![]() |
Intuitionistic Logic Explorer |
< Previous
Next >
Nearby theorems |
|
Mirrors > Home > ILE Home > Th. List > gausslemma2dlem7 | GIF version |
Description: Lemma 7 for gausslemma2d 15127. (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 15125 | . 2 ⊢ (𝜑 → ((!‘𝐻) mod 𝑃) = ((((-1↑𝑁) · (2↑𝐻)) · (!‘𝐻)) mod 𝑃)) |
7 | 1, 2 | gausslemma2dlem0b 15108 | . . . . . . . . . . 11 ⊢ (𝜑 → 𝐻 ∈ ℕ) |
8 | 7 | nnnn0d 9283 | . . . . . . . . . 10 ⊢ (𝜑 → 𝐻 ∈ ℕ0) |
9 | 8 | faccld 10794 | . . . . . . . . 9 ⊢ (𝜑 → (!‘𝐻) ∈ ℕ) |
10 | 9 | nncnd 8986 | . . . . . . . 8 ⊢ (𝜑 → (!‘𝐻) ∈ ℂ) |
11 | 10 | mullidd 8027 | . . . . . . 7 ⊢ (𝜑 → (1 · (!‘𝐻)) = (!‘𝐻)) |
12 | 11 | eqcomd 2199 | . . . . . 6 ⊢ (𝜑 → (!‘𝐻) = (1 · (!‘𝐻))) |
13 | 12 | oveq1d 5925 | . . . . 5 ⊢ (𝜑 → ((!‘𝐻) mod 𝑃) = ((1 · (!‘𝐻)) mod 𝑃)) |
14 | 13 | eqeq1d 2202 | . . . 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 9334 | . . . . 5 ⊢ (𝜑 → 1 ∈ ℤ) | |
16 | neg1z 9339 | . . . . . . 7 ⊢ -1 ∈ ℤ | |
17 | 1, 4, 2, 5 | gausslemma2dlem0h 15114 | . . . . . . 7 ⊢ (𝜑 → 𝑁 ∈ ℕ0) |
18 | zexpcl 10612 | . . . . . . 7 ⊢ ((-1 ∈ ℤ ∧ 𝑁 ∈ ℕ0) → (-1↑𝑁) ∈ ℤ) | |
19 | 16, 17, 18 | sylancr 414 | . . . . . 6 ⊢ (𝜑 → (-1↑𝑁) ∈ ℤ) |
20 | 2z 9335 | . . . . . . 7 ⊢ 2 ∈ ℤ | |
21 | zexpcl 10612 | . . . . . . 7 ⊢ ((2 ∈ ℤ ∧ 𝐻 ∈ ℕ0) → (2↑𝐻) ∈ ℤ) | |
22 | 20, 8, 21 | sylancr 414 | . . . . . 6 ⊢ (𝜑 → (2↑𝐻) ∈ ℤ) |
23 | 19, 22 | zmulcld 9435 | . . . . 5 ⊢ (𝜑 → ((-1↑𝑁) · (2↑𝐻)) ∈ ℤ) |
24 | 9 | nnzd 9428 | . . . . 5 ⊢ (𝜑 → (!‘𝐻) ∈ ℤ) |
25 | 1 | gausslemma2dlem0a 15107 | . . . . 5 ⊢ (𝜑 → 𝑃 ∈ ℕ) |
26 | 1, 2 | gausslemma2dlem0c 15109 | . . . . 5 ⊢ (𝜑 → ((!‘𝐻) gcd 𝑃) = 1) |
27 | cncongrcoprm 12231 | . . . . 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))) | |
28 | 15, 23, 24, 25, 26, 27 | syl32anc 1257 | . . . 4 ⊢ (𝜑 → (((1 · (!‘𝐻)) mod 𝑃) = ((((-1↑𝑁) · (2↑𝐻)) · (!‘𝐻)) mod 𝑃) ↔ (1 mod 𝑃) = (((-1↑𝑁) · (2↑𝐻)) mod 𝑃))) |
29 | 14, 28 | bitrd 188 | . . 3 ⊢ (𝜑 → (((!‘𝐻) mod 𝑃) = ((((-1↑𝑁) · (2↑𝐻)) · (!‘𝐻)) mod 𝑃) ↔ (1 mod 𝑃) = (((-1↑𝑁) · (2↑𝐻)) mod 𝑃))) |
