| Step | Hyp | Ref
| Expression |
| 1 | | stoweidlem14.1 |
. . . . . 6
⊢ 𝐴 = {𝑗 ∈ ℕ ∣ (1 / 𝐷) < 𝑗} |
| 2 | | ssrab2 4080 |
. . . . . . 7
⊢ {𝑗 ∈ ℕ ∣ (1 /
𝐷) < 𝑗} ⊆ ℕ |
| 3 | 2 | a1i 11 |
. . . . . 6
⊢ (𝜑 → {𝑗 ∈ ℕ ∣ (1 / 𝐷) < 𝑗} ⊆ ℕ) |
| 4 | 1, 3 | eqsstrid 4022 |
. . . . 5
⊢ (𝜑 → 𝐴 ⊆ ℕ) |
| 5 | | stoweidlem14.2 |
. . . . . . 7
⊢ (𝜑 → 𝐷 ∈
ℝ+) |
| 6 | 5 | rprecred 13088 |
. . . . . 6
⊢ (𝜑 → (1 / 𝐷) ∈ ℝ) |
| 7 | | arch 12523 |
. . . . . 6
⊢ ((1 /
𝐷) ∈ ℝ →
∃𝑘 ∈ ℕ (1
/ 𝐷) < 𝑘) |
| 8 | | breq2 5147 |
. . . . . . . . . . 11
⊢ (𝑗 = 𝑘 → ((1 / 𝐷) < 𝑗 ↔ (1 / 𝐷) < 𝑘)) |
| 9 | 8 | elrab 3692 |
. . . . . . . . . 10
⊢ (𝑘 ∈ {𝑗 ∈ ℕ ∣ (1 / 𝐷) < 𝑗} ↔ (𝑘 ∈ ℕ ∧ (1 / 𝐷) < 𝑘)) |
| 10 | 9 | biimpri 228 |
. . . . . . . . 9
⊢ ((𝑘 ∈ ℕ ∧ (1 / 𝐷) < 𝑘) → 𝑘 ∈ {𝑗 ∈ ℕ ∣ (1 / 𝐷) < 𝑗}) |
| 11 | 10, 1 | eleqtrrdi 2852 |
. . . . . . . 8
⊢ ((𝑘 ∈ ℕ ∧ (1 / 𝐷) < 𝑘) → 𝑘 ∈ 𝐴) |
| 12 | | simpr 484 |
. . . . . . . 8
⊢ ((𝑘 ∈ ℕ ∧ (1 / 𝐷) < 𝑘) → (1 / 𝐷) < 𝑘) |
| 13 | 11, 12 | jca 511 |
. . . . . . 7
⊢ ((𝑘 ∈ ℕ ∧ (1 / 𝐷) < 𝑘) → (𝑘 ∈ 𝐴 ∧ (1 / 𝐷) < 𝑘)) |
| 14 | 13 | reximi2 3079 |
. . . . . 6
⊢
(∃𝑘 ∈
ℕ (1 / 𝐷) < 𝑘 → ∃𝑘 ∈ 𝐴 (1 / 𝐷) < 𝑘) |
| 15 | | rexn0 4511 |
. . . . . 6
⊢
(∃𝑘 ∈
𝐴 (1 / 𝐷) < 𝑘 → 𝐴 ≠ ∅) |
| 16 | 6, 7, 14, 15 | 4syl 19 |
. . . . 5
⊢ (𝜑 → 𝐴 ≠ ∅) |
| 17 | | nnwo 12955 |
. . . . 5
⊢ ((𝐴 ⊆ ℕ ∧ 𝐴 ≠ ∅) →
∃𝑘 ∈ 𝐴 ∀𝑧 ∈ 𝐴 𝑘 ≤ 𝑧) |
| 18 | 4, 16, 17 | syl2anc 584 |
. . . 4
⊢ (𝜑 → ∃𝑘 ∈ 𝐴 ∀𝑧 ∈ 𝐴 𝑘 ≤ 𝑧) |
| 19 | | df-rex 3071 |
. . . 4
⊢
(∃𝑘 ∈
𝐴 ∀𝑧 ∈ 𝐴 𝑘 ≤ 𝑧 ↔ ∃𝑘(𝑘 ∈ 𝐴 ∧ ∀𝑧 ∈ 𝐴 𝑘 ≤ 𝑧)) |
| 20 | 18, 19 | sylib 218 |
. . 3
⊢ (𝜑 → ∃𝑘(𝑘 ∈ 𝐴 ∧ ∀𝑧 ∈ 𝐴 𝑘 ≤ 𝑧)) |
| 21 | 8, 1 | elrab2 3695 |
. . . . . . . 8
⊢ (𝑘 ∈ 𝐴 ↔ (𝑘 ∈ ℕ ∧ (1 / 𝐷) < 𝑘)) |
| 22 | 21 | simplbi 497 |
. . . . . . 7
⊢ (𝑘 ∈ 𝐴 → 𝑘 ∈ ℕ) |
| 23 | 22 | ad2antrl 728 |
. . . . . 6
⊢ ((𝜑 ∧ (𝑘 ∈ 𝐴 ∧ ∀𝑧 ∈ 𝐴 𝑘 ≤ 𝑧)) → 𝑘 ∈ ℕ) |
| 24 | | simpl 482 |
. . . . . . 7
⊢ ((𝜑 ∧ (𝑘 ∈ 𝐴 ∧ ∀𝑧 ∈ 𝐴 𝑘 ≤ 𝑧)) → 𝜑) |
| 25 | | simprl 771 |
. . . . . . 7
⊢ ((𝜑 ∧ (𝑘 ∈ 𝐴 ∧ ∀𝑧 ∈ 𝐴 𝑘 ≤ 𝑧)) → 𝑘 ∈ 𝐴) |
| 26 | | simprr 773 |
. . . . . . . 8
⊢ ((𝜑 ∧ (𝑘 ∈ 𝐴 ∧ ∀𝑧 ∈ 𝐴 𝑘 ≤ 𝑧)) → ∀𝑧 ∈ 𝐴 𝑘 ≤ 𝑧) |
| 27 | | nfcv 2905 |
. . . . . . . . 9
⊢
Ⅎ𝑧𝐴 |
| 28 | | nfrab1 3457 |
. . . . . . . . . 10
⊢
Ⅎ𝑗{𝑗 ∈ ℕ ∣ (1 / 𝐷) < 𝑗} |
| 29 | 1, 28 | nfcxfr 2903 |
. . . . . . . . 9
⊢
Ⅎ𝑗𝐴 |
| 30 | | nfv 1914 |
. . . . . . . . 9
⊢
Ⅎ𝑗 𝑘 ≤ 𝑧 |
| 31 | | nfv 1914 |
. . . . . . . . 9
⊢
Ⅎ𝑧 𝑘 ≤ 𝑗 |
| 32 | | breq2 5147 |
. . . . . . . . 9
