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Theorem rlimcnp2 20277
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 3402 . . . . . . . 8  |-  ( B  i^i  ( 1 [,) 
+oo ) )  C_  B
2 resmpt 5016 . . . . . . . 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 11 . . . . . . 7  |-  ( ph  ->  ( ( y  e.  B  |->  S )  |`  ( B  i^i  (
1 [,)  +oo ) ) )  =  ( y  e.  ( B  i^i  ( 1 [,)  +oo ) )  |->  S ) )
4 0xr 8894 . . . . . . . . . . 11  |-  0  e.  RR*
5 0lt1 9312 . . . . . . . . . . 11  |-  0  <  1
6 df-ioo 10676 . . . . . . . . . . . 12  |-  (,)  =  ( x  e.  RR* ,  y  e.  RR*  |->  { z  e.  RR*  |  (
x  <  z  /\  z  <  y ) } )
7 df-ico 10678 . . . . . . . . . . . 12  |-  [,)  =  ( x  e.  RR* ,  y  e.  RR*  |->  { z  e.  RR*  |  (
x  <_  z  /\  z  <  y ) } )
8 xrltletr 10504 . . . . . . . . . . . 12  |-  ( ( 0  e.  RR*  /\  1  e.  RR*  /\  w  e. 
RR* )  ->  (
( 0  <  1  /\  1  <_  w )  ->  0  <  w
) )
96, 7, 8ixxss1 10690 . . . . . . . . . . 11  |-  ( ( 0  e.  RR*  /\  0  <  1 )  ->  (
1 [,)  +oo )  C_  ( 0 (,)  +oo ) )
104, 5, 9mp2an 653 . . . . . . . . . 10  |-  ( 1 [,)  +oo )  C_  (
0 (,)  +oo )
11 ioorp 10743 . . . . . . . . . 10  |-  ( 0 (,)  +oo )  =  RR+
1210, 11sseqtri 3223 . . . . . . . . 9  |-  ( 1 [,)  +oo )  C_  RR+
13 sslin 3408 . . . . . . . . 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 5016 . . . . . . . 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 11 . . . . . . 7  |-  ( ph  ->  ( ( y  e.  ( B  i^i  RR+ )  |->  S )  |`  ( B  i^i  ( 1 [,) 
+oo ) ) )  =  ( y  e.  ( B  i^i  (
1 [,)  +oo ) ) 
|->  S ) )
173, 16eqtr4d 2331 . . . . . 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 4984 . . . . . 6  |-  ( ( ( y  e.  B  |->  S )  |`  B )  |`  ( 1 [,)  +oo ) )  =  ( ( y  e.  B  |->  S )  |`  ( B  i^i  ( 1 [,) 
+oo ) ) )
19 resres 4984 . . . . . 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 2353 . . . . 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 2296 . . . . . . . . 9  |-  ( y  e.  B  |->  S )  =  ( y  e.  B  |->  S )
2321, 22fmptd 5700 . . . . . . . 8  |-  ( ph  ->  ( y  e.  B  |->  S ) : B --> CC )
24 ffn 5405 . . . . . . . 8  |-  ( ( y  e.  B  |->  S ) : B --> CC  ->  ( y  e.  B  |->  S )  Fn  B )
2523, 24syl 15 . . . . . . 7  |-  ( ph  ->  ( y  e.  B  |->  S )  Fn  B
)
26 fnresdm 5369 . . . . . . 7  |-  ( ( y  e.  B  |->  S )  Fn  B  -> 
( ( y  e.  B  |->  S )  |`  B )  =  ( y  e.  B  |->  S ) )
