Metamath Proof Explorer

Theorem psrlmod

Description: The ring of power series is a left module. (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 psrlmod ( 𝜑 → 𝑆 ∈ LMod )

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 1 2 3 psrsca ⊢ ( 𝜑 → 𝑅 = ( Scalar ‘ 𝑆 ) )
7 eqidd ⊢ ( 𝜑 → ( ·𝑠 ‘ 𝑆 ) = ( ·𝑠 ‘ 𝑆 ) )
8 eqidd ⊢ ( 𝜑 → ( Base ‘ 𝑅 ) = ( Base ‘ 𝑅 ) )
9 eqidd ⊢ ( 𝜑 → ( +g ‘ 𝑅 ) = ( +g ‘ 𝑅 ) )
10 eqidd ⊢ ( 𝜑 → ( .r ‘ 𝑅 ) = ( .r ‘ 𝑅 ) )
11 eqidd ⊢ ( 𝜑 → ( 1r ‘ 𝑅 ) = ( 1r ‘ 𝑅 ) )
12 ringgrp ⊢ ( 𝑅 ∈ Ring → 𝑅 ∈ Grp )
13 3 12 syl ⊢ ( 𝜑 → 𝑅 ∈ Grp )
14 1 2 13 psrgrp ⊢ ( 𝜑 → 𝑆 ∈ Grp )
15 eqid ⊢ ( ·𝑠 ‘ 𝑆 ) = ( ·𝑠 ‘ 𝑆 )
16 eqid ⊢ ( Base ‘ 𝑅 ) = ( Base ‘ 𝑅 )
17 eqid ⊢ ( Base ‘ 𝑆 ) = ( Base ‘ 𝑆 )
18 3 3ad2ant1 ⊢ ( ( 𝜑 ∧ 𝑥 ∈ ( Base ‘ 𝑅 ) ∧ 𝑦 ∈ ( Base ‘ 𝑆 ) ) → 𝑅 ∈ Ring )
19 simp2 ⊢ ( ( 𝜑 ∧ 𝑥 ∈ ( Base ‘ 𝑅 ) ∧ 𝑦 ∈ ( Base ‘ 𝑆 ) ) → 𝑥 ∈ ( Base ‘ 𝑅 ) )
20 simp3 ⊢ ( ( 𝜑 ∧ 𝑥 ∈ ( Base ‘ 𝑅 ) ∧ 𝑦 ∈ ( Base ‘ 𝑆 ) ) → 𝑦 ∈ ( Base ‘ 𝑆 ) )
21 1 15 16 17 18 19 20 psrvscacl ⊢ ( ( 𝜑 ∧ 𝑥 ∈ ( Base ‘ 𝑅 ) ∧ 𝑦 ∈ ( Base ‘ 𝑆 ) ) → ( 𝑥 ( ·𝑠 ‘ 𝑆 ) 𝑦 ) ∈ ( Base ‘ 𝑆 ) )
22 ovex ⊢ ( ℕ0 ↑m 𝐼 ) ∈ V
23 22 rabex ⊢ { 𝑓 ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ 𝑓 “ ℕ ) ∈ Fin } ∈ V
24 23 a1i ⊢ ( ( 𝜑 ∧ ( 𝑥 ∈ ( Base ‘ 𝑅 ) ∧ 𝑦 ∈ ( Base ‘ 𝑆 ) ∧ 𝑧 ∈ ( Base ‘ 𝑆 ) ) ) → { 𝑓 ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ 𝑓 “ ℕ ) ∈ Fin } ∈ V )
25 simpr1 ⊢ ( ( 𝜑 ∧ ( 𝑥 ∈ ( Base ‘ 𝑅 ) ∧ 𝑦 ∈ ( Base ‘ 𝑆 ) ∧ 𝑧 ∈ ( Base ‘ 𝑆 ) ) ) → 𝑥 ∈ ( Base ‘ 𝑅 ) )
26 fconst6g ⊢ ( 𝑥 ∈ ( Base ‘ 𝑅 ) → ( { 𝑓 ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ 𝑓 “ ℕ ) ∈ Fin } × { 𝑥 } ) : { 𝑓 ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ 𝑓 “ ℕ ) ∈ Fin } ⟶ ( Base ‘ 𝑅 ) )
27 25 26 syl ⊢ ( ( 𝜑 ∧ ( 𝑥 ∈ ( Base ‘ 𝑅 ) ∧ 𝑦 ∈ ( Base ‘ 𝑆 ) ∧ 𝑧 ∈ ( Base ‘ 𝑆 ) ) ) → ( { 𝑓 ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ 𝑓 “ ℕ ) ∈ Fin } × { 𝑥 } ) : { 𝑓 ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ 𝑓 “ ℕ ) ∈ Fin } ⟶ ( Base ‘ 𝑅 ) )
28 eqid ⊢ { 𝑓 ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ 𝑓 “ ℕ ) ∈ Fin } = { 𝑓 ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ 𝑓 “ ℕ ) ∈ Fin }
