Theorem coprimeprodsq2 12149

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 9178	. . . . . 6 |- ( A e. ZZ -> A e. CC )
2		nn0cn 9106	. . . . . 6 |- ( B e. NN0 -> B e. CC )
3		mulcom 7864	. . . . . 6 |- ( ( A e. CC /\ B e. CC ) -> ( A x. B ) = ( B x. A ) )
4	1, 2, 3	syl2an 287	. . . . 5 |- ( ( A e. ZZ /\ B e. NN0 ) -> ( A x. B ) = ( B x. A ) )
5	4	3adant3 1002	. . . 4 |- ( ( A e. ZZ /\ B e. NN0 /\ C e. NN0 ) -> ( A x. B ) = ( B x. A ) )
6	5	adantr 274	. . 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 2169	. 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 986	. . 3 |- ( ( ( A e. ZZ /\ B e. NN0 /\ C e. NN0 ) /\ ( ( A gcd B ) gcd C ) = 1 ) -> B e. NN0 )
9		simpl1 985	. . 3 |- ( ( ( A e. ZZ /\ B e. NN0 /\ C e. NN0 ) /\ ( ( A gcd B ) gcd C ) = 1 ) -> A e. ZZ )
10		simpl3 987	. . 3 |- ( ( ( A e. ZZ /\ B e. NN0 /\ C e. NN0 ) /\ ( ( A gcd B ) gcd C ) = 1 ) -> C e. NN0 )
11		nn0z 9193	. . . . . 6 |- ( B e. NN0 -> B e. ZZ )
12		gcdcom 11873	. . . . . . . 8 |- ( ( A e. ZZ /\ B e. ZZ ) -> ( A gcd B ) = ( B gcd A ) )
13	12	oveq1d 5842	. . . . . . 7 |- ( ( A e. ZZ /\ B e. ZZ ) -> ( ( A gcd B ) gcd C ) = ( ( B gcd A ) gcd C ) )
14	13	eqeq1d 2166	. . . . . 6 |- ( ( A e. ZZ /\ B e. ZZ ) -> ( ( ( A gcd B ) gcd C ) = 1 <-> ( ( B gcd A ) gcd C ) = 1 ) )
15	11, 14	sylan2 284	. . . . 5 |- ( ( A e. ZZ /\ B e. NN0 ) -> ( ( ( A gcd B ) gcd C ) = 1 <-> ( ( B gcd A ) gcd C ) = 1 ) )
16	15	3adant3 1002	. . . 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 294	. . 3 |- ( ( ( A e. ZZ /\ B e. NN0 /\ C e. NN0 ) /\ ( ( A gcd B ) gcd C ) = 1 ) -> ( ( B gcd A ) gcd C ) = 1 )
18		coprimeprodsq 12148	. . 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 1223	. 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 149	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   /\ wa 103   <-> wb 104   /\ w3a 963   = wceq 1335   e. wcel 2128  (class class class)co 5827   CCcc 7733   1c1 7736   x. cmul 7740   2c2 8890   NN0cn0 9096   ZZcz 9173   ^cexp 10428   gcd cgcd 11842

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-in1 604  ax-in2 605  ax-io 699  ax-5 1427  ax-7 1428  ax-gen 1429  ax-ie1 1473  ax-ie2 1474  ax-8 1484  ax-10 1485  ax-11 1486  ax-i12 1487  ax-bndl 1489  ax-4 1490  ax-17 1506  ax-i9 1510  ax-ial 1514  ax-i5r 1515  ax-13 2130  ax-14 2131  ax-ext 2139  ax-coll 4082  ax-sep 4085  ax-nul 4093  ax-pow 4138  ax-pr 4172  ax-un 4396  ax-setind 4499  ax-iinf 4550  ax-cnex 7826  ax-resscn 7827  ax-1cn 7828  ax-1re 7829  ax-icn 7830  ax-addcl 7831  ax-addrcl 7832  ax-mulcl 7833  ax-mulrcl 7834  ax-addcom 7835  ax-mulcom 7836  ax-addass 7837  ax-mulass 7838  ax-distr 7839  ax-i2m1 7840  ax-0lt1 7841  ax-1rid 7842  ax-0id 7843  ax-rnegex 7844  ax-precex 7845  ax-cnre 7846  ax-pre-ltirr 7847  ax-pre-ltwlin 7848  ax-pre-lttrn 7849  ax-pre-apti 7850  ax-pre-ltadd 7851  ax-pre-mulgt0 7852  ax-pre-mulext 7853  ax-arch 7854  ax-caucvg 7855

This theorem depends on definitions:  df-bi 116  df-stab 817  df-dc 821  df-3or 964  df-3an 965  df-tru 1338  df-fal 1341  df-nf 1441  df-sb 1743  df-eu 2009  df-mo 2010  df-clab 2144  df-cleq 2150  df-clel 2153  df-nfc 2288  df-ne 2328  df-nel 2423  df-ral 2440  df-rex 2441  df-reu 2442  df-rmo 2443  df-rab 2444  df-v 2714  df-sbc 2938  df-csb 3032  df-dif 3104  df-un 3106  df-in 3108  df-ss 3115  df-nul 3396  df-if 3507  df-pw 3546  df-sn 3567  df-pr 3568  df-op 3570  df-uni 3775  df-int 3810  df-iun 3853  df-br 3968  df-opab 4029  df-mpt 4030  df-tr 4066  df-id 4256  df-po 4259  df-iso 4260  df-iord 4329  df-on 4331  df-ilim 4332  df-suc 4334  df-iom 4553  df-xp 4595  df-rel 4596  df-cnv 4597  df-co 4598  df-dm 4599  df-rn 4600  df-res 4601  df-ima 4602  df-iota 5138  df-fun 5175  df-fn 5176  df-f 5177  df-f1 5178  df-fo 5179  df-f1o 5180  df-fv 5181  df-riota 5783  df-ov 5830  df-oprab 5831  df-mpo 5832  df-1st 6091  df-2nd 6092  df-recs 6255  df-frec 6341  df-sup 6931  df-pnf 7917  df-mnf 7918  df-xr 7919  df-ltxr 7920  df-le 7921  df-sub 8053  df-neg 8054  df-reap 8455  df-ap 8462  df-div 8551  df-inn 8840  df-2 8898  df-3 8899  df-4 8900  df-n0 9097  df-z 9174  df-uz 9446  df-q 9536  df-rp 9568  df-fz 9920  df-fzo 10052  df-fl 10179  df-mod 10232  df-seqfrec 10355  df-exp 10429  df-cj 10754  df-re 10755  df-im 10756  df-rsqrt 10910  df-abs 10911  df-dvds 11696  df-gcd 11843

This theorem is referenced by:  pythagtriplem7  12162