Theorem 2sqlem5 14122

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 14118	. . 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 14118	. . 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 2646	. . 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 2646	. . . . 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 14121	. . . . . . 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 2602	. . . . 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 2602	. . 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 1353   e. wcel 2148   E.wrex 2456   |-> cmpt 4061   ran crn 4624   ` cfv 5212  (class class class)co 5869   + caddc 7805   x. cmul 7807   NNcn 8908   2c2 8959   ZZcz 9242   ^cexp 10505   abscabs 10990   Primecprime 12090   ZZ[_i]cgz 12350

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 614  ax-in2 615  ax-io 709  ax-5 1447  ax-7 1448  ax-gen 1449  ax-ie1 1493  ax-ie2 1494  ax-8 1504  ax-10 1505  ax-11 1506  ax-i12 1507  ax-bndl 1509  ax-4 1510  ax-17 1526  ax-i9 1530  ax-ial 1534  ax-i5r 1535  ax-13 2150  ax-14 2151  ax-ext 2159  ax-coll 4115  ax-sep 4118  ax-nul 4126  ax-pow 4171  ax-pr 4206  ax-un 4430  ax-setind 4533  ax-iinf 4584  ax-cnex 7893  ax-resscn 7894  ax-1cn 7895  ax-1re 7896  ax-icn 7897  ax-addcl 7898  ax-addrcl 7899  ax-mulcl 7900  ax-mulrcl 7901  ax-addcom 7902  ax-mulcom 7903  ax-addass 7904  ax-mulass 7905  ax-distr 7906  ax-i2m1 7907  ax-0lt1 7908  ax-1rid 7909  ax-0id 7910  ax-rnegex 7911  ax-precex 7912  ax-cnre 7913  ax-pre-ltirr 7914  ax-pre-ltwlin 7915  ax-pre-lttrn 7916  ax-pre-apti 7917  ax-pre-ltadd 7918  ax-pre-mulgt0 7919  ax-pre-mulext 7920  ax-arch 7921  ax-caucvg 7922

This theorem depends on definitions:  df-bi 117  df-dc 835  df-3or 979  df-3an 980  df-tru 1356  df-fal 1359  df-nf 1461  df-sb 1763  df-eu 2029  df-mo 2030  df-clab 2164  df-cleq 2170  df-clel 2173  df-nfc 2308  df-ne 2348  df-nel 2443  df-ral 2460  df-rex 2461  df-reu 2462  df-rmo 2463  df-rab 2464  df-v 2739  df-sbc 2963  df-csb 3058  df-dif 3131  df-un 3133  df-in 3135  df-ss 3142  df-nul 3423  df-if 3535  df-pw 3576  df-sn 3597  df-pr 3598  df-op 3600  df-uni 3808  df-int 3843  df-iun 3886  df-br 4001  df-opab 4062  df-mpt 4063  df-tr 4099  df-id 4290  df-po 4293  df-iso 4294  df-iord 4363  df-on 4365  df-ilim 4366  df-suc 4368  df-iom 4587  df-xp 4629  df-rel 4630  df-cnv 4631  df-co 4632  df-dm 4633  df-rn 4634  df-res 4635  df-ima 4636  df-iota 5174  df-fun 5214  df-fn 5215  df-f 5216  df-f1 5217  df-fo 5218  df-f1o 5219  df-fv 5220  df-riota 5825  df-ov 5872  df-oprab 5873  df-mpo 5874  df-1st 6135  df-2nd 6136  df-recs 6300  df-frec 6386  df-1o 6411  df-2o 6412  df-er 6529  df-en 6735  df-sup 6977  df-pnf 7984  df-mnf 7985  df-xr 7986  df-ltxr 7987  df-le 7988  df-sub 8120  df-neg 8121  df-reap 8522  df-ap 8529  df-div 8619  df-inn 8909  df-2 8967  df-3 8968  df-4 8969  df-n0 9166  df-z 9243  df-uz 9518  df-q 9609  df-rp 9641  df-fz 9996  df-fzo 10129  df-fl 10256  df-mod 10309  df-seqfrec 10432  df-exp 10506  df-cj 10835  df-re 10836  df-im 10837  df-rsqrt 10991  df-abs 10992  df-dvds 11779  df-gcd 11927  df-prm 12091  df-gz 12351

This theorem is referenced by:  2sqlem6  14123