Theorem iccf1o 12316

Description: Describe a bijection from [ 0 , 1 ] to an arbitrary nontrivial closed interval [ A , B ]. (Contributed by Mario Carneiro, 8-Sep-2015.)

Hypothesis

Ref	Expression
iccf1o.1	|- F = ( x e. ( 0 [,] 1 ) |-> ( ( x x. B ) + ( ( 1 - x ) x. A ) ) )

Assertion

Ref	Expression
iccf1o	|- ( ( 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 ) ) ) ) )

Distinct variable groups:   x, y, A   x, B, y

Allowed substitution hints:   F( x, y)

Proof of Theorem iccf1o

Step	Hyp	Ref	Expression
1		iccf1o.1	. 2 |- F = ( x e. ( 0 [,] 1 ) |-> ( ( x x. B ) + ( ( 1 - x ) x. A ) ) )
2		0re 10040	. . . . . . . . 9 |- 0 e. RR
3		1re 10039	. . . . . . . . 9 |- 1 e. RR
4	2, 3	elicc2i 12239	. . . . . . . 8 |- ( x e. ( 0 [,] 1 ) <-> ( x e. RR /\ 0 <_ x /\ x <_ 1 ) )
5	4	simp1bi 1076	. . . . . . 7 |- ( x e. ( 0 [,] 1 ) -> x e. RR )
6	5	adantl 482	. . . . . 6 |- ( ( ( A e. RR /\ B e. RR /\ A < B ) /\ x e. ( 0 [,] 1 ) ) -> x e. RR )
7	6	recnd 10068	. . . . 5 |- ( ( ( A e. RR /\ B e. RR /\ A < B ) /\ x e. ( 0 [,] 1 ) ) -> x e. CC )
8		simpl2 1065	. . . . . 6 |- ( ( ( A e. RR /\ B e. RR /\ A < B ) /\ x e. ( 0 [,] 1 ) ) -> B e. RR )
9	8	recnd 10068	. . . . 5 |- ( ( ( A e. RR /\ B e. RR /\ A < B ) /\ x e. ( 0 [,] 1 ) ) -> B e. CC )
10	7, 9	mulcld 10060	. . . 4 |- ( ( ( A e. RR /\ B e. RR /\ A < B ) /\ x e. ( 0 [,] 1 ) ) -> ( x x. B ) e. CC )
11		ax-1cn 9994	. . . . . 6 |- 1 e. CC
12		subcl 10280	. . . . . 6 |- ( ( 1 e. CC /\ x e. CC ) -> ( 1 - x ) e. CC )
13	11, 7, 12	sylancr 695	. . . . 5 |- ( ( ( A e. RR /\ B e. RR /\ A < B ) /\ x e. ( 0 [,] 1 ) ) -> ( 1 - x ) e. CC )
14		simpl1 1064	. . . . . 6 |- ( ( ( A e. RR /\ B e. RR /\ A < B ) /\ x e. ( 0 [,] 1 ) ) -> A e. RR )
15	14	recnd 10068	. . . . 5 |- ( ( ( A e. RR /\ B e. RR /\ A < B ) /\ x e. ( 0 [,] 1 ) ) -> A e. CC )
16	13, 15	mulcld 10060	. . . 4 |- ( ( ( A e. RR /\ B e. RR /\ A < B ) /\ x e. ( 0 [,] 1 ) ) -> ( ( 1 - x ) x. A ) e. CC )
17	10, 16	addcomd 10238	. . 3 |- ( ( ( A e. RR /\ B e. RR /\ A < B ) /\ x e. ( 0 [,] 1 ) ) -> ( ( x x. B ) + ( ( 1 - x ) x. A ) ) = ( ( ( 1 - x ) x. A ) + ( x x. B ) ) )
18		lincmb01cmp 12315	. . 3 |- ( ( ( A e. RR /\ B e. RR /\ A < B ) /\ x e. ( 0 [,] 1 ) ) -> ( ( ( 1 - x ) x. A ) + ( x x. B ) ) e. ( A [,] B ) )
19	17, 18	eqeltrd 2701	. 2 |- ( ( ( A e. RR /\ B e. RR /\ A < B ) /\ x e. ( 0 [,] 1 ) ) -> ( ( x x. B ) + ( ( 1 - x ) x. A ) ) e. ( A [,] B ) )
20		simpr 477	. . . . 5 |- ( ( ( A e. RR /\ B e. RR /\ A < B ) /\ y e. ( A [,] B ) ) -> y e. ( A [,] B ) )
21		simpl1 1064	. . . . . 6 |- ( ( ( A e. RR /\ B e. RR /\ A < B ) /\ y e. ( A [,] B ) ) -> A e. RR )
22		simpl2 1065	. . . . . 6 |- ( ( ( A e. RR /\ B e. RR /\ A < B ) /\ y e. ( A [,] B ) ) -> B e. RR )
23		elicc2 12238	. . . . . . . . 9 |- ( ( A e. RR /\ B e. RR ) -> ( y e. ( A [,] B ) <-> ( y e. RR /\ A <_ y /\ y <_ B ) ) )
