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Theorem rlimcnp2 23757
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 3688 . . . . . . . 8  |-  ( B  i^i  ( 1 [,) +oo ) )  C_  B
2 resmpt 5174 . . . . . . . 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 13 . . . . . . 7  |-  ( ph  ->  ( ( y  e.  B  |->  S )  |`  ( B  i^i  (
1 [,) +oo )
) )  =  ( y  e.  ( B  i^i  ( 1 [,) +oo ) )  |->  S ) )
4 0xr 9686 . . . . . . . . . . 11  |-  0  e.  RR*
5 0lt1 10135 . . . . . . . . . . 11  |-  0  <  1
6 df-ioo 11639 . . . . . . . . . . . 12  |-  (,)  =  ( x  e.  RR* ,  y  e.  RR*  |->  { z  e.  RR*  |  (
x  <  z  /\  z  <  y ) } )
7 df-ico 11641 . . . . . . . . . . . 12  |-  [,)  =  ( x  e.  RR* ,  y  e.  RR*  |->  { z  e.  RR*  |  (
x  <_  z  /\  z  <  y ) } )
8 xrltletr 11454 . . . . . . . . . . . 12  |-  ( ( 0  e.  RR*  /\  1  e.  RR*  /\  w  e. 
RR* )  ->  (
( 0  <  1  /\  1  <_  w )  ->  0  <  w
) )
96, 7, 8ixxss1 11653 . . . . . . . . . . 11  |-  ( ( 0  e.  RR*  /\  0  <  1 )  ->  (
1 [,) +oo )  C_  ( 0 (,) +oo ) )
104, 5, 9mp2an 676 . . . . . . . . . 10  |-  ( 1 [,) +oo )  C_  ( 0 (,) +oo )
11 ioorp 11712 . . . . . . . . . 10  |-  ( 0 (,) +oo )  = 
RR+
1210, 11sseqtri 3502 . . . . . . . . 9  |-  ( 1 [,) +oo )  C_  RR+
13 sslin 3694 . . . . . . . . 9  |-  ( ( 1 [,) +oo )  C_  RR+  ->  ( B  i^i  ( 1 [,) +oo ) )  C_  ( B  i^i  RR+ ) )
1412, 13ax-mp 5 . . . . . . . 8  |-  ( B  i^i  ( 1 [,) +oo ) )  C_  ( B  i^i  RR+ )
15 resmpt 5174 . . . . . . . 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 13 . . . . . . 7  |-  ( ph  ->  ( ( y  e.  ( B  i^i  RR+ )  |->  S )  |`  ( B  i^i  ( 1 [,) +oo ) ) )  =  ( y  e.  ( B  i^i  ( 1 [,) +oo ) ) 
|->  S ) )
173, 16eqtr4d 2473 . . . . . 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 5137 . . . . . 6  |-  ( ( ( y  e.  B  |->  S )  |`  B )  |`  ( 1 [,) +oo ) )  =  ( ( y  e.  B  |->  S )  |`  ( B  i^i  ( 1 [,) +oo ) ) )
19 resres 5137 . . . . . 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 2495 . . . . 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 2429 . . . . . . . . 9  |-  ( y  e.  B  |->  S )  =  ( y  e.  B  |->  S )
2321, 22fmptd 6061 . . . . . . . 8  |-  ( ph  ->  ( y  e.  B  |->  S ) : B --> CC )
24 ffn 5746 . . . . . . . 8  |-  ( ( y  e.  B  |->  S ) : B --> CC  ->  ( y  e.  B  |->  S )  Fn  B )
2523, 24syl 17 . . . . . . 7  |-  ( ph  ->  ( y  e.  B  |->  S )  Fn  B
)
26 fnresdm 5703 . . . . . . 7  |-  ( ( y  e.  B  |->  S )  Fn  B  -> 
( ( y  e.  B  |->  S )  |`  B )  =  ( y  e.  B  |->  S ) )
2725, 26syl 17 . . . . . 6  |-  ( ph  ->  ( ( y  e.  B  |->  S )  |`  B )  =  ( y  e.  B  |->  S ) )
