Theorem rngolz 38257

Description: The zero of a unital ring is a left-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
rngolz	⊢ ((𝑅 ∈ RingOps ∧ 𝐴 ∈ 𝑋) → (𝑍𝐻𝐴) = 𝑍)

Proof of Theorem rngolz

Step	Hyp	Ref	Expression
1		ringlz.3	. . . . . . 7 ⊢ 𝐺 = (1st ‘𝑅)
2	1	rngogrpo 38245	. . . . . 6 ⊢ (𝑅 ∈ RingOps → 𝐺 ∈ GrpOp)
3		ringlz.2	. . . . . . 7 ⊢ 𝑋 = ran 𝐺
4		ringlz.1	. . . . . . 7 ⊢ 𝑍 = (GId‘𝐺)
5	3, 4	grpoidcl 30600	. . . . . 6 ⊢ (𝐺 ∈ GrpOp → 𝑍 ∈ 𝑋)
6	3, 4	grpolid 30602	. . . . . 6 ⊢ ((𝐺 ∈ GrpOp ∧ 𝑍 ∈ 𝑋) → (𝑍𝐺𝑍) = 𝑍)
7	2, 5, 6	syl2anc2 586	. . . . 5 ⊢ (𝑅 ∈ RingOps → (𝑍𝐺𝑍) = 𝑍)
8	7	adantr 480	. . . 4 ⊢ ((𝑅 ∈ RingOps ∧ 𝐴 ∈ 𝑋) → (𝑍𝐺𝑍) = 𝑍)
9	8	oveq1d 7375	. . 3 ⊢ ((𝑅 ∈ RingOps ∧ 𝐴 ∈ 𝑋) → ((𝑍𝐺𝑍)𝐻𝐴) = (𝑍𝐻𝐴))
10	1, 3, 4	rngo0cl 38254	. . . . . 6 ⊢ (𝑅 ∈ RingOps → 𝑍 ∈ 𝑋)
11	10	adantr 480	. . . . 5 ⊢ ((𝑅 ∈ RingOps ∧ 𝐴 ∈ 𝑋) → 𝑍 ∈ 𝑋)
12		simpr 484	. . . . 5 ⊢ ((𝑅 ∈ RingOps ∧ 𝐴 ∈ 𝑋) → 𝐴 ∈ 𝑋)
13	11, 11, 12	3jca 1129	. . . 4 ⊢ ((𝑅 ∈ RingOps ∧ 𝐴 ∈ 𝑋) → (𝑍 ∈ 𝑋 ∧ 𝑍 ∈ 𝑋 ∧ 𝐴 ∈ 𝑋))
14		ringlz.4	. . . . 5 ⊢ 𝐻 = (2nd ‘𝑅)
15	1, 14, 3	rngodir 38240	. . . 4 ⊢ ((𝑅 ∈ RingOps ∧ (𝑍 ∈ 𝑋 ∧ 𝑍 ∈ 𝑋 ∧ 𝐴 ∈ 𝑋)) → ((𝑍𝐺𝑍)𝐻𝐴) = ((𝑍𝐻𝐴)𝐺(𝑍𝐻𝐴)))
16	13, 15	syldan 592	. . 3 ⊢ ((𝑅 ∈ RingOps ∧ 𝐴 ∈ 𝑋) → ((𝑍𝐺𝑍)𝐻𝐴) = ((𝑍𝐻𝐴)𝐺(𝑍𝐻𝐴)))
17	2	adantr 480	. . . 4 ⊢ ((𝑅 ∈ RingOps ∧ 𝐴 ∈ 𝑋) → 𝐺 ∈ GrpOp)
18		simpl 482	. . . . 5 ⊢ ((𝑅 ∈ RingOps ∧ 𝐴 ∈ 𝑋) → 𝑅 ∈ RingOps)
19	1, 14, 3	rngocl 38236	. . . . 5 ⊢ ((𝑅 ∈ RingOps ∧ 𝑍 ∈ 𝑋 ∧ 𝐴 ∈ 𝑋) → (𝑍𝐻𝐴) ∈ 𝑋)
20	18, 11, 12, 19	syl3anc 1374	. . . 4 ⊢ ((𝑅 ∈ RingOps ∧ 𝐴 ∈ 𝑋) → (𝑍𝐻𝐴) ∈ 𝑋)
21	3, 4	grporid 30603	. . . . 5 ⊢ ((𝐺 ∈ GrpOp ∧ (𝑍𝐻𝐴) ∈ 𝑋) → ((𝑍𝐻𝐴)𝐺𝑍) = (𝑍𝐻𝐴))
22	21	eqcomd 2743	. . . 4 ⊢ ((𝐺 ∈ GrpOp ∧ (𝑍𝐻𝐴) ∈ 𝑋) → (𝑍𝐻𝐴) = ((𝑍𝐻𝐴)𝐺𝑍))
23	17, 20, 22	syl2anc 585	. . 3 ⊢ ((𝑅 ∈ RingOps ∧ 𝐴 ∈ 𝑋) → (𝑍𝐻𝐴) = ((𝑍𝐻𝐴)𝐺𝑍))
24	9, 16, 23	3eqtr3d 2780	. 2 ⊢ ((𝑅 ∈ RingOps ∧ 𝐴 ∈ 𝑋) → ((𝑍𝐻𝐴)𝐺(𝑍𝐻𝐴)) = ((𝑍𝐻𝐴)𝐺𝑍))
25	3	grpolcan 30616	. . 3 ⊢ ((𝐺 ∈ GrpOp ∧ ((𝑍𝐻𝐴) ∈ 𝑋 ∧ 𝑍 ∈ 𝑋 ∧ (𝑍𝐻𝐴) ∈ 𝑋)) → (((𝑍𝐻𝐴)𝐺(𝑍𝐻𝐴)) = ((𝑍𝐻𝐴)𝐺𝑍) ↔ (𝑍𝐻𝐴) = 𝑍))
26	17, 20, 11, 20, 25	syl13anc 1375	. 2 ⊢ ((𝑅 ∈ RingOps ∧ 𝐴 ∈ 𝑋) → (((𝑍𝐻𝐴)𝐺(𝑍𝐻𝐴)) = ((𝑍𝐻𝐴)𝐺𝑍) ↔ (𝑍𝐻𝐴) = 𝑍))
27	24, 26	mpbid 232	1 ⊢ ((𝑅 ∈ RingOps ∧ 𝐴 ∈ 𝑋) → (𝑍𝐻𝐴) = 𝑍)

