Theorem bezoutlemle 12444

Description: Lemma for Bézout's identity. The number satisfying the greatest common divisor condition is the largest number which divides both A and B. (Contributed by Mario Carneiro and Jim Kingdon, 9-Jan-2022.)

Hypotheses

Ref	Expression
bezoutlemgcd.1	|- ( ph -> A e. ZZ )
bezoutlemgcd.2	|- ( ph -> B e. ZZ )
bezoutlemgcd.3	|- ( ph -> D e. NN0 )
bezoutlemgcd.4	|- ( ph -> A. z e. ZZ ( z || D <-> ( z || A /\ z || B ) ) )
bezoutlemgcd.5	|- ( ph -> -. ( A = 0 /\ B = 0 ) )

Assertion

Ref	Expression
bezoutlemle	|- ( ph -> A. z e. ZZ ( ( z || A /\ z || B ) -> z <_ D ) )

Distinct variable groups:   z, D   z, A   z, B   ph, z

Proof of Theorem bezoutlemle

Dummy variable w is distinct from all other variables.

Step	Hyp	Ref	Expression
1		simpr 110	. . . . 5 |- ( ( ( ph /\ z e. ZZ ) /\ ( z || A /\ z || B ) ) -> ( z || A /\ z || B ) )
2		breq1 4062	. . . . . . . 8 |- ( z = w -> ( z || D <-> w || D ) )
3		breq1 4062	. . . . . . . . 9 |- ( z = w -> ( z || A <-> w || A ) )
4		breq1 4062	. . . . . . . . 9 |- ( z = w -> ( z || B <-> w || B ) )
5	3, 4	anbi12d 473	. . . . . . . 8 |- ( z = w -> ( ( z || A /\ z || B ) <-> ( w || A /\ w || B ) ) )
6	2, 5	bibi12d 235	. . . . . . 7 |- ( z = w -> ( ( z || D <-> ( z || A /\ z || B ) ) <-> ( w || D <-> ( w || A /\ w || B ) ) ) )
7		equcom 1730	. . . . . . 7 |- ( z = w <-> w = z )
8		bicom 140	. . . . . . 7 |- ( ( ( z || D <-> ( z || A /\ z || B ) ) <-> ( w || D <-> ( w || A /\ w || B ) ) ) <-> ( ( w || D <-> ( w || A /\ w || B ) ) <-> ( z || D <-> ( z || A /\ z || B ) ) ) )
9	6, 7, 8	3imtr3i 200	. . . . . 6 |- ( w = z -> ( ( w || D <-> ( w || A /\ w || B ) ) <-> ( z || D <-> ( z || A /\ z || B ) ) ) )
10		bezoutlemgcd.4	. . . . . . . 8 |- ( ph -> A. z e. ZZ ( z || D <-> ( z || A /\ z || B ) ) )
11	6	cbvralv 2742	. . . . . . . 8 |- ( A. z e. ZZ ( z || D <-> ( z || A /\ z || B ) ) <-> A. w e. ZZ ( w || D <-> ( w || A /\ w || B ) ) )
12	10, 11	sylib 122	. . . . . . 7 |- ( ph -> A. w e. ZZ ( w || D <-> ( w || A /\ w || B ) ) )
13	12	ad2antrr 488	. . . . . 6 |- ( ( ( ph /\ z e. ZZ ) /\ ( z || A /\ z || B ) ) -> A. w e. ZZ ( w || D <-> ( w || A /\ w || B ) ) )
14		simplr 528	. . . . . 6 |- ( ( ( ph /\ z e. ZZ ) /\ ( z || A /\ z || B ) ) -> z e. ZZ )
15	9, 13, 14	rspcdva 2889	. . . . 5 |- ( ( ( ph /\ z e. ZZ ) /\ ( z || A /\ z || B ) ) -> ( z || D <-> ( z || A /\ z || B ) ) )
16	1, 15	mpbird 167	. . . 4 |- ( ( ( ph /\ z e. ZZ ) /\ ( z || A /\ z || B ) ) -> z || D )
17		bezoutlemgcd.3	. . . . . . 7 |- ( ph -> D e. NN0 )
18	17	ad2antrr 488	. . . . . 6 |- ( ( ( ph /\ z e. ZZ ) /\ ( z || A /\ z || B ) ) -> D e. NN0 )
19		bezoutlemgcd.5	. . . . . . . . 9 |- ( ph -> -. ( A = 0 /\ B = 0 ) )
20	19	ad2antrr 488	. . . . . . . 8 |- ( ( ( ph /\ z e. ZZ ) /\ ( z || A /\ z || B ) ) -> -. ( A = 0 /\ B = 0 ) )
21		breq1 4062	. . . . . . . . . . . 12 |- ( z = 0 -> ( z || D <-> 0 || D ) )