2827reseq1d 5124 . . . . 5  |-  ( ph  ->  ( ( ( y  e.  B  |->  S )  |`  B )  |`  (
1 [,) +oo )
)  =  ( ( y  e.  B  |->  S )  |`  ( 1 [,) +oo ) ) )
29 inss1 3688 . . . . . . . . . . 11  |-  ( B  i^i  RR+ )  C_  B
3029sseli 3466 . . . . . . . . . 10  |-  ( y  e.  ( B  i^i  RR+ )  ->  y  e.  B )
3130, 21sylan2 476 . . . . . . . . 9  |-  ( (
ph  /\  y  e.  ( B  i^i  RR+ )
)  ->  S  e.  CC )
32 eqid 2429 . . . . . . . . 9  |-  ( y  e.  ( B  i^i  RR+ )  |->  S )  =  ( y  e.  ( B  i^i  RR+ )  |->  S )
3331, 32fmptd 6061 . . . . . . . 8  |-  ( ph  ->  ( y  e.  ( B  i^i  RR+ )  |->  S ) : ( B  i^i  RR+ ) --> CC )
34 frel 5749 . . . . . . . 8  |-  ( ( y  e.  ( B  i^i  RR+ )  |->  S ) : ( B  i^i  RR+ ) --> CC  ->  Rel  ( y  e.  ( B  i^i  RR+ )  |->  S ) )
3533, 34syl 17 . . . . . . 7  |-  ( ph  ->  Rel  ( y  e.  ( B  i^i  RR+ )  |->  S ) )
3632, 31dmmptd 5726 . . . . . . . 8  |-  ( ph  ->  dom  ( y  e.  ( B  i^i  RR+ )  |->  S )  =  ( B  i^i  RR+ )
)
3736, 29syl6eqss 3520 . . . . . . 7  |-  ( ph  ->  dom  ( y  e.  ( B  i^i  RR+ )  |->  S )  C_  B
)
38 relssres 5162 . . . . . . 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 ) )
3935, 37, 38syl2anc 665 . . . . . 6  |-  ( ph  ->  ( ( y  e.  ( B  i^i  RR+ )  |->  S )  |`  B )  =  ( y  e.  ( B  i^i  RR+ )  |->  S ) )
4039reseq1d 5124 . . . . 5  |-  ( ph  ->  ( ( ( y  e.  ( B  i^i  RR+ )  |->  S )  |`  B )  |`  (
1 [,) +oo )
)  =  ( ( y  e.  ( B  i^i  RR+ )  |->  S )  |`  ( 1 [,) +oo ) ) )
4120, 28, 403eqtr3d 2478 . . . 4  |-  ( ph  ->  ( ( y  e.  B  |->  S )  |`  ( 1 [,) +oo ) )  =  ( ( y  e.  ( B  i^i  RR+ )  |->  S )  |`  (
1 [,) +oo )
) )
4241breq1d 4436 . . 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 9657 . . . 4  |-  ( ph  ->  1  e.  RR )
4523, 43, 44rlimresb 13607 . . 3  |-  ( ph  ->  ( ( y  e.  B  |->  S )  ~~> r  C  <->  ( ( y  e.  B  |->  S )  |`  (
1 [,) +oo )
)  ~~> r  C ) )
4629, 43syl5ss 3481 . . . 4  |-  ( ph  ->  ( B  i^i  RR+ )  C_  RR )
4733, 46, 44rlimresb 13607 . . 3  |-  ( ph  ->  ( ( y  e.  ( B  i^i  RR+ )  |->  S )  ~~> r  C  <->  ( ( y  e.  ( B  i^i  RR+ )  |->  S )  |`  (
1 [,) +oo )
)  ~~> r  C ) )
4842, 45, 473bitr4d 288 . 2  |-  ( ph  ->  ( ( y  e.  B  |->  S )  ~~> r  C  <->  ( y  e.  ( B  i^i  RR+ )  |->  S )  ~~> r  C ) )
49 inss2 3689 . . . . . . . . . . 11  |-  ( B  i^i  RR+ )  C_  RR+
5049a1i 11 . . . . . . . . . 10  |-  ( ph  ->  ( B  i^i  RR+ )  C_  RR+ )
5150sselda 3470 . . . . . . . . 9  |-  ( (
ph  /\  y  e.  ( B  i^i  RR+ )
)  ->  y  e.  RR+ )
5251rpreccld 11351 . . . . . . . 8  |-  ( (
ph  /\  y  e.  ( B  i^i  RR+ )
)  ->  ( 1  /  y )  e.  RR+ )
5352rpne0d 11346 . . . . . . 7  |-  ( (
ph  /\  y  e.  ( B  i^i  RR+ )
