Theorem coprimeprodsq2 15514

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, 17-Apr-2014.) (Revised by Mario Carneiro, 19-Apr-2014.)

Assertion

Ref	Expression
coprimeprodsq2	|- ( ( ( A e. ZZ /\ B e. NN0 /\ C e. NN0 ) /\ ( ( A gcd B ) gcd C ) = 1 ) -> ( ( C ^ 2 ) = ( A x. B ) -> B = ( ( B gcd C ) ^ 2 ) ) )

Proof of Theorem coprimeprodsq2

Step	Hyp	Ref	Expression
1		zcn 11382	. . . . . 6 |- ( A e. ZZ -> A e. CC )
2		nn0cn 11302	. . . . . 6 |- ( B e. NN0 -> B e. CC )
3		mulcom 10022	. . . . . 6 |- ( ( A e. CC /\ B e. CC ) -> ( A x. B ) = ( B x. A ) )
4	1, 2, 3	syl2an 494	. . . . 5 |- ( ( A e. ZZ /\ B e. NN0 ) -> ( A x. B ) = ( B x. A ) )
5	4	3adant3 1081	. . . 4 |- ( ( A e. ZZ /\ B e. NN0 /\ C e. NN0 ) -> ( A x. B ) = ( B x. A ) )
6	5	adantr 481	. . 3 |- ( ( ( A e. ZZ /\ B e. NN0 /\ C e. NN0 ) /\ ( ( A gcd B ) gcd C ) = 1 ) -> ( A x. B ) = ( B x. A ) )
7	6	eqeq2d 2632	. 2 |- ( ( ( A e. ZZ /\ B e. NN0 /\ C e. NN0 ) /\ ( ( A gcd B ) gcd C ) = 1 ) -> ( ( C ^ 2 ) = ( A x. B ) <-> ( C ^ 2 ) = ( B x. A ) ) )
8		simpl2 1065	. . 3 |- ( ( ( A e. ZZ /\ B e. NN0 /\ C e. NN0 ) /\ ( ( A gcd B ) gcd C ) = 1 ) -> B e. NN0 )
9		simpl1 1064	. . 3 |- ( ( ( A e. ZZ /\ B e. NN0 /\ C e. NN0 ) /\ ( ( A gcd B ) gcd C ) = 1 ) -> A e. ZZ )
10		simpl3 1066	. . 3 |- ( ( ( A e. ZZ /\ B e. NN0 /\ C e. NN0 ) /\ ( ( A gcd B ) gcd C ) = 1 ) -> C e. NN0 )
11		nn0z 11400	. . . . . 6 |- ( B e. NN0 -> B e. ZZ )
12		gcdcom 15235	. . . . . . . 8 |- ( ( A e. ZZ /\ B e. ZZ ) -> ( A gcd B ) = ( B gcd A ) )
13	12	oveq1d 6665	. . . . . . 7 |- ( ( A e. ZZ /\ B e. ZZ ) -> ( ( A gcd B ) gcd C ) = ( ( B gcd A ) gcd C ) )
14	13	eqeq1d 2624	. . . . . 6 |- ( ( A e. ZZ /\ B e. ZZ ) -> ( ( ( A gcd B ) gcd C ) = 1 <-> ( ( B gcd A ) gcd C ) = 1 ) )
15	11, 14	sylan2 491	. . . . 5 |- ( ( A e. ZZ /\ B e. NN0 ) -> ( ( ( A gcd B ) gcd C ) = 1 <-> ( ( B gcd A ) gcd C ) = 1 ) )
16	15	3adant3 1081	. . . 4 |- ( ( A e. ZZ /\ B e. NN0 /\ C e. NN0 ) -> ( ( ( A gcd B ) gcd C ) = 1 <-> ( ( B gcd A ) gcd C ) = 1 ) )
17	16	biimpa 501	. . 3 |- ( ( ( A e. ZZ /\ B e. NN0 /\ C e. NN0 ) /\ ( ( A gcd B ) gcd C ) = 1 ) -> ( ( B gcd A ) gcd C ) = 1 )
18		coprimeprodsq 15513	. . 3 |- ( ( ( B e. NN0 /\ A e. ZZ /\ C e. NN0 ) /\ ( ( B gcd A ) gcd C ) = 1 ) -> ( ( C ^ 2 ) = ( B x. A ) -> B = ( ( B gcd C ) ^ 2 ) ) )
19	8, 9, 10, 17, 18	syl31anc 1329	. 2 |- ( ( ( A e. ZZ /\ B e. NN0 /\ C e. NN0 ) /\ ( ( A gcd B ) gcd C ) = 1 ) -> ( ( C ^ 2 ) = ( B x. A ) -> B = ( ( B gcd C ) ^ 2 ) ) )
20	7, 19	sylbid 230	1 |- ( ( ( A e. ZZ /\ B e. NN0 /\ C e. NN0 ) /\ ( ( A gcd B ) gcd C ) = 1 ) -> ( ( C ^ 2 ) = ( A x. B ) -> B = ( ( B gcd C ) ^ 2 ) ) )

Syntax hints:   -> wi 4   <-> wb 196   /\ wa 384   /\ w3a 1037   = wceq 1483   e. wcel 1990  (class class class)co 6650   CCcc 9934   1c1 9937   x. cmul 9941   2c2 11070   NN0cn0 11292   ZZcz 11377   ^cexp 12860   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:  pythagtriplem7  15527