Theorem pythagtriplem6 12993

Description: Lemma for pythagtrip 13006. 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 9613	. . . . . . . . . . 11 |- ( C e. NN -> C e. ZZ )
2	1	3ad2ant3 1047	. . . . . . . . . 10 |- ( ( A e. NN /\ B e. NN /\ C e. NN ) -> C e. ZZ )
3		nnz 9613	. . . . . . . . . . 11 |- ( B e. NN -> B e. ZZ )
4	3	3ad2ant2 1046	. . . . . . . . . 10 |- ( ( A e. NN /\ B e. NN /\ C e. NN ) -> B e. ZZ )
5	2, 4	zsubcld 9723	. . . . . . . . 9 |- ( ( A e. NN /\ B e. NN /\ C e. NN ) -> ( C - B ) e. ZZ )
6	5	3ad2ant1 1045	. . . . . . . 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 12992	. . . . . . . . 9 |- ( ( ( A e. NN /\ B e. NN /\ C e. NN ) /\ ( ( A ^ 2 ) + ( B ^ 2 ) ) = ( C ^ 2 ) ) -> 0 < ( C - B ) )
8	7	3adant3 1044	. . . . . . . 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 9604	. . . . . . . 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 9570	. . . . . 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 1026	. . . . . . . . 9 |- ( ( A e. NN /\ B e. NN /\ C e. NN ) -> C e. NN )
13		simp2 1025	. . . . . . . . 9 |- ( ( A e. NN /\ B e. NN /\ C e. NN ) -> B e. NN )
14	12, 13	nnaddcld 9302	. . . . . . . 8 |- ( ( A e. NN /\ B e. NN /\ C e. NN ) -> ( C + B ) e. NN )
15	14	nnzd 9717	. . . . . . 7 |- ( ( A e. NN /\ B e. NN /\ C e. NN ) -> ( C + B ) e. ZZ )
16	15	3ad2ant1 1045	. . . . . 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 9520	. . . . . . . 8 |- ( A e. NN -> A e. NN0 )
18	17	3ad2ant1 1045	. . . . . . 7 |- ( ( A e. NN /\ B e. NN /\ C e. NN ) -> A e. NN0 )
19	18	3ad2ant1 1045	. . . . . 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 1204	. . . . 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 12991	. . . . . . 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 6073	. . . . . 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 9613	. . . . . . . . 9 |- ( A e. NN -> A e. ZZ )
24	23	3ad2ant1 1045	. . . . . . . 8 |- ( ( A e. NN /\ B e. NN /\ C e. NN ) -> A e. ZZ )
25	24	3ad2ant1 1045	. . . . . . 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 12713	. . . . . . 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 2267	. . . . 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 6065	. . . . . 6 |- ( ( ( A ^ 2 ) + ( B ^ 2 ) ) = ( C ^ 2 ) -> ( ( ( A ^ 2 ) + ( B ^ 2 ) ) - ( B ^ 2 ) ) = ( ( C ^ 2 ) - ( B ^ 2 ) ) )
31	30	3ad2ant2 1046	. . . . 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 9719	. . . . . . . 8 |- ( ( A e. NN /\ B e. NN /\ C e. NN ) -> A e. CC )
33	32	sqcld 11058	. . . . . . 7 |- ( ( A e. NN /\ B e. NN /\ C e. NN ) -> ( A ^ 2 ) e. CC )
34		nncn 9262	. . . . . . . . 9 |- ( B e. NN -> B e. CC )
35	34	3ad2ant2 1046	. . . . . . . 8 |- ( ( A e. NN /\ B e. NN /\ C e. NN ) -> B e. CC )
36	35	sqcld 11058	. . . . . . 7 |- ( ( A e. NN /\ B e. NN /\ C e. NN ) -> ( B ^ 2 ) e. CC )
37	33, 36	pncand 8601	. . . . . 6 |- ( ( A e. NN /\ B e. NN /\ C e. NN ) -> ( ( ( A ^ 2 ) + ( B ^ 2 ) ) - ( B ^ 2 ) ) = ( A ^ 2 ) )
38	37	3ad2ant1 1045	. . . . 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 9262	. . . . . . . . 9 |- ( C e. NN -> C e. CC )
40	39	3ad2ant3 1047	. . . . . . . 8 |- ( ( A e. NN /\ B e. NN /\ C e. NN ) -> C e. CC )
41		subsq 11032	. . . . . . . 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 9268	. . . . . . . 8 |- ( ( A e. NN /\ B e. NN /\ C e. NN ) -> ( C + B ) e. CC )
44	5	zcnd 9719	. . . . . . . 8 |- ( ( A e. NN /\ B e. NN /\ C e. NN ) -> ( C - B ) e. CC )
45	43, 44	mulcomd 8311	. . . . . . 7 |- ( ( A e. NN /\ B e. NN /\ C e. NN ) -> ( ( C + B ) x. ( C - B ) ) = ( ( C - B ) x. ( C + B ) ) )
46	42, 45	eqtrd 2267	. . . . . 6 |- ( ( A e. NN /\ B e. NN /\ C e. NN ) -> ( ( C ^ 2 ) - ( B ^ 2 ) ) = ( ( C - B ) x. ( C + B ) ) )
47	46	3ad2ant1 1045	. . . . 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 2275	. . . 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 12980	. . . 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 5679	. 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 12689	. . . 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 9571	. . 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 9573	. . 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 11875	. 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 2267	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 1005   = wceq 1398   e. wcel 2205   class class class wbr 4114   ` cfv 5357  (class class class)co 6058   CCcc 8141   0cc0 8143   1c1 8144   + caddc 8146   x. cmul 8148   < clt 8324   - cmin 8460   NNcn 9254   2c2 9305   NN0cn0 9513   ZZcz 9594   ^cexp 10924   sqrcsqrt 11706   || cdvds 12498   gcd cgcd 12674