⊢ (𝑧 = 𝑗 → (𝑘 ≤ 𝑧 ↔ 𝑘 ≤ 𝑗)) |
| 33 | 27, 29, 30, 31, 32 | cbvralfw 3304 |
. . . . . . . 8
⊢
(∀𝑧 ∈
𝐴 𝑘 ≤ 𝑧 ↔ ∀𝑗 ∈ 𝐴 𝑘 ≤ 𝑗) |
| 34 | 26, 33 | sylib 218 |
. . . . . . 7
⊢ ((𝜑 ∧ (𝑘 ∈ 𝐴 ∧ ∀𝑧 ∈ 𝐴 𝑘 ≤ 𝑧)) → ∀𝑗 ∈ 𝐴 𝑘 ≤ 𝑗) |
| 35 | 21 | simprbi 496 |
. . . . . . . . 9
⊢ (𝑘 ∈ 𝐴 → (1 / 𝐷) < 𝑘) |
| 36 | 35 | ad2antrl 728 |
. . . . . . . 8
⊢ ((𝜑 ∧ (𝑘 ∈ 𝐴 ∧ ∀𝑗 ∈ 𝐴 𝑘 ≤ 𝑗)) → (1 / 𝐷) < 𝑘) |
| 37 | 22 | ad2antrl 728 |
. . . . . . . . 9
⊢ ((𝜑 ∧ (𝑘 ∈ 𝐴 ∧ ∀𝑗 ∈ 𝐴 𝑘 ≤ 𝑗)) → 𝑘 ∈ ℕ) |
| 38 | | 1red 11262 |
. . . . . . . . . 10
⊢ ((𝜑 ∧ 𝑘 ∈ ℕ) → 1 ∈
ℝ) |
| 39 | | nnre 12273 |
. . . . . . . . . . 11
⊢ (𝑘 ∈ ℕ → 𝑘 ∈
ℝ) |
| 40 | 39 | adantl 481 |
. . . . . . . . . 10
⊢ ((𝜑 ∧ 𝑘 ∈ ℕ) → 𝑘 ∈ ℝ) |
| 41 | 5 | rpregt0d 13083 |
. . . . . . . . . . 11
⊢ (𝜑 → (𝐷 ∈ ℝ ∧ 0 < 𝐷)) |
| 42 | 41 | adantr 480 |
. . . . . . . . . 10
⊢ ((𝜑 ∧ 𝑘 ∈ ℕ) → (𝐷 ∈ ℝ ∧ 0 < 𝐷)) |
| 43 | | ltdivmul2 12145 |
. . . . . . . . . 10
⊢ ((1
∈ ℝ ∧ 𝑘
∈ ℝ ∧ (𝐷
∈ ℝ ∧ 0 < 𝐷)) → ((1 / 𝐷) < 𝑘 ↔ 1 < (𝑘 · 𝐷))) |
| 44 | 38, 40, 42, 43 | syl3anc 1373 |
. . . . . . . . 9
⊢ ((𝜑 ∧ 𝑘 ∈ ℕ) → ((1 / 𝐷) < 𝑘 ↔ 1 < (𝑘 · 𝐷))) |
| 45 | 37, 44 | syldan 591 |
. . . . . . . 8
⊢ ((𝜑 ∧ (𝑘 ∈ 𝐴 ∧ ∀𝑗 ∈ 𝐴 𝑘 ≤ 𝑗)) → ((1 / 𝐷) < 𝑘 ↔ 1 < (𝑘 · 𝐷))) |
| 46 | 36, 45 | mpbid 232 |
. . . . . . 7
⊢ ((𝜑 ∧ (𝑘 ∈ 𝐴 ∧ ∀𝑗 ∈ 𝐴 𝑘 ≤ 𝑗)) → 1 < (𝑘 · 𝐷)) |
| 47 | 24, 25, 34, 46 | syl12anc 837 |
. . . . . 6
⊢ ((𝜑 ∧ (𝑘 ∈ 𝐴 ∧ ∀𝑧 ∈ 𝐴 𝑘 ≤ 𝑧)) → 1 < (𝑘 · 𝐷)) |
| 48 | | oveq1 7438 |
. . . . . . . . . . . 12
⊢ (𝑘 = 1 → (𝑘 · 𝐷) = (1 · 𝐷)) |
| 49 | 48 | adantl 481 |
. . . . . . . . . . 11
⊢ ((𝜑 ∧ 𝑘 = 1) → (𝑘 · 𝐷) = (1 · 𝐷)) |
| 50 | 5 | rpcnd 13079 |
. . . . . . . . . . . . 13
⊢ (𝜑 → 𝐷 ∈ ℂ) |
| 51 | 50 | adantr 480 |
. . . . . . . . . . . 12
⊢ ((𝜑 ∧ 𝑘 = 1) → 𝐷 ∈ ℂ) |
| 52 | 51 | mullidd 11279 |
. . . . . . . . . . 11
⊢ ((𝜑 ∧ 𝑘 = 1) → (1 · 𝐷) = 𝐷) |
| 53 | 49, 52 | eqtrd 2777 |
. . . . . . . . . 10
⊢ ((𝜑 ∧ 𝑘 = 1) → (𝑘 · 𝐷) = 𝐷) |
| 54 | 53 | oveq1d 7446 |
. . . . . . . . 9
⊢ ((𝜑 ∧ 𝑘 = 1) → ((𝑘 · 𝐷) / 2) = (𝐷 / 2)) |
| 55 | 5 | rpred 13077 |
. . . . . . . . . . . 12
⊢ (𝜑 → 𝐷 ∈ ℝ) |
| 56 | 55 | rehalfcld 12513 |
. . . . . . . . . . 11
⊢ (𝜑 → (𝐷 / 2) ∈ ℝ) |
| 57 | | halfre 12480 |
. . . . . . . . . . . 12
⊢ (1 / 2)
∈ ℝ |
| 58 | 57 | a1i 11 |
. . . . . . . . . . 11
⊢ (𝜑 → (1 / 2) ∈
ℝ) |
| 59 | | 1red 11262 |
. . . . . . . . . . 11
⊢ (𝜑 → 1 ∈
ℝ) |
| 60 | | stoweidlem14.3 |
. . . . . . . . . . . 12
