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Theorem icchmeo 18439
Description: The natural bijection from  [ 0 ,  1 ] to an arbitrary nontrivial closed interval  [ A ,  B ] is a homeomorphism. (Contributed by Mario Carneiro, 8-Sep-2015.)
Hypotheses
Ref Expression
icchmeo.j  |-  J  =  ( TopOpen ` fld )
icchmeo.f  |-  F  =  ( x  e.  ( 0 [,] 1 ) 
|->  ( ( x  x.  B )  +  ( ( 1  -  x
)  x.  A ) ) )
Assertion
Ref Expression
icchmeo  |-  ( ( A  e.  RR  /\  B  e.  RR  /\  A  <  B )  ->  F  e.  ( II  Homeo  ( Jt  ( A [,] B ) ) ) )
Distinct variable groups:    x, A    x, B    x, J
Allowed substitution hint:    F( x)

Proof of Theorem icchmeo
Dummy variable  y is distinct from all other variables.
StepHypRef Expression
1 icchmeo.f . . . 4  |-  F  =  ( x  e.  ( 0 [,] 1 ) 
|->  ( ( x  x.  B )  +  ( ( 1  -  x
)  x.  A ) ) )
2 iitopon 18383 . . . . . 6  |-  II  e.  (TopOn `  ( 0 [,] 1 ) )
32a1i 10 . . . . 5  |-  ( ( A  e.  RR  /\  B  e.  RR  /\  A  <  B )  ->  II  e.  (TopOn `  ( 0 [,] 1 ) ) )
4 icchmeo.j . . . . . . . . . 10  |-  J  =  ( TopOpen ` fld )
54dfii3 18387 . . . . . . . . 9  |-  II  =  ( Jt  ( 0 [,] 1 ) )
65oveq2i 5869 . . . . . . . 8  |-  ( II 
Cn  II )  =  ( II  Cn  ( Jt  ( 0 [,] 1
) ) )
74cnfldtop 18293 . . . . . . . . 9  |-  J  e. 
Top
8 cnrest2r 17015 . . . . . . . . 9  |-  ( J  e.  Top  ->  (
II  Cn  ( Jt  (
0 [,] 1 ) ) )  C_  (
II  Cn  J )
)
97, 8ax-mp 8 . . . . . . . 8  |-  ( II 
Cn  ( Jt  ( 0 [,] 1 ) ) )  C_  ( II  Cn  J )
106, 9eqsstri 3208 . . . . . . 7  |-  ( II 
Cn  II )  C_  ( II  Cn  J
)
113cnmptid 17355 . . . . . . 7  |-  ( ( A  e.  RR  /\  B  e.  RR  /\  A  <  B )  ->  (
x  e.  ( 0 [,] 1 )  |->  x )  e.  ( II 
Cn  II ) )
1210, 11sseldi 3178 . . . . . 6  |-  ( ( A  e.  RR  /\  B  e.  RR  /\  A  <  B )  ->  (
x  e.  ( 0 [,] 1 )  |->  x )  e.  ( II 
Cn  J ) )
134cnfldtopon 18292 . . . . . . . 8  |-  J  e.  (TopOn `  CC )
1413a1i 10 . . . . . . 7  |-  ( ( A  e.  RR  /\  B  e.  RR  /\  A  <  B )  ->  J  e.  (TopOn `  CC )
)
15 simp2 956 . . . . . . . 8  |-  ( ( A  e.  RR  /\  B  e.  RR  /\  A  <  B )  ->  B  e.  RR )
1615recnd 8861 . . . . . . 7  |-  ( ( A  e.  RR  /\  B  e.  RR  /\  A  <  B )  ->  B  e.  CC )
173, 14, 16cnmptc 17356 . . . . . 6  |-  ( ( A  e.  RR  /\  B  e.  RR  /\  A  <  B )  ->  (
x  e.  ( 0 [,] 1 )  |->  B )  e.  ( II 
Cn  J ) )
184mulcn 18371 . . . . . . 7  |-  x.  e.  ( ( J  tX  J )  Cn  J
