Theorem hhssnv 9129

Description: Normed complex vector space property of a subspace.

Hypotheses

Ref	Expression
hhssnvt.1	|- W = <.<.( +h |` (H X. H)), ( .h |` (CC X. H))>., (normh |` H)>.
hhssnv.2	|- H e. SH

Assertion

Ref	Expression
hhssnv	|- W e. NrmCVec

<

Proof of Theorem hhssnv

Step	Hyp	Ref	Expression
1		hhssnv.2	. . . . 5 |- H e. SH
2	1	hhssabl 9127	. . . 4 |- ( +h |` (H X. H)) e. Abel
3		ablgrp 8098	. . . 4 |- (( +h |` (H X. H)) e. Abel -> ( +h |` (H X. H)) e. Grp)
4	2, 3	ax-mp 7	. . 3 |- ( +h |` (H X. H)) e. Grp
5	1	shssi 9076	. . . . . 6 |- H (_ H~
6		ssxp 3262	. . . . . 6 |- ((H (_ H~ /\ H (_ H~) -> (H X. H) (_ (H~ X. H~))
7	5, 5, 6	mp2an 699	. . . . 5 |- (H X. H) (_ (H~ X. H~)
8		ax-hfvadd 8865	. . . . . 6 |- +h :(H~ X. H~)-->H~
9	8	fdmi 3638	. . . . 5 |- dom +h = (H~ X. H~)
10	7, 9	sseqtr4 2097	. . . 4 |- (H X. H) (_ dom +h
11		ssdmres 3387	. . . 4 |- ((H X. H) (_ dom +h <-> dom ( +h |` (H X. H)) = (H X. H))
12	10, 11	mpbi 189	. . 3 |- dom ( +h |` (H X. H)) = (H X. H)
13	4, 12	grprn 8053	. 2 |- H = ran ( +h |` (H X. H))
14		sh0 9079	. . . . . 6 |- (H e. SH -> 0h e. H)
15	1, 14	ax-mp 7	. . . . 5 |- 0h e. H
16		oprvalres 4039	. . . . 5 |- ((0h e. H /\ 0h e. H) -> (0h( +h |` (H X. H))0h) = (0h +h 0h))
17	15, 15, 16	mp2an 699	. . . 4 |- (0h( +h |` (H X. H))0h) = (0h +h 0h)
18		ax-hv0cl 8868	. . . . 5 |- 0h e. H~
19	18	hvaddid2 8893	. . . 4 |- (0h +h 0h) = 0h
20	17, 19	eqtr 1498	. . 3 |- (0h( +h |` (H X. H))0h) = 0h
21		eqid 1478	. . . . 5 |- (Id` ( +h |` (H X. H))) = (Id` ( +h |` (H X. H)))
22	13, 21	grpid 8061	. . . 4 |- ((( +h |` (H X. H)) e. Grp /\ 0h e. H) -> (0h = (Id` ( +h |` (H X. H))) <-> (0h( +h |` (H X. H))0h) = 0h))
23	4, 15, 22	mp2an 699	. . 3 |- (0h = (Id` ( +h |` (H X. H))) <-> (0h( +h |` (H X. H))0h) = 0h)
24	20, 23	mpbir 190	. 2 |- 0h = (Id` ( +h |` (H X. H)))
25		df-f 3200	. . . 4 |- (( .h |` (CC X. H)):(CC X. H)-->H <-> (( .h |` (CC X. H)) Fn (CC X. H) /\ ran ( .h |` (CC X. H)) (_ H))
26		ax-hfvmul 8870	. . . . . 6 |- .h :(CC X. H~)-->H~
27		ffn 3633	. . . . . 6 |- ( .h :(CC X. H~)-->H~ -> .h Fn (CC X. H~))
28	26, 27	ax-mp 7	. . . . 5 |- .h Fn (CC X. H~)
29		ssid 2083	. . . . . 6 |- CC (_ CC
30		ssxp 3262	. . . . . 6 |- ((CC (_ CC /\ H (_ H~) -> (CC X. H) (_ (CC X. H~))
31	29, 5, 30	mp2an 699	. . . . 5 |- (CC X. H) (_ (CC X. H~)
32		fnssres 3606	. . . . 5 |- (( .h Fn (CC X. H~) /\ (CC X. H) (_ (CC X. H~)) -> ( .h |` (CC X. H)) Fn (CC X. H))
33	28, 31, 32	mp2an 699	. . . 4 |- ( .h |` (CC X. H)) Fn (CC X. H)
34		oprvalelrn 4045	. . . . . . 7 |- (( .h |` (CC X. H)) Fn (CC X. H) -> (z e. ran ( .h |` (CC X. H)) <-> E.x e. CC E.y e. H (x( .h |` (CC X. H))y) = z))
35	33, 34	ax-mp 7	. . . . . 6 |- (z e. ran ( .h |` (CC X. H)) <-> E.x e. CC E.y e. H (x( .h |` (CC X. H))y) = z)
36		eleq1 1537	. . . . . . . 8 |- ((x( .h |` (CC X. H))y) = z -> ((x( .h |` (CC X. H))y) e. H <-> z e. H))
37		oprvalres 4039	. . . . . . . . 9 |- ((x e. CC /\ y e. H) -> (x( .h |` (CC X. H))y) = (x .h y))
38		shmulclt 9082	. . . . . . . . . 10 |- ((H e. SH /\ x e. CC /\ y e. H) -> (x .h y) e. H)
39	1, 38	mp3an1 905	. . . . . . . . 9 |- ((x e. CC /\ y e. H) -> (x .h y) e. H)
40	37, 39	eqeltrd 1551	. . . . . . . 8 |- ((x e. CC /\ y e. H) -> (x( .h |` (CC X. H))y) e. H)
