rngurd - Mathbox for Thierry Arnoux

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Theorem rngurd 29119

Description: Deduce the unit of a ring from its properties. (Contributed by Thierry Arnoux, 6-Sep-2016.)

**Hypotheses**
Ref	Expression
rngurd.b	⊢ (𝜑 → 𝐵 = (Base‘𝑅))
rngurd.p	⊢ (𝜑 → · = (._r‘𝑅))
rngurd.z	⊢ (𝜑 → 1 ∈ 𝐵)
rngurd.i	⊢ ((𝜑 ∧ 𝑥 ∈ 𝐵) → ( 1 · 𝑥) = 𝑥)
rngurd.j	⊢ ((𝜑 ∧ 𝑥 ∈ 𝐵) → (𝑥 · 1 ) = 𝑥)

**Assertion**
Ref	Expression
rngurd	⊢ (𝜑 → 1 = (1_r‘𝑅))

Distinct variable groups: 𝑥,𝐵 𝑥,𝑅 𝑥, 1 𝑥, · 𝜑,𝑥

Proof of Theorem rngurd Dummy variable 𝑒 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		eqid 2610	. . 3 ⊢ (Base‘𝑅) = (Base‘𝑅)
2		eqid 2610	. . 3 ⊢ (._r‘𝑅) = (._r‘𝑅)
3		eqid 2610	. . 3 ⊢ (1_r‘𝑅) = (1_r‘𝑅)
4	1, 2, 3	dfur2 18327	. 2 ⊢ (1_r‘𝑅) = (℩𝑒(𝑒 ∈ (Base‘𝑅) ∧ ∀𝑥 ∈ (Base‘𝑅)((𝑒(._r‘𝑅)𝑥) = 𝑥 ∧ (𝑥(._r‘𝑅)𝑒) = 𝑥)))
5		rngurd.z	. . . 4 ⊢ (𝜑 → 1 ∈ 𝐵)
6		rngurd.b	. . . 4 ⊢ (𝜑 → 𝐵 = (Base‘𝑅))
7	5, 6	eleqtrd 2690	. . 3 ⊢ (𝜑 → 1 ∈ (Base‘𝑅))
8		rngurd.i	. . . . . 6 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐵) → ( 1 · 𝑥) = 𝑥)
9		rngurd.j	. . . . . 6 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐵) → (𝑥 · 1 ) = 𝑥)
10	8, 9	jca 553	. . . . 5 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐵) → (( 1 · 𝑥) = 𝑥 ∧ (𝑥 · 1 ) = 𝑥))
11	10	ralrimiva 2949	. . . 4 ⊢ (𝜑 → ∀𝑥 ∈ 𝐵 (( 1 · 𝑥) = 𝑥 ∧ (𝑥 · 1 ) = 𝑥))
12		rngurd.p	. . . . . . . . 9 ⊢ (𝜑 → · = (._r‘𝑅))
13	12	adantr 480	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐵) → · = (._r‘𝑅))
14	13	oveqd 6566	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐵) → ( 1 · 𝑥) = ( 1 (._r‘𝑅)𝑥))
15	14	eqeq1d 2612	. . . . . 6 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐵) → (( 1 · 𝑥) = 𝑥 ↔ ( 1 (._r‘𝑅)𝑥) = 𝑥))
16	13	oveqd 6566	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐵) → (𝑥 · 1 ) = (𝑥(._r‘𝑅) 1 ))
17	16	eqeq1d 2612	. . . . . 6 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐵) → ((𝑥 · 1 ) = 𝑥 ↔ (𝑥(._r‘𝑅) 1 ) = 𝑥))
