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Theorem rlimcnp2 20672
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 ` fld )
rlimcnp2.k  |-  K  =  ( Jt  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.
StepHypRef Expression
1 inss1 3504 . . . . . . . 8  |-  ( B  i^i  ( 1 [,) 
+oo ) )  C_  B
2 resmpt 5131 . . . . . . . 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 ) )
31, 2mp1i 12 . . . . . . 7  |-  ( ph  ->  ( ( y  e.  B  |->  S )  |`  ( B  i^i  (
1 [,)  +oo ) ) )  =  ( y  e.  ( B  i^i  ( 1 [,)  +oo ) )  |->  S ) )
4 0xr 9064 . . . . . . . . . . 11  |-  0  e.  RR*
5 0lt1 9482 . . . . . . . . . . 11  |-  0  <  1
6 df-ioo 10852 . . . . . . . . . . . 12  |-  (,)  =  ( x  e.  RR* ,  y  e.  RR*  |->  { z  e.  RR*  |  (
x  <  z  /\  z  <  y ) } )
7 df-ico 10854 . . . . . . . . . . . 12  |-  [,)  =  ( x  e.  RR* ,  y  e.  RR*  |->  { z  e.  RR*  |  (
x  <_  z  /\  z  <  y ) } )
8 xrltletr 10679 . . . . . . . . . . . 12  |-  ( ( 0  e.  RR*  /\  1  e.  RR*  /\  w  e. 
RR* )  ->  (
( 0  <  1  /\  1  <_  w )  ->  0  <  w
) )
96, 7, 8ixxss1 10866 . . . . . . . . . . 11  |-  ( ( 0  e.  RR*  /\  0  <  1 )  ->  (
1 [,)  +oo )  C_  ( 0 (,)  +oo ) )
104, 5, 9mp2an 654 . . . . . . . . . 10  |-  ( 1 [,)  +oo )  C_  (
0 (,)  +oo )
11 ioorp 10920 . . . . . . . . . 10  |-  ( 0 (,)  +oo )  =  RR+
1210, 11sseqtri 3323 . . . . . . . . 9  |-  ( 1 [,)  +oo )  C_  RR+
13 sslin 3510 . . . . . . . . 9  |-  ( ( 1 [,)  +oo )  C_  RR+  ->  ( B  i^i  ( 1 [,)  +oo ) )  C_  ( B  i^i  RR+ ) )
1412, 13ax-mp 8 . . . . . . . 8  |-  ( B  i^i  ( 1 [,) 
+oo ) )  C_  ( B  i^i  RR+ )
15 resmpt 5131 . . . . . . . 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 ) )
1614, 15mp1i 12 . . . . . . 7  |-  ( ph  ->  ( ( y  e.  ( B  i^i  RR+ )  |->  S )  |`  ( B  i^i  ( 1 [,) 
+oo ) ) )  =  ( y  e.  ( B  i^i  (
1 [,)  +oo ) ) 
|->  S ) )
173, 16eqtr4d 2422 . . . . . 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 5099 . . . . . 6  |-  ( ( ( y  e.  B  |->  S )  |`  B )  |`  ( 1 [,)  +oo ) )  =  ( ( y  e.  B  |->  S )  |`  ( B  i^i  ( 1 [,) 
+oo ) ) )
19 resres 5099 . . . . . 6  |-  ( ( ( y  e.  ( B  i^i  RR+ )  |->  S )  |`  B )  |`  ( 1 [,)  +oo ) )  =  ( ( y  e.  ( B  i^i  RR+ )  |->  S )  |`  ( B  i^i  ( 1 [,) 
+oo ) ) )
2017, 18, 193eqtr4g 2444 . . . . 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 2387 . . . . . . . . 9  |-  ( y  e.  B  |->  S )  =  ( y  e.  B  |->  S )
