Theorem 2tp1odd 12195

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 5952	. . . . . . . 8 |- ( k = A -> ( 2 x. k ) = ( 2 x. A ) )
3	2	oveq1d 5959	. . . . . . 7 |- ( k = A -> ( ( 2 x. k ) + 1 ) = ( ( 2 x. A ) + 1 ) )
4	3	eqeq1d 2214	. . . . . 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 2206	. . . . 5 |- ( A e. ZZ -> ( ( 2 x. A ) + 1 ) = ( ( 2 x. A ) + 1 ) )
7	1, 5, 6	rspcedvd 2883	. . . 4 |- ( A e. ZZ -> E. k e. ZZ ( ( 2 x. k ) + 1 ) = ( ( 2 x. A ) + 1 ) )
8		2z 9400	. . . . . . . 8 |- 2 e. ZZ
9	8	a1i 9	. . . . . . 7 |- ( A e. ZZ -> 2 e. ZZ )
10	9, 1	zmulcld 9501	. . . . . 6 |- ( A e. ZZ -> ( 2 x. A ) e. ZZ )
11	10	peano2zd 9498	. . . . 5 |- ( A e. ZZ -> ( ( 2 x. A ) + 1 ) e. ZZ )
12		odd2np1 12184	. . . . 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 4048	. . 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 675	1 |- ( ( A e. ZZ /\ B = ( ( 2 x. A ) + 1 ) ) -> -. 2 || B )

Syntax hints:   -. wn 3   -> wi 4   /\ wa 104   <-> wb 105   = wceq 1373   e. wcel 2176   E.wrex 2485   class class class wbr 4044  (class class class)co 5944   1c1 7926   + caddc 7928   x. cmul 7930   2c2 9087   ZZcz 9372   || cdvds 12098

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 615  ax-in2 616  ax-io 711  ax-5 1470  ax-7 1471  ax-gen 1472  ax-ie1 1516  ax-ie2 1517  ax-8 1527  ax-10 1528  ax-11 1529  ax-i12 1530  ax-bndl 1532  ax-4 1533  ax-17 1549  ax-i9 1553  ax-ial 1557  ax-i5r 1558  ax-13 2178  ax-14 2179  ax-ext 2187  ax-sep 4162  ax-pow 4218  ax-pr 4253  ax-un 4480  ax-setind 4585  ax-cnex 8016  ax-resscn 8017  ax-1cn 8018  ax-1re 8019  ax-icn 8020  ax-addcl 8021  ax-addrcl 8022  ax-mulcl 8023  ax-mulrcl 8024  ax-addcom 8025  ax-mulcom 8026  ax-addass 8027  ax-mulass 8028  ax-distr 8029  ax-i2m1 8030  ax-0lt1 8031  ax-1rid 8032  ax-0id 8033  ax-rnegex 8034  ax-precex 8035  ax-cnre 8036  ax-pre-ltirr 8037  ax-pre-ltwlin 8038  ax-pre-lttrn 8039  ax-pre-apti 8040  ax-pre-ltadd 8041  ax-pre-mulgt0 8042  ax-pre-mulext 8043

This theorem depends on definitions:  df-bi 117  df-3or 982  df-3an 983  df-tru 1376  df-fal 1379  df-xor 1396  df-nf 1484  df-sb 1786  df-eu 2057  df-mo 2058  df-clab 2192  df-cleq 2198  df-clel 2201  df-nfc 2337  df-ne 2377  df-nel 2472  df-ral 2489  df-rex 2490  df-reu 2491  df-rmo 2492  df-rab 2493  df-v 2774  df-sbc 2999  df-dif 3168  df-un 3170  df-in 3172  df-ss 3179  df-pw 3618  df-sn 3639  df-pr 3640  df-op 3642  df-uni 3851  df-int 3886  df-br 4045  df-opab 4106  df-id 4340  df-po 4343  df-iso 4344  df-xp 4681  df-rel 4682  df-cnv 4683  df-co 4684  df-dm 4685  df-iota 5232  df-fun 5273  df-fv 5279  df-riota 5899  df-ov 5947  df-oprab 5948  df-mpo 5949  df-pnf 8109  df-mnf 8110  df-xr 8111  df-ltxr 8112  df-le 8113  df-sub 8245  df-neg 8246  df-reap 8648  df-ap 8655  df-div 8746  df-inn 9037  df-2 9095  df-n0 9296  df-z 9373  df-dvds 12099

This theorem is referenced by:  2lgslem3b1  15575  2lgslem3c1  15576