29 simpr2 ⊢ ( ( 𝜑 ∧ ( 𝑥 ∈ ( Base ‘ 𝑅 ) ∧ 𝑦 ∈ ( Base ‘ 𝑆 ) ∧ 𝑧 ∈ ( Base ‘ 𝑆 ) ) ) → 𝑦 ∈ ( Base ‘ 𝑆 ) )
30 1 16 28 17 29 psrelbas ⊢ ( ( 𝜑 ∧ ( 𝑥 ∈ ( Base ‘ 𝑅 ) ∧ 𝑦 ∈ ( Base ‘ 𝑆 ) ∧ 𝑧 ∈ ( Base ‘ 𝑆 ) ) ) → 𝑦 : { 𝑓 ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ 𝑓 “ ℕ ) ∈ Fin } ⟶ ( Base ‘ 𝑅 ) )
31 simpr3 ⊢ ( ( 𝜑 ∧ ( 𝑥 ∈ ( Base ‘ 𝑅 ) ∧ 𝑦 ∈ ( Base ‘ 𝑆 ) ∧ 𝑧 ∈ ( Base ‘ 𝑆 ) ) ) → 𝑧 ∈ ( Base ‘ 𝑆 ) )
32 1 16 28 17 31 psrelbas ⊢ ( ( 𝜑 ∧ ( 𝑥 ∈ ( Base ‘ 𝑅 ) ∧ 𝑦 ∈ ( Base ‘ 𝑆 ) ∧ 𝑧 ∈ ( Base ‘ 𝑆 ) ) ) → 𝑧 : { 𝑓 ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ 𝑓 “ ℕ ) ∈ Fin } ⟶ ( Base ‘ 𝑅 ) )
33 3 adantr ⊢ ( ( 𝜑 ∧ ( 𝑥 ∈ ( Base ‘ 𝑅 ) ∧ 𝑦 ∈ ( Base ‘ 𝑆 ) ∧ 𝑧 ∈ ( Base ‘ 𝑆 ) ) ) → 𝑅 ∈ Ring )
34 eqid ⊢ ( +g ‘ 𝑅 ) = ( +g ‘ 𝑅 )
35 eqid ⊢ ( .r ‘ 𝑅 ) = ( .r ‘ 𝑅 )
36 16 34 35 ringdi ⊢ ( ( 𝑅 ∈ Ring ∧ ( 𝑟 ∈ ( Base ‘ 𝑅 ) ∧ 𝑠 ∈ ( Base ‘ 𝑅 ) ∧ 𝑡 ∈ ( Base ‘ 𝑅 ) ) ) → ( 𝑟 ( .r ‘ 𝑅 ) ( 𝑠 ( +g ‘ 𝑅 ) 𝑡 ) ) = ( ( 𝑟 ( .r ‘ 𝑅 ) 𝑠 ) ( +g ‘ 𝑅 ) ( 𝑟 ( .r ‘ 𝑅 ) 𝑡 ) ) )
37 33 36 sylan ⊢ ( ( ( 𝜑 ∧ ( 𝑥 ∈ ( Base ‘ 𝑅 ) ∧ 𝑦 ∈ ( Base ‘ 𝑆 ) ∧ 𝑧 ∈ ( Base ‘ 𝑆 ) ) ) ∧ ( 𝑟 ∈ ( Base ‘ 𝑅 ) ∧ 𝑠 ∈ ( Base ‘ 𝑅 ) ∧ 𝑡 ∈ ( Base ‘ 𝑅 ) ) ) → ( 𝑟 ( .r ‘ 𝑅 ) ( 𝑠 ( +g ‘ 𝑅 ) 𝑡 ) ) = ( ( 𝑟 ( .r ‘ 𝑅 ) 𝑠 ) ( +g ‘ 𝑅 ) ( 𝑟 ( .r ‘ 𝑅 ) 𝑡 ) ) )
38 24 27 30 32 37 caofdi ⊢ ( ( 𝜑 ∧ ( 𝑥 ∈ ( Base ‘ 𝑅 ) ∧ 𝑦 ∈ ( Base ‘ 𝑆 ) ∧ 𝑧 ∈ ( Base ‘ 𝑆 ) ) ) → ( ( { 𝑓 ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ 𝑓 “ ℕ ) ∈ Fin } × { 𝑥 } ) ∘f ( .r ‘ 𝑅 ) ( 𝑦 ∘f ( +g ‘ 𝑅 ) 𝑧 ) ) = ( ( ( { 𝑓 ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ 𝑓 “ ℕ ) ∈ Fin } × { 𝑥 } ) ∘f ( .r ‘ 𝑅 ) 𝑦 ) ∘f ( +g ‘ 𝑅 ) ( ( { 𝑓 ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ 𝑓 “ ℕ ) ∈ Fin } × { 𝑥 } ) ∘f ( .r ‘ 𝑅 ) 𝑧 ) ) )
39 eqid ⊢ ( +g ‘ 𝑆 ) = ( +g ‘ 𝑆 )
40 1 17 34 39 29 31 psradd ⊢ ( ( 𝜑 ∧ ( 𝑥 ∈ ( Base ‘ 𝑅 ) ∧ 𝑦 ∈ ( Base ‘ 𝑆 ) ∧ 𝑧 ∈ ( Base ‘ 𝑆 ) ) ) → ( 𝑦 ( +g ‘ 𝑆 ) 𝑧 ) = ( 𝑦 ∘f ( +g ‘ 𝑅 ) 𝑧 ) )
41 40 oveq2d ⊢ ( ( 𝜑 ∧ ( 𝑥 ∈ ( Base ‘ 𝑅 ) ∧ 𝑦 ∈ ( Base ‘ 𝑆 ) ∧ 𝑧 ∈ ( Base ‘ 𝑆 ) ) ) → ( ( { 𝑓 ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ 𝑓 “ ℕ ) ∈ Fin } × { 𝑥 } ) ∘f ( .r ‘ 𝑅 ) ( 𝑦 ( +g ‘ 𝑆 ) 𝑧 ) ) = ( ( { 𝑓 ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ 𝑓 “ ℕ ) ∈ Fin } × { 𝑥 } ) ∘f ( .r ‘ 𝑅 ) ( 𝑦 ∘f ( +g ‘ 𝑅 ) 𝑧 ) ) )