2321, 22fmptd 5832 . . . . . . . 8  |-  ( ph  ->  ( y  e.  B  |->  S ) : B --> CC )
24 ffn 5531 . . . . . . . 8  |-  ( ( y  e.  B  |->  S ) : B --> CC  ->  ( y  e.  B  |->  S )  Fn  B )
2523, 24syl 16 . . . . . . 7  |-  ( ph  ->  ( y  e.  B  |->  S )  Fn  B
)
26 fnresdm 5494 . . . . . . 7  |-  ( ( y  e.  B  |->  S )  Fn  B  -> 
( ( y  e.  B  |->  S )  |`  B )  =  ( y  e.  B  |->  S ) )
2725, 26syl 16 . . . . . 6  |-  ( ph  ->  ( ( y  e.  B  |->  S )  |`  B )  =  ( y  e.  B  |->  S ) )
2827reseq1d 5085 . . . . 5  |-  ( ph  ->  ( ( ( y  e.  B  |->  S )  |`  B )  |`  (
1 [,)  +oo ) )  =  ( ( y  e.  B  |->  S )  |`  ( 1 [,)  +oo ) ) )
29 inss1 3504 . . . . . . . . . . 11  |-  ( B  i^i  RR+ )  C_  B
3029sseli 3287 . . . . . . . . . 10  |-  ( y  e.  ( B  i^i  RR+ )  ->  y  e.  B )
3130, 21sylan2 461 . . . . . . . . 9  |-  ( (
ph  /\  y  e.  ( B  i^i  RR+ )
)  ->  S  e.  CC )
32 eqid 2387 . . . . . . . . 9  |-  ( y  e.  ( B  i^i  RR+ )  |->  S )  =  ( y  e.  ( B  i^i  RR+ )  |->  S )
3331, 32fmptd 5832 . . . . . . . 8  |-  ( ph  ->  ( y  e.  ( B  i^i  RR+ )  |->  S ) : ( B  i^i  RR+ ) --> CC )
34 frel 5534 . . . . . . . 8  |-  ( ( y  e.  ( B  i^i  RR+ )  |->  S ) : ( B  i^i  RR+ ) --> CC  ->  Rel  ( y  e.  ( B  i^i  RR+ )  |->  S ) )
3533, 34syl 16 . . . . . . 7  |-  ( ph  ->  Rel  ( y  e.  ( B  i^i  RR+ )  |->  S ) )
36 fdm 5535 . . . . . . . . 9  |-  ( ( y  e.  ( B  i^i  RR+ )  |->  S ) : ( B  i^i  RR+ ) --> CC  ->  dom  ( y  e.  ( B  i^i  RR+ )  |->  S )  =  ( B  i^i  RR+ )
)
3733, 36syl 16 . . . . . . . 8  |-  ( ph  ->  dom  ( y  e.  ( B  i^i  RR+ )  |->  S )  =  ( B  i^i  RR+ )
)
3837, 29syl6eqss 3341 . . . . . . 7  |-  ( ph  ->  dom  ( y  e.  ( B  i^i  RR+ )  |->  S )  C_  B
)
39 relssres 5123 . . . . . . 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 ) )
4035, 38, 39syl2anc 643 . . . . . 6  |-  ( ph  ->  ( ( y  e.  ( B  i^i  RR+ )  |->  S )  |`  B )  =  ( y  e.  ( B  i^i  RR+ )  |->  S ) )
4140reseq1d 5085 . . . . 5  |-  ( ph  ->  ( ( ( y  e.  ( B  i^i  RR+ )  |->  S )  |`  B )  |`  (
1 [,)  +oo ) )  =  ( ( y  e.  ( B  i^i  RR+ )  |->  S )  |`  ( 1 [,)  +oo ) ) )
4220, 28, 413eqtr3d 2427 . . . 4  |-  ( ph  ->  ( ( y  e.  B  |->  S )  |`  ( 1 [,)  +oo ) )  =  ( ( y  e.  ( B  i^i  RR+ )  |->  S )  |`  (
1 [,)  +oo ) ) )
4342breq1d 4163 . . 3  |-  ( ph  ->  ( ( ( y  e.  B  |->  S )  |`  ( 1 [,)  +oo ) )  ~~> r  C  <->  ( ( y  e.  ( B  i^i  RR+ )  |->  S )  |`  (
1 [,)  +oo ) )  ~~> r  C ) )
44 rlimcnp2.b . . . 4  |-  ( ph  ->  B  C_  RR )
