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Metamath Proof Explorer

Theorem subrngmcl

Description: A subring is closed under multiplication. (Contributed by Mario Carneiro, 2-Dec-2014) Generalization of subrgmcl . (Revised by AV, 14-Feb-2025)

Ref Expression
Hypothesis subrngmcl.p ⊢ · = ( .r ‘ 𝑅 )
Assertion subrngmcl ( ( 𝐴 ∈ ( SubRng ‘ 𝑅 ) ∧ 𝑋 ∈ 𝐴 ∧ 𝑌 ∈ 𝐴 ) → ( 𝑋 · 𝑌 ) ∈ 𝐴 )

Proof

Step Hyp Ref Expression
1 subrngmcl.p ⊢ · = ( .r ‘ 𝑅 )
2 eqid ⊢ ( 𝑅 ↾s 𝐴 ) = ( 𝑅 ↾s 𝐴 )
3 2 subrngrng ⊢ ( 𝐴 ∈ ( SubRng ‘ 𝑅 ) → ( 𝑅 ↾s 𝐴 ) ∈ Rng )
4 3 3ad2ant1 ⊢ ( ( 𝐴 ∈ ( SubRng ‘ 𝑅 ) ∧ 𝑋 ∈ 𝐴 ∧ 𝑌 ∈ 𝐴 ) → ( 𝑅 ↾s 𝐴 ) ∈ Rng )
5 simp2 ⊢ ( ( 𝐴 ∈ ( SubRng ‘ 𝑅 ) ∧ 𝑋 ∈ 𝐴 ∧ 𝑌 ∈ 𝐴 ) → 𝑋 ∈ 𝐴 )
6 2 subrngbas ⊢ ( 𝐴 ∈ ( SubRng ‘ 𝑅 ) → 𝐴 = ( Base ‘ ( 𝑅 ↾s 𝐴 ) ) )
7 6 3ad2ant1 ⊢ ( ( 𝐴 ∈ ( SubRng ‘ 𝑅 ) ∧ 𝑋 ∈ 𝐴 ∧ 𝑌 ∈ 𝐴 ) → 𝐴 = ( Base ‘ ( 𝑅 ↾s 𝐴 ) ) )
8 5 7 eleqtrd ⊢ ( ( 𝐴 ∈ ( SubRng ‘ 𝑅 ) ∧ 𝑋 ∈ 𝐴 ∧ 𝑌 ∈ 𝐴 ) → 𝑋 ∈ ( Base ‘ ( 𝑅 ↾s 𝐴 ) ) )
9 simp3 ⊢ ( ( 𝐴 ∈ ( SubRng ‘ 𝑅 ) ∧ 𝑋 ∈ 𝐴 ∧ 𝑌 ∈ 𝐴 ) → 𝑌 ∈ 𝐴 )
10 9 7 eleqtrd ⊢ ( ( 𝐴 ∈ ( SubRng ‘ 𝑅 ) ∧ 𝑋 ∈ 𝐴 ∧ 𝑌 ∈ 𝐴 ) → 𝑌 ∈ ( Base ‘ ( 𝑅 ↾s 𝐴 ) ) )
11 eqid ⊢ ( Base ‘ ( 𝑅 ↾s 𝐴 ) ) = ( Base ‘ ( 𝑅 ↾s 𝐴 ) )
12 eqid ⊢ ( .r ‘ ( 𝑅 ↾s 𝐴 ) ) = ( .r ‘ ( 𝑅 ↾s 𝐴 ) )
13 11 12 rngcl ⊢ ( ( ( 𝑅 ↾s 𝐴 ) ∈ Rng ∧ 𝑋 ∈ ( Base ‘ ( 𝑅 ↾s 𝐴 ) ) ∧ 𝑌 ∈ ( Base ‘ ( 𝑅 ↾s 𝐴 ) ) ) → ( 𝑋 ( .r ‘ ( 𝑅 ↾s 𝐴 ) ) 𝑌 ) ∈ ( Base ‘ ( 𝑅 ↾s 𝐴 ) ) )
14 4 8 10 13 syl3anc ⊢ ( ( 𝐴 ∈ ( SubRng ‘ 𝑅 ) ∧ 𝑋 ∈ 𝐴 ∧ 𝑌 ∈ 𝐴 ) → ( 𝑋 ( .r ‘ ( 𝑅 ↾s 𝐴 ) ) 𝑌 ) ∈ ( Base ‘ ( 𝑅 ↾s 𝐴 ) ) )
15 2 1 ressmulr ⊢ ( 𝐴 ∈ ( SubRng ‘ 𝑅 ) → · = ( .r ‘ ( 𝑅 ↾s 𝐴 ) ) )
16 15 3ad2ant1 ⊢ ( ( 𝐴 ∈ ( SubRng ‘ 𝑅 ) ∧ 𝑋 ∈ 𝐴 ∧ 𝑌 ∈ 𝐴 ) → · = ( .r ‘ ( 𝑅 ↾s 𝐴 ) ) )
17 16 oveqd ⊢ ( ( 𝐴 ∈ ( SubRng ‘ 𝑅 ) ∧ 𝑋 ∈ 𝐴 ∧ 𝑌 ∈ 𝐴 ) → ( 𝑋 · 𝑌 ) = ( 𝑋 ( .r ‘ ( 𝑅 ↾s 𝐴 ) ) 𝑌 ) )
18 14 17 7 3eltr4d ⊢ ( ( 𝐴 ∈ ( SubRng ‘ 𝑅 ) ∧ 𝑋 ∈ 𝐴 ∧ 𝑌 ∈ 𝐴 ) → ( 𝑋 · 𝑌 ) ∈ 𝐴 )