MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  omeo Structured version   Unicode version

Theorem omeo 13188
Description: The difference of an odd and an even is odd. (Contributed by Scott Fenton, 7-Apr-2014.) (Revised by Mario Carneiro, 19-Apr-2014.)
Assertion
Ref Expression
omeo  |-  ( ( ( A  e.  ZZ  /\ 
-.  2  ||  A
)  /\  ( B  e.  ZZ  /\  2  ||  B ) )  ->  -.  2  ||  ( A  -  B ) )

Proof of Theorem omeo
Dummy variables  a 
b  c are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 odd2np1 12908 . . . . . 6  |-  ( A  e.  ZZ  ->  ( -.  2  ||  A  <->  E. a  e.  ZZ  ( ( 2  x.  a )  +  1 )  =  A ) )
2 2z 10312 . . . . . . 7  |-  2  e.  ZZ
3 divides 12854 . . . . . . 7  |-  ( ( 2  e.  ZZ  /\  B  e.  ZZ )  ->  ( 2  ||  B  <->  E. b  e.  ZZ  (
b  x.  2 )  =  B ) )
42, 3mpan 652 . . . . . 6  |-  ( B  e.  ZZ  ->  (
2  ||  B  <->  E. b  e.  ZZ  ( b  x.  2 )  =  B ) )
51, 4bi2anan9 844 . . . . 5  |-  ( ( A  e.  ZZ  /\  B  e.  ZZ )  ->  ( ( -.  2  ||  A  /\  2  ||  B )  <->  ( E. a  e.  ZZ  (
( 2  x.  a
)  +  1 )  =  A  /\  E. b  e.  ZZ  (
b  x.  2 )  =  B ) ) )
6 reeanv 2875 . . . . . 6  |-  ( E. a  e.  ZZ  E. b  e.  ZZ  (
( ( 2  x.  a )  +  1 )  =  A  /\  ( b  x.  2 )  =  B )  <-> 
( E. a  e.  ZZ  ( ( 2  x.  a )  +  1 )  =  A  /\  E. b  e.  ZZ  ( b  x.  2 )  =  B ) )
7 zsubcl 10319 . . . . . . . . 9  |-  ( ( a  e.  ZZ  /\  b  e.  ZZ )  ->  ( a  -  b
)  e.  ZZ )
8 zcn 10287 . . . . . . . . . 10  |-  ( a  e.  ZZ  ->  a  e.  CC )
9 zcn 10287 . . . . . . . . . 10  |-  ( b  e.  ZZ  ->  b  e.  CC )
10 2cn 10070 . . . . . . . . . . . . 13  |-  2  e.  CC
11 subdi 9467 . . . . . . . . . . . . 13  |-  ( ( 2  e.  CC  /\  a  e.  CC  /\  b  e.  CC )  ->  (
2  x.  ( a  -  b ) )  =  ( ( 2  x.  a )  -  ( 2  x.  b
) ) )
1210, 11mp3an1 1266 . . . . . . . . . . . 12  |-  ( ( a  e.  CC  /\  b  e.  CC )  ->  ( 2  x.  (
a  -  b ) )  =  ( ( 2  x.  a )  -  ( 2  x.  b ) ) )
1312oveq1d 6096 . . . . . . . . . . 11  |-  ( ( a  e.  CC  /\  b  e.  CC )  ->  ( ( 2  x.  ( a  -  b
) )  +  1 )  =  ( ( ( 2  x.  a
)  -  ( 2  x.  b ) )  +  1 ) )
14 mulcl 9074 . . . . . . . . . . . . 13  |-  ( ( 2  e.  CC  /\  a  e.  CC )  ->  ( 2  x.  a
)  e.  CC )
1510, 14mpan 652 . . . . . . . . . . . 12  |-  ( a  e.  CC  ->  (
2  x.  a )  e.  CC )
16 mulcl 9074 . . . . . . . . . . . . 13  |-  ( ( 2  e.  CC  /\  b  e.  CC )  ->  ( 2  x.  b
