| Step | Hyp | Ref | Expression | 
|---|
| 1 |  | gsummonply1.f | . . 3
⊢ (𝜑 → (𝑘 ∈ ℕ0 ↦ 𝐴) finSupp 0 ) | 
| 2 |  | gsummonply1.a | . . . . . . 7
⊢ (𝜑 → ∀𝑘 ∈ ℕ0 𝐴 ∈ 𝐾) | 
| 3 | 2 | r19.21bi 3251 | . . . . . 6
⊢ ((𝜑 ∧ 𝑘 ∈ ℕ0) → 𝐴 ∈ 𝐾) | 
| 4 | 3 | fmpttd 7135 | . . . . 5
⊢ (𝜑 → (𝑘 ∈ ℕ0 ↦ 𝐴):ℕ0⟶𝐾) | 
| 5 |  | gsummonply1.k | . . . . . . . 8
⊢ 𝐾 = (Base‘𝑅) | 
| 6 | 5 | fvexi 6920 | . . . . . . 7
⊢ 𝐾 ∈ V | 
| 7 | 6 | a1i 11 | . . . . . 6
⊢ (𝜑 → 𝐾 ∈ V) | 
| 8 |  | nn0ex 12532 | . . . . . 6
⊢
ℕ0 ∈ V | 
| 9 |  | elmapg 8879 | . . . . . 6
⊢ ((𝐾 ∈ V ∧
ℕ0 ∈ V) → ((𝑘 ∈ ℕ0 ↦ 𝐴) ∈ (𝐾 ↑m ℕ0)
↔ (𝑘 ∈
ℕ0 ↦ 𝐴):ℕ0⟶𝐾)) | 
| 10 | 7, 8, 9 | sylancl 586 | . . . . 5
⊢ (𝜑 → ((𝑘 ∈ ℕ0 ↦ 𝐴) ∈ (𝐾 ↑m ℕ0)
↔ (𝑘 ∈
ℕ0 ↦ 𝐴):ℕ0⟶𝐾)) | 
| 11 | 4, 10 | mpbird 257 | . . . 4
⊢ (𝜑 → (𝑘 ∈ ℕ0 ↦ 𝐴) ∈ (𝐾 ↑m
ℕ0)) | 
| 12 |  | gsummonply1.0 | . . . . 5
⊢  0 =
(0g‘𝑅) | 
| 13 | 12 | fvexi 6920 | . . . 4
⊢  0 ∈
V | 
| 14 |  | fsuppmapnn0ub 14036 | . . . 4
⊢ (((𝑘 ∈ ℕ0
↦ 𝐴) ∈ (𝐾 ↑m
ℕ0) ∧ 0 ∈ V) → ((𝑘 ∈ ℕ0
↦ 𝐴) finSupp 0 →
∃𝑠 ∈
ℕ0 ∀𝑥 ∈ ℕ0 (𝑠 < 𝑥 → ((𝑘 ∈ ℕ0 ↦ 𝐴)‘𝑥) = 0 ))) | 
| 15 | 11, 13, 14 | sylancl 586 | . . 3
⊢ (𝜑 → ((𝑘 ∈ ℕ0 ↦ 𝐴) finSupp 0 → ∃𝑠 ∈ ℕ0
∀𝑥 ∈
ℕ0 (𝑠 <
𝑥 → ((𝑘 ∈ ℕ0
↦ 𝐴)‘𝑥) = 0 ))) | 
| 16 | 1, 15 | mpd 15 | . 2
⊢ (𝜑 → ∃𝑠 ∈ ℕ0 ∀𝑥 ∈ ℕ0
(𝑠 < 𝑥 → ((𝑘 ∈ ℕ0 ↦ 𝐴)‘𝑥) = 0 )) | 
| 17 |  | simpr 484 | . . . . . . . . 9
⊢ (((𝜑 ∧ 𝑠 ∈ ℕ0) ∧ 𝑥 ∈ ℕ0)
→ 𝑥 ∈
ℕ0) | 
| 18 | 2 | ad2antrr 726 | . . . . . . . . . 10
⊢ (((𝜑 ∧ 𝑠 ∈ ℕ0) ∧ 𝑥 ∈ ℕ0)
→ ∀𝑘 ∈
ℕ0 𝐴
∈ 𝐾) | 
| 19 |  | rspcsbela 4438 | . . . . . . . . . 10
⊢ ((𝑥 ∈ ℕ0
∧ ∀𝑘 ∈
ℕ0 𝐴
∈ 𝐾) →
⦋𝑥 / 𝑘⦌𝐴 ∈ 𝐾) | 
| 20 | 17, 18, 19 | syl2anc 584 | . . . . . . . . 9
⊢ (((𝜑 ∧ 𝑠 ∈ ℕ0) ∧ 𝑥 ∈ ℕ0)
→ ⦋𝑥 /
𝑘⦌𝐴 ∈ 𝐾) | 
| 21 |  | eqid 2737 | . . . . . . . . . 10
⊢ (𝑘 ∈ ℕ0
↦ 𝐴) = (𝑘 ∈ ℕ0
↦ 𝐴) | 
| 22 | 21 | fvmpts 7019 | . . . . . . . . 9
⊢ ((𝑥 ∈ ℕ0
∧ ⦋𝑥 /
𝑘⦌𝐴 ∈ 𝐾) → ((𝑘 ∈ ℕ0 ↦ 𝐴)‘𝑥) = ⦋𝑥 / 𝑘⦌𝐴) | 
| 23 | 17, 20, 22 | syl2anc 584 | . . . . . . . 8
⊢ (((𝜑 ∧ 𝑠 ∈ ℕ0) ∧ 𝑥 ∈ ℕ0)
→ ((𝑘 ∈
ℕ0 ↦ 𝐴)‘𝑥) = ⦋𝑥 / 𝑘⦌𝐴) | 
| 24 | 23 | eqeq1d 2739 | . . . . . . 7
⊢ (((𝜑 ∧ 𝑠 ∈ ℕ0) ∧ 𝑥 ∈ ℕ0)
→ (((𝑘 ∈
ℕ0 ↦ 𝐴)‘𝑥) = 0 ↔
⦋𝑥 / 𝑘⦌𝐴 = 0 )) | 
| 25 | 24 | imbi2d 340 | . . . . . 6
⊢ (((𝜑 ∧ 𝑠 ∈ ℕ0) ∧ 𝑥 ∈ ℕ0)
→ ((𝑠 < 𝑥 → ((𝑘 ∈ ℕ0 ↦ 𝐴)‘𝑥) = 0 ) ↔ (𝑠 < 𝑥 → ⦋𝑥 / 𝑘⦌𝐴 = 0 ))) | 
| 26 | 25 | biimpd 229 | . . . . 5
⊢ (((𝜑 ∧ 𝑠 ∈ ℕ0) ∧ 𝑥 ∈ ℕ0)
→ ((𝑠 < 𝑥 → ((𝑘 ∈ ℕ0 ↦ 𝐴)‘𝑥) = 0 ) → (𝑠 < 𝑥 → ⦋𝑥 / 𝑘⦌𝐴 = 0 ))) | 
| 27 | 26 | ralimdva 3167 | . . . 4
⊢ ((𝜑 ∧ 𝑠 ∈ ℕ0) →
(∀𝑥 ∈
ℕ0 (𝑠 <
𝑥 → ((𝑘 ∈ ℕ0
↦ 𝐴)‘𝑥) = 0 ) → ∀𝑥 ∈ ℕ0
(𝑠 < 𝑥 → ⦋𝑥 / 𝑘⦌𝐴 = 0 ))) | 
| 28 |  | gsummonply1.b | . . . . . . . . 9
⊢ 𝐵 = (Base‘𝑃) | 
| 29 |  | eqid 2737 | . . . . . . . . 9
⊢
(0g‘𝑃) = (0g‘𝑃) | 
| 30 |  | gsummonply1.r | . . . . . . . . . . 11
⊢ (𝜑 → 𝑅 ∈ Ring) | 
| 31 |  | gsummonply1.p | . . . . . . . . . . . 12
⊢ 𝑃 = (Poly1‘𝑅) | 
| 32 | 31 | ply1ring 22249 | . . . . . . . . . . 11
⊢ (𝑅 ∈ Ring → 𝑃 ∈ Ring) | 
| 33 |  | ringcmn 20279 | . . . . . . . . . . 11
⊢ (𝑃 ∈ Ring → 𝑃 ∈ CMnd) | 
| 34 | 30, 32, 33 | 3syl 18 | . . . . . . . . . 10
⊢ (𝜑 → 𝑃 ∈ CMnd) | 
| 35 | 34 | ad2antrr 726 | . . . . . . . . 9
⊢ (((𝜑 ∧ 𝑠 ∈ ℕ0) ∧
∀𝑥 ∈
ℕ0 (𝑠 <
𝑥 →
⦋𝑥 / 𝑘⦌𝐴 = 0 )) → 𝑃 ∈ CMnd) | 
| 36 | 30 | 3ad2ant1 1134 | . . . . . . . . . . . . . 14
⊢ ((𝜑 ∧ 𝑘 ∈ ℕ0 ∧ 𝐴 ∈ 𝐾) → 𝑅 ∈ Ring) | 
| 37 |  | simp3 1139 | . . . . . . . . . . . . . 14
⊢ ((𝜑 ∧ 𝑘 ∈ ℕ0 ∧ 𝐴 ∈ 𝐾) → 𝐴 ∈ 𝐾) | 
| 38 |  | simp2 1138 | . . . . . . . . . . . . . 14
⊢ ((𝜑 ∧ 𝑘 ∈ ℕ0 ∧ 𝐴 ∈ 𝐾) → 𝑘 ∈ ℕ0) | 
| 39 |  | gsummonply1.x | . . . . . . . . . . . . . . 15
⊢ 𝑋 = (var1‘𝑅) | 
| 40 |  | gsummonply1.m | . . . . . . . . . . . . . . 15
