Theorem divgcdodd 12142

Description: Either A / ( A gcd B ) is odd or B / ( A gcd B ) is odd. (Contributed by Scott Fenton, 19-Apr-2014.)

Assertion

Ref	Expression
divgcdodd	|- ( ( A e. NN /\ B e. NN ) -> ( -. 2 || ( A / ( A gcd B ) ) \/ -. 2 || ( B / ( A gcd B ) ) ) )

Proof of Theorem divgcdodd

Step	Hyp	Ref	Expression
1		n2dvds1 11916	. . . 4 |- -. 2 || 1
2		2z 9280	. . . . . . 7 |- 2 e. ZZ
3		nnz 9271	. . . . . . . . . 10 |- ( A e. NN -> A e. ZZ )
4		nnz 9271	. . . . . . . . . 10 |- ( B e. NN -> B e. ZZ )
5		gcddvds 11963	. . . . . . . . . 10 |- ( ( A e. ZZ /\ B e. ZZ ) -> ( ( A gcd B ) || A /\ ( A gcd B ) || B ) )
6	3, 4, 5	syl2an 289	. . . . . . . . 9 |- ( ( A e. NN /\ B e. NN ) -> ( ( A gcd B ) || A /\ ( A gcd B ) || B ) )
7	6	simpld 112	. . . . . . . 8 |- ( ( A e. NN /\ B e. NN ) -> ( A gcd B ) || A )
8		gcdnncl 11967	. . . . . . . . . 10 |- ( ( A e. NN /\ B e. NN ) -> ( A gcd B ) e. NN )
9	8	nnzd 9373	. . . . . . . . 9 |- ( ( A e. NN /\ B e. NN ) -> ( A gcd B ) e. ZZ )
10	8	nnne0d 8963	. . . . . . . . 9 |- ( ( A e. NN /\ B e. NN ) -> ( A gcd B ) =/= 0 )
11	3	adantr 276	. . . . . . . . 9 |- ( ( A e. NN /\ B e. NN ) -> A e. ZZ )
12		dvdsval2 11796	. . . . . . . . 9 |- ( ( ( A gcd B ) e. ZZ /\ ( A gcd B ) =/= 0 /\ A e. ZZ ) -> ( ( A gcd B ) || A <-> ( A / ( A gcd B ) ) e. ZZ ) )
13	9, 10, 11, 12	syl3anc 1238	. . . . . . . 8 |- ( ( A e. NN /\ B e. NN ) -> ( ( A gcd B ) || A <-> ( A / ( A gcd B ) ) e. ZZ ) )
14	7, 13	mpbid 147	. . . . . . 7 |- ( ( A e. NN /\ B e. NN ) -> ( A / ( A gcd B ) ) e. ZZ )
15	6	simprd 114	. . . . . . . 8 |- ( ( A e. NN /\ B e. NN ) -> ( A gcd B ) || B )
16	4	adantl 277	. . . . . . . . 9 |- ( ( A e. NN /\ B e. NN ) -> B e. ZZ )
17		dvdsval2 11796	. . . . . . . . 9 |- ( ( ( A gcd B ) e. ZZ /\ ( A gcd B ) =/= 0 /\ B e. ZZ ) -> ( ( A gcd B ) || B <-> ( B / ( A gcd B ) ) e. ZZ ) )
18	9, 10, 16, 17	syl3anc 1238	. . . . . . . 8 |- ( ( A e. NN /\ B e. NN ) -> ( ( A gcd B ) || B <-> ( B / ( A gcd B ) ) e. ZZ ) )
19	15, 18	mpbid 147	. . . . . . 7 |- ( ( A e. NN /\ B e. NN ) -> ( B / ( A gcd B ) ) e. ZZ )
20		dvdsgcdb 12013	. . . . . . 7 |- ( ( 2 e. ZZ /\ ( A / ( A gcd B ) ) e. ZZ /\ ( B / ( A gcd B ) ) e. ZZ ) -> ( ( 2 || ( A / ( A gcd B ) ) /\ 2 || ( B / ( A gcd B ) ) ) <-> 2 || ( ( A / ( A gcd B ) ) gcd ( B / ( A gcd B ) ) ) ) )
