Theorem bezoutlemzz 12757

Description: Lemma for Bézout's identity. Like bezoutlemex 12756 but where ' z ' is any integer, not just a nonnegative one. (Contributed by Mario Carneiro and Jim Kingdon, 8-Jan-2022.)

Assertion

Ref	Expression
bezoutlemzz	|- ( ( A e. NN0 /\ B e. NN0 ) -> E. d e. NN0 ( A. z e. ZZ ( z || d -> ( z || A /\ z || B ) ) /\ E. x e. ZZ E. y e. ZZ d = ( ( A x. x ) + ( B x. y ) ) ) )

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

Proof of Theorem bezoutlemzz

Dummy variable w is distinct from all other variables.

Step	Hyp	Ref	Expression
1		bezoutlemex 12756	. 2 |- ( ( A e. NN0 /\ B e. NN0 ) -> E. d e. NN0 ( A. z e. NN0 ( z || d -> ( z || A /\ z || B ) ) /\ E. x e. ZZ E. y e. ZZ d = ( ( A x. x ) + ( B x. y ) ) ) )
2		nfv 1581	. . . . . . 7 |- F/ z ( ( A e. NN0 /\ B e. NN0 ) /\ d e. NN0 )
3		nfra1 2581	. . . . . . 7 |- F/ z A. z e. NN0 ( z || d -> ( z || A /\ z || B ) )
4	2, 3	nfan 1618	. . . . . 6 |- F/ z ( ( ( A e. NN0 /\ B e. NN0 ) /\ d e. NN0 ) /\ A. z e. NN0 ( z || d -> ( z || A /\ z || B ) ) )
5		simpr 110	. . . . . . . . . 10 |- ( ( ( A. z e. NN0 ( z || d -> ( z || A /\ z || B ) ) /\ z e. ZZ ) /\ z e. NN0 ) -> z e. NN0 )
6		rsp 2597	. . . . . . . . . . 11 |- ( A. z e. NN0 ( z || d -> ( z || A /\ z || B ) ) -> ( z e. NN0 -> ( z || d -> ( z || A /\ z || B ) ) ) )
7	6	ad2antrr 492	. . . . . . . . . 10 |- ( ( ( A. z e. NN0 ( z || d -> ( z || A /\ z || B ) ) /\ z e. ZZ ) /\ z e. NN0 ) -> ( z e. NN0 -> ( z || d -> ( z || A /\ z || B ) ) ) )
8	5, 7	mpd 13	. . . . . . . . 9 |- ( ( ( A. z e. NN0 ( z || d -> ( z || A /\ z || B ) ) /\ z e. ZZ ) /\ z e. NN0 ) -> ( z || d -> ( z || A /\ z || B ) ) )
9	8	adantlll 484	. . . . . . . 8 |- ( ( ( ( ( ( A e. NN0 /\ B e. NN0 ) /\ d e. NN0 ) /\ A. z e. NN0 ( z || d -> ( z || A /\ z || B ) ) ) /\ z e. ZZ ) /\ z e. NN0 ) -> ( z || d -> ( z || A /\ z || B ) ) )
10		breq1 4128	. . . . . . . . . . . 12 |- ( w = -u z -> ( w || d <-> -u z || d ) )
11		breq1 4128	. . . . . . . . . . . . 13 |- ( w = -u z -> ( w || A <-> -u z || A ) )
12		breq1 4128	. . . . . . . . . . . . 13 |- ( w = -u z -> ( w || B <-> -u z || B ) )
13	11, 12	anbi12d 477	. . . . . . . . . . . 12 |- ( w = -u z -> ( ( w || A /\ w || B ) <-> ( -u z || A /\ -u z || B ) ) )
14	10, 13	imbi12d 234	. . . . . . . . . . 11 |- ( w = -u z -> ( ( w || d -> ( w || A /\ w || B ) ) <-> ( -u z || d -> ( -u z || A /\ -u z || B ) ) ) )
15		breq1 4128	. . . . . . . . . . . . . . 15 |- ( z = w -> ( z || d <-> w || d ) )
16		breq1 4128	. . . . . . . . . . . . . . . 16 |- ( z = w -> ( z || A <-> w || A ) )
17		breq1 4128	. . . . . . . . . . . . . . . 16 |- ( z = w -> ( z || B <-> w || B ) )
18	16, 17	anbi12d 477	. . . . . . . . . . . . . . 15 |- ( z = w -> ( ( z || A /\ z || B ) <-> ( w || A /\ w || B ) ) )
19	15, 18	imbi12d 234	. . . . . . . . . . . . . 14 |- ( z = w -> ( ( z || d -> ( z || A /\ z || B ) ) <-> ( w || d -> ( w || A /\ w || B ) ) ) )
20	19	cbvralv 2786	. . . . . . . . . . . . 13 |- ( A. z e. NN0 ( z || d -> ( z || A /\ z || B ) ) <-> A. w e. NN0 ( w || d -> ( w || A /\ w || B ) ) )