2725, 26syl 15 . . . . . 6  |-  ( ph  ->  ( ( y  e.  B  |->  S )  |`  B )  =  ( y  e.  B  |->  S ) )
2827reseq1d 4970 . . . . 5  |-  ( ph  ->  ( ( ( y  e.  B  |->  S )  |`  B )  |`  (
1 [,)  +oo ) )  =  ( ( y  e.  B  |->  S )  |`  ( 1 [,)  +oo ) ) )
29 inss1 3402 . . . . . . . . . . 11  |-  ( B  i^i  RR+ )  C_  B
3029sseli 3189 . . . . . . . . . 10  |-  ( y  e.  ( B  i^i  RR+ )  ->  y  e.  B )
3130, 21sylan2 460 . . . . . . . . 9  |-  ( (
ph  /\  y  e.  ( B  i^i  RR+ )
)  ->  S  e.  CC )
32 eqid 2296 . . . . . . . . 9  |-  ( y  e.  ( B  i^i  RR+ )  |->  S )  =  ( y  e.  ( B  i^i  RR+ )  |->  S )
3331, 32fmptd 5700 . . . . . . . 8  |-  ( ph  ->  ( y  e.  ( B  i^i  RR+ )  |->  S ) : ( B  i^i  RR+ ) --> CC )
34 frel 5408 . . . . . . . 8  |-  ( ( y  e.  ( B  i^i  RR+ )  |->  S ) : ( B  i^i  RR+ ) --> CC  ->  Rel  ( y  e.  ( B  i^i  RR+ )  |->  S ) )
3533, 34syl 15 . . . . . . 7  |-  ( ph  ->  Rel  ( y  e.  ( B  i^i  RR+ )  |->  S ) )
36 fdm 5409 . . . . . . . . 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 15 . . . . . . . 8  |-  ( ph  ->  dom  ( y  e.  ( B  i^i  RR+ )  |->  S )  =  ( B  i^i  RR+ )
)
3829a1i 10 . . . . . . . 8  |-  ( ph  ->  ( B  i^i  RR+ )  C_  B )
3937, 38eqsstrd 3225 . . . . . . 7  |-  ( ph  ->  dom  ( y  e.  ( B  i^i  RR+ )  |->  S )  C_  B
)
40 relssres 5008 . . . . . . 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 ) )
4135, 39, 40syl2anc 642 . . . . . 6  |-  ( ph  ->  ( ( y  e.  ( B  i^i  RR+ )  |->  S )  |`  B )  =  ( y  e.  ( B  i^i  RR+ )  |->  S ) )
4241reseq1d 4970 . . . . 5  |-  ( ph  ->  ( ( ( y  e.  ( B  i^i  RR+ )  |->  S )  |`  B )  |`  (
1 [,)  +oo ) )  =  ( ( y  e.  ( B  i^i  RR+ )  |->  S )  |`  ( 1 [,)  +oo ) ) )
4320, 28, 423eqtr3d 2336 . . . 4  |-  ( ph  ->  ( ( y  e.  B  |->  S )  |`  ( 1 [,)  +oo ) )  =  ( ( y  e.  ( B  i^i  RR+ )  |->  S )  |`  (
1 [,)  +oo ) ) )
4443breq1d 4049 . . 3  |-  ( ph  ->  ( ( ( y  e.  B  |->  S )  |`  ( 1 [,)  +oo ) )  ~~> r  C  <->  ( ( y  e.  ( B  i^i  RR+ )  |->  S )  |`  (
1 [,)  +oo ) )  ~~> r  C ) )
45 rlimcnp2.b . . . 4  |-  ( ph  ->  B  C_  RR )
46 1re 8853 . . . . 5  |-  1  e.  RR
4746a1i 10 . . . 4  |-  ( ph  ->  1  e.  RR )
4823, 45, 47rlimresb 12055 . . 3  |-  ( ph  ->  ( ( y  e.  B  |->  S )  ~~> r  C  <->  ( ( y  e.  B  |->  S )  |`  (
1 [,)  +oo ) )  ~~> r  C ) )
4929, 45syl5ss 3203 . . . 4  |-  ( ph  ->  ( B  i^i  RR+ )  C_  RR )
