Theorem nmlnop0iALT 32604

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 31734	. . . . . . . . . . . . . 14 ⊢ (𝑥 ∈ ℋ → (normℎ‘𝑥) ∈ ℝ)
2	1	recnd 11342	. . . . . . . . . . . . 13 ⊢ (𝑥 ∈ ℋ → (normℎ‘𝑥) ∈ ℂ)
3	2	adantr 486	. . . . . . . . . . . 12 ⊢ ((𝑥 ∈ ℋ ∧ (𝑇‘𝑥) ≠ 0ℎ) → (normℎ‘𝑥) ∈ ℂ)
4		norm-i 31738	. . . . . . . . . . . . . . 15 ⊢ (𝑥 ∈ ℋ → ((normℎ‘𝑥) = 0 ↔ 𝑥 = 0ℎ))
5		fveq2 6888	. . . . . . . . . . . . . . . 16 ⊢ (𝑥 = 0ℎ → (𝑇‘𝑥) = (𝑇‘0ℎ))
6		nmlnop0.1	. . . . . . . . . . . . . . . . 17 ⊢ 𝑇 ∈ LinOp
7	6	lnop0i 32579	. . . . . . . . . . . . . . . 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 12092	. . . . . . . . . . 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 12091	. . . . . . . . . . . . . 14 ⊢ ((𝑥 ∈ ℋ ∧ (𝑇‘𝑥) ≠ 0ℎ) → (1 / (normℎ‘𝑥)) ∈ ℂ)
15	6	lnopfi 32578	. . . . . . . . . . . . . . . 16 ⊢ 𝑇: ℋ⟶ ℋ
16	15	ffvelcdmi 7086	. . . . . . . . . . . . . . 15 ⊢ (𝑥 ∈ ℋ → (𝑇‘𝑥) ∈ ℋ)
17	16	adantr 486	. . . . . . . . . . . . . 14 ⊢ ((𝑥 ∈ ℋ ∧ (𝑇‘𝑥) ≠ 0ℎ) → (𝑇‘𝑥) ∈ ℋ)
18		hvmul0or 31634	. . . . . . . . . . . . . 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 31622	. . . . . . . . . . . 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 31736	. . . . . . . . . . 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 32473	. . . . . . . . . . . . . 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 11358	. . . . . . . . . . . . 13 ⊢ ℝ ⊆ ℝ*
34	32, 33	sstri 3918	. . . . . . . . . . . 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 31622	. . . . . . . . . . . . . . 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 31858	. . . . . . . . . . . . . . . . 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 11313	. . . . . . . . . . . . . . . 16 ⊢ 1 ∈ ℝ
42	40, 41	eqeltrdi 2868	. . . . . . . . . . . . . . 15 ⊢ ((𝑥 ∈ ℋ ∧ (𝑇‘𝑥) ≠ 0ℎ) → (normℎ‘((1 / (normℎ‘𝑥)) ·ℎ 𝑥)) ∈ ℝ)
43		eqle 11417	. . . . . . . . . . . . . . 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 32581	. . . . . . . . . . . . . . . . 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 6892	. . . . . . . . . . . . . 14 ⊢ ((𝑥 ∈ ℋ ∧ (𝑇‘𝑥) ≠ 0ℎ) → (normℎ‘((1 / (normℎ‘𝑥)) ·ℎ (𝑇‘𝑥))) = (normℎ‘(𝑇‘((1 / (normℎ‘𝑥)) ·ℎ 𝑥))))
49		fveq2 6888	. . . . . . . . . . . . . . . . 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 6888	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑧 = ((1 / (normℎ‘𝑥)) ·ℎ 𝑥) → (𝑇‘𝑧) = (𝑇‘((1 / (normℎ‘𝑥)) ·ℎ 𝑥)))
52	51	fveq2d 6892	. . . . . . . . . . . . . . . . 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 6901	. . . . . . . . . . . . . 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 3632	. . . . . . . . . . . . 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 13459	. . . . . . . . . . . 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 32465	. . . . . . . . . . . . . 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 5130	. . . . . . . . 9 ⊢ ((((normop‘𝑇) = 0 ∧ 𝑥 ∈ ℋ) ∧ (𝑇‘𝑥) ≠ 0ℎ) → (normℎ‘((1 / (normℎ‘𝑥)) ·ℎ (𝑇‘𝑥))) ≤ 0)
