Metamath Proof Explorer

Theorem psrridm

Description: The identity element of the ring of power series is a right identity. (Contributed by Mario Carneiro, 29-Dec-2014) (Proof shortened by AV, 8-Jul-2019)

Ref Expression
Hypotheses psrring.s ⊢ 𝑆 = ( 𝐼 mPwSer 𝑅 )
psrring.i ⊢ ( 𝜑 → 𝐼 ∈ 𝑉 )
psrring.r ⊢ ( 𝜑 → 𝑅 ∈ Ring )
psr1cl.d ⊢ 𝐷 = { 𝑓 ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ 𝑓 “ ℕ ) ∈ Fin }
psr1cl.z ⊢ 0 = ( 0g ‘ 𝑅 )
psr1cl.o ⊢ 1 = ( 1r ‘ 𝑅 )
psr1cl.u ⊢ 𝑈 = ( 𝑥 ∈ 𝐷 ↦ if ( 𝑥 = ( 𝐼 × { 0 } ) , 1 , 0 ) )
psr1cl.b ⊢ 𝐵 = ( Base ‘ 𝑆 )
psrlidm.t ⊢ · = ( .r ‘ 𝑆 )
psrlidm.x ⊢ ( 𝜑 → 𝑋 ∈ 𝐵 )
Assertion psrridm ( 𝜑 → ( 𝑋 · 𝑈 ) = 𝑋 )

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 psr1cl.d ⊢ 𝐷 = { 𝑓 ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ 𝑓 “ ℕ ) ∈ Fin }
5 psr1cl.z ⊢ 0 = ( 0g ‘ 𝑅 )
6 psr1cl.o ⊢ 1 = ( 1r ‘ 𝑅 )
7 psr1cl.u ⊢ 𝑈 = ( 𝑥 ∈ 𝐷 ↦ if ( 𝑥 = ( 𝐼 × { 0 } ) , 1 , 0 ) )
8 psr1cl.b ⊢ 𝐵 = ( Base ‘ 𝑆 )
9 psrlidm.t ⊢ · = ( .r ‘ 𝑆 )
10 psrlidm.x ⊢ ( 𝜑 → 𝑋 ∈ 𝐵 )
11 eqid ⊢ ( Base ‘ 𝑅 ) = ( Base ‘ 𝑅 )
12 1 2 3 4 5 6 7 8 psr1cl ⊢ ( 𝜑 → 𝑈 ∈ 𝐵 )
13 1 8 9 3 10 12 psrmulcl ⊢ ( 𝜑 → ( 𝑋 · 𝑈 ) ∈ 𝐵 )
14 1 11 4 8 13 psrelbas ⊢ ( 𝜑 → ( 𝑋 · 𝑈 ) : 𝐷 ⟶ ( Base ‘ 𝑅 ) )
15 14 ffnd ⊢ ( 𝜑 → ( 𝑋 · 𝑈 ) Fn 𝐷 )
16 1 11 4 8 10 psrelbas ⊢ ( 𝜑 → 𝑋 : 𝐷 ⟶ ( Base ‘ 𝑅 ) )
17 16 ffnd ⊢ ( 𝜑 → 𝑋 Fn 𝐷 )
18 eqid ⊢ ( .r ‘ 𝑅 ) = ( .r ‘ 𝑅 )
19 10 adantr ⊢ ( ( 𝜑 ∧ 𝑦 ∈ 𝐷 ) → 𝑋 ∈ 𝐵 )
20 12 adantr ⊢ ( ( 𝜑 ∧ 𝑦 ∈ 𝐷 ) → 𝑈 ∈ 𝐵 )
