Theorem coprimeprodsq 12426

Description: If three numbers are coprime, and the square of one is the product of the other two, then there is a formula for the other two in terms of gcd and square. (Contributed by Scott Fenton, 2-Apr-2014.) (Revised by Mario Carneiro, 19-Apr-2014.)

Assertion

Ref	Expression
coprimeprodsq	|- ( ( ( A e. NN0 /\ B e. ZZ /\ C e. NN0 ) /\ ( ( A gcd B ) gcd C ) = 1 ) -> ( ( C ^ 2 ) = ( A x. B ) -> A = ( ( A gcd C ) ^ 2 ) ) )

Proof of Theorem coprimeprodsq

Step	Hyp	Ref	Expression
1		nn0z 9346	. . . . . . . 8 |- ( A e. NN0 -> A e. ZZ )
2		nn0z 9346	. . . . . . . 8 |- ( C e. NN0 -> C e. ZZ )
3		gcdcl 12133	. . . . . . . 8 |- ( ( A e. ZZ /\ C e. ZZ ) -> ( A gcd C ) e. NN0 )
4	1, 2, 3	syl2an 289	. . . . . . 7 |- ( ( A e. NN0 /\ C e. NN0 ) -> ( A gcd C ) e. NN0 )
5	4	3adant2 1018	. . . . . 6 |- ( ( A e. NN0 /\ B e. ZZ /\ C e. NN0 ) -> ( A gcd C ) e. NN0 )
6	5	3ad2ant1 1020	. . . . 5 |- ( ( ( A e. NN0 /\ B e. ZZ /\ C e. NN0 ) /\ ( ( A gcd B ) gcd C ) = 1 /\ ( C ^ 2 ) = ( A x. B ) ) -> ( A gcd C ) e. NN0 )
7	6	nn0cnd 9304	. . . 4 |- ( ( ( A e. NN0 /\ B e. ZZ /\ C e. NN0 ) /\ ( ( A gcd B ) gcd C ) = 1 /\ ( C ^ 2 ) = ( A x. B ) ) -> ( A gcd C ) e. CC )
8	7	sqvald 10762	. . 3 |- ( ( ( A e. NN0 /\ B e. ZZ /\ C e. NN0 ) /\ ( ( A gcd B ) gcd C ) = 1 /\ ( C ^ 2 ) = ( A x. B ) ) -> ( ( A gcd C ) ^ 2 ) = ( ( A gcd C ) x. ( A gcd C ) ) )
9		simp13 1031	. . . . . . . . 9 |- ( ( ( A e. NN0 /\ B e. ZZ /\ C e. NN0 ) /\ ( ( A gcd B ) gcd C ) = 1 /\ ( C ^ 2 ) = ( A x. B ) ) -> C e. NN0 )
10	9	nn0cnd 9304	. . . . . . . 8 |- ( ( ( A e. NN0 /\ B e. ZZ /\ C e. NN0 ) /\ ( ( A gcd B ) gcd C ) = 1 /\ ( C ^ 2 ) = ( A x. B ) ) -> C e. CC )
11		nn0cn 9259	. . . . . . . . . 10 |- ( A e. NN0 -> A e. CC )
12	11	3ad2ant1 1020	. . . . . . . . 9 |- ( ( A e. NN0 /\ B e. ZZ /\ C e. NN0 ) -> A e. CC )
13	12	3ad2ant1 1020	. . . . . . . 8 |- ( ( ( A e. NN0 /\ B e. ZZ /\ C e. NN0 ) /\ ( ( A gcd B ) gcd C ) = 1 /\ ( C ^ 2 ) = ( A x. B ) ) -> A e. CC )
14	10, 13	mulcomd 8048	. . . . . . 7 |- ( ( ( A e. NN0 /\ B e. ZZ /\ C e. NN0 ) /\ ( ( A gcd B ) gcd C ) = 1 /\ ( C ^ 2 ) = ( A x. B ) ) -> ( C x. A ) = ( A x. C ) )
