Theorem 2tp1odd 12390

Description: A number which is twice an integer increased by 1 is odd. (Contributed by AV, 16-Jul-2021.)

Assertion

Ref	Expression
2tp1odd	|- ( ( A e. ZZ /\ B = ( ( 2 x. A ) + 1 ) ) -> -. 2 || B )

Proof of Theorem 2tp1odd

Dummy variable k is distinct from all other variables.

Step	Hyp	Ref	Expression
1		id 19	. . . . 5 |- ( A e. ZZ -> A e. ZZ )
2		oveq2 6008	. . . . . . . 8 |- ( k = A -> ( 2 x. k ) = ( 2 x. A ) )
3	2	oveq1d 6015	. . . . . . 7 |- ( k = A -> ( ( 2 x. k ) + 1 ) = ( ( 2 x. A ) + 1 ) )
4	3	eqeq1d 2238	. . . . . 6 |- ( k = A -> ( ( ( 2 x. k ) + 1 ) = ( ( 2 x. A ) + 1 ) <-> ( ( 2 x. A ) + 1 ) = ( ( 2 x. A ) + 1 ) ) )
5	4	adantl 277	. . . . 5 |- ( ( A e. ZZ /\ k = A ) -> ( ( ( 2 x. k ) + 1 ) = ( ( 2 x. A ) + 1 ) <-> ( ( 2 x. A ) + 1 ) = ( ( 2 x. A ) + 1 ) ) )
6		eqidd 2230	. . . . 5 |- ( A e. ZZ -> ( ( 2 x. A ) + 1 ) = ( ( 2 x. A ) + 1 ) )
7	1, 5, 6	rspcedvd 2913	. . . 4 |- ( A e. ZZ -> E. k e. ZZ ( ( 2 x. k ) + 1 ) = ( ( 2 x. A ) + 1 ) )
8		2z 9470	. . . . . . . 8 |- 2 e. ZZ
9	8	a1i 9	. . . . . . 7 |- ( A e. ZZ -> 2 e. ZZ )
10	9, 1	zmulcld 9571	. . . . . 6 |- ( A e. ZZ -> ( 2 x. A ) e. ZZ )
11	10	peano2zd 9568	. . . . 5 |- ( A e. ZZ -> ( ( 2 x. A ) + 1 ) e. ZZ )
12		odd2np1 12379	. . . . 5 |- ( ( ( 2 x. A ) + 1 ) e. ZZ -> ( -. 2 || ( ( 2 x. A ) + 1 ) <-> E. k e. ZZ ( ( 2 x. k ) + 1 ) = ( ( 2 x. A ) + 1 ) ) )
13	11, 12	syl 14	. . . 4 |- ( A e. ZZ -> ( -. 2 || ( ( 2 x. A ) + 1 ) <-> E. k e. ZZ ( ( 2 x. k ) + 1 ) = ( ( 2 x. A ) + 1 ) ) )
14	7, 13	mpbird 167	. . 3 |- ( A e. ZZ -> -. 2 || ( ( 2 x. A ) + 1 ) )
15	14	adantr 276	. 2 |- ( ( A e. ZZ /\ B = ( ( 2 x. A ) + 1 ) ) -> -. 2 || ( ( 2 x. A ) + 1 ) )
16		breq2 4086	. . 3 |- ( B = ( ( 2 x. A ) + 1 ) -> ( 2 || B <-> 2 || ( ( 2 x. A ) + 1 ) ) )
17	16	adantl 277	. 2 |- ( ( A e. ZZ /\ B = ( ( 2 x. A ) + 1 ) ) -> ( 2 || B <-> 2 || ( ( 2 x. A ) + 1 ) ) )
18	15, 17	mtbird 677	1 |- ( ( A e. ZZ /\ B = ( ( 2 x. A ) + 1 ) ) -> -. 2 || B )

Syntax hints:   -. wn 3   -> wi 4   /\ wa 104   <-> wb 105   = wceq 1395   e. wcel 2200   E.wrex 2509   class class class wbr 4082  (class class class)co 6000   1c1 7996   + caddc 7998   x. cmul 8000   2c2 9157   ZZcz 9442   || cdvds 12293

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 617  ax-in2 618  ax-io 714  ax-5 1493  ax-7 1494  ax-gen 1495  ax-ie1 1539  ax-ie2 1540  ax-8 1550  ax-10 1551  ax-11 1552  ax-i12 1553  ax-bndl 1555  ax-4 1556  ax-17 1572  ax-i9 1576  ax-ial 1580  ax-i5r 1581  ax-13 2202  ax-14 2203  ax-ext 2211  ax-sep 4201  ax-pow 4257  ax-pr 4292  ax-un 4523  ax-setind 4628  ax-cnex 8086  ax-resscn 8087  ax-1cn 8088  ax-1re 8089  ax-icn 8090  ax-addcl 8091  ax-addrcl 8092  ax-mulcl 8093  ax-mulrcl 8094  ax-addcom 8095  ax-mulcom 8096  ax-addass 8097  ax-mulass 8098  ax-distr 8099  ax-i2m1 8100  ax-0lt1 8101  ax-1rid 8102  ax-0id 8103  ax-rnegex 8104  ax-precex 8105  ax-cnre 8106  ax-pre-ltirr 8107  ax-pre-ltwlin 8108  ax-pre-lttrn 8109  ax-pre-apti 8110  ax-pre-ltadd 8111  ax-pre-mulgt0 8112  ax-pre-mulext 8113

This theorem depends on definitions:  df-bi 117  df-3or 1003  df-3an 1004  df-tru 1398  df-fal 1401  df-xor 1418  df-nf 1507  df-sb 1809  df-eu 2080  df-mo 2081  df-clab 2216  df-cleq 2222  df-clel 2225  df-nfc 2361  df-ne 2401  df-nel 2496  df-ral 2513  df-rex 2514  df-reu 2515  df-rmo 2516  df-rab 2517  df-v 2801  df-sbc 3029  df-dif 3199  df-un 3201  df-in 3203  df-ss 3210  df-pw 3651  df-sn 3672  df-pr 3673  df-op 3675  df-uni 3888  df-int 3923  df-br 4083  df-opab 4145  df-id 4383  df-po 4386  df-iso 4387  df-xp 4724  df-rel 4725  df-cnv 4726  df-co 4727  df-dm 4728  df-iota 5277  df-fun 5319  df-fv 5325  df-riota 5953  df-ov 6003  df-oprab 6004  df-mpo 6005  df-pnf 8179  df-mnf 8180  df-xr 8181  df-ltxr 8182  df-le 8183  df-sub 8315  df-neg 8316  df-reap 8718  df-ap 8725  df-div 8816  df-inn 9107  df-2 9165  df-n0 9366  df-z 9443  df-dvds 12294

This theorem is referenced by:  2lgslem3b1  15771  2lgslem3c1  15772