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Theorem stj 22831
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 22809 . . . 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 972 . . 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 3212 . . . . 5  |-  ( x  =  A  ->  (
x  C_  ( _|_ `  y )  <->  A  C_  ( _|_ `  y ) ) )
4 oveq1 5881 . . . . . . 7  |-  ( x  =  A  ->  (
x  vH  y )  =  ( A  vH  y ) )
54fveq2d 5545 . . . . . 6  |-  ( x  =  A  ->  ( S `  ( x  vH  y ) )  =  ( S `  ( A  vH  y ) ) )
6 fveq2 5541 . . . . . . 7  |-  ( x  =  A  ->  ( S `  x )  =  ( S `  A ) )
76oveq1d 5889 . . . . . 6  |-  ( x  =  A  ->  (
( S `  x
)  +  ( S `
 y ) )  =  ( ( S `
 A )  +  ( S `  y
) ) )
85, 7eqeq12d 2310 . . . . 5  |-  ( x  =  A  ->  (
( S `  (
x  vH  y )
)  =  ( ( S `  x )  +  ( S `  y ) )  <->  ( S `  ( A  vH  y
) )  =  ( ( S `  A
)  +  ( S `
 y ) ) ) )
93, 8imbi12d 311 . . . 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 5541 . . . . . 6  |-  ( y  =  B  ->  ( _|_ `  y )  =  ( _|_ `  B
) )
1110sseq2d 3219 . . . . 5  |-  ( y  =  B  ->  ( A  C_  ( _|_ `  y
)  <->  A  C_  ( _|_ `  B ) ) )
12 oveq2 5882 . . . . . . 7  |-  ( y  =  B  ->  ( A  vH  y )  =  ( A  vH  B
) )
1312fveq2d 5545 . . . . . 6  |-  ( y  =  B  ->  ( S `  ( A  vH  y ) )  =  ( S `  ( A  vH  B ) ) )
14 fveq2 5541 . . . . . . 7  |-  ( y  =  B  ->  ( S `  y )  =  ( S `  B ) )
1514oveq2d 5890 . . . . . 6  |-  ( y  =  B  ->  (
( S `  A
)  +  ( S `
 y ) )  =  ( ( S `
 A )  +  ( S `  B
) ) )
1613, 15eqeq12d 2310 . . . . 5  |-  ( y  =  B  ->  (
( S `  ( A  vH  y ) )  =  ( ( S `
 A )  +  ( S `  y
) )  <->  ( S `  ( A  vH  B
) )  =  ( ( S `  A
)  +  ( S `
 B ) ) ) )
1711, 16imbi12d 311 . . . 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 2903 . . 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 26 . 2  |-  ( S  e.  States  ->  ( ( A  e.  CH  /\  B  e.  CH )  ->  ( A  C_  ( _|_ `  B
)  ->  ( S `  ( A  vH  B
) )  =  ( ( S `  A
)  +  ( S `
 B ) ) ) ) )
2019imp3a 420 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 358    = wceq 1632    e. wcel 1696   A.wral 2556    C_ wss 3165   -->wf 5267   ` cfv 5271  (class class class)co 5874   0cc0 8753   1c1 8754    + caddc 8756   [,]cicc 10675   ~Hchil 21515   CHcch 21525   _|_cort 21526    vH chj 21529   Statescst 21558
This theorem is referenced by:  sto1i  22832  stlei  22836  stji1i  22838
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1536  ax-5 1547  ax-17 1606  ax-9 1644  ax-8 1661  ax-13 1698  ax-14 1700  ax-6 1715  ax-7 1720  ax-11 1727  ax-12 1878  ax-ext 2277  ax-sep 4157  ax-nul 4165  ax-pow 4204  ax-pr 4230  ax-un 4528  ax-hilex 21595
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 936  df-tru 1310  df-ex 1532  df-nf 1535  df-sb 1639  df-eu 2160  df-mo 2161  df-clab 2283  df-cleq 2289  df-clel 2292  df-nfc 2421  df-ne 2461  df-ral 2561  df-rex 2562  df-rab 2565  df-v 2803  df-sbc 3005  df-dif 3168  df-un 3170  df-in 3172  df-ss 3179  df-nul 3469  df-if 3579  df-pw 3640  df-sn 3659  df-pr 3660  df-op 3662  df-uni 3844  df-br 4040  df-opab 4094  df-id 4325  df-xp 4711  df-rel 4712  df-cnv 4713  df-co 4714  df-dm 4715  df-rn 4716  df-res 4717  df-ima 4718  df-iota 5235  df-fun 5273  df-fn 5274  df-f 5275  df-fv 5279  df-ov 5877  df-oprab 5878  df-mpt2 5879  df-map 6790  df-sh 21802  df-ch 21817  df-st 22807
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