ILE Home Intuitionistic Logic Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  ILE Home  >  Th. List  >  shftuz Unicode version

Theorem shftuz 10544
Description: A shift of the upper integers. (Contributed by Mario Carneiro, 5-Nov-2013.)
Assertion
Ref Expression
shftuz  |-  ( ( A  e.  ZZ  /\  B  e.  ZZ )  ->  { x  e.  CC  |  ( x  -  A )  e.  (
ZZ>= `  B ) }  =  ( ZZ>= `  ( B  +  A )
) )
Distinct variable groups:    x, A    x, B

Proof of Theorem shftuz
StepHypRef Expression
1 df-rab 2402 . 2  |-  { x  e.  CC  |  ( x  -  A )  e.  ( ZZ>= `  B ) }  =  { x  |  ( x  e.  CC  /\  ( x  -  A )  e.  ( ZZ>= `  B )
) }
2 simp2 967 . . . . . . . 8  |-  ( ( A  e.  ZZ  /\  x  e.  CC  /\  (
x  -  A )  e.  ( ZZ>= `  B
) )  ->  x  e.  CC )
3 zcn 9017 . . . . . . . . 9  |-  ( A  e.  ZZ  ->  A  e.  CC )
433ad2ant1 987 . . . . . . . 8  |-  ( ( A  e.  ZZ  /\  x  e.  CC  /\  (
x  -  A )  e.  ( ZZ>= `  B
) )  ->  A  e.  CC )
52, 4npcand 8045 . . . . . . 7  |-  ( ( A  e.  ZZ  /\  x  e.  CC  /\  (
x  -  A )  e.  ( ZZ>= `  B
) )  ->  (
( x  -  A
)  +  A )  =  x )
6 eluzadd 9310 . . . . . . . . 9  |-  ( ( ( x  -  A
)  e.  ( ZZ>= `  B )  /\  A  e.  ZZ )  ->  (
( x  -  A
)  +  A )  e.  ( ZZ>= `  ( B  +  A )
) )
76ancoms 266 . . . . . . . 8  |-  ( ( A  e.  ZZ  /\  ( x  -  A
)  e.  ( ZZ>= `  B ) )  -> 
( ( x  -  A )  +  A
)  e.  ( ZZ>= `  ( B  +  A
) ) )
873adant2 985 . . . . . . 7  |-  ( ( A  e.  ZZ  /\  x  e.  CC  /\  (
x  -  A )  e.  ( ZZ>= `  B
) )  ->  (
( x  -  A
)  +  A )  e.  ( ZZ>= `  ( B  +  A )
) )
95, 8eqeltrrd 2195 . . . . . 6  |-  ( ( A  e.  ZZ  /\  x  e.  CC  /\  (
x  -  A )  e.  ( ZZ>= `  B
) )  ->  x  e.  ( ZZ>= `  ( B  +  A ) ) )
1093expib 1169 . . . . 5  |-  ( A  e.  ZZ  ->  (
( x  e.  CC  /\  ( x  -  A
)  e.  ( ZZ>= `  B ) )  ->  x  e.  ( ZZ>= `  ( B  +  A
) ) ) )
1110adantr 274 . . . 4  |-  ( ( A  e.  ZZ  /\  B  e.  ZZ )  ->  ( ( x  e.  CC  /\  ( x  -  A )  e.  ( ZZ>= `  B )
)  ->  x  e.  ( ZZ>= `  ( B  +  A ) ) ) )
12 eluzelcn 9293 . . . . . 6  |-  ( x  e.  ( ZZ>= `  ( B  +  A )
)  ->  x  e.  CC )
1312a1i 9 . . . . 5  |-  ( ( A  e.  ZZ  /\  B  e.  ZZ )  ->  ( x  e.  (
ZZ>= `  ( B  +  A ) )  ->  x  e.  CC )
)
14 eluzsub 9311 . . . . . . 7  |-  ( ( B  e.  ZZ  /\  A  e.  ZZ  /\  x  e.  ( ZZ>= `  ( B  +  A ) ) )  ->  ( x  -  A )  e.  (
ZZ>= `  B ) )
15143expia 1168 . . . . . 6  |-  ( ( B  e.  ZZ  /\  A  e.  ZZ )  ->  ( x  e.  (
ZZ>= `  ( B  +  A ) )  -> 
( x  -  A
)  e.  ( ZZ>= `  B ) ) )
1615ancoms 266 . . . . 5  |-  ( ( A  e.  ZZ  /\  B  e.  ZZ )  ->  ( x  e.  (