45 1re 9023 . . . . 5  |-  1  e.  RR
4645a1i 11 . . . 4  |-  ( ph  ->  1  e.  RR )
4723, 44, 46rlimresb 12286 . . 3  |-  ( ph  ->  ( ( y  e.  B  |->  S )  ~~> r  C  <->  ( ( y  e.  B  |->  S )  |`  (
1 [,)  +oo ) )  ~~> r  C ) )
4829, 44syl5ss 3302 . . . 4  |-  ( ph  ->  ( B  i^i  RR+ )  C_  RR )
4933, 48, 46rlimresb 12286 . . 3  |-  ( ph  ->  ( ( y  e.  ( B  i^i  RR+ )  |->  S )  ~~> r  C  <->  ( ( y  e.  ( B  i^i  RR+ )  |->  S )  |`  (
1 [,)  +oo ) )  ~~> r  C ) )
5043, 47, 493bitr4d 277 . 2  |-  ( ph  ->  ( ( y  e.  B  |->  S )  ~~> r  C  <->  ( y  e.  ( B  i^i  RR+ )  |->  S )  ~~> r  C ) )
51 inss2 3505 . . . . . . . . . . 11  |-  ( B  i^i  RR+ )  C_  RR+
5251a1i 11 . . . . . . . . . 10  |-  ( ph  ->  ( B  i^i  RR+ )  C_  RR+ )
5352sselda 3291 . . . . . . . . 9  |-  ( (
ph  /\  y  e.  ( B  i^i  RR+ )
)  ->  y  e.  RR+ )
5453rpreccld 10590 . . . . . . . 8  |-  ( (
ph  /\  y  e.  ( B  i^i  RR+ )
)  ->  ( 1  /  y )  e.  RR+ )
5554rpne0d 10585 . . . . . . 7  |-  ( (
ph  /\  y  e.  ( B  i^i  RR+ )
)  ->  ( 1  /  y )  =/=  0 )
5655neneqd 2566 . . . . . 6  |-  ( (
ph  /\  y  e.  ( B  i^i  RR+ )
)  ->  -.  (
1  /  y )  =  0 )
57 iffalse 3689 . . . . . 6  |-  ( -.  ( 1  /  y
)  =  0  ->  if ( ( 1  / 
y )  =  0 ,  C ,  [_ ( 1  /  y
)  /  x ]_ R )  =  [_ ( 1  /  y
)  /  x ]_ R )
5856, 57syl 16 . . . . 5  |-  ( (
ph  /\  y  e.  ( B  i^i  RR+ )
)  ->  if (
( 1  /  y
)  =  0 ,  C ,  [_ (
1  /  y )  /  x ]_ R
)  =  [_ (
1  /  y )  /  x ]_ R
)
59 oveq2 6028 . . . . . . . . . 10  |-  ( x  =  ( 1  / 
y )  ->  (
1  /  x )  =  ( 1  / 
( 1  /  y
) ) )
60 rpcnne0 10561 . . . . . . . . . . 11  |-  ( y  e.  RR+  ->  ( y  e.  CC  /\  y  =/=  0 ) )
61 recrec 9643 . . . . . . . . . . 11  |-  ( ( y  e.  CC  /\  y  =/=  0 )  -> 
( 1  /  (
1  /  y ) )  =  y )
6253, 60, 613syl 19 . . . . . . . . . 10  |-  ( (
ph  /\  y  e.  ( B  i^i  RR+ )
)  ->  ( 1  /  ( 1  / 
y ) )  =  y )
6359, 62sylan9eqr 2441 . . . . . . . . 9  |-  ( ( ( ph  /\  y  e.  ( B  i^i  RR+ )
)  /\  x  =  ( 1  /  y
) )  ->  (
1  /  x )  =  y )
6463eqcomd 2392 . . . . . . . 8  |-  ( ( ( ph  /\  y  e.  ( B  i^i  RR+ )
)  /\  x  =  ( 1  /  y
) )  ->  y  =  ( 1  /  x ) )
65 rlimcnp2.s . . . . . . . 8  |-  ( y  =  ( 1  /  x )  ->  S  =  R )
6664, 65syl 16 . . . . . . 7  |-  ( ( ( ph  /\  y  e.  ( B  i^i  RR+ )
)  /\  x  =  ( 1  /  y
) )  ->  S  =  R )
6766eqcomd 2392 . . . . . 6  |-  ( ( ( ph  /\  y  e.  ( B  i^i  RR+ )
)  /\  x  =  ( 1  /  y
) )  ->  R  =  S )
