Theorem branmfnOLD 11676

Description: The norm of the bra function.

Assertion

Ref	Expression
branmfnOLD	|- (A e. ~H -> (normfn` (bra` A)) = (normh` A))

Proof of Theorem branmfnOLD

Step	Hyp	Ref	Expression
1		fveq2 4681	. . . 4 |- (A = 0h -> (bra` A) = (bra` 0h))
2	1	fveq2d 4685	. . 3 |- (A = 0h -> (normfn` (bra` A)) = (normfn` (bra` 0h)))
3		fveq2 4681	. . 3 |- (A = 0h -> (normh` A) = (normh` 0h))
4	2, 3	eqeq12d 1899	. 2 |- (A = 0h -> ((normfn` (bra` A)) = (normh` A) <-> (normfn` (bra` 0h)) = (normh` 0h)))
5		brafn 11508	. . . . 5 |- (A e. ~H -> (bra` A):~H-->CC)
6		nmfnval 11440	. . . . 5 |- ((bra` A):~H-->CC -> (normfn` (bra` A)) = sup({x | E.y e. ~H ((normh` y) <_ 1 /\ x = (abs` ((bra` A)` y)))}, RR*, < ))
7	5, 6	syl 12	. . . 4 |- (A e. ~H -> (normfn` (bra` A)) = sup({x | E.y e. ~H ((normh` y) <_ 1 /\ x = (abs` ((bra` A)` y)))}, RR*, < ))
8	7	adantr 425	. . 3 |- ((A e. ~H /\ A =/= 0h) -> (normfn` (bra` A)) = sup({x | E.y e. ~H ((normh` y) <_ 1 /\ x = (abs` ((bra` A)` y)))}, RR*, < ))
9		nmfnsetre 11441	. . . . . . . 8 |- ((bra` A):~H-->CC -> {x | E.y e. ~H ((normh` y) <_ 1 /\ x = (abs` ((bra` A)` y)))} C_ RR)
10	5, 9	syl 12	. . . . . . 7 |- (A e. ~H -> {x | E.y e. ~H ((normh` y) <_ 1 /\ x = (abs` ((bra` A)` y)))} C_ RR)
11		ressxr 6667	. . . . . . . 8 |- RR C_ RR*
12	11	a1i 8	. . . . . . 7 |- (A e. ~H -> RR C_ RR*)
13	10, 12	sstrd 2627	. . . . . 6 |- (A e. ~H -> {x | E.y e. ~H ((normh` y) <_ 1 /\ x = (abs` ((bra` A)` y)))} C_ RR*)
14		normcl 10624	. . . . . . 7 |- (A e. ~H -> (normh` A) e. RR)
15		rexr 6668	. . . . . . 7 |- ((normh` A) e. RR -> (normh` A) e. RR*)
16	14, 15	syl 12	. . . . . 6 |- (A e. ~H -> (normh` A) e. RR*)
17	13, 16	jca 310	. . . . 5 |- (A e. ~H -> ({x | E.y e. ~H ((normh` y) <_ 1 /\ x = (abs` ((bra` A)` y)))} C_ RR* /\ (normh` A) e. RR*))
18	17	adantr 425	. . . 4 |- ((A e. ~H /\ A =/= 0h) -> ({x | E.y e. ~H ((normh` y) <_ 1 /\ x = (abs` ((bra` A)` y)))} C_ RR* /\ (normh` A) e. RR*))
19		id 73	. . . . . . . . . . . . 13 |- (z = (abs` ((bra` A)` y)) -> z = (abs` ((bra` A)` y)))
20		bravalval 11505	. . . . . . . . . . . . . . 15 |- ((A e. ~H /\ y e. ~H) -> ((bra` A)` y) = (y .ih A))
21	20	fveq2d 4685	. . . . . . . . . . . . . 14 |- ((A e. ~H /\ y e. ~H) -> (abs` ((bra` A)` y)) = (abs` (y .ih A)))
22	21	adantr 425	. . . . . . . . . . . . 13 |- (((A e. ~H /\ y e. ~H) /\ (normh` y) <_ 1) -> (abs` ((bra` A)` y)) = (abs` (y .ih A)))
