Theorem htth 29280

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 7284	. . . . . . . 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 567	. . . . . . 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 2790	. . . . . 6 ⊢ (𝑈 = if(𝑈 ∈ CHilOLD, 𝑈, ⟨⟨ + , · ⟩, abs⟩) → 𝐿 = (if(𝑈 ∈ CHilOLD, 𝑈, ⟨⟨ + , · ⟩, abs⟩) LnOp if(𝑈 ∈ CHilOLD, 𝑈, ⟨⟨ + , · ⟩, abs⟩)))
5	4	eleq2d 2824	. . . . 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 6774	. . . . . . 7 ⊢ (𝑈 = if(𝑈 ∈ CHilOLD, 𝑈, ⟨⟨ + , · ⟩, abs⟩) → (BaseSet‘𝑈) = (BaseSet‘if(𝑈 ∈ CHilOLD, 𝑈, ⟨⟨ + , · ⟩, abs⟩)))
8	6, 7	eqtrid 2790	. . . . . 6 ⊢ (𝑈 = if(𝑈 ∈ CHilOLD, 𝑈, ⟨⟨ + , · ⟩, abs⟩) → 𝑋 = (BaseSet‘if(𝑈 ∈ CHilOLD, 𝑈, ⟨⟨ + , · ⟩, abs⟩)))
9		htth.2	. . . . . . . . . 10 ⊢ 𝑃 = (·𝑖OLD‘𝑈)
10		fveq2 6774	. . . . . . . . . 10 ⊢ (𝑈 = if(𝑈 ∈ CHilOLD, 𝑈, ⟨⟨ + , · ⟩, abs⟩) → (·𝑖OLD‘𝑈) = (·𝑖OLD‘if(𝑈 ∈ CHilOLD, 𝑈, ⟨⟨ + , · ⟩, abs⟩)))
11	9, 10	eqtrid 2790	. . . . . . . . 9 ⊢ (𝑈 = if(𝑈 ∈ CHilOLD, 𝑈, ⟨⟨ + , · ⟩, abs⟩) → 𝑃 = (·𝑖OLD‘if(𝑈 ∈ CHilOLD, 𝑈, ⟨⟨ + , · ⟩, abs⟩)))
12	11	oveqd 7292	. . . . . . . 8 ⊢ (𝑈 = if(𝑈 ∈ CHilOLD, 𝑈, ⟨⟨ + , · ⟩, abs⟩) → (𝑥𝑃(𝑇‘𝑦)) = (𝑥(·𝑖OLD‘if(𝑈 ∈ CHilOLD, 𝑈, ⟨⟨ + , · ⟩, abs⟩))(𝑇‘𝑦)))
13	11	oveqd 7292	. . . . . . . 8 ⊢ (𝑈 = if(𝑈 ∈ CHilOLD, 𝑈, ⟨⟨ + , · ⟩, abs⟩) → ((𝑇‘𝑥)𝑃𝑦) = ((𝑇‘𝑥)(·𝑖OLD‘if(𝑈 ∈ CHilOLD, 𝑈, ⟨⟨ + , · ⟩, abs⟩))𝑦))
14	12, 13	eqeq12d 2754	. . . . . . 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 3336	. . . . . 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 3336	. . . . 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 631	. . . 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 7284	. . . . . . 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 567	. . . . . 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 2790	. . . . 5 ⊢ (𝑈 = if(𝑈 ∈ CHilOLD, 𝑈, ⟨⟨ + , · ⟩, abs⟩) → 𝐵 = (if(𝑈 ∈ CHilOLD, 𝑈, ⟨⟨ + , · ⟩, abs⟩) BLnOp if(𝑈 ∈ CHilOLD, 𝑈, ⟨⟨ + , · ⟩, abs⟩)))
22	21	eleq2d 2824	. . . 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 345	. . 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 2738	. . . 4 ⊢ (BaseSet‘if(𝑈 ∈ CHilOLD, 𝑈, ⟨⟨ + , · ⟩, abs⟩)) = (BaseSet‘if(𝑈 ∈ CHilOLD, 𝑈, ⟨⟨ + , · ⟩, abs⟩))
25		eqid 2738	. . . 4 ⊢ (·𝑖OLD‘if(𝑈 ∈ CHilOLD, 𝑈, ⟨⟨ + , · ⟩, abs⟩)) = (·𝑖OLD‘if(𝑈 ∈ CHilOLD, 𝑈, ⟨⟨ + , · ⟩, abs⟩))
