Theorem nmlnop0iALT 31897

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 31027	. . . . . . . . . . . . . 14 ⊢ (𝑥 ∈ ℋ → (normℎ‘𝑥) ∈ ℝ)
2	1	recnd 11178	. . . . . . . . . . . . 13 ⊢ (𝑥 ∈ ℋ → (normℎ‘𝑥) ∈ ℂ)
3	2	adantr 480	. . . . . . . . . . . 12 ⊢ ((𝑥 ∈ ℋ ∧ (𝑇‘𝑥) ≠ 0ℎ) → (normℎ‘𝑥) ∈ ℂ)
4		norm-i 31031	. . . . . . . . . . . . . . 15 ⊢ (𝑥 ∈ ℋ → ((normℎ‘𝑥) = 0 ↔ 𝑥 = 0ℎ))
5		fveq2 6840	. . . . . . . . . . . . . . . 16 ⊢ (𝑥 = 0ℎ → (𝑇‘𝑥) = (𝑇‘0ℎ))
6		nmlnop0.1	. . . . . . . . . . . . . . . . 17 ⊢ 𝑇 ∈ LinOp
7	6	lnop0i 31872	. . . . . . . . . . . . . . . 16 ⊢ (𝑇‘0ℎ) = 0ℎ
8	5, 7	eqtrdi 2780	. . . . . . . . . . . . . . 15 ⊢ (𝑥 = 0ℎ → (𝑇‘𝑥) = 0ℎ)
9	4, 8	biimtrdi 253	. . . . . . . . . . . . . 14 ⊢ (𝑥 ∈ ℋ → ((normℎ‘𝑥) = 0 → (𝑇‘𝑥) = 0ℎ))
10	9	necon3d 2946	. . . . . . . . . . . . 13 ⊢ (𝑥 ∈ ℋ → ((𝑇‘𝑥) ≠ 0ℎ → (normℎ‘𝑥) ≠ 0))
11	10	imp 406	. . . . . . . . . . . 12 ⊢ ((𝑥 ∈ ℋ ∧ (𝑇‘𝑥) ≠ 0ℎ) → (normℎ‘𝑥) ≠ 0)
12	3, 11	recne0d 11928	. . . . . . . . . . 11 ⊢ ((𝑥 ∈ ℋ ∧ (𝑇‘𝑥) ≠ 0ℎ) → (1 / (normℎ‘𝑥)) ≠ 0)
13		simpr 484	. . . . . . . . . . 11 ⊢ ((𝑥 ∈ ℋ ∧ (𝑇‘𝑥) ≠ 0ℎ) → (𝑇‘𝑥) ≠ 0ℎ)
14	3, 11	reccld 11927	. . . . . . . . . . . . . 14 ⊢ ((𝑥 ∈ ℋ ∧ (𝑇‘𝑥) ≠ 0ℎ) → (1 / (normℎ‘𝑥)) ∈ ℂ)
15	6	lnopfi 31871	. . . . . . . . . . . . . . . 16 ⊢ 𝑇: ℋ⟶ ℋ
16	15	ffvelcdmi 7037	. . . . . . . . . . . . . . 15 ⊢ (𝑥 ∈ ℋ → (𝑇‘𝑥) ∈ ℋ)
17	16	adantr 480	. . . . . . . . . . . . . 14 ⊢ ((𝑥 ∈ ℋ ∧ (𝑇‘𝑥) ≠ 0ℎ) → (𝑇‘𝑥) ∈ ℋ)
18		hvmul0or 30927	. . . . . . . . . . . . . 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 584	. . . . . . . . . . . . 13 ⊢ ((𝑥 ∈ ℋ ∧ (𝑇‘𝑥) ≠ 0ℎ) → (((1 / (normℎ‘𝑥)) ·ℎ (𝑇‘𝑥)) = 0ℎ ↔ ((1 / (normℎ‘𝑥)) = 0 ∨ (𝑇‘𝑥) = 0ℎ)))
20	19	necon3abid 2961	. . . . . . . . . . . 12 ⊢ ((𝑥 ∈ ℋ ∧ (𝑇‘𝑥) ≠ 0ℎ) → (((1 / (normℎ‘𝑥)) ·ℎ (𝑇‘𝑥)) ≠ 0ℎ ↔ ¬ ((1 / (normℎ‘𝑥)) = 0 ∨ (𝑇‘𝑥) = 0ℎ)))
21		neanior 3018	. . . . . . . . . . . 12 ⊢ (((1 / (normℎ‘𝑥)) ≠ 0 ∧ (𝑇‘𝑥) ≠ 0ℎ) ↔ ¬ ((1 / (normℎ‘𝑥)) = 0 ∨ (𝑇‘𝑥) = 0ℎ))
22	20, 21	bitr4di 289	. . . . . . . . . . 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 713	. . . . . . . . . 10 ⊢ ((𝑥 ∈ ℋ ∧ (𝑇‘𝑥) ≠ 0ℎ) → ((1 / (normℎ‘𝑥)) ·ℎ (𝑇‘𝑥)) ≠ 0ℎ)
24		hvmulcl 30915	. . . . . . . . . . . 12 ⊢ (((1 / (normℎ‘𝑥)) ∈ ℂ ∧ (𝑇‘𝑥) ∈ ℋ) → ((1 / (normℎ‘𝑥)) ·ℎ (𝑇‘𝑥)) ∈ ℋ)
25	14, 17, 24	syl2anc 584	. . . . . . . . . . 11 ⊢ ((𝑥 ∈ ℋ ∧ (𝑇‘𝑥) ≠ 0ℎ) → ((1 / (normℎ‘𝑥)) ·ℎ (𝑇‘𝑥)) ∈ ℋ)