5033, 49, 47rlimresb 12055 . . 3  |-  ( ph  ->  ( ( y  e.  ( B  i^i  RR+ )  |->  S )  ~~> r  C  <->  ( ( y  e.  ( B  i^i  RR+ )  |->  S )  |`  (
1 [,)  +oo ) )  ~~> r  C ) )
5144, 48, 503bitr4d 276 . 2  |-  ( ph  ->  ( ( y  e.  B  |->  S )  ~~> r  C  <->  ( y  e.  ( B  i^i  RR+ )  |->  S )  ~~> r  C ) )
52 inss2 3403 . . . . . . . . . . 11  |-  ( B  i^i  RR+ )  C_  RR+
5352a1i 10 . . . . . . . . . 10  |-  ( ph  ->  ( B  i^i  RR+ )  C_  RR+ )
5453sselda 3193 . . . . . . . . 9  |-  ( (
ph  /\  y  e.  ( B  i^i  RR+ )
)  ->  y  e.  RR+ )
5554rpreccld 10416 . . . . . . . 8  |-  ( (
ph  /\  y  e.  ( B  i^i  RR+ )
)  ->  ( 1  /  y )  e.  RR+ )
5655rpne0d 10411 . . . . . . 7  |-  ( (
ph  /\  y  e.  ( B  i^i  RR+ )
)  ->  ( 1  /  y )  =/=  0 )
5756neneqd 2475 . . . . . 6  |-  ( (
ph  /\  y  e.  ( B  i^i  RR+ )
)  ->  -.  (
1  /  y )  =  0 )
58 iffalse 3585 . . . . . 6  |-  ( -.  ( 1  /  y
)  =  0  ->  if ( ( 1  / 
y )  =  0 ,  C ,  [_ ( 1  /  y
)  /  x ]_ R )  =  [_ ( 1  /  y
)  /  x ]_ R )
5957, 58syl 15 . . . . 5  |-  ( (
ph  /\  y  e.  ( B  i^i  RR+ )
)  ->  if (
( 1  /  y
)  =  0 ,  C ,  [_ (
1  /  y )  /  x ]_ R
)  =  [_ (
1  /  y )  /  x ]_ R
)
60 oveq2 5882 . . . . . . . . . 10  |-  ( x  =  ( 1  / 
y )  ->  (
1  /  x )  =  ( 1  / 
( 1  /  y
) ) )
61 rpcnne0 10387 . . . . . . . . . . 11  |-  ( y  e.  RR+  ->  ( y  e.  CC  /\  y  =/=  0 ) )
62 recrec 9473 . . . . . . . . . . 11  |-  ( ( y  e.  CC  /\  y  =/=  0 )  -> 
( 1  /  (
1  /  y ) )  =  y )
6354, 61, 623syl 18 . . . . . . . . . 10  |-  ( (
ph  /\  y  e.  ( B  i^i  RR+ )
)  ->  ( 1  /  ( 1  / 
y ) )  =  y )
6460, 63sylan9eqr 2350 . . . . . . . . 9  |-  ( ( ( ph  /\  y  e.  ( B  i^i  RR+ )
)  /\  x  =  ( 1  /  y
) )  ->  (
1  /  x )  =  y )
6564eqcomd 2301 . . . . . . . 8  |-  ( ( ( ph  /\  y  e.  ( B  i^i  RR+ )
)  /\  x  =  ( 1  /  y
) )  ->  y  =  ( 1  /  x ) )
66 rlimcnp2.s . . . . . . . 8  |-  ( y  =  ( 1  /  x )  ->  S  =  R )
6765, 66syl 15 . . . . . . 7  |-  ( ( ( ph  /\  y  e.  ( B  i^i  RR+ )
)  /\  x  =  ( 1  /  y
) )  ->  S  =  R )
6867eqcomd 2301 . . . . . 6  |-  ( ( ( ph  /\  y  e.  ( B  i^i  RR+ )
)  /\  x  =  ( 1  /  y
) )  ->  R  =  S )
6955, 68csbied 3136 . . . . 5  |-  ( (
ph  /\  y  e.  ( B  i^i  RR+ )
)  ->  [_ ( 1  /  y )  /  x ]_ R  =  S )
7059, 69eqtrd 2328 . . . 4  |-  ( (
ph  /\  y  e.  ( B  i^i  RR+ )
)  ->  if (
( 1  /  y
