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Theorem cntri 19211

Description: Defining property of the center of a group. (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 2729	. . . . 5 ⊢ (Cntz‘𝑀) = (Cntz‘𝑀)
4	2, 3	cntrval 19198	. . . 4 ⊢ ((Cntz‘𝑀)‘𝐵) = (Cntr‘𝑀)
5	1, 4	eqtr4i 2755	. . 3 ⊢ 𝑍 = ((Cntz‘𝑀)‘𝐵)
6	5	eleq2i 2820	. 2 ⊢ (𝑋 ∈ 𝑍 ↔ 𝑋 ∈ ((Cntz‘𝑀)‘𝐵))
7		cntri.p	. . 3 ⊢ + = (+_g‘𝑀)
8	7, 3	cntzi 19208	. 2 ⊢ ((𝑋 ∈ ((Cntz‘𝑀)‘𝐵) ∧ 𝑌 ∈ 𝐵) → (𝑋 + 𝑌) = (𝑌 + 𝑋))
9	6, 8	sylanb 581	1 ⊢ ((𝑋 ∈ 𝑍 ∧ 𝑌 ∈ 𝐵) → (𝑋 + 𝑌) = (𝑌 + 𝑋))

Colors of variables: wff setvar class

Syntax hints: → wi 4 ∧ wa 395 = wceq 1540 ∈ wcel 2109 ‘cfv 6482 (class class class)co 7349 Basecbs 17120 +_gcplusg 17161 Cntzccntz 19194 Cntrccntr 19195

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1795 ax-4 1809 ax-5 1910 ax-6 1967 ax-7 2008 ax-8 2111 ax-9 2119 ax-10 2142 ax-11 2158 ax-12 2178 ax-ext 2701 ax-rep 5218 ax-sep 5235 ax-nul 5245 ax-pow 5304 ax-pr 5371

This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3an 1088 df-tru 1543 df-fal 1553 df-ex 1780 df-nf 1784 df-sb 2066 df-mo 2533 df-eu 2562 df-clab 2708 df-cleq 2721 df-clel 2803 df-nfc 2878 df-ne 2926 df-ral 3045 df-rex 3054 df-reu 3344 df-rab 3395 df-v 3438 df-sbc 3743 df-csb 3852 df-dif 3906 df-un 3908 df-in 3910 df-ss 3920 df-nul 4285 df-if 4477 df-pw 4553 df-sn 4578 df-pr 4580 df-op 4584 df-uni 4859 df-iun 4943 df-br 5093 df-opab 5155 df-mpt 5174 df-id 5514 df-xp 5625 df-rel 5626 df-cnv 5627 df-co 5628 df-dm 5629 df-rn 5630 df-res 5631 df-ima 5632 df-iota 6438 df-fun 6484 df-fn 6485 df-f 6486 df-f1 6487 df-fo 6488 df-f1o 6489 df-fv 6490 df-ov 7352 df-cntz 19196 df-cntr 19197

This theorem is referenced by: cntrcmnd 19721 primefld 20690 sraassab 21775 zrhcntr 33952

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