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Theorem cntzcmnss 18952
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 18951 . 2 ((𝐺 ∈ CMnd ∧ 𝑆𝐵) → (𝑍𝑆) = 𝐵)
4 sseq2 3968 . . . . 5 (𝐵 = (𝑍𝑆) → (𝑆𝐵𝑆 ⊆ (𝑍𝑆)))
54eqcoms 2830 . . . 4 ((𝑍𝑆) = 𝐵 → (𝑆𝐵𝑆 ⊆ (𝑍𝑆)))
65biimpd 232 . . 3 ((𝑍𝑆) = 𝐵 → (𝑆𝐵𝑆 ⊆ (𝑍𝑆)))
76adantld 494 . 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 209  wa 399   = wceq 1538  wcel 2114  wss 3908  cfv 6334  Basecbs 16474  Cntzccntz 18436  CMndccmn 18897
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1911  ax-6 1970  ax-7 2015  ax-8 2116  ax-9 2124  ax-10 2145  ax-11 2161  ax-12 2178  ax-ext 2794  ax-rep 5166  ax-sep 5179  ax-nul 5186  ax-pow 5243  ax-pr 5307
This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-3an 1086  df-tru 1541  df-ex 1782  df-nf 1786  df-sb 2070  df-mo 2622  df-eu 2653  df-clab 2801  df-cleq 2815  df-clel 2894  df-nfc 2962  df-ne 3012  df-ral 3135  df-rex 3136  df-reu 3137  df-rab 3139  df-v 3471  df-sbc 3748  df-csb 3856  df-dif 3911  df-un 3913  df-in 3915  df-ss 3925  df-nul 4266  df-if 4440  df-pw 4513  df-sn 4540  df-pr 4542  df-op 4546  df-uni 4814  df-iun 4896  df-br 5043  df-opab 5105  df-mpt 5123  df-id 5437  df-xp 5538  df-rel 5539  df-cnv 5540  df-co 5541  df-dm 5542  df-rn 5543  df-res 5544  df-ima 5545  df-iota 6293  df-fun 6336  df-fn 6337  df-f 6338  df-f1 6339  df-fo 6340  df-f1o 6341  df-fv 6342  df-ov 7143  df-cntz 18438  df-cmn 18899
This theorem is referenced by:  smadiadetlem3lem2  21270
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