Theorem 4sqlem13m 12994

Description: Lemma for 4sq 13001. (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 2231	. . 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 2231	. . 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 12993	. 2 |- ( ph -> E. k e. ( 1 ... ( P - 1 ) ) E. u e. ZZ[_i] ( ( ( abs ` u ) ^ 2 ) + 1 ) = ( k x. P ) )
8		simplrl 537	. . . . . . . 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 10289	. . . . . . . 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 11650	. . . . . . . . . . . 12 |- ( abs ` 1 ) = 1
13	12	oveq1i 6028	. . . . . . . . . . 11 |- ( ( abs ` 1 ) ^ 2 ) = ( 1 ^ 2 )
14		sq1 10896	. . . . . . . . . . 11 |- ( 1 ^ 2 ) = 1
15	13, 14	eqtri 2252	. . . . . . . . . 10 |- ( ( abs ` 1 ) ^ 2 ) = 1
16	15	oveq2i 6029	. . . . . . . . 9 |- ( ( ( abs ` u ) ^ 2 ) + ( ( abs ` 1 ) ^ 2 ) ) = ( ( ( abs ` u ) ^ 2 ) + 1 )
17		simplrr 538	. . . . . . . . . 10 |- ( ( ( ph /\ ( k e. ( 1 ... ( P - 1 ) ) /\ u e. ZZ[_i] ) ) /\ ( ( ( abs ` u ) ^ 2 ) + 1 ) = ( k x. P ) ) -> u e. ZZ[_i] )
18		1z 9505	. . . . . . . . . . 11 |- 1 e. ZZ
19		zgz 12964	. . . . . . . . . . 11 |- ( 1 e. ZZ -> 1 e. ZZ[_i] )
20	18, 19	ax-mp 5	. . . . . . . . . 10 |- 1 e. ZZ[_i]
21	1	4sqlem4a 12982	. . . . . . . . . 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 2319	. . . . . . . 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 2309	. . . . . . 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 6025	. . . . . . . . 9 |- ( i = k -> ( i x. P ) = ( k x. P ) )
26	25	eleq1d 2300	. . . . . . . 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 2965	. . . . . . 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 2819	. . . . . 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 3313	. . . . . . . 8 |- T C_ NN
33		4sq.7	. . . . . . . . 9 |- M = inf ( T , RR , < )
34		1zzd 9506	. . . . . . . . . 10 |- ( ( ( ph /\ ( k e. ( 1 ... ( P - 1 ) ) /\ u e. ZZ[_i] ) ) /\ ( ( ( abs ` u ) ^ 2 ) + 1 ) = ( k x. P ) ) -> 1 e. ZZ )
35		nnuz 9792	. . . . . . . . . . . 12 |- NN = ( ZZ>= ` 1 )
36	35	rabeqi 2795	. . . . . . . . . . 11 |- { i e. NN | ( i x. P ) e. S } = { i e. ( ZZ>= ` 1 ) | ( i x. P ) e. S }
37	27, 36	eqtri 2252	. . . . . . . . . 10 |- T = { i e. ( ZZ>= ` 1 ) | ( i x. P ) e. S }
38		elfznn 10289	. . . . . . . . . . . . . 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 12700	. . . . . . . . . . . . . . 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 9192	. . . . . . . . . . . 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 9455	. . . . . . . . . . 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 12991	. . . . . . . . . . 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 10496	. . . . . . . . 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 2318	. . . . . . . 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 3225	. . . . . . 7 |- ( ( ( ph /\ ( k e. ( 1 ... ( P - 1 ) ) /\ u e. ZZ[_i] ) ) /\ ( ( ( abs ` u ) ^ 2 ) + 1 ) = ( k x. P ) ) -> M e. NN )
50	49	nnred 9156	. . . . . 6 |- ( ( ( ph /\ ( k e. ( 1 ... ( P - 1 ) ) /\ u e. ZZ[_i] ) ) /\ ( ( ( abs ` u ) ^ 2 ) + 1 ) = ( k x. P ) ) -> M e. RR )
51	10	nnred 9156	. . . . . 6 |- ( ( ( ph /\ ( k e. ( 1 ... ( P - 1 ) ) /\ u e. ZZ[_i] ) ) /\ ( ( ( abs ` u ) ^ 2 ) + 1 ) = ( k x. P ) ) -> k e. RR )
52	41	nnred 9156	. . . . . . 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 10495	. . . . . . 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 4123	. . . . . 6 |- ( ( ( ph /\ ( k e. ( 1 ... ( P - 1 ) ) /\ u e. ZZ[_i] ) ) /\ ( ( ( abs ` u ) ^ 2 ) + 1 ) = ( k x. P ) ) -> M <_ k )
56		prmz 12701	. . . . . . . . . . 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 10326	. . . . . . . . 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 1037	. . . . . 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 8304	. . . . 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 2658	. 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 841   /\ w3a 1004   = wceq 1397   E.wex 1540   e. wcel 2202   {cab 2217   E.wrex 2511   {crab 2514   C_ wss 3200   class class class wbr 4088   |-> cmpt 4150   ` cfv 5326  (class class class)co 6018  infcinf 7182   RRcr 8031   0cc0 8032   1c1 8033   + caddc 8035   x. cmul 8037   < clt 8214   <_ cle 8215   - cmin 8350   NNcn 9143   2c2 9194   NN0cn0 9402   ZZcz 9479   ZZ>=cuz 9755   ...cfz 10243   mod cmo 10585   ^cexp 10801   abscabs 11575   Primecprime 12697   ZZ[_i]cgz 12960

