Theorem nmlnop0iALT 32476

Description: A linear operator with a zero norm is identically zero. (Contributed by NM, 8-Feb-2006.) (New usage is discouraged.) (Proof modification is discouraged.)

Hypothesis

Ref	Expression
nmlnop0.1	⊢ 𝑇 ∈ LinOp

Assertion

Ref	Expression
nmlnop0iALT	⊢ ((normop‘𝑇) = 0 ↔ 𝑇 = 0hop )

Proof of Theorem nmlnop0iALT

Dummy variables 𝑥 𝑦 𝑧 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		normcl 31606	. . . . . . . . . . . . . 14 ⊢ (𝑥 ∈ ℋ → (normℎ‘𝑥) ∈ ℝ)
2	1	recnd 11261	. . . . . . . . . . . . 13 ⊢ (𝑥 ∈ ℋ → (normℎ‘𝑥) ∈ ℂ)
3	2	adantr 486	. . . . . . . . . . . 12 ⊢ ((𝑥 ∈ ℋ ∧ (𝑇‘𝑥) ≠ 0ℎ) → (normℎ‘𝑥) ∈ ℂ)
4		norm-i 31610	. . . . . . . . . . . . . . 15 ⊢ (𝑥 ∈ ℋ → ((normℎ‘𝑥) = 0 ↔ 𝑥 = 0ℎ))
5		fveq2 6878	. . . . . . . . . . . . . . . 16 ⊢ (𝑥 = 0ℎ → (𝑇‘𝑥) = (𝑇‘0ℎ))
6		nmlnop0.1	. . . . . . . . . . . . . . . . 17 ⊢ 𝑇 ∈ LinOp
7	6	lnop0i 32451	. . . . . . . . . . . . . . . 16 ⊢ (𝑇‘0ℎ) = 0ℎ
8	5, 7	eqtrdi 2811	. . . . . . . . . . . . . . 15 ⊢ (𝑥 = 0ℎ → (𝑇‘𝑥) = 0ℎ)
9	4, 8	biimtrdi 256	. . . . . . . . . . . . . 14 ⊢ (𝑥 ∈ ℋ → ((normℎ‘𝑥) = 0 → (𝑇‘𝑥) = 0ℎ))
10	9	necon3d 2976	. . . . . . . . . . . . 13 ⊢ (𝑥 ∈ ℋ → ((𝑇‘𝑥) ≠ 0ℎ → (normℎ‘𝑥) ≠ 0))
11	10	imp 412	. . . . . . . . . . . 12 ⊢ ((𝑥 ∈ ℋ ∧ (𝑇‘𝑥) ≠ 0ℎ) → (normℎ‘𝑥) ≠ 0)
12	3, 11	recne0d 12009	. . . . . . . . . . 11 ⊢ ((𝑥 ∈ ℋ ∧ (𝑇‘𝑥) ≠ 0ℎ) → (1 / (normℎ‘𝑥)) ≠ 0)
13		simpr 490	. . . . . . . . . . 11 ⊢ ((𝑥 ∈ ℋ ∧ (𝑇‘𝑥) ≠ 0ℎ) → (𝑇‘𝑥) ≠ 0ℎ)
14	3, 11	reccld 12008	. . . . . . . . . . . . . 14 ⊢ ((𝑥 ∈ ℋ ∧ (𝑇‘𝑥) ≠ 0ℎ) → (1 / (normℎ‘𝑥)) ∈ ℂ)
15	6	lnopfi 32450	. . . . . . . . . . . . . . . 16 ⊢ 𝑇: ℋ⟶ ℋ
16	15	ffvelcdmi 7076	. . . . . . . . . . . . . . 15 ⊢ (𝑥 ∈ ℋ → (𝑇‘𝑥) ∈ ℋ)
17	16	adantr 486	. . . . . . . . . . . . . 14 ⊢ ((𝑥 ∈ ℋ ∧ (𝑇‘𝑥) ≠ 0ℎ) → (𝑇‘𝑥) ∈ ℋ)
18		hvmul0or 31506	. . . . . . . . . . . . . 14 ⊢ (((1 / (normℎ‘𝑥)) ∈ ℂ ∧ (𝑇‘𝑥) ∈ ℋ) → (((1 / (normℎ‘𝑥)) ·ℎ (𝑇‘𝑥)) = 0ℎ ↔ ((1 / (normℎ‘𝑥)) = 0 ∨ (𝑇‘𝑥) = 0ℎ)))
