Theorem iccshftl 10377

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

Hypotheses

Ref	Expression
iccshftl.1	|- ( A - R ) = C
iccshftl.2	|- ( B - R ) = D

Assertion

Ref	Expression
iccshftl	|- ( ( ( A e. RR /\ B e. RR ) /\ ( X e. RR /\ R e. RR ) ) -> ( X e. ( A [,] B ) <-> ( X - R ) e. ( C [,] D ) ) )

Proof of Theorem iccshftl

Step	Hyp	Ref	Expression
1		simpl 109	. . . . 5 |- ( ( X e. RR /\ R e. RR ) -> X e. RR )
2		resubcl 8580	. . . . 5 |- ( ( X e. RR /\ R e. RR ) -> ( X - R ) e. RR )
3	1, 2	2thd 175	. . . 4 |- ( ( X e. RR /\ R e. RR ) -> ( X e. RR <-> ( X - R ) e. RR ) )
4	3	adantl 277	. . 3 |- ( ( ( A e. RR /\ B e. RR ) /\ ( X e. RR /\ R e. RR ) ) -> ( X e. RR <-> ( X - R ) e. RR ) )
5		lesub1 8774	. . . . . 6 |- ( ( A e. RR /\ X e. RR /\ R e. RR ) -> ( A <_ X <-> ( A - R ) <_ ( X - R ) ) )
6	5	3expb 1235	. . . . 5 |- ( ( A e. RR /\ ( X e. RR /\ R e. RR ) ) -> ( A <_ X <-> ( A - R ) <_ ( X - R ) ) )
7	6	adantlr 481	. . . 4 |- ( ( ( A e. RR /\ B e. RR ) /\ ( X e. RR /\ R e. RR ) ) -> ( A <_ X <-> ( A - R ) <_ ( X - R ) ) )
8		iccshftl.1	. . . . 5 |- ( A - R ) = C
9	8	breq1i 4132	. . . 4 |- ( ( A - R ) <_ ( X - R ) <-> C <_ ( X - R ) )
10	7, 9	bitrdi 196	. . 3 |- ( ( ( A e. RR /\ B e. RR ) /\ ( X e. RR /\ R e. RR ) ) -> ( A <_ X <-> C <_ ( X - R ) ) )
11		lesub1 8774	. . . . . . 7 |- ( ( X e. RR /\ B e. RR /\ R e. RR ) -> ( X <_ B <-> ( X - R ) <_ ( B - R ) ) )
12	11	3expb 1235	. . . . . 6 |- ( ( X e. RR /\ ( B e. RR /\ R e. RR ) ) -> ( X <_ B <-> ( X - R ) <_ ( B - R ) ) )
13	12	an12s 571	. . . . 5 |- ( ( B e. RR /\ ( X e. RR /\ R e. RR ) ) -> ( X <_ B <-> ( X - R ) <_ ( B - R ) ) )
14	13	adantll 480	. . . 4 |- ( ( ( A e. RR /\ B e. RR ) /\ ( X e. RR /\ R e. RR ) ) -> ( X <_ B <-> ( X - R ) <_ ( B - R ) ) )
15		iccshftl.2	. . . . 5 |- ( B - R ) = D
16	15	breq2i 4133	. . . 4 |- ( ( X - R ) <_ ( B - R ) <-> ( X - R ) <_ D )
17	14, 16	bitrdi 196	. . 3 |- ( ( ( A e. RR /\ B e. RR ) /\ ( X e. RR /\ R e. RR ) ) -> ( X <_ B <-> ( X - R ) <_ D ) )
18	4, 10, 17	3anbi123d 1353	. 2 |- ( ( ( A e. RR /\ B e. RR ) /\ ( X e. RR /\ R e. RR ) ) -> ( ( X e. RR /\ A <_ X /\ X <_ B ) <-> ( ( X - R ) e. RR /\ C <_ ( X - R ) /\ ( X - R ) <_ D ) ) )
19		elicc2 10319	. . 3 |- ( ( A e. RR /\ B e. RR ) -> ( X e. ( A [,] B ) <-> ( X e. RR /\ A <_ X /\ X <_ B ) ) )
20	19	adantr 276	. 2 |- ( ( ( A e. RR /\ B e. RR ) /\ ( X e. RR /\ R e. RR ) ) -> ( X e. ( A [,] B ) <-> ( X e. RR /\ A <_ X /\ X <_ B ) ) )
21		resubcl 8580	. . . . . 6 |- ( ( A e. RR /\ R e. RR ) -> ( A - R ) e. RR )
22	8, 21	eqeltrrid 2326	. . . . 5 |- ( ( A e. RR /\ R e. RR ) -> C e. RR )
23		resubcl 8580	. . . . . 6 |- ( ( B e. RR /\ R e. RR ) -> ( B - R ) e. RR )
24	15, 23	eqeltrrid 2326	. . . . 5 |- ( ( B e. RR /\ R e. RR ) -> D e. RR )
25		elicc2 10319	. . . . 5 |- ( ( C e. RR /\ D e. RR ) -> ( ( X - R ) e. ( C [,] D ) <-> ( ( X - R ) e. RR /\ C <_ ( X - R ) /\ ( X - R ) <_ D ) ) )
26	22, 24, 25	syl2an 289	. . . 4 |- ( ( ( A e. RR /\ R e. RR ) /\ ( B e. RR /\ R e. RR ) ) -> ( ( X - R ) e. ( C [,] D ) <-> ( ( X - R ) e. RR /\ C <_ ( X - R ) /\ ( X - R ) <_ D ) ) )
27	26	anandirs 601	. . 3 |- ( ( ( A e. RR /\ B e. RR ) /\ R e. RR ) -> ( ( X - R ) e. ( C [,] D ) <-> ( ( X - R ) e. RR /\ C <_ ( X - R ) /\ ( X - R ) <_ D ) ) )
28	27	adantrl 482	. 2 |- ( ( ( A e. RR /\ B e. RR ) /\ ( X e. RR /\ R e. RR ) ) -> ( ( X - R ) e. ( C [,] D ) <-> ( ( X - R ) e. RR /\ C <_ ( X - R ) /\ ( X - R ) <_ D ) ) )
29	18, 20, 28	3bitr4d 220	1 |- ( ( ( A e. RR /\ B e. RR ) /\ ( X e. RR /\ R e. RR ) ) -> ( X e. ( A [,] B ) <-> ( X - R ) e. ( C [,] D ) ) )

