Theorem nmlnop0iALT 32418

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 31548	. . . . . . . . . . . . . 14 ⊢ (𝑥 ∈ ℋ → (normℎ‘𝑥) ∈ ℝ)
2	1	recnd 11252	. . . . . . . . . . . . 13 ⊢ (𝑥 ∈ ℋ → (normℎ‘𝑥) ∈ ℂ)
3	2	adantr 486	. . . . . . . . . . . 12 ⊢ ((𝑥 ∈ ℋ ∧ (𝑇‘𝑥) ≠ 0ℎ) → (normℎ‘𝑥) ∈ ℂ)
4		norm-i 31552	. . . . . . . . . . . . . . 15 ⊢ (𝑥 ∈ ℋ → ((normℎ‘𝑥) = 0 ↔ 𝑥 = 0ℎ))
5		fveq2 6885	. . . . . . . . . . . . . . . 16 ⊢ (𝑥 = 0ℎ → (𝑇‘𝑥) = (𝑇‘0ℎ))
6		nmlnop0.1	. . . . . . . . . . . . . . . . 17 ⊢ 𝑇 ∈ LinOp
7	6	lnop0i 32393	. . . . . . . . . . . . . . . 16 ⊢ (𝑇‘0ℎ) = 0ℎ
8	5, 7	eqtrdi 2816	. . . . . . . . . . . . . . 15 ⊢ (𝑥 = 0ℎ → (𝑇‘𝑥) = 0ℎ)
9	4, 8	biimtrdi 256	. . . . . . . . . . . . . 14 ⊢ (𝑥 ∈ ℋ → ((normℎ‘𝑥) = 0 → (𝑇‘𝑥) = 0ℎ))
10	9	necon3d 2981	. . . . . . . . . . . . 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 12000	. . . . . . . . . . 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 11999	. . . . . . . . . . . . . 14 ⊢ ((𝑥 ∈ ℋ ∧ (𝑇‘𝑥) ≠ 0ℎ) → (1 / (normℎ‘𝑥)) ∈ ℂ)
15	6	lnopfi 32392	. . . . . . . . . . . . . . . 16 ⊢ 𝑇: ℋ⟶ ℋ
16	15	ffvelcdmi 7082	. . . . . . . . . . . . . . 15 ⊢ (𝑥 ∈ ℋ → (𝑇‘𝑥) ∈ ℋ)
17	16	adantr 486	. . . . . . . . . . . . . 14 ⊢ ((𝑥 ∈ ℋ ∧ (𝑇‘𝑥) ≠ 0ℎ) → (𝑇‘𝑥) ∈ ℋ)
18		hvmul0or 31448	. . . . . . . . . . . . . 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 2996	. . . . . . . . . . . 12 ⊢ ((𝑥 ∈ ℋ ∧ (𝑇‘𝑥) ≠ 0ℎ) → (((1 / (normℎ‘𝑥)) ·ℎ (𝑇‘𝑥)) ≠ 0ℎ ↔ ¬ ((1 / (normℎ‘𝑥)) = 0 ∨ (𝑇‘𝑥) = 0ℎ)))
21		neanior 3053	. . . . . . . . . . . 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 31436	. . . . . . . . . . . 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 31550	. . . . . . . . . . 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 32287	. . . . . . . . . . . . . 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 11268	. . . . . . . . . . . . 13 ⊢ ℝ ⊆ ℝ*
34	32, 33	sstri 3947	. . . . . . . . . . . 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 31436	. . . . . . . . . . . . . . 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 2992	. . . . . . . . . . . . . . . . 17 ⊢ ((𝑇‘𝑥) ≠ 0ℎ → 𝑥 ≠ 0ℎ)
39		norm1 31672	. . . . . . . . . . . . . . . . 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 11223	. . . . . . . . . . . . . . . 16 ⊢ 1 ∈ ℝ
42	40, 41	eqeltrdi 2873	. . . . . . . . . . . . . . 15 ⊢ ((𝑥 ∈ ℋ ∧ (𝑇‘𝑥) ≠ 0ℎ) → (normℎ‘((1 / (normℎ‘𝑥)) ·ℎ 𝑥)) ∈ ℝ)
43		eqle 11327	. . . . . . . . . . . . . . 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 32395	. . . . . . . . . . . . . . . . 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 2771	. . . . . . . . . . . . . . 15 ⊢ ((𝑥 ∈ ℋ ∧ (𝑇‘𝑥) ≠ 0ℎ) → ((1 / (normℎ‘𝑥)) ·ℎ (𝑇‘𝑥)) = (𝑇‘((1 / (normℎ‘𝑥)) ·ℎ 𝑥)))
