Theorem iccf1o 10126

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 8072	. . . . . . . . 9 |- 0 e. RR
3		1re 8071	. . . . . . . . 9 |- 1 e. RR
4	2, 3	elicc2i 10061	. . . . . . . 8 |- ( x e. ( 0 [,] 1 ) <-> ( x e. RR /\ 0 <_ x /\ x <_ 1 ) )
5	4	simp1bi 1015	. . . . . . 7 |- ( x e. ( 0 [,] 1 ) -> x e. RR )
6	5	adantl 277	. . . . . 6 |- ( ( ( A e. RR /\ B e. RR /\ A < B ) /\ x e. ( 0 [,] 1 ) ) -> x e. RR )
7	6	recnd 8101	. . . . 5 |- ( ( ( A e. RR /\ B e. RR /\ A < B ) /\ x e. ( 0 [,] 1 ) ) -> x e. CC )
8		simpl2 1004	. . . . . 6 |- ( ( ( A e. RR /\ B e. RR /\ A < B ) /\ x e. ( 0 [,] 1 ) ) -> B e. RR )
9	8	recnd 8101	. . . . 5 |- ( ( ( A e. RR /\ B e. RR /\ A < B ) /\ x e. ( 0 [,] 1 ) ) -> B e. CC )
10	7, 9	mulcld 8093	. . . 4 |- ( ( ( A e. RR /\ B e. RR /\ A < B ) /\ x e. ( 0 [,] 1 ) ) -> ( x x. B ) e. CC )
11		ax-1cn 8018	. . . . . 6 |- 1 e. CC
12		subcl 8271	. . . . . 6 |- ( ( 1 e. CC /\ x e. CC ) -> ( 1 - x ) e. CC )
13	11, 7, 12	sylancr 414	. . . . 5 |- ( ( ( A e. RR /\ B e. RR /\ A < B ) /\ x e. ( 0 [,] 1 ) ) -> ( 1 - x ) e. CC )
14		simpl1 1003	. . . . . 6 |- ( ( ( A e. RR /\ B e. RR /\ A < B ) /\ x e. ( 0 [,] 1 ) ) -> A e. RR )
15	14	recnd 8101	. . . . 5 |- ( ( ( A e. RR /\ B e. RR /\ A < B ) /\ x e. ( 0 [,] 1 ) ) -> A e. CC )
16	13, 15	mulcld 8093	. . . 4 |- ( ( ( A e. RR /\ B e. RR /\ A < B ) /\ x e. ( 0 [,] 1 ) ) -> ( ( 1 - x ) x. A ) e. CC )
17	10, 16	addcomd 8223	. . 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 10125	. . 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 2282	. 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 110	. . . . 5 |- ( ( ( A e. RR /\ B e. RR /\ A < B ) /\ y e. ( A [,] B ) ) -> y e. ( A [,] B ) )
21		simpl1 1003	. . . . . 6 |- ( ( ( A e. RR /\ B e. RR /\ A < B ) /\ y e. ( A [,] B ) ) -> A e. RR )
22		simpl2 1004	. . . . . 6 |- ( ( ( A e. RR /\ B e. RR /\ A < B ) /\ y e. ( A [,] B ) ) -> B e. RR )
23		elicc2 10060	. . . . . . . . 9 |- ( ( A e. RR /\ B e. RR ) -> ( y e. ( A [,] B ) <-> ( y e. RR /\ A <_ y /\ y <_ B ) ) )
24	23	3adant3 1020	. . . . . . . 8 |- ( ( A e. RR /\ B e. RR /\ A < B ) -> ( y e. ( A [,] B ) <-> ( y e. RR /\ A <_ y /\ y <_ B ) ) )
25	24	biimpa 296	. . . . . . 7 |- ( ( ( A e. RR /\ B e. RR /\ A < B ) /\ y e. ( A [,] B ) ) -> ( y e. RR /\ A <_ y /\ y <_ B ) )
26	25	simp1d 1012	. . . . . 6 |- ( ( ( A e. RR /\ B e. RR /\ A < B ) /\ y e. ( A [,] B ) ) -> y e. RR )
27		eqid 2205	. . . . . . 7 |- ( A - A ) = ( A - A )
28		eqid 2205	. . . . . . 7 |- ( B - A ) = ( B - A )
29	27, 28	iccshftl 10118	. . . . . 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 1251	. . . . 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 147	. . . 4 |- ( ( ( A e. RR /\ B e. RR /\ A < B ) /\ y e. ( A [,] B ) ) -> ( y - A ) e. ( ( A - A ) [,] ( B - A ) ) )
