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 ( 𝜑𝐼𝑉 )
psrring.r ( 𝜑𝑅 ∈ Ring )
psr1cl.d 𝐷 = { 𝑓 ∈ ( ℕ0m 𝐼 ) ∣ ( 𝑓 “ ℕ ) ∈ Fin }
psr1cl.z 0 = ( 0g𝑅 )
psr1cl.o 1 = ( 1r𝑅 )
psr1cl.u 𝑈 = ( 𝑥𝐷 ↦ if ( 𝑥 = ( 𝐼 × { 0 } ) , 1 , 0 ) )
psr1cl.b 𝐵 = ( Base ‘ 𝑆 )
psrlidm.t · = ( .r𝑆 )
psrlidm.x ( 𝜑𝑋𝐵 )
Assertion psrridm ( 𝜑 → ( 𝑋 · 𝑈 ) = 𝑋 )

Proof

Step Hyp Ref Expression
1 psrring.s 𝑆 = ( 𝐼 mPwSer 𝑅 )
2 psrring.i ( 𝜑𝐼𝑉 )
3 psrring.r ( 𝜑𝑅 ∈ Ring )
4 psr1cl.d 𝐷 = { 𝑓 ∈ ( ℕ0m 𝐼 ) ∣ ( 𝑓 “ ℕ ) ∈ Fin }
5 psr1cl.z 0 = ( 0g𝑅 )
6 psr1cl.o 1 = ( 1r𝑅 )
7 psr1cl.u 𝑈 = ( 𝑥𝐷 ↦ if ( 𝑥 = ( 𝐼 × { 0 } ) , 1 , 0 ) )
8 psr1cl.b 𝐵 = ( Base ‘ 𝑆 )
9 psrlidm.t · = ( .r𝑆 )
10 psrlidm.x ( 𝜑𝑋𝐵 )
11 eqid ( Base ‘ 𝑅 ) = ( Base ‘ 𝑅 )
12 1 2 3 4 5 6 7 8 psr1cl ( 𝜑𝑈𝐵 )
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 ( 𝜑𝑋 : 𝐷 ⟶ ( Base ‘ 𝑅 ) )
17 16 ffnd ( 𝜑𝑋 Fn 𝐷 )
18 eqid ( .r𝑅 ) = ( .r𝑅 )
19 10 adantr ( ( 𝜑𝑦𝐷 ) → 𝑋𝐵 )
20 12 adantr ( ( 𝜑𝑦𝐷 ) → 𝑈𝐵 )
21 simpr ( ( 𝜑𝑦𝐷 ) → 𝑦𝐷 )
22 1 8 18 9 4 19 20 21 psrmulval ( ( 𝜑𝑦𝐷 ) → ( ( 𝑋 · 𝑈 ) ‘ 𝑦 ) = ( 𝑅 Σg ( 𝑧 ∈ { 𝑔𝐷𝑔r𝑦 } ↦ ( ( 𝑋𝑧 ) ( .r𝑅 ) ( 𝑈 ‘ ( 𝑦f𝑧 ) ) ) ) ) )
23 breq1 ( 𝑔 = 𝑦 → ( 𝑔r𝑦𝑦r𝑦 ) )
24 2 adantr ( ( 𝜑𝑦𝐷 ) → 𝐼𝑉 )
25 4 psrbagf ( 𝑦𝐷𝑦 : 𝐼 ⟶ ℕ0 )
26 25 adantl ( ( 𝜑𝑦𝐷 ) → 𝑦 : 𝐼 ⟶ ℕ0 )
27 nn0re ( 𝑧 ∈ ℕ0𝑧 ∈ ℝ )
