Theorem isst 30161

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 7216	. . . 4 ⊢ (0[,]1) ∈ V
2		chex 29174	. . . 4 ⊢ Cℋ ∈ V
3	1, 2	elmap 8494	. . 3 ⊢ (𝑆 ∈ ((0[,]1) ↑m Cℋ ) ↔ 𝑆: Cℋ ⟶(0[,]1))
4	3	anbi1i 627	. 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 6686	. . . . 5 ⊢ (𝑓 = 𝑆 → (𝑓‘ ℋ) = (𝑆‘ ℋ))
6	5	eqeq1d 2741	. . . 4 ⊢ (𝑓 = 𝑆 → ((𝑓‘ ℋ) = 1 ↔ (𝑆‘ ℋ) = 1))
7		fveq1 6686	. . . . . . 7 ⊢ (𝑓 = 𝑆 → (𝑓‘(𝑥 ∨ℋ 𝑦)) = (𝑆‘(𝑥 ∨ℋ 𝑦)))
8		fveq1 6686	. . . . . . . 8 ⊢ (𝑓 = 𝑆 → (𝑓‘𝑥) = (𝑆‘𝑥))
9		fveq1 6686	. . . . . . . 8 ⊢ (𝑓 = 𝑆 → (𝑓‘𝑦) = (𝑆‘𝑦))
10	8, 9	oveq12d 7201	. . . . . . 7 ⊢ (𝑓 = 𝑆 → ((𝑓‘𝑥) + (𝑓‘𝑦)) = ((𝑆‘𝑥) + (𝑆‘𝑦)))
11	7, 10	eqeq12d 2755	. . . . . 6 ⊢ (𝑓 = 𝑆 → ((𝑓‘(𝑥 ∨ℋ 𝑦)) = ((𝑓‘𝑥) + (𝑓‘𝑦)) ↔ (𝑆‘(𝑥 ∨ℋ 𝑦)) = ((𝑆‘𝑥) + (𝑆‘𝑦))))
12	11	imbi2d 344	. . . . 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 3112	. . . 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 634	. . 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 30159	. . 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 3596	. 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 1096	. 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 306	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 209   ∧ wa 399   ∧ w3a 1088   = wceq 1542   ∈ wcel 2114  ∀wral 3054   ⊆ wss 3853  ⟶wf 6346  ‘cfv 6350  (class class class)co 7183   ↑_m cmap 8450  0cc0 10628  1c1 10629   + caddc 10631  [,]cicc 12837   ℋchba 28867   C_ℋ cch 28877  ⊥cort 28878   ∨_ℋ chj 28881  Statescst 28910

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1802  ax-4 1816  ax-5 1917  ax-6 1975  ax-7 2020  ax-8 2116  ax-9 2124  ax-10 2145  ax-11 2162  ax-12 2179  ax-ext 2711  ax-sep 5177  ax-nul 5184  ax-pow 5242  ax-pr 5306  ax-un 7492  ax-hilex 28947

This theorem depends on definitions:  df-bi 210  df-an 400  df-or 847  df-3an 1090  df-tru 1545  df-fal 1555  df-ex 1787  df-nf 1791  df-sb 2075  df-mo 2541  df-eu 2571  df-clab 2718  df-cleq 2731  df-clel 2812  df-nfc 2882  df-ral 3059  df-rex 3060  df-rab 3063  df-v 3402  df-sbc 3686  df-dif 3856  df-un 3858  df-in 3860  df-ss 3870  df-nul 4222  df-if 4425  df-pw 4500  df-sn 4527  df-pr 4529  df-op 4533  df-uni 4807  df-br 5041  df-opab 5103  df-id 5439  df-xp 5541  df-rel 5542  df-cnv 5543  df-co 5544  df-dm 5545  df-rn 5546  df-res 5547  df-ima 5548  df-iota 6308  df-fun 6352  df-fn 6353  df-f 6354  df-fv 6358  df-ov 7186  df-oprab 7187  df-mpo 7188  df-map 8452  df-sh 29155  df-ch 29169  df-st 30159

This theorem is referenced by:  sticl  30163  sthil  30182  stj  30183  strlem3a  30200