Theorem iccf1o 10238

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 8178	. . . . . . . . 9 |- 0 e. RR
3		1re 8177	. . . . . . . . 9 |- 1 e. RR
4	2, 3	elicc2i 10173	. . . . . . . 8 |- ( x e. ( 0 [,] 1 ) <-> ( x e. RR /\ 0 <_ x /\ x <_ 1 ) )
5	4	simp1bi 1038	. . . . . . 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 8207	. . . . 5 |- ( ( ( A e. RR /\ B e. RR /\ A < B ) /\ x e. ( 0 [,] 1 ) ) -> x e. CC )
8		simpl2 1027	. . . . . 6 |- ( ( ( A e. RR /\ B e. RR /\ A < B ) /\ x e. ( 0 [,] 1 ) ) -> B e. RR )
9	8	recnd 8207	. . . . 5 |- ( ( ( A e. RR /\ B e. RR /\ A < B ) /\ x e. ( 0 [,] 1 ) ) -> B e. CC )
10	7, 9	mulcld 8199	. . . 4 |- ( ( ( A e. RR /\ B e. RR /\ A < B ) /\ x e. ( 0 [,] 1 ) ) -> ( x x. B ) e. CC )
11		ax-1cn 8124	. . . . . 6 |- 1 e. CC
12		subcl 8377	. . . . . 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 1026	. . . . . 6 |- ( ( ( A e. RR /\ B e. RR /\ A < B ) /\ x e. ( 0 [,] 1 ) ) -> A e. RR )
15	14	recnd 8207	. . . . 5 |- ( ( ( A e. RR /\ B e. RR /\ A < B ) /\ x e. ( 0 [,] 1 ) ) -> A e. CC )
16	13, 15	mulcld 8199	. . . 4 |- ( ( ( A e. RR /\ B e. RR /\ A < B ) /\ x e. ( 0 [,] 1 ) ) -> ( ( 1 - x ) x. A ) e. CC )
17	10, 16	addcomd 8329	. . 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 10237	. . 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 2308	. 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 1026	. . . . . 6 |- ( ( ( A e. RR /\ B e. RR /\ A < B ) /\ y e. ( A [,] B ) ) -> A e. RR )
22		simpl2 1027	. . . . . 6 |- ( ( ( A e. RR /\ B e. RR /\ A < B ) /\ y e. ( A [,] B ) ) -> B e. RR )
23		elicc2 10172	. . . . . . . . 9 |- ( ( A e. RR /\ B e. RR ) -> ( y e. ( A [,] B ) <-> ( y e. RR /\ A <_ y /\ y <_ B ) ) )
24	23	3adant3 1043	. . . . . . . 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 1035	. . . . . 6 |- ( ( ( A e. RR /\ B e. RR /\ A < B ) /\ y e. ( A [,] B ) ) -> y e. RR )
27		eqid 2231	. . . . . . 7 |- ( A - A ) = ( A - A )
28		eqid 2231	. . . . . . 7 |- ( B - A ) = ( B - A )
29	27, 28	iccshftl 10230	. . . . . 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 1274	. . . . 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 8559	. . . . . 6 |- ( ( ( A e. RR /\ B e. RR /\ A < B ) /\ y e. ( A [,] B ) ) -> ( y - A ) e. RR )
33	32	recnd 8207	. . . . 5 |- ( ( ( A e. RR /\ B e. RR /\ A < B ) /\ y e. ( A [,] B ) ) -> ( y - A ) e. CC )
34		difrp 9926	. . . . . . . 8 |- ( ( A e. RR /\ B e. RR ) -> ( A < B <-> ( B - A ) e. RR+ ) )
35	34	biimp3a 1381	. . . . . . 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 9932	. . . . 5 |- ( ( ( A e. RR /\ B e. RR /\ A < B ) /\ y e. ( A [,] B ) ) -> ( B - A ) e. CC )
38		rpap0 9904	. . . . . 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 8970	. . . 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 8570	. . . . . 6 |- ( ( ( A e. RR /\ B e. RR /\ A < B ) /\ y e. ( A [,] B ) ) -> ( 0 x. ( B - A ) ) = 0 )
42	21	recnd 8207	. . . . . . 7 |- ( ( ( A e. RR /\ B e. RR /\ A < B ) /\ y e. ( A [,] B ) ) -> A e. CC )
43	42	subidd 8477	. . . . . 6 |- ( ( ( A e. RR /\ B e. RR /\ A < B ) /\ y e. ( A [,] B ) ) -> ( A - A ) = 0 )
44	41, 43	eqtr4d 2267	. . . . 5 |- ( ( ( A e. RR /\ B e. RR /\ A < B ) /\ y e. ( A [,] B ) ) -> ( 0 x. ( B - A ) ) = ( A - A ) )
45	37	mulid2d 8197	. . . . 5 |- ( ( ( A e. RR /\ B e. RR /\ A < B ) /\ y e. ( A [,] B ) ) -> ( 1 x. ( B - A ) ) = ( B - A ) )
46	44, 45	oveq12d 6035	. . . 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 2315	. . 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 8179	. . . 4 |- ( ( ( A e. RR /\ B e. RR /\ A < B ) /\ y e. ( A [,] B ) ) -> 0 e. RR )
49		1red 8193	. . . 4 |- ( ( ( A e. RR /\ B e. RR /\ A < B ) /\ y e. ( A [,] B ) ) -> 1 e. RR )
50	32, 36	rerpdivcld 9962	. . . 4 |- ( ( ( A e. RR /\ B e. RR /\ A < B ) /\ y e. ( A [,] B ) ) -> ( ( y - A ) / ( B - A ) ) e. RR )
51		eqid 2231	. . . . 5 |- ( 0 x. ( B - A ) ) = ( 0 x. ( B - A ) )
52		eqid 2231	. . . . 5 |- ( 1 x. ( B - A ) ) = ( 1 x. ( B - A ) )
53	51, 52	iccdil 10232	. . . 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 1274	. . 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 2233	. . . 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 9004	. . . 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 8207	. . . . 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 8559	. . . . . . . 8 |- ( ( ( A e. RR /\ B e. RR /\ A < B ) /\ x e. ( 0 [,] 1 ) ) -> ( B - A ) e. RR )
67	6, 66	remulcld 8209	. . . . . . 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 8207	. . . . 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 8508	. . . 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 2233	. . . 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 8199	. . . . . . 7 |- ( ( ( A e. RR /\ B e. RR /\ A < B ) /\ x e. ( 0 [,] 1 ) ) -> ( x x. A ) e. CC )
74	10, 73, 15	subadd23d 8511	. . . . . 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 8592	. . . . . . 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 6032	. . . . . 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 8194	. . . . . . . . 9 |- ( ( ( A e. RR /\ B e. RR /\ A < B ) /\ x e. ( 0 [,] 1 ) ) -> 1 e. CC )
78	77, 7, 15	subdird 8593	. . . . . . . 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 8197	. . . . . . . . 9 |- ( ( ( A e. RR /\ B e. RR /\ A < B ) /\ x e. ( 0 [,] 1 ) ) -> ( 1 x. A ) = A )
80	79	oveq1d 6032	. . . . . . . 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 2264	. . . . . . 7 |- ( ( ( A e. RR /\ B e. RR /\ A < B ) /\ x e. ( 0 [,] 1 ) ) -> ( ( 1 - x ) x. A ) = ( A - ( x x. A ) ) )
82	81	oveq2d 6033	. . . . . 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 2274	. . . . 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 2243	. . 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 6226	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 1004   = wceq 1397   e. wcel 2202   class class class wbr 4088   |-> cmpt 4150   `'ccnv 4724   -1-1-onto->wf1o 5325  (class class class)co 6017   CCcc 8029   RRcr 8030   0cc0 8031   1c1 8032   + caddc 8034   x. cmul 8036   < clt 8213   <_ cle 8214   - cmin 8349   # cap 8760   / cdiv 8851   RR+crp 9887   [,]cicc 10125

