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Theorem rngo2 37265
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 37260 . 2 ((𝑅 ∈ RingOps ∧ 𝐴𝑋) → ∃𝑥𝑋 ((𝑥𝐻𝐴) = 𝐴 ∧ (𝐴𝐻𝑥) = 𝐴))
5 oveq12 7410 . . . . . . 7 (((𝑥𝐻𝐴) = 𝐴 ∧ (𝑥𝐻𝐴) = 𝐴) → ((𝑥𝐻𝐴)𝐺(𝑥𝐻𝐴)) = (𝐴𝐺𝐴))
65anidms 566 . . . . . 6 ((𝑥𝐻𝐴) = 𝐴 → ((𝑥𝐻𝐴)𝐺(𝑥𝐻𝐴)) = (𝐴𝐺𝐴))
76eqcomd 2730 . . . . 5 ((𝑥𝐻𝐴) = 𝐴 → (𝐴𝐺𝐴) = ((𝑥𝐻𝐴)𝐺(𝑥𝐻𝐴)))
8 simpll 764 . . . . . . 7 (((𝑅 ∈ RingOps ∧ 𝐴𝑋) ∧ 𝑥𝑋) → 𝑅 ∈ RingOps)
9 simpr 484 . . . . . . 7 (((𝑅 ∈ RingOps ∧ 𝐴𝑋) ∧ 𝑥𝑋) → 𝑥𝑋)
10 simplr 766 . . . . . . 7 (((𝑅 ∈ RingOps ∧ 𝐴𝑋) ∧ 𝑥𝑋) → 𝐴𝑋)
111, 2, 3rngodir 37263 . . . . . . 7 ((𝑅 ∈ RingOps ∧ (𝑥𝑋𝑥𝑋𝐴𝑋)) → ((𝑥𝐺𝑥)𝐻𝐴) = ((𝑥𝐻𝐴)𝐺(𝑥𝐻𝐴)))
128, 9, 9, 10, 11syl13anc 1369 . . . . . 6 (((𝑅 ∈ RingOps ∧ 𝐴𝑋) ∧ 𝑥𝑋) → ((𝑥𝐺𝑥)𝐻𝐴) = ((𝑥𝐻𝐴)𝐺(𝑥𝐻𝐴)))
1312eqeq2d 2735 . . . . 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 491 . . 3 (((𝑅 ∈ RingOps ∧ 𝐴𝑋) ∧ 𝑥𝑋) → (((𝑥𝐻𝐴) = 𝐴 ∧ (𝐴𝐻𝑥) = 𝐴) → (𝐴𝐺𝐴) = ((𝑥𝐺𝑥)𝐻𝐴)))
1615reximdva 3160 . 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 395   = wceq 1533  wcel 2098  wrex 3062  ran crn 5667  cfv 6533  (class class class)co 7401  1st c1st 7966  2nd c2nd 7967  RingOpscrngo 37252
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 2695  ax-sep 5289  ax-nul 5296  ax-pr 5417  ax-un 7718
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 2526  df-eu 2555  df-clab 2702  df-cleq 2716  df-clel 2802  df-nfc 2877  df-ne 2933  df-ral 3054  df-rex 3063  df-rab 3425  df-v 3468  df-dif 3943  df-un 3945  df-in 3947  df-ss 3957  df-nul 4315  df-if 4521  df-sn 4621  df-pr 4623  df-op 4627  df-uni 4900  df-br 5139  df-opab 5201  df-mpt 5222  df-id 5564  df-xp 5672  df-rel 5673  df-cnv 5674  df-co 5675  df-dm 5676  df-rn 5677  df-iota 6485  df-fun 6535  df-fn 6536  df-f 6537  df-fv 6541  df-ov 7404  df-1st 7968  df-2nd 7969  df-rngo 37253
This theorem is referenced by: (None)
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