26		normgt0 31029	. . . . . . . . . . 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 17	. . . . . . . . . 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 232	. . . . . . . . 9 ⊢ ((𝑥 ∈ ℋ ∧ (𝑇‘𝑥) ≠ 0ℎ) → 0 < (normℎ‘((1 / (normℎ‘𝑥)) ·ℎ (𝑇‘𝑥))))
29	28	ex 412	. . . . . . . 8 ⊢ (𝑥 ∈ ℋ → ((𝑇‘𝑥) ≠ 0ℎ → 0 < (normℎ‘((1 / (normℎ‘𝑥)) ·ℎ (𝑇‘𝑥)))))
30	29	adantl 481	. . . . . . 7 ⊢ (((normop‘𝑇) = 0 ∧ 𝑥 ∈ ℋ) → ((𝑇‘𝑥) ≠ 0ℎ → 0 < (normℎ‘((1 / (normℎ‘𝑥)) ·ℎ (𝑇‘𝑥)))))
31		nmopsetretHIL 31766	. . . . . . . . . . . . . 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 11194	. . . . . . . . . . . . 13 ⊢ ℝ ⊆ ℝ*
34	32, 33	sstri 3953	. . . . . . . . . . . 12 ⊢ {𝑦 ∣ ∃𝑧 ∈ ℋ ((normℎ‘𝑧) ≤ 1 ∧ 𝑦 = (normℎ‘(𝑇‘𝑧)))} ⊆ ℝ*
35		simpl 482	. . . . . . . . . . . . . . 15 ⊢ ((𝑥 ∈ ℋ ∧ (𝑇‘𝑥) ≠ 0ℎ) → 𝑥 ∈ ℋ)
36		hvmulcl 30915	. . . . . . . . . . . . . . 15 ⊢ (((1 / (normℎ‘𝑥)) ∈ ℂ ∧ 𝑥 ∈ ℋ) → ((1 / (normℎ‘𝑥)) ·ℎ 𝑥) ∈ ℋ)
37	14, 35, 36	syl2anc 584	. . . . . . . . . . . . . 14 ⊢ ((𝑥 ∈ ℋ ∧ (𝑇‘𝑥) ≠ 0ℎ) → ((1 / (normℎ‘𝑥)) ·ℎ 𝑥) ∈ ℋ)
38	8	necon3i 2957	. . . . . . . . . . . . . . . . 17 ⊢ ((𝑇‘𝑥) ≠ 0ℎ → 𝑥 ≠ 0ℎ)
39		norm1 31151	. . . . . . . . . . . . . . . . 17 ⊢ ((𝑥 ∈ ℋ ∧ 𝑥 ≠ 0ℎ) → (normℎ‘((1 / (normℎ‘𝑥)) ·ℎ 𝑥)) = 1)
40	38, 39	sylan2 593	. . . . . . . . . . . . . . . 16 ⊢ ((𝑥 ∈ ℋ ∧ (𝑇‘𝑥) ≠ 0ℎ) → (normℎ‘((1 / (normℎ‘𝑥)) ·ℎ 𝑥)) = 1)
41		1re 11150	. . . . . . . . . . . . . . . 16 ⊢ 1 ∈ ℝ
42	40, 41	eqeltrdi 2836	. . . . . . . . . . . . . . 15 ⊢ ((𝑥 ∈ ℋ ∧ (𝑇‘𝑥) ≠ 0ℎ) → (normℎ‘((1 / (normℎ‘𝑥)) ·ℎ 𝑥)) ∈ ℝ)
43		eqle 11252	. . . . . . . . . . . . . . 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 584	. . . . . . . . . . . . . 14 ⊢ ((𝑥 ∈ ℋ ∧ (𝑇‘𝑥) ≠ 0ℎ) → (normℎ‘((1 / (normℎ‘𝑥)) ·ℎ 𝑥)) ≤ 1)
45	6	lnopmuli 31874	. . . . . . . . . . . . . . . . 17 ⊢ (((1 / (normℎ‘𝑥)) ∈ ℂ ∧ 𝑥 ∈ ℋ) → (𝑇‘((1 / (normℎ‘𝑥)) ·ℎ 𝑥)) = ((1 / (normℎ‘𝑥)) ·ℎ (𝑇‘𝑥)))
46	14, 35, 45	syl2anc 584	. . . . . . . . . . . . . . . 16 ⊢ ((𝑥 ∈ ℋ ∧ (𝑇‘𝑥) ≠ 0ℎ) → (𝑇‘((1 / (normℎ‘𝑥)) ·ℎ 𝑥)) = ((1 / (normℎ‘𝑥)) ·ℎ (𝑇‘𝑥)))
47	46	eqcomd 2735	. . . . . . . . . . . . . . 15 ⊢ ((𝑥 ∈ ℋ ∧ (𝑇‘𝑥) ≠ 0ℎ) → ((1 / (normℎ‘𝑥)) ·ℎ (𝑇‘𝑥)) = (𝑇‘((1 / (normℎ‘𝑥)) ·ℎ 𝑥)))
48	47	fveq2d 6844	. . . . . . . . . . . . . 14 ⊢ ((𝑥 ∈ ℋ ∧ (𝑇‘𝑥) ≠ 0ℎ) → (normℎ‘((1 / (normℎ‘𝑥)) ·ℎ (𝑇‘𝑥))) = (normℎ‘(𝑇‘((1 / (normℎ‘𝑥)) ·ℎ 𝑥))))
