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Mirrors > Home > MPE Home > Th. List > gausslemma2dlem7 | Structured version Visualization version GIF version |
Description: Lemma 7 for gausslemma2d 25668. (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 25666 | . 2 ⊢ (𝜑 → ((!‘𝐻) mod 𝑃) = ((((-1↑𝑁) · (2↑𝐻)) · (!‘𝐻)) mod 𝑃)) |
7 | 1, 2 | gausslemma2dlem0b 25651 | . . . . . . . . . . 11 ⊢ (𝜑 → 𝐻 ∈ ℕ) |
8 | 7 | nnnn0d 11766 | . . . . . . . . . 10 ⊢ (𝜑 → 𝐻 ∈ ℕ0) |
9 | 8 | faccld 13458 | . . . . . . . . 9 ⊢ (𝜑 → (!‘𝐻) ∈ ℕ) |
10 | 9 | nncnd 11456 | . . . . . . . 8 ⊢ (𝜑 → (!‘𝐻) ∈ ℂ) |
11 | 10 | mulid2d 10457 | . . . . . . 7 ⊢ (𝜑 → (1 · (!‘𝐻)) = (!‘𝐻)) |
12 | 11 | eqcomd 2779 | . . . . . 6 ⊢ (𝜑 → (!‘𝐻) = (1 · (!‘𝐻))) |
13 | 12 | oveq1d 6990 | . . . . 5 ⊢ (𝜑 → ((!‘𝐻) mod 𝑃) = ((1 · (!‘𝐻)) mod 𝑃)) |
14 | 13 | eqeq1d 2775 | . . . 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 11825 | . . . . 5 ⊢ (𝜑 → 1 ∈ ℤ) | |
16 | neg1z 11830 | . . . . . . 7 ⊢ -1 ∈ ℤ | |
17 | 1, 4, 2, 5 | gausslemma2dlem0h 25657 | . . . . . . 7 ⊢ (𝜑 → 𝑁 ∈ ℕ0) |
18 | zexpcl 13258 | . . . . . . 7 ⊢ ((-1 ∈ ℤ ∧ 𝑁 ∈ ℕ0) → (-1↑𝑁) ∈ ℤ) | |
19 | 16, 17, 18 | sylancr 579 | . . . . . 6 ⊢ (𝜑 → (-1↑𝑁) ∈ ℤ) |
20 | 2z 11826 | . . . . . . 7 ⊢ 2 ∈ ℤ | |
21 | zexpcl 13258 | . . . . . . 7 ⊢ ((2 ∈ ℤ ∧ 𝐻 ∈ ℕ0) → (2↑𝐻) ∈ ℤ) | |
22 | 20, 8, 21 | sylancr 579 | . . . . . 6 ⊢ (𝜑 → (2↑𝐻) ∈ ℤ) |
23 | 19, 22 | zmulcld 11905 | . . . . 5 ⊢ (𝜑 → ((-1↑𝑁) · (2↑𝐻)) ∈ ℤ) |
24 | 9 | nnzd 11898 | . . . . 5 ⊢ (𝜑 → (!‘𝐻) ∈ ℤ) |
25 | eldifi 3988 | . . . . . 6 ⊢ (𝑃 ∈ (ℙ ∖ {2}) → 𝑃 ∈ ℙ) | |
26 | prmnn 15873 | . . . . . 6 ⊢ (𝑃 ∈ ℙ → 𝑃 ∈ ℕ) | |
27 | 1, 25, 26 | 3syl 18 | . . . . 5 ⊢ (𝜑 → 𝑃 ∈ ℕ) |
28 | 1, 2 | gausslemma2dlem0c 25652 | . . . . 5 ⊢ (𝜑 → ((!‘𝐻) gcd 𝑃) = 1) |
29 | cncongrcoprm 15869 | . . . . 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 1359 | . . . 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 271 | . . 3 ⊢ (𝜑 → (((!‘𝐻) mod 𝑃) = ((((-1↑𝑁) · (2↑𝐻)) · (!‘𝐻)) mod 𝑃) ↔ (1 mod 𝑃) = (((-1↑𝑁) · (2↑𝐻)) mod 𝑃))) |
32 | simpr 477 | . . . . 5 ⊢ ((𝜑 ∧ (1 mod 𝑃) = (((-1↑𝑁) · (2↑𝐻)) mod 𝑃)) → (1 mod 𝑃) = (((-1↑𝑁) · (2↑𝐻)) mod 𝑃)) | |
33 | 26 | nnred 11455 | . . . . . . . . 9 ⊢ (𝑃 ∈ ℙ → 𝑃 ∈ ℝ) |
34 | prmgt1 15896 | . . . . . . . . 9 ⊢ (𝑃 ∈ ℙ → 1 < 𝑃) | |
35 | 33, 34 | jca 504 | . . . . . . . 8 ⊢ (𝑃 ∈ ℙ → (𝑃 ∈ ℝ ∧ 1 < 𝑃)) |
36 | 25, 35 | syl 17 | . . . . . . 7 ⊢ (𝑃 ∈ (ℙ ∖ {2}) → (𝑃 ∈ ℝ ∧ 1 < 𝑃)) |
37 | 1mod 13085 | . . . . . . 7 ⊢ ((𝑃 ∈ ℝ ∧ 1 < 𝑃) → (1 mod 𝑃) = 1) | |
38 | 1, 36, 37 | 3syl 18 | . . . . . 6 ⊢ (𝜑 → (1 mod 𝑃) = 1) |
39 | 38 | adantr 473 | . . . . 5 ⊢ ((𝜑 ∧ (1 mod 𝑃) = (((-1↑𝑁) · (2↑𝐻)) mod 𝑃)) → (1 mod 𝑃) = 1) |
40 | 32, 39 | eqtr3d 2811 | . . . 4 ⊢ ((𝜑 ∧ (1 mod 𝑃) = (((-1↑𝑁) · (2↑𝐻)) mod 𝑃)) → (((-1↑𝑁) · (2↑𝐻)) mod 𝑃) = 1) |
41 | 40 | ex 405 | . . 3 ⊢ (𝜑 → ((1 mod 𝑃) = (((-1↑𝑁) · (2↑𝐻)) mod 𝑃) → (((-1↑𝑁) · (2↑𝐻)) mod 𝑃) = 1)) |
