rngoaddneg1 - Mathbox for Jeff Madsen

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Theorem rngoaddneg1 37907

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 37889	. 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 30489	. 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 395 = wceq 1540 ∈ wcel 2109 ran crn 5624 ‘cfv 6486 (class class class)co 7353 1^st c1st 7929 GrpOpcgr 30451 GIdcgi 30452 invcgn 30453 RingOpscrngo 37873

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 5221 ax-sep 5238 ax-nul 5248 ax-pr 5374 ax-un 7675

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 3346 df-rab 3397 df-v 3440 df-sbc 3745 df-csb 3854 df-dif 3908 df-un 3910 df-in 3912 df-ss 3922 df-nul 4287 df-if 4479 df-sn 4580 df-pr 4582 df-op 4586 df-uni 4862 df-iun 4946 df-br 5096 df-opab 5158 df-mpt 5177 df-id 5518 df-xp 5629 df-rel 5630 df-cnv 5631 df-co 5632 df-dm 5633 df-rn 5634 df-res 5635 df-ima 5636 df-iota 6442 df-fun 6488 df-fn 6489 df-f 6490 df-f1 6491 df-fo 6492 df-f1o 6493 df-fv 6494 df-riota 7310 df-ov 7356 df-1st 7931 df-2nd 7932 df-grpo 30455 df-gid 30456 df-ginv 30457 df-ablo 30507 df-rngo 37874

This theorem is referenced by: rngonegmn1l 37920

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