15		simpl3 1004	. . . . . . . . . . 11 |- ( ( ( A e. NN0 /\ B e. ZZ /\ C e. NN0 ) /\ ( ( A gcd B ) gcd C ) = 1 ) -> C e. NN0 )
16	15	nn0cnd 9304	. . . . . . . . . 10 |- ( ( ( A e. NN0 /\ B e. ZZ /\ C e. NN0 ) /\ ( ( A gcd B ) gcd C ) = 1 ) -> C e. CC )
17	16	sqvald 10762	. . . . . . . . 9 |- ( ( ( A e. NN0 /\ B e. ZZ /\ C e. NN0 ) /\ ( ( A gcd B ) gcd C ) = 1 ) -> ( C ^ 2 ) = ( C x. C ) )
18	17	eqeq1d 2205	. . . . . . . 8 |- ( ( ( A e. NN0 /\ B e. ZZ /\ C e. NN0 ) /\ ( ( A gcd B ) gcd C ) = 1 ) -> ( ( C ^ 2 ) = ( A x. B ) <-> ( C x. C ) = ( A x. B ) ) )
19	18	biimp3a 1356	. . . . . . 7 |- ( ( ( A e. NN0 /\ B e. ZZ /\ C e. NN0 ) /\ ( ( A gcd B ) gcd C ) = 1 /\ ( C ^ 2 ) = ( A x. B ) ) -> ( C x. C ) = ( A x. B ) )
20	14, 19	oveq12d 5940	. . . . . 6 |- ( ( ( A e. NN0 /\ B e. ZZ /\ C e. NN0 ) /\ ( ( A gcd B ) gcd C ) = 1 /\ ( C ^ 2 ) = ( A x. B ) ) -> ( ( C x. A ) gcd ( C x. C ) ) = ( ( A x. C ) gcd ( A x. B ) ) )
21		simp11 1029	. . . . . . . 8 |- ( ( ( A e. NN0 /\ B e. ZZ /\ C e. NN0 ) /\ ( ( A gcd B ) gcd C ) = 1 /\ ( C ^ 2 ) = ( A x. B ) ) -> A e. NN0 )
22	21	nn0zd 9446	. . . . . . 7 |- ( ( ( A e. NN0 /\ B e. ZZ /\ C e. NN0 ) /\ ( ( A gcd B ) gcd C ) = 1 /\ ( C ^ 2 ) = ( A x. B ) ) -> A e. ZZ )
23	9	nn0zd 9446	. . . . . . 7 |- ( ( ( A e. NN0 /\ B e. ZZ /\ C e. NN0 ) /\ ( ( A gcd B ) gcd C ) = 1 /\ ( C ^ 2 ) = ( A x. B ) ) -> C e. ZZ )
24		mulgcd 12183	. . . . . . 7 |- ( ( C e. NN0 /\ A e. ZZ /\ C e. ZZ ) -> ( ( C x. A ) gcd ( C x. C ) ) = ( C x. ( A gcd C ) ) )
25	9, 22, 23, 24	syl3anc 1249	. . . . . 6 |- ( ( ( A e. NN0 /\ B e. ZZ /\ C e. NN0 ) /\ ( ( A gcd B ) gcd C ) = 1 /\ ( C ^ 2 ) = ( A x. B ) ) -> ( ( C x. A ) gcd ( C x. C ) ) = ( C x. ( A gcd C ) ) )
26		simp12 1030	. . . . . . 7 |- ( ( ( A e. NN0 /\ B e. ZZ /\ C e. NN0 ) /\ ( ( A gcd B ) gcd C ) = 1 /\ ( C ^ 2 ) = ( A x. B ) ) -> B e. ZZ )
27		mulgcd 12183	. . . . . . 7 |- ( ( A e. NN0 /\ C e. ZZ /\ B e. ZZ ) -> ( ( A x. C ) gcd ( A x. B ) ) = ( A x. ( C gcd B ) ) )
