Theorem 2sqlem7 16120

Description: Lemma for 2sq . (Contributed by Mario Carneiro, 19-Jun-2015.)

Hypotheses

Ref	Expression
2sq.1	|- S = ran ( w e. ZZ[_i] |-> ( ( abs ` w ) ^ 2 ) )
2sqlem7.2	|- Y = { z | E. x e. ZZ E. y e. ZZ ( ( x gcd y ) = 1 /\ z = ( ( x ^ 2 ) + ( y ^ 2 ) ) ) }

Assertion

Ref	Expression
2sqlem7	|- Y C_ ( S i^i NN )

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

Allowed substitution hints:   S( w)   Y( z, w)

Proof of Theorem 2sqlem7

Step	Hyp	Ref	Expression
1		2sqlem7.2	. 2 |- Y = { z | E. x e. ZZ E. y e. ZZ ( ( x gcd y ) = 1 /\ z = ( ( x ^ 2 ) + ( y ^ 2 ) ) ) }
2		simpr 110	. . . . . . 7 |- ( ( ( x gcd y ) = 1 /\ z = ( ( x ^ 2 ) + ( y ^ 2 ) ) ) -> z = ( ( x ^ 2 ) + ( y ^ 2 ) ) )
3	2	reximi 2641	. . . . . 6 |- ( E. y e. ZZ ( ( x gcd y ) = 1 /\ z = ( ( x ^ 2 ) + ( y ^ 2 ) ) ) -> E. y e. ZZ z = ( ( x ^ 2 ) + ( y ^ 2 ) ) )
4	3	reximi 2641	. . . . 5 |- ( E. x e. ZZ E. y e. ZZ ( ( x gcd y ) = 1 /\ z = ( ( x ^ 2 ) + ( y ^ 2 ) ) ) -> E. x e. ZZ E. y e. ZZ z = ( ( x ^ 2 ) + ( y ^ 2 ) ) )
5		2sq.1	. . . . . 6 |- S = ran ( w e. ZZ[_i] |-> ( ( abs ` w ) ^ 2 ) )
6	5	2sqlem2 16114	. . . . 5 |- ( z e. S <-> E. x e. ZZ E. y e. ZZ z = ( ( x ^ 2 ) + ( y ^ 2 ) ) )
7	4, 6	sylibr 134	. . . 4 |- ( E. x e. ZZ E. y e. ZZ ( ( x gcd y ) = 1 /\ z = ( ( x ^ 2 ) + ( y ^ 2 ) ) ) -> z e. S )
8		1ne0 9322	. . . . . . . . . 10 |- 1 =/= 0
9		gcdeq0 12698	. . . . . . . . . . . . 13 |- ( ( x e. ZZ /\ y e. ZZ ) -> ( ( x gcd y ) = 0 <-> ( x = 0 /\ y = 0 ) ) )
10	9	adantr 276	. . . . . . . . . . . 12 |- ( ( ( x e. ZZ /\ y e. ZZ ) /\ ( x gcd y ) = 1 ) -> ( ( x gcd y ) = 0 <-> ( x = 0 /\ y = 0 ) ) )
11		simpr 110	. . . . . . . . . . . . 13 |- ( ( ( x e. ZZ /\ y e. ZZ ) /\ ( x gcd y ) = 1 ) -> ( x gcd y ) = 1 )
12	11	eqeq1d 2243	. . . . . . . . . . . 12 |- ( ( ( x e. ZZ /\ y e. ZZ ) /\ ( x gcd y ) = 1 ) -> ( ( x gcd y ) = 0 <-> 1 = 0 ) )
13	10, 12	bitr3d 190	. . . . . . . . . . 11 |- ( ( ( x e. ZZ /\ y e. ZZ ) /\ ( x gcd y ) = 1 ) -> ( ( x = 0 /\ y = 0 ) <-> 1 = 0 ) )