18	15, 17	anbi12d 743	. . . . 5 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐵) → ((( 1 · 𝑥) = 𝑥 ∧ (𝑥 · 1 ) = 𝑥) ↔ (( 1 (._r‘𝑅)𝑥) = 𝑥 ∧ (𝑥(._r‘𝑅) 1 ) = 𝑥)))
19	6, 18	raleqbidva 3131	. . . 4 ⊢ (𝜑 → (∀𝑥 ∈ 𝐵 (( 1 · 𝑥) = 𝑥 ∧ (𝑥 · 1 ) = 𝑥) ↔ ∀𝑥 ∈ (Base‘𝑅)(( 1 (._r‘𝑅)𝑥) = 𝑥 ∧ (𝑥(._r‘𝑅) 1 ) = 𝑥)))
20	11, 19	mpbid 221	. . 3 ⊢ (𝜑 → ∀𝑥 ∈ (Base‘𝑅)(( 1 (._r‘𝑅)𝑥) = 𝑥 ∧ (𝑥(._r‘𝑅) 1 ) = 𝑥))
21	6	eleq2d 2673	. . . . . . . 8 ⊢ (𝜑 → (𝑒 ∈ 𝐵 ↔ 𝑒 ∈ (Base‘𝑅)))
22	13	oveqd 6566	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐵) → (𝑒 · 𝑥) = (𝑒(._r‘𝑅)𝑥))
23	22	eqeq1d 2612	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐵) → ((𝑒 · 𝑥) = 𝑥 ↔ (𝑒(._r‘𝑅)𝑥) = 𝑥))
24	13	oveqd 6566	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐵) → (𝑥 · 𝑒) = (𝑥(._r‘𝑅)𝑒))
25	24	eqeq1d 2612	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐵) → ((𝑥 · 𝑒) = 𝑥 ↔ (𝑥(._r‘𝑅)𝑒) = 𝑥))
26	23, 25	anbi12d 743	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐵) → (((𝑒 · 𝑥) = 𝑥 ∧ (𝑥 · 𝑒) = 𝑥) ↔ ((𝑒(._r‘𝑅)𝑥) = 𝑥 ∧ (𝑥(._r‘𝑅)𝑒) = 𝑥)))
27	6, 26	raleqbidva 3131	. . . . . . . 8 ⊢ (𝜑 → (∀𝑥 ∈ 𝐵 ((𝑒 · 𝑥) = 𝑥 ∧ (𝑥 · 𝑒) = 𝑥) ↔ ∀𝑥 ∈ (Base‘𝑅)((𝑒(._r‘𝑅)𝑥) = 𝑥 ∧ (𝑥(._r‘𝑅)𝑒) = 𝑥)))
28	21, 27	anbi12d 743	. . . . . . 7 ⊢ (𝜑 → ((𝑒 ∈ 𝐵 ∧ ∀𝑥 ∈ 𝐵 ((𝑒 · 𝑥) = 𝑥 ∧ (𝑥 · 𝑒) = 𝑥)) ↔ (𝑒 ∈ (Base‘𝑅) ∧ ∀𝑥 ∈ (Base‘𝑅)((𝑒(._r‘𝑅)𝑥) = 𝑥 ∧ (𝑥(._r‘𝑅)𝑒) = 𝑥))))
29	8	ralrimiva 2949	. . . . . . . . . . . 12 ⊢ (𝜑 → ∀𝑥 ∈ 𝐵 ( 1 · 𝑥) = 𝑥)
30	29	adantr 480	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑒 ∈ 𝐵) → ∀𝑥 ∈ 𝐵 ( 1 · 𝑥) = 𝑥)
31		simpr 476	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑒 ∈ 𝐵) → 𝑒 ∈ 𝐵)
32		simpr 476	. . . . . . . . . . . . . 14 ⊢ (((𝜑 ∧ 𝑒 ∈ 𝐵) ∧ 𝑥 = 𝑒) → 𝑥 = 𝑒)
33	32	oveq2d 6565	. . . . . . . . . . . . 13 ⊢ (((𝜑 ∧ 𝑒 ∈ 𝐵) ∧ 𝑥 = 𝑒) → ( 1 · 𝑥) = ( 1 · 𝑒))
34	33, 32	eqeq12d 2625	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ 𝑒 ∈ 𝐵) ∧ 𝑥 = 𝑒) → (( 1 · 𝑥) = 𝑥 ↔ ( 1 · 𝑒) = 𝑒))
