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Theorem climmulf 31146
Description: A version of climmul 13414 using bound-variable hypotheses instead of distinct variable conditions. (Contributed by Glauco Siliprandi, 29-Jun-2017.)
Hypotheses
Ref Expression
climmulf.1  |-  F/ k
ph
climmulf.2  |-  F/_ k F
climmulf.3  |-  F/_ k G
climmulf.4  |-  F/_ k H
climmulf.5  |-  Z  =  ( ZZ>= `  M )
climmulf.6  |-  ( ph  ->  M  e.  ZZ )
climmulf.7  |-  ( ph  ->  F  ~~>  A )
climmulf.8  |-  ( ph  ->  H  e.  X )
climmulf.9  |-  ( ph  ->  G  ~~>  B )
climmulf.10  |-  ( (
ph  /\  k  e.  Z )  ->  ( F `  k )  e.  CC )
climmulf.11  |-  ( (
ph  /\  k  e.  Z )  ->  ( G `  k )  e.  CC )
climmulf.12  |-  ( (
ph  /\  k  e.  Z )  ->  ( H `  k )  =  ( ( F `
 k )  x.  ( G `  k
) ) )
Assertion
Ref Expression
climmulf  |-  ( ph  ->  H  ~~>  ( A  x.  B ) )
Distinct variable group:    k, Z
Allowed substitution hints:    ph( k)    A( k)    B( k)    F( k)    G( k)    H( k)    M( k)    X( k)

Proof of Theorem climmulf
Dummy variable  j is distinct from all other variables.
StepHypRef Expression
1 climmulf.5 . 2  |-  Z  =  ( ZZ>= `  M )
2 climmulf.6 . 2  |-  ( ph  ->  M  e.  ZZ )
3 climmulf.7 . 2  |-  ( ph  ->  F  ~~>  A )
4 climmulf.8 . 2  |-  ( ph  ->  H  e.  X )
5 climmulf.9 . 2  |-  ( ph  ->  G  ~~>  B )
6 climmulf.1 . . . . 5  |-  F/ k
ph
7 nfcv 2629 . . . . . 6  |-  F/_ k
j
87nfel1 2645 . . . . 5  |-  F/ k  j  e.  Z
96, 8nfan 1875 . . . 4  |-  F/ k ( ph  /\  j  e.  Z )
10 climmulf.2 . . . . . 6  |-  F/_ k F
1110, 7nffv 5871 . . . . 5  |-  F/_ k
( F `  j
)
1211nfel1 2645 . . . 4  |-  F/ k ( F `  j
)  e.  CC
139, 12nfim 1867 . . 3  |-  F/ k ( ( ph  /\  j  e.  Z )  ->  ( F `  j
)  e.  CC )
14 eleq1 2539 . . . . 5  |-  ( k  =  j  ->  (
k  e.  Z  <->  j  e.  Z ) )
1514anbi2d 703 . . . 4  |-  ( k  =  j  ->  (
( ph  /\  k  e.  Z )  <->  ( ph  /\  j  e.  Z ) ) )
16 fveq2 5864 . . . . 5  |-  ( k  =  j  ->  ( F `  k )  =  ( F `  j ) )
1716eleq1d 2536 . . . 4  |-  ( k  =  j  ->  (
( F `  k
)  e.  CC  <->  ( F `  j )  e.  CC ) )
1815, 17imbi12d 320 . . 3  |-  ( k  =  j  ->  (
( ( ph  /\  k  e.  Z )  ->  ( F `  k
)  e.  CC )  <-> 
( ( ph  /\  j  e.  Z )  ->  ( F `  j
)  e.  CC ) ) )
19 climmulf.10 . . 3  |-  ( (
ph  /\  k  e.  Z )  ->  ( F `  k )  e.  CC )
2013, 18, 19chvar 1982 . 2  |-  ( (
ph  /\  j  e.  Z )  ->  ( F `  j )  e.  CC )
21 climmulf.3 . . . . . 6  |-  F/_ k G
2221, 7nffv 5871 . . . . 5  |-  F/_ k
( G `  j
)
2322nfel1 2645 . . . 4  |-  F/ k ( G `  j
)  e.  CC
249, 23nfim 1867 . . 3  |-  F/ k ( ( ph  /\  j  e.  Z )  ->  ( G `  j
)  e.  CC )
25 fveq2 5864 . . . . 5  |-  ( k  =  j  ->  ( G `  k )  =  ( G `  j ) )
2625eleq1d 2536 . . . 4  |-  ( k  =  j  ->  (
( G `  k
)  e.  CC  <->  ( G `  j )  e.  CC ) )
2715, 26imbi12d 320 . . 3  |-  ( k  =  j  ->  (
( ( ph  /\  k  e.  Z )  ->  ( G `  k
)  e.  CC )  <-> 
( ( ph  /\  j  e.  Z )  ->  ( G `  j
)  e.  CC ) ) )
28 climmulf.11 . . 3  |-  ( (
ph  /\  k  e.  Z )  ->  ( G `  k )  e.  CC )
2924, 27, 28chvar 1982 . 2  |-  ( (
ph  /\  j  e.  Z )  ->  ( G `  j )  e.  CC )
30 climmulf.4 . . . . . 6  |-  F/_ k H
3130, 7nffv 5871 . . . . 5  |-  F/_ k
( H `  j
)
32 nfcv 2629 . . . . . 6  |-  F/_ k  x.
