Theorem rngoid 35182

Description: 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.)

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 35179	. . . 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 772	. . 3 ⊢ (𝑅 ∈ RingOps → ∃𝑢 ∈ 𝑋 ∀𝑥 ∈ 𝑋 ((𝑢𝐻𝑥) = 𝑥 ∧ (𝑥𝐻𝑢) = 𝑥))
6		r19.12 3326	. . 3 ⊢ (∃𝑢 ∈ 𝑋 ∀𝑥 ∈ 𝑋 ((𝑢𝐻𝑥) = 𝑥 ∧ (𝑥𝐻𝑢) = 𝑥) → ∀𝑥 ∈ 𝑋 ∃𝑢 ∈ 𝑋 ((𝑢𝐻𝑥) = 𝑥 ∧ (𝑥𝐻𝑢) = 𝑥))
7	5, 6	syl 17	. 2 ⊢ (𝑅 ∈ RingOps → ∀𝑥 ∈ 𝑋 ∃𝑢 ∈ 𝑋 ((𝑢𝐻𝑥) = 𝑥 ∧ (𝑥𝐻𝑢) = 𝑥))
8		oveq2 7166	. . . . . 6 ⊢ (𝑥 = 𝐴 → (𝑢𝐻𝑥) = (𝑢𝐻𝐴))
9		id 22	. . . . . 6 ⊢ (𝑥 = 𝐴 → 𝑥 = 𝐴)
10	8, 9	eqeq12d 2839	. . . . 5 ⊢ (𝑥 = 𝐴 → ((𝑢𝐻𝑥) = 𝑥 ↔ (𝑢𝐻𝐴) = 𝐴))
11		oveq1 7165	. . . . . 6 ⊢ (𝑥 = 𝐴 → (𝑥𝐻𝑢) = (𝐴𝐻𝑢))
12	11, 9	eqeq12d 2839	. . . . 5 ⊢ (𝑥 = 𝐴 → ((𝑥𝐻𝑢) = 𝑥 ↔ (𝐴𝐻𝑢) = 𝐴))
13	10, 12	anbi12d 632	. . . 4 ⊢ (𝑥 = 𝐴 → (((𝑢𝐻𝑥) = 𝑥 ∧ (𝑥𝐻𝑢) = 𝑥) ↔ ((𝑢𝐻𝐴) = 𝐴 ∧ (𝐴𝐻𝑢) = 𝐴)))
14	13	rexbidv 3299	. . 3 ⊢ (𝑥 = 𝐴 → (∃𝑢 ∈ 𝑋 ((𝑢𝐻𝑥) = 𝑥 ∧ (𝑥𝐻𝑢) = 𝑥) ↔ ∃𝑢 ∈ 𝑋 ((𝑢𝐻𝐴) = 𝐴 ∧ (𝐴𝐻𝑢) = 𝐴)))
15	14	rspccva 3624	. 2 ⊢ ((∀𝑥 ∈ 𝑋 ∃𝑢 ∈ 𝑋 ((𝑢𝐻𝑥) = 𝑥 ∧ (𝑥𝐻𝑢) = 𝑥) ∧ 𝐴 ∈ 𝑋) → ∃𝑢 ∈ 𝑋 ((𝑢𝐻𝐴) = 𝐴 ∧ (𝐴𝐻𝑢) = 𝐴))
16	7, 15	sylan 582	1 ⊢ ((𝑅 ∈ RingOps ∧ 𝐴 ∈ 𝑋) → ∃𝑢 ∈ 𝑋 ((𝑢𝐻𝐴) = 𝐴 ∧ (𝐴𝐻𝑢) = 𝐴))

Syntax hints:   → wi 4   ∧ wa 398   ∧ w3a 1083   = wceq 1537   ∈ wcel 2114  ∀wral 3140  ∃wrex 3141   × cxp 5555  ran crn 5558  ⟶wf 6353  ‘cfv 6357  (class class class)co 7158  1^st c1st 7689  2^nd c2nd 7690  AbelOpcablo 28323  RingOpscrngo 35174

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1911  ax-6 1970  ax-7 2015  ax-8 2116  ax-9 2124  ax-10 2145  ax-11 2161  ax-12 2177  ax-ext 2795  ax-sep 5205  ax-nul 5212  ax-pow 5268  ax-pr 5332  ax-un 7463

This theorem depends on definitions:  df-bi 209  df-an 399  df-or 844  df-3an 1085  df-tru 1540  df-ex 1781  df-nf 1785  df-sb 2070  df-mo 2622  df-eu 2654  df-clab 2802  df-cleq 2816  df-clel 2895  df-nfc 2965  df-ral 3145  df-rex 3146  df-rab 3149  df-v 3498  df-sbc 3775  df-dif 3941  df-un 3943  df-in 3945  df-ss 3954  df-nul 4294  df-if 4470  df-sn 4570  df-pr 4572  df-op 4576  df-uni 4841  df-br 5069  df-opab 5131  df-mpt 5149  df-id 5462  df-xp 5563  df-rel 5564  df-cnv 5565  df-co 5566  df-dm 5567  df-rn 5568  df-iota 6316  df-fun 6359  df-fn 6360  df-f 6361  df-fv 6365  df-ov 7161  df-1st 7691  df-2nd 7692  df-rngo 35175

This theorem is referenced by:  rngo2  35187