Theorem xrge0iifcnv 29979

Description: Define a bijection from [ 0 , 1 ] to [ 0 , +oo ]. (Contributed by Thierry Arnoux, 29-Mar-2017.)

Hypothesis

Ref	Expression
xrge0iifhmeo.1	|- F = ( x e. ( 0 [,] 1 ) |-> if ( x = 0 , +oo , -u ( log ` x ) ) )

Assertion

Ref	Expression
xrge0iifcnv	|- ( F : ( 0 [,] 1 ) -1-1-onto-> ( 0 [,] +oo ) /\ `' F = ( y e. ( 0 [,] +oo ) |-> if ( y = +oo , 0 , ( exp ` -u y ) ) ) )

Distinct variable group:   x, y

Allowed substitution hints:   F( x, y)

Proof of Theorem xrge0iifcnv

Step	Hyp	Ref	Expression
1		xrge0iifhmeo.1	. . 3 |- F = ( x e. ( 0 [,] 1 ) |-> if ( x = 0 , +oo , -u ( log ` x ) ) )
2		0xr 10086	. . . . . . 7 |- 0 e. RR*
3		pnfxr 10092	. . . . . . 7 |- +oo e. RR*
4		0lepnf 11966	. . . . . . 7 |- 0 <_ +oo
5		ubicc2 12289	. . . . . . 7 |- ( ( 0 e. RR* /\ +oo e. RR* /\ 0 <_ +oo ) -> +oo e. ( 0 [,] +oo ) )
6	2, 3, 4, 5	mp3an 1424	. . . . . 6 |- +oo e. ( 0 [,] +oo )
7	6	a1i 11	. . . . 5 |- ( ( x e. ( 0 [,] 1 ) /\ x = 0 ) -> +oo e. ( 0 [,] +oo ) )
8		icossicc 12260	. . . . . 6 |- ( 0 [,) +oo ) C_ ( 0 [,] +oo )
9		uncom 3757	. . . . . . . . . . . . . 14 |- ( { 0 } u. ( 0 (,] 1 ) ) = ( ( 0 (,] 1 ) u. { 0 } )
10		1re 10039	. . . . . . . . . . . . . . . 16 |- 1 e. RR
11	10	rexri 10097	. . . . . . . . . . . . . . 15 |- 1 e. RR*
12		0le1 10551	. . . . . . . . . . . . . . 15 |- 0 <_ 1
13		snunioc 12300	. . . . . . . . . . . . . . 15 |- ( ( 0 e. RR* /\ 1 e. RR* /\ 0 <_ 1 ) -> ( { 0 } u. ( 0 (,] 1 ) ) = ( 0 [,] 1 ) )
14	2, 11, 12, 13	mp3an 1424	. . . . . . . . . . . . . 14 |- ( { 0 } u. ( 0 (,] 1 ) ) = ( 0 [,] 1 )
15	9, 14	eqtr3i 2646	. . . . . . . . . . . . 13 |- ( ( 0 (,] 1 ) u. { 0 } ) = ( 0 [,] 1 )
16	15	eleq2i 2693	. . . . . . . . . . . 12 |- ( x e. ( ( 0 (,] 1 ) u. { 0 } ) <-> x e. ( 0 [,] 1 ) )
17		elun 3753	. . . . . . . . . . . 12 |- ( x e. ( ( 0 (,] 1 ) u. { 0 } ) <-> ( x e. ( 0 (,] 1 ) \/ x e. { 0 } ) )
18	16, 17	bitr3i 266	. . . . . . . . . . 11 |- ( x e. ( 0 [,] 1 ) <-> ( x e. ( 0 (,] 1 ) \/ x e. { 0 } ) )
19		pm2.53 388	. . . . . . . . . . 11 |- ( ( x e. ( 0 (,] 1 ) \/ x e. { 0 } ) -> ( -. x e. ( 0 (,] 1 ) -> x e. { 0 } ) )
20	18, 19	sylbi 207	. . . . . . . . . 10 |- ( x e. ( 0 [,] 1 ) -> ( -. x e. ( 0 (,] 1 ) -> x e. { 0 } ) )
21		elsni 4194	. . . . . . . . . 10 |- ( x e. { 0 } -> x = 0 )
22	20, 21	syl6 35	. . . . . . . . 9 |- ( x e. ( 0 [,] 1 ) -> ( -. x e. ( 0 (,] 1 ) -> x = 0 ) )
23	22	con1d 139	. . . . . . . 8 |- ( x e. ( 0 [,] 1 ) -> ( -. x = 0 -> x e. ( 0 (,] 1 ) ) )
24	23	imp 445	. . . . . . 7 |- ( ( x e. ( 0 [,] 1 ) /\ -. x = 0 ) -> x e. ( 0 (,] 1 ) )
25		0le0 11110	. . . . . . . . . . . . . 14 |- 0 <_ 0
26		ltpnf 11954	. . . . . . . . . . . . . . 15 |- ( 1 e. RR -> 1 < +oo )