28 27 leidd ( 𝑧 ∈ ℕ0𝑧𝑧 )
29 28 adantl ( ( ( 𝜑𝑦𝐷 ) ∧ 𝑧 ∈ ℕ0 ) → 𝑧𝑧 )
30 24 26 29 caofref ( ( 𝜑𝑦𝐷 ) → 𝑦r𝑦 )
31 23 21 30 elrabd ( ( 𝜑𝑦𝐷 ) → 𝑦 ∈ { 𝑔𝐷𝑔r𝑦 } )
32 31 snssd ( ( 𝜑𝑦𝐷 ) → { 𝑦 } ⊆ { 𝑔𝐷𝑔r𝑦 } )
33 32 resmptd ( ( 𝜑𝑦𝐷 ) → ( ( 𝑧 ∈ { 𝑔𝐷𝑔r𝑦 } ↦ ( ( 𝑋𝑧 ) ( .r𝑅 ) ( 𝑈 ‘ ( 𝑦f𝑧 ) ) ) ) ↾ { 𝑦 } ) = ( 𝑧 ∈ { 𝑦 } ↦ ( ( 𝑋𝑧 ) ( .r𝑅 ) ( 𝑈 ‘ ( 𝑦f𝑧 ) ) ) ) )
34 33 oveq2d ( ( 𝜑𝑦𝐷 ) → ( 𝑅 Σg ( ( 𝑧 ∈ { 𝑔𝐷𝑔r𝑦 } ↦ ( ( 𝑋𝑧 ) ( .r𝑅 ) ( 𝑈 ‘ ( 𝑦f𝑧 ) ) ) ) ↾ { 𝑦 } ) ) = ( 𝑅 Σg ( 𝑧 ∈ { 𝑦 } ↦ ( ( 𝑋𝑧 ) ( .r𝑅 ) ( 𝑈 ‘ ( 𝑦f𝑧 ) ) ) ) ) )
35 ringcmn ( 𝑅 ∈ Ring → 𝑅 ∈ CMnd )
36 3 35 syl ( 𝜑𝑅 ∈ CMnd )
37 36 adantr ( ( 𝜑𝑦𝐷 ) → 𝑅 ∈ CMnd )
38 ovex ( ℕ0m 𝐼 ) ∈ V
39 4 38 rab2ex { 𝑔𝐷𝑔r𝑦 } ∈ V
40 39 a1i ( ( 𝜑𝑦𝐷 ) → { 𝑔𝐷𝑔r𝑦 } ∈ V )
41 3 ad2antrr ( ( ( 𝜑𝑦𝐷 ) ∧ 𝑧 ∈ { 𝑔𝐷𝑔r𝑦 } ) → 𝑅 ∈ Ring )
42 16 ad2antrr ( ( ( 𝜑𝑦𝐷 ) ∧ 𝑧 ∈ { 𝑔𝐷𝑔r𝑦 } ) → 𝑋 : 𝐷 ⟶ ( Base ‘ 𝑅 ) )
43 simpr ( ( ( 𝜑𝑦𝐷 ) ∧ 𝑧 ∈ { 𝑔𝐷𝑔r𝑦 } ) → 𝑧 ∈ { 𝑔𝐷𝑔r𝑦 } )
44 breq1 ( 𝑔 = 𝑧 → ( 𝑔r𝑦𝑧r𝑦 ) )
45 44 elrab ( 𝑧 ∈ { 𝑔𝐷𝑔r𝑦 } ↔ ( 𝑧𝐷𝑧r𝑦 ) )
46 43 45 sylib ( ( ( 𝜑𝑦𝐷 ) ∧ 𝑧 ∈ { 𝑔𝐷𝑔r𝑦 } ) → ( 𝑧𝐷𝑧r𝑦 ) )
47 46 simpld ( ( ( 𝜑𝑦𝐷 ) ∧ 𝑧 ∈ { 𝑔𝐷𝑔r𝑦 } ) → 𝑧𝐷 )
48 42 47 ffvelrnd ( ( ( 𝜑𝑦𝐷 ) ∧ 𝑧 ∈ { 𝑔𝐷𝑔r𝑦 } ) → ( 𝑋𝑧 ) ∈ ( Base ‘ 𝑅 ) )
49 1 11 4 8 20 psrelbas ( ( 𝜑𝑦𝐷 ) → 𝑈 : 𝐷 ⟶ ( Base ‘ 𝑅 ) )
