Theorem nmlnop0ALT 9915

Description: A linear operator with a zero norm is identically zero.

Hypothesis

Ref	Expression
nmlnop0.1	⊢ T ∈ LinOp

Assertion

Ref	Expression
nmlnop0ALT	⊢ ((normop ‘T) = 0 ↔ T = 0hop )

Proof of Theorem nmlnop0ALT

Step	Hyp	Ref	Expression
1		recne0t 5739	. . . . . . . . . . . . . 14 ⊢ (((normh ‘x) ∈ ℂ ⋀ (normh ‘x) ≠ 0) → (1 / (normh ‘x)) ≠ 0)
2		normclt 8986	. . . . . . . . . . . . . . . 16 ⊢ (x ∈ ℋ → (normh ‘x) ∈ ℝ)
3	2	recnd 5327	. . . . . . . . . . . . . . 15 ⊢ (x ∈ ℋ → (normh ‘x) ∈ ℂ)
4	3	adantr 391	. . . . . . . . . . . . . 14 ⊢ ((x ∈ ℋ ⋀ (T ‘x) ≠ 0h) → (normh ‘x) ∈ ℂ)
5		norm-it 8991	. . . . . . . . . . . . . . . . 17 ⊢ (x ∈ ℋ → ((normh ‘x) = 0 ↔ x = 0h))
6		fveq2 3730	. . . . . . . . . . . . . . . . . 18 ⊢ (x = 0h → (T ‘x) = (T ‘0h))
7		nmlnop0.1	. . . . . . . . . . . . . . . . . . 19 ⊢ T ∈ LinOp
8	7	lnop0 9889	. . . . . . . . . . . . . . . . . 18 ⊢ (T ‘0h) = 0h
9	6, 8	syl6eq 1526	. . . . . . . . . . . . . . . . 17 ⊢ (x = 0h → (T ‘x) = 0h)
10	5, 9	syl6bi 214	. . . . . . . . . . . . . . . 16 ⊢ (x ∈ ℋ → ((normh ‘x) = 0 → (T ‘x) = 0h))
11	10	necon3d 1607	. . . . . . . . . . . . . . 15 ⊢ (x ∈ ℋ → ((T ‘x) ≠ 0h → (normh ‘x) ≠ 0))
12	11	imp 350	. . . . . . . . . . . . . 14 ⊢ ((x ∈ ℋ ⋀ (T ‘x) ≠ 0h) → (normh ‘x) ≠ 0)
13	1, 4, 12	sylanc 473	. . . . . . . . . . . . 13 ⊢ ((x ∈ ℋ ⋀ (T ‘x) ≠ 0h) → (1 / (normh ‘x)) ≠ 0)
14		pm3.27 323	. . . . . . . . . . . . 13 ⊢ ((x ∈ ℋ ⋀ (T ‘x) ≠ 0h) → (T ‘x) ≠ 0h)
15	13, 14	jca 288	. . . . . . . . . . . 12 ⊢ ((x ∈ ℋ ⋀ (T ‘x) ≠ 0h) → ((1 / (normh ‘x)) ≠ 0 ⋀ (T ‘x) ≠ 0h))
16		hvmul0ort 8889	. . . . . . . . . . . . . . 15 ⊢ (((1 / (normh ‘x)) ∈ ℂ ⋀ (T ‘x) ∈ ℋ ) → (((1 / (normh ‘x)) ·h (T ‘x)) = 0h ↔ ((1 / (normh ‘x)) = 0 ⋁ (T ‘x) = 0h)))
17		recclt 5727	. . . . . . . . . . . . . . . 16 ⊢ (((normh ‘x) ∈ ℂ ⋀ (normh ‘x) ≠ 0) → (1 / (normh ‘x)) ∈ ℂ)
18	17, 4, 12	sylanc 473	. . . . . . . . . . . . . . 15 ⊢ ((x ∈ ℋ ⋀ (T ‘x) ≠ 0h) → (1 / (normh ‘x)) ∈ ℂ)
19	7	lnopf 9888	. . . . . . . . . . . . . . . . 17 ⊢ T: ℋ –→ ℋ
