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Theorem icchmeo 22024
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 21966 . . . . . 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 21970 . . . . . . . . 9  |-  II  =  ( Jt  ( 0 [,] 1 ) )
65oveq2i 6331 . . . . . . . 8  |-  ( II 
Cn  II )  =  ( II  Cn  ( Jt  ( 0 [,] 1
) ) )
74cnfldtop 21859 . . . . . . . . 9  |-  J  e. 
Top
8 cnrest2r 20358 . . . . . . . . 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 3474 . . . . . . 7  |-  ( II 
Cn  II )  C_  ( II  Cn  J
)
113cnmptid 20731 . . . . . . 7  |-  ( ( A  e.  RR  /\  B  e.  RR  /\  A  <  B )  ->  (
x  e.  ( 0 [,] 1 )  |->  x )  e.  ( II 
Cn  II ) )
1210, 11sseldi 3442 . . . . . 6  |-  ( ( A  e.  RR  /\  B  e.  RR  /\  A  <  B )  ->  (
x  e.  ( 0 [,] 1 )  |->  x )  e.  ( II 
Cn  J ) )
134cnfldtopon 21858 . . . . . . . 8  |-  J  e.  (TopOn `  CC )
1413a1i 11 . . . . . . 7  |-  ( ( A  e.  RR  /\  B  e.  RR  /\  A  <  B )  ->  J  e.  (TopOn `  CC )
)
15 simp2 1015 . . . . . . . 8  |-  ( ( A  e.  RR  /\  B  e.  RR  /\  A  <  B )  ->  B  e.  RR )
1615recnd 9700 . . . . . . 7  |-  ( ( A  e.  RR  /\  B  e.  RR  /\  A  <  B )  ->  B  e.  CC )
173, 14, 16cnmptc 20732 . . . . . 6  |-  ( ( A  e.  RR  /\  B  e.  RR  /\  A  <  B )  ->  (
x  e.  ( 0 [,] 1 )  |->  B )  e.  ( II 
Cn  J ) )
184mulcn 21954 . . . . . . 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 20736 . . . . 5  |-  ( ( A  e.  RR  /\  B  e.  RR  /\  A  <  B )  ->  (
x  e.  ( 0 [,] 1 )  |->  ( x  x.  B ) )  e.  ( II 
Cn  J ) )
21 1cnd 9690 . . . . . . . 8  |-  ( ( A  e.  RR  /\  B  e.  RR  /\  A  <  B )  ->  1  e.  CC )
223, 14, 21cnmptc 20732 . . . . . . 7  |-  ( ( A  e.  RR  /\  B  e.  RR  /\  A  <  B )  ->  (
x  e.  ( 0 [,] 1 )  |->  1 )  e.  ( II 
Cn  J ) )
234subcn 21953 . . . . . . . 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 20736 . . . . . 6  |-  ( ( A  e.  RR  /\  B  e.  RR  /\  A  <  B )  ->  (
x  e.  ( 0 [,] 1 )  |->  ( 1  -  x ) )  e.  ( II 
Cn  J ) )
26 simp1 1014 . . . . . . . 8  |-  ( ( A  e.  RR  /\  B  e.  RR  /\  A  <  B )  ->  A  e.  RR )
2726recnd 9700 . . . . . . 7  |-  ( ( A  e.  RR  /\  B  e.  RR  /\  A  <  B )  ->  A  e.  CC )
283, 14, 27cnmptc 20732 . . . . . 6  |-  ( ( A  e.  RR  /\  B  e.  RR  /\  A  <  B )  ->  (
x  e.  ( 0 [,] 1 )  |->  A )  e.  ( II 
Cn  J ) )
293, 25, 28, 19cnmpt12f 20736 . . . . 5  |-  ( ( A  e.  RR  /\  B  e.  RR  /\  A  <  B )  ->  (
x  e.  ( 0 [,] 1 )  |->  ( ( 1  -  x
)  x.  A ) )  e.  ( II 
Cn  J ) )
304addcn 21952 . . . . . 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 20736 . . . 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 2544 . . 3  |-  ( ( A  e.  RR  /\  B  e.  RR  /\  A  <  B )  ->  F  e.  ( II  Cn  J
) )
