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Theorem rngodm1dm2 37405
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 37383 . . 3 (𝑅 ∈ RingOps → 𝐺 ∈ GrpOp)
3 eqid 2728 . . . 4 ran 𝐺 = ran 𝐺
43grpofo 30322 . . 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 37373 . 2 (𝑅 ∈ RingOps → 𝐻:(ran 𝐺 × ran 𝐺)⟶ran 𝐺)
8 fof 6811 . . . 4 (𝐺:(ran 𝐺 × ran 𝐺)–onto→ran 𝐺𝐺:(ran 𝐺 × ran 𝐺)⟶ran 𝐺)
98fdmd 6733 . . 3 (𝐺:(ran 𝐺 × ran 𝐺)–onto→ran 𝐺 → dom 𝐺 = (ran 𝐺 × ran 𝐺))
10 fdm 6731 . . . 4 (𝐻:(ran 𝐺 × ran 𝐺)⟶ran 𝐺 → dom 𝐻 = (ran 𝐺 × ran 𝐺))
11 eqtr 2751 . . . . . . 7 ((dom 𝐺 = (ran 𝐺 × ran 𝐺) ∧ (ran 𝐺 × ran 𝐺) = dom 𝐻) → dom 𝐺 = dom 𝐻)
1211dmeqd 5908 . . . . . 6 ((dom 𝐺 = (ran 𝐺 × ran 𝐺) ∧ (ran 𝐺 × ran 𝐺) = dom 𝐻) → dom dom 𝐺 = dom dom 𝐻)
1312expcom 413 . . . . 5 ((ran 𝐺 × ran 𝐺) = dom 𝐻 → (dom 𝐺 = (ran 𝐺 × ran 𝐺) → dom dom 𝐺 = dom dom 𝐻))
1413eqcoms 2736 . . . 4 (dom 𝐻 = (ran 𝐺 × ran 𝐺) → (dom 𝐺 = (ran 𝐺 × ran 𝐺) → dom dom 𝐺 = dom dom 𝐻))
1510, 14syl 17 . . 3 (𝐻:(ran 𝐺 × ran 𝐺)⟶ran 𝐺 → (dom 𝐺 = (ran 𝐺 × ran 𝐺) → dom dom 𝐺 = dom dom 𝐻))
169, 15syl5com 31 . 2 (𝐺:(ran 𝐺 × ran 𝐺)–onto→ran 𝐺 → (𝐻:(ran 𝐺 × ran 𝐺)⟶ran 𝐺 → dom dom 𝐺 = dom dom 𝐻))
175, 7, 16sylc 65 1 (𝑅 ∈ RingOps → dom dom 𝐺 = dom dom 𝐻)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 395   = wceq 1534  wcel 2099   × cxp 5676  dom cdm 5678  ran crn 5679  wf 6544  ontowfo 6546  cfv 6548  1st c1st 7991  2nd c2nd 7992  GrpOpcgr 30312  RingOpscrngo 37367
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1790  ax-4 1804  ax-5 1906  ax-6 1964  ax-7 2004  ax-8 2101  ax-9 2109  ax-10 2130  ax-11 2147  ax-12 2167  ax-ext 2699  ax-sep 5299  ax-nul 5306  ax-pr 5429  ax-un 7740
This theorem depends on definitions:  df-bi 206  df-an 396  df-or 847  df-3an 1087  df-tru 1537  df-fal 1547  df-ex 1775  df-nf 1779  df-sb 2061  df-mo 2530  df-eu 2559  df-clab 2706  df-cleq 2720  df-clel 2806  df-nfc 2881  df-ne 2938  df-ral 3059  df-rex 3068  df-rab 3430  df-v 3473  df-sbc 3777  df-csb 3893  df-dif 3950  df-un 3952  df-in 3954  df-ss 3964  df-nul 4324  df-if 4530  df-sn 4630  df-pr 4632  df-op 4636  df-uni 4909  df-iun 4998  df-br 5149  df-opab 5211  df-mpt 5232  df-id 5576  df-xp 5684  df-rel 5685  df-cnv 5686  df-co 5687  df-dm 5688  df-rn 5689  df-iota 6500  df-fun 6550  df-fn 6551  df-f 6552  df-fo 6554  df-fv 6556  df-ov 7423  df-1st 7993  df-2nd 7994  df-grpo 30316  df-ablo 30368  df-rngo 37368
This theorem is referenced by:  rngorn1  37406
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