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Theorem cntzcmnf 18071
 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 (𝜑𝐺 ∈ CMnd)
cntzcmnf.f (𝜑𝐹:𝐴𝐵)
Assertion
Ref Expression
cntzcmnf (𝜑 → ran 𝐹 ⊆ (𝑍‘ran 𝐹))

Proof of Theorem cntzcmnf
StepHypRef Expression
1 cntzcmnf.f . . 3 (𝜑𝐹:𝐴𝐵)
2 frn 5966 . . 3 (𝐹:𝐴𝐵 → ran 𝐹𝐵)
31, 2syl 17 . 2 (𝜑 → ran 𝐹𝐵)
4 cntzcmnf.g . . 3 (𝜑𝐺 ∈ CMnd)
5 cntzcmnf.b . . . 4 𝐵 = (Base‘𝐺)
6 cntzcmnf.z . . . 4 𝑍 = (Cntz‘𝐺)
75, 6cntzcmn 18068 . . 3 ((𝐺 ∈ CMnd ∧ ran 𝐹𝐵) → (𝑍‘ran 𝐹) = 𝐵)
84, 3, 7syl2anc 691 . 2 (𝜑 → (𝑍‘ran 𝐹) = 𝐵)
93, 8sseqtr4d 3605 1 (𝜑 → ran 𝐹 ⊆ (𝑍‘ran 𝐹))
 Colors of variables: wff setvar class Syntax hints:   → wi 4   = wceq 1475   ∈ wcel 1977   ⊆ wss 3540  ran crn 5039  ⟶wf 5800  ‘cfv 5804  Basecbs 15695  Cntzccntz 17571  CMndccmn 18016 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1713  ax-4 1728  ax-5 1827  ax-6 1875  ax-7 1922  ax-8 1979  ax-9 1986  ax-10 2006  ax-11 2021  ax-12 2034  ax-13 2234  ax-ext 2590  ax-rep 4699  ax-sep 4709  ax-nul 4717  ax-pow 4769  ax-pr 4833 This theorem depends on definitions:  df-bi 196  df-or 384  df-an 385  df-3an 1033  df-tru 1478  df-ex 1696  df-nf 1701  df-sb 1868  df-eu 2462  df-mo 2463  df-clab 2597  df-cleq 2603  df-clel 2606  df-nfc 2740  df-ne 2782  df-ral 2901  df-rex 2902  df-reu 2903  df-rab 2905  df-v 3175  df-sbc 3403  df-csb 3500  df-dif 3543  df-un 3545  df-in 3547  df-ss 3554  df-nul 3875  df-if 4037  df-pw 4110  df-sn 4126  df-pr 4128  df-op 4132  df-uni 4373  df-iun 4457  df-br 4584  df-opab 4644  df-mpt 4645  df-id 4953  df-xp 5044  df-rel 5045  df-cnv 5046  df-co 5047  df-dm 5048  df-rn 5049  df-res 5050  df-ima 5051  df-iota 5768  df-fun 5806  df-fn 5807  df-f 5808  df-f1 5809  df-fo 5810  df-f1o 5811  df-fv 5812  df-ov 6552  df-cntz 17573  df-cmn 18018 This theorem is referenced by:  gsumres  18137  gsumcl2  18138  gsumf1o  18140  gsumsubmcl  18142  gsumsplit  18151  gsummhm  18161  gsumfsum  19632  wilthlem3  24596
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