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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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