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| Mirrors > Home > MPE Home > Th. List > lgsqrlem3 | Structured version Visualization version GIF version | ||
| Description: Lemma for lgsqr 27395. (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 3963 | . . . . . . . . 9 ⊢ (𝜑 → 𝑃 ∈ ℙ) |
| 3 | lgsqr.y | . . . . . . . . . 10 ⊢ 𝑌 = (ℤ/nℤ‘𝑃) | |
| 4 | 3 | znfld 21579 | . . . . . . . . 9 ⊢ (𝑃 ∈ ℙ → 𝑌 ∈ Field) |
| 5 | 2, 4 | syl 17 | . . . . . . . 8 ⊢ (𝜑 → 𝑌 ∈ Field) |
| 6 | fldidom 20771 | . . . . . . . 8 ⊢ (𝑌 ∈ Field → 𝑌 ∈ IDomn) | |
| 7 | 5, 6 | syl 17 | . . . . . . 7 ⊢ (𝜑 → 𝑌 ∈ IDomn) |
| 8 | isidom 20725 | . . . . . . . 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 20242 | . . . . . 6 ⊢ (𝑌 ∈ CRing → 𝑌 ∈ Ring) | |
| 12 | 10, 11 | syl 17 | . . . . 5 ⊢ (𝜑 → 𝑌 ∈ Ring) |
| 13 | lgsqr.l | . . . . . 6 ⊢ 𝐿 = (ℤRHom‘𝑌) | |
| 14 | 13 | zrhrhm 21522 | . . . . 5 ⊢ (𝑌 ∈ Ring → 𝐿 ∈ (ℤring RingHom 𝑌)) |
| 15 | 12, 14 | syl 17 | . . . 4 ⊢ (𝜑 → 𝐿 ∈ (ℤring RingHom 𝑌)) |
| 16 | zringbas 21464 | . . . . 5 ⊢ ℤ = (Base‘ℤring) | |
| 17 | eqid 2737 | . . . . 5 ⊢ (Base‘𝑌) = (Base‘𝑌) | |
| 18 | 16, 17 | rhmf 20485 | . . . 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 7105 | . 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 27360 | . . . . 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 7446 | . . . 4 ⊢ (𝜑 → ((𝐴 /L 𝑃) mod 𝑃) = (1 mod 𝑃)) |
| 35 | 32, 34 | eqtr3d 2779 | . . 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 27390 | . 2 ⊢ (𝜑 → ((𝑂‘𝑇)‘(𝐿‘𝐴)) = (0g‘𝑌)) |
| 37 | eqid 2737 | . . . . 5 ⊢ (𝑌 ↑s (Base‘𝑌)) = (𝑌 ↑s (Base‘𝑌)) | |
| 38 | eqid 2737 | . . . . 5 ⊢ (Base‘(𝑌 ↑s (Base‘𝑌))) = (Base‘(𝑌 ↑s (Base‘𝑌))) | |
| 39 | fvexd 6921 | . . . . 5 ⊢ (𝜑 → (Base‘𝑌) ∈ V) | |
| 40 | 25, 22, 37, 17 | evl1rhm 22336 | . . . . . . . 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 20485 | . . . . . . 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 22249 | . . . . . . . . . 10 ⊢ (𝑌 ∈ Ring → 𝑆 ∈ Ring) |
| 45 | 12, 44 | syl 17 | . . . . . . . . 9 ⊢ (𝜑 → 𝑆 ∈ Ring) |
| 46 | ringgrp 20235 | . . . . . . . . 9 ⊢ (𝑆 ∈ Ring → 𝑆 ∈ Grp) | |
| 47 | 45, 46 | syl 17 | . . . . . . . 8 ⊢ (𝜑 → 𝑆 ∈ Grp) |
| 48 | eqid 2737 | . . . . . . . . . 10 ⊢ (mulGrp‘𝑆) = (mulGrp‘𝑆) | |
| 49 | 48, 23 | mgpbas 20142 | . . . . . . . . 9 ⊢ 𝐵 = (Base‘(mulGrp‘𝑆)) |
| 50 | 48 | ringmgp 20236 | . . . . . . . . . 10 ⊢ (𝑆 ∈ Ring → (mulGrp‘𝑆) ∈ Mnd) |
| 51 | 45, 50 | syl 17 | . . . . . . . . 9 ⊢ (𝜑 → (mulGrp‘𝑆) ∈ Mnd) |
| 52 | oddprm 16848 | . . . . . . . . . . 11 ⊢ (𝑃 ∈ (ℙ ∖ {2}) → ((𝑃 − 1) / 2) ∈ ℕ) | |
| 53 | 1, 52 | syl 17 | . . . . . . . . . 10 ⊢ (𝜑 → ((𝑃 − 1) / 2) ∈ ℕ) |
| 54 | 53 | nnnn0d 12587 | . . . . . . . . 9 ⊢ (𝜑 → ((𝑃 − 1) / 2) ∈ ℕ0) |
| 55 | 27, 22, 23 | vr1cl 22219 | . . . . . . . . . 10 ⊢ (𝑌 ∈ Ring → 𝑋 ∈ 𝐵) |
