elcntr - Metamath Proof Explorer

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Theorem elcntr 19571

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 2763	. . . . 5 ⊢ (Cntz‘𝑀) = (Cntz‘𝑀)
4	2, 3	cntrval 19560	. . . 4 ⊢ ((Cntz‘𝑀)‘𝐵) = (Cntr‘𝑀)
5	1, 4	eqtr4i 2789	. . 3 ⊢ 𝑍 = ((Cntz‘𝑀)‘𝐵)
6	5	eleq2i 2856	. 2 ⊢ (𝐴 ∈ 𝑍 ↔ 𝐴 ∈ ((Cntz‘𝑀)‘𝐵))
7		ssid 3935	. . 3 ⊢ 𝐵 ⊆ 𝐵
8		elcntr.p	. . . 4 ⊢ + = (+_g‘𝑀)
9	2, 8, 3	elcntz 19563	. . 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 2147 ∀wral 3080 ⊆ wss 3900 ‘cfv 6555 (class class class)co 7440 Basecbs 17413 +_gcplusg 17454 Cntzccntz 19556 Cntrccntr 19557

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 1945 ax-6 2002 ax-7 2043 ax-8 2149 ax-9 2157 ax-10 2180 ax-11 2197 ax-12 2216 ax-ext 2735 ax-rep 5235 ax-sep 5252 ax-nul 5263 ax-pow 5331 ax-pr 5395

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 2102 df-mo 2567 df-eu 2597 df-clab 2742 df-cleq 2755 df-clel 2839 df-nfc 2913 df-ne 2960 df-ral 3081 df-rex 3091 df-reu 3370 df-rab 3417 df-v 3456 df-sbc 3743 df-csb 3852 df-ss 3901 df-dif 4050 df-in 4094 df-un 4154 df-nul 4282 df-if 4486 df-pw 4562 df-sn 4589 df-pr 4591 df-op 4839 df-uni 4871 df-iun 4956 df-br 5107 df-opab 5171 df-mpt 5190 df-id 5550 df-xp 5661 df-rel 5662 df-cnv 5663 df-co 5664 df-dm 5665 df-rn 5666 df-res 5667 df-ima 5668 df-iota 6509 df-fun 6557 df-fn 6558 df-f 6559 df-f1 6560 df-fo 6561 df-f1o 6562 df-fv 6563 df-ov 7443 df-cntz 19558 df-cntr 19559

This theorem is used by: sraassab 22203 cntrval2 33799 zrhcntr 34678 elmgpcntrd 50322

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