21	2, 14, 19, 20	mp3an2i 1342	. . . . . 6 |- ( ( A e. NN /\ B e. NN ) -> ( ( 2 || ( A / ( A gcd B ) ) /\ 2 || ( B / ( A gcd B ) ) ) <-> 2 || ( ( A / ( A gcd B ) ) gcd ( B / ( A gcd B ) ) ) ) )
22		gcddiv 12019	. . . . . . . . . 10 |- ( ( ( A e. ZZ /\ B e. ZZ /\ ( A gcd B ) e. NN ) /\ ( ( A gcd B ) || A /\ ( A gcd B ) || B ) ) -> ( ( A gcd B ) / ( A gcd B ) ) = ( ( A / ( A gcd B ) ) gcd ( B / ( A gcd B ) ) ) )
23	11, 16, 8, 6, 22	syl31anc 1241	. . . . . . . . 9 |- ( ( A e. NN /\ B e. NN ) -> ( ( A gcd B ) / ( A gcd B ) ) = ( ( A / ( A gcd B ) ) gcd ( B / ( A gcd B ) ) ) )
24	8	nncnd 8932	. . . . . . . . . 10 |- ( ( A e. NN /\ B e. NN ) -> ( A gcd B ) e. CC )
25	8	nnap0d 8964	. . . . . . . . . 10 |- ( ( A e. NN /\ B e. NN ) -> ( A gcd B ) # 0 )
26	24, 25	dividapd 8742	. . . . . . . . 9 |- ( ( A e. NN /\ B e. NN ) -> ( ( A gcd B ) / ( A gcd B ) ) = 1 )
27	23, 26	eqtr3d 2212	. . . . . . . 8 |- ( ( A e. NN /\ B e. NN ) -> ( ( A / ( A gcd B ) ) gcd ( B / ( A gcd B ) ) ) = 1 )
28	27	breq2d 4015	. . . . . . 7 |- ( ( A e. NN /\ B e. NN ) -> ( 2 || ( ( A / ( A gcd B ) ) gcd ( B / ( A gcd B ) ) ) <-> 2 || 1 ) )
29	28	biimpd 144	. . . . . 6 |- ( ( A e. NN /\ B e. NN ) -> ( 2 || ( ( A / ( A gcd B ) ) gcd ( B / ( A gcd B ) ) ) -> 2 || 1 ) )
30	21, 29	sylbid 150	. . . . 5 |- ( ( A e. NN /\ B e. NN ) -> ( ( 2 || ( A / ( A gcd B ) ) /\ 2 || ( B / ( A gcd B ) ) ) -> 2 || 1 ) )
31	30	expdimp 259	. . . 4 |- ( ( ( A e. NN /\ B e. NN ) /\ 2 || ( A / ( A gcd B ) ) ) -> ( 2 || ( B / ( A gcd B ) ) -> 2 || 1 ) )
32	1, 31	mtoi 664	. . 3 |- ( ( ( A e. NN /\ B e. NN ) /\ 2 || ( A / ( A gcd B ) ) ) -> -. 2 || ( B / ( A gcd B ) ) )
33	32	ex 115	. 2 |- ( ( A e. NN /\ B e. NN ) -> ( 2 || ( A / ( A gcd B ) ) -> -. 2 || ( B / ( A gcd B ) ) ) )
34		2nn 9079	. . . 4 |- 2 e. NN
35		dvdsdc 11804	. . . 4 |- ( ( 2 e. NN /\ ( A / ( A gcd B ) ) e. ZZ ) -> DECID 2 || ( A / ( A gcd B ) ) )
36	34, 14, 35	sylancr 414	. . 3 |- ( ( A e. NN /\ B e. NN ) -> DECID 2 || ( A / ( A gcd B ) ) )
37		imordc 897	. . 3 |- (DECID 2 || ( A / ( A gcd B ) ) -> ( ( 2 || ( A / ( A gcd B ) ) -> -. 2 || ( B / ( A gcd B ) ) ) <-> ( -. 2 || ( A / ( A gcd B ) ) \/ -. 2 || ( B / ( A gcd B ) ) ) ) )
38	36, 37	syl 14	. 2 |- ( ( A e. NN /\ B e. NN ) -> ( ( 2 || ( A / ( A gcd B ) ) -> -. 2 || ( B / ( A gcd B ) ) ) <-> ( -. 2 || ( A / ( A gcd B ) ) \/ -. 2 || ( B / ( A gcd B ) ) ) ) )
39	33, 38	mpbid 147	1 |- ( ( A e. NN /\ B e. NN ) -> ( -. 2 || ( A / ( A gcd B ) ) \/ -. 2 || ( B / ( A gcd B ) ) ) )

