gausslemma2d - Metamath Proof Explorer

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Theorem gausslemma2d 25952

Description: Gauss' Lemma (see also theorem 9.6 in [ApostolNT] p. 182) for integer 2: Let p be an odd prime. Let S={2,4,6,...,(p-1)}. Let n denote the number of elements of S whose least positive residue modulo p is greater than p/2. Then ( 2 | p ) = (-1)^n. (Contributed by AV, 14-Jul-2021.)

**Hypotheses**
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)

**Assertion**
Ref	Expression
gausslemma2d	⊢ (𝜑 → (2 /_L 𝑃) = (-1↑𝑁))

Distinct variable groups: 𝑥,𝐻 𝑥,𝑃 𝜑,𝑥 𝑥,𝑀 Allowed substitution hints: 𝑅(𝑥) 𝑁(𝑥)

**Proof of Theorem gausslemma2d**
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	gausslemma2dlem7 25951	. 2 ⊢ (𝜑 → (((-1↑𝑁) · (2↑𝐻)) mod 𝑃) = 1)
7		eldifi 4105	. . . . . . 7 ⊢ (𝑃 ∈ (ℙ ∖ {2}) → 𝑃 ∈ ℙ)
8		prmnn 16020	. . . . . . . . 9 ⊢ (𝑃 ∈ ℙ → 𝑃 ∈ ℕ)
9	8	nnred 11655	. . . . . . . 8 ⊢ (𝑃 ∈ ℙ → 𝑃 ∈ ℝ)
10		prmgt1 16043	. . . . . . . 8 ⊢ (𝑃 ∈ ℙ → 1 < 𝑃)
11	9, 10	jca 514	. . . . . . 7 ⊢ (𝑃 ∈ ℙ → (𝑃 ∈ ℝ ∧ 1 < 𝑃))
12	1, 7, 11	3syl 18	. . . . . 6 ⊢ (𝜑 → (𝑃 ∈ ℝ ∧ 1 < 𝑃))
13		1mod 13274	. . . . . 6 ⊢ ((𝑃 ∈ ℝ ∧ 1 < 𝑃) → (1 mod 𝑃) = 1)
14	12, 13	syl 17	. . . . 5 ⊢ (𝜑 → (1 mod 𝑃) = 1)
15	14	eqcomd 2829	. . . 4 ⊢ (𝜑 → 1 = (1 mod 𝑃))
16	15	eqeq2d 2834	. . 3 ⊢ (𝜑 → ((((-1↑𝑁) · (2↑𝐻)) mod 𝑃) = 1 ↔ (((-1↑𝑁) · (2↑𝐻)) mod 𝑃) = (1 mod 𝑃)))
17		neg1z 12021	. . . . . . . . . . 11 ⊢ -1 ∈ ℤ
18	1, 4, 2, 5	gausslemma2dlem0h 25941	. . . . . . . . . . 11 ⊢ (𝜑 → 𝑁 ∈ ℕ₀)
