Theorem rlimcnp2 24693

Description: Relate a limit of a real-valued sequence at infinity to the continuity of the function S ( y ) = R ( 1 / y ) at zero. (Contributed by Mario Carneiro, 1-Mar-2015.)

Hypotheses

Ref	Expression
rlimcnp2.a	|- ( ph -> A C_ ( 0 [,) +oo ) )
rlimcnp2.0	|- ( ph -> 0 e. A )
rlimcnp2.b	|- ( ph -> B C_ RR )
rlimcnp2.c	|- ( ph -> C e. CC )
rlimcnp2.r	|- ( ( ph /\ y e. B ) -> S e. CC )
rlimcnp2.d	|- ( ( ph /\ y e. RR+ ) -> ( y e. B <-> ( 1 / y ) e. A ) )
rlimcnp2.s	|- ( y = ( 1 / x ) -> S = R )
rlimcnp2.j	|- J = ( TopOpen ` ℂfld )
rlimcnp2.k	|- K = ( J ↾t A )

Assertion

Ref	Expression
rlimcnp2	|- ( ph -> ( ( y e. B |-> S ) ~~> r C <-> ( x e. A |-> if ( x = 0 , C , R ) ) e. ( ( K CnP J ) ` 0 ) ) )

Distinct variable groups:   x, y, A   x, B, y   x, C, y   ph, x, y   y, R   x, S

Allowed substitution hints:   R( x)   S( y)   J( x, y)   K( x, y)

Proof of Theorem rlimcnp2

Dummy variables w z are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		inss1 3833	. . . . . . . 8 |- ( B i^i ( 1 [,) +oo ) ) C_ B
2		resmpt 5449	. . . . . . . 8 |- ( ( B i^i ( 1 [,) +oo ) ) C_ B -> ( ( y e. B |-> S ) |` ( B i^i ( 1 [,) +oo ) ) ) = ( y e. ( B i^i ( 1 [,) +oo ) ) |-> S ) )
3	1, 2	mp1i 13	. . . . . . 7 |- ( ph -> ( ( y e. B |-> S ) |` ( B i^i ( 1 [,) +oo ) ) ) = ( y e. ( B i^i ( 1 [,) +oo ) ) |-> S ) )
4		0xr 10086	. . . . . . . . . . 11 |- 0 e. RR*
5		0lt1 10550	. . . . . . . . . . 11 |- 0 < 1
6		df-ioo 12179	. . . . . . . . . . . 12 |- (,) = ( x e. RR* , y e. RR* |-> { z e. RR* | ( x < z /\ z < y ) } )
7		df-ico 12181	. . . . . . . . . . . 12 |- [,) = ( x e. RR* , y e. RR* |-> { z e. RR* | ( x <_ z /\ z < y ) } )
8		xrltletr 11988	. . . . . . . . . . . 12 |- ( ( 0 e. RR* /\ 1 e. RR* /\ w e. RR* ) -> ( ( 0 < 1 /\ 1 <_ w ) -> 0 < w ) )
9	6, 7, 8	ixxss1 12193	. . . . . . . . . . 11 |- ( ( 0 e. RR* /\ 0 < 1 ) -> ( 1 [,) +oo ) C_ ( 0 (,) +oo ) )
10	4, 5, 9	mp2an 708	. . . . . . . . . 10 |- ( 1 [,) +oo ) C_ ( 0 (,) +oo )
11		ioorp 12251	. . . . . . . . . 10 |- ( 0 (,) +oo ) = RR+
12	10, 11	sseqtri 3637	. . . . . . . . 9 |- ( 1 [,) +oo ) C_ RR+
13		sslin 3839	. . . . . . . . 9 |- ( ( 1 [,) +oo ) C_ RR+ -> ( B i^i ( 1 [,) +oo ) ) C_ ( B i^i RR+ ) )