42 1 15 16 17 35 28 25 29 psrvsca ⊢ ( ( 𝜑 ∧ ( 𝑥 ∈ ( Base ‘ 𝑅 ) ∧ 𝑦 ∈ ( Base ‘ 𝑆 ) ∧ 𝑧 ∈ ( Base ‘ 𝑆 ) ) ) → ( 𝑥 ( ·𝑠 ‘ 𝑆 ) 𝑦 ) = ( ( { 𝑓 ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ 𝑓 “ ℕ ) ∈ Fin } × { 𝑥 } ) ∘f ( .r ‘ 𝑅 ) 𝑦 ) )
43 1 15 16 17 35 28 25 31 psrvsca ⊢ ( ( 𝜑 ∧ ( 𝑥 ∈ ( Base ‘ 𝑅 ) ∧ 𝑦 ∈ ( Base ‘ 𝑆 ) ∧ 𝑧 ∈ ( Base ‘ 𝑆 ) ) ) → ( 𝑥 ( ·𝑠 ‘ 𝑆 ) 𝑧 ) = ( ( { 𝑓 ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ 𝑓 “ ℕ ) ∈ Fin } × { 𝑥 } ) ∘f ( .r ‘ 𝑅 ) 𝑧 ) )
44 42 43 oveq12d ⊢ ( ( 𝜑 ∧ ( 𝑥 ∈ ( Base ‘ 𝑅 ) ∧ 𝑦 ∈ ( Base ‘ 𝑆 ) ∧ 𝑧 ∈ ( Base ‘ 𝑆 ) ) ) → ( ( 𝑥 ( ·𝑠 ‘ 𝑆 ) 𝑦 ) ∘f ( +g ‘ 𝑅 ) ( 𝑥 ( ·𝑠 ‘ 𝑆 ) 𝑧 ) ) = ( ( ( { 𝑓 ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ 𝑓 “ ℕ ) ∈ Fin } × { 𝑥 } ) ∘f ( .r ‘ 𝑅 ) 𝑦 ) ∘f ( +g ‘ 𝑅 ) ( ( { 𝑓 ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ 𝑓 “ ℕ ) ∈ Fin } × { 𝑥 } ) ∘f ( .r ‘ 𝑅 ) 𝑧 ) ) )
45 38 41 44 3eqtr4d ⊢ ( ( 𝜑 ∧ ( 𝑥 ∈ ( Base ‘ 𝑅 ) ∧ 𝑦 ∈ ( Base ‘ 𝑆 ) ∧ 𝑧 ∈ ( Base ‘ 𝑆 ) ) ) → ( ( { 𝑓 ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ 𝑓 “ ℕ ) ∈ Fin } × { 𝑥 } ) ∘f ( .r ‘ 𝑅 ) ( 𝑦 ( +g ‘ 𝑆 ) 𝑧 ) ) = ( ( 𝑥 ( ·𝑠 ‘ 𝑆 ) 𝑦 ) ∘f ( +g ‘ 𝑅 ) ( 𝑥 ( ·𝑠 ‘ 𝑆 ) 𝑧 ) ) )
46 13 grpmgmd ⊢ ( 𝜑 → 𝑅 ∈ Mgm )
47 46 adantr ⊢ ( ( 𝜑 ∧ ( 𝑥 ∈ ( Base ‘ 𝑅 ) ∧ 𝑦 ∈ ( Base ‘ 𝑆 ) ∧ 𝑧 ∈ ( Base ‘ 𝑆 ) ) ) → 𝑅 ∈ Mgm )
48 1 17 39 47 29 31 psraddcl ⊢ ( ( 𝜑 ∧ ( 𝑥 ∈ ( Base ‘ 𝑅 ) ∧ 𝑦 ∈ ( Base ‘ 𝑆 ) ∧ 𝑧 ∈ ( Base ‘ 𝑆 ) ) ) → ( 𝑦 ( +g ‘ 𝑆 ) 𝑧 ) ∈ ( Base ‘ 𝑆 ) )
49 1 15 16 17 35 28 25 48 psrvsca ⊢ ( ( 𝜑 ∧ ( 𝑥 ∈ ( Base ‘ 𝑅 ) ∧ 𝑦 ∈ ( Base ‘ 𝑆 ) ∧ 𝑧 ∈ ( Base ‘ 𝑆 ) ) ) → ( 𝑥 ( ·𝑠 ‘ 𝑆 ) ( 𝑦 ( +g ‘ 𝑆 ) 𝑧 ) ) = ( ( { 𝑓 ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ 𝑓 “ ℕ ) ∈ Fin } × { 𝑥 } ) ∘f ( .r ‘ 𝑅 ) ( 𝑦 ( +g ‘ 𝑆 ) 𝑧 ) ) )
50 21 3adant3r3 ⊢ ( ( 𝜑 ∧ ( 𝑥 ∈ ( Base ‘ 𝑅 ) ∧ 𝑦 ∈ ( Base ‘ 𝑆 ) ∧ 𝑧 ∈ ( Base ‘ 𝑆 ) ) ) → ( 𝑥 ( ·𝑠 ‘ 𝑆 ) 𝑦 ) ∈ ( Base ‘ 𝑆 ) )
51 1 15 16 17 33 25 31 psrvscacl ⊢ ( ( 𝜑 ∧ ( 𝑥 ∈ ( Base ‘ 𝑅 ) ∧ 𝑦 ∈ ( Base ‘ 𝑆 ) ∧ 𝑧 ∈ ( Base ‘ 𝑆 ) ) ) → ( 𝑥 ( ·𝑠 ‘ 𝑆 ) 𝑧 ) ∈ ( Base ‘ 𝑆 ) )
52 1 17 34 39 50 51 psradd ⊢ ( ( 𝜑 ∧ ( 𝑥 ∈ ( Base ‘ 𝑅 ) ∧ 𝑦 ∈ ( Base ‘ 𝑆 ) ∧ 𝑧 ∈ ( Base ‘ 𝑆 ) ) ) → ( ( 𝑥 ( ·𝑠 ‘ 𝑆 ) 𝑦 ) ( +g ‘ 𝑆 ) ( 𝑥 ( ·𝑠 ‘ 𝑆 ) 𝑧 ) ) = ( ( 𝑥 ( ·𝑠 ‘ 𝑆 ) 𝑦 ) ∘f ( +g ‘ 𝑅 ) ( 𝑥 ( ·𝑠 ‘ 𝑆 ) 𝑧 ) ) )
