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Theorem rngolz 34053
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 = (1st𝑅)
ringlz.4 𝐻 = (2nd𝑅)
Assertion
Ref Expression
rngolz ((𝑅 ∈ RingOps ∧ 𝐴𝑋) → (𝑍𝐻𝐴) = 𝑍)

Proof of Theorem rngolz
StepHypRef Expression
1 ringlz.3 . . . . . . 7 𝐺 = (1st𝑅)
21rngogrpo 34041 . . . . . 6 (𝑅 ∈ RingOps → 𝐺 ∈ GrpOp)
3 ringlz.2 . . . . . . . 8 𝑋 = ran 𝐺
4 ringlz.1 . . . . . . . 8 𝑍 = (GId‘𝐺)
53, 4grpoidcl 27708 . . . . . . 7 (𝐺 ∈ GrpOp → 𝑍𝑋)
63, 4grpolid 27710 . . . . . . 7 ((𝐺 ∈ GrpOp ∧ 𝑍𝑋) → (𝑍𝐺𝑍) = 𝑍)
75, 6mpdan 667 . . . . . 6 (𝐺 ∈ GrpOp → (𝑍𝐺𝑍) = 𝑍)
82, 7syl 17 . . . . 5 (𝑅 ∈ RingOps → (𝑍𝐺𝑍) = 𝑍)
98adantr 466 . . . 4 ((𝑅 ∈ RingOps ∧ 𝐴𝑋) → (𝑍𝐺𝑍) = 𝑍)
109oveq1d 6811 . . 3 ((𝑅 ∈ RingOps ∧ 𝐴𝑋) → ((𝑍𝐺𝑍)𝐻𝐴) = (𝑍𝐻𝐴))
111, 3, 4rngo0cl 34050 . . . . . 6 (𝑅 ∈ RingOps → 𝑍𝑋)
1211adantr 466 . . . . 5 ((𝑅 ∈ RingOps ∧ 𝐴𝑋) → 𝑍𝑋)
13 simpr 471 . . . . 5 ((𝑅 ∈ RingOps ∧ 𝐴𝑋) → 𝐴𝑋)
1412, 12, 133jca 1122 . . . 4 ((𝑅 ∈ RingOps ∧ 𝐴𝑋) → (𝑍𝑋𝑍𝑋𝐴𝑋))
15 ringlz.4 . . . . 5 𝐻 = (2nd𝑅)
161, 15, 3rngodir 34036 . . . 4 ((𝑅 ∈ RingOps ∧ (𝑍𝑋𝑍𝑋𝐴𝑋)) → ((𝑍𝐺𝑍)𝐻𝐴) = ((𝑍𝐻𝐴)𝐺(𝑍𝐻𝐴)))
1714, 16syldan 579 . . 3 ((𝑅 ∈ RingOps ∧ 𝐴𝑋) → ((𝑍𝐺𝑍)𝐻𝐴) = ((𝑍𝐻𝐴)𝐺(𝑍𝐻𝐴)))
182adantr 466 . . . 4 ((𝑅 ∈ RingOps ∧ 𝐴𝑋) → 𝐺 ∈ GrpOp)
19 simpl 468 . . . . 5 ((𝑅 ∈ RingOps ∧ 𝐴𝑋) → 𝑅 ∈ RingOps)
201, 15, 3rngocl 34032 . . . . 5 ((𝑅 ∈ RingOps ∧ 𝑍𝑋𝐴𝑋) → (𝑍𝐻𝐴) ∈ 𝑋)
2119, 12, 13, 20syl3anc 1476 . . . 4 ((𝑅 ∈ RingOps ∧ 𝐴𝑋) → (𝑍𝐻𝐴) ∈ 𝑋)
223, 4grporid 27711 . . . . 5 ((𝐺 ∈ GrpOp ∧ (𝑍𝐻𝐴) ∈ 𝑋) → ((𝑍𝐻𝐴)𝐺𝑍) = (𝑍𝐻𝐴))
2322eqcomd 2777 . . . 4 ((𝐺 ∈ GrpOp ∧ (𝑍𝐻𝐴) ∈ 𝑋) → (𝑍𝐻𝐴) = ((𝑍𝐻𝐴)𝐺𝑍))
2418, 21, 23syl2anc 573 . . 3 ((𝑅 ∈ RingOps ∧ 𝐴𝑋) → (𝑍𝐻𝐴) = ((𝑍𝐻𝐴)𝐺𝑍))
2510, 17, 243eqtr3d 2813 . 2 ((𝑅 ∈ RingOps ∧ 𝐴𝑋) → ((𝑍𝐻𝐴)𝐺(𝑍𝐻𝐴)) = ((𝑍𝐻𝐴)𝐺𝑍))
