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Theorem comffval 13618
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 13617 . . 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 5545 . . . . 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 6149 . . . . . . 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 2328 . . . 4  |-  ( (
ph  /\  ( x  =  <. X ,  Y >.  /\  z  =  Z ) )  ->  ( 2nd `  x )  =  Y )
15 simprr 733 . . . 4  |-  ( (
ph  /\  ( x  =  <. X ,  Y >.  /\  z  =  Z ) )  ->  z  =  Z )
1614, 15oveq12d 5892 . . 3  |-  ( (
ph  /\  ( x  =  <. X ,  Y >.  /\  z  =  Z ) )  ->  (
( 2nd `  x
) H z )  =  ( Y H Z ) )
177fveq2d 5545 . . . 4  |-  ( (
ph  /\  ( x  =  <. X ,  Y >.  /\  z  =  Z ) )  ->  ( H `  x )  =  ( H `  <. X ,  Y >. ) )
18 df-ov 5877 . . . 4  |-  ( X H Y )  =  ( H `  <. X ,  Y >. )
1917, 18syl6eqr 2346 . . 3  |-  ( (
ph  /\  ( x  =  <. X ,  Y >.  /\  z  =  Z ) )  ->  ( H `  x )  =  ( X H Y ) )
207, 15oveq12d 5892 . . . 4  |-  ( (
ph  /\  ( x  =  <. X ,  Y >.  /\  z  =  Z ) )  ->  (
x  .x.  z )  =  ( <. X ,  Y >.  .x.  Z )
)
2120oveqd 5891 . . 3  |-  ( (
ph  /\  ( x  =  <. X ,  Y >.  /\  z  =  Z ) )  ->  (
g ( x  .x.  z ) f )  =  ( g (
<. X ,  Y >.  .x. 
Z ) f ) )
2216, 19, 21mpt2eq123dv 5926 . 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 4737 . . 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 5899 . . . 4  |-  ( Y H Z )  e. 
_V
27 ovex 5899 . . . 4  |-  ( X H Y )  e. 
_V
2826, 27mpt2ex 6214 . . 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 5991 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 1632    e. wcel 1696   _Vcvv 2801   <.cop 3656    X. cxp 4703   ` cfv 5271  (class class class)co 5874    e. cmpt2 5876   2ndc2nd 6137   Basecbs 13164    Hom chom 13235  compcco 13236  compfccomf 13585
This theorem is referenced by:  comfval  13619  comffval2  13621  comffn  13624
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1536  ax-5 1547  ax-17 1606  ax-9 1644  ax-8 1661  ax-13 1698  ax-14 1700  ax-6 1715  ax-7 1720  ax-11 1727  ax-12 1878  ax-ext 2277  ax-rep 4147  ax-sep 4157  ax-nul 4165  ax-pow 4204  ax-pr 4230  ax-un 4528
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 936  df-tru 1310  df-ex 1532  df-nf 1535  df-sb 1639  df-eu 2160  df-mo 2161  df-clab 2283  df-cleq 2289  df-clel 2292  df-nfc 2421  df-ne 2461  df-ral 2561  df-rex 2562  df-reu 2563  df-rab 2565  df-v 2803  df-sbc 3005  df-csb 3095  df-dif 3168  df-un 3170  df-in 3172  df-ss 3179  df-nul 3469  df-if 3579  df-pw 3640  df-sn 3659  df-pr 3660  df-op 3662  df-uni 3844  df-iun 3923  df-br 4040  df-opab 4094  df-mpt 4095  df-id 4325  df-xp 4711  df-rel 4712  df-cnv 4713  df-co 4714  df-dm 4715  df-rn 4716  df-res 4717  df-ima 4718  df-iota 5235  df-fun 5273  df-fn 5274  df-f 5275  df-f1 5276  df-fo 5277  df-f1o 5278  df-fv 5279  df-ov 5877  df-oprab 5878  df-mpt2 5879  df-1st 6138  df-2nd 6139  df-comf 13589
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