| Step | Hyp | Ref
| Expression |
| 1 | | stoweidlem42.2 |
. 2
⊢
Ⅎ𝑡𝜑 |
| 2 | | 1red 11262 |
. . . . . . . 8
⊢ (𝜑 → 1 ∈
ℝ) |
| 3 | | stoweidlem42.11 |
. . . . . . . . 9
⊢ (𝜑 → 𝐸 ∈
ℝ+) |
| 4 | 3 | rpred 13077 |
. . . . . . . 8
⊢ (𝜑 → 𝐸 ∈ ℝ) |
| 5 | 2, 4 | resubcld 11691 |
. . . . . . 7
⊢ (𝜑 → (1 − 𝐸) ∈
ℝ) |
| 6 | 5 | adantr 480 |
. . . . . 6
⊢ ((𝜑 ∧ 𝑡 ∈ 𝐵) → (1 − 𝐸) ∈ ℝ) |
| 7 | | stoweidlem42.8 |
. . . . . . . . . 10
⊢ (𝜑 → 𝑀 ∈ ℕ) |
| 8 | 4, 7 | nndivred 12320 |
. . . . . . . . 9
⊢ (𝜑 → (𝐸 / 𝑀) ∈ ℝ) |
| 9 | 2, 8 | resubcld 11691 |
. . . . . . . 8
⊢ (𝜑 → (1 − (𝐸 / 𝑀)) ∈ ℝ) |
| 10 | 9 | adantr 480 |
. . . . . . 7
⊢ ((𝜑 ∧ 𝑡 ∈ 𝐵) → (1 − (𝐸 / 𝑀)) ∈ ℝ) |
| 11 | 7 | nnnn0d 12587 |
. . . . . . . 8
⊢ (𝜑 → 𝑀 ∈
ℕ0) |
| 12 | 11 | adantr 480 |
. . . . . . 7
⊢ ((𝜑 ∧ 𝑡 ∈ 𝐵) → 𝑀 ∈
ℕ0) |
| 13 | 10, 12 | reexpcld 14203 |
. . . . . 6
⊢ ((𝜑 ∧ 𝑡 ∈ 𝐵) → ((1 − (𝐸 / 𝑀))↑𝑀) ∈ ℝ) |
| 14 | | elnnuz 12922 |
. . . . . . . . 9
⊢ (𝑀 ∈ ℕ ↔ 𝑀 ∈
(ℤ≥‘1)) |
| 15 | 7, 14 | sylib 218 |
. . . . . . . 8
⊢ (𝜑 → 𝑀 ∈
(ℤ≥‘1)) |
| 16 | 15 | adantr 480 |
. . . . . . 7
⊢ ((𝜑 ∧ 𝑡 ∈ 𝐵) → 𝑀 ∈
(ℤ≥‘1)) |
| 17 | | stoweidlem42.1 |
. . . . . . . . . . 11
⊢
Ⅎ𝑖𝜑 |
| 18 | | nfv 1914 |
. . . . . . . . . . 11
⊢
Ⅎ𝑖 𝑡 ∈ 𝐵 |
| 19 | 17, 18 | nfan 1899 |
. . . . . . . . . 10
⊢
Ⅎ𝑖(𝜑 ∧ 𝑡 ∈ 𝐵) |
| 20 | | nfv 1914 |
. . . . . . . . . 10
⊢
Ⅎ𝑖 𝑎 ∈ (1...𝑀) |
| 21 | 19, 20 | nfan 1899 |
. . . . . . . . 9
⊢
Ⅎ𝑖((𝜑 ∧ 𝑡 ∈ 𝐵) ∧ 𝑎 ∈ (1...𝑀)) |
| 22 | | stoweidlem42.6 |
. . . . . . . . . . . . 13
⊢ 𝐹 = (𝑡 ∈ 𝑇 ↦ (𝑖 ∈ (1...𝑀) ↦ ((𝑈‘𝑖)‘𝑡))) |
| 23 | | nfcv 2905 |
. . . . . . . . . . . . . 14
⊢
Ⅎ𝑖𝑇 |
| 24 | | nfmpt1 5250 |
. . . . . . . . . . . . . 14
⊢
Ⅎ𝑖(𝑖 ∈ (1...𝑀) ↦ ((𝑈‘𝑖)‘𝑡)) |
| 25 | 23, 24 | nfmpt 5249 |
. . . . . . . . . . . . 13
⊢
