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Theorem div4p1lem1div2 8177
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 7996 . . . . . . 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 7531 . . . . 5  |-  ( N  e.  RR  ->  (
6  <_  N  <->  ( N  +  6 )  <_ 
( N  +  N
) ) )
54biimpa 280 . . . 4  |-  ( ( N  e.  RR  /\  6  <_  N )  -> 
( N  +  6 )  <_  ( N  +  N ) )
6 recn 7014 . . . . . 6  |-  ( N  e.  RR  ->  N  e.  CC )
76times2d 8168 . . . . 5  |-  ( N  e.  RR  ->  ( N  x.  2 )  =  ( N  +  N ) )
87adantr 261 . . . 4  |-  ( ( N  e.  RR  /\  6  <_  N )  -> 
( N  x.  2 )  =  ( N  +  N ) )
95, 8breqtrrd 3790 . . 3  |-  ( ( N  e.  RR  /\  6  <_  N )  -> 
( N  +  6 )  <_  ( N  x.  2 ) )
10 4cn 7993 . . . . . . . 8  |-  4  e.  CC
1110a1i 9 . . . . . . 7  |-  ( N  e.  RR  ->  4  e.  CC )
12 2cn 7986 . . . . . . . 8  |-  2  e.  CC
1312a1i 9 . . . . . . 7  |-  ( N  e.  RR  ->  2  e.  CC )
146, 11, 13addassd 7049 . . . . . 6  |-  ( N  e.  RR  ->  (
( N  +  4 )  +  2 )  =  ( N  +  ( 4  +  2 ) ) )
15 4p2e6 8054 . . . . . . 7  |-  ( 4  +  2 )  =  6
1615oveq2i 5523 . . . . . 6  |-  ( N  +  ( 4  +  2 ) )  =  ( N  +  6 )
1714, 16syl6eq 2088 . . . . 5  |-  ( N  e.  RR  ->  (
( N  +  4 )  +  2 )  =  ( N  + 
6 ) )
1817breq1d 3774 . . . 4  |-  ( N  e.  RR  ->  (
( ( N  + 
4 )  +  2 )  <_  ( N  x.  2 )  <->  ( N  +  6 )  <_ 
( N  x.  2 ) ) )
1918adantr 261 . . 3  |-  ( ( N  e.  RR  /\  6  <_  N )  -> 
( ( ( N  +  4 )  +  2 )  <_  ( N  x.  2 )  <-> 
( N  +  6 )  <_  ( N  x.  2 ) ) )
209, 19mpbird 156 . 2  |-  ( ( N  e.  RR  /\  6  <_  N )  -> 
( ( N  + 
4 )  +  2 )  <_  ( N  x.  2 ) )
21 4re 7992 . . . . . . . 8  |-  4  e.  RR
2221a1i 9 . . . . . . 7  |-  ( N  e.  RR  ->  4  e.  RR )
23 4ap0 8015 . . . . . . . 8  |-  4 #  0
2423a1i 9 . . . . . . 7  |-  ( N  e.  RR  ->  4 #  0 )
253, 22, 24redivclapd 7808 . . . . . 6  |-  ( N  e.  RR  ->  ( N  /  4 )  e.  RR )
26 peano2re 7149 . . . . . 6  |-  ( ( N  /  4 )  e.  RR  ->  (
( N  /  4
)  +  1 )  e.  RR )
2725, 26syl 14 . . . . 5  |-  ( N  e.  RR  ->  (
( N  /  4
)  +  1 )  e.  RR )
28 peano2rem 7278 . . . . . 6  |-  ( N  e.  RR  ->  ( N  -  1 )  e.  RR )
2928rehalfcld 8171 . . . . 5  |-  ( N  e.  RR  ->  (
( N  -  1 )  /  2 )  e.  RR )
30 4pos 8013 . . . . . . 7  |-  0  <  4
3121, 30pm3.2i 257 . . . . . 6  |-  ( 4  e.  RR  /\  0  <  4 )
3231a1i 9 . . . . 5  |-  ( N  e.  RR  ->  (
4  e.  RR  /\  0  <  4 ) )
33 lemul1 7584 . . . . 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 1135 . . . 4  |-  ( N  e.  RR  ->  (
( ( N  / 
4 )  +  1 )  <_  ( ( N  -  1 )  /  2 )  <->  ( (
( N  /  4
)  +  1 )  x.  4 )  <_ 
( ( ( N  -  1 )  / 
2 )  x.  4 ) ) )
3525recnd 7054 . . . . . 6  |-  ( N  e.  RR  ->  ( N  /  4 )  e.  CC )
36 1cnd 7043 . . . . . 6  |-  ( N  e.  RR  ->  1  e.  CC )
376, 11, 24divcanap1d 7766 . . . . . . 7  |-  ( N  e.  RR  ->  (
( N  /  4
)  x.  4 )  =  N )
3810mulid2i 7030 . . . . . . . 8  |-  ( 1  x.  4 )  =  4
3938a1i 9 . . . . . . 7  |-  ( N  e.  RR  ->  (
1  x.  4 )  =  4 )
