Theorem htth 31005

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 7377	. . . . . . . 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 566	. . . . . . 7 ⊢ (𝑈 = if(𝑈 ∈ CHilOLD, 𝑈, ⟨⟨ + , · ⟩, abs⟩) → (𝑈 LnOp 𝑈) = (if(𝑈 ∈ CHilOLD, 𝑈, ⟨⟨ + , · ⟩, abs⟩) LnOp if(𝑈 ∈ CHilOLD, 𝑈, ⟨⟨ + , · ⟩, abs⟩)))
4	1, 3	eqtrid 2784	. . . . . 6 ⊢ (𝑈 = if(𝑈 ∈ CHilOLD, 𝑈, ⟨⟨ + , · ⟩, abs⟩) → 𝐿 = (if(𝑈 ∈ CHilOLD, 𝑈, ⟨⟨ + , · ⟩, abs⟩) LnOp if(𝑈 ∈ CHilOLD, 𝑈, ⟨⟨ + , · ⟩, abs⟩)))
5	4	eleq2d 2823	. . . . 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 6842	. . . . . . 7 ⊢ (𝑈 = if(𝑈 ∈ CHilOLD, 𝑈, ⟨⟨ + , · ⟩, abs⟩) → (BaseSet‘𝑈) = (BaseSet‘if(𝑈 ∈ CHilOLD, 𝑈, ⟨⟨ + , · ⟩, abs⟩)))
8	6, 7	eqtrid 2784	. . . . . 6 ⊢ (𝑈 = if(𝑈 ∈ CHilOLD, 𝑈, ⟨⟨ + , · ⟩, abs⟩) → 𝑋 = (BaseSet‘if(𝑈 ∈ CHilOLD, 𝑈, ⟨⟨ + , · ⟩, abs⟩)))
9		htth.2	. . . . . . . . . 10 ⊢ 𝑃 = (·𝑖OLD‘𝑈)
10		fveq2 6842	. . . . . . . . . 10 ⊢ (𝑈 = if(𝑈 ∈ CHilOLD, 𝑈, ⟨⟨ + , · ⟩, abs⟩) → (·𝑖OLD‘𝑈) = (·𝑖OLD‘if(𝑈 ∈ CHilOLD, 𝑈, ⟨⟨ + , · ⟩, abs⟩)))
11	9, 10	eqtrid 2784	. . . . . . . . 9 ⊢ (𝑈 = if(𝑈 ∈ CHilOLD, 𝑈, ⟨⟨ + , · ⟩, abs⟩) → 𝑃 = (·𝑖OLD‘if(𝑈 ∈ CHilOLD, 𝑈, ⟨⟨ + , · ⟩, abs⟩)))
12	11	oveqd 7385	. . . . . . . 8 ⊢ (𝑈 = if(𝑈 ∈ CHilOLD, 𝑈, ⟨⟨ + , · ⟩, abs⟩) → (𝑥𝑃(𝑇‘𝑦)) = (𝑥(·𝑖OLD‘if(𝑈 ∈ CHilOLD, 𝑈, ⟨⟨ + , · ⟩, abs⟩))(𝑇‘𝑦)))
13	11	oveqd 7385	. . . . . . . 8 ⊢ (𝑈 = if(𝑈 ∈ CHilOLD, 𝑈, ⟨⟨ + , · ⟩, abs⟩) → ((𝑇‘𝑥)𝑃𝑦) = ((𝑇‘𝑥)(·𝑖OLD‘if(𝑈 ∈ CHilOLD, 𝑈, ⟨⟨ + , · ⟩, abs⟩))𝑦))
14	12, 13	eqeq12d 2753	. . . . . . 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 3318	. . . . . 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 3318	. . . . 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 633	. . . 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 7377	. . . . . . 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 566	. . . . . 6 ⊢ (𝑈 = if(𝑈 ∈ CHilOLD, 𝑈, ⟨⟨ + , · ⟩, abs⟩) → (𝑈 BLnOp 𝑈) = (if(𝑈 ∈ CHilOLD, 𝑈, ⟨⟨ + , · ⟩, abs⟩) BLnOp if(𝑈 ∈ CHilOLD, 𝑈, ⟨⟨ + , · ⟩, abs⟩)))
21	18, 20	eqtrid 2784	. . . . 5 ⊢ (𝑈 = if(𝑈 ∈ CHilOLD, 𝑈, ⟨⟨ + , · ⟩, abs⟩) → 𝐵 = (if(𝑈 ∈ CHilOLD, 𝑈, ⟨⟨ + , · ⟩, abs⟩) BLnOp if(𝑈 ∈ CHilOLD, 𝑈, ⟨⟨ + , · ⟩, abs⟩)))
22	21	eleq2d 2823	. . . 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 344	. . 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 2737	. . . 4 ⊢ (BaseSet‘if(𝑈 ∈ CHilOLD, 𝑈, ⟨⟨ + , · ⟩, abs⟩)) = (BaseSet‘if(𝑈 ∈ CHilOLD, 𝑈, ⟨⟨ + , · ⟩, abs⟩))
