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Mirrors > Home > MPE Home > Th. List > gausslemma2dlem5a | Structured version Visualization version GIF version |
Description: Lemma for gausslemma2dlem5 26519. (Contributed by AV, 8-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)) |
Ref | Expression |
---|---|
gausslemma2dlem5a | ⊢ (𝜑 → (∏𝑘 ∈ ((𝑀 + 1)...𝐻)(𝑅‘𝑘) mod 𝑃) = (∏𝑘 ∈ ((𝑀 + 1)...𝐻)(-1 · (𝑘 · 2)) mod 𝑃)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | gausslemma2d.p | . . . 4 ⊢ (𝜑 → 𝑃 ∈ (ℙ ∖ {2})) | |
2 | gausslemma2d.h | . . . 4 ⊢ 𝐻 = ((𝑃 − 1) / 2) | |
3 | gausslemma2d.r | . . . 4 ⊢ 𝑅 = (𝑥 ∈ (1...𝐻) ↦ if((𝑥 · 2) < (𝑃 / 2), (𝑥 · 2), (𝑃 − (𝑥 · 2)))) | |
4 | gausslemma2d.m | . . . 4 ⊢ 𝑀 = (⌊‘(𝑃 / 4)) | |
5 | 1, 2, 3, 4 | gausslemma2dlem3 26516 | . . 3 ⊢ (𝜑 → ∀𝑘 ∈ ((𝑀 + 1)...𝐻)(𝑅‘𝑘) = (𝑃 − (𝑘 · 2))) |
6 | prodeq2 15624 | . . . 4 ⊢ (∀𝑘 ∈ ((𝑀 + 1)...𝐻)(𝑅‘𝑘) = (𝑃 − (𝑘 · 2)) → ∏𝑘 ∈ ((𝑀 + 1)...𝐻)(𝑅‘𝑘) = ∏𝑘 ∈ ((𝑀 + 1)...𝐻)(𝑃 − (𝑘 · 2))) | |
7 | 6 | oveq1d 7290 | . . 3 ⊢ (∀𝑘 ∈ ((𝑀 + 1)...𝐻)(𝑅‘𝑘) = (𝑃 − (𝑘 · 2)) → (∏𝑘 ∈ ((𝑀 + 1)...𝐻)(𝑅‘𝑘) mod 𝑃) = (∏𝑘 ∈ ((𝑀 + 1)...𝐻)(𝑃 − (𝑘 · 2)) mod 𝑃)) |
8 | 5, 7 | syl 17 | . 2 ⊢ (𝜑 → (∏𝑘 ∈ ((𝑀 + 1)...𝐻)(𝑅‘𝑘) mod 𝑃) = (∏𝑘 ∈ ((𝑀 + 1)...𝐻)(𝑃 − (𝑘 · 2)) mod 𝑃)) |
9 | eldifi 4061 | . . 3 ⊢ (𝑃 ∈ (ℙ ∖ {2}) → 𝑃 ∈ ℙ) | |
10 | fzfid 13693 | . . . 4 ⊢ (𝑃 ∈ ℙ → ((𝑀 + 1)...𝐻) ∈ Fin) | |
11 | prmz 16380 | . . . . . 6 ⊢ (𝑃 ∈ ℙ → 𝑃 ∈ ℤ) | |
12 | 11 | adantr 481 | . . . . 5 ⊢ ((𝑃 ∈ ℙ ∧ 𝑘 ∈ ((𝑀 + 1)...𝐻)) → 𝑃 ∈ ℤ) |
13 | elfzelz 13256 | . . . . . . 7 ⊢ (𝑘 ∈ ((𝑀 + 1)...𝐻) → 𝑘 ∈ ℤ) | |
14 | 2z 12352 | . . . . . . . 8 ⊢ 2 ∈ ℤ | |
15 | 14 | a1i 11 | . . . . . . 7 ⊢ (𝑘 ∈ ((𝑀 + 1)...𝐻) → 2 ∈ ℤ) |
16 | 13, 15 | zmulcld 12432 | . . . . . 6 ⊢ (𝑘 ∈ ((𝑀 + 1)...𝐻) → (𝑘 · 2) ∈ ℤ) |
17 | 16 | adantl 482 | . . . . 5 ⊢ ((𝑃 ∈ ℙ ∧ 𝑘 ∈ ((𝑀 + 1)...𝐻)) → (𝑘 · 2) ∈ ℤ) |
18 | 12, 17 | zsubcld 12431 | . . . 4 ⊢ ((𝑃 ∈ ℙ ∧ 𝑘 ∈ ((𝑀 + 1)...𝐻)) → (𝑃 − (𝑘 · 2)) ∈ ℤ) |
19 | neg1z 12356 | . . . . . . 7 ⊢ -1 ∈ ℤ | |
20 | 19 | a1i 11 | . . . . . 6 ⊢ (𝑘 ∈ ((𝑀 + 1)...𝐻) → -1 ∈ ℤ) |