⊢  ∗ = (
·𝑠 ‘𝑃) | 
| 41 |  | eqid 2737 | . . . . . . . . . . . . . . 15
⊢
(mulGrp‘𝑃) =
(mulGrp‘𝑃) | 
| 42 |  | gsummonply1.e | . . . . . . . . . . . . . . 15
⊢  ↑ =
(.g‘(mulGrp‘𝑃)) | 
| 43 | 5, 31, 39, 40, 41, 42, 28 | ply1tmcl 22275 | . . . . . . . . . . . . . 14
⊢ ((𝑅 ∈ Ring ∧ 𝐴 ∈ 𝐾 ∧ 𝑘 ∈ ℕ0) → (𝐴 ∗ (𝑘 ↑ 𝑋)) ∈ 𝐵) | 
| 44 | 36, 37, 38, 43 | syl3anc 1373 | . . . . . . . . . . . . 13
⊢ ((𝜑 ∧ 𝑘 ∈ ℕ0 ∧ 𝐴 ∈ 𝐾) → (𝐴 ∗ (𝑘 ↑ 𝑋)) ∈ 𝐵) | 
| 45 | 44 | 3expia 1122 | . . . . . . . . . . . 12
⊢ ((𝜑 ∧ 𝑘 ∈ ℕ0) → (𝐴 ∈ 𝐾 → (𝐴 ∗ (𝑘 ↑ 𝑋)) ∈ 𝐵)) | 
| 46 | 45 | ralimdva 3167 | . . . . . . . . . . 11
⊢ (𝜑 → (∀𝑘 ∈ ℕ0
𝐴 ∈ 𝐾 → ∀𝑘 ∈ ℕ0 (𝐴 ∗ (𝑘 ↑ 𝑋)) ∈ 𝐵)) | 
| 47 | 2, 46 | mpd 15 | . . . . . . . . . 10
⊢ (𝜑 → ∀𝑘 ∈ ℕ0 (𝐴 ∗ (𝑘 ↑ 𝑋)) ∈ 𝐵) | 
| 48 | 47 | ad2antrr 726 | . . . . . . . . 9
⊢ (((𝜑 ∧ 𝑠 ∈ ℕ0) ∧
∀𝑥 ∈
ℕ0 (𝑠 <
𝑥 →
⦋𝑥 / 𝑘⦌𝐴 = 0 )) → ∀𝑘 ∈ ℕ0
(𝐴 ∗ (𝑘 ↑ 𝑋)) ∈ 𝐵) | 
| 49 |  | simplr 769 | . . . . . . . . 9
⊢ (((𝜑 ∧ 𝑠 ∈ ℕ0) ∧
∀𝑥 ∈
ℕ0 (𝑠 <
𝑥 →
⦋𝑥 / 𝑘⦌𝐴 = 0 )) → 𝑠 ∈
ℕ0) | 
| 50 |  | nfv 1914 | . . . . . . . . . . . . 13
⊢
Ⅎ𝑘 𝑠 < 𝑥 | 
| 51 |  | nfcsb1v 3923 | . . . . . . . . . . . . . 14
⊢
Ⅎ𝑘⦋𝑥 / 𝑘⦌𝐴 | 
| 52 | 51 | nfeq1 2921 | . . . . . . . . . . . . 13
⊢
Ⅎ𝑘⦋𝑥 / 𝑘⦌𝐴 = 0 | 
| 53 | 50, 52 | nfim 1896 | . . . . . . . . . . . 12
⊢
Ⅎ𝑘(𝑠 < 𝑥 → ⦋𝑥 / 𝑘⦌𝐴 = 0 ) | 
| 54 |  | nfv 1914 | . . . . . . . . . . . 12
⊢
Ⅎ𝑥(𝑠 < 𝑘 → ⦋𝑘 / 𝑘⦌𝐴 = 0 ) | 
| 55 |  | breq2 5147 | . . . . . . . . . . . . 13
⊢ (𝑥 = 𝑘 → (𝑠 < 𝑥 ↔ 𝑠 < 𝑘)) | 
| 56 |  | csbeq1 3902 | . . . . . . . . . . . . . 14
⊢ (𝑥 = 𝑘 → ⦋𝑥 / 𝑘⦌𝐴 = ⦋𝑘 / 𝑘⦌𝐴) | 
| 57 | 56 | eqeq1d 2739 | . . . . . . . . . . . . 13
⊢ (𝑥 = 𝑘 → (⦋𝑥 / 𝑘⦌𝐴 = 0 ↔
⦋𝑘 / 𝑘⦌𝐴 = 0 )) | 
| 58 | 55, 57 | imbi12d 344 | . . . . . . . . . . . 12
⊢ (𝑥 = 𝑘 → ((𝑠 < 𝑥 → ⦋𝑥 / 𝑘⦌𝐴 = 0 ) ↔ (𝑠 < 𝑘 → ⦋𝑘 / 𝑘⦌𝐴 = 0 ))) | 
| 59 | 53, 54, 58 | cbvralw 3306 | . . . . . . . . . . 11
⊢
(∀𝑥 ∈
ℕ0 (𝑠 <
𝑥 →
⦋𝑥 / 𝑘⦌𝐴 = 0 ) ↔ ∀𝑘 ∈ ℕ0
(𝑠 < 𝑘 → ⦋𝑘 / 𝑘⦌𝐴 = 0 )) | 
| 60 |  | csbid 3912 | . . . . . . . . . . . . . . 15
⊢
⦋𝑘 /
𝑘⦌𝐴 = 𝐴 | 
| 61 | 60 | eqeq1i 2742 | . . . . . . . . . . . . . 14
⊢
(⦋𝑘 /
𝑘⦌𝐴 = 0 ↔ 𝐴 = 0 ) | 
| 62 |  | oveq1 7438 | . . . . . . . . . . . . . . . 16
⊢ (𝐴 = 0 → (𝐴 ∗ (𝑘 ↑ 𝑋)) = ( 0 ∗ (𝑘 ↑ 𝑋))) | 
| 63 | 31 | ply1sca 22254 | . . . . . . . . . . . . . . . . . . . . . 22
⊢ (𝑅 ∈ Ring → 𝑅 = (Scalar‘𝑃)) | 
| 64 | 30, 63 | syl 17 | . . . . . . . . . . . . . . . . . . . . 21
⊢ (𝜑 → 𝑅 = (Scalar‘𝑃)) | 
| 65 | 64 | fveq2d 6910 | . . . . . . . . . . . . . . . . . . . 20
⊢ (𝜑 → (0g‘𝑅) =
(0g‘(Scalar‘𝑃))) | 
| 66 | 12, 65 | eqtrid 2789 | . . . . . . . . . . . . . . . . . . 19
⊢ (𝜑 → 0 =
(0g‘(Scalar‘𝑃))) | 
| 67 | 66 | ad2antrr 726 | . . . . . . . . . . . . . . . . . 18
⊢ (((𝜑 ∧ 𝑠 ∈ ℕ0) ∧ 𝑘 ∈ ℕ0)
→ 0
= (0g‘(Scalar‘𝑃))) | 
| 68 | 67 | oveq1d 7446 | . . . . . . . . . . . . . . . . 17
⊢ (((𝜑 ∧ 𝑠 ∈ ℕ0) ∧ 𝑘 ∈ ℕ0)
→ ( 0 ∗ (𝑘 ↑ 𝑋)) =
((0g‘(Scalar‘𝑃)) ∗ (𝑘 ↑ 𝑋))) | 
| 69 | 31 | ply1lmod 22253 | . . . . . . . . . . . . . . . . . . . 20
⊢ (𝑅 ∈ Ring → 𝑃 ∈ LMod) | 
| 70 | 30, 69 | syl 17 | . . . . . . . . . . . . . . . . . . 19
⊢ (𝜑 → 𝑃 ∈ LMod) | 
| 71 | 70 | ad2antrr 726 | . . . . . . . . . . . . . . . . . 18
⊢ (((𝜑 ∧ 𝑠 ∈ ℕ0) ∧ 𝑘 ∈ ℕ0)
→ 𝑃 ∈
LMod) | 
| 72 |  | eqid 2737 | . . . . . . . . . . . . . . . . . . . 20
⊢
(Base‘𝑃) =
(Base‘𝑃) | 
| 73 | 41, 72 | mgpbas 20142 | . . . . . . . . . . . . . . . . . . 19
⊢
(Base‘𝑃) =
(Base‘(mulGrp‘𝑃)) | 
| 74 | 41 | ringmgp 20236 | . . . . . . . . . . . . . . . . . . . . 21
⊢ (𝑃 ∈ Ring →
(mulGrp‘𝑃) ∈
Mnd) | 
| 75 | 30, 32, 74 | 3syl 18 | . . . . . . . . . . . . . . . . . . . 20