)
1918a1i 10 . . . . . 6  |-  ( ( A  e.  RR  /\  B  e.  RR  /\  A  <  B )  ->  x.  e.  ( ( J  tX  J )  Cn  J
) )
203, 12, 17, 19cnmpt12f 17360 . . . . 5  |-  ( ( A  e.  RR  /\  B  e.  RR  /\  A  <  B )  ->  (
x  e.  ( 0 [,] 1 )  |->  ( x  x.  B ) )  e.  ( II 
Cn  J ) )
21 ax-1cn 8795 . . . . . . . . 9  |-  1  e.  CC
2221a1i 10 . . . . . . . 8  |-  ( ( A  e.  RR  /\  B  e.  RR  /\  A  <  B )  ->  1  e.  CC )
233, 14, 22cnmptc 17356 . . . . . . 7  |-  ( ( A  e.  RR  /\  B  e.  RR  /\  A  <  B )  ->  (
x  e.  ( 0 [,] 1 )  |->  1 )  e.  ( II 
Cn  J ) )
244subcn 18370 . . . . . . . 8  |-  -  e.  ( ( J  tX  J )  Cn  J
)
2524a1i 10 . . . . . . 7  |-  ( ( A  e.  RR  /\  B  e.  RR  /\  A  <  B )  ->  -  e.  ( ( J  tX  J )  Cn  J
) )
263, 23, 12, 25cnmpt12f 17360 . . . . . 6  |-  ( ( A  e.  RR  /\  B  e.  RR  /\  A  <  B )  ->  (
x  e.  ( 0 [,] 1 )  |->  ( 1  -  x ) )  e.  ( II 
Cn  J ) )
27 simp1 955 . . . . . . . 8  |-  ( ( A  e.  RR  /\  B  e.  RR  /\  A  <  B )  ->  A  e.  RR )
2827recnd 8861 . . . . . . 7  |-  ( ( A  e.  RR  /\  B  e.  RR  /\  A  <  B )  ->  A  e.  CC )
293, 14, 28cnmptc 17356 . . . . . 6  |-  ( ( A  e.  RR  /\  B  e.  RR  /\  A  <  B )  ->  (
x  e.  ( 0 [,] 1 )  |->  A )  e.  ( II 
Cn  J ) )
303, 26, 29, 19cnmpt12f 17360 . . . . 5  |-  ( ( A  e.  RR  /\  B  e.  RR  /\  A  <  B )  ->  (
x  e.  ( 0 [,] 1 )  |->  ( ( 1  -  x
)  x.  A ) )  e.  ( II 
Cn  J ) )
314addcn 18369 . . . . . 6  |-  +  e.  ( ( J  tX  J )  Cn  J
)
3231a1i 10 . . . . 5  |-  ( ( A  e.  RR  /\  B  e.  RR  /\  A  <  B )  ->  +  e.  ( ( J  tX  J )  Cn  J
) )
333, 20, 30, 32cnmpt12f 17360 . . . 4  |-  ( ( A  e.  RR  /\  B  e.  RR  /\  A  <  B )  ->  (
x  e.  ( 0 [,] 1 )  |->  ( ( x  x.  B
)  +  ( ( 1  -  x )  x.  A ) ) )  e.  ( II 
Cn  J ) )
341, 33syl5eqel 2367 . . 3  |-  ( ( A  e.  RR  /\  B  e.  RR  /\  A  <  B )  ->  F  e.  ( II  Cn  J
) )
351iccf1o 10778 . . . . . 6  |-  ( ( A  e.  RR  /\  B  e.  RR  /\  A  <  B )  ->  ( F : ( 0 [,] 1 ) -1-1-onto-> ( A [,] B
)  /\  `' F  =  ( y  e.  ( A [,] B
)  |->  ( ( y  -  A )  / 
( B  -  A
) ) ) ) )
3635simpld 445 . . . . 5  |-  ( ( A  e.  RR  /\  B  e.  RR  /\  A  <  B )  ->  F : ( 0 [,] 1 ) -1-1-onto-> ( A [,] B
) )
37 f1of 5472 . . . . 5  |-  ( F : ( 0 [,] 1 ) -1-1-onto-> ( A [,] B
)  ->  F :
( 0 [,] 1
) --> ( A [,] B ) )