53 45 49 52 3eqtr4d ⊢ ( ( 𝜑 ∧ ( 𝑥 ∈ ( Base ‘ 𝑅 ) ∧ 𝑦 ∈ ( Base ‘ 𝑆 ) ∧ 𝑧 ∈ ( Base ‘ 𝑆 ) ) ) → ( 𝑥 ( ·𝑠 ‘ 𝑆 ) ( 𝑦 ( +g ‘ 𝑆 ) 𝑧 ) ) = ( ( 𝑥 ( ·𝑠 ‘ 𝑆 ) 𝑦 ) ( +g ‘ 𝑆 ) ( 𝑥 ( ·𝑠 ‘ 𝑆 ) 𝑧 ) ) )
54 simpr1 ⊢ ( ( 𝜑 ∧ ( 𝑥 ∈ ( Base ‘ 𝑅 ) ∧ 𝑦 ∈ ( Base ‘ 𝑅 ) ∧ 𝑧 ∈ ( Base ‘ 𝑆 ) ) ) → 𝑥 ∈ ( Base ‘ 𝑅 ) )
55 simpr3 ⊢ ( ( 𝜑 ∧ ( 𝑥 ∈ ( Base ‘ 𝑅 ) ∧ 𝑦 ∈ ( Base ‘ 𝑅 ) ∧ 𝑧 ∈ ( Base ‘ 𝑆 ) ) ) → 𝑧 ∈ ( Base ‘ 𝑆 ) )
56 1 15 16 17 35 28 54 55 psrvsca ⊢ ( ( 𝜑 ∧ ( 𝑥 ∈ ( Base ‘ 𝑅 ) ∧ 𝑦 ∈ ( Base ‘ 𝑅 ) ∧ 𝑧 ∈ ( Base ‘ 𝑆 ) ) ) → ( 𝑥 ( ·𝑠 ‘ 𝑆 ) 𝑧 ) = ( ( { 𝑓 ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ 𝑓 “ ℕ ) ∈ Fin } × { 𝑥 } ) ∘f ( .r ‘ 𝑅 ) 𝑧 ) )
57 simpr2 ⊢ ( ( 𝜑 ∧ ( 𝑥 ∈ ( Base ‘ 𝑅 ) ∧ 𝑦 ∈ ( Base ‘ 𝑅 ) ∧ 𝑧 ∈ ( Base ‘ 𝑆 ) ) ) → 𝑦 ∈ ( Base ‘ 𝑅 ) )
58 1 15 16 17 35 28 57 55 psrvsca ⊢ ( ( 𝜑 ∧ ( 𝑥 ∈ ( Base ‘ 𝑅 ) ∧ 𝑦 ∈ ( Base ‘ 𝑅 ) ∧ 𝑧 ∈ ( Base ‘ 𝑆 ) ) ) → ( 𝑦 ( ·𝑠 ‘ 𝑆 ) 𝑧 ) = ( ( { 𝑓 ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ 𝑓 “ ℕ ) ∈ Fin } × { 𝑦 } ) ∘f ( .r ‘ 𝑅 ) 𝑧 ) )
59 56 58 oveq12d ⊢ ( ( 𝜑 ∧ ( 𝑥 ∈ ( Base ‘ 𝑅 ) ∧ 𝑦 ∈ ( Base ‘ 𝑅 ) ∧ 𝑧 ∈ ( Base ‘ 𝑆 ) ) ) → ( ( 𝑥 ( ·𝑠 ‘ 𝑆 ) 𝑧 ) ∘f ( +g ‘ 𝑅 ) ( 𝑦 ( ·𝑠 ‘ 𝑆 ) 𝑧 ) ) = ( ( ( { 𝑓 ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ 𝑓 “ ℕ ) ∈ Fin } × { 𝑥 } ) ∘f ( .r ‘ 𝑅 ) 𝑧 ) ∘f ( +g ‘ 𝑅 ) ( ( { 𝑓 ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ 𝑓 “ ℕ ) ∈ Fin } × { 𝑦 } ) ∘f ( .r ‘ 𝑅 ) 𝑧 ) ) )
60 23 a1i ⊢ ( ( 𝜑 ∧ ( 𝑥 ∈ ( Base ‘ 𝑅 ) ∧ 𝑦 ∈ ( Base ‘ 𝑅 ) ∧ 𝑧 ∈ ( Base ‘ 𝑆 ) ) ) → { 𝑓 ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ 𝑓 “ ℕ ) ∈ Fin } ∈ V )
61 1 16 28 17 55 psrelbas ⊢ ( ( 𝜑 ∧ ( 𝑥 ∈ ( Base ‘ 𝑅 ) ∧ 𝑦 ∈ ( Base ‘ 𝑅 ) ∧ 𝑧 ∈ ( Base ‘ 𝑆 ) ) ) → 𝑧 : { 𝑓 ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ 𝑓 “ ℕ ) ∈ Fin } ⟶ ( Base ‘ 𝑅 ) )
62 54 26 syl ⊢ ( ( 𝜑 ∧ ( 𝑥 ∈ ( Base ‘ 𝑅 ) ∧ 𝑦 ∈ ( Base ‘ 𝑅 ) ∧ 𝑧 ∈ ( Base ‘ 𝑆 ) ) ) → ( { 𝑓 ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ 𝑓 “ ℕ ) ∈ Fin } × { 𝑥 } ) : { 𝑓 ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ 𝑓 “ ℕ ) ∈ Fin } ⟶ ( Base ‘ 𝑅 ) )
63 fconst6g ⊢ ( 𝑦 ∈ ( Base ‘ 𝑅 ) → ( { 𝑓 ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ 𝑓 “ ℕ ) ∈ Fin } × { 𝑦 } ) : { 𝑓 ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ 𝑓 “ ℕ ) ∈ Fin } ⟶ ( Base ‘ 𝑅 ) )
