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Theorem 2sqlem8a 23367
Description: Lemma for 2sqlem8 23368. (Contributed by Mario Carneiro, 4-Jun-2016.)
Hypotheses
Ref Expression
2sq.1  |-  S  =  ran  ( w  e.  ZZ[_i]  |->  ( ( abs `  w
) ^ 2 ) )
2sqlem7.2  |-  Y  =  { z  |  E. x  e.  ZZ  E. y  e.  ZZ  ( ( x  gcd  y )  =  1  /\  z  =  ( ( x ^
2 )  +  ( y ^ 2 ) ) ) }
2sqlem9.5  |-  ( ph  ->  A. b  e.  ( 1 ... ( M  -  1 ) ) A. a  e.  Y  ( b  ||  a  ->  b  e.  S ) )
2sqlem9.7  |-  ( ph  ->  M  ||  N )
2sqlem8.n  |-  ( ph  ->  N  e.  NN )
2sqlem8.m  |-  ( ph  ->  M  e.  ( ZZ>= ` 
2 ) )
2sqlem8.1  |-  ( ph  ->  A  e.  ZZ )
2sqlem8.2  |-  ( ph  ->  B  e.  ZZ )
2sqlem8.3  |-  ( ph  ->  ( A  gcd  B
)  =  1 )
2sqlem8.4  |-  ( ph  ->  N  =  ( ( A ^ 2 )  +  ( B ^
2 ) ) )
2sqlem8.c  |-  C  =  ( ( ( A  +  ( M  / 
2 ) )  mod 
M )  -  ( M  /  2 ) )
2sqlem8.d  |-  D  =  ( ( ( B  +  ( M  / 
2 ) )  mod 
M )  -  ( M  /  2 ) )
Assertion
Ref Expression
2sqlem8a  |-  ( ph  ->  ( C  gcd  D
)  e.  NN )
Distinct variable groups:    a, b, w, x, y, z    A, a, x, y, z    x, C    ph, x, y    B, a, b, x, y    M, a, b, x, y, z    S, a, b, x, y, z    x, D    x, N, y, z    Y, a, b, x, y
Allowed substitution hints:    ph( z, w, a, b)    A( w, b)    B( z, w)    C( y, z, w, a, b)    D( y, z, w, a, b)    S( w)    M( w)    N( w, a, b)    Y( z, w)

Proof of Theorem 2sqlem8a
StepHypRef Expression
1 2sqlem8.1 . . . 4  |-  ( ph  ->  A  e.  ZZ )
2 2sqlem8.m . . . . . 6  |-  ( ph  ->  M  e.  ( ZZ>= ` 
2 ) )
3 eluz2b3 11144 . . . . . 6  |-  ( M  e.  ( ZZ>= `  2
)  <->  ( M  e.  NN  /\  M  =/=  1 ) )
42, 3sylib 196 . . . . 5  |-  ( ph  ->  ( M  e.  NN  /\  M  =/=  1 ) )
54simpld 459 . . . 4  |-  ( ph  ->  M  e.  NN )
6 2sqlem8.c . . . 4  |-  C  =  ( ( ( A  +  ( M  / 
2 ) )  mod 
M )  -  ( M  /  2 ) )
71, 5, 64sqlem5 14308 . . 3  |-  ( ph  ->  ( C  e.  ZZ  /\  ( ( A  -  C )  /  M
)  e.  ZZ ) )
87simpld 459 . 2  |-  ( ph  ->  C  e.  ZZ )
9 2sqlem8.2 . . . 4  |-  ( ph  ->  B  e.  ZZ )
10 2sqlem8.d . . . 4  |-  D  =  ( ( ( B  +  ( M  / 
2 ) )  mod 
M )  -  ( M  /  2 ) )
119, 5, 104sqlem5 14308 . . 3  |-  ( ph  ->  ( D  e.  ZZ  /\  ( ( B  -  D )  /  M
)  e.  ZZ ) )
1211simpld 459 . 2  |-  ( ph  ->  D  e.  ZZ )
134simprd 463 . . . 4  |-  ( ph  ->  M  =/=  1 )
14 simpr 461 . . . . . . . . . 10  |-  ( (
ph  /\  ( C ^ 2 )  =  0 )  ->  ( C ^ 2 )  =  0 )
151, 5, 6, 144sqlem9 14312 . . . . . . . . 9  |-  ( (
ph  /\  ( C ^ 2 )  =  0 )  ->  ( M ^ 2 )  ||  ( A ^ 2 ) )
1615ex 434 . . . . . . . 8  |-  ( ph  ->  ( ( C ^
2 )  =  0  ->  ( M ^
2 )  ||  ( A ^ 2 ) ) )
17 eluzelz 11080 . . . . . . . . . 10  |-  ( M  e.  ( ZZ>= `  2
)  ->  M  e.  ZZ )
182, 17syl 16 . . . . . . . . 9  |-  ( ph  ->  M  e.  ZZ )
19 dvdssq 14046 . . . . . . . . 9  |-  ( ( M  e.  ZZ  /\  A  e.  ZZ )  ->  ( M  ||  A  <->  ( M ^ 2 ) 
||  ( A ^
2 ) ) )
2018, 1, 19syl2anc 661 . . . . . . . 8  |-  ( ph  ->  ( M  ||  A  <->  ( M ^ 2 ) 
||  ( A ^
2 ) ) )