6854, 67csbied 3236 . . . . 5  |-  ( (
ph  /\  y  e.  ( B  i^i  RR+ )
)  ->  [_ ( 1  /  y )  /  x ]_ R  =  S )
6958, 68eqtrd 2419 . . . 4  |-  ( (
ph  /\  y  e.  ( B  i^i  RR+ )
)  ->  if (
( 1  /  y
)  =  0 ,  C ,  [_ (
1  /  y )  /  x ]_ R
)  =  S )
7069mpteq2dva 4236 . . 3  |-  ( ph  ->  ( y  e.  ( B  i^i  RR+ )  |->  if ( ( 1  /  y )  =  0 ,  C ,  [_ ( 1  /  y
)  /  x ]_ R ) )  =  ( y  e.  ( B  i^i  RR+ )  |->  S ) )
7170breq1d 4163 . 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 ) )
72 rlimcnp2.a . . . 4  |-  ( ph  ->  A  C_  ( 0 [,)  +oo ) )
73 rlimcnp2.0 . . . 4  |-  ( ph  ->  0  e.  A )
74 rlimcnp2.c . . . . . 6  |-  ( ph  ->  C  e.  CC )
7574ad2antrr 707 . . . . 5  |-  ( ( ( ph  /\  w  e.  A )  /\  w  =  0 )  ->  C  e.  CC )
7672sselda 3291 . . . . . . . . . . . 12  |-  ( (
ph  /\  w  e.  A )  ->  w  e.  ( 0 [,)  +oo ) )
77 0re 9024 . . . . . . . . . . . . 13  |-  0  e.  RR
78 pnfxr 10645 . . . . . . . . . . . . 13  |-  +oo  e.  RR*
79 elico2 10906 . . . . . . . . . . . . 13  |-  ( ( 0  e.  RR  /\  +oo 
e.  RR* )  ->  (
w  e.  ( 0 [,)  +oo )  <->  ( w  e.  RR  /\  0  <_  w  /\  w  <  +oo ) ) )
8077, 78, 79mp2an 654 . . . . . . . . . . . 12  |-  ( w  e.  ( 0 [,) 
+oo )  <->  ( w  e.  RR  /\  0  <_  w  /\  w  <  +oo ) )
8176, 80sylib 189 . . . . . . . . . . 11  |-  ( (
ph  /\  w  e.  A )  ->  (
w  e.  RR  /\  0  <_  w  /\  w  <  +oo ) )
8281simp1d 969 . . . . . . . . . 10  |-  ( (
ph  /\  w  e.  A )  ->  w  e.  RR )
8382adantr 452 . . . . . . . . 9  |-  ( ( ( ph  /\  w  e.  A )  /\  -.  w  =  0 )  ->  w  e.  RR )
8481simp2d 970 . . . . . . . . . . . . . 14  |-  ( (
ph  /\  w  e.  A )  ->  0  <_  w )
85 leloe 9094 . . . . . . . . . . . . . . 15  |-  ( ( 0  e.  RR  /\  w  e.  RR )  ->  ( 0  <_  w  <->  ( 0  <  w  \/  0  =  w ) ) )
8677, 82, 85sylancr 645 . . . . . . . . . . . . . 14  |-  ( (
ph  /\  w  e.  A )  ->  (
0  <_  w  <->  ( 0  <  w  \/  0  =  w ) ) )
8784, 86mpbid 202 . . . . . . . . . . . . 13  |-  ( (
ph  /\  w  e.  A )  ->  (
0  <  w  \/  0  =  w )
)
8887ord 367 . . . . . . . . . . . 12  |-  ( (
ph  /\  w  e.  A )  ->  ( -.  0  <  w  -> 
0  =  w ) )
89 eqcom 2389 . . . . . . . . . . . 12  |-  ( 0  =  w  <->  w  = 
0 )
9088, 89syl6ib 218 . . . . . . . . . . 11  |-  ( (
ph  /\  w  e.  A )  ->  ( -.  0  <  w  ->  w  =  0 ) )
9190con1d 118 . . . . . . . . . 10  |-  ( (
ph  /\  w  e.  A )  ->  ( -.  w  =  0  ->  0  <  w ) )
9291imp 419 . . . . . . . . 9  |-  ( ( ( ph  /\  w  e.  A )  /\  -.  w  =  0 )  ->  0  <  w
