| Step | Hyp | Ref
| Expression |
| 1 | | rpvmasum2.g |
. . . 4
⊢ 𝐺 = (DChr‘𝑁) |
| 2 | | rpvmasum.z |
. . . 4
⊢ 𝑍 =
(ℤ/nℤ‘𝑁) |
| 3 | | rpvmasum2.d |
. . . 4
⊢ 𝐷 = (Base‘𝐺) |
| 4 | | eqid 2737 |
. . . 4
⊢
(Base‘𝑍) =
(Base‘𝑍) |
| 5 | | rpvmasum2.w |
. . . . . . 7
⊢ 𝑊 = {𝑦 ∈ (𝐷 ∖ { 1 }) ∣ Σ𝑚 ∈ ℕ ((𝑦‘(𝐿‘𝑚)) / 𝑚) = 0} |
| 6 | 5 | ssrab3 4082 |
. . . . . 6
⊢ 𝑊 ⊆ (𝐷 ∖ { 1 }) |
| 7 | | dchrisum0.b |
. . . . . 6
⊢ (𝜑 → 𝑋 ∈ 𝑊) |
| 8 | 6, 7 | sselid 3981 |
. . . . 5
⊢ (𝜑 → 𝑋 ∈ (𝐷 ∖ { 1 })) |
| 9 | 8 | eldifad 3963 |
. . . 4
⊢ (𝜑 → 𝑋 ∈ 𝐷) |
| 10 | 1, 2, 3, 4, 9 | dchrf 27286 |
. . 3
⊢ (𝜑 → 𝑋:(Base‘𝑍)⟶ℂ) |
| 11 | 10 | ffnd 6737 |
. 2
⊢ (𝜑 → 𝑋 Fn (Base‘𝑍)) |
| 12 | 10 | ffvelcdmda 7104 |
. . . 4
⊢ ((𝜑 ∧ 𝑥 ∈ (Base‘𝑍)) → (𝑋‘𝑥) ∈ ℂ) |
| 13 | | fvco3 7008 |
. . . . . 6
⊢ ((𝑋:(Base‘𝑍)⟶ℂ ∧ 𝑥 ∈ (Base‘𝑍)) → ((∗ ∘ 𝑋)‘𝑥) = (∗‘(𝑋‘𝑥))) |
| 14 | 10, 13 | sylan 580 |
. . . . 5
⊢ ((𝜑 ∧ 𝑥 ∈ (Base‘𝑍)) → ((∗ ∘ 𝑋)‘𝑥) = (∗‘(𝑋‘𝑥))) |
| 15 | | logno1 26678 |
. . . . . . . 8
⊢ ¬
(𝑥 ∈
ℝ+ ↦ (log‘𝑥)) ∈ 𝑂(1) |
| 16 | | 1red 11262 |
. . . . . . . . . . 11
⊢ ((𝜑 ∧ (∗ ∘ 𝑋) ≠ 𝑋) → 1 ∈ ℝ) |
| 17 | | rpvmasum.l |
. . . . . . . . . . . . 13
⊢ 𝐿 = (ℤRHom‘𝑍) |
| 18 | | rpvmasum.a |
. . . . . . . . . . . . 13
⊢ (𝜑 → 𝑁 ∈ ℕ) |
| 19 | | rpvmasum2.1 |
. . . . . . . . . . . . 13
⊢ 1 =
(0g‘𝐺) |
| 20 | | eqid 2737 |
. . . . . . . . . . . . 13
⊢
(Unit‘𝑍) =
(Unit‘𝑍) |
| 21 | 18 | nnnn0d 12587 |
. . . . . . . . . . . . . . . 16
⊢ (𝜑 → 𝑁 ∈
ℕ0) |
| 22 | 2 | zncrng 21563 |
. . . . . . . . . . . . . . . 16
⊢ (𝑁 ∈ ℕ0
→ 𝑍 ∈
CRing) |
| 23 | 21, 22 | syl 17 |
. . . . . . . . . . . . . . 15
⊢ (𝜑 → 𝑍 ∈ CRing) |
| 24 | | crngring 20242 |
. . . . . . . . . . . . . . 15
⊢ (𝑍 ∈ CRing → 𝑍 ∈ Ring) |
| 25 | 23, 24 | syl 17 |
. . . . . . . . . . . . . 14
⊢ (𝜑 → 𝑍 ∈ Ring) |
| 26 | | eqid 2737 |
. . . . . . . . . . . . . . 15
⊢
(1r‘𝑍) = (1r‘𝑍) |
| 27 | 20, 26 | 1unit 20374 |
. . . . . . . . . . . . . 14
⊢ (𝑍 ∈ Ring →
(1r‘𝑍)
∈ (Unit‘𝑍)) |
| 28 | 25, 27 | syl 17 |
. . . . . . . . . . . . 13
⊢ (𝜑 → (1r‘𝑍) ∈ (Unit‘𝑍)) |
| 29 | | eqid 2737 |
. . . . . . . . . . . . 13
⊢ (◡𝐿 “ {(1r‘𝑍)}) = (◡𝐿 “ {(1r‘𝑍)}) |
| 30 | | eqidd 2738 |
. . . . . . . . . . . . 13
⊢ ((𝜑 ∧ 𝑓 ∈ 𝑊) → (1r‘𝑍) = (1r‘𝑍)) |
| 31 | 2, 17, 18, 1, 3, 19, 5, 20, 28, 29, 30 | rpvmasum2 27556 |
. . . . . . . . . . . 12
⊢ (𝜑 → (𝑥 ∈ ℝ+ ↦
(((ϕ‘𝑁) ·
Σ𝑛 ∈
((1...(⌊‘𝑥))
∩ (◡𝐿 “ {(1r‘𝑍)}))((Λ‘𝑛) / 𝑛)) − ((log‘𝑥) · (1 − (♯‘𝑊))))) ∈
𝑂(1)) |
| 32 | 31 | adantr 480 |
. . . . . . . . . . 11
⊢ ((𝜑 ∧ (∗ ∘ 𝑋) ≠ 𝑋) → (𝑥 ∈ ℝ+ ↦
(((ϕ‘𝑁) ·
Σ𝑛 ∈
((1...(⌊‘𝑥))
∩ (◡𝐿 “ {(1r‘𝑍)}))((Λ‘𝑛) / 𝑛)) − ((log‘𝑥) · (1 − (♯‘𝑊))))) ∈
𝑂(1)) |
| 33 | 18 | phicld 16809 |
. . . . . . . . . . . . . . . . . 18
⊢ (𝜑 → (ϕ‘𝑁) ∈
ℕ) |
| 34 | 33 | nnnn0d 12587 |
. . . . . . . . . . . . . . . . 17
⊢ (𝜑 → (ϕ‘𝑁) ∈
ℕ0) |
| 35 | 34 | adantr 480 |
. . . . . . . . . . . . . . . 16
⊢ ((𝜑 ∧ 𝑥 ∈ ℝ+) →
(ϕ‘𝑁) ∈
ℕ0) |
| 36 | 35 | nn0red 12588 |
. . . . . . . . . . . . . . 15
⊢ ((𝜑 ∧ 𝑥 ∈ ℝ+) →
(ϕ‘𝑁) ∈
ℝ) |
| 37 | | fzfid 14014 |
. . . . . . . . . . . . . . . . 17
⊢ ((𝜑 ∧ 𝑥 ∈ ℝ+) →
(1...(⌊‘𝑥))
∈ Fin) |
| 38 | | inss1 4237 |
. . . . . . . . . . . . . . . . 17
⊢
((1...(⌊‘𝑥)) ∩ (◡𝐿 “ {(1r‘𝑍)})) ⊆
(1...(⌊‘𝑥)) |
| 39 | | ssfi 9213 |
. . . . . . . . . . . . . . . . 17
⊢
(((1...(⌊‘𝑥)) ∈ Fin ∧
((1...(⌊‘𝑥))
∩ (◡𝐿 “ {(1r‘𝑍)})) ⊆
(1...(⌊‘𝑥)))
→ ((1...(⌊‘𝑥)) ∩ (◡𝐿 “ {(1r‘𝑍)})) ∈
Fin) |
| 40 | 37, 38, 39 | sylancl 586 |
. . . . . . . . . . . . . . . 16
⊢ ((𝜑 ∧ 𝑥 ∈ ℝ+) →
((1...(⌊‘𝑥))
∩ (◡𝐿 “ {(1r‘𝑍)})) ∈
Fin) |
| 41 | | elinel1 4201 |
. . . . . . . . . . . . . . . . . 18
⊢ (𝑛 ∈
((1...(⌊‘𝑥))
∩ (◡𝐿 “ {(1r‘𝑍)})) → 𝑛 ∈ (1...(⌊‘𝑥))) |
| 42 | | elfznn 13593 |
. . . . . . . . . . . . . . . . . . 19
⊢ (𝑛 ∈
(1...(⌊‘𝑥))
→ 𝑛 ∈
ℕ) |
| 43 | 42 | adantl 481 |
. . . . . . . . . . . . . . . . . 18
⊢ (((𝜑 ∧ 𝑥 ∈ ℝ+) ∧ 𝑛 ∈
(1...(⌊‘𝑥)))
→ 𝑛 ∈
ℕ) |
| 44 | 41, 43 | sylan2 593 |
. . . . . . . . . . . . . . . . 17
⊢ (((𝜑 ∧ 𝑥 ∈ ℝ+) ∧ 𝑛 ∈
((1...(⌊‘𝑥))
∩ (◡𝐿 “ {(1r‘𝑍)}))) → 𝑛 ∈ ℕ) |
| 45 | | vmacl 27161 |
. . . . . . . . . . . . . . . . . 18
⊢ (𝑛 ∈ ℕ →
(Λ‘𝑛) ∈
ℝ) |
| 46 | | nndivre 12307 |
. . . . . . . . . . . . . . . . . 18
⊢
(((Λ‘𝑛)
∈ ℝ ∧ 𝑛
∈ ℕ) → ((Λ‘𝑛) / 𝑛) ∈ ℝ) |
| 47 | 45, 46 | mpancom 688 |