2116, 20sylibrd 234 . . . . . . 7  |-  ( ph  ->  ( ( C ^
2 )  =  0  ->  M  ||  A
) )
22 simpr 461 . . . . . . . . . 10  |-  ( (
ph  /\  ( D ^ 2 )  =  0 )  ->  ( D ^ 2 )  =  0 )
239, 5, 10, 224sqlem9 14312 . . . . . . . . 9  |-  ( (
ph  /\  ( D ^ 2 )  =  0 )  ->  ( M ^ 2 )  ||  ( B ^ 2 ) )
2423ex 434 . . . . . . . 8  |-  ( ph  ->  ( ( D ^
2 )  =  0  ->  ( M ^
2 )  ||  ( B ^ 2 ) ) )
25 dvdssq 14046 . . . . . . . . 9  |-  ( ( M  e.  ZZ  /\  B  e.  ZZ )  ->  ( M  ||  B  <->  ( M ^ 2 ) 
||  ( B ^
2 ) ) )
2618, 9, 25syl2anc 661 . . . . . . . 8  |-  ( ph  ->  ( M  ||  B  <->  ( M ^ 2 ) 
||  ( B ^
2 ) ) )
2724, 26sylibrd 234 . . . . . . 7  |-  ( ph  ->  ( ( D ^
2 )  =  0  ->  M  ||  B
) )
28 2sqlem8.3 . . . . . . . . . . 11  |-  ( ph  ->  ( A  gcd  B
)  =  1 )
29 ax-1ne0 9550 . . . . . . . . . . . 12  |-  1  =/=  0
3029a1i 11 . . . . . . . . . . 11  |-  ( ph  ->  1  =/=  0 )
3128, 30eqnetrd 2753 . . . . . . . . . 10  |-  ( ph  ->  ( A  gcd  B
)  =/=  0 )
3231neneqd 2662 . . . . . . . . 9  |-  ( ph  ->  -.  ( A  gcd  B )  =  0 )
33 gcdeq0 14007 . . . . . . . . . 10  |-  ( ( A  e.  ZZ  /\  B  e.  ZZ )  ->  ( ( A  gcd  B )  =  0  <->  ( A  =  0  /\  B  =  0 ) ) )
341, 9, 33syl2anc 661 . . . . . . . . 9  |-  ( ph  ->  ( ( A  gcd  B )  =  0  <->  ( A  =  0  /\  B  =  0 ) ) )
3532, 34mtbid 300 . . . . . . . 8  |-  ( ph  ->  -.  ( A  =  0  /\  B  =  0 ) )
36 dvdslegcd 14002 . . . . . . . 8  |-  ( ( ( M  e.  ZZ  /\  A  e.  ZZ  /\  B  e.  ZZ )  /\  -.  ( A  =  0  /\  B  =  0 ) )  -> 
( ( M  ||  A  /\  M  ||  B
)  ->  M  <_  ( A  gcd  B ) ) )
3718, 1, 9, 35, 36syl31anc 1226 . . . . . . 7  |-  ( ph  ->  ( ( M  ||  A  /\  M  ||  B
)  ->  M  <_  ( A  gcd  B ) ) )
3821, 27, 37syl2and 483 . . . . . 6  |-  ( ph  ->  ( ( ( C ^ 2 )  =  0  /\  ( D ^ 2 )  =  0 )  ->  M  <_  ( A  gcd  B
) ) )
3928breq2d 4452 . . . . . . 7  |-  ( ph  ->  ( M  <_  ( A  gcd  B )  <->  M  <_  1 ) )
40 nnle1eq1 10553 . . . . . . . 8  |-  ( M  e.  NN  ->  ( M  <_  1  <->  M  = 
1 ) )
415, 40syl 16 . . . . . . 7  |-  ( ph  ->  ( M  <_  1  <->  M  =  1 ) )
4239, 41bitrd 253 . . . . . 6  |-  ( ph  ->  ( M  <_  ( A  gcd  B )  <->  M  = 
1 ) )
4338, 42sylibd 214 . . . . 5  |-  ( ph  ->  ( ( ( C ^ 2 )  =  0  /\  ( D ^ 2 )  =  0 )  ->  M  =  1 ) )
4443necon3ad 2670 . . . 4  |-  ( ph  ->  ( M  =/=  1  ->  -.  ( ( C ^ 2 )  =  0  /\  ( D ^ 2 )  =  0 ) ) )
4513, 44mpd 15 . . 3  |-  ( ph  ->  -.  ( ( C ^ 2 )  =  0  /\  ( D ^ 2 )  =  0 ) )
468zcnd 10956 . . . . 5  |-  ( ph  ->  C  e.  CC )
47 sqeq0 12187 . . . . 5  |-  ( C  e.  CC  ->  (
( C ^ 2 )  =  0  <->  C  =  0 ) )
4846, 47syl 16 . . . 4  |-  ( ph  ->  ( ( C ^
2 )  =  0  <-> 
C  =  0 ) )
4912zcnd 10956 . . . . 5  |-  ( ph  ->  D  e.  CC )
50 sqeq0 12187 . . . . 5  |-  ( D  e.  CC  ->  (
( D ^ 2 )  =  0  <->  D  =  0 ) )
5149, 50syl 16 . . . 4  |-  ( ph  ->  ( ( D ^
2 )  =  0  <-> 
D  =  0 ) )
5248, 51anbi12d 710 . . 3  |-  ( ph  ->  ( ( ( C ^ 2 )  =  0  /\  ( D ^ 2 )  =  0 )  <->  ( C  =  0  /\  D  =  0 ) ) )