21 | 20, 16 | zmulcld 12432 | . . . . 5 ⊢ (𝑘 ∈ ((𝑀 + 1)...𝐻) → (-1 · (𝑘 · 2)) ∈ ℤ) |
22 | 21 | adantl 482 | . . . 4 ⊢ ((𝑃 ∈ ℙ ∧ 𝑘 ∈ ((𝑀 + 1)...𝐻)) → (-1 · (𝑘 · 2)) ∈ ℤ) |
23 | prmnn 16379 | . . . 4 ⊢ (𝑃 ∈ ℙ → 𝑃 ∈ ℕ) | |
24 | 16 | zcnd 12427 | . . . . . . . 8 ⊢ (𝑘 ∈ ((𝑀 + 1)...𝐻) → (𝑘 · 2) ∈ ℂ) |
25 | 24 | mulm1d 11427 | . . . . . . 7 ⊢ (𝑘 ∈ ((𝑀 + 1)...𝐻) → (-1 · (𝑘 · 2)) = -(𝑘 · 2)) |
26 | 25 | adantl 482 | . . . . . 6 ⊢ ((𝑃 ∈ ℙ ∧ 𝑘 ∈ ((𝑀 + 1)...𝐻)) → (-1 · (𝑘 · 2)) = -(𝑘 · 2)) |
27 | 26 | oveq1d 7290 | . . . . 5 ⊢ ((𝑃 ∈ ℙ ∧ 𝑘 ∈ ((𝑀 + 1)...𝐻)) → ((-1 · (𝑘 · 2)) mod 𝑃) = (-(𝑘 · 2) mod 𝑃)) |
28 | 16 | zred 12426 | . . . . . 6 ⊢ (𝑘 ∈ ((𝑀 + 1)...𝐻) → (𝑘 · 2) ∈ ℝ) |
29 | 23 | nnrpd 12770 | . . . . . 6 ⊢ (𝑃 ∈ ℙ → 𝑃 ∈ ℝ+) |
30 | negmod 13636 | . . . . . 6 ⊢ (((𝑘 · 2) ∈ ℝ ∧ 𝑃 ∈ ℝ+) → (-(𝑘 · 2) mod 𝑃) = ((𝑃 − (𝑘 · 2)) mod 𝑃)) | |
31 | 28, 29, 30 | syl2anr 597 | . . . . 5 ⊢ ((𝑃 ∈ ℙ ∧ 𝑘 ∈ ((𝑀 + 1)...𝐻)) → (-(𝑘 · 2) mod 𝑃) = ((𝑃 − (𝑘 · 2)) mod 𝑃)) |
32 | 27, 31 | eqtr2d 2779 | . . . 4 ⊢ ((𝑃 ∈ ℙ ∧ 𝑘 ∈ ((𝑀 + 1)...𝐻)) → ((𝑃 − (𝑘 · 2)) mod 𝑃) = ((-1 · (𝑘 · 2)) mod 𝑃)) |
33 | 10, 18, 22, 23, 32 | fprodmodd 15707 | . . 3 ⊢ (𝑃 ∈ ℙ → (∏𝑘 ∈ ((𝑀 + 1)...𝐻)(𝑃 − (𝑘 · 2)) mod 𝑃) = (∏𝑘 ∈ ((𝑀 + 1)...𝐻)(-1 · (𝑘 · 2)) mod 𝑃)) |
34 | 1, 9, 33 | 3syl 18 | . 2 ⊢ (𝜑 → (∏𝑘 ∈ ((𝑀 + 1)...𝐻)(𝑃 − (𝑘 · 2)) mod 𝑃) = (∏𝑘 ∈ ((𝑀 + 1)...𝐻)(-1 · (𝑘 · 2)) mod 𝑃)) |
35 | 8, 34 | eqtrd 2778 | 1 ⊢ (𝜑 → (∏𝑘 ∈ ((𝑀 + 1)...𝐻)(𝑅‘𝑘) mod 𝑃) = (∏𝑘 ∈ ((𝑀 + 1)...𝐻)(-1 · (𝑘 · 2)) mod 𝑃)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 396 = wceq 1539 ∈ wcel 2106 ∀wral 3064 ∖ cdif 3884 ifcif 4459 {csn 4561 class class class wbr 5074 ↦ cmpt 5157 ‘cfv 6433 (class class class)co 7275 ℝcr 10870 1c1 10872 + caddc 10874 · cmul 10876 < clt 11009 − cmin 11205 -cneg 11206 / cdiv 11632 2c2 12028 4c4 12030 ℤcz 12319 ℝ+crp 12730 ...cfz 13239 ⌊cfl 13510 mod cmo 13589 ∏cprod 15615 ℙcprime 16376 |
