Theorem divgcdodd 12146

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 11920	. . . 4 |- -. 2 || 1
2		2z 9284	. . . . . . 7 |- 2 e. ZZ
3		nnz 9275	. . . . . . . . . 10 |- ( A e. NN -> A e. ZZ )
4		nnz 9275	. . . . . . . . . 10 |- ( B e. NN -> B e. ZZ )
5		gcddvds 11967	. . . . . . . . . 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 11971	. . . . . . . . . 10 |- ( ( A e. NN /\ B e. NN ) -> ( A gcd B ) e. NN )
9	8	nnzd 9377	. . . . . . . . 9 |- ( ( A e. NN /\ B e. NN ) -> ( A gcd B ) e. ZZ )
10	8	nnne0d 8967	. . . . . . . . 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 11800	. . . . . . . . 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 11800	. . . . . . . . 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 12017	. . . . . . 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 12023	. . . . . . . . . 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 8936	. . . . . . . . . 10 |- ( ( A e. NN /\ B e. NN ) -> ( A gcd B ) e. CC )
25	8	nnap0d 8968	. . . . . . . . . 10 |- ( ( A e. NN /\ B e. NN ) -> ( A gcd B ) # 0 )
26	24, 25	dividapd 8746	. . . . . . . . 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 4017	. . . . . . 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 9083	. . . 4 |- 2 e. NN
35		dvdsdc 11808	. . . 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 4005  (class class class)co 5878   0cc0 7814   1c1 7815   / cdiv 8632   NNcn 8922   2c2 8973   ZZcz 9256   || cdvds 11797   gcd cgcd 11946

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 4120  ax-sep 4123  ax-nul 4131  ax-pow 4176  ax-pr 4211  ax-un 4435  ax-setind 4538  ax-iinf 4589  ax-cnex 7905  ax-resscn 7906  ax-1cn 7907  ax-1re 7908  ax-icn 7909  ax-addcl 7910  ax-addrcl 7911  ax-mulcl 7912  ax-mulrcl 7913  ax-addcom 7914  ax-mulcom 7915  ax-addass 7916  ax-mulass 7917  ax-distr 7918  ax-i2m1 7919  ax-0lt1 7920  ax-1rid 7921  ax-0id 7922  ax-rnegex 7923  ax-precex 7924  ax-cnre 7925  ax-pre-ltirr 7926  ax-pre-ltwlin 7927  ax-pre-lttrn 7928  ax-pre-apti 7929  ax-pre-ltadd 7930  ax-pre-mulgt0 7931  ax-pre-mulext 7932  ax-arch 7933  ax-caucvg 7934

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 2741  df-sbc 2965  df-csb 3060  df-dif 3133  df-un 3135  df-in 3137  df-ss 3144  df-nul 3425  df-if 3537  df-pw 3579  df-sn 3600  df-pr 3601  df-op 3603  df-uni 3812  df-int 3847  df-iun 3890  df-br 4006  df-opab 4067  df-mpt 4068  df-tr 4104  df-id 4295  df-po 4298  df-iso 4299  df-iord 4368  df-on 4370  df-ilim 4371  df-suc 4373  df-iom 4592  df-xp 4634  df-rel 4635  df-cnv 4636  df-co 4637  df-dm 4638  df-rn 4639  df-res 4640  df-ima 4641  df-iota 5180  df-fun 5220  df-fn 5221  df-f 5222  df-f1 5223  df-fo 5224  df-f1o 5225  df-fv 5226  df-riota 5834  df-ov 5881  df-oprab 5882  df-mpo 5883  df-1st 6144  df-2nd 6145  df-recs 6309  df-frec 6395  df-sup 6986  df-pnf 7997  df-mnf 7998  df-xr 7999  df-ltxr 8000  df-le 8001  df-sub 8133  df-neg 8134  df-reap 8535  df-ap 8542  df-div 8633  df-inn 8923  df-2 8981  df-3 8982  df-4 8983  df-n0 9180  df-z 9257  df-uz 9532  df-q 9623  df-rp 9657  df-fz 10012  df-fzo 10146  df-fl 10273  df-mod 10326  df-seqfrec 10449  df-exp 10523  df-cj 10854  df-re 10855  df-im 10856  df-rsqrt 11010  df-abs 11011  df-dvds 11798  df-gcd 11947

This theorem is referenced by:  pythagtrip  12286