Theorem dvne0 23774

Description: A function on a closed interval with nonzero derivative is either monotone increasing or monotone decreasing. (Contributed by Mario Carneiro, 19-Feb-2015.)

Hypotheses

Ref	Expression
dvne0.a	|- ( ph -> A e. RR )
dvne0.b	|- ( ph -> B e. RR )
dvne0.f	|- ( ph -> F e. ( ( A [,] B ) -cn-> RR ) )
dvne0.d	|- ( ph -> dom ( RR _D F ) = ( A (,) B ) )
dvne0.z	|- ( ph -> -. 0 e. ran ( RR _D F ) )

Assertion

Ref	Expression
dvne0	|- ( ph -> ( F Isom < , < ( ( A [,] B ) , ran F ) \/ F Isom < , `' < ( ( A [,] B ) , ran F ) ) )

Proof of Theorem dvne0

Dummy variables x y z are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		dvne0.z	. . . . . . . . . . . 12 |- ( ph -> -. 0 e. ran ( RR _D F ) )
2		eleq1 2689	. . . . . . . . . . . . 13 |- ( x = 0 -> ( x e. ran ( RR _D F ) <-> 0 e. ran ( RR _D F ) ) )
3	2	notbid 308	. . . . . . . . . . . 12 |- ( x = 0 -> ( -. x e. ran ( RR _D F ) <-> -. 0 e. ran ( RR _D F ) ) )
4	1, 3	syl5ibrcom 237	. . . . . . . . . . 11 |- ( ph -> ( x = 0 -> -. x e. ran ( RR _D F ) ) )
5	4	necon2ad 2809	. . . . . . . . . 10 |- ( ph -> ( x e. ran ( RR _D F ) -> x =/= 0 ) )
6	5	imp 445	. . . . . . . . 9 |- ( ( ph /\ x e. ran ( RR _D F ) ) -> x =/= 0 )
7		dvne0.f	. . . . . . . . . . . . . . 15 |- ( ph -> F e. ( ( A [,] B ) -cn-> RR ) )
8		cncff 22696	. . . . . . . . . . . . . . 15 |- ( F e. ( ( A [,] B ) -cn-> RR ) -> F : ( A [,] B ) --> RR )
9	7, 8	syl 17	. . . . . . . . . . . . . 14 |- ( ph -> F : ( A [,] B ) --> RR )
10		dvne0.a	. . . . . . . . . . . . . . 15 |- ( ph -> A e. RR )
11		dvne0.b	. . . . . . . . . . . . . . 15 |- ( ph -> B e. RR )
12		iccssre 12255	. . . . . . . . . . . . . . 15 |- ( ( A e. RR /\ B e. RR ) -> ( A [,] B ) C_ RR )
13	10, 11, 12	syl2anc 693	. . . . . . . . . . . . . 14 |- ( ph -> ( A [,] B ) C_ RR )
14		dvfre 23714	. . . . . . . . . . . . . 14 |- ( ( F : ( A [,] B ) --> RR /\ ( A [,] B ) C_ RR ) -> ( RR _D F ) : dom ( RR _D F ) --> RR )
15	9, 13, 14	syl2anc 693	. . . . . . . . . . . . 13 |- ( ph -> ( RR _D F ) : dom ( RR _D F ) --> RR )
16		frn 6053	. . . . . . . . . . . . 13 |- ( ( RR _D F ) : dom ( RR _D F ) --> RR -> ran ( RR _D F ) C_ RR )
17	15, 16	syl 17	. . . . . . . . . . . 12 |- ( ph -> ran ( RR _D F ) C_ RR )
18	17	sselda 3603	. . . . . . . . . . 11 |- ( ( ph /\ x e. ran ( RR _D F ) ) -> x e. RR )
19		0re 10040	. . . . . . . . . . 11 |- 0 e. RR
20		lttri2 10120	. . . . . . . . . . 11 |- ( ( x e. RR /\ 0 e. RR ) -> ( x =/= 0 <-> ( x < 0 \/ 0 < x ) ) )
21	18, 19, 20	sylancl 694	. . . . . . . . . 10 |- ( ( ph /\ x e. ran ( RR _D F ) ) -> ( x =/= 0 <-> ( x < 0 \/ 0 < x ) ) )
22		0xr 10086	. . . . . . . . . . . . . 14 |- 0 e. RR*
23		elioomnf 12268	. . . . . . . . . . . . . 14 |- ( 0 e. RR* -> ( x e. ( -oo (,) 0 ) <-> ( x e. RR /\ x < 0 ) ) )
24	22, 23	ax-mp 5	. . . . . . . . . . . . 13 |- ( x e. ( -oo (,) 0 ) <-> ( x e. RR /\ x < 0 ) )
25	24	baib 944	. . . . . . . . . . . 12 |- ( x e. RR -> ( x e. ( -oo (,) 0 ) <-> x < 0 ) )
