Theorem 4sqlem13m 12438

Description: Lemma for 4sq 12445. (Contributed by Mario Carneiro, 16-Jul-2014.) (Revised by AV, 14-Sep-2020.)

Hypotheses

Ref	Expression
4sqlem11.1	|- S = { n | E. x e. ZZ E. y e. ZZ E. z e. ZZ E. w e. ZZ n = ( ( ( x ^ 2 ) + ( y ^ 2 ) ) + ( ( z ^ 2 ) + ( w ^ 2 ) ) ) }
4sq.2	|- ( ph -> N e. NN )
4sq.3	|- ( ph -> P = ( ( 2 x. N ) + 1 ) )
4sq.4	|- ( ph -> P e. Prime )
4sq.5	|- ( ph -> ( 0 ... ( 2 x. N ) ) C_ S )
4sq.6	|- T = { i e. NN | ( i x. P ) e. S }
4sq.7	|- M = inf ( T , RR , < )

Assertion

Ref	Expression
4sqlem13m	|- ( ph -> ( E. j j e. T /\ M < P ) )

Distinct variable groups:   n, N   P, i, n, w, x, y, z   S, i, n   T, j   ph, i, n

Allowed substitution hints:   ph( x, y, z, w, j)   P( j)   S( x, y, z, w, j)   T( x, y, z, w, i, n)   M( x, y, z, w, i, j, n)   N( x, y, z, w, i, j)

