Theorem nn0gcdsq 12393

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 9268	. 2 |- ( A e. NN0 <-> ( A e. NN \/ A = 0 ) )
2		elnn0 9268	. 2 |- ( B e. NN0 <-> ( B e. NN \/ B = 0 ) )
3		sqgcd 12221	. . 3 |- ( ( A e. NN /\ B e. NN ) -> ( ( A gcd B ) ^ 2 ) = ( ( A ^ 2 ) gcd ( B ^ 2 ) ) )
4		nncn 9015	. . . . . . 7 |- ( B e. NN -> B e. CC )
5		abssq 11263	. . . . . . 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 9362	. . . . . . . 8 |- ( B e. NN -> B e. ZZ )
8		gcd0id 12171	. . . . . . . 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 5940	. . . . . 6 |- ( B e. NN -> ( ( 0 gcd B ) ^ 2 ) = ( ( abs ` B ) ^ 2 ) )
11		sq0 10739	. . . . . . . . 9 |- ( 0 ^ 2 ) = 0
12	11	a1i 9	. . . . . . . 8 |- ( B e. NN -> ( 0 ^ 2 ) = 0 )
13	12	oveq1d 5940	. . . . . . 7 |- ( B e. NN -> ( ( 0 ^ 2 ) gcd ( B ^ 2 ) ) = ( 0 gcd ( B ^ 2 ) ) )
14		zsqcl 10719	. . . . . . . 8 |- ( B e. ZZ -> ( B ^ 2 ) e. ZZ )
15		gcd0id 12171	. . . . . . . 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 2229	. . . . . 6 |- ( B e. NN -> ( ( 0 ^ 2 ) gcd ( B ^ 2 ) ) = ( abs ` ( B ^ 2 ) ) )
18	6, 10, 17	3eqtr4d 2239	. . . . 5 |- ( B e. NN -> ( ( 0 gcd B ) ^ 2 ) = ( ( 0 ^ 2 ) gcd ( B ^ 2 ) ) )
19	18	adantl 277	. . . 4 |- ( ( A = 0 /\ B e. NN ) -> ( ( 0 gcd B ) ^ 2 ) = ( ( 0 ^ 2 ) gcd ( B ^ 2 ) ) )
20		oveq1 5932	. . . . . . 7 |- ( A = 0 -> ( A gcd B ) = ( 0 gcd B ) )
21	20	oveq1d 5940	. . . . . 6 |- ( A = 0 -> ( ( A gcd B ) ^ 2 ) = ( ( 0 gcd B ) ^ 2 ) )
22		oveq1 5932	. . . . . . 7 |- ( A = 0 -> ( A ^ 2 ) = ( 0 ^ 2 ) )
23	22	oveq1d 5940	. . . . . 6 |- ( A = 0 -> ( ( A ^ 2 ) gcd ( B ^ 2 ) ) = ( ( 0 ^ 2 ) gcd ( B ^ 2 ) ) )
24	21, 23	eqeq12d 2211	. . . . 5 |- ( A = 0 -> ( ( ( A gcd B ) ^ 2 ) = ( ( A ^ 2 ) gcd ( B ^ 2 ) ) <-> ( ( 0 gcd B ) ^ 2 ) = ( ( 0 ^ 2 ) gcd ( B ^ 2 ) ) ) )
25	24	adantr 276	. . . 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 167	. . 3 |- ( ( A = 0 /\ B e. NN ) -> ( ( A gcd B ) ^ 2 ) = ( ( A ^ 2 ) gcd ( B ^ 2 ) ) )
27		nncn 9015	. . . . . . 7 |- ( A e. NN -> A e. CC )
28		abssq 11263	. . . . . . 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 9362	. . . . . . . 8 |- ( A e. NN -> A e. ZZ )
