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

Theorem cntzcmnf 16637
Description: Discharge the centralizer assumption in a commutative monoid. (Contributed by Mario Carneiro, 24-Apr-2016.)
Hypotheses
Ref Expression
cntzcmnf.b  |-  B  =  ( Base `  G
)
cntzcmnf.z  |-  Z  =  (Cntz `  G )
cntzcmnf.g  |-  ( ph  ->  G  e. CMnd )
cntzcmnf.f  |-  ( ph  ->  F : A --> B )
Assertion
Ref Expression
cntzcmnf  |-  ( ph  ->  ran  F  C_  ( Z `  ran  F ) )

Proof of Theorem cntzcmnf
StepHypRef Expression
1 cntzcmnf.f . . 3  |-  ( ph  ->  F : A --> B )
2 frn 5728 . . 3  |-  ( F : A --> B  ->  ran  F  C_  B )
31, 2syl 16 . 2  |-  ( ph  ->  ran  F  C_  B
)
4 cntzcmnf.g . . 3  |-  ( ph  ->  G  e. CMnd )
5 cntzcmnf.b . . . 4  |-  B  =  ( Base `  G
)
6 cntzcmnf.z . . . 4  |-  Z  =  (Cntz `  G )
75, 6cntzcmn 16634 . . 3  |-  ( ( G  e. CMnd  /\  ran  F 
C_  B )  -> 
( Z `  ran  F )  =  B )
84, 3, 7syl2anc 661 . 2  |-  ( ph  ->  ( Z `  ran  F )  =  B )
93, 8sseqtr4d 3534 1  |-  ( ph  ->  ran  F  C_  ( Z `  ran  F ) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    = wceq 1374    e. wcel 1762    C_ wss 3469   ran crn 4993   -->wf 5575   ` cfv 5579   Basecbs 14479  Cntzccntz 16141  CMndccmn 16587
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1596  ax-4 1607  ax-5 1675  ax-6 1714  ax-7 1734  ax-8 1764  ax-9 1766  ax-10 1781  ax-11 1786  ax-12 1798  ax-13 1961  ax-ext 2438  ax-rep 4551  ax-sep 4561  ax-nul 4569  ax-pow 4618  ax-pr 4679
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 970  df-tru 1377  df-ex 1592  df-nf 1595  df-sb 1707  df-eu 2272  df-mo 2273  df-clab 2446  df-cleq 2452  df-clel 2455  df-nfc 2610  df-ne 2657  df-ral 2812  df-rex 2813  df-reu 2814  df-rab 2816  df-v 3108  df-sbc 3325  df-csb 3429  df-dif 3472  df-un 3474  df-in 3476  df-ss 3483  df-nul 3779  df-if 3933  df-pw 4005  df-sn 4021  df-pr 4023  df-op 4027  df-uni 4239  df-iun 4320  df-br 4441  df-opab 4499  df-mpt 4500  df-id 4788  df-xp 4998  df-rel 4999  df-cnv 5000  df-co 5001  df-dm 5002  df-rn 5003  df-res 5004  df-ima 5005  df-iota 5542  df-fun 5581  df-fn 5582  df-f 5583  df-f1 5584  df-fo 5585  df-f1o 5586  df-fv 5587  df-ov 6278  df-cntz 16143  df-cmn 16589
This theorem is referenced by:  gsumres  16705  gsumcl2  16706  gsumf1o  16708  gsumresOLD  16709  gsumclOLD  16710  gsumf1oOLD  16711  gsumsubmcl  16714  gsumsubmclOLD  16715  gsumsplit  16730  gsumsplitOLD  16731  gsummhm  16743  gsummhmOLD  16744  gsumfsum  18245  wilthlem3  23065
  Copyright terms: Public domain W3C validator