Theorem pythagtriplem6 12836

Description: Lemma for pythagtrip 12849. Calculate ( sqr ` ( C - B ) ). (Contributed by Scott Fenton, 18-Apr-2014.) (Revised by Mario Carneiro, 19-Apr-2014.)

Assertion

Ref	Expression
pythagtriplem6	|- ( ( ( A e. NN /\ B e. NN /\ C e. NN ) /\ ( ( A ^ 2 ) + ( B ^ 2 ) ) = ( C ^ 2 ) /\ ( ( A gcd B ) = 1 /\ -. 2 || A ) ) -> ( sqr ` ( C - B ) ) = ( ( C - B ) gcd A ) )

Proof of Theorem pythagtriplem6

Step	Hyp	Ref	Expression
1		nnz 9491	. . . . . . . . . . 11 |- ( C e. NN -> C e. ZZ )
2	1	3ad2ant3 1044	. . . . . . . . . 10 |- ( ( A e. NN /\ B e. NN /\ C e. NN ) -> C e. ZZ )
3		nnz 9491	. . . . . . . . . . 11 |- ( B e. NN -> B e. ZZ )
4	3	3ad2ant2 1043	. . . . . . . . . 10 |- ( ( A e. NN /\ B e. NN /\ C e. NN ) -> B e. ZZ )
5	2, 4	zsubcld 9600	. . . . . . . . 9 |- ( ( A e. NN /\ B e. NN /\ C e. NN ) -> ( C - B ) e. ZZ )
6	5	3ad2ant1 1042	. . . . . . . 8 |- ( ( ( A e. NN /\ B e. NN /\ C e. NN ) /\ ( ( A ^ 2 ) + ( B ^ 2 ) ) = ( C ^ 2 ) /\ ( ( A gcd B ) = 1 /\ -. 2 || A ) ) -> ( C - B ) e. ZZ )
7		pythagtriplem10 12835	. . . . . . . . 9 |- ( ( ( A e. NN /\ B e. NN /\ C e. NN ) /\ ( ( A ^ 2 ) + ( B ^ 2 ) ) = ( C ^ 2 ) ) -> 0 < ( C - B ) )
8	7	3adant3 1041	. . . . . . . 8 |- ( ( ( A e. NN /\ B e. NN /\ C e. NN ) /\ ( ( A ^ 2 ) + ( B ^ 2 ) ) = ( C ^ 2 ) /\ ( ( A gcd B ) = 1 /\ -. 2 || A ) ) -> 0 < ( C - B ) )
9		elnnz 9482	. . . . . . . 8 |- ( ( C - B ) e. NN <-> ( ( C - B ) e. ZZ /\ 0 < ( C - B ) ) )
10	6, 8, 9	sylanbrc 417	. . . . . . 7 |- ( ( ( A e. NN /\ B e. NN /\ C e. NN ) /\ ( ( A ^ 2 ) + ( B ^ 2 ) ) = ( C ^ 2 ) /\ ( ( A gcd B ) = 1 /\ -. 2 || A ) ) -> ( C - B ) e. NN )
11	10	nnnn0d 9448	. . . . . 6 |- ( ( ( A e. NN /\ B e. NN /\ C e. NN ) /\ ( ( A ^ 2 ) + ( B ^ 2 ) ) = ( C ^ 2 ) /\ ( ( A gcd B ) = 1 /\ -. 2 || A ) ) -> ( C - B ) e. NN0 )
12		simp3 1023	. . . . . . . . 9 |- ( ( A e. NN /\ B e. NN /\ C e. NN ) -> C e. NN )
13		simp2 1022	. . . . . . . . 9 |- ( ( A e. NN /\ B e. NN /\ C e. NN ) -> B e. NN )
14	12, 13	nnaddcld 9184	. . . . . . . 8 |- ( ( A e. NN /\ B e. NN /\ C e. NN ) -> ( C + B ) e. NN )