14	12, 13	ax-mp 5	. . . . . . . 8 |- ( B i^i ( 1 [,) +oo ) ) C_ ( B i^i RR+ )
15		resmpt 5449	. . . . . . . 8 |- ( ( B i^i ( 1 [,) +oo ) ) C_ ( B i^i RR+ ) -> ( ( y e. ( B i^i RR+ ) |-> S ) |` ( B i^i ( 1 [,) +oo ) ) ) = ( y e. ( B i^i ( 1 [,) +oo ) ) |-> S ) )
16	14, 15	mp1i 13	. . . . . . 7 |- ( ph -> ( ( y e. ( B i^i RR+ ) |-> S ) |` ( B i^i ( 1 [,) +oo ) ) ) = ( y e. ( B i^i ( 1 [,) +oo ) ) |-> S ) )
17	3, 16	eqtr4d 2659	. . . . . 6 |- ( ph -> ( ( y e. B |-> S ) |` ( B i^i ( 1 [,) +oo ) ) ) = ( ( y e. ( B i^i RR+ ) |-> S ) |` ( B i^i ( 1 [,) +oo ) ) ) )
18		resres 5409	. . . . . 6 |- ( ( ( y e. B |-> S ) |` B ) |` ( 1 [,) +oo ) ) = ( ( y e. B |-> S ) |` ( B i^i ( 1 [,) +oo ) ) )
19		resres 5409	. . . . . 6 |- ( ( ( y e. ( B i^i RR+ ) |-> S ) |` B ) |` ( 1 [,) +oo ) ) = ( ( y e. ( B i^i RR+ ) |-> S ) |` ( B i^i ( 1 [,) +oo ) ) )
20	17, 18, 19	3eqtr4g 2681	. . . . 5 |- ( ph -> ( ( ( y e. B |-> S ) |` B ) |` ( 1 [,) +oo ) ) = ( ( ( y e. ( B i^i RR+ ) |-> S ) |` B ) |` ( 1 [,) +oo ) ) )
21		rlimcnp2.r	. . . . . . . . 9 |- ( ( ph /\ y e. B ) -> S e. CC )
22		eqid 2622	. . . . . . . . 9 |- ( y e. B |-> S ) = ( y e. B |-> S )
23	21, 22	fmptd 6385	. . . . . . . 8 |- ( ph -> ( y e. B |-> S ) : B --> CC )
24		ffn 6045	. . . . . . . 8 |- ( ( y e. B |-> S ) : B --> CC -> ( y e. B |-> S ) Fn B )
25	23, 24	syl 17	. . . . . . 7 |- ( ph -> ( y e. B |-> S ) Fn B )
26		fnresdm 6000	. . . . . . 7 |- ( ( y e. B |-> S ) Fn B -> ( ( y e. B |-> S ) |` B ) = ( y e. B |-> S ) )
27	25, 26	syl 17	. . . . . 6 |- ( ph -> ( ( y e. B |-> S ) |` B ) = ( y e. B |-> S ) )
28	27	reseq1d 5395	. . . . 5 |- ( ph -> ( ( ( y e. B |-> S ) |` B ) |` ( 1 [,) +oo ) ) = ( ( y e. B |-> S ) |` ( 1 [,) +oo ) ) )
29		inss1 3833	. . . . . . . . . . 11 |- ( B i^i RR+ ) C_ B
30	29	sseli 3599	. . . . . . . . . 10 |- ( y e. ( B i^i RR+ ) -> y e. B )
31	30, 21	sylan2 491	. . . . . . . . 9 |- ( ( ph /\ y e. ( B i^i RR+ ) ) -> S e. CC )
32		eqid 2622	. . . . . . . . 9 |- ( y e. ( B i^i RR+ ) |-> S ) = ( y e. ( B i^i RR+ ) |-> S )
33	31, 32	fmptd 6385	. . . . . . . 8 |- ( ph -> ( y e. ( B i^i RR+ ) |-> S ) : ( B i^i RR+ ) --> CC )
34		frel 6050	. . . . . . . 8 |- ( ( y e. ( B i^i RR+ ) |-> S ) : ( B i^i RR+ ) --> CC -> Rel ( y e. ( B i^i RR+ ) |-> S ) )
