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Theorem 2sqlem5 21144
Description: Lemma for 2sq 21152. 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 21140 . . 3  |-  ( P  e.  S  <->  E. p  e.  ZZ  E. q  e.  ZZ  P  =  ( ( p ^ 2 )  +  ( q ^ 2 ) ) )
41, 3sylib 189 . 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 21140 . . 3  |-  ( ( N  x.  P )  e.  S  <->  E. x  e.  ZZ  E. y  e.  ZZ  ( N  x.  P )  =  ( ( x ^ 2 )  +  ( y ^ 2 ) ) )
75, 6sylib 189 . 2  |-  ( ph  ->  E. x  e.  ZZ  E. y  e.  ZZ  ( N  x.  P )  =  ( ( x ^ 2 )  +  ( y ^ 2 ) ) )
8 reeanv 2867 . . 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 2867 . . . . 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 707 . . . . . . . 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 707 . . . . . . . 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 738 . . . . . . . 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 740 . . . . . . . 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 737 . . . . . . . 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 739 . . . . . . . 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 742 . . . . . . . 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 741 . . . . . . . 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 21143 . . . . . . 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 599 . . . . . 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 2829 . . . . 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 210 . . . 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 2829 . . 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 210 . 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 661 1  |-  ( ph  ->  N  e.  S )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 359    = wceq 1652    e. wcel 1725   E.wrex 2698    e. cmpt 4258   ran crn 4871   ` cfv 5446  (class class class)co 6073    + caddc 8985    x. cmul 8987   NNcn 9992   2c2 10041   ZZcz 10274   ^cexp 11374   abscabs 12031   Primecprime 13071   ZZ [ _i ]cgz 13289
This theorem is referenced by:  2sqlem6  21145
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1555  ax-5 1566  ax-17 1626  ax-9 1666  ax-8 1687  ax-13 1727  ax-14 1729  ax-6 1744  ax-7 1749  ax-11 1761  ax-12 1950  ax-ext 2416  ax-sep 4322  ax-nul 4330  ax-pow 4369  ax-pr 4395  ax-un 4693  ax-cnex 9038  ax-resscn 9039  ax-1cn 9040  ax-icn 9041  ax-addcl 9042  ax-addrcl 9043  ax-mulcl 9044  ax-mulrcl 9045  ax-mulcom 9046  ax-addass 9047  ax-mulass 9048  ax-distr 9049  ax-i2m1 9050  ax-1ne0 9051  ax-1rid 9052  ax-rnegex 9053  ax-rrecex 9054  ax-cnre 9055  ax-pre-lttri 9056  ax-pre-lttrn 9057  ax-pre-ltadd 9058  ax-pre-mulgt0 9059  ax-pre-sup 9060
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3or 937  df-3an 938  df-tru 1328  df-ex 1551  df-nf 1554  df-sb 1659  df-eu 2284  df-mo 2285  df-clab 2422  df-cleq 2428  df-clel 2431  df-nfc 2560  df-ne 2600  df-nel 2601  df-ral 2702  df-rex 2703  df-reu 2704  df-rmo 2705  df-rab 2706  df-v 2950  df-sbc 3154  df-csb 3244  df-dif 3315  df-un 3317  df-in 3319  df-ss 3326  df-pss 3328  df-nul 3621  df-if 3732  df-pw 3793  df-sn 3812  df-pr 3813  df-tp 3814  df-op 3815  df-uni 4008  df-int 4043  df-iun 4087  df-br 4205  df-opab 4259  df-mpt 4260  df-tr 4295  df-eprel 4486  df-id 4490  df-po 4495  df-so 4496  df-fr 4533  df-we 4535  df-ord 4576  df-on 4577  df-lim 4578  df-suc 4579  df-om 4838  df-xp 4876  df-rel 4877  df-cnv 4878  df-co 4879  df-dm 4880  df-rn 4881  df-res 4882  df-ima 4883  df-iota 5410  df-fun 5448  df-fn 5449  df-f 5450  df-f1 5451  df-fo 5452  df-f1o 5453  df-fv 5454  df-ov 6076  df-oprab 6077  df-mpt2 6078  df-2nd 6342  df-riota 6541  df-recs 6625  df-rdg 6660  df-1o 6716  df-2o 6717  df-oadd 6720  df-er 6897  df-en 7102  df-dom 7103  df-sdom 7104  df-fin 7105  df-sup 7438  df-pnf 9114  df-mnf 9115  df-xr 9116  df-ltxr 9117  df-le 9118  df-sub 9285  df-neg 9286  df-div 9670  df-nn 9993  df-2 10050  df-3 10051  df-n0 10214  df-z 10275  df-uz 10481  df-rp 10605  df-fl 11194  df-mod 11243  df-seq 11316  df-exp 11375  df-cj 11896  df-re 11897  df-im 11898  df-sqr 12032  df-abs 12033  df-dvds 12845  df-gcd 12999  df-prm 13072  df-gz 13290
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