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Mirrors > Home > MPE Home > Th. List > Mathboxes > flt4lem5d | Structured version Visualization version GIF version |
Description: Part 3 of Equation 2 of https://crypto.stanford.edu/pbc/notes/numberfield/fermatn4.html. (Contributed by SN, 23-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 |
---|---|
flt4lem5d | ⊢ (𝜑 → 𝑀 = ((𝑅↑2) + (𝑆↑2))) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | flt4lem5a.a | . 2 ⊢ (𝜑 → 𝐴 ∈ ℕ) | |
2 | 1 | nnsqcld 14214 | . . . 4 ⊢ (𝜑 → (𝐴↑2) ∈ ℕ) |
3 | flt4lem5a.b | . . . . 5 ⊢ (𝜑 → 𝐵 ∈ ℕ) | |
4 | 3 | nnsqcld 14214 | . . . 4 ⊢ (𝜑 → (𝐵↑2) ∈ ℕ) |
5 | flt4lem5a.c | . . . 4 ⊢ (𝜑 → 𝐶 ∈ ℕ) | |
6 | flt4lem5a.1 | . . . . 5 ⊢ (𝜑 → ¬ 2 ∥ 𝐴) | |
7 | 2prm 16636 | . . . . . 6 ⊢ 2 ∈ ℙ | |
8 | 1 | nnzd 12592 | . . . . . 6 ⊢ (𝜑 → 𝐴 ∈ ℤ) |
9 | prmdvdssq 16662 | . . . . . 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 12292 | . . . . . . 7 ⊢ 2 ∈ ℕ | |
14 | 13 | a1i 11 | . . . . . 6 ⊢ (𝜑 → 2 ∈ ℕ) |
15 | rplpwr 16506 | . . . . . 6 ⊢ ((𝐴 ∈ ℕ ∧ 𝐶 ∈ ℕ ∧ 2 ∈ ℕ) → ((𝐴 gcd 𝐶) = 1 → ((𝐴↑2) gcd 𝐶) = 1)) | |
16 | 1, 5, 14, 15 | syl3anc 1370 | . . . . 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 12235 | . . . . . . 7 ⊢ (𝜑 → 𝐴 ∈ ℂ) |
19 | 18 | flt4lem 41850 | . . . . . 6 ⊢ (𝜑 → (𝐴↑4) = ((𝐴↑2)↑2)) |
20 | 3 | nncnd 12235 | . . . . . . 7 ⊢ (𝜑 → 𝐵 ∈ ℂ) |
21 | 20 | flt4lem 41850 | . . . . . 6 ⊢ (𝜑 → (𝐵↑4) = ((𝐵↑2)↑2)) |
22 | 19, 21 | oveq12d 7430 | . . . . 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 2773 | . . . 4 ⊢ (𝜑 → (((𝐴↑2)↑2) + ((𝐵↑2)↑2)) = (𝐶↑2)) |
25 | 2, 4, 5, 11, 17, 24 | flt4lem1 41851 | . . 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 16767 | . . 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 16765 | . . 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 41857 | . 2 ⊢ (𝜑 → ((𝐴↑2) + (𝑁↑2)) = (𝑀↑2)) |
35 | 28 | nnzd 12592 | . . . 4 ⊢ (𝜑 → 𝑁 ∈ ℤ) |
36 | 8, 35 | gcdcomd 16462 | . . 3 ⊢ (𝜑 → (𝐴 gcd 𝑁) = (𝑁 gcd 𝐴)) |
37 | 31 | nnzd 12592 | . . . . . 6 ⊢ (𝜑 → 𝑀 ∈ ℤ) |
38 | 35, 37 | gcdcomd 16462 | . . . . 5 ⊢ (𝜑 → (𝑁 gcd 𝑀) = (𝑀 gcd 𝑁)) |
39 | 29, 26 | flt4lem5 41855 | . . . . . 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 2771 | . . . 4 ⊢ (𝜑 → (𝑁 gcd 𝑀) = 1) |
42 | 28 | nnsqcld 14214 | . . . . . . 7 ⊢ (𝜑 → (𝑁↑2) ∈ ℕ) |
43 | 42 | nncnd 12235 | . . . . . 6 ⊢ (𝜑 → (𝑁↑2) ∈ ℂ) |
44 | 2 | nncnd 12235 | . . . . . 6 ⊢ (𝜑 → (𝐴↑2) ∈ ℂ) |
45 | 43, 44 | addcomd 11423 | . . . . 5 ⊢ (𝜑 → ((𝑁↑2) + (𝐴↑2)) = ((𝐴↑2) + (𝑁↑2))) |
46 | 45, 34 | eqtrd 2771 | . . . 4 ⊢ (𝜑 → ((𝑁↑2) + (𝐴↑2)) = (𝑀↑2)) |
47 | 28, 1, 31, 41, 46 | fltabcoprm 41847 | . . 3 ⊢ (𝜑 → (𝑁 gcd 𝐴) = 1) |
48 | 36, 47 | eqtrd 2771 | . 2 ⊢ (𝜑 → (𝐴 gcd 𝑁) = 1) |