38 frn 5395 . . . . 5  |-  ( F : ( 0 [,] 1 ) --> ( A [,] B )  ->  ran  F  C_  ( A [,] B ) )
3936, 37, 383syl 18 . . . 4  |-  ( ( A  e.  RR  /\  B  e.  RR  /\  A  <  B )  ->  ran  F 
C_  ( A [,] B ) )
40 iccssre 10731 . . . . . 6  |-  ( ( A  e.  RR  /\  B  e.  RR )  ->  ( A [,] B
)  C_  RR )
41403adant3 975 . . . . 5  |-  ( ( A  e.  RR  /\  B  e.  RR  /\  A  <  B )  ->  ( A [,] B )  C_  RR )
42 ax-resscn 8794 . . . . 5  |-  RR  C_  CC
4341, 42syl6ss 3191 . . . 4  |-  ( ( A  e.  RR  /\  B  e.  RR  /\  A  <  B )  ->  ( A [,] B )  C_  CC )
44 cnrest2 17014 . . . 4  |-  ( ( J  e.  (TopOn `  CC )  /\  ran  F  C_  ( A [,] B
)  /\  ( A [,] B )  C_  CC )  ->  ( F  e.  ( II  Cn  J
)  <->  F  e.  (
II  Cn  ( Jt  ( A [,] B ) ) ) ) )
4514, 39, 43, 44syl3anc 1182 . . 3  |-  ( ( A  e.  RR  /\  B  e.  RR  /\  A  <  B )  ->  ( F  e.  ( II  Cn  J )  <->  F  e.  ( II  Cn  ( Jt  ( A [,] B ) ) ) ) )
4634, 45mpbid 201 . 2  |-  ( ( A  e.  RR  /\  B  e.  RR  /\  A  <  B )  ->  F  e.  ( II  Cn  ( Jt  ( A [,] B ) ) ) )
4735simprd 449 . . . . 5  |-  ( ( A  e.  RR  /\  B  e.  RR  /\  A  <  B )  ->  `' F  =  ( y  e.  ( A [,] B
)  |->  ( ( y  -  A )  / 
( B  -  A
) ) ) )
48 resttopon 16892 . . . . . . 7  |-  ( ( J  e.  (TopOn `  CC )  /\  ( A [,] B )  C_  CC )  ->  ( Jt  ( A [,] B ) )  e.  (TopOn `  ( A [,] B ) ) )
4913, 43, 48sylancr 644 . . . . . 6  |-  ( ( A  e.  RR  /\  B  e.  RR  /\  A  <  B )  ->  ( Jt  ( A [,] B ) )  e.  (TopOn `  ( A [,] B ) ) )
50 cnrest2r 17015 . . . . . . . . 9  |-  ( J  e.  Top  ->  (
( Jt  ( A [,] B ) )  Cn  ( Jt  ( A [,] B ) ) ) 
C_  ( ( Jt  ( A [,] B ) )  Cn  J ) )
517, 50ax-mp 8 . . . . . . . 8  |-  ( ( Jt  ( A [,] B
) )  Cn  ( Jt  ( A [,] B ) ) )  C_  (
( Jt  ( A [,] B ) )  Cn  J )
5249cnmptid 17355 . . . . . . . 8  |-  ( ( A  e.  RR  /\  B  e.  RR  /\  A  <  B )  ->  (
y  e.  ( A [,] B )  |->  y )  e.  ( ( Jt  ( A [,] B
) )  Cn  ( Jt  ( A [,] B ) ) ) )
5351, 52sseldi 3178 . . . . . . 7  |-  ( ( A  e.  RR  /\  B  e.  RR  /\  A  <  B )  ->  (
y  e.  ( A [,] B )  |->  y )  e.  ( ( Jt  ( A [,] B
) )  Cn  J
) )
5449, 14, 28cnmptc 17356 . . . . . . 7  |-  ( ( A  e.  RR  /\  B  e.  RR  /\  A  <  B )  ->  (
y  e.  ( A [,] B )  |->  A )  e.  ( ( Jt  ( A [,] B
) )  Cn  J
) )
5549, 53, 54, 25cnmpt12f 17360 . . . . . 6  |-  ( ( A  e.  RR  /\  B  e.  RR  /\  A  <  B )  ->  (
y  e.  ( A [,] B )  |->  ( y  -  A ) )  e.  ( ( Jt  ( A [,] B
) )  Cn  J
) )
