Theorem nn0gcdsq 12956

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 9544	. 2 |- ( A e. NN0 <-> ( A e. NN \/ A = 0 ) )
2		elnn0 9544	. 2 |- ( B e. NN0 <-> ( B e. NN \/ B = 0 ) )
3		sqgcd 12784	. . 3 |- ( ( A e. NN /\ B e. NN ) -> ( ( A gcd B ) ^ 2 ) = ( ( A ^ 2 ) gcd ( B ^ 2 ) ) )
4		nncn 9291	. . . . . . 7 |- ( B e. NN -> B e. CC )
5		abssq 11825	. . . . . . 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 9642	. . . . . . . 8 |- ( B e. NN -> B e. ZZ )
8		gcd0id 12734	. . . . . . . 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 6090	. . . . . 6 |- ( B e. NN -> ( ( 0 gcd B ) ^ 2 ) = ( ( abs ` B ) ^ 2 ) )
11		sq0 11045	. . . . . . . . 9 |- ( 0 ^ 2 ) = 0
12	11	a1i 9	. . . . . . . 8 |- ( B e. NN -> ( 0 ^ 2 ) = 0 )
13	12	oveq1d 6090	. . . . . . 7 |- ( B e. NN -> ( ( 0 ^ 2 ) gcd ( B ^ 2 ) ) = ( 0 gcd ( B ^ 2 ) ) )
14		zsqcl 11025	. . . . . . . 8 |- ( B e. ZZ -> ( B ^ 2 ) e. ZZ )
15		gcd0id 12734	. . . . . . . 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 2271	. . . . . 6 |- ( B e. NN -> ( ( 0 ^ 2 ) gcd ( B ^ 2 ) ) = ( abs ` ( B ^ 2 ) ) )
18	6, 10, 17	3eqtr4d 2281	. . . . 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 6082	. . . . . . 7 |- ( A = 0 -> ( A gcd B ) = ( 0 gcd B ) )
21	20	oveq1d 6090	. . . . . 6 |- ( A = 0 -> ( ( A gcd B ) ^ 2 ) = ( ( 0 gcd B ) ^ 2 ) )
22		oveq1 6082	. . . . . . 7 |- ( A = 0 -> ( A ^ 2 ) = ( 0 ^ 2 ) )
23	22	oveq1d 6090	. . . . . 6 |- ( A = 0 -> ( ( A ^ 2 ) gcd ( B ^ 2 ) ) = ( ( 0 ^ 2 ) gcd ( B ^ 2 ) ) )
24	21, 23	eqeq12d 2253	. . . . 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 9291	. . . . . . 7 |- ( A e. NN -> A e. CC )
28		abssq 11825	. . . . . . 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 9642	. . . . . . . 8 |- ( A e. NN -> A e. ZZ )
31		gcdid0 12735	. . . . . . . 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 6090	. . . . . 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 6091	. . . . . . 7 |- ( A e. NN -> ( ( A ^ 2 ) gcd ( 0 ^ 2 ) ) = ( ( A ^ 2 ) gcd 0 ) )
36		zsqcl 11025	. . . . . . . 8 |- ( A e. ZZ -> ( A ^ 2 ) e. ZZ )
37		gcdid0 12735	. . . . . . . 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 2271	. . . . . 6 |- ( A e. NN -> ( ( A ^ 2 ) gcd ( 0 ^ 2 ) ) = ( abs ` ( A ^ 2 ) ) )
40	29, 33, 39	3eqtr4d 2281	. . . . 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 6083	. . . . . . 7 |- ( B = 0 -> ( A gcd B ) = ( A gcd 0 ) )
43	42	oveq1d 6090	. . . . . 6 |- ( B = 0 -> ( ( A gcd B ) ^ 2 ) = ( ( A gcd 0 ) ^ 2 ) )
44		oveq1 6082	. . . . . . 7 |- ( B = 0 -> ( B ^ 2 ) = ( 0 ^ 2 ) )
45	44	oveq2d 6091	. . . . . 6 |- ( B = 0 -> ( ( A ^ 2 ) gcd ( B ^ 2 ) ) = ( ( A ^ 2 ) gcd ( 0 ^ 2 ) ) )
46	43, 45	eqeq12d 2253	. . . . 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 12715	. . . . . 6 |- ( 0 gcd 0 ) = 0
50	49	oveq1i 6085	. . . . 5 |- ( ( 0 gcd 0 ) ^ 2 ) = ( 0 ^ 2 )
51	11, 11	oveq12i 6087	. . . . . 6 |- ( ( 0 ^ 2 ) gcd ( 0 ^ 2 ) ) = ( 0 gcd 0 )
52	51, 49	eqtri 2259	. . . . 5 |- ( ( 0 ^ 2 ) gcd ( 0 ^ 2 ) ) = 0
53	11, 50, 52	3eqtr4i 2269	. . . 4 |- ( ( 0 gcd 0 ) ^ 2 ) = ( ( 0 ^ 2 ) gcd ( 0 ^ 2 ) )
54		oveq12 6084	. . . . 5 |- ( ( A = 0 /\ B = 0 ) -> ( A gcd B ) = ( 0 gcd 0 ) )
55	54	oveq1d 6090	. . . 4 |- ( ( A = 0 /\ B = 0 ) -> ( ( A gcd B ) ^ 2 ) = ( ( 0 gcd 0 ) ^ 2 ) )
56	22, 44	oveqan12d 6094	. . . 4 |- ( ( A = 0 /\ B = 0 ) -> ( ( A ^ 2 ) gcd ( B ^ 2 ) ) = ( ( 0 ^ 2 ) gcd ( 0 ^ 2 ) ) )
57	53, 55, 56	3eqtr4a 2297	. . 3 |- ( ( A = 0 /\ B = 0 ) -> ( ( A gcd B ) ^ 2 ) = ( ( A ^ 2 ) gcd ( B ^ 2 ) ) )
58	3, 26, 48, 57	ccase 977	. 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 720   = wceq 1402   e. wcel 2209   ` cfv 5372  (class class class)co 6075   CCcc 8167   0cc0 8169   NNcn 9283   2c2 9334   NN0cn0 9542   ZZcz 9623   ^cexp 10953   abscabs 11741   gcd cgcd 12708

