Proof of Theorem 4sqlem10
| Step | Hyp | Ref
| Expression |
| 1 | | 4sqlem5.3 |
. . . . . 6
⊢ (𝜑 → 𝑀 ∈ ℕ) |
| 2 | 1 | adantr 480 |
. . . . 5
⊢ ((𝜑 ∧ 𝜓) → 𝑀 ∈ ℕ) |
| 3 | 2 | nnzd 12640 |
. . . 4
⊢ ((𝜑 ∧ 𝜓) → 𝑀 ∈ ℤ) |
| 4 | | zsqcl 14169 |
. . . 4
⊢ (𝑀 ∈ ℤ → (𝑀↑2) ∈
ℤ) |
| 5 | 3, 4 | syl 17 |
. . 3
⊢ ((𝜑 ∧ 𝜓) → (𝑀↑2) ∈ ℤ) |
| 6 | | 4sqlem5.2 |
. . . . . 6
⊢ (𝜑 → 𝐴 ∈ ℤ) |
| 7 | 6 | adantr 480 |
. . . . 5
⊢ ((𝜑 ∧ 𝜓) → 𝐴 ∈ ℤ) |
| 8 | 2 | nnred 12281 |
. . . . . . . . 9
⊢ ((𝜑 ∧ 𝜓) → 𝑀 ∈ ℝ) |
| 9 | 8 | rehalfcld 12513 |
. . . . . . . 8
⊢ ((𝜑 ∧ 𝜓) → (𝑀 / 2) ∈ ℝ) |
| 10 | 9 | recnd 11289 |
. . . . . . 7
⊢ ((𝜑 ∧ 𝜓) → (𝑀 / 2) ∈ ℂ) |
| 11 | 10 | negnegd 11611 |
. . . . . 6
⊢ ((𝜑 ∧ 𝜓) → --(𝑀 / 2) = (𝑀 / 2)) |
| 12 | | 4sqlem5.4 |
. . . . . . . . . . . . . . 15
⊢ 𝐵 = (((𝐴 + (𝑀 / 2)) mod 𝑀) − (𝑀 / 2)) |
| 13 | 6, 1, 12 | 4sqlem5 16980 |
. . . . . . . . . . . . . 14
⊢ (𝜑 → (𝐵 ∈ ℤ ∧ ((𝐴 − 𝐵) / 𝑀) ∈ ℤ)) |
| 14 | 13 | adantr 480 |
. . . . . . . . . . . . 13
⊢ ((𝜑 ∧ 𝜓) → (𝐵 ∈ ℤ ∧ ((𝐴 − 𝐵) / 𝑀) ∈ ℤ)) |
| 15 | 14 | simpld 494 |
. . . . . . . . . . . 12
⊢ ((𝜑 ∧ 𝜓) → 𝐵 ∈ ℤ) |
| 16 | 15 | zred 12722 |
. . . . . . . . . . 11
⊢ ((𝜑 ∧ 𝜓) → 𝐵 ∈ ℝ) |
| 17 | 6, 1, 12 | 4sqlem6 16981 |
. . . . . . . . . . . . 13
⊢ (𝜑 → (-(𝑀 / 2) ≤ 𝐵 ∧ 𝐵 < (𝑀 / 2))) |
| 18 | 17 | adantr 480 |
. . . . . . . . . . . 12
⊢ ((𝜑 ∧ 𝜓) → (-(𝑀 / 2) ≤ 𝐵 ∧ 𝐵 < (𝑀 / 2))) |
| 19 | 18 | simprd 495 |
. . . . . . . . . . 11
⊢ ((𝜑 ∧ 𝜓) → 𝐵 < (𝑀 / 2)) |
| 20 | 16, 19 | ltned 11397 |
. . . . . . . . . 10
⊢ ((𝜑 ∧ 𝜓) → 𝐵 ≠ (𝑀 / 2)) |
| 21 | 20 | neneqd 2945 |
. . . . . . . . 9
⊢ ((𝜑 ∧ 𝜓) → ¬ 𝐵 = (𝑀 / 2)) |
