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Theorem cntri 19493

Description: Defining property of the center of a magma. (Contributed by Mario Carneiro, 22-Sep-2015.)

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

**Assertion**
Ref	Expression
cntri	⊢ ((𝑋 ∈ 𝑍 ∧ 𝑌 ∈ 𝐵) → (𝑋 + 𝑌) = (𝑌 + 𝑋))

**Proof of Theorem cntri**
Step	Hyp	Ref	Expression
1		cntri.z	. . . 4 ⊢ 𝑍 = (Cntr‘𝑀)
2		cntri.b	. . . . 5 ⊢ 𝐵 = (Base‘𝑀)
3		eqid 2760	. . . . 5 ⊢ (Cntz‘𝑀) = (Cntz‘𝑀)
4	2, 3	cntrval 19480	. . . 4 ⊢ ((Cntz‘𝑀)‘𝐵) = (Cntr‘𝑀)
5	1, 4	eqtr4i 2786	. . 3 ⊢ 𝑍 = ((Cntz‘𝑀)‘𝐵)
6	5	eleq2i 2852	. 2 ⊢ (𝑋 ∈ 𝑍 ↔ 𝑋 ∈ ((Cntz‘𝑀)‘𝐵))
7		cntri.p	. . 3 ⊢ + = (+_g‘𝑀)
8	7, 3	cntzi 19490	. 2 ⊢ ((𝑋 ∈ ((Cntz‘𝑀)‘𝐵) ∧ 𝑌 ∈ 𝐵) → (𝑋 + 𝑌) = (𝑌 + 𝑋))
9	6, 8	sylanb 593	1 ⊢ ((𝑋 ∈ 𝑍 ∧ 𝑌 ∈ 𝐵) → (𝑋 + 𝑌) = (𝑌 + 𝑋))

Colors of variables: wff setvar class

This proof depends on syntax axioms: → wi 4 ∧ wa 401 = wceq 1570 ∈ wcel 2145 ‘cfv 6528 (class class class)co 7409 Basecbs 17334 +_gcplusg 17375 Cntzccntz 19476 Cntrccntr 19477

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 5232 ax-sep 5249 ax-nul 5260 ax-pow 5327 ax-pr 5391

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 3740 df-csb 3848 df-dif 3902 df-un 3904 df-in 3906 df-ss 3916 df-nul 4280 df-if 4483 df-pw 4559 df-sn 4585 df-pr 4587 df-op 4591 df-uni 4868 df-iun 4953 df-br 5104 df-opab 5168 df-mpt 5187 df-id 5543 df-xp 5654 df-rel 5655 df-cnv 5656 df-co 5657 df-dm 5658 df-rn 5659 df-res 5660 df-ima 5661 df-iota 6484 df-fun 6530 df-fn 6531 df-f 6532 df-f1 6533 df-fo 6534 df-f1o 6535 df-fv 6536 df-ov 7412 df-cntz 19478 df-cntr 19479

This theorem is used by: cntrcmnd 20003 primefld 21009 sraassab 22123 zrhcntr 34530

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