49		fveq2 6840	. . . . . . . . . . . . . . . . 17 ⊢ (𝑧 = ((1 / (normℎ‘𝑥)) ·ℎ 𝑥) → (normℎ‘𝑧) = (normℎ‘((1 / (normℎ‘𝑥)) ·ℎ 𝑥)))
50	49	breq1d 5112	. . . . . . . . . . . . . . . 16 ⊢ (𝑧 = ((1 / (normℎ‘𝑥)) ·ℎ 𝑥) → ((normℎ‘𝑧) ≤ 1 ↔ (normℎ‘((1 / (normℎ‘𝑥)) ·ℎ 𝑥)) ≤ 1))
51		fveq2 6840	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑧 = ((1 / (normℎ‘𝑥)) ·ℎ 𝑥) → (𝑇‘𝑧) = (𝑇‘((1 / (normℎ‘𝑥)) ·ℎ 𝑥)))
52	51	fveq2d 6844	. . . . . . . . . . . . . . . . 17 ⊢ (𝑧 = ((1 / (normℎ‘𝑥)) ·ℎ 𝑥) → (normℎ‘(𝑇‘𝑧)) = (normℎ‘(𝑇‘((1 / (normℎ‘𝑥)) ·ℎ 𝑥))))
53	52	eqeq2d 2740	. . . . . . . . . . . . . . . 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 632	. . . . . . . . . . . . . . 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 3585	. . . . . . . . . . . . . 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 836	. . . . . . . . . . . . 13 ⊢ ((𝑥 ∈ ℋ ∧ (𝑇‘𝑥) ≠ 0ℎ) → ∃𝑧 ∈ ℋ ((normℎ‘𝑧) ≤ 1 ∧ (normℎ‘((1 / (normℎ‘𝑥)) ·ℎ (𝑇‘𝑥))) = (normℎ‘(𝑇‘𝑧))))
57		fvex 6853	. . . . . . . . . . . . . 14 ⊢ (normℎ‘((1 / (normℎ‘𝑥)) ·ℎ (𝑇‘𝑥))) ∈ V
58		eqeq1 2733	. . . . . . . . . . . . . . . 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 630	. . . . . . . . . . . . . . 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 3157	. . . . . . . . . . . . . 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 3643	. . . . . . . . . . . . 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 234	. . . . . . . . . . . 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 13260	. . . . . . . . . . . 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 587	. . . . . . . . . . 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 714	. . . . . . . . . 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 31758	. . . . . . . . . . . . . 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 2734	. . . . . . . . . . . 12 ⊢ ((normop‘𝑇) = 0 ↔ sup({𝑦 ∣ ∃𝑧 ∈ ℋ ((normℎ‘𝑧) ≤ 1 ∧ 𝑦 = (normℎ‘(𝑇‘𝑧)))}, ℝ*, < ) = 0)
69	68	biimpi 216	. . . . . . . . . . 11 ⊢ ((normop‘𝑇) = 0 → sup({𝑦 ∣ ∃𝑧 ∈ ℋ ((normℎ‘𝑧) ≤ 1 ∧ 𝑦 = (normℎ‘(𝑇‘𝑧)))}, ℝ*, < ) = 0)
70	69	ad2antrr 726	. . . . . . . . . 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 5128	. . . . . . . . 9 ⊢ ((((normop‘𝑇) = 0 ∧ 𝑥 ∈ ℋ) ∧ (𝑇‘𝑥) ≠ 0ℎ) → (normℎ‘((1 / (normℎ‘𝑥)) ·ℎ (𝑇‘𝑥))) ≤ 0)
