Theorem stjt 5676

Description: The value of a state on a join.

Assertion

Ref	Expression
stjt	|- (S e. States -> (((A e. CH /\ B e. CH) /\ A (_ (_|_` B)) -> (S` (A vH B)) = ((S` A) + (S` B))))

Proof of Theorem stjt

Step	Hyp	Ref	Expression
1		stelt 5671	. . . . 5 |- (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))))))
2	1	pm3.27bd 263	. . . 4 |- (S e. States -> ((S` H~) = 1 /\ A.x e. CH A.y e. CH (x (_ (_|_` y) -> (S` (x vH y)) = ((S` x) + (S` y)))))
3	2	pm3.27d 262	. . 3 |- (S e. States -> A.x e. CH A.y e. CH (x (_ (_|_` y) -> (S` (x vH y)) = ((S` x) + (S` y))))
4		sseq1 1521	. . . . 5 |- (x = A -> (x (_ (_|_` y) <-> A (_ (_|_` y)))
5		opreq1 3006	. . . . . . 7 |- (x = A -> (x vH y) = (A vH y))
6	5	fveq2d 2836	. . . . . 6 |- (x = A -> (S` (x vH y)) = (S` (A vH y)))
7		fveq2 2832	. . . . . . 7 |- (x = A -> (S` x) = (S` A))
8	7	opreq1d 3012	. . . . . 6 |- (x = A -> ((S` x) + (S` y)) = ((S` A) + (S` y)))
9	6, 8	cleq12d 1115	. . . . 5 |- (x = A -> ((S` (x vH y)) = ((S` x) + (S` y)) <-> (S` (A vH y)) = ((S` A) + (S` y))))
10	4, 9	imbi12d 474	. . . 4 |- (x = A -> ((x (_ (_|_` y) -> (S` (x vH y)) = ((S` x) + (S` y))) <-> (A (_ (_|_` y) -> (S` (A vH y)) = ((S` A) + (S` y)))))
11		fveq2 2832	. . . . . 6 |- (y = B -> (_|_` y) = (_|_` B))
12	11	sseq2d 1528	. . . . 5 |- (y = B -> (A (_ (_|_` y) <-> A (_ (_|_` B)))
13		opreq2 3007	. . . . . . 7 |- (y = B -> (A vH y) = (A vH B))
14	13	fveq2d 2836	. . . . . 6 |- (y = B -> (S` (A vH y)) = (S` (A vH B)))
15		fveq2 2832	. . . . . . 7 |- (y = B -> (S` y) = (S` B))
16	15	opreq2d 3013	. . . . . 6 |- (y = B -> ((S` A) + (S` y)) = ((S` A) + (S` B)))
17	14, 16	cleq12d 1115	. . . . 5 |- (y = B -> ((S` (A vH y)) = ((S` A) + (S` y)) <-> (S` (A vH B)) = ((S` A) + (S` B))))
18	12, 17	imbi12d 474	. . . 4 |- (y = B -> ((A (_ (_|_` y) -> (S` (A vH y)) = ((S` A) + (S` y))) <-> (A (_ (_|_` B) -> (S` (A vH B)) = ((S` A) + (S` B)))))
19	10, 18	rcla42v 1404	. . 3 |- (A.x e. CH A.y e. CH (x (_ (_|_` y) -> (S` (x vH y)) = ((S` x) + (S` y))) -> ((A e. CH /\ B e. CH) -> (A (_ (_|_` B) -> (S` (A vH B)) = ((S` A) + (S` B)))))
20	3, 19	syl 12	. 2 |- (S e. States -> ((A e. CH /\ B e. CH) -> (A (_ (_|_` B) -> (S` (A vH B)) = ((S` A) + (S` B)))))
21	20	imp3a 279	1 |- (S e. States -> (((A e. CH /\ B e. CH) /\ A (_ (_|_` B)) -> (S` (A vH B)) = ((S` A) + (S` B))))

Syntax hints:   -> wi 2   /\ wa 196   = wceq 1091   e. wcel 1092  A.wral 1201   (_ 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:  sto1 5677  stle 5681  stji1 5683

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

metamath.org