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| Mirrors > Home > MPE Home > Th. List > gausslemma2dlem7 | Structured version Visualization version GIF version | ||
| Description: Lemma 7 for gausslemma2d 27418. (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 27416 | . 2 ⊢ (𝜑 → ((!‘𝐻) mod 𝑃) = ((((-1↑𝑁) · (2↑𝐻)) · (!‘𝐻)) mod 𝑃)) |
| 7 | 1, 2 | gausslemma2dlem0b 27401 | . . . . . . . . . . 11 ⊢ (𝜑 → 𝐻 ∈ ℕ) |
| 8 | 7 | nnnn0d 12587 | . . . . . . . . . 10 ⊢ (𝜑 → 𝐻 ∈ ℕ0) |
| 9 | 8 | faccld 14323 | . . . . . . . . 9 ⊢ (𝜑 → (!‘𝐻) ∈ ℕ) |
| 10 | 9 | nncnd 12282 | . . . . . . . 8 ⊢ (𝜑 → (!‘𝐻) ∈ ℂ) |
| 11 | 10 | mullidd 11279 | . . . . . . 7 ⊢ (𝜑 → (1 · (!‘𝐻)) = (!‘𝐻)) |
| 12 | 11 | eqcomd 2743 | . . . . . 6 ⊢ (𝜑 → (!‘𝐻) = (1 · (!‘𝐻))) |
| 13 | 12 | oveq1d 7446 | . . . . 5 ⊢ (𝜑 → ((!‘𝐻) mod 𝑃) = ((1 · (!‘𝐻)) mod 𝑃)) |
| 14 | 13 | eqeq1d 2739 | . . . 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 12648 | . . . . 5 ⊢ (𝜑 → 1 ∈ ℤ) | |
| 16 | neg1z 12653 | . . . . . . 7 ⊢ -1 ∈ ℤ | |
| 17 | 1, 4, 2, 5 | gausslemma2dlem0h 27407 | . . . . . . 7 ⊢ (𝜑 → 𝑁 ∈ ℕ0) |
| 18 | zexpcl 14117 | . . . . . . 7 ⊢ ((-1 ∈ ℤ ∧ 𝑁 ∈ ℕ0) → (-1↑𝑁) ∈ ℤ) | |
| 19 | 16, 17, 18 | sylancr 587 | . . . . . 6 ⊢ (𝜑 → (-1↑𝑁) ∈ ℤ) |
| 20 | 2z 12649 | . . . . . . 7 ⊢ 2 ∈ ℤ | |
| 21 | zexpcl 14117 | . . . . . . 7 ⊢ ((2 ∈ ℤ ∧ 𝐻 ∈ ℕ0) → (2↑𝐻) ∈ ℤ) | |
| 22 | 20, 8, 21 | sylancr 587 | . . . . . 6 ⊢ (𝜑 → (2↑𝐻) ∈ ℤ) |
| 23 | 19, 22 | zmulcld 12728 | . . . . 5 ⊢ (𝜑 → ((-1↑𝑁) · (2↑𝐻)) ∈ ℤ) |
| 24 | 9 | nnzd 12640 | . . . . 5 ⊢ (𝜑 → (!‘𝐻) ∈ ℤ) |
| 25 | eldifi 4131 | . . . . . 6 ⊢ (𝑃 ∈ (ℙ ∖ {2}) → 𝑃 ∈ ℙ) | |
| 26 | prmnn 16711 | . . . . . 6 ⊢ (𝑃 ∈ ℙ → 𝑃 ∈ ℕ) | |
| 27 | 1, 25, 26 | 3syl 18 | . . . . 5 ⊢ (𝜑 → 𝑃 ∈ ℕ) |
| 28 | 1, 2 | gausslemma2dlem0c 27402 | . . . . 5 ⊢ (𝜑 → ((!‘𝐻) gcd 𝑃) = 1) |
| 29 | cncongrcoprm 16707 | . . . . 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 1380 | . . . 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 12281 | . . . . . . . 8 ⊢ (𝑃 ∈ ℙ → 𝑃 ∈ ℝ) |
| 34 | prmgt1 16734 | . . . . . . . 8 ⊢ (𝑃 ∈ ℙ → 1 < 𝑃) | |
| 35 | 33, 34 | jca 511 | . . . . . . 7 ⊢ (𝑃 ∈ ℙ → (𝑃 ∈ ℝ ∧ 1 < 𝑃)) |
| 36 | 1mod 13943 | . . . . . . 7 ⊢ ((𝑃 ∈ ℝ ∧ 1 < 𝑃) → (1 mod 𝑃) = 1) | |
| 37 | 1, 25, 35, 36 | 4syl 19 | . . . . . 6 ⊢ (𝜑 → (1 mod 𝑃) = 1) |
| 38 | 37 | adantr 480 | . . . . 5 ⊢ ((𝜑 ∧ (1 mod 𝑃) = (((-1↑𝑁) · (2↑𝐻)) mod 𝑃)) → (1 mod 𝑃) = 1) |
| 39 | 32, 38 | eqtr3d 2779 | . . . 4 ⊢ ((𝜑 ∧ (1 mod 𝑃) = (((-1↑𝑁) · (2↑𝐻)) mod 𝑃)) → (((-1↑𝑁) · (2↑𝐻)) mod 𝑃) = 1) |
| 40 | 39 | ex 412 | . . 3 ⊢ (𝜑 → ((1 mod 𝑃) = (((-1↑𝑁) · (2↑𝐻)) mod 𝑃) → (((-1↑𝑁) · (2↑𝐻)) mod 𝑃) = 1)) |
| 41 | 31, 40 | sylbid 240 | . 2 ⊢ (𝜑 → (((!‘𝐻) mod 𝑃) = ((((-1↑𝑁) · (2↑𝐻)) · (!‘𝐻)) mod 𝑃) → (((-1↑𝑁) · (2↑𝐻)) mod 𝑃) = 1)) |
| 42 | 6, 41 | mpd 15 | 1 ⊢ (𝜑 → (((-1↑𝑁) · (2↑𝐻)) mod 𝑃) = 1) |