Syntax hints:   → wi 4   ↔ wb 206   ∧ wa 395   ∧ w3a 1087   = wceq 1542   ∈ wcel 2114  ran crn 5625  ‘cfv 6492  (class class class)co 7360  1^st c1st 7933  2^nd c2nd 7934  GrpOpcgr 30575  GIdcgi 30576  RingOpscrngo 38229

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1912  ax-6 1969  ax-7 2010  ax-8 2116  ax-9 2124  ax-10 2147  ax-11 2163  ax-12 2185  ax-ext 2709  ax-rep 5212  ax-sep 5231  ax-nul 5241  ax-pr 5370  ax-un 7682

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3an 1089  df-tru 1545  df-fal 1555  df-ex 1782  df-nf 1786  df-sb 2069  df-mo 2540  df-eu 2570  df-clab 2716  df-cleq 2729  df-clel 2812  df-nfc 2886  df-ne 2934  df-ral 3053  df-rex 3063  df-reu 3344  df-rab 3391  df-v 3432  df-sbc 3730  df-csb 3839  df-dif 3893  df-un 3895  df-in 3897  df-ss 3907  df-nul 4275  df-if 4468  df-sn 4569  df-pr 4571  df-op 4575  df-uni 4852  df-iun 4936  df-br 5087  df-opab 5149  df-mpt 5168  df-id 5519  df-xp 5630  df-rel 5631  df-cnv 5632  df-co 5633  df-dm 5634  df-rn 5635  df-res 5636  df-ima 5637  df-iota 6448  df-fun 6494  df-fn 6495  df-f 6496  df-f1 6497  df-fo 6498  df-f1o 6499  df-fv 6500  df-riota 7317  df-ov 7363  df-1st 7935  df-2nd 7936  df-grpo 30579  df-gid 30580  df-ginv 30581  df-ablo 30631  df-rngo 38230

This theorem is referenced by:  rngonegmn1l  38276  isdrngo3  38294  0idl  38360  keridl  38367  prnc  38402