Theorem nvvcop 8209

Description: A normed complex vector space is a vector space.

Assertion

Ref	Expression
nvvcop	|- (<.<.G, S>., N>. e. NrmCVec -> <.G, S>. e. CVec)

Proof of Theorem nvvcop

Step	Hyp	Ref	Expression
1		df-nv 8207	. . . 4 |- 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		3simp1 790	. . . . . 6 |- ((<.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)
3	2	ssoprab2i 4014	. . . . 5 |- {<.<.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>., n>. | <.g, s>. e. CVec}
4		dfoprab2 3997	. . . . . . 7 |- {<.<.g, s>., n>. | <.g, s>. e. CVec} = {<.x, n>. | E.gE.s(x = <.g, s>. /\ <.g, s>. e. CVec)}
5		eleq1 1537	. . . . . . . . . . . 12 |- (x = <.g, s>. -> (x e. CVec <-> <.g, s>. e. CVec))
6	5	pm5.32i 647	. . . . . . . . . . 11 |- ((x = <.g, s>. /\ x e. CVec) <-> (x = <.g, s>. /\ <.g, s>. e. CVec))
7	6	2exbii 1054	. . . . . . . . . 10 |- (E.gE.s(x = <.g, s>. /\ x e. CVec) <-> E.gE.s(x = <.g, s>. /\ <.g, s>. e. CVec))
8		19.41vv 1308	. . . . . . . . . 10 |- (E.gE.s(x = <.g, s>. /\ x e. CVec) <-> (E.gE.s x = <.g, s>. /\ x e. CVec))
9	7, 8	bitr3 175	. . . . . . . . 9 |- (E.gE.s(x = <.g, s>. /\ <.g, s>. e. CVec) <-> (E.gE.s x = <.g, s>. /\ x e. CVec))
10	9	pm3.27bi 326	. . . . . . . 8 |- (E.gE.s(x = <.g, s>. /\ <.g, s>. e. CVec) -> x e. CVec)
11	10	ssopab2i 2829	. . . . . . 7 |- {<.x, n>. | E.gE.s(x = <.g, s>. /\ <.g, s>. e. CVec)} (_ {<.x, n>. | x e. CVec}
12	4, 11	eqsstr 2094	. . . . . 6 |- {<.<.g, s>., n>. | <.g, s>. e. CVec} (_ {<.x, n>. | x e. CVec}
13		df-opab 2672	. . . . . . 7 |- {<.x, n>. | x e. CVec} = {y | E.xE.n(y = <.x, n>. /\ x e. CVec)}
14		fveq2 3730	. . . . . . . . . . . . . 14 |- (y = <.x, n>. -> (1st` y) = (1st` <.x, n>.))
15		visset 1816	. . . . . . . . . . . . . . 15 |- x e. V
16	15	op1st 4091	. . . . . . . . . . . . . 14 |- (1st` <.x, n>.) = x
17	14, 16	syl6eq 1526	. . . . . . . . . . . . 13 |- (y = <.x, n>. -> (1st` y) = x)
18	17	eleq1d 1543	. . . . . . . . . . . 12 |- (y = <.x, n>. -> ((1st` y) e. CVec <-> x e. CVec))
19	18	pm5.32i 647	. . . . . . . . . . 11 |- ((y = <.x, n>. /\ (1st` y) e. CVec) <-> (y = <.x, n>. /\ x e. CVec))
20	19	2exbii 1054	. . . . . . . . . 10 |- (E.xE.n(y = <.x, n>. /\ (1st` y) e. CVec) <-> E.xE.n(y = <.x, n>. /\ x e. CVec))
21		19.41vv 1308	. . . . . . . . . 10 |- (E.xE.n(y = <.x, n>. /\ (1st` y) e. CVec) <-> (E.xE.n y = <.x, n>. /\ (1st` y) e. CVec))
22	20, 21	bitr3 175	. . . . . . . . 9 |- (E.xE.n(y = <.x, n>. /\ x e. CVec) <-> (E.xE.n y = <.x, n>. /\ (1st` y) e. CVec))
23	22	pm3.27bi 326	. . . . . . . 8 |- (E.xE.n(y = <.x, n>. /\ x e. CVec) -> (1st` y) e. CVec)
24	23	ss2abi 2123	. . . . . . 7 |- {y | E.xE.n(y = <.x, n>. /\ x e. CVec)} (_ {y | (1st` y) e. CVec}
25	13, 24	eqsstr 2094	. . . . . 6 |- {<.x, n>. | x e. CVec} (_ {y | (1st` y) e. CVec}
26	12, 25	sstri 2076	. . . . 5 |- {<.<.g, s>., n>. | <.g, s>. e. CVec} (_ {y | (1st` y) e. CVec}
27	3, 26	sstri 2076	. . . 4 |- {<.<.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))))} (_ {y | (1st` y) e. CVec}
28	1, 27	eqsstr 2094	. . 3 |- NrmCVec (_ {y | (1st` y) e. CVec}
29	28	sseli 2068	. 2 |- (<.<.G, S>., N>. e. NrmCVec -> <.<.G, S>., N>. e. {y | (1st` y) e. CVec})
30		opex 2788	. . 3 |- <.<.G, S>., N>. e. V
31		fveq2 3730	. . . . 5 |- (y = <.<.G, S>., N>. -> (1st` y) = (1st` <.<.G, S>., N>.))
32		opex 2788	. . . . . 6 |- <.G, S>. e. V
33	32	op1st 4091	. . . . 5 |- (1st` <.<.G, S>., N>.) = <.G, S>.
34	31, 33	syl6eq 1526	. . . 4 |- (y = <.<.G, S>., N>. -> (1st` y) = <.G, S>.)
35	34	eleq1d 1543	. . 3 |- (y = <.<.G, S>., N>. -> ((1st` y) e. CVec <-> <.G, S>. e. CVec))
36	30, 35	elab 1900	. 2 |- (<.<.G, S>., N>. e. {y | (1st` y) e. CVec} <-> <.G, S>. e. CVec)
37	29, 36	sylib 198	1 |- (<.<.G, S>., N>. e. NrmCVec -> <.G, S>. e. CVec)

