Theorem 2sqlem4 15797

Description: Lemma for 2sqlem5 15798. (Contributed by Mario Carneiro, 20-Jun-2015.)

Hypotheses

Ref	Expression
2sq.1	|- S = ran ( w e. ZZ[_i] |-> ( ( abs ` w ) ^ 2 ) )
2sqlem5.1	|- ( ph -> N e. NN )
2sqlem5.2	|- ( ph -> P e. Prime )
2sqlem4.3	|- ( ph -> A e. ZZ )
2sqlem4.4	|- ( ph -> B e. ZZ )
2sqlem4.5	|- ( ph -> C e. ZZ )
2sqlem4.6	|- ( ph -> D e. ZZ )
2sqlem4.7	|- ( ph -> ( N x. P ) = ( ( A ^ 2 ) + ( B ^ 2 ) ) )
2sqlem4.8	|- ( ph -> P = ( ( C ^ 2 ) + ( D ^ 2 ) ) )

Assertion

Ref	Expression
2sqlem4	|- ( ph -> N e. S )

Proof of Theorem 2sqlem4

Step	Hyp	Ref	Expression
1		2sq.1	. . 3 |- S = ran ( w e. ZZ[_i] |-> ( ( abs ` w ) ^ 2 ) )
2		2sqlem5.1	. . . 4 |- ( ph -> N e. NN )
3	2	adantr 276	. . 3 |- ( ( ph /\ P || ( ( C x. B ) + ( A x. D ) ) ) -> N e. NN )
4		2sqlem5.2	. . . 4 |- ( ph -> P e. Prime )
5	4	adantr 276	. . 3 |- ( ( ph /\ P || ( ( C x. B ) + ( A x. D ) ) ) -> P e. Prime )
6		2sqlem4.3	. . . 4 |- ( ph -> A e. ZZ )
7	6	adantr 276	. . 3 |- ( ( ph /\ P || ( ( C x. B ) + ( A x. D ) ) ) -> A e. ZZ )
8		2sqlem4.4	. . . 4 |- ( ph -> B e. ZZ )
9	8	adantr 276	. . 3 |- ( ( ph /\ P || ( ( C x. B ) + ( A x. D ) ) ) -> B e. ZZ )
10		2sqlem4.5	. . . 4 |- ( ph -> C e. ZZ )
11	10	adantr 276	. . 3 |- ( ( ph /\ P || ( ( C x. B ) + ( A x. D ) ) ) -> C e. ZZ )
12		2sqlem4.6	. . . 4 |- ( ph -> D e. ZZ )
13	12	adantr 276	. . 3 |- ( ( ph /\ P || ( ( C x. B ) + ( A x. D ) ) ) -> D e. ZZ )
14		2sqlem4.7	. . . 4 |- ( ph -> ( N x. P ) = ( ( A ^ 2 ) + ( B ^ 2 ) ) )
15	14	adantr 276	. . 3 |- ( ( ph /\ P || ( ( C x. B ) + ( A x. D ) ) ) -> ( N x. P ) = ( ( A ^ 2 ) + ( B ^ 2 ) ) )
16		2sqlem4.8	. . . 4 |- ( ph -> P = ( ( C ^ 2 ) + ( D ^ 2 ) ) )
17	16	adantr 276	. . 3 |- ( ( ph /\ P || ( ( C x. B ) + ( A x. D ) ) ) -> P = ( ( C ^ 2 ) + ( D ^ 2 ) ) )
18		simpr 110	. . 3 |- ( ( ph /\ P || ( ( C x. B ) + ( A x. D ) ) ) -> P || ( ( C x. B ) + ( A x. D ) ) )
19	1, 3, 5, 7, 9, 11, 13, 15, 17, 18	2sqlem3 15796	. 2 |- ( ( ph /\ P || ( ( C x. B ) + ( A x. D ) ) ) -> N e. S )
20	2	adantr 276	. . 3 |- ( ( ph /\ P || ( ( C x. B ) - ( A x. D ) ) ) -> N e. NN )
21	4	adantr 276	. . 3 |- ( ( ph /\ P || ( ( C x. B ) - ( A x. D ) ) ) -> P e. Prime )
22	6	znegcld 9571	. . . 4 |- ( ph -> -u A e. ZZ )
23	22	adantr 276	. . 3 |- ( ( ph /\ P || ( ( C x. B ) - ( A x. D ) ) ) -> -u A e. ZZ )
24	8	adantr 276	. . 3 |- ( ( ph /\ P || ( ( C x. B ) - ( A x. D ) ) ) -> B e. ZZ )
25	10	adantr 276	. . 3 |- ( ( ph /\ P || ( ( C x. B ) - ( A x. D ) ) ) -> C e. ZZ )
26	12	adantr 276	. . 3 |- ( ( ph /\ P || ( ( C x. B ) - ( A x. D ) ) ) -> D e. ZZ )
27	6	zcnd 9570	. . . . . . 7 |- ( ph -> A e. CC )
28		sqneg 10820	. . . . . . 7 |- ( A e. CC -> ( -u A ^ 2 ) = ( A ^ 2 ) )
