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Theorem vcoprne 26140
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 9559 . . . . 5  |-  1  =/=  0
21neii 2603 . . . 4  |-  -.  1  =  0
3 vcoprnelem 26139 . . . . . . . . 9  |-  ( <. G ,  G >.  e. 
CVecOLD  ->  G :
( CC  X.  CC )
--> CC )
4 cnex 9571 . . . . . . . . . 10  |-  CC  e.  _V
54, 4xpex 6553 . . . . . . . . 9  |-  ( CC 
X.  CC )  e. 
_V
6 fex 6097 . . . . . . . . 9  |-  ( ( G : ( CC 
X.  CC ) --> CC 
/\  ( CC  X.  CC )  e.  _V )  ->  G  e.  _V )
73, 5, 6sylancl 666 . . . . . . . 8  |-  ( <. G ,  G >.  e. 
CVecOLD  ->  G  e.  _V )
8 op1stg 6763 . . . . . . . 8  |-  ( ( G  e.  _V  /\  G  e.  _V )  ->  ( 1st `  <. G ,  G >. )  =  G )
97, 7, 8syl2anc 665 . . . . . . 7  |-  ( <. G ,  G >.  e. 
CVecOLD  ->  ( 1st ` 
<. G ,  G >. )  =  G )
109oveqd 6266 . . . . . 6  |-  ( <. G ,  G >.  e. 
CVecOLD  ->  ( 0 ( 1st `  <. G ,  G >. )
(GId `  ( 1st ` 
<. G ,  G >. ) ) )  =  ( 0 G (GId `  ( 1st `  <. G ,  G >. ) ) ) )
11 ax-1cn 9548 . . . . . . . . . . 11  |-  1  e.  CC
129rneqd 5024 . . . . . . . . . . . 12  |-  ( <. G ,  G >.  e. 
CVecOLD  ->  ran  ( 1st `  <. G ,  G >. )  =  ran  G
)
13 eqid 2428 . . . . . . . . . . . . . . 15  |-  ( 1st `  <. G ,  G >. )  =  ( 1st `  <. G ,  G >. )
1413vcgrp 26119 . . . . . . . . . . . . . 14  |-  ( <. G ,  G >.  e. 
CVecOLD  ->  ( 1st ` 
<. G ,  G >. )  e.  GrpOp )
159, 14eqeltrrd 2507 . . . . . . . . . . . . 13  |-  ( <. G ,  G >.  e. 
CVecOLD  ->  G  e.  GrpOp
)
16 grporndm 25880 . . . . . . . . . . . . 13  |-  ( G  e.  GrpOp  ->  ran  G  =  dom  dom  G )
1715, 16syl 17 . . . . . . . . . . . 12  |-  ( <. G ,  G >.  e. 
CVecOLD  ->  ran  G  =  dom  dom  G )
18 fdm 5693 . . . . . . . . . . . . . . 15  |-  ( G : ( CC  X.  CC ) --> CC  ->  dom  G  =  ( CC  X.  CC ) )
193, 18syl 17 . . . . . . . . . . . . . 14  |-  ( <. G ,  G >.  e. 
CVecOLD  ->  dom  G  =  ( CC  X.  CC ) )
2019dmeqd 4999 . . . . . . . . . . . . 13  |-  ( <. G ,  G >.  e. 
CVecOLD  ->  dom  dom  G  =  dom  ( CC  X.  CC ) )
21 dmxpid 5016 . . . . . . . . . . . . 13  |-  dom  ( CC  X.  CC )  =  CC
2220, 21syl6eq 2478 . . . . . . . . . . . 12  |-  ( <. G ,  G >.  e. 
CVecOLD  ->  dom  dom  G  =  CC )
2312, 17, 223eqtrd 2466 . . . . . . . . . . 11  |-  ( <. G ,  G >.  e. 
CVecOLD  ->  ran  ( 1st `  <. G ,  G >. )  =  CC )
2411, 23syl5eleqr 2513 . . . . . . . . . 10  |-  ( <. G ,  G >.  e. 
CVecOLD  ->  1  e.  ran  ( 1st `  <. G ,  G >. )
)
25 eqid 2428 . . . . . . . . . . 11  |-  ran  ( 1st `  <. G ,  G >. )  =  ran  ( 1st `  <. G ,  G >. )
26 eqid 2428 . . . . . . . . . . 11  |-  (GId `  ( 1st `  <. G ,  G >. ) )  =  (GId `  ( 1st ` 
<. G ,  G >. ) )
2713, 25, 26vc0rid 26128 . . . . . . . . . 10  |-  ( (
<. G ,  G >.  e. 
CVecOLD  /\  1  e. 
ran  ( 1st `  <. G ,  G >. )
)  ->  ( 1 ( 1st `  <. G ,  G >. )
(GId `  ( 1st ` 
<. G ,  G >. ) ) )  =  1 )
2824, 27mpdan 672 . . . . . . . . 9  |-  ( <. G ,  G >.  e. 