Proof of Theorem 4sqlem13m

Dummy variables k u m v are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		4sqlem11.1	. . 3 |- S = { n | E. x e. ZZ E. y e. ZZ E. z e. ZZ E. w e. ZZ n = ( ( ( x ^ 2 ) + ( y ^ 2 ) ) + ( ( z ^ 2 ) + ( w ^ 2 ) ) ) }
2		4sq.2	. . 3 |- ( ph -> N e. NN )
3		4sq.3	. . 3 |- ( ph -> P = ( ( 2 x. N ) + 1 ) )
4		4sq.4	. . 3 |- ( ph -> P e. Prime )
5		eqid 2189	. . 3 |- { u | E. m e. ( 0 ... N ) u = ( ( m ^ 2 ) mod P ) } = { u | E. m e. ( 0 ... N ) u = ( ( m ^ 2 ) mod P ) }
6		eqid 2189	. . 3 |- ( v e. { u | E. m e. ( 0 ... N ) u = ( ( m ^ 2 ) mod P ) } |-> ( ( P - 1 ) - v ) ) = ( v e. { u | E. m e. ( 0 ... N ) u = ( ( m ^ 2 ) mod P ) } |-> ( ( P - 1 ) - v ) )
7	1, 2, 3, 4, 5, 6	4sqlem12 12437	. 2 |- ( ph -> E. k e. ( 1 ... ( P - 1 ) ) E. u e. ZZ[_i] ( ( ( abs ` u ) ^ 2 ) + 1 ) = ( k x. P ) )
8		simplrl 535	. . . . . . . 8 |- ( ( ( ph /\ ( k e. ( 1 ... ( P - 1 ) ) /\ u e. ZZ[_i] ) ) /\ ( ( ( abs ` u ) ^ 2 ) + 1 ) = ( k x. P ) ) -> k e. ( 1 ... ( P - 1 ) ) )
9		elfznn 10086	. . . . . . . 8 |- ( k e. ( 1 ... ( P - 1 ) ) -> k e. NN )
10	8, 9	syl 14	. . . . . . 7 |- ( ( ( ph /\ ( k e. ( 1 ... ( P - 1 ) ) /\ u e. ZZ[_i] ) ) /\ ( ( ( abs ` u ) ^ 2 ) + 1 ) = ( k x. P ) ) -> k e. NN )
11		simpr 110	. . . . . . . 8 |- ( ( ( ph /\ ( k e. ( 1 ... ( P - 1 ) ) /\ u e. ZZ[_i] ) ) /\ ( ( ( abs ` u ) ^ 2 ) + 1 ) = ( k x. P ) ) -> ( ( ( abs ` u ) ^ 2 ) + 1 ) = ( k x. P ) )
12		abs1 11116	. . . . . . . . . . . 12 |- ( abs ` 1 ) = 1
13	12	oveq1i 5907	. . . . . . . . . . 11 |- ( ( abs ` 1 ) ^ 2 ) = ( 1 ^ 2 )
14		sq1 10648	. . . . . . . . . . 11 |- ( 1 ^ 2 ) = 1
15	13, 14	eqtri 2210	. . . . . . . . . 10 |- ( ( abs ` 1 ) ^ 2 ) = 1
16	15	oveq2i 5908	. . . . . . . . 9 |- ( ( ( abs ` u ) ^ 2 ) + ( ( abs ` 1 ) ^ 2 ) ) = ( ( ( abs ` u ) ^ 2 ) + 1 )
17		simplrr 536	. . . . . . . . . 10 |- ( ( ( ph /\ ( k e. ( 1 ... ( P - 1 ) ) /\ u e. ZZ[_i] ) ) /\ ( ( ( abs ` u ) ^ 2 ) + 1 ) = ( k x. P ) ) -> u e. ZZ[_i] )
18		1z 9310	. . . . . . . . . . 11 |- 1 e. ZZ
19		zgz 12408	. . . . . . . . . . 11 |- ( 1 e. ZZ -> 1 e. ZZ[_i] )
20	18, 19	ax-mp 5	. . . . . . . . . 10 |- 1 e. ZZ[_i]
21	1	4sqlem4a 12426	. . . . . . . . . 10 |- ( ( u e. ZZ[_i] /\ 1 e. ZZ[_i] ) -> ( ( ( abs ` u ) ^ 2 ) + ( ( abs ` 1 ) ^ 2 ) ) e. S )
22	17, 20, 21	sylancl 413	. . . . . . . . 9 |- ( ( ( ph /\ ( k e. ( 1 ... ( P - 1 ) ) /\ u e. ZZ[_i] ) ) /\ ( ( ( abs ` u ) ^ 2 ) + 1 ) = ( k x. P ) ) -> ( ( ( abs ` u ) ^ 2 ) + ( ( abs ` 1 ) ^ 2 ) ) e. S )
23	16, 22	eqeltrrid 2277	. . . . . . . 8 |- ( ( ( ph /\ ( k e. ( 1 ... ( P - 1 ) ) /\ u e. ZZ[_i] ) ) /\ ( ( ( abs ` u ) ^ 2 ) + 1 ) = ( k x. P ) ) -> ( ( ( abs ` u ) ^ 2 ) + 1 ) e. S )
24	11, 23	eqeltrrd 2267	. . . . . . 7 |- ( ( ( ph /\ ( k e. ( 1 ... ( P - 1 ) ) /\ u e. ZZ[_i] ) ) /\ ( ( ( abs ` u ) ^ 2 ) + 1 ) = ( k x. P ) ) -> ( k x. P ) e. S )
