Theorem 2sqlem4 13554

Description: Lemma for 2sqlem5 13555. (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 274	. . 3 |- ( ( ph /\ P || ( ( C x. B ) + ( A x. D ) ) ) -> N e. NN )
4		2sqlem5.2	. . . 4 |- ( ph -> P e. Prime )
5	4	adantr 274	. . 3 |- ( ( ph /\ P || ( ( C x. B ) + ( A x. D ) ) ) -> P e. Prime )
6		2sqlem4.3	. . . 4 |- ( ph -> A e. ZZ )
7	6	adantr 274	. . 3 |- ( ( ph /\ P || ( ( C x. B ) + ( A x. D ) ) ) -> A e. ZZ )
8		2sqlem4.4	. . . 4 |- ( ph -> B e. ZZ )
9	8	adantr 274	. . 3 |- ( ( ph /\ P || ( ( C x. B ) + ( A x. D ) ) ) -> B e. ZZ )
10		2sqlem4.5	. . . 4 |- ( ph -> C e. ZZ )
11	10	adantr 274	. . 3 |- ( ( ph /\ P || ( ( C x. B ) + ( A x. D ) ) ) -> C e. ZZ )
12		2sqlem4.6	. . . 4 |- ( ph -> D e. ZZ )
13	12	adantr 274	. . 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 274	. . 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 274	. . 3 |- ( ( ph /\ P || ( ( C x. B ) + ( A x. D ) ) ) -> P = ( ( C ^ 2 ) + ( D ^ 2 ) ) )
18		simpr 109	. . 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 13553	. 2 |- ( ( ph /\ P || ( ( C x. B ) + ( A x. D ) ) ) -> N e. S )
20	2	adantr 274	. . 3 |- ( ( ph /\ P || ( ( C x. B ) - ( A x. D ) ) ) -> N e. NN )
21	4	adantr 274	. . 3 |- ( ( ph /\ P || ( ( C x. B ) - ( A x. D ) ) ) -> P e. Prime )
22	6	znegcld 9311	. . . 4 |- ( ph -> -u A e. ZZ )
23	22	adantr 274	. . 3 |- ( ( ph /\ P || ( ( C x. B ) - ( A x. D ) ) ) -> -u A e. ZZ )
24	8	adantr 274	. . 3 |- ( ( ph /\ P || ( ( C x. B ) - ( A x. D ) ) ) -> B e. ZZ )
25	10	adantr 274	. . 3 |- ( ( ph /\ P || ( ( C x. B ) - ( A x. D ) ) ) -> C e. ZZ )
26	12	adantr 274	. . 3 |- ( ( ph /\ P || ( ( C x. B ) - ( A x. D ) ) ) -> D e. ZZ )
27	6	zcnd 9310	. . . . . . 7 |- ( ph -> A e. CC )
28		sqneg 10510	. . . . . . 7 |- ( A e. CC -> ( -u A ^ 2 ) = ( A ^ 2 ) )
29	27, 28	syl 14	. . . . . 6 |- ( ph -> ( -u A ^ 2 ) = ( A ^ 2 ) )
30	29	oveq1d 5856	. . . . 5 |- ( ph -> ( ( -u A ^ 2 ) + ( B ^ 2 ) ) = ( ( A ^ 2 ) + ( B ^ 2 ) ) )
31	14, 30	eqtr4d 2201	. . . 4 |- ( ph -> ( N x. P ) = ( ( -u A ^ 2 ) + ( B ^ 2 ) ) )
32	31	adantr 274	. . 3 |- ( ( ph /\ P || ( ( C x. B ) - ( A x. D ) ) ) -> ( N x. P ) = ( ( -u A ^ 2 ) + ( B ^ 2 ) ) )
33	16	adantr 274	. . 3 |- ( ( ph /\ P || ( ( C x. B ) - ( A x. D ) ) ) -> P = ( ( C ^ 2 ) + ( D ^ 2 ) ) )
