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

Theorem div4p1lem1div2 8235
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 8071 . . . . . . 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 7605 . . . . 5  |-  ( N  e.  RR  ->  (
6  <_  N  <->  ( N  +  6 )  <_ 
( N  +  N
) ) )
54biimpa 284 . . . 4  |-  ( ( N  e.  RR  /\  6  <_  N )  -> 
( N  +  6 )  <_  ( N  +  N ) )
6 recn 7072 . . . . . 6  |-  ( N  e.  RR  ->  N  e.  CC )
76times2d 8225 . . . . 5  |-  ( N  e.  RR  ->  ( N  x.  2 )  =  ( N  +  N ) )
87adantr 265 . . . 4  |-  ( ( N  e.  RR  /\  6  <_  N )  -> 
( N  x.  2 )  =  ( N  +  N ) )
95, 8breqtrrd 3818 . . 3  |-  ( ( N  e.  RR  /\  6  <_  N )  -> 
( N  +  6 )  <_  ( N  x.  2 ) )
10 4cn 8068 . . . . . . . 8  |-  4  e.  CC
1110a1i 9 . . . . . . 7  |-  ( N  e.  RR  ->  4  e.  CC )
12 2cn 8061 . . . . . . . 8  |-  2  e.  CC
1312a1i 9 . . . . . . 7  |-  ( N  e.  RR  ->  2  e.  CC )
146, 11, 13addassd 7107 . . . . . 6  |-  ( N  e.  RR  ->  (
( N  +  4 )  +  2 )  =  ( N  +  ( 4  +  2 ) ) )
15 4p2e6 8126 . . . . . . 7  |-  ( 4  +  2 )  =  6
1615oveq2i 5551 . . . . . 6  |-  ( N  +  ( 4  +  2 ) )  =  ( N  +  6 )
1714, 16syl6eq 2104 . . . . 5  |-  ( N  e.  RR  ->  (
( N  +  4 )  +  2 )  =  ( N  + 
6 ) )
1817breq1d 3802 . . . 4  |-  ( N  e.  RR  ->  (
( ( N  + 
4 )  +  2 )  <_  ( N  x.  2 )  <->  ( N  +  6 )  <_ 
( N  x.  2 ) ) )
1918adantr 265 . . 3  |-  ( ( N  e.  RR  /\  6  <_  N )  -> 
( ( ( N  +  4 )  +  2 )  <_  ( N  x.  2 )  <-> 
( N  +  6 )  <_  ( N  x.  2 ) ) )
209, 19mpbird 160 . 2  |-  ( ( N  e.  RR  /\  6  <_  N )  -> 
( ( N  + 
4 )  +  2 )  <_  ( N  x.  2 ) )
21 4re 8067 . . . . . . . 8  |-  4  e.  RR
2221a1i 9 . . . . . . 7  |-  ( N  e.  RR  ->  4  e.  RR )
23 4ap0 8089 . . . . . . . 8  |-  4 #  0
2423a1i 9 . . . . . . 7  |-  ( N  e.  RR  ->  4 #  0 )
253, 22, 24redivclapd 7883 . . . . . 6  |-  ( N  e.  RR  ->  ( N  /  4 )  e.  RR )
26 peano2re 7210 . . . . . 6  |-  ( ( N  /  4 )  e.  RR  ->  (
( N  /  4
)  +  1 )  e.  RR )
2725, 26syl 14 . . . . 5  |-  ( N  e.  RR  ->  (
( N  /  4
)  +  1 )  e.  RR )
28 peano2rem 7341 . . . . . 6  |-  ( N  e.  RR  ->  ( N  -  1 )  e.  RR )
2928rehalfcld 8228 . . . . 5  |-  ( N  e.  RR  ->  (
( N  -  1 )  /  2 )  e.  RR )
30 4pos 8087 . . . . . . 7  |-  0  <  4
3121, 30pm3.2i 261 . . . . . 6  |-  ( 4  e.  RR  /\  0  <  4 )
3231a1i 9 . . . . 5  |-  ( N  e.  RR  ->  (
4  e.  RR  /\  0  <  4 ) )
33 lemul1 7658 . . . . 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 1146 . . . 4  |-  ( N  e.  RR  ->  (
( ( N  / 
4 )  +  1 )  <_  ( ( N  -  1 )  /  2 )  <->  ( (
( N  /  4
)  +  1 )  x.  4 )  <_ 
( ( ( N  -  1 )  / 
2 )  x.  4 ) ) )
3525recnd 7113 . . . . . 6  |-  ( N  e.  RR  ->  ( N  /  4 )  e.  CC )
36 1cnd 7101 . . . . . 6  |-  ( N  e.  RR  ->  1  e.  CC )
376, 11, 24divcanap1d 7841 . . . . . . 7  |-  ( N  e.  RR  ->  (
( N  /  4
)  x.  4 )  =  N )
3810mulid2i 7088 . . . . . . . 8  |-  ( 1  x.  4 )  =  4
3938a1i 9 . . . . . . 7  |-  ( N  e.  RR  ->  (
1  x.  4 )  =  4 )
