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Theorem icchmeo 18919
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 18862 . . . . . 6  |-  II  e.  (TopOn `  ( 0 [,] 1 ) )
32a1i 11 . . . . 5  |-  ( ( A  e.  RR  /\  B  e.  RR  /\  A  <  B )  ->  II  e.  (TopOn `  ( 0 [,] 1 ) ) )
4 icchmeo.j . . . . . . . . . 10  |-  J  =  ( TopOpen ` fld )
54dfii3 18866 . . . . . . . . 9  |-  II  =  ( Jt  ( 0 [,] 1 ) )
65oveq2i 6051 . . . . . . . 8  |-  ( II 
Cn  II )  =  ( II  Cn  ( Jt  ( 0 [,] 1
) ) )
74cnfldtop 18771 . . . . . . . . 9  |-  J  e. 
Top
8 cnrest2r 17305 . . . . . . . . 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 3338 . . . . . . 7  |-  ( II 
Cn  II )  C_  ( II  Cn  J
)
113cnmptid 17646 . . . . . . 7  |-  ( ( A  e.  RR  /\  B  e.  RR  /\  A  <  B )  ->  (
x  e.  ( 0 [,] 1 )  |->  x )  e.  ( II 
Cn  II ) )
1210, 11sseldi 3306 . . . . . 6  |-  ( ( A  e.  RR  /\  B  e.  RR  /\  A  <  B )  ->  (
x  e.  ( 0 [,] 1 )  |->  x )  e.  ( II 
Cn  J ) )
134cnfldtopon 18770 . . . . . . . 8  |-  J  e.  (TopOn `  CC )
1413a1i 11 . . . . . . 7  |-  ( ( A  e.  RR  /\  B  e.  RR  /\  A  <  B )  ->  J  e.  (TopOn `  CC )
)
15 simp2 958 . . . . . . . 8  |-  ( ( A  e.  RR  /\  B  e.  RR  /\  A  <  B )  ->  B  e.  RR )
1615recnd 9070 . . . . . . 7  |-  ( ( A  e.  RR  /\  B  e.  RR  /\  A  <  B )  ->  B  e.  CC )
173, 14, 16cnmptc 17647 . . . . . 6  |-  ( ( A  e.  RR  /\  B  e.  RR  /\  A  <  B )  ->  (
x  e.  ( 0 [,] 1 )  |->  B )  e.  ( II 
Cn  J ) )
184mulcn 18850 . . . . . . 7  |-  x.  e.  ( ( J  tX  J )  Cn  J
)
1918a1i 11 . . . . . 6  |-  ( ( A  e.  RR  /\  B  e.  RR  /\  A  <  B )  ->  x.  e.  ( ( J  tX  J )  Cn  J
) )
203, 12, 17, 19cnmpt12f 17651 . . . . 5  |-  ( ( A  e.  RR  /\  B  e.  RR  /\  A  <  B )  ->  (
x  e.  ( 0 [,] 1 )  |->  ( x  x.  B ) )  e.  ( II 
Cn  J ) )
21 ax-1cn 9004 . . . . . . . . 9  |-  1  e.  CC
2221a1i 11 . . . . . . . 8  |-  ( ( A  e.  RR  /\  B  e.  RR  /\  A  <  B )  ->  1  e.  CC )
233, 14, 22cnmptc 17647 . . . . . . 7  |-  ( ( A  e.  RR  /\  B  e.  RR  /\  A  <  B )  ->  (
x  e.  ( 0 [,] 1 )  |->  1 )  e.  ( II 
Cn  J ) )
244subcn 18849 . . . . . . . 8  |-  -  e.  ( ( J  tX  J )  Cn  J
)
2524a1i 11 . . . . . . 7  |-  ( ( A  e.  RR  /\  B  e.  RR  /\  A  <  B )  ->  -  e.  ( ( J  tX  J )  Cn  J
) )
263, 23, 12, 25cnmpt12f 17651 . . . . . 6  |-  ( ( A  e.  RR  /\  B  e.  RR  /\  A  <  B )  ->  (
x  e.  ( 0 [,] 1 )  |->  ( 1  -  x ) )  e.  ( II 