28	21, 23, 26, 27	syl3anc 1249	. . . . . 6 |- ( ( ( A e. NN0 /\ B e. ZZ /\ C e. NN0 ) /\ ( ( A gcd B ) gcd C ) = 1 /\ ( C ^ 2 ) = ( A x. B ) ) -> ( ( A x. C ) gcd ( A x. B ) ) = ( A x. ( C gcd B ) ) )
29	20, 25, 28	3eqtr3d 2237	. . . . 5 |- ( ( ( A e. NN0 /\ B e. ZZ /\ C e. NN0 ) /\ ( ( A gcd B ) gcd C ) = 1 /\ ( C ^ 2 ) = ( A x. B ) ) -> ( C x. ( A gcd C ) ) = ( A x. ( C gcd B ) ) )
30	29	oveq2d 5938	. . . 4 |- ( ( ( A e. NN0 /\ B e. ZZ /\ C e. NN0 ) /\ ( ( A gcd B ) gcd C ) = 1 /\ ( C ^ 2 ) = ( A x. B ) ) -> ( ( A x. ( A gcd C ) ) gcd ( C x. ( A gcd C ) ) ) = ( ( A x. ( A gcd C ) ) gcd ( A x. ( C gcd B ) ) ) )
31		mulgcdr 12185	. . . . 5 |- ( ( A e. ZZ /\ C e. ZZ /\ ( A gcd C ) e. NN0 ) -> ( ( A x. ( A gcd C ) ) gcd ( C x. ( A gcd C ) ) ) = ( ( A gcd C ) x. ( A gcd C ) ) )
32	22, 23, 6, 31	syl3anc 1249	. . . 4 |- ( ( ( A e. NN0 /\ B e. ZZ /\ C e. NN0 ) /\ ( ( A gcd B ) gcd C ) = 1 /\ ( C ^ 2 ) = ( A x. B ) ) -> ( ( A x. ( A gcd C ) ) gcd ( C x. ( A gcd C ) ) ) = ( ( A gcd C ) x. ( A gcd C ) ) )
33	6	nn0zd 9446	. . . . 5 |- ( ( ( A e. NN0 /\ B e. ZZ /\ C e. NN0 ) /\ ( ( A gcd B ) gcd C ) = 1 /\ ( C ^ 2 ) = ( A x. B ) ) -> ( A gcd C ) e. ZZ )
34		gcdcl 12133	. . . . . . . . . 10 |- ( ( C e. ZZ /\ B e. ZZ ) -> ( C gcd B ) e. NN0 )
35	2, 34	sylan 283	. . . . . . . . 9 |- ( ( C e. NN0 /\ B e. ZZ ) -> ( C gcd B ) e. NN0 )
36	35	ancoms 268	. . . . . . . 8 |- ( ( B e. ZZ /\ C e. NN0 ) -> ( C gcd B ) e. NN0 )
37	36	3adant1 1017	. . . . . . 7 |- ( ( A e. NN0 /\ B e. ZZ /\ C e. NN0 ) -> ( C gcd B ) e. NN0 )
38	37	3ad2ant1 1020	. . . . . 6 |- ( ( ( A e. NN0 /\ B e. ZZ /\ C e. NN0 ) /\ ( ( A gcd B ) gcd C ) = 1 /\ ( C ^ 2 ) = ( A x. B ) ) -> ( C gcd B ) e. NN0 )
39	38	nn0zd 9446	. . . . 5 |- ( ( ( A e. NN0 /\ B e. ZZ /\ C e. NN0 ) /\ ( ( A gcd B ) gcd C ) = 1 /\ ( C ^ 2 ) = ( A x. B ) ) -> ( C gcd B ) e. ZZ )
40		mulgcd 12183	. . . . 5 |- ( ( A e. NN0 /\ ( A gcd C ) e. ZZ /\ ( C gcd B ) e. ZZ ) -> ( ( A x. ( A gcd C ) ) gcd ( A x. ( C gcd B ) ) ) = ( A x. ( ( A gcd C ) gcd ( C gcd B ) ) ) )
