Theorem htth 28999

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 7222	. . . . . . . 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 570	. . . . . . 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 2790	. . . . . 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 6717	. . . . . . 7 ⊢ (𝑈 = if(𝑈 ∈ CHilOLD, 𝑈, ⟨⟨ + , · ⟩, abs⟩) → (BaseSet‘𝑈) = (BaseSet‘if(𝑈 ∈ CHilOLD, 𝑈, ⟨⟨ + , · ⟩, abs⟩)))
8	6, 7	syl5eq 2790	. . . . . 6 ⊢ (𝑈 = if(𝑈 ∈ CHilOLD, 𝑈, ⟨⟨ + , · ⟩, abs⟩) → 𝑋 = (BaseSet‘if(𝑈 ∈ CHilOLD, 𝑈, ⟨⟨ + , · ⟩, abs⟩)))
9		htth.2	. . . . . . . . . 10 ⊢ 𝑃 = (·𝑖OLD‘𝑈)
10		fveq2 6717	. . . . . . . . . 10 ⊢ (𝑈 = if(𝑈 ∈ CHilOLD, 𝑈, ⟨⟨ + , · ⟩, abs⟩) → (·𝑖OLD‘𝑈) = (·𝑖OLD‘if(𝑈 ∈ CHilOLD, 𝑈, ⟨⟨ + , · ⟩, abs⟩)))
11	9, 10	syl5eq 2790	. . . . . . . . 9 ⊢ (𝑈 = if(𝑈 ∈ CHilOLD, 𝑈, ⟨⟨ + , · ⟩, abs⟩) → 𝑃 = (·𝑖OLD‘if(𝑈 ∈ CHilOLD, 𝑈, ⟨⟨ + , · ⟩, abs⟩)))
12	11	oveqd 7230	. . . . . . . 8 ⊢ (𝑈 = if(𝑈 ∈ CHilOLD, 𝑈, ⟨⟨ + , · ⟩, abs⟩) → (𝑥𝑃(𝑇‘𝑦)) = (𝑥(·𝑖OLD‘if(𝑈 ∈ CHilOLD, 𝑈, ⟨⟨ + , · ⟩, abs⟩))(𝑇‘𝑦)))
13	11	oveqd 7230	. . . . . . . 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 3313	. . . . . 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 3313	. . . . 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 634	. . . 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 7222	. . . . . . 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 570	. . . . . 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 2790	. . . . 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 348	. . 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 28997	. . . . 5 ⊢ ⟨⟨ + , · ⟩, abs⟩ ∈ CHilOLD
31	30	elimel 4508	. . . 4 ⊢ if(𝑈 ∈ CHilOLD, 𝑈, ⟨⟨ + , · ⟩, abs⟩) ∈ CHilOLD
32		simpl 486	. . . 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 488	. . . . 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 7220	. . . . . . 7 ⊢ (𝑥 = 𝑢 → (𝑥(·𝑖OLD‘if(𝑈 ∈ CHilOLD, 𝑈, ⟨⟨ + , · ⟩, abs⟩))(𝑇‘𝑦)) = (𝑢(·𝑖OLD‘if(𝑈 ∈ CHilOLD, 𝑈, ⟨⟨ + , · ⟩, abs⟩))(𝑇‘𝑦)))
35		fveq2 6717	. . . . . . . 8 ⊢ (𝑥 = 𝑢 → (𝑇‘𝑥) = (𝑇‘𝑢))
36	35	oveq1d 7228	. . . . . . 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 6717	. . . . . . . 8 ⊢ (𝑦 = 𝑣 → (𝑇‘𝑦) = (𝑇‘𝑣))
39	38	oveq2d 7229	. . . . . . 7 ⊢ (𝑦 = 𝑣 → (𝑢(·𝑖OLD‘if(𝑈 ∈ CHilOLD, 𝑈, ⟨⟨ + , · ⟩, abs⟩))(𝑇‘𝑦)) = (𝑢(·𝑖OLD‘if(𝑈 ∈ CHilOLD, 𝑈, ⟨⟨ + , · ⟩, abs⟩))(𝑇‘𝑣)))
40		oveq2 7221	. . . . . . 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 3371	. . . . 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 221	. . . 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 7220	. . . . . . 7 ⊢ (𝑦 = 𝑤 → (𝑦(·𝑖OLD‘if(𝑈 ∈ CHilOLD, 𝑈, ⟨⟨ + , · ⟩, abs⟩))(𝑇‘𝑥)) = (𝑤(·𝑖OLD‘if(𝑈 ∈ CHilOLD, 𝑈, ⟨⟨ + , · ⟩, abs⟩))(𝑇‘𝑥)))
45	44	cbvmptv 5158	. . . . . 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 6717	. . . . . . . 8 ⊢ (𝑥 = 𝑧 → (𝑇‘𝑥) = (𝑇‘𝑧))
47	46	oveq2d 7229	. . . . . . 7 ⊢ (𝑥 = 𝑧 → (𝑤(·𝑖OLD‘if(𝑈 ∈ CHilOLD, 𝑈, ⟨⟨ + , · ⟩, abs⟩))(𝑇‘𝑥)) = (𝑤(·𝑖OLD‘if(𝑈 ∈ CHilOLD, 𝑈, ⟨⟨ + , · ⟩, abs⟩))(𝑇‘𝑧)))
48	47	mpteq2dv 5151	. . . . . 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 2790	. . . . 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 5158	. . . 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 6717	. . . . . . 7 ⊢ (𝑥 = 𝑧 → ((normCV‘if(𝑈 ∈ CHilOLD, 𝑈, ⟨⟨ + , · ⟩, abs⟩))‘𝑥) = ((normCV‘if(𝑈 ∈ CHilOLD, 𝑈, ⟨⟨ + , · ⟩, abs⟩))‘𝑧))
52	51	breq1d 5063	. . . . . 6 ⊢ (𝑥 = 𝑧 → (((normCV‘if(𝑈 ∈ CHilOLD, 𝑈, ⟨⟨ + , · ⟩, abs⟩))‘𝑥) ≤ 1 ↔ ((normCV‘if(𝑈 ∈ CHilOLD, 𝑈, ⟨⟨ + , · ⟩, abs⟩))‘𝑧) ≤ 1))
53	52	cbvrabv 3402	. . . . 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 5927	. . . 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 28998	. . 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 4497	. 2 ⊢ (𝑈 ∈ CHilOLD → ((𝑇 ∈ 𝐿 ∧ ∀𝑥 ∈ 𝑋 ∀𝑦 ∈ 𝑋 (𝑥𝑃(𝑇‘𝑦)) = ((𝑇‘𝑥)𝑃𝑦)) → 𝑇 ∈ 𝐵))
57	56	3impib 1118	1 ⊢ ((𝑈 ∈ CHilOLD ∧ 𝑇 ∈ 𝐿 ∧ ∀𝑥 ∈ 𝑋 ∀𝑦 ∈ 𝑋 (𝑥𝑃(𝑇‘𝑦)) = ((𝑇‘𝑥)𝑃𝑦)) → 𝑇 ∈ 𝐵)

