Theorem pythagtriplem3 13076

Description: Lemma for pythagtrip 13092. 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 6096	. . . . . . 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 9675	. . . . . . . . . . 11 |- ( B e. NN -> B e. ZZ )
4		zsqcl 11069	. . . . . . . . . . 11 |- ( B e. ZZ -> ( B ^ 2 ) e. ZZ )
5	3, 4	syl 14	. . . . . . . . . 10 |- ( B e. NN -> ( B ^ 2 ) e. ZZ )
6	5	3ad2ant2 1050	. . . . . . . . 9 |- ( ( A e. NN /\ B e. NN /\ C e. NN ) -> ( B ^ 2 ) e. ZZ )
7		nnz 9675	. . . . . . . . . . 11 |- ( A e. NN -> A e. ZZ )
8		zsqcl 11069	. . . . . . . . . . 11 |- ( A e. ZZ -> ( A ^ 2 ) e. ZZ )
9	7, 8	syl 14	. . . . . . . . . 10 |- ( A e. NN -> ( A ^ 2 ) e. ZZ )
10	9	3ad2ant1 1049	. . . . . . . . 9 |- ( ( A e. NN /\ B e. NN /\ C e. NN ) -> ( A ^ 2 ) e. ZZ )
11		gcdadd 12788	. . . . . . . . 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 415	. . . . . . . 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 12777	. . . . . . . 8 |- ( ( A e. NN /\ B e. NN /\ C e. NN ) -> ( ( B ^ 2 ) gcd ( A ^ 2 ) ) = ( ( A ^ 2 ) gcd ( B ^ 2 ) ) )
14	12, 13	eqtr3d 2273	. . . . . . 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 2273	. . . . 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 1032	. . . . . 6 |- ( ( ( A e. NN /\ B e. NN /\ C e. NN ) /\ ( ( A ^ 2 ) + ( B ^ 2 ) ) = ( C ^ 2 ) ) -> B e. NN )
18		simpl3 1033	. . . . . 6 |- ( ( ( A e. NN /\ B e. NN /\ C e. NN ) /\ ( ( A ^ 2 ) + ( B ^ 2 ) ) = ( C ^ 2 ) ) -> C e. NN )
19		sqgcd 12832	. . . . . 6 |- ( ( B e. NN /\ C e. NN ) -> ( ( B gcd C ) ^ 2 ) = ( ( B ^ 2 ) gcd ( C ^ 2 ) ) )
20	17, 18, 19	syl2anc 415	. . . . 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 1031	. . . . . 6 |- ( ( ( A e. NN /\ B e. NN /\ C e. NN ) /\ ( ( A ^ 2 ) + ( B ^ 2 ) ) = ( C ^ 2 ) ) -> A e. NN )
22		sqgcd 12832	. . . . . 6 |- ( ( A e. NN /\ B e. NN ) -> ( ( A gcd B ) ^ 2 ) = ( ( A ^ 2 ) gcd ( B ^ 2 ) ) )
23	21, 17, 22	syl2anc 415	. . . . 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 2281	. . . 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 1048	. . 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 1056	. . . 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 6103	. . 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 2271	. 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 1050	. . . . . 6 |- ( ( A e. NN /\ B e. NN /\ C e. NN ) -> B e. ZZ )
30		nnz 9675	. . . . . . 7 |- ( C e. NN -> C e. ZZ )
31	30	3ad2ant3 1051	. . . . . 6 |- ( ( A e. NN /\ B e. NN /\ C e. NN ) -> C e. ZZ )
32	29, 31	gcdcld 12771	. . . . 5 |- ( ( A e. NN /\ B e. NN /\ C e. NN ) -> ( B gcd C ) e. NN0 )
33	32	nn0red 9633	. . . 4 |- ( ( A e. NN /\ B e. NN /\ C e. NN ) -> ( B gcd C ) e. RR )
34	33	3ad2ant1 1049	. . 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 9635	. . . 4 |- ( ( A e. NN /\ B e. NN /\ C e. NN ) -> 0 <_ ( B gcd C ) )
36	35	3ad2ant1 1049	. . 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 8332	. . . 4 |- 1 e. RR
38		0le1 8817	. . . 4 |- 0 <_ 1
39		sq11 11071	. . . 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 443	. . 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 415	. 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 )