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 619  ax-in2 620  ax-io 717  ax-5 1496  ax-7 1497  ax-gen 1498  ax-ie1 1542  ax-ie2 1543  ax-8 1553  ax-10 1554  ax-11 1555  ax-i12 1556  ax-bndl 1558  ax-4 1559  ax-17 1575  ax-i9 1579  ax-ial 1583  ax-i5r 1584  ax-13 2207  ax-14 2208  ax-ext 2216  ax-coll 4230  ax-sep 4233  ax-nul 4241  ax-pow 4292  ax-pr 4327  ax-un 4559  ax-setind 4664  ax-iinf 4715  ax-cnex 8234  ax-resscn 8235  ax-1cn 8236  ax-1re 8237  ax-icn 8238  ax-addcl 8239  ax-addrcl 8240  ax-mulcl 8241  ax-mulrcl 8242  ax-addcom 8243  ax-mulcom 8244  ax-addass 8245  ax-mulass 8246  ax-distr 8247  ax-i2m1 8248  ax-0lt1 8249  ax-1rid 8250  ax-0id 8251  ax-rnegex 8252  ax-precex 8253  ax-cnre 8254  ax-pre-ltirr 8255  ax-pre-ltwlin 8256  ax-pre-lttrn 8257  ax-pre-apti 8258  ax-pre-ltadd 8259  ax-pre-mulgt0 8260  ax-pre-mulext 8261  ax-arch 8262  ax-caucvg 8263

This theorem depends on definitions:  df-bi 117  df-stab 839  df-dc 843  df-3or 1006  df-3an 1007  df-tru 1401  df-fal 1404  df-nf 1510  df-sb 1812  df-eu 2085  df-mo 2086  df-clab 2221  df-cleq 2227  df-clel 2230  df-nfc 2375  df-ne 2415  df-nel 2510  df-ral 2527  df-rex 2528  df-reu 2529  df-rmo 2530  df-rab 2531  df-v 2817  df-sbc 3046  df-csb 3142  df-dif 3216  df-un 3218  df-in 3220  df-ss 3227  df-nul 3513  df-if 3625  df-pw 3676  df-sn 3700  df-pr 3701  df-op 3703  df-uni 3920  df-int 3955  df-iun 3998  df-br 4115  df-opab 4177  df-mpt 4178  df-tr 4214  df-id 4419  df-po 4422  df-iso 4423  df-iord 4492  df-on 4494  df-ilim 4495  df-suc 4497  df-iom 4718  df-xp 4760  df-rel 4761  df-cnv 4762  df-co 4763  df-dm 4764  df-rn 4765  df-res 4766  df-ima 4767  df-iota 5317  df-fun 5359  df-fn 5360  df-f 5361  df-f1 5362  df-fo 5363  df-f1o 5364  df-fv 5365  df-riota 6011  df-ov 6061  df-oprab 6062  df-mpo 6063  df-1st 6347  df-2nd 6348  df-recs 6549  df-frec 6635  df-1o 6660  df-2o 6661  df-er 6780  df-en 6989  df-sup 7288  df-pnf 8326  df-mnf 8327  df-xr 8328  df-ltxr 8329  df-le 8330  df-sub 8462  df-neg 8463  df-reap 8866  df-ap 8873  df-div 8964  df-inn 9255  df-2 9313  df-3 9314  df-4 9315  df-n0 9514  df-z 9595  df-uz 9872  df-q 9970  df-rp 10005  df-fz 10362  df-fzo 10499  df-fl 10654  df-mod 10709  df-seqfrec 10834  df-exp 10925  df-cj 11552  df-re 11553  df-im 11554  df-rsqrt 11708  df-abs 11709  df-dvds 12499  df-gcd 12675  df-prm 12830

This theorem is referenced by:  pythagtriplem8  12995  pythagtriplem11  12997  pythagtriplem13  12999