32	26, 21	resubcld 8453	. . . . . 6 |- ( ( ( A e. RR /\ B e. RR /\ A < B ) /\ y e. ( A [,] B ) ) -> ( y - A ) e. RR )
33	32	recnd 8101	. . . . 5 |- ( ( ( A e. RR /\ B e. RR /\ A < B ) /\ y e. ( A [,] B ) ) -> ( y - A ) e. CC )
34		difrp 9814	. . . . . . . 8 |- ( ( A e. RR /\ B e. RR ) -> ( A < B <-> ( B - A ) e. RR+ ) )
35	34	biimp3a 1358	. . . . . . 7 |- ( ( A e. RR /\ B e. RR /\ A < B ) -> ( B - A ) e. RR+ )
36	35	adantr 276	. . . . . 6 |- ( ( ( A e. RR /\ B e. RR /\ A < B ) /\ y e. ( A [,] B ) ) -> ( B - A ) e. RR+ )
37	36	rpcnd 9820	. . . . 5 |- ( ( ( A e. RR /\ B e. RR /\ A < B ) /\ y e. ( A [,] B ) ) -> ( B - A ) e. CC )
38		rpap0 9792	. . . . . 6 |- ( ( B - A ) e. RR+ -> ( B - A ) # 0 )
39	36, 38	syl 14	. . . . 5 |- ( ( ( A e. RR /\ B e. RR /\ A < B ) /\ y e. ( A [,] B ) ) -> ( B - A ) # 0 )
40	33, 37, 39	divcanap1d 8864	. . . 4 |- ( ( ( A e. RR /\ B e. RR /\ A < B ) /\ y e. ( A [,] B ) ) -> ( ( ( y - A ) / ( B - A ) ) x. ( B - A ) ) = ( y - A ) )
41	37	mul02d 8464	. . . . . 6 |- ( ( ( A e. RR /\ B e. RR /\ A < B ) /\ y e. ( A [,] B ) ) -> ( 0 x. ( B - A ) ) = 0 )
42	21	recnd 8101	. . . . . . 7 |- ( ( ( A e. RR /\ B e. RR /\ A < B ) /\ y e. ( A [,] B ) ) -> A e. CC )
43	42	subidd 8371	. . . . . 6 |- ( ( ( A e. RR /\ B e. RR /\ A < B ) /\ y e. ( A [,] B ) ) -> ( A - A ) = 0 )
44	41, 43	eqtr4d 2241	. . . . 5 |- ( ( ( A e. RR /\ B e. RR /\ A < B ) /\ y e. ( A [,] B ) ) -> ( 0 x. ( B - A ) ) = ( A - A ) )
45	37	mulid2d 8091	. . . . 5 |- ( ( ( A e. RR /\ B e. RR /\ A < B ) /\ y e. ( A [,] B ) ) -> ( 1 x. ( B - A ) ) = ( B - A ) )
46	44, 45	oveq12d 5962	. . . 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 ) ) )
47	31, 40, 46	3eltr4d 2289	. . 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 ) ) ) )
48		0red 8073	. . . 4 |- ( ( ( A e. RR /\ B e. RR /\ A < B ) /\ y e. ( A [,] B ) ) -> 0 e. RR )
49		1red 8087	. . . 4 |- ( ( ( A e. RR /\ B e. RR /\ A < B ) /\ y e. ( A [,] B ) ) -> 1 e. RR )
50	32, 36	rerpdivcld 9850	. . . 4 |- ( ( ( A e. RR /\ B e. RR /\ A < B ) /\ y e. ( A [,] B ) ) -> ( ( y - A ) / ( B - A ) ) e. RR )
51		eqid 2205	. . . . 5 |- ( 0 x. ( B - A ) ) = ( 0 x. ( B - A ) )
52		eqid 2205	. . . . 5 |- ( 1 x. ( B - A ) ) = ( 1 x. ( B - A ) )
53	51, 52	iccdil 10120	. . . 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 ) ) ) ) )
54	48, 49, 50, 36, 53	syl22anc 1251	. . 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 ) ) ) ) )
55	47, 54	mpbird 167	. 2 |- ( ( ( A e. RR /\ B e. RR /\ A < B ) /\ y e. ( A [,] B ) ) -> ( ( y - A ) / ( B - A ) ) e. ( 0 [,] 1 ) )
56		eqcom 2207	. . . 4 |- ( x = ( ( y - A ) / ( B - A ) ) <-> ( ( y - A ) / ( B - A ) ) = x )
57	33	adantrl 478	. . . . 5 |- ( ( ( A e. RR /\ B e. RR /\ A < B ) /\ ( x e. ( 0 [,] 1 ) /\ y e. ( A [,] B ) ) ) -> ( y - A ) e. CC )