19		zexpcl 13447	. . . . . . . . . . 11 ⊢ ((-1 ∈ ℤ ∧ 𝑁 ∈ ℕ₀) → (-1↑𝑁) ∈ ℤ)
20	17, 18, 19	sylancr 589	. . . . . . . . . 10 ⊢ (𝜑 → (-1↑𝑁) ∈ ℤ)
21		2nn 11713	. . . . . . . . . . . . 13 ⊢ 2 ∈ ℕ
22	21	a1i 11	. . . . . . . . . . . 12 ⊢ (𝜑 → 2 ∈ ℕ)
23	1, 2	gausslemma2dlem0b 25935	. . . . . . . . . . . . 13 ⊢ (𝜑 → 𝐻 ∈ ℕ)
24	23	nnnn0d 11958	. . . . . . . . . . . 12 ⊢ (𝜑 → 𝐻 ∈ ℕ₀)
25	22, 24	nnexpcld 13609	. . . . . . . . . . 11 ⊢ (𝜑 → (2↑𝐻) ∈ ℕ)
26	25	nnzd 12089	. . . . . . . . . 10 ⊢ (𝜑 → (2↑𝐻) ∈ ℤ)
27	20, 26	zmulcld 12096	. . . . . . . . 9 ⊢ (𝜑 → ((-1↑𝑁) · (2↑𝐻)) ∈ ℤ)
28	27	zred 12090	. . . . . . . 8 ⊢ (𝜑 → ((-1↑𝑁) · (2↑𝐻)) ∈ ℝ)
29		1red 10644	. . . . . . . 8 ⊢ (𝜑 → 1 ∈ ℝ)
30	28, 29	jca 514	. . . . . . 7 ⊢ (𝜑 → (((-1↑𝑁) · (2↑𝐻)) ∈ ℝ ∧ 1 ∈ ℝ))
31	30	adantr 483	. . . . . 6 ⊢ ((𝜑 ∧ (((-1↑𝑁) · (2↑𝐻)) mod 𝑃) = (1 mod 𝑃)) → (((-1↑𝑁) · (2↑𝐻)) ∈ ℝ ∧ 1 ∈ ℝ))
32	1	gausslemma2dlem0a 25934	. . . . . . . . 9 ⊢ (𝜑 → 𝑃 ∈ ℕ)
33	32	nnrpd 12432	. . . . . . . 8 ⊢ (𝜑 → 𝑃 ∈ ℝ⁺)
34	20, 33	jca 514	. . . . . . 7 ⊢ (𝜑 → ((-1↑𝑁) ∈ ℤ ∧ 𝑃 ∈ ℝ⁺))
35	34	adantr 483	. . . . . 6 ⊢ ((𝜑 ∧ (((-1↑𝑁) · (2↑𝐻)) mod 𝑃) = (1 mod 𝑃)) → ((-1↑𝑁) ∈ ℤ ∧ 𝑃 ∈ ℝ⁺))
36		simpr 487	. . . . . 6 ⊢ ((𝜑 ∧ (((-1↑𝑁) · (2↑𝐻)) mod 𝑃) = (1 mod 𝑃)) → (((-1↑𝑁) · (2↑𝐻)) mod 𝑃) = (1 mod 𝑃))
37		modmul1 13295	. . . . . 6 ⊢ (((((-1↑𝑁) · (2↑𝐻)) ∈ ℝ ∧ 1 ∈ ℝ) ∧ ((-1↑𝑁) ∈ ℤ ∧ 𝑃 ∈ ℝ⁺) ∧ (((-1↑𝑁) · (2↑𝐻)) mod 𝑃) = (1 mod 𝑃)) → ((((-1↑𝑁) · (2↑𝐻)) · (-1↑𝑁)) mod 𝑃) = ((1 · (-1↑𝑁)) mod 𝑃))
38	31, 35, 36, 37	syl3anc 1367	. . . . 5 ⊢ ((𝜑 ∧ (((-1↑𝑁) · (2↑𝐻)) mod 𝑃) = (1 mod 𝑃)) → ((((-1↑𝑁) · (2↑𝐻)) · (-1↑𝑁)) mod 𝑃) = ((1 · (-1↑𝑁)) mod 𝑃))