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 619  ax-in2 620  ax-io 716  ax-5 1495  ax-7 1496  ax-gen 1497  ax-ie1 1541  ax-ie2 1542  ax-8 1552  ax-10 1553  ax-11 1554  ax-i12 1555  ax-bndl 1557  ax-4 1558  ax-17 1574  ax-i9 1578  ax-ial 1582  ax-i5r 1583  ax-13 2204  ax-14 2205  ax-ext 2213  ax-coll 4204  ax-sep 4207  ax-nul 4215  ax-pow 4264  ax-pr 4299  ax-un 4530  ax-setind 4635  ax-iinf 4686  ax-cnex 8123  ax-resscn 8124  ax-1cn 8125  ax-1re 8126  ax-icn 8127  ax-addcl 8128  ax-addrcl 8129  ax-mulcl 8130  ax-mulrcl 8131  ax-addcom 8132  ax-mulcom 8133  ax-addass 8134  ax-mulass 8135  ax-distr 8136  ax-i2m1 8137  ax-0lt1 8138  ax-1rid 8139  ax-0id 8140  ax-rnegex 8141  ax-precex 8142  ax-cnre 8143  ax-pre-ltirr 8144  ax-pre-ltwlin 8145  ax-pre-lttrn 8146  ax-pre-apti 8147  ax-pre-ltadd 8148  ax-pre-mulgt0 8149  ax-pre-mulext 8150  ax-arch 8151  ax-caucvg 8152

This theorem depends on definitions:  df-bi 117  df-stab 838  df-dc 842  df-3or 1005  df-3an 1006  df-tru 1400  df-fal 1403  df-nf 1509  df-sb 1811  df-eu 2082  df-mo 2083  df-clab 2218  df-cleq 2224  df-clel 2227  df-nfc 2363  df-ne 2403  df-nel 2498  df-ral 2515  df-rex 2516  df-reu 2517  df-rmo 2518  df-rab 2519  df-v 2804  df-sbc 3032  df-csb 3128  df-dif 3202  df-un 3204  df-in 3206  df-ss 3213  df-nul 3495  df-if 3606  df-pw 3654  df-sn 3675  df-pr 3676  df-op 3678  df-uni 3894  df-int 3929  df-iun 3972  df-br 4089  df-opab 4151  df-mpt 4152  df-tr 4188  df-id 4390  df-po 4393  df-iso 4394  df-iord 4463  df-on 4465  df-ilim 4466  df-suc 4468  df-iom 4689  df-xp 4731  df-rel 4732  df-cnv 4733  df-co 4734  df-dm 4735  df-rn 4736  df-res 4737  df-ima 4738  df-iota 5286  df-fun 5328  df-fn 5329  df-f 5330  df-f1 5331  df-fo 5332  df-f1o 5333  df-fv 5334  df-isom 5335  df-riota 5971  df-ov 6021  df-oprab 6022  df-mpo 6023  df-1st 6303  df-2nd 6304  df-recs 6471  df-irdg 6536  df-frec 6557  df-1o 6582  df-2o 6583  df-oadd 6586  df-er 6702  df-en 6910  df-dom 6911  df-fin 6912  df-sup 7183  df-inf 7184  df-pnf 8216  df-mnf 8217  df-xr 8218  df-ltxr 8219  df-le 8220  df-sub 8352  df-neg 8353  df-reap 8755  df-ap 8762  df-div 8853  df-inn 9144  df-2 9202  df-3 9203  df-4 9204  df-n0 9403  df-z 9480  df-uz 9756  df-q 9854  df-rp 9889  df-fz 10244  df-fzo 10378  df-fl 10531  df-mod 10586  df-seqfrec 10711  df-exp 10802  df-ihash 11039  df-cj 11420  df-re 11421  df-im 11422  df-rsqrt 11576  df-abs 11577  df-dvds 12367  df-gcd 12543  df-prm 12698  df-gz 12961

This theorem is referenced by:  4sqlem14  12995  4sqlem17  12998  4sqlem18  12999