Ⅎ𝑖(𝑡 ∈ 𝑇 ↦ (𝑖 ∈ (1...𝑀) ↦ ((𝑈‘𝑖)‘𝑡))) |
| 26 | 22, 25 | nfcxfr 2903 |
. . . . . . . . . . . 12
⊢
Ⅎ𝑖𝐹 |
| 27 | | nfcv 2905 |
. . . . . . . . . . . 12
⊢
Ⅎ𝑖𝑡 |
| 28 | 26, 27 | nffv 6916 |
. . . . . . . . . . 11
⊢
Ⅎ𝑖(𝐹‘𝑡) |
| 29 | | nfcv 2905 |
. . . . . . . . . . 11
⊢
Ⅎ𝑖𝑎 |
| 30 | 28, 29 | nffv 6916 |
. . . . . . . . . 10
⊢
Ⅎ𝑖((𝐹‘𝑡)‘𝑎) |
| 31 | 30 | nfel1 2922 |
. . . . . . . . 9
⊢
Ⅎ𝑖((𝐹‘𝑡)‘𝑎) ∈ ℝ |
| 32 | 21, 31 | nfim 1896 |
. . . . . . . 8
⊢
Ⅎ𝑖(((𝜑 ∧ 𝑡 ∈ 𝐵) ∧ 𝑎 ∈ (1...𝑀)) → ((𝐹‘𝑡)‘𝑎) ∈ ℝ) |
| 33 | | eleq1 2829 |
. . . . . . . . . 10
⊢ (𝑖 = 𝑎 → (𝑖 ∈ (1...𝑀) ↔ 𝑎 ∈ (1...𝑀))) |
| 34 | 33 | anbi2d 630 |
. . . . . . . . 9
⊢ (𝑖 = 𝑎 → (((𝜑 ∧ 𝑡 ∈ 𝐵) ∧ 𝑖 ∈ (1...𝑀)) ↔ ((𝜑 ∧ 𝑡 ∈ 𝐵) ∧ 𝑎 ∈ (1...𝑀)))) |
| 35 | | fveq2 6906 |
. . . . . . . . . 10
⊢ (𝑖 = 𝑎 → ((𝐹‘𝑡)‘𝑖) = ((𝐹‘𝑡)‘𝑎)) |
| 36 | 35 | eleq1d 2826 |
. . . . . . . . 9
⊢ (𝑖 = 𝑎 → (((𝐹‘𝑡)‘𝑖) ∈ ℝ ↔ ((𝐹‘𝑡)‘𝑎) ∈ ℝ)) |
| 37 | 34, 36 | imbi12d 344 |
. . . . . . . 8
⊢ (𝑖 = 𝑎 → ((((𝜑 ∧ 𝑡 ∈ 𝐵) ∧ 𝑖 ∈ (1...𝑀)) → ((𝐹‘𝑡)‘𝑖) ∈ ℝ) ↔ (((𝜑 ∧ 𝑡 ∈ 𝐵) ∧ 𝑎 ∈ (1...𝑀)) → ((𝐹‘𝑡)‘𝑎) ∈ ℝ))) |
| 38 | | stoweidlem42.16 |
. . . . . . . . . . . 12
⊢ (𝜑 → 𝐵 ⊆ 𝑇) |
| 39 | 38 | sselda 3983 |
. . . . . . . . . . 11
⊢ ((𝜑 ∧ 𝑡 ∈ 𝐵) → 𝑡 ∈ 𝑇) |
| 40 | | ovex 7464 |
. . . . . . . . . . . 12
⊢
(1...𝑀) ∈
V |
| 41 | | mptexg 7241 |
. . . . . . . . . . . 12
⊢
((1...𝑀) ∈ V
→ (𝑖 ∈ (1...𝑀) ↦ ((𝑈‘𝑖)‘𝑡)) ∈ V) |
| 42 | 40, 41 | mp1i 13 |
. . . . . . . . . . 11
⊢ ((𝜑 ∧ 𝑡 ∈ 𝐵) → (𝑖 ∈ (1...𝑀) ↦ ((𝑈‘𝑖)‘𝑡)) ∈ V) |
| 43 | 22 | fvmpt2 7027 |
. . . . . . . . . . 11
⊢ ((𝑡 ∈ 𝑇 ∧ (𝑖 ∈ (1...𝑀) ↦ ((𝑈‘𝑖)‘𝑡)) ∈ V) → (𝐹‘𝑡) = (𝑖 ∈ (1...𝑀) ↦ ((𝑈‘𝑖)‘𝑡))) |
| 44 | 39, 42, 43 | syl2anc 584 |
. . . . . . . . . 10
⊢ ((𝜑 ∧ 𝑡 ∈ 𝐵) → (𝐹‘𝑡) = (𝑖 ∈ (1...𝑀) ↦ ((𝑈‘𝑖)‘𝑡))) |
| 45 | | stoweidlem42.9 |
