Theorem 4sqlem19 13286

Description: Lemma for 4sq 13287. The proof is by strong induction - we show that if all the integers less than k are in S, then k is as well. In this part of the proof we do the induction argument and dispense with all the cases except the odd prime case, which is sent to 4sqlem18 13285. If k is 0 , 1 , 2, we show k e. S directly; otherwise if k is composite, k is the product of two numbers less than it (and hence in S by assumption), so by mul4sq 13277 k e. S. (Contributed by Mario Carneiro, 14-Jul-2014.) (Revised by Mario Carneiro, 20-Jun-2015.)

Hypothesis

Ref	Expression
4sq.1	|- S = { n | E. x e. ZZ E. y e. ZZ E. z e. ZZ E. w e. ZZ n = ( ( ( x ^ 2 ) + ( y ^ 2 ) ) + ( ( z ^ 2 ) + ( w ^ 2 ) ) ) }

Assertion

Ref	Expression
4sqlem19	|- NN0 = S

Distinct variable groups:   w, n, x, y, z   S, n

Allowed substitution hints:   S( x, y, z, w)

Proof of Theorem 4sqlem19

Dummy variables j k i m are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		elnn0 10179	. . . 4 |- ( k e. NN0 <-> ( k e. NN \/ k = 0 ) )
2		eleq1 2464	. . . . . 6 |- ( j = 1 -> ( j e. S <-> 1 e. S ) )
3		eleq1 2464	. . . . . 6 |- ( j = m -> ( j e. S <-> m e. S ) )
4		eleq1 2464	. . . . . 6 |- ( j = i -> ( j e. S <-> i e. S ) )
5		eleq1 2464	. . . . . 6 |- ( j = ( m x. i ) -> ( j e. S <-> ( m x. i ) e. S ) )
6		eleq1 2464	. . . . . 6 |- ( j = k -> ( j e. S <-> k e. S ) )
7		abs1 12057	. . . . . . . . . . 11 |- ( abs ` 1 ) = 1
8	7	oveq1i 6050	. . . . . . . . . 10 |- ( ( abs ` 1 ) ^ 2 ) = ( 1 ^ 2 )
9		sq1 11431	. . . . . . . . . 10 |- ( 1 ^ 2 ) = 1
10	8, 9	eqtri 2424	. . . . . . . . 9 |- ( ( abs ` 1 ) ^ 2 ) = 1
11		abs0 12045	. . . . . . . . . . 11 |- ( abs ` 0 ) = 0
12	11	oveq1i 6050	. . . . . . . . . 10 |- ( ( abs ` 0 ) ^ 2 ) = ( 0 ^ 2 )
13		sq0 11428	. . . . . . . . . 10 |- ( 0 ^ 2 ) = 0
14	12, 13	eqtri 2424	. . . . . . . . 9 |- ( ( abs ` 0 ) ^ 2 ) = 0
15	10, 14	oveq12i 6052	. . . . . . . 8 |- ( ( ( abs ` 1 ) ^ 2 ) + ( ( abs ` 0 ) ^ 2 ) ) = ( 1 + 0 )
16		ax-1cn 9004	. . . . . . . . 9 |- 1 e. CC
17	16	addid1i 9209	. . . . . . . 8 |- ( 1 + 0 ) = 1
18	15, 17	eqtri 2424	. . . . . . 7 |- ( ( ( abs ` 1 ) ^ 2 ) + ( ( abs ` 0 ) ^ 2 ) ) = 1