CVecOLD  ->  ( 1 ( 1st `  <. G ,  G >. )
(GId `  ( 1st ` 
<. G ,  G >. ) ) )  =  1 )
29 eqid 2428 . . . . . . . . . . 11  |-  ( 2nd `  <. G ,  G >. )  =  ( 2nd `  <. G ,  G >. )
3013, 29, 25vcid 26112 . . . . . . . . . 10  |-  ( (
<. G ,  G >.  e. 
CVecOLD  /\  1  e. 
ran  ( 1st `  <. G ,  G >. )
)  ->  ( 1 ( 2nd `  <. G ,  G >. )
1 )  =  1 )
3124, 30mpdan 672 . . . . . . . . 9  |-  ( <. G ,  G >.  e. 
CVecOLD  ->  ( 1 ( 2nd `  <. G ,  G >. )
1 )  =  1 )
32 op2ndg 6764 . . . . . . . . . . . 12  |-  ( ( G  e.  _V  /\  G  e.  _V )  ->  ( 2nd `  <. G ,  G >. )  =  G )
337, 7, 32syl2anc 665 . . . . . . . . . . 11  |-  ( <. G ,  G >.  e. 
CVecOLD  ->  ( 2nd ` 
<. G ,  G >. )  =  G )
3433, 9eqtr4d 2465 . . . . . . . . . 10  |-  ( <. G ,  G >.  e. 
CVecOLD  ->  ( 2nd ` 
<. G ,  G >. )  =  ( 1st `  <. G ,  G >. )
)
3534oveqd 6266 . . . . . . . . 9  |-  ( <. G ,  G >.  e. 
CVecOLD  ->  ( 1 ( 2nd `  <. G ,  G >. )
1 )  =  ( 1 ( 1st `  <. G ,  G >. )
1 ) )
3628, 31, 353eqtr2d 2468 . . . . . . . 8  |-  ( <. G ,  G >.  e. 
CVecOLD  ->  ( 1 ( 1st `  <. G ,  G >. )
(GId `  ( 1st ` 
<. G ,  G >. ) ) )  =  ( 1 ( 1st `  <. G ,  G >. )
1 ) )
3713, 25, 26vczcl 26127 . . . . . . . . . 10  |-  ( <. G ,  G >.  e. 
CVecOLD  ->  (GId `  ( 1st `  <. G ,  G >. ) )  e.  ran  ( 1st `  <. G ,  G >. ) )
3837, 24, 243jca 1185 . . . . . . . . 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 26126 . . . . . . . . 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 672 . . . . . . . 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 213 . . . . . . 7  |-  ( <. G ,  G >.  e. 
CVecOLD  ->  (GId `  ( 1st `  <. G ,  G >. ) )  =  1 )
4241oveq2d 6265 . . . . . 6  |-  ( <. G ,  G >.  e. 
CVecOLD  ->  ( 0 G (GId `  ( 1st `  <. G ,  G >. ) ) )  =  ( 0 G 1 ) )
43 0cn 9586 . . . . . . . . 9  |-  0  e.  CC
4413, 29, 25, 26vcz 26131 . . . . . . . . 9  |-  ( (
<. G ,  G >.  e. 
CVecOLD  /\  0  e.  CC )  ->  (
0 ( 2nd `  <. G ,  G >. )
(GId `  ( 1st ` 
<. G ,  G >. ) ) )  =  (GId
`  ( 1st `  <. G ,  G >. )
) )
4543, 44mpan2 675 . . . . . . . 8  |-  ( <. G ,  G >.  e. 
CVecOLD  ->  ( 0 ( 2nd `  <. G ,  G >. )
(GId `  ( 1st ` 
<. G ,  G >. ) ) )  =  (GId
`  ( 1st `  <. G ,  G >. )
) )
4633oveqd 6266 . . . . . . . . 9  |-  ( <. G ,  G >.  e. 
CVecOLD  ->  ( 0 ( 2nd `  <. G ,  G >. )
(GId `  ( 1st ` 
<. G ,  G >. ) ) )  =  ( 0 G (GId `  ( 1st `  <. G ,  G >. ) ) ) )
4746, 42eqtrd 2462 . . . . . . . 8  |-  ( <. G ,  G >.  e. 
CVecOLD  ->  ( 0 ( 2nd `  <. G ,  G >. )
(GId `  ( 1st ` 
<. G ,  G >. ) ) )  =  ( 0 G 1 ) )
4845, 47eqtr3d 2464 . . . . . . 7  |-  ( <. G ,  G >.  e. 
CVecOLD  ->  (GId `  ( 1st `  <. G ,  G >. ) )  =  ( 0 G 1 ) )
4948, 41eqtr3d 2464 . . . . . 6  |-  ( <. G ,  G >.  e. 
CVecOLD  ->  ( 0 G 1 )  =  1 )
5010, 42, 493eqtrd 2466 . . . . 5  |-  ( <. G ,  G >.  e. 
