Theorem pythagtriplem7 12780

Description: Lemma for pythagtrip 12792. Calculate ( sqr ` ( C + B ) ). (Contributed by Scott Fenton, 18-Apr-2014.) (Revised by Mario Carneiro, 19-Apr-2014.)

Assertion

Ref	Expression
pythagtriplem7	|- ( ( ( 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 pythagtriplem7

Step	Hyp	Ref	Expression
1		simp3 1023	. . . . . . . . 9 |- ( ( A e. NN /\ B e. NN /\ C e. NN ) -> C e. NN )
2	1	nnzd 9556	. . . . . . . 8 |- ( ( A e. NN /\ B e. NN /\ C e. NN ) -> C e. ZZ )
3		simp2 1022	. . . . . . . . 9 |- ( ( A e. NN /\ B e. NN /\ C e. NN ) -> B e. NN )
4	3	nnzd 9556	. . . . . . . 8 |- ( ( A e. NN /\ B e. NN /\ C e. NN ) -> B e. ZZ )
5	2, 4	zsubcld 9562	. . . . . . 7 |- ( ( A e. NN /\ B e. NN /\ C e. NN ) -> ( C - B ) e. ZZ )
6	5	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 )
7	1, 3	nnaddcld 9146	. . . . . . . 8 |- ( ( A e. NN /\ B e. NN /\ C e. NN ) -> ( C + B ) e. NN )
8	7	nnnn0d 9410	. . . . . . 7 |- ( ( A e. NN /\ B e. NN /\ C e. NN ) -> ( C + B ) e. NN0 )
9	8	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. NN0 )
10		nnnn0 9364	. . . . . . . 8 |- ( A e. NN -> A e. NN0 )
11	10	3ad2ant1 1042	. . . . . . 7 |- ( ( A e. NN /\ B e. NN /\ C e. NN ) -> A e. NN0 )
12	11	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 )
13	6, 9, 12	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. ZZ /\ ( C + B ) e. NN0 /\ A e. NN0 ) )
14		pythagtriplem4 12777	. . . . . . 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 )
15	14	oveq1d 6009	. . . . . 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 ) )
16		nnz 9453	. . . . . . . . 9 |- ( A e. NN -> A e. ZZ )
17	16	3ad2ant1 1042	. . . . . . . 8 |- ( ( A e. NN /\ B e. NN /\ C e. NN ) -> A e. ZZ )
18	17	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 )
19		1gcd 12499	. . . . . . 7 |- ( A e. ZZ -> ( 1 gcd A ) = 1 )
20	18, 19	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 )
21	15, 20	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 )
22	13, 21	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. ZZ /\ ( C + B ) e. NN0 /\ A e. NN0 ) /\ ( ( ( C - B ) gcd ( C + B ) ) gcd A ) = 1 ) )
23		oveq1 6001	. . . . . 6 |- ( ( ( A ^ 2 ) + ( B ^ 2 ) ) = ( C ^ 2 ) -> ( ( ( A ^ 2 ) + ( B ^ 2 ) ) - ( B ^ 2 ) ) = ( ( C ^ 2 ) - ( B ^ 2 ) ) )
24	23	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 ) ) )
25		nncn 9106	. . . . . . . . 9 |- ( A e. NN -> A e. CC )
26	25	3ad2ant1 1042	. . . . . . . 8 |- ( ( A e. NN /\ B e. NN /\ C e. NN ) -> A e. CC )
27	26	sqcld 10880	. . . . . . 7 |- ( ( A e. NN /\ B e. NN /\ C e. NN ) -> ( A ^ 2 ) e. CC )
28	3	nncnd 9112	. . . . . . . 8 |- ( ( A e. NN /\ B e. NN /\ C e. NN ) -> B e. CC )
29	28	sqcld 10880	. . . . . . 7 |- ( ( A e. NN /\ B e. NN /\ C e. NN ) -> ( B ^ 2 ) e. CC )
30	27, 29	pncand 8446	. . . . . 6 |- ( ( A e. NN /\ B e. NN /\ C e. NN ) -> ( ( ( A ^ 2 ) + ( B ^ 2 ) ) - ( B ^ 2 ) ) = ( A ^ 2 ) )
31	30	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 ) )
32	1	nncnd 9112	. . . . . . . 8 |- ( ( A e. NN /\ B e. NN /\ C e. NN ) -> C e. CC )
33		subsq 10855	. . . . . . . 8 |- ( ( C e. CC /\ B e. CC ) -> ( ( C ^ 2 ) - ( B ^ 2 ) ) = ( ( C + B ) x. ( C - B ) ) )
34	32, 28, 33	syl2anc 411	. . . . . . 7 |- ( ( A e. NN /\ B e. NN /\ C e. NN ) -> ( ( C ^ 2 ) - ( B ^ 2 ) ) = ( ( C + B ) x. ( C - B ) ) )
35	7	nncnd 9112	. . . . . . . 8 |- ( ( A e. NN /\ B e. NN /\ C e. NN ) -> ( C + B ) e. CC )
36	5	zcnd 9558	. . . . . . . 8 |- ( ( A e. NN /\ B e. NN /\ C e. NN ) -> ( C - B ) e. CC )
37	35, 36	mulcomd 8156	. . . . . . 7 |- ( ( A e. NN /\ B e. NN /\ C e. NN ) -> ( ( C + B ) x. ( C - B ) ) = ( ( C - B ) x. ( C + B ) ) )
38	34, 37	eqtrd 2262	. . . . . 6 |- ( ( A e. NN /\ B e. NN /\ C e. NN ) -> ( ( C ^ 2 ) - ( B ^ 2 ) ) = ( ( C - B ) x. ( C + B ) ) )
39	38	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 ) ) )
40	24, 31, 39	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 ) ) )
41		coprimeprodsq2 12767	. . . 4 |- ( ( ( ( C - B ) e. ZZ /\ ( C + B ) e. NN0 /\ 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 ) ) )
42	22, 40, 41	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 ) )
43	42	fveq2d 5627	. 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 ) ) )
44	7	nnzd 9556	. . . . . 6 |- ( ( A e. NN /\ B e. NN /\ C e. NN ) -> ( C + B ) e. ZZ )
45	44	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 + B ) e. ZZ )
46	45, 18	gcdcld 12475	. . . 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 )
47	46	nn0red 9411	. . 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 )
48	46	nn0ge0d 9413	. . 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 ) )
49	47, 48	sqrtsqd 11662	. 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 ) )
50	43, 49	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 4082   ` cfv 5314  (class class class)co 5994   CCcc 7985   1c1 7988   + caddc 7990   x. cmul 7992   - cmin 8305   NNcn 9098   2c2 9149   NN0cn0 9357   ZZcz 9434   ^cexp 10747   sqrcsqrt 11493   || 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-1o 6552  df-2o 6553  df-er 6670  df-en 6878  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  df-prm 12616

This theorem is referenced by:  pythagtriplem9  12782  pythagtriplem11  12783  pythagtriplem13  12785