Theorem coprimeprodsq 15513

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 11400	. . . . . . . 8 |- ( A e. NN0 -> A e. ZZ )
2		nn0z 11400	. . . . . . . 8 |- ( C e. NN0 -> C e. ZZ )
3		gcdcl 15228	. . . . . . . 8 |- ( ( A e. ZZ /\ C e. ZZ ) -> ( A gcd C ) e. NN0 )
4	1, 2, 3	syl2an 494	. . . . . . 7 |- ( ( A e. NN0 /\ C e. NN0 ) -> ( A gcd C ) e. NN0 )
5	4	3adant2 1080	. . . . . 6 |- ( ( A e. NN0 /\ B e. ZZ /\ C e. NN0 ) -> ( A gcd C ) e. NN0 )
6	5	3ad2ant1 1082	. . . . 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 11353	. . . 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 13005	. . 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 1093	. . . . . . . . 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 11353	. . . . . . . 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 11302	. . . . . . . . . 10 |- ( A e. NN0 -> A e. CC )
12	11	3ad2ant1 1082	. . . . . . . . 9 |- ( ( A e. NN0 /\ B e. ZZ /\ C e. NN0 ) -> A e. CC )
13	12	3ad2ant1 1082	. . . . . . . 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 10061	. . . . . . 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 1066	. . . . . . . . . . 11 |- ( ( ( A e. NN0 /\ B e. ZZ /\ C e. NN0 ) /\ ( ( A gcd B ) gcd C ) = 1 ) -> C e. NN0 )
16	15	nn0cnd 11353	. . . . . . . . . 10 |- ( ( ( A e. NN0 /\ B e. ZZ /\ C e. NN0 ) /\ ( ( A gcd B ) gcd C ) = 1 ) -> C e. CC )
17	16	sqvald 13005	. . . . . . . . 9 |- ( ( ( A e. NN0 /\ B e. ZZ /\ C e. NN0 ) /\ ( ( A gcd B ) gcd C ) = 1 ) -> ( C ^ 2 ) = ( C x. C ) )
18	17	eqeq1d 2624	. . . . . . . 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 1432	. . . . . . 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 6668	. . . . . 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 1091	. . . . . . . 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 11480	. . . . . . 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 11480	. . . . . . 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 15265	. . . . . . 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 1326	. . . . . 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 1092	. . . . . . 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 15265	. . . . . . 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 1326	. . . . . 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 2664	. . . . 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 6666	. . . 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 15267	. . . . 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 1326	. . . 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 11480	. . . . 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 15228	. . . . . . . . . 10 |- ( ( C e. ZZ /\ B e. ZZ ) -> ( C gcd B ) e. NN0 )
35	2, 34	sylan 488	. . . . . . . . 9 |- ( ( C e. NN0 /\ B e. ZZ ) -> ( C gcd B ) e. NN0 )
36	35	ancoms 469	. . . . . . . 8 |- ( ( B e. ZZ /\ C e. NN0 ) -> ( C gcd B ) e. NN0 )
37	36	3adant1 1079	. . . . . . 7 |- ( ( A e. NN0 /\ B e. ZZ /\ C e. NN0 ) -> ( C gcd B ) e. NN0 )
