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Theorem dedalg 25846
Description: A deductive system is an "algebra". (Contributed by FL, 28-Oct-2007.)
Assertion
Ref Expression
dedalg  |-  ( T  e.  Ded  ->  T  e.  Alg  )

Proof of Theorem dedalg
Dummy variables  x  y  z are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 relded 25843 . . . . 5  |-  Rel  Ded
2 reldded 25844 . . . . 5  |-  Rel  dom  Ded
3 relrded 25845 . . . . 5  |-  Rel  ran  Ded
41, 2, 33pm3.2i 1130 . . . 4  |-  ( Rel 
Ded  /\  Rel  dom  Ded  /\ 
Rel  ran  Ded )
5 11st22nd 25148 . . . 4  |-  ( ( ( Rel  Ded  /\  Rel  dom  Ded  /\  Rel  ran  Ded )  /\  T  e. 
Ded )  ->  T  =  <. <. ( 1st `  ( 1st `  T ) ) ,  ( 2nd `  ( 1st `  T ) )
>. ,  <. ( 1st `  ( 2nd `  T
) ) ,  ( 2nd `  ( 2nd `  T ) ) >. >. )
64, 5mpan 651 . . 3  |-  ( T  e.  Ded  ->  T  =  <. <. ( 1st `  ( 1st `  T ) ) ,  ( 2nd `  ( 1st `  T ) )
>. ,  <. ( 1st `  ( 2nd `  T
) ) ,  ( 2nd `  ( 2nd `  T ) ) >. >. )
7 eqid 2296 . . . . . 6  |-  ( dom_ `  T )  =  (
dom_ `  T )
87domval 25826 . . . . 5  |-  ( dom_ `  T )  =  ( 1st `  ( 1st `  T ) )
9 eqid 2296 . . . . . 6  |-  ( cod_ `  T )  =  (
cod_ `  T )
109codval 25827 . . . . 5  |-  ( cod_ `  T )  =  ( 2nd `  ( 1st `  T ) )
118, 10opeq12i 3817 . . . 4  |-  <. ( dom_ `  T ) ,  ( cod_ `  T
) >.  =  <. ( 1st `  ( 1st `  T
) ) ,  ( 2nd `  ( 1st `  T ) ) >.
12 eqid 2296 . . . . . 6  |-  ( id_ `  T )  =  ( id_ `  T )
1312idval 25828 . . . . 5  |-  ( id_ `  T )  =  ( 1st `  ( 2nd `  T ) )
14 eqid 2296 . . . . . 6  |-  ( o_
`  T )  =  ( o_ `  T
)
1514cmpval 25829 . . . . 5  |-  ( o_
`  T )  =  ( 2nd `  ( 2nd `  T ) )
1613, 15opeq12i 3817 . . . 4  |-  <. ( id_ `  T ) ,  ( o_ `  T
) >.  =  <. ( 1st `  ( 2nd `  T
) ) ,  ( 2nd `  ( 2nd `  T ) ) >.
1711, 16opeq12i 3817 . . 3  |-  <. <. ( dom_ `  T ) ,  ( cod_ `  T
) >. ,  <. ( id_ `  T ) ,  ( o_ `  T
) >. >.  =  <. <. ( 1st `  ( 1st `  T
) ) ,  ( 2nd `  ( 1st `  T ) ) >. ,  <. ( 1st `  ( 2nd `  T ) ) ,  ( 2nd `  ( 2nd `  T ) )
>. >.