)  e.  CC )
1710, 16mpan 652 . . . . . . . . . . . 12  |-  ( b  e.  CC  ->  (
2  x.  b )  e.  CC )
18 ax-1cn 9048 . . . . . . . . . . . . 13  |-  1  e.  CC
19 addsub 9316 . . . . . . . . . . . . 13  |-  ( ( ( 2  x.  a
)  e.  CC  /\  1  e.  CC  /\  (
2  x.  b )  e.  CC )  -> 
( ( ( 2  x.  a )  +  1 )  -  (
2  x.  b ) )  =  ( ( ( 2  x.  a
)  -  ( 2  x.  b ) )  +  1 ) )
2018, 19mp3an2 1267 . . . . . . . . . . . 12  |-  ( ( ( 2  x.  a
)  e.  CC  /\  ( 2  x.  b
)  e.  CC )  ->  ( ( ( 2  x.  a )  +  1 )  -  ( 2  x.  b
) )  =  ( ( ( 2  x.  a )  -  (
2  x.  b ) )  +  1 ) )
2115, 17, 20syl2an 464 . . . . . . . . . . 11  |-  ( ( a  e.  CC  /\  b  e.  CC )  ->  ( ( ( 2  x.  a )  +  1 )  -  (
2  x.  b ) )  =  ( ( ( 2  x.  a
)  -  ( 2  x.  b ) )  +  1 ) )
22 mulcom 9076 . . . . . . . . . . . . . 14  |-  ( ( 2  e.  CC  /\  b  e.  CC )  ->  ( 2  x.  b
)  =  ( b  x.  2 ) )
2310, 22mpan 652 . . . . . . . . . . . . 13  |-  ( b  e.  CC  ->  (
2  x.  b )  =  ( b  x.  2 ) )
2423oveq2d 6097 . . . . . . . . . . . 12  |-  ( b  e.  CC  ->  (
( ( 2  x.  a )  +  1 )  -  ( 2  x.  b ) )  =  ( ( ( 2  x.  a )  +  1 )  -  ( b  x.  2 ) ) )
2524adantl 453 . . . . . . . . . . 11  |-  ( ( a  e.  CC  /\  b  e.  CC )  ->  ( ( ( 2  x.  a )  +  1 )  -  (
2  x.  b ) )  =  ( ( ( 2  x.  a
)  +  1 )  -  ( b  x.  2 ) ) )
2613, 21, 253eqtr2d 2474 . . . . . . . . . 10  |-  ( ( a  e.  CC  /\  b  e.  CC )  ->  ( ( 2  x.  ( a  -  b
) )  +  1 )  =  ( ( ( 2  x.  a
)  +  1 )  -  ( b  x.  2 ) ) )
278, 9, 26syl2an 464 . . . . . . . . 9  |-  ( ( a  e.  ZZ  /\  b  e.  ZZ )  ->  ( ( 2  x.  ( a  -  b
) )  +  1 )  =  ( ( ( 2  x.  a
)  +  1 )  -  ( b  x.  2 ) ) )
28 oveq2 6089 . . . . . . . . . . . 12  |-  ( c  =  ( a  -  b )  ->  (
2  x.  c )  =  ( 2  x.  ( a  -  b
) ) )
2928oveq1d 6096 . . . . . . . . . . 11  |-  ( c  =  ( a  -  b )  ->  (
( 2  x.  c
)  +  1 )  =  ( ( 2  x.  ( a  -  b ) )  +  1 ) )
3029eqeq1d 2444 . . . . . . . . . 10  |-  ( c  =  ( a  -  b )  ->  (
( ( 2  x.  c )  +  1 )  =  ( ( ( 2  x.  a
)  +  1 )  -  ( b  x.  2 ) )  <->  ( (
2  x.  ( a  -  b ) )  +  1 )  =  ( ( ( 2  x.  a )  +  1 )  -  (
b  x.  2 ) ) ) )
3130rspcev 3052 . . . . . . . . 9  |-  ( ( ( a  -  b
)  e.  ZZ  /\  ( ( 2  x.  ( a  -  b
) )  +  1 )  =  ( ( ( 2  x.  a
)  +  1 )  -  ( b  x.  2 ) ) )  ->  E. c  e.  ZZ  ( ( 2  x.  c )  +  1 )  =  ( ( ( 2  x.  a
