Theorem iccf1o 8642

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 6825	. . . . . . . . 9 |- 0 e. RR
3		1re 6824	. . . . . . . . 9 |- 1 e. RR
4	2, 3	elicc2i 8578	. . . . . . . 8 |- (x e. ( 0 [,] 1) <-> (x e. RR /\ 0 <_ x /\ x <_ 1))
5	4	simp1bi 918	. . . . . . 7 |- (x e. ( 0 [,] 1) -> x e. RR)
6	5	adantl 262	. . . . . 6 |- (((A e. RR /\ B e. RR /\ A < B) /\ x e. ( 0 [,] 1)) -> x e. RR)
7	6	recnd 6851	. . . . 5 |- (((A e. RR /\ B e. RR /\ A < B) /\ x e. ( 0 [,] 1)) -> x e. CC)
8		simpl2 907	. . . . . 6 |- (((A e. RR /\ B e. RR /\ A < B) /\ x e. ( 0 [,] 1)) -> B e. RR)
9	8	recnd 6851	. . . . 5 |- (((A e. RR /\ B e. RR /\ A < B) /\ x e. ( 0 [,] 1)) -> B e. CC)
10	7, 9	mulcld 6845	. . . 4 |- (((A e. RR /\ B e. RR /\ A < B) /\ x e. ( 0 [,] 1)) -> (x x. B) e. CC)
11		ax-1cn 6776	. . . . . 6 |- 1 e. CC
12		subcl 7007	. . . . . 6 |- (( 1 e. CC /\ x e. CC) -> ( 1 - x) e. CC)
13	11, 7, 12	sylancr 393	. . . . 5 |- (((A e. RR /\ B e. RR /\ A < B) /\ x e. ( 0 [,] 1)) -> ( 1 - x) e. CC)
14		simpl1 906	. . . . . 6 |- (((A e. RR /\ B e. RR /\ A < B) /\ x e. ( 0 [,] 1)) -> A e. RR)
15	14	recnd 6851	. . . . 5 |- (((A e. RR /\ B e. RR /\ A < B) /\ x e. ( 0 [,] 1)) -> A e. CC)
16	13, 15	mulcld 6845	. . . 4 |- (((A e. RR /\ B e. RR /\ A < B) /\ x e. ( 0 [,] 1)) -> (( 1 - x) x. A) e. CC)
17	10, 16	addcomd 6961	. . 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 8641	. . 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 2111	. 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 103	. . . . 5 |- (((A e. RR /\ B e. RR /\ A < B) /\ y e. (A [,]B)) -> y e. (A [,]B))
21		simpl1 906	. . . . . 6 |- (((A e. RR /\ B e. RR /\ A < B) /\ y e. (A [,]B)) -> A e. RR)
22		simpl2 907	. . . . . 6 |- (((A e. RR /\ B e. RR /\ A < B) /\ y e. (A [,]B)) -> B e. RR)
23		elicc2 8577	. . . . . . . . 9 |- ((A e. RR /\ B e. RR) -> (y e. (A [,]B) <-> (y e. RR /\ A <_ y /\ y <_ B)))
24	23	3adant3 923	. . . . . . . 8 |- ((A e. RR /\ B e. RR /\ A < B) -> (y e. (A [,]B) <-> (y e. RR /\ A <_ y /\ y <_ B)))
25	24	biimpa 280	. . . . . . 7 |- (((A e. RR /\ B e. RR /\ A < B) /\ y e. (A [,]B)) -> (y e. RR /\ A <_ y /\ y <_ B))
26	25	simp1d 915	. . . . . 6 |- (((A e. RR /\ B e. RR /\ A < B) /\ y e. (A [,]B)) -> y e. RR)
27		eqid 2037	. . . . . . 7 |- (A - A) = (A - A)
28		eqid 2037	. . . . . . 7 |- (B - A) = (B - A)
29	27, 28	iccshftl 8634	. . . . . 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 1135	. . . . 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 135	. . . 4 |- (((A e. RR /\ B e. RR /\ A < B) /\ y e. (A [,]B)) -> (y - A) e. ((A - A) [,](B - A)))
32	26, 21	resubcld 7175	. . . . . 6 |- (((A e. RR /\ B e. RR /\ A < B) /\ y e. (A [,]B)) -> (y - A) e. RR)
33	32	recnd 6851	. . . . 5 |- (((A e. RR /\ B e. RR /\ A < B) /\ y e. (A [,]B)) -> (y - A) e. CC)
34		difrp 8394	. . . . . . . 8 |- ((A e. RR /\ B e. RR) -> (A < B <-> (B - A) e. RR+))
35	34	biimp3a 1234	. . . . . . 7 |- ((A e. RR /\ B e. RR /\ A < B) -> (B - A) e. RR+)
36	35	adantr 261	. . . . . 6 |- (((A e. RR /\ B e. RR /\ A < B) /\ y e. (A [,]B)) -> (B - A) e. RR+)
37	36	rpcnd 8399	. . . . 5 |- (((A e. RR /\ B e. RR /\ A < B) /\ y e. (A [,]B)) -> (B - A) e. CC)
38		rpap0 8374	. . . . . 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 7548	. . . 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 7185	. . . . . 6 |- (((A e. RR /\ B e. RR /\ A < B) /\ y e. (A [,]B)) -> ( 0 x. (B - A)) = 0)
