Theorem 2tp1odd 12651

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 6093	. . . . . . . 8 |- ( k = A -> ( 2 x. k ) = ( 2 x. A ) )
3	2	oveq1d 6100	. . . . . . 7 |- ( k = A -> ( ( 2 x. k ) + 1 ) = ( ( 2 x. A ) + 1 ) )
4	3	eqeq1d 2247	. . . . . 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 2239	. . . . 5 |- ( A e. ZZ -> ( ( 2 x. A ) + 1 ) = ( ( 2 x. A ) + 1 ) )
7	1, 5, 6	rspcedvd 2935	. . . 4 |- ( A e. ZZ -> E. k e. ZZ ( ( 2 x. k ) + 1 ) = ( ( 2 x. A ) + 1 ) )
8		2z 9672	. . . . . . . 8 |- 2 e. ZZ
9	8	a1i 9	. . . . . . 7 |- ( A e. ZZ -> 2 e. ZZ )
10	9, 1	zmulcld 9774	. . . . . 6 |- ( A e. ZZ -> ( 2 x. A ) e. ZZ )
11	10	peano2zd 9771	. . . . 5 |- ( A e. ZZ -> ( ( 2 x. A ) + 1 ) e. ZZ )
12		odd2np1 12640	. . . . 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 4134	. . 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 684	1 |- ( ( A e. ZZ /\ B = ( ( 2 x. A ) + 1 ) ) -> -. 2 || B )

This proof depends on syntax axioms:   -. wn 3   -> wi 4   /\ wa 104   <-> wb 105   = wceq 1402   e. wcel 2209   E.wrex 2529   class class class wbr 4130  (class class class)co 6085   1c1 8180   + caddc 8182   x. cmul 8184   2c2 9355   ZZcz 9644   || cdvds 12554

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-sep 4249  ax-pow 4311  ax-pr 4346  ax-un 4578  ax-setind 4684  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

This proof depends on definitions:  df-bi 117  df-3or 1010  df-3an 1011  df-tru 1405  df-fal 1408  df-xor 1425  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-dif 3222  df-un 3224  df-in 3226  df-ss 3233  df-pw 3690  df-sn 3715  df-pr 3716  df-op 3718  df-uni 3936  df-int 3971  df-br 4131  df-opab 4193  df-id 4438  df-po 4441  df-iso 4442  df-xp 4780  df-rel 4781  df-cnv 4782  df-co 4783  df-dm 4784  df-iota 5337  df-fun 5379  df-fv 5385  df-riota 6038  df-ov 6088  df-oprab 6089  df-mpo 6090  df-pnf 8362  df-mnf 8363  df-xr 8364  df-ltxr 8365  df-le 8366  df-sub 8499  df-neg 8500  df-reap 8903  df-ap 8910  df-div 9003  df-inn 9305  df-2 9363  df-n0 9564  df-z 9645  df-dvds 12555

This theorem is used by:  2lgslem3b1  16217  2lgslem3c1  16218