Theorem nn0gcdsq 12132

Description: Squaring commutes with GCD, in particular two coprime numbers have coprime squares. (Contributed by Stefan O'Rear, 15-Sep-2014.)

Assertion

Ref	Expression
nn0gcdsq	|- ( ( A e. NN0 /\ B e. NN0 ) -> ( ( A gcd B ) ^ 2 ) = ( ( A ^ 2 ) gcd ( B ^ 2 ) ) )

Proof of Theorem nn0gcdsq

Step	Hyp	Ref	Expression
1		elnn0 9116	. 2 |- ( A e. NN0 <-> ( A e. NN \/ A = 0 ) )
2		elnn0 9116	. 2 |- ( B e. NN0 <-> ( B e. NN \/ B = 0 ) )
3		sqgcd 11962	. . 3 |- ( ( A e. NN /\ B e. NN ) -> ( ( A gcd B ) ^ 2 ) = ( ( A ^ 2 ) gcd ( B ^ 2 ) ) )
4		nncn 8865	. . . . . . 7 |- ( B e. NN -> B e. CC )
5		abssq 11023	. . . . . . 7 |- ( B e. CC -> ( ( abs ` B ) ^ 2 ) = ( abs ` ( B ^ 2 ) ) )
6	4, 5	syl 14	. . . . . 6 |- ( B e. NN -> ( ( abs ` B ) ^ 2 ) = ( abs ` ( B ^ 2 ) ) )
7		nnz 9210	. . . . . . . 8 |- ( B e. NN -> B e. ZZ )
8		gcd0id 11912	. . . . . . . 8 |- ( B e. ZZ -> ( 0 gcd B ) = ( abs ` B ) )
9	7, 8	syl 14	. . . . . . 7 |- ( B e. NN -> ( 0 gcd B ) = ( abs ` B ) )
10	9	oveq1d 5857	. . . . . 6 |- ( B e. NN -> ( ( 0 gcd B ) ^ 2 ) = ( ( abs ` B ) ^ 2 ) )
11		sq0 10545	. . . . . . . . 9 |- ( 0 ^ 2 ) = 0
12	11	a1i 9	. . . . . . . 8 |- ( B e. NN -> ( 0 ^ 2 ) = 0 )
13	12	oveq1d 5857	. . . . . . 7 |- ( B e. NN -> ( ( 0 ^ 2 ) gcd ( B ^ 2 ) ) = ( 0 gcd ( B ^ 2 ) ) )
14		zsqcl 10525	. . . . . . . 8 |- ( B e. ZZ -> ( B ^ 2 ) e. ZZ )
15		gcd0id 11912	. . . . . . . 8 |- ( ( B ^ 2 ) e. ZZ -> ( 0 gcd ( B ^ 2 ) ) = ( abs ` ( B ^ 2 ) ) )
16	7, 14, 15	3syl 17	. . . . . . 7 |- ( B e. NN -> ( 0 gcd ( B ^ 2 ) ) = ( abs ` ( B ^ 2 ) ) )
17	13, 16	eqtrd 2198	. . . . . 6 |- ( B e. NN -> ( ( 0 ^ 2 ) gcd ( B ^ 2 ) ) = ( abs ` ( B ^ 2 ) ) )
18	6, 10, 17	3eqtr4d 2208	. . . . 5 |- ( B e. NN -> ( ( 0 gcd B ) ^ 2 ) = ( ( 0 ^ 2 ) gcd ( B ^ 2 ) ) )
19	18	adantl 275	. . . 4 |- ( ( A = 0 /\ B e. NN ) -> ( ( 0 gcd B ) ^ 2 ) = ( ( 0 ^ 2 ) gcd ( B ^ 2 ) ) )