50 49 adantr ( ( ( 𝜑𝑦𝐷 ) ∧ 𝑧 ∈ { 𝑔𝐷𝑔r𝑦 } ) → 𝑈 : 𝐷 ⟶ ( Base ‘ 𝑅 ) )
51 21 adantr ( ( ( 𝜑𝑦𝐷 ) ∧ 𝑧 ∈ { 𝑔𝐷𝑔r𝑦 } ) → 𝑦𝐷 )
52 4 psrbagf ( 𝑧𝐷𝑧 : 𝐼 ⟶ ℕ0 )
53 47 52 syl ( ( ( 𝜑𝑦𝐷 ) ∧ 𝑧 ∈ { 𝑔𝐷𝑔r𝑦 } ) → 𝑧 : 𝐼 ⟶ ℕ0 )
54 46 simprd ( ( ( 𝜑𝑦𝐷 ) ∧ 𝑧 ∈ { 𝑔𝐷𝑔r𝑦 } ) → 𝑧r𝑦 )
55 4 psrbagcon ( ( 𝑦𝐷𝑧 : 𝐼 ⟶ ℕ0𝑧r𝑦 ) → ( ( 𝑦f𝑧 ) ∈ 𝐷 ∧ ( 𝑦f𝑧 ) ∘r𝑦 ) )
56 51 53 54 55 syl3anc ( ( ( 𝜑𝑦𝐷 ) ∧ 𝑧 ∈ { 𝑔𝐷𝑔r𝑦 } ) → ( ( 𝑦f𝑧 ) ∈ 𝐷 ∧ ( 𝑦f𝑧 ) ∘r𝑦 ) )
57 56 simpld ( ( ( 𝜑𝑦𝐷 ) ∧ 𝑧 ∈ { 𝑔𝐷𝑔r𝑦 } ) → ( 𝑦f𝑧 ) ∈ 𝐷 )
58 50 57 ffvelrnd ( ( ( 𝜑𝑦𝐷 ) ∧ 𝑧 ∈ { 𝑔𝐷𝑔r𝑦 } ) → ( 𝑈 ‘ ( 𝑦f𝑧 ) ) ∈ ( Base ‘ 𝑅 ) )
59 11 18 ringcl ( ( 𝑅 ∈ Ring ∧ ( 𝑋𝑧 ) ∈ ( Base ‘ 𝑅 ) ∧ ( 𝑈 ‘ ( 𝑦f𝑧 ) ) ∈ ( Base ‘ 𝑅 ) ) → ( ( 𝑋𝑧 ) ( .r𝑅 ) ( 𝑈 ‘ ( 𝑦f𝑧 ) ) ) ∈ ( Base ‘ 𝑅 ) )
60 41 48 58 59 syl3anc ( ( ( 𝜑𝑦𝐷 ) ∧ 𝑧 ∈ { 𝑔𝐷𝑔r𝑦 } ) → ( ( 𝑋𝑧 ) ( .r𝑅 ) ( 𝑈 ‘ ( 𝑦f𝑧 ) ) ) ∈ ( Base ‘ 𝑅 ) )
61 60 fmpttd ( ( 𝜑𝑦𝐷 ) → ( 𝑧 ∈ { 𝑔𝐷𝑔r𝑦 } ↦ ( ( 𝑋𝑧 ) ( .r𝑅 ) ( 𝑈 ‘ ( 𝑦f𝑧 ) ) ) ) : { 𝑔𝐷𝑔r𝑦 } ⟶ ( Base ‘ 𝑅 ) )
62 eldifi ( 𝑧 ∈ ( { 𝑔𝐷𝑔r𝑦 } ∖ { 𝑦 } ) → 𝑧 ∈ { 𝑔𝐷𝑔r𝑦 } )
63 62 57 sylan2 ( ( ( 𝜑𝑦𝐷 ) ∧ 𝑧 ∈ ( { 𝑔𝐷𝑔r𝑦 } ∖ { 𝑦 } ) ) → ( 𝑦f𝑧 ) ∈ 𝐷 )
64 eqeq1 ( 𝑥 = ( 𝑦f𝑧 ) → ( 𝑥 = ( 𝐼 × { 0 } ) ↔ ( 𝑦f𝑧 ) = ( 𝐼 × { 0 } ) ) )
65 64 ifbid ( 𝑥 = ( 𝑦f𝑧 ) → if ( 𝑥 = ( 𝐼 × { 0 } ) , 1 , 0 ) = if ( ( 𝑦f𝑧 ) = ( 𝐼 × { 0 } ) , 1 , 0 ) )
66 6 fvexi 1 ∈ V
67 5 fvexi 0 ∈ V
