Metamath Proof Explorer

Theorem cpmadugsumlemB

Description: Lemma B for cpmadugsum . (Contributed by AV, 2-Nov-2019)

Ref Expression
Hypotheses cpmadugsum.a ⊢ 𝐴 = ( 𝑁 Mat 𝑅 )
cpmadugsum.b ⊢ 𝐵 = ( Base ‘ 𝐴 )
cpmadugsum.p ⊢ 𝑃 = ( Poly1 ‘ 𝑅 )
cpmadugsum.y ⊢ 𝑌 = ( 𝑁 Mat 𝑃 )
cpmadugsum.t ⊢ 𝑇 = ( 𝑁 matToPolyMat 𝑅 )
cpmadugsum.x ⊢ 𝑋 = ( var1 ‘ 𝑅 )
cpmadugsum.e ⊢ ↑ = ( .g ‘ ( mulGrp ‘ 𝑃 ) )
cpmadugsum.m ⊢ · = ( ·𝑠 ‘ 𝑌 )
cpmadugsum.r ⊢ × = ( .r ‘ 𝑌 )
cpmadugsum.1 ⊢ 1 = ( 1r ‘ 𝑌 )
Assertion cpmadugsumlemB ( ( ( 𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵 ) ∧ ( 𝑠 ∈ ℕ0 ∧ 𝑏 ∈ ( 𝐵 ↑m ( 0 ... 𝑠 ) ) ) ) → ( ( 𝑋 · 1 ) × ( 𝑌 Σg ( 𝑖 ∈ ( 0 ... 𝑠 ) ↦ ( ( 𝑖 ↑ 𝑋 ) · ( 𝑇 ‘ ( 𝑏 ‘ 𝑖 ) ) ) ) ) ) = ( 𝑌 Σg ( 𝑖 ∈ ( 0 ... 𝑠 ) ↦ ( ( ( 𝑖 + 1 ) ↑ 𝑋 ) · ( 𝑇 ‘ ( 𝑏 ‘ 𝑖 ) ) ) ) ) )

Proof

Step Hyp Ref Expression
1 cpmadugsum.a ⊢ 𝐴 = ( 𝑁 Mat 𝑅 )
2 cpmadugsum.b ⊢ 𝐵 = ( Base ‘ 𝐴 )
3 cpmadugsum.p ⊢ 𝑃 = ( Poly1 ‘ 𝑅 )
4 cpmadugsum.y ⊢ 𝑌 = ( 𝑁 Mat 𝑃 )
5 cpmadugsum.t ⊢ 𝑇 = ( 𝑁 matToPolyMat 𝑅 )
6 cpmadugsum.x ⊢ 𝑋 = ( var1 ‘ 𝑅 )
7 cpmadugsum.e ⊢ ↑ = ( .g ‘ ( mulGrp ‘ 𝑃 ) )
8 cpmadugsum.m ⊢ · = ( ·𝑠 ‘ 𝑌 )
9 cpmadugsum.r ⊢ × = ( .r ‘ 𝑌 )
10 cpmadugsum.1 ⊢ 1 = ( 1r ‘ 𝑌 )
11 crngring ⊢ ( 𝑅 ∈ CRing → 𝑅 ∈ Ring )
12 3 ply1ring ⊢ ( 𝑅 ∈ Ring → 𝑃 ∈ Ring )
13 11 12 syl ⊢ ( 𝑅 ∈ CRing → 𝑃 ∈ Ring )
14 13 3ad2ant2 ⊢ ( ( 𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵 ) → 𝑃 ∈ Ring )
15 eqid ⊢ ( mulGrp ‘ 𝑃 ) = ( mulGrp ‘ 𝑃 )
16 15 ringmgp ⊢ ( 𝑃 ∈ Ring → ( mulGrp ‘ 𝑃 ) ∈ Mnd )
17 14 16 syl ⊢ ( ( 𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵 ) → ( mulGrp ‘ 𝑃 ) ∈ Mnd )
18 17 ad2antrr ⊢ ( ( ( ( 𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵 ) ∧ ( 𝑠 ∈ ℕ0 ∧ 𝑏 ∈ ( 𝐵 ↑m ( 0 ... 𝑠 ) ) ) ) ∧ 𝑖 ∈ ( 0 ... 𝑠 ) ) → ( mulGrp ‘ 𝑃 ) ∈ Mnd )
19 elfznn0 ⊢ ( 𝑖 ∈ ( 0 ... 𝑠 ) → 𝑖 ∈ ℕ0 )
20 19 adantl ⊢ ( ( ( ( 𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵 ) ∧ ( 𝑠 ∈ ℕ0 ∧ 𝑏 ∈ ( 𝐵 ↑m ( 0 ... 𝑠 ) ) ) ) ∧ 𝑖 ∈ ( 0 ... 𝑠 ) ) → 𝑖 ∈ ℕ0 )
21 1nn0 ⊢ 1 ∈ ℕ0
