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Theorem shftuz 12671
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 2805 . 2  |-  { x  e.  CC  |  ( x  -  A )  e.  ( ZZ>= `  B ) }  =  { x  |  ( x  e.  CC  /\  ( x  -  A )  e.  ( ZZ>= `  B )
) }
2 simp2 989 . . . . . . . 8  |-  ( ( A  e.  ZZ  /\  x  e.  CC  /\  (
x  -  A )  e.  ( ZZ>= `  B
) )  ->  x  e.  CC )
3 zcn 10757 . . . . . . . . 9  |-  ( A  e.  ZZ  ->  A  e.  CC )
433ad2ant1 1009 . . . . . . . 8  |-  ( ( A  e.  ZZ  /\  x  e.  CC  /\  (
x  -  A )  e.  ( ZZ>= `  B
) )  ->  A  e.  CC )
52, 4npcand 9829 . . . . . . 7  |-  ( ( A  e.  ZZ  /\  x  e.  CC  /\  (
x  -  A )  e.  ( ZZ>= `  B
) )  ->  (
( x  -  A
)  +  A )  =  x )
6 eluzadd 10995 . . . . . . . . 9  |-  ( ( ( x  -  A
)  e.  ( ZZ>= `  B )  /\  A  e.  ZZ )  ->  (
( x  -  A
)  +  A )  e.  ( ZZ>= `  ( B  +  A )
) )
76ancoms 453 . . . . . . . 8  |-  ( ( A  e.  ZZ  /\  ( x  -  A
)  e.  ( ZZ>= `  B ) )  -> 
( ( x  -  A )  +  A
)  e.  ( ZZ>= `  ( B  +  A
) ) )
873adant2 1007 . . . . . . 7  |-  ( ( A  e.  ZZ  /\  x  e.  CC  /\  (
x  -  A )  e.  ( ZZ>= `  B
) )  ->  (
( x  -  A
)  +  A )  e.  ( ZZ>= `  ( B  +  A )
) )
95, 8eqeltrrd 2541 . . . . . 6  |-  ( ( A  e.  ZZ  /\  x  e.  CC  /\  (
x  -  A )  e.  ( ZZ>= `  B
) )  ->  x  e.  ( ZZ>= `  ( B  +  A ) ) )
1093expib 1191 . . . . 5  |-  ( A  e.  ZZ  ->  (
( x  e.  CC  /\  ( x  -  A
)  e.  ( ZZ>= `  B ) )  ->  x  e.  ( ZZ>= `  ( B  +  A
) ) ) )
1110adantr 465 . . . 4  |-  ( ( A  e.  ZZ  /\  B  e.  ZZ )  ->  ( ( x  e.  CC  /\  ( x  -  A )  e.  ( ZZ>= `  B )
)  ->  x  e.  ( ZZ>= `  ( B  +  A ) ) ) )
12 eluzelz 10976 . . . . . . 7  |-  ( x  e.  ( ZZ>= `  ( B  +  A )
)  ->  x  e.  ZZ )
1312zcnd 10854 . . . . . 6  |-  ( x  e.  ( ZZ>= `  ( B  +  A )
)  ->  x  e.  CC )
1413a1i 11 . . . . 5  |-  ( ( A  e.  ZZ  /\  B  e.  ZZ )  ->  ( x  e.  (
ZZ>= `  ( B  +  A ) )  ->  x  e.  CC )
)
15 eluzsub 10996 . . . . . . 7  |-  ( ( B  e.  ZZ  /\  A  e.  ZZ  /\  x  e.  ( ZZ>= `  ( B  +  A ) ) )  ->  ( x  -  A )  e.  (
ZZ>= `  B ) )
16153expia 1190 . . . . . 6  |-  ( ( B  e.  ZZ  /\  A  e.  ZZ )  ->  ( x  e.  (
ZZ>= `  ( B  +  A ) )  -> 
( x  -  A
)  e.  ( ZZ>= `  B ) ) )
1716ancoms 453 . . . . 5  |-  ( ( A  e.  ZZ  /\  B  e.  ZZ )  ->  ( x  e.  (
ZZ>= `  ( B  +  A ) )  -> 
( x  -  A
)  e.  ( ZZ>= `  B ) ) )
1814, 17jcad 533 . . . 4  |-  ( ( A  e.  ZZ  /\  B  e.  ZZ )  ->  ( x  e.  (
ZZ>= `  ( B  +  A ) )  -> 
