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Theorem icchmeo 21878
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 21820 . . . . . 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 21824 . . . . . . . . 9  |-  II  =  ( Jt  ( 0 [,] 1 ) )
65oveq2i 6307 . . . . . . . 8  |-  ( II 
Cn  II )  =  ( II  Cn  ( Jt  ( 0 [,] 1
) ) )
74cnfldtop 21728 . . . . . . . . 9  |-  J  e. 
Top
8 cnrest2r 20227 . . . . . . . . 9  |-  ( J  e.  Top  ->  (
II  Cn  ( Jt  (
0 [,] 1 ) ) )  C_  (
II  Cn  J )
)
97, 8ax-mp 5 . . . . . . . 8  |-  ( II 
Cn  ( Jt  ( 0 [,] 1 ) ) )  C_  ( II  Cn  J )
106, 9eqsstri 3491 . . . . . . 7  |-  ( II 
Cn  II )  C_  ( II  Cn  J
)
113cnmptid 20600 . . . . . . 7  |-  ( ( A  e.  RR  /\  B  e.  RR  /\  A  <  B )  ->  (
x  e.  ( 0 [,] 1 )  |->  x )  e.  ( II 
Cn  II ) )
1210, 11sseldi 3459 . . . . . 6  |-  ( ( A  e.  RR  /\  B  e.  RR  /\  A  <  B )  ->  (
x  e.  ( 0 [,] 1 )  |->  x )  e.  ( II 
Cn  J ) )
134cnfldtopon 21727 . . . . . . . 8  |-  J  e.  (TopOn `  CC )
1413a1i 11 . . . . . . 7  |-  ( ( A  e.  RR  /\  B  e.  RR  /\  A  <  B )  ->  J  e.  (TopOn `  CC )
)
15 simp2 1006 . . . . . . . 8  |-  ( ( A  e.  RR  /\  B  e.  RR  /\  A  <  B )  ->  B  e.  RR )
1615recnd 9658 . . . . . . 7  |-  ( ( A  e.  RR  /\  B  e.  RR  /\  A  <  B )  ->  B  e.  CC )
173, 14, 16cnmptc 20601 . . . . . 6  |-  ( ( A  e.  RR  /\  B  e.  RR  /\  A  <  B )  ->  (
x  e.  ( 0 [,] 1 )  |->  B )  e.  ( II 
Cn  J ) )
184mulcn 21808 . . . . . . 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 20605 . . . . 5  |-  ( ( A  e.  RR  /\  B  e.  RR  /\  A  <  B )  ->  (
x  e.  ( 0 [,] 1 )  |->  ( x  x.  B ) )  e.  ( II 
Cn  J ) )
21 1cnd 9648 . . . . . . . 8  |-  ( ( A  e.  RR  /\  B  e.  RR  /\  A  <  B )  ->  1  e.  CC )
223, 14, 21cnmptc 20601 . . . . . . 7  |-  ( ( A  e.  RR  /\  B  e.  RR  /\  A  <  B )  ->  (
x  e.  ( 0 [,] 1 )  |->  1 )  e.  ( II 
Cn  J ) )
234subcn 21807 . . . . . . . 8  |-  -  e.  ( ( J  tX  J )  Cn  J
)
2423a1i 11 . . . . . . 7  |-  ( ( A  e.  RR  /\  B  e.  RR  /\  A  <  B )  ->  -  e.  ( ( J  tX  J )  Cn  J
) )
253, 22, 12, 24cnmpt12f 20605 . . . . . 6  |-  ( ( A  e.  RR  /\  B  e.  RR  /\  A  <  B )  ->  (
x  e.  ( 0 [,] 1 )  |->  ( 1  -  x ) )  e.  ( II 
Cn  J ) )
26 simp1 1005 . . . . . . . 8  |-  ( ( A  e.  RR  /\  B  e.  RR  /\  A  <  B )  ->  A  e.  RR )
2726recnd 9658 . . . . . . 7  |-  ( ( A  e.  RR  /\  B  e.  RR  /\  A  <  B )  ->  A  e.  CC )
283, 14, 27cnmptc 20601 . . . . . 6  |-  ( ( A  e.  RR  /\  B  e.  RR  /\  A  <  B )  ->  (
