Theorem divgcdodd 11997

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 11784	. . . 4 |- -. 2 || 1
2		2z 9178	. . . . . . 7 |- 2 e. ZZ
3		nnz 9169	. . . . . . . . . 10 |- ( A e. NN -> A e. ZZ )
4		nnz 9169	. . . . . . . . . 10 |- ( B e. NN -> B e. ZZ )
5		gcddvds 11827	. . . . . . . . . 10 |- ( ( A e. ZZ /\ B e. ZZ ) -> ( ( A gcd B ) || A /\ ( A gcd B ) || B ) )
6	3, 4, 5	syl2an 287	. . . . . . . . 9 |- ( ( A e. NN /\ B e. NN ) -> ( ( A gcd B ) || A /\ ( A gcd B ) || B ) )
7	6	simpld 111	. . . . . . . 8 |- ( ( A e. NN /\ B e. NN ) -> ( A gcd B ) || A )
8		gcdnncl 11831	. . . . . . . . . 10 |- ( ( A e. NN /\ B e. NN ) -> ( A gcd B ) e. NN )
9	8	nnzd 9268	. . . . . . . . 9 |- ( ( A e. NN /\ B e. NN ) -> ( A gcd B ) e. ZZ )
10	8	nnne0d 8861	. . . . . . . . 9 |- ( ( A e. NN /\ B e. NN ) -> ( A gcd B ) =/= 0 )
11	3	adantr 274	. . . . . . . . 9 |- ( ( A e. NN /\ B e. NN ) -> A e. ZZ )
12		dvdsval2 11668	. . . . . . . . 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 1220	. . . . . . . 8 |- ( ( A e. NN /\ B e. NN ) -> ( ( A gcd B ) || A <-> ( A / ( A gcd B ) ) e. ZZ ) )
14	7, 13	mpbid 146	. . . . . . 7 |- ( ( A e. NN /\ B e. NN ) -> ( A / ( A gcd B ) ) e. ZZ )
15	6	simprd 113	. . . . . . . 8 |- ( ( A e. NN /\ B e. NN ) -> ( A gcd B ) || B )
16	4	adantl 275	. . . . . . . . 9 |- ( ( A e. NN /\ B e. NN ) -> B e. ZZ )
17		dvdsval2 11668	. . . . . . . . 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 1220	. . . . . . . 8 |- ( ( A e. NN /\ B e. NN ) -> ( ( A gcd B ) || B <-> ( B / ( A gcd B ) ) e. ZZ ) )
19	15, 18	mpbid 146	. . . . . . 7 |- ( ( A e. NN /\ B e. NN ) -> ( B / ( A gcd B ) ) e. ZZ )
20		dvdsgcdb 11877	. . . . . . 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 1324	. . . . . 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 11883	. . . . . . . . . 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 1223	. . . . . . . . 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 8830	. . . . . . . . . 10 |- ( ( A e. NN /\ B e. NN ) -> ( A gcd B ) e. CC )
25	8	nnap0d 8862	. . . . . . . . . 10 |- ( ( A e. NN /\ B e. NN ) -> ( A gcd B ) # 0 )
26	24, 25	dividapd 8642	. . . . . . . . 9 |- ( ( A e. NN /\ B e. NN ) -> ( ( A gcd B ) / ( A gcd B ) ) = 1 )
27	23, 26	eqtr3d 2192	. . . . . . . 8 |- ( ( A e. NN /\ B e. NN ) -> ( ( A / ( A gcd B ) ) gcd ( B / ( A gcd B ) ) ) = 1 )
28	27	breq2d 3977	. . . . . . 7 |- ( ( A e. NN /\ B e. NN ) -> ( 2 || ( ( A / ( A gcd B ) ) gcd ( B / ( A gcd B ) ) ) <-> 2 || 1 ) )
29	28	biimpd 143	. . . . . 6 |- ( ( A e. NN /\ B e. NN ) -> ( 2 || ( ( A / ( A gcd B ) ) gcd ( B / ( A gcd B ) ) ) -> 2 || 1 ) )
30	21, 29	sylbid 149	. . . . 5 |- ( ( A e. NN /\ B e. NN ) -> ( ( 2 || ( A / ( A gcd B ) ) /\ 2 || ( B / ( A gcd B ) ) ) -> 2 || 1 ) )
31	30	expdimp 257	. . . 4 |- ( ( ( A e. NN /\ B e. NN ) /\ 2 || ( A / ( A gcd B ) ) ) -> ( 2 || ( B / ( A gcd B ) ) -> 2 || 1 ) )
32	1, 31	mtoi 654	. . 3 |- ( ( ( A e. NN /\ B e. NN ) /\ 2 || ( A / ( A gcd B ) ) ) -> -. 2 || ( B / ( A gcd B ) ) )
33	32	ex 114	. 2 |- ( ( A e. NN /\ B e. NN ) -> ( 2 || ( A / ( A gcd B ) ) -> -. 2 || ( B / ( A gcd B ) ) ) )
34		2nn 8977	. . . 4 |- 2 e. NN
35		dvdsdc 11676	. . . 4 |- ( ( 2 e. NN /\ ( A / ( A gcd B ) ) e. ZZ ) -> DECID 2 || ( A / ( A gcd B ) ) )
36	34, 14, 35	sylancr 411	. . 3 |- ( ( A e. NN /\ B e. NN ) -> DECID 2 || ( A / ( A gcd B ) ) )
37		imordc 883	. . 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 146	1 |- ( ( A e. NN /\ B e. NN ) -> ( -. 2 || ( A / ( A gcd B ) ) \/ -. 2 || ( B / ( A gcd B ) ) ) )