38	37	3ad2ant1 1082	. . . . . 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 11480	. . . . 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 15265	. . . . 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 1326	. . . 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 2664	. . 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 1084	. . . . . . . . . . . . . 14 |- ( ( A e. NN0 /\ B e. ZZ /\ C e. NN0 ) -> C e. ZZ )
44		gcdid 15248	. . . . . . . . . . . . . 14 |- ( C e. ZZ -> ( C gcd C ) = ( abs ` C ) )
45	43, 44	syl 17	. . . . . . . . . . . . 13 |- ( ( A e. NN0 /\ B e. ZZ /\ C e. NN0 ) -> ( C gcd C ) = ( abs ` C ) )
46	45	oveq1d 6665	. . . . . . . . . . . 12 |- ( ( A e. NN0 /\ B e. ZZ /\ C e. NN0 ) -> ( ( C gcd C ) gcd B ) = ( ( abs ` C ) gcd B ) )
47		simp2 1062	. . . . . . . . . . . . 13 |- ( ( A e. NN0 /\ B e. ZZ /\ C e. NN0 ) -> B e. ZZ )
48		gcdabs1 15251	. . . . . . . . . . . . 13 |- ( ( C e. ZZ /\ B e. ZZ ) -> ( ( abs ` C ) gcd B ) = ( C gcd B ) )
49	43, 47, 48	syl2anc 693	. . . . . . . . . . . 12 |- ( ( A e. NN0 /\ B e. ZZ /\ C e. NN0 ) -> ( ( abs ` C ) gcd B ) = ( C gcd B ) )
50	46, 49	eqtrd 2656	. . . . . . . . . . 11 |- ( ( A e. NN0 /\ B e. ZZ /\ C e. NN0 ) -> ( ( C gcd C ) gcd B ) = ( C gcd B ) )
51		gcdass 15264	. . . . . . . . . . . 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 1326	. . . . . . . . . . 11 |- ( ( A e. NN0 /\ B e. ZZ /\ C e. NN0 ) -> ( ( C gcd C ) gcd B ) = ( C gcd ( C gcd B ) ) )
53		gcdcom 15235	. . . . . . . . . . . 12 |- ( ( C e. ZZ /\ B e. ZZ ) -> ( C gcd B ) = ( B gcd C ) )
54	43, 47, 53	syl2anc 693	. . . . . . . . . . 11 |- ( ( A e. NN0 /\ B e. ZZ /\ C e. NN0 ) -> ( C gcd B ) = ( B gcd C ) )
55	50, 52, 54	3eqtr3d 2664	. . . . . . . . . 10 |- ( ( A e. NN0 /\ B e. ZZ /\ C e. NN0 ) -> ( C gcd ( C gcd B ) ) = ( B gcd C ) )
56	55	oveq2d 6666	. . . . . . . . 9 |- ( ( A e. NN0 /\ B e. ZZ /\ C e. NN0 ) -> ( A gcd ( C gcd ( C gcd B ) ) ) = ( A gcd ( B gcd C ) ) )
57	1	3ad2ant1 1082	. . . . . . . . . 10 |- ( ( A e. NN0 /\ B e. ZZ /\ C e. NN0 ) -> A e. ZZ )
58	37	nn0zd 11480	. . . . . . . . . 10 |- ( ( A e. NN0 /\ B e. ZZ /\ C e. NN0 ) -> ( C gcd B ) e. ZZ )
59		gcdass 15264	. . . . . . . . . 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 ) ) ) )
60	57, 43, 58, 59	syl3anc 1326	. . . . . . . . 9 |- ( ( A e. NN0 /\ B e. ZZ /\ C e. NN0 ) -> ( ( A gcd C ) gcd ( C gcd B ) ) = ( A gcd ( C gcd ( C gcd B ) ) ) )
61		gcdass 15264	. . . . . . . . . 10 |- ( ( A e. ZZ /\ B e. ZZ /\ C e. ZZ ) -> ( ( A gcd B ) gcd C ) = ( A gcd ( B gcd C ) ) )
62	57, 47, 43, 61	syl3anc 1326	. . . . . . . . 9 |- ( ( A e. NN0 /\ B e. ZZ /\ C e. NN0 ) -> ( ( A gcd B ) gcd C ) = ( A gcd ( B gcd C ) ) )
63	56, 60, 62	3eqtr4d 2666	. . . . . . . 8 |- ( ( A e. NN0 /\ B e. ZZ /\ C e. NN0 ) -> ( ( A gcd C ) gcd ( C gcd B ) ) = ( ( A gcd B ) gcd C ) )
64	63	eqeq1d 2624	. . . . . . 7 |- ( ( A e. NN0 /\ B e. ZZ /\ C e. NN0 ) -> ( ( ( A gcd C ) gcd ( C gcd B ) ) = 1 <-> ( ( A gcd B ) gcd C ) = 1 ) )