22		breq1 4062	. . . . . . . . . . . . 13 |- ( z = 0 -> ( z || A <-> 0 || A ) )
23		breq1 4062	. . . . . . . . . . . . 13 |- ( z = 0 -> ( z || B <-> 0 || B ) )
24	22, 23	anbi12d 473	. . . . . . . . . . . 12 |- ( z = 0 -> ( ( z || A /\ z || B ) <-> ( 0 || A /\ 0 || B ) ) )
25	21, 24	bibi12d 235	. . . . . . . . . . 11 |- ( z = 0 -> ( ( z || D <-> ( z || A /\ z || B ) ) <-> ( 0 || D <-> ( 0 || A /\ 0 || B ) ) ) )
26		0zd 9419	. . . . . . . . . . 11 |- ( ph -> 0 e. ZZ )
27	25, 10, 26	rspcdva 2889	. . . . . . . . . 10 |- ( ph -> ( 0 || D <-> ( 0 || A /\ 0 || B ) ) )
28	27	ad2antrr 488	. . . . . . . . 9 |- ( ( ( ph /\ z e. ZZ ) /\ ( z || A /\ z || B ) ) -> ( 0 || D <-> ( 0 || A /\ 0 || B ) ) )
29	18	nn0zd 9528	. . . . . . . . . 10 |- ( ( ( ph /\ z e. ZZ ) /\ ( z || A /\ z || B ) ) -> D e. ZZ )
30		0dvds 12237	. . . . . . . . . 10 |- ( D e. ZZ -> ( 0 || D <-> D = 0 ) )
31	29, 30	syl 14	. . . . . . . . 9 |- ( ( ( ph /\ z e. ZZ ) /\ ( z || A /\ z || B ) ) -> ( 0 || D <-> D = 0 ) )
32		bezoutlemgcd.1	. . . . . . . . . . . 12 |- ( ph -> A e. ZZ )
33	32	ad2antrr 488	. . . . . . . . . . 11 |- ( ( ( ph /\ z e. ZZ ) /\ ( z || A /\ z || B ) ) -> A e. ZZ )
34		0dvds 12237	. . . . . . . . . . 11 |- ( A e. ZZ -> ( 0 || A <-> A = 0 ) )
35	33, 34	syl 14	. . . . . . . . . 10 |- ( ( ( ph /\ z e. ZZ ) /\ ( z || A /\ z || B ) ) -> ( 0 || A <-> A = 0 ) )
36		bezoutlemgcd.2	. . . . . . . . . . . 12 |- ( ph -> B e. ZZ )
37	36	ad2antrr 488	. . . . . . . . . . 11 |- ( ( ( ph /\ z e. ZZ ) /\ ( z || A /\ z || B ) ) -> B e. ZZ )
38		0dvds 12237	. . . . . . . . . . 11 |- ( B e. ZZ -> ( 0 || B <-> B = 0 ) )
39	37, 38	syl 14	. . . . . . . . . 10 |- ( ( ( ph /\ z e. ZZ ) /\ ( z || A /\ z || B ) ) -> ( 0 || B <-> B = 0 ) )
40	35, 39	anbi12d 473	. . . . . . . . 9 |- ( ( ( ph /\ z e. ZZ ) /\ ( z || A /\ z || B ) ) -> ( ( 0 || A /\ 0 || B ) <-> ( A = 0 /\ B = 0 ) ) )
41	28, 31, 40	3bitr3d 218	. . . . . . . 8 |- ( ( ( ph /\ z e. ZZ ) /\ ( z || A /\ z || B ) ) -> ( D = 0 <-> ( A = 0 /\ B = 0 ) ) )
42	20, 41	mtbird 675	. . . . . . 7 |- ( ( ( ph /\ z e. ZZ ) /\ ( z || A /\ z || B ) ) -> -. D = 0 )
43	42	neqned 2385	. . . . . 6 |- ( ( ( ph /\ z e. ZZ ) /\ ( z || A /\ z || B ) ) -> D =/= 0 )
44		elnnne0 9344	. . . . . 6 |- ( D e. NN <-> ( D e. NN0 /\ D =/= 0 ) )
45	18, 43, 44	sylanbrc 417	. . . . 5 |- ( ( ( ph /\ z e. ZZ ) /\ ( z || A /\ z || B ) ) -> D e. NN )
46		dvdsle 12270	. . . . 5 |- ( ( z e. ZZ /\ D e. NN ) -> ( z || D -> z <_ D ) )
47	14, 45, 46	syl2anc 411	. . . 4 |- ( ( ( ph /\ z e. ZZ ) /\ ( z || A /\ z || B ) ) -> ( z || D -> z <_ D ) )
48	16, 47	mpd 13	. . 3 |- ( ( ( ph /\ z e. ZZ ) /\ ( z || A /\ z || B ) ) -> z <_ D )
49	48	ex 115	. 2 |- ( ( ph /\ z e. ZZ ) -> ( ( z || A /\ z || B ) -> z <_ D ) )
50	49	ralrimiva 2581	1 |- ( ph -> A. z e. ZZ ( ( z || A /\ z || B ) -> z <_ D ) )

