Theorem htth 28698

Description: Hellinger-Toeplitz Theorem: any self-adjoint linear operator defined on all of Hilbert space is bounded. Theorem 10.1-1 of [Kreyszig] p. 525. Discovered by E. Hellinger and O. Toeplitz in 1910, "it aroused both admiration and puzzlement since the theorem establishes a relation between properties of two different kinds, namely, the properties of being defined everywhere and being bounded." (Contributed by NM, 11-Jan-2008.) (Revised by Mario Carneiro, 23-Aug-2014.) (New usage is discouraged.)

Hypotheses

Ref	Expression
htth.1	⊢ 𝑋 = (BaseSet‘𝑈)
htth.2	⊢ 𝑃 = (·𝑖OLD‘𝑈)
htth.3	⊢ 𝐿 = (𝑈 LnOp 𝑈)
htth.4	⊢ 𝐵 = (𝑈 BLnOp 𝑈)

Assertion

Ref	Expression
htth	⊢ ((𝑈 ∈ CHilOLD ∧ 𝑇 ∈ 𝐿 ∧ ∀𝑥 ∈ 𝑋 ∀𝑦 ∈ 𝑋 (𝑥𝑃(𝑇‘𝑦)) = ((𝑇‘𝑥)𝑃𝑦)) → 𝑇 ∈ 𝐵)

Distinct variable groups:   𝑥,𝑦,𝑇   𝑥,𝑈,𝑦   𝑥,𝑋,𝑦

Allowed substitution hints:   𝐵(𝑥,𝑦)   𝑃(𝑥,𝑦)   𝐿(𝑥,𝑦)

Proof of Theorem htth

Dummy variables 𝑤 𝑧 𝑣 𝑢 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		htth.3	. . . . . . 7 ⊢ 𝐿 = (𝑈 LnOp 𝑈)
2		oveq12 7168	. . . . . . . 8 ⊢ ((𝑈 = if(𝑈 ∈ CHilOLD, 𝑈, ⟨⟨ + , · ⟩, abs⟩) ∧ 𝑈 = if(𝑈 ∈ CHilOLD, 𝑈, ⟨⟨ + , · ⟩, abs⟩)) → (𝑈 LnOp 𝑈) = (if(𝑈 ∈ CHilOLD, 𝑈, ⟨⟨ + , · ⟩, abs⟩) LnOp if(𝑈 ∈ CHilOLD, 𝑈, ⟨⟨ + , · ⟩, abs⟩)))
3	2	anidms 569	. . . . . . 7 ⊢ (𝑈 = if(𝑈 ∈ CHilOLD, 𝑈, ⟨⟨ + , · ⟩, abs⟩) → (𝑈 LnOp 𝑈) = (if(𝑈 ∈ CHilOLD, 𝑈, ⟨⟨ + , · ⟩, abs⟩) LnOp if(𝑈 ∈ CHilOLD, 𝑈, ⟨⟨ + , · ⟩, abs⟩)))
4	1, 3	syl5eq 2871	. . . . . 6 ⊢ (𝑈 = if(𝑈 ∈ CHilOLD, 𝑈, ⟨⟨ + , · ⟩, abs⟩) → 𝐿 = (if(𝑈 ∈ CHilOLD, 𝑈, ⟨⟨ + , · ⟩, abs⟩) LnOp if(𝑈 ∈ CHilOLD, 𝑈, ⟨⟨ + , · ⟩, abs⟩)))
5	4	eleq2d 2901	. . . . 5 ⊢ (𝑈 = if(𝑈 ∈ CHilOLD, 𝑈, ⟨⟨ + , · ⟩, abs⟩) → (𝑇 ∈ 𝐿 ↔ 𝑇 ∈ (if(𝑈 ∈ CHilOLD, 𝑈, ⟨⟨ + , · ⟩, abs⟩) LnOp if(𝑈 ∈ CHilOLD, 𝑈, ⟨⟨ + , · ⟩, abs⟩))))
6		htth.1	. . . . . . 7 ⊢ 𝑋 = (BaseSet‘𝑈)
7		fveq2 6673	. . . . . . 7 ⊢ (𝑈 = if(𝑈 ∈ CHilOLD, 𝑈, ⟨⟨ + , · ⟩, abs⟩) → (BaseSet‘𝑈) = (BaseSet‘if(𝑈 ∈ CHilOLD, 𝑈, ⟨⟨ + , · ⟩, abs⟩)))
8	6, 7	syl5eq 2871	. . . . . 6 ⊢ (𝑈 = if(𝑈 ∈ CHilOLD, 𝑈, ⟨⟨ + , · ⟩, abs⟩) → 𝑋 = (BaseSet‘if(𝑈 ∈ CHilOLD, 𝑈, ⟨⟨ + , · ⟩, abs⟩)))
9		htth.2	. . . . . . . . . 10 ⊢ 𝑃 = (·𝑖OLD‘𝑈)
