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Theorem stj 25639
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 25617 . . . 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 1005 . . 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 3377 . . . . 5  |-  ( x  =  A  ->  (
x  C_  ( _|_ `  y )  <->  A  C_  ( _|_ `  y ) ) )
4 oveq1 6098 . . . . . . 7  |-  ( x  =  A  ->  (
x  vH  y )  =  ( A  vH  y ) )
54fveq2d 5695 . . . . . 6  |-  ( x  =  A  ->  ( S `  ( x  vH  y ) )  =  ( S `  ( A  vH  y ) ) )
6 fveq2 5691 . . . . . . 7  |-  ( x  =  A  ->  ( S `  x )  =  ( S `  A ) )
76oveq1d 6106 . . . . . 6  |-  ( x  =  A  ->  (
( S `  x
)  +  ( S `
 y ) )  =  ( ( S `
 A )  +  ( S `  y
) ) )
85, 7eqeq12d 2457 . . . . 5  |-  ( x  =  A  ->  (
( S `  (
x  vH  y )
)  =  ( ( S `  x )  +  ( S `  y ) )  <->  ( S `  ( A  vH  y
) )  =  ( ( S `  A
)  +  ( S `
 y ) ) ) )
93, 8imbi12d 320 . . . 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 5691 . . . . . 6  |-  ( y  =  B  ->  ( _|_ `  y )  =  ( _|_ `  B
) )
1110sseq2d 3384 . . . . 5  |-  ( y  =  B  ->  ( A  C_  ( _|_ `  y
)  <->  A  C_  ( _|_ `  B ) ) )
12 oveq2 6099 . . . . . . 7  |-  ( y  =  B  ->  ( A  vH  y )  =  ( A  vH  B
) )
1312fveq2d 5695 . . . . . 6  |-  ( y  =  B  ->  ( S `  ( A  vH  y ) )  =  ( S `  ( A  vH  B ) ) )
14 fveq2 5691 . . . . . . 7  |-  ( y  =  B  ->  ( S `  y )  =  ( S `  B ) )
1514oveq2d 6107 . . . . . 6  |-  ( y  =  B  ->  (
( S `  A
)  +  ( S `
 y ) )  =  ( ( S `
 A )  +  ( S `  B
) ) )
1613, 15eqeq12d 2457 . . . . 5  |-  ( y  =  B  ->  (
( S `  ( A  vH  y ) )  =  ( ( S `
 A )  +  ( S `  y
) )  <->  ( S `  ( A  vH  B
) )  =  ( ( S `  A
)  +  ( S `
 B ) ) ) )
1711, 16imbi12d 320 . . . 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 3079 . . 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 30 . 2  |-  ( S  e.  States  ->  ( ( A  e.  CH  /\  B  e.  CH )  ->  ( A  C_  ( _|_ `  B
)  ->  ( S `  ( A  vH  B
) )  =  ( ( S `  A
)  +  ( S `
 B ) ) ) ) )
2019impd 431 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 setvar class
Syntax hints:    -> wi 4    /\ wa 369    = wceq 1369    e. wcel 1756   A.wral 2715    C_ wss 3328   -->wf 5414   ` cfv 5418  (class class class)co 6091   0cc0 9282   1c1 9283    + caddc 9285   [,]cicc 11303   ~Hchil 24321   CHcch 24331   _|_cort 24332    vH chj 24335   Statescst 24364
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1591  ax-4 1602  ax-5 1670  ax-6 1708  ax-7 1728  ax-8 1758  ax-9 1760  ax-10 1775  ax-11 1780  ax-12 1792  ax-13 1943  ax-ext 2423  ax-sep 4413  ax-nul 4421  ax-pow 4470  ax-pr 4531  ax-un 6372  ax-hilex 24401
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 967  df-tru 1372  df-ex 1587  df-nf 1590  df-sb 1701  df-eu 2257  df-mo 2258  df-clab 2430  df-cleq 2436  df-clel 2439  df-nfc 2568  df-ne 2608  df-ral 2720  df-rex 2721  df-rab 2724  df-v 2974  df-sbc 3187  df-dif 3331  df-un 3333  df-in 3335  df-ss 3342  df-nul 3638  df-if 3792  df-pw 3862  df-sn 3878  df-pr 3880  df-op 3884  df-uni 4092  df-br 4293  df-opab 4351  df-id 4636  df-xp 4846  df-rel 4847  df-cnv 4848  df-co 4849  df-dm 4850  df-rn 4851  df-res 4852  df-ima 4853  df-iota 5381  df-fun 5420  df-fn 5421  df-f 5422  df-fv 5426  df-ov 6094  df-oprab 6095  df-mpt2 6096  df-map 7216  df-sh 24609  df-ch 24624  df-st 25615
This theorem is referenced by:  sto1i  25640  stlei  25644  stji1i  25646
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