Theorem 4sqlem13m 12966

Description: Lemma for 4sq 12973. (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 2229	. . 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 2229	. . 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 12965	. 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 10279	. . . . . . . 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 11623	. . . . . . . . . . . 12 |- ( abs ` 1 ) = 1
13	12	oveq1i 6023	. . . . . . . . . . 11 |- ( ( abs ` 1 ) ^ 2 ) = ( 1 ^ 2 )
14		sq1 10885	. . . . . . . . . . 11 |- ( 1 ^ 2 ) = 1
15	13, 14	eqtri 2250	. . . . . . . . . 10 |- ( ( abs ` 1 ) ^ 2 ) = 1
16	15	oveq2i 6024	. . . . . . . . 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 9495	. . . . . . . . . . 11 |- 1 e. ZZ
19		zgz 12936	. . . . . . . . . . 11 |- ( 1 e. ZZ -> 1 e. ZZ[_i] )
20	18, 19	ax-mp 5	. . . . . . . . . 10 |- 1 e. ZZ[_i]
21	1	4sqlem4a 12954	. . . . . . . . . 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 2317	. . . . . . . 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 2307	. . . . . . 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 6020	. . . . . . . . 9 |- ( i = k -> ( i x. P ) = ( k x. P ) )
26	25	eleq1d 2298	. . . . . . . 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 2963	. . . . . . 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 2817	. . . . . 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 3311	. . . . . . . 8 |- T C_ NN
33		4sq.7	. . . . . . . . 9 |- M = inf ( T , RR , < )
34		1zzd 9496	. . . . . . . . . 10 |- ( ( ( ph /\ ( k e. ( 1 ... ( P - 1 ) ) /\ u e. ZZ[_i] ) ) /\ ( ( ( abs ` u ) ^ 2 ) + 1 ) = ( k x. P ) ) -> 1 e. ZZ )
35		nnuz 9782	. . . . . . . . . . . 12 |- NN = ( ZZ>= ` 1 )
36	35	rabeqi 2793	. . . . . . . . . . 11 |- { i e. NN | ( i x. P ) e. S } = { i e. ( ZZ>= ` 1 ) | ( i x. P ) e. S }
37	27, 36	eqtri 2250	. . . . . . . . . 10 |- T = { i e. ( ZZ>= ` 1 ) | ( i x. P ) e. S }
38		elfznn 10279	. . . . . . . . . . . . . 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 12672	. . . . . . . . . . . . . . 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 9182	. . . . . . . . . . . 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 9445	. . . . . . . . . . 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 12963	. . . . . . . . . . 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 10485	. . . . . . . . 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 2316	. . . . . . . 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 3223	. . . . . . 7 |- ( ( ( ph /\ ( k e. ( 1 ... ( P - 1 ) ) /\ u e. ZZ[_i] ) ) /\ ( ( ( abs ` u ) ^ 2 ) + 1 ) = ( k x. P ) ) -> M e. NN )
50	49	nnred 9146	. . . . . 6 |- ( ( ( ph /\ ( k e. ( 1 ... ( P - 1 ) ) /\ u e. ZZ[_i] ) ) /\ ( ( ( abs ` u ) ^ 2 ) + 1 ) = ( k x. P ) ) -> M e. RR )
51	10	nnred 9146	. . . . . 6 |- ( ( ( ph /\ ( k e. ( 1 ... ( P - 1 ) ) /\ u e. ZZ[_i] ) ) /\ ( ( ( abs ` u ) ^ 2 ) + 1 ) = ( k x. P ) ) -> k e. RR )
52	41	nnred 9146	. . . . . . 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 10484	. . . . . . 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 4121	. . . . . 6 |- ( ( ( ph /\ ( k e. ( 1 ... ( P - 1 ) ) /\ u e. ZZ[_i] ) ) /\ ( ( ( abs ` u ) ^ 2 ) + 1 ) = ( k x. P ) ) -> M <_ k )
56		prmz 12673	. . . . . . . . . . 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 10316	. . . . . . . . 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 1035	. . . . . 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 8294	. . . . 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 2656	. 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 839   /\ w3a 1002   = wceq 1395   E.wex 1538   e. wcel 2200   {cab 2215   E.wrex 2509   {crab 2512   C_ wss 3198   class class class wbr 4086   |-> cmpt 4148   ` cfv 5324  (class class class)co 6013  infcinf 7173   RRcr 8021   0cc0 8022   1c1 8023   + caddc 8025   x. cmul 8027   < clt 8204   <_ cle 8205   - cmin 8340   NNcn 9133   2c2 9184   NN0cn0 9392   ZZcz 9469   ZZ>=cuz 9745   ...cfz 10233   mod cmo 10574   ^cexp 10790   abscabs 11548   Primecprime 12669   ZZ[_i]cgz 12932