20	19	ffvelrni 3821	. . . . . . . . . . . . . . . 16 ⊢ (x ∈ ℋ → (T ‘x) ∈ ℋ )
21	20	adantr 391	. . . . . . . . . . . . . . 15 ⊢ ((x ∈ ℋ ⋀ (T ‘x) ≠ 0h) → (T ‘x) ∈ ℋ )
22	16, 18, 21	sylanc 473	. . . . . . . . . . . . . 14 ⊢ ((x ∈ ℋ ⋀ (T ‘x) ≠ 0h) → (((1 / (normh ‘x)) ·h (T ‘x)) = 0h ↔ ((1 / (normh ‘x)) = 0 ⋁ (T ‘x) = 0h)))
23	22	necon3abid 1602	. . . . . . . . . . . . 13 ⊢ ((x ∈ ℋ ⋀ (T ‘x) ≠ 0h) → (((1 / (normh ‘x)) ·h (T ‘x)) ≠ 0h ↔ ¬ ((1 / (normh ‘x)) = 0 ⋁ (T ‘x) = 0h)))
24		neanior 1642	. . . . . . . . . . . . 13 ⊢ (((1 / (normh ‘x)) ≠ 0 ⋀ (T ‘x) ≠ 0h) ↔ ¬ ((1 / (normh ‘x)) = 0 ⋁ (T ‘x) = 0h))
25	23, 24	syl6bbr 540	. . . . . . . . . . . 12 ⊢ ((x ∈ ℋ ⋀ (T ‘x) ≠ 0h) → (((1 / (normh ‘x)) ·h (T ‘x)) ≠ 0h ↔ ((1 / (normh ‘x)) ≠ 0 ⋀ (T ‘x) ≠ 0h)))
26	15, 25	mpbird 196	. . . . . . . . . . 11 ⊢ ((x ∈ ℋ ⋀ (T ‘x) ≠ 0h) → ((1 / (normh ‘x)) ·h (T ‘x)) ≠ 0h)
27		hvmulclt 8878	. . . . . . . . . . . . 13 ⊢ (((1 / (normh ‘x)) ∈ ℂ ⋀ (T ‘x) ∈ ℋ ) → ((1 / (normh ‘x)) ·h (T ‘x)) ∈ ℋ )
28	27, 18, 21	sylanc 473	. . . . . . . . . . . 12 ⊢ ((x ∈ ℋ ⋀ (T ‘x) ≠ 0h) → ((1 / (normh ‘x)) ·h (T ‘x)) ∈ ℋ )
29		normgt0t 8989	. . . . . . . . . . . 12 ⊢ (((1 / (normh ‘x)) ·h (T ‘x)) ∈ ℋ → (((1 / (normh ‘x)) ·h (T ‘x)) ≠ 0h ↔ 0 < (normh ‘((1 / (normh ‘x)) ·h (T ‘x)))))
30	28, 29	syl 10	. . . . . . . . . . 11 ⊢ ((x ∈ ℋ ⋀ (T ‘x) ≠ 0h) → (((1 / (normh ‘x)) ·h (T ‘x)) ≠ 0h ↔ 0 < (normh ‘((1 / (normh ‘x)) ·h (T ‘x)))))
31	26, 30	mpbid 195	. . . . . . . . . 10 ⊢ ((x ∈ ℋ ⋀ (T ‘x) ≠ 0h) → 0 < (normh ‘((1 / (normh ‘x)) ·h (T ‘x))))
32	31	ex 373	. . . . . . . . 9 ⊢ (x ∈ ℋ → ((T ‘x) ≠ 0h → 0 < (normh ‘((1 / (normh ‘x)) ·h (T ‘x)))))
33	32	adantl 390	. . . . . . . 8 ⊢ (((normop ‘T) = 0 ⋀ x ∈ ℋ ) → ((T ‘x) ≠ 0h → 0 < (normh ‘((1 / (normh ‘x)) ·h (T ‘x)))))
34		fveq2 3730	. . . . . . . . . . . . . . . . . 18 ⊢ (z = ((1 / (normh ‘x)) ·h x) → (normh ‘z) = (normh ‘((1 / (normh ‘x)) ·h x)))
35	34	breq1d 2634	. . . . . . . . . . . . . . . . 17 ⊢ (z = ((1 / (normh ‘x)) ·h x) → ((normh ‘z) ≤ 1 ↔ (normh ‘((1 / (normh ‘x)) ·h x)) ≤ 1))