21 simpr ⊢ ( ( 𝜑 ∧ 𝑦 ∈ 𝐷 ) → 𝑦 ∈ 𝐷 )
22 1 8 18 9 4 19 20 21 psrmulval ⊢ ( ( 𝜑 ∧ 𝑦 ∈ 𝐷 ) → ( ( 𝑋 · 𝑈 ) ‘ 𝑦 ) = ( 𝑅 Σg ( 𝑧 ∈ { 𝑔 ∈ 𝐷 ∣ 𝑔 ∘r ≤ 𝑦 } ↦ ( ( 𝑋 ‘ 𝑧 ) ( .r ‘ 𝑅 ) ( 𝑈 ‘ ( 𝑦 ∘f − 𝑧 ) ) ) ) ) )
23 breq1 ⊢ ( 𝑔 = 𝑦 → ( 𝑔 ∘r ≤ 𝑦 ↔ 𝑦 ∘r ≤ 𝑦 ) )
24 2 adantr ⊢ ( ( 𝜑 ∧ 𝑦 ∈ 𝐷 ) → 𝐼 ∈ 𝑉 )
25 4 psrbagf ⊢ ( 𝑦 ∈ 𝐷 → 𝑦 : 𝐼 ⟶ ℕ0 )
26 25 adantl ⊢ ( ( 𝜑 ∧ 𝑦 ∈ 𝐷 ) → 𝑦 : 𝐼 ⟶ ℕ0 )
27 nn0re ⊢ ( 𝑧 ∈ ℕ0 → 𝑧 ∈ ℝ )
28 27 leidd ⊢ ( 𝑧 ∈ ℕ0 → 𝑧 ≤ 𝑧 )
29 28 adantl ⊢ ( ( ( 𝜑 ∧ 𝑦 ∈ 𝐷 ) ∧ 𝑧 ∈ ℕ0 ) → 𝑧 ≤ 𝑧 )
30 24 26 29 caofref ⊢ ( ( 𝜑 ∧ 𝑦 ∈ 𝐷 ) → 𝑦 ∘r ≤ 𝑦 )
31 23 21 30 elrabd ⊢ ( ( 𝜑 ∧ 𝑦 ∈ 𝐷 ) → 𝑦 ∈ { 𝑔 ∈ 𝐷 ∣ 𝑔 ∘r ≤ 𝑦 } )
32 31 snssd ⊢ ( ( 𝜑 ∧ 𝑦 ∈ 𝐷 ) → { 𝑦 } ⊆ { 𝑔 ∈ 𝐷 ∣ 𝑔 ∘r ≤ 𝑦 } )
33 32 resmptd ⊢ ( ( 𝜑 ∧ 𝑦 ∈ 𝐷 ) → ( ( 𝑧 ∈ { 𝑔 ∈ 𝐷 ∣ 𝑔 ∘r ≤ 𝑦 } ↦ ( ( 𝑋 ‘ 𝑧 ) ( .r ‘ 𝑅 ) ( 𝑈 ‘ ( 𝑦 ∘f − 𝑧 ) ) ) ) ↾ { 𝑦 } ) = ( 𝑧 ∈ { 𝑦 } ↦ ( ( 𝑋 ‘ 𝑧 ) ( .r ‘ 𝑅 ) ( 𝑈 ‘ ( 𝑦 ∘f − 𝑧 ) ) ) ) )
34 33 oveq2d ⊢ ( ( 𝜑 ∧ 𝑦 ∈ 𝐷 ) → ( 𝑅 Σg ( ( 𝑧 ∈ { 𝑔 ∈ 𝐷 ∣ 𝑔 ∘r ≤ 𝑦 } ↦ ( ( 𝑋 ‘ 𝑧 ) ( .r ‘ 𝑅 ) ( 𝑈 ‘ ( 𝑦 ∘f − 𝑧 ) ) ) ) ↾ { 𝑦 } ) ) = ( 𝑅 Σg ( 𝑧 ∈ { 𝑦 } ↦ ( ( 𝑋 ‘ 𝑧 ) ( .r ‘ 𝑅 ) ( 𝑈 ‘ ( 𝑦 ∘f − 𝑧 ) ) ) ) ) )
35 ringcmn ⊢ ( 𝑅 ∈ Ring → 𝑅 ∈ CMnd )
36 3 35 syl ⊢ ( 𝜑 → 𝑅 ∈ CMnd )
37 36 adantr ⊢ ( ( 𝜑 ∧ 𝑦 ∈ 𝐷 ) → 𝑅 ∈ CMnd )
38 ovex ⊢ ( ℕ0 ↑m 𝐼 ) ∈ V
39 4 38 rab2ex ⊢ { 𝑔 ∈ 𝐷 ∣ 𝑔 ∘r ≤ 𝑦 } ∈ V
40 39 a1i ⊢ ( ( 𝜑 ∧ 𝑦 ∈ 𝐷 ) → { 𝑔 ∈ 𝐷 ∣ 𝑔 ∘r ≤ 𝑦 } ∈ V )
41 3 ad2antrr ⊢ ( ( ( 𝜑 ∧ 𝑦 ∈ 𝐷 ) ∧ 𝑧 ∈ { 𝑔 ∈ 𝐷 ∣ 𝑔 ∘r ≤ 𝑦 } ) → 𝑅 ∈ Ring )
