Theorem nmlnop0iALT 30258

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 29388	. . . . . . . . . . . . . 14 ⊢ (𝑥 ∈ ℋ → (normℎ‘𝑥) ∈ ℝ)
2	1	recnd 10934	. . . . . . . . . . . . 13 ⊢ (𝑥 ∈ ℋ → (normℎ‘𝑥) ∈ ℂ)
3	2	adantr 480	. . . . . . . . . . . 12 ⊢ ((𝑥 ∈ ℋ ∧ (𝑇‘𝑥) ≠ 0ℎ) → (normℎ‘𝑥) ∈ ℂ)
4		norm-i 29392	. . . . . . . . . . . . . . 15 ⊢ (𝑥 ∈ ℋ → ((normℎ‘𝑥) = 0 ↔ 𝑥 = 0ℎ))
5		fveq2 6756	. . . . . . . . . . . . . . . 16 ⊢ (𝑥 = 0ℎ → (𝑇‘𝑥) = (𝑇‘0ℎ))
6		nmlnop0.1	. . . . . . . . . . . . . . . . 17 ⊢ 𝑇 ∈ LinOp
7	6	lnop0i 30233	. . . . . . . . . . . . . . . 16 ⊢ (𝑇‘0ℎ) = 0ℎ
8	5, 7	eqtrdi 2795	. . . . . . . . . . . . . . 15 ⊢ (𝑥 = 0ℎ → (𝑇‘𝑥) = 0ℎ)
9	4, 8	syl6bi 252	. . . . . . . . . . . . . 14 ⊢ (𝑥 ∈ ℋ → ((normℎ‘𝑥) = 0 → (𝑇‘𝑥) = 0ℎ))
10	9	necon3d 2963	. . . . . . . . . . . . 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 11675	. . . . . . . . . . 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 11674	. . . . . . . . . . . . . 14 ⊢ ((𝑥 ∈ ℋ ∧ (𝑇‘𝑥) ≠ 0ℎ) → (1 / (normℎ‘𝑥)) ∈ ℂ)
15	6	lnopfi 30232	. . . . . . . . . . . . . . . 16 ⊢ 𝑇: ℋ⟶ ℋ
16	15	ffvelrni 6942	. . . . . . . . . . . . . . 15 ⊢ (𝑥 ∈ ℋ → (𝑇‘𝑥) ∈ ℋ)
17	16	adantr 480	. . . . . . . . . . . . . 14 ⊢ ((𝑥 ∈ ℋ ∧ (𝑇‘𝑥) ≠ 0ℎ) → (𝑇‘𝑥) ∈ ℋ)
18		hvmul0or 29288	. . . . . . . . . . . . . 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 583	. . . . . . . . . . . . 13 ⊢ ((𝑥 ∈ ℋ ∧ (𝑇‘𝑥) ≠ 0ℎ) → (((1 / (normℎ‘𝑥)) ·ℎ (𝑇‘𝑥)) = 0ℎ ↔ ((1 / (normℎ‘𝑥)) = 0 ∨ (𝑇‘𝑥) = 0ℎ)))
20	19	necon3abid 2979	. . . . . . . . . . . 12 ⊢ ((𝑥 ∈ ℋ ∧ (𝑇‘𝑥) ≠ 0ℎ) → (((1 / (normℎ‘𝑥)) ·ℎ (𝑇‘𝑥)) ≠ 0ℎ ↔ ¬ ((1 / (normℎ‘𝑥)) = 0 ∨ (𝑇‘𝑥) = 0ℎ)))
21		neanior 3036	. . . . . . . . . . . 12 ⊢ (((1 / (normℎ‘𝑥)) ≠ 0 ∧ (𝑇‘𝑥) ≠ 0ℎ) ↔ ¬ ((1 / (normℎ‘𝑥)) = 0 ∨ (𝑇‘𝑥) = 0ℎ))
22	20, 21	bitr4di 288	. . . . . . . . . . 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 709	. . . . . . . . . 10 ⊢ ((𝑥 ∈ ℋ ∧ (𝑇‘𝑥) ≠ 0ℎ) → ((1 / (normℎ‘𝑥)) ·ℎ (𝑇‘𝑥)) ≠ 0ℎ)
24		hvmulcl 29276	. . . . . . . . . . . 12 ⊢ (((1 / (normℎ‘𝑥)) ∈ ℂ ∧ (𝑇‘𝑥) ∈ ℋ) → ((1 / (normℎ‘𝑥)) ·ℎ (𝑇‘𝑥)) ∈ ℋ)
25	14, 17, 24	syl2anc 583	. . . . . . . . . . 11 ⊢ ((𝑥 ∈ ℋ ∧ (𝑇‘𝑥) ≠ 0ℎ) → ((1 / (normℎ‘𝑥)) ·ℎ (𝑇‘𝑥)) ∈ ℋ)