29	27, 28	syl 14	. . . . . 6 |- ( ph -> ( -u A ^ 2 ) = ( A ^ 2 ) )
30	29	oveq1d 6016	. . . . 5 |- ( ph -> ( ( -u A ^ 2 ) + ( B ^ 2 ) ) = ( ( A ^ 2 ) + ( B ^ 2 ) ) )
31	14, 30	eqtr4d 2265	. . . 4 |- ( ph -> ( N x. P ) = ( ( -u A ^ 2 ) + ( B ^ 2 ) ) )
32	31	adantr 276	. . 3 |- ( ( ph /\ P || ( ( C x. B ) - ( A x. D ) ) ) -> ( N x. P ) = ( ( -u A ^ 2 ) + ( B ^ 2 ) ) )
33	16	adantr 276	. . 3 |- ( ( ph /\ P || ( ( C x. B ) - ( A x. D ) ) ) -> P = ( ( C ^ 2 ) + ( D ^ 2 ) ) )
34	12	zcnd 9570	. . . . . . . 8 |- ( ph -> D e. CC )
35	27, 34	mulneg1d 8557	. . . . . . 7 |- ( ph -> ( -u A x. D ) = -u ( A x. D ) )
36	35	oveq2d 6017	. . . . . 6 |- ( ph -> ( ( C x. B ) + ( -u A x. D ) ) = ( ( C x. B ) + -u ( A x. D ) ) )
37	10, 8	zmulcld 9575	. . . . . . . 8 |- ( ph -> ( C x. B ) e. ZZ )
38	37	zcnd 9570	. . . . . . 7 |- ( ph -> ( C x. B ) e. CC )
39	6, 12	zmulcld 9575	. . . . . . . 8 |- ( ph -> ( A x. D ) e. ZZ )
40	39	zcnd 9570	. . . . . . 7 |- ( ph -> ( A x. D ) e. CC )
41	38, 40	negsubd 8463	. . . . . 6 |- ( ph -> ( ( C x. B ) + -u ( A x. D ) ) = ( ( C x. B ) - ( A x. D ) ) )
42	36, 41	eqtrd 2262	. . . . 5 |- ( ph -> ( ( C x. B ) + ( -u A x. D ) ) = ( ( C x. B ) - ( A x. D ) ) )
43	42	breq2d 4095	. . . 4 |- ( ph -> ( P || ( ( C x. B ) + ( -u A x. D ) ) <-> P || ( ( C x. B ) - ( A x. D ) ) ) )
44	43	biimpar 297	. . 3 |- ( ( ph /\ P || ( ( C x. B ) - ( A x. D ) ) ) -> P || ( ( C x. B ) + ( -u A x. D ) ) )
45	1, 20, 21, 23, 24, 25, 26, 32, 33, 44	2sqlem3 15796	. 2 |- ( ( ph /\ P || ( ( C x. B ) - ( A x. D ) ) ) -> N e. S )
46		prmz 12633	. . . . . 6 |- ( P e. Prime -> P e. ZZ )
47	4, 46	syl 14	. . . . 5 |- ( ph -> P e. ZZ )
48		zsqcl 10832	. . . . . . . 8 |- ( C e. ZZ -> ( C ^ 2 ) e. ZZ )
49	10, 48	syl 14	. . . . . . 7 |- ( ph -> ( C ^ 2 ) e. ZZ )
50	2	nnzd 9568	. . . . . . 7 |- ( ph -> N e. ZZ )
51	49, 50	zmulcld 9575	. . . . . 6 |- ( ph -> ( ( C ^ 2 ) x. N ) e. ZZ )
52		zsqcl 10832	. . . . . . 7 |- ( A e. ZZ -> ( A ^ 2 ) e. ZZ )
53	6, 52	syl 14	. . . . . 6 |- ( ph -> ( A ^ 2 ) e. ZZ )
54	51, 53	zsubcld 9574	. . . . 5 |- ( ph -> ( ( ( C ^ 2 ) x. N ) - ( A ^ 2 ) ) e. ZZ )
55		dvdsmul1 12324	. . . . 5 |- ( ( P e. ZZ /\ ( ( ( C ^ 2 ) x. N ) - ( A ^ 2 ) ) e. ZZ ) -> P || ( P x. ( ( ( C ^ 2 ) x. N ) - ( A ^ 2 ) ) ) )
56	47, 54, 55	syl2anc 411	. . . 4 |- ( ph -> P || ( P x. ( ( ( C ^ 2 ) x. N ) - ( A ^ 2 ) ) ) )
57	10, 6	zmulcld 9575	. . . . . . . . 9 |- ( ph -> ( C x. A ) e. ZZ )
58	57	zcnd 9570	. . . . . . . 8 |- ( ph -> ( C x. A ) e. CC )
59	58	sqcld 10893	. . . . . . 7 |- ( ph -> ( ( C x. A ) ^ 2 ) e. CC )
60	38	sqcld 10893	. . . . . . 7 |- ( ph -> ( ( C x. B ) ^ 2 ) e. CC )
61	40	sqcld 10893	. . . . . . 7 |- ( ph -> ( ( A x. D ) ^ 2 ) e. CC )
