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Mirrors > Home > MPE Home > Th. List > lgsqrlem3 | Structured version Visualization version GIF version |
Description: Lemma for lgsqr 27380. (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 3959 | . . . . . . . . 9 ⊢ (𝜑 → 𝑃 ∈ ℙ) |
3 | lgsqr.y | . . . . . . . . . 10 ⊢ 𝑌 = (ℤ/nℤ‘𝑃) | |
4 | 3 | znfld 21558 | . . . . . . . . 9 ⊢ (𝑃 ∈ ℙ → 𝑌 ∈ Field) |
5 | 2, 4 | syl 17 | . . . . . . . 8 ⊢ (𝜑 → 𝑌 ∈ Field) |
6 | fldidom 20749 | . . . . . . . 8 ⊢ (𝑌 ∈ Field → 𝑌 ∈ IDomn) | |
7 | 5, 6 | syl 17 | . . . . . . 7 ⊢ (𝜑 → 𝑌 ∈ IDomn) |
8 | isidom 20703 | . . . . . . . 8 ⊢ (𝑌 ∈ IDomn ↔ (𝑌 ∈ CRing ∧ 𝑌 ∈ Domn)) | |
9 | 8 | simplbi 496 | . . . . . . 7 ⊢ (𝑌 ∈ IDomn → 𝑌 ∈ CRing) |
10 | 7, 9 | syl 17 | . . . . . 6 ⊢ (𝜑 → 𝑌 ∈ CRing) |
11 | crngring 20228 | . . . . . 6 ⊢ (𝑌 ∈ CRing → 𝑌 ∈ Ring) | |
12 | 10, 11 | syl 17 | . . . . 5 ⊢ (𝜑 → 𝑌 ∈ Ring) |
13 | lgsqr.l | . . . . . 6 ⊢ 𝐿 = (ℤRHom‘𝑌) | |
14 | 13 | zrhrhm 21501 | . . . . 5 ⊢ (𝑌 ∈ Ring → 𝐿 ∈ (ℤring RingHom 𝑌)) |
15 | 12, 14 | syl 17 | . . . 4 ⊢ (𝜑 → 𝐿 ∈ (ℤring RingHom 𝑌)) |
16 | zringbas 21443 | . . . . 5 ⊢ ℤ = (Base‘ℤring) | |
17 | eqid 2726 | . . . . 5 ⊢ (Base‘𝑌) = (Base‘𝑌) | |
18 | 16, 17 | rhmf 20467 | . . . 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 7099 | . 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 27345 | . . . . 5 ⊢ ((𝐴 ∈ ℤ ∧ 𝑃 ∈ (ℙ ∖ {2})) → ((𝐴 /L 𝑃) mod 𝑃) = ((𝐴↑((𝑃 − 1) / 2)) mod 𝑃)) | |
32 | 20, 1, 31 | syl2anc 582 | . . . 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 7439 | . . . 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 27375 | . 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 6916 | . . . . 5 ⊢ (𝜑 → (Base‘𝑌) ∈ V) | |
40 | 25, 22, 37, 17 | evl1rhm 22323 | . . . . . . . 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 20467 | . . . . . . 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 22237 | . . . . . . . . . 10 ⊢ (𝑌 ∈ Ring → 𝑆 ∈ Ring) |
45 | 12, 44 | syl 17 | . . . . . . . . 9 ⊢ (𝜑 → 𝑆 ∈ Ring) |
46 | ringgrp 20221 | . . . . . . . . 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 20123 | . . . . . . . . 9 ⊢ 𝐵 = (Base‘(mulGrp‘𝑆)) |
50 | 48 | ringmgp 20222 | . . . . . . . . . 10 ⊢ (𝑆 ∈ Ring → (mulGrp‘𝑆) ∈ Mnd) |
51 | 45, 50 | syl 17 | . . . . . . . . 9 ⊢ (𝜑 → (mulGrp‘𝑆) ∈ Mnd) |
52 | oddprm 16812 | . . . . . . . . . . 11 ⊢ (𝑃 ∈ (ℙ ∖ {2}) → ((𝑃 − 1) / 2) ∈ ℕ) | |
53 | 1, 52 | syl 17 | . . . . . . . . . 10 ⊢ (𝜑 → ((𝑃 − 1) / 2) ∈ ℕ) |
54 | 53 | nnnn0d 12584 | . . . . . . . . 9 ⊢ (𝜑 → ((𝑃 − 1) / 2) ∈ ℕ0) |
55 | 27, 22, 23 | vr1cl 22207 | . . . . . . . . . 10 ⊢ (𝑌 ∈ Ring → 𝑋 ∈ 𝐵) |
