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Theorem isst 29904
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 7184 . . . 4 (0[,]1) ∈ V
2 chex 28917 . . . 4 C ∈ V
31, 2elmap 8428 . . 3 (𝑆 ∈ ((0[,]1) ↑m C ) ↔ 𝑆: C ⟶(0[,]1))
43anbi1i 623 . 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 6665 . . . . 5 (𝑓 = 𝑆 → (𝑓‘ ℋ) = (𝑆‘ ℋ))
65eqeq1d 2827 . . . 4 (𝑓 = 𝑆 → ((𝑓‘ ℋ) = 1 ↔ (𝑆‘ ℋ) = 1))
7 fveq1 6665 . . . . . . 7 (𝑓 = 𝑆 → (𝑓‘(𝑥 𝑦)) = (𝑆‘(𝑥 𝑦)))
8 fveq1 6665 . . . . . . . 8 (𝑓 = 𝑆 → (𝑓𝑥) = (𝑆𝑥))
9 fveq1 6665 . . . . . . . 8 (𝑓 = 𝑆 → (𝑓𝑦) = (𝑆𝑦))
108, 9oveq12d 7169 . . . . . . 7 (𝑓 = 𝑆 → ((𝑓𝑥) + (𝑓𝑦)) = ((𝑆𝑥) + (𝑆𝑦)))
117, 10eqeq12d 2841 . . . . . 6 (𝑓 = 𝑆 → ((𝑓‘(𝑥 𝑦)) = ((𝑓𝑥) + (𝑓𝑦)) ↔ (𝑆‘(𝑥 𝑦)) = ((𝑆𝑥) + (𝑆𝑦))))
1211imbi2d 342 . . . . 5 (𝑓 = 𝑆 → ((𝑥 ⊆ (⊥‘𝑦) → (𝑓‘(𝑥 𝑦)) = ((𝑓𝑥) + (𝑓𝑦))) ↔ (𝑥 ⊆ (⊥‘𝑦) → (𝑆‘(𝑥 𝑦)) = ((𝑆𝑥) + (𝑆𝑦)))))
13122ralbidv 3203 . . . 4 (𝑓 = 𝑆 → (∀𝑥C𝑦C (𝑥 ⊆ (⊥‘𝑦) → (𝑓‘(𝑥 𝑦)) = ((𝑓𝑥) + (𝑓𝑦))) ↔ ∀𝑥C𝑦C (𝑥 ⊆ (⊥‘𝑦) → (𝑆‘(𝑥 𝑦)) = ((𝑆𝑥) + (𝑆𝑦)))))
146, 13anbi12d 630 . . 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 29902 . . 3 States = {𝑓 ∈ ((0[,]1) ↑m C ) ∣ ((𝑓‘ ℋ) = 1 ∧ ∀𝑥C𝑦C (𝑥 ⊆ (⊥‘𝑦) → (𝑓‘(𝑥 𝑦)) = ((𝑓𝑥) + (𝑓𝑦))))}
1614, 15elrab2 3686 . 2 (𝑆 ∈ States ↔ (𝑆 ∈ ((0[,]1) ↑m C ) ∧ ((𝑆‘ ℋ) = 1 ∧ ∀𝑥C𝑦C (𝑥 ⊆ (⊥‘𝑦) → (𝑆‘(𝑥 𝑦)) = ((𝑆𝑥) + (𝑆𝑦))))))
17 3anass 1089 . 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 304 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 207  wa 396  w3a 1081   = wceq 1530  wcel 2107  wral 3142  wss 3939  wf 6347  cfv 6351  (class class class)co 7151  m cmap 8399  0cc0 10529  1c1 10530   + caddc 10532  [,]cicc 12734  chba 28610   C cch 28620  cort 28621   chj 28624  Statescst 28653
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1904  ax-6 1963  ax-7 2008  ax-8 2109  ax-9 2117  ax-10 2138  ax-11 2153  ax-12 2169  ax-ext 2797  ax-sep 5199  ax-nul 5206  ax-pow 5262  ax-pr 5325  ax-un 7454  ax-hilex 28690
This theorem depends on definitions:  df-bi 208  df-an 397  df-or 844  df-3an 1083  df-tru 1533  df-ex 1774  df-nf 1778  df-sb 2063  df-mo 2619  df-eu 2651  df-clab 2804  df-cleq 2818  df-clel 2897  df-nfc 2967  df-ral 3147  df-rex 3148  df-rab 3151  df-v 3501  df-sbc 3776  df-dif 3942  df-un 3944  df-in 3946  df-ss 3955  df-nul 4295  df-if 4470  df-pw 4543  df-sn 4564  df-pr 4566  df-op 4570  df-uni 4837  df-br 5063  df-opab 5125  df-id 5458  df-xp 5559  df-rel 5560  df-cnv 5561  df-co 5562  df-dm 5563  df-rn 5564  df-res 5565  df-ima 5566  df-iota 6311  df-fun 6353  df-fn 6354  df-f 6355  df-fv 6359  df-ov 7154  df-oprab 7155  df-mpo 7156  df-map 8401  df-sh 28898  df-ch 28912  df-st 29902
This theorem is referenced by:  sticl  29906  sthil  29925  stj  29926  strlem3a  29943
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