Theorem iccf1o 10386

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 8316	. . . . . . . . 9 |- 0 e. RR
3		1re 8315	. . . . . . . . 9 |- 1 e. RR
4	2, 3	elicc2i 10320	. . . . . . . 8 |- ( x e. ( 0 [,] 1 ) <-> ( x e. RR /\ 0 <_ x /\ x <_ 1 ) )
5	4	simp1bi 1043	. . . . . . 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 8344	. . . . 5 |- ( ( ( A e. RR /\ B e. RR /\ A < B ) /\ x e. ( 0 [,] 1 ) ) -> x e. CC )
8		simpl2 1032	. . . . . 6 |- ( ( ( A e. RR /\ B e. RR /\ A < B ) /\ x e. ( 0 [,] 1 ) ) -> B e. RR )
9	8	recnd 8344	. . . . 5 |- ( ( ( A e. RR /\ B e. RR /\ A < B ) /\ x e. ( 0 [,] 1 ) ) -> B e. CC )
10	7, 9	mulcld 8336	. . . 4 |- ( ( ( A e. RR /\ B e. RR /\ A < B ) /\ x e. ( 0 [,] 1 ) ) -> ( x x. B ) e. CC )
11		ax-1cn 8262	. . . . . 6 |- 1 e. CC
12		subcl 8515	. . . . . 6 |- ( ( 1 e. CC /\ x e. CC ) -> ( 1 - x ) e. CC )
13	11, 7, 12	sylancr 418	. . . . 5 |- ( ( ( A e. RR /\ B e. RR /\ A < B ) /\ x e. ( 0 [,] 1 ) ) -> ( 1 - x ) e. CC )
14		simpl1 1031	. . . . . 6 |- ( ( ( A e. RR /\ B e. RR /\ A < B ) /\ x e. ( 0 [,] 1 ) ) -> A e. RR )
15	14	recnd 8344	. . . . 5 |- ( ( ( A e. RR /\ B e. RR /\ A < B ) /\ x e. ( 0 [,] 1 ) ) -> A e. CC )
16	13, 15	mulcld 8336	. . . 4 |- ( ( ( A e. RR /\ B e. RR /\ A < B ) /\ x e. ( 0 [,] 1 ) ) -> ( ( 1 - x ) x. A ) e. CC )
17	10, 16	addcomd 8467	. . 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 10384	. . 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 2315	. 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 1031	. . . . . 6 |- ( ( ( A e. RR /\ B e. RR /\ A < B ) /\ y e. ( A [,] B ) ) -> A e. RR )
22		simpl2 1032	. . . . . 6 |- ( ( ( A e. RR /\ B e. RR /\ A < B ) /\ y e. ( A [,] B ) ) -> B e. RR )
23		elicc2 10319	. . . . . . . . 9 |- ( ( A e. RR /\ B e. RR ) -> ( y e. ( A [,] B ) <-> ( y e. RR /\ A <_ y /\ y <_ B ) ) )
24	23	3adant3 1048	. . . . . . . 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 1040	. . . . . 6 |- ( ( ( A e. RR /\ B e. RR /\ A < B ) /\ y e. ( A [,] B ) ) -> y e. RR )
27		eqid 2238	. . . . . . 7 |- ( A - A ) = ( A - A )
28		eqid 2238	. . . . . . 7 |- ( B - A ) = ( B - A )
29	27, 28	iccshftl 10377	. . . . . 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 1279	. . . . 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 8698	. . . . . 6 |- ( ( ( A e. RR /\ B e. RR /\ A < B ) /\ y e. ( A [,] B ) ) -> ( y - A ) e. RR )
33	32	recnd 8344	. . . . 5 |- ( ( ( A e. RR /\ B e. RR /\ A < B ) /\ y e. ( A [,] B ) ) -> ( y - A ) e. CC )
34		difrp 10072	. . . . . . . 8 |- ( ( A e. RR /\ B e. RR ) -> ( A < B <-> ( B - A ) e. RR+ ) )
35	34	biimp3a 1386	. . . . . . 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 10078	. . . . 5 |- ( ( ( A e. RR /\ B e. RR /\ A < B ) /\ y e. ( A [,] B ) ) -> ( B - A ) e. CC )
38		rpap0 10050	. . . . . 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 9111	. . . 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 8709	. . . . . 6 |- ( ( ( A e. RR /\ B e. RR /\ A < B ) /\ y e. ( A [,] B ) ) -> ( 0 x. ( B - A ) ) = 0 )