64 57 63 syl ⊢ ( ( 𝜑 ∧ ( 𝑥 ∈ ( Base ‘ 𝑅 ) ∧ 𝑦 ∈ ( Base ‘ 𝑅 ) ∧ 𝑧 ∈ ( Base ‘ 𝑆 ) ) ) → ( { 𝑓 ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ 𝑓 “ ℕ ) ∈ Fin } × { 𝑦 } ) : { 𝑓 ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ 𝑓 “ ℕ ) ∈ Fin } ⟶ ( Base ‘ 𝑅 ) )
65 3 adantr ⊢ ( ( 𝜑 ∧ ( 𝑥 ∈ ( Base ‘ 𝑅 ) ∧ 𝑦 ∈ ( Base ‘ 𝑅 ) ∧ 𝑧 ∈ ( Base ‘ 𝑆 ) ) ) → 𝑅 ∈ Ring )
66 16 34 35 ringdir ⊢ ( ( 𝑅 ∈ Ring ∧ ( 𝑟 ∈ ( Base ‘ 𝑅 ) ∧ 𝑠 ∈ ( Base ‘ 𝑅 ) ∧ 𝑡 ∈ ( Base ‘ 𝑅 ) ) ) → ( ( 𝑟 ( +g ‘ 𝑅 ) 𝑠 ) ( .r ‘ 𝑅 ) 𝑡 ) = ( ( 𝑟 ( .r ‘ 𝑅 ) 𝑡 ) ( +g ‘ 𝑅 ) ( 𝑠 ( .r ‘ 𝑅 ) 𝑡 ) ) )
67 65 66 sylan ⊢ ( ( ( 𝜑 ∧ ( 𝑥 ∈ ( Base ‘ 𝑅 ) ∧ 𝑦 ∈ ( Base ‘ 𝑅 ) ∧ 𝑧 ∈ ( Base ‘ 𝑆 ) ) ) ∧ ( 𝑟 ∈ ( Base ‘ 𝑅 ) ∧ 𝑠 ∈ ( Base ‘ 𝑅 ) ∧ 𝑡 ∈ ( Base ‘ 𝑅 ) ) ) → ( ( 𝑟 ( +g ‘ 𝑅 ) 𝑠 ) ( .r ‘ 𝑅 ) 𝑡 ) = ( ( 𝑟 ( .r ‘ 𝑅 ) 𝑡 ) ( +g ‘ 𝑅 ) ( 𝑠 ( .r ‘ 𝑅 ) 𝑡 ) ) )
68 60 61 62 64 67 caofdir ⊢ ( ( 𝜑 ∧ ( 𝑥 ∈ ( Base ‘ 𝑅 ) ∧ 𝑦 ∈ ( Base ‘ 𝑅 ) ∧ 𝑧 ∈ ( Base ‘ 𝑆 ) ) ) → ( ( ( { 𝑓 ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ 𝑓 “ ℕ ) ∈ Fin } × { 𝑥 } ) ∘f ( +g ‘ 𝑅 ) ( { 𝑓 ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ 𝑓 “ ℕ ) ∈ Fin } × { 𝑦 } ) ) ∘f ( .r ‘ 𝑅 ) 𝑧 ) = ( ( ( { 𝑓 ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ 𝑓 “ ℕ ) ∈ Fin } × { 𝑥 } ) ∘f ( .r ‘ 𝑅 ) 𝑧 ) ∘f ( +g ‘ 𝑅 ) ( ( { 𝑓 ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ 𝑓 “ ℕ ) ∈ Fin } × { 𝑦 } ) ∘f ( .r ‘ 𝑅 ) 𝑧 ) ) )
69 60 54 57 ofc12 ⊢ ( ( 𝜑 ∧ ( 𝑥 ∈ ( Base ‘ 𝑅 ) ∧ 𝑦 ∈ ( Base ‘ 𝑅 ) ∧ 𝑧 ∈ ( Base ‘ 𝑆 ) ) ) → ( ( { 𝑓 ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ 𝑓 “ ℕ ) ∈ Fin } × { 𝑥 } ) ∘f ( +g ‘ 𝑅 ) ( { 𝑓 ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ 𝑓 “ ℕ ) ∈ Fin } × { 𝑦 } ) ) = ( { 𝑓 ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ 𝑓 “ ℕ ) ∈ Fin } × { ( 𝑥 ( +g ‘ 𝑅 ) 𝑦 ) } ) )
70 69 oveq1d ⊢ ( ( 𝜑 ∧ ( 𝑥 ∈ ( Base ‘ 𝑅 ) ∧ 𝑦 ∈ ( Base ‘ 𝑅 ) ∧ 𝑧 ∈ ( Base ‘ 𝑆 ) ) ) → ( ( ( { 𝑓 ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ 𝑓 “ ℕ ) ∈ Fin } × { 𝑥 } ) ∘f ( +g ‘ 𝑅 ) ( { 𝑓 ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ 𝑓 “ ℕ ) ∈ Fin } × { 𝑦 } ) ) ∘f ( .r ‘ 𝑅 ) 𝑧 ) = ( ( { 𝑓 ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ 𝑓 “ ℕ ) ∈ Fin } × { ( 𝑥 ( +g ‘ 𝑅 ) 𝑦 ) } ) ∘f ( .r ‘ 𝑅 ) 𝑧 ) )