⊢ (𝜑 → (mulGrp‘𝑃) ∈ Mnd) | 
| 76 | 75 | ad2antrr 726 | . . . . . . . . . . . . . . . . . . 19
⊢ (((𝜑 ∧ 𝑠 ∈ ℕ0) ∧ 𝑘 ∈ ℕ0)
→ (mulGrp‘𝑃)
∈ Mnd) | 
| 77 |  | simpr 484 | . . . . . . . . . . . . . . . . . . 19
⊢ (((𝜑 ∧ 𝑠 ∈ ℕ0) ∧ 𝑘 ∈ ℕ0)
→ 𝑘 ∈
ℕ0) | 
| 78 | 39, 31, 72 | vr1cl 22219 | . . . . . . . . . . . . . . . . . . . . 21
⊢ (𝑅 ∈ Ring → 𝑋 ∈ (Base‘𝑃)) | 
| 79 | 30, 78 | syl 17 | . . . . . . . . . . . . . . . . . . . 20
⊢ (𝜑 → 𝑋 ∈ (Base‘𝑃)) | 
| 80 | 79 | ad2antrr 726 | . . . . . . . . . . . . . . . . . . 19
⊢ (((𝜑 ∧ 𝑠 ∈ ℕ0) ∧ 𝑘 ∈ ℕ0)
→ 𝑋 ∈
(Base‘𝑃)) | 
| 81 | 73, 42, 76, 77, 80 | mulgnn0cld 19113 | . . . . . . . . . . . . . . . . . 18
⊢ (((𝜑 ∧ 𝑠 ∈ ℕ0) ∧ 𝑘 ∈ ℕ0)
→ (𝑘 ↑ 𝑋) ∈ (Base‘𝑃)) | 
| 82 |  | eqid 2737 | . . . . . . . . . . . . . . . . . . 19
⊢
(Scalar‘𝑃) =
(Scalar‘𝑃) | 
| 83 |  | eqid 2737 | . . . . . . . . . . . . . . . . . . 19
⊢
(0g‘(Scalar‘𝑃)) =
(0g‘(Scalar‘𝑃)) | 
| 84 | 72, 82, 40, 83, 29 | lmod0vs 20893 | . . . . . . . . . . . . . . . . . 18
⊢ ((𝑃 ∈ LMod ∧ (𝑘 ↑ 𝑋) ∈ (Base‘𝑃)) →
((0g‘(Scalar‘𝑃)) ∗ (𝑘 ↑ 𝑋)) = (0g‘𝑃)) | 
| 85 | 71, 81, 84 | syl2anc 584 | . . . . . . . . . . . . . . . . 17
⊢ (((𝜑 ∧ 𝑠 ∈ ℕ0) ∧ 𝑘 ∈ ℕ0)
→ ((0g‘(Scalar‘𝑃)) ∗ (𝑘 ↑ 𝑋)) = (0g‘𝑃)) | 
| 86 | 68, 85 | eqtrd 2777 | . . . . . . . . . . . . . . . 16
⊢ (((𝜑 ∧ 𝑠 ∈ ℕ0) ∧ 𝑘 ∈ ℕ0)
→ ( 0 ∗ (𝑘 ↑ 𝑋)) = (0g‘𝑃)) | 
| 87 | 62, 86 | sylan9eqr 2799 | . . . . . . . . . . . . . . 15
⊢ ((((𝜑 ∧ 𝑠 ∈ ℕ0) ∧ 𝑘 ∈ ℕ0)
∧ 𝐴 = 0 ) →
(𝐴 ∗ (𝑘 ↑ 𝑋)) = (0g‘𝑃)) | 
| 88 | 87 | ex 412 | . . . . . . . . . . . . . 14
⊢ (((𝜑 ∧ 𝑠 ∈ ℕ0) ∧ 𝑘 ∈ ℕ0)
→ (𝐴 = 0 → (𝐴 ∗ (𝑘 ↑ 𝑋)) = (0g‘𝑃))) | 
| 89 | 61, 88 | biimtrid 242 | . . . . . . . . . . . . 13
⊢ (((𝜑 ∧ 𝑠 ∈ ℕ0) ∧ 𝑘 ∈ ℕ0)
→ (⦋𝑘 /
𝑘⦌𝐴 = 0 → (𝐴 ∗ (𝑘 ↑ 𝑋)) = (0g‘𝑃))) | 
| 90 | 89 | imim2d 57 | . . . . . . . . . . . 12
⊢ (((𝜑 ∧ 𝑠 ∈ ℕ0) ∧ 𝑘 ∈ ℕ0)
→ ((𝑠 < 𝑘 → ⦋𝑘 / 𝑘⦌𝐴 = 0 ) → (𝑠 < 𝑘 → (𝐴 ∗ (𝑘 ↑ 𝑋)) = (0g‘𝑃)))) | 
| 91 | 90 | ralimdva 3167 | . . . . . . . . . . 11
⊢ ((𝜑 ∧ 𝑠 ∈ ℕ0) →
(∀𝑘 ∈
ℕ0 (𝑠 <
𝑘 →
⦋𝑘 / 𝑘⦌𝐴 = 0 ) → ∀𝑘 ∈ ℕ0
(𝑠 < 𝑘 → (𝐴 ∗ (𝑘 ↑ 𝑋)) = (0g‘𝑃)))) | 
| 92 | 59, 91 | biimtrid 242 | . . . . . . . . . 10
⊢ ((𝜑 ∧ 𝑠 ∈ ℕ0) →
(∀𝑥 ∈
ℕ0 (𝑠 <
𝑥 →
⦋𝑥 / 𝑘⦌𝐴 = 0 ) → ∀𝑘 ∈ ℕ0
(𝑠 < 𝑘 → (𝐴 ∗ (𝑘 ↑ 𝑋)) = (0g‘𝑃)))) | 
| 93 | 92 | imp 406 | . . . . . . . . 9
⊢ (((𝜑 ∧ 𝑠 ∈ ℕ0) ∧
∀𝑥 ∈
ℕ0 (𝑠 <
𝑥 →
⦋𝑥 / 𝑘⦌𝐴 = 0 )) → ∀𝑘 ∈ ℕ0
(𝑠 < 𝑘 → (𝐴 ∗ (𝑘 ↑ 𝑋)) = (0g‘𝑃))) | 
| 94 | 28, 29, 35, 48, 49, 93 | gsummptnn0fz 20004 | . . . . . . . 8
⊢ (((𝜑 ∧ 𝑠 ∈ ℕ0) ∧
∀𝑥 ∈
ℕ0 (𝑠 <
𝑥 →
⦋𝑥 / 𝑘⦌𝐴 = 0 )) → (𝑃 Σg
(𝑘 ∈
ℕ0 ↦ (𝐴 ∗ (𝑘 ↑ 𝑋)))) = (𝑃 Σg (𝑘 ∈ (0...𝑠) ↦ (𝐴 ∗ (𝑘 ↑ 𝑋))))) | 
| 95 | 94 | fveq2d 6910 | . . . . . . 7
⊢ (((𝜑 ∧ 𝑠 ∈ ℕ0) ∧
∀𝑥 ∈
ℕ0 (𝑠 <
𝑥 →
⦋𝑥 / 𝑘⦌𝐴 = 0 )) →
(coe1‘(𝑃
Σg (𝑘 ∈ ℕ0 ↦ (𝐴 ∗ (𝑘 ↑ 𝑋))))) = (coe1‘(𝑃 Σg
(𝑘 ∈ (0...𝑠) ↦ (𝐴 ∗ (𝑘 ↑ 𝑋)))))) | 
| 96 | 95 | fveq1d 6908 | . . . . . 6
⊢ (((𝜑 ∧ 𝑠 ∈ ℕ0) ∧
∀𝑥 ∈
ℕ0 (𝑠 <
𝑥 →
⦋𝑥 / 𝑘⦌𝐴 = 0 )) →
((coe1‘(𝑃
Σg (𝑘 ∈ ℕ0 ↦ (𝐴 ∗ (𝑘 ↑ 𝑋)))))‘𝐿) = ((coe1‘(𝑃 Σg
(𝑘 ∈ (0...𝑠) ↦ (𝐴 ∗ (𝑘 ↑ 𝑋)))))‘𝐿)) | 
| 97 | 30 | ad2antrr 726 | . . . . . . 7
⊢ (((𝜑 ∧ 𝑠 ∈ ℕ0) ∧
∀𝑥 ∈
ℕ0 (𝑠 <
𝑥 →
⦋𝑥 / 𝑘⦌𝐴 = 0 )) → 𝑅 ∈ Ring) | 
| 98 |  | gsummonply1.l | . . . . . . . 8
⊢ (𝜑 → 𝐿 ∈
ℕ0) | 
| 99 | 98 | ad2antrr 726 | . . . . . . 7
⊢ (((𝜑 ∧ 𝑠 ∈ ℕ0) ∧
∀𝑥 ∈
ℕ0 (𝑠 <
𝑥 →
⦋𝑥 / 𝑘⦌𝐴 = 0 )) → 𝐿 ∈
ℕ0) | 
| 100 |  | elfznn0 13660 | . . . . . . . . . . 11
⊢ (𝑘 ∈ (0...𝑠) → 𝑘 ∈ ℕ0) | 
| 101 |  | simpll 767 | . . . . . . . . . . . 12
⊢ (((𝜑 ∧ 𝑠 ∈ ℕ0) ∧ 𝑘 ∈ ℕ0)
→ 𝜑) | 
| 102 | 3 | adantlr 715 | . . . . . . . . . . . 12
⊢ (((𝜑 ∧ 𝑠 ∈ ℕ0) ∧ 𝑘 ∈ ℕ0)
→ 𝐴 ∈ 𝐾) | 
| 103 | 101, 77, 102 | 3jca 1129 | . . . . . . . . . . 11
