Theorem 2sqlem8a 13752

Description: Lemma for 2sqlem8 13753. (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

Step	Hyp	Ref	Expression
1		2sqlem8.1	. . . 4 |- ( ph -> A e. ZZ )
2		2sqlem8.m	. . . . . 6 |- ( ph -> M e. ( ZZ>= ` 2 ) )
3		eluz2b3 9563	. . . . . 6 |- ( M e. ( ZZ>= ` 2 ) <-> ( M e. NN /\ M =/= 1 ) )
4	2, 3	sylib 121	. . . . 5 |- ( ph -> ( M e. NN /\ M =/= 1 ) )
5	4	simpld 111	. . . 4 |- ( ph -> M e. NN )
6		2sqlem8.c	. . . 4 |- C = ( ( ( A + ( M / 2 ) ) mod M ) - ( M / 2 ) )
7	1, 5, 6	4sqlem5 12334	. . 3 |- ( ph -> ( C e. ZZ /\ ( ( A - C ) / M ) e. ZZ ) )
8	7	simpld 111	. 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 ) )
11	9, 5, 10	4sqlem5 12334	. . 3 |- ( ph -> ( D e. ZZ /\ ( ( B - D ) / M ) e. ZZ ) )
12	11	simpld 111	. 2 |- ( ph -> D e. ZZ )
13	4	simprd 113	. . . 4 |- ( ph -> M =/= 1 )
14		simpr 109	. . . . . . . . . 10 |- ( ( ph /\ ( C ^ 2 ) = 0 ) -> ( C ^ 2 ) = 0 )
15	1, 5, 6, 14	4sqlem9 12338	. . . . . . . . 9 |- ( ( ph /\ ( C ^ 2 ) = 0 ) -> ( M ^ 2 ) || ( A ^ 2 ) )
16	15	ex 114	. . . . . . . 8 |- ( ph -> ( ( C ^ 2 ) = 0 -> ( M ^ 2 ) || ( A ^ 2 ) ) )
17		eluzelz 9496	. . . . . . . . . 10 |- ( M e. ( ZZ>= ` 2 ) -> M e. ZZ )
18	2, 17	syl 14	. . . . . . . . 9 |- ( ph -> M e. ZZ )
19		dvdssq 11986	. . . . . . . . 9 |- ( ( M e. ZZ /\ A e. ZZ ) -> ( M || A <-> ( M ^ 2 ) || ( A ^ 2 ) ) )
20	18, 1, 19	syl2anc 409	. . . . . . . 8 |- ( ph -> ( M || A <-> ( M ^ 2 ) || ( A ^ 2 ) ) )
21	16, 20	sylibrd 168	. . . . . . 7 |- ( ph -> ( ( C ^ 2 ) = 0 -> M || A ) )
22		simpr 109	. . . . . . . . . 10 |- ( ( ph /\ ( D ^ 2 ) = 0 ) -> ( D ^ 2 ) = 0 )
23	9, 5, 10, 22	4sqlem9 12338	. . . . . . . . 9 |- ( ( ph /\ ( D ^ 2 ) = 0 ) -> ( M ^ 2 ) || ( B ^ 2 ) )
24	23	ex 114	. . . . . . . 8 |- ( ph -> ( ( D ^ 2 ) = 0 -> ( M ^ 2 ) || ( B ^ 2 ) ) )
25		dvdssq 11986	. . . . . . . . 9 |- ( ( M e. ZZ /\ B e. ZZ ) -> ( M || B <-> ( M ^ 2 ) || ( B ^ 2 ) ) )
26	18, 9, 25	syl2anc 409	. . . . . . . 8 |- ( ph -> ( M || B <-> ( M ^ 2 ) || ( B ^ 2 ) ) )
27	24, 26	sylibrd 168	. . . . . . 7 |- ( ph -> ( ( D ^ 2 ) = 0 -> M || B ) )
28		2sqlem8.3	. . . . . . . . . . 11 |- ( ph -> ( A gcd B ) = 1 )
