Theorem icchmeo 22740

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 ` ℂfld )
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 ( J ↾t ( 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.

Step	Hyp	Ref	Expression
1		icchmeo.f	. . . 4 |- F = ( x e. ( 0 [,] 1 ) |-> ( ( x x. B ) + ( ( 1 - x ) x. A ) ) )
2		iitopon 22682	. . . . . 6 |- II e. (TopOn ` ( 0 [,] 1 ) )
3	2	a1i 11	. . . . 5 |- ( ( A e. RR /\ B e. RR /\ A < B ) -> II e. (TopOn ` ( 0 [,] 1 ) ) )
4		icchmeo.j	. . . . . . . . . 10 |- J = ( TopOpen ` ℂfld )
5	4	dfii3 22686	. . . . . . . . 9 |- II = ( J ↾t ( 0 [,] 1 ) )
6	5	oveq2i 6661	. . . . . . . 8 |- ( II Cn II ) = ( II Cn ( J ↾t ( 0 [,] 1 ) ) )
7	4	cnfldtop 22587	. . . . . . . . 9 |- J e. Top
8		cnrest2r 21091	. . . . . . . . 9 |- ( J e. Top -> ( II Cn ( J ↾t ( 0 [,] 1 ) ) ) C_ ( II Cn J ) )
9	7, 8	ax-mp 5	. . . . . . . 8 |- ( II Cn ( J ↾t ( 0 [,] 1 ) ) ) C_ ( II Cn J )
10	6, 9	eqsstri 3635	. . . . . . 7 |- ( II Cn II ) C_ ( II Cn J )
11	3	cnmptid 21464	. . . . . . 7 |- ( ( A e. RR /\ B e. RR /\ A < B ) -> ( x e. ( 0 [,] 1 ) |-> x ) e. ( II Cn II ) )
12	10, 11	sseldi 3601	. . . . . 6 |- ( ( A e. RR /\ B e. RR /\ A < B ) -> ( x e. ( 0 [,] 1 ) |-> x ) e. ( II Cn J ) )
13	4	cnfldtopon 22586	. . . . . . . 8 |- J e. (TopOn ` CC )
14	13	a1i 11	. . . . . . 7 |- ( ( A e. RR /\ B e. RR /\ A < B ) -> J e. (TopOn ` CC ) )
15		simp2 1062	. . . . . . . 8 |- ( ( A e. RR /\ B e. RR /\ A < B ) -> B e. RR )
16	15	recnd 10068	. . . . . . 7 |- ( ( A e. RR /\ B e. RR /\ A < B ) -> B e. CC )
17	3, 14, 16	cnmptc 21465	. . . . . 6 |- ( ( A e. RR /\ B e. RR /\ A < B ) -> ( x e. ( 0 [,] 1 ) |-> B ) e. ( II Cn J ) )
18	4	mulcn 22670	. . . . . . 7 |- x. e. ( ( J tX J ) Cn J )
19	18	a1i 11	. . . . . 6 |- ( ( A e. RR /\ B e. RR /\ A < B ) -> x. e. ( ( J tX J ) Cn J ) )
20	3, 12, 17, 19	cnmpt12f 21469	. . . . 5 |- ( ( A e. RR /\ B e. RR /\ A < B ) -> ( x e. ( 0 [,] 1 ) |-> ( x x. B ) ) e. ( II Cn J ) )
21		1cnd 10056	. . . . . . . 8 |- ( ( A e. RR /\ B e. RR /\ A < B ) -> 1 e. CC )
22	3, 14, 21	cnmptc 21465	. . . . . . 7 |- ( ( A e. RR /\ B e. RR /\ A < B ) -> ( x e. ( 0 [,] 1 ) |-> 1 ) e. ( II Cn J ) )
23	4	subcn 22669	. . . . . . . 8 |- - e. ( ( J tX J ) Cn J )
24	23	a1i 11	. . . . . . 7 |- ( ( A e. RR /\ B e. RR /\ A < B ) -> - e. ( ( J tX J ) Cn J ) )
25	3, 22, 12, 24	cnmpt12f 21469	. . . . . 6 |- ( ( A e. RR /\ B e. RR /\ A < B ) -> ( x e. ( 0 [,] 1 ) |-> ( 1 - x ) ) e. ( II Cn J ) )
