Metamath Proof Explorer

Theorem mhppwdeg

Description: Degree of a homogeneous polynomial raised to a power. General version of deg1pw . (Contributed by SN, 26-Jul-2024) Remove sethood hypothesis. (Revised by SN, 18-May-2025)

Ref Expression
Hypotheses mhppwdeg.h ⊢ 𝐻 = ( 𝐼 mHomP 𝑅 )
mhppwdeg.p ⊢ 𝑃 = ( 𝐼 mPoly 𝑅 )
mhppwdeg.t ⊢ 𝑇 = ( mulGrp ‘ 𝑃 )
mhppwdeg.e ⊢ ↑ = ( .g ‘ 𝑇 )
mhppwdeg.r ⊢ ( 𝜑 → 𝑅 ∈ Ring )
mhppwdeg.m ⊢ ( 𝜑 → 𝑀 ∈ ℕ0 )
mhppwdeg.n ⊢ ( 𝜑 → 𝑁 ∈ ℕ0 )
mhppwdeg.x ⊢ ( 𝜑 → 𝑋 ∈ ( 𝐻 ‘ 𝑀 ) )
Assertion mhppwdeg ( 𝜑 → ( 𝑁 ↑ 𝑋 ) ∈ ( 𝐻 ‘ ( 𝑀 · 𝑁 ) ) )

Proof

Step Hyp Ref Expression
1 mhppwdeg.h ⊢ 𝐻 = ( 𝐼 mHomP 𝑅 )
2 mhppwdeg.p ⊢ 𝑃 = ( 𝐼 mPoly 𝑅 )
3 mhppwdeg.t ⊢ 𝑇 = ( mulGrp ‘ 𝑃 )
4 mhppwdeg.e ⊢ ↑ = ( .g ‘ 𝑇 )
5 mhppwdeg.r ⊢ ( 𝜑 → 𝑅 ∈ Ring )
6 mhppwdeg.m ⊢ ( 𝜑 → 𝑀 ∈ ℕ0 )
7 mhppwdeg.n ⊢ ( 𝜑 → 𝑁 ∈ ℕ0 )
8 mhppwdeg.x ⊢ ( 𝜑 → 𝑋 ∈ ( 𝐻 ‘ 𝑀 ) )
9 oveq1 ⊢ ( 𝑥 = 0 → ( 𝑥 ↑ 𝑋 ) = ( 0 ↑ 𝑋 ) )
10 oveq2 ⊢ ( 𝑥 = 0 → ( 𝑀 · 𝑥 ) = ( 𝑀 · 0 ) )
11 10 fveq2d ⊢ ( 𝑥 = 0 → ( 𝐻 ‘ ( 𝑀 · 𝑥 ) ) = ( 𝐻 ‘ ( 𝑀 · 0 ) ) )
12 9 11 eleq12d ⊢ ( 𝑥 = 0 → ( ( 𝑥 ↑ 𝑋 ) ∈ ( 𝐻 ‘ ( 𝑀 · 𝑥 ) ) ↔ ( 0 ↑ 𝑋 ) ∈ ( 𝐻 ‘ ( 𝑀 · 0 ) ) ) )
13 oveq1 ⊢ ( 𝑥 = 𝑦 → ( 𝑥 ↑ 𝑋 ) = ( 𝑦 ↑ 𝑋 ) )
14 oveq2 ⊢ ( 𝑥 = 𝑦 → ( 𝑀 · 𝑥 ) = ( 𝑀 · 𝑦 ) )
15 14 fveq2d ⊢ ( 𝑥 = 𝑦 → ( 𝐻 ‘ ( 𝑀 · 𝑥 ) ) = ( 𝐻 ‘ ( 𝑀 · 𝑦 ) ) )
16 13 15 eleq12d ⊢ ( 𝑥 = 𝑦 → ( ( 𝑥 ↑ 𝑋 ) ∈ ( 𝐻 ‘ ( 𝑀 · 𝑥 ) ) ↔ ( 𝑦 ↑ 𝑋 ) ∈ ( 𝐻 ‘ ( 𝑀 · 𝑦 ) ) ) )
17 oveq1 ⊢ ( 𝑥 = ( 𝑦 + 1 ) → ( 𝑥 ↑ 𝑋 ) = ( ( 𝑦 + 1 ) ↑ 𝑋 ) )