72		normcl 31734	. . . . . . . . . . . 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 11315	. . . . . . . . . . 11 ⊢ 0 ∈ ℝ
75		lenlt 11393	. . . . . . . . . . 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 32359	. . . . . 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 6712	. . . . 5 ⊢ (𝑇: ℋ⟶ ℋ → 𝑇 Fn ℋ)
88	15, 87	ax-mp 5	. . . 4 ⊢ 𝑇 Fn ℋ
89		ho0f 32360	. . . . 5 ⊢ 0hop : ℋ⟶ ℋ
90		ffn 6712	. . . . 5 ⊢ ( 0hop : ℋ⟶ ℋ → 0hop Fn ℋ)
91	89, 90	ax-mp 5	. . . 4 ⊢ 0hop Fn ℋ
92		eqfnfv 7032	. . . 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 6888	. . 3 ⊢ (𝑇 = 0hop → (normop‘𝑇) = (normop‘ 0hop ))
96		nmop0 32595	. . 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 1569   ∈ wcel 2144  {cab 2738   ≠ wne 2955  ∀wral 3076  ∃wrex 3086   ⊆ wss 3896   class class class wbr 5102   Fn wfn 6536  ⟶wf 6537  ‘cfv 6541  (class class class)co 7422  supcsup 9437  ℂcc 11203  ℝcr 11204  0cc0 11205  1c1 11206  ℝ^*cxr 11347   < clt 11348   ≤ cle 11349   / cdiv 11978   ℋchba 31528   ·_ℎ csm 31530  norm_ℎcno 31532  0_ℎc0v 31533   0_hop ch0o 31552  norm_opcnop 31554  LinOpclo 31556

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1827  ax-4 1841  ax-5 1942  ax-6 1999  ax-7 2040  ax-8 2146  ax-9 2154  ax-10 2177  ax-11 2193  ax-12 2212  ax-ext 2732  ax-rep 5231  ax-sep 5248  ax-nul 5259  ax-pow 5326  ax-pr 5390  ax-un 7753  ax-inf2 9647  ax-cc 10518  ax-cnex 11261  ax-resscn 11262  ax-1cn 11263  ax-icn 11264  ax-addcl 11265  ax-addrcl 11266  ax-mulcl 11267  ax-mulrcl 11268  ax-mulcom 11269  ax-addass 11270  ax-mulass 11271  ax-distr 11272  ax-i2m1 11273  ax-1ne0 11274  ax-1rid 11275  ax-rnegex 11276  ax-rrecex 11277  ax-cnre 11278  ax-pre-lttri 11279  ax-pre-lttrn 11280  ax-pre-ltadd 11281  ax-pre-mulgt0 11282  ax-pre-sup 11283  ax-addf 11284  ax-mulf 11285  ax-hilex 31608  ax-hfvadd 31609  ax-hvcom 31610  ax-hvass 31611  ax-hv0cl 31612  ax-hvaddid 31613  ax-hfvmul 31614  ax-hvmulid 31615  ax-hvmulass 31616  ax-hvdistr1 31617  ax-hvdistr2 31618  ax-hvmul0 31619  ax-hfi 31688  ax-his1 31691  ax-his2 31692  ax-his3 31693  ax-his4 31694  ax-hcompl 31811

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3or 1104  df-3an 1105  df-tru 1572  df-fal 1582  df-ex 1812  df-nf 1816  df-sb 2099  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 3739  df-csb 3848  df-ss 3897  df-pss 4013  df-dif 4046  df-in 4090  df-un 4150  df-nul 4278  df-if 4482  df-pw 4558  df-sn 4585  df-pr 4587  df-tp 4589  df-op 4835  df-uni 4867  df-int 4907  df-iun 4952  df-iin 4953  df-br 5103  df-opab 5167  df-mpt 5186  df-tr 5212  df-id 5545  df-eprel 5550  df-po 5558  df-so 5559  df-fr 5603  df-se 5604  df-we 5605  df-xp 5656  