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Theorem vcoprne 24110
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 9463 . . . . 5  |-  1  =/=  0
2 df-ne 2650 . . . . 5  |-  ( 1  =/=  0  <->  -.  1  =  0 )
31, 2mpbi 208 . . . 4  |-  -.  1  =  0
4 vcoprnelem 24109 . . . . . . . . 9  |-  ( <. G ,  G >.  e. 
CVecOLD  ->  G :
( CC  X.  CC )
--> CC )
5 cnex 9475 . . . . . . . . . 10  |-  CC  e.  _V
65, 5xpex 6619 . . . . . . . . 9  |-  ( CC 
X.  CC )  e. 
_V
7 fex 6060 . . . . . . . . 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 6700 . . . . . . . 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 6218 . . . . . 6  |-  ( <. G ,  G >.  e. 
CVecOLD  ->  ( 0 ( 1st `  <. G ,  G >. )
(GId `  ( 1st ` 
<. G ,  G >. ) ) )  =  ( 0 G (GId `  ( 1st `  <. G ,  G >. ) ) ) )
12 ax-1cn 9452 . . . . . . . . . . 11  |-  1  e.  CC
1310rneqd 5176 . . . . . . . . . . . 12  |-  ( <. G ,  G >.  e. 
CVecOLD  ->  ran  ( 1st `  <. G ,  G >. )  =  ran  G
)
14 eqid 2454 . . . . . . . . . . . . . . 15  |-  ( 1st `  <. G ,  G >. )  =  ( 1st `  <. G ,  G >. )
1514vcgrp 24089 . . . . . . . . . . . . . 14  |-  ( <. G ,  G >.  e. 
CVecOLD  ->  ( 1st ` 
<. G ,  G >. )  e.  GrpOp )
1610, 15eqeltrrd 2543 . . . . . . . . . . . . 13  |-  ( <. G ,  G >.  e. 
CVecOLD  ->  G  e.  GrpOp
)
17 grporndm 23850 . . . . . . . . . . . . 13  |-  ( G  e.  GrpOp  ->  ran  G  =  dom  dom  G )
1816, 17syl 16 . . . . . . . . . . . 12  |-  ( <. G ,  G >.  e. 
CVecOLD  ->  ran  G  =  dom  dom  G )
19 fdm 5672 . . . . . . . . . . . . . . 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 5151 . . . . . . . . . . . . 13  |-  ( <. G ,  G >.  e. 
CVecOLD  ->  dom  dom  G  =  dom  ( CC  X.  CC ) )
22 dmxpid 5168 . . . . . . . . . . . . 13  |-  dom  ( CC  X.  CC )  =  CC
2321, 22syl6eq 2511 . . . . . . . . . . . 12  |-  ( <. G ,  G >.  e. 
CVecOLD  ->  dom  dom  G  =  CC )
2413, 18, 233eqtrd 2499 . . . . . . . . . . 11  |-  ( <. G ,  G >.  e. 
CVecOLD  ->  ran  ( 1st `  <. G ,  G >. )  =  CC )
2512, 24syl5eleqr 2549 . . . . . . . . . 10  |-  ( <. G ,  G >.  e. 
CVecOLD  ->  1  e.  ran  ( 1st `  <. G ,  G >. )
)
26 eqid 2454 . . . . . . . . . . 11  |-  ran  ( 1st `  <. G ,  G >. )  =  ran  ( 1st `  <. G ,  G >. )
27 eqid 2454 . . . . . . . . . . 11  |-  (GId `  ( 1st `  <. G ,  G >. ) )  =  (GId `  ( 1st ` 
<. G ,  G >. ) )
2814, 26, 27vc0rid 24098 . . . . . . . . . 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 2454 . . . . . . . . . . 11  |-  ( 2nd `  <. G ,  G >. )  =  ( 2nd `  <. G ,  G >. )
3114, 30, 26vcid 24082 . . . . . . . . . 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 6701 . . . . . . . . . . . 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 2498 . . . . . . . . . 10  |-  ( <. G ,  G >.  e. 
CVecOLD  ->  ( 2nd ` 
<. G ,  G >. )  =  ( 1st `  <. G ,  G >. )
)
3635oveqd 6218 . . . . . . . . 9  |-  ( <. G ,  G >.  e. 
CVecOLD  ->  ( 1 ( 2nd `  <. G ,  G >. )
1 )  =  ( 1 ( 1st `  <. G ,  G >. )
1 ) )
3729, 32, 363eqtr2d 2501 . . . . . . . 8  |-  ( <. G ,  G >.  e. 
CVecOLD  ->  ( 1 ( 1st `  <. G ,  G >. )
(GId `  ( 1st ` 
<. G ,  G >. ) ) )  =  ( 1 ( 1st `  <. G ,  G >. )
1 ) )
3814, 26, 27vczcl 24097 . . . . . . . . . 10  |-  ( <. G ,  G >.  e. 
CVecOLD  ->  (GId `  ( 1st `  <. G ,  G >. ) )  e.  ran  ( 1st `  <. G ,  G >. ) )
3938, 25, 253jca 1168 . . . . . . . . 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 24096 . . . . . . . . 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 6217 . . . . . 6  |-  ( <. G ,  G >.  e. 
CVecOLD  ->  ( 0 G (GId `  ( 1st `  <. G ,  G >. ) ) )  =  ( 0 G 1 ) )
44 0cn 9490 . . . . . . . . 9  |-  0  e.  CC
4514, 30, 26, 27vcz 24101 . . . . . . . . 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 6218 . . . . . . . . 9  |-  ( <. G ,  G >.  e. 