26		elrp 11834	. . . . . . . . . . . . 13 |- ( x e. RR+ <-> ( x e. RR /\ 0 < x ) )
27	26	baib 944	. . . . . . . . . . . 12 |- ( x e. RR -> ( x e. RR+ <-> 0 < x ) )
28	25, 27	orbi12d 746	. . . . . . . . . . 11 |- ( x e. RR -> ( ( x e. ( -oo (,) 0 ) \/ x e. RR+ ) <-> ( x < 0 \/ 0 < x ) ) )
29	18, 28	syl 17	. . . . . . . . . 10 |- ( ( ph /\ x e. ran ( RR _D F ) ) -> ( ( x e. ( -oo (,) 0 ) \/ x e. RR+ ) <-> ( x < 0 \/ 0 < x ) ) )
30	21, 29	bitr4d 271	. . . . . . . . 9 |- ( ( ph /\ x e. ran ( RR _D F ) ) -> ( x =/= 0 <-> ( x e. ( -oo (,) 0 ) \/ x e. RR+ ) ) )
31	6, 30	mpbid 222	. . . . . . . 8 |- ( ( ph /\ x e. ran ( RR _D F ) ) -> ( x e. ( -oo (,) 0 ) \/ x e. RR+ ) )
32		elun 3753	. . . . . . . 8 |- ( x e. ( ( -oo (,) 0 ) u. RR+ ) <-> ( x e. ( -oo (,) 0 ) \/ x e. RR+ ) )
33	31, 32	sylibr 224	. . . . . . 7 |- ( ( ph /\ x e. ran ( RR _D F ) ) -> x e. ( ( -oo (,) 0 ) u. RR+ ) )
34	33	ex 450	. . . . . 6 |- ( ph -> ( x e. ran ( RR _D F ) -> x e. ( ( -oo (,) 0 ) u. RR+ ) ) )
35	34	ssrdv 3609	. . . . 5 |- ( ph -> ran ( RR _D F ) C_ ( ( -oo (,) 0 ) u. RR+ ) )
36		disjssun 4036	. . . . 5 |- ( ( ran ( RR _D F ) i^i ( -oo (,) 0 ) ) = (/) -> ( ran ( RR _D F ) C_ ( ( -oo (,) 0 ) u. RR+ ) <-> ran ( RR _D F ) C_ RR+ ) )
37	35, 36	syl5ibcom 235	. . . 4 |- ( ph -> ( ( ran ( RR _D F ) i^i ( -oo (,) 0 ) ) = (/) -> ran ( RR _D F ) C_ RR+ ) )
38	37	imp 445	. . 3 |- ( ( ph /\ ( ran ( RR _D F ) i^i ( -oo (,) 0 ) ) = (/) ) -> ran ( RR _D F ) C_ RR+ )
39	10	adantr 481	. . . . 5 |- ( ( ph /\ ran ( RR _D F ) C_ RR+ ) -> A e. RR )
40	11	adantr 481	. . . . 5 |- ( ( ph /\ ran ( RR _D F ) C_ RR+ ) -> B e. RR )
41	7	adantr 481	. . . . 5 |- ( ( ph /\ ran ( RR _D F ) C_ RR+ ) -> F e. ( ( A [,] B ) -cn-> RR ) )
42		dvne0.d	. . . . . . . . . 10 |- ( ph -> dom ( RR _D F ) = ( A (,) B ) )
43	42	feq2d 6031	. . . . . . . . 9 |- ( ph -> ( ( RR _D F ) : dom ( RR _D F ) --> RR <-> ( RR _D F ) : ( A (,) B ) --> RR ) )
44	15, 43	mpbid 222	. . . . . . . 8 |- ( ph -> ( RR _D F ) : ( A (,) B ) --> RR )
45		ffn 6045	. . . . . . . 8 |- ( ( RR _D F ) : ( A (,) B ) --> RR -> ( RR _D F ) Fn ( A (,) B ) )
46	44, 45	syl 17	. . . . . . 7 |- ( ph -> ( RR _D F ) Fn ( A (,) B ) )
47	46	anim1i 592	. . . . . 6 |- ( ( ph /\ ran ( RR _D F ) C_ RR+ ) -> ( ( RR _D F ) Fn ( A (,) B ) /\ ran ( RR _D F ) C_ RR+ ) )
48		df-f 5892	. . . . . 6 |- ( ( RR _D F ) : ( A (,) B ) --> RR+ <-> ( ( RR _D F ) Fn ( A (,) B ) /\ ran ( RR _D F ) C_ RR+ ) )
49	47, 48	sylibr 224	. . . . 5 |- ( ( ph /\ ran ( RR _D F ) C_ RR+ ) -> ( RR _D F ) : ( A (,) B ) --> RR+ )
50	39, 40, 41, 49	dvgt0 23767	. . . 4 |- ( ( ph /\ ran ( RR _D F ) C_ RR+ ) -> F Isom < , < ( ( A [,] B ) , ran F ) )
51	50	orcd 407	. . 3 |- ( ( ph /\ ran ( RR _D F ) C_ RR+ ) -> ( F Isom < , < ( ( A [,] B ) , ran F ) \/ F Isom < , `' < ( ( A [,] B ) , ran F ) ) )
52	38, 51	syldan 487	. 2 |- ( ( ph /\ ( ran ( RR _D F ) i^i ( -oo (,) 0 ) ) = (/) ) -> ( F Isom < , < ( ( A [,] B ) , ran F ) \/ F Isom < , `' < ( ( A [,] B ) , ran F ) ) )
