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Mirrors > Home > MPE Home > Th. List > lgsqrlem3 | Structured version Visualization version GIF version |
Description: Lemma for lgsqr 27239. (Contributed by Mario Carneiro, 15-Jun-2015.) |
Ref | Expression |
---|---|
lgsqr.y | ⊢ 𝑌 = (ℤ/nℤ‘𝑃) |
lgsqr.s | ⊢ 𝑆 = (Poly1‘𝑌) |
lgsqr.b | ⊢ 𝐵 = (Base‘𝑆) |
lgsqr.d | ⊢ 𝐷 = ( deg1 ‘𝑌) |
lgsqr.o | ⊢ 𝑂 = (eval1‘𝑌) |
lgsqr.e | ⊢ ↑ = (.g‘(mulGrp‘𝑆)) |
lgsqr.x | ⊢ 𝑋 = (var1‘𝑌) |
lgsqr.m | ⊢ − = (-g‘𝑆) |
lgsqr.u | ⊢ 1 = (1r‘𝑆) |
lgsqr.t | ⊢ 𝑇 = ((((𝑃 − 1) / 2) ↑ 𝑋) − 1 ) |
lgsqr.l | ⊢ 𝐿 = (ℤRHom‘𝑌) |
lgsqr.1 | ⊢ (𝜑 → 𝑃 ∈ (ℙ ∖ {2})) |
lgsqr.g | ⊢ 𝐺 = (𝑦 ∈ (1...((𝑃 − 1) / 2)) ↦ (𝐿‘(𝑦↑2))) |
lgsqr.3 | ⊢ (𝜑 → 𝐴 ∈ ℤ) |
lgsqr.4 | ⊢ (𝜑 → (𝐴 /L 𝑃) = 1) |
Ref | Expression |
---|---|
lgsqrlem3 | ⊢ (𝜑 → (𝐿‘𝐴) ∈ (◡(𝑂‘𝑇) “ {(0g‘𝑌)})) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | lgsqr.1 | . . . . . . . . . 10 ⊢ (𝜑 → 𝑃 ∈ (ℙ ∖ {2})) | |
2 | 1 | eldifad 3955 | . . . . . . . . 9 ⊢ (𝜑 → 𝑃 ∈ ℙ) |
3 | lgsqr.y | . . . . . . . . . 10 ⊢ 𝑌 = (ℤ/nℤ‘𝑃) | |
4 | 3 | znfld 21455 | . . . . . . . . 9 ⊢ (𝑃 ∈ ℙ → 𝑌 ∈ Field) |
5 | 2, 4 | syl 17 | . . . . . . . 8 ⊢ (𝜑 → 𝑌 ∈ Field) |
6 | fldidom 21219 | . . . . . . . 8 ⊢ (𝑌 ∈ Field → 𝑌 ∈ IDomn) | |
7 | 5, 6 | syl 17 | . . . . . . 7 ⊢ (𝜑 → 𝑌 ∈ IDomn) |
8 | isidom 21216 | . . . . . . . 8 ⊢ (𝑌 ∈ IDomn ↔ (𝑌 ∈ CRing ∧ 𝑌 ∈ Domn)) | |
9 | 8 | simplbi 497 | . . . . . . 7 ⊢ (𝑌 ∈ IDomn → 𝑌 ∈ CRing) |
10 | 7, 9 | syl 17 | . . . . . 6 ⊢ (𝜑 → 𝑌 ∈ CRing) |
11 | crngring 20150 | . . . . . 6 ⊢ (𝑌 ∈ CRing → 𝑌 ∈ Ring) | |
12 | 10, 11 | syl 17 | . . . . 5 ⊢ (𝜑 → 𝑌 ∈ Ring) |
13 | lgsqr.l | . . . . . 6 ⊢ 𝐿 = (ℤRHom‘𝑌) | |
14 | 13 | zrhrhm 21398 | . . . . 5 ⊢ (𝑌 ∈ Ring → 𝐿 ∈ (ℤring RingHom 𝑌)) |
15 | 12, 14 | syl 17 | . . . 4 ⊢ (𝜑 → 𝐿 ∈ (ℤring RingHom 𝑌)) |
16 | zringbas 21340 | . . . . 5 ⊢ ℤ = (Base‘ℤring) | |
17 | eqid 2726 | . . . . 5 ⊢ (Base‘𝑌) = (Base‘𝑌) | |
18 | 16, 17 | rhmf 20387 | . . . 4 ⊢ (𝐿 ∈ (ℤring RingHom 𝑌) → 𝐿:ℤ⟶(Base‘𝑌)) |
19 | 15, 18 | syl 17 | . . 3 ⊢ (𝜑 → 𝐿:ℤ⟶(Base‘𝑌)) |