| 22 | | 2cnd 12344 |
. . . . . . . . . . . . . . 15
⊢ ((𝜑 ∧ 𝜓) → 2 ∈ ℂ) |
| 23 | 22 | sqvald 14183 |
. . . . . . . . . . . . . 14
⊢ ((𝜑 ∧ 𝜓) → (2↑2) = (2 ·
2)) |
| 24 | 23 | oveq2d 7447 |
. . . . . . . . . . . . 13
⊢ ((𝜑 ∧ 𝜓) → ((𝑀↑2) / (2↑2)) = ((𝑀↑2) / (2 · 2))) |
| 25 | 2 | nncnd 12282 |
. . . . . . . . . . . . . 14
⊢ ((𝜑 ∧ 𝜓) → 𝑀 ∈ ℂ) |
| 26 | | 2ne0 12370 |
. . . . . . . . . . . . . . 15
⊢ 2 ≠
0 |
| 27 | 26 | a1i 11 |
. . . . . . . . . . . . . 14
⊢ ((𝜑 ∧ 𝜓) → 2 ≠ 0) |
| 28 | 25, 22, 27 | sqdivd 14199 |
. . . . . . . . . . . . 13
⊢ ((𝜑 ∧ 𝜓) → ((𝑀 / 2)↑2) = ((𝑀↑2) / (2↑2))) |
| 29 | 25 | sqcld 14184 |
. . . . . . . . . . . . . 14
⊢ ((𝜑 ∧ 𝜓) → (𝑀↑2) ∈ ℂ) |
| 30 | 29, 22, 22, 27, 27 | divdiv1d 12074 |
. . . . . . . . . . . . 13
⊢ ((𝜑 ∧ 𝜓) → (((𝑀↑2) / 2) / 2) = ((𝑀↑2) / (2 · 2))) |
| 31 | 24, 28, 30 | 3eqtr4d 2787 |
. . . . . . . . . . . 12
⊢ ((𝜑 ∧ 𝜓) → ((𝑀 / 2)↑2) = (((𝑀↑2) / 2) / 2)) |
| 32 | 29 | halfcld 12511 |
. . . . . . . . . . . . . 14
⊢ ((𝜑 ∧ 𝜓) → ((𝑀↑2) / 2) ∈
ℂ) |
| 33 | 32 | halfcld 12511 |
. . . . . . . . . . . . 13
⊢ ((𝜑 ∧ 𝜓) → (((𝑀↑2) / 2) / 2) ∈
ℂ) |
| 34 | 15 | zcnd 12723 |
. . . . . . . . . . . . . 14
⊢ ((𝜑 ∧ 𝜓) → 𝐵 ∈ ℂ) |
| 35 | 34 | sqcld 14184 |
. . . . . . . . . . . . 13
⊢ ((𝜑 ∧ 𝜓) → (𝐵↑2) ∈ ℂ) |
| 36 | | 4sqlem10.5 |
. . . . . . . . . . . . 13
⊢ ((𝜑 ∧ 𝜓) → ((((𝑀↑2) / 2) / 2) − (𝐵↑2)) = 0) |
| 37 | 33, 35, 36 | subeq0d 11628 |
. . . . . . . . . . . 12
⊢ ((𝜑 ∧ 𝜓) → (((𝑀↑2) / 2) / 2) = (𝐵↑2)) |
| 38 | 31, 37 | eqtr2d 2778 |
. . . . . . . . . . 11
⊢ ((𝜑 ∧ 𝜓) → (𝐵↑2) = ((𝑀 / 2)↑2)) |
