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

Theorem comfeqd 13610
Description: Condition for two categories with the same hom-sets to have the same composition. (Contributed by Mario Carneiro, 4-Jan-2017.)
Hypotheses
Ref Expression
comfeqd.1  |-  ( ph  ->  (comp `  C )  =  (comp `  D )
)
comfeqd.2  |-  ( ph  ->  (  Homf 
`  C )  =  (  Homf 
`  D ) )
Assertion
Ref Expression
comfeqd  |-  ( ph  ->  (compf `  C )  =  (compf `  D ) )

Proof of Theorem comfeqd
Dummy variables  f 
g  x  y  z are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 comfeqd.1 . . . . . . . . 9  |-  ( ph  ->  (comp `  C )  =  (comp `  D )
)
21oveqd 5875 . . . . . . . 8  |-  ( ph  ->  ( <. x ,  y
>. (comp `  C )
z )  =  (
<. x ,  y >.
(comp `  D )
z ) )
32oveqd 5875 . . . . . . 7  |-  ( ph  ->  ( g ( <.
x ,  y >.
(comp `  C )
z ) f )  =  ( g (
<. x ,  y >.
(comp `  D )
z ) f ) )
43ralrimivw 2627 . . . . . 6  |-  ( ph  ->  A. g  e.  ( y (  Hom  `  C
) z ) ( g ( <. x ,  y >. (comp `  C ) z ) f )  =  ( g ( <. x ,  y >. (comp `  D ) z ) f ) )
54ralrimivw 2627 . . . . 5  |-  ( ph  ->  A. f  e.  ( x (  Hom  `  C
) y ) A. g  e.  ( y
(  Hom  `  C ) z ) ( g ( <. x ,  y
>. (comp `  C )
z ) f )  =  ( g (
<. x ,  y >.
(comp `  D )
z ) f ) )
65ralrimivw 2627 . . . 4  |-  ( ph  ->  A. z  e.  (
Base `  C ) A. f  e.  (
x (  Hom  `  C
) y ) A. g  e.  ( y
(  Hom  `  C ) z ) ( g ( <. x ,  y
>. (comp `  C )
z ) f )  =  ( g (
<. x ,  y >.
(comp `  D )
z ) f ) )
76ralrimivw 2627 . . 3  |-  ( ph  ->  A. y  e.  (
Base `  C ) A. z  e.  ( Base `  C ) A. f  e.  ( x
(  Hom  `  C ) y ) A. g  e.  ( y (  Hom  `  C ) z ) ( g ( <.
x ,  y >.
(comp `  C )
z ) f )  =  ( g (
<. x ,  y >.
(comp `  D )
z ) f ) )
87ralrimivw 2627 . 2  |-  ( ph  ->  A. x  e.  (
Base `  C ) A. y  e.  ( Base `  C ) A. z  e.  ( Base `  C ) A. f  e.  ( x (  Hom  `  C ) y ) A. g  e.  ( y (  Hom  `  C
) z ) ( g ( <. x ,  y >. (comp `  C ) z ) f )  =  ( g ( <. x ,  y >. (comp `  D ) z ) f ) )
9 eqid 2283 . . 3  |-  (comp `  C )  =  (comp `  C )
10 eqid 2283 . . 3  |-  (comp `  D )  =  (comp `  D )
11 eqid 2283 . . 3  |-  (  Hom  `  C )  =  (  Hom  `  C )
12 eqidd 2284 . . 3  |-  ( ph  ->  ( Base `  C
)  =  ( Base `  C ) )
13 comfeqd.2 . . . 4  |-  ( ph  ->  (  Homf 
`  C )  =  (  Homf 
`  D ) )
1413homfeqbas 13599 . . 3  |-  ( ph  ->  ( Base `  C
)  =  ( Base `  D ) )
159, 10, 11, 12, 14, 13comfeq 13609 . 2  |-  ( ph  ->  ( (compf `  C )  =  (compf `  D )  <->  A. x  e.  ( Base `  C
) A. y  e.  ( Base `  C
) A. z  e.  ( Base `  C
) A. f  e.  ( x (  Hom  `  C ) y ) A. g  e.  ( y (  Hom  `  C
) z ) ( g ( <. x ,  y >. (comp `  C ) z ) f )  =  ( g ( <. x ,  y >. (comp `  D ) z ) f ) ) )
168, 15mpbird 223 1  |-  ( ph  ->  (compf `  C )  =  (compf `  D ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    = wceq 1623   A.wral 2543   <.cop 3643   ` cfv 5255  (class class class)co 5858   Basecbs 13148    Hom chom 13219  compcco 13220    Homf chomf 13568  compfccomf 13569
This theorem is referenced by:  fullresc  13725
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-homf 13572  df-comf 13573
  Copyright terms: Public domain W3C validator