20 | lgsqr.3 | . . 3 ⊢ (𝜑 → 𝐴 ∈ ℤ) | |
21 | 19, 20 | ffvelcdmd 7081 | . 2 ⊢ (𝜑 → (𝐿‘𝐴) ∈ (Base‘𝑌)) |
22 | lgsqr.s | . . 3 ⊢ 𝑆 = (Poly1‘𝑌) | |
23 | lgsqr.b | . . 3 ⊢ 𝐵 = (Base‘𝑆) | |
24 | lgsqr.d | . . 3 ⊢ 𝐷 = ( deg1 ‘𝑌) | |
25 | lgsqr.o | . . 3 ⊢ 𝑂 = (eval1‘𝑌) | |
26 | lgsqr.e | . . 3 ⊢ ↑ = (.g‘(mulGrp‘𝑆)) | |
27 | lgsqr.x | . . 3 ⊢ 𝑋 = (var1‘𝑌) | |
28 | lgsqr.m | . . 3 ⊢ − = (-g‘𝑆) | |
29 | lgsqr.u | . . 3 ⊢ 1 = (1r‘𝑆) | |
30 | lgsqr.t | . . 3 ⊢ 𝑇 = ((((𝑃 − 1) / 2) ↑ 𝑋) − 1 ) | |
31 | lgsvalmod 27204 | . . . . 5 ⊢ ((𝐴 ∈ ℤ ∧ 𝑃 ∈ (ℙ ∖ {2})) → ((𝐴 /L 𝑃) mod 𝑃) = ((𝐴↑((𝑃 − 1) / 2)) mod 𝑃)) | |
32 | 20, 1, 31 | syl2anc 583 | . . . 4 ⊢ (𝜑 → ((𝐴 /L 𝑃) mod 𝑃) = ((𝐴↑((𝑃 − 1) / 2)) mod 𝑃)) |
33 | lgsqr.4 | . . . . 5 ⊢ (𝜑 → (𝐴 /L 𝑃) = 1) | |
34 | 33 | oveq1d 7420 | . . . 4 ⊢ (𝜑 → ((𝐴 /L 𝑃) mod 𝑃) = (1 mod 𝑃)) |
35 | 32, 34 | eqtr3d 2768 | . . 3 ⊢ (𝜑 → ((𝐴↑((𝑃 − 1) / 2)) mod 𝑃) = (1 mod 𝑃)) |
36 | 3, 22, 23, 24, 25, 26, 27, 28, 29, 30, 13, 1, 20, 35 | lgsqrlem1 27234 | . 2 ⊢ (𝜑 → ((𝑂‘𝑇)‘(𝐿‘𝐴)) = (0g‘𝑌)) |
37 | eqid 2726 | . . . . 5 ⊢ (𝑌 ↑s (Base‘𝑌)) = (𝑌 ↑s (Base‘𝑌)) | |
38 | eqid 2726 | . . . . 5 ⊢ (Base‘(𝑌 ↑s (Base‘𝑌))) = (Base‘(𝑌 ↑s (Base‘𝑌))) | |
39 | fvexd 6900 | . . . . 5 ⊢ (𝜑 → (Base‘𝑌) ∈ V) | |
40 | 25, 22, 37, 17 | evl1rhm 22206 | . . . . . . . 8 ⊢ (𝑌 ∈ CRing → 𝑂 ∈ (𝑆 RingHom (𝑌 ↑s (Base‘𝑌)))) |
41 | 10, 40 | syl 17 | . . . . . . 7 ⊢ (𝜑 → 𝑂 ∈ (𝑆 RingHom (𝑌 ↑s (Base‘𝑌)))) |
42 | 23, 38 | rhmf 20387 | . . . . . . 7 ⊢ (𝑂 ∈ (𝑆 RingHom (𝑌 ↑s (Base‘𝑌))) → 𝑂:𝐵⟶(Base‘(𝑌 ↑s (Base‘𝑌)))) |
43 | 41, 42 | syl 17 | . . . . . 6 ⊢ (𝜑 → 𝑂:𝐵⟶(Base‘(𝑌 ↑s (Base‘𝑌)))) |
44 | 22 | ply1ring 22121 | . . . . . . . . . 10 ⊢ (𝑌 ∈ Ring → 𝑆 ∈ Ring) |
45 | 12, 44 | syl 17 | . . . . . . . . 9 ⊢ (𝜑 → 𝑆 ∈ Ring) |
46 | ringgrp 20143 | . . . . . . . . 9 ⊢ (𝑆 ∈ Ring → 𝑆 ∈ Grp) | |
47 | 45, 46 | syl 17 | . . . . . . . 8 ⊢ (𝜑 → 𝑆 ∈ Grp) |
48 | eqid 2726 | . . . . . . . . . 10 ⊢ (mulGrp‘𝑆) = (mulGrp‘𝑆) | |
49 | 48, 23 | mgpbas 20045 | . . . . . . . . 9 ⊢ 𝐵 = (Base‘(mulGrp‘𝑆)) |