39	38	ex 415	. . . 4 ⊢ (𝜑 → ((((-1↑𝑁) · (2↑𝐻)) mod 𝑃) = (1 mod 𝑃) → ((((-1↑𝑁) · (2↑𝐻)) · (-1↑𝑁)) mod 𝑃) = ((1 · (-1↑𝑁)) mod 𝑃)))
40	20	zcnd 12091	. . . . . . . . 9 ⊢ (𝜑 → (-1↑𝑁) ∈ ℂ)
41	25	nncnd 11656	. . . . . . . . 9 ⊢ (𝜑 → (2↑𝐻) ∈ ℂ)
42	40, 41, 40	mul32d 10852	. . . . . . . 8 ⊢ (𝜑 → (((-1↑𝑁) · (2↑𝐻)) · (-1↑𝑁)) = (((-1↑𝑁) · (-1↑𝑁)) · (2↑𝐻)))
43	18	nn0cnd 11960	. . . . . . . . . . . . 13 ⊢ (𝜑 → 𝑁 ∈ ℂ)
44	43	2timesd 11883	. . . . . . . . . . . 12 ⊢ (𝜑 → (2 · 𝑁) = (𝑁 + 𝑁))
45	44	eqcomd 2829	. . . . . . . . . . 11 ⊢ (𝜑 → (𝑁 + 𝑁) = (2 · 𝑁))
46	45	oveq2d 7174	. . . . . . . . . 10 ⊢ (𝜑 → (-1↑(𝑁 + 𝑁)) = (-1↑(2 · 𝑁)))
47		neg1cn 11754	. . . . . . . . . . . 12 ⊢ -1 ∈ ℂ
48	47	a1i 11	. . . . . . . . . . 11 ⊢ (𝜑 → -1 ∈ ℂ)
49	48, 18, 18	expaddd 13515	. . . . . . . . . 10 ⊢ (𝜑 → (-1↑(𝑁 + 𝑁)) = ((-1↑𝑁) · (-1↑𝑁)))
50	18	nn0zd 12088	. . . . . . . . . . 11 ⊢ (𝜑 → 𝑁 ∈ ℤ)
51		m1expeven 13479	. . . . . . . . . . 11 ⊢ (𝑁 ∈ ℤ → (-1↑(2 · 𝑁)) = 1)
52	50, 51	syl 17	. . . . . . . . . 10 ⊢ (𝜑 → (-1↑(2 · 𝑁)) = 1)
53	46, 49, 52	3eqtr3d 2866	. . . . . . . . 9 ⊢ (𝜑 → ((-1↑𝑁) · (-1↑𝑁)) = 1)
54	53	oveq1d 7173	. . . . . . . 8 ⊢ (𝜑 → (((-1↑𝑁) · (-1↑𝑁)) · (2↑𝐻)) = (1 · (2↑𝐻)))
55	41	mulid2d 10661	. . . . . . . 8 ⊢ (𝜑 → (1 · (2↑𝐻)) = (2↑𝐻))
56	42, 54, 55	3eqtrd 2862	. . . . . . 7 ⊢ (𝜑 → (((-1↑𝑁) · (2↑𝐻)) · (-1↑𝑁)) = (2↑𝐻))
57	56	oveq1d 7173	. . . . . 6 ⊢ (𝜑 → ((((-1↑𝑁) · (2↑𝐻)) · (-1↑𝑁)) mod 𝑃) = ((2↑𝐻) mod 𝑃))
58	40	mulid2d 10661	. . . . . . 7 ⊢ (𝜑 → (1 · (-1↑𝑁)) = (-1↑𝑁))
59	58	oveq1d 7173	. . . . . 6 ⊢ (𝜑 → ((1 · (-1↑𝑁)) mod 𝑃) = ((-1↑𝑁) mod 𝑃))
