Theorem iccshftli 10330

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 10288	. . . 4 |- ( ( A e. RR /\ B e. RR ) -> ( A [,] B ) C_ RR )
4	1, 2, 3	mp2an 426	. . 3 |- ( A [,] B ) C_ RR
5	4	sseli 3234	. 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 10329	. . . . 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 436	. . . 4 |- ( ( X e. RR /\ R e. RR ) -> ( X e. ( A [,] B ) <-> ( X - R ) e. ( C [,] D ) ) )
11	6, 10	mpan2 425	. . 3 |- ( X e. RR -> ( X e. ( A [,] B ) <-> ( X - R ) e. ( C [,] D ) ) )
12	11	biimpd 144	. 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 104   <-> wb 105   = wceq 1398   e. wcel 2203   C_ wss 3211  (class class class)co 6050   RRcr 8126   - cmin 8444   [,]cicc 10224

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-icn 8222  ax-addcl 8223  ax-addrcl 8224  ax-mulcl 8225  ax-addcom 8227  ax-addass 8229  ax-distr 8231  ax-i2m1 8232  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-br 4110  df-opab 4172  df-id 4414  df-po 4417  df-iso 4418  df-xp 4755  df-rel 4756  df-cnv 4757  df-co 4758  df-dm 4759  df-iota 5312  df-fun 5354  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-icc 10228

This theorem is referenced by: (None)