Theorem lhop2 23778

Description: L'Hôpital's Rule for limits from the left. If F and G are differentiable real functions on ( A , B ), and F and G both approach 0 at B, and G ( x ) and G' ( x ) are not zero on ( A , B ), and the limit of F' ( x ) / G' ( x ) at B is C, then the limit F ( x ) / G ( x ) at B also exists and equals C. (Contributed by Mario Carneiro, 29-Dec-2016.)

Hypotheses

Ref	Expression
lhop2.a	|- ( ph -> A e. RR* )
lhop2.b	|- ( ph -> B e. RR )
lhop2.l	|- ( ph -> A < B )
lhop2.f	|- ( ph -> F : ( A (,) B ) --> RR )
lhop2.g	|- ( ph -> G : ( A (,) B ) --> RR )
lhop2.if	|- ( ph -> dom ( RR _D F ) = ( A (,) B ) )
lhop2.ig	|- ( ph -> dom ( RR _D G ) = ( A (,) B ) )
lhop2.f0	|- ( ph -> 0 e. ( F lim CC B ) )
lhop2.g0	|- ( ph -> 0 e. ( G lim CC B ) )
lhop2.gn0	|- ( ph -> -. 0 e. ran G )
lhop2.gd0	|- ( ph -> -. 0 e. ran ( RR _D G ) )
lhop2.c	|- ( ph -> C e. ( ( z e. ( A (,) B ) |-> ( ( ( RR _D F ) ` z ) / ( ( RR _D G ) ` z ) ) ) lim CC B ) )

Assertion

Ref	Expression
lhop2	|- ( ph -> C e. ( ( z e. ( A (,) B ) |-> ( ( F ` z ) / ( G ` z ) ) ) lim CC B ) )

Distinct variable groups:   z, A   z, B   z, C   ph, z   z, F   z, G

Proof of Theorem lhop2

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

Step	Hyp	Ref	Expression
1		qssre 11798	. . 3 |- QQ C_ RR
2		lhop2.a	. . . 4 |- ( ph -> A e. RR* )
3		lhop2.b	. . . . 5 |- ( ph -> B e. RR )
4	3	rexrd 10089	. . . 4 |- ( ph -> B e. RR* )
5		lhop2.l	. . . 4 |- ( ph -> A < B )
6		qbtwnxr 12031	. . . 4 |- ( ( A e. RR* /\ B e. RR* /\ A < B ) -> E. a e. QQ ( A < a /\ a < B ) )
7	2, 4, 5, 6	syl3anc 1326	. . 3 |- ( ph -> E. a e. QQ ( A < a /\ a < B ) )
8		ssrexv 3667	. . 3 |- ( QQ C_ RR -> ( E. a e. QQ ( A < a /\ a < B ) -> E. a e. RR ( A < a /\ a < B ) ) )
9	1, 7, 8	mpsyl 68	. 2 |- ( ph -> E. a e. RR ( A < a /\ a < B ) )
10		simpr 477	. . . . . 6 |- ( ( ( ph /\ ( a e. RR /\ ( A < a /\ a < B ) ) ) /\ z e. ( a (,) B ) ) -> z e. ( a (,) B ) )
11		simprl 794	. . . . . . . 8 |- ( ( ph /\ ( a e. RR /\ ( A < a /\ a < B ) ) ) -> a e. RR )
12	11	adantr 481	. . . . . . 7 |- ( ( ( ph /\ ( a e. RR /\ ( A < a /\ a < B ) ) ) /\ z e. ( a (,) B ) ) -> a e. RR )
13	3	ad2antrr 762	. . . . . . 7 |- ( ( ( ph /\ ( a e. RR /\ ( A < a /\ a < B ) ) ) /\ z e. ( a (,) B ) ) -> B e. RR )
14		elioore 12205	. . . . . . . 8 |- ( z e. ( a (,) B ) -> z e. RR )
15	14	adantl 482	. . . . . . 7 |- ( ( ( ph /\ ( a e. RR /\ ( A < a /\ a < B ) ) ) /\ z e. ( a (,) B ) ) -> z e. RR )
16		iooneg 12292	. . . . . . 7 |- ( ( a e. RR /\ B e. RR /\ z e. RR ) -> ( z e. ( a (,) B ) <-> -u z e. ( -u B (,) -u a ) ) )
17	12, 13, 15, 16	syl3anc 1326	. . . . . 6 |- ( ( ( ph /\ ( a e. RR /\ ( A < a /\ a < B ) ) ) /\ z e. ( a (,) B ) ) -> ( z e. ( a (,) B ) <-> -u z e. ( -u B (,) -u a ) ) )
18	10, 17	mpbid 222	. . . . 5 |- ( ( ( ph /\ ( a e. RR /\ ( A < a /\ a < B ) ) ) /\ z e. ( a (,) B ) ) -> -u z e. ( -u B (,) -u a ) )
19	18	adantrr 753	. . . 4 |- ( ( ( ph /\ ( a e. RR /\ ( A < a /\ a < B ) ) ) /\ ( z e. ( a (,) B ) /\ -u z =/= -u B ) ) -> -u z e. ( -u B (,) -u a ) )
20		lhop2.f	. . . . . . . 8 |- ( ph -> F : ( A (,) B ) --> RR )
21	20	ad2antrr 762	. . . . . . 7 |- ( ( ( ph /\ ( a e. RR /\ ( A < a /\ a < B ) ) ) /\ x e. ( -u B (,) -u a ) ) -> F : ( A (,) B ) --> RR )
22		elioore 12205	. . . . . . . . . . . . 13 |- ( x e. ( -u B (,) -u a ) -> x e. RR )
23	22	adantl 482	. . . . . . . . . . . 12 |- ( ( ( ph /\ ( a e. RR /\ ( A < a /\ a < B ) ) ) /\ x e. ( -u B (,) -u a ) ) -> x e. RR )
24	23	recnd 10068	. . . . . . . . . . 11 |- ( ( ( ph /\ ( a e. RR /\ ( A < a /\ a < B ) ) ) /\ x e. ( -u B (,) -u a ) ) -> x e. CC )
25	24	negnegd 10383	. . . . . . . . . 10 |- ( ( ( ph /\ ( a e. RR /\ ( A < a /\ a < B ) ) ) /\ x e. ( -u B (,) -u a ) ) -> -u -u x = x )
26		simpr 477	. . . . . . . . . 10 |- ( ( ( ph /\ ( a e. RR /\ ( A < a /\ a < B ) ) ) /\ x e. ( -u B (,) -u a ) ) -> x e. ( -u B (,) -u a ) )
27	25, 26	eqeltrd 2701	. . . . . . . . 9 |- ( ( ( ph /\ ( a e. RR /\ ( A < a /\ a < B ) ) ) /\ x e. ( -u B (,) -u a ) ) -> -u -u x e. ( -u B (,) -u a ) )
28	11	adantr 481	. . . . . . . . . 10 |- ( ( ( ph /\ ( a e. RR /\ ( A < a /\ a < B ) ) ) /\ x e. ( -u B (,) -u a ) ) -> a e. RR )
29	3	ad2antrr 762	. . . . . . . . . 10 |- ( ( ( ph /\ ( a e. RR /\ ( A < a /\ a < B ) ) ) /\ x e. ( -u B (,) -u a ) ) -> B e. RR )
30	23	renegcld 10457	. . . . . . . . . 10 |- ( ( ( ph /\ ( a e. RR /\ ( A < a /\ a < B ) ) ) /\ x e. ( -u B (,) -u a ) ) -> -u x e. RR )
31		iooneg 12292	. . . . . . . . . 10 |- ( ( a e. RR /\ B e. RR /\ -u x e. RR ) -> ( -u x e. ( a (,) B ) <-> -u -u x e. ( -u B (,) -u a ) ) )
32	28, 29, 30, 31	syl3anc 1326	. . . . . . . . 9 |- ( ( ( ph /\ ( a e. RR /\ ( A < a /\ a < B ) ) ) /\ x e. ( -u B (,) -u a ) ) -> ( -u x e. ( a (,) B ) <-> -u -u x e. ( -u B (,) -u a ) ) )
33	27, 32	mpbird 247	. . . . . . . 8 |- ( ( ( ph /\ ( a e. RR /\ ( A < a /\ a < B ) ) ) /\ x e. ( -u B (,) -u a ) ) -> -u x e. ( a (,) B ) )
34	2	adantr 481	. . . . . . . . . 10 |- ( ( ph /\ ( a e. RR /\ ( A < a /\ a < B ) ) ) -> A e. RR* )
35		simprrl 804	. . . . . . . . . . 11 |- ( ( ph /\ ( a e. RR /\ ( A < a /\ a < B ) ) ) -> A < a )
36	11	rexrd 10089	. . . . . . . . . . . 12 |- ( ( ph /\ ( a e. RR /\ ( A < a /\ a < B ) ) ) -> a e. RR* )
37		xrltle 11982	. . . . . . . . . . . 12 |- ( ( A e. RR* /\ a e. RR* ) -> ( A < a -> A <_ a ) )
38	34, 36, 37	syl2anc 693	. . . . . . . . . . 11 |- ( ( ph /\ ( a e. RR /\ ( A < a /\ a < B ) ) ) -> ( A < a -> A <_ a ) )
39	35, 38	mpd 15	. . . . . . . . . 10 |- ( ( ph /\ ( a e. RR /\ ( A < a /\ a < B ) ) ) -> A <_ a )
40		iooss1 12210	. . . . . . . . . 10 |- ( ( A e. RR* /\ A <_ a ) -> ( a (,) B ) C_ ( A (,) B ) )
41	34, 39, 40	syl2anc 693	. . . . . . . . 9 |- ( ( ph /\ ( a e. RR /\ ( A < a /\ a < B ) ) ) -> ( a (,) B ) C_ ( A (,) B ) )
42	41	sselda 3603	. . . . . . . 8 |- ( ( ( ph /\ ( a e. RR /\ ( A < a /\ a < B ) ) ) /\ -u x e. ( a (,) B ) ) -> -u x e. ( A (,) B ) )
43	33, 42	syldan 487	. . . . . . 7 |- ( ( ( ph /\ ( a e. RR /\ ( A < a /\ a < B ) ) ) /\ x e. ( -u B (,) -u a ) ) -> -u x e. ( A (,) B ) )
44	21, 43	ffvelrnd 6360	. . . . . 6 |- ( ( ( ph /\ ( a e. RR /\ ( A < a /\ a < B ) ) ) /\ x e. ( -u B (,) -u a ) ) -> ( F ` -u x ) e. RR )
45	44	recnd 10068	. . . . 5 |- ( ( ( ph /\ ( a e. RR /\ ( A < a /\ a < B ) ) ) /\ x e. ( -u B (,) -u a ) ) -> ( F ` -u x ) e. CC )
46		lhop2.g	. . . . . . . 8 |- ( ph -> G : ( A (,) B ) --> RR )
47	46	ad2antrr 762	. . . . . . 7 |- ( ( ( ph /\ ( a e. RR /\ ( A < a /\ a < B ) ) ) /\ x e. ( -u B (,) -u a ) ) -> G : ( A (,) B ) --> RR )
48	47, 43	ffvelrnd 6360	. . . . . 6 |- ( ( ( ph /\ ( a e. RR /\ ( A < a /\ a < B ) ) ) /\ x e. ( -u B (,) -u a ) ) -> ( G ` -u x ) e. RR )
49	48	recnd 10068	. . . . 5 |- ( ( ( ph /\ ( a e. RR /\ ( A < a /\ a < B ) ) ) /\ x e. ( -u B (,) -u a ) ) -> ( G ` -u x ) e. CC )
50		lhop2.gn0	. . . . . . 7 |- ( ph -> -. 0 e. ran G )
51	50	ad2antrr 762	. . . . . 6 |- ( ( ( ph /\ ( a e. RR /\ ( A < a /\ a < B ) ) ) /\ x e. ( -u B (,) -u a ) ) -> -. 0 e. ran G )
52	46	adantr 481	. . . . . . . . . . . 12 |- ( ( ph /\ ( a e. RR /\ ( A < a /\ a < B ) ) ) -> G : ( A (,) B ) --> RR )
53		ax-resscn 9993	. . . . . . . . . . . 12 |- RR C_ CC
54		fss 6056	. . . . . . . . . . . 12 |- ( ( G : ( A (,) B ) --> RR /\ RR C_ CC ) -> G : ( A (,) B ) --> CC )
55	52, 53, 54	sylancl 694	. . . . . . . . . . 11 |- ( ( ph /\ ( a e. RR /\ ( A < a /\ a < B ) ) ) -> G : ( A (,) B ) --> CC )