42 16 ad2antrr ⊢ ( ( ( 𝜑 ∧ 𝑦 ∈ 𝐷 ) ∧ 𝑧 ∈ { 𝑔 ∈ 𝐷 ∣ 𝑔 ∘r ≤ 𝑦 } ) → 𝑋 : 𝐷 ⟶ ( Base ‘ 𝑅 ) )
43 simpr ⊢ ( ( ( 𝜑 ∧ 𝑦 ∈ 𝐷 ) ∧ 𝑧 ∈ { 𝑔 ∈ 𝐷 ∣ 𝑔 ∘r ≤ 𝑦 } ) → 𝑧 ∈ { 𝑔 ∈ 𝐷 ∣ 𝑔 ∘r ≤ 𝑦 } )
44 breq1 ⊢ ( 𝑔 = 𝑧 → ( 𝑔 ∘r ≤ 𝑦 ↔ 𝑧 ∘r ≤ 𝑦 ) )
45 44 elrab ⊢ ( 𝑧 ∈ { 𝑔 ∈ 𝐷 ∣ 𝑔 ∘r ≤ 𝑦 } ↔ ( 𝑧 ∈ 𝐷 ∧ 𝑧 ∘r ≤ 𝑦 ) )
46 43 45 sylib ⊢ ( ( ( 𝜑 ∧ 𝑦 ∈ 𝐷 ) ∧ 𝑧 ∈ { 𝑔 ∈ 𝐷 ∣ 𝑔 ∘r ≤ 𝑦 } ) → ( 𝑧 ∈ 𝐷 ∧ 𝑧 ∘r ≤ 𝑦 ) )
47 46 simpld ⊢ ( ( ( 𝜑 ∧ 𝑦 ∈ 𝐷 ) ∧ 𝑧 ∈ { 𝑔 ∈ 𝐷 ∣ 𝑔 ∘r ≤ 𝑦 } ) → 𝑧 ∈ 𝐷 )
48 42 47 ffvelcdmd ⊢ ( ( ( 𝜑 ∧ 𝑦 ∈ 𝐷 ) ∧ 𝑧 ∈ { 𝑔 ∈ 𝐷 ∣ 𝑔 ∘r ≤ 𝑦 } ) → ( 𝑋 ‘ 𝑧 ) ∈ ( Base ‘ 𝑅 ) )
49 1 11 4 8 20 psrelbas ⊢ ( ( 𝜑 ∧ 𝑦 ∈ 𝐷 ) → 𝑈 : 𝐷 ⟶ ( Base ‘ 𝑅 ) )
50 49 adantr ⊢ ( ( ( 𝜑 ∧ 𝑦 ∈ 𝐷 ) ∧ 𝑧 ∈ { 𝑔 ∈ 𝐷 ∣ 𝑔 ∘r ≤ 𝑦 } ) → 𝑈 : 𝐷 ⟶ ( Base ‘ 𝑅 ) )
51 21 adantr ⊢ ( ( ( 𝜑 ∧ 𝑦 ∈ 𝐷 ) ∧ 𝑧 ∈ { 𝑔 ∈ 𝐷 ∣ 𝑔 ∘r ≤ 𝑦 } ) → 𝑦 ∈ 𝐷 )
52 4 psrbagf ⊢ ( 𝑧 ∈ 𝐷 → 𝑧 : 𝐼 ⟶ ℕ0 )
53 47 52 syl ⊢ ( ( ( 𝜑 ∧ 𝑦 ∈ 𝐷 ) ∧ 𝑧 ∈ { 𝑔 ∈ 𝐷 ∣ 𝑔 ∘r ≤ 𝑦 } ) → 𝑧 : 𝐼 ⟶ ℕ0 )
54 46 simprd ⊢ ( ( ( 𝜑 ∧ 𝑦 ∈ 𝐷 ) ∧ 𝑧 ∈ { 𝑔 ∈ 𝐷 ∣ 𝑔 ∘r ≤ 𝑦 } ) → 𝑧 ∘r ≤ 𝑦 )
55 4 psrbagcon ⊢ ( ( 𝑦 ∈ 𝐷 ∧ 𝑧 : 𝐼 ⟶ ℕ0 ∧ 𝑧 ∘r ≤ 𝑦 ) → ( ( 𝑦 ∘f − 𝑧 ) ∈ 𝐷 ∧ ( 𝑦 ∘f − 𝑧 ) ∘r ≤ 𝑦 ) )
56 51 53 54 55 syl3anc ⊢ ( ( ( 𝜑 ∧ 𝑦 ∈ 𝐷 ) ∧ 𝑧 ∈ { 𝑔 ∈ 𝐷 ∣ 𝑔 ∘r ≤ 𝑦 } ) → ( ( 𝑦 ∘f − 𝑧 ) ∈ 𝐷 ∧ ( 𝑦 ∘f − 𝑧 ) ∘r ≤ 𝑦 ) )
57 56 simpld ⊢ ( ( ( 𝜑 ∧ 𝑦 ∈ 𝐷 ) ∧ 𝑧 ∈ { 𝑔 ∈ 𝐷 ∣ 𝑔 ∘r ≤ 𝑦 } ) → ( 𝑦 ∘f − 𝑧 ) ∈ 𝐷 )
58 50 57 ffvelcdmd ⊢ ( ( ( 𝜑 ∧ 𝑦 ∈ 𝐷 ) ∧ 𝑧 ∈ { 𝑔 ∈ 𝐷 ∣ 𝑔 ∘r ≤ 𝑦 } ) → ( 𝑈 ‘ ( 𝑦 ∘f − 𝑧 ) ) ∈ ( Base ‘ 𝑅 ) )
