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Theorem climrecf 37308
Description: A version of climrec 37301 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 1751 . . . . 5  |-  F/ k  j  e.  Z
75, 6nfan 1983 . . . 4  |-  F/ k ( ph  /\  j  e.  Z )
8 climrecf.2 . . . . . 6  |-  F/_ k G
9 nfcv 2582 . . . . . 6  |-  F/_ k
j
108, 9nffv 5879 . . . . 5  |-  F/_ k
( G `  j
)
1110nfel1 2598 . . . 4  |-  F/ k ( G `  j
)  e.  ( CC 
\  { 0 } )
127, 11nfim 1975 . . 3  |-  F/ k ( ( ph  /\  j  e.  Z )  ->  ( G `  j
)  e.  ( CC 
\  { 0 } ) )
13 eleq1 2492 . . . . 5  |-  ( k  =  j  ->  (
k  e.  Z  <->  j  e.  Z ) )
1413anbi2d 708 . . . 4  |-  ( k  =  j  ->  (
( ph  /\  k  e.  Z )  <->  ( ph  /\  j  e.  Z ) ) )
15 fveq2 5872 . . . . 5  |-  ( k  =  j  ->  ( G `  k )  =  ( G `  j ) )
1615eleq1d 2489 . . . 4  |-  ( k  =  j  ->  (
( G `  k
)  e.  ( CC 
\  { 0 } )  <->  ( G `  j )  e.  ( CC  \  { 0 } ) ) )
1714, 16imbi12d 321 . . 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 2066 . 2  |-  ( (
ph  /\  j  e.  Z )  ->  ( G `  j )  e.  ( CC  \  {
0 } ) )
20 climrecf.3 . . . . . 6  |-  F/_ k H
2120, 9nffv 5879 . . . . 5  |-  F/_ k
( H `  j
)
22 nfcv 2582 . . . . . 6  |-  F/_ k
1
23 nfcv 2582 . . . . . 6  |-  F/_ k  /
2422, 23, 10nfov 6322 . . . . 5  |-  F/_ k
( 1  /  ( G `  j )
)
2521, 24nfeq 2593 . . . 4  |-  F/ k ( H `  j
)  =  ( 1  /  ( G `  j ) )
267, 25nfim 1975 . . 3  |-  F/ k ( ( ph  /\  j  e.  Z )  ->  ( H `  j
)  =  ( 1  /  ( G `  j ) ) )
27 fveq2 5872 . . . . 5  |-  ( k  =  j  ->  ( H `  k )  =  ( H `  j ) )
2815oveq2d 6312 . . . . 5  |-  ( k  =  j  ->  (
1  /  ( G `
 k ) )  =  ( 1  / 
( G `  j
) ) )
2927, 28eqeq12d 2442 . . . 4  |-  ( k  =  j  ->  (
( H `  k
)  =  ( 1  /  ( G `  k ) )  <->  ( H `  j )  =  ( 1  /  ( G `
 j ) ) ) )
3014, 29imbi12d 321 . . 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 2066 . 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 37301 1  |-  ( ph  ->  H  ~~>  ( 1  /  A ) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    /\ wa 370    = wceq 1437   F/wnf 1663    e. wcel 1867   F/_wnfc 2568    =/= wne 2616    \ cdif 3430   {csn 3993   class class class wbr 4417   ` cfv 5592  (class class class)co 6296   CCcc 9526   0cc0 9528   1c1 9529    / cdiv 10258   ZZcz 10926   ZZ>=cuz 11148    ~~> cli 13515
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1665  ax-4 1678  ax-5 1748  ax-6 1794  ax-7 1838  ax-8 1869  ax-9 1871  ax-10 1886  ax-11 1891  ax-12 1904  ax-13 2052  ax-ext 2398  ax-sep 4539  ax-nul 4547  ax-pow 4594  ax-pr 4652  ax-un 6588  ax-cnex 9584  ax-resscn 9585  ax-1cn 9586  ax-icn 9587  ax-addcl 9588  ax-addrcl 9589  ax-mulcl 9590  ax-mulrcl 9591  ax-mulcom 9592  ax-addass 9593  ax-mulass 9594  ax-distr 9595  ax-i2m1 9596  ax-1ne0 9597  ax-1rid 9598  ax-rnegex 9599  ax-rrecex 9600  ax-cnre 9601  ax-pre-lttri 9602  ax-pre-lttrn 9603  ax-pre-ltadd 9604  ax-pre-mulgt0 9605  ax-pre-sup 9606
This theorem depends on definitions:  df-bi 188  df-or 371  df-an 372  df-3or 983  df-3an 984  df-tru 1440  df-ex 1660  df-nf 1664  df-sb 1787  df-eu 2267  df-mo 2268  df-clab 2406  df-cleq 2412  df-clel 2415  df-nfc 2570  df-ne 2618  df-nel 2619  df-ral 2778  df-rex 2779  df-reu 2780  df-rmo 2781  df-rab 2782  df-v 3080  df-sbc 3297  df-csb 3393  df-dif 3436  df-un 3438  df-in 3440  df-ss 3447  df-pss 3449  df-nul 3759  df-if 3907  df-pw 3978  df-sn 3994  df-pr 3996  df-tp 3998  df-op 4000  df-uni 4214  df-iun 4295  df-br 4418  df-opab 4476  df-mpt 4477  df-tr 4512  df-eprel 4756  df-id 4760  df-po 4766  df-so 4767  df-fr 4804  df-we 4806  df-xp 4851  df-rel 4852  df-cnv 4853  df-co 4854  df-dm 4855  df-rn 4856  df-res 4857  df-ima 4858  df-pred 5390  df-ord 5436  df-on 5437  df-lim 5438  df-suc 5439  df-iota 5556  df-fun 5594  df-fn 5595  df-f 5596  df-f1 5597  df-fo 5598  df-f1o 5599  df-fv 5600  df-riota 6258  df-ov 6299  df-oprab 6300  df-mpt2 6301  df-om 6698  df-2nd 6799  df-wrecs 7027  df-recs 7089  df-rdg 7127  df-er 7362  df-en 7569  df-dom 7570  df-sdom 7571  df-sup 7953  df-pnf 9666  df-mnf 9667  df-xr 9668  df-ltxr 9669  df-le 9670  df-sub 9851  df-neg 9852  df-div 10259  df-nn 10599  df-2 10657  df-3 10658  df-n0 10859  df-z 10927  df-uz 11149  df-rp 11292  df-seq 12200  df-exp 12259  df-cj 13130  df-re 13131  df-im 13132  df-sqrt 13266  df-abs 13267  df-clim 13519
This theorem is referenced by:  climdivf  37312
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