Theorem 4sqlem19 14492

Description: Lemma for 4sq 14493. 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 14491. 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 14483 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 10818	. . . 4 |- ( k e. NN0 <-> ( k e. NN \/ k = 0 ) )
2		eleq1 2529	. . . . . 6 |- ( j = 1 -> ( j e. S <-> 1 e. S ) )
3		eleq1 2529	. . . . . 6 |- ( j = m -> ( j e. S <-> m e. S ) )
4		eleq1 2529	. . . . . 6 |- ( j = i -> ( j e. S <-> i e. S ) )
5		eleq1 2529	. . . . . 6 |- ( j = ( m x. i ) -> ( j e. S <-> ( m x. i ) e. S ) )
6		eleq1 2529	. . . . . 6 |- ( j = k -> ( j e. S <-> k e. S ) )
7		abs1 13141	. . . . . . . . . . 11 |- ( abs ` 1 ) = 1
8	7	oveq1i 6306	. . . . . . . . . 10 |- ( ( abs ` 1 ) ^ 2 ) = ( 1 ^ 2 )
9		sq1 12264	. . . . . . . . . 10 |- ( 1 ^ 2 ) = 1
10	8, 9	eqtri 2486	. . . . . . . . 9 |- ( ( abs ` 1 ) ^ 2 ) = 1
11		abs0 13129	. . . . . . . . . . 11 |- ( abs ` 0 ) = 0
12	11	oveq1i 6306	. . . . . . . . . 10 |- ( ( abs ` 0 ) ^ 2 ) = ( 0 ^ 2 )
13		sq0 12261	. . . . . . . . . 10 |- ( 0 ^ 2 ) = 0
14	12, 13	eqtri 2486	. . . . . . . . 9 |- ( ( abs ` 0 ) ^ 2 ) = 0
15	10, 14	oveq12i 6308	. . . . . . . 8 |- ( ( ( abs ` 1 ) ^ 2 ) + ( ( abs ` 0 ) ^ 2 ) ) = ( 1 + 0 )
16		1p0e1 10669	. . . . . . . 8 |- ( 1 + 0 ) = 1
17	15, 16	eqtri 2486	. . . . . . 7 |- ( ( ( abs ` 1 ) ^ 2 ) + ( ( abs ` 0 ) ^ 2 ) ) = 1
18		1z 10915	. . . . . . . . 9 |- 1 e. ZZ
19		zgz 14462	. . . . . . . . 9 |- ( 1 e. ZZ -> 1 e. ZZ[_i] )
20	18, 19	ax-mp 5	. . . . . . . 8 |- 1 e. ZZ[_i]
21		0z 10896	. . . . . . . . 9 |- 0 e. ZZ
22		zgz 14462	. . . . . . . . 9 |- ( 0 e. ZZ -> 0 e. ZZ[_i] )
23	21, 22	ax-mp 5	. . . . . . . 8 |- 0 e. ZZ[_i]
24		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 ) ) ) }
25	24	4sqlem4a 14480	. . . . . . . 8 |- ( ( 1 e. ZZ[_i] /\ 0 e. ZZ[_i] ) -> ( ( ( abs ` 1 ) ^ 2 ) + ( ( abs ` 0 ) ^ 2 ) ) e. S )
26	20, 23, 25	mp2an 672	. . . . . . 7 |- ( ( ( abs ` 1 ) ^ 2 ) + ( ( abs ` 0 ) ^ 2 ) ) e. S
27	17, 26	eqeltrri 2542	. . . . . 6 |- 1 e. S
28		eleq1 2529	. . . . . . 7 |- ( j = 2 -> ( j e. S <-> 2 e. S ) )
29		eldifsn 4157	. . . . . . . . 9 |- ( j e. ( Prime \ { 2 } ) <-> ( j e. Prime /\ j =/= 2 ) )
30		oddprm 14350	. . . . . . . . . . 11 |- ( j e. ( Prime \ { 2 } ) -> ( ( j - 1 ) / 2 ) e. NN )
31	30	adantr 465	. . . . . . . . . 10 |- ( ( j e. ( Prime \ { 2 } ) /\ A. m e. ( 1 ... ( j - 1 ) ) m e. S ) -> ( ( j - 1 ) / 2 ) e. NN )
32		eldifi 3622	. . . . . . . . . . . . . . . 16 |- ( j e. ( Prime \ { 2 } ) -> j e. Prime )