15	14	nnzd 9594	. . . . . . 7 |- ( ( A e. NN /\ B e. NN /\ C e. NN ) -> ( C + B ) e. ZZ )
16	15	3ad2ant1 1042	. . . . . 6 |- ( ( ( A e. NN /\ B e. NN /\ C e. NN ) /\ ( ( A ^ 2 ) + ( B ^ 2 ) ) = ( C ^ 2 ) /\ ( ( A gcd B ) = 1 /\ -. 2 || A ) ) -> ( C + B ) e. ZZ )
17		nnnn0 9402	. . . . . . . 8 |- ( A e. NN -> A e. NN0 )
18	17	3ad2ant1 1042	. . . . . . 7 |- ( ( A e. NN /\ B e. NN /\ C e. NN ) -> A e. NN0 )
19	18	3ad2ant1 1042	. . . . . 6 |- ( ( ( A e. NN /\ B e. NN /\ C e. NN ) /\ ( ( A ^ 2 ) + ( B ^ 2 ) ) = ( C ^ 2 ) /\ ( ( A gcd B ) = 1 /\ -. 2 || A ) ) -> A e. NN0 )
20	11, 16, 19	3jca 1201	. . . . 5 |- ( ( ( A e. NN /\ B e. NN /\ C e. NN ) /\ ( ( A ^ 2 ) + ( B ^ 2 ) ) = ( C ^ 2 ) /\ ( ( A gcd B ) = 1 /\ -. 2 || A ) ) -> ( ( C - B ) e. NN0 /\ ( C + B ) e. ZZ /\ A e. NN0 ) )
21		pythagtriplem4 12834	. . . . . . 7 |- ( ( ( A e. NN /\ B e. NN /\ C e. NN ) /\ ( ( A ^ 2 ) + ( B ^ 2 ) ) = ( C ^ 2 ) /\ ( ( A gcd B ) = 1 /\ -. 2 || A ) ) -> ( ( C - B ) gcd ( C + B ) ) = 1 )
22	21	oveq1d 6028	. . . . . 6 |- ( ( ( A e. NN /\ B e. NN /\ C e. NN ) /\ ( ( A ^ 2 ) + ( B ^ 2 ) ) = ( C ^ 2 ) /\ ( ( A gcd B ) = 1 /\ -. 2 || A ) ) -> ( ( ( C - B ) gcd ( C + B ) ) gcd A ) = ( 1 gcd A ) )
23		nnz 9491	. . . . . . . . 9 |- ( A e. NN -> A e. ZZ )
24	23	3ad2ant1 1042	. . . . . . . 8 |- ( ( A e. NN /\ B e. NN /\ C e. NN ) -> A e. ZZ )
25	24	3ad2ant1 1042	. . . . . . 7 |- ( ( ( A e. NN /\ B e. NN /\ C e. NN ) /\ ( ( A ^ 2 ) + ( B ^ 2 ) ) = ( C ^ 2 ) /\ ( ( A gcd B ) = 1 /\ -. 2 || A ) ) -> A e. ZZ )
26		1gcd 12556	. . . . . . 7 |- ( A e. ZZ -> ( 1 gcd A ) = 1 )
27	25, 26	syl 14	. . . . . 6 |- ( ( ( A e. NN /\ B e. NN /\ C e. NN ) /\ ( ( A ^ 2 ) + ( B ^ 2 ) ) = ( C ^ 2 ) /\ ( ( A gcd B ) = 1 /\ -. 2 || A ) ) -> ( 1 gcd A ) = 1 )
28	22, 27	eqtrd 2262	. . . . 5 |- ( ( ( A e. NN /\ B e. NN /\ C e. NN ) /\ ( ( A ^ 2 ) + ( B ^ 2 ) ) = ( C ^ 2 ) /\ ( ( A gcd B ) = 1 /\ -. 2 || A ) ) -> ( ( ( C - B ) gcd ( C + B ) ) gcd A ) = 1 )
