MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  dfceil2 Structured version   Unicode version

Theorem dfceil2 12067
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 12028 . 2  |- =  ( x  e.  RR  |->  -u ( |_ `  -u x
) )
2 zre 10941 . . . . . . 7  |-  ( z  e.  ZZ  ->  z  e.  RR )
3 lenegcon2 10119 . . . . . . . 8  |-  ( ( x  e.  RR  /\  z  e.  RR )  ->  ( x  <_  -u z  <->  z  <_  -u x ) )
4 peano2re 9806 . . . . . . . . . . . 12  |-  ( x  e.  RR  ->  (
x  +  1 )  e.  RR )
54anim2i 571 . . . . . . . . . . 11  |-  ( ( z  e.  RR  /\  x  e.  RR )  ->  ( z  e.  RR  /\  ( x  +  1 )  e.  RR ) )
65ancoms 454 . . . . . . . . . 10  |-  ( ( x  e.  RR  /\  z  e.  RR )  ->  ( z  e.  RR  /\  ( x  +  1 )  e.  RR ) )
7 ltnegcon1 10115 . . . . . . . . . 10  |-  ( ( z  e.  RR  /\  ( x  +  1
)  e.  RR )  ->  ( -u z  <  ( x  +  1 )  <->  -u ( x  + 
1 )  <  z
) )
86, 7syl 17 . . . . . . . . 9  |-  ( ( x  e.  RR  /\  z  e.  RR )  ->  ( -u z  < 
( x  +  1 )  <->  -u ( x  + 
1 )  <  z
) )
9 recn 9629 . . . . . . . . . . . 12  |-  ( x  e.  RR  ->  x  e.  CC )
10 1cnd 9659 . . . . . . . . . . . 12  |-  ( x  e.  RR  ->  1  e.  CC )
119, 10negdid 9999 . . . . . . . . . . 11  |-  ( x  e.  RR  ->  -u (
x  +  1 )  =  ( -u x  +  -u 1 ) )
1211adantr 466 . . . . . . . . . 10  |-  ( ( x  e.  RR  /\  z  e.  RR )  -> 
-u ( x  + 
1 )  =  (
-u x  +  -u
1 ) )
1312breq1d 4430 . . . . . . . . 9  |-  ( ( x  e.  RR  /\  z  e.  RR )  ->  ( -u ( x  +  1 )  < 
z  <->  ( -u x  +  -u 1 )  < 
z ) )
14 renegcl 9937 . . . . . . . . . . . 12  |-  ( x  e.  RR  ->  -u x  e.  RR )
1514adantr 466 . . . . . . . . . . 11  |-  ( ( x  e.  RR  /\  z  e.  RR )  -> 
-u x  e.  RR )
16 neg1rr 10714 . . . . . . . . . . . 12  |-  -u 1  e.  RR
1716a1i 11 . . . . . . . . . . 11  |-  ( ( x  e.  RR  /\  z  e.  RR )  -> 
-u 1  e.  RR )
18 simpr 462 . . . . . . . . . . 11  |-  ( ( x  e.  RR  /\  z  e.  RR )  ->  z  e.  RR )
1915, 17, 18ltaddsubd 10213 . . . . . . . . . 10  |-  ( ( x  e.  RR  /\  z  e.  RR )  ->  ( ( -u x  +  -u 1 )  < 
z  <->  -u x  <  (
z  -  -u 1
) ) )
20 recn 9629 . . . . . . . . . . . . 13  |-  ( z  e.  RR  ->  z  e.  CC )
21 1cnd 9659 . . . . . . . . . . . . 13  |-  ( z  e.  RR  ->  1  e.  CC )
2220, 21subnegd 9993 . . . . . . . . . . . 12  |-  ( z  e.  RR  ->  (
z  -  -u 1
)  =  ( z  +  1 ) )
2322adantl 467 . . . . . . . . . . 11  |-  ( ( x  e.  RR  /\  z  e.  RR )  ->  ( z  -  -u 1
)  =  ( z  +  1 ) )
2423breq2d 4432 . . . . . . . . . 10  |-  ( ( x  e.  RR  /\  z  e.  RR )  ->  ( -u x  < 
( z  -  -u 1
)  <->  -u x  <  (
z  +  1 ) ) )
2519, 24bitrd 256 . . . . . . . . 9  |-  ( ( x  e.  RR  /\  z  e.  RR )  ->  ( ( -u x  +  -u 1 )  < 
z  <->  -u x  <  (
z  +  1 ) ) )
268, 13, 253bitrd 282 . . . . . . . 8  |-  ( ( x  e.  RR  /\  z  e.  RR )  ->  ( -u z  < 
( x  +  1 )  <->  -u x  <  (
z  +  1 ) ) )
273, 26anbi12d 715 . . . . . . 7  |-  ( ( x  e.  RR  /\  z  e.  RR )  ->  ( ( x  <_  -u z  /\  -u z  <  ( x  +  1 ) )  <->  ( z  <_ 
-u x  /\  -u x  <  ( z  +  1 ) ) ) )
282, 27sylan2 476 . . . . . 6  |-  ( ( x  e.  RR  /\  z  e.  ZZ )  ->  ( ( x  <_  -u z  /\  -u z  <  ( x  +  1 ) )  <->  ( z  <_ 
-u x  /\  -u x  <  ( z  +  1 ) ) ) )
2928riotabidva 6279 . . . . 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 ) ) ) )
3029negeqd 9869 . . . 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 ) ) ) )