62	59, 60, 61	pnpcand 8494	. . . . . 6 |- ( ph -> ( ( ( ( C x. A ) ^ 2 ) + ( ( C x. B ) ^ 2 ) ) - ( ( ( C x. A ) ^ 2 ) + ( ( A x. D ) ^ 2 ) ) ) = ( ( ( C x. B ) ^ 2 ) - ( ( A x. D ) ^ 2 ) ) )
63	10	zcnd 9570	. . . . . . . . . . . 12 |- ( ph -> C e. CC )
64	63, 27	sqmuld 10907	. . . . . . . . . . 11 |- ( ph -> ( ( C x. A ) ^ 2 ) = ( ( C ^ 2 ) x. ( A ^ 2 ) ) )
65	8	zcnd 9570	. . . . . . . . . . . 12 |- ( ph -> B e. CC )
66	63, 65	sqmuld 10907	. . . . . . . . . . 11 |- ( ph -> ( ( C x. B ) ^ 2 ) = ( ( C ^ 2 ) x. ( B ^ 2 ) ) )
67	64, 66	oveq12d 6019	. . . . . . . . . 10 |- ( ph -> ( ( ( C x. A ) ^ 2 ) + ( ( C x. B ) ^ 2 ) ) = ( ( ( C ^ 2 ) x. ( A ^ 2 ) ) + ( ( C ^ 2 ) x. ( B ^ 2 ) ) ) )
68	63	sqcld 10893	. . . . . . . . . . 11 |- ( ph -> ( C ^ 2 ) e. CC )
69	53	zcnd 9570	. . . . . . . . . . 11 |- ( ph -> ( A ^ 2 ) e. CC )
70	65	sqcld 10893	. . . . . . . . . . 11 |- ( ph -> ( B ^ 2 ) e. CC )
71	68, 69, 70	adddid 8171	. . . . . . . . . 10 |- ( ph -> ( ( C ^ 2 ) x. ( ( A ^ 2 ) + ( B ^ 2 ) ) ) = ( ( ( C ^ 2 ) x. ( A ^ 2 ) ) + ( ( C ^ 2 ) x. ( B ^ 2 ) ) ) )
72	67, 71	eqtr4d 2265	. . . . . . . . 9 |- ( ph -> ( ( ( C x. A ) ^ 2 ) + ( ( C x. B ) ^ 2 ) ) = ( ( C ^ 2 ) x. ( ( A ^ 2 ) + ( B ^ 2 ) ) ) )
73	2	nncnd 9124	. . . . . . . . . . . . 13 |- ( ph -> N e. CC )
74	47	zcnd 9570	. . . . . . . . . . . . 13 |- ( ph -> P e. CC )
75	73, 74	mulcomd 8168	. . . . . . . . . . . 12 |- ( ph -> ( N x. P ) = ( P x. N ) )
76	14, 75	eqtr3d 2264	. . . . . . . . . . 11 |- ( ph -> ( ( A ^ 2 ) + ( B ^ 2 ) ) = ( P x. N ) )
77	76	oveq2d 6017	. . . . . . . . . 10 |- ( ph -> ( ( C ^ 2 ) x. ( ( A ^ 2 ) + ( B ^ 2 ) ) ) = ( ( C ^ 2 ) x. ( P x. N ) ) )
78	68, 74, 73	mul12d 8298	. . . . . . . . . 10 |- ( ph -> ( ( C ^ 2 ) x. ( P x. N ) ) = ( P x. ( ( C ^ 2 ) x. N ) ) )
79	77, 78	eqtrd 2262	. . . . . . . . 9 |- ( ph -> ( ( C ^ 2 ) x. ( ( A ^ 2 ) + ( B ^ 2 ) ) ) = ( P x. ( ( C ^ 2 ) x. N ) ) )
80	72, 79	eqtrd 2262	. . . . . . . 8 |- ( ph -> ( ( ( C x. A ) ^ 2 ) + ( ( C x. B ) ^ 2 ) ) = ( P x. ( ( C ^ 2 ) x. N ) ) )
81	27, 34	sqmuld 10907	. . . . . . . . . . . 12 |- ( ph -> ( ( A x. D ) ^ 2 ) = ( ( A ^ 2 ) x. ( D ^ 2 ) ) )
82	34	sqcld 10893	. . . . . . . . . . . . 13 |- ( ph -> ( D ^ 2 ) e. CC )
83	69, 82	mulcomd 8168	. . . . . . . . . . . 12 |- ( ph -> ( ( A ^ 2 ) x. ( D ^ 2 ) ) = ( ( D ^ 2 ) x. ( A ^ 2 ) ) )
84	81, 83	eqtrd 2262	. . . . . . . . . . 11 |- ( ph -> ( ( A x. D ) ^ 2 ) = ( ( D ^ 2 ) x. ( A ^ 2 ) ) )
85	64, 84	oveq12d 6019	. . . . . . . . . 10 |- ( ph -> ( ( ( C x. A ) ^ 2 ) + ( ( A x. D ) ^ 2 ) ) = ( ( ( C ^ 2 ) x. ( A ^ 2 ) ) + ( ( D ^ 2 ) x. ( A ^ 2 ) ) ) )
86	49	zcnd 9570	. . . . . . . . . . 11 |- ( ph -> ( C ^ 2 ) e. CC )