| 56 | 12, 55 | syl 17 | . . . . . . . . 9 ⊢ (𝜑 → 𝑋 ∈ 𝐵) |
| 57 | 49, 26, 51, 54, 56 | mulgnn0cld 19113 | . . . . . . . 8 ⊢ (𝜑 → (((𝑃 − 1) / 2) ↑ 𝑋) ∈ 𝐵) |
| 58 | 23, 29 | ringidcl 20262 | . . . . . . . . 9 ⊢ (𝑆 ∈ Ring → 1 ∈ 𝐵) |
| 59 | 45, 58 | syl 17 | . . . . . . . 8 ⊢ (𝜑 → 1 ∈ 𝐵) |
| 60 | 23, 28 | grpsubcl 19038 | . . . . . . . 8 ⊢ ((𝑆 ∈ Grp ∧ (((𝑃 − 1) / 2) ↑ 𝑋) ∈ 𝐵 ∧ 1 ∈ 𝐵) → ((((𝑃 − 1) / 2) ↑ 𝑋) − 1 ) ∈ 𝐵) |
| 61 | 47, 57, 59, 60 | syl3anc 1373 | . . . . . . 7 ⊢ (𝜑 → ((((𝑃 − 1) / 2) ↑ 𝑋) − 1 ) ∈ 𝐵) |
| 62 | 30, 61 | eqeltrid 2845 | . . . . . 6 ⊢ (𝜑 → 𝑇 ∈ 𝐵) |
| 63 | 43, 62 | ffvelcdmd 7105 | . . . . 5 ⊢ (𝜑 → (𝑂‘𝑇) ∈ (Base‘(𝑌 ↑s (Base‘𝑌)))) |
| 64 | 37, 17, 38, 5, 39, 63 | pwselbas 17534 | . . . 4 ⊢ (𝜑 → (𝑂‘𝑇):(Base‘𝑌)⟶(Base‘𝑌)) |
| 65 | 64 | ffnd 6737 | . . 3 ⊢ (𝜑 → (𝑂‘𝑇) Fn (Base‘𝑌)) |
| 66 | fniniseg 7080 | . . 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 713 | 1 ⊢ (𝜑 → (𝐿‘𝐴) ∈ (◡(𝑂‘𝑇) “ {(0g‘𝑌)})) |
| Colors of variables: wff setvar class |
| Syntax hints: → wi 4 ↔ wb 206 ∧ wa 395 = wceq 1540 ∈ wcel 2108 Vcvv 3480 ∖ cdif 3948 {csn 4626 ↦ cmpt 5225 ◡ccnv 5684 “ cima 5688 Fn wfn 6556 ⟶wf 6557 ‘cfv 6561 (class class class)co 7431 1c1 11156 − cmin 11492 / cdiv 11920 ℕcn 12266 2c2 12321 ℤcz 12613 ...cfz 13547 mod cmo 13909 ↑cexp 14102 ℙcprime 16708 Basecbs 17247 0gc0g 17484 ↑s cpws 17491 Mndcmnd 18747 Grpcgrp 18951 -gcsg 18953 .gcmg 19085 mulGrpcmgp 20137 1rcur 20178 Ringcrg 20230 CRingccrg 20231 RingHom crh 20469 Domncdomn 20692 IDomncidom 20693 Fieldcfield 20730 ℤringczring 21457 ℤRHomczrh 21510 ℤ/nℤczn 21513 var1cv1 22177 Poly1cpl1 22178 eval1ce1 22318 deg1cdg1 26093 /L clgs 27338 |
| This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1795 ax-4 1809 ax-5 1910 ax-6 1967 ax-7 2007 ax-8 2110 ax-9 2118 ax-10 2141 ax-11 2157 ax-12 2177 ax-ext 2708 ax-rep 5279 ax-sep 5296 ax-nul 5306 ax-pow 5365 ax-pr 5432 ax-un 7755 ax-cnex 11211 ax-resscn 11212 ax-1cn 11213 ax-icn 11214 ax-addcl 11215 ax-addrcl 11216 ax-mulcl 11217 ax-mulrcl 11218 ax-mulcom 11219 ax-addass 11220 ax-mulass 11221 ax-distr 11222 ax-i2m1 11223 ax-1ne0 11224 ax-1rid 11225 ax-rnegex 11226 ax-rrecex 11227 ax-cnre 11228 ax-pre-lttri 11229 ax-pre-lttrn 11230 ax-pre-ltadd 11231 ax-pre-mulgt0 11232 ax-pre-sup 11233 ax-addf 11234 ax-mulf 11235 |
| This theorem depends on definitions: df-bi 207 df-an 396 df-or 849 df-3or 1088 df-3an 1089 df-tru 1543 df-fal 1553 df-ex 1780 df-nf 1784 df-sb 2065 df-mo 2540 df-eu 2569 df-clab 2715 df-cleq 2729 df-clel 2816 df-nfc 2892 df-ne 2941 df-nel 3047 df-ral 3062 df-rex 3071 df-rmo 3380 df-reu 3381 df-rab 3437 df-v 3482 df-sbc 3789 df-csb 3900 df-dif 3954 df-un 3956 df-in 3958 df-ss 3968 df-pss 3971 df-nul 4334 df-if 4526 df-pw 4602 