. . . . . . . . . . . . . 14
⊢ (𝜑 → 𝑈:(1...𝑀)⟶𝑌) |
| 46 | 45 | ffvelcdmda 7104 |
. . . . . . . . . . . . 13
⊢ ((𝜑 ∧ 𝑖 ∈ (1...𝑀)) → (𝑈‘𝑖) ∈ 𝑌) |
| 47 | | simpl 482 |
. . . . . . . . . . . . . 14
⊢ ((𝜑 ∧ 𝑖 ∈ (1...𝑀)) → 𝜑) |
| 48 | 47, 46 | jca 511 |
. . . . . . . . . . . . 13
⊢ ((𝜑 ∧ 𝑖 ∈ (1...𝑀)) → (𝜑 ∧ (𝑈‘𝑖) ∈ 𝑌)) |
| 49 | | eleq1 2829 |
. . . . . . . . . . . . . . . 16
⊢ (𝑓 = (𝑈‘𝑖) → (𝑓 ∈ 𝑌 ↔ (𝑈‘𝑖) ∈ 𝑌)) |
| 50 | 49 | anbi2d 630 |
. . . . . . . . . . . . . . 15
⊢ (𝑓 = (𝑈‘𝑖) → ((𝜑 ∧ 𝑓 ∈ 𝑌) ↔ (𝜑 ∧ (𝑈‘𝑖) ∈ 𝑌))) |
| 51 | | feq1 6716 |
. . . . . . . . . . . . . . 15
⊢ (𝑓 = (𝑈‘𝑖) → (𝑓:𝑇⟶ℝ ↔ (𝑈‘𝑖):𝑇⟶ℝ)) |
| 52 | 50, 51 | imbi12d 344 |
. . . . . . . . . . . . . 14
⊢ (𝑓 = (𝑈‘𝑖) → (((𝜑 ∧ 𝑓 ∈ 𝑌) → 𝑓:𝑇⟶ℝ) ↔ ((𝜑 ∧ (𝑈‘𝑖) ∈ 𝑌) → (𝑈‘𝑖):𝑇⟶ℝ))) |
| 53 | | stoweidlem42.13 |
. . . . . . . . . . . . . 14
⊢ ((𝜑 ∧ 𝑓 ∈ 𝑌) → 𝑓:𝑇⟶ℝ) |
| 54 | 52, 53 | vtoclg 3554 |
. . . . . . . . . . . . 13
⊢ ((𝑈‘𝑖) ∈ 𝑌 → ((𝜑 ∧ (𝑈‘𝑖) ∈ 𝑌) → (𝑈‘𝑖):𝑇⟶ℝ)) |
| 55 | 46, 48, 54 | sylc 65 |
. . . . . . . . . . . 12
⊢ ((𝜑 ∧ 𝑖 ∈ (1...𝑀)) → (𝑈‘𝑖):𝑇⟶ℝ) |
| 56 | 55 | adantlr 715 |
. . . . . . . . . . 11
⊢ (((𝜑 ∧ 𝑡 ∈ 𝐵) ∧ 𝑖 ∈ (1...𝑀)) → (𝑈‘𝑖):𝑇⟶ℝ) |
| 57 | 39 | adantr 480 |
. . . . . . . . . . 11
⊢ (((𝜑 ∧ 𝑡 ∈ 𝐵) ∧ 𝑖 ∈ (1...𝑀)) → 𝑡 ∈ 𝑇) |
| 58 | 56, 57 | ffvelcdmd 7105 |
. . . . . . . . . 10
⊢ (((𝜑 ∧ 𝑡 ∈ 𝐵) ∧ 𝑖 ∈ (1...𝑀)) → ((𝑈‘𝑖)‘𝑡) ∈ ℝ) |
| 59 | 44, 58 | fvmpt2d 7029 |
. . . . . . . . 9
⊢ (((𝜑 ∧ 𝑡 ∈ 𝐵) ∧ 𝑖 ∈ (1...𝑀)) → ((𝐹‘𝑡)‘𝑖) = ((𝑈‘𝑖)‘𝑡)) |
| 60 | 59, 58 | eqeltrd 2841 |
. . . . . . . 8
⊢ (((𝜑 ∧ 𝑡 ∈ 𝐵) ∧ 𝑖 ∈ (1...𝑀)) → ((𝐹‘𝑡)‘𝑖) ∈ ℝ) |
| 61 | 32, 37, 60 | chvarfv 2240 |
. . . . . . 7
⊢ (((𝜑 ∧ 𝑡 ∈ 𝐵) ∧ 𝑎 ∈ (1...𝑀)) → ((𝐹‘𝑡)‘𝑎) ∈ ℝ) |
| 62 | | remulcl 11240 |
. . . . . . . 8
⊢ ((𝑎 ∈ ℝ ∧ 𝑗 ∈ ℝ) → (𝑎 · 𝑗) ∈ ℝ) |
| 63 | 62 | adantl 481 |
