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Theorem rlimcnp2 20806
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 3562 . . . . . . . 8  |-  ( B  i^i  ( 1 [,) 
+oo ) )  C_  B
2 resmpt 5192 . . . . . . . 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 9132 . . . . . . . . . . 11  |-  0  e.  RR*
5 0lt1 9551 . . . . . . . . . . 11  |-  0  <  1
6 df-ioo 10921 . . . . . . . . . . . 12  |-  (,)  =  ( x  e.  RR* ,  y  e.  RR*  |->  { z  e.  RR*  |  (
x  <  z  /\  z  <  y ) } )
7 df-ico 10923 . . . . . . . . . . . 12  |-  [,)  =  ( x  e.  RR* ,  y  e.  RR*  |->  { z  e.  RR*  |  (
x  <_  z  /\  z  <  y ) } )
8 xrltletr 10748 . . . . . . . . . . . 12  |-  ( ( 0  e.  RR*  /\  1  e.  RR*  /\  w  e. 
RR* )  ->  (
( 0  <  1  /\  1  <_  w )  ->  0  <  w
) )
96, 7, 8ixxss1 10935 . . . . . . . . . . 11  |-  ( ( 0  e.  RR*  /\  0  <  1 )  ->  (
1 [,)  +oo )  C_  ( 0 (,)  +oo ) )
104, 5, 9mp2an 655 . . . . . . . . . 10  |-  ( 1 [,)  +oo )  C_  (
0 (,)  +oo )
11 ioorp 10989 . . . . . . . . . 10  |-  ( 0 (,)  +oo )  =  RR+
1210, 11sseqtri 3381 . . . . . . . . 9  |-  ( 1 [,)  +oo )  C_  RR+
13 sslin 3568 . . . . . . . . 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 5192 . . . . . . . 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 2472 . . . . . 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 5160 . . . . . 6  |-  ( ( ( y  e.  B  |->  S )  |`  B )  |`  ( 1 [,)  +oo ) )  =  ( ( y  e.  B  |->  S )  |`  ( B  i^i  ( 1 [,) 
+oo ) ) )
19 resres 5160 . . . . . 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 2494 . . . . 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 2437 . . . . . . . . 9  |-  ( y  e.  B  |->  S )  =  ( y  e.  B  |->  S )
2321, 22fmptd 5894 . . . . . . . 8  |-  ( ph  ->  ( y  e.  B  |->  S ) : B --> CC )
24 ffn 5592 . . . . . . . 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 5555 . . . . . . 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 5146 . . . . 5  |-  ( ph  ->  ( ( ( y  e.  B  |->  S )  |`  B )  |`  (
1 [,)  +oo ) )  =  ( ( y  e.  B  |->  S )  |`  ( 1 [,)  +oo ) ) )
29 inss1 3562 . . . . . . . . . . 11  |-  ( B  i^i  RR+ )  C_  B
3029sseli 3345 . . . . . . . . . 10  |-  ( y  e.  ( B  i^i  RR+ )  ->  y  e.  B )
3130, 21sylan2 462 . . . . . . . . 9  |-  ( (
ph  /\  y  e.  ( B  i^i  RR+ )
)  ->  S  e.  CC )
32 eqid 2437 . . . . . . . . 9  |-  ( y  e.  ( B  i^i  RR+ )  |->  S )  =  ( y  e.  ( B  i^i  RR+ )  |->  S )
3331, 32fmptd 5894 . . . . . . . 8  |-  ( ph  ->  ( y  e.  ( B  i^i  RR+ )  |->  S ) : ( B  i^i  RR+ ) --> CC )
34 frel 5595 . . . . . . . 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 5596 . . . . . . . . 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 3399 . . . . . . 7  |-  ( ph  ->  dom  ( y  e.  ( B  i^i  RR+ )  |->  S )  C_  B
