Theorem divgcdodd 11667

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 11457	. . . 4 |- -. 2 || 1
2		2z 8986	. . . . . . 7 |- 2 e. ZZ
3		nnz 8977	. . . . . . . . . 10 |- ( A e. NN -> A e. ZZ )
4		nnz 8977	. . . . . . . . . 10 |- ( B e. NN -> B e. ZZ )
5		gcddvds 11500	. . . . . . . . . 10 |- ( ( A e. ZZ /\ B e. ZZ ) -> ( ( A gcd B ) || A /\ ( A gcd B ) || B ) )
6	3, 4, 5	syl2an 285	. . . . . . . . 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 11504	. . . . . . . . . 10 |- ( ( A e. NN /\ B e. NN ) -> ( A gcd B ) e. NN )
9	8	nnzd 9076	. . . . . . . . 9 |- ( ( A e. NN /\ B e. NN ) -> ( A gcd B ) e. ZZ )
10	8	nnne0d 8675	. . . . . . . . 9 |- ( ( A e. NN /\ B e. NN ) -> ( A gcd B ) =/= 0 )
11	3	adantr 272	. . . . . . . . 9 |- ( ( A e. NN /\ B e. NN ) -> A e. ZZ )
12		dvdsval2 11344	. . . . . . . . 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 1199	. . . . . . . 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 273	. . . . . . . . 9 |- ( ( A e. NN /\ B e. NN ) -> B e. ZZ )
17		dvdsval2 11344	. . . . . . . . 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 1199	. . . . . . . 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 11547	. . . . . . 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 1303	. . . . . 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 11553	. . . . . . . . . 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 1202	. . . . . . . . 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 8644	. . . . . . . . . 10 |- ( ( A e. NN /\ B e. NN ) -> ( A gcd B ) e. CC )
25	8	nnap0d 8676	. . . . . . . . . 10 |- ( ( A e. NN /\ B e. NN ) -> ( A gcd B ) # 0 )
26	24, 25	dividapd 8459	. . . . . . . . 9 |- ( ( A e. NN /\ B e. NN ) -> ( ( A gcd B ) / ( A gcd B ) ) = 1 )
27	23, 26	eqtr3d 2149	. . . . . . . 8 |- ( ( A e. NN /\ B e. NN ) -> ( ( A / ( A gcd B ) ) gcd ( B / ( A gcd B ) ) ) = 1 )
28	27	breq2d 3907	. . . . . . 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 636	. . 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 8785	. . . 4 |- 2 e. NN
35		dvdsdc 11349	. . . 4 |- ( ( 2 e. NN /\ ( A / ( A gcd B ) ) e. ZZ ) -> DECID 2 || ( A / ( A gcd B ) ) )
36	34, 14, 35	sylancr 408	. . 3 |- ( ( A e. NN /\ B e. NN ) -> DECID 2 || ( A / ( A gcd B ) ) )
37		imordc 865	. . 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 680  DECID wdc 802   = wceq 1314   e. wcel 1463   =/= wne 2282   class class class wbr 3895  (class class class)co 5728   0cc0 7547   1c1 7548   / cdiv 8345   NNcn 8630   2c2 8681   ZZcz 8958   || cdvds 11341   gcd cgcd 11483

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-in1 586  ax-in2 587  ax-io 681  ax-5 1406  ax-7 1407  ax-gen 1408  ax-ie1 1452  ax-ie2 1453  ax-8 1465  ax-10 1466  ax-11 1467  ax-i12 1468  ax-bndl 1469  ax-4 1470  ax-13 1474  ax-14 1475  ax-17 1489  ax-i9 1493  ax-ial 1497  ax-i5r 1498  ax-ext 2097  ax-coll 4003  ax-sep 4006  ax-nul 4014  ax-pow 4058  ax-pr 4091  ax-un 4315  ax-setind 4412  ax-iinf 4462  ax-cnex 7636  ax-resscn 7637  ax-1cn 7638  ax-1re 7639  ax-icn 7640  ax-addcl 7641  ax-addrcl 7642  ax-mulcl 7643  ax-mulrcl 7644  ax-addcom 7645  ax-mulcom 7646  ax-addass 7647  ax-mulass 7648  ax-distr 7649  ax-i2m1 7650  ax-0lt1 7651  ax-1rid 7652  ax-0id 7653  ax-rnegex 7654  ax-precex 7655  ax-cnre 7656  ax-pre-ltirr 7657  ax-pre-ltwlin 7658  ax-pre-lttrn 7659  ax-pre-apti 7660  ax-pre-ltadd 7661  ax-pre-mulgt0 7662  ax-pre-mulext 7663  ax-arch 7664  ax-caucvg 7665

This theorem depends on definitions:  df-bi 116  df-dc 803  df-3or 946  df-3an 947  df-tru 1317  df-fal 1320  df-nf 1420  df-sb 1719  df-eu 1978  df-mo 1979  df-clab 2102  df-cleq 2108  df-clel 2111  df-nfc 2244  df-ne 2283  df-nel 2378  df-ral 2395  df-rex 2396  df-reu 2397  df-rmo 2398  df-rab 2399  df-v 2659  df-sbc 2879  df-csb 2972  df-dif 3039  df-un 3041  df-in 3043  df-ss 3050  df-nul 3330  df-if 3441  df-pw 3478  df-sn 3499  df-pr 3500  df-op 3502  df-uni 3703  df-int 3738  df-iun 3781  df-br 3896  df-opab 3950  df-mpt 3951  df-tr 3987  df-id 4175  df-po 4178  df-iso 4179  df-iord 4248  df-on 4250  df-ilim 4251  df-suc 4253  df-iom 4465  df-xp 4505  df-rel 4506  df-cnv 4507  df-co 4508  df-dm 4509  df-rn 4510  df-res 4511  df-ima 4512  df-iota 5046  df-fun 5083  df-fn 5084  df-f 5085  df-f1 5086  df-fo 5087  df-f1o 5088  df-fv 5089  df-riota 5684  df-ov 5731  df-oprab 5732  df-mpo 5733  df-1st 5992  df-2nd 5993  df-recs 6156  df-frec 6242  df-sup 6823  df-pnf 7726  df-mnf 7727  df-xr 7728  df-ltxr 7729  df-le 7730  df-sub 7858  df-neg 7859  df-reap 8255  df-ap 8262  df-div 8346  df-inn 8631  df-2 8689  df-3 8690  df-4 8691  df-n0 8882  df-z 8959  df-uz 9229  df-q 9314  df-rp 9344  df-fz 9684  df-fzo 9813  df-fl 9936  df-mod 9989  df-seqfrec 10112  df-exp 10186  df-cj 10507  df-re 10508  df-im 10509  df-rsqrt 10662  df-abs 10663  df-dvds 11342  df-gcd 11484

This theorem is referenced by: (None)