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Theorem elo 25041
Description: The law of concretion for operation class abstraction. Compare with eloprabg 5935. This version is to be used with categories. (Contributed by FL, 14-Jul-2007.) (Revised by Mario Carneiro, 3-May-2015.)
Hypotheses
Ref Expression
elo.1  |-  ( y  =  A  ->  ( ph 
<->  ps ) )
elo.2  |-  ( z  =  B  ->  ( ps 
<->  ch ) )
elo.3  |-  ( v  =  C  ->  ( ch 
<->  th ) )
elo.4  |-  ( w  =  D  ->  ( th 
<->  ta ) )
Assertion
Ref Expression
elo  |-  ( ( ( A  e.  Q  /\  B  e.  R  /\  C  e.  S
)  /\  D  e.  T )  ->  ( <. <. A ,  B >. ,  <. C ,  D >. >.  e.  { x  |  E. y E. z E. v E. w ( x  =  <. <. y ,  z >. ,  <. v ,  w >. >.  /\  ph ) }  <->  ta ) )
Distinct variable groups:    v, A, w, y, z    v, B, w, y, z    v, C, w, y, z    v, D, w, y, z    ph, x    ta, v, w, y, z   
x, v, w, y, z
Allowed substitution hints:    ph( y, z, w, v)    ps( x, y, z, w, v)    ch( x, y, z, w, v)    th( x, y, z, w, v)    ta( x)    A( x)    B( x)    C( x)    D( x)    Q( x, y, z, w, v)    R( x, y, z, w, v)    S( x, y, z, w, v)    T( x, y, z, w, v)

Proof of Theorem elo
Dummy variable  u is distinct from all other variables.
StepHypRef Expression
1 elex 2796 . . 3  |-  ( A  e.  Q  ->  A  e.  _V )
2 elex 2796 . . 3  |-  ( B  e.  R  ->  B  e.  _V )
3 elex 2796 . . 3  |-  ( C  e.  S  ->  C  e.  _V )
41, 2, 33anim123i 1137 . 2  |-  ( ( A  e.  Q  /\  B  e.  R  /\  C  e.  S )  ->  ( A  e.  _V  /\  B  e.  _V  /\  C  e.  _V )
)
5 elex 2796 . 2  |-  ( D  e.  T  ->  D  e.  _V )
6 opex 4237 . . 3  |-  <. <. A ,  B >. ,  <. C ,  D >. >.  e.  _V
7 simpr 447 . . . . . . . . . 10  |-  ( ( ( ( A  e. 
_V  /\  B  e.  _V  /\  C  e.  _V )  /\  D  e.  _V )  /\  u  =  <. <. A ,  B >. , 
<. C ,  D >. >.
)  ->  u  =  <. <. A ,  B >. ,  <. C ,  D >. >. )
87eqeq1d 2291 . . . . . . . . 9  |-  ( ( ( ( A  e. 
_V  /\  B  e.  _V  /\  C  e.  _V )  /\  D  e.  _V )  /\  u  =  <. <. A ,  B >. , 
<. C ,  D >. >.
)  ->  ( u  =  <. <. y ,  z
>. ,  <. v ,  w >. >. 
<-> 
<. <. A ,  B >. ,  <. C ,  D >. >.  =  <. <. y ,  z >. ,  <. v ,  w >. >. )
)
9 vex 2791 . . . . . . . . . . . . 13  |-  y  e. 
_V
10 vex 2791 . . . . . . . . . . . . 13  |-  z  e. 
_V
119, 10opth 4245 . . . . . . . . . . . 12  |-  ( <.
y ,  z >.  =  <. A ,  B >.  <-> 
( y  =  A  /\  z  =  B ) )
12 vex 2791 . . . . . . . . . . . . 13  |-  v  e. 
_V
13 vex 2791 . . . . . . . . . . . . 13  |-  w  e. 
_V
1412, 13opth 4245 . . . . . . . . . . . 12  |-  ( <.
v ,  w >.  = 
<. C ,  D >.  <->  (
v  =  C  /\  w  =  D )
)
1511, 14anbi12i 678 . . . . . . . . . . 11  |-  ( (
<. y ,  z >.  =  <. A ,  B >.  /\  <. v ,  w >.  =  <. C ,  D >. )  <->  ( ( y  =  A  /\  z  =  B )  /\  (
v  =  C  /\  w  =  D )
) )
16 opex 4237 . . . . . . . . . . . 12  |-  <. y ,  z >.  e.  _V
17 opex 4237 . . . . . . . . . . . 12  |-  <. v ,  w >.  e.  _V
1816, 17opth 4245 . . . . . . . . . . 11  |-  ( <. <. y ,  z >. ,  <. v ,  w >. >.  =  <. <. A ,  B >. ,  <. C ,  D >. >. 
