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Theorem dedalg 25743
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 25740 . . . . 5  |-  Rel  Ded
2 reldded 25741 . . . . 5  |-  Rel  dom  Ded
3 relrded 25742 . . . . 5  |-  Rel  ran  Ded
41, 2, 33pm3.2i 1130 . . . 4  |-  ( Rel 
Ded  /\  Rel  dom  Ded  /\ 
Rel  ran  Ded )
5 11st22nd 25045 . . . 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 2283 . . . . . 6  |-  ( dom_ `  T )  =  (
dom_ `  T )
87domval 25723 . . . . 5  |-  ( dom_ `  T )  =  ( 1st `  ( 1st `  T ) )
9 eqid 2283 . . . . . 6  |-  ( cod_ `  T )  =  (
cod_ `  T )
109codval 25724 . . . . 5  |-  ( cod_ `  T )  =  ( 2nd `  ( 1st `  T ) )
118, 10opeq12i 3801 . . . 4  |-  <. ( dom_ `  T ) ,  ( cod_ `  T
) >.  =  <. ( 1st `  ( 1st `  T
) ) ,  ( 2nd `  ( 1st `  T ) ) >.
12 eqid 2283 . . . . . 6  |-  ( id_ `  T )  =  ( id_ `  T )
1312idval 25725 . . . . 5  |-  ( id_ `  T )  =  ( 1st `  ( 2nd `  T ) )
14 eqid 2283 . . . . . 6  |-  ( o_
`  T )  =  ( o_ `  T
)
1514cmpval 25726 . . . . 5  |-  ( o_
`  T )  =  ( 2nd `  ( 2nd `  T ) )
1613, 15opeq12i 3801 . . . 4  |-  <. ( id_ `  T ) ,  ( o_ `  T
) >.  =  <. ( 1st `  ( 2nd `  T
) ) ,  ( 2nd `  ( 2nd `  T ) ) >.
1711, 16opeq12i 3801 . . 3  |-  <. <. ( dom_ `  T ) ,  ( cod_ `  T
) >. ,  <. ( id_ `  T ) ,  ( o_ `  T
) >. >.  =  <. <. ( 1st `  ( 1st `  T
) ) ,  ( 2nd `  ( 1st `  T ) ) >. ,  <. ( 1st `  ( 2nd `  T ) ) ,  ( 2nd `  ( 2nd `  T ) )
>. >.
186, 17syl6eqr 2333 . 2  |-  ( T  e.  Ded  ->  T  =  <. <. ( dom_ `  T
) ,  ( cod_ `  T ) >. ,  <. ( id_ `  T ) ,  ( o_ `  T ) >. >. )
19 eqid 2283 . . . . 5  |-  dom  ( dom_ `  T )  =  dom  ( dom_ `  T
)
20 eqid 2283 . . . . 5  |-  dom  ( id_ `  T )  =  dom  ( id_ `  T
)
217, 9, 12, 14, 19, 20dedi 25737 . . . 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 2357 1  |-  ( T  e.  Ded  ->  T  e.  Alg  )
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 176    /\ wa 358    /\ w3a 934    = wceq 1623    e. wcel 1684   A.wral 2543   <.cop 3643   dom cdm 4689   ran crn 4690   Rel wrel 4694   ` cfv 5255  (class class class)co 5858   1stc1st 6120   2ndc2nd 6121    Alg calg 25711   dom_cdom_ 25712   cod_ccod_ 25713   id_cid_ 25714   o_co_ 25715   Dedcded 25734
This theorem is referenced by:  rdmob  25748  rcmob  25749  aidm2  25750  dmrngcmp  25751  domc  25765  codc  25766  idc  25767  cmppfc  25768  dualded  25783  dualcat2  25784  mrdmcd  25794  idsubidsup  25857
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1533  ax-5 1544  ax-17 1603  ax-9 1635  ax-8 1643  ax-13 1686  ax-14 1688  ax-6 1703  ax-7 1708  ax-11 1715  ax-12 1866  ax-ext 2264  ax-sep 4141  ax-nul 4149  ax-pow 4188  ax-pr 4214  ax-un 4512
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 936  df-tru 1310  df-ex 1529  df-nf 1532  df-sb 1630  df-eu 2147  df-mo 2148  df-clab 2270  df-cleq 2276  df-clel 2279  df-nfc 2408  df-ne 2448  df-ral 2548  df-rex 2549  df-rab 2552  df-v 2790  df-sbc 2992  df-dif 3155  df-un 3157  df-in 3159  df-ss 3166  df-nul 3456  df-if 3566  df-sn 3646  df-pr 3647  df-op 3649  df-uni 3828  df-int 3863  df-br 4024  df-opab 4078  df-mpt 4079  df-id 4309  df-xp 4695  df-rel 4696  df-cnv 4697  df-co 4698  df-dm 4699  df-rn 4700  df-res 4701  df-ima 4702  df-iota 5219  df-fun 5257  df-fn 5258  df-f 5259  df-fo 5261  df-fv 5263  df-ov 5861  df-1st 6122  df-2nd 6123  df-dom_ 25717  df-cod_ 25718  df-id_ 25719  df-cmpa 25720  df-ded 25735
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