48	47	fveq2d 6889	. . . . . . . . . . . . . 14 ⊢ ((𝑥 ∈ ℋ ∧ (𝑇‘𝑥) ≠ 0ℎ) → (normℎ‘((1 / (normℎ‘𝑥)) ·ℎ (𝑇‘𝑥))) = (normℎ‘(𝑇‘((1 / (normℎ‘𝑥)) ·ℎ 𝑥))))
49		fveq2 6885	. . . . . . . . . . . . . . . . 17 ⊢ (𝑧 = ((1 / (normℎ‘𝑥)) ·ℎ 𝑥) → (normℎ‘𝑧) = (normℎ‘((1 / (normℎ‘𝑥)) ·ℎ 𝑥)))
50	49	breq1d 5121	. . . . . . . . . . . . . . . 16 ⊢ (𝑧 = ((1 / (normℎ‘𝑥)) ·ℎ 𝑥) → ((normℎ‘𝑧) ≤ 1 ↔ (normℎ‘((1 / (normℎ‘𝑥)) ·ℎ 𝑥)) ≤ 1))
51		fveq2 6885	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑧 = ((1 / (normℎ‘𝑥)) ·ℎ 𝑥) → (𝑇‘𝑧) = (𝑇‘((1 / (normℎ‘𝑥)) ·ℎ 𝑥)))
52	51	fveq2d 6889	. . . . . . . . . . . . . . . . 17 ⊢ (𝑧 = ((1 / (normℎ‘𝑥)) ·ℎ 𝑥) → (normℎ‘(𝑇‘𝑧)) = (normℎ‘(𝑇‘((1 / (normℎ‘𝑥)) ·ℎ 𝑥))))
53	52	eqeq2d 2776	. . . . . . . . . . . . . . . 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 3583	. . . . . . . . . . . . . 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 6898	. . . . . . . . . . . . . 14 ⊢ (normℎ‘((1 / (normℎ‘𝑥)) ·ℎ (𝑇‘𝑥))) ∈ V
58		eqeq1 2769	. . . . . . . . . . . . . . . 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 3191	. . . . . . . . . . . . . 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 3640	. . . . . . . . . . . . 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 13366	. . . . . . . . . . . 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 32279	. . . . . . . . . . . . . 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 2770	. . . . . . . . . . . 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 5139	. . . . . . . . 9 ⊢ ((((normop‘𝑇) = 0 ∧ 𝑥 ∈ ℋ) ∧ (𝑇‘𝑥) ≠ 0ℎ) → (normℎ‘((1 / (normℎ‘𝑥)) ·ℎ (𝑇‘𝑥))) ≤ 0)
72		normcl 31548	. . . . . . . . . . . 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 11225	. . . . . . . . . . 11 ⊢ 0 ∈ ℝ
75		lenlt 11303	. . . . . . . . . . 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 2964	. . . . . 6 ⊢ (¬ (𝑇‘𝑥) ≠ 0ℎ ↔ (𝑇‘𝑥) = 0ℎ)
82	80, 81	sylib 221	. . . . 5 ⊢ (((normop‘𝑇) = 0 ∧ 𝑥 ∈ ℋ) → (𝑇‘𝑥) = 0ℎ)
83		ho0val 32173	. . . . . 6 ⊢ (𝑥 ∈ ℋ → ( 0hop ‘𝑥) = 0ℎ)
84	83	adantl 487	. . . . 5 ⊢ (((normop‘𝑇) = 0 ∧ 𝑥 ∈ ℋ) → ( 0hop ‘𝑥) = 0ℎ)
85	82, 84	eqtr4d 2803	. . . 4 ⊢ (((normop‘𝑇) = 0 ∧ 𝑥 ∈ ℋ) → (𝑇‘𝑥) = ( 0hop ‘𝑥))
86	85	ralrimiva 3159	. . 3 ⊢ ((normop‘𝑇) = 0 → ∀𝑥 ∈ ℋ (𝑇‘𝑥) = ( 0hop ‘𝑥))
87		ffn 6709	. . . . 5 ⊢ (𝑇: ℋ⟶ ℋ → 𝑇 Fn ℋ)
88	15, 87	ax-mp 5	. . . 4 ⊢ 𝑇 Fn ℋ
89		ho0f 32174	. . . . 5 ⊢ 0hop : ℋ⟶ ℋ
90		ffn 6709	. . . . 5 ⊢ ( 0hop : ℋ⟶ ℋ → 0hop Fn ℋ)
