Theorem iccshftli 9810

Description: Membership in a shifted interval. (Contributed by Jeff Madsen, 2-Sep-2009.)

Hypotheses

Ref	Expression
iccshftli.1	|- A e. RR
iccshftli.2	|- B e. RR
iccshftli.3	|- R e. RR
iccshftli.4	|- ( A - R ) = C
iccshftli.5	|- ( B - R ) = D

Assertion

Ref	Expression
iccshftli	|- ( X e. ( A [,] B ) -> ( X - R ) e. ( C [,] D ) )

Proof of Theorem iccshftli

Step	Hyp	Ref	Expression
1		iccshftli.1	. . . 4 |- A e. RR
2		iccshftli.2	. . . 4 |- B e. RR
3		iccssre 9768	. . . 4 |- ( ( A e. RR /\ B e. RR ) -> ( A [,] B ) C_ RR )
4	1, 2, 3	mp2an 423	. . 3 |- ( A [,] B ) C_ RR
5	4	sseli 3098	. 2 |- ( X e. ( A [,] B ) -> X e. RR )
6		iccshftli.3	. . . 4 |- R e. RR
7		iccshftli.4	. . . . . 6 |- ( A - R ) = C
8		iccshftli.5	. . . . . 6 |- ( B - R ) = D
9	7, 8	iccshftl 9809	. . . . 5 |- ( ( ( A e. RR /\ B e. RR ) /\ ( X e. RR /\ R e. RR ) ) -> ( X e. ( A [,] B ) <-> ( X - R ) e. ( C [,] D ) ) )
10	1, 2, 9	mpanl12 433	. . . 4 |- ( ( X e. RR /\ R e. RR ) -> ( X e. ( A [,] B ) <-> ( X - R ) e. ( C [,] D ) ) )
11	6, 10	mpan2 422	. . 3 |- ( X e. RR -> ( X e. ( A [,] B ) <-> ( X - R ) e. ( C [,] D ) ) )
12	11	biimpd 143	. 2 |- ( X e. RR -> ( X e. ( A [,] B ) -> ( X - R ) e. ( C [,] D ) ) )
13	5, 12	mpcom 36	1 |- ( X e. ( A [,] B ) -> ( X - R ) e. ( C [,] D ) )

Syntax hints:   -> wi 4   /\ wa 103   <-> wb 104   = wceq 1332   e. wcel 1481   C_ wss 3076  (class class class)co 5782   RRcr 7643   - cmin 7957   [,]cicc 9704

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 1424  ax-7 1425  ax-gen 1426  ax-ie1 1470  ax-ie2 1471  ax-8 1483  ax-10 1484  ax-11 1485  ax-i12 1486  ax-bndl 1487  ax-4 1488  ax-13 1492  ax-14 1493  ax-17 1507  ax-i9 1511  ax-ial 1515  ax-i5r 1516  ax-ext 2122  ax-sep 4054  ax-pow 4106  ax-pr 4139  ax-un 4363  ax-setind 4460  ax-cnex 7735  ax-resscn 7736  ax-1cn 7737  ax-icn 7739  ax-addcl 7740  ax-addrcl 7741  ax-mulcl 7742  ax-addcom 7744  ax-addass 7746  ax-distr 7748  ax-i2m1 7749  ax-0id 7752  ax-rnegex 7753  ax-cnre 7755  ax-pre-ltirr 7756  ax-pre-ltwlin 7757  ax-pre-lttrn 7758  ax-pre-ltadd 7760

This theorem depends on definitions:  df-bi 116  df-3or 964  df-3an 965  df-tru 1335  df-fal 1338  df-nf 1438  df-sb 1737  df-eu 2003  df-mo 2004  df-clab 2127  df-cleq 2133  df-clel 2136  df-nfc 2271  df-ne 2310  df-nel 2405  df-ral 2422  df-rex 2423  df-reu 2424  df-rab 2426  df-v 2691  df-sbc 2914  df-dif 3078  df-un 3080  df-in 3082  df-ss 3089  df-pw 3517  df-sn 3538  df-pr 3539  df-op 3541  df-uni 3745  df-br 3938  df-opab 3998  df-id 4223  df-po 4226  df-iso 4227  df-xp 4553  df-rel 4554  df-cnv 4555  df-co 4556  df-dm 4557  df-iota 5096  df-fun 5133  df-fv 5139  df-riota 5738  df-ov 5785  df-oprab 5786  df-mpo 5787  df-pnf 7826  df-mnf 7827  df-xr 7828  df-ltxr 7829  df-le 7830  df-sub 7959  df-neg 7960  df-icc 9708

This theorem is referenced by: (None)