18 oveq2 ⊢ ( 𝑥 = ( 𝑦 + 1 ) → ( 𝑀 · 𝑥 ) = ( 𝑀 · ( 𝑦 + 1 ) ) )
19 18 fveq2d ⊢ ( 𝑥 = ( 𝑦 + 1 ) → ( 𝐻 ‘ ( 𝑀 · 𝑥 ) ) = ( 𝐻 ‘ ( 𝑀 · ( 𝑦 + 1 ) ) ) )
20 17 19 eleq12d ⊢ ( 𝑥 = ( 𝑦 + 1 ) → ( ( 𝑥 ↑ 𝑋 ) ∈ ( 𝐻 ‘ ( 𝑀 · 𝑥 ) ) ↔ ( ( 𝑦 + 1 ) ↑ 𝑋 ) ∈ ( 𝐻 ‘ ( 𝑀 · ( 𝑦 + 1 ) ) ) ) )
21 oveq1 ⊢ ( 𝑥 = 𝑁 → ( 𝑥 ↑ 𝑋 ) = ( 𝑁 ↑ 𝑋 ) )
22 oveq2 ⊢ ( 𝑥 = 𝑁 → ( 𝑀 · 𝑥 ) = ( 𝑀 · 𝑁 ) )
23 22 fveq2d ⊢ ( 𝑥 = 𝑁 → ( 𝐻 ‘ ( 𝑀 · 𝑥 ) ) = ( 𝐻 ‘ ( 𝑀 · 𝑁 ) ) )
24 21 23 eleq12d ⊢ ( 𝑥 = 𝑁 → ( ( 𝑥 ↑ 𝑋 ) ∈ ( 𝐻 ‘ ( 𝑀 · 𝑥 ) ) ↔ ( 𝑁 ↑ 𝑋 ) ∈ ( 𝐻 ‘ ( 𝑀 · 𝑁 ) ) ) )
25 reldmmhp ⊢ Rel dom mHomP
26 25 1 8 elfvov1 ⊢ ( 𝜑 → 𝐼 ∈ V )
27 2 26 5 mplsca ⊢ ( 𝜑 → 𝑅 = ( Scalar ‘ 𝑃 ) )
28 27 fveq2d ⊢ ( 𝜑 → ( 1r ‘ 𝑅 ) = ( 1r ‘ ( Scalar ‘ 𝑃 ) ) )
29 28 fveq2d ⊢ ( 𝜑 → ( ( algSc ‘ 𝑃 ) ‘ ( 1r ‘ 𝑅 ) ) = ( ( algSc ‘ 𝑃 ) ‘ ( 1r ‘ ( Scalar ‘ 𝑃 ) ) ) )
30 eqid ⊢ ( algSc ‘ 𝑃 ) = ( algSc ‘ 𝑃 )
31 eqid ⊢ ( Scalar ‘ 𝑃 ) = ( Scalar ‘ 𝑃 )
32 2 26 5 mpllmodd ⊢ ( 𝜑 → 𝑃 ∈ LMod )
33 2 26 5 mplringd ⊢ ( 𝜑 → 𝑃 ∈ Ring )
34 30 31 32 33 ascl1 ⊢ ( 𝜑 → ( ( algSc ‘ 𝑃 ) ‘ ( 1r ‘ ( Scalar ‘ 𝑃 ) ) ) = ( 1r ‘ 𝑃 ) )
35 29 34 eqtrd ⊢ ( 𝜑 → ( ( algSc ‘ 𝑃 ) ‘ ( 1r ‘ 𝑅 ) ) = ( 1r ‘ 𝑃 ) )
36 eqid ⊢ ( Base ‘ 𝑅 ) = ( Base ‘ 𝑅 )
37 eqid ⊢ ( 1r ‘ 𝑅 ) = ( 1r ‘ 𝑅 )
38 36 37 ringidcl ⊢ ( 𝑅 ∈ Ring → ( 1r ‘ 𝑅 ) ∈ ( Base ‘ 𝑅 ) )
39 5 38 syl ⊢ ( 𝜑 → ( 1r ‘ 𝑅 ) ∈ ( Base ‘ 𝑅 ) )
40 1 2 30 36 26 5 39 mhpsclcl ⊢ ( 𝜑 → ( ( algSc ‘ 𝑃 ) ‘ ( 1r ‘ 𝑅 ) ) ∈ ( 𝐻 ‘ 0 ) )
41 35 40 eqeltrrd ⊢ ( 𝜑 → ( 1r ‘ 𝑃 ) ∈ ( 𝐻 ‘ 0 ) )