19		1z 10267	. . . . . . . . 9 |- 1 e. ZZ
20		zgz 13256	. . . . . . . . 9 |- ( 1 e. ZZ -> 1 e. ZZ [ _i ] )
21	19, 20	ax-mp 8	. . . . . . . 8 |- 1 e. ZZ [ _i ]
22		0z 10249	. . . . . . . . 9 |- 0 e. ZZ
23		zgz 13256	. . . . . . . . 9 |- ( 0 e. ZZ -> 0 e. ZZ [ _i ] )
24	22, 23	ax-mp 8	. . . . . . . 8 |- 0 e. ZZ [ _i ]
25		4sq.1	. . . . . . . . 9 |- S = { n | E. x e. ZZ E. y e. ZZ E. z e. ZZ E. w e. ZZ n = ( ( ( x ^ 2 ) + ( y ^ 2 ) ) + ( ( z ^ 2 ) + ( w ^ 2 ) ) ) }
26	25	4sqlem4a 13274	. . . . . . . 8 |- ( ( 1 e. ZZ [ _i ] /\ 0 e. ZZ [ _i ] ) -> ( ( ( abs ` 1 ) ^ 2 ) + ( ( abs ` 0 ) ^ 2 ) ) e. S )
27	21, 24, 26	mp2an 654	. . . . . . 7 |- ( ( ( abs ` 1 ) ^ 2 ) + ( ( abs ` 0 ) ^ 2 ) ) e. S
28	18, 27	eqeltrri 2475	. . . . . 6 |- 1 e. S
29		eleq1 2464	. . . . . . 7 |- ( j = 2 -> ( j e. S <-> 2 e. S ) )
30		eldifsn 3887	. . . . . . . . 9 |- ( j e. ( Prime \ { 2 } ) <-> ( j e. Prime /\ j =/= 2 ) )
31		oddprm 13144	. . . . . . . . . . 11 |- ( j e. ( Prime \ { 2 } ) -> ( ( j - 1 ) / 2 ) e. NN )
32	31	adantr 452	. . . . . . . . . 10 |- ( ( j e. ( Prime \ { 2 } ) /\ A. m e. ( 1 ... ( j - 1 ) ) m e. S ) -> ( ( j - 1 ) / 2 ) e. NN )
33		eldifi 3429	. . . . . . . . . . . . . . . 16 |- ( j e. ( Prime \ { 2 } ) -> j e. Prime )
34	33	adantr 452	. . . . . . . . . . . . . . 15 |- ( ( j e. ( Prime \ { 2 } ) /\ A. m e. ( 1 ... ( j - 1 ) ) m e. S ) -> j e. Prime )
35		prmnn 13037	. . . . . . . . . . . . . . 15 |- ( j e. Prime -> j e. NN )
36		nncn 9964	. . . . . . . . . . . . . . 15 |- ( j e. NN -> j e. CC )
37	34, 35, 36	3syl 19	. . . . . . . . . . . . . 14 |- ( ( j e. ( Prime \ { 2 } ) /\ A. m e. ( 1 ... ( j - 1 ) ) m e. S ) -> j e. CC )
38		subcl 9261	. . . . . . . . . . . . . 14 |- ( ( j e. CC /\ 1 e. CC ) -> ( j - 1 ) e. CC )
39	37, 16, 38	sylancl 644	. . . . . . . . . . . . 13 |- ( ( j e. ( Prime \ { 2 } ) /\ A. m e. ( 1 ... ( j - 1 ) ) m e. S ) -> ( j - 1 ) e. CC )
40		2cn 10026	. . . . . . . . . . . . . 14 |- 2 e. CC
41	40	a1i 11	. . . . . . . . . . . . 13 |- ( ( j e. ( Prime \ { 2 } ) /\ A. m e. ( 1 ... ( j - 1 ) ) m e. S ) -> 2 e. CC )
42		2ne0 10039	. . . . . . . . . . . . . 14 |- 2 =/= 0
