MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  cntzcmnf Structured version   Visualization version   GIF version

Theorem cntzcmnf 18957
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)
21frnd 6514 . 2 (𝜑 → ran 𝐹𝐵)
3 cntzcmnf.g . . 3 (𝜑𝐺 ∈ CMnd)
4 cntzcmnf.b . . . 4 𝐵 = (Base‘𝐺)
5 cntzcmnf.z . . . 4 𝑍 = (Cntz‘𝐺)
64, 5cntzcmn 18952 . . 3 ((𝐺 ∈ CMnd ∧ ran 𝐹𝐵) → (𝑍‘ran 𝐹) = 𝐵)
73, 2, 6syl2anc 586 . 2 (𝜑 → (𝑍‘ran 𝐹) = 𝐵)
82, 7sseqtrrd 4006 1 (𝜑 → ran 𝐹 ⊆ (𝑍‘ran 𝐹))
Colors of variables: wff setvar class
Syntax hints:  wi 4   = wceq 1531  wcel 2108  wss 3934  ran crn 5549  wf 6344  cfv 6348  Basecbs 16475  Cntzccntz 18437  CMndccmn 18898
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1790  ax-4 1804  ax-5 1905  ax-6 1964  ax-7 2009  ax-8 2110  ax-9 2118  ax-10 2139  ax-11 2154  ax-12 2170  ax-ext 2791  ax-rep 5181  ax-sep 5194  ax-nul 5201  ax-pow 5257  ax-pr 5320
This theorem depends on definitions:  df-bi 209  df-an 399  df-or 844  df-3an 1084  df-tru 1534  df-ex 1775  df-nf 1779  df-sb 2064  df-mo 2616  df-eu 2648  df-clab 2798  df-cleq 2812  df-clel 2891  df-nfc 2961  df-ne 3015  df-ral 3141  df-rex 3142  df-reu 3143  df-rab 3145  df-v 3495  df-sbc 3771  df-csb 3882  df-dif 3937  df-un 3939  df-in 3941  df-ss 3950  df-nul 4290  df-if 4466  df-pw 4539  df-sn 4560  df-pr 4562  df-op 4566  df-uni 4831  df-iun 4912  df-br 5058  df-opab 5120  df-mpt 5138  df-id 5453  df-xp 5554  df-rel 5555  df-cnv 5556  df-co 5557  df-dm 5558  df-rn 5559  df-res 5560  df-ima 5561  df-iota 6307  df-fun 6350  df-fn 6351  df-f 6352  df-f1 6353  df-fo 6354  df-f1o 6355  df-fv 6356  df-ov 7151  df-cntz 18439  df-cmn 18900
This theorem is referenced by:  gsumres  19025  gsumcl2  19026  gsumf1o  19028  gsumsubmcl  19031  gsumsplit  19040  gsummhm  19050  gsumfsum  20604  wilthlem3  25639
  Copyright terms: Public domain W3C validator