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Theorem comffval 13602
Description: Value of the functionalized composition operation. (Contributed by Mario Carneiro, 4-Jan-2017.)
Hypotheses
Ref Expression
comfffval.o  |-  O  =  (compf `  C )
comfffval.b  |-  B  =  ( Base `  C
)
comfffval.h  |-  H  =  (  Hom  `  C
)
comfffval.x  |-  .x.  =  (comp `  C )
comffval.x  |-  ( ph  ->  X  e.  B )
comffval.y  |-  ( ph  ->  Y  e.  B )
comffval.z  |-  ( ph  ->  Z  e.  B )
Assertion
Ref Expression
comffval  |-  ( ph  ->  ( <. X ,  Y >. O Z )  =  ( g  e.  ( Y H Z ) ,  f  e.  ( X H Y ) 
|->  ( g ( <. X ,  Y >.  .x. 
Z ) f ) ) )
Distinct variable groups:    f, g, C    ph, f, g    .x. , f,
g    f, X, g    f, Y, g    f, Z, g   
f, H, g
Allowed substitution hints:    B( f, g)    O( f, g)

Proof of Theorem comffval
Dummy variables  x  z are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 comfffval.o . . . 4  |-  O  =  (compf `  C )
2 comfffval.b . . . 4  |-  B  =  ( Base `  C
)
3 comfffval.h . . . 4  |-  H  =  (  Hom  `  C
)
4 comfffval.x . . . 4  |-  .x.  =  (comp `  C )
51, 2, 3, 4comfffval 13601 . . 3  |-  O  =  ( x  e.  ( B  X.  B ) ,  z  e.  B  |->  ( g  e.  ( ( 2nd `  x
) H z ) ,  f  e.  ( H `  x ) 
|->  ( g ( x 
.x.  z ) f ) ) )
65a1i 10 . 2  |-  ( ph  ->  O  =  ( x  e.  ( B  X.  B ) ,  z  e.  B  |->  ( g  e.  ( ( 2nd `  x ) H z ) ,  f  e.  ( H `  x
)  |->  ( g ( x  .x.  z ) f ) ) ) )
7 simprl 732 . . . . . 6  |-  ( (
ph  /\  ( x  =  <. X ,  Y >.  /\  z  =  Z ) )  ->  x  =  <. X ,  Y >. )
87fveq2d 5529 . . . . 5  |-  ( (
ph  /\  ( x  =  <. X ,  Y >.  /\  z  =  Z ) )  ->  ( 2nd `  x )  =  ( 2nd `  <. X ,  Y >. )
)
9 comffval.x . . . . . . 7  |-  ( ph  ->  X  e.  B )
10 comffval.y . . . . . . 7  |-  ( ph  ->  Y  e.  B )
11 op2ndg 6133 . . . . . . 7  |-  ( ( X  e.  B  /\  Y  e.  B )  ->  ( 2nd `  <. X ,  Y >. )  =  Y )
129, 10, 11syl2anc 642 . . . . . 6  |-  ( ph  ->  ( 2nd `  <. X ,  Y >. )  =  Y )
1312adantr 451 . . . . 5  |-  ( (
ph  /\  ( x  =  <. X ,  Y >.  /\  z  =  Z ) )  ->  ( 2nd `  <. X ,  Y >. )  =  Y )
148, 13eqtrd 2315 . . . 4  |-  ( (
ph  /\  ( x  =  <. X ,  Y >.  /\  z  =  Z ) )  ->  ( 2nd `  x )  =  Y )
15 simprr 733 . . . 4  |-  ( (
ph  /\  ( x  =  <. X ,  Y >.  /\  z  =  Z ) )  ->  z  =  Z )
1614, 15oveq12d 5876 . . 3  |-  ( (
ph  /\  ( x  =  <. X ,  Y >.  /\  z  =  Z ) )  ->  (
( 2nd `  x
) H z )  =  ( Y H Z ) )
177fveq2d 5529 . . . 4  |-  ( (
ph  /\  ( x  =  <. X ,  Y >.  /\  z  =  Z ) )  ->  ( H `  x )  =  ( H `  <. X ,  Y >. ) )
18 df-ov 5861 . . . 4  |-  ( X H Y )  =  ( H `  <. X ,  Y >. )
1917, 18syl6eqr 2333 . . 3  |-  ( (
ph  /\  ( x  =  <. X ,  Y >.  /\  z  =  Z ) )  ->  ( H `  x )  =  ( X H Y ) )
207, 15oveq12d 5876 . . . 4  |-  ( (
ph  /\  ( x  =  <. X ,  Y >.  /\  z  =  Z ) )  ->  (
x  .x.  z )  =  ( <. X ,  Y >.  .x.  Z )
)
2120oveqd 5875 . . 3  |-  ( (
ph  /\  ( x  =  <. X ,  Y >.  /\  z  =  Z ) )  ->  (
g ( x  .x.  z ) f )  =  ( g (
<. X ,  Y >.  .x. 