42 eqid ⊢ ( Base ‘ 𝑃 ) = ( Base ‘ 𝑃 )
43 1 2 42 26 5 6 8 mhpmpl ⊢ ( 𝜑 → 𝑋 ∈ ( Base ‘ 𝑃 ) )
44 3 42 mgpbas ⊢ ( Base ‘ 𝑃 ) = ( Base ‘ 𝑇 )
45 eqid ⊢ ( 1r ‘ 𝑃 ) = ( 1r ‘ 𝑃 )
46 3 45 ringidval ⊢ ( 1r ‘ 𝑃 ) = ( 0g ‘ 𝑇 )
47 44 46 4 mulg0 ⊢ ( 𝑋 ∈ ( Base ‘ 𝑃 ) → ( 0 ↑ 𝑋 ) = ( 1r ‘ 𝑃 ) )
48 43 47 syl ⊢ ( 𝜑 → ( 0 ↑ 𝑋 ) = ( 1r ‘ 𝑃 ) )
49 6 nn0cnd ⊢ ( 𝜑 → 𝑀 ∈ ℂ )
50 49 mul01d ⊢ ( 𝜑 → ( 𝑀 · 0 ) = 0 )
51 50 fveq2d ⊢ ( 𝜑 → ( 𝐻 ‘ ( 𝑀 · 0 ) ) = ( 𝐻 ‘ 0 ) )
52 41 48 51 3eltr4d ⊢ ( 𝜑 → ( 0 ↑ 𝑋 ) ∈ ( 𝐻 ‘ ( 𝑀 · 0 ) ) )
53 eqid ⊢ ( .r ‘ 𝑃 ) = ( .r ‘ 𝑃 )
54 5 ad2antrr ⊢ ( ( ( 𝜑 ∧ 𝑦 ∈ ℕ0 ) ∧ ( 𝑦 ↑ 𝑋 ) ∈ ( 𝐻 ‘ ( 𝑀 · 𝑦 ) ) ) → 𝑅 ∈ Ring )
55 6 ad2antrr ⊢ ( ( ( 𝜑 ∧ 𝑦 ∈ ℕ0 ) ∧ ( 𝑦 ↑ 𝑋 ) ∈ ( 𝐻 ‘ ( 𝑀 · 𝑦 ) ) ) → 𝑀 ∈ ℕ0 )
56 simplr ⊢ ( ( ( 𝜑 ∧ 𝑦 ∈ ℕ0 ) ∧ ( 𝑦 ↑ 𝑋 ) ∈ ( 𝐻 ‘ ( 𝑀 · 𝑦 ) ) ) → 𝑦 ∈ ℕ0 )
57 55 56 nn0mulcld ⊢ ( ( ( 𝜑 ∧ 𝑦 ∈ ℕ0 ) ∧ ( 𝑦 ↑ 𝑋 ) ∈ ( 𝐻 ‘ ( 𝑀 · 𝑦 ) ) ) → ( 𝑀 · 𝑦 ) ∈ ℕ0 )
58 simpr ⊢ ( ( ( 𝜑 ∧ 𝑦 ∈ ℕ0 ) ∧ ( 𝑦 ↑ 𝑋 ) ∈ ( 𝐻 ‘ ( 𝑀 · 𝑦 ) ) ) → ( 𝑦 ↑ 𝑋 ) ∈ ( 𝐻 ‘ ( 𝑀 · 𝑦 ) ) )
59 8 ad2antrr ⊢ ( ( ( 𝜑 ∧ 𝑦 ∈ ℕ0 ) ∧ ( 𝑦 ↑ 𝑋 ) ∈ ( 𝐻 ‘ ( 𝑀 · 𝑦 ) ) ) → 𝑋 ∈ ( 𝐻 ‘ 𝑀 ) )
60 1 2 53 54 57 55 58 59 mhpmulcl ⊢ ( ( ( 𝜑 ∧ 𝑦 ∈ ℕ0 ) ∧ ( 𝑦 ↑ 𝑋 ) ∈ ( 𝐻 ‘ ( 𝑀 · 𝑦 ) ) ) → ( ( 𝑦 ↑ 𝑋 ) ( .r ‘ 𝑃 ) 𝑋 ) ∈ ( 𝐻 ‘ ( ( 𝑀 · 𝑦 ) + 𝑀 ) ) )
61 3 ringmgp ⊢ ( 𝑃 ∈ Ring → 𝑇 ∈ Mnd )
62 33 61 syl ⊢ ( 𝜑 → 𝑇 ∈ Mnd )
63 62 ad2antrr ⊢ ( ( ( 𝜑 ∧ 𝑦 ∈ ℕ0 ) ∧ ( 𝑦 ↑ 𝑋 ) ∈ ( 𝐻 ‘ ( 𝑀 · 𝑦 ) ) ) → 𝑇 ∈ Mnd )
