Theorem vcoprne 8183

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 5267	. . . . 5 |- 1 =/= 0
2		df-ne 1586	. . . . 5 |- (1 =/= 0 <-> -. 1 = 0)
3	1, 2	mpbi 189	. . . 4 |- -. 1 = 0
4		vcoprnelem 8182	. . . . . . . . 9 |- (<.G, G>. e. CVec -> G:(CC X. CC)-->CC)
5		axcnex 5254	. . . . . . . . . . 11 |- CC e. V
6	5, 5	xpex 3257	. . . . . . . . . 10 |- (CC X. CC) e. V
7		fex 3649	. . . . . . . . . 10 |- ((G:(CC X. CC)-->CC /\ (CC X. CC) e. V) -> G e. V)
8	6, 7	mpan2 695	. . . . . . . . 9 |- (G:(CC X. CC)-->CC -> G e. V)
9	4, 8	syl 10	. . . . . . . 8 |- (<.G, G>. e. CVec -> G e. V)
10		op1stg 4084	. . . . . . . 8 |- (G e. V -> (1st` <.G, G>.) = G)
11	9, 10	syl 10	. . . . . . 7 |- (<.G, G>. e. CVec -> (1st` <.G, G>.) = G)
12	11	opreqd 3974	. . . . . 6 |- (<.G, G>. e. CVec -> (0(1st` <.G, G>.)(Id` (1st` <.G, G>.))) = (0G(Id` (1st` <.G, G>.))))
13	11	rneqd 3338	. . . . . . . . . . . 12 |- (<.G, G>. e. CVec -> ran (1st` <.G, G>.) = ran G)
14		eqid 1475	. . . . . . . . . . . . . . 15 |- (1st` <.G, G>.) = (1st` <.G, G>.)
15	14	vcgrp 8162	. . . . . . . . . . . . . 14 |- (<.G, G>. e. CVec -> (1st` <.G, G>.) e. Grp)
16	11, 15	eqeltrrd 1548	. . . . . . . . . . . . 13 |- (<.G, G>. e. CVec -> G e. Grp)
17		grprndm 8037	. . . . . . . . . . . . 13 |- (G e. Grp -> ran G = dom dom G)
18	16, 17	syl 10	. . . . . . . . . . . 12 |- (<.G, G>. e. CVec -> ran G = dom dom G)
19		fdm 3628	. . . . . . . . . . . . . . 15 |- (G:(CC X. CC)-->CC -> dom G = (CC X. CC))
20	4, 19	syl 10	. . . . . . . . . . . . . 14 |- (<.G, G>. e. CVec -> dom G = (CC X. CC))
21	20	dmeqd 3310	. . . . . . . . . . . . 13 |- (<.G, G>. e. CVec -> dom dom G = dom (CC X. CC))
22		dmxpid 3330	. . . . . . . . . . . . 13 |- dom (CC X. CC) = CC
23	21, 22	syl6eq 1522	. . . . . . . . . . . 12 |- (<.G, G>. e. CVec -> dom dom G = CC)
24	13, 18, 23	3eqtrd 1510	. . . . . . . . . . 11 |- (<.G, G>. e. CVec -> ran (1st` <.G, G>.) = CC)
25		ax1cn 5256	. . . . . . . . . . 11 |- 1 e. CC
26	24, 25	syl5eleqr 1554	. . . . . . . . . 10 |- (<.G, G>. e. CVec -> 1 e. ran (1st` <.G, G>.))
27		eqid 1475	. . . . . . . . . . 11 |- ran (1st` <.G, G>.) = ran (1st` <.G, G>.)
28		eqid 1475	. . . . . . . . . . 11 |- (Id` (1st` <.G, G>.)) = (Id` (1st` <.G, G>.))
29	14, 27, 28	vc0rid 8171	. . . . . . . . . 10 |- ((<.G, G>. e. CVec /\ 1 e. ran (1st` <.G, G>.)) -> (1(1st` <.G, G>.)(Id` (1st` <.G, G>.))) = 1)
30	26, 29	mpdan 703	. . . . . . . . 9 |- (<.G, G>. e. CVec -> (1(1st` <.G, G>.)(Id` (1st` <.G, G>.))) = 1)
31		eqid 1475	. . . . . . . . . . 11 |- (2nd` <.G, G>.) = (2nd` <.G, G>.)
32	14, 31, 27	vcid 8155	. . . . . . . . . 10 |- ((<.G, G>. e. CVec /\ 1 e. ran (1st` <.G, G>.)) -> (1(2nd` <.G, G>.)1) = 1)