)  =  0 ,  C ,  [_ (
1  /  y )  /  x ]_ R
)  =  S )
7170mpteq2dva 4122 . . 3  |-  ( ph  ->  ( y  e.  ( B  i^i  RR+ )  |->  if ( ( 1  /  y )  =  0 ,  C ,  [_ ( 1  /  y
)  /  x ]_ R ) )  =  ( y  e.  ( B  i^i  RR+ )  |->  S ) )
7271breq1d 4049 . 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 ) )
73 rlimcnp2.a . . . 4  |-  ( ph  ->  A  C_  ( 0 [,)  +oo ) )
74 rlimcnp2.0 . . . 4  |-  ( ph  ->  0  e.  A )
75 rlimcnp2.c . . . . . 6  |-  ( ph  ->  C  e.  CC )
7675ad2antrr 706 . . . . 5  |-  ( ( ( ph  /\  w  e.  A )  /\  w  =  0 )  ->  C  e.  CC )
7773sselda 3193 . . . . . . . . . . . 12  |-  ( (
ph  /\  w  e.  A )  ->  w  e.  ( 0 [,)  +oo ) )
78 0re 8854 . . . . . . . . . . . . 13  |-  0  e.  RR
79 pnfxr 10471 . . . . . . . . . . . . 13  |-  +oo  e.  RR*
80 elico2 10730 . . . . . . . . . . . . 13  |-  ( ( 0  e.  RR  /\  +oo 
e.  RR* )  ->  (
w  e.  ( 0 [,)  +oo )  <->  ( w  e.  RR  /\  0  <_  w  /\  w  <  +oo ) ) )
8178, 79, 80mp2an 653 . . . . . . . . . . . 12  |-  ( w  e.  ( 0 [,) 
+oo )  <->  ( w  e.  RR  /\  0  <_  w  /\  w  <  +oo ) )
8277, 81sylib 188 . . . . . . . . . . 11  |-  ( (
ph  /\  w  e.  A )  ->  (
w  e.  RR  /\  0  <_  w  /\  w  <  +oo ) )
8382simp1d 967 . . . . . . . . . 10  |-  ( (
ph  /\  w  e.  A )  ->  w  e.  RR )
8483adantr 451 . . . . . . . . 9  |-  ( ( ( ph  /\  w  e.  A )  /\  -.  w  =  0 )  ->  w  e.  RR )
8582simp2d 968 . . . . . . . . . . . . . 14  |-  ( (
ph  /\  w  e.  A )  ->  0  <_  w )
86 leloe 8924 . . . . . . . . . . . . . . 15  |-  ( ( 0  e.  RR  /\  w  e.  RR )  ->  ( 0  <_  w  <->  ( 0  <  w  \/  0  =  w ) ) )
8778, 83, 86sylancr 644 . . . . . . . . . . . . . 14  |-  ( (
ph  /\  w  e.  A )  ->  (
0  <_  w  <->  ( 0  <  w  \/  0  =  w ) ) )
8885, 87mpbid 201 . . . . . . . . . . . . 13  |-  ( (
ph  /\  w  e.  A )  ->  (
0  <  w  \/  0  =  w )
)
8988ord 366 . . . . . . . . . . . 12  |-  ( (
ph  /\  w  e.  A )  ->  ( -.  0  <  w  -> 
0  =  w ) )
90 eqcom 2298 . . . . . . . . . . . 12  |-  ( 0  =  w  <->  w  = 
0 )
9189, 90syl6ib 217 . . . . . . . . . . 11  |-  ( (
ph  /\  w  e.  A )  ->  ( -.  0  <  w  ->  w  =  0 ) )
9291con1d 116 . . . . . . . . . 10  |-  ( (
ph  /\  w  e.  A )  ->  ( -.  w  =  0  ->  0  <  w ) )
9392imp 418 . . . . . . . . 9  |-  ( ( ( ph  /\  w  e.  A )  /\  -.  w  =  0 )  ->  0  <  w
)
9484, 93elrpd 10404 . . . . . . . 8  |-  ( ( ( ph  /\  w  e.  A )  /\  -.  w  =  0 )  ->  w  e.  RR+ )