)  +  1 )  -  ( b  x.  2 ) ) )
327, 27, 31syl2anc 643 . . . . . . . 8  |-  ( ( a  e.  ZZ  /\  b  e.  ZZ )  ->  E. c  e.  ZZ  ( ( 2  x.  c )  +  1 )  =  ( ( ( 2  x.  a
)  +  1 )  -  ( b  x.  2 ) ) )
33 oveq12 6090 . . . . . . . . . 10  |-  ( ( ( ( 2  x.  a )  +  1 )  =  A  /\  ( b  x.  2 )  =  B )  ->  ( ( ( 2  x.  a )  +  1 )  -  ( b  x.  2 ) )  =  ( A  -  B ) )
3433eqeq2d 2447 . . . . . . . . 9  |-  ( ( ( ( 2  x.  a )  +  1 )  =  A  /\  ( b  x.  2 )  =  B )  ->  ( ( ( 2  x.  c )  +  1 )  =  ( ( ( 2  x.  a )  +  1 )  -  (
b  x.  2 ) )  <->  ( ( 2  x.  c )  +  1 )  =  ( A  -  B ) ) )
3534rexbidv 2726 . . . . . . . 8  |-  ( ( ( ( 2  x.  a )  +  1 )  =  A  /\  ( b  x.  2 )  =  B )  ->  ( E. c  e.  ZZ  ( ( 2  x.  c )  +  1 )  =  ( ( ( 2  x.  a )  +  1 )  -  ( b  x.  2 ) )  <->  E. c  e.  ZZ  ( ( 2  x.  c )  +  1 )  =  ( A  -  B ) ) )
3632, 35syl5ibcom 212 . . . . . . 7  |-  ( ( a  e.  ZZ  /\  b  e.  ZZ )  ->  ( ( ( ( 2  x.  a )  +  1 )  =  A  /\  ( b  x.  2 )  =  B )  ->  E. c  e.  ZZ  ( ( 2  x.  c )  +  1 )  =  ( A  -  B ) ) )
3736rexlimivv 2835 . . . . . 6  |-  ( E. a  e.  ZZ  E. b  e.  ZZ  (
( ( 2  x.  a )  +  1 )  =  A  /\  ( b  x.  2 )  =  B )  ->  E. c  e.  ZZ  ( ( 2  x.  c )  +  1 )  =  ( A  -  B ) )
386, 37sylbir 205 . . . . 5  |-  ( ( E. a  e.  ZZ  ( ( 2  x.  a )  +  1 )  =  A  /\  E. b  e.  ZZ  (
b  x.  2 )  =  B )  ->  E. c  e.  ZZ  ( ( 2  x.  c )  +  1 )  =  ( A  -  B ) )
395, 38syl6bi 220 . . . 4  |-  ( ( A  e.  ZZ  /\  B  e.  ZZ )  ->  ( ( -.  2  ||  A  /\  2  ||  B )  ->  E. c  e.  ZZ  ( ( 2  x.  c )  +  1 )  =  ( A  -  B ) ) )
4039imp 419 . . 3  |-  ( ( ( A  e.  ZZ  /\  B  e.  ZZ )  /\  ( -.  2  ||  A  /\  2  ||  B ) )  ->  E. c  e.  ZZ  ( ( 2  x.  c )  +  1 )  =  ( A  -  B ) )
4140an4s 800 . 2  |-  ( ( ( A  e.  ZZ  /\ 
-.  2  ||  A
)  /\  ( B  e.  ZZ  /\  2  ||  B ) )  ->  E. c  e.  ZZ  ( ( 2  x.  c )  +  1 )  =  ( A  -  B ) )
42 zsubcl 10319 . . . 4  |-  ( ( A  e.  ZZ  /\  B  e.  ZZ )  ->  ( A  -  B
)  e.  ZZ )
4342ad2ant2r 728 . . 3  |-  ( ( ( A  e.  ZZ  /\ 
-.  2  ||  A
)  /\  ( B  e.  ZZ  /\  2  ||  B ) )  -> 
( A  -  B
)  e.  ZZ )
44 odd2np1 12908 . . 3  |-  ( ( A  -  B )  e.  ZZ  ->  ( -.  2  ||  ( A  -  B )  <->  E. c  e.  ZZ  ( ( 2  x.  c )  +  1 )  =  ( A  -  B ) ) )
4543, 44syl 16 . 2  |-  ( ( ( A  e.  ZZ  /\ 
-.  2  ||  A
)  /\  ( B  e.  ZZ  /\  2  ||  B ) )  -> 
( -.  2  ||  ( A  -  B