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 619  ax-in2 620  ax-io 716  ax-5 1495  ax-7 1496  ax-gen 1497  ax-ie1 1541  ax-ie2 1542  ax-8 1552  ax-10 1553  ax-11 1554  ax-i12 1555  ax-bndl 1557  ax-4 1558  ax-17 1574  ax-i9 1578  ax-ial 1582  ax-i5r 1583  ax-13 2204  ax-14 2205  ax-ext 2213  ax-sep 4207  ax-pow 4264  ax-pr 4299  ax-un 4530  ax-setind 4635  ax-cnex 8122  ax-resscn 8123  ax-1cn 8124  ax-1re 8125  ax-icn 8126  ax-addcl 8127  ax-addrcl 8128  ax-mulcl 8129  ax-mulrcl 8130  ax-addcom 8131  ax-mulcom 8132  ax-addass 8133  ax-mulass 8134  ax-distr 8135  ax-i2m1 8136  ax-0lt1 8137  ax-1rid 8138  ax-0id 8139  ax-rnegex 8140  ax-precex 8141  ax-cnre 8142  ax-pre-ltirr 8143  ax-pre-ltwlin 8144  ax-pre-lttrn 8145  ax-pre-apti 8146  ax-pre-ltadd 8147  ax-pre-mulgt0 8148  ax-pre-mulext 8149

This theorem depends on definitions:  df-bi 117  df-3or 1005  df-3an 1006  df-tru 1400  df-fal 1403  df-nf 1509  df-sb 1811  df-eu 2082  df-mo 2083  df-clab 2218  df-cleq 2224  df-clel 2227  df-nfc 2363  df-ne 2403  df-nel 2498  df-ral 2515  df-rex 2516  df-reu 2517  df-rmo 2518  df-rab 2519  df-v 2804  df-sbc 3032  df-dif 3202  df-un 3204  df-in 3206  df-ss 3213  df-pw 3654  df-sn 3675  df-pr 3676  df-op 3678  df-uni 3894  df-br 4089  df-opab 4151  df-mpt 4152  df-id 4390  df-po 4393  df-iso 4394  df-xp 4731  df-rel 4732  df-cnv 4733  df-co 4734  df-dm 4735  df-rn 4736  df-iota 5286  df-fun 5328  df-fn 5329  df-f 5330  df-f1 5331  df-fo 5332  df-f1o 5333  df-fv 5334  df-riota 5970  df-ov 6020  df-oprab 6021  df-mpo 6022  df-pnf 8215  df-mnf 8216  df-xr 8217  df-ltxr 8218  df-le 8219  df-sub 8351  df-neg 8352  df-reap 8754  df-ap 8761  df-div 8852  df-rp 9888  df-icc 10129

This theorem is referenced by:  iccen  10240