36		fveq2 3730	. . . . . . . . . . . . . . . . . . 19 ⊢ (z = ((1 / (normh ‘x)) ·h x) → (T ‘z) = (T ‘((1 / (normh ‘x)) ·h x)))
37	36	fveq2d 3734	. . . . . . . . . . . . . . . . . 18 ⊢ (z = ((1 / (normh ‘x)) ·h x) → (normh ‘(T ‘z)) = (normh ‘(T ‘((1 / (normh ‘x)) ·h x))))
38	37	eqeq2d 1489	. . . . . . . . . . . . . . . . 17 ⊢ (z = ((1 / (normh ‘x)) ·h x) → ((normh ‘((1 / (normh ‘x)) ·h (T ‘x))) = (normh ‘(T ‘z)) ↔ (normh ‘((1 / (normh ‘x)) ·h (T ‘x))) = (normh ‘(T ‘((1 / (normh ‘x)) ·h x)))))
39	35, 38	anbi12d 630	. . . . . . . . . . . . . . . 16 ⊢ (z = ((1 / (normh ‘x)) ·h x) → (((normh ‘z) ≤ 1 ⋀ (normh ‘((1 / (normh ‘x)) ·h (T ‘x))) = (normh ‘(T ‘z))) ↔ ((normh ‘((1 / (normh ‘x)) ·h x)) ≤ 1 ⋀ (normh ‘((1 / (normh ‘x)) ·h (T ‘x))) = (normh ‘(T ‘((1 / (normh ‘x)) ·h x))))))
40	39	rcla4ev 1880	. . . . . . . . . . . . . . 15 ⊢ ((((1 / (normh ‘x)) ·h x) ∈ ℋ ⋀ ((normh ‘((1 / (normh ‘x)) ·h x)) ≤ 1 ⋀ (normh ‘((1 / (normh ‘x)) ·h (T ‘x))) = (normh ‘(T ‘((1 / (normh ‘x)) ·h x))))) → ∃z ∈ ℋ ((normh ‘z) ≤ 1 ⋀ (normh ‘((1 / (normh ‘x)) ·h (T ‘x))) = (normh ‘(T ‘z))))
41		hvmulclt 8878	. . . . . . . . . . . . . . . 16 ⊢ (((1 / (normh ‘x)) ∈ ℂ ⋀ x ∈ ℋ ) → ((1 / (normh ‘x)) ·h x) ∈ ℋ )
42		pm3.26 319	. . . . . . . . . . . . . . . 16 ⊢ ((x ∈ ℋ ⋀ (T ‘x) ≠ 0h) → x ∈ ℋ )
43	41, 18, 42	sylanc 473	. . . . . . . . . . . . . . 15 ⊢ ((x ∈ ℋ ⋀ (T ‘x) ≠ 0h) → ((1 / (normh ‘x)) ·h x) ∈ ℋ )
44		eqlet 5583	. . . . . . . . . . . . . . . . 17 ⊢ (((normh ‘((1 / (normh ‘x)) ·h x)) ∈ ℝ ⋀ (normh ‘((1 / (normh ‘x)) ·h x)) = 1) → (normh ‘((1 / (normh ‘x)) ·h x)) ≤ 1)
45		norm1t 9116	. . . . . . . . . . . . . . . . . . 19 ⊢ ((x ∈ ℋ ⋀ x ≠ 0h) → (normh ‘((1 / (normh ‘x)) ·h x)) = 1)
46	9	necon3i 1608	. . . . . . . . . . . . . . . . . . 19 ⊢ ((T ‘x) ≠ 0h → x ≠ 0h)
47	45, 46	sylan2 453	. . . . . . . . . . . . . . . . . 18 ⊢ ((x ∈ ℋ ⋀ (T ‘x) ≠ 0h) → (normh ‘((1 / (normh ‘x)) ·h x)) = 1)
48		1re 5447	. . . . . . . . . . . . . . . . . 18 ⊢ 1 ∈ ℝ
49	47, 48	syl6eqel 1559	. . . . . . . . . . . . . . . . 17 ⊢ ((x ∈ ℋ ⋀ (T ‘x) ≠ 0h) → (normh ‘((1 / (normh ‘x)) ·h x)) ∈ ℝ)
