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Theorem rlimcnp2 23168
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 3703 . . . . . . . 8  |-  ( B  i^i  ( 1 [,) +oo ) )  C_  B
2 resmpt 5313 . . . . . . . 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 9643 . . . . . . . . . . 11  |-  0  e.  RR*
5 0lt1 10081 . . . . . . . . . . 11  |-  0  <  1
6 df-ioo 11542 . . . . . . . . . . . 12  |-  (,)  =  ( x  e.  RR* ,  y  e.  RR*  |->  { z  e.  RR*  |  (
x  <  z  /\  z  <  y ) } )
7 df-ico 11544 . . . . . . . . . . . 12  |-  [,)  =  ( x  e.  RR* ,  y  e.  RR*  |->  { z  e.  RR*  |  (
x  <_  z  /\  z  <  y ) } )
8 xrltletr 11369 . . . . . . . . . . . 12  |-  ( ( 0  e.  RR*  /\  1  e.  RR*  /\  w  e. 
RR* )  ->  (
( 0  <  1  /\  1  <_  w )  ->  0  <  w
) )
96, 7, 8ixxss1 11556 . . . . . . . . . . 11  |-  ( ( 0  e.  RR*  /\  0  <  1 )  ->  (
1 [,) +oo )  C_  ( 0 (,) +oo ) )
104, 5, 9mp2an 672 . . . . . . . . . 10  |-  ( 1 [,) +oo )  C_  ( 0 (,) +oo )
11 ioorp 11611 . . . . . . . . . 10  |-  ( 0 (,) +oo )  = 
RR+
1210, 11sseqtri 3521 . . . . . . . . 9  |-  ( 1 [,) +oo )  C_  RR+
13 sslin 3709 . . . . . . . . 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 5313 . . . . . . . 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 2487 . . . . . 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 5276 . . . . . 6  |-  ( ( ( y  e.  B  |->  S )  |`  B )  |`  ( 1 [,) +oo ) )  =  ( ( y  e.  B  |->  S )  |`  ( B  i^i  ( 1 [,) +oo ) ) )
19 resres 5276 . . . . . 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 2509 . . . . 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 2443 . . . . . . . . 9  |-  ( y  e.  B  |->  S )  =  ( y  e.  B  |->  S )
2321, 22fmptd 6040 . . . . . . . 8  |-  ( ph  ->  ( y  e.  B  |->  S ) : B --> CC )
24 ffn 5721 . . . . . . . 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 5680 . . . . . . 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 5262 . . . . 5  |-  ( ph  ->  ( ( ( y  e.  B  |->  S )  |`  B )  |`  (
1 [,) +oo )
)  =  ( ( y  e.  B  |->  S )  |`  ( 1 [,) +oo ) ) )
29 inss1 3703 . . . . . . . . . . 11  |-  ( B  i^i  RR+ )  C_  B
3029sseli 3485 . . . . . . . . . 10  |-  ( y  e.  ( B  i^i  RR+ )  ->  y  e.  B )
3130, 21sylan2 474 . . . . . . . . 9  |-  ( (
ph  /\  y  e.  ( B  i^i  RR+ )
)  ->  S  e.  CC )
32 eqid 2443 . . . . . . . . 9  |-  ( y  e.  ( B  i^i  RR+ )  |->  S )  =  ( y  e.  ( B  i^i  RR+ )  |->  S )
3331, 32fmptd 6040 . . . . . . . 8  |-  ( ph  ->  ( y  e.  ( B  i^i  RR+ )  |->  S ) : ( B  i^i  RR+ ) --> CC )
34 frel 5724 . . . . . . . 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 ) )
3632, 31dmmptd 5701 . . . . . . . 8  |-  ( ph  ->  dom  ( y  e.  ( B  i^i  RR+ )  |->  S )  =  ( B  i^i  RR+ )
)
3736, 29syl6eqss 3539 . . . . . . 7  |-  ( ph  ->  dom  ( y  e.  ( B  i^i  RR+ )  |->  S )  C_  B