| Colors of variables: wff setvar class |
| Syntax hints: → wi 4 ↔ wb 206 ∧ wa 395 = wceq 1540 ∈ wcel 2108 ∖ cdif 3948 ifcif 4525 {csn 4626 class class class wbr 5143 ↦ cmpt 5225 ‘cfv 6561 (class class class)co 7431 ℝcr 11154 1c1 11156 · cmul 11160 < clt 11295 − cmin 11492 -cneg 11493 / cdiv 11920 ℕcn 12266 2c2 12321 4c4 12323 ℕ0cn0 12526 ℤcz 12613 ...cfz 13547 ⌊cfl 13830 mod cmo 13909 ↑cexp 14102 !cfa 14312 gcd cgcd 16531 ℙcprime 16708 |
| This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1795 ax-4 1809 ax-5 1910 ax-6 1967 ax-7 2007 ax-8 2110 ax-9 2118 ax-10 2141 ax-11 2157 ax-12 2177 ax-ext 2708 ax-rep 5279 ax-sep 5296 ax-nul 5306 ax-pow 5365 ax-pr 5432 ax-un 7755 ax-inf2 9681 ax-cnex 11211 ax-resscn 11212 ax-1cn 11213 ax-icn 11214 ax-addcl 11215 ax-addrcl 11216 ax-mulcl 11217 ax-mulrcl 11218 ax-mulcom 11219 ax-addass 11220 ax-mulass 11221 ax-distr 11222 ax-i2m1 11223 ax-1ne0 11224 ax-1rid 11225 ax-rnegex 11226 ax-rrecex 11227 ax-cnre 11228 ax-pre-lttri 11229 ax-pre-lttrn 11230 ax-pre-ltadd 11231 ax-pre-mulgt0 11232 ax-pre-sup 11233 |
| This theorem depends on definitions: df-bi 207 df-an 396 df-or 849 df-3or 1088 df-3an 1089 df-tru 1543 df-fal 1553 df-ex 1780 df-nf 1784 df-sb 2065 df-mo 2540 df-eu 2569 df-clab 2715 df-cleq 2729 df-clel 2816 df-nfc 2892 df-ne 2941 df-nel 3047 df-ral 3062 df-rex 3071 df-rmo 3380 df-reu 3381 df-rab 3437 df-v 3482 df-sbc 3789 df-csb 3900 df-dif 3954 df-un 3956 df-in 3958 df-ss 3968 df-pss 3971 df-nul 4334 df-if 4526 df-pw 4602 df-sn 4627 df-pr 4629 df-tp 4631 df-op 4633 df-uni 4908 df-int 4947 df-iun 4993 df-br 5144 df-opab 5206 df-mpt 5226 df-tr 5260 df-id 5578 df-eprel 5584 df-po 5592 df-so 5593 df-fr 5637 df-se 5638 df-we 5639 df-xp 5691 df-rel 5692 df-cnv 5693 df-co 5694 df-dm 5695 df-rn 5696 df-res 5697 df-ima 5698 df-pred 6321 df-ord 6387 df-on 6388 df-lim 6389 df-suc 6390 df-iota 6514 df-fun 6563 df-fn 6564 df-f 6565 df-f1 6566 df-fo 6567 df-f1o 6568 df-fv 6569 df-isom 6570 df-riota 7388 df-ov 7434 df-oprab 7435 df-mpo 7436 df-om 7888 df-1st 8014 df-2nd 8015 df-frecs 8306 df-wrecs 8337 df-recs 8411 df-rdg 8450 df-1o 8506 df-2o 8507 df-er 8745 df-en 8986 df-dom 8987 df-sdom 8988 df-fin 8989 df-sup 9482 df-inf 9483 df-oi 9550 df-card 9979 df-pnf 11297 df-mnf 11298 df-xr 11299 df-ltxr 11300 df-le 11301 df-sub 11494 df-neg 11495 df-div 11921 df-nn 12267 df-2 12329 df-3 12330 df-4 12331 df-5 12332 df-6 12333 df-n0 12527 df-z 12614 df-uz 12879 df-rp 13035 df-ioo 13391 df-fz 13548 df-fzo 13695 df-fl 13832 df-mod 13910 df-seq 14043 df-exp 14103 df-fac 14313 df-hash 14370 df-cj 15138 df-re 15139 df-im 15140 df-sqrt 15274 df-abs 15275 df-clim 15524 df-prod 15940 df-dvds 16291 df-gcd 16532 df-prm 16709 |
| This theorem is referenced by: gausslemma2d 27418 |
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