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Theorem cntzcmnf 17173
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 5719 . . 3  |-  ( F : A --> B  ->  ran  F  C_  B )
31, 2syl 17 . 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 17170 . . 3  |-  ( ( G  e. CMnd  /\  ran  F 
C_  B )  -> 
( Z `  ran  F )  =  B )
84, 3, 7syl2anc 659 . 2  |-  ( ph  ->  ( Z `  ran  F )  =  B )
93, 8sseqtr4d 3478 1  |-  ( ph  ->  ran  F  C_  ( Z `  ran  F ) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    = wceq 1405    e. wcel 1842    C_ wss 3413   ran crn 4823   -->wf 5564   ` cfv 5568   Basecbs 14839  Cntzccntz 16675  CMndccmn 17120
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1639  ax-4 1652  ax-5 1725  ax-6 1771  ax-7 1814  ax-8 1844  ax-9 1846  ax-10 1861  ax-11 1866  ax-12 1878  ax-13 2026  ax-ext 2380  ax-rep 4506  ax-sep 4516  ax-nul 4524  ax-pow 4571  ax-pr 4629
This theorem depends on definitions:  df-bi 185  df-or 368  df-an 369  df-3an 976  df-tru 1408  df-ex 1634  df-nf 1638  df-sb 1764  df-eu 2242  df-mo 2243  df-clab 2388  df-cleq 2394  df-clel 2397  df-nfc 2552  df-ne 2600  df-ral 2758  df-rex 2759  df-reu 2760  df-rab 2762  df-v 3060  df-sbc 3277  df-csb 3373  df-dif 3416  df-un 3418  df-in 3420  df-ss 3427  df-nul 3738  df-if 3885  df-pw 3956  df-sn 3972  df-pr 3974  df-op 3978  df-uni 4191  df-iun 4272  df-br 4395  df-opab 4453  df-mpt 4454  df-id 4737  df-xp 4828  df-rel 4829  df-cnv 4830  df-co 4831  df-dm 4832  df-rn 4833  df-res 4834  df-ima 4835  df-iota 5532  df-fun 5570  df-fn 5571  df-f 5572  df-f1 5573  df-fo 5574  df-f1o 5575  df-fv 5576  df-ov 6280  df-cntz 16677  df-cmn 17122
This theorem is referenced by:  gsumres  17243  gsumcl2  17244  gsumf1o  17246  gsumresOLD  17247  gsumclOLD  17248  gsumf1oOLD  17249  gsumsubmcl  17252  gsumsubmclOLD  17253  gsumsplit  17268  gsumsplitOLD  17269  gsummhm  17280  gsummhmOLD  17281  gsumfsum  18802  wilthlem3  23723
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