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Mirrors > Home > MPE Home > Th. List > lgsqrlem5 | Structured version Visualization version GIF version |
Description: Lemma for lgsqr 26783. (Contributed by Mario Carneiro, 15-Jun-2015.) |
Ref | Expression |
---|---|
lgsqrlem5 | ⊢ ((𝐴 ∈ ℤ ∧ 𝑃 ∈ (ℙ ∖ {2}) ∧ (𝐴 /L 𝑃) = 1) → ∃𝑥 ∈ ℤ 𝑃 ∥ ((𝑥↑2) − 𝐴)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | eqid 2732 | . 2 ⊢ (ℤ/nℤ‘𝑃) = (ℤ/nℤ‘𝑃) | |
2 | eqid 2732 | . 2 ⊢ (Poly1‘(ℤ/nℤ‘𝑃)) = (Poly1‘(ℤ/nℤ‘𝑃)) | |
3 | eqid 2732 | . 2 ⊢ (Base‘(Poly1‘(ℤ/nℤ‘𝑃))) = (Base‘(Poly1‘(ℤ/nℤ‘𝑃))) | |
4 | eqid 2732 | . 2 ⊢ ( deg1 ‘(ℤ/nℤ‘𝑃)) = ( deg1 ‘(ℤ/nℤ‘𝑃)) | |
5 | eqid 2732 | . 2 ⊢ (eval1‘(ℤ/nℤ‘𝑃)) = (eval1‘(ℤ/nℤ‘𝑃)) | |
6 | eqid 2732 | . 2 ⊢ (.g‘(mulGrp‘(Poly1‘(ℤ/nℤ‘𝑃)))) = (.g‘(mulGrp‘(Poly1‘(ℤ/nℤ‘𝑃)))) | |
7 | eqid 2732 | . 2 ⊢ (var1‘(ℤ/nℤ‘𝑃)) = (var1‘(ℤ/nℤ‘𝑃)) | |
8 | eqid 2732 | . 2 ⊢ (-g‘(Poly1‘(ℤ/nℤ‘𝑃))) = (-g‘(Poly1‘(ℤ/nℤ‘𝑃))) | |
9 | eqid 2732 | . 2 ⊢ (1r‘(Poly1‘(ℤ/nℤ‘𝑃))) = (1r‘(Poly1‘(ℤ/nℤ‘𝑃))) | |
10 | eqid 2732 | . 2 ⊢ ((((𝑃 − 1) / 2)(.g‘(mulGrp‘(Poly1‘(ℤ/nℤ‘𝑃))))(var1‘(ℤ/nℤ‘𝑃)))(-g‘(Poly1‘(ℤ/nℤ‘𝑃)))(1r‘(Poly1‘(ℤ/nℤ‘𝑃)))) = ((((𝑃 − 1) / 2)(.g‘(mulGrp‘(Poly1‘(ℤ/nℤ‘𝑃))))(var1‘(ℤ/nℤ‘𝑃)))(-g‘(Poly1‘(ℤ/nℤ‘𝑃)))(1r‘(Poly1‘(ℤ/nℤ‘𝑃)))) | |
11 | eqid 2732 | . 2 ⊢ (ℤRHom‘(ℤ/nℤ‘𝑃)) = (ℤRHom‘(ℤ/nℤ‘𝑃)) | |
12 | simp2 1137 | . 2 ⊢ ((𝐴 ∈ ℤ ∧ 𝑃 ∈ (ℙ ∖ {2}) ∧ (𝐴 /L 𝑃) = 1) → 𝑃 ∈ (ℙ ∖ {2})) | |
13 | eqid 2732 | . 2 ⊢ (𝑦 ∈ (1...((𝑃 − 1) / 2)) ↦ ((ℤRHom‘(ℤ/nℤ‘𝑃))‘(𝑦↑2))) = (𝑦 ∈ (1...((𝑃 − 1) / 2)) ↦ ((ℤRHom‘(ℤ/nℤ‘𝑃))‘(𝑦↑2))) | |
14 | simp1 1136 | . 2 ⊢ ((𝐴 ∈ ℤ ∧ 𝑃 ∈ (ℙ ∖ {2}) ∧ (𝐴 /L 𝑃) = 1) → 𝐴 ∈ ℤ) | |
15 | simp3 1138 | . 2 ⊢ ((𝐴 ∈ ℤ ∧ 𝑃 ∈ (ℙ ∖ {2}) ∧ (𝐴 /L 𝑃) = 1) → (𝐴 /L 𝑃) = 1) | |
16 | 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15 | lgsqrlem4 26781 | 1 ⊢ ((𝐴 ∈ ℤ ∧ 𝑃 ∈ (ℙ ∖ {2}) ∧ (𝐴 /L 𝑃) = 1) → ∃𝑥 ∈ ℤ 𝑃 ∥ ((𝑥↑2) − 𝐴)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ w3a 1087 = wceq 1541 ∈ wcel 2106 ∃wrex 3070 ∖ cdif 3942 {csn 4623 class class class wbr 5142 ↦ cmpt 5225 ‘cfv 6533 (class class class)co 7394 1c1 11095 − cmin 11428 / cdiv 11855 2c2 12251 ℤcz 12542 ...cfz 13468 ↑cexp 14011 ∥ cdvds 16181 ℙcprime 16592 Basecbs 17128 -gcsg 18798 .gcmg 18924 mulGrpcmgp 19948 1rcur 19965 ℤRHomczrh 20984 ℤ/nℤczn 20987 var1cv1 21631 Poly1cpl1 21632 eval1ce1 21764 deg1 cdg1 25500 /L clgs 26726 |
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 2703 ax-rep 5279 ax-sep 5293 ax-nul 5300 ax-pow 5357 ax-pr 5421 ax-un 7709 ax-cnex 11150 ax-resscn 11151 ax-1cn 11152 ax-icn 11153 ax-addcl 11154 ax-addrcl 11155 ax-mulcl 11156 ax-mulrcl 11157 ax-mulcom 11158 ax-addass 11159 ax-mulass 11160 ax-distr 11161 ax-i2m1 11162 ax-1ne0 11163 ax-1rid 11164 ax-rnegex 11165 ax-rrecex 11166 ax-cnre 11167 ax-pre-lttri 11168 ax-pre-lttrn 11169 ax-pre-ltadd 11170 ax-pre-mulgt0 11171 ax-pre-sup 11172 ax-addf 11173 ax-mulf 11174 |
