Theorem isst 32140

Description: Property of a state. (Contributed by NM, 23-Oct-1999.) (Revised by Mario Carneiro, 23-Dec-2013.) (New usage is discouraged.)

Assertion

Ref	Expression
isst	⊢ (𝑆 ∈ States ↔ (𝑆: Cℋ ⟶(0[,]1) ∧ (𝑆‘ ℋ) = 1 ∧ ∀𝑥 ∈ Cℋ ∀𝑦 ∈ Cℋ (𝑥 ⊆ (⊥‘𝑦) → (𝑆‘(𝑥 ∨ℋ 𝑦)) = ((𝑆‘𝑥) + (𝑆‘𝑦)))))

Distinct variable group:   𝑥,𝑦,𝑆

Proof of Theorem isst

Dummy variable 𝑓 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		ovex 7436	. . . 4 ⊢ (0[,]1) ∈ V
2		chex 31153	. . . 4 ⊢ Cℋ ∈ V
3	1, 2	elmap 8883	. . 3 ⊢ (𝑆 ∈ ((0[,]1) ↑m Cℋ ) ↔ 𝑆: Cℋ ⟶(0[,]1))
4	3	anbi1i 624	. 2 ⊢ ((𝑆 ∈ ((0[,]1) ↑m Cℋ ) ∧ ((𝑆‘ ℋ) = 1 ∧ ∀𝑥 ∈ Cℋ ∀𝑦 ∈ Cℋ (𝑥 ⊆ (⊥‘𝑦) → (𝑆‘(𝑥 ∨ℋ 𝑦)) = ((𝑆‘𝑥) + (𝑆‘𝑦))))) ↔ (𝑆: Cℋ ⟶(0[,]1) ∧ ((𝑆‘ ℋ) = 1 ∧ ∀𝑥 ∈ Cℋ ∀𝑦 ∈ Cℋ (𝑥 ⊆ (⊥‘𝑦) → (𝑆‘(𝑥 ∨ℋ 𝑦)) = ((𝑆‘𝑥) + (𝑆‘𝑦))))))
5		fveq1 6874	. . . . 5 ⊢ (𝑓 = 𝑆 → (𝑓‘ ℋ) = (𝑆‘ ℋ))
6	5	eqeq1d 2737	. . . 4 ⊢ (𝑓 = 𝑆 → ((𝑓‘ ℋ) = 1 ↔ (𝑆‘ ℋ) = 1))
7		fveq1 6874	. . . . . . 7 ⊢ (𝑓 = 𝑆 → (𝑓‘(𝑥 ∨ℋ 𝑦)) = (𝑆‘(𝑥 ∨ℋ 𝑦)))
8		fveq1 6874	. . . . . . . 8 ⊢ (𝑓 = 𝑆 → (𝑓‘𝑥) = (𝑆‘𝑥))
9		fveq1 6874	. . . . . . . 8 ⊢ (𝑓 = 𝑆 → (𝑓‘𝑦) = (𝑆‘𝑦))
10	8, 9	oveq12d 7421	. . . . . . 7 ⊢ (𝑓 = 𝑆 → ((𝑓‘𝑥) + (𝑓‘𝑦)) = ((𝑆‘𝑥) + (𝑆‘𝑦)))
11	7, 10	eqeq12d 2751	. . . . . 6 ⊢ (𝑓 = 𝑆 → ((𝑓‘(𝑥 ∨ℋ 𝑦)) = ((𝑓‘𝑥) + (𝑓‘𝑦)) ↔ (𝑆‘(𝑥 ∨ℋ 𝑦)) = ((𝑆‘𝑥) + (𝑆‘𝑦))))
12	11	imbi2d 340	. . . . 5 ⊢ (𝑓 = 𝑆 → ((𝑥 ⊆ (⊥‘𝑦) → (𝑓‘(𝑥 ∨ℋ 𝑦)) = ((𝑓‘𝑥) + (𝑓‘𝑦))) ↔ (𝑥 ⊆ (⊥‘𝑦) → (𝑆‘(𝑥 ∨ℋ 𝑦)) = ((𝑆‘𝑥) + (𝑆‘𝑦)))))
13	12	2ralbidv 3205	. . . 4 ⊢ (𝑓 = 𝑆 → (∀𝑥 ∈ Cℋ ∀𝑦 ∈ Cℋ (𝑥 ⊆ (⊥‘𝑦) → (𝑓‘(𝑥 ∨ℋ 𝑦)) = ((𝑓‘𝑥) + (𝑓‘𝑦))) ↔ ∀𝑥 ∈ Cℋ ∀𝑦 ∈ Cℋ (𝑥 ⊆ (⊥‘𝑦) → (𝑆‘(𝑥 ∨ℋ 𝑦)) = ((𝑆‘𝑥) + (𝑆‘𝑦)))))
14	6, 13	anbi12d 632	. . 3 ⊢ (𝑓 = 𝑆 → (((𝑓‘ ℋ) = 1 ∧ ∀𝑥 ∈ Cℋ ∀𝑦 ∈ Cℋ (𝑥 ⊆ (⊥‘𝑦) → (𝑓‘(𝑥 ∨ℋ 𝑦)) = ((𝑓‘𝑥) + (𝑓‘𝑦)))) ↔ ((𝑆‘ ℋ) = 1 ∧ ∀𝑥 ∈ Cℋ ∀𝑦 ∈ Cℋ (𝑥 ⊆ (⊥‘𝑦) → (𝑆‘(𝑥 ∨ℋ 𝑦)) = ((𝑆‘𝑥) + (𝑆‘𝑦))))))
15		df-st 32138	. . 3 ⊢ States = {𝑓 ∈ ((0[,]1) ↑m Cℋ ) ∣ ((𝑓‘ ℋ) = 1 ∧ ∀𝑥 ∈ Cℋ ∀𝑦 ∈ Cℋ (𝑥 ⊆ (⊥‘𝑦) → (𝑓‘(𝑥 ∨ℋ 𝑦)) = ((𝑓‘𝑥) + (𝑓‘𝑦))))}
16	14, 15	elrab2 3674	. 2 ⊢ (𝑆 ∈ States ↔ (𝑆 ∈ ((0[,]1) ↑m Cℋ ) ∧ ((𝑆‘ ℋ) = 1 ∧ ∀𝑥 ∈ Cℋ ∀𝑦 ∈ Cℋ (𝑥 ⊆ (⊥‘𝑦) → (𝑆‘(𝑥 ∨ℋ 𝑦)) = ((𝑆‘𝑥) + (𝑆‘𝑦))))))
17		3anass 1094	. 2 ⊢ ((𝑆: Cℋ ⟶(0[,]1) ∧ (𝑆‘ ℋ) = 1 ∧ ∀𝑥 ∈ Cℋ ∀𝑦 ∈ Cℋ (𝑥 ⊆ (⊥‘𝑦) → (𝑆‘(𝑥 ∨ℋ 𝑦)) = ((𝑆‘𝑥) + (𝑆‘𝑦)))) ↔ (𝑆: Cℋ ⟶(0[,]1) ∧ ((𝑆‘ ℋ) = 1 ∧ ∀𝑥 ∈ Cℋ ∀𝑦 ∈ Cℋ (𝑥 ⊆ (⊥‘𝑦) → (𝑆‘(𝑥 ∨ℋ 𝑦)) = ((𝑆‘𝑥) + (𝑆‘𝑦))))))
18	4, 16, 17	3bitr4i 303	1 ⊢ (𝑆 ∈ States ↔ (𝑆: Cℋ ⟶(0[,]1) ∧ (𝑆‘ ℋ) = 1 ∧ ∀𝑥 ∈ Cℋ ∀𝑦 ∈ Cℋ (𝑥 ⊆ (⊥‘𝑦) → (𝑆‘(𝑥 ∨ℋ 𝑦)) = ((𝑆‘𝑥) + (𝑆‘𝑦)))))

