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Theorem rngo2 36770
Description: A ring element plus itself is two times the element. (Contributed by Steve Rodriguez, 9-Sep-2007.) (Revised by Mario Carneiro, 22-Dec-2013.) (New usage is discouraged.)
Hypotheses
Ref Expression
ringi.1 𝐺 = (1st𝑅)
ringi.2 𝐻 = (2nd𝑅)
ringi.3 𝑋 = ran 𝐺
Assertion
Ref Expression
rngo2 ((𝑅 ∈ RingOps ∧ 𝐴𝑋) → ∃𝑥𝑋 (𝐴𝐺𝐴) = ((𝑥𝐺𝑥)𝐻𝐴))
Distinct variable groups:   𝑥,𝐺   𝑥,𝐻   𝑥,𝑋   𝑥,𝐴   𝑥,𝑅

Proof of Theorem rngo2
StepHypRef Expression
1 ringi.1 . . 3 𝐺 = (1st𝑅)
2 ringi.2 . . 3 𝐻 = (2nd𝑅)
3 ringi.3 . . 3 𝑋 = ran 𝐺
41, 2, 3rngoid 36765 . 2 ((𝑅 ∈ RingOps ∧ 𝐴𝑋) → ∃𝑥𝑋 ((𝑥𝐻𝐴) = 𝐴 ∧ (𝐴𝐻𝑥) = 𝐴))
5 oveq12 7417 . . . . . . 7 (((𝑥𝐻𝐴) = 𝐴 ∧ (𝑥𝐻𝐴) = 𝐴) → ((𝑥𝐻𝐴)𝐺(𝑥𝐻𝐴)) = (𝐴𝐺𝐴))
65anidms 567 . . . . . 6 ((𝑥𝐻𝐴) = 𝐴 → ((𝑥𝐻𝐴)𝐺(𝑥𝐻𝐴)) = (𝐴𝐺𝐴))
76eqcomd 2738 . . . . 5 ((𝑥𝐻𝐴) = 𝐴 → (𝐴𝐺𝐴) = ((𝑥𝐻𝐴)𝐺(𝑥𝐻𝐴)))
8 simpll 765 . . . . . . 7 (((𝑅 ∈ RingOps ∧ 𝐴𝑋) ∧ 𝑥𝑋) → 𝑅 ∈ RingOps)
9 simpr 485 . . . . . . 7 (((𝑅 ∈ RingOps ∧ 𝐴𝑋) ∧ 𝑥𝑋) → 𝑥𝑋)
10 simplr 767 . . . . . . 7 (((𝑅 ∈ RingOps ∧ 𝐴𝑋) ∧ 𝑥𝑋) → 𝐴𝑋)
111, 2, 3rngodir 36768 . . . . . . 7 ((𝑅 ∈ RingOps ∧ (𝑥𝑋𝑥𝑋𝐴𝑋)) → ((𝑥𝐺𝑥)𝐻𝐴) = ((𝑥𝐻𝐴)𝐺(𝑥𝐻𝐴)))
128, 9, 9, 10, 11syl13anc 1372 . . . . . 6 (((𝑅 ∈ RingOps ∧ 𝐴𝑋) ∧ 𝑥𝑋) → ((𝑥𝐺𝑥)𝐻𝐴) = ((𝑥𝐻𝐴)𝐺(𝑥𝐻𝐴)))
1312eqeq2d 2743 . . . . 5 (((𝑅 ∈ RingOps ∧ 𝐴𝑋) ∧ 𝑥𝑋) → ((𝐴𝐺𝐴) = ((𝑥𝐺𝑥)𝐻𝐴) ↔ (𝐴𝐺𝐴) = ((𝑥𝐻𝐴)𝐺(𝑥𝐻𝐴))))
147, 13imbitrrid 245 . . . 4 (((𝑅 ∈ RingOps ∧ 𝐴𝑋) ∧ 𝑥𝑋) → ((𝑥𝐻𝐴) = 𝐴 → (𝐴𝐺𝐴) = ((𝑥𝐺𝑥)𝐻𝐴)))
1514adantrd 492 . . 3 (((𝑅 ∈ RingOps ∧ 𝐴𝑋) ∧ 𝑥𝑋) → (((𝑥𝐻𝐴) = 𝐴 ∧ (𝐴𝐻𝑥) = 𝐴) → (𝐴𝐺𝐴) = ((𝑥𝐺𝑥)𝐻𝐴)))
1615reximdva 3168 . 2 ((𝑅 ∈ RingOps ∧ 𝐴𝑋) → (∃𝑥𝑋 ((𝑥𝐻𝐴) = 𝐴 ∧ (𝐴𝐻𝑥) = 𝐴) → ∃𝑥𝑋 (𝐴𝐺𝐴) = ((𝑥𝐺𝑥)𝐻𝐴)))
174, 16mpd 15 1 ((𝑅 ∈ RingOps ∧ 𝐴𝑋) → ∃𝑥𝑋 (𝐴𝐺𝐴) = ((𝑥𝐺𝑥)𝐻𝐴))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 396   = wceq 1541  wcel 2106  wrex 3070  ran crn 5677  cfv 6543  (class class class)co 7408  1st c1st 7972  2nd c2nd 7973  RingOpscrngo 36757
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 5299  ax-nul 5306  ax-pr 5427  ax-un 7724
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-dif 3951  df-un 3953  df-in 3955  df-ss 3965  df-nul 4323  df-if 4529  df-sn 4629  df-pr 4631  df-op 4635  df-uni 4909  df-br 5149  df-opab 5211  df-mpt 5232  df-id 5574  df-xp 5682  df-rel 5683  df-cnv 5684  df-co 5685  df-dm 5686  df-rn 5687  df-iota 6495  df-fun 6545  df-fn 6546  df-f 6547  df-fv 6551  df-ov 7411  df-1st 7974  df-2nd 7975  df-rngo 36758
This theorem is referenced by: (None)
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