56	55	adantr 481	. . . . . . . . . 10 |- ( ( ( ph /\ ( a e. RR /\ ( A < a /\ a < B ) ) ) /\ x e. ( -u B (,) -u a ) ) -> G : ( A (,) B ) --> CC )
57		ffn 6045	. . . . . . . . . 10 |- ( G : ( A (,) B ) --> CC -> G Fn ( A (,) B ) )
58	56, 57	syl 17	. . . . . . . . 9 |- ( ( ( ph /\ ( a e. RR /\ ( A < a /\ a < B ) ) ) /\ x e. ( -u B (,) -u a ) ) -> G Fn ( A (,) B ) )
59		fnfvelrn 6356	. . . . . . . . 9 |- ( ( G Fn ( A (,) B ) /\ -u x e. ( A (,) B ) ) -> ( G ` -u x ) e. ran G )
60	58, 43, 59	syl2anc 693	. . . . . . . 8 |- ( ( ( ph /\ ( a e. RR /\ ( A < a /\ a < B ) ) ) /\ x e. ( -u B (,) -u a ) ) -> ( G ` -u x ) e. ran G )
61		eleq1 2689	. . . . . . . 8 |- ( ( G ` -u x ) = 0 -> ( ( G ` -u x ) e. ran G <-> 0 e. ran G ) )
62	60, 61	syl5ibcom 235	. . . . . . 7 |- ( ( ( ph /\ ( a e. RR /\ ( A < a /\ a < B ) ) ) /\ x e. ( -u B (,) -u a ) ) -> ( ( G ` -u x ) = 0 -> 0 e. ran G ) )
63	62	necon3bd 2808	. . . . . 6 |- ( ( ( ph /\ ( a e. RR /\ ( A < a /\ a < B ) ) ) /\ x e. ( -u B (,) -u a ) ) -> ( -. 0 e. ran G -> ( G ` -u x ) =/= 0 ) )
64	51, 63	mpd 15	. . . . 5 |- ( ( ( ph /\ ( a e. RR /\ ( A < a /\ a < B ) ) ) /\ x e. ( -u B (,) -u a ) ) -> ( G ` -u x ) =/= 0 )
65	45, 49, 64	divcld 10801	. . . 4 |- ( ( ( ph /\ ( a e. RR /\ ( A < a /\ a < B ) ) ) /\ x e. ( -u B (,) -u a ) ) -> ( ( F ` -u x ) / ( G ` -u x ) ) e. CC )
66		limcresi 23649	. . . . . 6 |- ( ( z e. RR |-> -u z ) lim CC B ) C_ ( ( ( z e. RR |-> -u z ) |` ( a (,) B ) ) lim CC B )
67		ioossre 12235	. . . . . . . 8 |- ( a (,) B ) C_ RR
68		resmpt 5449	. . . . . . . 8 |- ( ( a (,) B ) C_ RR -> ( ( z e. RR |-> -u z ) |` ( a (,) B ) ) = ( z e. ( a (,) B ) |-> -u z ) )
69	67, 68	ax-mp 5	. . . . . . 7 |- ( ( z e. RR |-> -u z ) |` ( a (,) B ) ) = ( z e. ( a (,) B ) |-> -u z )
70	69	oveq1i 6660	. . . . . 6 |- ( ( ( z e. RR |-> -u z ) |` ( a (,) B ) ) lim CC B ) = ( ( z e. ( a (,) B ) |-> -u z ) lim CC B )
71	66, 70	sseqtri 3637	. . . . 5 |- ( ( z e. RR |-> -u z ) lim CC B ) C_ ( ( z e. ( a (,) B ) |-> -u z ) lim CC B )
72		eqid 2622	. . . . . . . 8 |- ( z e. RR |-> -u z ) = ( z e. RR |-> -u z )
73	72	negcncf 22721	. . . . . . 7 |- ( RR C_ CC -> ( z e. RR |-> -u z ) e. ( RR -cn-> CC ) )
74	53, 73	mp1i 13	. . . . . 6 |- ( ( ph /\ ( a e. RR /\ ( A < a /\ a < B ) ) ) -> ( z e. RR |-> -u z ) e. ( RR -cn-> CC ) )
75	3	adantr 481	. . . . . 6 |- ( ( ph /\ ( a e. RR /\ ( A < a /\ a < B ) ) ) -> B e. RR )
76		negeq 10273	. . . . . 6 |- ( z = B -> -u z = -u B )
77	74, 75, 76	cnmptlimc 23654	. . . . 5 |- ( ( ph /\ ( a e. RR /\ ( A < a /\ a < B ) ) ) -> -u B e. ( ( z e. RR |-> -u z ) lim CC B ) )
78	71, 77	sseldi 3601	. . . 4 |- ( ( ph /\ ( a e. RR /\ ( A < a /\ a < B ) ) ) -> -u B e. ( ( z e. ( a (,) B ) |-> -u z ) lim CC B ) )
79	75	renegcld 10457	. . . . . 6 |- ( ( ph /\ ( a e. RR /\ ( A < a /\ a < B ) ) ) -> -u B e. RR )
80	11	renegcld 10457	. . . . . . 7 |- ( ( ph /\ ( a e. RR /\ ( A < a /\ a < B ) ) ) -> -u a e. RR )
81	80	rexrd 10089	. . . . . 6 |- ( ( ph /\ ( a e. RR /\ ( A < a /\ a < B ) ) ) -> -u a e. RR* )
82		simprrr 805	. . . . . . 7 |- ( ( ph /\ ( a e. RR /\ ( A < a /\ a < B ) ) ) -> a < B )
83	11, 75	ltnegd 10605	. . . . . . 7 |- ( ( ph /\ ( a e. RR /\ ( A < a /\ a < B ) ) ) -> ( a < B <-> -u B < -u a ) )
84	82, 83	mpbid 222	. . . . . 6 |- ( ( ph /\ ( a e. RR /\ ( A < a /\ a < B ) ) ) -> -u B < -u a )
85		eqid 2622	. . . . . . 7 |- ( x e. ( -u B (,) -u a ) |-> ( F ` -u x ) ) = ( x e. ( -u B (,) -u a ) |-> ( F ` -u x ) )
86	44, 85	fmptd 6385	. . . . . 6 |- ( ( ph /\ ( a e. RR /\ ( A < a /\ a < B ) ) ) -> ( x e. ( -u B (,) -u a ) |-> ( F ` -u x ) ) : ( -u B (,) -u a ) --> RR )
87		eqid 2622	. . . . . . 7 |- ( x e. ( -u B (,) -u a ) |-> ( G ` -u x ) ) = ( x e. ( -u B (,) -u a ) |-> ( G ` -u x ) )
88	48, 87	fmptd 6385	. . . . . 6 |- ( ( ph /\ ( a e. RR /\ ( A < a /\ a < B ) ) ) -> ( x e. ( -u B (,) -u a ) |-> ( G ` -u x ) ) : ( -u B (,) -u a ) --> RR )
89		reelprrecn 10028	. . . . . . . . . . 11 |- RR e. { RR , CC }
90	89	a1i 11	. . . . . . . . . 10 |- ( ( ph /\ ( a e. RR /\ ( A < a /\ a < B ) ) ) -> RR e. { RR , CC } )
91		neg1cn 11124	. . . . . . . . . . 11 |- -u 1 e. CC
92	91	a1i 11	. . . . . . . . . 10 |- ( ( ( ph /\ ( a e. RR /\ ( A < a /\ a < B ) ) ) /\ x e. ( -u B (,) -u a ) ) -> -u 1 e. CC )
93	20	adantr 481	. . . . . . . . . . . 12 |- ( ( ph /\ ( a e. RR /\ ( A < a /\ a < B ) ) ) -> F : ( A (,) B ) --> RR )
94	93	ffvelrnda 6359	. . . . . . . . . . 11 |- ( ( ( ph /\ ( a e. RR /\ ( A < a /\ a < B ) ) ) /\ y e. ( A (,) B ) ) -> ( F ` y ) e. RR )
95	94	recnd 10068	. . . . . . . . . 10 |- ( ( ( ph /\ ( a e. RR /\ ( A < a /\ a < B ) ) ) /\ y e. ( A (,) B ) ) -> ( F ` y ) e. CC )
96		fvexd 6203	. . . . . . . . . 10 |- ( ( ( ph /\ ( a e. RR /\ ( A < a /\ a < B ) ) ) /\ y e. ( A (,) B ) ) -> ( ( RR _D F ) ` y ) e. _V )
97		1cnd 10056	. . . . . . . . . . 11 |- ( ( ( ph /\ ( a e. RR /\ ( A < a /\ a < B ) ) ) /\ x e. ( -u B (,) -u a ) ) -> 1 e. CC )
98		simpr 477	. . . . . . . . . . . . 13 |- ( ( ( ph /\ ( a e. RR /\ ( A < a /\ a < B ) ) ) /\ x e. RR ) -> x e. RR )
99	98	recnd 10068	. . . . . . . . . . . 12 |- ( ( ( ph /\ ( a e. RR /\ ( A < a /\ a < B ) ) ) /\ x e. RR ) -> x e. CC )
100		1cnd 10056	. . . . . . . . . . . 12 |- ( ( ( ph /\ ( a e. RR /\ ( A < a /\ a < B ) ) ) /\ x e. RR ) -> 1 e. CC )
101	90	dvmptid 23720	. . . . . . . . . . . 12 |- ( ( ph /\ ( a e. RR /\ ( A < a /\ a < B ) ) ) -> ( RR _D ( x e. RR |-> x ) ) = ( x e. RR |-> 1 ) )
102		ioossre 12235	. . . . . . . . . . . . 13 |- ( -u B (,) -u a ) C_ RR
103	102	a1i 11	. . . . . . . . . . . 12 |- ( ( ph /\ ( a e. RR /\ ( A < a /\ a < B ) ) ) -> ( -u B (,) -u a ) C_ RR )
104		eqid 2622	. . . . . . . . . . . . 13 |- ( TopOpen ` ℂfld ) = ( TopOpen ` ℂfld )
105	104	tgioo2 22606	. . . . . . . . . . . 12 |- ( topGen ` ran (,) ) = ( ( TopOpen ` ℂfld ) ↾t RR )
106		iooretop 22569	. . . . . . . . . . . . 13 |- ( -u B (,) -u a ) e. ( topGen ` ran (,) )
107	106	a1i 11	. . . . . . . . . . . 12 |- ( ( ph /\ ( a e. RR /\ ( A < a /\ a < B ) ) ) -> ( -u B (,) -u a ) e. ( topGen ` ran (,) ) )
108	90, 99, 100, 101, 103, 105, 104, 107	dvmptres 23726	. . . . . . . . . . 11 |- ( ( ph /\ ( a e. RR /\ ( A < a /\ a < B ) ) ) -> ( RR _D ( x e. ( -u B (,) -u a ) |-> x ) ) = ( x e. ( -u B (,) -u a ) |-> 1 ) )
109	90, 24, 97, 108	dvmptneg 23729	. . . . . . . . . 10 |- ( ( ph /\ ( a e. RR /\ ( A < a /\ a < B ) ) ) -> ( RR _D ( x e. ( -u B (,) -u a ) |-> -u x ) ) = ( x e. ( -u B (,) -u a ) |-> -u 1 ) )
110	93	feqmptd 6249	. . . . . . . . . . . 12 |- ( ( ph /\ ( a e. RR /\ ( A < a /\ a < B ) ) ) -> F = ( y e. ( A (,) B ) |-> ( F ` y ) ) )
111	110	oveq2d 6666	. . . . . . . . . . 11 |- ( ( ph /\ ( a e. RR /\ ( A < a /\ a < B ) ) ) -> ( RR _D F ) = ( RR _D ( y e. ( A (,) B ) |-> ( F ` y ) ) ) )
112		dvf 23671	. . . . . . . . . . . . 13 |- ( RR _D F ) : dom ( RR _D F ) --> CC
113		lhop2.if	. . . . . . . . . . . . . . 15 |- ( ph -> dom ( RR _D F ) = ( A (,) B ) )
114	113	adantr 481	. . . . . . . . . . . . . 14 |- ( ( ph /\ ( a e. RR /\ ( A < a /\ a < B ) ) ) -> dom ( RR _D F ) = ( A (,) B ) )
115	114	feq2d 6031	. . . . . . . . . . . . 13 |- ( ( ph /\ ( a e. RR /\ ( A < a /\ a < B ) ) ) -> ( ( RR _D F ) : dom ( RR _D F ) --> CC <-> ( RR _D F ) : ( A (,) B ) --> CC ) )
116	112, 115	mpbii 223	. . . . . . . . . . . 12 |- ( ( ph /\ ( a e. RR /\ ( A < a /\ a < B ) ) ) -> ( RR _D F ) : ( A (,) B ) --> CC )
117	116	feqmptd 6249	. . . . . . . . . . 11 |- ( ( ph /\ ( a e. RR /\ ( A < a /\ a < B ) ) ) -> ( RR _D F ) = ( y e. ( A (,) B ) |-> ( ( RR _D F ) ` y ) ) )
