Theorem icccntri 9023

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

Hypotheses

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

Assertion

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

Proof of Theorem icccntri

Step	Hyp	Ref	Expression
1		icccntri.1	. . . 4 |- A e. RR
2		icccntri.2	. . . 4 |- B e. RR
3		iccssre 8978	. . . 4 |- ( ( A e. RR /\ B e. RR ) -> ( A [,] B ) C_ RR )
4	1, 2, 3	mp2an 416	. . 3 |- ( A [,] B ) C_ RR
5	4	sseli 2995	. 2 |- ( X e. ( A [,] B ) -> X e. RR )
6		icccntri.3	. . . 4 |- R e. RR+
7		icccntri.4	. . . . . 6 |- ( A / R ) = C
8		icccntri.5	. . . . . 6 |- ( B / R ) = D
9	7, 8	icccntr 9022	. . . . 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 426	. . . 4 |- ( ( X e. RR /\ R e. RR+ ) -> ( X e. ( A [,] B ) <-> ( X / R ) e. ( C [,] D ) ) )
11	6, 10	mpan2 415	. . 3 |- ( X e. RR -> ( X e. ( A [,] B ) <-> ( X / R ) e. ( C [,] D ) ) )
12	11	biimpd 142	. 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 102   <-> wb 103   = wceq 1284   e. wcel 1433   C_ wss 2973  (class class class)co 5532   RRcr 6980   / cdiv 7760   RR+crp 8734   [,]cicc 8914

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-in1 576  ax-in2 577  ax-io 662  ax-5 1376  ax-7 1377  ax-gen 1378  ax-ie1 1422  ax-ie2 1423  ax-8 1435  ax-10 1436  ax-11 1437  ax-i12 1438  ax-bndl 1439  ax-4 1440  ax-13 1444  ax-14 1445  ax-17 1459  ax-i9 1463  ax-ial 1467  ax-i5r 1468  ax-ext 2063  ax-sep 3896  ax-pow 3948  ax-pr 3964  ax-un 4188  ax-setind 4280  ax-cnex 7067  ax-resscn 7068  ax-1cn 7069  ax-1re 7070  ax-icn 7071  ax-addcl 7072  ax-addrcl 7073  ax-mulcl 7074  ax-mulrcl 7075  ax-addcom 7076  ax-mulcom 7077  ax-addass 7078  ax-mulass 7079  ax-distr 7080  ax-i2m1 7081  ax-0lt1 7082  ax-1rid 7083  ax-0id 7084  ax-rnegex 7085  ax-precex 7086  ax-cnre 7087  ax-pre-ltirr 7088  ax-pre-ltwlin 7089  ax-pre-lttrn 7090  ax-pre-apti 7091  ax-pre-ltadd 7092  ax-pre-mulgt0 7093  ax-pre-mulext 7094

This theorem depends on definitions:  df-bi 115  df-3or 920  df-3an 921  df-tru 1287  df-fal 1290  df-nf 1390  df-sb 1686  df-eu 1944  df-mo 1945  df-clab 2068  df-cleq 2074  df-clel 2077  df-nfc 2208  df-ne 2246  df-nel 2340  df-ral 2353  df-rex 2354  df-reu 2355  df-rmo 2356  df-rab 2357  df-v 2603  df-sbc 2816  df-dif 2975  df-un 2977  df-in 2979  df-ss 2986  df-pw 3384  df-sn 3404  df-pr 3405  df-op 3407  df-uni 3602  df-br 3786  df-opab 3840  df-id 4048  df-po 4051  df-iso 4052  df-xp 4369  df-rel 4370  df-cnv 4371  df-co 4372  df-dm 4373  df-iota 4887  df-fun 4924  df-fv 4930  df-riota 5488  df-ov 5535  df-oprab 5536  df-mpt2 5537  df-pnf 7155  df-mnf 7156  df-xr 7157  df-ltxr 7158  df-le 7159  df-sub 7281  df-neg 7282  df-reap 7675  df-ap 7682  df-div 7761  df-rp 8735  df-icc 8918

This theorem is referenced by: (None)