263grpolcan 27724 . . 3 ((𝐺 ∈ GrpOp ∧ ((𝑍𝐻𝐴) ∈ 𝑋𝑍𝑋 ∧ (𝑍𝐻𝐴) ∈ 𝑋)) → (((𝑍𝐻𝐴)𝐺(𝑍𝐻𝐴)) = ((𝑍𝐻𝐴)𝐺𝑍) ↔ (𝑍𝐻𝐴) = 𝑍))
2718, 21, 12, 21, 26syl13anc 1478 . 2 ((𝑅 ∈ RingOps ∧ 𝐴𝑋) → (((𝑍𝐻𝐴)𝐺(𝑍𝐻𝐴)) = ((𝑍𝐻𝐴)𝐺𝑍) ↔ (𝑍𝐻𝐴) = 𝑍))
2825, 27mpbid 222 1 ((𝑅 ∈ RingOps ∧ 𝐴𝑋) → (𝑍𝐻𝐴) = 𝑍)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 196  wa 382  w3a 1071   = wceq 1631  wcel 2145  ran crn 5251  cfv 6030  (class class class)co 6796  1st c1st 7317  2nd c2nd 7318  GrpOpcgr 27683  GIdcgi 27684  RingOpscrngo 34025
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1870  ax-4 1885  ax-5 1991  ax-6 2057  ax-7 2093  ax-8 2147  ax-9 2154  ax-10 2174  ax-11 2190  ax-12 2203  ax-13 2408  ax-ext 2751  ax-rep 4905  ax-sep 4916  ax-nul 4924  ax-pow 4975  ax-pr 5035  ax-un 7100
This theorem depends on definitions:  df-bi 197  df-an 383  df-or 837  df-3an 1073  df-tru 1634  df-ex 1853  df-nf 1858  df-sb 2050  df-eu 2622  df-mo 2623  df-clab 2758  df-cleq 2764  df-clel 2767  df-nfc 2902  df-ne 2944  df-ral 3066  df-rex 3067  df-reu 3068  df-rab 3070  df-v 3353  df-sbc 3588  df-csb 3683  df-dif 3726  df-un 3728  df-in 3730  df-ss 3737  df-nul 4064  df-if 4227  df-sn 4318  df-pr 4320  df-op 4324  df-uni 4576  df-iun 4657  df-br 4788  df-opab 4848  df-mpt 4865  df-id 5158  df-xp 5256  df-rel 5257  df-cnv 5258  df-co 5259  df-dm 5260  df-rn 5261  df-res 5262  df-ima 5263  df-iota 5993  df-fun 6032  df-fn 6033  df-f 6034  df-f1 6035  df-fo 6036  df-f1o 6037  df-fv 6038  df-riota 6757  df-ov 6799  df-1st 7319  df-2nd 7320  df-grpo 27687  df-gid 27688  df-ginv 27689  df-ablo 27739  df-rngo 34026
This theorem is referenced by:  rngonegmn1l  34072  isdrngo3  34090  0idl  34156  keridl  34163  prnc  34198
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