Theorem stj 32038

Description: The value of a state on a join. (Contributed by NM, 23-Oct-1999.) (New usage is discouraged.)

Assertion

Ref	Expression
stj	⊢ (𝑆 ∈ States → (((𝐴 ∈ Cℋ ∧ 𝐵 ∈ Cℋ ) ∧ 𝐴 ⊆ (⊥‘𝐵)) → (𝑆‘(𝐴 ∨ℋ 𝐵)) = ((𝑆‘𝐴) + (𝑆‘𝐵))))

Proof of Theorem stj

Dummy variables 𝑥 𝑦 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		isst 32016	. . . 4 ⊢ (𝑆 ∈ States ↔ (𝑆: Cℋ ⟶(0[,]1) ∧ (𝑆‘ ℋ) = 1 ∧ ∀𝑥 ∈ Cℋ ∀𝑦 ∈ Cℋ (𝑥 ⊆ (⊥‘𝑦) → (𝑆‘(𝑥 ∨ℋ 𝑦)) = ((𝑆‘𝑥) + (𝑆‘𝑦)))))
2	1	simp3bi 1145	. . 3 ⊢ (𝑆 ∈ States → ∀𝑥 ∈ Cℋ ∀𝑦 ∈ Cℋ (𝑥 ⊆ (⊥‘𝑦) → (𝑆‘(𝑥 ∨ℋ 𝑦)) = ((𝑆‘𝑥) + (𝑆‘𝑦))))
3		sseq1 4003	. . . . 5 ⊢ (𝑥 = 𝐴 → (𝑥 ⊆ (⊥‘𝑦) ↔ 𝐴 ⊆ (⊥‘𝑦)))
4		fvoveq1 7437	. . . . . 6 ⊢ (𝑥 = 𝐴 → (𝑆‘(𝑥 ∨ℋ 𝑦)) = (𝑆‘(𝐴 ∨ℋ 𝑦)))
5		fveq2 6891	. . . . . . 7 ⊢ (𝑥 = 𝐴 → (𝑆‘𝑥) = (𝑆‘𝐴))
6	5	oveq1d 7429	. . . . . 6 ⊢ (𝑥 = 𝐴 → ((𝑆‘𝑥) + (𝑆‘𝑦)) = ((𝑆‘𝐴) + (𝑆‘𝑦)))
7	4, 6	eqeq12d 2743	. . . . 5 ⊢ (𝑥 = 𝐴 → ((𝑆‘(𝑥 ∨ℋ 𝑦)) = ((𝑆‘𝑥) + (𝑆‘𝑦)) ↔ (𝑆‘(𝐴 ∨ℋ 𝑦)) = ((𝑆‘𝐴) + (𝑆‘𝑦))))
8	3, 7	imbi12d 344	. . . 4 ⊢ (𝑥 = 𝐴 → ((𝑥 ⊆ (⊥‘𝑦) → (𝑆‘(𝑥 ∨ℋ 𝑦)) = ((𝑆‘𝑥) + (𝑆‘𝑦))) ↔ (𝐴 ⊆ (⊥‘𝑦) → (𝑆‘(𝐴 ∨ℋ 𝑦)) = ((𝑆‘𝐴) + (𝑆‘𝑦)))))
9		fveq2 6891	. . . . . 6 ⊢ (𝑦 = 𝐵 → (⊥‘𝑦) = (⊥‘𝐵))
10	9	sseq2d 4010	. . . . 5 ⊢ (𝑦 = 𝐵 → (𝐴 ⊆ (⊥‘𝑦) ↔ 𝐴 ⊆ (⊥‘𝐵)))
11		oveq2 7422	. . . . . . 7 ⊢ (𝑦 = 𝐵 → (𝐴 ∨ℋ 𝑦) = (𝐴 ∨ℋ 𝐵))
12	11	fveq2d 6895	. . . . . 6 ⊢ (𝑦 = 𝐵 → (𝑆‘(𝐴 ∨ℋ 𝑦)) = (𝑆‘(𝐴 ∨ℋ 𝐵)))
13		fveq2 6891	. . . . . . 7 ⊢ (𝑦 = 𝐵 → (𝑆‘𝑦) = (𝑆‘𝐵))
14	13	oveq2d 7430	. . . . . 6 ⊢ (𝑦 = 𝐵 → ((𝑆‘𝐴) + (𝑆‘𝑦)) = ((𝑆‘𝐴) + (𝑆‘𝐵)))
15	12, 14	eqeq12d 2743	. . . . 5 ⊢ (𝑦 = 𝐵 → ((𝑆‘(𝐴 ∨ℋ 𝑦)) = ((𝑆‘𝐴) + (𝑆‘𝑦)) ↔ (𝑆‘(𝐴 ∨ℋ 𝐵)) = ((𝑆‘𝐴) + (𝑆‘𝐵))))
16	10, 15	imbi12d 344	. . . 4 ⊢ (𝑦 = 𝐵 → ((𝐴 ⊆ (⊥‘𝑦) → (𝑆‘(𝐴 ∨ℋ 𝑦)) = ((𝑆‘𝐴) + (𝑆‘𝑦))) ↔ (𝐴 ⊆ (⊥‘𝐵) → (𝑆‘(𝐴 ∨ℋ 𝐵)) = ((𝑆‘𝐴) + (𝑆‘𝐵)))))
17	8, 16	rspc2v 3618	. . 3 ⊢ ((𝐴 ∈ Cℋ ∧ 𝐵 ∈ Cℋ ) → (∀𝑥 ∈ Cℋ ∀𝑦 ∈ Cℋ (𝑥 ⊆ (⊥‘𝑦) → (𝑆‘(𝑥 ∨ℋ 𝑦)) = ((𝑆‘𝑥) + (𝑆‘𝑦))) → (𝐴 ⊆ (⊥‘𝐵) → (𝑆‘(𝐴 ∨ℋ 𝐵)) = ((𝑆‘𝐴) + (𝑆‘𝐵)))))
18	2, 17	syl5com 31	. 2 ⊢ (𝑆 ∈ States → ((𝐴 ∈ Cℋ ∧ 𝐵 ∈ Cℋ ) → (𝐴 ⊆ (⊥‘𝐵) → (𝑆‘(𝐴 ∨ℋ 𝐵)) = ((𝑆‘𝐴) + (𝑆‘𝐵)))))
19	18	impd 410	1 ⊢ (𝑆 ∈ States → (((𝐴 ∈ Cℋ ∧ 𝐵 ∈ Cℋ ) ∧ 𝐴 ⊆ (⊥‘𝐵)) → (𝑆‘(𝐴 ∨ℋ 𝐵)) = ((𝑆‘𝐴) + (𝑆‘𝐵))))