71 59 68 70 3eqtr2rd ⊢ ( ( 𝜑 ∧ ( 𝑥 ∈ ( Base ‘ 𝑅 ) ∧ 𝑦 ∈ ( Base ‘ 𝑅 ) ∧ 𝑧 ∈ ( Base ‘ 𝑆 ) ) ) → ( ( { 𝑓 ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ 𝑓 “ ℕ ) ∈ Fin } × { ( 𝑥 ( +g ‘ 𝑅 ) 𝑦 ) } ) ∘f ( .r ‘ 𝑅 ) 𝑧 ) = ( ( 𝑥 ( ·𝑠 ‘ 𝑆 ) 𝑧 ) ∘f ( +g ‘ 𝑅 ) ( 𝑦 ( ·𝑠 ‘ 𝑆 ) 𝑧 ) ) )
72 16 34 ringacl ⊢ ( ( 𝑅 ∈ Ring ∧ 𝑥 ∈ ( Base ‘ 𝑅 ) ∧ 𝑦 ∈ ( Base ‘ 𝑅 ) ) → ( 𝑥 ( +g ‘ 𝑅 ) 𝑦 ) ∈ ( Base ‘ 𝑅 ) )
73 65 54 57 72 syl3anc ⊢ ( ( 𝜑 ∧ ( 𝑥 ∈ ( Base ‘ 𝑅 ) ∧ 𝑦 ∈ ( Base ‘ 𝑅 ) ∧ 𝑧 ∈ ( Base ‘ 𝑆 ) ) ) → ( 𝑥 ( +g ‘ 𝑅 ) 𝑦 ) ∈ ( Base ‘ 𝑅 ) )
74 1 15 16 17 35 28 73 55 psrvsca ⊢ ( ( 𝜑 ∧ ( 𝑥 ∈ ( Base ‘ 𝑅 ) ∧ 𝑦 ∈ ( Base ‘ 𝑅 ) ∧ 𝑧 ∈ ( Base ‘ 𝑆 ) ) ) → ( ( 𝑥 ( +g ‘ 𝑅 ) 𝑦 ) ( ·𝑠 ‘ 𝑆 ) 𝑧 ) = ( ( { 𝑓 ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ 𝑓 “ ℕ ) ∈ Fin } × { ( 𝑥 ( +g ‘ 𝑅 ) 𝑦 ) } ) ∘f ( .r ‘ 𝑅 ) 𝑧 ) )
75 1 15 16 17 65 54 55 psrvscacl ⊢ ( ( 𝜑 ∧ ( 𝑥 ∈ ( Base ‘ 𝑅 ) ∧ 𝑦 ∈ ( Base ‘ 𝑅 ) ∧ 𝑧 ∈ ( Base ‘ 𝑆 ) ) ) → ( 𝑥 ( ·𝑠 ‘ 𝑆 ) 𝑧 ) ∈ ( Base ‘ 𝑆 ) )
76 1 15 16 17 65 57 55 psrvscacl ⊢ ( ( 𝜑 ∧ ( 𝑥 ∈ ( Base ‘ 𝑅 ) ∧ 𝑦 ∈ ( Base ‘ 𝑅 ) ∧ 𝑧 ∈ ( Base ‘ 𝑆 ) ) ) → ( 𝑦 ( ·𝑠 ‘ 𝑆 ) 𝑧 ) ∈ ( Base ‘ 𝑆 ) )
77 1 17 34 39 75 76 psradd ⊢ ( ( 𝜑 ∧ ( 𝑥 ∈ ( Base ‘ 𝑅 ) ∧ 𝑦 ∈ ( Base ‘ 𝑅 ) ∧ 𝑧 ∈ ( Base ‘ 𝑆 ) ) ) → ( ( 𝑥 ( ·𝑠 ‘ 𝑆 ) 𝑧 ) ( +g ‘ 𝑆 ) ( 𝑦 ( ·𝑠 ‘ 𝑆 ) 𝑧 ) ) = ( ( 𝑥 ( ·𝑠 ‘ 𝑆 ) 𝑧 ) ∘f ( +g ‘ 𝑅 ) ( 𝑦 ( ·𝑠 ‘ 𝑆 ) 𝑧 ) ) )
78 71 74 77 3eqtr4d ⊢ ( ( 𝜑 ∧ ( 𝑥 ∈ ( Base ‘ 𝑅 ) ∧ 𝑦 ∈ ( Base ‘ 𝑅 ) ∧ 𝑧 ∈ ( Base ‘ 𝑆 ) ) ) → ( ( 𝑥 ( +g ‘ 𝑅 ) 𝑦 ) ( ·𝑠 ‘ 𝑆 ) 𝑧 ) = ( ( 𝑥 ( ·𝑠 ‘ 𝑆 ) 𝑧 ) ( +g ‘ 𝑆 ) ( 𝑦 ( ·𝑠 ‘ 𝑆 ) 𝑧 ) ) )
79 58 oveq2d ⊢ ( ( 𝜑 ∧ ( 𝑥 ∈ ( Base ‘ 𝑅 ) ∧ 𝑦 ∈ ( Base ‘ 𝑅 ) ∧ 𝑧 ∈ ( Base ‘ 𝑆 ) ) ) → ( ( { 𝑓 ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ 𝑓 “ ℕ ) ∈ Fin } × { 𝑥 } ) ∘f ( .r ‘ 𝑅 ) ( 𝑦 ( ·𝑠 ‘ 𝑆 ) 𝑧 ) ) = ( ( { 𝑓 ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ 𝑓 “ ℕ ) ∈ Fin } × { 𝑥 } ) ∘f ( .r ‘ 𝑅 ) ( ( { 𝑓 ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ 𝑓 “ ℕ ) ∈ Fin } × { 𝑦 } ) ∘f ( .r ‘ 𝑅 ) 𝑧 ) ) )
