 } ) )
63, 5grporn 20895 . . . 4  |-  ( CC 
\  { 0 } )  =  ran  (  x.  |`  ( ( CC 
\  { 0 } )  X.  ( CC 
\  { 0 } ) ) )
7 eqid 2296 . . . 4  |-  (GId `  (  x.  |`  ( ( CC  \  { 0 } )  X.  ( CC  \  { 0 } ) ) ) )  =  (GId `  (  x.  |`  ( ( CC 
\  { 0 } )  X.  ( CC 
\  { 0 } ) ) ) )
86, 7grpoidval 20899 . . 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 8 . 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 8811 . . . . . . 7  |-  1  e.  CC
11 ax-1ne0 8822 . . . . . . 7  |-  1  =/=  0
12 eldifsn 3762 . . . . . . 7  |-  ( 1  e.  ( CC  \  { 0 } )  <-> 
( 1  e.  CC  /\  1  =/=  0 ) )
1310, 11, 12mpbir2an 886 . . . . . 6  |-  1  e.  ( CC  \  {
0 } )
14 ovres 6003 . . . . . 6  |-  ( ( 1  e.  ( CC 
\  { 0 } )  /\  x  e.  ( CC  \  {
0 } ) )  ->  ( 1 (  x.  |`  ( ( CC  \  { 0 } )  X.  ( CC 
\  { 0 } ) ) ) x )  =  ( 1  x.  x ) )
1513, 14mpan 651 . . . . 5  |-  ( x  e.  ( CC  \  { 0 } )  ->  ( 1 (  x.  |`  ( ( CC  \  { 0 } )  X.  ( CC 
\  { 0 } ) ) ) x )  =  ( 1  x.  x ) )
16 eldifi 3311 . . . . . 6  |-  ( x  e.  ( CC  \  { 0 } )  ->  x  e.  CC )
1716mulid2d 8869 . . . . 5  |-  ( x  e.  ( CC  \  { 0 } )  ->  ( 1  x.  x )  =  x )
1815, 17eqtrd 2328 . . . 4  |-  ( x  e.  ( CC  \  { 0 } )  ->  ( 1 (  x.  |`  ( ( CC  \  { 0 } )  X.  ( CC 
\  { 0 } ) ) ) x )  =  x )
1918rgen 2621 . . 3  |-  A. x  e.  ( CC  \  {
0 } ) ( 1 (  x.  |`  (
( CC  \  {
0 } )  X.  ( CC  \  {
0 } ) ) ) x )  =  x
206grpoideu 20892 . . . . 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 8 . . . 4  |-  E! y  e.  ( CC  \  { 0 } ) A. x  e.  ( CC  \  { 0 } ) ( y (  x.  |`  (
( CC  \  {
0 } )  X.  ( CC  \  {
0 } ) ) ) x )  =  x
22 oveq1 5881 . . . . . . 7  |-  ( y  =  1  ->  (
y (  x.  |`  (
( CC  \  {
0 } )  X.  ( CC  \  {
0 } ) ) ) x )  =  ( 1 (  x.  |`  ( ( CC  \  { 0 } )  X.  ( CC  \  { 0 } ) ) ) x ) )
2322eqeq1d 2304 . . . . . 6  |-  ( y  =  1  ->  (
( y (  x.  |`  ( ( CC  \  { 0 } )  X.  ( CC  \  { 0 } ) ) ) x )  =  x  <->  ( 1 (  x.  |`  (
( CC  \  {
0 } )  X.  ( CC  \  {
0 } ) ) ) x )  =  x ) )
2423ralbidv 2576 . . . . 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 6343 . . . 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 653 . . 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 199 . 2  |-  ( iota_ y  e.  ( CC  \  { 0 } ) A. x  e.  ( CC  \  { 0 } ) ( y (  x.  |`  (
( CC  \  {
0 } )  X.  ( CC  \  {
0 } ) ) ) x )  =  x )  =  1
289, 27eqtri 2316 1  |-  (GId `  (  x.  |`  ( ( CC  \  { 0 } )  X.  ( CC  \  { 0 } ) ) ) )  =  1
Colors of variables: wff set class
Syntax hints:    <-> wb 176    = wceq 1632    e. wcel 1696    =/= wne 2459   A.wral 2556   E!wreu 2558    \ cdif 3162   {csn 3653    X. cxp 4703    |` cres 4707   ` cfv 5271  (class class class)co 5874   iota_crio 6313   CCcc 8751   0cc0 8753   1c1 8754    x. cmul 8758   GrpOpcgr 20869  GIdcgi 20870   AbelOpcablo 20964
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1536  ax-5 1547  ax-17 1606  ax-9 1644  ax-8 1661  ax-13 1698  ax-14 1700  ax-6 1715  ax-7 1720  ax-11 1727  ax-12 1878  ax-ext 2277  ax-rep 4147  ax-sep 4157  ax-nul 4165  ax-pow 4204  ax-pr 4230  ax-un 4528  ax-cnex 8809  ax-resscn 8810  ax-1cn 8811  ax-icn 8812  ax-addcl 8813  ax-addrcl 8814  ax-mulcl 8815  ax-mulrcl 8816  ax-mulcom 8817  ax-addass 8818  ax-mulass 8819  ax-distr 8820  ax-i2m1 8821  ax-1ne0 8822  ax-1rid 8823  ax-rnegex 8824  ax-rrecex 8825  ax-cnre 8826  ax-pre-lttri 8827  ax-pre-lttrn 8828  ax-pre-ltadd 8829  ax-pre-mulgt0 8830  ax-mulf 8833
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 1532  df-nf 1535  df-sb 1639  df-eu 2160  df-mo 2161  df-clab 2283  df-cleq 2289  df-clel 2292  df-nfc 2421  df-ne 2461  df-nel 2462  df-ral 2561  df-rex 2562  df-reu 2563  df-rmo 2564  df-rab 2565  df-v 2803  df-sbc 3005  df-csb 3095  df-dif 3168  df-un 3170  df-in 3172  df-ss 3179  df-nul 3469  df-if 3579  df-pw 3640  df-sn 3659  df-pr 3660  df-op 3662  df-uni 3844  df-iun 3923  df-br 4040  df-opab 4094  df-mpt 4095  df-id 4325  df-po 4330  df-so 4331  df-xp 4711  df-rel 4712  df-cnv 4713  df-co 4714  df-dm 4715  df-rn 4716  df-res 4717  df-ima 4718  df-iota 5235  df-fun 5273  df-fn 5274  df-f 5275  df-f1 5276  df-fo 5277  df-f1o 5278  df-fv 5279  df-ov 5877  df-oprab 5878  df-mpt2 5879  df-riota 6320  df-er 6676  df-en 6880  df-dom 6881  df-sdom 6882  df-pnf 8885  df-mnf 8886  df-xr 8887  df-ltxr 8888  df-le 8889  df-sub 9055  df-neg 9056  df-div 9440  df-grpo 20874  df-gid 20875  df-ablo 20965
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