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Theorem climrecf 31378
Description: A version of climrec 31372 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 1683 . . . . 5  |-  F/ k  j  e.  Z
75, 6nfan 1875 . . . 4  |-  F/ k ( ph  /\  j  e.  Z )
8 climrecf.2 . . . . . 6  |-  F/_ k G
9 nfcv 2629 . . . . . 6  |-  F/_ k
j
108, 9nffv 5873 . . . . 5  |-  F/_ k
( G `  j
)
1110nfel1 2645 . . . 4  |-  F/ k ( G `  j
)  e.  ( CC 
\  { 0 } )
127, 11nfim 1867 . . 3  |-  F/ k ( ( ph  /\  j  e.  Z )  ->  ( G `  j
)  e.  ( CC 
\  { 0 } ) )
13 eleq1 2539 . . . . 5  |-  ( k  =  j  ->  (
k  e.  Z  <->  j  e.  Z ) )
1413anbi2d 703 . . . 4  |-  ( k  =  j  ->  (
( ph  /\  k  e.  Z )  <->  ( ph  /\  j  e.  Z ) ) )
15 fveq2 5866 . . . . 5  |-  ( k  =  j  ->  ( G `  k )  =  ( G `  j ) )
1615eleq1d 2536 . . . 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 1982 . 2  |-  ( (
ph  /\  j  e.  Z )  ->  ( G `  j )  e.  ( CC  \  {
0 } ) )
20 climrecf.3 . . . . . 6  |-  F/_ k H
2120, 9nffv 5873 . . . . 5  |-  F/_ k
( H `  j
)
22 nfcv 2629 . . . . . 6  |-  F/_ k
1
23 nfcv 2629 . . . . . 6  |-  F/_ k  /
2422, 23, 10nfov 6308 . . . . 5  |-  F/_ k
( 1  /  ( G `  j )
)
2521, 24nfeq 2640 . . . 4  |-  F/ k ( H `  j
)  =  ( 1  /  ( G `  j ) )
267, 25nfim 1867 . . 3  |-  F/ k ( ( ph  /\  j  e.  Z )  ->  ( H `  j
)  =  ( 1  /  ( G `  j ) ) )
27 fveq2 5866 . . . . 5  |-  ( k  =  j  ->  ( H `  k )  =  ( H `  j ) )
2815oveq2d 6301 . . . . 5  |-  ( k  =  j  ->  (
1  /  ( G `
 k ) )  =  ( 1  / 
( G `  j
) ) )
2927, 28eqeq12d 2489 . . . 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 1982 . 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 31372 1  |-  ( ph  ->  H  ~~>  ( 1  /  A ) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    /\ wa 369    = wceq 1379   F/wnf 1599    e. wcel 1767   F/_wnfc 2615    =/= wne 2662    \ cdif 3473   {csn 4027   class class class wbr 4447   ` cfv 5588  (class class class)co 6285   CCcc 9491   0cc0 9493   1c1 9494    / cdiv 10207   ZZcz 10865   ZZ>=cuz 11083    ~~> cli 13273
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1601  ax-4 1612  ax-5 1680  ax-6 1719  ax-7 1739  ax-8 1769  ax-9 1771  ax-10 1786  ax-11 1791  ax-12 1803  ax-13 1968  ax-ext 2445  ax-sep 4568  ax-nul 4576  ax-pow 4625  ax-pr 4686  ax-un 6577  ax-cnex 9549  ax-resscn 9550  ax-1cn 9551  ax-icn 9552  ax-addcl 9553  ax-addrcl 9554  ax-mulcl 9555  ax-mulrcl 9556  ax-mulcom 9557  ax-addass 9558  ax-mulass 9559  ax-distr 9560  ax-i2m1 9561  ax-1ne0 9562  ax-1rid 9563  ax-rnegex 9564  ax-rrecex 9565  ax-cnre 9566  ax-pre-lttri 9567  ax-pre-lttrn 9568  ax-pre-ltadd 9569  ax-pre-mulgt0 9570  ax-pre-sup 9571
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 974  df-3an 975  df-tru 1382  df-ex 1597  df-nf 1600  df-sb 1712  df-eu 2279  df-mo 2280  df-clab 2453  df-cleq 2459  df-clel 2462  df-nfc 2617  df-ne 2664  df-nel 2665  df-ral 2819  df-rex 2820  df-reu 2821  df-rmo 2822  df-rab 2823  df-v 3115  df-sbc 3332  df-csb 3436  df-dif 3479  df-un 3481  df-in 3483  df-ss 3490  df-pss 3492  df-nul 3786  df-if 3940  df-pw 4012  df-sn 4028  df-pr 4030  df-tp 4032  df-op 4034  df-uni 4246  df-iun 4327  df-br 4448  df-opab 4506  df-mpt 4507  df-tr 4541  df-eprel 4791  df-id 4795  df-po 4800  df-so 4801  df-fr 4838  df-we 4840  df-ord 4881  df-on 4882  df-lim 4883  df-suc 4884  df-xp 5005  df-rel 5006  df-cnv 5007  df-co 5008  df-dm 5009  df-rn 5010  df-res 5011  df-ima 5012  df-iota 5551  df-fun 5590  df-fn 5591  df-f 5592  df-f1 5593  df-fo 5594  df-f1o 5595  df-fv 5596  df-riota 6246  df-ov 6288  df-oprab 6289  df-mpt2 6290  df-om 6686  df-2nd 6786  df-recs 7043  df-rdg 7077  df-er 7312  df-en 7518  df-dom 7519  df-sdom 7520  df-sup 7902  df-pnf 9631  df-mnf 9632  df-xr 9633  df-ltxr 9634  df-le 9635  df-sub 9808  df-neg 9809  df-div 10208  df-nn 10538  df-2 10595  df-3 10596  df-n0 10797  df-z 10866  df-uz 11084  df-rp 11222  df-seq 12077  df-exp 12136  df-cj 12898  df-re 12899  df-im 12900  df-sqrt 13034  df-abs 13035  df-clim 13277
This theorem is referenced by:  climdivf  31381
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