14	13	necon3bbid 2454	. . . . . . . . . 10 |- ( ( ( x e. ZZ /\ y e. ZZ ) /\ ( x gcd y ) = 1 ) -> ( -. ( x = 0 /\ y = 0 ) <-> 1 =/= 0 ) )
15	8, 14	mpbiri 168	. . . . . . . . 9 |- ( ( ( x e. ZZ /\ y e. ZZ ) /\ ( x gcd y ) = 1 ) -> -. ( x = 0 /\ y = 0 ) )
16		zsqcl2 11003	. . . . . . . . . . . . 13 |- ( x e. ZZ -> ( x ^ 2 ) e. NN0 )
17	16	ad2antrr 488	. . . . . . . . . . . 12 |- ( ( ( x e. ZZ /\ y e. ZZ ) /\ ( x gcd y ) = 1 ) -> ( x ^ 2 ) e. NN0 )
18	17	nn0red 9571	. . . . . . . . . . 11 |- ( ( ( x e. ZZ /\ y e. ZZ ) /\ ( x gcd y ) = 1 ) -> ( x ^ 2 ) e. RR )
19	17	nn0ge0d 9573	. . . . . . . . . . 11 |- ( ( ( x e. ZZ /\ y e. ZZ ) /\ ( x gcd y ) = 1 ) -> 0 <_ ( x ^ 2 ) )
20		zsqcl2 11003	. . . . . . . . . . . . 13 |- ( y e. ZZ -> ( y ^ 2 ) e. NN0 )
21	20	ad2antlr 489	. . . . . . . . . . . 12 |- ( ( ( x e. ZZ /\ y e. ZZ ) /\ ( x gcd y ) = 1 ) -> ( y ^ 2 ) e. NN0 )
22	21	nn0red 9571	. . . . . . . . . . 11 |- ( ( ( x e. ZZ /\ y e. ZZ ) /\ ( x gcd y ) = 1 ) -> ( y ^ 2 ) e. RR )
23	21	nn0ge0d 9573	. . . . . . . . . . 11 |- ( ( ( x e. ZZ /\ y e. ZZ ) /\ ( x gcd y ) = 1 ) -> 0 <_ ( y ^ 2 ) )
24		add20 8765	. . . . . . . . . . 11 |- ( ( ( ( x ^ 2 ) e. RR /\ 0 <_ ( x ^ 2 ) ) /\ ( ( y ^ 2 ) e. RR /\ 0 <_ ( y ^ 2 ) ) ) -> ( ( ( x ^ 2 ) + ( y ^ 2 ) ) = 0 <-> ( ( x ^ 2 ) = 0 /\ ( y ^ 2 ) = 0 ) ) )
25	18, 19, 22, 23, 24	syl22anc 1275	. . . . . . . . . 10 |- ( ( ( x e. ZZ /\ y e. ZZ ) /\ ( x gcd y ) = 1 ) -> ( ( ( x ^ 2 ) + ( y ^ 2 ) ) = 0 <-> ( ( x ^ 2 ) = 0 /\ ( y ^ 2 ) = 0 ) ) )
26		zcn 9599	. . . . . . . . . . . 12 |- ( x e. ZZ -> x e. CC )
27	26	ad2antrr 488	. . . . . . . . . . 11 |- ( ( ( x e. ZZ /\ y e. ZZ ) /\ ( x gcd y ) = 1 ) -> x e. CC )
28		zcn 9599	. . . . . . . . . . . 12 |- ( y e. ZZ -> y e. CC )
29	28	ad2antlr 489	. . . . . . . . . . 11 |- ( ( ( x e. ZZ /\ y e. ZZ ) /\ ( x gcd y ) = 1 ) -> y e. CC )
30		sqeq0 10988	. . . . . . . . . . . 12 |- ( x e. CC -> ( ( x ^ 2 ) = 0 <-> x = 0 ) )
31		sqeq0 10988	. . . . . . . . . . . 12 |- ( y e. CC -> ( ( y ^ 2 ) = 0 <-> y = 0 ) )
32	30, 31	bi2anan9 610	. . . . . . . . . . 11 |- ( ( x e. CC /\ y e. CC ) -> ( ( ( x ^ 2 ) = 0 /\ ( y ^ 2 ) = 0 ) <-> ( x = 0 /\ y = 0 ) ) )