23	19, 22	sylan9eqr 1951	. . . . . . . . . . . 12 |- ((((A e. ~H /\ y e. ~H) /\ (normh` y) <_ 1) /\ z = (abs` ((bra` A)` y))) -> z = (abs` (y .ih A)))
24		bcs2 10682	. . . . . . . . . . . . . . 15 |- ((y e. ~H /\ A e. ~H /\ (normh` y) <_ 1) -> (abs` (y .ih A)) <_ (normh` A))
25	24	3expa 1067	. . . . . . . . . . . . . 14 |- (((y e. ~H /\ A e. ~H) /\ (normh` y) <_ 1) -> (abs` (y .ih A)) <_ (normh` A))
26	25	ancom1s 548	. . . . . . . . . . . . 13 |- (((A e. ~H /\ y e. ~H) /\ (normh` y) <_ 1) -> (abs` (y .ih A)) <_ (normh` A))
27	26	adantr 425	. . . . . . . . . . . 12 |- ((((A e. ~H /\ y e. ~H) /\ (normh` y) <_ 1) /\ z = (abs` ((bra` A)` y))) -> (abs` (y .ih A)) <_ (normh` A))
28	23, 27	eqbrtrd 3357	. . . . . . . . . . 11 |- ((((A e. ~H /\ y e. ~H) /\ (normh` y) <_ 1) /\ z = (abs` ((bra` A)` y))) -> z <_ (normh` A))
29	28	exp41 413	. . . . . . . . . 10 |- (A e. ~H -> (y e. ~H -> ((normh` y) <_ 1 -> (z = (abs` ((bra` A)` y)) -> z <_ (normh` A)))))
30	29	imp4a 391	. . . . . . . . 9 |- (A e. ~H -> (y e. ~H -> (((normh` y) <_ 1 /\ z = (abs` ((bra` A)` y))) -> z <_ (normh` A))))
31	30	r19.23adv 2215	. . . . . . . 8 |- (A e. ~H -> (E.y e. ~H ((normh` y) <_ 1 /\ z = (abs` ((bra` A)` y))) -> z <_ (normh` A)))
32	31	imp 377	. . . . . . 7 |- ((A e. ~H /\ E.y e. ~H ((normh` y) <_ 1 /\ z = (abs` ((bra` A)` y)))) -> z <_ (normh` A))
33		visset 2295	. . . . . . . 8 |- z e. _V
34		eqeq1 1890	. . . . . . . . . 10 |- (x = z -> (x = (abs` ((bra` A)` y)) <-> z = (abs` ((bra` A)` y))))
35	34	anbi2d 678	. . . . . . . . 9 |- (x = z -> (((normh` y) <_ 1 /\ x = (abs` ((bra` A)` y))) <-> ((normh` y) <_ 1 /\ z = (abs` ((bra` A)` y)))))
36	35	rexbidv 2124	. . . . . . . 8 |- (x = z -> (E.y e. ~H ((normh` y) <_ 1 /\ x = (abs` ((bra` A)` y))) <-> E.y e. ~H ((normh` y) <_ 1 /\ z = (abs` ((bra` A)` y)))))
37	33, 36	elab 2403	. . . . . . 7 |- (z e. {x | E.y e. ~H ((normh` y) <_ 1 /\ x = (abs` ((bra` A)` y)))} <-> E.y e. ~H ((normh` y) <_ 1 /\ z = (abs` ((bra` A)` y))))
38	32, 37	sylan2b 501	. . . . . 6 |- ((A e. ~H /\ z e. {x | E.y e. ~H ((normh` y) <_ 1 /\ x = (abs` ((bra` A)` y)))}) -> z <_ (normh` A))
39	38	r19.21aiva 2176	. . . . 5 |- (A e. ~H -> A.z e. {x | E.y e. ~H ((normh` y) <_ 1 /\ x = (abs` ((bra` A)` y)))}z <_ (normh` A))
40	39	adantr 425	. . . 4 |- ((A e. ~H /\ A =/= 0h) -> A.z e. {x | E.y e. ~H ((normh` y) <_ 1 /\ x = (abs` ((bra` A)` y)))}z <_ (normh` A))