19	14, 17, 18	syl2anc 596	. . . . . . . . . . . . 13 ⊢ ((𝑥 ∈ ℋ ∧ (𝑇‘𝑥) ≠ 0ℎ) → (((1 / (normℎ‘𝑥)) ·ℎ (𝑇‘𝑥)) = 0ℎ ↔ ((1 / (normℎ‘𝑥)) = 0 ∨ (𝑇‘𝑥) = 0ℎ)))
20	19	necon3abid 2991	. . . . . . . . . . . 12 ⊢ ((𝑥 ∈ ℋ ∧ (𝑇‘𝑥) ≠ 0ℎ) → (((1 / (normℎ‘𝑥)) ·ℎ (𝑇‘𝑥)) ≠ 0ℎ ↔ ¬ ((1 / (normℎ‘𝑥)) = 0 ∨ (𝑇‘𝑥) = 0ℎ)))
21		neanior 3048	. . . . . . . . . . . 12 ⊢ (((1 / (normℎ‘𝑥)) ≠ 0 ∧ (𝑇‘𝑥) ≠ 0ℎ) ↔ ¬ ((1 / (normℎ‘𝑥)) = 0 ∨ (𝑇‘𝑥) = 0ℎ))
22	20, 21	bitr4di 292	. . . . . . . . . . 11 ⊢ ((𝑥 ∈ ℋ ∧ (𝑇‘𝑥) ≠ 0ℎ) → (((1 / (normℎ‘𝑥)) ·ℎ (𝑇‘𝑥)) ≠ 0ℎ ↔ ((1 / (normℎ‘𝑥)) ≠ 0 ∧ (𝑇‘𝑥) ≠ 0ℎ)))
23	12, 13, 22	mpbir2and 726	. . . . . . . . . 10 ⊢ ((𝑥 ∈ ℋ ∧ (𝑇‘𝑥) ≠ 0ℎ) → ((1 / (normℎ‘𝑥)) ·ℎ (𝑇‘𝑥)) ≠ 0ℎ)
24		hvmulcl 31494	. . . . . . . . . . . 12 ⊢ (((1 / (normℎ‘𝑥)) ∈ ℂ ∧ (𝑇‘𝑥) ∈ ℋ) → ((1 / (normℎ‘𝑥)) ·ℎ (𝑇‘𝑥)) ∈ ℋ)
25	14, 17, 24	syl2anc 596	. . . . . . . . . . 11 ⊢ ((𝑥 ∈ ℋ ∧ (𝑇‘𝑥) ≠ 0ℎ) → ((1 / (normℎ‘𝑥)) ·ℎ (𝑇‘𝑥)) ∈ ℋ)
26		normgt0 31608	. . . . . . . . . . 11 ⊢ (((1 / (normℎ‘𝑥)) ·ℎ (𝑇‘𝑥)) ∈ ℋ → (((1 / (normℎ‘𝑥)) ·ℎ (𝑇‘𝑥)) ≠ 0ℎ ↔ 0 < (normℎ‘((1 / (normℎ‘𝑥)) ·ℎ (𝑇‘𝑥)))))
27	25, 26	syl 18	. . . . . . . . . 10 ⊢ ((𝑥 ∈ ℋ ∧ (𝑇‘𝑥) ≠ 0ℎ) → (((1 / (normℎ‘𝑥)) ·ℎ (𝑇‘𝑥)) ≠ 0ℎ ↔ 0 < (normℎ‘((1 / (normℎ‘𝑥)) ·ℎ (𝑇‘𝑥)))))
28	23, 27	mpbid 235	. . . . . . . . 9 ⊢ ((𝑥 ∈ ℋ ∧ (𝑇‘𝑥) ≠ 0ℎ) → 0 < (normℎ‘((1 / (normℎ‘𝑥)) ·ℎ (𝑇‘𝑥))))
29	28	ex 418	. . . . . . . 8 ⊢ (𝑥 ∈ ℋ → ((𝑇‘𝑥) ≠ 0ℎ → 0 < (normℎ‘((1 / (normℎ‘𝑥)) ·ℎ (𝑇‘𝑥)))))
30	29	adantl 487	. . . . . . 7 ⊢ (((normop‘𝑇) = 0 ∧ 𝑥 ∈ ℋ) → ((𝑇‘𝑥) ≠ 0ℎ → 0 < (normℎ‘((1 / (normℎ‘𝑥)) ·ℎ (𝑇‘𝑥)))))
31		nmopsetretHIL 32345	. . . . . . . . . . . . . 14 ⊢ (𝑇: ℋ⟶ ℋ → {𝑦 ∣ ∃𝑧 ∈ ℋ ((normℎ‘𝑧) ≤ 1 ∧ 𝑦 = (normℎ‘(𝑇‘𝑧)))} ⊆ ℝ)
32	15, 31	ax-mp 5	. . . . . . . . . . . . 13 ⊢ {𝑦 ∣ ∃𝑧 ∈ ℋ ((normℎ‘𝑧) ≤ 1 ∧ 𝑦 = (normℎ‘(𝑇‘𝑧)))} ⊆ ℝ
33		ressxr 11277	. . . . . . . . . . . . 13 ⊢ ℝ ⊆ ℝ*
34	32, 33	sstri 3940	. . . . . . . . . . . 12 ⊢ {𝑦 ∣ ∃𝑧 ∈ ℋ ((normℎ‘𝑧) ≤ 1 ∧ 𝑦 = (normℎ‘(𝑇‘𝑧)))} ⊆ ℝ*
35		simpl 488	. . . . . . . . . . . . . . 15 ⊢ ((𝑥 ∈ ℋ ∧ (𝑇‘𝑥) ≠ 0ℎ) → 𝑥 ∈ ℋ)
