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Theorem dfceil2 11781
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 11744 . 2  |- =  ( x  e.  RR  |->  -u ( |_ `  -u x
) )
2 zre 10751 . . . . . . 7  |-  ( z  e.  ZZ  ->  z  e.  RR )
3 lenegcon2 9945 . . . . . . . 8  |-  ( ( x  e.  RR  /\  z  e.  RR )  ->  ( x  <_  -u z  <->  z  <_  -u x ) )
4 peano2re 9643 . . . . . . . . . . . 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 9941 . . . . . . . . . 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 9473 . . . . . . . . . . . 12  |-  ( x  e.  RR  ->  x  e.  CC )
10 ax-1cn 9441 . . . . . . . . . . . . 13  |-  1  e.  CC
1110a1i 11 . . . . . . . . . . . 12  |-  ( x  e.  RR  ->  1  e.  CC )
129, 11negdid 9833 . . . . . . . . . . 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 4400 . . . . . . . . 9  |-  ( ( x  e.  RR  /\  z  e.  RR )  ->  ( -u ( x  +  1 )  < 
z  <->  ( -u x  +  -u 1 )  < 
z ) )
15 renegcl 9773 . . . . . . . . . . . 12  |-  ( x  e.  RR  ->  -u x  e.  RR )
1615adantr 465 . . . . . . . . . . 11  |-  ( ( x  e.  RR  /\  z  e.  RR )  -> 
-u x  e.  RR )
17 neg1rr 10527 . . . . . . . . . . . 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 10040 . . . . . . . . . 10  |-  ( ( x  e.  RR  /\  z  e.  RR )  ->  ( ( -u x  +  -u 1 )  < 
z  <->  -u x  <  (
z  -  -u 1
) ) )
21 recn 9473 . . . . . . . . . . . . 13  |-  ( z  e.  RR  ->  z  e.  CC )
2210a1i 11 . . . . . . . . . . . . 13  |-  ( z  e.  RR  ->  1  e.  CC )
2321, 22subnegd 9827 . . . . . . . . . . . 12  |-  ( z  e.  RR  ->  (
z  -  -u 1
)  =  ( z  +  1 ) )
2423adantl 466 . . . . . . . . . . 11  |-  ( ( x  e.  RR  /\  z  e.  RR )  ->  ( z  -  -u 1
)  =  ( z  +  1 ) )
2524breq2d 4402 . . . . . . . . . 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 6168 . . . . 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 9705 . . . 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 11052 . . . . 5  |-  ( x  e.  RR  ->  E! y  e.  ZZ  (
x  <_  y  /\  y  <  ( x  + 
1 ) ) )
33 breq2 4394 . . . . . . 7  |-  ( y  =  -u z  ->  (
x  <_  y  <->  x  <_  -u z ) )
34 breq1 4393 . . . . . . 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 10855 . . . . 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 11745 . . . . . 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 9705 . . . 4  |-  ( x  e.  RR  ->  -u ( |_ `  -u x )  = 
-u ( iota_ z  e.  ZZ  ( z  <_  -u x  /\  -u x  <  ( z  +  1 ) ) ) )
4131, 37, 403eqtr4rd 2503 . . 3  |-  ( x  e.  RR  ->  -u ( |_ `  -u x )  =  ( iota_ y  e.  ZZ  ( x  <_  y  /\  y  <  ( x  + 
1 ) ) ) )
4241mpteq2ia 4472 . 2  |-  ( x  e.  RR  |->  -u ( |_ `  -u x ) )  =  ( x  e.  RR  |->  ( iota_ y  e.  ZZ  ( x  <_ 
y  /\  y  <  ( x  +  1 ) ) ) )
431, 42eqtri 2480 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 1370    e. wcel 1758   E!wreu 2797   class class class wbr 4390    |-> cmpt 4448   ` cfv 5516   iota_crio 6150  (class class class)co 6190   CCcc 9381   RRcr 9382   1c1 9384    + caddc 9386    < clt 9519    <_ cle 9520    - cmin 9696   -ucneg 9697   ZZcz 10747   |_cfl 11741  ⌈cceil 11742
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1592  ax-4 1603  ax-5 1671  ax-6 1710  ax-7 1730  ax-8 1760  ax-9 1762  ax-10 1777  ax-11 1782  ax-12 1794  ax-13 1952  ax-ext 2430  ax-sep 4511  ax-nul 4519  ax-pow 4568  ax-pr 4629  ax-un 6472  ax-cnex 9439  ax-resscn 9440  ax-1cn 9441  ax-icn 9442  ax-addcl 9443  ax-addrcl 9444  ax-mulcl 9445  ax-mulrcl 9446  ax-mulcom 9447  ax-addass 9448  ax-mulass 9449  ax-distr 9450  ax-i2m1 9451  ax-1ne0 9452  ax-1rid 9453  ax-rnegex 9454  ax-rrecex 9455  ax-cnre 9456  ax-pre-lttri 9457  ax-pre-lttrn 9458  ax-pre-ltadd 9459  ax-pre-mulgt0 9460  ax-pre-sup 9461
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 966  df-3an 967  df-tru 1373  df-ex 1588  df-nf 1591  df-sb 1703  df-eu 2264  df-mo 2265  df-clab 2437  df-cleq 2443  df-clel 2446  df-nfc 2601  df-ne 2646  df-nel 2647  df-ral 2800  df-rex 2801  df-reu 2802  df-rmo 2803  df-rab 2804  df-v 3070  df-sbc 3285  df-csb 3387  df-dif 3429  df-un 3431  df-in 3433  df-ss 3440  df-pss 3442  df-nul 3736  df-if 3890  df-pw 3960  df-sn 3976  df-pr 3978  df-tp 3980  df-op 3982  df-uni 4190  df-iun 4271  df-br 4391  df-opab 4449  df-mpt 4450  df-tr 4484  df-eprel 4730  df-id 4734  df-po 4739  df-so 4740  df-fr 4777  df-we 4779  df-ord 4820  df-on 4821  df-lim 4822  df-suc 4823  df-xp 4944  df-rel 4945  df-cnv 4946  df-co 4947  df-dm 4948  df-rn 4949  df-res 4950  df-ima 4951  df-iota 5479  df-fun 5518  df-fn 5519  df-f 5520  df-f1 5521  df-fo 5522  df-f1o 5523  df-fv 5524  df-riota 6151  df-ov 6193  df-oprab 6194  df-mpt2 6195  df-om 6577  df-recs 6932  df-rdg 6966  df-er 7201  df-en 7411  df-dom 7412  df-sdom 7413  df-sup 7792  df-pnf 9521  df-mnf 9522  df-xr 9523  df-ltxr 9524  df-le 9525  df-sub 9698  df-neg 9699  df-nn 10424  df-n0 10681  df-z 10748  df-uz 10963  df-fl 11743  df-ceil 11744
This theorem is referenced by:  ceilval2  11782
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