68 66 67 ifex if ( ( 𝑦f𝑧 ) = ( 𝐼 × { 0 } ) , 1 , 0 ) ∈ V
69 65 7 68 fvmpt ( ( 𝑦f𝑧 ) ∈ 𝐷 → ( 𝑈 ‘ ( 𝑦f𝑧 ) ) = if ( ( 𝑦f𝑧 ) = ( 𝐼 × { 0 } ) , 1 , 0 ) )
70 63 69 syl ( ( ( 𝜑𝑦𝐷 ) ∧ 𝑧 ∈ ( { 𝑔𝐷𝑔r𝑦 } ∖ { 𝑦 } ) ) → ( 𝑈 ‘ ( 𝑦f𝑧 ) ) = if ( ( 𝑦f𝑧 ) = ( 𝐼 × { 0 } ) , 1 , 0 ) )
71 eldifsni ( 𝑧 ∈ ( { 𝑔𝐷𝑔r𝑦 } ∖ { 𝑦 } ) → 𝑧𝑦 )
72 71 adantl ( ( ( 𝜑𝑦𝐷 ) ∧ 𝑧 ∈ ( { 𝑔𝐷𝑔r𝑦 } ∖ { 𝑦 } ) ) → 𝑧𝑦 )
73 72 necomd ( ( ( 𝜑𝑦𝐷 ) ∧ 𝑧 ∈ ( { 𝑔𝐷𝑔r𝑦 } ∖ { 𝑦 } ) ) → 𝑦𝑧 )
74 24 adantr ( ( ( 𝜑𝑦𝐷 ) ∧ 𝑧 ∈ { 𝑔𝐷𝑔r𝑦 } ) → 𝐼𝑉 )
75 nn0sscn 0 ⊆ ℂ
76 fss ( ( 𝑦 : 𝐼 ⟶ ℕ0 ∧ ℕ0 ⊆ ℂ ) → 𝑦 : 𝐼 ⟶ ℂ )
77 26 75 76 sylancl ( ( 𝜑𝑦𝐷 ) → 𝑦 : 𝐼 ⟶ ℂ )
78 77 adantr ( ( ( 𝜑𝑦𝐷 ) ∧ 𝑧 ∈ { 𝑔𝐷𝑔r𝑦 } ) → 𝑦 : 𝐼 ⟶ ℂ )
79 fss ( ( 𝑧 : 𝐼 ⟶ ℕ0 ∧ ℕ0 ⊆ ℂ ) → 𝑧 : 𝐼 ⟶ ℂ )
80 53 75 79 sylancl ( ( ( 𝜑𝑦𝐷 ) ∧ 𝑧 ∈ { 𝑔𝐷𝑔r𝑦 } ) → 𝑧 : 𝐼 ⟶ ℂ )
81 ofsubeq0 ( ( 𝐼𝑉𝑦 : 𝐼 ⟶ ℂ ∧ 𝑧 : 𝐼 ⟶ ℂ ) → ( ( 𝑦f𝑧 ) = ( 𝐼 × { 0 } ) ↔ 𝑦 = 𝑧 ) )
82 74 78 80 81 syl3anc ( ( ( 𝜑𝑦𝐷 ) ∧ 𝑧 ∈ { 𝑔𝐷𝑔r𝑦 } ) → ( ( 𝑦f𝑧 ) = ( 𝐼 × { 0 } ) ↔ 𝑦 = 𝑧 ) )
83 62 82 sylan2 ( ( ( 𝜑𝑦𝐷 ) ∧ 𝑧 ∈ ( { 𝑔𝐷𝑔r𝑦 } ∖ { 𝑦 } ) ) → ( ( 𝑦f𝑧 ) = ( 𝐼 × { 0 } ) ↔ 𝑦 = 𝑧 ) )
84 83 necon3bbid ( ( ( 𝜑𝑦𝐷 ) ∧ 𝑧 ∈ ( { 𝑔𝐷𝑔r𝑦 } ∖ { 𝑦 } ) ) → ( ¬ ( 𝑦f𝑧 ) = ( 𝐼 × { 0 } ) ↔ 𝑦𝑧 ) )
85 73 84 mpbird ( ( ( 𝜑𝑦𝐷 ) ∧ 𝑧 ∈ ( { 𝑔𝐷𝑔r𝑦 } ∖ { 𝑦 } ) ) → ¬ ( 𝑦f𝑧 ) = ( 𝐼 × { 0 } ) )