27	10, 26	ax-mp 5	. . . . . . . . . . . . . 14 |- 1 < +oo
28		iocssioo 12263	. . . . . . . . . . . . . 14 |- ( ( ( 0 e. RR* /\ +oo e. RR* ) /\ ( 0 <_ 0 /\ 1 < +oo ) ) -> ( 0 (,] 1 ) C_ ( 0 (,) +oo ) )
29	2, 3, 25, 27, 28	mp4an 709	. . . . . . . . . . . . 13 |- ( 0 (,] 1 ) C_ ( 0 (,) +oo )
30		ioorp 12251	. . . . . . . . . . . . 13 |- ( 0 (,) +oo ) = RR+
31	29, 30	sseqtri 3637	. . . . . . . . . . . 12 |- ( 0 (,] 1 ) C_ RR+
32	31	sseli 3599	. . . . . . . . . . 11 |- ( x e. ( 0 (,] 1 ) -> x e. RR+ )
33	32	relogcld 24369	. . . . . . . . . 10 |- ( x e. ( 0 (,] 1 ) -> ( log ` x ) e. RR )
34	33	renegcld 10457	. . . . . . . . 9 |- ( x e. ( 0 (,] 1 ) -> -u ( log ` x ) e. RR )
35	34	rexrd 10089	. . . . . . . 8 |- ( x e. ( 0 (,] 1 ) -> -u ( log ` x ) e. RR* )
36		elioc1 12217	. . . . . . . . . . . . 13 |- ( ( 0 e. RR* /\ 1 e. RR* ) -> ( x e. ( 0 (,] 1 ) <-> ( x e. RR* /\ 0 < x /\ x <_ 1 ) ) )
37	2, 11, 36	mp2an 708	. . . . . . . . . . . 12 |- ( x e. ( 0 (,] 1 ) <-> ( x e. RR* /\ 0 < x /\ x <_ 1 ) )
38	37	simp3bi 1078	. . . . . . . . . . 11 |- ( x e. ( 0 (,] 1 ) -> x <_ 1 )
39		1rp 11836	. . . . . . . . . . . . 13 |- 1 e. RR+
40	39	a1i 11	. . . . . . . . . . . 12 |- ( x e. ( 0 (,] 1 ) -> 1 e. RR+ )
41	32, 40	logled 24373	. . . . . . . . . . 11 |- ( x e. ( 0 (,] 1 ) -> ( x <_ 1 <-> ( log ` x ) <_ ( log ` 1 ) ) )
42	38, 41	mpbid 222	. . . . . . . . . 10 |- ( x e. ( 0 (,] 1 ) -> ( log ` x ) <_ ( log ` 1 ) )
43		log1 24332	. . . . . . . . . 10 |- ( log ` 1 ) = 0
44	42, 43	syl6breq 4694	. . . . . . . . 9 |- ( x e. ( 0 (,] 1 ) -> ( log ` x ) <_ 0 )
45	33	le0neg1d 10599	. . . . . . . . 9 |- ( x e. ( 0 (,] 1 ) -> ( ( log ` x ) <_ 0 <-> 0 <_ -u ( log ` x ) ) )
46	44, 45	mpbid 222	. . . . . . . 8 |- ( x e. ( 0 (,] 1 ) -> 0 <_ -u ( log ` x ) )
47		ltpnf 11954	. . . . . . . . 9 |- ( -u ( log ` x ) e. RR -> -u ( log ` x ) < +oo )
48	34, 47	syl 17	. . . . . . . 8 |- ( x e. ( 0 (,] 1 ) -> -u ( log ` x ) < +oo )
49		elico1 12218	. . . . . . . . 9 |- ( ( 0 e. RR* /\ +oo e. RR* ) -> ( -u ( log ` x ) e. ( 0 [,) +oo ) <-> ( -u ( log ` x ) e. RR* /\ 0 <_ -u ( log ` x ) /\ -u ( log ` x ) < +oo ) ) )
50	2, 3, 49	mp2an 708	. . . . . . . 8 |- ( -u ( log ` x ) e. ( 0 [,) +oo ) <-> ( -u ( log ` x ) e. RR* /\ 0 <_ -u ( log ` x ) /\ -u ( log ` x ) < +oo ) )
51	35, 46, 48, 50	syl3anbrc 1246	. . . . . . 7 |- ( x e. ( 0 (,] 1 ) -> -u ( log ` x ) e. ( 0 [,) +oo ) )
52	24, 51	syl 17	. . . . . 6 |- ( ( x e. ( 0 [,] 1 ) /\ -. x = 0 ) -> -u ( log ` x ) e. ( 0 [,) +oo ) )
53	8, 52	sseldi 3601	. . . . 5 |- ( ( x e. ( 0 [,] 1 ) /\ -. x = 0 ) -> -u ( log ` x ) e. ( 0 [,] +oo ) )
54	7, 53	ifclda 4120	. . . 4 |- ( x e. ( 0 [,] 1 ) -> if ( x = 0 , +oo , -u ( log ` x ) ) e. ( 0 [,] +oo ) )
