Theorem shftuz 10699

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

Step	Hyp	Ref	Expression
1		df-rab 2444	. 2 |- { x e. CC | ( x - A ) e. ( ZZ>= ` B ) } = { x | ( x e. CC /\ ( x - A ) e. ( ZZ>= ` B ) ) }
2		simp2 983	. . . . . . . 8 |- ( ( A e. ZZ /\ x e. CC /\ ( x - A ) e. ( ZZ>= ` B ) ) -> x e. CC )
3		zcn 9155	. . . . . . . . 9 |- ( A e. ZZ -> A e. CC )
4	3	3ad2ant1 1003	. . . . . . . 8 |- ( ( A e. ZZ /\ x e. CC /\ ( x - A ) e. ( ZZ>= ` B ) ) -> A e. CC )
5	2, 4	npcand 8173	. . . . . . 7 |- ( ( A e. ZZ /\ x e. CC /\ ( x - A ) e. ( ZZ>= ` B ) ) -> ( ( x - A ) + A ) = x )
6		eluzadd 9450	. . . . . . . . 9 |- ( ( ( x - A ) e. ( ZZ>= ` B ) /\ A e. ZZ ) -> ( ( x - A ) + A ) e. ( ZZ>= ` ( B + A ) ) )
7	6	ancoms 266	. . . . . . . 8 |- ( ( A e. ZZ /\ ( x - A ) e. ( ZZ>= ` B ) ) -> ( ( x - A ) + A ) e. ( ZZ>= ` ( B + A ) ) )
8	7	3adant2 1001	. . . . . . 7 |- ( ( A e. ZZ /\ x e. CC /\ ( x - A ) e. ( ZZ>= ` B ) ) -> ( ( x - A ) + A ) e. ( ZZ>= ` ( B + A ) ) )
9	5, 8	eqeltrrd 2235	. . . . . 6 |- ( ( A e. ZZ /\ x e. CC /\ ( x - A ) e. ( ZZ>= ` B ) ) -> x e. ( ZZ>= ` ( B + A ) ) )
10	9	3expib 1188	. . . . 5 |- ( A e. ZZ -> ( ( x e. CC /\ ( x - A ) e. ( ZZ>= ` B ) ) -> x e. ( ZZ>= ` ( B + A ) ) ) )
11	10	adantr 274	. . . 4 |- ( ( A e. ZZ /\ B e. ZZ ) -> ( ( x e. CC /\ ( x - A ) e. ( ZZ>= ` B ) ) -> x e. ( ZZ>= ` ( B + A ) ) ) )
12		eluzelcn 9433	. . . . . 6 |- ( x e. ( ZZ>= ` ( B + A ) ) -> x e. CC )
13	12	a1i 9	. . . . 5 |- ( ( A e. ZZ /\ B e. ZZ ) -> ( x e. ( ZZ>= ` ( B + A ) ) -> x e. CC ) )
14		eluzsub 9451	. . . . . . 7 |- ( ( B e. ZZ /\ A e. ZZ /\ x e. ( ZZ>= ` ( B + A ) ) ) -> ( x - A ) e. ( ZZ>= ` B ) )
15	14	3expia 1187	. . . . . 6 |- ( ( B e. ZZ /\ A e. ZZ ) -> ( x e. ( ZZ>= ` ( B + A ) ) -> ( x - A ) e. ( ZZ>= ` B ) ) )
16	15	ancoms 266	. . . . 5 |- ( ( A e. ZZ /\ B e. ZZ ) -> ( x e. ( ZZ>= ` ( B + A ) ) -> ( x - A ) e. ( ZZ>= ` B ) ) )
17	13, 16	jcad 305	. . . 4 |- ( ( A e. ZZ /\ B e. ZZ ) -> ( x e. ( ZZ>= ` ( B + A ) ) -> ( x e. CC /\ ( x - A ) e. ( ZZ>= ` B ) ) ) )
18	11, 17	impbid 128	. . 3 |- ( ( A e. ZZ /\ B e. ZZ ) -> ( ( x e. CC /\ ( x - A ) e. ( ZZ>= ` B ) ) <-> x e. ( ZZ>= ` ( B + A ) ) ) )
19	18	abbi1dv 2277	. 2 |- ( ( A e. ZZ /\ B e. ZZ ) -> { x | ( x e. CC /\ ( x - A ) e. ( ZZ>= ` B ) ) } = ( ZZ>= ` ( B + A ) ) )
20	1, 19	syl5eq 2202	1 |- ( ( A e. ZZ /\ B e. ZZ ) -> { x e. CC | ( x - A ) e. ( ZZ>= ` B ) } = ( ZZ>= ` ( B + A ) ) )

Syntax hints:   -> wi 4   /\ wa 103   /\ w3a 963   = wceq 1335   e. wcel 2128   {cab 2143   {crab 2439   ` cfv 5167  (class class class)co 5818   CCcc 7713   + caddc 7718   - cmin 8029   ZZcz 9150   ZZ>=cuz 9422

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 604  ax-in2 605  ax-io 699  ax-5 1427  ax-7 1428  ax-gen 1429  ax-ie1 1473  ax-ie2 1474  ax-8 1484  ax-10 1485  ax-11 1486  ax-i12 1487  ax-bndl 1489  ax-4 1490  ax-17 1506  ax-i9 1510  ax-ial 1514  ax-i5r 1515  ax-13 2130  ax-14 2131  ax-ext 2139  ax-sep 4082  ax-pow 4134  ax-pr 4168  ax-un 4392  ax-setind 4494  ax-cnex 7806  ax-resscn 7807  ax-1cn 7808  ax-1re 7809  ax-icn 7810  ax-addcl 7811  ax-addrcl 7812  ax-mulcl 7813  ax-addcom 7815  ax-addass 7817  ax-distr 7819  ax-i2m1 7820  ax-0lt1 7821  ax-0id 7823  ax-rnegex 7824  ax-cnre 7826  ax-pre-ltirr 7827  ax-pre-ltwlin 7828  ax-pre-lttrn 7829  ax-pre-ltadd 7831

This theorem depends on definitions:  df-bi 116  df-3or 964  df-3an 965  df-tru 1338  df-fal 1341  df-nf 1441  df-sb 1743  df-eu 2009  df-mo 2010  df-clab 2144  df-cleq 2150  df-clel 2153  df-nfc 2288  df-ne 2328  df-nel 2423  df-ral 2440  df-rex 2441  df-reu 2442  df-rab 2444  df-v 2714  df-sbc 2938  df-dif 3104  df-un 3106  df-in 3108  df-ss 3115  df-pw 3545  df-sn 3566  df-pr 3567  df-op 3569  df-uni 3773  df-int 3808  df-br 3966  df-opab 4026  df-mpt 4027  df-id 4252  df-xp 4589  df-rel 4590  df-cnv 4591  df-co 4592  df-dm 4593  df-rn 4594  df-res 4595  df-ima 4596  df-iota 5132  df-fun 5169  df-fn 5170  df-f 5171  df-fv 5175  df-riota 5774  df-ov 5821  df-oprab 5822  df-mpo 5823  df-pnf 7897  df-mnf 7898  df-xr 7899  df-ltxr 7900  df-le 7901  df-sub 8031  df-neg 8032  df-inn 8817  df-n0 9074  df-z 9151  df-uz 9423

This theorem is referenced by:  seq3shft  10720