Theorem nn0gcdsq 12202

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 9180	. 2 |- ( A e. NN0 <-> ( A e. NN \/ A = 0 ) )
2		elnn0 9180	. 2 |- ( B e. NN0 <-> ( B e. NN \/ B = 0 ) )
3		sqgcd 12032	. . 3 |- ( ( A e. NN /\ B e. NN ) -> ( ( A gcd B ) ^ 2 ) = ( ( A ^ 2 ) gcd ( B ^ 2 ) ) )
4		nncn 8929	. . . . . . 7 |- ( B e. NN -> B e. CC )
5		abssq 11092	. . . . . . 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 9274	. . . . . . . 8 |- ( B e. NN -> B e. ZZ )
8		gcd0id 11982	. . . . . . . 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 5892	. . . . . 6 |- ( B e. NN -> ( ( 0 gcd B ) ^ 2 ) = ( ( abs ` B ) ^ 2 ) )
11		sq0 10613	. . . . . . . . 9 |- ( 0 ^ 2 ) = 0
12	11	a1i 9	. . . . . . . 8 |- ( B e. NN -> ( 0 ^ 2 ) = 0 )
13	12	oveq1d 5892	. . . . . . 7 |- ( B e. NN -> ( ( 0 ^ 2 ) gcd ( B ^ 2 ) ) = ( 0 gcd ( B ^ 2 ) ) )
14		zsqcl 10593	. . . . . . . 8 |- ( B e. ZZ -> ( B ^ 2 ) e. ZZ )
15		gcd0id 11982	. . . . . . . 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 2210	. . . . . 6 |- ( B e. NN -> ( ( 0 ^ 2 ) gcd ( B ^ 2 ) ) = ( abs ` ( B ^ 2 ) ) )
18	6, 10, 17	3eqtr4d 2220	. . . . 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 5884	. . . . . . 7 |- ( A = 0 -> ( A gcd B ) = ( 0 gcd B ) )
21	20	oveq1d 5892	. . . . . 6 |- ( A = 0 -> ( ( A gcd B ) ^ 2 ) = ( ( 0 gcd B ) ^ 2 ) )
22		oveq1 5884	. . . . . . 7 |- ( A = 0 -> ( A ^ 2 ) = ( 0 ^ 2 ) )
23	22	oveq1d 5892	. . . . . 6 |- ( A = 0 -> ( ( A ^ 2 ) gcd ( B ^ 2 ) ) = ( ( 0 ^ 2 ) gcd ( B ^ 2 ) ) )
24	21, 23	eqeq12d 2192	. . . . 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 8929	. . . . . . 7 |- ( A e. NN -> A e. CC )
28		abssq 11092	. . . . . . 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 9274	. . . . . . . 8 |- ( A e. NN -> A e. ZZ )
31		gcdid0 11983	. . . . . . . 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 5892	. . . . . 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 5893	. . . . . . 7 |- ( A e. NN -> ( ( A ^ 2 ) gcd ( 0 ^ 2 ) ) = ( ( A ^ 2 ) gcd 0 ) )
36		zsqcl 10593	. . . . . . . 8 |- ( A e. ZZ -> ( A ^ 2 ) e. ZZ )
37		gcdid0 11983	. . . . . . . 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 2210	. . . . . 6 |- ( A e. NN -> ( ( A ^ 2 ) gcd ( 0 ^ 2 ) ) = ( abs ` ( A ^ 2 ) ) )
40	29, 33, 39	3eqtr4d 2220	. . . . 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 5885	. . . . . . 7 |- ( B = 0 -> ( A gcd B ) = ( A gcd 0 ) )
43	42	oveq1d 5892	. . . . . 6 |- ( B = 0 -> ( ( A gcd B ) ^ 2 ) = ( ( A gcd 0 ) ^ 2 ) )
44		oveq1 5884	. . . . . . 7 |- ( B = 0 -> ( B ^ 2 ) = ( 0 ^ 2 ) )
45	44	oveq2d 5893	. . . . . 6 |- ( B = 0 -> ( ( A ^ 2 ) gcd ( B ^ 2 ) ) = ( ( A ^ 2 ) gcd ( 0 ^ 2 ) ) )
46	43, 45	eqeq12d 2192	. . . . 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 11963	. . . . . 6 |- ( 0 gcd 0 ) = 0
50	49	oveq1i 5887	. . . . 5 |- ( ( 0 gcd 0 ) ^ 2 ) = ( 0 ^ 2 )
51	11, 11	oveq12i 5889	. . . . . 6 |- ( ( 0 ^ 2 ) gcd ( 0 ^ 2 ) ) = ( 0 gcd 0 )
52	51, 49	eqtri 2198	. . . . 5 |- ( ( 0 ^ 2 ) gcd ( 0 ^ 2 ) ) = 0
53	11, 50, 52	3eqtr4i 2208	. . . 4 |- ( ( 0 gcd 0 ) ^ 2 ) = ( ( 0 ^ 2 ) gcd ( 0 ^ 2 ) )
54		oveq12 5886	. . . . 5 |- ( ( A = 0 /\ B = 0 ) -> ( A gcd B ) = ( 0 gcd 0 ) )
55	54	oveq1d 5892	. . . 4 |- ( ( A = 0 /\ B = 0 ) -> ( ( A gcd B ) ^ 2 ) = ( ( 0 gcd 0 ) ^ 2 ) )
56	22, 44	oveqan12d 5896	. . . 4 |- ( ( A = 0 /\ B = 0 ) -> ( ( A ^ 2 ) gcd ( B ^ 2 ) ) = ( ( 0 ^ 2 ) gcd ( 0 ^ 2 ) ) )
57	53, 55, 56	3eqtr4a 2236	. . 3 |- ( ( A = 0 /\ B = 0 ) -> ( ( A gcd B ) ^ 2 ) = ( ( A ^ 2 ) gcd ( B ^ 2 ) ) )
58	3, 26, 48, 57	ccase 964	. 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 708   = wceq 1353   e. wcel 2148   ` cfv 5218  (class class class)co 5877   CCcc 7811   0cc0 7813   NNcn 8921   2c2 8972   NN0cn0 9178   ZZcz 9255   ^cexp 10521   abscabs 11008   gcd cgcd 11945