33	32	adantr 465	. . . . . . . . . . . . . . 15 |- ( ( j e. ( Prime \ { 2 } ) /\ A. m e. ( 1 ... ( j - 1 ) ) m e. S ) -> j e. Prime )
34		prmnn 14231	. . . . . . . . . . . . . . 15 |- ( j e. Prime -> j e. NN )
35		nncn 10564	. . . . . . . . . . . . . . 15 |- ( j e. NN -> j e. CC )
36	33, 34, 35	3syl 20	. . . . . . . . . . . . . 14 |- ( ( j e. ( Prime \ { 2 } ) /\ A. m e. ( 1 ... ( j - 1 ) ) m e. S ) -> j e. CC )
37		ax-1cn 9567	. . . . . . . . . . . . . 14 |- 1 e. CC
38		subcl 9838	. . . . . . . . . . . . . 14 |- ( ( j e. CC /\ 1 e. CC ) -> ( j - 1 ) e. CC )
39	36, 37, 38	sylancl 662	. . . . . . . . . . . . 13 |- ( ( j e. ( Prime \ { 2 } ) /\ A. m e. ( 1 ... ( j - 1 ) ) m e. S ) -> ( j - 1 ) e. CC )
40		2cnd 10629	. . . . . . . . . . . . 13 |- ( ( j e. ( Prime \ { 2 } ) /\ A. m e. ( 1 ... ( j - 1 ) ) m e. S ) -> 2 e. CC )
41		2ne0 10649	. . . . . . . . . . . . . 14 |- 2 =/= 0
42	41	a1i 11	. . . . . . . . . . . . 13 |- ( ( j e. ( Prime \ { 2 } ) /\ A. m e. ( 1 ... ( j - 1 ) ) m e. S ) -> 2 =/= 0 )
43	39, 40, 42	divcan2d 10343	. . . . . . . . . . . 12 |- ( ( j e. ( Prime \ { 2 } ) /\ A. m e. ( 1 ... ( j - 1 ) ) m e. S ) -> ( 2 x. ( ( j - 1 ) / 2 ) ) = ( j - 1 ) )
44	43	oveq1d 6311	. . . . . . . . . . 11 |- ( ( j e. ( Prime \ { 2 } ) /\ A. m e. ( 1 ... ( j - 1 ) ) m e. S ) -> ( ( 2 x. ( ( j - 1 ) / 2 ) ) + 1 ) = ( ( j - 1 ) + 1 ) )
45		npcan 9848	. . . . . . . . . . . 12 |- ( ( j e. CC /\ 1 e. CC ) -> ( ( j - 1 ) + 1 ) = j )
46	36, 37, 45	sylancl 662	. . . . . . . . . . 11 |- ( ( j e. ( Prime \ { 2 } ) /\ A. m e. ( 1 ... ( j - 1 ) ) m e. S ) -> ( ( j - 1 ) + 1 ) = j )
47	44, 46	eqtr2d 2499	. . . . . . . . . 10 |- ( ( j e. ( Prime \ { 2 } ) /\ A. m e. ( 1 ... ( j - 1 ) ) m e. S ) -> j = ( ( 2 x. ( ( j - 1 ) / 2 ) ) + 1 ) )
48	43	oveq2d 6312	. . . . . . . . . . . 12 |- ( ( j e. ( Prime \ { 2 } ) /\ A. m e. ( 1 ... ( j - 1 ) ) m e. S ) -> ( 0 ... ( 2 x. ( ( j - 1 ) / 2 ) ) ) = ( 0 ... ( j - 1 ) ) )
49		nnm1nn0 10858	. . . . . . . . . . . . . . 15 |- ( j e. NN -> ( j - 1 ) e. NN0 )
50	33, 34, 49	3syl 20	. . . . . . . . . . . . . 14 |- ( ( j e. ( Prime \ { 2 } ) /\ A. m e. ( 1 ... ( j - 1 ) ) m e. S ) -> ( j - 1 ) e. NN0 )
51		elnn0uz 11143	. . . . . . . . . . . . . 14 |- ( ( j - 1 ) e. NN0 <-> ( j - 1 ) e. ( ZZ>= ` 0 ) )
52	50, 51	sylib 196	. . . . . . . . . . . . 13 |- ( ( j e. ( Prime \ { 2 } ) /\ A. m e. ( 1 ... ( j - 1 ) ) m e. S ) -> ( j - 1 ) e. ( ZZ>= ` 0 ) )
53		eluzfz1 11718	. . . . . . . . . . . . 13 |- ( ( j - 1 ) e. ( ZZ>= ` 0 ) -> 0 e. ( 0 ... ( j - 1 ) ) )