20		oveq1 5849	. . . . . . 7 |- ( A = 0 -> ( A gcd B ) = ( 0 gcd B ) )
21	20	oveq1d 5857	. . . . . 6 |- ( A = 0 -> ( ( A gcd B ) ^ 2 ) = ( ( 0 gcd B ) ^ 2 ) )
22		oveq1 5849	. . . . . . 7 |- ( A = 0 -> ( A ^ 2 ) = ( 0 ^ 2 ) )
23	22	oveq1d 5857	. . . . . 6 |- ( A = 0 -> ( ( A ^ 2 ) gcd ( B ^ 2 ) ) = ( ( 0 ^ 2 ) gcd ( B ^ 2 ) ) )
24	21, 23	eqeq12d 2180	. . . . 5 |- ( A = 0 -> ( ( ( A gcd B ) ^ 2 ) = ( ( A ^ 2 ) gcd ( B ^ 2 ) ) <-> ( ( 0 gcd B ) ^ 2 ) = ( ( 0 ^ 2 ) gcd ( B ^ 2 ) ) ) )
25	24	adantr 274	. . . 4 |- ( ( A = 0 /\ B e. NN ) -> ( ( ( A gcd B ) ^ 2 ) = ( ( A ^ 2 ) gcd ( B ^ 2 ) ) <-> ( ( 0 gcd B ) ^ 2 ) = ( ( 0 ^ 2 ) gcd ( B ^ 2 ) ) ) )
26	19, 25	mpbird 166	. . 3 |- ( ( A = 0 /\ B e. NN ) -> ( ( A gcd B ) ^ 2 ) = ( ( A ^ 2 ) gcd ( B ^ 2 ) ) )
27		nncn 8865	. . . . . . 7 |- ( A e. NN -> A e. CC )
28		abssq 11023	. . . . . . 7 |- ( A e. CC -> ( ( abs ` A ) ^ 2 ) = ( abs ` ( A ^ 2 ) ) )
29	27, 28	syl 14	. . . . . 6 |- ( A e. NN -> ( ( abs ` A ) ^ 2 ) = ( abs ` ( A ^ 2 ) ) )
30		nnz 9210	. . . . . . . 8 |- ( A e. NN -> A e. ZZ )
31		gcdid0 11913	. . . . . . . 8 |- ( A e. ZZ -> ( A gcd 0 ) = ( abs ` A ) )
32	30, 31	syl 14	. . . . . . 7 |- ( A e. NN -> ( A gcd 0 ) = ( abs ` A ) )
33	32	oveq1d 5857	. . . . . 6 |- ( A e. NN -> ( ( A gcd 0 ) ^ 2 ) = ( ( abs ` A ) ^ 2 ) )
34	11	a1i 9	. . . . . . . 8 |- ( A e. NN -> ( 0 ^ 2 ) = 0 )
35	34	oveq2d 5858	. . . . . . 7 |- ( A e. NN -> ( ( A ^ 2 ) gcd ( 0 ^ 2 ) ) = ( ( A ^ 2 ) gcd 0 ) )
36		zsqcl 10525	. . . . . . . 8 |- ( A e. ZZ -> ( A ^ 2 ) e. ZZ )
37		gcdid0 11913	. . . . . . . 8 |- ( ( A ^ 2 ) e. ZZ -> ( ( A ^ 2 ) gcd 0 ) = ( abs ` ( A ^ 2 ) ) )
38	30, 36, 37	3syl 17	. . . . . . 7 |- ( A e. NN -> ( ( A ^ 2 ) gcd 0 ) = ( abs ` ( A ^ 2 ) ) )
39	35, 38	eqtrd 2198	. . . . . 6 |- ( A e. NN -> ( ( A ^ 2 ) gcd ( 0 ^ 2 ) ) = ( abs ` ( A ^ 2 ) ) )
40	29, 33, 39	3eqtr4d 2208	. . . . 5 |- ( A e. NN -> ( ( A gcd 0 ) ^ 2 ) = ( ( A ^ 2 ) gcd ( 0 ^ 2 ) ) )