10		fveq2 6673	. . . . . . . . . 10 ⊢ (𝑈 = if(𝑈 ∈ CHilOLD, 𝑈, ⟨⟨ + , · ⟩, abs⟩) → (·𝑖OLD‘𝑈) = (·𝑖OLD‘if(𝑈 ∈ CHilOLD, 𝑈, ⟨⟨ + , · ⟩, abs⟩)))
11	9, 10	syl5eq 2871	. . . . . . . . 9 ⊢ (𝑈 = if(𝑈 ∈ CHilOLD, 𝑈, ⟨⟨ + , · ⟩, abs⟩) → 𝑃 = (·𝑖OLD‘if(𝑈 ∈ CHilOLD, 𝑈, ⟨⟨ + , · ⟩, abs⟩)))
12	11	oveqd 7176	. . . . . . . 8 ⊢ (𝑈 = if(𝑈 ∈ CHilOLD, 𝑈, ⟨⟨ + , · ⟩, abs⟩) → (𝑥𝑃(𝑇‘𝑦)) = (𝑥(·𝑖OLD‘if(𝑈 ∈ CHilOLD, 𝑈, ⟨⟨ + , · ⟩, abs⟩))(𝑇‘𝑦)))
13	11	oveqd 7176	. . . . . . . 8 ⊢ (𝑈 = if(𝑈 ∈ CHilOLD, 𝑈, ⟨⟨ + , · ⟩, abs⟩) → ((𝑇‘𝑥)𝑃𝑦) = ((𝑇‘𝑥)(·𝑖OLD‘if(𝑈 ∈ CHilOLD, 𝑈, ⟨⟨ + , · ⟩, abs⟩))𝑦))
14	12, 13	eqeq12d 2840	. . . . . . 7 ⊢ (𝑈 = if(𝑈 ∈ CHilOLD, 𝑈, ⟨⟨ + , · ⟩, abs⟩) → ((𝑥𝑃(𝑇‘𝑦)) = ((𝑇‘𝑥)𝑃𝑦) ↔ (𝑥(·𝑖OLD‘if(𝑈 ∈ CHilOLD, 𝑈, ⟨⟨ + , · ⟩, abs⟩))(𝑇‘𝑦)) = ((𝑇‘𝑥)(·𝑖OLD‘if(𝑈 ∈ CHilOLD, 𝑈, ⟨⟨ + , · ⟩, abs⟩))𝑦)))
15	8, 14	raleqbidv 3404	. . . . . 6 ⊢ (𝑈 = if(𝑈 ∈ CHilOLD, 𝑈, ⟨⟨ + , · ⟩, abs⟩) → (∀𝑦 ∈ 𝑋 (𝑥𝑃(𝑇‘𝑦)) = ((𝑇‘𝑥)𝑃𝑦) ↔ ∀𝑦 ∈ (BaseSet‘if(𝑈 ∈ CHilOLD, 𝑈, ⟨⟨ + , · ⟩, abs⟩))(𝑥(·𝑖OLD‘if(𝑈 ∈ CHilOLD, 𝑈, ⟨⟨ + , · ⟩, abs⟩))(𝑇‘𝑦)) = ((𝑇‘𝑥)(·𝑖OLD‘if(𝑈 ∈ CHilOLD, 𝑈, ⟨⟨ + , · ⟩, abs⟩))𝑦)))
16	8, 15	raleqbidv 3404	. . . . 5 ⊢ (𝑈 = if(𝑈 ∈ CHilOLD, 𝑈, ⟨⟨ + , · ⟩, abs⟩) → (∀𝑥 ∈ 𝑋 ∀𝑦 ∈ 𝑋 (𝑥𝑃(𝑇‘𝑦)) = ((𝑇‘𝑥)𝑃𝑦) ↔ ∀𝑥 ∈ (BaseSet‘if(𝑈 ∈ CHilOLD, 𝑈, ⟨⟨ + , · ⟩, abs⟩))∀𝑦 ∈ (BaseSet‘if(𝑈 ∈ CHilOLD, 𝑈, ⟨⟨ + , · ⟩, abs⟩))(𝑥(·𝑖OLD‘if(𝑈 ∈ CHilOLD, 𝑈, ⟨⟨ + , · ⟩, abs⟩))(𝑇‘𝑦)) = ((𝑇‘𝑥)(·𝑖OLD‘if(𝑈 ∈ CHilOLD, 𝑈, ⟨⟨ + , · ⟩, abs⟩))𝑦)))
17	5, 16	anbi12d 632	. . . 4 ⊢ (𝑈 = if(𝑈 ∈ CHilOLD, 𝑈, ⟨⟨ + , · ⟩, abs⟩) → ((𝑇 ∈ 𝐿 ∧ ∀𝑥 ∈ 𝑋 ∀𝑦 ∈ 𝑋 (𝑥𝑃(𝑇‘𝑦)) = ((𝑇‘𝑥)𝑃𝑦)) ↔ (𝑇 ∈ (if(𝑈 ∈ CHilOLD, 𝑈, ⟨⟨ + , · ⟩, abs⟩) LnOp if(𝑈 ∈ CHilOLD, 𝑈, ⟨⟨ + , · ⟩, abs⟩)) ∧ ∀𝑥 ∈ (BaseSet‘if(𝑈 ∈ CHilOLD, 𝑈, ⟨⟨ + , · ⟩, abs⟩))∀𝑦 ∈ (BaseSet‘if(𝑈 ∈ CHilOLD, 𝑈, ⟨⟨ + , · ⟩, abs⟩))(𝑥(·𝑖OLD‘if(𝑈 ∈ CHilOLD, 𝑈, ⟨⟨ + , · ⟩, abs⟩))(𝑇‘𝑦)) = ((𝑇‘𝑥)(·𝑖OLD‘if(𝑈 ∈ CHilOLD, 𝑈, ⟨⟨ + , · ⟩, abs⟩))𝑦))))
18		htth.4	. . . . . 6 ⊢ 𝐵 = (𝑈 BLnOp 𝑈)
19		oveq12 7168	. . . . . . 7 ⊢ ((𝑈 = if(𝑈 ∈ CHilOLD, 𝑈, ⟨⟨ + , · ⟩, abs⟩) ∧ 𝑈 = if(𝑈 ∈ CHilOLD, 𝑈, ⟨⟨ + , · ⟩, abs⟩)) → (𝑈 BLnOp 𝑈) = (if(𝑈 ∈ CHilOLD, 𝑈, ⟨⟨ + , · ⟩, abs⟩) BLnOp if(𝑈 ∈ CHilOLD, 𝑈, ⟨⟨ + , · ⟩, abs⟩)))
20	19	anidms 569	. . . . . 6 ⊢ (𝑈 = if(𝑈 ∈ CHilOLD, 𝑈, ⟨⟨ + , · ⟩, abs⟩) → (𝑈 BLnOp 𝑈) = (if(𝑈 ∈ CHilOLD, 𝑈, ⟨⟨ + , · ⟩, abs⟩) BLnOp if(𝑈 ∈ CHilOLD, 𝑈, ⟨⟨ + , · ⟩, abs⟩)))
