Theorem 2sqlem9 13560

Description: Lemma for 2sq . (Contributed by Mario Carneiro, 19-Jun-2015.)

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 )
2sqlem9.6	|- ( ph -> M e. NN )
2sqlem9.4	|- ( ph -> N e. Y )

Assertion

Ref	Expression
2sqlem9	|- ( ph -> M e. S )

Distinct variable groups:   a, b, w, x, y, z   ph, x, y   M, a, b, x, y, z   S, a, b, x, y, z   x, N, y, z   Y, a, b, x, y

Allowed substitution hints:   ph( z, w, a, b)   S( w)   M( w)   N( w, a, b)   Y( z, w)

Proof of Theorem 2sqlem9

Dummy variables u v are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		2sqlem9.4	. . 3 |- ( ph -> N e. Y )
2		eqeq1 2172	. . . . . . . 8 |- ( z = N -> ( z = ( ( x ^ 2 ) + ( y ^ 2 ) ) <-> N = ( ( x ^ 2 ) + ( y ^ 2 ) ) ) )
3	2	anbi2d 460	. . . . . . 7 |- ( z = N -> ( ( ( x gcd y ) = 1 /\ z = ( ( x ^ 2 ) + ( y ^ 2 ) ) ) <-> ( ( x gcd y ) = 1 /\ N = ( ( x ^ 2 ) + ( y ^ 2 ) ) ) ) )
4	3	2rexbidv 2490	. . . . . 6 |- ( z = N -> ( E. x e. ZZ E. y e. ZZ ( ( x gcd y ) = 1 /\ z = ( ( x ^ 2 ) + ( y ^ 2 ) ) ) <-> E. x e. ZZ E. y e. ZZ ( ( x gcd y ) = 1 /\ N = ( ( x ^ 2 ) + ( y ^ 2 ) ) ) ) )
5		oveq1 5848	. . . . . . . . 9 |- ( x = u -> ( x gcd y ) = ( u gcd y ) )
6	5	eqeq1d 2174	. . . . . . . 8 |- ( x = u -> ( ( x gcd y ) = 1 <-> ( u gcd y ) = 1 ) )
7		oveq1 5848	. . . . . . . . . 10 |- ( x = u -> ( x ^ 2 ) = ( u ^ 2 ) )
8	7	oveq1d 5856	. . . . . . . . 9 |- ( x = u -> ( ( x ^ 2 ) + ( y ^ 2 ) ) = ( ( u ^ 2 ) + ( y ^ 2 ) ) )
9	8	eqeq2d 2177	. . . . . . . 8 |- ( x = u -> ( N = ( ( x ^ 2 ) + ( y ^ 2 ) ) <-> N = ( ( u ^ 2 ) + ( y ^ 2 ) ) ) )
10	6, 9	anbi12d 465	. . . . . . 7 |- ( x = u -> ( ( ( x gcd y ) = 1 /\ N = ( ( x ^ 2 ) + ( y ^ 2 ) ) ) <-> ( ( u gcd y ) = 1 /\ N = ( ( u ^ 2 ) + ( y ^ 2 ) ) ) ) )
11		oveq2 5849	. . . . . . . . 9 |- ( y = v -> ( u gcd y ) = ( u gcd v ) )
12	11	eqeq1d 2174	. . . . . . . 8 |- ( y = v -> ( ( u gcd y ) = 1 <-> ( u gcd v ) = 1 ) )
13		oveq1 5848	. . . . . . . . . 10 |- ( y = v -> ( y ^ 2 ) = ( v ^ 2 ) )
14	13	oveq2d 5857	. . . . . . . . 9 |- ( y = v -> ( ( u ^ 2 ) + ( y ^ 2 ) ) = ( ( u ^ 2 ) + ( v ^ 2 ) ) )
15	14	eqeq2d 2177	. . . . . . . 8 |- ( y = v -> ( N = ( ( u ^ 2 ) + ( y ^ 2 ) ) <-> N = ( ( u ^ 2 ) + ( v ^ 2 ) ) ) )
16	12, 15	anbi12d 465	. . . . . . 7 |- ( y = v -> ( ( ( u gcd y ) = 1 /\ N = ( ( u ^ 2 ) + ( y ^ 2 ) ) ) <-> ( ( u gcd v ) = 1 /\ N = ( ( u ^ 2 ) + ( v ^ 2 ) ) ) ) )
