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Mirrors > Home > MPE Home > Th. List > gausslemma2dlem7 | Structured version Visualization version GIF version |
Description: Lemma 7 for gausslemma2d 27320. (Contributed by AV, 13-Jul-2021.) |
Ref | Expression |
---|---|
gausslemma2d.p | ⊢ (𝜑 → 𝑃 ∈ (ℙ ∖ {2})) |
gausslemma2d.h | ⊢ 𝐻 = ((𝑃 − 1) / 2) |
gausslemma2d.r | ⊢ 𝑅 = (𝑥 ∈ (1...𝐻) ↦ if((𝑥 · 2) < (𝑃 / 2), (𝑥 · 2), (𝑃 − (𝑥 · 2)))) |
gausslemma2d.m | ⊢ 𝑀 = (⌊‘(𝑃 / 4)) |
gausslemma2d.n | ⊢ 𝑁 = (𝐻 − 𝑀) |
Ref | Expression |
---|---|
gausslemma2dlem7 | ⊢ (𝜑 → (((-1↑𝑁) · (2↑𝐻)) mod 𝑃) = 1) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | gausslemma2d.p | . . 3 ⊢ (𝜑 → 𝑃 ∈ (ℙ ∖ {2})) | |
2 | gausslemma2d.h | . . 3 ⊢ 𝐻 = ((𝑃 − 1) / 2) | |
3 | gausslemma2d.r | . . 3 ⊢ 𝑅 = (𝑥 ∈ (1...𝐻) ↦ if((𝑥 · 2) < (𝑃 / 2), (𝑥 · 2), (𝑃 − (𝑥 · 2)))) | |
4 | gausslemma2d.m | . . 3 ⊢ 𝑀 = (⌊‘(𝑃 / 4)) | |
5 | gausslemma2d.n | . . 3 ⊢ 𝑁 = (𝐻 − 𝑀) | |
6 | 1, 2, 3, 4, 5 | gausslemma2dlem6 27318 | . 2 ⊢ (𝜑 → ((!‘𝐻) mod 𝑃) = ((((-1↑𝑁) · (2↑𝐻)) · (!‘𝐻)) mod 𝑃)) |
7 | 1, 2 | gausslemma2dlem0b 27303 | . . . . . . . . . . 11 ⊢ (𝜑 → 𝐻 ∈ ℕ) |
8 | 7 | nnnn0d 12563 | . . . . . . . . . 10 ⊢ (𝜑 → 𝐻 ∈ ℕ0) |
9 | 8 | faccld 14276 | . . . . . . . . 9 ⊢ (𝜑 → (!‘𝐻) ∈ ℕ) |
10 | 9 | nncnd 12259 | . . . . . . . 8 ⊢ (𝜑 → (!‘𝐻) ∈ ℂ) |
11 | 10 | mullidd 11263 | . . . . . . 7 ⊢ (𝜑 → (1 · (!‘𝐻)) = (!‘𝐻)) |
12 | 11 | eqcomd 2734 | . . . . . 6 ⊢ (𝜑 → (!‘𝐻) = (1 · (!‘𝐻))) |
13 | 12 | oveq1d 7435 | . . . . 5 ⊢ (𝜑 → ((!‘𝐻) mod 𝑃) = ((1 · (!‘𝐻)) mod 𝑃)) |
14 | 13 | eqeq1d 2730 | . . . 4 ⊢ (𝜑 → (((!‘𝐻) mod 𝑃) = ((((-1↑𝑁) · (2↑𝐻)) · (!‘𝐻)) mod 𝑃) ↔ ((1 · (!‘𝐻)) mod 𝑃) = ((((-1↑𝑁) · (2↑𝐻)) · (!‘𝐻)) mod 𝑃))) |
15 | 1zzd 12624 | . . . . 5 ⊢ (𝜑 → 1 ∈ ℤ) | |
16 | neg1z 12629 | . . . . . . 7 ⊢ -1 ∈ ℤ | |
17 | 1, 4, 2, 5 | gausslemma2dlem0h 27309 | . . . . . . 7 ⊢ (𝜑 → 𝑁 ∈ ℕ0) |
18 | zexpcl 14074 | . . . . . . 7 ⊢ ((-1 ∈ ℤ ∧ 𝑁 ∈ ℕ0) → (-1↑𝑁) ∈ ℤ) | |
19 | 16, 17, 18 | sylancr 586 | . . . . . 6 ⊢ (𝜑 → (-1↑𝑁) ∈ ℤ) |
20 | 2z 12625 | . . . . . . 7 ⊢ 2 ∈ ℤ | |
21 | zexpcl 14074 | . . . . . . 7 ⊢ ((2 ∈ ℤ ∧ 𝐻 ∈ ℕ0) → (2↑𝐻) ∈ ℤ) | |
22 | 20, 8, 21 | sylancr 586 | . . . . . 6 ⊢ (𝜑 → (2↑𝐻) ∈ ℤ) |
23 | 19, 22 | zmulcld 12703 | . . . . 5 ⊢ (𝜑 → ((-1↑𝑁) · (2↑𝐻)) ∈ ℤ) |
