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| Mirrors > Home > MPE Home > Th. List > Mathboxes > flt4lem5c | Structured version Visualization version GIF version | ||
| Description: Part 2 of Equation 2 of https://crypto.stanford.edu/pbc/notes/numberfield/fermatn4.html. (Contributed by SN, 22-Aug-2024.) |
| Ref | Expression |
|---|---|
| flt4lem5a.m | ⊢ 𝑀 = (((√‘(𝐶 + (𝐵↑2))) + (√‘(𝐶 − (𝐵↑2)))) / 2) |
| flt4lem5a.n | ⊢ 𝑁 = (((√‘(𝐶 + (𝐵↑2))) − (√‘(𝐶 − (𝐵↑2)))) / 2) |
| flt4lem5a.r | ⊢ 𝑅 = (((√‘(𝑀 + 𝑁)) + (√‘(𝑀 − 𝑁))) / 2) |
| flt4lem5a.s | ⊢ 𝑆 = (((√‘(𝑀 + 𝑁)) − (√‘(𝑀 − 𝑁))) / 2) |
| flt4lem5a.a | ⊢ (𝜑 → 𝐴 ∈ ℕ) |
| flt4lem5a.b | ⊢ (𝜑 → 𝐵 ∈ ℕ) |
| flt4lem5a.c | ⊢ (𝜑 → 𝐶 ∈ ℕ) |
| flt4lem5a.1 | ⊢ (𝜑 → ¬ 2 ∥ 𝐴) |
| flt4lem5a.2 | ⊢ (𝜑 → (𝐴 gcd 𝐶) = 1) |
| flt4lem5a.3 | ⊢ (𝜑 → ((𝐴↑4) + (𝐵↑4)) = (𝐶↑2)) |
| Ref | Expression |
|---|---|
| flt4lem5c | ⊢ (𝜑 → 𝑁 = (2 · (𝑅 · 𝑆))) |
| Step | Hyp | Ref | Expression |
|---|---|---|---|
| 1 | flt4lem5a.a | . 2 ⊢ (𝜑 → 𝐴 ∈ ℕ) | |
| 2 | 1 | nnsqcld 14283 | . . . 4 ⊢ (𝜑 → (𝐴↑2) ∈ ℕ) |
| 3 | flt4lem5a.b | . . . . 5 ⊢ (𝜑 → 𝐵 ∈ ℕ) | |
| 4 | 3 | nnsqcld 14283 | . . . 4 ⊢ (𝜑 → (𝐵↑2) ∈ ℕ) |
| 5 | flt4lem5a.c | . . . 4 ⊢ (𝜑 → 𝐶 ∈ ℕ) | |
| 6 | flt4lem5a.1 | . . . . 5 ⊢ (𝜑 → ¬ 2 ∥ 𝐴) | |
| 7 | 2prm 16729 | . . . . . 6 ⊢ 2 ∈ ℙ | |
| 8 | 1 | nnzd 12640 | . . . . . 6 ⊢ (𝜑 → 𝐴 ∈ ℤ) |
| 9 | prmdvdssq 16755 | . . . . . 6 ⊢ ((2 ∈ ℙ ∧ 𝐴 ∈ ℤ) → (2 ∥ 𝐴 ↔ 2 ∥ (𝐴↑2))) | |
| 10 | 7, 8, 9 | sylancr 587 | . . . . 5 ⊢ (𝜑 → (2 ∥ 𝐴 ↔ 2 ∥ (𝐴↑2))) |
| 11 | 6, 10 | mtbid 324 | . . . 4 ⊢ (𝜑 → ¬ 2 ∥ (𝐴↑2)) |
| 12 | flt4lem5a.2 | . . . . 5 ⊢ (𝜑 → (𝐴 gcd 𝐶) = 1) | |
| 13 | 2nn 12339 | . . . . . . 7 ⊢ 2 ∈ ℕ | |
| 14 | 13 | a1i 11 | . . . . . 6 ⊢ (𝜑 → 2 ∈ ℕ) |
| 15 | rplpwr 16595 | . . . . . 6 ⊢ ((𝐴 ∈ ℕ ∧ 𝐶 ∈ ℕ ∧ 2 ∈ ℕ) → ((𝐴 gcd 𝐶) = 1 → ((𝐴↑2) gcd 𝐶) = 1)) | |
