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Theorem 2sqlem5 24026
Description: Lemma for 2sq 24034. 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 24022 . . 3  |-  ( P  e.  S  <->  E. p  e.  ZZ  E. q  e.  ZZ  P  =  ( ( p ^ 2 )  +  ( q ^ 2 ) ) )
41, 3sylib 198 . 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 24022 . . 3  |-  ( ( N  x.  P )  e.  S  <->  E. x  e.  ZZ  E. y  e.  ZZ  ( N  x.  P )  =  ( ( x ^ 2 )  +  ( y ^ 2 ) ) )
75, 6sylib 198 . 2  |-  ( ph  ->  E. x  e.  ZZ  E. y  e.  ZZ  ( N  x.  P )  =  ( ( x ^ 2 )  +  ( y ^ 2 ) ) )
8 reeanv 2977 . . 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 2977 . . . . 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 726 . . . . . . . 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 726 . . . . . . . 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 765 . . . . . . . 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 767 . . . . . . . 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 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 ) ) ) ) )  ->  p  e.  ZZ )
17 simprll 766 . . . . . . . 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 769 . . . . . . . 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 768 . . . . . . . 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 24025 . . . . . . 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 2905 . . . . 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 220 . . . 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 2905 . . 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 220 . 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 1407    e. wcel 1844   E.wrex 2757    |-> cmpt 4455   ran crn 4826   ` cfv 5571  (class class class)co 6280    + caddc 9527    x. cmul 9529   NNcn 10578   2c2 10628   ZZcz 10907   ^cexp 12212   abscabs 13218   Primecprime 14428   ZZ[_i]cgz 14658
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1641  ax-4 1654  ax-5 1727  ax-6 1773  ax-7 1816  ax-8 1846  ax-9 1848  ax-10 1863  ax-11 1868  ax-12 1880  ax-13 2028  ax-ext 2382  ax-sep 4519  ax-nul 4527  ax-pow 4574  ax-pr 4632  ax-un 6576  ax-cnex 9580  ax-resscn 9581  ax-1cn 9582  ax-icn 9583  ax-addcl 9584  ax-addrcl 9585  ax-mulcl 9586  ax-mulrcl 9587  ax-mulcom 9588  ax-addass 9589  ax-mulass 9590  ax-distr 9591  ax-i2m1 9592  ax-1ne0 9593  ax-1rid 9594  ax-rnegex 9595  ax-rrecex 9596  ax-cnre 9597  ax-pre-lttri 9598  ax-pre-lttrn 9599  ax-pre-ltadd 9600  ax-pre-mulgt0 9601  ax-pre-sup 9602
This theorem depends on definitions:  df-bi 187  df-or 370  df-an 371  df-3or 977  df-3an 978  df-tru 1410  df-ex 1636  df-nf 1640  df-sb 1766  df-eu 2244  df-mo 2245  df-clab 2390  df-cleq 2396  df-clel 2399  df-nfc 2554  df-ne 2602  df-nel 2603  df-ral 2761  df-rex 2762  df-reu 2763  df-rmo 2764  df-rab 2765  df-v 3063  df-sbc 3280  df-csb 3376  df-dif 3419  df-un 3421  df-in 3423  df-ss 3430  df-pss 3432  df-nul 3741  df-if 3888  df-pw 3959  df-sn 3975  df-pr 3977  df-tp 3979  df-op 3981  df-uni 4194  df-int 4230  df-iun 4275  df-br 4398  df-opab 4456  df-mpt 4457  df-tr 4492  df-eprel 4736  df-id 4740  df-po 4746  df-so 4747  df-fr 4784  df-we 4786  df-xp 4831  df-rel 4832  df-cnv 4833  df-co 4834  df-dm 4835  df-rn 4836  df-res 4837  df-ima 4838  df-pred 5369  df-ord 5415  df-on 5416  df-lim 5417  df-suc 5418  df-iota 5535  df-fun 5573  df-fn 5574  df-f 5575  df-f1 5576  df-fo 5577  df-f1o 5578  df-fv 5579  df-riota 6242  df-ov 6283  df-oprab 6284  df-mpt2 6285  df-om 6686  df-2nd 6787  df-wrecs 7015  df-recs 7077  df-rdg 7115  df-1o 7169  df-2o 7170  df-oadd 7173  df-er 7350  df-en 7557  df-dom 7558  df-sdom 7559  df-fin 7560  df-sup 7937  df-pnf 9662  df-mnf 9663  df-xr 9664  df-ltxr 9665  df-le 9666  df-sub 9845  df-neg 9846  df-div 10250  df-nn 10579  df-2 10637  df-3 10638  df-n0 10839  df-z 10908  df-uz 11130  df-rp 11268  df-fl 11968  df-mod 12037  df-seq 12154  df-exp 12213  df-cj 13083  df-re 13084  df-im 13085  df-sqrt 13219  df-abs 13220  df-dvds 14198  df-gcd 14356  df-prm 14429  df-gz 14659
This theorem is referenced by:  2sqlem6  24027
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