43	42	a1i 11	. . . . . . . . . . . . 13 |- ( ( j e. ( Prime \ { 2 } ) /\ A. m e. ( 1 ... ( j - 1 ) ) m e. S ) -> 2 =/= 0 )
44	39, 41, 43	divcan2d 9748	. . . . . . . . . . . 12 |- ( ( j e. ( Prime \ { 2 } ) /\ A. m e. ( 1 ... ( j - 1 ) ) m e. S ) -> ( 2 x. ( ( j - 1 ) / 2 ) ) = ( j - 1 ) )
45	44	oveq1d 6055	. . . . . . . . . . 11 |- ( ( j e. ( Prime \ { 2 } ) /\ A. m e. ( 1 ... ( j - 1 ) ) m e. S ) -> ( ( 2 x. ( ( j - 1 ) / 2 ) ) + 1 ) = ( ( j - 1 ) + 1 ) )
46		npcan 9270	. . . . . . . . . . . 12 |- ( ( j e. CC /\ 1 e. CC ) -> ( ( j - 1 ) + 1 ) = j )
47	37, 16, 46	sylancl 644	. . . . . . . . . . 11 |- ( ( j e. ( Prime \ { 2 } ) /\ A. m e. ( 1 ... ( j - 1 ) ) m e. S ) -> ( ( j - 1 ) + 1 ) = j )
48	45, 47	eqtr2d 2437	. . . . . . . . . 10 |- ( ( j e. ( Prime \ { 2 } ) /\ A. m e. ( 1 ... ( j - 1 ) ) m e. S ) -> j = ( ( 2 x. ( ( j - 1 ) / 2 ) ) + 1 ) )
49	44	oveq2d 6056	. . . . . . . . . . . 12 |- ( ( j e. ( Prime \ { 2 } ) /\ A. m e. ( 1 ... ( j - 1 ) ) m e. S ) -> ( 0 ... ( 2 x. ( ( j - 1 ) / 2 ) ) ) = ( 0 ... ( j - 1 ) ) )
50		nnm1nn0 10217	. . . . . . . . . . . . . . 15 |- ( j e. NN -> ( j - 1 ) e. NN0 )
51	34, 35, 50	3syl 19	. . . . . . . . . . . . . 14 |- ( ( j e. ( Prime \ { 2 } ) /\ A. m e. ( 1 ... ( j - 1 ) ) m e. S ) -> ( j - 1 ) e. NN0 )
52		elnn0uz 10479	. . . . . . . . . . . . . 14 |- ( ( j - 1 ) e. NN0 <-> ( j - 1 ) e. ( ZZ>= ` 0 ) )
53	51, 52	sylib 189	. . . . . . . . . . . . 13 |- ( ( j e. ( Prime \ { 2 } ) /\ A. m e. ( 1 ... ( j - 1 ) ) m e. S ) -> ( j - 1 ) e. ( ZZ>= ` 0 ) )
54		eluzfz1 11020	. . . . . . . . . . . . 13 |- ( ( j - 1 ) e. ( ZZ>= ` 0 ) -> 0 e. ( 0 ... ( j - 1 ) ) )
55		fzsplit 11033	. . . . . . . . . . . . 13 |- ( 0 e. ( 0 ... ( j - 1 ) ) -> ( 0 ... ( j - 1 ) ) = ( ( 0 ... 0 ) u. ( ( 0 + 1 ) ... ( j - 1 ) ) ) )
56	53, 54, 55	3syl 19	. . . . . . . . . . . 12 |- ( ( j e. ( Prime \ { 2 } ) /\ A. m e. ( 1 ... ( j - 1 ) ) m e. S ) -> ( 0 ... ( j - 1 ) ) = ( ( 0 ... 0 ) u. ( ( 0 + 1 ) ... ( j - 1 ) ) ) )
57	49, 56	eqtrd 2436	. . . . . . . . . . 11 |- ( ( j e. ( Prime \ { 2 } ) /\ A. m e. ( 1 ... ( j - 1 ) ) m e. S ) -> ( 0 ... ( 2 x. ( ( j - 1 ) / 2 ) ) ) = ( ( 0 ... 0 ) u. ( ( 0 + 1 ) ... ( j - 1 ) ) ) )
