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Theorem vcoprne 25599
Description: The operations of a complex vector space cannot be identical. (Contributed by NM, 31-May-2008.) (New usage is discouraged.)
Assertion
Ref Expression
vcoprne  |-  ( <. G ,  S >.  e. 
CVecOLD  ->  G  =/=  S )

Proof of Theorem vcoprne
StepHypRef Expression
1 ax-1ne0 9578 . . . . 5  |-  1  =/=  0
21neii 2656 . . . 4  |-  -.  1  =  0
3 vcoprnelem 25598 . . . . . . . . 9  |-  ( <. G ,  G >.  e. 
CVecOLD  ->  G :
( CC  X.  CC )
--> CC )
4 cnex 9590 . . . . . . . . . 10  |-  CC  e.  _V
54, 4xpex 6603 . . . . . . . . 9  |-  ( CC 
X.  CC )  e. 
_V
6 fex 6146 . . . . . . . . 9  |-  ( ( G : ( CC 
X.  CC ) --> CC 
/\  ( CC  X.  CC )  e.  _V )  ->  G  e.  _V )
73, 5, 6sylancl 662 . . . . . . . 8  |-  ( <. G ,  G >.  e. 
CVecOLD  ->  G  e.  _V )
8 op1stg 6811 . . . . . . . 8  |-  ( ( G  e.  _V  /\  G  e.  _V )  ->  ( 1st `  <. G ,  G >. )  =  G )
97, 7, 8syl2anc 661 . . . . . . 7  |-  ( <. G ,  G >.  e. 
CVecOLD  ->  ( 1st ` 
<. G ,  G >. )  =  G )
109oveqd 6313 . . . . . 6  |-  ( <. G ,  G >.  e. 
CVecOLD  ->  ( 0 ( 1st `  <. G ,  G >. )
(GId `  ( 1st ` 
<. G ,  G >. ) ) )  =  ( 0 G (GId `  ( 1st `  <. G ,  G >. ) ) ) )
11 ax-1cn 9567 . . . . . . . . . . 11  |-  1  e.  CC
129rneqd 5240 . . . . . . . . . . . 12  |-  ( <. G ,  G >.  e. 
CVecOLD  ->  ran  ( 1st `  <. G ,  G >. )  =  ran  G
)
13 eqid 2457 . . . . . . . . . . . . . . 15  |-  ( 1st `  <. G ,  G >. )  =  ( 1st `  <. G ,  G >. )
1413vcgrp 25578 . . . . . . . . . . . . . 14  |-  ( <. G ,  G >.  e. 
CVecOLD  ->  ( 1st ` 
<. G ,  G >. )  e.  GrpOp )
159, 14eqeltrrd 2546 . . . . . . . . . . . . 13  |-  ( <. G ,  G >.  e. 
CVecOLD  ->  G  e.  GrpOp
)
16 grporndm 25339 . . . . . . . . . . . . 13  |-  ( G  e.  GrpOp  ->  ran  G  =  dom  dom  G )
1715, 16syl 16 . . . . . . . . . . . 12  |-  ( <. G ,  G >.  e. 
CVecOLD  ->  ran  G  =  dom  dom  G )
18 fdm 5741 . . . . . . . . . . . . . . 15  |-  ( G : ( CC  X.  CC ) --> CC  ->  dom  G  =  ( CC  X.  CC ) )
193, 18syl 16 . . . . . . . . . . . . . 14  |-  ( <. G ,  G >.  e. 
CVecOLD  ->  dom  G  =  ( CC  X.  CC ) )
2019dmeqd 5215 . . . . . . . . . . . . 13  |-  ( <. G ,  G >.  e. 
CVecOLD  ->  dom  dom  G  =  dom  ( CC  X.  CC ) )
21 dmxpid 5232 . . . . . . . . . . . . 13  |-  dom  ( CC  X.  CC )  =  CC
2220, 21syl6eq 2514 . . . . . . . . . . . 12  |-  ( <. G ,  G >.  e. 
CVecOLD  ->  dom  dom  G  =  CC )
2312, 17, 223eqtrd 2502 . . . . . . . . . . 11  |-  ( <. G ,  G >.  e. 
