ILE Home Intuitionistic Logic Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  ILE Home  >  Th. List  >  div4p1lem1div2 Unicode version

Theorem div4p1lem1div2 8639
Description: An integer greater than 5, divided by 4 and increased by 1, is less than or equal to the half of the integer minus 1. (Contributed by AV, 8-Jul-2021.)
Assertion
Ref Expression
div4p1lem1div2  |-  ( ( N  e.  RR  /\  6  <_  N )  -> 
( ( N  / 
4 )  +  1 )  <_  ( ( N  -  1 )  /  2 ) )

Proof of Theorem div4p1lem1div2
StepHypRef Expression
1 6re 8474 . . . . . . 7  |-  6  e.  RR
21a1i 9 . . . . . 6  |-  ( N  e.  RR  ->  6  e.  RR )
3 id 19 . . . . . 6  |-  ( N  e.  RR  ->  N  e.  RR )
42, 3, 3leadd2d 7993 . . . . 5  |-  ( N  e.  RR  ->  (
6  <_  N  <->  ( N  +  6 )  <_ 
( N  +  N
) ) )
54biimpa 290 . . . 4  |-  ( ( N  e.  RR  /\  6  <_  N )  -> 
( N  +  6 )  <_  ( N  +  N ) )
6 recn 7454 . . . . . 6  |-  ( N  e.  RR  ->  N  e.  CC )
76times2d 8629 . . . . 5  |-  ( N  e.  RR  ->  ( N  x.  2 )  =  ( N  +  N ) )
87adantr 270 . . . 4  |-  ( ( N  e.  RR  /\  6  <_  N )  -> 
( N  x.  2 )  =  ( N  +  N ) )
95, 8breqtrrd 3863 . . 3  |-  ( ( N  e.  RR  /\  6  <_  N )  -> 
( N  +  6 )  <_  ( N  x.  2 ) )
10 4cn 8471 . . . . . . . 8  |-  4  e.  CC
1110a1i 9 . . . . . . 7  |-  ( N  e.  RR  ->  4  e.  CC )
12 2cn 8464 . . . . . . . 8  |-  2  e.  CC
1312a1i 9 . . . . . . 7  |-  ( N  e.  RR  ->  2  e.  CC )
146, 11, 13addassd 7489 . . . . . 6  |-  ( N  e.  RR  ->  (
( N  +  4 )  +  2 )  =  ( N  +  ( 4  +  2 ) ) )
15 4p2e6 8529 . . . . . . 7  |-  ( 4  +  2 )  =  6
1615oveq2i 5645 . . . . . 6  |-  ( N  +  ( 4  +  2 ) )  =  ( N  +  6 )
1714, 16syl6eq 2136 . . . . 5  |-  ( N  e.  RR  ->  (
( N  +  4 )  +  2 )  =  ( N  + 
6 ) )
1817breq1d 3847 . . . 4  |-  ( N  e.  RR  ->  (
( ( N  + 
4 )  +  2 )  <_  ( N  x.  2 )  <->  ( N  +  6 )  <_ 
( N  x.  2 ) ) )
1918adantr 270 . . 3  |-  ( ( N  e.  RR  /\  6  <_  N )  -> 
( ( ( N  +  4 )  +  2 )  <_  ( N  x.  2 )  <-> 
( N  +  6 )  <_  ( N  x.  2 ) ) )
209, 19mpbird 165 . 2  |-  ( ( N  e.  RR  /\  6  <_  N )  -> 
( ( N  + 
4 )  +  2 )  <_  ( N  x.  2 ) )
21 4re 8470 . . . . . . . 8  |-  4  e.  RR
2221a1i 9 . . . . . . 7  |-  ( N  e.  RR  ->  4  e.  RR )
23 4ap0 8492 . . . . . . . 8  |-  4 #  0
2423a1i 9 . . . . . . 7  |-  ( N  e.  RR  ->  4 #  0 )
253, 22, 24redivclapd 8273 . . . . . 6  |-  ( N  e.  RR  ->  ( N  /  4 )  e.  RR )
26 peano2re 7597 . . . . . 6  |-  ( ( N  /  4 )  e.  RR  ->  (
( N  /  4
)  +  1 )  e.  RR )
2725, 26syl 14 . . . . 5  |-  ( N  e.  RR  ->  (
( N  /  4
)  +  1 )  e.  RR )
28 peano2rem 7728 . . . . . 6  |-  ( N  e.  RR  ->  ( N  -  1 )  e.  RR )
2928rehalfcld 8632 . . . . 5  |-  ( N  e.  RR  ->  (