30 | simpr 110 | . . . . 5 ⊢ ((𝜑 ∧ (1 mod 𝑃) = (((-1↑𝑁) · (2↑𝐻)) mod 𝑃)) → (1 mod 𝑃) = (((-1↑𝑁) · (2↑𝐻)) mod 𝑃)) | |
31 | nnq 9688 | . . . . . . . 8 ⊢ (𝑃 ∈ ℕ → 𝑃 ∈ ℚ) | |
32 | 25, 31 | syl 14 | . . . . . . 7 ⊢ (𝜑 → 𝑃 ∈ ℚ) |
33 | 1 | eldifad 3164 | . . . . . . . 8 ⊢ (𝜑 → 𝑃 ∈ ℙ) |
34 | prmgt1 12257 | . . . . . . . 8 ⊢ (𝑃 ∈ ℙ → 1 < 𝑃) | |
35 | 33, 34 | syl 14 | . . . . . . 7 ⊢ (𝜑 → 1 < 𝑃) |
36 | q1mod 10417 | . . . . . . 7 ⊢ ((𝑃 ∈ ℚ ∧ 1 < 𝑃) → (1 mod 𝑃) = 1) | |
37 | 32, 35, 36 | syl2anc 411 | . . . . . 6 ⊢ (𝜑 → (1 mod 𝑃) = 1) |
38 | 37 | adantr 276 | . . . . 5 ⊢ ((𝜑 ∧ (1 mod 𝑃) = (((-1↑𝑁) · (2↑𝐻)) mod 𝑃)) → (1 mod 𝑃) = 1) |
39 | 30, 38 | eqtr3d 2228 | . . . 4 ⊢ ((𝜑 ∧ (1 mod 𝑃) = (((-1↑𝑁) · (2↑𝐻)) mod 𝑃)) → (((-1↑𝑁) · (2↑𝐻)) mod 𝑃) = 1) |
40 | 39 | ex 115 | . . 3 ⊢ (𝜑 → ((1 mod 𝑃) = (((-1↑𝑁) · (2↑𝐻)) mod 𝑃) → (((-1↑𝑁) · (2↑𝐻)) mod 𝑃) = 1)) |
41 | 29, 40 | sylbid 150 | . 2 ⊢ (𝜑 → (((!‘𝐻) mod 𝑃) = ((((-1↑𝑁) · (2↑𝐻)) · (!‘𝐻)) mod 𝑃) → (((-1↑𝑁) · (2↑𝐻)) mod 𝑃) = 1)) |
42 | 6, 41 | mpd 13 | 1 ⊢ (𝜑 → (((-1↑𝑁) · (2↑𝐻)) mod 𝑃) = 1) |
Colors of variables: wff set class |
Syntax hints: → wi 4 ∧ wa 104 ↔ wb 105 = wceq 1364 ∈ wcel 2164 ∖ cdif 3150 ifcif 3557 {csn 3618 class class class wbr 4029 ↦ cmpt 4090 ‘cfv 5246 (class class class)co 5910 1c1 7863 · cmul 7867 < clt 8044 − cmin 8180 -cneg 8181 / cdiv 8681 ℕcn 8972 2c2 9023 4c4 9025 ℕ0cn0 9230 ℤcz 9307 ℚcq 9674 ...cfz 10064 ⌊cfl 10327 mod cmo 10383 ↑cexp 10596 !cfa 10783 gcd cgcd 12066 ℙcprime 12232 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-ia1 106 ax-ia2 107 ax-ia3 108 ax-in1 615 ax-in2 616 ax-io 710 ax-5 1458 ax-7 1459 ax-gen 1460 ax-ie1 1504 ax-ie2 1505 ax-8 1515 ax-10 1516 ax-11 1517 ax-i12 1518 ax-bndl 1520 ax-4 1521 ax-17 1537 ax-i9 1541 ax-ial 1545 ax-i5r 1546 ax-13 2166 ax-14 2167 ax-ext 2175 ax-coll 4144 ax-sep 4147 ax-nul 4155 ax-pow 4203 ax-pr 4238 ax-un 4462 ax-setind 4565 ax-iinf 4616 ax-cnex 7953 ax-resscn 7954 ax-1cn 7955 ax-1re 7956 ax-icn 7957 ax-addcl 7958 ax-addrcl 7959 