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Theorem halfleoddlt 10987
Description: An integer is greater than half of an odd number iff it is greater than or equal to the half of the odd number. (Contributed by AV, 1-Jul-2021.)
Assertion
Ref Expression
halfleoddlt  |-  ( ( N  e.  ZZ  /\  -.  2  ||  N  /\  M  e.  ZZ )  ->  ( ( N  / 
2 )  <_  M  <->  ( N  /  2 )  <  M ) )

Proof of Theorem halfleoddlt
Dummy variable  n is distinct from all other variables.
StepHypRef Expression
1 odd2np1 10966 . . 3  |-  ( N  e.  ZZ  ->  ( -.  2  ||  N  <->  E. n  e.  ZZ  ( ( 2  x.  n )  +  1 )  =  N ) )
2 0xr 7513 . . . . . . . . . . . 12  |-  0  e.  RR*
3 1re 7466 . . . . . . . . . . . . 13  |-  1  e.  RR
43rexri 7524 . . . . . . . . . . . 12  |-  1  e.  RR*
5 halfre 8599 . . . . . . . . . . . . 13  |-  ( 1  /  2 )  e.  RR
65rexri 7524 . . . . . . . . . . . 12  |-  ( 1  /  2 )  e. 
RR*
72, 4, 63pm3.2i 1121 . . . . . . . . . . 11  |-  ( 0  e.  RR*  /\  1  e.  RR*  /\  ( 1  /  2 )  e. 
RR* )
8 halfgt0 8601 . . . . . . . . . . . 12  |-  0  <  ( 1  /  2
)
9 halflt1 8603 . . . . . . . . . . . 12  |-  ( 1  /  2 )  <  1
108, 9pm3.2i 266 . . . . . . . . . . 11  |-  ( 0  <  ( 1  / 
2 )  /\  (
1  /  2 )  <  1 )
11 elioo3g 9297 . . . . . . . . . . 11  |-  ( ( 1  /  2 )  e.  ( 0 (,) 1 )  <->  ( (
0  e.  RR*  /\  1  e.  RR*  /\  ( 1  /  2 )  e. 
RR* )  /\  (
0  <  ( 1  /  2 )  /\  ( 1  /  2
)  <  1 ) ) )
127, 10, 11mpbir2an 888 . . . . . . . . . 10  |-  ( 1  /  2 )  e.  ( 0 (,) 1
)
13 zltaddlt1le 9392 . . . . . . . . . 10  |-  ( ( n  e.  ZZ  /\  M  e.  ZZ  /\  (
1  /  2 )  e.  ( 0 (,) 1 ) )  -> 
( ( n  +  ( 1  /  2
) )  <  M  <->  ( n  +  ( 1  /  2 ) )  <_  M ) )
1412, 13mp3an3 1262 . . . . . . . . 9  |-  ( ( n  e.  ZZ  /\  M  e.  ZZ )  ->  ( ( n  +  ( 1  /  2
) )  <  M  <->  ( n  +  ( 1  /  2 ) )  <_  M ) )
15 zcn 8725 . . . . . . . . . . . 12  |-  ( n  e.  ZZ  ->  n  e.  CC )
1615adantr 270 . . . . . . . . . . 11  |-  ( ( n  e.  ZZ  /\  M  e.  ZZ )  ->  n  e.  CC )
17 1cnd 7483 . . . . . . . . . . 11  |-  ( ( n  e.  ZZ  /\  M  e.  ZZ )  ->  1  e.  CC )
18 2cn 8464 . . . . . . . . . . . . 13  |-  2  e.  CC
19 2ap0 8486 . . . . . . . . . . . . 13  |-  2 #  0
2018, 19pm3.2i 266 . . . . . . . . . . . 12  |-  ( 2  e.  CC  /\  2 #  0 )
2120a1i 9 . . . . . . . . . . 11  |-  ( ( n  e.  ZZ  /\  M  e.  ZZ )  ->  ( 2  e.  CC  /\  2 #  0 ) )
