Theorem rngo2 38665

Description: Obsolete theorem, use ringadd2 20423 instead. 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.) (Proof modification is discouraged.)

Hypotheses

Ref	Expression
ringi.1	⊢ 𝐺 = (1st ‘𝑅)
ringi.2	⊢ 𝐻 = (2nd ‘𝑅)
ringi.3	⊢ 𝑋 = ran 𝐺

Assertion

Ref	Expression
rngo2	⊢ ((𝑅 ∈ RingOps ∧ 𝐴 ∈ 𝑋) → ∃𝑥 ∈ 𝑋 (𝐴𝐺𝐴) = ((𝑥𝐺𝑥)𝐻𝐴))

Distinct variable groups:   𝑥,𝐺   𝑥,𝐻   𝑥,𝑋   𝑥,𝐴   𝑥,𝑅

Proof of Theorem rngo2

Step	Hyp	Ref	Expression
1		ringi.1	. . 3 ⊢ 𝐺 = (1st ‘𝑅)
2		ringi.2	. . 3 ⊢ 𝐻 = (2nd ‘𝑅)
3		ringi.3	. . 3 ⊢ 𝑋 = ran 𝐺
4	1, 2, 3	rngoid 38660	. 2 ⊢ ((𝑅 ∈ RingOps ∧ 𝐴 ∈ 𝑋) → ∃𝑥 ∈ 𝑋 ((𝑥𝐻𝐴) = 𝐴 ∧ (𝐴𝐻𝑥) = 𝐴))
5		oveq12 7426	. . . . . . 7 ⊢ (((𝑥𝐻𝐴) = 𝐴 ∧ (𝑥𝐻𝐴) = 𝐴) → ((𝑥𝐻𝐴)𝐺(𝑥𝐻𝐴)) = (𝐴𝐺𝐴))
6	5	anidms 577	. . . . . 6 ⊢ ((𝑥𝐻𝐴) = 𝐴 → ((𝑥𝐻𝐴)𝐺(𝑥𝐻𝐴)) = (𝐴𝐺𝐴))
7	6	eqcomd 2768	. . . . 5 ⊢ ((𝑥𝐻𝐴) = 𝐴 → (𝐴𝐺𝐴) = ((𝑥𝐻𝐴)𝐺(𝑥𝐻𝐴)))
8		simpll 779	. . . . . . 7 ⊢ (((𝑅 ∈ RingOps ∧ 𝐴 ∈ 𝑋) ∧ 𝑥 ∈ 𝑋) → 𝑅 ∈ RingOps)
9		simpr 490	. . . . . . 7 ⊢ (((𝑅 ∈ RingOps ∧ 𝐴 ∈ 𝑋) ∧ 𝑥 ∈ 𝑋) → 𝑥 ∈ 𝑋)
10		simplr 781	. . . . . . 7 ⊢ (((𝑅 ∈ RingOps ∧ 𝐴 ∈ 𝑋) ∧ 𝑥 ∈ 𝑋) → 𝐴 ∈ 𝑋)
11	1, 2, 3	rngodir 38663	. . . . . . 7 ⊢ ((𝑅 ∈ RingOps ∧ (𝑥 ∈ 𝑋 ∧ 𝑥 ∈ 𝑋 ∧ 𝐴 ∈ 𝑋)) → ((𝑥𝐺𝑥)𝐻𝐴) = ((𝑥𝐻𝐴)𝐺(𝑥𝐻𝐴)))
12	8, 9, 9, 10, 11	syl13anc 1399	. . . . . 6 ⊢ (((𝑅 ∈ RingOps ∧ 𝐴 ∈ 𝑋) ∧ 𝑥 ∈ 𝑋) → ((𝑥𝐺𝑥)𝐻𝐴) = ((𝑥𝐻𝐴)𝐺(𝑥𝐻𝐴)))
13	12	eqeq2d 2773	. . . . 5 ⊢ (((𝑅 ∈ RingOps ∧ 𝐴 ∈ 𝑋) ∧ 𝑥 ∈ 𝑋) → ((𝐴𝐺𝐴) = ((𝑥𝐺𝑥)𝐻𝐴) ↔ (𝐴𝐺𝐴) = ((𝑥𝐻𝐴)𝐺(𝑥𝐻𝐴))))
14	7, 13	imbitrrid 249	. . . 4 ⊢ (((𝑅 ∈ RingOps ∧ 𝐴 ∈ 𝑋) ∧ 𝑥 ∈ 𝑋) → ((𝑥𝐻𝐴) = 𝐴 → (𝐴𝐺𝐴) = ((𝑥𝐺𝑥)𝐻𝐴)))
15	14	adantrd 497	. . 3 ⊢ (((𝑅 ∈ RingOps ∧ 𝐴 ∈ 𝑋) ∧ 𝑥 ∈ 𝑋) → (((𝑥𝐻𝐴) = 𝐴 ∧ (𝐴𝐻𝑥) = 𝐴) → (𝐴𝐺𝐴) = ((𝑥𝐺𝑥)𝐻𝐴)))
16	15	reximdva 3177	. 2 ⊢ ((𝑅 ∈ RingOps ∧ 𝐴 ∈ 𝑋) → (∃𝑥 ∈ 𝑋 ((𝑥𝐻𝐴) = 𝐴 ∧ (𝐴𝐻𝑥) = 𝐴) → ∃𝑥 ∈ 𝑋 (𝐴𝐺𝐴) = ((𝑥𝐺𝑥)𝐻𝐴)))
17	4, 16	mpd 16	1 ⊢ ((𝑅 ∈ RingOps ∧ 𝐴 ∈ 𝑋) → ∃𝑥 ∈ 𝑋 (𝐴𝐺𝐴) = ((𝑥𝐺𝑥)𝐻𝐴))

This proof depends on syntax axioms:   → wi 4   ∧ wa 401   = wceq 1570   ∈ wcel 2145  ∃wrex 3088  ran crn 5660  ‘cfv 6537  (class class class)co 7417  1^st c1st 7988  2^nd c2nd 7989  RingOpscrngo 38652

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1828  ax-4 1842  ax-5 1943  ax-6 2000  ax-7 2041  ax-8 2147  ax-9 2155  ax-10 2178  ax-11 2194  ax-12 2215  ax-ext 2734  ax-sep 5255  ax-nul 5267  ax-pr 5402  ax-un 7740

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3an 1105  df-tru 1573  df-fal 1583  df-ex 1813  df-nf 1817  df-sb 2100  df-mo 2566  df-eu 2596  df-clab 2741  df-cleq 2754  df-clel 2837  df-nfc 2911  df-ne 2958  df-ral 3079  df-rex 3089  df-rab 3415  df-v 3455  df-dif 3905  df-un 3907  df-in 3909  df-ss 3919  df-nul 4283  df-if 4486  df-sn 4588  df-pr 4590  df-op 4594  df-uni 4871  df-br 5108  df-opab 5172  df-mpt 5191  df-id 5554  df-xp 5665  df-rel 5666  df-cnv 5667  df-co 5668  df-dm 5669  df-rn 5670  df-iota 6493  df-fun 6539  df-fn 6540  df-f 6541  df-fv 6545  df-ov 7420  df-1st 7990  df-2nd 7991  df-rngo 38653

This theorem is used by: (None)