118	111, 117	eqtr3d 2658	. . . . . . . . . 10 |- ( ( ph /\ ( a e. RR /\ ( A < a /\ a < B ) ) ) -> ( RR _D ( y e. ( A (,) B ) |-> ( F ` y ) ) ) = ( y e. ( A (,) B ) |-> ( ( RR _D F ) ` y ) ) )
119		fveq2 6191	. . . . . . . . . 10 |- ( y = -u x -> ( F ` y ) = ( F ` -u x ) )
120		fveq2 6191	. . . . . . . . . 10 |- ( y = -u x -> ( ( RR _D F ) ` y ) = ( ( RR _D F ) ` -u x ) )
121	90, 90, 43, 92, 95, 96, 109, 118, 119, 120	dvmptco 23735	. . . . . . . . 9 |- ( ( ph /\ ( a e. RR /\ ( A < a /\ a < B ) ) ) -> ( RR _D ( x e. ( -u B (,) -u a ) |-> ( F ` -u x ) ) ) = ( x e. ( -u B (,) -u a ) |-> ( ( ( RR _D F ) ` -u x ) x. -u 1 ) ) )
122	116	adantr 481	. . . . . . . . . . . . 13 |- ( ( ( ph /\ ( a e. RR /\ ( A < a /\ a < B ) ) ) /\ x e. ( -u B (,) -u a ) ) -> ( RR _D F ) : ( A (,) B ) --> CC )
123	122, 43	ffvelrnd 6360	. . . . . . . . . . . 12 |- ( ( ( ph /\ ( a e. RR /\ ( A < a /\ a < B ) ) ) /\ x e. ( -u B (,) -u a ) ) -> ( ( RR _D F ) ` -u x ) e. CC )
124	123, 92	mulcomd 10061	. . . . . . . . . . 11 |- ( ( ( ph /\ ( a e. RR /\ ( A < a /\ a < B ) ) ) /\ x e. ( -u B (,) -u a ) ) -> ( ( ( RR _D F ) ` -u x ) x. -u 1 ) = ( -u 1 x. ( ( RR _D F ) ` -u x ) ) )
125	123	mulm1d 10482	. . . . . . . . . . 11 |- ( ( ( ph /\ ( a e. RR /\ ( A < a /\ a < B ) ) ) /\ x e. ( -u B (,) -u a ) ) -> ( -u 1 x. ( ( RR _D F ) ` -u x ) ) = -u ( ( RR _D F ) ` -u x ) )
126	124, 125	eqtrd 2656	. . . . . . . . . 10 |- ( ( ( ph /\ ( a e. RR /\ ( A < a /\ a < B ) ) ) /\ x e. ( -u B (,) -u a ) ) -> ( ( ( RR _D F ) ` -u x ) x. -u 1 ) = -u ( ( RR _D F ) ` -u x ) )
127	126	mpteq2dva 4744	. . . . . . . . 9 |- ( ( ph /\ ( a e. RR /\ ( A < a /\ a < B ) ) ) -> ( x e. ( -u B (,) -u a ) |-> ( ( ( RR _D F ) ` -u x ) x. -u 1 ) ) = ( x e. ( -u B (,) -u a ) |-> -u ( ( RR _D F ) ` -u x ) ) )
128	121, 127	eqtrd 2656	. . . . . . . 8 |- ( ( ph /\ ( a e. RR /\ ( A < a /\ a < B ) ) ) -> ( RR _D ( x e. ( -u B (,) -u a ) |-> ( F ` -u x ) ) ) = ( x e. ( -u B (,) -u a ) |-> -u ( ( RR _D F ) ` -u x ) ) )
129	128	dmeqd 5326	. . . . . . 7 |- ( ( ph /\ ( a e. RR /\ ( A < a /\ a < B ) ) ) -> dom ( RR _D ( x e. ( -u B (,) -u a ) |-> ( F ` -u x ) ) ) = dom ( x e. ( -u B (,) -u a ) |-> -u ( ( RR _D F ) ` -u x ) ) )
130		negex 10279	. . . . . . . 8 |- -u ( ( RR _D F ) ` -u x ) e. _V
131		eqid 2622	. . . . . . . 8 |- ( x e. ( -u B (,) -u a ) |-> -u ( ( RR _D F ) ` -u x ) ) = ( x e. ( -u B (,) -u a ) |-> -u ( ( RR _D F ) ` -u x ) )
132	130, 131	dmmpti 6023	. . . . . . 7 |- dom ( x e. ( -u B (,) -u a ) |-> -u ( ( RR _D F ) ` -u x ) ) = ( -u B (,) -u a )
133	129, 132	syl6eq 2672	. . . . . 6 |- ( ( ph /\ ( a e. RR /\ ( A < a /\ a < B ) ) ) -> dom ( RR _D ( x e. ( -u B (,) -u a ) |-> ( F ` -u x ) ) ) = ( -u B (,) -u a ) )
134	52	ffvelrnda 6359	. . . . . . . . . . 11 |- ( ( ( ph /\ ( a e. RR /\ ( A < a /\ a < B ) ) ) /\ y e. ( A (,) B ) ) -> ( G ` y ) e. RR )
135	134	recnd 10068	. . . . . . . . . 10 |- ( ( ( ph /\ ( a e. RR /\ ( A < a /\ a < B ) ) ) /\ y e. ( A (,) B ) ) -> ( G ` y ) e. CC )
136		fvexd 6203	. . . . . . . . . 10 |- ( ( ( ph /\ ( a e. RR /\ ( A < a /\ a < B ) ) ) /\ y e. ( A (,) B ) ) -> ( ( RR _D G ) ` y ) e. _V )
137	52	feqmptd 6249	. . . . . . . . . . . 12 |- ( ( ph /\ ( a e. RR /\ ( A < a /\ a < B ) ) ) -> G = ( y e. ( A (,) B ) |-> ( G ` y ) ) )
138	137	oveq2d 6666	. . . . . . . . . . 11 |- ( ( ph /\ ( a e. RR /\ ( A < a /\ a < B ) ) ) -> ( RR _D G ) = ( RR _D ( y e. ( A (,) B ) |-> ( G ` y ) ) ) )
139		dvf 23671	. . . . . . . . . . . . 13 |- ( RR _D G ) : dom ( RR _D G ) --> CC
140		lhop2.ig	. . . . . . . . . . . . . . 15 |- ( ph -> dom ( RR _D G ) = ( A (,) B ) )
141	140	adantr 481	. . . . . . . . . . . . . 14 |- ( ( ph /\ ( a e. RR /\ ( A < a /\ a < B ) ) ) -> dom ( RR _D G ) = ( A (,) B ) )
142	141	feq2d 6031	. . . . . . . . . . . . 13 |- ( ( ph /\ ( a e. RR /\ ( A < a /\ a < B ) ) ) -> ( ( RR _D G ) : dom ( RR _D G ) --> CC <-> ( RR _D G ) : ( A (,) B ) --> CC ) )
143	139, 142	mpbii 223	. . . . . . . . . . . 12 |- ( ( ph /\ ( a e. RR /\ ( A < a /\ a < B ) ) ) -> ( RR _D G ) : ( A (,) B ) --> CC )
144	143	feqmptd 6249	. . . . . . . . . . 11 |- ( ( ph /\ ( a e. RR /\ ( A < a /\ a < B ) ) ) -> ( RR _D G ) = ( y e. ( A (,) B ) |-> ( ( RR _D G ) ` y ) ) )
145	138, 144	eqtr3d 2658	. . . . . . . . . 10 |- ( ( ph /\ ( a e. RR /\ ( A < a /\ a < B ) ) ) -> ( RR _D ( y e. ( A (,) B ) |-> ( G ` y ) ) ) = ( y e. ( A (,) B ) |-> ( ( RR _D G ) ` y ) ) )
146		fveq2 6191	. . . . . . . . . 10 |- ( y = -u x -> ( G ` y ) = ( G ` -u x ) )
147		fveq2 6191	. . . . . . . . . 10 |- ( y = -u x -> ( ( RR _D G ) ` y ) = ( ( RR _D G ) ` -u x ) )
148	90, 90, 43, 92, 135, 136, 109, 145, 146, 147	dvmptco 23735	. . . . . . . . 9 |- ( ( ph /\ ( a e. RR /\ ( A < a /\ a < B ) ) ) -> ( RR _D ( x e. ( -u B (,) -u a ) |-> ( G ` -u x ) ) ) = ( x e. ( -u B (,) -u a ) |-> ( ( ( RR _D G ) ` -u x ) x. -u 1 ) ) )
149	143	adantr 481	. . . . . . . . . . . . 13 |- ( ( ( ph /\ ( a e. RR /\ ( A < a /\ a < B ) ) ) /\ x e. ( -u B (,) -u a ) ) -> ( RR _D G ) : ( A (,) B ) --> CC )
150	149, 43	ffvelrnd 6360	. . . . . . . . . . . 12 |- ( ( ( ph /\ ( a e. RR /\ ( A < a /\ a < B ) ) ) /\ x e. ( -u B (,) -u a ) ) -> ( ( RR _D G ) ` -u x ) e. CC )
151	150, 92	mulcomd 10061	. . . . . . . . . . 11 |- ( ( ( ph /\ ( a e. RR /\ ( A < a /\ a < B ) ) ) /\ x e. ( -u B (,) -u a ) ) -> ( ( ( RR _D G ) ` -u x ) x. -u 1 ) = ( -u 1 x. ( ( RR _D G ) ` -u x ) ) )
152	150	mulm1d 10482	. . . . . . . . . . 11 |- ( ( ( ph /\ ( a e. RR /\ ( A < a /\ a < B ) ) ) /\ x e. ( -u B (,) -u a ) ) -> ( -u 1 x. ( ( RR _D G ) ` -u x ) ) = -u ( ( RR _D G ) ` -u x ) )
153	151, 152	eqtrd 2656	. . . . . . . . . 10 |- ( ( ( ph /\ ( a e. RR /\ ( A < a /\ a < B ) ) ) /\ x e. ( -u B (,) -u a ) ) -> ( ( ( RR _D G ) ` -u x ) x. -u 1 ) = -u ( ( RR _D G ) ` -u x ) )
154	153	mpteq2dva 4744	. . . . . . . . 9 |- ( ( ph /\ ( a e. RR /\ ( A < a /\ a < B ) ) ) -> ( x e. ( -u B (,) -u a ) |-> ( ( ( RR _D G ) ` -u x ) x. -u 1 ) ) = ( x e. ( -u B (,) -u a ) |-> -u ( ( RR _D G ) ` -u x ) ) )
155	148, 154	eqtrd 2656	. . . . . . . 8 |- ( ( ph /\ ( a e. RR /\ ( A < a /\ a < B ) ) ) -> ( RR _D ( x e. ( -u B (,) -u a ) |-> ( G ` -u x ) ) ) = ( x e. ( -u B (,) -u a ) |-> -u ( ( RR _D G ) ` -u x ) ) )
156	155	dmeqd 5326	. . . . . . 7 |- ( ( ph /\ ( a e. RR /\ ( A < a /\ a < B ) ) ) -> dom ( RR _D ( x e. ( -u B (,) -u a ) |-> ( G ` -u x ) ) ) = dom ( x e. ( -u B (,) -u a ) |-> -u ( ( RR _D G ) ` -u x ) ) )
157		negex 10279	. . . . . . . 8 |- -u ( ( RR _D G ) ` -u x ) e. _V
158		eqid 2622	. . . . . . . 8 |- ( x e. ( -u B (,) -u a ) |-> -u ( ( RR _D G ) ` -u x ) ) = ( x e. ( -u B (,) -u a ) |-> -u ( ( RR _D G ) ` -u x ) )
159	157, 158	dmmpti 6023	. . . . . . 7 |- dom ( x e. ( -u B (,) -u a ) |-> -u ( ( RR _D G ) ` -u x ) ) = ( -u B (,) -u a )
160	156, 159	syl6eq 2672	. . . . . 6 |- ( ( ph /\ ( a e. RR /\ ( A < a /\ a < B ) ) ) -> dom ( RR _D ( x e. ( -u B (,) -u a ) |-> ( G ` -u x ) ) ) = ( -u B (,) -u a ) )
161	43	adantrr 753	. . . . . . 7 |- ( ( ( ph /\ ( a e. RR /\ ( A < a /\ a < B ) ) ) /\ ( x e. ( -u B (,) -u a ) /\ -u x =/= B ) ) -> -u x e. ( A (,) B ) )
162		limcresi 23649	. . . . . . . . 9 |- ( ( x e. RR |-> -u x ) lim CC -u B ) C_ ( ( ( x e. RR |-> -u x ) |` ( -u B (,) -u a ) ) lim CC -u B )
163		resmpt 5449	. . . . . . . . . . 11 |- ( ( -u B (,) -u a ) C_ RR -> ( ( x e. RR |-> -u x ) |` ( -u B (,) -u a ) ) = ( x e. ( -u B (,) -u a ) |-> -u x ) )
