lgscllem - Intuitionistic Logic Explorer

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Theorem lgscllem 16040

Description: The Legendre symbol is an element of 𝑍. (Contributed by Mario Carneiro, 4-Feb-2015.)

**Hypotheses**
Ref	Expression
lgsval.1	⊢ 𝐹 = (𝑛 ∈ ℕ ↦ if(𝑛 ∈ ℙ, (if(𝑛 = 2, if(2 ∥ 𝐴, 0, if((𝐴 mod 8) ∈ {1, 7}, 1, -1)), ((((𝐴↑((𝑛 − 1) / 2)) + 1) mod 𝑛) − 1))↑(𝑛 pCnt 𝑁)), 1))
lgsfcl2.z	⊢ 𝑍 = {𝑥 ∈ ℤ ∣ (abs‘𝑥) ≤ 1}

**Assertion**
Ref	Expression
lgscllem	⊢ ((𝐴 ∈ ℤ ∧ 𝑁 ∈ ℤ) → (𝐴 /_L 𝑁) ∈ 𝑍)

Distinct variable groups: 𝐴,𝑛,𝑥 𝑥,𝐹 𝑛,𝑁,𝑥 𝑛,𝑍 Allowed substitution hints: 𝐹(𝑛) 𝑍(𝑥)

Proof of Theorem lgscllem Dummy variables 𝑦 𝑧 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		lgsval.1	. . 3 ⊢ 𝐹 = (𝑛 ∈ ℕ ↦ if(𝑛 ∈ ℙ, (if(𝑛 = 2, if(2 ∥ 𝐴, 0, if((𝐴 mod 8) ∈ {1, 7}, 1, -1)), ((((𝐴↑((𝑛 − 1) / 2)) + 1) mod 𝑛) − 1))↑(𝑛 pCnt 𝑁)), 1))
2	1	lgsval 16037	. 2 ⊢ ((𝐴 ∈ ℤ ∧ 𝑁 ∈ ℤ) → (𝐴 /_L 𝑁) = if(𝑁 = 0, if((𝐴↑2) = 1, 1, 0), (if((𝑁 < 0 ∧ 𝐴 < 0), -1, 1) · (seq1( · , 𝐹)‘(abs‘𝑁)))))
3		lgsfcl2.z	. . . . . . . 8 ⊢ 𝑍 = {𝑥 ∈ ℤ ∣ (abs‘𝑥) ≤ 1}
4	3	lgslem2 16034	. . . . . . 7 ⊢ (-1 ∈ 𝑍 ∧ 0 ∈ 𝑍 ∧ 1 ∈ 𝑍)
5	4	simp3i 1039	. . . . . 6 ⊢ 1 ∈ 𝑍
6	5	a1i 9	. . . . 5 ⊢ ((𝐴 ∈ ℤ ∧ 𝑁 ∈ ℤ) → 1 ∈ 𝑍)
7	4	simp2i 1038	. . . . . 6 ⊢ 0 ∈ 𝑍
8	7	a1i 9	. . . . 5 ⊢ ((𝐴 ∈ ℤ ∧ 𝑁 ∈ ℤ) → 0 ∈ 𝑍)
9		zsqcl 11025	. . . . . 6 ⊢ (𝐴 ∈ ℤ → (𝐴↑2) ∈ ℤ)
10		1zzd 9650	. . . . . 6 ⊢ ((𝐴 ∈ ℤ ∧ 𝑁 ∈ ℤ) → 1 ∈ ℤ)
