Theorem nvi 9565

Description: The properties of a normed complex vector space, which is a vector space accompanied by a norm.

Hypotheses

Ref	Expression
nvi.1	|- X = (BaseSet` U)
nvi.2	|- G = (+v` U)
nvi.4	|- S = (.s` U)
nvi.5	|- Z = (0v` U)
nvi.6	|- N = (norm` U)

Assertion

Ref	Expression
nvi	|- (U e. NrmCVec -> (<.G, S>. e. CVec /\ N:X-->RR /\ A.x e. X (((N` x) = 0 -> x = Z) /\ A.y e. CC (N` (ySx)) = ((abs` y) x. (N` x)) /\ A.y e. X (N` (xGy)) <_ ((N` x) + (N` y)))))

Distinct variable groups:   x,y,G   x,N,y   x,S,y   x,X,y   x,Z

Proof of Theorem nvi

Step	Hyp	Ref	Expression
1		df-nv 9543	. . 3 |- NrmCVec = {<.<.g, s>., n>. | (<.g, s>. e. CVec /\ n:ran g-->RR /\ A.x e. ran g(((n` x) = 0 -> x = (Id` g)) /\ A.y e. CC (n` (ysx)) = ((abs` y) x. (n` x)) /\ A.y e. ran g(n` (xgy)) <_ ((n` x) + (n` y))))}
2	1	eleq2i 1961	. 2 |- (U e. NrmCVec <-> U e. {<.<.g, s>., n>. | (<.g, s>. e. CVec /\ n:ran g-->RR /\ A.x e. ran g(((n` x) = 0 -> x = (Id` g)) /\ A.y e. CC (n` (ysx)) = ((abs` y) x. (n` x)) /\ A.y e. ran g(n` (xgy)) <_ ((n` x) + (n` y))))})
3		id 73	. . . . 5 |- (g = (1st` (1st` U)) -> g = (1st` (1st` U)))
4		nvi.2	. . . . . 6 |- G = (+v` U)
5	4	vafval 9554	. . . . 5 |- G = (1st` (1st` U))
6	3, 5	syl6eqr 1946	. . . 4 |- (g = (1st` (1st` U)) -> g = G)
7		opeq1 3158	. . . . . 6 |- (g = G -> <.g, s>. = <.G, s>.)
8	7	eleq1d 1963	. . . . 5 |- (g = G -> (<.g, s>. e. CVec <-> <.G, s>. e. CVec))
9		rneq 4186	. . . . . . 7 |- (g = G -> ran g = ran G)
10		nvi.1	. . . . . . . 8 |- X = (BaseSet` U)
11	10, 4	bafval 9555	. . . . . . 7 |- X = ran G
12	9, 11	syl6eqr 1946	. . . . . 6 |- (g = G -> ran g = X)
13	12	feq2d 4557	. . . . 5 |- (g = G -> (n:ran g-->RR <-> n:X-->RR))
14		fveq2 4681	. . . . . . . . . 10 |- (g = G -> (Id` g) = (Id` G))
15		nvi.5	. . . . . . . . . . 11 |- Z = (0v` U)
16	4, 15	0vfval 9557	. . . . . . . . . 10 |- Z = (Id` G)
17	14, 16	syl6eqr 1946	. . . . . . . . 9 |- (g = G -> (Id` g) = Z)
18	17	eqeq2d 1895	. . . . . . . 8 |- (g = G -> (x = (Id` g) <-> x = Z))
19	18	imbi2d 674	. . . . . . 7 |- (g = G -> (((n` x) = 0 -> x = (Id` g)) <-> ((n` x) = 0 -> x = Z)))
20		opreq 4888	. . . . . . . . . 10 |- (g = G -> (xgy) = (xGy))
21	20	fveq2d 4685	. . . . . . . . 9 |- (g = G -> (n` (xgy)) = (n` (xGy)))
22	21	breq1d 3348	. . . . . . . 8 |- (g = G -> ((n` (xgy)) <_ ((n` x) + (n` y)) <-> (n` (xGy)) <_ ((n` x) + (n` y))))
23	12, 22	raleqbidv 2274	. . . . . . 7 |- (g = G -> (A.y e. ran g(n` (xgy)) <_ ((n` x) + (n` y)) <-> A.y e. X (n` (xGy)) <_ ((n` x) + (n` y))))
24	19, 23	3anbi13d 1170	. . . . . 6 |- (g = G -> ((((n` x) = 0 -> x = (Id` g)) /\ A.y e. CC (n` (ysx)) = ((abs` y) x. (n` x)) /\ A.y e. ran g(n` (xgy)) <_ ((n` x) + (n` y))) <-> (((n` x) = 0 -> x = Z) /\ A.y e. CC (n` (ysx)) = ((abs` y) x. (n` x)) /\ A.y e. X (n` (xGy)) <_ ((n` x) + (n` y)))))
25	12, 24	raleqbidv 2274	. . . . 5 |- (g = G -> (A.x e. ran g(((n` x) = 0 -> x = (Id` g)) /\ A.y e. CC (n` (ysx)) = ((abs` y) x. (n` x)) /\ A.y e. ran g(n` (xgy)) <_ ((n` x) + (n` y))) <-> A.x e. X (((n` x) = 0 -> x = Z) /\ A.y e. CC (n` (ysx)) = ((abs` y) x. (n` x)) /\ A.y e. X (n` (xGy)) <_ ((n` x) + (n` y)))))
