Theorem 4sqlem13m 12572

Description: Lemma for 4sq 12579. (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 2196	. . 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 2196	. . 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 12571	. 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 10129	. . . . . . . 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 11237	. . . . . . . . . . . 12 |- ( abs ` 1 ) = 1
13	12	oveq1i 5932	. . . . . . . . . . 11 |- ( ( abs ` 1 ) ^ 2 ) = ( 1 ^ 2 )
14		sq1 10725	. . . . . . . . . . 11 |- ( 1 ^ 2 ) = 1
15	13, 14	eqtri 2217	. . . . . . . . . 10 |- ( ( abs ` 1 ) ^ 2 ) = 1
16	15	oveq2i 5933	. . . . . . . . 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 9352	. . . . . . . . . . 11 |- 1 e. ZZ
19		zgz 12542	. . . . . . . . . . 11 |- ( 1 e. ZZ -> 1 e. ZZ[_i] )
20	18, 19	ax-mp 5	. . . . . . . . . 10 |- 1 e. ZZ[_i]
21	1	4sqlem4a 12560	. . . . . . . . . 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 2284	. . . . . . . 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 2274	. . . . . . 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 5929	. . . . . . . . 9 |- ( i = k -> ( i x. P ) = ( k x. P ) )
26	25	eleq1d 2265	. . . . . . . 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 2923	. . . . . . 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 2779	. . . . . 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 3269	. . . . . . . 8 |- T C_ NN
33		4sq.7	. . . . . . . . 9 |- M = inf ( T , RR , < )
34		1zzd 9353	. . . . . . . . . 10 |- ( ( ( ph /\ ( k e. ( 1 ... ( P - 1 ) ) /\ u e. ZZ[_i] ) ) /\ ( ( ( abs ` u ) ^ 2 ) + 1 ) = ( k x. P ) ) -> 1 e. ZZ )
35		nnuz 9637	. . . . . . . . . . . 12 |- NN = ( ZZ>= ` 1 )
36	35	rabeqi 2756	. . . . . . . . . . 11 |- { i e. NN | ( i x. P ) e. S } = { i e. ( ZZ>= ` 1 ) | ( i x. P ) e. S }
37	27, 36	eqtri 2217	. . . . . . . . . 10 |- T = { i e. ( ZZ>= ` 1 ) | ( i x. P ) e. S }
38		elfznn 10129	. . . . . . . . . . . . . 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 12278	. . . . . . . . . . . . . . 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 9039	. . . . . . . . . . . 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 9302	. . . . . . . . . . 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 12569	. . . . . . . . . . 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 10325	. . . . . . . . 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 2283	. . . . . . . 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 3181	. . . . . . 7 |- ( ( ( ph /\ ( k e. ( 1 ... ( P - 1 ) ) /\ u e. ZZ[_i] ) ) /\ ( ( ( abs ` u ) ^ 2 ) + 1 ) = ( k x. P ) ) -> M e. NN )
50	49	nnred 9003	. . . . . 6 |- ( ( ( ph /\ ( k e. ( 1 ... ( P - 1 ) ) /\ u e. ZZ[_i] ) ) /\ ( ( ( abs ` u ) ^ 2 ) + 1 ) = ( k x. P ) ) -> M e. RR )
51	10	nnred 9003	. . . . . 6 |- ( ( ( ph /\ ( k e. ( 1 ... ( P - 1 ) ) /\ u e. ZZ[_i] ) ) /\ ( ( ( abs ` u ) ^ 2 ) + 1 ) = ( k x. P ) ) -> k e. RR )
52	41	nnred 9003	. . . . . . 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 10324	. . . . . . 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 4068	. . . . . 6 |- ( ( ( ph /\ ( k e. ( 1 ... ( P - 1 ) ) /\ u e. ZZ[_i] ) ) /\ ( ( ( abs ` u ) ^ 2 ) + 1 ) = ( k x. P ) ) -> M <_ k )
56		prmz 12279	. . . . . . . . . . 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 10166	. . . . . . . . 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 8151	. . . . 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 2622	. 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 1506   e. wcel 2167   {cab 2182   E.wrex 2476   {crab 2479   C_ wss 3157   class class class wbr 4033   |-> cmpt 4094   ` cfv 5258  (class class class)co 5922  infcinf 7049   RRcr 7878   0cc0 7879   1c1 7880   + caddc 7882   x. cmul 7884   < clt 8061   <_ cle 8062   - cmin 8197   NNcn 8990   2c2 9041   NN0cn0 9249   ZZcz 9326   ZZ>=cuz 9601   ...cfz 10083   mod cmo 10414   ^cexp 10630   abscabs 11162   Primecprime 12275   ZZ[_i]cgz 12538