31		gcdid0 12172	. . . . . . . 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 5940	. . . . . 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 5941	. . . . . . 7 |- ( A e. NN -> ( ( A ^ 2 ) gcd ( 0 ^ 2 ) ) = ( ( A ^ 2 ) gcd 0 ) )
36		zsqcl 10719	. . . . . . . 8 |- ( A e. ZZ -> ( A ^ 2 ) e. ZZ )
37		gcdid0 12172	. . . . . . . 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 2229	. . . . . 6 |- ( A e. NN -> ( ( A ^ 2 ) gcd ( 0 ^ 2 ) ) = ( abs ` ( A ^ 2 ) ) )
40	29, 33, 39	3eqtr4d 2239	. . . . 5 |- ( A e. NN -> ( ( A gcd 0 ) ^ 2 ) = ( ( A ^ 2 ) gcd ( 0 ^ 2 ) ) )
41	40	adantr 276	. . . 4 |- ( ( A e. NN /\ B = 0 ) -> ( ( A gcd 0 ) ^ 2 ) = ( ( A ^ 2 ) gcd ( 0 ^ 2 ) ) )
42		oveq2 5933	. . . . . . 7 |- ( B = 0 -> ( A gcd B ) = ( A gcd 0 ) )
43	42	oveq1d 5940	. . . . . 6 |- ( B = 0 -> ( ( A gcd B ) ^ 2 ) = ( ( A gcd 0 ) ^ 2 ) )
44		oveq1 5932	. . . . . . 7 |- ( B = 0 -> ( B ^ 2 ) = ( 0 ^ 2 ) )
45	44	oveq2d 5941	. . . . . 6 |- ( B = 0 -> ( ( A ^ 2 ) gcd ( B ^ 2 ) ) = ( ( A ^ 2 ) gcd ( 0 ^ 2 ) ) )
46	43, 45	eqeq12d 2211	. . . . 5 |- ( B = 0 -> ( ( ( A gcd B ) ^ 2 ) = ( ( A ^ 2 ) gcd ( B ^ 2 ) ) <-> ( ( A gcd 0 ) ^ 2 ) = ( ( A ^ 2 ) gcd ( 0 ^ 2 ) ) ) )
47	46	adantl 277	. . . 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 167	. . 3 |- ( ( A e. NN /\ B = 0 ) -> ( ( A gcd B ) ^ 2 ) = ( ( A ^ 2 ) gcd ( B ^ 2 ) ) )
49		gcd0val 12152	. . . . . 6 |- ( 0 gcd 0 ) = 0
50	49	oveq1i 5935	. . . . 5 |- ( ( 0 gcd 0 ) ^ 2 ) = ( 0 ^ 2 )
51	11, 11	oveq12i 5937	. . . . . 6 |- ( ( 0 ^ 2 ) gcd ( 0 ^ 2 ) ) = ( 0 gcd 0 )
52	51, 49	eqtri 2217	. . . . 5 |- ( ( 0 ^ 2 ) gcd ( 0 ^ 2 ) ) = 0
53	11, 50, 52	3eqtr4i 2227	. . . 4 |- ( ( 0 gcd 0 ) ^ 2 ) = ( ( 0 ^ 2 ) gcd ( 0 ^ 2 ) )
54		oveq12 5934	. . . . 5 |- ( ( A = 0 /\ B = 0 ) -> ( A gcd B ) = ( 0 gcd 0 ) )
55	54	oveq1d 5940	. . . 4 |- ( ( A = 0 /\ B = 0 ) -> ( ( A gcd B ) ^ 2 ) = ( ( 0 gcd 0 ) ^ 2 ) )
56	22, 44	oveqan12d 5944	. . . 4 |- ( ( A = 0 /\ B = 0 ) -> ( ( A ^ 2 ) gcd ( B ^ 2 ) ) = ( ( 0 ^ 2 ) gcd ( 0 ^ 2 ) ) )
57	53, 55, 56	3eqtr4a 2255	. . 3 |- ( ( A = 0 /\ B = 0 ) -> ( ( A gcd B ) ^ 2 ) = ( ( A ^ 2 ) gcd ( B ^ 2 ) ) )
58	3, 26, 48, 57	ccase 966	. 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 291	1 |- ( ( A e. NN0 /\ B e. NN0 ) -> ( ( A gcd B ) ^ 2 ) = ( ( A ^ 2 ) gcd ( B ^ 2 ) ) )