87	86, 82, 69	adddird 8172	. . . . . . . . . 10 |- ( ph -> ( ( ( C ^ 2 ) + ( D ^ 2 ) ) x. ( A ^ 2 ) ) = ( ( ( C ^ 2 ) x. ( A ^ 2 ) ) + ( ( D ^ 2 ) x. ( A ^ 2 ) ) ) )
88	85, 87	eqtr4d 2265	. . . . . . . . 9 |- ( ph -> ( ( ( C x. A ) ^ 2 ) + ( ( A x. D ) ^ 2 ) ) = ( ( ( C ^ 2 ) + ( D ^ 2 ) ) x. ( A ^ 2 ) ) )
89	16	oveq1d 6016	. . . . . . . . 9 |- ( ph -> ( P x. ( A ^ 2 ) ) = ( ( ( C ^ 2 ) + ( D ^ 2 ) ) x. ( A ^ 2 ) ) )
90	88, 89	eqtr4d 2265	. . . . . . . 8 |- ( ph -> ( ( ( C x. A ) ^ 2 ) + ( ( A x. D ) ^ 2 ) ) = ( P x. ( A ^ 2 ) ) )
91	80, 90	oveq12d 6019	. . . . . . 7 |- ( ph -> ( ( ( ( C x. A ) ^ 2 ) + ( ( C x. B ) ^ 2 ) ) - ( ( ( C x. A ) ^ 2 ) + ( ( A x. D ) ^ 2 ) ) ) = ( ( P x. ( ( C ^ 2 ) x. N ) ) - ( P x. ( A ^ 2 ) ) ) )
92	51	zcnd 9570	. . . . . . . 8 |- ( ph -> ( ( C ^ 2 ) x. N ) e. CC )
93	74, 92, 69	subdid 8560	. . . . . . 7 |- ( ph -> ( P x. ( ( ( C ^ 2 ) x. N ) - ( A ^ 2 ) ) ) = ( ( P x. ( ( C ^ 2 ) x. N ) ) - ( P x. ( A ^ 2 ) ) ) )
94	91, 93	eqtr4d 2265	. . . . . 6 |- ( ph -> ( ( ( ( C x. A ) ^ 2 ) + ( ( C x. B ) ^ 2 ) ) - ( ( ( C x. A ) ^ 2 ) + ( ( A x. D ) ^ 2 ) ) ) = ( P x. ( ( ( C ^ 2 ) x. N ) - ( A ^ 2 ) ) ) )
95	62, 94	eqtr3d 2264	. . . . 5 |- ( ph -> ( ( ( C x. B ) ^ 2 ) - ( ( A x. D ) ^ 2 ) ) = ( P x. ( ( ( C ^ 2 ) x. N ) - ( A ^ 2 ) ) ) )
96		subsq 10868	. . . . . 6 |- ( ( ( C x. B ) e. CC /\ ( A x. D ) e. CC ) -> ( ( ( C x. B ) ^ 2 ) - ( ( A x. D ) ^ 2 ) ) = ( ( ( C x. B ) + ( A x. D ) ) x. ( ( C x. B ) - ( A x. D ) ) ) )
97	38, 40, 96	syl2anc 411	. . . . 5 |- ( ph -> ( ( ( C x. B ) ^ 2 ) - ( ( A x. D ) ^ 2 ) ) = ( ( ( C x. B ) + ( A x. D ) ) x. ( ( C x. B ) - ( A x. D ) ) ) )
98	95, 97	eqtr3d 2264	. . . 4 |- ( ph -> ( P x. ( ( ( C ^ 2 ) x. N ) - ( A ^ 2 ) ) ) = ( ( ( C x. B ) + ( A x. D ) ) x. ( ( C x. B ) - ( A x. D ) ) ) )
99	56, 98	breqtrd 4109	. . 3 |- ( ph -> P || ( ( ( C x. B ) + ( A x. D ) ) x. ( ( C x. B ) - ( A x. D ) ) ) )
100	37, 39	zaddcld 9573	. . . 4 |- ( ph -> ( ( C x. B ) + ( A x. D ) ) e. ZZ )
101	37, 39	zsubcld 9574	. . . 4 |- ( ph -> ( ( C x. B ) - ( A x. D ) ) e. ZZ )
102		euclemma 12668	. . . 4 |- ( ( P e. Prime /\ ( ( C x. B ) + ( A x. D ) ) e. ZZ /\ ( ( C x. B ) - ( A x. D ) ) e. ZZ ) -> ( P || ( ( ( C x. B ) + ( A x. D ) ) x. ( ( C x. B ) - ( A x. D ) ) ) <-> ( P || ( ( C x. B ) + ( A x. D ) ) \/ P || ( ( C x. B ) - ( A x. D ) ) ) ) )
103	4, 100, 101, 102	syl3anc 1271	. . 3 |- ( ph -> ( P || ( ( ( C x. B ) + ( A x. D ) ) x. ( ( C x. B ) - ( A x. D ) ) ) <-> ( P || ( ( C x. B ) + ( A x. D ) ) \/ P || ( ( C x. B ) - ( A x. D ) ) ) ) )
104	99, 103	mpbid 147	. 2 |- ( ph -> ( P || ( ( C x. B ) + ( A x. D ) ) \/ P || ( ( C x. B ) - ( A x. D ) ) ) )
105	19, 45, 104	mpjaodan 803	1 |- ( ph -> N e. S )