26		eqid 2738	. . . 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 2738	. . . 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 2738	. . . 4 ⊢ (normCV‘if(𝑈 ∈ CHilOLD, 𝑈, ⟨⟨ + , · ⟩, abs⟩)) = (normCV‘if(𝑈 ∈ CHilOLD, 𝑈, ⟨⟨ + , · ⟩, abs⟩))
29		eqid 2738	. . . . . 6 ⊢ ⟨⟨ + , · ⟩, abs⟩ = ⟨⟨ + , · ⟩, abs⟩
30	29	cnchl 29278	. . . . 5 ⊢ ⟨⟨ + , · ⟩, abs⟩ ∈ CHilOLD
31	30	elimel 4528	. . . 4 ⊢ if(𝑈 ∈ CHilOLD, 𝑈, ⟨⟨ + , · ⟩, abs⟩) ∈ CHilOLD
32		simpl 483	. . . 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 485	. . . . 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 7282	. . . . . . 7 ⊢ (𝑥 = 𝑢 → (𝑥(·𝑖OLD‘if(𝑈 ∈ CHilOLD, 𝑈, ⟨⟨ + , · ⟩, abs⟩))(𝑇‘𝑦)) = (𝑢(·𝑖OLD‘if(𝑈 ∈ CHilOLD, 𝑈, ⟨⟨ + , · ⟩, abs⟩))(𝑇‘𝑦)))
35		fveq2 6774	. . . . . . . 8 ⊢ (𝑥 = 𝑢 → (𝑇‘𝑥) = (𝑇‘𝑢))
36	35	oveq1d 7290	. . . . . . 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 2754	. . . . . 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 6774	. . . . . . . 8 ⊢ (𝑦 = 𝑣 → (𝑇‘𝑦) = (𝑇‘𝑣))
39	38	oveq2d 7291	. . . . . . 7 ⊢ (𝑦 = 𝑣 → (𝑢(·𝑖OLD‘if(𝑈 ∈ CHilOLD, 𝑈, ⟨⟨ + , · ⟩, abs⟩))(𝑇‘𝑦)) = (𝑢(·𝑖OLD‘if(𝑈 ∈ CHilOLD, 𝑈, ⟨⟨ + , · ⟩, abs⟩))(𝑇‘𝑣)))
40		oveq2 7283	. . . . . . 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 2754	. . . . . 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 3396	. . . . 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 217	. . . 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 7282	. . . . . . 7 ⊢ (𝑦 = 𝑤 → (𝑦(·𝑖OLD‘if(𝑈 ∈ CHilOLD, 𝑈, ⟨⟨ + , · ⟩, abs⟩))(𝑇‘𝑥)) = (𝑤(·𝑖OLD‘if(𝑈 ∈ CHilOLD, 𝑈, ⟨⟨ + , · ⟩, abs⟩))(𝑇‘𝑥)))
45	44	cbvmptv 5187	. . . . . 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 6774	. . . . . . . 8 ⊢ (𝑥 = 𝑧 → (𝑇‘𝑥) = (𝑇‘𝑧))
47	46	oveq2d 7291	. . . . . . 7 ⊢ (𝑥 = 𝑧 → (𝑤(·𝑖OLD‘if(𝑈 ∈ CHilOLD, 𝑈, ⟨⟨ + , · ⟩, abs⟩))(𝑇‘𝑥)) = (𝑤(·𝑖OLD‘if(𝑈 ∈ CHilOLD, 𝑈, ⟨⟨ + , · ⟩, abs⟩))(𝑇‘𝑧)))
48	47	mpteq2dv 5176	. . . . . 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 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 5187	. . . 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 6774	. . . . . . 7 ⊢ (𝑥 = 𝑧 → ((normCV‘if(𝑈 ∈ CHilOLD, 𝑈, ⟨⟨ + , · ⟩, abs⟩))‘𝑥) = ((normCV‘if(𝑈 ∈ CHilOLD, 𝑈, ⟨⟨ + , · ⟩, abs⟩))‘𝑧))
52	51	breq1d 5084	. . . . . 6 ⊢ (𝑥 = 𝑧 → (((normCV‘if(𝑈 ∈ CHilOLD, 𝑈, ⟨⟨ + , · ⟩, abs⟩))‘𝑥) ≤ 1 ↔ ((normCV‘if(𝑈 ∈ CHilOLD, 𝑈, ⟨⟨ + , · ⟩, abs⟩))‘𝑧) ≤ 1))
