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Theorem rngorn1 37411
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 37388 . . 3 (𝑅 ∈ RingOps → 𝐺 ∈ GrpOp)
3 grporndm 30338 . . 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 37410 . 2 (𝑅 ∈ RingOps → dom dom 𝐺 = dom dom 𝐻)
74, 6eqtrd 2767 1 (𝑅 ∈ RingOps → ran 𝐺 = dom dom 𝐻)
Colors of variables: wff setvar class
Syntax hints:  wi 4   = wceq 1533  wcel 2098  dom cdm 5680  ran crn 5681  cfv 6551  1st c1st 7995  2nd c2nd 7996  GrpOpcgr 30317  RingOpscrngo 37372
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1905  ax-6 1963  ax-7 2003  ax-8 2100  ax-9 2108  ax-10 2129  ax-11 2146  ax-12 2166  ax-ext 2698  ax-sep 5301  ax-nul 5308  ax-pr 5431  ax-un 7744
This theorem depends on definitions:  df-bi 206  df-an 395  df-or 846  df-3an 1086  df-tru 1536  df-fal 1546  df-ex 1774  df-nf 1778  df-sb 2060  df-mo 2529  df-eu 2558  df-clab 2705  df-cleq 2719  df-clel 2805  df-nfc 2880  df-ne 2937  df-ral 3058  df-rex 3067  df-rab 3429  df-v 3473  df-sbc 3777  df-csb 3893  df-dif 3950  df-un 3952  df-in 3954  df-ss 3964  df-nul 4325  df-if 4531  df-sn 4631  df-pr 4633  df-op 4637  df-uni 4911  df-iun 5000  df-br 5151  df-opab 5213  df-mpt 5234  df-id 5578  df-xp 5686  df-rel 5687  df-cnv 5688  df-co 5689  df-dm 5690  df-rn 5691  df-iota 6503  df-fun 6553  df-fn 6554  df-f 6555  df-fo 6557  df-fv 6559  df-ov 7427  df-1st 7997  df-2nd 7998  df-grpo 30321  df-ablo 30373  df-rngo 37373
This theorem is referenced by:  rngomndo  37413
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