Theorem dvr1

Description: A cancellation law for division. ( div1 analog.) (Contributed by Mario Carneiro, 18-Jun-2015)

	Ref	Expression
Hypotheses	dvr1.b	⊢ 𝐵 = ( Base ‘ 𝑅 )
	dvr1.d	⊢ / = ( /r ‘ 𝑅 )
	dvr1.o	⊢ 1 = ( 1r ‘ 𝑅 )
Assertion	dvr1	⊢ ( ( 𝑅 ∈ Ring ∧ 𝑋 ∈ 𝐵 ) → ( 𝑋 / 1 ) = 𝑋 )

Proof

Step	Hyp	Ref	Expression
1		dvr1.b	⊢ 𝐵 = ( Base ‘ 𝑅 )
2		dvr1.d	⊢ / = ( /r ‘ 𝑅 )
3		dvr1.o	⊢ 1 = ( 1r ‘ 𝑅 )
4		id	⊢ ( 𝑋 ∈ 𝐵 → 𝑋 ∈ 𝐵 )
5		eqid	⊢ ( Unit ‘ 𝑅 ) = ( Unit ‘ 𝑅 )
6	5 3	1unit	⊢ ( 𝑅 ∈ Ring → 1 ∈ ( Unit ‘ 𝑅 ) )
7		eqid	⊢ ( .r ‘ 𝑅 ) = ( .r ‘ 𝑅 )
8		eqid	⊢ ( invr ‘ 𝑅 ) = ( invr ‘ 𝑅 )
9	1 7 5 8 2	dvrval	⊢ ( ( 𝑋 ∈ 𝐵 ∧ 1 ∈ ( Unit ‘ 𝑅 ) ) → ( 𝑋 / 1 ) = ( 𝑋 ( .r ‘ 𝑅 ) ( ( invr ‘ 𝑅 ) ‘ 1 ) ) )
10	4 6 9	syl2anr	⊢ ( ( 𝑅 ∈ Ring ∧ 𝑋 ∈ 𝐵 ) → ( 𝑋 / 1 ) = ( 𝑋 ( .r ‘ 𝑅 ) ( ( invr ‘ 𝑅 ) ‘ 1 ) ) )
11	8 3	1rinv	⊢ ( 𝑅 ∈ Ring → ( ( invr ‘ 𝑅 ) ‘ 1 ) = 1 )
12	11	adantr	⊢ ( ( 𝑅 ∈ Ring ∧ 𝑋 ∈ 𝐵 ) → ( ( invr ‘ 𝑅 ) ‘ 1 ) = 1 )
13	12	oveq2d	⊢ ( ( 𝑅 ∈ Ring ∧ 𝑋 ∈ 𝐵 ) → ( 𝑋 ( .r ‘ 𝑅 ) ( ( invr ‘ 𝑅 ) ‘ 1 ) ) = ( 𝑋 ( .r ‘ 𝑅 ) 1 ) )
14	1 7 3	ringridm	⊢ ( ( 𝑅 ∈ Ring ∧ 𝑋 ∈ 𝐵 ) → ( 𝑋 ( .r ‘ 𝑅 ) 1 ) = 𝑋 )
15	10 13 14	3eqtrd	⊢ ( ( 𝑅 ∈ Ring ∧ 𝑋 ∈ 𝐵 ) → ( 𝑋 / 1 ) = 𝑋 )