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Theorem rngodm1dm2 37458
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 37436 . . 3 (𝑅 ∈ RingOps β†’ 𝐺 ∈ GrpOp)
3 eqid 2725 . . . 4 ran 𝐺 = ran 𝐺
43grpofo 30348 . . 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 37426 . 2 (𝑅 ∈ RingOps β†’ 𝐻:(ran 𝐺 Γ— ran 𝐺)⟢ran 𝐺)
8 fof 6804 . . . 4 (𝐺:(ran 𝐺 Γ— ran 𝐺)–ontoβ†’ran 𝐺 β†’ 𝐺:(ran 𝐺 Γ— ran 𝐺)⟢ran 𝐺)
98fdmd 6727 . . 3 (𝐺:(ran 𝐺 Γ— ran 𝐺)–ontoβ†’ran 𝐺 β†’ dom 𝐺 = (ran 𝐺 Γ— ran 𝐺))
10 fdm 6726 . . . 4 (𝐻:(ran 𝐺 Γ— ran 𝐺)⟢ran 𝐺 β†’ dom 𝐻 = (ran 𝐺 Γ— ran 𝐺))
11 eqtr 2748 . . . . . . 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 412 . . . . 5 ((ran 𝐺 Γ— ran 𝐺) = dom 𝐻 β†’ (dom 𝐺 = (ran 𝐺 Γ— ran 𝐺) β†’ dom dom 𝐺 = dom dom 𝐻))
1413eqcoms 2733 . . . 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 394   = wceq 1533   ∈ wcel 2098   Γ— cxp 5671  dom cdm 5673  ran crn 5674  βŸΆwf 6539  β€“ontoβ†’wfo 6541  β€˜cfv 6543  1st c1st 7985  2nd c2nd 7986  GrpOpcgr 30338  RingOpscrngo 37420
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 2696  ax-sep 5295  ax-nul 5302  ax-pr 5424  ax-un 7735
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 2528  df-eu 2557  df-clab 2703  df-cleq 2717  df-clel 2802  df-nfc 2877  df-ne 2931  df-ral 3052  df-rex 3061  df-rab 3420  df-v 3465  df-sbc 3771  df-csb 3887  df-dif 3944  df-un 3946  df-in 3948  df-ss 3958  df-nul 4320  df-if 4526  df-sn 4626  df-pr 4628  df-op 4632  df-uni 4905  df-iun 4994  df-br 5145  df-opab 5207  df-mpt 5228  df-id 5571  df-xp 5679  df-rel 5680  df-cnv 5681  df-co 5682  df-dm 5683  df-rn 5684  df-iota 6495  df-fun 6545  df-fn 6546  df-f 6547  df-fo 6549  df-fv 6551  df-ov 7416  df-1st 7987  df-2nd 7988  df-grpo 30342  df-ablo 30394  df-rngo 37421
This theorem is referenced by:  rngorn1  37459
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