. . . . . . . . . . . . . . . . 17
⊢ (𝑛 ∈ ℕ →
((Λ‘𝑛) / 𝑛) ∈
ℝ) |
| 48 | 44, 47 | syl 17 |
. . . . . . . . . . . . . . . 16
⊢ (((𝜑 ∧ 𝑥 ∈ ℝ+) ∧ 𝑛 ∈
((1...(⌊‘𝑥))
∩ (◡𝐿 “ {(1r‘𝑍)}))) →
((Λ‘𝑛) / 𝑛) ∈
ℝ) |
| 49 | 40, 48 | fsumrecl 15770 |
. . . . . . . . . . . . . . 15
⊢ ((𝜑 ∧ 𝑥 ∈ ℝ+) →
Σ𝑛 ∈
((1...(⌊‘𝑥))
∩ (◡𝐿 “ {(1r‘𝑍)}))((Λ‘𝑛) / 𝑛) ∈ ℝ) |
| 50 | 36, 49 | remulcld 11291 |
. . . . . . . . . . . . . 14
⊢ ((𝜑 ∧ 𝑥 ∈ ℝ+) →
((ϕ‘𝑁) ·
Σ𝑛 ∈
((1...(⌊‘𝑥))
∩ (◡𝐿 “ {(1r‘𝑍)}))((Λ‘𝑛) / 𝑛)) ∈ ℝ) |
| 51 | | relogcl 26617 |
. . . . . . . . . . . . . . . 16
⊢ (𝑥 ∈ ℝ+
→ (log‘𝑥) ∈
ℝ) |
| 52 | 51 | adantl 481 |
. . . . . . . . . . . . . . 15
⊢ ((𝜑 ∧ 𝑥 ∈ ℝ+) →
(log‘𝑥) ∈
ℝ) |
| 53 | | 1re 11261 |
. . . . . . . . . . . . . . . . 17
⊢ 1 ∈
ℝ |
| 54 | 1, 3 | dchrfi 27299 |
. . . . . . . . . . . . . . . . . . . . 21
⊢ (𝑁 ∈ ℕ → 𝐷 ∈ Fin) |
| 55 | 18, 54 | syl 17 |
. . . . . . . . . . . . . . . . . . . 20
⊢ (𝜑 → 𝐷 ∈ Fin) |
| 56 | | difss 4136 |
. . . . . . . . . . . . . . . . . . . . 21
⊢ (𝐷 ∖ { 1 }) ⊆ 𝐷 |
| 57 | 6, 56 | sstri 3993 |
. . . . . . . . . . . . . . . . . . . 20
⊢ 𝑊 ⊆ 𝐷 |
| 58 | | ssfi 9213 |
. . . . . . . . . . . . . . . . . . . 20
⊢ ((𝐷 ∈ Fin ∧ 𝑊 ⊆ 𝐷) → 𝑊 ∈ Fin) |
| 59 | 55, 57, 58 | sylancl 586 |
. . . . . . . . . . . . . . . . . . 19
⊢ (𝜑 → 𝑊 ∈ Fin) |
| 60 | | hashcl 14395 |
. . . . . . . . . . . . . . . . . . 19
⊢ (𝑊 ∈ Fin →
(♯‘𝑊) ∈
ℕ0) |
| 61 | 59, 60 | syl 17 |
. . . . . . . . . . . . . . . . . 18
⊢ (𝜑 → (♯‘𝑊) ∈
ℕ0) |
| 62 | 61 | nn0red 12588 |
. . . . . . . . . . . . . . . . 17
⊢ (𝜑 → (♯‘𝑊) ∈
ℝ) |
| 63 | | resubcl 11573 |
. . . . . . . . . . . . . . . . 17
⊢ ((1
∈ ℝ ∧ (♯‘𝑊) ∈ ℝ) → (1 −
(♯‘𝑊)) ∈
ℝ) |
| 64 | 53, 62, 63 | sylancr 587 |
. . . . . . . . . . . . . . . 16
⊢ (𝜑 → (1 −
(♯‘𝑊)) ∈
ℝ) |
| 65 | 64 | adantr 480 |
. . . . . . . . . . . . . . 15
⊢ ((𝜑 ∧ 𝑥 ∈ ℝ+) → (1
− (♯‘𝑊))
∈ ℝ) |
| 66 | 52, 65 | remulcld 11291 |
. . . . . . . . . . . . . 14
⊢ ((𝜑 ∧ 𝑥 ∈ ℝ+) →
((log‘𝑥) · (1
− (♯‘𝑊)))
∈ ℝ) |
| 67 | 50, 66 | resubcld 11691 |
. . . . . . . . . . . . 13
⊢ ((𝜑 ∧ 𝑥 ∈ ℝ+) →
(((ϕ‘𝑁) ·
Σ𝑛 ∈
((1...(⌊‘𝑥))
∩ (◡𝐿 “ {(1r‘𝑍)}))((Λ‘𝑛) / 𝑛)) − ((log‘𝑥) · (1 − (♯‘𝑊)))) ∈
ℝ) |
| 68 | 67 | recnd 11289 |
. . . . . . . . . . . 12
⊢ ((𝜑 ∧ 𝑥 ∈ ℝ+) →
(((ϕ‘𝑁) ·
Σ𝑛 ∈
((1...(⌊‘𝑥))
∩ (◡𝐿 “ {(1r‘𝑍)}))((Λ‘𝑛) / 𝑛)) − ((log‘𝑥) · (1 − (♯‘𝑊)))) ∈
ℂ) |
| 69 | 68 | adantlr 715 |
. . . . . . . . . . 11
⊢ (((𝜑 ∧ (∗ ∘ 𝑋) ≠ 𝑋) ∧ 𝑥 ∈ ℝ+) →
(((ϕ‘𝑁) ·
Σ𝑛 ∈
((1...(⌊‘𝑥))
∩ (◡𝐿 “ {(1r‘𝑍)}))((Λ‘𝑛) / 𝑛)) − ((log‘𝑥) · (1 − (♯‘𝑊)))) ∈
ℂ) |
| 70 | 51 | adantl 481 |
. . . . . . . . . . . 12
⊢ (((𝜑 ∧ (∗ ∘ 𝑋) ≠ 𝑋) ∧ 𝑥 ∈ ℝ+) →
(log‘𝑥) ∈
ℝ) |
| 71 | 70 | recnd 11289 |
. . . . . . . . . . 11
⊢ (((𝜑 ∧ (∗ ∘ 𝑋) ≠ 𝑋) ∧ 𝑥 ∈ ℝ+) →
(log‘𝑥) ∈
ℂ) |
| 72 | 51 | ad2antrl 728 |
. . . . . . . . . . . . . . 15
⊢ (((𝜑 ∧ (∗ ∘ 𝑋) ≠ 𝑋) ∧ (𝑥 ∈ ℝ+ ∧ 1 ≤
𝑥)) → (log‘𝑥) ∈
ℝ) |
| 73 | 66 | ad2ant2r 747 |
. . . . . . . . . . . . . . 15
⊢ (((𝜑 ∧ (∗ ∘ 𝑋) ≠ 𝑋) ∧ (𝑥 ∈ ℝ+ ∧ 1 ≤
𝑥)) →
((log‘𝑥) · (1
− (♯‘𝑊)))
∈ ℝ) |
| 74 | 72, 73 | readdcld 11290 |
. . . . . . . . . . . . . 14
⊢ (((𝜑 ∧ (∗ ∘ 𝑋) ≠ 𝑋) ∧ (𝑥 ∈ ℝ+ ∧ 1 ≤
𝑥)) →
((log‘𝑥) +
((log‘𝑥) · (1
− (♯‘𝑊)))) ∈ ℝ) |
| 75 | | 0red 11264 |
. . . . . . . . . . . . . 14
⊢ (((𝜑 ∧ (∗ ∘ 𝑋) ≠ 𝑋) ∧ (𝑥 ∈ ℝ+ ∧ 1 ≤
𝑥)) → 0 ∈
ℝ) |
| 76 | 50 | ad2ant2r 747 |
. . . . . . . . . . . . . 14
⊢ (((𝜑 ∧ (∗ ∘ 𝑋) ≠ 𝑋) ∧ (𝑥 ∈ ℝ+ ∧ 1 ≤
𝑥)) →
((ϕ‘𝑁) ·
Σ𝑛 ∈
((1...(⌊‘𝑥))
∩ (◡𝐿 “ {(1r‘𝑍)}))((Λ‘𝑛) / 𝑛)) ∈ ℝ) |
| 77 | | 2re 12340 |
. . . . . . . . . . . . . . . . . 18
⊢ 2 ∈
ℝ |
| 78 | 77 | a1i 11 |
. . . . . . . . . . . . . . . . 17
⊢ (((𝜑 ∧ (∗ ∘ 𝑋) ≠ 𝑋) ∧ (𝑥 ∈ ℝ+ ∧ 1 ≤
𝑥)) → 2 ∈
ℝ) |
| 79 | 62 | ad2antrr 726 |
. . . . . . . . . . . . . . . . 17
⊢ (((𝜑 ∧ (∗ ∘ 𝑋) ≠ 𝑋) ∧ (𝑥 ∈ ℝ+ ∧ 1 ≤
𝑥)) →
(♯‘𝑊) ∈
ℝ) |
| 80 | 78, 79 | resubcld 11691 |
. . . . . . . . . . . . . . . 16
⊢ (((𝜑 ∧ (∗ ∘ 𝑋) ≠ 𝑋) ∧ (𝑥 ∈ ℝ+ ∧ 1 ≤
𝑥)) → (2 −
(♯‘𝑊)) ∈
ℝ) |
| 81 | | log1 26627 |
. . . . . . . . . . . . . . . . 17
⊢
(log‘1) = 0 |
| 82 | | simprr 773 |
. . . . . . . . . . . . . . . . . 18
⊢ (((𝜑 ∧ (∗ ∘ 𝑋) ≠ 𝑋) ∧ (𝑥 ∈ ℝ+ ∧ 1 ≤
𝑥)) → 1 ≤ 𝑥) |
| 83 | | 1rp 13038 |
. . . . . . . . . . . . . . . . . . 19
⊢ 1 ∈
ℝ+ |
| 84 | | simprl 771 |
. . . . . . . . . . . . . . . . . . 19
⊢ (((𝜑 ∧ (∗ ∘ 𝑋) ≠ 𝑋) ∧ (𝑥 ∈ ℝ+ ∧ 1 ≤
𝑥)) → 𝑥 ∈
