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Theorem ceim1l 11954
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 9894 . . . . . 6  |-  ( A  e.  RR  ->  -u A  e.  RR )
2 reflcl 11913 . . . . . 6  |-  ( -u A  e.  RR  ->  ( |_ `  -u A
)  e.  RR )
31, 2syl 16 . . . . 5  |-  ( A  e.  RR  ->  ( |_ `  -u A )  e.  RR )
43recnd 9634 . . . 4  |-  ( A  e.  RR  ->  ( |_ `  -u A )  e.  CC )
5 ax-1cn 9562 . . . 4  |-  1  e.  CC
6 negdi 9888 . . . 4  |-  ( ( ( |_ `  -u A
)  e.  CC  /\  1  e.  CC )  -> 
-u ( ( |_
`  -u A )  +  1 )  =  (
-u ( |_ `  -u A )  +  -u
1 ) )
74, 5, 6sylancl 662 . . 3  |-  ( A  e.  RR  ->  -u (
( |_ `  -u A
)  +  1 )  =  ( -u ( |_ `  -u A )  + 
-u 1 ) )
84negcld 9929 . . . 4  |-  ( A  e.  RR  ->  -u ( |_ `  -u A )  e.  CC )
9 negsub 9879 . . . 4  |-  ( (
-u ( |_ `  -u A )  e.  CC  /\  1  e.  CC )  ->  ( -u ( |_ `  -u A )  + 
-u 1 )  =  ( -u ( |_
`  -u A )  - 
1 ) )
108, 5, 9sylancl 662 . . 3  |-  ( A  e.  RR  ->  ( -u ( |_ `  -u A
)  +  -u 1
)  =  ( -u ( |_ `  -u A
)  -  1 ) )
117, 10eqtr2d 2509 . 2  |-  ( A  e.  RR  ->  ( -u ( |_ `  -u A
)  -  1 )  =  -u ( ( |_
`  -u A )  +  1 ) )
12 peano2re 9764 . . . 4  |-  ( ( |_ `  -u A
)  e.  RR  ->  ( ( |_ `  -u A
)  +  1 )  e.  RR )
133, 12syl 16 . . 3  |-  ( A  e.  RR  ->  (
( |_ `  -u A
)  +  1 )  e.  RR )
14 flltp1 11917 . . . . . 6  |-  ( -u A  e.  RR  ->  -u A  <  ( ( |_
`  -u A )  +  1 ) )
151, 14syl 16 . . . . 5  |-  ( A  e.  RR  ->  -u A  <  ( ( |_ `  -u A )  +  1 ) )
1615adantr 465 . . . 4  |-  ( ( A  e.  RR  /\  ( ( |_ `  -u A )  +  1 )  e.  RR )  ->  -u A  <  (
( |_ `  -u A
)  +  1 ) )
17 ltnegcon1 10065 . . . 4  |-  ( ( A  e.  RR  /\  ( ( |_ `  -u A )  +  1 )  e.  RR )  ->  ( -u A  <  ( ( |_ `  -u A )  +  1 )  <->  -u ( ( |_
`  -u A )  +  1 )  <  A
) )
1816, 17mpbid 210 . . 3  |-  ( ( A  e.  RR  /\  ( ( |_ `  -u A )  +  1 )  e.  RR )  ->  -u ( ( |_
`  -u A )  +  1 )  <  A
)
1913, 18mpdan 668 . 2  |-  ( A  e.  RR  ->  -u (
( |_ `  -u A
)  +  1 )  <  A )
2011, 19eqbrtrd 4473 1  |-  ( A  e.  RR  ->  ( -u ( |_ `  -u A
)  -  1 )  <  A )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    /\ wa 369    = wceq 1379    e. wcel 1767   class class class wbr 4453   ` cfv 5594  (class class class)co 6295   CCcc 9502   RRcr 9503   1c1 9505    + caddc 9507    < clt 9640    - cmin 9817   -ucneg 9818   |_cfl 11907
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1601  ax-4 1612  ax-5 1680  ax-6 1719  ax-7 1739  ax-8 1769  ax-9 1771  ax-10 1786  ax-11 1791  ax-12 1803  ax-13 1968  ax-ext 2445  ax-sep 4574  ax-nul 4582  ax-pow 4631  ax-pr 4692  ax-un 6587  ax-cnex 9560  ax-resscn 9561  ax-1cn 9562  ax-icn 9563  ax-addcl 9564  ax-addrcl 9565  ax-mulcl 9566  ax-mulrcl 9567  ax-mulcom 9568  ax-addass 9569  ax-mulass 9570  ax-distr 9571  ax-i2m1 9572  ax-1ne0 9573  ax-1rid 9574  ax-rnegex 9575  ax-rrecex 9576  ax-cnre 9577  ax-pre-lttri 9578  ax-pre-lttrn 9579  ax-pre-ltadd 9580  ax-pre-mulgt0 9581  ax-pre-sup 9582
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 974  df-3an 975  df-tru 1382  df-ex 1597  df-nf 1600  df-sb 1712  df-eu 2279  df-mo 2280  df-clab 2453  df-cleq 2459  df-clel 2462  df-nfc 2617  df-ne 2664  df-nel 2665  df-ral 2822  df-rex 2823  df-reu 2824  df-rmo 2825  df-rab 2826  df-v 3120  df-sbc 3337  df-csb 3441  df-dif 3484  df-un 3486  df-in 3488  df-ss 3495  df-pss 3497  df-nul 3791  df-if 3946  df-pw 4018  df-sn 4034  df-pr 4036  df-tp 4038  df-op 4040  df-uni 4252  df-iun 4333  df-br 4454  df-opab 4512  df-mpt 4513  df-tr 4547  df-eprel 4797  df-id 4801  df-po 4806  df-so 4807  df-fr 4844  df-we 4846  df-ord 4887  df-on 4888  df-lim 4889  df-suc 4890  df-xp 5011  df-rel 5012  df-cnv 5013  df-co 5014  df-dm 5015  df-rn 5016  df-res 5017  df-ima 5018  df-iota 5557  df-fun 5596  df-fn 5597  df-f 5598  df-f1 5599  df-fo 5600  df-f1o 5601  df-fv 5602  df-riota 6256  df-ov 6298  df-oprab 6299  df-mpt2 6300  df-om 6696  df-recs 7054  df-rdg 7088  df-er 7323  df-en 7529  df-dom 7530  df-sdom 7531  df-sup 7913  df-pnf 9642  df-mnf 9643  df-xr 9644  df-ltxr 9645  df-le 9646  df-sub 9819  df-neg 9820  df-nn 10549  df-n0 10808  df-z 10877  df-uz 11095  df-fl 11909
This theorem is referenced by:  ceilm1lt  11955  ceile  11956  ltflcei  29970
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