29	20, 28	jca 306	. . . 4 |- ( ( ( A e. NN /\ B e. NN /\ C e. NN ) /\ ( ( A ^ 2 ) + ( B ^ 2 ) ) = ( C ^ 2 ) /\ ( ( A gcd B ) = 1 /\ -. 2 || A ) ) -> ( ( ( C - B ) e. NN0 /\ ( C + B ) e. ZZ /\ A e. NN0 ) /\ ( ( ( C - B ) gcd ( C + B ) ) gcd A ) = 1 ) )
30		oveq1 6020	. . . . . 6 |- ( ( ( A ^ 2 ) + ( B ^ 2 ) ) = ( C ^ 2 ) -> ( ( ( A ^ 2 ) + ( B ^ 2 ) ) - ( B ^ 2 ) ) = ( ( C ^ 2 ) - ( B ^ 2 ) ) )
31	30	3ad2ant2 1043	. . . . 5 |- ( ( ( A e. NN /\ B e. NN /\ C e. NN ) /\ ( ( A ^ 2 ) + ( B ^ 2 ) ) = ( C ^ 2 ) /\ ( ( A gcd B ) = 1 /\ -. 2 || A ) ) -> ( ( ( A ^ 2 ) + ( B ^ 2 ) ) - ( B ^ 2 ) ) = ( ( C ^ 2 ) - ( B ^ 2 ) ) )
32	24	zcnd 9596	. . . . . . . 8 |- ( ( A e. NN /\ B e. NN /\ C e. NN ) -> A e. CC )
33	32	sqcld 10926	. . . . . . 7 |- ( ( A e. NN /\ B e. NN /\ C e. NN ) -> ( A ^ 2 ) e. CC )
34		nncn 9144	. . . . . . . . 9 |- ( B e. NN -> B e. CC )
35	34	3ad2ant2 1043	. . . . . . . 8 |- ( ( A e. NN /\ B e. NN /\ C e. NN ) -> B e. CC )
36	35	sqcld 10926	. . . . . . 7 |- ( ( A e. NN /\ B e. NN /\ C e. NN ) -> ( B ^ 2 ) e. CC )
37	33, 36	pncand 8484	. . . . . 6 |- ( ( A e. NN /\ B e. NN /\ C e. NN ) -> ( ( ( A ^ 2 ) + ( B ^ 2 ) ) - ( B ^ 2 ) ) = ( A ^ 2 ) )
38	37	3ad2ant1 1042	. . . . 5 |- ( ( ( A e. NN /\ B e. NN /\ C e. NN ) /\ ( ( A ^ 2 ) + ( B ^ 2 ) ) = ( C ^ 2 ) /\ ( ( A gcd B ) = 1 /\ -. 2 || A ) ) -> ( ( ( A ^ 2 ) + ( B ^ 2 ) ) - ( B ^ 2 ) ) = ( A ^ 2 ) )
39		nncn 9144	. . . . . . . . 9 |- ( C e. NN -> C e. CC )
40	39	3ad2ant3 1044	. . . . . . . 8 |- ( ( A e. NN /\ B e. NN /\ C e. NN ) -> C e. CC )
41		subsq 10901	. . . . . . . 8 |- ( ( C e. CC /\ B e. CC ) -> ( ( C ^ 2 ) - ( B ^ 2 ) ) = ( ( C + B ) x. ( C - B ) ) )
42	40, 35, 41	syl2anc 411	. . . . . . 7 |- ( ( A e. NN /\ B e. NN /\ C e. NN ) -> ( ( C ^ 2 ) - ( B ^ 2 ) ) = ( ( C + B ) x. ( C - B ) ) )
43	14	nncnd 9150	. . . . . . . 8 |- ( ( A e. NN /\ B e. NN /\ C e. NN ) -> ( C + B ) e. CC )
44	5	zcnd 9596	. . . . . . . 8 |- ( ( A e. NN /\ B e. NN /\ C e. NN ) -> ( C - B ) e. CC )
45	43, 44	mulcomd 8194	. . . . . . 7 |- ( ( A e. NN /\ B e. NN /\ C e. NN ) -> ( ( C + B ) x. ( C - B ) ) = ( ( C - B ) x. ( C + B ) ) )
