lgslem4 - Metamath Proof Explorer

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Theorem lgslem4 25025

Description: The function 𝐹 is closed in integers with absolute value less than 1 (namely {-1, 0, 1}, see zabsle1 25021). (Contributed by Mario Carneiro, 4-Feb-2015.)

**Hypothesis**
Ref	Expression
lgslem2.z	⊢ 𝑍 = {𝑥 ∈ ℤ ∣ (abs‘𝑥) ≤ 1}

**Assertion**
Ref	Expression
lgslem4	⊢ ((𝐴 ∈ ℤ ∧ 𝑃 ∈ (ℙ ∖ {2})) → ((((𝐴↑((𝑃 − 1) / 2)) + 1) mod 𝑃) − 1) ∈ 𝑍)

Distinct variable group: 𝑥,𝐴 Allowed substitution hints: 𝑃(𝑥) 𝑍(𝑥)

**Proof of Theorem lgslem4**
Step	Hyp	Ref	Expression
1		simpll 790	. . . . . . . . 9 ⊢ (((𝐴 ∈ ℤ ∧ 𝑃 ∈ (ℙ ∖ {2})) ∧ 𝑃 ∥ 𝐴) → 𝐴 ∈ ℤ)
2		oddprm 15515	. . . . . . . . . . 11 ⊢ (𝑃 ∈ (ℙ ∖ {2}) → ((𝑃 − 1) / 2) ∈ ℕ)
3	2	ad2antlr 763	. . . . . . . . . 10 ⊢ (((𝐴 ∈ ℤ ∧ 𝑃 ∈ (ℙ ∖ {2})) ∧ 𝑃 ∥ 𝐴) → ((𝑃 − 1) / 2) ∈ ℕ)
4	3	nnnn0d 11351	. . . . . . . . 9 ⊢ (((𝐴 ∈ ℤ ∧ 𝑃 ∈ (ℙ ∖ {2})) ∧ 𝑃 ∥ 𝐴) → ((𝑃 − 1) / 2) ∈ ℕ₀)
5		zexpcl 12875	. . . . . . . . 9 ⊢ ((𝐴 ∈ ℤ ∧ ((𝑃 − 1) / 2) ∈ ℕ₀) → (𝐴↑((𝑃 − 1) / 2)) ∈ ℤ)
6	1, 4, 5	syl2anc 693	. . . . . . . 8 ⊢ (((𝐴 ∈ ℤ ∧ 𝑃 ∈ (ℙ ∖ {2})) ∧ 𝑃 ∥ 𝐴) → (𝐴↑((𝑃 − 1) / 2)) ∈ ℤ)
7	6	zred 11482	. . . . . . 7 ⊢ (((𝐴 ∈ ℤ ∧ 𝑃 ∈ (ℙ ∖ {2})) ∧ 𝑃 ∥ 𝐴) → (𝐴↑((𝑃 − 1) / 2)) ∈ ℝ)
8		0red 10041	. . . . . . 7 ⊢ (((𝐴 ∈ ℤ ∧ 𝑃 ∈ (ℙ ∖ {2})) ∧ 𝑃 ∥ 𝐴) → 0 ∈ ℝ)
9		1red 10055	. . . . . . 7 ⊢ (((𝐴 ∈ ℤ ∧ 𝑃 ∈ (ℙ ∖ {2})) ∧ 𝑃 ∥ 𝐴) → 1 ∈ ℝ)
10		eldifi 3732	. . . . . . . . . . . 12 ⊢ (𝑃 ∈ (ℙ ∖ {2}) → 𝑃 ∈ ℙ)
11	10	ad2antlr 763	. . . . . . . . . . 11 ⊢ (((𝐴 ∈ ℤ ∧ 𝑃 ∈ (ℙ ∖ {2})) ∧ 𝑃 ∥ 𝐴) → 𝑃 ∈ ℙ)
12		prmuz2 15408	. . . . . . . . . . 11 ⊢ (𝑃 ∈ ℙ → 𝑃 ∈ (ℤ_≥‘2))
