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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 31197 | . . 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 1146 | . 2 ⊢ (𝑆 ∈ States → 𝑆: Cℋ ⟶(0[,]1)) |
3 | ffvelcdm 7037 | . . 3 ⊢ ((𝑆: Cℋ ⟶(0[,]1) ∧ 𝐴 ∈ Cℋ ) → (𝑆‘𝐴) ∈ (0[,]1)) | |
4 | 3 | ex 414 | . 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 1542 ∈ wcel 2107 ∀wral 3065 ⊆ wss 3915 ⟶wf 6497 ‘cfv 6501 (class class class)co 7362 0cc0 11058 1c1 11059 + caddc 11061 [,]cicc 13274 ℋchba 29903 Cℋ cch 29913 ⊥cort 29914 ∨ℋ chj 29917 Statescst 29946 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1798 ax-4 1812 ax-5 1914 ax-6 1972 ax-7 2012 ax-8 2109 ax-9 2117 ax-10 2138 ax-11 2155 ax-12 2172 ax-ext 2708 ax-sep 5261 ax-nul 5268 ax-pow 5325 ax-pr 5389 ax-un 7677 ax-hilex 29983 |
This theorem depends on definitions: df-bi 206 df-an 398 df-or 847 df-3an 1090 df-tru 1545 df-fal 1555 df-ex 1783 df-nf 1787 df-sb 2069 df-mo 2539 df-eu 2568 df-clab 2715 df-cleq 2729 df-clel 2815 df-nfc 2890 df-ne 2945 df-ral 3066 df-rex 3075 df-rab 3411 df-v 3450 df-sbc 3745 df-dif 3918 df-un 3920 df-in 3922 df-ss 3932 df-nul 4288 df-if 4492 df-pw 4567 df-sn 4592 df-pr 4594 df-op 4598 df-uni 4871 df-br 5111 df-opab 5173 df-id 5536 df-xp 5644 df-rel 5645 df-cnv 5646 df-co 5647 df-dm 5648 df-rn 5649 df-res 5650 df-ima 5651 df-iota 6453 df-fun 6503 df-fn 6504 df-f 6505 df-fv 6509 df-ov 7365 df-oprab 7366 df-mpo 7367 df-map 8774 df-sh 30191 df-ch 30205 df-st 31195 |
This theorem is referenced by: stcl 31200 stge0 31208 stle1 31209 |
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