Theorem stjt 10153

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 2080	. . . . 5 |- (x = A -> (x (_ (_|_` y) <-> A (_ (_|_` y)))
2		opreq1 3965	. . . . . . 7 |- (x = A -> (x vH y) = (A vH y))
3	2	fveq2d 3725	. . . . . 6 |- (x = A -> (S` (x vH y)) = (S` (A vH y)))
4		fveq2 3721	. . . . . . 7 |- (x = A -> (S` x) = (S` A))
5	4	opreq1d 3972	. . . . . 6 |- (x = A -> ((S` x) + (S` y)) = ((S` A) + (S` y)))
6	3, 5	eqeq12d 1488	. . . . 5 |- (x = A -> ((S` (x vH y)) = ((S` x) + (S` y)) <-> (S` (A vH y)) = ((S` A) + (S` y))))
7	1, 6	imbi12d 625	. . . 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 3721	. . . . . 6 |- (y = B -> (_|_` y) = (_|_` B))
9	8	sseq2d 2087	. . . . 5 |- (y = B -> (A (_ (_|_` y) <-> A (_ (_|_` B)))
10		opreq2 3966	. . . . . . 7 |- (y = B -> (A vH y) = (A vH B))
11	10	fveq2d 3725	. . . . . 6 |- (y = B -> (S` (A vH y)) = (S` (A vH B)))
12		fveq2 3721	. . . . . . 7 |- (y = B -> (S` y) = (S` B))
13	12	opreq2d 3973	. . . . . 6 |- (y = B -> ((S` A) + (S` y)) = ((S` A) + (S` B)))
14	11, 13	eqeq12d 1488	. . . . 5 |- (y = B -> ((S` (A vH y)) = ((S` A) + (S` y)) <-> (S` (A vH B)) = ((S` A) + (S` B))))
15	9, 14	imbi12d 625	. . . 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 1878	. . 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 10132	. . . . 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 955   e. wcel 957  A.wral 1644   (_ wss 2045   class class class wbr 2616  -->wf 3175  ` cfv 3179  (class class class)co 3960  RRcr 5220  0cc0 5221  1c1 5222   + caddc 5224   <_ cle 5282  H~chil 8772  CHcch 8782  _|_cort 8783   vH chj 8786  Statescst 8815

This theorem is referenced by:  sto1 10154  stle 10158  stji1 10160

This theorem was proved from axioms:  ax-1 4  ax-2 5  ax-3 6  ax-mp 7  ax-7 961  ax-gen 962  ax-8 963  ax-10 965  ax-11 966  ax-12 967  ax-13 968  ax-14 969  ax-17 970  ax-4 972  ax-5o 974  ax-6o 977  ax-9o 1122  ax-10o 1139  ax-16 1210  ax-11o 1218  ax-ext 1459  ax-rep 2690  ax-sep 2700  ax-pow 2739  ax-pr 2776  ax-un 2863  ax-hilex 8853

This theorem depends on definitions:  df-bi 147  df-or 224  df-an 225  df-ex 980  df-sb 1172  df-eu 1382  df-mo 1383  df-clab 1464  df-cleq 1469  df-clel 1472  df-ne 1586  df-ral 1648  df-rex 1649  df-v 1810  df-dif 2047  df-un 2048  df-in 2049  df-ss 2051  df-nul 2279  df-pw 2400  df-sn 2410  df-pr 2411  df-op 2414  df-uni 2501  df-br 2617  df-opab 2664  df-id 2832  df-xp 3181  df-rel 3182  df-cnv 3183  df-co 3184  df-dm 3185  df-rn 3186  df-res 3187  df-ima 3188  df-fun 3189  df-fn 3190  df-f 3191  df-fv 3195  df-opr 3962  df-sh 9064  df-ch 9080  df-st 10130

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