34	12	zcnd 9310	. . . . . . . 8 |- ( ph -> D e. CC )
35	27, 34	mulneg1d 8305	. . . . . . 7 |- ( ph -> ( -u A x. D ) = -u ( A x. D ) )
36	35	oveq2d 5857	. . . . . 6 |- ( ph -> ( ( C x. B ) + ( -u A x. D ) ) = ( ( C x. B ) + -u ( A x. D ) ) )
37	10, 8	zmulcld 9315	. . . . . . . 8 |- ( ph -> ( C x. B ) e. ZZ )
38	37	zcnd 9310	. . . . . . 7 |- ( ph -> ( C x. B ) e. CC )
39	6, 12	zmulcld 9315	. . . . . . . 8 |- ( ph -> ( A x. D ) e. ZZ )
40	39	zcnd 9310	. . . . . . 7 |- ( ph -> ( A x. D ) e. CC )
41	38, 40	negsubd 8211	. . . . . 6 |- ( ph -> ( ( C x. B ) + -u ( A x. D ) ) = ( ( C x. B ) - ( A x. D ) ) )
42	36, 41	eqtrd 2198	. . . . 5 |- ( ph -> ( ( C x. B ) + ( -u A x. D ) ) = ( ( C x. B ) - ( A x. D ) ) )
43	42	breq2d 3993	. . . 4 |- ( ph -> ( P || ( ( C x. B ) + ( -u A x. D ) ) <-> P || ( ( C x. B ) - ( A x. D ) ) ) )
44	43	biimpar 295	. . 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 13553	. 2 |- ( ( ph /\ P || ( ( C x. B ) - ( A x. D ) ) ) -> N e. S )
46		prmz 12039	. . . . . 6 |- ( P e. Prime -> P e. ZZ )
47	4, 46	syl 14	. . . . 5 |- ( ph -> P e. ZZ )
48		zsqcl 10521	. . . . . . . 8 |- ( C e. ZZ -> ( C ^ 2 ) e. ZZ )
49	10, 48	syl 14	. . . . . . 7 |- ( ph -> ( C ^ 2 ) e. ZZ )
50	2	nnzd 9308	. . . . . . 7 |- ( ph -> N e. ZZ )
51	49, 50	zmulcld 9315	. . . . . 6 |- ( ph -> ( ( C ^ 2 ) x. N ) e. ZZ )
52		zsqcl 10521	. . . . . . 7 |- ( A e. ZZ -> ( A ^ 2 ) e. ZZ )
53	6, 52	syl 14	. . . . . 6 |- ( ph -> ( A ^ 2 ) e. ZZ )
54	51, 53	zsubcld 9314	. . . . 5 |- ( ph -> ( ( ( C ^ 2 ) x. N ) - ( A ^ 2 ) ) e. ZZ )
55		dvdsmul1 11749	. . . . 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 409	. . . 4 |- ( ph -> P || ( P x. ( ( ( C ^ 2 ) x. N ) - ( A ^ 2 ) ) ) )
57	10, 6	zmulcld 9315	. . . . . . . . 9 |- ( ph -> ( C x. A ) e. ZZ )
58	57	zcnd 9310	. . . . . . . 8 |- ( ph -> ( C x. A ) e. CC )
59	58	sqcld 10582	. . . . . . 7 |- ( ph -> ( ( C x. A ) ^ 2 ) e. CC )
60	38	sqcld 10582	. . . . . . 7 |- ( ph -> ( ( C x. B ) ^ 2 ) e. CC )
61	40	sqcld 10582	. . . . . . 7 |- ( ph -> ( ( A x. D ) ^ 2 ) e. CC )
62	59, 60, 61	pnpcand 8242	. . . . . 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 9310	. . . . . . . . . . . 12 |- ( ph -> C e. CC )
64	63, 27	sqmuld 10596	. . . . . . . . . . 11 |- ( ph -> ( ( C x. A ) ^ 2 ) = ( ( C ^ 2 ) x. ( A ^ 2 ) ) )