| 16 | 1, 5, 14, 15 | syl3anc 1373 | . . . . 5 ⊢ (𝜑 → ((𝐴 gcd 𝐶) = 1 → ((𝐴↑2) gcd 𝐶) = 1)) |
| 17 | 12, 16 | mpd 15 | . . . 4 ⊢ (𝜑 → ((𝐴↑2) gcd 𝐶) = 1) |
| 18 | 1 | nncnd 12282 | . . . . . . 7 ⊢ (𝜑 → 𝐴 ∈ ℂ) |
| 19 | 18 | flt4lem 42655 | . . . . . 6 ⊢ (𝜑 → (𝐴↑4) = ((𝐴↑2)↑2)) |
| 20 | 3 | nncnd 12282 | . . . . . . 7 ⊢ (𝜑 → 𝐵 ∈ ℂ) |
| 21 | 20 | flt4lem 42655 | . . . . . 6 ⊢ (𝜑 → (𝐵↑4) = ((𝐵↑2)↑2)) |
| 22 | 19, 21 | oveq12d 7449 | . . . . 5 ⊢ (𝜑 → ((𝐴↑4) + (𝐵↑4)) = (((𝐴↑2)↑2) + ((𝐵↑2)↑2))) |
| 23 | flt4lem5a.3 | . . . . 5 ⊢ (𝜑 → ((𝐴↑4) + (𝐵↑4)) = (𝐶↑2)) | |
| 24 | 22, 23 | eqtr3d 2779 | . . . 4 ⊢ (𝜑 → (((𝐴↑2)↑2) + ((𝐵↑2)↑2)) = (𝐶↑2)) |
| 25 | 2, 4, 5, 11, 17, 24 | flt4lem1 42656 | . . 3 ⊢ (𝜑 → (((𝐴↑2) ∈ ℕ ∧ (𝐵↑2) ∈ ℕ ∧ 𝐶 ∈ ℕ) ∧ (((𝐴↑2)↑2) + ((𝐵↑2)↑2)) = (𝐶↑2) ∧ (((𝐴↑2) gcd (𝐵↑2)) = 1 ∧ ¬ 2 ∥ (𝐴↑2)))) |
| 26 | flt4lem5a.n | . . . 4 ⊢ 𝑁 = (((√‘(𝐶 + (𝐵↑2))) − (√‘(𝐶 − (𝐵↑2)))) / 2) | |
| 27 | 26 | pythagtriplem13 16865 | . . 3 ⊢ ((((𝐴↑2) ∈ ℕ ∧ (𝐵↑2) ∈ ℕ ∧ 𝐶 ∈ ℕ) ∧ (((𝐴↑2)↑2) + ((𝐵↑2)↑2)) = (𝐶↑2) ∧ (((𝐴↑2) gcd (𝐵↑2)) = 1 ∧ ¬ 2 ∥ (𝐴↑2))) → 𝑁 ∈ ℕ) |
| 28 | 25, 27 | syl 17 | . 2 ⊢ (𝜑 → 𝑁 ∈ ℕ) |
| 29 | flt4lem5a.m | . . . 4 ⊢ 𝑀 = (((√‘(𝐶 + (𝐵↑2))) + (√‘(𝐶 − (𝐵↑2)))) / 2) | |
| 30 | 29 | pythagtriplem11 16863 | . . 3 ⊢ ((((𝐴↑2) ∈ ℕ ∧ (𝐵↑2) ∈ ℕ ∧ 𝐶 ∈ ℕ) ∧ (((𝐴↑2)↑2) + ((𝐵↑2)↑2)) = (𝐶↑2) ∧ (((𝐴↑2) gcd (𝐵↑2)) = 1 ∧ ¬ 2 ∥ (𝐴↑2))) → 𝑀 ∈ ℕ) |
| 31 | 25, 30 | syl 17 | . 2 ⊢ (𝜑 → 𝑀 ∈ ℕ) |
| 32 | flt4lem5a.r | . . 3 ⊢ 𝑅 = (((√‘(𝑀 + 𝑁)) + (√‘(𝑀 − 𝑁))) / 2) | |
| 33 | flt4lem5a.s | . . 3 ⊢ 𝑆 = (((√‘(𝑀 + 𝑁)) − (√‘(𝑀 − 𝑁))) / 2) | |
| 34 | 29, 26, 32, 33, 1, 3, 5, 6, 12, 23 | flt4lem5a 42662 | . 2 ⊢ (𝜑 → ((𝐴↑2) + (𝑁↑2)) = (𝑀↑2)) |
| 35 | 28 | nnzd 12640 | . . . 4 ⊢ (𝜑 → 𝑁 ∈ ℤ) |
| 36 | 8, 35 | gcdcomd 16551 | . . 3 ⊢ (𝜑 → (𝐴 gcd 𝑁) = (𝑁 gcd 𝐴)) |
| 37 | 31 | nnzd 12640 | . . . . . 6 ⊢ (𝜑 → 𝑀 ∈ ℤ) |