)
38 relssres 5301 . . . . . . 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 661 . . . . . 6  |-  ( ph  ->  ( ( y  e.  ( B  i^i  RR+ )  |->  S )  |`  B )  =  ( y  e.  ( B  i^i  RR+ )  |->  S ) )
4039reseq1d 5262 . . . . 5  |-  ( ph  ->  ( ( ( y  e.  ( B  i^i  RR+ )  |->  S )  |`  B )  |`  (
1 [,) +oo )
)  =  ( ( y  e.  ( B  i^i  RR+ )  |->  S )  |`  ( 1 [,) +oo ) ) )
4120, 28, 403eqtr3d 2492 . . . 4  |-  ( ph  ->  ( ( y  e.  B  |->  S )  |`  ( 1 [,) +oo ) )  =  ( ( y  e.  ( B  i^i  RR+ )  |->  S )  |`  (
1 [,) +oo )
) )
4241breq1d 4447 . . 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 9614 . . . 4  |-  ( ph  ->  1  e.  RR )
4523, 43, 44rlimresb 13367 . . 3  |-  ( ph  ->  ( ( y  e.  B  |->  S )  ~~> r  C  <->  ( ( y  e.  B  |->  S )  |`  (
1 [,) +oo )
)  ~~> r  C ) )
4629, 43syl5ss 3500 . . . 4  |-  ( ph  ->  ( B  i^i  RR+ )  C_  RR )
4733, 46, 44rlimresb 13367 . . 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 285 . 2  |-  ( ph  ->  ( ( y  e.  B  |->  S )  ~~> r  C  <->  ( y  e.  ( B  i^i  RR+ )  |->  S )  ~~> r  C ) )
49 inss2 3704 . . . . . . . . . . 11  |-  ( B  i^i  RR+ )  C_  RR+
5049a1i 11 . . . . . . . . . 10  |-  ( ph  ->  ( B  i^i  RR+ )  C_  RR+ )
5150sselda 3489 . . . . . . . . 9  |-  ( (
ph  /\  y  e.  ( B  i^i  RR+ )
)  ->  y  e.  RR+ )
5251rpreccld 11275 . . . . . . . 8  |-  ( (
ph  /\  y  e.  ( B  i^i  RR+ )
)  ->  ( 1  /  y )  e.  RR+ )
5352rpne0d 11270 . . . . . . 7  |-  ( (
ph  /\  y  e.  ( B  i^i  RR+ )
)  ->  ( 1  /  y )  =/=  0 )
5453neneqd 2645 . . . . . 6  |-  ( (
ph  /\  y  e.  ( B  i^i  RR+ )
)  ->  -.  (
1  /  y )  =  0 )
5554iffalsed 3937 . . . . 5  |-  ( (
ph  /\  y  e.  ( B  i^i  RR+ )
)  ->  if (
( 1  /  y
)  =  0 ,  C ,  [_ (
1  /  y )  /  x ]_ R
)  =  [_ (
1  /  y )  /  x ]_ R
)
56 oveq2 6289 . . . . . . . . . 10  |-  ( x  =  ( 1  / 
y )  ->  (
1  /  x )  =  ( 1  / 
( 1  /  y
) ) )
57 rpcnne0 11246 . . . . . . . . . . 11  |-  ( y  e.  RR+  ->  ( y  e.  CC  /\  y  =/=  0 ) )
58 recrec 10247 . . . . . . . . . . 11  |-  ( ( y  e.  CC  /\  y  =/=  0 )  -> 
( 1  /  (
1  /  y ) )  =  y )
5951, 57, 583syl 20 . . . . . . . . . 10  |-  ( (
ph  /\  y  e.  ( B  i^i  RR+ )
)  ->  ( 1  /  ( 1  / 
y ) )  =  y )
6056, 59sylan9eqr 2506 . . . . . . . . 9  |-  ( ( ( ph  /\  y  e.  ( B  i^i  RR+ )
)  /\  x  =  ( 1  /  y
) )  ->  (
1  /  x )  =  y )
6160eqcomd 2451 . . . . . . . 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 16 . . . . . . 7  |-  ( ( ( ph  /\  y  e.  ( B  i^i  RR+ )
)  /\  x  =  ( 1  /  y
) )  ->  S  =  R )
6463eqcomd 2451 . . . . . 6  |-  ( ( ( ph  /\  y  e.  ( B  i^i  RR+ )
)  /\  x  =  ( 1  /  y
) )  ->  R  =  S )
6552, 64csbied 3447 . . . . 5  |-  ( (
ph  /\  y  e.  ( B  i^i  RR+ )
)  ->  [_ ( 1  /  y )  /  x ]_ R  =  S )
6655, 65eqtrd 2484 . . . 4  |-  ( (
