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Theorem stj 30625
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 30603 . . . 4 (𝑆 ∈ States ↔ (𝑆: C ⟶(0[,]1) ∧ (𝑆‘ ℋ) = 1 ∧ ∀𝑥C𝑦C (𝑥 ⊆ (⊥‘𝑦) → (𝑆‘(𝑥 𝑦)) = ((𝑆𝑥) + (𝑆𝑦)))))
21simp3bi 1145 . . 3 (𝑆 ∈ States → ∀𝑥C𝑦C (𝑥 ⊆ (⊥‘𝑦) → (𝑆‘(𝑥 𝑦)) = ((𝑆𝑥) + (𝑆𝑦))))
3 sseq1 3948 . . . . 5 (𝑥 = 𝐴 → (𝑥 ⊆ (⊥‘𝑦) ↔ 𝐴 ⊆ (⊥‘𝑦)))
4 fvoveq1 7318 . . . . . 6 (𝑥 = 𝐴 → (𝑆‘(𝑥 𝑦)) = (𝑆‘(𝐴 𝑦)))
5 fveq2 6792 . . . . . . 7 (𝑥 = 𝐴 → (𝑆𝑥) = (𝑆𝐴))
65oveq1d 7310 . . . . . 6 (𝑥 = 𝐴 → ((𝑆𝑥) + (𝑆𝑦)) = ((𝑆𝐴) + (𝑆𝑦)))
74, 6eqeq12d 2749 . . . . 5 (𝑥 = 𝐴 → ((𝑆‘(𝑥 𝑦)) = ((𝑆𝑥) + (𝑆𝑦)) ↔ (𝑆‘(𝐴 𝑦)) = ((𝑆𝐴) + (𝑆𝑦))))
83, 7imbi12d 344 . . . 4 (𝑥 = 𝐴 → ((𝑥 ⊆ (⊥‘𝑦) → (𝑆‘(𝑥 𝑦)) = ((𝑆𝑥) + (𝑆𝑦))) ↔ (𝐴 ⊆ (⊥‘𝑦) → (𝑆‘(𝐴 𝑦)) = ((𝑆𝐴) + (𝑆𝑦)))))
9 fveq2 6792 . . . . . 6 (𝑦 = 𝐵 → (⊥‘𝑦) = (⊥‘𝐵))
109sseq2d 3955 . . . . 5 (𝑦 = 𝐵 → (𝐴 ⊆ (⊥‘𝑦) ↔ 𝐴 ⊆ (⊥‘𝐵)))
11 oveq2 7303 . . . . . . 7 (𝑦 = 𝐵 → (𝐴 𝑦) = (𝐴 𝐵))
1211fveq2d 6796 . . . . . 6 (𝑦 = 𝐵 → (𝑆‘(𝐴 𝑦)) = (𝑆‘(𝐴 𝐵)))
13 fveq2 6792 . . . . . . 7 (𝑦 = 𝐵 → (𝑆𝑦) = (𝑆𝐵))
1413oveq2d 7311 . . . . . 6 (𝑦 = 𝐵 → ((𝑆𝐴) + (𝑆𝑦)) = ((𝑆𝐴) + (𝑆𝐵)))
1512, 14eqeq12d 2749 . . . . 5 (𝑦 = 𝐵 → ((𝑆‘(𝐴 𝑦)) = ((𝑆𝐴) + (𝑆𝑦)) ↔ (𝑆‘(𝐴 𝐵)) = ((𝑆𝐴) + (𝑆𝐵))))
1610, 15imbi12d 344 . . . 4 (𝑦 = 𝐵 → ((𝐴 ⊆ (⊥‘𝑦) → (𝑆‘(𝐴 𝑦)) = ((𝑆𝐴) + (𝑆𝑦))) ↔ (𝐴 ⊆ (⊥‘𝐵) → (𝑆‘(𝐴 𝐵)) = ((𝑆𝐴) + (𝑆𝐵)))))
178, 16rspc2v 3572 . . 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 410 1 (𝑆 ∈ States → (((𝐴C𝐵C ) ∧ 𝐴 ⊆ (⊥‘𝐵)) → (𝑆‘(𝐴 𝐵)) = ((𝑆𝐴) + (𝑆𝐵))))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 395   = wceq 1537  wcel 2101  wral 3059  wss 3889  wf 6443  cfv 6447  (class class class)co 7295  0cc0 10899  1c1 10900   + caddc 10902  [,]cicc 13110  chba 29309   C cch 29319  cort 29320   chj 29323  Statescst 29352
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1793  ax-4 1807  ax-5 1909  ax-6 1967  ax-7 2007  ax-8 2103  ax-9 2111  ax-10 2132  ax-11 2149  ax-12 2166  ax-ext 2704  ax-sep 5226  ax-nul 5233  ax-pow 5291  ax-pr 5355  ax-un 7608  ax-hilex 29389
This theorem depends on definitions:  df-bi 206  df-an 396  df-or 844  df-3an 1087  df-tru 1540  df-fal 1550  df-ex 1778  df-nf 1782  df-sb 2063  df-mo 2535  df-eu 2564  df-clab 2711  df-cleq 2725  df-clel 2811  df-nfc 2884  df-ne 2939  df-ral 3060  df-rex 3069  df-rab 3224  df-v 3436  df-sbc 3719  df-dif 3892  df-un 3894  df-in 3896  df-ss 3906  df-nul 4260  df-if 4463  df-pw 4538  df-sn 4565  df-pr 4567  df-op 4571  df-uni 4842  df-br 5078  df-opab 5140  df-id 5491  df-xp 5597  df-rel 5598  df-cnv 5599  df-co 5600  df-dm 5601  df-rn 5602  df-res 5603  df-ima 5604  df-iota 6399  df-fun 6449  df-fn 6450  df-f 6451  df-fv 6455  df-ov 7298  df-oprab 7299  df-mpo 7300  df-map 8637  df-sh 29597  df-ch 29611  df-st 30601
This theorem is referenced by:  sto1i  30626  stlei  30630  stji1i  30632
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