53		n0 3931	. . . 4 |- ( ( ran ( RR _D F ) i^i ( -oo (,) 0 ) ) =/= (/) <-> E. x x e. ( ran ( RR _D F ) i^i ( -oo (,) 0 ) ) )
54		elin 3796	. . . . . 6 |- ( x e. ( ran ( RR _D F ) i^i ( -oo (,) 0 ) ) <-> ( x e. ran ( RR _D F ) /\ x e. ( -oo (,) 0 ) ) )
55		fvelrnb 6243	. . . . . . . . 9 |- ( ( RR _D F ) Fn ( A (,) B ) -> ( x e. ran ( RR _D F ) <-> E. y e. ( A (,) B ) ( ( RR _D F ) ` y ) = x ) )
56	46, 55	syl 17	. . . . . . . 8 |- ( ph -> ( x e. ran ( RR _D F ) <-> E. y e. ( A (,) B ) ( ( RR _D F ) ` y ) = x ) )
57	10	adantr 481	. . . . . . . . . . . . 13 |- ( ( ph /\ ( y e. ( A (,) B ) /\ ( ( RR _D F ) ` y ) e. ( -oo (,) 0 ) ) ) -> A e. RR )
58	11	adantr 481	. . . . . . . . . . . . 13 |- ( ( ph /\ ( y e. ( A (,) B ) /\ ( ( RR _D F ) ` y ) e. ( -oo (,) 0 ) ) ) -> B e. RR )
59	7	adantr 481	. . . . . . . . . . . . 13 |- ( ( ph /\ ( y e. ( A (,) B ) /\ ( ( RR _D F ) ` y ) e. ( -oo (,) 0 ) ) ) -> F e. ( ( A [,] B ) -cn-> RR ) )
60	46	adantr 481	. . . . . . . . . . . . . 14 |- ( ( ph /\ ( y e. ( A (,) B ) /\ ( ( RR _D F ) ` y ) e. ( -oo (,) 0 ) ) ) -> ( RR _D F ) Fn ( A (,) B ) )
61	44	adantr 481	. . . . . . . . . . . . . . . . 17 |- ( ( ph /\ ( y e. ( A (,) B ) /\ ( ( RR _D F ) ` y ) e. ( -oo (,) 0 ) ) ) -> ( RR _D F ) : ( A (,) B ) --> RR )
62	61	ffvelrnda 6359	. . . . . . . . . . . . . . . 16 |- ( ( ( ph /\ ( y e. ( A (,) B ) /\ ( ( RR _D F ) ` y ) e. ( -oo (,) 0 ) ) ) /\ z e. ( A (,) B ) ) -> ( ( RR _D F ) ` z ) e. RR )
63	1	ad2antrr 762	. . . . . . . . . . . . . . . . . 18 |- ( ( ( ph /\ ( y e. ( A (,) B ) /\ ( ( RR _D F ) ` y ) e. ( -oo (,) 0 ) ) ) /\ z e. ( A (,) B ) ) -> -. 0 e. ran ( RR _D F ) )
64		simplrl 800	. . . . . . . . . . . . . . . . . . . . . 22 |- ( ( ( ph /\ ( y e. ( A (,) B ) /\ ( ( RR _D F ) ` y ) e. ( -oo (,) 0 ) ) ) /\ ( z e. ( A (,) B ) /\ 0 <_ ( ( RR _D F ) ` z ) ) ) -> y e. ( A (,) B ) )
65		simprl 794	. . . . . . . . . . . . . . . . . . . . . 22 |- ( ( ( ph /\ ( y e. ( A (,) B ) /\ ( ( RR _D F ) ` y ) e. ( -oo (,) 0 ) ) ) /\ ( z e. ( A (,) B ) /\ 0 <_ ( ( RR _D F ) ` z ) ) ) -> z e. ( A (,) B ) )
66		ioossicc 12259	. . . . . . . . . . . . . . . . . . . . . . . 24 |- ( A (,) B ) C_ ( A [,] B )
67		rescncf 22700	. . . . . . . . . . . . . . . . . . . . . . . 24 |- ( ( A (,) B ) C_ ( A [,] B ) -> ( F e. ( ( A [,] B ) -cn-> RR ) -> ( F |` ( A (,) B ) ) e. ( ( A (,) B ) -cn-> RR ) ) )
68	66, 7, 67	mpsyl 68	. . . . . . . . . . . . . . . . . . . . . . 23 |- ( ph -> ( F |` ( A (,) B ) ) e. ( ( A (,) B ) -cn-> RR ) )
69	68	ad2antrr 762	. . . . . . . . . . . . . . . . . . . . . 22 |- ( ( ( ph /\ ( y e. ( A (,) B ) /\ ( ( RR _D F ) ` y ) e. ( -oo (,) 0 ) ) ) /\ ( z e. ( A (,) B ) /\ 0 <_ ( ( RR _D F ) ` z ) ) ) -> ( F |` ( A (,) B ) ) e. ( ( A (,) B ) -cn-> RR ) )
70		ax-resscn 9993	. . . . . . . . . . . . . . . . . . . . . . . . . . . 28 |- RR C_ CC