41	40	adantr 274	. . . 4 |- ( ( A e. NN /\ B = 0 ) -> ( ( A gcd 0 ) ^ 2 ) = ( ( A ^ 2 ) gcd ( 0 ^ 2 ) ) )
42		oveq2 5850	. . . . . . 7 |- ( B = 0 -> ( A gcd B ) = ( A gcd 0 ) )
43	42	oveq1d 5857	. . . . . 6 |- ( B = 0 -> ( ( A gcd B ) ^ 2 ) = ( ( A gcd 0 ) ^ 2 ) )
44		oveq1 5849	. . . . . . 7 |- ( B = 0 -> ( B ^ 2 ) = ( 0 ^ 2 ) )
45	44	oveq2d 5858	. . . . . 6 |- ( B = 0 -> ( ( A ^ 2 ) gcd ( B ^ 2 ) ) = ( ( A ^ 2 ) gcd ( 0 ^ 2 ) ) )
46	43, 45	eqeq12d 2180	. . . . 5 |- ( B = 0 -> ( ( ( A gcd B ) ^ 2 ) = ( ( A ^ 2 ) gcd ( B ^ 2 ) ) <-> ( ( A gcd 0 ) ^ 2 ) = ( ( A ^ 2 ) gcd ( 0 ^ 2 ) ) ) )
47	46	adantl 275	. . . 4 |- ( ( A e. NN /\ B = 0 ) -> ( ( ( A gcd B ) ^ 2 ) = ( ( A ^ 2 ) gcd ( B ^ 2 ) ) <-> ( ( A gcd 0 ) ^ 2 ) = ( ( A ^ 2 ) gcd ( 0 ^ 2 ) ) ) )
48	41, 47	mpbird 166	. . 3 |- ( ( A e. NN /\ B = 0 ) -> ( ( A gcd B ) ^ 2 ) = ( ( A ^ 2 ) gcd ( B ^ 2 ) ) )
49		gcd0val 11893	. . . . . 6 |- ( 0 gcd 0 ) = 0
50	49	oveq1i 5852	. . . . 5 |- ( ( 0 gcd 0 ) ^ 2 ) = ( 0 ^ 2 )
51	11, 11	oveq12i 5854	. . . . . 6 |- ( ( 0 ^ 2 ) gcd ( 0 ^ 2 ) ) = ( 0 gcd 0 )
52	51, 49	eqtri 2186	. . . . 5 |- ( ( 0 ^ 2 ) gcd ( 0 ^ 2 ) ) = 0
53	11, 50, 52	3eqtr4i 2196	. . . 4 |- ( ( 0 gcd 0 ) ^ 2 ) = ( ( 0 ^ 2 ) gcd ( 0 ^ 2 ) )
54		oveq12 5851	. . . . 5 |- ( ( A = 0 /\ B = 0 ) -> ( A gcd B ) = ( 0 gcd 0 ) )
55	54	oveq1d 5857	. . . 4 |- ( ( A = 0 /\ B = 0 ) -> ( ( A gcd B ) ^ 2 ) = ( ( 0 gcd 0 ) ^ 2 ) )
56	22, 44	oveqan12d 5861	. . . 4 |- ( ( A = 0 /\ B = 0 ) -> ( ( A ^ 2 ) gcd ( B ^ 2 ) ) = ( ( 0 ^ 2 ) gcd ( 0 ^ 2 ) ) )
57	53, 55, 56	3eqtr4a 2225	. . 3 |- ( ( A = 0 /\ B = 0 ) -> ( ( A gcd B ) ^ 2 ) = ( ( A ^ 2 ) gcd ( B ^ 2 ) ) )
58	3, 26, 48, 57	ccase 954	. 2 |- ( ( ( A e. NN \/ A = 0 ) /\ ( B e. NN \/ B = 0 ) ) -> ( ( A gcd B ) ^ 2 ) = ( ( A ^ 2 ) gcd ( B ^ 2 ) ) )
59	1, 2, 58	syl2anb 289	1 |- ( ( A e. NN0 /\ B e. NN0 ) -> ( ( A gcd B ) ^ 2 ) = ( ( A ^ 2 ) gcd ( B ^ 2 ) ) )

