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Theorem 2sqlem5 22712
Description: Lemma for 2sq 22720. 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 22708 . . 3  |-  ( P  e.  S  <->  E. p  e.  ZZ  E. q  e.  ZZ  P  =  ( ( p ^ 2 )  +  ( q ^ 2 ) ) )
41, 3sylib 196 . 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 22708 . . 3  |-  ( ( N  x.  P )  e.  S  <->  E. x  e.  ZZ  E. y  e.  ZZ  ( N  x.  P )  =  ( ( x ^ 2 )  +  ( y ^ 2 ) ) )
75, 6sylib 196 . 2  |-  ( ph  ->  E. x  e.  ZZ  E. y  e.  ZZ  ( N  x.  P )  =  ( ( x ^ 2 )  +  ( y ^ 2 ) ) )
8 reeanv 2893 . . 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 2893 . . . . 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 725 . . . . . . . 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 725 . . . . . . . 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 760 . . . . . . . 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 762 . . . . . . . 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 759 . . . . . . . 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 761 . . . . . . . 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 764 . . . . . . . 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 763 . . . . . . . 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 22711 . . . . . . 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 615 . . . . . 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 2853 . . . . 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, 22syl5bir 218 . . . 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 2853 . . 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, 24syl5bir 218 . 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 679 1  |-  ( ph  ->  N  e.  S )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    /\ wa 369    = wceq 1369    e. wcel 1756   E.wrex 2721    e. cmpt 4355   ran crn 4846   ` cfv 5423  (class class class)co 6096    + caddc 9290    x. cmul 9292   NNcn 10327   2c2 10376   ZZcz 10651   ^cexp 11870   abscabs 12728   Primecprime 13768   ZZ[_i]cgz 13995
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1591  ax-4 1602  ax-5 1670  ax-6 1708  ax-7 1728  ax-8 1758  ax-9 1760  ax-10 1775  ax-11 1780  ax-12 1792  ax-13 1943  ax-ext 2423  ax-sep 4418  ax-nul 4426  ax-pow 4475  ax-pr 4536  ax-un 6377  ax-cnex 9343  ax-resscn 9344  ax-1cn 9345  ax-icn 9346  ax-addcl 9347  ax-addrcl 9348  ax-mulcl 9349  ax-mulrcl 9350  ax-mulcom 9351  ax-addass 9352  ax-mulass 9353  ax-distr 9354  ax-i2m1 9355  ax-1ne0 9356  ax-1rid 9357  ax-rnegex 9358  ax-rrecex 9359  ax-cnre 9360  ax-pre-lttri 9361  ax-pre-lttrn 9362  ax-pre-ltadd 9363  ax-pre-mulgt0 9364  ax-pre-sup 9365
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 966  df-3an 967  df-tru 1372  df-ex 1587  df-nf 1590  df-sb 1701  df-eu 2257  df-mo 2258  df-clab 2430  df-cleq 2436  df-clel 2439  df-nfc 2573  df-ne 2613  df-nel 2614  df-ral 2725  df-rex 2726  df-reu 2727  df-rmo 2728  df-rab 2729  df-v 2979  df-sbc 3192  df-csb 3294  df-dif 3336  df-un 3338  df-in 3340  df-ss 3347  df-pss 3349  df-nul 3643  df-if 3797  df-pw 3867  df-sn 3883  df-pr 3885  df-tp 3887  df-op 3889  df-uni 4097  df-int 4134  df-iun 4178  df-br 4298  df-opab 4356  df-mpt 4357  df-tr 4391  df-eprel 4637  df-id 4641  df-po 4646  df-so 4647  df-fr 4684  df-we 4686  df-ord 4727  df-on 4728  df-lim 4729  df-suc 4730  df-xp 4851  df-rel 4852  df-cnv 4853  df-co 4854  df-dm 4855  df-rn 4856  df-res 4857  df-ima 4858  df-iota 5386  df-fun 5425  df-fn 5426  df-f 5427  df-f1 5428  df-fo 5429  df-f1o 5430  df-fv 5431  df-riota 6057  df-ov 6099  df-oprab 6100  df-mpt2 6101  df-om 6482  df-2nd 6583  df-recs 6837  df-rdg 6871  df-1o 6925  df-2o 6926  df-oadd 6929  df-er 7106  df-en 7316  df-dom 7317  df-sdom 7318  df-fin 7319  df-sup 7696  df-pnf 9425  df-mnf 9426  df-xr 9427  df-ltxr 9428  df-le 9429  df-sub 9602  df-neg 9603  df-div 9999  df-nn 10328  df-2 10385  df-3 10386  df-n0 10585  df-z 10652  df-uz 10867  df-rp 10997  df-fl 11647  df-mod 11714  df-seq 11812  df-exp 11871  df-cj 12593  df-re 12594  df-im 12595  df-sqr 12729  df-abs 12730  df-dvds 13541  df-gcd 13696  df-prm 13769  df-gz 13996
This theorem is referenced by:  2sqlem6  22713
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