95 rpcnne0 10387 . . . . . . . . 9  |-  ( w  e.  RR+  ->  ( w  e.  CC  /\  w  =/=  0 ) )
96 recrec 9473 . . . . . . . . 9  |-  ( ( w  e.  CC  /\  w  =/=  0 )  -> 
( 1  /  (
1  /  w ) )  =  w )
9795, 96syl 15 . . . . . . . 8  |-  ( w  e.  RR+  ->  ( 1  /  ( 1  /  w ) )  =  w )
9894, 97syl 15 . . . . . . 7  |-  ( ( ( ph  /\  w  e.  A )  /\  -.  w  =  0 )  ->  ( 1  / 
( 1  /  w
) )  =  w )
9998csbeq1d 3100 . . . . . 6  |-  ( ( ( ph  /\  w  e.  A )  /\  -.  w  =  0 )  ->  [_ ( 1  / 
( 1  /  w
) )  /  x ]_ R  =  [_ w  /  x ]_ R )
100 simplr 731 . . . . . . . . 9  |-  ( ( ( ph  /\  w  e.  A )  /\  -.  w  =  0 )  ->  w  e.  A
)
101 simpll 730 . . . . . . . . . 10  |-  ( ( ( ph  /\  w  e.  A )  /\  -.  w  =  0 )  ->  ph )
102 rpreccl 10393 . . . . . . . . . . . . 13  |-  ( w  e.  RR+  ->  ( 1  /  w )  e.  RR+ )
103102adantl 452 . . . . . . . . . . . 12  |-  ( (
ph  /\  w  e.  RR+ )  ->  ( 1  /  w )  e.  RR+ )
104 rlimcnp2.d . . . . . . . . . . . . . 14  |-  ( (
ph  /\  y  e.  RR+ )  ->  ( y  e.  B  <->  ( 1  / 
y )  e.  A
) )
105104ralrimiva 2639 . . . . . . . . . . . . 13  |-  ( ph  ->  A. y  e.  RR+  ( y  e.  B  <->  ( 1  /  y )  e.  A ) )
106105adantr 451 . . . . . . . . . . . 12  |-  ( (
ph  /\  w  e.  RR+ )  ->  A. y  e.  RR+  ( y  e.  B  <->  ( 1  / 
y )  e.  A
) )
107 eleq1 2356 . . . . . . . . . . . . . 14  |-  ( y  =  ( 1  /  w )  ->  (
y  e.  B  <->  ( 1  /  w )  e.  B ) )
108 oveq2 5882 . . . . . . . . . . . . . . 15  |-  ( y  =  ( 1  /  w )  ->  (
1  /  y )  =  ( 1  / 
( 1  /  w
) ) )
109108eleq1d 2362 . . . . . . . . . . . . . 14  |-  ( y  =  ( 1  /  w )  ->  (
( 1  /  y
)  e.  A  <->  ( 1  /  ( 1  /  w ) )  e.  A ) )
110107, 109bibi12d 312 . . . . . . . . . . . . 13  |-  ( y  =  ( 1  /  w )  ->  (
( y  e.  B  <->  ( 1  /  y )  e.  A )  <->  ( (
1  /  w )  e.  B  <->  ( 1  /  ( 1  /  w ) )  e.  A ) ) )
111110rspcv 2893 . . . . . . . . . . . 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 ) ) )
112103, 106, 111sylc 56 . . . . . . . . . . 11  |-  ( (
ph  /\  w  e.  RR+ )  ->  ( (
1  /  w )  e.  B  <->  ( 1  /  ( 1  /  w ) )  e.  A ) )
11397adantl 452 . . . . . . . . . . . 12  |-  ( (
ph  /\  w  e.  RR+ )  ->  ( 1  /  ( 1  /  w ) )  =  w )
114113eleq1d 2362 . . . . . . . . . . 11  |-  ( (
ph  /\  w  e.  RR+ )  ->  ( (
1  /  ( 1  /  w ) )  e.  A  <->  w  e.  A ) )
115112, 114bitr2d 245 . . . . . . . . . 10  |-  ( (