35	31, 34	rspcdv 3285	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑒 ∈ 𝐵) → (∀𝑥 ∈ 𝐵 ( 1 · 𝑥) = 𝑥 → ( 1 · 𝑒) = 𝑒))
36	30, 35	mpd 15	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑒 ∈ 𝐵) → ( 1 · 𝑒) = 𝑒)
37	36	adantrr 749	. . . . . . . . 9 ⊢ ((𝜑 ∧ (𝑒 ∈ 𝐵 ∧ ∀𝑥 ∈ 𝐵 ((𝑒 · 𝑥) = 𝑥 ∧ (𝑥 · 𝑒) = 𝑥))) → ( 1 · 𝑒) = 𝑒)
38	5	adantr 480	. . . . . . . . . 10 ⊢ ((𝜑 ∧ (𝑒 ∈ 𝐵 ∧ ∀𝑥 ∈ 𝐵 ((𝑒 · 𝑥) = 𝑥 ∧ (𝑥 · 𝑒) = 𝑥))) → 1 ∈ 𝐵)
39		simprr 792	. . . . . . . . . 10 ⊢ ((𝜑 ∧ (𝑒 ∈ 𝐵 ∧ ∀𝑥 ∈ 𝐵 ((𝑒 · 𝑥) = 𝑥 ∧ (𝑥 · 𝑒) = 𝑥))) → ∀𝑥 ∈ 𝐵 ((𝑒 · 𝑥) = 𝑥 ∧ (𝑥 · 𝑒) = 𝑥))
40		oveq2 6557	. . . . . . . . . . . . . 14 ⊢ (𝑥 = 1 → (𝑒 · 𝑥) = (𝑒 · 1 ))
41		id 22	. . . . . . . . . . . . . 14 ⊢ (𝑥 = 1 → 𝑥 = 1 )
42	40, 41	eqeq12d 2625	. . . . . . . . . . . . 13 ⊢ (𝑥 = 1 → ((𝑒 · 𝑥) = 𝑥 ↔ (𝑒 · 1 ) = 1 ))
43		oveq1 6556	. . . . . . . . . . . . . 14 ⊢ (𝑥 = 1 → (𝑥 · 𝑒) = ( 1 · 𝑒))
44	43, 41	eqeq12d 2625	. . . . . . . . . . . . 13 ⊢ (𝑥 = 1 → ((𝑥 · 𝑒) = 𝑥 ↔ ( 1 · 𝑒) = 1 ))
45	42, 44	anbi12d 743	. . . . . . . . . . . 12 ⊢ (𝑥 = 1 → (((𝑒 · 𝑥) = 𝑥 ∧ (𝑥 · 𝑒) = 𝑥) ↔ ((𝑒 · 1 ) = 1 ∧ ( 1 · 𝑒) = 1 )))
46	45	rspcva 3280	. . . . . . . . . . 11 ⊢ (( 1 ∈ 𝐵 ∧ ∀𝑥 ∈ 𝐵 ((𝑒 · 𝑥) = 𝑥 ∧ (𝑥 · 𝑒) = 𝑥)) → ((𝑒 · 1 ) = 1 ∧ ( 1 · 𝑒) = 1 ))
47	46	simprd 478	. . . . . . . . . 10 ⊢ (( 1 ∈ 𝐵 ∧ ∀𝑥 ∈ 𝐵 ((𝑒 · 𝑥) = 𝑥 ∧ (𝑥 · 𝑒) = 𝑥)) → ( 1 · 𝑒) = 1 )
48	38, 39, 47	syl2anc 691	. . . . . . . . 9 ⊢ ((𝜑 ∧ (𝑒 ∈ 𝐵 ∧ ∀𝑥 ∈ 𝐵 ((𝑒 · 𝑥) = 𝑥 ∧ (𝑥 · 𝑒) = 𝑥))) → ( 1 · 𝑒) = 1 )
49	37, 48	eqtr3d 2646	. . . . . . . 8 ⊢ ((𝜑 ∧ (𝑒 ∈ 𝐵 ∧ ∀𝑥 ∈ 𝐵 ((𝑒 · 𝑥) = 𝑥 ∧ (𝑥 · 𝑒) = 𝑥))) → 𝑒 = 1 )
50	49	ex 449	. . . . . . 7 ⊢ (𝜑 → ((𝑒 ∈ 𝐵 ∧ ∀𝑥 ∈ 𝐵 ((𝑒 · 𝑥) = 𝑥 ∧ (𝑥 · 𝑒) = 𝑥)) → 𝑒 = 1 ))
51	28, 50	sylbird 249	. . . . . 6 ⊢ (𝜑 → ((𝑒 ∈ (Base‘𝑅) ∧ ∀𝑥 ∈ (Base‘𝑅)((𝑒(._r‘𝑅)𝑥) = 𝑥 ∧ (𝑥(._r‘𝑅)𝑒) = 𝑥)) → 𝑒 = 1 ))
52	51	alrimiv 1842	. . . . 5 ⊢ (𝜑 → ∀𝑒((𝑒 ∈ (Base‘𝑅) ∧ ∀𝑥 ∈ (Base‘𝑅)((𝑒(._r‘𝑅)𝑥) = 𝑥 ∧ (𝑥(._r‘𝑅)𝑒) = 𝑥)) → 𝑒 = 1 ))