29		1ne0 8946	. . . . . . . . . . . 12 |- 1 =/= 0
30	29	a1i 9	. . . . . . . . . . 11 |- ( ph -> 1 =/= 0 )
31	28, 30	eqnetrd 2364	. . . . . . . . . 10 |- ( ph -> ( A gcd B ) =/= 0 )
32	31	neneqd 2361	. . . . . . . . 9 |- ( ph -> -. ( A gcd B ) = 0 )
33		gcdeq0 11932	. . . . . . . . . 10 |- ( ( A e. ZZ /\ B e. ZZ ) -> ( ( A gcd B ) = 0 <-> ( A = 0 /\ B = 0 ) ) )
34	1, 9, 33	syl2anc 409	. . . . . . . . 9 |- ( ph -> ( ( A gcd B ) = 0 <-> ( A = 0 /\ B = 0 ) ) )
35	32, 34	mtbid 667	. . . . . . . 8 |- ( ph -> -. ( A = 0 /\ B = 0 ) )
36		dvdslegcd 11919	. . . . . . . 8 |- ( ( ( M e. ZZ /\ A e. ZZ /\ B e. ZZ ) /\ -. ( A = 0 /\ B = 0 ) ) -> ( ( M || A /\ M || B ) -> M <_ ( A gcd B ) ) )
37	18, 1, 9, 35, 36	syl31anc 1236	. . . . . . 7 |- ( ph -> ( ( M || A /\ M || B ) -> M <_ ( A gcd B ) ) )
38	21, 27, 37	syl2and 293	. . . . . 6 |- ( ph -> ( ( ( C ^ 2 ) = 0 /\ ( D ^ 2 ) = 0 ) -> M <_ ( A gcd B ) ) )
39	28	breq2d 4001	. . . . . . 7 |- ( ph -> ( M <_ ( A gcd B ) <-> M <_ 1 ) )
40		nnle1eq1 8902	. . . . . . . 8 |- ( M e. NN -> ( M <_ 1 <-> M = 1 ) )
41	5, 40	syl 14	. . . . . . 7 |- ( ph -> ( M <_ 1 <-> M = 1 ) )
42	39, 41	bitrd 187	. . . . . 6 |- ( ph -> ( M <_ ( A gcd B ) <-> M = 1 ) )
43	38, 42	sylibd 148	. . . . 5 |- ( ph -> ( ( ( C ^ 2 ) = 0 /\ ( D ^ 2 ) = 0 ) -> M = 1 ) )
44	43	necon3ad 2382	. . . 4 |- ( ph -> ( M =/= 1 -> -. ( ( C ^ 2 ) = 0 /\ ( D ^ 2 ) = 0 ) ) )
45	13, 44	mpd 13	. . 3 |- ( ph -> -. ( ( C ^ 2 ) = 0 /\ ( D ^ 2 ) = 0 ) )
46	8	zcnd 9335	. . . . 5 |- ( ph -> C e. CC )
47		sqeq0 10539	. . . . 5 |- ( C e. CC -> ( ( C ^ 2 ) = 0 <-> C = 0 ) )
48	46, 47	syl 14	. . . 4 |- ( ph -> ( ( C ^ 2 ) = 0 <-> C = 0 ) )
49	12	zcnd 9335	. . . . 5 |- ( ph -> D e. CC )
50		sqeq0 10539	. . . . 5 |- ( D e. CC -> ( ( D ^ 2 ) = 0 <-> D = 0 ) )
51	49, 50	syl 14	. . . 4 |- ( ph -> ( ( D ^ 2 ) = 0 <-> D = 0 ) )
52	48, 51	anbi12d 470	. . 3 |- ( ph -> ( ( ( C ^ 2 ) = 0 /\ ( D ^ 2 ) = 0 ) <-> ( C = 0 /\ D = 0 ) ) )
53	45, 52	mtbid 667	. 2 |- ( ph -> -. ( C = 0 /\ D = 0 ) )
54		gcdn0cl 11917	. 2 |- ( ( ( C e. ZZ /\ D e. ZZ ) /\ -. ( C = 0 /\ D = 0 ) ) -> ( C gcd D ) e. NN )
55	8, 12, 53, 54	syl21anc 1232	1 |- ( ph -> ( C gcd D ) e. NN )

