Proof of Theorem stoweidlem45
| Step | Hyp | Ref
| Expression |
| 1 | | stoweidlem45.1 |
. . 3
⊢
Ⅎ𝑡𝑃 |
| 2 | | stoweidlem45.2 |
. . 3
⊢
Ⅎ𝑡𝜑 |
| 3 | | stoweidlem45.4 |
. . 3
⊢ 𝑄 = (𝑡 ∈ 𝑇 ↦ ((1 − ((𝑃‘𝑡)↑𝑁))↑(𝐾↑𝑁))) |
| 4 | | eqid 2737 |
. . 3
⊢ (𝑡 ∈ 𝑇 ↦ (1 − ((𝑃‘𝑡)↑𝑁))) = (𝑡 ∈ 𝑇 ↦ (1 − ((𝑃‘𝑡)↑𝑁))) |
| 5 | | eqid 2737 |
. . 3
⊢ (𝑡 ∈ 𝑇 ↦ 1) = (𝑡 ∈ 𝑇 ↦ 1) |
| 6 | | eqid 2737 |
. . 3
⊢ (𝑡 ∈ 𝑇 ↦ ((𝑃‘𝑡)↑𝑁)) = (𝑡 ∈ 𝑇 ↦ ((𝑃‘𝑡)↑𝑁)) |
| 7 | | stoweidlem45.9 |
. . 3
⊢ (𝜑 → 𝑃 ∈ 𝐴) |
| 8 | | stoweidlem45.10 |
. . 3
⊢ (𝜑 → 𝑃:𝑇⟶ℝ) |
| 9 | | stoweidlem45.13 |
. . 3
⊢ ((𝜑 ∧ 𝑓 ∈ 𝐴) → 𝑓:𝑇⟶ℝ) |
| 10 | | stoweidlem45.14 |
. . 3
⊢ ((𝜑 ∧ 𝑓 ∈ 𝐴 ∧ 𝑔 ∈ 𝐴) → (𝑡 ∈ 𝑇 ↦ ((𝑓‘𝑡) + (𝑔‘𝑡))) ∈ 𝐴) |
| 11 | | stoweidlem45.15 |
. . 3
⊢ ((𝜑 ∧ 𝑓 ∈ 𝐴 ∧ 𝑔 ∈ 𝐴) → (𝑡 ∈ 𝑇 ↦ ((𝑓‘𝑡) · (𝑔‘𝑡))) ∈ 𝐴) |
| 12 | | stoweidlem45.16 |
. . 3
⊢ ((𝜑 ∧ 𝑥 ∈ ℝ) → (𝑡 ∈ 𝑇 ↦ 𝑥) ∈ 𝐴) |
| 13 | | stoweidlem45.5 |
. . 3
⊢ (𝜑 → 𝑁 ∈ ℕ) |
| 14 | | stoweidlem45.6 |
. . . 4
⊢ (𝜑 → 𝐾 ∈ ℕ) |
| 15 | 13 | nnnn0d 12587 |
. . . 4
⊢ (𝜑 → 𝑁 ∈
ℕ0) |
| 16 | 14, 15 | nnexpcld 14284 |
. . 3
⊢ (𝜑 → (𝐾↑𝑁) ∈ ℕ) |
| 17 | 1, 2, 3, 4, 5, 6, 7, 8, 9, 10,
11, 12, 13, 16 | stoweidlem40 46055 |
. 2
⊢ (𝜑 → 𝑄 ∈ 𝐴) |
| 18 | | 1red 11262 |
. . . . . . . 8
⊢ ((𝜑 ∧ 𝑡 ∈ 𝑇) → 1 ∈ ℝ) |
| 19 | 8 | ffvelcdmda 7104 |
. . . . . . . . 9
⊢ ((𝜑 ∧ 𝑡 ∈ 𝑇) → (𝑃‘𝑡) ∈ ℝ) |
| 20 | 15 | adantr 480 |
. . . . . . . . 9
⊢ ((𝜑 ∧ 𝑡 ∈ 𝑇) → 𝑁 ∈
ℕ0) |
| 21 | 19, 20 | reexpcld 14203 |
. . . . . . . 8
⊢ ((𝜑 ∧ 𝑡 ∈ 𝑇) → ((𝑃‘𝑡)↑𝑁) ∈ ℝ) |
| 22 | 18, 21 | resubcld 11691 |
. . . . . . 7
⊢ ((𝜑 ∧ 𝑡 ∈ 𝑇) → (1 − ((𝑃‘𝑡)↑𝑁)) ∈ ℝ) |
| 23 | 14 | nnnn0d 12587 |
