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Theorem climrecf 27702
Description: A version of climrec 27696 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 1629 . . . . 5  |-  F/ k  j  e.  Z
75, 6nfan 1846 . . . 4  |-  F/ k ( ph  /\  j  e.  Z )
8 climrecf.2 . . . . . 6  |-  F/_ k G
9 nfcv 2571 . . . . . 6  |-  F/_ k
j
108, 9nffv 5727 . . . . 5  |-  F/_ k
( G `  j
)
1110nfel1 2581 . . . 4  |-  F/ k ( G `  j
)  e.  ( CC 
\  { 0 } )
127, 11nfim 1832 . . 3  |-  F/ k ( ( ph  /\  j  e.  Z )  ->  ( G `  j
)  e.  ( CC 
\  { 0 } ) )
13 eleq1 2495 . . . . 5  |-  ( k  =  j  ->  (
k  e.  Z  <->  j  e.  Z ) )
1413anbi2d 685 . . . 4  |-  ( k  =  j  ->  (
( ph  /\  k  e.  Z )  <->  ( ph  /\  j  e.  Z ) ) )
15 fveq2 5720 . . . . 5  |-  ( k  =  j  ->  ( G `  k )  =  ( G `  j ) )
1615eleq1d 2501 . . . 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 1968 . 2  |-  ( (
ph  /\  j  e.  Z )  ->  ( G `  j )  e.  ( CC  \  {
0 } ) )
20 climrecf.3 . . . . . 6  |-  F/_ k H
2120, 9nffv 5727 . . . . 5  |-  F/_ k
( H `  j
)
22 nfcv 2571 . . . . . 6  |-  F/_ k
1
23 nfcv 2571 . . . . . 6  |-  F/_ k  /
2422, 23, 10nfov 6096 . . . . 5  |-  F/_ k
( 1  /  ( G `  j )
)
2521, 24nfeq 2578 . . . 4  |-  F/ k ( H `  j
)  =  ( 1  /  ( G `  j ) )
267, 25nfim 1832 . . 3  |-  F/ k ( ( ph  /\  j  e.  Z )  ->  ( H `  j
)  =  ( 1  /  ( G `  j ) ) )
27 fveq2 5720 . . . . 5  |-  ( k  =  j  ->  ( H `  k )  =  ( H `  j ) )
2815oveq2d 6089 . . . . 5  |-  ( k  =  j  ->  (
1  /  ( G `
 k ) )  =  ( 1  / 
( G `  j
) ) )
2927, 28eqeq12d 2449 . . . 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 1968 . 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 27696 1  |-  ( ph  ->  H  ~~>  ( 1  /  A ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 359   F/wnf 1553    = wceq 1652    e. wcel 1725   F/_wnfc 2558    =/= wne 2598    \ cdif 3309   {csn 3806   class class class wbr 4204   ` cfv 5446  (class class class)co 6073   CCcc 8980   0cc0 8982   1c1 8983    / cdiv 9669   ZZcz 10274   ZZ>=cuz 10480    ~~> cli 12270
This theorem is referenced by:  climdivf  27705
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1555  ax-5 1566  ax-17 1626  ax-9 1666  ax-8 1687  ax-13 1727  ax-14 1729  ax-6 1744  ax-7 1749  ax-11 1761  ax-12 1950  ax-ext 2416  ax-sep 4322  ax-nul 4330  ax-pow 4369  ax-pr 4395  ax-un 4693  ax-cnex 9038  ax-resscn 9039  ax-1cn 9040  ax-icn 9041  ax-addcl 9042  ax-addrcl 9043  ax-mulcl 9044  ax-mulrcl 9045  ax-mulcom 9046  ax-addass 9047  ax-mulass 9048  ax-distr 9049  ax-i2m1 9050  ax-1ne0 9051  ax-1rid 9052  ax-rnegex 9053  ax-rrecex 9054  ax-cnre 9055  ax-pre-lttri 9056  ax-pre-lttrn 9057  ax-pre-ltadd 9058  ax-pre-mulgt0 9059  ax-pre-sup 9060
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3or 937  df-3an 938  df-tru 1328  df-ex 1551  df-nf 1554  df-sb 1659  df-eu 2284  df-mo 2285  df-clab 2422  df-cleq 2428  df-clel 2431  df-nfc 2560  df-ne 2600  df-nel 2601  df-ral 2702  df-rex 2703  df-reu 2704  df-rmo 2705  df-rab 2706  df-v 2950  df-sbc 3154  df-csb 3244  df-dif 3315  df-un 3317  df-in 3319  df-ss 3326  df-pss 3328  df-nul 3621  df-if 3732  df-pw 3793  df-sn 3812  df-pr 3813  df-tp 3814  df-op 3815  df-uni 4008  df-iun 4087  df-br 4205  df-opab 4259  df-mpt 4260  df-tr 4295  df-eprel 4486  df-id 4490  df-po 4495  df-so 4496  df-fr 4533  df-we 4535  df-ord 4576  df-on 4577  df-lim 4578  df-suc 4579  df-om 4838  df-xp 4876  df-rel 4877  df-cnv 4878  df-co 4879  df-dm 4880  df-rn 4881  df-res 4882  df-ima 4883  df-iota 5410  df-fun 5448  df-fn 5449  df-f 5450  df-f1 5451  df-fo 5452  df-f1o 5453  df-fv 5454  df-ov 6076  df-oprab 6077  df-mpt2 6078  df-2nd 6342  df-riota 6541  df-recs 6625  df-rdg 6660  df-er 6897  df-en 7102  df-dom 7103  df-sdom 7104  df-sup 7438  df-pnf 9114  df-mnf 9115  df-xr 9116  df-ltxr 9117  df-le 9118  df-sub 9285  df-neg 9286  df-div 9670  df-nn 9993  df-2 10050  df-3 10051  df-n0 10214  df-z 10275  df-uz 10481  df-rp 10605  df-seq 11316  df-exp 11375  df-cj 11896  df-re 11897  df-im 11898  df-sqr 12032  df-abs 12033  df-clim 12274
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