Theorem isst 32149

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 7423	. . . 4 ⊢ (0[,]1) ∈ V
2		chex 31162	. . . 4 ⊢ Cℋ ∈ V
3	1, 2	elmap 8847	. . 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 6860	. . . . 5 ⊢ (𝑓 = 𝑆 → (𝑓‘ ℋ) = (𝑆‘ ℋ))
6	5	eqeq1d 2732	. . . 4 ⊢ (𝑓 = 𝑆 → ((𝑓‘ ℋ) = 1 ↔ (𝑆‘ ℋ) = 1))
7		fveq1 6860	. . . . . . 7 ⊢ (𝑓 = 𝑆 → (𝑓‘(𝑥 ∨ℋ 𝑦)) = (𝑆‘(𝑥 ∨ℋ 𝑦)))
8		fveq1 6860	. . . . . . . 8 ⊢ (𝑓 = 𝑆 → (𝑓‘𝑥) = (𝑆‘𝑥))
9		fveq1 6860	. . . . . . . 8 ⊢ (𝑓 = 𝑆 → (𝑓‘𝑦) = (𝑆‘𝑦))
10	8, 9	oveq12d 7408	. . . . . . 7 ⊢ (𝑓 = 𝑆 → ((𝑓‘𝑥) + (𝑓‘𝑦)) = ((𝑆‘𝑥) + (𝑆‘𝑦)))
11	7, 10	eqeq12d 2746	. . . . . 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 3202	. . . 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 32147	. . 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 3665	. 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 2109  ∀wral 3045   ⊆ wss 3917  ⟶wf 6510  ‘cfv 6514  (class class class)co 7390   ↑_m cmap 8802  0cc0 11075  1c1 11076   + caddc 11078  [,]cicc 13316   ℋchba 30855   C_ℋ cch 30865  ⊥cort 30866   ∨_ℋ chj 30869  Statescst 30898

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 2702  ax-sep 5254  ax-nul 5264  ax-pow 5323  ax-pr 5390  ax-un 7714  ax-hilex 30935

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 2534  df-eu 2563  df-clab 2709  df-cleq 2722  df-clel 2804  df-nfc 2879  df-ne 2927  df-ral 3046  df-rex 3055  df-rab 3409  df-v 3452  df-sbc 3757  df-dif 3920  df-un 3922  df-in 3924  df-ss 3934  df-nul 4300  df-if 4492  df-pw 4568  df-sn 4593  df-pr 4595  df-op 4599  df-uni 4875  df-br 5111  df-opab 5173  df-id 5536  df-xp 5647  df-rel 5648  df-cnv 5649  df-co 5650  df-dm 5651  df-rn 5652  df-res 5653  df-ima 5654  df-iota 6467  df-fun 6516  df-fn 6517  df-f 6518  df-fv 6522  df-ov 7393  df-oprab 7394  df-mpo 7395  df-map 8804  df-sh 31143  df-ch 31157  df-st 32147

This theorem is referenced by:  sticl  32151  sthil  32170  stj  32171  strlem3a  32188