)
9383, 92elrpd 10578 . . . . . . . 8  |-  ( ( ( ph  /\  w  e.  A )  /\  -.  w  =  0 )  ->  w  e.  RR+ )
94 rpcnne0 10561 . . . . . . . . 9  |-  ( w  e.  RR+  ->  ( w  e.  CC  /\  w  =/=  0 ) )
95 recrec 9643 . . . . . . . . 9  |-  ( ( w  e.  CC  /\  w  =/=  0 )  -> 
( 1  /  (
1  /  w ) )  =  w )
9694, 95syl 16 . . . . . . . 8  |-  ( w  e.  RR+  ->  ( 1  /  ( 1  /  w ) )  =  w )
9793, 96syl 16 . . . . . . 7  |-  ( ( ( ph  /\  w  e.  A )  /\  -.  w  =  0 )  ->  ( 1  / 
( 1  /  w
) )  =  w )
9897csbeq1d 3200 . . . . . 6  |-  ( ( ( ph  /\  w  e.  A )  /\  -.  w  =  0 )  ->  [_ ( 1  / 
( 1  /  w
) )  /  x ]_ R  =  [_ w  /  x ]_ R )
99 simplr 732 . . . . . . . . 9  |-  ( ( ( ph  /\  w  e.  A )  /\  -.  w  =  0 )  ->  w  e.  A
)
100 simpll 731 . . . . . . . . . 10  |-  ( ( ( ph  /\  w  e.  A )  /\  -.  w  =  0 )  ->  ph )
101 rpreccl 10567 . . . . . . . . . . . . 13  |-  ( w  e.  RR+  ->  ( 1  /  w )  e.  RR+ )
102101adantl 453 . . . . . . . . . . . 12  |-  ( (
ph  /\  w  e.  RR+ )  ->  ( 1  /  w )  e.  RR+ )
103 rlimcnp2.d . . . . . . . . . . . . . 14  |-  ( (
ph  /\  y  e.  RR+ )  ->  ( y  e.  B  <->  ( 1  / 
y )  e.  A
) )
104103ralrimiva 2732 . . . . . . . . . . . . 13  |-  ( ph  ->  A. y  e.  RR+  ( y  e.  B  <->  ( 1  /  y )  e.  A ) )
105104adantr 452 . . . . . . . . . . . 12  |-  ( (
ph  /\  w  e.  RR+ )  ->  A. y  e.  RR+  ( y  e.  B  <->  ( 1  / 
y )  e.  A
) )
106 eleq1 2447 . . . . . . . . . . . . . 14  |-  ( y  =  ( 1  /  w )  ->  (
y  e.  B  <->  ( 1  /  w )  e.  B ) )
107 oveq2 6028 . . . . . . . . . . . . . . 15  |-  ( y  =  ( 1  /  w )  ->  (
1  /  y )  =  ( 1  / 
( 1  /  w
) ) )
108107eleq1d 2453 . . . . . . . . . . . . . 14  |-  ( y  =  ( 1  /  w )  ->  (
( 1  /  y
)  e.  A  <->  ( 1  /  ( 1  /  w ) )  e.  A ) )
109106, 108bibi12d 313 . . . . . . . . . . . . 13  |-  ( y  =  ( 1  /  w )  ->  (
( y  e.  B  <->  ( 1  /  y )  e.  A )  <->  ( (
1  /  w )  e.  B  <->  ( 1  /  ( 1  /  w ) )  e.  A ) ) )
110109rspcv 2991 . . . . . . . . . . . 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 ) ) )
111102, 105, 110sylc 58 . . . . . . . . . . 11  |-  ( (
ph  /\  w  e.  RR+ )  ->  ( (
1  /  w )  e.  B  <->  ( 1  /  ( 1  /  w ) )  e.  A ) )
11296adantl 453 . . . . . . . . . . . 12  |-  ( (
ph  /\  w  e.  RR+ )  ->  ( 1  /  ( 1  /  w ) )  =  w )
113112eleq1d 2453 . . . . . . . . . . 11  |-  ( (
ph  /\  w  e.  RR+ )  ->  ( (
1  /  ( 1  /  w ) )  e.  A  <->  w  e.  A ) )