⊢ (𝜑 → 𝐷 < 1) |
| 61 | | 2re 12340 |
. . . . . . . . . . . . . 14
⊢ 2 ∈
ℝ |
| 62 | 61 | a1i 11 |
. . . . . . . . . . . . 13
⊢ (𝜑 → 2 ∈
ℝ) |
| 63 | | 2pos 12369 |
. . . . . . . . . . . . . 14
⊢ 0 <
2 |
| 64 | 63 | a1i 11 |
. . . . . . . . . . . . 13
⊢ (𝜑 → 0 < 2) |
| 65 | | ltdiv1 12132 |
. . . . . . . . . . . . 13
⊢ ((𝐷 ∈ ℝ ∧ 1 ∈
ℝ ∧ (2 ∈ ℝ ∧ 0 < 2)) → (𝐷 < 1 ↔ (𝐷 / 2) < (1 / 2))) |
| 66 | 55, 59, 62, 64, 65 | syl112anc 1376 |
. . . . . . . . . . . 12
⊢ (𝜑 → (𝐷 < 1 ↔ (𝐷 / 2) < (1 / 2))) |
| 67 | 60, 66 | mpbid 232 |
. . . . . . . . . . 11
⊢ (𝜑 → (𝐷 / 2) < (1 / 2)) |
| 68 | | halflt1 12484 |
. . . . . . . . . . . 12
⊢ (1 / 2)
< 1 |
| 69 | 68 | a1i 11 |
. . . . . . . . . . 11
⊢ (𝜑 → (1 / 2) <
1) |
| 70 | 56, 58, 59, 67, 69 | lttrd 11422 |
. . . . . . . . . 10
⊢ (𝜑 → (𝐷 / 2) < 1) |
| 71 | 70 | adantr 480 |
. . . . . . . . 9
⊢ ((𝜑 ∧ 𝑘 = 1) → (𝐷 / 2) < 1) |
| 72 | 54, 71 | eqbrtrd 5165 |
. . . . . . . 8
⊢ ((𝜑 ∧ 𝑘 = 1) → ((𝑘 · 𝐷) / 2) < 1) |
| 73 | 72 | adantlr 715 |
. . . . . . 7
⊢ (((𝜑 ∧ (𝑘 ∈ 𝐴 ∧ ∀𝑧 ∈ 𝐴 𝑘 ≤ 𝑧)) ∧ 𝑘 = 1) → ((𝑘 · 𝐷) / 2) < 1) |
| 74 | | simpll 767 |
. . . . . . . 8
⊢ (((𝜑 ∧ (𝑘 ∈ 𝐴 ∧ ∀𝑧 ∈ 𝐴 𝑘 ≤ 𝑧)) ∧ ¬ 𝑘 = 1) → 𝜑) |
| 75 | | simplrl 777 |
. . . . . . . . . 10
⊢ (((𝜑 ∧ (𝑘 ∈ 𝐴 ∧ ∀𝑧 ∈ 𝐴 𝑘 ≤ 𝑧)) ∧ ¬ 𝑘 = 1) → 𝑘 ∈ 𝐴) |
| 76 | 75, 22 | syl 17 |
. . . . . . . . 9
⊢ (((𝜑 ∧ (𝑘 ∈ 𝐴 ∧ ∀𝑧 ∈ 𝐴 𝑘 ≤ 𝑧)) ∧ ¬ 𝑘 = 1) → 𝑘 ∈ ℕ) |
| 77 | | neqne 2948 |
. . . . . . . . . 10
⊢ (¬
𝑘 = 1 → 𝑘 ≠ 1) |
| 78 | 77 | adantl 481 |
. . . . . . . . 9
⊢ (((𝜑 ∧ (𝑘 ∈ 𝐴 ∧ ∀𝑧 ∈ 𝐴 𝑘 ≤ 𝑧)) ∧ ¬ 𝑘 = 1) → 𝑘 ≠ 1) |
| 79 | | eluz2b3 12964 |
. . . . . . . . 9
⊢ (𝑘 ∈
(ℤ≥‘2) ↔ (𝑘 ∈ ℕ ∧ 𝑘 ≠ 1)) |
| 80 | 76, 78, 79 | sylanbrc 583 |
. . . . . . . 8
⊢ (((𝜑 ∧ (𝑘 ∈ 𝐴 ∧ ∀𝑧 ∈ 𝐴 𝑘 ≤ 𝑧)) ∧ ¬ 𝑘 = 1) → 𝑘 ∈
(ℤ≥‘2)) |
| 81 | | peano2rem 11576 |
. . . . . . . . . 10
⊢ (𝑘 ∈ ℝ → (𝑘 − 1) ∈
ℝ) |
| 82 | 75, 22, 39, 81 | 4syl 19 |
. . . . . . . . 9
⊢ (((𝜑 ∧ (𝑘 ∈ 𝐴 ∧ ∀𝑧 ∈ 𝐴 𝑘 ≤ 𝑧)) ∧ ¬ 𝑘 = 1) → (𝑘 − 1) ∈ ℝ) |
| 83 | 55 | ad2antrr 726 |
. . . . . . . . . 10
⊢ (((𝜑 ∧ (𝑘 ∈ 𝐴 ∧ ∀𝑧 ∈ 𝐴 𝑘 ≤ 𝑧)) ∧ ¬ 𝑘 = 1) → 𝐷 ∈ ℝ) |
| 84 | 5 | rpne0d 13082 |
. . . . . . . . . . 11
⊢ (𝜑 → 𝐷 ≠ 0) |
| 85 | 84 | ad2antrr 726 |
. . . . . . . . . 10
⊢ (((𝜑 ∧ (𝑘 ∈ 𝐴 ∧ ∀𝑧 ∈ 𝐴 𝑘 ≤ 𝑧)) ∧ ¬ 𝑘 = 1) → 𝐷 ≠ 0) |
| 86 | 83, 85 | rereccld 12094 |
. . . . . . . . 9
⊢ (((𝜑 ∧ (𝑘 ∈ 𝐴 ∧ ∀𝑧 ∈ 𝐴 𝑘 ≤ 𝑧)) ∧ ¬ 𝑘 = 1) → (1 / 𝐷) ∈ ℝ) |
| 87 | | 1zzd 12648 |
