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Theorem cntzcmnf 15546
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 5626 . . 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 15490 . . 3  |-  ( ( G  e. CMnd  /\  ran  F 
C_  B )  -> 
( Z `  ran  F )  =  B )
84, 3, 7syl2anc 644 . 2  |-  ( ph  ->  ( Z `  ran  F )  =  B )
93, 8sseqtr4d 3371 1  |-  ( ph  ->  ran  F  C_  ( Z `  ran  F ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    = wceq 1653    e. wcel 1727    C_ wss 3306   ran crn 4908   -->wf 5479   ` cfv 5483   Basecbs 13500  Cntzccntz 15145  CMndccmn 15443
This theorem is referenced by:  gsumres  15551  gsumcl  15552  gsumf1o  15553  gsumsubmcl  15555  gsumsplit  15561  gsummhm  15565  gsumfsum  16797  wilthlem3  20884
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1556  ax-5 1567  ax-17 1627  ax-9 1668  ax-8 1689  ax-13 1729  ax-14 1731  ax-6 1746  ax-7 1751  ax-11 1763  ax-12 1953  ax-ext 2423  ax-rep 4345  ax-sep 4355  ax-nul 4363  ax-pow 4406  ax-pr 4432
This theorem depends on definitions:  df-bi 179  df-or 361  df-an 362  df-3an 939  df-tru 1329  df-ex 1552  df-nf 1555  df-sb 1660  df-eu 2291  df-mo 2292  df-clab 2429  df-cleq 2435  df-clel 2438  df-nfc 2567  df-ne 2607  df-ral 2716  df-rex 2717  df-reu 2718  df-rab 2720  df-v 2964  df-sbc 3168  df-csb 3268  df-dif 3309  df-un 3311  df-in 3313  df-ss 3320  df-nul 3614  df-if 3764  df-pw 3825  df-sn 3844  df-pr 3845  df-op 3847  df-uni 4040  df-iun 4119  df-br 4238  df-opab 4292  df-mpt 4293  df-id 4527  df-xp 4913  df-rel 4914  df-cnv 4915  df-co 4916  df-dm 4917  df-rn 4918  df-res 4919  df-ima 4920  df-iota 5447  df-fun 5485  df-fn 5486  df-f 5487  df-f1 5488  df-fo 5489  df-f1o 5490  df-fv 5491  df-ov 6113  df-cntz 15147  df-cmn 15445
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