<->  ( <. y ,  z
>.  =  <. A ,  B >.  /\  <. v ,  w >.  =  <. C ,  D >. )
)
19 anass 630 . . . . . . . . . . 11  |-  ( ( ( ( y  =  A  /\  z  =  B )  /\  v  =  C )  /\  w  =  D )  <->  ( (
y  =  A  /\  z  =  B )  /\  ( v  =  C  /\  w  =  D ) ) )
2015, 18, 193bitr4i 268 . . . . . . . . . 10  |-  ( <. <. y ,  z >. ,  <. v ,  w >. >.  =  <. <. A ,  B >. ,  <. C ,  D >. >. 
<->  ( ( ( y  =  A  /\  z  =  B )  /\  v  =  C )  /\  w  =  D ) )
21 eqcom 2285 . . . . . . . . . 10  |-  ( <. <. A ,  B >. , 
<. C ,  D >. >.  =  <. <. y ,  z
>. ,  <. v ,  w >. >. 
<-> 
<. <. y ,  z
>. ,  <. v ,  w >. >.  =  <. <. A ,  B >. ,  <. C ,  D >. >. )
22 df-3an 936 . . . . . . . . . . 11  |-  ( ( y  =  A  /\  z  =  B  /\  v  =  C )  <->  ( ( y  =  A  /\  z  =  B )  /\  v  =  C ) )
2322anbi1i 676 . . . . . . . . . 10  |-  ( ( ( y  =  A  /\  z  =  B  /\  v  =  C )  /\  w  =  D )  <->  ( (
( y  =  A  /\  z  =  B )  /\  v  =  C )  /\  w  =  D ) )
2420, 21, 233bitr4i 268 . . . . . . . . 9  |-  ( <. <. A ,  B >. , 
<. C ,  D >. >.  =  <. <. y ,  z
>. ,  <. v ,  w >. >. 
<->  ( ( y  =  A  /\  z  =  B  /\  v  =  C )  /\  w  =  D ) )
258, 24syl6bb 252 . . . . . . . 8  |-  ( ( ( ( A  e. 
_V  /\  B  e.  _V  /\  C  e.  _V )  /\  D  e.  _V )  /\  u  =  <. <. A ,  B >. , 
<. C ,  D >. >.
)  ->  ( u  =  <. <. y ,  z
>. ,  <. v ,  w >. >. 
<->  ( ( y  =  A  /\  z  =  B  /\  v  =  C )  /\  w  =  D ) ) )
2625anbi1d 685 . . . . . . 7  |-  ( ( ( ( A  e. 
_V  /\  B  e.  _V  /\  C  e.  _V )  /\  D  e.  _V )  /\  u  =  <. <. A ,  B >. , 
<. C ,  D >. >.
)  ->  ( (
u  =  <. <. y ,  z >. ,  <. v ,  w >. >.  /\  ph ) 
<->  ( ( ( y  =  A  /\  z  =  B  /\  v  =  C )  /\  w  =  D )  /\  ph ) ) )
27 elo.1 . . . . . . . . . 10  |-  ( y  =  A  ->  ( ph 
<->  ps ) )
28 elo.2 . . . . . . . . . 10  |-  ( z  =  B  ->  ( ps 
<->  ch ) )
29 elo.3 . . . . . . . . . 10  |-  ( v  =  C  ->  ( ch 
<->  th ) )
3027, 28, 29syl3an9b 1250 . . . . . . . . 9  |-  ( ( y  =  A  /\  z  =  B  /\  v  =  C )  ->  ( ph  <->  th )
)
31 elo.4 . . . . . . . . 9  |-  ( w  =  D  ->  ( th 
<->  ta ) )
3230, 31sylan9bb 680 . . . . . . . 8  |-  ( ( ( y  =  A  /\  z  =  B  /\  v  =  C )  /\  w  =  D )  ->  ( ph 
<->  ta ) )
3332pm5.32i 618 . . . . . . 7  |-  ( ( ( ( y  =  A  /\  z  =  B  /\  v  =  C )  /\  w  =  D )  /\  ph ) 
<->  ( ( ( y  =  A  /\  z  =  B  /\  v  =  C )  /\  w  =  D )  /\  ta ) )
3426, 33syl6bb 252 . . . . . 6  |-  ( ( ( ( A  e. 
_V  /\  B  e.  _V  /\  C  e.  _V )  /\  D  e.  _V )  /\  u  =  <. <. A ,  B >. , 
<. C ,  D >. >.
)  ->  ( (
u  =  <. <. y ,  z >. ,  <. v ,  w >. >.  /\  ph ) 
<->  ( ( ( y  =  A  /\  z  =  B  /\  v  =  C )  /\  w  =  D )  /\  ta ) ) )
35344exbidv 1616 . . . . 5  |-  ( ( ( ( A  e. 