42 | 31, 41 | sylbid 232 | . 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 198 ∧ wa 387 = wceq 1508 ∈ wcel 2051 ∖ cdif 3821 ifcif 4345 {csn 4436 class class class wbr 4926 ↦ cmpt 5005 ‘cfv 6186 (class class class)co 6975 ℝcr 10333 1c1 10335 · cmul 10339 < clt 10473 − cmin 10669 -cneg 10670 / cdiv 11097 ℕcn 11438 2c2 11494 4c4 11496 ℕ0cn0 11706 ℤcz 11792 ...cfz 12707 ⌊cfl 12974 mod cmo 13051 ↑cexp 13243 !cfa 13447 gcd cgcd 15702 ℙcprime 15870 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1759 ax-4 1773 ax-5 1870 ax-6 1929 ax-7 1966 ax-8 2053 ax-9 2060 ax-10 2080 ax-11 2094 ax-12 2107 ax-13 2302 ax-ext 2745 ax-rep 5046 ax-sep 5057 ax-nul 5064 ax-pow 5116 ax-pr 5183 ax-un 7278 ax-inf2 8897 ax-cnex 10390 ax-resscn 10391 ax-1cn 10392 ax-icn 10393 ax-addcl 10394 ax-addrcl 10395 ax-mulcl 10396 ax-mulrcl 10397 ax-mulcom 10398 ax-addass 10399 ax-mulass 10400 ax-distr 10401 ax-i2m1 10402 ax-1ne0 10403 ax-1rid 10404 ax-rnegex 10405 ax-rrecex 10406 ax-cnre 10407 ax-pre-lttri 10408 ax-pre-lttrn 10409 ax-pre-ltadd 10410 ax-pre-mulgt0 10411 ax-pre-sup 10412 |
This theorem depends on definitions: df-bi 199 df-an 388 df-or 835 df-3or 1070 df-3an 1071 df-tru 1511 df-fal 1521 df-ex 1744 df-nf 1748 df-sb 2017 df-mo 2548 df-eu 2585 df-clab 2754 df-cleq 2766 df-clel 2841 df-nfc 2913 df-ne 2963 df-nel 3069 df-ral 3088 df-rex 3089 df-reu 3090 df-rmo 3091 df-rab 3092 df-v 3412 df-sbc 3677 df-csb 3782 df-dif 3827 df-un 3829 df-in 3831 df-ss 3838 df-pss 3840 df-nul 4174 df-if 4346 df-pw 4419 df-sn 4437 df-pr 4439 df-tp 4441 df-op 4443 df-uni 4710 df-int 4747 df-iun 4791 df-br 4927 df-opab 4989 df-mpt 5006 df-tr 5028 df-id 5309 df-eprel 5314 df-po 5323 df-so 5324 df-fr 5363 df-se 5364 df-we 5365 df-xp 5410 df-rel 5411 df-cnv 5412 df-co 5413 df-dm 5414 df-rn 5415 df-res 5416 df-ima 5417 df-pred 5984 df-ord 6030 df-on 6031 df-lim 6032 df-suc 6033 df-iota 6150 df-fun 6188 df-fn 6189 df-f 6190 df-f1 6191 df-fo 6192 df-f1o 6193 df-fv 6194 df-isom 6195 df-riota 6936 df-ov 6978 df-oprab 6979 df-mpo 6980 df-om 7396 df-1st 7500 df-2nd 7501 df-wrecs 7749 df-recs 7811 df-rdg 7849 df-1o 7904 df-2o 7905 df-oadd 7908 df-er 8088 df-en 8306 df-dom 8307 df-sdom 8308 df-fin 8309 df-sup 8700 df-inf 8701 df-oi 8768 df-card 9161 df-pnf 10475 df-mnf 10476 df-xr 10477 df-ltxr 10478 df-le 10479 df-sub 10671 df-neg 10672 df-div 11098 df-nn 11439 df-2 11502 df-3 11503 df-4 11504 df-5 11505 df-6 11506 df-n0 11707 df-z 11793 df-uz 12058 df-rp 12204 df-ioo 12557 df-fz 12708 df-fzo 12849 df-fl 12976 df-mod 13052 df-seq 13184 df-exp 13244 df-fac 13448 df-hash 13505 df-cj 14318 df-re 14319 df-im 14320 df-sqrt 14454 df-abs 14455 df-clim 14705 df-prod 15119 df-dvds 15467 df-gcd 15703 df-prm 15871 |
This theorem is referenced by: gausslemma2d 25668 |
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