Metamath Proof Explorer

Theorem psrgrp

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

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

Proof

Step Hyp Ref Expression
1 psrgrp.s ⊢ 𝑆 = ( 𝐼 mPwSer 𝑅 )
2 psrgrp.i ⊢ ( 𝜑 → 𝐼 ∈ 𝑉 )
3 psrgrp.r ⊢ ( 𝜑 → 𝑅 ∈ Grp )
4 eqidd ⊢ ( 𝜑 → ( Base ‘ 𝑆 ) = ( Base ‘ 𝑆 ) )
5 eqidd ⊢ ( 𝜑 → ( +g ‘ 𝑆 ) = ( +g ‘ 𝑆 ) )
6 eqid ⊢ ( Base ‘ 𝑆 ) = ( Base ‘ 𝑆 )
7 eqid ⊢ ( +g ‘ 𝑆 ) = ( +g ‘ 𝑆 )
8 3 3ad2ant1 ⊢ ( ( 𝜑 ∧ 𝑥 ∈ ( Base ‘ 𝑆 ) ∧ 𝑦 ∈ ( Base ‘ 𝑆 ) ) → 𝑅 ∈ Grp )
9 simp2 ⊢ ( ( 𝜑 ∧ 𝑥 ∈ ( Base ‘ 𝑆 ) ∧ 𝑦 ∈ ( Base ‘ 𝑆 ) ) → 𝑥 ∈ ( Base ‘ 𝑆 ) )
10 simp3 ⊢ ( ( 𝜑 ∧ 𝑥 ∈ ( Base ‘ 𝑆 ) ∧ 𝑦 ∈ ( Base ‘ 𝑆 ) ) → 𝑦 ∈ ( Base ‘ 𝑆 ) )
11 1 6 7 8 9 10 psraddcl ⊢ ( ( 𝜑 ∧ 𝑥 ∈ ( Base ‘ 𝑆 ) ∧ 𝑦 ∈ ( Base ‘ 𝑆 ) ) → ( 𝑥 ( +g ‘ 𝑆 ) 𝑦 ) ∈ ( Base ‘ 𝑆 ) )
12 ovex ⊢ ( ℕ0 ↑m 𝐼 ) ∈ V
13 12 rabex ⊢ { 𝑓 ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ 𝑓 “ ℕ ) ∈ Fin } ∈ V
14 13 a1i ⊢ ( ( 𝜑 ∧ ( 𝑥 ∈ ( Base ‘ 𝑆 ) ∧ 𝑦 ∈ ( Base ‘ 𝑆 ) ∧ 𝑧 ∈ ( Base ‘ 𝑆 ) ) ) → { 𝑓 ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ 𝑓 “ ℕ ) ∈ Fin } ∈ V )
15 eqid ⊢ ( Base ‘ 𝑅 ) = ( Base ‘ 𝑅 )
16 eqid ⊢ { 𝑓 ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ 𝑓 “ ℕ ) ∈ Fin } = { 𝑓 ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ 𝑓 “ ℕ ) ∈ Fin }
17 simpr1 ⊢ ( ( 𝜑 ∧ ( 𝑥 ∈ ( Base ‘ 𝑆 ) ∧ 𝑦 ∈ ( Base ‘ 𝑆 ) ∧ 𝑧 ∈ ( Base ‘ 𝑆 ) ) ) → 𝑥 ∈ ( Base ‘ 𝑆 ) )