341iccf1o 11811 . . . . . 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 465 . . . . 5  |-  ( ( A  e.  RR  /\  B  e.  RR  /\  A  <  B )  ->  F : ( 0 [,] 1 ) -1-1-onto-> ( A [,] B
) )
36 f1of 5841 . . . . 5  |-  ( F : ( 0 [,] 1 ) -1-1-onto-> ( A [,] B
)  ->  F :
( 0 [,] 1
) --> ( A [,] B ) )
37 frn 5762 . . . . 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 11750 . . . . . 6  |-  ( ( A  e.  RR  /\  B  e.  RR )  ->  ( A [,] B
)  C_  RR )
40393adant3 1034 . . . . 5  |-  ( ( A  e.  RR  /\  B  e.  RR  /\  A  <  B )  ->  ( A [,] B )  C_  RR )
41 ax-resscn 9627 . . . . 5  |-  RR  C_  CC
4240, 41syl6ss 3456 . . . 4  |-  ( ( A  e.  RR  /\  B  e.  RR  /\  A  <  B )  ->  ( A [,] B )  C_  CC )
43 cnrest2 20357 . . . 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 1276 . . 3  |-  ( ( A  e.  RR  /\  B  e.  RR  /\  A  <  B )  ->  ( F  e.  ( II  Cn  J )  <->  F  e.  ( II  Cn  ( Jt  ( A [,] B ) ) ) ) )
4533, 44mpbid 215 . 2  |-  ( ( A  e.  RR  /\  B  e.  RR  /\  A  <  B )  ->  F  e.  ( II  Cn  ( Jt  ( A [,] B ) ) ) )
4634simprd 469 . . . . 5  |-  ( ( A  e.  RR  /\  B  e.  RR  /\  A  <  B )  ->  `' F  =  ( y  e.  ( A [,] B
)  |->  ( ( y  -  A )  / 
( B  -  A
) ) ) )
47 resttopon 20232 . . . . . . 7  |-  ( ( J  e.  (TopOn `  CC )  /\  ( A [,] B )  C_  CC )  ->  ( Jt  ( A [,] B ) )  e.  (TopOn `  ( A [,] B ) ) )
4813, 42, 47sylancr 674 . . . . . 6  |-  ( ( A  e.  RR  /\  B  e.  RR  /\  A  <  B )  ->  ( Jt  ( A [,] B ) )  e.  (TopOn `  ( A [,] B ) ) )
49 cnrest2r 20358 . . . . . . . . 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 20731 . . . . . . . 8  |-  ( ( A  e.  RR  /\  B  e.  RR  /\  A  <  B )  ->  (
y  e.  ( A [,] B )  |->  y )  e.  ( ( Jt  ( A [,] B
) )  Cn  ( Jt  ( A [,] B ) ) ) )
5250, 51sseldi 3442 . . . . . . 7  |-  ( ( A  e.  RR  /\  B  e.  RR  /\  A  <  B )  ->  (
y  e.  ( A [,] B )  |->  y )  e.  ( ( Jt  ( A [,] B
) )  Cn  J
) )
5348, 14, 27cnmptc 20732 . . . . . . 7  |-  ( ( A  e.  RR  /\  B  e.  RR  /\  A  <  B )  ->  (
y  e.  ( A [,] B )  |->  A )  e.  ( ( Jt  ( A [,] B
) )  Cn  J
) )
5448, 52, 53, 24cnmpt12f 20736 . . . . . 6  |-  ( ( A  e.  RR  /\  B  e.  RR  /\  A  <  B )  ->  (
y  e.  ( A [,] B )  |->  ( y  -  A ) )  e.  ( ( Jt  ( A [,] B
) )  Cn  J
) )
55 difrp 11371 . . . . . . . . 9  |-  ( ( A  e.  RR  /\  B  e.  RR )  ->  ( A  <  B  <->  ( B  -  A )  e.  RR+ ) )
5655biimp3a 1379 . . . . . . . 8  |-  ( ( A  e.  RR  /\  B  e.  RR  /\  A  <  B )  ->  ( B  -  A )  e.  RR+ )
5756rpcnd 11377 . . . . . . 7  |-  ( ( A  e.  RR  /\  B  e.  RR  /\  A  <  B )  ->  ( B  -  A )  e.  CC )
5856rpne0d 11380 . . . . . . 7  |-  ( ( A  e.  RR  /\  B  e.  RR  /\  A  <  B )  ->  ( B  -  A )  =/=  0 )
594divccn 21960 . . . . . . 7  |-  ( ( ( B  -  A
)  e.  CC  /\  ( B  -  A
)  =/=  0 )  ->  ( x  e.  CC  |->  ( x  / 
( B  -  A
) ) )  e.  ( J  Cn  J
) )
6057, 58, 59syl2anc 671 . . . . . 6  |-  ( ( A  e.  RR  /\  B  e.  RR  /\  A  <  B )  ->  (
