Theorem 2sqlem5 15276

Description: Lemma for 2sq . If a number that is a sum of two squares is divisible by a prime that is a sum of two squares, then the quotient is a sum of two squares. (Contributed by Mario Carneiro, 20-Jun-2015.)

Hypotheses

Ref	Expression
2sq.1	|- S = ran ( w e. ZZ[_i] |-> ( ( abs ` w ) ^ 2 ) )
2sqlem5.1	|- ( ph -> N e. NN )
2sqlem5.2	|- ( ph -> P e. Prime )
2sqlem5.3	|- ( ph -> ( N x. P ) e. S )
2sqlem5.4	|- ( ph -> P e. S )

Assertion

Ref	Expression
2sqlem5	|- ( ph -> N e. S )

Proof of Theorem 2sqlem5

Dummy variables p q x y are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		2sqlem5.4	. . 3 |- ( ph -> P e. S )
2		2sq.1	. . . 4 |- S = ran ( w e. ZZ[_i] |-> ( ( abs ` w ) ^ 2 ) )
3	2	2sqlem2 15272	. . 3 |- ( P e. S <-> E. p e. ZZ E. q e. ZZ P = ( ( p ^ 2 ) + ( q ^ 2 ) ) )
4	1, 3	sylib 122	. 2 |- ( ph -> E. p e. ZZ E. q e. ZZ P = ( ( p ^ 2 ) + ( q ^ 2 ) ) )
5		2sqlem5.3	. . 3 |- ( ph -> ( N x. P ) e. S )
6	2	2sqlem2 15272	. . 3 |- ( ( N x. P ) e. S <-> E. x e. ZZ E. y e. ZZ ( N x. P ) = ( ( x ^ 2 ) + ( y ^ 2 ) ) )
7	5, 6	sylib 122	. 2 |- ( ph -> E. x e. ZZ E. y e. ZZ ( N x. P ) = ( ( x ^ 2 ) + ( y ^ 2 ) ) )
8		reeanv 2664	. . 3 |- ( E. p e. ZZ E. x e. ZZ ( E. q e. ZZ P = ( ( p ^ 2 ) + ( q ^ 2 ) ) /\ E. y e. ZZ ( N x. P ) = ( ( x ^ 2 ) + ( y ^ 2 ) ) ) <-> ( E. p e. ZZ E. q e. ZZ P = ( ( p ^ 2 ) + ( q ^ 2 ) ) /\ E. x e. ZZ E. y e. ZZ ( N x. P ) = ( ( x ^ 2 ) + ( y ^ 2 ) ) ) )
9		reeanv 2664	. . . . 5 |- ( E. q e. ZZ E. y e. ZZ ( P = ( ( p ^ 2 ) + ( q ^ 2 ) ) /\ ( N x. P ) = ( ( x ^ 2 ) + ( y ^ 2 ) ) ) <-> ( E. q e. ZZ P = ( ( p ^ 2 ) + ( q ^ 2 ) ) /\ E. y e. ZZ ( N x. P ) = ( ( x ^ 2 ) + ( y ^ 2 ) ) ) )
10		2sqlem5.1	. . . . . . . . 9 |- ( ph -> N e. NN )
11	10	ad2antrr 488	. . . . . . . 8 |- ( ( ( ph /\ ( p e. ZZ /\ x e. ZZ ) ) /\ ( ( q e. ZZ /\ y e. ZZ ) /\ ( P = ( ( p ^ 2 ) + ( q ^ 2 ) ) /\ ( N x. P ) = ( ( x ^ 2 ) + ( y ^ 2 ) ) ) ) ) -> N e. NN )
12		2sqlem5.2	. . . . . . . . 9 |- ( ph -> P e. Prime )
13	12	ad2antrr 488	. . . . . . . 8 |- ( ( ( ph /\ ( p e. ZZ /\ x e. ZZ ) ) /\ ( ( q e. ZZ /\ y e. ZZ ) /\ ( P = ( ( p ^ 2 ) + ( q ^ 2 ) ) /\ ( N x. P ) = ( ( x ^ 2 ) + ( y ^ 2 ) ) ) ) ) -> P e. Prime )
14		simplrr 536	. . . . . . . 8 |- ( ( ( ph /\ ( p e. ZZ /\ x e. ZZ ) ) /\ ( ( q e. ZZ /\ y e. ZZ ) /\ ( P = ( ( p ^ 2 ) + ( q ^ 2 ) ) /\ ( N x. P ) = ( ( x ^ 2 ) + ( y ^ 2 ) ) ) ) ) -> x e. ZZ )