33	26, 32	mpdan 703	. . . . . . . . 9 |- (<.G, G>. e. CVec -> (1(2nd` <.G, G>.)1) = 1)
34		op2ndg 4085	. . . . . . . . . . . . 13 |- ((G e. V /\ G e. V) -> (2nd` <.G, G>.) = G)
35	34	anidms 434	. . . . . . . . . . . 12 |- (G e. V -> (2nd` <.G, G>.) = G)
36	9, 35	syl 10	. . . . . . . . . . 11 |- (<.G, G>. e. CVec -> (2nd` <.G, G>.) = G)
37	36, 11	eqtr4d 1509	. . . . . . . . . 10 |- (<.G, G>. e. CVec -> (2nd` <.G, G>.) = (1st` <.G, G>.))
38	37	opreqd 3974	. . . . . . . . 9 |- (<.G, G>. e. CVec -> (1(2nd` <.G, G>.)1) = (1(1st` <.G, G>.)1))
39	30, 33, 38	3eqtr2d 1512	. . . . . . . 8 |- (<.G, G>. e. CVec -> (1(1st` <.G, G>.)(Id` (1st` <.G, G>.))) = (1(1st` <.G, G>.)1))
40	14, 27, 28	vczcl 8170	. . . . . . . . . 10 |- (<.G, G>. e. CVec -> (Id` (1st` <.G, G>.)) e. ran (1st` <.G, G>.))
41	40, 26, 26	3jca 818	. . . . . . . . 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>.)))
42	14, 27	vclcan 8169	. . . . . . . . 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))
43	41, 42	mpdan 703	. . . . . . . 8 |- (<.G, G>. e. CVec -> ((1(1st` <.G, G>.)(Id` (1st` <.G, G>.))) = (1(1st` <.G, G>.)1) <-> (Id` (1st` <.G, G>.)) = 1))
44	39, 43	mpbid 195	. . . . . . 7 |- (<.G, G>. e. CVec -> (Id` (1st` <.G, G>.)) = 1)
45	44	opreq2d 3973	. . . . . 6 |- (<.G, G>. e. CVec -> (0G(Id` (1st` <.G, G>.))) = (0G1))
46	12, 45	eqtr2d 1507	. . . . 5 |- (<.G, G>. e. CVec -> (0G1) = (0(1st` <.G, G>.)(Id` (1st` <.G, G>.))))
47		0cn 5315	. . . . . . 7 |- 0 e. CC
48	14, 31, 27, 28	vcz 8174	. . . . . . 7 |- ((<.G, G>. e. CVec /\ 0 e. CC) -> (0(2nd` <.G, G>.)(Id` (1st` <.G, G>.))) = (Id` (1st` <.G, G>.)))
49	47, 48	mpan2 695	. . . . . 6 |- (<.G, G>. e. CVec -> (0(2nd` <.G, G>.)(Id` (1st` <.G, G>.))) = (Id` (1st` <.G, G>.)))
50	36	opreqd 3974	. . . . . . 7 |- (<.G, G>. e. CVec -> (0(2nd` <.G, G>.)(Id` (1st` <.G, G>.))) = (0G(Id` (1st` <.G, G>.))))
51	50, 45	eqtrd 1506	. . . . . 6 |- (<.G, G>. e. CVec -> (0(2nd` <.G, G>.)(Id` (1st` <.G, G>.))) = (0G1))
52	49, 51, 44	3eqtr3d 1514	. . . . 5 |- (<.G, G>. e. CVec -> (0G1) = 1)
53	24, 47	syl5eleqr 1554	. . . . . 6 |- (<.G, G>. e. CVec -> 0 e. ran (1st` <.G, G>.))
54	14, 27, 28	vc0rid 8171	. . . . . 6 |- ((<.G, G>. e. CVec /\ 0 e. ran (1st` <.G, G>.)) -> (0(1st` <.G, G>.)(Id` (1st` <.G, G>.))) = 0)
55	53, 54	mpdan 703	. . . . 5 |- (<.G, G>. e. CVec -> (0(1st` <.G, G>.)(Id` (1st` <.G, G>.))) = 0)
56	46, 52, 55	3eqtr3d 1514	. . . 4 |- (<.G, G>. e. CVec -> 1 = 0)
57	3, 56	mto 106	. . 3 |- -. <.G, G>. e. CVec
58		opeq2 2486	. . . 4 |- (G = S -> <.G, G>. = <.G, S>.)
59	58	eleq1d 1539	. . 3 |- (G = S -> (<.G, G>. e. CVec <-> <.G, S>. e. CVec))
60	57, 59	mtbii 715	. 2 |- (G = S -> -. <.G, S>. e. CVec)
61	60	necon2ai 1610	1 |- (<.G, S>. e. CVec -> G =/= S)

