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Mirrors > Home > MPE Home > Th. List > lgsqrlem3 | Structured version Visualization version GIF version |
Description: Lemma for lgsqr 26699. (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 3922 | . . . . . . . . 9 ⊢ (𝜑 → 𝑃 ∈ ℙ) |
3 | lgsqr.y | . . . . . . . . . 10 ⊢ 𝑌 = (ℤ/nℤ‘𝑃) | |
4 | 3 | znfld 20967 | . . . . . . . . 9 ⊢ (𝑃 ∈ ℙ → 𝑌 ∈ Field) |
5 | 2, 4 | syl 17 | . . . . . . . 8 ⊢ (𝜑 → 𝑌 ∈ Field) |
6 | fldidom 20775 | . . . . . . . 8 ⊢ (𝑌 ∈ Field → 𝑌 ∈ IDomn) | |
7 | 5, 6 | syl 17 | . . . . . . 7 ⊢ (𝜑 → 𝑌 ∈ IDomn) |
8 | isidom 20774 | . . . . . . . 8 ⊢ (𝑌 ∈ IDomn ↔ (𝑌 ∈ CRing ∧ 𝑌 ∈ Domn)) | |
9 | 8 | simplbi 498 | . . . . . . 7 ⊢ (𝑌 ∈ IDomn → 𝑌 ∈ CRing) |
10 | 7, 9 | syl 17 | . . . . . 6 ⊢ (𝜑 → 𝑌 ∈ CRing) |
11 | crngring 19976 | . . . . . 6 ⊢ (𝑌 ∈ CRing → 𝑌 ∈ Ring) | |
12 | 10, 11 | syl 17 | . . . . 5 ⊢ (𝜑 → 𝑌 ∈ Ring) |
13 | lgsqr.l | . . . . . 6 ⊢ 𝐿 = (ℤRHom‘𝑌) | |
14 | 13 | zrhrhm 20912 | . . . . 5 ⊢ (𝑌 ∈ Ring → 𝐿 ∈ (ℤring RingHom 𝑌)) |
15 | 12, 14 | syl 17 | . . . 4 ⊢ (𝜑 → 𝐿 ∈ (ℤring RingHom 𝑌)) |
16 | zringbas 20875 | . . . . 5 ⊢ ℤ = (Base‘ℤring) | |
17 | eqid 2736 | . . . . 5 ⊢ (Base‘𝑌) = (Base‘𝑌) | |
18 | 16, 17 | rhmf 20158 | . . . 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 7036 | . 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 26664 | . . . . 5 ⊢ ((𝐴 ∈ ℤ ∧ 𝑃 ∈ (ℙ ∖ {2})) → ((𝐴 /L 𝑃) mod 𝑃) = ((𝐴↑((𝑃 − 1) / 2)) mod 𝑃)) | |
32 | 20, 1, 31 | syl2anc 584 | . . . 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 7372 | . . . 4 ⊢ (𝜑 → ((𝐴 /L 𝑃) mod 𝑃) = (1 mod 𝑃)) |
35 | 32, 34 | eqtr3d 2778 | . . 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 26694 | . 2 ⊢ (𝜑 → ((𝑂‘𝑇)‘(𝐿‘𝐴)) = (0g‘𝑌)) |
37 | eqid 2736 | . . . . 5 ⊢ (𝑌 ↑s (Base‘𝑌)) = (𝑌 ↑s (Base‘𝑌)) | |
38 | eqid 2736 | . . . . 5 ⊢ (Base‘(𝑌 ↑s (Base‘𝑌))) = (Base‘(𝑌 ↑s (Base‘𝑌))) | |
39 | fvexd 6857 | . . . . 5 ⊢ (𝜑 → (Base‘𝑌) ∈ V) | |
40 | 25, 22, 37, 17 | evl1rhm 21698 | . . . . . . . 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 20158 | . . . . . . 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 21619 | . . . . . . . . . 10 ⊢ (𝑌 ∈ Ring → 𝑆 ∈ Ring) |
45 | 12, 44 | syl 17 | . . . . . . . . 9 ⊢ (𝜑 → 𝑆 ∈ Ring) |
46 | ringgrp 19969 | . . . . . . . . 9 ⊢ (𝑆 ∈ Ring → 𝑆 ∈ Grp) | |
47 | 45, 46 | syl 17 | . . . . . . . 8 ⊢ (𝜑 → 𝑆 ∈ Grp) |
48 | eqid 2736 | . . . . . . . . . 10 ⊢ (mulGrp‘𝑆) = (mulGrp‘𝑆) | |
49 | 48, 23 | mgpbas 19902 | . . . . . . . . 9 ⊢ 𝐵 = (Base‘(mulGrp‘𝑆)) |
50 | 48 | ringmgp 19970 | . . . . . . . . . 10 ⊢ (𝑆 ∈ Ring → (mulGrp‘𝑆) ∈ Mnd) |
51 | 45, 50 | syl 17 | . . . . . . . . 9 ⊢ (𝜑 → (mulGrp‘𝑆) ∈ Mnd) |
52 | oddprm 16682 | . . . . . . . . . . 11 ⊢ (𝑃 ∈ (ℙ ∖ {2}) → ((𝑃 − 1) / 2) ∈ ℕ) | |
53 | 1, 52 | syl 17 | . . . . . . . . . 10 ⊢ (𝜑 → ((𝑃 − 1) / 2) ∈ ℕ) |
