Theorem stelt 10136

Description: Property of a state.

Assertion

Ref	Expression
stelt	⊢ (S ∈ States ↔ ((S: Cℋ –→ℝ ⋀ ∀x ∈ Cℋ (0 ≤ (S ‘x) ⋀ (S ‘x) ≤ 1)) ⋀ ((S ‘ ℋ ) = 1 ⋀ ∀x ∈ Cℋ ∀y ∈ Cℋ (x ⊆ (⊥ ‘y) → (S ‘(x ∨ℋ y)) = ((S ‘x) + (S ‘y))))))

Distinct variable group:   x,y,S

Proof of Theorem stelt

Step	Hyp	Ref	Expression
1		elisset 1820	. 2 ⊢ (S ∈ States → S ∈ V)
2		chex 9090	. . . 4 ⊢ Cℋ ∈ V
3		fex 3658	. . . 4 ⊢ ((S: Cℋ –→ℝ ⋀ Cℋ ∈ V) → S ∈ V)
4	2, 3	mpan2 698	. . 3 ⊢ (S: Cℋ –→ℝ → S ∈ V)
5	4	ad2antrr 406	. 2 ⊢ (((S: Cℋ –→ℝ ⋀ ∀x ∈ Cℋ (0 ≤ (S ‘x) ⋀ (S ‘x) ≤ 1)) ⋀ ((S ‘ ℋ ) = 1 ⋀ ∀x ∈ Cℋ ∀y ∈ Cℋ (x ⊆ (⊥ ‘y) → (S ‘(x ∨ℋ y)) = ((S ‘x) + (S ‘y))))) → S ∈ V)
6		feq1 3626	. . . . 5 ⊢ (f = S → (f: Cℋ –→ℝ ↔ S: Cℋ –→ℝ))
7		fveq1 3729	. . . . . . . 8 ⊢ (f = S → (f ‘x) = (S ‘x))
8	7	breq2d 2635	. . . . . . 7 ⊢ (f = S → (0 ≤ (f ‘x) ↔ 0 ≤ (S ‘x)))
9	7	breq1d 2634	. . . . . . 7 ⊢ (f = S → ((f ‘x) ≤ 1 ↔ (S ‘x) ≤ 1))
10	8, 9	anbi12d 630	. . . . . 6 ⊢ (f = S → ((0 ≤ (f ‘x) ⋀ (f ‘x) ≤ 1) ↔ (0 ≤ (S ‘x) ⋀ (S ‘x) ≤ 1)))
11	10	ralbidv 1666	. . . . 5 ⊢ (f = S → (∀x ∈ Cℋ (0 ≤ (f ‘x) ⋀ (f ‘x) ≤ 1) ↔ ∀x ∈ Cℋ (0 ≤ (S ‘x) ⋀ (S ‘x) ≤ 1)))
12	6, 11	anbi12d 630	. . . 4 ⊢ (f = S → ((f: Cℋ –→ℝ ⋀ ∀x ∈ Cℋ (0 ≤ (f ‘x) ⋀ (f ‘x) ≤ 1)) ↔ (S: Cℋ –→ℝ ⋀ ∀x ∈ Cℋ (0 ≤ (S ‘x) ⋀ (S ‘x) ≤ 1))))
13		fveq1 3729	. . . . . 6 ⊢ (f = S → (f ‘ ℋ ) = (S ‘ ℋ ))
14	13	eqeq1d 1486	. . . . 5 ⊢ (f = S → ((f ‘ ℋ ) = 1 ↔ (S ‘ ℋ ) = 1))
15		fveq1 3729	. . . . . . . 8 ⊢ (f = S → (f ‘(x ∨ℋ y)) = (S ‘(x ∨ℋ y)))
16		fveq1 3729	. . . . . . . . 9 ⊢ (f = S → (f ‘y) = (S ‘y))
17	7, 16	opreq12d 3984	. . . . . . . 8 ⊢ (f = S → ((f ‘x) + (f ‘y)) = ((S ‘x) + (S ‘y)))
18	15, 17	eqeq12d 1492	. . . . . . 7 ⊢ (f = S → ((f ‘(x ∨ℋ y)) = ((f ‘x) + (f ‘y)) ↔ (S ‘(x ∨ℋ y)) = ((S ‘x) + (S ‘y))))
19	18	imbi2d 614	. . . . . 6 ⊢ (f = S → ((x ⊆ (⊥ ‘y) → (f ‘(x ∨ℋ y)) = ((f ‘x) + (f ‘y))) ↔ (x ⊆ (⊥ ‘y) → (S ‘(x ∨ℋ y)) = ((S ‘x) + (S ‘y)))))
20	19	2ralbidv 1683	. . . . 5 ⊢ (f = S → (∀x ∈ Cℋ ∀y ∈ Cℋ (x ⊆ (⊥ ‘y) → (f ‘(x ∨ℋ y)) = ((f ‘x) + (f ‘y))) ↔ ∀x ∈ Cℋ ∀y ∈ Cℋ (x ⊆ (⊥ ‘y) → (S ‘(x ∨ℋ y)) = ((S ‘x) + (S ‘y)))))
21	14, 20	anbi12d 630	. . . 4 ⊢ (f = S → (((f ‘ ℋ ) = 1 ⋀ ∀x ∈ Cℋ ∀y ∈ Cℋ (x ⊆ (⊥ ‘y) → (f ‘(x ∨ℋ y)) = ((f ‘x) + (f ‘y)))) ↔ ((S ‘ ℋ ) = 1 ⋀ ∀x ∈ Cℋ ∀y ∈ Cℋ (x ⊆ (⊥ ‘y) → (S ‘(x ∨ℋ y)) = ((S ‘x) + (S ‘y))))))
22	12, 21	anbi12d 630	. . 3 ⊢ (f = S → (((f: Cℋ –→ℝ ⋀ ∀x ∈ Cℋ (0 ≤ (f ‘x) ⋀ (f ‘x) ≤ 1)) ⋀ ((f ‘ ℋ ) = 1 ⋀ ∀x ∈ Cℋ ∀y ∈ Cℋ (x ⊆ (⊥ ‘y) → (f ‘(x ∨ℋ y)) = ((f ‘x) + (f ‘y))))) ↔ ((S: Cℋ –→ℝ ⋀ ∀x ∈ Cℋ (0 ≤ (S ‘x) ⋀ (S ‘x) ≤ 1)) ⋀ ((S ‘ ℋ ) = 1 ⋀ ∀x ∈ Cℋ ∀y ∈ Cℋ (x ⊆ (⊥ ‘y) → (S ‘(x ∨ℋ y)) = ((S ‘x) + (S ‘y)))))))
23		df-st 10134	. . 3 ⊢ States = {f∣((f: Cℋ –→ℝ ⋀ ∀x ∈ Cℋ (0 ≤ (f ‘x) ⋀ (f ‘x) ≤ 1)) ⋀ ((f ‘ ℋ ) = 1 ⋀ ∀x ∈ Cℋ ∀y ∈ Cℋ (x ⊆ (⊥ ‘y) → (f ‘(x ∨ℋ y)) = ((f ‘x) + (f ‘y)))))}
24	22, 23	elab2g 1903	. 2 ⊢ (S ∈ V → (S ∈ States ↔ ((S: Cℋ –→ℝ ⋀ ∀x ∈ Cℋ (0 ≤ (S ‘x) ⋀ (S ‘x) ≤ 1)) ⋀ ((S ‘ ℋ ) = 1 ⋀ ∀x ∈ Cℋ ∀y ∈ Cℋ (x ⊆ (⊥ ‘y) → (S ‘(x ∨ℋ y)) = ((S ‘x) + (S ‘y)))))))
25	1, 5, 24	pm5.21nii 681	1 ⊢ (S ∈ States ↔ ((S: Cℋ –→ℝ ⋀ ∀x ∈ Cℋ (0 ≤ (S ‘x) ⋀ (S ‘x) ≤ 1)) ⋀ ((S ‘ ℋ ) = 1 ⋀ ∀x ∈ Cℋ ∀y ∈ Cℋ (x ⊆ (⊥ ‘y) → (S ‘(x ∨ℋ y)) = ((S ‘x) + (S ‘y))))))