Cn  J ) )
27 simp1 957 . . . . . . . 8  |-  ( ( A  e.  RR  /\  B  e.  RR  /\  A  <  B )  ->  A  e.  RR )
2827recnd 9070 . . . . . . 7  |-  ( ( A  e.  RR  /\  B  e.  RR  /\  A  <  B )  ->  A  e.  CC )
293, 14, 28cnmptc 17647 . . . . . 6  |-  ( ( A  e.  RR  /\  B  e.  RR  /\  A  <  B )  ->  (
x  e.  ( 0 [,] 1 )  |->  A )  e.  ( II 
Cn  J ) )
303, 26, 29, 19cnmpt12f 17651 . . . . 5  |-  ( ( A  e.  RR  /\  B  e.  RR  /\  A  <  B )  ->  (
x  e.  ( 0 [,] 1 )  |->  ( ( 1  -  x
)  x.  A ) )  e.  ( II 
Cn  J ) )
314addcn 18848 . . . . . 6  |-  +  e.  ( ( J  tX  J )  Cn  J
)
3231a1i 11 . . . . 5  |-  ( ( A  e.  RR  /\  B  e.  RR  /\  A  <  B )  ->  +  e.  ( ( J  tX  J )  Cn  J
) )
333, 20, 30, 32cnmpt12f 17651 . . . 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 2488 . . 3  |-  ( ( A  e.  RR  /\  B  e.  RR  /\  A  <  B )  ->  F  e.  ( II  Cn  J
) )
351iccf1o 10995 . . . . . 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 446 . . . . 5  |-  ( ( A  e.  RR  /\  B  e.  RR  /\  A  <  B )  ->  F : ( 0 [,] 1 ) -1-1-onto-> ( A [,] B
) )
37 f1of 5633 . . . . 5  |-  ( F : ( 0 [,] 1 ) -1-1-onto-> ( A [,] B
)  ->  F :
( 0 [,] 1
) --> ( A [,] B ) )
38 frn 5556 . . . . 5  |-  ( F : ( 0 [,] 1 ) --> ( A [,] B )  ->  ran  F  C_  ( A [,] B ) )
3936, 37, 383syl 19 . . . 4  |-  ( ( A  e.  RR  /\  B  e.  RR  /\  A  <  B )  ->  ran  F 
C_  ( A [,] B ) )
40 iccssre 10948 . . . . . 6  |-  ( ( A  e.  RR  /\  B  e.  RR )  ->  ( A [,] B
)  C_  RR )
41403adant3 977 . . . . 5  |-  ( ( A  e.  RR  /\  B  e.  RR  /\  A  <  B )  ->  ( A [,] B )  C_  RR )
42 ax-resscn 9003 . . . . 5  |-  RR  C_  CC
4341, 42syl6ss 3320 . . . 4  |-  ( ( A  e.  RR  /\  B  e.  RR  /\  A  <  B )  ->  ( A [,] B )  C_  CC )
44 cnrest2 17304 . . . 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 1184 . . 3  |-  ( ( A  e.  RR  /\  B  e.  RR  /\  A  <  B )  ->  ( F  e.  ( II  Cn  J )  <->  F  e.  ( II  Cn  ( Jt  ( A [,] B ) ) ) ) )
4634, 45mpbid 202 . 2  |-  ( ( A  e.  RR  /\  B  e.  RR  /\  A  <  B )  ->  F  e.  ( II  Cn  ( Jt  ( A [,] B ) ) ) )
4735simprd 450 . . . . 5  |-  ( ( A  e.  RR  /\  B  e.  RR  /\  A  <  B )  ->  `' F  =  ( y  e.  ( A [,] B
)  |->  ( ( y  -  A )  / 
( B  -  A
) ) ) )
48 resttopon 17179 . . . . . . 7  |-  ( ( J  e.  (TopOn `  CC )  /\  ( A [,] B )  C_  CC )  ->  ( Jt  ( A [,] B ) )  e.  (TopOn `  ( A [,] B ) ) )
4913, 43, 48sylancr 645 . . . . . 6  |-  ( ( A  e.  RR  /\  B  e.  RR  /\  A  <  B )  ->  ( Jt  ( A [,] B ) )  e.  (TopOn `  ( A [,] B ) ) )