. . . . . . 7
⊢ (((𝜑 ∧ 𝑡 ∈ 𝐵) ∧ (𝑎 ∈ ℝ ∧ 𝑗 ∈ ℝ)) → (𝑎 · 𝑗) ∈ ℝ) |
| 64 | 16, 61, 63 | seqcl 14063 |
. . . . . 6
⊢ ((𝜑 ∧ 𝑡 ∈ 𝐵) → (seq1( · , (𝐹‘𝑡))‘𝑀) ∈ ℝ) |
| 65 | 3 | rpcnd 13079 |
. . . . . . . . . . . 12
⊢ (𝜑 → 𝐸 ∈ ℂ) |
| 66 | 7 | nncnd 12282 |
. . . . . . . . . . . 12
⊢ (𝜑 → 𝑀 ∈ ℂ) |
| 67 | 7 | nnne0d 12316 |
. . . . . . . . . . . 12
⊢ (𝜑 → 𝑀 ≠ 0) |
| 68 | 65, 66, 67 | divcan1d 12044 |
. . . . . . . . . . 11
⊢ (𝜑 → ((𝐸 / 𝑀) · 𝑀) = 𝐸) |
| 69 | 68 | eqcomd 2743 |
. . . . . . . . . 10
⊢ (𝜑 → 𝐸 = ((𝐸 / 𝑀) · 𝑀)) |
| 70 | 69 | oveq2d 7447 |
. . . . . . . . 9
⊢ (𝜑 → (1 − 𝐸) = (1 − ((𝐸 / 𝑀) · 𝑀))) |
| 71 | | 1cnd 11256 |
. . . . . . . . . 10
⊢ (𝜑 → 1 ∈
ℂ) |
| 72 | 65, 66, 67 | divcld 12043 |
. . . . . . . . . . 11
⊢ (𝜑 → (𝐸 / 𝑀) ∈ ℂ) |
| 73 | 72, 66 | mulcld 11281 |
. . . . . . . . . 10
⊢ (𝜑 → ((𝐸 / 𝑀) · 𝑀) ∈ ℂ) |
| 74 | 71, 73 | negsubd 11626 |
. . . . . . . . 9
⊢ (𝜑 → (1 + -((𝐸 / 𝑀) · 𝑀)) = (1 − ((𝐸 / 𝑀) · 𝑀))) |
| 75 | 72, 66 | mulneg1d 11716 |
. . . . . . . . . . 11
⊢ (𝜑 → (-(𝐸 / 𝑀) · 𝑀) = -((𝐸 / 𝑀) · 𝑀)) |
| 76 | 75 | eqcomd 2743 |
. . . . . . . . . 10
⊢ (𝜑 → -((𝐸 / 𝑀) · 𝑀) = (-(𝐸 / 𝑀) · 𝑀)) |
| 77 | 76 | oveq2d 7447 |
. . . . . . . . 9
⊢ (𝜑 → (1 + -((𝐸 / 𝑀) · 𝑀)) = (1 + (-(𝐸 / 𝑀) · 𝑀))) |
| 78 | 70, 74, 77 | 3eqtr2d 2783 |
. . . . . . . 8
⊢ (𝜑 → (1 − 𝐸) = (1 + (-(𝐸 / 𝑀) · 𝑀))) |
| 79 | 8 | renegcld 11690 |
. . . . . . . . . 10
⊢ (𝜑 → -(𝐸 / 𝑀) ∈ ℝ) |
| 80 | 7 | nnred 12281 |
. . . . . . . . . . . . . 14
⊢ (𝜑 → 𝑀 ∈ ℝ) |
| 81 | | 3re 12346 |
. . . . . . . . . . . . . . . . . 18
⊢ 3 ∈
ℝ |
| 82 | 81 | a1i 11 |
. . . . . . . . . . . . . . . . 17
⊢ (𝜑 → 3 ∈
ℝ) |
| 83 | | 3ne0 12372 |
. . . . . . . . . . . . . . . . . 18
⊢ 3 ≠
0 |
| 84 | 83 | a1i 11 |