⊢ (((𝜑 ∧ 𝑠 ∈ ℕ0) ∧ 𝑘 ∈ ℕ0)
→ (𝜑 ∧ 𝑘 ∈ ℕ0
∧ 𝐴 ∈ 𝐾)) | 
| 104 | 100, 103 | sylan2 593 | . . . . . . . . . 10
⊢ (((𝜑 ∧ 𝑠 ∈ ℕ0) ∧ 𝑘 ∈ (0...𝑠)) → (𝜑 ∧ 𝑘 ∈ ℕ0 ∧ 𝐴 ∈ 𝐾)) | 
| 105 | 104, 44 | syl 17 | . . . . . . . . 9
⊢ (((𝜑 ∧ 𝑠 ∈ ℕ0) ∧ 𝑘 ∈ (0...𝑠)) → (𝐴 ∗ (𝑘 ↑ 𝑋)) ∈ 𝐵) | 
| 106 | 105 | ralrimiva 3146 | . . . . . . . 8
⊢ ((𝜑 ∧ 𝑠 ∈ ℕ0) →
∀𝑘 ∈ (0...𝑠)(𝐴 ∗ (𝑘 ↑ 𝑋)) ∈ 𝐵) | 
| 107 | 106 | adantr 480 | . . . . . . 7
⊢ (((𝜑 ∧ 𝑠 ∈ ℕ0) ∧
∀𝑥 ∈
ℕ0 (𝑠 <
𝑥 →
⦋𝑥 / 𝑘⦌𝐴 = 0 )) → ∀𝑘 ∈ (0...𝑠)(𝐴 ∗ (𝑘 ↑ 𝑋)) ∈ 𝐵) | 
| 108 |  | fzfid 14014 | . . . . . . 7
⊢ (((𝜑 ∧ 𝑠 ∈ ℕ0) ∧
∀𝑥 ∈
ℕ0 (𝑠 <
𝑥 →
⦋𝑥 / 𝑘⦌𝐴 = 0 )) → (0...𝑠) ∈ Fin) | 
| 109 | 31, 28, 97, 99, 107, 108 | coe1fzgsumd 22308 | . . . . . 6
⊢ (((𝜑 ∧ 𝑠 ∈ ℕ0) ∧
∀𝑥 ∈
ℕ0 (𝑠 <
𝑥 →
⦋𝑥 / 𝑘⦌𝐴 = 0 )) →
((coe1‘(𝑃
Σg (𝑘 ∈ (0...𝑠) ↦ (𝐴 ∗ (𝑘 ↑ 𝑋)))))‘𝐿) = (𝑅 Σg (𝑘 ∈ (0...𝑠) ↦ ((coe1‘(𝐴 ∗ (𝑘 ↑ 𝑋)))‘𝐿)))) | 
| 110 |  | nfv 1914 | . . . . . . . . . 10
⊢
Ⅎ𝑘(𝜑 ∧ 𝑠 ∈ ℕ0) | 
| 111 |  | nfcv 2905 | . . . . . . . . . . 11
⊢
Ⅎ𝑘ℕ0 | 
| 112 | 111, 53 | nfralw 3311 | . . . . . . . . . 10
⊢
Ⅎ𝑘∀𝑥 ∈ ℕ0 (𝑠 < 𝑥 → ⦋𝑥 / 𝑘⦌𝐴 = 0 ) | 
| 113 | 110, 112 | nfan 1899 | . . . . . . . . 9
⊢
Ⅎ𝑘((𝜑 ∧ 𝑠 ∈ ℕ0) ∧
∀𝑥 ∈
ℕ0 (𝑠 <
𝑥 →
⦋𝑥 / 𝑘⦌𝐴 = 0 )) | 
| 114 | 30 | ad3antrrr 730 | . . . . . . . . . . 11
⊢ ((((𝜑 ∧ 𝑠 ∈ ℕ0) ∧
∀𝑥 ∈
ℕ0 (𝑠 <
𝑥 →
⦋𝑥 / 𝑘⦌𝐴 = 0 )) ∧ 𝑘 ∈ (0...𝑠)) → 𝑅 ∈ Ring) | 
| 115 | 3 | expcom 413 | . . . . . . . . . . . . . 14
⊢ (𝑘 ∈ ℕ0
→ (𝜑 → 𝐴 ∈ 𝐾)) | 
| 116 | 115, 100 | syl11 33 | . . . . . . . . . . . . 13
⊢ (𝜑 → (𝑘 ∈ (0...𝑠) → 𝐴 ∈ 𝐾)) | 
| 117 | 116 | ad2antrr 726 | . . . . . . . . . . . 12
⊢ (((𝜑 ∧ 𝑠 ∈ ℕ0) ∧
∀𝑥 ∈
ℕ0 (𝑠 <
𝑥 →
⦋𝑥 / 𝑘⦌𝐴 = 0 )) → (𝑘 ∈ (0...𝑠) → 𝐴 ∈ 𝐾)) | 
| 118 | 117 | imp 406 | . . . . . . . . . . 11
⊢ ((((𝜑 ∧ 𝑠 ∈ ℕ0) ∧
∀𝑥 ∈
ℕ0 (𝑠 <
𝑥 →
⦋𝑥 / 𝑘⦌𝐴 = 0 )) ∧ 𝑘 ∈ (0...𝑠)) → 𝐴 ∈ 𝐾) | 
| 119 | 100 | adantl 481 | . . . . . . . . . . 11
⊢ ((((𝜑 ∧ 𝑠 ∈ ℕ0) ∧
∀𝑥 ∈
ℕ0 (𝑠 <
𝑥 →
⦋𝑥 / 𝑘⦌𝐴 = 0 )) ∧ 𝑘 ∈ (0...𝑠)) → 𝑘 ∈ ℕ0) | 
| 120 | 12, 5, 31, 39, 40, 41, 42 | coe1tm 22276 | . . . . . . . . . . 11
⊢ ((𝑅 ∈ Ring ∧ 𝐴 ∈ 𝐾 ∧ 𝑘 ∈ ℕ0) →
(coe1‘(𝐴
∗
(𝑘 ↑ 𝑋))) = (𝑛 ∈ ℕ0 ↦ if(𝑛 = 𝑘, 𝐴, 0 ))) | 
| 121 | 114, 118,
119, 120 | syl3anc 1373 | . . . . . . . . . 10
⊢ ((((𝜑 ∧ 𝑠 ∈ ℕ0) ∧
∀𝑥 ∈
ℕ0 (𝑠 <
𝑥 →
⦋𝑥 / 𝑘⦌𝐴 = 0 )) ∧ 𝑘 ∈ (0...𝑠)) → (coe1‘(𝐴 ∗ (𝑘 ↑ 𝑋))) = (𝑛 ∈ ℕ0 ↦ if(𝑛 = 𝑘, 𝐴, 0 ))) | 
| 122 |  | eqeq1 2741 | . . . . . . . . . . . 12
⊢ (𝑛 = 𝐿 → (𝑛 = 𝑘 ↔ 𝐿 = 𝑘)) | 
| 123 | 122 | ifbid 4549 | . . . . . . . . . . 11
⊢ (𝑛 = 𝐿 → if(𝑛 = 𝑘, 𝐴, 0 ) = if(𝐿 = 𝑘, 𝐴, 0 )) | 
| 124 | 123 | adantl 481 | . . . . . . . . . 10
⊢
(((((𝜑 ∧ 𝑠 ∈ ℕ0)
∧ ∀𝑥 ∈
ℕ0 (𝑠 <
𝑥 →
⦋𝑥 / 𝑘⦌𝐴 = 0 )) ∧ 𝑘 ∈ (0...𝑠)) ∧ 𝑛 = 𝐿) → if(𝑛 = 𝑘, 𝐴, 0 ) = if(𝐿 = 𝑘, 𝐴, 0 )) | 
| 125 | 98 | ad3antrrr 730 | . . . . . . . . . 10
⊢ ((((𝜑 ∧ 𝑠 ∈ ℕ0) ∧
∀𝑥 ∈
ℕ0 (𝑠 <
𝑥 →
⦋𝑥 / 𝑘⦌𝐴 = 0 )) ∧ 𝑘 ∈ (0...𝑠)) → 𝐿 ∈
ℕ0) | 
| 126 | 5, 12 | ring0cl 20264 | . . . . . . . . . . . . 13
⊢ (𝑅 ∈ Ring → 0 ∈ 𝐾) | 
| 127 | 30, 126 | syl 17 | . . . . . . . . . . . 12
⊢ (𝜑 → 0 ∈ 𝐾) | 
| 128 | 127 | ad3antrrr 730 | . . . . . . . . . . 11
⊢ ((((𝜑 ∧ 𝑠 ∈ ℕ0) ∧
∀𝑥 ∈
ℕ0 (𝑠 <
𝑥 →
⦋𝑥 / 𝑘⦌𝐴 = 0 )) ∧ 𝑘 ∈ (0...𝑠)) → 0 ∈ 𝐾) | 
| 129 | 118, 128 | ifcld 4572 | . . . . . . . . . 10
⊢ ((((𝜑 ∧ 𝑠 ∈ ℕ0) ∧
∀𝑥 ∈
ℕ0 (𝑠 <
𝑥 →
⦋𝑥 / 𝑘⦌𝐴 = 0 )) ∧ 𝑘 ∈ (0...𝑠)) → if(𝐿 = 𝑘, 𝐴, 0 ) ∈ 𝐾) | 
| 130 | 121, 124,
125, 129 | fvmptd 7023 | . . . . . . . . 9
⊢ ((((𝜑 ∧ 𝑠 ∈ ℕ0) ∧
∀𝑥 ∈
ℕ0 (𝑠 <
𝑥 →
⦋𝑥 / 𝑘⦌𝐴 = 0 )) ∧ 𝑘 ∈ (0...𝑠)) → ((coe1‘(𝐴 ∗ (𝑘 ↑ 𝑋)))‘𝐿) = if(𝐿 = 𝑘, 𝐴, 0 )) | 
| 131 | 113, 130 | mpteq2da 5240 | . . . . . . . 8
⊢ (((𝜑 ∧ 𝑠 ∈ ℕ0) ∧
∀𝑥 ∈
ℕ0 (𝑠 <
𝑥 →
⦋𝑥 / 𝑘⦌𝐴 = 0 )) → (𝑘 ∈ (0...𝑠) ↦ ((coe1‘(𝐴 ∗ (𝑘 ↑ 𝑋)))‘𝐿)) = (𝑘 ∈ (0...𝑠) ↦ if(𝐿 = 𝑘, 𝐴, 0 ))) | 