15		simprlr 538	. . . . . . . 8 |- ( ( ( ph /\ ( p e. ZZ /\ x e. ZZ ) ) /\ ( ( q e. ZZ /\ y e. ZZ ) /\ ( P = ( ( p ^ 2 ) + ( q ^ 2 ) ) /\ ( N x. P ) = ( ( x ^ 2 ) + ( y ^ 2 ) ) ) ) ) -> y e. ZZ )
16		simplrl 535	. . . . . . . 8 |- ( ( ( ph /\ ( p e. ZZ /\ x e. ZZ ) ) /\ ( ( q e. ZZ /\ y e. ZZ ) /\ ( P = ( ( p ^ 2 ) + ( q ^ 2 ) ) /\ ( N x. P ) = ( ( x ^ 2 ) + ( y ^ 2 ) ) ) ) ) -> p e. ZZ )
17		simprll 537	. . . . . . . 8 |- ( ( ( ph /\ ( p e. ZZ /\ x e. ZZ ) ) /\ ( ( q e. ZZ /\ y e. ZZ ) /\ ( P = ( ( p ^ 2 ) + ( q ^ 2 ) ) /\ ( N x. P ) = ( ( x ^ 2 ) + ( y ^ 2 ) ) ) ) ) -> q e. ZZ )
18		simprrr 540	. . . . . . . 8 |- ( ( ( ph /\ ( p e. ZZ /\ x e. ZZ ) ) /\ ( ( q e. ZZ /\ y e. ZZ ) /\ ( P = ( ( p ^ 2 ) + ( q ^ 2 ) ) /\ ( N x. P ) = ( ( x ^ 2 ) + ( y ^ 2 ) ) ) ) ) -> ( N x. P ) = ( ( x ^ 2 ) + ( y ^ 2 ) ) )
19		simprrl 539	. . . . . . . 8 |- ( ( ( ph /\ ( p e. ZZ /\ x e. ZZ ) ) /\ ( ( q e. ZZ /\ y e. ZZ ) /\ ( P = ( ( p ^ 2 ) + ( q ^ 2 ) ) /\ ( N x. P ) = ( ( x ^ 2 ) + ( y ^ 2 ) ) ) ) ) -> P = ( ( p ^ 2 ) + ( q ^ 2 ) ) )
20	2, 11, 13, 14, 15, 16, 17, 18, 19	2sqlem4 15275	. . . . . . 7 |- ( ( ( ph /\ ( p e. ZZ /\ x e. ZZ ) ) /\ ( ( q e. ZZ /\ y e. ZZ ) /\ ( P = ( ( p ^ 2 ) + ( q ^ 2 ) ) /\ ( N x. P ) = ( ( x ^ 2 ) + ( y ^ 2 ) ) ) ) ) -> N e. S )
21	20	expr 375	. . . . . 6 |- ( ( ( ph /\ ( p e. ZZ /\ x e. ZZ ) ) /\ ( q e. ZZ /\ y e. ZZ ) ) -> ( ( P = ( ( p ^ 2 ) + ( q ^ 2 ) ) /\ ( N x. P ) = ( ( x ^ 2 ) + ( y ^ 2 ) ) ) -> N e. S ) )
22	21	rexlimdvva 2619	. . . . 5 |- ( ( ph /\ ( p e. ZZ /\ x e. ZZ ) ) -> ( E. q e. ZZ E. y e. ZZ ( P = ( ( p ^ 2 ) + ( q ^ 2 ) ) /\ ( N x. P ) = ( ( x ^ 2 ) + ( y ^ 2 ) ) ) -> N e. S ) )
23	9, 22	biimtrrid 153	. . . 4 |- ( ( ph /\ ( p e. ZZ /\ x e. ZZ ) ) -> ( ( E. q e. ZZ P = ( ( p ^ 2 ) + ( q ^ 2 ) ) /\ E. y e. ZZ ( N x. P ) = ( ( x ^ 2 ) + ( y ^ 2 ) ) ) -> N e. S ) )
24	23	rexlimdvva 2619	. . 3 |- ( ph -> ( E. p e. ZZ E. x e. ZZ ( E. q e. ZZ P = ( ( p ^ 2 ) + ( q ^ 2 ) ) /\ E. y e. ZZ ( N x. P ) = ( ( x ^ 2 ) + ( y ^ 2 ) ) ) -> N e. S ) )
25	8, 24	biimtrrid 153	. 2 |- ( ph -> ( ( E. p e. ZZ E. q e. ZZ P = ( ( p ^ 2 ) + ( q ^ 2 ) ) /\ E. x e. ZZ E. y e. ZZ ( N x. P ) = ( ( x ^ 2 ) + ( y ^ 2 ) ) ) -> N e. S ) )
26	4, 7, 25	mp2and 433	1 |- ( ph -> N e. S )