26		normgt0 29390	. . . . . . . . . . 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 231	. . . . . . . . 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 30127	. . . . . . . . . . . . . 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 10950	. . . . . . . . . . . . 13 ⊢ ℝ ⊆ ℝ*
34	32, 33	sstri 3926	. . . . . . . . . . . 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 29276	. . . . . . . . . . . . . . 15 ⊢ (((1 / (normℎ‘𝑥)) ∈ ℂ ∧ 𝑥 ∈ ℋ) → ((1 / (normℎ‘𝑥)) ·ℎ 𝑥) ∈ ℋ)
37	14, 35, 36	syl2anc 583	. . . . . . . . . . . . . 14 ⊢ ((𝑥 ∈ ℋ ∧ (𝑇‘𝑥) ≠ 0ℎ) → ((1 / (normℎ‘𝑥)) ·ℎ 𝑥) ∈ ℋ)
38	8	necon3i 2975	. . . . . . . . . . . . . . . . 17 ⊢ ((𝑇‘𝑥) ≠ 0ℎ → 𝑥 ≠ 0ℎ)
39		norm1 29512	. . . . . . . . . . . . . . . . 17 ⊢ ((𝑥 ∈ ℋ ∧ 𝑥 ≠ 0ℎ) → (normℎ‘((1 / (normℎ‘𝑥)) ·ℎ 𝑥)) = 1)
40	38, 39	sylan2 592	. . . . . . . . . . . . . . . 16 ⊢ ((𝑥 ∈ ℋ ∧ (𝑇‘𝑥) ≠ 0ℎ) → (normℎ‘((1 / (normℎ‘𝑥)) ·ℎ 𝑥)) = 1)
41		1re 10906	. . . . . . . . . . . . . . . 16 ⊢ 1 ∈ ℝ
42	40, 41	eqeltrdi 2847	. . . . . . . . . . . . . . 15 ⊢ ((𝑥 ∈ ℋ ∧ (𝑇‘𝑥) ≠ 0ℎ) → (normℎ‘((1 / (normℎ‘𝑥)) ·ℎ 𝑥)) ∈ ℝ)
43		eqle 11007	. . . . . . . . . . . . . . 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 583	. . . . . . . . . . . . . 14 ⊢ ((𝑥 ∈ ℋ ∧ (𝑇‘𝑥) ≠ 0ℎ) → (normℎ‘((1 / (normℎ‘𝑥)) ·ℎ 𝑥)) ≤ 1)
45	6	lnopmuli 30235	. . . . . . . . . . . . . . . . 17 ⊢ (((1 / (normℎ‘𝑥)) ∈ ℂ ∧ 𝑥 ∈ ℋ) → (𝑇‘((1 / (normℎ‘𝑥)) ·ℎ 𝑥)) = ((1 / (normℎ‘𝑥)) ·ℎ (𝑇‘𝑥)))
46	14, 35, 45	syl2anc 583	. . . . . . . . . . . . . . . 16 ⊢ ((𝑥 ∈ ℋ ∧ (𝑇‘𝑥) ≠ 0ℎ) → (𝑇‘((1 / (normℎ‘𝑥)) ·ℎ 𝑥)) = ((1 / (normℎ‘𝑥)) ·ℎ (𝑇‘𝑥)))
47	46	eqcomd 2744	. . . . . . . . . . . . . . 15 ⊢ ((𝑥 ∈ ℋ ∧ (𝑇‘𝑥) ≠ 0ℎ) → ((1 / (normℎ‘𝑥)) ·ℎ (𝑇‘𝑥)) = (𝑇‘((1 / (normℎ‘𝑥)) ·ℎ 𝑥)))
48	47	fveq2d 6760	. . . . . . . . . . . . . 14 ⊢ ((𝑥 ∈ ℋ ∧ (𝑇‘𝑥) ≠ 0ℎ) → (normℎ‘((1 / (normℎ‘𝑥)) ·ℎ (𝑇‘𝑥))) = (normℎ‘(𝑇‘((1 / (normℎ‘𝑥)) ·ℎ 𝑥))))
49		fveq2 6756	. . . . . . . . . . . . . . . . 17 ⊢ (𝑧 = ((1 / (normℎ‘𝑥)) ·ℎ 𝑥) → (normℎ‘𝑧) = (normℎ‘((1 / (normℎ‘𝑥)) ·ℎ 𝑥)))
50	49	breq1d 5080	. . . . . . . . . . . . . . . 16 ⊢ (𝑧 = ((1 / (normℎ‘𝑥)) ·ℎ 𝑥) → ((normℎ‘𝑧) ≤ 1 ↔ (normℎ‘((1 / (normℎ‘𝑥)) ·ℎ 𝑥)) ≤ 1))