CVecOLD  ->  ran  ( 1st `  <. G ,  G >. )  =  CC )
2411, 23syl5eleqr 2552 . . . . . . . . . 10  |-  ( <. G ,  G >.  e. 
CVecOLD  ->  1  e.  ran  ( 1st `  <. G ,  G >. )
)
25 eqid 2457 . . . . . . . . . . 11  |-  ran  ( 1st `  <. G ,  G >. )  =  ran  ( 1st `  <. G ,  G >. )
26 eqid 2457 . . . . . . . . . . 11  |-  (GId `  ( 1st `  <. G ,  G >. ) )  =  (GId `  ( 1st ` 
<. G ,  G >. ) )
2713, 25, 26vc0rid 25587 . . . . . . . . . 10  |-  ( (
<. G ,  G >.  e. 
CVecOLD  /\  1  e. 
ran  ( 1st `  <. G ,  G >. )
)  ->  ( 1 ( 1st `  <. G ,  G >. )
(GId `  ( 1st ` 
<. G ,  G >. ) ) )  =  1 )
2824, 27mpdan 668 . . . . . . . . 9  |-  ( <. G ,  G >.  e. 
CVecOLD  ->  ( 1 ( 1st `  <. G ,  G >. )
(GId `  ( 1st ` 
<. G ,  G >. ) ) )  =  1 )
29 eqid 2457 . . . . . . . . . . 11  |-  ( 2nd `  <. G ,  G >. )  =  ( 2nd `  <. G ,  G >. )
3013, 29, 25vcid 25571 . . . . . . . . . 10  |-  ( (
<. G ,  G >.  e. 
CVecOLD  /\  1  e. 
ran  ( 1st `  <. G ,  G >. )
)  ->  ( 1 ( 2nd `  <. G ,  G >. )
1 )  =  1 )
3124, 30mpdan 668 . . . . . . . . 9  |-  ( <. G ,  G >.  e. 
CVecOLD  ->  ( 1 ( 2nd `  <. G ,  G >. )
1 )  =  1 )
32 op2ndg 6812 . . . . . . . . . . . 12  |-  ( ( G  e.  _V  /\  G  e.  _V )  ->  ( 2nd `  <. G ,  G >. )  =  G )
337, 7, 32syl2anc 661 . . . . . . . . . . 11  |-  ( <. G ,  G >.  e. 
CVecOLD  ->  ( 2nd ` 
<. G ,  G >. )  =  G )
3433, 9eqtr4d 2501 . . . . . . . . . 10  |-  ( <. G ,  G >.  e. 
CVecOLD  ->  ( 2nd ` 
<. G ,  G >. )  =  ( 1st `  <. G ,  G >. )
)
3534oveqd 6313 . . . . . . . . 9  |-  ( <. G ,  G >.  e. 
CVecOLD  ->  ( 1 ( 2nd `  <. G ,  G >. )
1 )  =  ( 1 ( 1st `  <. G ,  G >. )
1 ) )
3628, 31, 353eqtr2d 2504 . . . . . . . 8  |-  ( <. G ,  G >.  e. 
CVecOLD  ->  ( 1 ( 1st `  <. G ,  G >. )
(GId `  ( 1st ` 
<. G ,  G >. ) ) )  =  ( 1 ( 1st `  <. G ,  G >. )
1 ) )
3713, 25, 26vczcl 25586 . . . . . . . . . 10  |-  ( <. G ,  G >.  e. 
CVecOLD  ->  (GId `  ( 1st `  <. G ,  G >. ) )  e.  ran  ( 1st `  <. G ,  G >. ) )
3837, 24, 243jca 1176 . . . . . . . . 9  |-  ( <. G ,  G >.  e. 
CVecOLD  ->  ( (GId `  ( 1st `  <. G ,  G >. )
)  e.  ran  ( 1st `  <. G ,  G >. )  /\  1  e. 
ran  ( 1st `  <. G ,  G >. )  /\  1  e.  ran  ( 1st `  <. G ,  G >. ) ) )
3913, 25vclcan 25585 . . . . . . . . 9  |-  ( (
<. G ,  G >.  e. 