Syntax hints:   -> wi 4   /\ wa 103   <-> wb 104   \/ wo 698   = wceq 1343   e. wcel 2136   ` cfv 5188  (class class class)co 5842   CCcc 7751   0cc0 7753   NNcn 8857   2c2 8908   NN0cn0 9114   ZZcz 9191   ^cexp 10454   abscabs 10939   gcd cgcd 11875

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-in1 604  ax-in2 605  ax-io 699  ax-5 1435  ax-7 1436  ax-gen 1437  ax-ie1 1481  ax-ie2 1482  ax-8 1492  ax-10 1493  ax-11 1494  ax-i12 1495  ax-bndl 1497  ax-4 1498  ax-17 1514  ax-i9 1518  ax-ial 1522  ax-i5r 1523  ax-13 2138  ax-14 2139  ax-ext 2147  ax-coll 4097  ax-sep 4100  ax-nul 4108  ax-pow 4153  ax-pr 4187  ax-un 4411  ax-setind 4514  ax-iinf 4565  ax-cnex 7844  ax-resscn 7845  ax-1cn 7846  ax-1re 7847  ax-icn 7848  ax-addcl 7849  ax-addrcl 7850  ax-mulcl 7851  ax-mulrcl 7852  ax-addcom 7853  ax-mulcom 7854  ax-addass 7855  ax-mulass 7856  ax-distr 7857  ax-i2m1 7858  ax-0lt1 7859  ax-1rid 7860  ax-0id 7861  ax-rnegex 7862  ax-precex 7863  ax-cnre 7864  ax-pre-ltirr 7865  ax-pre-ltwlin 7866  ax-pre-lttrn 7867  ax-pre-apti 7868  ax-pre-ltadd 7869  ax-pre-mulgt0 7870  ax-pre-mulext 7871  ax-arch 7872  ax-caucvg 7873

This theorem depends on definitions:  df-bi 116  df-stab 821  df-dc 825  df-3or 969  df-3an 970  df-tru 1346  df-fal 1349  df-nf 1449  df-sb 1751  df-eu 2017  df-mo 2018  df-clab 2152  df-cleq 2158  df-clel 2161  df-nfc 2297  df-ne 2337  df-nel 2432  df-ral 2449  df-rex 2450  df-reu 2451  df-rmo 2452  df-rab 2453  df-v 2728  df-sbc 2952  df-csb 3046  df-dif 3118  df-un 3120  df-in 3122  df-ss 3129  df-nul 3410  df-if 3521  df-pw 3561  df-sn 3582  df-pr 3583  df-op 3585  df-uni 3790  df-int 3825  df-iun 3868  df-br 3983  df-opab 4044  df-mpt 4045  df-tr 4081  df-id 4271  df-po 4274  df-iso 4275  df-iord 4344  df-on 4346  df-ilim 4347  df-suc 4349  df-iom 4568  df-xp 4610  df-rel 4611  df-cnv 4612  df-co 4613  df-dm 4614  df-rn 4615  df-res 4616  df-ima 4617  df-iota 5153  df-fun 5190  df-fn 5191  df-f 5192  df-f1 5193  df-fo 5194  df-f1o 5195  df-fv 5196  df-riota 5798  df-ov 5845  df-oprab 5846  df-mpo 5847  df-1st 6108  df-2nd 6109  df-recs 6273  df-frec 6359  df-sup 6949  df-pnf 7935  df-mnf 7936  df-xr 7937  df-ltxr 7938  df-le 7939  df-sub 8071  df-neg 8072  df-reap 8473  df-ap 8480  df-div 8569  df-inn 8858  df-2 8916  df-3 8917  df-4 8918  df-n0 9115  df-z 9192  df-uz 9467  df-q 9558  df-rp 9590  df-fz 9945  df-fzo 10078  df-fl 10205  df-mod 10258  df-seqfrec 10381  df-exp 10455  df-cj 10784  df-re 10785  df-im 10786  df-rsqrt 10940  df-abs 10941  df-dvds 11728  df-gcd 11876

This theorem is referenced by:  zgcdsq  12133