17	10, 16	cbvrex2vw 2703	. . . . . 6 |- ( E. x e. ZZ E. y e. ZZ ( ( x gcd y ) = 1 /\ N = ( ( x ^ 2 ) + ( y ^ 2 ) ) ) <-> E. u e. ZZ E. v e. ZZ ( ( u gcd v ) = 1 /\ N = ( ( u ^ 2 ) + ( v ^ 2 ) ) ) )
18	4, 17	bitrdi 195	. . . . 5 |- ( z = N -> ( E. x e. ZZ E. y e. ZZ ( ( x gcd y ) = 1 /\ z = ( ( x ^ 2 ) + ( y ^ 2 ) ) ) <-> E. u e. ZZ E. v e. ZZ ( ( u gcd v ) = 1 /\ N = ( ( u ^ 2 ) + ( v ^ 2 ) ) ) ) )
19		2sqlem7.2	. . . . 5 |- Y = { z | E. x e. ZZ E. y e. ZZ ( ( x gcd y ) = 1 /\ z = ( ( x ^ 2 ) + ( y ^ 2 ) ) ) }
20	18, 19	elab2g 2872	. . . 4 |- ( N e. Y -> ( N e. Y <-> E. u e. ZZ E. v e. ZZ ( ( u gcd v ) = 1 /\ N = ( ( u ^ 2 ) + ( v ^ 2 ) ) ) ) )
21	20	ibi 175	. . 3 |- ( N e. Y -> E. u e. ZZ E. v e. ZZ ( ( u gcd v ) = 1 /\ N = ( ( u ^ 2 ) + ( v ^ 2 ) ) ) )
22	1, 21	syl 14	. 2 |- ( ph -> E. u e. ZZ E. v e. ZZ ( ( u gcd v ) = 1 /\ N = ( ( u ^ 2 ) + ( v ^ 2 ) ) ) )
23		simpr 109	. . . . . 6 |- ( ( ( ( ph /\ ( u e. ZZ /\ v e. ZZ ) ) /\ ( ( u gcd v ) = 1 /\ N = ( ( u ^ 2 ) + ( v ^ 2 ) ) ) ) /\ M = 1 ) -> M = 1 )
24		1z 9213	. . . . . . . . 9 |- 1 e. ZZ
25		zgz 12299	. . . . . . . . 9 |- ( 1 e. ZZ -> 1 e. ZZ[_i] )
26	24, 25	ax-mp 5	. . . . . . . 8 |- 1 e. ZZ[_i]
27		sq1 10544	. . . . . . . . 9 |- ( 1 ^ 2 ) = 1
28	27	eqcomi 2169	. . . . . . . 8 |- 1 = ( 1 ^ 2 )
29		fveq2 5485	. . . . . . . . . . 11 |- ( x = 1 -> ( abs ` x ) = ( abs ` 1 ) )
30		abs1 11010	. . . . . . . . . . 11 |- ( abs ` 1 ) = 1
31	29, 30	eqtrdi 2214	. . . . . . . . . 10 |- ( x = 1 -> ( abs ` x ) = 1 )
32	31	oveq1d 5856	. . . . . . . . 9 |- ( x = 1 -> ( ( abs ` x ) ^ 2 ) = ( 1 ^ 2 ) )
33	32	rspceeqv 2847	. . . . . . . 8 |- ( ( 1 e. ZZ[_i] /\ 1 = ( 1 ^ 2 ) ) -> E. x e. ZZ[_i] 1 = ( ( abs ` x ) ^ 2 ) )
34	26, 28, 33	mp2an 423	. . . . . . 7 |- E. x e. ZZ[_i] 1 = ( ( abs ` x ) ^ 2 )
35		2sq.1	. . . . . . . 8 |- S = ran ( w e. ZZ[_i] |-> ( ( abs ` w ) ^ 2 ) )
36	35	2sqlem1 13550	. . . . . . 7 |- ( 1 e. S <-> E. x e. ZZ[_i] 1 = ( ( abs ` x ) ^ 2 ) )
37	34, 36	mpbir 145	. . . . . 6 |- 1 e. S
38	23, 37	eqeltrdi 2256	. . . . 5 |- ( ( ( ( ph /\ ( u e. ZZ /\ v e. ZZ ) ) /\ ( ( u gcd v ) = 1 /\ N = ( ( u ^ 2 ) + ( v ^ 2 ) ) ) ) /\ M = 1 ) -> M e. S )
39		2sqlem9.5	. . . . . . . 8 |- ( ph -> A. b e. ( 1 ... ( M - 1 ) ) A. a e. Y ( b || a -> b e. S ) )
40	39	ad2antrr 480	. . . . . . 7 |- ( ( ( ph /\ ( u e. ZZ /\ v e. ZZ ) ) /\ ( ( ( u gcd v ) = 1 /\ N = ( ( u ^ 2 ) + ( v ^ 2 ) ) ) /\ M =/= 1 ) ) -> A. b e. ( 1 ... ( M - 1 ) ) A. a e. Y ( b || a -> b e. S ) )