CVecOLD  /\  ( (GId
`  ( 1st `  <. G ,  G >. )
)  e.  ran  ( 1st `  <. G ,  G >. )  /\  1  e. 
ran  ( 1st `  <. G ,  G >. )  /\  1  e.  ran  ( 1st `  <. G ,  G >. ) ) )  ->  ( ( 1 ( 1st `  <. G ,  G >. )
(GId `  ( 1st ` 
<. G ,  G >. ) ) )  =  ( 1 ( 1st `  <. G ,  G >. )
1 )  <->  (GId `  ( 1st `  <. G ,  G >. ) )  =  1 ) )
4038, 39mpdan 668 . . . . . . . 8  |-  ( <. G ,  G >.  e. 
CVecOLD  ->  ( (
1 ( 1st `  <. G ,  G >. )
(GId `  ( 1st ` 
<. G ,  G >. ) ) )  =  ( 1 ( 1st `  <. G ,  G >. )
1 )  <->  (GId `  ( 1st `  <. G ,  G >. ) )  =  1 ) )
4136, 40mpbid 210 . . . . . . 7  |-  ( <. G ,  G >.  e. 
CVecOLD  ->  (GId `  ( 1st `  <. G ,  G >. ) )  =  1 )
4241oveq2d 6312 . . . . . 6  |-  ( <. G ,  G >.  e. 
CVecOLD  ->  ( 0 G (GId `  ( 1st `  <. G ,  G >. ) ) )  =  ( 0 G 1 ) )
43 0cn 9605 . . . . . . . . 9  |-  0  e.  CC
4413, 29, 25, 26vcz 25590 . . . . . . . . 9  |-  ( (
<. G ,  G >.  e. 
CVecOLD  /\  0  e.  CC )  ->  (
0 ( 2nd `  <. G ,  G >. )
(GId `  ( 1st ` 
<. G ,  G >. ) ) )  =  (GId
`  ( 1st `  <. G ,  G >. )
) )
4543, 44mpan2 671 . . . . . . . 8  |-  ( <. G ,  G >.  e. 
CVecOLD  ->  ( 0 ( 2nd `  <. G ,  G >. )
(GId `  ( 1st ` 
<. G ,  G >. ) ) )  =  (GId
`  ( 1st `  <. G ,  G >. )
) )
4633oveqd 6313 . . . . . . . . 9  |-  ( <. G ,  G >.  e. 
CVecOLD  ->  ( 0 ( 2nd `  <. G ,  G >. )
(GId `  ( 1st ` 
<. G ,  G >. ) ) )  =  ( 0 G (GId `  ( 1st `  <. G ,  G >. ) ) ) )
4746, 42eqtrd 2498 . . . . . . . 8  |-  ( <. G ,  G >.  e. 
CVecOLD  ->  ( 0 ( 2nd `  <. G ,  G >. )
(GId `  ( 1st ` 
<. G ,  G >. ) ) )  =  ( 0 G 1 ) )
4845, 47eqtr3d 2500 . . . . . . 7  |-  ( <. G ,  G >.  e. 
CVecOLD  ->  (GId `  ( 1st `  <. G ,  G >. ) )  =  ( 0 G 1 ) )
4948, 41eqtr3d 2500 . . . . . 6  |-  ( <. G ,  G >.  e. 
CVecOLD  ->  ( 0 G 1 )  =  1 )
5010, 42, 493eqtrd 2502 . . . . 5  |-  ( <. G ,  G >.  e. 
CVecOLD  ->  ( 0 ( 1st `  <. G ,  G >. )
(GId `  ( 1st ` 
<. G ,  G >. ) ) )  =  1 )
5143, 23syl5eleqr 2552 . . . . . 6  |-  ( <. G ,  G >.  e. 
CVecOLD  ->  0  e.  ran  ( 1st `  <. G ,  G >. )
)
5213, 25, 26vc0rid 25587 . . . . . 6  |-  ( (
<. G ,  G >.  e. 
