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Theorem divgcdoddALTV 41593
Description: Either A / ( A gcd B ) is odd or B / ( A gcd B ) is odd. (Contributed by Scott Fenton, 19-Apr-2014.) (Revised by AV, 21-Jun-2020.)
Assertion
Ref Expression
divgcdoddALTV |- ( ( A e. NN /\ B e. NN ) -> ( ( A / ( A gcd B ) ) e. Odd \/ ( B / ( A gcd B ) ) e. Odd ) )
Proof of Theorem divgcdoddALTV
StepHypRef Expression
1 divgcdodd 15422 . . 3 |- ( ( A e. NN /\ B e. NN ) -> ( -. 2 || ( A / ( A gcd B ) ) \/ -. 2 || ( B / ( A gcd B ) ) ) )
2 nnz 11399 . . . . . . . 8 |- ( A e. NN -> A e. ZZ )
3 nnz 11399 . . . . . . . 8 |- ( B e. NN -> B e. ZZ )
4 gcddvds 15225 . . . . . . . 8 |- ( ( A e. ZZ /\ B e. ZZ ) -> ( ( A gcd B ) || A /\ ( A gcd B ) || B ) )
52, 3, 4syl2an 494 . . . . . . 7 |- ( ( A e. NN /\ B e. NN ) -> ( ( A gcd B ) || A /\ ( A gcd B ) || B ) )
65simpld 475 . . . . . 6 |- ( ( A e. NN /\ B e. NN ) -> ( A gcd B ) || A )
72, 3anim12i 590 . . . . . . . . 9 |- ( ( A e. NN /\ B e. NN ) -> ( A e. ZZ /\ B e. ZZ ) )
8 nnne0 11053 . . . . . . . . . . . 12 |- ( A e. NN -> A =/= 0 )
98neneqd 2799 . . . . . . . . . . 11 |- ( A e. NN -> -. A = 0 )
109intnanrd 963 . . . . . . . . . 10 |- ( A e. NN -> -. ( A = 0 /\ B = 0 ) )
1110adantr 481 . . . . . . . . 9 |- ( ( A e. NN /\ B e. NN ) -> -. ( A = 0 /\ B = 0 ) )
12 gcdn0cl 15224 . . . . . . . . 9 |- ( ( ( A e. ZZ /\ B e. ZZ ) /\ -. ( A = 0 /\ B = 0 ) ) -> ( A gcd B ) e. NN )
137, 11, 12syl2anc 693 . . . . . . . 8 |- ( ( A e. NN /\ B e. NN ) -> ( A gcd B ) e. NN )
1413nnzd 11481 . . . . . . 7 |- ( ( A e. NN /\ B e. NN ) -> ( A gcd B ) e. ZZ )
1513nnne0d 11065 . . . . . . 7 |- ( ( A e. NN /\ B e. NN ) -> ( A gcd B ) =/= 0 )
162adantr 481 . . . . . . 7 |- ( ( A e. NN /\ B e. NN ) -> A e. ZZ )
17 dvdsval2 14986 . . . . . . 7 |- ( ( ( A gcd B ) e. ZZ /\ ( A gcd B ) =/= 0 /\ A e. ZZ ) -> ( ( A gcd B ) || A <-> ( A / ( A gcd B ) ) e. ZZ ) )
1814, 15, 16, 17syl3anc 1326 . . . . . 6 |- ( ( A e. NN /\ B e. NN ) -> ( ( A gcd B ) || A <-> ( A / ( A gcd B ) ) e. ZZ ) )
196, 18mpbid 222 . . . . 5 |- ( ( A e. NN /\ B e. NN ) -> ( A / ( A gcd B ) ) e. ZZ )
2019biantrurd 529 . . . 4 |- ( ( A e. NN /\ B e. NN ) -> ( -. 2 || ( A / ( A gcd B ) ) <-> ( ( A / ( A gcd B ) ) e. ZZ /\ -. 2 || ( A / ( A gcd B ) ) ) ) )
215simprd 479 . . . . . 6 |- ( ( A e. NN /\ B e. NN ) -> ( A gcd B ) || B )
223adantl 482 . . . . . . 7 |- ( ( A e. NN /\ B e. NN ) -> B e. ZZ )
23 dvdsval2 14986 . . . . . . 7 |- ( ( ( A gcd B ) e. ZZ /\ ( A gcd B ) =/= 0 /\ B e. ZZ ) -> ( ( A gcd B ) || B <-> ( B / ( A gcd B ) ) e. ZZ ) )
2414, 15, 22, 23syl3anc 1326 . . . . . 6 |- ( ( A e. NN /\ B e. NN ) -> ( ( A gcd B ) || B <-> ( B / ( A gcd B ) ) e. ZZ ) )
2521, 24mpbid 222 . . . . 5 |- ( ( A e. NN /\ B e. NN ) -> ( B / ( A gcd B ) ) e. ZZ )
2625biantrurd 529 . . . 4 |- ( ( A e. NN /\ B e. NN ) -> ( -. 2 || ( B / ( A gcd B ) ) <-> ( ( B / ( A gcd B ) ) e. ZZ /\ -. 2 || ( B / ( A gcd B ) ) ) ) )
2720, 26orbi12d 746 . . 3 |- ( ( A e. NN /\ B e. NN ) -> ( ( -. 2 || ( A / ( A gcd B ) ) \/ -. 2 || ( B / ( A gcd B ) ) ) <-> ( ( ( A / ( A gcd B ) ) e. ZZ /\ -. 2 || ( A / ( A gcd B ) ) ) \/ ( ( B / ( A gcd B ) ) e. ZZ /\ -. 2 || ( B / ( A gcd B ) ) ) ) ) )