. . . . . . . . . . . . . . . . 17
⊢ (𝜑 → 3 ≠ 0) |
| 85 | 82, 84 | rereccld 12094 |
. . . . . . . . . . . . . . . 16
⊢ (𝜑 → (1 / 3) ∈
ℝ) |
| 86 | | stoweidlem42.12 |
. . . . . . . . . . . . . . . 16
⊢ (𝜑 → 𝐸 < (1 / 3)) |
| 87 | | 1lt3 12439 |
. . . . . . . . . . . . . . . . . . 19
⊢ 1 <
3 |
| 88 | 87 | a1i 11 |
. . . . . . . . . . . . . . . . . 18
⊢ (𝜑 → 1 < 3) |
| 89 | | 0lt1 11785 |
. . . . . . . . . . . . . . . . . . . 20
⊢ 0 <
1 |
| 90 | 89 | a1i 11 |
. . . . . . . . . . . . . . . . . . 19
⊢ (𝜑 → 0 < 1) |
| 91 | | 3pos 12371 |
. . . . . . . . . . . . . . . . . . . 20
⊢ 0 <
3 |
| 92 | 91 | a1i 11 |
. . . . . . . . . . . . . . . . . . 19
⊢ (𝜑 → 0 < 3) |
| 93 | | ltdiv2 12154 |
. . . . . . . . . . . . . . . . . . 19
⊢ (((1
∈ ℝ ∧ 0 < 1) ∧ (3 ∈ ℝ ∧ 0 < 3) ∧ (1
∈ ℝ ∧ 0 < 1)) → (1 < 3 ↔ (1 / 3) < (1 /
1))) |
| 94 | 2, 90, 82, 92, 2, 90, 93 | syl222anc 1388 |
. . . . . . . . . . . . . . . . . 18
⊢ (𝜑 → (1 < 3 ↔ (1 / 3)
< (1 / 1))) |
| 95 | 88, 94 | mpbid 232 |
. . . . . . . . . . . . . . . . 17
⊢ (𝜑 → (1 / 3) < (1 /
1)) |
| 96 | | 1div1e1 11958 |
. . . . . . . . . . . . . . . . 17
⊢ (1 / 1) =
1 |
| 97 | 95, 96 | breqtrdi 5184 |
. . . . . . . . . . . . . . . 16
⊢ (𝜑 → (1 / 3) <
1) |
| 98 | 4, 85, 2, 86, 97 | lttrd 11422 |
. . . . . . . . . . . . . . 15
⊢ (𝜑 → 𝐸 < 1) |
| 99 | 7 | nnge1d 12314 |
. . . . . . . . . . . . . . 15
⊢ (𝜑 → 1 ≤ 𝑀) |
| 100 | 4, 2, 80, 98, 99 | ltletrd 11421 |
. . . . . . . . . . . . . 14
⊢ (𝜑 → 𝐸 < 𝑀) |
| 101 | 4, 80, 100 | ltled 11409 |
. . . . . . . . . . . . 13
⊢ (𝜑 → 𝐸 ≤ 𝑀) |
| 102 | 3 | rpregt0d 13083 |
. . . . . . . . . . . . . 14
⊢ (𝜑 → (𝐸 ∈ ℝ ∧ 0 < 𝐸)) |
| 103 | 7 | nngt0d 12315 |
. . . . . . . . . . . . . 14
⊢ (𝜑 → 0 < 𝑀) |
| 104 | | lediv2 12158 |