21	18, 20	syl5eq 2871	. . . . 5 ⊢ (𝑈 = if(𝑈 ∈ CHilOLD, 𝑈, ⟨⟨ + , · ⟩, abs⟩) → 𝐵 = (if(𝑈 ∈ CHilOLD, 𝑈, ⟨⟨ + , · ⟩, abs⟩) BLnOp if(𝑈 ∈ CHilOLD, 𝑈, ⟨⟨ + , · ⟩, abs⟩)))
22	21	eleq2d 2901	. . . 4 ⊢ (𝑈 = if(𝑈 ∈ CHilOLD, 𝑈, ⟨⟨ + , · ⟩, abs⟩) → (𝑇 ∈ 𝐵 ↔ 𝑇 ∈ (if(𝑈 ∈ CHilOLD, 𝑈, ⟨⟨ + , · ⟩, abs⟩) BLnOp if(𝑈 ∈ CHilOLD, 𝑈, ⟨⟨ + , · ⟩, abs⟩))))
23	17, 22	imbi12d 347	. . 3 ⊢ (𝑈 = if(𝑈 ∈ CHilOLD, 𝑈, ⟨⟨ + , · ⟩, abs⟩) → (((𝑇 ∈ 𝐿 ∧ ∀𝑥 ∈ 𝑋 ∀𝑦 ∈ 𝑋 (𝑥𝑃(𝑇‘𝑦)) = ((𝑇‘𝑥)𝑃𝑦)) → 𝑇 ∈ 𝐵) ↔ ((𝑇 ∈ (if(𝑈 ∈ CHilOLD, 𝑈, ⟨⟨ + , · ⟩, abs⟩) LnOp if(𝑈 ∈ CHilOLD, 𝑈, ⟨⟨ + , · ⟩, abs⟩)) ∧ ∀𝑥 ∈ (BaseSet‘if(𝑈 ∈ CHilOLD, 𝑈, ⟨⟨ + , · ⟩, abs⟩))∀𝑦 ∈ (BaseSet‘if(𝑈 ∈ CHilOLD, 𝑈, ⟨⟨ + , · ⟩, abs⟩))(𝑥(·𝑖OLD‘if(𝑈 ∈ CHilOLD, 𝑈, ⟨⟨ + , · ⟩, abs⟩))(𝑇‘𝑦)) = ((𝑇‘𝑥)(·𝑖OLD‘if(𝑈 ∈ CHilOLD, 𝑈, ⟨⟨ + , · ⟩, abs⟩))𝑦)) → 𝑇 ∈ (if(𝑈 ∈ CHilOLD, 𝑈, ⟨⟨ + , · ⟩, abs⟩) BLnOp if(𝑈 ∈ CHilOLD, 𝑈, ⟨⟨ + , · ⟩, abs⟩)))))
24		eqid 2824	. . . 4 ⊢ (BaseSet‘if(𝑈 ∈ CHilOLD, 𝑈, ⟨⟨ + , · ⟩, abs⟩)) = (BaseSet‘if(𝑈 ∈ CHilOLD, 𝑈, ⟨⟨ + , · ⟩, abs⟩))
25		eqid 2824	. . . 4 ⊢ (·𝑖OLD‘if(𝑈 ∈ CHilOLD, 𝑈, ⟨⟨ + , · ⟩, abs⟩)) = (·𝑖OLD‘if(𝑈 ∈ CHilOLD, 𝑈, ⟨⟨ + , · ⟩, abs⟩))
26		eqid 2824	. . . 4 ⊢ (if(𝑈 ∈ CHilOLD, 𝑈, ⟨⟨ + , · ⟩, abs⟩) LnOp if(𝑈 ∈ CHilOLD, 𝑈, ⟨⟨ + , · ⟩, abs⟩)) = (if(𝑈 ∈ CHilOLD, 𝑈, ⟨⟨ + , · ⟩, abs⟩) LnOp if(𝑈 ∈ CHilOLD, 𝑈, ⟨⟨ + , · ⟩, abs⟩))
27		eqid 2824	. . . 4 ⊢ (if(𝑈 ∈ CHilOLD, 𝑈, ⟨⟨ + , · ⟩, abs⟩) BLnOp if(𝑈 ∈ CHilOLD, 𝑈, ⟨⟨ + , · ⟩, abs⟩)) = (if(𝑈 ∈ CHilOLD, 𝑈, ⟨⟨ + , · ⟩, abs⟩) BLnOp if(𝑈 ∈ CHilOLD, 𝑈, ⟨⟨ + , · ⟩, abs⟩))
28		eqid 2824	. . . 4 ⊢ (normCV‘if(𝑈 ∈ CHilOLD, 𝑈, ⟨⟨ + , · ⟩, abs⟩)) = (normCV‘if(𝑈 ∈ CHilOLD, 𝑈, ⟨⟨ + , · ⟩, abs⟩))
29		eqid 2824	. . . . . 6 ⊢ ⟨⟨ + , · ⟩, abs⟩ = ⟨⟨ + , · ⟩, abs⟩
30	29	cnchl 28696	. . . . 5 ⊢ ⟨⟨ + , · ⟩, abs⟩ ∈ CHilOLD
31	30	elimel 4537	. . . 4 ⊢ if(𝑈 ∈ CHilOLD, 𝑈, ⟨⟨ + , · ⟩, abs⟩) ∈ CHilOLD
32		simpl 485	. . . 4 ⊢ ((𝑇 ∈ (if(𝑈 ∈ CHilOLD, 𝑈, ⟨⟨ + , · ⟩, abs⟩) LnOp if(𝑈 ∈ CHilOLD, 𝑈, ⟨⟨ + , · ⟩, abs⟩)) ∧ ∀𝑥 ∈ (BaseSet‘if(𝑈 ∈ CHilOLD, 𝑈, ⟨⟨ + , · ⟩, abs⟩))∀𝑦 ∈ (BaseSet‘if(𝑈 ∈ CHilOLD, 𝑈, ⟨⟨ + , · ⟩, abs⟩))(𝑥(·𝑖OLD‘if(𝑈 ∈ CHilOLD, 𝑈, ⟨⟨ + , · ⟩, abs⟩))(𝑇‘𝑦)) = ((𝑇‘𝑥)(·𝑖OLD‘if(𝑈 ∈ CHilOLD, 𝑈, ⟨⟨ + , · ⟩, abs⟩))𝑦)) → 𝑇 ∈ (if(𝑈 ∈ CHilOLD, 𝑈, ⟨⟨ + , · ⟩, abs⟩) LnOp if(𝑈 ∈ CHilOLD, 𝑈, ⟨⟨ + , · ⟩, abs⟩)))