41		breq2 3342	. . . . . . . . 9 |- (w = (normh` A) -> (z < w <-> z < (normh` A)))
42	41	rcla4ev 2381	. . . . . . . 8 |- (((normh` A) e. {x | E.y e. ~H ((normh` y) <_ 1 /\ x = (abs` ((bra` A)` y)))} /\ z < (normh` A)) -> E.w e. {x | E.y e. ~H ((normh` y) <_ 1 /\ x = (abs` ((bra` A)` y)))}z < w)
43	14	recnd 6468	. . . . . . . . . . . . . 14 |- (A e. ~H -> (normh` A) e. CC)
44	43	adantr 425	. . . . . . . . . . . . 13 |- ((A e. ~H /\ A =/= 0h) -> (normh` A) e. CC)
45		normne0 10630	. . . . . . . . . . . . . 14 |- (A e. ~H -> ((normh` A) =/= 0 <-> A =/= 0h))
46	45	biimpar 461	. . . . . . . . . . . . 13 |- ((A e. ~H /\ A =/= 0h) -> (normh` A) =/= 0)
47		reccl 6904	. . . . . . . . . . . . 13 |- (((normh` A) e. CC /\ (normh` A) =/= 0) -> (1 / (normh` A)) e. CC)
48	44, 46, 47	syl11anc 524	. . . . . . . . . . . 12 |- ((A e. ~H /\ A =/= 0h) -> (1 / (normh` A)) e. CC)
49		simpl 346	. . . . . . . . . . . 12 |- ((A e. ~H /\ A =/= 0h) -> A e. ~H)
50		hvmulcl 10515	. . . . . . . . . . . 12 |- (((1 / (normh` A)) e. CC /\ A e. ~H) -> ((1 / (normh` A)) .h A) e. ~H)
51	48, 49, 50	syl11anc 524	. . . . . . . . . . 11 |- ((A e. ~H /\ A =/= 0h) -> ((1 / (normh` A)) .h A) e. ~H)
52		norm1 10754	. . . . . . . . . . . 12 |- ((A e. ~H /\ A =/= 0h) -> (normh` ((1 / (normh` A)) .h A)) = 1)
53		1re 6598	. . . . . . . . . . . . 13 |- 1 e. RR
54	53	leidi 6790	. . . . . . . . . . . 12 |- 1 <_ 1
55	52, 54	syl6eqbr 3374	. . . . . . . . . . 11 |- ((A e. ~H /\ A =/= 0h) -> (normh` ((1 / (normh` A)) .h A)) <_ 1)
56		ax-his3 10584	. . . . . . . . . . . . 13 |- (((1 / (normh` A)) e. CC /\ A e. ~H /\ A e. ~H) -> (((1 / (normh` A)) .h A) .ih A) = ((1 / (normh` A)) x. (A .ih A)))
57	48, 49, 49, 56	syl111anc 1100	. . . . . . . . . . . 12 |- ((A e. ~H /\ A =/= 0h) -> (((1 / (normh` A)) .h A) .ih A) = ((1 / (normh` A)) x. (A .ih A)))
58	14	adantr 425	. . . . . . . . . . . . . . . 16 |- ((A e. ~H /\ A =/= 0h) -> (normh` A) e. RR)
59		rereccl 6981	. . . . . . . . . . . . . . . 16 |- (((normh` A) e. RR /\ (normh` A) =/= 0) -> (1 / (normh` A)) e. RR)
60	58, 46, 59	syl11anc 524	. . . . . . . . . . . . . . 15 |- ((A e. ~H /\ A =/= 0h) -> (1 / (normh` A)) e. RR)
61		hiidrcl 10594	. . . . . . . . . . . . . . . 16 |- (A e. ~H -> (A .ih A) e. RR)
62	61	adantr 425	. . . . . . . . . . . . . . 15 |- ((A e. ~H /\ A =/= 0h) -> (A .ih A) e. RR)