Syntax hints:   -> wi 4   /\ wa 104   = wceq 1364   e. wcel 2164   E.wrex 2473   |-> cmpt 4091   ran crn 4661   ` cfv 5255  (class class class)co 5919   + caddc 7877   x. cmul 7879   NNcn 8984   2c2 9035   ZZcz 9320   ^cexp 10612   abscabs 11144   Primecprime 12248   ZZ[_i]cgz 12510

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 2166  ax-14 2167  ax-ext 2175  ax-coll 4145  ax-sep 4148  ax-nul 4156  ax-pow 4204  ax-pr 4239  ax-un 4465  ax-setind 4570  ax-iinf 4621  ax-cnex 7965  ax-resscn 7966  ax-1cn 7967  ax-1re 7968  ax-icn 7969  ax-addcl 7970  ax-addrcl 7971  ax-mulcl 7972  ax-mulrcl 7973  ax-addcom 7974  ax-mulcom 7975  ax-addass 7976  ax-mulass 7977  ax-distr 7978  ax-i2m1 7979  ax-0lt1 7980  ax-1rid 7981  ax-0id 7982  ax-rnegex 7983  ax-precex 7984  ax-cnre 7985  ax-pre-ltirr 7986  ax-pre-ltwlin 7987  ax-pre-lttrn 7988  ax-pre-apti 7989  ax-pre-ltadd 7990  ax-pre-mulgt0 7991  ax-pre-mulext 7992  ax-arch 7993  ax-caucvg 7994

This theorem depends on definitions:  df-bi 117  df-dc 836  df-3or 981  df-3an 982  df-tru 1367  df-fal 1370  df-nf 1472  df-sb 1774  df-eu 2045  df-mo 2046  df-clab 2180  df-cleq 2186  df-clel 2189  df-nfc 2325  df-ne 2365  df-nel 2460  df-ral 2477  df-rex 2478  df-reu 2479  df-rmo 2480  df-rab 2481  df-v 2762  df-sbc 2987  df-csb 3082  df-dif 3156  df-un 3158  df-in 3160  df-ss 3167  df-nul 3448  df-if 3559  df-pw 3604  df-sn 3625  df-pr 3626  df-op 3628  df-uni 3837  df-int 3872  df-iun 3915  df-br 4031  df-opab 4092  df-mpt 4093  df-tr 4129  df-id 4325  df-po 4328  df-iso 4329  df-iord 4398  df-on 4400  df-ilim 4401  df-suc 4403  df-iom 4624  df-xp 4666  df-rel 4667  df-cnv 4668  df-co 4669  df-dm 4670  df-rn 4671  df-res 4672  df-ima 4673  df-iota 5216  df-fun 5257  df-fn 5258  df-f 5259  df-f1 5260  df-fo 5261  df-f1o 5262  df-fv 5263  df-riota 5874  df-ov 5922  df-oprab 5923  df-mpo 5924  df-1st 6195  df-2nd 6196  df-recs 6360  df-frec 6446  df-1o 6471  df-2o 6472  df-er 6589  df-en 6797  df-sup 7045  df-pnf 8058  df-mnf 8059  df-xr 8060  df-ltxr 8061  df-le 8062  df-sub 8194  df-neg 8195  df-reap 8596  df-ap 8603  df-div 8694  df-inn 8985  df-2 9043  df-3 9044  df-4 9045  df-n0 9244  df-z 9321  df-uz 9596  df-q 9688  df-rp 9723  df-fz 10078  df-fzo 10212  df-fl 10342  df-mod 10397  df-seqfrec 10522  df-exp 10613  df-cj 10989  df-re 10990  df-im 10991  df-rsqrt 11145  df-abs 11146  df-dvds 11934  df-gcd 12083  df-prm 12249  df-gz 12511

This theorem is referenced by:  2sqlem6  15277