MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  icchmeo Structured version   Unicode version

Theorem icchmeo 20518
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 20460 . . . . . 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 20464 . . . . . . . . 9  |-  II  =  ( Jt  ( 0 [,] 1 ) )
65oveq2i 6107 . . . . . . . 8  |-  ( II 
Cn  II )  =  ( II  Cn  ( Jt  ( 0 [,] 1
) ) )
74cnfldtop 20368 . . . . . . . . 9  |-  J  e. 
Top
8 cnrest2r 18896 . . . . . . . . 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 3391 . . . . . . 7  |-  ( II 
Cn  II )  C_  ( II  Cn  J
)
113cnmptid 19239 . . . . . . 7  |-  ( ( A  e.  RR  /\  B  e.  RR  /\  A  <  B )  ->  (
x  e.  ( 0 [,] 1 )  |->  x )  e.  ( II 
Cn  II ) )
1210, 11sseldi 3359 . . . . . 6  |-  ( ( A  e.  RR  /\  B  e.  RR  /\  A  <  B )  ->  (
x  e.  ( 0 [,] 1 )  |->  x )  e.  ( II 
Cn  J ) )
134cnfldtopon 20367 . . . . . . . 8  |-  J  e.  (TopOn `  CC )
1413a1i 11 . . . . . . 7  |-  ( ( A  e.  RR  /\  B  e.  RR  /\  A  <  B )  ->  J  e.  (TopOn `  CC )
)
15 simp2 989 . . . . . . . 8  |-  ( ( A  e.  RR  /\  B  e.  RR  /\  A  <  B )  ->  B  e.  RR )
1615recnd 9417 . . . . . . 7  |-  ( ( A  e.  RR  /\  B  e.  RR  /\  A  <  B )  ->  B  e.  CC )
173, 14, 16cnmptc 19240 . . . . . 6  |-  ( ( A  e.  RR  /\  B  e.  RR  /\  A  <  B )  ->  (
x  e.  ( 0 [,] 1 )  |->  B )  e.  ( II 
Cn  J ) )
184mulcn 20448 . . . . . . 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 19244 . . . . 5  |-  ( ( A  e.  RR  /\  B  e.  RR  /\  A  <  B )  ->  (
x  e.  ( 0 [,] 1 )  |->  ( x  x.  B ) )  e.  ( II 
Cn  J ) )
21 1cnd 9407 . . . . . . . 8  |-  ( ( A  e.  RR  /\  B  e.  RR  /\  A  <  B )  ->  1  e.  CC )
223, 14, 21cnmptc 19240 . . . . . . 7  |-  ( ( A  e.  RR  /\  B  e.  RR  /\  A  <  B )  ->  (
x  e.  ( 0 [,] 1 )  |->  1 )  e.  ( II 
Cn  J ) )
234subcn 20447 . . . . . . . 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 19244 . . . . . 6  |-  ( ( A  e.  RR  /\  B  e.  RR  /\  A  <  B )  ->  (
x  e.  ( 0 [,] 1 )  |->  ( 1  -  x ) )  e.  ( II 
Cn  J ) )
26 simp1 988 . . . . . . . 8  |-  ( ( A  e.  RR  /\  B  e.  RR  /\  A  <  B )  ->  A  e.  RR )
2726recnd 9417 . . . . . . 7  |-  ( ( A  e.  RR  /\  B  e.  RR  /\  A  <  B )  ->  A  e.  CC )
283, 14, 27cnmptc 19240 . . . . . 6  |-  ( ( A  e.  RR  /\  B  e.  RR  /\  A  <  B )  ->  (
x  e.  ( 0 [,] 1 )  |->  A )  e.  ( II 
Cn  J ) )
293, 25, 28, 19cnmpt12f 19244 . . . . 5  |-  ( ( A  e.  RR  /\  B  e.  RR  /\  A  <  B )  ->  (
x  e.  ( 0 [,] 1 )  |->  ( ( 1  -  x
)  x.  A ) )  e.  ( II 
Cn  J ) )
304addcn 20446 . . . . . 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 19244 . . . 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 2527 . . 3  |-  ( ( A  e.  RR  /\  B  e.  RR  /\  A  <  B )  ->  F  e.  ( II  Cn  J
) )