71	70	a1i 11	. . . . . . . . . . . . . . . . . . . . . . . . . . 27 |- ( ph -> RR C_ CC )
72		fss 6056	. . . . . . . . . . . . . . . . . . . . . . . . . . . 28 |- ( ( F : ( A [,] B ) --> RR /\ RR C_ CC ) -> F : ( A [,] B ) --> CC )
73	9, 70, 72	sylancl 694	. . . . . . . . . . . . . . . . . . . . . . . . . . 27 |- ( ph -> F : ( A [,] B ) --> CC )
74	66, 13	syl5ss 3614	. . . . . . . . . . . . . . . . . . . . . . . . . . 27 |- ( ph -> ( A (,) B ) C_ RR )
75		eqid 2622	. . . . . . . . . . . . . . . . . . . . . . . . . . . 28 |- ( TopOpen ` ℂfld ) = ( TopOpen ` ℂfld )
76	75	tgioo2 22606	. . . . . . . . . . . . . . . . . . . . . . . . . . . 28 |- ( topGen ` ran (,) ) = ( ( TopOpen ` ℂfld ) ↾t RR )
77	75, 76	dvres 23675	. . . . . . . . . . . . . . . . . . . . . . . . . . 27 |- ( ( ( RR C_ CC /\ F : ( A [,] B ) --> CC ) /\ ( ( A [,] B ) C_ RR /\ ( A (,) B ) C_ RR ) ) -> ( RR _D ( F |` ( A (,) B ) ) ) = ( ( RR _D F ) |` ( ( int ` ( topGen ` ran (,) ) ) ` ( A (,) B ) ) ) )
78	71, 73, 13, 74, 77	syl22anc 1327	. . . . . . . . . . . . . . . . . . . . . . . . . 26 |- ( ph -> ( RR _D ( F |` ( A (,) B ) ) ) = ( ( RR _D F ) |` ( ( int ` ( topGen ` ran (,) ) ) ` ( A (,) B ) ) ) )
79		retop 22565	. . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 |- ( topGen ` ran (,) ) e. Top
80		iooretop 22569	. . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 |- ( A (,) B ) e. ( topGen ` ran (,) )
81		isopn3i 20886	. . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 |- ( ( ( topGen ` ran (,) ) e. Top /\ ( A (,) B ) e. ( topGen ` ran (,) ) ) -> ( ( int ` ( topGen ` ran (,) ) ) ` ( A (,) B ) ) = ( A (,) B ) )
82	79, 80, 81	mp2an 708	. . . . . . . . . . . . . . . . . . . . . . . . . . . 28 |- ( ( int ` ( topGen ` ran (,) ) ) ` ( A (,) B ) ) = ( A (,) B )
83	82	reseq2i 5393	. . . . . . . . . . . . . . . . . . . . . . . . . . 27 |- ( ( RR _D F ) |` ( ( int ` ( topGen ` ran (,) ) ) ` ( A (,) B ) ) ) = ( ( RR _D F ) |` ( A (,) B ) )
84		fnresdm 6000	. . . . . . . . . . . . . . . . . . . . . . . . . . . 28 |- ( ( RR _D F ) Fn ( A (,) B ) -> ( ( RR _D F ) |` ( A (,) B ) ) = ( RR _D F ) )
85	46, 84	syl 17	. . . . . . . . . . . . . . . . . . . . . . . . . . 27 |- ( ph -> ( ( RR _D F ) |` ( A (,) B ) ) = ( RR _D F ) )
86	83, 85	syl5eq 2668	. . . . . . . . . . . . . . . . . . . . . . . . . 26 |- ( ph -> ( ( RR _D F ) |` ( ( int ` ( topGen ` ran (,) ) ) ` ( A (,) B ) ) ) = ( RR _D F ) )
87	78, 86	eqtrd 2656	. . . . . . . . . . . . . . . . . . . . . . . . 25 |- ( ph -> ( RR _D ( F |` ( A (,) B ) ) ) = ( RR _D F ) )
88	87	dmeqd 5326	. . . . . . . . . . . . . . . . . . . . . . . 24 |- ( ph -> dom ( RR _D ( F |` ( A (,) B ) ) ) = dom ( RR _D F ) )
89	88, 42	eqtrd 2656	. . . . . . . . . . . . . . . . . . . . . . 23 |- ( ph -> dom ( RR _D ( F |` ( A (,) B ) ) ) = ( A (,) B ) )
90	89	ad2antrr 762	. . . . . . . . . . . . . . . . . . . . . 22 |- ( ( ( ph /\ ( y e. ( A (,) B ) /\ ( ( RR _D F ) ` y ) e. ( -oo (,) 0 ) ) ) /\ ( z e. ( A (,) B ) /\ 0 <_ ( ( RR _D F ) ` z ) ) ) -> dom ( RR _D ( F |` ( A (,) B ) ) ) = ( A (,) B ) )