51		fveq2 6756	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑧 = ((1 / (normℎ‘𝑥)) ·ℎ 𝑥) → (𝑇‘𝑧) = (𝑇‘((1 / (normℎ‘𝑥)) ·ℎ 𝑥)))
52	51	fveq2d 6760	. . . . . . . . . . . . . . . . 17 ⊢ (𝑧 = ((1 / (normℎ‘𝑥)) ·ℎ 𝑥) → (normℎ‘(𝑇‘𝑧)) = (normℎ‘(𝑇‘((1 / (normℎ‘𝑥)) ·ℎ 𝑥))))
53	52	eqeq2d 2749	. . . . . . . . . . . . . . . 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 630	. . . . . . . . . . . . . . 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 3552	. . . . . . . . . . . . . 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 833	. . . . . . . . . . . . 13 ⊢ ((𝑥 ∈ ℋ ∧ (𝑇‘𝑥) ≠ 0ℎ) → ∃𝑧 ∈ ℋ ((normℎ‘𝑧) ≤ 1 ∧ (normℎ‘((1 / (normℎ‘𝑥)) ·ℎ (𝑇‘𝑥))) = (normℎ‘(𝑇‘𝑧))))
57		fvex 6769	. . . . . . . . . . . . . 14 ⊢ (normℎ‘((1 / (normℎ‘𝑥)) ·ℎ (𝑇‘𝑥))) ∈ V
58		eqeq1 2742	. . . . . . . . . . . . . . . 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 628	. . . . . . . . . . . . . . 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 3225	. . . . . . . . . . . . . 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 3602	. . . . . . . . . . . . 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 233	. . . . . . . . . . . 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 12987	. . . . . . . . . . . 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 586	. . . . . . . . . . 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 710	. . . . . . . . . 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 30119	. . . . . . . . . . . . . 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 2743	. . . . . . . . . . . 12 ⊢ ((normop‘𝑇) = 0 ↔ sup({𝑦 ∣ ∃𝑧 ∈ ℋ ((normℎ‘𝑧) ≤ 1 ∧ 𝑦 = (normℎ‘(𝑇‘𝑧)))}, ℝ*, < ) = 0)
69	68	biimpi 215	. . . . . . . . . . 11 ⊢ ((normop‘𝑇) = 0 → sup({𝑦 ∣ ∃𝑧 ∈ ℋ ((normℎ‘𝑧) ≤ 1 ∧ 𝑦 = (normℎ‘(𝑇‘𝑧)))}, ℝ*, < ) = 0)
70	69	ad2antrr 722	. . . . . . . . . 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 5096	. . . . . . . . 9 ⊢ ((((normop‘𝑇) = 0 ∧ 𝑥 ∈ ℋ) ∧ (𝑇‘𝑥) ≠ 0ℎ) → (normℎ‘((1 / (normℎ‘𝑥)) ·ℎ (𝑇‘𝑥))) ≤ 0)
72		normcl 29388	. . . . . . . . . . . 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 10908	. . . . . . . . . . 11 ⊢ 0 ∈ ℝ