55	54	adantl 482	. . 3 |- ( ( T. /\ x e. ( 0 [,] 1 ) ) -> if ( x = 0 , +oo , -u ( log ` x ) ) e. ( 0 [,] +oo ) )
56		0elunit 12290	. . . . . 6 |- 0 e. ( 0 [,] 1 )
57	56	a1i 11	. . . . 5 |- ( ( y e. ( 0 [,] +oo ) /\ y = +oo ) -> 0 e. ( 0 [,] 1 ) )
58		iocssicc 12261	. . . . . 6 |- ( 0 (,] 1 ) C_ ( 0 [,] 1 )
59		snunico 12299	. . . . . . . . . . . . . 14 |- ( ( 0 e. RR* /\ +oo e. RR* /\ 0 <_ +oo ) -> ( ( 0 [,) +oo ) u. { +oo } ) = ( 0 [,] +oo ) )
60	2, 3, 4, 59	mp3an 1424	. . . . . . . . . . . . 13 |- ( ( 0 [,) +oo ) u. { +oo } ) = ( 0 [,] +oo )
61	60	eleq2i 2693	. . . . . . . . . . . 12 |- ( y e. ( ( 0 [,) +oo ) u. { +oo } ) <-> y e. ( 0 [,] +oo ) )
62		elun 3753	. . . . . . . . . . . 12 |- ( y e. ( ( 0 [,) +oo ) u. { +oo } ) <-> ( y e. ( 0 [,) +oo ) \/ y e. { +oo } ) )
63	61, 62	bitr3i 266	. . . . . . . . . . 11 |- ( y e. ( 0 [,] +oo ) <-> ( y e. ( 0 [,) +oo ) \/ y e. { +oo } ) )
64		pm2.53 388	. . . . . . . . . . 11 |- ( ( y e. ( 0 [,) +oo ) \/ y e. { +oo } ) -> ( -. y e. ( 0 [,) +oo ) -> y e. { +oo } ) )
65	63, 64	sylbi 207	. . . . . . . . . 10 |- ( y e. ( 0 [,] +oo ) -> ( -. y e. ( 0 [,) +oo ) -> y e. { +oo } ) )
66		elsni 4194	. . . . . . . . . 10 |- ( y e. { +oo } -> y = +oo )
67	65, 66	syl6 35	. . . . . . . . 9 |- ( y e. ( 0 [,] +oo ) -> ( -. y e. ( 0 [,) +oo ) -> y = +oo ) )
68	67	con1d 139	. . . . . . . 8 |- ( y e. ( 0 [,] +oo ) -> ( -. y = +oo -> y e. ( 0 [,) +oo ) ) )
69	68	imp 445	. . . . . . 7 |- ( ( y e. ( 0 [,] +oo ) /\ -. y = +oo ) -> y e. ( 0 [,) +oo ) )
70		rge0ssre 12280	. . . . . . . . . . . 12 |- ( 0 [,) +oo ) C_ RR
71	70	sseli 3599	. . . . . . . . . . 11 |- ( y e. ( 0 [,) +oo ) -> y e. RR )
72	71	renegcld 10457	. . . . . . . . . 10 |- ( y e. ( 0 [,) +oo ) -> -u y e. RR )
73	72	reefcld 14818	. . . . . . . . 9 |- ( y e. ( 0 [,) +oo ) -> ( exp ` -u y ) e. RR )
74	73	rexrd 10089	. . . . . . . 8 |- ( y e. ( 0 [,) +oo ) -> ( exp ` -u y ) e. RR* )
75		efgt0 14833	. . . . . . . . 9 |- ( -u y e. RR -> 0 < ( exp ` -u y ) )
76	72, 75	syl 17	. . . . . . . 8 |- ( y e. ( 0 [,) +oo ) -> 0 < ( exp ` -u y ) )
77		elico1 12218	. . . . . . . . . . . . 13 |- ( ( 0 e. RR* /\ +oo e. RR* ) -> ( y e. ( 0 [,) +oo ) <-> ( y e. RR* /\ 0 <_ y /\ y < +oo ) ) )
78	2, 3, 77	mp2an 708	. . . . . . . . . . . 12 |- ( y e. ( 0 [,) +oo ) <-> ( y e. RR* /\ 0 <_ y /\ y < +oo ) )
79	78	simp2bi 1077	. . . . . . . . . . 11 |- ( y e. ( 0 [,) +oo ) -> 0 <_ y )
80	71	le0neg2d 10600	. . . . . . . . . . 11 |- ( y e. ( 0 [,) +oo ) -> ( 0 <_ y <-> -u y <_ 0 ) )
81	79, 80	mpbid 222	. . . . . . . . . 10 |- ( y e. ( 0 [,) +oo ) -> -u y <_ 0 )
82		0re 10040	. . . . . . . . . . 11 |- 0 e. RR