33	27, 29, 32	syl2anc 411	. . . . . . . . . 10 |- ( ( ( x e. ZZ /\ y e. ZZ ) /\ ( x gcd y ) = 1 ) -> ( ( ( x ^ 2 ) = 0 /\ ( y ^ 2 ) = 0 ) <-> ( x = 0 /\ y = 0 ) ) )
34	25, 33	bitrd 188	. . . . . . . . 9 |- ( ( ( x e. ZZ /\ y e. ZZ ) /\ ( x gcd y ) = 1 ) -> ( ( ( x ^ 2 ) + ( y ^ 2 ) ) = 0 <-> ( x = 0 /\ y = 0 ) ) )
35	15, 34	mtbird 680	. . . . . . . 8 |- ( ( ( x e. ZZ /\ y e. ZZ ) /\ ( x gcd y ) = 1 ) -> -. ( ( x ^ 2 ) + ( y ^ 2 ) ) = 0 )
36		nn0addcl 9548	. . . . . . . . . . 11 |- ( ( ( x ^ 2 ) e. NN0 /\ ( y ^ 2 ) e. NN0 ) -> ( ( x ^ 2 ) + ( y ^ 2 ) ) e. NN0 )
37	16, 20, 36	syl2an 289	. . . . . . . . . 10 |- ( ( x e. ZZ /\ y e. ZZ ) -> ( ( x ^ 2 ) + ( y ^ 2 ) ) e. NN0 )
38	37	adantr 276	. . . . . . . . 9 |- ( ( ( x e. ZZ /\ y e. ZZ ) /\ ( x gcd y ) = 1 ) -> ( ( x ^ 2 ) + ( y ^ 2 ) ) e. NN0 )
39		elnn0 9515	. . . . . . . . 9 |- ( ( ( x ^ 2 ) + ( y ^ 2 ) ) e. NN0 <-> ( ( ( x ^ 2 ) + ( y ^ 2 ) ) e. NN \/ ( ( x ^ 2 ) + ( y ^ 2 ) ) = 0 ) )
40	38, 39	sylib 122	. . . . . . . 8 |- ( ( ( x e. ZZ /\ y e. ZZ ) /\ ( x gcd y ) = 1 ) -> ( ( ( x ^ 2 ) + ( y ^ 2 ) ) e. NN \/ ( ( x ^ 2 ) + ( y ^ 2 ) ) = 0 ) )
41	35, 40	ecased 1386	. . . . . . 7 |- ( ( ( x e. ZZ /\ y e. ZZ ) /\ ( x gcd y ) = 1 ) -> ( ( x ^ 2 ) + ( y ^ 2 ) ) e. NN )
42		eleq1 2297	. . . . . . 7 |- ( z = ( ( x ^ 2 ) + ( y ^ 2 ) ) -> ( z e. NN <-> ( ( x ^ 2 ) + ( y ^ 2 ) ) e. NN ) )
43	41, 42	syl5ibrcom 157	. . . . . 6 |- ( ( ( x e. ZZ /\ y e. ZZ ) /\ ( x gcd y ) = 1 ) -> ( z = ( ( x ^ 2 ) + ( y ^ 2 ) ) -> z e. NN ) )
44	43	expimpd 363	. . . . 5 |- ( ( x e. ZZ /\ y e. ZZ ) -> ( ( ( x gcd y ) = 1 /\ z = ( ( x ^ 2 ) + ( y ^ 2 ) ) ) -> z e. NN ) )
45	44	rexlimivv 2668	. . . 4 |- ( E. x e. ZZ E. y e. ZZ ( ( x gcd y ) = 1 /\ z = ( ( x ^ 2 ) + ( y ^ 2 ) ) ) -> z e. NN )
46	7, 45	elind 3408	. . 3 |- ( E. x e. ZZ E. y e. ZZ ( ( x gcd y ) = 1 /\ z = ( ( x ^ 2 ) + ( y ^ 2 ) ) ) -> z e. ( S i^i NN ) )
47	46	abssi 3317	. 2 |- { z | E. x e. ZZ E. y e. ZZ ( ( x gcd y ) = 1 /\ z = ( ( x ^ 2 ) + ( y ^ 2 ) ) ) } C_ ( S i^i NN )
48	1, 47	eqsstri 3274	1 |- Y C_ ( S i^i NN )