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 1913 ax-6 1971 ax-7 2011 ax-8 2108 ax-9 2116 ax-10 2137 ax-11 2154 ax-12 2171 ax-ext 2709 ax-rep 5209 ax-sep 5223 ax-nul 5230 ax-pow 5288 ax-pr 5352 ax-un 7588 ax-inf2 9399 ax-cnex 10927 ax-resscn 10928 ax-1cn 10929 ax-icn 10930 ax-addcl 10931 ax-addrcl 10932 ax-mulcl 10933 ax-mulrcl 10934 ax-mulcom 10935 ax-addass 10936 ax-mulass 10937 ax-distr 10938 ax-i2m1 10939 ax-1ne0 10940 ax-1rid 10941 ax-rnegex 10942 ax-rrecex 10943 ax-cnre 10944 ax-pre-lttri 10945 ax-pre-lttrn 10946 ax-pre-ltadd 10947 ax-pre-mulgt0 10948 ax-pre-sup 10949 |
This theorem depends on definitions: df-bi 206 df-an 397 df-or 845 df-3or 1087 df-3an 1088 df-tru 1542 df-fal 1552 df-ex 1783 df-nf 1787 df-sb 2068 df-mo 2540 df-eu 2569 df-clab 2716 df-cleq 2730 df-clel 2816 df-nfc 2889 df-ne 2944 df-nel 3050 df-ral 3069 df-rex 3070 df-rmo 3071 df-reu 3072 df-rab 3073 df-v 3434 df-sbc 3717 df-csb 3833 df-dif 3890 df-un 3892 df-in 3894 df-ss 3904 df-pss 3906 df-nul 4257 df-if 4460 df-pw 4535 df-sn 4562 df-pr 4564 df-op 4568 df-uni 4840 df-int 4880 df-iun 4926 df-br 5075 df-opab 5137 df-mpt 5158 df-tr 5192 df-id 5489 df-eprel 5495 df-po 5503 df-so 5504 df-fr 5544 df-se 5545 df-we 5546 df-xp 5595 df-rel 5596 df-cnv 5597 df-co 5598 df-dm 5599 df-rn 5600 df-res 5601 df-ima 5602 df-pred 6202 df-ord 6269 df-on 6270 df-lim 6271 df-suc 6272 df-iota 6391 df-fun 6435 df-fn 6436 df-f 6437 df-f1 6438 df-fo 6439 df-f1o 6440 df-fv 6441 df-isom 6442 df-riota 7232 df-ov 7278 df-oprab 7279 df-mpo 7280 df-om 7713 df-1st 7831 df-2nd 7832 df-frecs 8097 df-wrecs 8128 df-recs 8202 df-rdg 8241 df-1o 8297 df-2o 8298 df-er 8498 df-en 8734 df-dom 8735 df-sdom 8736 df-fin 8737 df-sup 9201 df-inf 9202 df-oi 9269 df-card 9697 df-pnf 11011 df-mnf 11012 df-xr 11013 df-ltxr 11014 df-le 11015 df-sub 11207 df-neg 11208 df-div 11633 df-nn 11974 df-2 12036 df-3 12037 df-4 12038 df-n0 12234 df-z 12320 df-uz 12583 df-rp 12731 df-fz 13240 df-fzo 13383 df-fl 13512 df-mod 13590 df-seq 13722 df-exp 13783 df-hash 14045 df-cj 14810 df-re 14811 df-im 14812 df-sqrt 14946 df-abs 14947 df-clim 15197 df-prod 15616 df-dvds 15964 df-prm 16377 |
This theorem is referenced by: gausslemma2dlem5 26519 |
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