56 | 12, 55 | syl 17 | . . . . . . . . 9 ⊢ (𝜑 → 𝑋 ∈ 𝐵) |
57 | 49, 26, 51, 54, 56 | mulgnn0cld 19089 | . . . . . . . 8 ⊢ (𝜑 → (((𝑃 − 1) / 2) ↑ 𝑋) ∈ 𝐵) |
58 | 23, 29 | ringidcl 20245 | . . . . . . . . 9 ⊢ (𝑆 ∈ Ring → 1 ∈ 𝐵) |
59 | 45, 58 | syl 17 | . . . . . . . 8 ⊢ (𝜑 → 1 ∈ 𝐵) |
60 | 23, 28 | grpsubcl 19014 | . . . . . . . 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 2830 | . . . . . 6 ⊢ (𝜑 → 𝑇 ∈ 𝐵) |
63 | 43, 62 | ffvelcdmd 7099 | . . . . 5 ⊢ (𝜑 → (𝑂‘𝑇) ∈ (Base‘(𝑌 ↑s (Base‘𝑌)))) |
64 | 37, 17, 38, 5, 39, 63 | pwselbas 17504 | . . . 4 ⊢ (𝜑 → (𝑂‘𝑇):(Base‘𝑌)⟶(Base‘𝑌)) |
65 | 64 | ffnd 6729 | . . 3 ⊢ (𝜑 → (𝑂‘𝑇) Fn (Base‘𝑌)) |
66 | fniniseg 7073 | . . 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 394 = wceq 1534 ∈ wcel 2099 Vcvv 3462 ∖ cdif 3944 {csn 4633 ↦ cmpt 5236 ◡ccnv 5681 “ cima 5685 Fn wfn 6549 ⟶wf 6550 ‘cfv 6554 (class class class)co 7424 1c1 11159 − cmin 11494 / cdiv 11921 ℕcn 12264 2c2 12319 ℤcz 12610 ...cfz 13538 mod cmo 13889 ↑cexp 14081 ℙcprime 16672 Basecbs 17213 0gc0g 17454 ↑s cpws 17461 Mndcmnd 18727 Grpcgrp 18928 -gcsg 18930 .gcmg 19061 mulGrpcmgp 20117 1rcur 20164 Ringcrg 20216 CRingccrg 20217 RingHom crh 20451 Domncdomn 20670 IDomncidom 20671 Fieldcfield 20708 ℤringczring 21436 ℤRHomczrh 21489 ℤ/nℤczn 21492 var1cv1 22165 Poly1cpl1 22166 eval1ce1 22305 deg1cdg1 26078 /L clgs 27323 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1790 ax-4 1804 ax-5 1906 ax-6 1964 ax-7 2004 ax-8 2101 ax-9 2109 ax-10 2130 ax-11 2147 ax-12 2167 ax-ext 2697 ax-rep 5290 ax-sep 5304 ax-nul 5311 ax-pow 5369 ax-pr 5433 ax-un 7746 ax-cnex 11214 ax-resscn 11215 ax-1cn 11216 ax-icn 11217 ax-addcl 11218 ax-addrcl 11219 ax-mulcl 11220 ax-mulrcl 11221 ax-mulcom 11222 ax-addass 11223 ax-mulass 11224 ax-distr 11225 ax-i2m1 11226 ax-1ne0 11227 ax-1rid 11228 ax-rnegex 11229 ax-rrecex 11230 ax-cnre 11231 ax-pre-lttri 11232 ax-pre-lttrn 11233 ax-pre-ltadd 11234 ax-pre-mulgt0 11235 ax-pre-sup 11236 ax-addf 11237 ax-mulf 11238 |
This theorem depends on definitions: df-bi 206 df-an 395 df-or 846 df-3or 1085 df-3an 1086 df-tru 1537 df-fal 1547 df-ex 1775 df-nf 1779 df-sb 2061 df-mo 2529 df-eu 2558 df-clab 2704 df-cleq 2718 df-clel 2803 df-nfc 2878 df-ne 2931 df-nel 3037 df-ral 3052 df-rex 3061 df-rmo 3364 df-reu 3365 df-rab 3420 df-v 3464 df-sbc 3777 df-csb 3893 df-dif 3950 df-un 3952 df-in 3954 df-ss 3964 df-pss 3967 df-nul 4326 df-if 4534 df-pw 4609 df-sn 4634 df-pr 4636 df-tp 4638 df-op 4640 df-uni 4914 df-int 4955 df-iun 5003 df-iin 5004 df-br 5154 df-opab 5216 df-mpt 5237 df-tr 5271 df-id 5580 