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Theorem mulid 25758
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 25757 . . . 4  |-  (  x.  |`  ( ( CC  \  { 0 } )  X.  ( CC  \  { 0 } ) ) )  e.  AbelOp
2 ablogrpo 25686 . . . 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 10235 . . . . . 6  |-  (  x.  |`  ( ( CC  \  { 0 } )  X.  ( CC  \  { 0 } ) ) ) : ( ( CC  \  {
0 } )  X.  ( CC  \  {
0 } ) ) --> ( CC  \  {
0 } )
54fdmi 5718 . . . . 5  |-  dom  (  x.  |`  ( ( CC 
\  { 0 } )  X.  ( CC 
\  { 0 } ) ) )  =  ( ( CC  \  { 0 } )  X.  ( CC  \  { 0 } ) )
63, 5grporn 25614 . . . 4  |-  ( CC 
\  { 0 } )  =  ran  (  x.  |`  ( ( CC 
\  { 0 } )  X.  ( CC 
\  { 0 } ) ) )
7 eqid 2402 . . . 4  |-  (GId `  (  x.  |`  ( ( CC  \  { 0 } )  X.  ( CC  \  { 0 } ) ) ) )  =  (GId `  (  x.  |`  ( ( CC 
\  { 0 } )  X.  ( CC 
\  { 0 } ) ) ) )
86, 7grpoidval 25618 . . 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 9579 . . . . . . 7  |-  1  e.  CC
11 ax-1ne0 9590 . . . . . . 7  |-  1  =/=  0
12 eldifsn 4096 . . . . . . 7  |-  ( 1  e.  ( CC  \  { 0 } )  <-> 
( 1  e.  CC  /\  1  =/=  0 ) )
1310, 11, 12mpbir2an 921 . . . . . 6  |-  1  e.  ( CC  \  {
0 } )
14 ovres 6422 . . . . . 6  |-  ( ( 1  e.  ( CC 
\  { 0 } )  /\  x  e.  ( CC  \  {
0 } ) )  ->  ( 1 (  x.  |`  ( ( CC  \  { 0 } )  X.  ( CC 
\  { 0 } ) ) ) x )  =  ( 1  x.  x ) )
1513, 14mpan 668 . . . . 5  |-  ( x  e.  ( CC  \  { 0 } )  ->  ( 1 (  x.  |`  ( ( CC  \  { 0 } )  X.  ( CC 
\  { 0 } ) ) ) x )  =  ( 1  x.  x ) )
16 eldifi 3564 . . . . . 6  |-  ( x  e.  ( CC  \  { 0 } )  ->  x  e.  CC )
1716mulid2d 9643 . . . . 5  |-  ( x  e.  ( CC  \  { 0 } )  ->  ( 1  x.  x )  =  x )
1815, 17eqtrd 2443 . . . 4  |-  ( x  e.  ( CC  \  { 0 } )  ->  ( 1 (  x.  |`  ( ( CC  \  { 0 } )  X.  ( CC 
\  { 0 } ) ) ) x )  =  x )
1918rgen 2763 . . 3  |-  A. x  e.  ( CC  \  {
0 } ) ( 1 (  x.  |`  (
( CC  \  {
0 } )  X.  ( CC  \  {
0 } ) ) ) x )  =  x
206grpoideu 25611 . . . . 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 6284 . . . . . . 7  |-  ( y  =  1  ->  (
y (  x.  |`  (
( CC  \  {
0 } )  X.  ( CC  \  {
0 } ) ) ) x )  =  ( 1 (  x.  |`  ( ( CC  \  { 0 } )  X.  ( CC  \  { 0 } ) ) ) x ) )
2322eqeq1d 2404 . . . . . 6  |-  ( y  =  1  ->  (
( y (  x.  |`  ( ( CC  \  { 0 } )  X.  ( CC  \  { 0 } ) ) ) x )  =  x  <->  ( 1 (  x.  |`  (
( CC  \  {
0 } )  X.  ( CC  \  {
0 } ) ) ) x )  =  x ) )
2423ralbidv 2842 . . . . 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 6261 . . . 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 670 . . 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 2431 1  |-  (GId `  (  x.  |`  ( ( CC  \  { 0 } )  X.  ( CC  \  { 0 } ) ) ) )  =  1
Colors of variables: wff setvar class
Syntax hints:    <-> wb 184    = wceq 1405    e. wcel 1842    =/= wne 2598   A.wral 2753   E!wreu 2755    \ cdif 3410   {csn 3971    X. cxp 4820    |` cres 4824   ` cfv 5568   iota_crio 6238  (class class class)co 6277   CCcc 9519   0cc0 9521   1c1 9522    x. cmul 9526   GrpOpcgr 25588  GIdcgi 25589   AbelOpcablo 25683
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1639  ax-4 1652  ax-5 1725  ax-6 1771  ax-7 1814  ax-8 1844  ax-9 1846  ax-10 1861  ax-11 1866  ax-12 1878  ax-13 2026  ax-ext 2380  ax-rep 4506  ax-sep 4516  ax-nul 4524  ax-pow 4571  ax-pr 4629  ax-un 6573  ax-cnex 9577  ax-resscn 9578  ax-1cn 9579  ax-icn 9580  ax-addcl 9581  ax-addrcl 9582  ax-mulcl 9583  ax-mulrcl 9584  ax-mulcom 9585  ax-addass 9586  ax-mulass 9587  ax-distr 9588  ax-i2m1 9589  ax-1ne0 9590  ax-1rid 9591  ax-rnegex 9592  ax-rrecex 9593  ax-cnre 9594  ax-pre-lttri 9595  ax-pre-lttrn 9596  ax-pre-ltadd 9597  ax-pre-mulgt0 9598  ax-mulf 9601
This theorem depends on definitions:  df-bi 185  df-or 368  df-an 369  df-3or 975  df-3an 976  df-tru 1408  df-ex 1634  df-nf 1638  df-sb 1764  df-eu 2242  df-mo 2243  df-clab 2388  df-cleq 2394  df-clel 2397  df-nfc 2552  df-ne 2600  df-nel 2601  df-ral 2758  df-rex 2759  df-reu 2760  df-rmo 2761  df-rab 2762  df-v 3060  df-sbc 3277  df-csb 3373  df-dif 3416  df-un 3418  df-in 3420  df-ss 3427  df-nul 3738  df-if 3885  df-pw 3956  df-sn 3972  df-pr 3974  df-op 3978  df-uni 4191  df-iun 4272  df-br 4395  df-opab 4453  df-mpt 4454  df-id 4737  df-po 4743  df-so 4744  df-xp 4828  df-rel 4829  df-cnv 4830  df-co 4831  df-dm 4832  df-rn 4833  df-res 4834  df-ima 4835  df-iota 5532  df-fun 5570  df-fn 5571  df-f 5572  df-f1 5573  df-fo 5574  df-f1o 5575  df-fv 5576  df-riota 6239  df-ov 6280  df-oprab 6281  df-mpt2 6282  df-er 7347  df-en 7554  df-dom 7555  df-sdom 7556  df-pnf 9659  df-mnf 9660  df-xr 9661  df-ltxr 9662  df-le 9663  df-sub 9842  df-neg 9843  df-div 10247  df-grpo 25593  df-gid 25594  df-ablo 25684
This theorem is referenced by: (None)
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