41		2sqlem9.7	. . . . . . . 8 |- ( ph -> M || N )
42	41	ad2antrr 480	. . . . . . 7 |- ( ( ( ph /\ ( u e. ZZ /\ v e. ZZ ) ) /\ ( ( ( u gcd v ) = 1 /\ N = ( ( u ^ 2 ) + ( v ^ 2 ) ) ) /\ M =/= 1 ) ) -> M || N )
43	35, 19	2sqlem7 13557	. . . . . . . . . 10 |- Y C_ ( S i^i NN )
44		inss2 3342	. . . . . . . . . 10 |- ( S i^i NN ) C_ NN
45	43, 44	sstri 3150	. . . . . . . . 9 |- Y C_ NN
46	45, 1	sselid 3139	. . . . . . . 8 |- ( ph -> N e. NN )
47	46	ad2antrr 480	. . . . . . 7 |- ( ( ( ph /\ ( u e. ZZ /\ v e. ZZ ) ) /\ ( ( ( u gcd v ) = 1 /\ N = ( ( u ^ 2 ) + ( v ^ 2 ) ) ) /\ M =/= 1 ) ) -> N e. NN )
48		2sqlem9.6	. . . . . . . . 9 |- ( ph -> M e. NN )
49	48	ad2antrr 480	. . . . . . . 8 |- ( ( ( ph /\ ( u e. ZZ /\ v e. ZZ ) ) /\ ( ( ( u gcd v ) = 1 /\ N = ( ( u ^ 2 ) + ( v ^ 2 ) ) ) /\ M =/= 1 ) ) -> M e. NN )
50		simprr 522	. . . . . . . 8 |- ( ( ( ph /\ ( u e. ZZ /\ v e. ZZ ) ) /\ ( ( ( u gcd v ) = 1 /\ N = ( ( u ^ 2 ) + ( v ^ 2 ) ) ) /\ M =/= 1 ) ) -> M =/= 1 )
51		eluz2b3 9538	. . . . . . . 8 |- ( M e. ( ZZ>= ` 2 ) <-> ( M e. NN /\ M =/= 1 ) )
52	49, 50, 51	sylanbrc 414	. . . . . . 7 |- ( ( ( ph /\ ( u e. ZZ /\ v e. ZZ ) ) /\ ( ( ( u gcd v ) = 1 /\ N = ( ( u ^ 2 ) + ( v ^ 2 ) ) ) /\ M =/= 1 ) ) -> M e. ( ZZ>= ` 2 ) )
53		simplrl 525	. . . . . . 7 |- ( ( ( ph /\ ( u e. ZZ /\ v e. ZZ ) ) /\ ( ( ( u gcd v ) = 1 /\ N = ( ( u ^ 2 ) + ( v ^ 2 ) ) ) /\ M =/= 1 ) ) -> u e. ZZ )
54		simplrr 526	. . . . . . 7 |- ( ( ( ph /\ ( u e. ZZ /\ v e. ZZ ) ) /\ ( ( ( u gcd v ) = 1 /\ N = ( ( u ^ 2 ) + ( v ^ 2 ) ) ) /\ M =/= 1 ) ) -> v e. ZZ )
55		simprll 527	. . . . . . 7 |- ( ( ( ph /\ ( u e. ZZ /\ v e. ZZ ) ) /\ ( ( ( u gcd v ) = 1 /\ N = ( ( u ^ 2 ) + ( v ^ 2 ) ) ) /\ M =/= 1 ) ) -> ( u gcd v ) = 1 )
56		simprlr 528	. . . . . . 7 |- ( ( ( ph /\ ( u e. ZZ /\ v e. ZZ ) ) /\ ( ( ( u gcd v ) = 1 /\ N = ( ( u ^ 2 ) + ( v ^ 2 ) ) ) /\ M =/= 1 ) ) -> N = ( ( u ^ 2 ) + ( v ^ 2 ) ) )
57		eqid 2165	. . . . . . 7 |- ( ( ( u + ( M / 2 ) ) mod M ) - ( M / 2 ) ) = ( ( ( u + ( M / 2 ) ) mod M ) - ( M / 2 ) )
58		eqid 2165	. . . . . . 7 |- ( ( ( v + ( M / 2 ) ) mod M ) - ( M / 2 ) ) = ( ( ( v + ( M / 2 ) ) mod M ) - ( M / 2 ) )
59		eqid 2165	. . . . . . 7 |- ( ( ( ( u + ( M / 2 ) ) mod M ) - ( M / 2 ) ) / ( ( ( ( u + ( M / 2 ) ) mod M ) - ( M / 2 ) ) gcd ( ( ( v + ( M / 2 ) ) mod M ) - ( M / 2 ) ) ) ) = ( ( ( ( u + ( M / 2 ) ) mod M ) - ( M / 2 ) ) / ( ( ( ( u + ( M / 2 ) ) mod M ) - ( M / 2 ) ) gcd ( ( ( v + ( M / 2 ) ) mod M ) - ( M / 2 ) ) ) )
