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Theorem cntzcmnss 18466
Description: Any subset in a commutative monoid is a subset of its centralizer. (Contributed by AV, 12-Jan-2019.)
Hypotheses
Ref Expression
cntzcmnss.b 𝐵 = (Base‘𝐺)
cntzcmnss.z 𝑍 = (Cntz‘𝐺)
Assertion
Ref Expression
cntzcmnss ((𝐺 ∈ CMnd ∧ 𝑆𝐵) → 𝑆 ⊆ (𝑍𝑆))

Proof of Theorem cntzcmnss
StepHypRef Expression
1 cntzcmnss.b . . 3 𝐵 = (Base‘𝐺)
2 cntzcmnss.z . . 3 𝑍 = (Cntz‘𝐺)
31, 2cntzcmn 18465 . 2 ((𝐺 ∈ CMnd ∧ 𝑆𝐵) → (𝑍𝑆) = 𝐵)
4 sseq2 3768 . . . . 5 (𝐵 = (𝑍𝑆) → (𝑆𝐵𝑆 ⊆ (𝑍𝑆)))
54eqcoms 2768 . . . 4 ((𝑍𝑆) = 𝐵 → (𝑆𝐵𝑆 ⊆ (𝑍𝑆)))
65biimpd 219 . . 3 ((𝑍𝑆) = 𝐵 → (𝑆𝐵𝑆 ⊆ (𝑍𝑆)))
76adantld 484 . 2 ((𝑍𝑆) = 𝐵 → ((𝐺 ∈ CMnd ∧ 𝑆𝐵) → 𝑆 ⊆ (𝑍𝑆)))
83, 7mpcom 38 1 ((𝐺 ∈ CMnd ∧ 𝑆𝐵) → 𝑆 ⊆ (𝑍𝑆))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 196  wa 383   = wceq 1632  wcel 2139  wss 3715  cfv 6049  Basecbs 16079  Cntzccntz 17968  CMndccmn 18413
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1871  ax-4 1886  ax-5 1988  ax-6 2054  ax-7 2090  ax-8 2141  ax-9 2148  ax-10 2168  ax-11 2183  ax-12 2196  ax-13 2391  ax-ext 2740  ax-rep 4923  ax-sep 4933  ax-nul 4941  ax-pow 4992  ax-pr 5055
This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-3an 1074  df-tru 1635  df-ex 1854  df-nf 1859  df-sb 2047  df-eu 2611  df-mo 2612  df-clab 2747  df-cleq 2753  df-clel 2756  df-nfc 2891  df-ne 2933  df-ral 3055  df-rex 3056  df-reu 3057  df-rab 3059  df-v 3342  df-sbc 3577  df-csb 3675  df-dif 3718  df-un 3720  df-in 3722  df-ss 3729  df-nul 4059  df-if 4231  df-pw 4304  df-sn 4322  df-pr 4324  df-op 4328  df-uni 4589  df-iun 4674  df-br 4805  df-opab 4865  df-mpt 4882  df-id 5174  df-xp 5272  df-rel 5273  df-cnv 5274  df-co 5275  df-dm 5276  df-rn 5277  df-res 5278  df-ima 5279  df-iota 6012  df-fun 6051  df-fn 6052  df-f 6053  df-f1 6054  df-fo 6055  df-f1o 6056  df-fv 6057  df-ov 6817  df-cntz 17970  df-cmn 18415
This theorem is referenced by:  smadiadetlem3lem2  20695
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