stcltr1i - Hilbert Space Explorer

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Theorem stcltr1i 32566

Description: Property of a strong classical state. (Contributed by NM, 24-Oct-1999.) (New usage is discouraged.)

**Hypotheses**
Ref	Expression
stcltr1.1	⊢ (𝜑 ↔ (𝑆 ∈ States ∧ ∀𝑥 ∈ C_ℋ ∀𝑦 ∈ C_ℋ (((𝑆‘𝑥) = 1 → (𝑆‘𝑦) = 1) → 𝑥 ⊆ 𝑦)))
stcltr1.2	⊢ 𝐴 ∈ C_ℋ
stcltr1.3	⊢ 𝐵 ∈ C_ℋ

**Assertion**
Ref	Expression
stcltr1i	⊢ (𝜑 → (((𝑆‘𝐴) = 1 → (𝑆‘𝐵) = 1) → 𝐴 ⊆ 𝐵))

Distinct variable groups: 𝑥,𝑦,𝐴 𝑥,𝐵,𝑦 𝑥,𝑆,𝑦 Allowed substitution hints: 𝜑(𝑥,𝑦)

**Proof of Theorem stcltr1i**
Step	Hyp	Ref	Expression
1		stcltr1.1	. 2 ⊢ (𝜑 ↔ (𝑆 ∈ States ∧ ∀𝑥 ∈ C_ℋ ∀𝑦 ∈ C_ℋ (((𝑆‘𝑥) = 1 → (𝑆‘𝑦) = 1) → 𝑥 ⊆ 𝑦)))
2		stcltr1.2	. . 3 ⊢ 𝐴 ∈ C_ℋ
3		stcltr1.3	. . 3 ⊢ 𝐵 ∈ C_ℋ
4		fveqeq2 6891	. . . . . 6 ⊢ (𝑥 = 𝐴 → ((𝑆‘𝑥) = 1 ↔ (𝑆‘𝐴) = 1))
5	4	imbi1d 344	. . . . 5 ⊢ (𝑥 = 𝐴 → (((𝑆‘𝑥) = 1 → (𝑆‘𝑦) = 1) ↔ ((𝑆‘𝐴) = 1 → (𝑆‘𝑦) = 1)))
6		sseq1 3970	. . . . 5 ⊢ (𝑥 = 𝐴 → (𝑥 ⊆ 𝑦 ↔ 𝐴 ⊆ 𝑦))
7	5, 6	imbi12d 347	. . . 4 ⊢ (𝑥 = 𝐴 → ((((𝑆‘𝑥) = 1 → (𝑆‘𝑦) = 1) → 𝑥 ⊆ 𝑦) ↔ (((𝑆‘𝐴) = 1 → (𝑆‘𝑦) = 1) → 𝐴 ⊆ 𝑦)))
8		fveqeq2 6891	. . . . . 6 ⊢ (𝑦 = 𝐵 → ((𝑆‘𝑦) = 1 ↔ (𝑆‘𝐵) = 1))
9	8	imbi2d 343	. . . . 5 ⊢ (𝑦 = 𝐵 → (((𝑆‘𝐴) = 1 → (𝑆‘𝑦) = 1) ↔ ((𝑆‘𝐴) = 1 → (𝑆‘𝐵) = 1)))
10		sseq2 3971	. . . . 5 ⊢ (𝑦 = 𝐵 → (𝐴 ⊆ 𝑦 ↔ 𝐴 ⊆ 𝐵))
11	9, 10	imbi12d 347	. . . 4 ⊢ (𝑦 = 𝐵 → ((((𝑆‘𝐴) = 1 → (𝑆‘𝑦) = 1) → 𝐴 ⊆ 𝑦) ↔ (((𝑆‘𝐴) = 1 → (𝑆‘𝐵) = 1) → 𝐴 ⊆ 𝐵)))
12	7, 11	rspc2v 3601	. . 3 ⊢ ((𝐴 ∈ C_ℋ ∧ 𝐵 ∈ C_ℋ ) → (∀𝑥 ∈ C_ℋ ∀𝑦 ∈ C_ℋ (((𝑆‘𝑥) = 1 → (𝑆‘𝑦) = 1) → 𝑥 ⊆ 𝑦) → (((𝑆‘𝐴) = 1 → (𝑆‘𝐵) = 1) → 𝐴 ⊆ 𝐵)))
13	2, 3, 12	mp2an 704	. 2 ⊢ (∀𝑥 ∈ C_ℋ ∀𝑦 ∈ C_ℋ (((𝑆‘𝑥) = 1 → (𝑆‘𝑦) = 1) → 𝑥 ⊆ 𝑦) → (((𝑆‘𝐴) = 1 → (𝑆‘𝐵) = 1) → 𝐴 ⊆ 𝐵))
14	1, 13	simplbiim 513	1 ⊢ (𝜑 → (((𝑆‘𝐴) = 1 → (𝑆‘𝐵) = 1) → 𝐴 ⊆ 𝐵))

Colors of variables: wff setvar class

Syntax hints: → wi 4 ↔ wb 209 ∧ wa 400 = wceq 1567 ∈ wcel 2149 ∀wral 3085 ⊆ wss 3913 ‘cfv 6537 1c1 11100 C_ℋ cch 31221 Statescst 31254

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1822 ax-4 1836 ax-5 1937 ax-6 1994 ax-7 2035 ax-8 2151 ax-9 2159 ax-ext 2741

This theorem depends on definitions: df-bi 210 df-an 401 df-or 861 df-3an 1103 df-tru 1570 df-fal 1580 df-ex 1807 df-sb 2098 df-clab 2748 df-cleq 2761 df-clel 2844 df-ral 3086 df-rab 3424 df-v 3465 df-dif 3916 df-un 3918 df-ss 3930 df-nul 4295 df-if 4493 df-sn 4595 df-pr 4597 df-op 4601 df-uni 4877 df-br 5114 df-iota 6493 df-fv 6545

This theorem is referenced by: stcltr2i 32567 stcltrlem2 32569

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