22 21 a1i ⊢ ( ( ( ( 𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵 ) ∧ ( 𝑠 ∈ ℕ0 ∧ 𝑏 ∈ ( 𝐵 ↑m ( 0 ... 𝑠 ) ) ) ) ∧ 𝑖 ∈ ( 0 ... 𝑠 ) ) → 1 ∈ ℕ0 )
23 11 3ad2ant2 ⊢ ( ( 𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵 ) → 𝑅 ∈ Ring )
24 eqid ⊢ ( Base ‘ 𝑃 ) = ( Base ‘ 𝑃 )
25 6 3 24 vr1cl ⊢ ( 𝑅 ∈ Ring → 𝑋 ∈ ( Base ‘ 𝑃 ) )
26 23 25 syl ⊢ ( ( 𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵 ) → 𝑋 ∈ ( Base ‘ 𝑃 ) )
27 26 ad2antrr ⊢ ( ( ( ( 𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵 ) ∧ ( 𝑠 ∈ ℕ0 ∧ 𝑏 ∈ ( 𝐵 ↑m ( 0 ... 𝑠 ) ) ) ) ∧ 𝑖 ∈ ( 0 ... 𝑠 ) ) → 𝑋 ∈ ( Base ‘ 𝑃 ) )
28 15 24 mgpbas ⊢ ( Base ‘ 𝑃 ) = ( Base ‘ ( mulGrp ‘ 𝑃 ) )
29 eqid ⊢ ( .r ‘ 𝑃 ) = ( .r ‘ 𝑃 )
30 15 29 mgpplusg ⊢ ( .r ‘ 𝑃 ) = ( +g ‘ ( mulGrp ‘ 𝑃 ) )
31 28 7 30 mulgnn0dir ⊢ ( ( ( mulGrp ‘ 𝑃 ) ∈ Mnd ∧ ( 𝑖 ∈ ℕ0 ∧ 1 ∈ ℕ0 ∧ 𝑋 ∈ ( Base ‘ 𝑃 ) ) ) → ( ( 𝑖 + 1 ) ↑ 𝑋 ) = ( ( 𝑖 ↑ 𝑋 ) ( .r ‘ 𝑃 ) ( 1 ↑ 𝑋 ) ) )
32 18 20 22 27 31 syl13anc ⊢ ( ( ( ( 𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵 ) ∧ ( 𝑠 ∈ ℕ0 ∧ 𝑏 ∈ ( 𝐵 ↑m ( 0 ... 𝑠 ) ) ) ) ∧ 𝑖 ∈ ( 0 ... 𝑠 ) ) → ( ( 𝑖 + 1 ) ↑ 𝑋 ) = ( ( 𝑖 ↑ 𝑋 ) ( .r ‘ 𝑃 ) ( 1 ↑ 𝑋 ) ) )
33 3 ply1crng ⊢ ( 𝑅 ∈ CRing → 𝑃 ∈ CRing )
34 33 anim2i ⊢ ( ( 𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ) → ( 𝑁 ∈ Fin ∧ 𝑃 ∈ CRing ) )
35 34 3adant3 ⊢ ( ( 𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵 ) → ( 𝑁 ∈ Fin ∧ 𝑃 ∈ CRing ) )
36 4 matsca2 ⊢ ( ( 𝑁 ∈ Fin ∧ 𝑃 ∈ CRing ) → 𝑃 = ( Scalar ‘ 𝑌 ) )
37 35 36 syl ⊢ ( ( 𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵 ) → 𝑃 = ( Scalar ‘ 𝑌 ) )
38 37 ad2antrr ⊢ ( ( ( ( 𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵 ) ∧ ( 𝑠 ∈ ℕ0 ∧ 𝑏 ∈ ( 𝐵 ↑m ( 0 ... 𝑠 ) ) ) ) ∧ 𝑖 ∈ ( 0 ... 𝑠 ) ) → 𝑃 = ( Scalar ‘ 𝑌 ) )
39 38 fveq2d ⊢ ( ( ( ( 𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵 ) ∧ ( 𝑠 ∈ ℕ0 ∧ 𝑏 ∈ ( 𝐵 ↑m ( 0 ... 𝑠 ) ) ) ) ∧ 𝑖 ∈ ( 0 ... 𝑠 ) ) → ( .r ‘ 𝑃 ) = ( .r ‘ ( Scalar ‘ 𝑌 ) ) )
40 eqidd ⊢ ( ( ( ( 𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵 ) ∧ ( 𝑠 ∈ ℕ0 ∧ 𝑏 ∈ ( 𝐵 ↑m ( 0 ... 𝑠 ) ) ) ) ∧ 𝑖 ∈ ( 0 ... 𝑠 ) ) → ( 𝑖 ↑ 𝑋 ) = ( 𝑖 ↑ 𝑋 ) )
41 28 7 mulg1 ⊢ ( 𝑋 ∈ ( Base ‘ 𝑃 ) → ( 1 ↑ 𝑋 ) = 𝑋 )