| 132 | 131 | oveq2d 7447 | . . . . . . 7
⊢ (((𝜑 ∧ 𝑠 ∈ ℕ0) ∧
∀𝑥 ∈
ℕ0 (𝑠 <
𝑥 →
⦋𝑥 / 𝑘⦌𝐴 = 0 )) → (𝑅 Σg
(𝑘 ∈ (0...𝑠) ↦
((coe1‘(𝐴
∗
(𝑘 ↑ 𝑋)))‘𝐿))) = (𝑅 Σg (𝑘 ∈ (0...𝑠) ↦ if(𝐿 = 𝑘, 𝐴, 0 )))) | 
| 133 |  | breq2 5147 | . . . . . . . . . . . . . . . 16
⊢ (𝑥 = 𝐿 → (𝑠 < 𝑥 ↔ 𝑠 < 𝐿)) | 
| 134 |  | csbeq1 3902 | . . . . . . . . . . . . . . . . 17
⊢ (𝑥 = 𝐿 → ⦋𝑥 / 𝑘⦌𝐴 = ⦋𝐿 / 𝑘⦌𝐴) | 
| 135 | 134 | eqeq1d 2739 | . . . . . . . . . . . . . . . 16
⊢ (𝑥 = 𝐿 → (⦋𝑥 / 𝑘⦌𝐴 = 0 ↔
⦋𝐿 / 𝑘⦌𝐴 = 0 )) | 
| 136 | 133, 135 | imbi12d 344 | . . . . . . . . . . . . . . 15
⊢ (𝑥 = 𝐿 → ((𝑠 < 𝑥 → ⦋𝑥 / 𝑘⦌𝐴 = 0 ) ↔ (𝑠 < 𝐿 → ⦋𝐿 / 𝑘⦌𝐴 = 0 ))) | 
| 137 | 136 | rspcva 3620 | . . . . . . . . . . . . . 14
⊢ ((𝐿 ∈ ℕ0
∧ ∀𝑥 ∈
ℕ0 (𝑠 <
𝑥 →
⦋𝑥 / 𝑘⦌𝐴 = 0 )) → (𝑠 < 𝐿 → ⦋𝐿 / 𝑘⦌𝐴 = 0 )) | 
| 138 |  | nfv 1914 | . . . . . . . . . . . . . . . . . . . . . . 23
⊢
Ⅎ𝑘(𝜑 ∧ (𝑠 ∈ ℕ0 ∧ 𝑠 < 𝐿)) | 
| 139 |  | nfcsb1v 3923 | . . . . . . . . . . . . . . . . . . . . . . . 24
⊢
Ⅎ𝑘⦋𝐿 / 𝑘⦌𝐴 | 
| 140 | 139 | nfeq1 2921 | . . . . . . . . . . . . . . . . . . . . . . 23
⊢
Ⅎ𝑘⦋𝐿 / 𝑘⦌𝐴 = 0 | 
| 141 | 138, 140 | nfan 1899 | . . . . . . . . . . . . . . . . . . . . . 22
⊢
Ⅎ𝑘((𝜑 ∧ (𝑠 ∈ ℕ0 ∧ 𝑠 < 𝐿)) ∧ ⦋𝐿 / 𝑘⦌𝐴 = 0 ) | 
| 142 |  | elfz2nn0 13658 | . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
32
⊢ (𝑘 ∈ (0...𝑠) ↔ (𝑘 ∈ ℕ0 ∧ 𝑠 ∈ ℕ0
∧ 𝑘 ≤ 𝑠)) | 
| 143 |  | nn0re 12535 | . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
. . . . . . 40
⊢ (𝑘 ∈ ℕ0
→ 𝑘 ∈
ℝ) | 
| 144 | 143 | ad2antrr 726 | . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
. . . . . 39
⊢ (((𝑘 ∈ ℕ0
∧ 𝑠 ∈
ℕ0) ∧ 𝐿 ∈ ℕ0) → 𝑘 ∈
ℝ) | 
| 145 |  | nn0re 12535 | . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
. . . . . . . 41
⊢ (𝑠 ∈ ℕ0
→ 𝑠 ∈
ℝ) | 
| 146 | 145 | adantl 481 | . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
. . . . . . 40
⊢ ((𝑘 ∈ ℕ0
∧ 𝑠 ∈
ℕ0) → 𝑠 ∈ ℝ) | 
| 147 | 146 | adantr 480 | . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
. . . . . 39
⊢ (((𝑘 ∈ ℕ0
∧ 𝑠 ∈
ℕ0) ∧ 𝐿 ∈ ℕ0) → 𝑠 ∈
ℝ) | 
| 148 |  | nn0re 12535 | . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
. . . . . . 40
⊢ (𝐿 ∈ ℕ0
→ 𝐿 ∈
ℝ) | 
| 149 | 148 | adantl 481 | . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
. . . . . 39
⊢ (((𝑘 ∈ ℕ0
∧ 𝑠 ∈
ℕ0) ∧ 𝐿 ∈ ℕ0) → 𝐿 ∈
ℝ) | 
| 150 |  | lelttr 11351 | . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
. . . . . 39
⊢ ((𝑘 ∈ ℝ ∧ 𝑠 ∈ ℝ ∧ 𝐿 ∈ ℝ) → ((𝑘 ≤ 𝑠 ∧ 𝑠 < 𝐿) → 𝑘 < 𝐿)) | 
| 151 | 144, 147,
149, 150 | syl3anc 1373 | . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
. . . . 38
⊢ (((𝑘 ∈ ℕ0
∧ 𝑠 ∈
ℕ0) ∧ 𝐿 ∈ ℕ0) → ((𝑘 ≤ 𝑠 ∧ 𝑠 < 𝐿) → 𝑘 < 𝐿)) | 
| 152 |  | animorr 981 | . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
. . . . . . 40
⊢ ((((𝑘 ∈ ℕ0
∧ 𝑠 ∈
ℕ0) ∧ 𝐿 ∈ ℕ0) ∧ 𝑘 < 𝐿) → (𝐿 < 𝑘 ∨ 𝑘 < 𝐿)) | 
| 153 |  | df-ne 2941 | . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
. . . . . . . 41
⊢ (𝐿 ≠ 𝑘 ↔ ¬ 𝐿 = 𝑘) | 
| 154 | 143 | adantr 480 | . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
. . . . . . . . . 43
⊢ ((𝑘 ∈ ℕ0
∧ 𝑠 ∈
ℕ0) → 𝑘 ∈ ℝ) | 
| 155 |  | lttri2 11343 | . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
. . . . . . . . . 43
⊢ ((𝐿 ∈ ℝ ∧ 𝑘 ∈ ℝ) → (𝐿 ≠ 𝑘 ↔ (𝐿 < 𝑘 ∨ 𝑘 < 𝐿))) | 
| 156 | 148, 154,
155 | syl2anr 597 | . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
. . . . . . . . 42
⊢ (((𝑘 ∈ ℕ0
∧ 𝑠 ∈
ℕ0) ∧ 𝐿 ∈ ℕ0) → (𝐿 ≠ 𝑘 ↔ (𝐿 < 𝑘 ∨ 𝑘 < 𝐿))) | 
| 157 | 156 | adantr 480 | . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
. . . . . . . 41
⊢ ((((𝑘 ∈ ℕ0
∧ 𝑠 ∈
