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Theorem elcntr 19244

Description: Elementhood in the center of a magma. (Contributed by SN, 21-Mar-2025.)

**Hypotheses**
Ref	Expression
elcntr.b	⊢ 𝐵 = (Base‘𝑀)
elcntr.p	⊢ + = (+_g‘𝑀)
elcntr.z	⊢ 𝑍 = (Cntr‘𝑀)

**Assertion**
Ref	Expression
elcntr	⊢ (𝐴 ∈ 𝑍 ↔ (𝐴 ∈ 𝐵 ∧ ∀𝑦 ∈ 𝐵 (𝐴 + 𝑦) = (𝑦 + 𝐴)))

Distinct variable groups: 𝑦, + 𝑦,𝐴 𝑦,𝐵 𝑦,𝑀 Allowed substitution hint: 𝑍(𝑦)

**Proof of Theorem elcntr**
Step	Hyp	Ref	Expression
1		elcntr.z	. . . 4 ⊢ 𝑍 = (Cntr‘𝑀)
2		elcntr.b	. . . . 5 ⊢ 𝐵 = (Base‘𝑀)
3		eqid 2729	. . . . 5 ⊢ (Cntz‘𝑀) = (Cntz‘𝑀)
4	2, 3	cntrval 19233	. . . 4 ⊢ ((Cntz‘𝑀)‘𝐵) = (Cntr‘𝑀)
5	1, 4	eqtr4i 2755	. . 3 ⊢ 𝑍 = ((Cntz‘𝑀)‘𝐵)
6	5	eleq2i 2820	. 2 ⊢ (𝐴 ∈ 𝑍 ↔ 𝐴 ∈ ((Cntz‘𝑀)‘𝐵))
7		ssid 3966	. . 3 ⊢ 𝐵 ⊆ 𝐵
8		elcntr.p	. . . 4 ⊢ + = (+_g‘𝑀)
9	2, 8, 3	elcntz 19236	. . 3 ⊢ (𝐵 ⊆ 𝐵 → (𝐴 ∈ ((Cntz‘𝑀)‘𝐵) ↔ (𝐴 ∈ 𝐵 ∧ ∀𝑦 ∈ 𝐵 (𝐴 + 𝑦) = (𝑦 + 𝐴))))
10	7, 9	ax-mp 5	. 2 ⊢ (𝐴 ∈ ((Cntz‘𝑀)‘𝐵) ↔ (𝐴 ∈ 𝐵 ∧ ∀𝑦 ∈ 𝐵 (𝐴 + 𝑦) = (𝑦 + 𝐴)))
11	6, 10	bitri 275	1 ⊢ (𝐴 ∈ 𝑍 ↔ (𝐴 ∈ 𝐵 ∧ ∀𝑦 ∈ 𝐵 (𝐴 + 𝑦) = (𝑦 + 𝐴)))

Colors of variables: wff setvar class

Syntax hints: ↔ wb 206 ∧ wa 395 = wceq 1540 ∈ wcel 2109 ∀wral 3044 ⊆ wss 3911 ‘cfv 6499 (class class class)co 7369 Basecbs 17155 +_gcplusg 17196 Cntzccntz 19229 Cntrccntr 19230

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1795 ax-4 1809 ax-5 1910 ax-6 1967 ax-7 2008 ax-8 2111 ax-9 2119 ax-10 2142 ax-11 2158 ax-12 2178 ax-ext 2701 ax-rep 5229 ax-sep 5246 ax-nul 5256 ax-pow 5315 ax-pr 5382

This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3an 1088 df-tru 1543 df-fal 1553 df-ex 1780 df-nf 1784 df-sb 2066 df-mo 2533 df-eu 2562 df-clab 2708 df-cleq 2721 df-clel 2803 df-nfc 2878 df-ne 2926 df-ral 3045 df-rex 3054 df-reu 3352 df-rab 3403 df-v 3446 df-sbc 3751 df-csb 3860 df-dif 3914 df-un 3916 df-in 3918 df-ss 3928 df-nul 4293 df-if 4485 df-pw 4561 df-sn 4586 df-pr 4588 df-op 4592 df-uni 4868 df-iun 4953 df-br 5103 df-opab 5165 df-mpt 5184 df-id 5526 df-xp 5637 df-rel 5638 df-cnv 5639 df-co 5640 df-dm 5641 df-rn 5642 df-res 5643 df-ima 5644 df-iota 6452 df-fun 6501 df-fn 6502 df-f 6503 df-f1 6504 df-fo 6505 df-f1o 6506 df-fv 6507 df-ov 7372 df-cntz 19231 df-cntr 19232

This theorem is referenced by: sraassab 21810 cntrval2 33143 zrhcntr 33962 elmgpcntrd 48986

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