ℝ+) |
| 85 | | logleb 26645 |
. . . . . . . . . . . . . . . . . . 19
⊢ ((1
∈ ℝ+ ∧ 𝑥 ∈ ℝ+) → (1 ≤
𝑥 ↔ (log‘1) ≤
(log‘𝑥))) |
| 86 | 83, 84, 85 | sylancr 587 |
. . . . . . . . . . . . . . . . . 18
⊢ (((𝜑 ∧ (∗ ∘ 𝑋) ≠ 𝑋) ∧ (𝑥 ∈ ℝ+ ∧ 1 ≤
𝑥)) → (1 ≤ 𝑥 ↔ (log‘1) ≤
(log‘𝑥))) |
| 87 | 82, 86 | mpbid 232 |
. . . . . . . . . . . . . . . . 17
⊢ (((𝜑 ∧ (∗ ∘ 𝑋) ≠ 𝑋) ∧ (𝑥 ∈ ℝ+ ∧ 1 ≤
𝑥)) → (log‘1)
≤ (log‘𝑥)) |
| 88 | 81, 87 | eqbrtrrid 5179 |
. . . . . . . . . . . . . . . 16
⊢ (((𝜑 ∧ (∗ ∘ 𝑋) ≠ 𝑋) ∧ (𝑥 ∈ ℝ+ ∧ 1 ≤
𝑥)) → 0 ≤
(log‘𝑥)) |
| 89 | 59 | ad2antrr 726 |
. . . . . . . . . . . . . . . . . 18
⊢ (((𝜑 ∧ (∗ ∘ 𝑋) ≠ 𝑋) ∧ (𝑥 ∈ ℝ+ ∧ 1 ≤
𝑥)) → 𝑊 ∈ Fin) |
| 90 | | eqid 2737 |
. . . . . . . . . . . . . . . . . . . . . . 23
⊢
(invg‘𝐺) = (invg‘𝐺) |
| 91 | 1, 3, 9, 90 | dchrinv 27305 |
. . . . . . . . . . . . . . . . . . . . . 22
⊢ (𝜑 →
((invg‘𝐺)‘𝑋) = (∗ ∘ 𝑋)) |
| 92 | 1 | dchrabl 27298 |
. . . . . . . . . . . . . . . . . . . . . . . . 25
⊢ (𝑁 ∈ ℕ → 𝐺 ∈ Abel) |
| 93 | 18, 92 | syl 17 |
. . . . . . . . . . . . . . . . . . . . . . . 24
⊢ (𝜑 → 𝐺 ∈ Abel) |
| 94 | | ablgrp 19803 |
. . . . . . . . . . . . . . . . . . . . . . . 24
⊢ (𝐺 ∈ Abel → 𝐺 ∈ Grp) |
| 95 | 93, 94 | syl 17 |
. . . . . . . . . . . . . . . . . . . . . . 23
⊢ (𝜑 → 𝐺 ∈ Grp) |
| 96 | 3, 90 | grpinvcl 19005 |
. . . . . . . . . . . . . . . . . . . . . . 23
⊢ ((𝐺 ∈ Grp ∧ 𝑋 ∈ 𝐷) → ((invg‘𝐺)‘𝑋) ∈ 𝐷) |
| 97 | 95, 9, 96 | syl2anc 584 |
. . . . . . . . . . . . . . . . . . . . . 22
⊢ (𝜑 →
((invg‘𝐺)‘𝑋) ∈ 𝐷) |
| 98 | 91, 97 | eqeltrrd 2842 |
. . . . . . . . . . . . . . . . . . . . 21
⊢ (𝜑 → (∗ ∘ 𝑋) ∈ 𝐷) |
| 99 | | eldifsni 4790 |
. . . . . . . . . . . . . . . . . . . . . . . 24
⊢ (𝑋 ∈ (𝐷 ∖ { 1 }) → 𝑋 ≠ 1 ) |
| 100 | 8, 99 | syl 17 |
. . . . . . . . . . . . . . . . . . . . . . 23
⊢ (𝜑 → 𝑋 ≠ 1 ) |
| 101 | 3, 19 | grpidcl 18983 |
. . . . . . . . . . . . . . . . . . . . . . . . . 26
⊢ (𝐺 ∈ Grp → 1 ∈ 𝐷) |
| 102 | 95, 101 | syl 17 |
. . . . . . . . . . . . . . . . . . . . . . . . 25
⊢ (𝜑 → 1 ∈ 𝐷) |
| 103 | 3, 90, 95, 9, 102 | grpinv11 19025 |
. . . . . . . . . . . . . . . . . . . . . . . 24
⊢ (𝜑 →
(((invg‘𝐺)‘𝑋) = ((invg‘𝐺)‘ 1 ) ↔ 𝑋 = 1 )) |
| 104 | 103 | necon3bid 2985 |
. . . . . . . . . . . . . . . . . . . . . . 23
⊢ (𝜑 →
(((invg‘𝐺)‘𝑋) ≠ ((invg‘𝐺)‘ 1 ) ↔ 𝑋 ≠ 1 )) |
| 105 | 100, 104 | mpbird 257 |
. . . . . . . . . . . . . . . . . . . . . 22
⊢ (𝜑 →
((invg‘𝐺)‘𝑋) ≠ ((invg‘𝐺)‘ 1 )) |
| 106 | 19, 90 | grpinvid 19017 |
. . . . . . . . . . . . . . . . . . . . . . 23
⊢ (𝐺 ∈ Grp →
((invg‘𝐺)‘ 1 ) = 1 ) |
| 107 | 95, 106 | syl 17 |
. . . . . . . . . . . . . . . . . . . . . 22
⊢ (𝜑 →
((invg‘𝐺)‘ 1 ) = 1 ) |
| 108 | 105, 91, 107 | 3netr3d 3017 |
. . . . . . . . . . . . . . . . . . . . 21
⊢ (𝜑 → (∗ ∘ 𝑋) ≠ 1 ) |
| 109 | | eldifsn 4786 |
. . . . . . . . . . . . . . . . . . . . 21
⊢
((∗ ∘ 𝑋) ∈ (𝐷 ∖ { 1 }) ↔ ((∗
∘ 𝑋) ∈ 𝐷 ∧ (∗ ∘ 𝑋) ≠ 1 )) |
| 110 | 98, 108, 109 | sylanbrc 583 |
. . . . . . . . . . . . . . . . . . . 20
⊢ (𝜑 → (∗ ∘ 𝑋) ∈ (𝐷 ∖ { 1 })) |
| 111 | | nnuz 12921 |
. . . . . . . . . . . . . . . . . . . . 21
⊢ ℕ =
(ℤ≥‘1) |
| 112 | | 1zzd 12648 |
. . . . . . . . . . . . . . . . . . . . 21
⊢ (𝜑 → 1 ∈
ℤ) |
| 113 | | 2fveq3 6911 |
. . . . . . . . . . . . . . . . . . . . . . . . . 26
⊢ (𝑛 = 𝑚 → (𝑋‘(𝐿‘𝑛)) = (𝑋‘(𝐿‘𝑚))) |
| 114 | | id 22 |
. . . . . . . . . . . . . . . . . . . . . . . . . 26
⊢ (𝑛 = 𝑚 → 𝑛 = 𝑚) |
| 115 | 113, 114 | oveq12d 7449 |
. . . . . . . . . . . . . . . . . . . . . . . . 25
⊢ (𝑛 = 𝑚 → ((𝑋‘(𝐿‘𝑛)) / 𝑛) = ((𝑋‘(𝐿‘𝑚)) / 𝑚)) |
| 116 | 115 | fveq2d 6910 |
. . . . . . . . . . . . . . . . . . . . . . . 24
⊢ (𝑛 = 𝑚 → (∗‘((𝑋‘(𝐿‘𝑛)) / 𝑛)) = (∗‘((𝑋‘(𝐿‘𝑚)) / 𝑚))) |
| 117 | | eqid 2737 |
. . . . . . . . . . . . . . . . . . . . . . . 24
⊢ (𝑛 ∈ ℕ ↦
(∗‘((𝑋‘(𝐿‘𝑛)) / 𝑛))) = (𝑛 ∈ ℕ ↦
(∗‘((𝑋‘(𝐿‘𝑛)) / 𝑛))) |
| 118 | | fvex 6919 |
. . . . . . . . . . . . . . . . . . . . . . . 24
⊢
(∗‘((𝑋‘(𝐿‘𝑚)) / 𝑚)) ∈ V |
| 119 | 116, 117,
118 | fvmpt 7016 |
. . . . . . . . . . . . . . . . . . . . . . 23
⊢ (𝑚 ∈ ℕ → ((𝑛 ∈ ℕ ↦
(∗‘((𝑋‘(𝐿‘𝑛)) / 𝑛)))‘𝑚) = (∗‘((𝑋‘(𝐿‘𝑚)) / 𝑚))) |
| 120 | 119 | adantl 481 |
. . . . . . . . . . . . . . . . . . . . . 22
⊢ ((𝜑 ∧ 𝑚 ∈ ℕ) → ((𝑛 ∈ ℕ ↦
(∗‘((𝑋‘(𝐿‘𝑛)) / 𝑛)))‘𝑚) = (∗‘((𝑋‘(𝐿‘𝑚)) / 𝑚))) |
