Theorem 2tp1odd 12444

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 6025	. . . . . . . 8 |- ( k = A -> ( 2 x. k ) = ( 2 x. A ) )
3	2	oveq1d 6032	. . . . . . 7 |- ( k = A -> ( ( 2 x. k ) + 1 ) = ( ( 2 x. A ) + 1 ) )
4	3	eqeq1d 2240	. . . . . 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 2232	. . . . 5 |- ( A e. ZZ -> ( ( 2 x. A ) + 1 ) = ( ( 2 x. A ) + 1 ) )
7	1, 5, 6	rspcedvd 2916	. . . 4 |- ( A e. ZZ -> E. k e. ZZ ( ( 2 x. k ) + 1 ) = ( ( 2 x. A ) + 1 ) )
8		2z 9506	. . . . . . . 8 |- 2 e. ZZ
9	8	a1i 9	. . . . . . 7 |- ( A e. ZZ -> 2 e. ZZ )
10	9, 1	zmulcld 9607	. . . . . 6 |- ( A e. ZZ -> ( 2 x. A ) e. ZZ )
11	10	peano2zd 9604	. . . . 5 |- ( A e. ZZ -> ( ( 2 x. A ) + 1 ) e. ZZ )
12		odd2np1 12433	. . . . 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 4092	. . 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 679	1 |- ( ( A e. ZZ /\ B = ( ( 2 x. A ) + 1 ) ) -> -. 2 || B )

Syntax hints:   -. wn 3   -> wi 4   /\ wa 104   <-> wb 105   = wceq 1397   e. wcel 2202   E.wrex 2511   class class class wbr 4088  (class class class)co 6017   1c1 8032   + caddc 8034   x. cmul 8036   2c2 9193   ZZcz 9478   || cdvds 12347

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 619  ax-in2 620  ax-io 716  ax-5 1495  ax-7 1496  ax-gen 1497  ax-ie1 1541  ax-ie2 1542  ax-8 1552  ax-10 1553  ax-11 1554  ax-i12 1555  ax-bndl 1557  ax-4 1558  ax-17 1574  ax-i9 1578  ax-ial 1582  ax-i5r 1583  ax-13 2204  ax-14 2205  ax-ext 2213  ax-sep 4207  ax-pow 4264  ax-pr 4299  ax-un 4530  ax-setind 4635  ax-cnex 8122  ax-resscn 8123  ax-1cn 8124  ax-1re 8125  ax-icn 8126  ax-addcl 8127  ax-addrcl 8128  ax-mulcl 8129  ax-mulrcl 8130  ax-addcom 8131  ax-mulcom 8132  ax-addass 8133  ax-mulass 8134  ax-distr 8135  ax-i2m1 8136  ax-0lt1 8137  ax-1rid 8138  ax-0id 8139  ax-rnegex 8140  ax-precex 8141  ax-cnre 8142  ax-pre-ltirr 8143  ax-pre-ltwlin 8144  ax-pre-lttrn 8145  ax-pre-apti 8146  ax-pre-ltadd 8147  ax-pre-mulgt0 8148  ax-pre-mulext 8149

This theorem depends on definitions:  df-bi 117  df-3or 1005  df-3an 1006  df-tru 1400  df-fal 1403  df-xor 1420  df-nf 1509  df-sb 1811  df-eu 2082  df-mo 2083  df-clab 2218  df-cleq 2224  df-clel 2227  df-nfc 2363  df-ne 2403  df-nel 2498  df-ral 2515  df-rex 2516  df-reu 2517  df-rmo 2518  df-rab 2519  df-v 2804  df-sbc 3032  df-dif 3202  df-un 3204  df-in 3206  df-ss 3213  df-pw 3654  df-sn 3675  df-pr 3676  df-op 3678  df-uni 3894  df-int 3929  df-br 4089  df-opab 4151  df-id 4390  df-po 4393  df-iso 4394  df-xp 4731  df-rel 4732  df-cnv 4733  df-co 4734  df-dm 4735  df-iota 5286  df-fun 5328  df-fv 5334  df-riota 5970  df-ov 6020  df-oprab 6021  df-mpo 6022  df-pnf 8215  df-mnf 8216  df-xr 8217  df-ltxr 8218  df-le 8219  df-sub 8351  df-neg 8352  df-reap 8754  df-ap 8761  df-div 8852  df-inn 9143  df-2 9201  df-n0 9402  df-z 9479  df-dvds 12348

This theorem is referenced by:  2lgslem3b1  15826  2lgslem3c1  15827