Theorem shftuz 11502

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 2529	. 2 |- { x e. CC | ( x - A ) e. ( ZZ>= ` B ) } = { x | ( x e. CC /\ ( x - A ) e. ( ZZ>= ` B ) ) }
2		simp2 1025	. . . . . . . 8 |- ( ( A e. ZZ /\ x e. CC /\ ( x - A ) e. ( ZZ>= ` B ) ) -> x e. CC )
3		zcn 9582	. . . . . . . . 9 |- ( A e. ZZ -> A e. CC )
4	3	3ad2ant1 1045	. . . . . . . 8 |- ( ( A e. ZZ /\ x e. CC /\ ( x - A ) e. ( ZZ>= ` B ) ) -> A e. CC )
5	2, 4	npcand 8588	. . . . . . 7 |- ( ( A e. ZZ /\ x e. CC /\ ( x - A ) e. ( ZZ>= ` B ) ) -> ( ( x - A ) + A ) = x )
6		eluzadd 9883	. . . . . . . . 9 |- ( ( ( x - A ) e. ( ZZ>= ` B ) /\ A e. ZZ ) -> ( ( x - A ) + A ) e. ( ZZ>= ` ( B + A ) ) )
7	6	ancoms 268	. . . . . . . 8 |- ( ( A e. ZZ /\ ( x - A ) e. ( ZZ>= ` B ) ) -> ( ( x - A ) + A ) e. ( ZZ>= ` ( B + A ) ) )
8	7	3adant2 1043	. . . . . . 7 |- ( ( A e. ZZ /\ x e. CC /\ ( x - A ) e. ( ZZ>= ` B ) ) -> ( ( x - A ) + A ) e. ( ZZ>= ` ( B + A ) ) )
9	5, 8	eqeltrrd 2310	. . . . . 6 |- ( ( A e. ZZ /\ x e. CC /\ ( x - A ) e. ( ZZ>= ` B ) ) -> x e. ( ZZ>= ` ( B + A ) ) )
10	9	3expib 1233	. . . . 5 |- ( A e. ZZ -> ( ( x e. CC /\ ( x - A ) e. ( ZZ>= ` B ) ) -> x e. ( ZZ>= ` ( B + A ) ) ) )
11	10	adantr 276	. . . 4 |- ( ( A e. ZZ /\ B e. ZZ ) -> ( ( x e. CC /\ ( x - A ) e. ( ZZ>= ` B ) ) -> x e. ( ZZ>= ` ( B + A ) ) ) )
12		eluzelcn 9865	. . . . . 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 9884	. . . . . . 7 |- ( ( B e. ZZ /\ A e. ZZ /\ x e. ( ZZ>= ` ( B + A ) ) ) -> ( x - A ) e. ( ZZ>= ` B ) )
15	14	3expia 1232	. . . . . 6 |- ( ( B e. ZZ /\ A e. ZZ ) -> ( x e. ( ZZ>= ` ( B + A ) ) -> ( x - A ) e. ( ZZ>= ` B ) ) )
16	15	ancoms 268	. . . . 5 |- ( ( A e. ZZ /\ B e. ZZ ) -> ( x e. ( ZZ>= ` ( B + A ) ) -> ( x - A ) e. ( ZZ>= ` B ) ) )
17	13, 16	jcad 307	. . . 4 |- ( ( A e. ZZ /\ B e. ZZ ) -> ( x e. ( ZZ>= ` ( B + A ) ) -> ( x e. CC /\ ( x - A ) e. ( ZZ>= ` B ) ) ) )
18	11, 17	impbid 129	. . 3 |- ( ( A e. ZZ /\ B e. ZZ ) -> ( ( x e. CC /\ ( x - A ) e. ( ZZ>= ` B ) ) <-> x e. ( ZZ>= ` ( B + A ) ) ) )
19	18	abbi1dv 2354	. 2 |- ( ( A e. ZZ /\ B e. ZZ ) -> { x | ( x e. CC /\ ( x - A ) e. ( ZZ>= ` B ) ) } = ( ZZ>= ` ( B + A ) ) )
20	1, 19	eqtrid 2277	1 |- ( ( A e. ZZ /\ B e. ZZ ) -> { x e. CC | ( x - A ) e. ( ZZ>= ` B ) } = ( ZZ>= ` ( B + A ) ) )

Syntax hints:   -> wi 4   /\ wa 104   /\ w3a 1005   = wceq 1398   e. wcel 2203   {cab 2218   {crab 2524   ` cfv 5352  (class class class)co 6050   CCcc 8125   + caddc 8130   - cmin 8444   ZZcz 9577   ZZ>=cuz 9853

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-in1 619  ax-in2 620  ax-io 717  ax-5 1496  ax-7 1497  ax-gen 1498  ax-ie1 1542  ax-ie2 1543  ax-8 1553  ax-10 1554  ax-11 1555  ax-i12 1556  ax-bndl 1558  ax-4 1559  ax-17 1575  ax-i9 1579  ax-ial 1583  ax-i5r 1584  ax-13 2205  ax-14 2206  ax-ext 2214  ax-sep 4228  ax-pow 4287  ax-pr 4322  ax-un 4554  ax-setind 4659  ax-cnex 8218  ax-resscn 8219  ax-1cn 8220  ax-1re 8221  ax-icn 8222  ax-addcl 8223  ax-addrcl 8224  ax-mulcl 8225  ax-addcom 8227  ax-addass 8229  ax-distr 8231  ax-i2m1 8232  ax-0lt1 8233  ax-0id 8235  ax-rnegex 8236  ax-cnre 8238  ax-pre-ltirr 8239  ax-pre-ltwlin 8240  ax-pre-lttrn 8241  ax-pre-ltadd 8243

This theorem depends on definitions:  df-bi 117  df-3or 1006  df-3an 1007  df-tru 1401  df-fal 1404  df-nf 1510  df-sb 1812  df-eu 2083  df-mo 2084  df-clab 2219  df-cleq 2225  df-clel 2228  df-nfc 2373  df-ne 2413  df-nel 2508  df-ral 2525  df-rex 2526  df-reu 2527  df-rab 2529  df-v 2815  df-sbc 3043  df-dif 3213  df-un 3215  df-in 3217  df-ss 3224  df-pw 3671  df-sn 3695  df-pr 3696  df-op 3698  df-uni 3915  df-int 3950  df-br 4110  df-opab 4172  df-mpt 4173  df-id 4414  df-xp 4755  df-rel 4756  df-cnv 4757  df-co 4758  df-dm 4759  df-rn 4760  df-res 4761  df-ima 4762  df-iota 5312  df-fun 5354  df-fn 5355  df-f 5356  df-fv 5360  df-riota 6003  df-ov 6053  df-oprab 6054  df-mpo 6055  df-pnf 8310  df-mnf 8311  df-xr 8312  df-ltxr 8313  df-le 8314  df-sub 8446  df-neg 8447  df-inn 9238  df-n0 9497  df-z 9578  df-uz 9854

This theorem is referenced by:  seq3shft  11523