56 difrp 10387 . . . . . . . . 9  |-  ( ( A  e.  RR  /\  B  e.  RR )  ->  ( A  <  B  <->  ( B  -  A )  e.  RR+ ) )
5756biimp3a 1281 . . . . . . . 8  |-  ( ( A  e.  RR  /\  B  e.  RR  /\  A  <  B )  ->  ( B  -  A )  e.  RR+ )
5857rpcnd 10392 . . . . . . 7  |-  ( ( A  e.  RR  /\  B  e.  RR  /\  A  <  B )  ->  ( B  -  A )  e.  CC )
5957rpne0d 10395 . . . . . . 7  |-  ( ( A  e.  RR  /\  B  e.  RR  /\  A  <  B )  ->  ( B  -  A )  =/=  0 )
604divccn 18377 . . . . . . 7  |-  ( ( ( B  -  A
)  e.  CC  /\  ( B  -  A
)  =/=  0 )  ->  ( x  e.  CC  |->  ( x  / 
( B  -  A
) ) )  e.  ( J  Cn  J
) )
6158, 59, 60syl2anc 642 . . . . . 6  |-  ( ( A  e.  RR  /\  B  e.  RR  /\  A  <  B )  ->  (
x  e.  CC  |->  ( x  /  ( B  -  A ) ) )  e.  ( J  Cn  J ) )
62 oveq1 5865 . . . . . 6  |-  ( x  =  ( y  -  A )  ->  (
x  /  ( B  -  A ) )  =  ( ( y  -  A )  / 
( B  -  A
) ) )
6349, 55, 14, 61, 62cnmpt11 17357 . . . . 5  |-  ( ( A  e.  RR  /\  B  e.  RR  /\  A  <  B )  ->  (
y  e.  ( A [,] B )  |->  ( ( y  -  A
)  /  ( B  -  A ) ) )  e.  ( ( Jt  ( A [,] B
) )  Cn  J
) )
6447, 63eqeltrd 2357 . . . 4  |-  ( ( A  e.  RR  /\  B  e.  RR  /\  A  <  B )  ->  `' F  e.  ( ( Jt  ( A [,] B ) )  Cn  J ) )
65 dfdm4 4872 . . . . . . 7  |-  dom  F  =  ran  `' F
6665eqimss2i 3233 . . . . . 6  |-  ran  `' F  C_  dom  F
67 f1odm 5476 . . . . . . 7  |-  ( F : ( 0 [,] 1 ) -1-1-onto-> ( A [,] B
)  ->  dom  F  =  ( 0 [,] 1
) )
6836, 67syl 15 . . . . . 6  |-  ( ( A  e.  RR  /\  B  e.  RR  /\  A  <  B )  ->  dom  F  =  ( 0 [,] 1 ) )
6966, 68syl5sseq 3226 . . . . 5  |-  ( ( A  e.  RR  /\  B  e.  RR  /\  A  <  B )  ->  ran  `' F  C_  ( 0 [,] 1 ) )
70 0re 8838 . . . . . . . 8  |-  0  e.  RR
71 1re 8837 . . . . . . . 8  |-  1  e.  RR
72 iccssre 10731 . . . . . . . 8  |-  ( ( 0  e.  RR  /\  1  e.  RR )  ->  ( 0 [,] 1
)  C_  RR )
7370, 71, 72mp2an 653 . . . . . . 7  |-  ( 0 [,] 1 )  C_  RR
7473a1i 10 . . . . . 6  |-  ( ( A  e.  RR  /\  B  e.  RR  /\  A  <  B )  ->  (
0 [,] 1 ) 
C_  RR )
7574, 42syl6ss 3191 . . . . 5  |-  ( ( A  e.  RR  /\  B  e.  RR  /\  A  <  B )  ->  (
0 [,] 1 ) 
C_  CC )
76 cnrest2 17014 . . . . 5  |-  ( ( J  e.  (TopOn `  CC )  /\  ran  `' F  C_  ( 0 [,] 1 )  /\  (
0 [,] 1 ) 
C_  CC )  -> 
( `' F  e.  ( ( Jt  ( A [,] B ) )  Cn  J )  <->  `' F  e.  ( ( Jt  ( A [,] B ) )  Cn  ( Jt  ( 0 [,] 1 ) ) ) ) )