26		simp1 1061	. . . . . . . 8 |- ( ( A e. RR /\ B e. RR /\ A < B ) -> A e. RR )
27	26	recnd 10068	. . . . . . 7 |- ( ( A e. RR /\ B e. RR /\ A < B ) -> A e. CC )
28	3, 14, 27	cnmptc 21465	. . . . . 6 |- ( ( A e. RR /\ B e. RR /\ A < B ) -> ( x e. ( 0 [,] 1 ) |-> A ) e. ( II Cn J ) )
29	3, 25, 28, 19	cnmpt12f 21469	. . . . 5 |- ( ( A e. RR /\ B e. RR /\ A < B ) -> ( x e. ( 0 [,] 1 ) |-> ( ( 1 - x ) x. A ) ) e. ( II Cn J ) )
30	4	addcn 22668	. . . . . 6 |- + e. ( ( J tX J ) Cn J )
31	30	a1i 11	. . . . 5 |- ( ( A e. RR /\ B e. RR /\ A < B ) -> + e. ( ( J tX J ) Cn J ) )
32	3, 20, 29, 31	cnmpt12f 21469	. . . 4 |- ( ( A e. RR /\ B e. RR /\ A < B ) -> ( x e. ( 0 [,] 1 ) |-> ( ( x x. B ) + ( ( 1 - x ) x. A ) ) ) e. ( II Cn J ) )
33	1, 32	syl5eqel 2705	. . 3 |- ( ( A e. RR /\ B e. RR /\ A < B ) -> F e. ( II Cn J ) )
34	1	iccf1o 12316	. . . . . 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 ) ) ) ) )
35	34	simpld 475	. . . . 5 |- ( ( A e. RR /\ B e. RR /\ A < B ) -> F : ( 0 [,] 1 ) -1-1-onto-> ( A [,] B ) )
36		f1of 6137	. . . . 5 |- ( F : ( 0 [,] 1 ) -1-1-onto-> ( A [,] B ) -> F : ( 0 [,] 1 ) --> ( A [,] B ) )
37		frn 6053	. . . . 5 |- ( F : ( 0 [,] 1 ) --> ( A [,] B ) -> ran F C_ ( A [,] B ) )
38	35, 36, 37	3syl 18	. . . 4 |- ( ( A e. RR /\ B e. RR /\ A < B ) -> ran F C_ ( A [,] B ) )
39		iccssre 12255	. . . . . 6 |- ( ( A e. RR /\ B e. RR ) -> ( A [,] B ) C_ RR )
40	39	3adant3 1081	. . . . 5 |- ( ( A e. RR /\ B e. RR /\ A < B ) -> ( A [,] B ) C_ RR )
41		ax-resscn 9993	. . . . 5 |- RR C_ CC
42	40, 41	syl6ss 3615	. . . 4 |- ( ( A e. RR /\ B e. RR /\ A < B ) -> ( A [,] B ) C_ CC )
43		cnrest2 21090	. . . 4 |- ( ( J e. (TopOn ` CC ) /\ ran F C_ ( A [,] B ) /\ ( A [,] B ) C_ CC ) -> ( F e. ( II Cn J ) <-> F e. ( II Cn ( J ↾t ( A [,] B ) ) ) ) )
44	14, 38, 42, 43	syl3anc 1326	. . 3 |- ( ( A e. RR /\ B e. RR /\ A < B ) -> ( F e. ( II Cn J ) <-> F e. ( II Cn ( J ↾t ( A [,] B ) ) ) ) )
45	33, 44	mpbid 222	. 2 |- ( ( A e. RR /\ B e. RR /\ A < B ) -> F e. ( II Cn ( J ↾t ( A [,] B ) ) ) )
46	34	simprd 479	. . . . 5 |- ( ( A e. RR /\ B e. RR /\ A < B ) -> `' F = ( y e. ( A [,] B ) |-> ( ( y - A ) / ( B - A ) ) ) )
47		resttopon 20965	. . . . . . 7 |- ( ( J e. (TopOn ` CC ) /\ ( A [,] B ) C_ CC ) -> ( J ↾t ( A [,] B ) ) e. (TopOn ` ( A [,] B ) ) )
48	13, 42, 47	sylancr 695	. . . . . 6 |- ( ( A e. RR /\ B e. RR /\ A < B ) -> ( J ↾t ( A [,] B ) ) e. (TopOn ` ( A [,] B ) ) )
