Theorem 2sqlem8a 13558

Description: Lemma for 2sqlem8 13559. (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 9538	. . . . . 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 12308	. . 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 12308	. . 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 12312	. . . . . . . . 9 |- ( ( ph /\ ( C ^ 2 ) = 0 ) -> ( M ^ 2 ) || ( A ^ 2 ) )
16	15	ex 114	. . . . . . . 8 |- ( ph -> ( ( C ^ 2 ) = 0 -> ( M ^ 2 ) || ( A ^ 2 ) ) )
17		eluzelz 9471	. . . . . . . . . 10 |- ( M e. ( ZZ>= ` 2 ) -> M e. ZZ )
18	2, 17	syl 14	. . . . . . . . 9 |- ( ph -> M e. ZZ )
19		dvdssq 11960	. . . . . . . . 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 12312	. . . . . . . . 9 |- ( ( ph /\ ( D ^ 2 ) = 0 ) -> ( M ^ 2 ) || ( B ^ 2 ) )
24	23	ex 114	. . . . . . . 8 |- ( ph -> ( ( D ^ 2 ) = 0 -> ( M ^ 2 ) || ( B ^ 2 ) ) )
25		dvdssq 11960	. . . . . . . . 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 8921	. . . . . . . . . . . 12 |- 1 =/= 0
30	29	a1i 9	. . . . . . . . . . 11 |- ( ph -> 1 =/= 0 )
31	28, 30	eqnetrd 2359	. . . . . . . . . 10 |- ( ph -> ( A gcd B ) =/= 0 )
32	31	neneqd 2356	. . . . . . . . 9 |- ( ph -> -. ( A gcd B ) = 0 )
33		gcdeq0 11906	. . . . . . . . . 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 662	. . . . . . . 8 |- ( ph -> -. ( A = 0 /\ B = 0 ) )
36		dvdslegcd 11893	. . . . . . . 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 1231	. . . . . . 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 3993	. . . . . . 7 |- ( ph -> ( M <_ ( A gcd B ) <-> M <_ 1 ) )
40		nnle1eq1 8877	. . . . . . . 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 2377	. . . 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 9310	. . . . 5 |- ( ph -> C e. CC )
47		sqeq0 10514	. . . . 5 |- ( C e. CC -> ( ( C ^ 2 ) = 0 <-> C = 0 ) )
48	46, 47	syl 14	. . . 4 |- ( ph -> ( ( C ^ 2 ) = 0 <-> C = 0 ) )
49	12	zcnd 9310	. . . . 5 |- ( ph -> D e. CC )
50		sqeq0 10514	. . . . 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 465	. . 3 |- ( ph -> ( ( ( C ^ 2 ) = 0 /\ ( D ^ 2 ) = 0 ) <-> ( C = 0 /\ D = 0 ) ) )
53	45, 52	mtbid 662	. 2 |- ( ph -> -. ( C = 0 /\ D = 0 ) )
54		gcdn0cl 11891	. 2 |- ( ( ( C e. ZZ /\ D e. ZZ ) /\ -. ( C = 0 /\ D = 0 ) ) -> ( C gcd D ) e. NN )
55	8, 12, 53, 54	syl21anc 1227	1 |- ( ph -> ( C gcd D ) e. NN )

Syntax hints:   -. wn 3   -> wi 4   /\ wa 103   <-> wb 104   = wceq 1343   e. wcel 2136   {cab 2151   =/= wne 2335   A.wral 2443   E.wrex 2444   class class class wbr 3981   |-> cmpt 4042   ran crn 4604   ` cfv 5187  (class class class)co 5841   CCcc 7747   0cc0 7749   1c1 7750   + caddc 7752   <_ cle 7930   - cmin 8065   / cdiv 8564   NNcn 8853   2c2 8904   ZZcz 9187   ZZ>=cuz 9462   ...cfz 9940   mod cmo 10253   ^cexp 10450   abscabs 10935   || cdvds 11723   gcd cgcd 11871   ZZ[_i]cgz 12295

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 604  ax-in2 605  ax-io 699  ax-5 1435  ax-7 1436  ax-gen 1437  ax-ie1 1481  ax-ie2 1482  ax-8 1492  ax-10 1493  ax-11 1494  ax-i12 1495  ax-bndl 1497  ax-4 1498  ax-17 1514  ax-i9 1518  ax-ial 1522  ax-i5r 1523  ax-13 2138  ax-14 2139  ax-ext 2147  ax-coll 4096  ax-sep 4099  ax-nul 4107  ax-pow 4152  ax-pr 4186  ax-un 4410  ax-setind 4513  ax-iinf 4564  ax-cnex 7840  ax-resscn 7841  ax-1cn 7842  ax-1re 7843  ax-icn 7844  ax-addcl 7845  ax-addrcl 7846  ax-mulcl 7847  ax-mulrcl 7848  ax-addcom 7849  ax-mulcom 7850  ax-addass 7851  ax-mulass 7852  ax-distr 7853  ax-i2m1 7854  ax-0lt1 7855  ax-1rid 7856  ax-0id 7857  ax-rnegex 7858  ax-precex 7859  ax-cnre 7860  ax-pre-ltirr 7861  ax-pre-ltwlin 7862  ax-pre-lttrn 7863  ax-pre-apti 7864  ax-pre-ltadd 7865  ax-pre-mulgt0 7866  ax-pre-mulext 7867  ax-arch 7868  ax-caucvg 7869

This theorem depends on definitions:  df-bi 116  df-stab 821  df-dc 825  df-3or 969  df-3an 970  df-tru 1346  df-fal 1349  df-nf 1449  df-sb 1751  df-eu 2017  df-mo 2018  df-clab 2152  df-cleq 2158  df-clel 2161  df-nfc 2296  df-ne 2336  df-nel 2431  df-ral 2448  df-rex 2449  df-reu 2450  df-rmo 2451  df-rab 2452  df-v 2727  df-sbc 2951  df-csb 3045  df-dif 3117  df-un 3119  df-in 3121  df-ss 3128  df-nul 3409  df-if 3520  df-pw 3560  df-sn 3581  df-pr 3582  df-op 3584  df-uni 3789  df-int 3824  df-iun 3867  df-br 3982  df-opab 4043  df-mpt 4044  df-tr 4080  df-id 4270  df-po 4273  df-iso 4274  df-iord 4343  df-on 4345  df-ilim 4346  df-suc 4348  df-iom 4567  df-xp 4609  df-rel 4610  df-cnv 4611  df-co 4612  df-dm 4613  df-rn 4614  df-res 4615  df-ima 4616  df-iota 5152  df-fun 5189  df-fn 5190  df-f 5191  df-f1 5192  df-fo 5193  df-f1o 5194  df-fv 5195  df-riota 5797  df-ov 5844  df-oprab 5845  df-mpo 5846  df-1st 6105  df-2nd 6106  df-recs 6269  df-frec 6355  df-sup 6945  df-pnf 7931  df-mnf 7932  df-xr 7933  df-ltxr 7934  df-le 7935  df-sub 8067  df-neg 8068  df-reap 8469  df-ap 8476  df-div 8565  df-inn 8854  df-2 8912  df-3 8913  df-4 8914  df-n0 9111  df-z 9188  df-uz 9463  df-q 9554  df-rp 9586  df-fz 9941  df-fzo 10074  df-fl 10201  df-mod 10254  df-seqfrec 10377  df-exp 10451  df-cj 10780  df-re 10781  df-im 10782  df-rsqrt 10936  df-abs 10937  df-dvds 11724  df-gcd 11872

This theorem is referenced by:  2sqlem8  13559