75		lenlt 10984	. . . . . . . . . . 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 585	. . . . . . . . . 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 710	. . . . . . . . 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 231	. . . . . . . 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 195	. . . . . 6 ⊢ (((normop‘𝑇) = 0 ∧ 𝑥 ∈ ℋ) → ¬ (𝑇‘𝑥) ≠ 0ℎ)
81		nne 2946	. . . . . 6 ⊢ (¬ (𝑇‘𝑥) ≠ 0ℎ ↔ (𝑇‘𝑥) = 0ℎ)
82	80, 81	sylib 217	. . . . 5 ⊢ (((normop‘𝑇) = 0 ∧ 𝑥 ∈ ℋ) → (𝑇‘𝑥) = 0ℎ)
83		ho0val 30013	. . . . . 6 ⊢ (𝑥 ∈ ℋ → ( 0hop ‘𝑥) = 0ℎ)
84	83	adantl 481	. . . . 5 ⊢ (((normop‘𝑇) = 0 ∧ 𝑥 ∈ ℋ) → ( 0hop ‘𝑥) = 0ℎ)
85	82, 84	eqtr4d 2781	. . . 4 ⊢ (((normop‘𝑇) = 0 ∧ 𝑥 ∈ ℋ) → (𝑇‘𝑥) = ( 0hop ‘𝑥))
86	85	ralrimiva 3107	. . 3 ⊢ ((normop‘𝑇) = 0 → ∀𝑥 ∈ ℋ (𝑇‘𝑥) = ( 0hop ‘𝑥))
87		ffn 6584	. . . . 5 ⊢ (𝑇: ℋ⟶ ℋ → 𝑇 Fn ℋ)
88	15, 87	ax-mp 5	. . . 4 ⊢ 𝑇 Fn ℋ
89		ho0f 30014	. . . . 5 ⊢ 0hop : ℋ⟶ ℋ
90		ffn 6584	. . . . 5 ⊢ ( 0hop : ℋ⟶ ℋ → 0hop Fn ℋ)
91	89, 90	ax-mp 5	. . . 4 ⊢ 0hop Fn ℋ
92		eqfnfv 6891	. . . 4 ⊢ ((𝑇 Fn ℋ ∧ 0hop Fn ℋ) → (𝑇 = 0hop ↔ ∀𝑥 ∈ ℋ (𝑇‘𝑥) = ( 0hop ‘𝑥)))
93	88, 91, 92	mp2an 688	. . 3 ⊢ (𝑇 = 0hop ↔ ∀𝑥 ∈ ℋ (𝑇‘𝑥) = ( 0hop ‘𝑥))
94	86, 93	sylibr 233	. 2 ⊢ ((normop‘𝑇) = 0 → 𝑇 = 0hop )
95		fveq2 6756	. . 3 ⊢ (𝑇 = 0hop → (normop‘𝑇) = (normop‘ 0hop ))
96		nmop0 30249	. . 3 ⊢ (normop‘ 0hop ) = 0
97	95, 96	eqtrdi 2795	. 2 ⊢ (𝑇 = 0hop → (normop‘𝑇) = 0)
98	94, 97	impbii 208	1 ⊢ ((normop‘𝑇) = 0 ↔ 𝑇 = 0hop )

Syntax hints:  ¬ wn 3   → wi 4   ↔ wb 205   ∧ wa 395   ∨ wo 843   = wceq 1539   ∈ wcel 2108  {cab 2715   ≠ wne 2942  ∀wral 3063  ∃wrex 3064   ⊆ wss 3883   class class class wbr 5070   Fn wfn 6413  ⟶wf 6414  ‘cfv 6418  (class class class)co 7255  supcsup 9129  ℂcc 10800  ℝcr 10801  0cc0 10802  1c1 10803  ℝ^*cxr 10939   < clt 10940   ≤ cle 10941   / cdiv 11562   ℋchba 29182   ·_ℎ csm 29184  norm_ℎcno 29186  0_ℎc0v 29187   0_hop ch0o 29206  norm_opcnop 29208  LinOpclo 29210

