Theorem 2sqlem4 15937

Description: Lemma for 2sqlem5 15938. (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 15936	. 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 9665	. . . 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 9664	. . . . . . 7 |- ( ph -> A e. CC )
28		sqneg 10923	. . . . . . 7 |- ( A e. CC -> ( -u A ^ 2 ) = ( A ^ 2 ) )
29	27, 28	syl 14	. . . . . 6 |- ( ph -> ( -u A ^ 2 ) = ( A ^ 2 ) )
30	29	oveq1d 6043	. . . . 5 |- ( ph -> ( ( -u A ^ 2 ) + ( B ^ 2 ) ) = ( ( A ^ 2 ) + ( B ^ 2 ) ) )
31	14, 30	eqtr4d 2267	. . . 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 9664	. . . . . . . 8 |- ( ph -> D e. CC )
35	27, 34	mulneg1d 8649	. . . . . . 7 |- ( ph -> ( -u A x. D ) = -u ( A x. D ) )
36	35	oveq2d 6044	. . . . . 6 |- ( ph -> ( ( C x. B ) + ( -u A x. D ) ) = ( ( C x. B ) + -u ( A x. D ) ) )
37	10, 8	zmulcld 9669	. . . . . . . 8 |- ( ph -> ( C x. B ) e. ZZ )
38	37	zcnd 9664	. . . . . . 7 |- ( ph -> ( C x. B ) e. CC )
39	6, 12	zmulcld 9669	. . . . . . . 8 |- ( ph -> ( A x. D ) e. ZZ )
40	39	zcnd 9664	. . . . . . 7 |- ( ph -> ( A x. D ) e. CC )
41	38, 40	negsubd 8555	. . . . . 6 |- ( ph -> ( ( C x. B ) + -u ( A x. D ) ) = ( ( C x. B ) - ( A x. D ) ) )
42	36, 41	eqtrd 2264	. . . . 5 |- ( ph -> ( ( C x. B ) + ( -u A x. D ) ) = ( ( C x. B ) - ( A x. D ) ) )
43	42	breq2d 4105	. . . 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 15936	. 2 |- ( ( ph /\ P || ( ( C x. B ) - ( A x. D ) ) ) -> N e. S )
46		prmz 12763	. . . . . 6 |- ( P e. Prime -> P e. ZZ )
47	4, 46	syl 14	. . . . 5 |- ( ph -> P e. ZZ )
48		zsqcl 10935	. . . . . . . 8 |- ( C e. ZZ -> ( C ^ 2 ) e. ZZ )
49	10, 48	syl 14	. . . . . . 7 |- ( ph -> ( C ^ 2 ) e. ZZ )
50	2	nnzd 9662	. . . . . . 7 |- ( ph -> N e. ZZ )
51	49, 50	zmulcld 9669	. . . . . 6 |- ( ph -> ( ( C ^ 2 ) x. N ) e. ZZ )
52		zsqcl 10935	. . . . . . 7 |- ( A e. ZZ -> ( A ^ 2 ) e. ZZ )
53	6, 52	syl 14	. . . . . 6 |- ( ph -> ( A ^ 2 ) e. ZZ )
54	51, 53	zsubcld 9668	. . . . 5 |- ( ph -> ( ( ( C ^ 2 ) x. N ) - ( A ^ 2 ) ) e. ZZ )
55		dvdsmul1 12454	. . . . 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 9669	. . . . . . . . 9 |- ( ph -> ( C x. A ) e. ZZ )
58	57	zcnd 9664	. . . . . . . 8 |- ( ph -> ( C x. A ) e. CC )
59	58	sqcld 10996	. . . . . . 7 |- ( ph -> ( ( C x. A ) ^ 2 ) e. CC )
60	38	sqcld 10996	. . . . . . 7 |- ( ph -> ( ( C x. B ) ^ 2 ) e. CC )
