Theorem pythagtriplem15 12261

Description: Lemma for pythagtrip 12266. Show the relationship between M, N, and A. (Contributed by Scott Fenton, 17-Apr-2014.) (Revised by Mario Carneiro, 19-Apr-2014.)

Hypotheses

Ref	Expression
pythagtriplem15.1	|- M = ( ( ( sqr ` ( C + B ) ) + ( sqr ` ( C - B ) ) ) / 2 )
pythagtriplem15.2	|- N = ( ( ( sqr ` ( C + B ) ) - ( sqr ` ( C - B ) ) ) / 2 )

Assertion

Ref	Expression
pythagtriplem15	|- ( ( ( A e. NN /\ B e. NN /\ C e. NN ) /\ ( ( A ^ 2 ) + ( B ^ 2 ) ) = ( C ^ 2 ) /\ ( ( A gcd B ) = 1 /\ -. 2 || A ) ) -> A = ( ( M ^ 2 ) - ( N ^ 2 ) ) )

Proof of Theorem pythagtriplem15

Step	Hyp	Ref	Expression
1		pythagtriplem15.1	. . . . 5 |- M = ( ( ( sqr ` ( C + B ) ) + ( sqr ` ( C - B ) ) ) / 2 )
2	1	pythagtriplem12 12258	. . . 4 |- ( ( ( A e. NN /\ B e. NN /\ C e. NN ) /\ ( ( A ^ 2 ) + ( B ^ 2 ) ) = ( C ^ 2 ) /\ ( ( A gcd B ) = 1 /\ -. 2 || A ) ) -> ( M ^ 2 ) = ( ( C + A ) / 2 ) )
3		pythagtriplem15.2	. . . . 5 |- N = ( ( ( sqr ` ( C + B ) ) - ( sqr ` ( C - B ) ) ) / 2 )
4	3	pythagtriplem14 12260	. . . 4 |- ( ( ( A e. NN /\ B e. NN /\ C e. NN ) /\ ( ( A ^ 2 ) + ( B ^ 2 ) ) = ( C ^ 2 ) /\ ( ( A gcd B ) = 1 /\ -. 2 || A ) ) -> ( N ^ 2 ) = ( ( C - A ) / 2 ) )
5	2, 4	oveq12d 5887	. . 3 |- ( ( ( A e. NN /\ B e. NN /\ C e. NN ) /\ ( ( A ^ 2 ) + ( B ^ 2 ) ) = ( C ^ 2 ) /\ ( ( A gcd B ) = 1 /\ -. 2 || A ) ) -> ( ( M ^ 2 ) - ( N ^ 2 ) ) = ( ( ( C + A ) / 2 ) - ( ( C - A ) / 2 ) ) )
6		simp3 999	. . . . . . 7 |- ( ( A e. NN /\ B e. NN /\ C e. NN ) -> C e. NN )
7		simp1 997	. . . . . . 7 |- ( ( A e. NN /\ B e. NN /\ C e. NN ) -> A e. NN )
8	6, 7	nnaddcld 8956	. . . . . 6 |- ( ( A e. NN /\ B e. NN /\ C e. NN ) -> ( C + A ) e. NN )
9	8	nncnd 8922	. . . . 5 |- ( ( A e. NN /\ B e. NN /\ C e. NN ) -> ( C + A ) e. CC )
10	9	3ad2ant1 1018	. . . 4 |- ( ( ( A e. NN /\ B e. NN /\ C e. NN ) /\ ( ( A ^ 2 ) + ( B ^ 2 ) ) = ( C ^ 2 ) /\ ( ( A gcd B ) = 1 /\ -. 2 || A ) ) -> ( C + A ) e. CC )
11		nnz 9261	. . . . . . . 8 |- ( C e. NN -> C e. ZZ )
12	11	3ad2ant3 1020	. . . . . . 7 |- ( ( A e. NN /\ B e. NN /\ C e. NN ) -> C e. ZZ )
13		nnz 9261	. . . . . . . 8 |- ( A e. NN -> A e. ZZ )
14	13	3ad2ant1 1018	. . . . . . 7 |- ( ( A e. NN /\ B e. NN /\ C e. NN ) -> A e. ZZ )
15	12, 14	zsubcld 9369	. . . . . 6 |- ( ( A e. NN /\ B e. NN /\ C e. NN ) -> ( C - A ) e. ZZ )
16	15	zcnd 9365	. . . . 5 |- ( ( A e. NN /\ B e. NN /\ C e. NN ) -> ( C - A ) e. CC )
17	16	3ad2ant1 1018	. . . 4 |- ( ( ( A e. NN /\ B e. NN /\ C e. NN ) /\ ( ( A ^ 2 ) + ( B ^ 2 ) ) = ( C ^ 2 ) /\ ( ( A gcd B ) = 1 /\ -. 2 || A ) ) -> ( C - A ) e. CC )