42 26 41 syl ⊢ ( ( 𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵 ) → ( 1 ↑ 𝑋 ) = 𝑋 )
43 42 ad2antrr ⊢ ( ( ( ( 𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵 ) ∧ ( 𝑠 ∈ ℕ0 ∧ 𝑏 ∈ ( 𝐵 ↑m ( 0 ... 𝑠 ) ) ) ) ∧ 𝑖 ∈ ( 0 ... 𝑠 ) ) → ( 1 ↑ 𝑋 ) = 𝑋 )
44 39 40 43 oveq123d ⊢ ( ( ( ( 𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵 ) ∧ ( 𝑠 ∈ ℕ0 ∧ 𝑏 ∈ ( 𝐵 ↑m ( 0 ... 𝑠 ) ) ) ) ∧ 𝑖 ∈ ( 0 ... 𝑠 ) ) → ( ( 𝑖 ↑ 𝑋 ) ( .r ‘ 𝑃 ) ( 1 ↑ 𝑋 ) ) = ( ( 𝑖 ↑ 𝑋 ) ( .r ‘ ( Scalar ‘ 𝑌 ) ) 𝑋 ) )
45 32 44 eqtrd ⊢ ( ( ( ( 𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵 ) ∧ ( 𝑠 ∈ ℕ0 ∧ 𝑏 ∈ ( 𝐵 ↑m ( 0 ... 𝑠 ) ) ) ) ∧ 𝑖 ∈ ( 0 ... 𝑠 ) ) → ( ( 𝑖 + 1 ) ↑ 𝑋 ) = ( ( 𝑖 ↑ 𝑋 ) ( .r ‘ ( Scalar ‘ 𝑌 ) ) 𝑋 ) )
46 13 anim2i ⊢ ( ( 𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ) → ( 𝑁 ∈ Fin ∧ 𝑃 ∈ Ring ) )
47 46 3adant3 ⊢ ( ( 𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵 ) → ( 𝑁 ∈ Fin ∧ 𝑃 ∈ Ring ) )
48 4 matring ⊢ ( ( 𝑁 ∈ Fin ∧ 𝑃 ∈ Ring ) → 𝑌 ∈ Ring )
49 47 48 syl ⊢ ( ( 𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵 ) → 𝑌 ∈ Ring )
50 49 ad2antrr ⊢ ( ( ( ( 𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵 ) ∧ ( 𝑠 ∈ ℕ0 ∧ 𝑏 ∈ ( 𝐵 ↑m ( 0 ... 𝑠 ) ) ) ) ∧ 𝑖 ∈ ( 0 ... 𝑠 ) ) → 𝑌 ∈ Ring )
51 simpll1 ⊢ ( ( ( ( 𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵 ) ∧ ( 𝑠 ∈ ℕ0 ∧ 𝑏 ∈ ( 𝐵 ↑m ( 0 ... 𝑠 ) ) ) ) ∧ 𝑖 ∈ ( 0 ... 𝑠 ) ) → 𝑁 ∈ Fin )
52 23 ad2antrr ⊢ ( ( ( ( 𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵 ) ∧ ( 𝑠 ∈ ℕ0 ∧ 𝑏 ∈ ( 𝐵 ↑m ( 0 ... 𝑠 ) ) ) ) ∧ 𝑖 ∈ ( 0 ... 𝑠 ) ) → 𝑅 ∈ Ring )
53 simplrl ⊢ ( ( ( ( 𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵 ) ∧ ( 𝑠 ∈ ℕ0 ∧ 𝑏 ∈ ( 𝐵 ↑m ( 0 ... 𝑠 ) ) ) ) ∧ 𝑖 ∈ ( 0 ... 𝑠 ) ) → 𝑠 ∈ ℕ0 )
54 simprr ⊢ ( ( ( 𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵 ) ∧ ( 𝑠 ∈ ℕ0 ∧ 𝑏 ∈ ( 𝐵 ↑m ( 0 ... 𝑠 ) ) ) ) → 𝑏 ∈ ( 𝐵 ↑m ( 0 ... 𝑠 ) ) )
55 54 anim1i ⊢ ( ( ( ( 𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵 ) ∧ ( 𝑠 ∈ ℕ0 ∧ 𝑏 ∈ ( 𝐵 ↑m ( 0 ... 𝑠 ) ) ) ) ∧ 𝑖 ∈ ( 0 ... 𝑠 ) ) → ( 𝑏 ∈ ( 𝐵 ↑m ( 0 ... 𝑠 ) ) ∧ 𝑖 ∈ ( 0 ... 𝑠 ) ) )