| 121 | | nnre 12273 |
. . . . . . . . . . . . . . . . . . . . . . . . . 26
⊢ (𝑚 ∈ ℕ → 𝑚 ∈
ℝ) |
| 122 | 121 | adantl 481 |
. . . . . . . . . . . . . . . . . . . . . . . . 25
⊢ ((𝜑 ∧ 𝑚 ∈ ℕ) → 𝑚 ∈ ℝ) |
| 123 | 122 | cjred 15265 |
. . . . . . . . . . . . . . . . . . . . . . . 24
⊢ ((𝜑 ∧ 𝑚 ∈ ℕ) → (∗‘𝑚) = 𝑚) |
| 124 | 123 | oveq2d 7447 |
. . . . . . . . . . . . . . . . . . . . . . 23
⊢ ((𝜑 ∧ 𝑚 ∈ ℕ) →
((∗‘(𝑋‘(𝐿‘𝑚))) / (∗‘𝑚)) = ((∗‘(𝑋‘(𝐿‘𝑚))) / 𝑚)) |
| 125 | 10 | adantr 480 |
. . . . . . . . . . . . . . . . . . . . . . . . 25
⊢ ((𝜑 ∧ 𝑚 ∈ ℕ) → 𝑋:(Base‘𝑍)⟶ℂ) |
| 126 | 2, 4, 17 | znzrhfo 21566 |
. . . . . . . . . . . . . . . . . . . . . . . . . . . 28
⊢ (𝑁 ∈ ℕ0
→ 𝐿:ℤ–onto→(Base‘𝑍)) |
| 127 | 21, 126 | syl 17 |
. . . . . . . . . . . . . . . . . . . . . . . . . . 27
⊢ (𝜑 → 𝐿:ℤ–onto→(Base‘𝑍)) |
| 128 | | fof 6820 |
. . . . . . . . . . . . . . . . . . . . . . . . . . 27
⊢ (𝐿:ℤ–onto→(Base‘𝑍) → 𝐿:ℤ⟶(Base‘𝑍)) |
| 129 | 127, 128 | syl 17 |
. . . . . . . . . . . . . . . . . . . . . . . . . 26
⊢ (𝜑 → 𝐿:ℤ⟶(Base‘𝑍)) |
| 130 | | nnz 12634 |
. . . . . . . . . . . . . . . . . . . . . . . . . 26
⊢ (𝑚 ∈ ℕ → 𝑚 ∈
ℤ) |
| 131 | | ffvelcdm 7101 |
. . . . . . . . . . . . . . . . . . . . . . . . . 26
⊢ ((𝐿:ℤ⟶(Base‘𝑍) ∧ 𝑚 ∈ ℤ) → (𝐿‘𝑚) ∈ (Base‘𝑍)) |
| 132 | 129, 130,
131 | syl2an 596 |
. . . . . . . . . . . . . . . . . . . . . . . . 25
⊢ ((𝜑 ∧ 𝑚 ∈ ℕ) → (𝐿‘𝑚) ∈ (Base‘𝑍)) |
| 133 | 125, 132 | ffvelcdmd 7105 |
. . . . . . . . . . . . . . . . . . . . . . . 24
⊢ ((𝜑 ∧ 𝑚 ∈ ℕ) → (𝑋‘(𝐿‘𝑚)) ∈ ℂ) |
| 134 | | nncn 12274 |
. . . . . . . . . . . . . . . . . . . . . . . . 25
⊢ (𝑚 ∈ ℕ → 𝑚 ∈
ℂ) |
| 135 | 134 | adantl 481 |
. . . . . . . . . . . . . . . . . . . . . . . 24
⊢ ((𝜑 ∧ 𝑚 ∈ ℕ) → 𝑚 ∈ ℂ) |
| 136 | | nnne0 12300 |
. . . . . . . . . . . . . . . . . . . . . . . . 25
⊢ (𝑚 ∈ ℕ → 𝑚 ≠ 0) |
| 137 | 136 | adantl 481 |
. . . . . . . . . . . . . . . . . . . . . . . 24
⊢ ((𝜑 ∧ 𝑚 ∈ ℕ) → 𝑚 ≠ 0) |
| 138 | 133, 135,
137 | cjdivd 15262 |
. . . . . . . . . . . . . . . . . . . . . . 23
⊢ ((𝜑 ∧ 𝑚 ∈ ℕ) →
(∗‘((𝑋‘(𝐿‘𝑚)) / 𝑚)) = ((∗‘(𝑋‘(𝐿‘𝑚))) / (∗‘𝑚))) |
| 139 | | fvco3 7008 |
. . . . . . . . . . . . . . . . . . . . . . . . 25
⊢ ((𝑋:(Base‘𝑍)⟶ℂ ∧ (𝐿‘𝑚) ∈ (Base‘𝑍)) → ((∗ ∘ 𝑋)‘(𝐿‘𝑚)) = (∗‘(𝑋‘(𝐿‘𝑚)))) |
| 140 | 125, 132,
139 | syl2anc 584 |
. . . . . . . . . . . . . . . . . . . . . . . 24
⊢ ((𝜑 ∧ 𝑚 ∈ ℕ) → ((∗ ∘
𝑋)‘(𝐿‘𝑚)) = (∗‘(𝑋‘(𝐿‘𝑚)))) |
| 141 | 140 | oveq1d 7446 |
. . . . . . . . . . . . . . . . . . . . . . 23
⊢ ((𝜑 ∧ 𝑚 ∈ ℕ) → (((∗ ∘
𝑋)‘(𝐿‘𝑚)) / 𝑚) = ((∗‘(𝑋‘(𝐿‘𝑚))) / 𝑚)) |
| 142 | 124, 138,
141 | 3eqtr4d 2787 |
. . . . . . . . . . . . . . . . . . . . . 22
⊢ ((𝜑 ∧ 𝑚 ∈ ℕ) →
(∗‘((𝑋‘(𝐿‘𝑚)) / 𝑚)) = (((∗ ∘ 𝑋)‘(𝐿‘𝑚)) / 𝑚)) |
| 143 | 120, 142 | eqtrd 2777 |
. . . . . . . . . . . . . . . . . . . . 21
⊢ ((𝜑 ∧ 𝑚 ∈ ℕ) → ((𝑛 ∈ ℕ ↦
(∗‘((𝑋‘(𝐿‘𝑛)) / 𝑛)))‘𝑚) = (((∗ ∘ 𝑋)‘(𝐿‘𝑚)) / 𝑚)) |
| 144 | 133 | cjcld 15235 |
. . . . . . . . . . . . . . . . . . . . . . 23
⊢ ((𝜑 ∧ 𝑚 ∈ ℕ) →
(∗‘(𝑋‘(𝐿‘𝑚))) ∈ ℂ) |
| 145 | 144, 135,
137 | divcld 12043 |
. . . . . . . . . . . . . . . . . . . . . 22
⊢ ((𝜑 ∧ 𝑚 ∈ ℕ) →
((∗‘(𝑋‘(𝐿‘𝑚))) / 𝑚) ∈ ℂ) |
| 146 | 141, 145 | eqeltrd 2841 |
. . . . . . . . . . . . . . . . . . . . 21
⊢ ((𝜑 ∧ 𝑚 ∈ ℕ) → (((∗ ∘
𝑋)‘(𝐿‘𝑚)) / 𝑚) ∈ ℂ) |
| 147 | | eqid 2737 |
. . . . . . . . . . . . . . . . . . . . . . . . 25
⊢ (𝑎 ∈ ℕ ↦ ((𝑋‘(𝐿‘𝑎)) / 𝑎)) = (𝑎 ∈ ℕ ↦ ((𝑋‘(𝐿‘𝑎)) / 𝑎)) |
| 148 | 2, 17, 18, 1, 3, 19, 9, 100, 147 | dchrmusumlema 27537 |
. . . . . . . . . . . . . . . . . . . . . . . 24
⊢ (𝜑 → ∃𝑡∃𝑐 ∈ (0[,)+∞)(seq1( + , (𝑎 ∈ ℕ ↦ ((𝑋‘(𝐿‘𝑎)) / 𝑎))) ⇝ 𝑡 ∧ ∀𝑦 ∈ (1[,)+∞)(abs‘((seq1( + ,
(𝑎 ∈ ℕ ↦
((𝑋‘(𝐿‘𝑎)) / 𝑎)))‘(⌊‘𝑦)) − 𝑡)) ≤ (𝑐 / 𝑦))) |
| 149 | | simprrl 781 |
. . . . . . . . . . . . . . . . . . . . . . . . . . 27
⊢ ((𝜑 ∧ (𝑐 ∈ (0[,)+∞) ∧ (seq1( + ,
(𝑎 ∈ ℕ ↦
((𝑋‘(𝐿‘𝑎)) / 𝑎))) ⇝ 𝑡 ∧ ∀𝑦 ∈ (1[,)+∞)(abs‘((seq1( + ,
(𝑎 ∈ ℕ ↦
((𝑋‘(𝐿‘𝑎)) / 𝑎)))‘(⌊‘𝑦)) − 𝑡)) ≤ (𝑐 / 𝑦)))) → seq1( + , (𝑎 ∈ ℕ ↦ ((𝑋‘(𝐿‘𝑎)) / 𝑎))) ⇝ 𝑡) |
| 150 | 7 | adantr 480 |
. . . . . . . . . . . . . . . . . . . . . . . . . . . 28
⊢ ((𝜑 ∧ (𝑐 ∈ (0[,)+∞) ∧ (seq1( + ,
(𝑎 ∈ ℕ ↦
((𝑋‘(𝐿‘𝑎)) / 𝑎))) ⇝ 𝑡 ∧ ∀𝑦 ∈ (1[,)+∞)(abs‘((seq1( + ,
(𝑎 ∈ ℕ ↦
((𝑋‘(𝐿‘𝑎)) / 𝑎)))‘(⌊‘𝑦)) − 𝑡)) ≤ (𝑐 / 𝑦)))) → 𝑋 ∈ 𝑊) |
| 151 | 18 | adantr 480 |