53	52	cbvrabv 3426	. . . . 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 5967	. . . 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 29279	. . 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 4517	. 2 ⊢ (𝑈 ∈ CHilOLD → ((𝑇 ∈ 𝐿 ∧ ∀𝑥 ∈ 𝑋 ∀𝑦 ∈ 𝑋 (𝑥𝑃(𝑇‘𝑦)) = ((𝑇‘𝑥)𝑃𝑦)) → 𝑇 ∈ 𝐵))
57	56	3impib 1115	1 ⊢ ((𝑈 ∈ CHilOLD ∧ 𝑇 ∈ 𝐿 ∧ ∀𝑥 ∈ 𝑋 ∀𝑦 ∈ 𝑋 (𝑥𝑃(𝑇‘𝑦)) = ((𝑇‘𝑥)𝑃𝑦)) → 𝑇 ∈ 𝐵)

Syntax hints:   → wi 4   ∧ wa 396   ∧ w3a 1086   = wceq 1539   ∈ wcel 2106  ∀wral 3064  {crab 3068  ifcif 4459  ⟨cop 4567   class class class wbr 5074   ↦ cmpt 5157   “ cima 5592  ‘cfv 6433  (class class class)co 7275  1c1 10872   + caddc 10874   · cmul 10876   ≤ cle 11010  abscabs 14945  BaseSetcba 28948  norm_CVcnmcv 28952  ·_𝑖OLDcdip 29062   LnOp clno 29102   BLnOp cblo 29104  CHil_OLDchlo 29247

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-10 2137  ax-11 2154  ax-12 2171  ax-ext 2709  ax-rep 5209  ax-sep 5223  ax-nul 5230  ax-pow 5288  ax-pr 5352  ax-un 7588  ax-inf2 9399  ax-dc 10202  ax-cnex 10927  ax-resscn 10928  ax-1cn 10929  ax-icn 10930  ax-addcl 10931  ax-addrcl 10932  ax-mulcl 10933  ax-mulrcl 10934  ax-mulcom 10935  ax-addass 10936  ax-mulass 10937  ax-distr 10938  ax-i2m1 10939  ax-1ne0 10940  ax-1rid 10941  ax-rnegex 10942  ax-rrecex 10943  ax-cnre 10944  ax-pre-lttri 10945  ax-pre-lttrn 10946  ax-pre-ltadd 10947  ax-pre-mulgt0 10948  ax-pre-sup 10949  ax-addf 10950  ax-mulf 10951

This theorem depends on definitions:  df-bi 206  df-an 397  df-or 845  df-3or 1087  df-3an 1088  df-tru 1542  df-fal 1552  df-ex 1783  df-nf 1787  df-sb 2068  df-mo 2540  df-eu 2569  df-clab 2716  df-cleq 2730  df-clel 2816  df-nfc 2889  df-ne 2944  df-nel 3050  df-ral 3069  df-rex 3070  df-rmo 3071  df-reu 3072  df-rab 3073  df-v 3434  df-sbc 3717  df-csb 3833  df-dif 3890  df-un 3892  df-in 3894  df-ss 3904  df-pss 3906  df-nul 4257  df-if 4460  df-pw 4535  df-sn 4562  df-pr 4564  df-tp 4566  df-op 4568  df-uni 4840  df-int 4880  df-iun 4926  df-iin 4927  df-br 5075  df-opab 5137  df-mpt 5158  df-tr 5192  df-id 5489  df-eprel 5495  df-po 5503  df-so 5504  df-fr 5544  df-se 5545  df-we 5546  df-xp 5595  df-rel 5596  df-cnv 5597  df-co 5598  df-dm 5599  df-rn 5600  df-res 5601  df-ima 5602  df-pred 6202  df-ord 6269  df-on 6270  df-lim 6271  df-suc 6272  df-iota 6391  df-fun 6435  df-fn 6436  df-f 6437  df-f1 6438  df-fo 6439  df-f1o 6440  df-fv 6441  df-isom 6442  df-riota 7232  df-ov 7278  df-oprab 7279  