56 1 2 3 4 5 m2pmfzmap ⊢ ( ( ( 𝑁 ∈ Fin ∧ 𝑅 ∈ Ring ∧ 𝑠 ∈ ℕ0 ) ∧ ( 𝑏 ∈ ( 𝐵 ↑m ( 0 ... 𝑠 ) ) ∧ 𝑖 ∈ ( 0 ... 𝑠 ) ) ) → ( 𝑇 ‘ ( 𝑏 ‘ 𝑖 ) ) ∈ ( Base ‘ 𝑌 ) )
57 51 52 53 55 56 syl31anc ⊢ ( ( ( ( 𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵 ) ∧ ( 𝑠 ∈ ℕ0 ∧ 𝑏 ∈ ( 𝐵 ↑m ( 0 ... 𝑠 ) ) ) ) ∧ 𝑖 ∈ ( 0 ... 𝑠 ) ) → ( 𝑇 ‘ ( 𝑏 ‘ 𝑖 ) ) ∈ ( Base ‘ 𝑌 ) )
58 eqid ⊢ ( Base ‘ 𝑌 ) = ( Base ‘ 𝑌 )
59 58 9 10 ringlidm ⊢ ( ( 𝑌 ∈ Ring ∧ ( 𝑇 ‘ ( 𝑏 ‘ 𝑖 ) ) ∈ ( Base ‘ 𝑌 ) ) → ( 1 × ( 𝑇 ‘ ( 𝑏 ‘ 𝑖 ) ) ) = ( 𝑇 ‘ ( 𝑏 ‘ 𝑖 ) ) )
60 50 57 59 syl2anc ⊢ ( ( ( ( 𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵 ) ∧ ( 𝑠 ∈ ℕ0 ∧ 𝑏 ∈ ( 𝐵 ↑m ( 0 ... 𝑠 ) ) ) ) ∧ 𝑖 ∈ ( 0 ... 𝑠 ) ) → ( 1 × ( 𝑇 ‘ ( 𝑏 ‘ 𝑖 ) ) ) = ( 𝑇 ‘ ( 𝑏 ‘ 𝑖 ) ) )
61 60 eqcomd ⊢ ( ( ( ( 𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵 ) ∧ ( 𝑠 ∈ ℕ0 ∧ 𝑏 ∈ ( 𝐵 ↑m ( 0 ... 𝑠 ) ) ) ) ∧ 𝑖 ∈ ( 0 ... 𝑠 ) ) → ( 𝑇 ‘ ( 𝑏 ‘ 𝑖 ) ) = ( 1 × ( 𝑇 ‘ ( 𝑏 ‘ 𝑖 ) ) ) )
62 45 61 oveq12d ⊢ ( ( ( ( 𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵 ) ∧ ( 𝑠 ∈ ℕ0 ∧ 𝑏 ∈ ( 𝐵 ↑m ( 0 ... 𝑠 ) ) ) ) ∧ 𝑖 ∈ ( 0 ... 𝑠 ) ) → ( ( ( 𝑖 + 1 ) ↑ 𝑋 ) · ( 𝑇 ‘ ( 𝑏 ‘ 𝑖 ) ) ) = ( ( ( 𝑖 ↑ 𝑋 ) ( .r ‘ ( Scalar ‘ 𝑌 ) ) 𝑋 ) · ( 1 × ( 𝑇 ‘ ( 𝑏 ‘ 𝑖 ) ) ) ) )
63 4 matassa ⊢ ( ( 𝑁 ∈ Fin ∧ 𝑃 ∈ CRing ) → 𝑌 ∈ AssAlg )
64 34 63 syl ⊢ ( ( 𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ) → 𝑌 ∈ AssAlg )
65 64 3adant3 ⊢ ( ( 𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵 ) → 𝑌 ∈ AssAlg )
66 65 ad2antrr ⊢ ( ( ( ( 𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵 ) ∧ ( 𝑠 ∈ ℕ0 ∧ 𝑏 ∈ ( 𝐵 ↑m ( 0 ... 𝑠 ) ) ) ) ∧ 𝑖 ∈ ( 0 ... 𝑠 ) ) → 𝑌 ∈ AssAlg )
67 37 eqcomd ⊢ ( ( 𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵 ) → ( Scalar ‘ 𝑌 ) = 𝑃 )
68 67 fveq2d ⊢ ( ( 𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵 ) → ( Base ‘ ( Scalar ‘ 𝑌 ) ) = ( Base ‘ 𝑃 ) )
69 26 68 eleqtrrd ⊢ ( ( 𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵 ) → 𝑋 ∈ ( Base ‘ ( Scalar ‘ 𝑌 ) ) )
70 69 ad2antrr ⊢ ( ( ( ( 𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵 ) ∧ ( 𝑠 ∈ ℕ0 ∧ 𝑏 ∈ ( 𝐵 ↑m ( 0 ... 𝑠 ) ) ) ) ∧ 𝑖 ∈ ( 0 ... 𝑠 ) ) → 𝑋 ∈ ( Base ‘ ( Scalar ‘ 𝑌 ) ) )
