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Theorem div4p1lem1div2 9161
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 8989 . . . . . . 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 8487 . . . . 5  |-  ( N  e.  RR  ->  (
6  <_  N  <->  ( N  +  6 )  <_ 
( N  +  N
) ) )
54biimpa 296 . . . 4  |-  ( ( N  e.  RR  /\  6  <_  N )  -> 
( N  +  6 )  <_  ( N  +  N ) )
6 recn 7935 . . . . . 6  |-  ( N  e.  RR  ->  N  e.  CC )
76times2d 9151 . . . . 5  |-  ( N  e.  RR  ->  ( N  x.  2 )  =  ( N  +  N ) )
87adantr 276 . . . 4  |-  ( ( N  e.  RR  /\  6  <_  N )  -> 
( N  x.  2 )  =  ( N  +  N ) )
95, 8breqtrrd 4028 . . 3  |-  ( ( N  e.  RR  /\  6  <_  N )  -> 
( N  +  6 )  <_  ( N  x.  2 ) )
10 4cn 8986 . . . . . . . 8  |-  4  e.  CC
1110a1i 9 . . . . . . 7  |-  ( N  e.  RR  ->  4  e.  CC )
12 2cn 8979 . . . . . . . 8  |-  2  e.  CC
1312a1i 9 . . . . . . 7  |-  ( N  e.  RR  ->  2  e.  CC )
146, 11, 13addassd 7970 . . . . . 6  |-  ( N  e.  RR  ->  (
( N  +  4 )  +  2 )  =  ( N  +  ( 4  +  2 ) ) )
15 4p2e6 9051 . . . . . . 7  |-  ( 4  +  2 )  =  6
1615oveq2i 5880 . . . . . 6  |-  ( N  +  ( 4  +  2 ) )  =  ( N  +  6 )
1714, 16eqtrdi 2226 . . . . 5  |-  ( N  e.  RR  ->  (
( N  +  4 )  +  2 )  =  ( N  + 
6 ) )
1817breq1d 4010 . . . 4  |-  ( N  e.  RR  ->  (
( ( N  + 
4 )  +  2 )  <_  ( N  x.  2 )  <->  ( N  +  6 )  <_ 
( N  x.  2 ) ) )
1918adantr 276 . . 3  |-  ( ( N  e.  RR  /\  6  <_  N )  -> 
( ( ( N  +  4 )  +  2 )  <_  ( N  x.  2 )  <-> 
( N  +  6 )  <_  ( N  x.  2 ) ) )
209, 19mpbird 167 . 2  |-  ( ( N  e.  RR  /\  6  <_  N )  -> 
( ( N  + 
4 )  +  2 )  <_  ( N  x.  2 ) )
21 4re 8985 . . . . . . . 8  |-  4  e.  RR
2221a1i 9 . . . . . . 7  |-  ( N  e.  RR  ->  4  e.  RR )
23 4ap0 9007 . . . . . . . 8  |-  4 #  0
2423a1i 9 . . . . . . 7  |-  ( N  e.  RR  ->  4 #  0 )
253, 22, 24redivclapd 8781 . . . . . 6  |-  ( N  e.  RR  ->  ( N  /  4 )  e.  RR )
26 peano2re 8083 . . . . . 6  |-  ( ( N  /  4 )  e.  RR  ->  (
( N  /  4
)  +  1 )  e.  RR )
2725, 26syl 14 . . . . 5  |-  ( N  e.  RR  ->  (
( N  /  4
)  +  1 )  e.  RR )
28 peano2rem 8214 . . . . . 6  |-  ( N  e.  RR  ->  ( N  -  1 )  e.  RR )
2928rehalfcld 9154 . . . . 5  |-  ( N  e.  RR  ->  (
( N  -  1 )  /  2 )  e.  RR )
30 4pos 9005 . . . . . . 7  |-  0  <  4
3121, 30pm3.2i 272 . . . . . 6  |-  ( 4  e.  RR  /\  0  <  4 )
3231a1i 9 . . . . 5  |-  ( N  e.  RR  ->  (
4  e.  RR  /\  0  <  4 ) )
33 lemul1 8540 . . . . 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 1238 . . . 4  |-  ( N  e.  RR  ->  (
( ( N  / 
4 )  +  1 )  <_  ( ( N  -  1 )  /  2 )  <->  ( (
( N  /  4
)  +  1 )  x.  4 )  <_ 
( ( ( N  -  1 )  / 
2 )  x.  4 ) ) )
3525recnd 7976 . . . . . 6  |-  ( N  e.  RR  ->  ( N  /  4 )  e.  CC )
