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Mirrors > Home > MPE Home > Th. List > lgsqrlem3 | Structured version Visualization version GIF version |
Description: Lemma for lgsqr 26204. (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 3869 | . . . . . . . . 9 ⊢ (𝜑 → 𝑃 ∈ ℙ) |
3 | lgsqr.y | . . . . . . . . . 10 ⊢ 𝑌 = (ℤ/nℤ‘𝑃) | |
4 | 3 | znfld 20497 | . . . . . . . . 9 ⊢ (𝑃 ∈ ℙ → 𝑌 ∈ Field) |
5 | 2, 4 | syl 17 | . . . . . . . 8 ⊢ (𝜑 → 𝑌 ∈ Field) |
6 | fldidom 20315 | . . . . . . . 8 ⊢ (𝑌 ∈ Field → 𝑌 ∈ IDomn) | |
7 | 5, 6 | syl 17 | . . . . . . 7 ⊢ (𝜑 → 𝑌 ∈ IDomn) |
8 | isidom 20314 | . . . . . . . 8 ⊢ (𝑌 ∈ IDomn ↔ (𝑌 ∈ CRing ∧ 𝑌 ∈ Domn)) | |
9 | 8 | simplbi 501 | . . . . . . 7 ⊢ (𝑌 ∈ IDomn → 𝑌 ∈ CRing) |
10 | 7, 9 | syl 17 | . . . . . 6 ⊢ (𝜑 → 𝑌 ∈ CRing) |
11 | crngring 19546 | . . . . . 6 ⊢ (𝑌 ∈ CRing → 𝑌 ∈ Ring) | |
12 | 10, 11 | syl 17 | . . . . 5 ⊢ (𝜑 → 𝑌 ∈ Ring) |
13 | lgsqr.l | . . . . . 6 ⊢ 𝐿 = (ℤRHom‘𝑌) | |
14 | 13 | zrhrhm 20450 | . . . . 5 ⊢ (𝑌 ∈ Ring → 𝐿 ∈ (ℤring RingHom 𝑌)) |
15 | 12, 14 | syl 17 | . . . 4 ⊢ (𝜑 → 𝐿 ∈ (ℤring RingHom 𝑌)) |
16 | zringbas 20413 | . . . . 5 ⊢ ℤ = (Base‘ℤring) | |
17 | eqid 2734 | . . . . 5 ⊢ (Base‘𝑌) = (Base‘𝑌) | |
18 | 16, 17 | rhmf 19718 | . . . 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 | ffvelrnd 6894 | . 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 26169 | . . . . 5 ⊢ ((𝐴 ∈ ℤ ∧ 𝑃 ∈ (ℙ ∖ {2})) → ((𝐴 /L 𝑃) mod 𝑃) = ((𝐴↑((𝑃 − 1) / 2)) mod 𝑃)) | |
32 | 20, 1, 31 | syl2anc 587 | . . . 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 7217 | . . . 4 ⊢ (𝜑 → ((𝐴 /L 𝑃) mod 𝑃) = (1 mod 𝑃)) |
35 | 32, 34 | eqtr3d 2776 | . . 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 26199 | . 2 ⊢ (𝜑 → ((𝑂‘𝑇)‘(𝐿‘𝐴)) = (0g‘𝑌)) |
37 | eqid 2734 | . . . . 5 ⊢ (𝑌 ↑s (Base‘𝑌)) = (𝑌 ↑s (Base‘𝑌)) | |
38 | eqid 2734 | . . . . 5 ⊢ (Base‘(𝑌 ↑s (Base‘𝑌))) = (Base‘(𝑌 ↑s (Base‘𝑌))) | |
39 | fvexd 6721 | . . . . 5 ⊢ (𝜑 → (Base‘𝑌) ∈ V) | |
40 | 25, 22, 37, 17 | evl1rhm 21220 | . . . . . . . 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 19718 | . . . . . . 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 21141 | . . . . . . . . . 10 ⊢ (𝑌 ∈ Ring → 𝑆 ∈ Ring) |