25		eqid 2737	. . . 4 ⊢ (·𝑖OLD‘if(𝑈 ∈ CHilOLD, 𝑈, ⟨⟨ + , · ⟩, abs⟩)) = (·𝑖OLD‘if(𝑈 ∈ CHilOLD, 𝑈, ⟨⟨ + , · ⟩, abs⟩))
26		eqid 2737	. . . 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 2737	. . . 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 2737	. . . 4 ⊢ (normCV‘if(𝑈 ∈ CHilOLD, 𝑈, ⟨⟨ + , · ⟩, abs⟩)) = (normCV‘if(𝑈 ∈ CHilOLD, 𝑈, ⟨⟨ + , · ⟩, abs⟩))
29		eqid 2737	. . . . . 6 ⊢ ⟨⟨ + , · ⟩, abs⟩ = ⟨⟨ + , · ⟩, abs⟩
30	29	cnchl 31003	. . . . 5 ⊢ ⟨⟨ + , · ⟩, abs⟩ ∈ CHilOLD
31	30	elimel 4551	. . . 4 ⊢ if(𝑈 ∈ CHilOLD, 𝑈, ⟨⟨ + , · ⟩, abs⟩) ∈ CHilOLD
32		simpl 482	. . . 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 484	. . . . 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 7375	. . . . . . 7 ⊢ (𝑥 = 𝑢 → (𝑥(·𝑖OLD‘if(𝑈 ∈ CHilOLD, 𝑈, ⟨⟨ + , · ⟩, abs⟩))(𝑇‘𝑦)) = (𝑢(·𝑖OLD‘if(𝑈 ∈ CHilOLD, 𝑈, ⟨⟨ + , · ⟩, abs⟩))(𝑇‘𝑦)))
35		fveq2 6842	. . . . . . . 8 ⊢ (𝑥 = 𝑢 → (𝑇‘𝑥) = (𝑇‘𝑢))
36	35	oveq1d 7383	. . . . . . 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 2753	. . . . . 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 6842	. . . . . . . 8 ⊢ (𝑦 = 𝑣 → (𝑇‘𝑦) = (𝑇‘𝑣))
39	38	oveq2d 7384	. . . . . . 7 ⊢ (𝑦 = 𝑣 → (𝑢(·𝑖OLD‘if(𝑈 ∈ CHilOLD, 𝑈, ⟨⟨ + , · ⟩, abs⟩))(𝑇‘𝑦)) = (𝑢(·𝑖OLD‘if(𝑈 ∈ CHilOLD, 𝑈, ⟨⟨ + , · ⟩, abs⟩))(𝑇‘𝑣)))
40		oveq2 7376	. . . . . . 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 2753	. . . . . 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 3220	. . . . 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 218	. . . 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 7375	. . . . . . 7 ⊢ (𝑦 = 𝑤 → (𝑦(·𝑖OLD‘if(𝑈 ∈ CHilOLD, 𝑈, ⟨⟨ + , · ⟩, abs⟩))(𝑇‘𝑥)) = (𝑤(·𝑖OLD‘if(𝑈 ∈ CHilOLD, 𝑈, ⟨⟨ + , · ⟩, abs⟩))(𝑇‘𝑥)))
45	44	cbvmptv 5204	. . . . . 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 6842	. . . . . . . 8 ⊢ (𝑥 = 𝑧 → (𝑇‘𝑥) = (𝑇‘𝑧))
47	46	oveq2d 7384	. . . . . . 7 ⊢ (𝑥 = 𝑧 → (𝑤(·𝑖OLD‘if(𝑈 ∈ CHilOLD, 𝑈, ⟨⟨ + , · ⟩, abs⟩))(𝑇‘𝑥)) = (𝑤(·𝑖OLD‘if(𝑈 ∈ CHilOLD, 𝑈, ⟨⟨ + , · ⟩, abs⟩))(𝑇‘𝑧)))
48	47	mpteq2dv 5194	. . . . . 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	eqtrid 2784	. . . . 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 5204	. . . 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 6842	. . . . . . 7 ⊢ (𝑥 = 𝑧 → ((normCV‘if(𝑈 ∈ CHilOLD, 𝑈, ⟨⟨ + , · ⟩, abs⟩))‘𝑥) = ((normCV‘if(𝑈 ∈ CHilOLD, 𝑈, ⟨⟨ + , · ⟩, abs⟩))‘𝑧))
52	51	breq1d 5110	. . . . . 6 ⊢ (𝑥 = 𝑧 → (((normCV‘if(𝑈 ∈ CHilOLD, 𝑈, ⟨⟨ + , · ⟩, abs⟩))‘𝑥) ≤ 1 ↔ ((normCV‘if(𝑈 ∈ CHilOLD, 𝑈, ⟨⟨ + , · ⟩, abs⟩))‘𝑧) ≤ 1))
