Theorem nn0gcdsq 12771

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 9403	. 2 |- ( A e. NN0 <-> ( A e. NN \/ A = 0 ) )
2		elnn0 9403	. 2 |- ( B e. NN0 <-> ( B e. NN \/ B = 0 ) )
3		sqgcd 12599	. . 3 |- ( ( A e. NN /\ B e. NN ) -> ( ( A gcd B ) ^ 2 ) = ( ( A ^ 2 ) gcd ( B ^ 2 ) ) )
4		nncn 9150	. . . . . . 7 |- ( B e. NN -> B e. CC )
5		abssq 11641	. . . . . . 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 9497	. . . . . . . 8 |- ( B e. NN -> B e. ZZ )
8		gcd0id 12549	. . . . . . . 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 6032	. . . . . 6 |- ( B e. NN -> ( ( 0 gcd B ) ^ 2 ) = ( ( abs ` B ) ^ 2 ) )
11		sq0 10891	. . . . . . . . 9 |- ( 0 ^ 2 ) = 0
12	11	a1i 9	. . . . . . . 8 |- ( B e. NN -> ( 0 ^ 2 ) = 0 )
13	12	oveq1d 6032	. . . . . . 7 |- ( B e. NN -> ( ( 0 ^ 2 ) gcd ( B ^ 2 ) ) = ( 0 gcd ( B ^ 2 ) ) )
14		zsqcl 10871	. . . . . . . 8 |- ( B e. ZZ -> ( B ^ 2 ) e. ZZ )
15		gcd0id 12549	. . . . . . . 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 2264	. . . . . 6 |- ( B e. NN -> ( ( 0 ^ 2 ) gcd ( B ^ 2 ) ) = ( abs ` ( B ^ 2 ) ) )
18	6, 10, 17	3eqtr4d 2274	. . . . 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 6024	. . . . . . 7 |- ( A = 0 -> ( A gcd B ) = ( 0 gcd B ) )
21	20	oveq1d 6032	. . . . . 6 |- ( A = 0 -> ( ( A gcd B ) ^ 2 ) = ( ( 0 gcd B ) ^ 2 ) )
22		oveq1 6024	. . . . . . 7 |- ( A = 0 -> ( A ^ 2 ) = ( 0 ^ 2 ) )
23	22	oveq1d 6032	. . . . . 6 |- ( A = 0 -> ( ( A ^ 2 ) gcd ( B ^ 2 ) ) = ( ( 0 ^ 2 ) gcd ( B ^ 2 ) ) )
24	21, 23	eqeq12d 2246	. . . . 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 9150	. . . . . . 7 |- ( A e. NN -> A e. CC )
28		abssq 11641	. . . . . . 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 9497	. . . . . . . 8 |- ( A e. NN -> A e. ZZ )
31		gcdid0 12550	. . . . . . . 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 6032	. . . . . 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 6033	. . . . . . 7 |- ( A e. NN -> ( ( A ^ 2 ) gcd ( 0 ^ 2 ) ) = ( ( A ^ 2 ) gcd 0 ) )
36		zsqcl 10871	. . . . . . . 8 |- ( A e. ZZ -> ( A ^ 2 ) e. ZZ )
37		gcdid0 12550	. . . . . . . 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 2264	. . . . . 6 |- ( A e. NN -> ( ( A ^ 2 ) gcd ( 0 ^ 2 ) ) = ( abs ` ( A ^ 2 ) ) )
40	29, 33, 39	3eqtr4d 2274	. . . . 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 6025	. . . . . . 7 |- ( B = 0 -> ( A gcd B ) = ( A gcd 0 ) )
43	42	oveq1d 6032	. . . . . 6 |- ( B = 0 -> ( ( A gcd B ) ^ 2 ) = ( ( A gcd 0 ) ^ 2 ) )
44		oveq1 6024	. . . . . . 7 |- ( B = 0 -> ( B ^ 2 ) = ( 0 ^ 2 ) )
45	44	oveq2d 6033	. . . . . 6 |- ( B = 0 -> ( ( A ^ 2 ) gcd ( B ^ 2 ) ) = ( ( A ^ 2 ) gcd ( 0 ^ 2 ) ) )
46	43, 45	eqeq12d 2246	. . . . 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 12530	. . . . . 6 |- ( 0 gcd 0 ) = 0
50	49	oveq1i 6027	. . . . 5 |- ( ( 0 gcd 0 ) ^ 2 ) = ( 0 ^ 2 )
51	11, 11	oveq12i 6029	. . . . . 6 |- ( ( 0 ^ 2 ) gcd ( 0 ^ 2 ) ) = ( 0 gcd 0 )
52	51, 49	eqtri 2252	. . . . 5 |- ( ( 0 ^ 2 ) gcd ( 0 ^ 2 ) ) = 0
53	11, 50, 52	3eqtr4i 2262	. . . 4 |- ( ( 0 gcd 0 ) ^ 2 ) = ( ( 0 ^ 2 ) gcd ( 0 ^ 2 ) )
54		oveq12 6026	. . . . 5 |- ( ( A = 0 /\ B = 0 ) -> ( A gcd B ) = ( 0 gcd 0 ) )
55	54	oveq1d 6032	. . . 4 |- ( ( A = 0 /\ B = 0 ) -> ( ( A gcd B ) ^ 2 ) = ( ( 0 gcd 0 ) ^ 2 ) )
56	22, 44	oveqan12d 6036	. . . 4 |- ( ( A = 0 /\ B = 0 ) -> ( ( A ^ 2 ) gcd ( B ^ 2 ) ) = ( ( 0 ^ 2 ) gcd ( 0 ^ 2 ) ) )
57	53, 55, 56	3eqtr4a 2290	. . 3 |- ( ( A = 0 /\ B = 0 ) -> ( ( A gcd B ) ^ 2 ) = ( ( A ^ 2 ) gcd ( B ^ 2 ) ) )
58	3, 26, 48, 57	ccase 972	. 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 715   = wceq 1397   e. wcel 2202   ` cfv 5326  (class class class)co 6017   CCcc 8029   0cc0 8031   NNcn 9142   2c2 9193   NN0cn0 9401   ZZcz 9478   ^cexp 10799   abscabs 11557   gcd cgcd 12523