83		efle 14848	. . . . . . . . . . 11 |- ( ( -u y e. RR /\ 0 e. RR ) -> ( -u y <_ 0 <-> ( exp ` -u y ) <_ ( exp ` 0 ) ) )
84	72, 82, 83	sylancl 694	. . . . . . . . . 10 |- ( y e. ( 0 [,) +oo ) -> ( -u y <_ 0 <-> ( exp ` -u y ) <_ ( exp ` 0 ) ) )
85	81, 84	mpbid 222	. . . . . . . . 9 |- ( y e. ( 0 [,) +oo ) -> ( exp ` -u y ) <_ ( exp ` 0 ) )
86		ef0 14821	. . . . . . . . 9 |- ( exp ` 0 ) = 1
87	85, 86	syl6breq 4694	. . . . . . . 8 |- ( y e. ( 0 [,) +oo ) -> ( exp ` -u y ) <_ 1 )
88		elioc1 12217	. . . . . . . . 9 |- ( ( 0 e. RR* /\ 1 e. RR* ) -> ( ( exp ` -u y ) e. ( 0 (,] 1 ) <-> ( ( exp ` -u y ) e. RR* /\ 0 < ( exp ` -u y ) /\ ( exp ` -u y ) <_ 1 ) ) )
89	2, 11, 88	mp2an 708	. . . . . . . 8 |- ( ( exp ` -u y ) e. ( 0 (,] 1 ) <-> ( ( exp ` -u y ) e. RR* /\ 0 < ( exp ` -u y ) /\ ( exp ` -u y ) <_ 1 ) )
90	74, 76, 87, 89	syl3anbrc 1246	. . . . . . 7 |- ( y e. ( 0 [,) +oo ) -> ( exp ` -u y ) e. ( 0 (,] 1 ) )
91	69, 90	syl 17	. . . . . 6 |- ( ( y e. ( 0 [,] +oo ) /\ -. y = +oo ) -> ( exp ` -u y ) e. ( 0 (,] 1 ) )
92	58, 91	sseldi 3601	. . . . 5 |- ( ( y e. ( 0 [,] +oo ) /\ -. y = +oo ) -> ( exp ` -u y ) e. ( 0 [,] 1 ) )
93	57, 92	ifclda 4120	. . . 4 |- ( y e. ( 0 [,] +oo ) -> if ( y = +oo , 0 , ( exp ` -u y ) ) e. ( 0 [,] 1 ) )
94	93	adantl 482	. . 3 |- ( ( T. /\ y e. ( 0 [,] +oo ) ) -> if ( y = +oo , 0 , ( exp ` -u y ) ) e. ( 0 [,] 1 ) )
95		eqeq2 2633	. . . . . 6 |- ( 0 = if ( y = +oo , 0 , ( exp ` -u y ) ) -> ( x = 0 <-> x = if ( y = +oo , 0 , ( exp ` -u y ) ) ) )
96	95	bibi1d 333	. . . . 5 |- ( 0 = if ( y = +oo , 0 , ( exp ` -u y ) ) -> ( ( x = 0 <-> y = if ( x = 0 , +oo , -u ( log ` x ) ) ) <-> ( x = if ( y = +oo , 0 , ( exp ` -u y ) ) <-> y = if ( x = 0 , +oo , -u ( log ` x ) ) ) ) )
97		eqeq2 2633	. . . . . 6 |- ( ( exp ` -u y ) = if ( y = +oo , 0 , ( exp ` -u y ) ) -> ( x = ( exp ` -u y ) <-> x = if ( y = +oo , 0 , ( exp ` -u y ) ) ) )
98	97	bibi1d 333	. . . . 5 |- ( ( exp ` -u y ) = if ( y = +oo , 0 , ( exp ` -u y ) ) -> ( ( x = ( exp ` -u y ) <-> y = if ( x = 0 , +oo , -u ( log ` x ) ) ) <-> ( x = if ( y = +oo , 0 , ( exp ` -u y ) ) <-> y = if ( x = 0 , +oo , -u ( log ` x ) ) ) ) )
99		simpr 477	. . . . . . 7 |- ( ( ( x e. ( 0 [,] 1 ) /\ y e. ( 0 [,] +oo ) ) /\ y = +oo ) -> y = +oo )
100		iftrue 4092	. . . . . . . 8 |- ( x = 0 -> if ( x = 0 , +oo , -u ( log ` x ) ) = +oo )
101	100	eqeq2d 2632	. . . . . . 7 |- ( x = 0 -> ( y = if ( x = 0 , +oo , -u ( log ` x ) ) <-> y = +oo ) )
102	99, 101	syl5ibrcom 237	. . . . . 6 |- ( ( ( x e. ( 0 [,] 1 ) /\ y e. ( 0 [,] +oo ) ) /\ y = +oo ) -> ( x = 0 -> y = if ( x = 0 , +oo , -u ( log ` x ) ) ) )
103		ubico 29537	. . . . . . . . . . 11 |- ( ( 0 e. RR /\ +oo e. RR* ) -> -. +oo e. ( 0 [,) +oo ) )
