Theorem rngoideu 37911

Description: The unity element of a ring is unique. (Contributed by NM, 4-Apr-2009.) (Revised by Mario Carneiro, 21-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
rngoideu	⊢ (𝑅 ∈ RingOps → ∃!𝑢 ∈ 𝑋 ∀𝑥 ∈ 𝑋 ((𝑢𝐻𝑥) = 𝑥 ∧ (𝑥𝐻𝑢) = 𝑥))

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

Proof of Theorem rngoideu

Dummy variable 𝑦 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		ringi.1	. . . 4 ⊢ 𝐺 = (1st ‘𝑅)
2		ringi.2	. . . 4 ⊢ 𝐻 = (2nd ‘𝑅)
3		ringi.3	. . . 4 ⊢ 𝑋 = ran 𝐺
4	1, 2, 3	rngoi 37907	. . 3 ⊢ (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 773	. 2 ⊢ (𝑅 ∈ RingOps → ∃𝑢 ∈ 𝑋 ∀𝑥 ∈ 𝑋 ((𝑢𝐻𝑥) = 𝑥 ∧ (𝑥𝐻𝑢) = 𝑥))
6		simpl 482	. . . . . . 7 ⊢ (((𝑢𝐻𝑥) = 𝑥 ∧ (𝑥𝐻𝑢) = 𝑥) → (𝑢𝐻𝑥) = 𝑥)
7	6	ralimi 3082	. . . . . 6 ⊢ (∀𝑥 ∈ 𝑋 ((𝑢𝐻𝑥) = 𝑥 ∧ (𝑥𝐻𝑢) = 𝑥) → ∀𝑥 ∈ 𝑋 (𝑢𝐻𝑥) = 𝑥)
8		oveq2 7440	. . . . . . . 8 ⊢ (𝑥 = 𝑦 → (𝑢𝐻𝑥) = (𝑢𝐻𝑦))
9		id 22	. . . . . . . 8 ⊢ (𝑥 = 𝑦 → 𝑥 = 𝑦)
10	8, 9	eqeq12d 2752	. . . . . . 7 ⊢ (𝑥 = 𝑦 → ((𝑢𝐻𝑥) = 𝑥 ↔ (𝑢𝐻𝑦) = 𝑦))
11	10	rspcv 3617	. . . . . 6 ⊢ (𝑦 ∈ 𝑋 → (∀𝑥 ∈ 𝑋 (𝑢𝐻𝑥) = 𝑥 → (𝑢𝐻𝑦) = 𝑦))
12	7, 11	syl5 34	. . . . 5 ⊢ (𝑦 ∈ 𝑋 → (∀𝑥 ∈ 𝑋 ((𝑢𝐻𝑥) = 𝑥 ∧ (𝑥𝐻𝑢) = 𝑥) → (𝑢𝐻𝑦) = 𝑦))
13		simpr 484	. . . . . . 7 ⊢ (((𝑦𝐻𝑥) = 𝑥 ∧ (𝑥𝐻𝑦) = 𝑥) → (𝑥𝐻𝑦) = 𝑥)
14	13	ralimi 3082	. . . . . 6 ⊢ (∀𝑥 ∈ 𝑋 ((𝑦𝐻𝑥) = 𝑥 ∧ (𝑥𝐻𝑦) = 𝑥) → ∀𝑥 ∈ 𝑋 (𝑥𝐻𝑦) = 𝑥)
15		oveq1 7439	. . . . . . . 8 ⊢ (𝑥 = 𝑢 → (𝑥𝐻𝑦) = (𝑢𝐻𝑦))
16		id 22	. . . . . . . 8 ⊢ (𝑥 = 𝑢 → 𝑥 = 𝑢)
17	15, 16	eqeq12d 2752	. . . . . . 7 ⊢ (𝑥 = 𝑢 → ((𝑥𝐻𝑦) = 𝑥 ↔ (𝑢𝐻𝑦) = 𝑢))
18	17	rspcv 3617	. . . . . 6 ⊢ (𝑢 ∈ 𝑋 → (∀𝑥 ∈ 𝑋 (𝑥𝐻𝑦) = 𝑥 → (𝑢𝐻𝑦) = 𝑢))
19	14, 18	syl5 34	. . . . 5 ⊢ (𝑢 ∈ 𝑋 → (∀𝑥 ∈ 𝑋 ((𝑦𝐻𝑥) = 𝑥 ∧ (𝑥𝐻𝑦) = 𝑥) → (𝑢𝐻𝑦) = 𝑢))
20	12, 19	im2anan9r 621	. . . 4 ⊢ ((𝑢 ∈ 𝑋 ∧ 𝑦 ∈ 𝑋) → ((∀𝑥 ∈ 𝑋 ((𝑢𝐻𝑥) = 𝑥 ∧ (𝑥𝐻𝑢) = 𝑥) ∧ ∀𝑥 ∈ 𝑋 ((𝑦𝐻𝑥) = 𝑥 ∧ (𝑥𝐻𝑦) = 𝑥)) → ((𝑢𝐻𝑦) = 𝑦 ∧ (𝑢𝐻𝑦) = 𝑢)))
21		eqtr2 2760	. . . . 5 ⊢ (((𝑢𝐻𝑦) = 𝑦 ∧ (𝑢𝐻𝑦) = 𝑢) → 𝑦 = 𝑢)
22	21	equcomd 2017	. . . 4 ⊢ (((𝑢𝐻𝑦) = 𝑦 ∧ (𝑢𝐻𝑦) = 𝑢) → 𝑢 = 𝑦)
23	20, 22	syl6 35	. . 3 ⊢ ((𝑢 ∈ 𝑋 ∧ 𝑦 ∈ 𝑋) → ((∀𝑥 ∈ 𝑋 ((𝑢𝐻𝑥) = 𝑥 ∧ (𝑥𝐻𝑢) = 𝑥) ∧ ∀𝑥 ∈ 𝑋 ((𝑦𝐻𝑥) = 𝑥 ∧ (𝑥𝐻𝑦) = 𝑥)) → 𝑢 = 𝑦))
24	23	rgen2 3198	. 2 ⊢ ∀𝑢 ∈ 𝑋 ∀𝑦 ∈ 𝑋 ((∀𝑥 ∈ 𝑋 ((𝑢𝐻𝑥) = 𝑥 ∧ (𝑥𝐻𝑢) = 𝑥) ∧ ∀𝑥 ∈ 𝑋 ((𝑦𝐻𝑥) = 𝑥 ∧ (𝑥𝐻𝑦) = 𝑥)) → 𝑢 = 𝑦)
25		oveq1 7439	. . . . 5 ⊢ (𝑢 = 𝑦 → (𝑢𝐻𝑥) = (𝑦𝐻𝑥))
26	25	eqeq1d 2738	. . . 4 ⊢ (𝑢 = 𝑦 → ((𝑢𝐻𝑥) = 𝑥 ↔ (𝑦𝐻𝑥) = 𝑥))
27	26	ovanraleqv 7456	. . 3 ⊢ (𝑢 = 𝑦 → (∀𝑥 ∈ 𝑋 ((𝑢𝐻𝑥) = 𝑥 ∧ (𝑥𝐻𝑢) = 𝑥) ↔ ∀𝑥 ∈ 𝑋 ((𝑦𝐻𝑥) = 𝑥 ∧ (𝑥𝐻𝑦) = 𝑥)))
28	27	reu4 3736	. 2 ⊢ (∃!𝑢 ∈ 𝑋 ∀𝑥 ∈ 𝑋 ((𝑢𝐻𝑥) = 𝑥 ∧ (𝑥𝐻𝑢) = 𝑥) ↔ (∃𝑢 ∈ 𝑋 ∀𝑥 ∈ 𝑋 ((𝑢𝐻𝑥) = 𝑥 ∧ (𝑥𝐻𝑢) = 𝑥) ∧ ∀𝑢 ∈ 𝑋 ∀𝑦 ∈ 𝑋 ((∀𝑥 ∈ 𝑋 ((𝑢𝐻𝑥) = 𝑥 ∧ (𝑥𝐻𝑢) = 𝑥) ∧ ∀𝑥 ∈ 𝑋 ((𝑦𝐻𝑥) = 𝑥 ∧ (𝑥𝐻𝑦) = 𝑥)) → 𝑢 = 𝑦)))
29	5, 24, 28	sylanblrc 590	1 ⊢ (𝑅 ∈ RingOps → ∃!𝑢 ∈ 𝑋 ∀𝑥 ∈ 𝑋 ((𝑢𝐻𝑥) = 𝑥 ∧ (𝑥𝐻𝑢) = 𝑥))

