Theorem rngoid 38929

Description: Obsolete theorem, use ringid 20508 instead. The multiplication operation of a unital ring has (one or more) identity elements. (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
rngoid	⊢ ((𝑅 ∈ RingOps ∧ 𝐴 ∈ 𝑋) → ∃𝑢 ∈ 𝑋 ((𝑢𝐻𝐴) = 𝐴 ∧ (𝐴𝐻𝑢) = 𝐴))

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

Proof of Theorem rngoid

Dummy variables 𝑥 𝑦 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		ringi.1	. . . . 5 ⊢ 𝐺 = (1st ‘𝑅)
2		ringi.2	. . . . 5 ⊢ 𝐻 = (2nd ‘𝑅)
3		ringi.3	. . . . 5 ⊢ 𝑋 = ran 𝐺
4	1, 2, 3	rngoi 38926	. . . 4 ⊢ (𝑅 ∈ RingOps → ((𝐺 ∈ AbelOp ∧ 𝐻:(𝑋 × 𝑋)⟶𝑋) ∧ (∀𝑢 ∈ 𝑋 ∀𝑥 ∈ 𝑋 ∀𝑦 ∈ 𝑋 (((𝑢𝐻𝑥)𝐻𝑦) = (𝑢𝐻(𝑥𝐻𝑦)) ∧ (𝑢𝐻(𝑥𝐺𝑦)) = ((𝑢𝐻𝑥)𝐺(𝑢𝐻𝑦)) ∧ ((𝑢𝐺𝑥)𝐻𝑦) = ((𝑢𝐻𝑦)𝐺(𝑥𝐻𝑦))) ∧ ∃𝑢 ∈ 𝑋 ∀𝑥 ∈ 𝑋 ((𝑢𝐻𝑥) = 𝑥 ∧ (𝑥𝐻𝑢) = 𝑥))))
5	4	simprrd 786	. . 3 ⊢ (𝑅 ∈ RingOps → ∃𝑢 ∈ 𝑋 ∀𝑥 ∈ 𝑋 ((𝑢𝐻𝑥) = 𝑥 ∧ (𝑥𝐻𝑢) = 𝑥))
6		r19.12 3311	. . 3 ⊢ (∃𝑢 ∈ 𝑋 ∀𝑥 ∈ 𝑋 ((𝑢𝐻𝑥) = 𝑥 ∧ (𝑥𝐻𝑢) = 𝑥) → ∀𝑥 ∈ 𝑋 ∃𝑢 ∈ 𝑋 ((𝑢𝐻𝑥) = 𝑥 ∧ (𝑥𝐻𝑢) = 𝑥))
7	5, 6	syl 18	. 2 ⊢ (𝑅 ∈ RingOps → ∀𝑥 ∈ 𝑋 ∃𝑢 ∈ 𝑋 ((𝑢𝐻𝑥) = 𝑥 ∧ (𝑥𝐻𝑢) = 𝑥))
8		oveq2 7428	. . . . . 6 ⊢ (𝑥 = 𝐴 → (𝑢𝐻𝑥) = (𝑢𝐻𝐴))
9		id 23	. . . . . 6 ⊢ (𝑥 = 𝐴 → 𝑥 = 𝐴)
10	8, 9	eqeq12d 2776	. . . . 5 ⊢ (𝑥 = 𝐴 → ((𝑢𝐻𝑥) = 𝑥 ↔ (𝑢𝐻𝐴) = 𝐴))
11		oveq1 7427	. . . . . 6 ⊢ (𝑥 = 𝐴 → (𝑥𝐻𝑢) = (𝐴𝐻𝑢))
12	11, 9	eqeq12d 2776	. . . . 5 ⊢ (𝑥 = 𝐴 → ((𝑥𝐻𝑢) = 𝑥 ↔ (𝐴𝐻𝑢) = 𝐴))
13	10, 12	anbi12d 644	. . . 4 ⊢ (𝑥 = 𝐴 → (((𝑢𝐻𝑥) = 𝑥 ∧ (𝑥𝐻𝑢) = 𝑥) ↔ ((𝑢𝐻𝐴) = 𝐴 ∧ (𝐴𝐻𝑢) = 𝐴)))
14	13	rexbidv 3186	. . 3 ⊢ (𝑥 = 𝐴 → (∃𝑢 ∈ 𝑋 ((𝑢𝐻𝑥) = 𝑥 ∧ (𝑥𝐻𝑢) = 𝑥) ↔ ∃𝑢 ∈ 𝑋 ((𝑢𝐻𝐴) = 𝐴 ∧ (𝐴𝐻𝑢) = 𝐴)))
15	14	rspccva 3575	. 2 ⊢ ((∀𝑥 ∈ 𝑋 ∃𝑢 ∈ 𝑋 ((𝑢𝐻𝑥) = 𝑥 ∧ (𝑥𝐻𝑢) = 𝑥) ∧ 𝐴 ∈ 𝑋) → ∃𝑢 ∈ 𝑋 ((𝑢𝐻𝐴) = 𝐴 ∧ (𝐴𝐻𝑢) = 𝐴))
16	7, 15	sylan 592	1 ⊢ ((𝑅 ∈ RingOps ∧ 𝐴 ∈ 𝑋) → ∃𝑢 ∈ 𝑋 ((𝑢𝐻𝐴) = 𝐴 ∧ (𝐴𝐻𝑢) = 𝐴))

This proof depends on syntax axioms:   → wi 4   ∧ wa 401   ∧ w3a 1103   = wceq 1569   ∈ wcel 2144  ∀wral 3076  ∃wrex 3086   × cxp 5648  ran crn 5651  ⟶wf 6534  ‘cfv 6538  (class class class)co 7420  1^st c1st 7999  2^nd c2nd 8000  AbelOpcablo 31151  RingOpscrngo 38921

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1827  ax-4 1841  ax-5 1942  ax-6 1999  ax-7 2040  ax-8 2146  ax-9 2154  ax-10 2177  ax-11 2193  ax-12 2212  ax-ext 2732  ax-sep 5248  ax-nul 5259  ax-pr 5390  ax-un 7751

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3an 1105  df-tru 1572  df-fal 1582  df-ex 1812  df-nf 1816  df-sb 2099  df-mo 2564  df-eu 2594  df-clab 2739  df-cleq 2752  df-clel 2835  df-nfc 2909  df-ne 2956  df-ral 3077  df-rex 3087  df-rab 3413  df-v 3452  df-ss 3896  df-dif 4047  df-in 4091  df-un 4151  df-nul 4279  df-if 4482  df-sn 4585  df-pr 4587  df-op 4835  df-uni 4867  df-br 5103  df-opab 5167  df-mpt 5186  df-id 5545  df-xp 5656  df-rel 5657  df-cnv 5658  df-co 5659  df-dm 5660  df-rn 5661  df-iota 6494  df-fun 6540  df-fn 6541  df-f 6542  df-fv 6546  df-ov 7423  df-1st 8001  df-2nd 8002  df-rngo 38922

This theorem is used by:  rngo2  38934