35	33, 34	syl 17	. . . . . . 7 |- ( ph -> Rel ( y e. ( B i^i RR+ ) |-> S ) )
36	32, 31	dmmptd 6024	. . . . . . . 8 |- ( ph -> dom ( y e. ( B i^i RR+ ) |-> S ) = ( B i^i RR+ ) )
37	36, 29	syl6eqss 3655	. . . . . . 7 |- ( ph -> dom ( y e. ( B i^i RR+ ) |-> S ) C_ B )
38		relssres 5437	. . . . . . 7 |- ( ( Rel ( y e. ( B i^i RR+ ) |-> S ) /\ dom ( y e. ( B i^i RR+ ) |-> S ) C_ B ) -> ( ( y e. ( B i^i RR+ ) |-> S ) |` B ) = ( y e. ( B i^i RR+ ) |-> S ) )
39	35, 37, 38	syl2anc 693	. . . . . 6 |- ( ph -> ( ( y e. ( B i^i RR+ ) |-> S ) |` B ) = ( y e. ( B i^i RR+ ) |-> S ) )
40	39	reseq1d 5395	. . . . 5 |- ( ph -> ( ( ( y e. ( B i^i RR+ ) |-> S ) |` B ) |` ( 1 [,) +oo ) ) = ( ( y e. ( B i^i RR+ ) |-> S ) |` ( 1 [,) +oo ) ) )
41	20, 28, 40	3eqtr3d 2664	. . . 4 |- ( ph -> ( ( y e. B |-> S ) |` ( 1 [,) +oo ) ) = ( ( y e. ( B i^i RR+ ) |-> S ) |` ( 1 [,) +oo ) ) )
42	41	breq1d 4663	. . 3 |- ( ph -> ( ( ( y e. B |-> S ) |` ( 1 [,) +oo ) ) ~~> r C <-> ( ( y e. ( B i^i RR+ ) |-> S ) |` ( 1 [,) +oo ) ) ~~> r C ) )
43		rlimcnp2.b	. . . 4 |- ( ph -> B C_ RR )
44		1red 10055	. . . 4 |- ( ph -> 1 e. RR )
45	23, 43, 44	rlimresb 14296	. . 3 |- ( ph -> ( ( y e. B |-> S ) ~~> r C <-> ( ( y e. B |-> S ) |` ( 1 [,) +oo ) ) ~~> r C ) )
46	29, 43	syl5ss 3614	. . . 4 |- ( ph -> ( B i^i RR+ ) C_ RR )
47	33, 46, 44	rlimresb 14296	. . 3 |- ( ph -> ( ( y e. ( B i^i RR+ ) |-> S ) ~~> r C <-> ( ( y e. ( B i^i RR+ ) |-> S ) |` ( 1 [,) +oo ) ) ~~> r C ) )
48	42, 45, 47	3bitr4d 300	. 2 |- ( ph -> ( ( y e. B |-> S ) ~~> r C <-> ( y e. ( B i^i RR+ ) |-> S ) ~~> r C ) )
49		inss2 3834	. . . . . . . . . . 11 |- ( B i^i RR+ ) C_ RR+
50	49	a1i 11	. . . . . . . . . 10 |- ( ph -> ( B i^i RR+ ) C_ RR+ )
51	50	sselda 3603	. . . . . . . . 9 |- ( ( ph /\ y e. ( B i^i RR+ ) ) -> y e. RR+ )
52	51	rpreccld 11882	. . . . . . . 8 |- ( ( ph /\ y e. ( B i^i RR+ ) ) -> ( 1 / y ) e. RR+ )
53	52	rpne0d 11877	. . . . . . 7 |- ( ( ph /\ y e. ( B i^i RR+ ) ) -> ( 1 / y ) =/= 0 )
54	53	neneqd 2799	. . . . . 6 |- ( ( ph /\ y e. ( B i^i RR+ ) ) -> -. ( 1 / y ) = 0 )
55	54	iffalsed 4097	. . . . 5 |- ( ( ph /\ y e. ( B i^i RR+ ) ) -> if ( ( 1 / y ) = 0 , C , [_ ( 1 / y ) / x ]_ R ) = [_ ( 1 / y ) / x ]_ R )