Syntax hints:   → wi 4   ∧ wa 395   = wceq 1534   ∈ wcel 2099  ∀wral 3056   ⊆ wss 3944  ⟶wf 6538  ‘cfv 6542  (class class class)co 7414  0cc0 11132  1c1 11133   + caddc 11135  [,]cicc 13353   ℋchba 30722   C_ℋ cch 30732  ⊥cort 30733   ∨_ℋ chj 30736  Statescst 30765

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1790  ax-4 1804  ax-5 1906  ax-6 1964  ax-7 2004  ax-8 2101  ax-9 2109  ax-10 2130  ax-11 2147  ax-12 2164  ax-ext 2698  ax-sep 5293  ax-nul 5300  ax-pow 5359  ax-pr 5423  ax-un 7734  ax-hilex 30802

This theorem depends on definitions:  df-bi 206  df-an 396  df-or 847  df-3an 1087  df-tru 1537  df-fal 1547  df-ex 1775  df-nf 1779  df-sb 2061  df-mo 2529  df-eu 2558  df-clab 2705  df-cleq 2719  df-clel 2805  df-nfc 2880  df-ne 2936  df-ral 3057  df-rex 3066  df-rab 3428  df-v 3471  df-sbc 3775  df-dif 3947  df-un 3949  df-in 3951  df-ss 3961  df-nul 4319  df-if 4525  df-pw 4600  df-sn 4625  df-pr 4627  df-op 4631  df-uni 4904  df-br 5143  df-opab 5205  df-id 5570  df-xp 5678  df-rel 5679  df-cnv 5680  df-co 5681  df-dm 5682  df-rn 5683  df-res 5684  df-ima 5685  df-iota 6494  df-fun 6544  df-fn 6545  df-f 6546  df-fv 6550  df-ov 7417  df-oprab 7418  df-mpo 7419  df-map 8840  df-sh 31010  df-ch 31024  df-st 32014

This theorem is referenced by:  sto1i  32039  stlei  32043  stji1i  32045