36		hvmulcl 31494	. . . . . . . . . . . . . . 15 ⊢ (((1 / (normℎ‘𝑥)) ∈ ℂ ∧ 𝑥 ∈ ℋ) → ((1 / (normℎ‘𝑥)) ·ℎ 𝑥) ∈ ℋ)
37	14, 35, 36	syl2anc 596	. . . . . . . . . . . . . 14 ⊢ ((𝑥 ∈ ℋ ∧ (𝑇‘𝑥) ≠ 0ℎ) → ((1 / (normℎ‘𝑥)) ·ℎ 𝑥) ∈ ℋ)
38	8	necon3i 2987	. . . . . . . . . . . . . . . . 17 ⊢ ((𝑇‘𝑥) ≠ 0ℎ → 𝑥 ≠ 0ℎ)
39		norm1 31730	. . . . . . . . . . . . . . . . 17 ⊢ ((𝑥 ∈ ℋ ∧ 𝑥 ≠ 0ℎ) → (normℎ‘((1 / (normℎ‘𝑥)) ·ℎ 𝑥)) = 1)
40	38, 39	sylan2 605	. . . . . . . . . . . . . . . 16 ⊢ ((𝑥 ∈ ℋ ∧ (𝑇‘𝑥) ≠ 0ℎ) → (normℎ‘((1 / (normℎ‘𝑥)) ·ℎ 𝑥)) = 1)
41		1re 11232	. . . . . . . . . . . . . . . 16 ⊢ 1 ∈ ℝ
42	40, 41	eqeltrdi 2868	. . . . . . . . . . . . . . 15 ⊢ ((𝑥 ∈ ℋ ∧ (𝑇‘𝑥) ≠ 0ℎ) → (normℎ‘((1 / (normℎ‘𝑥)) ·ℎ 𝑥)) ∈ ℝ)
43		eqle 11336	. . . . . . . . . . . . . . 15 ⊢ (((normℎ‘((1 / (normℎ‘𝑥)) ·ℎ 𝑥)) ∈ ℝ ∧ (normℎ‘((1 / (normℎ‘𝑥)) ·ℎ 𝑥)) = 1) → (normℎ‘((1 / (normℎ‘𝑥)) ·ℎ 𝑥)) ≤ 1)
44	42, 40, 43	syl2anc 596	. . . . . . . . . . . . . 14 ⊢ ((𝑥 ∈ ℋ ∧ (𝑇‘𝑥) ≠ 0ℎ) → (normℎ‘((1 / (normℎ‘𝑥)) ·ℎ 𝑥)) ≤ 1)
45	6	lnopmuli 32453	. . . . . . . . . . . . . . . . 17 ⊢ (((1 / (normℎ‘𝑥)) ∈ ℂ ∧ 𝑥 ∈ ℋ) → (𝑇‘((1 / (normℎ‘𝑥)) ·ℎ 𝑥)) = ((1 / (normℎ‘𝑥)) ·ℎ (𝑇‘𝑥)))
46	14, 35, 45	syl2anc 596	. . . . . . . . . . . . . . . 16 ⊢ ((𝑥 ∈ ℋ ∧ (𝑇‘𝑥) ≠ 0ℎ) → (𝑇‘((1 / (normℎ‘𝑥)) ·ℎ 𝑥)) = ((1 / (normℎ‘𝑥)) ·ℎ (𝑇‘𝑥)))
47	46	eqcomd 2766	. . . . . . . . . . . . . . 15 ⊢ ((𝑥 ∈ ℋ ∧ (𝑇‘𝑥) ≠ 0ℎ) → ((1 / (normℎ‘𝑥)) ·ℎ (𝑇‘𝑥)) = (𝑇‘((1 / (normℎ‘𝑥)) ·ℎ 𝑥)))
48	47	fveq2d 6882	. . . . . . . . . . . . . 14 ⊢ ((𝑥 ∈ ℋ ∧ (𝑇‘𝑥) ≠ 0ℎ) → (normℎ‘((1 / (normℎ‘𝑥)) ·ℎ (𝑇‘𝑥))) = (normℎ‘(𝑇‘((1 / (normℎ‘𝑥)) ·ℎ 𝑥))))
49		fveq2 6878	. . . . . . . . . . . . . . . . 17 ⊢ (𝑧 = ((1 / (normℎ‘𝑥)) ·ℎ 𝑥) → (normℎ‘𝑧) = (normℎ‘((1 / (normℎ‘𝑥)) ·ℎ 𝑥)))
50	49	breq1d 5113	. . . . . . . . . . . . . . . 16 ⊢ (𝑧 = ((1 / (normℎ‘𝑥)) ·ℎ 𝑥) → ((normℎ‘𝑧) ≤ 1 ↔ (normℎ‘((1 / (normℎ‘𝑥)) ·ℎ 𝑥)) ≤ 1))
