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Theorem rngorz 34054
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 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)
109oveq2d 6809 . . 3 ((𝑅 ∈ RingOps ∧ 𝐴𝑋) → (𝐴𝐻(𝑍𝐺𝑍)) = (𝐴𝐻𝑍))
11 simpr 471 . . . . 5 ((𝑅 ∈ RingOps ∧ 𝐴𝑋) → 𝐴𝑋)
121, 3, 4rngo0cl 34050 . . . . . 6 (𝑅 ∈ RingOps → 𝑍𝑋)
1312adantr 466 . . . . 5 ((𝑅 ∈ RingOps ∧ 𝐴𝑋) → 𝑍𝑋)
1411, 13, 133jca 1122 . . . 4 ((𝑅 ∈ RingOps ∧ 𝐴𝑋) → (𝐴𝑋𝑍𝑋𝑍𝑋))
15 ringlz.4 . . . . 5 𝐻 = (2nd𝑅)
161, 15, 3rngodi 34035 . . . 4 ((𝑅 ∈ RingOps ∧ (𝐴𝑋𝑍𝑋𝑍𝑋)) → (𝐴𝐻(𝑍𝐺𝑍)) = ((𝐴𝐻𝑍)𝐺(𝐴𝐻𝑍)))
1714, 16syldan 579 . . 3 ((𝑅 ∈ RingOps ∧ 𝐴𝑋) → (𝐴𝐻(𝑍𝐺𝑍)) = ((𝐴𝐻𝑍)𝐺(𝐴𝐻𝑍)))
182adantr 466 . . . 4 ((𝑅 ∈ RingOps ∧ 𝐴𝑋) → 𝐺 ∈ GrpOp)
191, 15, 3rngocl 34032 . . . . 5 ((𝑅 ∈ RingOps ∧ 𝐴𝑋𝑍𝑋) → (𝐴𝐻𝑍) ∈ 𝑋)
2013, 19mpd3an3 1573 . . . 4 ((𝑅 ∈ RingOps ∧ 𝐴𝑋) → (𝐴𝐻𝑍) ∈ 𝑋)
213, 4grpolid 27710 . . . . 5 ((𝐺 ∈ GrpOp ∧ (𝐴𝐻𝑍) ∈ 𝑋) → (𝑍𝐺(𝐴𝐻𝑍)) = (𝐴𝐻𝑍))
2221eqcomd 2777 . . . 4 ((𝐺 ∈ GrpOp ∧ (𝐴𝐻𝑍) ∈ 𝑋) → (𝐴𝐻𝑍) = (𝑍𝐺(𝐴𝐻𝑍)))
2318, 20, 22syl2anc 573 . . 3 ((𝑅 ∈ RingOps ∧ 𝐴𝑋) → (𝐴𝐻𝑍) = (𝑍𝐺(𝐴𝐻𝑍)))
2410, 17, 233eqtr3d 2813 . 2 ((𝑅 ∈ RingOps ∧ 𝐴𝑋) → ((𝐴𝐻𝑍)𝐺(𝐴𝐻𝑍)) = (𝑍𝐺(𝐴𝐻𝑍)))
253grporcan 27712 . . 3 ((𝐺 ∈ GrpOp ∧ ((𝐴𝐻𝑍) ∈ 𝑋𝑍𝑋 ∧ (𝐴𝐻𝑍) ∈ 𝑋)) → (((𝐴𝐻𝑍)𝐺(𝐴𝐻𝑍)) = (𝑍𝐺(𝐴𝐻𝑍)) ↔ (𝐴𝐻𝑍) = 𝑍))
2618, 20, 13, 20, 25syl13anc 1478 . 2 ((𝑅 ∈ RingOps ∧ 𝐴𝑋) → (((𝐴𝐻𝑍)𝐺(𝐴𝐻𝑍)) = (𝑍𝐺(𝐴𝐻𝑍)) ↔ (𝐴𝐻𝑍) = 𝑍))
2724, 26mpbid 222 1 ((𝑅 ∈ RingOps ∧ 𝐴𝑋) → (𝐴𝐻𝑍) = 𝑍)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 196  wa 382  w3a 1071   = wceq 1631  wcel 2145  ran crn 5250  cfv 6031  (class class class)co 6793  1st c1st 7313  2nd c2nd 7314  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-sep 4915  ax-nul 4923  ax-pow 4974  ax-pr 5034  ax-un 7096
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-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 4226  df-sn 4317  df-pr 4319  df-op 4323  df-uni 4575  df-iun 4656  df-br 4787  df-opab 4847  df-mpt 4864  df-id 5157  df-xp 5255  df-rel 5256  df-cnv 5257  df-co 5258  df-dm 5259  df-rn 5260  df-iota 5994  df-fun 6033  df-fn 6034  df-f 6035  df-fo 6037  df-fv 6039  df-riota 6754  df-ov 6796  df-1st 7315  df-2nd 7316  df-grpo 27687  df-gid 27688  df-ablo 27739  df-rngo 34026
This theorem is referenced by:  rngoueqz  34071  rngonegmn1r  34073  zerdivemp1x  34078  0idl  34156  keridl  34163
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