71 28 7 mulgnn0cl ⊢ ( ( ( mulGrp ‘ 𝑃 ) ∈ Mnd ∧ 𝑖 ∈ ℕ0 ∧ 𝑋 ∈ ( Base ‘ 𝑃 ) ) → ( 𝑖 ↑ 𝑋 ) ∈ ( Base ‘ 𝑃 ) )
72 18 20 27 71 syl3anc ⊢ ( ( ( ( 𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵 ) ∧ ( 𝑠 ∈ ℕ0 ∧ 𝑏 ∈ ( 𝐵 ↑m ( 0 ... 𝑠 ) ) ) ) ∧ 𝑖 ∈ ( 0 ... 𝑠 ) ) → ( 𝑖 ↑ 𝑋 ) ∈ ( Base ‘ 𝑃 ) )
73 68 ad2antrr ⊢ ( ( ( ( 𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵 ) ∧ ( 𝑠 ∈ ℕ0 ∧ 𝑏 ∈ ( 𝐵 ↑m ( 0 ... 𝑠 ) ) ) ) ∧ 𝑖 ∈ ( 0 ... 𝑠 ) ) → ( Base ‘ ( Scalar ‘ 𝑌 ) ) = ( Base ‘ 𝑃 ) )
74 72 73 eleqtrrd ⊢ ( ( ( ( 𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵 ) ∧ ( 𝑠 ∈ ℕ0 ∧ 𝑏 ∈ ( 𝐵 ↑m ( 0 ... 𝑠 ) ) ) ) ∧ 𝑖 ∈ ( 0 ... 𝑠 ) ) → ( 𝑖 ↑ 𝑋 ) ∈ ( Base ‘ ( Scalar ‘ 𝑌 ) ) )
75 46 48 syl ⊢ ( ( 𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ) → 𝑌 ∈ Ring )
76 75 3adant3 ⊢ ( ( 𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵 ) → 𝑌 ∈ Ring )
77 58 10 ringidcl ⊢ ( 𝑌 ∈ Ring → 1 ∈ ( Base ‘ 𝑌 ) )
78 76 77 syl ⊢ ( ( 𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵 ) → 1 ∈ ( Base ‘ 𝑌 ) )
79 78 ad2antrr ⊢ ( ( ( ( 𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵 ) ∧ ( 𝑠 ∈ ℕ0 ∧ 𝑏 ∈ ( 𝐵 ↑m ( 0 ... 𝑠 ) ) ) ) ∧ 𝑖 ∈ ( 0 ... 𝑠 ) ) → 1 ∈ ( Base ‘ 𝑌 ) )
80 eqid ⊢ ( Scalar ‘ 𝑌 ) = ( Scalar ‘ 𝑌 )
81 eqid ⊢ ( Base ‘ ( Scalar ‘ 𝑌 ) ) = ( Base ‘ ( Scalar ‘ 𝑌 ) )
82 eqid ⊢ ( .r ‘ ( Scalar ‘ 𝑌 ) ) = ( .r ‘ ( Scalar ‘ 𝑌 ) )
83 58 80 81 82 8 9 assa2ass ⊢ ( ( 𝑌 ∈ AssAlg ∧ ( 𝑋 ∈ ( Base ‘ ( Scalar ‘ 𝑌 ) ) ∧ ( 𝑖 ↑ 𝑋 ) ∈ ( Base ‘ ( Scalar ‘ 𝑌 ) ) ) ∧ ( 1 ∈ ( Base ‘ 𝑌 ) ∧ ( 𝑇 ‘ ( 𝑏 ‘ 𝑖 ) ) ∈ ( Base ‘ 𝑌 ) ) ) → ( ( 𝑋 · 1 ) × ( ( 𝑖 ↑ 𝑋 ) · ( 𝑇 ‘ ( 𝑏 ‘ 𝑖 ) ) ) ) = ( ( ( 𝑖 ↑ 𝑋 ) ( .r ‘ ( Scalar ‘ 𝑌 ) ) 𝑋 ) · ( 1 × ( 𝑇 ‘ ( 𝑏 ‘ 𝑖 ) ) ) ) )
84 66 70 74 79 57 83 syl122anc ⊢ ( ( ( ( 𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵 ) ∧ ( 𝑠 ∈ ℕ0 ∧ 𝑏 ∈ ( 𝐵 ↑m ( 0 ... 𝑠 ) ) ) ) ∧ 𝑖 ∈ ( 0 ... 𝑠 ) ) → ( ( 𝑋 · 1 ) × ( ( 𝑖 ↑ 𝑋 ) · ( 𝑇 ‘ ( 𝑏 ‘ 𝑖 ) ) ) ) = ( ( ( 𝑖 ↑ 𝑋 ) ( .r ‘ ( Scalar ‘ 𝑌 ) ) 𝑋 ) · ( 1 × ( 𝑇 ‘ ( 𝑏 ‘ 𝑖 ) ) ) ) )