51		fveq2 6878	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑧 = ((1 / (normℎ‘𝑥)) ·ℎ 𝑥) → (𝑇‘𝑧) = (𝑇‘((1 / (normℎ‘𝑥)) ·ℎ 𝑥)))
52	51	fveq2d 6882	. . . . . . . . . . . . . . . . 17 ⊢ (𝑧 = ((1 / (normℎ‘𝑥)) ·ℎ 𝑥) → (normℎ‘(𝑇‘𝑧)) = (normℎ‘(𝑇‘((1 / (normℎ‘𝑥)) ·ℎ 𝑥))))
53	52	eqeq2d 2771	. . . . . . . . . . . . . . . 16 ⊢ (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)))))
54	50, 53	anbi12d 644	. . . . . . . . . . . . . . 15 ⊢ (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))))))
55	54	rspcev 3576	. . . . . . . . . . . . . 14 ⊢ ((((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))))
56	37, 44, 48, 55	syl12anc 850	. . . . . . . . . . . . 13 ⊢ ((𝑥 ∈ ℋ ∧ (𝑇‘𝑥) ≠ 0ℎ) → ∃𝑧 ∈ ℋ ((normℎ‘𝑧) ≤ 1 ∧ (normℎ‘((1 / (normℎ‘𝑥)) ·ℎ (𝑇‘𝑥))) = (normℎ‘(𝑇‘𝑧))))
57		fvex 6891	. . . . . . . . . . . . . 14 ⊢ (normℎ‘((1 / (normℎ‘𝑥)) ·ℎ (𝑇‘𝑥))) ∈ V
58		eqeq1 2764	. . . . . . . . . . . . . . . 16 ⊢ (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 642	. . . . . . . . . . . . . . 15 ⊢ (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 3186	. . . . . . . . . . . . . 14 ⊢ (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 3633	. . . . . . . . . . . . 13 ⊢ ((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 237	. . . . . . . . . . . 12 ⊢ ((𝑥 ∈ ℋ ∧ (𝑇‘𝑥) ≠ 0ℎ) → (normℎ‘((1 / (normℎ‘𝑥)) ·ℎ (𝑇‘𝑥))) ∈ {𝑦 ∣ ∃𝑧 ∈ ℋ ((normℎ‘𝑧) ≤ 1 ∧ 𝑦 = (normℎ‘(𝑇‘𝑧)))})
63		supxrub 13376	. . . . . . . . . . . 12 ⊢ (({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*, < ))
64	34, 62, 63	sylancr 599	. . . . . . . . . . 11 ⊢ ((𝑥 ∈ ℋ ∧ (𝑇‘𝑥) ≠ 0ℎ) → (normℎ‘((1 / (normℎ‘𝑥)) ·ℎ (𝑇‘𝑥))) ≤ sup({𝑦 ∣ ∃𝑧 ∈ ℋ ((normℎ‘𝑧) ≤ 1 ∧ 𝑦 = (normℎ‘(𝑇‘𝑧)))}, ℝ*, < ))
65	64	adantll 727	. . . . . . . . . 10 ⊢ ((((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*, < ))
66		nmopval 32337	. . . . . . . . . . . . . 14 ⊢ (𝑇: ℋ⟶ ℋ → (normop‘𝑇) = sup({𝑦 ∣ ∃𝑧 ∈ ℋ ((normℎ‘𝑧) ≤ 1 ∧ 𝑦 = (normℎ‘(𝑇‘𝑧)))}, ℝ*, < ))
67	15, 66	ax-mp 5	. . . . . . . . . . . . 13 ⊢ (normop‘𝑇) = sup({𝑦 ∣ ∃𝑧 ∈ ℋ ((normℎ‘𝑧) ≤ 1 ∧ 𝑦 = (normℎ‘(𝑇‘𝑧)))}, ℝ*, < )