49 | 32, 33 | pythagtriplem17 16771 | . 2 ⊢ (((𝐴 ∈ ℕ ∧ 𝑁 ∈ ℕ ∧ 𝑀 ∈ ℕ) ∧ ((𝐴↑2) + (𝑁↑2)) = (𝑀↑2) ∧ ((𝐴 gcd 𝑁) = 1 ∧ ¬ 2 ∥ 𝐴)) → 𝑀 = ((𝑅↑2) + (𝑆↑2))) |
50 | 1, 28, 31, 34, 48, 6, 49 | syl312anc 1390 | 1 ⊢ (𝜑 → 𝑀 = ((𝑅↑2) + (𝑆↑2))) |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 → wi 4 ↔ wb 205 ∧ wa 395 ∧ w3a 1086 = wceq 1540 ∈ wcel 2105 class class class wbr 5148 ‘cfv 6543 (class class class)co 7412 1c1 11117 + caddc 11119 − cmin 11451 / cdiv 11878 ℕcn 12219 2c2 12274 4c4 12276 ℤcz 12565 ↑cexp 14034 √csqrt 15187 ∥ cdvds 16204 gcd cgcd 16442 ℙcprime 16615 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1796 ax-4 1810 ax-5 1912 ax-6 1970 ax-7 2010 ax-8 2107 ax-9 2115 ax-10 2136 ax-11 2153 ax-12 2170 ax-ext 2702 ax-sep 5299 ax-nul 5306 ax-pow 5363 ax-pr 5427 ax-un 7729 ax-cnex 11172 ax-resscn 11173 ax-1cn 11174 ax-icn 11175 ax-addcl 11176 ax-addrcl 11177 ax-mulcl 11178 ax-mulrcl 11179 ax-mulcom 11180 ax-addass 11181 ax-mulass 11182 ax-distr 11183 ax-i2m1 11184 ax-1ne0 11185 ax-1rid 11186 ax-rnegex 11187 ax-rrecex 11188 ax-cnre 11189 ax-pre-lttri 11190 ax-pre-lttrn 11191 ax-pre-ltadd 11192 ax-pre-mulgt0 11193 ax-pre-sup 11194 |
This theorem depends on definitions: df-bi 206 df-an 396 df-or 845 df-3or 1087 df-3an 1088 df-tru 1543 df-fal 1553 df-ex 1781 df-nf 1785 df-sb 2067 df-mo 2533 df-eu 2562 df-clab 2709 df-cleq 2723 df-clel 2809 df-nfc 2884 df-ne 2940 df-nel 3046 df-ral 3061 df-rex 3070 df-rmo 3375 df-reu 3376 df-rab 3432 df-v 3475 df-sbc 3778 df-csb 3894 df-dif 3951 df-un 3953 df-in 3955 df-ss 3965 df-pss 3967 df-nul 4323 df-if 4529 df-pw 4604 df-sn 4629 df-pr 4631 df-op 4635 df-uni 4909 df-iun 4999 df-br 5149 df-opab 5211 df-mpt 5232 df-tr 5266 df-id 5574 df-eprel 5580 df-po 5588 df-so 5589 df-fr 5631 df-we 5633 df-xp 5682 df-rel 5683 df-cnv 5684 df-co 5685 df-dm 5686 df-rn 5687 df-res 5688 df-ima 5689 df-pred 6300 df-ord 6367 df-on 6368 df-lim 6369 df-suc 6370 df-iota 6495 df-fun 6545 df-fn 6546 df-f 6547 df-f1 6548 df-fo 6549 df-f1o 6550 df-fv 6551 df-riota 7368 df-ov 7415 df-oprab 7416 df-mpo 7417 df-om 7860 df-1st 7979 df-2nd 7980 df-frecs 8272 df-wrecs 8303 df-recs 8377 df-rdg 8416 df-1o 8472 df-2o 8473 df-er 8709 df-en 8946 df-dom 8947 df-sdom 8948 df-fin 8949 df-sup 9443 df-inf 9444 df-pnf 11257 df-mnf 11258 df-xr 11259 df-ltxr 11260 df-le 11261 df-sub 11453 df-neg 11454 df-div 11879 df-nn 12220 df-2 12282 df-3 12283 df-4 12284 df-n0 12480 df-z 12566 df-uz 12830 df-rp 12982 df-fz 13492 df-fl 13764 df-mod 13842 df-seq 13974 df-exp 14035 df-cj 15053 df-re 15054 df-im 15055 df-sqrt 15189 df-abs 15190 df-dvds 16205 df-gcd 16443 df-prm 16616 |
This theorem is referenced by: flt4lem5e 41861 flt4lem5f 41862 |
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