11		zdceq 9699	. . . . . 6 ⊢ (((𝐴↑2) ∈ ℤ ∧ 1 ∈ ℤ) → DECID (𝐴↑2) = 1)
12	9, 10, 11	syl2an2r 603	. . . . 5 ⊢ ((𝐴 ∈ ℤ ∧ 𝑁 ∈ ℤ) → DECID (𝐴↑2) = 1)
13	6, 8, 12	ifcldcd 3675	. . . 4 ⊢ ((𝐴 ∈ ℤ ∧ 𝑁 ∈ ℤ) → if((𝐴↑2) = 1, 1, 0) ∈ 𝑍)
14	13	adantr 276	. . 3 ⊢ (((𝐴 ∈ ℤ ∧ 𝑁 ∈ ℤ) ∧ 𝑁 = 0) → if((𝐴↑2) = 1, 1, 0) ∈ 𝑍)
15	4	simp1i 1037	. . . . . 6 ⊢ -1 ∈ 𝑍
16	15	a1i 9	. . . . 5 ⊢ ((𝐴 ∈ ℤ ∧ 𝑁 ∈ ℤ) → -1 ∈ 𝑍)
17		simpr 110	. . . . . . 7 ⊢ ((𝐴 ∈ ℤ ∧ 𝑁 ∈ ℤ) → 𝑁 ∈ ℤ)
18		0zd 9635	. . . . . . 7 ⊢ ((𝐴 ∈ ℤ ∧ 𝑁 ∈ ℤ) → 0 ∈ ℤ)
19		zdclt 9701	. . . . . . 7 ⊢ ((𝑁 ∈ ℤ ∧ 0 ∈ ℤ) → DECID 𝑁 < 0)
20	17, 18, 19	syl2anc 415	. . . . . 6 ⊢ ((𝐴 ∈ ℤ ∧ 𝑁 ∈ ℤ) → DECID 𝑁 < 0)
21		zdclt 9701	. . . . . . 7 ⊢ ((𝐴 ∈ ℤ ∧ 0 ∈ ℤ) → DECID 𝐴 < 0)
22	18, 21	syldan 282	. . . . . 6 ⊢ ((𝐴 ∈ ℤ ∧ 𝑁 ∈ ℤ) → DECID 𝐴 < 0)
23		dcan2 947	. . . . . 6 ⊢ (DECID 𝑁 < 0 → (DECID 𝐴 < 0 → DECID (𝑁 < 0 ∧ 𝐴 < 0)))
24	20, 22, 23	sylc 62	. . . . 5 ⊢ ((𝐴 ∈ ℤ ∧ 𝑁 ∈ ℤ) → DECID (𝑁 < 0 ∧ 𝐴 < 0))
25	16, 6, 24	ifcldcd 3675	. . . 4 ⊢ ((𝐴 ∈ ℤ ∧ 𝑁 ∈ ℤ) → if((𝑁 < 0 ∧ 𝐴 < 0), -1, 1) ∈ 𝑍)
26		nnuz 9937	. . . . . 6 ⊢ ℕ = (ℤ_≥‘1)
27		1zzd 9650	. . . . . 6 ⊢ (((𝐴 ∈ ℤ ∧ 𝑁 ∈ ℤ) ∧ ¬ 𝑁 = 0) → 1 ∈ ℤ)
28		df-ne 2421	. . . . . . . 8 ⊢ (𝑁 ≠ 0 ↔ ¬ 𝑁 = 0)
29	1, 3	lgsfcl2 16039	. . . . . . . . 9 ⊢ ((𝐴 ∈ ℤ ∧ 𝑁 ∈ ℤ ∧ 𝑁 ≠ 0) → 𝐹:ℕ⟶𝑍)
30	29	3expa 1234	. . . . . . . 8 ⊢ (((𝐴 ∈ ℤ ∧ 𝑁 ∈ ℤ) ∧ 𝑁 ≠ 0) → 𝐹:ℕ⟶𝑍)
31	28, 30	sylan2br 288	. . . . . . 7 ⊢ (((𝐴 ∈ ℤ ∧ 𝑁 ∈ ℤ) ∧ ¬ 𝑁 = 0) → 𝐹:ℕ⟶𝑍)