| 38 | 35, 37 | gcdcomd 16551 | . . . . 5 ⊢ (𝜑 → (𝑁 gcd 𝑀) = (𝑀 gcd 𝑁)) |
| 39 | 29, 26 | flt4lem5 42660 | . . . . . 6 ⊢ ((((𝐴↑2) ∈ ℕ ∧ (𝐵↑2) ∈ ℕ ∧ 𝐶 ∈ ℕ) ∧ (((𝐴↑2)↑2) + ((𝐵↑2)↑2)) = (𝐶↑2) ∧ (((𝐴↑2) gcd (𝐵↑2)) = 1 ∧ ¬ 2 ∥ (𝐴↑2))) → (𝑀 gcd 𝑁) = 1) |
| 40 | 25, 39 | syl 17 | . . . . 5 ⊢ (𝜑 → (𝑀 gcd 𝑁) = 1) |
| 41 | 38, 40 | eqtrd 2777 | . . . 4 ⊢ (𝜑 → (𝑁 gcd 𝑀) = 1) |
| 42 | 28 | nnsqcld 14283 | . . . . . . 7 ⊢ (𝜑 → (𝑁↑2) ∈ ℕ) |
| 43 | 42 | nncnd 12282 | . . . . . 6 ⊢ (𝜑 → (𝑁↑2) ∈ ℂ) |
| 44 | 2 | nncnd 12282 | . . . . . 6 ⊢ (𝜑 → (𝐴↑2) ∈ ℂ) |
| 45 | 43, 44 | addcomd 11463 | . . . . 5 ⊢ (𝜑 → ((𝑁↑2) + (𝐴↑2)) = ((𝐴↑2) + (𝑁↑2))) |
| 46 | 45, 34 | eqtrd 2777 | . . . 4 ⊢ (𝜑 → ((𝑁↑2) + (𝐴↑2)) = (𝑀↑2)) |
| 47 | 28, 1, 31, 41, 46 | fltabcoprm 42652 | . . 3 ⊢ (𝜑 → (𝑁 gcd 𝐴) = 1) |
| 48 | 36, 47 | eqtrd 2777 | . 2 ⊢ (𝜑 → (𝐴 gcd 𝑁) = 1) |
| 49 | 32, 33 | pythagtriplem16 16868 | . 2 ⊢ (((𝐴 ∈ ℕ ∧ 𝑁 ∈ ℕ ∧ 𝑀 ∈ ℕ) ∧ ((𝐴↑2) + (𝑁↑2)) = (𝑀↑2) ∧ ((𝐴 gcd 𝑁) = 1 ∧ ¬ 2 ∥ 𝐴)) → 𝑁 = (2 · (𝑅 · 𝑆))) |
| 50 | 1, 28, 31, 34, 48, 6, 49 | syl312anc 1393 | 1 ⊢ (𝜑 → 𝑁 = (2 · (𝑅 · 𝑆))) |
| Colors of variables: wff setvar class |
| Syntax hints: ¬ wn 3 → wi 4 ↔ wb 206 ∧ wa 395 ∧ w3a 1087 = wceq 1540 ∈ wcel 2108 class class class wbr 5143 ‘cfv 6561 (class class class)co 7431 1c1 11156 + caddc 11158 · cmul 11160 − cmin 11492 / cdiv 11920 ℕcn 12266 2c2 12321 4c4 12323 ℤcz 12613 ↑cexp 14102 √csqrt 15272 ∥ cdvds 16290 gcd cgcd 16531 ℙcprime 16708 |
| This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1795 ax-4 1809 ax-5 1910 ax-6 1967 ax-7 2007 ax-8 2110 ax-9 2118 ax-10 2141 ax-11 2157 ax-12 2177 ax-ext 2708 ax-sep 5296 ax-nul 5306 ax-pow 5365 ax-pr 5432 ax-un 7755 ax-cnex 11211 ax-resscn 11212 ax-1cn 11213 ax-icn 11214 ax-addcl 11215 ax-addrcl 11216 ax-mulcl 11217 ax-mulrcl 11218 ax-mulcom 11219 ax-addass 11220 ax-mulass 11221 ax-distr 11222 ax-i2m1 11223 ax-1ne0 11224 ax-1rid 11225 ax-rnegex 11226 ax-rrecex 11227 ax-cnre 11228 ax-pre-lttri 11229 ax-pre-lttrn 11230 ax-pre-ltadd 11231 ax-pre-mulgt0 11232 ax-pre-sup 11233 |
