Theorem bezoutlemzz 12653

Description: Lemma for Bézout's identity. Like bezoutlemex 12652 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 12652	. 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 1577	. . . . . . 7 |- F/ z ( ( A e. NN0 /\ B e. NN0 ) /\ d e. NN0 )
3		nfra1 2564	. . . . . . 7 |- F/ z A. z e. NN0 ( z || d -> ( z || A /\ z || B ) )
4	2, 3	nfan 1614	. . . . . 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 2580	. . . . . . . . . . 11 |- ( A. z e. NN0 ( z || d -> ( z || A /\ z || B ) ) -> ( z e. NN0 -> ( z || d -> ( z || A /\ z || B ) ) ) )
7	6	ad2antrr 488	. . . . . . . . . 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 480	. . . . . . . 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 4096	. . . . . . . . . . . 12 |- ( w = -u z -> ( w || d <-> -u z || d ) )
11		breq1 4096	. . . . . . . . . . . . 13 |- ( w = -u z -> ( w || A <-> -u z || A ) )
12		breq1 4096	. . . . . . . . . . . . 13 |- ( w = -u z -> ( w || B <-> -u z || B ) )
13	11, 12	anbi12d 473	. . . . . . . . . . . 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 4096	. . . . . . . . . . . . . . 15 |- ( z = w -> ( z || d <-> w || d ) )
16		breq1 4096	. . . . . . . . . . . . . . . 16 |- ( z = w -> ( z || A <-> w || A ) )
17		breq1 4096	. . . . . . . . . . . . . . . 16 |- ( z = w -> ( z || B <-> w || B ) )
18	16, 17	anbi12d 473	. . . . . . . . . . . . . . 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 2768	. . . . . . . . . . . . 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 488	. . . . . . . . . . 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 2916	. . . . . . . . . 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 480	. . . . . . . . 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 529	. . . . . . . . . 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 536	. . . . . . . . . . . 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 9661	. . . . . . . . . 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 12448	. . . . . . . . . 10 |- ( ( z e. ZZ /\ d e. ZZ ) -> ( z || d <-> -u z || d ) )
31	26, 29, 30	syl2anc 411	. . . . . . . . 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 535	. . . . . . . . . . . . 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 488	. . . . . . . . . . . 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 9661	. . . . . . . . . . 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 12448	. . . . . . . . . . 11 |- ( ( z e. ZZ /\ A e. ZZ ) -> ( z || A <-> -u z || A ) )
36	26, 34, 35	syl2anc 411	. . . . . . . . . 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 536	. . . . . . . . . . . . 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 488	. . . . . . . . . . . 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 9661	. . . . . . . . . . 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 12448	. . . . . . . . . . 11 |- ( ( z e. ZZ /\ B e. ZZ ) -> ( z || B <-> -u z || B ) )
41	26, 39, 40	syl2anc 411	. . . . . . . . . 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 473	. . . . . . . . 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 9555	. . . . . . . . . 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 806	. . . . . . 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 2604	. . . . 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 2635	. 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 716   = wceq 1398   e. wcel 2202   A.wral 2511   E.wrex 2512   class class class wbr 4093  (class class class)co 6028   RRcr 8091   + caddc 8095   x. cmul 8097   -ucneg 8410   NN0cn0 9461   ZZcz 9540   || cdvds 12428

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 2204  ax-14 2205  ax-ext 2213  ax-coll 4209  ax-sep 4212  ax-nul 4220  ax-pow 4270  ax-pr 4305  ax-un 4536  ax-setind 4641  ax-iinf 4692  ax-cnex 8183  ax-resscn 8184  ax-1cn 8185  ax-1re 8186  ax-icn 8187  ax-addcl 8188  ax-addrcl 8189  ax-mulcl 8190  ax-mulrcl 8191  ax-addcom 8192  ax-mulcom 8193  ax-addass 8194  ax-mulass 8195  ax-distr 8196  ax-i2m1 8197  ax-0lt1 8198  ax-1rid 8199  ax-0id 8200  ax-rnegex 8201  ax-precex 8202  ax-cnre 8203  ax-pre-ltirr 8204  ax-pre-ltwlin 8205  ax-pre-lttrn 8206  ax-pre-apti 8207  ax-pre-ltadd 8208  ax-pre-mulgt0 8209  ax-pre-mulext 8210  ax-arch 8211

This theorem depends on definitions:  df-bi 117  df-dc 843  df-3or 1006  df-3an 1007  df-tru 1401  df-fal 1404  df-nf 1510  df-sb 1811  df-eu 2082  df-mo 2083  df-clab 2218  df-cleq 2224  df-clel 2227  df-nfc 2364  df-ne 2404  df-nel 2499  df-ral 2516  df-rex 2517  df-reu 2518  df-rmo 2519  df-rab 2520  df-v 2805  df-sbc 3033  df-csb 3129  df-dif 3203  df-un 3205  df-in 3207  df-ss 3214  df-nul 3497  df-if 3608  df-pw 3658  df-sn 3679  df-pr 3680  df-op 3682  df-uni 3899  df-int 3934  df-iun 3977  df-br 4094  df-opab 4156  df-mpt 4157  df-tr 4193  df-id 4396  df-po 4399  df-iso 4400  df-iord 4469  df-on 4471  df-ilim 4472  df-suc 4474  df-iom 4695  df-xp 4737  df-rel 4738  df-cnv 4739  df-co 4740  df-dm 4741  df-rn 4742  df-res 4743  df-ima 4744  df-iota 5293  df-fun 5335  df-fn 5336  df-f 5337  df-f1 5338  df-fo 5339  df-f1o 5340  df-fv 5341  df-riota 5981  df-ov 6031  df-oprab 6032  df-mpo 6033  df-1st 6312  df-2nd 6313  df-recs 6514  df-frec 6600  df-pnf 8275  df-mnf 8276  df-xr 8277  df-ltxr 8278  df-le 8279  df-sub 8411  df-neg 8412  df-reap 8814  df-ap 8821  df-div 8912  df-inn 9203  df-2 9261  df-n0 9462  df-z 9541  df-uz 9817  df-q 9915  df-rp 9950  df-fz 10306  df-fl 10593  df-mod 10648  df-seqfrec 10773  df-exp 10864  df-cj 11482  df-re 11483  df-im 11484  df-rsqrt 11638  df-abs 11639  df-dvds 12429

This theorem is referenced by:  bezoutlemaz  12654