4037, 39oveq12d 5530 . . . . . 6  |-  ( N  e.  RR  ->  (
( ( N  / 
4 )  x.  4 )  +  ( 1  x.  4 ) )  =  ( N  + 
4 ) )
4135, 11, 36, 40joinlmuladdmuld 7053 . . . . 5  |-  ( N  e.  RR  ->  (
( ( N  / 
4 )  +  1 )  x.  4 )  =  ( N  + 
4 ) )
42 2t2e4 8069 . . . . . . . . 9  |-  ( 2  x.  2 )  =  4
4342eqcomi 2044 . . . . . . . 8  |-  4  =  ( 2  x.  2 )
4443a1i 9 . . . . . . 7  |-  ( N  e.  RR  ->  4  =  ( 2  x.  2 ) )
4544oveq2d 5528 . . . . . 6  |-  ( N  e.  RR  ->  (
( ( N  - 
1 )  /  2
)  x.  4 )  =  ( ( ( N  -  1 )  /  2 )  x.  ( 2  x.  2 ) ) )
4629recnd 7054 . . . . . . 7  |-  ( N  e.  RR  ->  (
( N  -  1 )  /  2 )  e.  CC )
47 mulass 7012 . . . . . . . 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 2045 . . . . . . 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 1135 . . . . . 6  |-  ( N  e.  RR  ->  (
( ( N  - 
1 )  /  2
)  x.  ( 2  x.  2 ) )  =  ( ( ( ( N  -  1 )  /  2 )  x.  2 )  x.  2 ) )
5028recnd 7054 . . . . . . . . 9  |-  ( N  e.  RR  ->  ( N  -  1 )  e.  CC )
51 2ap0 8009 . . . . . . . . . 10  |-  2 #  0
5251a1i 9 . . . . . . . . 9  |-  ( N  e.  RR  ->  2 #  0 )
5350, 13, 52divcanap1d 7766 . . . . . . . 8  |-  ( N  e.  RR  ->  (
( ( N  - 
1 )  /  2
)  x.  2 )  =  ( N  - 
1 ) )
5453oveq1d 5527 . . . . . . 7  |-  ( N  e.  RR  ->  (
( ( ( N  -  1 )  / 
2 )  x.  2 )  x.  2 )  =  ( ( N  -  1 )  x.  2 ) )
556, 36, 13subdird 7412 . . . . . . 7  |-  ( N  e.  RR  ->  (
( N  -  1 )  x.  2 )  =  ( ( N  x.  2 )  -  ( 1  x.  2 ) ) )
5612mulid2i 7030 . . . . . . . . 9  |-  ( 1  x.  2 )  =  2
5756a1i 9 . . . . . . . 8  |-  ( N  e.  RR  ->  (
1  x.  2 )  =  2 )
5857oveq2d 5528 . . . . . . 7  |-  ( N  e.  RR  ->  (
( N  x.  2 )  -  ( 1  x.  2 ) )  =  ( ( N  x.  2 )  - 
2 ) )
5954, 55, 583eqtrd 2076 . . . . . 6  |-  ( N  e.  RR  ->  (
( ( ( N  -  1 )  / 
2 )  x.  2 )  x.  2 )  =  ( ( N  x.  2 )  - 
2 ) )
6045, 49, 593eqtrd 2076 . . . . 5  |-  ( N  e.  RR  ->  (
( ( N  - 
1 )  /  2
)  x.  4 )  =  ( ( N  x.  2 )  - 
2 ) )
6141, 60breq12d 3777 . . . 4  |-  ( N  e.  RR  ->  (
( ( ( N  /  4 )  +  1 )  x.  4 )  <_  ( (
( N  -  1 )  /  2 )  x.  4 )  <->  ( N  +  4 )  <_ 
( ( N  x.  2 )  -  2 ) ) )
623, 22readdcld 7055 . . . . 5  |-  ( N  e.  RR  ->  ( N  +  4 )  e.  RR )
63 2re 7985 . . . . . 6  |-  2  e.  RR
6463a1i 9 . . . . 5  |-  ( N  e.  RR  ->  2  e.  RR )
653, 64remulcld 7056 . . . . 5  |-  ( N  e.  RR  ->  ( N  x.  2 )  e.  RR )
66 leaddsub 7433 . . . . . 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 129 . . . . 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 1135 . . . 4  |-  ( N  e.  RR  ->  (
( N  +  4 )  <_  ( ( N  x.  2 )  -  2 )  <->  ( ( N  +  4 )  +  2 )  <_ 
( N  x.  2 ) ) )
6934, 61, 683bitrd 203 . . 3  |-  ( N  e.  RR  ->  (
( ( N  / 
4 )  +  1 )  <_  ( ( N  -  1 )  /  2 )  <->  ( ( N  +  4 )  +  2 )  <_ 
( N  x.  2 ) ) )
7069adantr 261 . 2  |-  ( ( N  e.  RR  /\  6  <_  N )  -> 
( ( ( N  /  4 )  +  1 )  <_  (
( N  -  1 )  /  2 )  <-> 
( ( N  + 
4 )  +  2 )  <_  ( N  x.  2 ) ) )
7120, 70mpbird 156 1  |-  ( ( N  e.  RR  /\  6  <_  N )  -> 
( ( N  / 
