Theorem dvdsabseq 12592

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 12537	. . 3 |- ( M || N -> ( M e. ZZ /\ N e. ZZ ) )
2		simpr 110	. . . . . . 7 |- ( ( M || N /\ N || M ) -> N || M )
3		breq1 4128	. . . . . . . . 9 |- ( N = 0 -> ( N || M <-> 0 || M ) )
4		0dvds 12556	. . . . . . . . . . 11 |- ( M e. ZZ -> ( 0 || M <-> M = 0 ) )
5	4	adantr 276	. . . . . . . . . 10 |- ( ( M e. ZZ /\ N e. ZZ ) -> ( 0 || M <-> M = 0 ) )
6		zcn 9628	. . . . . . . . . . . . 13 |- ( M e. ZZ -> M e. CC )
7	6	abs00ad 11809	. . . . . . . . . . . 12 |- ( M e. ZZ -> ( ( abs ` M ) = 0 <-> M = 0 ) )
8	7	bicomd 141	. . . . . . . . . . 11 |- ( M e. ZZ -> ( M = 0 <-> ( abs ` M ) = 0 ) )
9	8	adantr 276	. . . . . . . . . 10 |- ( ( M e. ZZ /\ N e. ZZ ) -> ( M = 0 <-> ( abs ` M ) = 0 ) )
10	5, 9	bitrd 188	. . . . . . . . 9 |- ( ( M e. ZZ /\ N e. ZZ ) -> ( 0 || M <-> ( abs ` M ) = 0 ) )
11	3, 10	sylan9bb 466	. . . . . . . 8 |- ( ( N = 0 /\ ( M e. ZZ /\ N e. ZZ ) ) -> ( N || M <-> ( abs ` M ) = 0 ) )
12		fveq2 5690	. . . . . . . . . . 11 |- ( N = 0 -> ( abs ` N ) = ( abs ` 0 ) )
13		abs0 11802	. . . . . . . . . . 11 |- ( abs ` 0 ) = 0
14	12, 13	eqtrdi 2287	. . . . . . . . . 10 |- ( N = 0 -> ( abs ` N ) = 0 )
15	14	adantr 276	. . . . . . . . 9 |- ( ( N = 0 /\ ( M e. ZZ /\ N e. ZZ ) ) -> ( abs ` N ) = 0 )
16	15	eqeq2d 2250	. . . . . . . 8 |- ( ( N = 0 /\ ( M e. ZZ /\ N e. ZZ ) ) -> ( ( abs ` M ) = ( abs ` N ) <-> ( abs ` M ) = 0 ) )
17	11, 16	bitr4d 191	. . . . . . 7 |- ( ( N = 0 /\ ( M e. ZZ /\ N e. ZZ ) ) -> ( N || M <-> ( abs ` M ) = ( abs ` N ) ) )
18	2, 17	imbitrid 154	. . . . . 6 |- ( ( N = 0 /\ ( M e. ZZ /\ N e. ZZ ) ) -> ( ( M || N /\ N || M ) -> ( abs ` M ) = ( abs ` N ) ) )
19	18	expd 258	. . . . 5 |- ( ( N = 0 /\ ( M e. ZZ /\ N e. ZZ ) ) -> ( M || N -> ( N || M -> ( abs ` M ) = ( abs ` N ) ) ) )
20	19	expcom 116	. . . 4 |- ( ( M e. ZZ /\ N e. ZZ ) -> ( N = 0 -> ( M || N -> ( N || M -> ( abs ` M ) = ( abs ` N ) ) ) ) )
21		simprl 535	. . . . . . 7 |- ( ( -. N = 0 /\ ( M e. ZZ /\ N e. ZZ ) ) -> M e. ZZ )
22		simpr 110	. . . . . . . 8 |- ( ( M e. ZZ /\ N e. ZZ ) -> N e. ZZ )
23	22	adantl 277	. . . . . . 7 |- ( ( -. N = 0 /\ ( M e. ZZ /\ N e. ZZ ) ) -> N e. ZZ )
24		neqne 2428	. . . . . . . 8 |- ( -. N = 0 -> N =/= 0 )
25	24	adantr 276	. . . . . . 7 |- ( ( -. N = 0 /\ ( M e. ZZ /\ N e. ZZ ) ) -> N =/= 0 )
26		dvdsleabs2 12591	. . . . . . 7 |- ( ( M e. ZZ /\ N e. ZZ /\ N =/= 0 ) -> ( M || N -> ( abs ` M ) <_ ( abs ` N ) ) )