50 | 48 | ringmgp 20144 | . . . . . . . . . 10 ⊢ (𝑆 ∈ Ring → (mulGrp‘𝑆) ∈ Mnd) |
51 | 45, 50 | syl 17 | . . . . . . . . 9 ⊢ (𝜑 → (mulGrp‘𝑆) ∈ Mnd) |
52 | oddprm 16752 | . . . . . . . . . . 11 ⊢ (𝑃 ∈ (ℙ ∖ {2}) → ((𝑃 − 1) / 2) ∈ ℕ) | |
53 | 1, 52 | syl 17 | . . . . . . . . . 10 ⊢ (𝜑 → ((𝑃 − 1) / 2) ∈ ℕ) |
54 | 53 | nnnn0d 12536 | . . . . . . . . 9 ⊢ (𝜑 → ((𝑃 − 1) / 2) ∈ ℕ0) |
55 | 27, 22, 23 | vr1cl 22091 | . . . . . . . . . 10 ⊢ (𝑌 ∈ Ring → 𝑋 ∈ 𝐵) |
56 | 12, 55 | syl 17 | . . . . . . . . 9 ⊢ (𝜑 → 𝑋 ∈ 𝐵) |
57 | 49, 26, 51, 54, 56 | mulgnn0cld 19022 | . . . . . . . 8 ⊢ (𝜑 → (((𝑃 − 1) / 2) ↑ 𝑋) ∈ 𝐵) |
58 | 23, 29 | ringidcl 20165 | . . . . . . . . 9 ⊢ (𝑆 ∈ Ring → 1 ∈ 𝐵) |
59 | 45, 58 | syl 17 | . . . . . . . 8 ⊢ (𝜑 → 1 ∈ 𝐵) |
60 | 23, 28 | grpsubcl 18948 | . . . . . . . 8 ⊢ ((𝑆 ∈ Grp ∧ (((𝑃 − 1) / 2) ↑ 𝑋) ∈ 𝐵 ∧ 1 ∈ 𝐵) → ((((𝑃 − 1) / 2) ↑ 𝑋) − 1 ) ∈ 𝐵) |
61 | 47, 57, 59, 60 | syl3anc 1368 | . . . . . . 7 ⊢ (𝜑 → ((((𝑃 − 1) / 2) ↑ 𝑋) − 1 ) ∈ 𝐵) |
62 | 30, 61 | eqeltrid 2831 | . . . . . 6 ⊢ (𝜑 → 𝑇 ∈ 𝐵) |
63 | 43, 62 | ffvelcdmd 7081 | . . . . 5 ⊢ (𝜑 → (𝑂‘𝑇) ∈ (Base‘(𝑌 ↑s (Base‘𝑌)))) |
64 | 37, 17, 38, 5, 39, 63 | pwselbas 17444 | . . . 4 ⊢ (𝜑 → (𝑂‘𝑇):(Base‘𝑌)⟶(Base‘𝑌)) |
65 | 64 | ffnd 6712 | . . 3 ⊢ (𝜑 → (𝑂‘𝑇) Fn (Base‘𝑌)) |
66 | fniniseg 7055 | . . 3 ⊢ ((𝑂‘𝑇) Fn (Base‘𝑌) → ((𝐿‘𝐴) ∈ (◡(𝑂‘𝑇) “ {(0g‘𝑌)}) ↔ ((𝐿‘𝐴) ∈ (Base‘𝑌) ∧ ((𝑂‘𝑇)‘(𝐿‘𝐴)) = (0g‘𝑌)))) | |
67 | 65, 66 | syl 17 | . 2 ⊢ (𝜑 → ((𝐿‘𝐴) ∈ (◡(𝑂‘𝑇) “ {(0g‘𝑌)}) ↔ ((𝐿‘𝐴) ∈ (Base‘𝑌) ∧ ((𝑂‘𝑇)‘(𝐿‘𝐴)) = (0g‘𝑌)))) |
68 | 21, 36, 67 | mpbir2and 710 | 1 ⊢ (𝜑 → (𝐿‘𝐴) ∈ (◡(𝑂‘𝑇) “ {(0g‘𝑌)})) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 205 ∧ wa 395 = wceq 1533 ∈ wcel 2098 Vcvv 3468 ∖ cdif 3940 {csn 4623 ↦ cmpt 5224 ◡ccnv 5668 “ cima 5672 Fn wfn 6532 ⟶wf 6533 ‘cfv 6537 (class class class)co 7405 1c1 11113 − cmin 11448 / cdiv 11875 ℕcn 12216 2c2 12271 ℤcz 12562 ...cfz 13490 mod cmo 13840 ↑cexp 14032 ℙcprime 16615 Basecbs 17153 0gc0g 17394 ↑s cpws 17401 Mndcmnd 18667 Grpcgrp 18863 -gcsg 18865 .gcmg 18995 mulGrpcmgp 20039 1rcur 