72		normcl 31027	. . . . . . . . . . . 12 ⊢ (((1 / (normℎ‘𝑥)) ·ℎ (𝑇‘𝑥)) ∈ ℋ → (normℎ‘((1 / (normℎ‘𝑥)) ·ℎ (𝑇‘𝑥))) ∈ ℝ)
73	25, 72	syl 17	. . . . . . . . . . 11 ⊢ ((𝑥 ∈ ℋ ∧ (𝑇‘𝑥) ≠ 0ℎ) → (normℎ‘((1 / (normℎ‘𝑥)) ·ℎ (𝑇‘𝑥))) ∈ ℝ)
74		0re 11152	. . . . . . . . . . 11 ⊢ 0 ∈ ℝ
75		lenlt 11228	. . . . . . . . . . 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 586	. . . . . . . . . 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 714	. . . . . . . . 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 232	. . . . . . . 8 ⊢ ((((normop‘𝑇) = 0 ∧ 𝑥 ∈ ℋ) ∧ (𝑇‘𝑥) ≠ 0ℎ) → ¬ 0 < (normℎ‘((1 / (normℎ‘𝑥)) ·ℎ (𝑇‘𝑥))))
79	78	ex 412	. . . . . . 7 ⊢ (((normop‘𝑇) = 0 ∧ 𝑥 ∈ ℋ) → ((𝑇‘𝑥) ≠ 0ℎ → ¬ 0 < (normℎ‘((1 / (normℎ‘𝑥)) ·ℎ (𝑇‘𝑥)))))
80	30, 79	pm2.65d 196	. . . . . 6 ⊢ (((normop‘𝑇) = 0 ∧ 𝑥 ∈ ℋ) → ¬ (𝑇‘𝑥) ≠ 0ℎ)
81		nne 2929	. . . . . 6 ⊢ (¬ (𝑇‘𝑥) ≠ 0ℎ ↔ (𝑇‘𝑥) = 0ℎ)
82	80, 81	sylib 218	. . . . 5 ⊢ (((normop‘𝑇) = 0 ∧ 𝑥 ∈ ℋ) → (𝑇‘𝑥) = 0ℎ)
83		ho0val 31652	. . . . . 6 ⊢ (𝑥 ∈ ℋ → ( 0hop ‘𝑥) = 0ℎ)
84	83	adantl 481	. . . . 5 ⊢ (((normop‘𝑇) = 0 ∧ 𝑥 ∈ ℋ) → ( 0hop ‘𝑥) = 0ℎ)
85	82, 84	eqtr4d 2767	. . . 4 ⊢ (((normop‘𝑇) = 0 ∧ 𝑥 ∈ ℋ) → (𝑇‘𝑥) = ( 0hop ‘𝑥))
86	85	ralrimiva 3125	. . 3 ⊢ ((normop‘𝑇) = 0 → ∀𝑥 ∈ ℋ (𝑇‘𝑥) = ( 0hop ‘𝑥))
87		ffn 6670	. . . . 5 ⊢ (𝑇: ℋ⟶ ℋ → 𝑇 Fn ℋ)
88	15, 87	ax-mp 5	. . . 4 ⊢ 𝑇 Fn ℋ
89		ho0f 31653	. . . . 5 ⊢ 0hop : ℋ⟶ ℋ
90		ffn 6670	. . . . 5 ⊢ ( 0hop : ℋ⟶ ℋ → 0hop Fn ℋ)
91	89, 90	ax-mp 5	. . . 4 ⊢ 0hop Fn ℋ
92		eqfnfv 6985	. . . 4 ⊢ ((𝑇 Fn ℋ ∧ 0hop Fn ℋ) → (𝑇 = 0hop ↔ ∀𝑥 ∈ ℋ (𝑇‘𝑥) = ( 0hop ‘𝑥)))
93	88, 91, 92	mp2an 692	. . 3 ⊢ (𝑇 = 0hop ↔ ∀𝑥 ∈ ℋ (𝑇‘𝑥) = ( 0hop ‘𝑥))
94	86, 93	sylibr 234	. 2 ⊢ ((normop‘𝑇) = 0 → 𝑇 = 0hop )
95		fveq2 6840	. . 3 ⊢ (𝑇 = 0hop → (normop‘𝑇) = (normop‘ 0hop ))
96		nmop0 31888	. . 3 ⊢ (normop‘ 0hop ) = 0
97	95, 96	eqtrdi 2780	. 2 ⊢ (𝑇 = 0hop → (normop‘𝑇) = 0)
98	94, 97	impbii 209	1 ⊢ ((normop‘𝑇) = 0 ↔ 𝑇 = 0hop )

Syntax hints:  ¬ wn 3   → wi 4   ↔ wb 206   ∧ wa 395   ∨ wo 847   = wceq 1540   ∈ wcel 2109  {cab 2707   ≠ wne 2925  ∀wral 3044  ∃wrex 3053   ⊆ wss 3911   class class class wbr 5102   Fn wfn 6494  ⟶wf 6495  ‘cfv 6499  (class class class)co 7369  supcsup 9367  ℂcc 11042  ℝcr 11043  0cc0 11044  1c1 11045  ℝ^*cxr 11183   < clt 11184   ≤ cle 11185   / cdiv 11811   ℋchba 30821   ·_ℎ csm 30823  norm_ℎcno 30825  0_ℎc0v 30826   0_hop ch0o 30845  norm_opcnop 30847  LinOpclo 30849