13	11, 12	syl 17	. . . . . . . . . 10 ⊢ (((𝐴 ∈ ℤ ∧ 𝑃 ∈ (ℙ ∖ {2})) ∧ 𝑃 ∥ 𝐴) → 𝑃 ∈ (ℤ_≥‘2))
14		eluz2b2 11761	. . . . . . . . . 10 ⊢ (𝑃 ∈ (ℤ_≥‘2) ↔ (𝑃 ∈ ℕ ∧ 1 < 𝑃))
15	13, 14	sylib 208	. . . . . . . . 9 ⊢ (((𝐴 ∈ ℤ ∧ 𝑃 ∈ (ℙ ∖ {2})) ∧ 𝑃 ∥ 𝐴) → (𝑃 ∈ ℕ ∧ 1 < 𝑃))
16	15	simpld 475	. . . . . . . 8 ⊢ (((𝐴 ∈ ℤ ∧ 𝑃 ∈ (ℙ ∖ {2})) ∧ 𝑃 ∥ 𝐴) → 𝑃 ∈ ℕ)
17	16	nnrpd 11870	. . . . . . 7 ⊢ (((𝐴 ∈ ℤ ∧ 𝑃 ∈ (ℙ ∖ {2})) ∧ 𝑃 ∥ 𝐴) → 𝑃 ∈ ℝ⁺)
18		0zd 11389	. . . . . . . . 9 ⊢ (((𝐴 ∈ ℤ ∧ 𝑃 ∈ (ℙ ∖ {2})) ∧ 𝑃 ∥ 𝐴) → 0 ∈ ℤ)
19		simpr 477	. . . . . . . . . . 11 ⊢ (((𝐴 ∈ ℤ ∧ 𝑃 ∈ (ℙ ∖ {2})) ∧ 𝑃 ∥ 𝐴) → 𝑃 ∥ 𝐴)
20		dvdsval3 14987	. . . . . . . . . . . 12 ⊢ ((𝑃 ∈ ℕ ∧ 𝐴 ∈ ℤ) → (𝑃 ∥ 𝐴 ↔ (𝐴 mod 𝑃) = 0))
21	16, 1, 20	syl2anc 693	. . . . . . . . . . 11 ⊢ (((𝐴 ∈ ℤ ∧ 𝑃 ∈ (ℙ ∖ {2})) ∧ 𝑃 ∥ 𝐴) → (𝑃 ∥ 𝐴 ↔ (𝐴 mod 𝑃) = 0))
22	19, 21	mpbid 222	. . . . . . . . . 10 ⊢ (((𝐴 ∈ ℤ ∧ 𝑃 ∈ (ℙ ∖ {2})) ∧ 𝑃 ∥ 𝐴) → (𝐴 mod 𝑃) = 0)
23		0mod 12701	. . . . . . . . . . 11 ⊢ (𝑃 ∈ ℝ⁺ → (0 mod 𝑃) = 0)
24	17, 23	syl 17	. . . . . . . . . 10 ⊢ (((𝐴 ∈ ℤ ∧ 𝑃 ∈ (ℙ ∖ {2})) ∧ 𝑃 ∥ 𝐴) → (0 mod 𝑃) = 0)
25	22, 24	eqtr4d 2659	. . . . . . . . 9 ⊢ (((𝐴 ∈ ℤ ∧ 𝑃 ∈ (ℙ ∖ {2})) ∧ 𝑃 ∥ 𝐴) → (𝐴 mod 𝑃) = (0 mod 𝑃))
26		modexp 12999	. . . . . . . . 9 ⊢ (((𝐴 ∈ ℤ ∧ 0 ∈ ℤ) ∧ (((𝑃 − 1) / 2) ∈ ℕ₀ ∧ 𝑃 ∈ ℝ⁺) ∧ (𝐴 mod 𝑃) = (0 mod 𝑃)) → ((𝐴↑((𝑃 − 1) / 2)) mod 𝑃) = ((0↑((𝑃 − 1) / 2)) mod 𝑃))
27	1, 18, 4, 17, 25, 26	syl221anc 1337	. . . . . . . 8 ⊢ (((𝐴 ∈ ℤ ∧ 𝑃 ∈ (ℙ ∖ {2})) ∧ 𝑃 ∥ 𝐴) → ((𝐴↑((𝑃 − 1) / 2)) mod 𝑃) = ((0↑((𝑃 − 1) / 2)) mod 𝑃))
28	3	0expd 13024	. . . . . . . . 9 ⊢ (((𝐴 ∈ ℤ ∧ 𝑃 ∈ (ℙ ∖ {2})) ∧ 𝑃 ∥ 𝐴) → (0↑((𝑃 − 1) / 2)) = 0)
