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Theorem climmulf 27730
Description: A version of climmul 12106 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 simpr 447 . . 3  |-  ( (
ph  /\  j  e.  Z )  ->  j  e.  Z )
7 id 19 . . 3  |-  ( (
ph  /\  j  e.  Z )  ->  ( ph  /\  j  e.  Z
) )
8 nfcv 2419 . . . 4  |-  F/_ k
j
9 climmulf.1 . . . . . 6  |-  F/ k
ph
10 nfcv 2419 . . . . . . 7  |-  F/_ k Z
118, 10nfel 2427 . . . . . 6  |-  F/ k  j  e.  Z
129, 11nfan 1771 . . . . 5  |-  F/ k ( ph  /\  j  e.  Z )
13 climmulf.2 . . . . . . 7  |-  F/_ k F
1413, 8nffv 5532 . . . . . 6  |-  F/_ k
( F `  j
)
15 nfcv 2419 . . . . . 6  |-  F/_ k CC
1614, 15nfel 2427 . . . . 5  |-  F/ k ( F `  j
)  e.  CC
1712, 16nfim 1769 . . . 4  |-  F/ k ( ( ph  /\  j  e.  Z )  ->  ( F `  j
)  e.  CC )
18 eleq1 2343 . . . . . 6  |-  ( k  =  j  ->  (
k  e.  Z  <->  j  e.  Z ) )
1918anbi2d 684 . . . . 5  |-  ( k  =  j  ->  (
( ph  /\  k  e.  Z )  <->  ( ph  /\  j  e.  Z ) ) )
20 fveq2 5525 . . . . . 6  |-  ( k  =  j  ->  ( F `  k )  =  ( F `  j ) )
2120eleq1d 2349 . . . . 5  |-  ( k  =  j  ->  (
( F `  k
)  e.  CC  <->  ( F `  j )  e.  CC ) )
2219, 21imbi12d 311 . . . 4  |-  ( k  =  j  ->  (
( ( ph  /\  k  e.  Z )  ->  ( F `  k
)  e.  CC )  <-> 
( ( ph  /\  j  e.  Z )  ->  ( F `  j
)  e.  CC ) ) )
23 climmulf.10 . . . . 5  |-  ( (
ph  /\  k  e.  Z )  ->  ( F `  k )  e.  CC )
2423a1i 10 . . . 4  |-  ( k  e.  Z  ->  (
( ph  /\  k  e.  Z )  ->  ( F `  k )  e.  CC ) )
258, 17, 22, 24vtoclgaf 2848 . . 3  |-  ( j  e.  Z  ->  (
( ph  /\  j  e.  Z )  ->  ( F `  j )  e.  CC ) )
266, 7, 25sylc 56 . 2  |-  ( (
ph  /\  j  e.  Z )  ->  ( F `  j )  e.  CC )
27 climmulf.3 . . . . . . 7  |-  F/_ k G
2827, 8nffv 5532 . . . . . 6  |-  F/_ k
( G `  j
)
2928, 15nfel 2427 . . . . 5  |-  F/ k ( G `  j
)  e.  CC
3012, 29nfim 1769 . . . 4  |-  F/ k ( ( ph  /\  j  e.  Z )  ->  ( G `  j
)  e.  CC )
31 fveq2 5525 . . . . . 6  |-  ( k  =  j  ->  ( G `  k )  =  ( G `  j ) )
3231eleq1d 2349 . . . . 5  |-  ( k  =  j  ->  (
( G `  k
)  e.  CC  <->  ( G `  j )  e.  CC ) )
3319, 32imbi12d 311 . . . 4  |-  ( k  =  j  ->  (
( ( ph  /\  k  e.  Z )  ->  ( G `  k
)  e.  CC )  <-> 
( ( ph  /\  j  e.  Z )  ->  ( G `  j
)  e.  CC ) ) )
34 climmulf.11 . . . . 5  |-  ( (
ph  /\  k  e.  Z )  ->  ( G `  k )  e.  CC )
3534a1i 10 . . . 4  |-  ( k  e.  Z  ->  (
( ph  /\  k  e.  Z )  ->  ( G `  k )  e.  CC ) )
368, 30, 33, 35vtoclgaf 2848 . . 3  |-  ( j  e.  Z  ->  (
( ph  /\  j  e.  Z )  ->  ( G `  j )  e.  CC ) )
376, 7, 36sylc 56 . 2  |-  ( (
ph  /\  j  e.  Z )  ->  ( G `  j )  e.  CC )
38 climmulf.4 . . . . . . 7  |-  F/_ k H
3938, 8nffv 5532 . . . . . 6  |-  F/_ k
( H `  j
)
40 nfcv 2419 . . . . . . 7  |-  F/_ k  x.
