Theorem pythagtriplem11 12908

Description: Lemma for pythagtrip 12917. 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 12907	. . . . . 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 9644	. . . . 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 9550	. . . . . . . . . 10 |- 2 e. ZZ
6		nnz 9541	. . . . . . . . . . . . 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 9541	. . . . . . . . . . . . 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 9649	. . . . . . . . . . 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 9541	. . . . . . . . . . . 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 12645	. . . . . . . . . 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 12905	. . . . . . 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 4105	. . . . . 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 12906	. . . . . 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 9644	. . . . 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 9650	. . . . . . . . . . 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 12645	. . . . . . . . . 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 12904	. . . . . . 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 4105	. . . . . 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 12517	. . . . 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 9234	. . . . . 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 9644	. . . . 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 12511	. . . . 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 9199	. . . . 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 9199	. . . . 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 9230	. . . . 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 9230	. . . . 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 8744	. . . 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 9199	. . . . 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 9417	. . . . 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 9532	. . 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 1005   = wceq 1398   e. wcel 2202   class class class wbr 4093   ` cfv 5333  (class class class)co 6028   RRcr 8074   0cc0 8075   1c1 8076   + caddc 8078   < clt 8257   - cmin 8393   / cdiv 8895   NNcn 9186   2c2 9237   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-xor 1421  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:  pythagtriplem18  12915