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Mirrors > Home > HSE Home > Th. List > stj | Structured version Visualization version GIF version |
Description: The value of a state on a join. (Contributed by NM, 23-Oct-1999.) (New usage is discouraged.) |
Ref | Expression |
---|---|
stj | ⊢ (𝑆 ∈ States → (((𝐴 ∈ Cℋ ∧ 𝐵 ∈ Cℋ ) ∧ 𝐴 ⊆ (⊥‘𝐵)) → (𝑆‘(𝐴 ∨ℋ 𝐵)) = ((𝑆‘𝐴) + (𝑆‘𝐵)))) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | isst 32017 | . . . 4 ⊢ (𝑆 ∈ States ↔ (𝑆: Cℋ ⟶(0[,]1) ∧ (𝑆‘ ℋ) = 1 ∧ ∀𝑥 ∈ Cℋ ∀𝑦 ∈ Cℋ (𝑥 ⊆ (⊥‘𝑦) → (𝑆‘(𝑥 ∨ℋ 𝑦)) = ((𝑆‘𝑥) + (𝑆‘𝑦))))) | |
2 | 1 | simp3bi 1145 | . . 3 ⊢ (𝑆 ∈ States → ∀𝑥 ∈ Cℋ ∀𝑦 ∈ Cℋ (𝑥 ⊆ (⊥‘𝑦) → (𝑆‘(𝑥 ∨ℋ 𝑦)) = ((𝑆‘𝑥) + (𝑆‘𝑦)))) |
3 | sseq1 4004 | . . . . 5 ⊢ (𝑥 = 𝐴 → (𝑥 ⊆ (⊥‘𝑦) ↔ 𝐴 ⊆ (⊥‘𝑦))) | |
4 | fvoveq1 7438 | . . . . . 6 ⊢ (𝑥 = 𝐴 → (𝑆‘(𝑥 ∨ℋ 𝑦)) = (𝑆‘(𝐴 ∨ℋ 𝑦))) | |
5 | fveq2 6892 | . . . . . . 7 ⊢ (𝑥 = 𝐴 → (𝑆‘𝑥) = (𝑆‘𝐴)) | |
6 | 5 | oveq1d 7430 | . . . . . 6 ⊢ (𝑥 = 𝐴 → ((𝑆‘𝑥) + (𝑆‘𝑦)) = ((𝑆‘𝐴) + (𝑆‘𝑦))) |
7 | 4, 6 | eqeq12d 2744 | . . . . 5 ⊢ (𝑥 = 𝐴 → ((𝑆‘(𝑥 ∨ℋ 𝑦)) = ((𝑆‘𝑥) + (𝑆‘𝑦)) ↔ (𝑆‘(𝐴 ∨ℋ 𝑦)) = ((𝑆‘𝐴) + (𝑆‘𝑦)))) |
8 | 3, 7 | imbi12d 344 | . . . 4 ⊢ (𝑥 = 𝐴 → ((𝑥 ⊆ (⊥‘𝑦) → (𝑆‘(𝑥 ∨ℋ 𝑦)) = ((𝑆‘𝑥) + (𝑆‘𝑦))) ↔ (𝐴 ⊆ (⊥‘𝑦) → (𝑆‘(𝐴 ∨ℋ 𝑦)) = ((𝑆‘𝐴) + (𝑆‘𝑦))))) |
9 | fveq2 6892 | . . . . . 6 ⊢ (𝑦 = 𝐵 → (⊥‘𝑦) = (⊥‘𝐵)) | |
10 | 9 | sseq2d 4011 | . . . . 5 ⊢ (𝑦 = 𝐵 → (𝐴 ⊆ (⊥‘𝑦) ↔ 𝐴 ⊆ (⊥‘𝐵))) |
11 | oveq2 7423 | . . . . . . 7 ⊢ (𝑦 = 𝐵 → (𝐴 ∨ℋ 𝑦) = (𝐴 ∨ℋ 𝐵)) | |
12 | 11 | fveq2d 6896 | . . . . . 6 ⊢ (𝑦 = 𝐵 → (𝑆‘(𝐴 ∨ℋ 𝑦)) = (𝑆‘(𝐴 ∨ℋ 𝐵))) |
13 | fveq2 6892 | . . . . . . 7 ⊢ (𝑦 = 𝐵 → (𝑆‘𝑦) = (𝑆‘𝐵)) | |
14 | 13 | oveq2d 7431 | . . . . . 6 ⊢ (𝑦 = 𝐵 → ((𝑆‘𝐴) + (𝑆‘𝑦)) = ((𝑆‘𝐴) + (𝑆‘𝐵))) |
15 | 12, 14 | eqeq12d 2744 | . . . . 5 ⊢ (𝑦 = 𝐵 → ((𝑆‘(𝐴 ∨ℋ 𝑦)) = ((𝑆‘𝐴) + (𝑆‘𝑦)) ↔ (𝑆‘(𝐴 ∨ℋ 𝐵)) = ((𝑆‘𝐴) + (𝑆‘𝐵)))) |