59 11 18 ringcl ⊢ ( ( 𝑅 ∈ Ring ∧ ( 𝑋 ‘ 𝑧 ) ∈ ( Base ‘ 𝑅 ) ∧ ( 𝑈 ‘ ( 𝑦 ∘f − 𝑧 ) ) ∈ ( Base ‘ 𝑅 ) ) → ( ( 𝑋 ‘ 𝑧 ) ( .r ‘ 𝑅 ) ( 𝑈 ‘ ( 𝑦 ∘f − 𝑧 ) ) ) ∈ ( Base ‘ 𝑅 ) )
60 41 48 58 59 syl3anc ⊢ ( ( ( 𝜑 ∧ 𝑦 ∈ 𝐷 ) ∧ 𝑧 ∈ { 𝑔 ∈ 𝐷 ∣ 𝑔 ∘r ≤ 𝑦 } ) → ( ( 𝑋 ‘ 𝑧 ) ( .r ‘ 𝑅 ) ( 𝑈 ‘ ( 𝑦 ∘f − 𝑧 ) ) ) ∈ ( Base ‘ 𝑅 ) )
61 60 fmpttd ⊢ ( ( 𝜑 ∧ 𝑦 ∈ 𝐷 ) → ( 𝑧 ∈ { 𝑔 ∈ 𝐷 ∣ 𝑔 ∘r ≤ 𝑦 } ↦ ( ( 𝑋 ‘ 𝑧 ) ( .r ‘ 𝑅 ) ( 𝑈 ‘ ( 𝑦 ∘f − 𝑧 ) ) ) ) : { 𝑔 ∈ 𝐷 ∣ 𝑔 ∘r ≤ 𝑦 } ⟶ ( Base ‘ 𝑅 ) )
62 eldifi ⊢ ( 𝑧 ∈ ( { 𝑔 ∈ 𝐷 ∣ 𝑔 ∘r ≤ 𝑦 } ∖ { 𝑦 } ) → 𝑧 ∈ { 𝑔 ∈ 𝐷 ∣ 𝑔 ∘r ≤ 𝑦 } )
63 62 57 sylan2 ⊢ ( ( ( 𝜑 ∧ 𝑦 ∈ 𝐷 ) ∧ 𝑧 ∈ ( { 𝑔 ∈ 𝐷 ∣ 𝑔 ∘r ≤ 𝑦 } ∖ { 𝑦 } ) ) → ( 𝑦 ∘f − 𝑧 ) ∈ 𝐷 )
64 eqeq1 ⊢ ( 𝑥 = ( 𝑦 ∘f − 𝑧 ) → ( 𝑥 = ( 𝐼 × { 0 } ) ↔ ( 𝑦 ∘f − 𝑧 ) = ( 𝐼 × { 0 } ) ) )
65 64 ifbid ⊢ ( 𝑥 = ( 𝑦 ∘f − 𝑧 ) → if ( 𝑥 = ( 𝐼 × { 0 } ) , 1 , 0 ) = if ( ( 𝑦 ∘f − 𝑧 ) = ( 𝐼 × { 0 } ) , 1 , 0 ) )
66 6 fvexi ⊢ 1 ∈ V
67 5 fvexi ⊢ 0 ∈ V
68 66 67 ifex ⊢ if ( ( 𝑦 ∘f − 𝑧 ) = ( 𝐼 × { 0 } ) , 1 , 0 ) ∈ V
69 65 7 68 fvmpt ⊢ ( ( 𝑦 ∘f − 𝑧 ) ∈ 𝐷 → ( 𝑈 ‘ ( 𝑦 ∘f − 𝑧 ) ) = if ( ( 𝑦 ∘f − 𝑧 ) = ( 𝐼 × { 0 } ) , 1 , 0 ) )
70 63 69 syl ⊢ ( ( ( 𝜑 ∧ 𝑦 ∈ 𝐷 ) ∧ 𝑧 ∈ ( { 𝑔 ∈ 𝐷 ∣ 𝑔 ∘r ≤ 𝑦 } ∖ { 𝑦 } ) ) → ( 𝑈 ‘ ( 𝑦 ∘f − 𝑧 ) ) = if ( ( 𝑦 ∘f − 𝑧 ) = ( 𝐼 × { 0 } ) , 1 , 0 ) )
71 eldifsni ⊢ ( 𝑧 ∈ ( { 𝑔 ∈ 𝐷 ∣ 𝑔 ∘r ≤ 𝑦 } ∖ { 𝑦 } ) → 𝑧 ≠ 𝑦 )
72 71 adantl ⊢ ( ( ( 𝜑 ∧ 𝑦 ∈ 𝐷 ) ∧ 𝑧 ∈ ( { 𝑔 ∈ 𝐷 ∣ 𝑔 ∘r ≤ 𝑦 } ∖ { 𝑦 } ) ) → 𝑧 ≠ 𝑦 )
73 72 necomd ⊢ ( ( ( 𝜑 ∧ 𝑦 ∈ 𝐷 ) ∧ 𝑧 ∈ ( { 𝑔 ∈ 𝐷 ∣ 𝑔 ∘r ≤ 𝑦 } ∖ { 𝑦 } ) ) → 𝑦 ≠ 𝑧 )
74 24 adantr ⊢ ( ( ( 𝜑 ∧ 𝑦 ∈ 𝐷 ) ∧ 𝑧 ∈ { 𝑔 ∈ 𝐷 ∣ 𝑔 ∘r ≤ 𝑦 } ) → 𝐼 ∈ 𝑉 )