114111, 113bitr2d 246 . . . . . . . . . 10  |-  ( (
ph  /\  w  e.  RR+ )  ->  ( w  e.  A  <->  ( 1  /  w )  e.  B
) )
115100, 93, 114syl2anc 643 . . . . . . . . 9  |-  ( ( ( ph  /\  w  e.  A )  /\  -.  w  =  0 )  ->  ( w  e.  A  <->  ( 1  /  w )  e.  B
) )
11699, 115mpbid 202 . . . . . . . 8  |-  ( ( ( ph  /\  w  e.  A )  /\  -.  w  =  0 )  ->  ( 1  /  w )  e.  B
)
11793rpreccld 10590 . . . . . . . 8  |-  ( ( ( ph  /\  w  e.  A )  /\  -.  w  =  0 )  ->  ( 1  /  w )  e.  RR+ )
118 elin 3473 . . . . . . . 8  |-  ( ( 1  /  w )  e.  ( B  i^i  RR+ )  <->  ( ( 1  /  w )  e.  B  /\  ( 1  /  w )  e.  RR+ ) )
119116, 117, 118sylanbrc 646 . . . . . . 7  |-  ( ( ( ph  /\  w  e.  A )  /\  -.  w  =  0 )  ->  ( 1  /  w )  e.  ( B  i^i  RR+ )
)
12068, 31eqeltrd 2461 . . . . . . . . 9  |-  ( (
ph  /\  y  e.  ( B  i^i  RR+ )
)  ->  [_ ( 1  /  y )  /  x ]_ R  e.  CC )
121120ralrimiva 2732 . . . . . . . 8  |-  ( ph  ->  A. y  e.  ( B  i^i  RR+ ) [_ ( 1  /  y
)  /  x ]_ R  e.  CC )
122121ad2antrr 707 . . . . . . 7  |-  ( ( ( ph  /\  w  e.  A )  /\  -.  w  =  0 )  ->  A. y  e.  ( B  i^i  RR+ ) [_ ( 1  /  y
)  /  x ]_ R  e.  CC )
123107csbeq1d 3200 . . . . . . . . 9  |-  ( y  =  ( 1  /  w )  ->  [_ (
1  /  y )  /  x ]_ R  =  [_ ( 1  / 
( 1  /  w
) )  /  x ]_ R )
124123eleq1d 2453 . . . . . . . 8  |-  ( y  =  ( 1  /  w )  ->  ( [_ ( 1  /  y
)  /  x ]_ R  e.  CC  <->  [_ ( 1  /  ( 1  /  w ) )  /  x ]_ R  e.  CC ) )
125124rspcv 2991 . . . . . . 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 ) )
126119, 122, 125sylc 58 . . . . . 6  |-  ( ( ( ph  /\  w  e.  A )  /\  -.  w  =  0 )  ->  [_ ( 1  / 
( 1  /  w
) )  /  x ]_ R  e.  CC )
12798, 126eqeltrrd 2462 . . . . 5  |-  ( ( ( ph  /\  w  e.  A )  /\  -.  w  =  0 )  ->  [_ w  /  x ]_ R  e.  CC )
12875, 127ifclda 3709 . . . 4  |-  ( (
ph  /\  w  e.  A )  ->  if ( w  =  0 ,  C ,  [_ w  /  x ]_ R )  e.  CC )
129102biantrud 494 . . . . . 6  |-  ( (
ph  /\  w  e.  RR+ )  ->  ( (
1  /  w )  e.  B  <->  ( (
1  /  w )  e.  B  /\  (
1  /  w )  e.  RR+ ) ) )
130114, 129bitrd 245 . . . . 5  |-  ( (
ph  /\  w  e.  RR+ )  ->  ( w  e.  A  <->  ( ( 1  /  w )  e.  B  /\  ( 1  /  w )  e.  RR+ ) ) )
131130, 118syl6bbr 255 . . . 4  |-  ( (
ph  /\  w  e.  RR+ )  ->  ( w  e.  A  <->  ( 1  /  w )  e.  ( B  i^i  RR+ )
) )
132 iftrue 3688 . . . 4  |-  ( w  =  0  ->  if ( w  =  0 ,  C ,  [_ w  /  x ]_ R )  =  C )
133 eqeq1 2393 . . . . 5  |-  ( w  =  ( 1  / 
y )  ->  (
w  =  0  <->  (
