Theorem sticl 32301

Description: [0, 1] closure of the value of a state. (Contributed by Mario Carneiro, 23-Dec-2013.) (New usage is discouraged.)

Assertion

Ref	Expression
sticl	⊢ (𝑆 ∈ States → (𝐴 ∈ Cℋ → (𝑆‘𝐴) ∈ (0[,]1)))

Proof of Theorem sticl

Dummy variables 𝑥 𝑦 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		isst 32299	. . 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 7027	. . 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)))

Syntax hints:   → wi 4   = wceq 1542   ∈ wcel 2114  ∀wral 3052   ⊆ wss 3890  ⟶wf 6488  ‘cfv 6492  (class class class)co 7360  0cc0 11029  1c1 11030   + caddc 11032  [,]cicc 13292   ℋchba 31005   C_ℋ cch 31015  ⊥cort 31016   ∨_ℋ chj 31019  Statescst 31048

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1912  ax-6 1969  ax-7 2010  ax-8 2116  ax-9 2124  ax-10 2147  ax-11 2163  ax-12 2185  ax-ext 2709  ax-sep 5231  ax-nul 5241  ax-pow 5302  ax-pr 5370  ax-un 7682  ax-hilex 31085

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3an 1089  df-tru 1545  df-fal 1555  df-ex 1782  df-nf 1786  df-sb 2069  df-mo 2540  df-eu 2570  df-clab 2716  df-cleq 2729  df-clel 2812  df-nfc 2886  df-ne 2934  df-ral 3053  df-rex 3063  df-rab 3391  df-v 3432  df-sbc 3730  df-dif 3893  df-un 3895  df-in 3897  df-ss 3907  df-nul 4275  df-if 4468  df-pw 4544  df-sn 4569  df-pr 4571  df-op 4575  df-uni 4852  df-br 5087  df-opab 5149  df-id 5519  df-xp 5630  df-rel 5631  df-cnv 5632  df-co 5633  df-dm 5634  df-rn 5635  df-res 5636  df-ima 5637  df-iota 6448  df-fun 6494  df-fn 6495  df-f 6496  df-fv 6500  df-ov 7363  df-oprab 7364  df-mpo 7365  df-map 8768  df-sh 31293  df-ch 31307  df-st 32297

This theorem is referenced by:  stcl  32302  stge0  32310  stle1  32311