61	40	sqcld 10996	. . . . . . 7 |- ( ph -> ( ( A x. D ) ^ 2 ) e. CC )
62	59, 60, 61	pnpcand 8586	. . . . . 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 9664	. . . . . . . . . . . 12 |- ( ph -> C e. CC )
64	63, 27	sqmuld 11010	. . . . . . . . . . 11 |- ( ph -> ( ( C x. A ) ^ 2 ) = ( ( C ^ 2 ) x. ( A ^ 2 ) ) )
65	8	zcnd 9664	. . . . . . . . . . . 12 |- ( ph -> B e. CC )
66	63, 65	sqmuld 11010	. . . . . . . . . . 11 |- ( ph -> ( ( C x. B ) ^ 2 ) = ( ( C ^ 2 ) x. ( B ^ 2 ) ) )
67	64, 66	oveq12d 6046	. . . . . . . . . 10 |- ( ph -> ( ( ( C x. A ) ^ 2 ) + ( ( C x. B ) ^ 2 ) ) = ( ( ( C ^ 2 ) x. ( A ^ 2 ) ) + ( ( C ^ 2 ) x. ( B ^ 2 ) ) ) )
68	63	sqcld 10996	. . . . . . . . . . 11 |- ( ph -> ( C ^ 2 ) e. CC )
69	53	zcnd 9664	. . . . . . . . . . 11 |- ( ph -> ( A ^ 2 ) e. CC )
70	65	sqcld 10996	. . . . . . . . . . 11 |- ( ph -> ( B ^ 2 ) e. CC )
71	68, 69, 70	adddid 8263	. . . . . . . . . 10 |- ( ph -> ( ( C ^ 2 ) x. ( ( A ^ 2 ) + ( B ^ 2 ) ) ) = ( ( ( C ^ 2 ) x. ( A ^ 2 ) ) + ( ( C ^ 2 ) x. ( B ^ 2 ) ) ) )
72	67, 71	eqtr4d 2267	. . . . . . . . 9 |- ( ph -> ( ( ( C x. A ) ^ 2 ) + ( ( C x. B ) ^ 2 ) ) = ( ( C ^ 2 ) x. ( ( A ^ 2 ) + ( B ^ 2 ) ) ) )
73	2	nncnd 9216	. . . . . . . . . . . . 13 |- ( ph -> N e. CC )
74	47	zcnd 9664	. . . . . . . . . . . . 13 |- ( ph -> P e. CC )
75	73, 74	mulcomd 8260	. . . . . . . . . . . 12 |- ( ph -> ( N x. P ) = ( P x. N ) )
76	14, 75	eqtr3d 2266	. . . . . . . . . . 11 |- ( ph -> ( ( A ^ 2 ) + ( B ^ 2 ) ) = ( P x. N ) )
77	76	oveq2d 6044	. . . . . . . . . 10 |- ( ph -> ( ( C ^ 2 ) x. ( ( A ^ 2 ) + ( B ^ 2 ) ) ) = ( ( C ^ 2 ) x. ( P x. N ) ) )
78	68, 74, 73	mul12d 8390	. . . . . . . . . 10 |- ( ph -> ( ( C ^ 2 ) x. ( P x. N ) ) = ( P x. ( ( C ^ 2 ) x. N ) ) )
79	77, 78	eqtrd 2264	. . . . . . . . 9 |- ( ph -> ( ( C ^ 2 ) x. ( ( A ^ 2 ) + ( B ^ 2 ) ) ) = ( P x. ( ( C ^ 2 ) x. N ) ) )
80	72, 79	eqtrd 2264	. . . . . . . 8 |- ( ph -> ( ( ( C x. A ) ^ 2 ) + ( ( C x. B ) ^ 2 ) ) = ( P x. ( ( C ^ 2 ) x. N ) ) )
81	27, 34	sqmuld 11010	. . . . . . . . . . . 12 |- ( ph -> ( ( A x. D ) ^ 2 ) = ( ( A ^ 2 ) x. ( D ^ 2 ) ) )
82	34	sqcld 10996	. . . . . . . . . . . . 13 |- ( ph -> ( D ^ 2 ) e. CC )
83	69, 82	mulcomd 8260	. . . . . . . . . . . 12 |- ( ph -> ( ( A ^ 2 ) x. ( D ^ 2 ) ) = ( ( D ^ 2 ) x. ( A ^ 2 ) ) )