25		oveq1 5904	. . . . . . . . 9 |- ( i = k -> ( i x. P ) = ( k x. P ) )
26	25	eleq1d 2258	. . . . . . . 8 |- ( i = k -> ( ( i x. P ) e. S <-> ( k x. P ) e. S ) )
27		4sq.6	. . . . . . . 8 |- T = { i e. NN | ( i x. P ) e. S }
28	26, 27	elrab2 2911	. . . . . . 7 |- ( k e. T <-> ( k e. NN /\ ( k x. P ) e. S ) )
29	10, 24, 28	sylanbrc 417	. . . . . 6 |- ( ( ( ph /\ ( k e. ( 1 ... ( P - 1 ) ) /\ u e. ZZ[_i] ) ) /\ ( ( ( abs ` u ) ^ 2 ) + 1 ) = ( k x. P ) ) -> k e. T )
30		elex2 2768	. . . . . 6 |- ( k e. T -> E. j j e. T )
31	29, 30	syl 14	. . . . 5 |- ( ( ( ph /\ ( k e. ( 1 ... ( P - 1 ) ) /\ u e. ZZ[_i] ) ) /\ ( ( ( abs ` u ) ^ 2 ) + 1 ) = ( k x. P ) ) -> E. j j e. T )
32	27	ssrab3 3256	. . . . . . . 8 |- T C_ NN
33		4sq.7	. . . . . . . . 9 |- M = inf ( T , RR , < )
34		1zzd 9311	. . . . . . . . . 10 |- ( ( ( ph /\ ( k e. ( 1 ... ( P - 1 ) ) /\ u e. ZZ[_i] ) ) /\ ( ( ( abs ` u ) ^ 2 ) + 1 ) = ( k x. P ) ) -> 1 e. ZZ )
35		nnuz 9595	. . . . . . . . . . . 12 |- NN = ( ZZ>= ` 1 )
36	35	rabeqi 2745	. . . . . . . . . . 11 |- { i e. NN | ( i x. P ) e. S } = { i e. ( ZZ>= ` 1 ) | ( i x. P ) e. S }
37	27, 36	eqtri 2210	. . . . . . . . . 10 |- T = { i e. ( ZZ>= ` 1 ) | ( i x. P ) e. S }
38		elfznn 10086	. . . . . . . . . . . . . 14 |- ( i e. ( 1 ... k ) -> i e. NN )
39	38	adantl 277	. . . . . . . . . . . . 13 |- ( ( ( ( ph /\ ( k e. ( 1 ... ( P - 1 ) ) /\ u e. ZZ[_i] ) ) /\ ( ( ( abs ` u ) ^ 2 ) + 1 ) = ( k x. P ) ) /\ i e. ( 1 ... k ) ) -> i e. NN )
40		prmnn 12145	. . . . . . . . . . . . . . 15 |- ( P e. Prime -> P e. NN )
41	4, 40	syl 14	. . . . . . . . . . . . . 14 |- ( ph -> P e. NN )
42	41	ad3antrrr 492	. . . . . . . . . . . . 13 |- ( ( ( ( ph /\ ( k e. ( 1 ... ( P - 1 ) ) /\ u e. ZZ[_i] ) ) /\ ( ( ( abs ` u ) ^ 2 ) + 1 ) = ( k x. P ) ) /\ i e. ( 1 ... k ) ) -> P e. NN )
43	39, 42	nnmulcld 8999	. . . . . . . . . . . 12 |- ( ( ( ( ph /\ ( k e. ( 1 ... ( P - 1 ) ) /\ u e. ZZ[_i] ) ) /\ ( ( ( abs ` u ) ^ 2 ) + 1 ) = ( k x. P ) ) /\ i e. ( 1 ... k ) ) -> ( i x. P ) e. NN )
44	43	nnnn0d 9260	. . . . . . . . . . 11 |- ( ( ( ( ph /\ ( k e. ( 1 ... ( P - 1 ) ) /\ u e. ZZ[_i] ) ) /\ ( ( ( abs ` u ) ^ 2 ) + 1 ) = ( k x. P ) ) /\ i e. ( 1 ... k ) ) -> ( i x. P ) e. NN0 )
45	1	4sqlemsdc 12435	. . . . . . . . . . 11 |- ( ( i x. P ) e. NN0 -> DECID ( i x. P ) e. S )
46	44, 45	syl 14	. . . . . . . . . 10 |- ( ( ( ( ph /\ ( k e. ( 1 ... ( P - 1 ) ) /\ u e. ZZ[_i] ) ) /\ ( ( ( abs ` u ) ^ 2 ) + 1 ) = ( k x. P ) ) /\ i e. ( 1 ... k ) ) -> DECID ( i x. P ) e. S )
47	34, 37, 29, 46	infssuzcldc 11987	. . . . . . . . 9 |- ( ( ( ph /\ ( k e. ( 1 ... ( P - 1 ) ) /\ u e. ZZ[_i] ) ) /\ ( ( ( abs ` u ) ^ 2 ) + 1 ) = ( k x. P ) ) -> inf ( T , RR , < ) e. T )