63		remulcl 6457	. . . . . . . . . . . . . . 15 |- (((1 / (normh` A)) e. RR /\ (A .ih A) e. RR) -> ((1 / (normh` A)) x. (A .ih A)) e. RR)
64	60, 62, 63	syl11anc 524	. . . . . . . . . . . . . 14 |- ((A e. ~H /\ A =/= 0h) -> ((1 / (normh` A)) x. (A .ih A)) e. RR)
65	57, 64	eqeltrd 1971	. . . . . . . . . . . . 13 |- ((A e. ~H /\ A =/= 0h) -> (((1 / (normh` A)) .h A) .ih A) e. RR)
66		normgt0 10627	. . . . . . . . . . . . . . . . . 18 |- (A e. ~H -> (A =/= 0h <-> 0 < (normh` A)))
67	66	biimpa 460	. . . . . . . . . . . . . . . . 17 |- ((A e. ~H /\ A =/= 0h) -> 0 < (normh` A))
68		recgt0 7043	. . . . . . . . . . . . . . . . 17 |- (((normh` A) e. RR /\ 0 < (normh` A)) -> 0 < (1 / (normh` A)))
69	58, 67, 68	syl11anc 524	. . . . . . . . . . . . . . . 16 |- ((A e. ~H /\ A =/= 0h) -> 0 < (1 / (normh` A)))
70		0re 6603	. . . . . . . . . . . . . . . . 17 |- 0 e. RR
71		ltle 6690	. . . . . . . . . . . . . . . . 17 |- ((0 e. RR /\ (1 / (normh` A)) e. RR) -> (0 < (1 / (normh` A)) -> 0 <_ (1 / (normh` A))))
72	70, 71	mpan 759	. . . . . . . . . . . . . . . 16 |- ((1 / (normh` A)) e. RR -> (0 < (1 / (normh` A)) -> 0 <_ (1 / (normh` A))))
73	60, 69, 72	sylc 83	. . . . . . . . . . . . . . 15 |- ((A e. ~H /\ A =/= 0h) -> 0 <_ (1 / (normh` A)))
74		hiidge0 10597	. . . . . . . . . . . . . . . 16 |- (A e. ~H -> 0 <_ (A .ih A))
75	74	adantr 425	. . . . . . . . . . . . . . 15 |- ((A e. ~H /\ A =/= 0h) -> 0 <_ (A .ih A))
76		mulge0OLD 6869	. . . . . . . . . . . . . . 15 |- ((((1 / (normh` A)) e. RR /\ (A .ih A) e. RR) /\ (0 <_ (1 / (normh` A)) /\ 0 <_ (A .ih A))) -> 0 <_ ((1 / (normh` A)) x. (A .ih A)))
77	60, 62, 73, 75, 76	syl22anc 1101	. . . . . . . . . . . . . 14 |- ((A e. ~H /\ A =/= 0h) -> 0 <_ ((1 / (normh` A)) x. (A .ih A)))
78	77, 57	breqtrrd 3363	. . . . . . . . . . . . 13 |- ((A e. ~H /\ A =/= 0h) -> 0 <_ (((1 / (normh` A)) .h A) .ih A))
79		absid 8113	. . . . . . . . . . . . 13 |- (((((1 / (normh` A)) .h A) .ih A) e. RR /\ 0 <_ (((1 / (normh` A)) .h A) .ih A)) -> (abs` (((1 / (normh` A)) .h A) .ih A)) = (((1 / (normh` A)) .h A) .ih A))
80	65, 78, 79	syl11anc 524	. . . . . . . . . . . 12 |- ((A e. ~H /\ A =/= 0h) -> (abs` (((1 / (normh` A)) .h A) .ih A)) = (((1 / (normh` A)) .h A) .ih A))
81		mulcom 6459	. . . . . . . . . . . . . . . . 17 |- (((1 / (normh` A)) e. CC /\ (normh` A) e. CC) -> ((1 / (normh` A)) x. (normh` A)) = ((normh` A) x. (1 / (normh` A))))