75 nn0sscn ⊢ ℕ0 ⊆ ℂ
76 fss ⊢ ( ( 𝑦 : 𝐼 ⟶ ℕ0 ∧ ℕ0 ⊆ ℂ ) → 𝑦 : 𝐼 ⟶ ℂ )
77 26 75 76 sylancl ⊢ ( ( 𝜑 ∧ 𝑦 ∈ 𝐷 ) → 𝑦 : 𝐼 ⟶ ℂ )
78 77 adantr ⊢ ( ( ( 𝜑 ∧ 𝑦 ∈ 𝐷 ) ∧ 𝑧 ∈ { 𝑔 ∈ 𝐷 ∣ 𝑔 ∘r ≤ 𝑦 } ) → 𝑦 : 𝐼 ⟶ ℂ )
79 fss ⊢ ( ( 𝑧 : 𝐼 ⟶ ℕ0 ∧ ℕ0 ⊆ ℂ ) → 𝑧 : 𝐼 ⟶ ℂ )
80 53 75 79 sylancl ⊢ ( ( ( 𝜑 ∧ 𝑦 ∈ 𝐷 ) ∧ 𝑧 ∈ { 𝑔 ∈ 𝐷 ∣ 𝑔 ∘r ≤ 𝑦 } ) → 𝑧 : 𝐼 ⟶ ℂ )
81 ofsubeq0 ⊢ ( ( 𝐼 ∈ 𝑉 ∧ 𝑦 : 𝐼 ⟶ ℂ ∧ 𝑧 : 𝐼 ⟶ ℂ ) → ( ( 𝑦 ∘f − 𝑧 ) = ( 𝐼 × { 0 } ) ↔ 𝑦 = 𝑧 ) )
82 74 78 80 81 syl3anc ⊢ ( ( ( 𝜑 ∧ 𝑦 ∈ 𝐷 ) ∧ 𝑧 ∈ { 𝑔 ∈ 𝐷 ∣ 𝑔 ∘r ≤ 𝑦 } ) → ( ( 𝑦 ∘f − 𝑧 ) = ( 𝐼 × { 0 } ) ↔ 𝑦 = 𝑧 ) )
83 62 82 sylan2 ⊢ ( ( ( 𝜑 ∧ 𝑦 ∈ 𝐷 ) ∧ 𝑧 ∈ ( { 𝑔 ∈ 𝐷 ∣ 𝑔 ∘r ≤ 𝑦 } ∖ { 𝑦 } ) ) → ( ( 𝑦 ∘f − 𝑧 ) = ( 𝐼 × { 0 } ) ↔ 𝑦 = 𝑧 ) )
84 83 necon3bbid ⊢ ( ( ( 𝜑 ∧ 𝑦 ∈ 𝐷 ) ∧ 𝑧 ∈ ( { 𝑔 ∈ 𝐷 ∣ 𝑔 ∘r ≤ 𝑦 } ∖ { 𝑦 } ) ) → ( ¬ ( 𝑦 ∘f − 𝑧 ) = ( 𝐼 × { 0 } ) ↔ 𝑦 ≠ 𝑧 ) )
85 73 84 mpbird ⊢ ( ( ( 𝜑 ∧ 𝑦 ∈ 𝐷 ) ∧ 𝑧 ∈ ( { 𝑔 ∈ 𝐷 ∣ 𝑔 ∘r ≤ 𝑦 } ∖ { 𝑦 } ) ) → ¬ ( 𝑦 ∘f − 𝑧 ) = ( 𝐼 × { 0 } ) )
86 85 iffalsed ⊢ ( ( ( 𝜑 ∧ 𝑦 ∈ 𝐷 ) ∧ 𝑧 ∈ ( { 𝑔 ∈ 𝐷 ∣ 𝑔 ∘r ≤ 𝑦 } ∖ { 𝑦 } ) ) → if ( ( 𝑦 ∘f − 𝑧 ) = ( 𝐼 × { 0 } ) , 1 , 0 ) = 0 )
87 70 86 eqtrd ⊢ ( ( ( 𝜑 ∧ 𝑦 ∈ 𝐷 ) ∧ 𝑧 ∈ ( { 𝑔 ∈ 𝐷 ∣ 𝑔 ∘r ≤ 𝑦 } ∖ { 𝑦 } ) ) → ( 𝑈 ‘ ( 𝑦 ∘f − 𝑧 ) ) = 0 )
88 87 oveq2d ⊢ ( ( ( 𝜑 ∧ 𝑦 ∈ 𝐷 ) ∧ 𝑧 ∈ ( { 𝑔 ∈ 𝐷 ∣ 𝑔 ∘r ≤ 𝑦 } ∖ { 𝑦 } ) ) → ( ( 𝑋 ‘ 𝑧 ) ( .r ‘ 𝑅 ) ( 𝑈 ‘ ( 𝑦 ∘f − 𝑧 ) ) ) = ( ( 𝑋 ‘ 𝑧 ) ( .r ‘ 𝑅 ) 0 ) )
89 11 18 5 ringrz ⊢ ( ( 𝑅 ∈ Ring ∧ ( 𝑋 ‘ 𝑧 ) ∈ ( Base ‘ 𝑅 ) ) → ( ( 𝑋 ‘ 𝑧 ) ( .r ‘ 𝑅 ) 0 ) = 0 )
