Theorem pythagtriplem11 12852

Description: Lemma for pythagtrip 12861. 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 12851	. . . . . 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 9601	. . . . 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 1052	. . . . . . 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 9507	. . . . . . . . . 10 |- 2 e. ZZ
6		nnz 9498	. . . . . . . . . . . . 13 |- ( C e. NN -> C e. ZZ )
7	6	3ad2ant3 1046	. . . . . . . . . . . 12 |- ( ( A e. NN /\ B e. NN /\ C e. NN ) -> C e. ZZ )
8		nnz 9498	. . . . . . . . . . . . 13 |- ( B e. NN -> B e. ZZ )
9	8	3ad2ant2 1045	. . . . . . . . . . . 12 |- ( ( A e. NN /\ B e. NN /\ C e. NN ) -> B e. ZZ )
10	7, 9	zaddcld 9606	. . . . . . . . . . 11 |- ( ( A e. NN /\ B e. NN /\ C e. NN ) -> ( C + B ) e. ZZ )
11	10	3ad2ant1 1044	. . . . . . . . . 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 9498	. . . . . . . . . . . 12 |- ( A e. NN -> A e. ZZ )
13	12	3ad2ant1 1044	. . . . . . . . . . 11 |- ( ( A e. NN /\ B e. NN /\ C e. NN ) -> A e. ZZ )
14	13	3ad2ant1 1044	. . . . . . . . . 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 12589	. . . . . . . . . 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 1378	. . . . . . . . 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 12849	. . . . . . 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 4100	. . . . . 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 679	. . . . 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 12850	. . . . . 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 9601	. . . . 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 9607	. . . . . . . . . . 11 |- ( ( A e. NN /\ B e. NN /\ C e. NN ) -> ( C - B ) e. ZZ )
26	25	3ad2ant1 1044	. . . . . . . . . 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 12589	. . . . . . . . . 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 1378	. . . . . . . . 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 12848	. . . . . . 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 4100	. . . . . 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 679	. . . . 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 12461	. . . . 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 1274	. . . 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 9191	. . . . . 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 9601	. . . . 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 12455	. . . . 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 9156	. . . . 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 9156	. . . . 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 9187	. . . . 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 9187	. . . . 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 8701	. . . 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 9156	. . . . 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 9374	. . . . 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 9489	. . 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 2318	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 1004   = wceq 1397   e. wcel 2202   class class class wbr 4088   ` cfv 5326  (class class class)co 6018   RRcr 8031   0cc0 8032   1c1 8033   + caddc 8035   < clt 8214   - cmin 8350   / cdiv 8852   NNcn 9143   2c2 9194   ZZcz 9479   ^cexp 10801   sqrcsqrt 11561   || cdvds 12353   gcd cgcd 12529

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 716  ax-5 1495  ax-7 1496  ax-gen 1497  ax-ie1 1541  ax-ie2 1542  ax-8 1552  ax-10 1553  ax-11 1554  ax-i12 1555  ax-bndl 1557  ax-4 1558  ax-17 1574  ax-i9 1578  ax-ial 1582  ax-i5r 1583  ax-13 2204  ax-14 2205  ax-ext 2213  ax-coll 4204  ax-sep 4207  ax-nul 4215  ax-pow 4264  ax-pr 4299  ax-un 4530  ax-setind 4635  ax-iinf 4686  ax-cnex 8123  ax-resscn 8124  ax-1cn 8125  ax-1re 8126  ax-icn 8127  ax-addcl 8128  ax-addrcl 8129  ax-mulcl 8130  ax-mulrcl 8131  ax-addcom 8132  ax-mulcom 8133  ax-addass 8134  ax-mulass 8135  ax-distr 8136  ax-i2m1 8137  ax-0lt1 8138  ax-1rid 8139  ax-0id 8140  ax-rnegex 8141  ax-precex 8142  ax-cnre 8143  ax-pre-ltirr 8144  ax-pre-ltwlin 8145  ax-pre-lttrn 8146  ax-pre-apti 8147  ax-pre-ltadd 8148  ax-pre-mulgt0 8149  ax-pre-mulext 8150  ax-arch 8151  ax-caucvg 8152

This theorem depends on definitions:  df-bi 117  df-stab 838  df-dc 842  df-3or 1005  df-3an 1006  df-tru 1400  df-fal 1403  df-xor 1420  df-nf 1509  df-sb 1811  df-eu 2082  df-mo 2083  df-clab 2218  df-cleq 2224  df-clel 2227  df-nfc 2363  df-ne 2403  df-nel 2498  df-ral 2515  df-rex 2516  df-reu 2517  df-rmo 2518  df-rab 2519  df-v 2804  df-sbc 3032  df-csb 3128  df-dif 3202  df-un 3204  df-in 3206  df-ss 3213  df-nul 3495  df-if 3606  df-pw 3654  df-sn 3675  df-pr 3676  df-op 3678  df-uni 3894  df-int 3929  df-iun 3972  df-br 4089  df-opab 4151  df-mpt 4152  df-tr 4188  df-id 4390  df-po 4393  df-iso 4394  df-iord 4463  df-on 4465  df-ilim 4466  df-suc 4468  df-iom 4689  df-xp 4731  df-rel 4732  df-cnv 4733  df-co 4734  df-dm 4735  df-rn 4736  df-res 4737  df-ima 4738  df-iota 5286  df-fun 5328  df-fn 5329  df-f 5330  df-f1 5331  df-fo 5332  df-f1o 5333  df-fv 5334  df-riota 5971  df-ov 6021  df-oprab 6022  df-mpo 6023  df-1st 6303  df-2nd 6304  df-recs 6471  df-frec 6557  df-1o 6582  df-2o 6583  df-er 6702  df-en 6910  df-sup 7183  df-pnf 8216  df-mnf 8217  df-xr 8218  df-ltxr 8219  df-le 8220  df-sub 8352  df-neg 8353  df-reap 8755  df-ap 8762  df-div 8853  df-inn 9144  df-2 9202  df-3 9203  df-4 9204  df-n0 9403  df-z 9480  df-uz 9756  df-q 9854  df-rp 9889  df-fz 10244  df-fzo 10378  df-fl 10531  df-mod 10586  df-seqfrec 10711  df-exp 10802  df-cj 11407  df-re 11408  df-im 11409  df-rsqrt 11563  df-abs 11564  df-dvds 12354  df-gcd 12530  df-prm 12685

This theorem is referenced by:  pythagtriplem18  12859