82	48, 44, 81	syl11anc 524	. . . . . . . . . . . . . . . 16 |- ((A e. ~H /\ A =/= 0h) -> ((1 / (normh` A)) x. (normh` A)) = ((normh` A) x. (1 / (normh` A))))
83		recid 6918	. . . . . . . . . . . . . . . . 17 |- (((normh` A) e. CC /\ (normh` A) =/= 0) -> ((normh` A) x. (1 / (normh` A))) = 1)
84	44, 46, 83	syl11anc 524	. . . . . . . . . . . . . . . 16 |- ((A e. ~H /\ A =/= 0h) -> ((normh` A) x. (1 / (normh` A))) = 1)
85	82, 84	eqtrd 1925	. . . . . . . . . . . . . . 15 |- ((A e. ~H /\ A =/= 0h) -> ((1 / (normh` A)) x. (normh` A)) = 1)
86	85	opreq2d 4898	. . . . . . . . . . . . . 14 |- ((A e. ~H /\ A =/= 0h) -> ((normh` A) x. ((1 / (normh` A)) x. (normh` A))) = ((normh` A) x. 1))
87		ax1id 6435	. . . . . . . . . . . . . . . 16 |- ((normh` A) e. CC -> ((normh` A) x. 1) = (normh` A))
88	43, 87	syl 12	. . . . . . . . . . . . . . 15 |- (A e. ~H -> ((normh` A) x. 1) = (normh` A))
89	88	adantr 425	. . . . . . . . . . . . . 14 |- ((A e. ~H /\ A =/= 0h) -> ((normh` A) x. 1) = (normh` A))
90	86, 89	eqtr2d 1926	. . . . . . . . . . . . 13 |- ((A e. ~H /\ A =/= 0h) -> (normh` A) = ((normh` A) x. ((1 / (normh` A)) x. (normh` A))))
91		mul12 6579	. . . . . . . . . . . . . 14 |- (((normh` A) e. CC /\ (1 / (normh` A)) e. CC /\ (normh` A) e. CC) -> ((normh` A) x. ((1 / (normh` A)) x. (normh` A))) = ((1 / (normh` A)) x. ((normh` A) x. (normh` A))))
92	44, 48, 44, 91	syl111anc 1100	. . . . . . . . . . . . 13 |- ((A e. ~H /\ A =/= 0h) -> ((normh` A) x. ((1 / (normh` A)) x. (normh` A))) = ((1 / (normh` A)) x. ((normh` A) x. (normh` A))))
93		sqval 7854	. . . . . . . . . . . . . . . . 17 |- ((normh` A) e. CC -> ((normh` A)^2) = ((normh` A) x. (normh` A)))
94	43, 93	syl 12	. . . . . . . . . . . . . . . 16 |- (A e. ~H -> ((normh` A)^2) = ((normh` A) x. (normh` A)))
95		normsq 10634	. . . . . . . . . . . . . . . 16 |- (A e. ~H -> ((normh` A)^2) = (A .ih A))
96	94, 95	eqtr3d 1927	. . . . . . . . . . . . . . 15 |- (A e. ~H -> ((normh` A) x. (normh` A)) = (A .ih A))
97	96	adantr 425	. . . . . . . . . . . . . 14 |- ((A e. ~H /\ A =/= 0h) -> ((normh` A) x. (normh` A)) = (A .ih A))
98	97	opreq2d 4898	. . . . . . . . . . . . 13 |- ((A e. ~H /\ A =/= 0h) -> ((1 / (normh` A)) x. ((normh` A) x. (normh` A))) = ((1 / (normh` A)) x. (A .ih A)))
99	90, 92, 98	3eqtrd 1929	. . . . . . . . . . . 12 |- ((A e. ~H /\ A =/= 0h) -> (normh` A) = ((1 / (normh` A)) x. (A .ih A)))