29	28	oveq1d 6665	. . . . . . . 8 ⊢ (((𝐴 ∈ ℤ ∧ 𝑃 ∈ (ℙ ∖ {2})) ∧ 𝑃 ∥ 𝐴) → ((0↑((𝑃 − 1) / 2)) mod 𝑃) = (0 mod 𝑃))
30	27, 29	eqtrd 2656	. . . . . . 7 ⊢ (((𝐴 ∈ ℤ ∧ 𝑃 ∈ (ℙ ∖ {2})) ∧ 𝑃 ∥ 𝐴) → ((𝐴↑((𝑃 − 1) / 2)) mod 𝑃) = (0 mod 𝑃))
31		modadd1 12707	. . . . . . 7 ⊢ ((((𝐴↑((𝑃 − 1) / 2)) ∈ ℝ ∧ 0 ∈ ℝ) ∧ (1 ∈ ℝ ∧ 𝑃 ∈ ℝ⁺) ∧ ((𝐴↑((𝑃 − 1) / 2)) mod 𝑃) = (0 mod 𝑃)) → (((𝐴↑((𝑃 − 1) / 2)) + 1) mod 𝑃) = ((0 + 1) mod 𝑃))
32	7, 8, 9, 17, 30, 31	syl221anc 1337	. . . . . 6 ⊢ (((𝐴 ∈ ℤ ∧ 𝑃 ∈ (ℙ ∖ {2})) ∧ 𝑃 ∥ 𝐴) → (((𝐴↑((𝑃 − 1) / 2)) + 1) mod 𝑃) = ((0 + 1) mod 𝑃))
33		0p1e1 11132	. . . . . . 7 ⊢ (0 + 1) = 1
34	33	oveq1i 6660	. . . . . 6 ⊢ ((0 + 1) mod 𝑃) = (1 mod 𝑃)
35	32, 34	syl6eq 2672	. . . . 5 ⊢ (((𝐴 ∈ ℤ ∧ 𝑃 ∈ (ℙ ∖ {2})) ∧ 𝑃 ∥ 𝐴) → (((𝐴↑((𝑃 − 1) / 2)) + 1) mod 𝑃) = (1 mod 𝑃))
36	16	nnred 11035	. . . . . 6 ⊢ (((𝐴 ∈ ℤ ∧ 𝑃 ∈ (ℙ ∖ {2})) ∧ 𝑃 ∥ 𝐴) → 𝑃 ∈ ℝ)
37	15	simprd 479	. . . . . 6 ⊢ (((𝐴 ∈ ℤ ∧ 𝑃 ∈ (ℙ ∖ {2})) ∧ 𝑃 ∥ 𝐴) → 1 < 𝑃)
38		1mod 12702	. . . . . 6 ⊢ ((𝑃 ∈ ℝ ∧ 1 < 𝑃) → (1 mod 𝑃) = 1)
39	36, 37, 38	syl2anc 693	. . . . 5 ⊢ (((𝐴 ∈ ℤ ∧ 𝑃 ∈ (ℙ ∖ {2})) ∧ 𝑃 ∥ 𝐴) → (1 mod 𝑃) = 1)
40	35, 39	eqtrd 2656	. . . 4 ⊢ (((𝐴 ∈ ℤ ∧ 𝑃 ∈ (ℙ ∖ {2})) ∧ 𝑃 ∥ 𝐴) → (((𝐴↑((𝑃 − 1) / 2)) + 1) mod 𝑃) = 1)
41	40	oveq1d 6665	. . 3 ⊢ (((𝐴 ∈ ℤ ∧ 𝑃 ∈ (ℙ ∖ {2})) ∧ 𝑃 ∥ 𝐴) → ((((𝐴↑((𝑃 − 1) / 2)) + 1) mod 𝑃) − 1) = (1 − 1))
42		1m1e0 11089	. . . 4 ⊢ (1 − 1) = 0
43		lgslem2.z	. . . . . 6 ⊢ 𝑍 = {𝑥 ∈ ℤ ∣ (abs‘𝑥) ≤ 1}
44	43	lgslem2 25023	. . . . 5 ⊢ (-1 ∈ 𝑍 ∧ 0 ∈ 𝑍 ∧ 1 ∈ 𝑍)
45	44	simp2i 1071	. . . 4 ⊢ 0 ∈ 𝑍
46	42, 45	eqeltri 2697	. . 3 ⊢ (1 − 1) ∈ 𝑍