Syntax hints:   → wi 4   ∧ wa 399   ∧ w3a 1089   = wceq 1543   ∈ wcel 2110  ∀wral 3061  {crab 3065  ifcif 4439  ⟨cop 4547   class class class wbr 5053   ↦ cmpt 5135   “ cima 5554  ‘cfv 6380  (class class class)co 7213  1c1 10730   + caddc 10732   · cmul 10734   ≤ cle 10868  abscabs 14797  BaseSetcba 28667  norm_CVcnmcv 28671  ·_𝑖OLDcdip 28781   LnOp clno 28821   BLnOp cblo 28823  CHil_OLDchlo 28966

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1803  ax-4 1817  ax-5 1918  ax-6 1976  ax-7 2016  ax-8 2112  ax-9 2120  ax-10 2141  ax-11 2158  ax-12 2175  ax-ext 2708  ax-rep 5179  ax-sep 5192  ax-nul 5199  ax-pow 5258  ax-pr 5322  ax-un 7523  ax-inf2 9256  ax-dc 10060  ax-cnex 10785  ax-resscn 10786  ax-1cn 10787  ax-icn 10788  ax-addcl 10789  ax-addrcl 10790  ax-mulcl 10791  ax-mulrcl 10792  ax-mulcom 10793  ax-addass 10794  ax-mulass 10795  ax-distr 10796  ax-i2m1 10797  ax-1ne0 10798  ax-1rid 10799  ax-rnegex 10800  ax-rrecex 10801  ax-cnre 10802  ax-pre-lttri 10803  ax-pre-lttrn 10804  ax-pre-ltadd 10805  ax-pre-mulgt0 10806  ax-pre-sup 10807  ax-addf 10808  ax-mulf 10809