56		oveq2 6658	. . . . . . . . . 10 |- ( x = ( 1 / y ) -> ( 1 / x ) = ( 1 / ( 1 / y ) ) )
57		rpcnne0 11850	. . . . . . . . . . 11 |- ( y e. RR+ -> ( y e. CC /\ y =/= 0 ) )
58		recrec 10722	. . . . . . . . . . 11 |- ( ( y e. CC /\ y =/= 0 ) -> ( 1 / ( 1 / y ) ) = y )
59	51, 57, 58	3syl 18	. . . . . . . . . 10 |- ( ( ph /\ y e. ( B i^i RR+ ) ) -> ( 1 / ( 1 / y ) ) = y )
60	56, 59	sylan9eqr 2678	. . . . . . . . 9 |- ( ( ( ph /\ y e. ( B i^i RR+ ) ) /\ x = ( 1 / y ) ) -> ( 1 / x ) = y )
61	60	eqcomd 2628	. . . . . . . 8 |- ( ( ( ph /\ y e. ( B i^i RR+ ) ) /\ x = ( 1 / y ) ) -> y = ( 1 / x ) )
62		rlimcnp2.s	. . . . . . . 8 |- ( y = ( 1 / x ) -> S = R )
63	61, 62	syl 17	. . . . . . 7 |- ( ( ( ph /\ y e. ( B i^i RR+ ) ) /\ x = ( 1 / y ) ) -> S = R )
64	63	eqcomd 2628	. . . . . 6 |- ( ( ( ph /\ y e. ( B i^i RR+ ) ) /\ x = ( 1 / y ) ) -> R = S )
65	52, 64	csbied 3560	. . . . 5 |- ( ( ph /\ y e. ( B i^i RR+ ) ) -> [_ ( 1 / y ) / x ]_ R = S )
66	55, 65	eqtrd 2656	. . . 4 |- ( ( ph /\ y e. ( B i^i RR+ ) ) -> if ( ( 1 / y ) = 0 , C , [_ ( 1 / y ) / x ]_ R ) = S )
67	66	mpteq2dva 4744	. . 3 |- ( ph -> ( y e. ( B i^i RR+ ) |-> if ( ( 1 / y ) = 0 , C , [_ ( 1 / y ) / x ]_ R ) ) = ( y e. ( B i^i RR+ ) |-> S ) )
68	67	breq1d 4663	. 2 |- ( ph -> ( ( y e. ( B i^i RR+ ) |-> if ( ( 1 / y ) = 0 , C , [_ ( 1 / y ) / x ]_ R ) ) ~~> r C <-> ( y e. ( B i^i RR+ ) |-> S ) ~~> r C ) )
69		rlimcnp2.a	. . . 4 |- ( ph -> A C_ ( 0 [,) +oo ) )
70		rlimcnp2.0	. . . 4 |- ( ph -> 0 e. A )
71		rlimcnp2.c	. . . . . 6 |- ( ph -> C e. CC )
72	71	ad2antrr 762	. . . . 5 |- ( ( ( ph /\ w e. A ) /\ w = 0 ) -> C e. CC )
73	69	sselda 3603	. . . . . . . . . . . 12 |- ( ( ph /\ w e. A ) -> w e. ( 0 [,) +oo ) )
74		0re 10040	. . . . . . . . . . . . 13 |- 0 e. RR
75		pnfxr 10092	. . . . . . . . . . . . 13 |- +oo e. RR*
76		elico2 12237	. . . . . . . . . . . . 13 |- ( ( 0 e. RR /\ +oo e. RR* ) -> ( w e. ( 0 [,) +oo ) <-> ( w e. RR /\ 0 <_ w /\ w < +oo ) ) )
77	74, 75, 76	mp2an 708	. . . . . . . . . . . 12 |- ( w e. ( 0 [,) +oo ) <-> ( w e. RR /\ 0 <_ w /\ w < +oo ) )
78	73, 77	sylib 208	. . . . . . . . . . 11 |- ( ( ph /\ w e. A ) -> ( w e. RR /\ 0 <_ w /\ w < +oo ) )