. . . . . . . . . . 11
⊢ (((𝜑 ∧ (𝑘 ∈ 𝐴 ∧ ∀𝑧 ∈ 𝐴 𝑘 ≤ 𝑧)) ∧ ¬ 𝑘 = 1) → 1 ∈
ℤ) |
| 88 | | df-2 12329 |
. . . . . . . . . . . . . . 15
⊢ 2 = (1 +
1) |
| 89 | 88 | fveq2i 6909 |
. . . . . . . . . . . . . 14
⊢
(ℤ≥‘2) = (ℤ≥‘(1 +
1)) |
| 90 | 89 | eleq2i 2833 |
. . . . . . . . . . . . 13
⊢ (𝑘 ∈
(ℤ≥‘2) ↔ 𝑘 ∈ (ℤ≥‘(1 +
1))) |
| 91 | | eluzsub 12908 |
. . . . . . . . . . . . 13
⊢ ((1
∈ ℤ ∧ 1 ∈ ℤ ∧ 𝑘 ∈ (ℤ≥‘(1 +
1))) → (𝑘 − 1)
∈ (ℤ≥‘1)) |
| 92 | 90, 91 | syl3an3b 1407 |
. . . . . . . . . . . 12
⊢ ((1
∈ ℤ ∧ 1 ∈ ℤ ∧ 𝑘 ∈ (ℤ≥‘2))
→ (𝑘 − 1) ∈
(ℤ≥‘1)) |
| 93 | | nnuz 12921 |
. . . . . . . . . . . 12
⊢ ℕ =
(ℤ≥‘1) |
| 94 | 92, 93 | eleqtrrdi 2852 |
. . . . . . . . . . 11
⊢ ((1
∈ ℤ ∧ 1 ∈ ℤ ∧ 𝑘 ∈ (ℤ≥‘2))
→ (𝑘 − 1) ∈
ℕ) |
| 95 | 87, 87, 80, 94 | syl3anc 1373 |
. . . . . . . . . 10
⊢ (((𝜑 ∧ (𝑘 ∈ 𝐴 ∧ ∀𝑧 ∈ 𝐴 𝑘 ≤ 𝑧)) ∧ ¬ 𝑘 = 1) → (𝑘 − 1) ∈ ℕ) |
| 96 | 22, 39 | syl 17 |
. . . . . . . . . . . . . . . . . . . . . 22
⊢ (𝑘 ∈ 𝐴 → 𝑘 ∈ ℝ) |
| 97 | 96 | adantl 481 |
. . . . . . . . . . . . . . . . . . . . 21
⊢ (((𝑘 − 1) ∈ 𝐴 ∧ 𝑘 ∈ 𝐴) → 𝑘 ∈ ℝ) |
| 98 | 97, 81 | syl 17 |
. . . . . . . . . . . . . . . . . . . 20
⊢ (((𝑘 − 1) ∈ 𝐴 ∧ 𝑘 ∈ 𝐴) → (𝑘 − 1) ∈ ℝ) |
| 99 | | simpr 484 |
. . . . . . . . . . . . . . . . . . . . . 22
⊢ (((𝑘 − 1) ∈ ℝ ∧
𝑘 ∈ ℝ) →
𝑘 ∈
ℝ) |
| 100 | 99 | ltm1d 12200 |
. . . . . . . . . . . . . . . . . . . . 21
⊢ (((𝑘 − 1) ∈ ℝ ∧
𝑘 ∈ ℝ) →
(𝑘 − 1) < 𝑘) |
| 101 | | ltnle 11340 |
. . . . . . . . . . . . . . . . . . . . 21
⊢ (((𝑘 − 1) ∈ ℝ ∧
𝑘 ∈ ℝ) →
((𝑘 − 1) < 𝑘 ↔ ¬ 𝑘 ≤ (𝑘 − 1))) |
| 102 | 100, 101 | mpbid 232 |
. . . . . . . . . . . . . . . . . . . 20
⊢ (((𝑘 − 1) ∈ ℝ ∧
𝑘 ∈ ℝ) →
¬ 𝑘 ≤ (𝑘 − 1)) |
| 103 | 98, 97, 102 | syl2anc 584 |
. . . . . . . . . . . . . . . . . . 19
⊢ (((𝑘 − 1) ∈ 𝐴 ∧ 𝑘 ∈ 𝐴) → ¬ 𝑘 ≤ (𝑘 − 1)) |
| 104 | | breq2 5147 |
. . . . . . . . . . . . . . . . . . . . 21
⊢ (𝑧 = (𝑘 − 1) → (𝑘 ≤ 𝑧 ↔ 𝑘 ≤ (𝑘 − 1))) |
| 105 | 104 | notbid 318 |
. . . . . . . . . . . . . . . . . . . 20
⊢ (𝑧 = (𝑘 − 1) → (¬ 𝑘 ≤ 𝑧 ↔ ¬ 𝑘 ≤ (𝑘 − 1))) |
| 106 | 105 | rspcev 3622 |
. . . . . . . . . . . . . . . . . . 19
⊢ (((𝑘 − 1) ∈ 𝐴 ∧ ¬ 𝑘 ≤ (𝑘 − 1)) → ∃𝑧 ∈ 𝐴 ¬ 𝑘 ≤ 𝑧) |
| 107 | 103, 106 | syldan 591 |
. . . . . . . . . . . . . . . . . 18
⊢ (((𝑘 − 1) ∈ 𝐴 ∧ 𝑘 ∈ 𝐴) → ∃𝑧 ∈ 𝐴 ¬ 𝑘 ≤ 𝑧) |
| 108 | | rexnal 3100 |
. . . . . . . . . . . . . . . . . 18
⊢
(∃𝑧 ∈
𝐴 ¬ 𝑘 ≤ 𝑧 ↔ ¬ ∀𝑧 ∈ 𝐴 𝑘 ≤ 𝑧) |
| 109 | 107, 108 | sylib 218 |