90 41 48 89 syl2anc ⊢ ( ( ( 𝜑 ∧ 𝑦 ∈ 𝐷 ) ∧ 𝑧 ∈ { 𝑔 ∈ 𝐷 ∣ 𝑔 ∘r ≤ 𝑦 } ) → ( ( 𝑋 ‘ 𝑧 ) ( .r ‘ 𝑅 ) 0 ) = 0 )
91 62 90 sylan2 ⊢ ( ( ( 𝜑 ∧ 𝑦 ∈ 𝐷 ) ∧ 𝑧 ∈ ( { 𝑔 ∈ 𝐷 ∣ 𝑔 ∘r ≤ 𝑦 } ∖ { 𝑦 } ) ) → ( ( 𝑋 ‘ 𝑧 ) ( .r ‘ 𝑅 ) 0 ) = 0 )
92 88 91 eqtrd ⊢ ( ( ( 𝜑 ∧ 𝑦 ∈ 𝐷 ) ∧ 𝑧 ∈ ( { 𝑔 ∈ 𝐷 ∣ 𝑔 ∘r ≤ 𝑦 } ∖ { 𝑦 } ) ) → ( ( 𝑋 ‘ 𝑧 ) ( .r ‘ 𝑅 ) ( 𝑈 ‘ ( 𝑦 ∘f − 𝑧 ) ) ) = 0 )
93 92 40 suppss2 ⊢ ( ( 𝜑 ∧ 𝑦 ∈ 𝐷 ) → ( ( 𝑧 ∈ { 𝑔 ∈ 𝐷 ∣ 𝑔 ∘r ≤ 𝑦 } ↦ ( ( 𝑋 ‘ 𝑧 ) ( .r ‘ 𝑅 ) ( 𝑈 ‘ ( 𝑦 ∘f − 𝑧 ) ) ) ) supp 0 ) ⊆ { 𝑦 } )
94 40 mptexd ⊢ ( ( 𝜑 ∧ 𝑦 ∈ 𝐷 ) → ( 𝑧 ∈ { 𝑔 ∈ 𝐷 ∣ 𝑔 ∘r ≤ 𝑦 } ↦ ( ( 𝑋 ‘ 𝑧 ) ( .r ‘ 𝑅 ) ( 𝑈 ‘ ( 𝑦 ∘f − 𝑧 ) ) ) ) ∈ V )
95 funmpt ⊢ Fun ( 𝑧 ∈ { 𝑔 ∈ 𝐷 ∣ 𝑔 ∘r ≤ 𝑦 } ↦ ( ( 𝑋 ‘ 𝑧 ) ( .r ‘ 𝑅 ) ( 𝑈 ‘ ( 𝑦 ∘f − 𝑧 ) ) ) )
96 95 a1i ⊢ ( ( 𝜑 ∧ 𝑦 ∈ 𝐷 ) → Fun ( 𝑧 ∈ { 𝑔 ∈ 𝐷 ∣ 𝑔 ∘r ≤ 𝑦 } ↦ ( ( 𝑋 ‘ 𝑧 ) ( .r ‘ 𝑅 ) ( 𝑈 ‘ ( 𝑦 ∘f − 𝑧 ) ) ) ) )
97 67 a1i ⊢ ( ( 𝜑 ∧ 𝑦 ∈ 𝐷 ) → 0 ∈ V )
98 snfi ⊢ { 𝑦 } ∈ Fin
99 98 a1i ⊢ ( ( 𝜑 ∧ 𝑦 ∈ 𝐷 ) → { 𝑦 } ∈ Fin )
100 suppssfifsupp ⊢ ( ( ( ( 𝑧 ∈ { 𝑔 ∈ 𝐷 ∣ 𝑔 ∘r ≤ 𝑦 } ↦ ( ( 𝑋 ‘ 𝑧 ) ( .r ‘ 𝑅 ) ( 𝑈 ‘ ( 𝑦 ∘f − 𝑧 ) ) ) ) ∈ V ∧ Fun ( 𝑧 ∈ { 𝑔 ∈ 𝐷 ∣ 𝑔 ∘r ≤ 𝑦 } ↦ ( ( 𝑋 ‘ 𝑧 ) ( .r ‘ 𝑅 ) ( 𝑈 ‘ ( 𝑦 ∘f − 𝑧 ) ) ) ) ∧ 0 ∈ V ) ∧ ( { 𝑦 } ∈ Fin ∧ ( ( 𝑧 ∈ { 𝑔 ∈ 𝐷 ∣ 𝑔 ∘r ≤ 𝑦 } ↦ ( ( 𝑋 ‘ 𝑧 ) ( .r ‘ 𝑅 ) ( 𝑈 ‘ ( 𝑦 ∘f − 𝑧 ) ) ) ) supp 0 ) ⊆ { 𝑦 } ) ) → ( 𝑧 ∈ { 𝑔 ∈ 𝐷 ∣ 𝑔 ∘r ≤ 𝑦 } ↦ ( ( 𝑋 ‘ 𝑧 ) ( .r ‘ 𝑅 ) ( 𝑈 ‘ ( 𝑦 ∘f − 𝑧 ) ) ) ) finSupp 0 )
101 94 96 97 99 93 100 syl32anc ⊢ ( ( 𝜑 ∧ 𝑦 ∈ 𝐷 ) → ( 𝑧 ∈ { 𝑔 ∈ 𝐷 ∣ 𝑔 ∘r ≤ 𝑦 } ↦ ( ( 𝑋 ‘ 𝑧 ) ( .r ‘ 𝑅 ) ( 𝑈 ‘ ( 𝑦 ∘f − 𝑧 ) ) ) ) finSupp 0 )