50	44, 49, 47	sylanc 473	. . . . . . . . . . . . . . . 16 ⊢ ((x ∈ ℋ ⋀ (T ‘x) ≠ 0h) → (normh ‘((1 / (normh ‘x)) ·h x)) ≤ 1)
51	7	lnopmul 9891	. . . . . . . . . . . . . . . . . . 19 ⊢ (((1 / (normh ‘x)) ∈ ℂ ⋀ x ∈ ℋ ) → (T ‘((1 / (normh ‘x)) ·h x)) = ((1 / (normh ‘x)) ·h (T ‘x)))
52	51, 18, 42	sylanc 473	. . . . . . . . . . . . . . . . . 18 ⊢ ((x ∈ ℋ ⋀ (T ‘x) ≠ 0h) → (T ‘((1 / (normh ‘x)) ·h x)) = ((1 / (normh ‘x)) ·h (T ‘x)))
53	52	eqcomd 1483	. . . . . . . . . . . . . . . . 17 ⊢ ((x ∈ ℋ ⋀ (T ‘x) ≠ 0h) → ((1 / (normh ‘x)) ·h (T ‘x)) = (T ‘((1 / (normh ‘x)) ·h x)))
54	53	fveq2d 3734	. . . . . . . . . . . . . . . 16 ⊢ ((x ∈ ℋ ⋀ (T ‘x) ≠ 0h) → (normh ‘((1 / (normh ‘x)) ·h (T ‘x))) = (normh ‘(T ‘((1 / (normh ‘x)) ·h x))))
55	50, 54	jca 288	. . . . . . . . . . . . . . 15 ⊢ ((x ∈ ℋ ⋀ (T ‘x) ≠ 0h) → ((normh ‘((1 / (normh ‘x)) ·h x)) ≤ 1 ⋀ (normh ‘((1 / (normh ‘x)) ·h (T ‘x))) = (normh ‘(T ‘((1 / (normh ‘x)) ·h x)))))
56	40, 43, 55	sylanc 473	. . . . . . . . . . . . . 14 ⊢ ((x ∈ ℋ ⋀ (T ‘x) ≠ 0h) → ∃z ∈ ℋ ((normh ‘z) ≤ 1 ⋀ (normh ‘((1 / (normh ‘x)) ·h (T ‘x))) = (normh ‘(T ‘z))))
57		fvex 3738	. . . . . . . . . . . . . . 15 ⊢ (normh ‘((1 / (normh ‘x)) ·h (T ‘x))) ∈ V
58		eqeq1 1484	. . . . . . . . . . . . . . . . 17 ⊢ (y = (normh ‘((1 / (normh ‘x)) ·h (T ‘x))) → (y = (normh ‘(T ‘z)) ↔ (normh ‘((1 / (normh ‘x)) ·h (T ‘x))) = (normh ‘(T ‘z))))
59	58	anbi2d 618	. . . . . . . . . . . . . . . 16 ⊢ (y = (normh ‘((1 / (normh ‘x)) ·h (T ‘x))) → (((normh ‘z) ≤ 1 ⋀ y = (normh ‘(T ‘z))) ↔ ((normh ‘z) ≤ 1 ⋀ (normh ‘((1 / (normh ‘x)) ·h (T ‘x))) = (normh ‘(T ‘z)))))
60	59	rexbidv 1667	. . . . . . . . . . . . . . 15 ⊢ (y = (normh ‘((1 / (normh ‘x)) ·h (T ‘x))) → (∃z ∈ ℋ ((normh ‘z) ≤ 1 ⋀ y = (normh ‘(T ‘z))) ↔ ∃z ∈ ℋ ((normh ‘z) ≤ 1 ⋀ (normh ‘((1 / (normh ‘x)) ·h (T ‘x))) = (normh ‘(T ‘z)))))
61	57, 60	elab 1900	. . . . . . . . . . . . . 14 ⊢ ((normh ‘((1 / (normh ‘x)) ·h (T ‘x))) ∈ {y∣∃z ∈ ℋ ((normh ‘z) ≤ 1 ⋀ y = (normh ‘(T ‘z)))} ↔ ∃z ∈ ℋ ((normh ‘z) ≤ 1 ⋀ (normh ‘((1 / (normh ‘x)) ·h (T ‘x))) = (normh ‘(T ‘z))))