ℕ0) ∧ 𝐿 ∈ ℕ0) ∧ 𝑘 < 𝐿) → (𝐿 ≠ 𝑘 ↔ (𝐿 < 𝑘 ∨ 𝑘 < 𝐿))) | 
| 158 | 153, 157 | bitr3id 285 | . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
. . . . . . 40
⊢ ((((𝑘 ∈ ℕ0
∧ 𝑠 ∈
ℕ0) ∧ 𝐿 ∈ ℕ0) ∧ 𝑘 < 𝐿) → (¬ 𝐿 = 𝑘 ↔ (𝐿 < 𝑘 ∨ 𝑘 < 𝐿))) | 
| 159 | 152, 158 | mpbird 257 | . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
. . . . . 39
⊢ ((((𝑘 ∈ ℕ0
∧ 𝑠 ∈
ℕ0) ∧ 𝐿 ∈ ℕ0) ∧ 𝑘 < 𝐿) → ¬ 𝐿 = 𝑘) | 
| 160 | 159 | ex 412 | . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
. . . . 38
⊢ (((𝑘 ∈ ℕ0
∧ 𝑠 ∈
ℕ0) ∧ 𝐿 ∈ ℕ0) → (𝑘 < 𝐿 → ¬ 𝐿 = 𝑘)) | 
| 161 | 151, 160 | syld 47 | . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
. . . 37
⊢ (((𝑘 ∈ ℕ0
∧ 𝑠 ∈
ℕ0) ∧ 𝐿 ∈ ℕ0) → ((𝑘 ≤ 𝑠 ∧ 𝑠 < 𝐿) → ¬ 𝐿 = 𝑘)) | 
| 162 | 161 | exp4b 430 | . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
. . 36
⊢ ((𝑘 ∈ ℕ0
∧ 𝑠 ∈
ℕ0) → (𝐿 ∈ ℕ0 → (𝑘 ≤ 𝑠 → (𝑠 < 𝐿 → ¬ 𝐿 = 𝑘)))) | 
| 163 | 162 | expimpd 453 | . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
. 35
⊢ (𝑘 ∈ ℕ0
→ ((𝑠 ∈
ℕ0 ∧ 𝐿
∈ ℕ0) → (𝑘 ≤ 𝑠 → (𝑠 < 𝐿 → ¬ 𝐿 = 𝑘)))) | 
| 164 | 163 | com23 86 | . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
34
⊢ (𝑘 ∈ ℕ0
→ (𝑘 ≤ 𝑠 → ((𝑠 ∈ ℕ0 ∧ 𝐿 ∈ ℕ0)
→ (𝑠 < 𝐿 → ¬ 𝐿 = 𝑘)))) | 
| 165 | 164 | imp 406 | . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
33
⊢ ((𝑘 ∈ ℕ0
∧ 𝑘 ≤ 𝑠) → ((𝑠 ∈ ℕ0 ∧ 𝐿 ∈ ℕ0)
→ (𝑠 < 𝐿 → ¬ 𝐿 = 𝑘))) | 
| 166 | 165 | 3adant2 1132 | . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
32
⊢ ((𝑘 ∈ ℕ0
∧ 𝑠 ∈
ℕ0 ∧ 𝑘
≤ 𝑠) → ((𝑠 ∈ ℕ0
∧ 𝐿 ∈
ℕ0) → (𝑠 < 𝐿 → ¬ 𝐿 = 𝑘))) | 
| 167 | 142, 166 | sylbi 217 | . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
31
⊢ (𝑘 ∈ (0...𝑠) → ((𝑠 ∈ ℕ0 ∧ 𝐿 ∈ ℕ0)
→ (𝑠 < 𝐿 → ¬ 𝐿 = 𝑘))) | 
| 168 | 167 | expd 415 | . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
30
⊢ (𝑘 ∈ (0...𝑠) → (𝑠 ∈ ℕ0 → (𝐿 ∈ ℕ0
→ (𝑠 < 𝐿 → ¬ 𝐿 = 𝑘)))) | 
| 169 | 98, 168 | syl7 74 | . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29
⊢ (𝑘 ∈ (0...𝑠) → (𝑠 ∈ ℕ0 → (𝜑 → (𝑠 < 𝐿 → ¬ 𝐿 = 𝑘)))) | 
| 170 | 169 | com12 32 | . . . . . . . . . . . . . . . . . . . . . . . . . . . 28
⊢ (𝑠 ∈ ℕ0
→ (𝑘 ∈ (0...𝑠) → (𝜑 → (𝑠 < 𝐿 → ¬ 𝐿 = 𝑘)))) | 
| 171 | 170 | com24 95 | . . . . . . . . . . . . . . . . . . . . . . . . . . 27
⊢ (𝑠 ∈ ℕ0
→ (𝑠 < 𝐿 → (𝜑 → (𝑘 ∈ (0...𝑠) → ¬ 𝐿 = 𝑘)))) | 
| 172 | 171 | imp 406 | . . . . . . . . . . . . . . . . . . . . . . . . . 26
⊢ ((𝑠 ∈ ℕ0
∧ 𝑠 < 𝐿) → (𝜑 → (𝑘 ∈ (0...𝑠) → ¬ 𝐿 = 𝑘))) | 
| 173 | 172 | impcom 407 | . . . . . . . . . . . . . . . . . . . . . . . . 25
⊢ ((𝜑 ∧ (𝑠 ∈ ℕ0 ∧ 𝑠 < 𝐿)) → (𝑘 ∈ (0...𝑠) → ¬ 𝐿 = 𝑘)) | 
| 174 | 173 | adantr 480 | . . . . . . . . . . . . . . . . . . . . . . . 24
⊢ (((𝜑 ∧ (𝑠 ∈ ℕ0 ∧ 𝑠 < 𝐿)) ∧ ⦋𝐿 / 𝑘⦌𝐴 = 0 ) → (𝑘 ∈ (0...𝑠) → ¬ 𝐿 = 𝑘)) | 
| 175 | 174 | imp 406 | . . . . . . . . . . . . . . . . . . . . . . 23
⊢ ((((𝜑 ∧ (𝑠 ∈ ℕ0 ∧ 𝑠 < 𝐿)) ∧ ⦋𝐿 / 𝑘⦌𝐴 = 0 ) ∧ 𝑘 ∈ (0...𝑠)) → ¬ 𝐿 = 𝑘) | 
| 176 | 175 | iffalsed 4536 | . . . . . . . . . . . . . . . . . . . . . 22
⊢ ((((𝜑 ∧ (𝑠 ∈ ℕ0 ∧ 𝑠 < 𝐿)) ∧ ⦋𝐿 / 𝑘⦌𝐴 = 0 ) ∧ 𝑘 ∈ (0...𝑠)) → if(𝐿 = 𝑘, 𝐴, 0 ) = 0 ) | 
| 177 | 141, 176 | mpteq2da 5240 | . . . . . . . . . . . . . . . . . . . . 21
⊢ (((𝜑 ∧ (𝑠 ∈ ℕ0 ∧ 𝑠 < 𝐿)) ∧ ⦋𝐿 / 𝑘⦌𝐴 = 0 ) → (𝑘 ∈ (0...𝑠) ↦ if(𝐿 = 𝑘, 𝐴, 0 )) = (𝑘 ∈ (0...𝑠) ↦ 0 )) | 
| 178 | 177 | oveq2d 7447 | . . . . . . . . . . . . . . . . . . . 20
⊢ (((𝜑 ∧ (𝑠 ∈ ℕ0 ∧ 𝑠 < 𝐿)) ∧ ⦋𝐿 / 𝑘⦌𝐴 = 0 ) → (𝑅 Σg
(𝑘 ∈ (0...𝑠) ↦ if(𝐿 = 𝑘, 𝐴, 0 ))) = (𝑅 Σg (𝑘 ∈ (0...𝑠) ↦ 0 ))) | 
| 179 |  | ringmnd 20240 | . . . . . . . . . . . . . . . . . . . . . . . 24
⊢ (𝑅 ∈ Ring → 𝑅 ∈ Mnd) | 
| 180 | 30, 179 | syl 17 | . . . . . . . . . . . . . . . . . . . . . . 23