24 | 9 | nnzd 12616 | . . . . 5 ⊢ (𝜑 → (!‘𝐻) ∈ ℤ) |
25 | eldifi 4125 | . . . . . 6 ⊢ (𝑃 ∈ (ℙ ∖ {2}) → 𝑃 ∈ ℙ) | |
26 | prmnn 16645 | . . . . . 6 ⊢ (𝑃 ∈ ℙ → 𝑃 ∈ ℕ) | |
27 | 1, 25, 26 | 3syl 18 | . . . . 5 ⊢ (𝜑 → 𝑃 ∈ ℕ) |
28 | 1, 2 | gausslemma2dlem0c 27304 | . . . . 5 ⊢ (𝜑 → ((!‘𝐻) gcd 𝑃) = 1) |
29 | cncongrcoprm 16641 | . . . . 5 ⊢ (((1 ∈ ℤ ∧ ((-1↑𝑁) · (2↑𝐻)) ∈ ℤ ∧ (!‘𝐻) ∈ ℤ) ∧ (𝑃 ∈ ℕ ∧ ((!‘𝐻) gcd 𝑃) = 1)) → (((1 · (!‘𝐻)) mod 𝑃) = ((((-1↑𝑁) · (2↑𝐻)) · (!‘𝐻)) mod 𝑃) ↔ (1 mod 𝑃) = (((-1↑𝑁) · (2↑𝐻)) mod 𝑃))) | |
30 | 15, 23, 24, 27, 28, 29 | syl32anc 1376 | . . . 4 ⊢ (𝜑 → (((1 · (!‘𝐻)) mod 𝑃) = ((((-1↑𝑁) · (2↑𝐻)) · (!‘𝐻)) mod 𝑃) ↔ (1 mod 𝑃) = (((-1↑𝑁) · (2↑𝐻)) mod 𝑃))) |
31 | 14, 30 | bitrd 279 | . . 3 ⊢ (𝜑 → (((!‘𝐻) mod 𝑃) = ((((-1↑𝑁) · (2↑𝐻)) · (!‘𝐻)) mod 𝑃) ↔ (1 mod 𝑃) = (((-1↑𝑁) · (2↑𝐻)) mod 𝑃))) |
32 | simpr 484 | . . . . 5 ⊢ ((𝜑 ∧ (1 mod 𝑃) = (((-1↑𝑁) · (2↑𝐻)) mod 𝑃)) → (1 mod 𝑃) = (((-1↑𝑁) · (2↑𝐻)) mod 𝑃)) | |
33 | 26 | nnred 12258 | . . . . . . . . 9 ⊢ (𝑃 ∈ ℙ → 𝑃 ∈ ℝ) |
34 | prmgt1 16668 | . . . . . . . . 9 ⊢ (𝑃 ∈ ℙ → 1 < 𝑃) | |
35 | 33, 34 | jca 511 | . . . . . . . 8 ⊢ (𝑃 ∈ ℙ → (𝑃 ∈ ℝ ∧ 1 < 𝑃)) |
36 | 25, 35 | syl 17 | . . . . . . 7 ⊢ (𝑃 ∈ (ℙ ∖ {2}) → (𝑃 ∈ ℝ ∧ 1 < 𝑃)) |
37 | 1mod 13901 | . . . . . . 7 ⊢ ((𝑃 ∈ ℝ ∧ 1 < 𝑃) → (1 mod 𝑃) = 1) | |
38 | 1, 36, 37 | 3syl 18 | . . . . . 6 ⊢ (𝜑 → (1 mod 𝑃) = 1) |
39 | 38 | adantr 480 | . . . . 5 ⊢ ((𝜑 ∧ (1 mod 𝑃) = (((-1↑𝑁) · (2↑𝐻)) mod 𝑃)) → (1 mod 𝑃) = 1) |
40 | 32, 39 | eqtr3d 2770 | . . . 4 ⊢ ((𝜑 ∧ (1 mod 𝑃) = (((-1↑𝑁) · (2↑𝐻)) mod 𝑃)) → (((-1↑𝑁) · (2↑𝐻)) mod 𝑃) = 1) |
41 | 40 | ex 412 | . . 3 ⊢ (𝜑 → ((1 mod 𝑃) = (((-1↑𝑁) · (2↑𝐻)) mod 𝑃) → (((-1↑𝑁) · (2↑𝐻)) mod 𝑃) = 1)) |
42 | 31, 41 | sylbid 239 | . 2 ⊢ (𝜑 → (((!‘𝐻) mod 𝑃) = ((((-1↑𝑁) · (2↑𝐻)) · (!‘𝐻)) mod 𝑃) → (((-1↑𝑁) · (2↑𝐻)) mod 𝑃) = 1)) |
43 | 6, 42 | mpd 15 | 1 ⊢ (𝜑 → (((-1↑𝑁) · (2↑𝐻)) mod 𝑃) = 1) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 205 ∧ wa 395 = wceq 1534 ∈ wcel 2099 ∖ cdif 3944 ifcif 4529 {csn 4629 class class class wbr 5148 ↦ cmpt 5231 ‘cfv 6548 (class class class)co 7420 ℝcr 11138 1c1 11140 · cmul 11144 < clt 11279 − cmin 11475 -cneg 11476 / cdiv 11902 ℕcn 12243 2c2 12298 4c4 12300 ℕ0cn0 12503 ℤcz 12589 ...cfz 13517 ⌊cfl 13788 mod cmo 13867 ↑cexp 14059 !cfa 14265 gcd cgcd 16469 ℙcprime 16642 |
