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Theorem rngodm1dm2 32901
 Description: In a unital ring the domain of the first variable 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
rngodm1dm2 (𝑅 ∈ RingOps → dom dom 𝐺 = dom dom 𝐻)

Proof of Theorem rngodm1dm2
StepHypRef Expression
1 rnplrnml0.2 . . . 4 𝐺 = (1st𝑅)
21rngogrpo 32879 . . 3 (𝑅 ∈ RingOps → 𝐺 ∈ GrpOp)
3 eqid 2610 . . . 4 ran 𝐺 = ran 𝐺
43grpofo 26737 . . 3 (𝐺 ∈ GrpOp → 𝐺:(ran 𝐺 × ran 𝐺)–onto→ran 𝐺)
52, 4syl 17 . 2 (𝑅 ∈ RingOps → 𝐺:(ran 𝐺 × ran 𝐺)–onto→ran 𝐺)
6 rnplrnml0.1 . . 3 𝐻 = (2nd𝑅)
71, 6, 3rngosm 32869 . 2 (𝑅 ∈ RingOps → 𝐻:(ran 𝐺 × ran 𝐺)⟶ran 𝐺)
8 fof 6028 . . . 4 (𝐺:(ran 𝐺 × ran 𝐺)–onto→ran 𝐺𝐺:(ran 𝐺 × ran 𝐺)⟶ran 𝐺)
9 fdm 5964 . . . 4 (𝐺:(ran 𝐺 × ran 𝐺)⟶ran 𝐺 → dom 𝐺 = (ran 𝐺 × ran 𝐺))
108, 9syl 17 . . 3 (𝐺:(ran 𝐺 × ran 𝐺)–onto→ran 𝐺 → dom 𝐺 = (ran 𝐺 × ran 𝐺))
11 fdm 5964 . . . 4 (𝐻:(ran 𝐺 × ran 𝐺)⟶ran 𝐺 → dom 𝐻 = (ran 𝐺 × ran 𝐺))
12 eqtr 2629 . . . . . . 7 ((dom 𝐺 = (ran 𝐺 × ran 𝐺) ∧ (ran 𝐺 × ran 𝐺) = dom 𝐻) → dom 𝐺 = dom 𝐻)
1312dmeqd 5248 . . . . . 6 ((dom 𝐺 = (ran 𝐺 × ran 𝐺) ∧ (ran 𝐺 × ran 𝐺) = dom 𝐻) → dom dom 𝐺 = dom dom 𝐻)
1413expcom 450 . . . . 5 ((ran 𝐺 × ran 𝐺) = dom 𝐻 → (dom 𝐺 = (ran 𝐺 × ran 𝐺) → dom dom 𝐺 = dom dom 𝐻))
1514eqcoms 2618 . . . 4 (dom 𝐻 = (ran 𝐺 × ran 𝐺) → (dom 𝐺 = (ran 𝐺 × ran 𝐺) → dom dom 𝐺 = dom dom 𝐻))
1611, 15syl 17 . . 3 (𝐻:(ran 𝐺 × ran 𝐺)⟶ran 𝐺 → (dom 𝐺 = (ran 𝐺 × ran 𝐺) → dom dom 𝐺 = dom dom 𝐻))
1710, 16syl5com 31 . 2 (𝐺:(ran 𝐺 × ran 𝐺)–onto→ran 𝐺 → (𝐻:(ran 𝐺 × ran 𝐺)⟶ran 𝐺 → dom dom 𝐺 = dom dom 𝐻))
185, 7, 17sylc 63 1 (𝑅 ∈ RingOps → dom dom 𝐺 = dom dom 𝐻)
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ∧ wa 383   = wceq 1475   ∈ wcel 1977   × cxp 5036  dom cdm 5038  ran crn 5039  ⟶wf 5800  –onto→wfo 5802  ‘cfv 5804  1st c1st 7057  2nd c2nd 7058  GrpOpcgr 26727  RingOpscrngo 32863 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1713  ax-4 1728  ax-5 1827  ax-6 1875  ax-7 1922  ax-8 1979  ax-9 1986  ax-10 2006  ax-11 2021  ax-12 2034  ax-13 2234  ax-ext 2590  ax-sep 4709  ax-nul 4717  ax-pow 4769  ax-pr 4833  ax-un 6847 This theorem depends on definitions:  df-bi 196  df-or 384  df-an 385  df-3an 1033  df-tru 1478  df-ex 1696  df-nf 1701  df-sb 1868  df-eu 2462  df-mo 2463  df-clab 2597  df-cleq 2603  df-clel 2606  df-nfc 2740  df-ral 2901  df-rex 2902  df-rab 2905  df-v 3175  df-sbc 3403  df-csb 3500  df-dif 3543  df-un 3545  df-in 3547  df-ss 3554  df-nul 3875  df-if 4037  df-sn 4126  df-pr 4128  df-op 4132  df-uni 4373  df-iun 4457  df-br 4584  df-opab 4644  df-mpt 4645  df-id 4953  df-xp 5044  df-rel 5045  df-cnv 5046  df-co 5047  df-dm 5048  df-rn 5049  df-iota 5768  df-fun 5806  df-fn 5807  df-f 5808  df-fo 5810  df-fv 5812  df-ov 6552  df-1st 7059  df-2nd 7060  df-grpo 26731  df-ablo 26783  df-rngo 32864 This theorem is referenced by:  rngorn1  32902
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