58	7	adantrr 479	. . . . 5 |- ( ( ( A e. RR /\ B e. RR /\ A < B ) /\ ( x e. ( 0 [,] 1 ) /\ y e. ( A [,] B ) ) ) -> x e. CC )
59	37	adantrl 478	. . . . 5 |- ( ( ( A e. RR /\ B e. RR /\ A < B ) /\ ( x e. ( 0 [,] 1 ) /\ y e. ( A [,] B ) ) ) -> ( B - A ) e. CC )
60	39	adantrl 478	. . . . 5 |- ( ( ( A e. RR /\ B e. RR /\ A < B ) /\ ( x e. ( 0 [,] 1 ) /\ y e. ( A [,] B ) ) ) -> ( B - A ) # 0 )
61	57, 58, 59, 60	divmulap3d 8898	. . . 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 ) ) ) )
62	56, 61	bitrid 192	. . 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 ) ) ) )
63	26	adantrl 478	. . . . . 6 |- ( ( ( A e. RR /\ B e. RR /\ A < B ) /\ ( x e. ( 0 [,] 1 ) /\ y e. ( A [,] B ) ) ) -> y e. RR )
64	63	recnd 8101	. . . . 5 |- ( ( ( A e. RR /\ B e. RR /\ A < B ) /\ ( x e. ( 0 [,] 1 ) /\ y e. ( A [,] B ) ) ) -> y e. CC )
65	42	adantrl 478	. . . . 5 |- ( ( ( A e. RR /\ B e. RR /\ A < B ) /\ ( x e. ( 0 [,] 1 ) /\ y e. ( A [,] B ) ) ) -> A e. CC )
66	8, 14	resubcld 8453	. . . . . . . 8 |- ( ( ( A e. RR /\ B e. RR /\ A < B ) /\ x e. ( 0 [,] 1 ) ) -> ( B - A ) e. RR )
67	6, 66	remulcld 8103	. . . . . . 7 |- ( ( ( A e. RR /\ B e. RR /\ A < B ) /\ x e. ( 0 [,] 1 ) ) -> ( x x. ( B - A ) ) e. RR )
68	67	adantrr 479	. . . . . 6 |- ( ( ( A e. RR /\ B e. RR /\ A < B ) /\ ( x e. ( 0 [,] 1 ) /\ y e. ( A [,] B ) ) ) -> ( x x. ( B - A ) ) e. RR )
69	68	recnd 8101	. . . . 5 |- ( ( ( A e. RR /\ B e. RR /\ A < B ) /\ ( x e. ( 0 [,] 1 ) /\ y e. ( A [,] B ) ) ) -> ( x x. ( B - A ) ) e. CC )
70	64, 65, 69	subadd2d 8402	. . . 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 ) )
71		eqcom 2207	. . . 4 |- ( ( ( x x. ( B - A ) ) + A ) = y <-> y = ( ( x x. ( B - A ) ) + A ) )
72	70, 71	bitrdi 196	. . 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 ) ) )
73	7, 15	mulcld 8093	. . . . . . 7 |- ( ( ( A e. RR /\ B e. RR /\ A < B ) /\ x e. ( 0 [,] 1 ) ) -> ( x x. A ) e. CC )
74	10, 73, 15	subadd23d 8405	. . . . . 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 ) ) ) )
75	7, 9, 15	subdid 8486	. . . . . . 7 |- ( ( ( A e. RR /\ B e. RR /\ A < B ) /\ x e. ( 0 [,] 1 ) ) -> ( x x. ( B - A ) ) = ( ( x x. B ) - ( x x. A ) ) )
76	75	oveq1d 5959	. . . . . 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 ) )
77		1cnd 8088	. . . . . . . . 9 |- ( ( ( A e. RR /\ B e. RR /\ A < B ) /\ x e. ( 0 [,] 1 ) ) -> 1 e. CC )
78	77, 7, 15	subdird 8487	. . . . . . . 8 |- ( ( ( A e. RR /\ B e. RR /\ A < B ) /\ x e. ( 0 [,] 1 ) ) -> ( ( 1 - x ) x. A ) = ( ( 1 x. A ) - ( x x. A ) ) )
79	15	mulid2d 8091	. . . . . . . . 9 |- ( ( ( A e. RR /\ B e. RR /\ A < B ) /\ x e. ( 0 [,] 1 ) ) -> ( 1 x. A ) = A )
80	79	oveq1d 5959	. . . . . . . 8 |- ( ( ( A e. RR /\ B e. RR /\ A < B ) /\ x e. ( 0 [,] 1 ) ) -> ( ( 1 x. A ) - ( x x. A ) ) = ( A - ( x x. A ) ) )