58		fzsn 11050	. . . . . . . . . . . . . . 15 |- ( 0 e. ZZ -> ( 0 ... 0 ) = { 0 } )
59	22, 58	ax-mp 8	. . . . . . . . . . . . . 14 |- ( 0 ... 0 ) = { 0 }
60	14, 14	oveq12i 6052	. . . . . . . . . . . . . . . . 17 |- ( ( ( abs ` 0 ) ^ 2 ) + ( ( abs ` 0 ) ^ 2 ) ) = ( 0 + 0 )
61		00id 9197	. . . . . . . . . . . . . . . . 17 |- ( 0 + 0 ) = 0
62	60, 61	eqtri 2424	. . . . . . . . . . . . . . . 16 |- ( ( ( abs ` 0 ) ^ 2 ) + ( ( abs ` 0 ) ^ 2 ) ) = 0
63	25	4sqlem4a 13274	. . . . . . . . . . . . . . . . 17 |- ( ( 0 e. ZZ [ _i ] /\ 0 e. ZZ [ _i ] ) -> ( ( ( abs ` 0 ) ^ 2 ) + ( ( abs ` 0 ) ^ 2 ) ) e. S )
64	24, 24, 63	mp2an 654	. . . . . . . . . . . . . . . 16 |- ( ( ( abs ` 0 ) ^ 2 ) + ( ( abs ` 0 ) ^ 2 ) ) e. S
65	62, 64	eqeltrri 2475	. . . . . . . . . . . . . . 15 |- 0 e. S
66		snssi 3902	. . . . . . . . . . . . . . 15 |- ( 0 e. S -> { 0 } C_ S )
67	65, 66	ax-mp 8	. . . . . . . . . . . . . 14 |- { 0 } C_ S
68	59, 67	eqsstri 3338	. . . . . . . . . . . . 13 |- ( 0 ... 0 ) C_ S
69	68	a1i 11	. . . . . . . . . . . 12 |- ( ( j e. ( Prime \ { 2 } ) /\ A. m e. ( 1 ... ( j - 1 ) ) m e. S ) -> ( 0 ... 0 ) C_ S )
70		0p1e1 10049	. . . . . . . . . . . . . 14 |- ( 0 + 1 ) = 1
71	70	oveq1i 6050	. . . . . . . . . . . . 13 |- ( ( 0 + 1 ) ... ( j - 1 ) ) = ( 1 ... ( j - 1 ) )
72		simpr 448	. . . . . . . . . . . . . 14 |- ( ( j e. ( Prime \ { 2 } ) /\ A. m e. ( 1 ... ( j - 1 ) ) m e. S ) -> A. m e. ( 1 ... ( j - 1 ) ) m e. S )
73		dfss3 3298	. . . . . . . . . . . . . 14 |- ( ( 1 ... ( j - 1 ) ) C_ S <-> A. m e. ( 1 ... ( j - 1 ) ) m e. S )
74	72, 73	sylibr 204	. . . . . . . . . . . . 13 |- ( ( j e. ( Prime \ { 2 } ) /\ A. m e. ( 1 ... ( j - 1 ) ) m e. S ) -> ( 1 ... ( j - 1 ) ) C_ S )
75	71, 74	syl5eqss 3352	. . . . . . . . . . . 12 |- ( ( j e. ( Prime \ { 2 } ) /\ A. m e. ( 1 ... ( j - 1 ) ) m e. S ) -> ( ( 0 + 1 ) ... ( j - 1 ) ) C_ S )
76	69, 75	unssd 3483	. . . . . . . . . . 11 |- ( ( j e. ( Prime \ { 2 } ) /\ A. m e. ( 1 ... ( j - 1 ) ) m e. S ) -> ( ( 0 ... 0 ) u. ( ( 0 + 1 ) ... ( j - 1 ) ) ) C_ S )
77	57, 76	eqsstrd 3342	. . . . . . . . . 10 |- ( ( j e. ( Prime \ { 2 } ) /\ A. m e. ( 1 ... ( j - 1 ) ) m e. S ) -> ( 0 ... ( 2 x. ( ( j - 1 ) / 2 ) ) ) C_ S )
