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Theorem dfceil2 11948
Description: Alternative definition of the ceiling function using restricted iota. (Contributed by AV, 1-Dec-2018.)
Assertion
Ref Expression
dfceil2  |- =  ( x  e.  RR  |->  (
iota_ y  e.  ZZ  ( x  <_  y  /\  y  <  ( x  + 
1 ) ) ) )
Distinct variable group:    x, y

Proof of Theorem dfceil2
Dummy variable  z is distinct from all other variables.
StepHypRef Expression
1 df-ceil 11910 . 2  |- =  ( x  e.  RR  |->  -u ( |_ `  -u x
) )
2 zre 10880 . . . . . . 7  |-  ( z  e.  ZZ  ->  z  e.  RR )
3 lenegcon2 10069 . . . . . . . 8  |-  ( ( x  e.  RR  /\  z  e.  RR )  ->  ( x  <_  -u z  <->  z  <_  -u x ) )
4 peano2re 9764 . . . . . . . . . . . 12  |-  ( x  e.  RR  ->  (
x  +  1 )  e.  RR )
54anim2i 569 . . . . . . . . . . 11  |-  ( ( z  e.  RR  /\  x  e.  RR )  ->  ( z  e.  RR  /\  ( x  +  1 )  e.  RR ) )
65ancoms 453 . . . . . . . . . 10  |-  ( ( x  e.  RR  /\  z  e.  RR )  ->  ( z  e.  RR  /\  ( x  +  1 )  e.  RR ) )
7 ltnegcon1 10065 . . . . . . . . . 10  |-  ( ( z  e.  RR  /\  ( x  +  1
)  e.  RR )  ->  ( -u z  <  ( x  +  1 )  <->  -u ( x  + 
1 )  <  z
) )
86, 7syl 16 . . . . . . . . 9  |-  ( ( x  e.  RR  /\  z  e.  RR )  ->  ( -u z  < 
( x  +  1 )  <->  -u ( x  + 
1 )  <  z
) )
9 recn 9594 . . . . . . . . . . . 12  |-  ( x  e.  RR  ->  x  e.  CC )
10 ax-1cn 9562 . . . . . . . . . . . . 13  |-  1  e.  CC
1110a1i 11 . . . . . . . . . . . 12  |-  ( x  e.  RR  ->  1  e.  CC )
129, 11negdid 9955 . . . . . . . . . . 11  |-  ( x  e.  RR  ->  -u (
x  +  1 )  =  ( -u x  +  -u 1 ) )
1312adantr 465 . . . . . . . . . 10  |-  ( ( x  e.  RR  /\  z  e.  RR )  -> 
-u ( x  + 
1 )  =  (
-u x  +  -u
1 ) )
1413breq1d 4463 . . . . . . . . 9  |-  ( ( x  e.  RR  /\  z  e.  RR )  ->  ( -u ( x  +  1 )  < 
z  <->  ( -u x  +  -u 1 )  < 
z ) )
15 renegcl 9894 . . . . . . . . . . . 12  |-  ( x  e.  RR  ->  -u x  e.  RR )
1615adantr 465 . . . . . . . . . . 11  |-  ( ( x  e.  RR  /\  z  e.  RR )  -> 
-u x  e.  RR )
17 neg1rr 10652 . . . . . . . . . . . 12  |-  -u 1  e.  RR
1817a1i 11 . . . . . . . . . . 11  |-  ( ( x  e.  RR  /\  z  e.  RR )  -> 
-u 1  e.  RR )
19 simpr 461 . . . . . . . . . . 11  |-  ( ( x  e.  RR  /\  z  e.  RR )  ->  z  e.  RR )
2016, 18, 19ltaddsubd 10164 . . . . . . . . . 10  |-  ( ( x  e.  RR  /\  z  e.  RR )  ->  ( ( -u x  +  -u 1 )  < 
z  <->  -u x  <  (
z  -  -u 1
) ) )
21 recn 9594 . . . . . . . . . . . . 13  |-  ( z  e.  RR  ->  z  e.  CC )
2210a1i 11 . . . . . . . . . . . . 13  |-  ( z  e.  RR  ->  1  e.  CC )
2321, 22subnegd 9949 . . . . . . . . . . . 12  |-  ( z  e.  RR  ->  (
z  -  -u 1
)  =  ( z  +  1 ) )
2423adantl 466 . . . . . . . . . . 11  |-  ( ( x  e.  RR  /\  z  e.  RR )  ->  ( z  -  -u 1
)  =  ( z  +  1 ) )
2524breq2d 4465 . . . . . . . . . 10  |-  ( ( x  e.  RR  /\  z  e.  RR )  ->  ( -u x  < 
( z  -  -u 1
)  <->  -u x  <  (
z  +  1 ) ) )
2620, 25bitrd 253 . . . . . . . . 9  |-  ( ( x  e.  RR  /\  z  e.  RR )  ->  ( ( -u x  +  -u 1 )  < 
z  <->  -u x  <  (
z  +  1 ) ) )