_V  /\  B  e.  _V  /\  C  e.  _V )  /\  D  e.  _V )  /\  u  =  <. <. A ,  B >. , 
<. C ,  D >. >.
)  ->  ( E. y E. z E. v E. w ( u  = 
<. <. y ,  z
>. ,  <. v ,  w >. >.  /\  ph )  <->  E. y E. z E. v E. w ( ( ( y  =  A  /\  z  =  B  /\  v  =  C )  /\  w  =  D
)  /\  ta )
) )
36 vex 2791 . . . . . . . 8  |-  u  e. 
_V
37 eqeq1 2289 . . . . . . . . . 10  |-  ( x  =  u  ->  (
x  =  <. <. y ,  z >. ,  <. v ,  w >. >.  <->  u  =  <. <. y ,  z
>. ,  <. v ,  w >. >. ) )
3837anbi1d 685 . . . . . . . . 9  |-  ( x  =  u  ->  (
( x  =  <. <.
y ,  z >. ,  <. v ,  w >. >.  /\  ph )  <->  ( u  =  <. <. y ,  z
>. ,  <. v ,  w >. >.  /\  ph ) ) )
39384exbidv 1616 . . . . . . . 8  |-  ( x  =  u  ->  ( E. y E. z E. v E. w ( x  =  <. <. y ,  z >. ,  <. v ,  w >. >.  /\  ph ) 
<->  E. y E. z E. v E. w ( u  =  <. <. y ,  z >. ,  <. v ,  w >. >.  /\  ph ) ) )
4036, 39elab 2914 . . . . . . 7  |-  ( u  e.  { x  |  E. y E. z E. v E. w ( x  =  <. <. y ,  z >. ,  <. v ,  w >. >.  /\  ph ) }  <->  E. y E. z E. v E. w ( u  =  <. <. y ,  z >. ,  <. v ,  w >. >.  /\  ph ) )
41 eleq1 2343 . . . . . . 7  |-  ( u  =  <. <. A ,  B >. ,  <. C ,  D >. >.  ->  ( u  e.  { x  |  E. y E. z E. v E. w ( x  = 
<. <. y ,  z
>. ,  <. v ,  w >. >.  /\  ph ) }  <->  <. <. A ,  B >. ,  <. C ,  D >. >.  e.  { x  |  E. y E. z E. v E. w ( x  =  <. <. y ,  z >. ,  <. v ,  w >. >.  /\  ph ) } ) )
4240, 41syl5bbr 250 . . . . . 6  |-  ( u  =  <. <. A ,  B >. ,  <. C ,  D >. >.  ->  ( E. y E. z E. v E. w ( u  = 
<. <. y ,  z
>. ,  <. v ,  w >. >.  /\  ph )  <->  <. <. A ,  B >. ,  <. C ,  D >. >.  e.  { x  |  E. y E. z E. v E. w ( x  =  <. <. y ,  z >. ,  <. v ,  w >. >.  /\  ph ) } ) )
4342adantl 452 . . . . 5  |-  ( ( ( ( A  e. 
_V  /\  B  e.  _V  /\  C  e.  _V )  /\  D  e.  _V )  /\  u  =  <. <. A ,  B >. , 
<. C ,  D >. >.
)  ->  ( E. y E. z E. v E. w ( u  = 
<. <. y ,  z
>. ,  <. v ,  w >. >.  /\  ph )  <->  <. <. A ,  B >. ,  <. C ,  D >. >.  e.  { x  |  E. y E. z E. v E. w ( x  =  <. <. y ,  z >. ,  <. v ,  w >. >.  /\  ph ) } ) )
44 elisset 2798 . . . . . . . . . . 11  |-  ( A  e.  _V  ->  E. y 
y  =  A )
45 elisset 2798 . . . . . . . . . . 11  |-  ( B  e.  _V  ->  E. z 
z  =  B )
46 elisset 2798 . . . . . . . . . . 11  |-  ( C  e.  _V  ->  E. v 
v  =  C )
4744, 45, 463anim123i 1137 . . . . . . . . . 10  |-  ( ( A  e.  _V  /\  B  e.  _V  /\  C  e.  _V )  ->  ( E. y  y  =  A  /\  E. z  z  =  B  /\  E. v  v  =  C
) )
48 elisset 2798 . . . . . . . . . 10  |-  ( D  e.  _V  ->  E. w  w  =  D )
4947, 48anim12i 549 . . . . . . . . 9  |-  ( ( ( A  e.  _V  /\  B  e.  _V  /\  C  e.  _V )  /\  D  e.  _V )  ->  ( ( E. y  y  =  A  /\  E. z  z  =  B  /\  E. v  v  =  C
)  /\  E. w  w  =  D )
)
50 eeeeanv 24944 . . . . . . . . 9  |-  ( E. y E. z E. v E. w ( ( y  =  A  /\  z  =  B  /\  v  =  C )  /\  w  =  D )  <->  ( ( E. y  y  =  A  /\  E. z  z  =  B  /\  E. v  v  =  C