84	81, 83	eqtrd 2264	. . . . . . . . . . 11 |- ( ph -> ( ( A x. D ) ^ 2 ) = ( ( D ^ 2 ) x. ( A ^ 2 ) ) )
85	64, 84	oveq12d 6046	. . . . . . . . . 10 |- ( ph -> ( ( ( C x. A ) ^ 2 ) + ( ( A x. D ) ^ 2 ) ) = ( ( ( C ^ 2 ) x. ( A ^ 2 ) ) + ( ( D ^ 2 ) x. ( A ^ 2 ) ) ) )
86	49	zcnd 9664	. . . . . . . . . . 11 |- ( ph -> ( C ^ 2 ) e. CC )
87	86, 82, 69	adddird 8264	. . . . . . . . . 10 |- ( ph -> ( ( ( C ^ 2 ) + ( D ^ 2 ) ) x. ( A ^ 2 ) ) = ( ( ( C ^ 2 ) x. ( A ^ 2 ) ) + ( ( D ^ 2 ) x. ( A ^ 2 ) ) ) )
88	85, 87	eqtr4d 2267	. . . . . . . . 9 |- ( ph -> ( ( ( C x. A ) ^ 2 ) + ( ( A x. D ) ^ 2 ) ) = ( ( ( C ^ 2 ) + ( D ^ 2 ) ) x. ( A ^ 2 ) ) )
89	16	oveq1d 6043	. . . . . . . . 9 |- ( ph -> ( P x. ( A ^ 2 ) ) = ( ( ( C ^ 2 ) + ( D ^ 2 ) ) x. ( A ^ 2 ) ) )
90	88, 89	eqtr4d 2267	. . . . . . . 8 |- ( ph -> ( ( ( C x. A ) ^ 2 ) + ( ( A x. D ) ^ 2 ) ) = ( P x. ( A ^ 2 ) ) )
91	80, 90	oveq12d 6046	. . . . . . 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 9664	. . . . . . . 8 |- ( ph -> ( ( C ^ 2 ) x. N ) e. CC )
93	74, 92, 69	subdid 8652	. . . . . . 7 |- ( ph -> ( P x. ( ( ( C ^ 2 ) x. N ) - ( A ^ 2 ) ) ) = ( ( P x. ( ( C ^ 2 ) x. N ) ) - ( P x. ( A ^ 2 ) ) ) )
94	91, 93	eqtr4d 2267	. . . . . 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 2266	. . . . 5 |- ( ph -> ( ( ( C x. B ) ^ 2 ) - ( ( A x. D ) ^ 2 ) ) = ( P x. ( ( ( C ^ 2 ) x. N ) - ( A ^ 2 ) ) ) )
96		subsq 10971	. . . . . 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 2266	. . . 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 4119	. . 3 |- ( ph -> P || ( ( ( C x. B ) + ( A x. D ) ) x. ( ( C x. B ) - ( A x. D ) ) ) )
100	37, 39	zaddcld 9667	. . . 4 |- ( ph -> ( ( C x. B ) + ( A x. D ) ) e. ZZ )
101	37, 39	zsubcld 9668	. . . 4 |- ( ph -> ( ( C x. B ) - ( A x. D ) ) e. ZZ )
102		euclemma 12798	. . . 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 1274	. . 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 806	1 |- ( ph -> N e. S )

Syntax hints:   -> wi 4   /\ wa 104   <-> wb 105   \/ wo 716   = wceq 1398   e. wcel 2202   class class class wbr 4093   |-> cmpt 4155   ran crn 4732   ` cfv 5333  (class class class)co 6028   CCcc 8090   + caddc 8095   x. cmul 8097   - cmin 8409   -ucneg 8410   NNcn 9202   2c2 9253   ZZcz 9540   ^cexp 10863   abscabs 11637   || cdvds 12428   Primecprime 12759   ZZ[_i]cgz 13022