| 39 | | sqeqor 14255 |
. . . . . . . . . . . 12
⊢ ((𝐵 ∈ ℂ ∧ (𝑀 / 2) ∈ ℂ) →
((𝐵↑2) = ((𝑀 / 2)↑2) ↔ (𝐵 = (𝑀 / 2) ∨ 𝐵 = -(𝑀 / 2)))) |
| 40 | 34, 10, 39 | syl2anc 584 |
. . . . . . . . . . 11
⊢ ((𝜑 ∧ 𝜓) → ((𝐵↑2) = ((𝑀 / 2)↑2) ↔ (𝐵 = (𝑀 / 2) ∨ 𝐵 = -(𝑀 / 2)))) |
| 41 | 38, 40 | mpbid 232 |
. . . . . . . . . 10
⊢ ((𝜑 ∧ 𝜓) → (𝐵 = (𝑀 / 2) ∨ 𝐵 = -(𝑀 / 2))) |
| 42 | 41 | ord 865 |
. . . . . . . . 9
⊢ ((𝜑 ∧ 𝜓) → (¬ 𝐵 = (𝑀 / 2) → 𝐵 = -(𝑀 / 2))) |
| 43 | 21, 42 | mpd 15 |
. . . . . . . 8
⊢ ((𝜑 ∧ 𝜓) → 𝐵 = -(𝑀 / 2)) |
| 44 | 43, 15 | eqeltrrd 2842 |
. . . . . . 7
⊢ ((𝜑 ∧ 𝜓) → -(𝑀 / 2) ∈ ℤ) |
| 45 | 44 | znegcld 12724 |
. . . . . 6
⊢ ((𝜑 ∧ 𝜓) → --(𝑀 / 2) ∈ ℤ) |
| 46 | 11, 45 | eqeltrrd 2842 |
. . . . 5
⊢ ((𝜑 ∧ 𝜓) → (𝑀 / 2) ∈ ℤ) |
| 47 | 7, 46 | zaddcld 12726 |
. . . 4
⊢ ((𝜑 ∧ 𝜓) → (𝐴 + (𝑀 / 2)) ∈ ℤ) |
| 48 | | zsqcl 14169 |
. . . 4
⊢ ((𝐴 + (𝑀 / 2)) ∈ ℤ → ((𝐴 + (𝑀 / 2))↑2) ∈
ℤ) |
| 49 | 47, 48 | syl 17 |
. . 3
⊢ ((𝜑 ∧ 𝜓) → ((𝐴 + (𝑀 / 2))↑2) ∈
ℤ) |
| 50 | 47, 3 | zmulcld 12728 |
. . 3
⊢ ((𝜑 ∧ 𝜓) → ((𝐴 + (𝑀 / 2)) · 𝑀) ∈ ℤ) |
| 51 | 47 | zred 12722 |
. . . . . . . 8
⊢ ((𝜑 ∧ 𝜓) → (𝐴 + (𝑀 / 2)) ∈ ℝ) |
| 52 | 2 | nnrpd 13075 |
. . . . . . . 8
⊢ ((𝜑 ∧ 𝜓) → 𝑀 ∈
ℝ+) |
| 53 | 51, 52 | modcld 13915 |
. . . . . . 7
⊢ ((𝜑 ∧ 𝜓) → ((𝐴 + (𝑀 / 2)) mod 𝑀) ∈ ℝ) |
| 54 | 53 | recnd 11289 |
. . . . . 6
⊢ ((𝜑 ∧ 𝜓) → ((𝐴 + (𝑀 / 2)) mod 𝑀) ∈ ℂ) |
| 55 | | 0cnd 11254 |
. . . . . 6
⊢ ((𝜑 ∧ 𝜓) → 0 ∈ ℂ) |
| 56 | | df-neg 11495 |
. . . . . . 7
⊢ -(𝑀 / 2) = (0 − (𝑀 / 2)) |
| 57 | 43, 12, 56 | 3eqtr3g 2800 |
. . . . . 6
⊢ ((𝜑 ∧ 𝜓) → (((𝐴 + (𝑀 / 2)) mod 𝑀) − (𝑀 / 2)) = (0 − (𝑀 / 2))) |