42	21	recnd 8344	. . . . . . 7 |- ( ( ( A e. RR /\ B e. RR /\ A < B ) /\ y e. ( A [,] B ) ) -> A e. CC )
43	42	subidd 8615	. . . . . 6 |- ( ( ( A e. RR /\ B e. RR /\ A < B ) /\ y e. ( A [,] B ) ) -> ( A - A ) = 0 )
44	41, 43	eqtr4d 2274	. . . . 5 |- ( ( ( A e. RR /\ B e. RR /\ A < B ) /\ y e. ( A [,] B ) ) -> ( 0 x. ( B - A ) ) = ( A - A ) )
45	37	mullidd 8334	. . . . 5 |- ( ( ( A e. RR /\ B e. RR /\ A < B ) /\ y e. ( A [,] B ) ) -> ( 1 x. ( B - A ) ) = ( B - A ) )
46	44, 45	oveq12d 6093	. . . 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 2322	. . 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 8317	. . . 4 |- ( ( ( A e. RR /\ B e. RR /\ A < B ) /\ y e. ( A [,] B ) ) -> 0 e. RR )
49		1red 8331	. . . 4 |- ( ( ( A e. RR /\ B e. RR /\ A < B ) /\ y e. ( A [,] B ) ) -> 1 e. RR )
50	32, 36	rerpdivcld 10108	. . . 4 |- ( ( ( A e. RR /\ B e. RR /\ A < B ) /\ y e. ( A [,] B ) ) -> ( ( y - A ) / ( B - A ) ) e. RR )
51		eqid 2238	. . . . 5 |- ( 0 x. ( B - A ) ) = ( 0 x. ( B - A ) )
52		eqid 2238	. . . . 5 |- ( 1 x. ( B - A ) ) = ( 1 x. ( B - A ) )
53	51, 52	iccdil 10379	. . . 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 1279	. . 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 2240	. . . 4 |- ( x = ( ( y - A ) / ( B - A ) ) <-> ( ( y - A ) / ( B - A ) ) = x )
57	33	adantrl 482	. . . . 5 |- ( ( ( A e. RR /\ B e. RR /\ A < B ) /\ ( x e. ( 0 [,] 1 ) /\ y e. ( A [,] B ) ) ) -> ( y - A ) e. CC )
58	7	adantrr 483	. . . . 5 |- ( ( ( A e. RR /\ B e. RR /\ A < B ) /\ ( x e. ( 0 [,] 1 ) /\ y e. ( A [,] B ) ) ) -> x e. CC )
59	37	adantrl 482	. . . . 5 |- ( ( ( A e. RR /\ B e. RR /\ A < B ) /\ ( x e. ( 0 [,] 1 ) /\ y e. ( A [,] B ) ) ) -> ( B - A ) e. CC )
60	39	adantrl 482	. . . . 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 9145	. . . 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 482	. . . . . 6 |- ( ( ( A e. RR /\ B e. RR /\ A < B ) /\ ( x e. ( 0 [,] 1 ) /\ y e. ( A [,] B ) ) ) -> y e. RR )
64	63	recnd 8344	. . . . 5 |- ( ( ( A e. RR /\ B e. RR /\ A < B ) /\ ( x e. ( 0 [,] 1 ) /\ y e. ( A [,] B ) ) ) -> y e. CC )
65	42	adantrl 482	. . . . 5 |- ( ( ( A e. RR /\ B e. RR /\ A < B ) /\ ( x e. ( 0 [,] 1 ) /\ y e. ( A [,] B ) ) ) -> A e. CC )
66	8, 14	resubcld 8698	. . . . . . . 8 |- ( ( ( A e. RR /\ B e. RR /\ A < B ) /\ x e. ( 0 [,] 1 ) ) -> ( B - A ) e. RR )
67	6, 66	remulcld 8346	. . . . . . 7 |- ( ( ( A e. RR /\ B e. RR /\ A < B ) /\ x e. ( 0 [,] 1 ) ) -> ( x x. ( B - A ) ) e. RR )
68	67	adantrr 483	. . . . . 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 8344	. . . . 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 8646	. . . 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 2240	. . . 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 8336	. . . . . . 7 |- ( ( ( A e. RR /\ B e. RR /\ A < B ) /\ x e. ( 0 [,] 1 ) ) -> ( x x. A ) e. CC )
