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Theorem 2tp1odd 11843
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.
StepHypRef Expression
1 id 19 . . . . 5  |-  ( A  e.  ZZ  ->  A  e.  ZZ )
2 oveq2 5861 . . . . . . . 8  |-  ( k  =  A  ->  (
2  x.  k )  =  ( 2  x.  A ) )
32oveq1d 5868 . . . . . . 7  |-  ( k  =  A  ->  (
( 2  x.  k
)  +  1 )  =  ( ( 2  x.  A )  +  1 ) )
43eqeq1d 2179 . . . . . 6  |-  ( k  =  A  ->  (
( ( 2  x.  k )  +  1 )  =  ( ( 2  x.  A )  +  1 )  <->  ( (
2  x.  A )  +  1 )  =  ( ( 2  x.  A )  +  1 ) ) )
54adantl 275 . . . . 5  |-  ( ( A  e.  ZZ  /\  k  =  A )  ->  ( ( ( 2  x.  k )  +  1 )  =  ( ( 2  x.  A
)  +  1 )  <-> 
( ( 2  x.  A )  +  1 )  =  ( ( 2  x.  A )  +  1 ) ) )
6 eqidd 2171 . . . . 5  |-  ( A  e.  ZZ  ->  (
( 2  x.  A
)  +  1 )  =  ( ( 2  x.  A )  +  1 ) )
71, 5, 6rspcedvd 2840 . . . 4  |-  ( A  e.  ZZ  ->  E. k  e.  ZZ  ( ( 2  x.  k )  +  1 )  =  ( ( 2  x.  A
)  +  1 ) )
8 2z 9240 . . . . . . . 8  |-  2  e.  ZZ
98a1i 9 . . . . . . 7  |-  ( A  e.  ZZ  ->  2  e.  ZZ )
109, 1zmulcld 9340 . . . . . 6  |-  ( A  e.  ZZ  ->  (
2  x.  A )  e.  ZZ )
1110peano2zd 9337 . . . . 5  |-  ( A  e.  ZZ  ->  (
( 2  x.  A
)  +  1 )  e.  ZZ )
12 odd2np1 11832 . . . . 5  |-  ( ( ( 2  x.  A
)  +  1 )  e.  ZZ  ->  ( -.  2  ||  ( ( 2  x.  A )  +  1 )  <->  E. k  e.  ZZ  ( ( 2  x.  k )  +  1 )  =  ( ( 2  x.  A
)  +  1 ) ) )
1311, 12syl 14 . . . 4  |-  ( A  e.  ZZ  ->  ( -.  2  ||  ( ( 2  x.  A )  +  1 )  <->  E. k  e.  ZZ  ( ( 2  x.  k )  +  1 )  =  ( ( 2  x.  A
)  +  1 ) ) )
147, 13mpbird 166 . . 3  |-  ( A  e.  ZZ  ->  -.  2  ||  ( ( 2  x.  A )  +  1 ) )
1514adantr 274 . 2  |-  ( ( A  e.  ZZ  /\  B  =  ( (
2  x.  A )  +  1 ) )  ->  -.  2  ||  ( ( 2  x.  A )  +  1 ) )
16 breq2 3993 . . 3  |-  ( B  =  ( ( 2  x.  A )  +  1 )  ->  (
2  ||  B  <->  2  ||  ( ( 2  x.  A )  +  1 ) ) )
1716adantl 275 . 2  |-  ( ( A  e.  ZZ  /\  B  =  ( (
2  x.  A )  +  1 ) )  ->  ( 2  ||  B 
<->  2  ||  ( ( 2  x.  A )  +  1 ) ) )
1815, 17mtbird 668 1  |-  ( ( A  e.  ZZ  /\  B  =  ( (
2  x.  A )  +  1 ) )  ->  -.  2  ||  B )
Colors of variables: wff set class
Syntax hints:   -. wn 3    -> wi 4    /\ wa 103    <-> wb 104    = wceq 1348    e. wcel 2141   E.wrex 2449   class class class wbr 3989  (class class class)co 5853   1c1 7775    + caddc 7777    x. cmul 7779   2c2 8929   ZZcz 9212    || cdvds 11749
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-in1 609  ax-in2 610  ax-io 704  ax-5 1440  ax-7 1441  ax-gen 1442  ax-ie1 1486  ax-ie2 1487  ax-8 1497  ax-10 1498  ax-11 1499  ax-i12 1500  ax-bndl 1502  ax-4 1503  ax-17 1519  ax-i9 1523  ax-ial 1527  ax-i5r 1528  ax-13 2143  ax-14 2144  ax-ext 2152  ax-sep 4107  ax-pow 4160  ax-pr 4194  ax-un 4418  ax-setind 4521  ax-cnex 7865  ax-resscn 7866  ax-1cn 7867  ax-1re 7868  ax-icn 7869  ax-addcl 7870  ax-addrcl 7871  ax-mulcl 7872  ax-mulrcl 7873  ax-addcom 7874  ax-mulcom 7875  ax-addass 7876  ax-mulass 7877  ax-distr 7878  ax-i2m1 7879  ax-0lt1 7880  ax-1rid 7881  ax-0id 7882  ax-rnegex 7883  ax-precex 7884  ax-cnre 7885  ax-pre-ltirr 7886  ax-pre-ltwlin 7887  ax-pre-lttrn 7888  ax-pre-apti 7889  ax-pre-ltadd 7890  ax-pre-mulgt0 7891  ax-pre-mulext 7892
This theorem depends on definitions:  df-bi 116  df-3or 974  df-3an 975  df-tru 1351  df-fal 1354  df-xor 1371  df-nf 1454  df-sb 1756  df-eu 2022  df-mo 2023  df-clab 2157  df-cleq 2163  df-clel 2166  df-nfc 2301  df-ne 2341  df-nel 2436  df-ral 2453  df-rex 2454  df-reu 2455  df-rmo 2456  df-rab 2457  df-v 2732  df-sbc 2956  df-dif 3123  df-un 3125  df-in 3127  df-ss 3134  df-pw 3568  df-sn 3589  df-pr 3590  df-op 3592  df-uni 3797  df-int 3832  df-br 3990  df-opab 4051  df-id 4278  df-po 4281  df-iso 4282  df-xp 4617  df-rel 4618  df-cnv 4619  df-co 4620  df-dm 4621  df-iota 5160  df-fun 5200  df-fv 5206  df-riota 5809  df-ov 5856  df-oprab 5857  df-mpo 5858  df-pnf 7956  df-mnf 7957  df-xr 7958  df-ltxr 7959  df-le 7960  df-sub 8092  df-neg 8093  df-reap 8494  df-ap 8501  df-div 8590  df-inn 8879  df-2 8937  df-n0 9136  df-z 9213  df-dvds 11750
This theorem is referenced by: (None)
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