Theorem rngorz 37910

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 37897	. . . . . 6 ⊢ (𝑅 ∈ RingOps → 𝐺 ∈ GrpOp)
3		ringlz.2	. . . . . . 7 ⊢ 𝑋 = ran 𝐺
4		ringlz.1	. . . . . . 7 ⊢ 𝑍 = (GId‘𝐺)
5	3, 4	grpoidcl 30493	. . . . . 6 ⊢ (𝐺 ∈ GrpOp → 𝑍 ∈ 𝑋)
6	3, 4	grpolid 30495	. . . . . 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 7385	. . 3 ⊢ ((𝑅 ∈ RingOps ∧ 𝐴 ∈ 𝑋) → (𝐴𝐻(𝑍𝐺𝑍)) = (𝐴𝐻𝑍))
10		simpr 484	. . . . 5 ⊢ ((𝑅 ∈ RingOps ∧ 𝐴 ∈ 𝑋) → 𝐴 ∈ 𝑋)
11	1, 3, 4	rngo0cl 37906	. . . . . 6 ⊢ (𝑅 ∈ RingOps → 𝑍 ∈ 𝑋)
12	11	adantr 480	. . . . 5 ⊢ ((𝑅 ∈ RingOps ∧ 𝐴 ∈ 𝑋) → 𝑍 ∈ 𝑋)
13	10, 12, 12	3jca 1128	. . . 4 ⊢ ((𝑅 ∈ RingOps ∧ 𝐴 ∈ 𝑋) → (𝐴 ∈ 𝑋 ∧ 𝑍 ∈ 𝑋 ∧ 𝑍 ∈ 𝑋))
14		ringlz.4	. . . . 5 ⊢ 𝐻 = (2nd ‘𝑅)
15	1, 14, 3	rngodi 37891	. . . 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 37888	. . . . 5 ⊢ ((𝑅 ∈ RingOps ∧ 𝐴 ∈ 𝑋 ∧ 𝑍 ∈ 𝑋) → (𝐴𝐻𝑍) ∈ 𝑋)
19	12, 18	mpd3an3 1464	. . . 4 ⊢ ((𝑅 ∈ RingOps ∧ 𝐴 ∈ 𝑋) → (𝐴𝐻𝑍) ∈ 𝑋)
20	3, 4	grpolid 30495	. . . . 5 ⊢ ((𝐺 ∈ GrpOp ∧ (𝐴𝐻𝑍) ∈ 𝑋) → (𝑍𝐺(𝐴𝐻𝑍)) = (𝐴𝐻𝑍))
21	20	eqcomd 2735	. . . 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 2772	. 2 ⊢ ((𝑅 ∈ RingOps ∧ 𝐴 ∈ 𝑋) → ((𝐴𝐻𝑍)𝐺(𝐴𝐻𝑍)) = (𝑍𝐺(𝐴𝐻𝑍)))
24	3	grporcan 30497	. . 3 ⊢ ((𝐺 ∈ GrpOp ∧ ((𝐴𝐻𝑍) ∈ 𝑋 ∧ 𝑍 ∈ 𝑋 ∧ (𝐴𝐻𝑍) ∈ 𝑋)) → (((𝐴𝐻𝑍)𝐺(𝐴𝐻𝑍)) = (𝑍𝐺(𝐴𝐻𝑍)) ↔ (𝐴𝐻𝑍) = 𝑍))
25	17, 19, 12, 19, 24	syl13anc 1374	. 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 1540   ∈ wcel 2109  ran crn 5632  ‘cfv 6499  (class class class)co 7369  1^st c1st 7945  2^nd c2nd 7946  GrpOpcgr 30468  GIdcgi 30469  RingOpscrngo 37881

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-sep 5246  ax-nul 5256  ax-pr 5382  ax-un 7691

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 3352  df-rab 3403  df-v 3446  df-sbc 3751  df-csb 3860  df-dif 3914  df-un 3916  df-in 3918  df-ss 3928  df-nul 4293  df-if 4485  df-sn 4586  df-pr 4588  df-op 4592  df-uni 4868  df-iun 4953  df-br 5103  df-opab 5165  df-mpt 5184  df-id 5526  df-xp 5637  df-rel 5638  df-cnv 5639  df-co 5640  df-dm 5641  df-rn 5642  df-iota 6452  df-fun 6501  df-fn 6502  df-f 6503  df-fo 6505  df-fv 6507  df-riota 7326  df-ov 7372  df-1st 7947  df-2nd 7948  df-grpo 30472  df-gid 30473  df-ablo 30524  df-rngo 37882

This theorem is referenced by:  rngoueqz  37927  rngonegmn1r  37929  zerdivemp1x  37934  0idl  38012  keridl  38019