| 58 | 54, 55, 10, 57 | subcan2d 11662 |
. . . . 5
⊢ ((𝜑 ∧ 𝜓) → ((𝐴 + (𝑀 / 2)) mod 𝑀) = 0) |
| 59 | | dvdsval3 16294 |
. . . . . 6
⊢ ((𝑀 ∈ ℕ ∧ (𝐴 + (𝑀 / 2)) ∈ ℤ) → (𝑀 ∥ (𝐴 + (𝑀 / 2)) ↔ ((𝐴 + (𝑀 / 2)) mod 𝑀) = 0)) |
| 60 | 2, 47, 59 | syl2anc 584 |
. . . . 5
⊢ ((𝜑 ∧ 𝜓) → (𝑀 ∥ (𝐴 + (𝑀 / 2)) ↔ ((𝐴 + (𝑀 / 2)) mod 𝑀) = 0)) |
| 61 | 58, 60 | mpbird 257 |
. . . 4
⊢ ((𝜑 ∧ 𝜓) → 𝑀 ∥ (𝐴 + (𝑀 / 2))) |
| 62 | | dvdssq 16604 |
. . . . 5
⊢ ((𝑀 ∈ ℤ ∧ (𝐴 + (𝑀 / 2)) ∈ ℤ) → (𝑀 ∥ (𝐴 + (𝑀 / 2)) ↔ (𝑀↑2) ∥ ((𝐴 + (𝑀 / 2))↑2))) |
| 63 | 3, 47, 62 | syl2anc 584 |
. . . 4
⊢ ((𝜑 ∧ 𝜓) → (𝑀 ∥ (𝐴 + (𝑀 / 2)) ↔ (𝑀↑2) ∥ ((𝐴 + (𝑀 / 2))↑2))) |
| 64 | 61, 63 | mpbid 232 |
. . 3
⊢ ((𝜑 ∧ 𝜓) → (𝑀↑2) ∥ ((𝐴 + (𝑀 / 2))↑2)) |
| 65 | 25 | sqvald 14183 |
. . . 4
⊢ ((𝜑 ∧ 𝜓) → (𝑀↑2) = (𝑀 · 𝑀)) |
| 66 | 2 | nnne0d 12316 |
. . . . . 6
⊢ ((𝜑 ∧ 𝜓) → 𝑀 ≠ 0) |
| 67 | | dvdsmulcr 16323 |
. . . . . 6
⊢ ((𝑀 ∈ ℤ ∧ (𝐴 + (𝑀 / 2)) ∈ ℤ ∧ (𝑀 ∈ ℤ ∧ 𝑀 ≠ 0)) → ((𝑀 · 𝑀) ∥ ((𝐴 + (𝑀 / 2)) · 𝑀) ↔ 𝑀 ∥ (𝐴 + (𝑀 / 2)))) |
| 68 | 3, 47, 3, 66, 67 | syl112anc 1376 |
. . . . 5
⊢ ((𝜑 ∧ 𝜓) → ((𝑀 · 𝑀) ∥ ((𝐴 + (𝑀 / 2)) · 𝑀) ↔ 𝑀 ∥ (𝐴 + (𝑀 / 2)))) |
| 69 | 61, 68 | mpbird 257 |
. . . 4
⊢ ((𝜑 ∧ 𝜓) → (𝑀 · 𝑀) ∥ ((𝐴 + (𝑀 / 2)) · 𝑀)) |
| 70 | 65, 69 | eqbrtrd 5165 |
. . 3
⊢ ((𝜑 ∧ 𝜓) → (𝑀↑2) ∥ ((𝐴 + (𝑀 / 2)) · 𝑀)) |
| 71 | 5, 49, 50, 64, 70 | dvds2subd 16330 |
. 2
⊢ ((𝜑 ∧ 𝜓) → (𝑀↑2) ∥ (((𝐴 + (𝑀 / 2))↑2) − ((𝐴 + (𝑀 / 2)) · 𝑀))) |
| 72 | 47 | zcnd 12723 |
. . . . 5
⊢ ((𝜑 ∧ 𝜓) → (𝐴 + (𝑀 / 2)) ∈ ℂ) |
| 73 | 72 | sqvald 14183 |
. . . 4
⊢ ((𝜑 ∧ 𝜓) → ((𝐴 + (𝑀 / 2))↑2) = ((𝐴 + (𝑀 / 2)) · (𝐴 + (𝑀 / 2)))) |