54 | 53 | nnnn0d 12473 | . . . . . . . . 9 ⊢ (𝜑 → ((𝑃 − 1) / 2) ∈ ℕ0) |
55 | 27, 22, 23 | vr1cl 21588 | . . . . . . . . . 10 ⊢ (𝑌 ∈ Ring → 𝑋 ∈ 𝐵) |
56 | 12, 55 | syl 17 | . . . . . . . . 9 ⊢ (𝜑 → 𝑋 ∈ 𝐵) |
57 | 49, 26, 51, 54, 56 | mulgnn0cld 18897 | . . . . . . . 8 ⊢ (𝜑 → (((𝑃 − 1) / 2) ↑ 𝑋) ∈ 𝐵) |
58 | 23, 29 | ringidcl 19989 | . . . . . . . . 9 ⊢ (𝑆 ∈ Ring → 1 ∈ 𝐵) |
59 | 45, 58 | syl 17 | . . . . . . . 8 ⊢ (𝜑 → 1 ∈ 𝐵) |
60 | 23, 28 | grpsubcl 18827 | . . . . . . . 8 ⊢ ((𝑆 ∈ Grp ∧ (((𝑃 − 1) / 2) ↑ 𝑋) ∈ 𝐵 ∧ 1 ∈ 𝐵) → ((((𝑃 − 1) / 2) ↑ 𝑋) − 1 ) ∈ 𝐵) |
61 | 47, 57, 59, 60 | syl3anc 1371 | . . . . . . 7 ⊢ (𝜑 → ((((𝑃 − 1) / 2) ↑ 𝑋) − 1 ) ∈ 𝐵) |
62 | 30, 61 | eqeltrid 2842 | . . . . . 6 ⊢ (𝜑 → 𝑇 ∈ 𝐵) |
63 | 43, 62 | ffvelcdmd 7036 | . . . . 5 ⊢ (𝜑 → (𝑂‘𝑇) ∈ (Base‘(𝑌 ↑s (Base‘𝑌)))) |
64 | 37, 17, 38, 5, 39, 63 | pwselbas 17371 | . . . 4 ⊢ (𝜑 → (𝑂‘𝑇):(Base‘𝑌)⟶(Base‘𝑌)) |
65 | 64 | ffnd 6669 | . . 3 ⊢ (𝜑 → (𝑂‘𝑇) Fn (Base‘𝑌)) |
66 | fniniseg 7010 | . . 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 711 | 1 ⊢ (𝜑 → (𝐿‘𝐴) ∈ (◡(𝑂‘𝑇) “ {(0g‘𝑌)})) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 205 ∧ wa 396 = wceq 1541 ∈ wcel 2106 Vcvv 3445 ∖ cdif 3907 {csn 4586 ↦ cmpt 5188 ◡ccnv 5632 “ cima 5636 Fn wfn 6491 ⟶wf 6492 ‘cfv 6496 (class class class)co 7357 1c1 11052 − cmin 11385 / cdiv 11812 ℕcn 12153 2c2 12208 ℤcz 12499 ...cfz 13424 mod cmo 13774 ↑cexp 13967 ℙcprime 16547 Basecbs 17083 0gc0g 17321 ↑s cpws 17328 Mndcmnd 18556 Grpcgrp 18748 -gcsg 18750 .gcmg 18872 mulGrpcmgp 19896 1rcur 19913 Ringcrg 19964 CRingccrg 19965 RingHom crh 20143 Fieldcfield 20186 Domncdomn 20750 IDomncidom 20751 ℤringczring 20869 ℤRHomczrh 20900 ℤ/nℤczn 20903 var1cv1 21547 Poly1cpl1 21548 eval1ce1 21680 deg1 cdg1 25416 /L clgs 26642 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1797 ax-4 1811 ax-5 1913 ax-6 1971 ax-7 2011 ax-8 2108 ax-9 2116 ax-10 2137 ax-11 2154 ax-12 2171 ax-ext 2707 ax-rep 5242 ax-sep 5256 ax-nul 5263 ax-pow 5320 ax-pr 5384 ax-un 7672 ax-cnex 11107 ax-resscn 11108 ax-1cn 11109 ax-icn 11110 ax-addcl 11111 ax-addrcl 11112 ax-mulcl 11113 ax-mulrcl 11114 ax-mulcom 11115 ax-addass 11116 ax-mulass 11117 ax-distr 11118 ax-i2m1 11119 ax-1ne0 11120 ax-1rid 11121 ax-rnegex 11122 ax-rrecex 11123 ax-cnre 11124 ax-pre-lttri 11125 ax-pre-lttrn 11126 ax-pre-ltadd 11127 ax-pre-mulgt0 11128 ax-pre-sup 11129 ax-addf 11130 ax-mulf 11131 |