46	42, 45	eqtrd 2262	. . . . . 6 |- ( ( A e. NN /\ B e. NN /\ C e. NN ) -> ( ( C ^ 2 ) - ( B ^ 2 ) ) = ( ( C - B ) x. ( C + B ) ) )
47	46	3ad2ant1 1042	. . . . 5 |- ( ( ( A e. NN /\ B e. NN /\ C e. NN ) /\ ( ( A ^ 2 ) + ( B ^ 2 ) ) = ( C ^ 2 ) /\ ( ( A gcd B ) = 1 /\ -. 2 || A ) ) -> ( ( C ^ 2 ) - ( B ^ 2 ) ) = ( ( C - B ) x. ( C + B ) ) )
48	31, 38, 47	3eqtr3d 2270	. . . 4 |- ( ( ( A e. NN /\ B e. NN /\ C e. NN ) /\ ( ( A ^ 2 ) + ( B ^ 2 ) ) = ( C ^ 2 ) /\ ( ( A gcd B ) = 1 /\ -. 2 || A ) ) -> ( A ^ 2 ) = ( ( C - B ) x. ( C + B ) ) )
49		coprimeprodsq 12823	. . . 4 |- ( ( ( ( C - B ) e. NN0 /\ ( C + B ) e. ZZ /\ A e. NN0 ) /\ ( ( ( C - B ) gcd ( C + B ) ) gcd A ) = 1 ) -> ( ( A ^ 2 ) = ( ( C - B ) x. ( C + B ) ) -> ( C - B ) = ( ( ( C - B ) gcd A ) ^ 2 ) ) )
50	29, 48, 49	sylc 62	. . 3 |- ( ( ( A e. NN /\ B e. NN /\ C e. NN ) /\ ( ( A ^ 2 ) + ( B ^ 2 ) ) = ( C ^ 2 ) /\ ( ( A gcd B ) = 1 /\ -. 2 || A ) ) -> ( C - B ) = ( ( ( C - B ) gcd A ) ^ 2 ) )
51	50	fveq2d 5639	. 2 |- ( ( ( A e. NN /\ B e. NN /\ C e. NN ) /\ ( ( A ^ 2 ) + ( B ^ 2 ) ) = ( C ^ 2 ) /\ ( ( A gcd B ) = 1 /\ -. 2 || A ) ) -> ( sqr ` ( C - B ) ) = ( sqr ` ( ( ( C - B ) gcd A ) ^ 2 ) ) )
52	6, 25	gcdcld 12532	. . . 4 |- ( ( ( A e. NN /\ B e. NN /\ C e. NN ) /\ ( ( A ^ 2 ) + ( B ^ 2 ) ) = ( C ^ 2 ) /\ ( ( A gcd B ) = 1 /\ -. 2 || A ) ) -> ( ( C - B ) gcd A ) e. NN0 )
53	52	nn0red 9449	. . 3 |- ( ( ( A e. NN /\ B e. NN /\ C e. NN ) /\ ( ( A ^ 2 ) + ( B ^ 2 ) ) = ( C ^ 2 ) /\ ( ( A gcd B ) = 1 /\ -. 2 || A ) ) -> ( ( C - B ) gcd A ) e. RR )
54	52	nn0ge0d 9451	. . 3 |- ( ( ( A e. NN /\ B e. NN /\ C e. NN ) /\ ( ( A ^ 2 ) + ( B ^ 2 ) ) = ( C ^ 2 ) /\ ( ( A gcd B ) = 1 /\ -. 2 || A ) ) -> 0 <_ ( ( C - B ) gcd A ) )
55	53, 54	sqrtsqd 11719	. 2 |- ( ( ( A e. NN /\ B e. NN /\ C e. NN ) /\ ( ( A ^ 2 ) + ( B ^ 2 ) ) = ( C ^ 2 ) /\ ( ( A gcd B ) = 1 /\ -. 2 || A ) ) -> ( sqr ` ( ( ( C - B ) gcd A ) ^ 2 ) ) = ( ( C - B ) gcd A ) )
56	51, 55	eqtrd 2262	1 |- ( ( ( A e. NN /\ B e. NN /\ C e. NN ) /\ ( ( A ^ 2 ) + ( B ^ 2 ) ) = ( C ^ 2 ) /\ ( ( A gcd B ) = 1 /\ -. 2 || A ) ) -> ( sqr ` ( C - B ) ) = ( ( C - B ) gcd A ) )