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1799  ax-4 1813  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2110  ax-9 2118  ax-10 2139  ax-11 2156  ax-12 2173  ax-ext 2709  ax-rep 5205  ax-sep 5218  ax-nul 5225  ax-pow 5283  ax-pr 5347  ax-un 7566  ax-inf2 9329  ax-cc 10122  ax-cnex 10858  ax-resscn 10859  ax-1cn 10860  ax-icn 10861  ax-addcl 10862  ax-addrcl 10863  ax-mulcl 10864  ax-mulrcl 10865  ax-mulcom 10866  ax-addass 10867  ax-mulass 10868  ax-distr 10869  ax-i2m1 10870  ax-1ne0 10871  ax-1rid 10872  ax-rnegex 10873  ax-rrecex 10874  ax-cnre 10875  ax-pre-lttri 10876  ax-pre-lttrn 10877  ax-pre-ltadd 10878  ax-pre-mulgt0 10879  ax-pre-sup 10880  ax-addf 10881  ax-mulf 10882  ax-hilex 29262  ax-hfvadd 29263  ax-hvcom 29264  ax-hvass 29265  ax-hv0cl 29266  ax-hvaddid 29267  ax-hfvmul 29268  ax-hvmulid 29269  ax-hvmulass 29270  ax-hvdistr1 29271  ax-hvdistr2 29272  ax-hvmul0 29273  ax-hfi 29342  ax-his1 29345  ax-his2 29346  ax-his3 29347  ax-his4 29348  ax-hcompl 29465

This theorem depends on definitions:  df-bi 206  df-an 396  df-or 844  df-3or 1086  df-3an 1087  df-tru 1542  df-fal 1552  df-ex 1784  df-nf 1788  df-sb 2069  df-mo 2540  df-eu 2569  df-clab 2716  df-cleq 2730  df-clel 2817  df-nfc 2888  df-ne 2943  df-nel 3049  df-ral 3068  df-rex 3069  df-reu 3070  df-rmo 3071  df-rab 3072  df-v 3424  df-sbc 3712  df-csb 3829  df-dif 3886  df-un 3888  df-in 3890  df-ss 3900  df-pss 3902  df-nul 4254  df-if 4457  df-pw 4532  df-sn 4559  df-pr 4561  df-tp 4563  df-op 4565  df-uni 4837  df-int 4877  df-iun 4923  df-iin 4924  df-br 5071  df-opab 5133  df-mpt 5154  df-tr 5188  df-id 5480  df-eprel 5486  df-po 5494  df-so 5495  df-fr 5535  df-se 5536  df-we 5537  df-xp 5586  df-rel 5587  df-cnv 5588  df-co 5589  df-dm 5590  df-rn 5591  df-res 5592  df-ima 5593  df-pred 6191  df-ord 6254  df-on 6255  df-lim 6256  df-suc 6257  df-iota 6376  df-fun 6420  df-fn 6421  df-f 6422  df-f1 6423  df-fo 6424  df-f1o 6425  df-fv 6426  df-isom 6427  df-riota 7212  df-ov 7258  df-oprab 7259  df-mpo 7260  df-of 7511  df-om 7688  df-1st 7804  df-2nd 7805  df-supp 7949  df-frecs 8068  df-wrecs 8099  df-recs 8173  df-rdg 8212  df-1o 8267  df-2o 8268  df-oadd 8271  df-omul 8272  df-er 8456  df-map 8575  df-pm 8576  df-ixp 8644  df-en 8692  df-dom 8693  df-sdom 8694  df-fin 8695  df-fsupp 9059  df-fi 9100  df-sup 9131  df-inf 9132  df-oi 9199  df-card 