31 zbtwnre 11262 . . . . 5  |-  ( x  e.  RR  ->  E! y  e.  ZZ  (
x  <_  y  /\  y  <  ( x  + 
1 ) ) )
32 breq2 4424 . . . . . . 7  |-  ( y  =  -u z  ->  (
x  <_  y  <->  x  <_  -u z ) )
33 breq1 4423 . . . . . . 7  |-  ( y  =  -u z  ->  (
y  <  ( x  +  1 )  <->  -u z  < 
( x  +  1 ) ) )
3432, 33anbi12d 715 . . . . . 6  |-  ( y  =  -u z  ->  (
( x  <_  y  /\  y  <  ( x  +  1 ) )  <-> 
( x  <_  -u z  /\  -u z  <  (
x  +  1 ) ) ) )
3534zriotaneg 11049 . . . . 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 ) ) ) )
3631, 35syl 17 . . . 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 ) ) ) )
37 flval 12029 . . . . . 6  |-  ( -u x  e.  RR  ->  ( |_ `  -u x
)  =  ( iota_ z  e.  ZZ  ( z  <_  -u x  /\  -u x  <  ( z  +  1 ) ) ) )
3814, 37syl 17 . . . . 5  |-  ( x  e.  RR  ->  ( |_ `  -u x )  =  ( iota_ z  e.  ZZ  ( z  <_  -u x  /\  -u x  <  (
z  +  1 ) ) ) )
3938negeqd 9869 . . . 4  |-  ( x  e.  RR  ->  -u ( |_ `  -u x )  = 
-u ( iota_ z  e.  ZZ  ( z  <_  -u x  /\  -u x  <  ( z  +  1 ) ) ) )
4030, 36, 393eqtr4rd 2474 . . 3  |-  ( x  e.  RR  ->  -u ( |_ `  -u x )  =  ( iota_ y  e.  ZZ  ( x  <_  y  /\  y  <  ( x  + 
1 ) ) ) )
4140mpteq2ia 4503 . 2  |-  ( x  e.  RR  |->  -u ( |_ `  -u x ) )  =  ( x  e.  RR  |->  ( iota_ y  e.  ZZ  ( x  <_ 
y  /\  y  <  ( x  +  1 ) ) ) )
421, 41eqtri 2451 1  |- =  ( x  e.  RR  |->  (
iota_ y  e.  ZZ  ( x  <_  y  /\  y  <  ( x  + 
1 ) ) ) )
Colors of variables: wff setvar class
Syntax hints:    <-> wb 187    /\ wa 370    = wceq 1437    e. wcel 1868   E!wreu 2777   class class class wbr 4420    |-> cmpt 4479   ` cfv 5597   iota_crio 6262  (class class class)co 6301   RRcr 9538   1c1 9540    + caddc 9542    < clt 9675    <_ cle 9676    - cmin 9860   -ucneg 9861   ZZcz 10937   |_cfl 12025  ⌈cceil 12026
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1665  ax-4 1678  ax-5 1748  ax-6 1794  ax-7 1839  ax-8 1870  ax-9 1872  ax-10 1887  ax-11 1892  ax-12 1905  ax-13 2053  ax-ext 2400  ax-sep 4543  ax-nul 4551  ax-pow 4598  ax-pr 4656  ax-un 6593  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 188  df-or 371  df-an 372  df-3or 983  df-3an 984  df-tru 1440  df-ex 1660  df-nf 1664  df-sb 1787  df-eu 2269  df-mo 2270  df-clab 2408  df-cleq 2414  df-clel 2417  df-nfc 2572  df-ne 2620  df-nel 2621  df-ral 2780  df-rex 2781  df-reu 2782  df-rmo 2783  df-rab 2784  df-v 3083  df-sbc 3300  df-csb 3396  df-dif 3439  df-un 3441  df-in 3443  df-ss 3450  df-pss 3452  df-nul 3762  df-if 3910  df-pw 3981  df-sn 3997  df-pr 3999  df-tp 4001  df-op 4003  df-uni 4217  df-iun 4298  df-br 4421  df-opab 4480  df-mpt 4481  df-tr 4516  df-eprel 4760  df-id 4764  df-po 4770  df-so 4771  df-fr 4808  df-we 4810  df-xp 4855  df-rel 4856  df-cnv 4857  df-co 4858  df-dm 4859  df-rn 4860  df-res 4861  df-ima 4862  df-pred 5395  df-ord 5441  df-on 5442  df-lim 5443  df-suc 5444  df-iota 5561  df-fun 5599  df-fn 5600  df-f 5601  df-f1 5602  df-fo 5603  df-f1o 5604  df-fv 5605  df-riota 6263  df-ov 6304  df-oprab 6305  df-mpt2 6306  df-om 6703  df-wrecs 7032  df-recs 7094  df-rdg 7132  df-er 7367  df-en 7574  df-dom 7575  df-sdom 7576  df-sup 7958  df-inf 7959  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 12027  df-ceil 12028
This theorem is referenced by:  ceilval2  12068
  Copyright terms: Public domain W3C validator