Theorem rngoideu 35183

Description: The unit 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 35179	. . 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 772	. 2 ⊢ (𝑅 ∈ RingOps → ∃𝑢 ∈ 𝑋 ∀𝑥 ∈ 𝑋 ((𝑢𝐻𝑥) = 𝑥 ∧ (𝑥𝐻𝑢) = 𝑥))
6		simpl 485	. . . . . . 7 ⊢ (((𝑢𝐻𝑥) = 𝑥 ∧ (𝑥𝐻𝑢) = 𝑥) → (𝑢𝐻𝑥) = 𝑥)
7	6	ralimi 3162	. . . . . 6 ⊢ (∀𝑥 ∈ 𝑋 ((𝑢𝐻𝑥) = 𝑥 ∧ (𝑥𝐻𝑢) = 𝑥) → ∀𝑥 ∈ 𝑋 (𝑢𝐻𝑥) = 𝑥)
8		oveq2 7166	. . . . . . . 8 ⊢ (𝑥 = 𝑦 → (𝑢𝐻𝑥) = (𝑢𝐻𝑦))
9		id 22	. . . . . . . 8 ⊢ (𝑥 = 𝑦 → 𝑥 = 𝑦)
10	8, 9	eqeq12d 2839	. . . . . . 7 ⊢ (𝑥 = 𝑦 → ((𝑢𝐻𝑥) = 𝑥 ↔ (𝑢𝐻𝑦) = 𝑦))
11	10	rspcv 3620	. . . . . 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 487	. . . . . . 7 ⊢ (((𝑦𝐻𝑥) = 𝑥 ∧ (𝑥𝐻𝑦) = 𝑥) → (𝑥𝐻𝑦) = 𝑥)
14	13	ralimi 3162	. . . . . 6 ⊢ (∀𝑥 ∈ 𝑋 ((𝑦𝐻𝑥) = 𝑥 ∧ (𝑥𝐻𝑦) = 𝑥) → ∀𝑥 ∈ 𝑋 (𝑥𝐻𝑦) = 𝑥)
15		oveq1 7165	. . . . . . . 8 ⊢ (𝑥 = 𝑢 → (𝑥𝐻𝑦) = (𝑢𝐻𝑦))
16		id 22	. . . . . . . 8 ⊢ (𝑥 = 𝑢 → 𝑥 = 𝑢)
17	15, 16	eqeq12d 2839	. . . . . . 7 ⊢ (𝑥 = 𝑢 → ((𝑥𝐻𝑦) = 𝑥 ↔ (𝑢𝐻𝑦) = 𝑢))
18	17	rspcv 3620	. . . . . 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 622	. . . 4 ⊢ ((𝑢 ∈ 𝑋 ∧ 𝑦 ∈ 𝑋) → ((∀𝑥 ∈ 𝑋 ((𝑢𝐻𝑥) = 𝑥 ∧ (𝑥𝐻𝑢) = 𝑥) ∧ ∀𝑥 ∈ 𝑋 ((𝑦𝐻𝑥) = 𝑥 ∧ (𝑥𝐻𝑦) = 𝑥)) → ((𝑢𝐻𝑦) = 𝑦 ∧ (𝑢𝐻𝑦) = 𝑢)))
21		eqtr2 2844	. . . . 5 ⊢ (((𝑢𝐻𝑦) = 𝑦 ∧ (𝑢𝐻𝑦) = 𝑢) → 𝑦 = 𝑢)
22	21	equcomd 2026	. . . 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 3205	. 2 ⊢ ∀𝑢 ∈ 𝑋 ∀𝑦 ∈ 𝑋 ((∀𝑥 ∈ 𝑋 ((𝑢𝐻𝑥) = 𝑥 ∧ (𝑥𝐻𝑢) = 𝑥) ∧ ∀𝑥 ∈ 𝑋 ((𝑦𝐻𝑥) = 𝑥 ∧ (𝑥𝐻𝑦) = 𝑥)) → 𝑢 = 𝑦)
25		oveq1 7165	. . . . 5 ⊢ (𝑢 = 𝑦 → (𝑢𝐻𝑥) = (𝑦𝐻𝑥))
26	25	eqeq1d 2825	. . . 4 ⊢ (𝑢 = 𝑦 → ((𝑢𝐻𝑥) = 𝑥 ↔ (𝑦𝐻𝑥) = 𝑥))
27	26	ovanraleqv 7182	. . 3 ⊢ (𝑢 = 𝑦 → (∀𝑥 ∈ 𝑋 ((𝑢𝐻𝑥) = 𝑥 ∧ (𝑥𝐻𝑢) = 𝑥) ↔ ∀𝑥 ∈ 𝑋 ((𝑦𝐻𝑥) = 𝑥 ∧ (𝑥𝐻𝑦) = 𝑥)))
28	27	reu4 3724	. 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 592	1 ⊢ (𝑅 ∈ RingOps → ∃!𝑢 ∈ 𝑋 ∀𝑥 ∈ 𝑋 ((𝑢𝐻𝑥) = 𝑥 ∧ (𝑥𝐻𝑢) = 𝑥))

Syntax hints:   → wi 4   ∧ wa 398   ∧ w3a 1083   = wceq 1537   ∈ wcel 2114  ∀wral 3140  ∃wrex 3141  ∃!wreu 3142   × 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-reu 3147  df-rmo 3148  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: (None)