Theorem dvdsabseq 15035

Description: If two integers divide each other, they must be equal, up to a difference in sign. Theorem 1.1(j) in [ApostolNT] p. 14. (Contributed by Mario Carneiro, 30-May-2014.) (Revised by AV, 7-Aug-2021.)

Assertion

Ref	Expression
dvdsabseq	|- ( ( M || N /\ N || M ) -> ( abs ` M ) = ( abs ` N ) )

Proof of Theorem dvdsabseq

Step	Hyp	Ref	Expression
1		dvdszrcl 14988	. . 3 |- ( M || N -> ( M e. ZZ /\ N e. ZZ ) )
2		simpr 477	. . . . . 6 |- ( ( M || N /\ N || M ) -> N || M )
3		breq1 4656	. . . . . . . 8 |- ( N = 0 -> ( N || M <-> 0 || M ) )
4		0dvds 15002	. . . . . . . . . 10 |- ( M e. ZZ -> ( 0 || M <-> M = 0 ) )
5	4	adantr 481	. . . . . . . . 9 |- ( ( M e. ZZ /\ N e. ZZ ) -> ( 0 || M <-> M = 0 ) )
6		zcn 11382	. . . . . . . . . . . 12 |- ( M e. ZZ -> M e. CC )
7	6	abs00ad 14030	. . . . . . . . . . 11 |- ( M e. ZZ -> ( ( abs ` M ) = 0 <-> M = 0 ) )
8	7	bicomd 213	. . . . . . . . . 10 |- ( M e. ZZ -> ( M = 0 <-> ( abs ` M ) = 0 ) )
9	8	adantr 481	. . . . . . . . 9 |- ( ( M e. ZZ /\ N e. ZZ ) -> ( M = 0 <-> ( abs ` M ) = 0 ) )
10	5, 9	bitrd 268	. . . . . . . 8 |- ( ( M e. ZZ /\ N e. ZZ ) -> ( 0 || M <-> ( abs ` M ) = 0 ) )
11	3, 10	sylan9bb 736	. . . . . . 7 |- ( ( N = 0 /\ ( M e. ZZ /\ N e. ZZ ) ) -> ( N || M <-> ( abs ` M ) = 0 ) )
12		fveq2 6191	. . . . . . . . . 10 |- ( N = 0 -> ( abs ` N ) = ( abs ` 0 ) )
13		abs0 14025	. . . . . . . . . 10 |- ( abs ` 0 ) = 0
14	12, 13	syl6eq 2672	. . . . . . . . 9 |- ( N = 0 -> ( abs ` N ) = 0 )
15	14	adantr 481	. . . . . . . 8 |- ( ( N = 0 /\ ( M e. ZZ /\ N e. ZZ ) ) -> ( abs ` N ) = 0 )
16	15	eqeq2d 2632	. . . . . . 7 |- ( ( N = 0 /\ ( M e. ZZ /\ N e. ZZ ) ) -> ( ( abs ` M ) = ( abs ` N ) <-> ( abs ` M ) = 0 ) )
17	11, 16	bitr4d 271	. . . . . 6 |- ( ( N = 0 /\ ( M e. ZZ /\ N e. ZZ ) ) -> ( N || M <-> ( abs ` M ) = ( abs ` N ) ) )
18	2, 17	syl5ib 234	. . . . 5 |- ( ( N = 0 /\ ( M e. ZZ /\ N e. ZZ ) ) -> ( ( M || N /\ N || M ) -> ( abs ` M ) = ( abs ` N ) ) )
19	18	expd 452	. . . 4 |- ( ( N = 0 /\ ( M e. ZZ /\ N e. ZZ ) ) -> ( M || N -> ( N || M -> ( abs ` M ) = ( abs ` N ) ) ) )
20		simprl 794	. . . . . 6 |- ( ( -. N = 0 /\ ( M e. ZZ /\ N e. ZZ ) ) -> M e. ZZ )
21		simpr 477	. . . . . . 7 |- ( ( M e. ZZ /\ N e. ZZ ) -> N e. ZZ )
22	21	adantl 482	. . . . . 6 |- ( ( -. N = 0 /\ ( M e. ZZ /\ N e. ZZ ) ) -> N e. ZZ )
23		neqne 2802	. . . . . . 7 |- ( -. N = 0 -> N =/= 0 )
24	23	adantr 481	. . . . . 6 |- ( ( -. N = 0 /\ ( M e. ZZ /\ N e. ZZ ) ) -> N =/= 0 )
25		dvdsleabs2 15034	. . . . . 6 |- ( ( M e. ZZ /\ N e. ZZ /\ N =/= 0 ) -> ( M || N -> ( abs ` M ) <_ ( abs ` N ) ) )
26	20, 22, 24, 25	syl3anc 1326	. . . . 5 |- ( ( -. N = 0 /\ ( M e. ZZ /\ N e. ZZ ) ) -> ( M || N -> ( abs ` M ) <_ ( abs ` N ) ) )