41	21, 33, 39, 40	syl3anc 1249	. . . 4 |- ( ( ( A e. NN0 /\ B e. ZZ /\ C e. NN0 ) /\ ( ( A gcd B ) gcd C ) = 1 /\ ( C ^ 2 ) = ( A x. B ) ) -> ( ( A x. ( A gcd C ) ) gcd ( A x. ( C gcd B ) ) ) = ( A x. ( ( A gcd C ) gcd ( C gcd B ) ) ) )
42	30, 32, 41	3eqtr3d 2237	. . 3 |- ( ( ( A e. NN0 /\ B e. ZZ /\ C e. NN0 ) /\ ( ( A gcd B ) gcd C ) = 1 /\ ( C ^ 2 ) = ( A x. B ) ) -> ( ( A gcd C ) x. ( A gcd C ) ) = ( A x. ( ( A gcd C ) gcd ( C gcd B ) ) ) )
43	2	3ad2ant3 1022	. . . . . . . . . . . . . 14 |- ( ( A e. NN0 /\ B e. ZZ /\ C e. NN0 ) -> C e. ZZ )
44		gcdid 12153	. . . . . . . . . . . . . 14 |- ( C e. ZZ -> ( C gcd C ) = ( abs ` C ) )
45	43, 44	syl 14	. . . . . . . . . . . . 13 |- ( ( A e. NN0 /\ B e. ZZ /\ C e. NN0 ) -> ( C gcd C ) = ( abs ` C ) )
46	45	oveq1d 5937	. . . . . . . . . . . 12 |- ( ( A e. NN0 /\ B e. ZZ /\ C e. NN0 ) -> ( ( C gcd C ) gcd B ) = ( ( abs ` C ) gcd B ) )
47		simp2 1000	. . . . . . . . . . . . 13 |- ( ( A e. NN0 /\ B e. ZZ /\ C e. NN0 ) -> B e. ZZ )
48		gcdabs1 12156	. . . . . . . . . . . . 13 |- ( ( C e. ZZ /\ B e. ZZ ) -> ( ( abs ` C ) gcd B ) = ( C gcd B ) )
49	43, 47, 48	syl2anc 411	. . . . . . . . . . . 12 |- ( ( A e. NN0 /\ B e. ZZ /\ C e. NN0 ) -> ( ( abs ` C ) gcd B ) = ( C gcd B ) )
50	46, 49	eqtrd 2229	. . . . . . . . . . 11 |- ( ( A e. NN0 /\ B e. ZZ /\ C e. NN0 ) -> ( ( C gcd C ) gcd B ) = ( C gcd B ) )
51		gcdass 12182	. . . . . . . . . . . 12 |- ( ( C e. ZZ /\ C e. ZZ /\ B e. ZZ ) -> ( ( C gcd C ) gcd B ) = ( C gcd ( C gcd B ) ) )
52	43, 43, 47, 51	syl3anc 1249	. . . . . . . . . . 11 |- ( ( A e. NN0 /\ B e. ZZ /\ C e. NN0 ) -> ( ( C gcd C ) gcd B ) = ( C gcd ( C gcd B ) ) )
53	43, 47	gcdcomd 12141	. . . . . . . . . . 11 |- ( ( A e. NN0 /\ B e. ZZ /\ C e. NN0 ) -> ( C gcd B ) = ( B gcd C ) )
54	50, 52, 53	3eqtr3d 2237	. . . . . . . . . 10 |- ( ( A e. NN0 /\ B e. ZZ /\ C e. NN0 ) -> ( C gcd ( C gcd B ) ) = ( B gcd C ) )
55	54	oveq2d 5938	. . . . . . . . 9 |- ( ( A e. NN0 /\ B e. ZZ /\ C e. NN0 ) -> ( A gcd ( C gcd ( C gcd B ) ) ) = ( A gcd ( B gcd C ) ) )
56	1	3ad2ant1 1020	. . . . . . . . . 10 |- ( ( A e. NN0 /\ B e. ZZ /\ C e. NN0 ) -> A e. ZZ )
