Theorem icccntri 9928

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 9882	. . . 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 3133	. 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 9927	. . . . 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 1342   e. wcel 2135   C_ wss 3111  (class class class)co 5836   RRcr 7743   / cdiv 8559   RR+crp 9580   [,]cicc 9818

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 1434  ax-7 1435  ax-gen 1436  ax-ie1 1480  ax-ie2 1481  ax-8 1491  ax-10 1492  ax-11 1493  ax-i12 1494  ax-bndl 1496  ax-4 1497  ax-17 1513  ax-i9 1517  ax-ial 1521  ax-i5r 1522  ax-13 2137  ax-14 2138  ax-ext 2146  ax-sep 4094  ax-pow 4147  ax-pr 4181  ax-un 4405  ax-setind 4508  ax-cnex 7835  ax-resscn 7836  ax-1cn 7837  ax-1re 7838  ax-icn 7839  ax-addcl 7840  ax-addrcl 7841  ax-mulcl 7842  ax-mulrcl 7843  ax-addcom 7844  ax-mulcom 7845  ax-addass 7846  ax-mulass 7847  ax-distr 7848  ax-i2m1 7849  ax-0lt1 7850  ax-1rid 7851  ax-0id 7852  ax-rnegex 7853  ax-precex 7854  ax-cnre 7855  ax-pre-ltirr 7856  ax-pre-ltwlin 7857  ax-pre-lttrn 7858  ax-pre-apti 7859  ax-pre-ltadd 7860  ax-pre-mulgt0 7861  ax-pre-mulext 7862

This theorem depends on definitions:  df-bi 116  df-3or 968  df-3an 969  df-tru 1345  df-fal 1348  df-nf 1448  df-sb 1750  df-eu 2016  df-mo 2017  df-clab 2151  df-cleq 2157  df-clel 2160  df-nfc 2295  df-ne 2335  df-nel 2430  df-ral 2447  df-rex 2448  df-reu 2449  df-rmo 2450  df-rab 2451  df-v 2723  df-sbc 2947  df-dif 3113  df-un 3115  df-in 3117  df-ss 3124  df-pw 3555  df-sn 3576  df-pr 3577  df-op 3579  df-uni 3784  df-br 3977  df-opab 4038  df-id 4265  df-po 4268  df-iso 4269  df-xp 4604  df-rel 4605  df-cnv 4606  df-co 4607  df-dm 4608  df-iota 5147  df-fun 5184  df-fv 5190  df-riota 5792  df-ov 5839  df-oprab 5840  df-mpo 5841  df-pnf 7926  df-mnf 7927  df-xr 7928  df-ltxr 7929  df-le 7930  df-sub 8062  df-neg 8063  df-reap 8464  df-ap 8471  df-div 8560  df-rp 9581  df-icc 9822

This theorem is referenced by: (None)