60		eqid 2165	. . . . . . 7 |- ( ( ( ( v + ( M / 2 ) ) mod M ) - ( M / 2 ) ) / ( ( ( ( u + ( M / 2 ) ) mod M ) - ( M / 2 ) ) gcd ( ( ( v + ( M / 2 ) ) mod M ) - ( M / 2 ) ) ) ) = ( ( ( ( v + ( M / 2 ) ) mod M ) - ( M / 2 ) ) / ( ( ( ( u + ( M / 2 ) ) mod M ) - ( M / 2 ) ) gcd ( ( ( v + ( M / 2 ) ) mod M ) - ( M / 2 ) ) ) )
61	35, 19, 40, 42, 47, 52, 53, 54, 55, 56, 57, 58, 59, 60	2sqlem8 13559	. . . . . 6 |- ( ( ( ph /\ ( u e. ZZ /\ v e. ZZ ) ) /\ ( ( ( u gcd v ) = 1 /\ N = ( ( u ^ 2 ) + ( v ^ 2 ) ) ) /\ M =/= 1 ) ) -> M e. S )
62	61	anassrs 398	. . . . 5 |- ( ( ( ( ph /\ ( u e. ZZ /\ v e. ZZ ) ) /\ ( ( u gcd v ) = 1 /\ N = ( ( u ^ 2 ) + ( v ^ 2 ) ) ) ) /\ M =/= 1 ) -> M e. S )
63	48	nnzd 9308	. . . . . . . 8 |- ( ph -> M e. ZZ )
64	63	ad2antrr 480	. . . . . . 7 |- ( ( ( ph /\ ( u e. ZZ /\ v e. ZZ ) ) /\ ( ( u gcd v ) = 1 /\ N = ( ( u ^ 2 ) + ( v ^ 2 ) ) ) ) -> M e. ZZ )
65		zdceq 9262	. . . . . . 7 |- ( ( M e. ZZ /\ 1 e. ZZ ) -> DECID M = 1 )
66	64, 24, 65	sylancl 410	. . . . . 6 |- ( ( ( ph /\ ( u e. ZZ /\ v e. ZZ ) ) /\ ( ( u gcd v ) = 1 /\ N = ( ( u ^ 2 ) + ( v ^ 2 ) ) ) ) -> DECID M = 1 )
67		dcne 2346	. . . . . 6 |- (DECID M = 1 <-> ( M = 1 \/ M =/= 1 ) )
68	66, 67	sylib 121	. . . . 5 |- ( ( ( ph /\ ( u e. ZZ /\ v e. ZZ ) ) /\ ( ( u gcd v ) = 1 /\ N = ( ( u ^ 2 ) + ( v ^ 2 ) ) ) ) -> ( M = 1 \/ M =/= 1 ) )
69	38, 62, 68	mpjaodan 788	. . . 4 |- ( ( ( ph /\ ( u e. ZZ /\ v e. ZZ ) ) /\ ( ( u gcd v ) = 1 /\ N = ( ( u ^ 2 ) + ( v ^ 2 ) ) ) ) -> M e. S )
70	69	ex 114	. . 3 |- ( ( ph /\ ( u e. ZZ /\ v e. ZZ ) ) -> ( ( ( u gcd v ) = 1 /\ N = ( ( u ^ 2 ) + ( v ^ 2 ) ) ) -> M e. S ) )
71	70	rexlimdvva 2590	. 2 |- ( ph -> ( E. u e. ZZ E. v e. ZZ ( ( u gcd v ) = 1 /\ N = ( ( u ^ 2 ) + ( v ^ 2 ) ) ) -> M e. S ) )
72	22, 71	mpd 13	1 |- ( ph -> M e. S )

Syntax hints:   -> wi 4   /\ wa 103   \/ wo 698  DECID wdc 824   = wceq 1343   e. wcel 2136   {cab 2151   =/= wne 2335   A.wral 2443   E.wrex 2444   i^i cin 3114   class class class wbr 3981   |-> cmpt 4042   ran crn 4604   ` cfv 5187  (class class class)co 5841   1c1 7750   + caddc 7752   - 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-1o 6380  df-2o 6381  df-er 6497  df-en 6703  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  df-prm 12036  df-gz 12296

This theorem is referenced by:  2sqlem10  13561