Theorem 4sqlem13m 12841

Description: Lemma for 4sq 12848. (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 2207	. . 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 2207	. . 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 12840	. 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 10211	. . . . . . . 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 11498	. . . . . . . . . . . 12 |- ( abs ` 1 ) = 1
13	12	oveq1i 5977	. . . . . . . . . . 11 |- ( ( abs ` 1 ) ^ 2 ) = ( 1 ^ 2 )
14		sq1 10815	. . . . . . . . . . 11 |- ( 1 ^ 2 ) = 1
15	13, 14	eqtri 2228	. . . . . . . . . 10 |- ( ( abs ` 1 ) ^ 2 ) = 1
16	15	oveq2i 5978	. . . . . . . . 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 9433	. . . . . . . . . . 11 |- 1 e. ZZ
19		zgz 12811	. . . . . . . . . . 11 |- ( 1 e. ZZ -> 1 e. ZZ[_i] )
20	18, 19	ax-mp 5	. . . . . . . . . 10 |- 1 e. ZZ[_i]
21	1	4sqlem4a 12829	. . . . . . . . . 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 2295	. . . . . . . 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 2285	. . . . . . 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 5974	. . . . . . . . 9 |- ( i = k -> ( i x. P ) = ( k x. P ) )
26	25	eleq1d 2276	. . . . . . . 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 2939	. . . . . . 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 2793	. . . . . 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 3287	. . . . . . . 8 |- T C_ NN
33		4sq.7	. . . . . . . . 9 |- M = inf ( T , RR , < )
34		1zzd 9434	. . . . . . . . . 10 |- ( ( ( ph /\ ( k e. ( 1 ... ( P - 1 ) ) /\ u e. ZZ[_i] ) ) /\ ( ( ( abs ` u ) ^ 2 ) + 1 ) = ( k x. P ) ) -> 1 e. ZZ )
35		nnuz 9719	. . . . . . . . . . . 12 |- NN = ( ZZ>= ` 1 )
36	35	rabeqi 2769	. . . . . . . . . . 11 |- { i e. NN | ( i x. P ) e. S } = { i e. ( ZZ>= ` 1 ) | ( i x. P ) e. S }
37	27, 36	eqtri 2228	. . . . . . . . . 10 |- T = { i e. ( ZZ>= ` 1 ) | ( i x. P ) e. S }
38		elfznn 10211	. . . . . . . . . . . . . 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 12547	. . . . . . . . . . . . . . 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 9120	. . . . . . . . . . . 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 9383	. . . . . . . . . . 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 12838	. . . . . . . . . . 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 10415	. . . . . . . . 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 2294	. . . . . . . 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 3199	. . . . . . 7 |- ( ( ( ph /\ ( k e. ( 1 ... ( P - 1 ) ) /\ u e. ZZ[_i] ) ) /\ ( ( ( abs ` u ) ^ 2 ) + 1 ) = ( k x. P ) ) -> M e. NN )
50	49	nnred 9084	. . . . . 6 |- ( ( ( ph /\ ( k e. ( 1 ... ( P - 1 ) ) /\ u e. ZZ[_i] ) ) /\ ( ( ( abs ` u ) ^ 2 ) + 1 ) = ( k x. P ) ) -> M e. RR )
51	10	nnred 9084	. . . . . 6 |- ( ( ( ph /\ ( k e. ( 1 ... ( P - 1 ) ) /\ u e. ZZ[_i] ) ) /\ ( ( ( abs ` u ) ^ 2 ) + 1 ) = ( k x. P ) ) -> k e. RR )
52	41	nnred 9084	. . . . . . 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 10414	. . . . . . 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 4094	. . . . . 6 |- ( ( ( ph /\ ( k e. ( 1 ... ( P - 1 ) ) /\ u e. ZZ[_i] ) ) /\ ( ( ( abs ` u ) ^ 2 ) + 1 ) = ( k x. P ) ) -> M <_ k )
56		prmz 12548	. . . . . . . . . . 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 10248	. . . . . . . . 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 1014	. . . . . 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 8232	. . . . 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 2633	. 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 836   /\ w3a 981   = wceq 1373   E.wex 1516   e. wcel 2178   {cab 2193   E.wrex 2487   {crab 2490   C_ wss 3174   class class class wbr 4059   |-> cmpt 4121   ` cfv 5290  (class class class)co 5967  infcinf 7111   RRcr 7959   0cc0 7960   1c1 7961   + caddc 7963   x. cmul 7965   < clt 8142   <_ cle 8143   - cmin 8278   NNcn 9071   2c2 9122   NN0cn0 9330   ZZcz 9407   ZZ>=cuz 9683   ...cfz 10165   mod cmo 10504   ^cexp 10720   abscabs 11423   Primecprime 12544   ZZ[_i]cgz 12807

