Theorem isst 29072

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 6678	. . . 4 ⊢ (0[,]1) ∈ V
2		chex 28083	. . . 4 ⊢ Cℋ ∈ V
3	1, 2	elmap 7886	. . 3 ⊢ (𝑆 ∈ ((0[,]1) ↑𝑚 Cℋ ) ↔ 𝑆: Cℋ ⟶(0[,]1))
4	3	anbi1i 731	. 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 6190	. . . . 5 ⊢ (𝑓 = 𝑆 → (𝑓‘ ℋ) = (𝑆‘ ℋ))
6	5	eqeq1d 2624	. . . 4 ⊢ (𝑓 = 𝑆 → ((𝑓‘ ℋ) = 1 ↔ (𝑆‘ ℋ) = 1))
7		fveq1 6190	. . . . . . 7 ⊢ (𝑓 = 𝑆 → (𝑓‘(𝑥 ∨ℋ 𝑦)) = (𝑆‘(𝑥 ∨ℋ 𝑦)))
8		fveq1 6190	. . . . . . . 8 ⊢ (𝑓 = 𝑆 → (𝑓‘𝑥) = (𝑆‘𝑥))
9		fveq1 6190	. . . . . . . 8 ⊢ (𝑓 = 𝑆 → (𝑓‘𝑦) = (𝑆‘𝑦))
10	8, 9	oveq12d 6668	. . . . . . 7 ⊢ (𝑓 = 𝑆 → ((𝑓‘𝑥) + (𝑓‘𝑦)) = ((𝑆‘𝑥) + (𝑆‘𝑦)))
11	7, 10	eqeq12d 2637	. . . . . 6 ⊢ (𝑓 = 𝑆 → ((𝑓‘(𝑥 ∨ℋ 𝑦)) = ((𝑓‘𝑥) + (𝑓‘𝑦)) ↔ (𝑆‘(𝑥 ∨ℋ 𝑦)) = ((𝑆‘𝑥) + (𝑆‘𝑦))))
12	11	imbi2d 330	. . . . 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 2989	. . . 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 747	. . 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 29070	. . 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 3366	. 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 1042	. 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 292	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 196   ∧ wa 384   ∧ w3a 1037   = wceq 1483   ∈ wcel 1990  ∀wral 2912   ⊆ wss 3574  ⟶wf 5884  ‘cfv 5888  (class class class)co 6650   ↑_𝑚 cmap 7857  0cc0 9936  1c1 9937   + caddc 9939  [,]cicc 12178   ℋchil 27776   C_ℋ cch 27786  ⊥cort 27787   ∨_ℋ chj 27790  Statescst 27819

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-8 1992  ax-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246  ax-ext 2602  ax-sep 4781  ax-nul 4789  ax-pow 4843  ax-pr 4906  ax-un 6949  ax-hilex 27856

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3an 1039  df-tru 1486  df-ex 1705  df-nf 1710  df-sb 1881  df-eu 2474  df-mo 2475  df-clab 2609  df-cleq 2615  df-clel 2618  df-nfc 2753  df-ral 2917  df-rex 2918  df-rab 2921  df-v 3202  df-sbc 3436  df-dif 3577  df-un 3579  df-in 3581  df-ss 3588  df-nul 3916  df-if 4087  df-pw 4160  df-sn 4178  df-pr 4180  df-op 4184  df-uni 4437  df-br 4654  df-opab 4713  df-id 5024  df-xp 5120  df-rel 5121  df-cnv 5122  df-co 5123  df-dm 5124  df-rn 5125  df-res 5126  df-ima 5127  df-iota 5851  df-fun 5890  df-fn 5891  df-f 5892  df-fv 5896  df-ov 6653  df-oprab 6654  df-mpt2 6655  df-map 7859  df-sh 28064  df-ch 28078  df-st 29070

This theorem is referenced by:  sticl  29074  sthil  29093  stj  29094  strlem3a  29111