48	33, 47	eqeltrid 2276	. . . . . . . 8 |- ( ( ( ph /\ ( k e. ( 1 ... ( P - 1 ) ) /\ u e. ZZ[_i] ) ) /\ ( ( ( abs ` u ) ^ 2 ) + 1 ) = ( k x. P ) ) -> M e. T )
49	32, 48	sselid 3168	. . . . . . 7 |- ( ( ( ph /\ ( k e. ( 1 ... ( P - 1 ) ) /\ u e. ZZ[_i] ) ) /\ ( ( ( abs ` u ) ^ 2 ) + 1 ) = ( k x. P ) ) -> M e. NN )
50	49	nnred 8963	. . . . . 6 |- ( ( ( ph /\ ( k e. ( 1 ... ( P - 1 ) ) /\ u e. ZZ[_i] ) ) /\ ( ( ( abs ` u ) ^ 2 ) + 1 ) = ( k x. P ) ) -> M e. RR )
51	10	nnred 8963	. . . . . 6 |- ( ( ( ph /\ ( k e. ( 1 ... ( P - 1 ) ) /\ u e. ZZ[_i] ) ) /\ ( ( ( abs ` u ) ^ 2 ) + 1 ) = ( k x. P ) ) -> k e. RR )
52	41	nnred 8963	. . . . . . 7 |- ( ph -> P e. RR )
53	52	ad2antrr 488	. . . . . 6 |- ( ( ( ph /\ ( k e. ( 1 ... ( P - 1 ) ) /\ u e. ZZ[_i] ) ) /\ ( ( ( abs ` u ) ^ 2 ) + 1 ) = ( k x. P ) ) -> P e. RR )
54	34, 37, 29, 46	infssuzledc 11986	. . . . . . 7 |- ( ( ( ph /\ ( k e. ( 1 ... ( P - 1 ) ) /\ u e. ZZ[_i] ) ) /\ ( ( ( abs ` u ) ^ 2 ) + 1 ) = ( k x. P ) ) -> inf ( T , RR , < ) <_ k )
55	33, 54	eqbrtrid 4053	. . . . . 6 |- ( ( ( ph /\ ( k e. ( 1 ... ( P - 1 ) ) /\ u e. ZZ[_i] ) ) /\ ( ( ( abs ` u ) ^ 2 ) + 1 ) = ( k x. P ) ) -> M <_ k )
56		prmz 12146	. . . . . . . . . . 11 |- ( P e. Prime -> P e. ZZ )
57	4, 56	syl 14	. . . . . . . . . 10 |- ( ph -> P e. ZZ )
58	57	ad2antrr 488	. . . . . . . . 9 |- ( ( ( ph /\ ( k e. ( 1 ... ( P - 1 ) ) /\ u e. ZZ[_i] ) ) /\ ( ( ( abs ` u ) ^ 2 ) + 1 ) = ( k x. P ) ) -> P e. ZZ )
59		elfzm11 10123	. . . . . . . . 9 |- ( ( 1 e. ZZ /\ P e. ZZ ) -> ( k e. ( 1 ... ( P - 1 ) ) <-> ( k e. ZZ /\ 1 <_ k /\ k < P ) ) )
60	18, 58, 59	sylancr 414	. . . . . . . 8 |- ( ( ( ph /\ ( k e. ( 1 ... ( P - 1 ) ) /\ u e. ZZ[_i] ) ) /\ ( ( ( abs ` u ) ^ 2 ) + 1 ) = ( k x. P ) ) -> ( k e. ( 1 ... ( P - 1 ) ) <-> ( k e. ZZ /\ 1 <_ k /\ k < P ) ) )
61	8, 60	mpbid 147	. . . . . . 7 |- ( ( ( ph /\ ( k e. ( 1 ... ( P - 1 ) ) /\ u e. ZZ[_i] ) ) /\ ( ( ( abs ` u ) ^ 2 ) + 1 ) = ( k x. P ) ) -> ( k e. ZZ /\ 1 <_ k /\ k < P ) )
62	61	simp3d 1013	. . . . . 6 |- ( ( ( ph /\ ( k e. ( 1 ... ( P - 1 ) ) /\ u e. ZZ[_i] ) ) /\ ( ( ( abs ` u ) ^ 2 ) + 1 ) = ( k x. P ) ) -> k < P )
63	50, 51, 53, 55, 62	lelttrd 8113	. . . . 5 |- ( ( ( ph /\ ( k e. ( 1 ... ( P - 1 ) ) /\ u e. ZZ[_i] ) ) /\ ( ( ( abs ` u ) ^ 2 ) + 1 ) = ( k x. P ) ) -> M < P )
64	31, 63	jca 306	. . . 4 |- ( ( ( ph /\ ( k e. ( 1 ... ( P - 1 ) ) /\ u e. ZZ[_i] ) ) /\ ( ( ( abs ` u ) ^ 2 ) + 1 ) = ( k x. P ) ) -> ( E. j j e. T /\ M < P ) )
65	64	ex 115	. . 3 |- ( ( ph /\ ( k e. ( 1 ... ( P - 1 ) ) /\ u e. ZZ[_i] ) ) -> ( ( ( ( abs ` u ) ^ 2 ) + 1 ) = ( k x. P ) -> ( E. j j e. T /\ M < P ) ) )
66	65	rexlimdvva 2615	. 2 |- ( ph -> ( E. k e. ( 1 ... ( P - 1 ) ) E. u e. ZZ[_i] ( ( ( abs ` u ) ^ 2 ) + 1 ) = ( k x. P ) -> ( E. j j e. T /\ M < P ) ) )
67	7, 66	mpd 13	1 |- ( ph -> ( E. j j e. T /\ M < P ) )