68	67	eqeq1i 2765	. . . . . . . . . . . 12 ⊢ ((normop‘𝑇) = 0 ↔ sup({𝑦 ∣ ∃𝑧 ∈ ℋ ((normℎ‘𝑧) ≤ 1 ∧ 𝑦 = (normℎ‘(𝑇‘𝑧)))}, ℝ*, < ) = 0)
69	68	biimpi 219	. . . . . . . . . . 11 ⊢ ((normop‘𝑇) = 0 → sup({𝑦 ∣ ∃𝑧 ∈ ℋ ((normℎ‘𝑧) ≤ 1 ∧ 𝑦 = (normℎ‘(𝑇‘𝑧)))}, ℝ*, < ) = 0)
70	69	ad2antrr 739	. . . . . . . . . 10 ⊢ ((((normop‘𝑇) = 0 ∧ 𝑥 ∈ ℋ) ∧ (𝑇‘𝑥) ≠ 0ℎ) → sup({𝑦 ∣ ∃𝑧 ∈ ℋ ((normℎ‘𝑧) ≤ 1 ∧ 𝑦 = (normℎ‘(𝑇‘𝑧)))}, ℝ*, < ) = 0)
71	65, 70	breqtrd 5131	. . . . . . . . 9 ⊢ ((((normop‘𝑇) = 0 ∧ 𝑥 ∈ ℋ) ∧ (𝑇‘𝑥) ≠ 0ℎ) → (normℎ‘((1 / (normℎ‘𝑥)) ·ℎ (𝑇‘𝑥))) ≤ 0)
72		normcl 31606	. . . . . . . . . . . 12 ⊢ (((1 / (normℎ‘𝑥)) ·ℎ (𝑇‘𝑥)) ∈ ℋ → (normℎ‘((1 / (normℎ‘𝑥)) ·ℎ (𝑇‘𝑥))) ∈ ℝ)
73	25, 72	syl 18	. . . . . . . . . . 11 ⊢ ((𝑥 ∈ ℋ ∧ (𝑇‘𝑥) ≠ 0ℎ) → (normℎ‘((1 / (normℎ‘𝑥)) ·ℎ (𝑇‘𝑥))) ∈ ℝ)
74		0re 11234	. . . . . . . . . . 11 ⊢ 0 ∈ ℝ
75		lenlt 11312	. . . . . . . . . . 11 ⊢ (((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)))))
76	73, 74, 75	sylancl 598	. . . . . . . . . 10 ⊢ ((𝑥 ∈ ℋ ∧ (𝑇‘𝑥) ≠ 0ℎ) → ((normℎ‘((1 / (normℎ‘𝑥)) ·ℎ (𝑇‘𝑥))) ≤ 0 ↔ ¬ 0 < (normℎ‘((1 / (normℎ‘𝑥)) ·ℎ (𝑇‘𝑥)))))
77	76	adantll 727	. . . . . . . . 9 ⊢ ((((normop‘𝑇) = 0 ∧ 𝑥 ∈ ℋ) ∧ (𝑇‘𝑥) ≠ 0ℎ) → ((normℎ‘((1 / (normℎ‘𝑥)) ·ℎ (𝑇‘𝑥))) ≤ 0 ↔ ¬ 0 < (normℎ‘((1 / (normℎ‘𝑥)) ·ℎ (𝑇‘𝑥)))))
78	71, 77	mpbid 235	. . . . . . . 8 ⊢ ((((normop‘𝑇) = 0 ∧ 𝑥 ∈ ℋ) ∧ (𝑇‘𝑥) ≠ 0ℎ) → ¬ 0 < (normℎ‘((1 / (normℎ‘𝑥)) ·ℎ (𝑇‘𝑥))))
79	78	ex 418	. . . . . . 7 ⊢ (((normop‘𝑇) = 0 ∧ 𝑥 ∈ ℋ) → ((𝑇‘𝑥) ≠ 0ℎ → ¬ 0 < (normℎ‘((1 / (normℎ‘𝑥)) ·ℎ (𝑇‘𝑥)))))
80	30, 79	pm2.65d 199	. . . . . 6 ⊢ (((normop‘𝑇) = 0 ∧ 𝑥 ∈ ℋ) → ¬ (𝑇‘𝑥) ≠ 0ℎ)
81		nne 2959	. . . . . 6 ⊢ (¬ (𝑇‘𝑥) ≠ 0ℎ ↔ (𝑇‘𝑥) = 0ℎ)
82	80, 81	sylib 221	. . . . 5 ⊢ (((normop‘𝑇) = 0 ∧ 𝑥 ∈ ℋ) → (𝑇‘𝑥) = 0ℎ)
83		ho0val 32231	. . . . . 6 ⊢ (𝑥 ∈ ℋ → ( 0hop ‘𝑥) = 0ℎ)
84	83	adantl 487	. . . . 5 ⊢ (((normop‘𝑇) = 0 ∧ 𝑥 ∈ ℋ) → ( 0hop ‘𝑥) = 0ℎ)
85	82, 84	eqtr4d 2798	. . . 4 ⊢ (((normop‘𝑇) = 0 ∧ 𝑥 ∈ ℋ) → (𝑇‘𝑥) = ( 0hop ‘𝑥))
