Theorem rngorz 37909

Description: The zero of a unital ring is a right-absorbing element. (Contributed by FL, 31-Aug-2009.) (New usage is discouraged.)

Hypotheses

Ref	Expression
ringlz.1	⊢ 𝑍 = (GId‘𝐺)
ringlz.2	⊢ 𝑋 = ran 𝐺
ringlz.3	⊢ 𝐺 = (1st ‘𝑅)
ringlz.4	⊢ 𝐻 = (2nd ‘𝑅)

Assertion

Ref	Expression
rngorz	⊢ ((𝑅 ∈ RingOps ∧ 𝐴 ∈ 𝑋) → (𝐴𝐻𝑍) = 𝑍)

Proof of Theorem rngorz

Step	Hyp	Ref	Expression
1		ringlz.3	. . . . . . 7 ⊢ 𝐺 = (1st ‘𝑅)
2	1	rngogrpo 37896	. . . . . 6 ⊢ (𝑅 ∈ RingOps → 𝐺 ∈ GrpOp)
3		ringlz.2	. . . . . . 7 ⊢ 𝑋 = ran 𝐺
4		ringlz.1	. . . . . . 7 ⊢ 𝑍 = (GId‘𝐺)
5	3, 4	grpoidcl 30542	. . . . . 6 ⊢ (𝐺 ∈ GrpOp → 𝑍 ∈ 𝑋)
6	3, 4	grpolid 30544	. . . . . 6 ⊢ ((𝐺 ∈ GrpOp ∧ 𝑍 ∈ 𝑋) → (𝑍𝐺𝑍) = 𝑍)
7	2, 5, 6	syl2anc2 585	. . . . 5 ⊢ (𝑅 ∈ RingOps → (𝑍𝐺𝑍) = 𝑍)
8	7	adantr 480	. . . 4 ⊢ ((𝑅 ∈ RingOps ∧ 𝐴 ∈ 𝑋) → (𝑍𝐺𝑍) = 𝑍)
9	8	oveq2d 7446	. . 3 ⊢ ((𝑅 ∈ RingOps ∧ 𝐴 ∈ 𝑋) → (𝐴𝐻(𝑍𝐺𝑍)) = (𝐴𝐻𝑍))
10		simpr 484	. . . . 5 ⊢ ((𝑅 ∈ RingOps ∧ 𝐴 ∈ 𝑋) → 𝐴 ∈ 𝑋)
11	1, 3, 4	rngo0cl 37905	. . . . . 6 ⊢ (𝑅 ∈ RingOps → 𝑍 ∈ 𝑋)
12	11	adantr 480	. . . . 5 ⊢ ((𝑅 ∈ RingOps ∧ 𝐴 ∈ 𝑋) → 𝑍 ∈ 𝑋)
13	10, 12, 12	3jca 1127	. . . 4 ⊢ ((𝑅 ∈ RingOps ∧ 𝐴 ∈ 𝑋) → (𝐴 ∈ 𝑋 ∧ 𝑍 ∈ 𝑋 ∧ 𝑍 ∈ 𝑋))
14		ringlz.4	. . . . 5 ⊢ 𝐻 = (2nd ‘𝑅)
15	1, 14, 3	rngodi 37890	. . . 4 ⊢ ((𝑅 ∈ RingOps ∧ (𝐴 ∈ 𝑋 ∧ 𝑍 ∈ 𝑋 ∧ 𝑍 ∈ 𝑋)) → (𝐴𝐻(𝑍𝐺𝑍)) = ((𝐴𝐻𝑍)𝐺(𝐴𝐻𝑍)))
16	13, 15	syldan 591	. . 3 ⊢ ((𝑅 ∈ RingOps ∧ 𝐴 ∈ 𝑋) → (𝐴𝐻(𝑍𝐺𝑍)) = ((𝐴𝐻𝑍)𝐺(𝐴𝐻𝑍)))
17	2	adantr 480	. . . 4 ⊢ ((𝑅 ∈ RingOps ∧ 𝐴 ∈ 𝑋) → 𝐺 ∈ GrpOp)
18	1, 14, 3	rngocl 37887	. . . . 5 ⊢ ((𝑅 ∈ RingOps ∧ 𝐴 ∈ 𝑋 ∧ 𝑍 ∈ 𝑋) → (𝐴𝐻𝑍) ∈ 𝑋)
19	12, 18	mpd3an3 1461	. . . 4 ⊢ ((𝑅 ∈ RingOps ∧ 𝐴 ∈ 𝑋) → (𝐴𝐻𝑍) ∈ 𝑋)
20	3, 4	grpolid 30544	. . . . 5 ⊢ ((𝐺 ∈ GrpOp ∧ (𝐴𝐻𝑍) ∈ 𝑋) → (𝑍𝐺(𝐴𝐻𝑍)) = (𝐴𝐻𝑍))
21	20	eqcomd 2740	. . . 4 ⊢ ((𝐺 ∈ GrpOp ∧ (𝐴𝐻𝑍) ∈ 𝑋) → (𝐴𝐻𝑍) = (𝑍𝐺(𝐴𝐻𝑍)))
22	17, 19, 21	syl2anc 584	. . 3 ⊢ ((𝑅 ∈ RingOps ∧ 𝐴 ∈ 𝑋) → (𝐴𝐻𝑍) = (𝑍𝐺(𝐴𝐻𝑍)))
23	9, 16, 22	3eqtr3d 2782	. 2 ⊢ ((𝑅 ∈ RingOps ∧ 𝐴 ∈ 𝑋) → ((𝐴𝐻𝑍)𝐺(𝐴𝐻𝑍)) = (𝑍𝐺(𝐴𝐻𝑍)))
24	3	grporcan 30546	. . 3 ⊢ ((𝐺 ∈ GrpOp ∧ ((𝐴𝐻𝑍) ∈ 𝑋 ∧ 𝑍 ∈ 𝑋 ∧ (𝐴𝐻𝑍) ∈ 𝑋)) → (((𝐴𝐻𝑍)𝐺(𝐴𝐻𝑍)) = (𝑍𝐺(𝐴𝐻𝑍)) ↔ (𝐴𝐻𝑍) = 𝑍))
25	17, 19, 12, 19, 24	syl13anc 1371	. 2 ⊢ ((𝑅 ∈ RingOps ∧ 𝐴 ∈ 𝑋) → (((𝐴𝐻𝑍)𝐺(𝐴𝐻𝑍)) = (𝑍𝐺(𝐴𝐻𝑍)) ↔ (𝐴𝐻𝑍) = 𝑍))
26	23, 25	mpbid 232	1 ⊢ ((𝑅 ∈ RingOps ∧ 𝐴 ∈ 𝑋) → (𝐴𝐻𝑍) = 𝑍)