80 16 35 ringass ⊢ ( ( 𝑅 ∈ Ring ∧ ( 𝑟 ∈ ( Base ‘ 𝑅 ) ∧ 𝑠 ∈ ( Base ‘ 𝑅 ) ∧ 𝑡 ∈ ( Base ‘ 𝑅 ) ) ) → ( ( 𝑟 ( .r ‘ 𝑅 ) 𝑠 ) ( .r ‘ 𝑅 ) 𝑡 ) = ( 𝑟 ( .r ‘ 𝑅 ) ( 𝑠 ( .r ‘ 𝑅 ) 𝑡 ) ) )
81 65 80 sylan ⊢ ( ( ( 𝜑 ∧ ( 𝑥 ∈ ( Base ‘ 𝑅 ) ∧ 𝑦 ∈ ( Base ‘ 𝑅 ) ∧ 𝑧 ∈ ( Base ‘ 𝑆 ) ) ) ∧ ( 𝑟 ∈ ( Base ‘ 𝑅 ) ∧ 𝑠 ∈ ( Base ‘ 𝑅 ) ∧ 𝑡 ∈ ( Base ‘ 𝑅 ) ) ) → ( ( 𝑟 ( .r ‘ 𝑅 ) 𝑠 ) ( .r ‘ 𝑅 ) 𝑡 ) = ( 𝑟 ( .r ‘ 𝑅 ) ( 𝑠 ( .r ‘ 𝑅 ) 𝑡 ) ) )
82 60 62 64 61 81 caofass ⊢ ( ( 𝜑 ∧ ( 𝑥 ∈ ( Base ‘ 𝑅 ) ∧ 𝑦 ∈ ( Base ‘ 𝑅 ) ∧ 𝑧 ∈ ( Base ‘ 𝑆 ) ) ) → ( ( ( { 𝑓 ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ 𝑓 “ ℕ ) ∈ Fin } × { 𝑥 } ) ∘f ( .r ‘ 𝑅 ) ( { 𝑓 ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ 𝑓 “ ℕ ) ∈ Fin } × { 𝑦 } ) ) ∘f ( .r ‘ 𝑅 ) 𝑧 ) = ( ( { 𝑓 ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ 𝑓 “ ℕ ) ∈ Fin } × { 𝑥 } ) ∘f ( .r ‘ 𝑅 ) ( ( { 𝑓 ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ 𝑓 “ ℕ ) ∈ Fin } × { 𝑦 } ) ∘f ( .r ‘ 𝑅 ) 𝑧 ) ) )
83 60 54 57 ofc12 ⊢ ( ( 𝜑 ∧ ( 𝑥 ∈ ( Base ‘ 𝑅 ) ∧ 𝑦 ∈ ( Base ‘ 𝑅 ) ∧ 𝑧 ∈ ( Base ‘ 𝑆 ) ) ) → ( ( { 𝑓 ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ 𝑓 “ ℕ ) ∈ Fin } × { 𝑥 } ) ∘f ( .r ‘ 𝑅 ) ( { 𝑓 ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ 𝑓 “ ℕ ) ∈ Fin } × { 𝑦 } ) ) = ( { 𝑓 ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ 𝑓 “ ℕ ) ∈ Fin } × { ( 𝑥 ( .r ‘ 𝑅 ) 𝑦 ) } ) )
84 83 oveq1d ⊢ ( ( 𝜑 ∧ ( 𝑥 ∈ ( Base ‘ 𝑅 ) ∧ 𝑦 ∈ ( Base ‘ 𝑅 ) ∧ 𝑧 ∈ ( Base ‘ 𝑆 ) ) ) → ( ( ( { 𝑓 ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ 𝑓 “ ℕ ) ∈ Fin } × { 𝑥 } ) ∘f ( .r ‘ 𝑅 ) ( { 𝑓 ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ 𝑓 “ ℕ ) ∈ Fin } × { 𝑦 } ) ) ∘f ( .r ‘ 𝑅 ) 𝑧 ) = ( ( { 𝑓 ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ 𝑓 “ ℕ ) ∈ Fin } × { ( 𝑥 ( .r ‘ 𝑅 ) 𝑦 ) } ) ∘f ( .r ‘ 𝑅 ) 𝑧 ) )
85 79 82 84 3eqtr2rd ⊢ ( ( 𝜑 ∧ ( 𝑥 ∈ ( Base ‘ 𝑅 ) ∧ 𝑦 ∈ ( Base ‘ 𝑅 ) ∧ 𝑧 ∈ ( Base ‘ 𝑆 ) ) ) → ( ( { 𝑓 ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ 𝑓 “ ℕ ) ∈ Fin } × { ( 𝑥 ( .r ‘ 𝑅 ) 𝑦 ) } ) ∘f ( .r ‘ 𝑅 ) 𝑧 ) = ( ( { 𝑓 ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ 𝑓 “ ℕ ) ∈ Fin } × { 𝑥 } ) ∘f ( .r ‘ 𝑅 ) ( 𝑦 ( ·𝑠 ‘ 𝑆 ) 𝑧 ) ) )
86 16 35 ringcl ⊢ ( ( 𝑅 ∈ Ring ∧ 𝑥 ∈ ( Base ‘ 𝑅 ) ∧ 𝑦 ∈ ( Base ‘ 𝑅 ) ) → ( 𝑥 ( .r ‘ 𝑅 ) 𝑦 ) ∈ ( Base ‘ 𝑅 ) )