)
39 relssres 5184 . . . . . . 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 644 . . . . . 6  |-  ( ph  ->  ( ( y  e.  ( B  i^i  RR+ )  |->  S )  |`  B )  =  ( y  e.  ( B  i^i  RR+ )  |->  S ) )
4140reseq1d 5146 . . . . 5  |-  ( ph  ->  ( ( ( y  e.  ( B  i^i  RR+ )  |->  S )  |`  B )  |`  (
1 [,)  +oo ) )  =  ( ( y  e.  ( B  i^i  RR+ )  |->  S )  |`  ( 1 [,)  +oo ) ) )
4220, 28, 413eqtr3d 2477 . . . 4  |-  ( ph  ->  ( ( y  e.  B  |->  S )  |`  ( 1 [,)  +oo ) )  =  ( ( y  e.  ( B  i^i  RR+ )  |->  S )  |`  (
1 [,)  +oo ) ) )
4342breq1d 4223 . . 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 9091 . . . . 5  |-  1  e.  RR
4645a1i 11 . . . 4  |-  ( ph  ->  1  e.  RR )
4723, 44, 46rlimresb 12360 . . 3  |-  ( ph  ->  ( ( y  e.  B  |->  S )  ~~> r  C  <->  ( ( y  e.  B  |->  S )  |`  (
1 [,)  +oo ) )  ~~> r  C ) )
4829, 44syl5ss 3360 . . . 4  |-  ( ph  ->  ( B  i^i  RR+ )  C_  RR )
4933, 48, 46rlimresb 12360 . . 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 278 . 2  |-  ( ph  ->  ( ( y  e.  B  |->  S )  ~~> r  C  <->  ( y  e.  ( B  i^i  RR+ )  |->  S )  ~~> r  C ) )
51 inss2 3563 . . . . . . . . . . 11  |-  ( B  i^i  RR+ )  C_  RR+
5251a1i 11 . . . . . . . . . 10  |-  ( ph  ->  ( B  i^i  RR+ )  C_  RR+ )
5352sselda 3349 . . . . . . . . 9  |-  ( (
ph  /\  y  e.  ( B  i^i  RR+ )
)  ->  y  e.  RR+ )
5453rpreccld 10659 . . . . . . . 8  |-  ( (
ph  /\  y  e.  ( B  i^i  RR+ )
)  ->  ( 1  /  y )  e.  RR+ )
5554rpne0d 10654 . . . . . . 7  |-  ( (
ph  /\  y  e.  ( B  i^i  RR+ )
)  ->  ( 1  /  y )  =/=  0 )
5655neneqd 2618 . . . . . 6  |-  ( (
ph  /\  y  e.  ( B  i^i  RR+ )
)  ->  -.  (
1  /  y )  =  0 )
57 iffalse 3747 . . . . . 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 6090 . . . . . . . . . 10  |-  ( x  =  ( 1  / 
y )  ->  (
1  /  x )  =  ( 1  / 
( 1  /  y
) ) )
60 rpcnne0 10630 . . . . . . . . . . 11  |-  ( y  e.  RR+  ->  ( y  e.  CC  /\  y  =/=  0 ) )
61 recrec 9712 . . . . . . . . . . 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 2491 . . . . . . . . 9  |-  ( ( ( ph  /\  y  e.  ( B  i^i  RR+ )
)  /\  x  =  ( 1  /  y
) )  ->  (
1  /  x )  =  y )
6463eqcomd 2442 . . . . . . . 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 2442 . . . . . 6  |-  ( ( ( ph  /\  y  e.  ( B  i^i  RR+ )
)  /\  x  =  ( 1  /  y
) )  ->  R  =  S )
6854, 67csbied 3294 . . . . 5  |-  ( (
ph  /\  y  e.  ( B  i^i  RR+ )
)  ->  [_ ( 1  /  y )  /  x ]_ R  =  S )
6958, 68eqtrd 2469 . . . 4  |-  ( (
ph  /\  y  e.  ( B  i^i  RR+ )
)  ->  if (
( 1  /  y
)  =  0 ,  C ,  [_ (
1  /  y )  /  x ]_ R
)  =  S )