26	8, 13, 25	3anbi123d 1168	. . . 4 |- (g = G -> ((<.g, s>. e. CVec /\ n:ran g-->RR /\ A.x e. ran g(((n` x) = 0 -> x = (Id` g)) /\ A.y e. CC (n` (ysx)) = ((abs` y) x. (n` x)) /\ A.y e. ran g(n` (xgy)) <_ ((n` x) + (n` y)))) <-> (<.G, s>. e. CVec /\ n:X-->RR /\ A.x e. X (((n` x) = 0 -> x = Z) /\ A.y e. CC (n` (ysx)) = ((abs` y) x. (n` x)) /\ A.y e. X (n` (xGy)) <_ ((n` x) + (n` y))))))
27	6, 26	syl 12	. . 3 |- (g = (1st` (1st` U)) -> ((<.g, s>. e. CVec /\ n:ran g-->RR /\ A.x e. ran g(((n` x) = 0 -> x = (Id` g)) /\ A.y e. CC (n` (ysx)) = ((abs` y) x. (n` x)) /\ A.y e. ran g(n` (xgy)) <_ ((n` x) + (n` y)))) <-> (<.G, s>. e. CVec /\ n:X-->RR /\ A.x e. X (((n` x) = 0 -> x = Z) /\ A.y e. CC (n` (ysx)) = ((abs` y) x. (n` x)) /\ A.y e. X (n` (xGy)) <_ ((n` x) + (n` y))))))
28		id 73	. . . . 5 |- (s = (2nd` (1st` U)) -> s = (2nd` (1st` U)))
29		nvi.4	. . . . . 6 |- S = (.s` U)
30	29	smfval 9556	. . . . 5 |- S = (2nd` (1st` U))
31	28, 30	syl6eqr 1946	. . . 4 |- (s = (2nd` (1st` U)) -> s = S)
32		opeq2 3159	. . . . . 6 |- (s = S -> <.G, s>. = <.G, S>.)
33	32	eleq1d 1963	. . . . 5 |- (s = S -> (<.G, s>. e. CVec <-> <.G, S>. e. CVec))
34		opreq 4888	. . . . . . . . . 10 |- (s = S -> (ysx) = (ySx))
35	34	fveq2d 4685	. . . . . . . . 9 |- (s = S -> (n` (ysx)) = (n` (ySx)))
36	35	eqeq1d 1892	. . . . . . . 8 |- (s = S -> ((n` (ysx)) = ((abs` y) x. (n` x)) <-> (n` (ySx)) = ((abs` y) x. (n` x))))
37	36	ralbidv 2123	. . . . . . 7 |- (s = S -> (A.y e. CC (n` (ysx)) = ((abs` y) x. (n` x)) <-> A.y e. CC (n` (ySx)) = ((abs` y) x. (n` x))))
38	37	3anbi2d 1173	. . . . . 6 |- (s = S -> ((((n` x) = 0 -> x = Z) /\ A.y e. CC (n` (ysx)) = ((abs` y) x. (n` x)) /\ A.y e. X (n` (xGy)) <_ ((n` x) + (n` y))) <-> (((n` x) = 0 -> x = Z) /\ A.y e. CC (n` (ySx)) = ((abs` y) x. (n` x)) /\ A.y e. X (n` (xGy)) <_ ((n` x) + (n` y)))))
39	38	ralbidv 2123	. . . . 5 |- (s = S -> (A.x e. X (((n` x) = 0 -> x = Z) /\ A.y e. CC (n` (ysx)) = ((abs` y) x. (n` x)) /\ A.y e. X (n` (xGy)) <_ ((n` x) + (n` y))) <-> A.x e. X (((n` x) = 0 -> x = Z) /\ A.y e. CC (n` (ySx)) = ((abs` y) x. (n` x)) /\ A.y e. X (n` (xGy)) <_ ((n` x) + (n` y)))))
40	33, 39	3anbi13d 1170	. . . 4 |- (s = S -> ((<.G, s>. e. CVec /\ n:X-->RR /\ A.x e. X (((n` x) = 0 -> x = Z) /\ A.y e. CC (n` (ysx)) = ((abs` y) x. (n` x)) /\ A.y e. X (n` (xGy)) <_ ((n` x) + (n` y)))) <-> (<.G, S>. e. CVec /\ n:X-->RR /\ A.x e. X (((n` x) = 0 -> x = Z) /\ A.y e. CC (n` (ySx)) = ((abs` y) x. (n` x)) /\ A.y e. X (n` (xGy)) <_ ((n` x) + (n` y))))))
41	31, 40	syl 12	. . 3 |- (s = (2nd` (1st` U)) -> ((<.G, s>. e. CVec /\ n:X-->RR /\ A.x e. X (((n` x) = 0 -> x = Z) /\ A.y e. CC (n` (ysx)) = ((abs` y) x. (n` x)) /\ A.y e. X (n` (xGy)) <_ ((n` x) + (n` y)))) <-> (<.G, S>. e. CVec /\ n:X-->RR /\ A.x e. X (((n` x) = 0 -> x = Z) /\ A.y e. CC (n` (ySx)) = ((abs` y) x. (n` x)) /\ A.y e. X (n` (xGy)) <_ ((n` x) + (n` y))))))
42		id 73	. . . . 5 |- (n = (2nd` U) -> n = (2nd` U))
43		nvi.6	. . . . . 6 |- N = (norm` U)
44	43	nmfval 9558	. . . . 5 |- N = (2nd` U)
45	42, 44	syl6eqr 1946	. . . 4 |- (n = (2nd` U) -> n = N)
46		feq1 4551	. . . . 5 |- (n = N -> (n:X-->RR <-> N:X-->RR))