32	31	ffvelcdmda 5834	. . . . . 6 ⊢ ((((𝐴 ∈ ℤ ∧ 𝑁 ∈ ℤ) ∧ ¬ 𝑁 = 0) ∧ 𝑦 ∈ ℕ) → (𝐹‘𝑦) ∈ 𝑍)
33	3	lgslem3 16035	. . . . . . 7 ⊢ ((𝑦 ∈ 𝑍 ∧ 𝑧 ∈ 𝑍) → (𝑦 · 𝑧) ∈ 𝑍)
34	33	adantl 277	. . . . . 6 ⊢ ((((𝐴 ∈ ℤ ∧ 𝑁 ∈ ℤ) ∧ ¬ 𝑁 = 0) ∧ (𝑦 ∈ 𝑍 ∧ 𝑧 ∈ 𝑍)) → (𝑦 · 𝑧) ∈ 𝑍)
35	26, 27, 32, 34	seqf 10879	. . . . 5 ⊢ (((𝐴 ∈ ℤ ∧ 𝑁 ∈ ℤ) ∧ ¬ 𝑁 = 0) → seq1( · , 𝐹):ℕ⟶𝑍)
36		simplr 533	. . . . . 6 ⊢ (((𝐴 ∈ ℤ ∧ 𝑁 ∈ ℤ) ∧ ¬ 𝑁 = 0) → 𝑁 ∈ ℤ)
37		simpr 110	. . . . . . 7 ⊢ (((𝐴 ∈ ℤ ∧ 𝑁 ∈ ℤ) ∧ ¬ 𝑁 = 0) → ¬ 𝑁 = 0)
38	37	neqned 2427	. . . . . 6 ⊢ (((𝐴 ∈ ℤ ∧ 𝑁 ∈ ℤ) ∧ ¬ 𝑁 = 0) → 𝑁 ≠ 0)
39		nnabscl 11844	. . . . . 6 ⊢ ((𝑁 ∈ ℤ ∧ 𝑁 ≠ 0) → (abs‘𝑁) ∈ ℕ)
40	36, 38, 39	syl2anc 415	. . . . 5 ⊢ (((𝐴 ∈ ℤ ∧ 𝑁 ∈ ℤ) ∧ ¬ 𝑁 = 0) → (abs‘𝑁) ∈ ℕ)
41	35, 40	ffvelcdmd 5835	. . . 4 ⊢ (((𝐴 ∈ ℤ ∧ 𝑁 ∈ ℤ) ∧ ¬ 𝑁 = 0) → (seq1( · , 𝐹)‘(abs‘𝑁)) ∈ 𝑍)
42	3	lgslem3 16035	. . . 4 ⊢ ((if((𝑁 < 0 ∧ 𝐴 < 0), -1, 1) ∈ 𝑍 ∧ (seq1( · , 𝐹)‘(abs‘𝑁)) ∈ 𝑍) → (if((𝑁 < 0 ∧ 𝐴 < 0), -1, 1) · (seq1( · , 𝐹)‘(abs‘𝑁))) ∈ 𝑍)
43	25, 41, 42	syl2an2r 603	. . 3 ⊢ (((𝐴 ∈ ℤ ∧ 𝑁 ∈ ℤ) ∧ ¬ 𝑁 = 0) → (if((𝑁 < 0 ∧ 𝐴 < 0), -1, 1) · (seq1( · , 𝐹)‘(abs‘𝑁))) ∈ 𝑍)
44		zdceq 9699	. . . 4 ⊢ ((𝑁 ∈ ℤ ∧ 0 ∈ ℤ) → DECID 𝑁 = 0)
45	17, 18, 44	syl2anc 415	. . 3 ⊢ ((𝐴 ∈ ℤ ∧ 𝑁 ∈ ℤ) → DECID 𝑁 = 0)
46	14, 43, 45	ifcldadc 3667	. 2 ⊢ ((𝐴 ∈ ℤ ∧ 𝑁 ∈ ℤ) → if(𝑁 = 0, if((𝐴↑2) = 1, 1, 0), (if((𝑁 < 0 ∧ 𝐴 < 0), -1, 1) · (seq1( · , 𝐹)‘(abs‘𝑁)))) ∈ 𝑍)
47	2, 46	eqeltrd 2315	1 ⊢ ((𝐴 ∈ ℤ ∧ 𝑁 ∈ ℤ) → (𝐴 /_L 𝑁) ∈ 𝑍)