65	64	biimpar 502	. . . . . 6 |- ( ( ( A e. NN0 /\ B e. ZZ /\ C e. NN0 ) /\ ( ( A gcd B ) gcd C ) = 1 ) -> ( ( A gcd C ) gcd ( C gcd B ) ) = 1 )
66	65	oveq2d 6666	. . . . 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 ) )
67	66	3adant3 1081	. . . 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 ) )
68	13	mulid1d 10057	. . . 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 )
69	67, 68	eqtrd 2656	. . 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 )
70	8, 42, 69	3eqtrrd 2661	. 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 ) )
71	70	3expia 1267	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 384   /\ w3a 1037   = wceq 1483   e. wcel 1990   ` cfv 5888  (class class class)co 6650   CCcc 9934   1c1 9937   x. cmul 9941   2c2 11070   NN0cn0 11292   ZZcz 11377   ^cexp 12860   abscabs 13974   gcd cgcd 15216

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-8 1992  ax-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246  ax-ext 2602  ax-sep 4781  ax-nul 4789  ax-pow 4843  ax-pr 4906  ax-un 6949  ax-cnex 9992  ax-resscn 9993  ax-1cn 9994  ax-icn 9995  ax-addcl 9996  ax-addrcl 9997  ax-mulcl 9998  ax-mulrcl 9999  ax-mulcom 10000  ax-addass 10001  ax-mulass 10002  ax-distr 10003  ax-i2m1 10004  ax-1ne0 10005  ax-1rid 10006  ax-rnegex 10007  ax-rrecex 10008  ax-cnre 10009  ax-pre-lttri 10010  ax-pre-lttrn 10011  ax-pre-ltadd 10012  ax-pre-mulgt0 10013  ax-pre-sup 10014

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3or 1038  df-3an 1039  df-tru 1486  df-ex 1705  df-nf 1710  df-sb 1881  df-eu 2474  df-mo 2475  df-clab 2609  df-cleq 2615  df-clel 2618  df-nfc 2753  df-ne 2795  df-nel 2898  df-ral 2917  df-rex 2918  df-reu 2919  df-rmo 2920  df-rab 2921  df-v 3202  df-sbc 3436  df-csb 3534  df-dif 3577  df-un 3579  df-in 3581  df-ss 3588  df-pss 3590  df-nul 3916  df-if 4087  df-pw 4160  df-sn 4178  df-pr 4180  df-tp 4182  df-op 4184  df-uni 4437  df-iun 4522  df-br 4654  df-opab 4713  df-mpt 4730  df-tr 4753  df-id 5024  df-eprel 5029  df-po 5035  df-so 5036  df-fr 5073  df-we 5075  df-xp 5120  df-rel 5121  df-cnv 5122  df-co 5123  df-dm 5124  df-rn 5125  df-res 5126  df-ima 5127  df-pred 5680  df-ord 5726  df-on 5727  df-lim 5728  df-suc 5729  df-iota 5851  df-fun 5890  df-fn 5891  df-f 5892  df-f1 5893  df-fo 5894  df-f1o 5895  df-fv 5896  df-riota 6611  df-ov 6653  df-oprab 6654  df-mpt2 6655  df-om 7066  df-2nd 7169  df-wrecs 7407  df-recs 7468  df-rdg 7506  df-er 7742  df-en 7956  df-dom 7957  df-sdom 7958  df-sup 8348  df-inf 8349  df-pnf 10076  df-mnf 10077  df-xr 10078  df-ltxr 10079  df-le 10080  df-sub 10268  df-neg 10269  df-div 10685  df-nn 11021  df-2 11079  df-3 11080  df-n0 11293  df-z 11378  df-uz 11688  df-rp 11833  df-fl 12593  df-mod 12669  df-seq 12802  df-exp 12861  df-cj 13839  df-re 13840  df-im 13841  df-sqrt 13975  df-abs 13976  df-dvds 14984  df-gcd 15217

This theorem is referenced by:  coprimeprodsq2  15514  pythagtriplem6  15526