50 cnrest2r 17305 . . . . . . . . 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 17646 . . . . . . . 8  |-  ( ( A  e.  RR  /\  B  e.  RR  /\  A  <  B )  ->  (
y  e.  ( A [,] B )  |->  y )  e.  ( ( Jt  ( A [,] B
) )  Cn  ( Jt  ( A [,] B ) ) ) )
5351, 52sseldi 3306 . . . . . . 7  |-  ( ( A  e.  RR  /\  B  e.  RR  /\  A  <  B )  ->  (
y  e.  ( A [,] B )  |->  y )  e.  ( ( Jt  ( A [,] B
) )  Cn  J
) )
5449, 14, 28cnmptc 17647 . . . . . . 7  |-  ( ( A  e.  RR  /\  B  e.  RR  /\  A  <  B )  ->  (
y  e.  ( A [,] B )  |->  A )  e.  ( ( Jt  ( A [,] B
) )  Cn  J
) )
5549, 53, 54, 25cnmpt12f 17651 . . . . . 6  |-  ( ( A  e.  RR  /\  B  e.  RR  /\  A  <  B )  ->  (
y  e.  ( A [,] B )  |->  ( y  -  A ) )  e.  ( ( Jt  ( A [,] B
) )  Cn  J
) )
56 difrp 10601 . . . . . . . . 9  |-  ( ( A  e.  RR  /\  B  e.  RR )  ->  ( A  <  B  <->  ( B  -  A )  e.  RR+ ) )
5756biimp3a 1283 . . . . . . . 8  |-  ( ( A  e.  RR  /\  B  e.  RR  /\  A  <  B )  ->  ( B  -  A )  e.  RR+ )
5857rpcnd 10606 . . . . . . 7  |-  ( ( A  e.  RR  /\  B  e.  RR  /\  A  <  B )  ->  ( B  -  A )  e.  CC )
5957rpne0d 10609 . . . . . . 7  |-  ( ( A  e.  RR  /\  B  e.  RR  /\  A  <  B )  ->  ( B  -  A )  =/=  0 )
604divccn 18856 . . . . . . 7  |-  ( ( ( B  -  A
)  e.  CC  /\  ( B  -  A
)  =/=  0 )  ->  ( x  e.  CC  |->  ( x  / 
( B  -  A
) ) )  e.  ( J  Cn  J
) )
6158, 59, 60syl2anc 643 . . . . . 6  |-  ( ( A  e.  RR  /\  B  e.  RR  /\  A  <  B )  ->  (
x  e.  CC  |->  ( x  /  ( B  -  A ) ) )  e.  ( J  Cn  J ) )
62 oveq1 6047 . . . . . 6  |-  ( x  =  ( y  -  A )  ->  (
x  /  ( B  -  A ) )  =  ( ( y  -  A )  / 
( B  -  A
) ) )
6349, 55, 14, 61, 62cnmpt11 17648 . . . . 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 2478 . . . 4  |-  ( ( A  e.  RR  /\  B  e.  RR  /\  A  <  B )  ->  `' F  e.  ( ( Jt  ( A [,] B ) )  Cn  J ) )
65 dfdm4 5022 . . . . . . 7  |-  dom  F  =  ran  `' F
6665eqimss2i 3363 . . . . . 6  |-  ran  `' F  C_  dom  F
67 f1odm 5637 . . . . . . 7  |-  ( F : ( 0 [,] 1 ) -1-1-onto-> ( A [,] B
)  ->  dom  F  =  ( 0 [,] 1
) )
6836, 67syl 16 . . . . . 6  |-  ( ( A  e.  RR  /\  B  e.  RR  /\  A  <  B )  ->  dom  F  =  ( 0 [,] 1 ) )
6966, 68syl5sseq 3356 . . . . 5  |-  ( ( A  e.  RR  /\  B  e.  RR  /\  A  <  B )  ->  ran  `' F  C_  ( 0 [,] 1 ) )
70 unitssre 10998 . . . . . . 7  |-  ( 0 [,] 1 )  C_  RR
7170a1i 11 . . . . . 6  |-  ( ( A  e.  RR  /\  B  e.  RR  /\  A  <  B )  ->  (