Syntax hints:   -> wi 4   /\ wa 104   <-> wb 105   \/ wo 709   = wceq 1364   e. wcel 2167   ` cfv 5259  (class class class)co 5925   CCcc 7894   0cc0 7896   NNcn 9007   2c2 9058   NN0cn0 9266   ZZcz 9343   ^cexp 10647   abscabs 11179   gcd cgcd 12145

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 710  ax-5 1461  ax-7 1462  ax-gen 1463  ax-ie1 1507  ax-ie2 1508  ax-8 1518  ax-10 1519  ax-11 1520  ax-i12 1521  ax-bndl 1523  ax-4 1524  ax-17 1540  ax-i9 1544  ax-ial 1548  ax-i5r 1549  ax-13 2169  ax-14 2170  ax-ext 2178  ax-coll 4149  ax-sep 4152  ax-nul 4160  ax-pow 4208  ax-pr 4243  ax-un 4469  ax-setind 4574  ax-iinf 4625  ax-cnex 7987  ax-resscn 7988  ax-1cn 7989  ax-1re 7990  ax-icn 7991  ax-addcl 7992  ax-addrcl 7993  ax-mulcl 7994  ax-mulrcl 7995  ax-addcom 7996  ax-mulcom 7997  ax-addass 7998  ax-mulass 7999  ax-distr 8000  ax-i2m1 8001  ax-0lt1 8002  ax-1rid 8003  ax-0id 8004  ax-rnegex 8005  ax-precex 8006  ax-cnre 8007  ax-pre-ltirr 8008  ax-pre-ltwlin 8009  ax-pre-lttrn 8010  ax-pre-apti 8011  ax-pre-ltadd 8012  ax-pre-mulgt0 8013  ax-pre-mulext 8014  ax-arch 8015  ax-caucvg 8016

This theorem depends on definitions:  df-bi 117  df-stab 832  df-dc 836  df-3or 981  df-3an 982  df-tru 1367  df-fal 1370  df-nf 1475  df-sb 1777  df-eu 2048  df-mo 2049  df-clab 2183  df-cleq 2189  df-clel 2192  df-nfc 2328  df-ne 2368  df-nel 2463  df-ral 2480  df-rex 2481  df-reu 2482  df-rmo 2483  df-rab 2484  df-v 2765  df-sbc 2990  df-csb 3085  df-dif 3159  df-un 3161  df-in 3163  df-ss 3170  df-nul 3452  df-if 3563  df-pw 3608  df-sn 3629  df-pr 3630  df-op 3632  df-uni 3841  df-int 3876  df-iun 3919  df-br 4035  df-opab 4096  df-mpt 4097  df-tr 4133  df-id 4329  df-po 4332  df-iso 4333  df-iord 4402  df-on 4404  df-ilim 4405  df-suc 4407  df-iom 4628  df-xp 4670  df-rel 4671  df-cnv 4672  df-co 4673  df-dm 4674  df-rn 4675  df-res 4676  df-ima 4677  df-iota 5220  df-fun 5261  df-fn 5262  df-f 5263  df-f1 5264  df-fo 5265  df-f1o 5266  df-fv 5267  df-riota 5880  df-ov 5928  df-oprab 5929  df-mpo 5930  df-1st 6207  df-2nd 6208  df-recs 6372  df-frec 6458  df-sup 7059  df-pnf 8080  df-mnf 8081  df-xr 8082  df-ltxr 8083  df-le 8084  df-sub 8216  df-neg 8217  df-reap 8619  df-ap 8626  df-div 8717  df-inn 9008  df-2 9066  df-3 9067  df-4 9068  df-n0 9267  df-z 9344  df-uz 9619  df-q 9711  df-rp 9746  df-fz 10101  df-fzo 10235  df-fl 10377  df-mod 10432  df-seqfrec 10557  df-exp 10648  df-cj 11024  df-re 11025  df-im 11026  df-rsqrt 11180  df-abs 11181  df-dvds 11970  df-gcd 12146

This theorem is referenced by:  zgcdsq  12394