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Theorem 2sqlem5 15798
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.
StepHypRef Expression
1 2sqlem5.4 . . 3 |- ( ph -> P e. S )
2 2sq.1 . . . 4 |- S = ran ( w e. ZZ[_i] |-> ( ( abs ` w ) ^ 2 ) )
322sqlem2 15794 . . 3 |- ( P e. S <-> E. p e. ZZ E. q e. ZZ P = ( ( p ^ 2 ) + ( q ^ 2 ) ) )
41, 3sylib 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 )
622sqlem2 15794 . . 3 |- ( ( N x. P ) e. S <-> E. x e. ZZ E. y e. ZZ ( N x. P ) = ( ( x ^ 2 ) + ( y ^ 2 ) ) )
75, 6sylib 122 . 2 |- ( ph -> E. x e. ZZ E. y e. ZZ ( N x. P ) = ( ( x ^ 2 ) + ( y ^ 2 ) ) )
8 reeanv 2701 . . 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 2701 . . . . 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 )
1110ad2antrr 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 )
1312ad2antrr 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 ) ) )
202, 11, 13, 14, 15, 16, 17, 18, 192sqlem4 15797 . . . . . . 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 )
2120expr 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 ) )
2221rexlimdvva 2656 . . . . 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 ) )
239, 22biimtrrid 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 ) )
2423rexlimdvva 2656 . . 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 ) )
258, 24biimtrrid 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 ) )
264, 7, 25mp2and 433 1 |- ( ph -> N e. S )
Colors of variables: wff set class
Syntax hints:   -> wi 4   /\ wa 104   = wceq 1395   e. wcel 2200   E.wrex 2509   |-> cmpt 4145   ran crn 4720   ` cfv 5318  (class class class)co 6001   + caddc 8002   x. cmul 8004   NNcn 9110   2c2 9161   ZZcz 9446   ^cexp 10760   abscabs 11508   Primecprime 12629   ZZ[_i]cgz 12892
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 617  ax-in2 618  ax-io 714  ax-5 1493  ax-7 1494  ax-gen 1495  ax-ie1 1539  ax-ie2 1540  ax-8 1550  ax-10 1551  ax-11 1552  ax-i12 1553  ax-bndl 1555  ax-4 1556  ax-17 1572  ax-i9 1576  ax-ial 1580  ax-i5r 1581  ax-13 2202  ax-14 2203  ax-ext 2211  ax-coll 4199  ax-sep 4202  ax-nul 4210  ax-pow 4258  ax-pr 4293  ax-un 4524  ax-setind 4629  ax-iinf 4680  ax-cnex 8090  ax-resscn 8091  ax-1cn 8092  ax-1re 8093  ax-icn 8094  ax-addcl 8095  ax-addrcl 8096  ax-mulcl 8097  ax-mulrcl 8098  ax-addcom 8099  ax-mulcom 8100  ax-addass 8101  ax-mulass 8102  ax-distr 8103  ax-i2m1 8104  ax-0lt1 8105  ax-1rid 8106  ax-0id 8107  ax-rnegex 8108  ax-precex 8109  ax-cnre 8110  ax-pre-ltirr 8111  ax-pre-ltwlin 8112  ax-pre-lttrn 8113  ax-pre-apti 8114  ax-pre-ltadd 8115  ax-pre-mulgt0 8116  ax-pre-mulext 8117  ax-arch 8118  ax-caucvg 8119
This theorem depends on definitions:  df-bi 117  df-dc 840  df-3or 1003  df-3an 1004  df-tru 1398  df-fal 1401  df-nf 1507  df-sb 1809  df-eu 2080  df-mo 2081  df-clab 2216  df-cleq 2222  df-clel 2225  df-nfc 2361  df-ne 2401  df-nel 2496  df-ral 2513  df-rex 2514  df-reu 2515  df-rmo 2516  df-rab 2517  df-v 2801  df-sbc 3029  df-csb 3125  df-dif 3199  df-un 3201  df-in 3203  df-ss 3210  df-nul 3492  df-if 3603  df-pw 3651  df-sn 3672  df-pr 3673  df-op 3675  df-uni 3889  df-int 3924  df-iun 3967  df-br 4084  df-opab 4146  df-mpt 4147  df-tr 4183  df-id 4384  df-po 4387  df-iso 4388  df-iord 4457  df-on 4459  df-ilim 4460  df-suc 4462  df-iom 4683  df-xp 4725  df-rel 4726  df-cnv 4727  df-co 4728  df-dm 4729  df-rn 4730  df-res 4731  df-ima 4732  df-iota 5278  df-fun 5320  df-fn 5321  df-f 5322  df-f1 5323  df-fo 5324  df-f1o 5325  df-fv 5326  df-riota 5954  df-ov 6004  df-oprab 6005  df-mpo 6006  df-1st 6286  df-2nd 6287  df-recs 6451  df-frec 6537  df-1o 6562  df-2o 6563  df-er 6680  df-en 6888  df-sup 7151  df-pnf 8183  df-mnf 8184  df-xr 8185  df-ltxr 8186  df-le 8187  df-sub 8319  df-neg 8320  df-reap 8722  df-ap 8729  df-div 8820  df-inn 9111  df-2 9169  df-3 9170  df-4 9171  df-n0 9370  df-z 9447  df-uz 9723  df-q 9815  df-rp 9850  df-fz 10205  df-fzo 10339  df-fl 10490  df-mod 10545  df-seqfrec 10670  df-exp 10761  df-cj 11353  df-re 11354  df-im 11355  df-rsqrt 11509  df-abs 11510  df-dvds 12299  df-gcd 12475  df-prm 12630  df-gz 12893
This theorem is referenced by:  2sqlem6  15799
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