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Theorem cntzcmnf 15402
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 5501 . . 3  |-  ( F : A --> B  ->  ran  F  C_  B )
31, 2syl 15 . 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 15346 . . 3  |-  ( ( G  e. CMnd  /\  ran  F 
C_  B )  -> 
( Z `  ran  F )  =  B )
84, 3, 7syl2anc 642 . 2  |-  ( ph  ->  ( Z `  ran  F )  =  B )
93, 8sseqtr4d 3301 1  |-  ( ph  ->  ran  F  C_  ( Z `  ran  F ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    = wceq 1647    e. wcel 1715    C_ wss 3238   ran crn 4793   -->wf 5354   ` cfv 5358   Basecbs 13356  Cntzccntz 15001  CMndccmn 15299
This theorem is referenced by:  gsumres  15407  gsumcl  15408  gsumf1o  15409  gsumsubmcl  15411  gsumsplit  15417  gsummhm  15421  gsumfsum  16656  wilthlem3  20531
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1551  ax-5 1562  ax-17 1621  ax-9 1659  ax-8 1680  ax-13 1717  ax-14 1719  ax-6 1734  ax-7 1739  ax-11 1751  ax-12 1937  ax-ext 2347  ax-rep 4233  ax-sep 4243  ax-nul 4251  ax-pow 4290  ax-pr 4316
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 937  df-tru 1324  df-ex 1547  df-nf 1550  df-sb 1654  df-eu 2221  df-mo 2222  df-clab 2353  df-cleq 2359  df-clel 2362  df-nfc 2491  df-ne 2531  df-ral 2633  df-rex 2634  df-reu 2635  df-rab 2637  df-v 2875  df-sbc 3078  df-csb 3168  df-dif 3241  df-un 3243  df-in 3245  df-ss 3252  df-nul 3544  df-if 3655  df-pw 3716  df-sn 3735  df-pr 3736  df-op 3738  df-uni 3930  df-iun 4009  df-br 4126  df-opab 4180  df-mpt 4181  df-id 4412  df-xp 4798  df-rel 4799  df-cnv 4800  df-co 4801  df-dm 4802  df-rn 4803  df-res 4804  df-ima 4805  df-iota 5322  df-fun 5360  df-fn 5361  df-f 5362  df-f1 5363  df-fo 5364  df-f1o 5365  df-fv 5366  df-ov 5984  df-cntz 15003  df-cmn 15301
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