. . . . . . . . 9
⊢ (𝜑 → 𝐾 ∈
ℕ0) |
| 24 | 23, 15 | nn0expcld 14285 |
. . . . . . . 8
⊢ (𝜑 → (𝐾↑𝑁) ∈
ℕ0) |
| 25 | 24 | adantr 480 |
. . . . . . 7
⊢ ((𝜑 ∧ 𝑡 ∈ 𝑇) → (𝐾↑𝑁) ∈
ℕ0) |
| 26 | | 1m1e0 12338 |
. . . . . . . 8
⊢ (1
− 1) = 0 |
| 27 | | stoweidlem45.11 |
. . . . . . . . . . . 12
⊢ (𝜑 → ∀𝑡 ∈ 𝑇 (0 ≤ (𝑃‘𝑡) ∧ (𝑃‘𝑡) ≤ 1)) |
| 28 | 27 | r19.21bi 3251 |
. . . . . . . . . . 11
⊢ ((𝜑 ∧ 𝑡 ∈ 𝑇) → (0 ≤ (𝑃‘𝑡) ∧ (𝑃‘𝑡) ≤ 1)) |
| 29 | 28 | simpld 494 |
. . . . . . . . . 10
⊢ ((𝜑 ∧ 𝑡 ∈ 𝑇) → 0 ≤ (𝑃‘𝑡)) |
| 30 | 28 | simprd 495 |
. . . . . . . . . 10
⊢ ((𝜑 ∧ 𝑡 ∈ 𝑇) → (𝑃‘𝑡) ≤ 1) |
| 31 | | exple1 14216 |
. . . . . . . . . 10
⊢ ((((𝑃‘𝑡) ∈ ℝ ∧ 0 ≤ (𝑃‘𝑡) ∧ (𝑃‘𝑡) ≤ 1) ∧ 𝑁 ∈ ℕ0) → ((𝑃‘𝑡)↑𝑁) ≤ 1) |
| 32 | 19, 29, 30, 20, 31 | syl31anc 1375 |
. . . . . . . . 9
⊢ ((𝜑 ∧ 𝑡 ∈ 𝑇) → ((𝑃‘𝑡)↑𝑁) ≤ 1) |
| 33 | 21, 18, 18, 32 | lesub2dd 11880 |
. . . . . . . 8
⊢ ((𝜑 ∧ 𝑡 ∈ 𝑇) → (1 − 1) ≤ (1 −
((𝑃‘𝑡)↑𝑁))) |
| 34 | 26, 33 | eqbrtrrid 5179 |
. . . . . . 7
⊢ ((𝜑 ∧ 𝑡 ∈ 𝑇) → 0 ≤ (1 − ((𝑃‘𝑡)↑𝑁))) |
| 35 | 22, 25, 34 | expge0d 14204 |
. . . . . 6
⊢ ((𝜑 ∧ 𝑡 ∈ 𝑇) → 0 ≤ ((1 − ((𝑃‘𝑡)↑𝑁))↑(𝐾↑𝑁))) |
| 36 | 3, 8, 15, 23 | stoweidlem12 46027 |
. . . . . 6
⊢ ((𝜑 ∧ 𝑡 ∈ 𝑇) → (𝑄‘𝑡) = ((1 − ((𝑃‘𝑡)↑𝑁))↑(𝐾↑𝑁))) |
| 37 | 35, 36 | breqtrrd 5171 |
. . . . 5
⊢ ((𝜑 ∧ 𝑡 ∈ 𝑇) → 0 ≤ (𝑄‘𝑡)) |
| 38 | | 0red 11264 |
. . . . . . . . 9
⊢ ((𝜑 ∧ 𝑡 ∈ 𝑇) → 0 ∈ ℝ) |
| 39 | 19, 20, 29 | expge0d 14204 |
. . . . . . . . 9
⊢ ((𝜑 ∧ 𝑡 ∈ 𝑇) → 0 ≤ ((𝑃‘𝑡)↑𝑁)) |
| 40 | 38, 21, 18, 39 | lesub2dd 11880 |
. . . . . . . 8
⊢ ((𝜑 ∧ 𝑡 ∈ 𝑇) → (1 − ((𝑃‘𝑡)↑𝑁)) ≤ (1 − 0)) |
| 41 | | 1m0e1 12387 |
. . . . . . . 8