df-sn 4627 df-pr 4629 df-tp 4631 df-op 4633 df-uni 4908 df-int 4947 df-iun 4993 df-iin 4994 df-br 5144 df-opab 5206 df-mpt 5226 df-tr 5260 df-id 5578 df-eprel 5584 df-po 5592 df-so 5593 df-fr 5637 df-se 5638 df-we 5639 df-xp 5691 df-rel 5692 df-cnv 5693 df-co 5694 df-dm 5695 df-rn 5696 df-res 5697 df-ima 5698 df-pred 6321 df-ord 6387 df-on 6388 df-lim 6389 df-suc 6390 df-iota 6514 df-fun 6563 df-fn 6564 df-f 6565 df-f1 6566 df-fo 6567 df-f1o 6568 df-fv 6569 df-isom 6570 df-riota 7388 df-ov 7434 df-oprab 7435 df-mpo 7436 df-of 7697 df-ofr 7698 df-om 7888 df-1st 8014 df-2nd 8015 df-supp 8186 df-tpos 8251 df-frecs 8306 df-wrecs 8337 df-recs 8411 df-rdg 8450 df-1o 8506 df-2o 8507 df-oadd 8510 df-er 8745 df-ec 8747 df-qs 8751 df-map 8868 df-pm 8869 df-ixp 8938 df-en 8986 df-dom 8987 df-sdom 8988 df-fin 8989 df-fsupp 9402 df-sup 9482 df-inf 9483 df-oi 9550 df-dju 9941 df-card 9979 df-pnf 11297 df-mnf 11298 df-xr 11299 df-ltxr 11300 df-le 11301 df-sub 11494 df-neg 11495 df-div 11921 df-nn 12267 df-2 12329 df-3 12330 df-4 12331 df-5 12332 df-6 12333 df-7 12334 df-8 12335 df-9 12336 df-n0 12527 df-xnn0 12600 df-z 12614 df-dec 12734 df-uz 12879 df-q 12991 df-rp 13035 df-fz 13548 df-fzo 13695 df-fl 13832 df-mod 13910 df-seq 14043 df-exp 14103 df-hash 14370 df-cj 15138 df-re 15139 df-im 15140 df-sqrt 15274 df-abs 15275 df-dvds 16291 df-gcd 16532 df-prm 16709 df-phi 16803 df-pc 16875 df-struct 17184 df-sets 17201 df-slot 17219 df-ndx 17231 df-base 17248 df-ress 17275 df-plusg 17310 df-mulr 17311 df-starv 17312 df-sca 17313 df-vsca 17314 df-ip 17315 df-tset 17316 df-ple 17317 df-ds 17319 df-unif 17320 df-hom 17321 df-cco 17322 df-0g 17486 df-gsum 17487 df-prds 17492 df-pws 17494 df-imas 17553 df-qus 17554 df-mre 17629 df-mrc 17630 df-acs 17632 df-mgm 18653 df-sgrp 18732 df-mnd 18748 df-mhm 18796 df-submnd 18797 df-grp 18954 df-minusg 18955 df-sbg 18956 df-mulg 19086 df-subg 19141 df-nsg 19142 df-eqg 19143 df-ghm 19231 df-cntz 19335 df-cmn 19800 df-abl 19801 df-mgp 20138 df-rng 20150 df-ur 20179 df-srg 20184 df-ring 20232 df-cring 20233 df-oppr 20334 df-dvdsr 20357 df-unit 20358 df-invr 20388 df-dvr 20401 df-rhm 20472 df-nzr 20513 df-subrng 20546 df-subrg 20570 df-rlreg 20694 df-domn 20695 df-idom 20696 df-drng 20731 df-field 20732 df-lmod 20860 df-lss 20930 df-lsp 20970 df-sra 21172 df-rgmod 21173 df-lidl 21218 df-rsp 21219 df-2idl 21260 df-cnfld 21365 df-zring 21458 df-zrh 21514 df-zn 21517 df-assa 21873 df-asp 21874 df-ascl 21875 df-psr 21929 df-mvr 21930 df-mpl 21931 df-opsr 21933 df-evls 22098 df-evl 22099 df-psr1 22181 df-vr1 22182 df-ply1 22183 df-evl1 22320 df-lgs 27339 |
| This theorem is referenced by: lgsqrlem4 27393 |
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