ph  /\  w  e.  RR+ )  ->  ( w  e.  A  <->  ( 1  /  w )  e.  B
) )
116101, 94, 115syl2anc 642 . . . . . . . . 9  |-  ( ( ( ph  /\  w  e.  A )  /\  -.  w  =  0 )  ->  ( w  e.  A  <->  ( 1  /  w )  e.  B
) )
117100, 116mpbid 201 . . . . . . . 8  |-  ( ( ( ph  /\  w  e.  A )  /\  -.  w  =  0 )  ->  ( 1  /  w )  e.  B
)
11894rpreccld 10416 . . . . . . . 8  |-  ( ( ( ph  /\  w  e.  A )  /\  -.  w  =  0 )  ->  ( 1  /  w )  e.  RR+ )
119 elin 3371 . . . . . . . 8  |-  ( ( 1  /  w )  e.  ( B  i^i  RR+ )  <->  ( ( 1  /  w )  e.  B  /\  ( 1  /  w )  e.  RR+ ) )
120117, 118, 119sylanbrc 645 . . . . . . 7  |-  ( ( ( ph  /\  w  e.  A )  /\  -.  w  =  0 )  ->  ( 1  /  w )  e.  ( B  i^i  RR+ )
)
12169, 31eqeltrd 2370 . . . . . . . . 9  |-  ( (
ph  /\  y  e.  ( B  i^i  RR+ )
)  ->  [_ ( 1  /  y )  /  x ]_ R  e.  CC )
122121ralrimiva 2639 . . . . . . . 8  |-  ( ph  ->  A. y  e.  ( B  i^i  RR+ ) [_ ( 1  /  y
)  /  x ]_ R  e.  CC )
123122ad2antrr 706 . . . . . . 7  |-  ( ( ( ph  /\  w  e.  A )  /\  -.  w  =  0 )  ->  A. y  e.  ( B  i^i  RR+ ) [_ ( 1  /  y
)  /  x ]_ R  e.  CC )
124108csbeq1d 3100 . . . . . . . . 9  |-  ( y  =  ( 1  /  w )  ->  [_ (
1  /  y )  /  x ]_ R  =  [_ ( 1  / 
( 1  /  w
) )  /  x ]_ R )
125124eleq1d 2362 . . . . . . . 8  |-  ( y  =  ( 1  /  w )  ->  ( [_ ( 1  /  y
)  /  x ]_ R  e.  CC  <->  [_ ( 1  /  ( 1  /  w ) )  /  x ]_ R  e.  CC ) )
126125rspcv 2893 . . . . . . 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 ) )
127120, 123, 126sylc 56 . . . . . 6  |-  ( ( ( ph  /\  w  e.  A )  /\  -.  w  =  0 )  ->  [_ ( 1  / 
( 1  /  w
) )  /  x ]_ R  e.  CC )
12899, 127eqeltrrd 2371 . . . . 5  |-  ( ( ( ph  /\  w  e.  A )  /\  -.  w  =  0 )  ->  [_ w  /  x ]_ R  e.  CC )
12976, 128ifclda 3605 . . . 4  |-  ( (
ph  /\  w  e.  A )  ->  if ( w  =  0 ,  C ,  [_ w  /  x ]_ R )  e.  CC )
130103biantrud 493 . . . . . 6  |-  ( (
ph  /\  w  e.  RR+ )  ->  ( (
1  /  w )  e.  B  <->  ( (
1  /  w )  e.  B  /\  (
1  /  w )  e.  RR+ ) ) )
131115, 130bitrd 244 . . . . 5  |-  ( (
ph  /\  w  e.  RR+ )  ->  ( w  e.  A  <->  ( ( 1  /  w )  e.  B  /\  ( 1  /  w )  e.  RR+ ) ) )
132131, 119syl6bbr 254 . . . 4  |-  ( (
ph  /\  w  e.  RR+ )  ->  ( w  e.  A  <->  ( 1  /  w )  e.  ( B  i^i  RR+ )
) )
133 iftrue 3584 . . . 4  |-  ( w  =  0  ->  if ( w  =  0 ,  C ,  [_ w  /  x ]_ R )  =  C )
134 eqeq1 2302 . . . . 5  |-  ( w  =  ( 1  / 
y )  ->  (