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 614  ax-in2 615  ax-io 709  ax-5 1447  ax-7 1448  ax-gen 1449  ax-ie1 1493  ax-ie2 1494  ax-8 1504  ax-10 1505  ax-11 1506  ax-i12 1507  ax-bndl 1509  ax-4 1510  ax-17 1526  ax-i9 1530  ax-ial 1534  ax-i5r 1535  ax-13 2150  ax-14 2151  ax-ext 2159  ax-coll 4120  ax-sep 4123  ax-nul 4131  ax-pow 4176  ax-pr 4211  ax-un 4435  ax-setind 4538  ax-iinf 4589  ax-cnex 7904  ax-resscn 7905  ax-1cn 7906  ax-1re 7907  ax-icn 7908  ax-addcl 7909  ax-addrcl 7910  ax-mulcl 7911  ax-mulrcl 7912  ax-addcom 7913  ax-mulcom 7914  ax-addass 7915  ax-mulass 7916  ax-distr 7917  ax-i2m1 7918  ax-0lt1 7919  ax-1rid 7920  ax-0id 7921  ax-rnegex 7922  ax-precex 7923  ax-cnre 7924  ax-pre-ltirr 7925  ax-pre-ltwlin 7926  ax-pre-lttrn 7927  ax-pre-apti 7928  ax-pre-ltadd 7929  ax-pre-mulgt0 7930  ax-pre-mulext 7931  ax-arch 7932  ax-caucvg 7933

This theorem depends on definitions:  df-bi 117  df-stab 831  df-dc 835  df-3or 979  df-3an 980  df-tru 1356  df-fal 1359  df-nf 1461  df-sb 1763  df-eu 2029  df-mo 2030  df-clab 2164  df-cleq 2170  df-clel 2173  df-nfc 2308  df-ne 2348  df-nel 2443  df-ral 2460  df-rex 2461  df-reu 2462  df-rmo 2463  df-rab 2464  df-v 2741  df-sbc 2965  df-csb 3060  df-dif 3133  df-un 3135  df-in 3137  df-ss 3144  df-nul 3425  df-if 3537  df-pw 3579  df-sn 3600  df-pr 3601  df-op 3603  df-uni 3812  df-int 3847  df-iun 3890  df-br 4006  df-opab 4067  df-mpt 4068  df-tr 4104  df-id 4295  df-po 4298  df-iso 4299  df-iord 4368  df-on 4370  df-ilim 4371  df-suc 4373  df-iom 4592  df-xp 4634  df-rel 4635  df-cnv 4636  df-co 4637  df-dm 4638  df-rn 4639  df-res 4640  df-ima 4641  df-iota 5180  df-fun 5220  df-fn 5221  df-f 5222  df-f1 5223  df-fo 5224  df-f1o 5225  df-fv 5226  df-riota 5833  df-ov 5880  df-oprab 5881  df-mpo 5882  df-1st 6143  df-2nd 6144  df-recs 6308  df-frec 6394  df-sup 6985  df-pnf 7996  df-mnf 7997  df-xr 7998  df-ltxr 7999  df-le 8000  df-sub 8132  df-neg 8133  df-reap 8534  df-ap 8541  df-div 8632  df-inn 8922  df-2 8980  df-3 8981  df-4 8982  df-n0 9179  df-z 9256  df-uz 9531  df-q 9622  df-rp 9656  df-fz 10011  df-fzo 10145  df-fl 10272  df-mod 10325  df-seqfrec 10448  df-exp 10522  df-cj 10853  df-re 10854  df-im 10855  df-rsqrt 11009  df-abs 11010  df-dvds 11797  df-gcd 11946

This theorem is referenced by:  zgcdsq  12203