85 84 eqcomd ⊢ ( ( ( ( 𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵 ) ∧ ( 𝑠 ∈ ℕ0 ∧ 𝑏 ∈ ( 𝐵 ↑m ( 0 ... 𝑠 ) ) ) ) ∧ 𝑖 ∈ ( 0 ... 𝑠 ) ) → ( ( ( 𝑖 ↑ 𝑋 ) ( .r ‘ ( Scalar ‘ 𝑌 ) ) 𝑋 ) · ( 1 × ( 𝑇 ‘ ( 𝑏 ‘ 𝑖 ) ) ) ) = ( ( 𝑋 · 1 ) × ( ( 𝑖 ↑ 𝑋 ) · ( 𝑇 ‘ ( 𝑏 ‘ 𝑖 ) ) ) ) )
86 62 85 eqtrd ⊢ ( ( ( ( 𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵 ) ∧ ( 𝑠 ∈ ℕ0 ∧ 𝑏 ∈ ( 𝐵 ↑m ( 0 ... 𝑠 ) ) ) ) ∧ 𝑖 ∈ ( 0 ... 𝑠 ) ) → ( ( ( 𝑖 + 1 ) ↑ 𝑋 ) · ( 𝑇 ‘ ( 𝑏 ‘ 𝑖 ) ) ) = ( ( 𝑋 · 1 ) × ( ( 𝑖 ↑ 𝑋 ) · ( 𝑇 ‘ ( 𝑏 ‘ 𝑖 ) ) ) ) )
87 86 mpteq2dva ⊢ ( ( ( 𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵 ) ∧ ( 𝑠 ∈ ℕ0 ∧ 𝑏 ∈ ( 𝐵 ↑m ( 0 ... 𝑠 ) ) ) ) → ( 𝑖 ∈ ( 0 ... 𝑠 ) ↦ ( ( ( 𝑖 + 1 ) ↑ 𝑋 ) · ( 𝑇 ‘ ( 𝑏 ‘ 𝑖 ) ) ) ) = ( 𝑖 ∈ ( 0 ... 𝑠 ) ↦ ( ( 𝑋 · 1 ) × ( ( 𝑖 ↑ 𝑋 ) · ( 𝑇 ‘ ( 𝑏 ‘ 𝑖 ) ) ) ) ) )
88 87 oveq2d ⊢ ( ( ( 𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵 ) ∧ ( 𝑠 ∈ ℕ0 ∧ 𝑏 ∈ ( 𝐵 ↑m ( 0 ... 𝑠 ) ) ) ) → ( 𝑌 Σg ( 𝑖 ∈ ( 0 ... 𝑠 ) ↦ ( ( ( 𝑖 + 1 ) ↑ 𝑋 ) · ( 𝑇 ‘ ( 𝑏 ‘ 𝑖 ) ) ) ) ) = ( 𝑌 Σg ( 𝑖 ∈ ( 0 ... 𝑠 ) ↦ ( ( 𝑋 · 1 ) × ( ( 𝑖 ↑ 𝑋 ) · ( 𝑇 ‘ ( 𝑏 ‘ 𝑖 ) ) ) ) ) ) )
89 eqid ⊢ ( 0g ‘ 𝑌 ) = ( 0g ‘ 𝑌 )
90 eqid ⊢ ( +g ‘ 𝑌 ) = ( +g ‘ 𝑌 )
91 76 adantr ⊢ ( ( ( 𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵 ) ∧ ( 𝑠 ∈ ℕ0 ∧ 𝑏 ∈ ( 𝐵 ↑m ( 0 ... 𝑠 ) ) ) ) → 𝑌 ∈ Ring )
92 ovexd ⊢ ( ( ( 𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵 ) ∧ ( 𝑠 ∈ ℕ0 ∧ 𝑏 ∈ ( 𝐵 ↑m ( 0 ... 𝑠 ) ) ) ) → ( 0 ... 𝑠 ) ∈ V )
93 4 matlmod ⊢ ( ( 𝑁 ∈ Fin ∧ 𝑃 ∈ Ring ) → 𝑌 ∈ LMod )
94 46 93 syl ⊢ ( ( 𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ) → 𝑌 ∈ LMod )
95 94 3adant3 ⊢ ( ( 𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵 ) → 𝑌 ∈ LMod )
96 11 adantl ⊢ ( ( 𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ) → 𝑅 ∈ Ring )
97 96 25 syl ⊢ ( ( 𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ) → 𝑋 ∈ ( Base ‘ 𝑃 ) )
98 34 36 syl ⊢ ( ( 𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ) → 𝑃 = ( Scalar ‘ 𝑌 ) )
99 98 eqcomd ⊢ ( ( 𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ) → ( Scalar ‘ 𝑌 ) = 𝑃 )
