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Mirrors > Home > MPE Home > Th. List > Mathboxes > flt4lem5a | Structured version Visualization version GIF version |
Description: Part 1 of Equation 1 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 |
---|---|
flt4lem5a | ⊢ (𝜑 → ((𝐴↑2) + (𝑁↑2)) = (𝑀↑2)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | flt4lem5a.a | . . . . . . 7 ⊢ (𝜑 → 𝐴 ∈ ℕ) | |
2 | 1 | nnsqcld 13669 | . . . . . 6 ⊢ (𝜑 → (𝐴↑2) ∈ ℕ) |
3 | flt4lem5a.b | . . . . . . 7 ⊢ (𝜑 → 𝐵 ∈ ℕ) | |
4 | 3 | nnsqcld 13669 | . . . . . 6 ⊢ (𝜑 → (𝐵↑2) ∈ ℕ) |
5 | flt4lem5a.c | . . . . . 6 ⊢ (𝜑 → 𝐶 ∈ ℕ) | |
6 | flt4lem5a.1 | . . . . . . 7 ⊢ (𝜑 → ¬ 2 ∥ 𝐴) | |
7 | 2prm 16103 | . . . . . . . 8 ⊢ 2 ∈ ℙ | |
8 | 1 | nnzd 12139 | . . . . . . . 8 ⊢ (𝜑 → 𝐴 ∈ ℤ) |
9 | prmdvdssq 16129 | . . . . . . . 8 ⊢ ((2 ∈ ℙ ∧ 𝐴 ∈ ℤ) → (2 ∥ 𝐴 ↔ 2 ∥ (𝐴↑2))) | |
10 | 7, 8, 9 | sylancr 590 | . . . . . . 7 ⊢ (𝜑 → (2 ∥ 𝐴 ↔ 2 ∥ (𝐴↑2))) |
11 | 6, 10 | mtbid 327 | . . . . . 6 ⊢ (𝜑 → ¬ 2 ∥ (𝐴↑2)) |
12 | flt4lem5a.2 | . . . . . . 7 ⊢ (𝜑 → (𝐴 gcd 𝐶) = 1) | |
13 | 2nn 11761 | . . . . . . . . 9 ⊢ 2 ∈ ℕ | |
14 | 13 | a1i 11 | . . . . . . . 8 ⊢ (𝜑 → 2 ∈ ℕ) |
15 | rplpwr 15973 | . . . . . . . 8 ⊢ ((𝐴 ∈ ℕ ∧ 𝐶 ∈ ℕ ∧ 2 ∈ ℕ) → ((𝐴 gcd 𝐶) = 1 → ((𝐴↑2) gcd 𝐶) = 1)) | |
16 | 1, 5, 14, 15 | syl3anc 1369 | . . . . . . 7 ⊢ (𝜑 → ((𝐴 gcd 𝐶) = 1 → ((𝐴↑2) gcd 𝐶) = 1)) |
17 | 12, 16 | mpd 15 | . . . . . 6 ⊢ (𝜑 → ((𝐴↑2) gcd 𝐶) = 1) |
18 | 1 | nncnd 11704 | . . . . . . . . 9 ⊢ (𝜑 → 𝐴 ∈ ℂ) |
19 | 18 | flt4lem 40020 | . . . . . . . 8 ⊢ (𝜑 → (𝐴↑4) = ((𝐴↑2)↑2)) |
20 | 3 | nncnd 11704 | . . . . . . . . 9 ⊢ (𝜑 → 𝐵 ∈ ℂ) |
21 | 20 | flt4lem 40020 | . . . . . . . 8 ⊢ (𝜑 → (𝐵↑4) = ((𝐵↑2)↑2)) |
22 | 19, 21 | oveq12d 7175 | . . . . . . 7 ⊢ (𝜑 → ((𝐴↑4) + (𝐵↑4)) = (((𝐴↑2)↑2) + ((𝐵↑2)↑2))) |
23 | flt4lem5a.3 | . . . . . . 7 ⊢ (𝜑 → ((𝐴↑4) + (𝐵↑4)) = (𝐶↑2)) | |
24 | 22, 23 | eqtr3d 2796 | . . . . . 6 ⊢ (𝜑 → (((𝐴↑2)↑2) + ((𝐵↑2)↑2)) = (𝐶↑2)) |
25 | 2, 4, 5, 11, 17, 24 | flt4lem1 40021 | . . . . 5 ⊢ (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.m | . . . . . 6 ⊢ 𝑀 = (((√‘(𝐶 + (𝐵↑2))) + (√‘(𝐶 − (𝐵↑2)))) / 2) | |
27 | 26 | pythagtriplem11 16232 | . . . . 5 ⊢ ((((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) |
28 | 25, 27 | syl 17 | . . . 4 ⊢ (𝜑 → 𝑀 ∈ ℕ) |
29 | 28 | nnsqcld 13669 | . . 3 ⊢ (𝜑 → (𝑀↑2) ∈ ℕ) |
30 | 29 | nncnd 11704 | . 2 ⊢ (𝜑 → (𝑀↑2) ∈ ℂ) |
31 | flt4lem5a.n | . . . . . 6 ⊢ 𝑁 = (((√‘(𝐶 + (𝐵↑2))) − (√‘(𝐶 − (𝐵↑2)))) / 2) | |
32 | 31 | pythagtriplem13 16234 | . . . . 5 ⊢ ((((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) |
33 | 25, 32 | syl 17 | . . . 4 ⊢ (𝜑 → 𝑁 ∈ ℕ) |
34 | 33 | nnsqcld 13669 | . . 3 ⊢ (𝜑 → (𝑁↑2) ∈ ℕ) |
35 | 34 | nncnd 11704 | . 2 ⊢ (𝜑 → (𝑁↑2) ∈ ℂ) |
