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Theorem comffval 15187
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 15186 . . 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 754 . . . . . 6  |-  ( (
ph  /\  ( x  =  <. X ,  Y >.  /\  z  =  Z ) )  ->  x  =  <. X ,  Y >. )
87fveq2d 5852 . . . . 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 6786 . . . . . . 7  |-  ( ( X  e.  B  /\  Y  e.  B )  ->  ( 2nd `  <. X ,  Y >. )  =  Y )
129, 10, 11syl2anc 659 . . . . . 6  |-  ( ph  ->  ( 2nd `  <. X ,  Y >. )  =  Y )
1312adantr 463 . . . . 5  |-  ( (
ph  /\  ( x  =  <. X ,  Y >.  /\  z  =  Z ) )  ->  ( 2nd `  <. X ,  Y >. )  =  Y )
148, 13eqtrd 2495 . . . 4  |-  ( (
ph  /\  ( x  =  <. X ,  Y >.  /\  z  =  Z ) )  ->  ( 2nd `  x )  =  Y )
15 simprr 755 . . . 4  |-  ( (
ph  /\  ( x  =  <. X ,  Y >.  /\  z  =  Z ) )  ->  z  =  Z )
1614, 15oveq12d 6288 . . 3  |-  ( (
ph  /\  ( x  =  <. X ,  Y >.  /\  z  =  Z ) )  ->  (
( 2nd `  x
) H z )  =  ( Y H Z ) )
177fveq2d 5852 . . . 4  |-  ( (
ph  /\  ( x  =  <. X ,  Y >.  /\  z  =  Z ) )  ->  ( H `  x )  =  ( H `  <. X ,  Y >. ) )
18 df-ov 6273 . . . 4  |-  ( X H Y )  =  ( H `  <. X ,  Y >. )
1917, 18syl6eqr 2513 . . 3  |-  ( (
ph  /\  ( x  =  <. X ,  Y >.  /\  z  =  Z ) )  ->  ( H `  x )  =  ( X H Y ) )
207, 15oveq12d 6288 . . . 4  |-  ( (
ph  /\  ( x  =  <. X ,  Y >.  /\  z  =  Z ) )  ->  (
x  .x.  z )  =  ( <. X ,  Y >.  .x.  Z )
)
2120oveqd 6287 . . 3  |-  ( (
ph  /\  ( x  =  <. X ,  Y >.  /\  z  =  Z ) )  ->  (
g ( x  .x.  z ) f )  =  ( g (
<. X ,  Y >.  .x. 
Z ) f ) )
2216, 19, 21mpt2eq123dv 6332 . 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 5020 . . 3  |-  ( ( X  e.  B  /\  Y  e.  B )  -> 
<. X ,  Y >.  e.  ( B  X.  B
) )
249, 10, 23syl2anc 659 . 2  |-  ( ph  -> 
<. X ,  Y >.  e.  ( B  X.  B
) )
25 comffval.z . 2  |-  ( ph  ->  Z  e.  B )
26 ovex 6298 . . . 4  |-  ( Y H Z )  e. 
_V
27 ovex 6298 . . . 4  |-  ( X H Y )  e. 
_V
2826, 27mpt2ex 6850 . . 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 6403 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 367    = wceq 1398    e. wcel 1823   _Vcvv 3106   <.cop 4022    X. cxp 4986   ` cfv 5570  (class class class)co 6270    |-> cmpt2 6272   2ndc2nd 6772   Basecbs 14716   Hom chom 14795  compcco 14796  compfccomf 15156
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1623  ax-4 1636  ax-5 1709  ax-6 1752  ax-7 1795  ax-8 1825  ax-9 1827  ax-10 1842  ax-11 1847  ax-12 1859  ax-13 2004  ax-ext 2432  ax-rep 4550  ax-sep 4560  ax-nul 4568  ax-pow 4615  ax-pr 4676  ax-un 6565
This theorem depends on definitions:  df-bi 185  df-or 368  df-an 369  df-3an 973  df-tru 1401  df-ex 1618  df-nf 1622  df-sb 1745  df-eu 2288  df-mo 2289  df-clab 2440  df-cleq 2446  df-clel 2449  df-nfc 2604  df-ne 2651  df-ral 2809  df-rex 2810  df-reu 2811  df-rab 2813  df-v 3108  df-sbc 3325  df-csb 3421  df-dif 3464  df-un 3466  df-in 3468  df-ss 3475  df-nul 3784  df-if 3930  df-pw 4001  df-sn 4017  df-pr 4019  df-op 4023  df-uni 4236  df-iun 4317  df-br 4440  df-opab 4498  df-mpt 4499  df-id 4784  df-xp 4994  df-rel 4995  df-cnv 4996  df-co 4997  df-dm 4998  df-rn 4999  df-res 5000  df-ima 5001  df-iota 5534  df-fun 5572  df-fn 5573  df-f 5574  df-f1 5575  df-fo 5576  df-f1o 5577  df-fv 5578  df-ov 6273  df-oprab 6274  df-mpt2 6275  df-1st 6773  df-2nd 6774  df-comf 15160
This theorem is referenced by:  comfval  15188  comffval2  15190  comffn  15193
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