Syntax hints:   -> wi 4   /\ wa 104   <-> wb 105   \/ wo 713   = wceq 1395   e. wcel 2200   class class class wbr 4083   |-> cmpt 4145   ran crn 4720   ` cfv 5318  (class class class)co 6001   CCcc 7997   + caddc 8002   x. cmul 8004   - cmin 8317   -ucneg 8318   NNcn 9110   2c2 9161   ZZcz 9446   ^cexp 10760   abscabs 11508   || cdvds 12298   Primecprime 12629   ZZ[_i]cgz 12892

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 617  ax-in2 618  ax-io 714  ax-5 1493  ax-7 1494  ax-gen 1495  ax-ie1 1539  ax-ie2 1540  ax-8 1550  ax-10 1551  ax-11 1552  ax-i12 1553  ax-bndl 1555  ax-4 1556  ax-17 1572  ax-i9 1576  ax-ial 1580  ax-i5r 1581  ax-13 2202  ax-14 2203  ax-ext 2211  ax-coll 4199  ax-sep 4202  ax-nul 4210  ax-pow 4258  ax-pr 4293  ax-un 4524  ax-setind 4629  ax-iinf 4680  ax-cnex 8090  ax-resscn 8091  ax-1cn 8092  ax-1re 8093  ax-icn 8094  ax-addcl 8095  ax-addrcl 8096  ax-mulcl 8097  ax-mulrcl 8098  ax-addcom 8099  ax-mulcom 8100  ax-addass 8101  ax-mulass 8102  ax-distr 8103  ax-i2m1 8104  ax-0lt1 8105  ax-1rid 8106  ax-0id 8107  ax-rnegex 8108  ax-precex 8109  ax-cnre 8110  ax-pre-ltirr 8111  ax-pre-ltwlin 8112  ax-pre-lttrn 8113  ax-pre-apti 8114  ax-pre-ltadd 8115  ax-pre-mulgt0 8116  ax-pre-mulext 8117  ax-arch 8118  ax-caucvg 8119