18 1 15 16 6 17 psrelbas ⊢ ( ( 𝜑 ∧ ( 𝑥 ∈ ( Base ‘ 𝑆 ) ∧ 𝑦 ∈ ( Base ‘ 𝑆 ) ∧ 𝑧 ∈ ( Base ‘ 𝑆 ) ) ) → 𝑥 : { 𝑓 ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ 𝑓 “ ℕ ) ∈ Fin } ⟶ ( Base ‘ 𝑅 ) )
19 simpr2 ⊢ ( ( 𝜑 ∧ ( 𝑥 ∈ ( Base ‘ 𝑆 ) ∧ 𝑦 ∈ ( Base ‘ 𝑆 ) ∧ 𝑧 ∈ ( Base ‘ 𝑆 ) ) ) → 𝑦 ∈ ( Base ‘ 𝑆 ) )
20 1 15 16 6 19 psrelbas ⊢ ( ( 𝜑 ∧ ( 𝑥 ∈ ( Base ‘ 𝑆 ) ∧ 𝑦 ∈ ( Base ‘ 𝑆 ) ∧ 𝑧 ∈ ( Base ‘ 𝑆 ) ) ) → 𝑦 : { 𝑓 ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ 𝑓 “ ℕ ) ∈ Fin } ⟶ ( Base ‘ 𝑅 ) )
21 simpr3 ⊢ ( ( 𝜑 ∧ ( 𝑥 ∈ ( Base ‘ 𝑆 ) ∧ 𝑦 ∈ ( Base ‘ 𝑆 ) ∧ 𝑧 ∈ ( Base ‘ 𝑆 ) ) ) → 𝑧 ∈ ( Base ‘ 𝑆 ) )
22 1 15 16 6 21 psrelbas ⊢ ( ( 𝜑 ∧ ( 𝑥 ∈ ( Base ‘ 𝑆 ) ∧ 𝑦 ∈ ( Base ‘ 𝑆 ) ∧ 𝑧 ∈ ( Base ‘ 𝑆 ) ) ) → 𝑧 : { 𝑓 ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ 𝑓 “ ℕ ) ∈ Fin } ⟶ ( Base ‘ 𝑅 ) )
23 3 adantr ⊢ ( ( 𝜑 ∧ ( 𝑥 ∈ ( Base ‘ 𝑆 ) ∧ 𝑦 ∈ ( Base ‘ 𝑆 ) ∧ 𝑧 ∈ ( Base ‘ 𝑆 ) ) ) → 𝑅 ∈ Grp )
24 eqid ⊢ ( +g ‘ 𝑅 ) = ( +g ‘ 𝑅 )
25 15 24 grpass ⊢ ( ( 𝑅 ∈ Grp ∧ ( 𝑟 ∈ ( Base ‘ 𝑅 ) ∧ 𝑠 ∈ ( Base ‘ 𝑅 ) ∧ 𝑡 ∈ ( Base ‘ 𝑅 ) ) ) → ( ( 𝑟 ( +g ‘ 𝑅 ) 𝑠 ) ( +g ‘ 𝑅 ) 𝑡 ) = ( 𝑟 ( +g ‘ 𝑅 ) ( 𝑠 ( +g ‘ 𝑅 ) 𝑡 ) ) )
26 23 25 sylan ⊢ ( ( ( 𝜑 ∧ ( 𝑥 ∈ ( Base ‘ 𝑆 ) ∧ 𝑦 ∈ ( Base ‘ 𝑆 ) ∧ 𝑧 ∈ ( Base ‘ 𝑆 ) ) ) ∧ ( 𝑟 ∈ ( Base ‘ 𝑅 ) ∧ 𝑠 ∈ ( Base ‘ 𝑅 ) ∧ 𝑡 ∈ ( Base ‘ 𝑅 ) ) ) → ( ( 𝑟 ( +g ‘ 𝑅 ) 𝑠 ) ( +g ‘ 𝑅 ) 𝑡 ) = ( 𝑟 ( +g ‘ 𝑅 ) ( 𝑠 ( +g ‘ 𝑅 ) 𝑡 ) ) )