49		cnrest2r 21091	. . . . . . . . 9 |- ( J e. Top -> ( ( J ↾t ( A [,] B ) ) Cn ( J ↾t ( A [,] B ) ) ) C_ ( ( J ↾t ( A [,] B ) ) Cn J ) )
50	7, 49	ax-mp 5	. . . . . . . 8 |- ( ( J ↾t ( A [,] B ) ) Cn ( J ↾t ( A [,] B ) ) ) C_ ( ( J ↾t ( A [,] B ) ) Cn J )
51	48	cnmptid 21464	. . . . . . . 8 |- ( ( A e. RR /\ B e. RR /\ A < B ) -> ( y e. ( A [,] B ) |-> y ) e. ( ( J ↾t ( A [,] B ) ) Cn ( J ↾t ( A [,] B ) ) ) )
52	50, 51	sseldi 3601	. . . . . . 7 |- ( ( A e. RR /\ B e. RR /\ A < B ) -> ( y e. ( A [,] B ) |-> y ) e. ( ( J ↾t ( A [,] B ) ) Cn J ) )
53	48, 14, 27	cnmptc 21465	. . . . . . 7 |- ( ( A e. RR /\ B e. RR /\ A < B ) -> ( y e. ( A [,] B ) |-> A ) e. ( ( J ↾t ( A [,] B ) ) Cn J ) )
54	48, 52, 53, 24	cnmpt12f 21469	. . . . . 6 |- ( ( A e. RR /\ B e. RR /\ A < B ) -> ( y e. ( A [,] B ) |-> ( y - A ) ) e. ( ( J ↾t ( A [,] B ) ) Cn J ) )
55		difrp 11868	. . . . . . . . 9 |- ( ( A e. RR /\ B e. RR ) -> ( A < B <-> ( B - A ) e. RR+ ) )
56	55	biimp3a 1432	. . . . . . . 8 |- ( ( A e. RR /\ B e. RR /\ A < B ) -> ( B - A ) e. RR+ )
57	56	rpcnd 11874	. . . . . . 7 |- ( ( A e. RR /\ B e. RR /\ A < B ) -> ( B - A ) e. CC )
58	56	rpne0d 11877	. . . . . . 7 |- ( ( A e. RR /\ B e. RR /\ A < B ) -> ( B - A ) =/= 0 )
59	4	divccn 22676	. . . . . . 7 |- ( ( ( B - A ) e. CC /\ ( B - A ) =/= 0 ) -> ( x e. CC |-> ( x / ( B - A ) ) ) e. ( J Cn J ) )
60	57, 58, 59	syl2anc 693	. . . . . 6 |- ( ( A e. RR /\ B e. RR /\ A < B ) -> ( x e. CC |-> ( x / ( B - A ) ) ) e. ( J Cn J ) )
61		oveq1 6657	. . . . . 6 |- ( x = ( y - A ) -> ( x / ( B - A ) ) = ( ( y - A ) / ( B - A ) ) )
62	48, 54, 14, 60, 61	cnmpt11 21466	. . . . 5 |- ( ( A e. RR /\ B e. RR /\ A < B ) -> ( y e. ( A [,] B ) |-> ( ( y - A ) / ( B - A ) ) ) e. ( ( J ↾t ( A [,] B ) ) Cn J ) )
63	46, 62	eqeltrd 2701	. . . 4 |- ( ( A e. RR /\ B e. RR /\ A < B ) -> `' F e. ( ( J ↾t ( A [,] B ) ) Cn J ) )
64		dfdm4 5316	. . . . . . 7 |- dom F = ran `' F
65	64	eqimss2i 3660	. . . . . 6 |- ran `' F C_ dom F
66		f1odm 6141	. . . . . . 7 |- ( F : ( 0 [,] 1 ) -1-1-onto-> ( A [,] B ) -> dom F = ( 0 [,] 1 ) )
67	35, 66	syl 17	. . . . . 6 |- ( ( A e. RR /\ B e. RR /\ A < B ) -> dom F = ( 0 [,] 1 ) )
68	65, 67	syl5sseq 3653	. . . . 5 |- ( ( A e. RR /\ B e. RR /\ A < B ) -> ran `' F C_ ( 0 [,] 1 ) )
69		unitssre 12319	. . . . . . 7 |- ( 0 [,] 1 ) C_ RR
70	69	a1i 11	. . . . . 6 |- ( ( A e. RR /\ B e. RR /\ A < B ) -> ( 0 [,] 1 ) C_ RR )
71	70, 41	syl6ss 3615	. . . . 5 |- ( ( A e. RR /\ B e. RR /\ A < B ) -> ( 0 [,] 1 ) C_ CC )
