Theorem pythagtriplem3 12776

Description: Lemma for pythagtrip 12792. Show that C and B are relatively prime under some conditions. (Contributed by Scott Fenton, 8-Apr-2014.) (Revised by Mario Carneiro, 19-Apr-2014.)

Assertion

Ref	Expression
pythagtriplem3	|- ( ( ( A e. NN /\ B e. NN /\ C e. NN ) /\ ( ( A ^ 2 ) + ( B ^ 2 ) ) = ( C ^ 2 ) /\ ( ( A gcd B ) = 1 /\ -. 2 || A ) ) -> ( B gcd C ) = 1 )

Proof of Theorem pythagtriplem3

Step	Hyp	Ref	Expression
1		oveq2 6002	. . . . . . 7 |- ( ( ( A ^ 2 ) + ( B ^ 2 ) ) = ( C ^ 2 ) -> ( ( B ^ 2 ) gcd ( ( A ^ 2 ) + ( B ^ 2 ) ) ) = ( ( B ^ 2 ) gcd ( C ^ 2 ) ) )
2	1	adantl 277	. . . . . 6 |- ( ( ( A e. NN /\ B e. NN /\ C e. NN ) /\ ( ( A ^ 2 ) + ( B ^ 2 ) ) = ( C ^ 2 ) ) -> ( ( B ^ 2 ) gcd ( ( A ^ 2 ) + ( B ^ 2 ) ) ) = ( ( B ^ 2 ) gcd ( C ^ 2 ) ) )
3		nnz 9453	. . . . . . . . . . 11 |- ( B e. NN -> B e. ZZ )
4		zsqcl 10819	. . . . . . . . . . 11 |- ( B e. ZZ -> ( B ^ 2 ) e. ZZ )
5	3, 4	syl 14	. . . . . . . . . 10 |- ( B e. NN -> ( B ^ 2 ) e. ZZ )
6	5	3ad2ant2 1043	. . . . . . . . 9 |- ( ( A e. NN /\ B e. NN /\ C e. NN ) -> ( B ^ 2 ) e. ZZ )
7		nnz 9453	. . . . . . . . . . 11 |- ( A e. NN -> A e. ZZ )
8		zsqcl 10819	. . . . . . . . . . 11 |- ( A e. ZZ -> ( A ^ 2 ) e. ZZ )
9	7, 8	syl 14	. . . . . . . . . 10 |- ( A e. NN -> ( A ^ 2 ) e. ZZ )
10	9	3ad2ant1 1042	. . . . . . . . 9 |- ( ( A e. NN /\ B e. NN /\ C e. NN ) -> ( A ^ 2 ) e. ZZ )
11		gcdadd 12492	. . . . . . . . 9 |- ( ( ( B ^ 2 ) e. ZZ /\ ( A ^ 2 ) e. ZZ ) -> ( ( B ^ 2 ) gcd ( A ^ 2 ) ) = ( ( B ^ 2 ) gcd ( ( A ^ 2 ) + ( B ^ 2 ) ) ) )
12	6, 10, 11	syl2anc 411	. . . . . . . 8 |- ( ( A e. NN /\ B e. NN /\ C e. NN ) -> ( ( B ^ 2 ) gcd ( A ^ 2 ) ) = ( ( B ^ 2 ) gcd ( ( A ^ 2 ) + ( B ^ 2 ) ) ) )
13	6, 10	gcdcomd 12481	. . . . . . . 8 |- ( ( A e. NN /\ B e. NN /\ C e. NN ) -> ( ( B ^ 2 ) gcd ( A ^ 2 ) ) = ( ( A ^ 2 ) gcd ( B ^ 2 ) ) )
