Theorem pythagtriplem11 12997

Description: Lemma for pythagtrip 13006. Show that M (which will eventually be closely related to the m in the final statement) is a natural. (Contributed by Scott Fenton, 17-Apr-2014.) (Revised by Mario Carneiro, 19-Apr-2014.)

Hypothesis

Ref	Expression
pythagtriplem11.1	|- M = ( ( ( sqr ` ( C + B ) ) + ( sqr ` ( C - B ) ) ) / 2 )

Assertion

Ref	Expression
pythagtriplem11	|- ( ( ( A e. NN /\ B e. NN /\ C e. NN ) /\ ( ( A ^ 2 ) + ( B ^ 2 ) ) = ( C ^ 2 ) /\ ( ( A gcd B ) = 1 /\ -. 2 || A ) ) -> M e. NN )

Proof of Theorem pythagtriplem11

Step	Hyp	Ref	Expression
1		pythagtriplem11.1	. 2 |- M = ( ( ( sqr ` ( C + B ) ) + ( sqr ` ( C - B ) ) ) / 2 )
2		pythagtriplem9 12996	. . . . . 6 |- ( ( ( A e. NN /\ B e. NN /\ C e. NN ) /\ ( ( A ^ 2 ) + ( B ^ 2 ) ) = ( C ^ 2 ) /\ ( ( A gcd B ) = 1 /\ -. 2 || A ) ) -> ( sqr ` ( C + B ) ) e. NN )
3	2	nnzd 9717	. . . . 5 |- ( ( ( A e. NN /\ B e. NN /\ C e. NN ) /\ ( ( A ^ 2 ) + ( B ^ 2 ) ) = ( C ^ 2 ) /\ ( ( A gcd B ) = 1 /\ -. 2 || A ) ) -> ( sqr ` ( C + B ) ) e. ZZ )
4		simp3r 1053	. . . . . . 7 |- ( ( ( A e. NN /\ B e. NN /\ C e. NN ) /\ ( ( A ^ 2 ) + ( B ^ 2 ) ) = ( C ^ 2 ) /\ ( ( A gcd B ) = 1 /\ -. 2 || A ) ) -> -. 2 || A )
5		2z 9622	. . . . . . . . . 10 |- 2 e. ZZ
6		nnz 9613	. . . . . . . . . . . . 13 |- ( C e. NN -> C e. ZZ )
7	6	3ad2ant3 1047	. . . . . . . . . . . 12 |- ( ( A e. NN /\ B e. NN /\ C e. NN ) -> C e. ZZ )
8		nnz 9613	. . . . . . . . . . . . 13 |- ( B e. NN -> B e. ZZ )
9	8	3ad2ant2 1046	. . . . . . . . . . . 12 |- ( ( A e. NN /\ B e. NN /\ C e. NN ) -> B e. ZZ )
10	7, 9	zaddcld 9722	. . . . . . . . . . 11 |- ( ( A e. NN /\ B e. NN /\ C e. NN ) -> ( C + B ) e. ZZ )
11	10	3ad2ant1 1045	. . . . . . . . . 10 |- ( ( ( 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 )
12		nnz 9613	. . . . . . . . . . . 12 |- ( A e. NN -> A e. ZZ )
13	12	3ad2ant1 1045	. . . . . . . . . . 11 |- ( ( A e. NN /\ B e. NN /\ C e. NN ) -> A e. ZZ )
14	13	3ad2ant1 1045	. . . . . . . . . 10 |- ( ( ( A e. NN /\ B e. NN /\ C e. NN ) /\ ( ( A ^ 2 ) + ( B ^ 2 ) ) = ( C ^ 2 ) /\ ( ( A gcd B ) = 1 /\ -. 2 || A ) ) -> A e. ZZ )
15		dvdsgcdb 12734	. . . . . . . . . 10 |- ( ( 2 e. ZZ /\ ( C + B ) e. ZZ /\ A e. ZZ ) -> ( ( 2 || ( C + B ) /\ 2 || A ) <-> 2 || ( ( C + B ) gcd A ) ) )