104	82, 3, 103	mp2an 708	. . . . . . . . . 10 |- -. +oo e. ( 0 [,) +oo )
105	104	nelir 2900	. . . . . . . . 9 |- +oo e/ ( 0 [,) +oo )
106		neleq1 2902	. . . . . . . . . 10 |- ( y = +oo -> ( y e/ ( 0 [,) +oo ) <-> +oo e/ ( 0 [,) +oo ) ) )
107	106	adantl 482	. . . . . . . . 9 |- ( ( ( x e. ( 0 [,] 1 ) /\ y e. ( 0 [,] +oo ) ) /\ y = +oo ) -> ( y e/ ( 0 [,) +oo ) <-> +oo e/ ( 0 [,) +oo ) ) )
108	105, 107	mpbiri 248	. . . . . . . 8 |- ( ( ( x e. ( 0 [,] 1 ) /\ y e. ( 0 [,] +oo ) ) /\ y = +oo ) -> y e/ ( 0 [,) +oo ) )
109		neleq1 2902	. . . . . . . 8 |- ( y = if ( x = 0 , +oo , -u ( log ` x ) ) -> ( y e/ ( 0 [,) +oo ) <-> if ( x = 0 , +oo , -u ( log ` x ) ) e/ ( 0 [,) +oo ) ) )
110	108, 109	syl5ibcom 235	. . . . . . 7 |- ( ( ( x e. ( 0 [,] 1 ) /\ y e. ( 0 [,] +oo ) ) /\ y = +oo ) -> ( y = if ( x = 0 , +oo , -u ( log ` x ) ) -> if ( x = 0 , +oo , -u ( log ` x ) ) e/ ( 0 [,) +oo ) ) )
111		df-nel 2898	. . . . . . . 8 |- ( if ( x = 0 , +oo , -u ( log ` x ) ) e/ ( 0 [,) +oo ) <-> -. if ( x = 0 , +oo , -u ( log ` x ) ) e. ( 0 [,) +oo ) )
112		iffalse 4095	. . . . . . . . . . . . 13 |- ( -. x = 0 -> if ( x = 0 , +oo , -u ( log ` x ) ) = -u ( log ` x ) )
113	112	adantl 482	. . . . . . . . . . . 12 |- ( ( x e. ( 0 [,] 1 ) /\ -. x = 0 ) -> if ( x = 0 , +oo , -u ( log ` x ) ) = -u ( log ` x ) )
114	113, 52	eqeltrd 2701	. . . . . . . . . . 11 |- ( ( x e. ( 0 [,] 1 ) /\ -. x = 0 ) -> if ( x = 0 , +oo , -u ( log ` x ) ) e. ( 0 [,) +oo ) )
115	114	ex 450	. . . . . . . . . 10 |- ( x e. ( 0 [,] 1 ) -> ( -. x = 0 -> if ( x = 0 , +oo , -u ( log ` x ) ) e. ( 0 [,) +oo ) ) )
116	115	ad2antrr 762	. . . . . . . . 9 |- ( ( ( x e. ( 0 [,] 1 ) /\ y e. ( 0 [,] +oo ) ) /\ y = +oo ) -> ( -. x = 0 -> if ( x = 0 , +oo , -u ( log ` x ) ) e. ( 0 [,) +oo ) ) )
117	116	con1d 139	. . . . . . . 8 |- ( ( ( x e. ( 0 [,] 1 ) /\ y e. ( 0 [,] +oo ) ) /\ y = +oo ) -> ( -. if ( x = 0 , +oo , -u ( log ` x ) ) e. ( 0 [,) +oo ) -> x = 0 ) )
118	111, 117	syl5bi 232	. . . . . . 7 |- ( ( ( x e. ( 0 [,] 1 ) /\ y e. ( 0 [,] +oo ) ) /\ y = +oo ) -> ( if ( x = 0 , +oo , -u ( log ` x ) ) e/ ( 0 [,) +oo ) -> x = 0 ) )
119	110, 118	syld 47	. . . . . 6 |- ( ( ( x e. ( 0 [,] 1 ) /\ y e. ( 0 [,] +oo ) ) /\ y = +oo ) -> ( y = if ( x = 0 , +oo , -u ( log ` x ) ) -> x = 0 ) )
120	102, 119	impbid 202	. . . . 5 |- ( ( ( x e. ( 0 [,] 1 ) /\ y e. ( 0 [,] +oo ) ) /\ y = +oo ) -> ( x = 0 <-> y = if ( x = 0 , +oo , -u ( log ` x ) ) ) )
121		eqeq2 2633	. . . . . . 7 |- ( +oo = if ( x = 0 , +oo , -u ( log ` x ) ) -> ( y = +oo <-> y = if ( x = 0 , +oo , -u ( log ` x ) ) ) )
