Theorem stjt 10162

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		sseq1 2082	. . . . 5 |- (x = A -> (x (_ (_|_` y) <-> A (_ (_|_` y)))
2		opreq1 3968	. . . . . . 7 |- (x = A -> (x vH y) = (A vH y))
3	2	fveq2d 3728	. . . . . 6 |- (x = A -> (S` (x vH y)) = (S` (A vH y)))
4		fveq2 3724	. . . . . . 7 |- (x = A -> (S` x) = (S` A))
5	4	opreq1d 3975	. . . . . 6 |- (x = A -> ((S` x) + (S` y)) = ((S` A) + (S` y)))
6	3, 5	eqeq12d 1489	. . . . 5 |- (x = A -> ((S` (x vH y)) = ((S` x) + (S` y)) <-> (S` (A vH y)) = ((S` A) + (S` y))))
7	1, 6	imbi12d 626	. . . 4 |- (x = A -> ((x (_ (_|_` y) -> (S` (x vH y)) = ((S` x) + (S` y))) <-> (A (_ (_|_` y) -> (S` (A vH y)) = ((S` A) + (S` y)))))
8		fveq2 3724	. . . . . 6 |- (y = B -> (_|_` y) = (_|_` B))
9	8	sseq2d 2089	. . . . 5 |- (y = B -> (A (_ (_|_` y) <-> A (_ (_|_` B)))
10		opreq2 3969	. . . . . . 7 |- (y = B -> (A vH y) = (A vH B))
11	10	fveq2d 3728	. . . . . 6 |- (y = B -> (S` (A vH y)) = (S` (A vH B)))
12		fveq2 3724	. . . . . . 7 |- (y = B -> (S` y) = (S` B))
13	12	opreq2d 3976	. . . . . 6 |- (y = B -> ((S` A) + (S` y)) = ((S` A) + (S` B)))
14	11, 13	eqeq12d 1489	. . . . 5 |- (y = B -> ((S` (A vH y)) = ((S` A) + (S` y)) <-> (S` (A vH B)) = ((S` A) + (S` B))))
15	9, 14	imbi12d 626	. . . 4 |- (y = B -> ((A (_ (_|_` y) -> (S` (A vH y)) = ((S` A) + (S` y))) <-> (A (_ (_|_` B) -> (S` (A vH B)) = ((S` A) + (S` B)))))
16	7, 15	rcla42v 1880	. . 3 |- ((A e. CH /\ B e. CH) -> (A.x e. CH A.y e. CH (x (_ (_|_` y) -> (S` (x vH y)) = ((S` x) + (S` y))) -> (A (_ (_|_` B) -> (S` (A vH B)) = ((S` A) + (S` B)))))
17		stelt 10141	. . . . 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))))))
18	17	pm3.27bi 326	. . . 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)))))
19	18	pm3.27d 325	. . 3 |- (S e. States -> A.x e. CH A.y e. CH (x (_ (_|_` y) -> (S` (x vH y)) = ((S` x) + (S` y))))
20	16, 19	syl5com 52	. 2 |- (S e. States -> ((A e. CH /\ B e. CH) -> (A (_ (_|_` B) -> (S` (A vH B)) = ((S` A) + (S` B)))))
21	20	imp3a 361	1 |- (S e. States -> (((A e. CH /\ B e. CH) /\ A (_ (_|_` B)) -> (S` (A vH B)) = ((S` A) + (S` B))))

Syntax hints:   -> wi 3   /\ wa 223   = wceq 956   e. wcel 958  A.wral 1645   (_ wss 2047   class class class wbr 2619  -->wf 3178  ` cfv 3182  (class class class)co 3963  RRcr 5233  0cc0 5234  1c1 5235   + caddc 5237   <_ cle 5295  H~chil 8788  CHcch 8798  _|_cort 8799   vH chj 8802  Statescst 8831

This theorem is referenced by:  sto1 10163  stle 10167  stji1 10169

This theorem was proved from axioms:  ax-1 4  ax-2 5  ax-3 6  ax-mp 7  ax-7 962  ax-gen 963  ax-8 964  ax-10 966  ax-11 967  ax-12 968  ax-13 969  ax-14 970  ax-17 971  ax-4 973  ax-5o 975  ax-6o 978  ax-9o 1123  ax-10o 1140  ax-16 1210  ax-11o 1218  ax-ext 1459  ax-rep 2693  ax-sep 2703  ax-pow 2742  ax-pr 2779  ax-un 2866  ax-hilex 8869

This theorem depends on definitions:  df-bi 147  df-or 224  df-an 225  df-ex 981  df-sb 1172  df-eu 1382  df-mo 1383  df-clab 1464  df-cleq 1469  df-clel 1472  df-ne 1587  df-ral 1649  df-rex 1650  df-v 1812  df-dif 2049  df-un 2050  df-in 2051  df-ss 2053  df-nul 2281  df-pw 2402  df-sn 2412  df-pr 2413  df-op 2416  df-uni 2504  df-br 2620  df-opab 2667  df-id 2835  df-xp 3184  df-rel 3185  df-cnv 3186  df-co 3187  df-dm 3188  df-rn 3189  df-res 3190  df-ima 3191  df-fun 3192  df-fn 3193  df-f 3194  df-fv 3198  df-opr 3965  df-sh 9076  df-ch 9092  df-st 10139

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