| This theorem depends on definitions: df-bi 207 df-an 396 df-or 849 df-3or 1088 df-3an 1089 df-tru 1543 df-fal 1553 df-ex 1780 df-nf 1784 df-sb 2065 df-mo 2540 df-eu 2569 df-clab 2715 df-cleq 2729 df-clel 2816 df-nfc 2892 df-ne 2941 df-nel 3047 df-ral 3062 df-rex 3071 df-rmo 3380 df-reu 3381 df-rab 3437 df-v 3482 df-sbc 3789 df-csb 3900 df-dif 3954 df-un 3956 df-in 3958 df-ss 3968 df-pss 3971 df-nul 4334 df-if 4526 df-pw 4602 df-sn 4627 df-pr 4629 df-op 4633 df-uni 4908 df-iun 4993 df-br 5144 df-opab 5206 df-mpt 5226 df-tr 5260 df-id 5578 df-eprel 5584 df-po 5592 df-so 5593 df-fr 5637 df-we 5639 df-xp 5691 df-rel 5692 df-cnv 5693 df-co 5694 df-dm 5695 df-rn 5696 df-res 5697 df-ima 5698 df-pred 6321 df-ord 6387 df-on 6388 df-lim 6389 df-suc 6390 df-iota 6514 df-fun 6563 df-fn 6564 df-f 6565 df-f1 6566 df-fo 6567 df-f1o 6568 df-fv 6569 df-riota 7388 df-ov 7434 df-oprab 7435 df-mpo 7436 df-om 7888 df-1st 8014 df-2nd 8015 df-frecs 8306 df-wrecs 8337 df-recs 8411 df-rdg 8450 df-1o 8506 df-2o 8507 df-er 8745 df-en 8986 df-dom 8987 df-sdom 8988 df-fin 8989 df-sup 9482 df-inf 9483 df-pnf 11297 df-mnf 11298 df-xr 11299 df-ltxr 11300 df-le 11301 df-sub 11494 df-neg 11495 df-div 11921 df-nn 12267 df-2 12329 df-3 12330 df-4 12331 df-n0 12527 df-z 12614 df-uz 12879 df-rp 13035 df-fz 13548 df-fl 13832 df-mod 13910 df-seq 14043 df-exp 14103 df-cj 15138 df-re 15139 df-im 15140 df-sqrt 15274 df-abs 15275 df-dvds 16291 df-gcd 16532 df-prm 16709 |
| This theorem is referenced by: flt4lem5e 42666 |
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