Syntax hints:  -. wn 2   -> wi 3   <-> wb 146   /\ w3a 774   = wceq 955   e. wcel 957   =/= wne 1584  Vcvv 1809  <.cop 2409   X. cxp 3165  dom cdm 3167  ran crn 3168  -->wf 3175  ` cfv 3179  (class class class)co 3960  1stc1st 4074  2ndc2nd 4075  CCcc 5219  0cc0 5221  1c1 5222  Grpcgr 8016  Idcgi 8017  CVeccvc 8149

This theorem is referenced by:  vcex 8184  nvex 8215  nvoprne 8291

This theorem was proved from axioms:  ax-1 4  ax-2 5  ax-3 6  ax-mp 7  ax-7 961  ax-gen 962  ax-8 963  ax-9 964  ax-10 965  ax-11 966  ax-12 967  ax-13 968  ax-14 969  ax-17 970  ax-4 972  ax-5o 974  ax-6o 977  ax-9o 1122  ax-10o 1139  ax-16 1210  ax-11o 1218  ax-ext 1459  ax-rep 2690  ax-sep 2700  ax-nul 2707  ax-pow 2739  ax-pr 2776  ax-un 2863  ax-inf2 4612

This theorem depends on definitions:  df-bi 147  df-or 224  df-an 225  df-3or 775  df-3an 776  df-ex 980  df-sb 1172  df-eu 1382  df-mo 1383  df-clab 1464  df-cleq 1469  df-clel 1472  df-ne 1586  df-ral 1648  df-rex 1649  df-reu 1650  df-rab 1651  df-v 1810  df-sbc 1940  df-csb 2000  df-dif 2047  df-un 2048  df-in 2049  df-ss 2051  df-pss 2053  df-nul 2279  df-if 2360  df-pw 2400  df-sn 2410  df-pr 2411  df-tp 2413  df-op 2414  df-uni 2501  df-int 2531  df-iun 2565  df-br 2617  df-opab 2664  df-tr 2678  df-eprel 2829  df-id 2832  df-po 2837  df-so 2847  df-fr 2914  df-we 2931  df-ord 2948  df-on 2949  df-lim 2950  df-suc 2951  df-om 3129  df-xp 3181  df-rel 3182  df-cnv 3183  df-co 3184  df-dm 3185  df-rn 3186  df-res 3187  df-ima 3188  df-fun 3189  df-fn 3190  df-f 3191  df-fo 3193  df-fv 3195  df-rdg 3929  df-opr 3962  df-oprab 3963  df-1st 4076  df-2nd 4077  df-1o 4130  df-oadd 4132  df-omul 4133  df-er 4258  df-ec 4260  df-qs 4263  df-ni 4987  df-pli 4988  df-mi 4989  df-lti 4990  df-plpq 5022  df-mpq 5023  df-enq 5024  df-nq 5025  df-plq 5026  df-mq 5027  df-rq 5028  df-ltq 5029  df-1q 5030  df-np 5073  df-1p 5074  df-plp 5075  df-mp 5076  df-ltp 5077  df-plpr 5151  df-mpr 5152  df-enr 5153  df-nr 5154  df-plr 5155  df-mr 5156  df-ltr 5157  df-0r 5158  df-1r 5159  df-m1r 5160  df-c 5227  df-0 5228  df-1 5229  df-i 5230  df-r 5231  df-plus 5232  df-mul 5233  df-sub 5343  df-neg 5345  df-grp 8020  df-gid 8021  df-ginv 8022  df-abl 8084  df-vc 8150

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