100 99 fveq2d ⊢ ( ( 𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ) → ( Base ‘ ( Scalar ‘ 𝑌 ) ) = ( Base ‘ 𝑃 ) )
101 97 100 eleqtrrd ⊢ ( ( 𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ) → 𝑋 ∈ ( Base ‘ ( Scalar ‘ 𝑌 ) ) )
102 101 3adant3 ⊢ ( ( 𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵 ) → 𝑋 ∈ ( Base ‘ ( Scalar ‘ 𝑌 ) ) )
103 49 77 syl ⊢ ( ( 𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵 ) → 1 ∈ ( Base ‘ 𝑌 ) )
104 58 80 8 81 lmodvscl ⊢ ( ( 𝑌 ∈ LMod ∧ 𝑋 ∈ ( Base ‘ ( Scalar ‘ 𝑌 ) ) ∧ 1 ∈ ( Base ‘ 𝑌 ) ) → ( 𝑋 · 1 ) ∈ ( Base ‘ 𝑌 ) )
105 95 102 103 104 syl3anc ⊢ ( ( 𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵 ) → ( 𝑋 · 1 ) ∈ ( Base ‘ 𝑌 ) )
106 105 adantr ⊢ ( ( ( 𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵 ) ∧ ( 𝑠 ∈ ℕ0 ∧ 𝑏 ∈ ( 𝐵 ↑m ( 0 ... 𝑠 ) ) ) ) → ( 𝑋 · 1 ) ∈ ( Base ‘ 𝑌 ) )
107 95 ad2antrr ⊢ ( ( ( ( 𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵 ) ∧ ( 𝑠 ∈ ℕ0 ∧ 𝑏 ∈ ( 𝐵 ↑m ( 0 ... 𝑠 ) ) ) ) ∧ 𝑖 ∈ ( 0 ... 𝑠 ) ) → 𝑌 ∈ LMod )
108 36 eqcomd ⊢ ( ( 𝑁 ∈ Fin ∧ 𝑃 ∈ CRing ) → ( Scalar ‘ 𝑌 ) = 𝑃 )
109 108 fveq2d ⊢ ( ( 𝑁 ∈ Fin ∧ 𝑃 ∈ CRing ) → ( Base ‘ ( Scalar ‘ 𝑌 ) ) = ( Base ‘ 𝑃 ) )
110 35 109 syl ⊢ ( ( 𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵 ) → ( Base ‘ ( Scalar ‘ 𝑌 ) ) = ( Base ‘ 𝑃 ) )
111 110 eleq2d ⊢ ( ( 𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵 ) → ( ( 𝑖 ↑ 𝑋 ) ∈ ( Base ‘ ( Scalar ‘ 𝑌 ) ) ↔ ( 𝑖 ↑ 𝑋 ) ∈ ( Base ‘ 𝑃 ) ) )
112 111 ad2antrr ⊢ ( ( ( ( 𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵 ) ∧ ( 𝑠 ∈ ℕ0 ∧ 𝑏 ∈ ( 𝐵 ↑m ( 0 ... 𝑠 ) ) ) ) ∧ 𝑖 ∈ ( 0 ... 𝑠 ) ) → ( ( 𝑖 ↑ 𝑋 ) ∈ ( Base ‘ ( Scalar ‘ 𝑌 ) ) ↔ ( 𝑖 ↑ 𝑋 ) ∈ ( Base ‘ 𝑃 ) ) )
113 72 112 mpbird ⊢ ( ( ( ( 𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵 ) ∧ ( 𝑠 ∈ ℕ0 ∧ 𝑏 ∈ ( 𝐵 ↑m ( 0 ... 𝑠 ) ) ) ) ∧ 𝑖 ∈ ( 0 ... 𝑠 ) ) → ( 𝑖 ↑ 𝑋 ) ∈ ( Base ‘ ( Scalar ‘ 𝑌 ) ) )
114 58 80 8 81 lmodvscl ⊢ ( ( 𝑌 ∈ LMod ∧ ( 𝑖 ↑ 𝑋 ) ∈ ( Base ‘ ( Scalar ‘ 𝑌 ) ) ∧ ( 𝑇 ‘ ( 𝑏 ‘ 𝑖 ) ) ∈ ( Base ‘ 𝑌 ) ) → ( ( 𝑖 ↑ 𝑋 ) · ( 𝑇 ‘ ( 𝑏 ‘ 𝑖 ) ) ) ∈ ( Base ‘ 𝑌 ) )
