Theorem lgsqrmodndvds 25078

Description: If the Legendre symbol of an integer A for an odd prime is 1, then the number is a quadratic residue mod P with a solution x of the congruence ( x ^ 2 ) == A (mod P) which is not divisible by the prime. (Contributed by AV, 20-Aug-2021.)

Assertion

Ref	Expression
lgsqrmodndvds	|- ( ( A e. ZZ /\ P e. ( Prime \ { 2 } ) ) -> ( ( A /L P ) = 1 -> E. x e. ZZ ( ( ( x ^ 2 ) mod P ) = ( A mod P ) /\ -. P || x ) ) )

Distinct variable groups:   x, A   x, P

Proof of Theorem lgsqrmodndvds

Step	Hyp	Ref	Expression
1		lgsqrmod 25077	. . . 4 |- ( ( A e. ZZ /\ P e. ( Prime \ { 2 } ) ) -> ( ( A /L P ) = 1 -> E. x e. ZZ ( ( x ^ 2 ) mod P ) = ( A mod P ) ) )
2	1	imp 445	. . 3 |- ( ( ( A e. ZZ /\ P e. ( Prime \ { 2 } ) ) /\ ( A /L P ) = 1 ) -> E. x e. ZZ ( ( x ^ 2 ) mod P ) = ( A mod P ) )
3		eldifi 3732	. . . . . . . . 9 |- ( P e. ( Prime \ { 2 } ) -> P e. Prime )
4		prmnn 15388	. . . . . . . . 9 |- ( P e. Prime -> P e. NN )
5	3, 4	syl 17	. . . . . . . 8 |- ( P e. ( Prime \ { 2 } ) -> P e. NN )
6	5	ad3antlr 767	. . . . . . 7 |- ( ( ( ( A e. ZZ /\ P e. ( Prime \ { 2 } ) ) /\ ( A /L P ) = 1 ) /\ x e. ZZ ) -> P e. NN )
7		zsqcl 12934	. . . . . . . 8 |- ( x e. ZZ -> ( x ^ 2 ) e. ZZ )
8	7	adantl 482	. . . . . . 7 |- ( ( ( ( A e. ZZ /\ P e. ( Prime \ { 2 } ) ) /\ ( A /L P ) = 1 ) /\ x e. ZZ ) -> ( x ^ 2 ) e. ZZ )
9		simplll 798	. . . . . . 7 |- ( ( ( ( A e. ZZ /\ P e. ( Prime \ { 2 } ) ) /\ ( A /L P ) = 1 ) /\ x e. ZZ ) -> A e. ZZ )
10		moddvds 14991	. . . . . . 7 |- ( ( P e. NN /\ ( x ^ 2 ) e. ZZ /\ A e. ZZ ) -> ( ( ( x ^ 2 ) mod P ) = ( A mod P ) <-> P || ( ( x ^ 2 ) - A ) ) )
11	6, 8, 9, 10	syl3anc 1326	. . . . . 6 |- ( ( ( ( A e. ZZ /\ P e. ( Prime \ { 2 } ) ) /\ ( A /L P ) = 1 ) /\ x e. ZZ ) -> ( ( ( x ^ 2 ) mod P ) = ( A mod P ) <-> P || ( ( x ^ 2 ) - A ) ) )
12	5	nnzd 11481	. . . . . . . . . . . . . 14 |- ( P e. ( Prime \ { 2 } ) -> P e. ZZ )
13	12	ad3antlr 767	. . . . . . . . . . . . 13 |- ( ( ( ( A e. ZZ /\ P e. ( Prime \ { 2 } ) ) /\ ( A /L P ) = 1 ) /\ x e. ZZ ) -> P e. ZZ )
14	13, 8, 9	3jca 1242	. . . . . . . . . . . 12 |- ( ( ( ( A e. ZZ /\ P e. ( Prime \ { 2 } ) ) /\ ( A /L P ) = 1 ) /\ x e. ZZ ) -> ( P e. ZZ /\ ( x ^ 2 ) e. ZZ /\ A e. ZZ ) )
15	14	adantl 482	. . . . . . . . . . 11 |- ( ( P || x /\ ( ( ( A e. ZZ /\ P e. ( Prime \ { 2 } ) ) /\ ( A /L P ) = 1 ) /\ x e. ZZ ) ) -> ( P e. ZZ /\ ( x ^ 2 ) e. ZZ /\ A e. ZZ ) )
16		dvdssub2 15023	. . . . . . . . . . 11 |- ( ( ( P e. ZZ /\ ( x ^ 2 ) e. ZZ /\ A e. ZZ ) /\ P || ( ( x ^ 2 ) - A ) ) -> ( P || ( x ^ 2 ) <-> P || A ) )
17	15, 16	sylan 488	. . . . . . . . . 10 |- ( ( ( P || x /\ ( ( ( A e. ZZ /\ P e. ( Prime \ { 2 } ) ) /\ ( A /L P ) = 1 ) /\ x e. ZZ ) ) /\ P || ( ( x ^ 2 ) - A ) ) -> ( P || ( x ^ 2 ) <-> P || A ) )
18	17	ex 450	. . . . . . . . 9 |- ( ( P || x /\ ( ( ( A e. ZZ /\ P e. ( Prime \ { 2 } ) ) /\ ( A /L P ) = 1 ) /\ x e. ZZ ) ) -> ( P || ( ( x ^ 2 ) - A ) -> ( P || ( x ^ 2 ) <-> P || A ) ) )
