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Metamath Proof Explorer

Theorem psrring

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

Ref Expression
Hypotheses psrring.s ⊢ 𝑆 = ( 𝐼 mPwSer 𝑅 )
psrring.i ⊢ ( 𝜑 → 𝐼 ∈ 𝑉 )
psrring.r ⊢ ( 𝜑 → 𝑅 ∈ Ring )
Assertion psrring ( 𝜑 → 𝑆 ∈ Ring )

Proof

Step Hyp Ref Expression
1 psrring.s ⊢ 𝑆 = ( 𝐼 mPwSer 𝑅 )
2 psrring.i ⊢ ( 𝜑 → 𝐼 ∈ 𝑉 )
3 psrring.r ⊢ ( 𝜑 → 𝑅 ∈ Ring )
4 eqidd ⊢ ( 𝜑 → ( Base ‘ 𝑆 ) = ( Base ‘ 𝑆 ) )
5 eqidd ⊢ ( 𝜑 → ( +g ‘ 𝑆 ) = ( +g ‘ 𝑆 ) )
6 eqidd ⊢ ( 𝜑 → ( .r ‘ 𝑆 ) = ( .r ‘ 𝑆 ) )
7 ringgrp ⊢ ( 𝑅 ∈ Ring → 𝑅 ∈ Grp )
8 3 7 syl ⊢ ( 𝜑 → 𝑅 ∈ Grp )
9 1 2 8 psrgrp ⊢ ( 𝜑 → 𝑆 ∈ Grp )
10 eqid ⊢ ( Base ‘ 𝑆 ) = ( Base ‘ 𝑆 )
11 eqid ⊢ ( .r ‘ 𝑆 ) = ( .r ‘ 𝑆 )
12 3 3ad2ant1 ⊢ ( ( 𝜑 ∧ 𝑥 ∈ ( Base ‘ 𝑆 ) ∧ 𝑦 ∈ ( Base ‘ 𝑆 ) ) → 𝑅 ∈ Ring )
13 simp2 ⊢ ( ( 𝜑 ∧ 𝑥 ∈ ( Base ‘ 𝑆 ) ∧ 𝑦 ∈ ( Base ‘ 𝑆 ) ) → 𝑥 ∈ ( Base ‘ 𝑆 ) )
14 simp3 ⊢ ( ( 𝜑 ∧ 𝑥 ∈ ( Base ‘ 𝑆 ) ∧ 𝑦 ∈ ( Base ‘ 𝑆 ) ) → 𝑦 ∈ ( Base ‘ 𝑆 ) )
15 1 10 11 12 13 14 psrmulcl ⊢ ( ( 𝜑 ∧ 𝑥 ∈ ( Base ‘ 𝑆 ) ∧ 𝑦 ∈ ( Base ‘ 𝑆 ) ) → ( 𝑥 ( .r ‘ 𝑆 ) 𝑦 ) ∈ ( Base ‘ 𝑆 ) )
16 2 adantr ⊢ ( ( 𝜑 ∧ ( 𝑥 ∈ ( Base ‘ 𝑆 ) ∧ 𝑦 ∈ ( Base ‘ 𝑆 ) ∧ 𝑧 ∈ ( Base ‘ 𝑆 ) ) ) → 𝐼 ∈ 𝑉 )
17 3 adantr ⊢ ( ( 𝜑 ∧ ( 𝑥 ∈ ( Base ‘ 𝑆 ) ∧ 𝑦 ∈ ( Base ‘ 𝑆 ) ∧ 𝑧 ∈ ( Base ‘ 𝑆 ) ) ) → 𝑅 ∈ Ring )
18 eqid ⊢ { 𝑓 ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ 𝑓 “ ℕ ) ∈ Fin } = { 𝑓 ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ 𝑓 “ ℕ ) ∈ Fin }
19 simpr1 ⊢ ( ( 𝜑 ∧ ( 𝑥 ∈ ( Base ‘ 𝑆 ) ∧ 𝑦 ∈ ( Base ‘ 𝑆 ) ∧ 𝑧 ∈ ( Base ‘ 𝑆 ) ) ) → 𝑥 ∈ ( Base ‘ 𝑆 ) )
20 simpr2 ⊢ ( ( 𝜑 ∧ ( 𝑥 ∈ ( Base ‘ 𝑆 ) ∧ 𝑦 ∈ ( Base ‘ 𝑆 ) ∧ 𝑧 ∈ ( Base ‘ 𝑆 ) ) ) → 𝑦 ∈ ( Base ‘ 𝑆 ) )
21 simpr3 ⊢ ( ( 𝜑 ∧ ( 𝑥 ∈ ( Base ‘ 𝑆 ) ∧ 𝑦 ∈ ( Base ‘ 𝑆 ) ∧ 𝑧 ∈ ( Base ‘ 𝑆 ) ) ) → 𝑧 ∈ ( Base ‘ 𝑆 ) )