53	52	cbvrabv 3411	. . . . 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 6025	. . . 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 31004	. . 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 4540	. 2 ⊢ (𝑈 ∈ CHilOLD → ((𝑇 ∈ 𝐿 ∧ ∀𝑥 ∈ 𝑋 ∀𝑦 ∈ 𝑋 (𝑥𝑃(𝑇‘𝑦)) = ((𝑇‘𝑥)𝑃𝑦)) → 𝑇 ∈ 𝐵))
57	56	3impib 1117	1 ⊢ ((𝑈 ∈ CHilOLD ∧ 𝑇 ∈ 𝐿 ∧ ∀𝑥 ∈ 𝑋 ∀𝑦 ∈ 𝑋 (𝑥𝑃(𝑇‘𝑦)) = ((𝑇‘𝑥)𝑃𝑦)) → 𝑇 ∈ 𝐵)

Syntax hints:   → wi 4   ∧ wa 395   ∧ w3a 1087   = wceq 1542   ∈ wcel 2114  ∀wral 3052  {crab 3401  ifcif 4481  ⟨cop 4588   class class class wbr 5100   ↦ cmpt 5181   “ cima 5635  ‘cfv 6500  (class class class)co 7368  1c1 11039   + caddc 11041   · cmul 11043   ≤ cle 11179  abscabs 15169  BaseSetcba 30673  norm_CVcnmcv 30677  ·_𝑖OLDcdip 30787   LnOp clno 30827   BLnOp cblo 30829  CHil_OLDchlo 30972

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1912  ax-6 1969  ax-7 2010  ax-8 2116  ax-9 2124  ax-10 2147  ax-11 2163  ax-12 2185  ax-ext 2709  ax-rep 5226  ax-sep 5243  ax-nul 5253  ax-pow 5312  ax-pr 5379  ax-un 7690  ax-inf2 9562  ax-dc 10368  ax-cnex 11094  ax-resscn 11095  ax-1cn 11096  ax-icn 11097  ax-addcl 11098  ax-addrcl 11099  ax-mulcl 11100  ax-mulrcl 11101  ax-mulcom 11102  ax-addass 11103  ax-mulass 11104  ax-distr 11105  ax-i2m1 11106  ax-1ne0 11107  ax-1rid 11108  ax-rnegex 11109  ax-rrecex 11110  ax-cnre 11111  ax-pre-lttri 11112  ax-pre-lttrn 11113  ax-pre-ltadd 11114  ax-pre-mulgt0 11115  ax-pre-sup 11116  ax-addf 11117  ax-mulf 11118

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3or 1088  df-3an 1089  df-tru 1545  df-fal 1555  df-ex 1782  df-nf 1786  df-sb 2069  df-mo 2540  df-eu 2570  df-clab 2716  df-cleq 2729  df-clel 2812  df-nfc 2886  df-ne 2934  df-nel 3038  df-ral 3053  df-rex 3063  df-rmo 3352  df-reu 3353  df-rab 3402  df-v 3444  df-sbc 3743  df-csb 3852  df-dif 3906  df-un 3908  df-in 3910  df-ss 3920  df-pss 3923  df-nul 4288  df-if 4482  df-pw 4558  df-sn 4583  df-pr 4585  df-tp 4587  df-op 4589  df-uni 4866  df-int 4905  df-iun 4950  df-iin 4951  df-br 5101  df-opab 5163  df-mpt 5182  df-tr 5208  df-id 5527  df-eprel 5532  df-po 5540  df-so 5541  df-fr 5585  df-se 5586  df-we 5587  df-xp 5638  df-rel 5639  df-cnv 5640  df-co 5641  df-dm 5642  df-rn 5643  df-res 5644  df-ima 5645  df-pred 6267  df-ord 6328  df-on 6329  df-lim 6330  df-suc 6331  df-iota 6456  df-fun 6502  df-fn 6503  df-f 6504  df-f1 6505  df-fo 6506  df-f1o 6507  