60	57, 59	eqeq12d 2839	. . . . 5 ⊢ (𝜑 → (((((-1↑𝑁) · (2↑𝐻)) · (-1↑𝑁)) mod 𝑃) = ((1 · (-1↑𝑁)) mod 𝑃) ↔ ((2↑𝐻) mod 𝑃) = ((-1↑𝑁) mod 𝑃)))
61	2	oveq2i 7169	. . . . . . . 8 ⊢ (2↑𝐻) = (2↑((𝑃 − 1) / 2))
62	61	oveq1i 7168	. . . . . . 7 ⊢ ((2↑𝐻) mod 𝑃) = ((2↑((𝑃 − 1) / 2)) mod 𝑃)
63	62	eqeq1i 2828	. . . . . 6 ⊢ (((2↑𝐻) mod 𝑃) = ((-1↑𝑁) mod 𝑃) ↔ ((2↑((𝑃 − 1) / 2)) mod 𝑃) = ((-1↑𝑁) mod 𝑃))
64		2z 12017	. . . . . . . . . 10 ⊢ 2 ∈ ℤ
65		lgsvalmod 25894	. . . . . . . . . 10 ⊢ ((2 ∈ ℤ ∧ 𝑃 ∈ (ℙ ∖ {2})) → ((2 /_L 𝑃) mod 𝑃) = ((2↑((𝑃 − 1) / 2)) mod 𝑃))
66	64, 1, 65	sylancr 589	. . . . . . . . 9 ⊢ (𝜑 → ((2 /_L 𝑃) mod 𝑃) = ((2↑((𝑃 − 1) / 2)) mod 𝑃))
67	66	eqcomd 2829	. . . . . . . 8 ⊢ (𝜑 → ((2↑((𝑃 − 1) / 2)) mod 𝑃) = ((2 /_L 𝑃) mod 𝑃))
68	67	eqeq1d 2825	. . . . . . 7 ⊢ (𝜑 → (((2↑((𝑃 − 1) / 2)) mod 𝑃) = ((-1↑𝑁) mod 𝑃) ↔ ((2 /_L 𝑃) mod 𝑃) = ((-1↑𝑁) mod 𝑃)))
69	1, 4, 2, 5	gausslemma2dlem0i 25942	. . . . . . 7 ⊢ (𝜑 → (((2 /_L 𝑃) mod 𝑃) = ((-1↑𝑁) mod 𝑃) → (2 /_L 𝑃) = (-1↑𝑁)))
70	68, 69	sylbid 242	. . . . . 6 ⊢ (𝜑 → (((2↑((𝑃 − 1) / 2)) mod 𝑃) = ((-1↑𝑁) mod 𝑃) → (2 /_L 𝑃) = (-1↑𝑁)))
71	63, 70	syl5bi 244	. . . . 5 ⊢ (𝜑 → (((2↑𝐻) mod 𝑃) = ((-1↑𝑁) mod 𝑃) → (2 /_L 𝑃) = (-1↑𝑁)))
72	60, 71	sylbid 242	. . . 4 ⊢ (𝜑 → (((((-1↑𝑁) · (2↑𝐻)) · (-1↑𝑁)) mod 𝑃) = ((1 · (-1↑𝑁)) mod 𝑃) → (2 /_L 𝑃) = (-1↑𝑁)))
73	39, 72	syld 47	. . 3 ⊢ (𝜑 → ((((-1↑𝑁) · (2↑𝐻)) mod 𝑃) = (1 mod 𝑃) → (2 /_L 𝑃) = (-1↑𝑁)))
74	16, 73	sylbid 242	. 2 ⊢ (𝜑 → ((((-1↑𝑁) · (2↑𝐻)) mod 𝑃) = 1 → (2 /_L 𝑃) = (-1↑𝑁)))
75	6, 74	mpd 15	1 ⊢ (𝜑 → (2 /_L 𝑃) = (-1↑𝑁))