CVecOLD  ->  ( 0 ( 2nd `  <. G ,  G >. )
(GId `  ( 1st ` 
<. G ,  G >. ) ) )  =  ( 0 G (GId `  ( 1st `  <. G ,  G >. ) ) ) )
4847, 43eqtrd 2495 . . . . . . . 8  |-  ( <. G ,  G >.  e. 
CVecOLD  ->  ( 0 ( 2nd `  <. G ,  G >. )
(GId `  ( 1st ` 
<. G ,  G >. ) ) )  =  ( 0 G 1 ) )
4946, 48eqtr3d 2497 . . . . . . 7  |-  ( <. G ,  G >.  e. 
CVecOLD  ->  (GId `  ( 1st `  <. G ,  G >. ) )  =  ( 0 G 1 ) )
5049, 42eqtr3d 2497 . . . . . 6  |-  ( <. G ,  G >.  e. 
CVecOLD  ->  ( 0 G 1 )  =  1 )
5111, 43, 503eqtrd 2499 . . . . 5  |-  ( <. G ,  G >.  e. 
CVecOLD  ->  ( 0 ( 1st `  <. G ,  G >. )
(GId `  ( 1st ` 
<. G ,  G >. ) ) )  =  1 )
5244, 24syl5eleqr 2549 . . . . . 6  |-  ( <. G ,  G >.  e. 
CVecOLD  ->  0  e.  ran  ( 1st `  <. G ,  G >. )
)
5314, 26, 27vc0rid 24098 . . . . . 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 2497 . . . 4  |-  ( <. G ,  G >.  e. 
CVecOLD  ->  1  = 
0 )
563, 55mto 176 . . 3  |-  -.  <. G ,  G >.  e.  CVecOLD
57 opeq2 4169 . . . 4  |-  ( G  =  S  ->  <. G ,  G >.  =  <. G ,  S >. )
5857eleq1d 2523 . . 3  |-  ( G  =  S  ->  ( <. G ,  G >.  e. 
CVecOLD  <->  <. G ,  S >.  e.  CVecOLD ) )
5956, 58mtbii 302 . 2  |-  ( G  =  S  ->  -.  <. G ,  S >.  e. 
CVecOLD )
6059necon2ai 2687 1  |-  ( <. G ,  S >.  e. 
CVecOLD  ->  G  =/=  S )
Colors of variables: wff setvar class
Syntax hints:   -. wn 3    -> wi 4    <-> wb 184    /\ w3a 965    = wceq 1370    e. wcel 1758    =/= wne 2648   _Vcvv 3078   <.cop 3992    X. cxp 4947   dom cdm 4949   ran crn 4950   -->wf 5523   ` cfv 5527  (class class class)co 6201   1stc1st 6686   2ndc2nd 6687   CCcc 9392   0cc0 9394   1c1 9395   GrpOpcgr 23826  GIdcgi 23827   CVecOLDcvc 24076
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1592  ax-4 1603  ax-5 1671  ax-6 1710  ax-7 1730  ax-8 1760  ax-9 1762  ax-10 1777  ax-11 1782  ax-12 1794  ax-13 1955  ax-ext 2432  ax-rep 4512  ax-sep 4522  ax-nul 4530  ax-pow 4579  ax-pr 4640  ax-un 6483  ax-cnex 9450  ax-resscn 9451  ax-1cn 9452  ax-icn 9453  ax-addcl 9454  ax-addrcl 9455  ax-mulcl 9456  ax-mulrcl 9457  ax-mulcom 9458  ax-addass 9459  ax-mulass 9460  ax-distr 9461  ax-i2m1 9462  ax-1ne0 9463  ax-1rid 9464  ax-rnegex 9465  ax-rrecex 9466  ax-cnre 9467  ax-pre-lttri 9468  ax-pre-lttrn 9469  ax-pre-ltadd 9470
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 966  df-3an 967  df-tru 1373  df-ex 1588  df-nf 1591  df-sb 1703  df-eu 2266  df-mo 2267  df-clab 2440  df-cleq 2446  df-clel 2449  df-nfc 2604  df-ne 2650  df-nel 2651  df-ral 2804  df-rex 2805  df-reu 2806  df-rab 2808  df-v 3080  df-sbc 3295  df-csb 3397  df-dif 3440  df-un 3442  df-in 3444  df-ss 3451  df-nul 3747  df-if 3901  df-pw 3971  df-sn 3987  df-pr 3989  df-op 3993  df-uni 4201  df-iun 4282  df-br 4402  df-opab 4460  df-mpt 4461  df-id 4745  df-po 4750  df-so 4751  df-xp 4955  df-rel 4956  df-cnv 4957  df-co 4958  df-dm 4959  df-rn 4960  df-res 4961  df-ima 4962  df-iota 5490  df-fun 5529  df-fn 5530  df-f 5531  df-f1 5532  df-fo 5533  df-f1o 5534  df-fv 5535  df-riota 6162  df-ov 6204  df-1st 6688  df-2nd 6689  df-er 7212  df-en 7422  df-dom 7423  df-sdom 7424  df-pnf 9532  df-mnf 9533  df-ltxr 9535  df-grpo 23831  df-gid 23832  df-ginv 23833  df-ablo 23922  df-vc 24077
This theorem is referenced by:  nvoprne  24219
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