Theorem 2sqlem5 15938

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 15934	. . 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 15934	. . 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 2704	. . 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 2704	. . . . 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 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 ) ) ) ) ) -> x e. ZZ )
15		simprlr 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 ) ) ) ) ) -> y e. ZZ )
16		simplrl 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 ) ) ) ) ) -> p e. ZZ )
17		simprll 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 ) ) ) ) ) -> q e. ZZ )
18		simprrr 542	. . . . . . . 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 541	. . . . . . . 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 15937	. . . . . . 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 2659	. . . . 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 2659	. . 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 1398   e. wcel 2202   E.wrex 2512   |-> cmpt 4155   ran crn 4732   ` cfv 5333  (class class class)co 6028   + caddc 8095   x. cmul 8097   NNcn 9202   2c2 9253   ZZcz 9540   ^cexp 10863   abscabs 11637   Primecprime 12759   ZZ[_i]cgz 13022

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 619  ax-in2 620  ax-io 717  ax-5 1496  ax-7 1497  ax-gen 1498  ax-ie1 1542  ax-ie2 1543  ax-8 1553  ax-10 1554  ax-11 1555  ax-i12 1556  ax-bndl 1558  ax-4 1559  ax-17 1575  ax-i9 1579  ax-ial 1583  ax-i5r 1584  ax-13 2204  ax-14 2205  ax-ext 2213  ax-coll 4209  ax-sep 4212  ax-nul 4220  ax-pow 4270  ax-pr 4305  ax-un 4536  ax-setind 4641  ax-iinf 4692  ax-cnex 8183  ax-resscn 8184  ax-1cn 8185  ax-1re 8186  ax-icn 8187  ax-addcl 8188  ax-addrcl 8189  ax-mulcl 8190  ax-mulrcl 8191  ax-addcom 8192  ax-mulcom 8193  ax-addass 8194  ax-mulass 8195  ax-distr 8196  ax-i2m1 8197  ax-0lt1 8198  ax-1rid 8199  ax-0id 8200  ax-rnegex 8201  ax-precex 8202  ax-cnre 8203  ax-pre-ltirr 8204  ax-pre-ltwlin 8205  ax-pre-lttrn 8206  ax-pre-apti 8207  ax-pre-ltadd 8208  ax-pre-mulgt0 8209  ax-pre-mulext 8210  ax-arch 8211  ax-caucvg 8212

This theorem depends on definitions:  df-bi 117  df-dc 843  df-3or 1006  df-3an 1007  df-tru 1401  df-fal 1404  df-nf 1510  df-sb 1811  df-eu 2082  df-mo 2083  df-clab 2218  df-cleq 2224  df-clel 2227  df-nfc 2364  df-ne 2404  df-nel 2499  df-ral 2516  df-rex 2517  df-reu 2518  df-rmo 2519  df-rab 2520  df-v 2805  df-sbc 3033  df-csb 3129  df-dif 3203  df-un 3205  df-in 3207  df-ss 3214  df-nul 3497  df-if 3608  df-pw 3658  df-sn 3679  df-pr 3680  df-op 3682  df-uni 3899  df-int 3934  df-iun 3977  df-br 4094  df-opab 4156  df-mpt 4157  df-tr 4193  df-id 4396  df-po 4399  df-iso 4400  df-iord 4469  df-on 4471  df-ilim 4472  df-suc 4474  df-iom 4695  df-xp 4737  df-rel 4738  df-cnv 4739  df-co 4740  df-dm 4741  df-rn 4742  df-res 4743  df-ima 4744  df-iota 5293  df-fun 5335  df-fn 5336  df-f 5337  df-f1 5338  df-fo 5339  df-f1o 5340  df-fv 5341  df-riota 5981  df-ov 6031  df-oprab 6032  df-mpo 6033  df-1st 6312  df-2nd 6313  df-recs 6514  df-frec 6600  df-1o 6625  df-2o 6626  df-er 6745  df-en 6953  df-sup 7243  df-pnf 8275  df-mnf 8276  df-xr 8277  df-ltxr 8278  df-le 8279  df-sub 8411  df-neg 8412  df-reap 8814  df-ap 8821  df-div 8912  df-inn 9203  df-2 9261  df-3 9262  df-4 9263  df-n0 9462  df-z 9541  df-uz 9817  df-q 9915  df-rp 9950  df-fz 10306  df-fzo 10440  df-fl 10593  df-mod 10648  df-seqfrec 10773  df-exp 10864  df-cj 11482  df-re 11483  df-im 11484  df-rsqrt 11638  df-abs 11639  df-dvds 12429  df-gcd 12605  df-prm 12760  df-gz 13023

This theorem is referenced by:  2sqlem6  15939