![]() |
Mathbox for Steven Nguyen |
< Previous
Next >
Nearby theorems |
|
Mirrors > Home > MPE Home > Th. List > Mathboxes > flt4lem5b | Structured version Visualization version GIF version |
Description: Part 2 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 |
---|---|
flt4lem5b | ⊢ (𝜑 → (2 · (𝑀 · 𝑁)) = (𝐵↑2)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | flt4lem5a.a | . . . . 5 ⊢ (𝜑 → 𝐴 ∈ ℕ) | |
2 | 1 | nnsqcld 14244 | . . . 4 ⊢ (𝜑 → (𝐴↑2) ∈ ℕ) |
3 | flt4lem5a.b | . . . . 5 ⊢ (𝜑 → 𝐵 ∈ ℕ) | |
4 | 3 | nnsqcld 14244 | . . . 4 ⊢ (𝜑 → (𝐵↑2) ∈ ℕ) |
5 | flt4lem5a.c | . . . 4 ⊢ (𝜑 → 𝐶 ∈ ℕ) | |
6 | flt4lem5a.1 | . . . . 5 ⊢ (𝜑 → ¬ 2 ∥ 𝐴) | |
7 | 2prm 16668 | . . . . . 6 ⊢ 2 ∈ ℙ | |
8 | 1 | nnzd 12621 | . . . . . 6 ⊢ (𝜑 → 𝐴 ∈ ℤ) |
9 | prmdvdssq 16694 | . . . . . 6 ⊢ ((2 ∈ ℙ ∧ 𝐴 ∈ ℤ) → (2 ∥ 𝐴 ↔ 2 ∥ (𝐴↑2))) | |
10 | 7, 8, 9 | sylancr 585 | . . . . 5 ⊢ (𝜑 → (2 ∥ 𝐴 ↔ 2 ∥ (𝐴↑2))) |
11 | 6, 10 | mtbid 323 | . . . 4 ⊢ (𝜑 → ¬ 2 ∥ (𝐴↑2)) |
12 | flt4lem5a.2 | . . . . 5 ⊢ (𝜑 → (𝐴 gcd 𝐶) = 1) | |
13 | 2nn 12321 | . . . . . . 7 ⊢ 2 ∈ ℕ | |
14 | 13 | a1i 11 | . . . . . 6 ⊢ (𝜑 → 2 ∈ ℕ) |
15 | rplpwr 16538 | . . . . . 6 ⊢ ((𝐴 ∈ ℕ ∧ 𝐶 ∈ ℕ ∧ 2 ∈ ℕ) → ((𝐴 gcd 𝐶) = 1 → ((𝐴↑2) gcd 𝐶) = 1)) | |
16 | 1, 5, 14, 15 | syl3anc 1368 | . . . . 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 12264 | . . . . . . 7 ⊢ (𝜑 → 𝐴 ∈ ℂ) |
19 | 18 | flt4lem 42072 | . . . . . 6 ⊢ (𝜑 → (𝐴↑4) = ((𝐴↑2)↑2)) |
20 | 3 | nncnd 12264 | . . . . . . 7 ⊢ (𝜑 → 𝐵 ∈ ℂ) |
21 | 20 | flt4lem 42072 | . . . . . 6 ⊢ (𝜑 → (𝐵↑4) = ((𝐵↑2)↑2)) |
22 | 19, 21 | oveq12d 7442 | . . . . 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 42073 | . . 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.m | . . . 4 ⊢ 𝑀 = (((√‘(𝐶 + (𝐵↑2))) + (√‘(𝐶 − (𝐵↑2)))) / 2) | |
27 | flt4lem5a.n | . . . 4 ⊢ 𝑁 = (((√‘(𝐶 + (𝐵↑2))) − (√‘(𝐶 − (𝐵↑2)))) / 2) | |
28 | 26, 27 | pythagtriplem16 16804 | . . 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))) → (𝐵↑2) = (2 · (𝑀 · 𝑁))) |
29 | 25, 28 | syl 17 | . 2 ⊢ (𝜑 → (𝐵↑2) = (2 · (𝑀 · 𝑁))) |