)  ->  ( 1  /  y )  =/=  0 )
5453neneqd 2632 . . . . . 6  |-  ( (
ph  /\  y  e.  ( B  i^i  RR+ )
)  ->  -.  (
1  /  y )  =  0 )
5554iffalsed 3926 . . . . 5  |-  ( (
ph  /\  y  e.  ( B  i^i  RR+ )
)  ->  if (
( 1  /  y
)  =  0 ,  C ,  [_ (
1  /  y )  /  x ]_ R
)  =  [_ (
1  /  y )  /  x ]_ R
)
56 oveq2 6313 . . . . . . . . . 10  |-  ( x  =  ( 1  / 
y )  ->  (
1  /  x )  =  ( 1  / 
( 1  /  y
) ) )
57 rpcnne0 11319 . . . . . . . . . . 11  |-  ( y  e.  RR+  ->  ( y  e.  CC  /\  y  =/=  0 ) )
58 recrec 10303 . . . . . . . . . . 11  |-  ( ( y  e.  CC  /\  y  =/=  0 )  -> 
( 1  /  (
1  /  y ) )  =  y )
5951, 57, 583syl 18 . . . . . . . . . 10  |-  ( (
ph  /\  y  e.  ( B  i^i  RR+ )
)  ->  ( 1  /  ( 1  / 
y ) )  =  y )
6056, 59sylan9eqr 2492 . . . . . . . . 9  |-  ( ( ( ph  /\  y  e.  ( B  i^i  RR+ )
)  /\  x  =  ( 1  /  y
) )  ->  (
1  /  x )  =  y )
6160eqcomd 2437 . . . . . . . 8  |-  ( ( ( ph  /\  y  e.  ( B  i^i  RR+ )
)  /\  x  =  ( 1  /  y
) )  ->  y  =  ( 1  /  x ) )
62 rlimcnp2.s . . . . . . . 8  |-  ( y  =  ( 1  /  x )  ->  S  =  R )
6361, 62syl 17 . . . . . . 7  |-  ( ( ( ph  /\  y  e.  ( B  i^i  RR+ )
)  /\  x  =  ( 1  /  y
) )  ->  S  =  R )
6463eqcomd 2437 . . . . . 6  |-  ( ( ( ph  /\  y  e.  ( B  i^i  RR+ )
)  /\  x  =  ( 1  /  y
) )  ->  R  =  S )
6552, 64csbied 3428 . . . . 5  |-  ( (
ph  /\  y  e.  ( B  i^i  RR+ )
)  ->  [_ ( 1  /  y )  /  x ]_ R  =  S )
6655, 65eqtrd 2470 . . . 4  |-  ( (
ph  /\  y  e.  ( B  i^i  RR+ )
)  ->  if (
( 1  /  y
)  =  0 ,  C ,  [_ (
1  /  y )  /  x ]_ R
)  =  S )
6766mpteq2dva 4512 . . 3  |-  ( ph  ->  ( y  e.  ( B  i^i  RR+ )  |->  if ( ( 1  /  y )  =  0 ,  C ,  [_ ( 1  /  y
)  /  x ]_ R ) )  =  ( y  e.  ( B  i^i  RR+ )  |->  S ) )
6867breq1d 4436 . 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 )
7271ad2antrr 730 . . . . 5  |-  ( ( ( ph  /\  w  e.  A )  /\  w  =  0 )  ->  C  e.  CC )
7369sselda 3470 . . . . . . . . . . . 12  |-  ( (
ph  /\  w  e.  A )  ->  w  e.  ( 0 [,) +oo ) )
74 0re 9642 . . . . . . . . . . . . 13  |-  0  e.  RR
75 pnfxr 11412 . . . . . . . . . . . . 13  |- +oo  e.  RR*
76 elico2 11698 . . . . . . . . . . . . 13  |-  ( ( 0  e.  RR  /\ +oo  e.  RR* )  ->  (
w  e.  ( 0 [,) +oo )  <->  ( w  e.  RR  /\  0  <_  w  /\  w  < +oo ) ) )
7774, 75, 76mp2an 676 . . . . . . . . . . . 12  |-  ( w  e.  ( 0 [,) +oo )  <->  ( w  e.  RR  /\  0  <_  w  /\  w  < +oo ) )
7873, 77sylib 199 . . . . . . . . . . 11  |-  ( (
ph  /\  w  e.  A )  ->  (
w  e.  RR  /\  0  <_  w  /\  w  < +oo ) )
7978simp1d 1017 . . . . . . . . . 10  |-  ( (
ph  /\  w  e.  A )  ->  w  e.  RR )
8079adantr 466 . . . . . . . . 9  |-  ( ( ( ph  /\  w  e.  A )  /\  -.  w  =  0 )  ->  w  e.  RR )