64 43 ad2antrr ⊢ ( ( ( 𝜑 ∧ 𝑦 ∈ ℕ0 ) ∧ ( 𝑦 ↑ 𝑋 ) ∈ ( 𝐻 ‘ ( 𝑀 · 𝑦 ) ) ) → 𝑋 ∈ ( Base ‘ 𝑃 ) )
65 3 53 mgpplusg ⊢ ( .r ‘ 𝑃 ) = ( +g ‘ 𝑇 )
66 44 4 65 mulgnn0p1 ⊢ ( ( 𝑇 ∈ Mnd ∧ 𝑦 ∈ ℕ0 ∧ 𝑋 ∈ ( Base ‘ 𝑃 ) ) → ( ( 𝑦 + 1 ) ↑ 𝑋 ) = ( ( 𝑦 ↑ 𝑋 ) ( .r ‘ 𝑃 ) 𝑋 ) )
67 63 56 64 66 syl3anc ⊢ ( ( ( 𝜑 ∧ 𝑦 ∈ ℕ0 ) ∧ ( 𝑦 ↑ 𝑋 ) ∈ ( 𝐻 ‘ ( 𝑀 · 𝑦 ) ) ) → ( ( 𝑦 + 1 ) ↑ 𝑋 ) = ( ( 𝑦 ↑ 𝑋 ) ( .r ‘ 𝑃 ) 𝑋 ) )
68 49 ad2antrr ⊢ ( ( ( 𝜑 ∧ 𝑦 ∈ ℕ0 ) ∧ ( 𝑦 ↑ 𝑋 ) ∈ ( 𝐻 ‘ ( 𝑀 · 𝑦 ) ) ) → 𝑀 ∈ ℂ )
69 56 nn0cnd ⊢ ( ( ( 𝜑 ∧ 𝑦 ∈ ℕ0 ) ∧ ( 𝑦 ↑ 𝑋 ) ∈ ( 𝐻 ‘ ( 𝑀 · 𝑦 ) ) ) → 𝑦 ∈ ℂ )
70 1cnd ⊢ ( ( ( 𝜑 ∧ 𝑦 ∈ ℕ0 ) ∧ ( 𝑦 ↑ 𝑋 ) ∈ ( 𝐻 ‘ ( 𝑀 · 𝑦 ) ) ) → 1 ∈ ℂ )
71 68 69 70 adddid ⊢ ( ( ( 𝜑 ∧ 𝑦 ∈ ℕ0 ) ∧ ( 𝑦 ↑ 𝑋 ) ∈ ( 𝐻 ‘ ( 𝑀 · 𝑦 ) ) ) → ( 𝑀 · ( 𝑦 + 1 ) ) = ( ( 𝑀 · 𝑦 ) + ( 𝑀 · 1 ) ) )
72 68 mulridd ⊢ ( ( ( 𝜑 ∧ 𝑦 ∈ ℕ0 ) ∧ ( 𝑦 ↑ 𝑋 ) ∈ ( 𝐻 ‘ ( 𝑀 · 𝑦 ) ) ) → ( 𝑀 · 1 ) = 𝑀 )
73 72 oveq2d ⊢ ( ( ( 𝜑 ∧ 𝑦 ∈ ℕ0 ) ∧ ( 𝑦 ↑ 𝑋 ) ∈ ( 𝐻 ‘ ( 𝑀 · 𝑦 ) ) ) → ( ( 𝑀 · 𝑦 ) + ( 𝑀 · 1 ) ) = ( ( 𝑀 · 𝑦 ) + 𝑀 ) )
74 71 73 eqtrd ⊢ ( ( ( 𝜑 ∧ 𝑦 ∈ ℕ0 ) ∧ ( 𝑦 ↑ 𝑋 ) ∈ ( 𝐻 ‘ ( 𝑀 · 𝑦 ) ) ) → ( 𝑀 · ( 𝑦 + 1 ) ) = ( ( 𝑀 · 𝑦 ) + 𝑀 ) )
75 74 fveq2d ⊢ ( ( ( 𝜑 ∧ 𝑦 ∈ ℕ0 ) ∧ ( 𝑦 ↑ 𝑋 ) ∈ ( 𝐻 ‘ ( 𝑀 · 𝑦 ) ) ) → ( 𝐻 ‘ ( 𝑀 · ( 𝑦 + 1 ) ) ) = ( 𝐻 ‘ ( ( 𝑀 · 𝑦 ) + 𝑀 ) ) )
76 60 67 75 3eltr4d ⊢ ( ( ( 𝜑 ∧ 𝑦 ∈ ℕ0 ) ∧ ( 𝑦 ↑ 𝑋 ) ∈ ( 𝐻 ‘ ( 𝑀 · 𝑦 ) ) ) → ( ( 𝑦 + 1 ) ↑ 𝑋 ) ∈ ( 𝐻 ‘ ( 𝑀 · ( 𝑦 + 1 ) ) ) )
77 12 16 20 24 52 76 nn0indd ⊢ ( ( 𝜑 ∧ 𝑁 ∈ ℕ0 ) → ( 𝑁 ↑ 𝑋 ) ∈ ( 𝐻 ‘ ( 𝑀 · 𝑁 ) ) )
78 7 77 mpdan ⊢ ( 𝜑 → ( 𝑁 ↑ 𝑋 ) ∈ ( 𝐻 ‘ ( 𝑀 · 𝑁 ) ) )