| 74 | 73 | oveq1d 7446 |
. . 3
⊢ ((𝜑 ∧ 𝜓) → (((𝐴 + (𝑀 / 2))↑2) − ((𝐴 + (𝑀 / 2)) · 𝑀)) = (((𝐴 + (𝑀 / 2)) · (𝐴 + (𝑀 / 2))) − ((𝐴 + (𝑀 / 2)) · 𝑀))) |
| 75 | 72, 72, 25 | subdid 11719 |
. . 3
⊢ ((𝜑 ∧ 𝜓) → ((𝐴 + (𝑀 / 2)) · ((𝐴 + (𝑀 / 2)) − 𝑀)) = (((𝐴 + (𝑀 / 2)) · (𝐴 + (𝑀 / 2))) − ((𝐴 + (𝑀 / 2)) · 𝑀))) |
| 76 | 25 | 2halvesd 12512 |
. . . . . . 7
⊢ ((𝜑 ∧ 𝜓) → ((𝑀 / 2) + (𝑀 / 2)) = 𝑀) |
| 77 | 76 | oveq2d 7447 |
. . . . . 6
⊢ ((𝜑 ∧ 𝜓) → ((𝐴 + (𝑀 / 2)) − ((𝑀 / 2) + (𝑀 / 2))) = ((𝐴 + (𝑀 / 2)) − 𝑀)) |
| 78 | 7 | zcnd 12723 |
. . . . . . 7
⊢ ((𝜑 ∧ 𝜓) → 𝐴 ∈ ℂ) |
| 79 | 78, 10, 10 | pnpcan2d 11658 |
. . . . . 6
⊢ ((𝜑 ∧ 𝜓) → ((𝐴 + (𝑀 / 2)) − ((𝑀 / 2) + (𝑀 / 2))) = (𝐴 − (𝑀 / 2))) |
| 80 | 77, 79 | eqtr3d 2779 |
. . . . 5
⊢ ((𝜑 ∧ 𝜓) → ((𝐴 + (𝑀 / 2)) − 𝑀) = (𝐴 − (𝑀 / 2))) |
| 81 | 80 | oveq2d 7447 |
. . . 4
⊢ ((𝜑 ∧ 𝜓) → ((𝐴 + (𝑀 / 2)) · ((𝐴 + (𝑀 / 2)) − 𝑀)) = ((𝐴 + (𝑀 / 2)) · (𝐴 − (𝑀 / 2)))) |
| 82 | | subsq 14249 |
. . . . 5
⊢ ((𝐴 ∈ ℂ ∧ (𝑀 / 2) ∈ ℂ) →
((𝐴↑2) − ((𝑀 / 2)↑2)) = ((𝐴 + (𝑀 / 2)) · (𝐴 − (𝑀 / 2)))) |
| 83 | 78, 10, 82 | syl2anc 584 |
. . . 4
⊢ ((𝜑 ∧ 𝜓) → ((𝐴↑2) − ((𝑀 / 2)↑2)) = ((𝐴 + (𝑀 / 2)) · (𝐴 − (𝑀 / 2)))) |
| 84 | 31 | oveq2d 7447 |
. . . 4
⊢ ((𝜑 ∧ 𝜓) → ((𝐴↑2) − ((𝑀 / 2)↑2)) = ((𝐴↑2) − (((𝑀↑2) / 2) / 2))) |
| 85 | 81, 83, 84 | 3eqtr2d 2783 |
. . 3
⊢ ((𝜑 ∧ 𝜓) → ((𝐴 + (𝑀 / 2)) · ((𝐴 + (𝑀 / 2)) − 𝑀)) = ((𝐴↑2) − (((𝑀↑2) / 2) / 2))) |
| 86 | 74, 75, 85 | 3eqtr2d 2783 |
. 2
⊢ ((𝜑 ∧ 𝜓) → (((𝐴 + (𝑀 / 2))↑2) − ((𝐴 + (𝑀 / 2)) · 𝑀)) = ((𝐴↑2) − (((𝑀↑2) / 2) / 2))) |
| 87 | 71, 86 | breqtrd 5169 |
1
⊢ ((𝜑 ∧ 𝜓) → (𝑀↑2) ∥ ((𝐴↑2) − (((𝑀↑2) / 2) / 2))) |