102 11 5 37 40 61 93 101 gsumres ⊢ ( ( 𝜑 ∧ 𝑦 ∈ 𝐷 ) → ( 𝑅 Σg ( ( 𝑧 ∈ { 𝑔 ∈ 𝐷 ∣ 𝑔 ∘r ≤ 𝑦 } ↦ ( ( 𝑋 ‘ 𝑧 ) ( .r ‘ 𝑅 ) ( 𝑈 ‘ ( 𝑦 ∘f − 𝑧 ) ) ) ) ↾ { 𝑦 } ) ) = ( 𝑅 Σg ( 𝑧 ∈ { 𝑔 ∈ 𝐷 ∣ 𝑔 ∘r ≤ 𝑦 } ↦ ( ( 𝑋 ‘ 𝑧 ) ( .r ‘ 𝑅 ) ( 𝑈 ‘ ( 𝑦 ∘f − 𝑧 ) ) ) ) ) )
103 3 adantr ⊢ ( ( 𝜑 ∧ 𝑦 ∈ 𝐷 ) → 𝑅 ∈ Ring )
104 ringmnd ⊢ ( 𝑅 ∈ Ring → 𝑅 ∈ Mnd )
105 103 104 syl ⊢ ( ( 𝜑 ∧ 𝑦 ∈ 𝐷 ) → 𝑅 ∈ Mnd )
106 eqid ⊢ 𝑦 = 𝑦
107 ofsubeq0 ⊢ ( ( 𝐼 ∈ 𝑉 ∧ 𝑦 : 𝐼 ⟶ ℂ ∧ 𝑦 : 𝐼 ⟶ ℂ ) → ( ( 𝑦 ∘f − 𝑦 ) = ( 𝐼 × { 0 } ) ↔ 𝑦 = 𝑦 ) )
108 24 77 77 107 syl3anc ⊢ ( ( 𝜑 ∧ 𝑦 ∈ 𝐷 ) → ( ( 𝑦 ∘f − 𝑦 ) = ( 𝐼 × { 0 } ) ↔ 𝑦 = 𝑦 ) )
109 106 108 mpbiri ⊢ ( ( 𝜑 ∧ 𝑦 ∈ 𝐷 ) → ( 𝑦 ∘f − 𝑦 ) = ( 𝐼 × { 0 } ) )
110 109 fveq2d ⊢ ( ( 𝜑 ∧ 𝑦 ∈ 𝐷 ) → ( 𝑈 ‘ ( 𝑦 ∘f − 𝑦 ) ) = ( 𝑈 ‘ ( 𝐼 × { 0 } ) ) )
111 fconstmpt ⊢ ( 𝐼 × { 0 } ) = ( 𝑤 ∈ 𝐼 ↦ 0 )
112 4 fczpsrbag ⊢ ( 𝐼 ∈ 𝑉 → ( 𝑤 ∈ 𝐼 ↦ 0 ) ∈ 𝐷 )
113 2 112 syl ⊢ ( 𝜑 → ( 𝑤 ∈ 𝐼 ↦ 0 ) ∈ 𝐷 )
114 111 113 eqeltrid ⊢ ( 𝜑 → ( 𝐼 × { 0 } ) ∈ 𝐷 )
115 114 adantr ⊢ ( ( 𝜑 ∧ 𝑦 ∈ 𝐷 ) → ( 𝐼 × { 0 } ) ∈ 𝐷 )
116 iftrue ⊢ ( 𝑥 = ( 𝐼 × { 0 } ) → if ( 𝑥 = ( 𝐼 × { 0 } ) , 1 , 0 ) = 1 )
117 116 7 66 fvmpt ⊢ ( ( 𝐼 × { 0 } ) ∈ 𝐷 → ( 𝑈 ‘ ( 𝐼 × { 0 } ) ) = 1 )
118 115 117 syl ⊢ ( ( 𝜑 ∧ 𝑦 ∈ 𝐷 ) → ( 𝑈 ‘ ( 𝐼 × { 0 } ) ) = 1 )
119 110 118 eqtrd ⊢ ( ( 𝜑 ∧ 𝑦 ∈ 𝐷 ) → ( 𝑈 ‘ ( 𝑦 ∘f − 𝑦 ) ) = 1 )
120 119 oveq2d ⊢ ( ( 𝜑 ∧ 𝑦 ∈ 𝐷 ) → ( ( 𝑋 ‘ 𝑦 ) ( .r ‘ 𝑅 ) ( 𝑈 ‘ ( 𝑦 ∘f − 𝑦 ) ) ) = ( ( 𝑋 ‘ 𝑦 ) ( .r ‘ 𝑅 ) 1 ) )
121 16 ffvelcdmda ⊢ ( ( 𝜑 ∧ 𝑦 ∈ 𝐷 ) → ( 𝑋 ‘ 𝑦 ) ∈ ( Base ‘ 𝑅 ) )