20086 Ringcrg 20138 CRingccrg 20139 RingHom crh 20371 Fieldcfield 20588 Domncdomn 21190 IDomncidom 21191 ℤringczring 21333 ℤRHomczrh 21386 ℤ/nℤczn 21389 var1cv1 22050 Poly1cpl1 22051 eval1ce1 22188 deg1 cdg1 25942 /L clgs 27182 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1789 ax-4 1803 ax-5 1905 ax-6 1963 ax-7 2003 ax-8 2100 ax-9 2108 ax-10 2129 ax-11 2146 ax-12 2163 ax-ext 2697 ax-rep 5278 ax-sep 5292 ax-nul 5299 ax-pow 5356 ax-pr 5420 ax-un 7722 ax-cnex 11168 ax-resscn 11169 ax-1cn 11170 ax-icn 11171 ax-addcl 11172 ax-addrcl 11173 ax-mulcl 11174 ax-mulrcl 11175 ax-mulcom 11176 ax-addass 11177 ax-mulass 11178 ax-distr 11179 ax-i2m1 11180 ax-1ne0 11181 ax-1rid 11182 ax-rnegex 11183 ax-rrecex 11184 ax-cnre 11185 ax-pre-lttri 11186 ax-pre-lttrn 11187 ax-pre-ltadd 11188 ax-pre-mulgt0 11189 ax-pre-sup 11190 ax-addf 11191 ax-mulf 11192 |
This theorem depends on definitions: df-bi 206 df-an 396 df-or 845 df-3or 1085 df-3an 1086 df-tru 1536 df-fal 1546 df-ex 1774 df-nf 1778 df-sb 2060 df-mo 2528 df-eu 2557 df-clab 2704 df-cleq 2718 df-clel 2804 df-nfc 2879 df-ne 2935 df-nel 3041 df-ral 3056 df-rex 3065 df-rmo 3370 df-reu 3371 df-rab 3427 df-v 3470 df-sbc 3773 df-csb 3889 df-dif 3946 df-un 3948 df-in 3950 df-ss 3960 df-pss 3962 df-nul 4318 df-if 4524 df-pw 4599 df-sn 4624 df-pr 4626 df-tp 4628 df-op 4630 df-uni 4903 df-int 4944 df-iun 4992 df-iin 4993 df-br 5142 df-opab 5204 df-mpt 5225 df-tr 5259 df-id 5567 df-eprel 5573 df-po 5581 df-so 5582 df-fr 5624 df-se 5625 df-we 5626 df-xp 5675 df-rel 5676 df-cnv 5677 df-co 5678 df-dm 5679 df-rn 5680 df-res 5681 df-ima 5682 df-pred 6294 df-ord 6361 df-on 6362 df-lim 6363 df-suc 6364 df-iota 6489 df-fun 6539 df-fn 6540 df-f 6541 df-f1 6542 df-fo 6543 df-f1o 6544 df-fv 6545 df-isom 6546 df-riota 7361 df-ov 7408 df-oprab 7409 df-mpo 7410 df-of 7667 df-ofr 7668 df-om 7853 df-1st 7974 df-2nd 7975 df-supp 8147 df-tpos 8212 df-frecs 8267 df-wrecs 8298 df-recs 8372 df-rdg 8411 df-1o 8467 df-2o 8468 df-oadd 8471 df-er 8705 df-ec 