47		fveq1 4680	. . . . . . . . 9 |- (n = N -> (n` x) = (N` x))
48	47	eqeq1d 1892	. . . . . . . 8 |- (n = N -> ((n` x) = 0 <-> (N` x) = 0))
49	48	imbi1d 675	. . . . . . 7 |- (n = N -> (((n` x) = 0 -> x = Z) <-> ((N` x) = 0 -> x = Z)))
50		fveq1 4680	. . . . . . . . 9 |- (n = N -> (n` (ySx)) = (N` (ySx)))
51	47	opreq2d 4898	. . . . . . . . 9 |- (n = N -> ((abs` y) x. (n` x)) = ((abs` y) x. (N` x)))
52	50, 51	eqeq12d 1899	. . . . . . . 8 |- (n = N -> ((n` (ySx)) = ((abs` y) x. (n` x)) <-> (N` (ySx)) = ((abs` y) x. (N` x))))
53	52	ralbidv 2123	. . . . . . 7 |- (n = N -> (A.y e. CC (n` (ySx)) = ((abs` y) x. (n` x)) <-> A.y e. CC (N` (ySx)) = ((abs` y) x. (N` x))))
54		fveq1 4680	. . . . . . . . 9 |- (n = N -> (n` (xGy)) = (N` (xGy)))
55		fveq1 4680	. . . . . . . . . 10 |- (n = N -> (n` y) = (N` y))
56	47, 55	opreq12d 4900	. . . . . . . . 9 |- (n = N -> ((n` x) + (n` y)) = ((N` x) + (N` y)))
57	54, 56	breq12d 3351	. . . . . . . 8 |- (n = N -> ((n` (xGy)) <_ ((n` x) + (n` y)) <-> (N` (xGy)) <_ ((N` x) + (N` y))))
58	57	ralbidv 2123	. . . . . . 7 |- (n = N -> (A.y e. X (n` (xGy)) <_ ((n` x) + (n` y)) <-> A.y e. X (N` (xGy)) <_ ((N` x) + (N` y))))
59	49, 53, 58	3anbi123d 1168	. . . . . 6 |- (n = N -> ((((n` x) = 0 -> x = Z) /\ A.y e. CC (n` (ySx)) = ((abs` y) x. (n` x)) /\ A.y e. X (n` (xGy)) <_ ((n` x) + (n` y))) <-> (((N` x) = 0 -> x = Z) /\ A.y e. CC (N` (ySx)) = ((abs` y) x. (N` x)) /\ A.y e. X (N` (xGy)) <_ ((N` x) + (N` y)))))
60	59	ralbidv 2123	. . . . 5 |- (n = N -> (A.x e. X (((n` x) = 0 -> x = Z) /\ A.y e. CC (n` (ySx)) = ((abs` y) x. (n` x)) /\ A.y e. X (n` (xGy)) <_ ((n` x) + (n` y))) <-> A.x e. X (((N` x) = 0 -> x = Z) /\ A.y e. CC (N` (ySx)) = ((abs` y) x. (N` x)) /\ A.y e. X (N` (xGy)) <_ ((N` x) + (N` y)))))
61	46, 60	3anbi23d 1171	. . . 4 |- (n = N -> ((<.G, S>. e. CVec /\ n:X-->RR /\ A.x e. X (((n` x) = 0 -> x = Z) /\ A.y e. CC (n` (ySx)) = ((abs` y) x. (n` x)) /\ A.y e. X (n` (xGy)) <_ ((n` x) + (n` y)))) <-> (<.G, S>. e. CVec /\ N:X-->RR /\ A.x e. X (((N` x) = 0 -> x = Z) /\ A.y e. CC (N` (ySx)) = ((abs` y) x. (N` x)) /\ A.y e. X (N` (xGy)) <_ ((N` x) + (N` y))))))
62	45, 61	syl 12	. . 3 |- (n = (2nd` U) -> ((<.G, S>. e. CVec /\ n:X-->RR /\ A.x e. X (((n` x) = 0 -> x = Z) /\ A.y e. CC (n` (ySx)) = ((abs` y) x. (n` x)) /\ A.y e. X (n` (xGy)) <_ ((n` x) + (n` y)))) <-> (<.G, S>. e. CVec /\ N:X-->RR /\ A.x e. X (((N` x) = 0 -> x = Z) /\ A.y e. CC (N` (ySx)) = ((abs` y) x. (N` x)) /\ A.y e. X (N` (xGy)) <_ ((N` x) + (N` y))))))
63	27, 41, 62	eloprabi 5060	. 2 |- (U e. {<.<.g, s>., n>. | (<.g, s>. e. CVec /\ n:ran g-->RR /\ A.x e. ran g(((n` x) = 0 -> x = (Id` g)) /\ A.y e. CC (n` (ysx)) = ((abs` y) x. (n` x)) /\ A.y e. ran g(n` (xgy)) <_ ((n` x) + (n` y))))} -> (<.G, S>. e. CVec /\ N:X-->RR /\ A.x e. X (((N` x) = 0 -> x = Z) /\ A.y e. CC (N` (ySx)) = ((abs` y) x. (N` x)) /\ A.y e. X (N` (xGy)) <_ ((N` x) + (N` y)))))
64	2, 63	sylbi 216	1 |- (U e. NrmCVec -> (<.G, S>. e. CVec /\ N:X-->RR /\ A.x e. X (((N` x) = 0 -> x = Z) /\ A.y e. CC (N` (ySx)) = ((abs` y) x. (N` x)) /\ A.y e. X (N` (xGy)) <_ ((N` x) + (N` y)))))