0 [,] 1 ) 
C_  RR )
7271, 42syl6ss 3320 . . . . 5  |-  ( ( A  e.  RR  /\  B  e.  RR  /\  A  <  B )  ->  (
0 [,] 1 ) 
C_  CC )
73 cnrest2 17304 . . . . 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 ) ) ) ) )
7414, 69, 72, 73syl3anc 1184 . . . 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 ) ) ) ) )
7564, 74mpbid 202 . . 3  |-  ( ( A  e.  RR  /\  B  e.  RR  /\  A  <  B )  ->  `' F  e.  ( ( Jt  ( A [,] B ) )  Cn  ( Jt  ( 0 [,] 1 ) ) ) )
765oveq2i 6051 . . 3  |-  ( ( Jt  ( A [,] B
) )  Cn  II )  =  ( ( Jt  ( A [,] B ) )  Cn  ( Jt  ( 0 [,] 1 ) ) )
7775, 76syl6eleqr 2495 . 2  |-  ( ( A  e.  RR  /\  B  e.  RR  /\  A  <  B )  ->  `' F  e.  ( ( Jt  ( A [,] B ) )  Cn  II ) )
78 ishmeo 17744 . 2  |-  ( F  e.  ( II  Homeo  ( Jt  ( A [,] B
) ) )  <->  ( F  e.  ( II  Cn  ( Jt  ( A [,] B ) ) )  /\  `' F  e.  ( ( Jt  ( A [,] B ) )  Cn  II ) ) )
7946, 77, 78sylanbrc 646 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 177    /\ w3a 936    = wceq 1649    e. wcel 1721    =/= wne 2567    C_ wss 3280   class class class wbr 4172    e. cmpt 4226   `'ccnv 4836   dom cdm 4837   ran crn 4838   -->wf 5409   -1-1-onto->wf1o 5412   ` cfv 5413  (class class class)co 6040   CCcc 8944   RRcr 8945   0cc0 8946   1c1 8947    + caddc 8949    x. cmul 8951    < clt 9076    - cmin 9247    / cdiv 9633   RR+crp 10568   [,]cicc 10875   ↾t crest 13603   TopOpenctopn 13604  ℂfldccnfld 16658   Topctop 16913  TopOnctopon 16914    Cn ccn 17242    tX ctx 17545    Homeo chmeo 17738   IIcii 18858
This theorem is referenced by:  xrhmph  18925
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1552  ax-5 1563  ax-17 1623  ax-9 1662  ax-8 1683  ax-13 1723  ax-14 1725  ax-6 1740  ax-7 1745  ax-11 1757  ax-12 1946  ax-ext 2385  ax-rep 4280  ax-sep 4290  ax-nul 4298  ax-pow 4337  ax-pr 4363  ax-un 4660  ax-inf2 7552  ax-cnex 9002  ax-resscn 9003  ax-1cn 9004  ax-icn 9005  ax-addcl 9006  ax-addrcl 9007  ax-mulcl 9008  ax-mulrcl 9009  ax-mulcom 9010  ax-addass 9011  ax-mulass 9012  ax-distr 9013  ax-i2m1 9014  ax-1ne0 9015  ax-1rid 9016  ax-rnegex 9017  ax-rrecex 9018  ax-cnre 9019  ax-pre-lttri 9020  ax-pre-lttrn 9021  ax-pre-ltadd 9022  ax-pre-mulgt0 9023  ax-pre-sup 9024  ax-addf 9025  ax-mulf 9026
