Users' Mathboxes Mathbox for Glauco Siliprandi < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >   Mathboxes  >  climrecf Unicode version

Theorem climrecf 27405
Description: A version of climrec 27399 using bound-variable hypotheses instead of distinct variable conditions. (Contributed by Glauco Siliprandi, 29-Jun-2017.)
Hypotheses
Ref Expression
climrecf.1  |-  F/ k
ph
climrecf.2  |-  F/_ k G
climrecf.3  |-  F/_ k H
climrecf.4  |-  Z  =  ( ZZ>= `  M )
climrecf.5  |-  ( ph  ->  M  e.  ZZ )
climrecf.6  |-  ( ph  ->  G  ~~>  A )
climrecf.7  |-  ( ph  ->  A  =/=  0 )
climrecf.8  |-  ( (
ph  /\  k  e.  Z )  ->  ( G `  k )  e.  ( CC  \  {
0 } ) )
climrecf.9  |-  ( (
ph  /\  k  e.  Z )  ->  ( H `  k )  =  ( 1  / 
( G `  k
) ) )
climrecf.10  |-  ( ph  ->  H  e.  W )
Assertion
Ref Expression
climrecf  |-  ( ph  ->  H  ~~>  ( 1  /  A ) )
Distinct variable group:    k, Z
Allowed substitution hints:    ph( k)    A( k)    G( k)    H( k)    M( k)    W( k)

Proof of Theorem climrecf
Dummy variable  j is distinct from all other variables.
StepHypRef Expression
1 climrecf.4 . 2  |-  Z  =  ( ZZ>= `  M )
2 climrecf.5 . 2  |-  ( ph  ->  M  e.  ZZ )
3 climrecf.6 . 2  |-  ( ph  ->  G  ~~>  A )
4 climrecf.7 . 2  |-  ( ph  ->  A  =/=  0 )
5 climrecf.1 . . . . 5  |-  F/ k
ph
6 nfv 1626 . . . . 5  |-  F/ k  j  e.  Z
75, 6nfan 1836 . . . 4  |-  F/ k ( ph  /\  j  e.  Z )
8 climrecf.2 . . . . . 6  |-  F/_ k G
9 nfcv 2525 . . . . . 6  |-  F/_ k
j
108, 9nffv 5677 . . . . 5  |-  F/_ k
( G `  j
)
1110nfel1 2535 . . . 4  |-  F/ k ( G `  j
)  e.  ( CC 
\  { 0 } )
127, 11nfim 1822 . . 3  |-  F/ k ( ( ph  /\  j  e.  Z )  ->  ( G `  j
)  e.  ( CC 
\  { 0 } ) )
13 eleq1 2449 . . . . 5  |-  ( k  =  j  ->  (
k  e.  Z  <->  j  e.  Z ) )
1413anbi2d 685 . . . 4  |-  ( k  =  j  ->  (
( ph  /\  k  e.  Z )  <->  ( ph  /\  j  e.  Z ) ) )
15 fveq2 5670 . . . . 5  |-  ( k  =  j  ->  ( G `  k )  =  ( G `  j ) )
1615eleq1d 2455 . . . 4  |-  ( k  =  j  ->  (
( G `  k
)  e.  ( CC 
\  { 0 } )  <->  ( G `  j )  e.  ( CC  \  { 0 } ) ) )
1714, 16imbi12d 312 . . 3  |-  ( k  =  j  ->  (
( ( ph  /\  k  e.  Z )  ->  ( G `  k
)  e.  ( CC 
\  { 0 } ) )  <->  ( ( ph  /\  j  e.  Z
)  ->  ( G `  j )  e.  ( CC  \  { 0 } ) ) ) )
18 climrecf.8 . . 3  |-  ( (
ph  /\  k  e.  Z )  ->  ( G `  k )  e.  ( CC  \  {
0 } ) )
1912, 17, 18chvar 2026 . 2  |-  ( (
ph  /\  j  e.  Z )  ->  ( G `  j )  e.  ( CC  \  {
0 } ) )
20 climrecf.3 . . . . . 6  |-  F/_ k H
2120, 9nffv 5677 . . . . 5  |-  F/_ k
( H `  j
)
22 nfcv 2525 . . . . . 6  |-  F/_ k
1
23 nfcv 2525 . . . . . 6  |-  F/_ k  /
2422, 23, 10nfov 6045 . . . . 5  |-  F/_ k
( 1  /  ( G `  j )
)
2521, 24nfeq 2532 . . . 4  |-  F/ k ( H `  j
)  =  ( 1  /  ( G `  j ) )
267, 25nfim 1822 . . 3  |-  F/ k ( ( ph  /\  j  e.  Z )  ->  ( H `  j
)  =  ( 1  /  ( G `  j ) ) )
27 fveq2 5670 . . . . 5  |-  ( k  =  j  ->  ( H `  k )  =  ( H `  j ) )