341iccf1o 11434 . . . . . 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 459 . . . . 5  |-  ( ( A  e.  RR  /\  B  e.  RR  /\  A  <  B )  ->  F : ( 0 [,] 1 ) -1-1-onto-> ( A [,] B
) )
36 f1of 5646 . . . . 5  |-  ( F : ( 0 [,] 1 ) -1-1-onto-> ( A [,] B
)  ->  F :
( 0 [,] 1
) --> ( A [,] B ) )
37 frn 5570 . . . . 5  |-  ( F : ( 0 [,] 1 ) --> ( A [,] B )  ->  ran  F  C_  ( A [,] B ) )
3835, 36, 373syl 20 . . . 4  |-  ( ( A  e.  RR  /\  B  e.  RR  /\  A  <  B )  ->  ran  F 
C_  ( A [,] B ) )
39 iccssre 11382 . . . . . 6  |-  ( ( A  e.  RR  /\  B  e.  RR )  ->  ( A [,] B
)  C_  RR )
40393adant3 1008 . . . . 5  |-  ( ( A  e.  RR  /\  B  e.  RR  /\  A  <  B )  ->  ( A [,] B )  C_  RR )
41 ax-resscn 9344 . . . . 5  |-  RR  C_  CC
4240, 41syl6ss 3373 . . . 4  |-  ( ( A  e.  RR  /\  B  e.  RR  /\  A  <  B )  ->  ( A [,] B )  C_  CC )
43 cnrest2 18895 . . . 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 1218 . . 3  |-  ( ( A  e.  RR  /\  B  e.  RR  /\  A  <  B )  ->  ( F  e.  ( II  Cn  J )  <->  F  e.  ( II  Cn  ( Jt  ( A [,] B ) ) ) ) )
4533, 44mpbid 210 . 2  |-  ( ( A  e.  RR  /\  B  e.  RR  /\  A  <  B )  ->  F  e.  ( II  Cn  ( Jt  ( A [,] B ) ) ) )
4634simprd 463 . . . . 5  |-  ( ( A  e.  RR  /\  B  e.  RR  /\  A  <  B )  ->  `' F  =  ( y  e.  ( A [,] B
)  |->  ( ( y  -  A )  / 
( B  -  A
) ) ) )
47 resttopon 18770 . . . . . . 7  |-  ( ( J  e.  (TopOn `  CC )  /\  ( A [,] B )  C_  CC )  ->  ( Jt  ( A [,] B ) )  e.  (TopOn `  ( A [,] B ) ) )
4813, 42, 47sylancr 663 . . . . . 6  |-  ( ( A  e.  RR  /\  B  e.  RR  /\  A  <  B )  ->  ( Jt  ( A [,] B ) )  e.  (TopOn `  ( A [,] B ) ) )
49 cnrest2r 18896 . . . . . . . . 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 19239 . . . . . . . 8  |-  ( ( A  e.  RR  /\  B  e.  RR  /\  A  <  B )  ->  (
y  e.  ( A [,] B )  |->  y )  e.  ( ( Jt  ( A [,] B
) )  Cn  ( Jt  ( A [,] B ) ) ) )
5250, 51sseldi 3359 . . . . . . 7  |-  ( ( A  e.  RR  /\  B  e.  RR  /\  A  <  B )  ->  (
y  e.  ( A [,] B )  |->  y )  e.  ( ( Jt  ( A [,] B
) )  Cn  J
) )
5348, 14, 27cnmptc 19240 . . . . . . 7  |-  ( ( A  e.  RR  /\  B  e.  RR  /\  A  <  B )  ->  (
y  e.  ( A [,] B )  |->  A )  e.  ( ( Jt  ( A [,] B
) )  Cn  J
) )
5448, 52, 53, 24cnmpt12f 19244 . . . . . 6  |-  ( ( A  e.  RR  /\  B  e.  RR  /\  A  <  B )  ->  (
y  e.  ( A [,] B )  |->  ( y  -  A ) )  e.  ( ( Jt  ( A [,] B
) )  Cn  J
) )
55 difrp 11029 . . . . . . . . 9  |-  ( ( A  e.  RR  /\  B  e.  RR )  ->  ( A  <  B  <->  ( B  -  A )  e.  RR+ ) )
5655biimp3a 1318 . . . . . . . 8  |-  ( ( A  e.  RR  /\  B  e.  RR  /\  A  <  B )  ->  ( B  -  A )  e.  RR+ )
5756rpcnd 11034 . . . . . . 7  |-  ( ( A  e.  RR  /\  B  e.  RR  /\  A  <  B )  ->  ( B  -  A )  e.  CC )
5856rpne0d 11037 . . . . . . 7  |-  ( ( A  e.  RR  /\  B  e.  RR  /\  A  <  B )  ->  ( B  -  A )  =/=  0 )
594divccn 20454 . . . . . . 7  |-  ( ( ( B  -  A
)  e.  CC  /\  ( B  -  A
)  =/=  0 )  ->  ( x  e.  CC  |->  ( x  / 
( B  -  A