53		eleq1 2676	. . . . . . 7 ⊢ (𝑒 = 1 → (𝑒 ∈ (Base‘𝑅) ↔ 1 ∈ (Base‘𝑅)))
54		oveq1 6556	. . . . . . . . . 10 ⊢ (𝑒 = 1 → (𝑒(._r‘𝑅)𝑥) = ( 1 (._r‘𝑅)𝑥))
55	54	eqeq1d 2612	. . . . . . . . 9 ⊢ (𝑒 = 1 → ((𝑒(._r‘𝑅)𝑥) = 𝑥 ↔ ( 1 (._r‘𝑅)𝑥) = 𝑥))
56		oveq2 6557	. . . . . . . . . 10 ⊢ (𝑒 = 1 → (𝑥(._r‘𝑅)𝑒) = (𝑥(._r‘𝑅) 1 ))
57	56	eqeq1d 2612	. . . . . . . . 9 ⊢ (𝑒 = 1 → ((𝑥(._r‘𝑅)𝑒) = 𝑥 ↔ (𝑥(._r‘𝑅) 1 ) = 𝑥))
58	55, 57	anbi12d 743	. . . . . . . 8 ⊢ (𝑒 = 1 → (((𝑒(._r‘𝑅)𝑥) = 𝑥 ∧ (𝑥(._r‘𝑅)𝑒) = 𝑥) ↔ (( 1 (._r‘𝑅)𝑥) = 𝑥 ∧ (𝑥(._r‘𝑅) 1 ) = 𝑥)))
59	58	ralbidv 2969	. . . . . . 7 ⊢ (𝑒 = 1 → (∀𝑥 ∈ (Base‘𝑅)((𝑒(._r‘𝑅)𝑥) = 𝑥 ∧ (𝑥(._r‘𝑅)𝑒) = 𝑥) ↔ ∀𝑥 ∈ (Base‘𝑅)(( 1 (._r‘𝑅)𝑥) = 𝑥 ∧ (𝑥(._r‘𝑅) 1 ) = 𝑥)))
60	53, 59	anbi12d 743	. . . . . 6 ⊢ (𝑒 = 1 → ((𝑒 ∈ (Base‘𝑅) ∧ ∀𝑥 ∈ (Base‘𝑅)((𝑒(._r‘𝑅)𝑥) = 𝑥 ∧ (𝑥(._r‘𝑅)𝑒) = 𝑥)) ↔ ( 1 ∈ (Base‘𝑅) ∧ ∀𝑥 ∈ (Base‘𝑅)(( 1 (._r‘𝑅)𝑥) = 𝑥 ∧ (𝑥(._r‘𝑅) 1 ) = 𝑥))))
61	60	eqeu 3344	. . . . 5 ⊢ (( 1 ∈ (Base‘𝑅) ∧ ( 1 ∈ (Base‘𝑅) ∧ ∀𝑥 ∈ (Base‘𝑅)(( 1 (._r‘𝑅)𝑥) = 𝑥 ∧ (𝑥(._r‘𝑅) 1 ) = 𝑥)) ∧ ∀𝑒((𝑒 ∈ (Base‘𝑅) ∧ ∀𝑥 ∈ (Base‘𝑅)((𝑒(._r‘𝑅)𝑥) = 𝑥 ∧ (𝑥(._r‘𝑅)𝑒) = 𝑥)) → 𝑒 = 1 )) → ∃!𝑒(𝑒 ∈ (Base‘𝑅) ∧ ∀𝑥 ∈ (Base‘𝑅)((𝑒(._r‘𝑅)𝑥) = 𝑥 ∧ (𝑥(._r‘𝑅)𝑒) = 𝑥)))
62	7, 7, 20, 52, 61	syl121anc 1323	. . . 4 ⊢ (𝜑 → ∃!𝑒(𝑒 ∈ (Base‘𝑅) ∧ ∀𝑥 ∈ (Base‘𝑅)((𝑒(._r‘𝑅)𝑥) = 𝑥 ∧ (𝑥(._r‘𝑅)𝑒) = 𝑥)))
63	60	iota2 5794	. . . 4 ⊢ (( 1 ∈ 𝐵 ∧ ∃!𝑒(𝑒 ∈ (Base‘𝑅) ∧ ∀𝑥 ∈ (Base‘𝑅)((𝑒(._r‘𝑅)𝑥) = 𝑥 ∧ (𝑥(._r‘𝑅)𝑒) = 𝑥))) → (( 1 ∈ (Base‘𝑅) ∧ ∀𝑥 ∈ (Base‘𝑅)(( 1 (._r‘𝑅)𝑥) = 𝑥 ∧ (𝑥(._r‘𝑅) 1 ) = 𝑥)) ↔ (℩𝑒(𝑒 ∈ (Base‘𝑅) ∧ ∀𝑥 ∈ (Base‘𝑅)((𝑒(._r‘𝑅)𝑥) = 𝑥 ∧ (𝑥(._r‘𝑅)𝑒) = 𝑥))) = 1 ))
64	5, 62, 63	syl2anc 691	. . 3 ⊢ (𝜑 → (( 1 ∈ (Base‘𝑅) ∧ ∀𝑥 ∈ (Base‘𝑅)(( 1 (._r‘𝑅)𝑥) = 𝑥 ∧ (𝑥(._r‘𝑅) 1 ) = 𝑥)) ↔ (℩𝑒(𝑒 ∈ (Base‘𝑅) ∧ ∀𝑥 ∈ (Base‘𝑅)((𝑒(._r‘𝑅)𝑥) = 𝑥 ∧ (𝑥(._r‘𝑅)𝑒) = 𝑥))) = 1 ))
65	7, 20, 64	mpbi2and 958	. 2 ⊢ (𝜑 → (℩𝑒(𝑒 ∈ (Base‘𝑅) ∧ ∀𝑥 ∈ (Base‘𝑅)((𝑒(._r‘𝑅)𝑥) = 𝑥 ∧ (𝑥(._r‘𝑅)𝑒) = 𝑥))) = 1 )
66	4, 65	syl5req 2657	1 ⊢ (𝜑 → 1 = (1_r‘𝑅))