Syntax hints:   -. wn 3   -> wi 4   /\ wa 104   <-> wb 105   \/ wo 708  DECID wdc 834   = wceq 1353   e. wcel 2148   =/= wne 2347   class class class wbr 4003  (class class class)co 5874   0cc0 7810   1c1 7811   / cdiv 8628   NNcn 8918   2c2 8969   ZZcz 9252   || cdvds 11793   gcd cgcd 11942

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 4118  ax-sep 4121  ax-nul 4129  ax-pow 4174  ax-pr 4209  ax-un 4433  ax-setind 4536  ax-iinf 4587  ax-cnex 7901  ax-resscn 7902  ax-1cn 7903  ax-1re 7904  ax-icn 7905  ax-addcl 7906  ax-addrcl 7907  ax-mulcl 7908  ax-mulrcl 7909  ax-addcom 7910  ax-mulcom 7911  ax-addass 7912  ax-mulass 7913  ax-distr 7914  ax-i2m1 7915  ax-0lt1 7916  ax-1rid 7917  ax-0id 7918  ax-rnegex 7919  ax-precex 7920  ax-cnre 7921  ax-pre-ltirr 7922  ax-pre-ltwlin 7923  ax-pre-lttrn 7924  ax-pre-apti 7925  ax-pre-ltadd 7926  ax-pre-mulgt0 7927  ax-pre-mulext 7928  ax-arch 7929  ax-caucvg 7930

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 3577  df-sn 3598  df-pr 3599  df-op 3601  df-uni 3810  df-int 3845  df-iun 3888  df-br 4004  df-opab 4065  df-mpt 4066  df-tr 4102  df-id 4293  df-po 4296  df-iso 4297  df-iord 4366  df-on 4368  df-ilim 4369  df-suc 4371  df-iom 4590  df-xp 4632  df-rel 4633  df-cnv 4634  df-co 4635  df-dm 4636  df-rn 4637  df-res 4638  df-ima 4639  df-iota 5178  df-fun 5218  df-fn 5219  df-f 5220  df-f1 5221  df-fo 5222  df-f1o 5223  df-fv 5224  df-riota 5830  df-ov 5877  df-oprab 5878  df-mpo 5879  df-1st 6140  df-2nd 6141  df-recs 6305  df-frec 6391  df-sup 6982  df-pnf 7993  df-mnf 7994  df-xr 7995  df-ltxr 7996  df-le 7997  df-sub 8129  df-neg 8130  df-reap 8531  df-ap 8538  df-div 8629  df-inn 8919  df-2 8977  df-3 8978  df-4 8979  df-n0 9176  df-z 9253  df-uz 9528  df-q 9619  df-rp 9653  df-fz 10008  df-fzo 10142  df-fl 10269  df-mod 10322  df-seqfrec 10445  df-exp 10519  df-cj 10850  df-re 10851  df-im 10852  df-rsqrt 11006  df-abs 11007  df-dvds 11794  df-gcd 11943

This theorem is referenced by:  pythagtrip  12282