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Theorem rngodm1dm2 37313
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 37291 . . 3 (𝑅 ∈ RingOps β†’ 𝐺 ∈ GrpOp)
3 eqid 2726 . . . 4 ran 𝐺 = ran 𝐺
43grpofo 30261 . . 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 37281 . 2 (𝑅 ∈ RingOps β†’ 𝐻:(ran 𝐺 Γ— ran 𝐺)⟢ran 𝐺)
8 fof 6799 . . . 4 (𝐺:(ran 𝐺 Γ— ran 𝐺)–ontoβ†’ran 𝐺 β†’ 𝐺:(ran 𝐺 Γ— ran 𝐺)⟢ran 𝐺)
98fdmd 6722 . . 3 (𝐺:(ran 𝐺 Γ— ran 𝐺)–ontoβ†’ran 𝐺 β†’ dom 𝐺 = (ran 𝐺 Γ— ran 𝐺))
10 fdm 6720 . . . 4 (𝐻:(ran 𝐺 Γ— ran 𝐺)⟢ran 𝐺 β†’ dom 𝐻 = (ran 𝐺 Γ— ran 𝐺))
11 eqtr 2749 . . . . . . 7 ((dom 𝐺 = (ran 𝐺 Γ— ran 𝐺) ∧ (ran 𝐺 Γ— ran 𝐺) = dom 𝐻) β†’ dom 𝐺 = dom 𝐻)
1211dmeqd 5899 . . . . . 6 ((dom 𝐺 = (ran 𝐺 Γ— ran 𝐺) ∧ (ran 𝐺 Γ— ran 𝐺) = dom 𝐻) β†’ dom dom 𝐺 = dom dom 𝐻)
1312expcom 413 . . . . 5 ((ran 𝐺 Γ— ran 𝐺) = dom 𝐻 β†’ (dom 𝐺 = (ran 𝐺 Γ— ran 𝐺) β†’ dom dom 𝐺 = dom dom 𝐻))
1413eqcoms 2734 . . . 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 395   = wceq 1533   ∈ wcel 2098   Γ— cxp 5667  dom cdm 5669  ran crn 5670  βŸΆwf 6533  β€“ontoβ†’wfo 6535  β€˜cfv 6537  1st c1st 7972  2nd c2nd 7973  GrpOpcgr 30251  RingOpscrngo 37275
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 2163  ax-ext 2697  ax-sep 5292  ax-nul 5299  ax-pr 5420  ax-un 7722
This theorem depends on definitions:  df-bi 206  df-an 396  df-or 845  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 2704  df-cleq 2718  df-clel 2804  df-nfc 2879  df-ne 2935  df-ral 3056  df-rex 3065  df-rab 3427  df-v 3470  df-sbc 3773  df-csb 3889  df-dif 3946  df-un 3948  df-in 3950  df-ss 3960  df-nul 4318  df-if 4524  df-sn 4624  df-pr 4626  df-op 4630  df-uni 4903  df-iun 4992  df-br 5142  df-opab 5204  df-mpt 5225  df-id 5567  df-xp 5675  df-rel 5676  df-cnv 5677  df-co 5678  df-dm 5679  df-rn 5680  df-iota 6489  df-fun 6539  df-fn 6540  df-f 6541  df-fo 6543  df-fv 6545  df-ov 7408  df-1st 7974  df-2nd 7975  df-grpo 30255  df-ablo 30307  df-rngo 37276
This theorem is referenced by:  rngorn1  37314
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