⊢ (𝜑 → 𝑅 ∈ Mnd) | 
| 181 | 180 | adantr 480 | . . . . . . . . . . . . . . . . . . . . . 22
⊢ ((𝜑 ∧ (𝑠 ∈ ℕ0 ∧ 𝑠 < 𝐿)) → 𝑅 ∈ Mnd) | 
| 182 |  | ovex 7464 | . . . . . . . . . . . . . . . . . . . . . 22
⊢
(0...𝑠) ∈
V | 
| 183 | 12 | gsumz 18849 | . . . . . . . . . . . . . . . . . . . . . 22
⊢ ((𝑅 ∈ Mnd ∧ (0...𝑠) ∈ V) → (𝑅 Σg
(𝑘 ∈ (0...𝑠) ↦ 0 )) = 0 ) | 
| 184 | 181, 182,
183 | sylancl 586 | . . . . . . . . . . . . . . . . . . . . 21
⊢ ((𝜑 ∧ (𝑠 ∈ ℕ0 ∧ 𝑠 < 𝐿)) → (𝑅 Σg (𝑘 ∈ (0...𝑠) ↦ 0 )) = 0 ) | 
| 185 | 184 | adantr 480 | . . . . . . . . . . . . . . . . . . . 20
⊢ (((𝜑 ∧ (𝑠 ∈ ℕ0 ∧ 𝑠 < 𝐿)) ∧ ⦋𝐿 / 𝑘⦌𝐴 = 0 ) → (𝑅 Σg
(𝑘 ∈ (0...𝑠) ↦ 0 )) = 0 ) | 
| 186 |  | id 22 | . . . . . . . . . . . . . . . . . . . . . 22
⊢
(⦋𝐿 /
𝑘⦌𝐴 = 0 →
⦋𝐿 / 𝑘⦌𝐴 = 0 ) | 
| 187 | 186 | eqcomd 2743 | . . . . . . . . . . . . . . . . . . . . 21
⊢
(⦋𝐿 /
𝑘⦌𝐴 = 0 → 0 = ⦋𝐿 / 𝑘⦌𝐴) | 
| 188 | 187 | adantl 481 | . . . . . . . . . . . . . . . . . . . 20
⊢ (((𝜑 ∧ (𝑠 ∈ ℕ0 ∧ 𝑠 < 𝐿)) ∧ ⦋𝐿 / 𝑘⦌𝐴 = 0 ) → 0 =
⦋𝐿 / 𝑘⦌𝐴) | 
| 189 | 178, 185,
188 | 3eqtrd 2781 | . . . . . . . . . . . . . . . . . . 19
⊢ (((𝜑 ∧ (𝑠 ∈ ℕ0 ∧ 𝑠 < 𝐿)) ∧ ⦋𝐿 / 𝑘⦌𝐴 = 0 ) → (𝑅 Σg
(𝑘 ∈ (0...𝑠) ↦ if(𝐿 = 𝑘, 𝐴, 0 ))) = ⦋𝐿 / 𝑘⦌𝐴) | 
| 190 | 189 | ex 412 | . . . . . . . . . . . . . . . . . 18
⊢ ((𝜑 ∧ (𝑠 ∈ ℕ0 ∧ 𝑠 < 𝐿)) → (⦋𝐿 / 𝑘⦌𝐴 = 0 → (𝑅 Σg (𝑘 ∈ (0...𝑠) ↦ if(𝐿 = 𝑘, 𝐴, 0 ))) = ⦋𝐿 / 𝑘⦌𝐴)) | 
| 191 | 190 | expr 456 | . . . . . . . . . . . . . . . . 17
⊢ ((𝜑 ∧ 𝑠 ∈ ℕ0) → (𝑠 < 𝐿 → (⦋𝐿 / 𝑘⦌𝐴 = 0 → (𝑅 Σg (𝑘 ∈ (0...𝑠) ↦ if(𝐿 = 𝑘, 𝐴, 0 ))) = ⦋𝐿 / 𝑘⦌𝐴))) | 
| 192 | 191 | a2d 29 | . . . . . . . . . . . . . . . 16
⊢ ((𝜑 ∧ 𝑠 ∈ ℕ0) → ((𝑠 < 𝐿 → ⦋𝐿 / 𝑘⦌𝐴 = 0 ) → (𝑠 < 𝐿 → (𝑅 Σg (𝑘 ∈ (0...𝑠) ↦ if(𝐿 = 𝑘, 𝐴, 0 ))) = ⦋𝐿 / 𝑘⦌𝐴))) | 
| 193 | 192 | ex 412 | . . . . . . . . . . . . . . 15
⊢ (𝜑 → (𝑠 ∈ ℕ0 → ((𝑠 < 𝐿 → ⦋𝐿 / 𝑘⦌𝐴 = 0 ) → (𝑠 < 𝐿 → (𝑅 Σg (𝑘 ∈ (0...𝑠) ↦ if(𝐿 = 𝑘, 𝐴, 0 ))) = ⦋𝐿 / 𝑘⦌𝐴)))) | 
| 194 | 193 | com13 88 | . . . . . . . . . . . . . 14
⊢ ((𝑠 < 𝐿 → ⦋𝐿 / 𝑘⦌𝐴 = 0 ) → (𝑠 ∈ ℕ0
→ (𝜑 → (𝑠 < 𝐿 → (𝑅 Σg (𝑘 ∈ (0...𝑠) ↦ if(𝐿 = 𝑘, 𝐴, 0 ))) = ⦋𝐿 / 𝑘⦌𝐴)))) | 
| 195 | 137, 194 | syl 17 | . . . . . . . . . . . . 13
⊢ ((𝐿 ∈ ℕ0
∧ ∀𝑥 ∈
ℕ0 (𝑠 <
𝑥 →
⦋𝑥 / 𝑘⦌𝐴 = 0 )) → (𝑠 ∈ ℕ0
→ (𝜑 → (𝑠 < 𝐿 → (𝑅 Σg (𝑘 ∈ (0...𝑠) ↦ if(𝐿 = 𝑘, 𝐴, 0 ))) = ⦋𝐿 / 𝑘⦌𝐴)))) | 
| 196 | 195 | ex 412 | . . . . . . . . . . . 12
⊢ (𝐿 ∈ ℕ0
→ (∀𝑥 ∈
ℕ0 (𝑠 <
𝑥 →
⦋𝑥 / 𝑘⦌𝐴 = 0 ) → (𝑠 ∈ ℕ0
→ (𝜑 → (𝑠 < 𝐿 → (𝑅 Σg (𝑘 ∈ (0...𝑠) ↦ if(𝐿 = 𝑘, 𝐴, 0 ))) = ⦋𝐿 / 𝑘⦌𝐴))))) | 
| 197 | 196 | com24 95 | . . . . . . . . . . 11
⊢ (𝐿 ∈ ℕ0
→ (𝜑 → (𝑠 ∈ ℕ0
→ (∀𝑥 ∈
ℕ0 (𝑠 <
𝑥 →
⦋𝑥 / 𝑘⦌𝐴 = 0 ) → (𝑠 < 𝐿 → (𝑅 Σg (𝑘 ∈ (0...𝑠) ↦ if(𝐿 = 𝑘, 𝐴, 0 ))) = ⦋𝐿 / 𝑘⦌𝐴))))) | 
| 198 | 98, 197 | mpcom 38 | . . . . . . . . . 10
⊢ (𝜑 → (𝑠 ∈ ℕ0 →
(∀𝑥 ∈
ℕ0 (𝑠 <
𝑥 →
⦋𝑥 / 𝑘⦌𝐴 = 0 ) → (𝑠 < 𝐿 → (𝑅 Σg (𝑘 ∈ (0...𝑠) ↦ if(𝐿 = 𝑘, 𝐴, 0 ))) = ⦋𝐿 / 𝑘⦌𝐴)))) | 
| 199 | 198 | imp31 417 | . . . . . . . . 9
⊢ (((𝜑 ∧ 𝑠 ∈ ℕ0) ∧
∀𝑥 ∈
ℕ0 (𝑠 <
𝑥 →
⦋𝑥 / 𝑘⦌𝐴 = 0 )) → (𝑠 < 𝐿 → (𝑅 Σg (𝑘 ∈ (0...𝑠) ↦ if(𝐿 = 𝑘, 𝐴, 0 ))) = ⦋𝐿 / 𝑘⦌𝐴)) | 
| 200 | 199 | com12 32 | . . . . . . . 8
⊢ (𝑠 < 𝐿 → (((𝜑 ∧ 𝑠 ∈ ℕ0) ∧
∀𝑥 ∈
ℕ0 (𝑠 <
𝑥 →
⦋𝑥 / 𝑘⦌𝐴 = 0 )) → (𝑅 Σg
(𝑘 ∈ (0...𝑠) ↦ if(𝐿 = 𝑘, 𝐴, 0 ))) = ⦋𝐿 / 𝑘⦌𝐴)) | 
| 201 |  | pm3.2 469 | . . . . . . . . . 10
⊢ ((𝜑 ∧ 𝑠 ∈ ℕ0) → (¬
𝑠 < 𝐿 → ((𝜑 ∧ 𝑠 ∈ ℕ0) ∧ ¬
𝑠 < 𝐿))) | 
| 202 | 201 | adantr 480 | . . . . . . . . 9
⊢ (((𝜑 ∧ 𝑠 ∈ ℕ0) ∧
∀𝑥 ∈
ℕ0 (𝑠 <
𝑥 →
⦋𝑥 / 𝑘⦌𝐴 = 0 )) → (¬ 𝑠 < 𝐿 → ((𝜑 ∧ 𝑠 ∈ ℕ0) ∧ ¬
𝑠 < 𝐿))) | 
| 203 | 180 | ad2antrr 726 | . . . . . . . . . 10
⊢ (((𝜑 ∧ 𝑠 ∈ ℕ0) ∧ ¬
𝑠 < 𝐿) → 𝑅 ∈ Mnd) | 
| 204 | 182 | a1i 11 | . . . . . . . . . 10
⊢ (((𝜑 ∧ 𝑠 ∈ ℕ0) ∧ ¬