4 )  +  1 )  <_  ( ( N  -  1 )  /  2 ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 97    <-> wb 98    /\ w3a 885    = wceq 1243    e. wcel 1393   class class class wbr 3764  (class class class)co 5512   CCcc 6887   RRcr 6888   0cc0 6889   1c1 6890    + caddc 6892    x. cmul 6894    < clt 7060    <_ cle 7061    - cmin 7182   # cap 7572    / cdiv 7651   2c2 7964   4c4 7966   6c6 7968
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 99  ax-ia2 100  ax-ia3 101  ax-in1 544  ax-in2 545  ax-io 630  ax-5 1336  ax-7 1337  ax-gen 1338  ax-ie1 1382  ax-ie2 1383  ax-8 1395  ax-10 1396  ax-11 1397  ax-i12 1398  ax-bndl 1399  ax-4 1400  ax-13 1404  ax-14 1405  ax-17 1419  ax-i9 1423  ax-ial 1427  ax-i5r 1428  ax-ext 2022  ax-coll 3872  ax-sep 3875  ax-nul 3883  ax-pow 3927  ax-pr 3944  ax-un 4170  ax-setind 4262  ax-iinf 4311  ax-cnex 6975  ax-resscn 6976  ax-1cn 6977  ax-1re 6978  ax-icn 6979  ax-addcl 6980  ax-addrcl 6981  ax-mulcl 6982  ax-mulrcl 6983  ax-addcom 6984  ax-mulcom 6985  ax-addass 6986  ax-mulass 6987  ax-distr 6988  ax-i2m1 6989  ax-1rid 6991  ax-0id 6992  ax-rnegex 6993  ax-precex 6994  ax-cnre 6995  ax-pre-ltirr 6996  ax-pre-ltwlin 6997  ax-pre-lttrn 6998  ax-pre-apti 6999  ax-pre-ltadd 7000  ax-pre-mulgt0 7001  ax-pre-mulext 7002
This theorem depends on definitions:  df-bi 110  df-dc 743  df-3or 886  df-3an 887  df-tru 1246  df-fal 1249  df-nf 1350  df-sb 1646  df-eu 1903  df-mo 1904  df-clab 2027  df-cleq 2033  df-clel 2036  df-nfc 2167  df-ne 2206  df-nel 2207  df-ral 2311  df-rex 2312  df-reu 2313  df-rmo 2314  df-rab 2315  df-v 2559  df-sbc 2765  df-csb 2853  df-dif 2920  df-un 2922  df-in 2924  df-ss 2931  df-nul 3225  df-pw 3361  df-sn 3381  df-pr 3382  df-op 3384  df-uni 3581  df-int 3616  df-iun 3659  df-br 3765  df-opab 3819  df-mpt 3820  df-tr 3855  df-eprel 4026  df-id 4030  df-po 4033  df-iso 4034  df-iord 4103  df-on 4105  df-suc 4108  df-iom 4314  df-xp 4351  df-rel 4352  df-cnv 4353  df-co 4354  df-dm 4355  df-rn 4356  df-res 4357  df-ima 4358  df-iota 4867  df-fun 4904  df-fn 4905  df-f 4906  df-f1 4907  df-fo 4908  df-f1o 4909  df-fv 4910  df-riota 5468  df-ov 5515  df-oprab 5516  df-mpt2 5517  df-1st 5767  df-2nd 5768  df-recs 5920  df-irdg 5957  df-1o 6001  df-2o 6002  df-oadd 6005  df-omul 6006  df-er 6106  df-ec 6108  df-qs 6112  df-ni 6402  df-pli 6403  df-mi 6404  df-lti 6405  df-plpq 6442  df-mpq 6443  df-enq 6445  df-nqqs 6446  df-plqqs 6447  df-mqqs 6448  df-1nqqs 6449  df-rq 6450  df-ltnqqs 6451  df-enq0 6522  df-nq0 6523  df-0nq0 6524  df-plq0 6525  df-mq0 6526  df-inp 6564  df-i1p 6565  df-iplp 6566  df-iltp 6568  df-enr 6811  df-nr 6812  df-ltr 6815  df-0r 6816  df-1r 6817  df-0 6896  df-1 6897  df-r 6899  df-lt 6902  df-pnf 7062  df-mnf 7063  df-xr 7064  df-ltxr 7065  df-le 7066  df-sub 7184  df-neg 7185  df-reap 7566  df-ap 7573  df-div 7652  df-2 7973  df-3 7974  df-4 7975  df-5 7976  df-6 7977
This theorem is referenced by:  fldiv4p1lem1div2  9147
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