122 11 18 6 ringridm ⊢ ( ( 𝑅 ∈ Ring ∧ ( 𝑋 ‘ 𝑦 ) ∈ ( Base ‘ 𝑅 ) ) → ( ( 𝑋 ‘ 𝑦 ) ( .r ‘ 𝑅 ) 1 ) = ( 𝑋 ‘ 𝑦 ) )
123 103 121 122 syl2anc ⊢ ( ( 𝜑 ∧ 𝑦 ∈ 𝐷 ) → ( ( 𝑋 ‘ 𝑦 ) ( .r ‘ 𝑅 ) 1 ) = ( 𝑋 ‘ 𝑦 ) )
124 120 123 eqtrd ⊢ ( ( 𝜑 ∧ 𝑦 ∈ 𝐷 ) → ( ( 𝑋 ‘ 𝑦 ) ( .r ‘ 𝑅 ) ( 𝑈 ‘ ( 𝑦 ∘f − 𝑦 ) ) ) = ( 𝑋 ‘ 𝑦 ) )
125 124 121 eqeltrd ⊢ ( ( 𝜑 ∧ 𝑦 ∈ 𝐷 ) → ( ( 𝑋 ‘ 𝑦 ) ( .r ‘ 𝑅 ) ( 𝑈 ‘ ( 𝑦 ∘f − 𝑦 ) ) ) ∈ ( Base ‘ 𝑅 ) )
126 fveq2 ⊢ ( 𝑧 = 𝑦 → ( 𝑋 ‘ 𝑧 ) = ( 𝑋 ‘ 𝑦 ) )
127 oveq2 ⊢ ( 𝑧 = 𝑦 → ( 𝑦 ∘f − 𝑧 ) = ( 𝑦 ∘f − 𝑦 ) )
128 127 fveq2d ⊢ ( 𝑧 = 𝑦 → ( 𝑈 ‘ ( 𝑦 ∘f − 𝑧 ) ) = ( 𝑈 ‘ ( 𝑦 ∘f − 𝑦 ) ) )
129 126 128 oveq12d ⊢ ( 𝑧 = 𝑦 → ( ( 𝑋 ‘ 𝑧 ) ( .r ‘ 𝑅 ) ( 𝑈 ‘ ( 𝑦 ∘f − 𝑧 ) ) ) = ( ( 𝑋 ‘ 𝑦 ) ( .r ‘ 𝑅 ) ( 𝑈 ‘ ( 𝑦 ∘f − 𝑦 ) ) ) )
130 11 129 gsumsn ⊢ ( ( 𝑅 ∈ Mnd ∧ 𝑦 ∈ 𝐷 ∧ ( ( 𝑋 ‘ 𝑦 ) ( .r ‘ 𝑅 ) ( 𝑈 ‘ ( 𝑦 ∘f − 𝑦 ) ) ) ∈ ( Base ‘ 𝑅 ) ) → ( 𝑅 Σg ( 𝑧 ∈ { 𝑦 } ↦ ( ( 𝑋 ‘ 𝑧 ) ( .r ‘ 𝑅 ) ( 𝑈 ‘ ( 𝑦 ∘f − 𝑧 ) ) ) ) ) = ( ( 𝑋 ‘ 𝑦 ) ( .r ‘ 𝑅 ) ( 𝑈 ‘ ( 𝑦 ∘f − 𝑦 ) ) ) )
131 105 21 125 130 syl3anc ⊢ ( ( 𝜑 ∧ 𝑦 ∈ 𝐷 ) → ( 𝑅 Σg ( 𝑧 ∈ { 𝑦 } ↦ ( ( 𝑋 ‘ 𝑧 ) ( .r ‘ 𝑅 ) ( 𝑈 ‘ ( 𝑦 ∘f − 𝑧 ) ) ) ) ) = ( ( 𝑋 ‘ 𝑦 ) ( .r ‘ 𝑅 ) ( 𝑈 ‘ ( 𝑦 ∘f − 𝑦 ) ) ) )
132 34 102 131 3eqtr3d ⊢ ( ( 𝜑 ∧ 𝑦 ∈ 𝐷 ) → ( 𝑅 Σg ( 𝑧 ∈ { 𝑔 ∈ 𝐷 ∣ 𝑔 ∘r ≤ 𝑦 } ↦ ( ( 𝑋 ‘ 𝑧 ) ( .r ‘ 𝑅 ) ( 𝑈 ‘ ( 𝑦 ∘f − 𝑧 ) ) ) ) ) = ( ( 𝑋 ‘ 𝑦 ) ( .r ‘ 𝑅 ) ( 𝑈 ‘ ( 𝑦 ∘f − 𝑦 ) ) ) )
133 22 132 124 3eqtrd ⊢ ( ( 𝜑 ∧ 𝑦 ∈ 𝐷 ) → ( ( 𝑋 · 𝑈 ) ‘ 𝑦 ) = ( 𝑋 ‘ 𝑦 ) )
134 15 17 133 eqfnfvd ⊢ ( 𝜑 → ( 𝑋 · 𝑈 ) = 𝑋 )