Theorem 2sqlem9 15852

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 2238	. . . . . . . 8 |- ( z = N -> ( z = ( ( x ^ 2 ) + ( y ^ 2 ) ) <-> N = ( ( x ^ 2 ) + ( y ^ 2 ) ) ) )
3	2	anbi2d 464	. . . . . . 7 |- ( z = N -> ( ( ( x gcd y ) = 1 /\ z = ( ( x ^ 2 ) + ( y ^ 2 ) ) ) <-> ( ( x gcd y ) = 1 /\ N = ( ( x ^ 2 ) + ( y ^ 2 ) ) ) ) )
4	3	2rexbidv 2557	. . . . . 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 6024	. . . . . . . . 9 |- ( x = u -> ( x gcd y ) = ( u gcd y ) )
6	5	eqeq1d 2240	. . . . . . . 8 |- ( x = u -> ( ( x gcd y ) = 1 <-> ( u gcd y ) = 1 ) )
7		oveq1 6024	. . . . . . . . . 10 |- ( x = u -> ( x ^ 2 ) = ( u ^ 2 ) )
8	7	oveq1d 6032	. . . . . . . . 9 |- ( x = u -> ( ( x ^ 2 ) + ( y ^ 2 ) ) = ( ( u ^ 2 ) + ( y ^ 2 ) ) )
9	8	eqeq2d 2243	. . . . . . . 8 |- ( x = u -> ( N = ( ( x ^ 2 ) + ( y ^ 2 ) ) <-> N = ( ( u ^ 2 ) + ( y ^ 2 ) ) ) )
10	6, 9	anbi12d 473	. . . . . . 7 |- ( x = u -> ( ( ( x gcd y ) = 1 /\ N = ( ( x ^ 2 ) + ( y ^ 2 ) ) ) <-> ( ( u gcd y ) = 1 /\ N = ( ( u ^ 2 ) + ( y ^ 2 ) ) ) ) )
11		oveq2 6025	. . . . . . . . 9 |- ( y = v -> ( u gcd y ) = ( u gcd v ) )
12	11	eqeq1d 2240	. . . . . . . 8 |- ( y = v -> ( ( u gcd y ) = 1 <-> ( u gcd v ) = 1 ) )
13		oveq1 6024	. . . . . . . . . 10 |- ( y = v -> ( y ^ 2 ) = ( v ^ 2 ) )
14	13	oveq2d 6033	. . . . . . . . 9 |- ( y = v -> ( ( u ^ 2 ) + ( y ^ 2 ) ) = ( ( u ^ 2 ) + ( v ^ 2 ) ) )
15	14	eqeq2d 2243	. . . . . . . 8 |- ( y = v -> ( N = ( ( u ^ 2 ) + ( y ^ 2 ) ) <-> N = ( ( u ^ 2 ) + ( v ^ 2 ) ) ) )
16	12, 15	anbi12d 473	. . . . . . 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 2779	. . . . . 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 196	. . . . 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 2953	. . . 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 176	. . 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 110	. . . . . 6 |- ( ( ( ( ph /\ ( u e. ZZ /\ v e. ZZ ) ) /\ ( ( u gcd v ) = 1 /\ N = ( ( u ^ 2 ) + ( v ^ 2 ) ) ) ) /\ M = 1 ) -> M = 1 )
24		1z 9504	. . . . . . . . 9 |- 1 e. ZZ
25		zgz 12945	. . . . . . . . 9 |- ( 1 e. ZZ -> 1 e. ZZ[_i] )
26	24, 25	ax-mp 5	. . . . . . . 8 |- 1 e. ZZ[_i]
27		sq1 10894	. . . . . . . . 9 |- ( 1 ^ 2 ) = 1
28	27	eqcomi 2235	. . . . . . . 8 |- 1 = ( 1 ^ 2 )
29		fveq2 5639	. . . . . . . . . . 11 |- ( x = 1 -> ( abs ` x ) = ( abs ` 1 ) )
30		abs1 11632	. . . . . . . . . . 11 |- ( abs ` 1 ) = 1
31	29, 30	eqtrdi 2280	. . . . . . . . . 10 |- ( x = 1 -> ( abs ` x ) = 1 )
32	31	oveq1d 6032	. . . . . . . . 9 |- ( x = 1 -> ( ( abs ` x ) ^ 2 ) = ( 1 ^ 2 ) )
33	32	rspceeqv 2928	. . . . . . . 8 |- ( ( 1 e. ZZ[_i] /\ 1 = ( 1 ^ 2 ) ) -> E. x e. ZZ[_i] 1 = ( ( abs ` x ) ^ 2 ) )
34	26, 28, 33	mp2an 426	. . . . . . 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 15842	. . . . . . 7 |- ( 1 e. S <-> E. x e. ZZ[_i] 1 = ( ( abs ` x ) ^ 2 ) )
37	34, 36	mpbir 146	. . . . . 6 |- 1 e. S