df-rel 5657  df-cnv 5658  df-co 5659  df-dm 5660  df-rn 5661  df-res 5662  df-ima 5663  df-pred 6306  df-ord 6367  df-on 6368  df-lim 6369  df-suc 6370  df-iota 6496  df-fun 6543  df-fn 6544  df-f 6545  df-f1 6546  df-fo 6547  df-f1o 6548  df-fv 6549  df-isom 6550  df-riota 7379  df-ov 7425  df-oprab 7426  df-mpo 7427  df-of 7695  df-om 7880  df-1st 8003  df-2nd 8004  df-supp 8182  df-frecs 8304  df-wrecs 8335  df-recs 8384  df-rdg 8423  df-1o 8481  df-2o 8482  df-oadd 8485  df-omul 8486  df-er 8722  df-map 8854  df-pm 8855  df-ixp 8931  df-en 8979  df-dom 8980  df-sdom 8981  df-fin 8982  df-fsupp 9359  df-fi 9408  df-sup 9439  df-inf 9440  df-oi 9509  df-card 10025  df-acn 10028  df-pnf 11350  df-mnf 11351  df-xr 11352  df-ltxr 11353  df-le 11354  df-sub 11548  df-neg 11549  df-div 11979  df-nn 12341  df-2 12410  df-3 12411  df-4 12412  df-5 12413  df-6 12414  df-7 12415  df-8 12416  df-9 12417  df-n0 12612  df-z 12699  df-dec 12820  df-uz 12971  df-q 13081  df-rp 13126  df-xneg 13246  df-xadd 13247  df-xmul 13248  df-ioo 13485  df-ico 13487  df-icc 13488  df-fz 13645  df-fzo 13794  df-fl 13937  df-seq 14150  df-exp 14210  df-hash 14480  df-cj 15273  df-re 15274  df-im 15275  df-sqrt 15409  df-abs 15410  df-clim 15662  df-rlim 15663  df-sum 15861  df-struct 17332  df-sets 17349  df-slot 17367  df-ndx 17379  df-base 17395  df-ress 17416  df-plusg 17448  df-mulr 17449  df-starv 17450  df-sca 17451  df-vsca 17452  df-ip 17453  df-tset 17454  df-ple 17455  df-ds 17457  df-unif 17458  df-hom 17459  df-cco 17460  df-rest 17600  df-topn 17601  df-0g 17619  df-gsum 17620  df-topgen 17621  df-pt 17622  df-prds 17625  df-xrs 17681  df-qtop 17686  df-imas 17687  df-xps 17689  df-mre 17763  df-mrc 17764  df-acs 17766  df-mgm 18823  df-sgrp 18915  df-mnd 18931  df-submnd 18986  df-mulg 19285  df-cntz 19538  df-cmn 20003  df-psmet 21677  df-xmet 21678  df-met 21679  df-bl 21680  df-mopn 21681  df-fbas 21682  df-fg 21683  df-cnfld 21686  df-top 23219  df-topon 23236  df-topsp 23258  df-bases 23271  df-cld 23344  df-ntr 23345  df-cls 23346  df-nei 23423  df-cn 23552  df-cnp 23553  df-lm 23554  df-haus 23640  df-tx 23888  df-hmeo 24081  df-fil 24172  df-fm 24264  df-flim 24265  df-flf 24266  df-xms 24646  df-ms 24647  df-tms 24648  df-cfil 25583  df-cau 25584  df-cmet 25585  df-grpo 31102  df-gid 31103  df-ginv 31104  df-gdiv 31105  df-ablo 31154  df-vc 31168  df-nv 31201  df-va 31204  df-ba 31205  df-sm 31206  df-0v 31207  df-vs 31208  df-nmcv 31209  df-ims 31210  df-dip 31310  df-ssp 31331  df-ph 31422  df-cbn 31472  df-hnorm 31577  df-hba 31578  df-hvsub 31580  df-hlim 31581  df-hcau 31582  df-sh 31816  df-ch 31830  df-oc 31861  df-ch0 31862  df-shs 31917  df-pjh 32004  df-h0op 32357  df-nmop 32448  df-lnop 32450

This theorem is used by: (None)