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Mirrors > Home > HSE Home > Th. List > sticl | Structured version Visualization version GIF version |
Description: [0, 1] closure of the value of a state. (Contributed by Mario Carneiro, 23-Dec-2013.) (New usage is discouraged.) |
Ref | Expression |
---|---|
sticl | ⊢ (𝑆 ∈ States → (𝐴 ∈ Cℋ → (𝑆‘𝐴) ∈ (0[,]1))) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | isst 32036 | . . 3 ⊢ (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 | simp1bi 1143 | . 2 ⊢ (𝑆 ∈ States → 𝑆: Cℋ ⟶(0[,]1)) |
3 | ffvelcdm 7091 | . . 3 ⊢ ((𝑆: Cℋ ⟶(0[,]1) ∧ 𝐴 ∈ Cℋ ) → (𝑆‘𝐴) ∈ (0[,]1)) | |
4 | 3 | ex 412 | . 2 ⊢ (𝑆: Cℋ ⟶(0[,]1) → (𝐴 ∈ Cℋ → (𝑆‘𝐴) ∈ (0[,]1))) |
5 | 2, 4 | syl 17 | 1 ⊢ (𝑆 ∈ States → (𝐴 ∈ Cℋ → (𝑆‘𝐴) ∈ (0[,]1))) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 = wceq 1534 ∈ wcel 2099 ∀wral 3058 ⊆ wss 3947 ⟶wf 6544 ‘cfv 6548 (class class class)co 7420 0cc0 11139 1c1 11140 + caddc 11142 [,]cicc 13360 ℋchba 30742 Cℋ cch 30752 ⊥cort 30753 ∨ℋ chj 30756 Statescst 30785 |
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 5299 ax-nul 5306 ax-pow 5365 ax-pr 5429 ax-un 7740 ax-hilex 30822 |
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 2938 df-ral 3059 df-rex 3068 df-rab 3430 df-v 3473 df-sbc 3777 df-dif 3950 df-un 3952 df-in 3954 df-ss 3964 df-nul 4324 df-if 4530 df-pw 4605 df-sn 4630 df-pr 4632 df-op 4636 df-uni 4909 df-br 5149 df-opab 5211 df-id 5576 df-xp 5684 df-rel 5685 df-cnv 5686 df-co 5687 df-dm 5688 df-rn 5689 df-res 5690 df-ima 5691 df-iota 6500 df-fun 6550 df-fn 6551 df-f 6552 df-fv 6556 df-ov 7423 df-oprab 7424 df-mpo 7425 df-map 8847 df-sh 31030 df-ch 31044 df-st 32034 |
This theorem is referenced by: stcl 32039 stge0 32047 stle1 32048 |
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