Syntax hints:   → wi 4   ↔ wb 206   ∧ wa 395   ∧ w3a 1086   = wceq 1536   ∈ wcel 2105  ran crn 5689  ‘cfv 6562  (class class class)co 7430  1^st c1st 8010  2^nd c2nd 8011  GrpOpcgr 30517  GIdcgi 30518  RingOpscrngo 37880

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1791  ax-4 1805  ax-5 1907  ax-6 1964  ax-7 2004  ax-8 2107  ax-9 2115  ax-10 2138  ax-11 2154  ax-12 2174  ax-ext 2705  ax-sep 5301  ax-nul 5311  ax-pr 5437  ax-un 7753

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1539  df-fal 1549  df-ex 1776  df-nf 1780  df-sb 2062  df-mo 2537  df-eu 2566  df-clab 2712  df-cleq 2726  df-clel 2813  df-nfc 2889  df-ne 2938  df-ral 3059  df-rex 3068  df-reu 3378  df-rab 3433  df-v 3479  df-sbc 3791  df-csb 3908  df-dif 3965  df-un 3967  df-in 3969  df-ss 3979  df-nul 4339  df-if 4531  df-sn 4631  df-pr 4633  df-op 4637  df-uni 4912  df-iun 4997  df-br 5148  df-opab 5210  df-mpt 5231  df-id 5582  df-xp 5694  df-rel 5695  df-cnv 5696  df-co 5697  df-dm 5698  df-rn 5699  df-iota 6515  df-fun 6564  df-fn 6565  df-f 6566  df-fo 6568  df-fv 6570  df-riota 7387  df-ov 7433  df-1st 8012  df-2nd 8013  df-grpo 30521  df-gid 30522  df-ablo 30573  df-rngo 37881

This theorem is referenced by:  rngoueqz  37926  rngonegmn1r  37928  zerdivemp1x  37933  0idl  38011  keridl  38018