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Theorem vcoprne 25245
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 9562 . . . . 5  |-  1  =/=  0
2 df-ne 2664 . . . . 5  |-  ( 1  =/=  0  <->  -.  1  =  0 )
31, 2mpbi 208 . . . 4  |-  -.  1  =  0
4 vcoprnelem 25244 . . . . . . . . 9  |-  ( <. G ,  G >.  e. 
CVecOLD  ->  G :
( CC  X.  CC )
--> CC )
5 cnex 9574 . . . . . . . . . 10  |-  CC  e.  _V
65, 5xpex 6589 . . . . . . . . 9  |-  ( CC 
X.  CC )  e. 
_V
7 fex 6134 . . . . . . . . 9  |-  ( ( G : ( CC 
X.  CC ) --> CC 
/\  ( CC  X.  CC )  e.  _V )  ->  G  e.  _V )
84, 6, 7sylancl 662 . . . . . . . 8  |-  ( <. G ,  G >.  e. 
CVecOLD  ->  G  e.  _V )
9 op1stg 6797 . . . . . . . 8  |-  ( ( G  e.  _V  /\  G  e.  _V )  ->  ( 1st `  <. G ,  G >. )  =  G )
108, 8, 9syl2anc 661 . . . . . . 7  |-  ( <. G ,  G >.  e. 
CVecOLD  ->  ( 1st ` 
<. G ,  G >. )  =  G )
1110oveqd 6302 . . . . . 6  |-  ( <. G ,  G >.  e. 
CVecOLD  ->  ( 0 ( 1st `  <. G ,  G >. )
(GId `  ( 1st ` 
<. G ,  G >. ) ) )  =  ( 0 G (GId `  ( 1st `  <. G ,  G >. ) ) ) )
12 ax-1cn 9551 . . . . . . . . . . 11  |-  1  e.  CC
1310rneqd 5230 . . . . . . . . . . . 12  |-  ( <. G ,  G >.  e. 
CVecOLD  ->  ran  ( 1st `  <. G ,  G >. )  =  ran  G
)
14 eqid 2467 . . . . . . . . . . . . . . 15  |-  ( 1st `  <. G ,  G >. )  =  ( 1st `  <. G ,  G >. )
1514vcgrp 25224 . . . . . . . . . . . . . 14  |-  ( <. G ,  G >.  e. 
CVecOLD  ->  ( 1st ` 
<. G ,  G >. )  e.  GrpOp )
1610, 15eqeltrrd 2556 . . . . . . . . . . . . 13  |-  ( <. G ,  G >.  e. 
CVecOLD  ->  G  e.  GrpOp
)
17 grporndm 24985 . . . . . . . . . . . . 13  |-  ( G  e.  GrpOp  ->  ran  G  =  dom  dom  G )
1816, 17syl 16 . . . . . . . . . . . 12  |-  ( <. G ,  G >.  e. 
CVecOLD  ->  ran  G  =  dom  dom  G )
19 fdm 5735 . . . . . . . . . . . . . . 15  |-  ( G : ( CC  X.  CC ) --> CC  ->  dom  G  =  ( CC  X.  CC ) )
204, 19syl 16 . . . . . . . . . . . . . 14  |-  ( <. G ,  G >.  e. 
CVecOLD  ->  dom  G  =  ( CC  X.  CC ) )
2120dmeqd 5205 . . . . . . . . . . . . 13  |-  ( <. G ,  G >.  e. 
CVecOLD  ->  dom  dom  G  =  dom  ( CC  X.  CC ) )
22 dmxpid 5222 . . . . . . . . . . . . 13  |-  dom  ( CC  X.  CC )  =  CC
2321, 22syl6eq 2524 . . . . . . . . . . . 12  |-  ( <. G ,  G >.  e. 
CVecOLD  ->  dom  dom  G  =  CC )
2413, 18, 233eqtrd 2512 . . . . . . . . . . 11  |-  ( <. G ,  G >.  e. 
CVecOLD  ->  ran  ( 1st `  <. G ,  G >. )  =  CC )
2512, 24syl5eleqr 2562 . . . . . . . . . 10  |-  ( <. G ,  G >.  e. 
CVecOLD  ->  1  e.  ran  ( 1st `  <. G ,  G >. )
)
26 eqid 2467 . . . . . . . . . . 11  |-  ran  ( 1st `  <. G ,  G >. )  =  ran  ( 1st `  <. G ,  G >. )
27 eqid 2467 . . . . . . . . . . 11  |-  (GId `  ( 1st `  <. G ,  G >. ) )  =  (GId `  ( 1st ` 
<. G ,  G >. ) )
2814, 26, 27vc0rid 25233 . . . . . . . . . 10  |-  ( (
<. G ,  G >.  e. 
CVecOLD  /\  1  e. 
ran  ( 1st `  <. G ,  G >. )
)  ->  ( 1 ( 1st `  <. G ,  G >. )
(GId `  ( 1st ` 
<. G ,  G >. ) ) )  =  1 )
2925, 28mpdan 668 . . . . . . . . 9  |-  ( <. G ,  G >.  e. 