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 619  ax-in2 620  ax-io 716  ax-5 1495  ax-7 1496  ax-gen 1497  ax-ie1 1541  ax-ie2 1542  ax-8 1552  ax-10 1553  ax-11 1554  ax-i12 1555  ax-bndl 1557  ax-4 1558  ax-17 1574  ax-i9 1578  ax-ial 1582  ax-i5r 1583  ax-13 2204  ax-14 2205  ax-ext 2213  ax-coll 4204  ax-sep 4207  ax-nul 4215  ax-pow 4264  ax-pr 4299  ax-un 4530  ax-setind 4635  ax-iinf 4686  ax-cnex 8122  ax-resscn 8123  ax-1cn 8124  ax-1re 8125  ax-icn 8126  ax-addcl 8127  ax-addrcl 8128  ax-mulcl 8129  ax-mulrcl 8130  ax-addcom 8131  ax-mulcom 8132  ax-addass 8133  ax-mulass 8134  ax-distr 8135  ax-i2m1 8136  ax-0lt1 8137  ax-1rid 8138  ax-0id 8139  ax-rnegex 8140  ax-precex 8141  ax-cnre 8142  ax-pre-ltirr 8143  ax-pre-ltwlin 8144  ax-pre-lttrn 8145  ax-pre-apti 8146  ax-pre-ltadd 8147  ax-pre-mulgt0 8148  ax-pre-mulext 8149  ax-arch 8150  ax-caucvg 8151

This theorem depends on definitions:  df-bi 117  df-stab 838  df-dc 842  df-3or 1005  df-3an 1006  df-tru 1400  df-fal 1403  df-nf 1509  df-sb 1811  df-eu 2082  df-mo 2083  df-clab 2218  df-cleq 2224  df-clel 2227  df-nfc 2363  df-ne 2403  df-nel 2498  df-ral 2515  df-rex 2516  df-reu 2517  df-rmo 2518  df-rab 2519  df-v 2804  df-sbc 3032  df-csb 3128  df-dif 3202  df-un 3204  df-in 3206  df-ss 3213  df-nul 3495  df-if 3606  df-pw 3654  df-sn 3675  df-pr 3676  df-op 3678  df-uni 3894  df-int 3929  df-iun 3972  df-br 4089  df-opab 4151  df-mpt 4152  df-tr 4188  df-id 4390  df-po 4393  df-iso 4394  df-iord 4463  df-on 4465  df-ilim 4466  df-suc 4468  df-iom 4689  df-xp 4731  df-rel 4732  df-cnv 4733  df-co 4734  df-dm 4735  df-rn 4736  df-res 4737  df-ima 4738  df-iota 5286  df-fun 5328  df-fn 5329  df-f 5330  df-f1 5331  df-fo 5332  df-f1o 5333  df-fv 5334  df-riota 5970  df-ov 6020  df-oprab 6021  df-mpo 6022  df-1st 6302  df-2nd 6303  df-recs 6470  df-frec 6556  df-sup 7182  df-pnf 8215  df-mnf 8216  df-xr 8217  df-ltxr 8218  df-le 8219  df-sub 8351  df-neg 8352  df-reap 8754  df-ap 8761  df-div 8852  df-inn 9143  df-2 9201  df-3 9202  df-4 9203  df-n0 9402  df-z 9479  df-uz 9755  df-q 9853  df-rp 9888  df-fz 10243  df-fzo 10377  df-fl 10529  df-mod 10584  df-seqfrec 10709  df-exp 10800  df-cj 11402  df-re 11403  df-im 11404  df-rsqrt 11558  df-abs 11559  df-dvds 12348  df-gcd 12524

This theorem is referenced by:  zgcdsq  12772