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Theorem rngodm1dm2 36437
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 36415 . . 3 (𝑅 ∈ RingOps β†’ 𝐺 ∈ GrpOp)
3 eqid 2733 . . . 4 ran 𝐺 = ran 𝐺
43grpofo 29483 . . 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 36405 . 2 (𝑅 ∈ RingOps β†’ 𝐻:(ran 𝐺 Γ— ran 𝐺)⟢ran 𝐺)
8 fof 6757 . . . 4 (𝐺:(ran 𝐺 Γ— ran 𝐺)–ontoβ†’ran 𝐺 β†’ 𝐺:(ran 𝐺 Γ— ran 𝐺)⟢ran 𝐺)
98fdmd 6680 . . 3 (𝐺:(ran 𝐺 Γ— ran 𝐺)–ontoβ†’ran 𝐺 β†’ dom 𝐺 = (ran 𝐺 Γ— ran 𝐺))
10 fdm 6678 . . . 4 (𝐻:(ran 𝐺 Γ— ran 𝐺)⟢ran 𝐺 β†’ dom 𝐻 = (ran 𝐺 Γ— ran 𝐺))
11 eqtr 2756 . . . . . . 7 ((dom 𝐺 = (ran 𝐺 Γ— ran 𝐺) ∧ (ran 𝐺 Γ— ran 𝐺) = dom 𝐻) β†’ dom 𝐺 = dom 𝐻)
1211dmeqd 5862 . . . . . 6 ((dom 𝐺 = (ran 𝐺 Γ— ran 𝐺) ∧ (ran 𝐺 Γ— ran 𝐺) = dom 𝐻) β†’ dom dom 𝐺 = dom dom 𝐻)
1312expcom 415 . . . . 5 ((ran 𝐺 Γ— ran 𝐺) = dom 𝐻 β†’ (dom 𝐺 = (ran 𝐺 Γ— ran 𝐺) β†’ dom dom 𝐺 = dom dom 𝐻))
1413eqcoms 2741 . . . 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 397   = wceq 1542   ∈ wcel 2107   Γ— cxp 5632  dom cdm 5634  ran crn 5635  βŸΆwf 6493  β€“ontoβ†’wfo 6495  β€˜cfv 6497  1st c1st 7920  2nd c2nd 7921  GrpOpcgr 29473  RingOpscrngo 36399
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 1914  ax-6 1972  ax-7 2012  ax-8 2109  ax-9 2117  ax-10 2138  ax-11 2155  ax-12 2172  ax-ext 2704  ax-sep 5257  ax-nul 5264  ax-pr 5385  ax-un 7673
This theorem depends on definitions:  df-bi 206  df-an 398  df-or 847  df-3an 1090  df-tru 1545  df-fal 1555  df-ex 1783  df-nf 1787  df-sb 2069  df-mo 2535  df-eu 2564  df-clab 2711  df-cleq 2725  df-clel 2811  df-nfc 2886  df-ne 2941  df-ral 3062  df-rex 3071  df-rab 3407  df-v 3446  df-sbc 3741  df-csb 3857  df-dif 3914  df-un 3916  df-in 3918  df-ss 3928  df-nul 4284  df-if 4488  df-sn 4588  df-pr 4590  df-op 4594  df-uni 4867  df-iun 4957  df-br 5107  df-opab 5169  df-mpt 5190  df-id 5532  df-xp 5640  df-rel 5641  df-cnv 5642  df-co 5643  df-dm 5644  df-rn 5645  df-iota 6449  df-fun 6499  df-fn 6500  df-f 6501  df-fo 6503  df-fv 6505  df-ov 7361  df-1st 7922  df-2nd 7923  df-grpo 29477  df-ablo 29529  df-rngo 36400
This theorem is referenced by:  rngorn1  36438
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