Syntax hints:   -. wn 3   -> wi 4   /\ wa 104   /\ w3a 1002   = wceq 1395   e. wcel 2200   class class class wbr 4086   ` cfv 5324  (class class class)co 6013   CCcc 8023   0cc0 8025   1c1 8026   + caddc 8028   x. cmul 8030   < clt 8207   - cmin 8343   NNcn 9136   2c2 9187   NN0cn0 9395   ZZcz 9472   ^cexp 10793   sqrcsqrt 11550   || cdvds 12341   gcd cgcd 12517

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 4202  ax-sep 4205  ax-nul 4213  ax-pow 4262  ax-pr 4297  ax-un 4528  ax-setind 4633  ax-iinf 4684  ax-cnex 8116  ax-resscn 8117  ax-1cn 8118  ax-1re 8119  ax-icn 8120  ax-addcl 8121  ax-addrcl 8122  ax-mulcl 8123  ax-mulrcl 8124  ax-addcom 8125  ax-mulcom 8126  ax-addass 8127  ax-mulass 8128  ax-distr 8129  ax-i2m1 8130  ax-0lt1 8131  ax-1rid 8132  ax-0id 8133  ax-rnegex 8134  ax-precex 8135  ax-cnre 8136  ax-pre-ltirr 8137  ax-pre-ltwlin 8138  ax-pre-lttrn 8139  ax-pre-apti 8140  ax-pre-ltadd 8141  ax-pre-mulgt0 8142  ax-pre-mulext 8143  ax-arch 8144  ax-caucvg 8145

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 2802  df-sbc 3030  df-csb 3126  df-dif 3200  df-un 3202  df-in 3204  df-ss 3211  df-nul 3493  df-if 3604  df-pw 3652  df-sn 3673  df-pr 3674  df-op 3676  df-uni 3892  df-int 3927  df-iun 3970  df-br 4087  df-opab 4149  df-mpt 4150  df-tr 4186  df-id 4388  df-po 4391  df-iso 4392  df-iord 4461  df-on 4463  df-ilim 4464  df-suc 4466  df-iom 4687  df-xp 4729  df-rel 4730  df-cnv 4731  df-co 4732  df-dm 4733  df-rn 4734  df-res 4735  df-ima 4736  df-iota 5284  df-fun 5326  df-fn 5327  df-f 5328  df-f1 5329  df-fo 5330  df-f1o 5331  df-fv 5332  df-riota 5966  df-ov 6016  df-oprab 6017  df-mpo 6018  df-1st 6298  df-2nd 6299  df-recs 6466  df-frec 6552  df-1o 6577  df-2o 6578  df-er 6697  df-en 6905  df-sup 7177  df-pnf 8209  df-mnf 8210  df-xr 8211  df-ltxr 8212  df-le 8213  df-sub 8345  df-neg 8346  df-reap 8748  df-ap 8755  df-div 8846  df-inn 9137  df-2 9195  df-3 9196  df-4 9197  df-n0 9396  df-z 9473  df-uz 9749  df-q 9847  df-rp 9882  df-fz 10237  df-fzo 10371  df-fl 10523  df-mod 10578  df-seqfrec 10703  df-exp 10794  df-cj 11396  df-re 11397  df-im 11398  df-rsqrt 11552  df-abs 11553  df-dvds 12342  df-gcd 12518  df-prm 12673

This theorem is referenced by:  pythagtriplem8  12838  pythagtriplem11  12840  pythagtriplem13  12842