45 | 12, 44 | syl 17 | . . . . . . . . 9 ⊢ (𝜑 → 𝑆 ∈ Ring) |
46 | ringgrp 19539 | . . . . . . . . 9 ⊢ (𝑆 ∈ Ring → 𝑆 ∈ Grp) | |
47 | 45, 46 | syl 17 | . . . . . . . 8 ⊢ (𝜑 → 𝑆 ∈ Grp) |
48 | eqid 2734 | . . . . . . . . . . 11 ⊢ (mulGrp‘𝑆) = (mulGrp‘𝑆) | |
49 | 48 | ringmgp 19540 | . . . . . . . . . 10 ⊢ (𝑆 ∈ Ring → (mulGrp‘𝑆) ∈ Mnd) |
50 | 45, 49 | syl 17 | . . . . . . . . 9 ⊢ (𝜑 → (mulGrp‘𝑆) ∈ Mnd) |
51 | oddprm 16344 | . . . . . . . . . . 11 ⊢ (𝑃 ∈ (ℙ ∖ {2}) → ((𝑃 − 1) / 2) ∈ ℕ) | |
52 | 1, 51 | syl 17 | . . . . . . . . . 10 ⊢ (𝜑 → ((𝑃 − 1) / 2) ∈ ℕ) |
53 | 52 | nnnn0d 12133 | . . . . . . . . 9 ⊢ (𝜑 → ((𝑃 − 1) / 2) ∈ ℕ0) |
54 | 27, 22, 23 | vr1cl 21110 | . . . . . . . . . 10 ⊢ (𝑌 ∈ Ring → 𝑋 ∈ 𝐵) |
55 | 12, 54 | syl 17 | . . . . . . . . 9 ⊢ (𝜑 → 𝑋 ∈ 𝐵) |
56 | 48, 23 | mgpbas 19482 | . . . . . . . . . 10 ⊢ 𝐵 = (Base‘(mulGrp‘𝑆)) |
57 | 56, 26 | mulgnn0cl 18480 | . . . . . . . . 9 ⊢ (((mulGrp‘𝑆) ∈ Mnd ∧ ((𝑃 − 1) / 2) ∈ ℕ0 ∧ 𝑋 ∈ 𝐵) → (((𝑃 − 1) / 2) ↑ 𝑋) ∈ 𝐵) |
58 | 50, 53, 55, 57 | syl3anc 1373 | . . . . . . . 8 ⊢ (𝜑 → (((𝑃 − 1) / 2) ↑ 𝑋) ∈ 𝐵) |
59 | 23, 29 | ringidcl 19558 | . . . . . . . . 9 ⊢ (𝑆 ∈ Ring → 1 ∈ 𝐵) |
60 | 45, 59 | syl 17 | . . . . . . . 8 ⊢ (𝜑 → 1 ∈ 𝐵) |
61 | 23, 28 | grpsubcl 18415 | . . . . . . . 8 ⊢ ((𝑆 ∈ Grp ∧ (((𝑃 − 1) / 2) ↑ 𝑋) ∈ 𝐵 ∧ 1 ∈ 𝐵) → ((((𝑃 − 1) / 2) ↑ 𝑋) − 1 ) ∈ 𝐵) |
62 | 47, 58, 60, 61 | syl3anc 1373 | . . . . . . 7 ⊢ (𝜑 → ((((𝑃 − 1) / 2) ↑ 𝑋) − 1 ) ∈ 𝐵) |
63 | 30, 62 | eqeltrid 2838 | . . . . . 6 ⊢ (𝜑 → 𝑇 ∈ 𝐵) |
64 | 43, 63 | ffvelrnd 6894 | . . . . 5 ⊢ (𝜑 → (𝑂‘𝑇) ∈ (Base‘(𝑌 ↑s (Base‘𝑌)))) |
65 | 37, 17, 38, 5, 39, 64 | pwselbas 16966 | . . . 4 ⊢ (𝜑 → (𝑂‘𝑇):(Base‘𝑌)⟶(Base‘𝑌)) |
66 | 65 | ffnd 6535 | . . 3 ⊢ (𝜑 → (𝑂‘𝑇) Fn (Base‘𝑌)) |
67 | fniniseg 6869 | . . 3 ⊢ ((𝑂‘𝑇) Fn (Base‘𝑌) → ((𝐿‘𝐴) ∈ (◡(𝑂‘𝑇) “ {(0g‘𝑌)}) ↔ ((𝐿‘𝐴) ∈ (Base‘𝑌) ∧ ((𝑂‘𝑇)‘(𝐿‘𝐴)) = (0g‘𝑌)))) | |
68 | 66, 67 | syl 17 | . 2 ⊢ (𝜑 → ((𝐿‘𝐴) ∈ (◡(𝑂‘𝑇) “ {(0g‘𝑌)}) ↔ ((𝐿‘𝐴) ∈ (Base‘𝑌) ∧ ((𝑂‘𝑇)‘(𝐿‘𝐴)) = (0g‘𝑌)))) |