78		oveq1 6047	. . . . . . . . . . . 12 |- ( k = i -> ( k x. j ) = ( i x. j ) )
79	78	eleq1d 2470	. . . . . . . . . . 11 |- ( k = i -> ( ( k x. j ) e. S <-> ( i x. j ) e. S ) )
80	79	cbvrabv 2915	. . . . . . . . . 10 |- { k e. NN | ( k x. j ) e. S } = { i e. NN | ( i x. j ) e. S }
81		eqid 2404	. . . . . . . . . 10 |- sup ( { k e. NN | ( k x. j ) e. S } , RR , `' < ) = sup ( { k e. NN | ( k x. j ) e. S } , RR , `' < )
82	25, 32, 48, 34, 77, 80, 81	4sqlem18 13285	. . . . . . . . 9 |- ( ( j e. ( Prime \ { 2 } ) /\ A. m e. ( 1 ... ( j - 1 ) ) m e. S ) -> j e. S )
83	30, 82	sylanbr 460	. . . . . . . 8 |- ( ( ( j e. Prime /\ j =/= 2 ) /\ A. m e. ( 1 ... ( j - 1 ) ) m e. S ) -> j e. S )
84	83	an32s 780	. . . . . . 7 |- ( ( ( j e. Prime /\ A. m e. ( 1 ... ( j - 1 ) ) m e. S ) /\ j =/= 2 ) -> j e. S )
85	10, 10	oveq12i 6052	. . . . . . . . . 10 |- ( ( ( abs ` 1 ) ^ 2 ) + ( ( abs ` 1 ) ^ 2 ) ) = ( 1 + 1 )
86		df-2 10014	. . . . . . . . . 10 |- 2 = ( 1 + 1 )
87	85, 86	eqtr4i 2427	. . . . . . . . 9 |- ( ( ( abs ` 1 ) ^ 2 ) + ( ( abs ` 1 ) ^ 2 ) ) = 2
88	25	4sqlem4a 13274	. . . . . . . . . 10 |- ( ( 1 e. ZZ [ _i ] /\ 1 e. ZZ [ _i ] ) -> ( ( ( abs ` 1 ) ^ 2 ) + ( ( abs ` 1 ) ^ 2 ) ) e. S )
89	21, 21, 88	mp2an 654	. . . . . . . . 9 |- ( ( ( abs ` 1 ) ^ 2 ) + ( ( abs ` 1 ) ^ 2 ) ) e. S
90	87, 89	eqeltrri 2475	. . . . . . . 8 |- 2 e. S
91	90	a1i 11	. . . . . . 7 |- ( ( j e. Prime /\ A. m e. ( 1 ... ( j - 1 ) ) m e. S ) -> 2 e. S )
92	29, 84, 91	pm2.61ne 2642	. . . . . 6 |- ( ( j e. Prime /\ A. m e. ( 1 ... ( j - 1 ) ) m e. S ) -> j e. S )
93	25	mul4sq 13277	. . . . . . 7 |- ( ( m e. S /\ i e. S ) -> ( m x. i ) e. S )
94	93	a1i 11	. . . . . 6 |- ( ( m e. ( ZZ>= ` 2 ) /\ i e. ( ZZ>= ` 2 ) ) -> ( ( m e. S /\ i e. S ) -> ( m x. i ) e. S ) )
95	2, 3, 4, 5, 6, 28, 92, 94	prmind2 13045	. . . . 5 |- ( k e. NN -> k e. S )
96		id 20	. . . . . 6 |- ( k = 0 -> k = 0 )
97	96, 65	syl6eqel 2492	. . . . 5 |- ( k = 0 -> k e. S )
98	95, 97	jaoi 369	. . . 4 |- ( ( k e. NN \/ k = 0 ) -> k e. S )
99	1, 98	sylbi 188	. . 3 |- ( k e. NN0 -> k e. S )
100	99	ssriv 3312	. 2 |- NN0 C_ S
101	25	4sqlem1 13271	. 2 |- S C_ NN0
102	100, 101	eqssi 3324	1 |- NN0 = S

