Theorem stj 29094

Description: The value of a state on a join. (Contributed by NM, 23-Oct-1999.) (New usage is discouraged.)

Assertion

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

Proof of Theorem stj

Dummy variables x y are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		isst 29072	. . . 4 |- ( S e. States <-> ( S : CH --> ( 0 [,] 1 ) /\ ( S ` ~H ) = 1 /\ A. x e. CH A. y e. CH ( x C_ ( _|_ ` y ) -> ( S ` ( x vH y ) ) = ( ( S ` x ) + ( S ` y ) ) ) ) )
2	1	simp3bi 1078	. . 3 |- ( S e. States -> A. x e. CH A. y e. CH ( x C_ ( _|_ ` y ) -> ( S ` ( x vH y ) ) = ( ( S ` x ) + ( S ` y ) ) ) )
3		sseq1 3626	. . . . 5 |- ( x = A -> ( x C_ ( _|_ ` y ) <-> A C_ ( _|_ ` y ) ) )
4		oveq1 6657	. . . . . . 7 |- ( x = A -> ( x vH y ) = ( A vH y ) )
5	4	fveq2d 6195	. . . . . 6 |- ( x = A -> ( S ` ( x vH y ) ) = ( S ` ( A vH y ) ) )
6		fveq2 6191	. . . . . . 7 |- ( x = A -> ( S ` x ) = ( S ` A ) )
7	6	oveq1d 6665	. . . . . 6 |- ( x = A -> ( ( S ` x ) + ( S ` y ) ) = ( ( S ` A ) + ( S ` y ) ) )
8	5, 7	eqeq12d 2637	. . . . 5 |- ( x = A -> ( ( S ` ( x vH y ) ) = ( ( S ` x ) + ( S ` y ) ) <-> ( S ` ( A vH y ) ) = ( ( S ` A ) + ( S ` y ) ) ) )
9	3, 8	imbi12d 334	. . . 4 |- ( x = A -> ( ( x C_ ( _|_ ` y ) -> ( S ` ( x vH y ) ) = ( ( S ` x ) + ( S ` y ) ) ) <-> ( A C_ ( _|_ ` y ) -> ( S ` ( A vH y ) ) = ( ( S ` A ) + ( S ` y ) ) ) ) )
10		fveq2 6191	. . . . . 6 |- ( y = B -> ( _|_ ` y ) = ( _|_ ` B ) )
11	10	sseq2d 3633	. . . . 5 |- ( y = B -> ( A C_ ( _|_ ` y ) <-> A C_ ( _|_ ` B ) ) )
12		oveq2 6658	. . . . . . 7 |- ( y = B -> ( A vH y ) = ( A vH B ) )
13	12	fveq2d 6195	. . . . . 6 |- ( y = B -> ( S ` ( A vH y ) ) = ( S ` ( A vH B ) ) )
14		fveq2 6191	. . . . . . 7 |- ( y = B -> ( S ` y ) = ( S ` B ) )
15	14	oveq2d 6666	. . . . . 6 |- ( y = B -> ( ( S ` A ) + ( S ` y ) ) = ( ( S ` A ) + ( S ` B ) ) )
16	13, 15	eqeq12d 2637	. . . . 5 |- ( y = B -> ( ( S ` ( A vH y ) ) = ( ( S ` A ) + ( S ` y ) ) <-> ( S ` ( A vH B ) ) = ( ( S ` A ) + ( S ` B ) ) ) )
17	11, 16	imbi12d 334	. . . 4 |- ( y = B -> ( ( A C_ ( _|_ ` y ) -> ( S ` ( A vH y ) ) = ( ( S ` A ) + ( S ` y ) ) ) <-> ( A C_ ( _|_ ` B ) -> ( S ` ( A vH B ) ) = ( ( S ` A ) + ( S ` B ) ) ) ) )
18	9, 17	rspc2v 3322	. . 3 |- ( ( A e. CH /\ B e. CH ) -> ( A. x e. CH A. y e. CH ( x C_ ( _|_ ` y ) -> ( S ` ( x vH y ) ) = ( ( S ` x ) + ( S ` y ) ) ) -> ( A C_ ( _|_ ` B ) -> ( S ` ( A vH B ) ) = ( ( S ` A ) + ( S ` B ) ) ) ) )
19	2, 18	syl5com 31	. 2 |- ( S e. States -> ( ( A e. CH /\ B e. CH ) -> ( A C_ ( _|_ ` B ) -> ( S ` ( A vH B ) ) = ( ( S ` A ) + ( S ` B ) ) ) ) )
20	19	impd 447	1 |- ( S e. States -> ( ( ( A e. CH /\ B e. CH ) /\ A C_ ( _|_ ` B ) ) -> ( S ` ( A vH B ) ) = ( ( S ` A ) + ( S ` B ) ) ) )

Syntax hints:   -> wi 4   /\ wa 384   = wceq 1483   e. wcel 1990   A.wral 2912   C_ wss 3574   -->wf 5884   ` cfv 5888  (class class class)co 6650   0cc0 9936   1c1 9937   + caddc 9939   [,]cicc 12178   ~Hchil 27776   CHcch 27786   _|_cort 27787   vH chj 27790   Statescst 27819

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-8 1992  ax-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246  ax-ext 2602  ax-sep 4781  ax-nul 4789  ax-pow 4843  ax-pr 4906  ax-un 6949  ax-hilex 27856

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3an 1039  df-tru 1486  df-ex 1705  df-nf 1710  df-sb 1881  df-eu 2474  df-mo 2475  df-clab 2609  df-cleq 2615  df-clel 2618  df-nfc 2753  df-ral 2917  df-rex 2918  df-rab 2921  df-v 3202  df-sbc 3436  df-dif 3577  df-un 3579  df-in 3581  df-ss 3588  df-nul 3916  df-if 4087  df-pw 4160  df-sn 4178  df-pr 4180  df-op 4184  df-uni 4437  df-br 4654  df-opab 4713  df-id 5024  df-xp 5120  df-rel 5121  df-cnv 5122  df-co 5123  df-dm 5124  df-rn 5125  df-res 5126  df-ima 5127  df-iota 5851  df-fun 5890  df-fn 5891  df-f 5892  df-fv 5896  df-ov 6653  df-oprab 6654  df-mpt2 6655  df-map 7859  df-sh 28064  df-ch 28078  df-st 29070

This theorem is referenced by:  sto1i  29095  stlei  29099  stji1i  29101