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Theorem stj 27280
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 27258 . . . 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 1013 . . 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 3520 . . . . 5  |-  ( x  =  A  ->  (
x  C_  ( _|_ `  y )  <->  A  C_  ( _|_ `  y ) ) )
4 oveq1 6303 . . . . . . 7  |-  ( x  =  A  ->  (
x  vH  y )  =  ( A  vH  y ) )
54fveq2d 5876 . . . . . 6  |-  ( x  =  A  ->  ( S `  ( x  vH  y ) )  =  ( S `  ( A  vH  y ) ) )
6 fveq2 5872 . . . . . . 7  |-  ( x  =  A  ->  ( S `  x )  =  ( S `  A ) )
76oveq1d 6311 . . . . . 6  |-  ( x  =  A  ->  (
( S `  x
)  +  ( S `
 y ) )  =  ( ( S `
 A )  +  ( S `  y
) ) )
85, 7eqeq12d 2479 . . . . 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 5872 . . . . . 6  |-  ( y  =  B  ->  ( _|_ `  y )  =  ( _|_ `  B
) )
1110sseq2d 3527 . . . . 5  |-  ( y  =  B  ->  ( A  C_  ( _|_ `  y
)  <->  A  C_  ( _|_ `  B ) ) )
12 oveq2 6304 . . . . . . 7  |-  ( y  =  B  ->  ( A  vH  y )  =  ( A  vH  B
) )
1312fveq2d 5876 . . . . . 6  |-  ( y  =  B  ->  ( S `  ( A  vH  y ) )  =  ( S `  ( A  vH  B ) ) )
14 fveq2 5872 . . . . . . 7  |-  ( y  =  B  ->  ( S `  y )  =  ( S `  B ) )
1514oveq2d 6312 . . . . . 6  |-  ( y  =  B  ->  (
( S `  A
)  +  ( S `
 y ) )  =  ( ( S `
 A )  +  ( S `  B
) ) )
1613, 15eqeq12d 2479 . . . . 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 3219 . . 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 1395    e. wcel 1819   A.wral 2807    C_ wss 3471   -->wf 5590   ` cfv 5594  (class class class)co 6296   0cc0 9509   1c1 9510    + caddc 9512   [,]cicc 11557   ~Hchil 25962   CHcch 25972   _|_cort 25973    vH chj 25976   Statescst 26005
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1619  ax-4 1632  ax-5 1705  ax-6 1748  ax-7 1791  ax-8 1821  ax-9 1823  ax-10 1838  ax-11 1843  ax-12 1855  ax-13 2000  ax-ext 2435  ax-sep 4578  ax-nul 4586  ax-pow 4634  ax-pr 4695  ax-un 6591  ax-hilex 26042
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 975  df-tru 1398  df-ex 1614  df-nf 1618  df-sb 1741  df-eu 2287  df-mo 2288  df-clab 2443  df-cleq 2449  df-clel 2452  df-nfc 2607  df-ne 2654  df-ral 2812  df-rex 2813  df-rab 2816  df-v 3111  df-sbc 3328  df-dif 3474  df-un 3476  df-in 3478  df-ss 3485  df-nul 3794  df-if 3945  df-pw 4017  df-sn 4033  df-pr 4035  df-op 4039  df-uni 4252  df-br 4457  df-opab 4516  df-id 4804  df-xp 5014  df-rel 5015  df-cnv 5016  df-co 5017  df-dm 5018  df-rn 5019  df-res 5020  df-ima 5021  df-iota 5557  df-fun 5596  df-fn 5597  df-f 5598  df-fv 5602  df-ov 6299  df-oprab 6300  df-mpt2 6301  df-map 7440  df-sh 26250  df-ch 26265  df-st 27256
This theorem is referenced by:  sto1i  27281  stlei  27285  stji1i  27287
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