This theorem depends on definitions: df-bi 206 df-an 397 df-or 846 df-3or 1088 df-3an 1089 df-tru 1544 df-fal 1554 df-ex 1782 df-nf 1786 df-sb 2068 df-mo 2538 df-eu 2567 df-clab 2714 df-cleq 2728 df-clel 2814 df-nfc 2889 df-ne 2944 df-nel 3050 df-ral 3065 df-rex 3074 df-rmo 3353 df-reu 3354 df-rab 3408 df-v 3447 df-sbc 3740 df-csb 3856 df-dif 3913 df-un 3915 df-in 3917 df-ss 3927 df-pss 3929 df-nul 4283 df-if 4487 df-pw 4562 df-sn 4587 df-pr 4589 df-tp 4591 df-op 4593 df-uni 4866 df-int 4908 df-iun 4956 df-iin 4957 df-br 5106 df-opab 5168 df-mpt 5189 df-tr 5223 df-id 5531 df-eprel 5537 df-po 5545 df-so 5546 df-fr 5588 df-se 5589 df-we 5590 df-xp 5639 df-rel 5640 df-cnv 5641 df-co 5642 df-dm 5643 df-rn 5644 df-res 5645 df-ima 5646 df-pred 6253 df-ord 6320 df-on 6321 df-lim 6322 df-suc 6323 df-iota 6448 df-fun 6498 df-fn 6499 df-f 6500 df-f1 6501 df-fo 6502 df-f1o 6503 df-fv 6504 df-isom 6505 df-riota 7313 df-ov 7360 df-oprab 7361 df-mpo 7362 df-of 7617 df-ofr 7618 df-om 7803 df-1st 7921 df-2nd 7922 df-supp 8093 df-tpos 8157 df-frecs 8212 df-wrecs 8243 df-recs 8317 df-rdg 8356 df-1o 8412 df-2o 8413 df-oadd 8416 df-er 8648 df-ec 8650 df-qs 8654 df-map 8767 df-pm 8768 df-ixp 8836 df-en 8884 df-dom 8885 df-sdom 8886 df-fin 8887 df-fsupp 9306 df-sup 9378 df-inf 9379 df-oi 9446 df-dju 9837 df-card 9875 df-pnf 11191 df-mnf 11192 df-xr 11193 df-ltxr 11194 df-le 11195 df-sub 11387 df-neg 11388 df-div 11813 df-nn 12154 df-2 12216 df-3 12217 df-4 12218 df-5 12219 df-6 12220 df-7 12221 df-8 12222 df-9 12223 df-n0 12414 df-xnn0 12486 df-z 12500 df-dec 12619 df-uz 12764 df-q 12874 df-rp 12916 df-fz 13425 df-fzo 13568 df-fl 13697 df-mod 13775 df-seq 13907 df-exp 13968 df-hash 14231 df-cj 14984 df-re 14985 df-im 14986 df-sqrt 15120 df-abs 15121 df-dvds 16137 df-gcd 16375 df-prm 16548 df-phi 16638 df-pc 16709 df-struct 17019 df-sets 17036 df-slot 17054 df-ndx 17066 df-base 17084 df-ress 17113 df-plusg 17146 df-mulr 17147 df-starv 17148 df-sca 17149 df-vsca 17150 df-ip 17151 df-tset 17152 df-ple 17153 df-ds 17155 df-unif 17156 df-hom 17157 df-cco 17158 df-0g 17323 df-gsum 17324 df-prds 17329 df-pws 17331 df-imas 17390 df-qus 17391 df-mre 17466 df-mrc 17467 df-acs 17469 df-mgm 18497 df-sgrp 18546 df-mnd 18557 df-mhm 18601 df-submnd 18602 df-grp 18751 df-minusg 18752 df-sbg 18753 df-mulg 18873 df-subg 18925 df-nsg 18926 df-eqg 18927 df-ghm 19006 df-cntz 19097 df-cmn 19564 df-abl 19565 df-mgp 19897 df-ur 19914 df-srg 19918 df-ring 19966 df-cring 19967 df-oppr 20049 df-dvdsr 20070 df-unit 20071 df-invr 20101 df-dvr 20112 df-rnghom 20146 df-drng 20187 df-field 20188 df-subrg 20220 df-lmod 20324 df-lss 20393 df-lsp 20433 df-sra 20633 df-rgmod 20634 df-lidl 20635 df-rsp 20636 df-2idl 20702 df-nzr 20728 df-rlreg 20753 df-domn 20754 df-idom 20755 df-cnfld 20797 df-zring 20870 df-zrh 20904 df-zn 20907 df-assa 21259 df-asp 21260 df-ascl 21261 df-psr 21311 df-mvr 21312 df-mpl 21313 df-opsr 21315 df-evls 21482 df-evl 21483 df-psr1 21551 df-vr1 21552 df-ply1 21553 df-evl1 21682 df-lgs 26643 |
This theorem is referenced by: lgsqrlem4 26697 |
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