Theorem coprimeprodsq 12695

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 9427	. . . . . . . 8 |- ( A e. NN0 -> A e. ZZ )
2		nn0z 9427	. . . . . . . 8 |- ( C e. NN0 -> C e. ZZ )
3		gcdcl 12402	. . . . . . . 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 1019	. . . . . 6 |- ( ( A e. NN0 /\ B e. ZZ /\ C e. NN0 ) -> ( A gcd C ) e. NN0 )
6	5	3ad2ant1 1021	. . . . 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 9385	. . . 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 10852	. . 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 1032	. . . . . . . . 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 9385	. . . . . . . 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 9340	. . . . . . . . . 10 |- ( A e. NN0 -> A e. CC )
12	11	3ad2ant1 1021	. . . . . . . . 9 |- ( ( A e. NN0 /\ B e. ZZ /\ C e. NN0 ) -> A e. CC )
13	12	3ad2ant1 1021	. . . . . . . 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 8129	. . . . . . 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 1005	. . . . . . . . . . 11 |- ( ( ( A e. NN0 /\ B e. ZZ /\ C e. NN0 ) /\ ( ( A gcd B ) gcd C ) = 1 ) -> C e. NN0 )
16	15	nn0cnd 9385	. . . . . . . . . 10 |- ( ( ( A e. NN0 /\ B e. ZZ /\ C e. NN0 ) /\ ( ( A gcd B ) gcd C ) = 1 ) -> C e. CC )
17	16	sqvald 10852	. . . . . . . . 9 |- ( ( ( A e. NN0 /\ B e. ZZ /\ C e. NN0 ) /\ ( ( A gcd B ) gcd C ) = 1 ) -> ( C ^ 2 ) = ( C x. C ) )
18	17	eqeq1d 2216	. . . . . . . 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 1358	. . . . . . 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 5985	. . . . . 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 1030	. . . . . . . 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 9528	. . . . . . 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 9528	. . . . . . 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 12452	. . . . . . 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 1250	. . . . . 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 1031	. . . . . . 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 12452	. . . . . . 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 1250	. . . . . 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 2248	. . . . 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 5983	. . . 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 12454	. . . . 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 1250	. . . 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 9528	. . . . 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 12402	. . . . . . . . . 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 1018	. . . . . . 7 |- ( ( A e. NN0 /\ B e. ZZ /\ C e. NN0 ) -> ( C gcd B ) e. NN0 )
38	37	3ad2ant1 1021	. . . . . 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 9528	. . . . 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 12452	. . . . 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 1250	. . . 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 2248	. . 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 1023	. . . . . . . . . . . . . 14 |- ( ( A e. NN0 /\ B e. ZZ /\ C e. NN0 ) -> C e. ZZ )
44		gcdid 12422	. . . . . . . . . . . . . 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 5982	. . . . . . . . . . . 12 |- ( ( A e. NN0 /\ B e. ZZ /\ C e. NN0 ) -> ( ( C gcd C ) gcd B ) = ( ( abs ` C ) gcd B ) )
47		simp2 1001	. . . . . . . . . . . . 13 |- ( ( A e. NN0 /\ B e. ZZ /\ C e. NN0 ) -> B e. ZZ )
48		gcdabs1 12425	. . . . . . . . . . . . 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 2240	. . . . . . . . . . 11 |- ( ( A e. NN0 /\ B e. ZZ /\ C e. NN0 ) -> ( ( C gcd C ) gcd B ) = ( C gcd B ) )
51		gcdass 12451	. . . . . . . . . . . 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 1250	. . . . . . . . . . 11 |- ( ( A e. NN0 /\ B e. ZZ /\ C e. NN0 ) -> ( ( C gcd C ) gcd B ) = ( C gcd ( C gcd B ) ) )
53	43, 47	gcdcomd 12410	. . . . . . . . . . 11 |- ( ( A e. NN0 /\ B e. ZZ /\ C e. NN0 ) -> ( C gcd B ) = ( B gcd C ) )
54	50, 52, 53	3eqtr3d 2248	. . . . . . . . . 10 |- ( ( A e. NN0 /\ B e. ZZ /\ C e. NN0 ) -> ( C gcd ( C gcd B ) ) = ( B gcd C ) )
55	54	oveq2d 5983	. . . . . . . . 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 1021	. . . . . . . . . 10 |- ( ( A e. NN0 /\ B e. ZZ /\ C e. NN0 ) -> A e. ZZ )
57	37	nn0zd 9528	. . . . . . . . . 10 |- ( ( A e. NN0 /\ B e. ZZ /\ C e. NN0 ) -> ( C gcd B ) e. ZZ )
58		gcdass 12451	. . . . . . . . . 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 1250	. . . . . . . . 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 12451	. . . . . . . . . 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 1250	. . . . . . . . 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 2250	. . . . . . . 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 2216	. . . . . . 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 5983	. . . . 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 1020	. . . 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 8124	. . . 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 2240	. . 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 2245	. 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 1208	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 981   = wceq 1373   e. wcel 2178   ` cfv 5290  (class class class)co 5967   CCcc 7958   1c1 7961   x. cmul 7965   2c2 9122   NN0cn0 9330   ZZcz 9407   ^cexp 10720   abscabs 11423   gcd cgcd 12389

