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Theorem dfceil2 11676
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 11639 . 2  |- =  ( x  e.  RR  |->  -u ( |_ `  -u x
) )
2 zre 10646 . . . . . . 7  |-  ( z  e.  ZZ  ->  z  e.  RR )
3 lenegcon2 9840 . . . . . . . 8  |-  ( ( x  e.  RR  /\  z  e.  RR )  ->  ( x  <_  -u z  <->  z  <_  -u x ) )
4 peano2re 9538 . . . . . . . . . . . 12  |-  ( x  e.  RR  ->  (
x  +  1 )  e.  RR )
54anim2i 566 . . . . . . . . . . 11  |-  ( ( z  e.  RR  /\  x  e.  RR )  ->  ( z  e.  RR  /\  ( x  +  1 )  e.  RR ) )
65ancoms 450 . . . . . . . . . 10  |-  ( ( x  e.  RR  /\  z  e.  RR )  ->  ( z  e.  RR  /\  ( x  +  1 )  e.  RR ) )
7 ltnegcon1 9836 . . . . . . . . . 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 9368 . . . . . . . . . . . 12  |-  ( x  e.  RR  ->  x  e.  CC )
10 ax-1cn 9336 . . . . . . . . . . . . 13  |-  1  e.  CC
1110a1i 11 . . . . . . . . . . . 12  |-  ( x  e.  RR  ->  1  e.  CC )
129, 11negdid 9728 . . . . . . . . . . 11  |-  ( x  e.  RR  ->  -u (
x  +  1 )  =  ( -u x  +  -u 1 ) )
1312adantr 462 . . . . . . . . . 10  |-  ( ( x  e.  RR  /\  z  e.  RR )  -> 
-u ( x  + 
1 )  =  (
-u x  +  -u
1 ) )
1413breq1d 4299 . . . . . . . . 9  |-  ( ( x  e.  RR  /\  z  e.  RR )  ->  ( -u ( x  +  1 )  < 
z  <->  ( -u x  +  -u 1 )  < 
z ) )
15 renegcl 9668 . . . . . . . . . . . 12  |-  ( x  e.  RR  ->  -u x  e.  RR )
1615adantr 462 . . . . . . . . . . 11  |-  ( ( x  e.  RR  /\  z  e.  RR )  -> 
-u x  e.  RR )
17 neg1rr 10422 . . . . . . . . . . . 12  |-  -u 1  e.  RR
1817a1i 11 . . . . . . . . . . 11  |-  ( ( x  e.  RR  /\  z  e.  RR )  -> 
-u 1  e.  RR )
19 simpr 458 . . . . . . . . . . 11  |-  ( ( x  e.  RR  /\  z  e.  RR )  ->  z  e.  RR )
2016, 18, 19ltaddsubd 9935 . . . . . . . . . 10  |-  ( ( x  e.  RR  /\  z  e.  RR )  ->  ( ( -u x  +  -u 1 )  < 
z  <->  -u x  <  (
z  -  -u 1
) ) )
21 recn 9368 . . . . . . . . . . . . 13  |-  ( z  e.  RR  ->  z  e.  CC )
2210a1i 11 . . . . . . . . . . . . 13  |-  ( z  e.  RR  ->  1  e.  CC )
2321, 22subnegd 9722 . . . . . . . . . . . 12  |-  ( z  e.  RR  ->  (
z  -  -u 1
)  =  ( z  +  1 ) )
2423adantl 463 . . . . . . . . . . 11  |-  ( ( x  e.  RR  /\  z  e.  RR )  ->  ( z  -  -u 1
)  =  ( z  +  1 ) )
2524breq2d 4301 . . . . . . . . . 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 705 . . . . . . 7  |-  ( ( x  e.  RR  /\  z  e.  RR )  ->  ( ( x  <_  -u z  /\  -u z  <  ( x  +  1 ) )  <->  ( z  <_ 
-u x  /\  -u x  <  ( z  +  1 ) ) ) )
292, 28sylan2 471 . . . . . 6  |-  ( ( x  e.  RR  /\  z  e.  ZZ )  ->  ( ( x  <_  -u z  /\  -u z  <  ( x  +  1 ) )  <->  ( z  <_ 
-u x  /\  -u x  <  ( z  +  1 ) ) ) )
3029riotabidva 6067 . . . . 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 9600 . . . 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 10947 . . . . 5  |-  ( x  e.  RR  ->  E! y  e.  ZZ  (
x  <_  y  /\  y  <  ( x  + 
1 ) ) )
33 breq2 4293 . . . . . . 7  |-  ( y  =  -u z  ->  (
x  <_  y  <->  x  <_  -u z ) )
34 breq1 4292 . . . . . . 7  |-  ( y  =  -u z  ->  (