164	102, 163	ax-mp 5	. . . . . . . . . 10 |- ( ( x e. RR |-> -u x ) |` ( -u B (,) -u a ) ) = ( x e. ( -u B (,) -u a ) |-> -u x )
165	164	oveq1i 6660	. . . . . . . . 9 |- ( ( ( x e. RR |-> -u x ) |` ( -u B (,) -u a ) ) lim CC -u B ) = ( ( x e. ( -u B (,) -u a ) |-> -u x ) lim CC -u B )
166	162, 165	sseqtri 3637	. . . . . . . 8 |- ( ( x e. RR |-> -u x ) lim CC -u B ) C_ ( ( x e. ( -u B (,) -u a ) |-> -u x ) lim CC -u B )
167	75	recnd 10068	. . . . . . . . . 10 |- ( ( ph /\ ( a e. RR /\ ( A < a /\ a < B ) ) ) -> B e. CC )
168	167	negnegd 10383	. . . . . . . . 9 |- ( ( ph /\ ( a e. RR /\ ( A < a /\ a < B ) ) ) -> -u -u B = B )
169		eqid 2622	. . . . . . . . . . . 12 |- ( x e. RR |-> -u x ) = ( x e. RR |-> -u x )
170	169	negcncf 22721	. . . . . . . . . . 11 |- ( RR C_ CC -> ( x e. RR |-> -u x ) e. ( RR -cn-> CC ) )
171	53, 170	mp1i 13	. . . . . . . . . 10 |- ( ( ph /\ ( a e. RR /\ ( A < a /\ a < B ) ) ) -> ( x e. RR |-> -u x ) e. ( RR -cn-> CC ) )
172		negeq 10273	. . . . . . . . . 10 |- ( x = -u B -> -u x = -u -u B )
173	171, 79, 172	cnmptlimc 23654	. . . . . . . . 9 |- ( ( ph /\ ( a e. RR /\ ( A < a /\ a < B ) ) ) -> -u -u B e. ( ( x e. RR |-> -u x ) lim CC -u B ) )
174	168, 173	eqeltrrd 2702	. . . . . . . 8 |- ( ( ph /\ ( a e. RR /\ ( A < a /\ a < B ) ) ) -> B e. ( ( x e. RR |-> -u x ) lim CC -u B ) )
175	166, 174	sseldi 3601	. . . . . . 7 |- ( ( ph /\ ( a e. RR /\ ( A < a /\ a < B ) ) ) -> B e. ( ( x e. ( -u B (,) -u a ) |-> -u x ) lim CC -u B ) )
176		lhop2.f0	. . . . . . . . 9 |- ( ph -> 0 e. ( F lim CC B ) )
177	176	adantr 481	. . . . . . . 8 |- ( ( ph /\ ( a e. RR /\ ( A < a /\ a < B ) ) ) -> 0 e. ( F lim CC B ) )
178	110	oveq1d 6665	. . . . . . . 8 |- ( ( ph /\ ( a e. RR /\ ( A < a /\ a < B ) ) ) -> ( F lim CC B ) = ( ( y e. ( A (,) B ) |-> ( F ` y ) ) lim CC B ) )
179	177, 178	eleqtrd 2703	. . . . . . 7 |- ( ( ph /\ ( a e. RR /\ ( A < a /\ a < B ) ) ) -> 0 e. ( ( y e. ( A (,) B ) |-> ( F ` y ) ) lim CC B ) )
180		eliooord 12233	. . . . . . . . . . . . . 14 |- ( x e. ( -u B (,) -u a ) -> ( -u B < x /\ x < -u a ) )
181	180	adantl 482	. . . . . . . . . . . . 13 |- ( ( ( ph /\ ( a e. RR /\ ( A < a /\ a < B ) ) ) /\ x e. ( -u B (,) -u a ) ) -> ( -u B < x /\ x < -u a ) )
182	181	simpld 475	. . . . . . . . . . . 12 |- ( ( ( ph /\ ( a e. RR /\ ( A < a /\ a < B ) ) ) /\ x e. ( -u B (,) -u a ) ) -> -u B < x )
183	29, 23, 182	ltnegcon1d 10607	. . . . . . . . . . 11 |- ( ( ( ph /\ ( a e. RR /\ ( A < a /\ a < B ) ) ) /\ x e. ( -u B (,) -u a ) ) -> -u x < B )
184	30, 183	ltned 10173	. . . . . . . . . 10 |- ( ( ( ph /\ ( a e. RR /\ ( A < a /\ a < B ) ) ) /\ x e. ( -u B (,) -u a ) ) -> -u x =/= B )
185	184	neneqd 2799	. . . . . . . . 9 |- ( ( ( ph /\ ( a e. RR /\ ( A < a /\ a < B ) ) ) /\ x e. ( -u B (,) -u a ) ) -> -. -u x = B )
186	185	pm2.21d 118	. . . . . . . 8 |- ( ( ( ph /\ ( a e. RR /\ ( A < a /\ a < B ) ) ) /\ x e. ( -u B (,) -u a ) ) -> ( -u x = B -> ( F ` -u x ) = 0 ) )
187	186	impr 649	. . . . . . 7 |- ( ( ( ph /\ ( a e. RR /\ ( A < a /\ a < B ) ) ) /\ ( x e. ( -u B (,) -u a ) /\ -u x = B ) ) -> ( F ` -u x ) = 0 )
188	161, 95, 175, 179, 119, 187	limcco 23657	. . . . . 6 |- ( ( ph /\ ( a e. RR /\ ( A < a /\ a < B ) ) ) -> 0 e. ( ( x e. ( -u B (,) -u a ) |-> ( F ` -u x ) ) lim CC -u B ) )
189		lhop2.g0	. . . . . . . . 9 |- ( ph -> 0 e. ( G lim CC B ) )
190	189	adantr 481	. . . . . . . 8 |- ( ( ph /\ ( a e. RR /\ ( A < a /\ a < B ) ) ) -> 0 e. ( G lim CC B ) )
191	137	oveq1d 6665	. . . . . . . 8 |- ( ( ph /\ ( a e. RR /\ ( A < a /\ a < B ) ) ) -> ( G lim CC B ) = ( ( y e. ( A (,) B ) |-> ( G ` y ) ) lim CC B ) )
192	190, 191	eleqtrd 2703	. . . . . . 7 |- ( ( ph /\ ( a e. RR /\ ( A < a /\ a < B ) ) ) -> 0 e. ( ( y e. ( A (,) B ) |-> ( G ` y ) ) lim CC B ) )
193	185	pm2.21d 118	. . . . . . . 8 |- ( ( ( ph /\ ( a e. RR /\ ( A < a /\ a < B ) ) ) /\ x e. ( -u B (,) -u a ) ) -> ( -u x = B -> ( G ` -u x ) = 0 ) )
194	193	impr 649	. . . . . . 7 |- ( ( ( ph /\ ( a e. RR /\ ( A < a /\ a < B ) ) ) /\ ( x e. ( -u B (,) -u a ) /\ -u x = B ) ) -> ( G ` -u x ) = 0 )
195	161, 135, 175, 192, 146, 194	limcco 23657	. . . . . 6 |- ( ( ph /\ ( a e. RR /\ ( A < a /\ a < B ) ) ) -> 0 e. ( ( x e. ( -u B (,) -u a ) |-> ( G ` -u x ) ) lim CC -u B ) )
196	60, 87	fmptd 6385	. . . . . . . 8 |- ( ( ph /\ ( a e. RR /\ ( A < a /\ a < B ) ) ) -> ( x e. ( -u B (,) -u a ) |-> ( G ` -u x ) ) : ( -u B (,) -u a ) --> ran G )
197		frn 6053	. . . . . . . 8 |- ( ( x e. ( -u B (,) -u a ) |-> ( G ` -u x ) ) : ( -u B (,) -u a ) --> ran G -> ran ( x e. ( -u B (,) -u a ) |-> ( G ` -u x ) ) C_ ran G )
198	196, 197	syl 17	. . . . . . 7 |- ( ( ph /\ ( a e. RR /\ ( A < a /\ a < B ) ) ) -> ran ( x e. ( -u B (,) -u a ) |-> ( G ` -u x ) ) C_ ran G )
199	50	adantr 481	. . . . . . 7 |- ( ( ph /\ ( a e. RR /\ ( A < a /\ a < B ) ) ) -> -. 0 e. ran G )
200	198, 199	ssneldd 3606	. . . . . 6 |- ( ( ph /\ ( a e. RR /\ ( A < a /\ a < B ) ) ) -> -. 0 e. ran ( x e. ( -u B (,) -u a ) |-> ( G ` -u x ) ) )
201		lhop2.gd0	. . . . . . . 8 |- ( ph -> -. 0 e. ran ( RR _D G ) )
202	201	adantr 481	. . . . . . 7 |- ( ( ph /\ ( a e. RR /\ ( A < a /\ a < B ) ) ) -> -. 0 e. ran ( RR _D G ) )
203	155	rneqd 5353	. . . . . . . . 9 |- ( ( ph /\ ( a e. RR /\ ( A < a /\ a < B ) ) ) -> ran ( RR _D ( x e. ( -u B (,) -u a ) |-> ( G ` -u x ) ) ) = ran ( x e. ( -u B (,) -u a ) |-> -u ( ( RR _D G ) ` -u x ) ) )
204	203	eleq2d 2687	. . . . . . . 8 |- ( ( ph /\ ( a e. RR /\ ( A < a /\ a < B ) ) ) -> ( 0 e. ran ( RR _D ( x e. ( -u B (,) -u a ) |-> ( G ` -u x ) ) ) <-> 0 e. ran ( x e. ( -u B (,) -u a ) |-> -u ( ( RR _D G ) ` -u x ) ) ) )
205	158, 157	elrnmpti 5376	. . . . . . . . 9 |- ( 0 e. ran ( x e. ( -u B (,) -u a ) |-> -u ( ( RR _D G ) ` -u x ) ) <-> E. x e. ( -u B (,) -u a ) 0 = -u ( ( RR _D G ) ` -u x ) )
206		eqcom 2629	. . . . . . . . . . 11 |- ( 0 = -u ( ( RR _D G ) ` -u x ) <-> -u ( ( RR _D G ) ` -u x ) = 0 )
207	150	negeq0d 10384	. . . . . . . . . . . 12 |- ( ( ( ph /\ ( a e. RR /\ ( A < a /\ a < B ) ) ) /\ x e. ( -u B (,) -u a ) ) -> ( ( ( RR _D G ) ` -u x ) = 0 <-> -u ( ( RR _D G ) ` -u x ) = 0 ) )
208		ffn 6045	. . . . . . . . . . . . . . 15 |- ( ( RR _D G ) : ( A (,) B ) --> CC -> ( RR _D G ) Fn ( A (,) B ) )
209	149, 208	syl 17	. . . . . . . . . . . . . 14 |- ( ( ( ph /\ ( a e. RR /\ ( A < a /\ a < B ) ) ) /\ x e. ( -u B (,) -u a ) ) -> ( RR _D G ) Fn ( A (,) B ) )
210		fnfvelrn 6356	. . . . . . . . . . . . . 14 |- ( ( ( RR _D G ) Fn ( A (,) B ) /\ -u x e. ( A (,) B ) ) -> ( ( RR _D G ) ` -u x ) e. ran ( RR _D G ) )
211	209, 43, 210	syl2anc 693	. . . . . . . . . . . . 13 |- ( ( ( ph /\ ( a e. RR /\ ( A < a /\ a < B ) ) ) /\ x e. ( -u B (,) -u a ) ) -> ( ( RR _D G ) ` -u x ) e. ran ( RR _D G ) )
212		eleq1 2689	. . . . . . . . . . . . 13 |- ( ( ( RR _D G ) ` -u x ) = 0 -> ( ( ( RR _D G ) ` -u x ) e. ran ( RR _D G ) <-> 0 e. ran ( RR _D G ) ) )
213	211, 212	syl5ibcom 235	. . . . . . . . . . . 12 |- ( ( ( ph /\ ( a e. RR /\ ( A < a /\ a < B ) ) ) /\ x e. ( -u B (,) -u a ) ) -> ( ( ( RR _D G ) ` -u x ) = 0 -> 0 e. ran ( RR _D G ) ) )
214	207, 213	sylbird 250	. . . . . . . . . . 11 |- ( ( ( ph /\ ( a e. RR /\ ( A < a /\ a < B ) ) ) /\ x e. ( -u B (,) -u a ) ) -> ( -u ( ( RR _D G ) ` -u x ) = 0 -> 0 e. ran ( RR _D G ) ) )
215	206, 214	syl5bi 232	. . . . . . . . . 10 |- ( ( ( ph /\ ( a e. RR /\ ( A < a /\ a < B ) ) ) /\ x e. ( -u B (,) -u a ) ) -> ( 0 = -u ( ( RR _D G ) ` -u x ) -> 0 e. ran ( RR _D G ) ) )
