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Theorem rngorz 35234
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 = (1st𝑅)
ringlz.4 𝐻 = (2nd𝑅)
Assertion
Ref Expression
rngorz ((𝑅 ∈ RingOps ∧ 𝐴𝑋) → (𝐴𝐻𝑍) = 𝑍)

Proof of Theorem rngorz
StepHypRef Expression
1 ringlz.3 . . . . . . 7 𝐺 = (1st𝑅)
21rngogrpo 35221 . . . . . 6 (𝑅 ∈ RingOps → 𝐺 ∈ GrpOp)
3 ringlz.2 . . . . . . 7 𝑋 = ran 𝐺
4 ringlz.1 . . . . . . 7 𝑍 = (GId‘𝐺)
53, 4grpoidcl 28273 . . . . . 6 (𝐺 ∈ GrpOp → 𝑍𝑋)
63, 4grpolid 28275 . . . . . 6 ((𝐺 ∈ GrpOp ∧ 𝑍𝑋) → (𝑍𝐺𝑍) = 𝑍)
72, 5, 6syl2anc2 587 . . . . 5 (𝑅 ∈ RingOps → (𝑍𝐺𝑍) = 𝑍)
87adantr 483 . . . 4 ((𝑅 ∈ RingOps ∧ 𝐴𝑋) → (𝑍𝐺𝑍) = 𝑍)
98oveq2d 7147 . . 3 ((𝑅 ∈ RingOps ∧ 𝐴𝑋) → (𝐴𝐻(𝑍𝐺𝑍)) = (𝐴𝐻𝑍))
10 simpr 487 . . . . 5 ((𝑅 ∈ RingOps ∧ 𝐴𝑋) → 𝐴𝑋)
111, 3, 4rngo0cl 35230 . . . . . 6 (𝑅 ∈ RingOps → 𝑍𝑋)
1211adantr 483 . . . . 5 ((𝑅 ∈ RingOps ∧ 𝐴𝑋) → 𝑍𝑋)
1310, 12, 123jca 1124 . . . 4 ((𝑅 ∈ RingOps ∧ 𝐴𝑋) → (𝐴𝑋𝑍𝑋𝑍𝑋))
14 ringlz.4 . . . . 5 𝐻 = (2nd𝑅)
151, 14, 3rngodi 35215 . . . 4 ((𝑅 ∈ RingOps ∧ (𝐴𝑋𝑍𝑋𝑍𝑋)) → (𝐴𝐻(𝑍𝐺𝑍)) = ((𝐴𝐻𝑍)𝐺(𝐴𝐻𝑍)))
1613, 15syldan 593 . . 3 ((𝑅 ∈ RingOps ∧ 𝐴𝑋) → (𝐴𝐻(𝑍𝐺𝑍)) = ((𝐴𝐻𝑍)𝐺(𝐴𝐻𝑍)))
172adantr 483 . . . 4 ((𝑅 ∈ RingOps ∧ 𝐴𝑋) → 𝐺 ∈ GrpOp)
181, 14, 3rngocl 35212 . . . . 5 ((𝑅 ∈ RingOps ∧ 𝐴𝑋𝑍𝑋) → (𝐴𝐻𝑍) ∈ 𝑋)
1912, 18mpd3an3 1458 . . . 4 ((𝑅 ∈ RingOps ∧ 𝐴𝑋) → (𝐴𝐻𝑍) ∈ 𝑋)
203, 4grpolid 28275 . . . . 5 ((𝐺 ∈ GrpOp ∧ (𝐴𝐻𝑍) ∈ 𝑋) → (𝑍𝐺(𝐴𝐻𝑍)) = (𝐴𝐻𝑍))
2120eqcomd 2826 . . . 4 ((𝐺 ∈ GrpOp ∧ (𝐴𝐻𝑍) ∈ 𝑋) → (𝐴𝐻𝑍) = (𝑍𝐺(𝐴𝐻𝑍)))
2217, 19, 21syl2anc 586 . . 3 ((𝑅 ∈ RingOps ∧ 𝐴𝑋) → (𝐴𝐻𝑍) = (𝑍𝐺(𝐴𝐻𝑍)))
239, 16, 223eqtr3d 2863 . 2 ((𝑅 ∈ RingOps ∧ 𝐴𝑋) → ((𝐴𝐻𝑍)𝐺(𝐴𝐻𝑍)) = (𝑍𝐺(𝐴𝐻𝑍)))
243grporcan 28277 . . 3 ((𝐺 ∈ GrpOp ∧ ((𝐴𝐻𝑍) ∈ 𝑋𝑍𝑋 ∧ (𝐴𝐻𝑍) ∈ 𝑋)) → (((𝐴𝐻𝑍)𝐺(𝐴𝐻𝑍)) = (𝑍𝐺(𝐴𝐻𝑍)) ↔ (𝐴𝐻𝑍) = 𝑍))
2517, 19, 12, 19, 24syl13anc 1368 . 2 ((𝑅 ∈ RingOps ∧ 𝐴𝑋) → (((𝐴𝐻𝑍)𝐺(𝐴𝐻𝑍)) = (𝑍𝐺(𝐴𝐻𝑍)) ↔ (𝐴𝐻𝑍) = 𝑍))
2623, 25mpbid 234 1 ((𝑅 ∈ RingOps ∧ 𝐴𝑋) → (𝐴𝐻𝑍) = 𝑍)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 208  wa 398  w3a 1083   = wceq 1537  wcel 2114  ran crn 5530  cfv 6329  (class class class)co 7131  1st c1st 7663  2nd c2nd 7664  GrpOpcgr 28248  GIdcgi 28249  RingOpscrngo 35205
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1911  ax-6 1970  ax-7 2015  ax-8 2116  ax-9 2124  ax-10 2145  ax-11 2161  ax-12 2177  ax-ext 2792  ax-sep 5177  ax-nul 5184  ax-pow 5240  ax-pr 5304  ax-un 7437
This theorem depends on definitions:  df-bi 209  df-an 399  df-or 844  df-3an 1085  df-tru 1540  df-ex 1781  df-nf 1785  df-sb 2070  df-mo 2622  df-eu 2653  df-clab 2799  df-cleq 2813  df-clel 2891  df-nfc 2959  df-ral 3130  df-rex 3131  df-reu 3132  df-rab 3134  df-v 3475  df-sbc 3752  df-csb 3860  df-dif 3915  df-un 3917  df-in 3919  df-ss 3928  df-nul 4268  df-if 4442  df-sn 4542  df-pr 4544  df-op 4548  df-uni 4813  df-iun 4895  df-br 5041  df-opab 5103  df-mpt 5121  df-id 5434  df-xp 5535  df-rel 5536  df-cnv 5537  df-co 5538  df-dm 5539  df-rn 5540  df-iota 6288  df-fun 6331  df-fn 6332  df-f 6333  df-fo 6335  df-fv 6337  df-riota 7089  df-ov 7134  df-1st 7665  df-2nd 7666  df-grpo 28252  df-gid 28253  df-ablo 28304  df-rngo 35206
This theorem is referenced by:  rngoueqz  35251  rngonegmn1r  35253  zerdivemp1x  35258  0idl  35336  keridl  35343
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