Theorem bezoutlem1 15256

Description: Lemma for bezout 15260. (Contributed by Mario Carneiro, 15-Mar-2014.)

Hypotheses

Ref	Expression
bezout.1	|- M = { z e. NN | E. x e. ZZ E. y e. ZZ z = ( ( A x. x ) + ( B x. y ) ) }
bezout.3	|- ( ph -> A e. ZZ )
bezout.4	|- ( ph -> B e. ZZ )

Assertion

Ref	Expression
bezoutlem1	|- ( ph -> ( A =/= 0 -> ( abs ` A ) e. M ) )

Distinct variable groups:   x, y, z, A   x, B, y, z   ph, x, y, z

Allowed substitution hints:   M( x, y, z)

Proof of Theorem bezoutlem1

Step	Hyp	Ref	Expression
1		bezout.3	. . . 4 |- ( ph -> A e. ZZ )
2		fveq2 6191	. . . . . . 7 |- ( z = A -> ( abs ` z ) = ( abs ` A ) )
3		oveq1 6657	. . . . . . 7 |- ( z = A -> ( z x. x ) = ( A x. x ) )
4	2, 3	eqeq12d 2637	. . . . . 6 |- ( z = A -> ( ( abs ` z ) = ( z x. x ) <-> ( abs ` A ) = ( A x. x ) ) )
5	4	rexbidv 3052	. . . . 5 |- ( z = A -> ( E. x e. ZZ ( abs ` z ) = ( z x. x ) <-> E. x e. ZZ ( abs ` A ) = ( A x. x ) ) )
6		zre 11381	. . . . . 6 |- ( z e. ZZ -> z e. RR )
7		1z 11407	. . . . . . . . 9 |- 1 e. ZZ
8		ax-1rid 10006	. . . . . . . . . 10 |- ( z e. RR -> ( z x. 1 ) = z )
9	8	eqcomd 2628	. . . . . . . . 9 |- ( z e. RR -> z = ( z x. 1 ) )
10		oveq2 6658	. . . . . . . . . . 11 |- ( x = 1 -> ( z x. x ) = ( z x. 1 ) )
11	10	eqeq2d 2632	. . . . . . . . . 10 |- ( x = 1 -> ( z = ( z x. x ) <-> z = ( z x. 1 ) ) )
12	11	rspcev 3309	. . . . . . . . 9 |- ( ( 1 e. ZZ /\ z = ( z x. 1 ) ) -> E. x e. ZZ z = ( z x. x ) )
13	7, 9, 12	sylancr 695	. . . . . . . 8 |- ( z e. RR -> E. x e. ZZ z = ( z x. x ) )
14		eqeq1 2626	. . . . . . . . 9 |- ( ( abs ` z ) = z -> ( ( abs ` z ) = ( z x. x ) <-> z = ( z x. x ) ) )
15	14	rexbidv 3052	. . . . . . . 8 |- ( ( abs ` z ) = z -> ( E. x e. ZZ ( abs ` z ) = ( z x. x ) <-> E. x e. ZZ z = ( z x. x ) ) )
16	13, 15	syl5ibrcom 237	. . . . . . 7 |- ( z e. RR -> ( ( abs ` z ) = z -> E. x e. ZZ ( abs ` z ) = ( z x. x ) ) )
17		neg1z 11413	. . . . . . . . 9 |- -u 1 e. ZZ
18		recn 10026	. . . . . . . . . . 11 |- ( z e. RR -> z e. CC )
19	18	mulm1d 10482	. . . . . . . . . 10 |- ( z e. RR -> ( -u 1 x. z ) = -u z )
20		neg1cn 11124	. . . . . . . . . . 11 |- -u 1 e. CC
21		mulcom 10022	. . . . . . . . . . 11 |- ( ( -u 1 e. CC /\ z e. CC ) -> ( -u 1 x. z ) = ( z x. -u 1 ) )
22	20, 18, 21	sylancr 695	. . . . . . . . . 10 |- ( z e. RR -> ( -u 1 x. z ) = ( z x. -u 1 ) )
23	19, 22	eqtr3d 2658	. . . . . . . . 9 |- ( z e. RR -> -u z = ( z x. -u 1 ) )
24		oveq2 6658	. . . . . . . . . . 11 |- ( x = -u 1 -> ( z x. x ) = ( z x. -u 1 ) )
25	24	eqeq2d 2632	. . . . . . . . . 10 |- ( x = -u 1 -> ( -u z = ( z x. x ) <-> -u z = ( z x. -u 1 ) ) )
26	25	rspcev 3309	. . . . . . . . 9 |- ( ( -u 1 e. ZZ /\ -u z = ( z x. -u 1 ) ) -> E. x e. ZZ -u z = ( z x. x ) )
27	17, 23, 26	sylancr 695	. . . . . . . 8 |- ( z e. RR -> E. x e. ZZ -u z = ( z x. x ) )
28		eqeq1 2626	. . . . . . . . 9 |- ( ( abs ` z ) = -u z -> ( ( abs ` z ) = ( z x. x ) <-> -u z = ( z x. x ) ) )
29	28	rexbidv 3052	. . . . . . . 8 |- ( ( abs ` z ) = -u z -> ( E. x e. ZZ ( abs ` z ) = ( z x. x ) <-> E. x e. ZZ -u z = ( z x. x ) ) )
30	27, 29	syl5ibrcom 237	. . . . . . 7 |- ( z e. RR -> ( ( abs ` z ) = -u z -> E. x e. ZZ ( abs ` z ) = ( z x. x ) ) )
31		absor 14040	. . . . . . 7 |- ( z e. RR -> ( ( abs ` z ) = z \/ ( abs ` z ) = -u z ) )
