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Theorem rngolz 37623
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 37611 . . . . . 6 (𝑅 ∈ RingOps → 𝐺 ∈ GrpOp)
3 ringlz.2 . . . . . . 7 𝑋 = ran 𝐺
4 ringlz.1 . . . . . . 7 𝑍 = (GId‘𝐺)
53, 4grpoidcl 30447 . . . . . 6 (𝐺 ∈ GrpOp → 𝑍𝑋)
63, 4grpolid 30449 . . . . . 6 ((𝐺 ∈ GrpOp ∧ 𝑍𝑋) → (𝑍𝐺𝑍) = 𝑍)
72, 5, 6syl2anc2 583 . . . . 5 (𝑅 ∈ RingOps → (𝑍𝐺𝑍) = 𝑍)
87adantr 479 . . . 4 ((𝑅 ∈ RingOps ∧ 𝐴𝑋) → (𝑍𝐺𝑍) = 𝑍)
98oveq1d 7439 . . 3 ((𝑅 ∈ RingOps ∧ 𝐴𝑋) → ((𝑍𝐺𝑍)𝐻𝐴) = (𝑍𝐻𝐴))
101, 3, 4rngo0cl 37620 . . . . . 6 (𝑅 ∈ RingOps → 𝑍𝑋)
1110adantr 479 . . . . 5 ((𝑅 ∈ RingOps ∧ 𝐴𝑋) → 𝑍𝑋)
12 simpr 483 . . . . 5 ((𝑅 ∈ RingOps ∧ 𝐴𝑋) → 𝐴𝑋)
1311, 11, 123jca 1125 . . . 4 ((𝑅 ∈ RingOps ∧ 𝐴𝑋) → (𝑍𝑋𝑍𝑋𝐴𝑋))
14 ringlz.4 . . . . 5 𝐻 = (2nd𝑅)
151, 14, 3rngodir 37606 . . . 4 ((𝑅 ∈ RingOps ∧ (𝑍𝑋𝑍𝑋𝐴𝑋)) → ((𝑍𝐺𝑍)𝐻𝐴) = ((𝑍𝐻𝐴)𝐺(𝑍𝐻𝐴)))
1613, 15syldan 589 . . 3 ((𝑅 ∈ RingOps ∧ 𝐴𝑋) → ((𝑍𝐺𝑍)𝐻𝐴) = ((𝑍𝐻𝐴)𝐺(𝑍𝐻𝐴)))
172adantr 479 . . . 4 ((𝑅 ∈ RingOps ∧ 𝐴𝑋) → 𝐺 ∈ GrpOp)
18 simpl 481 . . . . 5 ((𝑅 ∈ RingOps ∧ 𝐴𝑋) → 𝑅 ∈ RingOps)
191, 14, 3rngocl 37602 . . . . 5 ((𝑅 ∈ RingOps ∧ 𝑍𝑋𝐴𝑋) → (𝑍𝐻𝐴) ∈ 𝑋)
2018, 11, 12, 19syl3anc 1368 . . . 4 ((𝑅 ∈ RingOps ∧ 𝐴𝑋) → (𝑍𝐻𝐴) ∈ 𝑋)
213, 4grporid 30450 . . . . 5 ((𝐺 ∈ GrpOp ∧ (𝑍𝐻𝐴) ∈ 𝑋) → ((𝑍𝐻𝐴)𝐺𝑍) = (𝑍𝐻𝐴))
2221eqcomd 2732 . . . 4 ((𝐺 ∈ GrpOp ∧ (𝑍𝐻𝐴) ∈ 𝑋) → (𝑍𝐻𝐴) = ((𝑍𝐻𝐴)𝐺𝑍))
2317, 20, 22syl2anc 582 . . 3 ((𝑅 ∈ RingOps ∧ 𝐴𝑋) → (𝑍𝐻𝐴) = ((𝑍𝐻𝐴)𝐺𝑍))
249, 16, 233eqtr3d 2774 . 2 ((𝑅 ∈ RingOps ∧ 𝐴𝑋) → ((𝑍𝐻𝐴)𝐺(𝑍𝐻𝐴)) = ((𝑍𝐻𝐴)𝐺𝑍))
253grpolcan 30463 . . 3 ((𝐺 ∈ GrpOp ∧ ((𝑍𝐻𝐴) ∈ 𝑋𝑍𝑋 ∧ (𝑍𝐻𝐴) ∈ 𝑋)) → (((𝑍𝐻𝐴)𝐺(𝑍𝐻𝐴)) = ((𝑍𝐻𝐴)𝐺𝑍) ↔ (𝑍𝐻𝐴) = 𝑍))
2617, 20, 11, 20, 25syl13anc 1369 . 2 ((𝑅 ∈ RingOps ∧ 𝐴𝑋) → (((𝑍𝐻𝐴)𝐺(𝑍𝐻𝐴)) = ((𝑍𝐻𝐴)𝐺𝑍) ↔ (𝑍𝐻𝐴) = 𝑍))
2724, 26mpbid 231 1 ((𝑅 ∈ RingOps ∧ 𝐴𝑋) → (𝑍𝐻𝐴) = 𝑍)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 205  wa 394  w3a 1084   = wceq 1534  wcel 2099  ran crn 5683  cfv 6554  (class class class)co 7424  1st c1st 8001  2nd c2nd 8002  GrpOpcgr 30422  GIdcgi 30423  RingOpscrngo 37595
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 2697  ax-rep 5290  ax-sep 5304  ax-nul 5311  ax-pr 5433  ax-un 7746
This theorem depends on definitions:  df-bi 206  df-an 395  df-or 846  df-3an 1086  df-tru 1537  df-fal 1547  df-ex 1775  df-nf 1779  df-sb 2061  df-mo 2529  df-eu 2558  df-clab 2704  df-cleq 2718  df-clel 2803  df-nfc 2878  df-ne 2931  df-ral 3052  df-rex 3061  df-reu 3365  df-rab 3420  df-v 3464  df-sbc 3777  df-csb 3893  df-dif 3950  df-un 3952  df-in 3954  df-ss 3964  df-nul 4326  df-if 4534  df-sn 4634  df-pr 4636  df-op 4640  df-uni 4914  df-iun 5003  df-br 5154  df-opab 5216  df-mpt 5237  df-id 5580  df-xp 5688  df-rel 5689  df-cnv 5690  df-co 5691  df-dm 5692  df-rn 5693  df-res 5694  df-ima 5695  df-iota 6506  df-fun 6556  df-fn 6557  df-f 6558  df-f1 6559  df-fo 6560  df-f1o 6561  df-fv 6562  df-riota 7380  df-ov 7427  df-1st 8003  df-2nd 8004  df-grpo 30426  df-gid 30427  df-ginv 30428  df-ablo 30478  df-rngo 37596
This theorem is referenced by:  rngonegmn1l  37642  isdrngo3  37660  0idl  37726  keridl  37733  prnc  37768
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