7069mpteq2dva 4296 . . 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 4223 . 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 708 . . . . 5  |-  ( ( ( ph  /\  w  e.  A )  /\  w  =  0 )  ->  C  e.  CC )
7672sselda 3349 . . . . . . . . . . . 12  |-  ( (
ph  /\  w  e.  A )  ->  w  e.  ( 0 [,)  +oo ) )
77 0re 9092 . . . . . . . . . . . . 13  |-  0  e.  RR
78 pnfxr 10714 . . . . . . . . . . . . 13  |-  +oo  e.  RR*
79 elico2 10975 . . . . . . . . . . . . 13  |-  ( ( 0  e.  RR  /\  +oo 
e.  RR* )  ->  (
w  e.  ( 0 [,)  +oo )  <->  ( w  e.  RR  /\  0  <_  w  /\  w  <  +oo ) ) )
8077, 78, 79mp2an 655 . . . . . . . . . . . 12  |-  ( w  e.  ( 0 [,) 
+oo )  <->  ( w  e.  RR  /\  0  <_  w  /\  w  <  +oo ) )
8176, 80sylib 190 . . . . . . . . . . 11  |-  ( (
ph  /\  w  e.  A )  ->  (
w  e.  RR  /\  0  <_  w  /\  w  <  +oo ) )
8281simp1d 970 . . . . . . . . . 10  |-  ( (
ph  /\  w  e.  A )  ->  w  e.  RR )
8382adantr 453 . . . . . . . . 9  |-  ( ( ( ph  /\  w  e.  A )  /\  -.  w  =  0 )  ->  w  e.  RR )
8481simp2d 971 . . . . . . . . . . . . . 14  |-  ( (
ph  /\  w  e.  A )  ->  0  <_  w )
85 leloe 9162 . . . . . . . . . . . . . . 15  |-  ( ( 0  e.  RR  /\  w  e.  RR )  ->  ( 0  <_  w  <->  ( 0  <  w  \/  0  =  w ) ) )
8677, 82, 85sylancr 646 . . . . . . . . . . . . . 14  |-  ( (
ph  /\  w  e.  A )  ->  (
0  <_  w  <->  ( 0  <  w  \/  0  =  w ) ) )
8784, 86mpbid 203 . . . . . . . . . . . . 13  |-  ( (
ph  /\  w  e.  A )  ->  (
0  <  w  \/  0  =  w )
)
8887ord 368 . . . . . . . . . . . 12  |-  ( (
ph  /\  w  e.  A )  ->  ( -.  0  <  w  -> 
0  =  w ) )
89 eqcom 2439 . . . . . . . . . . . 12  |-  ( 0  =  w  <->  w  = 
0 )
9088, 89syl6ib 219 . . . . . . . . . . 11  |-  ( (
ph  /\  w  e.  A )  ->  ( -.  0  <  w  ->  w  =  0 ) )
9190con1d 119 . . . . . . . . . 10  |-  ( (
ph  /\  w  e.  A )  ->  ( -.  w  =  0  ->  0  <  w ) )
9291imp 420 . . . . . . . . 9  |-  ( ( ( ph  /\  w  e.  A )  /\  -.  w  =  0 )  ->  0  <  w
)
9383, 92elrpd 10647 . . . . . . . 8  |-  ( ( ( ph  /\  w  e.  A )  /\  -.  w  =  0 )  ->  w  e.  RR+ )
94 rpcnne0 10630 . . . . . . . . 9  |-  ( w  e.  RR+  ->  ( w  e.  CC  /\  w  =/=  0 ) )
95 recrec 9712 . . . . . . . . 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 3258 . . . . . 6  |-  ( ( ( ph  /\  w  e.  A )  /\  -.  w  =  0 )  ->  [_ ( 1  / 
( 1  /  w
) )  /  x ]_ R  =  [_ w  /  x ]_ R )
99 simplr 733 . . . . . . . . 9  |-  ( ( ( ph  /\  w  e.  A )  /\  -.  w  =  0 )  ->  w  e.  A
)
100 simpll 732 . . . . . . . . . 10  |-  ( ( ( ph  /\  w  e.  A )  /\  -.  w  =  0 )  ->  ph )
101 rpreccl 10636 . . . . . . . . . . . . 13  |-  ( w  e.  RR+  ->  ( 1  /  w )  e.  RR+ )
102101adantl 454 . . . . . . . . . . . 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 2790 . . . . . . . . . . . . 13  |-  ( ph  ->  A. y  e.  RR+  ( y  e.  B  <->  ( 1  /  y )  e.  A ) )