8178simp2d 1018 . . . . . . . . . . . . . 14  |-  ( (
ph  /\  w  e.  A )  ->  0  <_  w )
82 leloe 9719 . . . . . . . . . . . . . . 15  |-  ( ( 0  e.  RR  /\  w  e.  RR )  ->  ( 0  <_  w  <->  ( 0  <  w  \/  0  =  w ) ) )
8374, 79, 82sylancr 667 . . . . . . . . . . . . . 14  |-  ( (
ph  /\  w  e.  A )  ->  (
0  <_  w  <->  ( 0  <  w  \/  0  =  w ) ) )
8481, 83mpbid 213 . . . . . . . . . . . . 13  |-  ( (
ph  /\  w  e.  A )  ->  (
0  <  w  \/  0  =  w )
)
8584ord 378 . . . . . . . . . . . 12  |-  ( (
ph  /\  w  e.  A )  ->  ( -.  0  <  w  -> 
0  =  w ) )
86 eqcom 2438 . . . . . . . . . . . 12  |-  ( 0  =  w  <->  w  = 
0 )
8785, 86syl6ib 229 . . . . . . . . . . 11  |-  ( (
ph  /\  w  e.  A )  ->  ( -.  0  <  w  ->  w  =  0 ) )
8887con1d 127 . . . . . . . . . 10  |-  ( (
ph  /\  w  e.  A )  ->  ( -.  w  =  0  ->  0  <  w ) )
8988imp 430 . . . . . . . . 9  |-  ( ( ( ph  /\  w  e.  A )  /\  -.  w  =  0 )  ->  0  <  w
)
9080, 89elrpd 11338 . . . . . . . 8  |-  ( ( ( ph  /\  w  e.  A )  /\  -.  w  =  0 )  ->  w  e.  RR+ )
91 rpcnne0 11319 . . . . . . . . 9  |-  ( w  e.  RR+  ->  ( w  e.  CC  /\  w  =/=  0 ) )
92 recrec 10303 . . . . . . . . 9  |-  ( ( w  e.  CC  /\  w  =/=  0 )  -> 
( 1  /  (
1  /  w ) )  =  w )
9391, 92syl 17 . . . . . . . 8  |-  ( w  e.  RR+  ->  ( 1  /  ( 1  /  w ) )  =  w )
9490, 93syl 17 . . . . . . 7  |-  ( ( ( ph  /\  w  e.  A )  /\  -.  w  =  0 )  ->  ( 1  / 
( 1  /  w
) )  =  w )
9594csbeq1d 3408 . . . . . 6  |-  ( ( ( ph  /\  w  e.  A )  /\  -.  w  =  0 )  ->  [_ ( 1  / 
( 1  /  w
) )  /  x ]_ R  =  [_ w  /  x ]_ R )
96 simplr 760 . . . . . . . . 9  |-  ( ( ( ph  /\  w  e.  A )  /\  -.  w  =  0 )  ->  w  e.  A
)
97 simpll 758 . . . . . . . . . 10  |-  ( ( ( ph  /\  w  e.  A )  /\  -.  w  =  0 )  ->  ph )
98 rpreccl 11326 . . . . . . . . . . . . 13  |-  ( w  e.  RR+  ->  ( 1  /  w )  e.  RR+ )
9998adantl 467 . . . . . . . . . . . 12  |-  ( (
ph  /\  w  e.  RR+ )  ->  ( 1  /  w )  e.  RR+ )
100 rlimcnp2.d . . . . . . . . . . . . . 14  |-  ( (
ph  /\  y  e.  RR+ )  ->  ( y  e.  B  <->  ( 1  / 
y )  e.  A
) )
101100ralrimiva 2846 . . . . . . . . . . . . 13  |-  ( ph  ->  A. y  e.  RR+  ( y  e.  B  <->  ( 1  /  y )  e.  A ) )
102101adantr 466 . . . . . . . . . . . 12  |-  ( (
ph  /\  w  e.  RR+ )  ->  A. y  e.  RR+  ( y  e.  B  <->  ( 1  / 
y )  e.  A
) )
103 eleq1 2501 . . . . . . . . . . . . . 14  |-  ( y  =  ( 1  /  w )  ->  (
y  e.  B  <->  ( 1  /  w )  e.  B ) )
104 oveq2 6313 . . . . . . . . . . . . . . 15  |-  ( y  =  ( 1  /  w )  ->  (
1  /  y )  =  ( 1  / 
( 1  /  w
) ) )
105104eleq1d 2498 . . . . . . . . . . . . . 14  |-  ( y  =  ( 1  /  w )  ->  (