x  e.  CC  |->  ( x  /  ( B  -  A ) ) )  e.  ( J  Cn  J ) )
61 oveq1 6327 . . . . . 6  |-  ( x  =  ( y  -  A )  ->  (
x  /  ( B  -  A ) )  =  ( ( y  -  A )  / 
( B  -  A
) ) )
6248, 54, 14, 60, 61cnmpt11 20733 . . . . 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 2540 . . . 4  |-  ( ( A  e.  RR  /\  B  e.  RR  /\  A  <  B )  ->  `' F  e.  ( ( Jt  ( A [,] B ) )  Cn  J ) )
64 dfdm4 5049 . . . . . . 7  |-  dom  F  =  ran  `' F
6564eqimss2i 3499 . . . . . 6  |-  ran  `' F  C_  dom  F
66 f1odm 5845 . . . . . . 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 3492 . . . . 5  |-  ( ( A  e.  RR  /\  B  e.  RR  /\  A  <  B )  ->  ran  `' F  C_  ( 0 [,] 1 ) )
69 unitssre 11814 . . . . . . 7  |-  ( 0 [,] 1 )  C_  RR
7069a1i 11 . . . . . 6  |-  ( ( A  e.  RR  /\  B  e.  RR  /\  A  <  B )  ->  (
0 [,] 1 ) 
C_  RR )
7170, 41syl6ss 3456 . . . . 5  |-  ( ( A  e.  RR  /\  B  e.  RR  /\  A  <  B )  ->  (
0 [,] 1 ) 
C_  CC )
72 cnrest2 20357 . . . . 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 1276 . . . 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 215 . . 3  |-  ( ( A  e.  RR  /\  B  e.  RR  /\  A  <  B )  ->  `' F  e.  ( ( Jt  ( A [,] B ) )  Cn  ( Jt  ( 0 [,] 1 ) ) ) )
755oveq2i 6331 . . 3  |-  ( ( Jt  ( A [,] B
) )  Cn  II )  =  ( ( Jt  ( A [,] B ) )  Cn  ( Jt  ( 0 [,] 1 ) ) )
7674, 75syl6eleqr 2551 . 2  |-  ( ( A  e.  RR  /\  B  e.  RR  /\  A  <  B )  ->  `' F  e.  ( ( Jt  ( A [,] B ) )  Cn  II ) )
77 ishmeo 20829 . 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 675 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 189    /\ w3a 991    = wceq 1455    e. wcel 1898    =/= wne 2633    C_ wss 3416   class class class wbr 4418    |-> cmpt 4477   `'ccnv 4855   dom cdm 4856   ran crn 4857   -->wf 5601   -1-1-onto->wf1o 5604   ` cfv 5605  (class class class)co 6320   CCcc 9568   RRcr 9569   0cc0 9570   1c1 9571    + caddc 9573    x. cmul 9575    < clt 9706    - cmin 9891    / cdiv 10302   RR+crp 11336   [,]cicc 11672   ↾t crest 15374   TopOpenctopn 15375  ℂfldccnfld 19025   Topctop 19972  TopOnctopon 19973    Cn ccn 20295    tX ctx 20630   Homeochmeo 20823   IIcii 21962
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1680  ax-4 1693  ax-5 1769  ax-6 1816  ax-7 1862  ax-8 1900  ax-9 1907  ax-10 1926  ax-11 1931  ax-12 1944  ax-13 2102  ax-ext 2442  ax-rep 4531  ax-sep 4541  ax-nul 4550  ax-pow 4598  ax-pr 4656  ax-un 6615  ax-inf2 8177  ax-cnex 9626  ax-resscn 9627  ax-1cn 9628  ax-icn 9629  ax-addcl 9630  ax-addrcl 9631  ax-mulcl 9632  ax-mulrcl 9633  ax-mulcom 9634  ax-addass 9635  ax-mulass 9636  ax-distr 9637  ax-i2m1 9638  ax-1ne0 9639  ax-1rid 9640  ax-rnegex 9641  ax-rrecex 9642  ax-cnre 9643  ax-pre-lttri 9644  ax-pre-lttrn 9645  ax-pre-ltadd 9646  ax-pre-mulgt0 9647  ax-pre-sup 9648  ax-addf 9649  ax-mulf 9650