3311, 32, 22nfov 6305 . . . . 5  |-  F/_ k
( ( F `  j )  x.  ( G `  j )
)
3431, 33nfeq 2640 . . . 4  |-  F/ k ( H `  j
)  =  ( ( F `  j )  x.  ( G `  j ) )
359, 34nfim 1867 . . 3  |-  F/ k ( ( ph  /\  j  e.  Z )  ->  ( H `  j
)  =  ( ( F `  j )  x.  ( G `  j ) ) )
36 fveq2 5864 . . . . 5  |-  ( k  =  j  ->  ( H `  k )  =  ( H `  j ) )
3716, 25oveq12d 6300 . . . . 5  |-  ( k  =  j  ->  (
( F `  k
)  x.  ( G `
 k ) )  =  ( ( F `
 j )  x.  ( G `  j
) ) )
3836, 37eqeq12d 2489 . . . 4  |-  ( k  =  j  ->  (
( H `  k
)  =  ( ( F `  k )  x.  ( G `  k ) )  <->  ( H `  j )  =  ( ( F `  j
)  x.  ( G `
 j ) ) ) )
3915, 38imbi12d 320 . . 3  |-  ( k  =  j  ->  (
( ( ph  /\  k  e.  Z )  ->  ( H `  k
)  =  ( ( F `  k )  x.  ( G `  k ) ) )  <-> 
( ( ph  /\  j  e.  Z )  ->  ( H `  j
)  =  ( ( F `  j )  x.  ( G `  j ) ) ) ) )
40 climmulf.12 . . 3  |-  ( (
ph  /\  k  e.  Z )  ->  ( H `  k )  =  ( ( F `
 k )  x.  ( G `  k
) ) )
4135, 39, 40chvar 1982 . 2  |-  ( (
ph  /\  j  e.  Z )  ->  ( H `  j )  =  ( ( F `
 j )  x.  ( G `  j
) ) )
421, 2, 3, 4, 5, 20, 29, 41climmul 13414 1  |-  ( ph  ->  H  ~~>  ( A  x.  B ) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    /\ wa 369    = wceq 1379   F/wnf 1599    e. wcel 1767   F/_wnfc 2615   class class class wbr 4447   ` cfv 5586  (class class class)co 6282   CCcc 9486    x. cmul 9493   ZZcz 10860   ZZ>=cuz 11078    ~~> cli 13266
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 6574  ax-cnex 9544  ax-resscn 9545  ax-1cn 9546  ax-icn 9547  ax-addcl 9548  ax-addrcl 9549  ax-mulcl 9550  ax-mulrcl 9551  ax-mulcom 9552  ax-addass 9553  ax-mulass 9554  ax-distr 9555  ax-i2m1 9556  ax-1ne0 9557  ax-1rid 9558  ax-rnegex 9559  ax-rrecex 9560  ax-cnre 9561  ax-pre-lttri 9562  ax-pre-lttrn 9563  ax-pre-ltadd 9564  ax-pre-mulgt0 9565  ax-pre-sup 9566
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 5549  df-fun 5588  df-fn 5589  df-f 5590  df-f1 5591  df-fo 5592  df-f1o 5593  df-fv 5594  df-riota 6243  df-ov 6285  df-oprab 6286  df-mpt2 6287  df-om 6679  df-2nd 6782  df-recs 7039  df-rdg 7073  df-er 7308  df-en 7514  df-dom 7515  df-sdom 7516  df-sup 7897  df-pnf 9626  df-mnf 9627  df-xr 9628  df-ltxr 9629  df-le 9630  df-sub 9803  df-neg 9804  df-div 10203  df-nn 10533  df-2 10590  df-3 10591  df-n0 10792  df-z 10861  df-uz 11079  df-rp 11217  df-seq 12072  df-exp 12131  df-cj 12891  df-re 12892  df-im 12893  df-sqrt 13027  df-abs 13028  df-clim 13270
This theorem is referenced by:  climneg  31152  climdivf  31154  stirlinglem15  31388
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