14	12, 13	eqtr3d 2264	. . . . . . 7 |- ( ( A e. NN /\ B e. NN /\ C e. NN ) -> ( ( B ^ 2 ) gcd ( ( A ^ 2 ) + ( B ^ 2 ) ) ) = ( ( A ^ 2 ) gcd ( B ^ 2 ) ) )
15	14	adantr 276	. . . . . 6 |- ( ( ( A e. NN /\ B e. NN /\ C e. NN ) /\ ( ( A ^ 2 ) + ( B ^ 2 ) ) = ( C ^ 2 ) ) -> ( ( B ^ 2 ) gcd ( ( A ^ 2 ) + ( B ^ 2 ) ) ) = ( ( A ^ 2 ) gcd ( B ^ 2 ) ) )
16	2, 15	eqtr3d 2264	. . . . 5 |- ( ( ( A e. NN /\ B e. NN /\ C e. NN ) /\ ( ( A ^ 2 ) + ( B ^ 2 ) ) = ( C ^ 2 ) ) -> ( ( B ^ 2 ) gcd ( C ^ 2 ) ) = ( ( A ^ 2 ) gcd ( B ^ 2 ) ) )
17		simpl2 1025	. . . . . 6 |- ( ( ( A e. NN /\ B e. NN /\ C e. NN ) /\ ( ( A ^ 2 ) + ( B ^ 2 ) ) = ( C ^ 2 ) ) -> B e. NN )
18		simpl3 1026	. . . . . 6 |- ( ( ( A e. NN /\ B e. NN /\ C e. NN ) /\ ( ( A ^ 2 ) + ( B ^ 2 ) ) = ( C ^ 2 ) ) -> C e. NN )
19		sqgcd 12536	. . . . . 6 |- ( ( B e. NN /\ C e. NN ) -> ( ( B gcd C ) ^ 2 ) = ( ( B ^ 2 ) gcd ( C ^ 2 ) ) )
20	17, 18, 19	syl2anc 411	. . . . 5 |- ( ( ( A e. NN /\ B e. NN /\ C e. NN ) /\ ( ( A ^ 2 ) + ( B ^ 2 ) ) = ( C ^ 2 ) ) -> ( ( B gcd C ) ^ 2 ) = ( ( B ^ 2 ) gcd ( C ^ 2 ) ) )
21		simpl1 1024	. . . . . 6 |- ( ( ( A e. NN /\ B e. NN /\ C e. NN ) /\ ( ( A ^ 2 ) + ( B ^ 2 ) ) = ( C ^ 2 ) ) -> A e. NN )
22		sqgcd 12536	. . . . . 6 |- ( ( A e. NN /\ B e. NN ) -> ( ( A gcd B ) ^ 2 ) = ( ( A ^ 2 ) gcd ( B ^ 2 ) ) )
23	21, 17, 22	syl2anc 411	. . . . 5 |- ( ( ( A e. NN /\ B e. NN /\ C e. NN ) /\ ( ( A ^ 2 ) + ( B ^ 2 ) ) = ( C ^ 2 ) ) -> ( ( A gcd B ) ^ 2 ) = ( ( A ^ 2 ) gcd ( B ^ 2 ) ) )
24	16, 20, 23	3eqtr4d 2272	. . . 4 |- ( ( ( A e. NN /\ B e. NN /\ C e. NN ) /\ ( ( A ^ 2 ) + ( B ^ 2 ) ) = ( C ^ 2 ) ) -> ( ( B gcd C ) ^ 2 ) = ( ( A gcd B ) ^ 2 ) )
25	24	3adant3 1041	. . 3 |- ( ( ( A e. NN /\ B e. NN /\ C e. NN ) /\ ( ( A ^ 2 ) + ( B ^ 2 ) ) = ( C ^ 2 ) /\ ( ( A gcd B ) = 1 /\ -. 2 || A ) ) -> ( ( B gcd C ) ^ 2 ) = ( ( A gcd B ) ^ 2 ) )
26		simp3l 1049	. . . 4 |- ( ( ( A e. NN /\ B e. NN /\ C e. NN ) /\ ( ( A ^ 2 ) + ( B ^ 2 ) ) = ( C ^ 2 ) /\ ( ( A gcd B ) = 1 /\ -. 2 || A ) ) -> ( A gcd B ) = 1 )