33		simpr 487	. . . . 5 ⊢ ((𝑇 ∈ (if(𝑈 ∈ CHilOLD, 𝑈, ⟨⟨ + , · ⟩, abs⟩) LnOp if(𝑈 ∈ CHilOLD, 𝑈, ⟨⟨ + , · ⟩, abs⟩)) ∧ ∀𝑥 ∈ (BaseSet‘if(𝑈 ∈ CHilOLD, 𝑈, ⟨⟨ + , · ⟩, abs⟩))∀𝑦 ∈ (BaseSet‘if(𝑈 ∈ CHilOLD, 𝑈, ⟨⟨ + , · ⟩, abs⟩))(𝑥(·𝑖OLD‘if(𝑈 ∈ CHilOLD, 𝑈, ⟨⟨ + , · ⟩, abs⟩))(𝑇‘𝑦)) = ((𝑇‘𝑥)(·𝑖OLD‘if(𝑈 ∈ CHilOLD, 𝑈, ⟨⟨ + , · ⟩, abs⟩))𝑦)) → ∀𝑥 ∈ (BaseSet‘if(𝑈 ∈ CHilOLD, 𝑈, ⟨⟨ + , · ⟩, abs⟩))∀𝑦 ∈ (BaseSet‘if(𝑈 ∈ CHilOLD, 𝑈, ⟨⟨ + , · ⟩, abs⟩))(𝑥(·𝑖OLD‘if(𝑈 ∈ CHilOLD, 𝑈, ⟨⟨ + , · ⟩, abs⟩))(𝑇‘𝑦)) = ((𝑇‘𝑥)(·𝑖OLD‘if(𝑈 ∈ CHilOLD, 𝑈, ⟨⟨ + , · ⟩, abs⟩))𝑦))
34		oveq1 7166	. . . . . . 7 ⊢ (𝑥 = 𝑢 → (𝑥(·𝑖OLD‘if(𝑈 ∈ CHilOLD, 𝑈, ⟨⟨ + , · ⟩, abs⟩))(𝑇‘𝑦)) = (𝑢(·𝑖OLD‘if(𝑈 ∈ CHilOLD, 𝑈, ⟨⟨ + , · ⟩, abs⟩))(𝑇‘𝑦)))
35		fveq2 6673	. . . . . . . 8 ⊢ (𝑥 = 𝑢 → (𝑇‘𝑥) = (𝑇‘𝑢))
36	35	oveq1d 7174	. . . . . . 7 ⊢ (𝑥 = 𝑢 → ((𝑇‘𝑥)(·𝑖OLD‘if(𝑈 ∈ CHilOLD, 𝑈, ⟨⟨ + , · ⟩, abs⟩))𝑦) = ((𝑇‘𝑢)(·𝑖OLD‘if(𝑈 ∈ CHilOLD, 𝑈, ⟨⟨ + , · ⟩, abs⟩))𝑦))
37	34, 36	eqeq12d 2840	. . . . . 6 ⊢ (𝑥 = 𝑢 → ((𝑥(·𝑖OLD‘if(𝑈 ∈ CHilOLD, 𝑈, ⟨⟨ + , · ⟩, abs⟩))(𝑇‘𝑦)) = ((𝑇‘𝑥)(·𝑖OLD‘if(𝑈 ∈ CHilOLD, 𝑈, ⟨⟨ + , · ⟩, abs⟩))𝑦) ↔ (𝑢(·𝑖OLD‘if(𝑈 ∈ CHilOLD, 𝑈, ⟨⟨ + , · ⟩, abs⟩))(𝑇‘𝑦)) = ((𝑇‘𝑢)(·𝑖OLD‘if(𝑈 ∈ CHilOLD, 𝑈, ⟨⟨ + , · ⟩, abs⟩))𝑦)))
38		fveq2 6673	. . . . . . . 8 ⊢ (𝑦 = 𝑣 → (𝑇‘𝑦) = (𝑇‘𝑣))
39	38	oveq2d 7175	. . . . . . 7 ⊢ (𝑦 = 𝑣 → (𝑢(·𝑖OLD‘if(𝑈 ∈ CHilOLD, 𝑈, ⟨⟨ + , · ⟩, abs⟩))(𝑇‘𝑦)) = (𝑢(·𝑖OLD‘if(𝑈 ∈ CHilOLD, 𝑈, ⟨⟨ + , · ⟩, abs⟩))(𝑇‘𝑣)))
40		oveq2 7167	. . . . . . 7 ⊢ (𝑦 = 𝑣 → ((𝑇‘𝑢)(·𝑖OLD‘if(𝑈 ∈ CHilOLD, 𝑈, ⟨⟨ + , · ⟩, abs⟩))𝑦) = ((𝑇‘𝑢)(·𝑖OLD‘if(𝑈 ∈ CHilOLD, 𝑈, ⟨⟨ + , · ⟩, abs⟩))𝑣))
41	39, 40	eqeq12d 2840	. . . . . 6 ⊢ (𝑦 = 𝑣 → ((𝑢(·𝑖OLD‘if(𝑈 ∈ CHilOLD, 𝑈, ⟨⟨ + , · ⟩, abs⟩))(𝑇‘𝑦)) = ((𝑇‘𝑢)(·𝑖OLD‘if(𝑈 ∈ CHilOLD, 𝑈, ⟨⟨ + , · ⟩, abs⟩))𝑦) ↔ (𝑢(·𝑖OLD‘if(𝑈 ∈ CHilOLD, 𝑈, ⟨⟨ + , · ⟩, abs⟩))(𝑇‘𝑣)) = ((𝑇‘𝑢)(·𝑖OLD‘if(𝑈 ∈ CHilOLD, 𝑈, ⟨⟨ + , · ⟩, abs⟩))𝑣)))