Syntax hints:   → wi 4   ∧ wa 395   ∧ w3a 1086   = wceq 1539   ∈ wcel 2107  ∀wral 3060  ∃wrex 3069  ∃!wreu 3377   × cxp 5682  ran crn 5685  ⟶wf 6556  ‘cfv 6560  (class class class)co 7432  1^st c1st 8013  2^nd c2nd 8014  AbelOpcablo 30564  RingOpscrngo 37902

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1794  ax-4 1808  ax-5 1909  ax-6 1966  ax-7 2006  ax-8 2109  ax-9 2117  ax-10 2140  ax-11 2156  ax-12 2176  ax-ext 2707  ax-sep 5295  ax-nul 5305  ax-pr 5431  ax-un 7756

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1542  df-fal 1552  df-ex 1779  df-nf 1783  df-sb 2064  df-mo 2539  df-eu 2568  df-clab 2714  df-cleq 2728  df-clel 2815  df-nfc 2891  df-ne 2940  df-ral 3061  df-rex 3070  df-rmo 3379  df-reu 3380  df-rab 3436  df-v 3481  df-dif 3953  df-un 3955  df-in 3957  df-ss 3967  df-nul 4333  df-if 4525  df-sn 4626  df-pr 4628  df-op 4632  df-uni 4907  df-br 5143  df-opab 5205  df-mpt 5225  df-id 5577  df-xp 5690  df-rel 5691  df-cnv 5692  df-co 5693  df-dm 5694  df-rn 5695  df-iota 6513  df-fun 6562  df-fn 6563  df-f 6564  df-fv 6568  df-ov 7435  df-1st 8015  df-2nd 8016  df-rngo 37903

This theorem is referenced by: (None)