Theorem iccdili 9994

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 9950	. . . 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 3151	. 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 9993	. . . . 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 1353   e. wcel 2148   C_ wss 3129  (class class class)co 5871   RRcr 7806   x. cmul 7812   RR+crp 9648   [,]cicc 9886

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 614  ax-in2 615  ax-io 709  ax-5 1447  ax-7 1448  ax-gen 1449  ax-ie1 1493  ax-ie2 1494  ax-8 1504  ax-10 1505  ax-11 1506  ax-i12 1507  ax-bndl 1509  ax-4 1510  ax-17 1526  ax-i9 1530  ax-ial 1534  ax-i5r 1535  ax-13 2150  ax-14 2151  ax-ext 2159  ax-sep 4120  ax-pow 4173  ax-pr 4208  ax-un 4432  ax-setind 4535  ax-cnex 7898  ax-resscn 7899  ax-1cn 7900  ax-1re 7901  ax-icn 7902  ax-addcl 7903  ax-addrcl 7904  ax-mulcl 7905  ax-mulrcl 7906  ax-addcom 7907  ax-mulcom 7908  ax-addass 7909  ax-mulass 7910  ax-distr 7911  ax-i2m1 7912  ax-1rid 7914  ax-0id 7915  ax-rnegex 7916  ax-precex 7917  ax-cnre 7918  ax-pre-ltirr 7919  ax-pre-ltwlin 7920  ax-pre-lttrn 7921  ax-pre-ltadd 7923  ax-pre-mulgt0 7924

This theorem depends on definitions:  df-bi 117  df-3or 979  df-3an 980  df-tru 1356  df-fal 1359  df-nf 1461  df-sb 1763  df-eu 2029  df-mo 2030  df-clab 2164  df-cleq 2170  df-clel 2173  df-nfc 2308  df-ne 2348  df-nel 2443  df-ral 2460  df-rex 2461  df-reu 2462  df-rab 2464  df-v 2739  df-sbc 2963  df-dif 3131  df-un 3133  df-in 3135  df-ss 3142  df-pw 3577  df-sn 3598  df-pr 3599  df-op 3601  df-uni 3810  df-br 4003  df-opab 4064  df-id 4292  df-po 4295  df-iso 4296  df-xp 4631  df-rel 4632  df-cnv 4633  df-co 4634  df-dm 4635  df-iota 5176  df-fun 5216  df-fv 5222  df-riota 5827  df-ov 5874  df-oprab 5875  df-mpo 5876  df-pnf 7989  df-mnf 7990  df-xr 7991  df-ltxr 7992  df-le 7993  df-sub 8125  df-neg 8126  df-rp 9649  df-icc 9890

This theorem is referenced by: (None)