19		bicom 212	. . . . . . . . . 10 |- ( ( P || ( x ^ 2 ) <-> P || A ) <-> ( P || A <-> P || ( x ^ 2 ) ) )
20	3	ad3antlr 767	. . . . . . . . . . . . 13 |- ( ( ( ( A e. ZZ /\ P e. ( Prime \ { 2 } ) ) /\ ( A /L P ) = 1 ) /\ x e. ZZ ) -> P e. Prime )
21		simpr 477	. . . . . . . . . . . . 13 |- ( ( ( ( A e. ZZ /\ P e. ( Prime \ { 2 } ) ) /\ ( A /L P ) = 1 ) /\ x e. ZZ ) -> x e. ZZ )
22		2nn 11185	. . . . . . . . . . . . . 14 |- 2 e. NN
23	22	a1i 11	. . . . . . . . . . . . 13 |- ( ( ( ( A e. ZZ /\ P e. ( Prime \ { 2 } ) ) /\ ( A /L P ) = 1 ) /\ x e. ZZ ) -> 2 e. NN )
24		prmdvdsexp 15427	. . . . . . . . . . . . 13 |- ( ( P e. Prime /\ x e. ZZ /\ 2 e. NN ) -> ( P || ( x ^ 2 ) <-> P || x ) )
25	20, 21, 23, 24	syl3anc 1326	. . . . . . . . . . . 12 |- ( ( ( ( A e. ZZ /\ P e. ( Prime \ { 2 } ) ) /\ ( A /L P ) = 1 ) /\ x e. ZZ ) -> ( P || ( x ^ 2 ) <-> P || x ) )
26	25	biimparc 504	. . . . . . . . . . 11 |- ( ( P || x /\ ( ( ( A e. ZZ /\ P e. ( Prime \ { 2 } ) ) /\ ( A /L P ) = 1 ) /\ x e. ZZ ) ) -> P || ( x ^ 2 ) )
27		bianir 1009	. . . . . . . . . . . . . 14 |- ( ( P || ( x ^ 2 ) /\ ( P || A <-> P || ( x ^ 2 ) ) ) -> P || A )
28	5	anim2i 593	. . . . . . . . . . . . . . . . . . . . 21 |- ( ( A e. ZZ /\ P e. ( Prime \ { 2 } ) ) -> ( A e. ZZ /\ P e. NN ) )
29	28	ancomd 467	. . . . . . . . . . . . . . . . . . . 20 |- ( ( A e. ZZ /\ P e. ( Prime \ { 2 } ) ) -> ( P e. NN /\ A e. ZZ ) )
30		dvdsval3 14987	. . . . . . . . . . . . . . . . . . . 20 |- ( ( P e. NN /\ A e. ZZ ) -> ( P || A <-> ( A mod P ) = 0 ) )
31	29, 30	syl 17	. . . . . . . . . . . . . . . . . . 19 |- ( ( A e. ZZ /\ P e. ( Prime \ { 2 } ) ) -> ( P || A <-> ( A mod P ) = 0 ) )
32		lgsprme0 25064	. . . . . . . . . . . . . . . . . . . . 21 |- ( ( A e. ZZ /\ P e. Prime ) -> ( ( A /L P ) = 0 <-> ( A mod P ) = 0 ) )
33	3, 32	sylan2 491	. . . . . . . . . . . . . . . . . . . 20 |- ( ( A e. ZZ /\ P e. ( Prime \ { 2 } ) ) -> ( ( A /L P ) = 0 <-> ( A mod P ) = 0 ) )
34		eqeq1 2626	. . . . . . . . . . . . . . . . . . . . 21 |- ( ( A /L P ) = 0 -> ( ( A /L P ) = 1 <-> 0 = 1 ) )
35		0ne1 11088	. . . . . . . . . . . . . . . . . . . . . . 23 |- 0 =/= 1
36		eqneqall 2805	. . . . . . . . . . . . . . . . . . . . . . 23 |- ( 0 = 1 -> ( 0 =/= 1 -> -. P || x ) )
37	35, 36	mpi 20	. . . . . . . . . . . . . . . . . . . . . 22 |- ( 0 = 1 -> -. P || x )
38	37	a1i 11	. . . . . . . . . . . . . . . . . . . . 21 |- ( ( A /L P ) = 0 -> ( 0 = 1 -> -. P || x ) )
39	34, 38	sylbid 230	. . . . . . . . . . . . . . . . . . . 20 |- ( ( A /L P ) = 0 -> ( ( A /L P ) = 1 -> -. P || x ) )
40	33, 39	syl6bir 244	. . . . . . . . . . . . . . . . . . 19 |- ( ( A e. ZZ /\ P e. ( Prime \ { 2 } ) ) -> ( ( A mod P ) = 0 -> ( ( A /L P ) = 1 -> -. P || x ) ) )
41	31, 40	sylbid 230	. . . . . . . . . . . . . . . . . 18 |- ( ( A e. ZZ /\ P e. ( Prime \ { 2 } ) ) -> ( P || A -> ( ( A /L P ) = 1 -> -. P || x ) ) )
42	41	com23 86	. . . . . . . . . . . . . . . . 17 |- ( ( A e. ZZ /\ P e. ( Prime \ { 2 } ) ) -> ( ( A /L P ) = 1 -> ( P || A -> -. P || x ) ) )