22 muldivdirap 8148 . . . . . . . . . . 11  |-  ( ( n  e.  CC  /\  1  e.  CC  /\  (
2  e.  CC  /\  2 #  0 ) )  -> 
( ( ( 2  x.  n )  +  1 )  /  2
)  =  ( n  +  ( 1  / 
2 ) ) )
2316, 17, 21, 22syl3anc 1174 . . . . . . . . . 10  |-  ( ( n  e.  ZZ  /\  M  e.  ZZ )  ->  ( ( ( 2  x.  n )  +  1 )  /  2
)  =  ( n  +  ( 1  / 
2 ) ) )
2423breq1d 3847 . . . . . . . . 9  |-  ( ( n  e.  ZZ  /\  M  e.  ZZ )  ->  ( ( ( ( 2  x.  n )  +  1 )  / 
2 )  <  M  <->  ( n  +  ( 1  /  2 ) )  <  M ) )
2523breq1d 3847 . . . . . . . . 9  |-  ( ( n  e.  ZZ  /\  M  e.  ZZ )  ->  ( ( ( ( 2  x.  n )  +  1 )  / 
2 )  <_  M  <->  ( n  +  ( 1  /  2 ) )  <_  M ) )
2614, 24, 253bitr4rd 219 . . . . . . . 8  |-  ( ( n  e.  ZZ  /\  M  e.  ZZ )  ->  ( ( ( ( 2  x.  n )  +  1 )  / 
2 )  <_  M  <->  ( ( ( 2  x.  n )  +  1 )  /  2 )  <  M ) )
27 oveq1 5641 . . . . . . . . . 10  |-  ( ( ( 2  x.  n
)  +  1 )  =  N  ->  (
( ( 2  x.  n )  +  1 )  /  2 )  =  ( N  / 
2 ) )
2827breq1d 3847 . . . . . . . . 9  |-  ( ( ( 2  x.  n
)  +  1 )  =  N  ->  (
( ( ( 2  x.  n )  +  1 )  /  2
)  <_  M  <->  ( N  /  2 )  <_  M ) )
2927breq1d 3847 . . . . . . . . 9  |-  ( ( ( 2  x.  n
)  +  1 )  =  N  ->  (
( ( ( 2  x.  n )  +  1 )  /  2
)  <  M  <->  ( N  /  2 )  < 
M ) )
3028, 29bibi12d 233 . . . . . . . 8  |-  ( ( ( 2  x.  n
)  +  1 )  =  N  ->  (
( ( ( ( 2  x.  n )  +  1 )  / 
2 )  <_  M  <->  ( ( ( 2  x.  n )  +  1 )  /  2 )  <  M )  <->  ( ( N  /  2 )  <_  M 
<->  ( N  /  2
)  <  M )
) )
3126, 30syl5ibcom 153 . . . . . . 7  |-  ( ( n  e.  ZZ  /\  M  e.  ZZ )  ->  ( ( ( 2  x.  n )  +  1 )  =  N  ->  ( ( N  /  2 )  <_  M 
<->  ( N  /  2
)  <  M )
) )
3231ex 113 . . . . . 6  |-  ( n  e.  ZZ  ->  ( M  e.  ZZ  ->  ( ( ( 2  x.  n )  +  1 )  =  N  -> 
( ( N  / 
2 )  <_  M  <->  ( N  /  2 )  <  M ) ) ) )
3332adantl 271 . . . . 5  |-  ( ( N  e.  ZZ  /\  n  e.  ZZ )  ->  ( M  e.  ZZ  ->  ( ( ( 2  x.  n )  +  1 )  =  N  ->  ( ( N  /  2 )  <_  M 
<->  ( N  /  2
)  <  M )
) ) )
3433com23 77 . . . 4  |-  ( ( N  e.  ZZ  /\  n  e.  ZZ )  ->  ( ( ( 2  x.  n )  +  1 )  =  N  ->  ( M  e.  ZZ  ->  ( ( N  /  2 )  <_  M 
<->  ( N  /  2
)  <  M )
) ) )
3534rexlimdva 2489 . . 3  |-  ( N  e.  ZZ  ->  ( E. n  e.  ZZ  ( ( 2  x.  n )  +  1 )  =  N  -> 
( M  e.  ZZ  ->  ( ( N  / 
2 )  <_  M  <->  ( N  /  2 )  <  M ) ) ) )
361, 35sylbid 148 . 2  |-  ( N  e.  ZZ  ->  ( -.  2  ||  N  -> 
( M  e.  ZZ  ->  ( ( N  / 