27		simpr 477	. . . . . . . . . . 11 |- ( ( N || M /\ M || N ) -> M || N )
28		breq1 4656	. . . . . . . . . . . . 13 |- ( M = 0 -> ( M || N <-> 0 || N ) )
29		0dvds 15002	. . . . . . . . . . . . . . 15 |- ( N e. ZZ -> ( 0 || N <-> N = 0 ) )
30		eqcom 2629	. . . . . . . . . . . . . . . 16 |- ( ( abs ` N ) = 0 <-> 0 = ( abs ` N ) )
31		zcn 11382	. . . . . . . . . . . . . . . . 17 |- ( N e. ZZ -> N e. CC )
32	31	abs00ad 14030	. . . . . . . . . . . . . . . 16 |- ( N e. ZZ -> ( ( abs ` N ) = 0 <-> N = 0 ) )
33	30, 32	syl5rbbr 275	. . . . . . . . . . . . . . 15 |- ( N e. ZZ -> ( N = 0 <-> 0 = ( abs ` N ) ) )
34	29, 33	bitrd 268	. . . . . . . . . . . . . 14 |- ( N e. ZZ -> ( 0 || N <-> 0 = ( abs ` N ) ) )
35	34	adantl 482	. . . . . . . . . . . . 13 |- ( ( M e. ZZ /\ N e. ZZ ) -> ( 0 || N <-> 0 = ( abs ` N ) ) )
36	28, 35	sylan9bb 736	. . . . . . . . . . . 12 |- ( ( M = 0 /\ ( M e. ZZ /\ N e. ZZ ) ) -> ( M || N <-> 0 = ( abs ` N ) ) )
37		fveq2 6191	. . . . . . . . . . . . . . 15 |- ( M = 0 -> ( abs ` M ) = ( abs ` 0 ) )
38	37, 13	syl6eq 2672	. . . . . . . . . . . . . 14 |- ( M = 0 -> ( abs ` M ) = 0 )
39	38	adantr 481	. . . . . . . . . . . . 13 |- ( ( M = 0 /\ ( M e. ZZ /\ N e. ZZ ) ) -> ( abs ` M ) = 0 )
40	39	eqeq1d 2624	. . . . . . . . . . . 12 |- ( ( M = 0 /\ ( M e. ZZ /\ N e. ZZ ) ) -> ( ( abs ` M ) = ( abs ` N ) <-> 0 = ( abs ` N ) ) )
41	36, 40	bitr4d 271	. . . . . . . . . . 11 |- ( ( M = 0 /\ ( M e. ZZ /\ N e. ZZ ) ) -> ( M || N <-> ( abs ` M ) = ( abs ` N ) ) )
42	27, 41	syl5ib 234	. . . . . . . . . 10 |- ( ( M = 0 /\ ( M e. ZZ /\ N e. ZZ ) ) -> ( ( N || M /\ M || N ) -> ( abs ` M ) = ( abs ` N ) ) )
43	42	a1dd 50	. . . . . . . . 9 |- ( ( M = 0 /\ ( M e. ZZ /\ N e. ZZ ) ) -> ( ( N || M /\ M || N ) -> ( ( abs ` M ) <_ ( abs ` N ) -> ( abs ` M ) = ( abs ` N ) ) ) )
44	43	expcomd 454	. . . . . . . 8 |- ( ( M = 0 /\ ( M e. ZZ /\ N e. ZZ ) ) -> ( M || N -> ( N || M -> ( ( abs ` M ) <_ ( abs ` N ) -> ( abs ` M ) = ( abs ` N ) ) ) ) )
45	21	adantl 482	. . . . . . . . . . 11 |- ( ( -. M = 0 /\ ( M e. ZZ /\ N e. ZZ ) ) -> N e. ZZ )
46		simprl 794	. . . . . . . . . . 11 |- ( ( -. M = 0 /\ ( M e. ZZ /\ N e. ZZ ) ) -> M e. ZZ )
47		neqne 2802	. . . . . . . . . . . 12 |- ( -. M = 0 -> M =/= 0 )
48	47	adantr 481	. . . . . . . . . . 11 |- ( ( -. M = 0 /\ ( M e. ZZ /\ N e. ZZ ) ) -> M =/= 0 )
49		dvdsleabs2 15034	. . . . . . . . . . 11 |- ( ( N e. ZZ /\ M e. ZZ /\ M =/= 0 ) -> ( N || M -> ( abs ` N ) <_ ( abs ` M ) ) )
50	45, 46, 48, 49	syl3anc 1326	. . . . . . . . . 10 |- ( ( -. M = 0 /\ ( M e. ZZ /\ N e. ZZ ) ) -> ( N || M -> ( abs ` N ) <_ ( abs ` M ) ) )
51		eqcom 2629	. . . . . . . . . . . . . 14 |- ( ( abs ` M ) = ( abs ` N ) <-> ( abs ` N ) = ( abs ` M ) )
52	31	abscld 14175	. . . . . . . . . . . . . . 15 |- ( N e. ZZ -> ( abs ` N ) e. RR )