CVecOLD  ->  ( 1 ( 1st `  <. G ,  G >. )
(GId `  ( 1st ` 
<. G ,  G >. ) ) )  =  1 )
30 eqid 2467 . . . . . . . . . . 11  |-  ( 2nd `  <. G ,  G >. )  =  ( 2nd `  <. G ,  G >. )
3114, 30, 26vcid 25217 . . . . . . . . . 10  |-  ( (
<. G ,  G >.  e. 
CVecOLD  /\  1  e. 
ran  ( 1st `  <. G ,  G >. )
)  ->  ( 1 ( 2nd `  <. G ,  G >. )
1 )  =  1 )
3225, 31mpdan 668 . . . . . . . . 9  |-  ( <. G ,  G >.  e. 
CVecOLD  ->  ( 1 ( 2nd `  <. G ,  G >. )
1 )  =  1 )
33 op2ndg 6798 . . . . . . . . . . . 12  |-  ( ( G  e.  _V  /\  G  e.  _V )  ->  ( 2nd `  <. G ,  G >. )  =  G )
348, 8, 33syl2anc 661 . . . . . . . . . . 11  |-  ( <. G ,  G >.  e. 
CVecOLD  ->  ( 2nd ` 
<. G ,  G >. )  =  G )
3534, 10eqtr4d 2511 . . . . . . . . . 10  |-  ( <. G ,  G >.  e. 
CVecOLD  ->  ( 2nd ` 
<. G ,  G >. )  =  ( 1st `  <. G ,  G >. )
)
3635oveqd 6302 . . . . . . . . 9  |-  ( <. G ,  G >.  e. 
CVecOLD  ->  ( 1 ( 2nd `  <. G ,  G >. )
1 )  =  ( 1 ( 1st `  <. G ,  G >. )
1 ) )
3729, 32, 363eqtr2d 2514 . . . . . . . 8  |-  ( <. G ,  G >.  e. 
CVecOLD  ->  ( 1 ( 1st `  <. G ,  G >. )
(GId `  ( 1st ` 
<. G ,  G >. ) ) )  =  ( 1 ( 1st `  <. G ,  G >. )
1 ) )
3814, 26, 27vczcl 25232 . . . . . . . . . 10  |-  ( <. G ,  G >.  e. 
CVecOLD  ->  (GId `  ( 1st `  <. G ,  G >. ) )  e.  ran  ( 1st `  <. G ,  G >. ) )
3938, 25, 253jca 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 >. ) ) )
4014, 26vclcan 25231 . . . . . . . . 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 ) )
4139, 40mpdan 668 . . . . . . . 8  |-  ( <. G ,  G >.  e. 
CVecOLD  ->  ( (
1 ( 1st `  <. G ,  G >. )
(GId `  ( 1st ` 
<. G ,  G >. ) ) )  =  ( 1 ( 1st `  <. G ,  G >. )
1 )  <->  (GId `  ( 1st `  <. G ,  G >. ) )  =  1 ) )
4237, 41mpbid 210 . . . . . . 7  |-  ( <. G ,  G >.  e. 
CVecOLD  ->  (GId `  ( 1st `  <. G ,  G >. ) )  =  1 )
4342oveq2d 6301 . . . . . 6  |-  ( <. G ,  G >.  e. 
CVecOLD  ->  ( 0 G (GId `  ( 1st `  <. G ,  G >. ) ) )  =  ( 0 G 1 ) )
44 0cn 9589 . . . . . . . . 9  |-  0  e.  CC
4514, 30, 26, 27vcz 25236 . . . . . . . . 9  |-  ( (
<. G ,  G >.  e. 
CVecOLD  /\  0  e.  CC )  ->  (
0 ( 2nd `  <. G ,  G >. )
(GId `  ( 1st ` 
<. G ,  G >. ) ) )  =  (GId
`  ( 1st `  <. G ,  G >. )
) )
4644, 45mpan2 671 . . . . . . . 8  |-  ( <. G ,  G >.  e. 
CVecOLD  ->  ( 0 ( 2nd `  <. G ,  G >. )
(GId `  ( 1st ` 
<. G ,  G >. ) ) )  =  (GId
`  ( 1st `  <. G ,  G >. )
) )
4734oveqd 6302 . . . . . . . . 9  |-  ( <. G ,  G >.  e. 
CVecOLD  ->  ( 0 ( 2nd `  <. G ,  G >. )
(GId `  ( 1st ` 
<. G ,  G >. ) ) )  =  ( 0 G (GId `  ( 1st `  <. G ,  G >. ) ) ) )
4847, 43eqtrd 2508 . . . . . . . 8  |-  ( <. G ,  G >.  e. 
CVecOLD  ->  ( 0 ( 2nd `  <. G ,  G >. )
(GId `  ( 1st ` 
<. G ,  G >. ) ) )  =  ( 0 G 1 ) )
4946, 48eqtr3d 2510 . . . . . . 7  |-  ( <. G ,  G >.  e. 
CVecOLD  ->  (GId `  ( 1st `  <. G ,  G >. ) )  =  ( 0 G 1 ) )
5049, 42eqtr3d 2510 . . . . . 6  |-  ( <. G ,  G >.  e. 