Colors of variables: wff setvar class

Syntax hints: → wi 4 ∧ wa 398 = wceq 1537 ∈ wcel 2114 ∖ cdif 3935 ifcif 4469 {csn 4569 class class class wbr 5068 ↦ cmpt 5148 ‘cfv 6357 (class class class)co 7158 ℂcc 10537 ℝcr 10538 1c1 10540 + caddc 10542 · cmul 10544 < clt 10677 − cmin 10872 -cneg 10873 / cdiv 11299 ℕcn 11640 2c2 11695 4c4 11697 ℕ₀cn0 11900 ℤcz 11984 ℝ⁺crp 12392 ...cfz 12895 ⌊cfl 13163 mod cmo 13240 ↑cexp 13432 ℙcprime 16017 /_L clgs 25872

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1796 ax-4 1810 ax-5 1911 ax-6 1970 ax-7 2015 ax-8 2116 ax-9 2124 ax-10 2145 ax-11 2161 ax-12 2177 ax-ext 2795 ax-rep 5192 ax-sep 5205 ax-nul 5212 ax-pow 5268 ax-pr 5332 ax-un 7463 ax-inf2 9106 ax-cnex 10595 ax-resscn 10596 ax-1cn 10597 ax-icn 10598 ax-addcl 10599 ax-addrcl 10600 ax-mulcl 10601 ax-mulrcl 10602 ax-mulcom 10603 ax-addass 10604 ax-mulass 10605 ax-distr 10606 ax-i2m1 10607 ax-1ne0 10608 ax-1rid 10609 ax-rnegex 10610 ax-rrecex 10611 ax-cnre 10612 ax-pre-lttri 10613 ax-pre-lttrn 10614 ax-pre-ltadd 10615 ax-pre-mulgt0 10616 ax-pre-sup 10617

This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3or 1084 df-3an 1085 df-tru 1540 df-fal 1550 df-ex 1781 df-nf 1785 df-sb 2070 df-mo 2622 df-eu 2654 df-clab 2802 df-cleq 2816 df-clel 2895 df-nfc 2965 df-ne 3019 df-nel 3126 df-ral 3145 df-rex 3146 df-reu 3147 df-rmo 3148 df-rab 3149 df-v 3498 df-sbc 3775 df-csb 3886 df-dif 3941 df-un 3943 df-in 3945 df-ss 3954 df-pss 3956 df-nul 4294 df-if 4470 df-pw 4543 df-sn 4570 df-pr 4572 df-tp 4574 df-op 4576 df-uni 4841 df-int 4879 df-iun 4923 df-br 5069 df-opab 5131 df-mpt 5149 df-tr 5175 df-id 5462 df-eprel 5467 df-po 5476 df-so 5477 df-fr 5516 df-se 5517 df-we 5518 df-xp 5563 df-rel 5564 df-cnv 5565 df-co 5566 df-dm 5567 df-rn 5568 df-res 5569 df-ima 5570 df-pred 6150 df-ord 6196 df-on 6197 df-lim 6198 df-suc 6199 df-iota 6316 df-fun 6359 df-fn 6360 df-f 6361 df-f1 6362 df-fo 6363 df-f1o 6364 df-fv 6365 df-isom 6366 df-riota 7116 df-ov 7161 df-oprab 7162 df-mpo 7163 df-om 7583 df-1st 7691 df-2nd 7692 df-wrecs 7949 df-recs 8010 df-rdg 8048 df-1o 8104 df-2o 8105 df-oadd 8108 df-er 8291 df-map 8410 df-en 8512 df-dom 8513 df-sdom 8514 df-fin 8515 df-sup 8908 df-inf 8909 df-oi 8976 df-dju 9332 df-card 9370 df-pnf 10679 df-mnf 10680 df-xr 10681 df-ltxr 10682 df-le 10683 df-sub 10874 df-neg 10875 df-div 11300 df-nn 11641 df-2 11703 df-3 11704 df-4 11705 df-5 11706 df-6 11707 df-n0 11901 df-xnn0 11971 df-z 11985 df-uz 12247 df-q 12352 df-rp 12393 df-ioo 12745 df-fz 12896 df-fzo 13037 df-fl 13165 df-mod 13241 df-seq 13373 df-exp 13433 df-fac 13637 df-hash 13694 df-cj 14460 df-re 14461 df-im 14462 df-sqrt 14596 df-abs 14597 df-clim 14847 df-prod 15262 df-dvds 15610 df-gcd 15846 df-prm 16018 df-phi 16105 df-pc 16176 df-lgs 25873

This theorem is referenced by: 2lgs 25985

W3C validator