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Theorem ceim1l 12074
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 9937 . . . . . 6  |-  ( A  e.  RR  ->  -u A  e.  RR )
2 reflcl 12032 . . . . . 6  |-  ( -u A  e.  RR  ->  ( |_ `  -u A
)  e.  RR )
31, 2syl 17 . . . . 5  |-  ( A  e.  RR  ->  ( |_ `  -u A )  e.  RR )
43recnd 9669 . . . 4  |-  ( A  e.  RR  ->  ( |_ `  -u A )  e.  CC )
5 ax-1cn 9597 . . . 4  |-  1  e.  CC
6 negdi 9931 . . . 4  |-  ( ( ( |_ `  -u A
)  e.  CC  /\  1  e.  CC )  -> 
-u ( ( |_
`  -u A )  +  1 )  =  (
-u ( |_ `  -u A )  +  -u
1 ) )
74, 5, 6sylancl 668 . . 3  |-  ( A  e.  RR  ->  -u (
( |_ `  -u A
)  +  1 )  =  ( -u ( |_ `  -u A )  + 
-u 1 ) )
84negcld 9973 . . . 4  |-  ( A  e.  RR  ->  -u ( |_ `  -u A )  e.  CC )
9 negsub 9922 . . . 4  |-  ( (
-u ( |_ `  -u A )  e.  CC  /\  1  e.  CC )  ->  ( -u ( |_ `  -u A )  + 
-u 1 )  =  ( -u ( |_
`  -u A )  - 
1 ) )
108, 5, 9sylancl 668 . . 3  |-  ( A  e.  RR  ->  ( -u ( |_ `  -u A
)  +  -u 1
)  =  ( -u ( |_ `  -u A
)  -  1 ) )
117, 10eqtr2d 2486 . 2  |-  ( A  e.  RR  ->  ( -u ( |_ `  -u A
)  -  1 )  =  -u ( ( |_
`  -u A )  +  1 ) )
12 peano2re 9806 . . . 4  |-  ( ( |_ `  -u A
)  e.  RR  ->  ( ( |_ `  -u A
)  +  1 )  e.  RR )
133, 12syl 17 . . 3  |-  ( A  e.  RR  ->  (
( |_ `  -u A
)  +  1 )  e.  RR )
14 flltp1 12036 . . . . . 6  |-  ( -u A  e.  RR  ->  -u A  <  ( ( |_
`  -u A )  +  1 ) )
151, 14syl 17 . . . . 5  |-  ( A  e.  RR  ->  -u A  <  ( ( |_ `  -u A )  +  1 ) )
1615adantr 467 . . . 4  |-  ( ( A  e.  RR  /\  ( ( |_ `  -u A )  +  1 )  e.  RR )  ->  -u A  <  (
( |_ `  -u A
)  +  1 ) )
17 ltnegcon1 10115 . . . 4  |-  ( ( A  e.  RR  /\  ( ( |_ `  -u A )  +  1 )  e.  RR )  ->  ( -u A  <  ( ( |_ `  -u A )  +  1 )  <->  -u ( ( |_
`  -u A )  +  1 )  <  A
) )
1816, 17mpbid 214 . . 3  |-  ( ( A  e.  RR  /\  ( ( |_ `  -u A )  +  1 )  e.  RR )  ->  -u ( ( |_
`  -u A )  +  1 )  <  A
)
1913, 18mpdan 674 . 2  |-  ( A  e.  RR  ->  -u (
( |_ `  -u A
)  +  1 )  <  A )
2011, 19eqbrtrd 4423 1  |-  ( A  e.  RR  ->  ( -u ( |_ `  -u A
)  -  1 )  <  A )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    /\ wa 371    = wceq 1444    e. wcel 1887   class class class wbr 4402   ` cfv 5582  (class class class)co 6290   CCcc 9537   RRcr 9538   1c1 9540    + caddc 9542    < clt 9675    - cmin 9860   -ucneg 9861   |_cfl 12026
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1669  ax-4 1682  ax-5 1758  ax-6 1805  ax-7 1851  ax-8 1889  ax-9 1896  ax-10 1915  ax-11 1920  ax-12 1933  ax-13 2091  ax-ext 2431  ax-sep 4525  ax-nul 4534  ax-pow 4581  ax-pr 4639  ax-un 6583  ax-cnex 9595  ax-resscn 9596  ax-1cn 9597  ax-icn 9598  ax-addcl 9599  ax-addrcl 9600  ax-mulcl 9601  ax-mulrcl 9602  ax-mulcom 9603  ax-addass 9604  ax-mulass 9605  ax-distr 9606  ax-i2m1 9607  ax-1ne0 9608  ax-1rid 9609  ax-rnegex 9610  ax-rrecex 9611  ax-cnre 9612  ax-pre-lttri 9613  ax-pre-lttrn 9614  ax-pre-ltadd 9615  ax-pre-mulgt0 9616  ax-pre-sup 9617
This theorem depends on definitions:  df-bi 189  df-or 372  df-an 373  df-3or 986  df-3an 987  df-tru 1447  df-ex 1664  df-nf 1668  df-sb 1798  df-eu 2303  df-mo 2304  df-clab 2438  df-cleq 2444  df-clel 2447  df-nfc 2581  df-ne 2624  df-nel 2625  df-ral 2742  df-rex 2743  df-reu 2744  df-rmo 2745  df-rab 2746  df-v 3047  df-sbc 3268  df-csb 3364  df-dif 3407  df-un 3409  df-in 3411  df-ss 3418  df-pss 3420  df-nul 3732  df-if 3882  df-pw 3953  df-sn 3969  df-pr 3971  df-tp 3973  df-op 3975  df-uni 4199  df-iun 4280  df-br 4403  df-opab 4462  df-mpt 4463  df-tr 4498  df-eprel 4745  df-id 4749  df-po 4755  df-so 4756  df-fr 4793  df-we 4795  df-xp 4840  df-rel 4841  df-cnv 4842  df-co 4843  df-dm 4844  df-rn 4845  df-res 4846  df-ima 4847  df-pred 5380  df-ord 5426  df-on 5427  df-lim 5428  df-suc 5429  df-iota 5546  df-fun 5584  df-fn 5585  df-f 5586  df-f1 5587  df-fo 5588  df-f1o 5589  df-fv 5590  df-riota 6252  df-ov 6293  df-oprab 6294  df-mpt2 6295  df-om 6693  df-wrecs 7028  df-recs 7090  df-rdg 7128  df-er 7363  df-en 7570  df-dom 7571  df-sdom 7572  df-sup 7956  df-inf 7957  df-pnf 9677  df-mnf 9678  df-xr 9679  df-ltxr 9680  df-le 9681  df-sub 9862  df-neg 9863  df-nn 10610  df-n0 10870  df-z 10938  df-uz 11160  df-fl 12028
This theorem is referenced by:  ceilm1lt  12075  ceile  12076  ltflcei  31933
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