91	64, 65, 69, 90	dvivth 23773	. . . . . . . . . . . . . . . . . . . . 21 |- ( ( ( ph /\ ( y e. ( A (,) B ) /\ ( ( RR _D F ) ` y ) e. ( -oo (,) 0 ) ) ) /\ ( z e. ( A (,) B ) /\ 0 <_ ( ( RR _D F ) ` z ) ) ) -> ( ( ( RR _D ( F |` ( A (,) B ) ) ) ` y ) [,] ( ( RR _D ( F |` ( A (,) B ) ) ) ` z ) ) C_ ran ( RR _D ( F |` ( A (,) B ) ) ) )
92	87	ad2antrr 762	. . . . . . . . . . . . . . . . . . . . . . 23 |- ( ( ( ph /\ ( y e. ( A (,) B ) /\ ( ( RR _D F ) ` y ) e. ( -oo (,) 0 ) ) ) /\ ( z e. ( A (,) B ) /\ 0 <_ ( ( RR _D F ) ` z ) ) ) -> ( RR _D ( F |` ( A (,) B ) ) ) = ( RR _D F ) )
93	92	fveq1d 6193	. . . . . . . . . . . . . . . . . . . . . 22 |- ( ( ( ph /\ ( y e. ( A (,) B ) /\ ( ( RR _D F ) ` y ) e. ( -oo (,) 0 ) ) ) /\ ( z e. ( A (,) B ) /\ 0 <_ ( ( RR _D F ) ` z ) ) ) -> ( ( RR _D ( F |` ( A (,) B ) ) ) ` y ) = ( ( RR _D F ) ` y ) )
94	92	fveq1d 6193	. . . . . . . . . . . . . . . . . . . . . 22 |- ( ( ( ph /\ ( y e. ( A (,) B ) /\ ( ( RR _D F ) ` y ) e. ( -oo (,) 0 ) ) ) /\ ( z e. ( A (,) B ) /\ 0 <_ ( ( RR _D F ) ` z ) ) ) -> ( ( RR _D ( F |` ( A (,) B ) ) ) ` z ) = ( ( RR _D F ) ` z ) )
95	93, 94	oveq12d 6668	. . . . . . . . . . . . . . . . . . . . 21 |- ( ( ( ph /\ ( y e. ( A (,) B ) /\ ( ( RR _D F ) ` y ) e. ( -oo (,) 0 ) ) ) /\ ( z e. ( A (,) B ) /\ 0 <_ ( ( RR _D F ) ` z ) ) ) -> ( ( ( RR _D ( F |` ( A (,) B ) ) ) ` y ) [,] ( ( RR _D ( F |` ( A (,) B ) ) ) ` z ) ) = ( ( ( RR _D F ) ` y ) [,] ( ( RR _D F ) ` z ) ) )
96	92	rneqd 5353	. . . . . . . . . . . . . . . . . . . . 21 |- ( ( ( ph /\ ( y e. ( A (,) B ) /\ ( ( RR _D F ) ` y ) e. ( -oo (,) 0 ) ) ) /\ ( z e. ( A (,) B ) /\ 0 <_ ( ( RR _D F ) ` z ) ) ) -> ran ( RR _D ( F |` ( A (,) B ) ) ) = ran ( RR _D F ) )
97	91, 95, 96	3sstr3d 3647	. . . . . . . . . . . . . . . . . . . 20 |- ( ( ( ph /\ ( y e. ( A (,) B ) /\ ( ( RR _D F ) ` y ) e. ( -oo (,) 0 ) ) ) /\ ( z e. ( A (,) B ) /\ 0 <_ ( ( RR _D F ) ` z ) ) ) -> ( ( ( RR _D F ) ` y ) [,] ( ( RR _D F ) ` z ) ) C_ ran ( RR _D F ) )
98	19	a1i 11	. . . . . . . . . . . . . . . . . . . . 21 |- ( ( ( ph /\ ( y e. ( A (,) B ) /\ ( ( RR _D F ) ` y ) e. ( -oo (,) 0 ) ) ) /\ ( z e. ( A (,) B ) /\ 0 <_ ( ( RR _D F ) ` z ) ) ) -> 0 e. RR )
99		simplrr 801	. . . . . . . . . . . . . . . . . . . . . . . 24 |- ( ( ( ph /\ ( y e. ( A (,) B ) /\ ( ( RR _D F ) ` y ) e. ( -oo (,) 0 ) ) ) /\ ( z e. ( A (,) B ) /\ 0 <_ ( ( RR _D F ) ` z ) ) ) -> ( ( RR _D F ) ` y ) e. ( -oo (,) 0 ) )
100		elioomnf 12268	. . . . . . . . . . . . . . . . . . . . . . . . 25 |- ( 0 e. RR* -> ( ( ( RR _D F ) ` y ) e. ( -oo (,) 0 ) <-> ( ( ( RR _D F ) ` y ) e. RR /\ ( ( RR _D F ) ` y ) < 0 ) ) )
101	22, 100	ax-mp 5	. . . . . . . . . . . . . . . . . . . . . . . 24 |- ( ( ( RR _D F ) ` y ) e. ( -oo (,) 0 ) <-> ( ( ( RR _D F ) ` y ) e. RR /\ ( ( RR _D F ) ` y ) < 0 ) )
