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Theorem rngolz 36410
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 36398 . . . . . 6 (𝑅 ∈ RingOps → 𝐺 ∈ GrpOp)
3 ringlz.2 . . . . . . 7 𝑋 = ran 𝐺
4 ringlz.1 . . . . . . 7 𝑍 = (GId‘𝐺)
53, 4grpoidcl 29498 . . . . . 6 (𝐺 ∈ GrpOp → 𝑍𝑋)
63, 4grpolid 29500 . . . . . 6 ((𝐺 ∈ GrpOp ∧ 𝑍𝑋) → (𝑍𝐺𝑍) = 𝑍)
72, 5, 6syl2anc2 586 . . . . 5 (𝑅 ∈ RingOps → (𝑍𝐺𝑍) = 𝑍)
87adantr 482 . . . 4 ((𝑅 ∈ RingOps ∧ 𝐴𝑋) → (𝑍𝐺𝑍) = 𝑍)
98oveq1d 7377 . . 3 ((𝑅 ∈ RingOps ∧ 𝐴𝑋) → ((𝑍𝐺𝑍)𝐻𝐴) = (𝑍𝐻𝐴))
101, 3, 4rngo0cl 36407 . . . . . 6 (𝑅 ∈ RingOps → 𝑍𝑋)
1110adantr 482 . . . . 5 ((𝑅 ∈ RingOps ∧ 𝐴𝑋) → 𝑍𝑋)
12 simpr 486 . . . . 5 ((𝑅 ∈ RingOps ∧ 𝐴𝑋) → 𝐴𝑋)
1311, 11, 123jca 1129 . . . 4 ((𝑅 ∈ RingOps ∧ 𝐴𝑋) → (𝑍𝑋𝑍𝑋𝐴𝑋))
14 ringlz.4 . . . . 5 𝐻 = (2nd𝑅)
151, 14, 3rngodir 36393 . . . 4 ((𝑅 ∈ RingOps ∧ (𝑍𝑋𝑍𝑋𝐴𝑋)) → ((𝑍𝐺𝑍)𝐻𝐴) = ((𝑍𝐻𝐴)𝐺(𝑍𝐻𝐴)))
1613, 15syldan 592 . . 3 ((𝑅 ∈ RingOps ∧ 𝐴𝑋) → ((𝑍𝐺𝑍)𝐻𝐴) = ((𝑍𝐻𝐴)𝐺(𝑍𝐻𝐴)))
172adantr 482 . . . 4 ((𝑅 ∈ RingOps ∧ 𝐴𝑋) → 𝐺 ∈ GrpOp)
18 simpl 484 . . . . 5 ((𝑅 ∈ RingOps ∧ 𝐴𝑋) → 𝑅 ∈ RingOps)
191, 14, 3rngocl 36389 . . . . 5 ((𝑅 ∈ RingOps ∧ 𝑍𝑋𝐴𝑋) → (𝑍𝐻𝐴) ∈ 𝑋)
2018, 11, 12, 19syl3anc 1372 . . . 4 ((𝑅 ∈ RingOps ∧ 𝐴𝑋) → (𝑍𝐻𝐴) ∈ 𝑋)
213, 4grporid 29501 . . . . 5 ((𝐺 ∈ GrpOp ∧ (𝑍𝐻𝐴) ∈ 𝑋) → ((𝑍𝐻𝐴)𝐺𝑍) = (𝑍𝐻𝐴))
2221eqcomd 2743 . . . 4 ((𝐺 ∈ GrpOp ∧ (𝑍𝐻𝐴) ∈ 𝑋) → (𝑍𝐻𝐴) = ((𝑍𝐻𝐴)𝐺𝑍))
2317, 20, 22syl2anc 585 . . 3 ((𝑅 ∈ RingOps ∧ 𝐴𝑋) → (𝑍𝐻𝐴) = ((𝑍𝐻𝐴)𝐺𝑍))
249, 16, 233eqtr3d 2785 . 2 ((𝑅 ∈ RingOps ∧ 𝐴𝑋) → ((𝑍𝐻𝐴)𝐺(𝑍𝐻𝐴)) = ((𝑍𝐻𝐴)𝐺𝑍))
253grpolcan 29514 . . 3 ((𝐺 ∈ GrpOp ∧ ((𝑍𝐻𝐴) ∈ 𝑋𝑍𝑋 ∧ (𝑍𝐻𝐴) ∈ 𝑋)) → (((𝑍𝐻𝐴)𝐺(𝑍𝐻𝐴)) = ((𝑍𝐻𝐴)𝐺𝑍) ↔ (𝑍𝐻𝐴) = 𝑍))
2617, 20, 11, 20, 25syl13anc 1373 . 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 397  w3a 1088   = wceq 1542  wcel 2107  ran crn 5639  cfv 6501  (class class class)co 7362  1st c1st 7924  2nd c2nd 7925  GrpOpcgr 29473  GIdcgi 29474  RingOpscrngo 36382
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2109  ax-9 2117  ax-10 2138  ax-11 2155  ax-12 2172  ax-ext 2708  ax-rep 5247  ax-sep 5261  ax-nul 5268  ax-pr 5389  ax-un 7677
This theorem depends on definitions:  df-bi 206  df-an 398  df-or 847  df-3an 1090  df-tru 1545  df-fal 1555  df-ex 1783  df-nf 1787  df-sb 2069  df-mo 2539  df-eu 2568  df-clab 2715  df-cleq 2729  df-clel 2815  df-nfc 2890  df-ne 2945  df-ral 3066  df-rex 3075  df-reu 3357  df-rab 3411  df-v 3450  df-sbc 3745  df-csb 3861  df-dif 3918  df-un 3920  df-in 3922  df-ss 3932  df-nul 4288  df-if 4492  df-sn 4592  df-pr 4594  df-op 4598  df-uni 4871  df-iun 4961  df-br 5111  df-opab 5173  df-mpt 5194  df-id 5536  df-xp 5644  df-rel 5645  df-cnv 5646  df-co 5647  df-dm 5648  df-rn 5649  df-res 5650  df-ima 5651  df-iota 6453  df-fun 6503  df-fn 6504  df-f 6505  df-f1 6506  df-fo 6507  df-f1o 6508  df-fv 6509  df-riota 7318  df-ov 7365  df-1st 7926  df-2nd 7927  df-grpo 29477  df-gid 29478  df-ginv 29479  df-ablo 29529  df-rngo 36383
This theorem is referenced by:  rngonegmn1l  36429  isdrngo3  36447  0idl  36513  keridl  36520  prnc  36555
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