30 | 29 | eqcomd 2733 | 1 ⊢ (𝜑 → (2 · (𝑀 · 𝑁)) = (𝐵↑2)) |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 → wi 4 ↔ wb 205 ∧ wa 394 ∧ w3a 1084 = wceq 1533 ∈ wcel 2098 class class class wbr 5150 ‘cfv 6551 (class class class)co 7424 1c1 11145 + caddc 11147 · cmul 11149 − cmin 11480 / cdiv 11907 ℕcn 12248 2c2 12303 4c4 12305 ℤcz 12594 ↑cexp 14064 √csqrt 15218 ∥ cdvds 16236 gcd cgcd 16474 ℙcprime 16647 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1789 ax-4 1803 ax-5 1905 ax-6 1963 ax-7 2003 ax-8 2100 ax-9 2108 ax-10 2129 ax-11 2146 ax-12 2166 ax-ext 2698 ax-sep 5301 ax-nul 5308 ax-pow 5367 ax-pr 5431 ax-un 7744 ax-cnex 11200 ax-resscn 11201 ax-1cn 11202 ax-icn 11203 ax-addcl 11204 ax-addrcl 11205 ax-mulcl 11206 ax-mulrcl 11207 ax-mulcom 11208 ax-addass 11209 ax-mulass 11210 ax-distr 11211 ax-i2m1 11212 ax-1ne0 11213 ax-1rid 11214 ax-rnegex 11215 ax-rrecex 11216 ax-cnre 11217 ax-pre-lttri 11218 ax-pre-lttrn 11219 ax-pre-ltadd 11220 ax-pre-mulgt0 11221 ax-pre-sup 11222 |
This theorem depends on definitions: df-bi 206 df-an 395 df-or 846 df-3or 1085 df-3an 1086 df-tru 1536 df-fal 1546 df-ex 1774 df-nf 1778 df-sb 2060 df-mo 2529 df-eu 2558 df-clab 2705 df-cleq 2719 df-clel 2805 df-nfc 2880 df-ne 2937 df-nel 3043 df-ral 3058 df-rex 3067 df-rmo 3372 df-reu 3373 df-rab 3429 df-v 3473 df-sbc 3777 df-csb 3893 df-dif 3950 df-un 3952 df-in 3954 df-ss 3964 df-pss 3966 df-nul 4325 df-if 4531 df-pw 4606 df-sn 4631 df-pr 4633 df-op 4637 df-uni 4911 df-iun 5000 df-br 5151 df-opab 5213 df-mpt 5234 df-tr 5268 df-id 5578 df-eprel 5584 df-po 5592 df-so 5593 df-fr 5635 df-we 5637 df-xp 5686 df-rel 5687 df-cnv 5688 df-co 5689 df-dm 5690 df-rn 5691 df-res 5692 df-ima 5693 df-pred 6308 df-ord 6375 df-on 6376 df-lim 6377 df-suc 6378 df-iota 6503 df-fun 6553 df-fn 6554 df-f 6555 df-f1 6556 df-fo 6557 df-f1o 6558 df-fv 6559 df-riota 7380 df-ov 7427 df-oprab 7428 df-mpo 7429 df-om 7875 df-1st 7997 df-2nd 7998 df-frecs 8291 df-wrecs 8322 df-recs 8396 df-rdg 8435 df-1o 8491 df-2o 8492 df-er 8729 df-en 8969 df-dom 8970 df-sdom 8971 df-fin 8972 df-sup 9471 df-inf 9472 df-pnf 11286 df-mnf 11287 df-xr 11288 df-ltxr 11289 df-le 11290 df-sub 11482 df-neg 11483 df-div 11908 df-nn 12249 df-2 12311 df-3 12312 df-4 12313 df-n0 12509 df-z 12595 df-uz 12859 df-rp 13013 df-fz 13523 df-fl 13795 df-mod 13873 df-seq 14005 df-exp 14065 df-cj 15084 df-re 15085 df-im 15086 df-sqrt 15220 df-abs 15221 df-dvds 16237 df-gcd 16475 df-prm 16648 |
This theorem is referenced by: flt4lem5e 42083 |
Copyright terms: Public domain | W3C validator |