Syntax hints:   -> wi 4   \/ wo 358   /\ wa 359   = wceq 1649   e. wcel 1721   {cab 2390   =/= wne 2567   A.wral 2666   E.wrex 2667   {crab 2670   \ cdif 3277   u. cun 3278   C_ wss 3280   {csn 3774   `'ccnv 4836   ` cfv 5413  (class class class)co 6040   supcsup 7403   CCcc 8944   RRcr 8945   0cc0 8946   1c1 8947   + caddc 8949   x. cmul 8951   < clt 9076   - cmin 9247   / cdiv 9633   NNcn 9956   2c2 10005   NN0cn0 10177   ZZcz 10238   ZZ>=cuz 10444   ...cfz 10999   ^cexp 11337   abscabs 11994   Primecprime 13034   ZZ [ _i ]cgz 13252

This theorem is referenced by:  4sq  13287

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1552  ax-5 1563  ax-17 1623  ax-9 1662  ax-8 1683  ax-13 1723  ax-14 1725  ax-6 1740  ax-7 1745  ax-11 1757  ax-12 1946  ax-ext 2385  ax-rep 4280  ax-sep 4290  ax-nul 4298  ax-pow 4337  ax-pr 4363  ax-un 4660  ax-cnex 9002  ax-resscn 9003  ax-1cn 9004  ax-icn 9005  ax-addcl 9006  ax-addrcl 9007  ax-mulcl 9008  ax-mulrcl 9009  ax-mulcom 9010  ax-addass 9011  ax-mulass 9012  ax-distr 9013  ax-i2m1 9014  ax-1ne0 9015  ax-1rid 9016  ax-rnegex 9017  ax-rrecex 9018  ax-cnre 9019  ax-pre-lttri 9020  ax-pre-lttrn 9021  ax-pre-ltadd 9022  ax-pre-mulgt0 9023  ax-pre-sup 9024

This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3or 937  df-3an 938  df-tru 1325  df-ex 1548  df-nf 1551  df-sb 1656  df-eu 2258  df-mo 2259  df-clab 2391  df-cleq 2397  df-clel 2400  df-nfc 2529  df-ne 2569  df-nel 2570  df-ral 2671  df-rex 2672  df-reu 2673  df-rmo 2674  df-rab 2675  df-v 2918  df-sbc 3122  df-csb 3212  df-dif 3283  df-un 3285  df-in 3287  df-ss 3294  df-pss 3296  df-nul 3589  df-if 3700  df-pw 3761  df-sn 3780  df-pr 3781  df-tp 3782  df-op 3783  df-uni 3976  df-int 4011  df-iun 4055  df-br 4173  df-opab 4227  df-mpt 4228  df-tr 4263  df-eprel 4454  df-id 4458  df-po 4463  df-so 4464  df-fr 4501  df-we 4503  df-ord 4544  df-on 4545  df-lim 4546  df-suc 4547  df-om 4805  df-xp 4843  df-rel 4844  df-cnv 4845  df-co 4846  df-dm 4847  df-rn 4848  df-res 4849  df-ima 4850  df-iota 5377  df-fun 5415  df-fn 5416  df-f 5417  df-f1 5418  df-fo 5419  df-f1o 5420  df-fv 5421  df-ov 6043  df-oprab 6044  df-mpt2 6045  df-1st 6308  df-2nd 6309  df-riota 6508  df-recs 6592  df-rdg 6627  df-1o 6683  df-2o 6684  df-oadd 6687  df-er 6864  df-en 7069  df-dom 7070  df-sdom 7071  df-fin 7072  df-sup 7404  df-card 7782  df-cda 8004  df-pnf 9078  df-mnf 9079  df-xr 9080  df-ltxr 9081  df-le 9082  df-sub 9249  df-neg 9250  df-div 9634  df-nn 9957  df-2 10014  df-3 10015  df-4 10016  df-n0 10178  df-z 10239  df-uz 10445  df-rp 10569  df-fz 11000  df-fl 11157  df-mod 11206  df-seq 11279  df-exp 11338  df-hash 11574  df-cj 11859  df-re 11860  df-im 11861  df-sqr 11995  df-abs 11996  df-dvds 12808  df-gcd 12962  df-prm 13035  df-gz 13253