Syntax hints:   -> wi 3   <-> wb 163   /\ w3a 858   = wceq 1298   e. wcel 1300  A.wral 2105  <.cop 3046   class class class wbr 3338  ran crn 3987  -->wf 3994  ` cfv 3998  (class class class)co 4884  {copab2 4885  1stc1st 5018  2ndc2nd 5019  CCcc 6384  RRcr 6385  0cc0 6386   + caddc 6389   x. cmul 6391   <_ cle 6448  abscabs 8000  Idcgi 9312  CVeccvc 9496  NrmCVeccnv 9535  +vcpv 9536  BaseSetcba 9537  .scns 9538  0vcn0v 9539  normcnm 9541

This theorem is referenced by:  nvvc 9566  nvf 9618  nvs 9622  nvz 9629  nvtri 9630

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-sep 3438  ax-nul 3445  ax-pow 3481  ax-pr 3524  ax-un 3790

This theorem depends on definitions:  df-bi 164  df-or 241  df-an 242  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-rab 2112  df-v 2294  df-sbc 2454  df-csb 2541  df-dif 2597  df-un 2600  df-in 2603  df-ss 2605  df-nul 2876  df-pw 3035  df-sn 3049  df-pr 3050  df-op 3053  df-uni 3178  df-br 3339  df-opab 3396  df-id 3586  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-fo 4012  df-fv 4014  df-opr 4886  df-oprab 4887  df-1st 5020  df-2nd 5021  df-gid 9317  df-nv 9543  df-va 9546  df-ba 9547  df-sm 9548  df-0v 9549  df-nm 9551

Copyright terms: Public domain