Theorem rngorz 38602

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 38589	. . . . . 6 ⊢ (𝑅 ∈ RingOps → 𝐺 ∈ GrpOp)
3		ringlz.2	. . . . . . 7 ⊢ 𝑋 = ran 𝐺
4		ringlz.1	. . . . . . 7 ⊢ 𝑍 = (GId‘𝐺)
5	3, 4	grpoidcl 30877	. . . . . 6 ⊢ (𝐺 ∈ GrpOp → 𝑍 ∈ 𝑋)
6	3, 4	grpolid 30879	. . . . . 6 ⊢ ((𝐺 ∈ GrpOp ∧ 𝑍 ∈ 𝑋) → (𝑍𝐺𝑍) = 𝑍)
7	2, 5, 6	syl2anc2 596	. . . . 5 ⊢ (𝑅 ∈ RingOps → (𝑍𝐺𝑍) = 𝑍)
8	7	adantr 485	. . . 4 ⊢ ((𝑅 ∈ RingOps ∧ 𝐴 ∈ 𝑋) → (𝑍𝐺𝑍) = 𝑍)
9	8	oveq2d 7428	. . 3 ⊢ ((𝑅 ∈ RingOps ∧ 𝐴 ∈ 𝑋) → (𝐴𝐻(𝑍𝐺𝑍)) = (𝐴𝐻𝑍))
10		simpr 489	. . . . 5 ⊢ ((𝑅 ∈ RingOps ∧ 𝐴 ∈ 𝑋) → 𝐴 ∈ 𝑋)
11	1, 3, 4	rngo0cl 38598	. . . . . 6 ⊢ (𝑅 ∈ RingOps → 𝑍 ∈ 𝑋)
12	11	adantr 485	. . . . 5 ⊢ ((𝑅 ∈ RingOps ∧ 𝐴 ∈ 𝑋) → 𝑍 ∈ 𝑋)
13	10, 12, 12	3jca 1145	. . . 4 ⊢ ((𝑅 ∈ RingOps ∧ 𝐴 ∈ 𝑋) → (𝐴 ∈ 𝑋 ∧ 𝑍 ∈ 𝑋 ∧ 𝑍 ∈ 𝑋))
14		ringlz.4	. . . . 5 ⊢ 𝐻 = (2nd ‘𝑅)
15	1, 14, 3	rngodi 38583	. . . 4 ⊢ ((𝑅 ∈ RingOps ∧ (𝐴 ∈ 𝑋 ∧ 𝑍 ∈ 𝑋 ∧ 𝑍 ∈ 𝑋)) → (𝐴𝐻(𝑍𝐺𝑍)) = ((𝐴𝐻𝑍)𝐺(𝐴𝐻𝑍)))
16	13, 15	syldan 602	. . 3 ⊢ ((𝑅 ∈ RingOps ∧ 𝐴 ∈ 𝑋) → (𝐴𝐻(𝑍𝐺𝑍)) = ((𝐴𝐻𝑍)𝐺(𝐴𝐻𝑍)))
17	2	adantr 485	. . . 4 ⊢ ((𝑅 ∈ RingOps ∧ 𝐴 ∈ 𝑋) → 𝐺 ∈ GrpOp)
18	1, 14, 3	rngocl 38580	. . . . 5 ⊢ ((𝑅 ∈ RingOps ∧ 𝐴 ∈ 𝑋 ∧ 𝑍 ∈ 𝑋) → (𝐴𝐻𝑍) ∈ 𝑋)
19	12, 18	mpd3an3 1490	. . . 4 ⊢ ((𝑅 ∈ RingOps ∧ 𝐴 ∈ 𝑋) → (𝐴𝐻𝑍) ∈ 𝑋)
20	3, 4	grpolid 30879	. . . . 5 ⊢ ((𝐺 ∈ GrpOp ∧ (𝐴𝐻𝑍) ∈ 𝑋) → (𝑍𝐺(𝐴𝐻𝑍)) = (𝐴𝐻𝑍))
21	20	eqcomd 2768	. . . 4 ⊢ ((𝐺 ∈ GrpOp ∧ (𝐴𝐻𝑍) ∈ 𝑋) → (𝐴𝐻𝑍) = (𝑍𝐺(𝐴𝐻𝑍)))
22	17, 19, 21	syl2anc 595	. . 3 ⊢ ((𝑅 ∈ RingOps ∧ 𝐴 ∈ 𝑋) → (𝐴𝐻𝑍) = (𝑍𝐺(𝐴𝐻𝑍)))
23	9, 16, 22	3eqtr3d 2805	. 2 ⊢ ((𝑅 ∈ RingOps ∧ 𝐴 ∈ 𝑋) → ((𝐴𝐻𝑍)𝐺(𝐴𝐻𝑍)) = (𝑍𝐺(𝐴𝐻𝑍)))
24	3	grporcan 30881	. . 3 ⊢ ((𝐺 ∈ GrpOp ∧ ((𝐴𝐻𝑍) ∈ 𝑋 ∧ 𝑍 ∈ 𝑋 ∧ (𝐴𝐻𝑍) ∈ 𝑋)) → (((𝐴𝐻𝑍)𝐺(𝐴𝐻𝑍)) = (𝑍𝐺(𝐴𝐻𝑍)) ↔ (𝐴𝐻𝑍) = 𝑍))
25	17, 19, 12, 19, 24	syl13anc 1398	. 2 ⊢ ((𝑅 ∈ RingOps ∧ 𝐴 ∈ 𝑋) → (((𝐴𝐻𝑍)𝐺(𝐴𝐻𝑍)) = (𝑍𝐺(𝐴𝐻𝑍)) ↔ (𝐴𝐻𝑍) = 𝑍))
26	23, 25	mpbid 235	1 ⊢ ((𝑅 ∈ RingOps ∧ 𝐴 ∈ 𝑋) → (𝐴𝐻𝑍) = 𝑍)