. . . . . . . . . . . . . . . . 17
⊢ (((𝑘 − 1) ∈ 𝐴 ∧ 𝑘 ∈ 𝐴) → ¬ ∀𝑧 ∈ 𝐴 𝑘 ≤ 𝑧) |
| 110 | 109 | ex 412 |
. . . . . . . . . . . . . . . 16
⊢ ((𝑘 − 1) ∈ 𝐴 → (𝑘 ∈ 𝐴 → ¬ ∀𝑧 ∈ 𝐴 𝑘 ≤ 𝑧)) |
| 111 | | imnan 399 |
. . . . . . . . . . . . . . . 16
⊢ ((𝑘 ∈ 𝐴 → ¬ ∀𝑧 ∈ 𝐴 𝑘 ≤ 𝑧) ↔ ¬ (𝑘 ∈ 𝐴 ∧ ∀𝑧 ∈ 𝐴 𝑘 ≤ 𝑧)) |
| 112 | 110, 111 | sylib 218 |
. . . . . . . . . . . . . . 15
⊢ ((𝑘 − 1) ∈ 𝐴 → ¬ (𝑘 ∈ 𝐴 ∧ ∀𝑧 ∈ 𝐴 𝑘 ≤ 𝑧)) |
| 113 | 112 | con2i 139 |
. . . . . . . . . . . . . 14
⊢ ((𝑘 ∈ 𝐴 ∧ ∀𝑧 ∈ 𝐴 𝑘 ≤ 𝑧) → ¬ (𝑘 − 1) ∈ 𝐴) |
| 114 | 113 | ad2antlr 727 |
. . . . . . . . . . . . 13
⊢ (((𝜑 ∧ (𝑘 ∈ 𝐴 ∧ ∀𝑧 ∈ 𝐴 𝑘 ≤ 𝑧)) ∧ ¬ 𝑘 = 1) → ¬ (𝑘 − 1) ∈ 𝐴) |
| 115 | | breq2 5147 |
. . . . . . . . . . . . . 14
⊢ (𝑗 = (𝑘 − 1) → ((1 / 𝐷) < 𝑗 ↔ (1 / 𝐷) < (𝑘 − 1))) |
| 116 | 115, 1 | elrab2 3695 |
. . . . . . . . . . . . 13
⊢ ((𝑘 − 1) ∈ 𝐴 ↔ ((𝑘 − 1) ∈ ℕ ∧ (1 / 𝐷) < (𝑘 − 1))) |
| 117 | 114, 116 | sylnib 328 |
. . . . . . . . . . . 12
⊢ (((𝜑 ∧ (𝑘 ∈ 𝐴 ∧ ∀𝑧 ∈ 𝐴 𝑘 ≤ 𝑧)) ∧ ¬ 𝑘 = 1) → ¬ ((𝑘 − 1) ∈ ℕ ∧ (1 / 𝐷) < (𝑘 − 1))) |
| 118 | | ianor 984 |
. . . . . . . . . . . 12
⊢ (¬
((𝑘 − 1) ∈
ℕ ∧ (1 / 𝐷) <
(𝑘 − 1)) ↔
(¬ (𝑘 − 1) ∈
ℕ ∨ ¬ (1 / 𝐷)
< (𝑘 −
1))) |
| 119 | 117, 118 | sylib 218 |
. . . . . . . . . . 11
⊢ (((𝜑 ∧ (𝑘 ∈ 𝐴 ∧ ∀𝑧 ∈ 𝐴 𝑘 ≤ 𝑧)) ∧ ¬ 𝑘 = 1) → (¬ (𝑘 − 1) ∈ ℕ ∨ ¬ (1 /
𝐷) < (𝑘 − 1))) |
| 120 | | imor 854 |
. . . . . . . . . . 11
⊢ (((𝑘 − 1) ∈ ℕ
→ ¬ (1 / 𝐷) <
(𝑘 − 1)) ↔
(¬ (𝑘 − 1) ∈
ℕ ∨ ¬ (1 / 𝐷)
< (𝑘 −
1))) |
| 121 | 119, 120 | sylibr 234 |
. . . . . . . . . 10
⊢ (((𝜑 ∧ (𝑘 ∈ 𝐴 ∧ ∀𝑧 ∈ 𝐴 𝑘 ≤ 𝑧)) ∧ ¬ 𝑘 = 1) → ((𝑘 − 1) ∈ ℕ → ¬ (1 /
𝐷) < (𝑘 − 1))) |
| 122 | 95, 121 | mpd 15 |
. . . . . . . . 9
⊢ (((𝜑 ∧ (𝑘 ∈ 𝐴 ∧ ∀𝑧 ∈ 𝐴 𝑘 ≤ 𝑧)) ∧ ¬ 𝑘 = 1) → ¬ (1 / 𝐷) < (𝑘 − 1)) |
| 123 | 82, 86, 122 | nltled 11411 |
. . . . . . . 8
⊢ (((𝜑 ∧ (𝑘 ∈ 𝐴 ∧ ∀𝑧 ∈ 𝐴 𝑘 ≤ 𝑧)) ∧ ¬ 𝑘 = 1) → (𝑘 − 1) ≤ (1 / 𝐷)) |
| 124 | | eluzelre 12889 |
. . . . . . . . . . . . 13
⊢ (𝑘 ∈
(ℤ≥‘2) → 𝑘 ∈ ℝ) |
| 125 | 124 | adantl 481 |
. . . . . . . . . . . 12
⊢ ((𝜑 ∧ 𝑘 ∈ (ℤ≥‘2))
→ 𝑘 ∈
ℝ) |
| 126 | 55 | adantr 480 |
. . . . . . . . . . . 12
⊢ ((𝜑 ∧ 𝑘 ∈ (ℤ≥‘2))
→ 𝐷 ∈
ℝ) |
| 127 | 125, 126 | remulcld 11291 |
. . . . . . . . . . 11
⊢ ((𝜑 ∧ 𝑘 ∈ (ℤ≥‘2))
→ (𝑘 · 𝐷) ∈
ℝ) |
| 128 | 127 | rehalfcld 12513 |
. . . . . . . . . 10
⊢ ((𝜑 ∧ 𝑘 ∈ (ℤ≥‘2))
→ ((𝑘 · 𝐷) / 2) ∈
ℝ) |
| 129 | 128 | 3adant3 1133 |
. . . . . . . . 9
⊢ ((𝜑 ∧ 𝑘 ∈ (ℤ≥‘2)
∧ (𝑘 − 1) ≤ (1
/ 𝐷)) → ((𝑘 · 𝐷) / 2) ∈ ℝ) |