62	56, 61	sylibr 200	. . . . . . . . . . . . 13 ⊢ ((x ∈ ℋ ⋀ (T ‘x) ≠ 0h) → (normh ‘((1 / (normh ‘x)) ·h (T ‘x))) ∈ {y∣∃z ∈ ℋ ((normh ‘z) ≤ 1 ⋀ y = (normh ‘(T ‘z)))})
63		nmopsetretHIL 9786	. . . . . . . . . . . . . . . 16 ⊢ (T: ℋ –→ ℋ → {y∣∃z ∈ ℋ ((normh ‘z) ≤ 1 ⋀ y = (normh ‘(T ‘z)))} ⊆ ℝ)
64	19, 63	ax-mp 7	. . . . . . . . . . . . . . 15 ⊢ {y∣∃z ∈ ℋ ((normh ‘z) ≤ 1 ⋀ y = (normh ‘(T ‘z)))} ⊆ ℝ
65		ressxr 5510	. . . . . . . . . . . . . . 15 ⊢ ℝ ⊆ ℝ*
66	64, 65	sstri 2076	. . . . . . . . . . . . . 14 ⊢ {y∣∃z ∈ ℋ ((normh ‘z) ≤ 1 ⋀ y = (normh ‘(T ‘z)))} ⊆ ℝ*
67		supxrub 6100	. . . . . . . . . . . . . 14 ⊢ (({y∣∃z ∈ ℋ ((normh ‘z) ≤ 1 ⋀ y = (normh ‘(T ‘z)))} ⊆ ℝ* ⋀ (normh ‘((1 / (normh ‘x)) ·h (T ‘x))) ∈ {y∣∃z ∈ ℋ ((normh ‘z) ≤ 1 ⋀ y = (normh ‘(T ‘z)))}) → (normh ‘((1 / (normh ‘x)) ·h (T ‘x))) ≤ sup({y∣∃z ∈ ℋ ((normh ‘z) ≤ 1 ⋀ y = (normh ‘(T ‘z)))}, ℝ*, < ))
68	66, 67	mpan 697	. . . . . . . . . . . . 13 ⊢ ((normh ‘((1 / (normh ‘x)) ·h (T ‘x))) ∈ {y∣∃z ∈ ℋ ((normh ‘z) ≤ 1 ⋀ y = (normh ‘(T ‘z)))} → (normh ‘((1 / (normh ‘x)) ·h (T ‘x))) ≤ sup({y∣∃z ∈ ℋ ((normh ‘z) ≤ 1 ⋀ y = (normh ‘(T ‘z)))}, ℝ*, < ))
69	62, 68	syl 10	. . . . . . . . . . . 12 ⊢ ((x ∈ ℋ ⋀ (T ‘x) ≠ 0h) → (normh ‘((1 / (normh ‘x)) ·h (T ‘x))) ≤ sup({y∣∃z ∈ ℋ ((normh ‘z) ≤ 1 ⋀ y = (normh ‘(T ‘z)))}, ℝ*, < ))
70	69	adantll 394	. . . . . . . . . . 11 ⊢ ((((normop ‘T) = 0 ⋀ x ∈ ℋ ) ⋀ (T ‘x) ≠ 0h) → (normh ‘((1 / (normh ‘x)) ·h (T ‘x))) ≤ sup({y∣∃z ∈ ℋ ((normh ‘z) ≤ 1 ⋀ y = (normh ‘(T ‘z)))}, ℝ*, < ))
71		nmopvalt 9777	. . . . . . . . . . . . . . 15 ⊢ (T: ℋ –→ ℋ → (normop ‘T) = sup({y∣∃z ∈ ℋ ((normh ‘z) ≤ 1 ⋀ y = (normh ‘(T ‘z)))}, ℝ*, < ))
72	19, 71	ax-mp 7	. . . . . . . . . . . . . 14 ⊢ (normop ‘T) = sup({y∣∃z ∈ ℋ ((normh ‘z) ≤ 1 ⋀ y = (normh ‘(T ‘z)))}, ℝ*, < )
73	72	eqeq1i 1485	. . . . . . . . . . . . 13 ⊢ ((normop ‘T) = 0 ↔ sup({y∣∃z ∈ ℋ ((normh ‘z) ≤ 1 ⋀ y = (normh ‘(T ‘z)))}, ℝ*, < ) = 0)
74	73	biimp 151	. . . . . . . . . . . 12 ⊢ ((normop ‘T) = 0 → sup({y∣∃z ∈ ℋ ((normh ‘z) ≤ 1 ⋀ y = (normh ‘(T ‘z)))}, ℝ*, < ) = 0)