Syntax hints:   -. wn 3   -> wi 4   /\ wa 103   <-> wb 104   \/ wo 698  DECID wdc 820   = wceq 1335   e. wcel 2128   =/= wne 2327   class class class wbr 3965  (class class class)co 5818   0cc0 7715   1c1 7716   / cdiv 8528   NNcn 8816   2c2 8867   ZZcz 9150   || cdvds 11665   gcd cgcd 11810

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-in1 604  ax-in2 605  ax-io 699  ax-5 1427  ax-7 1428  ax-gen 1429  ax-ie1 1473  ax-ie2 1474  ax-8 1484  ax-10 1485  ax-11 1486  ax-i12 1487  ax-bndl 1489  ax-4 1490  ax-17 1506  ax-i9 1510  ax-ial 1514  ax-i5r 1515  ax-13 2130  ax-14 2131  ax-ext 2139  ax-coll 4079  ax-sep 4082  ax-nul 4090  ax-pow 4134  ax-pr 4168  ax-un 4392  ax-setind 4494  ax-iinf 4545  ax-cnex 7806  ax-resscn 7807  ax-1cn 7808  ax-1re 7809  ax-icn 7810  ax-addcl 7811  ax-addrcl 7812  ax-mulcl 7813  ax-mulrcl 7814  ax-addcom 7815  ax-mulcom 7816  ax-addass 7817  ax-mulass 7818  ax-distr 7819  ax-i2m1 7820  ax-0lt1 7821  ax-1rid 7822  ax-0id 7823  ax-rnegex 7824  ax-precex 7825  ax-cnre 7826  ax-pre-ltirr 7827  ax-pre-ltwlin 7828  ax-pre-lttrn 7829  ax-pre-apti 7830  ax-pre-ltadd 7831  ax-pre-mulgt0 7832  ax-pre-mulext 7833  ax-arch 7834  ax-caucvg 7835

This theorem depends on definitions:  df-bi 116  df-dc 821  df-3or 964  df-3an 965  df-tru 1338  df-fal 1341  df-nf 1441  df-sb 1743  df-eu 2009  df-mo 2010  df-clab 2144  df-cleq 2150  df-clel 2153  df-nfc 2288  df-ne 2328  df-nel 2423  df-ral 2440  df-rex 2441  df-reu 2442  df-rmo 2443  df-rab 2444  df-v 2714  df-sbc 2938  df-csb 3032  df-dif 3104  df-un 3106  df-in 3108  df-ss 3115  df-nul 3395  df-if 3506  df-pw 3545  df-sn 3566  df-pr 3567  df-op 3569  df-uni 3773  df-int 3808  df-iun 3851  df-br 3966  df-opab 4026  df-mpt 4027  df-tr 4063  df-id 4252  df-po 4255  df-iso 4256  df-iord 4325  df-on 4327  df-ilim 4328  df-suc 4330  df-iom 4548  df-xp 4589  df-rel 4590  df-cnv 4591  df-co 4592  df-dm 4593  df-rn 4594  df-res 4595  df-ima 4596  df-iota 5132  df-fun 5169  df-fn 5170  df-f 5171  df-f1 5172  df-fo 5173  df-f1o 5174  df-fv 5175  df-riota 5774  df-ov 5821  df-oprab 5822  df-mpo 5823  df-1st 6082  df-2nd 6083  df-recs 6246  df-frec 6332  df-sup 6920  df-pnf 7897  df-mnf 7898  df-xr 7899  df-ltxr 7900  df-le 7901  df-sub 8031  df-neg 8032  df-reap 8433  df-ap 8440  df-div 8529  df-inn 8817  df-2 8875  df-3 8876  df-4 8877  df-n0 9074  df-z 9151  df-uz 9423  df-q 9511  df-rp 9543  df-fz 9895  df-fzo 10024  df-fl 10151  df-mod 10204  df-seqfrec 10327  df-exp 10401  df-cj 10724  df-re 10725  df-im 10726  df-rsqrt 10880  df-abs 10881  df-dvds 11666  df-gcd 11811

This theorem is referenced by: (None)