79	78	simp1d 1073	. . . . . . . . . 10 |- ( ( ph /\ w e. A ) -> w e. RR )
80	79	adantr 481	. . . . . . . . 9 |- ( ( ( ph /\ w e. A ) /\ -. w = 0 ) -> w e. RR )
81	78	simp2d 1074	. . . . . . . . . . . . . 14 |- ( ( ph /\ w e. A ) -> 0 <_ w )
82		leloe 10124	. . . . . . . . . . . . . . 15 |- ( ( 0 e. RR /\ w e. RR ) -> ( 0 <_ w <-> ( 0 < w \/ 0 = w ) ) )
83	74, 79, 82	sylancr 695	. . . . . . . . . . . . . 14 |- ( ( ph /\ w e. A ) -> ( 0 <_ w <-> ( 0 < w \/ 0 = w ) ) )
84	81, 83	mpbid 222	. . . . . . . . . . . . 13 |- ( ( ph /\ w e. A ) -> ( 0 < w \/ 0 = w ) )
85	84	ord 392	. . . . . . . . . . . 12 |- ( ( ph /\ w e. A ) -> ( -. 0 < w -> 0 = w ) )
86		eqcom 2629	. . . . . . . . . . . 12 |- ( 0 = w <-> w = 0 )
87	85, 86	syl6ib 241	. . . . . . . . . . 11 |- ( ( ph /\ w e. A ) -> ( -. 0 < w -> w = 0 ) )
88	87	con1d 139	. . . . . . . . . 10 |- ( ( ph /\ w e. A ) -> ( -. w = 0 -> 0 < w ) )
89	88	imp 445	. . . . . . . . 9 |- ( ( ( ph /\ w e. A ) /\ -. w = 0 ) -> 0 < w )
90	80, 89	elrpd 11869	. . . . . . . 8 |- ( ( ( ph /\ w e. A ) /\ -. w = 0 ) -> w e. RR+ )
91		rpcnne0 11850	. . . . . . . . 9 |- ( w e. RR+ -> ( w e. CC /\ w =/= 0 ) )
92		recrec 10722	. . . . . . . . 9 |- ( ( w e. CC /\ w =/= 0 ) -> ( 1 / ( 1 / w ) ) = w )
93	91, 92	syl 17	. . . . . . . 8 |- ( w e. RR+ -> ( 1 / ( 1 / w ) ) = w )
94	90, 93	syl 17	. . . . . . 7 |- ( ( ( ph /\ w e. A ) /\ -. w = 0 ) -> ( 1 / ( 1 / w ) ) = w )
95	94	csbeq1d 3540	. . . . . 6 |- ( ( ( ph /\ w e. A ) /\ -. w = 0 ) -> [_ ( 1 / ( 1 / w ) ) / x ]_ R = [_ w / x ]_ R )
96		simplr 792	. . . . . . . . 9 |- ( ( ( ph /\ w e. A ) /\ -. w = 0 ) -> w e. A )
97		simpll 790	. . . . . . . . . 10 |- ( ( ( ph /\ w e. A ) /\ -. w = 0 ) -> ph )
98		rpreccl 11857	. . . . . . . . . . . . 13 |- ( w e. RR+ -> ( 1 / w ) e. RR+ )
99	98	adantl 482	. . . . . . . . . . . 12 |- ( ( ph /\ w e. RR+ ) -> ( 1 / w ) e. RR+ )
100		rlimcnp2.d	. . . . . . . . . . . . . 14 |- ( ( ph /\ y e. RR+ ) -> ( y e. B <-> ( 1 / y ) e. A ) )
101	100	ralrimiva 2966	. . . . . . . . . . . . 13 |- ( ph -> A. y e. RR+ ( y e. B <-> ( 1 / y ) e. A ) )
102	101	adantr 481	. . . . . . . . . . . 12 |- ( ( ph /\ w e. RR+ ) -> A. y e. RR+ ( y e. B <-> ( 1 / y ) e. A ) )