24	23	3adant3 1081	. . . . . . . 8 |- ( ( A e. RR /\ B e. RR /\ A < B ) -> ( y e. ( A [,] B ) <-> ( y e. RR /\ A <_ y /\ y <_ B ) ) )
25	24	biimpa 501	. . . . . . 7 |- ( ( ( A e. RR /\ B e. RR /\ A < B ) /\ y e. ( A [,] B ) ) -> ( y e. RR /\ A <_ y /\ y <_ B ) )
26	25	simp1d 1073	. . . . . 6 |- ( ( ( A e. RR /\ B e. RR /\ A < B ) /\ y e. ( A [,] B ) ) -> y e. RR )
27		eqid 2622	. . . . . . 7 |- ( A - A ) = ( A - A )
28		eqid 2622	. . . . . . 7 |- ( B - A ) = ( B - A )
29	27, 28	iccshftl 12308	. . . . . 6 |- ( ( ( A e. RR /\ B e. RR ) /\ ( y e. RR /\ A e. RR ) ) -> ( y e. ( A [,] B ) <-> ( y - A ) e. ( ( A - A ) [,] ( B - A ) ) ) )
30	21, 22, 26, 21, 29	syl22anc 1327	. . . . 5 |- ( ( ( A e. RR /\ B e. RR /\ A < B ) /\ y e. ( A [,] B ) ) -> ( y e. ( A [,] B ) <-> ( y - A ) e. ( ( A - A ) [,] ( B - A ) ) ) )
31	20, 30	mpbid 222	. . . 4 |- ( ( ( A e. RR /\ B e. RR /\ A < B ) /\ y e. ( A [,] B ) ) -> ( y - A ) e. ( ( A - A ) [,] ( B - A ) ) )
32	26, 21	resubcld 10458	. . . . . 6 |- ( ( ( A e. RR /\ B e. RR /\ A < B ) /\ y e. ( A [,] B ) ) -> ( y - A ) e. RR )
33	32	recnd 10068	. . . . 5 |- ( ( ( A e. RR /\ B e. RR /\ A < B ) /\ y e. ( A [,] B ) ) -> ( y - A ) e. CC )
34		difrp 11868	. . . . . . . 8 |- ( ( A e. RR /\ B e. RR ) -> ( A < B <-> ( B - A ) e. RR+ ) )
35	34	biimp3a 1432	. . . . . . 7 |- ( ( A e. RR /\ B e. RR /\ A < B ) -> ( B - A ) e. RR+ )
36	35	adantr 481	. . . . . 6 |- ( ( ( A e. RR /\ B e. RR /\ A < B ) /\ y e. ( A [,] B ) ) -> ( B - A ) e. RR+ )
37	36	rpcnd 11874	. . . . 5 |- ( ( ( A e. RR /\ B e. RR /\ A < B ) /\ y e. ( A [,] B ) ) -> ( B - A ) e. CC )
38	36	rpne0d 11877	. . . . 5 |- ( ( ( A e. RR /\ B e. RR /\ A < B ) /\ y e. ( A [,] B ) ) -> ( B - A ) =/= 0 )
39	33, 37, 38	divcan1d 10802	. . . 4 |- ( ( ( A e. RR /\ B e. RR /\ A < B ) /\ y e. ( A [,] B ) ) -> ( ( ( y - A ) / ( B - A ) ) x. ( B - A ) ) = ( y - A ) )
40	37	mul02d 10234	. . . . . 6 |- ( ( ( A e. RR /\ B e. RR /\ A < B ) /\ y e. ( A [,] B ) ) -> ( 0 x. ( B - A ) ) = 0 )
41	21	recnd 10068	. . . . . . 7 |- ( ( ( A e. RR /\ B e. RR /\ A < B ) /\ y e. ( A [,] B ) ) -> A e. CC )
42	41	subidd 10380	. . . . . 6 |- ( ( ( A e. RR /\ B e. RR /\ A < B ) /\ y e. ( A [,] B ) ) -> ( A - A ) = 0 )
43	40, 42	eqtr4d 2659	. . . . 5 |- ( ( ( A e. RR /\ B e. RR /\ A < B ) /\ y e. ( A [,] B ) ) -> ( 0 x. ( B - A ) ) = ( A - A ) )
44	37	mulid2d 10058	. . . . 5 |- ( ( ( A e. RR /\ B e. RR /\ A < B ) /\ y e. ( A [,] B ) ) -> ( 1 x. ( B - A ) ) = ( B - A ) )
45	43, 44	oveq12d 6668	. . . 4 |- ( ( ( A e. RR /\ B e. RR /\ A < B ) /\ y e. ( A [,] B ) ) -> ( ( 0 x. ( B - A ) ) [,] ( 1 x. ( B - A ) ) ) = ( ( A - A ) [,] ( B - A ) ) )
46	31, 39, 45	3eltr4d 2716	. . 3 |- ( ( ( A e. RR /\ B e. RR /\ A < B ) /\ y e. ( A [,] B ) ) -> ( ( ( y - A ) / ( B - A ) ) x. ( B - A ) ) e. ( ( 0 x. ( B - A ) ) [,] ( 1 x. ( B - A ) ) ) )
47		0red 10041	. . . 4 |- ( ( ( A e. RR /\ B e. RR /\ A < B ) /\ y e. ( A [,] B ) ) -> 0 e. RR )