CVecOLD  ->  ( 0 G 1 )  =  1 )
5111, 43, 503eqtrd 2512 . . . . 5  |-  ( <. G ,  G >.  e. 
CVecOLD  ->  ( 0 ( 1st `  <. G ,  G >. )
(GId `  ( 1st ` 
<. G ,  G >. ) ) )  =  1 )
5244, 24syl5eleqr 2562 . . . . . 6  |-  ( <. G ,  G >.  e. 
CVecOLD  ->  0  e.  ran  ( 1st `  <. G ,  G >. )
)
5314, 26, 27vc0rid 25233 . . . . . 6  |-  ( (
<. G ,  G >.  e. 
CVecOLD  /\  0  e. 
ran  ( 1st `  <. G ,  G >. )
)  ->  ( 0 ( 1st `  <. G ,  G >. )
(GId `  ( 1st ` 
<. G ,  G >. ) ) )  =  0 )
5452, 53mpdan 668 . . . . 5  |-  ( <. G ,  G >.  e. 
CVecOLD  ->  ( 0 ( 1st `  <. G ,  G >. )
(GId `  ( 1st ` 
<. G ,  G >. ) ) )  =  0 )
5551, 54eqtr3d 2510 . . . 4  |-  ( <. G ,  G >.  e. 
CVecOLD  ->  1  = 
0 )
563, 55mto 176 . . 3  |-  -.  <. G ,  G >.  e.  CVecOLD
57 opeq2 4214 . . . 4  |-  ( G  =  S  ->  <. G ,  G >.  =  <. G ,  S >. )
5857eleq1d 2536 . . 3  |-  ( G  =  S  ->  ( <. G ,  G >.  e. 
CVecOLD  <->  <. G ,  S >.  e.  CVecOLD ) )
5956, 58mtbii 302 . 2  |-  ( G  =  S  ->  -.  <. G ,  S >.  e. 
CVecOLD )
6059necon2ai 2702 1  |-  ( <. G ,  S >.  e. 
CVecOLD  ->  G  =/=  S )
Colors of variables: wff setvar class
Syntax hints:   -. wn 3    -> wi 4    <-> wb 184    /\ w3a 973    = wceq 1379    e. wcel 1767    =/= wne 2662   _Vcvv 3113   <.cop 4033    X. cxp 4997   dom cdm 4999   ran crn 5000   -->wf 5584   ` cfv 5588  (class class class)co 6285   1stc1st 6783   2ndc2nd 6784   CCcc 9491   0cc0 9493   1c1 9494   GrpOpcgr 24961  GIdcgi 24962   CVecOLDcvc 25211
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1601  ax-4 1612  ax-5 1680  ax-6 1719  ax-7 1739  ax-8 1769  ax-9 1771  ax-10 1786  ax-11 1791  ax-12 1803  ax-13 1968  ax-ext 2445  ax-rep 4558  ax-sep 4568  ax-nul 4576  ax-pow 4625  ax-pr 4686  ax-un 6577  ax-cnex 9549  ax-resscn 9550  ax-1cn 9551  ax-icn 9552  ax-addcl 9553  ax-addrcl 9554  ax-mulcl 9555  ax-mulrcl 9556  ax-mulcom 9557  ax-addass 9558  ax-mulass 9559  ax-distr 9560  ax-i2m1 9561  ax-1ne0 9562  ax-1rid 9563  ax-rnegex 9564  ax-rrecex 9565  ax-cnre 9566  ax-pre-lttri 9567  ax-pre-lttrn 9568  ax-pre-ltadd 9569
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 974  df-3an 975  df-tru 1382  df-ex 1597  df-nf 1600  df-sb 1712  df-eu 2279  df-mo 2280  df-clab 2453  df-cleq 2459  df-clel 2462  df-nfc 2617  df-ne 2664  df-nel 2665  df-ral 2819  df-rex 2820  df-reu 2821  df-rab 2823  df-v 3115  df-sbc 3332  df-csb 3436  df-dif 3479  df-un 3481  df-in 3483  df-ss 3490  df-nul 3786  df-if 3940  df-pw 4012  df-sn 4028  df-pr 4030  df-op 4034  df-uni 4246  df-iun 4327  df-br 4448  df-opab 4506  df-mpt 4507  df-id 4795  df-po 4800  df-so 4801  df-xp 5005  df-rel 5006  df-cnv 5007  df-co 5008  df-dm 5009  df-rn 5010  df-res 5011  df-ima 5012  df-iota 5551  df-fun 5590  df-fn 5591  df-f 5592  df-f1 5593  df-fo 5594  df-f1o 5595  df-fv 5596  df-riota 6246  df-ov 6288  df-1st 6785  df-2nd 6786  df-er 7312  df-en 7518  df-dom 7519  df-sdom 7520  df-pnf 9631  df-mnf 9632  df-ltxr 9634  df-grpo 24966  df-gid 24967  df-ginv 24968  df-ablo 25057  df-vc 25212
This theorem is referenced by:  nvoprne  25354
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