Theorem vcoprne 9530

Description: The operations of a complex vector space cannot be identical.

Assertion

Ref	Expression
vcoprne	|- (<.G, S>. e. CVec -> G =/= S)

Proof of Theorem vcoprne

Step	Hyp	Ref	Expression
1		ax1ne0 6433	. . . . 5 |- 1 =/= 0
2		df-ne 2019	. . . . 5 |- (1 =/= 0 <-> -. 1 = 0)
3	1, 2	mpbi 206	. . . 4 |- -. 1 = 0
4		fex 4595	. . . . . . . . 9 |- ((G:(CC X. CC)-->CC /\ (CC X. CC) e. _V) -> G e. _V)
5		vcoprnelem 9529	. . . . . . . . 9 |- (<.G, G>. e. CVec -> G:(CC X. CC)-->CC)
6		axcnex 6419	. . . . . . . . . 10 |- CC e. _V
7	6, 6	xpex 4096	. . . . . . . . 9 |- (CC X. CC) e. _V
8	4, 5, 7	sylancl 525	. . . . . . . 8 |- (<.G, G>. e. CVec -> G e. _V)
9		op1stg 5028	. . . . . . . 8 |- (G e. _V -> (1st` <.G, G>.) = G)
10	8, 9	syl 12	. . . . . . 7 |- (<.G, G>. e. CVec -> (1st` <.G, G>.) = G)
11	10	opreqd 4899	. . . . . 6 |- (<.G, G>. e. CVec -> (0(1st` <.G, G>.)(Id` (1st` <.G, G>.))) = (0G(Id` (1st` <.G, G>.))))
12	10	rneqd 4188	. . . . . . . . . . . 12 |- (<.G, G>. e. CVec -> ran (1st` <.G, G>.) = ran G)
13		eqid 1884	. . . . . . . . . . . . . . 15 |- (1st` <.G, G>.) = (1st` <.G, G>.)
14	13	vcgrp 9509	. . . . . . . . . . . . . 14 |- (<.G, G>. e. CVec -> (1st` <.G, G>.) e. Grp)
15	10, 14	eqeltrrd 1972	. . . . . . . . . . . . 13 |- (<.G, G>. e. CVec -> G e. Grp)
16		grprndm 9334	. . . . . . . . . . . . 13 |- (G e. Grp -> ran G = dom dom G)
17	15, 16	syl 12	. . . . . . . . . . . 12 |- (<.G, G>. e. CVec -> ran G = dom dom G)
18		fdm 4567	. . . . . . . . . . . . . . 15 |- (G:(CC X. CC)-->CC -> dom G = (CC X. CC))
19	5, 18	syl 12	. . . . . . . . . . . . . 14 |- (<.G, G>. e. CVec -> dom G = (CC X. CC))
20	19	dmeqd 4159	. . . . . . . . . . . . 13 |- (<.G, G>. e. CVec -> dom dom G = dom (CC X. CC))
21		dmxpid 4179	. . . . . . . . . . . . 13 |- dom (CC X. CC) = CC
22	20, 21	syl6eq 1944	. . . . . . . . . . . 12 |- (<.G, G>. e. CVec -> dom dom G = CC)
23	12, 17, 22	3eqtrd 1929	. . . . . . . . . . 11 |- (<.G, G>. e. CVec -> ran (1st` <.G, G>.) = CC)
24		ax1cn 6422	. . . . . . . . . . 11 |- 1 e. CC
25	23, 24	syl5eleqr 1978	. . . . . . . . . 10 |- (<.G, G>. e. CVec -> 1 e. ran (1st` <.G, G>.))
26		eqid 1884	. . . . . . . . . . 11 |- ran (1st` <.G, G>.) = ran (1st` <.G, G>.)
27		eqid 1884	. . . . . . . . . . 11 |- (Id` (1st` <.G, G>.)) = (Id` (1st` <.G, G>.))
28	13, 26, 27	vc0rid 9518	. . . . . . . . . 10 |- ((<.G, G>. e. CVec /\ 1 e. ran (1st` <.G, G>.)) -> (1(1st` <.G, G>.)(Id` (1st` <.G, G>.))) = 1)
29	25, 28	mpdan 768	. . . . . . . . 9 |- (<.G, G>. e. CVec -> (1(1st` <.G, G>.)(Id` (1st` <.G, G>.))) = 1)
30		eqid 1884	. . . . . . . . . . 11 |- (2nd` <.G, G>.) = (2nd` <.G, G>.)
31	13, 30, 26	vcid 9502	. . . . . . . . . 10 |- ((<.G, G>. e. CVec /\ 1 e. ran (1st` <.G, G>.)) -> (1(2nd` <.G, G>.)1) = 1)
32	25, 31	mpdan 768	. . . . . . . . 9 |- (<.G, G>. e. CVec -> (1(2nd` <.G, G>.)1) = 1)