| 130 | 59, 55 | readdcld 11290 |
. . . . . . . . . . . 12
⊢ (𝜑 → (1 + 𝐷) ∈ ℝ) |
| 131 | 130 | adantr 480 |
. . . . . . . . . . 11
⊢ ((𝜑 ∧ 𝑘 ∈ (ℤ≥‘2))
→ (1 + 𝐷) ∈
ℝ) |
| 132 | 131 | rehalfcld 12513 |
. . . . . . . . . 10
⊢ ((𝜑 ∧ 𝑘 ∈ (ℤ≥‘2))
→ ((1 + 𝐷) / 2) ∈
ℝ) |
| 133 | 132 | 3adant3 1133 |
. . . . . . . . 9
⊢ ((𝜑 ∧ 𝑘 ∈ (ℤ≥‘2)
∧ (𝑘 − 1) ≤ (1
/ 𝐷)) → ((1 + 𝐷) / 2) ∈
ℝ) |
| 134 | | 1red 11262 |
. . . . . . . . 9
⊢ ((𝜑 ∧ 𝑘 ∈ (ℤ≥‘2)
∧ (𝑘 − 1) ≤ (1
/ 𝐷)) → 1 ∈
ℝ) |
| 135 | | eluzelcn 12890 |
. . . . . . . . . . . . . . 15
⊢ (𝑘 ∈
(ℤ≥‘2) → 𝑘 ∈ ℂ) |
| 136 | 135 | adantl 481 |
. . . . . . . . . . . . . 14
⊢ ((𝜑 ∧ 𝑘 ∈ (ℤ≥‘2))
→ 𝑘 ∈
ℂ) |
| 137 | 50 | adantr 480 |
. . . . . . . . . . . . . 14
⊢ ((𝜑 ∧ 𝑘 ∈ (ℤ≥‘2))
→ 𝐷 ∈
ℂ) |
| 138 | 136, 137 | mulcld 11281 |
. . . . . . . . . . . . 13
⊢ ((𝜑 ∧ 𝑘 ∈ (ℤ≥‘2))
→ (𝑘 · 𝐷) ∈
ℂ) |
| 139 | 138 | 3adant3 1133 |
. . . . . . . . . . . 12
⊢ ((𝜑 ∧ 𝑘 ∈ (ℤ≥‘2)
∧ (𝑘 − 1) ≤ (1
/ 𝐷)) → (𝑘 · 𝐷) ∈ ℂ) |
| 140 | 50 | 3ad2ant1 1134 |
. . . . . . . . . . . 12
⊢ ((𝜑 ∧ 𝑘 ∈ (ℤ≥‘2)
∧ (𝑘 − 1) ≤ (1
/ 𝐷)) → 𝐷 ∈
ℂ) |
| 141 | 139, 140 | npcand 11624 |
. . . . . . . . . . 11
⊢ ((𝜑 ∧ 𝑘 ∈ (ℤ≥‘2)
∧ (𝑘 − 1) ≤ (1
/ 𝐷)) → (((𝑘 · 𝐷) − 𝐷) + 𝐷) = (𝑘 · 𝐷)) |
| 142 | 127, 126 | resubcld 11691 |
. . . . . . . . . . . . 13
⊢ ((𝜑 ∧ 𝑘 ∈ (ℤ≥‘2))
→ ((𝑘 · 𝐷) − 𝐷) ∈ ℝ) |
| 143 | 142 | 3adant3 1133 |
. . . . . . . . . . . 12
⊢ ((𝜑 ∧ 𝑘 ∈ (ℤ≥‘2)
∧ (𝑘 − 1) ≤ (1
/ 𝐷)) → ((𝑘 · 𝐷) − 𝐷) ∈ ℝ) |
| 144 | 55 | 3ad2ant1 1134 |
. . . . . . . . . . . 12
⊢ ((𝜑 ∧ 𝑘 ∈ (ℤ≥‘2)
∧ (𝑘 − 1) ≤ (1
/ 𝐷)) → 𝐷 ∈
ℝ) |
| 145 | | simp3 1139 |
. . . . . . . . . . . . . 14
⊢ ((𝜑 ∧ 𝑘 ∈ (ℤ≥‘2)
∧ (𝑘 − 1) ≤ (1
/ 𝐷)) → (𝑘 − 1) ≤ (1 / 𝐷)) |
| 146 | | 1red 11262 |
. . . . . . . . . . . . . . . . 17
⊢ (𝑘 ∈
(ℤ≥‘2) → 1 ∈ ℝ) |
| 147 | 124, 146 | resubcld 11691 |
. . . . . . . . . . . . . . . 16
⊢ (𝑘 ∈
(ℤ≥‘2) → (𝑘 − 1) ∈ ℝ) |
| 148 | 147 | 3ad2ant2 1135 |
. . . . . . . . . . . . . . 15
⊢ ((𝜑 ∧ 𝑘 ∈ (ℤ≥‘2)
∧ (𝑘 − 1) ≤ (1
/ 𝐷)) → (𝑘 − 1) ∈
ℝ) |
| 149 | 6 | 3ad2ant1 1134 |
. . . . . . . . . . . . . . 15
⊢ ((𝜑 ∧ 𝑘 ∈ (ℤ≥‘2)
∧ (𝑘 − 1) ≤ (1
/ 𝐷)) → (1 / 𝐷) ∈
ℝ) |
| 150 | 41 | 3ad2ant1 1134 |
. . . . . . . . . . . . . . 15
⊢ ((𝜑 ∧ 𝑘 ∈ (ℤ≥‘2)
∧ (𝑘 − 1) ≤ (1
/ 𝐷)) → (𝐷 ∈ ℝ ∧ 0 <
𝐷)) |
| 151 | | lemul1 12119 |
. . . . . . . . . . . . . . 15
⊢ (((𝑘 − 1) ∈ ℝ ∧
(1 / 𝐷) ∈ ℝ
∧ (𝐷 ∈ ℝ
∧ 0 < 𝐷)) →
((𝑘 − 1) ≤ (1 /
𝐷) ↔ ((𝑘 − 1) · 𝐷) ≤ ((1 / 𝐷) · 𝐷))) |