115 107 113 57 114 syl3anc ⊢ ( ( ( ( 𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵 ) ∧ ( 𝑠 ∈ ℕ0 ∧ 𝑏 ∈ ( 𝐵 ↑m ( 0 ... 𝑠 ) ) ) ) ∧ 𝑖 ∈ ( 0 ... 𝑠 ) ) → ( ( 𝑖 ↑ 𝑋 ) · ( 𝑇 ‘ ( 𝑏 ‘ 𝑖 ) ) ) ∈ ( Base ‘ 𝑌 ) )
116 simpl1 ⊢ ( ( ( 𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵 ) ∧ ( 𝑠 ∈ ℕ0 ∧ 𝑏 ∈ ( 𝐵 ↑m ( 0 ... 𝑠 ) ) ) ) → 𝑁 ∈ Fin )
117 23 adantr ⊢ ( ( ( 𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵 ) ∧ ( 𝑠 ∈ ℕ0 ∧ 𝑏 ∈ ( 𝐵 ↑m ( 0 ... 𝑠 ) ) ) ) → 𝑅 ∈ Ring )
118 simprl ⊢ ( ( ( 𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵 ) ∧ ( 𝑠 ∈ ℕ0 ∧ 𝑏 ∈ ( 𝐵 ↑m ( 0 ... 𝑠 ) ) ) ) → 𝑠 ∈ ℕ0 )
119 eqid ⊢ ( 𝑖 ∈ ( 0 ... 𝑠 ) ↦ ( ( 𝑖 ↑ 𝑋 ) · ( 𝑇 ‘ ( 𝑏 ‘ 𝑖 ) ) ) ) = ( 𝑖 ∈ ( 0 ... 𝑠 ) ↦ ( ( 𝑖 ↑ 𝑋 ) · ( 𝑇 ‘ ( 𝑏 ‘ 𝑖 ) ) ) )
120 fzfid ⊢ ( ( ( 𝑁 ∈ Fin ∧ 𝑅 ∈ Ring ∧ 𝑠 ∈ ℕ0 ) ∧ 𝑏 ∈ ( 𝐵 ↑m ( 0 ... 𝑠 ) ) ) → ( 0 ... 𝑠 ) ∈ Fin )
121 ovexd ⊢ ( ( ( ( 𝑁 ∈ Fin ∧ 𝑅 ∈ Ring ∧ 𝑠 ∈ ℕ0 ) ∧ 𝑏 ∈ ( 𝐵 ↑m ( 0 ... 𝑠 ) ) ) ∧ 𝑖 ∈ ( 0 ... 𝑠 ) ) → ( ( 𝑖 ↑ 𝑋 ) · ( 𝑇 ‘ ( 𝑏 ‘ 𝑖 ) ) ) ∈ V )
122 fvexd ⊢ ( ( ( 𝑁 ∈ Fin ∧ 𝑅 ∈ Ring ∧ 𝑠 ∈ ℕ0 ) ∧ 𝑏 ∈ ( 𝐵 ↑m ( 0 ... 𝑠 ) ) ) → ( 0g ‘ 𝑌 ) ∈ V )
123 119 120 121 122 fsuppmptdm ⊢ ( ( ( 𝑁 ∈ Fin ∧ 𝑅 ∈ Ring ∧ 𝑠 ∈ ℕ0 ) ∧ 𝑏 ∈ ( 𝐵 ↑m ( 0 ... 𝑠 ) ) ) → ( 𝑖 ∈ ( 0 ... 𝑠 ) ↦ ( ( 𝑖 ↑ 𝑋 ) · ( 𝑇 ‘ ( 𝑏 ‘ 𝑖 ) ) ) ) finSupp ( 0g ‘ 𝑌 ) )
124 116 117 118 54 123 syl31anc ⊢ ( ( ( 𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵 ) ∧ ( 𝑠 ∈ ℕ0 ∧ 𝑏 ∈ ( 𝐵 ↑m ( 0 ... 𝑠 ) ) ) ) → ( 𝑖 ∈ ( 0 ... 𝑠 ) ↦ ( ( 𝑖 ↑ 𝑋 ) · ( 𝑇 ‘ ( 𝑏 ‘ 𝑖 ) ) ) ) finSupp ( 0g ‘ 𝑌 ) )
125 58 89 90 9 91 92 106 115 124 gsummulc2 ⊢ ( ( ( 𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵 ) ∧ ( 𝑠 ∈ ℕ0 ∧ 𝑏 ∈ ( 𝐵 ↑m ( 0 ... 𝑠 ) ) ) ) → ( 𝑌 Σg ( 𝑖 ∈ ( 0 ... 𝑠 ) ↦ ( ( 𝑋 · 1 ) × ( ( 𝑖 ↑ 𝑋 ) · ( 𝑇 ‘ ( 𝑏 ‘ 𝑖 ) ) ) ) ) ) = ( ( 𝑋 · 1 ) × ( 𝑌 Σg ( 𝑖 ∈ ( 0 ... 𝑠 ) ↦ ( ( 𝑖 ↑ 𝑋 ) · ( 𝑇 ‘ ( 𝑏 ‘ 𝑖 ) ) ) ) ) ) )
126 88 125 eqtr2d ⊢ ( ( ( 𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵 ) ∧ ( 𝑠 ∈ ℕ0 ∧ 𝑏 ∈ ( 𝐵 ↑m ( 0 ... 𝑠 ) ) ) ) → ( ( 𝑋 · 1 ) × ( 𝑌 Σg ( 𝑖 ∈ ( 0 ... 𝑠 ) ↦ ( ( 𝑖 ↑ 𝑋 ) · ( 𝑇 ‘ ( 𝑏 ‘ 𝑖 ) ) ) ) ) ) = ( 𝑌 Σg ( 𝑖 ∈ ( 0 ... 𝑠 ) ↦ ( ( ( 𝑖 + 1 ) ↑ 𝑋 ) · ( 𝑇 ‘ ( 𝑏 ‘ 𝑖 ) ) ) ) ) )