( 1  /  y
)  e.  A  <->  ( 1  /  ( 1  /  w ) )  e.  A ) )
106103, 105bibi12d 322 . . . . . . . . . . . . 13  |-  ( y  =  ( 1  /  w )  ->  (
( y  e.  B  <->  ( 1  /  y )  e.  A )  <->  ( (
1  /  w )  e.  B  <->  ( 1  /  ( 1  /  w ) )  e.  A ) ) )
107106rspcv 3184 . . . . . . . . . . . 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 ) ) )
10899, 102, 107sylc 62 . . . . . . . . . . 11  |-  ( (
ph  /\  w  e.  RR+ )  ->  ( (
1  /  w )  e.  B  <->  ( 1  /  ( 1  /  w ) )  e.  A ) )
10993adantl 467 . . . . . . . . . . . 12  |-  ( (
ph  /\  w  e.  RR+ )  ->  ( 1  /  ( 1  /  w ) )  =  w )
110109eleq1d 2498 . . . . . . . . . . 11  |-  ( (
ph  /\  w  e.  RR+ )  ->  ( (
1  /  ( 1  /  w ) )  e.  A  <->  w  e.  A ) )
111108, 110bitr2d 257 . . . . . . . . . 10  |-  ( (
ph  /\  w  e.  RR+ )  ->  ( w  e.  A  <->  ( 1  /  w )  e.  B
) )
11297, 90, 111syl2anc 665 . . . . . . . . 9  |-  ( ( ( ph  /\  w  e.  A )  /\  -.  w  =  0 )  ->  ( w  e.  A  <->  ( 1  /  w )  e.  B
) )
11396, 112mpbid 213 . . . . . . . 8  |-  ( ( ( ph  /\  w  e.  A )  /\  -.  w  =  0 )  ->  ( 1  /  w )  e.  B
)
11490rpreccld 11351 . . . . . . . 8  |-  ( ( ( ph  /\  w  e.  A )  /\  -.  w  =  0 )  ->  ( 1  /  w )  e.  RR+ )
115113, 114elind 3656 . . . . . . 7  |-  ( ( ( ph  /\  w  e.  A )  /\  -.  w  =  0 )  ->  ( 1  /  w )  e.  ( B  i^i  RR+ )
)
11665, 31eqeltrd 2517 . . . . . . . . 9  |-  ( (
ph  /\  y  e.  ( B  i^i  RR+ )
)  ->  [_ ( 1  /  y )  /  x ]_ R  e.  CC )
117116ralrimiva 2846 . . . . . . . 8  |-  ( ph  ->  A. y  e.  ( B  i^i  RR+ ) [_ ( 1  /  y
)  /  x ]_ R  e.  CC )
118117ad2antrr 730 . . . . . . 7  |-  ( ( ( ph  /\  w  e.  A )  /\  -.  w  =  0 )  ->  A. y  e.  ( B  i^i  RR+ ) [_ ( 1  /  y
)  /  x ]_ R  e.  CC )
119104csbeq1d 3408 . . . . . . . . 9  |-  ( y  =  ( 1  /  w )  ->  [_ (
1  /  y )  /  x ]_ R  =  [_ ( 1  / 
( 1  /  w
) )  /  x ]_ R )
120119eleq1d 2498 . . . . . . . 8  |-  ( y  =  ( 1  /  w )  ->  ( [_ ( 1  /  y
)  /  x ]_ R  e.  CC  <->  [_ ( 1  /  ( 1  /  w ) )  /  x ]_ R  e.  CC ) )
121120rspcv 3184 . . . . . . 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 ) )
122115, 118, 121sylc 62 . . . . . 6  |-  ( ( ( ph  /\  w  e.  A )  /\  -.  w  =  0 )  ->  [_ ( 1  / 
( 1  /  w
) )  /  x ]_ R  e.  CC )
12395, 122eqeltrrd 2518 . . . . 5  |-  ( ( ( ph  /\  w  e.  A )  /\  -.  w  =  0 )  ->  [_ w  /  x ]_ R  e.  CC )
12472, 123ifclda 3947 . . . 4  |-  ( (
ph  /\  w  e.  A )  ->  if ( w  =  0 ,  C ,  [_ w  /  x ]_ R )  e.  CC )
12599biantrud 509 . . . . . 6  |-  ( (
ph  /\  w  e.  RR+ )  ->  ( (
1  /  w )  e.  B  <->  ( (
1  /  w )  e.  B  /\  (
1  /  w )  e.  RR+ ) ) )