Syntax hints:   -. wn 3   -> wi 4   /\ wa 103   <-> wb 104   = wceq 1348   e. wcel 2141   {cab 2156   =/= wne 2340   A.wral 2448   E.wrex 2449   class class class wbr 3989   |-> cmpt 4050   ran crn 4612   ` cfv 5198  (class class class)co 5853   CCcc 7772   0cc0 7774   1c1 7775   + caddc 7777   <_ cle 7955   - cmin 8090   / cdiv 8589   NNcn 8878   2c2 8929   ZZcz 9212   ZZ>=cuz 9487   ...cfz 9965   mod cmo 10278   ^cexp 10475   abscabs 10961   || cdvds 11749   gcd cgcd 11897   ZZ[_i]cgz 12321

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-in1 609  ax-in2 610  ax-io 704  ax-5 1440  ax-7 1441  ax-gen 1442  ax-ie1 1486  ax-ie2 1487  ax-8 1497  ax-10 1498  ax-11 1499  ax-i12 1500  ax-bndl 1502  ax-4 1503  ax-17 1519  ax-i9 1523  ax-ial 1527  ax-i5r 1528  ax-13 2143  ax-14 2144  ax-ext 2152  ax-coll 4104  ax-sep 4107  ax-nul 4115  ax-pow 4160  ax-pr 4194  ax-un 4418  ax-setind 4521  ax-iinf 4572  ax-cnex 7865  ax-resscn 7866  ax-1cn 7867  ax-1re 7868  ax-icn 7869  ax-addcl 7870  ax-addrcl 7871  ax-mulcl 7872  ax-mulrcl 7873  ax-addcom 7874  ax-mulcom 7875  ax-addass 7876  ax-mulass 7877  ax-distr 7878  ax-i2m1 7879  ax-0lt1 7880  ax-1rid 7881  ax-0id 7882  ax-rnegex 7883  ax-precex 7884  ax-cnre 7885  ax-pre-ltirr 7886  ax-pre-ltwlin 7887  ax-pre-lttrn 7888  ax-pre-apti 7889  ax-pre-ltadd 7890  ax-pre-mulgt0 7891  ax-pre-mulext 7892  ax-arch 7893  ax-caucvg 7894

This theorem depends on definitions:  df-bi 116  df-stab 826  df-dc 830  df-3or 974  df-3an 975  df-tru 1351  df-fal 1354  df-nf 1454  df-sb 1756  df-eu 2022  df-mo 2023  df-clab 2157  df-cleq 2163  df-clel 2166  df-nfc 2301  df-ne 2341  df-nel 2436  df-ral 2453  df-rex 2454  df-reu 2455  df-rmo 2456  df-rab 2457  df-v 2732  df-sbc 2956  df-csb 3050  df-dif 3123  df-un 3125  df-in 3127  df-ss 3134  df-nul 3415  df-if 3527  df-pw 3568  df-sn 3589  df-pr 3590  df-op 3592  df-uni 3797  df-int 3832  df-iun 3875  df-br 3990  df-opab 4051  df-mpt 4052  df-tr 4088  df-id 4278  df-po 4281  df-iso 4282  df-iord 4351  df-on 4353  df-ilim 4354  df-suc 4356  df-iom 4575  df-xp 4617  df-rel 4618  df-cnv 4619  df-co 4620  df-dm 4621  df-rn 4622  df-res 4623  df-ima 4624  df-iota 5160  df-fun 5200  df-fn 5201  df-f 5202  df-f1 5203  df-fo 5204  df-f1o 5205  df-fv 5206  df-riota 5809  df-ov 5856  df-oprab 5857  df-mpo 5858  df-1st 6119  df-2nd 6120  df-recs 6284  df-frec 6370  df-sup 6961  df-pnf 7956  df-mnf 7957  df-xr 7958  df-ltxr 7959  df-le 7960  df-sub 8092  df-neg 8093  df-reap 8494  df-ap 8501  df-div 8590  df-inn 8879  df-2 8937  df-3 8938  df-4 8939  df-n0 9136  df-z 9213  df-uz 9488  df-q 9579  df-rp 9611  df-fz 9966  df-fzo 10099  df-fl 10226  df-mod 10279  df-seqfrec 10402  df-exp 10476  df-cj 10806  df-re 10807  df-im 10808  df-rsqrt 10962  df-abs 10963  df-dvds 11750  df-gcd 11898

This theorem is referenced by:  2sqlem8  13753