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Theorem 2tp1odd 10475
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 5572 . . . . . . . 8  |-  ( k  =  A  ->  (
2  x.  k )  =  ( 2  x.  A ) )
32oveq1d 5579 . . . . . . 7  |-  ( k  =  A  ->  (
( 2  x.  k
)  +  1 )  =  ( ( 2  x.  A )  +  1 ) )
43eqeq1d 2091 . . . . . 6  |-  ( k  =  A  ->  (
( ( 2  x.  k )  +  1 )  =  ( ( 2  x.  A )  +  1 )  <->  ( (
2  x.  A )  +  1 )  =  ( ( 2  x.  A )  +  1 ) ) )
54adantl 271 . . . . 5  |-  ( ( A  e.  ZZ  /\  k  =  A )  ->  ( ( ( 2  x.  k )  +  1 )  =  ( ( 2  x.  A
)  +  1 )  <-> 
( ( 2  x.  A )  +  1 )  =  ( ( 2  x.  A )  +  1 ) ) )
6 eqidd 2084 . . . . 5  |-  ( A  e.  ZZ  ->  (
( 2  x.  A
)  +  1 )  =  ( ( 2  x.  A )  +  1 ) )
71, 5, 6rspcedvd 2717 . . . 4  |-  ( A  e.  ZZ  ->  E. k  e.  ZZ  ( ( 2  x.  k )  +  1 )  =  ( ( 2  x.  A
)  +  1 ) )
8 2z 8496 . . . . . . . 8  |-  2  e.  ZZ
98a1i 9 . . . . . . 7  |-  ( A  e.  ZZ  ->  2  e.  ZZ )
109, 1zmulcld 8592 . . . . . 6  |-  ( A  e.  ZZ  ->  (
2  x.  A )  e.  ZZ )
1110peano2zd 8589 . . . . 5  |-  ( A  e.  ZZ  ->  (
( 2  x.  A
)  +  1 )  e.  ZZ )
12 odd2np1 10464 . . . . 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 165 . . 3  |-  ( A  e.  ZZ  ->  -.  2  ||  ( ( 2  x.  A )  +  1 ) )
1514adantr 270 . 2  |-  ( ( A  e.  ZZ  /\  B  =  ( (
2  x.  A )  +  1 ) )  ->  -.  2  ||  ( ( 2  x.  A )  +  1 ) )
16 breq2 3810 . . 3  |-  ( B  =  ( ( 2  x.  A )  +  1 )  ->  (
2  ||  B  <->  2  ||  ( ( 2  x.  A )  +  1 ) ) )
1716adantl 271 . 2  |-  ( ( A  e.  ZZ  /\  B  =  ( (
2  x.  A )  +  1 ) )  ->  ( 2  ||  B 
<->  2  ||  ( ( 2  x.  A )  +  1 ) ) )
1815, 17mtbird 631 1  |-  ( ( A  e.  ZZ  /\  B  =  ( (
2  x.  A )  +  1 ) )  ->  -.  2  ||  B )
Colors of variables: wff set class
Syntax hints:   -. wn 3    -> wi 4    /\ wa 102    <-> wb 103    = wceq 1285    e. wcel 1434   E.wrex 2354   class class class wbr 3806  (class class class)co 5564   1c1 7080    + caddc 7082    x. cmul 7084   2c2 8192   ZZcz 8468    || cdvds 10387
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-in1 577  ax-in2 578  ax-io 663  ax-5 1377  ax-7 1378  ax-gen 1379  ax-ie1 1423  ax-ie2 1424  ax-8 1436  ax-10 1437  ax-11 1438  ax-i12 1439  ax-bndl 1440  ax-4 1441  ax-13 1445  ax-14 1446  ax-17 1460  ax-i9 1464  ax-ial 1468  ax-i5r 1469  ax-ext 2065  ax-sep 3917  ax-pow 3969  ax-pr 3993  ax-un 4217  ax-setind 4309  ax-cnex 7165  ax-resscn 7166  ax-1cn 7167  ax-1re 7168  ax-icn 7169  ax-addcl 7170  ax-addrcl 7171  ax-mulcl 7172  ax-mulrcl 7173  ax-addcom 7174  ax-mulcom 7175  ax-addass 7176  ax-mulass 7177  ax-distr 7178  ax-i2m1 7179  ax-0lt1 7180  ax-1rid 7181  ax-0id 7182  ax-rnegex 7183  ax-precex 7184  ax-cnre 7185  ax-pre-ltirr 7186  ax-pre-ltwlin 7187  ax-pre-lttrn 7188  ax-pre-apti 7189  ax-pre-ltadd 7190  ax-pre-mulgt0 7191  ax-pre-mulext 7192
This theorem depends on definitions:  df-bi 115  df-3or 921  df-3an 922  df-tru 1288  df-fal 1291  df-xor 1308  df-nf 1391  df-sb 1688  df-eu 1946  df-mo 1947  df-clab 2070  df-cleq 2076  df-clel 2079  df-nfc 2212  df-ne 2250  df-nel 2345  df-ral 2358  df-rex 2359  df-reu 2360  df-rmo 2361  df-rab 2362  df-v 2612  df-sbc 2826  df-dif 2985  df-un 2987  df-in 2989  df-ss 2996  df-pw 3403  df-sn 3423  df-pr 3424  df-op 3426  df-uni 3623  df-int 3658  df-br 3807  df-opab 3861  df-id 4077  df-po 4080  df-iso 4081  df-xp 4398  df-rel 4399  df-cnv 4400  df-co 4401  df-dm 4402  df-iota 4918  df-fun 4955  df-fv 4961  df-riota 5520  df-ov 5567  df-oprab 5568  df-mpt2 5569  df-pnf 7253  df-mnf 7254  df-xr 7255  df-ltxr 7256  df-le 7257  df-sub 7384  df-neg 7385  df-reap 7778  df-ap 7785  df-div 7864  df-inn 8143  df-2 8201  df-n0 8392  df-z 8469  df-dvds 10388
This theorem is referenced by: (None)
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