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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 14232 | . . . 4 ⊢ (𝜑 → (𝐴↑2) ∈ ℕ) |
3 | flt4lem5a.b | . . . . 5 ⊢ (𝜑 → 𝐵 ∈ ℕ) | |
4 | 3 | nnsqcld 14232 | . . . 4 ⊢ (𝜑 → (𝐵↑2) ∈ ℕ) |
5 | flt4lem5a.c | . . . 4 ⊢ (𝜑 → 𝐶 ∈ ℕ) | |
6 | flt4lem5a.1 | . . . . 5 ⊢ (𝜑 → ¬ 2 ∥ 𝐴) | |
7 | 2prm 16656 | . . . . . 6 ⊢ 2 ∈ ℙ | |
8 | 1 | nnzd 12609 | . . . . . 6 ⊢ (𝜑 → 𝐴 ∈ ℤ) |
9 | prmdvdssq 16682 | . . . . . 6 ⊢ ((2 ∈ ℙ ∧ 𝐴 ∈ ℤ) → (2 ∥ 𝐴 ↔ 2 ∥ (𝐴↑2))) | |
10 | 7, 8, 9 | sylancr 586 | . . . . 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 12309 | . . . . . . 7 ⊢ 2 ∈ ℕ | |
14 | 13 | a1i 11 | . . . . . 6 ⊢ (𝜑 → 2 ∈ ℕ) |
15 | rplpwr 16526 | . . . . . 6 ⊢ ((𝐴 ∈ ℕ ∧ 𝐶 ∈ ℕ ∧ 2 ∈ ℕ) → ((𝐴 gcd 𝐶) = 1 → ((𝐴↑2) gcd 𝐶) = 1)) | |
16 | 1, 5, 14, 15 | syl3anc 1369 | . . . . 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 12252 | . . . . . . 7 ⊢ (𝜑 → 𝐴 ∈ ℂ) |
19 | 18 | flt4lem 42041 | . . . . . 6 ⊢ (𝜑 → (𝐴↑4) = ((𝐴↑2)↑2)) |
20 | 3 | nncnd 12252 | . . . . . . 7 ⊢ (𝜑 → 𝐵 ∈ ℂ) |
21 | 20 | flt4lem 42041 | . . . . . 6 ⊢ (𝜑 → (𝐵↑4) = ((𝐵↑2)↑2)) |
22 | 19, 21 | oveq12d 7432 | . . . . 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 2769 | . . . 4 ⊢ (𝜑 → (((𝐴↑2)↑2) + ((𝐵↑2)↑2)) = (𝐶↑2)) |
25 | 2, 4, 5, 11, 17, 24 | flt4lem1 42042 | . . 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 16789 | . . 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 16787 | . . 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 42048 | . 2 ⊢ (𝜑 → ((𝐴↑2) + (𝑁↑2)) = (𝑀↑2)) |
35 | 28 | nnzd 12609 | . . . 4 ⊢ (𝜑 → 𝑁 ∈ ℤ) |
36 | 8, 35 | gcdcomd 16482 | . . 3 ⊢ (𝜑 → (𝐴 gcd 𝑁) = (𝑁 gcd 𝐴)) |
37 | 31 | nnzd 12609 | . . . . . 6 ⊢ (𝜑 → 𝑀 ∈ ℤ) |
38 | 35, 37 | gcdcomd 16482 | . . . . 5 ⊢ (𝜑 → (𝑁 gcd 𝑀) = (𝑀 gcd 𝑁)) |
39 | 29, 26 | flt4lem5 42046 | . . . . . 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 2767 | . . . 4 ⊢ (𝜑 → (𝑁 gcd 𝑀) = 1) |
42 | 28 | nnsqcld 14232 | . . . . . . 7 ⊢ (𝜑 → (𝑁↑2) ∈ ℕ) |
43 | 42 | nncnd 12252 | . . . . . 6 ⊢ (𝜑 → (𝑁↑2) ∈ ℂ) |
44 | 2 | nncnd 12252 | . . . . . 6 ⊢ (𝜑 → (𝐴↑2) ∈ ℂ) |
45 | 43, 44 | addcomd 11440 | . . . . 5 ⊢ (𝜑 → ((𝑁↑2) + (𝐴↑2)) = ((𝐴↑2) + (𝑁↑2))) |
46 | 45, 34 | eqtrd 2767 | . . . 4 ⊢ (𝜑 → ((𝑁↑2) + (𝐴↑2)) = (𝑀↑2)) |
47 | 28, 1, 31, 41, 46 | fltabcoprm 42038 | . . 3 ⊢ (𝜑 → (𝑁 gcd 𝐴) = 1) |
48 | 36, 47 | eqtrd 2767 | . 2 ⊢ (𝜑 → (𝐴 gcd 𝑁) = 1) |
