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Theorem climrecf 29931
Description: A version of climrec 29925 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 1674 . . . . 5  |-  F/ k  j  e.  Z
75, 6nfan 1866 . . . 4  |-  F/ k ( ph  /\  j  e.  Z )
8 climrecf.2 . . . . . 6  |-  F/_ k G
9 nfcv 2616 . . . . . 6  |-  F/_ k
j
108, 9nffv 5807 . . . . 5  |-  F/_ k
( G `  j
)
1110nfel1 2632 . . . 4  |-  F/ k ( G `  j
)  e.  ( CC 
\  { 0 } )
127, 11nfim 1858 . . 3  |-  F/ k ( ( ph  /\  j  e.  Z )  ->  ( G `  j
)  e.  ( CC 
\  { 0 } ) )
13 eleq1 2526 . . . . 5  |-  ( k  =  j  ->  (
k  e.  Z  <->  j  e.  Z ) )
1413anbi2d 703 . . . 4  |-  ( k  =  j  ->  (
( ph  /\  k  e.  Z )  <->  ( ph  /\  j  e.  Z ) ) )
15 fveq2 5800 . . . . 5  |-  ( k  =  j  ->  ( G `  k )  =  ( G `  j ) )
1615eleq1d 2523 . . . 4  |-  ( k  =  j  ->  (
( G `  k
)  e.  ( CC 
\  { 0 } )  <->  ( G `  j )  e.  ( CC  \  { 0 } ) ) )
1714, 16imbi12d 320 . . 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 1969 . 2  |-  ( (
ph  /\  j  e.  Z )  ->  ( G `  j )  e.  ( CC  \  {
0 } ) )
20 climrecf.3 . . . . . 6  |-  F/_ k H
2120, 9nffv 5807 . . . . 5  |-  F/_ k
( H `  j
)
22 nfcv 2616 . . . . . 6  |-  F/_ k
1
23 nfcv 2616 . . . . . 6  |-  F/_ k  /
2422, 23, 10nfov 6224 . . . . 5  |-  F/_ k
( 1  /  ( G `  j )
)
2521, 24nfeq 2627 . . . 4  |-  F/ k ( H `  j
)  =  ( 1  /  ( G `  j ) )
267, 25nfim 1858 . . 3  |-  F/ k ( ( ph  /\  j  e.  Z )  ->  ( H `  j
)  =  ( 1  /  ( G `  j ) ) )
27 fveq2 5800 . . . . 5  |-  ( k  =  j  ->  ( H `  k )  =  ( H `  j ) )
2815oveq2d 6217 . . . . 5  |-  ( k  =  j  ->  (
1  /  ( G `
 k ) )  =  ( 1  / 
( G `  j
) ) )
2927, 28eqeq12d 2476 . . . 4  |-  ( k  =  j  ->  (
( H `  k
)  =  ( 1  /  ( G `  k ) )  <->  ( H `  j )  =  ( 1  /  ( G `
 j ) ) ) )
3014, 29imbi12d 320 . . 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 1969 . 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 29925 1  |-  ( ph  ->  H  ~~>  ( 1  /  A ) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    /\ wa 369    = wceq 1370   F/wnf 1590    e. wcel 1758   F/_wnfc 2602    =/= wne 2648    \ cdif 3434   {csn 3986   class class class wbr 4401   ` cfv 5527  (class class class)co 6201   CCcc 9392   0cc0 9394   1c1 9395    / cdiv 10105   ZZcz 10758   ZZ>=cuz 10973    ~~> cli 13081
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1592  ax-4 1603  ax-5 1671  ax-6 1710  ax-7 1730  ax-8 1760  ax-9 1762  ax-10 1777  ax-11 1782  ax-12 1794  ax-13 1955  ax-ext 2432  ax-sep 4522  ax-nul 4530  ax-pow 4579  ax-pr 4640  ax-un 6483  ax-cnex 9450  ax-resscn 9451  ax-1cn 9452  ax-icn 9453  ax-addcl 9454  ax-addrcl 9455  ax-mulcl 9456  ax-mulrcl 9457  ax-mulcom 9458  ax-addass 9459  ax-mulass 9460  ax-distr 9461  ax-i2m1 9462  ax-1ne0 9463  ax-1rid 9464  ax-rnegex 9465  ax-rrecex 9466  ax-cnre 9467  ax-pre-lttri 9468  ax-pre-lttrn 9469  ax-pre-ltadd 9470  ax-pre-mulgt0 9471  ax-pre-sup 9472
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 966  df-3an 967  df-tru 1373  df-ex 1588  df-nf 1591  df-sb 1703  df-eu 2266  df-mo 2267  df-clab 2440  df-cleq 2446  df-clel 2449  df-nfc 2604  df-ne 2650  df-nel 2651  df-ral 2804  df-rex 2805  df-reu 2806  df-rmo 2807  df-rab 2808  df-v 3080  df-sbc 3295  df-csb 3397  df-dif 3440  df-un 3442  df-in 3444  df-ss 3451  df-pss 3453  df-nul 3747  df-if 3901  df-pw 3971  df-sn 3987  df-pr 3989  df-tp 3991  df-op 3993  df-uni 4201  df-iun 4282  df-br 4402  df-opab 4460  df-mpt 4461  df-tr 4495  df-eprel 4741  df-id 4745  df-po 4750  df-so 4751  df-fr 4788  df-we 4790  df-ord 4831  df-on 4832  df-lim 4833  df-suc 4834  df-xp 4955  df-rel 4956  df-cnv 4957  df-co 4958  df-dm 4959  df-rn 4960  df-res 4961  df-ima 4962  df-iota 5490  df-fun 5529  df-fn 5530  df-f 5531  df-f1 5532  df-fo 5533  df-f1o 5534  df-fv 5535  df-riota 6162  df-ov 6204  df-oprab 6205  df-mpt2 6206  df-om 6588  df-2nd 6689  df-recs 6943  df-rdg 6977  df-er 7212  df-en 7422  df-dom 7423  df-sdom 7424  df-sup 7803  df-pnf 9532  df-mnf 9533  df-xr 9534  df-ltxr 9535  df-le 9536  df-sub 9709  df-neg 9710  df-div 10106  df-nn 10435  df-2 10492  df-3 10493  df-n0 10692  df-z 10759  df-uz 10974  df-rp 11104  df-seq 11925  df-exp 11984  df-cj 12707  df-re 12708  df-im 12709  df-sqr 12843  df-abs 12844  df-clim 13085
This theorem is referenced by:  climdivf  29934
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