27	26	oveq1d 6009	. . 3 |- ( ( ( A e. NN /\ B e. NN /\ C e. NN ) /\ ( ( A ^ 2 ) + ( B ^ 2 ) ) = ( C ^ 2 ) /\ ( ( A gcd B ) = 1 /\ -. 2 || A ) ) -> ( ( A gcd B ) ^ 2 ) = ( 1 ^ 2 ) )
28	25, 27	eqtrd 2262	. 2 |- ( ( ( A e. NN /\ B e. NN /\ C e. NN ) /\ ( ( A ^ 2 ) + ( B ^ 2 ) ) = ( C ^ 2 ) /\ ( ( A gcd B ) = 1 /\ -. 2 || A ) ) -> ( ( B gcd C ) ^ 2 ) = ( 1 ^ 2 ) )
29	3	3ad2ant2 1043	. . . . . 6 |- ( ( A e. NN /\ B e. NN /\ C e. NN ) -> B e. ZZ )
30		nnz 9453	. . . . . . 7 |- ( C e. NN -> C e. ZZ )
31	30	3ad2ant3 1044	. . . . . 6 |- ( ( A e. NN /\ B e. NN /\ C e. NN ) -> C e. ZZ )
32	29, 31	gcdcld 12475	. . . . 5 |- ( ( A e. NN /\ B e. NN /\ C e. NN ) -> ( B gcd C ) e. NN0 )
33	32	nn0red 9411	. . . 4 |- ( ( A e. NN /\ B e. NN /\ C e. NN ) -> ( B gcd C ) e. RR )
34	33	3ad2ant1 1042	. . 3 |- ( ( ( A e. NN /\ B e. NN /\ C e. NN ) /\ ( ( A ^ 2 ) + ( B ^ 2 ) ) = ( C ^ 2 ) /\ ( ( A gcd B ) = 1 /\ -. 2 || A ) ) -> ( B gcd C ) e. RR )
35	32	nn0ge0d 9413	. . . 4 |- ( ( A e. NN /\ B e. NN /\ C e. NN ) -> 0 <_ ( B gcd C ) )
36	35	3ad2ant1 1042	. . 3 |- ( ( ( A e. NN /\ B e. NN /\ C e. NN ) /\ ( ( A ^ 2 ) + ( B ^ 2 ) ) = ( C ^ 2 ) /\ ( ( A gcd B ) = 1 /\ -. 2 || A ) ) -> 0 <_ ( B gcd C ) )
37		1re 8133	. . . 4 |- 1 e. RR
38		0le1 8616	. . . 4 |- 0 <_ 1
39		sq11 10821	. . . 4 |- ( ( ( ( B gcd C ) e. RR /\ 0 <_ ( B gcd C ) ) /\ ( 1 e. RR /\ 0 <_ 1 ) ) -> ( ( ( B gcd C ) ^ 2 ) = ( 1 ^ 2 ) <-> ( B gcd C ) = 1 ) )
40	37, 38, 39	mpanr12 439	. . 3 |- ( ( ( B gcd C ) e. RR /\ 0 <_ ( B gcd C ) ) -> ( ( ( B gcd C ) ^ 2 ) = ( 1 ^ 2 ) <-> ( B gcd C ) = 1 ) )
41	34, 36, 40	syl2anc 411	. 2 |- ( ( ( A e. NN /\ B e. NN /\ C e. NN ) /\ ( ( A ^ 2 ) + ( B ^ 2 ) ) = ( C ^ 2 ) /\ ( ( A gcd B ) = 1 /\ -. 2 || A ) ) -> ( ( ( B gcd C ) ^ 2 ) = ( 1 ^ 2 ) <-> ( B gcd C ) = 1 ) )
42	28, 41	mpbid 147	1 |- ( ( ( A e. NN /\ B e. NN /\ C e. NN ) /\ ( ( A ^ 2 ) + ( B ^ 2 ) ) = ( C ^ 2 ) /\ ( ( A gcd B ) = 1 /\ -. 2 || A ) ) -> ( B gcd C ) = 1 )