102	99, 101	sylib 208	. . . . . . . . . . . . . . . . . . . . . . 23 |- ( ( ( ph /\ ( y e. ( A (,) B ) /\ ( ( RR _D F ) ` y ) e. ( -oo (,) 0 ) ) ) /\ ( z e. ( A (,) B ) /\ 0 <_ ( ( RR _D F ) ` z ) ) ) -> ( ( ( RR _D F ) ` y ) e. RR /\ ( ( RR _D F ) ` y ) < 0 ) )
103	102	simprd 479	. . . . . . . . . . . . . . . . . . . . . 22 |- ( ( ( ph /\ ( y e. ( A (,) B ) /\ ( ( RR _D F ) ` y ) e. ( -oo (,) 0 ) ) ) /\ ( z e. ( A (,) B ) /\ 0 <_ ( ( RR _D F ) ` z ) ) ) -> ( ( RR _D F ) ` y ) < 0 )
104	102	simpld 475	. . . . . . . . . . . . . . . . . . . . . . 23 |- ( ( ( ph /\ ( y e. ( A (,) B ) /\ ( ( RR _D F ) ` y ) e. ( -oo (,) 0 ) ) ) /\ ( z e. ( A (,) B ) /\ 0 <_ ( ( RR _D F ) ` z ) ) ) -> ( ( RR _D F ) ` y ) e. RR )
105		ltle 10126	. . . . . . . . . . . . . . . . . . . . . . 23 |- ( ( ( ( RR _D F ) ` y ) e. RR /\ 0 e. RR ) -> ( ( ( RR _D F ) ` y ) < 0 -> ( ( RR _D F ) ` y ) <_ 0 ) )
106	104, 19, 105	sylancl 694	. . . . . . . . . . . . . . . . . . . . . 22 |- ( ( ( ph /\ ( y e. ( A (,) B ) /\ ( ( RR _D F ) ` y ) e. ( -oo (,) 0 ) ) ) /\ ( z e. ( A (,) B ) /\ 0 <_ ( ( RR _D F ) ` z ) ) ) -> ( ( ( RR _D F ) ` y ) < 0 -> ( ( RR _D F ) ` y ) <_ 0 ) )
107	103, 106	mpd 15	. . . . . . . . . . . . . . . . . . . . 21 |- ( ( ( ph /\ ( y e. ( A (,) B ) /\ ( ( RR _D F ) ` y ) e. ( -oo (,) 0 ) ) ) /\ ( z e. ( A (,) B ) /\ 0 <_ ( ( RR _D F ) ` z ) ) ) -> ( ( RR _D F ) ` y ) <_ 0 )
108		simprr 796	. . . . . . . . . . . . . . . . . . . . 21 |- ( ( ( ph /\ ( y e. ( A (,) B ) /\ ( ( RR _D F ) ` y ) e. ( -oo (,) 0 ) ) ) /\ ( z e. ( A (,) B ) /\ 0 <_ ( ( RR _D F ) ` z ) ) ) -> 0 <_ ( ( RR _D F ) ` z ) )
109	65, 62	syldan 487	. . . . . . . . . . . . . . . . . . . . . 22 |- ( ( ( ph /\ ( y e. ( A (,) B ) /\ ( ( RR _D F ) ` y ) e. ( -oo (,) 0 ) ) ) /\ ( z e. ( A (,) B ) /\ 0 <_ ( ( RR _D F ) ` z ) ) ) -> ( ( RR _D F ) ` z ) e. RR )
110		elicc2 12238	. . . . . . . . . . . . . . . . . . . . . 22 |- ( ( ( ( RR _D F ) ` y ) e. RR /\ ( ( RR _D F ) ` z ) e. RR ) -> ( 0 e. ( ( ( RR _D F ) ` y ) [,] ( ( RR _D F ) ` z ) ) <-> ( 0 e. RR /\ ( ( RR _D F ) ` y ) <_ 0 /\ 0 <_ ( ( RR _D F ) ` z ) ) ) )
111	104, 109, 110	syl2anc 693	. . . . . . . . . . . . . . . . . . . . 21 |- ( ( ( ph /\ ( y e. ( A (,) B ) /\ ( ( RR _D F ) ` y ) e. ( -oo (,) 0 ) ) ) /\ ( z e. ( A (,) B ) /\ 0 <_ ( ( RR _D F ) ` z ) ) ) -> ( 0 e. ( ( ( RR _D F ) ` y ) [,] ( ( RR _D F ) ` z ) ) <-> ( 0 e. RR /\ ( ( RR _D F ) ` y ) <_ 0 /\ 0 <_ ( ( RR _D F ) ` z ) ) ) )
112	98, 107, 108, 111	mpbir3and 1245	. . . . . . . . . . . . . . . . . . . 20 |- ( ( ( ph /\ ( y e. ( A (,) B ) /\ ( ( RR _D F ) ` y ) e. ( -oo (,) 0 ) ) ) /\ ( z e. ( A (,) B ) /\ 0 <_ ( ( RR _D F ) ` z ) ) ) -> 0 e. ( ( ( RR _D F ) ` y ) [,] ( ( RR _D F ) ` z ) ) )
113	97, 112	sseldd 3604	. . . . . . . . . . . . . . . . . . 19 |- ( ( ( ph /\ ( y e. ( A (,) B ) /\ ( ( RR _D F ) ` y ) e. ( -oo (,) 0 ) ) ) /\ ( z e. ( A (,) B ) /\ 0 <_ ( ( RR _D F ) ` z ) ) ) -> 0 e. ran ( RR _D F ) )