48		1red 10055	. . . 4 |- ( ( ( A e. RR /\ B e. RR /\ A < B ) /\ y e. ( A [,] B ) ) -> 1 e. RR )
49	32, 36	rerpdivcld 11903	. . . 4 |- ( ( ( A e. RR /\ B e. RR /\ A < B ) /\ y e. ( A [,] B ) ) -> ( ( y - A ) / ( B - A ) ) e. RR )
50		eqid 2622	. . . . 5 |- ( 0 x. ( B - A ) ) = ( 0 x. ( B - A ) )
51		eqid 2622	. . . . 5 |- ( 1 x. ( B - A ) ) = ( 1 x. ( B - A ) )
52	50, 51	iccdil 12310	. . . 4 |- ( ( ( 0 e. RR /\ 1 e. RR ) /\ ( ( ( y - A ) / ( B - A ) ) e. RR /\ ( B - A ) e. RR+ ) ) -> ( ( ( y - A ) / ( B - A ) ) e. ( 0 [,] 1 ) <-> ( ( ( y - A ) / ( B - A ) ) x. ( B - A ) ) e. ( ( 0 x. ( B - A ) ) [,] ( 1 x. ( B - A ) ) ) ) )
53	47, 48, 49, 36, 52	syl22anc 1327	. . 3 |- ( ( ( A e. RR /\ B e. RR /\ A < B ) /\ y e. ( A [,] B ) ) -> ( ( ( y - A ) / ( B - A ) ) e. ( 0 [,] 1 ) <-> ( ( ( y - A ) / ( B - A ) ) x. ( B - A ) ) e. ( ( 0 x. ( B - A ) ) [,] ( 1 x. ( B - A ) ) ) ) )
54	46, 53	mpbird 247	. 2 |- ( ( ( A e. RR /\ B e. RR /\ A < B ) /\ y e. ( A [,] B ) ) -> ( ( y - A ) / ( B - A ) ) e. ( 0 [,] 1 ) )
55		eqcom 2629	. . . 4 |- ( x = ( ( y - A ) / ( B - A ) ) <-> ( ( y - A ) / ( B - A ) ) = x )
56	33	adantrl 752	. . . . 5 |- ( ( ( A e. RR /\ B e. RR /\ A < B ) /\ ( x e. ( 0 [,] 1 ) /\ y e. ( A [,] B ) ) ) -> ( y - A ) e. CC )
57	7	adantrr 753	. . . . 5 |- ( ( ( A e. RR /\ B e. RR /\ A < B ) /\ ( x e. ( 0 [,] 1 ) /\ y e. ( A [,] B ) ) ) -> x e. CC )
58	37	adantrl 752	. . . . 5 |- ( ( ( A e. RR /\ B e. RR /\ A < B ) /\ ( x e. ( 0 [,] 1 ) /\ y e. ( A [,] B ) ) ) -> ( B - A ) e. CC )
59	38	adantrl 752	. . . . 5 |- ( ( ( A e. RR /\ B e. RR /\ A < B ) /\ ( x e. ( 0 [,] 1 ) /\ y e. ( A [,] B ) ) ) -> ( B - A ) =/= 0 )
60	56, 57, 58, 59	divmul3d 10835	. . . 4 |- ( ( ( A e. RR /\ B e. RR /\ A < B ) /\ ( x e. ( 0 [,] 1 ) /\ y e. ( A [,] B ) ) ) -> ( ( ( y - A ) / ( B - A ) ) = x <-> ( y - A ) = ( x x. ( B - A ) ) ) )
61	55, 60	syl5bb 272	. . 3 |- ( ( ( A e. RR /\ B e. RR /\ A < B ) /\ ( x e. ( 0 [,] 1 ) /\ y e. ( A [,] B ) ) ) -> ( x = ( ( y - A ) / ( B - A ) ) <-> ( y - A ) = ( x x. ( B - A ) ) ) )
62	26	adantrl 752	. . . . . 6 |- ( ( ( A e. RR /\ B e. RR /\ A < B ) /\ ( x e. ( 0 [,] 1 ) /\ y e. ( A [,] B ) ) ) -> y e. RR )
63	62	recnd 10068	. . . . 5 |- ( ( ( A e. RR /\ B e. RR /\ A < B ) /\ ( x e. ( 0 [,] 1 ) /\ y e. ( A [,] B ) ) ) -> y e. CC )
64	41	adantrl 752	. . . . 5 |- ( ( ( A e. RR /\ B e. RR /\ A < B ) /\ ( x e. ( 0 [,] 1 ) /\ y e. ( A [,] B ) ) ) -> A e. CC )
65	8, 14	resubcld 10458	. . . . . . . 8 |- ( ( ( A e. RR /\ B e. RR /\ A < B ) /\ x e. ( 0 [,] 1 ) ) -> ( B - A ) e. RR )
66	6, 65	remulcld 10070	. . . . . . 7 |- ( ( ( A e. RR /\ B e. RR /\ A < B ) /\ x e. ( 0 [,] 1 ) ) -> ( x x. ( B - A ) ) e. RR )
67	66	adantrr 753	. . . . . 6 |- ( ( ( A e. RR /\ B e. RR /\ A < B ) /\ ( x e. ( 0 [,] 1 ) /\ y e. ( A [,] B ) ) ) -> ( x x. ( B - A ) ) e. RR )
