Theorem sticl 32142

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 32140	. . 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 1145	. 2 ⊢ (𝑆 ∈ States → 𝑆: Cℋ ⟶(0[,]1))
3		ffvelcdm 7070	. . 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 1540   ∈ wcel 2108  ∀wral 3051   ⊆ wss 3926  ⟶wf 6526  ‘cfv 6530  (class class class)co 7403  0cc0 11127  1c1 11128   + caddc 11130  [,]cicc 13363   ℋchba 30846   C_ℋ cch 30856  ⊥cort 30857   ∨_ℋ chj 30860  Statescst 30889

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-10 2141  ax-11 2157  ax-12 2177  ax-ext 2707  ax-sep 5266  ax-nul 5276  ax-pow 5335  ax-pr 5402  ax-un 7727  ax-hilex 30926

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2065  df-mo 2539  df-eu 2568  df-clab 2714  df-cleq 2727  df-clel 2809  df-nfc 2885  df-ne 2933  df-ral 3052  df-rex 3061  df-rab 3416  df-v 3461  df-sbc 3766  df-dif 3929  df-un 3931  df-in 3933  df-ss 3943  df-nul 4309  df-if 4501  df-pw 4577  df-sn 4602  df-pr 4604  df-op 4608  df-uni 4884  df-br 5120  df-opab 5182  df-id 5548  df-xp 5660  df-rel 5661  df-cnv 5662  df-co 5663  df-dm 5664  df-rn 5665  df-res 5666  df-ima 5667  df-iota 6483  df-fun 6532  df-fn 6533  df-f 6534  df-fv 6538  df-ov 7406  df-oprab 7407  df-mpo 7408  df-map 8840  df-sh 31134  df-ch 31148  df-st 32138

This theorem is referenced by:  stcl  32143  stge0  32151  stle1  32152