ph  /\  y  e.  ( B  i^i  RR+ )
)  ->  if (
( 1  /  y
)  =  0 ,  C ,  [_ (
1  /  y )  /  x ]_ R
)  =  S )
6766mpteq2dva 4523 . . 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 4447 . 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 725 . . . . 5  |-  ( ( ( ph  /\  w  e.  A )  /\  w  =  0 )  ->  C  e.  CC )
7369sselda 3489 . . . . . . . . . . . 12  |-  ( (
ph  /\  w  e.  A )  ->  w  e.  ( 0 [,) +oo ) )
74 0re 9599 . . . . . . . . . . . . 13  |-  0  e.  RR
75 pnfxr 11330 . . . . . . . . . . . . 13  |- +oo  e.  RR*
76 elico2 11597 . . . . . . . . . . . . 13  |-  ( ( 0  e.  RR  /\ +oo  e.  RR* )  ->  (
w  e.  ( 0 [,) +oo )  <->  ( w  e.  RR  /\  0  <_  w  /\  w  < +oo ) ) )
7774, 75, 76mp2an 672 . . . . . . . . . . . 12  |-  ( w  e.  ( 0 [,) +oo )  <->  ( w  e.  RR  /\  0  <_  w  /\  w  < +oo ) )
7873, 77sylib 196 . . . . . . . . . . 11  |-  ( (
ph  /\  w  e.  A )  ->  (
w  e.  RR  /\  0  <_  w  /\  w  < +oo ) )
7978simp1d 1009 . . . . . . . . . 10  |-  ( (
ph  /\  w  e.  A )  ->  w  e.  RR )
8079adantr 465 . . . . . . . . 9  |-  ( ( ( ph  /\  w  e.  A )  /\  -.  w  =  0 )  ->  w  e.  RR )
8178simp2d 1010 . . . . . . . . . . . . . 14  |-  ( (
ph  /\  w  e.  A )  ->  0  <_  w )
82 leloe 9674 . . . . . . . . . . . . . . 15  |-  ( ( 0  e.  RR  /\  w  e.  RR )  ->  ( 0  <_  w  <->  ( 0  <  w  \/  0  =  w ) ) )
8374, 79, 82sylancr 663 . . . . . . . . . . . . . 14  |-  ( (
ph  /\  w  e.  A )  ->  (
0  <_  w  <->  ( 0  <  w  \/  0  =  w ) ) )
8481, 83mpbid 210 . . . . . . . . . . . . 13  |-  ( (
ph  /\  w  e.  A )  ->  (
0  <  w  \/  0  =  w )
)
8584ord 377 . . . . . . . . . . . 12  |-  ( (
ph  /\  w  e.  A )  ->  ( -.  0  <  w  -> 
0  =  w ) )
86 eqcom 2452 . . . . . . . . . . . 12  |-  ( 0  =  w  <->  w  = 
0 )
8785, 86syl6ib 226 . . . . . . . . . . 11  |-  ( (
ph  /\  w  e.  A )  ->  ( -.  0  <  w  ->  w  =  0 ) )
8887con1d 124 . . . . . . . . . 10  |-  ( (
ph  /\  w  e.  A )  ->  ( -.  w  =  0  ->  0  <  w ) )
8988imp 429 . . . . . . . . 9  |-  ( ( ( ph  /\  w  e.  A )  /\  -.  w  =  0 )  ->  0  <  w
)
9080, 89elrpd 11263 . . . . . . . 8  |-  ( ( ( ph  /\  w  e.  A )  /\  -.  w  =  0 )  ->  w  e.  RR+ )
91 rpcnne0 11246 . . . . . . . . 9  |-  ( w  e.  RR+  ->  ( w  e.  CC  /\  w  =/=  0 ) )
92 recrec 10247 . . . . . . . . 9  |-  ( ( w  e.  CC  /\  w  =/=  0 )  -> 
( 1  /  (
1  /  w ) )  =  w )
9391, 92syl 16 . . . . . . . 8  |-  ( w  e.  RR+  ->  ( 1  /  ( 1  /  w ) )  =  w )
9490, 93syl 16 . . . . . . 7  |-  ( ( ( ph  /\  w  e.  A )  /\  -.  w  =  0 )  ->  ( 1  / 
( 1  /  w
) )  =  w )
9594csbeq1d 3427 . . . . . 6  |-  ( ( ( ph  /\  w  e.  A )  /\  -.  w  =  0 )  ->  [_ ( 1  / 
( 1  /  w
) )  /  x ]_ R  =  [_ w  /  x ]_ R )
96 simplr 755 . . . . . . . . 9  |-  ( ( ( ph  /\  w  e.  A )  /\  -.  w  =  0 )  ->  w  e.  A
)
