Theorem icccntri 10382

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 10336	. . . 4 |- ( ( A e. RR /\ B e. RR ) -> ( A [,] B ) C_ RR )
4	1, 2, 3	mp2an 430	. . 3 |- ( A [,] B ) C_ RR
5	4	sseli 3244	. 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 10381	. . . . 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 440	. . . 4 |- ( ( X e. RR /\ R e. RR+ ) -> ( X e. ( A [,] B ) <-> ( X / R ) e. ( C [,] D ) ) )
11	6, 10	mpan2 429	. . 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 1402   e. wcel 2209   C_ wss 3220  (class class class)co 6075   RRcr 8168   / cdiv 8992   RR+crp 10033   [,]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-1re 8263  ax-icn 8264  ax-addcl 8265  ax-addrcl 8266  ax-mulcl 8267  ax-mulrcl 8268  ax-addcom 8269  ax-mulcom 8270  ax-addass 8271  ax-mulass 8272  ax-distr 8273  ax-i2m1 8274  ax-0lt1 8275  ax-1rid 8276  ax-0id 8277  ax-rnegex 8278  ax-precex 8279  ax-cnre 8280  ax-pre-ltirr 8281  ax-pre-ltwlin 8282  ax-pre-lttrn 8283  ax-pre-apti 8284  ax-pre-ltadd 8285  ax-pre-mulgt0 8286  ax-pre-mulext 8287

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-rmo 2536  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-reap 8893  df-ap 8900  df-div 8993  df-rp 10034  df-icc 10276

This theorem is referenced by: (None)