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Theorem cntzcmnf 15478
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 5564 . . 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 15422 . . 3  |-  ( ( G  e. CMnd  /\  ran  F 
C_  B )  -> 
( Z `  ran  F )  =  B )
84, 3, 7syl2anc 643 . 2  |-  ( ph  ->  ( Z `  ran  F )  =  B )
93, 8sseqtr4d 3353 1  |-  ( ph  ->  ran  F  C_  ( Z `  ran  F ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    = wceq 1649    e. wcel 1721    C_ wss 3288   ran crn 4846   -->wf 5417   ` cfv 5421   Basecbs 13432  Cntzccntz 15077  CMndccmn 15375
This theorem is referenced by:  gsumres  15483  gsumcl  15484  gsumf1o  15485  gsumsubmcl  15487  gsumsplit  15493  gsummhm  15497  gsumfsum  16729  wilthlem3  20814
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1552  ax-5 1563  ax-17 1623  ax-9 1662  ax-8 1683  ax-13 1723  ax-14 1725  ax-6 1740  ax-7 1745  ax-11 1757  ax-12 1946  ax-ext 2393  ax-rep 4288  ax-sep 4298  ax-nul 4306  ax-pow 4345  ax-pr 4371
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3an 938  df-tru 1325  df-ex 1548  df-nf 1551  df-sb 1656  df-eu 2266  df-mo 2267  df-clab 2399  df-cleq 2405  df-clel 2408  df-nfc 2537  df-ne 2577  df-ral 2679  df-rex 2680  df-reu 2681  df-rab 2683  df-v 2926  df-sbc 3130  df-csb 3220  df-dif 3291  df-un 3293  df-in 3295  df-ss 3302  df-nul 3597  df-if 3708  df-pw 3769  df-sn 3788  df-pr 3789  df-op 3791  df-uni 3984  df-iun 4063  df-br 4181  df-opab 4235  df-mpt 4236  df-id 4466  df-xp 4851  df-rel 4852  df-cnv 4853  df-co 4854  df-dm 4855  df-rn 4856  df-res 4857  df-ima 4858  df-iota 5385  df-fun 5423  df-fn 5424  df-f 5425  df-f1 5426  df-fo 5427  df-f1o 5428  df-fv 5429  df-ov 6051  df-cntz 15079  df-cmn 15377
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