36 1cnd 7964 . . . . . 6  |-  ( N  e.  RR  ->  1  e.  CC )
376, 11, 24divcanap1d 8737 . . . . . . 7  |-  ( N  e.  RR  ->  (
( N  /  4
)  x.  4 )  =  N )
3810mulid2i 7951 . . . . . . . 8  |-  ( 1  x.  4 )  =  4
3938a1i 9 . . . . . . 7  |-  ( N  e.  RR  ->  (
1  x.  4 )  =  4 )
4037, 39oveq12d 5887 . . . . . 6  |-  ( N  e.  RR  ->  (
( ( N  / 
4 )  x.  4 )  +  ( 1  x.  4 ) )  =  ( N  + 
4 ) )
4135, 11, 36, 40joinlmuladdmuld 7975 . . . . 5  |-  ( N  e.  RR  ->  (
( ( N  / 
4 )  +  1 )  x.  4 )  =  ( N  + 
4 ) )
42 2t2e4 9062 . . . . . . . . 9  |-  ( 2  x.  2 )  =  4
4342eqcomi 2181 . . . . . . . 8  |-  4  =  ( 2  x.  2 )
4443a1i 9 . . . . . . 7  |-  ( N  e.  RR  ->  4  =  ( 2  x.  2 ) )
4544oveq2d 5885 . . . . . 6  |-  ( N  e.  RR  ->  (
( ( N  - 
1 )  /  2
)  x.  4 )  =  ( ( ( N  -  1 )  /  2 )  x.  ( 2  x.  2 ) ) )
4629recnd 7976 . . . . . . 7  |-  ( N  e.  RR  ->  (
( N  -  1 )  /  2 )  e.  CC )
47 mulass 7933 . . . . . . . 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 2183 . . . . . . 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 1238 . . . . . 6  |-  ( N  e.  RR  ->  (
( ( N  - 
1 )  /  2
)  x.  ( 2  x.  2 ) )  =  ( ( ( ( N  -  1 )  /  2 )  x.  2 )  x.  2 ) )
5028recnd 7976 . . . . . . . . 9  |-  ( N  e.  RR  ->  ( N  -  1 )  e.  CC )
51 2ap0 9001 . . . . . . . . . 10  |-  2 #  0
5251a1i 9 . . . . . . . . 9  |-  ( N  e.  RR  ->  2 #  0 )
5350, 13, 52divcanap1d 8737 . . . . . . . 8  |-  ( N  e.  RR  ->  (
( ( N  - 
1 )  /  2
)  x.  2 )  =  ( N  - 
1 ) )
5453oveq1d 5884 . . . . . . 7  |-  ( N  e.  RR  ->  (
( ( ( N  -  1 )  / 
2 )  x.  2 )  x.  2 )  =  ( ( N  -  1 )  x.  2 ) )
556, 36, 13subdird 8362 . . . . . . 7  |-  ( N  e.  RR  ->  (
( N  -  1 )  x.  2 )  =  ( ( N  x.  2 )  -  ( 1  x.  2 ) ) )
5612mulid2i 7951 . . . . . . . . 9  |-  ( 1  x.  2 )  =  2
5756a1i 9 . . . . . . . 8  |-  ( N  e.  RR  ->  (
1  x.  2 )  =  2 )
5857oveq2d 5885 . . . . . . 7  |-  ( N  e.  RR  ->  (
( N  x.  2 )  -  ( 1  x.  2 ) )  =  ( ( N  x.  2 )  - 
2 ) )
5954, 55, 583eqtrd 2214 . . . . . 6  |-  ( N  e.  RR  ->  (
( ( ( N  -  1 )  / 
2 )  x.  2 )  x.  2 )  =  ( ( N  x.  2 )  - 
2 ) )
6045, 49, 593eqtrd 2214 . . . . 5  |-  ( N  e.  RR  ->  (
( ( N  - 
1 )  /  2
)  x.  4 )  =  ( ( N  x.  2 )  - 
2 ) )
6141, 60breq12d 4013 . . . 4  |-  ( N  e.  RR  ->  (
( ( ( N  /  4 )  +  1 )  x.  4 )  <_  ( (
( N  -  1 )  /  2 )  x.  4 )  <->  ( N  +  4 )  <_ 
( ( N  x.  2 )  -  2 ) ) )
623, 22readdcld 7977 . . . . 5  |-  ( N  e.  RR  ->  ( N  +  4 )  e.  RR )
63 2re 8978 . . . . . 6  |-  2  e.  RR
6463a1i 9 . . . . 5  |-  ( N  e.  RR  ->  2  e.  RR )
653, 64remulcld 7978 . . . . 5  |-  ( N  e.  RR  ->  ( N  x.  2 )  e.  RR )
