Theorem stelt 10051

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:   x,y,S

Proof of Theorem stelt

Step	Hyp	Ref	Expression
1		elisset 1808	. 2 |- (S e. States -> S e. V)
2		chex 9016	. . . 4 |- CH e. V
3		fex 3637	. . . 4 |- ((S:CH-->RR /\ CH e. V) -> S e. V)
4	2, 3	mpan2 694	. . 3 |- (S:CH-->RR -> S e. V)
5	4	ad2antrr 404	. 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 3606	. . . . 5 |- (f = S -> (f:CH-->RR <-> S:CH-->RR))
7		fveq1 3708	. . . . . . . 8 |- (f = S -> (f` x) = (S` x))
8	7	breq2d 2620	. . . . . . 7 |- (f = S -> (0 <_ (f` x) <-> 0 <_ (S` x)))
9	7	breq1d 2619	. . . . . . 7 |- (f = S -> ((f` x) <_ 1 <-> (S` x) <_ 1))
10	8, 9	anbi12d 626	. . . . . 6 |- (f = S -> ((0 <_ (f` x) /\ (f` x) <_ 1) <-> (0 <_ (S` x) /\ (S` x) <_ 1)))
11	10	ralbidv 1655	. . . . 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 626	. . . 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 3708	. . . . . 6 |- (f = S -> (f` H~) = (S` H~))
14	13	eqeq1d 1475	. . . . 5 |- (f = S -> ((f` H~) = 1 <-> (S` H~) = 1))
15		fveq1 3708	. . . . . . . 8 |- (f = S -> (f` (x vH y)) = (S` (x vH y)))
16		fveq1 3708	. . . . . . . . 9 |- (f = S -> (f` y) = (S` y))
17	7, 16	opreq12d 3963	. . . . . . . 8 |- (f = S -> ((f` x) + (f` y)) = ((S` x) + (S` y)))
18	15, 17	eqeq12d 1481	. . . . . . 7 |- (f = S -> ((f` (x vH y)) = ((f` x) + (f` y)) <-> (S` (x vH y)) = ((S` x) + (S` y))))
19	18	imbi2d 610	. . . . . 6 |- (f = S -> ((x (_ (_|_` y) -> (f` (x vH y)) = ((f` x) + (f` y))) <-> (x (_ (_|_` y) -> (S` (x vH y)) = ((S` x) + (S` y)))))
20	19	2ralbidv 1672	. . . . 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)))))
21	14, 20	anbi12d 626	. . . 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))))))
22	12, 21	anbi12d 626	. . 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)))))))
23		df-st 10049	. . 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)))))}
24	22, 23	elab2g 1891	. 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)))))))
25	1, 5, 24	pm5.21nii 677	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 3   <-> wb 146   /\ wa 223   = wceq 953   e. wcel 955  A.wral 1637  Vcvv 1802   (_ wss 2037   class class class wbr 2609  -->wf 3168  ` cfv 3172  (class class class)co 3948  RRcr 5205  0cc0 5206  1c1 5207   + caddc 5209   <_ cle 5267  H~chil 8727  CHcch 8737  _|_cort 8738   vH chj 8741  Statescst 8770

This theorem is referenced by:  stclt 10053  stge0t 10061  stle1t 10062  sthil 10071  stjt 10072  strlem3a 10089

This theorem was proved from axioms:  ax-1 4  ax-2 5  ax-3 6  ax-mp 7  ax-7 959  ax-gen 960  ax-8 961  ax-10 963  ax-11 964  ax-12 965  ax-13 966  ax-14 967  ax-17 968  ax-4 970  ax-5o 972  ax-6o 975  ax-9o 1119  ax-10o 1136  ax-16 1206  ax-11o 1213  ax-ext 1452  ax-rep 2683  ax-sep 2693  ax-pow 2732  ax-pr 2769  ax-un 2857  ax-hilex 8790

This theorem depends on definitions:  df-bi 147  df-or 224  df-an 225  df-ex 978  df-sb 1168  df-eu 1375  df-mo 1376  df-clab 1457  df-cleq 1462  df-clel 1465  df-ne 1579  df-ral 1641  df-rex 1642  df-v 1803  df-dif 2039  df-un 2040  df-in 2041  df-ss 2043  df-nul 2271  df-pw 2392  df-sn 2402  df-pr 2403  df-op 2406  df-uni 2494  df-br 2610  df-opab 2657  df-id 2824  df-xp 3174  df-rel 3175  df-cnv 3176  df-co 3177  df-dm 3178  df-rn 3179  df-res 3180  df-ima 3181  df-fun 3182  df-fn 3183  df-f 3184  df-fv 3188  df-opr 3950  df-sh 8997  df-ch 9013  df-st 10049

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