32	16, 30, 31	mpjaod 396	. . . . . 6 |- ( z e. RR -> E. x e. ZZ ( abs ` z ) = ( z x. x ) )
33	6, 32	syl 17	. . . . 5 |- ( z e. ZZ -> E. x e. ZZ ( abs ` z ) = ( z x. x ) )
34	5, 33	vtoclga 3272	. . . 4 |- ( A e. ZZ -> E. x e. ZZ ( abs ` A ) = ( A x. x ) )
35	1, 34	syl 17	. . 3 |- ( ph -> E. x e. ZZ ( abs ` A ) = ( A x. x ) )
36		bezout.4	. . . . . . . . . . 11 |- ( ph -> B e. ZZ )
37	36	zcnd 11483	. . . . . . . . . 10 |- ( ph -> B e. CC )
38	37	adantr 481	. . . . . . . . 9 |- ( ( ph /\ x e. ZZ ) -> B e. CC )
39	38	mul01d 10235	. . . . . . . 8 |- ( ( ph /\ x e. ZZ ) -> ( B x. 0 ) = 0 )
40	39	oveq2d 6666	. . . . . . 7 |- ( ( ph /\ x e. ZZ ) -> ( ( A x. x ) + ( B x. 0 ) ) = ( ( A x. x ) + 0 ) )
41	1	zcnd 11483	. . . . . . . . 9 |- ( ph -> A e. CC )
42		zcn 11382	. . . . . . . . 9 |- ( x e. ZZ -> x e. CC )
43		mulcl 10020	. . . . . . . . 9 |- ( ( A e. CC /\ x e. CC ) -> ( A x. x ) e. CC )
44	41, 42, 43	syl2an 494	. . . . . . . 8 |- ( ( ph /\ x e. ZZ ) -> ( A x. x ) e. CC )
45	44	addid1d 10236	. . . . . . 7 |- ( ( ph /\ x e. ZZ ) -> ( ( A x. x ) + 0 ) = ( A x. x ) )
46	40, 45	eqtrd 2656	. . . . . 6 |- ( ( ph /\ x e. ZZ ) -> ( ( A x. x ) + ( B x. 0 ) ) = ( A x. x ) )
47	46	eqeq2d 2632	. . . . 5 |- ( ( ph /\ x e. ZZ ) -> ( ( abs ` A ) = ( ( A x. x ) + ( B x. 0 ) ) <-> ( abs ` A ) = ( A x. x ) ) )
48		0z 11388	. . . . . 6 |- 0 e. ZZ
49		oveq2 6658	. . . . . . . . 9 |- ( y = 0 -> ( B x. y ) = ( B x. 0 ) )
50	49	oveq2d 6666	. . . . . . . 8 |- ( y = 0 -> ( ( A x. x ) + ( B x. y ) ) = ( ( A x. x ) + ( B x. 0 ) ) )
51	50	eqeq2d 2632	. . . . . . 7 |- ( y = 0 -> ( ( abs ` A ) = ( ( A x. x ) + ( B x. y ) ) <-> ( abs ` A ) = ( ( A x. x ) + ( B x. 0 ) ) ) )
52	51	rspcev 3309	. . . . . 6 |- ( ( 0 e. ZZ /\ ( abs ` A ) = ( ( A x. x ) + ( B x. 0 ) ) ) -> E. y e. ZZ ( abs ` A ) = ( ( A x. x ) + ( B x. y ) ) )
53	48, 52	mpan 706	. . . . 5 |- ( ( abs ` A ) = ( ( A x. x ) + ( B x. 0 ) ) -> E. y e. ZZ ( abs ` A ) = ( ( A x. x ) + ( B x. y ) ) )
54	47, 53	syl6bir 244	. . . 4 |- ( ( ph /\ x e. ZZ ) -> ( ( abs ` A ) = ( A x. x ) -> E. y e. ZZ ( abs ` A ) = ( ( A x. x ) + ( B x. y ) ) ) )
55	54	reximdva 3017	. . 3 |- ( ph -> ( E. x e. ZZ ( abs ` A ) = ( A x. x ) -> E. x e. ZZ E. y e. ZZ ( abs ` A ) = ( ( A x. x ) + ( B x. y ) ) ) )
56	35, 55	mpd 15	. 2 |- ( ph -> E. x e. ZZ E. y e. ZZ ( abs ` A ) = ( ( A x. x ) + ( B x. y ) ) )
57		nnabscl 14065	. . . 4 |- ( ( A e. ZZ /\ A =/= 0 ) -> ( abs ` A ) e. NN )
58	57	ex 450	. . 3 |- ( A e. ZZ -> ( A =/= 0 -> ( abs ` A ) e. NN ) )
59	1, 58	syl 17	. 2 |- ( ph -> ( A =/= 0 -> ( abs ` A ) e. NN ) )
60		eqeq1 2626	. . . . 5 |- ( z = ( abs ` A ) -> ( z = ( ( A x. x ) + ( B x. y ) ) <-> ( abs ` A ) = ( ( A x. x ) + ( B x. y ) ) ) )
61	60	2rexbidv 3057	. . . 4 |- ( z = ( abs ` A ) -> ( E. x e. ZZ E. y e. ZZ z = ( ( A x. x ) + ( B x. y ) ) <-> E. x e. ZZ E. y e. ZZ ( abs ` A ) = ( ( A x. x ) + ( B x. y ) ) ) )
62		bezout.1	. . . 4 |- M = { z e. NN | E. x e. ZZ E. y e. ZZ z = ( ( A x. x ) + ( B x. y ) ) }
63	61, 62	elrab2 3366	. . 3 |- ( ( abs ` A ) e. M <-> ( ( abs ` A ) e. NN /\ E. x e. ZZ E. y e. ZZ ( abs ` A ) = ( ( A x. x ) + ( B x. y ) ) ) )
64	63	simplbi2com 657	. 2 |- ( E. x e. ZZ E. y e. ZZ ( abs ` A ) = ( ( A x. x ) + ( B x. y ) ) -> ( ( abs ` A ) e. NN -> ( abs ` A ) e. M ) )
65	56, 59, 64	sylsyld 61	1 |- ( ph -> ( A =/= 0 -> ( abs ` A ) e. M ) )