16	5, 11, 14, 15	mp3an2i 1379	. . . . . . . . 9 |- ( ( ( A e. NN /\ B e. NN /\ C e. NN ) /\ ( ( A ^ 2 ) + ( B ^ 2 ) ) = ( C ^ 2 ) /\ ( ( A gcd B ) = 1 /\ -. 2 || A ) ) -> ( ( 2 || ( C + B ) /\ 2 || A ) <-> 2 || ( ( C + B ) gcd A ) ) )
17	16	biimpar 297	. . . . . . . 8 |- ( ( ( ( A e. NN /\ B e. NN /\ C e. NN ) /\ ( ( A ^ 2 ) + ( B ^ 2 ) ) = ( C ^ 2 ) /\ ( ( A gcd B ) = 1 /\ -. 2 || A ) ) /\ 2 || ( ( C + B ) gcd A ) ) -> ( 2 || ( C + B ) /\ 2 || A ) )
18	17	simprd 114	. . . . . . 7 |- ( ( ( ( A e. NN /\ B e. NN /\ C e. NN ) /\ ( ( A ^ 2 ) + ( B ^ 2 ) ) = ( C ^ 2 ) /\ ( ( A gcd B ) = 1 /\ -. 2 || A ) ) /\ 2 || ( ( C + B ) gcd A ) ) -> 2 || A )
19	4, 18	mtand 671	. . . . . 6 |- ( ( ( A e. NN /\ B e. NN /\ C e. NN ) /\ ( ( A ^ 2 ) + ( B ^ 2 ) ) = ( C ^ 2 ) /\ ( ( A gcd B ) = 1 /\ -. 2 || A ) ) -> -. 2 || ( ( C + B ) gcd A ) )
20		pythagtriplem7 12994	. . . . . . 7 |- ( ( ( 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 ) )
21	20	breq2d 4126	. . . . . 6 |- ( ( ( A e. NN /\ B e. NN /\ C e. NN ) /\ ( ( A ^ 2 ) + ( B ^ 2 ) ) = ( C ^ 2 ) /\ ( ( A gcd B ) = 1 /\ -. 2 || A ) ) -> ( 2 || ( sqr ` ( C + B ) ) <-> 2 || ( ( C + B ) gcd A ) ) )
22	19, 21	mtbird 680	. . . . 5 |- ( ( ( A e. NN /\ B e. NN /\ C e. NN ) /\ ( ( A ^ 2 ) + ( B ^ 2 ) ) = ( C ^ 2 ) /\ ( ( A gcd B ) = 1 /\ -. 2 || A ) ) -> -. 2 || ( sqr ` ( C + B ) ) )
23		pythagtriplem8 12995	. . . . . 6 |- ( ( ( A e. NN /\ B e. NN /\ C e. NN ) /\ ( ( A ^ 2 ) + ( B ^ 2 ) ) = ( C ^ 2 ) /\ ( ( A gcd B ) = 1 /\ -. 2 || A ) ) -> ( sqr ` ( C - B ) ) e. NN )
24	23	nnzd 9717	. . . . 5 |- ( ( ( A e. NN /\ B e. NN /\ C e. NN ) /\ ( ( A ^ 2 ) + ( B ^ 2 ) ) = ( C ^ 2 ) /\ ( ( A gcd B ) = 1 /\ -. 2 || A ) ) -> ( sqr ` ( C - B ) ) e. ZZ )
25	7, 9	zsubcld 9723	. . . . . . . . . . 11 |- ( ( A e. NN /\ B e. NN /\ C e. NN ) -> ( C - B ) e. ZZ )
26	25	3ad2ant1 1045	. . . . . . . . . 10 |- ( ( ( 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 )
27		dvdsgcdb 12734	. . . . . . . . . 10 |- ( ( 2 e. ZZ /\ ( C - B ) e. ZZ /\ A e. ZZ ) -> ( ( 2 || ( C - B ) /\ 2 || A ) <-> 2 || ( ( C - B ) gcd A ) ) )
28	5, 26, 14, 27	mp3an2i 1379	. . . . . . . . 9 |- ( ( ( A e. NN /\ B e. NN /\ C e. NN ) /\ ( ( A ^ 2 ) + ( B ^ 2 ) ) = ( C ^ 2 ) /\ ( ( A gcd B ) = 1 /\ -. 2 || A ) ) -> ( ( 2 || ( C - B ) /\ 2 || A ) <-> 2 || ( ( C - B ) gcd A ) ) )
29	28	biimpar 297	. . . . . . . 8 |- ( ( ( ( A e. NN /\ B e. NN /\ C e. NN ) /\ ( ( A ^ 2 ) + ( B ^ 2 ) ) = ( C ^ 2 ) /\ ( ( A gcd B ) = 1 /\ -. 2 || A ) ) /\ 2 || ( ( C - B ) gcd A ) ) -> ( 2 || ( C - B ) /\ 2 || A ) )
