Theorem stelt 5671

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(s):   x,y,S

Proof of Theorem stelt

Step	Hyp	Ref	Expression
1		elisset 1354	. 2 ⊢ (S ∈ States → S ∈ V)
2		chex 5130	. . . 4 ⊢ Cℋ ∈ V
3		fex 2771	. . . 4 ⊢ ( Cℋ ∈ V → (S: Cℋ –→ℝ → S ∈ V))
4	2, 3	ax-mp 6	. . 3 ⊢ (S: Cℋ –→ℝ → S ∈ V)
5	4	ad2antll 320	. 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 2748	. . . . 5 ⊢ (f = S → (f: Cℋ –→ℝ ↔ S: Cℋ –→ℝ))
7		fveq1 2831	. . . . . . . 8 ⊢ (f = S → (f ‘x) = (S ‘x))
8	7	breq2d 2072	. . . . . . 7 ⊢ (f = S → (0 ≤ (f ‘x) ↔ 0 ≤ (S ‘x)))
9	7	breq1d 2071	. . . . . . 7 ⊢ (f = S → ((f ‘x) ≤ 1 ↔ (S ‘x) ≤ 1))
10	8, 9	anbi12d 476	. . . . . 6 ⊢ (f = S → ((0 ≤ (f ‘x) ∧ (f ‘x) ≤ 1) ↔ (0 ≤ (S ‘x) ∧ (S ‘x) ≤ 1)))
11	10	biraldv 1219	. . . . 5 ⊢ (f = S → (∀x ∈ Cℋ (0 ≤ (f ‘x) ∧ (f ‘x) ≤ 1) ↔ ∀x ∈ Cℋ (0 ≤ (S ‘x) ∧ (S ‘x) ≤ 1)))
12	6, 11	anbi12d 476	. . . 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 2831	. . . . . 6 ⊢ (f = S → (f ‘ ℋ ) = (S ‘ ℋ ))
14	13	cleq1d 1109	. . . . 5 ⊢ (f = S → ((f ‘ ℋ ) = 1 ↔ (S ‘ ℋ ) = 1))
15		fveq1 2831	. . . . . . . . 9 ⊢ (f = S → (f ‘(x ∨ℋ y)) = (S ‘(x ∨ℋ y)))
16		fveq1 2831	. . . . . . . . . 10 ⊢ (f = S → (f ‘y) = (S ‘y))
17	7, 16	opreq12d 3014	. . . . . . . . 9 ⊢ (f = S → ((f ‘x) + (f ‘y)) = ((S ‘x) + (S ‘y)))
18	15, 17	cleq12d 1115	. . . . . . . 8 ⊢ (f = S → ((f ‘(x ∨ℋ y)) = ((f ‘x) + (f ‘y)) ↔ (S ‘(x ∨ℋ y)) = ((S ‘x) + (S ‘y))))
19	18	imbi2d 464	. . . . . . 7 ⊢ (f = S → ((x ⊆ (⊥ ‘y) → (f ‘(x ∨ℋ y)) = ((f ‘x) + (f ‘y))) ↔ (x ⊆ (⊥ ‘y) → (S ‘(x ∨ℋ y)) = ((S ‘x) + (S ‘y)))))
20	19	biraldv 1219	. . . . . 6 ⊢ (f = S → (∀y ∈ Cℋ (x ⊆ (⊥ ‘y) → (f ‘(x ∨ℋ y)) = ((f ‘x) + (f ‘y))) ↔ ∀y ∈ Cℋ (x ⊆ (⊥ ‘y) → (S ‘(x ∨ℋ y)) = ((S ‘x) + (S ‘y)))))
21	20	biraldv 1219	. . . . 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)))))
22	14, 21	anbi12d 476	. . . 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))))))
23	12, 22	anbi12d 476	. . 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)))))))
24		df-st 5670	. . 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)))))}
25	23, 24	elab2g 1418	. 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)))))))
26	1, 5, 25	pm5.21nii 504	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 2   ↔ wb 127   ∧ wa 196   = wceq 1091   ∈ wcel 1092  ∀wral 1201  Vcvv 1348   ⊆ wss 1487   class class class wbr 2054  –→wf 2418   ‘cfv 2422  (class class class)co 3001  ℝcr 4027  0cc0 4028  1c1 4029   + caddc 4031   ≤ cle 4092   ℋ chil 4958   C_ℋ cch 4968  ⊥cort 4969   ∨_ℋ chj 4972  Statescst 4979

This theorem is referenced by:  stclt 5672  stge0t 5673  stle1t 5674  sthil 5675  stjt 5676  strlem3a 5693

This theorem was proved from axioms:  ax-1 3  ax-2 4  ax-3 5  ax-mp 6  ax-4 673  ax-5 674  ax-6 675  ax-7 676  ax-gen 677  ax-8 798  ax-9 799  ax-10 800  ax-11 801  ax-12 802  ax-13 804  ax-14 805  ax-16 922  ax-17 925  ax-ext 1074  ax-rep 1075  ax-un 1076  ax-pow 1077  ax-hilex 4983

This theorem depends on definitions:  df-bi 128  df-or 197  df-an 198  df-ex 679  df-sb 853  df-eu 1009  df-mo 1010  df-clab 1093  df-cleq 1097  df-clel 1099  df-ral 1205  df-rex 1206  df-v 1349  df-dif 1489  df-un 1490  df-in 1491  df-ss 1492  df-nul 1708  df-pw 1799  df-sn 1811  df-pr 1812  df-op 1815  df-uni 1920  df-br 2063  df-opab 2098  df-id 2125  df-xp 2424  df-rel 2425  df-cnv 2426  df-co 2427  df-dm 2428  df-rn 2429  df-res 2430  df-ima 2431  df-fun 2432  df-fn 2433  df-f 2434  df-fv 2438  df-opr 3003  df-sh 5114  df-ch 5127  df-st 5670

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