216	215	rexlimdva 3031	. . . . . . . . 9 |- ( ( ph /\ ( a e. RR /\ ( A < a /\ a < B ) ) ) -> ( E. x e. ( -u B (,) -u a ) 0 = -u ( ( RR _D G ) ` -u x ) -> 0 e. ran ( RR _D G ) ) )
217	205, 216	syl5bi 232	. . . . . . . 8 |- ( ( ph /\ ( a e. RR /\ ( A < a /\ a < B ) ) ) -> ( 0 e. ran ( x e. ( -u B (,) -u a ) |-> -u ( ( RR _D G ) ` -u x ) ) -> 0 e. ran ( RR _D G ) ) )
218	204, 217	sylbid 230	. . . . . . 7 |- ( ( ph /\ ( a e. RR /\ ( A < a /\ a < B ) ) ) -> ( 0 e. ran ( RR _D ( x e. ( -u B (,) -u a ) |-> ( G ` -u x ) ) ) -> 0 e. ran ( RR _D G ) ) )
219	202, 218	mtod 189	. . . . . 6 |- ( ( ph /\ ( a e. RR /\ ( A < a /\ a < B ) ) ) -> -. 0 e. ran ( RR _D ( x e. ( -u B (,) -u a ) |-> ( G ` -u x ) ) ) )
220	116	ffvelrnda 6359	. . . . . . . . 9 |- ( ( ( ph /\ ( a e. RR /\ ( A < a /\ a < B ) ) ) /\ z e. ( A (,) B ) ) -> ( ( RR _D F ) ` z ) e. CC )
221	143	ffvelrnda 6359	. . . . . . . . 9 |- ( ( ( ph /\ ( a e. RR /\ ( A < a /\ a < B ) ) ) /\ z e. ( A (,) B ) ) -> ( ( RR _D G ) ` z ) e. CC )
222	201	ad2antrr 762	. . . . . . . . . 10 |- ( ( ( ph /\ ( a e. RR /\ ( A < a /\ a < B ) ) ) /\ z e. ( A (,) B ) ) -> -. 0 e. ran ( RR _D G ) )
223	143, 208	syl 17	. . . . . . . . . . . . 13 |- ( ( ph /\ ( a e. RR /\ ( A < a /\ a < B ) ) ) -> ( RR _D G ) Fn ( A (,) B ) )
224		fnfvelrn 6356	. . . . . . . . . . . . 13 |- ( ( ( RR _D G ) Fn ( A (,) B ) /\ z e. ( A (,) B ) ) -> ( ( RR _D G ) ` z ) e. ran ( RR _D G ) )
225	223, 224	sylan 488	. . . . . . . . . . . 12 |- ( ( ( ph /\ ( a e. RR /\ ( A < a /\ a < B ) ) ) /\ z e. ( A (,) B ) ) -> ( ( RR _D G ) ` z ) e. ran ( RR _D G ) )
226		eleq1 2689	. . . . . . . . . . . 12 |- ( ( ( RR _D G ) ` z ) = 0 -> ( ( ( RR _D G ) ` z ) e. ran ( RR _D G ) <-> 0 e. ran ( RR _D G ) ) )
227	225, 226	syl5ibcom 235	. . . . . . . . . . 11 |- ( ( ( ph /\ ( a e. RR /\ ( A < a /\ a < B ) ) ) /\ z e. ( A (,) B ) ) -> ( ( ( RR _D G ) ` z ) = 0 -> 0 e. ran ( RR _D G ) ) )
228	227	necon3bd 2808	. . . . . . . . . 10 |- ( ( ( ph /\ ( a e. RR /\ ( A < a /\ a < B ) ) ) /\ z e. ( A (,) B ) ) -> ( -. 0 e. ran ( RR _D G ) -> ( ( RR _D G ) ` z ) =/= 0 ) )
229	222, 228	mpd 15	. . . . . . . . 9 |- ( ( ( ph /\ ( a e. RR /\ ( A < a /\ a < B ) ) ) /\ z e. ( A (,) B ) ) -> ( ( RR _D G ) ` z ) =/= 0 )
230	220, 221, 229	divcld 10801	. . . . . . . 8 |- ( ( ( ph /\ ( a e. RR /\ ( A < a /\ a < B ) ) ) /\ z e. ( A (,) B ) ) -> ( ( ( RR _D F ) ` z ) / ( ( RR _D G ) ` z ) ) e. CC )
231		lhop2.c	. . . . . . . . 9 |- ( ph -> C e. ( ( z e. ( A (,) B ) |-> ( ( ( RR _D F ) ` z ) / ( ( RR _D G ) ` z ) ) ) lim CC B ) )
232	231	adantr 481	. . . . . . . 8 |- ( ( ph /\ ( a e. RR /\ ( A < a /\ a < B ) ) ) -> C e. ( ( z e. ( A (,) B ) |-> ( ( ( RR _D F ) ` z ) / ( ( RR _D G ) ` z ) ) ) lim CC B ) )
233		fveq2 6191	. . . . . . . . 9 |- ( z = -u x -> ( ( RR _D F ) ` z ) = ( ( RR _D F ) ` -u x ) )
234		fveq2 6191	. . . . . . . . 9 |- ( z = -u x -> ( ( RR _D G ) ` z ) = ( ( RR _D G ) ` -u x ) )
235	233, 234	oveq12d 6668	. . . . . . . 8 |- ( z = -u x -> ( ( ( RR _D F ) ` z ) / ( ( RR _D G ) ` z ) ) = ( ( ( RR _D F ) ` -u x ) / ( ( RR _D G ) ` -u x ) ) )
236	185	pm2.21d 118	. . . . . . . . 9 |- ( ( ( ph /\ ( a e. RR /\ ( A < a /\ a < B ) ) ) /\ x e. ( -u B (,) -u a ) ) -> ( -u x = B -> ( ( ( RR _D F ) ` -u x ) / ( ( RR _D G ) ` -u x ) ) = C ) )
237	236	impr 649	. . . . . . . 8 |- ( ( ( ph /\ ( a e. RR /\ ( A < a /\ a < B ) ) ) /\ ( x e. ( -u B (,) -u a ) /\ -u x = B ) ) -> ( ( ( RR _D F ) ` -u x ) / ( ( RR _D G ) ` -u x ) ) = C )
238	161, 230, 175, 232, 235, 237	limcco 23657	. . . . . . 7 |- ( ( ph /\ ( a e. RR /\ ( A < a /\ a < B ) ) ) -> C e. ( ( x e. ( -u B (,) -u a ) |-> ( ( ( RR _D F ) ` -u x ) / ( ( RR _D G ) ` -u x ) ) ) lim CC -u B ) )
239		nfcv 2764	. . . . . . . . . . . . 13 |- F/_ x RR
240		nfcv 2764	. . . . . . . . . . . . 13 |- F/_ x _D
241		nfmpt1 4747	. . . . . . . . . . . . 13 |- F/_ x ( x e. ( -u B (,) -u a ) |-> ( F ` -u x ) )
242	239, 240, 241	nfov 6676	. . . . . . . . . . . 12 |- F/_ x ( RR _D ( x e. ( -u B (,) -u a ) |-> ( F ` -u x ) ) )
243		nfcv 2764	. . . . . . . . . . . 12 |- F/_ x y
244	242, 243	nffv 6198	. . . . . . . . . . 11 |- F/_ x ( ( RR _D ( x e. ( -u B (,) -u a ) |-> ( F ` -u x ) ) ) ` y )
245		nfcv 2764	. . . . . . . . . . 11 |- F/_ x /
246		nfmpt1 4747	. . . . . . . . . . . . 13 |- F/_ x ( x e. ( -u B (,) -u a ) |-> ( G ` -u x ) )
247	239, 240, 246	nfov 6676	. . . . . . . . . . . 12 |- F/_ x ( RR _D ( x e. ( -u B (,) -u a ) |-> ( G ` -u x ) ) )
248	247, 243	nffv 6198	. . . . . . . . . . 11 |- F/_ x ( ( RR _D ( x e. ( -u B (,) -u a ) |-> ( G ` -u x ) ) ) ` y )
249	244, 245, 248	nfov 6676	. . . . . . . . . 10 |- F/_ x ( ( ( RR _D ( x e. ( -u B (,) -u a ) |-> ( F ` -u x ) ) ) ` y ) / ( ( RR _D ( x e. ( -u B (,) -u a ) |-> ( G ` -u x ) ) ) ` y ) )
250		nfcv 2764	. . . . . . . . . 10 |- F/_ y ( ( ( RR _D ( x e. ( -u B (,) -u a ) |-> ( F ` -u x ) ) ) ` x ) / ( ( RR _D ( x e. ( -u B (,) -u a ) |-> ( G ` -u x ) ) ) ` x ) )
251		fveq2 6191	. . . . . . . . . . 11 |- ( y = x -> ( ( RR _D ( x e. ( -u B (,) -u a ) |-> ( F ` -u x ) ) ) ` y ) = ( ( RR _D ( x e. ( -u B (,) -u a ) |-> ( F ` -u x ) ) ) ` x ) )
252		fveq2 6191	. . . . . . . . . . 11 |- ( y = x -> ( ( RR _D ( x e. ( -u B (,) -u a ) |-> ( G ` -u x ) ) ) ` y ) = ( ( RR _D ( x e. ( -u B (,) -u a ) |-> ( G ` -u x ) ) ) ` x ) )
253	251, 252	oveq12d 6668	. . . . . . . . . 10 |- ( y = x -> ( ( ( RR _D ( x e. ( -u B (,) -u a ) |-> ( F ` -u x ) ) ) ` y ) / ( ( RR _D ( x e. ( -u B (,) -u a ) |-> ( G ` -u x ) ) ) ` y ) ) = ( ( ( RR _D ( x e. ( -u B (,) -u a ) |-> ( F ` -u x ) ) ) ` x ) / ( ( RR _D ( x e. ( -u B (,) -u a ) |-> ( G ` -u x ) ) ) ` x ) ) )
254	249, 250, 253	cbvmpt 4749	. . . . . . . . 9 |- ( y e. ( -u B (,) -u a ) |-> ( ( ( RR _D ( x e. ( -u B (,) -u a ) |-> ( F ` -u x ) ) ) ` y ) / ( ( RR _D ( x e. ( -u B (,) -u a ) |-> ( G ` -u x ) ) ) ` y ) ) ) = ( x e. ( -u B (,) -u a ) |-> ( ( ( RR _D ( x e. ( -u B (,) -u a ) |-> ( F ` -u x ) ) ) ` x ) / ( ( RR _D ( x e. ( -u B (,) -u a ) |-> ( G ` -u x ) ) ) ` x ) ) )
255	128	fveq1d 6193	. . . . . . . . . . . . 13 |- ( ( ph /\ ( a e. RR /\ ( A < a /\ a < B ) ) ) -> ( ( RR _D ( x e. ( -u B (,) -u a ) |-> ( F ` -u x ) ) ) ` x ) = ( ( x e. ( -u B (,) -u a ) |-> -u ( ( RR _D F ) ` -u x ) ) ` x ) )
256	131	fvmpt2 6291	. . . . . . . . . . . . . 14 |- ( ( x e. ( -u B (,) -u a ) /\ -u ( ( RR _D F ) ` -u x ) e. _V ) -> ( ( x e. ( -u B (,) -u a ) |-> -u ( ( RR _D F ) ` -u x ) ) ` x ) = -u ( ( RR _D F ) ` -u x ) )
257	130, 256	mpan2 707	. . . . . . . . . . . . 13 |- ( x e. ( -u B (,) -u a ) -> ( ( x e. ( -u B (,) -u a ) |-> -u ( ( RR _D F ) ` -u x ) ) ` x ) = -u ( ( RR _D F ) ` -u x ) )
258	255, 257	sylan9eq 2676	. . . . . . . . . . . 12 |- ( ( ( ph /\ ( a e. RR /\ ( A < a /\ a < B ) ) ) /\ x e. ( -u B (,) -u a ) ) -> ( ( RR _D ( x e. ( -u B (,) -u a ) |-> ( F ` -u x ) ) ) ` x ) = -u ( ( RR _D F ) ` -u x ) )
259	155	fveq1d 6193	. . . . . . . . . . . . 13 |- ( ( ph /\ ( a e. RR /\ ( A < a /\ a < B ) ) ) -> ( ( RR _D ( x e. ( -u B (,) -u a ) |-> ( G ` -u x ) ) ) ` x ) = ( ( x e. ( -u B (,) -u a ) |-> -u ( ( RR _D G ) ` -u x ) ) ` x ) )
260	158	fvmpt2 6291	. . . . . . . . . . . . . 14 |- ( ( x e. ( -u B (,) -u a ) /\ -u ( ( RR _D G ) ` -u x ) e. _V ) -> ( ( x e. ( -u B (,) -u a ) |-> -u ( ( RR _D G ) ` -u x ) ) ` x ) = -u ( ( RR _D G ) ` -u x ) )
261	157, 260	mpan2 707	. . . . . . . . . . . . 13 |- ( x e. ( -u B (,) -u a ) -> ( ( x e. ( -u B (,) -u a ) |-> -u ( ( RR _D G ) ` -u x ) ) ` x ) = -u ( ( RR _D G ) ` -u x ) )