75	74	ad2antrr 406	. . . . . . . . . . 11 ⊢ ((((normop ‘T) = 0 ⋀ x ∈ ℋ ) ⋀ (T ‘x) ≠ 0h) → sup({y∣∃z ∈ ℋ ((normh ‘z) ≤ 1 ⋀ y = (normh ‘(T ‘z)))}, ℝ*, < ) = 0)
76	70, 75	breqtrd 2644	. . . . . . . . . 10 ⊢ ((((normop ‘T) = 0 ⋀ x ∈ ℋ ) ⋀ (T ‘x) ≠ 0h) → (normh ‘((1 / (normh ‘x)) ·h (T ‘x))) ≤ 0)
77		normclt 8986	. . . . . . . . . . . . 13 ⊢ (((1 / (normh ‘x)) ·h (T ‘x)) ∈ ℋ → (normh ‘((1 / (normh ‘x)) ·h (T ‘x))) ∈ ℝ)
78	28, 77	syl 10	. . . . . . . . . . . 12 ⊢ ((x ∈ ℋ ⋀ (T ‘x) ≠ 0h) → (normh ‘((1 / (normh ‘x)) ·h (T ‘x))) ∈ ℝ)
79		0re 5452	. . . . . . . . . . . . 13 ⊢ 0 ∈ ℝ
80		lenltt 5522	. . . . . . . . . . . . 13 ⊢ (((normh ‘((1 / (normh ‘x)) ·h (T ‘x))) ∈ ℝ ⋀ 0 ∈ ℝ) → ((normh ‘((1 / (normh ‘x)) ·h (T ‘x))) ≤ 0 ↔ ¬ 0 < (normh ‘((1 / (normh ‘x)) ·h (T ‘x)))))
81	79, 80	mpan2 698	. . . . . . . . . . . 12 ⊢ ((normh ‘((1 / (normh ‘x)) ·h (T ‘x))) ∈ ℝ → ((normh ‘((1 / (normh ‘x)) ·h (T ‘x))) ≤ 0 ↔ ¬ 0 < (normh ‘((1 / (normh ‘x)) ·h (T ‘x)))))
82	78, 81	syl 10	. . . . . . . . . . 11 ⊢ ((x ∈ ℋ ⋀ (T ‘x) ≠ 0h) → ((normh ‘((1 / (normh ‘x)) ·h (T ‘x))) ≤ 0 ↔ ¬ 0 < (normh ‘((1 / (normh ‘x)) ·h (T ‘x)))))
83	82	adantll 394	. . . . . . . . . 10 ⊢ ((((normop ‘T) = 0 ⋀ x ∈ ℋ ) ⋀ (T ‘x) ≠ 0h) → ((normh ‘((1 / (normh ‘x)) ·h (T ‘x))) ≤ 0 ↔ ¬ 0 < (normh ‘((1 / (normh ‘x)) ·h (T ‘x)))))
84	76, 83	mpbid 195	. . . . . . . . 9 ⊢ ((((normop ‘T) = 0 ⋀ x ∈ ℋ ) ⋀ (T ‘x) ≠ 0h) → ¬ 0 < (normh ‘((1 / (normh ‘x)) ·h (T ‘x))))
85	84	ex 373	. . . . . . . 8 ⊢ (((normop ‘T) = 0 ⋀ x ∈ ℋ ) → ((T ‘x) ≠ 0h → ¬ 0 < (normh ‘((1 / (normh ‘x)) ·h (T ‘x)))))
86	33, 85	pm2.65d 136	. . . . . . 7 ⊢ (((normop ‘T) = 0 ⋀ x ∈ ℋ ) → ¬ (T ‘x) ≠ 0h)
87		nne 1592	. . . . . . 7 ⊢ (¬ (T ‘x) ≠ 0h ↔ (T ‘x) = 0h)
88	86, 87	sylib 198	. . . . . 6 ⊢ (((normop ‘T) = 0 ⋀ x ∈ ℋ ) → (T ‘x) = 0h)
89		ho0valt 9671	. . . . . . 7 ⊢ (x ∈ ℋ → ( 0hop ‘x) = 0h)
90	89	adantl 390	. . . . . 6 ⊢ (((normop ‘T) = 0 ⋀ x ∈ ℋ ) → ( 0hop ‘x) = 0h)
91	88, 90	eqtr4d 1513	. . . . 5 ⊢ (((normop ‘T) = 0 ⋀ x ∈ ℋ ) → (T ‘x) = ( 0hop ‘x))
92	91	r19.21aiva 1717	. . . 4 ⊢ ((normop ‘T) = 0 → ∀x ∈ ℋ (T ‘x) = ( 0hop ‘x))