27 14 18 20 22 26 caofass ⊢ ( ( 𝜑 ∧ ( 𝑥 ∈ ( Base ‘ 𝑆 ) ∧ 𝑦 ∈ ( Base ‘ 𝑆 ) ∧ 𝑧 ∈ ( Base ‘ 𝑆 ) ) ) → ( ( 𝑥 ∘f ( +g ‘ 𝑅 ) 𝑦 ) ∘f ( +g ‘ 𝑅 ) 𝑧 ) = ( 𝑥 ∘f ( +g ‘ 𝑅 ) ( 𝑦 ∘f ( +g ‘ 𝑅 ) 𝑧 ) ) )
28 1 6 24 7 17 19 psradd ⊢ ( ( 𝜑 ∧ ( 𝑥 ∈ ( Base ‘ 𝑆 ) ∧ 𝑦 ∈ ( Base ‘ 𝑆 ) ∧ 𝑧 ∈ ( Base ‘ 𝑆 ) ) ) → ( 𝑥 ( +g ‘ 𝑆 ) 𝑦 ) = ( 𝑥 ∘f ( +g ‘ 𝑅 ) 𝑦 ) )
29 28 oveq1d ⊢ ( ( 𝜑 ∧ ( 𝑥 ∈ ( Base ‘ 𝑆 ) ∧ 𝑦 ∈ ( Base ‘ 𝑆 ) ∧ 𝑧 ∈ ( Base ‘ 𝑆 ) ) ) → ( ( 𝑥 ( +g ‘ 𝑆 ) 𝑦 ) ∘f ( +g ‘ 𝑅 ) 𝑧 ) = ( ( 𝑥 ∘f ( +g ‘ 𝑅 ) 𝑦 ) ∘f ( +g ‘ 𝑅 ) 𝑧 ) )
30 1 6 24 7 19 21 psradd ⊢ ( ( 𝜑 ∧ ( 𝑥 ∈ ( Base ‘ 𝑆 ) ∧ 𝑦 ∈ ( Base ‘ 𝑆 ) ∧ 𝑧 ∈ ( Base ‘ 𝑆 ) ) ) → ( 𝑦 ( +g ‘ 𝑆 ) 𝑧 ) = ( 𝑦 ∘f ( +g ‘ 𝑅 ) 𝑧 ) )
31 30 oveq2d ⊢ ( ( 𝜑 ∧ ( 𝑥 ∈ ( Base ‘ 𝑆 ) ∧ 𝑦 ∈ ( Base ‘ 𝑆 ) ∧ 𝑧 ∈ ( Base ‘ 𝑆 ) ) ) → ( 𝑥 ∘f ( +g ‘ 𝑅 ) ( 𝑦 ( +g ‘ 𝑆 ) 𝑧 ) ) = ( 𝑥 ∘f ( +g ‘ 𝑅 ) ( 𝑦 ∘f ( +g ‘ 𝑅 ) 𝑧 ) ) )
32 27 29 31 3eqtr4d ⊢ ( ( 𝜑 ∧ ( 𝑥 ∈ ( Base ‘ 𝑆 ) ∧ 𝑦 ∈ ( Base ‘ 𝑆 ) ∧ 𝑧 ∈ ( Base ‘ 𝑆 ) ) ) → ( ( 𝑥 ( +g ‘ 𝑆 ) 𝑦 ) ∘f ( +g ‘ 𝑅 ) 𝑧 ) = ( 𝑥 ∘f ( +g ‘ 𝑅 ) ( 𝑦 ( +g ‘ 𝑆 ) 𝑧 ) ) )
33 11 3adant3r3 ⊢ ( ( 𝜑 ∧ ( 𝑥 ∈ ( Base ‘ 𝑆 ) ∧ 𝑦 ∈ ( Base ‘ 𝑆 ) ∧ 𝑧 ∈ ( Base ‘ 𝑆 ) ) ) → ( 𝑥 ( +g ‘ 𝑆 ) 𝑦 ) ∈ ( Base ‘ 𝑆 ) )
34 1 6 24 7 33 21 psradd ⊢ ( ( 𝜑 ∧ ( 𝑥 ∈ ( Base ‘ 𝑆 ) ∧ 𝑦 ∈ ( Base ‘ 𝑆 ) ∧ 𝑧 ∈ ( Base ‘ 𝑆 ) ) ) → ( ( 𝑥 ( +g ‘ 𝑆 ) 𝑦 ) ( +g ‘ 𝑆 ) 𝑧 ) = ( ( 𝑥 ( +g ‘ 𝑆 ) 𝑦 ) ∘f ( +g ‘ 𝑅 ) 𝑧 ) )