16 | 10, 15 | imbi12d 344 | . . . 4 ⊢ (𝑦 = 𝐵 → ((𝐴 ⊆ (⊥‘𝑦) → (𝑆‘(𝐴 ∨ℋ 𝑦)) = ((𝑆‘𝐴) + (𝑆‘𝑦))) ↔ (𝐴 ⊆ (⊥‘𝐵) → (𝑆‘(𝐴 ∨ℋ 𝐵)) = ((𝑆‘𝐴) + (𝑆‘𝐵))))) |
17 | 8, 16 | rspc2v 3619 | . . 3 ⊢ ((𝐴 ∈ Cℋ ∧ 𝐵 ∈ Cℋ ) → (∀𝑥 ∈ Cℋ ∀𝑦 ∈ Cℋ (𝑥 ⊆ (⊥‘𝑦) → (𝑆‘(𝑥 ∨ℋ 𝑦)) = ((𝑆‘𝑥) + (𝑆‘𝑦))) → (𝐴 ⊆ (⊥‘𝐵) → (𝑆‘(𝐴 ∨ℋ 𝐵)) = ((𝑆‘𝐴) + (𝑆‘𝐵))))) |
18 | 2, 17 | syl5com 31 | . 2 ⊢ (𝑆 ∈ States → ((𝐴 ∈ Cℋ ∧ 𝐵 ∈ Cℋ ) → (𝐴 ⊆ (⊥‘𝐵) → (𝑆‘(𝐴 ∨ℋ 𝐵)) = ((𝑆‘𝐴) + (𝑆‘𝐵))))) |
19 | 18 | impd 410 | 1 ⊢ (𝑆 ∈ States → (((𝐴 ∈ Cℋ ∧ 𝐵 ∈ Cℋ ) ∧ 𝐴 ⊆ (⊥‘𝐵)) → (𝑆‘(𝐴 ∨ℋ 𝐵)) = ((𝑆‘𝐴) + (𝑆‘𝐵)))) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 395 = wceq 1534 ∈ wcel 2099 ∀wral 3057 ⊆ wss 3945 ⟶wf 6539 ‘cfv 6543 (class class class)co 7415 0cc0 11133 1c1 11134 + caddc 11136 [,]cicc 13354 ℋchba 30723 Cℋ cch 30733 ⊥cort 30734 ∨ℋ chj 30737 Statescst 30766 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1790 ax-4 1804 ax-5 1906 ax-6 1964 ax-7 2004 ax-8 2101 ax-9 2109 ax-10 2130 ax-11 2147 ax-12 2167 ax-ext 2699 ax-sep 5294 ax-nul 5301 ax-pow 5360 ax-pr 5424 ax-un 7735 ax-hilex 30803 |
This theorem depends on definitions: df-bi 206 df-an 396 df-or 847 df-3an 1087 df-tru 1537 df-fal 1547 df-ex 1775 df-nf 1779 df-sb 2061 df-mo 2530 df-eu 2559 df-clab 2706 df-cleq 2720 df-clel 2806 df-nfc 2881 df-ne 2937 df-ral 3058 df-rex 3067 df-rab 3429 df-v 3472 df-sbc 3776 df-dif 3948 df-un 3950 df-in 3952 df-ss 3962 df-nul 4320 df-if 4526 df-pw 4601 df-sn 4626 df-pr 4628 df-op 4632 df-uni 4905 df-br 5144 df-opab 5206 df-id 5571 df-xp 5679 df-rel 5680 df-cnv 5681 df-co 5682 df-dm 5683 df-rn 5684 df-res 5685 df-ima 5686 df-iota 6495 df-fun 6545 df-fn 6546 df-f 6547 df-fv 6551 df-ov 7418 df-oprab 7419 df-mpo 7420 df-map 8841 df-sh 31011 df-ch 31025 df-st 32015 |
This theorem is referenced by: sto1i 32040 stlei 32044 stji1i 32046 |
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