df-mpo 7280  df-of 7533  df-om 7713  df-1st 7831  df-2nd 7832  df-supp 7978  df-frecs 8097  df-wrecs 8128  df-recs 8202  df-rdg 8241  df-1o 8297  df-2o 8298  df-er 8498  df-map 8617  df-pm 8618  df-ixp 8686  df-en 8734  df-dom 8735  df-sdom 8736  df-fin 8737  df-fsupp 9129  df-fi 9170  df-sup 9201  df-inf 9202  df-oi 9269  df-card 9697  df-pnf 11011  df-mnf 11012  df-xr 11013  df-ltxr 11014  df-le 11015  df-sub 11207  df-neg 11208  df-div 11633  df-nn 11974  df-2 12036  df-3 12037  df-4 12038  df-5 12039  df-6 12040  df-7 12041  df-8 12042  df-9 12043  df-n0 12234  df-z 12320  df-dec 12438  df-uz 12583  df-q 12689  df-rp 12731  df-xneg 12848  df-xadd 12849  df-xmul 12850  df-ioo 13083  df-ico 13085  df-icc 13086  df-fz 13240  df-fzo 13383  df-seq 13722  df-exp 13783  df-hash 14045  df-cj 14810  df-re 14811  df-im 14812  df-sqrt 14946  df-abs 14947  df-clim 15197  df-sum 15398  df-struct 16848  df-sets 16865  df-slot 16883  df-ndx 16895  df-base 16913  df-ress 16942  df-plusg 16975  df-mulr 16976  df-starv 16977  df-sca 16978  df-vsca 16979  df-ip 16980  df-tset 16981  df-ple 16982  df-ds 16984  df-unif 16985  df-hom 16986  df-cco 16987  df-rest 17133  df-topn 17134  df-0g 17152  df-gsum 17153  df-topgen 17154  df-pt 17155  df-prds 17158  df-xrs 17213  df-qtop 17218  df-imas 17219  df-xps 17221  df-mre 17295  df-mrc 17296  df-acs 17298  df-mgm 18326  df-sgrp 18375  df-mnd 18386  df-submnd 18431  df-mulg 18701  df-cntz 18923  df-cmn 19388  df-psmet 20589  df-xmet 20590  df-met 20591  df-bl 20592  df-mopn 20593  df-fbas 20594  df-fg 20595  df-cnfld 20598  df-top 22043  df-topon 22060  df-topsp 22082  df-bases 22096  df-cld 22170  df-ntr 22171  df-cls 22172  df-nei 22249  df-cn 22378  df-cnp 22379  df-lm 22380  df-t1 22465  df-haus 22466  df-cmp 22538  df-tx 22713  df-hmeo 22906  df-fil 22997  df-fm 23089  df-flim 23090  df-flf 23091  df-fcls 23092  df-xms 23473  df-ms 23474  df-tms 23475  df-cncf 24041  df-cfil 24419  df-cau 24420  df-cmet 24421  df-grpo 28855  df-gid 28856  df-ginv 28857  df-gdiv 28858  df-ablo 28907  df-vc 28921  df-nv 28954  df-va 28957  df-ba 28958  df-sm 28959  df-0v 28960  df-vs 28961  df-nmcv 28962  df-ims 28963  df-dip 29063  df-lno 29106  df-nmoo 29107  df-blo 29108  df-0o 29109  df-ph 29175  df-cbn 29225  df-hlo 29248

This theorem is referenced by:  hmopbdoptHIL  30350