47	41, 46	syl6eqel 2709	. 2 ⊢ (((𝐴 ∈ ℤ ∧ 𝑃 ∈ (ℙ ∖ {2})) ∧ 𝑃 ∥ 𝐴) → ((((𝐴↑((𝑃 − 1) / 2)) + 1) mod 𝑃) − 1) ∈ 𝑍)
48		lgslem1 25022	. . . 4 ⊢ ((𝐴 ∈ ℤ ∧ 𝑃 ∈ (ℙ ∖ {2}) ∧ ¬ 𝑃 ∥ 𝐴) → (((𝐴↑((𝑃 − 1) / 2)) + 1) mod 𝑃) ∈ {0, 2})
49		elpri 4197	. . . 4 ⊢ ((((𝐴↑((𝑃 − 1) / 2)) + 1) mod 𝑃) ∈ {0, 2} → ((((𝐴↑((𝑃 − 1) / 2)) + 1) mod 𝑃) = 0 ∨ (((𝐴↑((𝑃 − 1) / 2)) + 1) mod 𝑃) = 2))
50		oveq1 6657	. . . . . 6 ⊢ ((((𝐴↑((𝑃 − 1) / 2)) + 1) mod 𝑃) = 0 → ((((𝐴↑((𝑃 − 1) / 2)) + 1) mod 𝑃) − 1) = (0 − 1))
51		df-neg 10269	. . . . . . 7 ⊢ -1 = (0 − 1)
52	44	simp1i 1070	. . . . . . 7 ⊢ -1 ∈ 𝑍
53	51, 52	eqeltrri 2698	. . . . . 6 ⊢ (0 − 1) ∈ 𝑍
54	50, 53	syl6eqel 2709	. . . . 5 ⊢ ((((𝐴↑((𝑃 − 1) / 2)) + 1) mod 𝑃) = 0 → ((((𝐴↑((𝑃 − 1) / 2)) + 1) mod 𝑃) − 1) ∈ 𝑍)
55		oveq1 6657	. . . . . 6 ⊢ ((((𝐴↑((𝑃 − 1) / 2)) + 1) mod 𝑃) = 2 → ((((𝐴↑((𝑃 − 1) / 2)) + 1) mod 𝑃) − 1) = (2 − 1))
56		2m1e1 11135	. . . . . . 7 ⊢ (2 − 1) = 1
57	44	simp3i 1072	. . . . . . 7 ⊢ 1 ∈ 𝑍
58	56, 57	eqeltri 2697	. . . . . 6 ⊢ (2 − 1) ∈ 𝑍
59	55, 58	syl6eqel 2709	. . . . 5 ⊢ ((((𝐴↑((𝑃 − 1) / 2)) + 1) mod 𝑃) = 2 → ((((𝐴↑((𝑃 − 1) / 2)) + 1) mod 𝑃) − 1) ∈ 𝑍)
60	54, 59	jaoi 394	. . . 4 ⊢ (((((𝐴↑((𝑃 − 1) / 2)) + 1) mod 𝑃) = 0 ∨ (((𝐴↑((𝑃 − 1) / 2)) + 1) mod 𝑃) = 2) → ((((𝐴↑((𝑃 − 1) / 2)) + 1) mod 𝑃) − 1) ∈ 𝑍)
61	48, 49, 60	3syl 18	. . 3 ⊢ ((𝐴 ∈ ℤ ∧ 𝑃 ∈ (ℙ ∖ {2}) ∧ ¬ 𝑃 ∥ 𝐴) → ((((𝐴↑((𝑃 − 1) / 2)) + 1) mod 𝑃) − 1) ∈ 𝑍)
62	61	3expa 1265	. 2 ⊢ (((𝐴 ∈ ℤ ∧ 𝑃 ∈ (ℙ ∖ {2})) ∧ ¬ 𝑃 ∥ 𝐴) → ((((𝐴↑((𝑃 − 1) / 2)) + 1) mod 𝑃) − 1) ∈ 𝑍)
63	47, 62	pm2.61dan 832	1 ⊢ ((𝐴 ∈ ℤ ∧ 𝑃 ∈ (ℙ ∖ {2})) → ((((𝐴↑((𝑃 − 1) / 2)) + 1) mod 𝑃) − 1) ∈ 𝑍)