105104adantr 453 . . . . . . . . . . . 12  |-  ( (
ph  /\  w  e.  RR+ )  ->  A. y  e.  RR+  ( y  e.  B  <->  ( 1  / 
y )  e.  A
) )
106 eleq1 2497 . . . . . . . . . . . . . 14  |-  ( y  =  ( 1  /  w )  ->  (
y  e.  B  <->  ( 1  /  w )  e.  B ) )
107 oveq2 6090 . . . . . . . . . . . . . . 15  |-  ( y  =  ( 1  /  w )  ->  (
1  /  y )  =  ( 1  / 
( 1  /  w
) ) )
108107eleq1d 2503 . . . . . . . . . . . . . 14  |-  ( y  =  ( 1  /  w )  ->  (
( 1  /  y
)  e.  A  <->  ( 1  /  ( 1  /  w ) )  e.  A ) )
109106, 108bibi12d 314 . . . . . . . . . . . . 13  |-  ( y  =  ( 1  /  w )  ->  (
( y  e.  B  <->  ( 1  /  y )  e.  A )  <->  ( (
1  /  w )  e.  B  <->  ( 1  /  ( 1  /  w ) )  e.  A ) ) )
110109rspcv 3049 . . . . . . . . . . . 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 59 . . . . . . . . . . 11  |-  ( (
ph  /\  w  e.  RR+ )  ->  ( (
1  /  w )  e.  B  <->  ( 1  /  ( 1  /  w ) )  e.  A ) )
11296adantl 454 . . . . . . . . . . . 12  |-  ( (
ph  /\  w  e.  RR+ )  ->  ( 1  /  ( 1  /  w ) )  =  w )
113112eleq1d 2503 . . . . . . . . . . 11  |-  ( (
ph  /\  w  e.  RR+ )  ->  ( (
1  /  ( 1  /  w ) )  e.  A  <->  w  e.  A ) )
114111, 113bitr2d 247 . . . . . . . . . 10  |-  ( (
ph  /\  w  e.  RR+ )  ->  ( w  e.  A  <->  ( 1  /  w )  e.  B
) )
115100, 93, 114syl2anc 644 . . . . . . . . 9  |-  ( ( ( ph  /\  w  e.  A )  /\  -.  w  =  0 )  ->  ( w  e.  A  <->  ( 1  /  w )  e.  B
) )
11699, 115mpbid 203 . . . . . . . 8  |-  ( ( ( ph  /\  w  e.  A )  /\  -.  w  =  0 )  ->  ( 1  /  w )  e.  B
)
11793rpreccld 10659 . . . . . . . 8  |-  ( ( ( ph  /\  w  e.  A )  /\  -.  w  =  0 )  ->  ( 1  /  w )  e.  RR+ )
118 elin 3531 . . . . . . . 8  |-  ( ( 1  /  w )  e.  ( B  i^i  RR+ )  <->  ( ( 1  /  w )  e.  B  /\  ( 1  /  w )  e.  RR+ ) )
119116, 117, 118sylanbrc 647 . . . . . . 7  |-  ( ( ( ph  /\  w  e.  A )  /\  -.  w  =  0 )  ->  ( 1  /  w )  e.  ( B  i^i  RR+ )
)
12068, 31eqeltrd 2511 . . . . . . . . 9  |-  ( (
ph  /\  y  e.  ( B  i^i  RR+ )
)  ->  [_ ( 1  /  y )  /  x ]_ R  e.  CC )
121120ralrimiva 2790 . . . . . . . 8  |-  ( ph  ->  A. y  e.  ( B  i^i  RR+ ) [_ ( 1  /  y
)  /  x ]_ R  e.  CC )
122121ad2antrr 708 . . . . . . 7  |-  ( ( ( ph  /\  w  e.  A )  /\  -.  w  =  0 )  ->  A. y  e.  ( B  i^i  RR+ ) [_ ( 1  /  y
)  /  x ]_ R  e.  CC )
123107csbeq1d 3258 . . . . . . . . 9  |-  ( y  =  ( 1  /  w )  ->  [_ (
1  /  y )  /  x ]_ R  =  [_ ( 1  / 
( 1  /  w
) )  /  x ]_ R )
124123eleq1d 2503 . . . . . . . 8  |-  ( y  =  ( 1  /  w )  ->  ( [_ ( 1  /  y
)  /  x ]_ R  e.  CC  <->  [_ ( 1  /  ( 1  /  w ) )  /  x ]_ R  e.  CC ) )
125124rspcv 3049 . . . . . . 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 59 . . . . . 6  |-  ( ( ( ph  /\  w  e.  A )  /\  -.  w  =  0 )  ->  [_ ( 1  / 