57	37	nn0zd 9446	. . . . . . . . . 10 |- ( ( A e. NN0 /\ B e. ZZ /\ C e. NN0 ) -> ( C gcd B ) e. ZZ )
58		gcdass 12182	. . . . . . . . . 10 |- ( ( A e. ZZ /\ C e. ZZ /\ ( C gcd B ) e. ZZ ) -> ( ( A gcd C ) gcd ( C gcd B ) ) = ( A gcd ( C gcd ( C gcd B ) ) ) )
59	56, 43, 57, 58	syl3anc 1249	. . . . . . . . 9 |- ( ( A e. NN0 /\ B e. ZZ /\ C e. NN0 ) -> ( ( A gcd C ) gcd ( C gcd B ) ) = ( A gcd ( C gcd ( C gcd B ) ) ) )
60		gcdass 12182	. . . . . . . . . 10 |- ( ( A e. ZZ /\ B e. ZZ /\ C e. ZZ ) -> ( ( A gcd B ) gcd C ) = ( A gcd ( B gcd C ) ) )
61	56, 47, 43, 60	syl3anc 1249	. . . . . . . . 9 |- ( ( A e. NN0 /\ B e. ZZ /\ C e. NN0 ) -> ( ( A gcd B ) gcd C ) = ( A gcd ( B gcd C ) ) )
62	55, 59, 61	3eqtr4d 2239	. . . . . . . 8 |- ( ( A e. NN0 /\ B e. ZZ /\ C e. NN0 ) -> ( ( A gcd C ) gcd ( C gcd B ) ) = ( ( A gcd B ) gcd C ) )
63	62	eqeq1d 2205	. . . . . . 7 |- ( ( A e. NN0 /\ B e. ZZ /\ C e. NN0 ) -> ( ( ( A gcd C ) gcd ( C gcd B ) ) = 1 <-> ( ( A gcd B ) gcd C ) = 1 ) )
64	63	biimpar 297	. . . . . 6 |- ( ( ( A e. NN0 /\ B e. ZZ /\ C e. NN0 ) /\ ( ( A gcd B ) gcd C ) = 1 ) -> ( ( A gcd C ) gcd ( C gcd B ) ) = 1 )
65	64	oveq2d 5938	. . . . 5 |- ( ( ( A e. NN0 /\ B e. ZZ /\ C e. NN0 ) /\ ( ( A gcd B ) gcd C ) = 1 ) -> ( A x. ( ( A gcd C ) gcd ( C gcd B ) ) ) = ( A x. 1 ) )
66	65	3adant3 1019	. . . 4 |- ( ( ( A e. NN0 /\ B e. ZZ /\ C e. NN0 ) /\ ( ( A gcd B ) gcd C ) = 1 /\ ( C ^ 2 ) = ( A x. B ) ) -> ( A x. ( ( A gcd C ) gcd ( C gcd B ) ) ) = ( A x. 1 ) )
67	13	mulridd 8043	. . . 4 |- ( ( ( A e. NN0 /\ B e. ZZ /\ C e. NN0 ) /\ ( ( A gcd B ) gcd C ) = 1 /\ ( C ^ 2 ) = ( A x. B ) ) -> ( A x. 1 ) = A )
68	66, 67	eqtrd 2229	. . 3 |- ( ( ( A e. NN0 /\ B e. ZZ /\ C e. NN0 ) /\ ( ( A gcd B ) gcd C ) = 1 /\ ( C ^ 2 ) = ( A x. B ) ) -> ( A x. ( ( A gcd C ) gcd ( C gcd B ) ) ) = A )
69	8, 42, 68	3eqtrrd 2234	. 2 |- ( ( ( A e. NN0 /\ B e. ZZ /\ C e. NN0 ) /\ ( ( A gcd B ) gcd C ) = 1 /\ ( C ^ 2 ) = ( A x. B ) ) -> A = ( ( A gcd C ) ^ 2 ) )
70	69	3expia 1207	1 |- ( ( ( A e. NN0 /\ B e. ZZ /\ C e. NN0 ) /\ ( ( A gcd B ) gcd C ) = 1 ) -> ( ( C ^ 2 ) = ( A x. B ) -> A = ( ( A gcd C ) ^ 2 ) ) )