262	259, 261	sylan9eq 2676	. . . . . . . . . . . 12 |- ( ( ( ph /\ ( a e. RR /\ ( A < a /\ a < B ) ) ) /\ x e. ( -u B (,) -u a ) ) -> ( ( RR _D ( x e. ( -u B (,) -u a ) |-> ( G ` -u x ) ) ) ` x ) = -u ( ( RR _D G ) ` -u x ) )
263	258, 262	oveq12d 6668	. . . . . . . . . . 11 |- ( ( ( ph /\ ( a e. RR /\ ( A < a /\ a < B ) ) ) /\ x e. ( -u B (,) -u a ) ) -> ( ( ( RR _D ( x e. ( -u B (,) -u a ) |-> ( F ` -u x ) ) ) ` x ) / ( ( RR _D ( x e. ( -u B (,) -u a ) |-> ( G ` -u x ) ) ) ` x ) ) = ( -u ( ( RR _D F ) ` -u x ) / -u ( ( RR _D G ) ` -u x ) ) )
264	201	ad2antrr 762	. . . . . . . . . . . . 13 |- ( ( ( ph /\ ( a e. RR /\ ( A < a /\ a < B ) ) ) /\ x e. ( -u B (,) -u a ) ) -> -. 0 e. ran ( RR _D G ) )
265	213	necon3bd 2808	. . . . . . . . . . . . 13 |- ( ( ( ph /\ ( a e. RR /\ ( A < a /\ a < B ) ) ) /\ x e. ( -u B (,) -u a ) ) -> ( -. 0 e. ran ( RR _D G ) -> ( ( RR _D G ) ` -u x ) =/= 0 ) )
266	264, 265	mpd 15	. . . . . . . . . . . 12 |- ( ( ( ph /\ ( a e. RR /\ ( A < a /\ a < B ) ) ) /\ x e. ( -u B (,) -u a ) ) -> ( ( RR _D G ) ` -u x ) =/= 0 )
267	123, 150, 266	div2negd 10816	. . . . . . . . . . 11 |- ( ( ( ph /\ ( a e. RR /\ ( A < a /\ a < B ) ) ) /\ x e. ( -u B (,) -u a ) ) -> ( -u ( ( RR _D F ) ` -u x ) / -u ( ( RR _D G ) ` -u x ) ) = ( ( ( RR _D F ) ` -u x ) / ( ( RR _D G ) ` -u x ) ) )
268	263, 267	eqtrd 2656	. . . . . . . . . 10 |- ( ( ( ph /\ ( a e. RR /\ ( A < a /\ a < B ) ) ) /\ x e. ( -u B (,) -u a ) ) -> ( ( ( RR _D ( x e. ( -u B (,) -u a ) |-> ( F ` -u x ) ) ) ` x ) / ( ( RR _D ( x e. ( -u B (,) -u a ) |-> ( G ` -u x ) ) ) ` x ) ) = ( ( ( RR _D F ) ` -u x ) / ( ( RR _D G ) ` -u x ) ) )
269	268	mpteq2dva 4744	. . . . . . . . 9 |- ( ( ph /\ ( a e. RR /\ ( A < a /\ a < B ) ) ) -> ( x e. ( -u B (,) -u a ) |-> ( ( ( RR _D ( x e. ( -u B (,) -u a ) |-> ( F ` -u x ) ) ) ` x ) / ( ( RR _D ( x e. ( -u B (,) -u a ) |-> ( G ` -u x ) ) ) ` x ) ) ) = ( x e. ( -u B (,) -u a ) |-> ( ( ( RR _D F ) ` -u x ) / ( ( RR _D G ) ` -u x ) ) ) )
270	254, 269	syl5eq 2668	. . . . . . . 8 |- ( ( ph /\ ( a e. RR /\ ( A < a /\ a < B ) ) ) -> ( y e. ( -u B (,) -u a ) |-> ( ( ( RR _D ( x e. ( -u B (,) -u a ) |-> ( F ` -u x ) ) ) ` y ) / ( ( RR _D ( x e. ( -u B (,) -u a ) |-> ( G ` -u x ) ) ) ` y ) ) ) = ( x e. ( -u B (,) -u a ) |-> ( ( ( RR _D F ) ` -u x ) / ( ( RR _D G ) ` -u x ) ) ) )
271	270	oveq1d 6665	. . . . . . 7 |- ( ( ph /\ ( a e. RR /\ ( A < a /\ a < B ) ) ) -> ( ( y e. ( -u B (,) -u a ) |-> ( ( ( RR _D ( x e. ( -u B (,) -u a ) |-> ( F ` -u x ) ) ) ` y ) / ( ( RR _D ( x e. ( -u B (,) -u a ) |-> ( G ` -u x ) ) ) ` y ) ) ) lim CC -u B ) = ( ( x e. ( -u B (,) -u a ) |-> ( ( ( RR _D F ) ` -u x ) / ( ( RR _D G ) ` -u x ) ) ) lim CC -u B ) )
272	238, 271	eleqtrrd 2704	. . . . . 6 |- ( ( ph /\ ( a e. RR /\ ( A < a /\ a < B ) ) ) -> C e. ( ( y e. ( -u B (,) -u a ) |-> ( ( ( RR _D ( x e. ( -u B (,) -u a ) |-> ( F ` -u x ) ) ) ` y ) / ( ( RR _D ( x e. ( -u B (,) -u a ) |-> ( G ` -u x ) ) ) ` y ) ) ) lim CC -u B ) )
273	79, 81, 84, 86, 88, 133, 160, 188, 195, 200, 219, 272	lhop1 23777	. . . . 5 |- ( ( ph /\ ( a e. RR /\ ( A < a /\ a < B ) ) ) -> C e. ( ( y e. ( -u B (,) -u a ) |-> ( ( ( x e. ( -u B (,) -u a ) |-> ( F ` -u x ) ) ` y ) / ( ( x e. ( -u B (,) -u a ) |-> ( G ` -u x ) ) ` y ) ) ) lim CC -u B ) )
274		nffvmpt1 6199	. . . . . . . . 9 |- F/_ x ( ( x e. ( -u B (,) -u a ) |-> ( F ` -u x ) ) ` y )
275		nffvmpt1 6199	. . . . . . . . 9 |- F/_ x ( ( x e. ( -u B (,) -u a ) |-> ( G ` -u x ) ) ` y )
276	274, 245, 275	nfov 6676	. . . . . . . 8 |- F/_ x ( ( ( x e. ( -u B (,) -u a ) |-> ( F ` -u x ) ) ` y ) / ( ( x e. ( -u B (,) -u a ) |-> ( G ` -u x ) ) ` y ) )
277		nfcv 2764	. . . . . . . 8 |- F/_ y ( ( ( x e. ( -u B (,) -u a ) |-> ( F ` -u x ) ) ` x ) / ( ( x e. ( -u B (,) -u a ) |-> ( G ` -u x ) ) ` x ) )
278		fveq2 6191	. . . . . . . . 9 |- ( y = x -> ( ( x e. ( -u B (,) -u a ) |-> ( F ` -u x ) ) ` y ) = ( ( x e. ( -u B (,) -u a ) |-> ( F ` -u x ) ) ` x ) )
279		fveq2 6191	. . . . . . . . 9 |- ( y = x -> ( ( x e. ( -u B (,) -u a ) |-> ( G ` -u x ) ) ` y ) = ( ( x e. ( -u B (,) -u a ) |-> ( G ` -u x ) ) ` x ) )
280	278, 279	oveq12d 6668	. . . . . . . 8 |- ( y = x -> ( ( ( x e. ( -u B (,) -u a ) |-> ( F ` -u x ) ) ` y ) / ( ( x e. ( -u B (,) -u a ) |-> ( G ` -u x ) ) ` y ) ) = ( ( ( x e. ( -u B (,) -u a ) |-> ( F ` -u x ) ) ` x ) / ( ( x e. ( -u B (,) -u a ) |-> ( G ` -u x ) ) ` x ) ) )
281	276, 277, 280	cbvmpt 4749	. . . . . . 7 |- ( y e. ( -u B (,) -u a ) |-> ( ( ( x e. ( -u B (,) -u a ) |-> ( F ` -u x ) ) ` y ) / ( ( x e. ( -u B (,) -u a ) |-> ( G ` -u x ) ) ` y ) ) ) = ( x e. ( -u B (,) -u a ) |-> ( ( ( x e. ( -u B (,) -u a ) |-> ( F ` -u x ) ) ` x ) / ( ( x e. ( -u B (,) -u a ) |-> ( G ` -u x ) ) ` x ) ) )
282		fvex 6201	. . . . . . . . . 10 |- ( F ` -u x ) e. _V
283	85	fvmpt2 6291	. . . . . . . . . 10 |- ( ( x e. ( -u B (,) -u a ) /\ ( F ` -u x ) e. _V ) -> ( ( x e. ( -u B (,) -u a ) |-> ( F ` -u x ) ) ` x ) = ( F ` -u x ) )
284	26, 282, 283	sylancl 694	. . . . . . . . 9 |- ( ( ( ph /\ ( a e. RR /\ ( A < a /\ a < B ) ) ) /\ x e. ( -u B (,) -u a ) ) -> ( ( x e. ( -u B (,) -u a ) |-> ( F ` -u x ) ) ` x ) = ( F ` -u x ) )
285		fvex 6201	. . . . . . . . . 10 |- ( G ` -u x ) e. _V
286	87	fvmpt2 6291	. . . . . . . . . 10 |- ( ( x e. ( -u B (,) -u a ) /\ ( G ` -u x ) e. _V ) -> ( ( x e. ( -u B (,) -u a ) |-> ( G ` -u x ) ) ` x ) = ( G ` -u x ) )
287	26, 285, 286	sylancl 694	. . . . . . . . 9 |- ( ( ( ph /\ ( a e. RR /\ ( A < a /\ a < B ) ) ) /\ x e. ( -u B (,) -u a ) ) -> ( ( x e. ( -u B (,) -u a ) |-> ( G ` -u x ) ) ` x ) = ( G ` -u x ) )
288	284, 287	oveq12d 6668	. . . . . . . 8 |- ( ( ( ph /\ ( a e. RR /\ ( A < a /\ a < B ) ) ) /\ x e. ( -u B (,) -u a ) ) -> ( ( ( x e. ( -u B (,) -u a ) |-> ( F ` -u x ) ) ` x ) / ( ( x e. ( -u B (,) -u a ) |-> ( G ` -u x ) ) ` x ) ) = ( ( F ` -u x ) / ( G ` -u x ) ) )
289	288	mpteq2dva 4744	. . . . . . 7 |- ( ( ph /\ ( a e. RR /\ ( A < a /\ a < B ) ) ) -> ( x e. ( -u B (,) -u a ) |-> ( ( ( x e. ( -u B (,) -u a ) |-> ( F ` -u x ) ) ` x ) / ( ( x e. ( -u B (,) -u a ) |-> ( G ` -u x ) ) ` x ) ) ) = ( x e. ( -u B (,) -u a ) |-> ( ( F ` -u x ) / ( G ` -u x ) ) ) )
290	281, 289	syl5eq 2668	. . . . . 6 |- ( ( ph /\ ( a e. RR /\ ( A < a /\ a < B ) ) ) -> ( y e. ( -u B (,) -u a ) |-> ( ( ( x e. ( -u B (,) -u a ) |-> ( F ` -u x ) ) ` y ) / ( ( x e. ( -u B (,) -u a ) |-> ( G ` -u x ) ) ` y ) ) ) = ( x e. ( -u B (,) -u a ) |-> ( ( F ` -u x ) / ( G ` -u x ) ) ) )
291	290	oveq1d 6665	. . . . 5 |- ( ( ph /\ ( a e. RR /\ ( A < a /\ a < B ) ) ) -> ( ( y e. ( -u B (,) -u a ) |-> ( ( ( x e. ( -u B (,) -u a ) |-> ( F ` -u x ) ) ` y ) / ( ( x e. ( -u B (,) -u a ) |-> ( G ` -u x ) ) ` y ) ) ) lim CC -u B ) = ( ( x e. ( -u B (,) -u a ) |-> ( ( F ` -u x ) / ( G ` -u x ) ) ) lim CC -u B ) )
292	273, 291	eleqtrd 2703	. . . 4 |- ( ( ph /\ ( a e. RR /\ ( A < a /\ a < B ) ) ) -> C e. ( ( x e. ( -u B (,) -u a ) |-> ( ( F ` -u x ) / ( G ` -u x ) ) ) lim CC -u B ) )
293		negeq 10273	. . . . . 6 |- ( x = -u z -> -u x = -u -u z )
294	293	fveq2d 6195	. . . . 5 |- ( x = -u z -> ( F ` -u x ) = ( F ` -u -u z ) )
295	293	fveq2d 6195	. . . . 5 |- ( x = -u z -> ( G ` -u x ) = ( G ` -u -u z ) )
296	294, 295	oveq12d 6668	. . . 4 |- ( x = -u z -> ( ( F ` -u x ) / ( G ` -u x ) ) = ( ( F ` -u -u z ) / ( G ` -u -u z ) ) )
297	79	adantr 481	. . . . . . . 8 |- ( ( ( ph /\ ( a e. RR /\ ( A < a /\ a < B ) ) ) /\ z e. ( a (,) B ) ) -> -u B e. RR )
298		eliooord 12233	. . . . . . . . . . 11 |- ( z e. ( a (,) B ) -> ( a < z /\ z < B ) )