278, 14, 263bitrd 279 . . . . . . . 8  |-  ( ( x  e.  RR  /\  z  e.  RR )  ->  ( -u z  < 
( x  +  1 )  <->  -u x  <  (
z  +  1 ) ) )
283, 27anbi12d 710 . . . . . . 7  |-  ( ( x  e.  RR  /\  z  e.  RR )  ->  ( ( x  <_  -u z  /\  -u z  <  ( x  +  1 ) )  <->  ( z  <_ 
-u x  /\  -u x  <  ( z  +  1 ) ) ) )
292, 28sylan2 474 . . . . . 6  |-  ( ( x  e.  RR  /\  z  e.  ZZ )  ->  ( ( x  <_  -u z  /\  -u z  <  ( x  +  1 ) )  <->  ( z  <_ 
-u x  /\  -u x  <  ( z  +  1 ) ) ) )
3029riotabidva 6273 . . . . 5  |-  ( x  e.  RR  ->  ( iota_ z  e.  ZZ  (
x  <_  -u z  /\  -u z  <  ( x  +  1 ) ) )  =  ( iota_ z  e.  ZZ  ( z  <_  -u x  /\  -u x  <  ( z  +  1 ) ) ) )
3130negeqd 9826 . . . 4  |-  ( x  e.  RR  ->  -u ( iota_ z  e.  ZZ  (
x  <_  -u z  /\  -u z  <  ( x  +  1 ) ) )  =  -u ( iota_ z  e.  ZZ  (
z  <_  -u x  /\  -u x  <  ( z  +  1 ) ) ) )
32 zbtwnre 11192 . . . . 5  |-  ( x  e.  RR  ->  E! y  e.  ZZ  (
x  <_  y  /\  y  <  ( x  + 
1 ) ) )
33 breq2 4457 . . . . . . 7  |-  ( y  =  -u z  ->  (
x  <_  y  <->  x  <_  -u z ) )
34 breq1 4456 . . . . . . 7  |-  ( y  =  -u z  ->  (
y  <  ( x  +  1 )  <->  -u z  < 
( x  +  1 ) ) )
3533, 34anbi12d 710 . . . . . 6  |-  ( y  =  -u z  ->  (
( x  <_  y  /\  y  <  ( x  +  1 ) )  <-> 
( x  <_  -u z  /\  -u z  <  (
x  +  1 ) ) ) )
3635zriotaneg 10986 . . . . 5  |-  ( E! y  e.  ZZ  (
x  <_  y  /\  y  <  ( x  + 
1 ) )  -> 
( iota_ y  e.  ZZ  ( x  <_  y  /\  y  <  ( x  + 
1 ) ) )  =  -u ( iota_ z  e.  ZZ  ( x  <_  -u z  /\  -u z  <  ( x  +  1 ) ) ) )
3732, 36syl 16 . . . 4  |-  ( x  e.  RR  ->  ( iota_ y  e.  ZZ  (
x  <_  y  /\  y  <  ( x  + 
1 ) ) )  =  -u ( iota_ z  e.  ZZ  ( x  <_  -u z  /\  -u z  <  ( x  +  1 ) ) ) )
38 flval 11911 . . . . . 6  |-  ( -u x  e.  RR  ->  ( |_ `  -u x
)  =  ( iota_ z  e.  ZZ  ( z  <_  -u x  /\  -u x  <  ( z  +  1 ) ) ) )
3915, 38syl 16 . . . . 5  |-  ( x  e.  RR  ->  ( |_ `  -u x )  =  ( iota_ z  e.  ZZ  ( z  <_  -u x  /\  -u x  <  (
z  +  1 ) ) ) )
4039negeqd 9826 . . . 4  |-  ( x  e.  RR  ->  -u ( |_ `  -u x )  = 
-u ( iota_ z  e.  ZZ  ( z  <_  -u x  /\  -u x  <  ( z  +  1 ) ) ) )
4131, 37, 403eqtr4rd 2519 . . 3  |-  ( x  e.  RR  ->  -u ( |_ `  -u x )  =  ( iota_ y  e.  ZZ  ( x  <_  y  /\  y  <  ( x  + 
1 ) ) ) )
4241mpteq2ia 4535 . 2  |-  ( x  e.  RR  |->  -u ( |_ `  -u x ) )  =  ( x  e.  RR  |->  ( iota_ y  e.  ZZ  ( x  <_ 
y  /\  y  <  ( x  +  1 ) ) ) )
431, 42eqtri 2496 1  |- =  ( x  e.  RR  |->  (
iota_ y  e.  ZZ  ( x  <_  y  /\  y  <  ( x  + 
1 ) ) ) )
Colors of variables: wff setvar class
Syntax hints:    <-> wb 184    /\ wa 369    = wceq 1379    e. wcel 1767   E!wreu 2819   class class class wbr 4453    |-> cmpt 4511   ` cfv 5594   iota_crio 6255  (class class class)co 6295   CCcc 9502   RRcr 9503   1c1 9505    + caddc 9507    < clt 9640    <_ cle 9641    - cmin 9817   -ucneg 9818   ZZcz 10876   |_cfl 11907  ⌈cceil 11908
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  df-ceil 11910
This theorem is referenced by:  ceilval2  11949
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