69 | 21, 36, 68 | mpbir2and 713 | 1 ⊢ (𝜑 → (𝐿‘𝐴) ∈ (◡(𝑂‘𝑇) “ {(0g‘𝑌)})) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 209 ∧ wa 399 = wceq 1543 ∈ wcel 2110 Vcvv 3401 ∖ cdif 3854 {csn 4531 ↦ cmpt 5124 ◡ccnv 5539 “ cima 5543 Fn wfn 6364 ⟶wf 6365 ‘cfv 6369 (class class class)co 7202 1c1 10713 − cmin 11045 / cdiv 11472 ℕcn 11813 2c2 11868 ℕ0cn0 12073 ℤcz 12159 ...cfz 13078 mod cmo 13425 ↑cexp 13618 ℙcprime 16209 Basecbs 16684 0gc0g 16916 ↑s cpws 16923 Mndcmnd 18145 Grpcgrp 18337 -gcsg 18339 .gcmg 18460 mulGrpcmgp 19476 1rcur 19488 Ringcrg 19534 CRingccrg 19535 RingHom crh 19704 Fieldcfield 19740 Domncdomn 20290 IDomncidom 20291 ℤringzring 20407 ℤRHomczrh 20438 ℤ/nℤczn 20441 var1cv1 21069 Poly1cpl1 21070 eval1ce1 21202 deg1 cdg1 24921 /L clgs 26147 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1803 ax-4 1817 ax-5 1918 ax-6 1976 ax-7 2016 ax-8 2112 ax-9 2120 ax-10 2141 ax-11 2158 ax-12 2175 ax-ext 2706 ax-rep 5168 ax-sep 5181 ax-nul 5188 ax-pow 5247 ax-pr 5311 ax-un 7512 ax-cnex 10768 ax-resscn 10769 ax-1cn 10770 ax-icn 10771 ax-addcl 10772 ax-addrcl 10773 ax-mulcl 10774 ax-mulrcl 10775 ax-mulcom 10776 ax-addass 10777 ax-mulass 10778 ax-distr 10779 ax-i2m1 10780 ax-1ne0 10781 ax-1rid 10782 ax-rnegex 10783 ax-rrecex 10784 ax-cnre 10785 ax-pre-lttri 10786 ax-pre-lttrn 10787 ax-pre-ltadd 10788 ax-pre-mulgt0 10789 ax-pre-sup 10790 ax-addf 10791 ax-mulf 10792 |
This theorem depends on definitions: df-bi 210 df-an 400 df-or 848 df-3or 1090 df-3an 1091 df-tru 1546 df-fal 1556 df-ex 1788 df-nf 1792 df-sb 2071 df-mo 2537 df-eu 2566 df-clab 2713 df-cleq 2726 df-clel 2812 df-nfc 2882 df-ne 2936 df-nel 3040 df-ral 3059 df-rex 3060 df-reu 3061 df-rmo 3062 df-rab 3063 df-v 3403 df-sbc 3688 df-csb 3803 df-dif 3860 df-un 3862 df-in 3864 df-ss 3874 df-pss 3876 df-nul 4228 df-if 4430 df-pw 4505 df-sn 4532 df-pr 4534 df-tp 4536 df-op 4538 df-uni 4810 df-int 4850 df-iun 4896 df-iin 4897 df-br 5044 df-opab 5106 df-mpt 5125 df-tr 5151 df-id 5444 df-eprel 5449 df-po 5457 df-so 5458 df-fr 5498 df-se 5499 df-we 5500 df-xp 5546 df-rel 5547 df-cnv 5548 df-co 5549 df-dm 5550 df-rn 5551 df-res 5552 df-ima 5553 df-pred 6149 df-ord 6205 df-on 6206 df-lim 6207 df-suc 6208 df-iota 6327 df-fun 6371 df-fn 6372 df-f 6373 df-f1 6374 df-fo 6375 df-f1o 6376 df-fv 6377 df-isom 6378 df-riota 7159 df-ov 7205 df-oprab 7206 df-mpo 7207 df-of 