)  /\  E. w  w  =  D )
)
5149, 50sylibr 203 . . . . . . . 8  |-  ( ( ( A  e.  _V  /\  B  e.  _V  /\  C  e.  _V )  /\  D  e.  _V )  ->  E. y E. z E. v E. w ( ( y  =  A  /\  z  =  B  /\  v  =  C )  /\  w  =  D ) )
5251biantrurd 494 . . . . . . 7  |-  ( ( ( A  e.  _V  /\  B  e.  _V  /\  C  e.  _V )  /\  D  e.  _V )  ->  ( ta  <->  ( E. y E. z E. v E. w ( ( y  =  A  /\  z  =  B  /\  v  =  C )  /\  w  =  D )  /\  ta ) ) )
53 19.41vvvv 1845 . . . . . . 7  |-  ( E. y E. z E. v E. w ( ( ( y  =  A  /\  z  =  B  /\  v  =  C )  /\  w  =  D )  /\  ta ) 
<->  ( E. y E. z E. v E. w ( ( y  =  A  /\  z  =  B  /\  v  =  C )  /\  w  =  D )  /\  ta ) )
5452, 53syl6rbbr 255 . . . . . 6  |-  ( ( ( A  e.  _V  /\  B  e.  _V  /\  C  e.  _V )  /\  D  e.  _V )  ->  ( E. y E. z E. v E. w ( ( ( y  =  A  /\  z  =  B  /\  v  =  C )  /\  w  =  D
)  /\  ta )  <->  ta ) )
5554adantr 451 . . . . 5  |-  ( ( ( ( A  e. 
_V  /\  B  e.  _V  /\  C  e.  _V )  /\  D  e.  _V )  /\  u  =  <. <. A ,  B >. , 
<. C ,  D >. >.
)  ->  ( E. y E. z E. v E. w ( ( ( y  =  A  /\  z  =  B  /\  v  =  C )  /\  w  =  D
)  /\  ta )  <->  ta ) )
5635, 43, 553bitr3d 274 . . . 4  |-  ( ( ( ( A  e. 
_V  /\  B  e.  _V  /\  C  e.  _V )  /\  D  e.  _V )  /\  u  =  <. <. A ,  B >. , 
<. C ,  D >. >.
)  ->  ( <. <. A ,  B >. , 
<. C ,  D >. >.  e.  { x  |  E. y E. z E. v E. w ( x  = 
<. <. y ,  z
>. ,  <. v ,  w >. >.  /\  ph ) }  <->  ta ) )
5756expcom 424 . . 3  |-  ( u  =  <. <. A ,  B >. ,  <. C ,  D >. >.  ->  ( (
( A  e.  _V  /\  B  e.  _V  /\  C  e.  _V )  /\  D  e.  _V )  ->  ( <. <. A ,  B >. ,  <. C ,  D >. >.  e.  { x  |  E. y E. z E. v E. w ( x  =  <. <. y ,  z >. ,  <. v ,  w >. >.  /\  ph ) }  <->  ta ) ) )
586, 57vtocle 2857 . 2  |-  ( ( ( A  e.  _V  /\  B  e.  _V  /\  C  e.  _V )  /\  D  e.  _V )  ->  ( <. <. A ,  B >. ,  <. C ,  D >. >.  e.  { x  |  E. y E. z E. v E. w ( x  =  <. <. y ,  z >. ,  <. v ,  w >. >.  /\  ph ) }  <->  ta ) )
594, 5, 58syl2an 463 1  |-  ( ( ( A  e.  Q  /\  B  e.  R  /\  C  e.  S
)  /\  D  e.  T )  ->  ( <. <. A ,  B >. ,  <. C ,  D >. >.  e.  { x  |  E. y E. z E. v E. w ( x  =  <. <. y ,  z >. ,  <. v ,  w >. >.  /\  ph ) }  <->  ta ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 176    /\ wa 358    /\ w3a 934   E.wex 1528    = wceq 1623    e. wcel 1684   {cab 2269   _Vcvv 2788   <.cop 3643
This theorem is referenced by:  eloi  25086  vecval1b  25451  isalg  25721  isded  25736  iscatOLD  25754
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-14 1688  ax-6 1703  ax-7 1708  ax-11 1715  ax-12 1866  ax-ext 2264  ax-sep 4141  ax-nul 4149  ax-pr 4214
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-clab 2270  df-cleq 2276  df-clel 2279  df-nfc 2408  df-ne 2448  df-rab 2552  df-v 2790  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
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