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Theorem isst 32232
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: C ⟶(0[,]1) ∧ (𝑆‘ ℋ) = 1 ∧ ∀𝑥C𝑦C (𝑥 ⊆ (⊥‘𝑦) → (𝑆‘(𝑥 𝑦)) = ((𝑆𝑥) + (𝑆𝑦)))))
Distinct variable group:   𝑥,𝑦,𝑆

Proof of Theorem isst
Dummy variable 𝑓 is distinct from all other variables.
StepHypRef Expression
1 ovex 7464 . . . 4 (0[,]1) ∈ V
2 chex 31245 . . . 4 C ∈ V
31, 2elmap 8911 . . 3 (𝑆 ∈ ((0[,]1) ↑m C ) ↔ 𝑆: C ⟶(0[,]1))
43anbi1i 624 . 2 ((𝑆 ∈ ((0[,]1) ↑m C ) ∧ ((𝑆‘ ℋ) = 1 ∧ ∀𝑥C𝑦C (𝑥 ⊆ (⊥‘𝑦) → (𝑆‘(𝑥 𝑦)) = ((𝑆𝑥) + (𝑆𝑦))))) ↔ (𝑆: C ⟶(0[,]1) ∧ ((𝑆‘ ℋ) = 1 ∧ ∀𝑥C𝑦C (𝑥 ⊆ (⊥‘𝑦) → (𝑆‘(𝑥 𝑦)) = ((𝑆𝑥) + (𝑆𝑦))))))
5 fveq1 6905 . . . . 5 (𝑓 = 𝑆 → (𝑓‘ ℋ) = (𝑆‘ ℋ))
65eqeq1d 2739 . . . 4 (𝑓 = 𝑆 → ((𝑓‘ ℋ) = 1 ↔ (𝑆‘ ℋ) = 1))
7 fveq1 6905 . . . . . . 7 (𝑓 = 𝑆 → (𝑓‘(𝑥 𝑦)) = (𝑆‘(𝑥 𝑦)))
8 fveq1 6905 . . . . . . . 8 (𝑓 = 𝑆 → (𝑓𝑥) = (𝑆𝑥))
9 fveq1 6905 . . . . . . . 8 (𝑓 = 𝑆 → (𝑓𝑦) = (𝑆𝑦))
108, 9oveq12d 7449 . . . . . . 7 (𝑓 = 𝑆 → ((𝑓𝑥) + (𝑓𝑦)) = ((𝑆𝑥) + (𝑆𝑦)))
117, 10eqeq12d 2753 . . . . . 6 (𝑓 = 𝑆 → ((𝑓‘(𝑥 𝑦)) = ((𝑓𝑥) + (𝑓𝑦)) ↔ (𝑆‘(𝑥 𝑦)) = ((𝑆𝑥) + (𝑆𝑦))))
1211imbi2d 340 . . . . 5 (𝑓 = 𝑆 → ((𝑥 ⊆ (⊥‘𝑦) → (𝑓‘(𝑥 𝑦)) = ((𝑓𝑥) + (𝑓𝑦))) ↔ (𝑥 ⊆ (⊥‘𝑦) → (𝑆‘(𝑥 𝑦)) = ((𝑆𝑥) + (𝑆𝑦)))))
13122ralbidv 3221 . . . 4 (𝑓 = 𝑆 → (∀𝑥C𝑦C (𝑥 ⊆ (⊥‘𝑦) → (𝑓‘(𝑥 𝑦)) = ((𝑓𝑥) + (𝑓𝑦))) ↔ ∀𝑥C𝑦C (𝑥 ⊆ (⊥‘𝑦) → (𝑆‘(𝑥 𝑦)) = ((𝑆𝑥) + (𝑆𝑦)))))
146, 13anbi12d 632 . . 3 (𝑓 = 𝑆 → (((𝑓‘ ℋ) = 1 ∧ ∀𝑥C𝑦C (𝑥 ⊆ (⊥‘𝑦) → (𝑓‘(𝑥 𝑦)) = ((𝑓𝑥) + (𝑓𝑦)))) ↔ ((𝑆‘ ℋ) = 1 ∧ ∀𝑥C𝑦C (𝑥 ⊆ (⊥‘𝑦) → (𝑆‘(𝑥 𝑦)) = ((𝑆𝑥) + (𝑆𝑦))))))
15 df-st 32230 . . 3 States = {𝑓 ∈ ((0[,]1) ↑m C ) ∣ ((𝑓‘ ℋ) = 1 ∧ ∀𝑥C𝑦C (𝑥 ⊆ (⊥‘𝑦) → (𝑓‘(𝑥 𝑦)) = ((𝑓𝑥) + (𝑓𝑦))))}
1614, 15elrab2 3695 . 2 (𝑆 ∈ States ↔ (𝑆 ∈ ((0[,]1) ↑m C ) ∧ ((𝑆‘ ℋ) = 1 ∧ ∀𝑥C𝑦C (𝑥 ⊆ (⊥‘𝑦) → (𝑆‘(𝑥 𝑦)) = ((𝑆𝑥) + (𝑆𝑦))))))
17 3anass 1095 . 2 ((𝑆: C ⟶(0[,]1) ∧ (𝑆‘ ℋ) = 1 ∧ ∀𝑥C𝑦C (𝑥 ⊆ (⊥‘𝑦) → (𝑆‘(𝑥 𝑦)) = ((𝑆𝑥) + (𝑆𝑦)))) ↔ (𝑆: C ⟶(0[,]1) ∧ ((𝑆‘ ℋ) = 1 ∧ ∀𝑥C𝑦C (𝑥 ⊆ (⊥‘𝑦) → (𝑆‘(𝑥 𝑦)) = ((𝑆𝑥) + (𝑆𝑦))))))
184, 16, 173bitr4i 303 1 (𝑆 ∈ States ↔ (𝑆: C ⟶(0[,]1) ∧ (𝑆‘ ℋ) = 1 ∧ ∀𝑥C𝑦C (𝑥 ⊆ (⊥‘𝑦) → (𝑆‘(𝑥 𝑦)) = ((𝑆𝑥) + (𝑆𝑦)))))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 206  wa 395  w3a 1087   = wceq 1540  wcel 2108  wral 3061  wss 3951  wf 6557  cfv 6561  (class class class)co 7431  m cmap 8866  0cc0 11155  1c1 11156   + caddc 11158  [,]cicc 13390  chba 30938   C cch 30948  cort 30949   chj 30952  Statescst 30981
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 2708  ax-sep 5296  ax-nul 5306  ax-pow 5365  ax-pr 5432  ax-un 7755  ax-hilex 31018
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3an 1089  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2065  df-mo 2540  df-eu 2569  df-clab 2715  df-cleq 2729  df-clel 2816  df-nfc 2892  df-ne 2941  df-ral 3062  df-rex 3071  df-rab 3437  df-v 3482  df-sbc 3789  df-dif 3954  df-un 3956  df-in 3958  df-ss 3968  df-nul 4334  df-if 4526  df-pw 4602  df-sn 4627  df-pr 4629  df-op 4633  df-uni 4908  df-br 5144  df-opab 5206  df-id 5578  df-xp 5691  df-rel 5692  df-cnv 5693  df-co 5694  df-dm 5695  df-rn 5696  df-res 5697  df-ima 5698  df-iota 6514  df-fun 6563  df-fn 6564  df-f 6565  df-fv 6569  df-ov 7434  df-oprab 7435  df-mpo 7436  df-map 8868  df-sh 31226  df-ch 31240  df-st 32230
This theorem is referenced by:  sticl  32234  sthil  32253  stj  32254  strlem3a  32271
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