Theorem stelt 5671

Description: Property of a state.

Assertion

Ref	Expression
stelt	|- (S e. States <-> ((S:CH-->RR /\ A.x e. CH (0 <_ (S` x) /\ (S` x) <_ 1)) /\ ((S` H~) = 1 /\ A.x e. CH A.y e. CH (x (_ (_|_` y) -> (S` (x vH 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 e. States -> S e. V)
2		chex 5130	. . . 4 |- CH e. V
3		fex 2771	. . . 4 |- (CH e. V -> (S:CH-->RR -> S e. V))
4	2, 3	ax-mp 6	. . 3 |- (S:CH-->RR -> S e. V)
5	4	ad2antll 320	. 2 |- (((S:CH-->RR /\ A.x e. CH (0 <_ (S` x) /\ (S` x) <_ 1)) /\ ((S` H~) = 1 /\ A.x e. CH A.y e. CH (x (_ (_|_` y) -> (S` (x vH y)) = ((S` x) + (S` y))))) -> S e. V)
6		feq1 2748	. . . . 5 |- (f = S -> (f:CH-->RR <-> S:CH-->RR))
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 -> (A.x e. CH (0 <_ (f` x) /\ (f` x) <_ 1) <-> A.x e. CH (0 <_ (S` x) /\ (S` x) <_ 1)))
12	6, 11	anbi12d 476	. . . 4 |- (f = S -> ((f:CH-->RR /\ A.x e. CH (0 <_ (f` x) /\ (f` x) <_ 1)) <-> (S:CH-->RR /\ A.x e. 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 vH y)) = (S` (x vH 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 vH y)) = ((f` x) + (f` y)) <-> (S` (x vH y)) = ((S` x) + (S` y))))
19	18	imbi2d 464	. . . . . . 7 |- (f = S -> ((x (_ (_|_` y) -> (f` (x vH y)) = ((f` x) + (f` y))) <-> (x (_ (_|_` y) -> (S` (x vH y)) = ((S` x) + (S` y)))))
20	19	biraldv 1219	. . . . . 6 |- (f = S -> (A.y e. CH (x (_ (_|_` y) -> (f` (x vH y)) = ((f` x) + (f` y))) <-> A.y e. CH (x (_ (_|_` y) -> (S` (x vH y)) = ((S` x) + (S` y)))))
21	20	biraldv 1219	. . . . 5 |- (f = S -> (A.x e. CH A.y e. CH (x (_ (_|_` y) -> (f` (x vH y)) = ((f` x) + (f` y))) <-> A.x e. CH A.y e. CH (x (_ (_|_` y) -> (S` (x vH y)) = ((S` x) + (S` y)))))
22	14, 21	anbi12d 476	. . . 4 |- (f = S -> (((f` H~) = 1 /\ A.x e. CH A.y e. CH (x (_ (_|_` y) -> (f` (x vH y)) = ((f` x) + (f` y)))) <-> ((S` H~) = 1 /\ A.x e. CH A.y e. CH (x (_ (_|_` y) -> (S` (x vH y)) = ((S` x) + (S` y))))))
23	12, 22	anbi12d 476	. . 3 |- (f = S -> (((f:CH-->RR /\ A.x e. CH (0 <_ (f` x) /\ (f` x) <_ 1)) /\ ((f` H~) = 1 /\ A.x e. CH A.y e. CH (x (_ (_|_` y) -> (f` (x vH y)) = ((f` x) + (f` y))))) <-> ((S:CH-->RR /\ A.x e. CH (0 <_ (S` x) /\ (S` x) <_ 1)) /\ ((S` H~) = 1 /\ A.x e. CH A.y e. CH (x (_ (_|_` y) -> (S` (x vH y)) = ((S` x) + (S` y)))))))
24		df-st 5670	. . 3 |- States = {f | ((f:CH-->RR /\ A.x e. CH (0 <_ (f` x) /\ (f` x) <_ 1)) /\ ((f` H~) = 1 /\ A.x e. CH A.y e. CH (x (_ (_|_` y) -> (f` (x vH y)) = ((f` x) + (f` y)))))}
25	23, 24	elab2g 1418	. 2 |- (S e. V -> (S e. States <-> ((S:CH-->RR /\ A.x e. CH (0 <_ (S` x) /\ (S` x) <_ 1)) /\ ((S` H~) = 1 /\ A.x e. CH A.y e. CH (x (_ (_|_` y) -> (S` (x vH y)) = ((S` x) + (S` y)))))))
26	1, 5, 25	pm5.21nii 504	1 |- (S e. States <-> ((S:CH-->RR /\ A.x e. CH (0 <_ (S` x) /\ (S` x) <_ 1)) /\ ((S` H~) = 1 /\ A.x e. CH A.y e. CH (x (_ (_|_` y) -> (S` (x vH y)) = ((S` x) + (S` y))))))

Syntax hints:   -> wi 2   <-> wb 127   /\ wa 196   = wceq 1091   e. wcel 1092  A.wral 1201  Vcvv 1348   (_ wss 1487   class class class wbr 2054  -->wf 2418  ` cfv 2422  (class class class)co 3001  RRcr 4027  0cc0 4028  1c1 4029   + caddc 4031   <_ cle 4092  H~chil 4958  CHcch 4968  _|_cort 4969   vH 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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