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Mirrors > Home > ILE Home > Th. List > lgslem4 | GIF version |
Description: Lemma for lgsfcl2 14074. (Contributed by Mario Carneiro, 4-Feb-2015.) (Proof shortened by AV, 19-Mar-2022.) |
Ref | Expression |
---|---|
lgslem2.z | ⊢ 𝑍 = {𝑥 ∈ ℤ ∣ (abs‘𝑥) ≤ 1} |
Ref | Expression |
---|---|
lgslem4 | ⊢ ((𝐴 ∈ ℤ ∧ 𝑃 ∈ (ℙ ∖ {2})) → ((((𝐴↑((𝑃 − 1) / 2)) + 1) mod 𝑃) − 1) ∈ 𝑍) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | eldifi 3257 | . . . . . . . 8 ⊢ (𝑃 ∈ (ℙ ∖ {2}) → 𝑃 ∈ ℙ) | |
2 | 1 | adantl 277 | . . . . . . 7 ⊢ ((𝐴 ∈ ℤ ∧ 𝑃 ∈ (ℙ ∖ {2})) → 𝑃 ∈ ℙ) |
3 | simpl 109 | . . . . . . 7 ⊢ ((𝐴 ∈ ℤ ∧ 𝑃 ∈ (ℙ ∖ {2})) → 𝐴 ∈ ℤ) | |
4 | oddprm 12242 | . . . . . . . 8 ⊢ (𝑃 ∈ (ℙ ∖ {2}) → ((𝑃 − 1) / 2) ∈ ℕ) | |
5 | 4 | adantl 277 | . . . . . . 7 ⊢ ((𝐴 ∈ ℤ ∧ 𝑃 ∈ (ℙ ∖ {2})) → ((𝑃 − 1) / 2) ∈ ℕ) |
6 | prmdvdsexp 12131 | . . . . . . 7 ⊢ ((𝑃 ∈ ℙ ∧ 𝐴 ∈ ℤ ∧ ((𝑃 − 1) / 2) ∈ ℕ) → (𝑃 ∥ (𝐴↑((𝑃 − 1) / 2)) ↔ 𝑃 ∥ 𝐴)) | |
7 | 2, 3, 5, 6 | syl3anc 1238 | . . . . . 6 ⊢ ((𝐴 ∈ ℤ ∧ 𝑃 ∈ (ℙ ∖ {2})) → (𝑃 ∥ (𝐴↑((𝑃 − 1) / 2)) ↔ 𝑃 ∥ 𝐴)) |
8 | 7 | biimpar 297 | . . . . 5 ⊢ (((𝐴 ∈ ℤ ∧ 𝑃 ∈ (ℙ ∖ {2})) ∧ 𝑃 ∥ 𝐴) → 𝑃 ∥ (𝐴↑((𝑃 − 1) / 2))) |
9 | prmgt1 12115 | . . . . . . 7 ⊢ (𝑃 ∈ ℙ → 1 < 𝑃) | |
10 | 1, 9 | syl 14 | . . . . . 6 ⊢ (𝑃 ∈ (ℙ ∖ {2}) → 1 < 𝑃) |
11 | 10 | ad2antlr 489 | . . . . 5 ⊢ (((𝐴 ∈ ℤ ∧ 𝑃 ∈ (ℙ ∖ {2})) ∧ 𝑃 ∥ 𝐴) → 1 < 𝑃) |
12 | p1modz1 11785 | . . . . 5 ⊢ ((𝑃 ∥ (𝐴↑((𝑃 − 1) / 2)) ∧ 1 < 𝑃) → (((𝐴↑((𝑃 − 1) / 2)) + 1) mod 𝑃) = 1) | |
13 | 8, 11, 12 | syl2anc 411 | . . . 4 ⊢ (((𝐴 ∈ ℤ ∧ 𝑃 ∈ (ℙ ∖ {2})) ∧ 𝑃 ∥ 𝐴) → (((𝐴↑((𝑃 − 1) / 2)) + 1) mod 𝑃) = 1) |
14 | 13 | oveq1d 5884 | . . 3 ⊢ (((𝐴 ∈ ℤ ∧ 𝑃 ∈ (ℙ ∖ {2})) ∧ 𝑃 ∥ 𝐴) → ((((𝐴↑((𝑃 − 1) / 2)) + 1) mod 𝑃) − 1) = (1 − 1)) |
15 | 1m1e0 8977 | . . . 4 ⊢ (1 − 1) = 0 | |
16 | lgslem2.z | . . . . . 6 ⊢ 𝑍 = {𝑥 ∈ ℤ ∣ (abs‘𝑥) ≤ 1} | |
17 | 16 | lgslem2 14069 | . . . . 5 ⊢ (-1 ∈ 𝑍 ∧ 0 ∈ 𝑍 ∧ 1 ∈ 𝑍) |
18 | 17 | simp2i 1007 | . . . 4 ⊢ 0 ∈ 𝑍 |
19 | 15, 18 | eqeltri 2250 | . . 3 ⊢ (1 − 1) ∈ 𝑍 |
20 | 14, 19 | eqeltrdi 2268 | . 2 ⊢ (((𝐴 ∈ ℤ ∧ 𝑃 ∈ (ℙ ∖ {2})) ∧ 𝑃 ∥ 𝐴) → ((((𝐴↑((𝑃 − 1) / 2)) + 1) mod 𝑃) − 1) ∈ 𝑍) |
21 | lgslem1 14068 | . . . 4 ⊢ ((𝐴 ∈ ℤ ∧ 𝑃 ∈ (ℙ ∖ {2}) ∧ ¬ 𝑃 ∥ 𝐴) → (((𝐴↑((𝑃 − 1) / 2)) + 1) mod 𝑃) ∈ {0, 2}) | |
22 | elpri 3614 | . . . 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)) | |