86 85 iffalsed ( ( ( 𝜑𝑦𝐷 ) ∧ 𝑧 ∈ ( { 𝑔𝐷𝑔r𝑦 } ∖ { 𝑦 } ) ) → if ( ( 𝑦f𝑧 ) = ( 𝐼 × { 0 } ) , 1 , 0 ) = 0 )
87 70 86 eqtrd ( ( ( 𝜑𝑦𝐷 ) ∧ 𝑧 ∈ ( { 𝑔𝐷𝑔r𝑦 } ∖ { 𝑦 } ) ) → ( 𝑈 ‘ ( 𝑦f𝑧 ) ) = 0 )
88 87 oveq2d ( ( ( 𝜑𝑦𝐷 ) ∧ 𝑧 ∈ ( { 𝑔𝐷𝑔r𝑦 } ∖ { 𝑦 } ) ) → ( ( 𝑋𝑧 ) ( .r𝑅 ) ( 𝑈 ‘ ( 𝑦f𝑧 ) ) ) = ( ( 𝑋𝑧 ) ( .r𝑅 ) 0 ) )
89 11 18 5 ringrz ( ( 𝑅 ∈ Ring ∧ ( 𝑋𝑧 ) ∈ ( Base ‘ 𝑅 ) ) → ( ( 𝑋𝑧 ) ( .r𝑅 ) 0 ) = 0 )
90 41 48 89 syl2anc ( ( ( 𝜑𝑦𝐷 ) ∧ 𝑧 ∈ { 𝑔𝐷𝑔r𝑦 } ) → ( ( 𝑋𝑧 ) ( .r𝑅 ) 0 ) = 0 )
91 62 90 sylan2 ( ( ( 𝜑𝑦𝐷 ) ∧ 𝑧 ∈ ( { 𝑔𝐷𝑔r𝑦 } ∖ { 𝑦 } ) ) → ( ( 𝑋𝑧 ) ( .r𝑅 ) 0 ) = 0 )
92 88 91 eqtrd ( ( ( 𝜑𝑦𝐷 ) ∧ 𝑧 ∈ ( { 𝑔𝐷𝑔r𝑦 } ∖ { 𝑦 } ) ) → ( ( 𝑋𝑧 ) ( .r𝑅 ) ( 𝑈 ‘ ( 𝑦f𝑧 ) ) ) = 0 )
93 92 40 suppss2 ( ( 𝜑𝑦𝐷 ) → ( ( 𝑧 ∈ { 𝑔𝐷𝑔r𝑦 } ↦ ( ( 𝑋𝑧 ) ( .r𝑅 ) ( 𝑈 ‘ ( 𝑦f𝑧 ) ) ) ) supp 0 ) ⊆ { 𝑦 } )
94 40 mptexd ( ( 𝜑𝑦𝐷 ) → ( 𝑧 ∈ { 𝑔𝐷𝑔r𝑦 } ↦ ( ( 𝑋𝑧 ) ( .r𝑅 ) ( 𝑈 ‘ ( 𝑦f𝑧 ) ) ) ) ∈ V )
95 funmpt Fun ( 𝑧 ∈ { 𝑔𝐷𝑔r𝑦 } ↦ ( ( 𝑋𝑧 ) ( .r𝑅 ) ( 𝑈 ‘ ( 𝑦f𝑧 ) ) ) )
96 95 a1i ( ( 𝜑𝑦𝐷 ) → Fun ( 𝑧 ∈ { 𝑔𝐷𝑔r𝑦 } ↦ ( ( 𝑋𝑧 ) ( .r𝑅 ) ( 𝑈 ‘ ( 𝑦f𝑧 ) ) ) ) )
97 67 a1i ( ( 𝜑𝑦𝐷 ) → 0 ∈ V )
98 snfi { 𝑦 } ∈ Fin
99 98 a1i ( ( 𝜑𝑦𝐷 ) → { 𝑦 } ∈ Fin )
100 suppssfifsupp ( ( ( ( 𝑧 ∈ { 𝑔𝐷𝑔r𝑦 } ↦ ( ( 𝑋𝑧 ) ( .r𝑅 ) ( 𝑈 ‘ ( 𝑦f𝑧 ) ) ) ) ∈ V ∧ Fun ( 𝑧 ∈ { 𝑔𝐷𝑔r𝑦 } ↦ ( ( 𝑋𝑧 ) ( .r𝑅 ) ( 𝑈 ‘ ( 𝑦f𝑧 ) ) ) ) ∧ 0 ∈ V ) ∧ ( { 𝑦 } ∈ Fin ∧ ( ( 𝑧 ∈ { 𝑔𝐷𝑔r𝑦 } ↦ ( ( 𝑋𝑧 ) ( .r𝑅 ) ( 𝑈 ‘ ( 𝑦f𝑧 ) ) ) ) supp 0 ) ⊆ { 𝑦 } ) ) → ( 𝑧 ∈ { 𝑔𝐷𝑔r𝑦 } ↦ ( ( 𝑋𝑧 ) ( .r𝑅 ) ( 𝑈 ‘ ( 𝑦f𝑧 ) ) ) ) finSupp 0 )
