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Theorem rngorn1 34041
Description: In a unital ring the range of the addition equals the domain of the first variable of the multiplication. (Contributed by FL, 24-Jan-2010.) (New usage is discouraged.)
Hypotheses
Ref Expression
rnplrnml0.1 𝐻 = (2nd𝑅)
rnplrnml0.2 𝐺 = (1st𝑅)
Assertion
Ref Expression
rngorn1 (𝑅 ∈ RingOps → ran 𝐺 = dom dom 𝐻)

Proof of Theorem rngorn1
StepHypRef Expression
1 rnplrnml0.2 . . . 4 𝐺 = (1st𝑅)
21rngogrpo 34018 . . 3 (𝑅 ∈ RingOps → 𝐺 ∈ GrpOp)
3 grporndm 27669 . . 3 (𝐺 ∈ GrpOp → ran 𝐺 = dom dom 𝐺)
42, 3syl 17 . 2 (𝑅 ∈ RingOps → ran 𝐺 = dom dom 𝐺)
5 rnplrnml0.1 . . 3 𝐻 = (2nd𝑅)
65, 1rngodm1dm2 34040 . 2 (𝑅 ∈ RingOps → dom dom 𝐺 = dom dom 𝐻)
74, 6eqtrd 2790 1 (𝑅 ∈ RingOps → ran 𝐺 = dom dom 𝐻)
Colors of variables: wff setvar class
Syntax hints:  wi 4   = wceq 1628  wcel 2135  dom cdm 5262  ran crn 5263  cfv 6045  1st c1st 7327  2nd c2nd 7328  GrpOpcgr 27648  RingOpscrngo 34002
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1867  ax-4 1882  ax-5 1984  ax-6 2050  ax-7 2086  ax-8 2137  ax-9 2144  ax-10 2164  ax-11 2179  ax-12 2192  ax-13 2387  ax-ext 2736  ax-sep 4929  ax-nul 4937  ax-pow 4988  ax-pr 5051  ax-un 7110
This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-3an 1074  df-tru 1631  df-ex 1850  df-nf 1855  df-sb 2043  df-eu 2607  df-mo 2608  df-clab 2743  df-cleq 2749  df-clel 2752  df-nfc 2887  df-ne 2929  df-ral 3051  df-rex 3052  df-rab 3055  df-v 3338  df-sbc 3573  df-csb 3671  df-dif 3714  df-un 3716  df-in 3718  df-ss 3725  df-nul 4055  df-if 4227  df-sn 4318  df-pr 4320  df-op 4324  df-uni 4585  df-iun 4670  df-br 4801  df-opab 4861  df-mpt 4878  df-id 5170  df-xp 5268  df-rel 5269  df-cnv 5270  df-co 5271  df-dm 5272  df-rn 5273  df-iota 6008  df-fun 6047  df-fn 6048  df-f 6049  df-fo 6051  df-fv 6053  df-ov 6812  df-1st 7329  df-2nd 7330  df-grpo 27652  df-ablo 27704  df-rngo 34003
This theorem is referenced by:  rngomndo  34043
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