23 | oveq1 5876 | . . . . . 6 ⊢ ((((𝐴↑((𝑃 − 1) / 2)) + 1) mod 𝑃) = 0 → ((((𝐴↑((𝑃 − 1) / 2)) + 1) mod 𝑃) − 1) = (0 − 1)) | |
24 | df-neg 8121 | . . . . . . 7 ⊢ -1 = (0 − 1) | |
25 | 17 | simp1i 1006 | . . . . . . 7 ⊢ -1 ∈ 𝑍 |
26 | 24, 25 | eqeltrri 2251 | . . . . . 6 ⊢ (0 − 1) ∈ 𝑍 |
27 | 23, 26 | eqeltrdi 2268 | . . . . 5 ⊢ ((((𝐴↑((𝑃 − 1) / 2)) + 1) mod 𝑃) = 0 → ((((𝐴↑((𝑃 − 1) / 2)) + 1) mod 𝑃) − 1) ∈ 𝑍) |
28 | oveq1 5876 | . . . . . 6 ⊢ ((((𝐴↑((𝑃 − 1) / 2)) + 1) mod 𝑃) = 2 → ((((𝐴↑((𝑃 − 1) / 2)) + 1) mod 𝑃) − 1) = (2 − 1)) | |
29 | 2m1e1 9026 | . . . . . . 7 ⊢ (2 − 1) = 1 | |
30 | 17 | simp3i 1008 | . . . . . . 7 ⊢ 1 ∈ 𝑍 |
31 | 29, 30 | eqeltri 2250 | . . . . . 6 ⊢ (2 − 1) ∈ 𝑍 |
32 | 28, 31 | eqeltrdi 2268 | . . . . 5 ⊢ ((((𝐴↑((𝑃 − 1) / 2)) + 1) mod 𝑃) = 2 → ((((𝐴↑((𝑃 − 1) / 2)) + 1) mod 𝑃) − 1) ∈ 𝑍) |
33 | 27, 32 | jaoi 716 | . . . 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) |
34 | 21, 22, 33 | 3syl 17 | . . 3 ⊢ ((𝐴 ∈ ℤ ∧ 𝑃 ∈ (ℙ ∖ {2}) ∧ ¬ 𝑃 ∥ 𝐴) → ((((𝐴↑((𝑃 − 1) / 2)) + 1) mod 𝑃) − 1) ∈ 𝑍) |
35 | 34 | 3expa 1203 | . 2 ⊢ (((𝐴 ∈ ℤ ∧ 𝑃 ∈ (ℙ ∖ {2})) ∧ ¬ 𝑃 ∥ 𝐴) → ((((𝐴↑((𝑃 − 1) / 2)) + 1) mod 𝑃) − 1) ∈ 𝑍) |
36 | prmnn 12093 | . . . . . 6 ⊢ (𝑃 ∈ ℙ → 𝑃 ∈ ℕ) | |
37 | 1, 36 | syl 14 | . . . . 5 ⊢ (𝑃 ∈ (ℙ ∖ {2}) → 𝑃 ∈ ℕ) |
38 | 37 | adantl 277 | . . . 4 ⊢ ((𝐴 ∈ ℤ ∧ 𝑃 ∈ (ℙ ∖ {2})) → 𝑃 ∈ ℕ) |
39 | dvdsdc 11789 | . . . 4 ⊢ ((𝑃 ∈ ℕ ∧ 𝐴 ∈ ℤ) → DECID 𝑃 ∥ 𝐴) | |
40 | 38, 3, 39 | syl2anc 411 | . . 3 ⊢ ((𝐴 ∈ ℤ ∧ 𝑃 ∈ (ℙ ∖ {2})) → DECID 𝑃 ∥ 𝐴) |
41 | exmiddc 836 | . . 3 ⊢ (DECID 𝑃 ∥ 𝐴 → (𝑃 ∥ 𝐴 ∨ ¬ 𝑃 ∥ 𝐴)) | |
42 | 40, 41 | syl 14 | . 2 ⊢ ((𝐴 ∈ ℤ ∧ 𝑃 ∈ (ℙ ∖ {2})) → (𝑃 ∥ 𝐴 ∨ ¬ 𝑃 ∥ 𝐴)) |
43 | 20, 35, 42 | mpjaodan 798 | 1 ⊢ ((𝐴 ∈ ℤ ∧ 𝑃 ∈ (ℙ ∖ {2})) → ((((𝐴↑((𝑃 − 1) / 2)) + 1) mod 𝑃) − 1) ∈ 𝑍) |
Colors of variables: wff set class |
Syntax hints: ¬ wn 3 → wi 4 ∧ wa 104 ↔ wb 105 ∨ wo 708 DECID wdc 834 ∧ w3a 978 = wceq 1353 ∈ wcel 2148 {crab 2459 ∖ cdif 3126 {csn 3591 {cpr 3592 class class class wbr 4000 ‘cfv 5212 (class class class)co 5869 0cc0 7802 1c1 7803 + caddc 7805 < clt 7982 ≤ cle 7983 − cmin 8118 -cneg 8119 / cdiv 8618 ℕcn 8908 2c2 8959 ℤcz 9242 mod cmo 10308 ↑cexp 10505 abscabs 10990 ∥ cdvds 11778 ℙcprime 12090 |