36 | 26, 31 | pythagtriplem15 16236 | . . 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))) → (𝐴↑2) = ((𝑀↑2) − (𝑁↑2))) |
37 | 25, 36 | syl 17 | . 2 ⊢ (𝜑 → (𝐴↑2) = ((𝑀↑2) − (𝑁↑2))) |
38 | 30, 35, 37 | mvrrsubd 39845 | 1 ⊢ (𝜑 → ((𝐴↑2) + (𝑁↑2)) = (𝑀↑2)) |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 → wi 4 ↔ wb 209 ∧ wa 399 ∧ w3a 1085 = wceq 1539 ∈ wcel 2112 class class class wbr 5037 ‘cfv 6341 (class class class)co 7157 1c1 10590 + caddc 10592 − cmin 10922 / cdiv 11349 ℕcn 11688 2c2 11743 4c4 11745 ℤcz 12034 ↑cexp 13493 √csqrt 14654 ∥ cdvds 15669 gcd cgcd 15907 ℙcprime 16082 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1798 ax-4 1812 ax-5 1912 ax-6 1971 ax-7 2016 ax-8 2114 ax-9 2122 ax-10 2143 ax-11 2159 ax-12 2176 ax-ext 2730 ax-sep 5174 ax-nul 5181 ax-pow 5239 ax-pr 5303 ax-un 7466 ax-cnex 10645 ax-resscn 10646 ax-1cn 10647 ax-icn 10648 ax-addcl 10649 ax-addrcl 10650 ax-mulcl 10651 ax-mulrcl 10652 ax-mulcom 10653 ax-addass 10654 ax-mulass 10655 ax-distr 10656 ax-i2m1 10657 ax-1ne0 10658 ax-1rid 10659 ax-rnegex 10660 ax-rrecex 10661 ax-cnre 10662 ax-pre-lttri 10663 ax-pre-lttrn 10664 ax-pre-ltadd 10665 ax-pre-mulgt0 10666 ax-pre-sup 10667 |
This theorem depends on definitions: df-bi 210 df-an 400 df-or 845 df-3or 1086 df-3an 1087 df-tru 1542 df-fal 1552 df-ex 1783 df-nf 1787 df-sb 2071 df-mo 2558 df-eu 2589 df-clab 2737 df-cleq 2751 df-clel 2831 df-nfc 2902 df-ne 2953 df-nel 3057 df-ral 3076 df-rex 3077 df-reu 3078 df-rmo 3079 df-rab 3080 df-v 3412 df-sbc 3700 df-csb 3809 df-dif 3864 df-un 3866 df-in 3868 df-ss 3878 df-pss 3880 df-nul 4229 df-if 4425 df-pw 4500 df-sn 4527 df-pr 4529 df-tp 4531 df-op 4533 df-uni 4803 df-iun 4889 df-br 5038 df-opab 5100 df-mpt 5118 df-tr 5144 df-id 5435 df-eprel 5440 df-po 5448 df-so 5449 df-fr 5488 df-we 5490 df-xp 5535 df-rel 5536 df-cnv 5537 df-co 5538 df-dm 5539 df-rn 5540 df-res 5541 df-ima 5542 df-pred 6132 df-ord 6178 df-on 6179 df-lim 6180 df-suc 6181 df-iota 6300 df-fun 6343 df-fn 6344 df-f 6345 df-f1 6346 df-fo 6347 df-f1o 6348 df-fv 6349 df-riota 7115 df-ov 7160 df-oprab 7161 df-mpo 7162 df-om 7587 df-1st 7700 df-2nd 7701 df-wrecs 7964 df-recs 8025 df-rdg 8063 df-1o 8119 df-2o 8120 df-er 8306 df-en 8542 df-dom 8543 df-sdom 8544 df-fin 8545 df-sup 8953 df-inf 8954 df-pnf 10729 df-mnf 10730 df-xr 10731 df-ltxr 10732 df-le 10733 df-sub 10924 df-neg 10925 df-div 11350 df-nn 11689 df-2 11751 df-3 11752 df-4 11753 df-n0 11949 df-z 12035 df-uz 12297 df-rp 12445 df-fz 12954 df-fl 13225 df-mod 13301 df-seq 13433 df-exp 13494 df-cj 14520 df-re 14521 df-im 14522 df-sqrt 14656 df-abs 14657 df-dvds 15670 df-gcd 15908 df-prm 16083 |
This theorem is referenced by: flt4lem5c 40029 flt4lem5d 40030 flt4lem5e 40031 |
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