. . . . . . . . . . . . . 14
⊢ (((𝐸 ∈ ℝ ∧ 0 <
𝐸) ∧ (𝑀 ∈ ℝ ∧ 0 < 𝑀) ∧ (𝐸 ∈ ℝ ∧ 0 < 𝐸)) → (𝐸 ≤ 𝑀 ↔ (𝐸 / 𝑀) ≤ (𝐸 / 𝐸))) |
| 105 | 102, 80, 103, 102, 104 | syl121anc 1377 |
. . . . . . . . . . . . 13
⊢ (𝜑 → (𝐸 ≤ 𝑀 ↔ (𝐸 / 𝑀) ≤ (𝐸 / 𝐸))) |
| 106 | 101, 105 | mpbid 232 |
. . . . . . . . . . . 12
⊢ (𝜑 → (𝐸 / 𝑀) ≤ (𝐸 / 𝐸)) |
| 107 | 3 | rpcnne0d 13086 |
. . . . . . . . . . . . 13
⊢ (𝜑 → (𝐸 ∈ ℂ ∧ 𝐸 ≠ 0)) |
| 108 | | divid 11953 |
. . . . . . . . . . . . 13
⊢ ((𝐸 ∈ ℂ ∧ 𝐸 ≠ 0) → (𝐸 / 𝐸) = 1) |
| 109 | 107, 108 | syl 17 |
. . . . . . . . . . . 12
⊢ (𝜑 → (𝐸 / 𝐸) = 1) |
| 110 | 106, 109 | breqtrd 5169 |
. . . . . . . . . . 11
⊢ (𝜑 → (𝐸 / 𝑀) ≤ 1) |
| 111 | 8, 2 | lenegd 11842 |
. . . . . . . . . . 11
⊢ (𝜑 → ((𝐸 / 𝑀) ≤ 1 ↔ -1 ≤ -(𝐸 / 𝑀))) |
| 112 | 110, 111 | mpbid 232 |
. . . . . . . . . 10
⊢ (𝜑 → -1 ≤ -(𝐸 / 𝑀)) |
| 113 | | bernneq 14268 |
. . . . . . . . . 10
⊢ ((-(𝐸 / 𝑀) ∈ ℝ ∧ 𝑀 ∈ ℕ0 ∧ -1 ≤
-(𝐸 / 𝑀)) → (1 + (-(𝐸 / 𝑀) · 𝑀)) ≤ ((1 + -(𝐸 / 𝑀))↑𝑀)) |
| 114 | 79, 11, 112, 113 | syl3anc 1373 |
. . . . . . . . 9
⊢ (𝜑 → (1 + (-(𝐸 / 𝑀) · 𝑀)) ≤ ((1 + -(𝐸 / 𝑀))↑𝑀)) |
| 115 | 71, 72 | negsubd 11626 |
. . . . . . . . . 10
⊢ (𝜑 → (1 + -(𝐸 / 𝑀)) = (1 − (𝐸 / 𝑀))) |
| 116 | 115 | oveq1d 7446 |
. . . . . . . . 9
⊢ (𝜑 → ((1 + -(𝐸 / 𝑀))↑𝑀) = ((1 − (𝐸 / 𝑀))↑𝑀)) |
| 117 | 114, 116 | breqtrd 5169 |
. . . . . . . 8
⊢ (𝜑 → (1 + (-(𝐸 / 𝑀) · 𝑀)) ≤ ((1 − (𝐸 / 𝑀))↑𝑀)) |
| 118 | 78, 117 | eqbrtrd 5165 |
. . . . . . 7
⊢ (𝜑 → (1 − 𝐸) ≤ ((1 − (𝐸 / 𝑀))↑𝑀)) |
| 119 | 118 | adantr 480 |
. . . . . 6
⊢ ((𝜑 ∧ 𝑡 ∈ 𝐵) → (1 − 𝐸) ≤ ((1 − (𝐸 / 𝑀))↑𝑀)) |
| 120 | | eqid 2737 |
. . . . . . 7
⊢ seq1(
· , (𝐹‘𝑡)) = seq1( · , (𝐹‘𝑡)) |
| 121 | 7 | adantr 480 |
. . . . . . 7
⊢ ((𝜑 ∧ 𝑡 ∈ 𝐵) → 𝑀 ∈ ℕ) |
| 122 | | eqid 2737 |
. . . . . . . . 9
