elcntr - Metamath Proof Explorer

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Theorem elcntr 19505

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 2760	. . . . 5 ⊢ (Cntz‘𝑀) = (Cntz‘𝑀)
4	2, 3	cntrval 19494	. . . 4 ⊢ ((Cntz‘𝑀)‘𝐵) = (Cntr‘𝑀)
5	1, 4	eqtr4i 2786	. . 3 ⊢ 𝑍 = ((Cntz‘𝑀)‘𝐵)
6	5	eleq2i 2852	. 2 ⊢ (𝐴 ∈ 𝑍 ↔ 𝐴 ∈ ((Cntz‘𝑀)‘𝐵))
7		ssid 3952	. . 3 ⊢ 𝐵 ⊆ 𝐵
8		elcntr.p	. . . 4 ⊢ + = (+_g‘𝑀)
9	2, 8, 3	elcntz 19497	. . 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 278	1 ⊢ (𝐴 ∈ 𝑍 ↔ (𝐴 ∈ 𝐵 ∧ ∀𝑦 ∈ 𝐵 (𝐴 + 𝑦) = (𝑦 + 𝐴)))

Colors of variables: wff setvar class

This proof depends on syntax axioms: ↔ wb 209 ∧ wa 401 = wceq 1570 ∈ wcel 2145 ∀wral 3076 ⊆ wss 3898 ‘cfv 6527 (class class class)co 7408 Basecbs 17348 +_gcplusg 17389 Cntzccntz 19490 Cntrccntr 19491

This proof depends on axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1828 ax-4 1842 ax-5 1943 ax-6 2000 ax-7 2041 ax-8 2147 ax-9 2155 ax-10 2178 ax-11 2194 ax-12 2213 ax-ext 2732 ax-rep 5231 ax-sep 5248 ax-nul 5259 ax-pow 5326 ax-pr 5390

This proof depends on definitions: df-bi 210 df-an 402 df-or 862 df-3an 1105 df-tru 1573 df-fal 1583 df-ex 1813 df-nf 1817 df-sb 2100 df-mo 2564 df-eu 2594 df-clab 2739 df-cleq 2752 df-clel 2835 df-nfc 2909 df-ne 2956 df-ral 3077 df-rex 3087 df-reu 3366 df-rab 3413 df-v 3452 df-sbc 3739 df-csb 3847 df-dif 3901 df-un 3903 df-in 3905 df-ss 3915 df-nul 4279 df-if 4482 df-pw 4558 df-sn 4584 df-pr 4586 df-op 4590 df-uni 4867 df-iun 4952 df-br 5103 df-opab 5167 df-mpt 5186 df-id 5542 df-xp 5653 df-rel 5654 df-cnv 5655 df-co 5656 df-dm 5657 df-rn 5658 df-res 5659 df-ima 5660 df-iota 6483 df-fun 6529 df-fn 6530 df-f 6531 df-f1 6532 df-fo 6533 df-f1o 6534 df-fv 6535 df-ov 7411 df-cntz 19492 df-cntr 19493

This theorem is used by: sraassab 22137 cntrval2 33665 zrhcntr 34544 elmgpcntrd 50035

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