. . . . . . . . . . . . . . . . . . . . . . . . . . . . 29
⊢ ((𝜑 ∧ (𝑐 ∈ (0[,)+∞) ∧ (seq1( + ,
(𝑎 ∈ ℕ ↦
((𝑋‘(𝐿‘𝑎)) / 𝑎))) ⇝ 𝑡 ∧ ∀𝑦 ∈ (1[,)+∞)(abs‘((seq1( + ,
(𝑎 ∈ ℕ ↦
((𝑋‘(𝐿‘𝑎)) / 𝑎)))‘(⌊‘𝑦)) − 𝑡)) ≤ (𝑐 / 𝑦)))) → 𝑁 ∈ ℕ) |
| 152 | 9 | adantr 480 |
. . . . . . . . . . . . . . . . . . . . . . . . . . . . 29
⊢ ((𝜑 ∧ (𝑐 ∈ (0[,)+∞) ∧ (seq1( + ,
(𝑎 ∈ ℕ ↦
((𝑋‘(𝐿‘𝑎)) / 𝑎))) ⇝ 𝑡 ∧ ∀𝑦 ∈ (1[,)+∞)(abs‘((seq1( + ,
(𝑎 ∈ ℕ ↦
((𝑋‘(𝐿‘𝑎)) / 𝑎)))‘(⌊‘𝑦)) − 𝑡)) ≤ (𝑐 / 𝑦)))) → 𝑋 ∈ 𝐷) |
| 153 | 100 | adantr 480 |
. . . . . . . . . . . . . . . . . . . . . . . . . . . . 29
⊢ ((𝜑 ∧ (𝑐 ∈ (0[,)+∞) ∧ (seq1( + ,
(𝑎 ∈ ℕ ↦
((𝑋‘(𝐿‘𝑎)) / 𝑎))) ⇝ 𝑡 ∧ ∀𝑦 ∈ (1[,)+∞)(abs‘((seq1( + ,
(𝑎 ∈ ℕ ↦
((𝑋‘(𝐿‘𝑎)) / 𝑎)))‘(⌊‘𝑦)) − 𝑡)) ≤ (𝑐 / 𝑦)))) → 𝑋 ≠ 1 ) |
| 154 | | simprl 771 |
. . . . . . . . . . . . . . . . . . . . . . . . . . . . 29
⊢ ((𝜑 ∧ (𝑐 ∈ (0[,)+∞) ∧ (seq1( + ,
(𝑎 ∈ ℕ ↦
((𝑋‘(𝐿‘𝑎)) / 𝑎))) ⇝ 𝑡 ∧ ∀𝑦 ∈ (1[,)+∞)(abs‘((seq1( + ,
(𝑎 ∈ ℕ ↦
((𝑋‘(𝐿‘𝑎)) / 𝑎)))‘(⌊‘𝑦)) − 𝑡)) ≤ (𝑐 / 𝑦)))) → 𝑐 ∈ (0[,)+∞)) |
| 155 | | simprrr 782 |
. . . . . . . . . . . . . . . . . . . . . . . . . . . . 29
⊢ ((𝜑 ∧ (𝑐 ∈ (0[,)+∞) ∧ (seq1( + ,
(𝑎 ∈ ℕ ↦
((𝑋‘(𝐿‘𝑎)) / 𝑎))) ⇝ 𝑡 ∧ ∀𝑦 ∈ (1[,)+∞)(abs‘((seq1( + ,
(𝑎 ∈ ℕ ↦
((𝑋‘(𝐿‘𝑎)) / 𝑎)))‘(⌊‘𝑦)) − 𝑡)) ≤ (𝑐 / 𝑦)))) → ∀𝑦 ∈ (1[,)+∞)(abs‘((seq1( + ,
(𝑎 ∈ ℕ ↦
((𝑋‘(𝐿‘𝑎)) / 𝑎)))‘(⌊‘𝑦)) − 𝑡)) ≤ (𝑐 / 𝑦)) |
| 156 | 2, 17, 151, 1, 3, 19, 152, 153, 147, 154, 149, 155, 5 | dchrvmaeq0 27548 |
. . . . . . . . . . . . . . . . . . . . . . . . . . . 28
⊢ ((𝜑 ∧ (𝑐 ∈ (0[,)+∞) ∧ (seq1( + ,
(𝑎 ∈ ℕ ↦
((𝑋‘(𝐿‘𝑎)) / 𝑎))) ⇝ 𝑡 ∧ ∀𝑦 ∈ (1[,)+∞)(abs‘((seq1( + ,
(𝑎 ∈ ℕ ↦
((𝑋‘(𝐿‘𝑎)) / 𝑎)))‘(⌊‘𝑦)) − 𝑡)) ≤ (𝑐 / 𝑦)))) → (𝑋 ∈ 𝑊 ↔ 𝑡 = 0)) |
| 157 | 150, 156 | mpbid 232 |
. . . . . . . . . . . . . . . . . . . . . . . . . . 27
⊢ ((𝜑 ∧ (𝑐 ∈ (0[,)+∞) ∧ (seq1( + ,
(𝑎 ∈ ℕ ↦
((𝑋‘(𝐿‘𝑎)) / 𝑎))) ⇝ 𝑡 ∧ ∀𝑦 ∈ (1[,)+∞)(abs‘((seq1( + ,
(𝑎 ∈ ℕ ↦
((𝑋‘(𝐿‘𝑎)) / 𝑎)))‘(⌊‘𝑦)) − 𝑡)) ≤ (𝑐 / 𝑦)))) → 𝑡 = 0) |
| 158 | 149, 157 | breqtrd 5169 |
. . . . . . . . . . . . . . . . . . . . . . . . . 26
⊢ ((𝜑 ∧ (𝑐 ∈ (0[,)+∞) ∧ (seq1( + ,
(𝑎 ∈ ℕ ↦
((𝑋‘(𝐿‘𝑎)) / 𝑎))) ⇝ 𝑡 ∧ ∀𝑦 ∈ (1[,)+∞)(abs‘((seq1( + ,
(𝑎 ∈ ℕ ↦
((𝑋‘(𝐿‘𝑎)) / 𝑎)))‘(⌊‘𝑦)) − 𝑡)) ≤ (𝑐 / 𝑦)))) → seq1( + , (𝑎 ∈ ℕ ↦ ((𝑋‘(𝐿‘𝑎)) / 𝑎))) ⇝ 0) |
| 159 | 158 | rexlimdvaa 3156 |
. . . . . . . . . . . . . . . . . . . . . . . . 25
⊢ (𝜑 → (∃𝑐 ∈ (0[,)+∞)(seq1( + , (𝑎 ∈ ℕ ↦ ((𝑋‘(𝐿‘𝑎)) / 𝑎))) ⇝ 𝑡 ∧ ∀𝑦 ∈ (1[,)+∞)(abs‘((seq1( + ,
(𝑎 ∈ ℕ ↦
((𝑋‘(𝐿‘𝑎)) / 𝑎)))‘(⌊‘𝑦)) − 𝑡)) ≤ (𝑐 / 𝑦)) → seq1( + , (𝑎 ∈ ℕ ↦ ((𝑋‘(𝐿‘𝑎)) / 𝑎))) ⇝ 0)) |
| 160 | 159 | exlimdv 1933 |
. . . . . . . . . . . . . . . . . . . . . . . 24
⊢ (𝜑 → (∃𝑡∃𝑐 ∈ (0[,)+∞)(seq1( + , (𝑎 ∈ ℕ ↦ ((𝑋‘(𝐿‘𝑎)) / 𝑎))) ⇝ 𝑡 ∧ ∀𝑦 ∈ (1[,)+∞)(abs‘((seq1( + ,
(𝑎 ∈ ℕ ↦
((𝑋‘(𝐿‘𝑎)) / 𝑎)))‘(⌊‘𝑦)) − 𝑡)) ≤ (𝑐 / 𝑦)) → seq1( + , (𝑎 ∈ ℕ ↦ ((𝑋‘(𝐿‘𝑎)) / 𝑎))) ⇝ 0)) |
| 161 | 148, 160 | mpd 15 |
. . . . . . . . . . . . . . . . . . . . . . 23
⊢ (𝜑 → seq1( + , (𝑎 ∈ ℕ ↦ ((𝑋‘(𝐿‘𝑎)) / 𝑎))) ⇝ 0) |
| 162 | | seqex 14044 |
. . . . . . . . . . . . . . . . . . . . . . . 24
⊢ seq1( + ,
(𝑛 ∈ ℕ ↦
(∗‘((𝑋‘(𝐿‘𝑛)) / 𝑛)))) ∈ V |
| 163 | 162 | a1i 11 |
. . . . . . . . . . . . . . . . . . . . . . 23
⊢ (𝜑 → seq1( + , (𝑛 ∈ ℕ ↦
(∗‘((𝑋‘(𝐿‘𝑛)) / 𝑛)))) ∈ V) |
| 164 | | 2fveq3 6911 |
. . . . . . . . . . . . . . . . . . . . . . . . . . . . 29
⊢ (𝑎 = 𝑚 → (𝑋‘(𝐿‘𝑎)) = (𝑋‘(𝐿‘𝑚))) |
| 165 | | id 22 |
. . . . . . . . . . . . . . . . . . . . . . . . . . . . 29
⊢ (𝑎 = 𝑚 → 𝑎 = 𝑚) |
| 166 | 164, 165 | oveq12d 7449 |
. . . . . . . . . . . . . . . . . . . . . . . . . . . 28
⊢ (𝑎 = 𝑚 → ((𝑋‘(𝐿‘𝑎)) / 𝑎) = ((𝑋‘(𝐿‘𝑚)) / 𝑚)) |
| 167 | | ovex 7464 |
. . . . . . . . . . . . . . . . . . . . . . . . . . . 28
⊢ ((𝑋‘(𝐿‘𝑚)) / 𝑚) ∈ V |
| 168 | 166, 147,
167 | fvmpt 7016 |
. . . . . . . . . . . . . . . . . . . . . . . . . . 27
⊢ (𝑚 ∈ ℕ → ((𝑎 ∈ ℕ ↦ ((𝑋‘(𝐿‘𝑎)) / 𝑎))‘𝑚) = ((𝑋‘(𝐿‘𝑚)) / 𝑚)) |
| 169 | 168 | adantl 481 |
. . . . . . . . . . . . . . . . . . . . . . . . . 26
⊢ ((𝜑 ∧ 𝑚 ∈ ℕ) → ((𝑎 ∈ ℕ ↦ ((𝑋‘(𝐿‘𝑎)) / 𝑎))‘𝑚) = ((𝑋‘(𝐿‘𝑚)) / 𝑚)) |
| 170 | 133, 135,
137 | divcld 12043 |
. . . . . . . . . . . . . . . . . . . . . . . . . 26
⊢ ((𝜑 ∧ 𝑚 ∈ ℕ) → ((𝑋‘(𝐿‘𝑚)) / 𝑚) ∈ ℂ) |
| 171 | 169, 170 | eqeltrd 2841 |
. . . . . . . . . . . . . . . . . . . . . . . . 25