7458 df-ofr 7459 df-om 7634 df-1st 7750 df-2nd 7751 df-supp 7893 df-tpos 7957 df-wrecs 8036 df-recs 8097 df-rdg 8135 df-1o 8191 df-2o 8192 df-oadd 8195 df-er 8380 df-ec 8382 df-qs 8386 df-map 8499 df-pm 8500 df-ixp 8568 df-en 8616 df-dom 8617 df-sdom 8618 df-fin 8619 df-fsupp 8975 df-sup 9047 df-inf 9048 df-oi 9115 df-dju 9500 df-card 9538 df-pnf 10852 df-mnf 10853 df-xr 10854 df-ltxr 10855 df-le 10856 df-sub 11047 df-neg 11048 df-div 11473 df-nn 11814 df-2 11876 df-3 11877 df-4 11878 df-5 11879 df-6 11880 df-7 11881 df-8 11882 df-9 11883 df-n0 12074 df-xnn0 12146 df-z 12160 df-dec 12277 df-uz 12422 df-q 12528 df-rp 12570 df-fz 13079 df-fzo 13222 df-fl 13350 df-mod 13426 df-seq 13558 df-exp 13619 df-hash 13880 df-cj 14645 df-re 14646 df-im 14647 df-sqrt 14781 df-abs 14782 df-dvds 15797 df-gcd 16035 df-prm 16210 df-phi 16300 df-pc 16371 df-struct 16686 df-ndx 16687 df-slot 16688 df-base 16690 df-sets 16691 df-ress 16692 df-plusg 16780 df-mulr 16781 df-starv 16782 df-sca 16783 df-vsca 16784 df-ip 16785 df-tset 16786 df-ple 16787 df-ds 16789 df-unif 16790 df-hom 16791 df-cco 16792 df-0g 16918 df-gsum 16919 df-prds 16924 df-pws 16926 df-imas 16985 df-qus 16986 df-mre 17061 df-mrc 17062 df-acs 17064 df-mgm 18086 df-sgrp 18135 df-mnd 18146 df-mhm 18190 df-submnd 18191 df-grp 18340 df-minusg 18341 df-sbg 18342 df-mulg 18461 df-subg 18512 df-nsg 18513 df-eqg 18514 df-ghm 18592 df-cntz 18683 df-cmn 19144 df-abl 19145 df-mgp 19477 df-ur 19489 df-srg 19493 df-ring 19536 df-cring 19537 df-oppr 19613 df-dvdsr 19631 df-unit 19632 df-invr 19662 df-dvr 19673 df-rnghom 19707 df-drng 19741 df-field 19742 df-subrg 19770 df-lmod 19873 df-lss 19941 df-lsp 19981 df-sra 20181 df-rgmod 20182 df-lidl 20183 df-rsp 20184 df-2idl 20242 df-nzr 20268 df-rlreg 20293 df-domn 20294 df-idom 20295 df-cnfld 20336 df-zring 20408 df-zrh 20442 df-zn 20445 df-assa 20787 df-asp 20788 df-ascl 20789 df-psr 20840 df-mvr 20841 df-mpl 20842 df-opsr 20844 df-evls 21004 df-evl 21005 df-psr1 21073 df-vr1 21074 df-ply1 21075 df-evl1 21204 df-lgs 26148 |
This theorem is referenced by: lgsqrlem4 26202 |
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