Syntax hints:   -. wn 3   /\ wa 104   <-> wb 105   \/ wo 716   = wceq 1398   e. wcel 2205   {cab 2220   =/= wne 2414   E.wrex 2523   i^i cin 3213   C_ wss 3214   class class class wbr 4114   |-> cmpt 4176   ran crn 4755   ` cfv 5357  (class class class)co 6058   CCcc 8141   RRcr 8142   0cc0 8143   1c1 8144   + caddc 8146   <_ cle 8325   NNcn 9254   2c2 9305   NN0cn0 9513   ZZcz 9594   ^cexp 10924   abscabs 11707   gcd cgcd 12674   ZZ[_i]cgz 13092

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 2207  ax-14 2208  ax-ext 2216  ax-coll 4230  ax-sep 4233  ax-nul 4241  ax-pow 4292  ax-pr 4327  ax-un 4559  ax-setind 4664  ax-iinf 4715  ax-cnex 8234  ax-resscn 8235  ax-1cn 8236  ax-1re 8237  ax-icn 8238  ax-addcl 8239  ax-addrcl 8240  ax-mulcl 8241  ax-mulrcl 8242  ax-addcom 8243  ax-mulcom 8244  ax-addass 8245  ax-mulass 8246  ax-distr 8247  ax-i2m1 8248  ax-0lt1 8249  ax-1rid 8250  ax-0id 8251  ax-rnegex 8252  ax-precex 8253  ax-cnre 8254  ax-pre-ltirr 8255  ax-pre-ltwlin 8256  ax-pre-lttrn 8257  ax-pre-apti 8258  ax-pre-ltadd 8259  ax-pre-mulgt0 8260  ax-pre-mulext 8261  ax-arch 8262  ax-caucvg 8263

This theorem depends on definitions:  df-bi 117  df-stab 839  df-dc 843  df-3or 1006  df-3an 1007  df-tru 1401  df-fal 1404  df-nf 1510  df-sb 1812  df-eu 2085  df-mo 2086  df-clab 2221  df-cleq 2227  df-clel 2230  df-nfc 2375  df-ne 2415  df-nel 2510  df-ral 2527  df-rex 2528  df-reu 2529  df-rmo 2530  df-rab 2531  df-v 2817  df-sbc 3046  df-csb 3142  df-dif 3216  df-un 3218  df-in 3220  df-ss 3227  df-nul 3513  df-if 3625  df-pw 3676  df-sn 3700  df-pr 3701  df-op 3703  df-uni 3920  df-int 3955  df-iun 3998  df-br 4115  df-opab 4177  df-mpt 4178  df-tr 4214  df-id 4419  df-po 4422  df-iso 4423  df-iord 4492  df-on 4494  df-ilim 4495  df-suc 4497  df-iom 4718  df-xp 4760  df-rel 4761  df-cnv 4762  df-co 4763  df-dm 4764  df-rn 4765  df-res 4766  df-ima 4767  df-iota 5317  df-fun 5359  df-fn 5360  df-f 5361  df-f1 5362  df-fo 5363  df-f1o 5364  df-fv 5365  df-riota 6011  df-ov 6061  df-oprab 6062  df-mpo 6063  df-1st 6347  df-2nd 6348  df-recs 6549  df-frec 6635  df-sup 7288  df-pnf 8326  df-mnf 8327  df-xr 8328  df-ltxr 8329  df-le 8330  df-sub 8462  df-neg 8463  df-reap 8866  df-ap 8873  df-div 8964  df-inn 9255  df-2 9313  df-3 9314  df-4 9315  df-n0 9514  df-z 9595  df-uz 9872  df-q 9970  df-rp 10005  df-fz 10362  df-fzo 10499  df-fl 10654  df-mod 10709  df-seqfrec 10834  df-exp 10925  df-cj 11552  df-re 11553  df-im 11554  df-rsqrt 11708  df-abs 11709  df-dvds 12499  df-gcd 12675  df-gz 13093

This theorem is referenced by:  2sqlem8  16122  2sqlem9  16123