18		2cnd 8981	. . . 4 |- ( ( ( A e. NN /\ B e. NN /\ C e. NN ) /\ ( ( A ^ 2 ) + ( B ^ 2 ) ) = ( C ^ 2 ) /\ ( ( A gcd B ) = 1 /\ -. 2 || A ) ) -> 2 e. CC )
19		2ap0 9001	. . . . 5 |- 2 # 0
20	19	a1i 9	. . . 4 |- ( ( ( A e. NN /\ B e. NN /\ C e. NN ) /\ ( ( A ^ 2 ) + ( B ^ 2 ) ) = ( C ^ 2 ) /\ ( ( A gcd B ) = 1 /\ -. 2 || A ) ) -> 2 # 0 )
21	10, 17, 18, 20	divsubdirapd 8776	. . 3 |- ( ( ( A e. NN /\ B e. NN /\ C e. NN ) /\ ( ( A ^ 2 ) + ( B ^ 2 ) ) = ( C ^ 2 ) /\ ( ( A gcd B ) = 1 /\ -. 2 || A ) ) -> ( ( ( C + A ) - ( C - A ) ) / 2 ) = ( ( ( C + A ) / 2 ) - ( ( C - A ) / 2 ) ) )
22	5, 21	eqtr4d 2213	. 2 |- ( ( ( A e. NN /\ B e. NN /\ C e. NN ) /\ ( ( A ^ 2 ) + ( B ^ 2 ) ) = ( C ^ 2 ) /\ ( ( A gcd B ) = 1 /\ -. 2 || A ) ) -> ( ( M ^ 2 ) - ( N ^ 2 ) ) = ( ( ( C + A ) - ( C - A ) ) / 2 ) )
23		nncn 8916	. . . . . . 7 |- ( C e. NN -> C e. CC )
24	23	3ad2ant3 1020	. . . . . 6 |- ( ( A e. NN /\ B e. NN /\ C e. NN ) -> C e. CC )
25	24	3ad2ant1 1018	. . . . 5 |- ( ( ( A e. NN /\ B e. NN /\ C e. NN ) /\ ( ( A ^ 2 ) + ( B ^ 2 ) ) = ( C ^ 2 ) /\ ( ( A gcd B ) = 1 /\ -. 2 || A ) ) -> C e. CC )
26		nncn 8916	. . . . . . 7 |- ( A e. NN -> A e. CC )
27	26	3ad2ant1 1018	. . . . . 6 |- ( ( A e. NN /\ B e. NN /\ C e. NN ) -> A e. CC )
28	27	3ad2ant1 1018	. . . . 5 |- ( ( ( A e. NN /\ B e. NN /\ C e. NN ) /\ ( ( A ^ 2 ) + ( B ^ 2 ) ) = ( C ^ 2 ) /\ ( ( A gcd B ) = 1 /\ -. 2 || A ) ) -> A e. CC )
29	25, 28, 28	pnncand 8297	. . . 4 |- ( ( ( A e. NN /\ B e. NN /\ C e. NN ) /\ ( ( A ^ 2 ) + ( B ^ 2 ) ) = ( C ^ 2 ) /\ ( ( A gcd B ) = 1 /\ -. 2 || A ) ) -> ( ( C + A ) - ( C - A ) ) = ( A + A ) )
30	28	2timesd 9150	. . . 4 |- ( ( ( A e. NN /\ B e. NN /\ C e. NN ) /\ ( ( A ^ 2 ) + ( B ^ 2 ) ) = ( C ^ 2 ) /\ ( ( A gcd B ) = 1 /\ -. 2 || A ) ) -> ( 2 x. A ) = ( A + A ) )
31	29, 30	eqtr4d 2213	. . 3 |- ( ( ( A e. NN /\ B e. NN /\ C e. NN ) /\ ( ( A ^ 2 ) + ( B ^ 2 ) ) = ( C ^ 2 ) /\ ( ( A gcd B ) = 1 /\ -. 2 || A ) ) -> ( ( C + A ) - ( C - A ) ) = ( 2 x. A ) )
32	31	oveq1d 5884	. 2 |- ( ( ( A e. NN /\ B e. NN /\ C e. NN ) /\ ( ( A ^ 2 ) + ( B ^ 2 ) ) = ( C ^ 2 ) /\ ( ( A gcd B ) = 1 /\ -. 2 || A ) ) -> ( ( ( C + A ) - ( C - A ) ) / 2 ) = ( ( 2 x. A ) / 2 ) )
33	28, 18, 20	divcanap3d 8741	. 2 |- ( ( ( A e. NN /\ B e. NN /\ C e. NN ) /\ ( ( A ^ 2 ) + ( B ^ 2 ) ) = ( C ^ 2 ) /\ ( ( A gcd B ) = 1 /\ -. 2 || A ) ) -> ( ( 2 x. A ) / 2 ) = A )
34	22, 32, 33	3eqtrrd 2215	1 |- ( ( ( A e. NN /\ B e. NN /\ C e. NN ) /\ ( ( A ^ 2 ) + ( B ^ 2 ) ) = ( C ^ 2 ) /\ ( ( A gcd B ) = 1 /\ -. 2 || A ) ) -> A = ( ( M ^ 2 ) - ( N ^ 2 ) ) )