Syntax hints:   -> wi 4   /\ wa 104   /\ w3a 980   = wceq 1364   e. wcel 2167   ` cfv 5258  (class class class)co 5922   CCcc 7877   1c1 7880   x. cmul 7884   2c2 9041   NN0cn0 9249   ZZcz 9326   ^cexp 10630   abscabs 11162   gcd cgcd 12120

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 4148  ax-sep 4151  ax-nul 4159  ax-pow 4207  ax-pr 4242  ax-un 4468  ax-setind 4573  ax-iinf 4624  ax-cnex 7970  ax-resscn 7971  ax-1cn 7972  ax-1re 7973  ax-icn 7974  ax-addcl 7975  ax-addrcl 7976  ax-mulcl 7977  ax-mulrcl 7978  ax-addcom 7979  ax-mulcom 7980  ax-addass 7981  ax-mulass 7982  ax-distr 7983  ax-i2m1 7984  ax-0lt1 7985  ax-1rid 7986  ax-0id 7987  ax-rnegex 7988  ax-precex 7989  ax-cnre 7990  ax-pre-ltirr 7991  ax-pre-ltwlin 7992  ax-pre-lttrn 7993  ax-pre-apti 7994  ax-pre-ltadd 7995  ax-pre-mulgt0 7996  ax-pre-mulext 7997  ax-arch 7998  ax-caucvg 7999

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 3451  df-if 3562  df-pw 3607  df-sn 3628  df-pr 3629  df-op 3631  df-uni 3840  df-int 3875  df-iun 3918  df-br 4034  df-opab 4095  df-mpt 4096  df-tr 4132  df-id 4328  df-po 4331  df-iso 4332  df-iord 4401  df-on 4403  df-ilim 4404  df-suc 4406  df-iom 4627  df-xp 4669  df-rel 4670  df-cnv 4671  df-co 4672  df-dm 4673  df-rn 4674  df-res 4675  df-ima 4676  df-iota 5219  df-fun 5260  df-fn 5261  df-f 5262  df-f1 5263  df-fo 5264  df-f1o 5265  df-fv 5266  df-riota 5877  df-ov 5925  df-oprab 5926  df-mpo 5927  df-1st 6198  df-2nd 6199  df-recs 6363  df-frec 6449  df-sup 7050  df-pnf 8063  df-mnf 8064  df-xr 8065  df-ltxr 8066  df-le 8067  df-sub 8199  df-neg 8200  df-reap 8602  df-ap 8609  df-div 8700  df-inn 8991  df-2 9049  df-3 9050  df-4 9051  df-n0 9250  df-z 9327  df-uz 9602  df-q 9694  df-rp 9729  df-fz 10084  df-fzo 10218  df-fl 10360  df-mod 10415  df-seqfrec 10540  df-exp 10631  df-cj 11007  df-re 11008  df-im 11009  df-rsqrt 11163  df-abs 11164  df-dvds 11953  df-gcd 12121

This theorem is referenced by:  coprimeprodsq2  12427  pythagtriplem6  12439