Syntax hints:   -> wi 3   /\ wa 223   /\ w3a 777   = wceq 958   e. wcel 960  E.wex 982  {cab 1466  A.wral 1648  <.cop 2415   class class class wbr 2624  {copab 2671  ran crn 3177  -->wf 3184  ` cfv 3188  (class class class)co 3969  {copab2 3970  1stc1st 4083  CCcc 5244  RRcr 5245  0cc0 5246   + caddc 5249   x. cmul 5251   <_ cle 5307  abscabs 6751  Idcgi 8031  CVeccvc 8160  NrmCVeccnv 8199

This theorem is referenced by:  nvex 8226  h2hsm 8839

This theorem was proved from axioms:  ax-1 4  ax-2 5  ax-3 6  ax-mp 7  ax-7 964  ax-gen 965  ax-8 966  ax-9 967  ax-10 968  ax-11 969  ax-12 970  ax-13 971  ax-14 972  ax-17 973  ax-4 975  ax-5o 977  ax-6o 980  ax-9o 1125  ax-10o 1142  ax-16 1212  ax-11o 1220  ax-ext 1462  ax-sep 2708  ax-nul 2715  ax-pow 2748  ax-pr 2785  ax-un 2872

This theorem depends on definitions:  df-bi 147  df-or 224  df-an 225  df-3an 779  df-ex 983  df-sb 1174  df-eu 1384  df-mo 1385  df-clab 1467  df-cleq 1472  df-clel 1475  df-ne 1590  df-ral 1652  df-rex 1653  df-v 1815  df-dif 2052  df-un 2053  df-in 2054  df-ss 2056  df-nul 2284  df-pw 2406  df-sn 2416  df-pr 2417  df-op 2420  df-uni 2508  df-br 2625  df-opab 2672  df-id 2841  df-xp 3190  df-rel 3191  df-cnv 3192  df-co 3193  df-dm 3194  df-rn 3195  df-res 3196  df-ima 3197  df-fun 3198  df-fv 3204  df-oprab 3972  df-1st 4085  df-nv 8207

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