87 65 54 57 86 syl3anc ⊢ ( ( 𝜑 ∧ ( 𝑥 ∈ ( Base ‘ 𝑅 ) ∧ 𝑦 ∈ ( Base ‘ 𝑅 ) ∧ 𝑧 ∈ ( Base ‘ 𝑆 ) ) ) → ( 𝑥 ( .r ‘ 𝑅 ) 𝑦 ) ∈ ( Base ‘ 𝑅 ) )
88 1 15 16 17 35 28 87 55 psrvsca ⊢ ( ( 𝜑 ∧ ( 𝑥 ∈ ( Base ‘ 𝑅 ) ∧ 𝑦 ∈ ( Base ‘ 𝑅 ) ∧ 𝑧 ∈ ( Base ‘ 𝑆 ) ) ) → ( ( 𝑥 ( .r ‘ 𝑅 ) 𝑦 ) ( ·𝑠 ‘ 𝑆 ) 𝑧 ) = ( ( { 𝑓 ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ 𝑓 “ ℕ ) ∈ Fin } × { ( 𝑥 ( .r ‘ 𝑅 ) 𝑦 ) } ) ∘f ( .r ‘ 𝑅 ) 𝑧 ) )
89 1 15 16 17 35 28 54 76 psrvsca ⊢ ( ( 𝜑 ∧ ( 𝑥 ∈ ( Base ‘ 𝑅 ) ∧ 𝑦 ∈ ( Base ‘ 𝑅 ) ∧ 𝑧 ∈ ( Base ‘ 𝑆 ) ) ) → ( 𝑥 ( ·𝑠 ‘ 𝑆 ) ( 𝑦 ( ·𝑠 ‘ 𝑆 ) 𝑧 ) ) = ( ( { 𝑓 ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ 𝑓 “ ℕ ) ∈ Fin } × { 𝑥 } ) ∘f ( .r ‘ 𝑅 ) ( 𝑦 ( ·𝑠 ‘ 𝑆 ) 𝑧 ) ) )
90 85 88 89 3eqtr4d ⊢ ( ( 𝜑 ∧ ( 𝑥 ∈ ( Base ‘ 𝑅 ) ∧ 𝑦 ∈ ( Base ‘ 𝑅 ) ∧ 𝑧 ∈ ( Base ‘ 𝑆 ) ) ) → ( ( 𝑥 ( .r ‘ 𝑅 ) 𝑦 ) ( ·𝑠 ‘ 𝑆 ) 𝑧 ) = ( 𝑥 ( ·𝑠 ‘ 𝑆 ) ( 𝑦 ( ·𝑠 ‘ 𝑆 ) 𝑧 ) ) )
91 3 adantr ⊢ ( ( 𝜑 ∧ 𝑥 ∈ ( Base ‘ 𝑆 ) ) → 𝑅 ∈ Ring )
92 eqid ⊢ ( 1r ‘ 𝑅 ) = ( 1r ‘ 𝑅 )
93 16 92 ringidcl ⊢ ( 𝑅 ∈ Ring → ( 1r ‘ 𝑅 ) ∈ ( Base ‘ 𝑅 ) )
94 91 93 syl ⊢ ( ( 𝜑 ∧ 𝑥 ∈ ( Base ‘ 𝑆 ) ) → ( 1r ‘ 𝑅 ) ∈ ( Base ‘ 𝑅 ) )
95 simpr ⊢ ( ( 𝜑 ∧ 𝑥 ∈ ( Base ‘ 𝑆 ) ) → 𝑥 ∈ ( Base ‘ 𝑆 ) )
96 1 15 16 17 35 28 94 95 psrvsca ⊢ ( ( 𝜑 ∧ 𝑥 ∈ ( Base ‘ 𝑆 ) ) → ( ( 1r ‘ 𝑅 ) ( ·𝑠 ‘ 𝑆 ) 𝑥 ) = ( ( { 𝑓 ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ 𝑓 “ ℕ ) ∈ Fin } × { ( 1r ‘ 𝑅 ) } ) ∘f ( .r ‘ 𝑅 ) 𝑥 ) )
97 23 a1i ⊢ ( ( 𝜑 ∧ 𝑥 ∈ ( Base ‘ 𝑆 ) ) → { 𝑓 ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ 𝑓 “ ℕ ) ∈ Fin } ∈ V )
98 1 16 28 17 95 psrelbas ⊢ ( ( 𝜑 ∧ 𝑥 ∈ ( Base ‘ 𝑆 ) ) → 𝑥 : { 𝑓 ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ 𝑓 “ ℕ ) ∈ Fin } ⟶ ( Base ‘ 𝑅 ) )
99 16 35 92 ringlidm ⊢ ( ( 𝑅 ∈ Ring ∧ 𝑟 ∈ ( Base ‘ 𝑅 ) ) → ( ( 1r ‘ 𝑅 ) ( .r ‘ 𝑅 ) 𝑟 ) = 𝑟 )
100 91 99 sylan ⊢ ( ( ( 𝜑 ∧ 𝑥 ∈ ( Base ‘ 𝑆 ) ) ∧ 𝑟 ∈ ( Base ‘ 𝑅 ) ) → ( ( 1r ‘ 𝑅 ) ( .r ‘ 𝑅 ) 𝑟 ) = 𝑟 )
101 97 98 94 100 caofid0l ⊢ ( ( 𝜑 ∧ 𝑥 ∈ ( Base ‘ 𝑆 ) ) → ( ( { 𝑓 ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ 𝑓 “ ℕ ) ∈ Fin } × { ( 1r ‘ 𝑅 ) } ) ∘f ( .r ‘ 𝑅 ) 𝑥 ) = 𝑥 )
102 96 101 eqtrd ⊢ ( ( 𝜑 ∧ 𝑥 ∈ ( Base ‘ 𝑆 ) ) → ( ( 1r ‘ 𝑅 ) ( ·𝑠 ‘ 𝑆 ) 𝑥 ) = 𝑥 )
103 4 5 6 7 8 9 10 11 3 14 21 53 78 90 102 islmodd ⊢ ( 𝜑 → 𝑆 ∈ LMod )