42	37, 41	cbvral2vw 3464	. . . . 5 ⊢ (∀𝑥 ∈ (BaseSet‘if(𝑈 ∈ CHilOLD, 𝑈, ⟨⟨ + , · ⟩, abs⟩))∀𝑦 ∈ (BaseSet‘if(𝑈 ∈ CHilOLD, 𝑈, ⟨⟨ + , · ⟩, abs⟩))(𝑥(·𝑖OLD‘if(𝑈 ∈ CHilOLD, 𝑈, ⟨⟨ + , · ⟩, abs⟩))(𝑇‘𝑦)) = ((𝑇‘𝑥)(·𝑖OLD‘if(𝑈 ∈ CHilOLD, 𝑈, ⟨⟨ + , · ⟩, abs⟩))𝑦) ↔ ∀𝑢 ∈ (BaseSet‘if(𝑈 ∈ CHilOLD, 𝑈, ⟨⟨ + , · ⟩, abs⟩))∀𝑣 ∈ (BaseSet‘if(𝑈 ∈ CHilOLD, 𝑈, ⟨⟨ + , · ⟩, abs⟩))(𝑢(·𝑖OLD‘if(𝑈 ∈ CHilOLD, 𝑈, ⟨⟨ + , · ⟩, abs⟩))(𝑇‘𝑣)) = ((𝑇‘𝑢)(·𝑖OLD‘if(𝑈 ∈ CHilOLD, 𝑈, ⟨⟨ + , · ⟩, abs⟩))𝑣))
43	33, 42	sylib 220	. . . 4 ⊢ ((𝑇 ∈ (if(𝑈 ∈ CHilOLD, 𝑈, ⟨⟨ + , · ⟩, abs⟩) LnOp if(𝑈 ∈ CHilOLD, 𝑈, ⟨⟨ + , · ⟩, abs⟩)) ∧ ∀𝑥 ∈ (BaseSet‘if(𝑈 ∈ CHilOLD, 𝑈, ⟨⟨ + , · ⟩, abs⟩))∀𝑦 ∈ (BaseSet‘if(𝑈 ∈ CHilOLD, 𝑈, ⟨⟨ + , · ⟩, abs⟩))(𝑥(·𝑖OLD‘if(𝑈 ∈ CHilOLD, 𝑈, ⟨⟨ + , · ⟩, abs⟩))(𝑇‘𝑦)) = ((𝑇‘𝑥)(·𝑖OLD‘if(𝑈 ∈ CHilOLD, 𝑈, ⟨⟨ + , · ⟩, abs⟩))𝑦)) → ∀𝑢 ∈ (BaseSet‘if(𝑈 ∈ CHilOLD, 𝑈, ⟨⟨ + , · ⟩, abs⟩))∀𝑣 ∈ (BaseSet‘if(𝑈 ∈ CHilOLD, 𝑈, ⟨⟨ + , · ⟩, abs⟩))(𝑢(·𝑖OLD‘if(𝑈 ∈ CHilOLD, 𝑈, ⟨⟨ + , · ⟩, abs⟩))(𝑇‘𝑣)) = ((𝑇‘𝑢)(·𝑖OLD‘if(𝑈 ∈ CHilOLD, 𝑈, ⟨⟨ + , · ⟩, abs⟩))𝑣))
44		oveq1 7166	. . . . . . 7 ⊢ (𝑦 = 𝑤 → (𝑦(·𝑖OLD‘if(𝑈 ∈ CHilOLD, 𝑈, ⟨⟨ + , · ⟩, abs⟩))(𝑇‘𝑥)) = (𝑤(·𝑖OLD‘if(𝑈 ∈ CHilOLD, 𝑈, ⟨⟨ + , · ⟩, abs⟩))(𝑇‘𝑥)))
45	44	cbvmptv 5172	. . . . . 6 ⊢ (𝑦 ∈ (BaseSet‘if(𝑈 ∈ CHilOLD, 𝑈, ⟨⟨ + , · ⟩, abs⟩)) ↦ (𝑦(·𝑖OLD‘if(𝑈 ∈ CHilOLD, 𝑈, ⟨⟨ + , · ⟩, abs⟩))(𝑇‘𝑥))) = (𝑤 ∈ (BaseSet‘if(𝑈 ∈ CHilOLD, 𝑈, ⟨⟨ + , · ⟩, abs⟩)) ↦ (𝑤(·𝑖OLD‘if(𝑈 ∈ CHilOLD, 𝑈, ⟨⟨ + , · ⟩, abs⟩))(𝑇‘𝑥)))
46		fveq2 6673	. . . . . . . 8 ⊢ (𝑥 = 𝑧 → (𝑇‘𝑥) = (𝑇‘𝑧))
47	46	oveq2d 7175	. . . . . . 7 ⊢ (𝑥 = 𝑧 → (𝑤(·𝑖OLD‘if(𝑈 ∈ CHilOLD, 𝑈, ⟨⟨ + , · ⟩, abs⟩))(𝑇‘𝑥)) = (𝑤(·𝑖OLD‘if(𝑈 ∈ CHilOLD, 𝑈, ⟨⟨ + , · ⟩, abs⟩))(𝑇‘𝑧)))
48	47	mpteq2dv 5165	. . . . . 6 ⊢ (𝑥 = 𝑧 → (𝑤 ∈ (BaseSet‘if(𝑈 ∈ CHilOLD, 𝑈, ⟨⟨ + , · ⟩, abs⟩)) ↦ (𝑤(·𝑖OLD‘if(𝑈 ∈ CHilOLD, 𝑈, ⟨⟨ + , · ⟩, abs⟩))(𝑇‘𝑥))) = (𝑤 ∈ (BaseSet‘if(𝑈 ∈ CHilOLD, 𝑈, ⟨⟨ + , · ⟩, abs⟩)) ↦ (𝑤(·𝑖OLD‘if(𝑈 ∈ CHilOLD, 𝑈, ⟨⟨ + , · ⟩, abs⟩))(𝑇‘𝑧))))
49	45, 48	syl5eq 2871	. . . . 5 ⊢ (𝑥 = 𝑧 → (𝑦 ∈ (BaseSet‘if(𝑈 ∈ CHilOLD, 𝑈, ⟨⟨ + , · ⟩, abs⟩)) ↦ (𝑦(·𝑖OLD‘if(𝑈 ∈ CHilOLD, 𝑈, ⟨⟨ + , · ⟩, abs⟩))(𝑇‘𝑥))) = (𝑤 ∈ (BaseSet‘if(𝑈 ∈ CHilOLD, 𝑈, ⟨⟨ + , · ⟩, abs⟩)) ↦ (𝑤(·𝑖OLD‘if(𝑈 ∈ CHilOLD, 𝑈, ⟨⟨ + , · ⟩, abs⟩))(𝑇‘𝑧))))
50	49	cbvmptv 5172	. . . 4 ⊢ (𝑥 ∈ (BaseSet‘if(𝑈 ∈ CHilOLD, 𝑈, ⟨⟨ + , · ⟩, abs⟩)) ↦ (𝑦 ∈ (BaseSet‘if(𝑈 ∈ CHilOLD, 𝑈, ⟨⟨ + , · ⟩, abs⟩)) ↦ (𝑦(·𝑖OLD‘if(𝑈 ∈ CHilOLD, 𝑈, ⟨⟨ + , · ⟩, abs⟩))(𝑇‘𝑥)))) = (𝑧 ∈ (BaseSet‘if(𝑈 ∈ CHilOLD, 𝑈, ⟨⟨ + , · ⟩, abs⟩)) ↦ (𝑤 ∈ (BaseSet‘if(𝑈 ∈ CHilOLD, 𝑈, ⟨⟨ + , · ⟩, abs⟩)) ↦ (𝑤(·𝑖OLD‘if(𝑈 ∈ CHilOLD, 𝑈, ⟨⟨ + , · ⟩, abs⟩))(𝑇‘𝑧))))