Syntax hints:   -> wi 4   /\ wa 384   = wceq 1483   e. wcel 1990   =/= wne 2794   E.wrex 2913   {crab 2916   ` cfv 5888  (class class class)co 6650   CCcc 9934   RRcr 9935   0cc0 9936   1c1 9937   + caddc 9939   x. cmul 9941   -ucneg 10267   NNcn 11020   ZZcz 11377   abscabs 13974

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-8 1992  ax-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246  ax-ext 2602  ax-sep 4781  ax-nul 4789  ax-pow 4843  ax-pr 4906  ax-un 6949  ax-cnex 9992  ax-resscn 9993  ax-1cn 9994  ax-icn 9995  ax-addcl 9996  ax-addrcl 9997  ax-mulcl 9998  ax-mulrcl 9999  ax-mulcom 10000  ax-addass 10001  ax-mulass 10002  ax-distr 10003  ax-i2m1 10004  ax-1ne0 10005  ax-1rid 10006  ax-rnegex 10007  ax-rrecex 10008  ax-cnre 10009  ax-pre-lttri 10010  ax-pre-lttrn 10011  ax-pre-ltadd 10012  ax-pre-mulgt0 10013  ax-pre-sup 10014

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3or 1038  df-3an 1039  df-tru 1486  df-ex 1705  df-nf 1710  df-sb 1881  df-eu 2474  df-mo 2475  df-clab 2609  df-cleq 2615  df-clel 2618  df-nfc 2753  df-ne 2795  df-nel 2898  df-ral 2917  df-rex 2918  df-reu 2919  df-rmo 2920  df-rab 2921  df-v 3202  df-sbc 3436  df-csb 3534  df-dif 3577  df-un 3579  df-in 3581  df-ss 3588  df-pss 3590  df-nul 3916  df-if 4087  df-pw 4160  df-sn 4178  df-pr 4180  df-tp 4182  df-op 4184  df-uni 4437  df-iun 4522  df-br 4654  df-opab 4713  df-mpt 4730  df-tr 4753  df-id 5024  df-eprel 5029  df-po 5035  df-so 5036  df-fr 5073  df-we 5075  df-xp 5120  df-rel 5121  df-cnv 5122  df-co 5123  df-dm 5124  df-rn 5125  df-res 5126  df-ima 5127  df-pred 5680  df-ord 5726  df-on 5727  df-lim 5728  df-suc 5729  df-iota 5851  df-fun 5890  df-fn 5891  df-f 5892  df-f1 5893  df-fo 5894  df-f1o 5895  df-fv 5896  df-riota 6611  df-ov 6653  df-oprab 6654  df-mpt2 6655  df-om 7066  df-2nd 7169  df-wrecs 7407  df-recs 7468  df-rdg 7506  df-er 7742  df-en 7956  df-dom 7957  df-sdom 7958  df-sup 8348  df-pnf 10076  df-mnf 10077  df-xr 10078  df-ltxr 10079  df-le 10080  df-sub 10268  df-neg 10269  df-div 10685  df-nn 11021  df-2 11079  df-3 11080  df-n0 11293  df-z 11378  df-uz 11688  df-rp 11833  df-seq 12802  df-exp 12861  df-cj 13839  df-re 13840  df-im 13841  df-sqrt 13975  df-abs 13976

This theorem is referenced by:  bezoutlem2  15257  bezoutlem4  15259