54		fzsplit 11736	. . . . . . . . . . . . 13 |- ( 0 e. ( 0 ... ( j - 1 ) ) -> ( 0 ... ( j - 1 ) ) = ( ( 0 ... 0 ) u. ( ( 0 + 1 ) ... ( j - 1 ) ) ) )
55	52, 53, 54	3syl 20	. . . . . . . . . . . 12 |- ( ( j e. ( Prime \ { 2 } ) /\ A. m e. ( 1 ... ( j - 1 ) ) m e. S ) -> ( 0 ... ( j - 1 ) ) = ( ( 0 ... 0 ) u. ( ( 0 + 1 ) ... ( j - 1 ) ) ) )
56	48, 55	eqtrd 2498	. . . . . . . . . . 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 ) ) ) )
57		fzsn 11750	. . . . . . . . . . . . . . 15 |- ( 0 e. ZZ -> ( 0 ... 0 ) = { 0 } )
58	21, 57	ax-mp 5	. . . . . . . . . . . . . 14 |- ( 0 ... 0 ) = { 0 }
59	14, 14	oveq12i 6308	. . . . . . . . . . . . . . . . 17 |- ( ( ( abs ` 0 ) ^ 2 ) + ( ( abs ` 0 ) ^ 2 ) ) = ( 0 + 0 )
60		00id 9772	. . . . . . . . . . . . . . . . 17 |- ( 0 + 0 ) = 0
61	59, 60	eqtri 2486	. . . . . . . . . . . . . . . 16 |- ( ( ( abs ` 0 ) ^ 2 ) + ( ( abs ` 0 ) ^ 2 ) ) = 0
62	24	4sqlem4a 14480	. . . . . . . . . . . . . . . . 17 |- ( ( 0 e. ZZ[_i] /\ 0 e. ZZ[_i] ) -> ( ( ( abs ` 0 ) ^ 2 ) + ( ( abs ` 0 ) ^ 2 ) ) e. S )
63	23, 23, 62	mp2an 672	. . . . . . . . . . . . . . . 16 |- ( ( ( abs ` 0 ) ^ 2 ) + ( ( abs ` 0 ) ^ 2 ) ) e. S
64	61, 63	eqeltrri 2542	. . . . . . . . . . . . . . 15 |- 0 e. S
65		snssi 4176	. . . . . . . . . . . . . . 15 |- ( 0 e. S -> { 0 } C_ S )
66	64, 65	ax-mp 5	. . . . . . . . . . . . . 14 |- { 0 } C_ S
67	58, 66	eqsstri 3529	. . . . . . . . . . . . 13 |- ( 0 ... 0 ) C_ S
68	67	a1i 11	. . . . . . . . . . . 12 |- ( ( j e. ( Prime \ { 2 } ) /\ A. m e. ( 1 ... ( j - 1 ) ) m e. S ) -> ( 0 ... 0 ) C_ S )
69		0p1e1 10668	. . . . . . . . . . . . . 14 |- ( 0 + 1 ) = 1
70	69	oveq1i 6306	. . . . . . . . . . . . 13 |- ( ( 0 + 1 ) ... ( j - 1 ) ) = ( 1 ... ( j - 1 ) )
71		simpr 461	. . . . . . . . . . . . . 14 |- ( ( j e. ( Prime \ { 2 } ) /\ A. m e. ( 1 ... ( j - 1 ) ) m e. S ) -> A. m e. ( 1 ... ( j - 1 ) ) m e. S )
72		dfss3 3489	. . . . . . . . . . . . . 14 |- ( ( 1 ... ( j - 1 ) ) C_ S <-> A. m e. ( 1 ... ( j - 1 ) ) m e. S )
73	71, 72	sylibr 212	. . . . . . . . . . . . 13 |- ( ( j e. ( Prime \ { 2 } ) /\ A. m e. ( 1 ... ( j - 1 ) ) m e. S ) -> ( 1 ... ( j - 1 ) ) C_ S )
74	70, 73	syl5eqss 3543	. . . . . . . . . . . 12 |- ( ( j e. ( Prime \ { 2 } ) /\ A. m e. ( 1 ... ( j - 1 ) ) m e. S ) -> ( ( 0 + 1 ) ... ( j - 1 ) ) C_ S )
75	68, 74	unssd 3676	. . . . . . . . . . 11 |- ( ( j e. ( Prime \ { 2 } ) /\ A. m e. ( 1 ... ( j - 1 ) ) m e. S ) -> ( ( 0 ... 0 ) u. ( ( 0 + 1 ) ... ( j - 1 ) ) ) C_ S )