) ) )  e.  ( J  Cn  J
) )
6057, 58, 59syl2anc 661 . . . . . 6  |-  ( ( A  e.  RR  /\  B  e.  RR  /\  A  <  B )  ->  (
x  e.  CC  |->  ( x  /  ( B  -  A ) ) )  e.  ( J  Cn  J ) )
61 oveq1 6103 . . . . . 6  |-  ( x  =  ( y  -  A )  ->  (
x  /  ( B  -  A ) )  =  ( ( y  -  A )  / 
( B  -  A
) ) )
6248, 54, 14, 60, 61cnmpt11 19241 . . . . 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 2517 . . . 4  |-  ( ( A  e.  RR  /\  B  e.  RR  /\  A  <  B )  ->  `' F  e.  ( ( Jt  ( A [,] B ) )  Cn  J ) )
64 dfdm4 5037 . . . . . . 7  |-  dom  F  =  ran  `' F
6564eqimss2i 3416 . . . . . 6  |-  ran  `' F  C_  dom  F
66 f1odm 5650 . . . . . . 7  |-  ( F : ( 0 [,] 1 ) -1-1-onto-> ( A [,] B
)  ->  dom  F  =  ( 0 [,] 1
) )
6735, 66syl 16 . . . . . 6  |-  ( ( A  e.  RR  /\  B  e.  RR  /\  A  <  B )  ->  dom  F  =  ( 0 [,] 1 ) )
6865, 67syl5sseq 3409 . . . . 5  |-  ( ( A  e.  RR  /\  B  e.  RR  /\  A  <  B )  ->  ran  `' F  C_  ( 0 [,] 1 ) )
69 unitssre 11437 . . . . . . 7  |-  ( 0 [,] 1 )  C_  RR
7069a1i 11 . . . . . 6  |-  ( ( A  e.  RR  /\  B  e.  RR  /\  A  <  B )  ->  (
0 [,] 1 ) 
C_  RR )
7170, 41syl6ss 3373 . . . . 5  |-  ( ( A  e.  RR  /\  B  e.  RR  /\  A  <  B )  ->  (
0 [,] 1 ) 
C_  CC )
72 cnrest2 18895 . . . . 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 1218 . . . 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 210 . . 3  |-  ( ( A  e.  RR  /\  B  e.  RR  /\  A  <  B )  ->  `' F  e.  ( ( Jt  ( A [,] B ) )  Cn  ( Jt  ( 0 [,] 1 ) ) ) )
755oveq2i 6107 . . 3  |-  ( ( Jt  ( A [,] B
) )  Cn  II )  =  ( ( Jt  ( A [,] B ) )  Cn  ( Jt  ( 0 [,] 1 ) ) )
7674, 75syl6eleqr 2534 . 2  |-  ( ( A  e.  RR  /\  B  e.  RR  /\  A  <  B )  ->  `' F  e.  ( ( Jt  ( A [,] B ) )  Cn  II ) )
77 ishmeo 19337 . 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 664 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 184    /\ w3a 965    = wceq 1369    e. wcel 1756    =/= wne 2611    C_ wss 3333   class class class wbr 4297    e. cmpt 4355   `'ccnv 4844   dom cdm 4845   ran crn 4846   -->wf 5419   -1-1-onto->wf1o 5422   ` cfv 5423  (class class class)co 6096   CCcc 9285   RRcr 9286   0cc0 9287   1c1 9288    + caddc 9290    x. cmul 9292    < clt 9423    - cmin 9600    / cdiv 9998   RR+crp 10996   [,]cicc 11308   ↾t crest 14364   TopOpenctopn 14365  ℂfldccnfld 17823   Topctop 18503  TopOnctopon 18504    Cn ccn 18833    tX ctx 19138   Homeochmeo 19331   IIcii 20456
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1591  ax-4 1602  ax-5 1670  ax-6 1708  ax-7 1728  ax-8 1758  ax-9 1760  ax-10 1775  ax-11 1780  ax-12 1792  ax-13 1943  ax-ext 2423  ax-rep 4408  ax-sep 4418  ax-nul 4426  ax-pow 4475  ax-pr 4536  ax-un 6377  ax-inf2 7852  ax-cnex 9343  ax-resscn 9344  ax-1cn 9345  ax-icn 9346  ax-addcl 9347  ax-addrcl 9348  ax-mulcl 9349  ax-mulrcl 9350  ax-mulcom 9351  ax-addass 9352  ax-mulass 9353  ax-distr 9354  ax-i2m1 9355  ax-1ne0 9356  ax-1rid 9357  ax-rnegex 9358  ax-rrecex 9359  ax-cnre 9360  ax-pre-lttri 9361  ax-pre-lttrn 9362  ax-pre-ltadd 9363  ax-pre-mulgt0 9364  ax-pre-sup 9365  ax-addf 9366  ax-mulf 9367