21	20	biimpi 120	. . . . . . . . . . . 12 |- ( A. z e. NN0 ( z || d -> ( z || A /\ z || B ) ) -> A. w e. NN0 ( w || d -> ( w || A /\ w || B ) ) )
22	21	ad2antrr 492	. . . . . . . . . . 11 |- ( ( ( A. z e. NN0 ( z || d -> ( z || A /\ z || B ) ) /\ z e. ZZ ) /\ -u z e. NN0 ) -> A. w e. NN0 ( w || d -> ( w || A /\ w || B ) ) )
23		simpr 110	. . . . . . . . . . 11 |- ( ( ( A. z e. NN0 ( z || d -> ( z || A /\ z || B ) ) /\ z e. ZZ ) /\ -u z e. NN0 ) -> -u z e. NN0 )
24	14, 22, 23	rspcdva 2934	. . . . . . . . . 10 |- ( ( ( A. z e. NN0 ( z || d -> ( z || A /\ z || B ) ) /\ z e. ZZ ) /\ -u z e. NN0 ) -> ( -u z || d -> ( -u z || A /\ -u z || B ) ) )
25	24	adantlll 484	. . . . . . . . 9 |- ( ( ( ( ( ( A e. NN0 /\ B e. NN0 ) /\ d e. NN0 ) /\ A. z e. NN0 ( z || d -> ( z || A /\ z || B ) ) ) /\ z e. ZZ ) /\ -u z e. NN0 ) -> ( -u z || d -> ( -u z || A /\ -u z || B ) ) )
26		simplr 533	. . . . . . . . . 10 |- ( ( ( ( ( ( A e. NN0 /\ B e. NN0 ) /\ d e. NN0 ) /\ A. z e. NN0 ( z || d -> ( z || A /\ z || B ) ) ) /\ z e. ZZ ) /\ -u z e. NN0 ) -> z e. ZZ )
27		simpllr 540	. . . . . . . . . . . 12 |- ( ( ( ( ( A e. NN0 /\ B e. NN0 ) /\ d e. NN0 ) /\ A. z e. NN0 ( z || d -> ( z || A /\ z || B ) ) ) /\ z e. ZZ ) -> d e. NN0 )
28	27	adantr 276	. . . . . . . . . . 11 |- ( ( ( ( ( ( A e. NN0 /\ B e. NN0 ) /\ d e. NN0 ) /\ A. z e. NN0 ( z || d -> ( z || A /\ z || B ) ) ) /\ z e. ZZ ) /\ -u z e. NN0 ) -> d e. NN0 )
29	28	nn0zd 9745	. . . . . . . . . 10 |- ( ( ( ( ( ( A e. NN0 /\ B e. NN0 ) /\ d e. NN0 ) /\ A. z e. NN0 ( z || d -> ( z || A /\ z || B ) ) ) /\ z e. ZZ ) /\ -u z e. NN0 ) -> d e. ZZ )
30		negdvdsb 12552	. . . . . . . . . 10 |- ( ( z e. ZZ /\ d e. ZZ ) -> ( z || d <-> -u z || d ) )
31	26, 29, 30	syl2anc 415	. . . . . . . . 9 |- ( ( ( ( ( ( A e. NN0 /\ B e. NN0 ) /\ d e. NN0 ) /\ A. z e. NN0 ( z || d -> ( z || A /\ z || B ) ) ) /\ z e. ZZ ) /\ -u z e. NN0 ) -> ( z || d <-> -u z || d ) )
32		simplll 539	. . . . . . . . . . . . 13 |- ( ( ( ( A e. NN0 /\ B e. NN0 ) /\ d e. NN0 ) /\ A. z e. NN0 ( z || d -> ( z || A /\ z || B ) ) ) -> A e. NN0 )
33	32	ad2antrr 492	. . . . . . . . . . . 12 |- ( ( ( ( ( ( A e. NN0 /\ B e. NN0 ) /\ d e. NN0 ) /\ A. z e. NN0 ( z || d -> ( z || A /\ z || B ) ) ) /\ z e. ZZ ) /\ -u z e. NN0 ) -> A e. NN0 )
34	33	nn0zd 9745	. . . . . . . . . . 11 |- ( ( ( ( ( ( A e. NN0 /\ B e. NN0 ) /\ d e. NN0 ) /\ A. z e. NN0 ( z || d -> ( z || A /\ z || B ) ) ) /\ z e. ZZ ) /\ -u z e. NN0 ) -> A e. ZZ )
35		negdvdsb 12552	. . . . . . . . . . 11 |- ( ( z e. ZZ /\ A e. ZZ ) -> ( z || A <-> -u z || A ) )
36	26, 34, 35	syl2anc 415	. . . . . . . . . 10 |- ( ( ( ( ( ( A e. NN0 /\ B e. NN0 ) /\ d e. NN0 ) /\ A. z e. NN0 ( z || d -> ( z || A /\ z || B ) ) ) /\ z e. ZZ ) /\ -u z e. NN0 ) -> ( z || A <-> -u z || A ) )
37		simpllr 540	. . . . . . . . . . . . 13 |- ( ( ( ( A e. NN0 /\ B e. NN0 ) /\ d e. NN0 ) /\ A. z e. NN0 ( z || d -> ( z || A /\ z || B ) ) ) -> B e. NN0 )