186, 17syl6eqr 2346 . 2  |-  ( T  e.  Ded  ->  T  =  <. <. ( dom_ `  T
) ,  ( cod_ `  T ) >. ,  <. ( id_ `  T ) ,  ( o_ `  T ) >. >. )
19 eqid 2296 . . . . 5  |-  dom  ( dom_ `  T )  =  dom  ( dom_ `  T
)
20 eqid 2296 . . . . 5  |-  dom  ( id_ `  T )  =  dom  ( id_ `  T
)
217, 9, 12, 14, 19, 20dedi 25840 . . . 4  |-  ( T  e.  Ded  ->  (
( <. <. ( dom_ `  T
) ,  ( cod_ `  T ) >. ,  <. ( id_ `  T ) ,  ( o_ `  T ) >. >.  e.  Alg  /\ 
A. z  e.  dom  ( id_ `  T ) ( ( ( dom_ `  T ) `  (
( id_ `  T
) `  z )
)  =  z  /\  ( ( cod_ `  T
) `  ( ( id_ `  T ) `  z ) )  =  z )  /\  A. x  e.  dom  ( dom_ `  T ) A. y  e.  dom  ( dom_ `  T
) ( <. y ,  x >.  e.  dom  ( o_ `  T )  <-> 
( ( dom_ `  T
) `  y )  =  ( ( cod_ `  T ) `  x
) ) )  /\  ( A. x  e.  dom  ( dom_ `  T ) A. y  e.  dom  ( dom_ `  T )
( ( ( dom_ `  T ) `  y
)  =  ( (
cod_ `  T ) `  x )  ->  (
( dom_ `  T ) `  ( y ( o_
`  T ) x ) )  =  ( ( dom_ `  T
) `  x )
)  /\  A. x  e.  dom  ( dom_ `  T
) A. y  e. 
dom  ( dom_ `  T
) ( ( (
dom_ `  T ) `  y )  =  ( ( cod_ `  T
) `  x )  ->  ( ( cod_ `  T
) `  ( y
( o_ `  T
) x ) )  =  ( ( cod_ `  T ) `  y
) ) ) ) )
2221simpld 445 . . 3  |-  ( T  e.  Ded  ->  ( <. <. ( dom_ `  T
) ,  ( cod_ `  T ) >. ,  <. ( id_ `  T ) ,  ( o_ `  T ) >. >.  e.  Alg  /\ 
A. z  e.  dom  ( id_ `  T ) ( ( ( dom_ `  T ) `  (
( id_ `  T
) `  z )
)  =  z  /\  ( ( cod_ `  T
) `  ( ( id_ `  T ) `  z ) )  =  z )  /\  A. x  e.  dom  ( dom_ `  T ) A. y  e.  dom  ( dom_ `  T
) ( <. y ,  x >.  e.  dom  ( o_ `  T )  <-> 
( ( dom_ `  T
) `  y )  =  ( ( cod_ `  T ) `  x
) ) ) )
2322simp1d 967 . 2  |-  ( T  e.  Ded  ->  <. <. ( dom_ `  T ) ,  ( cod_ `  T
) >. ,  <. ( id_ `  T ) ,  ( o_ `  T
) >. >.  e.  Alg  )
2418, 23eqeltrd 2370 1  |-  ( T  e.  Ded  ->  T  e.  Alg  )
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 176    /\ wa 358    /\ w3a 934    = wceq 1632    e. wcel 1696   A.wral 2556   <.cop 3656   dom cdm 4705   ran crn 4706   Rel wrel 4710   ` cfv 5271  (class class class)co 5874   1stc1st 6136   2ndc2nd 6137    Alg calg 25814   dom_cdom_ 25815   cod_ccod_ 25816   id_cid_ 25817   o_co_ 25818   Dedcded 25837
This theorem is referenced by:  rdmob  25851  rcmob  25852  aidm2  25853  dmrngcmp  25854  domc  25868  codc  25869  idc  25870  cmppfc  25871  dualded  25886  dualcat2  25887  mrdmcd  25897  idsubidsup  25960
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
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-sn 3659  df-pr 3660  df-op 3662  df-uni 3844  df-int 3879  df-br 4040  df-opab 4094  df-mpt 4095  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-fo 5277  df-fv 5279  df-ov 5877  df-1st 6138  df-2nd 6139  df-dom_ 25820  df-cod_ 25821  df-id_ 25822  df-cmpa 25823  df-ded 25838
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