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 717  ax-5 1496  ax-7 1497  ax-gen 1498  ax-ie1 1542  ax-ie2 1543  ax-8 1553  ax-10 1554  ax-11 1555  ax-i12 1556  ax-bndl 1558  ax-4 1559  ax-17 1575  ax-i9 1579  ax-ial 1583  ax-i5r 1584  ax-13 2204  ax-14 2205  ax-ext 2213  ax-coll 4209  ax-sep 4212  ax-nul 4220  ax-pow 4270  ax-pr 4305  ax-un 4536  ax-setind 4641  ax-iinf 4692  ax-cnex 8183  ax-resscn 8184  ax-1cn 8185  ax-1re 8186  ax-icn 8187  ax-addcl 8188  ax-addrcl 8189  ax-mulcl 8190  ax-mulrcl 8191  ax-addcom 8192  ax-mulcom 8193  ax-addass 8194  ax-mulass 8195  ax-distr 8196  ax-i2m1 8197  ax-0lt1 8198  ax-1rid 8199  ax-0id 8200  ax-rnegex 8201  ax-precex 8202  ax-cnre 8203  ax-pre-ltirr 8204  ax-pre-ltwlin 8205  ax-pre-lttrn 8206  ax-pre-apti 8207  ax-pre-ltadd 8208  ax-pre-mulgt0 8209  ax-pre-mulext 8210  ax-arch 8211  ax-caucvg 8212

This theorem depends on definitions:  df-bi 117  df-dc 843  df-3or 1006  df-3an 1007  df-tru 1401  df-fal 1404  df-nf 1510  df-sb 1811  df-eu 2082  df-mo 2083  df-clab 2218  df-cleq 2224  df-clel 2227  df-nfc 2364  df-ne 2404  df-nel 2499  df-ral 2516  df-rex 2517  df-reu 2518  df-rmo 2519  df-rab 2520  df-v 2805  df-sbc 3033  df-csb 3129  df-dif 3203  df-un 3205  df-in 3207  df-ss 3214  df-nul 3497  df-if 3608  df-pw 3658  df-sn 3679  df-pr 3680  df-op 3682  df-uni 3899  df-int 3934  df-iun 3977  df-br 4094  df-opab 4156  df-mpt 4157  df-tr 4193  df-id 4396  df-po 4399  df-iso 4400  df-iord 4469  df-on 4471  df-ilim 4472  df-suc 4474  df-iom 4695  df-xp 4737  df-rel 4738  df-cnv 4739  df-co 4740  df-dm 4741  df-rn 4742  df-res 4743  df-ima 4744  df-iota 5293  df-fun 5335  df-fn 5336  df-f 5337  df-f1 5338  df-fo 5339  df-f1o 5340  df-fv 5341  df-riota 5981  df-ov 6031  df-oprab 6032  df-mpo 6033  df-1st 6312  df-2nd 6313  df-recs 6514  df-frec 6600  df-1o 6625  df-2o 6626  df-er 6745  df-en 6953  df-sup 7243  df-pnf 8275  df-mnf 8276  df-xr 8277  df-ltxr 8278  df-le 8279  df-sub 8411  df-neg 8412  df-reap 8814  df-ap 8821  df-div 8912  df-inn 9203  df-2 9261  df-3 9262  df-4 9263  df-n0 9462  df-z 9541  df-uz 9817  df-q 9915  df-rp 9950  df-fz 10306  df-fzo 10440  df-fl 10593  df-mod 10648  df-seqfrec 10773  df-exp 10864  df-cj 11482  df-re 11483  df-im 11484  df-rsqrt 11638  df-abs 11639  df-dvds 12429  df-gcd 12605  df-prm 12760  df-gz 13023

This theorem is referenced by:  2sqlem5  15938