51		fveq2 6673	. . . . . . 7 ⊢ (𝑥 = 𝑧 → ((normCV‘if(𝑈 ∈ CHilOLD, 𝑈, ⟨⟨ + , · ⟩, abs⟩))‘𝑥) = ((normCV‘if(𝑈 ∈ CHilOLD, 𝑈, ⟨⟨ + , · ⟩, abs⟩))‘𝑧))
52	51	breq1d 5079	. . . . . 6 ⊢ (𝑥 = 𝑧 → (((normCV‘if(𝑈 ∈ CHilOLD, 𝑈, ⟨⟨ + , · ⟩, abs⟩))‘𝑥) ≤ 1 ↔ ((normCV‘if(𝑈 ∈ CHilOLD, 𝑈, ⟨⟨ + , · ⟩, abs⟩))‘𝑧) ≤ 1))
53	52	cbvrabv 3494	. . . . 5 ⊢ {𝑥 ∈ (BaseSet‘if(𝑈 ∈ CHilOLD, 𝑈, ⟨⟨ + , · ⟩, abs⟩)) ∣ ((normCV‘if(𝑈 ∈ CHilOLD, 𝑈, ⟨⟨ + , · ⟩, abs⟩))‘𝑥) ≤ 1} = {𝑧 ∈ (BaseSet‘if(𝑈 ∈ CHilOLD, 𝑈, ⟨⟨ + , · ⟩, abs⟩)) ∣ ((normCV‘if(𝑈 ∈ CHilOLD, 𝑈, ⟨⟨ + , · ⟩, abs⟩))‘𝑧) ≤ 1}
54	53	imaeq2i 5930	. . . 4 ⊢ ((𝑥 ∈ (BaseSet‘if(𝑈 ∈ CHilOLD, 𝑈, ⟨⟨ + , · ⟩, abs⟩)) ↦ (𝑦 ∈ (BaseSet‘if(𝑈 ∈ CHilOLD, 𝑈, ⟨⟨ + , · ⟩, abs⟩)) ↦ (𝑦(·𝑖OLD‘if(𝑈 ∈ CHilOLD, 𝑈, ⟨⟨ + , · ⟩, abs⟩))(𝑇‘𝑥)))) “ {𝑥 ∈ (BaseSet‘if(𝑈 ∈ CHilOLD, 𝑈, ⟨⟨ + , · ⟩, abs⟩)) ∣ ((normCV‘if(𝑈 ∈ CHilOLD, 𝑈, ⟨⟨ + , · ⟩, abs⟩))‘𝑥) ≤ 1}) = ((𝑥 ∈ (BaseSet‘if(𝑈 ∈ CHilOLD, 𝑈, ⟨⟨ + , · ⟩, abs⟩)) ↦ (𝑦 ∈ (BaseSet‘if(𝑈 ∈ CHilOLD, 𝑈, ⟨⟨ + , · ⟩, abs⟩)) ↦ (𝑦(·𝑖OLD‘if(𝑈 ∈ CHilOLD, 𝑈, ⟨⟨ + , · ⟩, abs⟩))(𝑇‘𝑥)))) “ {𝑧 ∈ (BaseSet‘if(𝑈 ∈ CHilOLD, 𝑈, ⟨⟨ + , · ⟩, abs⟩)) ∣ ((normCV‘if(𝑈 ∈ CHilOLD, 𝑈, ⟨⟨ + , · ⟩, abs⟩))‘𝑧) ≤ 1})
55	24, 25, 26, 27, 28, 31, 29, 32, 43, 50, 54	htthlem 28697	. . 3 ⊢ ((𝑇 ∈ (if(𝑈 ∈ CHilOLD, 𝑈, ⟨⟨ + , · ⟩, abs⟩) LnOp if(𝑈 ∈ CHilOLD, 𝑈, ⟨⟨ + , · ⟩, abs⟩)) ∧ ∀𝑥 ∈ (BaseSet‘if(𝑈 ∈ CHilOLD, 𝑈, ⟨⟨ + , · ⟩, abs⟩))∀𝑦 ∈ (BaseSet‘if(𝑈 ∈ CHilOLD, 𝑈, ⟨⟨ + , · ⟩, abs⟩))(𝑥(·𝑖OLD‘if(𝑈 ∈ CHilOLD, 𝑈, ⟨⟨ + , · ⟩, abs⟩))(𝑇‘𝑦)) = ((𝑇‘𝑥)(·𝑖OLD‘if(𝑈 ∈ CHilOLD, 𝑈, ⟨⟨ + , · ⟩, abs⟩))𝑦)) → 𝑇 ∈ (if(𝑈 ∈ CHilOLD, 𝑈, ⟨⟨ + , · ⟩, abs⟩) BLnOp if(𝑈 ∈ CHilOLD, 𝑈, ⟨⟨ + , · ⟩, abs⟩)))
56	23, 55	dedth 4526	. 2 ⊢ (𝑈 ∈ CHilOLD → ((𝑇 ∈ 𝐿 ∧ ∀𝑥 ∈ 𝑋 ∀𝑦 ∈ 𝑋 (𝑥𝑃(𝑇‘𝑦)) = ((𝑇‘𝑥)𝑃𝑦)) → 𝑇 ∈ 𝐵))
57	56	3impib 1112	1 ⊢ ((𝑈 ∈ CHilOLD ∧ 𝑇 ∈ 𝐿 ∧ ∀𝑥 ∈ 𝑋 ∀𝑦 ∈ 𝑋 (𝑥𝑃(𝑇‘𝑦)) = ((𝑇‘𝑥)𝑃𝑦)) → 𝑇 ∈ 𝐵)