( x  e.  CC  /\  ( x  -  A
)  e.  ( ZZ>= `  B ) ) ) )
1911, 18impbid 191 . . 3  |-  ( ( A  e.  ZZ  /\  B  e.  ZZ )  ->  ( ( x  e.  CC  /\  ( x  -  A )  e.  ( ZZ>= `  B )
)  <->  x  e.  ( ZZ>=
`  ( B  +  A ) ) ) )
2019abbi1dv 2590 . 2  |-  ( ( A  e.  ZZ  /\  B  e.  ZZ )  ->  { x  |  ( x  e.  CC  /\  ( x  -  A
)  e.  ( ZZ>= `  B ) ) }  =  ( ZZ>= `  ( B  +  A )
) )
211, 20syl5eq 2505 1  |-  ( ( A  e.  ZZ  /\  B  e.  ZZ )  ->  { x  e.  CC  |  ( x  -  A )  e.  (
ZZ>= `  B ) }  =  ( ZZ>= `  ( B  +  A )
) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    /\ wa 369    /\ w3a 965    = wceq 1370    e. wcel 1758   {cab 2437   {crab 2800   ` cfv 5521  (class class class)co 6195   CCcc 9386    + caddc 9391    - cmin 9701   ZZcz 10752   ZZ>=cuz 10967
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 1954  ax-ext 2431  ax-sep 4516  ax-nul 4524  ax-pow 4573  ax-pr 4634  ax-un 6477  ax-cnex 9444  ax-resscn 9445  ax-1cn 9446  ax-icn 9447  ax-addcl 9448  ax-addrcl 9449  ax-mulcl 9450  ax-mulrcl 9451  ax-mulcom 9452  ax-addass 9453  ax-mulass 9454  ax-distr 9455  ax-i2m1 9456  ax-1ne0 9457  ax-1rid 9458  ax-rnegex 9459  ax-rrecex 9460  ax-cnre 9461  ax-pre-lttri 9462  ax-pre-lttrn 9463  ax-pre-ltadd 9464  ax-pre-mulgt0 9465
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 2265  df-mo 2266  df-clab 2438  df-cleq 2444  df-clel 2447  df-nfc 2602  df-ne 2647  df-nel 2648  df-ral 2801  df-rex 2802  df-reu 2803  df-rab 2805  df-v 3074  df-sbc 3289  df-csb 3391  df-dif 3434  df-un 3436  df-in 3438  df-ss 3445  df-pss 3447  df-nul 3741  df-if 3895  df-pw 3965  df-sn 3981  df-pr 3983  df-tp 3985  df-op 3987  df-uni 4195  df-iun 4276  df-br 4396  df-opab 4454  df-mpt 4455  df-tr 4489  df-eprel 4735  df-id 4739  df-po 4744  df-so 4745  df-fr 4782  df-we 4784  df-ord 4825  df-on 4826  df-lim 4827  df-suc 4828  df-xp 4949  df-rel 4950  df-cnv 4951  df-co 4952  df-dm 4953  df-rn 4954  df-res 4955  df-ima 4956  df-iota 5484  df-fun 5523  df-fn 5524  df-f 5525  df-f1 5526  df-fo 5527  df-f1o 5528  df-fv 5529  df-riota 6156  df-ov 6198  df-oprab 6199  df-mpt2 6200  df-om 6582  df-recs 6937  df-rdg 6971  df-er 7206  df-en 7416  df-dom 7417  df-sdom 7418  df-pnf 9526  df-mnf 9527  df-xr 9528  df-ltxr 9529  df-le 9530  df-sub 9703  df-neg 9704  df-nn 10429  df-n0 10686  df-z 10753  df-uz 10968
This theorem is referenced by:  seqshft  12687
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