114	113	expr 643	. . . . . . . . . . . . . . . . . 18 |- ( ( ( ph /\ ( y e. ( A (,) B ) /\ ( ( RR _D F ) ` y ) e. ( -oo (,) 0 ) ) ) /\ z e. ( A (,) B ) ) -> ( 0 <_ ( ( RR _D F ) ` z ) -> 0 e. ran ( RR _D F ) ) )
115	63, 114	mtod 189	. . . . . . . . . . . . . . . . 17 |- ( ( ( ph /\ ( y e. ( A (,) B ) /\ ( ( RR _D F ) ` y ) e. ( -oo (,) 0 ) ) ) /\ z e. ( A (,) B ) ) -> -. 0 <_ ( ( RR _D F ) ` z ) )
116		ltnle 10117	. . . . . . . . . . . . . . . . . 18 |- ( ( ( ( RR _D F ) ` z ) e. RR /\ 0 e. RR ) -> ( ( ( RR _D F ) ` z ) < 0 <-> -. 0 <_ ( ( RR _D F ) ` z ) ) )
117	62, 19, 116	sylancl 694	. . . . . . . . . . . . . . . . 17 |- ( ( ( ph /\ ( y e. ( A (,) B ) /\ ( ( RR _D F ) ` y ) e. ( -oo (,) 0 ) ) ) /\ z e. ( A (,) B ) ) -> ( ( ( RR _D F ) ` z ) < 0 <-> -. 0 <_ ( ( RR _D F ) ` z ) ) )
118	115, 117	mpbird 247	. . . . . . . . . . . . . . . 16 |- ( ( ( ph /\ ( y e. ( A (,) B ) /\ ( ( RR _D F ) ` y ) e. ( -oo (,) 0 ) ) ) /\ z e. ( A (,) B ) ) -> ( ( RR _D F ) ` z ) < 0 )
119		elioomnf 12268	. . . . . . . . . . . . . . . . 17 |- ( 0 e. RR* -> ( ( ( RR _D F ) ` z ) e. ( -oo (,) 0 ) <-> ( ( ( RR _D F ) ` z ) e. RR /\ ( ( RR _D F ) ` z ) < 0 ) ) )
120	22, 119	ax-mp 5	. . . . . . . . . . . . . . . 16 |- ( ( ( RR _D F ) ` z ) e. ( -oo (,) 0 ) <-> ( ( ( RR _D F ) ` z ) e. RR /\ ( ( RR _D F ) ` z ) < 0 ) )
121	62, 118, 120	sylanbrc 698	. . . . . . . . . . . . . . 15 |- ( ( ( ph /\ ( y e. ( A (,) B ) /\ ( ( RR _D F ) ` y ) e. ( -oo (,) 0 ) ) ) /\ z e. ( A (,) B ) ) -> ( ( RR _D F ) ` z ) e. ( -oo (,) 0 ) )
122	121	ralrimiva 2966	. . . . . . . . . . . . . 14 |- ( ( ph /\ ( y e. ( A (,) B ) /\ ( ( RR _D F ) ` y ) e. ( -oo (,) 0 ) ) ) -> A. z e. ( A (,) B ) ( ( RR _D F ) ` z ) e. ( -oo (,) 0 ) )
123		ffnfv 6388	. . . . . . . . . . . . . 14 |- ( ( RR _D F ) : ( A (,) B ) --> ( -oo (,) 0 ) <-> ( ( RR _D F ) Fn ( A (,) B ) /\ A. z e. ( A (,) B ) ( ( RR _D F ) ` z ) e. ( -oo (,) 0 ) ) )
124	60, 122, 123	sylanbrc 698	. . . . . . . . . . . . 13 |- ( ( ph /\ ( y e. ( A (,) B ) /\ ( ( RR _D F ) ` y ) e. ( -oo (,) 0 ) ) ) -> ( RR _D F ) : ( A (,) B ) --> ( -oo (,) 0 ) )
125	57, 58, 59, 124	dvlt0 23768	. . . . . . . . . . . 12 |- ( ( ph /\ ( y e. ( A (,) B ) /\ ( ( RR _D F ) ` y ) e. ( -oo (,) 0 ) ) ) -> F Isom < , `' < ( ( A [,] B ) , ran F ) )
126	125	olcd 408	. . . . . . . . . . 11 |- ( ( ph /\ ( y e. ( A (,) B ) /\ ( ( RR _D F ) ` y ) e. ( -oo (,) 0 ) ) ) -> ( F Isom < , < ( ( A [,] B ) , ran F ) \/ F Isom < , `' < ( ( A [,] B ) , ran F ) ) )
127	126	expr 643	. . . . . . . . . 10 |- ( ( ph /\ y e. ( A (,) B ) ) -> ( ( ( RR _D F ) ` y ) e. ( -oo (,) 0 ) -> ( F Isom < , < ( ( A [,] B ) , ran F ) \/ F Isom < , `' < ( ( A [,] B ) , ran F ) ) ) )
128		eleq1 2689	. . . . . . . . . . 11 |- ( ( ( RR _D F ) ` y ) = x -> ( ( ( RR _D F ) ` y ) e. ( -oo (,) 0 ) <-> x e. ( -oo (,) 0 ) ) )
129	128	imbi1d 331	. . . . . . . . . 10 |- ( ( ( RR _D F ) ` y ) = x -> ( ( ( ( RR _D F ) ` y ) e. ( -oo (,) 0 ) -> ( F Isom < , < ( ( A [,] B ) , ran F ) \/ F Isom < , `' < ( ( A [,] B ) , ran F ) ) ) <-> ( x e. ( -oo (,) 0 ) -> ( F Isom < , < ( ( A [,] B ) , ran F ) \/ F Isom < , `' < ( ( A [,] B ) , ran F ) ) ) ) )