91	89, 90	ax-mp 5	. . . 4 ⊢ 0hop Fn ℋ
92		eqfnfv 7029	. . . 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 6885	. . 3 ⊢ (𝑇 = 0hop → (normop‘𝑇) = (normop‘ 0hop ))
96		nmop0 32409	. . 3 ⊢ (normop‘ 0hop ) = 0
97	95, 96	eqtrdi 2816	. 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 2146  {cab 2743   ≠ wne 2960  ∀wral 3081  ∃wrex 3091   ⊆ wss 3906   class class class wbr 5111   Fn wfn 6535  ⟶wf 6536  ‘cfv 6540  (class class class)co 7419  supcsup 9407  ℂcc 11113  ℝcr 11114  0cc0 11115  1c1 11116  ℝ^*cxr 11257   < clt 11258   ≤ cle 11259   / cdiv 11886   ℋchba 31342   ·_ℎ csm 31344  norm_ℎcno 31346  0_ℎc0v 31347   0_hop ch0o 31366  norm_opcnop 31368  LinOpclo 31370

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 2148  ax-9 2156  ax-10 2179  ax-11 2195  ax-12 2216  ax-ext 2737  ax-rep 5240  ax-sep 5259  ax-nul 5271  ax-pow 5338  ax-pr 5406  ax-un 7742  ax-inf2 9617  ax-cc 10434  ax-cnex 11171  ax-resscn 11172  ax-1cn 11173  ax-icn 11174  ax-addcl 11175  ax-addrcl 11176  ax-mulcl 11177  ax-mulrcl 11178  ax-mulcom 11179  ax-addass 11180  ax-mulass 11181  ax-distr 11182  ax-i2m1 11183  ax-1ne0 11184  ax-1rid 11185  ax-rnegex 11186  ax-rrecex 11187  ax-cnre 11188  ax-pre-lttri 11189  ax-pre-lttrn 11190  ax-pre-ltadd 11191  ax-pre-mulgt0 11192  ax-pre-sup 11193  ax-addf 11194  ax-mulf 11195  ax-hilex 31422  ax-hfvadd 31423  ax-hvcom 31424  ax-hvass 31425  ax-hv0cl 31426  ax-hvaddid 31427  ax-hfvmul 31428  ax-hvmulid 31429  ax-hvmulass 31430  ax-hvdistr1 31431  ax-hvdistr2 31432  ax-hvmul0 31433  ax-hfi 31502  ax-his1 31505  ax-his2 31506  ax-his3 31507  ax-his4 31508  ax-hcompl 31625

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 2569  df-eu 2599  df-clab 2744  df-cleq 2757  df-clel 2840  df-nfc 2914  df-ne 2961  df-nel 3067  df-ral 3082  df-rex 3092  df-rmo 3371  df-reu 3372  df-rab 3419  df-v 3459  df-sbc 3747  df-csb 3855  df-dif 3909  df-un 3911  df-in 3913  df-ss 3923  df-pss 3926  df-nul 4287  df-if 4490  df-pw 4566  df-sn 4592  df-pr 4594  df-tp 4596  df-op 4598  df-uni 4875  df-int 4915  df-iun 4960  df-iin 4961  df-br 5112  df-opab 5176  df-mpt 5195  df-tr 5221  df-id 5558  df-eprel 5563  df-po 5571  df-so 5572  df-fr 5616  df-se 5617  df-we 5618  df-xp 5669  df-rel 5670  df-cnv 5671  df-co 5672  df-dm 5673  df-rn 5674  df-res 5675  df-ima 5676  df-pred 6306  df-ord 6367  df-on 6368  df-lim 6369  df-suc 6370  df-iota 6496  df-fun 6542  df-fn 6543  df-f 6544  df-f1 6545  df-fo 6546  df-f1o 6547  df-fv 6548  df-isom 6549  df-riota 