65	8	zcnd 9310	. . . . . . . . . . . 12 |- ( ph -> B e. CC )
66	63, 65	sqmuld 10596	. . . . . . . . . . 11 |- ( ph -> ( ( C x. B ) ^ 2 ) = ( ( C ^ 2 ) x. ( B ^ 2 ) ) )
67	64, 66	oveq12d 5859	. . . . . . . . . 10 |- ( ph -> ( ( ( C x. A ) ^ 2 ) + ( ( C x. B ) ^ 2 ) ) = ( ( ( C ^ 2 ) x. ( A ^ 2 ) ) + ( ( C ^ 2 ) x. ( B ^ 2 ) ) ) )
68	63	sqcld 10582	. . . . . . . . . . 11 |- ( ph -> ( C ^ 2 ) e. CC )
69	53	zcnd 9310	. . . . . . . . . . 11 |- ( ph -> ( A ^ 2 ) e. CC )
70	65	sqcld 10582	. . . . . . . . . . 11 |- ( ph -> ( B ^ 2 ) e. CC )
71	68, 69, 70	adddid 7919	. . . . . . . . . 10 |- ( ph -> ( ( C ^ 2 ) x. ( ( A ^ 2 ) + ( B ^ 2 ) ) ) = ( ( ( C ^ 2 ) x. ( A ^ 2 ) ) + ( ( C ^ 2 ) x. ( B ^ 2 ) ) ) )
72	67, 71	eqtr4d 2201	. . . . . . . . 9 |- ( ph -> ( ( ( C x. A ) ^ 2 ) + ( ( C x. B ) ^ 2 ) ) = ( ( C ^ 2 ) x. ( ( A ^ 2 ) + ( B ^ 2 ) ) ) )
73	2	nncnd 8867	. . . . . . . . . . . . 13 |- ( ph -> N e. CC )
74	47	zcnd 9310	. . . . . . . . . . . . 13 |- ( ph -> P e. CC )
75	73, 74	mulcomd 7916	. . . . . . . . . . . 12 |- ( ph -> ( N x. P ) = ( P x. N ) )
76	14, 75	eqtr3d 2200	. . . . . . . . . . 11 |- ( ph -> ( ( A ^ 2 ) + ( B ^ 2 ) ) = ( P x. N ) )
77	76	oveq2d 5857	. . . . . . . . . 10 |- ( ph -> ( ( C ^ 2 ) x. ( ( A ^ 2 ) + ( B ^ 2 ) ) ) = ( ( C ^ 2 ) x. ( P x. N ) ) )
78	68, 74, 73	mul12d 8046	. . . . . . . . . 10 |- ( ph -> ( ( C ^ 2 ) x. ( P x. N ) ) = ( P x. ( ( C ^ 2 ) x. N ) ) )
79	77, 78	eqtrd 2198	. . . . . . . . 9 |- ( ph -> ( ( C ^ 2 ) x. ( ( A ^ 2 ) + ( B ^ 2 ) ) ) = ( P x. ( ( C ^ 2 ) x. N ) ) )
80	72, 79	eqtrd 2198	. . . . . . . 8 |- ( ph -> ( ( ( C x. A ) ^ 2 ) + ( ( C x. B ) ^ 2 ) ) = ( P x. ( ( C ^ 2 ) x. N ) ) )
81	27, 34	sqmuld 10596	. . . . . . . . . . . 12 |- ( ph -> ( ( A x. D ) ^ 2 ) = ( ( A ^ 2 ) x. ( D ^ 2 ) ) )
82	34	sqcld 10582	. . . . . . . . . . . . 13 |- ( ph -> ( D ^ 2 ) e. CC )
83	69, 82	mulcomd 7916	. . . . . . . . . . . 12 |- ( ph -> ( ( A ^ 2 ) x. ( D ^ 2 ) ) = ( ( D ^ 2 ) x. ( A ^ 2 ) ) )
84	81, 83	eqtrd 2198	. . . . . . . . . . 11 |- ( ph -> ( ( A x. D ) ^ 2 ) = ( ( D ^ 2 ) x. ( A ^ 2 ) ) )
85	64, 84	oveq12d 5859	. . . . . . . . . 10 |- ( ph -> ( ( ( C x. A ) ^ 2 ) + ( ( A x. D ) ^ 2 ) ) = ( ( ( C ^ 2 ) x. ( A ^ 2 ) ) + ( ( D ^ 2 ) x. ( A ^ 2 ) ) ) )
86	49	zcnd 9310	. . . . . . . . . . 11 |- ( ph -> ( C ^ 2 ) e. CC )
