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Theorem cmnbascntr 19838
Description: The base set of a commutative monoid is its center. (Contributed by SN, 21-Mar-2025.)
Hypotheses
Ref Expression
cmnbascntr.b 𝐵 = (Base‘𝐺)
cmnbascntr.z 𝑍 = (Cntr‘𝐺)
Assertion
Ref Expression
cmnbascntr (𝐺 ∈ CMnd → 𝐵 = 𝑍)

Proof of Theorem cmnbascntr
Dummy variables 𝑥 𝑦 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 cmnbascntr.z . . 3 𝑍 = (Cntr‘𝐺)
2 cmnbascntr.b . . . 4 𝐵 = (Base‘𝐺)
3 eqid 2735 . . . 4 (Cntz‘𝐺) = (Cntz‘𝐺)
42, 3cntrval 19350 . . 3 ((Cntz‘𝐺)‘𝐵) = (Cntr‘𝐺)
5 ssid 4018 . . . 4 𝐵𝐵
6 eqid 2735 . . . . 5 (+g𝐺) = (+g𝐺)
72, 6, 3cntzval 19352 . . . 4 (𝐵𝐵 → ((Cntz‘𝐺)‘𝐵) = {𝑥𝐵 ∣ ∀𝑦𝐵 (𝑥(+g𝐺)𝑦) = (𝑦(+g𝐺)𝑥)})
85, 7ax-mp 5 . . 3 ((Cntz‘𝐺)‘𝐵) = {𝑥𝐵 ∣ ∀𝑦𝐵 (𝑥(+g𝐺)𝑦) = (𝑦(+g𝐺)𝑥)}
91, 4, 83eqtr2i 2769 . 2 𝑍 = {𝑥𝐵 ∣ ∀𝑦𝐵 (𝑥(+g𝐺)𝑦) = (𝑦(+g𝐺)𝑥)}
102, 6cmncom 19831 . . . . 5 ((𝐺 ∈ CMnd ∧ 𝑥𝐵𝑦𝐵) → (𝑥(+g𝐺)𝑦) = (𝑦(+g𝐺)𝑥))
11103expa 1117 . . . 4 (((𝐺 ∈ CMnd ∧ 𝑥𝐵) ∧ 𝑦𝐵) → (𝑥(+g𝐺)𝑦) = (𝑦(+g𝐺)𝑥))
1211ralrimiva 3144 . . 3 ((𝐺 ∈ CMnd ∧ 𝑥𝐵) → ∀𝑦𝐵 (𝑥(+g𝐺)𝑦) = (𝑦(+g𝐺)𝑥))
1312rabeqcda 3445 . 2 (𝐺 ∈ CMnd → {𝑥𝐵 ∣ ∀𝑦𝐵 (𝑥(+g𝐺)𝑦) = (𝑦(+g𝐺)𝑥)} = 𝐵)
149, 13eqtr2id 2788 1 (𝐺 ∈ CMnd → 𝐵 = 𝑍)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 395   = wceq 1537  wcel 2106  wral 3059  {crab 3433  wss 3963  cfv 6563  (class class class)co 7431  Basecbs 17245  +gcplusg 17298  Cntzccntz 19346  Cntrccntr 19347  CMndccmn 19813
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1792  ax-4 1806  ax-5 1908  ax-6 1965  ax-7 2005  ax-8 2108  ax-9 2116  ax-10 2139  ax-11 2155  ax-12 2175  ax-ext 2706  ax-rep 5285  ax-sep 5302  ax-nul 5312  ax-pow 5371  ax-pr 5438
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1540  df-fal 1550  df-ex 1777  df-nf 1781  df-sb 2063  df-mo 2538  df-eu 2567  df-clab 2713  df-cleq 2727  df-clel 2814  df-nfc 2890  df-ne 2939  df-ral 3060  df-rex 3069  df-reu 3379  df-rab 3434  df-v 3480  df-sbc 3792  df-csb 3909  df-dif 3966  df-un 3968  df-in 3970  df-ss 3980  df-nul 4340  df-if 4532  df-pw 4607  df-sn 4632  df-pr 4634  df-op 4638  df-uni 4913  df-iun 4998  df-br 5149  df-opab 5211  df-mpt 5232  df-id 5583  df-xp 5695  df-rel 5696  df-cnv 5697  df-co 5698  df-dm 5699  df-rn 5700  df-res 5701  df-ima 5702  df-iota 6516  df-fun 6565  df-fn 6566  df-f 6567  df-f1 6568  df-fo 6569  df-f1o 6570  df-fv 6571  df-ov 7434  df-cntz 19348  df-cntr 19349  df-cmn 19815
This theorem is referenced by:  crngbascntr  20274
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