86	85	ralrimiva 3154	. . 3 ⊢ ((normop‘𝑇) = 0 → ∀𝑥 ∈ ℋ (𝑇‘𝑥) = ( 0hop ‘𝑥))
87		ffn 6702	. . . . 5 ⊢ (𝑇: ℋ⟶ ℋ → 𝑇 Fn ℋ)
88	15, 87	ax-mp 5	. . . 4 ⊢ 𝑇 Fn ℋ
89		ho0f 32232	. . . . 5 ⊢ 0hop : ℋ⟶ ℋ
90		ffn 6702	. . . . 5 ⊢ ( 0hop : ℋ⟶ ℋ → 0hop Fn ℋ)
91	89, 90	ax-mp 5	. . . 4 ⊢ 0hop Fn ℋ
92		eqfnfv 7022	. . . 4 ⊢ ((𝑇 Fn ℋ ∧ 0hop Fn ℋ) → (𝑇 = 0hop ↔ ∀𝑥 ∈ ℋ (𝑇‘𝑥) = ( 0hop ‘𝑥)))
93	88, 91, 92	mp2an 705	. . 3 ⊢ (𝑇 = 0hop ↔ ∀𝑥 ∈ ℋ (𝑇‘𝑥) = ( 0hop ‘𝑥))
94	86, 93	sylibr 237	. 2 ⊢ ((normop‘𝑇) = 0 → 𝑇 = 0hop )
95		fveq2 6878	. . 3 ⊢ (𝑇 = 0hop → (normop‘𝑇) = (normop‘ 0hop ))
96		nmop0 32467	. . 3 ⊢ (normop‘ 0hop ) = 0
97	95, 96	eqtrdi 2811	. 2 ⊢ (𝑇 = 0hop → (normop‘𝑇) = 0)
98	94, 97	impbii 212	1 ⊢ ((normop‘𝑇) = 0 ↔ 𝑇 = 0hop )

This proof depends on syntax axioms:  ¬ wn 3   → wi 4   ↔ wb 209   ∧ wa 401   ∨ wo 861   = wceq 1570   ∈ wcel 2145  {cab 2738   ≠ wne 2955  ∀wral 3076  ∃wrex 3086   ⊆ wss 3899   class class class wbr 5103   Fn wfn 6528  ⟶wf 6529  ‘cfv 6533  (class class class)co 7413  supcsup 9410  ℂcc 11122  ℝcr 11123  0cc0 11124  1c1 11125  ℝ^*cxr 11266   < clt 11267   ≤ cle 11268   / cdiv 11895   ℋchba 31400   ·_ℎ csm 31402  norm_ℎcno 31404  0_ℎc0v 31405   0_hop ch0o 31424  norm_opcnop 31426  LinOpclo 31428

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1828  ax-4 1842  ax-5 1943  ax-6 2000  ax-7 2041  ax-8 2147  ax-9 2155  ax-10 2178  ax-11 2194  ax-12 2213  ax-ext 2732  ax-rep 5232  ax-sep 5251  ax-nul 5263  ax-pow 5330  ax-pr 5398  ax-un 7736  ax-inf2 9620  ax-cc 10437  ax-cnex 11180  ax-resscn 11181  ax-1cn 11182  ax-icn 11183  ax-addcl 11184  ax-addrcl 11185  ax-mulcl 11186  ax-mulrcl 11187  ax-mulcom 11188  ax-addass 11189  ax-mulass 11190  ax-distr 11191  ax-i2m1 11192  ax-1ne0 11193  ax-1rid 11194  ax-rnegex 11195  ax-rrecex 11196  ax-cnre 11197  ax-pre-lttri 11198  ax-pre-lttrn 11199  ax-pre-ltadd 11200  ax-pre-mulgt0 11201  ax-pre-sup 