2 )  <_  M  <->  ( N  /  2 )  <  M ) ) ) )
37363imp 1137 1  |-  ( ( N  e.  ZZ  /\  -.  2  ||  N  /\  M  e.  ZZ )  ->  ( ( N  / 
2 )  <_  M  <->  ( N  /  2 )  <  M ) )
Colors of variables: wff set class
Syntax hints:   -. wn 3    -> wi 4    /\ wa 102    <-> wb 103    /\ w3a 924    = wceq 1289    e. wcel 1438   E.wrex 2360   class class class wbr 3837  (class class class)co 5634   CCcc 7327   0cc0 7329   1c1 7330    + caddc 7332    x. cmul 7334   RR*cxr 7500    < clt 7501    <_ cle 7502   # cap 8034    / cdiv 8113   2c2 8444   ZZcz 8720   (,)cioo 9275    || cdvds 10889
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 579  ax-in2 580  ax-io 665  ax-5 1381  ax-7 1382  ax-gen 1383  ax-ie1 1427  ax-ie2 1428  ax-8 1440  ax-10 1441  ax-11 1442  ax-i12 1443  ax-bndl 1444  ax-4 1445  ax-13 1449  ax-14 1450  ax-17 1464  ax-i9 1468  ax-ial 1472  ax-i5r 1473  ax-ext 2070  ax-sep 3949  ax-pow 4001  ax-pr 4027  ax-un 4251  ax-setind 4343  ax-cnex 7415  ax-resscn 7416  ax-1cn 7417  ax-1re 7418  ax-icn 7419  ax-addcl 7420  ax-addrcl 7421  ax-mulcl 7422  ax-mulrcl 7423  ax-addcom 7424  ax-mulcom 7425  ax-addass 7426  ax-mulass 7427  ax-distr 7428  ax-i2m1 7429  ax-0lt1 7430  ax-1rid 7431  ax-0id 7432  ax-rnegex 7433  ax-precex 7434  ax-cnre 7435  ax-pre-ltirr 7436  ax-pre-ltwlin 7437  ax-pre-lttrn 7438  ax-pre-apti 7439  ax-pre-ltadd 7440  ax-pre-mulgt0 7441  ax-pre-mulext 7442
This theorem depends on definitions:  df-bi 115  df-3or 925  df-3an 926  df-tru 1292  df-fal 1295  df-xor 1312  df-nf 1395  df-sb 1693  df-eu 1951  df-mo 1952  df-clab 2075  df-cleq 2081  df-clel 2084  df-nfc 2217  df-ne 2256  df-nel 2351  df-ral 2364  df-rex 2365  df-reu 2366  df-rmo 2367  df-rab 2368  df-v 2621  df-sbc 2839  df-dif 2999  df-un 3001  df-in 3003  df-ss 3010  df-pw 3427  df-sn 3447  df-pr 3448  df-op 3450  df-uni 3649  df-int 3684  df-br 3838  df-opab 3892  df-id 4111  df-po 4114  df-iso 4115  df-xp 4434  df-rel 4435  df-cnv 4436  df-co 4437  df-dm 4438  df-iota 4967  df-fun 5004  df-fv 5010  df-riota 5590  df-ov 5637  df-oprab 5638  df-mpt2 5639  df-pnf 7503  df-mnf 7504  df-xr 7505  df-ltxr 7506  df-le 7507  df-sub 7634  df-neg 7635  df-reap 8028  df-ap 8035  df-div 8114  df-inn 8395  df-2 8452  df-n0 8644  df-z 8721  df-rp 9104  df-ioo 9279  df-dvds 10890
This theorem is referenced by: (None)
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