This proof depends on syntax axioms:   -. wn 3   -> wi 4   /\ wa 104   <-> wb 105   /\ w3a 1009   = wceq 1402   e. wcel 2209   class class class wbr 4131  (class class class)co 6088   RRcr 8185   0cc0 8186   1c1 8187   + caddc 8189   <_ cle 8368   NNcn 9314   2c2 9365   ZZcz 9656   ^cexp 10997   || cdvds 12580   gcd cgcd 12756

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-in1 623  ax-in2 624  ax-io 721  ax-5 1500  ax-7 1501  ax-gen 1502  ax-ie1 1546  ax-ie2 1547  ax-8 1557  ax-10 1558  ax-11 1559  ax-i12 1560  ax-bndl 1562  ax-4 1563  ax-17 1579  ax-i9 1583  ax-ial 1587  ax-i5r 1588  ax-14 2212  ax-ext 2220  ax-coll 4248  ax-sep 4251  ax-nul 4261  ax-pow 4313  ax-pr 4348  ax-un 4580  ax-setind 4686  ax-iinf 4737  ax-cnex 8277  ax-resscn 8278  ax-1cn 8279  ax-1re 8280  ax-icn 8281  ax-addcl 8282  ax-addrcl 8283  ax-mulcl 8284  ax-mulrcl 8285  ax-addcom 8286  ax-mulcom 8287  ax-addass 8288  ax-mulass 8289  ax-distr 8290  ax-i2m1 8291  ax-0lt1 8292  ax-1rid 8293  ax-0id 8294  ax-rnegex 8295  ax-precex 8296  ax-cnre 8297  ax-pre-ltirr 8298  ax-pre-ltwlin 8299  ax-pre-lttrn 8300  ax-pre-apti 8301  ax-pre-ltadd 8302  ax-pre-mulgt0 8303  ax-pre-mulext 8304  ax-arch 8305  ax-caucvg 8306

This proof depends on definitions:  df-bi 117  df-stab 843  df-dc 847  df-3or 1010  df-3an 1011  df-tru 1405  df-fal 1408  df-nf 1514  df-sb 1816  df-eu 2089  df-mo 2090  df-clab 2225  df-cleq 2231  df-clel 2234  df-nfc 2381  df-ne 2421  df-nel 2516  df-ral 2533  df-rex 2534  df-reu 2535  df-rmo 2536  df-rab 2537  df-v 2823  df-sbc 3052  df-csb 3148  df-dif 3222  df-un 3224  df-in 3226  df-ss 3233  df-nul 3521  df-if 3639  df-pw 3690  df-sn 3715  df-pr 3716  df-op 3718  df-uni 3936  df-int 3971  df-iun 4014  df-br 4132  df-opab 4194  df-mpt 4195  df-tr 4232  df-id 4440  df-po 4443  df-iso 4444  df-iord 4513  df-on 4515  df-ilim 4516  df-suc 4518  df-iom 4740  df-xp 4782  df-rel 4783  df-cnv 4784  df-co 4785  df-dm 4786  df-rn 4787  df-res 4788  df-ima 4789  df-iota 5340  df-fun 5382  df-fn 5383  df-f 5384  df-f1 5385  df-fo 5386  df-f1o 5387  df-fv 5388  df-riota 6041  df-ov 6091  df-oprab 6092  df-mpo 6093  df-1st 6378  df-2nd 6379  df-recs 6580  df-frec 6666  df-sup 7331  df-pnf 8369  df-mnf 8370  df-xr 8371  df-ltxr 8372  df-le 8373  df-sub 8507  df-neg 8508  df-reap 8912  df-ap 8919  df-div 9012  df-inn 9315  df-2 9373  df-3 9374  df-4 9375  df-n0 9576  df-z 9657  df-uz 9939  df-q 10037  df-rp 10073  df-fz 10430  df-fzo 10568  df-fl 10723  df-mod 10782  df-seqfrec 10907  df-exp 10998  df-cj 11630  df-re 11631  df-im 11632  df-rsqrt 11787  df-abs 11788  df-dvds 12581  df-gcd 12757

This theorem is used by:  pythagtriplem4  13077