38	23, 37	eqeltrdi 2322	. . . . 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 488	. . . . . . 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 488	. . . . . . 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 15849	. . . . . . . . . 10 |- Y C_ ( S i^i NN )
44		inss2 3428	. . . . . . . . . 10 |- ( S i^i NN ) C_ NN
45	43, 44	sstri 3236	. . . . . . . . 9 |- Y C_ NN
46	45, 1	sselid 3225	. . . . . . . 8 |- ( ph -> N e. NN )
47	46	ad2antrr 488	. . . . . . 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 488	. . . . . . . 8 |- ( ( ( ph /\ ( u e. ZZ /\ v e. ZZ ) ) /\ ( ( ( u gcd v ) = 1 /\ N = ( ( u ^ 2 ) + ( v ^ 2 ) ) ) /\ M =/= 1 ) ) -> M e. NN )
50		simprr 533	. . . . . . . 8 |- ( ( ( ph /\ ( u e. ZZ /\ v e. ZZ ) ) /\ ( ( ( u gcd v ) = 1 /\ N = ( ( u ^ 2 ) + ( v ^ 2 ) ) ) /\ M =/= 1 ) ) -> M =/= 1 )
51		eluz2b3 9837	. . . . . . . 8 |- ( M e. ( ZZ>= ` 2 ) <-> ( M e. NN /\ M =/= 1 ) )
52	49, 50, 51	sylanbrc 417	. . . . . . 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 537	. . . . . . 7 |- ( ( ( ph /\ ( u e. ZZ /\ v e. ZZ ) ) /\ ( ( ( u gcd v ) = 1 /\ N = ( ( u ^ 2 ) + ( v ^ 2 ) ) ) /\ M =/= 1 ) ) -> u e. ZZ )
54		simplrr 538	. . . . . . 7 |- ( ( ( ph /\ ( u e. ZZ /\ v e. ZZ ) ) /\ ( ( ( u gcd v ) = 1 /\ N = ( ( u ^ 2 ) + ( v ^ 2 ) ) ) /\ M =/= 1 ) ) -> v e. ZZ )
55		simprll 539	. . . . . . 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 540	. . . . . . 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 2231	. . . . . . 7 |- ( ( ( u + ( M / 2 ) ) mod M ) - ( M / 2 ) ) = ( ( ( u + ( M / 2 ) ) mod M ) - ( M / 2 ) )
58		eqid 2231	. . . . . . 7 |- ( ( ( v + ( M / 2 ) ) mod M ) - ( M / 2 ) ) = ( ( ( v + ( M / 2 ) ) mod M ) - ( M / 2 ) )
59		eqid 2231	. . . . . . 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 2231	. . . . . . 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 15851	. . . . . 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 400	. . . . 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 9600	. . . . . . . 8 |- ( ph -> M e. ZZ )
64	63	ad2antrr 488	. . . . . . 7 |- ( ( ( ph /\ ( u e. ZZ /\ v e. ZZ ) ) /\ ( ( u gcd v ) = 1 /\ N = ( ( u ^ 2 ) + ( v ^ 2 ) ) ) ) -> M e. ZZ )
65		zdceq 9554	. . . . . . 7 |- ( ( M e. ZZ /\ 1 e. ZZ ) -> DECID M = 1 )
66	64, 24, 65	sylancl 413	. . . . . 6 |- ( ( ( ph /\ ( u e. ZZ /\ v e. ZZ ) ) /\ ( ( u gcd v ) = 1 /\ N = ( ( u ^ 2 ) + ( v ^ 2 ) ) ) ) -> DECID M = 1 )
67		dcne 2413	. . . . . 6 |- (DECID M = 1 <-> ( M = 1 \/ M =/= 1 ) )
68	66, 67	sylib 122	. . . . 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 805	. . . 4 |- ( ( ( ph /\ ( u e. ZZ /\ v e. ZZ ) ) /\ ( ( u gcd v ) = 1 /\ N = ( ( u ^ 2 ) + ( v ^ 2 ) ) ) ) -> M e. S )
70	69	ex 115	. . 3 |- ( ( ph /\ ( u e. ZZ /\ v e. ZZ ) ) -> ( ( ( u gcd v ) = 1 /\ N = ( ( u ^ 2 ) + ( v ^ 2 ) ) ) -> M e. S ) )
