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Theorem elcntr 19302

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 2737	. . . . 5 ⊢ (Cntz‘𝑀) = (Cntz‘𝑀)
4	2, 3	cntrval 19291	. . . 4 ⊢ ((Cntz‘𝑀)‘𝐵) = (Cntr‘𝑀)
5	1, 4	eqtr4i 2763	. . 3 ⊢ 𝑍 = ((Cntz‘𝑀)‘𝐵)
6	5	eleq2i 2829	. 2 ⊢ (𝐴 ∈ 𝑍 ↔ 𝐴 ∈ ((Cntz‘𝑀)‘𝐵))
7		ssid 3945	. . 3 ⊢ 𝐵 ⊆ 𝐵
8		elcntr.p	. . . 4 ⊢ + = (+_g‘𝑀)
9	2, 8, 3	elcntz 19294	. . 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 1542 ∈ wcel 2114 ∀wral 3052 ⊆ wss 3890 ‘cfv 6496 (class class class)co 7364 Basecbs 17176 +_gcplusg 17217 Cntzccntz 19287 Cntrccntr 19288

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 1912 ax-6 1969 ax-7 2010 ax-8 2116 ax-9 2124 ax-10 2147 ax-11 2163 ax-12 2185 ax-ext 2709 ax-rep 5213 ax-sep 5232 ax-nul 5242 ax-pow 5306 ax-pr 5374

This theorem depends on definitions: df-bi 207 df-an 396 df-or 849 df-3an 1089 df-tru 1545 df-fal 1555 df-ex 1782 df-nf 1786 df-sb 2069 df-mo 2540 df-eu 2570 df-clab 2716 df-cleq 2729 df-clel 2812 df-nfc 2886 df-ne 2934 df-ral 3053 df-rex 3063 df-reu 3344 df-rab 3391 df-v 3432 df-sbc 3730 df-csb 3839 df-dif 3893 df-un 3895 df-in 3897 df-ss 3907 df-nul 4275 df-if 4468 df-pw 4544 df-sn 4569 df-pr 4571 df-op 4575 df-uni 4852 df-iun 4936 df-br 5087 df-opab 5149 df-mpt 5168 df-id 5523 df-xp 5634 df-rel 5635 df-cnv 5636 df-co 5637 df-dm 5638 df-rn 5639 df-res 5640 df-ima 5641 df-iota 6452 df-fun 6498 df-fn 6499 df-f 6500 df-f1 6501 df-fo 6502 df-f1o 6503 df-fv 6504 df-ov 7367 df-cntz 19289 df-cntr 19290

This theorem is referenced by: sraassab 21864 cntrval2 33253 zrhcntr 34145 elmgpcntrd 49500

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