38	37	ad2antrr 492	. . . . . . . . . . . 12 |- ( ( ( ( ( ( A e. NN0 /\ B e. NN0 ) /\ d e. NN0 ) /\ A. z e. NN0 ( z || d -> ( z || A /\ z || B ) ) ) /\ z e. ZZ ) /\ -u z e. NN0 ) -> B e. NN0 )
39	38	nn0zd 9745	. . . . . . . . . . 11 |- ( ( ( ( ( ( A e. NN0 /\ B e. NN0 ) /\ d e. NN0 ) /\ A. z e. NN0 ( z || d -> ( z || A /\ z || B ) ) ) /\ z e. ZZ ) /\ -u z e. NN0 ) -> B e. ZZ )
40		negdvdsb 12552	. . . . . . . . . . 11 |- ( ( z e. ZZ /\ B e. ZZ ) -> ( z || B <-> -u z || B ) )
41	26, 39, 40	syl2anc 415	. . . . . . . . . 10 |- ( ( ( ( ( ( A e. NN0 /\ B e. NN0 ) /\ d e. NN0 ) /\ A. z e. NN0 ( z || d -> ( z || A /\ z || B ) ) ) /\ z e. ZZ ) /\ -u z e. NN0 ) -> ( z || B <-> -u z || B ) )
42	36, 41	anbi12d 477	. . . . . . . . 9 |- ( ( ( ( ( ( A e. NN0 /\ B e. NN0 ) /\ d e. NN0 ) /\ A. z e. NN0 ( z || d -> ( z || A /\ z || B ) ) ) /\ z e. ZZ ) /\ -u z e. NN0 ) -> ( ( z || A /\ z || B ) <-> ( -u z || A /\ -u z || B ) ) )
43	25, 31, 42	3imtr4d 203	. . . . . . . 8 |- ( ( ( ( ( ( A e. NN0 /\ B e. NN0 ) /\ d e. NN0 ) /\ A. z e. NN0 ( z || d -> ( z || A /\ z || B ) ) ) /\ z e. ZZ ) /\ -u z e. NN0 ) -> ( z || d -> ( z || A /\ z || B ) ) )
44		elznn0 9638	. . . . . . . . . 10 |- ( z e. ZZ <-> ( z e. RR /\ ( z e. NN0 \/ -u z e. NN0 ) ) )
45	44	simprbi 275	. . . . . . . . 9 |- ( z e. ZZ -> ( z e. NN0 \/ -u z e. NN0 ) )
46	45	adantl 277	. . . . . . . 8 |- ( ( ( ( ( A e. NN0 /\ B e. NN0 ) /\ d e. NN0 ) /\ A. z e. NN0 ( z || d -> ( z || A /\ z || B ) ) ) /\ z e. ZZ ) -> ( z e. NN0 \/ -u z e. NN0 ) )
47	9, 43, 46	mpjaodan 810	. . . . . . 7 |- ( ( ( ( ( A e. NN0 /\ B e. NN0 ) /\ d e. NN0 ) /\ A. z e. NN0 ( z || d -> ( z || A /\ z || B ) ) ) /\ z e. ZZ ) -> ( z || d -> ( z || A /\ z || B ) ) )
48	47	ex 115	. . . . . 6 |- ( ( ( ( A e. NN0 /\ B e. NN0 ) /\ d e. NN0 ) /\ A. z e. NN0 ( z || d -> ( z || A /\ z || B ) ) ) -> ( z e. ZZ -> ( z || d -> ( z || A /\ z || B ) ) ) )
49	4, 48	ralrimi 2621	. . . . 5 |- ( ( ( ( A e. NN0 /\ B e. NN0 ) /\ d e. NN0 ) /\ A. z e. NN0 ( z || d -> ( z || A /\ z || B ) ) ) -> A. z e. ZZ ( z || d -> ( z || A /\ z || B ) ) )
50	49	ex 115	. . . 4 |- ( ( ( A e. NN0 /\ B e. NN0 ) /\ d e. NN0 ) -> ( A. z e. NN0 ( z || d -> ( z || A /\ z || B ) ) -> A. z e. ZZ ( z || d -> ( z || A /\ z || B ) ) ) )
51	50	anim1d 336	. . 3 |- ( ( ( A e. NN0 /\ B e. NN0 ) /\ d e. NN0 ) -> ( ( A. z e. NN0 ( z || d -> ( z || A /\ z || B ) ) /\ E. x e. ZZ E. y e. ZZ d = ( ( A x. x ) + ( B x. y ) ) ) -> ( A. z e. ZZ ( z || d -> ( z || A /\ z || B ) ) /\ E. x e. ZZ E. y e. ZZ d = ( ( A x. x ) + ( B x. y ) ) ) ) )
52	51	reximdva 2652	. 2 |- ( ( A e. NN0 /\ B e. NN0 ) -> ( E. d e. NN0 ( A. z e. NN0 ( z || d -> ( z || A /\ z || B ) ) /\ E. x e. ZZ E. y e. ZZ d = ( ( A x. x ) + ( B x. y ) ) ) -> E. d e. NN0 ( A. z e. ZZ ( z || d -> ( z || A /\ z || B ) ) /\ E. x e. ZZ E. y e. ZZ d = ( ( A x. x ) + ( B x. y ) ) ) ) )
53	1, 52	mpd 13	1 |- ( ( A e. NN0 /\ B e. NN0 ) -> E. d e. NN0 ( A. z e. ZZ ( z || d -> ( z || A /\ z || B ) ) /\ E. x e. ZZ E. y e. ZZ d = ( ( A x. x ) + ( B x. y ) ) ) )