72		cnrest2 21090	. . . . 5 |- ( ( J e. (TopOn ` CC ) /\ ran `' F C_ ( 0 [,] 1 ) /\ ( 0 [,] 1 ) C_ CC ) -> ( `' F e. ( ( J ↾t ( A [,] B ) ) Cn J ) <-> `' F e. ( ( J ↾t ( A [,] B ) ) Cn ( J ↾t ( 0 [,] 1 ) ) ) ) )
73	14, 68, 71, 72	syl3anc 1326	. . . 4 |- ( ( A e. RR /\ B e. RR /\ A < B ) -> ( `' F e. ( ( J ↾t ( A [,] B ) ) Cn J ) <-> `' F e. ( ( J ↾t ( A [,] B ) ) Cn ( J ↾t ( 0 [,] 1 ) ) ) ) )
74	63, 73	mpbid 222	. . 3 |- ( ( A e. RR /\ B e. RR /\ A < B ) -> `' F e. ( ( J ↾t ( A [,] B ) ) Cn ( J ↾t ( 0 [,] 1 ) ) ) )
75	5	oveq2i 6661	. . 3 |- ( ( J ↾t ( A [,] B ) ) Cn II ) = ( ( J ↾t ( A [,] B ) ) Cn ( J ↾t ( 0 [,] 1 ) ) )
76	74, 75	syl6eleqr 2712	. 2 |- ( ( A e. RR /\ B e. RR /\ A < B ) -> `' F e. ( ( J ↾t ( A [,] B ) ) Cn II ) )
77		ishmeo 21562	. 2 |- ( F e. ( II Homeo ( J ↾t ( A [,] B ) ) ) <-> ( F e. ( II Cn ( J ↾t ( A [,] B ) ) ) /\ `' F e. ( ( J ↾t ( A [,] B ) ) Cn II ) ) )
78	45, 76, 77	sylanbrc 698	1 |- ( ( A e. RR /\ B e. RR /\ A < B ) -> F e. ( II Homeo ( J ↾t ( A [,] B ) ) ) )

Syntax hints:   -> wi 4   <-> wb 196   /\ w3a 1037   = wceq 1483   e. wcel 1990   =/= wne 2794   C_ wss 3574   class class class wbr 4653   |-> cmpt 4729   `'ccnv 5113   dom cdm 5114   ran crn 5115   -->wf 5884   -1-1-onto->wf1o 5887   ` cfv 5888  (class class class)co 6650   CCcc 9934   RRcr 9935   0cc0 9936   1c1 9937   + caddc 9939   x. cmul 9941   < clt 10074   - cmin 10266   / cdiv 10684   RR+crp 11832   [,]cicc 12178   ↾_t crest 16081   TopOpenctopn 16082  ℂ_fldccnfld 19746   Topctop 20698  TopOnctopon 20715   Cn ccn 21028   tX ctx 21363   Homeochmeo 21556   IIcii 22678

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-8 1992  ax-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246  ax-ext 2602  ax-rep 4771  ax-sep 4781  ax-nul 4789  ax-pow 4843  ax-pr 4906  ax-un 6949  ax-inf2 8538  ax-cnex 9992  ax-resscn 9993  ax-1cn 9994  ax-icn 9995  ax-addcl 9996  ax-addrcl 9997  ax-mulcl 9998  ax-mulrcl 9999  ax-mulcom 10000  ax-addass 10001  ax-mulass 10002  ax-distr 10003  ax-i2m1 10004  ax-1ne0 10005  ax-1rid 10006  ax-rnegex 10007  ax-rrecex 10008  ax-cnre 10009  ax-pre-lttri 10010  ax-pre-lttrn 10011  ax-pre-ltadd 10012  ax-pre-mulgt0 10013  ax-pre-sup 10014  ax-addf 10015  ax-mulf 10016