43	42	imp 445	. . . . . . . . . . . . . . . 16 |- ( ( ( A e. ZZ /\ P e. ( Prime \ { 2 } ) ) /\ ( A /L P ) = 1 ) -> ( P || A -> -. P || x ) )
44	43	ad2antrl 764	. . . . . . . . . . . . . . 15 |- ( ( P || x /\ ( ( ( A e. ZZ /\ P e. ( Prime \ { 2 } ) ) /\ ( A /L P ) = 1 ) /\ x e. ZZ ) ) -> ( P || A -> -. P || x ) )
45	44	com12 32	. . . . . . . . . . . . . 14 |- ( P || A -> ( ( P || x /\ ( ( ( A e. ZZ /\ P e. ( Prime \ { 2 } ) ) /\ ( A /L P ) = 1 ) /\ x e. ZZ ) ) -> -. P || x ) )
46	27, 45	syl 17	. . . . . . . . . . . . 13 |- ( ( P || ( x ^ 2 ) /\ ( P || A <-> P || ( x ^ 2 ) ) ) -> ( ( P || x /\ ( ( ( A e. ZZ /\ P e. ( Prime \ { 2 } ) ) /\ ( A /L P ) = 1 ) /\ x e. ZZ ) ) -> -. P || x ) )
47	46	ex 450	. . . . . . . . . . . 12 |- ( P || ( x ^ 2 ) -> ( ( P || A <-> P || ( x ^ 2 ) ) -> ( ( P || x /\ ( ( ( A e. ZZ /\ P e. ( Prime \ { 2 } ) ) /\ ( A /L P ) = 1 ) /\ x e. ZZ ) ) -> -. P || x ) ) )
48	47	com23 86	. . . . . . . . . . 11 |- ( P || ( x ^ 2 ) -> ( ( P || x /\ ( ( ( A e. ZZ /\ P e. ( Prime \ { 2 } ) ) /\ ( A /L P ) = 1 ) /\ x e. ZZ ) ) -> ( ( P || A <-> P || ( x ^ 2 ) ) -> -. P || x ) ) )
49	26, 48	mpcom 38	. . . . . . . . . 10 |- ( ( P || x /\ ( ( ( A e. ZZ /\ P e. ( Prime \ { 2 } ) ) /\ ( A /L P ) = 1 ) /\ x e. ZZ ) ) -> ( ( P || A <-> P || ( x ^ 2 ) ) -> -. P || x ) )
50	19, 49	syl5bi 232	. . . . . . . . 9 |- ( ( P || x /\ ( ( ( A e. ZZ /\ P e. ( Prime \ { 2 } ) ) /\ ( A /L P ) = 1 ) /\ x e. ZZ ) ) -> ( ( P || ( x ^ 2 ) <-> P || A ) -> -. P || x ) )
51	18, 50	syld 47	. . . . . . . 8 |- ( ( P || x /\ ( ( ( A e. ZZ /\ P e. ( Prime \ { 2 } ) ) /\ ( A /L P ) = 1 ) /\ x e. ZZ ) ) -> ( P || ( ( x ^ 2 ) - A ) -> -. P || x ) )
52	51	ex 450	. . . . . . 7 |- ( P || x -> ( ( ( ( A e. ZZ /\ P e. ( Prime \ { 2 } ) ) /\ ( A /L P ) = 1 ) /\ x e. ZZ ) -> ( P || ( ( x ^ 2 ) - A ) -> -. P || x ) ) )
53		2a1 28	. . . . . . 7 |- ( -. P || x -> ( ( ( ( A e. ZZ /\ P e. ( Prime \ { 2 } ) ) /\ ( A /L P ) = 1 ) /\ x e. ZZ ) -> ( P || ( ( x ^ 2 ) - A ) -> -. P || x ) ) )
54	52, 53	pm2.61i 176	. . . . . 6 |- ( ( ( ( A e. ZZ /\ P e. ( Prime \ { 2 } ) ) /\ ( A /L P ) = 1 ) /\ x e. ZZ ) -> ( P || ( ( x ^ 2 ) - A ) -> -. P || x ) )
55	11, 54	sylbid 230	. . . . 5 |- ( ( ( ( A e. ZZ /\ P e. ( Prime \ { 2 } ) ) /\ ( A /L P ) = 1 ) /\ x e. ZZ ) -> ( ( ( x ^ 2 ) mod P ) = ( A mod P ) -> -. P || x ) )
56	55	ancld 576	. . . 4 |- ( ( ( ( A e. ZZ /\ P e. ( Prime \ { 2 } ) ) /\ ( A /L P ) = 1 ) /\ x e. ZZ ) -> ( ( ( x ^ 2 ) mod P ) = ( A mod P ) -> ( ( ( x ^ 2 ) mod P ) = ( A mod P ) /\ -. P || x ) ) )
57	56	reximdva 3017	. . 3 |- ( ( ( A e. ZZ /\ P e. ( Prime \ { 2 } ) ) /\ ( A /L P ) = 1 ) -> ( E. x e. ZZ ( ( x ^ 2 ) mod P ) = ( A mod P ) -> E. x e. ZZ ( ( ( x ^ 2 ) mod P ) = ( A mod P ) /\ -. P || x ) ) )
58	2, 57	mpd 15	. 2 |- ( ( ( A e. ZZ /\ P e. ( Prime \ { 2 } ) ) /\ ( A /L P ) = 1 ) -> E. x e. ZZ ( ( ( x ^ 2 ) mod P ) = ( A mod P ) /\ -. P || x ) )
59	58	ex 450	1 |- ( ( A e. ZZ /\ P e. ( Prime \ { 2 } ) ) -> ( ( A /L P ) = 1 -> E. x e. ZZ ( ( ( x ^ 2 ) mod P ) = ( A mod P ) /\ -. P || x ) ) )

