Theorem pythagtriplem6 12904

Description: Lemma for pythagtrip 12917. 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 9541	. . . . . . . . . . 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 9541	. . . . . . . . . . 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 9650	. . . . . . . . 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 12903	. . . . . . . . 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 9532	. . . . . . . 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 9498	. . . . . 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 9234	. . . . . . . 8 |- ( ( A e. NN /\ B e. NN /\ C e. NN ) -> ( C + B ) e. NN )
15	14	nnzd 9644	. . . . . . 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 9452	. . . . . . . 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 12902	. . . . . . 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 6043	. . . . . 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 9541	. . . . . . . . 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 12624	. . . . . . 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 2264	. . . . 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 6035	. . . . . 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 9646	. . . . . . . 8 |- ( ( A e. NN /\ B e. NN /\ C e. NN ) -> A e. CC )
33	32	sqcld 10977	. . . . . . 7 |- ( ( A e. NN /\ B e. NN /\ C e. NN ) -> ( A ^ 2 ) e. CC )
34		nncn 9194	. . . . . . . . 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 10977	. . . . . . 7 |- ( ( A e. NN /\ B e. NN /\ C e. NN ) -> ( B ^ 2 ) e. CC )
37	33, 36	pncand 8534	. . . . . 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 9194	. . . . . . . . 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 10952	. . . . . . . 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 9200	. . . . . . . 8 |- ( ( A e. NN /\ B e. NN /\ C e. NN ) -> ( C + B ) e. CC )
44	5	zcnd 9646	. . . . . . . 8 |- ( ( A e. NN /\ B e. NN /\ C e. NN ) -> ( C - B ) e. CC )
45	43, 44	mulcomd 8244	. . . . . . 7 |- ( ( A e. NN /\ B e. NN /\ C e. NN ) -> ( ( C + B ) x. ( C - B ) ) = ( ( C - B ) x. ( C + B ) ) )
46	42, 45	eqtrd 2264	. . . . . 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 2272	. . . 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 12891	. . . 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 5652	. 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 12600	. . . 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 9499	. . 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 9501	. . 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 11786	. 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 2264	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 2202   class class class wbr 4093   ` cfv 5333  (class class class)co 6028   CCcc 8073   0cc0 8075   1c1 8076   + caddc 8078   x. cmul 8080   < clt 8257   - cmin 8393   NNcn 9186   2c2 9237   NN0cn0 9445   ZZcz 9522   ^cexp 10844   sqrcsqrt 11617   || cdvds 12409   gcd cgcd 12585

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 2204  ax-14 2205  ax-ext 2213  ax-coll 4209  ax-sep 4212  ax-nul 4220  ax-pow 4270  ax-pr 4305  ax-un 4536  ax-setind 4641  ax-iinf 4692  ax-cnex 8166  ax-resscn 8167  ax-1cn 8168  ax-1re 8169  ax-icn 8170  ax-addcl 8171  ax-addrcl 8172  ax-mulcl 8173  ax-mulrcl 8174  ax-addcom 8175  ax-mulcom 8176  ax-addass 8177  ax-mulass 8178  ax-distr 8179  ax-i2m1 8180  ax-0lt1 8181  ax-1rid 8182  ax-0id 8183  ax-rnegex 8184  ax-precex 8185  ax-cnre 8186  ax-pre-ltirr 8187  ax-pre-ltwlin 8188  ax-pre-lttrn 8189  ax-pre-apti 8190  ax-pre-ltadd 8191  ax-pre-mulgt0 8192  ax-pre-mulext 8193  ax-arch 8194  ax-caucvg 8195

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 1811  df-eu 2082  df-mo 2083  df-clab 2218  df-cleq 2224  df-clel 2227  df-nfc 2364  df-ne 2404  df-nel 2499  df-ral 2516  df-rex 2517  df-reu 2518  df-rmo 2519  df-rab 2520  df-v 2805  df-sbc 3033  df-csb 3129  df-dif 3203  df-un 3205  df-in 3207  df-ss 3214  df-nul 3497  df-if 3608  df-pw 3658  df-sn 3679  df-pr 3680  df-op 3682  df-uni 3899  df-int 3934  df-iun 3977  df-br 4094  df-opab 4156  df-mpt 4157  df-tr 4193  df-id 4396  df-po 4399  df-iso 4400  df-iord 4469  df-on 4471  df-ilim 4472  df-suc 4474  df-iom 4695  df-xp 4737  df-rel 4738  df-cnv 4739  df-co 4740  df-dm 4741  df-rn 4742  df-res 4743  df-ima 4744  df-iota 5293  df-fun 5335  df-fn 5336  df-f 5337  df-f1 5338  df-fo 5339  df-f1o 5340  df-fv 5341  df-riota 5981  df-ov 6031  df-oprab 6032  df-mpo 6033  df-1st 6312  df-2nd 6313  df-recs 6514  df-frec 6600  df-1o 6625  df-2o 6626  df-er 6745  df-en 6953  df-sup 7226  df-pnf 8259  df-mnf 8260  df-xr 8261  df-ltxr 8262  df-le 8263  df-sub 8395  df-neg 8396  df-reap 8798  df-ap 8805  df-div 8896  df-inn 9187  df-2 9245  df-3 9246  df-4 9247  df-n0 9446  df-z 9523  df-uz 9799  df-q 9897  df-rp 9932  df-fz 10287  df-fzo 10421  df-fl 10574  df-mod 10629  df-seqfrec 10754  df-exp 10845  df-cj 11463  df-re 11464  df-im 11465  df-rsqrt 11619  df-abs 11620  df-dvds 12410  df-gcd 12586  df-prm 12741

This theorem is referenced by:  pythagtriplem8  12906  pythagtriplem11  12908  pythagtriplem13  12910