Syntax hints:   -> wi 4   /\ wa 104   <-> wb 105  DECID wdc 835   /\ w3a 980   = wceq 1364   E.wex 1503   e. wcel 2160   {cab 2175   E.wrex 2469   {crab 2472   C_ wss 3144   class class class wbr 4018   |-> cmpt 4079   ` cfv 5235  (class class class)co 5897  infcinf 7013   RRcr 7841   0cc0 7842   1c1 7843   + caddc 7845   x. cmul 7847   < clt 8023   <_ cle 8024   - cmin 8159   NNcn 8950   2c2 9001   NN0cn0 9207   ZZcz 9284   ZZ>=cuz 9559   ...cfz 10040   mod cmo 10355   ^cexp 10553   abscabs 11041   Primecprime 12142   ZZ[_i]cgz 12404

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-in1 615  ax-in2 616  ax-io 710  ax-5 1458  ax-7 1459  ax-gen 1460  ax-ie1 1504  ax-ie2 1505  ax-8 1515  ax-10 1516  ax-11 1517  ax-i12 1518  ax-bndl 1520  ax-4 1521  ax-17 1537  ax-i9 1541  ax-ial 1545  ax-i5r 1546  ax-13 2162  ax-14 2163  ax-ext 2171  ax-coll 4133  ax-sep 4136  ax-nul 4144  ax-pow 4192  ax-pr 4227  ax-un 4451  ax-setind 4554  ax-iinf 4605  ax-cnex 7933  ax-resscn 7934  ax-1cn 7935  ax-1re 7936  ax-icn 7937  ax-addcl 7938  ax-addrcl 7939  ax-mulcl 7940  ax-mulrcl 7941  ax-addcom 7942  ax-mulcom 7943  ax-addass 7944  ax-mulass 7945  ax-distr 7946  ax-i2m1 7947  ax-0lt1 7948  ax-1rid 7949  ax-0id 7950  ax-rnegex 7951  ax-precex 7952  ax-cnre 7953  ax-pre-ltirr 7954  ax-pre-ltwlin 7955  ax-pre-lttrn 7956  ax-pre-apti 7957  ax-pre-ltadd 7958  ax-pre-mulgt0 7959  ax-pre-mulext 7960  ax-arch 7961  ax-caucvg 7962

This theorem depends on definitions:  df-bi 117  df-stab 832  df-dc 836  df-3or 981  df-3an 982  df-tru 1367  df-fal 1370  df-nf 1472  df-sb 1774  df-eu 2041  df-mo 2042  df-clab 2176  df-cleq 2182  df-clel 2185  df-nfc 2321  df-ne 2361  df-nel 2456  df-ral 2473  df-rex 2474  df-reu 2475  df-rmo 2476  df-rab 2477  df-v 2754  df-sbc 2978  df-csb 3073  df-dif 3146  df-un 3148  df-in 3150  df-ss 3157  df-nul 3438  df-if 3550  df-pw 3592  df-sn 3613  df-pr 3614  df-op 3616  df-uni 3825  df-int 3860  df-iun 3903  df-br 4019  df-opab 4080  df-mpt 4081  df-tr 4117  df-id 4311  df-po 4314  df-iso 4315  df-iord 4384  df-on 4386  df-ilim 4387  df-suc 4389  df-iom 4608  df-xp 4650  df-rel 4651  df-cnv 4652  df-co 4653  df-dm 4654  df-rn 4655  df-res 4656  df-ima 4657  df-iota 5196  df-fun 5237  df-fn 5238  df-f 5239  df-f1 5240  df-fo 5241  df-f1o 5242  df-fv 5243  df-isom 5244  df-riota 5852  df-ov 5900  df-oprab 5901  df-mpo 5902  df-1st 6166  df-2nd 6167  df-recs 6331  df-irdg 6396  df-frec 6417  df-1o 6442  df-2o 6443  df-oadd 6446  df-er 6560  df-en 6768  df-dom 6769  df-fin 6770  df-sup 7014  df-inf 7015  df-pnf 8025  df-mnf 8026  df-xr 8027  df-ltxr 8028  df-le 8029  df-sub 8161  df-neg 8162  df-reap 8563  df-ap 8570  df-div 8661  df-inn 8951  df-2 9009  df-3 9010  df-4 9011  df-n0 9208  df-z 9285  df-uz 9560  df-q 9652  df-rp 9686  df-fz 10041  df-fzo 10175  df-fl 10303  df-mod 10356  df-seqfrec 10479  df-exp 10554  df-ihash 10791  df-cj 10886  df-re 10887  df-im 10888  df-rsqrt 11042  df-abs 11043  df-dvds 11830  df-gcd 11979  df-prm 12143  df-gz 12405

This theorem is referenced by:  4sqlem14  12439  4sqlem17  12442  4sqlem18  12443