22 1 16 17 18 11 10 19 20 21 psrass1 ⊢ ( ( 𝜑 ∧ ( 𝑥 ∈ ( Base ‘ 𝑆 ) ∧ 𝑦 ∈ ( Base ‘ 𝑆 ) ∧ 𝑧 ∈ ( Base ‘ 𝑆 ) ) ) → ( ( 𝑥 ( .r ‘ 𝑆 ) 𝑦 ) ( .r ‘ 𝑆 ) 𝑧 ) = ( 𝑥 ( .r ‘ 𝑆 ) ( 𝑦 ( .r ‘ 𝑆 ) 𝑧 ) ) )
23 eqid ⊢ ( +g ‘ 𝑆 ) = ( +g ‘ 𝑆 )
24 1 16 17 18 11 10 19 20 21 23 psrdi ⊢ ( ( 𝜑 ∧ ( 𝑥 ∈ ( Base ‘ 𝑆 ) ∧ 𝑦 ∈ ( Base ‘ 𝑆 ) ∧ 𝑧 ∈ ( Base ‘ 𝑆 ) ) ) → ( 𝑥 ( .r ‘ 𝑆 ) ( 𝑦 ( +g ‘ 𝑆 ) 𝑧 ) ) = ( ( 𝑥 ( .r ‘ 𝑆 ) 𝑦 ) ( +g ‘ 𝑆 ) ( 𝑥 ( .r ‘ 𝑆 ) 𝑧 ) ) )
25 1 16 17 18 11 10 19 20 21 23 psrdir ⊢ ( ( 𝜑 ∧ ( 𝑥 ∈ ( Base ‘ 𝑆 ) ∧ 𝑦 ∈ ( Base ‘ 𝑆 ) ∧ 𝑧 ∈ ( Base ‘ 𝑆 ) ) ) → ( ( 𝑥 ( +g ‘ 𝑆 ) 𝑦 ) ( .r ‘ 𝑆 ) 𝑧 ) = ( ( 𝑥 ( .r ‘ 𝑆 ) 𝑧 ) ( +g ‘ 𝑆 ) ( 𝑦 ( .r ‘ 𝑆 ) 𝑧 ) ) )
26 eqid ⊢ ( 0g ‘ 𝑅 ) = ( 0g ‘ 𝑅 )
27 eqid ⊢ ( 1r ‘ 𝑅 ) = ( 1r ‘ 𝑅 )
28 eqid ⊢ ( 𝑟 ∈ { 𝑓 ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ 𝑓 “ ℕ ) ∈ Fin } ↦ if ( 𝑟 = ( 𝐼 × { 0 } ) , ( 1r ‘ 𝑅 ) , ( 0g ‘ 𝑅 ) ) ) = ( 𝑟 ∈ { 𝑓 ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ 𝑓 “ ℕ ) ∈ Fin } ↦ if ( 𝑟 = ( 𝐼 × { 0 } ) , ( 1r ‘ 𝑅 ) , ( 0g ‘ 𝑅 ) ) )
29 1 2 3 18 26 27 28 10 psr1cl ⊢ ( 𝜑 → ( 𝑟 ∈ { 𝑓 ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ 𝑓 “ ℕ ) ∈ Fin } ↦ if ( 𝑟 = ( 𝐼 × { 0 } ) , ( 1r ‘ 𝑅 ) , ( 0g ‘ 𝑅 ) ) ) ∈ ( Base ‘ 𝑆 ) )
30 2 adantr ⊢ ( ( 𝜑 ∧ 𝑥 ∈ ( Base ‘ 𝑆 ) ) → 𝐼 ∈ 𝑉 )
31 3 adantr ⊢ ( ( 𝜑 ∧ 𝑥 ∈ ( Base ‘ 𝑆 ) ) → 𝑅 ∈ Ring )
32 simpr ⊢ ( ( 𝜑 ∧ 𝑥 ∈ ( Base ‘ 𝑆 ) ) → 𝑥 ∈ ( Base ‘ 𝑆 ) )
33 1 30 31 18 26 27 28 10 11 32 psrlidm ⊢ ( ( 𝜑 ∧ 𝑥 ∈ ( Base ‘ 𝑆 ) ) → ( ( 𝑟 ∈ { 𝑓 ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ 𝑓 “ ℕ ) ∈ Fin } ↦ if ( 𝑟 = ( 𝐼 × { 0 } ) , ( 1r ‘ 𝑅 ) , ( 0g ‘ 𝑅 ) ) ) ( .r ‘ 𝑆 ) 𝑥 ) = 𝑥 )
34 1 30 31 18 26 27 28 10 11 32 psrridm ⊢ ( ( 𝜑 ∧ 𝑥 ∈ ( Base ‘ 𝑆 ) ) → ( 𝑥 ( .r ‘ 𝑆 ) ( 𝑟 ∈ { 𝑓 ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ 𝑓 “ ℕ ) ∈ Fin } ↦ if ( 𝑟 = ( 𝐼 × { 0 } ) , ( 1r ‘ 𝑅 ) , ( 0g ‘ 𝑅 ) ) ) ) = 𝑥 )
35 4 5 6 9 15 22 24 25 29 33 34 isringd ⊢ ( 𝜑 → 𝑆 ∈ Ring )