Syntax hints:   -> wi 4   /\ wa 104   <-> wb 105   /\ w3a 1009   = wceq 1402   e. wcel 2209   class class class wbr 4125  (class class class)co 6075   RRcr 8168   <_ cle 8351   - cmin 8487   [,]cicc 10272

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 623  ax-in2 624  ax-io 721  ax-5 1500  ax-7 1501  ax-gen 1502  ax-ie1 1546  ax-ie2 1547  ax-8 1557  ax-10 1558  ax-11 1559  ax-i12 1560  ax-bndl 1562  ax-4 1563  ax-17 1579  ax-i9 1583  ax-ial 1587  ax-i5r 1588  ax-14 2212  ax-ext 2220  ax-sep 4244  ax-pow 4306  ax-pr 4341  ax-un 4573  ax-setind 4679  ax-cnex 8260  ax-resscn 8261  ax-1cn 8262  ax-icn 8264  ax-addcl 8265  ax-addrcl 8266  ax-mulcl 8267  ax-addcom 8269  ax-addass 8271  ax-distr 8273  ax-i2m1 8274  ax-0id 8277  ax-rnegex 8278  ax-cnre 8280  ax-pre-ltirr 8281  ax-pre-ltwlin 8282  ax-pre-lttrn 8283  ax-pre-ltadd 8285

This theorem depends on definitions:  df-bi 117  df-3or 1010  df-3an 1011  df-tru 1405  df-fal 1408  df-nf 1514  df-sb 1816  df-eu 2089  df-mo 2090  df-clab 2225  df-cleq 2231  df-clel 2234  df-nfc 2381  df-ne 2421  df-nel 2516  df-ral 2533  df-rex 2534  df-reu 2535  df-rab 2537  df-v 2823  df-sbc 3052  df-dif 3222  df-un 3224  df-in 3226  df-ss 3233  df-pw 3687  df-sn 3711  df-pr 3712  df-op 3714  df-uni 3931  df-br 4126  df-opab 4188  df-id 4433  df-po 4436  df-iso 4437  df-xp 4775  df-rel 4776  df-cnv 4777  df-co 4778  df-dm 4779  df-iota 5332  df-fun 5374  df-fv 5380  df-riota 6028  df-ov 6078  df-oprab 6079  df-mpo 6080  df-pnf 8352  df-mnf 8353  df-xr 8354  df-ltxr 8355  df-le 8356  df-sub 8489  df-neg 8490  df-icc 10276

This theorem is referenced by:  iccshftli  10378  iccf1o  10386