ZZ>= `  ( B  +  A ) )  -> 
( x  -  A
)  e.  ( ZZ>= `  B ) ) )
1713, 16jcad 305 . . . 4  |-  ( ( A  e.  ZZ  /\  B  e.  ZZ )  ->  ( x  e.  (
ZZ>= `  ( B  +  A ) )  -> 
( x  e.  CC  /\  ( x  -  A
)  e.  ( ZZ>= `  B ) ) ) )
1811, 17impbid 128 . . 3  |-  ( ( A  e.  ZZ  /\  B  e.  ZZ )  ->  ( ( x  e.  CC  /\  ( x  -  A )  e.  ( ZZ>= `  B )
)  <->  x  e.  ( ZZ>=
`  ( B  +  A ) ) ) )
1918abbi1dv 2237 . 2  |-  ( ( A  e.  ZZ  /\  B  e.  ZZ )  ->  { x  |  ( x  e.  CC  /\  ( x  -  A
)  e.  ( ZZ>= `  B ) ) }  =  ( ZZ>= `  ( B  +  A )
) )
201, 19syl5eq 2162 1  |-  ( ( A  e.  ZZ  /\  B  e.  ZZ )  ->  { x  e.  CC  |  ( x  -  A )  e.  (
ZZ>= `  B ) }  =  ( ZZ>= `  ( B  +  A )
) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 103    /\ w3a 947    = wceq 1316    e. wcel 1465   {cab 2103   {crab 2397   ` cfv 5093  (class class class)co 5742   CCcc 7586    + caddc 7591    - cmin 7901   ZZcz 9012   ZZ>=cuz 9282
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-in1 588  ax-in2 589  ax-io 683  ax-5 1408  ax-7 1409  ax-gen 1410  ax-ie1 1454  ax-ie2 1455  ax-8 1467  ax-10 1468  ax-11 1469  ax-i12 1470  ax-bndl 1471  ax-4 1472  ax-13 1476  ax-14 1477  ax-17 1491  ax-i9 1495  ax-ial 1499  ax-i5r 1500  ax-ext 2099  ax-sep 4016  ax-pow 4068  ax-pr 4101  ax-un 4325  ax-setind 4422  ax-cnex 7679  ax-resscn 7680  ax-1cn 7681  ax-1re 7682  ax-icn 7683  ax-addcl 7684  ax-addrcl 7685  ax-mulcl 7686  ax-addcom 7688  ax-addass 7690  ax-distr 7692  ax-i2m1 7693  ax-0lt1 7694  ax-0id 7696  ax-rnegex 7697  ax-cnre 7699  ax-pre-ltirr 7700  ax-pre-ltwlin 7701  ax-pre-lttrn 7702  ax-pre-ltadd 7704
This theorem depends on definitions:  df-bi 116  df-3or 948  df-3an 949  df-tru 1319  df-fal 1322  df-nf 1422  df-sb 1721  df-eu 1980  df-mo 1981  df-clab 2104  df-cleq 2110  df-clel 2113  df-nfc 2247  df-ne 2286  df-nel 2381  df-ral 2398  df-rex 2399  df-reu 2400  df-rab 2402  df-v 2662  df-sbc 2883  df-dif 3043  df-un 3045  df-in 3047  df-ss 3054  df-pw 3482  df-sn 3503  df-pr 3504  df-op 3506  df-uni 3707  df-int 3742  df-br 3900  df-opab 3960  df-mpt 3961  df-id 4185  df-xp 4515  df-rel 4516  df-cnv 4517  df-co 4518  df-dm 4519  df-rn 4520  df-res 4521  df-ima 4522  df-iota 5058  df-fun 5095  df-fn 5096  df-f 5097  df-fv 5101  df-riota 5698  df-ov 5745  df-oprab 5746  df-mpo 5747  df-pnf 7770  df-mnf 7771  df-xr 7772  df-ltxr 7773  df-le 7774  df-sub 7903  df-neg 7904  df-inn 8685  df-n0 8936  df-z 9013  df-uz 9283
This theorem is referenced by:  seq3shft  10565
  Copyright terms: Public domain W3C validator