100	57, 80, 99	3eqtr4rd 1939	. . . . . . . . . . 11 |- ((A e. ~H /\ A =/= 0h) -> (normh` A) = (abs` (((1 / (normh` A)) .h A) .ih A)))
101		fveq2 4681	. . . . . . . . . . . . . 14 |- (y = ((1 / (normh` A)) .h A) -> (normh` y) = (normh` ((1 / (normh` A)) .h A)))
102	101	breq1d 3348	. . . . . . . . . . . . 13 |- (y = ((1 / (normh` A)) .h A) -> ((normh` y) <_ 1 <-> (normh` ((1 / (normh` A)) .h A)) <_ 1))
103		opreq1 4889	. . . . . . . . . . . . . . 15 |- (y = ((1 / (normh` A)) .h A) -> (y .ih A) = (((1 / (normh` A)) .h A) .ih A))
104	103	fveq2d 4685	. . . . . . . . . . . . . 14 |- (y = ((1 / (normh` A)) .h A) -> (abs` (y .ih A)) = (abs` (((1 / (normh` A)) .h A) .ih A)))
105	104	eqeq2d 1895	. . . . . . . . . . . . 13 |- (y = ((1 / (normh` A)) .h A) -> ((normh` A) = (abs` (y .ih A)) <-> (normh` A) = (abs` (((1 / (normh` A)) .h A) .ih A))))
106	102, 105	anbi12d 690	. . . . . . . . . . . 12 |- (y = ((1 / (normh` A)) .h A) -> (((normh` y) <_ 1 /\ (normh` A) = (abs` (y .ih A))) <-> ((normh` ((1 / (normh` A)) .h A)) <_ 1 /\ (normh` A) = (abs` (((1 / (normh` A)) .h A) .ih A)))))
107	106	rcla4ev 2381	. . . . . . . . . . 11 |- ((((1 / (normh` A)) .h A) e. ~H /\ ((normh` ((1 / (normh` A)) .h A)) <_ 1 /\ (normh` A) = (abs` (((1 / (normh` A)) .h A) .ih A)))) -> E.y e. ~H ((normh` y) <_ 1 /\ (normh` A) = (abs` (y .ih A))))
108	51, 55, 100, 107	syl12anc 1098	. . . . . . . . . 10 |- ((A e. ~H /\ A =/= 0h) -> E.y e. ~H ((normh` y) <_ 1 /\ (normh` A) = (abs` (y .ih A))))
109	21	eqeq2d 1895	. . . . . . . . . . . . 13 |- ((A e. ~H /\ y e. ~H) -> ((normh` A) = (abs` ((bra` A)` y)) <-> (normh` A) = (abs` (y .ih A))))
110	109	anbi2d 678	. . . . . . . . . . . 12 |- ((A e. ~H /\ y e. ~H) -> (((normh` y) <_ 1 /\ (normh` A) = (abs` ((bra` A)` y))) <-> ((normh` y) <_ 1 /\ (normh` A) = (abs` (y .ih A)))))
111	110	rexbidva 2120	. . . . . . . . . . 11 |- (A e. ~H -> (E.y e. ~H ((normh` y) <_ 1 /\ (normh` A) = (abs` ((bra` A)` y))) <-> E.y e. ~H ((normh` y) <_ 1 /\ (normh` A) = (abs` (y .ih A)))))
112	111	adantr 425	. . . . . . . . . 10 |- ((A e. ~H /\ A =/= 0h) -> (E.y e. ~H ((normh` y) <_ 1 /\ (normh` A) = (abs` ((bra` A)` y))) <-> E.y e. ~H ((normh` y) <_ 1 /\ (normh` A) = (abs` (y .ih A)))))
113	108, 112	mpbird 213	. . . . . . . . 9 |- ((A e. ~H /\ A =/= 0h) -> E.y e. ~H ((normh` y) <_ 1 /\ (normh` A) = (abs` ((bra` A)` y))))
114		fvex 4689	. . . . . . . . . 10 |- (normh` A) e. _V
