Theorem iooshf 8821

Description: Shift the arguments of the open interval function. (Contributed by NM, 17-Aug-2008.)

Assertion

Ref	Expression
iooshf	|- ( ( ( A e. RR /\ B e. RR ) /\ ( C e. RR /\ D e. RR ) ) -> ( ( A - B ) e. ( C (,) D ) <-> A e. ( ( C + B ) (,) ( D + B ) ) ) )

Proof of Theorem iooshf

Step	Hyp	Ref	Expression
1		ltaddsub 7431	. . . . . 6 |- ( ( C e. RR /\ B e. RR /\ A e. RR ) -> ( ( C + B ) < A <-> C < ( A - B ) ) )
2	1	3com13 1109	. . . . 5 |- ( ( A e. RR /\ B e. RR /\ C e. RR ) -> ( ( C + B ) < A <-> C < ( A - B ) ) )
3	2	3expa 1104	. . . 4 |- ( ( ( A e. RR /\ B e. RR ) /\ C e. RR ) -> ( ( C + B ) < A <-> C < ( A - B ) ) )
4	3	adantrr 448	. . 3 |- ( ( ( A e. RR /\ B e. RR ) /\ ( C e. RR /\ D e. RR ) ) -> ( ( C + B ) < A <-> C < ( A - B ) ) )
5		ltsubadd 7427	. . . . . 6 |- ( ( A e. RR /\ B e. RR /\ D e. RR ) -> ( ( A - B ) < D <-> A < ( D + B ) ) )
6	5	bicomd 129	. . . . 5 |- ( ( A e. RR /\ B e. RR /\ D e. RR ) -> ( A < ( D + B ) <-> ( A - B ) < D ) )
7	6	3expa 1104	. . . 4 |- ( ( ( A e. RR /\ B e. RR ) /\ D e. RR ) -> ( A < ( D + B ) <-> ( A - B ) < D ) )
8	7	adantrl 447	. . 3 |- ( ( ( A e. RR /\ B e. RR ) /\ ( C e. RR /\ D e. RR ) ) -> ( A < ( D + B ) <-> ( A - B ) < D ) )
9	4, 8	anbi12d 442	. 2 |- ( ( ( A e. RR /\ B e. RR ) /\ ( C e. RR /\ D e. RR ) ) -> ( ( ( C + B ) < A /\ A < ( D + B ) ) <-> ( C < ( A - B ) /\ ( A - B ) < D ) ) )
10		readdcl 7007	. . . . . 6 |- ( ( C e. RR /\ B e. RR ) -> ( C + B ) e. RR )
11	10	rexrd 7075	. . . . 5 |- ( ( C e. RR /\ B e. RR ) -> ( C + B ) e. RR* )
12	11	ad2ant2rl 480	. . . 4 |- ( ( ( C e. RR /\ D e. RR ) /\ ( A e. RR /\ B e. RR ) ) -> ( C + B ) e. RR* )
13		readdcl 7007	. . . . . 6 |- ( ( D e. RR /\ B e. RR ) -> ( D + B ) e. RR )
14	13	rexrd 7075	. . . . 5 |- ( ( D e. RR /\ B e. RR ) -> ( D + B ) e. RR* )
15	14	ad2ant2l 477	. . . 4 |- ( ( ( C e. RR /\ D e. RR ) /\ ( A e. RR /\ B e. RR ) ) -> ( D + B ) e. RR* )
16		rexr 7071	. . . . 5 |- ( A e. RR -> A e. RR* )
17	16	ad2antrl 459	. . . 4 |- ( ( ( C e. RR /\ D e. RR ) /\ ( A e. RR /\ B e. RR ) ) -> A e. RR* )
18		elioo5 8802	. . . 4 |- ( ( ( C + B ) e. RR* /\ ( D + B ) e. RR* /\ A e. RR* ) -> ( A e. ( ( C + B ) (,) ( D + B ) ) <-> ( ( C + B ) < A /\ A < ( D + B ) ) ) )
19	12, 15, 17, 18	syl3anc 1135	. . 3 |- ( ( ( C e. RR /\ D e. RR ) /\ ( A e. RR /\ B e. RR ) ) -> ( A e. ( ( C + B ) (,) ( D + B ) ) <-> ( ( C + B ) < A /\ A < ( D + B ) ) ) )
20	19	ancoms 255	. 2 |- ( ( ( A e. RR /\ B e. RR ) /\ ( C e. RR /\ D e. RR ) ) -> ( A e. ( ( C + B ) (,) ( D + B ) ) <-> ( ( C + B ) < A /\ A < ( D + B ) ) ) )
21		rexr 7071	. . . 4 |- ( C e. RR -> C e. RR* )
22	21	ad2antrl 459	. . 3 |- ( ( ( A e. RR /\ B e. RR ) /\ ( C e. RR /\ D e. RR ) ) -> C e. RR* )
23		rexr 7071	. . . 4 |- ( D e. RR -> D e. RR* )
24	23	ad2antll 460	. . 3 |- ( ( ( A e. RR /\ B e. RR ) /\ ( C e. RR /\ D e. RR ) ) -> D e. RR* )
25		resubcl 7275	. . . . 5 |- ( ( A e. RR /\ B e. RR ) -> ( A - B ) e. RR )
26	25	rexrd 7075	. . . 4 |- ( ( A e. RR /\ B e. RR ) -> ( A - B ) e. RR* )
27	26	adantr 261	. . 3 |- ( ( ( A e. RR /\ B e. RR ) /\ ( C e. RR /\ D e. RR ) ) -> ( A - B ) e. RR* )
28		elioo5 8802	. . 3 |- ( ( C e. RR* /\ D e. RR* /\ ( A - B ) e. RR* ) -> ( ( A - B ) e. ( C (,) D ) <-> ( C < ( A - B ) /\ ( A - B ) < D ) ) )
29	22, 24, 27, 28	syl3anc 1135	. 2 |- ( ( ( A e. RR /\ B e. RR ) /\ ( C e. RR /\ D e. RR ) ) -> ( ( A - B ) e. ( C (,) D ) <-> ( C < ( A - B ) /\ ( A - B ) < D ) ) )
30	9, 20, 29	3bitr4rd 210	1 |- ( ( ( A e. RR /\ B e. RR ) /\ ( C e. RR /\ D e. RR ) ) -> ( ( A - B ) e. ( C (,) D ) <-> A e. ( ( C + B ) (,) ( D + B ) ) ) )