101 94 96 97 99 93 100 syl32anc ( ( 𝜑𝑦𝐷 ) → ( 𝑧 ∈ { 𝑔𝐷𝑔r𝑦 } ↦ ( ( 𝑋𝑧 ) ( .r𝑅 ) ( 𝑈 ‘ ( 𝑦f𝑧 ) ) ) ) finSupp 0 )
102 11 5 37 40 61 93 101 gsumres ( ( 𝜑𝑦𝐷 ) → ( 𝑅 Σg ( ( 𝑧 ∈ { 𝑔𝐷𝑔r𝑦 } ↦ ( ( 𝑋𝑧 ) ( .r𝑅 ) ( 𝑈 ‘ ( 𝑦f𝑧 ) ) ) ) ↾ { 𝑦 } ) ) = ( 𝑅 Σg ( 𝑧 ∈ { 𝑔𝐷𝑔r𝑦 } ↦ ( ( 𝑋𝑧 ) ( .r𝑅 ) ( 𝑈 ‘ ( 𝑦f𝑧 ) ) ) ) ) )
103 3 adantr ( ( 𝜑𝑦𝐷 ) → 𝑅 ∈ Ring )
104 ringmnd ( 𝑅 ∈ Ring → 𝑅 ∈ Mnd )
105 103 104 syl ( ( 𝜑𝑦𝐷 ) → 𝑅 ∈ Mnd )
106 eqid 𝑦 = 𝑦
107 ofsubeq0 ( ( 𝐼𝑉𝑦 : 𝐼 ⟶ ℂ ∧ 𝑦 : 𝐼 ⟶ ℂ ) → ( ( 𝑦f𝑦 ) = ( 𝐼 × { 0 } ) ↔ 𝑦 = 𝑦 ) )
108 24 77 77 107 syl3anc ( ( 𝜑𝑦𝐷 ) → ( ( 𝑦f𝑦 ) = ( 𝐼 × { 0 } ) ↔ 𝑦 = 𝑦 ) )
109 106 108 mpbiri ( ( 𝜑𝑦𝐷 ) → ( 𝑦f𝑦 ) = ( 𝐼 × { 0 } ) )
110 109 fveq2d ( ( 𝜑𝑦𝐷 ) → ( 𝑈 ‘ ( 𝑦f𝑦 ) ) = ( 𝑈 ‘ ( 𝐼 × { 0 } ) ) )
111 fconstmpt ( 𝐼 × { 0 } ) = ( 𝑤𝐼 ↦ 0 )
112 4 fczpsrbag ( 𝐼𝑉 → ( 𝑤𝐼 ↦ 0 ) ∈ 𝐷 )
113 2 112 syl ( 𝜑 → ( 𝑤𝐼 ↦ 0 ) ∈ 𝐷 )
114 111 113 eqeltrid ( 𝜑 → ( 𝐼 × { 0 } ) ∈ 𝐷 )
115 114 adantr ( ( 𝜑𝑦𝐷 ) → ( 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 ( ( 𝜑𝑦𝐷 ) → ( 𝑈 ‘ ( 𝐼 × { 0 } ) ) = 1 )
119 110 118 eqtrd ( ( 𝜑𝑦𝐷 ) → ( 𝑈 ‘ ( 𝑦f𝑦 ) ) = 1 )