This proof depends on syntax axioms:   → wi 4   ↔ wb 209   ∧ wa 400   ∧ w3a 1102   = wceq 1569   ∈ wcel 2142  ran crn 5661  ‘cfv 6536  (class class class)co 7412  1^st c1st 7982  2^nd c2nd 7983  GrpOpcgr 30852  GIdcgi 30853  RingOpscrngo 38573

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1824  ax-4 1838  ax-5 1939  ax-6 1996  ax-7 2037  ax-8 2144  ax-9 2152  ax-10 2175  ax-11 2191  ax-12 2212  ax-ext 2734  ax-sep 5256  ax-nul 5268  ax-pr 5403  ax-un 7734

This proof depends on definitions:  df-bi 210  df-an 401  df-or 861  df-3an 1104  df-tru 1572  df-fal 1582  df-ex 1809  df-nf 1813  df-sb 2096  df-mo 2566  df-eu 2596  df-clab 2741  df-cleq 2754  df-clel 2837  df-nfc 2911  df-ne 2958  df-ral 3079  df-rex 3089  df-reu 3369  df-rab 3416  df-v 3456  df-sbc 3744  df-csb 3853  df-dif 3907  df-un 3909  df-in 3911  df-ss 3921  df-nul 4286  df-if 4487  df-sn 4589  df-pr 4591  df-op 4595  df-uni 4872  df-iun 4957  df-br 5109  df-opab 5173  df-mpt 5192  df-id 5555  df-xp 5666  df-rel 5667  df-cnv 5668  df-co 5669  df-dm 5670  df-rn 5671  df-iota 6492  df-fun 6538  df-fn 6539  df-f 6540  df-fo 6542  df-fv 6544  df-riota 7369  df-ov 7415  df-1st 7984  df-2nd 7985  df-grpo 30856  df-gid 30857  df-ablo 30908  df-rngo 38574

This theorem is used by:  rngoueqz  38619  rngonegmn1r  38621  zerdivemp1x  38626  0idl  38704  keridl  38711