⊢ ((𝜑 ∧ 𝑚 ∈ ℕ) → ((𝑎 ∈ ℕ ↦ ((𝑋‘(𝐿‘𝑎)) / 𝑎))‘𝑚) ∈ ℂ) |
| 172 | 111, 112,
171 | serf 14071 |
. . . . . . . . . . . . . . . . . . . . . . . 24
⊢ (𝜑 → seq1( + , (𝑎 ∈ ℕ ↦ ((𝑋‘(𝐿‘𝑎)) / 𝑎))):ℕ⟶ℂ) |
| 173 | 172 | ffvelcdmda 7104 |
. . . . . . . . . . . . . . . . . . . . . . 23
⊢ ((𝜑 ∧ 𝑘 ∈ ℕ) → (seq1( + , (𝑎 ∈ ℕ ↦ ((𝑋‘(𝐿‘𝑎)) / 𝑎)))‘𝑘) ∈ ℂ) |
| 174 | | fzfid 14014 |
. . . . . . . . . . . . . . . . . . . . . . . . 25
⊢ ((𝜑 ∧ 𝑘 ∈ ℕ) → (1...𝑘) ∈ Fin) |
| 175 | | simpl 482 |
. . . . . . . . . . . . . . . . . . . . . . . . . 26
⊢ ((𝜑 ∧ 𝑘 ∈ ℕ) → 𝜑) |
| 176 | | elfznn 13593 |
. . . . . . . . . . . . . . . . . . . . . . . . . 26
⊢ (𝑚 ∈ (1...𝑘) → 𝑚 ∈ ℕ) |
| 177 | 175, 176,
170 | syl2an 596 |
. . . . . . . . . . . . . . . . . . . . . . . . 25
⊢ (((𝜑 ∧ 𝑘 ∈ ℕ) ∧ 𝑚 ∈ (1...𝑘)) → ((𝑋‘(𝐿‘𝑚)) / 𝑚) ∈ ℂ) |
| 178 | 174, 177 | fsumcj 15846 |
. . . . . . . . . . . . . . . . . . . . . . . 24
⊢ ((𝜑 ∧ 𝑘 ∈ ℕ) →
(∗‘Σ𝑚
∈ (1...𝑘)((𝑋‘(𝐿‘𝑚)) / 𝑚)) = Σ𝑚 ∈ (1...𝑘)(∗‘((𝑋‘(𝐿‘𝑚)) / 𝑚))) |
| 179 | 175, 176,
169 | syl2an 596 |
. . . . . . . . . . . . . . . . . . . . . . . . . 26
⊢ (((𝜑 ∧ 𝑘 ∈ ℕ) ∧ 𝑚 ∈ (1...𝑘)) → ((𝑎 ∈ ℕ ↦ ((𝑋‘(𝐿‘𝑎)) / 𝑎))‘𝑚) = ((𝑋‘(𝐿‘𝑚)) / 𝑚)) |
| 180 | | simpr 484 |
. . . . . . . . . . . . . . . . . . . . . . . . . . 27
⊢ ((𝜑 ∧ 𝑘 ∈ ℕ) → 𝑘 ∈ ℕ) |
| 181 | 180, 111 | eleqtrdi 2851 |
. . . . . . . . . . . . . . . . . . . . . . . . . 26
⊢ ((𝜑 ∧ 𝑘 ∈ ℕ) → 𝑘 ∈
(ℤ≥‘1)) |
| 182 | 179, 181,
177 | fsumser 15766 |
. . . . . . . . . . . . . . . . . . . . . . . . 25
⊢ ((𝜑 ∧ 𝑘 ∈ ℕ) → Σ𝑚 ∈ (1...𝑘)((𝑋‘(𝐿‘𝑚)) / 𝑚) = (seq1( + , (𝑎 ∈ ℕ ↦ ((𝑋‘(𝐿‘𝑎)) / 𝑎)))‘𝑘)) |
| 183 | 182 | fveq2d 6910 |
. . . . . . . . . . . . . . . . . . . . . . . 24
⊢ ((𝜑 ∧ 𝑘 ∈ ℕ) →
(∗‘Σ𝑚
∈ (1...𝑘)((𝑋‘(𝐿‘𝑚)) / 𝑚)) = (∗‘(seq1( + , (𝑎 ∈ ℕ ↦ ((𝑋‘(𝐿‘𝑎)) / 𝑎)))‘𝑘))) |
| 184 | 175, 176,
120 | syl2an 596 |
. . . . . . . . . . . . . . . . . . . . . . . . 25
⊢ (((𝜑 ∧ 𝑘 ∈ ℕ) ∧ 𝑚 ∈ (1...𝑘)) → ((𝑛 ∈ ℕ ↦
(∗‘((𝑋‘(𝐿‘𝑛)) / 𝑛)))‘𝑚) = (∗‘((𝑋‘(𝐿‘𝑚)) / 𝑚))) |
| 185 | 170 | cjcld 15235 |
. . . . . . . . . . . . . . . . . . . . . . . . . 26
⊢ ((𝜑 ∧ 𝑚 ∈ ℕ) →
(∗‘((𝑋‘(𝐿‘𝑚)) / 𝑚)) ∈ ℂ) |
| 186 | 175, 176,
185 | syl2an 596 |
. . . . . . . . . . . . . . . . . . . . . . . . 25
⊢ (((𝜑 ∧ 𝑘 ∈ ℕ) ∧ 𝑚 ∈ (1...𝑘)) → (∗‘((𝑋‘(𝐿‘𝑚)) / 𝑚)) ∈ ℂ) |
| 187 | 184, 181,
186 | fsumser 15766 |
. . . . . . . . . . . . . . . . . . . . . . . 24
⊢ ((𝜑 ∧ 𝑘 ∈ ℕ) → Σ𝑚 ∈ (1...𝑘)(∗‘((𝑋‘(𝐿‘𝑚)) / 𝑚)) = (seq1( + , (𝑛 ∈ ℕ ↦
(∗‘((𝑋‘(𝐿‘𝑛)) / 𝑛))))‘𝑘)) |
| 188 | 178, 183,
187 | 3eqtr3rd 2786 |
. . . . . . . . . . . . . . . . . . . . . . 23
⊢ ((𝜑 ∧ 𝑘 ∈ ℕ) → (seq1( + , (𝑛 ∈ ℕ ↦
(∗‘((𝑋‘(𝐿‘𝑛)) / 𝑛))))‘𝑘) = (∗‘(seq1( + , (𝑎 ∈ ℕ ↦ ((𝑋‘(𝐿‘𝑎)) / 𝑎)))‘𝑘))) |
| 189 | 111, 161,
163, 112, 173, 188 | climcj 15641 |
. . . . . . . . . . . . . . . . . . . . . 22
⊢ (𝜑 → seq1( + , (𝑛 ∈ ℕ ↦
(∗‘((𝑋‘(𝐿‘𝑛)) / 𝑛)))) ⇝
(∗‘0)) |
| 190 | | cj0 15197 |
. . . . . . . . . . . . . . . . . . . . . 22
⊢
(∗‘0) = 0 |
| 191 | 189, 190 | breqtrdi 5184 |
. . . . . . . . . . . . . . . . . . . . 21
⊢ (𝜑 → seq1( + , (𝑛 ∈ ℕ ↦
(∗‘((𝑋‘(𝐿‘𝑛)) / 𝑛)))) ⇝ 0) |
| 192 | 111, 112,
143, 146, 191 | isumclim 15793 |
. . . . . . . . . . . . . . . . . . . 20
⊢ (𝜑 → Σ𝑚 ∈ ℕ (((∗ ∘ 𝑋)‘(𝐿‘𝑚)) / 𝑚) = 0) |
| 193 | | fveq1 6905 |
. . . . . . . . . . . . . . . . . . . . . . . 24
⊢ (𝑦 = (∗ ∘ 𝑋) → (𝑦‘(𝐿‘𝑚)) = ((∗ ∘ 𝑋)‘(𝐿‘𝑚))) |
| 194 | 193 | oveq1d 7446 |
. . . . . . . . . . . . . . . . . . . . . . 23
⊢ (𝑦 = (∗ ∘ 𝑋) → ((𝑦‘(𝐿‘𝑚)) / 𝑚) = (((∗ ∘ 𝑋)‘(𝐿‘𝑚)) / 𝑚)) |
| 195 | 194 | sumeq2sdv 15739 |
. . . . . . . . . . . . . . . . . . . . . 22
⊢ (𝑦 = (∗ ∘ 𝑋) → Σ𝑚 ∈ ℕ ((𝑦‘(𝐿‘𝑚)) / 𝑚) = Σ𝑚 ∈ ℕ (((∗ ∘ 𝑋)‘(𝐿‘𝑚)) / 𝑚)) |
| 196 | 195 | eqeq1d 2739 |
. . . . . . . . . . . . . . . . . . . . 21
⊢ (𝑦 = (∗ ∘ 𝑋) → (Σ𝑚 ∈ ℕ ((𝑦‘(𝐿‘𝑚)) / 𝑚) = 0 ↔ Σ𝑚 ∈ ℕ (((∗ ∘ 𝑋)‘(𝐿‘𝑚)) / 𝑚) = 0)) |
| 197 | 196, 5 | elrab2 3695 |
. . . . . . . . . . . . . . . . . . . 20
⊢
((∗ ∘ 𝑋) ∈ 𝑊 ↔ ((∗ ∘ 𝑋) ∈ (𝐷 ∖ { 1 }) ∧ Σ𝑚 ∈ ℕ (((∗
∘ 𝑋)‘(𝐿‘𝑚)) / 𝑚) = 0)) |
| 198 | 110, 192,
197 | sylanbrc 583 |
. . . . . . . . . . . . . . . . . . 19
⊢ (𝜑 → (∗ ∘ 𝑋) ∈ 𝑊) |
| 199 | 198 | ad2antrr 726 |
. . . . . . . . . . . . . . . . . 18
⊢ (((𝜑 ∧ (∗ ∘ 𝑋) ≠ 𝑋) ∧ (𝑥 ∈ ℝ+ ∧ 1 ≤
𝑥)) → (∗
∘ 𝑋) ∈ 𝑊) |
| 200 | 7 | ad2antrr 726 |
. . . . . . . . . . . . . . . . . 18
⊢ (((𝜑 ∧ (∗ ∘ 𝑋) ≠ 𝑋) ∧ (𝑥 ∈ ℝ+ ∧ 1 ≤