103		eleq1 2689	. . . . . . . . . . . . . 14 |- ( y = ( 1 / w ) -> ( y e. B <-> ( 1 / w ) e. B ) )
104		oveq2 6658	. . . . . . . . . . . . . . 15 |- ( y = ( 1 / w ) -> ( 1 / y ) = ( 1 / ( 1 / w ) ) )
105	104	eleq1d 2686	. . . . . . . . . . . . . 14 |- ( y = ( 1 / w ) -> ( ( 1 / y ) e. A <-> ( 1 / ( 1 / w ) ) e. A ) )
106	103, 105	bibi12d 335	. . . . . . . . . . . . 13 |- ( y = ( 1 / w ) -> ( ( y e. B <-> ( 1 / y ) e. A ) <-> ( ( 1 / w ) e. B <-> ( 1 / ( 1 / w ) ) e. A ) ) )
107	106	rspcv 3305	. . . . . . . . . . . 12 |- ( ( 1 / w ) e. RR+ -> ( A. y e. RR+ ( y e. B <-> ( 1 / y ) e. A ) -> ( ( 1 / w ) e. B <-> ( 1 / ( 1 / w ) ) e. A ) ) )
108	99, 102, 107	sylc 65	. . . . . . . . . . 11 |- ( ( ph /\ w e. RR+ ) -> ( ( 1 / w ) e. B <-> ( 1 / ( 1 / w ) ) e. A ) )
109	93	adantl 482	. . . . . . . . . . . 12 |- ( ( ph /\ w e. RR+ ) -> ( 1 / ( 1 / w ) ) = w )
110	109	eleq1d 2686	. . . . . . . . . . 11 |- ( ( ph /\ w e. RR+ ) -> ( ( 1 / ( 1 / w ) ) e. A <-> w e. A ) )
111	108, 110	bitr2d 269	. . . . . . . . . 10 |- ( ( ph /\ w e. RR+ ) -> ( w e. A <-> ( 1 / w ) e. B ) )
112	97, 90, 111	syl2anc 693	. . . . . . . . 9 |- ( ( ( ph /\ w e. A ) /\ -. w = 0 ) -> ( w e. A <-> ( 1 / w ) e. B ) )
113	96, 112	mpbid 222	. . . . . . . 8 |- ( ( ( ph /\ w e. A ) /\ -. w = 0 ) -> ( 1 / w ) e. B )
114	90	rpreccld 11882	. . . . . . . 8 |- ( ( ( ph /\ w e. A ) /\ -. w = 0 ) -> ( 1 / w ) e. RR+ )
115	113, 114	elind 3798	. . . . . . 7 |- ( ( ( ph /\ w e. A ) /\ -. w = 0 ) -> ( 1 / w ) e. ( B i^i RR+ ) )
116	65, 31	eqeltrd 2701	. . . . . . . . 9 |- ( ( ph /\ y e. ( B i^i RR+ ) ) -> [_ ( 1 / y ) / x ]_ R e. CC )
117	116	ralrimiva 2966	. . . . . . . 8 |- ( ph -> A. y e. ( B i^i RR+ ) [_ ( 1 / y ) / x ]_ R e. CC )
118	117	ad2antrr 762	. . . . . . 7 |- ( ( ( ph /\ w e. A ) /\ -. w = 0 ) -> A. y e. ( B i^i RR+ ) [_ ( 1 / y ) / x ]_ R e. CC )
119	104	csbeq1d 3540	. . . . . . . . 9 |- ( y = ( 1 / w ) -> [_ ( 1 / y ) / x ]_ R = [_ ( 1 / ( 1 / w ) ) / x ]_ R )
120	119	eleq1d 2686	. . . . . . . 8 |- ( y = ( 1 / w ) -> ( [_ ( 1 / y ) / x ]_ R e. CC <-> [_ ( 1 / ( 1 / w ) ) / x ]_ R e. CC ) )
121	120	rspcv 3305	. . . . . . 7 |- ( ( 1 / w ) e. ( B i^i RR+ ) -> ( A. y e. ( B i^i RR+ ) [_ ( 1 / y ) / x ]_ R e. CC -> [_ ( 1 / ( 1 / w ) ) / x ]_ R e. CC ) )