1  /  y )  =  0 ) )
134 csbeq1 3197 . . . . 5  |-  ( w  =  ( 1  / 
y )  ->  [_ w  /  x ]_ R  = 
[_ ( 1  / 
y )  /  x ]_ R )
135133, 134ifbieq2d 3702 . . . 4  |-  ( w  =  ( 1  / 
y )  ->  if ( w  =  0 ,  C ,  [_ w  /  x ]_ R )  =  if ( ( 1  /  y )  =  0 ,  C ,  [_ ( 1  / 
y )  /  x ]_ R ) )
136 rlimcnp2.j . . . 4  |-  J  =  ( TopOpen ` fld )
137 rlimcnp2.k . . . 4  |-  K  =  ( Jt  A )
13872, 73, 52, 128, 131, 132, 135, 136, 137rlimcnp 20671 . . 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 )
) )
139 nfcv 2523 . . . . 5  |-  F/_ w if ( x  =  0 ,  C ,  R
)
140 nfv 1626 . . . . . 6  |-  F/ x  w  =  0
141 nfcv 2523 . . . . . 6  |-  F/_ x C
142 nfcsb1v 3226 . . . . . 6  |-  F/_ x [_ w  /  x ]_ R
143140, 141, 142nfif 3706 . . . . 5  |-  F/_ x if ( w  =  0 ,  C ,  [_ w  /  x ]_ R
)
144 eqeq1 2393 . . . . . 6  |-  ( x  =  w  ->  (
x  =  0  <->  w  =  0 ) )
145 csbeq1a 3202 . . . . . 6  |-  ( x  =  w  ->  R  =  [_ w  /  x ]_ R )
146144, 145ifbieq2d 3702 . . . . 5  |-  ( x  =  w  ->  if ( x  =  0 ,  C ,  R )  =  if ( w  =  0 ,  C ,  [_ w  /  x ]_ R ) )
147139, 143, 146cbvmpt 4240 . . . 4  |-  ( x  e.  A  |->  if ( x  =  0 ,  C ,  R ) )  =  ( w  e.  A  |->  if ( w  =  0 ,  C ,  [_ w  /  x ]_ R ) )
148147eleq1i 2450 . . 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 )
)
149138, 148syl6bbr 255 . 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 )
) )
15050, 71, 1493bitr2d 273 1  |-  ( ph  ->  ( ( y  e.  B  |->  S )  ~~> r  C  <->  ( x  e.  A  |->  if ( x  =  0 ,  C ,  R
) )  e.  ( ( K  CnP  J
) `  0 )
) )
Colors of variables: wff set class
Syntax hints:   -. wn 3    -> wi 4    <-> wb 177    \/ wo 358    /\ wa 359    /\ w3a 936    = wceq 1649    e. wcel 1717    =/= wne 2550   A.wral 2649   [_csb 3194    i^i cin 3262    C_ wss 3263   ifcif 3682   class class class wbr 4153    e. cmpt 4207   dom cdm 4818    |` cres 4820   Rel wrel 4823    Fn wfn 5389   -->wf 5390   ` cfv 5394  (class class class)co 6020   CCcc 8921   RRcr 8922   0cc0 8923   1c1 8924    +oocpnf 9050   RR*cxr 9052    < clt 9053    <_ cle 9054    / cdiv 9609   RR+crp 10544   (,)cioo 10848   [,)cico 10850    ~~> r crli 12206   ↾t crest 13575   TopOpenctopn 13576  ℂfldccnfld 16626    CnP ccnp 17211
This theorem is referenced by:  rlimcnp3  20673