93		eqid 1478	. . . 4 ⊢ ℋ = ℋ
94	92, 93	jctil 292	. . 3 ⊢ ((normop ‘T) = 0 → ( ℋ = ℋ ⋀ ∀x ∈ ℋ (T ‘x) = ( 0hop ‘x)))
95		ffn 3633	. . . . 5 ⊢ (T: ℋ –→ ℋ → T Fn ℋ )
96	19, 95	ax-mp 7	. . . 4 ⊢ T Fn ℋ
97		ho0f 9672	. . . . 5 ⊢ 0hop : ℋ –→ ℋ
98		ffn 3633	. . . . 5 ⊢ ( 0hop : ℋ –→ ℋ → 0hop Fn ℋ )
99	97, 98	ax-mp 7	. . . 4 ⊢ 0hop Fn ℋ
100		eqfnfv 3803	. . . 4 ⊢ ((T Fn ℋ ⋀ 0hop Fn ℋ ) → (T = 0hop ↔ ( ℋ = ℋ ⋀ ∀x ∈ ℋ (T ‘x) = ( 0hop ‘x))))
101	96, 99, 100	mp2an 699	. . 3 ⊢ (T = 0hop ↔ ( ℋ = ℋ ⋀ ∀x ∈ ℋ (T ‘x) = ( 0hop ‘x)))
102	94, 101	sylibr 200	. 2 ⊢ ((normop ‘T) = 0 → T = 0hop )
103		fveq2 3730	. . 3 ⊢ (T = 0hop → (normop ‘T) = (normop ‘ 0hop ))
104		nmop0 9905	. . 3 ⊢ (normop ‘ 0hop ) = 0
105	103, 104	syl6eq 1526	. 2 ⊢ (T = 0hop → (normop ‘T) = 0)
106	102, 105	impbi 157	1 ⊢ ((normop ‘T) = 0 ↔ T = 0hop )

Syntax hints:  ¬ wn 2   → wi 3   ↔ wb 146   ⋁ wo 222   ⋀ wa 223   = wceq 958   ∈ wcel 960  {cab 1466   ≠ wne 1588  ∀wral 1648  ∃wrex 1649   ⊆ wss 2050   class class class wbr 2624   Fn wfn 3183  –→wf 3184   ‘cfv 3188  (class class class)co 3969  supcsup 4582  ℂcc 5244  ℝcr 5245  0cc0 5246  1c1 5247   / cdiv 5306   ≤ cle 5307  ℝ^*cxr 5497   < clt 5498   ℋ chil 8783   ·_h csm 8785  0_hc0v 8786  norm_hcno 8789   0_hop ch0o 8807  norm_opcnop 8809  LinOpclo 8811

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-rep 2698  ax-sep 2708  ax-nul 2715  ax-pow 2748  ax-pr 2785  ax-un 2872  ax-reg 4602  ax-inf2 4634  ax-ac 4754  ax-hilex 8864  ax-hfvadd 8865  ax-hvcom 8866  ax-hvass 8867  ax-hv0cl 8868  ax-hvaddid 8869  ax-hfvmul 8870  ax-hvmulid 8871  ax-hvmulass 8872  ax-hvdistr1 8873  ax-hvdistr2 8874  ax-hvmul0 8875  ax-hfi 8941  ax-his1 8944  ax-his2 8945  ax-his3 8946  ax-his4 8947  ax-hcompl 9066

This theorem depends on definitions:  df-bi 147  df-or 224  df-an 225  df-3or 778  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-nel 1591  df-ral 1652  df-rex 1653  df-reu 1654  df-rab 1655  df-v 1815  df-sbc 1945  df-csb 2005  df-dif 2052  df-un 2053  df-in 2054  df-ss 2056  df-pss 2058  df-nul 2284  df-if 2366  