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2008  ax-8 2111  ax-9 2119  ax-10 2142  ax-11 2158  ax-12 2178  ax-ext 2701  ax-rep 5229  ax-sep 5246  ax-nul 5256  ax-pow 5315  ax-pr 5382  ax-un 7691  ax-inf2 9570  ax-cc 10364  ax-cnex 11100  ax-resscn 11101  ax-1cn 11102  ax-icn 11103  ax-addcl 11104  ax-addrcl 11105  ax-mulcl 11106  ax-mulrcl 11107  ax-mulcom 11108  ax-addass 11109  ax-mulass 11110  ax-distr 11111  ax-i2m1 11112  ax-1ne0 11113  ax-1rid 11114  ax-rnegex 11115  ax-rrecex 11116  ax-cnre 11117  ax-pre-lttri 11118  ax-pre-lttrn 11119  ax-pre-ltadd 11120  ax-pre-mulgt0 11121  ax-pre-sup 11122  ax-addf 11123  ax-mulf 11124  ax-hilex 30901  ax-hfvadd 30902  ax-hvcom 30903  ax-hvass 30904  ax-hv0cl 30905  ax-hvaddid 30906  ax-hfvmul 30907  ax-hvmulid 30908  ax-hvmulass 30909  ax-hvdistr1 30910  ax-hvdistr2 30911  ax-hvmul0 30912  ax-hfi 30981  ax-his1 30984  ax-his2 30985  ax-his3 30986  ax-his4 30987  ax-hcompl 31104

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3or 1087  df-3an 1088  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2066  df-mo 2533  df-eu 2562  df-clab 2708  df-cleq 2721  df-clel 2803  df-nfc 2878  df-ne 2926  df-nel 3030  df-ral 3045  df-rex 3054  df-rmo 3351  df-reu 3352  df-rab 3403  df-v 3446  df-sbc 3751  df-csb 3860  df-dif 3914  df-un 3916  df-in 3918  df-ss 3928  df-pss 3931  df-nul 4293  df-if 4485  df-pw 4561  df-sn 4586  df-pr 4588  df-tp 4590  df-op 4592  df-uni 4868  df-int 4907  df-iun 4953  df-iin 4954  df-br 5103  df-opab 5165  df-mpt 5184  df-tr 5210  df-id 5526  df-eprel 5531  df-po 5539  df-so 5540  df-fr 5584  df-se 5585  df-we 5586  df-xp 5637  df-rel 5638  df-cnv 5639  df-co 5640  df-dm 5641  df-rn 5642  df-res 5643  df-ima 5644  df-pred 6262  df-ord 6323  df-on 6324  df-lim 6325  df-suc 6326  df-iota 6452  df-fun 6501  df-fn 6502  df-f 6503  df-f1 6504  df-fo 6505  df-f1o 6506  df-fv 6507  df-isom 6508  df-riota 7326  df-ov 7372  df-oprab 7373  df-mpo 7374  df-of 7633  df-om 7823  df-1st 7947  df-2nd 7948  df-supp 8117  df-frecs 8237  df-wrecs 8268  df-recs 8317  df-rdg 8355  df-1o 8411  df-2o 8412  df-oadd 8415  df-omul 8416  df-er 8648  df-map 8778  df-pm 8779  