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Theorem rngorz 37396
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 37383 . . . . . 6 (𝑅 ∈ RingOps → 𝐺 ∈ GrpOp)
3 ringlz.2 . . . . . . 7 𝑋 = ran 𝐺
4 ringlz.1 . . . . . . 7 𝑍 = (GId‘𝐺)
53, 4grpoidcl 30337 . . . . . 6 (𝐺 ∈ GrpOp → 𝑍𝑋)
63, 4grpolid 30339 . . . . . 6 ((𝐺 ∈ GrpOp ∧ 𝑍𝑋) → (𝑍𝐺𝑍) = 𝑍)
72, 5, 6syl2anc2 584 . . . . 5 (𝑅 ∈ RingOps → (𝑍𝐺𝑍) = 𝑍)
87adantr 480 . . . 4 ((𝑅 ∈ RingOps ∧ 𝐴𝑋) → (𝑍𝐺𝑍) = 𝑍)
98oveq2d 7436 . . 3 ((𝑅 ∈ RingOps ∧ 𝐴𝑋) → (𝐴𝐻(𝑍𝐺𝑍)) = (𝐴𝐻𝑍))
10 simpr 484 . . . . 5 ((𝑅 ∈ RingOps ∧ 𝐴𝑋) → 𝐴𝑋)
111, 3, 4rngo0cl 37392 . . . . . 6 (𝑅 ∈ RingOps → 𝑍𝑋)
1211adantr 480 . . . . 5 ((𝑅 ∈ RingOps ∧ 𝐴𝑋) → 𝑍𝑋)
1310, 12, 123jca 1126 . . . 4 ((𝑅 ∈ RingOps ∧ 𝐴𝑋) → (𝐴𝑋𝑍𝑋𝑍𝑋))
14 ringlz.4 . . . . 5 𝐻 = (2nd𝑅)
151, 14, 3rngodi 37377 . . . 4 ((𝑅 ∈ RingOps ∧ (𝐴𝑋𝑍𝑋𝑍𝑋)) → (𝐴𝐻(𝑍𝐺𝑍)) = ((𝐴𝐻𝑍)𝐺(𝐴𝐻𝑍)))
1613, 15syldan 590 . . 3 ((𝑅 ∈ RingOps ∧ 𝐴𝑋) → (𝐴𝐻(𝑍𝐺𝑍)) = ((𝐴𝐻𝑍)𝐺(𝐴𝐻𝑍)))
172adantr 480 . . . 4 ((𝑅 ∈ RingOps ∧ 𝐴𝑋) → 𝐺 ∈ GrpOp)
181, 14, 3rngocl 37374 . . . . 5 ((𝑅 ∈ RingOps ∧ 𝐴𝑋𝑍𝑋) → (𝐴𝐻𝑍) ∈ 𝑋)
1912, 18mpd3an3 1459 . . . 4 ((𝑅 ∈ RingOps ∧ 𝐴𝑋) → (𝐴𝐻𝑍) ∈ 𝑋)
203, 4grpolid 30339 . . . . 5 ((𝐺 ∈ GrpOp ∧ (𝐴𝐻𝑍) ∈ 𝑋) → (𝑍𝐺(𝐴𝐻𝑍)) = (𝐴𝐻𝑍))
2120eqcomd 2734 . . . 4 ((𝐺 ∈ GrpOp ∧ (𝐴𝐻𝑍) ∈ 𝑋) → (𝐴𝐻𝑍) = (𝑍𝐺(𝐴𝐻𝑍)))
2217, 19, 21syl2anc 583 . . 3 ((𝑅 ∈ RingOps ∧ 𝐴𝑋) → (𝐴𝐻𝑍) = (𝑍𝐺(𝐴𝐻𝑍)))
239, 16, 223eqtr3d 2776 . 2 ((𝑅 ∈ RingOps ∧ 𝐴𝑋) → ((𝐴𝐻𝑍)𝐺(𝐴𝐻𝑍)) = (𝑍𝐺(𝐴𝐻𝑍)))
243grporcan 30341 . . 3 ((𝐺 ∈ GrpOp ∧ ((𝐴𝐻𝑍) ∈ 𝑋𝑍𝑋 ∧ (𝐴𝐻𝑍) ∈ 𝑋)) → (((𝐴𝐻𝑍)𝐺(𝐴𝐻𝑍)) = (𝑍𝐺(𝐴𝐻𝑍)) ↔ (𝐴𝐻𝑍) = 𝑍))
2517, 19, 12, 19, 24syl13anc 1370 . 2 ((𝑅 ∈ RingOps ∧ 𝐴𝑋) → (((𝐴𝐻𝑍)𝐺(𝐴𝐻𝑍)) = (𝑍𝐺(𝐴𝐻𝑍)) ↔ (𝐴𝐻𝑍) = 𝑍))
2623, 25mpbid 231 1 ((𝑅 ∈ RingOps ∧ 𝐴𝑋) → (𝐴𝐻𝑍) = 𝑍)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 205  wa 395  w3a 1085   = wceq 1534  wcel 2099  ran crn 5679  cfv 6548  (class class class)co 7420  1st c1st 7991  2nd c2nd 7992  GrpOpcgr 30312  GIdcgi 30313  RingOpscrngo 37367
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1790  ax-4 1804  ax-5 1906  ax-6 1964  ax-7 2004  ax-8 2101  ax-9 2109  ax-10 2130  ax-11 2147  ax-12 2167  ax-ext 2699  ax-sep 5299  ax-nul 5306  ax-pr 5429  ax-un 7740
This theorem depends on definitions:  df-bi 206  df-an 396  df-or 847  df-3an 1087  df-tru 1537  df-fal 1547  df-ex 1775  df-nf 1779  df-sb 2061  df-mo 2530  df-eu 2559  df-clab 2706  df-cleq 2720  df-clel 2806  df-nfc 2881  df-ne 2938  df-ral 3059  df-rex 3068  df-reu 3374  df-rab 3430  df-v 3473  df-sbc 3777  df-csb 3893  df-dif 3950  df-un 3952  df-in 3954  df-ss 3964  df-nul 4324  df-if 4530  df-sn 4630  df-pr 4632  df-op 4636  df-uni 4909  df-iun 4998  df-br 5149  df-opab 5211  df-mpt 5232  df-id 5576  df-xp 5684  df-rel 5685  df-cnv 5686  df-co 5687  df-dm 5688  df-rn 5689  df-iota 6500  df-fun 6550  df-fn 6551  df-f 6552  df-fo 6554  df-fv 6556  df-riota 7376  df-ov 7423  df-1st 7993  df-2nd 7994  df-grpo 30316  df-gid 30317  df-ablo 30368  df-rngo 37368
This theorem is referenced by:  rngoueqz  37413  rngonegmn1r  37415  zerdivemp1x  37420  0idl  37498  keridl  37505
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