Theorem dvdssq 15280

Description: Two numbers are divisible iff their squares are. (Contributed by Scott Fenton, 18-Apr-2014.) (Revised by Mario Carneiro, 19-Apr-2014.)

Assertion

Ref	Expression
dvdssq	|- ( ( M e. ZZ /\ N e. ZZ ) -> ( M || N <-> ( M ^ 2 ) || ( N ^ 2 ) ) )

Proof of Theorem dvdssq

Step	Hyp	Ref	Expression
1		breq1 4656	. . 3 |- ( M = 0 -> ( M || N <-> 0 || N ) )
2		sq0i 12956	. . . 4 |- ( M = 0 -> ( M ^ 2 ) = 0 )
3	2	breq1d 4663	. . 3 |- ( M = 0 -> ( ( M ^ 2 ) || ( N ^ 2 ) <-> 0 || ( N ^ 2 ) ) )
4	1, 3	bibi12d 335	. 2 |- ( M = 0 -> ( ( M || N <-> ( M ^ 2 ) || ( N ^ 2 ) ) <-> ( 0 || N <-> 0 || ( N ^ 2 ) ) ) )
5		nnabscl 14065	. . . . 5 |- ( ( M e. ZZ /\ M =/= 0 ) -> ( abs ` M ) e. NN )
6		breq2 4657	. . . . . . 7 |- ( N = 0 -> ( ( abs ` M ) || N <-> ( abs ` M ) || 0 ) )
7		sq0i 12956	. . . . . . . 8 |- ( N = 0 -> ( N ^ 2 ) = 0 )
8	7	breq2d 4665	. . . . . . 7 |- ( N = 0 -> ( ( ( abs ` M ) ^ 2 ) || ( N ^ 2 ) <-> ( ( abs ` M ) ^ 2 ) || 0 ) )
9	6, 8	bibi12d 335	. . . . . 6 |- ( N = 0 -> ( ( ( abs ` M ) || N <-> ( ( abs ` M ) ^ 2 ) || ( N ^ 2 ) ) <-> ( ( abs ` M ) || 0 <-> ( ( abs ` M ) ^ 2 ) || 0 ) ) )
10		nnabscl 14065	. . . . . . . . 9 |- ( ( N e. ZZ /\ N =/= 0 ) -> ( abs ` N ) e. NN )
11		dvdssqlem 15279	. . . . . . . . 9 |- ( ( ( abs ` M ) e. NN /\ ( abs ` N ) e. NN ) -> ( ( abs ` M ) || ( abs ` N ) <-> ( ( abs ` M ) ^ 2 ) || ( ( abs ` N ) ^ 2 ) ) )
12	10, 11	sylan2 491	. . . . . . . 8 |- ( ( ( abs ` M ) e. NN /\ ( N e. ZZ /\ N =/= 0 ) ) -> ( ( abs ` M ) || ( abs ` N ) <-> ( ( abs ` M ) ^ 2 ) || ( ( abs ` N ) ^ 2 ) ) )
13		nnz 11399	. . . . . . . . 9 |- ( ( abs ` M ) e. NN -> ( abs ` M ) e. ZZ )
14		simpl 473	. . . . . . . . 9 |- ( ( N e. ZZ /\ N =/= 0 ) -> N e. ZZ )
15		dvdsabsb 15001	. . . . . . . . 9 |- ( ( ( abs ` M ) e. ZZ /\ N e. ZZ ) -> ( ( abs ` M ) || N <-> ( abs ` M ) || ( abs ` N ) ) )
16	13, 14, 15	syl2an 494	. . . . . . . 8 |- ( ( ( abs ` M ) e. NN /\ ( N e. ZZ /\ N =/= 0 ) ) -> ( ( abs ` M ) || N <-> ( abs ` M ) || ( abs ` N ) ) )
17		nnsqcl 12933	. . . . . . . . . . 11 |- ( ( abs ` M ) e. NN -> ( ( abs ` M ) ^ 2 ) e. NN )
18	17	nnzd 11481	. . . . . . . . . 10 |- ( ( abs ` M ) e. NN -> ( ( abs ` M ) ^ 2 ) e. ZZ )
19		zsqcl 12934	. . . . . . . . . . 11 |- ( N e. ZZ -> ( N ^ 2 ) e. ZZ )
20	19	adantr 481	. . . . . . . . . 10 |- ( ( N e. ZZ /\ N =/= 0 ) -> ( N ^ 2 ) e. ZZ )
21		dvdsabsb 15001	. . . . . . . . . 10 |- ( ( ( ( abs ` M ) ^ 2 ) e. ZZ /\ ( N ^ 2 ) e. ZZ ) -> ( ( ( abs ` M ) ^ 2 ) || ( N ^ 2 ) <-> ( ( abs ` M ) ^ 2 ) || ( abs ` ( N ^ 2 ) ) ) )
22	18, 20, 21	syl2an 494	. . . . . . . . 9 |- ( ( ( abs ` M ) e. NN /\ ( N e. ZZ /\ N =/= 0 ) ) -> ( ( ( abs ` M ) ^ 2 ) || ( N ^ 2 ) <-> ( ( abs ` M ) ^ 2 ) || ( abs ` ( N ^ 2 ) ) ) )