Syntax hints:   -. wn 3   -> wi 4   /\ wa 104   <-> wb 105   /\ w3a 1002   = wceq 1395   e. wcel 2200   class class class wbr 4082  (class class class)co 5994   RRcr 7986   0cc0 7987   1c1 7988   + caddc 7990   <_ cle 8170   NNcn 9098   2c2 9149   ZZcz 9434   ^cexp 10747   || cdvds 12284   gcd cgcd 12460

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 617  ax-in2 618  ax-io 714  ax-5 1493  ax-7 1494  ax-gen 1495  ax-ie1 1539  ax-ie2 1540  ax-8 1550  ax-10 1551  ax-11 1552  ax-i12 1553  ax-bndl 1555  ax-4 1556  ax-17 1572  ax-i9 1576  ax-ial 1580  ax-i5r 1581  ax-13 2202  ax-14 2203  ax-ext 2211  ax-coll 4198  ax-sep 4201  ax-nul 4209  ax-pow 4257  ax-pr 4292  ax-un 4521  ax-setind 4626  ax-iinf 4677  ax-cnex 8078  ax-resscn 8079  ax-1cn 8080  ax-1re 8081  ax-icn 8082  ax-addcl 8083  ax-addrcl 8084  ax-mulcl 8085  ax-mulrcl 8086  ax-addcom 8087  ax-mulcom 8088  ax-addass 8089  ax-mulass 8090  ax-distr 8091  ax-i2m1 8092  ax-0lt1 8093  ax-1rid 8094  ax-0id 8095  ax-rnegex 8096  ax-precex 8097  ax-cnre 8098  ax-pre-ltirr 8099  ax-pre-ltwlin 8100  ax-pre-lttrn 8101  ax-pre-apti 8102  ax-pre-ltadd 8103  ax-pre-mulgt0 8104  ax-pre-mulext 8105  ax-arch 8106  ax-caucvg 8107

This theorem depends on definitions:  df-bi 117  df-stab 836  df-dc 840  df-3or 1003  df-3an 1004  df-tru 1398  df-fal 1401  df-nf 1507  df-sb 1809  df-eu 2080  df-mo 2081  df-clab 2216  df-cleq 2222  df-clel 2225  df-nfc 2361  df-ne 2401  df-nel 2496  df-ral 2513  df-rex 2514  df-reu 2515  df-rmo 2516  df-rab 2517  df-v 2801  df-sbc 3029  df-csb 3125  df-dif 3199  df-un 3201  df-in 3203  df-ss 3210  df-nul 3492  df-if 3603  df-pw 3651  df-sn 3672  df-pr 3673  df-op 3675  df-uni 3888  df-int 3923  df-iun 3966  df-br 4083  df-opab 4145  df-mpt 4146  df-tr 4182  df-id 4381  df-po 4384  df-iso 4385  df-iord 4454  df-on 4456  df-ilim 4457  df-suc 4459  df-iom 4680  df-xp 4722  df-rel 4723  df-cnv 4724  df-co 4725  df-dm 4726  df-rn 4727  df-res 4728  df-ima 4729  df-iota 5274  df-fun 5316  df-fn 5317  df-f 5318  df-f1 5319  df-fo 5320  df-f1o 5321  df-fv 5322  df-riota 5947  df-ov 5997  df-oprab 5998  df-mpo 5999  df-1st 6276  df-2nd 6277  df-recs 6441  df-frec 6527  df-sup 7139  df-pnf 8171  df-mnf 8172  df-xr 8173  df-ltxr 8174  df-le 8175  df-sub 8307  df-neg 8308  df-reap 8710  df-ap 8717  df-div 8808  df-inn 9099  df-2 9157  df-3 9158  df-4 9159  df-n0 9358  df-z 9435  df-uz 9711  df-q 9803  df-rp 9838  df-fz 10193  df-fzo 10327  df-fl 10477  df-mod 10532  df-seqfrec 10657  df-exp 10748  df-cj 11339  df-re 11340  df-im 11341  df-rsqrt 11495  df-abs 11496  df-dvds 12285  df-gcd 12461

This theorem is referenced by:  pythagtriplem4  12777