11202  ax-addf 11203  ax-mulf 11204  ax-hilex 31480  ax-hfvadd 31481  ax-hvcom 31482  ax-hvass 31483  ax-hv0cl 31484  ax-hvaddid 31485  ax-hfvmul 31486  ax-hvmulid 31487  ax-hvmulass 31488  ax-hvdistr1 31489  ax-hvdistr2 31490  ax-hvmul0 31491  ax-hfi 31560  ax-his1 31563  ax-his2 31564  ax-his3 31565  ax-his4 31566  ax-hcompl 31683

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3or 1104  df-3an 1105  df-tru 1573  df-fal 1583  df-ex 1813  df-nf 1817  df-sb 2100  df-mo 2564  df-eu 2594  df-clab 2739  df-cleq 2752  df-clel 2835  df-nfc 2909  df-ne 2956  df-nel 3062  df-ral 3077  df-rex 3087  df-rmo 3365  df-reu 3366  df-rab 3413  df-v 3452  df-sbc 3740  df-csb 3848  df-dif 3902  df-un 3904  df-in 3906  df-ss 3916  df-pss 3919  df-nul 4280  df-if 4483  df-pw 4559  df-sn 4585  df-pr 4587  df-tp 4589  df-op 4591  df-uni 4868  df-int 4908  df-iun 4953  df-iin 4954  df-br 5104  df-opab 5168  df-mpt 5187  df-tr 5213  df-id 5550  df-eprel 5555  df-po 5563  df-so 5564  df-fr 5608  df-se 5609  df-we 5610  df-xp 5661  df-rel 5662  df-cnv 5663  df-co 5664  df-dm 5665  df-rn 5666  df-res 5667  df-ima 5668  df-pred 6299  df-ord 6360  df-on 6361  df-lim 6362  df-suc 6363  df-iota 6489  df-fun 6535  df-fn 6536  df-f 6537  df-f1 6538  df-fo 6539  df-f1o 6540  df-fv 6541  df-isom 6542  df-riota 7370  df-ov 7416  df-oprab 7417  df-mpo 7418  df-of 7678  df-om 7863  df-1st 7986  df-2nd 7987  df-supp 8159  df-frecs 8280  df-wrecs 8311  df-recs 8360  df-rdg 8399  df-1o 8455  df-2o 8456  df-oadd 8459  df-omul 8460  df-er 8696  df-map 8828  df-pm 8829  df-ixp 8905  df-en 8953  df-dom 8954  df-sdom 8955  df-fin 8956  df-fsupp 9332  df-fi 9381  df-sup 9412  df-inf 9413  df-oi 9482  df-card 9944  df-acn 9947  df-pnf 11269  df-mnf 11270  df-xr 11271  df-ltxr 11272  df-le 11273  df-sub 11467  df-neg 11468  df-div 11896  df-nn 12258  df-2 12327  df-3 12328  df-4 12329  df-5 12330  df-6 12331  df-7 