30	29	simprd 114	. . . . . . 7 |- ( ( ( ( A e. NN /\ B e. NN /\ C e. NN ) /\ ( ( A ^ 2 ) + ( B ^ 2 ) ) = ( C ^ 2 ) /\ ( ( A gcd B ) = 1 /\ -. 2 || A ) ) /\ 2 || ( ( C - B ) gcd A ) ) -> 2 || A )
31	4, 30	mtand 671	. . . . . 6 |- ( ( ( A e. NN /\ B e. NN /\ C e. NN ) /\ ( ( A ^ 2 ) + ( B ^ 2 ) ) = ( C ^ 2 ) /\ ( ( A gcd B ) = 1 /\ -. 2 || A ) ) -> -. 2 || ( ( C - B ) gcd A ) )
32		pythagtriplem6 12993	. . . . . . 7 |- ( ( ( 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 ) )
33	32	breq2d 4126	. . . . . 6 |- ( ( ( A e. NN /\ B e. NN /\ C e. NN ) /\ ( ( A ^ 2 ) + ( B ^ 2 ) ) = ( C ^ 2 ) /\ ( ( A gcd B ) = 1 /\ -. 2 || A ) ) -> ( 2 || ( sqr ` ( C - B ) ) <-> 2 || ( ( C - B ) gcd A ) ) )
34	31, 33	mtbird 680	. . . . 5 |- ( ( ( A e. NN /\ B e. NN /\ C e. NN ) /\ ( ( A ^ 2 ) + ( B ^ 2 ) ) = ( C ^ 2 ) /\ ( ( A gcd B ) = 1 /\ -. 2 || A ) ) -> -. 2 || ( sqr ` ( C - B ) ) )
35		opoe 12606	. . . . 5 |- ( ( ( ( sqr ` ( C + B ) ) e. ZZ /\ -. 2 || ( sqr ` ( C + B ) ) ) /\ ( ( sqr ` ( C - B ) ) e. ZZ /\ -. 2 || ( sqr ` ( C - B ) ) ) ) -> 2 || ( ( sqr ` ( C + B ) ) + ( sqr ` ( C - B ) ) ) )
36	3, 22, 24, 34, 35	syl22anc 1275	. . . 4 |- ( ( ( A e. NN /\ B e. NN /\ C e. NN ) /\ ( ( A ^ 2 ) + ( B ^ 2 ) ) = ( C ^ 2 ) /\ ( ( A gcd B ) = 1 /\ -. 2 || A ) ) -> 2 || ( ( sqr ` ( C + B ) ) + ( sqr ` ( C - B ) ) ) )
37	2, 23	nnaddcld 9302	. . . . . 6 |- ( ( ( 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 ) ) ) e. NN )
38	37	nnzd 9717	. . . . 5 |- ( ( ( 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 ) ) ) e. ZZ )
39		evend2 12600	. . . . 5 |- ( ( ( sqr ` ( C + B ) ) + ( sqr ` ( C - B ) ) ) e. ZZ -> ( 2 || ( ( sqr ` ( C + B ) ) + ( sqr ` ( C - B ) ) ) <-> ( ( ( sqr ` ( C + B ) ) + ( sqr ` ( C - B ) ) ) / 2 ) e. ZZ ) )
40	38, 39	syl 14	. . . 4 |- ( ( ( A e. NN /\ B e. NN /\ C e. NN ) /\ ( ( A ^ 2 ) + ( B ^ 2 ) ) = ( C ^ 2 ) /\ ( ( A gcd B ) = 1 /\ -. 2 || A ) ) -> ( 2 || ( ( sqr ` ( C + B ) ) + ( sqr ` ( C - B ) ) ) <-> ( ( ( sqr ` ( C + B ) ) + ( sqr ` ( C - B ) ) ) / 2 ) e. ZZ ) )
41	36, 40	mpbid 147	. . 3 |- ( ( ( 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 ) ) ) / 2 ) e. ZZ )
42	2	nnred 9267	. . . . 5 |- ( ( ( A e. NN /\ B e. NN /\ C e. NN ) /\ ( ( A ^ 2 ) + ( B ^ 2 ) ) = ( C ^ 2 ) /\ ( ( A gcd B ) = 1 /\ -. 2 || A ) ) -> ( sqr ` ( C + B ) ) e. RR )