y  <  ( x  +  1 )  <->  -u z  < 
( x  +  1 ) ) )
3533, 34anbi12d 705 . . . . . 6  |-  ( y  =  -u z  ->  (
( x  <_  y  /\  y  <  ( x  +  1 ) )  <-> 
( x  <_  -u z  /\  -u z  <  (
x  +  1 ) ) ) )
3635zriotaneg 10750 . . . . 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 11640 . . . . . 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 9600 . . . 4  |-  ( x  e.  RR  ->  -u ( |_ `  -u x )  = 
-u ( iota_ z  e.  ZZ  ( z  <_  -u x  /\  -u x  <  ( z  +  1 ) ) ) )
4131, 37, 403eqtr4rd 2484 . . 3  |-  ( x  e.  RR  ->  -u ( |_ `  -u x )  =  ( iota_ y  e.  ZZ  ( x  <_  y  /\  y  <  ( x  + 
1 ) ) ) )
4241mpteq2ia 4371 . 2  |-  ( x  e.  RR  |->  -u ( |_ `  -u x ) )  =  ( x  e.  RR  |->  ( iota_ y  e.  ZZ  ( x  <_ 
y  /\  y  <  ( x  +  1 ) ) ) )
431, 42eqtri 2461 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 1364    e. wcel 1761   E!wreu 2715   class class class wbr 4289    e. cmpt 4347   ` cfv 5415   iota_crio 6048  (class class class)co 6090   CCcc 9276   RRcr 9277   1c1 9279    + caddc 9281    < clt 9414    <_ cle 9415    - cmin 9591   -ucneg 9592   ZZcz 10642   |_cfl 11636  ⌈cceil 11637
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1596  ax-4 1607  ax-5 1675  ax-6 1713  ax-7 1733  ax-8 1763  ax-9 1765  ax-10 1780  ax-11 1785  ax-12 1797  ax-13 1948  ax-ext 2422  ax-sep 4410  ax-nul 4418  ax-pow 4467  ax-pr 4528  ax-un 6371  ax-cnex 9334  ax-resscn 9335  ax-1cn 9336  ax-icn 9337  ax-addcl 9338  ax-addrcl 9339  ax-mulcl 9340  ax-mulrcl 9341  ax-mulcom 9342  ax-addass 9343  ax-mulass 9344  ax-distr 9345  ax-i2m1 9346  ax-1ne0 9347  ax-1rid 9348  ax-rnegex 9349  ax-rrecex 9350  ax-cnre 9351  ax-pre-lttri 9352  ax-pre-lttrn 9353  ax-pre-ltadd 9354  ax-pre-mulgt0 9355  ax-pre-sup 9356
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 961  df-3an 962  df-tru 1367  df-ex 1592  df-nf 1595  df-sb 1706  df-eu 2261  df-mo 2262  df-clab 2428  df-cleq 2434  df-clel 2437  df-nfc 2566  df-ne 2606  df-nel 2607  df-ral 2718  df-rex 2719  df-reu 2720  df-rmo 2721  df-rab 2722  df-v 2972  df-sbc 3184  df-csb 3286  df-dif 3328  df-un 3330  df-in 3332  df-ss 3339  df-pss 3341  df-nul 3635  df-if 3789  df-pw 3859  df-sn 3875  df-pr 3877  df-tp 3879  df-op 3881  df-uni 4089  df-iun 4170  df-br 4290  df-opab 4348  df-mpt 4349  df-tr 4383  df-eprel 4628  df-id 4632  df-po 4637  df-so 4638  df-fr 4675  df-we 4677  df-ord 4718  df-on 4719  df-lim 4720  df-suc 4721  df-xp 4842  df-rel 4843  df-cnv 4844  df-co 4845  df-dm 4846  df-rn 4847  df-res 4848  df-ima 4849  df-iota 5378  df-fun 5417  df-fn 5418  df-f 5419  df-f1 5420  df-fo 5421  df-f1o 5422  df-fv 5423  df-riota 6049  df-ov 6093  df-oprab 6094  df-mpt2 6095  df-om 6476  df-recs 6828  df-rdg 6862  df-er 7097  df-en 7307  df-dom 7308  df-sdom 7309  df-sup 7687  df-pnf 9416  df-mnf 9417  df-xr 9418  df-ltxr 9419  df-le 9420  df-sub 9593  df-neg 9594  df-nn 10319  df-n0 10576  df-z 10643  df-uz 10858  df-fl 11638  df-ceil 11639
This theorem is referenced by:  ceilval2  11677
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