122	115, 118, 121	sylc 65	. . . . . 6 |- ( ( ( ph /\ w e. A ) /\ -. w = 0 ) -> [_ ( 1 / ( 1 / w ) ) / x ]_ R e. CC )
123	95, 122	eqeltrrd 2702	. . . . 5 |- ( ( ( ph /\ w e. A ) /\ -. w = 0 ) -> [_ w / x ]_ R e. CC )
124	72, 123	ifclda 4120	. . . 4 |- ( ( ph /\ w e. A ) -> if ( w = 0 , C , [_ w / x ]_ R ) e. CC )
125	99	biantrud 528	. . . . . 6 |- ( ( ph /\ w e. RR+ ) -> ( ( 1 / w ) e. B <-> ( ( 1 / w ) e. B /\ ( 1 / w ) e. RR+ ) ) )
126	111, 125	bitrd 268	. . . . 5 |- ( ( ph /\ w e. RR+ ) -> ( w e. A <-> ( ( 1 / w ) e. B /\ ( 1 / w ) e. RR+ ) ) )
127		elin 3796	. . . . 5 |- ( ( 1 / w ) e. ( B i^i RR+ ) <-> ( ( 1 / w ) e. B /\ ( 1 / w ) e. RR+ ) )
128	126, 127	syl6bbr 278	. . . 4 |- ( ( ph /\ w e. RR+ ) -> ( w e. A <-> ( 1 / w ) e. ( B i^i RR+ ) ) )
129		iftrue 4092	. . . 4 |- ( w = 0 -> if ( w = 0 , C , [_ w / x ]_ R ) = C )
130		eqeq1 2626	. . . . 5 |- ( w = ( 1 / y ) -> ( w = 0 <-> ( 1 / y ) = 0 ) )
131		csbeq1 3536	. . . . 5 |- ( w = ( 1 / y ) -> [_ w / x ]_ R = [_ ( 1 / y ) / x ]_ R )
132	130, 131	ifbieq2d 4111	. . . 4 |- ( w = ( 1 / y ) -> if ( w = 0 , C , [_ w / x ]_ R ) = if ( ( 1 / y ) = 0 , C , [_ ( 1 / y ) / x ]_ R ) )
133		rlimcnp2.j	. . . 4 |- J = ( TopOpen ` ℂfld )
134		rlimcnp2.k	. . . 4 |- K = ( J ↾t A )
135	69, 70, 50, 124, 128, 129, 132, 133, 134	rlimcnp 24692	. . 3 |- ( ph -> ( ( y e. ( B i^i RR+ ) |-> if ( ( 1 / y ) = 0 , C , [_ ( 1 / y ) / x ]_ R ) ) ~~> r C <-> ( w e. A |-> if ( w = 0 , C , [_ w / x ]_ R ) ) e. ( ( K CnP J ) ` 0 ) ) )
136		nfcv 2764	. . . . 5 |- F/_ w if ( x = 0 , C , R )
137		nfv 1843	. . . . . 6 |- F/ x w = 0
138		nfcv 2764	. . . . . 6 |- F/_ x C
139		nfcsb1v 3549	. . . . . 6 |- F/_ x [_ w / x ]_ R
140	137, 138, 139	nfif 4115	. . . . 5 |- F/_ x if ( w = 0 , C , [_ w / x ]_ R )
141		eqeq1 2626	. . . . . 6 |- ( x = w -> ( x = 0 <-> w = 0 ) )
142		csbeq1a 3542	. . . . . 6 |- ( x = w -> R = [_ w / x ]_ R )
143	141, 142	ifbieq2d 4111	. . . . 5 |- ( x = w -> if ( x = 0 , C , R ) = if ( w = 0 , C , [_ w / x ]_ R ) )
144	136, 140, 143	cbvmpt 4749	. . . 4 |- ( x e. A |-> if ( x = 0 , C , R ) ) = ( w e. A |-> if ( w = 0 , C , [_ w / x ]_ R ) )