| 152 | 148, 149,
150, 151 | syl3anc 1373 |
. . . . . . . . . . . . . 14
⊢ ((𝜑 ∧ 𝑘 ∈ (ℤ≥‘2)
∧ (𝑘 − 1) ≤ (1
/ 𝐷)) → ((𝑘 − 1) ≤ (1 / 𝐷) ↔ ((𝑘 − 1) · 𝐷) ≤ ((1 / 𝐷) · 𝐷))) |
| 153 | 145, 152 | mpbid 232 |
. . . . . . . . . . . . 13
⊢ ((𝜑 ∧ 𝑘 ∈ (ℤ≥‘2)
∧ (𝑘 − 1) ≤ (1
/ 𝐷)) → ((𝑘 − 1) · 𝐷) ≤ ((1 / 𝐷) · 𝐷)) |
| 154 | | 1cnd 11256 |
. . . . . . . . . . . . . . . 16
⊢ ((𝜑 ∧ 𝑘 ∈ (ℤ≥‘2))
→ 1 ∈ ℂ) |
| 155 | 136, 154,
137 | subdird 11720 |
. . . . . . . . . . . . . . 15
⊢ ((𝜑 ∧ 𝑘 ∈ (ℤ≥‘2))
→ ((𝑘 − 1)
· 𝐷) = ((𝑘 · 𝐷) − (1 · 𝐷))) |
| 156 | 137 | mullidd 11279 |
. . . . . . . . . . . . . . . 16
⊢ ((𝜑 ∧ 𝑘 ∈ (ℤ≥‘2))
→ (1 · 𝐷) =
𝐷) |
| 157 | 156 | oveq2d 7447 |
. . . . . . . . . . . . . . 15
⊢ ((𝜑 ∧ 𝑘 ∈ (ℤ≥‘2))
→ ((𝑘 · 𝐷) − (1 · 𝐷)) = ((𝑘 · 𝐷) − 𝐷)) |
| 158 | 155, 157 | eqtrd 2777 |
. . . . . . . . . . . . . 14
⊢ ((𝜑 ∧ 𝑘 ∈ (ℤ≥‘2))
→ ((𝑘 − 1)
· 𝐷) = ((𝑘 · 𝐷) − 𝐷)) |
| 159 | 158 | 3adant3 1133 |
. . . . . . . . . . . . 13
⊢ ((𝜑 ∧ 𝑘 ∈ (ℤ≥‘2)
∧ (𝑘 − 1) ≤ (1
/ 𝐷)) → ((𝑘 − 1) · 𝐷) = ((𝑘 · 𝐷) − 𝐷)) |
| 160 | | 1cnd 11256 |
. . . . . . . . . . . . . . . 16
⊢ (𝜑 → 1 ∈
ℂ) |
| 161 | 160, 50, 84 | 3jca 1129 |
. . . . . . . . . . . . . . 15
⊢ (𝜑 → (1 ∈ ℂ ∧
𝐷 ∈ ℂ ∧
𝐷 ≠ 0)) |
| 162 | 161 | 3ad2ant1 1134 |
. . . . . . . . . . . . . 14
⊢ ((𝜑 ∧ 𝑘 ∈ (ℤ≥‘2)
∧ (𝑘 − 1) ≤ (1
/ 𝐷)) → (1 ∈
ℂ ∧ 𝐷 ∈
ℂ ∧ 𝐷 ≠
0)) |
| 163 | | divcan1 11931 |
. . . . . . . . . . . . . 14
⊢ ((1
∈ ℂ ∧ 𝐷
∈ ℂ ∧ 𝐷 ≠
0) → ((1 / 𝐷) ·
𝐷) = 1) |
| 164 | 162, 163 | syl 17 |
. . . . . . . . . . . . 13
⊢ ((𝜑 ∧ 𝑘 ∈ (ℤ≥‘2)
∧ (𝑘 − 1) ≤ (1
/ 𝐷)) → ((1 / 𝐷) · 𝐷) = 1) |
| 165 | 153, 159,
164 | 3brtr3d 5174 |
. . . . . . . . . . . 12
⊢ ((𝜑 ∧ 𝑘 ∈ (ℤ≥‘2)
∧ (𝑘 − 1) ≤ (1
/ 𝐷)) → ((𝑘 · 𝐷) − 𝐷) ≤ 1) |
| 166 | 143, 134,
144, 165 | leadd1dd 11877 |
. . . . . . . . . . 11
⊢ ((𝜑 ∧ 𝑘 ∈ (ℤ≥‘2)
∧ (𝑘 − 1) ≤ (1
/ 𝐷)) → (((𝑘 · 𝐷) − 𝐷) + 𝐷) ≤ (1 + 𝐷)) |
| 167 | 141, 166 | eqbrtrrd 5167 |
. . . . . . . . . 10
⊢ ((𝜑 ∧ 𝑘 ∈ (ℤ≥‘2)
∧ (𝑘 − 1) ≤ (1
/ 𝐷)) → (𝑘 · 𝐷) ≤ (1 + 𝐷)) |
| 168 | 127 | 3adant3 1133 |
. . . . . . . . . . 11
⊢ ((𝜑 ∧ 𝑘 ∈ (ℤ≥‘2)
∧ (𝑘 − 1) ≤ (1
/ 𝐷)) → (𝑘 · 𝐷) ∈ ℝ) |
| 169 | 130 | 3ad2ant1 1134 |
. . . . . . . . . . 11
⊢ ((𝜑 ∧ 𝑘 ∈ (ℤ≥‘2)
∧ (𝑘 − 1) ≤ (1
/ 𝐷)) → (1 + 𝐷) ∈
ℝ) |
| 170 | 61, 63 | pm3.2i 470 |
. . . . . . . . . . . 12
⊢ (2 ∈
ℝ ∧ 0 < 2) |
| 171 | 170 | a1i 11 |
. . . . . . . . . . 11
⊢ ((𝜑 ∧ 𝑘 ∈ (ℤ≥‘2)
∧ (𝑘 − 1) ≤ (1
/ 𝐷)) → (2 ∈
ℝ ∧ 0 < 2)) |