299	298	adantl 482	. . . . . . . . . 10 |- ( ( ( ph /\ ( a e. RR /\ ( A < a /\ a < B ) ) ) /\ z e. ( a (,) B ) ) -> ( a < z /\ z < B ) )
300	299	simprd 479	. . . . . . . . 9 |- ( ( ( ph /\ ( a e. RR /\ ( A < a /\ a < B ) ) ) /\ z e. ( a (,) B ) ) -> z < B )
301	15, 13	ltnegd 10605	. . . . . . . . 9 |- ( ( ( ph /\ ( a e. RR /\ ( A < a /\ a < B ) ) ) /\ z e. ( a (,) B ) ) -> ( z < B <-> -u B < -u z ) )
302	300, 301	mpbid 222	. . . . . . . 8 |- ( ( ( ph /\ ( a e. RR /\ ( A < a /\ a < B ) ) ) /\ z e. ( a (,) B ) ) -> -u B < -u z )
303	297, 302	gtned 10172	. . . . . . 7 |- ( ( ( ph /\ ( a e. RR /\ ( A < a /\ a < B ) ) ) /\ z e. ( a (,) B ) ) -> -u z =/= -u B )
304	303	neneqd 2799	. . . . . 6 |- ( ( ( ph /\ ( a e. RR /\ ( A < a /\ a < B ) ) ) /\ z e. ( a (,) B ) ) -> -. -u z = -u B )
305	304	pm2.21d 118	. . . . 5 |- ( ( ( ph /\ ( a e. RR /\ ( A < a /\ a < B ) ) ) /\ z e. ( a (,) B ) ) -> ( -u z = -u B -> ( ( F ` -u -u z ) / ( G ` -u -u z ) ) = C ) )
306	305	impr 649	. . . 4 |- ( ( ( ph /\ ( a e. RR /\ ( A < a /\ a < B ) ) ) /\ ( z e. ( a (,) B ) /\ -u z = -u B ) ) -> ( ( F ` -u -u z ) / ( G ` -u -u z ) ) = C )
307	19, 65, 78, 292, 296, 306	limcco 23657	. . 3 |- ( ( ph /\ ( a e. RR /\ ( A < a /\ a < B ) ) ) -> C e. ( ( z e. ( a (,) B ) |-> ( ( F ` -u -u z ) / ( G ` -u -u z ) ) ) lim CC B ) )
308	15	recnd 10068	. . . . . . . . 9 |- ( ( ( ph /\ ( a e. RR /\ ( A < a /\ a < B ) ) ) /\ z e. ( a (,) B ) ) -> z e. CC )
309	308	negnegd 10383	. . . . . . . 8 |- ( ( ( ph /\ ( a e. RR /\ ( A < a /\ a < B ) ) ) /\ z e. ( a (,) B ) ) -> -u -u z = z )
310	309	fveq2d 6195	. . . . . . 7 |- ( ( ( ph /\ ( a e. RR /\ ( A < a /\ a < B ) ) ) /\ z e. ( a (,) B ) ) -> ( F ` -u -u z ) = ( F ` z ) )
311	309	fveq2d 6195	. . . . . . 7 |- ( ( ( ph /\ ( a e. RR /\ ( A < a /\ a < B ) ) ) /\ z e. ( a (,) B ) ) -> ( G ` -u -u z ) = ( G ` z ) )
312	310, 311	oveq12d 6668	. . . . . 6 |- ( ( ( ph /\ ( a e. RR /\ ( A < a /\ a < B ) ) ) /\ z e. ( a (,) B ) ) -> ( ( F ` -u -u z ) / ( G ` -u -u z ) ) = ( ( F ` z ) / ( G ` z ) ) )
313	312	mpteq2dva 4744	. . . . 5 |- ( ( ph /\ ( a e. RR /\ ( A < a /\ a < B ) ) ) -> ( z e. ( a (,) B ) |-> ( ( F ` -u -u z ) / ( G ` -u -u z ) ) ) = ( z e. ( a (,) B ) |-> ( ( F ` z ) / ( G ` z ) ) ) )
314	313	oveq1d 6665	. . . 4 |- ( ( ph /\ ( a e. RR /\ ( A < a /\ a < B ) ) ) -> ( ( z e. ( a (,) B ) |-> ( ( F ` -u -u z ) / ( G ` -u -u z ) ) ) lim CC B ) = ( ( z e. ( a (,) B ) |-> ( ( F ` z ) / ( G ` z ) ) ) lim CC B ) )
315	41	resmptd 5452	. . . . 5 |- ( ( ph /\ ( a e. RR /\ ( A < a /\ a < B ) ) ) -> ( ( z e. ( A (,) B ) |-> ( ( F ` z ) / ( G ` z ) ) ) |` ( a (,) B ) ) = ( z e. ( a (,) B ) |-> ( ( F ` z ) / ( G ` z ) ) ) )
316	315	oveq1d 6665	. . . 4 |- ( ( ph /\ ( a e. RR /\ ( A < a /\ a < B ) ) ) -> ( ( ( z e. ( A (,) B ) |-> ( ( F ` z ) / ( G ` z ) ) ) |` ( a (,) B ) ) lim CC B ) = ( ( z e. ( a (,) B ) |-> ( ( F ` z ) / ( G ` z ) ) ) lim CC B ) )
317		fss 6056	. . . . . . . . 9 |- ( ( F : ( A (,) B ) --> RR /\ RR C_ CC ) -> F : ( A (,) B ) --> CC )
318	93, 53, 317	sylancl 694	. . . . . . . 8 |- ( ( ph /\ ( a e. RR /\ ( A < a /\ a < B ) ) ) -> F : ( A (,) B ) --> CC )
319	318	ffvelrnda 6359	. . . . . . 7 |- ( ( ( ph /\ ( a e. RR /\ ( A < a /\ a < B ) ) ) /\ z e. ( A (,) B ) ) -> ( F ` z ) e. CC )
320	55	ffvelrnda 6359	. . . . . . 7 |- ( ( ( ph /\ ( a e. RR /\ ( A < a /\ a < B ) ) ) /\ z e. ( A (,) B ) ) -> ( G ` z ) e. CC )
321	50	ad2antrr 762	. . . . . . . 8 |- ( ( ( ph /\ ( a e. RR /\ ( A < a /\ a < B ) ) ) /\ z e. ( A (,) B ) ) -> -. 0 e. ran G )
322	55, 57	syl 17	. . . . . . . . . . 11 |- ( ( ph /\ ( a e. RR /\ ( A < a /\ a < B ) ) ) -> G Fn ( A (,) B ) )
323		fnfvelrn 6356	. . . . . . . . . . 11 |- ( ( G Fn ( A (,) B ) /\ z e. ( A (,) B ) ) -> ( G ` z ) e. ran G )
324	322, 323	sylan 488	. . . . . . . . . 10 |- ( ( ( ph /\ ( a e. RR /\ ( A < a /\ a < B ) ) ) /\ z e. ( A (,) B ) ) -> ( G ` z ) e. ran G )
325		eleq1 2689	. . . . . . . . . 10 |- ( ( G ` z ) = 0 -> ( ( G ` z ) e. ran G <-> 0 e. ran G ) )
326	324, 325	syl5ibcom 235	. . . . . . . . 9 |- ( ( ( ph /\ ( a e. RR /\ ( A < a /\ a < B ) ) ) /\ z e. ( A (,) B ) ) -> ( ( G ` z ) = 0 -> 0 e. ran G ) )
327	326	necon3bd 2808	. . . . . . . 8 |- ( ( ( ph /\ ( a e. RR /\ ( A < a /\ a < B ) ) ) /\ z e. ( A (,) B ) ) -> ( -. 0 e. ran G -> ( G ` z ) =/= 0 ) )
328	321, 327	mpd 15	. . . . . . 7 |- ( ( ( ph /\ ( a e. RR /\ ( A < a /\ a < B ) ) ) /\ z e. ( A (,) B ) ) -> ( G ` z ) =/= 0 )
329	319, 320, 328	divcld 10801	. . . . . 6 |- ( ( ( ph /\ ( a e. RR /\ ( A < a /\ a < B ) ) ) /\ z e. ( A (,) B ) ) -> ( ( F ` z ) / ( G ` z ) ) e. CC )
330		eqid 2622	. . . . . 6 |- ( z e. ( A (,) B ) |-> ( ( F ` z ) / ( G ` z ) ) ) = ( z e. ( A (,) B ) |-> ( ( F ` z ) / ( G ` z ) ) )
331	329, 330	fmptd 6385	. . . . 5 |- ( ( ph /\ ( a e. RR /\ ( A < a /\ a < B ) ) ) -> ( z e. ( A (,) B ) |-> ( ( F ` z ) / ( G ` z ) ) ) : ( A (,) B ) --> CC )
332		ioossre 12235	. . . . . . 7 |- ( A (,) B ) C_ RR
333	332, 53	sstri 3612	. . . . . 6 |- ( A (,) B ) C_ CC
334	333	a1i 11	. . . . 5 |- ( ( ph /\ ( a e. RR /\ ( A < a /\ a < B ) ) ) -> ( A (,) B ) C_ CC )
335		eqid 2622	. . . . 5 |- ( ( TopOpen ` ℂfld ) ↾t ( ( A (,) B ) u. { B } ) ) = ( ( TopOpen ` ℂfld ) ↾t ( ( A (,) B ) u. { B } ) )
336		ssun2 3777	. . . . . . 7 |- { B } C_ ( ( a (,) B ) u. { B } )
337		snssg 4327	. . . . . . . 8 |- ( B e. RR -> ( B e. ( ( a (,) B ) u. { B } ) <-> { B } C_ ( ( a (,) B ) u. { B } ) ) )
338	75, 337	syl 17	. . . . . . 7 |- ( ( ph /\ ( a e. RR /\ ( A < a /\ a < B ) ) ) -> ( B e. ( ( a (,) B ) u. { B } ) <-> { B } C_ ( ( a (,) B ) u. { B } ) ) )
339	336, 338	mpbiri 248	. . . . . 6 |- ( ( ph /\ ( a e. RR /\ ( A < a /\ a < B ) ) ) -> B e. ( ( a (,) B ) u. { B } ) )
340	104	cnfldtopon 22586	. . . . . . . . 9 |- ( TopOpen ` ℂfld ) e. (TopOn ` CC )
341	332	a1i 11	. . . . . . . . . . 11 |- ( ( ph /\ ( a e. RR /\ ( A < a /\ a < B ) ) ) -> ( A (,) B ) C_ RR )
342	75	snssd 4340	. . . . . . . . . . 11 |- ( ( ph /\ ( a e. RR /\ ( A < a /\ a < B ) ) ) -> { B } C_ RR )
343	341, 342	unssd 3789	. . . . . . . . . 10 |- ( ( ph /\ ( a e. RR /\ ( A < a /\ a < B ) ) ) -> ( ( A (,) B ) u. { B } ) C_ RR )
344	343, 53	syl6ss 3615	. . . . . . . . 9 |- ( ( ph /\ ( a e. RR /\ ( A < a /\ a < B ) ) ) -> ( ( A (,) B ) u. { B } ) C_ CC )
345		resttopon 20965	. . . . . . . . 9 |- ( ( ( TopOpen ` ℂfld ) e. (TopOn ` CC ) /\ ( ( A (,) B ) u. { B } ) C_ CC ) -> ( ( TopOpen ` ℂfld ) ↾t ( ( A (,) B ) u. { B } ) ) e. (TopOn ` ( ( A (,) B ) u. { B } ) ) )
346	340, 344, 345	sylancr 695	. . . . . . . 8 |- ( ( ph /\ ( a e. RR /\ ( A < a /\ a < B ) ) ) -> ( ( TopOpen ` ℂfld ) ↾t ( ( A (,) B ) u. { B } ) ) e. (TopOn ` ( ( A (,) B ) u. { B } ) ) )
347		topontop 20718	. . . . . . . 8 |- ( ( ( TopOpen ` ℂfld ) ↾t ( ( A (,) B ) u. { B } ) ) e. (TopOn ` ( ( A (,) B ) u. { B } ) ) -> ( ( TopOpen ` ℂfld ) ↾t ( ( A (,) B ) u. { B } ) ) e. Top )
348	346, 347	syl 17	. . . . . . 7 |- ( ( ph /\ ( a e. RR /\ ( A < a /\ a < B ) ) ) -> ( ( TopOpen ` ℂfld ) ↾t ( ( A (,) B ) u. { B } ) ) e. Top )
349		indi 3873	. . . . . . . . . 10 |- ( ( a (,) +oo ) i^i ( ( A (,) B ) u. { B } ) ) = ( ( ( a (,) +oo ) i^i ( A (,) B ) ) u. ( ( a (,) +oo ) i^i { B } ) )
350		pnfxr 10092	. . . . . . . . . . . . . 14 |- +oo e. RR*
351	350	a1i 11	. . . . . . . . . . . . 13 |- ( ( ph /\ ( a e. RR /\ ( A < a /\ a < B ) ) ) -> +oo e. RR* )
352	4	adantr 481	. . . . . . . . . . . . 13 |- ( ( ph /\ ( a e. RR /\ ( A < a /\ a < B ) ) ) -> B e. RR* )
