elcntr - Intuitionistic Logic Explorer

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Theorem elcntr 14157

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 2238	. . . . 5 ⊢ (Cntz‘𝑀) = (Cntz‘𝑀)
4	2, 3	cntrval 14145	. . . 4 ⊢ ((Cntz‘𝑀)‘𝐵) = (Cntr‘𝑀)
5	1, 4	eqtr4i 2262	. . 3 ⊢ 𝑍 = ((Cntz‘𝑀)‘𝐵)
6	5	eleq2i 2305	. 2 ⊢ (𝐴 ∈ 𝑍 ↔ 𝐴 ∈ ((Cntz‘𝑀)‘𝐵))
7		ssid 3268	. . 3 ⊢ 𝐵 ⊆ 𝐵
8		elcntr.p	. . . 4 ⊢ + = (+_g‘𝑀)
9	2, 8, 3	elcntz 14148	. . 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 184	1 ⊢ (𝐴 ∈ 𝑍 ↔ (𝐴 ∈ 𝐵 ∧ ∀𝑦 ∈ 𝐵 (𝐴 + 𝑦) = (𝑦 + 𝐴)))

Colors of variables: wff set class

This proof depends on syntax axioms: ∧ wa 104 ↔ wb 105 = wceq 1402 ∈ wcel 2209 ∀wral 2528 ⊆ wss 3220 ‘cfv 5377 (class class class)co 6085 Basecbs 13404 +_gcplusg 13484 Cntzccntz 14140 Cntrccntr 14141

This proof depends on axioms: ax-mp 5 ax-1 6 ax-2 7 ax-ia1 106 ax-ia2 107 ax-ia3 108 ax-io 721 ax-5 1500 ax-7 1501 ax-gen 1502 ax-ie1 1546 ax-ie2 1547 ax-8 1557 ax-10 1558 ax-11 1559 ax-i12 1560 ax-bndl 1562 ax-4 1563 ax-17 1579 ax-i9 1583 ax-ial 1587 ax-i5r 1588 ax-14 2212 ax-ext 2220 ax-coll 4246 ax-sep 4249 ax-pow 4311 ax-pr 4346 ax-un 4578 ax-cnex 8271 ax-resscn 8272 ax-1re 8274 ax-addrcl 8277

This proof depends on definitions: df-bi 117 df-3an 1011 df-tru 1405 df-nf 1514 df-sb 1816 df-eu 2089 df-mo 2090 df-clab 2225 df-cleq 2231 df-clel 2234 df-nfc 2381 df-ral 2533 df-rex 2534 df-reu 2535 df-rab 2537 df-v 2823 df-sbc 3052 df-csb 3148 df-un 3224 df-in 3226 df-ss 3233 df-pw 3690 df-sn 3715 df-pr 3716 df-op 3718 df-uni 3936 df-int 3971 df-iun 4014 df-br 4131 df-opab 4193 df-mpt 4194 df-id 4438 df-xp 4780 df-rel 4781 df-cnv 4782 df-co 4783 df-dm 4784 df-rn 4785 df-res 4786 df-ima 4787 df-iota 5337 df-fun 5379 df-fn 5380 df-f 5381 df-f1 5382 df-fo 5383 df-f1o 5384 df-fv 5385 df-ov 6088 df-inn 9308 df-ndx 13407 df-slot 13408 df-base 13410 df-cntz 14142 df-cntr 14143

This theorem is used by: (None)

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