74	10, 73, 15	subadd23d 8649	. . . . . 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 8731	. . . . . . 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 6090	. . . . . 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 8332	. . . . . . . . 9 |- ( ( ( A e. RR /\ B e. RR /\ A < B ) /\ x e. ( 0 [,] 1 ) ) -> 1 e. CC )
78	77, 7, 15	subdird 8732	. . . . . . . 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	mullidd 8334	. . . . . . . . 9 |- ( ( ( A e. RR /\ B e. RR /\ A < B ) /\ x e. ( 0 [,] 1 ) ) -> ( 1 x. A ) = A )
80	79	oveq1d 6090	. . . . . . . 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 2271	. . . . . . 7 |- ( ( ( A e. RR /\ B e. RR /\ A < B ) /\ x e. ( 0 [,] 1 ) ) -> ( ( 1 - x ) x. A ) = ( A - ( x x. A ) ) )
82	81	oveq2d 6091	. . . . . 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 2281	. . . . 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 483	. . . 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 2250	. . 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 6284	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 1009   = wceq 1402   e. wcel 2209   class class class wbr 4125   |-> cmpt 4187   `'ccnv 4768   -1-1-onto->wf1o 5371  (class class class)co 6075   CCcc 8167   RRcr 8168   0cc0 8169   1c1 8170   + caddc 8172   x. cmul 8174   < clt 8350   <_ cle 8351   - cmin 8487   # cap 8899   / cdiv 8992   RR+crp 10033   [,]cicc 10272

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 623  ax-in2 624  ax-io 721  ax-5 1500  ax-7 1501  ax-gen 1502  ax-ie1 1546  ax-ie2 1547  ax-8 1557  ax-10 1558  ax-11 1559  ax-i12 1560  ax-bndl 1562  ax-4 1563  ax-17 1579  ax-i9 1583  ax-ial 1587  ax-i5r 1588  ax-14 2212  ax-ext 2220  ax-sep 4244  ax-pow 4306  ax-pr 4341  ax-un 4573  ax-setind 4679  ax-cnex 8260  ax-resscn 8261  ax-1cn 8262  ax-1re 8263  ax-icn 8264  ax-addcl 8265  ax-addrcl 8266  ax-mulcl 8267  ax-mulrcl 8268  ax-addcom 8269  ax-mulcom 8270  ax-addass 8271  ax-mulass 8272  ax-distr 8273  ax-i2m1 8274  ax-0lt1 8275  ax-1rid 8276  ax-0id 8277  ax-rnegex 8278  ax-precex 8279  ax-cnre 8280  ax-pre-ltirr 8281  ax-pre-ltwlin 8282  ax-pre-lttrn 8283  ax-pre-apti 8284  ax-pre-ltadd 8285  ax-pre-mulgt0 8286  ax-pre-mulext 8287

This theorem depends on definitions:  df-bi 117  df-3or 1010  df-3an 1011  df-tru 1405  df-fal 1408  df-nf 1514  df-sb 1816  df-eu 2089  df-mo 2090  df-clab 2225  df-cleq 2231  df-clel 2234  df-nfc 2381  df-ne 2421  df-nel 2516  df-ral 2533  df-rex 2534  df-reu 2535  df-rmo 2536  df-rab 2537  df-v 2823  df-sbc 3052  df-dif 3222  df-un 3224  df-in 3226  df-ss 3233  df-pw 3687  df-sn 3711  df-pr 3712  df-op 3714  df-uni 3931  df-br 4126  df-opab 4188  df-mpt 4189  df-id 4433  df-po 4436  df-iso 4437  df-xp 4775  df-rel 4776  df-cnv 4777  df-co 4778  df-dm 4779  df-rn 4780  df-iota 5332  df-fun 5374  df-fn 5375  df-f 5376  df-f1 5377  df-fo 5378  df-f1o 5379  df-fv 5380  df-riota 6028  df-ov 6078  df-oprab 6079  df-mpo 6080  df-pnf 8352  df-mnf 8353  df-xr 8354  df-ltxr 8355  df-le 8356  df-sub 8489  df-neg 8490  df-reap 8893  df-ap 8900  df-div 8993  df-rp 10034  df-icc 10276

This theorem is referenced by:  iccen  10388