68	67	recnd 10068	. . . . 5 |- ( ( ( A e. RR /\ B e. RR /\ A < B ) /\ ( x e. ( 0 [,] 1 ) /\ y e. ( A [,] B ) ) ) -> ( x x. ( B - A ) ) e. CC )
69	63, 64, 68	subadd2d 10411	. . . 4 |- ( ( ( A e. RR /\ B e. RR /\ A < B ) /\ ( x e. ( 0 [,] 1 ) /\ y e. ( A [,] B ) ) ) -> ( ( y - A ) = ( x x. ( B - A ) ) <-> ( ( x x. ( B - A ) ) + A ) = y ) )
70		eqcom 2629	. . . 4 |- ( ( ( x x. ( B - A ) ) + A ) = y <-> y = ( ( x x. ( B - A ) ) + A ) )
71	69, 70	syl6bb 276	. . 3 |- ( ( ( A e. RR /\ B e. RR /\ A < B ) /\ ( x e. ( 0 [,] 1 ) /\ y e. ( A [,] B ) ) ) -> ( ( y - A ) = ( x x. ( B - A ) ) <-> y = ( ( x x. ( B - A ) ) + A ) ) )
72	7, 15	mulcld 10060	. . . . . . 7 |- ( ( ( A e. RR /\ B e. RR /\ A < B ) /\ x e. ( 0 [,] 1 ) ) -> ( x x. A ) e. CC )
73	10, 72, 15	subadd23d 10414	. . . . . 6 |- ( ( ( A e. RR /\ B e. RR /\ A < B ) /\ x e. ( 0 [,] 1 ) ) -> ( ( ( x x. B ) - ( x x. A ) ) + A ) = ( ( x x. B ) + ( A - ( x x. A ) ) ) )
74	7, 9, 15	subdid 10486	. . . . . . 7 |- ( ( ( A e. RR /\ B e. RR /\ A < B ) /\ x e. ( 0 [,] 1 ) ) -> ( x x. ( B - A ) ) = ( ( x x. B ) - ( x x. A ) ) )
75	74	oveq1d 6665	. . . . . 6 |- ( ( ( A e. RR /\ B e. RR /\ A < B ) /\ x e. ( 0 [,] 1 ) ) -> ( ( x x. ( B - A ) ) + A ) = ( ( ( x x. B ) - ( x x. A ) ) + A ) )
76		1cnd 10056	. . . . . . . . 9 |- ( ( ( A e. RR /\ B e. RR /\ A < B ) /\ x e. ( 0 [,] 1 ) ) -> 1 e. CC )
77	76, 7, 15	subdird 10487	. . . . . . . 8 |- ( ( ( A e. RR /\ B e. RR /\ A < B ) /\ x e. ( 0 [,] 1 ) ) -> ( ( 1 - x ) x. A ) = ( ( 1 x. A ) - ( x x. A ) ) )
78	15	mulid2d 10058	. . . . . . . . 9 |- ( ( ( A e. RR /\ B e. RR /\ A < B ) /\ x e. ( 0 [,] 1 ) ) -> ( 1 x. A ) = A )
79	78	oveq1d 6665	. . . . . . . 8 |- ( ( ( A e. RR /\ B e. RR /\ A < B ) /\ x e. ( 0 [,] 1 ) ) -> ( ( 1 x. A ) - ( x x. A ) ) = ( A - ( x x. A ) ) )
80	77, 79	eqtrd 2656	. . . . . . 7 |- ( ( ( A e. RR /\ B e. RR /\ A < B ) /\ x e. ( 0 [,] 1 ) ) -> ( ( 1 - x ) x. A ) = ( A - ( x x. A ) ) )
81	80	oveq2d 6666	. . . . . 6 |- ( ( ( A e. RR /\ B e. RR /\ A < B ) /\ x e. ( 0 [,] 1 ) ) -> ( ( x x. B ) + ( ( 1 - x ) x. A ) ) = ( ( x x. B ) + ( A - ( x x. A ) ) ) )
82	73, 75, 81	3eqtr4d 2666	. . . . 5 |- ( ( ( A e. RR /\ B e. RR /\ A < B ) /\ x e. ( 0 [,] 1 ) ) -> ( ( x x. ( B - A ) ) + A ) = ( ( x x. B ) + ( ( 1 - x ) x. A ) ) )
83	82	adantrr 753	. . . 4 |- ( ( ( A e. RR /\ B e. RR /\ A < B ) /\ ( x e. ( 0 [,] 1 ) /\ y e. ( A [,] B ) ) ) -> ( ( x x. ( B - A ) ) + A ) = ( ( x x. B ) + ( ( 1 - x ) x. A ) ) )
84	83	eqeq2d 2632	. . 3 |- ( ( ( A e. RR /\ B e. RR /\ A < B ) /\ ( x e. ( 0 [,] 1 ) /\ y e. ( A [,] B ) ) ) -> ( y = ( ( x x. ( B - A ) ) + A ) <-> y = ( ( x x. B ) + ( ( 1 - x ) x. A ) ) ) )
85	61, 71, 84	3bitrd 294	. 2 |- ( ( ( A e. RR /\ B e. RR /\ A < B ) /\ ( x e. ( 0 [,] 1 ) /\ y e. ( A [,] B ) ) ) -> ( x = ( ( y - A ) / ( B - A ) ) <-> y = ( ( x x. B ) + ( ( 1 - x ) x. A ) ) ) )
86	1, 19, 54, 85	f1ocnv2d 6886	1 |- ( ( 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 ) ) ) ) )

