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

Theorem homdmcoa 13915
Description: If  F : X --> Y and  G : Y
--> Z, then  G and  F are composable. (Contributed by Mario Carneiro, 11-Jan-2017.)
Hypotheses
Ref Expression
homdmcoa.o  |-  .x.  =  (compa `  C )
homdmcoa.h  |-  H  =  (Homa
`  C )
homdmcoa.f  |-  ( ph  ->  F  e.  ( X H Y ) )
homdmcoa.g  |-  ( ph  ->  G  e.  ( Y H Z ) )
Assertion
Ref Expression
homdmcoa  |-  ( ph  ->  G dom  .x.  F
)

Proof of Theorem homdmcoa
StepHypRef Expression
1 eqid 2296 . . . 4  |-  (Nat `  C )  =  (Nat
`  C )
2 homdmcoa.h . . . 4  |-  H  =  (Homa
`  C )
31, 2homarw 13894 . . 3  |-  ( X H Y )  C_  (Nat `  C )
4 homdmcoa.f . . 3  |-  ( ph  ->  F  e.  ( X H Y ) )
53, 4sseldi 3191 . 2  |-  ( ph  ->  F  e.  (Nat `  C ) )
61, 2homarw 13894 . . 3  |-  ( Y H Z )  C_  (Nat `  C )
7 homdmcoa.g . . 3  |-  ( ph  ->  G  e.  ( Y H Z ) )
86, 7sseldi 3191 . 2  |-  ( ph  ->  G  e.  (Nat `  C ) )
92homacd 13889 . . . 4  |-  ( F  e.  ( X H Y )  ->  (coda `  F
)  =  Y )
104, 9syl 15 . . 3  |-  ( ph  ->  (coda
`  F )  =  Y )
112homadm 13888 . . . 4  |-  ( G  e.  ( Y H Z )  ->  (domA `  G )  =  Y )
127, 11syl 15 . . 3  |-  ( ph  ->  (domA `  G )  =  Y )
1310, 12eqtr4d 2331 . 2  |-  ( ph  ->  (coda
`  F )  =  (domA `  G ) )
14 homdmcoa.o . . 3  |-  .x.  =  (compa `  C )
1514, 1eldmcoa 13913 . 2  |-  ( G dom  .x.  F  <->  ( F  e.  (Nat `  C )  /\  G  e.  (Nat `  C )  /\  (coda `  F
)  =  (domA `  G ) ) )
165, 8, 13, 15syl3anbrc 1136 1  |-  ( ph  ->  G dom  .x.  F
)
Colors of variables: wff set class
Syntax hints:    -> wi 4    = wceq 1632    e. wcel 1696   class class class wbr 4039   dom cdm 4705   ` cfv 5271  (class class class)co 5874  domAcdoma 13868  codaccoda 13869  Natcarw 13870  Homachoma 13871  compaccoa 13902
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-ot 3663  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-doma 13872  df-coda 13873  df-homa 13874  df-arw 13875  df-coa 13904
  Copyright terms: Public domain W3C validator