12332  df-8 12333  df-9 12334  df-n0 12529  df-z 12616  df-dec 12737  df-uz 12888  df-q 12998  df-rp 13043  df-xneg 13163  df-xadd 13164  df-xmul 13165  df-ioo 13402  df-ico 13404  df-icc 13405  df-fz 13562  df-fzo 13710  df-fl 13853  df-seq 14066  df-exp 14126  df-hash 14395  df-cj 15186  df-re 15187  df-im 15188  df-sqrt 15322  df-abs 15323  df-clim 15575  df-rlim 15576  df-sum 15774  df-struct 17239  df-sets 17256  df-slot 17274  df-ndx 17286  df-base 17302  df-ress 17323  df-plusg 17355  df-mulr 17356  df-starv 17357  df-sca 17358  df-vsca 17359  df-ip 17360  df-tset 17361  df-ple 17362  df-ds 17364  df-unif 17365  df-hom 17366  df-cco 17367  df-rest 17507  df-topn 17508  df-0g 17526  df-gsum 17527  df-topgen 17528  df-pt 17529  df-prds 17532  df-xrs 17588  df-qtop 17593  df-imas 17594  df-xps 17596  df-mre 17670  df-mrc 17671  df-acs 17673  df-mgm 18730  df-sgrp 18821  df-mnd 18837  df-submnd 18892  df-mulg 19191  df-cntz 19444  df-cmn 19909  df-psmet 21577  df-xmet 21578  df-met 21579  df-bl 21580  df-mopn 21581  df-fbas 21582  df-fg 21583  df-cnfld 21586  df-top 23119  df-topon 23136  df-topsp 23158  df-bases 23171  df-cld 23244  df-ntr 23245  df-cls 23246  df-nei 23323  df-cn 23452  df-cnp 23453  df-lm 23454  df-haus 23540  df-tx 23788  df-hmeo 23981  df-fil 24072  df-fm 24164  df-flim 24165  df-flf 24166  df-xms 24546  df-ms 24547  df-tms 24548  df-cfil 25483  df-cau 25484  df-cmet 25485  df-grpo 30974  df-gid 30975  df-ginv 30976  df-gdiv 30977  df-ablo 31026  df-vc 31040  df-nv 31073  df-va 31076  df-ba 31077  df-sm 31078  df-0v 31079  df-vs 31080  df-nmcv 31081  df-ims 31082  df-dip 31182  df-ssp 31203  df-ph 31294  df-cbn 31344  df-hnorm 31449  df-hba 31450  df-hvsub 31452  df-hlim 31453  df-hcau 31454  df-sh 31688  df-ch 31702  df-oc 31733  df-ch0 31734  df-shs 31789  df-pjh 31876  df-h0op 32229  df-nmop 32320  df-lnop 32322

This theorem is used by: (None)