⊢ (𝑖 ∈ (1...𝑀) ↦ ((𝑈‘𝑖)‘𝑡)) = (𝑖 ∈ (1...𝑀) ↦ ((𝑈‘𝑖)‘𝑡)) |
| 123 | 19, 58, 122 | fmptdf 7137 |
. . . . . . . 8
⊢ ((𝜑 ∧ 𝑡 ∈ 𝐵) → (𝑖 ∈ (1...𝑀) ↦ ((𝑈‘𝑖)‘𝑡)):(1...𝑀)⟶ℝ) |
| 124 | 44 | feq1d 6720 |
. . . . . . . 8
⊢ ((𝜑 ∧ 𝑡 ∈ 𝐵) → ((𝐹‘𝑡):(1...𝑀)⟶ℝ ↔ (𝑖 ∈ (1...𝑀) ↦ ((𝑈‘𝑖)‘𝑡)):(1...𝑀)⟶ℝ)) |
| 125 | 123, 124 | mpbird 257 |
. . . . . . 7
⊢ ((𝜑 ∧ 𝑡 ∈ 𝐵) → (𝐹‘𝑡):(1...𝑀)⟶ℝ) |
| 126 | | stoweidlem42.10 |
. . . . . . . . . 10
⊢ ((𝜑 ∧ 𝑖 ∈ (1...𝑀)) → ∀𝑡 ∈ 𝐵 (1 − (𝐸 / 𝑀)) < ((𝑈‘𝑖)‘𝑡)) |
| 127 | 126 | r19.21bi 3251 |
. . . . . . . . 9
⊢ (((𝜑 ∧ 𝑖 ∈ (1...𝑀)) ∧ 𝑡 ∈ 𝐵) → (1 − (𝐸 / 𝑀)) < ((𝑈‘𝑖)‘𝑡)) |
| 128 | 127 | an32s 652 |
. . . . . . . 8
⊢ (((𝜑 ∧ 𝑡 ∈ 𝐵) ∧ 𝑖 ∈ (1...𝑀)) → (1 − (𝐸 / 𝑀)) < ((𝑈‘𝑖)‘𝑡)) |
| 129 | 128, 59 | breqtrrd 5171 |
. . . . . . 7
⊢ (((𝜑 ∧ 𝑡 ∈ 𝐵) ∧ 𝑖 ∈ (1...𝑀)) → (1 − (𝐸 / 𝑀)) < ((𝐹‘𝑡)‘𝑖)) |
| 130 | 72 | addlidd 11462 |
. . . . . . . . . . 11
⊢ (𝜑 → (0 + (𝐸 / 𝑀)) = (𝐸 / 𝑀)) |
| 131 | | lediv2 12158 |
. . . . . . . . . . . . . . 15
⊢ (((1
∈ ℝ ∧ 0 < 1) ∧ (𝑀 ∈ ℝ ∧ 0 < 𝑀) ∧ (𝐸 ∈ ℝ ∧ 0 < 𝐸)) → (1 ≤ 𝑀 ↔ (𝐸 / 𝑀) ≤ (𝐸 / 1))) |
| 132 | 2, 90, 80, 103, 102, 131 | syl221anc 1383 |
. . . . . . . . . . . . . 14
⊢ (𝜑 → (1 ≤ 𝑀 ↔ (𝐸 / 𝑀) ≤ (𝐸 / 1))) |
| 133 | 99, 132 | mpbid 232 |
. . . . . . . . . . . . 13
⊢ (𝜑 → (𝐸 / 𝑀) ≤ (𝐸 / 1)) |
| 134 | 65 | div1d 12035 |
. . . . . . . . . . . . 13
⊢ (𝜑 → (𝐸 / 1) = 𝐸) |
| 135 | 133, 134 | breqtrd 5169 |
. . . . . . . . . . . 12
⊢ (𝜑 → (𝐸 / 𝑀) ≤ 𝐸) |
| 136 | 8, 4, 2, 135, 98 | lelttrd 11419 |
. . . . . . . . . . 11
⊢ (𝜑 → (𝐸 / 𝑀) < 1) |
| 137 | 130, 136 | eqbrtrd 5165 |
. . . . . . . . . 10
⊢ (𝜑 → (0 + (𝐸 / 𝑀)) < 1) |
| 138 | | 0red 11264 |
. . . . . . . . . . 11
⊢ (𝜑 → 0 ∈
ℝ) |
| 139 | 138, 8, 2 | ltaddsubd 11863 |
. . . . . . . . . 10