CVecOLD  /\  0  e. 
ran  ( 1st `  <. G ,  G >. )
)  ->  ( 0 ( 1st `  <. G ,  G >. )
(GId `  ( 1st ` 
<. G ,  G >. ) ) )  =  0 )
5351, 52mpdan 668 . . . . 5  |-  ( <. G ,  G >.  e. 
CVecOLD  ->  ( 0 ( 1st `  <. G ,  G >. )
(GId `  ( 1st ` 
<. G ,  G >. ) ) )  =  0 )
5450, 53eqtr3d 2500 . . . 4  |-  ( <. G ,  G >.  e. 
CVecOLD  ->  1  = 
0 )
552, 54mto 176 . . 3  |-  -.  <. G ,  G >.  e.  CVecOLD
56 opeq2 4220 . . . 4  |-  ( G  =  S  ->  <. G ,  G >.  =  <. G ,  S >. )
5756eleq1d 2526 . . 3  |-  ( G  =  S  ->  ( <. G ,  G >.  e. 
CVecOLD  <->  <. G ,  S >.  e.  CVecOLD ) )
5855, 57mtbii 302 . 2  |-  ( G  =  S  ->  -.  <. G ,  S >.  e. 
CVecOLD )
5958necon2ai 2692 1  |-  ( <. G ,  S >.  e. 
CVecOLD  ->  G  =/=  S )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    <-> wb 184    /\ w3a 973    = wceq 1395    e. wcel 1819    =/= wne 2652   _Vcvv 3109   <.cop 4038    X. cxp 5006   dom cdm 5008   ran crn 5009   -->wf 5590   ` cfv 5594  (class class class)co 6296   1stc1st 6797   2ndc2nd 6798   CCcc 9507   0cc0 9509   1c1 9510   GrpOpcgr 25315  GIdcgi 25316   CVecOLDcvc 25565
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1619  ax-4 1632  ax-5 1705  ax-6 1748  ax-7 1791  ax-8 1821  ax-9 1823  ax-10 1838  ax-11 1843  ax-12 1855  ax-13 2000  ax-ext 2435  ax-rep 4568  ax-sep 4578  ax-nul 4586  ax-pow 4634  ax-pr 4695  ax-un 6591  ax-cnex 9565  ax-resscn 9566  ax-1cn 9567  ax-icn 9568  ax-addcl 9569  ax-addrcl 9570  ax-mulcl 9571  ax-mulrcl 9572  ax-mulcom 9573  ax-addass 9574  ax-mulass 9575  ax-distr 9576  ax-i2m1 9577  ax-1ne0 9578  ax-1rid 9579  ax-rnegex 9580  ax-rrecex 9581  ax-cnre 9582  ax-pre-lttri 9583  ax-pre-lttrn 9584  ax-pre-ltadd 9585
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 974  df-3an 975  df-tru 1398  df-ex 1614  df-nf 1618  df-sb 1741  df-eu 2287  df-mo 2288  df-clab 2443  df-cleq 2449  df-clel 2452  df-nfc 2607  df-ne 2654  df-nel 2655  df-ral 2812  df-rex 2813  df-reu 2814  df-rab 2816  df-v 3111  df-sbc 3328  df-csb 3431  df-dif 3474  df-un 3476  df-in 3478  df-ss 3485  df-nul 3794  df-if 3945  df-pw 4017  df-sn 4033  df-pr 4035  df-op 4039  df-uni 4252  df-iun 4334  df-br 4457  df-opab 4516  df-mpt 4517  df-id 4804  df-po 4809  df-so 4810  df-xp 5014  df-rel 5015  df-cnv 5016  df-co 5017  df-dm 5018  df-rn 5019  df-res 5020  df-ima 5021  df-iota 5557  df-fun 5596  df-fn 5597  df-f 5598  df-f1 5599  df-fo 5600  df-f1o 5601  df-fv 5602  df-riota 6258  df-ov 6299  df-1st 6799  df-2nd 6800  df-er 7329  df-en 7536  df-dom 7537  df-sdom 7538  df-pnf 9647  df-mnf 9648  df-ltxr 9650  df-grpo 25320  df-gid 25321  df-ginv 25322  df-ablo 25411  df-vc 25566
This theorem is referenced by:  nvoprne  25708
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