33		op2ndg 5029	. . . . . . . . . . . . 13 |- ((G e. _V /\ G e. _V) -> (2nd` <.G, G>.) = G)
34	33	anidms 480	. . . . . . . . . . . 12 |- (G e. _V -> (2nd` <.G, G>.) = G)
35	8, 34	syl 12	. . . . . . . . . . 11 |- (<.G, G>. e. CVec -> (2nd` <.G, G>.) = G)
36	35, 10	eqtr4d 1928	. . . . . . . . . 10 |- (<.G, G>. e. CVec -> (2nd` <.G, G>.) = (1st` <.G, G>.))
37	36	opreqd 4899	. . . . . . . . 9 |- (<.G, G>. e. CVec -> (1(2nd` <.G, G>.)1) = (1(1st` <.G, G>.)1))
38	29, 32, 37	3eqtr2d 1932	. . . . . . . 8 |- (<.G, G>. e. CVec -> (1(1st` <.G, G>.)(Id` (1st` <.G, G>.))) = (1(1st` <.G, G>.)1))
39	13, 26, 27	vczcl 9517	. . . . . . . . . 10 |- (<.G, G>. e. CVec -> (Id` (1st` <.G, G>.)) e. ran (1st` <.G, G>.))
40	39, 25, 25	3jca 1050	. . . . . . . . 9 |- (<.G, G>. e. CVec -> ((Id` (1st` <.G, G>.)) e. ran (1st` <.G, G>.) /\ 1 e. ran (1st` <.G, G>.) /\ 1 e. ran (1st` <.G, G>.)))
41	13, 26	vclcan 9516	. . . . . . . . 9 |- ((<.G, G>. e. CVec /\ ((Id` (1st` <.G, G>.)) e. ran (1st` <.G, G>.) /\ 1 e. ran (1st` <.G, G>.) /\ 1 e. ran (1st` <.G, G>.))) -> ((1(1st` <.G, G>.)(Id` (1st` <.G, G>.))) = (1(1st` <.G, G>.)1) <-> (Id` (1st` <.G, G>.)) = 1))
42	40, 41	mpdan 768	. . . . . . . 8 |- (<.G, G>. e. CVec -> ((1(1st` <.G, G>.)(Id` (1st` <.G, G>.))) = (1(1st` <.G, G>.)1) <-> (Id` (1st` <.G, G>.)) = 1))
43	38, 42	mpbid 212	. . . . . . 7 |- (<.G, G>. e. CVec -> (Id` (1st` <.G, G>.)) = 1)
44	43	opreq2d 4898	. . . . . 6 |- (<.G, G>. e. CVec -> (0G(Id` (1st` <.G, G>.))) = (0G1))
45	11, 44	eqtr2d 1926	. . . . 5 |- (<.G, G>. e. CVec -> (0G1) = (0(1st` <.G, G>.)(Id` (1st` <.G, G>.))))
46		0cn 6481	. . . . . . 7 |- 0 e. CC
47	13, 30, 26, 27	vcz 9521	. . . . . . 7 |- ((<.G, G>. e. CVec /\ 0 e. CC) -> (0(2nd` <.G, G>.)(Id` (1st` <.G, G>.))) = (Id` (1st` <.G, G>.)))
48	46, 47	mpan2 760	. . . . . 6 |- (<.G, G>. e. CVec -> (0(2nd` <.G, G>.)(Id` (1st` <.G, G>.))) = (Id` (1st` <.G, G>.)))
49	35	opreqd 4899	. . . . . . 7 |- (<.G, G>. e. CVec -> (0(2nd` <.G, G>.)(Id` (1st` <.G, G>.))) = (0G(Id` (1st` <.G, G>.))))
50	49, 44	eqtrd 1925	. . . . . 6 |- (<.G, G>. e. CVec -> (0(2nd` <.G, G>.)(Id` (1st` <.G, G>.))) = (0G1))
51	48, 50, 43	3eqtr3d 1934	. . . . 5 |- (<.G, G>. e. CVec -> (0G1) = 1)
52	23, 46	syl5eleqr 1978	. . . . . 6 |- (<.G, G>. e. CVec -> 0 e. ran (1st` <.G, G>.))
53	13, 26, 27	vc0rid 9518	. . . . . 6 |- ((<.G, G>. e. CVec /\ 0 e. ran (1st` <.G, G>.)) -> (0(1st` <.G, G>.)(Id` (1st` <.G, G>.))) = 0)
54	52, 53	mpdan 768	. . . . 5 |- (<.G, G>. e. CVec -> (0(1st` <.G, G>.)(Id` (1st` <.G, G>.))) = 0)
55	45, 51, 54	3eqtr3d 1934	. . . 4 |- (<.G, G>. e. CVec -> 1 = 0)
56	3, 55	mto 121	. . 3 |- -. <.G, G>. e. CVec
57		opeq2 3159	. . . 4 |- (G = S -> <.G, G>. = <.G, S>.)
58	57	eleq1d 1963	. . 3 |- (G = S -> (<.G, G>. e. CVec <-> <.G, S>. e. CVec))
59	56, 58	mtbii 784	. 2 |- (G = S -> -. <.G, S>. e. CVec)
60	59	necon2ai 2051	1 |- (<.G, S>. e. CVec -> G =/= S)