⊢ (𝜑 → ((0 + (𝐸 / 𝑀)) < 1 ↔ 0 < (1 − (𝐸 / 𝑀)))) |
| 140 | 137, 139 | mpbid 232 |
. . . . . . . . 9
⊢ (𝜑 → 0 < (1 − (𝐸 / 𝑀))) |
| 141 | 9, 140 | elrpd 13074 |
. . . . . . . 8
⊢ (𝜑 → (1 − (𝐸 / 𝑀)) ∈
ℝ+) |
| 142 | 141 | adantr 480 |
. . . . . . 7
⊢ ((𝜑 ∧ 𝑡 ∈ 𝐵) → (1 − (𝐸 / 𝑀)) ∈
ℝ+) |
| 143 | 28, 19, 120, 121, 125, 129, 142 | stoweidlem3 46018 |
. . . . . 6
⊢ ((𝜑 ∧ 𝑡 ∈ 𝐵) → ((1 − (𝐸 / 𝑀))↑𝑀) < (seq1( · , (𝐹‘𝑡))‘𝑀)) |
| 144 | 6, 13, 64, 119, 143 | lelttrd 11419 |
. . . . 5
⊢ ((𝜑 ∧ 𝑡 ∈ 𝐵) → (1 − 𝐸) < (seq1( · , (𝐹‘𝑡))‘𝑀)) |
| 145 | | stoweidlem42.7 |
. . . . . . 7
⊢ 𝑍 = (𝑡 ∈ 𝑇 ↦ (seq1( · , (𝐹‘𝑡))‘𝑀)) |
| 146 | 145 | fvmpt2 7027 |
. . . . . 6
⊢ ((𝑡 ∈ 𝑇 ∧ (seq1( · , (𝐹‘𝑡))‘𝑀) ∈ ℝ) → (𝑍‘𝑡) = (seq1( · , (𝐹‘𝑡))‘𝑀)) |
| 147 | 39, 64, 146 | syl2anc 584 |
. . . . 5
⊢ ((𝜑 ∧ 𝑡 ∈ 𝐵) → (𝑍‘𝑡) = (seq1( · , (𝐹‘𝑡))‘𝑀)) |
| 148 | 144, 147 | breqtrrd 5171 |
. . . 4
⊢ ((𝜑 ∧ 𝑡 ∈ 𝐵) → (1 − 𝐸) < (𝑍‘𝑡)) |
| 149 | | simpl 482 |
. . . . 5
⊢ ((𝜑 ∧ 𝑡 ∈ 𝐵) → 𝜑) |
| 150 | | stoweidlem42.3 |
. . . . . 6
⊢
Ⅎ𝑡𝑌 |
| 151 | | stoweidlem42.4 |
. . . . . 6
⊢ 𝑃 = (𝑓 ∈ 𝑌, 𝑔 ∈ 𝑌 ↦ (𝑡 ∈ 𝑇 ↦ ((𝑓‘𝑡) · (𝑔‘𝑡)))) |
| 152 | | stoweidlem42.5 |
. . . . . 6
⊢ 𝑋 = (seq1(𝑃, 𝑈)‘𝑀) |
| 153 | | stoweidlem42.15 |
. . . . . 6
⊢ (𝜑 → 𝑇 ∈ V) |
| 154 | | stoweidlem42.14 |
. . . . . 6
⊢ ((𝜑 ∧ 𝑓 ∈ 𝑌 ∧ 𝑔 ∈ 𝑌) → (𝑡 ∈ 𝑇 ↦ ((𝑓‘𝑡) · (𝑔‘𝑡))) ∈ 𝑌) |
| 155 | 17, 150, 151, 152, 22, 145, 153, 7, 45, 53, 154 | fmuldfeq 45598 |
. . . . 5
⊢ ((𝜑 ∧ 𝑡 ∈ 𝑇) → (𝑋‘𝑡) = (𝑍‘𝑡)) |
| 156 | 149, 39, 155 | syl2anc 584 |
. . . 4
⊢ ((𝜑 ∧ 𝑡 ∈ 𝐵) → (𝑋‘𝑡) = (𝑍‘𝑡)) |
| 157 | 148, 156 | breqtrrd 5171 |
. . 3
⊢ ((𝜑 ∧ 𝑡 ∈ 𝐵) → (1 − 𝐸) < (𝑋‘𝑡)) |
| 158 | 157 | ex 412 |
. 2
⊢ (𝜑 → (𝑡 ∈ 𝐵 → (1 − 𝐸) < (𝑋‘𝑡))) |
| 159 | 1, 158 | ralrimi 3257 |
1
⊢ (𝜑 → ∀𝑡 ∈ 𝐵 (1 − 𝐸) < (𝑋‘𝑡)) |