Theorem iccdili 10121

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

Hypotheses

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

Assertion

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

Proof of Theorem iccdili

Step	Hyp	Ref	Expression
1		iccdili.1	. . . 4 |- A e. RR
2		iccdili.2	. . . 4 |- B e. RR
3		iccssre 10077	. . . 4 |- ( ( A e. RR /\ B e. RR ) -> ( A [,] B ) C_ RR )
4	1, 2, 3	mp2an 426	. . 3 |- ( A [,] B ) C_ RR
5	4	sseli 3189	. 2 |- ( X e. ( A [,] B ) -> X e. RR )
6		iccdili.3	. . . 4 |- R e. RR+
7		iccdili.4	. . . . . 6 |- ( A x. R ) = C
8		iccdili.5	. . . . . 6 |- ( B x. R ) = D
9	7, 8	iccdil 10120	. . . . 5 |- ( ( ( A e. RR /\ B e. RR ) /\ ( X e. RR /\ R e. RR+ ) ) -> ( X e. ( A [,] B ) <-> ( X x. R ) e. ( C [,] D ) ) )
10	1, 2, 9	mpanl12 436	. . . 4 |- ( ( X e. RR /\ R e. RR+ ) -> ( X e. ( A [,] B ) <-> ( X x. R ) e. ( C [,] D ) ) )
11	6, 10	mpan2 425	. . 3 |- ( X e. RR -> ( X e. ( A [,] B ) <-> ( X x. R ) e. ( C [,] D ) ) )
12	11	biimpd 144	. 2 |- ( X e. RR -> ( X e. ( A [,] B ) -> ( X x. R ) e. ( C [,] D ) ) )
13	5, 12	mpcom 36	1 |- ( X e. ( A [,] B ) -> ( X x. R ) e. ( C [,] D ) )

Syntax hints:   -> wi 4   /\ wa 104   <-> wb 105   = wceq 1373   e. wcel 2176   C_ wss 3166  (class class class)co 5944   RRcr 7924   x. cmul 7930   RR+crp 9775   [,]cicc 10013

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 615  ax-in2 616  ax-io 711  ax-5 1470  ax-7 1471  ax-gen 1472  ax-ie1 1516  ax-ie2 1517  ax-8 1527  ax-10 1528  ax-11 1529  ax-i12 1530  ax-bndl 1532  ax-4 1533  ax-17 1549  ax-i9 1553  ax-ial 1557  ax-i5r 1558  ax-13 2178  ax-14 2179  ax-ext 2187  ax-sep 4162  ax-pow 4218  ax-pr 4253  ax-un 4480  ax-setind 4585  ax-cnex 8016  ax-resscn 8017  ax-1cn 8018  ax-1re 8019  ax-icn 8020  ax-addcl 8021  ax-addrcl 8022  ax-mulcl 8023  ax-mulrcl 8024  ax-addcom 8025  ax-mulcom 8026  ax-addass 8027  ax-mulass 8028  ax-distr 8029  ax-i2m1 8030  ax-1rid 8032  ax-0id 8033  ax-rnegex 8034  ax-precex 8035  ax-cnre 8036  ax-pre-ltirr 8037  ax-pre-ltwlin 8038  ax-pre-lttrn 8039  ax-pre-ltadd 8041  ax-pre-mulgt0 8042

This theorem depends on definitions:  df-bi 117  df-3or 982  df-3an 983  df-tru 1376  df-fal 1379  df-nf 1484  df-sb 1786  df-eu 2057  df-mo 2058  df-clab 2192  df-cleq 2198  df-clel 2201  df-nfc 2337  df-ne 2377  df-nel 2472  df-ral 2489  df-rex 2490  df-reu 2491  df-rab 2493  df-v 2774  df-sbc 2999  df-dif 3168  df-un 3170  df-in 3172  df-ss 3179  df-pw 3618  df-sn 3639  df-pr 3640  df-op 3642  df-uni 3851  df-br 4045  df-opab 4106  df-id 4340  df-po 4343  df-iso 4344  df-xp 4681  df-rel 4682  df-cnv 4683  df-co 4684  df-dm 4685  df-iota 5232  df-fun 5273  df-fv 5279  df-riota 5899  df-ov 5947  df-oprab 5948  df-mpo 5949  df-pnf 8109  df-mnf 8110  df-xr 8111  df-ltxr 8112  df-le 8113  df-sub 8245  df-neg 8246  df-rp 9776  df-icc 10017

This theorem is referenced by: (None)