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Theorem comffval 14659
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 14658 . . 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 11 . 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 755 . . . . . 6  |-  ( (
ph  /\  ( x  =  <. X ,  Y >.  /\  z  =  Z ) )  ->  x  =  <. X ,  Y >. )
87fveq2d 5716 . . . . 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 6611 . . . . . . 7  |-  ( ( X  e.  B  /\  Y  e.  B )  ->  ( 2nd `  <. X ,  Y >. )  =  Y )
129, 10, 11syl2anc 661 . . . . . 6  |-  ( ph  ->  ( 2nd `  <. X ,  Y >. )  =  Y )
1312adantr 465 . . . . 5  |-  ( (
ph  /\  ( x  =  <. X ,  Y >.  /\  z  =  Z ) )  ->  ( 2nd `  <. X ,  Y >. )  =  Y )
148, 13eqtrd 2475 . . . 4  |-  ( (
ph  /\  ( x  =  <. X ,  Y >.  /\  z  =  Z ) )  ->  ( 2nd `  x )  =  Y )
15 simprr 756 . . . 4  |-  ( (
ph  /\  ( x  =  <. X ,  Y >.  /\  z  =  Z ) )  ->  z  =  Z )
1614, 15oveq12d 6130 . . 3  |-  ( (
ph  /\  ( x  =  <. X ,  Y >.  /\  z  =  Z ) )  ->  (
( 2nd `  x
) H z )  =  ( Y H Z ) )
177fveq2d 5716 . . . 4  |-  ( (
ph  /\  ( x  =  <. X ,  Y >.  /\  z  =  Z ) )  ->  ( H `  x )  =  ( H `  <. X ,  Y >. ) )
18 df-ov 6115 . . . 4  |-  ( X H Y )  =  ( H `  <. X ,  Y >. )
1917, 18syl6eqr 2493 . . 3  |-  ( (
ph  /\  ( x  =  <. X ,  Y >.  /\  z  =  Z ) )  ->  ( H `  x )  =  ( X H Y ) )
207, 15oveq12d 6130 . . . 4  |-  ( (
ph  /\  ( x  =  <. X ,  Y >.  /\  z  =  Z ) )  ->  (
x  .x.  z )  =  ( <. X ,  Y >.  .x.  Z )
)
2120oveqd 6129 . . 3  |-  ( (
ph  /\  ( x  =  <. X ,  Y >.  /\  z  =  Z ) )  ->  (
g ( x  .x.  z ) f )  =  ( g (
<. X ,  Y >.  .x. 
Z ) f ) )
2216, 19, 21mpt2eq123dv 6169 . 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 4892 . . 3  |-  ( ( X  e.  B  /\  Y  e.  B )  -> 
<. X ,  Y >.  e.  ( B  X.  B
) )
249, 10, 23syl2anc 661 . 2  |-  ( ph  -> 
<. X ,  Y >.  e.  ( B  X.  B
) )
25 comffval.z . 2  |-  ( ph  ->  Z  e.  B )
26 ovex 6137 . . . 4  |-  ( Y H Z )  e. 
_V
27 ovex 6137 . . . 4  |-  ( X H Y )  e. 
_V
2826, 27mpt2ex 6671 . . 3  |-  ( g  e.  ( Y H Z ) ,  f  e.  ( X H Y )  |->  ( g ( <. X ,  Y >.  .x.  Z ) f ) )  e.  _V
2928a1i 11 . 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 6239 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 setvar class
Syntax hints:    -> wi 4    /\ wa 369    = wceq 1369    e. wcel 1756   _Vcvv 2993   <.cop 3904    X. cxp 4859   ` cfv 5439  (class class class)co 6112    e. cmpt2 6114   2ndc2nd 6597   Basecbs 14195   Hom chom 14270  compcco 14271  compfccomf 14626
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1591  ax-4 1602  ax-5 1670  ax-6 1708  ax-7 1728  ax-8 1758  ax-9 1760  ax-10 1775  ax-11 1780  ax-12 1792  ax-13 1943  ax-ext 2423  ax-rep 4424  ax-sep 4434  ax-nul 4442  ax-pow 4491  ax-pr 4552  ax-un 6393
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 967  df-tru 1372  df-ex 1587  df-nf 1590  df-sb 1701  df-eu 2257  df-mo 2258  df-clab 2430  df-cleq 2436  df-clel 2439  df-nfc 2577  df-ne 2622  df-ral 2741  df-rex 2742  df-reu 2743  df-rab 2745  df-v 2995  df-sbc 3208  df-csb 3310  df-dif 3352  df-un 3354  df-in 3356  df-ss 3363  df-nul 3659  df-if 3813  df-pw 3883  df-sn 3899  df-pr 3901  df-op 3905  df-uni 4113  df-iun 4194  df-br 4314  df-opab 4372  df-mpt 4373  df-id 4657  df-xp 4867  df-rel 4868  df-cnv 4869  df-co 4870  df-dm 4871  df-rn 4872  df-res 4873  df-ima 4874  df-iota 5402  df-fun 5441  df-fn 5442  df-f 5443  df-f1 5444  df-fo 5445  df-f1o 5446  df-fv 5447  df-ov 6115  df-oprab 6116  df-mpt2 6117  df-1st 6598  df-2nd 6599  df-comf 14630
This theorem is referenced by:  comfval  14660  comffval2  14662  comffn  14665
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