23		zcn 11382	. . . . . . . . . . . . 13 |- ( N e. ZZ -> N e. CC )
24	23	adantr 481	. . . . . . . . . . . 12 |- ( ( N e. ZZ /\ N =/= 0 ) -> N e. CC )
25		abssq 14046	. . . . . . . . . . . 12 |- ( N e. CC -> ( ( abs ` N ) ^ 2 ) = ( abs ` ( N ^ 2 ) ) )
26	24, 25	syl 17	. . . . . . . . . . 11 |- ( ( N e. ZZ /\ N =/= 0 ) -> ( ( abs ` N ) ^ 2 ) = ( abs ` ( N ^ 2 ) ) )
27	26	breq2d 4665	. . . . . . . . . 10 |- ( ( N e. ZZ /\ N =/= 0 ) -> ( ( ( abs ` M ) ^ 2 ) || ( ( abs ` N ) ^ 2 ) <-> ( ( abs ` M ) ^ 2 ) || ( abs ` ( N ^ 2 ) ) ) )
28	27	adantl 482	. . . . . . . . 9 |- ( ( ( abs ` M ) e. NN /\ ( N e. ZZ /\ N =/= 0 ) ) -> ( ( ( abs ` M ) ^ 2 ) || ( ( abs ` N ) ^ 2 ) <-> ( ( abs ` M ) ^ 2 ) || ( abs ` ( N ^ 2 ) ) ) )
29	22, 28	bitr4d 271	. . . . . . . 8 |- ( ( ( abs ` M ) e. NN /\ ( N e. ZZ /\ N =/= 0 ) ) -> ( ( ( abs ` M ) ^ 2 ) || ( N ^ 2 ) <-> ( ( abs ` M ) ^ 2 ) || ( ( abs ` N ) ^ 2 ) ) )
30	12, 16, 29	3bitr4d 300	. . . . . . 7 |- ( ( ( abs ` M ) e. NN /\ ( N e. ZZ /\ N =/= 0 ) ) -> ( ( abs ` M ) || N <-> ( ( abs ` M ) ^ 2 ) || ( N ^ 2 ) ) )
31	30	anassrs 680	. . . . . 6 |- ( ( ( ( abs ` M ) e. NN /\ N e. ZZ ) /\ N =/= 0 ) -> ( ( abs ` M ) || N <-> ( ( abs ` M ) ^ 2 ) || ( N ^ 2 ) ) )
32		dvds0 14997	. . . . . . . . 9 |- ( ( abs ` M ) e. ZZ -> ( abs ` M ) || 0 )
33		zsqcl 12934	. . . . . . . . . 10 |- ( ( abs ` M ) e. ZZ -> ( ( abs ` M ) ^ 2 ) e. ZZ )
34		dvds0 14997	. . . . . . . . . 10 |- ( ( ( abs ` M ) ^ 2 ) e. ZZ -> ( ( abs ` M ) ^ 2 ) || 0 )
35	33, 34	syl 17	. . . . . . . . 9 |- ( ( abs ` M ) e. ZZ -> ( ( abs ` M ) ^ 2 ) || 0 )
36	32, 35	2thd 255	. . . . . . . 8 |- ( ( abs ` M ) e. ZZ -> ( ( abs ` M ) || 0 <-> ( ( abs ` M ) ^ 2 ) || 0 ) )
37	13, 36	syl 17	. . . . . . 7 |- ( ( abs ` M ) e. NN -> ( ( abs ` M ) || 0 <-> ( ( abs ` M ) ^ 2 ) || 0 ) )
38	37	adantr 481	. . . . . 6 |- ( ( ( abs ` M ) e. NN /\ N e. ZZ ) -> ( ( abs ` M ) || 0 <-> ( ( abs ` M ) ^ 2 ) || 0 ) )
39	9, 31, 38	pm2.61ne 2879	. . . . 5 |- ( ( ( abs ` M ) e. NN /\ N e. ZZ ) -> ( ( abs ` M ) || N <-> ( ( abs ` M ) ^ 2 ) || ( N ^ 2 ) ) )
40	5, 39	sylan 488	. . . 4 |- ( ( ( M e. ZZ /\ M =/= 0 ) /\ N e. ZZ ) -> ( ( abs ` M ) || N <-> ( ( abs ` M ) ^ 2 ) || ( N ^ 2 ) ) )
41		absdvdsb 15000	. . . . 5 |- ( ( M e. ZZ /\ N e. ZZ ) -> ( M || N <-> ( abs ` M ) || N ) )
42	41	adantlr 751	. . . 4 |- ( ( ( M e. ZZ /\ M =/= 0 ) /\ N e. ZZ ) -> ( M || N <-> ( abs ` M ) || N ) )
43		zsqcl 12934	. . . . . . 7 |- ( M e. ZZ -> ( M ^ 2 ) e. ZZ )
44	43	adantr 481	. . . . . 6 |- ( ( M e. ZZ /\ M =/= 0 ) -> ( M ^ 2 ) e. ZZ )
45		absdvdsb 15000	. . . . . 6 |- ( ( ( M ^ 2 ) e. ZZ /\ ( N ^ 2 ) e. ZZ ) -> ( ( M ^ 2 ) || ( N ^ 2 ) <-> ( abs ` ( M ^ 2 ) ) || ( N ^ 2 ) ) )