122	121	bibi2d 332	. . . . . 6 |- ( +oo = if ( x = 0 , +oo , -u ( log ` x ) ) -> ( ( x = ( exp ` -u y ) <-> y = +oo ) <-> ( x = ( exp ` -u y ) <-> y = if ( x = 0 , +oo , -u ( log ` x ) ) ) ) )
123		eqeq2 2633	. . . . . . 7 |- ( -u ( log ` x ) = if ( x = 0 , +oo , -u ( log ` x ) ) -> ( y = -u ( log ` x ) <-> y = if ( x = 0 , +oo , -u ( log ` x ) ) ) )
124	123	bibi2d 332	. . . . . 6 |- ( -u ( log ` x ) = if ( x = 0 , +oo , -u ( log ` x ) ) -> ( ( x = ( exp ` -u y ) <-> y = -u ( log ` x ) ) <-> ( x = ( exp ` -u y ) <-> y = if ( x = 0 , +oo , -u ( log ` x ) ) ) ) )
125	82	a1i 11	. . . . . . . . . . . 12 |- ( ( y e. ( 0 [,] +oo ) /\ -. y = +oo ) -> 0 e. RR )
126	69, 76	syl 17	. . . . . . . . . . . 12 |- ( ( y e. ( 0 [,] +oo ) /\ -. y = +oo ) -> 0 < ( exp ` -u y ) )
127	125, 126	ltned 10173	. . . . . . . . . . 11 |- ( ( y e. ( 0 [,] +oo ) /\ -. y = +oo ) -> 0 =/= ( exp ` -u y ) )
128	127	adantll 750	. . . . . . . . . 10 |- ( ( ( x e. ( 0 [,] 1 ) /\ y e. ( 0 [,] +oo ) ) /\ -. y = +oo ) -> 0 =/= ( exp ` -u y ) )
129	128	neneqd 2799	. . . . . . . . 9 |- ( ( ( x e. ( 0 [,] 1 ) /\ y e. ( 0 [,] +oo ) ) /\ -. y = +oo ) -> -. 0 = ( exp ` -u y ) )
130		eqeq1 2626	. . . . . . . . . 10 |- ( x = 0 -> ( x = ( exp ` -u y ) <-> 0 = ( exp ` -u y ) ) )
131	130	notbid 308	. . . . . . . . 9 |- ( x = 0 -> ( -. x = ( exp ` -u y ) <-> -. 0 = ( exp ` -u y ) ) )
132	129, 131	syl5ibrcom 237	. . . . . . . 8 |- ( ( ( x e. ( 0 [,] 1 ) /\ y e. ( 0 [,] +oo ) ) /\ -. y = +oo ) -> ( x = 0 -> -. x = ( exp ` -u y ) ) )
133	132	imp 445	. . . . . . 7 |- ( ( ( ( x e. ( 0 [,] 1 ) /\ y e. ( 0 [,] +oo ) ) /\ -. y = +oo ) /\ x = 0 ) -> -. x = ( exp ` -u y ) )
134		simplr 792	. . . . . . 7 |- ( ( ( ( x e. ( 0 [,] 1 ) /\ y e. ( 0 [,] +oo ) ) /\ -. y = +oo ) /\ x = 0 ) -> -. y = +oo )
135	133, 134	2falsed 366	. . . . . 6 |- ( ( ( ( x e. ( 0 [,] 1 ) /\ y e. ( 0 [,] +oo ) ) /\ -. y = +oo ) /\ x = 0 ) -> ( x = ( exp ` -u y ) <-> y = +oo ) )
136		eqcom 2629	. . . . . . . . . . 11 |- ( x = ( exp ` -u y ) <-> ( exp ` -u y ) = x )
137	136	a1i 11	. . . . . . . . . 10 |- ( ( x e. ( 0 (,] 1 ) /\ y e. ( 0 [,) +oo ) ) -> ( x = ( exp ` -u y ) <-> ( exp ` -u y ) = x ) )
138		relogeftb 24331	. . . . . . . . . . 11 |- ( ( x e. RR+ /\ -u y e. RR ) -> ( ( log ` x ) = -u y <-> ( exp ` -u y ) = x ) )
139	32, 72, 138	syl2an 494	. . . . . . . . . 10 |- ( ( x e. ( 0 (,] 1 ) /\ y e. ( 0 [,) +oo ) ) -> ( ( log ` x ) = -u y <-> ( exp ` -u y ) = x ) )
140	33	recnd 10068	. . . . . . . . . . 11 |- ( x e. ( 0 (,] 1 ) -> ( log ` x ) e. CC )
141	71	recnd 10068	. . . . . . . . . . 11 |- ( y e. ( 0 [,) +oo ) -> y e. CC )
142		negcon2 10334	. . . . . . . . . . 11 |- ( ( ( log ` x ) e. CC /\ y e. CC ) -> ( ( log ` x ) = -u y <-> y = -u ( log ` x ) ) )