4037, 39oveq12d 5558 . . . . . 6  |-  ( N  e.  RR  ->  (
( ( N  / 
4 )  x.  4 )  +  ( 1  x.  4 ) )  =  ( N  + 
4 ) )
4135, 11, 36, 40joinlmuladdmuld 7112 . . . . 5  |-  ( N  e.  RR  ->  (
( ( N  / 
4 )  +  1 )  x.  4 )  =  ( N  + 
4 ) )
42 2t2e4 8137 . . . . . . . . 9  |-  ( 2  x.  2 )  =  4
4342eqcomi 2060 . . . . . . . 8  |-  4  =  ( 2  x.  2 )
4443a1i 9 . . . . . . 7  |-  ( N  e.  RR  ->  4  =  ( 2  x.  2 ) )
4544oveq2d 5556 . . . . . 6  |-  ( N  e.  RR  ->  (
( ( N  - 
1 )  /  2
)  x.  4 )  =  ( ( ( N  -  1 )  /  2 )  x.  ( 2  x.  2 ) ) )
4629recnd 7113 . . . . . . 7  |-  ( N  e.  RR  ->  (
( N  -  1 )  /  2 )  e.  CC )
47 mulass 7070 . . . . . . . 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 2061 . . . . . . 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 1146 . . . . . 6  |-  ( N  e.  RR  ->  (
( ( N  - 
1 )  /  2
)  x.  ( 2  x.  2 ) )  =  ( ( ( ( N  -  1 )  /  2 )  x.  2 )  x.  2 ) )
5028recnd 7113 . . . . . . . . 9  |-  ( N  e.  RR  ->  ( N  -  1 )  e.  CC )
51 2ap0 8083 . . . . . . . . . 10  |-  2 #  0
5251a1i 9 . . . . . . . . 9  |-  ( N  e.  RR  ->  2 #  0 )
5350, 13, 52divcanap1d 7841 . . . . . . . 8  |-  ( N  e.  RR  ->  (
( ( N  - 
1 )  /  2
)  x.  2 )  =  ( N  - 
1 ) )
5453oveq1d 5555 . . . . . . 7  |-  ( N  e.  RR  ->  (
( ( ( N  -  1 )  / 
2 )  x.  2 )  x.  2 )  =  ( ( N  -  1 )  x.  2 ) )
556, 36, 13subdird 7484 . . . . . . 7  |-  ( N  e.  RR  ->  (
( N  -  1 )  x.  2 )  =  ( ( N  x.  2 )  -  ( 1  x.  2 ) ) )
5612mulid2i 7088 . . . . . . . . 9  |-  ( 1  x.  2 )  =  2
5756a1i 9 . . . . . . . 8  |-  ( N  e.  RR  ->  (
1  x.  2 )  =  2 )
5857oveq2d 5556 . . . . . . 7  |-  ( N  e.  RR  ->  (
( N  x.  2 )  -  ( 1  x.  2 ) )  =  ( ( N  x.  2 )  - 
2 ) )
5954, 55, 583eqtrd 2092 . . . . . 6  |-  ( N  e.  RR  ->  (
( ( ( N  -  1 )  / 
2 )  x.  2 )  x.  2 )  =  ( ( N  x.  2 )  - 
2 ) )
6045, 49, 593eqtrd 2092 . . . . 5  |-  ( N  e.  RR  ->  (
( ( N  - 
1 )  /  2
)  x.  4 )  =  ( ( N  x.  2 )  - 
2 ) )
6141, 60breq12d 3805 . . . 4  |-  ( N  e.  RR  ->  (
( ( ( N  /  4 )  +  1 )  x.  4 )  <_  ( (
( N  -  1 )  /  2 )  x.  4 )  <->  ( N  +  4 )  <_ 
( ( N  x.  2 )  -  2 ) ) )
623, 22readdcld 7114 . . . . 5  |-  ( N  e.  RR  ->  ( N  +  4 )  e.  RR )
63 2re 8060 . . . . . 6  |-  2  e.  RR
6463a1i 9 . . . . 5  |-  ( N  e.  RR  ->  2  e.  RR )
653, 64remulcld 7115 . . . . 5  |-  ( N  e.  RR  ->  ( N  x.  2 )  e.  RR )
66 leaddsub 7507 . . . . . 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 133 . . . . 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 1146 . . . 4  |-  ( N  e.  RR  ->  (
( N  +  4 )  <_  ( ( N  x.  2 )  -  2 )  <->  ( ( N  +  4 )  +  2 )  <_ 
( N  x.  2 ) ) )
6934, 61, 683bitrd 207 . . 3  |-  ( N  e.  RR  ->  (
( ( N  / 
4 )  +  1 )  <_  ( ( N  -  1 )  /  2 )  <->  ( ( N  +  4 )  +  2 )  <_ 
( N  x.  2 ) ) )
7069adantr 265 . 2  |-  ( ( N  e.  RR  /\  6  <_  N )  -> 
( ( ( N  /  4 )  +  1 )  <_  (
( N  -  1 )  /  2 )  <-> 
( ( N  + 
4 )  +  2 )  <_  ( N  x.  2 ) ) )
7120, 70mpbird 160 1  |-  ( ( N  e.  RR  /\  6  <_  N )  -> 
