Theorem sticl 32151

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 32149	. . 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 7056	. . 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 2109  ∀wral 3045   ⊆ wss 3917  ⟶wf 6510  ‘cfv 6514  (class class class)co 7390  0cc0 11075  1c1 11076   + caddc 11078  [,]cicc 13316   ℋchba 30855   C_ℋ cch 30865  ⊥cort 30866   ∨_ℋ chj 30869  Statescst 30898

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 2008  ax-8 2111  ax-9 2119  ax-10 2142  ax-11 2158  ax-12 2178  ax-ext 2702  ax-sep 5254  ax-nul 5264  ax-pow 5323  ax-pr 5390  ax-un 7714  ax-hilex 30935

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 2066  df-mo 2534  df-eu 2563  df-clab 2709  df-cleq 2722  df-clel 2804  df-nfc 2879  df-ne 2927  df-ral 3046  df-rex 3055  df-rab 3409  df-v 3452  df-sbc 3757  df-dif 3920  df-un 3922  df-in 3924  df-ss 3934  df-nul 4300  df-if 4492  df-pw 4568  df-sn 4593  df-pr 4595  df-op 4599  df-uni 4875  df-br 5111  df-opab 5173  df-id 5536  df-xp 5647  df-rel 5648  df-cnv 5649  df-co 5650  df-dm 5651  df-rn 5652  df-res 5653  df-ima 5654  df-iota 6467  df-fun 6516  df-fn 6517  df-f 6518  df-fv 6522  df-ov 7393  df-oprab 7394  df-mpo 7395  df-map 8804  df-sh 31143  df-ch 31157  df-st 32147

This theorem is referenced by:  stcl  32152  stge0  32160  stle1  32161