Z ) f ) )
2216, 19, 21mpt2eq123dv 5910 . 2  |-  ( (
ph  /\  ( x  =  <. X ,  Y >.  /\  z  =  Z ) )  ->  (
g  e.  ( ( 2nd `  x ) H z ) ,  f  e.  ( H `
 x )  |->  ( g ( x  .x.  z ) f ) )  =  ( g  e.  ( Y H Z ) ,  f  e.  ( X H Y )  |->  ( g ( <. X ,  Y >.  .x.  Z ) f ) ) )
23 opelxpi 4721 . . 3  |-  ( ( X  e.  B  /\  Y  e.  B )  -> 
<. X ,  Y >.  e.  ( B  X.  B
) )
249, 10, 23syl2anc 642 . 2  |-  ( ph  -> 
<. X ,  Y >.  e.  ( B  X.  B
) )
25 comffval.z . 2  |-  ( ph  ->  Z  e.  B )
26 ovex 5883 . . . 4  |-  ( Y H Z )  e. 
_V
27 ovex 5883 . . . 4  |-  ( X H Y )  e. 
_V
2826, 27mpt2ex 6198 . . 3  |-  ( g  e.  ( Y H Z ) ,  f  e.  ( X H Y )  |->  ( g ( <. X ,  Y >.  .x.  Z ) f ) )  e.  _V
2928a1i 10 . 2  |-  ( ph  ->  ( g  e.  ( Y H Z ) ,  f  e.  ( X H Y ) 
|->  ( g ( <. X ,  Y >.  .x. 
Z ) f ) )  e.  _V )
306, 22, 24, 25, 29ovmpt2d 5975 1  |-  ( ph  ->  ( <. X ,  Y >. O Z )  =  ( g  e.  ( Y H Z ) ,  f  e.  ( X H Y ) 
|->  ( g ( <. X ,  Y >.  .x. 
Z ) f ) ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 358    = wceq 1623    e. wcel 1684   _Vcvv 2788   <.cop 3643    X. cxp 4687   ` cfv 5255  (class class class)co 5858    e. cmpt2 5860   2ndc2nd 6121   Basecbs 13148    Hom chom 13219  compcco 13220  compfccomf 13569
This theorem is referenced by:  comfval  13603  comffval2  13605  comffn  13608
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-rep 4131  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-reu 2550  df-rab 2552  df-v 2790  df-sbc 2992  df-csb 3082  df-dif 3155  df-un 3157  df-in 3159  df-ss 3166  df-nul 3456  df-if 3566  df-pw 3627  df-sn 3646  df-pr 3647  df-op 3649  df-uni 3828  df-iun 3907  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-f1 5260  df-fo 5261  df-f1o 5262  df-fv 5263  df-ov 5861  df-oprab 5862  df-mpt2 5863  df-1st 6122  df-2nd 6123  df-comf 13573
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