Syntax hints:   -. wn 3   -> wi 4   /\ wa 104   /\ w3a 978   = wceq 1353   e. wcel 2148   class class class wbr 4000   ` cfv 5212  (class class class)co 5869   CCcc 7800   0cc0 7802   1c1 7803   + caddc 7805   x. cmul 7807   - cmin 8118   # cap 8528   / cdiv 8618   NNcn 8908   2c2 8959   ZZcz 9242   ^cexp 10505   sqrcsqrt 10989   || cdvds 11778   gcd cgcd 11926

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 614  ax-in2 615  ax-io 709  ax-5 1447  ax-7 1448  ax-gen 1449  ax-ie1 1493  ax-ie2 1494  ax-8 1504  ax-10 1505  ax-11 1506  ax-i12 1507  ax-bndl 1509  ax-4 1510  ax-17 1526  ax-i9 1530  ax-ial 1534  ax-i5r 1535  ax-13 2150  ax-14 2151  ax-ext 2159  ax-coll 4115  ax-sep 4118  ax-nul 4126  ax-pow 4171  ax-pr 4206  ax-un 4430  ax-setind 4533  ax-iinf 4584  ax-cnex 7893  ax-resscn 7894  ax-1cn 7895  ax-1re 7896  ax-icn 7897  ax-addcl 7898  ax-addrcl 7899  ax-mulcl 7900  ax-mulrcl 7901  ax-addcom 7902  ax-mulcom 7903  ax-addass 7904  ax-mulass 7905  ax-distr 7906  ax-i2m1 7907  ax-0lt1 7908  ax-1rid 7909  ax-0id 7910  ax-rnegex 7911  ax-precex 7912  ax-cnre 7913  ax-pre-ltirr 7914  ax-pre-ltwlin 7915  ax-pre-lttrn 7916  ax-pre-apti 7917  ax-pre-ltadd 7918  ax-pre-mulgt0 7919  ax-pre-mulext 7920  ax-arch 7921  ax-caucvg 7922

This theorem depends on definitions:  df-bi 117  df-dc 835  df-3or 979  df-3an 980  df-tru 1356  df-fal 1359  df-nf 1461  df-sb 1763  df-eu 2029  df-mo 2030  df-clab 2164  df-cleq 2170  df-clel 2173  df-nfc 2308  df-ne 2348  df-nel 2443  df-ral 2460  df-rex 2461  df-reu 2462  df-rmo 2463  df-rab 2464  df-v 2739  df-sbc 2963  df-csb 3058  df-dif 3131  df-un 3133  df-in 3135  df-ss 3142  df-nul 3423  df-if 3535  df-pw 3576  df-sn 3597  df-pr 3598  df-op 3600  df-uni 3808  df-int 3843  df-iun 3886  df-br 4001  df-opab 4062  df-mpt 4063  df-tr 4099  df-id 4290  df-po 4293  df-iso 4294  df-iord 4363  df-on 4365  df-ilim 4366  df-suc 4368  df-iom 4587  df-xp 4629  df-rel 4630  df-cnv 4631  df-co 4632  df-dm 4633  df-rn 4634  df-res 4635  df-ima 4636  df-iota 5174  df-fun 5214  df-fn 5215  df-f 5216  df-f1 5217  df-fo 5218  df-f1o 5219  df-fv 5220  df-riota 5825  df-ov 5872  df-oprab 5873  df-mpo 5874  df-1st 6135  df-2nd 6136  df-recs 6300  df-frec 6386  df-pnf 7984  df-mnf 7985  df-xr 7986  df-ltxr 7987  df-le 7988  df-sub 8120  df-neg 8121  df-reap 8522  df-ap 8529  df-div 8619  df-inn 8909  df-2 8967  df-3 8968  df-4 8969  df-n0 9166  df-z 9243  df-uz 9518  df-rp 9641  df-seqfrec 10432  df-exp 10506  df-rsqrt 10991

This theorem is referenced by:  pythagtriplem18  12264