Theorem 2sqlem9 14029

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 2182	. . . . . . . 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 2500	. . . . . 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 5872	. . . . . . . . 9 |- ( x = u -> ( x gcd y ) = ( u gcd y ) )
6	5	eqeq1d 2184	. . . . . . . 8 |- ( x = u -> ( ( x gcd y ) = 1 <-> ( u gcd y ) = 1 ) )
7		oveq1 5872	. . . . . . . . . 10 |- ( x = u -> ( x ^ 2 ) = ( u ^ 2 ) )
8	7	oveq1d 5880	. . . . . . . . 9 |- ( x = u -> ( ( x ^ 2 ) + ( y ^ 2 ) ) = ( ( u ^ 2 ) + ( y ^ 2 ) ) )
9	8	eqeq2d 2187	. . . . . . . 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 5873	. . . . . . . . 9 |- ( y = v -> ( u gcd y ) = ( u gcd v ) )
12	11	eqeq1d 2184	. . . . . . . 8 |- ( y = v -> ( ( u gcd y ) = 1 <-> ( u gcd v ) = 1 ) )
13		oveq1 5872	. . . . . . . . . 10 |- ( y = v -> ( y ^ 2 ) = ( v ^ 2 ) )
14	13	oveq2d 5881	. . . . . . . . 9 |- ( y = v -> ( ( u ^ 2 ) + ( y ^ 2 ) ) = ( ( u ^ 2 ) + ( v ^ 2 ) ) )
15	14	eqeq2d 2187	. . . . . . . 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 2713	. . . . . 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 2882	. . . 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 9250	. . . . . . . . 9 |- 1 e. ZZ
25		zgz 12336	. . . . . . . . 9 |- ( 1 e. ZZ -> 1 e. ZZ[_i] )
26	24, 25	ax-mp 5	. . . . . . . 8 |- 1 e. ZZ[_i]
27		sq1 10581	. . . . . . . . 9 |- ( 1 ^ 2 ) = 1
28	27	eqcomi 2179	. . . . . . . 8 |- 1 = ( 1 ^ 2 )
29		fveq2 5507	. . . . . . . . . . 11 |- ( x = 1 -> ( abs ` x ) = ( abs ` 1 ) )
30		abs1 11047	. . . . . . . . . . 11 |- ( abs ` 1 ) = 1
31	29, 30	eqtrdi 2224	. . . . . . . . . 10 |- ( x = 1 -> ( abs ` x ) = 1 )
32	31	oveq1d 5880	. . . . . . . . 9 |- ( x = 1 -> ( ( abs ` x ) ^ 2 ) = ( 1 ^ 2 ) )
33	32	rspceeqv 2857	. . . . . . . 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 14019	. . . . . . 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 2266	. . . . 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 14026	. . . . . . . . . 10 |- Y C_ ( S i^i NN )
44		inss2 3354	. . . . . . . . . 10 |- ( S i^i NN ) C_ NN
45	43, 44	sstri 3162	. . . . . . . . 9 |- Y C_ NN
46	45, 1	sselid 3151	. . . . . . . 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 531	. . . . . . . 8 |- ( ( ( ph /\ ( u e. ZZ /\ v e. ZZ ) ) /\ ( ( ( u gcd v ) = 1 /\ N = ( ( u ^ 2 ) + ( v ^ 2 ) ) ) /\ M =/= 1 ) ) -> M =/= 1 )
51		eluz2b3 9575	. . . . . . . 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 535	. . . . . . 7 |- ( ( ( ph /\ ( u e. ZZ /\ v e. ZZ ) ) /\ ( ( ( u gcd v ) = 1 /\ N = ( ( u ^ 2 ) + ( v ^ 2 ) ) ) /\ M =/= 1 ) ) -> u e. ZZ )
54		simplrr 536	. . . . . . 7 |- ( ( ( ph /\ ( u e. ZZ /\ v e. ZZ ) ) /\ ( ( ( u gcd v ) = 1 /\ N = ( ( u ^ 2 ) + ( v ^ 2 ) ) ) /\ M =/= 1 ) ) -> v e. ZZ )
55		simprll 537	. . . . . . 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 538	. . . . . . 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 2175	. . . . . . 7 |- ( ( ( u + ( M / 2 ) ) mod M ) - ( M / 2 ) ) = ( ( ( u + ( M / 2 ) ) mod M ) - ( M / 2 ) )
58		eqid 2175	. . . . . . 7 |- ( ( ( v + ( M / 2 ) ) mod M ) - ( M / 2 ) ) = ( ( ( v + ( M / 2 ) ) mod M ) - ( M / 2 ) )
59		eqid 2175	. . . . . . 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 2175	. . . . . . 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 14028	. . . . . 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 9345	. . . . . . . 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 9299	. . . . . . 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 2356	. . . . . 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 798	. . . 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 2600	. 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 708  DECID wdc 834   = wceq 1353   e. wcel 2146   {cab 2161   =/= wne 2345   A.wral 2453   E.wrex 2454   i^i cin 3126   class class class wbr 3998   |-> cmpt 4059   ran crn 4621   ` cfv 5208  (class class class)co 5865   1c1 7787   + caddc 7789   - cmin 8102   / cdiv 8601   NNcn 8890   2c2 8941   ZZcz 9224   ZZ>=cuz 9499   ...cfz 9977   mod cmo 10290   ^cexp 10487   abscabs 10972   || cdvds 11760   gcd cgcd 11908   ZZ[_i]cgz 12332

