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Theorem ceim1l 12646
Description: One less than the ceiling of a real number is strictly less than that number. (Contributed by Jeff Hankins, 10-Jun-2007.)
Assertion
Ref Expression
ceim1l  |-  ( A  e.  RR  ->  ( -u ( |_ `  -u A
)  -  1 )  <  A )

Proof of Theorem ceim1l
StepHypRef Expression
1 renegcl 10344 . . . . . 6  |-  ( A  e.  RR  ->  -u A  e.  RR )
2 reflcl 12597 . . . . . 6  |-  ( -u A  e.  RR  ->  ( |_ `  -u A
)  e.  RR )
31, 2syl 17 . . . . 5  |-  ( A  e.  RR  ->  ( |_ `  -u A )  e.  RR )
43recnd 10068 . . . 4  |-  ( A  e.  RR  ->  ( |_ `  -u A )  e.  CC )
5 ax-1cn 9994 . . . 4  |-  1  e.  CC
6 negdi 10338 . . . 4  |-  ( ( ( |_ `  -u A
)  e.  CC  /\  1  e.  CC )  -> 
-u ( ( |_
`  -u A )  +  1 )  =  (
-u ( |_ `  -u A )  +  -u
1 ) )
74, 5, 6sylancl 694 . . 3  |-  ( A  e.  RR  ->  -u (
( |_ `  -u A
)  +  1 )  =  ( -u ( |_ `  -u A )  + 
-u 1 ) )
84negcld 10379 . . . 4  |-  ( A  e.  RR  ->  -u ( |_ `  -u A )  e.  CC )
9 negsub 10329 . . . 4  |-  ( (
-u ( |_ `  -u A )  e.  CC  /\  1  e.  CC )  ->  ( -u ( |_ `  -u A )  + 
-u 1 )  =  ( -u ( |_
`  -u A )  - 
1 ) )
108, 5, 9sylancl 694 . . 3  |-  ( A  e.  RR  ->  ( -u ( |_ `  -u A
)  +  -u 1
)  =  ( -u ( |_ `  -u A
)  -  1 ) )
117, 10eqtr2d 2657 . 2  |-  ( A  e.  RR  ->  ( -u ( |_ `  -u A
)  -  1 )  =  -u ( ( |_
`  -u A )  +  1 ) )
12 peano2re 10209 . . . 4  |-  ( ( |_ `  -u A
)  e.  RR  ->  ( ( |_ `  -u A
)  +  1 )  e.  RR )
133, 12syl 17 . . 3  |-  ( A  e.  RR  ->  (
( |_ `  -u A
)  +  1 )  e.  RR )
14 flltp1 12601 . . . . . 6  |-  ( -u A  e.  RR  ->  -u A  <  ( ( |_
`  -u A )  +  1 ) )
151, 14syl 17 . . . . 5  |-  ( A  e.  RR  ->  -u A  <  ( ( |_ `  -u A )  +  1 ) )
1615adantr 481 . . . 4  |-  ( ( A  e.  RR  /\  ( ( |_ `  -u A )  +  1 )  e.  RR )  ->  -u A  <  (
( |_ `  -u A
)  +  1 ) )
17 ltnegcon1 10529 . . . 4  |-  ( ( A  e.  RR  /\  ( ( |_ `  -u A )  +  1 )  e.  RR )  ->  ( -u A  <  ( ( |_ `  -u A )  +  1 )  <->  -u ( ( |_
`  -u A )  +  1 )  <  A
) )
1816, 17mpbid 222 . . 3  |-  ( ( A  e.  RR  /\  ( ( |_ `  -u A )  +  1 )  e.  RR )  ->  -u ( ( |_
`  -u A )  +  1 )  <  A
)
1913, 18mpdan 702 . 2  |-  ( A  e.  RR  ->  -u (
( |_ `  -u A
)  +  1 )  <  A )
2011, 19eqbrtrd 4675 1  |-  ( A  e.  RR  ->  ( -u ( |_ `  -u A
)  -  1 )  <  A )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    /\ wa 384    = wceq 1483    e. wcel 1990   class class class wbr 4653   ` cfv 5888  (class class class)co 6650   CCcc 9934   RRcr 9935   1c1 9937    + caddc 9939    < clt 10074    - cmin 10266   -ucneg 10267   |_cfl 12591
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-8 1992  ax-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246  ax-ext 2602  ax-sep 4781  ax-nul 4789  ax-pow 4843  ax-pr 4906  ax-un 6949  ax-cnex 9992  ax-resscn 9993  ax-1cn 9994  ax-icn 9995  ax-addcl 9996  ax-addrcl 9997  ax-mulcl 9998  ax-mulrcl 9999  ax-mulcom 10000  ax-addass 10001  ax-mulass 10002  ax-distr 10003  ax-i2m1 10004  ax-1ne0 10005  ax-1rid 10006  ax-rnegex 10007  ax-rrecex 10008  ax-cnre 10009  ax-pre-lttri 10010  ax-pre-lttrn 10011  ax-pre-ltadd 10012  ax-pre-mulgt0 10013  ax-pre-sup 10014
This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3or 1038  df-3an 1039  df-tru 1486  df-ex 1705  df-nf 1710  df-sb 1881  df-eu 2474  df-mo 2475  df-clab 2609  df-cleq 2615  df-clel 2618  df-nfc 2753  df-ne 2795  df-nel 2898  df-ral 2917  df-rex 2918  df-reu 2919  df-rmo 2920  df-rab 2921  df-v 3202  df-sbc 3436  df-csb 3534  df-dif 3577  df-un 3579  df-in 3581  df-ss 3588  df-pss 3590  df-nul 3916  df-if 4087  df-pw 4160  df-sn 4178  df-pr 4180  df-tp 4182  df-op 4184  df-uni 4437  df-iun 4522  df-br 4654  df-opab 4713  df-mpt 4730  df-tr 4753  df-id 5024  df-eprel 5029  df-po 5035  df-so 5036  df-fr 5073  df-we 5075  df-xp 5120  df-rel 5121  df-cnv 5122  df-co 5123  df-dm 5124  df-rn 5125  df-res 5126  df-ima 5127  df-pred 5680  df-ord 5726  df-on 5727  df-lim 5728  df-suc 5729  df-iota 5851  df-fun 5890  df-fn 5891  df-f 5892  df-f1 5893  df-fo 5894  df-f1o 5895  df-fv 5896  df-riota 6611  df-ov 6653  df-oprab 6654  df-mpt2 6655  df-om 7066  df-wrecs 7407  df-recs 7468  df-rdg 7506  df-er 7742  df-en 7956  df-dom 7957  df-sdom 7958  df-sup 8348  df-inf 8349  df-pnf 10076  df-mnf 10077  df-xr 10078  df-ltxr 10079  df-le 10080  df-sub 10268  df-neg 10269  df-nn 11021  df-n0 11293  df-z 11378  df-uz 11688  df-fl 12593
This theorem is referenced by:  ceilm1lt  12647  ceile  12648  ltflcei  33397
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