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3or 1038  df-3an 1039  df-tru 1486  df-ex 1705  df-nf 1710  df-sb 1881  df-eu 2474  df-mo 2475  df-clab 2609  df-cleq 2615  df-clel 2618  df-nfc 2753  df-ne 2795  df-nel 2898  df-ral 2917  df-rex 2918  df-reu 2919  df-rmo 2920  df-rab 2921  df-v 3202  df-sbc 3436  df-csb 3534  df-dif 3577  df-un 3579  df-in 3581  df-ss 3588  df-pss 3590  df-nul 3916  df-if 4087  df-pw 4160  df-sn 4178  df-pr 4180  df-tp 4182  df-op 4184  df-uni 4437  df-int 4476  df-iun 4522  df-iin 4523  df-br 4654  df-opab 4713  df-mpt 4730  df-tr 4753  df-id 5024  df-eprel 5029  df-po 5035  df-so 5036  df-fr 5073  df-se 5074  df-we 5075  df-xp 5120  df-rel 5121  df-cnv 5122  df-co 5123  df-dm 5124  df-rn 5125  df-res 5126  df-ima 5127  df-pred 5680  df-ord 5726  df-on 5727  df-lim 5728  df-suc 5729  df-iota 5851  df-fun 5890  df-fn 5891  df-f 5892  df-f1 5893  df-fo 5894  df-f1o 5895  df-fv 5896  df-isom 5897  df-riota 6611  df-ov 6653  df-oprab 6654  df-mpt2 6655  df-of 6897  df-om 7066  df-1st 7168  df-2nd 7169  df-supp 7296  df-wrecs 7407  df-recs 7468  df-rdg 7506  df-1o 7560  df-2o 7561  df-oadd 7564  df-er 7742  df-map 7859  df-ixp 7909  df-en 7956  df-dom 7957  df-sdom 7958  df-fin 7959  df-fsupp 8276  df-fi 8317  df-sup 8348  df-inf 8349  df-oi 8415  df-card 8765  df-cda 8990  df-pnf 10076  df-mnf 10077  df-xr 10078  df-ltxr 10079  df-le 10080  df-sub 10268  df-neg 10269  df-div 10685  df-nn 11021  df-2 11079  df-3 11080  df-4 11081  df-5 11082  df-6 11083  df-7 11084  df-8 11085  df-9 11086  df-n0 11293  df-z 11378  df-dec 11494  df-uz 11688  df-q 11789  df-rp 11833  df-xneg 11946  df-xadd 11947  df-xmul 11948  df-icc 12182  df-fz 12327  df-fzo 12466  df-seq 12802  df-exp 12861  df-hash 13118  df-cj 13839  df-re 13840  df-im 13841  df-sqrt 13975  df-abs 13976  df-struct 15859  df-ndx 15860  df-slot 15861  df-base 15863  df-sets 15864  df-ress 15865  df-plusg 15954  df-mulr 15955  df-starv 15956  df-sca 15957  df-vsca 15958  df-ip 15959  df-tset 15960  df-ple 15961  df-ds 15964  df-unif 15965  df-hom 15966  df-cco 15967  df-rest 16083  df-topn 16084  df-0g 16102  df-gsum 16103  df-topgen 16104  df-pt 16105  df-prds 16108  df-xrs 16162  df-qtop 16167  df-imas 16168  df-xps 16170  df-mre 16246  df-mrc 16247  df-acs 16249  df-mgm 17242  df-sgrp 17284  df-mnd 17295  df-submnd 17336  df-mulg 17541  df-cntz 17750  df-cmn 18195  df-psmet 19738  df-xmet 19739  df-met 19740  df-bl 19741  df-mopn 19742  df-cnfld 19747  df-top 20699  df-topon 20716  df-topsp 20737  df-bases 20750  df-cn 21031  df-cnp 21032  df-tx 21365  df-hmeo 21558  df-xms 22125  df-ms 22126  df-tms 22127  df-ii 22680

This theorem is referenced by:  xrhmph  22746