Syntax hints:   -> wi 4   /\ wa 97   <-> wb 98   /\ w3a 885   e. wcel 1393   class class class wbr 3764  (class class class)co 5512   RRcr 6888   + caddc 6892   RR*cxr 7059   < clt 7060   - cmin 7182   (,)cioo 8757

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 99  ax-ia2 100  ax-ia3 101  ax-in1 544  ax-in2 545  ax-io 630  ax-5 1336  ax-7 1337  ax-gen 1338  ax-ie1 1382  ax-ie2 1383  ax-8 1395  ax-10 1396  ax-11 1397  ax-i12 1398  ax-bndl 1399  ax-4 1400  ax-13 1404  ax-14 1405  ax-17 1419  ax-i9 1423  ax-ial 1427  ax-i5r 1428  ax-ext 2022  ax-sep 3875  ax-pow 3927  ax-pr 3944  ax-un 4170  ax-setind 4262  ax-cnex 6975  ax-resscn 6976  ax-1cn 6977  ax-icn 6979  ax-addcl 6980  ax-addrcl 6981  ax-mulcl 6982  ax-addcom 6984  ax-addass 6986  ax-distr 6988  ax-i2m1 6989  ax-0id 6992  ax-rnegex 6993  ax-cnre 6995  ax-pre-ltadd 7000

This theorem depends on definitions:  df-bi 110  df-3an 887  df-tru 1246  df-fal 1249  df-nf 1350  df-sb 1646  df-eu 1903  df-mo 1904  df-clab 2027  df-cleq 2033  df-clel 2036  df-nfc 2167  df-ne 2206  df-nel 2207  df-ral 2311  df-rex 2312  df-reu 2313  df-rab 2315  df-v 2559  df-sbc 2765  df-dif 2920  df-un 2922  df-in 2924  df-ss 2931  df-pw 3361  df-sn 3381  df-pr 3382  df-op 3384  df-uni 3581  df-br 3765  df-opab 3819  df-id 4030  df-xp 4351  df-rel 4352  df-cnv 4353  df-co 4354  df-dm 4355  df-iota 4867  df-fun 4904  df-fv 4910  df-riota 5468  df-ov 5515  df-oprab 5516  df-mpt2 5517  df-pnf 7062  df-mnf 7063  df-xr 7064  df-ltxr 7065  df-sub 7184  df-neg 7185  df-ioo 8761

This theorem is referenced by: (None)