⊢ (1
− 0) = 1 |
| 42 | 40, 41 | breqtrdi 5184 |
. . . . . . 7
⊢ ((𝜑 ∧ 𝑡 ∈ 𝑇) → (1 − ((𝑃‘𝑡)↑𝑁)) ≤ 1) |
| 43 | | exple1 14216 |
. . . . . . 7
⊢ ((((1
− ((𝑃‘𝑡)↑𝑁)) ∈ ℝ ∧ 0 ≤ (1 −
((𝑃‘𝑡)↑𝑁)) ∧ (1 − ((𝑃‘𝑡)↑𝑁)) ≤ 1) ∧ (𝐾↑𝑁) ∈ ℕ0) → ((1
− ((𝑃‘𝑡)↑𝑁))↑(𝐾↑𝑁)) ≤ 1) |
| 44 | 22, 34, 42, 25, 43 | syl31anc 1375 |
. . . . . 6
⊢ ((𝜑 ∧ 𝑡 ∈ 𝑇) → ((1 − ((𝑃‘𝑡)↑𝑁))↑(𝐾↑𝑁)) ≤ 1) |
| 45 | 36, 44 | eqbrtrd 5165 |
. . . . 5
⊢ ((𝜑 ∧ 𝑡 ∈ 𝑇) → (𝑄‘𝑡) ≤ 1) |
| 46 | 37, 45 | jca 511 |
. . . 4
⊢ ((𝜑 ∧ 𝑡 ∈ 𝑇) → (0 ≤ (𝑄‘𝑡) ∧ (𝑄‘𝑡) ≤ 1)) |
| 47 | 46 | ex 412 |
. . 3
⊢ (𝜑 → (𝑡 ∈ 𝑇 → (0 ≤ (𝑄‘𝑡) ∧ (𝑄‘𝑡) ≤ 1))) |
| 48 | 2, 47 | ralrimi 3257 |
. 2
⊢ (𝜑 → ∀𝑡 ∈ 𝑇 (0 ≤ (𝑄‘𝑡) ∧ (𝑄‘𝑡) ≤ 1)) |
| 49 | | stoweidlem45.3 |
. . . . 5
⊢ 𝑉 = {𝑡 ∈ 𝑇 ∣ (𝑃‘𝑡) < (𝐷 / 2)} |
| 50 | | stoweidlem45.7 |
. . . . 5
⊢ (𝜑 → 𝐷 ∈
ℝ+) |
| 51 | | stoweidlem45.17 |
. . . . 5
⊢ (𝜑 → 𝐸 ∈
ℝ+) |
| 52 | | stoweidlem45.18 |
. . . . 5
⊢ (𝜑 → (1 − 𝐸) < (1 − (((𝐾 · 𝐷) / 2)↑𝑁))) |
| 53 | 49, 3, 8, 15, 23, 50, 51, 52, 27 | stoweidlem24 46039 |
. . . 4
⊢ ((𝜑 ∧ 𝑡 ∈ 𝑉) → (1 − 𝐸) < (𝑄‘𝑡)) |
| 54 | 53 | ex 412 |
. . 3
⊢ (𝜑 → (𝑡 ∈ 𝑉 → (1 − 𝐸) < (𝑄‘𝑡))) |
| 55 | 2, 54 | ralrimi 3257 |
. 2
⊢ (𝜑 → ∀𝑡 ∈ 𝑉 (1 − 𝐸) < (𝑄‘𝑡)) |
| 56 | | stoweidlem45.12 |
. . . . 5
⊢ (𝜑 → ∀𝑡 ∈ (𝑇 ∖ 𝑈)𝐷 ≤ (𝑃‘𝑡)) |
| 57 | | stoweidlem45.19 |
. . . . 5
⊢ (𝜑 → (1 / ((𝐾 · 𝐷)↑𝑁)) < 𝐸) |
| 58 | 3, 13, 14, 50, 8, 27, 56, 51, 57 | stoweidlem25 46040 |
. . . 4
⊢ ((𝜑 ∧ 𝑡 ∈ (𝑇 ∖ 𝑈)) → (𝑄‘𝑡) < 𝐸) |
| 59 | 58 | ex 412 |
. . 3
⊢ (𝜑 → (𝑡 ∈ (𝑇 ∖ 𝑈) → (𝑄‘𝑡) < 𝐸)) |
| 60 | 2, 59 | ralrimi 3257 |
. 2
⊢ (𝜑 → ∀𝑡 ∈ (𝑇 ∖ 𝑈)(𝑄‘𝑡) < 𝐸) |
| 61 | | nfmpt1 5250 |