42	21	recnd 6851	. . . . . . 7 |- (((A e. RR /\ B e. RR /\ A < B) /\ y e. (A [,]B)) -> A e. CC)
43	42	subidd 7106	. . . . . 6 |- (((A e. RR /\ B e. RR /\ A < B) /\ y e. (A [,]B)) -> (A - A) = 0)
44	41, 43	eqtr4d 2072	. . . . 5 |- (((A e. RR /\ B e. RR /\ A < B) /\ y e. (A [,]B)) -> ( 0 x. (B - A)) = (A - A))
45	37	mulid2d 6843	. . . . 5 |- (((A e. RR /\ B e. RR /\ A < B) /\ y e. (A [,]B)) -> ( 1 x. (B - A)) = (B - A))
46	44, 45	oveq12d 5473	. . . 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 2118	. . 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 6826	. . . 4 |- (((A e. RR /\ B e. RR /\ A < B) /\ y e. (A [,]B)) -> 0 e. RR)
49		1red 6840	. . . 4 |- (((A e. RR /\ B e. RR /\ A < B) /\ y e. (A [,]B)) -> 1 e. RR)
50	32, 36	rerpdivcld 8424	. . . 4 |- (((A e. RR /\ B e. RR /\ A < B) /\ y e. (A [,]B)) -> ((y - A) / (B - A)) e. RR)
51		eqid 2037	. . . . 5 |- ( 0 x. (B - A)) = ( 0 x. (B - A))
52		eqid 2037	. . . . 5 |- ( 1 x. (B - A)) = ( 1 x. (B - A))
53	51, 52	iccdil 8636	. . . 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 1135	. . 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 156	. 2 |- (((A e. RR /\ B e. RR /\ A < B) /\ y e. (A [,]B)) -> ((y - A) / (B - A)) e. ( 0 [,] 1))
56		eqcom 2039	. . . 4 |- (x = ((y - A) / (B - A)) <-> ((y - A) / (B - A)) = x)
57	33	adantrl 447	. . . . 5 |- (((A e. RR /\ B e. RR /\ A < B) /\ (x e. ( 0 [,] 1) /\ y e. (A [,]B))) -> (y - A) e. CC)
58	7	adantrr 448	. . . . 5 |- (((A e. RR /\ B e. RR /\ A < B) /\ (x e. ( 0 [,] 1) /\ y e. (A [,]B))) -> x e. CC)
59	37	adantrl 447	. . . . 5 |- (((A e. RR /\ B e. RR /\ A < B) /\ (x e. ( 0 [,] 1) /\ y e. (A [,]B))) -> (B - A) e. CC)
60	39	adantrl 447	. . . . 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 7581	. . . 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	syl5bb 181	. . 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 447	. . . . . 6 |- (((A e. RR /\ B e. RR /\ A < B) /\ (x e. ( 0 [,] 1) /\ y e. (A [,]B))) -> y e. RR)
64	63	recnd 6851	. . . . 5 |- (((A e. RR /\ B e. RR /\ A < B) /\ (x e. ( 0 [,] 1) /\ y e. (A [,]B))) -> y e. CC)
65	42	adantrl 447	. . . . 5 |- (((A e. RR /\ B e. RR /\ A < B) /\ (x e. ( 0 [,] 1) /\ y e. (A [,]B))) -> A e. CC)
66	8, 14	resubcld 7175	. . . . . . . 8 |- (((A e. RR /\ B e. RR /\ A < B) /\ x e. ( 0 [,] 1)) -> (B - A) e. RR)
67	6, 66	remulcld 6853	. . . . . . 7 |- (((A e. RR /\ B e. RR /\ A < B) /\ x e. ( 0 [,] 1)) -> (x x. (B - A)) e. RR)
68	67	adantrr 448	. . . . . 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 6851	. . . . 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 7137	. . . 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 2039	. . . 4 |- (((x x. (B - A)) + A) = y <-> y = ((x x. (B - A)) + A))
72	70, 71	syl6bb 185	. . 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 6845	. . . . . . 7 |- (((A e. RR /\ B e. RR /\ A < B) /\ x e. ( 0 [,] 1)) -> (x x. A) e. CC)
74	10, 73, 15	subadd23d 7140	. . . . . 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 7207	. . . . . . 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 5470	. . . . . 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 6841	. . . . . . . . 9 |- (((A e. RR /\ B e. RR /\ A < B) /\ x e. ( 0 [,] 1)) -> 1 e. CC)
78	77, 7, 15	subdird 7208	. . . . . . . 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 6843	. . . . . . . . 9 |- (((A e. RR /\ B e. RR /\ A < B) /\ x e. ( 0 [,] 1)) -> ( 1 x. A) = A)