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3or 937  df-3an 938  df-tru 1325  df-ex 1548  df-nf 1551  df-sb 1656  df-eu 2258  df-mo 2259  df-clab 2391  df-cleq 2397  df-clel 2400  df-nfc 2529  df-ne 2569  df-nel 2570  df-ral 2671  df-rex 2672  df-reu 2673  df-rmo 2674  df-rab 2675  df-v 2918  df-sbc 3122  df-csb 3212  df-dif 3283  df-un 3285  df-in 3287  df-ss 3294  df-pss 3296  df-nul 3589  df-if 3700  df-pw 3761  df-sn 3780  df-pr 3781  df-tp 3782  df-op 3783  df-uni 3976  df-int 4011  df-iun 4055  df-iin 4056  df-br 4173  df-opab 4227  df-mpt 4228  df-tr 4263  df-eprel 4454  df-id 4458  df-po 4463  df-so 4464  df-fr 4501  df-se 4502  df-we 4503  df-ord 4544  df-on 4545  df-lim 4546  df-suc 4547  df-om 4805  df-xp 4843  df-rel 4844  df-cnv 4845  df-co 4846  df-dm 4847  df-rn 4848  df-res 4849  df-ima 4850  df-iota 5377  df-fun 5415  df-fn 5416  df-f 5417  df-f1 5418  df-fo 5419  df-f1o 5420  df-fv 5421  df-isom 5422  df-ov 6043  df-oprab 6044  df-mpt2 6045  df-of 6264  df-1st 6308  df-2nd 6309  df-riota 6508  df-recs 6592  df-rdg 6627  df-1o 6683  df-2o 6684  df-oadd 6687  df-er 6864  df-map 6979  df-ixp 7023  df-en 7069  df-dom 7070  df-sdom 7071  df-fin 7072  df-fi 7374  df-sup 7404  df-oi 7435  df-card 7782  df-cda 8004  df-pnf 9078  df-mnf 9079  df-xr 9080  df-ltxr 9081  df-le 9082  df-sub 9249  df-neg 9250  df-div 9634  df-nn 9957  df-2 10014  df-3 10015  df-4 10016  df-5 10017  df-6 10018  df-7 10019  df-8 10020  df-9 10021  df-10 10022  df-n0 10178  df-z 10239  df-dec 10339  df-uz 10445  df-q 10531  df-rp 10569  df-xneg 10666  df-xadd 10667  df-xmul 10668  df-icc 10879  df-fz 11000  df-fzo 11091  df-seq 11279  df-exp 11338  df-hash 11574  df-cj 11859  df-re 11860  df-im 11861  df-sqr 11995  df-abs 11996  df-struct 13426  df-ndx 13427  df-slot 13428  df-base 13429  df-sets 13430  df-ress 13431  df-plusg 13497  df-mulr 13498  df-starv 13499  df-sca 13500  df-vsca 13501  df-tset 13503  df-ple 13504  df-ds 13506  df-unif 13507  df-hom 13508  df-cco 13509  df-rest 13605  df-topn 13606  df-topgen 13622  df-pt 13623  df-prds 13626  df-xrs 13681  df-0g 13682  df-gsum 13683  df-qtop 13688  df-imas 13689  df-xps 13691  df-mre 13766  df-mrc 13767  df-acs 13769  df-mnd 14645  df-submnd 14694  df-mulg 14770  df-cntz 15071  df-cmn 15369  df-psmet 16649  df-xmet 16650  df-met 16651  df-bl 16652  df-mopn 16653  df-cnfld 16659  df-top 16918  df-bases 16920  df-topon 16921  df-topsp 16922  df-cn 17245  df-cnp 17246  df-tx 17547  df-hmeo 17740  df-xms 18303  df-ms 18304  df-tms 18305  df-ii 18860
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