w  =  0  <->  (
1  /  y )  =  0 ) )
135 csbeq1 3097 . . . . 5  |-  ( w  =  ( 1  / 
y )  ->  [_ w  /  x ]_ R  = 
[_ ( 1  / 
y )  /  x ]_ R )
136134, 135ifbieq2d 3598 . . . 4  |-  ( w  =  ( 1  / 
y )  ->  if ( w  =  0 ,  C ,  [_ w  /  x ]_ R )  =  if ( ( 1  /  y )  =  0 ,  C ,  [_ ( 1  / 
y )  /  x ]_ R ) )
137 rlimcnp2.j . . . 4  |-  J  =  ( TopOpen ` fld )
138 rlimcnp2.k . . . 4  |-  K  =  ( Jt  A )
13973, 74, 53, 129, 132, 133, 136, 137, 138rlimcnp 20276 . . 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 )
) )
140 nfcv 2432 . . . . 5  |-  F/_ w if ( x  =  0 ,  C ,  R
)
141 nfv 1609 . . . . . 6  |-  F/ x  w  =  0
142 nfcv 2432 . . . . . 6  |-  F/_ x C
143 nfcsb1v 3126 . . . . . 6  |-  F/_ x [_ w  /  x ]_ R
144141, 142, 143nfif 3602 . . . . 5  |-  F/_ x if ( w  =  0 ,  C ,  [_ w  /  x ]_ R
)
145 eqeq1 2302 . . . . . 6  |-  ( x  =  w  ->  (
x  =  0  <->  w  =  0 ) )
146 csbeq1a 3102 . . . . . 6  |-  ( x  =  w  ->  R  =  [_ w  /  x ]_ R )
147145, 146ifbieq2d 3598 . . . . 5  |-  ( x  =  w  ->  if ( x  =  0 ,  C ,  R )  =  if ( w  =  0 ,  C ,  [_ w  /  x ]_ R ) )
148140, 144, 147cbvmpt 4126 . . . 4  |-  ( x  e.  A  |->  if ( x  =  0 ,  C ,  R ) )  =  ( w  e.  A  |->  if ( w  =  0 ,  C ,  [_ w  /  x ]_ R ) )
149148eleq1i 2359 . . 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 )
)
150139, 149syl6bbr 254 . 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 )
) )
15151, 72, 1503bitr2d 272 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 176    \/ wo 357    /\ wa 358    /\ w3a 934    = wceq 1632    e. wcel 1696    =/= wne 2459   A.wral 2556   [_csb 3094    i^i cin 3164    C_ wss 3165   ifcif 3578   class class class wbr 4039    e. cmpt 4093   dom cdm 4705    |` cres 4707   Rel wrel 4710    Fn wfn 5266   -->wf 5267   ` cfv 5271  (class class class)co 5874   CCcc 8751   RRcr 8752   0cc0 8753   1c1 8754    +oocpnf 8880   RR*cxr 8882    < clt 8883    <_ cle 8884    / cdiv 9439   RR+crp 10370   (,)cioo 10672   [,)cico 10674    ~~> r crli 11975   ↾t crest 13341   TopOpenctopn 13342  ℂfldccnfld 16393    CnP ccnp 16971
This theorem is referenced by:  rlimcnp3  20278