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 711  ax-5 1471  ax-7 1472  ax-gen 1473  ax-ie1 1517  ax-ie2 1518  ax-8 1528  ax-10 1529  ax-11 1530  ax-i12 1531  ax-bndl 1533  ax-4 1534  ax-17 1550  ax-i9 1554  ax-ial 1558  ax-i5r 1559  ax-13 2180  ax-14 2181  ax-ext 2189  ax-coll 4175  ax-sep 4178  ax-nul 4186  ax-pow 4234  ax-pr 4269  ax-un 4498  ax-setind 4603  ax-iinf 4654  ax-cnex 8051  ax-resscn 8052  ax-1cn 8053  ax-1re 8054  ax-icn 8055  ax-addcl 8056  ax-addrcl 8057  ax-mulcl 8058  ax-mulrcl 8059  ax-addcom 8060  ax-mulcom 8061  ax-addass 8062  ax-mulass 8063  ax-distr 8064  ax-i2m1 8065  ax-0lt1 8066  ax-1rid 8067  ax-0id 8068  ax-rnegex 8069  ax-precex 8070  ax-cnre 8071  ax-pre-ltirr 8072  ax-pre-ltwlin 8073  ax-pre-lttrn 8074  ax-pre-apti 8075  ax-pre-ltadd 8076  ax-pre-mulgt0 8077  ax-pre-mulext 8078  ax-arch 8079  ax-caucvg 8080

This theorem depends on definitions:  df-bi 117  df-stab 833  df-dc 837  df-3or 982  df-3an 983  df-tru 1376  df-fal 1379  df-nf 1485  df-sb 1787  df-eu 2058  df-mo 2059  df-clab 2194  df-cleq 2200  df-clel 2203  df-nfc 2339  df-ne 2379  df-nel 2474  df-ral 2491  df-rex 2492  df-reu 2493  df-rmo 2494  df-rab 2495  df-v 2778  df-sbc 3006  df-csb 3102  df-dif 3176  df-un 3178  df-in 3180  df-ss 3187  df-nul 3469  df-if 3580  df-pw 3628  df-sn 3649  df-pr 3650  df-op 3652  df-uni 3865  df-int 3900  df-iun 3943  df-br 4060  df-opab 4122  df-mpt 4123  df-tr 4159  df-id 4358  df-po 4361  df-iso 4362  df-iord 4431  df-on 4433  df-ilim 4434  df-suc 4436  df-iom 4657  df-xp 4699  df-rel 4700  df-cnv 4701  df-co 4702  df-dm 4703  df-rn 4704  df-res 4705  df-ima 4706  df-iota 5251  df-fun 5292  df-fn 5293  df-f 5294  df-f1 5295  df-fo 5296  df-f1o 5297  df-fv 5298  df-isom 5299  df-riota 5922  df-ov 5970  df-oprab 5971  df-mpo 5972  df-1st 6249  df-2nd 6250  df-recs 6414  df-irdg 6479  df-frec 6500  df-1o 6525  df-2o 6526  df-oadd 6529  df-er 6643  df-en 6851  df-dom 6852  df-fin 6853  df-sup 7112  df-inf 7113  df-pnf 8144  df-mnf 8145  df-xr 8146  df-ltxr 8147  df-le 8148  df-sub 8280  df-neg 8281  df-reap 8683  df-ap 8690  df-div 8781  df-inn 9072  df-2 9130  df-3 9131  df-4 9132  df-n0 9331  df-z 9408  df-uz 9684  df-q 9776  df-rp 9811  df-fz 10166  df-fzo 10300  df-fl 10450  df-mod 10505  df-seqfrec 10630  df-exp 10721  df-ihash 10958  df-cj 11268  df-re 11269  df-im 11270  df-rsqrt 11424  df-abs 11425  df-dvds 12214  df-gcd 12390  df-prm 12545  df-gz 12808

This theorem is referenced by:  4sqlem14  12842  4sqlem17  12845  4sqlem18  12846