120 119 oveq2d ( ( 𝜑𝑦𝐷 ) → ( ( 𝑋𝑦 ) ( .r𝑅 ) ( 𝑈 ‘ ( 𝑦f𝑦 ) ) ) = ( ( 𝑋𝑦 ) ( .r𝑅 ) 1 ) )
121 16 ffvelrnda ( ( 𝜑𝑦𝐷 ) → ( 𝑋𝑦 ) ∈ ( Base ‘ 𝑅 ) )
122 11 18 6 ringridm ( ( 𝑅 ∈ Ring ∧ ( 𝑋𝑦 ) ∈ ( Base ‘ 𝑅 ) ) → ( ( 𝑋𝑦 ) ( .r𝑅 ) 1 ) = ( 𝑋𝑦 ) )
123 103 121 122 syl2anc ( ( 𝜑𝑦𝐷 ) → ( ( 𝑋𝑦 ) ( .r𝑅 ) 1 ) = ( 𝑋𝑦 ) )
124 120 123 eqtrd ( ( 𝜑𝑦𝐷 ) → ( ( 𝑋𝑦 ) ( .r𝑅 ) ( 𝑈 ‘ ( 𝑦f𝑦 ) ) ) = ( 𝑋𝑦 ) )
125 124 121 eqeltrd ( ( 𝜑𝑦𝐷 ) → ( ( 𝑋𝑦 ) ( .r𝑅 ) ( 𝑈 ‘ ( 𝑦f𝑦 ) ) ) ∈ ( Base ‘ 𝑅 ) )
126 fveq2 ( 𝑧 = 𝑦 → ( 𝑋𝑧 ) = ( 𝑋𝑦 ) )
127 oveq2 ( 𝑧 = 𝑦 → ( 𝑦f𝑧 ) = ( 𝑦f𝑦 ) )
128 127 fveq2d ( 𝑧 = 𝑦 → ( 𝑈 ‘ ( 𝑦f𝑧 ) ) = ( 𝑈 ‘ ( 𝑦f𝑦 ) ) )
129 126 128 oveq12d ( 𝑧 = 𝑦 → ( ( 𝑋𝑧 ) ( .r𝑅 ) ( 𝑈 ‘ ( 𝑦f𝑧 ) ) ) = ( ( 𝑋𝑦 ) ( .r𝑅 ) ( 𝑈 ‘ ( 𝑦f𝑦 ) ) ) )
130 11 129 gsumsn ( ( 𝑅 ∈ Mnd ∧ 𝑦𝐷 ∧ ( ( 𝑋𝑦 ) ( .r𝑅 ) ( 𝑈 ‘ ( 𝑦f𝑦 ) ) ) ∈ ( Base ‘ 𝑅 ) ) → ( 𝑅 Σg ( 𝑧 ∈ { 𝑦 } ↦ ( ( 𝑋𝑧 ) ( .r𝑅 ) ( 𝑈 ‘ ( 𝑦f𝑧 ) ) ) ) ) = ( ( 𝑋𝑦 ) ( .r𝑅 ) ( 𝑈 ‘ ( 𝑦f𝑦 ) ) ) )
131 105 21 125 130 syl3anc ( ( 𝜑𝑦𝐷 ) → ( 𝑅 Σg ( 𝑧 ∈ { 𝑦 } ↦ ( ( 𝑋𝑧 ) ( .r𝑅 ) ( 𝑈 ‘ ( 𝑦f𝑧 ) ) ) ) ) = ( ( 𝑋𝑦 ) ( .r𝑅 ) ( 𝑈 ‘ ( 𝑦f𝑦 ) ) ) )
132 34 102 131 3eqtr3d ( ( 𝜑𝑦𝐷 ) → ( 𝑅 Σg ( 𝑧 ∈ { 𝑔𝐷𝑔r𝑦 } ↦ ( ( 𝑋𝑧 ) ( .r𝑅 ) ( 𝑈 ‘ ( 𝑦f𝑧 ) ) ) ) ) = ( ( 𝑋𝑦 ) ( .r𝑅 ) ( 𝑈 ‘ ( 𝑦f𝑦 ) ) ) )
133 22 132 124 3eqtrd ( ( 𝜑𝑦𝐷 ) → ( ( 𝑋 · 𝑈 ) ‘ 𝑦 ) = ( 𝑋𝑦 ) )
134 15 17 133 eqfnfvd ( 𝜑 → ( 𝑋 · 𝑈 ) = 𝑋 )