5345, 52mtbid 300 . 2  |-  ( ph  ->  -.  ( C  =  0  /\  D  =  0 ) )
54 gcdn0cl 14000 . 2  |-  ( ( ( C  e.  ZZ  /\  D  e.  ZZ )  /\  -.  ( C  =  0  /\  D  =  0 ) )  ->  ( C  gcd  D )  e.  NN )
558, 12, 53, 54syl21anc 1222 1  |-  ( ph  ->  ( C  gcd  D
)  e.  NN )
Colors of variables: wff setvar class
Syntax hints:   -. wn 3    -> wi 4    <-> wb 184    /\ wa 369    = wceq 1374    e. wcel 1762   {cab 2445    =/= wne 2655   A.wral 2807   E.wrex 2808   class class class wbr 4440    |-> cmpt 4498   ran crn 4993   ` cfv 5579  (class class class)co 6275   CCcc 9479   0cc0 9481   1c1 9482    + caddc 9484    <_ cle 9618    - cmin 9794    / cdiv 10195   NNcn 10525   2c2 10574   ZZcz 10853   ZZ>=cuz 11071   ...cfz 11661    mod cmo 11952   ^cexp 12122   abscabs 13017    || cdivides 13836    gcd cgcd 13992   ZZ[_i]cgz 14295
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1596  ax-4 1607  ax-5 1675  ax-6 1714  ax-7 1734  ax-8 1764  ax-9 1766  ax-10 1781  ax-11 1786  ax-12 1798  ax-13 1961  ax-ext 2438  ax-sep 4561  ax-nul 4569  ax-pow 4618  ax-pr 4679  ax-un 6567  ax-cnex 9537  ax-resscn 9538  ax-1cn 9539  ax-icn 9540  ax-addcl 9541  ax-addrcl 9542  ax-mulcl 9543  ax-mulrcl 9544  ax-mulcom 9545  ax-addass 9546  ax-mulass 9547  ax-distr 9548  ax-i2m1 9549  ax-1ne0 9550  ax-1rid 9551  ax-rnegex 9552  ax-rrecex 9553  ax-cnre 9554  ax-pre-lttri 9555  ax-pre-lttrn 9556  ax-pre-ltadd 9557  ax-pre-mulgt0 9558  ax-pre-sup 9559
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 969  df-3an 970  df-tru 1377  df-ex 1592  df-nf 1595  df-sb 1707  df-eu 2272  df-mo 2273  df-clab 2446  df-cleq 2452  df-clel 2455  df-nfc 2610  df-ne 2657  df-nel 2658  df-ral 2812  df-rex 2813  df-reu 2814  df-rmo 2815  df-rab 2816  df-v 3108  df-sbc 3325  df-csb 3429  df-dif 3472  df-un 3474  df-in 3476  df-ss 3483  df-pss 3485  df-nul 3779  df-if 3933  df-pw 4005  df-sn 4021  df-pr 4023  df-tp 4025  df-op 4027  df-uni 4239  df-iun 4320  df-br 4441  df-opab 4499  df-mpt 4500  df-tr 4534  df-eprel 4784  df-id 4788  df-po 4793  df-so 4794  df-fr 4831  df-we 4833  df-ord 4874  df-on 4875  df-lim 4876  df-suc 4877  df-xp 4998  df-rel 4999  df-cnv 5000  df-co 5001  df-dm 5002  df-rn 5003  df-res 5004  df-ima 5005  df-iota 5542  df-fun 5581  df-fn 5582  df-f 5583  df-f1 5584  df-fo 5585  df-f1o 5586  df-fv 5587  df-riota 6236  df-ov 6278  df-oprab 6279  df-mpt2 6280  df-om 6672  df-2nd 6775  df-recs 7032  df-rdg 7066  df-er 7301  df-en 7507  df-dom 7508  df-sdom 7509  df-sup 7890  df-pnf 9619  df-mnf 9620  df-xr 9621  df-ltxr 9622  df-le 9623  df-sub 9796  df-neg 9797  df-div 10196  df-nn 10526  df-2 10583  df-3 10584  df-n0 10785  df-z 10854  df-uz 11072  df-rp 11210  df-fl 11886  df-mod 11953  df-seq 12064  df-exp 12123  df-cj 12882  df-re 12883  df-im 12884  df-sqr 13018  df-abs 13019  df-dvds 13837  df-gcd 13993
This theorem is referenced by:  2sqlem8  23368
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