df-ixp 8848  df-en 8896  df-dom 8897  df-sdom 8898  df-fin 8899  df-fsupp 9289  df-fi 9338  df-sup 9369  df-inf 9370  df-oi 9439  df-card 9868  df-acn 9871  df-pnf 11186  df-mnf 11187  df-xr 11188  df-ltxr 11189  df-le 11190  df-sub 11383  df-neg 11384  df-div 11812  df-nn 12163  df-2 12225  df-3 12226  df-4 12227  df-5 12228  df-6 12229  df-7 12230  df-8 12231  df-9 12232  df-n0 12419  df-z 12506  df-dec 12626  df-uz 12770  df-q 12884  df-rp 12928  df-xneg 13048  df-xadd 13049  df-xmul 13050  df-ioo 13286  df-ico 13288  df-icc 13289  df-fz 13445  df-fzo 13592  df-fl 13730  df-seq 13943  df-exp 14003  df-hash 14272  df-cj 15041  df-re 15042  df-im 15043  df-sqrt 15177  df-abs 15178  df-clim 15430  df-rlim 15431  df-sum 15629  df-struct 17093  df-sets 17110  df-slot 17128  df-ndx 17140  df-base 17156  df-ress 17177  df-plusg 17209  df-mulr 17210  df-starv 17211  df-sca 17212  df-vsca 17213  df-ip 17214  df-tset 17215  df-ple 17216  df-ds 17218  df-unif 17219  df-hom 17220  df-cco 17221  df-rest 17361  df-topn 17362  df-0g 17380  df-gsum 17381  df-topgen 17382  df-pt 17383  df-prds 17386  df-xrs 17441  df-qtop 17446  df-imas 17447  df-xps 17449  df-mre 17523  df-mrc 17524  df-acs 17526  df-mgm 18543  df-sgrp 18622  df-mnd 18638  df-submnd 18687  df-mulg 18976  df-cntz 19225  df-cmn 19688  df-psmet 21232  df-xmet 21233  df-met 21234  df-bl 21235  df-mopn 21236  df-fbas 21237  df-fg 21238  df-cnfld 21241  df-top 22757  df-topon 22774  df-topsp 22796  df-bases 22809  df-cld 22882  df-ntr 22883  df-cls 22884  df-nei 22961  df-cn 23090  df-cnp 23091  df-lm 23092  df-haus 23178  df-tx 23425  df-hmeo 23618  df-fil 23709  df-fm 23801  df-flim 23802  df-flf 23803  df-xms 24184  df-ms 24185  df-tms 24186  df-cfil 25131  df-cau 25132  df-cmet 25133  df-grpo 30395  df-gid 30396  df-ginv 30397  df-gdiv 30398  df-ablo 30447  df-vc 30461  df-nv 30494  df-va 30497  df-ba 30498  df-sm 30499  df-0v 30500  df-vs 30501  df-nmcv 30502  df-ims 30503  df-dip 30603  df-ssp 30624  df-ph 30715  df-cbn 30765  df-hnorm 30870  df-hba 30871  df-hvsub 30873  df-hlim 30874  df-hcau 30875  df-sh 31109  df-ch 31123  df-oc 31154  df-ch0 31155  df-shs 31210  df-pjh 31297  df-h0op 31650  df-nmop 31741  df-lnop 31743

This theorem is referenced by: (None)