This theorem depends on definitions:  df-bi 210  df-an 400  df-or 848  df-3or 1090  df-3an 1091  df-tru 1546  df-fal 1556  df-ex 1788  df-nf 1792  df-sb 2071  df-mo 2539  df-eu 2568  df-clab 2715  df-cleq 2729  df-clel 2816  df-nfc 2886  df-ne 2941  df-nel 3047  df-ral 3066  df-rex 3067  df-reu 3068  df-rmo 3069  df-rab 3070  df-v 3410  df-sbc 3695  df-csb 3812  df-dif 3869  df-un 3871  df-in 3873  df-ss 3883  df-pss 3885  df-nul 4238  df-if 4440  df-pw 4515  df-sn 4542  df-pr 4544  df-tp 4546  df-op 4548  df-uni 4820  df-int 4860  df-iun 4906  df-iin 4907  df-br 5054  df-opab 5116  df-mpt 5136  df-tr 5162  df-id 5455  df-eprel 5460  df-po 5468  df-so 5469  df-fr 5509  df-se 5510  df-we 5511  df-xp 5557  df-rel 5558  df-cnv 5559  df-co 5560  df-dm 5561  df-rn 5562  df-res 5563  df-ima 5564  df-pred 6160  df-ord 6216  df-on 6217  df-lim 6218  df-suc 6219  df-iota 6338  df-fun 6382  df-fn 6383  df-f 6384  df-f1 6385  df-fo 6386  df-f1o 6387  df-fv 6388  df-isom 6389  df-riota 7170  df-ov 7216  df-oprab 7217  df-mpo 7218  df-of 7469  df-om 7645  df-1st 7761  df-2nd 7762  df-supp 7904  df-wrecs 8047  df-recs 8108  df-rdg 8146  df-1o 8202  df-2o 8203  df-er 8391  df-map 8510  df-pm 8511  df-ixp 8579  df-en 8627  df-dom 8628  df-sdom 8629  df-fin 8630  df-fsupp 8986  df-fi 9027  df-sup 9058  df-inf 9059  df-oi 9126  df-card 9555  df-pnf 10869  df-mnf 10870  df-xr 10871  df-ltxr 10872  df-le 10873  df-sub 11064  df-neg 11065  df-div 11490  df-nn 11831  df-2 11893  df-3 11894  df-4 11895  df-5 11896  df-6 11897  df-7 11898  df-8 11899  df-9 11900  df-n0 12091  df-z 12177  df-dec 12294  df-uz 12439  df-q 12545  df-rp 12587  df-xneg 12704  df-xadd 12705  df-xmul 12706  df-ioo 12939  df-ico 12941  df-icc 12942  df-fz 13096  df-fzo 13239  df-seq 13575  df-exp 13636  df-hash 13897  df-cj 14662  df-re 14663  df-im 14664  df-sqrt 14798  df-abs 14799  df-clim 15049  df-sum 15250  df-struct 16700  df-sets 16717  df-slot 16735  df-ndx 16745  df-base 16761  df-ress 16785  df-plusg 16815  df-mulr 16816  df-starv 16817  df-sca 16818  df-vsca 16819  df-ip 16820  df-tset 16821  df-ple 16822  df-ds 16824  df-unif 16825  df-hom 16826  df-cco 16827  df-rest 16927  df-topn 16928  df-0g 16946  df-gsum 16947  df-topgen 16948  df-pt 16949  df-prds 16952  df-xrs 17007  df-qtop 17012  df-imas 17013  df-xps 17015  df-mre 17089  df-mrc 17090  df-acs 17092  df-mgm 18114  df-sgrp 18163  df-mnd 18174  df-submnd 18219  df-mulg 18489  df-cntz 18711  df-cmn 19172  df-psmet 20355  df-xmet 20356  df-met 20357  df-bl 20358  df-mopn 20359  df-fbas 20360  df-fg 20361  df-cnfld 20364  df-top 21791  df-topon 21808  df-topsp 21830  df-bases 21843  df-cld 21916  df-ntr 21917  df-cls 21918  df-nei 21995  df-cn 22124  df-cnp 22125  df-lm 22126  df-t1 22211  df-haus 22212  df-cmp 22284  df-tx 22459  df-hmeo 22652  df-fil 22743  df-fm 22835  df-flim 22836  df-flf 22837  df-fcls 22838  df-xms 23218  df-ms 23219  df-tms 23220  df-cncf 23775  df-cfil 24152  df-cau 24153  df-cmet 24154  df-grpo 28574  df-gid 28575  df-ginv 28576  df-gdiv 28577  df-ablo 28626  df-vc 28640  df-nv 28673  df-va 28676  df-ba 28677  df-sm 28678  df-0v 28679  df-vs 28680  df-nmcv 28681  df-ims 28682  df-dip 28782  df-lno 28825  df-nmoo 28826  df-blo 28827  df-0o 28828  df-ph 28894  df-cbn 28944  df-hlo 28967

This theorem is referenced by:  hmopbdoptHIL  30069