𝑥)) → 𝑋 ∈ 𝑊) |
| 201 | | simplr 769 |
. . . . . . . . . . . . . . . . . 18
⊢ (((𝜑 ∧ (∗ ∘ 𝑋) ≠ 𝑋) ∧ (𝑥 ∈ ℝ+ ∧ 1 ≤
𝑥)) → (∗
∘ 𝑋) ≠ 𝑋) |
| 202 | 89, 199, 200, 201 | nehash2 14513 |
. . . . . . . . . . . . . . . . 17
⊢ (((𝜑 ∧ (∗ ∘ 𝑋) ≠ 𝑋) ∧ (𝑥 ∈ ℝ+ ∧ 1 ≤
𝑥)) → 2 ≤
(♯‘𝑊)) |
| 203 | | suble0 11777 |
. . . . . . . . . . . . . . . . . 18
⊢ ((2
∈ ℝ ∧ (♯‘𝑊) ∈ ℝ) → ((2 −
(♯‘𝑊)) ≤ 0
↔ 2 ≤ (♯‘𝑊))) |
| 204 | 77, 79, 203 | sylancr 587 |
. . . . . . . . . . . . . . . . 17
⊢ (((𝜑 ∧ (∗ ∘ 𝑋) ≠ 𝑋) ∧ (𝑥 ∈ ℝ+ ∧ 1 ≤
𝑥)) → ((2 −
(♯‘𝑊)) ≤ 0
↔ 2 ≤ (♯‘𝑊))) |
| 205 | 202, 204 | mpbird 257 |
. . . . . . . . . . . . . . . 16
⊢ (((𝜑 ∧ (∗ ∘ 𝑋) ≠ 𝑋) ∧ (𝑥 ∈ ℝ+ ∧ 1 ≤
𝑥)) → (2 −
(♯‘𝑊)) ≤
0) |
| 206 | 80, 75, 72, 88, 205 | lemul2ad 12208 |
. . . . . . . . . . . . . . 15
⊢ (((𝜑 ∧ (∗ ∘ 𝑋) ≠ 𝑋) ∧ (𝑥 ∈ ℝ+ ∧ 1 ≤
𝑥)) →
((log‘𝑥) · (2
− (♯‘𝑊)))
≤ ((log‘𝑥)
· 0)) |
| 207 | | df-2 12329 |
. . . . . . . . . . . . . . . . . . 19
⊢ 2 = (1 +
1) |
| 208 | 207 | oveq1i 7441 |
. . . . . . . . . . . . . . . . . 18
⊢ (2
− (♯‘𝑊))
= ((1 + 1) − (♯‘𝑊)) |
| 209 | | 1cnd 11256 |
. . . . . . . . . . . . . . . . . . 19
⊢ (((𝜑 ∧ (∗ ∘ 𝑋) ≠ 𝑋) ∧ (𝑥 ∈ ℝ+ ∧ 1 ≤
𝑥)) → 1 ∈
ℂ) |
| 210 | 79 | recnd 11289 |
. . . . . . . . . . . . . . . . . . 19
⊢ (((𝜑 ∧ (∗ ∘ 𝑋) ≠ 𝑋) ∧ (𝑥 ∈ ℝ+ ∧ 1 ≤
𝑥)) →
(♯‘𝑊) ∈
ℂ) |
| 211 | 209, 209,
210 | addsubassd 11640 |
. . . . . . . . . . . . . . . . . 18
⊢ (((𝜑 ∧ (∗ ∘ 𝑋) ≠ 𝑋) ∧ (𝑥 ∈ ℝ+ ∧ 1 ≤
𝑥)) → ((1 + 1) −
(♯‘𝑊)) = (1 +
(1 − (♯‘𝑊)))) |
| 212 | 208, 211 | eqtrid 2789 |
. . . . . . . . . . . . . . . . 17
⊢ (((𝜑 ∧ (∗ ∘ 𝑋) ≠ 𝑋) ∧ (𝑥 ∈ ℝ+ ∧ 1 ≤
𝑥)) → (2 −
(♯‘𝑊)) = (1 +
(1 − (♯‘𝑊)))) |
| 213 | 212 | oveq2d 7447 |
. . . . . . . . . . . . . . . 16
⊢ (((𝜑 ∧ (∗ ∘ 𝑋) ≠ 𝑋) ∧ (𝑥 ∈ ℝ+ ∧ 1 ≤
𝑥)) →
((log‘𝑥) · (2
− (♯‘𝑊)))
= ((log‘𝑥) ·
(1 + (1 − (♯‘𝑊))))) |
| 214 | 71 | adantrr 717 |
. . . . . . . . . . . . . . . . 17
⊢ (((𝜑 ∧ (∗ ∘ 𝑋) ≠ 𝑋) ∧ (𝑥 ∈ ℝ+ ∧ 1 ≤
𝑥)) → (log‘𝑥) ∈
ℂ) |
| 215 | 64 | ad2antrr 726 |
. . . . . . . . . . . . . . . . . 18
⊢ (((𝜑 ∧ (∗ ∘ 𝑋) ≠ 𝑋) ∧ (𝑥 ∈ ℝ+ ∧ 1 ≤
𝑥)) → (1 −
(♯‘𝑊)) ∈
ℝ) |
| 216 | 215 | recnd 11289 |
. . . . . . . . . . . . . . . . 17
⊢ (((𝜑 ∧ (∗ ∘ 𝑋) ≠ 𝑋) ∧ (𝑥 ∈ ℝ+ ∧ 1 ≤
𝑥)) → (1 −
(♯‘𝑊)) ∈
ℂ) |
| 217 | 214, 209,
216 | adddid 11285 |
. . . . . . . . . . . . . . . 16
⊢ (((𝜑 ∧ (∗ ∘ 𝑋) ≠ 𝑋) ∧ (𝑥 ∈ ℝ+ ∧ 1 ≤
𝑥)) →
((log‘𝑥) · (1
+ (1 − (♯‘𝑊)))) = (((log‘𝑥) · 1) + ((log‘𝑥) · (1 −
(♯‘𝑊))))) |
| 218 | 214 | mulridd 11278 |
. . . . . . . . . . . . . . . . 17
⊢ (((𝜑 ∧ (∗ ∘ 𝑋) ≠ 𝑋) ∧ (𝑥 ∈ ℝ+ ∧ 1 ≤
𝑥)) →
((log‘𝑥) · 1)
= (log‘𝑥)) |
| 219 | 218 | oveq1d 7446 |
. . . . . . . . . . . . . . . 16
⊢ (((𝜑 ∧ (∗ ∘ 𝑋) ≠ 𝑋) ∧ (𝑥 ∈ ℝ+ ∧ 1 ≤
𝑥)) →
(((log‘𝑥) · 1)
+ ((log‘𝑥) ·
(1 − (♯‘𝑊)))) = ((log‘𝑥) + ((log‘𝑥) · (1 − (♯‘𝑊))))) |
| 220 | 213, 217,
219 | 3eqtrd 2781 |
. . . . . . . . . . . . . . 15
⊢ (((𝜑 ∧ (∗ ∘ 𝑋) ≠ 𝑋) ∧ (𝑥 ∈ ℝ+ ∧ 1 ≤
𝑥)) →
((log‘𝑥) · (2
− (♯‘𝑊)))
= ((log‘𝑥) +
((log‘𝑥) · (1
− (♯‘𝑊))))) |
| 221 | 214 | mul01d 11460 |
. . . . . . . . . . . . . . 15
⊢ (((𝜑 ∧ (∗ ∘ 𝑋) ≠ 𝑋) ∧ (𝑥 ∈ ℝ+ ∧ 1 ≤
𝑥)) →
((log‘𝑥) · 0)
= 0) |
| 222 | 206, 220,
221 | 3brtr3d 5174 |
. . . . . . . . . . . . . 14
⊢ (((𝜑 ∧ (∗ ∘ 𝑋) ≠ 𝑋) ∧ (𝑥 ∈ ℝ+ ∧ 1 ≤
𝑥)) →
((log‘𝑥) +
((log‘𝑥) · (1
− (♯‘𝑊)))) ≤ 0) |
| 223 | 33 | nnred 12281 |
. . . . . . . . . . . . . . . 16
⊢ (𝜑 → (ϕ‘𝑁) ∈
ℝ) |
| 224 | 223 | ad2antrr 726 |
. . . . . . . . . . . . . . 15
⊢ (((𝜑 ∧ (∗ ∘ 𝑋) ≠ 𝑋) ∧ (𝑥 ∈ ℝ+ ∧ 1 ≤
𝑥)) →
(ϕ‘𝑁) ∈
ℝ) |
| 225 | 49 | ad2ant2r 747 |
. . . . . . . . . . . . . . 15
⊢ (((𝜑 ∧ (∗ ∘ 𝑋) ≠ 𝑋) ∧ (𝑥 ∈ ℝ+ ∧ 1 ≤
𝑥)) → Σ𝑛 ∈
((1...(⌊‘𝑥))
∩ (◡𝐿 “ {(1r‘𝑍)}))((Λ‘𝑛) / 𝑛) ∈ ℝ) |
| 226 | 34 | ad2antrr 726 |
. . . . . . . . . . . . . . . 16
⊢ (((𝜑 ∧ (∗ ∘ 𝑋) ≠ 𝑋) ∧ (𝑥 ∈ ℝ+ ∧ 1 ≤
𝑥)) →
(ϕ‘𝑁) ∈
ℕ0) |
| 227 | 226 | nn0ge0d 12590 |
. . . . . . . . . . . . . . 15
⊢ (((𝜑 ∧ (∗ ∘ 𝑋) ≠ 𝑋) ∧ (𝑥 ∈ ℝ+ ∧ 1 ≤
𝑥)) → 0 ≤
(ϕ‘𝑁)) |
| 228 | 44, 45 | syl 17 |
. . . . . . . . . . . . . . . . . 18
⊢ (((𝜑 ∧ 𝑥 ∈ ℝ+) ∧ 𝑛 ∈
((1...(⌊‘𝑥))
∩ (◡𝐿 “ {(1r‘𝑍)}))) →
(Λ‘𝑛) ∈
ℝ) |
| 229 | | vmage0 27164 |
. . . . . . . . . . . . . . . . . . 19
⊢ (𝑛 ∈ ℕ → 0 ≤
(Λ‘𝑛)) |
| 230 | 44, 229 | syl 17 |
. . . . . . . . . . . . . . . . . 18
⊢ (((𝜑 ∧ 𝑥 ∈ ℝ+) ∧ 𝑛 ∈
((1...(⌊‘𝑥))
∩ (◡𝐿 “ {(1r‘𝑍)}))) → 0 ≤
(Λ‘𝑛)) |
| 231 | 44 | nnred 12281 |
