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Theorem 2sqlem5 20607
Description: Lemma for 2sq 20615. 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 20603 . . 3  |-  ( P  e.  S  <->  E. p  e.  ZZ  E. q  e.  ZZ  P  =  ( ( p ^ 2 )  +  ( q ^ 2 ) ) )
41, 3sylib 188 . 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 20603 . . 3  |-  ( ( N  x.  P )  e.  S  <->  E. x  e.  ZZ  E. y  e.  ZZ  ( N  x.  P )  =  ( ( x ^ 2 )  +  ( y ^ 2 ) ) )
75, 6sylib 188 . 2  |-  ( ph  ->  E. x  e.  ZZ  E. y  e.  ZZ  ( N  x.  P )  =  ( ( x ^ 2 )  +  ( y ^ 2 ) ) )
8 reeanv 2707 . . 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 2707 . . . . 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 706 . . . . . . . 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 706 . . . . . . . 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 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 ) ) ) ) )  ->  x  e.  ZZ )
15 simprlr 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 ) ) ) ) )  -> 
y  e.  ZZ )
16 simplrl 736 . . . . . . . 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 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 ) ) ) ) )  -> 
q  e.  ZZ )
18 simprrr 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 ) ) ) ) )  -> 
( N  x.  P
)  =  ( ( x ^ 2 )  +  ( y ^
2 ) ) )
19 simprrl 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 ) ) ) ) )  ->  P  =  ( (
p ^ 2 )  +  ( q ^
2 ) ) )
202, 11, 13, 14, 15, 16, 17, 18, 192sqlem4 20606 . . . . . . 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 598 . . . . . 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 2674 . . . . 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 209 . . . 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 2674 . . 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 209 . 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 660 1  |-  ( ph  ->  N  e.  S )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 358    = wceq 1623    e. wcel 1684   E.wrex 2544    e. cmpt 4077   ran crn 4690   ` cfv 5255  (class class class)co 5858    + caddc 8740    x. cmul 8742   NNcn 9746   2c2 9795   ZZcz 10024   ^cexp 11104   abscabs 11719   Primecprime 12758   ZZ [ _i ]cgz 12976
This theorem is referenced by:  2sqlem6  20608
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1533  ax-5 1544  ax-17 1603  ax-9 1635  ax-8 1643  ax-13 1686  ax-14 1688  ax-6 1703  ax-7 1708  ax-11 1715  ax-12 1866  ax-ext 2264  ax-sep 4141  ax-nul 4149  ax-pow 4188  ax-pr 4214  ax-un 4512  ax-cnex 8793  ax-resscn 8794  ax-1cn 8795  ax-icn 8796  ax-addcl 8797  ax-addrcl 8798  ax-mulcl 8799  ax-mulrcl 8800  ax-mulcom 8801  ax-addass 8802  ax-mulass 8803  ax-distr 8804  ax-i2m1 8805  ax-1ne0 8806  ax-1rid 8807  ax-rnegex 8808  ax-rrecex 8809  ax-cnre 8810  ax-pre-lttri 8811  ax-pre-lttrn 8812  ax-pre-ltadd 8813  ax-pre-mulgt0 8814  ax-pre-sup 8815
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3or 935  df-3an 936  df-tru 1310  df-ex 1529  df-nf 1532  df-sb 1630  df-eu 2147  df-mo 2148  df-clab 2270  df-cleq 2276  df-clel 2279  df-nfc 2408  df-ne 2448  df-nel 2449  df-ral 2548  df-rex 2549  df-reu 2550  df-rmo 2551  df-rab 2552  df-v 2790  df-sbc 2992  df-csb 3082  df-dif 3155  df-un 3157  df-in 3159  df-ss 3166  df-pss 3168  df-nul 3456  df-if 3566  df-pw 3627  df-sn 3646  df-pr 3647  df-tp 3648  df-op 3649  df-uni 3828  df-int 3863  df-iun 3907  df-br 4024  df-opab 4078  df-mpt 4079  df-tr 4114  df-eprel 4305  df-id 4309  df-po 4314  df-so 4315  df-fr 4352  df-we 4354  df-ord 4395  df-on 4396  df-lim 4397  df-suc 4398  df-om 4657  df-xp 4695  df-rel 4696  df-cnv 4697  df-co 4698  df-dm 4699  df-rn 4700  df-res 4701  df-ima 4702  df-iota 5219  df-fun 5257  df-fn 5258  df-f 5259  df-f1 5260  df-fo 5261  df-f1o 5262  df-fv 5263  df-ov 5861  df-oprab 5862  df-mpt2 5863  df-2nd 6123  df-riota 6304  df-recs 6388  df-rdg 6423  df-1o 6479  df-2o 6480  df-oadd 6483  df-er 6660  df-en 6864  df-dom 6865  df-sdom 6866  df-fin 6867  df-sup 7194  df-pnf 8869  df-mnf 8870  df-xr 8871  df-ltxr 8872  df-le 8873  df-sub 9039  df-neg 9040  df-div 9424  df-nn 9747  df-2 9804  df-3 9805  df-n0 9966  df-z 10025  df-uz 10231  df-rp 10355  df-fl 10925  df-mod 10974  df-seq 11047  df-exp 11105  df-cj 11584  df-re 11585  df-im 11586  df-sqr 11720  df-abs 11721  df-dvds 12532  df-gcd 12686  df-prm 12759  df-gz 12977
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