97 simpll 753 . . . . . . . . . 10  |-  ( ( ( ph  /\  w  e.  A )  /\  -.  w  =  0 )  ->  ph )
98 rpreccl 11252 . . . . . . . . . . . . 13  |-  ( w  e.  RR+  ->  ( 1  /  w )  e.  RR+ )
9998adantl 466 . . . . . . . . . . . 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 2857 . . . . . . . . . . . . 13  |-  ( ph  ->  A. y  e.  RR+  ( y  e.  B  <->  ( 1  /  y )  e.  A ) )
102101adantr 465 . . . . . . . . . . . 12  |-  ( (
ph  /\  w  e.  RR+ )  ->  A. y  e.  RR+  ( y  e.  B  <->  ( 1  / 
y )  e.  A
) )
103 eleq1 2515 . . . . . . . . . . . . . 14  |-  ( y  =  ( 1  /  w )  ->  (
y  e.  B  <->  ( 1  /  w )  e.  B ) )
104 oveq2 6289 . . . . . . . . . . . . . . 15  |-  ( y  =  ( 1  /  w )  ->  (
1  /  y )  =  ( 1  / 
( 1  /  w
) ) )
105104eleq1d 2512 . . . . . . . . . . . . . 14  |-  ( y  =  ( 1  /  w )  ->  (
( 1  /  y
)  e.  A  <->  ( 1  /  ( 1  /  w ) )  e.  A ) )
106103, 105bibi12d 321 . . . . . . . . . . . . 13  |-  ( y  =  ( 1  /  w )  ->  (
( y  e.  B  <->  ( 1  /  y )  e.  A )  <->  ( (
1  /  w )  e.  B  <->  ( 1  /  ( 1  /  w ) )  e.  A ) ) )
107106rspcv 3192 . . . . . . . . . . . 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 60 . . . . . . . . . . 11  |-  ( (
ph  /\  w  e.  RR+ )  ->  ( (
1  /  w )  e.  B  <->  ( 1  /  ( 1  /  w ) )  e.  A ) )
10993adantl 466 . . . . . . . . . . . 12  |-  ( (
ph  /\  w  e.  RR+ )  ->  ( 1  /  ( 1  /  w ) )  =  w )
110109eleq1d 2512 . . . . . . . . . . 11  |-  ( (
ph  /\  w  e.  RR+ )  ->  ( (
1  /  ( 1  /  w ) )  e.  A  <->  w  e.  A ) )
111108, 110bitr2d 254 . . . . . . . . . 10  |-  ( (
ph  /\  w  e.  RR+ )  ->  ( w  e.  A  <->  ( 1  /  w )  e.  B
) )
11297, 90, 111syl2anc 661 . . . . . . . . 9  |-  ( ( ( ph  /\  w  e.  A )  /\  -.  w  =  0 )  ->  ( w  e.  A  <->  ( 1  /  w )  e.  B
) )
11396, 112mpbid 210 . . . . . . . 8  |-  ( ( ( ph  /\  w  e.  A )  /\  -.  w  =  0 )  ->  ( 1  /  w )  e.  B
)
11490rpreccld 11275 . . . . . . . 8  |-  ( ( ( ph  /\  w  e.  A )  /\  -.  w  =  0 )  ->  ( 1  /  w )  e.  RR+ )
115113, 114elind 3673 . . . . . . 7  |-  ( ( ( ph  /\  w  e.  A )  /\  -.  w  =  0 )  ->  ( 1  /  w )  e.  ( B  i^i  RR+ )
)
11665, 31eqeltrd 2531 . . . . . . . . 9  |-  ( (
ph  /\  y  e.  ( B  i^i  RR+ )
)  ->  [_ ( 1  /  y )  /  x ]_ R  e.  CC )
117116ralrimiva 2857 . . . . . . . 8  |-  ( ph  ->  A. y  e.  ( B  i^i  RR+ ) [_ ( 1  /  y
)  /  x ]_ R  e.  CC )
118117ad2antrr 725 . . . . . . 7  |-  ( ( ( ph  /\  w  e.  A )  /\  -.  w  =  0 )  ->  A. y  e.  ( B  i^i  RR+ ) [_ ( 1  /  y
)  /  x ]_ R  e.  CC )
119104csbeq1d 3427 . . . . . . . . 9  |-  ( y  =  ( 1  /  w )  ->  [_ (
1  /  y )  /  x ]_ R  =  [_ ( 1  / 
( 1  /  w
) )  /  x ]_ R )
120119eleq1d 2512 . . . . . . . 8  |-  ( y  =  ( 1  /  w )  ->  ( [_ ( 1  /  y
)  /  x ]_ R  e.  CC  <->  [_ ( 1  /  ( 1  /  w ) )  /  x ]_ R  e.  CC ) )