87	86, 82, 69	adddird 7920	. . . . . . . . . 10 |- ( ph -> ( ( ( C ^ 2 ) + ( D ^ 2 ) ) x. ( A ^ 2 ) ) = ( ( ( C ^ 2 ) x. ( A ^ 2 ) ) + ( ( D ^ 2 ) x. ( A ^ 2 ) ) ) )
88	85, 87	eqtr4d 2201	. . . . . . . . 9 |- ( ph -> ( ( ( C x. A ) ^ 2 ) + ( ( A x. D ) ^ 2 ) ) = ( ( ( C ^ 2 ) + ( D ^ 2 ) ) x. ( A ^ 2 ) ) )
89	16	oveq1d 5856	. . . . . . . . 9 |- ( ph -> ( P x. ( A ^ 2 ) ) = ( ( ( C ^ 2 ) + ( D ^ 2 ) ) x. ( A ^ 2 ) ) )
90	88, 89	eqtr4d 2201	. . . . . . . 8 |- ( ph -> ( ( ( C x. A ) ^ 2 ) + ( ( A x. D ) ^ 2 ) ) = ( P x. ( A ^ 2 ) ) )
91	80, 90	oveq12d 5859	. . . . . . 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 9310	. . . . . . . 8 |- ( ph -> ( ( C ^ 2 ) x. N ) e. CC )
93	74, 92, 69	subdid 8308	. . . . . . 7 |- ( ph -> ( P x. ( ( ( C ^ 2 ) x. N ) - ( A ^ 2 ) ) ) = ( ( P x. ( ( C ^ 2 ) x. N ) ) - ( P x. ( A ^ 2 ) ) ) )
94	91, 93	eqtr4d 2201	. . . . . 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 2200	. . . . 5 |- ( ph -> ( ( ( C x. B ) ^ 2 ) - ( ( A x. D ) ^ 2 ) ) = ( P x. ( ( ( C ^ 2 ) x. N ) - ( A ^ 2 ) ) ) )
96		subsq 10557	. . . . . 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 409	. . . . 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 2200	. . . 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 4007	. . 3 |- ( ph -> P || ( ( ( C x. B ) + ( A x. D ) ) x. ( ( C x. B ) - ( A x. D ) ) ) )
100	37, 39	zaddcld 9313	. . . 4 |- ( ph -> ( ( C x. B ) + ( A x. D ) ) e. ZZ )
101	37, 39	zsubcld 9314	. . . 4 |- ( ph -> ( ( C x. B ) - ( A x. D ) ) e. ZZ )
102		euclemma 12074	. . . 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 1228	. . 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 146	. 2 |- ( ph -> ( P || ( ( C x. B ) + ( A x. D ) ) \/ P || ( ( C x. B ) - ( A x. D ) ) ) )
105	19, 45, 104	mpjaodan 788	1 |- ( ph -> N e. S )

Syntax hints:   -> wi 4   /\ wa 103   <-> wb 104   \/ wo 698   = wceq 1343   e. wcel 2136   class class class wbr 3981   |-> cmpt 4042   ran crn 4604   ` cfv 5187  (class class class)co 5841   CCcc 7747   + caddc 7752   x. cmul 7754   - cmin 8065   -ucneg 8066   NNcn 8853   2c2 8904   ZZcz 9187   ^cexp 10450   abscabs 10935   || cdvds 11723   Primecprime 12035   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-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:  2sqlem5  13555