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 2534 df-eu 2563 df-clab 2710 df-cleq 2724 df-clel 2810 df-nfc 2885 df-ne 2941 df-nel 3047 df-ral 3062 df-rex 3071 df-rmo 3376 df-reu 3377 df-rab 3433 df-v 3476 df-sbc 3775 df-csb 3891 df-dif 3948 df-un 3950 df-in 3952 df-ss 3962 df-pss 3964 df-nul 4320 df-if 4524 df-pw 4599 df-sn 4624 df-pr 4626 df-tp 4628 df-op 4630 df-uni 4903 df-int 4945 df-iun 4993 df-iin 4994 df-br 5143 df-opab 5205 df-mpt 5226 df-tr 5260 df-id 5568 df-eprel 5574 df-po 5582 df-so 5583 df-fr 5625 df-se 5626 df-we 5627 df-xp 5676 df-rel 5677 df-cnv 5678 df-co 5679 df-dm 5680 df-rn 5681 df-res 5682 df-ima 5683 df-pred 6290 df-ord 6357 df-on 6358 df-lim 6359 df-suc 6360 df-iota 6485 df-fun 6535 df-fn 6536 df-f 6537 df-f1 6538 df-fo 6539 df-f1o 6540 df-fv 6541 df-isom 6542 df-riota 7350 df-ov 7397 df-oprab 7398 df-mpo 7399 df-of 7654 df-ofr 7655 df-om 7840 df-1st 7959 df-2nd 7960 df-supp 8131 df-tpos 8195 df-frecs 8250 df-wrecs 8281 df-recs 8355 df-rdg 8394 df-1o 8450 df-2o 8451 df-oadd 8454 df-er 8688 df-ec 8690 df-qs 8694 df-map 8807 df-pm 8808 df-ixp 8877 df-en 8925 df-dom 8926 df-sdom 8927 df-fin 8928 df-fsupp 9347 df-sup 9421 df-inf 9422 df-oi 9489 df-dju 9880 df-card 9918 df-pnf 11234 df-mnf 11235 df-xr 11236 df-ltxr 11237 df-le 11238 df-sub 11430 df-neg 11431 df-div 11856 df-nn 12197 df-2 12259 df-3 12260 df-4 12261 df-5 12262 df-6 12263 df-7 12264 df-8 12265 df-9 12266 df-n0 12457 df-xnn0 12529 df-z 12543 df-dec 12662 df-uz 12807 df-q 12917 df-rp 12959 df-fz 13469 df-fzo 13612 df-fl 13741 df-mod 13819 df-seq 13951 df-exp 14012 df-hash 14275 df-cj 15030 df-re 15031 df-im 15032 df-sqrt 15166 df-abs 15167 df-dvds 16182 df-gcd 16420 df-prm 16593 df-phi 16683 df-pc 16754 df-struct 17064 df-sets 17081 df-slot 17099 df-ndx 17111 df-base 17129 df-ress 17158 df-plusg 17194 df-mulr 17195 df-starv 17196 df-sca 17197 df-vsca 17198 df-ip 17199 df-tset 17200 df-ple 17201 df-ds 17203 df-unif 17204 df-hom 17205 df-cco 17206 df-0g 17371 df-gsum 17372 df-prds 17377 df-pws 17379 df-imas 17438 df-qus 17439 df-mre 17514 df-mrc 17515 df-acs 17517 df-mgm 18545 df-sgrp 18594 df-mnd 18605 df-mhm 18649 df-submnd 18650 df-grp 18799 df-minusg 18800 df-sbg 18801 df-mulg 18925 df-subg 18977 df-nsg 18978 df-eqg 18979 df-ghm 19058 df-cntz 19149 df-cmn 19616 df-abl 19617 df-mgp 19949 df-ur 19966 df-srg 19970 df-ring 20018 df-cring 20019 df-oppr 20104 df-dvdsr 20125 df-unit 20126 df-invr 20156 df-dvr 20167 df-rnghom 20203 df-nzr 20244 df-drng 20269 df-field 20270 df-subrg 20312 df-lmod 20424 df-lss 20494 df-lsp 20534 df-sra 20736 df-rgmod 20737 df-lidl 20738 df-rsp 20739 df-2idl 20805 df-rlreg 20837 df-domn 20838 df-idom 20839 df-cnfld 20881 df-zring 20954 df-zrh 20988 df-zn 20991 df-assa 21343 df-asp 21344 df-ascl 21345 df-psr 21395 df-mvr 21396 df-mpl 21397 df-opsr 21399 df-evls 21566 df-evl 21567 df-psr1 21635 df-vr1 21636 df-ply1 21637 df-coe1 21638 df-evl1 21766 df-mdeg 25501 df-deg1 25502 df-mon1 25579 df-uc1p 25580 df-q1p 25581 df-r1p 25582 df-lgs 26727 |
This theorem is referenced by: lgsqr 26783 |
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