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 623  ax-in2 624  ax-io 721  ax-5 1500  ax-7 1501  ax-gen 1502  ax-ie1 1546  ax-ie2 1547  ax-8 1557  ax-10 1558  ax-11 1559  ax-i12 1560  ax-bndl 1562  ax-4 1563  ax-17 1579  ax-i9 1583  ax-ial 1587  ax-i5r 1588  ax-14 2212  ax-ext 2220  ax-coll 4241  ax-sep 4244  ax-nul 4254  ax-pow 4306  ax-pr 4341  ax-un 4573  ax-setind 4679  ax-iinf 4730  ax-cnex 8260  ax-resscn 8261  ax-1cn 8262  ax-1re 8263  ax-icn 8264  ax-addcl 8265  ax-addrcl 8266  ax-mulcl 8267  ax-mulrcl 8268  ax-addcom 8269  ax-mulcom 8270  ax-addass 8271  ax-mulass 8272  ax-distr 8273  ax-i2m1 8274  ax-0lt1 8275  ax-1rid 8276  ax-0id 8277  ax-rnegex 8278  ax-precex 8279  ax-cnre 8280  ax-pre-ltirr 8281  ax-pre-ltwlin 8282  ax-pre-lttrn 8283  ax-pre-apti 8284  ax-pre-ltadd 8285  ax-pre-mulgt0 8286  ax-pre-mulext 8287  ax-arch 8288  ax-caucvg 8289

This theorem depends on definitions:  df-bi 117  df-stab 843  df-dc 847  df-3or 1010  df-3an 1011  df-tru 1405  df-fal 1408  df-nf 1514  df-sb 1816  df-eu 2089  df-mo 2090  df-clab 2225  df-cleq 2231  df-clel 2234  df-nfc 2381  df-ne 2421  df-nel 2516  df-ral 2533  df-rex 2534  df-reu 2535  df-rmo 2536  df-rab 2537  df-v 2823  df-sbc 3052  df-csb 3148  df-dif 3222  df-un 3224  df-in 3226  df-ss 3233  df-nul 3521  df-if 3636  df-pw 3687  df-sn 3711  df-pr 3712  df-op 3714  df-uni 3931  df-int 3966  df-iun 4009  df-br 4126  df-opab 4188  df-mpt 4189  df-tr 4225  df-id 4433  df-po 4436  df-iso 4437  df-iord 4506  df-on 4508  df-ilim 4509  df-suc 4511  df-iom 4733  df-xp 4775  df-rel 4776  df-cnv 4777  df-co 4778  df-dm 4779  df-rn 4780  df-res 4781  df-ima 4782  df-iota 5332  df-fun 5374  df-fn 5375  df-f 5376  df-f1 5377  df-fo 5378  df-f1o 5379  df-fv 5380  df-riota 6028  df-ov 6078  df-oprab 6079  df-mpo 6080  df-1st 6364  df-2nd 6365  df-recs 6566  df-frec 6652  df-sup 7314  df-pnf 8352  df-mnf 8353  df-xr 8354  df-ltxr 8355  df-le 8356  df-sub 8489  df-neg 8490  df-reap 8893  df-ap 8900  df-div 8993  df-inn 9284  df-2 9342  df-3 9343  df-4 9344  df-n0 9543  df-z 9624  df-uz 9901  df-q 9999  df-rp 10034  df-fz 10391  df-fzo 10528  df-fl 10683  df-mod 10738  df-seqfrec 10863  df-exp 10954  df-cj 11585  df-re 11586  df-im 11587  df-rsqrt 11742  df-abs 11743  df-dvds 12533  df-gcd 12709

This theorem is referenced by:  zgcdsq  12957