27	21, 23, 25, 26	syl3anc 1278	. . . . . 6 |- ( ( -. N = 0 /\ ( M e. ZZ /\ N e. ZZ ) ) -> ( M || N -> ( abs ` M ) <_ ( abs ` N ) ) )
28		simpr 110	. . . . . . . . . . . . 13 |- ( ( N || M /\ M || N ) -> M || N )
29		breq1 4128	. . . . . . . . . . . . . . 15 |- ( M = 0 -> ( M || N <-> 0 || N ) )
30		0dvds 12556	. . . . . . . . . . . . . . . . 17 |- ( N e. ZZ -> ( 0 || N <-> N = 0 ) )
31		zcn 9628	. . . . . . . . . . . . . . . . . . 19 |- ( N e. ZZ -> N e. CC )
32	31	abs00ad 11809	. . . . . . . . . . . . . . . . . 18 |- ( N e. ZZ -> ( ( abs ` N ) = 0 <-> N = 0 ) )
33		eqcom 2240	. . . . . . . . . . . . . . . . . 18 |- ( ( abs ` N ) = 0 <-> 0 = ( abs ` N ) )
34	32, 33	bitr3di 195	. . . . . . . . . . . . . . . . 17 |- ( N e. ZZ -> ( N = 0 <-> 0 = ( abs ` N ) ) )
35	30, 34	bitrd 188	. . . . . . . . . . . . . . . 16 |- ( N e. ZZ -> ( 0 || N <-> 0 = ( abs ` N ) ) )
36	35	adantl 277	. . . . . . . . . . . . . . 15 |- ( ( M e. ZZ /\ N e. ZZ ) -> ( 0 || N <-> 0 = ( abs ` N ) ) )
37	29, 36	sylan9bb 466	. . . . . . . . . . . . . 14 |- ( ( M = 0 /\ ( M e. ZZ /\ N e. ZZ ) ) -> ( M || N <-> 0 = ( abs ` N ) ) )
38		fveq2 5690	. . . . . . . . . . . . . . . . 17 |- ( M = 0 -> ( abs ` M ) = ( abs ` 0 ) )
39	38, 13	eqtrdi 2287	. . . . . . . . . . . . . . . 16 |- ( M = 0 -> ( abs ` M ) = 0 )
40	39	adantr 276	. . . . . . . . . . . . . . 15 |- ( ( M = 0 /\ ( M e. ZZ /\ N e. ZZ ) ) -> ( abs ` M ) = 0 )
41	40	eqeq1d 2247	. . . . . . . . . . . . . 14 |- ( ( M = 0 /\ ( M e. ZZ /\ N e. ZZ ) ) -> ( ( abs ` M ) = ( abs ` N ) <-> 0 = ( abs ` N ) ) )
42	37, 41	bitr4d 191	. . . . . . . . . . . . 13 |- ( ( M = 0 /\ ( M e. ZZ /\ N e. ZZ ) ) -> ( M || N <-> ( abs ` M ) = ( abs ` N ) ) )
43	28, 42	imbitrid 154	. . . . . . . . . . . 12 |- ( ( M = 0 /\ ( M e. ZZ /\ N e. ZZ ) ) -> ( ( N || M /\ M || N ) -> ( abs ` M ) = ( abs ` N ) ) )
44	43	a1dd 48	. . . . . . . . . . 11 |- ( ( M = 0 /\ ( M e. ZZ /\ N e. ZZ ) ) -> ( ( N || M /\ M || N ) -> ( ( abs ` M ) <_ ( abs ` N ) -> ( abs ` M ) = ( abs ` N ) ) ) )
45	44	expcomd 1491	. . . . . . . . . 10 |- ( ( M = 0 /\ ( M e. ZZ /\ N e. ZZ ) ) -> ( M || N -> ( N || M -> ( ( abs ` M ) <_ ( abs ` N ) -> ( abs ` M ) = ( abs ` N ) ) ) ) )
46	45	expcom 116	. . . . . . . . 9 |- ( ( M e. ZZ /\ N e. ZZ ) -> ( M = 0 -> ( M || N -> ( N || M -> ( ( abs ` M ) <_ ( abs ` N ) -> ( abs ` M ) = ( abs ` N ) ) ) ) ) )