115		eqeq1 1890	. . . . . . . . . . . 12 |- (x = (normh` A) -> (x = (abs` ((bra` A)` y)) <-> (normh` A) = (abs` ((bra` A)` y))))
116	115	anbi2d 678	. . . . . . . . . . 11 |- (x = (normh` A) -> (((normh` y) <_ 1 /\ x = (abs` ((bra` A)` y))) <-> ((normh` y) <_ 1 /\ (normh` A) = (abs` ((bra` A)` y)))))
117	116	rexbidv 2124	. . . . . . . . . 10 |- (x = (normh` A) -> (E.y e. ~H ((normh` y) <_ 1 /\ x = (abs` ((bra` A)` y))) <-> E.y e. ~H ((normh` y) <_ 1 /\ (normh` A) = (abs` ((bra` A)` y)))))
118	114, 117	elab 2403	. . . . . . . . 9 |- ((normh` A) e. {x | E.y e. ~H ((normh` y) <_ 1 /\ x = (abs` ((bra` A)` y)))} <-> E.y e. ~H ((normh` y) <_ 1 /\ (normh` A) = (abs` ((bra` A)` y))))
119	113, 118	sylibr 217	. . . . . . . 8 |- ((A e. ~H /\ A =/= 0h) -> (normh` A) e. {x | E.y e. ~H ((normh` y) <_ 1 /\ x = (abs` ((bra` A)` y)))})
120	42, 119	sylan 497	. . . . . . 7 |- (((A e. ~H /\ A =/= 0h) /\ z < (normh` A)) -> E.w e. {x | E.y e. ~H ((normh` y) <_ 1 /\ x = (abs` ((bra` A)` y)))}z < w)
121	120	adantlr 429	. . . . . 6 |- ((((A e. ~H /\ A =/= 0h) /\ z e. RR) /\ z < (normh` A)) -> E.w e. {x | E.y e. ~H ((normh` y) <_ 1 /\ x = (abs` ((bra` A)` y)))}z < w)
122	121	ex 402	. . . . 5 |- (((A e. ~H /\ A =/= 0h) /\ z e. RR) -> (z < (normh` A) -> E.w e. {x | E.y e. ~H ((normh` y) <_ 1 /\ x = (abs` ((bra` A)` y)))}z < w))
123	122	r19.21aiva 2176	. . . 4 |- ((A e. ~H /\ A =/= 0h) -> A.z e. RR (z < (normh` A) -> E.w e. {x | E.y e. ~H ((normh` y) <_ 1 /\ x = (abs` ((bra` A)` y)))}z < w))
124		supxr2 7291	. . . 4 |- ((({x | E.y e. ~H ((normh` y) <_ 1 /\ x = (abs` ((bra` A)` y)))} C_ RR* /\ (normh` A) e. RR*) /\ (A.z e. {x | E.y e. ~H ((normh` y) <_ 1 /\ x = (abs` ((bra` A)` y)))}z <_ (normh` A) /\ A.z e. RR (z < (normh` A) -> E.w e. {x | E.y e. ~H ((normh` y) <_ 1 /\ x = (abs` ((bra` A)` y)))}z < w))) -> sup({x | E.y e. ~H ((normh` y) <_ 1 /\ x = (abs` ((bra` A)` y)))}, RR*, < ) = (normh` A))
125	18, 40, 123, 124	syl12anc 1098	. . 3 |- ((A e. ~H /\ A =/= 0h) -> sup({x | E.y e. ~H ((normh` y) <_ 1 /\ x = (abs` ((bra` A)` y)))}, RR*, < ) = (normh` A))
126	8, 125	eqtrd 1925	. 2 |- ((A e. ~H /\ A =/= 0h) -> (normfn` (bra` A)) = (normh` A))
127		nmfn0 11548	. . . 4 |- (normfn` (~H X. {0})) = 0
128		bra0 11511	. . . . 5 |- (bra` 0h) = (~H X. {0})
129	128	fveq2i 4684	. . . 4 |- (normfn` (bra` 0h)) = (normfn` (~H X. {0}))
130		norm0 10628	. . . 4 |- (normh` 0h) = 0
131	127, 129, 130	3eqtr4i 1921	. . 3 |- (normfn` (bra` 0h)) = (normh` 0h)
