Theorem coprimeprodsq 12613

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 9394	. . . . . . . 8 |- ( A e. NN0 -> A e. ZZ )
2		nn0z 9394	. . . . . . . 8 |- ( C e. NN0 -> C e. ZZ )
3		gcdcl 12320	. . . . . . . 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 9352	. . . 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 10817	. . 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 9352	. . . . . . . 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 9307	. . . . . . . . . 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 8096	. . . . . . 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 9352	. . . . . . . . . 10 |- ( ( ( A e. NN0 /\ B e. ZZ /\ C e. NN0 ) /\ ( ( A gcd B ) gcd C ) = 1 ) -> C e. CC )
17	16	sqvald 10817	. . . . . . . . 9 |- ( ( ( A e. NN0 /\ B e. ZZ /\ C e. NN0 ) /\ ( ( A gcd B ) gcd C ) = 1 ) -> ( C ^ 2 ) = ( C x. C ) )
18	17	eqeq1d 2214	. . . . . . . 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 5964	. . . . . 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 9495	. . . . . . 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 9495	. . . . . . 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 12370	. . . . . . 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 12370	. . . . . . 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 2246	. . . . 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 5962	. . . 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 12372	. . . . 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 9495	. . . . 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 12320	. . . . . . . . . 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 9495	. . . . 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 12370	. . . . 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 2246	. . 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 12340	. . . . . . . . . . . . . 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 5961	. . . . . . . . . . . 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 12343	. . . . . . . . . . . . 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 2238	. . . . . . . . . . 11 |- ( ( A e. NN0 /\ B e. ZZ /\ C e. NN0 ) -> ( ( C gcd C ) gcd B ) = ( C gcd B ) )
51		gcdass 12369	. . . . . . . . . . . 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 12328	. . . . . . . . . . 11 |- ( ( A e. NN0 /\ B e. ZZ /\ C e. NN0 ) -> ( C gcd B ) = ( B gcd C ) )
54	50, 52, 53	3eqtr3d 2246	. . . . . . . . . 10 |- ( ( A e. NN0 /\ B e. ZZ /\ C e. NN0 ) -> ( C gcd ( C gcd B ) ) = ( B gcd C ) )
55	54	oveq2d 5962	. . . . . . . . 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 9495	. . . . . . . . . 10 |- ( ( A e. NN0 /\ B e. ZZ /\ C e. NN0 ) -> ( C gcd B ) e. ZZ )
58		gcdass 12369	. . . . . . . . . 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 12369	. . . . . . . . . 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 2248	. . . . . . . 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 2214	. . . . . . 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 5962	. . . . 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 8091	. . . 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 2238	. . 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 2243	. 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 2176   ` cfv 5272  (class class class)co 5946   CCcc 7925   1c1 7928   x. cmul 7932   2c2 9089   NN0cn0 9297   ZZcz 9374   ^cexp 10685   abscabs 11341   gcd cgcd 12307

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 1470  ax-7 1471  ax-gen 1472  ax-ie1 1516  ax-ie2 1517  ax-8 1527  ax-10 1528  ax-11 1529  ax-i12 1530  ax-bndl 1532  ax-4 1533  ax-17 1549  ax-i9 1553  ax-ial 1557  ax-i5r 1558  ax-13 2178  ax-14 2179  ax-ext 2187  ax-coll 4160  ax-sep 4163  ax-nul 4171  ax-pow 4219  ax-pr 4254  ax-un 4481  ax-setind 4586  ax-iinf 4637  ax-cnex 8018  ax-resscn 8019  ax-1cn 8020  ax-1re 8021  ax-icn 8022  ax-addcl 8023  ax-addrcl 8024  ax-mulcl 8025  ax-mulrcl 8026  ax-addcom 8027  ax-mulcom 8028  ax-addass 8029  ax-mulass 8030  ax-distr 8031  ax-i2m1 8032  ax-0lt1 8033  ax-1rid 8034  ax-0id 8035  ax-rnegex 8036  ax-precex 8037  ax-cnre 8038  ax-pre-ltirr 8039  ax-pre-ltwlin 8040  ax-pre-lttrn 8041  ax-pre-apti 8042  ax-pre-ltadd 8043  ax-pre-mulgt0 8044  ax-pre-mulext 8045  ax-arch 8046  ax-caucvg 8047

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 1484  df-sb 1786  df-eu 2057  df-mo 2058  df-clab 2192  df-cleq 2198  df-clel 2201  df-nfc 2337  df-ne 2377  df-nel 2472  df-ral 2489  df-rex 2490  df-reu 2491  df-rmo 2492  df-rab 2493  df-v 2774  df-sbc 2999  df-csb 3094  df-dif 3168  df-un 3170  df-in 3172  df-ss 3179  df-nul 3461  df-if 3572  df-pw 3618  df-sn 3639  df-pr 3640  df-op 3642  df-uni 3851  df-int 3886  df-iun 3929  df-br 4046  df-opab 4107  df-mpt 4108  df-tr 4144  df-id 4341  df-po 4344  df-iso 4345  df-iord 4414  df-on 4416  df-ilim 4417  df-suc 4419  df-iom 4640  df-xp 4682  df-rel 4683  df-cnv 4684  df-co 4685  df-dm 4686  df-rn 4687  df-res 4688  df-ima 4689  df-iota 5233  df-fun 5274  df-fn 5275  df-f 5276  df-f1 5277  df-fo 5278  df-f1o 5279  df-fv 5280  df-riota 5901  df-ov 5949  df-oprab 5950  df-mpo 5951  df-1st 6228  df-2nd 6229  df-recs 6393  df-frec 6479  df-sup 7088  df-pnf 8111  df-mnf 8112  df-xr 8113  df-ltxr 8114  df-le 8115  df-sub 8247  df-neg 8248  df-reap 8650  df-ap 8657  df-div 8748  df-inn 9039  df-2 9097  df-3 9098  df-4 9099  df-n0 9298  df-z 9375  df-uz 9651  df-q 9743  df-rp 9778  df-fz 10133  df-fzo 10267  df-fl 10415  df-mod 10470  df-seqfrec 10595  df-exp 10686  df-cj 11186  df-re 11187  df-im 11188  df-rsqrt 11342  df-abs 11343  df-dvds 12132  df-gcd 12308

This theorem is referenced by:  coprimeprodsq2  12614  pythagtriplem6  12626