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 614 ax-in2 615 ax-io 709 ax-5 1447 ax-7 1448 ax-gen 1449 ax-ie1 1493 ax-ie2 1494 ax-8 1504 ax-10 1505 ax-11 1506 ax-i12 1507 ax-bndl 1509 ax-4 1510 ax-17 1526 ax-i9 1530 ax-ial 1534 ax-i5r 1535 ax-13 2150 ax-14 2151 ax-ext 2159 ax-coll 4115 ax-sep 4118 ax-nul 4126 ax-pow 4171 ax-pr 4206 ax-un 4430 ax-setind 4533 ax-iinf 4584 ax-cnex 7893 ax-resscn 7894 ax-1cn 7895 ax-1re 7896 ax-icn 7897 ax-addcl 7898 ax-addrcl 7899 ax-mulcl 7900 ax-mulrcl 7901 ax-addcom 7902 ax-mulcom 7903 ax-addass 7904 ax-mulass 7905 ax-distr 7906 ax-i2m1 7907 ax-0lt1 7908 ax-1rid 7909 ax-0id 7910 ax-rnegex 7911 ax-precex 7912 ax-cnre 7913 ax-pre-ltirr 7914 ax-pre-ltwlin 7915 ax-pre-lttrn 7916 ax-pre-apti 7917 ax-pre-ltadd 7918 ax-pre-mulgt0 7919 ax-pre-mulext 7920 ax-arch 7921 ax-caucvg 7922 |
This theorem depends on definitions: df-bi 117 df-stab 831 df-dc 835 df-3or 979 df-3an 980 df-tru 1356 df-fal 1359 df-xor 1376 df-nf 1461 df-sb 1763 df-eu 2029 df-mo 2030 df-clab 2164 df-cleq 2170 df-clel 2173 df-nfc 2308 df-ne 2348 df-nel 2443 df-ral 2460 df-rex 2461 df-reu 2462 df-rmo 2463 df-rab 2464 df-v 2739 df-sbc 2963 df-csb 3058 df-dif 3131 df-un 3133 df-in 3135 df-ss 3142 df-nul 3423 df-if 3535 df-pw 3576 df-sn 3597 df-pr 3598 df-op 3600 df-uni 3808 df-int 3843 df-iun 3886 df-br 4001 df-opab 4062 df-mpt 4063 df-tr 4099 df-id 4290 df-po 4293 df-iso 4294 df-iord 4363 df-on 4365 df-ilim 4366 df-suc 4368 df-iom 4587 df-xp 4629 df-rel 4630 df-cnv 4631 df-co 4632 df-dm 4633 df-rn 4634 df-res 4635 df-ima 4636 df-iota 5174 df-fun 5214 df-fn 5215 df-f 5216 df-f1 5217 df-fo 5218 df-f1o 5219 df-fv 5220 df-isom 5221 df-riota 5825 df-ov 5872 df-oprab 5873 df-mpo 5874 df-1st 6135 df-2nd 6136 df-recs 6300 df-irdg 6365 df-frec 6386 df-1o 6411 df-2o 6412 df-oadd 6415 df-er 6529 df-en 6735 df-dom 6736 df-fin 6737 df-sup 6977 df-pnf 7984 df-mnf 7985 df-xr 7986 df-ltxr 7987 df-le 7988 df-sub 8120 df-neg 8121 df-reap 8522 df-ap 8529 df-div 8619 df-inn 8909 df-2 8967 df-3 8968 df-4 8969 df-n0 9166 df-z 9243 df-uz 9518 df-q 9609 df-rp 9641 df-fz 9996 df-fzo 10129 df-fl 10256 df-mod 10309 df-seqfrec 10432 df-exp 10506 df-ihash 10740 df-cj 10835 df-re 10836 df-im 10837 df-rsqrt 10991 df-abs 10992 df-clim 11271 df-proddc 11543 df-dvds 11779 df-gcd 11927 df-prm 12091 df-phi 12194 |
This theorem is referenced by: lgsfcl2 14074 |
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