)  <->  E. c  e.  ZZ  ( ( 2  x.  c )  +  1 )  =  ( A  -  B ) ) )
4641, 45mpbird 224 1  |-  ( ( ( A  e.  ZZ  /\ 
-.  2  ||  A
)  /\  ( B  e.  ZZ  /\  2  ||  B ) )  ->  -.  2  ||  ( A  -  B ) )
Colors of variables: wff set class
Syntax hints:   -. wn 3    -> wi 4    <-> wb 177    /\ wa 359    = wceq 1652    e. wcel 1725   E.wrex 2706   class class class wbr 4212  (class class class)co 6081   CCcc 8988   1c1 8991    + caddc 8993    x. cmul 8995    - cmin 9291   2c2 10049   ZZcz 10282    || cdivides 12852
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1555  ax-5 1566  ax-17 1626  ax-9 1666  ax-8 1687  ax-13 1727  ax-14 1729  ax-6 1744  ax-7 1749  ax-11 1761  ax-12 1950  ax-ext 2417  ax-sep 4330  ax-nul 4338  ax-pow 4377  ax-pr 4403  ax-un 4701  ax-resscn 9047  ax-1cn 9048  ax-icn 9049  ax-addcl 9050  ax-addrcl 9051  ax-mulcl 9052  ax-mulrcl 9053  ax-mulcom 9054  ax-addass 9055  ax-mulass 9056  ax-distr 9057  ax-i2m1 9058  ax-1ne0 9059  ax-1rid 9060  ax-rnegex 9061  ax-rrecex 9062  ax-cnre 9063  ax-pre-lttri 9064  ax-pre-lttrn 9065  ax-pre-ltadd 9066  ax-pre-mulgt0 9067
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3or 937  df-3an 938  df-tru 1328  df-ex 1551  df-nf 1554  df-sb 1659  df-eu 2285  df-mo 2286  df-clab 2423  df-cleq 2429  df-clel 2432  df-nfc 2561  df-ne 2601  df-nel 2602  df-ral 2710  df-rex 2711  df-reu 2712  df-rmo 2713  df-rab 2714  df-v 2958  df-sbc 3162  df-csb 3252  df-dif 3323  df-un 3325  df-in 3327  df-ss 3334  df-pss 3336  df-nul 3629  df-if 3740  df-pw 3801  df-sn 3820  df-pr 3821  df-tp 3822  df-op 3823  df-uni 4016  df-iun 4095  df-br 4213  df-opab 4267  df-mpt 4268  df-tr 4303  df-eprel 4494  df-id 4498  df-po 4503  df-so 4504  df-fr 4541  df-we 4543  df-ord 4584  df-on 4585  df-lim 4586  df-suc 4587  df-om 4846  df-xp 4884  df-rel 4885  df-cnv 4886  df-co 4887  df-dm 4888  df-rn 4889  df-res 4890  df-ima 4891  df-iota 5418  df-fun 5456  df-fn 5457  df-f 5458  df-f1 5459  df-fo 5460  df-f1o 5461  df-fv 5462  df-ov 6084  df-oprab 6085  df-mpt2 6086  df-riota 6549  df-recs 6633  df-rdg 6668  df-er 6905  df-en 7110  df-dom 7111  df-sdom 7112  df-pnf 9122  df-mnf 9123  df-xr 9124  df-ltxr 9125  df-le 9126  df-sub 9293  df-neg 9294  df-div 9678  df-nn 10001  df-2 10058  df-n0 10222  df-z 10283  df-dvds 12853
  Copyright terms: Public domain W3C validator