Colors of variables: wff setvar class

Syntax hints: → wi 4 ↔ wb 195 ∧ wa 383 ∀wal 1473 = wceq 1475 ∈ wcel 1977 ∃!weu 2458 ∀wral 2896 ℩cio 5766 ‘cfv 5804 (class class class)co 6549 Basecbs 15695 ._rcmulr 15769 1_rcur 18324

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1713 ax-4 1728 ax-5 1827 ax-6 1875 ax-7 1922 ax-8 1979 ax-9 1986 ax-10 2006 ax-11 2021 ax-12 2034 ax-13 2234 ax-ext 2590 ax-sep 4709 ax-nul 4717 ax-pow 4769 ax-pr 4833 ax-un 6847 ax-cnex 9871 ax-resscn 9872 ax-1cn 9873 ax-icn 9874 ax-addcl 9875 ax-addrcl 9876 ax-mulcl 9877 ax-mulrcl 9878 ax-mulcom 9879 ax-addass 9880 ax-mulass 9881 ax-distr 9882 ax-i2m1 9883 ax-1ne0 9884 ax-1rid 9885 ax-rnegex 9886 ax-rrecex 9887 ax-cnre 9888 ax-pre-lttri 9889 ax-pre-lttrn 9890 ax-pre-ltadd 9891 ax-pre-mulgt0 9892

This theorem depends on definitions: df-bi 196 df-or 384 df-an 385 df-3or 1032 df-3an 1033 df-tru 1478 df-ex 1696 df-nf 1701 df-sb 1868 df-eu 2462 df-mo 2463 df-clab 2597 df-cleq 2603 df-clel 2606 df-nfc 2740 df-ne 2782 df-nel 2783 df-ral 2901 df-rex 2902 df-reu 2903 df-rab 2905 df-v 3175 df-sbc 3403 df-csb 3500 df-dif 3543 df-un 3545 df-in 3547 df-ss 3554 df-pss 3556 df-nul 3875 df-if 4037 df-pw 4110 df-sn 4126 df-pr 4128 df-tp 4130 df-op 4132 df-uni 4373 df-iun 4457 df-br 4584 df-opab 4644 df-mpt 4645 df-tr 4681 df-eprel 4949 df-id 4953 df-po 4959 df-so 4960 df-fr 4997 df-we 4999 df-xp 5044 df-rel 5045 df-cnv 5046 df-co 5047 df-dm 5048 df-rn 5049 df-res 5050 df-ima 5051 df-pred 5597 df-ord 5643 df-on 5644 df-lim 5645 df-suc 5646 df-iota 5768 df-fun 5806 df-fn 5807 df-f 5808 df-f1 5809 df-fo 5810 df-f1o 5811 df-fv 5812 df-riota 6511 df-ov 6552 df-oprab 6553 df-mpt2 6554 df-om 6958 df-wrecs 7294 df-recs 7355 df-rdg 7393 df-er 7629 df-en 7842 df-dom 7843 df-sdom 7844 df-pnf 9955 df-mnf 9956 df-xr 9957 df-ltxr 9958 df-le 9959 df-sub 10147 df-neg 10148 df-nn 10898 df-2 10956 df-ndx 15698 df-slot 15699 df-base 15700 df-sets 15701 df-plusg 15781 df-0g 15925 df-mgp 18313 df-ur 18325

This theorem is referenced by: ress1r 29120

W3C validator