76	56, 75	eqsstrd 3533	. . . . . . . . . 10 |- ( ( j e. ( Prime \ { 2 } ) /\ A. m e. ( 1 ... ( j - 1 ) ) m e. S ) -> ( 0 ... ( 2 x. ( ( j - 1 ) / 2 ) ) ) C_ S )
77		oveq1 6303	. . . . . . . . . . . 12 |- ( k = i -> ( k x. j ) = ( i x. j ) )
78	77	eleq1d 2526	. . . . . . . . . . 11 |- ( k = i -> ( ( k x. j ) e. S <-> ( i x. j ) e. S ) )
79	78	cbvrabv 3108	. . . . . . . . . 10 |- { k e. NN | ( k x. j ) e. S } = { i e. NN | ( i x. j ) e. S }
80		eqid 2457	. . . . . . . . . 10 |- sup ( { k e. NN | ( k x. j ) e. S } , RR , `' < ) = sup ( { k e. NN | ( k x. j ) e. S } , RR , `' < )
81	24, 31, 47, 33, 76, 79, 80	4sqlem18 14491	. . . . . . . . 9 |- ( ( j e. ( Prime \ { 2 } ) /\ A. m e. ( 1 ... ( j - 1 ) ) m e. S ) -> j e. S )
82	29, 81	sylanbr 473	. . . . . . . 8 |- ( ( ( j e. Prime /\ j =/= 2 ) /\ A. m e. ( 1 ... ( j - 1 ) ) m e. S ) -> j e. S )
83	82	an32s 804	. . . . . . 7 |- ( ( ( j e. Prime /\ A. m e. ( 1 ... ( j - 1 ) ) m e. S ) /\ j =/= 2 ) -> j e. S )
84	10, 10	oveq12i 6308	. . . . . . . . . 10 |- ( ( ( abs ` 1 ) ^ 2 ) + ( ( abs ` 1 ) ^ 2 ) ) = ( 1 + 1 )
85		df-2 10615	. . . . . . . . . 10 |- 2 = ( 1 + 1 )
86	84, 85	eqtr4i 2489	. . . . . . . . 9 |- ( ( ( abs ` 1 ) ^ 2 ) + ( ( abs ` 1 ) ^ 2 ) ) = 2
87	24	4sqlem4a 14480	. . . . . . . . . 10 |- ( ( 1 e. ZZ[_i] /\ 1 e. ZZ[_i] ) -> ( ( ( abs ` 1 ) ^ 2 ) + ( ( abs ` 1 ) ^ 2 ) ) e. S )
88	20, 20, 87	mp2an 672	. . . . . . . . 9 |- ( ( ( abs ` 1 ) ^ 2 ) + ( ( abs ` 1 ) ^ 2 ) ) e. S
89	86, 88	eqeltrri 2542	. . . . . . . 8 |- 2 e. S
90	89	a1i 11	. . . . . . 7 |- ( ( j e. Prime /\ A. m e. ( 1 ... ( j - 1 ) ) m e. S ) -> 2 e. S )
91	28, 83, 90	pm2.61ne 2772	. . . . . 6 |- ( ( j e. Prime /\ A. m e. ( 1 ... ( j - 1 ) ) m e. S ) -> j e. S )
92	24	mul4sq 14483	. . . . . . 7 |- ( ( m e. S /\ i e. S ) -> ( m x. i ) e. S )
93	92	a1i 11	. . . . . 6 |- ( ( m e. ( ZZ>= ` 2 ) /\ i e. ( ZZ>= ` 2 ) ) -> ( ( m e. S /\ i e. S ) -> ( m x. i ) e. S ) )
94	2, 3, 4, 5, 6, 27, 91, 93	prmind2 14239	. . . . 5 |- ( k e. NN -> k e. S )
95		id 22	. . . . . 6 |- ( k = 0 -> k = 0 )
96	95, 64	syl6eqel 2553	. . . . 5 |- ( k = 0 -> k e. S )
97	94, 96	jaoi 379	. . . 4 |- ( ( k e. NN \/ k = 0 ) -> k e. S )
98	1, 97	sylbi 195	. . 3 |- ( k e. NN0 -> k e. S )
99	98	ssriv 3503	. 2 |- NN0 C_ S
100	24	4sqlem1 14477	. 2 |- S C_ NN0
101	99, 100	eqssi 3515	1 |- NN0 = S