Syntax hints:   -. wn 3   -> wi 4   <-> wb 196   /\ wa 384   /\ w3a 1037   = wceq 1483   e. wcel 1990   =/= wne 2794   E.wrex 2913   \ cdif 3571   {csn 4177   class class class wbr 4653  (class class class)co 6650   0cc0 9936   1c1 9937   - cmin 10266   NNcn 11020   2c2 11070   ZZcz 11377   mod cmo 12668   ^cexp 12860   || cdvds 14983   Primecprime 15385   /Lclgs 25019

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-8 1992  ax-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246  ax-ext 2602  ax-rep 4771  ax-sep 4781  ax-nul 4789  ax-pow 4843  ax-pr 4906  ax-un 6949  ax-inf2 8538  ax-cnex 9992  ax-resscn 9993  ax-1cn 9994  ax-icn 9995  ax-addcl 9996  ax-addrcl 9997  ax-mulcl 9998  ax-mulrcl 9999  ax-mulcom 10000  ax-addass 10001  ax-mulass 10002  ax-distr 10003  ax-i2m1 10004  ax-1ne0 10005  ax-1rid 10006  ax-rnegex 10007  ax-rrecex 10008  ax-cnre 10009  ax-pre-lttri 10010  ax-pre-lttrn 10011  ax-pre-ltadd 10012  ax-pre-mulgt0 10013  ax-pre-sup 10014  ax-addf 10015  ax-mulf 10016