( 1  /  w
) )  /  x ]_ R  e.  CC )
12798, 126eqeltrrd 2512 . . . . 5  |-  ( ( ( ph  /\  w  e.  A )  /\  -.  w  =  0 )  ->  [_ w  /  x ]_ R  e.  CC )
12875, 127ifclda 3767 . . . 4  |-  ( (
ph  /\  w  e.  A )  ->  if ( w  =  0 ,  C ,  [_ w  /  x ]_ R )  e.  CC )
129102biantrud 495 . . . . . 6  |-  ( (
ph  /\  w  e.  RR+ )  ->  ( (
1  /  w )  e.  B  <->  ( (
1  /  w )  e.  B  /\  (
1  /  w )  e.  RR+ ) ) )
130114, 129bitrd 246 . . . . 5  |-  ( (
ph  /\  w  e.  RR+ )  ->  ( w  e.  A  <->  ( ( 1  /  w )  e.  B  /\  ( 1  /  w )  e.  RR+ ) ) )
131130, 118syl6bbr 256 . . . 4  |-  ( (
ph  /\  w  e.  RR+ )  ->  ( w  e.  A  <->  ( 1  /  w )  e.  ( B  i^i  RR+ )
) )
132 iftrue 3746 . . . 4  |-  ( w  =  0  ->  if ( w  =  0 ,  C ,  [_ w  /  x ]_ R )  =  C )
133 eqeq1 2443 . . . . 5  |-  ( w  =  ( 1  / 
y )  ->  (
w  =  0  <->  (
1  /  y )  =  0 ) )
134 csbeq1 3255 . . . . 5  |-  ( w  =  ( 1  / 
y )  ->  [_ w  /  x ]_ R  = 
[_ ( 1  / 
y )  /  x ]_ R )
135133, 134ifbieq2d 3760 . . . 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 20805 . . 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 2573 . . . . 5  |-  F/_ w if ( x  =  0 ,  C ,  R
)
140 nfv 1630 . . . . . 6  |-  F/ x  w  =  0
141 nfcv 2573 . . . . . 6  |-  F/_ x C
142 nfcsb1v 3284 . . . . . 6  |-  F/_ x [_ w  /  x ]_ R
143140, 141, 142nfif 3764 . . . . 5  |-  F/_ x if ( w  =  0 ,  C ,  [_ w  /  x ]_ R
)
144 eqeq1 2443 . . . . . 6  |-  ( x  =  w  ->  (
x  =  0  <->  w  =  0 ) )
145 csbeq1a 3260 . . . . . 6  |-  ( x  =  w  ->  R  =  [_ w  /  x ]_ R )
146144, 145ifbieq2d 3760 . . . . 5  |-  ( x  =  w  ->  if ( x  =  0 ,  C ,  R )  =  if ( w  =  0 ,  C ,  [_ w  /  x ]_ R ) )
147139, 143, 146cbvmpt 4300 . . . 4  |-  ( x  e.  A  |->  if ( x  =  0 ,  C ,  R ) )  =  ( w  e.  A  |->  if ( w  =  0 ,  C ,  [_ w  /  x ]_ R ) )
148147eleq1i 2500 . . 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 256 . 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 274 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 178    \/ wo 359    /\ wa 360    /\ w3a 937    = wceq 1653    e. wcel 1726    =/= wne 2600   A.wral 2706   [_csb 3252    i^i cin 3320    C_ wss 3321   ifcif 3740   class class class wbr 4213    e. cmpt 4267   dom cdm 4879    |` cres 4881   Rel wrel 4884    Fn wfn 5450   -->wf 5451   ` cfv 5455  (class class class)co 6082   CCcc 8989   RRcr 8990   0cc0 8991   1c1 8992    +oocpnf 9118   RR*cxr 9120    < clt 9121    <_ cle 9122    / cdiv 9678   RR+crp 10613   (,)cioo 10917   [,)cico 10919    ~~> r crli 12280   ↾t crest 13649   TopOpenctopn 13650  ℂfldccnfld 16704    CnP ccnp 17290
This theorem is referenced by:  rlimcnp3  20807