7714, 69, 75, 76syl3anc 1182 . . . 4  |-  ( ( A  e.  RR  /\  B  e.  RR  /\  A  <  B )  ->  ( `' F  e.  (
( Jt  ( A [,] B ) )  Cn  J )  <->  `' F  e.  ( ( Jt  ( A [,] B ) )  Cn  ( Jt  ( 0 [,] 1 ) ) ) ) )
7864, 77mpbid 201 . . 3  |-  ( ( A  e.  RR  /\  B  e.  RR  /\  A  <  B )  ->  `' F  e.  ( ( Jt  ( A [,] B ) )  Cn  ( Jt  ( 0 [,] 1 ) ) ) )
795oveq2i 5869 . . 3  |-  ( ( Jt  ( A [,] B
) )  Cn  II )  =  ( ( Jt  ( A [,] B ) )  Cn  ( Jt  ( 0 [,] 1 ) ) )
8078, 79syl6eleqr 2374 . 2  |-  ( ( A  e.  RR  /\  B  e.  RR  /\  A  <  B )  ->  `' F  e.  ( ( Jt  ( A [,] B ) )  Cn  II ) )
81 ishmeo 17450 . 2  |-  ( F  e.  ( II  Homeo  ( Jt  ( A [,] B
) ) )  <->  ( F  e.  ( II  Cn  ( Jt  ( A [,] B ) ) )  /\  `' F  e.  ( ( Jt  ( A [,] B ) )  Cn  II ) ) )
8246, 80, 81sylanbrc 645 1  |-  ( ( A  e.  RR  /\  B  e.  RR  /\  A  <  B )  ->  F  e.  ( II  Homeo  ( Jt  ( A [,] B ) ) ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 176    /\ w3a 934    = wceq 1623    e. wcel 1684    =/= wne 2446    C_ wss 3152   class class class wbr 4023    e. cmpt 4077   `'ccnv 4688   dom cdm 4689   ran crn 4690   -->wf 5251   -1-1-onto->wf1o 5254   ` cfv 5255  (class class class)co 5858   CCcc 8735   RRcr 8736   0cc0 8737   1c1 8738    + caddc 8740    x. cmul 8742    < clt 8867    - cmin 9037    / cdiv 9423   RR+crp 10354   [,]cicc 10659   ↾t crest 13325   TopOpenctopn 13326  ℂfldccnfld 16377   Topctop 16631  TopOnctopon 16632    Cn ccn 16954    tX ctx 17255    Homeo chmeo 17444   IIcii 18379
This theorem is referenced by:  xrhmph  18445
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1533  ax-5 1544  ax-17 1603  ax-9 1635  ax-8 1643  ax-13 1686  ax-14 1688  ax-6 1703  ax-7 1708  ax-11 1715  ax-12 1866  ax-ext 2264  ax-rep 4131  ax-sep 4141  ax-nul 4149  ax-pow 4188  ax-pr 4214  ax-un 4512  ax-inf2 7342  ax-cnex 8793  ax-resscn 8794  ax-1cn 8795  ax-icn 8796  ax-addcl 8797  ax-addrcl 8798  ax-mulcl 8799  ax-mulrcl 8800  ax-mulcom 8801  ax-addass 8802  ax-mulass 8803  ax-distr 8804  ax-i2m1 8805  ax-1ne0 8806  ax-1rid 8807  ax-rnegex 8808  ax-rrecex 8809  ax-cnre 8810  ax-pre-lttri 8811  ax-pre-lttrn 8812  ax-pre-ltadd 8813  ax-pre-mulgt0 8814  ax-pre-sup 8815  ax-addf 8816  ax-mulf 8817