53	6	abscld 14175	. . . . . . . . . . . . . . 15 |- ( M e. ZZ -> ( abs ` M ) e. RR )
54		letri3 10123	. . . . . . . . . . . . . . 15 |- ( ( ( abs ` N ) e. RR /\ ( abs ` M ) e. RR ) -> ( ( abs ` N ) = ( abs ` M ) <-> ( ( abs ` N ) <_ ( abs ` M ) /\ ( abs ` M ) <_ ( abs ` N ) ) ) )
55	52, 53, 54	syl2anr 495	. . . . . . . . . . . . . 14 |- ( ( M e. ZZ /\ N e. ZZ ) -> ( ( abs ` N ) = ( abs ` M ) <-> ( ( abs ` N ) <_ ( abs ` M ) /\ ( abs ` M ) <_ ( abs ` N ) ) ) )
56	51, 55	syl5bb 272	. . . . . . . . . . . . 13 |- ( ( M e. ZZ /\ N e. ZZ ) -> ( ( abs ` M ) = ( abs ` N ) <-> ( ( abs ` N ) <_ ( abs ` M ) /\ ( abs ` M ) <_ ( abs ` N ) ) ) )
57	56	biimprd 238	. . . . . . . . . . . 12 |- ( ( M e. ZZ /\ N e. ZZ ) -> ( ( ( abs ` N ) <_ ( abs ` M ) /\ ( abs ` M ) <_ ( abs ` N ) ) -> ( abs ` M ) = ( abs ` N ) ) )
58	57	expd 452	. . . . . . . . . . 11 |- ( ( M e. ZZ /\ N e. ZZ ) -> ( ( abs ` N ) <_ ( abs ` M ) -> ( ( abs ` M ) <_ ( abs ` N ) -> ( abs ` M ) = ( abs ` N ) ) ) )
59	58	adantl 482	. . . . . . . . . 10 |- ( ( -. M = 0 /\ ( M e. ZZ /\ N e. ZZ ) ) -> ( ( abs ` N ) <_ ( abs ` M ) -> ( ( abs ` M ) <_ ( abs ` N ) -> ( abs ` M ) = ( abs ` N ) ) ) )
60	50, 59	syld 47	. . . . . . . . 9 |- ( ( -. M = 0 /\ ( M e. ZZ /\ N e. ZZ ) ) -> ( N || M -> ( ( abs ` M ) <_ ( abs ` N ) -> ( abs ` M ) = ( abs ` N ) ) ) )
61	60	a1d 25	. . . . . . . 8 |- ( ( -. M = 0 /\ ( M e. ZZ /\ N e. ZZ ) ) -> ( M || N -> ( N || M -> ( ( abs ` M ) <_ ( abs ` N ) -> ( abs ` M ) = ( abs ` N ) ) ) ) )
62	44, 61	pm2.61ian 831	. . . . . . 7 |- ( ( M e. ZZ /\ N e. ZZ ) -> ( M || N -> ( N || M -> ( ( abs ` M ) <_ ( abs ` N ) -> ( abs ` M ) = ( abs ` N ) ) ) ) )
63	62	com34 91	. . . . . 6 |- ( ( M e. ZZ /\ N e. ZZ ) -> ( M || N -> ( ( abs ` M ) <_ ( abs ` N ) -> ( N || M -> ( abs ` M ) = ( abs ` N ) ) ) ) )
64	63	adantl 482	. . . . 5 |- ( ( -. N = 0 /\ ( M e. ZZ /\ N e. ZZ ) ) -> ( M || N -> ( ( abs ` M ) <_ ( abs ` N ) -> ( N || M -> ( abs ` M ) = ( abs ` N ) ) ) ) )
65	26, 64	mpdd 43	. . . 4 |- ( ( -. N = 0 /\ ( M e. ZZ /\ N e. ZZ ) ) -> ( M || N -> ( N || M -> ( abs ` M ) = ( abs ` N ) ) ) )
66	19, 65	pm2.61ian 831	. . 3 |- ( ( M e. ZZ /\ N e. ZZ ) -> ( M || N -> ( N || M -> ( abs ` M ) = ( abs ` N ) ) ) )
67	1, 66	mpcom 38	. 2 |- ( M || N -> ( N || M -> ( abs ` M ) = ( abs ` N ) ) )
68	67	imp 445	1 |- ( ( M || N /\ N || M ) -> ( abs ` M ) = ( abs ` N ) )

Syntax hints:   -. wn 3   -> wi 4   <-> wb 196   /\ wa 384   = wceq 1483   e. wcel 1990   =/= wne 2794   class class class wbr 4653   ` cfv 5888   RRcr 9935   0cc0 9936   <_ cle 10075   ZZcz 11377   abscabs 13974   || cdvds 14983

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  df-dvds 14984

This theorem is referenced by:  dvdseq  15036