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1556  ax-5 1567  ax-17 1627  ax-9 1667  ax-8 1688  ax-13 1728  ax-14 1730  ax-6 1745  ax-7 1750  ax-11 1762  ax-12 1951  ax-ext 2418  ax-rep 4321  ax-sep 4331  ax-nul 4339  ax-pow 4378  ax-pr 4404  ax-un 4702  ax-cnex 9047  ax-resscn 9048  ax-1cn 9049  ax-icn 9050  ax-addcl 9051  ax-addrcl 9052  ax-mulcl 9053  ax-mulrcl 9054  ax-mulcom 9055  ax-addass 9056  ax-mulass 9057  ax-distr 9058  ax-i2m1 9059  ax-1ne0 9060  ax-1rid 9061  ax-rnegex 9062  ax-rrecex 9063  ax-cnre 9064  ax-pre-lttri 9065  ax-pre-lttrn 9066  ax-pre-ltadd 9067  ax-pre-mulgt0 9068  ax-pre-sup 9069
This theorem depends on definitions:  df-bi 179  df-or 361  df-an 362  df-3or 938  df-3an 939  df-tru 1329  df-ex 1552  df-nf 1555  df-sb 1660  df-eu 2286  df-mo 2287  df-clab 2424  df-cleq 2430  df-clel 2433  df-nfc 2562  df-ne 2602  df-nel 2603  df-ral 2711  df-rex 2712  df-reu 2713  df-rmo 2714  df-rab 2715  df-v 2959  df-sbc 3163  df-csb 3253  df-dif 3324  df-un 3326  df-in 3328  df-ss 3335  df-pss 3337  df-nul 3630  df-if 3741  df-pw 3802  df-sn 3821  df-pr 3822  df-tp 3823  df-op 3824  df-uni 4017  df-int 4052  df-iun 4096  df-br 4214  df-opab 4268  df-mpt 4269  df-tr 4304  df-eprel 4495  df-id 4499  df-po 4504  df-so 4505  df-fr 4542  df-we 4544  df-ord 4585  df-on 4586  df-lim 4587  df-suc 4588  df-om 4847  df-xp 4885  df-rel 4886  df-cnv 4887  df-co 4888  df-dm 4889  df-rn 4890  df-res 4891  df-ima 4892  df-iota 5419  df-fun 5457  df-fn 5458  df-f 5459  df-f1 5460  df-fo 5461  df-f1o 5462  df-fv 5463  df-ov 6085  df-oprab 6086  df-mpt2 6087  df-1st 6350  df-2nd 6351  df-riota 6550  df-recs 6634  df-rdg 6669  df-1o 6725  df-oadd 6729  df-er 6906  df-map 7021  df-pm 7022  df-en 7111  df-dom 7112  df-sdom 7113  df-fin 7114  df-sup 7447  df-pnf 9123  df-mnf 9124  df-xr 9125  df-ltxr 9126  df-le 9127  df-sub 9294  df-neg 9295  df-div 9679  df-nn 10002  df-2 10059  df-3 10060  df-4 10061  df-5 10062  df-6 10063  df-7 10064  df-8 10065  df-9 10066  df-10 10067  df-n0 10223  df-z 10284  df-dec 10384  df-uz 10490  df-q 10576  df-rp 10614  df-xneg 10711  df-xadd 10712  df-xmul 10713  df-ioo 10921  df-ico 10923  df-fz 11045  df-seq 11325  df-exp 11384  df-cj 11905  df-re 11906  df-im 11907  df-sqr 12041  df-abs 12042  df-rlim 12284  df-struct 13472  df-ndx 13473  df-slot 13474  df-base 13475  df-plusg 13543  df-mulr 13544  df-starv 13545  df-tset 13549  df-ple 13550  df-ds 13552  df-unif 13553  df-rest 13651  df-topn 13652  df-topgen 13668  df-psmet 16695  df-xmet 16696  df-met 16697  df-bl 16698  df-mopn 16699  df-cnfld 16705  df-top 16964  df-bases 16966  df-topon 16967  df-cnp 17293
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