Theorem coprimeprodsq 12259

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 9275	. . . . . . . 8 |- ( A e. NN0 -> A e. ZZ )
2		nn0z 9275	. . . . . . . 8 |- ( C e. NN0 -> C e. ZZ )
3		gcdcl 11969	. . . . . . . 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 1016	. . . . . 6 |- ( ( A e. NN0 /\ B e. ZZ /\ C e. NN0 ) -> ( A gcd C ) e. NN0 )
6	5	3ad2ant1 1018	. . . . 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 9233	. . . 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 10653	. . 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 1029	. . . . . . . . 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 9233	. . . . . . . 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 9188	. . . . . . . . . 10 |- ( A e. NN0 -> A e. CC )
12	11	3ad2ant1 1018	. . . . . . . . 9 |- ( ( A e. NN0 /\ B e. ZZ /\ C e. NN0 ) -> A e. CC )
13	12	3ad2ant1 1018	. . . . . . . 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 7981	. . . . . . 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 1002	. . . . . . . . . . 11 |- ( ( ( A e. NN0 /\ B e. ZZ /\ C e. NN0 ) /\ ( ( A gcd B ) gcd C ) = 1 ) -> C e. NN0 )
16	15	nn0cnd 9233	. . . . . . . . . 10 |- ( ( ( A e. NN0 /\ B e. ZZ /\ C e. NN0 ) /\ ( ( A gcd B ) gcd C ) = 1 ) -> C e. CC )
17	16	sqvald 10653	. . . . . . . . 9 |- ( ( ( A e. NN0 /\ B e. ZZ /\ C e. NN0 ) /\ ( ( A gcd B ) gcd C ) = 1 ) -> ( C ^ 2 ) = ( C x. C ) )
18	17	eqeq1d 2186	. . . . . . . 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 1345	. . . . . . 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 5895	. . . . . 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 1027	. . . . . . . 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 9375	. . . . . . 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 9375	. . . . . . 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 12019	. . . . . . 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 1238	. . . . . 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 1028	. . . . . . 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 12019	. . . . . . 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 1238	. . . . . 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 2218	. . . . 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 5893	. . . 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 12021	. . . . 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 1238	. . . 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 9375	. . . . 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 11969	. . . . . . . . . 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 1015	. . . . . . 7 |- ( ( A e. NN0 /\ B e. ZZ /\ C e. NN0 ) -> ( C gcd B ) e. NN0 )
38	37	3ad2ant1 1018	. . . . . 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 9375	. . . . 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 12019	. . . . 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 1238	. . . 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 2218	. . 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 1020	. . . . . . . . . . . . . 14 |- ( ( A e. NN0 /\ B e. ZZ /\ C e. NN0 ) -> C e. ZZ )
44		gcdid 11989	. . . . . . . . . . . . . 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 5892	. . . . . . . . . . . 12 |- ( ( A e. NN0 /\ B e. ZZ /\ C e. NN0 ) -> ( ( C gcd C ) gcd B ) = ( ( abs ` C ) gcd B ) )
47		simp2 998	. . . . . . . . . . . . 13 |- ( ( A e. NN0 /\ B e. ZZ /\ C e. NN0 ) -> B e. ZZ )
48		gcdabs1 11992	. . . . . . . . . . . . 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 2210	. . . . . . . . . . 11 |- ( ( A e. NN0 /\ B e. ZZ /\ C e. NN0 ) -> ( ( C gcd C ) gcd B ) = ( C gcd B ) )
51		gcdass 12018	. . . . . . . . . . . 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 1238	. . . . . . . . . . 11 |- ( ( A e. NN0 /\ B e. ZZ /\ C e. NN0 ) -> ( ( C gcd C ) gcd B ) = ( C gcd ( C gcd B ) ) )
53	43, 47	gcdcomd 11977	. . . . . . . . . . 11 |- ( ( A e. NN0 /\ B e. ZZ /\ C e. NN0 ) -> ( C gcd B ) = ( B gcd C ) )
54	50, 52, 53	3eqtr3d 2218	. . . . . . . . . 10 |- ( ( A e. NN0 /\ B e. ZZ /\ C e. NN0 ) -> ( C gcd ( C gcd B ) ) = ( B gcd C ) )
55	54	oveq2d 5893	. . . . . . . . 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 1018	. . . . . . . . . 10 |- ( ( A e. NN0 /\ B e. ZZ /\ C e. NN0 ) -> A e. ZZ )
57	37	nn0zd 9375	. . . . . . . . . 10 |- ( ( A e. NN0 /\ B e. ZZ /\ C e. NN0 ) -> ( C gcd B ) e. ZZ )
58		gcdass 12018	. . . . . . . . . 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 1238	. . . . . . . . 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 12018	. . . . . . . . . 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 1238	. . . . . . . . 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 2220	. . . . . . . 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 2186	. . . . . . 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 5893	. . . . 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 1017	. . . 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 7976	. . . 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 2210	. . 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 2215	. 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 1205	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 978   = wceq 1353   e. wcel 2148   ` cfv 5218  (class class class)co 5877   CCcc 7811   1c1 7814   x. cmul 7818   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:  coprimeprodsq2  12260  pythagtriplem6  12272