x  e.  ( 0 [,] 1 )  |->  A )  e.  ( II 
Cn  J ) )
293, 25, 28, 19cnmpt12f 20605 . . . . 5  |-  ( ( A  e.  RR  /\  B  e.  RR  /\  A  <  B )  ->  (
x  e.  ( 0 [,] 1 )  |->  ( ( 1  -  x
)  x.  A ) )  e.  ( II 
Cn  J ) )
304addcn 21806 . . . . . 6  |-  +  e.  ( ( J  tX  J )  Cn  J
)
3130a1i 11 . . . . 5  |-  ( ( A  e.  RR  /\  B  e.  RR  /\  A  <  B )  ->  +  e.  ( ( J  tX  J )  Cn  J
) )
323, 20, 29, 31cnmpt12f 20605 . . . 4  |-  ( ( A  e.  RR  /\  B  e.  RR  /\  A  <  B )  ->  (
x  e.  ( 0 [,] 1 )  |->  ( ( x  x.  B
)  +  ( ( 1  -  x )  x.  A ) ) )  e.  ( II 
Cn  J ) )
331, 32syl5eqel 2512 . . 3  |-  ( ( A  e.  RR  /\  B  e.  RR  /\  A  <  B )  ->  F  e.  ( II  Cn  J
) )
341iccf1o 11763 . . . . . 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
) ) ) ) )
3534simpld 460 . . . . 5  |-  ( ( A  e.  RR  /\  B  e.  RR  /\  A  <  B )  ->  F : ( 0 [,] 1 ) -1-1-onto-> ( A [,] B
) )
36 f1of 5822 . . . . 5  |-  ( F : ( 0 [,] 1 ) -1-1-onto-> ( A [,] B
)  ->  F :
( 0 [,] 1
) --> ( A [,] B ) )
37 frn 5743 . . . . 5  |-  ( F : ( 0 [,] 1 ) --> ( A [,] B )  ->  ran  F  C_  ( A [,] B ) )
3835, 36, 373syl 18 . . . 4  |-  ( ( A  e.  RR  /\  B  e.  RR  /\  A  <  B )  ->  ran  F 
C_  ( A [,] B ) )
39 iccssre 11705 . . . . . 6  |-  ( ( A  e.  RR  /\  B  e.  RR )  ->  ( A [,] B
)  C_  RR )
40393adant3 1025 . . . . 5  |-  ( ( A  e.  RR  /\  B  e.  RR  /\  A  <  B )  ->  ( A [,] B )  C_  RR )
41 ax-resscn 9585 . . . . 5  |-  RR  C_  CC
4240, 41syl6ss 3473 . . . 4  |-  ( ( A  e.  RR  /\  B  e.  RR  /\  A  <  B )  ->  ( A [,] B )  C_  CC )
43 cnrest2 20226 . . . 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 ) ) ) ) )
4414, 38, 42, 43syl3anc 1264 . . 3  |-  ( ( A  e.  RR  /\  B  e.  RR  /\  A  <  B )  ->  ( F  e.  ( II  Cn  J )  <->  F  e.  ( II  Cn  ( Jt  ( A [,] B ) ) ) ) )
4533, 44mpbid 213 . 2  |-  ( ( A  e.  RR  /\  B  e.  RR  /\  A  <  B )  ->  F  e.  ( II  Cn  ( Jt  ( A [,] B ) ) ) )
4634simprd 464 . . . . 5  |-  ( ( A  e.  RR  /\  B  e.  RR  /\  A  <  B )  ->  `' F  =  ( y  e.  ( A [,] B
)  |->  ( ( y  -  A )  / 
( B  -  A
) ) ) )
47 resttopon 20101 . . . . . . 7  |-  ( ( J  e.  (TopOn `  CC )  /\  ( A [,] B )  C_  CC )  ->  ( Jt  ( A [,] B ) )  e.  (TopOn `  ( A [,] B ) ) )
4813, 42, 47sylancr 667 . . . . . 6  |-  ( ( A  e.  RR  /\  B  e.  RR  /\  A  <  B )  ->  ( Jt  ( A [,] B ) )  e.  (TopOn `  ( A [,] B ) ) )
49 cnrest2r 20227 . . . . . . . . 9  |-  ( J  e.  Top  ->  (
( Jt  ( A [,] B ) )  Cn  ( Jt  ( A [,] B ) ) ) 
C_  ( ( Jt  ( A [,] B ) )  Cn  J ) )
507, 49ax-mp 5 . . . . . . . 8  |-  ( ( Jt  ( A [,] B