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 617  ax-in2 618  ax-io 714  ax-5 1493  ax-7 1494  ax-gen 1495  ax-ie1 1539  ax-ie2 1540  ax-8 1550  ax-10 1551  ax-11 1552  ax-i12 1553  ax-bndl 1555  ax-4 1556  ax-17 1572  ax-i9 1576  ax-ial 1580  ax-i5r 1581  ax-13 2202  ax-14 2203  ax-ext 2211  ax-coll 4202  ax-sep 4205  ax-nul 4213  ax-pow 4262  ax-pr 4297  ax-un 4528  ax-setind 4633  ax-iinf 4684  ax-cnex 8113  ax-resscn 8114  ax-1cn 8115  ax-1re 8116  ax-icn 8117  ax-addcl 8118  ax-addrcl 8119  ax-mulcl 8120  ax-mulrcl 8121  ax-addcom 8122  ax-mulcom 8123  ax-addass 8124  ax-mulass 8125  ax-distr 8126  ax-i2m1 8127  ax-0lt1 8128  ax-1rid 8129  ax-0id 8130  ax-rnegex 8131  ax-precex 8132  ax-cnre 8133  ax-pre-ltirr 8134  ax-pre-ltwlin 8135  ax-pre-lttrn 8136  ax-pre-apti 8137  ax-pre-ltadd 8138  ax-pre-mulgt0 8139  ax-pre-mulext 8140  ax-arch 8141  ax-caucvg 8142

This theorem depends on definitions:  df-bi 117  df-stab 836  df-dc 840  df-3or 1003  df-3an 1004  df-tru 1398  df-fal 1401  df-nf 1507  df-sb 1809  df-eu 2080  df-mo 2081  df-clab 2216  df-cleq 2222  df-clel 2225  df-nfc 2361  df-ne 2401  df-nel 2496  df-ral 2513  df-rex 2514  df-reu 2515  df-rmo 2516  df-rab 2517  df-v 2802  df-sbc 3030  df-csb 3126  df-dif 3200  df-un 3202  df-in 3204  df-ss 3211  df-nul 3493  df-if 3604  df-pw 3652  df-sn 3673  df-pr 3674  df-op 3676  df-uni 3892  df-int 3927  df-iun 3970  df-br 4087  df-opab 4149  df-mpt 4150  df-tr 4186  df-id 4388  df-po 4391  df-iso 4392  df-iord 4461  df-on 4463  df-ilim 4464  df-suc 4466  df-iom 4687  df-xp 4729  df-rel 4730  df-cnv 4731  df-co 4732  df-dm 4733  df-rn 4734  df-res 4735  df-ima 4736  df-iota 5284  df-fun 5326  df-fn 5327  df-f 5328  df-f1 5329  df-fo 5330  df-f1o 5331  df-fv 5332  df-isom 5333  df-riota 5966  df-ov 6016  df-oprab 6017  df-mpo 6018  df-1st 6298  df-2nd 6299  df-recs 6466  df-irdg 6531  df-frec 6552  df-1o 6577  df-2o 6578  df-oadd 6581  df-er 6697  df-en 6905  df-dom 6906  df-fin 6907  df-sup 7174  df-inf 7175  df-pnf 8206  df-mnf 8207  df-xr 8208  df-ltxr 8209  df-le 8210  df-sub 8342  df-neg 8343  df-reap 8745  df-ap 8752  df-div 8843  df-inn 9134  df-2 9192  df-3 9193  df-4 9194  df-n0 9393  df-z 9470  df-uz 9746  df-q 9844  df-rp 9879  df-fz 10234  df-fzo 10368  df-fl 10520  df-mod 10575  df-seqfrec 10700  df-exp 10791  df-ihash 11028  df-cj 11393  df-re 11394  df-im 11395  df-rsqrt 11549  df-abs 11550  df-dvds 12339  df-gcd 12515  df-prm 12670  df-gz 12933

This theorem is referenced by:  4sqlem14  12967  4sqlem17  12970  4sqlem18  12971