Metamath Proof Explorer


Theorem 1idl

Description: Two ways of expressing the unit ideal. (Contributed by Jeff Madsen, 10-Jun-2010)

Ref Expression
Hypotheses 1idl.1 𝐺 = ( 1st𝑅 )
1idl.2 𝐻 = ( 2nd𝑅 )
1idl.3 𝑋 = ran 𝐺
1idl.4 𝑈 = ( GId ‘ 𝐻 )
Assertion 1idl ( ( 𝑅 ∈ RingOps ∧ 𝐼 ∈ ( Idl ‘ 𝑅 ) ) → ( 𝑈𝐼𝐼 = 𝑋 ) )

Proof

Step Hyp Ref Expression
1 1idl.1 𝐺 = ( 1st𝑅 )
2 1idl.2 𝐻 = ( 2nd𝑅 )
3 1idl.3 𝑋 = ran 𝐺
4 1idl.4 𝑈 = ( GId ‘ 𝐻 )
5 1 3 idlss ( ( 𝑅 ∈ RingOps ∧ 𝐼 ∈ ( Idl ‘ 𝑅 ) ) → 𝐼𝑋 )
6 5 adantr ( ( ( 𝑅 ∈ RingOps ∧ 𝐼 ∈ ( Idl ‘ 𝑅 ) ) ∧ 𝑈𝐼 ) → 𝐼𝑋 )
7 1 rneqi ran 𝐺 = ran ( 1st𝑅 )
8 3 7 eqtri 𝑋 = ran ( 1st𝑅 )
9 2 8 4 rngolidm ( ( 𝑅 ∈ RingOps ∧ 𝑥𝑋 ) → ( 𝑈 𝐻 𝑥 ) = 𝑥 )
10 9 ad2ant2rl ( ( ( 𝑅 ∈ RingOps ∧ 𝐼 ∈ ( Idl ‘ 𝑅 ) ) ∧ ( 𝑈𝐼𝑥𝑋 ) ) → ( 𝑈 𝐻 𝑥 ) = 𝑥 )
11 1 2 3 idlrmulcl ( ( ( 𝑅 ∈ RingOps ∧ 𝐼 ∈ ( Idl ‘ 𝑅 ) ) ∧ ( 𝑈𝐼𝑥𝑋 ) ) → ( 𝑈 𝐻 𝑥 ) ∈ 𝐼 )
12 10 11 eqeltrrd ( ( ( 𝑅 ∈ RingOps ∧ 𝐼 ∈ ( Idl ‘ 𝑅 ) ) ∧ ( 𝑈𝐼𝑥𝑋 ) ) → 𝑥𝐼 )
13 12 expr ( ( ( 𝑅 ∈ RingOps ∧ 𝐼 ∈ ( Idl ‘ 𝑅 ) ) ∧ 𝑈𝐼 ) → ( 𝑥𝑋𝑥𝐼 ) )
14 13 ssrdv ( ( ( 𝑅 ∈ RingOps ∧ 𝐼 ∈ ( Idl ‘ 𝑅 ) ) ∧ 𝑈𝐼 ) → 𝑋𝐼 )
15 6 14 eqssd ( ( ( 𝑅 ∈ RingOps ∧ 𝐼 ∈ ( Idl ‘ 𝑅 ) ) ∧ 𝑈𝐼 ) → 𝐼 = 𝑋 )
16 15 ex ( ( 𝑅 ∈ RingOps ∧ 𝐼 ∈ ( Idl ‘ 𝑅 ) ) → ( 𝑈𝐼𝐼 = 𝑋 ) )
17 8 2 4 rngo1cl ( 𝑅 ∈ RingOps → 𝑈𝑋 )
18 17 adantr ( ( 𝑅 ∈ RingOps ∧ 𝐼 ∈ ( Idl ‘ 𝑅 ) ) → 𝑈𝑋 )
19 eleq2 ( 𝐼 = 𝑋 → ( 𝑈𝐼𝑈𝑋 ) )
20 18 19 syl5ibrcom ( ( 𝑅 ∈ RingOps ∧ 𝐼 ∈ ( Idl ‘ 𝑅 ) ) → ( 𝐼 = 𝑋𝑈𝐼 ) )
21 16 20 impbid ( ( 𝑅 ∈ RingOps ∧ 𝐼 ∈ ( Idl ‘ 𝑅 ) ) → ( 𝑈𝐼𝐼 = 𝑋 ) )