8707 df-qs 8711 df-map 8824 df-pm 8825 df-ixp 8894 df-en 8942 df-dom 8943 df-sdom 8944 df-fin 8945 df-fsupp 9364 df-sup 9439 df-inf 9440 df-oi 9507 df-dju 9898 df-card 9936 df-pnf 11254 df-mnf 11255 df-xr 11256 df-ltxr 11257 df-le 11258 df-sub 11450 df-neg 11451 df-div 11876 df-nn 12217 df-2 12279 df-3 12280 df-4 12281 df-5 12282 df-6 12283 df-7 12284 df-8 12285 df-9 12286 df-n0 12477 df-xnn0 12549 df-z 12563 df-dec 12682 df-uz 12827 df-q 12937 df-rp 12981 df-fz 13491 df-fzo 13634 df-fl 13763 df-mod 13841 df-seq 13973 df-exp 14033 df-hash 14296 df-cj 15052 df-re 15053 df-im 15054 df-sqrt 15188 df-abs 15189 df-dvds 16205 df-gcd 16443 df-prm 16616 df-phi 16708 df-pc 16779 df-struct 17089 df-sets 17106 df-slot 17124 df-ndx 17136 df-base 17154 df-ress 17183 df-plusg 17219 df-mulr 17220 df-starv 17221 df-sca 17222 df-vsca 17223 df-ip 17224 df-tset 17225 df-ple 17226 df-ds 17228 df-unif 17229 df-hom 17230 df-cco 17231 df-0g 17396 df-gsum 17397 df-prds 17402 df-pws 17404 df-imas 17463 df-qus 17464 df-mre 17539 df-mrc 17540 df-acs 17542 df-mgm 18573 df-sgrp 18652 df-mnd 18668 df-mhm 18713 df-submnd 18714 df-grp 18866 df-minusg 18867 df-sbg 18868 df-mulg 18996 df-subg 19050 df-nsg 19051 df-eqg 19052 df-ghm 19139 df-cntz 19233 df-cmn 19702 df-abl 19703 df-mgp 20040 df-rng 20058 df-ur 20087 df-srg 20092 df-ring 20140 df-cring 20141 df-oppr 20236 df-dvdsr 20259 df-unit 20260 df-invr 20290 df-dvr 20303 df-rhm 20374 df-nzr 20415 df-subrng 20446 df-subrg 20471 df-drng 20589 df-field 20590 df-lmod 20708 df-lss 20779 df-lsp 20819 df-sra 21021 df-rgmod 21022 df-lidl 21067 df-rsp 21068 df-2idl 21107 df-rlreg 21193 df-domn 21194 df-idom 21195 df-cnfld 21241 df-zring 21334 df-zrh 21390 df-zn 21393 df-assa 21748 df-asp 21749 df-ascl 21750 df-psr 21803 df-mvr 21804 df-mpl 21805 df-opsr 21807 df-evls 21977 df-evl 21978 df-psr1 22054 df-vr1 22055 df-ply1 22056 df-evl1 22190 df-lgs 27183 |
This theorem is referenced by: lgsqrlem4 27237 |
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