145	144	eleq1i 2692	. . 3 |- ( ( x e. A |-> if ( x = 0 , C , R ) ) e. ( ( K CnP J ) ` 0 ) <-> ( w e. A |-> if ( w = 0 , C , [_ w / x ]_ R ) ) e. ( ( K CnP J ) ` 0 ) )
146	135, 145	syl6bbr 278	. 2 |- ( ph -> ( ( y e. ( B i^i RR+ ) |-> if ( ( 1 / y ) = 0 , C , [_ ( 1 / y ) / x ]_ R ) ) ~~> r C <-> ( x e. A |-> if ( x = 0 , C , R ) ) e. ( ( K CnP J ) ` 0 ) ) )
147	48, 68, 146	3bitr2d 296	1 |- ( ph -> ( ( y e. B |-> S ) ~~> r C <-> ( x e. A |-> if ( x = 0 , C , R ) ) e. ( ( K CnP J ) ` 0 ) ) )

Syntax hints:   -. wn 3   -> wi 4   <-> wb 196   \/ wo 383   /\ wa 384   /\ w3a 1037   = wceq 1483   e. wcel 1990   =/= wne 2794   A.wral 2912   [_csb 3533   i^i cin 3573   C_ wss 3574   ifcif 4086   class class class wbr 4653   |-> cmpt 4729   dom cdm 5114   |` cres 5116   Rel wrel 5119   Fn wfn 5883   -->wf 5884   ` cfv 5888  (class class class)co 6650   CCcc 9934   RRcr 9935   0cc0 9936   1c1 9937   +oocpnf 10071   RR*cxr 10073   < clt 10074   <_ cle 10075   / cdiv 10684   RR+crp 11832   (,)cioo 12175   [,)cico 12177   ~~> r crli 14216   ↾_t crest 16081   TopOpenctopn 16082  ℂ_fldccnfld 19746   CnP ccnp 21029

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-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

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-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-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-riota 6611  df-ov 6653  df-oprab 6654  df-mpt2 6655  df-om 7066  df-1st 7168  df-2nd 7169  df-wrecs 7407  df-recs 7468  df-rdg 7506  df-1o 7560  df-oadd 7564  df-er 7742  df-map 7859  df-pm 7860  df-en 7956  df-dom 7957  df-sdom 7958  df-fin 7959  df-sup 8348  df-inf 8349  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-fz 12327  df-seq 12802  df-exp 12861  df-cj 13839  df-re 13840  df-im 13841  df-sqrt 13975  df-abs 13976  df-rlim 14220  df-struct 15859  df-ndx 15860  df-slot 15861  df-base 15863  df-plusg 15954  df-mulr 15955  df-starv 15956  df-tset 15960  df-ple 15961  df-ds 15964  df-unif 15965  df-rest 16083  df-topn 16084  df-topgen 16104  df-psmet 19738  df-xmet 19739  df-met 19740  df-bl 19741  df-mopn 19742  df-cnfld 19747  df-top 20699  df-topon 20716  df-bases 20750  df-cnp 21032

This theorem is referenced by:  rlimcnp3  24694