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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 14224 | . . . . . 6 ⊢ (𝜑 → (𝐴↑2) ∈ ℕ) |
3 | flt4lem5a.b | . . . . . . 7 ⊢ (𝜑 → 𝐵 ∈ ℕ) | |
4 | 3 | nnsqcld 14224 | . . . . . 6 ⊢ (𝜑 → (𝐵↑2) ∈ ℕ) |
5 | flt4lem5a.c | . . . . . 6 ⊢ (𝜑 → 𝐶 ∈ ℕ) | |
6 | flt4lem5a.1 | . . . . . . 7 ⊢ (𝜑 → ¬ 2 ∥ 𝐴) | |
7 | 2prm 16648 | . . . . . . . 8 ⊢ 2 ∈ ℙ | |
8 | 1 | nnzd 12601 | . . . . . . . 8 ⊢ (𝜑 → 𝐴 ∈ ℤ) |
9 | prmdvdssq 16674 | . . . . . . . 8 ⊢ ((2 ∈ ℙ ∧ 𝐴 ∈ ℤ) → (2 ∥ 𝐴 ↔ 2 ∥ (𝐴↑2))) | |
10 | 7, 8, 9 | sylancr 586 | . . . . . . 7 ⊢ (𝜑 → (2 ∥ 𝐴 ↔ 2 ∥ (𝐴↑2))) |
11 | 6, 10 | mtbid 324 | . . . . . 6 ⊢ (𝜑 → ¬ 2 ∥ (𝐴↑2)) |
12 | flt4lem5a.2 | . . . . . . 7 ⊢ (𝜑 → (𝐴 gcd 𝐶) = 1) | |
13 | 2nn 12301 | . . . . . . . . 9 ⊢ 2 ∈ ℕ | |
14 | 13 | a1i 11 | . . . . . . . 8 ⊢ (𝜑 → 2 ∈ ℕ) |
15 | rplpwr 16518 | . . . . . . . 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 12244 | . . . . . . . . 9 ⊢ (𝜑 → 𝐴 ∈ ℂ) |
19 | 18 | flt4lem 41981 | . . . . . . . 8 ⊢ (𝜑 → (𝐴↑4) = ((𝐴↑2)↑2)) |
20 | 3 | nncnd 12244 | . . . . . . . . 9 ⊢ (𝜑 → 𝐵 ∈ ℂ) |
21 | 20 | flt4lem 41981 | . . . . . . . 8 ⊢ (𝜑 → (𝐵↑4) = ((𝐵↑2)↑2)) |
22 | 19, 21 | oveq12d 7432 | . . . . . . 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 2769 | . . . . . 6 ⊢ (𝜑 → (((𝐴↑2)↑2) + ((𝐵↑2)↑2)) = (𝐶↑2)) |
25 | 2, 4, 5, 11, 17, 24 | flt4lem1 41982 | . . . . 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 16779 | . . . . 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 14224 | . . 3 ⊢ (𝜑 → (𝑀↑2) ∈ ℕ) |
30 | 29 | nncnd 12244 | . 2 ⊢ (𝜑 → (𝑀↑2) ∈ ℂ) |
31 | flt4lem5a.n | . . . . . 6 ⊢ 𝑁 = (((√‘(𝐶 + (𝐵↑2))) − (√‘(𝐶 − (𝐵↑2)))) / 2) | |
32 | 31 | pythagtriplem13 16781 | . . . . 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 14224 | . . 3 ⊢ (𝜑 → (𝑁↑2) ∈ ℕ) |
35 | 34 | nncnd 12244 | . 2 ⊢ (𝜑 → (𝑁↑2) ∈ ℂ) |
36 | 26, 31 | pythagtriplem15 16783 | . . 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 41761 | 1 ⊢ (𝜑 → ((𝐴↑2) + (𝑁↑2)) = (𝑀↑2)) |
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 11125 + caddc 11127 − cmin 11460 / cdiv 11887 ℕcn 12228 2c2 12283 4c4 12285 ℤcz 12574 ↑cexp 14044 √csqrt 15198 ∥ cdvds 16216 gcd cgcd 16454 ℙcprime 16627 |
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 7732 ax-cnex 11180 ax-resscn 11181 ax-1cn 11182 ax-icn 11183 ax-addcl 11184 ax-addrcl 11185 ax-mulcl 11186 ax-mulrcl 11187 ax-mulcom 11188 ax-addass 11189 ax-mulass 11190 ax-distr 11191 ax-i2m1 11192 ax-1ne0 11193 ax-1rid 11194 ax-rnegex 11195 ax-rrecex 11196 ax-cnre 11197 ax-pre-lttri 11198 ax-pre-lttrn 11199 ax-pre-ltadd 11200 ax-pre-mulgt0 11201 ax-pre-sup 11202 |
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 7863 df-1st 7985 df-2nd 7986 df-frecs 8278 df-wrecs 8309 df-recs 8383 df-rdg 8422 df-1o 8478 df-2o 8479 df-er 8716 df-en 8954 df-dom 8955 df-sdom 8956 df-fin 8957 df-sup 9451 df-inf 9452 df-pnf 11266 df-mnf 11267 df-xr 11268 df-ltxr 11269 df-le 11270 df-sub 11462 df-neg 11463 df-div 11888 df-nn 12229 df-2 12291 df-3 12292 df-4 12293 df-n0 12489 df-z 12575 df-uz 12839 df-rp 12993 df-fz 13503 df-fl 13775 df-mod 13853 df-seq 13985 df-exp 14045 df-cj 15064 df-re 15065 df-im 15066 df-sqrt 15200 df-abs 15201 df-dvds 16217 df-gcd 16455 df-prm 16628 |
This theorem is referenced by: flt4lem5c 41990 flt4lem5d 41991 flt4lem5e 41992 |
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