121120rspcv 3192 . . . . . . 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 60 . . . . . 6  |-  ( ( ( ph  /\  w  e.  A )  /\  -.  w  =  0 )  ->  [_ ( 1  / 
( 1  /  w
) )  /  x ]_ R  e.  CC )
12395, 122eqeltrrd 2532 . . . . 5  |-  ( ( ( ph  /\  w  e.  A )  /\  -.  w  =  0 )  ->  [_ w  /  x ]_ R  e.  CC )
12472, 123ifclda 3958 . . . 4  |-  ( (
ph  /\  w  e.  A )  ->  if ( w  =  0 ,  C ,  [_ w  /  x ]_ R )  e.  CC )
12599biantrud 507 . . . . . 6  |-  ( (
ph  /\  w  e.  RR+ )  ->  ( (
1  /  w )  e.  B  <->  ( (
1  /  w )  e.  B  /\  (
1  /  w )  e.  RR+ ) ) )
126111, 125bitrd 253 . . . . 5  |-  ( (
ph  /\  w  e.  RR+ )  ->  ( w  e.  A  <->  ( ( 1  /  w )  e.  B  /\  ( 1  /  w )  e.  RR+ ) ) )
127 elin 3672 . . . . 5  |-  ( ( 1  /  w )  e.  ( B  i^i  RR+ )  <->  ( ( 1  /  w )  e.  B  /\  ( 1  /  w )  e.  RR+ ) )
128126, 127syl6bbr 263 . . . 4  |-  ( (
ph  /\  w  e.  RR+ )  ->  ( w  e.  A  <->  ( 1  /  w )  e.  ( B  i^i  RR+ )
) )
129 iftrue 3932 . . . 4  |-  ( w  =  0  ->  if ( w  =  0 ,  C ,  [_ w  /  x ]_ R )  =  C )
130 eqeq1 2447 . . . . 5  |-  ( w  =  ( 1  / 
y )  ->  (
w  =  0  <->  (
1  /  y )  =  0 ) )
131 csbeq1 3423 . . . . 5  |-  ( w  =  ( 1  / 
y )  ->  [_ w  /  x ]_ R  = 
[_ ( 1  / 
y )  /  x ]_ R )
132130, 131ifbieq2d 3951 . . . 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 23167 . . 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 2605 . . . . 5  |-  F/_ w if ( x  =  0 ,  C ,  R
)
137 nfv 1694 . . . . . 6  |-  F/ x  w  =  0
138 nfcv 2605 . . . . . 6  |-  F/_ x C
139 nfcsb1v 3436 . . . . . 6  |-  F/_ x [_ w  /  x ]_ R
140137, 138, 139nfif 3955 . . . . 5  |-  F/_ x if ( w  =  0 ,  C ,  [_ w  /  x ]_ R
)
141 eqeq1 2447 . . . . . 6  |-  ( x  =  w  ->  (
x  =  0  <->  w  =  0 ) )
142 csbeq1a 3429 . . . . . 6  |-  ( x  =  w  ->  R  =  [_ w  /  x ]_ R )
143141, 142ifbieq2d 3951 . . . . 5  |-  ( x  =  w  ->  if ( x  =  0 ,  C ,  R )  =  if ( w  =  0 ,  C ,  [_ w  /  x ]_ R ) )
144136, 140, 143cbvmpt 4527 . . . 4  |-  ( x  e.  A  |->  if ( x  =  0 ,  C ,  R ) )  =  ( w  e.  A  |->  if ( w  =  0 ,  C ,  [_ w  /  x ]_ R ) )
145144eleq1i 2520 . . 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 263 . 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 281 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 184    \/ wo 368    /\ wa 369    /\ w3a 974    = wceq 1383    e. wcel 1804    =/= wne 2638   A.wral 2793   [_csb 3420    i^i cin 3460    C_ wss 3461   ifcif 3926   class class class wbr 4437    |-> cmpt 4495   dom cdm 4989    |` cres 4991   Rel wrel 4994    Fn wfn 5573   -->wf 5574   ` cfv 5578  (class class class)co 6281   CCcc 9493   RRcr 9494   0cc0 9495   1c1 9496   +oocpnf 9628   RR*cxr 9630    < clt 9631    <_ cle 9632    / cdiv 10212   RR+crp 11229   (,)cioo 11538   [,)cico 11540    ~~> r crli 13287   ↾t crest 14695   TopOpenctopn 14696  ℂfldccnfld 18294    CnP ccnp 19599
