Theorem divgcdodd 12923

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 12681	. . . 4 |- -. 2 || 1
2		2z 9674	. . . . . . 7 |- 2 e. ZZ
3		nnz 9665	. . . . . . . . . 10 |- ( A e. NN -> A e. ZZ )
4		nnz 9665	. . . . . . . . . 10 |- ( B e. NN -> B e. ZZ )
5		gcddvds 12742	. . . . . . . . . 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 12746	. . . . . . . . . 10 |- ( ( A e. NN /\ B e. NN ) -> ( A gcd B ) e. NN )
9	8	nnzd 9769	. . . . . . . . 9 |- ( ( A e. NN /\ B e. NN ) -> ( A gcd B ) e. ZZ )
10	8	nnne0d 9350	. . . . . . . . 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 12559	. . . . . . . . 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 1278	. . . . . . . 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 12559	. . . . . . . . 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 1278	. . . . . . . 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 12792	. . . . . . 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 1383	. . . . . 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 12798	. . . . . . . . . 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 1281	. . . . . . . . 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 9319	. . . . . . . . . 10 |- ( ( A e. NN /\ B e. NN ) -> ( A gcd B ) e. CC )
25	8	nnap0d 9351	. . . . . . . . . 10 |- ( ( A e. NN /\ B e. NN ) -> ( A gcd B ) # 0 )
26	24, 25	dividapd 9117	. . . . . . . . 9 |- ( ( A e. NN /\ B e. NN ) -> ( ( A gcd B ) / ( A gcd B ) ) = 1 )
27	23, 26	eqtr3d 2273	. . . . . . . 8 |- ( ( A e. NN /\ B e. NN ) -> ( ( A / ( A gcd B ) ) gcd ( B / ( A gcd B ) ) ) = 1 )
28	27	breq2d 4142	. . . . . . 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 674	. . 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 9468	. . . 4 |- 2 e. NN
35		dvdsdc 12567	. . . 4 |- ( ( 2 e. NN /\ ( A / ( A gcd B ) ) e. ZZ ) -> DECID 2 || ( A / ( A gcd B ) ) )
36	34, 14, 35	sylancr 418	. . 3 |- ( ( A e. NN /\ B e. NN ) -> DECID 2 || ( A / ( A gcd B ) ) )
37		imordc 909	. . 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 ) ) ) )

This proof depends on syntax axioms:   -. wn 3   -> wi 4   /\ wa 104   <-> wb 105   \/ wo 720  DECID wdc 846   = wceq 1402   e. wcel 2209   =/= wne 2420   class class class wbr 4130  (class class class)co 6085   0cc0 8179   1c1 8180   / cdiv 9003   NNcn 9305   2c2 9356   ZZcz 9646   || cdvds 12556   gcd cgcd 12732

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-in1 623  ax-in2 624  ax-io 721  ax-5 1500  ax-7 1501  ax-gen 1502  ax-ie1 1546  ax-ie2 1547  ax-8 1557  ax-10 1558  ax-11 1559  ax-i12 1560  ax-bndl 1562  ax-4 1563  ax-17 1579  ax-i9 1583  ax-ial 1587  ax-i5r 1588  ax-14 2212  ax-ext 2220  ax-coll 4246  ax-sep 4249  ax-nul 4259  ax-pow 4311  ax-pr 4346  ax-un 4578  ax-setind 4684  ax-iinf 4735  ax-cnex 8270  ax-resscn 8271  ax-1cn 8272  ax-1re 8273  ax-icn 8274  ax-addcl 8275  ax-addrcl 8276  ax-mulcl 8277  ax-mulrcl 8278  ax-addcom 8279  ax-mulcom 8280  ax-addass 8281  ax-mulass 8282  ax-distr 8283  ax-i2m1 8284  ax-0lt1 8285  ax-1rid 8286  ax-0id 8287  ax-rnegex 8288  ax-precex 8289  ax-cnre 8290  ax-pre-ltirr 8291  ax-pre-ltwlin 8292  ax-pre-lttrn 8293  ax-pre-apti 8294  ax-pre-ltadd 8295  ax-pre-mulgt0 8296  ax-pre-mulext 8297  ax-arch 8298  ax-caucvg 8299

This proof depends on definitions:  df-bi 117  df-dc 847  df-3or 1010  df-3an 1011  df-tru 1405  df-fal 1408  df-nf 1514  df-sb 1816  df-eu 2089  df-mo 2090  df-clab 2225  df-cleq 2231  df-clel 2234  df-nfc 2381  df-ne 2421  df-nel 2516  df-ral 2533  df-rex 2534  df-reu 2535  df-rmo 2536  df-rab 2537  df-v 2823  df-sbc 3052  df-csb 3148  df-dif 3222  df-un 3224  df-in 3226  df-ss 3233  df-nul 3521  df-if 3639  df-pw 3690  df-sn 3715  df-pr 3716  df-op 3718  df-uni 3936  df-int 3971  df-iun 4014  df-br 4131  df-opab 4193  df-mpt 4194  df-tr 4230  df-id 4438  df-po 4441  df-iso 4442  df-iord 4511  df-on 4513  df-ilim 4514  df-suc 4516  df-iom 4738  df-xp 4780  df-rel 4781  df-cnv 4782  df-co 4783  df-dm 4784  df-rn 4785  df-res 4786  df-ima 4787  df-iota 5337  df-fun 5379  df-fn 5380  df-f 5381  df-f1 5382  df-fo 5383  df-f1o 5384  df-fv 5385  df-riota 6038  df-ov 6088  df-oprab 6089  df-mpo 6090  df-1st 6374  df-2nd 6375  df-recs 6576  df-frec 6662  df-sup 7324  df-pnf 8362  df-mnf 8363  df-xr 8364  df-ltxr 8365  df-le 8366  df-sub 8499  df-neg 8500  df-reap 8904  df-ap 8911  df-div 9004  df-inn 9306  df-2 9364  df-3 9365  df-4 9366  df-n0 9566  df-z 9647  df-uz 9924  df-q 10022  df-rp 10057  df-fz 10414  df-fzo 10552  df-fl 10707  df-mod 10762  df-seqfrec 10887  df-exp 10978  df-cj 11609  df-re 11610  df-im 11611  df-rsqrt 11766  df-abs 11767  df-dvds 12557  df-gcd 12733

This theorem is used by:  pythagtrip  13064