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Theorem stj 29638
Description: The value of a state on a join. (Contributed by NM, 23-Oct-1999.) (New usage is discouraged.)
Assertion
Ref Expression
stj (𝑆 ∈ States → (((𝐴C𝐵C ) ∧ 𝐴 ⊆ (⊥‘𝐵)) → (𝑆‘(𝐴 𝐵)) = ((𝑆𝐴) + (𝑆𝐵))))

Proof of Theorem stj
Dummy variables 𝑥 𝑦 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 isst 29616 . . . 4 (𝑆 ∈ States ↔ (𝑆: C ⟶(0[,]1) ∧ (𝑆‘ ℋ) = 1 ∧ ∀𝑥C𝑦C (𝑥 ⊆ (⊥‘𝑦) → (𝑆‘(𝑥 𝑦)) = ((𝑆𝑥) + (𝑆𝑦)))))
21simp3bi 1181 . . 3 (𝑆 ∈ States → ∀𝑥C𝑦C (𝑥 ⊆ (⊥‘𝑦) → (𝑆‘(𝑥 𝑦)) = ((𝑆𝑥) + (𝑆𝑦))))
3 sseq1 3851 . . . . 5 (𝑥 = 𝐴 → (𝑥 ⊆ (⊥‘𝑦) ↔ 𝐴 ⊆ (⊥‘𝑦)))
4 fvoveq1 6928 . . . . . 6 (𝑥 = 𝐴 → (𝑆‘(𝑥 𝑦)) = (𝑆‘(𝐴 𝑦)))
5 fveq2 6433 . . . . . . 7 (𝑥 = 𝐴 → (𝑆𝑥) = (𝑆𝐴))
65oveq1d 6920 . . . . . 6 (𝑥 = 𝐴 → ((𝑆𝑥) + (𝑆𝑦)) = ((𝑆𝐴) + (𝑆𝑦)))
74, 6eqeq12d 2840 . . . . 5 (𝑥 = 𝐴 → ((𝑆‘(𝑥 𝑦)) = ((𝑆𝑥) + (𝑆𝑦)) ↔ (𝑆‘(𝐴 𝑦)) = ((𝑆𝐴) + (𝑆𝑦))))
83, 7imbi12d 336 . . . 4 (𝑥 = 𝐴 → ((𝑥 ⊆ (⊥‘𝑦) → (𝑆‘(𝑥 𝑦)) = ((𝑆𝑥) + (𝑆𝑦))) ↔ (𝐴 ⊆ (⊥‘𝑦) → (𝑆‘(𝐴 𝑦)) = ((𝑆𝐴) + (𝑆𝑦)))))
9 fveq2 6433 . . . . . 6 (𝑦 = 𝐵 → (⊥‘𝑦) = (⊥‘𝐵))
109sseq2d 3858 . . . . 5 (𝑦 = 𝐵 → (𝐴 ⊆ (⊥‘𝑦) ↔ 𝐴 ⊆ (⊥‘𝐵)))
11 oveq2 6913 . . . . . . 7 (𝑦 = 𝐵 → (𝐴 𝑦) = (𝐴 𝐵))
1211fveq2d 6437 . . . . . 6 (𝑦 = 𝐵 → (𝑆‘(𝐴 𝑦)) = (𝑆‘(𝐴 𝐵)))
13 fveq2 6433 . . . . . . 7 (𝑦 = 𝐵 → (𝑆𝑦) = (𝑆𝐵))
1413oveq2d 6921 . . . . . 6 (𝑦 = 𝐵 → ((𝑆𝐴) + (𝑆𝑦)) = ((𝑆𝐴) + (𝑆𝐵)))
1512, 14eqeq12d 2840 . . . . 5 (𝑦 = 𝐵 → ((𝑆‘(𝐴 𝑦)) = ((𝑆𝐴) + (𝑆𝑦)) ↔ (𝑆‘(𝐴 𝐵)) = ((𝑆𝐴) + (𝑆𝐵))))
1610, 15imbi12d 336 . . . 4 (𝑦 = 𝐵 → ((𝐴 ⊆ (⊥‘𝑦) → (𝑆‘(𝐴 𝑦)) = ((𝑆𝐴) + (𝑆𝑦))) ↔ (𝐴 ⊆ (⊥‘𝐵) → (𝑆‘(𝐴 𝐵)) = ((𝑆𝐴) + (𝑆𝐵)))))
178, 16rspc2v 3539 . . 3 ((𝐴C𝐵C ) → (∀𝑥C𝑦C (𝑥 ⊆ (⊥‘𝑦) → (𝑆‘(𝑥 𝑦)) = ((𝑆𝑥) + (𝑆𝑦))) → (𝐴 ⊆ (⊥‘𝐵) → (𝑆‘(𝐴 𝐵)) = ((𝑆𝐴) + (𝑆𝐵)))))
182, 17syl5com 31 . 2 (𝑆 ∈ States → ((𝐴C𝐵C ) → (𝐴 ⊆ (⊥‘𝐵) → (𝑆‘(𝐴 𝐵)) = ((𝑆𝐴) + (𝑆𝐵)))))
1918impd 400 1 (𝑆 ∈ States → (((𝐴C𝐵C ) ∧ 𝐴 ⊆ (⊥‘𝐵)) → (𝑆‘(𝐴 𝐵)) = ((𝑆𝐴) + (𝑆𝐵))))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 386   = wceq 1656  wcel 2164  wral 3117  wss 3798  wf 6119  cfv 6123  (class class class)co 6905  0cc0 10252  1c1 10253   + caddc 10255  [,]cicc 12466  chba 28320   C cch 28330  cort 28331   chj 28334  Statescst 28363
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1894  ax-4 1908  ax-5 2009  ax-6 2075  ax-7 2112  ax-8 2166  ax-9 2173  ax-10 2192  ax-11 2207  ax-12 2220  ax-13 2389  ax-ext 2803  ax-sep 5005  ax-nul 5013  ax-pow 5065  ax-pr 5127  ax-un 7209  ax-hilex 28400
This theorem depends on definitions:  df-bi 199  df-an 387  df-or 879  df-3an 1113  df-tru 1660  df-ex 1879  df-nf 1883  df-sb 2068  df-mo 2605  df-eu 2640  df-clab 2812  df-cleq 2818  df-clel 2821  df-nfc 2958  df-ral 3122  df-rex 3123  df-rab 3126  df-v 3416  df-sbc 3663  df-dif 3801  df-un 3803  df-in 3805  df-ss 3812  df-nul 4145  df-if 4307  df-pw 4380  df-sn 4398  df-pr 4400  df-op 4404  df-uni 4659  df-br 4874  df-opab 4936  df-id 5250  df-xp 5348  df-rel 5349  df-cnv 5350  df-co 5351  df-dm 5352  df-rn 5353  df-res 5354  df-ima 5355  df-iota 6086  df-fun 6125  df-fn 6126  df-f 6127  df-fv 6131  df-ov 6908  df-oprab 6909  df-mpt2 6910  df-map 8124  df-sh 28608  df-ch 28622  df-st 29614
This theorem is referenced by:  sto1i  29639  stlei  29643  stji1i  29645
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