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Theorem mulid 26084
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 26083 . . . 4  |-  (  x.  |`  ( ( CC  \  { 0 } )  X.  ( CC  \  { 0 } ) ) )  e.  AbelOp
2 ablogrpo 26012 . . . 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 10258 . . . . . 6  |-  (  x.  |`  ( ( CC  \  { 0 } )  X.  ( CC  \  { 0 } ) ) ) : ( ( CC  \  {
0 } )  X.  ( CC  \  {
0 } ) ) --> ( CC  \  {
0 } )
54fdmi 5734 . . . . 5  |-  dom  (  x.  |`  ( ( CC 
\  { 0 } )  X.  ( CC 
\  { 0 } ) ) )  =  ( ( CC  \  { 0 } )  X.  ( CC  \  { 0 } ) )
63, 5grporn 25940 . . . 4  |-  ( CC 
\  { 0 } )  =  ran  (  x.  |`  ( ( CC 
\  { 0 } )  X.  ( CC 
\  { 0 } ) ) )
7 eqid 2451 . . . 4  |-  (GId `  (  x.  |`  ( ( CC  \  { 0 } )  X.  ( CC  \  { 0 } ) ) ) )  =  (GId `  (  x.  |`  ( ( CC 
\  { 0 } )  X.  ( CC 
\  { 0 } ) ) ) )
86, 7grpoidval 25944 . . 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 9597 . . . . . . 7  |-  1  e.  CC
11 ax-1ne0 9608 . . . . . . 7  |-  1  =/=  0
12 eldifsn 4097 . . . . . . 7  |-  ( 1  e.  ( CC  \  { 0 } )  <-> 
( 1  e.  CC  /\  1  =/=  0 ) )
1310, 11, 12mpbir2an 931 . . . . . 6  |-  1  e.  ( CC  \  {
0 } )
14 ovres 6436 . . . . . 6  |-  ( ( 1  e.  ( CC 
\  { 0 } )  /\  x  e.  ( CC  \  {
0 } ) )  ->  ( 1 (  x.  |`  ( ( CC  \  { 0 } )  X.  ( CC 
\  { 0 } ) ) ) x )  =  ( 1  x.  x ) )
1513, 14mpan 676 . . . . 5  |-  ( x  e.  ( CC  \  { 0 } )  ->  ( 1 (  x.  |`  ( ( CC  \  { 0 } )  X.  ( CC 
\  { 0 } ) ) ) x )  =  ( 1  x.  x ) )
16 eldifi 3555 . . . . . 6  |-  ( x  e.  ( CC  \  { 0 } )  ->  x  e.  CC )
1716mulid2d 9661 . . . . 5  |-  ( x  e.  ( CC  \  { 0 } )  ->  ( 1  x.  x )  =  x )
1815, 17eqtrd 2485 . . . 4  |-  ( x  e.  ( CC  \  { 0 } )  ->  ( 1 (  x.  |`  ( ( CC  \  { 0 } )  X.  ( CC 
\  { 0 } ) ) ) x )  =  x )
1918rgen 2747 . . 3  |-  A. x  e.  ( CC  \  {
0 } ) ( 1 (  x.  |`  (
( CC  \  {
0 } )  X.  ( CC  \  {
0 } ) ) ) x )  =  x
206grpoideu 25937 . . . . 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 6297 . . . . . . 7  |-  ( y  =  1  ->  (
y (  x.  |`  (
( CC  \  {
0 } )  X.  ( CC  \  {
0 } ) ) ) x )  =  ( 1 (  x.  |`  ( ( CC  \  { 0 } )  X.  ( CC  \  { 0 } ) ) ) x ) )
2322eqeq1d 2453 . . . . . 6  |-  ( y  =  1  ->  (
( y (  x.  |`  ( ( CC  \  { 0 } )  X.  ( CC  \  { 0 } ) ) ) x )  =  x  <->  ( 1 (  x.  |`  (
( CC  \  {
0 } )  X.  ( CC  \  {
0 } ) ) ) x )  =  x ) )
2423ralbidv 2827 . . . . 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 6274 . . . 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 678 . . 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 212 . 2  |-  ( iota_ y  e.  ( CC  \  { 0 } ) A. x  e.  ( CC  \  { 0 } ) ( y (  x.  |`  (
( CC  \  {
0 } )  X.  ( CC  \  {
0 } ) ) ) x )  =  x )  =  1
289, 27eqtri 2473 1  |-  (GId `  (  x.  |`  ( ( CC  \  { 0 } )  X.  ( CC  \  { 0 } ) ) ) )  =  1
Colors of variables: wff setvar class
Syntax hints:    <-> wb 188    = wceq 1444    e. wcel 1887    =/= wne 2622   A.wral 2737   E!wreu 2739    \ cdif 3401   {csn 3968    X. cxp 4832    |` cres 4836   ` cfv 5582   iota_crio 6251  (class class class)co 6290   CCcc 9537   0cc0 9539   1c1 9540    x. cmul 9544   GrpOpcgr 25914  GIdcgi 25915   AbelOpcablo 26009
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1669  ax-4 1682  ax-5 1758  ax-6 1805  ax-7 1851  ax-8 1889  ax-9 1896  ax-10 1915  ax-11 1920  ax-12 1933  ax-13 2091  ax-ext 2431  ax-rep 4515  ax-sep 4525  ax-nul 4534  ax-pow 4581  ax-pr 4639  ax-un 6583  ax-cnex 9595  ax-resscn 9596  ax-1cn 9597  ax-icn 9598  ax-addcl 9599  ax-addrcl 9600  ax-mulcl 9601  ax-mulrcl 9602  ax-mulcom 9603  ax-addass 9604  ax-mulass 9605  ax-distr 9606  ax-i2m1 9607  ax-1ne0 9608  ax-1rid 9609  ax-rnegex 9610  ax-rrecex 9611  ax-cnre 9612  ax-pre-lttri 9613  ax-pre-lttrn 9614  ax-pre-ltadd 9615  ax-pre-mulgt0 9616  ax-mulf 9619
This theorem depends on definitions:  df-bi 189  df-or 372  df-an 373  df-3or 986  df-3an 987  df-tru 1447  df-ex 1664  df-nf 1668  df-sb 1798  df-eu 2303  df-mo 2304  df-clab 2438  df-cleq 2444  df-clel 2447  df-nfc 2581  df-ne 2624  df-nel 2625  df-ral 2742  df-rex 2743  df-reu 2744  df-rmo 2745  df-rab 2746  df-v 3047  df-sbc 3268  df-csb 3364  df-dif 3407  df-un 3409  df-in 3411  df-ss 3418  df-nul 3732  df-if 3882  df-pw 3953  df-sn 3969  df-pr 3971  df-op 3975  df-uni 4199  df-iun 4280  df-br 4403  df-opab 4462  df-mpt 4463  df-id 4749  df-po 4755  df-so 4756  df-xp 4840  df-rel 4841  df-cnv 4842  df-co 4843  df-dm 4844  df-rn 4845  df-res 4846  df-ima 4847  df-iota 5546  df-fun 5584  df-fn 5585  df-f 5586  df-f1 5587  df-fo 5588  df-f1o 5589  df-fv 5590  df-riota 6252  df-ov 6293  df-oprab 6294  df-mpt2 6295  df-er 7363  df-en 7570  df-dom 7571  df-sdom 7572  df-pnf 9677  df-mnf 9678  df-xr 9679  df-ltxr 9680  df-le 9681  df-sub 9862  df-neg 9863  df-div 10270  df-grpo 25919  df-gid 25920  df-ablo 26010
This theorem is referenced by: (None)
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