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 711  ax-5 1471  ax-7 1472  ax-gen 1473  ax-ie1 1517  ax-ie2 1518  ax-8 1528  ax-10 1529  ax-11 1530  ax-i12 1531  ax-bndl 1533  ax-4 1534  ax-17 1550  ax-i9 1554  ax-ial 1558  ax-i5r 1559  ax-13 2180  ax-14 2181  ax-ext 2189  ax-coll 4175  ax-sep 4178  ax-nul 4186  ax-pow 4234  ax-pr 4269  ax-un 4498  ax-setind 4603  ax-iinf 4654  ax-cnex 8051  ax-resscn 8052  ax-1cn 8053  ax-1re 8054  ax-icn 8055  ax-addcl 8056  ax-addrcl 8057  ax-mulcl 8058  ax-mulrcl 8059  ax-addcom 8060  ax-mulcom 8061  ax-addass 8062  ax-mulass 8063  ax-distr 8064  ax-i2m1 8065  ax-0lt1 8066  ax-1rid 8067  ax-0id 8068  ax-rnegex 8069  ax-precex 8070  ax-cnre 8071  ax-pre-ltirr 8072  ax-pre-ltwlin 8073  ax-pre-lttrn 8074  ax-pre-apti 8075  ax-pre-ltadd 8076  ax-pre-mulgt0 8077  ax-pre-mulext 8078  ax-arch 8079  ax-caucvg 8080

This theorem depends on definitions:  df-bi 117  df-stab 833  df-dc 837  df-3or 982  df-3an 983  df-tru 1376  df-fal 1379  df-nf 1485  df-sb 1787  df-eu 2058  df-mo 2059  df-clab 2194  df-cleq 2200  df-clel 2203  df-nfc 2339  df-ne 2379  df-nel 2474  df-ral 2491  df-rex 2492  df-reu 2493  df-rmo 2494  df-rab 2495  df-v 2778  df-sbc 3006  df-csb 3102  df-dif 3176  df-un 3178  df-in 3180  df-ss 3187  df-nul 3469  df-if 3580  df-pw 3628  df-sn 3649  df-pr 3650  df-op 3652  df-uni 3865  df-int 3900  df-iun 3943  df-br 4060  df-opab 4122  df-mpt 4123  df-tr 4159  df-id 4358  df-po 4361  df-iso 4362  df-iord 4431  df-on 4433  df-ilim 4434  df-suc 4436  df-iom 4657  df-xp 4699  df-rel 4700  df-cnv 4701  df-co 4702  df-dm 4703  df-rn 4704  df-res 4705  df-ima 4706  df-iota 5251  df-fun 5292  df-fn 5293  df-f 5294  df-f1 5295  df-fo 5296  df-f1o 5297  df-fv 5298  df-riota 5922  df-ov 5970  df-oprab 5971  df-mpo 5972  df-1st 6249  df-2nd 6250  df-recs 6414  df-frec 6500  df-sup 7112  df-pnf 8144  df-mnf 8145  df-xr 8146  df-ltxr 8147  df-le 8148  df-sub 8280  df-neg 8281  df-reap 8683  df-ap 8690  df-div 8781  df-inn 9072  df-2 9130  df-3 9131  df-4 9132  df-n0 9331  df-z 9408  df-uz 9684  df-q 9776  df-rp 9811  df-fz 10166  df-fzo 10300  df-fl 10450  df-mod 10505  df-seqfrec 10630  df-exp 10721  df-cj 11268  df-re 11269  df-im 11270  df-rsqrt 11424  df-abs 11425  df-dvds 12214  df-gcd 12390

This theorem is referenced by:  coprimeprodsq2  12696  pythagtriplem6  12708