rngoaddneg1 - Mathbox for Jeff Madsen

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Theorem rngoaddneg1 36086

Description: Adding the negative in a ring gives zero. (Contributed by Jeff Madsen, 10-Jun-2010.)

**Hypotheses**
Ref	Expression
ringnegcl.1	⊢ 𝐺 = (1^st ‘𝑅)
ringnegcl.2	⊢ 𝑋 = ran 𝐺
ringnegcl.3	⊢ 𝑁 = (inv‘𝐺)
ringaddneg.4	⊢ 𝑍 = (GId‘𝐺)

**Assertion**
Ref	Expression
rngoaddneg1	⊢ ((𝑅 ∈ RingOps ∧ 𝐴 ∈ 𝑋) → (𝐴𝐺(𝑁‘𝐴)) = 𝑍)

**Proof of Theorem rngoaddneg1**
Step	Hyp	Ref	Expression
1		ringnegcl.1	. . 3 ⊢ 𝐺 = (1^st ‘𝑅)
2	1	rngogrpo 36068	. 2 ⊢ (𝑅 ∈ RingOps → 𝐺 ∈ GrpOp)
3		ringnegcl.2	. . 3 ⊢ 𝑋 = ran 𝐺
4		ringaddneg.4	. . 3 ⊢ 𝑍 = (GId‘𝐺)
5		ringnegcl.3	. . 3 ⊢ 𝑁 = (inv‘𝐺)
6	3, 4, 5	grporinv 28889	. 2 ⊢ ((𝐺 ∈ GrpOp ∧ 𝐴 ∈ 𝑋) → (𝐴𝐺(𝑁‘𝐴)) = 𝑍)
7	2, 6	sylan 580	1 ⊢ ((𝑅 ∈ RingOps ∧ 𝐴 ∈ 𝑋) → (𝐴𝐺(𝑁‘𝐴)) = 𝑍)

Colors of variables: wff setvar class

Syntax hints: → wi 4 ∧ wa 396 = wceq 1539 ∈ wcel 2106 ran crn 5590 ‘cfv 6433 (class class class)co 7275 1^st c1st 7829 GrpOpcgr 28851 GIdcgi 28852 invcgn 28853 RingOpscrngo 36052

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1798 ax-4 1812 ax-5 1913 ax-6 1971 ax-7 2011 ax-8 2108 ax-9 2116 ax-10 2137 ax-11 2154 ax-12 2171 ax-ext 2709 ax-rep 5209 ax-sep 5223 ax-nul 5230 ax-pr 5352 ax-un 7588

This theorem depends on definitions: df-bi 206 df-an 397 df-or 845 df-3an 1088 df-tru 1542 df-fal 1552 df-ex 1783 df-nf 1787 df-sb 2068 df-mo 2540 df-eu 2569 df-clab 2716 df-cleq 2730 df-clel 2816 df-nfc 2889 df-ne 2944 df-ral 3069 df-rex 3070 df-reu 3072 df-rab 3073 df-v 3434 df-sbc 3717 df-csb 3833 df-dif 3890 df-un 3892 df-in 3894 df-ss 3904 df-nul 4257 df-if 4460 df-sn 4562 df-pr 4564 df-op 4568 df-uni 4840 df-iun 4926 df-br 5075 df-opab 5137 df-mpt 5158 df-id 5489 df-xp 5595 df-rel 5596 df-cnv 5597 df-co 5598 df-dm 5599 df-rn 5600 df-res 5601 df-ima 5602 df-iota 6391 df-fun 6435 df-fn 6436 df-f 6437 df-f1 6438 df-fo 6439 df-f1o 6440 df-fv 6441 df-riota 7232 df-ov 7278 df-1st 7831 df-2nd 7832 df-grpo 28855 df-gid 28856 df-ginv 28857 df-ablo 28907 df-rngo 36053

This theorem is referenced by: rngonegmn1l 36099

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