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Theorem rngorn1 36088
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 36065 . . 3 (𝑅 ∈ RingOps → 𝐺 ∈ GrpOp)
3 grporndm 28869 . . 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 36087 . 2 (𝑅 ∈ RingOps → dom dom 𝐺 = dom dom 𝐻)
74, 6eqtrd 2778 1 (𝑅 ∈ RingOps → ran 𝐺 = dom dom 𝐻)
Colors of variables: wff setvar class
Syntax hints:  wi 4   = wceq 1539  wcel 2106  dom cdm 5591  ran crn 5592  cfv 6435  1st c1st 7829  2nd c2nd 7830  GrpOpcgr 28848  RingOpscrngo 36049
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 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-10 2137  ax-11 2154  ax-12 2171  ax-ext 2709  ax-sep 5225  ax-nul 5232  ax-pr 5354  ax-un 7588
This theorem depends on definitions:  df-bi 206  df-an 397  df-or 845  df-3an 1088  df-tru 1542  df-fal 1552  df-ex 1783  df-nf 1787  df-sb 2068  df-mo 2540  df-eu 2569  df-clab 2716  df-cleq 2730  df-clel 2816  df-nfc 2889  df-ne 2944  df-ral 3069  df-rex 3070  df-rab 3073  df-v 3433  df-sbc 3718  df-csb 3834  df-dif 3891  df-un 3893  df-in 3895  df-ss 3905  df-nul 4259  df-if 4462  df-sn 4564  df-pr 4566  df-op 4570  df-uni 4842  df-iun 4928  df-br 5077  df-opab 5139  df-mpt 5160  df-id 5491  df-xp 5597  df-rel 5598  df-cnv 5599  df-co 5600  df-dm 5601  df-rn 5602  df-iota 6393  df-fun 6437  df-fn 6438  df-f 6439  df-fo 6441  df-fv 6443  df-ov 7280  df-1st 7831  df-2nd 7832  df-grpo 28852  df-ablo 28904  df-rngo 36050
This theorem is referenced by:  rngomndo  36090
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