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 2167 ax-ext 2699 ax-rep 5285 ax-sep 5299 ax-nul 5306 ax-pow 5365 ax-pr 5429 ax-un 7740 ax-inf2 9665 ax-cnex 11195 ax-resscn 11196 ax-1cn 11197 ax-icn 11198 ax-addcl 11199 ax-addrcl 11200 ax-mulcl 11201 ax-mulrcl 11202 ax-mulcom 11203 ax-addass 11204 ax-mulass 11205 ax-distr 11206 ax-i2m1 11207 ax-1ne0 11208 ax-1rid 11209 ax-rnegex 11210 ax-rrecex 11211 ax-cnre 11212 ax-pre-lttri 11213 ax-pre-lttrn 11214 ax-pre-ltadd 11215 ax-pre-mulgt0 11216 ax-pre-sup 11217 |
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 2530 df-eu 2559 df-clab 2706 df-cleq 2720 df-clel 2806 df-nfc 2881 df-ne 2938 df-nel 3044 df-ral 3059 df-rex 3068 df-rmo 3373 df-reu 3374 df-rab 3430 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 4324 df-if 4530 df-pw 4605 df-sn 4630 df-pr 4632 df-tp 4634 df-op 4636 df-uni 4909 df-int 4950 df-iun 4998 df-br 5149 df-opab 5211 df-mpt 5232 df-tr 5266 df-id 5576 df-eprel 5582 df-po 5590 df-so 5591 df-fr 5633 df-se 5634 df-we 5635 df-xp 5684 df-rel 5685 df-cnv 5686 df-co 5687 df-dm 5688 df-rn 5689 df-res 5690 df-ima 5691 df-pred 6305 df-ord 6372 df-on 6373 df-lim 6374 df-suc 6375 df-iota 6500 df-fun 6550 df-fn 6551 df-f 6552 df-f1 6553 df-fo 6554 df-f1o 6555 df-fv 6556 df-isom 6557 df-riota 7376 df-ov 7423 df-oprab 7424 df-mpo 7425 df-om 7871 df-1st 7993 df-2nd 7994 df-frecs 8287 df-wrecs 8318 df-recs 8392 df-rdg 8431 df-1o 8487 df-2o 8488 df-er 8725 df-en 8965 df-dom 8966 df-sdom 8967 df-fin 8968 df-sup 9466 df-inf 9467 df-oi 9534 df-card 9963 df-pnf 11281 df-mnf 11282 df-xr 11283 df-ltxr 11284 df-le 11285 df-sub 11477 df-neg 11478 df-div 11903 df-nn 12244 df-2 12306 df-3 12307 df-4 12308 df-5 12309 df-6 12310 df-n0 12504 df-z 12590 df-uz 12854 df-rp 13008 df-ioo 13361 df-fz 13518 df-fzo 13661 df-fl 13790 df-mod 13868 df-seq 14000 df-exp 14060 df-fac 14266 df-hash 14323 df-cj 15079 df-re 15080 df-im 15081 df-sqrt 15215 df-abs 15216 df-clim 15465 df-prod 15883 df-dvds 16232 df-gcd 16470 df-prm 16643 |
This theorem is referenced by: gausslemma2d 27320 |
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