130	127, 129	syl5ibcom 235	. . . . . . . . 9 |- ( ( ph /\ y e. ( A (,) B ) ) -> ( ( ( RR _D F ) ` y ) = x -> ( x e. ( -oo (,) 0 ) -> ( F Isom < , < ( ( A [,] B ) , ran F ) \/ F Isom < , `' < ( ( A [,] B ) , ran F ) ) ) ) )
131	130	rexlimdva 3031	. . . . . . . 8 |- ( ph -> ( E. y e. ( A (,) B ) ( ( RR _D F ) ` y ) = x -> ( x e. ( -oo (,) 0 ) -> ( F Isom < , < ( ( A [,] B ) , ran F ) \/ F Isom < , `' < ( ( A [,] B ) , ran F ) ) ) ) )
132	56, 131	sylbid 230	. . . . . . 7 |- ( ph -> ( x e. ran ( RR _D F ) -> ( x e. ( -oo (,) 0 ) -> ( F Isom < , < ( ( A [,] B ) , ran F ) \/ F Isom < , `' < ( ( A [,] B ) , ran F ) ) ) ) )
133	132	impd 447	. . . . . 6 |- ( ph -> ( ( x e. ran ( RR _D F ) /\ x e. ( -oo (,) 0 ) ) -> ( F Isom < , < ( ( A [,] B ) , ran F ) \/ F Isom < , `' < ( ( A [,] B ) , ran F ) ) ) )
134	54, 133	syl5bi 232	. . . . 5 |- ( ph -> ( x e. ( ran ( RR _D F ) i^i ( -oo (,) 0 ) ) -> ( F Isom < , < ( ( A [,] B ) , ran F ) \/ F Isom < , `' < ( ( A [,] B ) , ran F ) ) ) )
135	134	exlimdv 1861	. . . 4 |- ( ph -> ( E. x x e. ( ran ( RR _D F ) i^i ( -oo (,) 0 ) ) -> ( F Isom < , < ( ( A [,] B ) , ran F ) \/ F Isom < , `' < ( ( A [,] B ) , ran F ) ) ) )
136	53, 135	syl5bi 232	. . 3 |- ( ph -> ( ( ran ( RR _D F ) i^i ( -oo (,) 0 ) ) =/= (/) -> ( F Isom < , < ( ( A [,] B ) , ran F ) \/ F Isom < , `' < ( ( A [,] B ) , ran F ) ) ) )
137	136	imp 445	. 2 |- ( ( ph /\ ( ran ( RR _D F ) i^i ( -oo (,) 0 ) ) =/= (/) ) -> ( F Isom < , < ( ( A [,] B ) , ran F ) \/ F Isom < , `' < ( ( A [,] B ) , ran F ) ) )
138	52, 137	pm2.61dane 2881	1 |- ( ph -> ( F Isom < , < ( ( A [,] B ) , ran F ) \/ F Isom < , `' < ( ( A [,] B ) , ran F ) ) )

Syntax hints:   -. wn 3   -> wi 4   <-> wb 196   \/ wo 383   /\ wa 384   /\ w3a 1037   = wceq 1483   E.wex 1704   e. wcel 1990   =/= wne 2794   A.wral 2912   E.wrex 2913   u. cun 3572   i^i cin 3573   C_ wss 3574   (/)c0 3915   class class class wbr 4653   `'ccnv 5113   dom cdm 5114   ran crn 5115   |` cres 5116   Fn wfn 5883   -->wf 5884   ` cfv 5888   Isom wiso 5889  (class class class)co 6650   CCcc 9934   RRcr 9935   0cc0 9936   -oocmnf 10072   RR*cxr 10073   < clt 10074   <_ cle 10075   RR+crp 11832   (,)cioo 12175   [,]cicc 12178   TopOpenctopn 16082   topGenctg 16098  ℂ_fldccnfld 19746   Topctop 20698   intcnt 20821   -cn->ccncf 22679   _D cdv 23627

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-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-ico 12181  df-icc 12182  df-fz 12327  df-fzo 12466  df-seq 12802  df-exp 12861  df-hash 13118  df-cj 13839  df-re 13840  df-im 13841  df-sqrt 13975  df-abs 13976  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-cmp 21190  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

This theorem is referenced by:  dvne0f1  23775  dvcnvrelem1  23780