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1536  ax-5 1547  ax-17 1606  ax-9 1644  ax-8 1661  ax-13 1698  ax-14 1700  ax-6 1715  ax-7 1720  ax-11 1727  ax-12 1878  ax-ext 2277  ax-rep 4147  ax-sep 4157  ax-nul 4165  ax-pow 4204  ax-pr 4230  ax-un 4528  ax-cnex 8809  ax-resscn 8810  ax-1cn 8811  ax-icn 8812  ax-addcl 8813  ax-addrcl 8814  ax-mulcl 8815  ax-mulrcl 8816  ax-mulcom 8817  ax-addass 8818  ax-mulass 8819  ax-distr 8820  ax-i2m1 8821  ax-1ne0 8822  ax-1rid 8823  ax-rnegex 8824  ax-rrecex 8825  ax-cnre 8826  ax-pre-lttri 8827  ax-pre-lttrn 8828  ax-pre-ltadd 8829  ax-pre-mulgt0 8830  ax-pre-sup 8831
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3or 935  df-3an 936  df-tru 1310  df-ex 1532  df-nf 1535  df-sb 1639  df-eu 2160  df-mo 2161  df-clab 2283  df-cleq 2289  df-clel 2292  df-nfc 2421  df-ne 2461  df-nel 2462  df-ral 2561  df-rex 2562  df-reu 2563  df-rmo 2564  df-rab 2565  df-v 2803  df-sbc 3005  df-csb 3095  df-dif 3168  df-un 3170  df-in 3172  df-ss 3179  df-pss 3181  df-nul 3469  df-if 3579  df-pw 3640  df-sn 3659  df-pr 3660  df-tp 3661  df-op 3662  df-uni 3844  df-int 3879  df-iun 3923  df-br 4040  df-opab 4094  df-mpt 4095  df-tr 4130  df-eprel 4321  df-id 4325  df-po 4330  df-so 4331  df-fr 4368  df-we 4370  df-ord 4411  df-on 4412  df-lim 4413  df-suc 4414  df-om 4673  df-xp 4711  df-rel 4712  df-cnv 4713  df-co 4714  df-dm 4715  df-rn 4716  df-res 4717  df-ima 4718  df-iota 5235  df-fun 5273  df-fn 5274  df-f 5275  df-f1 5276  df-fo 5277  df-f1o 5278  df-fv 5279  df-ov 5877  df-oprab 5878  df-mpt2 5879  df-1st 6138  df-2nd 6139  df-riota 6320  df-recs 6404  df-rdg 6439  df-1o 6495  df-oadd 6499  df-er 6676  df-map 6790  df-pm 6791  df-en 6880  df-dom 6881  df-sdom 6882  df-fin 6883  df-sup 7210  df-pnf 8885  df-mnf 8886  df-xr 8887  df-ltxr 8888  df-le 8889  df-sub 9055  df-neg 9056  df-div 9440  df-nn 9763  df-2 9820  df-3 9821  df-4 9822  df-5 9823  df-6 9824  df-7 9825  df-8 9826  df-9 9827  df-10 9828  df-n0 9982  df-z 10041  df-dec 10141  df-uz 10247  df-q 10333  df-rp 10371  df-xneg 10468  df-xadd 10469  df-xmul 10470  df-ioo 10676  df-ico 10678  df-fz 10799  df-seq 11063  df-exp 11121  df-cj 11600  df-re 11601  df-im 11602  df-sqr 11736  df-abs 11737  df-rlim 11979  df-struct 13166  df-ndx 13167  df-slot 13168  df-base 13169  df-plusg 13237  df-mulr 13238  df-starv 13239  df-tset 13243  df-ple 13244  df-ds 13246  df-rest 13343  df-topn 13344  df-topgen 13360  df-xmet 16389  df-met 16390  df-bl 16391  df-mopn 16392  df-cnfld 16394  df-top 16652  df-bases 16654  df-topon 16655  df-cnp 16974
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