Syntax hints:   → wi 4   ↔ wb 206   ∧ wa 395   ∧ w3a 1086   = wceq 1540   ∈ wcel 2108  ∀wral 3051   ⊆ wss 3926  ⟶wf 6526  ‘cfv 6530  (class class class)co 7403   ↑_m cmap 8838  0cc0 11127  1c1 11128   + caddc 11130  [,]cicc 13363   ℋchba 30846   C_ℋ cch 30856  ⊥cort 30857   ∨_ℋ chj 30860  Statescst 30889

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 2007  ax-8 2110  ax-9 2118  ax-10 2141  ax-11 2157  ax-12 2177  ax-ext 2707  ax-sep 5266  ax-nul 5276  ax-pow 5335  ax-pr 5402  ax-un 7727  ax-hilex 30926

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 2065  df-mo 2539  df-eu 2568  df-clab 2714  df-cleq 2727  df-clel 2809  df-nfc 2885  df-ne 2933  df-ral 3052  df-rex 3061  df-rab 3416  df-v 3461  df-sbc 3766  df-dif 3929  df-un 3931  df-in 3933  df-ss 3943  df-nul 4309  df-if 4501  df-pw 4577  df-sn 4602  df-pr 4604  df-op 4608  df-uni 4884  df-br 5120  df-opab 5182  df-id 5548  df-xp 5660  df-rel 5661  df-cnv 5662  df-co 5663  df-dm 5664  df-rn 5665  df-res 5666  df-ima 5667  df-iota 6483  df-fun 6532  df-fn 6533  df-f 6534  df-fv 6538  df-ov 7406  df-oprab 7407  df-mpo 7408  df-map 8840  df-sh 31134  df-ch 31148  df-st 32138

This theorem is referenced by:  sticl  32142  sthil  32161  stj  32162  strlem3a  32179