41	36, 40	syl5cbi 209	. . . . . . 7 |- ((x e. CC /\ y e. H) -> ((x( .h |` (CC X. H))y) = z -> z e. H))
42	41	r19.23aivv 1751	. . . . . 6 |- (E.x e. CC E.y e. H (x( .h |` (CC X. H))y) = z -> z e. H)
43	35, 42	sylbi 199	. . . . 5 |- (z e. ran ( .h |` (CC X. H)) -> z e. H)
44	43	ssriv 2072	. . . 4 |- ran ( .h |` (CC X. H)) (_ H
45	25, 33, 44	mpbir2an 732	. . 3 |- ( .h |` (CC X. H)):(CC X. H)-->H
46		ax1cn 5281	. . . . 5 |- 1 e. CC
47		oprvalres 4039	. . . . 5 |- ((1 e. CC /\ x e. H) -> (1( .h |` (CC X. H))x) = (1 .h x))
48	46, 47	mpan 697	. . . 4 |- (x e. H -> (1( .h |` (CC X. H))x) = (1 .h x))
49	1	shel 9077	. . . . 5 |- (x e. H -> x e. H~)
50		ax-hvmulid 8871	. . . . 5 |- (x e. H~ -> (1 .h x) = x)
51	49, 50	syl 10	. . . 4 |- (x e. H -> (1 .h x) = x)
52	48, 51	eqtrd 1510	. . 3 |- (x e. H -> (1( .h |` (CC X. H))x) = x)
53		ax-hvdistr1 8873	. . . . 5 |- ((y e. CC /\ x e. H~ /\ z e. H~) -> (y .h (x +h z)) = ((y .h x) +h (y .h z)))
54		id 59	. . . . 5 |- (y e. CC -> y e. CC)
55	1	shel 9077	. . . . 5 |- (z e. H -> z e. H~)
56	53, 54, 49, 55	syl3an 870	. . . 4 |- ((y e. CC /\ x e. H /\ z e. H) -> (y .h (x +h z)) = ((y .h x) +h (y .h z)))
57		oprvalres 4039	. . . . . . 7 |- ((x e. H /\ z e. H) -> (x( +h |` (H X. H))z) = (x +h z))
58	57	3adant1 799	. . . . . 6 |- ((y e. CC /\ x e. H /\ z e. H) -> (x( +h |` (H X. H))z) = (x +h z))
59	58	opreq2d 3982	. . . . 5 |- ((y e. CC /\ x e. H /\ z e. H) -> (y( .h |` (CC X. H))(x( +h |` (H X. H))z)) = (y( .h |` (CC X. H))(x +h z)))
60		oprvalres 4039	. . . . . . 7 |- ((y e. CC /\ (x +h z) e. H) -> (y( .h |` (CC X. H))(x +h z)) = (y .h (x +h z)))
61		shaddclt 9080	. . . . . . . 8 |- ((H e. SH /\ x e. H /\ z e. H) -> (x +h z) e. H)
62	1, 61	mp3an1 905	. . . . . . 7 |- ((x e. H /\ z e. H) -> (x +h z) e. H)
63	60, 62	sylan2 453	. . . . . 6 |- ((y e. CC /\ (x e. H /\ z e. H)) -> (y( .h |` (CC X. H))(x +h z)) = (y .h (x +h z)))
64	63	3impb 831	. . . . 5 |- ((y e. CC /\ x e. H /\ z e. H) -> (y( .h |` (CC X. H))(x +h z)) = (y .h (x +h z)))
65	59, 64	eqtrd 1510	. . . 4 |- ((y e. CC /\ x e. H /\ z e. H) -> (y( .h |` (CC X. H))(x( +h |` (H X. H))z)) = (y .h (x +h z)))
66		oprvalres 4039	. . . . . . 7 |- ((y e. CC /\ x e. H) -> (y( .h |` (CC X. H))x) = (y .h x))
67	66	3adant3 801	. . . . . 6 |- ((y e. CC /\ x e. H /\ z e. H) -> (y( .h |` (CC X. H))x) = (y .h x))
68		oprvalres 4039	. . . . . . 7 |- ((y e. CC /\ z e. H) -> (y( .h |` (CC X. H))z) = (y .h z))
69	68	3adant2 800	. . . . . 6 |- ((y e. CC /\ x e. H /\ z e. H) -> (y( .h |` (CC X. H))z) = (y .h z))
70	67, 69	opreq12d 3984	. . . . 5 |- ((y e. CC /\ x e. H /\ z e. H) -> ((y( .h |` (CC X. H))x)( +h |` (H X. H))(y( .h |` (CC X. H))z)) = ((y .h x)( +h |` (H X. H))(y .h z)))
71		oprvalres 4039	. . . . . 6 |- (((y .h x) e. H /\ (y .h z) e. H) -> ((y .h x)( +h |` (H X. H))(y .h z)) = ((y .h x) +h (y .h z)))
72		shmulclt 9082	. . . . . . . 8 |- ((H e. SH /\ y e. CC /\ x e. H) -> (y .h x) e. H)
73	1, 72	mp3an1 905	. . . . . . 7 |- ((y e. CC /\ x e. H) -> (y .h x) e. H)
74	73	3adant3 801	. . . . . 6 |- ((y e. CC /\ x e. H /\ z e. H) -> (y .h x) e. H)
75		shmulclt 9082	. . . . . . . 8 |- ((H e. SH /\ y e. CC /\ z e. H) -> (y .h z) e. H)
76	1, 75	mp3an1 905	. . . . . . 7 |- ((y e. CC /\ z e. H) -> (y .h z) e. H)