Metamath Proof Explorer

Theorem psrassa

Description: The ring of power series is an associative algebra. (Contributed by Mario Carneiro, 29-Dec-2014)

Ref Expression
Hypotheses psrcnrg.s ⊢ 𝑆 = ( 𝐼 mPwSer 𝑅 )
psrcnrg.i ⊢ ( 𝜑 → 𝐼 ∈ 𝑉 )
psrcnrg.r ⊢ ( 𝜑 → 𝑅 ∈ CRing )
Assertion psrassa ( 𝜑 → 𝑆 ∈ AssAlg )

Proof

Step Hyp Ref Expression
1 psrcnrg.s ⊢ 𝑆 = ( 𝐼 mPwSer 𝑅 )
2 psrcnrg.i ⊢ ( 𝜑 → 𝐼 ∈ 𝑉 )
3 psrcnrg.r ⊢ ( 𝜑 → 𝑅 ∈ CRing )
4 eqidd ⊢ ( 𝜑 → ( Base ‘ 𝑆 ) = ( Base ‘ 𝑆 ) )
5 1 2 3 psrsca ⊢ ( 𝜑 → 𝑅 = ( Scalar ‘ 𝑆 ) )
6 eqidd ⊢ ( 𝜑 → ( Base ‘ 𝑅 ) = ( Base ‘ 𝑅 ) )
7 eqidd ⊢ ( 𝜑 → ( ·𝑠 ‘ 𝑆 ) = ( ·𝑠 ‘ 𝑆 ) )
8 eqidd ⊢ ( 𝜑 → ( .r ‘ 𝑆 ) = ( .r ‘ 𝑆 ) )
9 3 crngringd ⊢ ( 𝜑 → 𝑅 ∈ Ring )
10 1 2 9 psrlmod ⊢ ( 𝜑 → 𝑆 ∈ LMod )
11 1 2 9 psrring ⊢ ( 𝜑 → 𝑆 ∈ Ring )
12 2 adantr ⊢ ( ( 𝜑 ∧ ( 𝑥 ∈ ( Base ‘ 𝑅 ) ∧ 𝑦 ∈ ( Base ‘ 𝑆 ) ∧ 𝑧 ∈ ( Base ‘ 𝑆 ) ) ) → 𝐼 ∈ 𝑉 )
13 9 adantr ⊢ ( ( 𝜑 ∧ ( 𝑥 ∈ ( Base ‘ 𝑅 ) ∧ 𝑦 ∈ ( Base ‘ 𝑆 ) ∧ 𝑧 ∈ ( Base ‘ 𝑆 ) ) ) → 𝑅 ∈ Ring )
14 eqid ⊢ { 𝑓 ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ 𝑓 “ ℕ ) ∈ Fin } = { 𝑓 ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ 𝑓 “ ℕ ) ∈ Fin }
15 eqid ⊢ ( .r ‘ 𝑆 ) = ( .r ‘ 𝑆 )
16 eqid ⊢ ( Base ‘ 𝑆 ) = ( Base ‘ 𝑆 )
17 simpr2 ⊢ ( ( 𝜑 ∧ ( 𝑥 ∈ ( Base ‘ 𝑅 ) ∧ 𝑦 ∈ ( Base ‘ 𝑆 ) ∧ 𝑧 ∈ ( Base ‘ 𝑆 ) ) ) → 𝑦 ∈ ( Base ‘ 𝑆 ) )
18 simpr3 ⊢ ( ( 𝜑 ∧ ( 𝑥 ∈ ( Base ‘ 𝑅 ) ∧ 𝑦 ∈ ( Base ‘ 𝑆 ) ∧ 𝑧 ∈ ( Base ‘ 𝑆 ) ) ) → 𝑧 ∈ ( Base ‘ 𝑆 ) )
19 3 adantr ⊢ ( ( 𝜑 ∧ ( 𝑥 ∈ ( Base ‘ 𝑅 ) ∧ 𝑦 ∈ ( Base ‘ 𝑆 ) ∧ 𝑧 ∈ ( Base ‘ 𝑆 ) ) ) → 𝑅 ∈ CRing )
20 eqid ⊢ ( Base ‘ 𝑅 ) = ( Base ‘ 𝑅 )
21 eqid ⊢ ( ·𝑠 ‘ 𝑆 ) = ( ·𝑠 ‘ 𝑆 )
22 simpr1 ⊢ ( ( 𝜑 ∧ ( 𝑥 ∈ ( Base ‘ 𝑅 ) ∧ 𝑦 ∈ ( Base ‘ 𝑆 ) ∧ 𝑧 ∈ ( Base ‘ 𝑆 ) ) ) → 𝑥 ∈ ( Base ‘ 𝑅 ) )
23 1 12 13 14 15 16 17 18 19 20 21 22 psrass23 ⊢ ( ( 𝜑 ∧ ( 𝑥 ∈ ( Base ‘ 𝑅 ) ∧ 𝑦 ∈ ( Base ‘ 𝑆 ) ∧ 𝑧 ∈ ( Base ‘ 𝑆 ) ) ) → ( ( ( 𝑥 ( ·𝑠 ‘ 𝑆 ) 𝑦 ) ( .r ‘ 𝑆 ) 𝑧 ) = ( 𝑥 ( ·𝑠 ‘ 𝑆 ) ( 𝑦 ( .r ‘ 𝑆 ) 𝑧 ) ) ∧ ( 𝑦 ( .r ‘ 𝑆 ) ( 𝑥 ( ·𝑠 ‘ 𝑆 ) 𝑧 ) ) = ( 𝑥 ( ·𝑠 ‘ 𝑆 ) ( 𝑦 ( .r ‘ 𝑆 ) 𝑧 ) ) ) )
24 23 simpld ⊢ ( ( 𝜑 ∧ ( 𝑥 ∈ ( Base ‘ 𝑅 ) ∧ 𝑦 ∈ ( Base ‘ 𝑆 ) ∧ 𝑧 ∈ ( Base ‘ 𝑆 ) ) ) → ( ( 𝑥 ( ·𝑠 ‘ 𝑆 ) 𝑦 ) ( .r ‘ 𝑆 ) 𝑧 ) = ( 𝑥 ( ·𝑠 ‘ 𝑆 ) ( 𝑦 ( .r ‘ 𝑆 ) 𝑧 ) ) )
25 23 simprd ⊢ ( ( 𝜑 ∧ ( 𝑥 ∈ ( Base ‘ 𝑅 ) ∧ 𝑦 ∈ ( Base ‘ 𝑆 ) ∧ 𝑧 ∈ ( Base ‘ 𝑆 ) ) ) → ( 𝑦 ( .r ‘ 𝑆 ) ( 𝑥 ( ·𝑠 ‘ 𝑆 ) 𝑧 ) ) = ( 𝑥 ( ·𝑠 ‘ 𝑆 ) ( 𝑦 ( .r ‘ 𝑆 ) 𝑧 ) ) )
26 4 5 6 7 8 10 11 24 25 isassad ⊢ ( 𝜑 → 𝑆 ∈ AssAlg )