132	131	a1i 8	. 2 |- (A e. ~H -> (normfn` (bra` 0h)) = (normh` 0h))
133	4, 126, 132	pm2.61ne 2087	1 |- (A e. ~H -> (normfn` (bra` A)) = (normh` A))

Syntax hints:   -> wi 3   <-> wb 163   /\ wa 240   = wceq 1298   e. wcel 1300  {cab 1871   =/= wne 2017  A.wral 2105  E.wrex 2106   C_ wss 2593  {csn 3044   class class class wbr 3338   X. cxp 3984  -->wf 3994  ` cfv 3998  (class class class)co 4884  supcsup 5663  CCcc 6384  RRcr 6385  0cc0 6386  1c1 6387   x. cmul 6391   / cdiv 6447   <_ cle 6448  RR*cxr 6652   < clt 6653  2c2 7145  ^cexp 7811  abscabs 8000  ~Hchil 10420   .h csm 10422  0hc0v 10423   .ih csp 10425  normhcno 10426  norm_fncnmf 10452  bracbr 10457

This theorem was proved from axioms:  ax-1 4  ax-2 5  ax-3 6  ax-mp 7  ax-5 1302  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-reg 5695  ax-inf2 5731  ax-ac 5906  ax-hilex 10501  ax-hfvadd 10502  ax-hvcom 10503  ax-hvass 10504  ax-hv0cl 10505  ax-hvaddid 10506  ax-hfvmul 10507  ax-hvmulid 10508  ax-hvmulass 10509  ax-hvdistr1 10510  ax-hvdistr2 10511  ax-hvmul0 10512  ax-hfi 10579  ax-his1 10582  ax-his2 10583  ax-his3 10584  ax-his4 10585

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-nel 2020  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-iin 3258  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-map 5383  df-en 5427  df-dom 5428  df-sdom 5429  df-undef 5556  df-riota 5560  df-sup 5664  df-r1 5750  df-rank 5751  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-lt 6399  df-sub 6511  df-neg 6513  df-pnf 6654  df-mnf 6655  df-xr 6656  df-ltxr 6657  df-le 6658  df-div 6892  df-n 7108  df-2 7154  df-3 7155  df-4 7156  df-n0 7309  df-z 7345  df-q 7436  df-fl 7463  df-ioo 7528  df-uz 7587  df-fz 7638  df-seq1 7721  df-shft 7754  df-seqz 7776  df-exp 7812  df-sqr 7920  df-re 8001  df-im 8002  df-cj 8003  df-abs 8004  df-clim 8235  df-sum 8240  df-top 8861  df-bases 8863  df-topgen 8864  df-cld 8939  df-ntr 8940  df-cls 8941  df-cn 9030  df-cnp 9031  df-haus 9059  df-met 9070  df-bl 9072  df-opn 9073  df-lm 9200  df-grp 9316  df-gid 9317  df-ginv 9318  df-gdiv 9319  df-abl 9408  df-vc 9497  df-nv 9543  df-va 9546  df-ba 9547  df-sm 9548  df-0v 9549  df-vs 9550  df-nm 9551  df-ims 9552  df-ip 9689  df-ph 9813  df-hnorm 10469  df-hvsub 10472  df-nmfn 11408  df-lnfn 11411  df-bra 11413

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