Syntax hints:   -> wi 4   /\ wa 104   <-> wb 105   \/ wo 720   = wceq 1402   e. wcel 2209   A.wral 2528   E.wrex 2529   class class class wbr 4125  (class class class)co 6075   RRcr 8168   + caddc 8172   x. cmul 8174   -ucneg 8488   NN0cn0 9542   ZZcz 9623   || cdvds 12532

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 623  ax-in2 624  ax-io 721  ax-5 1500  ax-7 1501  ax-gen 1502  ax-ie1 1546  ax-ie2 1547  ax-8 1557  ax-10 1558  ax-11 1559  ax-i12 1560  ax-bndl 1562  ax-4 1563  ax-17 1579  ax-i9 1583  ax-ial 1587  ax-i5r 1588  ax-14 2212  ax-ext 2220  ax-coll 4241  ax-sep 4244  ax-nul 4254  ax-pow 4306  ax-pr 4341  ax-un 4573  ax-setind 4679  ax-iinf 4730  ax-cnex 8260  ax-resscn 8261  ax-1cn 8262  ax-1re 8263  ax-icn 8264  ax-addcl 8265  ax-addrcl 8266  ax-mulcl 8267  ax-mulrcl 8268  ax-addcom 8269  ax-mulcom 8270  ax-addass 8271  ax-mulass 8272  ax-distr 8273  ax-i2m1 8274  ax-0lt1 8275  ax-1rid 8276  ax-0id 8277  ax-rnegex 8278  ax-precex 8279  ax-cnre 8280  ax-pre-ltirr 8281  ax-pre-ltwlin 8282  ax-pre-lttrn 8283  ax-pre-apti 8284  ax-pre-ltadd 8285  ax-pre-mulgt0 8286  ax-pre-mulext 8287  ax-arch 8288