126111, 125bitrd 256 . . . . 5  |-  ( (
ph  /\  w  e.  RR+ )  ->  ( w  e.  A  <->  ( ( 1  /  w )  e.  B  /\  ( 1  /  w )  e.  RR+ ) ) )
127 elin 3655 . . . . 5  |-  ( ( 1  /  w )  e.  ( B  i^i  RR+ )  <->  ( ( 1  /  w )  e.  B  /\  ( 1  /  w )  e.  RR+ ) )
128126, 127syl6bbr 266 . . . 4  |-  ( (
ph  /\  w  e.  RR+ )  ->  ( w  e.  A  <->  ( 1  /  w )  e.  ( B  i^i  RR+ )
) )
129 iftrue 3921 . . . 4  |-  ( w  =  0  ->  if ( w  =  0 ,  C ,  [_ w  /  x ]_ R )  =  C )
130 eqeq1 2433 . . . . 5  |-  ( w  =  ( 1  / 
y )  ->  (
w  =  0  <->  (
1  /  y )  =  0 ) )
131 csbeq1 3404 . . . . 5  |-  ( w  =  ( 1  / 
y )  ->  [_ w  /  x ]_ R  = 
[_ ( 1  / 
y )  /  x ]_ R )
132130, 131ifbieq2d 3940 . . . 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 ` fld )
134 rlimcnp2.k . . . 4  |-  K  =  ( Jt  A )
13569, 70, 50, 124, 128, 129, 132, 133, 134rlimcnp 23756 . . 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 2591 . . . . 5  |-  F/_ w if ( x  =  0 ,  C ,  R
)
137 nfv 1754 . . . . . 6  |-  F/ x  w  =  0
138 nfcv 2591 . . . . . 6  |-  F/_ x C
139 nfcsb1v 3417 . . . . . 6  |-  F/_ x [_ w  /  x ]_ R
140137, 138, 139nfif 3944 . . . . 5  |-  F/_ x if ( w  =  0 ,  C ,  [_ w  /  x ]_ R
)
141 eqeq1 2433 . . . . . 6  |-  ( x  =  w  ->  (
x  =  0  <->  w  =  0 ) )
142 csbeq1a 3410 . . . . . 6  |-  ( x  =  w  ->  R  =  [_ w  /  x ]_ R )
143141, 142ifbieq2d 3940 . . . . 5  |-  ( x  =  w  ->  if ( x  =  0 ,  C ,  R )  =  if ( w  =  0 ,  C ,  [_ w  /  x ]_ R ) )
144136, 140, 143cbvmpt 4517 . . . 4  |-  ( x  e.  A  |->  if ( x  =  0 ,  C ,  R ) )  =  ( w  e.  A  |->  if ( w  =  0 ,  C ,  [_ w  /  x ]_ R ) )
145144eleq1i 2506 . . 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 )
)
146135, 145syl6bbr 266 . 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 )
) )
14748, 68, 1463bitr2d 284 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 setvar class
Syntax hints:   -. wn 3    -> wi 4    <-> wb 187    \/ wo 369    /\ wa 370    /\ w3a 982    = wceq 1437    e. wcel 1870    =/= wne 2625   A.wral 2782   [_csb 3401    i^i cin 3441    C_ wss 3442   ifcif 3915   class class class wbr 4426    |-> cmpt 4484   dom cdm 4854    |` cres 4856   Rel wrel 4859    Fn wfn 5596   -->wf 5597   ` cfv 5601  (class class class)co 6305   CCcc 9536   RRcr 9537   0cc0 9538   1c1 9539   +oocpnf 9671   RR*cxr 9673    < clt 9674    <_ cle 9675    / cdiv 10268   RR+crp 11302   (,)cioo 11635   [,)cico 11637    ~~> r crli 13527   ↾t crest 15278   TopOpenctopn 15279  ℂfldccnfld 18905    CnP ccnp 20172