df-fv 6508  df-isom 6509  df-riota 7325  df-ov 7371  df-oprab 7372  df-mpo 7373  df-of 7632  df-om 7819  df-1st 7943  df-2nd 7944  df-supp 8113  df-frecs 8233  df-wrecs 8264  df-recs 8313  df-rdg 8351  df-1o 8407  df-2o 8408  df-er 8645  df-map 8777  df-pm 8778  df-ixp 8848  df-en 8896  df-dom 8897  df-sdom 8898  df-fin 8899  df-fsupp 9277  df-fi 9326  df-sup 9357  df-inf 9358  df-oi 9427  df-card 9863  df-pnf 11180  df-mnf 11181  df-xr 11182  df-ltxr 11183  df-le 11184  df-sub 11378  df-neg 11379  df-div 11807  df-nn 12158  df-2 12220  df-3 12221  df-4 12222  df-5 12223  df-6 12224  df-7 12225  df-8 12226  df-9 12227  df-n0 12414  df-z 12501  df-dec 12620  df-uz 12764  df-q 12874  df-rp 12918  df-xneg 13038  df-xadd 13039  df-xmul 13040  df-ioo 13277  df-ico 13279  df-icc 13280  df-fz 13436  df-fzo 13583  df-seq 13937  df-exp 13997  df-hash 14266  df-cj 15034  df-re 15035  df-im 15036  df-sqrt 15170  df-abs 15171  df-clim 15423  df-sum 15622  df-struct 17086  df-sets 17103  df-slot 17121  df-ndx 17133  df-base 17149  df-ress 17170  df-plusg 17202  df-mulr 17203  df-starv 17204  df-sca 17205  df-vsca 17206  df-ip 17207  df-tset 17208  df-ple 17209  df-ds 17211  df-unif 17212  df-hom 17213  df-cco 17214  df-rest 17354  df-topn 17355  df-0g 17373  df-gsum 17374  df-topgen 17375  df-pt 17376  df-prds 17379  df-xrs 17435  df-qtop 17440  df-imas 17441  df-xps 17443  df-mre 17517  df-mrc 17518  df-acs 17520  df-mgm 18577  df-sgrp 18656  df-mnd 18672  df-submnd 18721  df-mulg 19010  df-cntz 19258  df-cmn 19723  df-psmet 21313  df-xmet 21314  df-met 21315  df-bl 21316  df-mopn 21317  df-fbas 21318  df-fg 21319  df-cnfld 21322  df-top 22850  df-topon 22867  df-topsp 22889  df-bases 22902  df-cld 22975  df-ntr 22976  df-cls 22977  df-nei 23054  df-cn 23183  df-cnp 23184  df-lm 23185  df-t1 23270  df-haus 23271  df-cmp 23343  df-tx 23518  df-hmeo 23711  df-fil 23802  df-fm 23894  df-flim 23895  df-flf 23896  df-fcls 23897  df-xms 24276  df-ms 24277  df-tms 24278  df-cncf 24839  df-cfil 25223  df-cau 25224  df-cmet 25225  df-grpo 30580  df-gid 30581  df-ginv 30582  df-gdiv 30583  df-ablo 30632  df-vc 30646  df-nv 30679  df-va 30682  df-ba 30683  df-sm 30684  df-0v 30685  df-vs 30686  df-nmcv 30687  df-ims 30688  df-dip 30788  df-lno 30831  df-nmoo 30832  df-blo 30833  df-0o 30834  df-ph 30900  df-cbn 30950  df-hlo 30973

This theorem is referenced by:  hmopbdoptHIL  32075