71	70	rexlimdvva 2658	. 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 104   \/ wo 715  DECID wdc 841   = wceq 1397   e. wcel 2202   {cab 2217   =/= wne 2402   A.wral 2510   E.wrex 2511   i^i cin 3199   class class class wbr 4088   |-> cmpt 4150   ran crn 4726   ` cfv 5326  (class class class)co 6017   1c1 8032   + caddc 8034   - cmin 8349   / cdiv 8851   NNcn 9142   2c2 9193   ZZcz 9478   ZZ>=cuz 9754   ...cfz 10242   mod cmo 10583   ^cexp 10799   abscabs 11557   || cdvds 12347   gcd cgcd 12523   ZZ[_i]cgz 12941

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-in1 619  ax-in2 620  ax-io 716  ax-5 1495  ax-7 1496  ax-gen 1497  ax-ie1 1541  ax-ie2 1542  ax-8 1552  ax-10 1553  ax-11 1554  ax-i12 1555  ax-bndl 1557  ax-4 1558  ax-17 1574  ax-i9 1578  ax-ial 1582  ax-i5r 1583  ax-13 2204  ax-14 2205  ax-ext 2213  ax-coll 4204  ax-sep 4207  ax-nul 4215  ax-pow 4264  ax-pr 4299  ax-un 4530  ax-setind 4635  ax-iinf 4686  ax-cnex 8122  ax-resscn 8123  ax-1cn 8124  ax-1re 8125  ax-icn 8126  ax-addcl 8127  ax-addrcl 8128  ax-mulcl 8129  ax-mulrcl 8130  ax-addcom 8131  ax-mulcom 8132  ax-addass 8133  ax-mulass 8134  ax-distr 8135  ax-i2m1 8136  ax-0lt1 8137  ax-1rid 8138  ax-0id 8139  ax-rnegex 8140  ax-precex 8141  ax-cnre 8142  ax-pre-ltirr 8143  ax-pre-ltwlin 8144  ax-pre-lttrn 8145  ax-pre-apti 8146  ax-pre-ltadd 8147  ax-pre-mulgt0 8148  ax-pre-mulext 8149  ax-arch 8150  ax-caucvg 8151

This theorem depends on definitions:  df-bi 117  df-stab 838  df-dc 842  df-3or 1005  df-3an 1006  df-tru 1400  df-fal 1403  df-nf 1509  df-sb 1811  df-eu 2082  df-mo 2083  df-clab 2218  df-cleq 2224  df-clel 2227  df-nfc 2363  df-ne 2403  df-nel 2498  df-ral 2515  df-rex 2516  df-reu 2517  df-rmo 2518  df-rab 2519  df-v 2804  df-sbc 3032  df-csb 3128  df-dif 3202  df-un 3204  df-in 3206  df-ss 3213  df-nul 3495  df-if 3606  df-pw 3654  df-sn 3675  df-pr 3676  df-op 3678  df-uni 3894  df-int 3929  df-iun 3972  df-br 4089  df-opab 4151  df-mpt 4152  df-tr 4188  df-id 4390  df-po 4393  df-iso 4394  df-iord 4463  df-on 4465  df-ilim 4466  df-suc 4468  df-iom 4689  df-xp 4731  df-rel 4732  df-cnv 4733  df-co 4734  df-dm 4735  df-rn 4736  df-res 4737  df-ima 4738  df-iota 5286  df-fun 5328  df-fn 5329  df-f 5330  df-f1 5331  df-fo 5332  df-f1o 5333  df-fv 5334  df-riota 5970  df-ov 6020  df-oprab 6021  df-mpo 6022  df-1st 6302  df-2nd 6303  df-recs 6470  df-frec 6556  df-1o 6581  df-2o 6582  df-er 6701  df-en 6909  df-sup 7182  df-pnf 8215  df-mnf 8216  df-xr 8217  df-ltxr 8218  df-le 8219  df-sub 8351  df-neg 8352  df-reap 8754  df-ap 8761  df-div 8852  df-inn 9143  df-2 9201  df-3 9202  df-4 9203  df-n0 9402  df-z 9479  df-uz 9755  df-q 9853  df-rp 9888  df-fz 10243  df-fzo 10377  df-fl 10529  df-mod 10584  df-seqfrec 10709  df-exp 10800  df-cj 11402  df-re 11403  df-im 11404  df-rsqrt 11558  df-abs 11559  df-dvds 12348  df-gcd 12524  df-prm 12679  df-gz 12942

This theorem is referenced by:  2sqlem10  15853