| 172 | | lediv1 12133 |
. . . . . . . . . . 11
⊢ (((𝑘 · 𝐷) ∈ ℝ ∧ (1 + 𝐷) ∈ ℝ ∧ (2 ∈
ℝ ∧ 0 < 2)) → ((𝑘 · 𝐷) ≤ (1 + 𝐷) ↔ ((𝑘 · 𝐷) / 2) ≤ ((1 + 𝐷) / 2))) |
| 173 | 168, 169,
171, 172 | syl3anc 1373 |
. . . . . . . . . 10
⊢ ((𝜑 ∧ 𝑘 ∈ (ℤ≥‘2)
∧ (𝑘 − 1) ≤ (1
/ 𝐷)) → ((𝑘 · 𝐷) ≤ (1 + 𝐷) ↔ ((𝑘 · 𝐷) / 2) ≤ ((1 + 𝐷) / 2))) |
| 174 | 167, 173 | mpbid 232 |
. . . . . . . . 9
⊢ ((𝜑 ∧ 𝑘 ∈ (ℤ≥‘2)
∧ (𝑘 − 1) ≤ (1
/ 𝐷)) → ((𝑘 · 𝐷) / 2) ≤ ((1 + 𝐷) / 2)) |
| 175 | 55, 59, 59, 60 | ltadd2dd 11420 |
. . . . . . . . . . . . 13
⊢ (𝜑 → (1 + 𝐷) < (1 + 1)) |
| 176 | | 1p1e2 12391 |
. . . . . . . . . . . . 13
⊢ (1 + 1) =
2 |
| 177 | 175, 176 | breqtrdi 5184 |
. . . . . . . . . . . 12
⊢ (𝜑 → (1 + 𝐷) < 2) |
| 178 | | ltdiv1 12132 |
. . . . . . . . . . . . 13
⊢ (((1 +
𝐷) ∈ ℝ ∧ 2
∈ ℝ ∧ (2 ∈ ℝ ∧ 0 < 2)) → ((1 + 𝐷) < 2 ↔ ((1 + 𝐷) / 2) < (2 /
2))) |
| 179 | 130, 62, 62, 64, 178 | syl112anc 1376 |
. . . . . . . . . . . 12
⊢ (𝜑 → ((1 + 𝐷) < 2 ↔ ((1 + 𝐷) / 2) < (2 / 2))) |
| 180 | 177, 179 | mpbid 232 |
. . . . . . . . . . 11
⊢ (𝜑 → ((1 + 𝐷) / 2) < (2 / 2)) |
| 181 | | 2div2e1 12407 |
. . . . . . . . . . 11
⊢ (2 / 2) =
1 |
| 182 | 180, 181 | breqtrdi 5184 |
. . . . . . . . . 10
⊢ (𝜑 → ((1 + 𝐷) / 2) < 1) |
| 183 | 182 | 3ad2ant1 1134 |
. . . . . . . . 9
⊢ ((𝜑 ∧ 𝑘 ∈ (ℤ≥‘2)
∧ (𝑘 − 1) ≤ (1
/ 𝐷)) → ((1 + 𝐷) / 2) < 1) |
| 184 | 129, 133,
134, 174, 183 | lelttrd 11419 |
. . . . . . . 8
⊢ ((𝜑 ∧ 𝑘 ∈ (ℤ≥‘2)
∧ (𝑘 − 1) ≤ (1
/ 𝐷)) → ((𝑘 · 𝐷) / 2) < 1) |
| 185 | 74, 80, 123, 184 | syl3anc 1373 |
. . . . . . 7
⊢ (((𝜑 ∧ (𝑘 ∈ 𝐴 ∧ ∀𝑧 ∈ 𝐴 𝑘 ≤ 𝑧)) ∧ ¬ 𝑘 = 1) → ((𝑘 · 𝐷) / 2) < 1) |
| 186 | 73, 185 | pm2.61dan 813 |
. . . . . 6
⊢ ((𝜑 ∧ (𝑘 ∈ 𝐴 ∧ ∀𝑧 ∈ 𝐴 𝑘 ≤ 𝑧)) → ((𝑘 · 𝐷) / 2) < 1) |
| 187 | 23, 47, 186 | jca32 515 |
. . . . 5
⊢ ((𝜑 ∧ (𝑘 ∈ 𝐴 ∧ ∀𝑧 ∈ 𝐴 𝑘 ≤ 𝑧)) → (𝑘 ∈ ℕ ∧ (1 < (𝑘 · 𝐷) ∧ ((𝑘 · 𝐷) / 2) < 1))) |
| 188 | 187 | ex 412 |
. . . 4
⊢ (𝜑 → ((𝑘 ∈ 𝐴 ∧ ∀𝑧 ∈ 𝐴 𝑘 ≤ 𝑧) → (𝑘 ∈ ℕ ∧ (1 < (𝑘 · 𝐷) ∧ ((𝑘 · 𝐷) / 2) < 1)))) |
| 189 | 188 | eximdv 1917 |
. . 3
⊢ (𝜑 → (∃𝑘(𝑘 ∈ 𝐴 ∧ ∀𝑧 ∈ 𝐴 𝑘 ≤ 𝑧) → ∃𝑘(𝑘 ∈ ℕ ∧ (1 < (𝑘 · 𝐷) ∧ ((𝑘 · 𝐷) / 2) < 1)))) |
| 190 | 20, 189 | mpd 15 |
. 2
⊢ (𝜑 → ∃𝑘(𝑘 ∈ ℕ ∧ (1 < (𝑘 · 𝐷) ∧ ((𝑘 · 𝐷) / 2) < 1))) |
| 191 | | df-rex 3071 |
. 2
⊢
(∃𝑘 ∈
ℕ (1 < (𝑘 ·
𝐷) ∧ ((𝑘 · 𝐷) / 2) < 1) ↔ ∃𝑘(𝑘 ∈ ℕ ∧ (1 < (𝑘 · 𝐷) ∧ ((𝑘 · 𝐷) / 2) < 1))) |
| 192 | 190, 191 | sylibr 234 |
1
⊢ (𝜑 → ∃𝑘 ∈ ℕ (1 < (𝑘 · 𝐷) ∧ ((𝑘 · 𝐷) / 2) < 1)) |