Theorem isringrng

Description: The predicate "is a unital ring" as extension of the predicate "is a non-unital ring". (Contributed by AV, 17-Feb-2020)

	Ref	Expression
Hypotheses	isringrng.b	⊢ 𝐵 = ( Base ‘ 𝑅 )
	isringrng.t	⊢ · = ( .r ‘ 𝑅 )
Assertion	isringrng	⊢ ( 𝑅 ∈ Ring ↔ ( 𝑅 ∈ Rng ∧ ∃ 𝑥 ∈ 𝐵 ∀ 𝑦 ∈ 𝐵 ( ( 𝑥 · 𝑦 ) = 𝑦 ∧ ( 𝑦 · 𝑥 ) = 𝑦 ) ) )

Proof

Step	Hyp	Ref	Expression
1		isringrng.b	⊢ 𝐵 = ( Base ‘ 𝑅 )
2		isringrng.t	⊢ · = ( .r ‘ 𝑅 )
3		ringrng	⊢ ( 𝑅 ∈ Ring → 𝑅 ∈ Rng )
4	1 2	ringideu	⊢ ( 𝑅 ∈ Ring → ∃! 𝑥 ∈ 𝐵 ∀ 𝑦 ∈ 𝐵 ( ( 𝑥 · 𝑦 ) = 𝑦 ∧ ( 𝑦 · 𝑥 ) = 𝑦 ) )
5		reurex	⊢ ( ∃! 𝑥 ∈ 𝐵 ∀ 𝑦 ∈ 𝐵 ( ( 𝑥 · 𝑦 ) = 𝑦 ∧ ( 𝑦 · 𝑥 ) = 𝑦 ) → ∃ 𝑥 ∈ 𝐵 ∀ 𝑦 ∈ 𝐵 ( ( 𝑥 · 𝑦 ) = 𝑦 ∧ ( 𝑦 · 𝑥 ) = 𝑦 ) )
6	4 5	syl	⊢ ( 𝑅 ∈ Ring → ∃ 𝑥 ∈ 𝐵 ∀ 𝑦 ∈ 𝐵 ( ( 𝑥 · 𝑦 ) = 𝑦 ∧ ( 𝑦 · 𝑥 ) = 𝑦 ) )
7	3 6	jca	⊢ ( 𝑅 ∈ Ring → ( 𝑅 ∈ Rng ∧ ∃ 𝑥 ∈ 𝐵 ∀ 𝑦 ∈ 𝐵 ( ( 𝑥 · 𝑦 ) = 𝑦 ∧ ( 𝑦 · 𝑥 ) = 𝑦 ) ) )
8		rngabl	⊢ ( 𝑅 ∈ Rng → 𝑅 ∈ Abel )
9		ablgrp	⊢ ( 𝑅 ∈ Abel → 𝑅 ∈ Grp )
10	8 9	syl	⊢ ( 𝑅 ∈ Rng → 𝑅 ∈ Grp )
11	10	adantr	⊢ ( ( 𝑅 ∈ Rng ∧ ∃ 𝑥 ∈ 𝐵 ∀ 𝑦 ∈ 𝐵 ( ( 𝑥 · 𝑦 ) = 𝑦 ∧ ( 𝑦 · 𝑥 ) = 𝑦 ) ) → 𝑅 ∈ Grp )
12		eqid	⊢ ( mulGrp ‘ 𝑅 ) = ( mulGrp ‘ 𝑅 )
13	12	rngmgp	⊢ ( 𝑅 ∈ Rng → ( mulGrp ‘ 𝑅 ) ∈ Smgrp )
14	13	anim1i	⊢ ( ( 𝑅 ∈ Rng ∧ ∃ 𝑥 ∈ 𝐵 ∀ 𝑦 ∈ 𝐵 ( ( 𝑥 · 𝑦 ) = 𝑦 ∧ ( 𝑦 · 𝑥 ) = 𝑦 ) ) → ( ( mulGrp ‘ 𝑅 ) ∈ Smgrp ∧ ∃ 𝑥 ∈ 𝐵 ∀ 𝑦 ∈ 𝐵 ( ( 𝑥 · 𝑦 ) = 𝑦 ∧ ( 𝑦 · 𝑥 ) = 𝑦 ) ) )
15	12 1	mgpbas	⊢ 𝐵 = ( Base ‘ ( mulGrp ‘ 𝑅 ) )
16	12 2	mgpplusg	⊢ · = ( +g ‘ ( mulGrp ‘ 𝑅 ) )