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1665  ax-4 1678  ax-5 1751  ax-6 1797  ax-7 1841  ax-8 1872  ax-9 1874  ax-10 1889  ax-11 1894  ax-12 1907  ax-13 2055  ax-ext 2407  ax-rep 4538  ax-sep 4548  ax-nul 4556  ax-pow 4603  ax-pr 4661  ax-un 6597  ax-cnex 9594  ax-resscn 9595  ax-1cn 9596  ax-icn 9597  ax-addcl 9598  ax-addrcl 9599  ax-mulcl 9600  ax-mulrcl 9601  ax-mulcom 9602  ax-addass 9603  ax-mulass 9604  ax-distr 9605  ax-i2m1 9606  ax-1ne0 9607  ax-1rid 9608  ax-rnegex 9609  ax-rrecex 9610  ax-cnre 9611  ax-pre-lttri 9612  ax-pre-lttrn 9613  ax-pre-ltadd 9614  ax-pre-mulgt0 9615  ax-pre-sup 9616
This theorem depends on definitions:  df-bi 188  df-or 371  df-an 372  df-3or 983  df-3an 984  df-tru 1440  df-ex 1660  df-nf 1664  df-sb 1790  df-eu 2270  df-mo 2271  df-clab 2415  df-cleq 2421  df-clel 2424  df-nfc 2579  df-ne 2627  df-nel 2628  df-ral 2787  df-rex 2788  df-reu 2789  df-rmo 2790  df-rab 2791  df-v 3089  df-sbc 3306  df-csb 3402  df-dif 3445  df-un 3447  df-in 3449  df-ss 3456  df-pss 3458  df-nul 3768  df-if 3916  df-pw 3987  df-sn 4003  df-pr 4005  df-tp 4007  df-op 4009  df-uni 4223  df-int 4259  df-iun 4304  df-br 4427  df-opab 4485  df-mpt 4486  df-tr 4521  df-eprel 4765  df-id 4769  df-po 4775  df-so 4776  df-fr 4813  df-we 4815  df-xp 4860  df-rel 4861  df-cnv 4862  df-co 4863  df-dm 4864  df-rn 4865  df-res 4866  df-ima 4867  df-pred 5399  df-ord 5445  df-on 5446  df-lim 5447  df-suc 5448  df-iota 5565  df-fun 5603  df-fn 5604  df-f 5605  df-f1 5606  df-fo 5607  df-f1o 5608  df-fv 5609  df-riota 6267  df-ov 6308  df-oprab 6309  df-mpt2 6310  df-om 6707  df-1st 6807  df-2nd 6808  df-wrecs 7036  df-recs 7098  df-rdg 7136  df-1o 7190  df-oadd 7194  df-er 7371  df-map 7482  df-pm 7483  df-en 7578  df-dom 7579  df-sdom 7580  df-fin 7581  df-sup 7962  df-pnf 9676  df-mnf 9677  df-xr 9678  df-ltxr 9679  df-le 9680  df-sub 9861  df-neg 9862  df-div 10269  df-nn 10610  df-2 10668  df-3 10669  df-4 10670  df-5 10671  df-6 10672  df-7 10673  df-8 10674  df-9 10675  df-10 10676  df-n0 10870  df-z 10938  df-dec 11052  df-uz 11160  df-q 11265  df-rp 11303  df-xneg 11409  df-xadd 11410  df-xmul 11411  df-ioo 11639  df-ico 11641  df-fz 11783  df-seq 12211  df-exp 12270  df-cj 13141  df-re 13142  df-im 13143  df-sqrt 13277  df-abs 13278  df-rlim 13531  df-struct 15086  df-ndx 15087  df-slot 15088  df-base 15089  df-plusg 15165  df-mulr 15166  df-starv 15167  df-tset 15171  df-ple 15172  df-ds 15174  df-unif 15175  df-rest 15280  df-topn 15281  df-topgen 15301  df-psmet 18897  df-xmet 18898  df-met 18899  df-bl 18900  df-mopn 18901  df-cnfld 18906  df-top 19852  df-bases 19853  df-topon 19854  df-cnp 20175
This theorem is referenced by:  rlimcnp3  23758
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