) )  Cn  ( Jt  ( A [,] B ) ) )  C_  (
( Jt  ( A [,] B ) )  Cn  J )
5148cnmptid 20600 . . . . . . . 8  |-  ( ( A  e.  RR  /\  B  e.  RR  /\  A  <  B )  ->  (
y  e.  ( A [,] B )  |->  y )  e.  ( ( Jt  ( A [,] B
) )  Cn  ( Jt  ( A [,] B ) ) ) )
5250, 51sseldi 3459 . . . . . . 7  |-  ( ( A  e.  RR  /\  B  e.  RR  /\  A  <  B )  ->  (
y  e.  ( A [,] B )  |->  y )  e.  ( ( Jt  ( A [,] B
) )  Cn  J
) )
5348, 14, 27cnmptc 20601 . . . . . . 7  |-  ( ( A  e.  RR  /\  B  e.  RR  /\  A  <  B )  ->  (
y  e.  ( A [,] B )  |->  A )  e.  ( ( Jt  ( A [,] B
) )  Cn  J
) )
5448, 52, 53, 24cnmpt12f 20605 . . . . . 6  |-  ( ( A  e.  RR  /\  B  e.  RR  /\  A  <  B )  ->  (
y  e.  ( A [,] B )  |->  ( y  -  A ) )  e.  ( ( Jt  ( A [,] B
) )  Cn  J
) )
55 difrp 11326 . . . . . . . . 9  |-  ( ( A  e.  RR  /\  B  e.  RR )  ->  ( A  <  B  <->  ( B  -  A )  e.  RR+ ) )
5655biimp3a 1364 . . . . . . . 8  |-  ( ( A  e.  RR  /\  B  e.  RR  /\  A  <  B )  ->  ( B  -  A )  e.  RR+ )
5756rpcnd 11332 . . . . . . 7  |-  ( ( A  e.  RR  /\  B  e.  RR  /\  A  <  B )  ->  ( B  -  A )  e.  CC )
5856rpne0d 11335 . . . . . . 7  |-  ( ( A  e.  RR  /\  B  e.  RR  /\  A  <  B )  ->  ( B  -  A )  =/=  0 )
594divccn 21814 . . . . . . 7  |-  ( ( ( B  -  A
)  e.  CC  /\  ( B  -  A
)  =/=  0 )  ->  ( x  e.  CC  |->  ( x  / 
( B  -  A
) ) )  e.  ( J  Cn  J
) )
6057, 58, 59syl2anc 665 . . . . . 6  |-  ( ( A  e.  RR  /\  B  e.  RR  /\  A  <  B )  ->  (
x  e.  CC  |->  ( x  /  ( B  -  A ) ) )  e.  ( J  Cn  J ) )
61 oveq1 6303 . . . . . 6  |-  ( x  =  ( y  -  A )  ->  (
x  /  ( B  -  A ) )  =  ( ( y  -  A )  / 
( B  -  A
) ) )
6248, 54, 14, 60, 61cnmpt11 20602 . . . . 5  |-  ( ( A  e.  RR  /\  B  e.  RR  /\  A  <  B )  ->  (
y  e.  ( A [,] B )  |->  ( ( y  -  A
)  /  ( B  -  A ) ) )  e.  ( ( Jt  ( A [,] B
) )  Cn  J
) )
6346, 62eqeltrd 2508 . . . 4  |-  ( ( A  e.  RR  /\  B  e.  RR  /\  A  <  B )  ->  `' F  e.  ( ( Jt  ( A [,] B ) )  Cn  J ) )
64 dfdm4 5038 . . . . . . 7  |-  dom  F  =  ran  `' F
6564eqimss2i 3516 . . . . . 6  |-  ran  `' F  C_  dom  F
66 f1odm 5826 . . . . . . 7  |-  ( F : ( 0 [,] 1 ) -1-1-onto-> ( A [,] B
)  ->  dom  F  =  ( 0 [,] 1
) )
6735, 66syl 17 . . . . . 6  |-  ( ( A  e.  RR  /\  B  e.  RR  /\  A  <  B )  ->  dom  F  =  ( 0 [,] 1 ) )
6865, 67syl5sseq 3509 . . . . 5  |-  ( ( A  e.  RR  /\  B  e.  RR  /\  A  <  B )  ->  ran  `' F  C_  ( 0 [,] 1 ) )
69 unitssre 11766 . . . . . . 7  |-  ( 0 [,] 1 )  C_  RR
7069a1i 11 . . . . . 6  |-  ( ( A  e.  RR  /\  B  e.  RR  /\  A  <  B )  ->  (