Syntax hints:   → wi 4   ∧ wa 398   ∧ w3a 1083   = wceq 1536   ∈ wcel 2113  ∀wral 3141  {crab 3145  ifcif 4470  ⟨cop 4576   class class class wbr 5069   ↦ cmpt 5149   “ cima 5561  ‘cfv 6358  (class class class)co 7159  1c1 10541   + caddc 10543   · cmul 10545   ≤ cle 10679  abscabs 14596  BaseSetcba 28366  norm_CVcnmcv 28370  ·_𝑖OLDcdip 28480   LnOp clno 28520   BLnOp cblo 28522  CHil_OLDchlo 28665

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 1969  ax-7 2014  ax-8 2115  ax-9 2123  ax-10 2144  ax-11 2160  ax-12 2176  ax-ext 2796  ax-rep 5193  ax-sep 5206  ax-nul 5213  ax-pow 5269  ax-pr 5333  ax-un 7464  ax-inf2 9107  ax-dc 9871  ax-cnex 10596  ax-resscn 10597  ax-1cn 10598  ax-icn 10599  ax-addcl 10600  ax-addrcl 10601  ax-mulcl 10602  ax-mulrcl 10603  ax-mulcom 10604  ax-addass 10605  ax-mulass 10606  ax-distr 10607  ax-i2m1 10608  ax-1ne0 10609  ax-1rid 10610  ax-rnegex 10611  ax-rrecex 10612  ax-cnre 10613  ax-pre-lttri 10614  ax-pre-lttrn 10615  ax-pre-ltadd 10616  ax-pre-mulgt0 10617  ax-pre-sup 10618  ax-addf 10619  ax-mulf 10620

This theorem depends on definitions:  df-bi 209  df-an 399  df-or 844  df-3or 1084  df-3an 1085  df-tru 1539  df-fal 1549  df-ex 1780  df-nf 1784  df-sb 2069  df-mo 2621  df-eu 2653  df-clab 2803  df-cleq 2817  df-clel 2896  df-nfc 2966  df-ne 3020  df-nel 3127  df-ral 3146  df-rex 3147  df-reu 3148  df-rmo 3149  df-rab 3150  df-v 3499  df-sbc 3776  df-csb 3887  df-dif 3942  df-un 3944  df-in 3946  df-ss 3955  df-pss 3957  df-nul 4295  df-if 4471  df-pw 4544  df-sn 4571  df-pr 4573  df-tp 4575  df-op 4577  df-uni 4842  df-int 4880  df-iun 4924  df-iin 4925  df-br 5070  df-opab 5132  df-mpt 5150  df-tr 5176  df-id 5463  df-eprel 5468  df-po 5477  df-so 5478  df-fr 5517  df-se 5518  df-we 5519  df-xp 5564  df-rel 5565  df-cnv 5566  df-co 5567  df-dm 5568  df-rn 5569  df-res 5570  df-ima 5571  df-pred 6151  df-ord 6197  df-on 6198  df-lim 6199  df-suc 6200  df-iota 6317  df-fun 6360  df-fn 6361  df-f 6362  df-f1 6363  df-fo 6364  df-f1o 6365  df-fv 6366  df-isom 6367  df-riota 7117  df-ov 7162  df-oprab 7163  df-mpo 7164  df-of 7412  df-om 7584  df-1st 7692  df-2nd 7693  df-supp 7834  df-wrecs 7950  df-recs 8011  df-rdg 8049  df-1o 8105  df-2o 8106  df-oadd 8109  df-er 8292  df-map 8411  df-pm 8412  df-ixp 8465  