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 1461  ax-7 1462  ax-gen 1463  ax-ie1 1507  ax-ie2 1508  ax-8 1518  ax-10 1519  ax-11 1520  ax-i12 1521  ax-bndl 1523  ax-4 1524  ax-17 1540  ax-i9 1544  ax-ial 1548  ax-i5r 1549  ax-13 2169  ax-14 2170  ax-ext 2178  ax-coll 4148  ax-sep 4151  ax-nul 4159  ax-pow 4207  ax-pr 4242  ax-un 4468  ax-setind 4573  ax-iinf 4624  ax-cnex 7970  ax-resscn 7971  ax-1cn 7972  ax-1re 7973  ax-icn 7974  ax-addcl 7975  ax-addrcl 7976  ax-mulcl 7977  ax-mulrcl 7978  ax-addcom 7979  ax-mulcom 7980  ax-addass 7981  ax-mulass 7982  ax-distr 7983  ax-i2m1 7984  ax-0lt1 7985  ax-1rid 7986  ax-0id 7987  ax-rnegex 7988  ax-precex 7989  ax-cnre 7990  ax-pre-ltirr 7991  ax-pre-ltwlin 7992  ax-pre-lttrn 7993  ax-pre-apti 7994  ax-pre-ltadd 7995  ax-pre-mulgt0 7996  ax-pre-mulext 7997  ax-arch 7998  ax-caucvg 7999

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 1475  df-sb 1777  df-eu 2048  df-mo 2049  df-clab 2183  df-cleq 2189  df-clel 2192  df-nfc 2328  df-ne 2368  df-nel 2463  df-ral 2480  df-rex 2481  df-reu 2482  df-rmo 2483  df-rab 2484  df-v 2765  df-sbc 2990  df-csb 3085  df-dif 3159  df-un 3161  df-in 3163  df-ss 3170  df-nul 3451  df-if 3562  df-pw 3607  df-sn 3628  df-pr 3629  df-op 3631  df-uni 3840  df-int 3875  df-iun 3918  df-br 4034  df-opab 4095  df-mpt 4096  df-tr 4132  df-id 4328  df-po 4331  df-iso 4332  df-iord 4401  df-on 4403  df-ilim 4404  df-suc 4406  df-iom 4627  df-xp 4669  df-rel 4670  df-cnv 4671  df-co 4672  df-dm 4673  df-rn 4674  df-res 4675  df-ima 4676  df-iota 5219  df-fun 5260  df-fn 5261  df-f 5262  df-f1 5263  df-fo 5264  df-f1o 5265  df-fv 5266  df-isom 5267  df-riota 5877  df-ov 5925  df-oprab 5926  df-mpo 5927  df-1st 6198  df-2nd 6199  df-recs 6363  df-irdg 6428  df-frec 6449  df-1o 6474  df-2o 6475  df-oadd 6478  df-er 6592  df-en 6800  df-dom 6801  df-fin 6802  df-sup 7050  df-inf 7051  df-pnf 8063  df-mnf 8064  df-xr 8065  df-ltxr 8066  df-le 8067  df-sub 8199  df-neg 8200  df-reap 8602  df-ap 8609  df-div 8700  df-inn 8991  df-2 9049  df-3 9050  df-4 9051  df-n0 9250  df-z 9327  df-uz 9602  df-q 9694  df-rp 9729  df-fz 10084  df-fzo 10218  df-fl 10360  df-mod 10415  df-seqfrec 10540  df-exp 10631  df-ihash 10868  df-cj 11007  df-re 11008  df-im 11009  df-rsqrt 11163  df-abs 11164  df-dvds 11953  df-gcd 12121  df-prm 12276  df-gz 12539

This theorem is referenced by:  4sqlem14  12573  4sqlem17  12576  4sqlem18  12577