0 [,] 1 ) 
C_  RR )
7170, 41syl6ss 3473 . . . . 5  |-  ( ( A  e.  RR  /\  B  e.  RR  /\  A  <  B )  ->  (
0 [,] 1 ) 
C_  CC )
72 cnrest2 20226 . . . . 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 ) ) ) ) )
7314, 68, 71, 72syl3anc 1264 . . . 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 ) ) ) ) )
7463, 73mpbid 213 . . 3  |-  ( ( A  e.  RR  /\  B  e.  RR  /\  A  <  B )  ->  `' F  e.  ( ( Jt  ( A [,] B ) )  Cn  ( Jt  ( 0 [,] 1 ) ) ) )
755oveq2i 6307 . . 3  |-  ( ( Jt  ( A [,] B
) )  Cn  II )  =  ( ( Jt  ( A [,] B ) )  Cn  ( Jt  ( 0 [,] 1 ) ) )
7674, 75syl6eleqr 2519 . 2  |-  ( ( A  e.  RR  /\  B  e.  RR  /\  A  <  B )  ->  `' F  e.  ( ( Jt  ( A [,] B ) )  Cn  II ) )
77 ishmeo 20698 . 2  |-  ( F  e.  ( II Homeo ( Jt  ( A [,] B
) ) )  <->  ( F  e.  ( II  Cn  ( Jt  ( A [,] B ) ) )  /\  `' F  e.  ( ( Jt  ( A [,] B ) )  Cn  II ) ) )
7845, 76, 77sylanbrc 668 1  |-  ( ( A  e.  RR  /\  B  e.  RR  /\  A  <  B )  ->  F  e.  ( II Homeo ( Jt  ( A [,] B ) ) ) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    <-> wb 187    /\ w3a 982    = wceq 1437    e. wcel 1867    =/= wne 2616    C_ wss 3433   class class class wbr 4417    |-> cmpt 4475   `'ccnv 4844   dom cdm 4845   ran crn 4846   -->wf 5588   -1-1-onto->wf1o 5591   ` cfv 5592  (class class class)co 6296   CCcc 9526   RRcr 9527   0cc0 9528   1c1 9529    + caddc 9531    x. cmul 9533    < clt 9664    - cmin 9849    / cdiv 10258   RR+crp 11291   [,]cicc 11627   ↾t crest 15271   TopOpenctopn 15272  ℂfldccnfld 18898   Topctop 19841  TopOnctopon 19842    Cn ccn 20164    tX ctx 20499   Homeochmeo 20692   IIcii 21816
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1665  ax-4 1678  ax-5 1748  ax-6 1794  ax-7 1838  ax-8 1869  ax-9 1871  ax-10 1886  ax-11 1891  ax-12 1904  ax-13 2052  ax-ext 2398  ax-rep 4529  ax-sep 4539  ax-nul 4547  ax-pow 4594  ax-pr 4652  ax-un 6588  ax-inf2 8137  ax-cnex 9584  ax-resscn 9585  ax-1cn 9586  ax-icn 9587  ax-addcl 9588  ax-addrcl 9589  ax-mulcl 9590  ax-mulrcl 9591  ax-mulcom 9592  ax-addass 9593  ax-mulass 9594  ax-distr 9595  ax-i2m1 9596  ax-1ne0 9597  ax-1rid 9598  ax-rnegex 9599  ax-rrecex 9600  ax-cnre 9601  ax-pre-lttri 9602  ax-pre-lttrn 9603  ax-pre-ltadd 9604  ax-pre-mulgt0 9605  ax-pre-sup 9606  ax-addf 9607  ax-mulf 9608