281, 27mpbid 222 . 2 |- ( ( A e. NN /\ B e. NN ) -> ( ( ( A / ( A gcd B ) ) e. ZZ /\ -. 2 || ( A / ( A gcd B ) ) ) \/ ( ( B / ( A gcd B ) ) e. ZZ /\ -. 2 || ( B / ( A gcd B ) ) ) ) )
29 isodd3 41565 . . 3 |- ( ( A / ( A gcd B ) ) e. Odd <-> ( ( A / ( A gcd B ) ) e. ZZ /\ -. 2 || ( A / ( A gcd B ) ) ) )
30 isodd3 41565 . . 3 |- ( ( B / ( A gcd B ) ) e. Odd <-> ( ( B / ( A gcd B ) ) e. ZZ /\ -. 2 || ( B / ( A gcd B ) ) ) )
3129, 30orbi12i 543 . 2 |- ( ( ( A / ( A gcd B ) ) e. Odd \/ ( B / ( A gcd B ) ) e. Odd ) <-> ( ( ( A / ( A gcd B ) ) e. ZZ /\ -. 2 || ( A / ( A gcd B ) ) ) \/ ( ( B / ( A gcd B ) ) e. ZZ /\ -. 2 || ( B / ( A gcd B ) ) ) ) )
3228, 31sylibr 224 1 |- ( ( A e. NN /\ B e. NN ) -> ( ( A / ( A gcd B ) ) e. Odd \/ ( B / ( A gcd B ) ) e. Odd ) )
Colors of variables: wff setvar class
Syntax hints:   -. wn 3   -> wi 4   <-> wb 196   \/ wo 383   /\ wa 384   = wceq 1483   e. wcel 1990   =/= wne 2794   class class class wbr 4653  (class class class)co 6650   0cc0 9936   / cdiv 10684   NNcn 11020   2c2 11070   ZZcz 11377   || cdvds 14983   gcd cgcd 15216   Odd codd 41538
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-8 1992  ax-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246  ax-ext 2602  ax-sep 4781  ax-nul 4789  ax-pow 4843  ax-pr 4906  ax-un 6949  ax-cnex 9992  ax-resscn 9993  ax-1cn 9994  ax-icn 9995  ax-addcl 9996  ax-addrcl 9997  ax-mulcl 9998  ax-mulrcl 9999  ax-mulcom 10000  ax-addass 10001  ax-mulass 10002  ax-distr 10003  ax-i2m1 10004  ax-1ne0 10005  ax-1rid 10006  ax-rnegex 10007  ax-rrecex 10008  ax-cnre 10009  ax-pre-lttri 10010  ax-pre-lttrn 10011  ax-pre-ltadd 10012  ax-pre-mulgt0 10013  ax-pre-sup 10014
This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3or 1038  df-3an 1039  df-tru 1486  df-ex 1705  df-nf 1710  df-sb 1881  df-eu 2474  df-mo 2475  df-clab 2609  df-cleq 2615  df-clel 2618  df-nfc 2753  df-ne 2795  df-nel 2898  df-ral 2917  df-rex 2918  df-reu 2919  df-rmo 2920  df-rab 2921  df-v 3202  df-sbc 3436  df-csb 3534  df-dif 3577  df-un 3579  df-in 3581  df-ss 3588  df-pss 3590  df-nul 3916  df-if 4087  df-pw 4160  df-sn 4178  df-pr 4180  df-tp 4182  df-op 4184  df-uni 4437  df-iun 4522  df-br 4654  df-opab 4713  df-mpt 4730  df-tr 4753  df-id 5024  df-eprel 5029  df-po 5035  df-so 5036  df-fr 5073  df-we 5075  df-xp 5120  df-rel 5121  df-cnv 5122  df-co 5123  df-dm 5124  df-rn 5125  df-res 5126  df-ima 5127  df-pred 5680  df-ord 5726  df-on 5727  df-lim 5728  df-suc 5729  df-iota 5851  df-fun 5890  df-fn 5891  df-f 5892  df-f1 5893  df-fo 5894  df-f1o 5895  df-fv 5896  df-riota 6611  df-ov 6653  df-oprab 6654  df-mpt2 6655  df-om 7066  df-2nd 7169  df-wrecs 7407  df-recs 7468  df-rdg 7506  df-er 7742  df-en 7956  df-dom 7957  df-sdom 7958  df-sup 8348  df-inf 8349  df-pnf 10076  df-mnf 10077  df-xr 10078  df-ltxr 10079  df-le 10080  df-sub 10268  df-neg 10269  df-div 10685  df-nn 11021  df-2 11079  df-3 11080  df-n0 11293  df-z 11378  df-uz 11688  df-rp 11833  df-fl 12593  df-mod 12669  df-seq 12802  df-exp 12861  df-cj 13839  df-re 13840  df-im 13841  df-sqrt 13975  df-abs 13976  df-dvds 14984  df-gcd 15217  df-odd 41540
This theorem is referenced by: (None)
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