Colors of variables: wff setvar class

Syntax hints: ¬ wn 3 → wi 4 ↔ wb 196 ∨ wo 383 ∧ wa 384 ∧ w3a 1037 = wceq 1483 ∈ wcel 1990 {crab 2916 ∖ cdif 3571 {csn 4177 {cpr 4179 class class class wbr 4653 ‘cfv 5888 (class class class)co 6650 ℝcr 9935 0cc0 9936 1c1 9937 + caddc 9939 < clt 10074 ≤ cle 10075 − cmin 10266 -cneg 10267 / cdiv 10684 ℕcn 11020 2c2 11070 ℕ₀cn0 11292 ℤcz 11377 ℤ_≥cuz 11687 ℝ⁺crp 11832 mod cmo 12668 ↑cexp 12860 abscabs 13974 ∥ cdvds 14983 ℙcprime 15385

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1722 ax-4 1737 ax-5 1839 ax-6 1888 ax-7 1935 ax-8 1992 ax-9 1999 ax-10 2019 ax-11 2034 ax-12 2047 ax-13 2246 ax-ext 2602 ax-rep 4771 ax-sep 4781 ax-nul 4789 ax-pow 4843 ax-pr 4906 ax-un 6949 ax-cnex 9992 ax-resscn 9993 ax-1cn 9994 ax-icn 9995 ax-addcl 9996 ax-addrcl 9997 ax-mulcl 9998 ax-mulrcl 9999 ax-mulcom 10000 ax-addass 10001 ax-mulass 10002 ax-distr 10003 ax-i2m1 10004 ax-1ne0 10005 ax-1rid 10006 ax-rnegex 10007 ax-rrecex 10008 ax-cnre 10009 ax-pre-lttri 10010 ax-pre-lttrn 10011 ax-pre-ltadd 10012 ax-pre-mulgt0 10013 ax-pre-sup 10014

This theorem depends on definitions: df-bi 197 df-or 385 df-an 386 df-3or 1038 df-3an 1039 df-tru 1486 df-ex 1705 df-nf 1710 df-sb 1881 df-eu 2474 df-mo 2475 df-clab 2609 df-cleq 2615 df-clel 2618 df-nfc 2753 df-ne 2795 df-nel 2898 df-ral 2917 df-rex 2918 df-reu 2919 df-rmo 2920 df-rab 2921 df-v 3202 df-sbc 3436 df-csb 3534 df-dif 3577 df-un 3579 df-in 3581 df-ss 3588 df-pss 3590 df-nul 3916 df-if 4087 df-pw 4160 df-sn 4178 df-pr 4180 df-tp 4182 df-op 4184 df-uni 4437 df-int 4476 df-iun 4522 df-br 4654 df-opab 4713 df-mpt 4730 df-tr 4753 df-id 5024 df-eprel 5029 df-po 5035 df-so 5036 df-fr 5073 df-we 5075 df-xp 5120 df-rel 5121 df-cnv 5122 df-co 5123 df-dm 5124 df-rn 5125 df-res 5126 df-ima 5127 df-pred 5680 df-ord 5726 df-on 5727 df-lim 5728 df-suc 5729 df-iota 5851 df-fun 5890 df-fn 5891 df-f 5892 df-f1 5893 df-fo 5894 df-f1o 5895 df-fv 5896 df-riota 6611 df-ov 6653 df-oprab 6654 df-mpt2 6655 df-om 7066 df-1st 7168 df-2nd 7169 df-wrecs 7407 df-recs 7468 df-rdg 7506 df-1o 7560 df-2o 7561 df-oadd 7564 df-er 7742 df-map 7859 df-en 7956 df-dom 7957 df-sdom 7958 df-fin 7959 df-sup 8348 df-inf 8349 df-card 8765 df-cda 8990 df-pnf 10076 df-mnf 10077 df-xr 10078 df-ltxr 10079 df-le 10080 df-sub 10268 df-neg 10269 df-div 10685 df-nn 11021 df-2 11079 df-3 11080 df-n0 11293 df-xnn0 11364 df-z 11378 df-uz 11688 df-rp 11833 df-fz 12327 df-fzo 12466 df-fl 12593 df-mod 12669 df-seq 12802 df-exp 12861 df-hash 13118 df-cj 13839 df-re 13840 df-im 13841 df-sqrt 13975 df-abs 13976 df-dvds 14984 df-gcd 15217 df-prm 15386 df-phi 15471

This theorem is referenced by: lgsfcl2 25028

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