81	78, 80	eqtrd 2238	. . . . . . 7 |- ( ( ( A e. RR /\ B e. RR /\ A < B ) /\ x e. ( 0 [,] 1 ) ) -> ( ( 1 - x ) x. A ) = ( A - ( x x. A ) ) )
82	81	oveq2d 5960	. . . . . 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 ) ) ) )
83	74, 76, 82	3eqtr4d 2248	. . . . 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 ) ) )
84	83	adantrr 479	. . . 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 ) ) )
85	84	eqeq2d 2217	. . 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 ) ) ) )
86	62, 72, 85	3bitrd 214	. 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 ) ) ) )
87	1, 19, 55, 86	f1ocnv2d 6150	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   /\ wa 104   <-> wb 105   /\ w3a 981   = wceq 1373   e. wcel 2176   class class class wbr 4044   |-> cmpt 4105   `'ccnv 4674   -1-1-onto->wf1o 5270  (class class class)co 5944   CCcc 7923   RRcr 7924   0cc0 7925   1c1 7926   + caddc 7928   x. cmul 7930   < clt 8107   <_ cle 8108   - cmin 8243   # cap 8654   / cdiv 8745   RR+crp 9775   [,]cicc 10013

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-in1 615  ax-in2 616  ax-io 711  ax-5 1470  ax-7 1471  ax-gen 1472  ax-ie1 1516  ax-ie2 1517  ax-8 1527  ax-10 1528  ax-11 1529  ax-i12 1530  ax-bndl 1532  ax-4 1533  ax-17 1549  ax-i9 1553  ax-ial 1557  ax-i5r 1558  ax-13 2178  ax-14 2179  ax-ext 2187  ax-sep 4162  ax-pow 4218  ax-pr 4253  ax-un 4480  ax-setind 4585  ax-cnex 8016  ax-resscn 8017  ax-1cn 8018  ax-1re 8019  ax-icn 8020  ax-addcl 8021  ax-addrcl 8022  ax-mulcl 8023  ax-mulrcl 8024  ax-addcom 8025  ax-mulcom 8026  ax-addass 8027  ax-mulass 8028  ax-distr 8029  ax-i2m1 8030  ax-0lt1 8031  ax-1rid 8032  ax-0id 8033  ax-rnegex 8034  ax-precex 8035  ax-cnre 8036  ax-pre-ltirr 8037  ax-pre-ltwlin 8038  ax-pre-lttrn 8039  ax-pre-apti 8040  ax-pre-ltadd 8041  ax-pre-mulgt0 8042  ax-pre-mulext 8043

This theorem depends on definitions:  df-bi 117  df-3or 982  df-3an 983  df-tru 1376  df-fal 1379  df-nf 1484  df-sb 1786  df-eu 2057  df-mo 2058  df-clab 2192  df-cleq 2198  df-clel 2201  df-nfc 2337  df-ne 2377  df-nel 2472  df-ral 2489  df-rex 2490  df-reu 2491  df-rmo 2492  df-rab 2493  df-v 2774  df-sbc 2999  df-dif 3168  df-un 3170  df-in 3172  df-ss 3179  df-pw 3618  df-sn 3639  df-pr 3640  df-op 3642  df-uni 3851  df-br 4045  df-opab 4106  df-mpt 4107  df-id 4340  df-po 4343  df-iso 4344  df-xp 4681  df-rel 4682  df-cnv 4683  df-co 4684  df-dm 4685  df-rn 4686  df-iota 5232  df-fun 5273  df-fn 5274  df-f 5275  df-f1 5276  df-fo 5277  df-f1o 5278  df-fv 5279  df-riota 5899  df-ov 5947  df-oprab 5948  df-mpo 5949  df-pnf 8109  df-mnf 8110  df-xr 8111  df-ltxr 8112  df-le 8113  df-sub 8245  df-neg 8246  df-reap 8648  df-ap 8655  df-div 8746  df-rp 9776  df-icc 10017

This theorem is referenced by:  iccen  10128