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Theorem rngo2 36416
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 36411 . 2 ((𝑅 ∈ RingOps ∧ 𝐴𝑋) → ∃𝑥𝑋 ((𝑥𝐻𝐴) = 𝐴 ∧ (𝐴𝐻𝑥) = 𝐴))
5 oveq12 7370 . . . . . . 7 (((𝑥𝐻𝐴) = 𝐴 ∧ (𝑥𝐻𝐴) = 𝐴) → ((𝑥𝐻𝐴)𝐺(𝑥𝐻𝐴)) = (𝐴𝐺𝐴))
65anidms 568 . . . . . 6 ((𝑥𝐻𝐴) = 𝐴 → ((𝑥𝐻𝐴)𝐺(𝑥𝐻𝐴)) = (𝐴𝐺𝐴))
76eqcomd 2739 . . . . 5 ((𝑥𝐻𝐴) = 𝐴 → (𝐴𝐺𝐴) = ((𝑥𝐻𝐴)𝐺(𝑥𝐻𝐴)))
8 simpll 766 . . . . . . 7 (((𝑅 ∈ RingOps ∧ 𝐴𝑋) ∧ 𝑥𝑋) → 𝑅 ∈ RingOps)
9 simpr 486 . . . . . . 7 (((𝑅 ∈ RingOps ∧ 𝐴𝑋) ∧ 𝑥𝑋) → 𝑥𝑋)
10 simplr 768 . . . . . . 7 (((𝑅 ∈ RingOps ∧ 𝐴𝑋) ∧ 𝑥𝑋) → 𝐴𝑋)
111, 2, 3rngodir 36414 . . . . . . 7 ((𝑅 ∈ RingOps ∧ (𝑥𝑋𝑥𝑋𝐴𝑋)) → ((𝑥𝐺𝑥)𝐻𝐴) = ((𝑥𝐻𝐴)𝐺(𝑥𝐻𝐴)))
128, 9, 9, 10, 11syl13anc 1373 . . . . . 6 (((𝑅 ∈ RingOps ∧ 𝐴𝑋) ∧ 𝑥𝑋) → ((𝑥𝐺𝑥)𝐻𝐴) = ((𝑥𝐻𝐴)𝐺(𝑥𝐻𝐴)))
1312eqeq2d 2744 . . . . 5 (((𝑅 ∈ RingOps ∧ 𝐴𝑋) ∧ 𝑥𝑋) → ((𝐴𝐺𝐴) = ((𝑥𝐺𝑥)𝐻𝐴) ↔ (𝐴𝐺𝐴) = ((𝑥𝐻𝐴)𝐺(𝑥𝐻𝐴))))
147, 13syl5ibr 246 . . . 4 (((𝑅 ∈ RingOps ∧ 𝐴𝑋) ∧ 𝑥𝑋) → ((𝑥𝐻𝐴) = 𝐴 → (𝐴𝐺𝐴) = ((𝑥𝐺𝑥)𝐻𝐴)))
1514adantrd 493 . . 3 (((𝑅 ∈ RingOps ∧ 𝐴𝑋) ∧ 𝑥𝑋) → (((𝑥𝐻𝐴) = 𝐴 ∧ (𝐴𝐻𝑥) = 𝐴) → (𝐴𝐺𝐴) = ((𝑥𝐺𝑥)𝐻𝐴)))
1615reximdva 3162 . 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 397   = wceq 1542  wcel 2107  wrex 3070  ran crn 5638  cfv 6500  (class class class)co 7361  1st c1st 7923  2nd c2nd 7924  RingOpscrngo 36403
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 5260  ax-nul 5267  ax-pr 5388  ax-un 7676
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 3449  df-dif 3917  df-un 3919  df-in 3921  df-ss 3931  df-nul 4287  df-if 4491  df-sn 4591  df-pr 4593  df-op 4597  df-uni 4870  df-br 5110  df-opab 5172  df-mpt 5193  df-id 5535  df-xp 5643  df-rel 5644  df-cnv 5645  df-co 5646  df-dm 5647  df-rn 5648  df-iota 6452  df-fun 6502  df-fn 6503  df-f 6504  df-fv 6508  df-ov 7364  df-1st 7925  df-2nd 7926  df-rngo 36404
This theorem is referenced by: (None)
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