MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  comffval Structured version   Unicode version

Theorem comffval 13925
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 13924 . . 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 733 . . . . . 6  |-  ( (
ph  /\  ( x  =  <. X ,  Y >.  /\  z  =  Z ) )  ->  x  =  <. X ,  Y >. )
87fveq2d 5732 . . . . 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 6360 . . . . . . 7  |-  ( ( X  e.  B  /\  Y  e.  B )  ->  ( 2nd `  <. X ,  Y >. )  =  Y )
129, 10, 11syl2anc 643 . . . . . 6  |-  ( ph  ->  ( 2nd `  <. X ,  Y >. )  =  Y )
1312adantr 452 . . . . 5  |-  ( (
ph  /\  ( x  =  <. X ,  Y >.  /\  z  =  Z ) )  ->  ( 2nd `  <. X ,  Y >. )  =  Y )
148, 13eqtrd 2468 . . . 4  |-  ( (
ph  /\  ( x  =  <. X ,  Y >.  /\  z  =  Z ) )  ->  ( 2nd `  x )  =  Y )
15 simprr 734 . . . 4  |-  ( (
ph  /\  ( x  =  <. X ,  Y >.  /\  z  =  Z ) )  ->  z  =  Z )
1614, 15oveq12d 6099 . . 3  |-  ( (
ph  /\  ( x  =  <. X ,  Y >.  /\  z  =  Z ) )  ->  (
( 2nd `  x
) H z )  =  ( Y H Z ) )
177fveq2d 5732 . . . 4  |-  ( (
ph  /\  ( x  =  <. X ,  Y >.  /\  z  =  Z ) )  ->  ( H `  x )  =  ( H `  <. X ,  Y >. ) )
18 df-ov 6084 . . . 4  |-  ( X H Y )  =  ( H `  <. X ,  Y >. )
1917, 18syl6eqr 2486 . . 3  |-  ( (
ph  /\  ( x  =  <. X ,  Y >.  /\  z  =  Z ) )  ->  ( H `  x )  =  ( X H Y ) )
207, 15oveq12d 6099 . . . 4  |-  ( (
ph  /\  ( x  =  <. X ,  Y >.  /\  z  =  Z ) )  ->  (
x  .x.  z )  =  ( <. X ,  Y >.  .x.  Z )
)
2120oveqd 6098 . . 3  |-  ( (
ph  /\  ( x  =  <. X ,  Y >.  /\  z  =  Z ) )  ->  (
g ( x  .x.  z ) f )  =  ( g (
<. X ,  Y >.  .x. 
Z ) f ) )
2216, 19, 21mpt2eq123dv 6136 . 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 4910 . . 3  |-  ( ( X  e.  B  /\  Y  e.  B )  -> 
<. X ,  Y >.  e.  ( B  X.  B
) )
249, 10, 23syl2anc 643 . 2  |-  ( ph  -> 
<. X ,  Y >.  e.  ( B  X.  B
) )
25 comffval.z . 2  |-  ( ph  ->  Z  e.  B )
26 ovex 6106 . . . 4  |-  ( Y H Z )  e. 
_V
27 ovex 6106 . . . 4  |-  ( X H Y )  e. 
_V
2826, 27mpt2ex 6425 . . 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 6201 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 359    = wceq 1652    e. wcel 1725   _Vcvv 2956   <.cop 3817    X. cxp 4876   ` cfv 5454  (class class class)co 6081    e. cmpt2 6083   2ndc2nd 6348   Basecbs 13469    Hom chom 13540  compcco 13541  compfccomf 13892
This theorem is referenced by:  comfval  13926  comffval2  13928  comffn  13931
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1555  ax-5 1566  ax-17 1626  ax-9 1666  ax-8 1687  ax-13 1727  ax-14 1729  ax-6 1744  ax-7 1749  ax-11 1761  ax-12 1950  ax-ext 2417  ax-rep 4320  ax-sep 4330  ax-nul 4338  ax-pow 4377  ax-pr 4403  ax-un 4701
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3an 938  df-tru 1328  df-ex 1551  df-nf 1554  df-sb 1659  df-eu 2285  df-mo 2286  df-clab 2423  df-cleq 2429  df-clel 2432  df-nfc 2561  df-ne 2601  df-ral 2710  df-rex 2711  df-reu 2712  df-rab 2714  df-v 2958  df-sbc 3162  df-csb 3252  df-dif 3323  df-un 3325  df-in 3327  df-ss 3334  df-nul 3629  df-if 3740  df-pw 3801  df-sn 3820  df-pr 3821  df-op 3823  df-uni 4016  df-iun 4095  df-br 4213  df-opab 4267  df-mpt 4268  df-id 4498  df-xp 4884  df-rel 4885  df-cnv 4886  df-co 4887  df-dm 4888  df-rn 4889  df-res 4890  df-ima 4891  df-iota 5418  df-fun 5456  df-fn 5457  df-f 5458  df-f1 5459  df-fo 5460  df-f1o 5461  df-fv 5462  df-ov 6084  df-oprab 6085  df-mpt2 6086  df-1st 6349  df-2nd 6350  df-comf 13896
  Copyright terms: Public domain W3C validator