143	140, 141, 142	syl2an 494	. . . . . . . . . 10 |- ( ( x e. ( 0 (,] 1 ) /\ y e. ( 0 [,) +oo ) ) -> ( ( log ` x ) = -u y <-> y = -u ( log ` x ) ) )
144	137, 139, 143	3bitr2d 296	. . . . . . . . 9 |- ( ( x e. ( 0 (,] 1 ) /\ y e. ( 0 [,) +oo ) ) -> ( x = ( exp ` -u y ) <-> y = -u ( log ` x ) ) )
145	24, 69, 144	syl2an 494	. . . . . . . 8 |- ( ( ( x e. ( 0 [,] 1 ) /\ -. x = 0 ) /\ ( y e. ( 0 [,] +oo ) /\ -. y = +oo ) ) -> ( x = ( exp ` -u y ) <-> y = -u ( log ` x ) ) )
146	145	an4s 869	. . . . . . 7 |- ( ( ( x e. ( 0 [,] 1 ) /\ y e. ( 0 [,] +oo ) ) /\ ( -. x = 0 /\ -. y = +oo ) ) -> ( x = ( exp ` -u y ) <-> y = -u ( log ` x ) ) )
147	146	anass1rs 849	. . . . . 6 |- ( ( ( ( x e. ( 0 [,] 1 ) /\ y e. ( 0 [,] +oo ) ) /\ -. y = +oo ) /\ -. x = 0 ) -> ( x = ( exp ` -u y ) <-> y = -u ( log ` x ) ) )
148	122, 124, 135, 147	ifbothda 4123	. . . . 5 |- ( ( ( x e. ( 0 [,] 1 ) /\ y e. ( 0 [,] +oo ) ) /\ -. y = +oo ) -> ( x = ( exp ` -u y ) <-> y = if ( x = 0 , +oo , -u ( log ` x ) ) ) )
149	96, 98, 120, 148	ifbothda 4123	. . . 4 |- ( ( x e. ( 0 [,] 1 ) /\ y e. ( 0 [,] +oo ) ) -> ( x = if ( y = +oo , 0 , ( exp ` -u y ) ) <-> y = if ( x = 0 , +oo , -u ( log ` x ) ) ) )
150	149	adantl 482	. . 3 |- ( ( T. /\ ( x e. ( 0 [,] 1 ) /\ y e. ( 0 [,] +oo ) ) ) -> ( x = if ( y = +oo , 0 , ( exp ` -u y ) ) <-> y = if ( x = 0 , +oo , -u ( log ` x ) ) ) )
151	1, 55, 94, 150	f1ocnv2d 6886	. 2 |- ( T. -> ( F : ( 0 [,] 1 ) -1-1-onto-> ( 0 [,] +oo ) /\ `' F = ( y e. ( 0 [,] +oo ) |-> if ( y = +oo , 0 , ( exp ` -u y ) ) ) ) )
152	151	trud 1493	1 |- ( F : ( 0 [,] 1 ) -1-1-onto-> ( 0 [,] +oo ) /\ `' F = ( y e. ( 0 [,] +oo ) |-> if ( y = +oo , 0 , ( exp ` -u y ) ) ) )

Syntax hints:   -. wn 3   -> wi 4   <-> wb 196   \/ wo 383   /\ wa 384   /\ w3a 1037   = wceq 1483   T. wtru 1484   e. wcel 1990   =/= wne 2794   e/ wnel 2897   u. cun 3572   C_ wss 3574   ifcif 4086   {csn 4177   class class class wbr 4653   |-> cmpt 4729   `'ccnv 5113   -1-1-onto->wf1o 5887   ` cfv 5888  (class class class)co 6650   CCcc 9934   RRcr 9935   0cc0 9936   1c1 9937   +oocpnf 10071   RR*cxr 10073   < clt 10074   <_ cle 10075   -ucneg 10267   RR+crp 11832   (,)cioo 12175   (,]cioc 12176   [,)cico 12177   [,]cicc 12178   expce 14792   logclog 24301

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-rep 4771  ax-sep 4781  ax-nul 4789  ax-pow 4843  ax-pr 4906  ax-un 6949  ax-inf2 8538  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  ax-pre-sup 10014  ax-addf 10015  ax-mulf 10016