66 leaddsub 8385 . . . . . 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 141 . . . . 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 1238 . . . 4  |-  ( N  e.  RR  ->  (
( N  +  4 )  <_  ( ( N  x.  2 )  -  2 )  <->  ( ( N  +  4 )  +  2 )  <_ 
( N  x.  2 ) ) )
6934, 61, 683bitrd 214 . . 3  |-  ( N  e.  RR  ->  (
( ( N  / 
4 )  +  1 )  <_  ( ( N  -  1 )  /  2 )  <->  ( ( N  +  4 )  +  2 )  <_ 
( N  x.  2 ) ) )
7069adantr 276 . 2  |-  ( ( N  e.  RR  /\  6  <_  N )  -> 
( ( ( N  /  4 )  +  1 )  <_  (
( N  -  1 )  /  2 )  <-> 
( ( N  + 
4 )  +  2 )  <_  ( N  x.  2 ) ) )
7120, 70mpbird 167 1  |-  ( ( N  e.  RR  /\  6  <_  N )  -> 
( ( N  / 
4 )  +  1 )  <_  ( ( N  -  1 )  /  2 ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 104    <-> wb 105    /\ w3a 978    = wceq 1353    e. wcel 2148   class class class wbr 4000  (class class class)co 5869   CCcc 7800   RRcr 7801   0cc0 7802   1c1 7803    + caddc 7805    x. cmul 7807    < clt 7982    <_ cle 7983    - cmin 8118   # cap 8528    / cdiv 8618   2c2 8959   4c4 8961   6c6 8963
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-in1 614  ax-in2 615  ax-io 709  ax-5 1447  ax-7 1448  ax-gen 1449  ax-ie1 1493  ax-ie2 1494  ax-8 1504  ax-10 1505  ax-11 1506  ax-i12 1507  ax-bndl 1509  ax-4 1510  ax-17 1526  ax-i9 1530  ax-ial 1534  ax-i5r 1535  ax-13 2150  ax-14 2151  ax-ext 2159  ax-sep 4118  ax-pow 4171  ax-pr 4206  ax-un 4430  ax-setind 4533  ax-cnex 7893  ax-resscn 7894  ax-1cn 7895  ax-1re 7896  ax-icn 7897  ax-addcl 7898  ax-addrcl 7899  ax-mulcl 7900  ax-mulrcl 7901  ax-addcom 7902  ax-mulcom 7903  ax-addass 7904  ax-mulass 7905  ax-distr 7906  ax-i2m1 7907  ax-0lt1 7908  ax-1rid 7909  ax-0id 7910  ax-rnegex 7911  ax-precex 7912  ax-cnre 7913  ax-pre-ltirr 7914  ax-pre-ltwlin 7915  ax-pre-lttrn 7916  ax-pre-apti 7917  ax-pre-ltadd 7918  ax-pre-mulgt0 7919  ax-pre-mulext 7920
This theorem depends on definitions:  df-bi 117  df-3an 980  df-tru 1356  df-fal 1359  df-nf 1461  df-sb 1763  df-eu 2029  df-mo 2030  df-clab 2164  df-cleq 2170  df-clel 2173  df-nfc 2308  df-ne 2348  df-nel 2443  df-ral 2460  df-rex 2461  df-reu 2462  df-rmo 2463  df-rab 2464  df-v 2739  df-sbc 2963  df-dif 3131  df-un 3133  df-in 3135  df-ss 3142  df-pw 3576  df-sn 3597  df-pr 3598  df-op 3600  df-uni 3808  df-br 4001  df-opab 4062  df-id 4290  df-po 4293  df-iso 4294  df-xp 4629  df-rel 4630  df-cnv 4631  df-co 4632  df-dm 4633  df-iota 5174  df-fun 5214  df-fv 5220  df-riota 5825  df-ov 5872  df-oprab 5873  df-mpo 5874  df-pnf 7984  df-mnf 7985  df-xr 7986  df-ltxr 7987  df-le 7988  df-sub 8120  df-neg 8121  df-reap 8522  df-ap 8529  df-div 8619  df-2 8967  df-3 8968  df-4 8969  df-5 8970  df-6 8971
This theorem is referenced by:  fldiv4p1lem1div2  10291
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