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Theorem climmulf 27706
Description: A version of climmul 12426 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 2572 . . . . . 6  |-  F/_ k
j
87nfel1 2582 . . . . 5  |-  F/ k  j  e.  Z
96, 8nfan 1846 . . . 4  |-  F/ k ( ph  /\  j  e.  Z )
10 climmulf.2 . . . . . 6  |-  F/_ k F
1110, 7nffv 5735 . . . . 5  |-  F/_ k
( F `  j
)
1211nfel1 2582 . . . 4  |-  F/ k ( F `  j
)  e.  CC
139, 12nfim 1832 . . 3  |-  F/ k ( ( ph  /\  j  e.  Z )  ->  ( F `  j
)  e.  CC )
14 eleq1 2496 . . . . 5  |-  ( k  =  j  ->  (
k  e.  Z  <->  j  e.  Z ) )
1514anbi2d 685 . . . 4  |-  ( k  =  j  ->  (
( ph  /\  k  e.  Z )  <->  ( ph  /\  j  e.  Z ) ) )
16 fveq2 5728 . . . . 5  |-  ( k  =  j  ->  ( F `  k )  =  ( F `  j ) )
1716eleq1d 2502 . . . 4  |-  ( k  =  j  ->  (
( F `  k
)  e.  CC  <->  ( F `  j )  e.  CC ) )
1815, 17imbi12d 312 . . 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 1968 . 2  |-  ( (
ph  /\  j  e.  Z )  ->  ( F `  j )  e.  CC )
21 climmulf.3 . . . . . 6  |-  F/_ k G
2221, 7nffv 5735 . . . . 5  |-  F/_ k
( G `  j
)
2322nfel1 2582 . . . 4  |-  F/ k ( G `  j
)  e.  CC
249, 23nfim 1832 . . 3  |-  F/ k ( ( ph  /\  j  e.  Z )  ->  ( G `  j
)  e.  CC )
25 fveq2 5728 . . . . 5  |-  ( k  =  j  ->  ( G `  k )  =  ( G `  j ) )
2625eleq1d 2502 . . . 4  |-  ( k  =  j  ->  (
( G `  k
)  e.  CC  <->  ( G `  j )  e.  CC ) )
2715, 26imbi12d 312 . . 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 1968 . 2  |-  ( (
ph  /\  j  e.  Z )  ->  ( G `  j )  e.  CC )
30 climmulf.4 . . . . . 6  |-  F/_ k H
3130, 7nffv 5735 . . . . 5  |-  F/_ k
( H `  j
)
32 nfcv 2572 . . . . . 6  |-  F/_ k  x.
3311, 32, 22nfov 6104 . . . . 5  |-  F/_ k
( ( F `  j )  x.  ( G `  j )
)
3431, 33nfeq 2579 . . . 4  |-  F/ k ( H `  j
)  =  ( ( F `  j )  x.  ( G `  j ) )
359, 34nfim 1832 . . 3  |-  F/ k ( ( ph  /\  j  e.  Z )  ->  ( H `  j
)  =  ( ( F `  j )  x.  ( G `  j ) ) )
36 fveq2 5728 . . . . 5  |-  ( k  =  j  ->  ( H `  k )  =  ( H `  j ) )
3716, 25oveq12d 6099 . . . . 5  |-  ( k  =  j  ->  (
( F `  k
)  x.  ( G `
 k ) )  =  ( ( F `
 j )  x.  ( G `  j
) ) )
3836, 37eqeq12d 2450 . . . 4  |-  ( k  =  j  ->  (
( H `  k
)  =  ( ( F `  k )  x.  ( G `  k ) )  <->  ( H `  j )  =  ( ( F `  j
)  x.  ( G `
 j ) ) ) )
3915, 38imbi12d 312 . . 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 1968 . 2  |-  ( (
ph  /\  j  e.  Z )  ->  ( H `  j )  =  ( ( F `
 j )  x.  ( G `  j
) ) )
421, 2, 3, 4, 5, 20, 29, 41climmul 12426 1  |-  ( ph  ->  H  ~~>  ( A  x.  B ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 359   F/wnf 1553    = wceq 1652    e. wcel 1725   F/_wnfc 2559   class class class wbr 4212   ` cfv 5454  (class class class)co 6081   CCcc 8988    x. cmul 8995   ZZcz 10282   ZZ>=cuz 10488    ~~> cli 12278
This theorem is referenced by:  climneg  27712  climdivf  27714  stirlinglem15  27813
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 2417  ax-sep 4330  ax-nul 4338  ax-pow 4377  ax-pr 4403  ax-un 4701  ax-cnex 9046  ax-resscn 9047  ax-1cn 9048  ax-icn 9049  ax-addcl 9050  ax-addrcl 9051  ax-mulcl 9052  ax-mulrcl 9053  ax-mulcom 9054  ax-addass 9055  ax-mulass 9056  ax-distr 9057  ax-i2m1 9058  ax-1ne0 9059  ax-1rid 9060  ax-rnegex 9061  ax-rrecex 9062  ax-cnre 9063  ax-pre-lttri 9064  ax-pre-lttrn 9065  ax-pre-ltadd 9066  ax-pre-mulgt0 9067  ax-pre-sup 9068
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 2285  df-mo 2286  df-clab 2423  df-cleq 2429  df-clel 2432  df-nfc 2561  df-ne 2601  df-nel 2602  df-ral 2710  df-rex 2711  df-reu 2712  df-rmo 2713  df-rab 2714  df-v 2958  df-sbc 3162  df-csb 3252  df-dif 3323  df-un 3325  df-in 3327  df-ss 3334  df-pss 3336  df-nul 3629  df-if 3740  df-pw 3801  df-sn 3820  df-pr 3821  df-tp 3822  df-op 3823  df-uni 4016  df-iun 4095  df-br 4213  df-opab 4267  df-mpt 4268  df-tr 4303  df-eprel 4494  df-id 4498  df-po 4503  df-so 4504  df-fr 4541  df-we 4543  df-ord 4584  df-on 4585  df-lim 4586  df-suc 4587  df-om 4846  df-xp 4884  df-rel 4885  df-cnv 4886  df-co 4887  df-dm 4888  df-rn 4889  df-res 4890  df-ima 4891  df-iota 5418  df-fun 5456  df-fn 5457  df-f 5458  df-f1 5459  df-fo 5460  df-f1o 5461  df-fv 5462  df-ov 6084  df-oprab 6085  df-mpt2 6086  df-2nd 6350  df-riota 6549  df-recs 6633  df-rdg 6668  df-er 6905  df-en 7110  df-dom 7111  df-sdom 7112  df-sup 7446  df-pnf 9122  df-mnf 9123  df-xr 9124  df-ltxr 9125  df-le 9126  df-sub 9293  df-neg 9294  df-div 9678  df-nn 10001  df-2 10058  df-3 10059  df-n0 10222  df-z 10283  df-uz 10489  df-rp 10613  df-seq 11324  df-exp 11383  df-cj 11904  df-re 11905  df-im 11906  df-sqr 12040  df-abs 12041  df-clim 12282
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