9628  df-acn 9631  df-pnf 10942  df-mnf 10943  df-xr 10944  df-ltxr 10945  df-le 10946  df-sub 11137  df-neg 11138  df-div 11563  df-nn 11904  df-2 11966  df-3 11967  df-4 11968  df-5 11969  df-6 11970  df-7 11971  df-8 11972  df-9 11973  df-n0 12164  df-z 12250  df-dec 12367  df-uz 12512  df-q 12618  df-rp 12660  df-xneg 12777  df-xadd 12778  df-xmul 12779  df-ioo 13012  df-ico 13014  df-icc 13015  df-fz 13169  df-fzo 13312  df-fl 13440  df-seq 13650  df-exp 13711  df-hash 13973  df-cj 14738  df-re 14739  df-im 14740  df-sqrt 14874  df-abs 14875  df-clim 15125  df-rlim 15126  df-sum 15326  df-struct 16776  df-sets 16793  df-slot 16811  df-ndx 16823  df-base 16841  df-ress 16868  df-plusg 16901  df-mulr 16902  df-starv 16903  df-sca 16904  df-vsca 16905  df-ip 16906  df-tset 16907  df-ple 16908  df-ds 16910  df-unif 16911  df-hom 16912  df-cco 16913  df-rest 17050  df-topn 17051  df-0g 17069  df-gsum 17070  df-topgen 17071  df-pt 17072  df-prds 17075  df-xrs 17130  df-qtop 17135  df-imas 17136  df-xps 17138  df-mre 17212  df-mrc 17213  df-acs 17215  df-mgm 18241  df-sgrp 18290  df-mnd 18301  df-submnd 18346  df-mulg 18616  df-cntz 18838  df-cmn 19303  df-psmet 20502  df-xmet 20503  df-met 20504  df-bl 20505  df-mopn 20506  df-fbas 20507  df-fg 20508  df-cnfld 20511  df-top 21951  df-topon 21968  df-topsp 21990  df-bases 22004  df-cld 22078  df-ntr 22079  df-cls 22080  df-nei 22157  df-cn 22286  df-cnp 22287  df-lm 22288  df-haus 22374  df-tx 22621  df-hmeo 22814  df-fil 22905  df-fm 22997  df-flim 22998  df-flf 22999  df-xms 23381  df-ms 23382  df-tms 23383  df-cfil 24324  df-cau 24325  df-cmet 24326  df-grpo 28756  df-gid 28757  df-ginv 28758  df-gdiv 28759  df-ablo 28808  df-vc 28822  df-nv 28855  df-va 28858  df-ba 28859  df-sm 28860  df-0v 28861  df-vs 28862  df-nmcv 28863  df-ims 28864  df-dip 28964  df-ssp 28985  df-ph 29076  df-cbn 29126  df-hnorm 29231  df-hba 29232  df-hvsub 29234  df-hlim 29235  df-hcau 29236  df-sh 29470  df-ch 29484  df-oc 29515  df-ch0 29516  df-shs 29571  df-pjh 29658  df-h0op 30011  df-nmop 30102  df-lnop 30104

This theorem is referenced by: (None)