This theorem depends on definitions:  df-bi 117  df-dc 847  df-3or 1010  df-3an 1011  df-tru 1405  df-fal 1408  df-nf 1514  df-sb 1816  df-eu 2089  df-mo 2090  df-clab 2225  df-cleq 2231  df-clel 2234  df-nfc 2381  df-ne 2421  df-nel 2516  df-ral 2533  df-rex 2534  df-reu 2535  df-rmo 2536  df-rab 2537  df-v 2823  df-sbc 3052  df-csb 3148  df-dif 3222  df-un 3224  df-in 3226  df-ss 3233  df-nul 3521  df-if 3636  df-pw 3687  df-sn 3711  df-pr 3712  df-op 3714  df-uni 3931  df-int 3966  df-iun 4009  df-br 4126  df-opab 4188  df-mpt 4189  df-tr 4225  df-id 4433  df-po 4436  df-iso 4437  df-iord 4506  df-on 4508  df-ilim 4509  df-suc 4511  df-iom 4733  df-xp 4775  df-rel 4776  df-cnv 4777  df-co 4778  df-dm 4779  df-rn 4780  df-res 4781  df-ima 4782  df-iota 5332  df-fun 5374  df-fn 5375  df-f 5376  df-f1 5377  df-fo 5378  df-f1o 5379  df-fv 5380  df-riota 6028  df-ov 6078  df-oprab 6079  df-mpo 6080  df-1st 6364  df-2nd 6365  df-recs 6566  df-frec 6652  df-pnf 8352  df-mnf 8353  df-xr 8354  df-ltxr 8355  df-le 8356  df-sub 8489  df-neg 8490  df-reap 8893  df-ap 8900  df-div 8993  df-inn 9284  df-2 9342  df-n0 9543  df-z 9624  df-uz 9901  df-q 9999  df-rp 10034  df-fz 10391  df-fl 10683  df-mod 10738  df-seqfrec 10863  df-exp 10954  df-cj 11585  df-re 11586  df-im 11587  df-rsqrt 11742  df-abs 11743  df-dvds 12533

This theorem is referenced by:  bezoutlemaz  12758