2815oveq2d 6038 . . . . 5  |-  ( k  =  j  ->  (
1  /  ( G `
 k ) )  =  ( 1  / 
( G `  j
) ) )
2927, 28eqeq12d 2403 . . . 4  |-  ( k  =  j  ->  (
( H `  k
)  =  ( 1  /  ( G `  k ) )  <->  ( H `  j )  =  ( 1  /  ( G `
 j ) ) ) )
3014, 29imbi12d 312 . . 3  |-  ( k  =  j  ->  (
( ( ph  /\  k  e.  Z )  ->  ( H `  k
)  =  ( 1  /  ( G `  k ) ) )  <-> 
( ( ph  /\  j  e.  Z )  ->  ( H `  j
)  =  ( 1  /  ( G `  j ) ) ) ) )
31 climrecf.9 . . 3  |-  ( (
ph  /\  k  e.  Z )  ->  ( H `  k )  =  ( 1  / 
( G `  k
) ) )
3226, 30, 31chvar 2026 . 2  |-  ( (
ph  /\  j  e.  Z )  ->  ( H `  j )  =  ( 1  / 
( G `  j
) ) )
33 climrecf.10 . 2  |-  ( ph  ->  H  e.  W )
341, 2, 3, 4, 19, 32, 33climrec 27399 1  |-  ( ph  ->  H  ~~>  ( 1  /  A ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 359   F/wnf 1550    = wceq 1649    e. wcel 1717   F/_wnfc 2512    =/= wne 2552    \ cdif 3262   {csn 3759   class class class wbr 4155   ` cfv 5396  (class class class)co 6022   CCcc 8923   0cc0 8925   1c1 8926    / cdiv 9611   ZZcz 10216   ZZ>=cuz 10422    ~~> cli 12207
This theorem is referenced by:  climdivf  27408
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 2370  ax-sep 4273  ax-nul 4281  ax-pow 4320  ax-pr 4346  ax-un 4643  ax-cnex 8981  ax-resscn 8982  ax-1cn 8983  ax-icn 8984  ax-addcl 8985  ax-addrcl 8986  ax-mulcl 8987  ax-mulrcl 8988  ax-mulcom 8989  ax-addass 8990  ax-mulass 8991  ax-distr 8992  ax-i2m1 8993  ax-1ne0 8994  ax-1rid 8995  ax-rnegex 8996  ax-rrecex 8997  ax-cnre 8998  ax-pre-lttri 8999  ax-pre-lttrn 9000  ax-pre-ltadd 9001  ax-pre-mulgt0 9002  ax-pre-sup 9003
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 2244  df-mo 2245  df-clab 2376  df-cleq 2382  df-clel 2385  df-nfc 2514  df-ne 2554  df-nel 2555  df-ral 2656  df-rex 2657  df-reu 2658  df-rmo 2659  df-rab 2660  df-v 2903  df-sbc 3107  df-csb 3197  df-dif 3268  df-un 3270  df-in 3272  df-ss 3279  df-pss 3281  df-nul 3574  df-if 3685  df-pw 3746  df-sn 3765  df-pr 3766  df-tp 3767  df-op 3768  df-uni 3960  df-iun 4039  df-br 4156  df-opab 4210  df-mpt 4211  df-tr 4246  df-eprel 4437  df-id 4441  df-po 4446  df-so 4447  df-fr 4484  df-we 4486  df-ord 4527  df-on 4528  df-lim 4529  df-suc 4530  df-om 4788  df-xp 4826  df-rel 4827  df-cnv 4828  df-co 4829  df-dm 4830  df-rn 4831  df-res 4832  df-ima 4833  df-iota 5360  df-fun 5398  df-fn 5399  df-f 5400  df-f1 5401  df-fo 5402  df-f1o 5403  df-fv 5404  df-ov 6025  df-oprab 6026  df-mpt2 6027  df-2nd 6291  df-riota 6487  df-recs 6571  df-rdg 6606  df-er 6843  df-en 7048  df-dom 7049  df-sdom 7050  df-sup 7383  df-pnf 9057  df-mnf 9058  df-xr 9059  df-ltxr 9060  df-le 9061  df-sub 9227  df-neg 9228  df-div 9612  df-nn 9935  df-2 9992  df-3 9993  df-n0 10156  df-z 10217  df-uz 10423  df-rp 10547  df-seq 11253  df-exp 11312  df-cj 11833  df-re 11834  df-im 11835  df-sqr 11969  df-abs 11970  df-clim 12211
  Copyright terms: Public domain W3C validator