Users' Mathboxes Mathbox for Jeff Madsen < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >   Mathboxes  >  rngodm1dm2 Structured version   Visualization version   GIF version
Theorem rngodm1dm2 36788
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 = (2nd β€˜π‘…)
rnplrnml0.2 𝐺 = (1st β€˜π‘…)
Assertion
Ref Expression
rngodm1dm2 (𝑅 ∈ RingOps β†’ dom dom 𝐺 = dom dom 𝐻)
Proof of Theorem rngodm1dm2
StepHypRef Expression
1 rnplrnml0.2 . . . 4 𝐺 = (1st β€˜π‘…)
21rngogrpo 36766 . . 3 (𝑅 ∈ RingOps β†’ 𝐺 ∈ GrpOp)
3 eqid 2732 . . . 4 ran 𝐺 = ran 𝐺
43grpofo 29739 . . 3 (𝐺 ∈ GrpOp β†’ 𝐺:(ran 𝐺 Γ— ran 𝐺)–ontoβ†’ran 𝐺)
52, 4syl 17 . 2 (𝑅 ∈ RingOps β†’ 𝐺:(ran 𝐺 Γ— ran 𝐺)–ontoβ†’ran 𝐺)
6 rnplrnml0.1 . . 3 𝐻 = (2nd β€˜π‘…)
71, 6, 3rngosm 36756 . 2 (𝑅 ∈ RingOps β†’ 𝐻:(ran 𝐺 Γ— ran 𝐺)⟢ran 𝐺)
8 fof 6802 . . . 4 (𝐺:(ran 𝐺 Γ— ran 𝐺)–ontoβ†’ran 𝐺 β†’ 𝐺:(ran 𝐺 Γ— ran 𝐺)⟢ran 𝐺)
98fdmd 6725 . . 3 (𝐺:(ran 𝐺 Γ— ran 𝐺)–ontoβ†’ran 𝐺 β†’ dom 𝐺 = (ran 𝐺 Γ— ran 𝐺))
10 fdm 6723 . . . 4 (𝐻:(ran 𝐺 Γ— ran 𝐺)⟢ran 𝐺 β†’ dom 𝐻 = (ran 𝐺 Γ— ran 𝐺))
11 eqtr 2755 . . . . . . 7 ((dom 𝐺 = (ran 𝐺 Γ— ran 𝐺) ∧ (ran 𝐺 Γ— ran 𝐺) = dom 𝐻) β†’ dom 𝐺 = dom 𝐻)
1211dmeqd 5903 . . . . . 6 ((dom 𝐺 = (ran 𝐺 Γ— ran 𝐺) ∧ (ran 𝐺 Γ— ran 𝐺) = dom 𝐻) β†’ dom dom 𝐺 = dom dom 𝐻)
1312expcom 414 . . . . 5 ((ran 𝐺 Γ— ran 𝐺) = dom 𝐻 β†’ (dom 𝐺 = (ran 𝐺 Γ— ran 𝐺) β†’ dom dom 𝐺 = dom dom 𝐻))
1413eqcoms 2740 . . . 4 (dom 𝐻 = (ran 𝐺 Γ— ran 𝐺) β†’ (dom 𝐺 = (ran 𝐺 Γ— ran 𝐺) β†’ dom dom 𝐺 = dom dom 𝐻))
1510, 14syl 17 . . 3 (𝐻:(ran 𝐺 Γ— ran 𝐺)⟢ran 𝐺 β†’ (dom 𝐺 = (ran 𝐺 Γ— ran 𝐺) β†’ dom dom 𝐺 = dom dom 𝐻))
169, 15syl5com 31 . 2 (𝐺:(ran 𝐺 Γ— ran 𝐺)–ontoβ†’ran 𝐺 β†’ (𝐻:(ran 𝐺 Γ— ran 𝐺)⟢ran 𝐺 β†’ dom dom 𝐺 = dom dom 𝐻))
175, 7, 16sylc 65 1 (𝑅 ∈ RingOps β†’ dom dom 𝐺 = dom dom 𝐻)
Colors of variables: wff setvar class
Syntax hints:   β†’ wi 4   ∧ wa 396   = wceq 1541   ∈ wcel 2106   Γ— cxp 5673  dom cdm 5675  ran crn 5676  βŸΆwf 6536  β€“ontoβ†’wfo 6538  β€˜cfv 6540  1st c1st 7969  2nd c2nd 7970  GrpOpcgr 29729  RingOpscrngo 36750
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  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 2703  ax-sep 5298  ax-nul 5305  ax-pr 5426  ax-un 7721
This theorem depends on definitions:  df-bi 206  df-an 397  df-or 846  df-3an 1089  df-tru 1544  df-fal 1554  df-ex 1782  df-nf 1786  df-sb 2068  df-mo 2534  df-eu 2563  df-clab 2710  df-cleq 2724  df-clel 2810  df-nfc 2885  df-ne 2941  df-ral 3062  df-rex 3071  df-rab 3433  df-v 3476  df-sbc 3777  df-csb 3893  df-dif 3950  df-un 3952  df-in 3954  df-ss 3964  df-nul 4322  df-if 4528  df-sn 4628  df-pr 4630  df-op 4634  df-uni 4908  df-iun 4998  df-br 5148  df-opab 5210  df-mpt 5231  df-id 5573  df-xp 5681  df-rel 5682  df-cnv 5683  df-co 5684  df-dm 5685  df-rn 5686  df-iota 6492  df-fun 6542  df-fn 6543  df-f 6544  df-fo 6546  df-fv 6548  df-ov 7408  df-1st 7971  df-2nd 7972  df-grpo 29733  df-ablo 29785  df-rngo 36751
This theorem is referenced by:  rngorn1  36789
  Copyright terms: Public domain W3C validator