This theorem depends on definitions:  df-bi 188  df-or 371  df-an 372  df-3or 983  df-3an 984  df-tru 1440  df-ex 1660  df-nf 1664  df-sb 1787  df-eu 2267  df-mo 2268  df-clab 2406  df-cleq 2412  df-clel 2415  df-nfc 2570  df-ne 2618  df-nel 2619  df-ral 2778  df-rex 2779  df-reu 2780  df-rmo 2781  df-rab 2782  df-v 3080  df-sbc 3297  df-csb 3393  df-dif 3436  df-un 3438  df-in 3440  df-ss 3447  df-pss 3449  df-nul 3759  df-if 3907  df-pw 3978  df-sn 3994  df-pr 3996  df-tp 3998  df-op 4000  df-uni 4214  df-int 4250  df-iun 4295  df-iin 4296  df-br 4418  df-opab 4476  df-mpt 4477  df-tr 4512  df-eprel 4756  df-id 4760  df-po 4766  df-so 4767  df-fr 4804  df-se 4805  df-we 4806  df-xp 4851  df-rel 4852  df-cnv 4853  df-co 4854  df-dm 4855  df-rn 4856  df-res 4857  df-ima 4858  df-pred 5390  df-ord 5436  df-on 5437  df-lim 5438  df-suc 5439  df-iota 5556  df-fun 5594  df-fn 5595  df-f 5596  df-f1 5597  df-fo 5598  df-f1o 5599  df-fv 5600  df-isom 5601  df-riota 6258  df-ov 6299  df-oprab 6300  df-mpt2 6301  df-of 6536  df-om 6698  df-1st 6798  df-2nd 6799  df-supp 6917  df-wrecs 7027  df-recs 7089  df-rdg 7127  df-1o 7181  df-2o 7182  df-oadd 7185  df-er 7362  df-map 7473  df-ixp 7522  df-en 7569  df-dom 7570  df-sdom 7571  df-fin 7572  df-fsupp 7881  df-fi 7922  df-sup 7953  df-oi 8016  df-card 8363  df-cda 8587  df-pnf 9666  df-mnf 9667  df-xr 9668  df-ltxr 9669  df-le 9670  df-sub 9851  df-neg 9852  df-div 10259  df-nn 10599  df-2 10657  df-3 10658  df-4 10659  df-5 10660  df-6 10661  df-7 10662  df-8 10663  df-9 10664  df-10 10665  df-n0 10859  df-z 10927  df-dec 11041  df-uz 11149  df-q 11254  df-rp 11292  df-xneg 11398  df-xadd 11399  df-xmul 11400  df-icc 11631  df-fz 11772  df-fzo 11903  df-seq 12200  df-exp 12259  df-hash 12502  df-cj 13130  df-re 13131  df-im 13132  df-sqrt 13266  df-abs 13267  df-struct 15075  df-ndx 15076  df-slot 15077  df-base 15078  df-sets 15079  df-ress 15080  df-plusg 15155  df-mulr 15156  df-starv 15157  df-sca 15158  df-vsca 15159  df-ip 15160  df-tset 15161  df-ple 15162  df-ds 15164  df-unif 15165  df-hom 15166  df-cco 15167  df-rest 15273  df-topn 15274  df-0g 15292  df-gsum 15293  df-topgen 15294  df-pt 15295  df-prds 15298  df-xrs 15352  df-qtop 15357  df-imas 15358  df-xps 15360  df-mre 15436  df-mrc 15437  df-acs 15439  df-mgm 16432  df-sgrp 16471  df-mnd 16481  df-submnd 16527  df-mulg 16620  df-cntz 16915  df-cmn 17360  df-psmet 18890  df-xmet 18891  df-met 18892  df-bl 18893  df-mopn 18894  df-cnfld 18899  df-top 19845  df-bases 19846  df-topon 19847  df-topsp 19848  df-cn 20167  df-cnp 20168  df-tx 20501  df-hmeo 20694  df-xms 21259  df-ms 21260  df-tms 21261  df-ii 21818
This theorem is referenced by:  xrhmph  21884
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