df-pw 2406  df-sn 2416  df-pr 2417  df-tp 2419  df-op 2420  df-uni 2508  df-int 2538  df-iun 2572  df-iin 2573  df-br 2625  df-opab 2672  df-tr 2686  df-eprel 2838  df-id 2841  df-po 2846  df-so 2856  df-fr 2923  df-we 2940  df-ord 2957  df-on 2958  df-lim 2959  df-suc 2960  df-om 3138  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-fn 3199  df-f 3200  df-f1 3201  df-fo 3202  df-f1o 3203  df-fv 3204  df-rdg 3938  df-opr 3971  df-oprab 3972  df-1st 4085  df-2nd 4086  df-1o 4139  df-oadd 4141  df-omul 4142  df-er 4267  df-ec 4269  df-qs 4272  df-map 4330  df-en 4374  df-dom 4375  df-sdom 4376  df-sup 4583  df-r1 4653  df-rank 4654  df-ni 5012  df-pli 5013  df-mi 5014  df-lti 5015  df-plpq 5047  df-mpq 5048  df-enq 5049  df-nq 5050  df-plq 5051  df-mq 5052  df-rq 5053  df-ltq 5054  df-1q 5055  df-np 5098  df-1p 5099  df-plp 5100  df-mp 5101  df-ltp 5102  df-plpr 5176  df-mpr 5177  df-enr 5178  df-nr 5179  df-plr 5180  df-mr 5181  df-ltr 5182  df-0r 5183  df-1r 5184  df-m1r 5185  df-c 5252  df-0 5253  df-1 5254  df-i 5255  df-r 5256  df-plus 5257  df-mul 5258  df-lt 5259  df-sub 5368  df-neg 5370  df-pnf 5499  df-mnf 5500  df-xr 5501  df-ltxr 5502  df-le 5503  df-div 5715  df-n 5927  df-2 5972  df-3 5973  df-4 5974  df-n0 6102  df-z 6138  df-fl 6226  df-q 6257  df-seq1 6309  df-shft 6342  df-ioo 6362  df-uz 6419  df-fz 6469  df-seqz 6534  df-exp 6570  df-sqr 6671  df-re 6752  df-im 6753  df-cj 6754  df-abs 6755  df-clim 6975  df-sum 6980  df-top 7594  df-bases 7596  df-topgen 7597  df-cld 7660  df-ntr 7661  df-cls 7662  df-cn 7751  df-cnp 7752  df-haus 7779  df-met 7790  df-bl 7792  df-opn 7793  df-lm 7919  df-grp 8034  df-gid 8035  df-ginv 8036  df-gdiv 8037  df-abl 8096  df-vc 8161  df-nv 8207  df-va 8210  df-ba 8211  df-sm 8212  df-0v 8213  df-vs 8214  df-nm 8215  df-ims 8216  df-ip 8346  df-ph 8468  df-hnorm 8832  df-hvsub 8835  df-hlim 8836  df-hcau 8837  df-sh 9071  df-ch 9087  df-oc 9119  df-ch0 9120  df-pj 9232  df-h0op 9669  df-nmop 9760  df-lnop 9762

Copyright terms: Public domain