Syntax hints:   -. wn 3   -> wi 4   /\ wa 104   <-> wb 105   = wceq 1373   e. wcel 2178   =/= wne 2378   A.wral 2486   class class class wbr 4059   0cc0 7960   <_ cle 8143   NNcn 9071   NN0cn0 9330   ZZcz 9407   || cdvds 12213

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 615  ax-in2 616  ax-io 711  ax-5 1471  ax-7 1472  ax-gen 1473  ax-ie1 1517  ax-ie2 1518  ax-8 1528  ax-10 1529  ax-11 1530  ax-i12 1531  ax-bndl 1533  ax-4 1534  ax-17 1550  ax-i9 1554  ax-ial 1558  ax-i5r 1559  ax-13 2180  ax-14 2181  ax-ext 2189  ax-sep 4178  ax-pow 4234  ax-pr 4269  ax-un 4498  ax-setind 4603  ax-cnex 8051  ax-resscn 8052  ax-1cn 8053  ax-1re 8054  ax-icn 8055  ax-addcl 8056  ax-addrcl 8057  ax-mulcl 8058  ax-mulrcl 8059  ax-addcom 8060  ax-mulcom 8061  ax-addass 8062  ax-mulass 8063  ax-distr 8064  ax-i2m1 8065  ax-0lt1 8066  ax-1rid 8067  ax-0id 8068  ax-rnegex 8069  ax-precex 8070  ax-cnre 8071  ax-pre-ltirr 8072  ax-pre-ltwlin 8073  ax-pre-lttrn 8074  ax-pre-apti 8075  ax-pre-ltadd 8076  ax-pre-mulgt0 8077  ax-pre-mulext 8078

This theorem depends on definitions:  df-bi 117  df-3or 982  df-3an 983  df-tru 1376  df-fal 1379  df-nf 1485  df-sb 1787  df-eu 2058  df-mo 2059  df-clab 2194  df-cleq 2200  df-clel 2203  df-nfc 2339  df-ne 2379  df-nel 2474  df-ral 2491  df-rex 2492  df-reu 2493  df-rmo 2494  df-rab 2495  df-v 2778  df-sbc 3006  df-csb 3102  df-dif 3176  df-un 3178  df-in 3180  df-ss 3187  df-nul 3469  df-pw 3628  df-sn 3649  df-pr 3650  df-op 3652  df-uni 3865  df-int 3900  df-iun 3943  df-br 4060  df-opab 4122  df-mpt 4123  df-id 4358  df-po 4361  df-iso 4362  df-xp 4699  df-rel 4700  df-cnv 4701  df-co 4702  df-dm 4703  df-rn 4704  df-res 4705  df-ima 4706  df-iota 5251  df-fun 5292  df-fn 5293  df-f 5294  df-fv 5298  df-riota 5922  df-ov 5970  df-oprab 5971  df-mpo 5972  df-1st 6249  df-2nd 6250  df-pnf 8144  df-mnf 8145  df-xr 8146  df-ltxr 8147  df-le 8148  df-sub 8280  df-neg 8281  df-reap 8683  df-ap 8690  df-div 8781  df-inn 9072  df-n0 9331  df-z 9408  df-q 9776  df-dvds 12214

This theorem is referenced by:  bezoutlemsup  12445