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Theorem cntzcmnf 19765
Description: Discharge the centralizer assumption in a commutative monoid. (Contributed by Mario Carneiro, 24-Apr-2016.)
Hypotheses
Ref Expression
cntzcmnf.b 𝐡 = (Baseβ€˜πΊ)
cntzcmnf.z 𝑍 = (Cntzβ€˜πΊ)
cntzcmnf.g (πœ‘ β†’ 𝐺 ∈ CMnd)
cntzcmnf.f (πœ‘ β†’ 𝐹:𝐴⟢𝐡)
Assertion
Ref Expression
cntzcmnf (πœ‘ β†’ ran 𝐹 βŠ† (π‘β€˜ran 𝐹))

Proof of Theorem cntzcmnf
StepHypRef Expression
1 cntzcmnf.f . . 3 (πœ‘ β†’ 𝐹:𝐴⟢𝐡)
21frnd 6719 . 2 (πœ‘ β†’ ran 𝐹 βŠ† 𝐡)
3 cntzcmnf.g . . 3 (πœ‘ β†’ 𝐺 ∈ CMnd)
4 cntzcmnf.b . . . 4 𝐡 = (Baseβ€˜πΊ)
5 cntzcmnf.z . . . 4 𝑍 = (Cntzβ€˜πΊ)
64, 5cntzcmn 19760 . . 3 ((𝐺 ∈ CMnd ∧ ran 𝐹 βŠ† 𝐡) β†’ (π‘β€˜ran 𝐹) = 𝐡)
73, 2, 6syl2anc 583 . 2 (πœ‘ β†’ (π‘β€˜ran 𝐹) = 𝐡)
82, 7sseqtrrd 4018 1 (πœ‘ β†’ ran 𝐹 βŠ† (π‘β€˜ran 𝐹))
Colors of variables: wff setvar class
Syntax hints:   β†’ wi 4   = wceq 1533   ∈ wcel 2098   βŠ† wss 3943  ran crn 5670  βŸΆwf 6533  β€˜cfv 6537  Basecbs 17153  Cntzccntz 19231  CMndccmn 19700
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1905  ax-6 1963  ax-7 2003  ax-8 2100  ax-9 2108  ax-10 2129  ax-11 2146  ax-12 2163  ax-ext 2697  ax-rep 5278  ax-sep 5292  ax-nul 5299  ax-pow 5356  ax-pr 5420
This theorem depends on definitions:  df-bi 206  df-an 396  df-or 845  df-3an 1086  df-tru 1536  df-fal 1546  df-ex 1774  df-nf 1778  df-sb 2060  df-mo 2528  df-eu 2557  df-clab 2704  df-cleq 2718  df-clel 2804  df-nfc 2879  df-ne 2935  df-ral 3056  df-rex 3065  df-reu 3371  df-rab 3427  df-v 3470  df-sbc 3773  df-csb 3889  df-dif 3946  df-un 3948  df-in 3950  df-ss 3960  df-nul 4318  df-if 4524  df-pw 4599  df-sn 4624  df-pr 4626  df-op 4630  df-uni 4903  df-iun 4992  df-br 5142  df-opab 5204  df-mpt 5225  df-id 5567  df-xp 5675  df-rel 5676  df-cnv 5677  df-co 5678  df-dm 5679  df-rn 5680  df-res 5681  df-ima 5682  df-iota 6489  df-fun 6539  df-fn 6540  df-f 6541  df-f1 6542  df-fo 6543  df-f1o 6544  df-fv 6545  df-ov 7408  df-cntz 19233  df-cmn 19702
This theorem is referenced by:  gsumres  19833  gsumcl2  19834  gsumf1o  19836  gsumsubmcl  19839  gsumsplit  19848  gsummhm  19858  gsumfsum  21328  wilthlem3  26957
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