( ( N  / 
4 )  +  1 )  <_  ( ( N  -  1 )  /  2 ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 101    <-> wb 102    /\ w3a 896    = wceq 1259    e. wcel 1409   class class class wbr 3792  (class class class)co 5540   CCcc 6945   RRcr 6946   0cc0 6947   1c1 6948    + caddc 6950    x. cmul 6952    < clt 7119    <_ cle 7120    - cmin 7245   # cap 7646    / cdiv 7725   2c2 8040   4c4 8042   6c6 8044
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 103  ax-ia2 104  ax-ia3 105  ax-in1 554  ax-in2 555  ax-io 640  ax-5 1352  ax-7 1353  ax-gen 1354  ax-ie1 1398  ax-ie2 1399  ax-8 1411  ax-10 1412  ax-11 1413  ax-i12 1414  ax-bndl 1415  ax-4 1416  ax-13 1420  ax-14 1421  ax-17 1435  ax-i9 1439  ax-ial 1443  ax-i5r 1444  ax-ext 2038  ax-coll 3900  ax-sep 3903  ax-nul 3911  ax-pow 3955  ax-pr 3972  ax-un 4198  ax-setind 4290  ax-iinf 4339  ax-cnex 7033  ax-resscn 7034  ax-1cn 7035  ax-1re 7036  ax-icn 7037  ax-addcl 7038  ax-addrcl 7039  ax-mulcl 7040  ax-mulrcl 7041  ax-addcom 7042  ax-mulcom 7043  ax-addass 7044  ax-mulass 7045  ax-distr 7046  ax-i2m1 7047  ax-1rid 7049  ax-0id 7050  ax-rnegex 7051  ax-precex 7052  ax-cnre 7053  ax-pre-ltirr 7054  ax-pre-ltwlin 7055  ax-pre-lttrn 7056  ax-pre-apti 7057  ax-pre-ltadd 7058  ax-pre-mulgt0 7059  ax-pre-mulext 7060
This theorem depends on definitions:  df-bi 114  df-dc 754  df-3or 897  df-3an 898  df-tru 1262  df-fal 1265  df-nf 1366  df-sb 1662  df-eu 1919  df-mo 1920  df-clab 2043  df-cleq 2049  df-clel 2052  df-nfc 2183  df-ne 2221  df-nel 2315  df-ral 2328  df-rex 2329  df-reu 2330  df-rmo 2331  df-rab 2332  df-v 2576  df-sbc 2788  df-csb 2881  df-dif 2948  df-un 2950  df-in 2952  df-ss 2959  df-nul 3253  df-pw 3389  df-sn 3409  df-pr 3410  df-op 3412  df-uni 3609  df-int 3644  df-iun 3687  df-br 3793  df-opab 3847  df-mpt 3848  df-tr 3883  df-eprel 4054  df-id 4058  df-po 4061  df-iso 4062  df-iord 4131  df-on 4133  df-suc 4136  df-iom 4342  df-xp 4379  df-rel 4380  df-cnv 4381  df-co 4382  df-dm 4383  df-rn 4384  df-res 4385  df-ima 4386  df-iota 4895  df-fun 4932  df-fn 4933  df-f 4934  df-f1 4935  df-fo 4936  df-f1o 4937  df-fv 4938  df-riota 5496  df-ov 5543  df-oprab 5544  df-mpt2 5545  df-1st 5795  df-2nd 5796  df-recs 5951  df-irdg 5988  df-1o 6032  df-2o 6033  df-oadd 6036  df-omul 6037  df-er 6137  df-ec 6139  df-qs 6143  df-ni 6460  df-pli 6461  df-mi 6462  df-lti 6463  df-plpq 6500  df-mpq 6501  df-enq 6503  df-nqqs 6504  df-plqqs 6505  df-mqqs 6506  df-1nqqs 6507  df-rq 6508  df-ltnqqs 6509  df-enq0 6580  df-nq0 6581  df-0nq0 6582  df-plq0 6583  df-mq0 6584  df-inp 6622  df-i1p 6623  df-iplp 6624  df-iltp 6626  df-enr 6869  df-nr 6870  df-ltr 6873  df-0r 6874  df-1r 6875  df-0 6954  df-1 6955  df-r 6957  df-lt 6960  df-pnf 7121  df-mnf 7122  df-xr 7123  df-ltxr 7124  df-le 7125  df-sub 7247  df-neg 7248  df-reap 7640  df-ap 7647  df-div 7726  df-2 8049  df-3 8050  df-4 8051  df-5 8052  df-6 8053
This theorem is referenced by:  fldiv4p1lem1div2  9255
  Copyright terms: Public domain W3C validator