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3or 1038  df-3an 1039  df-tru 1486  df-fal 1489  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-pss 3590  df-nul 3916  df-if 4087  df-pw 4160  df-sn 4178  df-pr 4180  df-tp 4182  df-op 4184  df-uni 4437  df-int 4476  df-iun 4522  df-iin 4523  df-br 4654  df-opab 4713  df-mpt 4730  df-tr 4753  df-id 5024  df-eprel 5029  df-po 5035  df-so 5036  df-fr 5073  df-se 5074  df-we 5075  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-pred 5680  df-ord 5726  df-on 5727  df-lim 5728  df-suc 5729  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-isom 5897  df-riota 6611  df-ov 6653  df-oprab 6654  df-mpt2 6655  df-of 6897  df-om 7066  df-1st 7168  df-2nd 7169  df-supp 7296  df-wrecs 7407  df-recs 7468  df-rdg 7506  df-1o 7560  df-2o 7561  df-oadd 7564  df-er 7742  df-map 7859  df-pm 7860  df-ixp 7909  df-en 7956  df-dom 7957  df-sdom 7958  df-fin 7959  df-fsupp 8276  df-fi 8317  df-sup 8348  df-inf 8349  df-oi 8415  df-card 8765  df-cda 8990  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-nn 11021  df-2 11079  df-3 11080  df-4 11081  df-5 11082  df-6 11083  df-7 11084  df-8 11085  df-9 11086  df-n0 11293  df-z 11378  df-dec 11494  df-uz 11688  df-q 11789  df-rp 11833  df-xneg 11946  df-xadd 11947  df-xmul 11948  df-ioo 12179  df-ioc 12180  df-ico 12181  df-icc 12182  df-fz 12327  df-fzo 12466  df-fl 12593  df-mod 12669  df-seq 12802  df-exp 12861  df-fac 13061  df-bc 13090  df-hash 13118  df-shft 13807  df-cj 13839  df-re 13840  df-im 13841  df-sqrt 13975  df-abs 13976  df-limsup 14202  df-clim 14219  df-rlim 14220  df-sum 14417  df-ef 14798  df-sin 14800  df-cos 14801  df-pi 14803  df-struct 15859  df-ndx 15860  df-slot 15861  df-base 15863  df-sets 15864  df-ress 15865  df-plusg 15954  df-mulr 15955  df-starv 15956  df-sca 15957  df-vsca 15958  df-ip 15959  df-tset 15960  df-ple 15961  df-ds 15964  df-unif 15965  df-hom 15966  df-cco 15967  df-rest 16083  df-topn 16084  df-0g 16102  df-gsum 16103  df-topgen 16104  df-pt 16105  df-prds 16108  df-xrs 16162  df-qtop 16167  df-imas 16168  df-xps 16170  df-mre 16246  df-mrc 16247  df-acs 16249  df-mgm 17242  df-sgrp 17284  df-mnd 17295  df-submnd 17336  df-mulg 17541  df-cntz 17750  df-cmn 18195  df-psmet 19738  df-xmet 19739  df-met 19740  df-bl 19741  df-mopn 19742  df-fbas 19743  df-fg 19744  df-cnfld 19747  df-top 20699  df-topon 20716  df-topsp 20737  df-bases 20750  df-cld 20823  df-ntr 20824  df-cls 20825  df-nei 20902  df-lp 20940  df-perf 20941  df-cn 21031  df-cnp 21032  df-haus 21119  df-tx 21365  df-hmeo 21558  df-fil 21650  df-fm 21742  df-flim 21743  df-flf 21744  df-xms 22125  df-ms 22126  df-tms 22127  df-cncf 22681  df-limc 23630  df-dv 23631  df-log 24303

This theorem is referenced by:  xrge0iifiso  29981  xrge0iifmhm  29985  xrge0pluscn  29986