7376  df-ov 7422  df-oprab 7423  df-mpo 7424  df-of 7684  df-om 7869  df-1st 7992  df-2nd 7993  df-supp 8163  df-frecs 8284  df-wrecs 8315  df-recs 8364  df-rdg 8403  df-1o 8459  df-2o 8460  df-oadd 8463  df-omul 8464  df-er 8700  df-map 8832  df-pm 8833  df-ixp 8902  df-en 8950  df-dom 8951  df-sdom 8952  df-fin 8953  df-fsupp 9329  df-fi 9378  df-sup 9409  df-inf 9410  df-oi 9479  df-card 9941  df-acn 9944  df-pnf 11260  df-mnf 11261  df-xr 11262  df-ltxr 11263  df-le 11264  df-sub 11458  df-neg 11459  df-div 11887  df-nn 12249  df-2 12318  df-3 12319  df-4 12320  df-5 12321  df-6 12322  df-7 12323  df-8 12324  df-9 12325  df-n0 12520  df-z 12607  df-dec 12728  df-uz 12879  df-q 12989  df-rp 13033  df-xneg 13153  df-xadd 13154  df-xmul 13155  df-ioo 13392  df-ico 13394  df-icc 13395  df-fz 13552  df-fzo 13700  df-fl 13843  df-seq 14056  df-exp 14116  df-hash 14385  df-cj 15174  df-re 15175  df-im 15176  df-sqrt 15310  df-abs 15311  df-clim 15563  df-rlim 15564  df-sum 15762  df-struct 17229  df-sets 17246  df-slot 17264  df-ndx 17276  df-base 17292  df-ress 17313  df-plusg 17345  df-mulr 17346  df-starv 17347  df-sca 17348  df-vsca 17349  df-ip 17350  df-tset 17351  df-ple 17352  df-ds 17354  df-unif 17355  df-hom 17356  df-cco 17357  df-rest 17497  df-topn 17498  df-0g 17516  df-gsum 17517  df-topgen 17518  df-pt 17519  df-prds 17522  df-xrs 17578  df-qtop 17583  df-imas 17584  df-xps 17586  df-mre 17660  df-mrc 17661  df-acs 17663  df-mgm 18720  df-sgrp 18809  df-mnd 18825  df-submnd 18879  df-mulg 19178  df-cntz 19431  df-cmn 19896  df-psmet 21564  df-xmet 21565  df-met 21566  df-bl 21567  df-mopn 21568  df-fbas 21569  df-fg 21570  df-cnfld 21573  df-top 23101  df-topon 23118  df-topsp 23140  df-bases 23153  df-cld 23226  df-ntr 23227  df-cls 23228  df-nei 23305  df-cn 23434  df-cnp 23435  df-lm 23436  df-haus 23522  df-tx 23770  df-hmeo 23963  df-fil 24054  df-fm 24146  df-flim 24147  df-flf 24148  df-xms 24528  df-ms 24529  df-tms 24530  df-cfil 25465  df-cau 25466  df-cmet 25467  df-grpo 30916  df-gid 30917  df-ginv 30918  df-gdiv 30919  df-ablo 30968  df-vc 30982  df-nv 31015  df-va 31018  df-ba 31019  df-sm 31020  df-0v 31021  df-vs 31022  df-nmcv 31023  df-ims 31024  df-dip 31124  df-ssp 31145  df-ph 31236  df-cbn 31286  df-hnorm 31391  df-hba 31392  df-hvsub 31394  df-hlim 31395  df-hcau 31396  df-sh 31630  df-ch 31644  df-oc 31675  df-ch0 31676  df-shs 31731  df-pjh 31818  df-h0op 32171  df-nmop 32262  df-lnop 32264

This theorem is used by: (None)