Syntax hints:   -> wi 4   \/ wo 368   /\ wa 369   = wceq 1395   e. wcel 1819   {cab 2442   =/= wne 2652   A.wral 2807   E.wrex 2808   {crab 2811   \ cdif 3468   u. cun 3469   C_ wss 3471   {csn 4032   `'ccnv 5007   ` cfv 5594  (class class class)co 6296   supcsup 7918   CCcc 9507   RRcr 9508   0cc0 9509   1c1 9510   + caddc 9512   x. cmul 9514   < clt 9645   - cmin 9824   / cdiv 10227   NNcn 10556   2c2 10606   NN0cn0 10816   ZZcz 10885   ZZ>=cuz 11106   ...cfz 11697   ^cexp 12168   abscabs 13078   Primecprime 14228   ZZ[_i]cgz 14458

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1619  ax-4 1632  ax-5 1705  ax-6 1748  ax-7 1791  ax-8 1821  ax-9 1823  ax-10 1838  ax-11 1843  ax-12 1855  ax-13 2000  ax-ext 2435  ax-rep 4568  ax-sep 4578  ax-nul 4586  ax-pow 4634  ax-pr 4695  ax-un 6591  ax-cnex 9565  ax-resscn 9566  ax-1cn 9567  ax-icn 9568  ax-addcl 9569  ax-addrcl 9570  ax-mulcl 9571  ax-mulrcl 9572  ax-mulcom 9573  ax-addass 9574  ax-mulass 9575  ax-distr 9576  ax-i2m1 9577  ax-1ne0 9578  ax-1rid 9579  ax-rnegex 9580  ax-rrecex 9581  ax-cnre 9582  ax-pre-lttri 9583  ax-pre-lttrn 9584  ax-pre-ltadd 9585  ax-pre-mulgt0 9586  ax-pre-sup 9587

This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 974  df-3an 975  df-tru 1398  df-ex 1614  df-nf 1618  df-sb 1741  df-eu 2287  df-mo 2288  df-clab 2443  df-cleq 2449  df-clel 2452  df-nfc 2607  df-ne 2654  df-nel 2655  df-ral 2812  df-rex 2813  df-reu 2814  df-rmo 2815  df-rab 2816  df-v 3111  df-sbc 3328  df-csb 3431  df-dif 3474  df-un 3476  df-in 3478  df-ss 3485  df-pss 3487  df-nul 3794  df-if 3945  df-pw 4017  df-sn 4033  df-pr 4035  df-tp 4037  df-op 4039  df-uni 4252  df-int 4289  df-iun 4334  df-br 4457  df-opab 4516  df-mpt 4517  df-tr 4551  df-eprel 4800  df-id 4804  df-po 4809  df-so 4810  df-fr 4847  df-we 4849  df-ord 4890  df-on 4891  df-lim 4892  df-suc 4893  df-xp 5014  df-rel 5015  df-cnv 5016  df-co 5017  df-dm 5018  df-rn 5019  df-res 5020  df-ima 5021  df-iota 5557  df-fun 5596  df-fn 5597  df-f 5598  df-f1 5599  df-fo 5600  df-f1o 5601  df-fv 5602  df-riota 6258  df-ov 6299  df-oprab 6300  df-mpt2 6301  df-om 6700  df-1st 6799  df-2nd 6800  df-recs 7060  df-rdg 7094  df-1o 7148  df-2o 7149  df-oadd 7152  df-er 7329  df-en 7536  df-dom 7537  df-sdom 7538  df-fin 7539  df-sup 7919  df-card 8337  df-cda 8565  df-pnf 9647  df-mnf 9648  df-xr 9649  df-ltxr 9650  df-le 9651  df-sub 9826  df-neg 9827  df-div 10228  df-nn 10557  df-2 10615  df-3 10616  df-4 10617  df-n0 10817  df-z 10886  df-uz 11107  df-rp 11246  df-fz 11698  df-fl 11931  df-mod 11999  df-seq 12110  df-exp 12169  df-hash 12408  df-cj 12943  df-re 12944  df-im 12945  df-sqrt 13079  df-abs 13080  df-dvds 13998  df-gcd 14156  df-prm 14229  df-gz 14459

This theorem is referenced by:  4sq  14493