17	15 16	ismnddef	⊢ ( ( mulGrp ‘ 𝑅 ) ∈ Mnd ↔ ( ( mulGrp ‘ 𝑅 ) ∈ Smgrp ∧ ∃ 𝑥 ∈ 𝐵 ∀ 𝑦 ∈ 𝐵 ( ( 𝑥 · 𝑦 ) = 𝑦 ∧ ( 𝑦 · 𝑥 ) = 𝑦 ) ) )
18	14 17	sylibr	⊢ ( ( 𝑅 ∈ Rng ∧ ∃ 𝑥 ∈ 𝐵 ∀ 𝑦 ∈ 𝐵 ( ( 𝑥 · 𝑦 ) = 𝑦 ∧ ( 𝑦 · 𝑥 ) = 𝑦 ) ) → ( mulGrp ‘ 𝑅 ) ∈ Mnd )
19		eqid	⊢ ( +g ‘ 𝑅 ) = ( +g ‘ 𝑅 )
20	1 12 19 2	isrng	⊢ ( 𝑅 ∈ Rng ↔ ( 𝑅 ∈ Abel ∧ ( mulGrp ‘ 𝑅 ) ∈ Smgrp ∧ ∀ 𝑥 ∈ 𝐵 ∀ 𝑦 ∈ 𝐵 ∀ 𝑧 ∈ 𝐵 ( ( 𝑥 · ( 𝑦 ( +g ‘ 𝑅 ) 𝑧 ) ) = ( ( 𝑥 · 𝑦 ) ( +g ‘ 𝑅 ) ( 𝑥 · 𝑧 ) ) ∧ ( ( 𝑥 ( +g ‘ 𝑅 ) 𝑦 ) · 𝑧 ) = ( ( 𝑥 · 𝑧 ) ( +g ‘ 𝑅 ) ( 𝑦 · 𝑧 ) ) ) ) )
21	20	simp3bi	⊢ ( 𝑅 ∈ Rng → ∀ 𝑥 ∈ 𝐵 ∀ 𝑦 ∈ 𝐵 ∀ 𝑧 ∈ 𝐵 ( ( 𝑥 · ( 𝑦 ( +g ‘ 𝑅 ) 𝑧 ) ) = ( ( 𝑥 · 𝑦 ) ( +g ‘ 𝑅 ) ( 𝑥 · 𝑧 ) ) ∧ ( ( 𝑥 ( +g ‘ 𝑅 ) 𝑦 ) · 𝑧 ) = ( ( 𝑥 · 𝑧 ) ( +g ‘ 𝑅 ) ( 𝑦 · 𝑧 ) ) ) )
22	21	adantr	⊢ ( ( 𝑅 ∈ Rng ∧ ∃ 𝑥 ∈ 𝐵 ∀ 𝑦 ∈ 𝐵 ( ( 𝑥 · 𝑦 ) = 𝑦 ∧ ( 𝑦 · 𝑥 ) = 𝑦 ) ) → ∀ 𝑥 ∈ 𝐵 ∀ 𝑦 ∈ 𝐵 ∀ 𝑧 ∈ 𝐵 ( ( 𝑥 · ( 𝑦 ( +g ‘ 𝑅 ) 𝑧 ) ) = ( ( 𝑥 · 𝑦 ) ( +g ‘ 𝑅 ) ( 𝑥 · 𝑧 ) ) ∧ ( ( 𝑥 ( +g ‘ 𝑅 ) 𝑦 ) · 𝑧 ) = ( ( 𝑥 · 𝑧 ) ( +g ‘ 𝑅 ) ( 𝑦 · 𝑧 ) ) ) )
23	1 12 19 2	isring	⊢ ( 𝑅 ∈ Ring ↔ ( 𝑅 ∈ Grp ∧ ( mulGrp ‘ 𝑅 ) ∈ Mnd ∧ ∀ 𝑥 ∈ 𝐵 ∀ 𝑦 ∈ 𝐵 ∀ 𝑧 ∈ 𝐵 ( ( 𝑥 · ( 𝑦 ( +g ‘ 𝑅 ) 𝑧 ) ) = ( ( 𝑥 · 𝑦 ) ( +g ‘ 𝑅 ) ( 𝑥 · 𝑧 ) ) ∧ ( ( 𝑥 ( +g ‘ 𝑅 ) 𝑦 ) · 𝑧 ) = ( ( 𝑥 · 𝑧 ) ( +g ‘ 𝑅 ) ( 𝑦 · 𝑧 ) ) ) ) )
24	11 18 22 23	syl3anbrc	⊢ ( ( 𝑅 ∈ Rng ∧ ∃ 𝑥 ∈ 𝐵 ∀ 𝑦 ∈ 𝐵 ( ( 𝑥 · 𝑦 ) = 𝑦 ∧ ( 𝑦 · 𝑥 ) = 𝑦 ) ) → 𝑅 ∈ Ring )
25	7 24	impbii	⊢ ( 𝑅 ∈ Ring ↔ ( 𝑅 ∈ Rng ∧ ∃ 𝑥 ∈ 𝐵 ∀ 𝑦 ∈ 𝐵 ( ( 𝑥 · 𝑦 ) = 𝑦 ∧ ( 𝑦 · 𝑥 ) = 𝑦 ) ) )