80	79	oveq1d 5470	. . . . . . . 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 2069	. . . . . . 7 |- (((A e. RR /\ B e. RR /\ A < B) /\ x e. ( 0 [,] 1)) -> (( 1 - x) x. A) = (A - (x x. A)))
82	81	oveq2d 5471	. . . . . 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 2079	. . . . 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 448	. . . 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 2048	. . 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 203	. 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 5646	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 97   <-> wb 98   /\ w3a 884   = wceq 1242   e. wcel 1390   class class class wbr 3755   |-> cmpt 3809   `'ccnv 4287   -1-1-onto->wf1o 4844  (class class class)co 5455   CCcc 6709   RRcr 6710   0cc0 6711   1c1 6712   + caddc 6714   x. cmul 6716   < clt 6857   <_ cle 6858   - cmin 6979   # cap 7365   / cdiv 7433   RR+crp 8358   [,]cicc 8530

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 99  ax-ia2 100  ax-ia3 101  ax-in1 544  ax-in2 545  ax-io 629  ax-5 1333  ax-7 1334  ax-gen 1335  ax-ie1 1379  ax-ie2 1380  ax-8 1392  ax-10 1393  ax-11 1394  ax-i12 1395  ax-bndl 1396  ax-4 1397  ax-13 1401  ax-14 1402  ax-17 1416  ax-i9 1420  ax-ial 1424  ax-i5r 1425  ax-ext 2019  ax-coll 3863  ax-sep 3866  ax-nul 3874  ax-pow 3918  ax-pr 3935  ax-un 4136  ax-setind 4220  ax-iinf 4254  ax-cnex 6774  ax-resscn 6775  ax-1cn 6776  ax-1re 6777  ax-icn 6778  ax-addcl 6779  ax-addrcl 6780  ax-mulcl 6781  ax-mulrcl 6782  ax-addcom 6783  ax-mulcom 6784  ax-addass 6785  ax-mulass 6786  ax-distr 6787  ax-i2m1 6788  ax-1rid 6790  ax-0id 6791  ax-rnegex 6792  ax-precex 6793  ax-cnre 6794  ax-pre-ltirr 6795  ax-pre-ltwlin 6796  ax-pre-lttrn 6797  ax-pre-apti 6798  ax-pre-ltadd 6799  ax-pre-mulgt0 6800  ax-pre-mulext 6801

This theorem depends on definitions:  df-bi 110  df-dc 742  df-3or 885  df-3an 886  df-tru 1245  df-fal 1248  df-nf 1347  df-sb 1643  df-eu 1900  df-mo 1901  df-clab 2024  df-cleq 2030  df-clel 2033  df-nfc 2164  df-ne 2203  df-nel 2204  df-ral 2305  df-rex 2306  df-reu 2307  df-rmo 2308  df-rab 2309  df-v 2553  df-sbc 2759  df-csb 2847  df-dif 2914  df-un 2916  df-in 2918  df-ss 2925  df-nul 3219  df-pw 3353  df-sn 3373  df-pr 3374  df-op 3376  df-uni 3572  df-int 3607  df-iun 3650  df-br 3756  df-opab 3810  df-mpt 3811  df-tr 3846  df-eprel 4017  df-id 4021  df-po 4024  df-iso 4025  df-iord 4069  df-on 4071  df-suc 4074  df-iom 4257  df-xp 4294  df-rel 4295  df-cnv 4296  df-co 4297  df-dm 4298  df-rn 4299  df-res 4300  df-ima 4301  df-iota 4810  df-fun 4847  df-fn 4848  df-f 4849  df-f1 4850  df-fo 4851  df-f1o 4852  df-fv 4853  df-riota 5411  df-ov 5458  df-oprab 5459  df-mpt2 5460  df-1st 5709  df-2nd 5710  df-recs 5861  df-irdg 5897  df-1o 5940  df-2o 5941  df-oadd 5944  df-omul 5945  df-er 6042  df-ec 6044  df-qs 6048  df-ni 6288  df-pli 6289  df-mi 6290  df-lti 6291  df-plpq 6328  df-mpq 6329  df-enq 6331  df-nqqs 6332  df-plqqs 6333  df-mqqs 6334  df-1nqqs 6335  df-rq 6336  df-ltnqqs 6337  df-enq0 6407  df-nq0 6408  df-0nq0 6409  df-plq0 6410  df-mq0 6411  df-inp 6449  df-i1p 6450  df-iplp 6451  df-iltp 6453  df-enr 6654  df-nr 6655  df-ltr 6658  df-0r 6659  df-1r 6660  df-0 6718  df-1 6719  df-r 6721  df-lt 6724  df-pnf 6859  df-mnf 6860  df-xr 6861  df-ltxr 6862  df-le 6863  df-sub 6981  df-neg 6982  df-reap 7359  df-ap 7366  df-div 7434  df-rp 8359  df-icc 8534

This theorem is referenced by: (None)