𝑠 < 𝐿) → (0...𝑠) ∈ V) | 
| 205 | 98 | nn0red 12588 | . . . . . . . . . . . . 13
⊢ (𝜑 → 𝐿 ∈ ℝ) | 
| 206 |  | lenlt 11339 | . . . . . . . . . . . . 13
⊢ ((𝐿 ∈ ℝ ∧ 𝑠 ∈ ℝ) → (𝐿 ≤ 𝑠 ↔ ¬ 𝑠 < 𝐿)) | 
| 207 | 205, 145,
206 | syl2an 596 | . . . . . . . . . . . 12
⊢ ((𝜑 ∧ 𝑠 ∈ ℕ0) → (𝐿 ≤ 𝑠 ↔ ¬ 𝑠 < 𝐿)) | 
| 208 | 98 | ad2antrr 726 | . . . . . . . . . . . . . 14
⊢ (((𝜑 ∧ 𝑠 ∈ ℕ0) ∧ 𝐿 ≤ 𝑠) → 𝐿 ∈
ℕ0) | 
| 209 |  | simplr 769 | . . . . . . . . . . . . . 14
⊢ (((𝜑 ∧ 𝑠 ∈ ℕ0) ∧ 𝐿 ≤ 𝑠) → 𝑠 ∈ ℕ0) | 
| 210 |  | simpr 484 | . . . . . . . . . . . . . 14
⊢ (((𝜑 ∧ 𝑠 ∈ ℕ0) ∧ 𝐿 ≤ 𝑠) → 𝐿 ≤ 𝑠) | 
| 211 |  | elfz2nn0 13658 | . . . . . . . . . . . . . 14
⊢ (𝐿 ∈ (0...𝑠) ↔ (𝐿 ∈ ℕ0 ∧ 𝑠 ∈ ℕ0
∧ 𝐿 ≤ 𝑠)) | 
| 212 | 208, 209,
210, 211 | syl3anbrc 1344 | . . . . . . . . . . . . 13
⊢ (((𝜑 ∧ 𝑠 ∈ ℕ0) ∧ 𝐿 ≤ 𝑠) → 𝐿 ∈ (0...𝑠)) | 
| 213 | 212 | ex 412 | . . . . . . . . . . . 12
⊢ ((𝜑 ∧ 𝑠 ∈ ℕ0) → (𝐿 ≤ 𝑠 → 𝐿 ∈ (0...𝑠))) | 
| 214 | 207, 213 | sylbird 260 | . . . . . . . . . . 11
⊢ ((𝜑 ∧ 𝑠 ∈ ℕ0) → (¬
𝑠 < 𝐿 → 𝐿 ∈ (0...𝑠))) | 
| 215 | 214 | imp 406 | . . . . . . . . . 10
⊢ (((𝜑 ∧ 𝑠 ∈ ℕ0) ∧ ¬
𝑠 < 𝐿) → 𝐿 ∈ (0...𝑠)) | 
| 216 |  | eqcom 2744 | . . . . . . . . . . . 12
⊢ (𝐿 = 𝑘 ↔ 𝑘 = 𝐿) | 
| 217 |  | ifbi 4548 | . . . . . . . . . . . 12
⊢ ((𝐿 = 𝑘 ↔ 𝑘 = 𝐿) → if(𝐿 = 𝑘, 𝐴, 0 ) = if(𝑘 = 𝐿, 𝐴, 0 )) | 
| 218 | 216, 217 | ax-mp 5 | . . . . . . . . . . 11
⊢ if(𝐿 = 𝑘, 𝐴, 0 ) = if(𝑘 = 𝐿, 𝐴, 0 ) | 
| 219 | 218 | mpteq2i 5247 | . . . . . . . . . 10
⊢ (𝑘 ∈ (0...𝑠) ↦ if(𝐿 = 𝑘, 𝐴, 0 )) = (𝑘 ∈ (0...𝑠) ↦ if(𝑘 = 𝐿, 𝐴, 0 )) | 
| 220 | 3, 5 | eleqtrdi 2851 | . . . . . . . . . . . . . . 15
⊢ ((𝜑 ∧ 𝑘 ∈ ℕ0) → 𝐴 ∈ (Base‘𝑅)) | 
| 221 | 220 | ex 412 | . . . . . . . . . . . . . 14
⊢ (𝜑 → (𝑘 ∈ ℕ0 → 𝐴 ∈ (Base‘𝑅))) | 
| 222 | 221 | adantr 480 | . . . . . . . . . . . . 13
⊢ ((𝜑 ∧ 𝑠 ∈ ℕ0) → (𝑘 ∈ ℕ0
→ 𝐴 ∈
(Base‘𝑅))) | 
| 223 | 222, 100 | impel 505 | . . . . . . . . . . . 12
⊢ (((𝜑 ∧ 𝑠 ∈ ℕ0) ∧ 𝑘 ∈ (0...𝑠)) → 𝐴 ∈ (Base‘𝑅)) | 
| 224 | 223 | ralrimiva 3146 | . . . . . . . . . . 11
⊢ ((𝜑 ∧ 𝑠 ∈ ℕ0) →
∀𝑘 ∈ (0...𝑠)𝐴 ∈ (Base‘𝑅)) | 
| 225 | 224 | adantr 480 | . . . . . . . . . 10
⊢ (((𝜑 ∧ 𝑠 ∈ ℕ0) ∧ ¬
𝑠 < 𝐿) → ∀𝑘 ∈ (0...𝑠)𝐴 ∈ (Base‘𝑅)) | 
| 226 | 12, 203, 204, 215, 219, 225 | gsummpt1n0 19983 | . . . . . . . . 9
⊢ (((𝜑 ∧ 𝑠 ∈ ℕ0) ∧ ¬
𝑠 < 𝐿) → (𝑅 Σg (𝑘 ∈ (0...𝑠) ↦ if(𝐿 = 𝑘, 𝐴, 0 ))) = ⦋𝐿 / 𝑘⦌𝐴) | 
| 227 | 202, 226 | syl6com 37 | . . . . . . . 8
⊢ (¬
𝑠 < 𝐿 → (((𝜑 ∧ 𝑠 ∈ ℕ0) ∧
∀𝑥 ∈
ℕ0 (𝑠 <
𝑥 →
⦋𝑥 / 𝑘⦌𝐴 = 0 )) → (𝑅 Σg
(𝑘 ∈ (0...𝑠) ↦ if(𝐿 = 𝑘, 𝐴, 0 ))) = ⦋𝐿 / 𝑘⦌𝐴)) | 
| 228 | 200, 227 | pm2.61i 182 | . . . . . . 7
⊢ (((𝜑 ∧ 𝑠 ∈ ℕ0) ∧
∀𝑥 ∈
ℕ0 (𝑠 <
𝑥 →
⦋𝑥 / 𝑘⦌𝐴 = 0 )) → (𝑅 Σg
(𝑘 ∈ (0...𝑠) ↦ if(𝐿 = 𝑘, 𝐴, 0 ))) = ⦋𝐿 / 𝑘⦌𝐴) | 
| 229 | 132, 228 | eqtrd 2777 | . . . . . 6
⊢ (((𝜑 ∧ 𝑠 ∈ ℕ0) ∧
∀𝑥 ∈
ℕ0 (𝑠 <
𝑥 →
⦋𝑥 / 𝑘⦌𝐴 = 0 )) → (𝑅 Σg
(𝑘 ∈ (0...𝑠) ↦
((coe1‘(𝐴
∗
(𝑘 ↑ 𝑋)))‘𝐿))) = ⦋𝐿 / 𝑘⦌𝐴) | 
| 230 | 96, 109, 229 | 3eqtrd 2781 | . . . . 5
⊢ (((𝜑 ∧ 𝑠 ∈ ℕ0) ∧
∀𝑥 ∈
ℕ0 (𝑠 <
𝑥 →
⦋𝑥 / 𝑘⦌𝐴 = 0 )) →
((coe1‘(𝑃
Σg (𝑘 ∈ ℕ0 ↦ (𝐴 ∗ (𝑘 ↑ 𝑋)))))‘𝐿) = ⦋𝐿 / 𝑘⦌𝐴) | 
| 231 | 230 | ex 412 | . . . 4
⊢ ((𝜑 ∧ 𝑠 ∈ ℕ0) →
(∀𝑥 ∈
ℕ0 (𝑠 <
𝑥 →
⦋𝑥 / 𝑘⦌𝐴 = 0 ) →
((coe1‘(𝑃
Σg (𝑘 ∈ ℕ0 ↦ (𝐴 ∗ (𝑘 ↑ 𝑋)))))‘𝐿) = ⦋𝐿 / 𝑘⦌𝐴)) | 
| 232 | 27, 231 | syld 47 | . . 3
⊢ ((𝜑 ∧ 𝑠 ∈ ℕ0) →
(∀𝑥 ∈
ℕ0 (𝑠 <
𝑥 → ((𝑘 ∈ ℕ0
↦ 𝐴)‘𝑥) = 0 ) →
((coe1‘(𝑃
Σg (𝑘 ∈ ℕ0 ↦ (𝐴 ∗ (𝑘 ↑ 𝑋)))))‘𝐿) = ⦋𝐿 / 𝑘⦌𝐴)) | 
| 233 | 232 | rexlimdva 3155 | . 2
⊢ (𝜑 → (∃𝑠 ∈ ℕ0 ∀𝑥 ∈ ℕ0
(𝑠 < 𝑥 → ((𝑘 ∈ ℕ0 ↦ 𝐴)‘𝑥) = 0 ) →
((coe1‘(𝑃
Σg (𝑘 ∈ ℕ0 ↦ (𝐴 ∗ (𝑘 ↑ 𝑋)))))‘𝐿) = ⦋𝐿 / 𝑘⦌𝐴)) | 
| 234 | 16, 233 | mpd 15 | 1
⊢ (𝜑 →
((coe1‘(𝑃
Σg (𝑘 ∈ ℕ0 ↦ (𝐴 ∗ (𝑘 ↑ 𝑋)))))‘𝐿) = ⦋𝐿 / 𝑘⦌𝐴) |