. . . . . . . . . . . . . . . . . 18
⊢ (((𝜑 ∧ 𝑥 ∈ ℝ+) ∧ 𝑛 ∈
((1...(⌊‘𝑥))
∩ (◡𝐿 “ {(1r‘𝑍)}))) → 𝑛 ∈ ℝ) |
| 232 | 44 | nngt0d 12315 |
. . . . . . . . . . . . . . . . . 18
⊢ (((𝜑 ∧ 𝑥 ∈ ℝ+) ∧ 𝑛 ∈
((1...(⌊‘𝑥))
∩ (◡𝐿 “ {(1r‘𝑍)}))) → 0 < 𝑛) |
| 233 | | divge0 12137 |
. . . . . . . . . . . . . . . . . 18
⊢
((((Λ‘𝑛) ∈ ℝ ∧ 0 ≤
(Λ‘𝑛)) ∧
(𝑛 ∈ ℝ ∧ 0
< 𝑛)) → 0 ≤
((Λ‘𝑛) / 𝑛)) |
| 234 | 228, 230,
231, 232, 233 | syl22anc 839 |
. . . . . . . . . . . . . . . . 17
⊢ (((𝜑 ∧ 𝑥 ∈ ℝ+) ∧ 𝑛 ∈
((1...(⌊‘𝑥))
∩ (◡𝐿 “ {(1r‘𝑍)}))) → 0 ≤
((Λ‘𝑛) / 𝑛)) |
| 235 | 40, 48, 234 | fsumge0 15831 |
. . . . . . . . . . . . . . . 16
⊢ ((𝜑 ∧ 𝑥 ∈ ℝ+) → 0 ≤
Σ𝑛 ∈
((1...(⌊‘𝑥))
∩ (◡𝐿 “ {(1r‘𝑍)}))((Λ‘𝑛) / 𝑛)) |
| 236 | 235 | ad2ant2r 747 |
. . . . . . . . . . . . . . 15
⊢ (((𝜑 ∧ (∗ ∘ 𝑋) ≠ 𝑋) ∧ (𝑥 ∈ ℝ+ ∧ 1 ≤
𝑥)) → 0 ≤
Σ𝑛 ∈
((1...(⌊‘𝑥))
∩ (◡𝐿 “ {(1r‘𝑍)}))((Λ‘𝑛) / 𝑛)) |
| 237 | 224, 225,
227, 236 | mulge0d 11840 |
. . . . . . . . . . . . . 14
⊢ (((𝜑 ∧ (∗ ∘ 𝑋) ≠ 𝑋) ∧ (𝑥 ∈ ℝ+ ∧ 1 ≤
𝑥)) → 0 ≤
((ϕ‘𝑁) ·
Σ𝑛 ∈
((1...(⌊‘𝑥))
∩ (◡𝐿 “ {(1r‘𝑍)}))((Λ‘𝑛) / 𝑛))) |
| 238 | 74, 75, 76, 222, 237 | letrd 11418 |
. . . . . . . . . . . . 13
⊢ (((𝜑 ∧ (∗ ∘ 𝑋) ≠ 𝑋) ∧ (𝑥 ∈ ℝ+ ∧ 1 ≤
𝑥)) →
((log‘𝑥) +
((log‘𝑥) · (1
− (♯‘𝑊)))) ≤ ((ϕ‘𝑁) · Σ𝑛 ∈ ((1...(⌊‘𝑥)) ∩ (◡𝐿 “ {(1r‘𝑍)}))((Λ‘𝑛) / 𝑛))) |
| 239 | | leaddsub 11739 |
. . . . . . . . . . . . . 14
⊢
(((log‘𝑥)
∈ ℝ ∧ ((log‘𝑥) · (1 − (♯‘𝑊))) ∈ ℝ ∧
((ϕ‘𝑁) ·
Σ𝑛 ∈
((1...(⌊‘𝑥))
∩ (◡𝐿 “ {(1r‘𝑍)}))((Λ‘𝑛) / 𝑛)) ∈ ℝ) → (((log‘𝑥) + ((log‘𝑥) · (1 −
(♯‘𝑊)))) ≤
((ϕ‘𝑁) ·
Σ𝑛 ∈
((1...(⌊‘𝑥))
∩ (◡𝐿 “ {(1r‘𝑍)}))((Λ‘𝑛) / 𝑛)) ↔ (log‘𝑥) ≤ (((ϕ‘𝑁) · Σ𝑛 ∈ ((1...(⌊‘𝑥)) ∩ (◡𝐿 “ {(1r‘𝑍)}))((Λ‘𝑛) / 𝑛)) − ((log‘𝑥) · (1 − (♯‘𝑊)))))) |
| 240 | 72, 73, 76, 239 | syl3anc 1373 |
. . . . . . . . . . . . 13
⊢ (((𝜑 ∧ (∗ ∘ 𝑋) ≠ 𝑋) ∧ (𝑥 ∈ ℝ+ ∧ 1 ≤
𝑥)) →
(((log‘𝑥) +
((log‘𝑥) · (1
− (♯‘𝑊)))) ≤ ((ϕ‘𝑁) · Σ𝑛 ∈ ((1...(⌊‘𝑥)) ∩ (◡𝐿 “ {(1r‘𝑍)}))((Λ‘𝑛) / 𝑛)) ↔ (log‘𝑥) ≤ (((ϕ‘𝑁) · Σ𝑛 ∈ ((1...(⌊‘𝑥)) ∩ (◡𝐿 “ {(1r‘𝑍)}))((Λ‘𝑛) / 𝑛)) − ((log‘𝑥) · (1 − (♯‘𝑊)))))) |
| 241 | 238, 240 | mpbid 232 |
. . . . . . . . . . . 12
⊢ (((𝜑 ∧ (∗ ∘ 𝑋) ≠ 𝑋) ∧ (𝑥 ∈ ℝ+ ∧ 1 ≤
𝑥)) → (log‘𝑥) ≤ (((ϕ‘𝑁) · Σ𝑛 ∈
((1...(⌊‘𝑥))
∩ (◡𝐿 “ {(1r‘𝑍)}))((Λ‘𝑛) / 𝑛)) − ((log‘𝑥) · (1 − (♯‘𝑊))))) |
| 242 | 72, 88 | absidd 15461 |
. . . . . . . . . . . 12
⊢ (((𝜑 ∧ (∗ ∘ 𝑋) ≠ 𝑋) ∧ (𝑥 ∈ ℝ+ ∧ 1 ≤
𝑥)) →
(abs‘(log‘𝑥)) =
(log‘𝑥)) |
| 243 | 67 | ad2ant2r 747 |
. . . . . . . . . . . . 13
⊢ (((𝜑 ∧ (∗ ∘ 𝑋) ≠ 𝑋) ∧ (𝑥 ∈ ℝ+ ∧ 1 ≤
𝑥)) →
(((ϕ‘𝑁) ·
Σ𝑛 ∈
((1...(⌊‘𝑥))
∩ (◡𝐿 “ {(1r‘𝑍)}))((Λ‘𝑛) / 𝑛)) − ((log‘𝑥) · (1 − (♯‘𝑊)))) ∈
ℝ) |
| 244 | 75, 72, 243, 88, 241 | letrd 11418 |
. . . . . . . . . . . . 13
⊢ (((𝜑 ∧ (∗ ∘ 𝑋) ≠ 𝑋) ∧ (𝑥 ∈ ℝ+ ∧ 1 ≤
𝑥)) → 0 ≤
(((ϕ‘𝑁) ·
Σ𝑛 ∈
((1...(⌊‘𝑥))
∩ (◡𝐿 “ {(1r‘𝑍)}))((Λ‘𝑛) / 𝑛)) − ((log‘𝑥) · (1 − (♯‘𝑊))))) |
| 245 | 243, 244 | absidd 15461 |
. . . . . . . . . . . 12
⊢ (((𝜑 ∧ (∗ ∘ 𝑋) ≠ 𝑋) ∧ (𝑥 ∈ ℝ+ ∧ 1 ≤
𝑥)) →
(abs‘(((ϕ‘𝑁) · Σ𝑛 ∈ ((1...(⌊‘𝑥)) ∩ (◡𝐿 “ {(1r‘𝑍)}))((Λ‘𝑛) / 𝑛)) − ((log‘𝑥) · (1 − (♯‘𝑊))))) = (((ϕ‘𝑁) · Σ𝑛 ∈
((1...(⌊‘𝑥))
∩ (◡𝐿 “ {(1r‘𝑍)}))((Λ‘𝑛) / 𝑛)) − ((log‘𝑥) · (1 − (♯‘𝑊))))) |
| 246 | 241, 242,
245 | 3brtr4d 5175 |
. . . . . . . . . . 11
⊢ (((𝜑 ∧ (∗ ∘ 𝑋) ≠ 𝑋) ∧ (𝑥 ∈ ℝ+ ∧ 1 ≤
𝑥)) →
(abs‘(log‘𝑥))
≤ (abs‘(((ϕ‘𝑁) · Σ𝑛 ∈ ((1...(⌊‘𝑥)) ∩ (◡𝐿 “ {(1r‘𝑍)}))((Λ‘𝑛) / 𝑛)) − ((log‘𝑥) · (1 − (♯‘𝑊)))))) |
| 247 | 16, 32, 69, 71, 246 | o1le 15689 |
. . . . . . . . . 10
⊢ ((𝜑 ∧ (∗ ∘ 𝑋) ≠ 𝑋) → (𝑥 ∈ ℝ+ ↦
(log‘𝑥)) ∈
𝑂(1)) |
| 248 | 247 | ex 412 |
. . . . . . . . 9
⊢ (𝜑 → ((∗ ∘ 𝑋) ≠ 𝑋 → (𝑥 ∈ ℝ+ ↦
(log‘𝑥)) ∈
𝑂(1))) |
| 249 | 248 | necon1bd 2958 |
. . . . . . . 8
⊢ (𝜑 → (¬ (𝑥 ∈ ℝ+ ↦
(log‘𝑥)) ∈
𝑂(1) → (∗ ∘ 𝑋) = 𝑋)) |
| 250 | 15, 249 | mpi 20 |
. . . . . . 7
⊢ (𝜑 → (∗ ∘ 𝑋) = 𝑋) |
| 251 | 250 | adantr 480 |
. . . . . 6
⊢ ((𝜑 ∧ 𝑥 ∈ (Base‘𝑍)) → (∗ ∘ 𝑋) = 𝑋) |
| 252 | 251 | fveq1d 6908 |
. . . . 5
⊢ ((𝜑 ∧ 𝑥 ∈ (Base‘𝑍)) → ((∗ ∘ 𝑋)‘𝑥) = (𝑋‘𝑥)) |
| 253 | 14, 252 | eqtr3d 2779 |
. . . 4
⊢ ((𝜑 ∧ 𝑥 ∈ (Base‘𝑍)) → (∗‘(𝑋‘𝑥)) = (𝑋‘𝑥)) |
| 254 | 12, 253 | cjrebd 15241 |
. . 3
⊢ ((𝜑 ∧ 𝑥 ∈ (Base‘𝑍)) → (𝑋‘𝑥) ∈ ℝ) |
| 255 | 254 | ralrimiva 3146 |
. 2
⊢ (𝜑 → ∀𝑥 ∈ (Base‘𝑍)(𝑋‘𝑥) ∈ ℝ) |
| 256 | | ffnfv 7139 |
. 2
⊢ (𝑋:(Base‘𝑍)⟶ℝ ↔ (𝑋 Fn (Base‘𝑍) ∧ ∀𝑥 ∈ (Base‘𝑍)(𝑋‘𝑥) ∈ ℝ)) |
| 257 | 11, 255, 256 | sylanbrc 583 |
1
⊢ (𝜑 → 𝑋:(Base‘𝑍)⟶ℝ) |