46	44, 19, 45	syl2an 494	. . . . 5 |- ( ( ( M e. ZZ /\ M =/= 0 ) /\ N e. ZZ ) -> ( ( M ^ 2 ) || ( N ^ 2 ) <-> ( abs ` ( M ^ 2 ) ) || ( N ^ 2 ) ) )
47		zcn 11382	. . . . . . . . . 10 |- ( M e. ZZ -> M e. CC )
48		abssq 14046	. . . . . . . . . 10 |- ( M e. CC -> ( ( abs ` M ) ^ 2 ) = ( abs ` ( M ^ 2 ) ) )
49	47, 48	syl 17	. . . . . . . . 9 |- ( M e. ZZ -> ( ( abs ` M ) ^ 2 ) = ( abs ` ( M ^ 2 ) ) )
50	49	eqcomd 2628	. . . . . . . 8 |- ( M e. ZZ -> ( abs ` ( M ^ 2 ) ) = ( ( abs ` M ) ^ 2 ) )
51	50	adantr 481	. . . . . . 7 |- ( ( M e. ZZ /\ M =/= 0 ) -> ( abs ` ( M ^ 2 ) ) = ( ( abs ` M ) ^ 2 ) )
52	51	breq1d 4663	. . . . . 6 |- ( ( M e. ZZ /\ M =/= 0 ) -> ( ( abs ` ( M ^ 2 ) ) || ( N ^ 2 ) <-> ( ( abs ` M ) ^ 2 ) || ( N ^ 2 ) ) )
53	52	adantr 481	. . . . 5 |- ( ( ( M e. ZZ /\ M =/= 0 ) /\ N e. ZZ ) -> ( ( abs ` ( M ^ 2 ) ) || ( N ^ 2 ) <-> ( ( abs ` M ) ^ 2 ) || ( N ^ 2 ) ) )
54	46, 53	bitrd 268	. . . 4 |- ( ( ( M e. ZZ /\ M =/= 0 ) /\ N e. ZZ ) -> ( ( M ^ 2 ) || ( N ^ 2 ) <-> ( ( abs ` M ) ^ 2 ) || ( N ^ 2 ) ) )
55	40, 42, 54	3bitr4d 300	. . 3 |- ( ( ( M e. ZZ /\ M =/= 0 ) /\ N e. ZZ ) -> ( M || N <-> ( M ^ 2 ) || ( N ^ 2 ) ) )
56	55	an32s 846	. 2 |- ( ( ( M e. ZZ /\ N e. ZZ ) /\ M =/= 0 ) -> ( M || N <-> ( M ^ 2 ) || ( N ^ 2 ) ) )
57		0dvds 15002	. . . . 5 |- ( N e. ZZ -> ( 0 || N <-> N = 0 ) )
58		sqeq0 12927	. . . . . 6 |- ( N e. CC -> ( ( N ^ 2 ) = 0 <-> N = 0 ) )
59	23, 58	syl 17	. . . . 5 |- ( N e. ZZ -> ( ( N ^ 2 ) = 0 <-> N = 0 ) )
60	57, 59	bitr4d 271	. . . 4 |- ( N e. ZZ -> ( 0 || N <-> ( N ^ 2 ) = 0 ) )
61		0dvds 15002	. . . . 5 |- ( ( N ^ 2 ) e. ZZ -> ( 0 || ( N ^ 2 ) <-> ( N ^ 2 ) = 0 ) )
62	19, 61	syl 17	. . . 4 |- ( N e. ZZ -> ( 0 || ( N ^ 2 ) <-> ( N ^ 2 ) = 0 ) )
63	60, 62	bitr4d 271	. . 3 |- ( N e. ZZ -> ( 0 || N <-> 0 || ( N ^ 2 ) ) )
64	63	adantl 482	. 2 |- ( ( M e. ZZ /\ N e. ZZ ) -> ( 0 || N <-> 0 || ( N ^ 2 ) ) )
65	4, 56, 64	pm2.61ne 2879	1 |- ( ( M e. ZZ /\ N e. ZZ ) -> ( M || N <-> ( M ^ 2 ) || ( N ^ 2 ) ) )

Syntax hints:   -> wi 4   <-> wb 196   /\ wa 384   = wceq 1483   e. wcel 1990   =/= wne 2794   class class class wbr 4653   ` cfv 5888  (class class class)co 6650   CCcc 9934   0cc0 9936   NNcn 11020   2c2 11070   ZZcz 11377   ^cexp 12860   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-inf 8349  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-fl 12593  df-mod 12669  df-seq 12802  df-exp 12861  df-cj 13839  df-re 13840  df-im 13841  df-sqrt 13975  df-abs 13976  df-dvds 14984  df-gcd 15217

This theorem is referenced by:  pythagtriplem19  15538  4sqlem9  15650  4sqlem10  15651  lgsdir  25057  2sqlem8a  25150