47	22	adantl 277	. . . . . . . . . . . . 13 |- ( ( -. M = 0 /\ ( M e. ZZ /\ N e. ZZ ) ) -> N e. ZZ )
48		simprl 535	. . . . . . . . . . . . 13 |- ( ( -. M = 0 /\ ( M e. ZZ /\ N e. ZZ ) ) -> M e. ZZ )
49		neqne 2428	. . . . . . . . . . . . . 14 |- ( -. M = 0 -> M =/= 0 )
50	49	adantr 276	. . . . . . . . . . . . 13 |- ( ( -. M = 0 /\ ( M e. ZZ /\ N e. ZZ ) ) -> M =/= 0 )
51		dvdsleabs2 12591	. . . . . . . . . . . . 13 |- ( ( N e. ZZ /\ M e. ZZ /\ M =/= 0 ) -> ( N || M -> ( abs ` N ) <_ ( abs ` M ) ) )
52	47, 48, 50, 51	syl3anc 1278	. . . . . . . . . . . 12 |- ( ( -. M = 0 /\ ( M e. ZZ /\ N e. ZZ ) ) -> ( N || M -> ( abs ` N ) <_ ( abs ` M ) ) )
53		eqcom 2240	. . . . . . . . . . . . . . . 16 |- ( ( abs ` M ) = ( abs ` N ) <-> ( abs ` N ) = ( abs ` M ) )
54	31	abscld 11925	. . . . . . . . . . . . . . . . 17 |- ( N e. ZZ -> ( abs ` N ) e. RR )
55	6	abscld 11925	. . . . . . . . . . . . . . . . 17 |- ( M e. ZZ -> ( abs ` M ) e. RR )
56		letri3 8396	. . . . . . . . . . . . . . . . 17 |- ( ( ( abs ` N ) e. RR /\ ( abs ` M ) e. RR ) -> ( ( abs ` N ) = ( abs ` M ) <-> ( ( abs ` N ) <_ ( abs ` M ) /\ ( abs ` M ) <_ ( abs ` N ) ) ) )
57	54, 55, 56	syl2anr 290	. . . . . . . . . . . . . . . 16 |- ( ( M e. ZZ /\ N e. ZZ ) -> ( ( abs ` N ) = ( abs ` M ) <-> ( ( abs ` N ) <_ ( abs ` M ) /\ ( abs ` M ) <_ ( abs ` N ) ) ) )
58	53, 57	bitrid 192	. . . . . . . . . . . . . . 15 |- ( ( M e. ZZ /\ N e. ZZ ) -> ( ( abs ` M ) = ( abs ` N ) <-> ( ( abs ` N ) <_ ( abs ` M ) /\ ( abs ` M ) <_ ( abs ` N ) ) ) )
59	58	biimprd 158	. . . . . . . . . . . . . 14 |- ( ( M e. ZZ /\ N e. ZZ ) -> ( ( ( abs ` N ) <_ ( abs ` M ) /\ ( abs ` M ) <_ ( abs ` N ) ) -> ( abs ` M ) = ( abs ` N ) ) )
60	59	expd 258	. . . . . . . . . . . . 13 |- ( ( M e. ZZ /\ N e. ZZ ) -> ( ( abs ` N ) <_ ( abs ` M ) -> ( ( abs ` M ) <_ ( abs ` N ) -> ( abs ` M ) = ( abs ` N ) ) ) )
61	60	adantl 277	. . . . . . . . . . . 12 |- ( ( -. M = 0 /\ ( M e. ZZ /\ N e. ZZ ) ) -> ( ( abs ` N ) <_ ( abs ` M ) -> ( ( abs ` M ) <_ ( abs ` N ) -> ( abs ` M ) = ( abs ` N ) ) ) )
62	52, 61	syld 45	. . . . . . . . . . 11 |- ( ( -. M = 0 /\ ( M e. ZZ /\ N e. ZZ ) ) -> ( N || M -> ( ( abs ` M ) <_ ( abs ` N ) -> ( abs ` M ) = ( abs ` N ) ) ) )
63	62	a1d 22	. . . . . . . . . 10 |- ( ( -. M = 0 /\ ( M e. ZZ /\ N e. ZZ ) ) -> ( M || N -> ( N || M -> ( ( abs ` M ) <_ ( abs ` N ) -> ( abs ` M ) = ( abs ` N ) ) ) ) )
64	63	expcom 116	. . . . . . . . 9 |- ( ( M e. ZZ /\ N e. ZZ ) -> ( -. M = 0 -> ( M || N -> ( N || M -> ( ( abs ` M ) <_ ( abs ` N ) -> ( abs ` M ) = ( abs ` N ) ) ) ) ) )