Syntax hints:   → wi 3   ↔ wb 146   ⋀ wa 223   = wceq 958   ∈ wcel 960  ∀wral 1648  Vcvv 1814   ⊆ wss 2050   class class class wbr 2624  –→wf 3184   ‘cfv 3188  (class class class)co 3969  ℝcr 5245  0cc0 5246  1c1 5247   + caddc 5249   ≤ cle 5307   ℋ chil 8783   C_ℋ cch 8793  ⊥cort 8794   ∨_ℋ chj 8797  Statescst 8826

This theorem is referenced by:  stclt 10138  stge0t 10146  stle1t 10147  sthil 10156  stjt 10157  strlem3a 10174

This theorem was proved from axioms:  ax-1 4  ax-2 5  ax-3 6  ax-mp 7  ax-7 964  ax-gen 965  ax-8 966  ax-10 968  ax-11 969  ax-12 970  ax-13 971  ax-14 972  ax-17 973  ax-4 975  ax-5o 977  ax-6o 980  ax-9o 1125  ax-10o 1142  ax-16 1212  ax-11o 1220  ax-ext 1462  ax-rep 2698  ax-sep 2708  ax-pow 2748  ax-pr 2785  ax-un 2872  ax-hilex 8864

This theorem depends on definitions:  df-bi 147  df-or 224  df-an 225  df-ex 983  df-sb 1174  df-eu 1384  df-mo 1385  df-clab 1467  df-cleq 1472  df-clel 1475  df-ne 1590  df-ral 1652  df-rex 1653  df-v 1815  df-dif 2052  df-un 2053  df-in 2054  df-ss 2056  df-nul 2284  df-pw 2406  df-sn 2416  df-pr 2417  df-op 2420  df-uni 2508  df-br 2625  df-opab 2672  df-id 2841  df-xp 3190  df-rel 3191  df-cnv 3192  df-co 3193  df-dm 3194  df-rn 3195  df-res 3196  df-ima 3197  df-fun 3198  df-fn 3199  df-f 3200  df-fv 3204  df-opr 3971  df-sh 9071  df-ch 9087  df-st 10134

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