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 614  ax-in2 615  ax-io 709  ax-5 1445  ax-7 1446  ax-gen 1447  ax-ie1 1491  ax-ie2 1492  ax-8 1502  ax-10 1503  ax-11 1504  ax-i12 1505  ax-bndl 1507  ax-4 1508  ax-17 1524  ax-i9 1528  ax-ial 1532  ax-i5r 1533  ax-13 2148  ax-14 2149  ax-ext 2157  ax-coll 4113  ax-sep 4116  ax-nul 4124  ax-pow 4169  ax-pr 4203  ax-un 4427  ax-setind 4530  ax-iinf 4581  ax-cnex 7877  ax-resscn 7878  ax-1cn 7879  ax-1re 7880  ax-icn 7881  ax-addcl 7882  ax-addrcl 7883  ax-mulcl 7884  ax-mulrcl 7885  ax-addcom 7886  ax-mulcom 7887  ax-addass 7888  ax-mulass 7889  ax-distr 7890  ax-i2m1 7891  ax-0lt1 7892  ax-1rid 7893  ax-0id 7894  ax-rnegex 7895  ax-precex 7896  ax-cnre 7897  ax-pre-ltirr 7898  ax-pre-ltwlin 7899  ax-pre-lttrn 7900  ax-pre-apti 7901  ax-pre-ltadd 7902  ax-pre-mulgt0 7903  ax-pre-mulext 7904  ax-arch 7905  ax-caucvg 7906

This theorem depends on definitions:  df-bi 117  df-stab 831  df-dc 835  df-3or 979  df-3an 980  df-tru 1356  df-fal 1359  df-nf 1459  df-sb 1761  df-eu 2027  df-mo 2028  df-clab 2162  df-cleq 2168  df-clel 2171  df-nfc 2306  df-ne 2346  df-nel 2441  df-ral 2458  df-rex 2459  df-reu 2460  df-rmo 2461  df-rab 2462  df-v 2737  df-sbc 2961  df-csb 3056  df-dif 3129  df-un 3131  df-in 3133  df-ss 3140  df-nul 3421  df-if 3533  df-pw 3574  df-sn 3595  df-pr 3596  df-op 3598  df-uni 3806  df-int 3841  df-iun 3884  df-br 3999  df-opab 4060  df-mpt 4061  df-tr 4097  df-id 4287  df-po 4290  df-iso 4291  df-iord 4360  df-on 4362  df-ilim 4363  df-suc 4365  df-iom 4584  df-xp 4626  df-rel 4627  df-cnv 4628  df-co 4629  df-dm 4630  df-rn 4631  df-res 4632  df-ima 4633  df-iota 5170  df-fun 5210  df-fn 5211  df-f 5212  df-f1 5213  df-fo 5214  df-f1o 5215  df-fv 5216  df-riota 5821  df-ov 5868  df-oprab 5869  df-mpo 5870  df-1st 6131  df-2nd 6132  df-recs 6296  df-frec 6382  df-1o 6407  df-2o 6408  df-er 6525  df-en 6731  df-sup 6973  df-pnf 7968  df-mnf 7969  df-xr 7970  df-ltxr 7971  df-le 7972  df-sub 8104  df-neg 8105  df-reap 8506  df-ap 8513  df-div 8602  df-inn 8891  df-2 8949  df-3 8950  df-4 8951  df-n0 9148  df-z 9225  df-uz 9500  df-q 9591  df-rp 9623  df-fz 9978  df-fzo 10111  df-fl 10238  df-mod 10291  df-seqfrec 10414  df-exp 10488  df-cj 10817  df-re 10818  df-im 10819  df-rsqrt 10973  df-abs 10974  df-dvds 11761  df-gcd 11909  df-prm 12073  df-gz 12333

This theorem is referenced by:  2sqlem10  14030