df-en 8513  df-dom 8514  df-sdom 8515  df-fin 8516  df-fsupp 8837  df-fi 8878  df-sup 8909  df-inf 8910  df-oi 8977  df-card 9371  df-pnf 10680  df-mnf 10681  df-xr 10682  df-ltxr 10683  df-le 10684  df-sub 10875  df-neg 10876  df-div 11301  df-nn 11642  df-2 11703  df-3 11704  df-4 11705  df-5 11706  df-6 11707  df-7 11708  df-8 11709  df-9 11710  df-n0 11901  df-z 11985  df-dec 12102  df-uz 12247  df-q 12352  df-rp 12393  df-xneg 12510  df-xadd 12511  df-xmul 12512  df-ioo 12745  df-ico 12747  df-icc 12748  df-fz 12896  df-fzo 13037  df-seq 13373  df-exp 13433  df-hash 13694  df-cj 14461  df-re 14462  df-im 14463  df-sqrt 14597  df-abs 14598  df-clim 14848  df-sum 15046  df-struct 16488  df-ndx 16489  df-slot 16490  df-base 16492  df-sets 16493  df-ress 16494  df-plusg 16581  df-mulr 16582  df-starv 16583  df-sca 16584  df-vsca 16585  df-ip 16586  df-tset 16587  df-ple 16588  df-ds 16590  df-unif 16591  df-hom 16592  df-cco 16593  df-rest 16699  df-topn 16700  df-0g 16718  df-gsum 16719  df-topgen 16720  df-pt 16721  df-prds 16724  df-xrs 16778  df-qtop 16783  df-imas 16784  df-xps 16786  df-mre 16860  df-mrc 16861  df-acs 16863  df-mgm 17855  df-sgrp 17904  df-mnd 17915  df-submnd 17960  df-mulg 18228  df-cntz 18450  df-cmn 18911  df-psmet 20540  df-xmet 20541  df-met 20542  df-bl 20543  df-mopn 20544  df-fbas 20545  df-fg 20546  df-cnfld 20549  df-top 21505  df-topon 21522  df-topsp 21544  df-bases 21557  df-cld 21630  df-ntr 21631  df-cls 21632  df-nei 21709  df-cn 21838  df-cnp 21839  df-lm 21840  df-t1 21925  df-haus 21926  df-cmp 21998  df-tx 22173  df-hmeo 22366  df-fil 22457  df-fm 22549  df-flim 22550  df-flf 22551  df-fcls 22552  df-xms 22933  df-ms 22934  df-tms 22935  df-cncf 23489  df-cfil 23861  df-cau 23862  df-cmet 23863  df-grpo 28273  df-gid 28274  df-ginv 28275  df-gdiv 28276  df-ablo 28325  df-vc 28339  df-nv 28372  df-va 28375  df-ba 28376  df-sm 28377  df-0v 28378  df-vs 28379  df-nmcv 28380  df-ims 28381  df-dip 28481  df-lno 28524  df-nmoo 28525  df-blo 28526  df-0o 28527  df-ph 28593  df-cbn 28643  df-hlo 28666

This theorem is referenced by:  hmopbdoptHIL  29768