This theorem depends on definitions:  df-bi 190  df-or 376  df-an 377  df-3or 992  df-3an 993  df-tru 1458  df-ex 1675  df-nf 1679  df-sb 1809  df-eu 2314  df-mo 2315  df-clab 2449  df-cleq 2455  df-clel 2458  df-nfc 2592  df-ne 2635  df-nel 2636  df-ral 2754  df-rex 2755  df-reu 2756  df-rmo 2757  df-rab 2758  df-v 3059  df-sbc 3280  df-csb 3376  df-dif 3419  df-un 3421  df-in 3423  df-ss 3430  df-pss 3432  df-nul 3744  df-if 3894  df-pw 3965  df-sn 3981  df-pr 3983  df-tp 3985  df-op 3987  df-uni 4213  df-int 4249  df-iun 4294  df-iin 4295  df-br 4419  df-opab 4478  df-mpt 4479  df-tr 4514  df-eprel 4767  df-id 4771  df-po 4777  df-so 4778  df-fr 4815  df-se 4816  df-we 4817  df-xp 4862  df-rel 4863  df-cnv 4864  df-co 4865  df-dm 4866  df-rn 4867  df-res 4868  df-ima 4869  df-pred 5403  df-ord 5449  df-on 5450  df-lim 5451  df-suc 5452  df-iota 5569  df-fun 5607  df-fn 5608  df-f 5609  df-f1 5610  df-fo 5611  df-f1o 5612  df-fv 5613  df-isom 5614  df-riota 6282  df-ov 6323  df-oprab 6324  df-mpt2 6325  df-of 6563  df-om 6725  df-1st 6825  df-2nd 6826  df-supp 6947  df-wrecs 7059  df-recs 7121  df-rdg 7159  df-1o 7213  df-2o 7214  df-oadd 7217  df-er 7394  df-map 7505  df-ixp 7554  df-en 7601  df-dom 7602  df-sdom 7603  df-fin 7604  df-fsupp 7915  df-fi 7956  df-sup 7987  df-inf 7988  df-oi 8056  df-card 8404  df-cda 8629  df-pnf 9708  df-mnf 9709  df-xr 9710  df-ltxr 9711  df-le 9712  df-sub 9893  df-neg 9894  df-div 10303  df-nn 10643  df-2 10701  df-3 10702  df-4 10703  df-5 10704  df-6 10705  df-7 10706  df-8 10707  df-9 10708  df-10 10709  df-n0 10904  df-z 10972  df-dec 11086  df-uz 11194  df-q 11299  df-rp 11337  df-xneg 11443  df-xadd 11444  df-xmul 11445  df-icc 11676  df-fz 11820  df-fzo 11953  df-seq 12252  df-exp 12311  df-hash 12554  df-cj 13217  df-re 13218  df-im 13219  df-sqrt 13353  df-abs 13354  df-struct 15178  df-ndx 15179  df-slot 15180  df-base 15181  df-sets 15182  df-ress 15183  df-plusg 15258  df-mulr 15259  df-starv 15260  df-sca 15261  df-vsca 15262  df-ip 15263  df-tset 15264  df-ple 15265  df-ds 15267  df-unif 15268  df-hom 15269  df-cco 15270  df-rest 15376  df-topn 15377  df-0g 15395  df-gsum 15396  df-topgen 15397  df-pt 15398  df-prds 15401  df-xrs 15455  df-qtop 15461  df-imas 15462  df-xps 15465  df-mre 15547  df-mrc 15548  df-acs 15550  df-mgm 16543  df-sgrp 16582  df-mnd 16592  df-submnd 16638  df-mulg 16731  df-cntz 17026  df-cmn 17487  df-psmet 19017  df-xmet 19018  df-met 19019  df-bl 19020  df-mopn 19021  df-cnfld 19026  df-top 19976  df-bases 19977  df-topon 19978  df-topsp 19979  df-cn 20298  df-cnp 20299  df-tx 20632  df-hmeo 20825  df-xms 21390  df-ms 21391  df-tms 21392  df-ii 21964
This theorem is referenced by:  xrhmph  22030
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