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 966  df-3an 967  df-tru 1372  df-ex 1587  df-nf 1590  df-sb 1701  df-eu 2257  df-mo 2258  df-clab 2430  df-cleq 2436  df-clel 2439  df-nfc 2573  df-ne 2613  df-nel 2614  df-ral 2725  df-rex 2726  df-reu 2727  df-rmo 2728  df-rab 2729  df-v 2979  df-sbc 3192  df-csb 3294  df-dif 3336  df-un 3338  df-in 3340  df-ss 3347  df-pss 3349  df-nul 3643  df-if 3797  df-pw 3867  df-sn 3883  df-pr 3885  df-tp 3887  df-op 3889  df-uni 4097  df-int 4134  df-iun 4178  df-iin 4179  df-br 4298  df-opab 4356  df-mpt 4357  df-tr 4391  df-eprel 4637  df-id 4641  df-po 4646  df-so 4647  df-fr 4684  df-se 4685  df-we 4686  df-ord 4727  df-on 4728  df-lim 4729  df-suc 4730  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-iota 5386  df-fun 5425  df-fn 5426  df-f 5427  df-f1 5428  df-fo 5429  df-f1o 5430  df-fv 5431  df-isom 5432  df-riota 6057  df-ov 6099  df-oprab 6100  df-mpt2 6101  df-of 6325  df-om 6482  df-1st 6582  df-2nd 6583  df-supp 6696  df-recs 6837  df-rdg 6871  df-1o 6925  df-2o 6926  df-oadd 6929  df-er 7106  df-map 7221  df-ixp 7269  df-en 7316  df-dom 7317  df-sdom 7318  df-fin 7319  df-fsupp 7626  df-fi 7666  df-sup 7696  df-oi 7729  df-card 8114  df-cda 8342  df-pnf 9425  df-mnf 9426  df-xr 9427  df-ltxr 9428  df-le 9429  df-sub 9602  df-neg 9603  df-div 9999  df-nn 10328  df-2 10385  df-3 10386  df-4 10387  df-5 10388  df-6 10389  df-7 10390  df-8 10391  df-9 10392  df-10 10393  df-n0 10585  df-z 10652  df-dec 10761  df-uz 10867  df-q 10959  df-rp 10997  df-xneg 11094  df-xadd 11095  df-xmul 11096  df-icc 11312  df-fz 11443  df-fzo 11554  df-seq 11812  df-exp 11871  df-hash 12109  df-cj 12593  df-re 12594  df-im 12595  df-sqr 12729  df-abs 12730  df-struct 14181  df-ndx 14182  df-slot 14183  df-base 14184  df-sets 14185  df-ress 14186  df-plusg 14256  df-mulr 14257  df-starv 14258  df-sca 14259  df-vsca 14260  df-ip 14261  df-tset 14262  df-ple 14263  df-ds 14265  df-unif 14266  df-hom 14267  df-cco 14268  df-rest 14366  df-topn 14367  df-0g 14385  df-gsum 14386  df-topgen 14387  df-pt 14388  df-prds 14391  df-xrs 14445  df-qtop 14450  df-imas 14451  df-xps 14453  df-mre 14529  df-mrc 14530  df-acs 14532  df-mnd 15420  df-submnd 15470  df-mulg 15553  df-cntz 15840  df-cmn 16284  df-psmet 17814  df-xmet 17815  df-met 17816  df-bl 17817  df-mopn 17818  df-cnfld 17824  df-top 18508  df-bases 18510  df-topon 18511  df-topsp 18512  df-cn 18836  df-cnp 18837  df-tx 19140  df-hmeo 19333  df-xms 19900  df-ms 19901  df-tms 19902  df-ii 20458
This theorem is referenced by:  xrhmph  20524
  Copyright terms: Public domain W3C validator