65		0z 9634	. . . . . . . . . . . 12 |- 0 e. ZZ
66		zdceq 9699	. . . . . . . . . . . 12 |- ( ( M e. ZZ /\ 0 e. ZZ ) -> DECID M = 0 )
67	65, 66	mpan2 429	. . . . . . . . . . 11 |- ( M e. ZZ -> DECID M = 0 )
68		exmiddc 848	. . . . . . . . . . 11 |- (DECID M = 0 -> ( M = 0 \/ -. M = 0 ) )
69	67, 68	syl 14	. . . . . . . . . 10 |- ( M e. ZZ -> ( M = 0 \/ -. M = 0 ) )
70	69	adantr 276	. . . . . . . . 9 |- ( ( M e. ZZ /\ N e. ZZ ) -> ( M = 0 \/ -. M = 0 ) )
71	46, 64, 70	mpjaod 730	. . . . . . . 8 |- ( ( M e. ZZ /\ N e. ZZ ) -> ( M || N -> ( N || M -> ( ( abs ` M ) <_ ( abs ` N ) -> ( abs ` M ) = ( abs ` N ) ) ) ) )
72	71	com34 83	. . . . . . 7 |- ( ( M e. ZZ /\ N e. ZZ ) -> ( M || N -> ( ( abs ` M ) <_ ( abs ` N ) -> ( N || M -> ( abs ` M ) = ( abs ` N ) ) ) ) )
73	72	adantl 277	. . . . . 6 |- ( ( -. N = 0 /\ ( M e. ZZ /\ N e. ZZ ) ) -> ( M || N -> ( ( abs ` M ) <_ ( abs ` N ) -> ( N || M -> ( abs ` M ) = ( abs ` N ) ) ) ) )
74	27, 73	mpdd 41	. . . . 5 |- ( ( -. N = 0 /\ ( M e. ZZ /\ N e. ZZ ) ) -> ( M || N -> ( N || M -> ( abs ` M ) = ( abs ` N ) ) ) )
75	74	expcom 116	. . . 4 |- ( ( M e. ZZ /\ N e. ZZ ) -> ( -. N = 0 -> ( M || N -> ( N || M -> ( abs ` M ) = ( abs ` N ) ) ) ) )
76		zdceq 9699	. . . . . . 7 |- ( ( N e. ZZ /\ 0 e. ZZ ) -> DECID N = 0 )
77	65, 76	mpan2 429	. . . . . 6 |- ( N e. ZZ -> DECID N = 0 )
78		exmiddc 848	. . . . . 6 |- (DECID N = 0 -> ( N = 0 \/ -. N = 0 ) )
79	77, 78	syl 14	. . . . 5 |- ( N e. ZZ -> ( N = 0 \/ -. N = 0 ) )
80	79	adantl 277	. . . 4 |- ( ( M e. ZZ /\ N e. ZZ ) -> ( N = 0 \/ -. N = 0 ) )
81	20, 75, 80	mpjaod 730	. . 3 |- ( ( M e. ZZ /\ N e. ZZ ) -> ( M || N -> ( N || M -> ( abs ` M ) = ( abs ` N ) ) ) )
82	1, 81	mpcom 36	. 2 |- ( M || N -> ( N || M -> ( abs ` M ) = ( abs ` N ) ) )
83	82	imp 124	1 |- ( ( M || N /\ N || M ) -> ( abs ` M ) = ( abs ` N ) )

Syntax hints:   -. wn 3   -> wi 4   /\ wa 104   <-> wb 105   \/ wo 720  DECID wdc 846   = wceq 1402   e. wcel 2209   =/= wne 2420   class class class wbr 4125   ` cfv 5372   RRcr 8168   0cc0 8169   <_ cle 8351   ZZcz 9623   abscabs 11741   || 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  ax-caucvg 8289

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-3 9343  df-4 9344  df-n0 9543  df-z 9624  df-uz 9901  df-q 9999  df-rp 10034  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:  dvdseq  12593