Theorem shftuz 10781

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 2457	. 2 |- { x e. CC | ( x - A ) e. ( ZZ>= ` B ) } = { x | ( x e. CC /\ ( x - A ) e. ( ZZ>= ` B ) ) }
2		simp2 993	. . . . . . . 8 |- ( ( A e. ZZ /\ x e. CC /\ ( x - A ) e. ( ZZ>= ` B ) ) -> x e. CC )
3		zcn 9217	. . . . . . . . 9 |- ( A e. ZZ -> A e. CC )
4	3	3ad2ant1 1013	. . . . . . . 8 |- ( ( A e. ZZ /\ x e. CC /\ ( x - A ) e. ( ZZ>= ` B ) ) -> A e. CC )
5	2, 4	npcand 8234	. . . . . . 7 |- ( ( A e. ZZ /\ x e. CC /\ ( x - A ) e. ( ZZ>= ` B ) ) -> ( ( x - A ) + A ) = x )
6		eluzadd 9515	. . . . . . . . 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 1011	. . . . . . 7 |- ( ( A e. ZZ /\ x e. CC /\ ( x - A ) e. ( ZZ>= ` B ) ) -> ( ( x - A ) + A ) e. ( ZZ>= ` ( B + A ) ) )
9	5, 8	eqeltrrd 2248	. . . . . 6 |- ( ( A e. ZZ /\ x e. CC /\ ( x - A ) e. ( ZZ>= ` B ) ) -> x e. ( ZZ>= ` ( B + A ) ) )
10	9	3expib 1201	. . . . 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 9498	. . . . . 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 9516	. . . . . . 7 |- ( ( B e. ZZ /\ A e. ZZ /\ x e. ( ZZ>= ` ( B + A ) ) ) -> ( x - A ) e. ( ZZ>= ` B ) )
15	14	3expia 1200	. . . . . 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 2290	. 2 |- ( ( A e. ZZ /\ B e. ZZ ) -> { x | ( x e. CC /\ ( x - A ) e. ( ZZ>= ` B ) ) } = ( ZZ>= ` ( B + A ) ) )
20	1, 19	eqtrid 2215	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 973   = wceq 1348   e. wcel 2141   {cab 2156   {crab 2452   ` cfv 5198  (class class class)co 5853   CCcc 7772   + caddc 7777   - cmin 8090   ZZcz 9212   ZZ>=cuz 9487

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 609  ax-in2 610  ax-io 704  ax-5 1440  ax-7 1441  ax-gen 1442  ax-ie1 1486  ax-ie2 1487  ax-8 1497  ax-10 1498  ax-11 1499  ax-i12 1500  ax-bndl 1502  ax-4 1503  ax-17 1519  ax-i9 1523  ax-ial 1527  ax-i5r 1528  ax-13 2143  ax-14 2144  ax-ext 2152  ax-sep 4107  ax-pow 4160  ax-pr 4194  ax-un 4418  ax-setind 4521  ax-cnex 7865  ax-resscn 7866  ax-1cn 7867  ax-1re 7868  ax-icn 7869  ax-addcl 7870  ax-addrcl 7871  ax-mulcl 7872  ax-addcom 7874  ax-addass 7876  ax-distr 7878  ax-i2m1 7879  ax-0lt1 7880  ax-0id 7882  ax-rnegex 7883  ax-cnre 7885  ax-pre-ltirr 7886  ax-pre-ltwlin 7887  ax-pre-lttrn 7888  ax-pre-ltadd 7890

This theorem depends on definitions:  df-bi 116  df-3or 974  df-3an 975  df-tru 1351  df-fal 1354  df-nf 1454  df-sb 1756  df-eu 2022  df-mo 2023  df-clab 2157  df-cleq 2163  df-clel 2166  df-nfc 2301  df-ne 2341  df-nel 2436  df-ral 2453  df-rex 2454  df-reu 2455  df-rab 2457  df-v 2732  df-sbc 2956  df-dif 3123  df-un 3125  df-in 3127  df-ss 3134  df-pw 3568  df-sn 3589  df-pr 3590  df-op 3592  df-uni 3797  df-int 3832  df-br 3990  df-opab 4051  df-mpt 4052  df-id 4278  df-xp 4617  df-rel 4618  df-cnv 4619  df-co 4620  df-dm 4621  df-rn 4622  df-res 4623  df-ima 4624  df-iota 5160  df-fun 5200  df-fn 5201  df-f 5202  df-fv 5206  df-riota 5809  df-ov 5856  df-oprab 5857  df-mpo 5858  df-pnf 7956  df-mnf 7957  df-xr 7958  df-ltxr 7959  df-le 7960  df-sub 8092  df-neg 8093  df-inn 8879  df-n0 9136  df-z 9213  df-uz 9488

This theorem is referenced by:  seq3shft  10802