Syntax hints:  -. wn 2   -> wi 3   <-> wb 163   /\ w3a 858   = wceq 1298   e. wcel 1300   =/= wne 2017  _Vcvv 2292  <.cop 3046   X. cxp 3984  dom cdm 3986  ran crn 3987  -->wf 3994  ` cfv 3998  (class class class)co 4884  1stc1st 5018  2ndc2nd 5019  CCcc 6384  0cc0 6386  1c1 6387  Grpcgr 9311  Idcgi 9312  CVeccvc 9496

This theorem is referenced by:  vcex 9531  nvex 9562  nvoprne 9638

This theorem was proved from axioms:  ax-1 4  ax-2 5  ax-3 6  ax-mp 7  ax-7 1304  ax-gen 1305  ax-8 1306  ax-9 1307  ax-10 1308  ax-11 1309  ax-12 1310  ax-13 1311  ax-14 1312  ax-17 1317  ax-4 1319  ax-5o 1321  ax-6o 1324  ax-9o 1481  ax-10o 1500  ax-16 1580  ax-11o 1588  ax-ext 1865  ax-rep 3428  ax-sep 3438  ax-nul 3445  ax-pow 3481  ax-pr 3524  ax-un 3790  ax-inf2 5731

This theorem depends on definitions:  df-bi 164  df-or 241  df-an 242  df-3or 859  df-3an 860  df-ex 1327  df-sb 1536  df-eu 1775  df-mo 1776  df-clab 1872  df-cleq 1877  df-clel 1880  df-ne 2019  df-ral 2109  df-rex 2110  df-reu 2111  df-rab 2112  df-v 2294  df-sbc 2454  df-csb 2541  df-dif 2597  df-un 2600  df-in 2603  df-ss 2605  df-pss 2607  df-nul 2876  df-if 2983  df-pw 3035  df-sn 3049  df-pr 3050  df-tp 3052  df-op 3053  df-uni 3178  df-int 3215  df-iun 3257  df-br 3339  df-opab 3396  df-tr 3412  df-eprel 3583  df-id 3586  df-po 3591  df-so 3604  df-fr 3625  df-we 3644  df-ord 3660  df-on 3661  df-lim 3662  df-suc 3663  df-om 3950  df-xp 4000  df-rel 4001  df-cnv 4002  df-co 4003  df-dm 4004  df-rn 4005  df-res 4006  df-ima 4007  df-fun 4008  df-fn 4009  df-f 4010  df-f1 4011  df-fo 4012  df-f1o 4013  df-fv 4014  df-opr 4886  df-oprab 4887  df-mpt 5006  df-1st 5020  df-2nd 5021  df-iota 5089  df-rdg 5140  df-1o 5177  df-oadd 5179  df-omul 5180  df-er 5318  df-ec 5320  df-qs 5323  df-en 5427  df-dom 5428  df-sdom 5429  df-undef 5556  df-riota 5560  df-ni 6152  df-pli 6153  df-mi 6154  df-lti 6155  df-plpq 6187  df-mpq 6188  df-enq 6189  df-nq 6190  df-plq 6191  df-mq 6192  df-rq 6193  df-ltq 6194  df-1q 6195  df-np 6238  df-1p 6239  df-plp 6240  df-mp 6241  df-ltp 6242  df-plpr 6316  df-mpr 6317  df-enr 6318  df-nr 6319  df-plr 6320  df-mr 6321  df-ltr 6322  df-0r 6323  df-1r 6324  df-m1r 6325  df-c 6392  df-0 6393  df-1 6394  df-i 6395  df-r 6396  df-plus 6397  df-mul 6398  df-sub 6511  df-neg 6513  df-grp 9316  df-gid 9317  df-ginv 9318  df-abl 9408  df-vc 9497

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