49 | 32, 33 | pythagtriplem16 16792 | . 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 1389 | 1 ⊢ (𝜑 → 𝑁 = (2 · (𝑅 · 𝑆))) |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 → wi 4 ↔ wb 205 ∧ wa 395 ∧ w3a 1085 = wceq 1534 ∈ wcel 2099 class class class wbr 5142 ‘cfv 6542 (class class class)co 7414 1c1 11133 + caddc 11135 · cmul 11137 − cmin 11468 / cdiv 11895 ℕcn 12236 2c2 12291 4c4 12293 ℤcz 12582 ↑cexp 14052 √csqrt 15206 ∥ cdvds 16224 gcd cgcd 16462 ℙcprime 16635 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1790 ax-4 1804 ax-5 1906 ax-6 1964 ax-7 2004 ax-8 2101 ax-9 2109 ax-10 2130 ax-11 2147 ax-12 2164 ax-ext 2698 ax-sep 5293 ax-nul 5300 ax-pow 5359 ax-pr 5423 ax-un 7734 ax-cnex 11188 ax-resscn 11189 ax-1cn 11190 ax-icn 11191 ax-addcl 11192 ax-addrcl 11193 ax-mulcl 11194 ax-mulrcl 11195 ax-mulcom 11196 ax-addass 11197 ax-mulass 11198 ax-distr 11199 ax-i2m1 11200 ax-1ne0 11201 ax-1rid 11202 ax-rnegex 11203 ax-rrecex 11204 ax-cnre 11205 ax-pre-lttri 11206 ax-pre-lttrn 11207 ax-pre-ltadd 11208 ax-pre-mulgt0 11209 ax-pre-sup 11210 |
This theorem depends on definitions: df-bi 206 df-an 396 df-or 847 df-3or 1086 df-3an 1087 df-tru 1537 df-fal 1547 df-ex 1775 df-nf 1779 df-sb 2061 df-mo 2529 df-eu 2558 df-clab 2705 df-cleq 2719 df-clel 2805 df-nfc 2880 df-ne 2936 df-nel 3042 df-ral 3057 df-rex 3066 df-rmo 3371 df-reu 3372 df-rab 3428 df-v 3471 df-sbc 3775 df-csb 3890 df-dif 3947 df-un 3949 df-in 3951 df-ss 3961 df-pss 3963 df-nul 4319 df-if 4525 df-pw 4600 df-sn 4625 df-pr 4627 df-op 4631 df-uni 4904 df-iun 4993 df-br 5143 df-opab 5205 df-mpt 5226 df-tr 5260 df-id 5570 df-eprel 5576 df-po 5584 df-so 5585 df-fr 5627 df-we 5629 df-xp 5678 df-rel 5679 df-cnv 5680 df-co 5681 df-dm 5682 df-rn 5683 df-res 5684 df-ima 5685 df-pred 6299 df-ord 6366 df-on 6367 df-lim 6368 df-suc 6369 df-iota 6494 df-fun 6544 df-fn 6545 df-f 6546 df-f1 6547 df-fo 6548 df-f1o 6549 df-fv 6550 df-riota 7370 df-ov 7417 df-oprab 7418 df-mpo 7419 df-om 7865 df-1st 7987 df-2nd 7988 df-frecs 8280 df-wrecs 8311 df-recs 8385 df-rdg 8424 df-1o 8480 df-2o 8481 df-er 8718 df-en 8958 df-dom 8959 df-sdom 8960 df-fin 8961 df-sup 9459 df-inf 9460 df-pnf 11274 df-mnf 11275 df-xr 11276 df-ltxr 11277 df-le 11278 df-sub 11470 df-neg 11471 df-div 11896 df-nn 12237 df-2 12299 df-3 12300 df-4 12301 df-n0 12497 df-z 12583 df-uz 12847 df-rp 13001 df-fz 13511 df-fl 13783 df-mod 13861 df-seq 13993 df-exp 14053 df-cj 15072 df-re 15073 df-im 15074 df-sqrt 15208 df-abs 15209 df-dvds 16225 df-gcd 16463 df-prm 16636 |
This theorem is referenced by: flt4lem5e 42052 |
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