df-eprel 5586 df-po 5594 df-so 5595 df-fr 5637 df-se 5638 df-we 5639 df-xp 5688 df-rel 5689 df-cnv 5690 df-co 5691 df-dm 5692 df-rn 5693 df-res 5694 df-ima 5695 df-pred 6312 df-ord 6379 df-on 6380 df-lim 6381 df-suc 6382 df-iota 6506 df-fun 6556 df-fn 6557 df-f 6558 df-f1 6559 df-fo 6560 df-f1o 6561 df-fv 6562 df-isom 6563 df-riota 7380 df-ov 7427 df-oprab 7428 df-mpo 7429 df-of 7690 df-ofr 7691 df-om 7877 df-1st 8003 df-2nd 8004 df-supp 8175 df-tpos 8241 df-frecs 8296 df-wrecs 8327 df-recs 8401 df-rdg 8440 df-1o 8496 df-2o 8497 df-oadd 8500 df-er 8734 df-ec 8736 df-qs 8740 df-map 8857 df-pm 8858 df-ixp 8927 df-en 8975 df-dom 8976 df-sdom 8977 df-fin 8978 df-fsupp 9406 df-sup 9485 df-inf 9486 df-oi 9553 df-dju 9944 df-card 9982 df-pnf 11300 df-mnf 11301 df-xr 11302 df-ltxr 11303 df-le 11304 df-sub 11496 df-neg 11497 df-div 11922 df-nn 12265 df-2 12327 df-3 12328 df-4 12329 df-5 12330 df-6 12331 df-7 12332 df-8 12333 df-9 12334 df-n0 12525 df-xnn0 12597 df-z 12611 df-dec 12730 df-uz 12875 df-q 12985 df-rp 13029 df-fz 13539 df-fzo 13682 df-fl 13812 df-mod 13890 df-seq 14022 df-exp 14082 df-hash 14348 df-cj 15104 df-re 15105 df-im 15106 df-sqrt 15240 df-abs 15241 df-dvds 16257 df-gcd 16495 df-prm 16673 df-phi 16768 df-pc 16839 df-struct 17149 df-sets 17166 df-slot 17184 df-ndx 17196 df-base 17214 df-ress 17243 df-plusg 17279 df-mulr 17280 df-starv 17281 df-sca 17282 df-vsca 17283 df-ip 17284 df-tset 17285 df-ple 17286 df-ds 17288 df-unif 17289 df-hom 17290 df-cco 17291 df-0g 17456 df-gsum 17457 df-prds 17462 df-pws 17464 df-imas 17523 df-qus 17524 df-mre 17599 df-mrc 17600 df-acs 17602 df-mgm 18633 df-sgrp 18712 df-mnd 18728 df-mhm 18773 df-submnd 18774 df-grp 18931 df-minusg 18932 df-sbg 18933 df-mulg 19062 df-subg 19117 df-nsg 19118 df-eqg 19119 df-ghm 19207 df-cntz 19311 df-cmn 19780 df-abl 19781 df-mgp 20118 df-rng 20136 df-ur 20165 df-srg 20170 df-ring 20218 df-cring 20219 df-oppr 20316 df-dvdsr 20339 df-unit 20340 df-invr 20370 df-dvr 20383 df-rhm 20454 df-nzr 20495 df-subrng 20528 df-subrg 20553 df-rlreg 20672 df-domn 20673 df-idom 20674 df-drng 20709 df-field 20710 df-lmod 20838 df-lss 20909 df-lsp 20949 df-sra 21151 df-rgmod 21152 df-lidl 21197 df-rsp 21198 df-2idl 21239 df-cnfld 21344 df-zring 21437 df-zrh 21493 df-zn 21496 df-assa 21851 df-asp 21852 df-ascl 21853 df-psr 21906 df-mvr 21907 df-mpl 21908 df-opsr 21910 df-evls 22087 df-evl 22088 df-psr1 22169 df-vr1 22170 df-ply1 22171 df-evl1 22307 df-lgs 27324 |
This theorem is referenced by: lgsqrlem4 27378 |
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