353		iooin 12209	. . . . . . . . . . . . 13 |- ( ( ( a e. RR* /\ +oo e. RR* ) /\ ( A e. RR* /\ B e. RR* ) ) -> ( ( a (,) +oo ) i^i ( A (,) B ) ) = ( if ( a <_ A , A , a ) (,) if ( +oo <_ B , +oo , B ) ) )
354	36, 351, 34, 352, 353	syl22anc 1327	. . . . . . . . . . . 12 |- ( ( ph /\ ( a e. RR /\ ( A < a /\ a < B ) ) ) -> ( ( a (,) +oo ) i^i ( A (,) B ) ) = ( if ( a <_ A , A , a ) (,) if ( +oo <_ B , +oo , B ) ) )
355		xrltnle 10105	. . . . . . . . . . . . . . . 16 |- ( ( A e. RR* /\ a e. RR* ) -> ( A < a <-> -. a <_ A ) )
356	34, 36, 355	syl2anc 693	. . . . . . . . . . . . . . 15 |- ( ( ph /\ ( a e. RR /\ ( A < a /\ a < B ) ) ) -> ( A < a <-> -. a <_ A ) )
357	35, 356	mpbid 222	. . . . . . . . . . . . . 14 |- ( ( ph /\ ( a e. RR /\ ( A < a /\ a < B ) ) ) -> -. a <_ A )
358	357	iffalsed 4097	. . . . . . . . . . . . 13 |- ( ( ph /\ ( a e. RR /\ ( A < a /\ a < B ) ) ) -> if ( a <_ A , A , a ) = a )
359		ltpnf 11954	. . . . . . . . . . . . . . . 16 |- ( B e. RR -> B < +oo )
360	75, 359	syl 17	. . . . . . . . . . . . . . 15 |- ( ( ph /\ ( a e. RR /\ ( A < a /\ a < B ) ) ) -> B < +oo )
361		xrltnle 10105	. . . . . . . . . . . . . . . 16 |- ( ( B e. RR* /\ +oo e. RR* ) -> ( B < +oo <-> -. +oo <_ B ) )
362	352, 350, 361	sylancl 694	. . . . . . . . . . . . . . 15 |- ( ( ph /\ ( a e. RR /\ ( A < a /\ a < B ) ) ) -> ( B < +oo <-> -. +oo <_ B ) )
363	360, 362	mpbid 222	. . . . . . . . . . . . . 14 |- ( ( ph /\ ( a e. RR /\ ( A < a /\ a < B ) ) ) -> -. +oo <_ B )
364	363	iffalsed 4097	. . . . . . . . . . . . 13 |- ( ( ph /\ ( a e. RR /\ ( A < a /\ a < B ) ) ) -> if ( +oo <_ B , +oo , B ) = B )
365	358, 364	oveq12d 6668	. . . . . . . . . . . 12 |- ( ( ph /\ ( a e. RR /\ ( A < a /\ a < B ) ) ) -> ( if ( a <_ A , A , a ) (,) if ( +oo <_ B , +oo , B ) ) = ( a (,) B ) )
366	354, 365	eqtrd 2656	. . . . . . . . . . 11 |- ( ( ph /\ ( a e. RR /\ ( A < a /\ a < B ) ) ) -> ( ( a (,) +oo ) i^i ( A (,) B ) ) = ( a (,) B ) )
367		elioopnf 12267	. . . . . . . . . . . . . . 15 |- ( a e. RR* -> ( B e. ( a (,) +oo ) <-> ( B e. RR /\ a < B ) ) )
368	36, 367	syl 17	. . . . . . . . . . . . . 14 |- ( ( ph /\ ( a e. RR /\ ( A < a /\ a < B ) ) ) -> ( B e. ( a (,) +oo ) <-> ( B e. RR /\ a < B ) ) )
369	75, 82, 368	mpbir2and 957	. . . . . . . . . . . . 13 |- ( ( ph /\ ( a e. RR /\ ( A < a /\ a < B ) ) ) -> B e. ( a (,) +oo ) )
370	369	snssd 4340	. . . . . . . . . . . 12 |- ( ( ph /\ ( a e. RR /\ ( A < a /\ a < B ) ) ) -> { B } C_ ( a (,) +oo ) )
371		sseqin2 3817	. . . . . . . . . . . 12 |- ( { B } C_ ( a (,) +oo ) <-> ( ( a (,) +oo ) i^i { B } ) = { B } )
372	370, 371	sylib 208	. . . . . . . . . . 11 |- ( ( ph /\ ( a e. RR /\ ( A < a /\ a < B ) ) ) -> ( ( a (,) +oo ) i^i { B } ) = { B } )
373	366, 372	uneq12d 3768	. . . . . . . . . 10 |- ( ( ph /\ ( a e. RR /\ ( A < a /\ a < B ) ) ) -> ( ( ( a (,) +oo ) i^i ( A (,) B ) ) u. ( ( a (,) +oo ) i^i { B } ) ) = ( ( a (,) B ) u. { B } ) )
374	349, 373	syl5eq 2668	. . . . . . . . 9 |- ( ( ph /\ ( a e. RR /\ ( A < a /\ a < B ) ) ) -> ( ( a (,) +oo ) i^i ( ( A (,) B ) u. { B } ) ) = ( ( a (,) B ) u. { B } ) )
375		retop 22565	. . . . . . . . . . 11 |- ( topGen ` ran (,) ) e. Top
376	375	a1i 11	. . . . . . . . . 10 |- ( ( ph /\ ( a e. RR /\ ( A < a /\ a < B ) ) ) -> ( topGen ` ran (,) ) e. Top )
377		reex 10027	. . . . . . . . . . . 12 |- RR e. _V
378	377	ssex 4802	. . . . . . . . . . 11 |- ( ( ( A (,) B ) u. { B } ) C_ RR -> ( ( A (,) B ) u. { B } ) e. _V )
379	343, 378	syl 17	. . . . . . . . . 10 |- ( ( ph /\ ( a e. RR /\ ( A < a /\ a < B ) ) ) -> ( ( A (,) B ) u. { B } ) e. _V )
380		iooretop 22569	. . . . . . . . . . 11 |- ( a (,) +oo ) e. ( topGen ` ran (,) )
381	380	a1i 11	. . . . . . . . . 10 |- ( ( ph /\ ( a e. RR /\ ( A < a /\ a < B ) ) ) -> ( a (,) +oo ) e. ( topGen ` ran (,) ) )
382		elrestr 16089	. . . . . . . . . 10 |- ( ( ( topGen ` ran (,) ) e. Top /\ ( ( A (,) B ) u. { B } ) e. _V /\ ( a (,) +oo ) e. ( topGen ` ran (,) ) ) -> ( ( a (,) +oo ) i^i ( ( A (,) B ) u. { B } ) ) e. ( ( topGen ` ran (,) ) ↾t ( ( A (,) B ) u. { B } ) ) )
383	376, 379, 381, 382	syl3anc 1326	. . . . . . . . 9 |- ( ( ph /\ ( a e. RR /\ ( A < a /\ a < B ) ) ) -> ( ( a (,) +oo ) i^i ( ( A (,) B ) u. { B } ) ) e. ( ( topGen ` ran (,) ) ↾t ( ( A (,) B ) u. { B } ) ) )
384	374, 383	eqeltrrd 2702	. . . . . . . 8 |- ( ( ph /\ ( a e. RR /\ ( A < a /\ a < B ) ) ) -> ( ( a (,) B ) u. { B } ) e. ( ( topGen ` ran (,) ) ↾t ( ( A (,) B ) u. { B } ) ) )
385		eqid 2622	. . . . . . . . . 10 |- ( topGen ` ran (,) ) = ( topGen ` ran (,) )
386	104, 385	rerest 22607	. . . . . . . . 9 |- ( ( ( A (,) B ) u. { B } ) C_ RR -> ( ( TopOpen ` ℂfld ) ↾t ( ( A (,) B ) u. { B } ) ) = ( ( topGen ` ran (,) ) ↾t ( ( A (,) B ) u. { B } ) ) )
387	343, 386	syl 17	. . . . . . . 8 |- ( ( ph /\ ( a e. RR /\ ( A < a /\ a < B ) ) ) -> ( ( TopOpen ` ℂfld ) ↾t ( ( A (,) B ) u. { B } ) ) = ( ( topGen ` ran (,) ) ↾t ( ( A (,) B ) u. { B } ) ) )
388	384, 387	eleqtrrd 2704	. . . . . . 7 |- ( ( ph /\ ( a e. RR /\ ( A < a /\ a < B ) ) ) -> ( ( a (,) B ) u. { B } ) e. ( ( TopOpen ` ℂfld ) ↾t ( ( A (,) B ) u. { B } ) ) )
389		isopn3i 20886	. . . . . . 7 |- ( ( ( ( TopOpen ` ℂfld ) ↾t ( ( A (,) B ) u. { B } ) ) e. Top /\ ( ( a (,) B ) u. { B } ) e. ( ( TopOpen ` ℂfld ) ↾t ( ( A (,) B ) u. { B } ) ) ) -> ( ( int ` ( ( TopOpen ` ℂfld ) ↾t ( ( A (,) B ) u. { B } ) ) ) ` ( ( a (,) B ) u. { B } ) ) = ( ( a (,) B ) u. { B } ) )
390	348, 388, 389	syl2anc 693	. . . . . 6 |- ( ( ph /\ ( a e. RR /\ ( A < a /\ a < B ) ) ) -> ( ( int ` ( ( TopOpen ` ℂfld ) ↾t ( ( A (,) B ) u. { B } ) ) ) ` ( ( a (,) B ) u. { B } ) ) = ( ( a (,) B ) u. { B } ) )
391	339, 390	eleqtrrd 2704	. . . . 5 |- ( ( ph /\ ( a e. RR /\ ( A < a /\ a < B ) ) ) -> B e. ( ( int ` ( ( TopOpen ` ℂfld ) ↾t ( ( A (,) B ) u. { B } ) ) ) ` ( ( a (,) B ) u. { B } ) ) )
392	331, 41, 334, 104, 335, 391	limcres 23650	. . . 4 |- ( ( ph /\ ( a e. RR /\ ( A < a /\ a < B ) ) ) -> ( ( ( z e. ( A (,) B ) |-> ( ( F ` z ) / ( G ` z ) ) ) |` ( a (,) B ) ) lim CC B ) = ( ( z e. ( A (,) B ) |-> ( ( F ` z ) / ( G ` z ) ) ) lim CC B ) )
393	314, 316, 392	3eqtr2d 2662	. . 3 |- ( ( ph /\ ( a e. RR /\ ( A < a /\ a < B ) ) ) -> ( ( z e. ( a (,) B ) |-> ( ( F ` -u -u z ) / ( G ` -u -u z ) ) ) lim CC B ) = ( ( z e. ( A (,) B ) |-> ( ( F ` z ) / ( G ` z ) ) ) lim CC B ) )
394	307, 393	eleqtrd 2703	. 2 |- ( ( ph /\ ( a e. RR /\ ( A < a /\ a < B ) ) ) -> C e. ( ( z e. ( A (,) B ) |-> ( ( F ` z ) / ( G ` z ) ) ) lim CC B ) )
395	9, 394	rexlimddv 3035	1 |- ( ph -> C e. ( ( z e. ( A (,) B ) |-> ( ( F ` z ) / ( G ` z ) ) ) lim CC B ) )

Syntax hints:   -. wn 3   -> wi 4   <-> wb 196   /\ wa 384   = wceq 1483   e. wcel 1990   =/= wne 2794   E.wrex 2913   _Vcvv 3200   u. cun 3572   i^i cin 3573   C_ wss 3574   ifcif 4086   {csn 4177   {cpr 4179   class class class wbr 4653   |-> cmpt 4729   dom cdm 5114   ran crn 5115   |` cres 5116   Fn wfn 5883   -->wf 5884   ` cfv 5888  (class class class)co 6650   CCcc 9934   RRcr 9935   0cc0 9936   1c1 9937   x. cmul 9941   +oocpnf 10071   RR*cxr 10073   < clt 10074   <_ cle 10075   -ucneg 10267   / cdiv 10684   QQcq 11788   (,)cioo 12175   ↾_t crest 16081   TopOpenctopn 16082   topGenctg 16098  ℂ_fldccnfld 19746   Topctop 20698  TopOnctopon 20715   intcnt 20821   -cn->ccncf 22679   lim CC climc 23626   _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-ioc 12180  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:  lhop  23779  fourierdlem60  40383