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Theorem stj 23739
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.
StepHypRef Expression
1 isst 23717 . . . 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 ) ) ) ) )
21simp3bi 975 . . 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 3370 . . . . 5  |-  ( x  =  A  ->  (
x  C_  ( _|_ `  y )  <->  A  C_  ( _|_ `  y ) ) )
4 oveq1 6089 . . . . . . 7  |-  ( x  =  A  ->  (
x  vH  y )  =  ( A  vH  y ) )
54fveq2d 5733 . . . . . 6  |-  ( x  =  A  ->  ( S `  ( x  vH  y ) )  =  ( S `  ( A  vH  y ) ) )
6 fveq2 5729 . . . . . . 7  |-  ( x  =  A  ->  ( S `  x )  =  ( S `  A ) )
76oveq1d 6097 . . . . . 6  |-  ( x  =  A  ->  (
( S `  x
)  +  ( S `
 y ) )  =  ( ( S `
 A )  +  ( S `  y
) ) )
85, 7eqeq12d 2451 . . . . 5  |-  ( x  =  A  ->  (
( S `  (
x  vH  y )
)  =  ( ( S `  x )  +  ( S `  y ) )  <->  ( S `  ( A  vH  y
) )  =  ( ( S `  A
)  +  ( S `
 y ) ) ) )
93, 8imbi12d 313 . . . 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 5729 . . . . . 6  |-  ( y  =  B  ->  ( _|_ `  y )  =  ( _|_ `  B
) )
1110sseq2d 3377 . . . . 5  |-  ( y  =  B  ->  ( A  C_  ( _|_ `  y
)  <->  A  C_  ( _|_ `  B ) ) )
12 oveq2 6090 . . . . . . 7  |-  ( y  =  B  ->  ( A  vH  y )  =  ( A  vH  B
) )
1312fveq2d 5733 . . . . . 6  |-  ( y  =  B  ->  ( S `  ( A  vH  y ) )  =  ( S `  ( A  vH  B ) ) )
14 fveq2 5729 . . . . . . 7  |-  ( y  =  B  ->  ( S `  y )  =  ( S `  B ) )
1514oveq2d 6098 . . . . . 6  |-  ( y  =  B  ->  (
( S `  A
)  +  ( S `
 y ) )  =  ( ( S `
 A )  +  ( S `  B
) ) )
1613, 15eqeq12d 2451 . . . . 5  |-  ( y  =  B  ->  (
( S `  ( A  vH  y ) )  =  ( ( S `
 A )  +  ( S `  y
) )  <->  ( S `  ( A  vH  B
) )  =  ( ( S `  A
)  +  ( S `
 B ) ) ) )
1711, 16imbi12d 313 . . . 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 ) ) ) ) )
189, 17rspc2v 3059 . . 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 ) ) ) ) )
192, 18syl5com 29 . 2  |-  ( S  e.  States  ->  ( ( A  e.  CH  /\  B  e.  CH )  ->  ( A  C_  ( _|_ `  B
)  ->  ( S `  ( A  vH  B
) )  =  ( ( S `  A
)  +  ( S `
 B ) ) ) ) )
2019imp3a 422 1  |-  ( S  e.  States  ->  ( ( ( A  e.  CH  /\  B  e.  CH )  /\  A  C_  ( _|_ `  B ) )  -> 
( S `  ( A  vH  B ) )  =  ( ( S `
 A )  +  ( S `  B
) ) ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 360    = wceq 1653    e. wcel 1726   A.wral 2706    C_ wss 3321   -->wf 5451   ` cfv 5455  (class class class)co 6082   0cc0 8991   1c1 8992    + caddc 8994   [,]cicc 10920   ~Hchil 22423   CHcch 22433   _|_cort 22434    vH chj 22437   Statescst 22466
This theorem is referenced by:  sto1i  23740  stlei  23744  stji1i  23746
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1556  ax-5 1567  ax-17 1627  ax-9 1667  ax-8 1688  ax-13 1728  ax-14 1730  ax-6 1745  ax-7 1750  ax-11 1762  ax-12 1951  ax-ext 2418  ax-sep 4331  ax-nul 4339  ax-pow 4378  ax-pr 4404  ax-un 4702  ax-hilex 22503
This theorem depends on definitions:  df-bi 179  df-or 361  df-an 362  df-3an 939  df-tru 1329  df-ex 1552  df-nf 1555  df-sb 1660  df-eu 2286  df-mo 2287  df-clab 2424  df-cleq 2430  df-clel 2433  df-nfc 2562  df-ne 2602  df-ral 2711  df-rex 2712  df-rab 2715  df-v 2959  df-sbc 3163  df-dif 3324  df-un 3326  df-in 3328  df-ss 3335  df-nul 3630  df-if 3741  df-pw 3802  df-sn 3821  df-pr 3822  df-op 3824  df-uni 4017  df-br 4214  df-opab 4268  df-id 4499  df-xp 4885  df-rel 4886  df-cnv 4887  df-co 4888  df-dm 4889  df-rn 4890  df-res 4891  df-ima 4892  df-iota 5419  df-fun 5457  df-fn 5458  df-f 5459  df-fv 5463  df-ov 6085  df-oprab 6086  df-mpt2 6087  df-map 7021  df-sh 22710  df-ch 22725  df-st 23715
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