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1552  ax-5 1563  ax-17 1623  ax-9 1661  ax-8 1682  ax-13 1719  ax-14 1721  ax-6 1736  ax-7 1741  ax-11 1753  ax-12 1939  ax-ext 2368  ax-rep 4261  ax-sep 4271  ax-nul 4279  ax-pow 4318  ax-pr 4344  ax-un 4641  ax-cnex 8979  ax-resscn 8980  ax-1cn 8981  ax-icn 8982  ax-addcl 8983  ax-addrcl 8984  ax-mulcl 8985  ax-mulrcl 8986  ax-mulcom 8987  ax-addass 8988  ax-mulass 8989  ax-distr 8990  ax-i2m1 8991  ax-1ne0 8992  ax-1rid 8993  ax-rnegex 8994  ax-rrecex 8995  ax-cnre 8996  ax-pre-lttri 8997  ax-pre-lttrn 8998  ax-pre-ltadd 8999  ax-pre-mulgt0 9000  ax-pre-sup 9001
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3or 937  df-3an 938  df-tru 1325  df-ex 1548  df-nf 1551  df-sb 1656  df-eu 2242  df-mo 2243  df-clab 2374  df-cleq 2380  df-clel 2383  df-nfc 2512  df-ne 2552  df-nel 2553  df-ral 2654  df-rex 2655  df-reu 2656  df-rmo 2657  df-rab 2658  df-v 2901  df-sbc 3105  df-csb 3195  df-dif 3266  df-un 3268  df-in 3270  df-ss 3277  df-pss 3279  df-nul 3572  df-if 3683  df-pw 3744  df-sn 3763  df-pr 3764  df-tp 3765  df-op 3766  df-uni 3958  df-int 3993  df-iun 4037  df-br 4154  df-opab 4208  df-mpt 4209  df-tr 4244  df-eprel 4435  df-id 4439  df-po 4444  df-so 4445  df-fr 4482  df-we 4484  df-ord 4525  df-on 4526  df-lim 4527  df-suc 4528  df-om 4786  df-xp 4824  df-rel 4825  df-cnv 4826  df-co 4827  df-dm 4828  df-rn 4829  df-res 4830  df-ima 4831  df-iota 5358  df-fun 5396  df-fn 5397  df-f 5398  df-f1 5399  df-fo 5400  df-f1o 5401  df-fv 5402  df-ov 6023  df-oprab 6024  df-mpt2 6025  df-1st 6288  df-2nd 6289  df-riota 6485  df-recs 6569  df-rdg 6604  df-1o 6660  df-oadd 6664  df-er 6841  df-map 6956  df-pm 6957  df-en 7046  df-dom 7047  df-sdom 7048  df-fin 7049  df-sup 7381  df-pnf 9055  df-mnf 9056  df-xr 9057  df-ltxr 9058  df-le 9059  df-sub 9225  df-neg 9226  df-div 9610  df-nn 9933  df-2 9990  df-3 9991  df-4 9992  df-5 9993  df-6 9994  df-7 9995  df-8 9996  df-9 9997  df-10 9998  df-n0 10154  df-z 10215  df-dec 10315  df-uz 10421  df-q 10507  df-rp 10545  df-xneg 10642  df-xadd 10643  df-xmul 10644  df-ioo 10852  df-ico 10854  df-fz 10976  df-seq 11251  df-exp 11310  df-cj 11831  df-re 11832  df-im 11833  df-sqr 11967  df-abs 11968  df-rlim 12210  df-struct 13398  df-ndx 13399  df-slot 13400  df-base 13401  df-plusg 13469  df-mulr 13470  df-starv 13471  df-tset 13475  df-ple 13476  df-ds 13478  df-unif 13479  df-rest 13577  df-topn 13578  df-topgen 13594  df-xmet 16619  df-met 16620  df-bl 16621  df-mopn 16622  df-cnfld 16627  df-top 16886  df-bases 16888  df-topon 16889  df-cnp 17214
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