Syntax hints:   -> wi 4   <-> wb 196   /\ wa 384   /\ w3a 1037   = wceq 1483   e. wcel 1990   =/= wne 2794   class class class wbr 4653   |-> cmpt 4729   `'ccnv 5113   -1-1-onto->wf1o 5887  (class class class)co 6650   CCcc 9934   RRcr 9935   0cc0 9936   1c1 9937   + caddc 9939   x. cmul 9941   < clt 10074   <_ cle 10075   - cmin 10266   / cdiv 10684   RR+crp 11832   [,]cicc 12178

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-sep 4781  ax-nul 4789  ax-pow 4843  ax-pr 4906  ax-un 6949  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

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-nul 3916  df-if 4087  df-pw 4160  df-sn 4178  df-pr 4180  df-op 4184  df-uni 4437  df-br 4654  df-opab 4713  df-mpt 4730  df-id 5024  df-po 5035  df-so 5036  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-iota 5851  df-fun 5890  df-fn 5891  df-f 5892  df-f1 5893  df-fo 5894  df-f1o 5895  df-fv 5896  df-riota 6611  df-ov 6653  df-oprab 6654  df-mpt2 6655  df-er 7742  df-en 7956  df-dom 7957  df-sdom 7958  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-rp 11833  df-icc 12182

This theorem is referenced by:  iccen  12317  icchmeo  22740