43	23	nnred 9267	. . . . 5 |- ( ( ( A e. NN /\ B e. NN /\ C e. NN ) /\ ( ( A ^ 2 ) + ( B ^ 2 ) ) = ( C ^ 2 ) /\ ( ( A gcd B ) = 1 /\ -. 2 || A ) ) -> ( sqr ` ( C - B ) ) e. RR )
44	2	nngt0d 9298	. . . . 5 |- ( ( ( A e. NN /\ B e. NN /\ C e. NN ) /\ ( ( A ^ 2 ) + ( B ^ 2 ) ) = ( C ^ 2 ) /\ ( ( A gcd B ) = 1 /\ -. 2 || A ) ) -> 0 < ( sqr ` ( C + B ) ) )
45	23	nngt0d 9298	. . . . 5 |- ( ( ( A e. NN /\ B e. NN /\ C e. NN ) /\ ( ( A ^ 2 ) + ( B ^ 2 ) ) = ( C ^ 2 ) /\ ( ( A gcd B ) = 1 /\ -. 2 || A ) ) -> 0 < ( sqr ` ( C - B ) ) )
46	42, 43, 44, 45	addgt0d 8812	. . . 4 |- ( ( ( A e. NN /\ B e. NN /\ C e. NN ) /\ ( ( A ^ 2 ) + ( B ^ 2 ) ) = ( C ^ 2 ) /\ ( ( A gcd B ) = 1 /\ -. 2 || A ) ) -> 0 < ( ( sqr ` ( C + B ) ) + ( sqr ` ( C - B ) ) ) )
47	37	nnred 9267	. . . . 5 |- ( ( ( 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 ) ) ) e. RR )
48		halfpos2 9485	. . . . 5 |- ( ( ( sqr ` ( C + B ) ) + ( sqr ` ( C - B ) ) ) e. RR -> ( 0 < ( ( sqr ` ( C + B ) ) + ( sqr ` ( C - B ) ) ) <-> 0 < ( ( ( sqr ` ( C + B ) ) + ( sqr ` ( C - B ) ) ) / 2 ) ) )
49	47, 48	syl 14	. . . 4 |- ( ( ( A e. NN /\ B e. NN /\ C e. NN ) /\ ( ( A ^ 2 ) + ( B ^ 2 ) ) = ( C ^ 2 ) /\ ( ( A gcd B ) = 1 /\ -. 2 || A ) ) -> ( 0 < ( ( sqr ` ( C + B ) ) + ( sqr ` ( C - B ) ) ) <-> 0 < ( ( ( sqr ` ( C + B ) ) + ( sqr ` ( C - B ) ) ) / 2 ) ) )
50	46, 49	mpbid 147	. . 3 |- ( ( ( A e. NN /\ B e. NN /\ C e. NN ) /\ ( ( A ^ 2 ) + ( B ^ 2 ) ) = ( C ^ 2 ) /\ ( ( A gcd B ) = 1 /\ -. 2 || A ) ) -> 0 < ( ( ( sqr ` ( C + B ) ) + ( sqr ` ( C - B ) ) ) / 2 ) )
51		elnnz 9604	. . 3 |- ( ( ( ( sqr ` ( C + B ) ) + ( sqr ` ( C - B ) ) ) / 2 ) e. NN <-> ( ( ( ( sqr ` ( C + B ) ) + ( sqr ` ( C - B ) ) ) / 2 ) e. ZZ /\ 0 < ( ( ( sqr ` ( C + B ) ) + ( sqr ` ( C - B ) ) ) / 2 ) ) )
52	41, 50, 51	sylanbrc 417	. 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 ) ) ) / 2 ) e. NN )
53	1, 52	eqeltrid 2321	1 |- ( ( ( A e. NN /\ B e. NN /\ C e. NN ) /\ ( ( A ^ 2 ) + ( B ^ 2 ) ) = ( C ^ 2 ) /\ ( ( A gcd B ) = 1 /\ -. 2 || A ) ) -> M e. NN )

Syntax hints:   -. wn 3   -> wi 4   /\ wa 104   <-> wb 105   /\ w3a 1005   = wceq 1398   e. wcel 2205   class class class wbr 4114   ` cfv 5357  (class class class)co 6058   RRcr 8142   0cc0 8143   1c1 8144   + caddc 8146   < clt 8324   - cmin 8460   / cdiv 8963   NNcn 9254   2c2 9305   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-xor 1421  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:  pythagtriplem18  13004