( N  -  1 )  /  2 )  e.  RR )
30 4pos 8490 . . . . . . 7  |-  0  <  4
3121, 30pm3.2i 266 . . . . . 6  |-  ( 4  e.  RR  /\  0  <  4 )
3231a1i 9 . . . . 5  |-  ( N  e.  RR  ->  (
4  e.  RR  /\  0  <  4 ) )
33 lemul1 8046 . . . . 5  |-  ( ( ( ( N  / 
4 )  +  1 )  e.  RR  /\  ( ( N  - 
1 )  /  2
)  e.  RR  /\  ( 4  e.  RR  /\  0  <  4 ) )  ->  ( (
( N  /  4
)  +  1 )  <_  ( ( N  -  1 )  / 
2 )  <->  ( (
( N  /  4
)  +  1 )  x.  4 )  <_ 
( ( ( N  -  1 )  / 
2 )  x.  4 ) ) )
3427, 29, 32, 33syl3anc 1174 . . . 4  |-  ( N  e.  RR  ->  (
( ( N  / 
4 )  +  1 )  <_  ( ( N  -  1 )  /  2 )  <->  ( (
( N  /  4
)  +  1 )  x.  4 )  <_ 
( ( ( N  -  1 )  / 
2 )  x.  4 ) ) )
3525recnd 7495 . . . . . 6  |-  ( N  e.  RR  ->  ( N  /  4 )  e.  CC )
36 1cnd 7483 . . . . . 6  |-  ( N  e.  RR  ->  1  e.  CC )
376, 11, 24divcanap1d 8231 . . . . . . 7  |-  ( N  e.  RR  ->  (
( N  /  4
)  x.  4 )  =  N )
3810mulid2i 7470 . . . . . . . 8  |-  ( 1  x.  4 )  =  4
3938a1i 9 . . . . . . 7  |-  ( N  e.  RR  ->  (
1  x.  4 )  =  4 )
4037, 39oveq12d 5652 . . . . . 6  |-  ( N  e.  RR  ->  (
( ( N  / 
4 )  x.  4 )  +  ( 1  x.  4 ) )  =  ( N  + 
4 ) )
4135, 11, 36, 40joinlmuladdmuld 7494 . . . . 5  |-  ( N  e.  RR  ->  (
( ( N  / 
4 )  +  1 )  x.  4 )  =  ( N  + 
4 ) )
42 2t2e4 8540 . . . . . . . . 9  |-  ( 2  x.  2 )  =  4
4342eqcomi 2092 . . . . . . . 8  |-  4  =  ( 2  x.  2 )
4443a1i 9 . . . . . . 7  |-  ( N  e.  RR  ->  4  =  ( 2  x.  2 ) )
4544oveq2d 5650 . . . . . 6  |-  ( N  e.  RR  ->  (
( ( N  - 
1 )  /  2
)  x.  4 )  =  ( ( ( N  -  1 )  /  2 )  x.  ( 2  x.  2 ) ) )
4629recnd 7495 . . . . . . 7  |-  ( N  e.  RR  ->  (
( N  -  1 )  /  2 )  e.  CC )
47 mulass 7452 . . . . . . . 8  |-  ( ( ( ( N  - 
1 )  /  2
)  e.  CC  /\  2  e.  CC  /\  2  e.  CC )  ->  (
( ( ( N  -  1 )  / 
2 )  x.  2 )  x.  2 )  =  ( ( ( N  -  1 )  /  2 )  x.  ( 2  x.  2 ) ) )
4847eqcomd 2093 . . . . . . 7  |-  ( ( ( ( N  - 
1 )  /  2
)  e.  CC  /\  2  e.  CC  /\  2  e.  CC )  ->  (
( ( N  - 
1 )  /  2
)  x.  ( 2  x.  2 ) )  =  ( ( ( ( N  -  1 )  /  2 )  x.  2 )  x.  2 ) )
4946, 13, 13, 48syl3anc 1174 . . . . . 6  |-  ( N  e.  RR  ->  (
( ( N  - 
1 )  /  2
)  x.  ( 2  x.  2 ) )  =  ( ( ( ( N  -  1 )  /  2 )  x.  2 )  x.  2 ) )
5028recnd 7495 . . . . . . . . 9  |-  ( N  e.  RR  ->  ( N  -  1 )  e.  CC )
51 2ap0 8486 . . . . . . . . . 10  |-  2 #  0
5251a1i 9 . . . . . . . . 9  |-  ( N  e.  RR  ->  2 #  0 )
5350, 13, 52divcanap1d 8231 . . . . . . . 8  |-  ( N  e.  RR  ->  (
( ( N  - 
1 )  /  2
)  x.  2 )  =  ( N  - 
1 ) )
5453oveq1d 5649 . . . . . . 7  |-  ( N  e.  RR  ->  (
( ( ( N  -  1 )  / 
2 )  x.  2 )  x.  2 )  =  ( ( N  -  1 )  x.  2 ) )
556, 36, 13subdird 7872 . . . . . . 7  |-  ( N  e.  RR  ->  (
