Theorem stj 32137

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 32115	. . . 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 1147	. . 3 ⊢ (𝑆 ∈ States → ∀𝑥 ∈ Cℋ ∀𝑦 ∈ Cℋ (𝑥 ⊆ (⊥‘𝑦) → (𝑆‘(𝑥 ∨ℋ 𝑦)) = ((𝑆‘𝑥) + (𝑆‘𝑦))))
3		sseq1 3969	. . . . 5 ⊢ (𝑥 = 𝐴 → (𝑥 ⊆ (⊥‘𝑦) ↔ 𝐴 ⊆ (⊥‘𝑦)))
4		fvoveq1 7392	. . . . . 6 ⊢ (𝑥 = 𝐴 → (𝑆‘(𝑥 ∨ℋ 𝑦)) = (𝑆‘(𝐴 ∨ℋ 𝑦)))
5		fveq2 6840	. . . . . . 7 ⊢ (𝑥 = 𝐴 → (𝑆‘𝑥) = (𝑆‘𝐴))
6	5	oveq1d 7384	. . . . . 6 ⊢ (𝑥 = 𝐴 → ((𝑆‘𝑥) + (𝑆‘𝑦)) = ((𝑆‘𝐴) + (𝑆‘𝑦)))
7	4, 6	eqeq12d 2745	. . . . 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 6840	. . . . . 6 ⊢ (𝑦 = 𝐵 → (⊥‘𝑦) = (⊥‘𝐵))
10	9	sseq2d 3976	. . . . 5 ⊢ (𝑦 = 𝐵 → (𝐴 ⊆ (⊥‘𝑦) ↔ 𝐴 ⊆ (⊥‘𝐵)))
11		oveq2 7377	. . . . . . 7 ⊢ (𝑦 = 𝐵 → (𝐴 ∨ℋ 𝑦) = (𝐴 ∨ℋ 𝐵))
12	11	fveq2d 6844	. . . . . 6 ⊢ (𝑦 = 𝐵 → (𝑆‘(𝐴 ∨ℋ 𝑦)) = (𝑆‘(𝐴 ∨ℋ 𝐵)))
13		fveq2 6840	. . . . . . 7 ⊢ (𝑦 = 𝐵 → (𝑆‘𝑦) = (𝑆‘𝐵))
14	13	oveq2d 7385	. . . . . 6 ⊢ (𝑦 = 𝐵 → ((𝑆‘𝐴) + (𝑆‘𝑦)) = ((𝑆‘𝐴) + (𝑆‘𝐵)))
15	12, 14	eqeq12d 2745	. . . . 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 3596	. . 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 1540   ∈ wcel 2109  ∀wral 3044   ⊆ wss 3911  ⟶wf 6495  ‘cfv 6499  (class class class)co 7369  0cc0 11044  1c1 11045   + caddc 11047  [,]cicc 13285   ℋchba 30821   C_ℋ cch 30831  ⊥cort 30832   ∨_ℋ chj 30835  Statescst 30864

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2008  ax-8 2111  ax-9 2119  ax-10 2142  ax-11 2158  ax-12 2178  ax-ext 2701  ax-sep 5246  ax-nul 5256  ax-pow 5315  ax-pr 5382  ax-un 7691  ax-hilex 30901

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2066  df-mo 2533  df-eu 2562  df-clab 2708  df-cleq 2721  df-clel 2803  df-nfc 2878  df-ne 2926  df-ral 3045  df-rex 3054  df-rab 3403  df-v 3446  df-sbc 3751  df-dif 3914  df-un 3916  df-in 3918  df-ss 3928  df-nul 4293  df-if 4485  df-pw 4561  df-sn 4586  df-pr 4588  df-op 4592  df-uni 4868  df-br 5103  df-opab 5165  df-id 5526  df-xp 5637  df-rel 5638  df-cnv 5639  df-co 5640  df-dm 5641  df-rn 5642  df-res 5643  df-ima 5644  df-iota 6452  df-fun 6501  df-fn 6502  df-f 6503  df-fv 6507  df-ov 7372  df-oprab 7373  df-mpo 7374  df-map 8778  df-sh 31109  df-ch 31123  df-st 32113

This theorem is referenced by:  sto1i  32138  stlei  32142  stji1i  32144