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1605  ax-4 1618  ax-5 1691  ax-6 1734  ax-7 1776  ax-8 1806  ax-9 1808  ax-10 1823  ax-11 1828  ax-12 1840  ax-13 1985  ax-ext 2421  ax-rep 4548  ax-sep 4558  ax-nul 4566  ax-pow 4615  ax-pr 4676  ax-un 6577  ax-cnex 9551  ax-resscn 9552  ax-1cn 9553  ax-icn 9554  ax-addcl 9555  ax-addrcl 9556  ax-mulcl 9557  ax-mulrcl 9558  ax-mulcom 9559  ax-addass 9560  ax-mulass 9561  ax-distr 9562  ax-i2m1 9563  ax-1ne0 9564  ax-1rid 9565  ax-rnegex 9566  ax-rrecex 9567  ax-cnre 9568  ax-pre-lttri 9569  ax-pre-lttrn 9570  ax-pre-ltadd 9571  ax-pre-mulgt0 9572  ax-pre-sup 9573
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 975  df-3an 976  df-tru 1386  df-ex 1600  df-nf 1604  df-sb 1727  df-eu 2272  df-mo 2273  df-clab 2429  df-cleq 2435  df-clel 2438  df-nfc 2593  df-ne 2640  df-nel 2641  df-ral 2798  df-rex 2799  df-reu 2800  df-rmo 2801  df-rab 2802  df-v 3097  df-sbc 3314  df-csb 3421  df-dif 3464  df-un 3466  df-in 3468  df-ss 3475  df-pss 3477  df-nul 3771  df-if 3927  df-pw 3999  df-sn 4015  df-pr 4017  df-tp 4019  df-op 4021  df-uni 4235  df-int 4272  df-iun 4317  df-br 4438  df-opab 4496  df-mpt 4497  df-tr 4531  df-eprel 4781  df-id 4785  df-po 4790  df-so 4791  df-fr 4828  df-we 4830  df-ord 4871  df-on 4872  df-lim 4873  df-suc 4874  df-xp 4995  df-rel 4996  df-cnv 4997  df-co 4998  df-dm 4999  df-rn 5000  df-res 5001  df-ima 5002  df-iota 5541  df-fun 5580  df-fn 5581  df-f 5582  df-f1 5583  df-fo 5584  df-f1o 5585  df-fv 5586  df-riota 6242  df-ov 6284  df-oprab 6285  df-mpt2 6286  df-om 6686  df-1st 6785  df-2nd 6786  df-recs 7044  df-rdg 7078  df-1o 7132  df-oadd 7136  df-er 7313  df-map 7424  df-pm 7425  df-en 7519  df-dom 7520  df-sdom 7521  df-fin 7522  df-sup 7903  df-pnf 9633  df-mnf 9634  df-xr 9635  df-ltxr 9636  df-le 9637  df-sub 9812  df-neg 9813  df-div 10213  df-nn 10543  df-2 10600  df-3 10601  df-4 10602  df-5 10603  df-6 10604  df-7 10605  df-8 10606  df-9 10607  df-10 10608  df-n0 10802  df-z 10871  df-dec 10985  df-uz 11091  df-q 11192  df-rp 11230  df-xneg 11327  df-xadd 11328  df-xmul 11329  df-ioo 11542  df-ico 11544  df-fz 11682  df-seq 12087  df-exp 12146  df-cj 12911  df-re 12912  df-im 12913  df-sqrt 13047  df-abs 13048  df-rlim 13291  df-struct 14511  df-ndx 14512  df-slot 14513  df-base 14514  df-plusg 14587  df-mulr 14588  df-starv 14589  df-tset 14593  df-ple 14594  df-ds 14596  df-unif 14597  df-rest 14697  df-topn 14698  df-topgen 14718  df-psmet 18285  df-xmet 18286  df-met 18287  df-bl 18288  df-mopn 18289  df-cnfld 18295  df-top 19272  df-bases 19274  df-topon 19275  df-cnp 19602
This theorem is referenced by:  rlimcnp3  23169
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