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3or 1038  df-3an 1039  df-tru 1486  df-ex 1705  df-nf 1710  df-sb 1881  df-eu 2474  df-mo 2475  df-clab 2609  df-cleq 2615  df-clel 2618  df-nfc 2753  df-ne 2795  df-nel 2898  df-ral 2917  df-rex 2918  df-reu 2919  df-rmo 2920  df-rab 2921  df-v 3202  df-sbc 3436  df-csb 3534  df-dif 3577  df-un 3579  df-in 3581  df-ss 3588  df-pss 3590  df-nul 3916  df-if 4087  df-pw 4160  df-sn 4178  df-pr 4180  df-tp 4182  df-op 4184  df-uni 4437  df-int 4476  df-iun 4522  df-iin 4523  df-br 4654  df-opab 4713  df-mpt 4730  df-tr 4753  df-id 5024  df-eprel 5029  df-po 5035  df-so 5036  df-fr 5073  df-se 5074  df-we 5075  df-xp 5120  df-rel 5121  df-cnv 5122  df-co 5123  df-dm 5124  df-rn 5125  df-res 5126  df-ima 5127  df-pred 5680  df-ord 5726  df-on 5727  df-lim 5728  df-suc 5729  df-iota 5851  df-fun 5890  df-fn 5891  df-f 5892  df-f1 5893  df-fo 5894  df-f1o 5895  df-fv 5896  df-isom 5897  df-riota 6611  df-ov 6653  df-oprab 6654  df-mpt2 6655  df-of 6897  df-ofr 6898  df-om 7066  df-1st 7168  df-2nd 7169  df-supp 7296  df-tpos 7352  df-wrecs 7407  df-recs 7468  df-rdg 7506  df-1o 7560  df-2o 7561  df-oadd 7564  df-er 7742  df-ec 7744  df-qs 7748  df-map 7859  df-pm 7860  df-ixp 7909  df-en 7956  df-dom 7957  df-sdom 7958  df-fin 7959  df-fsupp 8276  df-sup 8348  df-inf 8349  df-oi 8415  df-card 8765  df-cda 8990  df-pnf 10076  df-mnf 10077  df-xr 10078  df-ltxr 10079  df-le 10080  df-sub 10268  df-neg 10269  df-div 10685  df-nn 11021  df-2 11079  df-3 11080  df-4 11081  df-5 11082  df-6 11083  df-7 11084  df-8 11085  df-9 11086  df-n0 11293  df-xnn0 11364  df-z 11378  df-dec 11494  df-uz 11688  df-q 11789  df-rp 11833  df-fz 12327  df-fzo 12466  df-fl 12593  df-mod 12669  df-seq 12802  df-exp 12861  df-hash 13118  df-cj 13839  df-re 13840  df-im 13841  df-sqrt 13975  df-abs 13976  df-dvds 14984  df-gcd 15217  df-prm 15386  df-phi 15471  df-pc 15542  df-struct 15859  df-ndx 15860  df-slot 15861  df-base 15863  df-sets 15864  df-ress 15865  df-plusg 15954  df-mulr 15955  df-starv 15956  df-sca 15957  df-vsca 15958  df-ip 15959  df-tset 15960  df-ple 15961  df-ds 15964  df-unif 15965  df-hom 15966  df-cco 15967  df-0g 16102  df-gsum 16103  df-prds 16108  df-pws 16110  df-imas 16168  df-qus 16169  df-mre 16246  df-mrc 16247  df-acs 16249  df-mgm 17242  df-sgrp 17284  df-mnd 17295  df-mhm 17335  df-submnd 17336  df-grp 17425  df-minusg 17426  df-sbg 17427  df-mulg 17541  df-subg 17591  df-nsg 17592  df-eqg 17593  df-ghm 17658  df-cntz 17750  df-cmn 18195  df-abl 18196  df-mgp 18490  df-ur 18502  df-srg 18506  df-ring 18549  df-cring 18550  df-oppr 18623  df-dvdsr 18641  df-unit 18642  df-invr 18672  df-dvr 18683  df-rnghom 18715  df-drng 18749  df-field 18750  df-subrg 18778  df-lmod 18865  df-lss 18933  df-lsp 18972  df-sra 19172  df-rgmod 19173  df-lidl 19174  df-rsp 19175  df-2idl 19232  df-nzr 19258  df-rlreg 19283  df-domn 19284  df-idom 19285  df-assa 19312  df-asp 19313  df-ascl 19314  df-psr 19356  df-mvr 19357  df-mpl 19358  df-opsr 19360  df-evls 19506  df-evl 19507  df-psr1 19550  df-vr1 19551  df-ply1 19552  df-coe1 19553  df-evl1 19681  df-cnfld 19747  df-zring 19819  df-zrh 19852  df-zn 19855  df-mdeg 23815  df-deg1 23816  df-mon1 23890  df-uc1p 23891  df-q1p 23892  df-r1p 23893  df-lgs 25020

This theorem is referenced by:  sfprmdvdsmersenne  41520