CVecOLD  ->  ( 0 ( 1st `  <. G ,  G >. )
(GId `  ( 1st ` 
<. G ,  G >. ) ) )  =  1 )
5143, 23syl5eleqr 2513 . . . . . 6  |-  ( <. G ,  G >.  e. 
CVecOLD  ->  0  e.  ran  ( 1st `  <. G ,  G >. )
)
5213, 25, 26vc0rid 26128 . . . . . 6  |-  ( (
<. G ,  G >.  e. 
CVecOLD  /\  0  e. 
ran  ( 1st `  <. G ,  G >. )
)  ->  ( 0 ( 1st `  <. G ,  G >. )
(GId `  ( 1st ` 
<. G ,  G >. ) ) )  =  0 )
5351, 52mpdan 672 . . . . 5  |-  ( <. G ,  G >.  e. 
CVecOLD  ->  ( 0 ( 1st `  <. G ,  G >. )
(GId `  ( 1st ` 
<. G ,  G >. ) ) )  =  0 )
5450, 53eqtr3d 2464 . . . 4  |-  ( <. G ,  G >.  e. 
CVecOLD  ->  1  = 
0 )
552, 54mto 179 . . 3  |-  -.  <. G ,  G >.  e.  CVecOLD
56 opeq2 4131 . . . 4  |-  ( G  =  S  ->  <. G ,  G >.  =  <. G ,  S >. )
5756eleq1d 2490 . . 3  |-  ( G  =  S  ->  ( <. G ,  G >.  e. 
CVecOLD  <->  <. G ,  S >.  e.  CVecOLD ) )
5855, 57mtbii 303 . 2  |-  ( G  =  S  ->  -.  <. G ,  S >.  e. 
CVecOLD )
5958necon2ai 2630 1  |-  ( <. G ,  S >.  e. 
CVecOLD  ->  G  =/=  S )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    <-> wb 187    /\ w3a 982    = wceq 1437    e. wcel 1872    =/= wne 2599   _Vcvv 3022   <.cop 3947    X. cxp 4794   dom cdm 4796   ran crn 4797   -->wf 5540   ` cfv 5544  (class class class)co 6249   1stc1st 6749   2ndc2nd 6750   CCcc 9488   0cc0 9490   1c1 9491   GrpOpcgr 25856  GIdcgi 25857   CVecOLDcvc 26106
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1663  ax-4 1676  ax-5 1752  ax-6 1798  ax-7 1843  ax-8 1874  ax-9 1876  ax-10 1891  ax-11 1896  ax-12 1909  ax-13 2063  ax-ext 2408  ax-rep 4479  ax-sep 4489  ax-nul 4498  ax-pow 4545  ax-pr 4603  ax-un 6541  ax-cnex 9546  ax-resscn 9547  ax-1cn 9548  ax-icn 9549  ax-addcl 9550  ax-addrcl 9551  ax-mulcl 9552  ax-mulrcl 9553  ax-mulcom 9554  ax-addass 9555  ax-mulass 9556  ax-distr 9557  ax-i2m1 9558  ax-1ne0 9559  ax-1rid 9560  ax-rnegex 9561  ax-rrecex 9562  ax-cnre 9563  ax-pre-lttri 9564  ax-pre-lttrn 9565  ax-pre-ltadd 9566
This theorem depends on definitions:  df-bi 188  df-or 371  df-an 372  df-3or 983  df-3an 984  df-tru 1440  df-ex 1658  df-nf 1662  df-sb 1791  df-eu 2280  df-mo 2281  df-clab 2415  df-cleq 2421  df-clel 2424  df-nfc 2558  df-ne 2601  df-nel 2602  df-ral 2719  df-rex 2720  df-reu 2721  df-rab 2723  df-v 3024  df-sbc 3243  df-csb 3339  df-dif 3382  df-un 3384  df-in 3386  df-ss 3393  df-nul 3705  df-if 3855  df-pw 3926  df-sn 3942  df-pr 3944  df-op 3948  df-uni 4163  df-iun 4244  df-br 4367  df-opab 4426  df-mpt 4427  df-id 4711  df-po 4717  df-so 4718  df-xp 4802  df-rel 4803  df-cnv 4804  df-co 4805  df-dm 4806  df-rn 4807  df-res 4808  df-ima 4809  df-iota 5508  df-fun 5546  df-fn 5547  df-f 5548  df-f1 5549  df-fo 5550  df-f1o 5551  df-fv 5552  df-riota 6211  df-ov 6252  df-1st 6751  df-2nd 6752  df-er 7318  df-en 7525  df-dom 7526  df-sdom 7527  df-pnf 9628  df-mnf 9629  df-ltxr 9631  df-grpo 25861  df-gid 25862  df-ginv 25863  df-ablo 25952  df-vc 26107
This theorem is referenced by:  nvoprne  26249
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