cntri - Metamath Proof Explorer

	Metamath Proof Explorer		< Previous Next > Nearby theorems
Mirrors > Home > MPE Home > Th. List > cntri		Structured version Visualization version GIF version

Theorem cntri 19302

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 2741	. . . . 5 ⊢ (Cntz‘𝑀) = (Cntz‘𝑀)
4	2, 3	cntrval 19289	. . . 4 ⊢ ((Cntz‘𝑀)‘𝐵) = (Cntr‘𝑀)
5	1, 4	eqtr4i 2767	. . 3 ⊢ 𝑍 = ((Cntz‘𝑀)‘𝐵)
6	5	eleq2i 2833	. 2 ⊢ (𝑋 ∈ 𝑍 ↔ 𝑋 ∈ ((Cntz‘𝑀)‘𝐵))
7		cntri.p	. . 3 ⊢ + = (+_g‘𝑀)
8	7, 3	cntzi 19299	. 2 ⊢ ((𝑋 ∈ ((Cntz‘𝑀)‘𝐵) ∧ 𝑌 ∈ 𝐵) → (𝑋 + 𝑌) = (𝑌 + 𝑋))
9	6, 8	sylanb 588	1 ⊢ ((𝑋 ∈ 𝑍 ∧ 𝑌 ∈ 𝐵) → (𝑋 + 𝑌) = (𝑌 + 𝑋))

Colors of variables: wff setvar class

Syntax hints: → wi 4 ∧ wa 397 = wceq 1548 ∈ wcel 2121 ‘cfv 6489 (class class class)co 7360 Basecbs 17174 +_gcplusg 17215 Cntzccntz 19285 Cntrccntr 19286

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1803 ax-4 1817 ax-5 1918 ax-6 1975 ax-7 2016 ax-8 2123 ax-9 2131 ax-10 2154 ax-11 2170 ax-12 2191 ax-ext 2713 ax-rep 5202 ax-sep 5221 ax-nul 5231 ax-pow 5297 ax-pr 5365

This theorem depends on definitions: df-bi 209 df-an 398 df-or 855 df-3an 1095 df-tru 1551 df-fal 1561 df-ex 1788 df-nf 1792 df-sb 2075 df-mo 2545 df-eu 2575 df-clab 2720 df-cleq 2733 df-clel 2816 df-nfc 2890 df-ne 2937 df-ral 3056 df-rex 3066 df-reu 3347 df-rab 3394 df-v 3435 df-sbc 3726 df-csb 3834 df-dif 3888 df-un 3890 df-in 3892 df-ss 3902 df-nul 4265 df-if 4458 df-pw 4534 df-sn 4559 df-pr 4561 df-op 4565 df-uni 4842 df-iun 4926 df-br 5076 df-opab 5138 df-mpt 5157 df-id 5516 df-xp 5627 df-rel 5628 df-cnv 5629 df-co 5630 df-dm 5631 df-rn 5632 df-res 5633 df-ima 5634 df-iota 6445 df-fun 6491 df-fn 6492 df-f 6493 df-f1 6494 df-fo 6495 df-f1o 6496 df-fv 6497 df-ov 7363 df-cntz 19287 df-cntr 19288

This theorem is referenced by: cntrcmnd 19812 primefld 20781 sraassab 21847 zrhcntr 34175

W3C validator