. . . . . . 7
⊢
Ⅎ𝑡(𝑡 ∈ 𝑇 ↦ ((1 − ((𝑃‘𝑡)↑𝑁))↑(𝐾↑𝑁))) |
| 62 | 3, 61 | nfcxfr 2903 |
. . . . . 6
⊢
Ⅎ𝑡𝑄 |
| 63 | 62 | nfeq2 2923 |
. . . . 5
⊢
Ⅎ𝑡 𝑦 = 𝑄 |
| 64 | | fveq1 6905 |
. . . . . . 7
⊢ (𝑦 = 𝑄 → (𝑦‘𝑡) = (𝑄‘𝑡)) |
| 65 | 64 | breq2d 5155 |
. . . . . 6
⊢ (𝑦 = 𝑄 → (0 ≤ (𝑦‘𝑡) ↔ 0 ≤ (𝑄‘𝑡))) |
| 66 | 64 | breq1d 5153 |
. . . . . 6
⊢ (𝑦 = 𝑄 → ((𝑦‘𝑡) ≤ 1 ↔ (𝑄‘𝑡) ≤ 1)) |
| 67 | 65, 66 | anbi12d 632 |
. . . . 5
⊢ (𝑦 = 𝑄 → ((0 ≤ (𝑦‘𝑡) ∧ (𝑦‘𝑡) ≤ 1) ↔ (0 ≤ (𝑄‘𝑡) ∧ (𝑄‘𝑡) ≤ 1))) |
| 68 | 63, 67 | ralbid 3273 |
. . . 4
⊢ (𝑦 = 𝑄 → (∀𝑡 ∈ 𝑇 (0 ≤ (𝑦‘𝑡) ∧ (𝑦‘𝑡) ≤ 1) ↔ ∀𝑡 ∈ 𝑇 (0 ≤ (𝑄‘𝑡) ∧ (𝑄‘𝑡) ≤ 1))) |
| 69 | 64 | breq2d 5155 |
. . . . 5
⊢ (𝑦 = 𝑄 → ((1 − 𝐸) < (𝑦‘𝑡) ↔ (1 − 𝐸) < (𝑄‘𝑡))) |
| 70 | 63, 69 | ralbid 3273 |
. . . 4
⊢ (𝑦 = 𝑄 → (∀𝑡 ∈ 𝑉 (1 − 𝐸) < (𝑦‘𝑡) ↔ ∀𝑡 ∈ 𝑉 (1 − 𝐸) < (𝑄‘𝑡))) |
| 71 | 64 | breq1d 5153 |
. . . . 5
⊢ (𝑦 = 𝑄 → ((𝑦‘𝑡) < 𝐸 ↔ (𝑄‘𝑡) < 𝐸)) |
| 72 | 63, 71 | ralbid 3273 |
. . . 4
⊢ (𝑦 = 𝑄 → (∀𝑡 ∈ (𝑇 ∖ 𝑈)(𝑦‘𝑡) < 𝐸 ↔ ∀𝑡 ∈ (𝑇 ∖ 𝑈)(𝑄‘𝑡) < 𝐸)) |
| 73 | 68, 70, 72 | 3anbi123d 1438 |
. . 3
⊢ (𝑦 = 𝑄 → ((∀𝑡 ∈ 𝑇 (0 ≤ (𝑦‘𝑡) ∧ (𝑦‘𝑡) ≤ 1) ∧ ∀𝑡 ∈ 𝑉 (1 − 𝐸) < (𝑦‘𝑡) ∧ ∀𝑡 ∈ (𝑇 ∖ 𝑈)(𝑦‘𝑡) < 𝐸) ↔ (∀𝑡 ∈ 𝑇 (0 ≤ (𝑄‘𝑡) ∧ (𝑄‘𝑡) ≤ 1) ∧ ∀𝑡 ∈ 𝑉 (1 − 𝐸) < (𝑄‘𝑡) ∧ ∀𝑡 ∈ (𝑇 ∖ 𝑈)(𝑄‘𝑡) < 𝐸))) |
| 74 | 73 | rspcev 3622 |
. 2
⊢ ((𝑄 ∈ 𝐴 ∧ (∀𝑡 ∈ 𝑇 (0 ≤ (𝑄‘𝑡) ∧ (𝑄‘𝑡) ≤ 1) ∧ ∀𝑡 ∈ 𝑉 (1 − 𝐸) < (𝑄‘𝑡) ∧ ∀𝑡 ∈ (𝑇 ∖ 𝑈)(𝑄‘𝑡) < 𝐸)) → ∃𝑦 ∈ 𝐴 (∀𝑡 ∈ 𝑇 (0 ≤ (𝑦‘𝑡) ∧ (𝑦‘𝑡) ≤ 1) ∧ ∀𝑡 ∈ 𝑉 (1 − 𝐸) < (𝑦‘𝑡) ∧ ∀𝑡 ∈ (𝑇 ∖ 𝑈)(𝑦‘𝑡) < 𝐸)) |
| 75 | 17, 48, 55, 60, 74 | syl13anc 1374 |
1
⊢ (𝜑 → ∃𝑦 ∈ 𝐴 (∀𝑡 ∈ 𝑇 (0 ≤ (𝑦‘𝑡) ∧ (𝑦‘𝑡) ≤ 1) ∧ ∀𝑡 ∈ 𝑉 (1 − 𝐸) < (𝑦‘𝑡) ∧ ∀𝑡 ∈ (𝑇 ∖ 𝑈)(𝑦‘𝑡) < 𝐸)) |