35 1 6 7 23 19 21 psraddcl ⊢ ( ( 𝜑 ∧ ( 𝑥 ∈ ( Base ‘ 𝑆 ) ∧ 𝑦 ∈ ( Base ‘ 𝑆 ) ∧ 𝑧 ∈ ( Base ‘ 𝑆 ) ) ) → ( 𝑦 ( +g ‘ 𝑆 ) 𝑧 ) ∈ ( Base ‘ 𝑆 ) )
36 1 6 24 7 17 35 psradd ⊢ ( ( 𝜑 ∧ ( 𝑥 ∈ ( Base ‘ 𝑆 ) ∧ 𝑦 ∈ ( Base ‘ 𝑆 ) ∧ 𝑧 ∈ ( Base ‘ 𝑆 ) ) ) → ( 𝑥 ( +g ‘ 𝑆 ) ( 𝑦 ( +g ‘ 𝑆 ) 𝑧 ) ) = ( 𝑥 ∘f ( +g ‘ 𝑅 ) ( 𝑦 ( +g ‘ 𝑆 ) 𝑧 ) ) )
37 32 34 36 3eqtr4d ⊢ ( ( 𝜑 ∧ ( 𝑥 ∈ ( Base ‘ 𝑆 ) ∧ 𝑦 ∈ ( Base ‘ 𝑆 ) ∧ 𝑧 ∈ ( Base ‘ 𝑆 ) ) ) → ( ( 𝑥 ( +g ‘ 𝑆 ) 𝑦 ) ( +g ‘ 𝑆 ) 𝑧 ) = ( 𝑥 ( +g ‘ 𝑆 ) ( 𝑦 ( +g ‘ 𝑆 ) 𝑧 ) ) )
38 eqid ⊢ ( 0g ‘ 𝑅 ) = ( 0g ‘ 𝑅 )
39 1 2 3 16 38 6 psr0cl ⊢ ( 𝜑 → ( { 𝑓 ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ 𝑓 “ ℕ ) ∈ Fin } × { ( 0g ‘ 𝑅 ) } ) ∈ ( Base ‘ 𝑆 ) )
40 2 adantr ⊢ ( ( 𝜑 ∧ 𝑥 ∈ ( Base ‘ 𝑆 ) ) → 𝐼 ∈ 𝑉 )
41 3 adantr ⊢ ( ( 𝜑 ∧ 𝑥 ∈ ( Base ‘ 𝑆 ) ) → 𝑅 ∈ Grp )
42 simpr ⊢ ( ( 𝜑 ∧ 𝑥 ∈ ( Base ‘ 𝑆 ) ) → 𝑥 ∈ ( Base ‘ 𝑆 ) )
43 1 40 41 16 38 6 7 42 psr0lid ⊢ ( ( 𝜑 ∧ 𝑥 ∈ ( Base ‘ 𝑆 ) ) → ( ( { 𝑓 ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ 𝑓 “ ℕ ) ∈ Fin } × { ( 0g ‘ 𝑅 ) } ) ( +g ‘ 𝑆 ) 𝑥 ) = 𝑥 )
44 eqid ⊢ ( invg ‘ 𝑅 ) = ( invg ‘ 𝑅 )
45 1 40 41 16 44 6 42 psrnegcl ⊢ ( ( 𝜑 ∧ 𝑥 ∈ ( Base ‘ 𝑆 ) ) → ( ( invg ‘ 𝑅 ) ∘ 𝑥 ) ∈ ( Base ‘ 𝑆 ) )
46 1 40 41 16 44 6 42 38 7 psrlinv ⊢ ( ( 𝜑 ∧ 𝑥 ∈ ( Base ‘ 𝑆 ) ) → ( ( ( invg ‘ 𝑅 ) ∘ 𝑥 ) ( +g ‘ 𝑆 ) 𝑥 ) = ( { 𝑓 ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ 𝑓 “ ℕ ) ∈ Fin } × { ( 0g ‘ 𝑅 ) } ) )
47 4 5 11 37 39 43 45 46 isgrpd ⊢ ( 𝜑 → 𝑆 ∈ Grp )