This theorem depends on definitions:  df-bi 117  df-dc 840  df-3or 1003  df-3an 1004  df-tru 1398  df-fal 1401  df-nf 1507  df-sb 1809  df-eu 2080  df-mo 2081  df-clab 2216  df-cleq 2222  df-clel 2225  df-nfc 2361  df-ne 2401  df-nel 2496  df-ral 2513  df-rex 2514  df-reu 2515  df-rmo 2516  df-rab 2517  df-v 2801  df-sbc 3029  df-csb 3125  df-dif 3199  df-un 3201  df-in 3203  df-ss 3210  df-nul 3492  df-if 3603  df-pw 3651  df-sn 3672  df-pr 3673  df-op 3675  df-uni 3889  df-int 3924  df-iun 3967  df-br 4084  df-opab 4146  df-mpt 4147  df-tr 4183  df-id 4384  df-po 4387  df-iso 4388  df-iord 4457  df-on 4459  df-ilim 4460  df-suc 4462  df-iom 4683  df-xp 4725  df-rel 4726  df-cnv 4727  df-co 4728  df-dm 4729  df-rn 4730  df-res 4731  df-ima 4732  df-iota 5278  df-fun 5320  df-fn 5321  df-f 5322  df-f1 5323  df-fo 5324  df-f1o 5325  df-fv 5326  df-riota 5954  df-ov 6004  df-oprab 6005  df-mpo 6006  df-1st 6286  df-2nd 6287  df-recs 6451  df-frec 6537  df-1o 6562  df-2o 6563  df-er 6680  df-en 6888  df-sup 7151  df-pnf 8183  df-mnf 8184  df-xr 8185  df-ltxr 8186  df-le 8187  df-sub 8319  df-neg 8320  df-reap 8722  df-ap 8729  df-div 8820  df-inn 9111  df-2 9169  df-3 9170  df-4 9171  df-n0 9370  df-z 9447  df-uz 9723  df-q 9815  df-rp 9850  df-fz 10205  df-fzo 10339  df-fl 10490  df-mod 10545  df-seqfrec 10670  df-exp 10761  df-cj 11353  df-re 11354  df-im 11355  df-rsqrt 11509  df-abs 11510  df-dvds 12299  df-gcd 12475  df-prm 12630  df-gz 12893

This theorem is referenced by:  2sqlem5  15798