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3or 935  df-3an 936  df-tru 1310  df-ex 1529  df-nf 1532  df-sb 1630  df-eu 2147  df-mo 2148  df-clab 2270  df-cleq 2276  df-clel 2279  df-nfc 2408  df-ne 2448  df-nel 2449  df-ral 2548  df-rex 2549  df-reu 2550  df-rmo 2551  df-rab 2552  df-v 2790  df-sbc 2992  df-csb 3082  df-dif 3155  df-un 3157  df-in 3159  df-ss 3166  df-pss 3168  df-nul 3456  df-if 3566  df-pw 3627  df-sn 3646  df-pr 3647  df-tp 3648  df-op 3649  df-uni 3828  df-int 3863  df-iun 3907  df-iin 3908  df-br 4024  df-opab 4078  df-mpt 4079  df-tr 4114  df-eprel 4305  df-id 4309  df-po 4314  df-so 4315  df-fr 4352  df-se 4353  df-we 4354  df-ord 4395  df-on 4396  df-lim 4397  df-suc 4398  df-om 4657  df-xp 4695  df-rel 4696  df-cnv 4697  df-co 4698  df-dm 4699  df-rn 4700  df-res 4701  df-ima 4702  df-iota 5219  df-fun 5257  df-fn 5258  df-f 5259  df-f1 5260  df-fo 5261  df-f1o 5262  df-fv 5263  df-isom 5264  df-ov 5861  df-oprab 5862  df-mpt2 5863  df-of 6078  df-1st 6122  df-2nd 6123  df-riota 6304  df-recs 6388  df-rdg 6423  df-1o 6479  df-2o 6480  df-oadd 6483  df-er 6660  df-map 6774  df-ixp 6818  df-en 6864  df-dom 6865  df-sdom 6866  df-fin 6867  df-fi 7165  df-sup 7194  df-oi 7225  df-card 7572  df-cda 7794  df-pnf 8869  df-mnf 8870  df-xr 8871  df-ltxr 8872  df-le 8873  df-sub 9039  df-neg 9040  df-div 9424  df-nn 9747  df-2 9804  df-3 9805  df-4 9806  df-5 9807  df-6 9808  df-7 9809  df-8 9810  df-9 9811  df-10 9812  df-n0 9966  df-z 10025  df-dec 10125  df-uz 10231  df-q 10317  df-rp 10355  df-xneg 10452  df-xadd 10453  df-xmul 10454  df-icc 10663  df-fz 10783  df-fzo 10871  df-seq 11047  df-exp 11105  df-hash 11338  df-cj 11584  df-re 11585  df-im 11586  df-sqr 11720  df-abs 11721  df-struct 13150  df-ndx 13151  df-slot 13152  df-base 13153  df-sets 13154  df-ress 13155  df-plusg 13221  df-mulr 13222  df-starv 13223  df-sca 13224  df-vsca 13225  df-tset 13227  df-ple 13228  df-ds 13230  df-hom 13232  df-cco 13233  df-rest 13327  df-topn 13328  df-topgen 13344  df-pt 13345  df-prds 13348  df-xrs 13403  df-0g 13404  df-gsum 13405  df-qtop 13410  df-imas 13411  df-xps 13413  df-mre 13488  df-mrc 13489  df-acs 13491  df-mnd 14367  df-submnd 14416  df-mulg 14492  df-cntz 14793  df-cmn 15091  df-xmet 16373  df-met 16374  df-bl 16375  df-mopn 16376  df-cnfld 16378  df-top 16636  df-bases 16638  df-topon 16639  df-topsp 16640  df-cn 16957  df-cnp 16958  df-tx 17257  df-hmeo 17446  df-xms 17885  df-ms 17886  df-tms 17887  df-ii 18381
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