ax-mulcl 7960 ax-mulrcl 7961 ax-addcom 7962 ax-mulcom 7963 ax-addass 7964 ax-mulass 7965 ax-distr 7966 ax-i2m1 7967 ax-0lt1 7968 ax-1rid 7969 ax-0id 7970 ax-rnegex 7971 ax-precex 7972 ax-cnre 7973 ax-pre-ltirr 7974 ax-pre-ltwlin 7975 ax-pre-lttrn 7976 ax-pre-apti 7977 ax-pre-ltadd 7978 ax-pre-mulgt0 7979 ax-pre-mulext 7980 ax-arch 7981 ax-caucvg 7982 |
This theorem depends on definitions: df-bi 117 df-stab 832 df-dc 836 df-3or 981 df-3an 982 df-tru 1367 df-fal 1370 df-xor 1387 df-nf 1472 df-sb 1774 df-eu 2045 df-mo 2046 df-clab 2180 df-cleq 2186 df-clel 2189 df-nfc 2325 df-ne 2365 df-nel 2460 df-ral 2477 df-rex 2478 df-reu 2479 df-rmo 2480 df-rab 2481 df-v 2762 df-sbc 2986 df-csb 3081 df-dif 3155 df-un 3157 df-in 3159 df-ss 3166 df-nul 3447 df-if 3558 df-pw 3603 df-sn 3624 df-pr 3625 df-tp 3626 df-op 3627 df-uni 3836 df-int 3871 df-iun 3914 df-br 4030 df-opab 4091 df-mpt 4092 df-tr 4128 df-id 4322 df-po 4325 df-iso 4326 df-iord 4395 df-on 4397 df-ilim 4398 df-suc 4400 df-iom 4619 df-xp 4661 df-rel 4662 df-cnv 4663 df-co 4664 df-dm 4665 df-rn 4666 df-res 4667 df-ima 4668 df-iota 5207 df-fun 5248 df-fn 5249 df-f 5250 df-f1 5251 df-fo 5252 df-f1o 5253 df-fv 5254 df-isom 5255 df-riota 5865 df-ov 5913 df-oprab 5914 df-mpo 5915 df-1st 6184 df-2nd 6185 df-recs 6349 df-irdg 6414 df-frec 6435 df-1o 6460 df-2o 6461 df-oadd 6464 df-er 6578 df-en 6786 df-dom 6787 df-fin 6788 df-sup 7033 df-pnf 8046 df-mnf 8047 df-xr 8048 df-ltxr 8049 df-le 8050 df-sub 8182 df-neg 8183 df-reap 8584 df-ap 8591 df-div 8682 df-inn 8973 df-2 9031 df-3 9032 df-4 9033 df-5 9034 df-6 9035 df-n0 9231 df-z 9308 df-uz 9583 df-q 9675 df-rp 9710 df-ioo 9948 df-fz 10065 df-fzo 10199 df-fl 10329 df-mod 10384 df-seqfrec 10509 df-exp 10597 df-fac 10784 df-ihash 10834 df-cj 10973 df-re 10974 df-im 10975 df-rsqrt 11129 df-abs 11130 df-clim 11409 df-proddc 11681 df-dvds 11918 df-gcd 12067 df-prm 12233 |
This theorem is referenced by: gausslemma2d 15127 |
Copyright terms: Public domain | W3C validator |