( N  -  1 )  x.  2 )  =  ( ( N  x.  2 )  -  ( 1  x.  2 ) ) )
5612mulid2i 7470 . . . . . . . . 9  |-  ( 1  x.  2 )  =  2
5756a1i 9 . . . . . . . 8  |-  ( N  e.  RR  ->  (
1  x.  2 )  =  2 )
5857oveq2d 5650 . . . . . . 7  |-  ( N  e.  RR  ->  (
( N  x.  2 )  -  ( 1  x.  2 ) )  =  ( ( N  x.  2 )  - 
2 ) )
5954, 55, 583eqtrd 2124 . . . . . 6  |-  ( N  e.  RR  ->  (
( ( ( N  -  1 )  / 
2 )  x.  2 )  x.  2 )  =  ( ( N  x.  2 )  - 
2 ) )
6045, 49, 593eqtrd 2124 . . . . 5  |-  ( N  e.  RR  ->  (
( ( N  - 
1 )  /  2
)  x.  4 )  =  ( ( N  x.  2 )  - 
2 ) )
6141, 60breq12d 3850 . . . 4  |-  ( N  e.  RR  ->  (
( ( ( N  /  4 )  +  1 )  x.  4 )  <_  ( (
( N  -  1 )  /  2 )  x.  4 )  <->  ( N  +  4 )  <_ 
( ( N  x.  2 )  -  2 ) ) )
623, 22readdcld 7496 . . . . 5  |-  ( N  e.  RR  ->  ( N  +  4 )  e.  RR )
63 2re 8463 . . . . . 6  |-  2  e.  RR
6463a1i 9 . . . . 5  |-  ( N  e.  RR  ->  2  e.  RR )
653, 64remulcld 7497 . . . . 5  |-  ( N  e.  RR  ->  ( N  x.  2 )  e.  RR )
66 leaddsub 7895 . . . . . 6  |-  ( ( ( N  +  4 )  e.  RR  /\  2  e.  RR  /\  ( N  x.  2 )  e.  RR )  -> 
( ( ( N  +  4 )  +  2 )  <_  ( N  x.  2 )  <-> 
( N  +  4 )  <_  ( ( N  x.  2 )  -  2 ) ) )
6766bicomd 139 . . . . 5  |-  ( ( ( N  +  4 )  e.  RR  /\  2  e.  RR  /\  ( N  x.  2 )  e.  RR )  -> 
( ( N  + 
4 )  <_  (
( N  x.  2 )  -  2 )  <-> 
( ( N  + 
4 )  +  2 )  <_  ( N  x.  2 ) ) )
6862, 64, 65, 67syl3anc 1174 . . . 4  |-  ( N  e.  RR  ->  (
( N  +  4 )  <_  ( ( N  x.  2 )  -  2 )  <->  ( ( N  +  4 )  +  2 )  <_ 
( N  x.  2 ) ) )
6934, 61, 683bitrd 212 . . 3  |-  ( N  e.  RR  ->  (
( ( N  / 
4 )  +  1 )  <_  ( ( N  -  1 )  /  2 )  <->  ( ( N  +  4 )  +  2 )  <_ 
( N  x.  2 ) ) )
7069adantr 270 . 2  |-  ( ( N  e.  RR  /\  6  <_  N )  -> 
( ( ( N  /  4 )  +  1 )  <_  (
( N  -  1 )  /  2 )  <-> 
( ( N  + 
4 )  +  2 )  <_  ( N  x.  2 ) ) )
7120, 70mpbird 165 1  |-  ( ( N  e.  RR  /\  6  <_  N )  -> 
( ( N  / 
4 )  +  1 )  <_  ( ( N  -  1 )  /  2 ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 102    <-> wb 103    /\ w3a 924    = wceq 1289    e. wcel 1438   class class class wbr 3837  (class class class)co 5634   CCcc 7327   RRcr 7328   0cc0 7329   1c1 7330    + caddc 7332    x. cmul 7334    < clt 7501    <_ cle 7502    - cmin 7632   # cap 8034    / cdiv 8113   2c2 8444   4c4 8446   6c6 8448
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-3an 926  df-tru 1292  df-fal 1295  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-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-2 8452  df-3 8453  df-4 8454  df-5 8455  df-6 8456
This theorem is referenced by:  fldiv4p1lem1div2  9677
  Copyright terms: Public domain W3C validator