Colors of variables: wff set class

Syntax hints: ¬ wn 3 → wi 4 ∧ wa 104 DECID wdc 846 = wceq 1402 ∈ wcel 2209 ≠ wne 2420 {crab 2532 ifcif 3635 {cpr 3706 class class class wbr 4125 ↦ cmpt 4187 ⟶wf 5368 ‘cfv 5372 (class class class)co 6075 0cc0 8169 1c1 8170 + caddc 8172 · cmul 8174 < clt 8350 ≤ cle 8351 − cmin 8487 -cneg 8488 / cdiv 8992 ℕcn 9283 2c2 9334 7c7 9339 8c8 9340 ℤcz 9623 mod cmo 10737 seqcseq 10862 ↑cexp 10953 abscabs 11741 ∥ cdvds 12532 ℙcprime 12863 pCnt cpc 13041 /_L clgs 16030

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 623 ax-in2 624 ax-io 721 ax-5 1500 ax-7 1501 ax-gen 1502 ax-ie1 1546 ax-ie2 1547 ax-8 1557 ax-10 1558 ax-11 1559 ax-i12 1560 ax-bndl 1562 ax-4 1563 ax-17 1579 ax-i9 1583 ax-ial 1587 ax-i5r 1588 ax-14 2212 ax-ext 2220 ax-coll 4241 ax-sep 4244 ax-nul 4254 ax-pow 4306 ax-pr 4341 ax-un 4573 ax-setind 4679 ax-iinf 4730 ax-cnex 8260 ax-resscn 8261 ax-1cn 8262 ax-1re 8263 ax-icn 8264 ax-addcl 8265 ax-addrcl 8266 ax-mulcl 8267 ax-mulrcl 8268 ax-addcom 8269 ax-mulcom 8270 ax-addass 8271 ax-mulass 8272 ax-distr 8273 ax-i2m1 8274 ax-0lt1 8275 ax-1rid 8276 ax-0id 8277 ax-rnegex 8278 ax-precex 8279 ax-cnre 8280 ax-pre-ltirr 8281 ax-pre-ltwlin 8282 ax-pre-lttrn 8283 ax-pre-apti 8284 ax-pre-ltadd 8285 ax-pre-mulgt0 8286 ax-pre-mulext 8287 ax-arch 8288 ax-caucvg 8289

This theorem depends on definitions: df-bi 117 df-stab 843 df-dc 847 df-3or 1010 df-3an 1011 df-tru 1405 df-fal 1408 df-xor 1425 df-nf 1514 df-sb 1816 df-eu 2089 df-mo 2090 df-clab 2225 df-cleq 2231 df-clel 2234 df-nfc 2381 df-ne 2421 df-nel 2516 df-ral 2533 df-rex 2534 df-reu 2535 df-rmo 2536 df-rab 2537 df-v 2823 df-sbc 3052 df-csb 3148 df-dif 3222 df-un 3224 df-in 3226 df-ss 3233 df-nul 3521 df-if 3636 df-pw 3687 df-sn 3711 df-pr 3712 df-op 3714 df-uni 3931 df-int 3966 df-iun 4009 df-br 4126 df-opab 4188 df-mpt 4189 df-tr 4225 df-id 4433 df-po 4436 df-iso 4437 df-iord 4506 df-on 4508 df-ilim 4509 df-suc 4511 df-iom 4733 df-xp 4775 df-rel 4776 df-cnv 4777 df-co 4778 df-dm 4779 df-rn 4780 df-res 4781 df-ima 4782 df-iota 5332 df-fun 5374 df-fn 5375 df-f 5376 df-f1 5377 df-fo 5378 df-f1o 5379 df-fv 5380 df-isom 5381 df-riota 6028 df-ov 6078 df-oprab 6079 df-mpo 6080 df-1st 6364 df-2nd 6365 df-recs 6566 df-irdg 6631 df-frec 6652 df-1o 6677 df-2o 6678 df-oadd 6681 df-er 6797 df-en 7013 df-dom 7014 df-fin 7015 df-sup 7314 df-inf 7315 df-pnf 8352 df-mnf 8353 df-xr 8354 df-ltxr 8355 df-le 8356 df-sub 8489 df-neg 8490 df-reap 8893 df-ap 8900 df-div 8993 df-inn 9284 df-2 9342 df-3 9343 df-4 9344 df-5 9345 df-6 9346 df-7 9347 df-8 9348 df-n0 9543 df-z 9624 df-uz 9901 df-q 9999 df-rp 10034 df-fz 10391 df-fzo 10528 df-fl 10683 df-mod 10738 df-seqfrec 10863 df-exp 10954 df-ihash 11193 df-cj 11585 df-re 11586 df-im 11587 df-rsqrt 11742 df-abs 11743 df-clim 12023 df-proddc 12296 df-dvds 12533 df-gcd 12709 df-prm 12864 df-phi 12967 df-pc 13042 df-lgs 16031

This theorem is referenced by: lgscl2 16045

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