4114, 40, 28nfov 5881 . . . . . 6  |-  F/_ k
( ( F `  j )  x.  ( G `  j )
)
4239, 41nfeq 2426 . . . . 5  |-  F/ k ( H `  j
)  =  ( ( F `  j )  x.  ( G `  j ) )
4312, 42nfim 1769 . . . 4  |-  F/ k ( ( ph  /\  j  e.  Z )  ->  ( H `  j
)  =  ( ( F `  j )  x.  ( G `  j ) ) )
44 fveq2 5525 . . . . . 6  |-  ( k  =  j  ->  ( H `  k )  =  ( H `  j ) )
4520, 31oveq12d 5876 . . . . . 6  |-  ( k  =  j  ->  (
( F `  k
)  x.  ( G `
 k ) )  =  ( ( F `
 j )  x.  ( G `  j
) ) )
4644, 45eqeq12d 2297 . . . . 5  |-  ( k  =  j  ->  (
( H `  k
)  =  ( ( F `  k )  x.  ( G `  k ) )  <->  ( H `  j )  =  ( ( F `  j
)  x.  ( G `
 j ) ) ) )
4719, 46imbi12d 311 . . . 4  |-  ( k  =  j  ->  (
( ( ph  /\  k  e.  Z )  ->  ( H `  k
)  =  ( ( F `  k )  x.  ( G `  k ) ) )  <-> 
( ( ph  /\  j  e.  Z )  ->  ( H `  j
)  =  ( ( F `  j )  x.  ( G `  j ) ) ) ) )
48 climmulf.12 . . . . 5  |-  ( (
ph  /\  k  e.  Z )  ->  ( H `  k )  =  ( ( F `
 k )  x.  ( G `  k
) ) )
4948a1i 10 . . . 4  |-  ( k  e.  Z  ->  (
( ph  /\  k  e.  Z )  ->  ( H `  k )  =  ( ( F `
 k )  x.  ( G `  k
) ) ) )
508, 43, 47, 49vtoclgaf 2848 . . 3  |-  ( j  e.  Z  ->  (
( ph  /\  j  e.  Z )  ->  ( H `  j )  =  ( ( F `
 j )  x.  ( G `  j
) ) ) )
516, 7, 50sylc 56 . 2  |-  ( (
ph  /\  j  e.  Z )  ->  ( H `  j )  =  ( ( F `
 j )  x.  ( G `  j
) ) )
521, 2, 3, 4, 5, 26, 37, 51climmul 12106 1  |-  ( ph  ->  H  ~~>  ( A  x.  B ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 358   F/wnf 1531    = wceq 1623    e. wcel 1684   F/_wnfc 2406   class class class wbr 4023   ` cfv 5255  (class class class)co 5858   CCcc 8735    x. cmul 8742   ZZcz 10024   ZZ>=cuz 10230    ~~> cli 11958
This theorem is referenced by:  climneg  27736  climdivf  27738  stirlinglem15  27837
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1533  ax-5 1544  ax-17 1603  ax-9 1635  ax-8 1643  ax-13 1686  ax-14 1688  ax-6 1703  ax-7 1708  ax-11 1715  ax-12 1866  ax-ext 2264  ax-sep 4141  ax-nul 4149  ax-pow 4188  ax-pr 4214  ax-un 4512  ax-cnex 8793  ax-resscn 8794  ax-1cn 8795  ax-icn 8796  ax-addcl 8797  ax-addrcl 8798  ax-mulcl 8799  ax-mulrcl 8800  ax-mulcom 8801  ax-addass 8802  ax-mulass 8803  ax-distr 8804  ax-i2m1 8805  ax-1ne0 8806  ax-1rid 8807  ax-rnegex 8808  ax-rrecex 8809  ax-cnre 8810  ax-pre-lttri 8811  ax-pre-lttrn 8812  ax-pre-ltadd 8813  ax-pre-mulgt0 8814  ax-pre-sup 8815
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3or 935  df-3an 936  df-tru 1310  df-ex 1529  df-nf 1532  df-sb 1630  df-eu 2147  df-mo 2148  df-clab 2270  df-cleq 2276  df-clel 2279  df-nfc 2408  df-ne 2448  df-nel 2449  df-ral 2548  df-rex 2549  df-reu 2550  df-rmo 2551  df-rab 2552  df-v 2790  df-sbc 2992  df-csb 3082  df-dif 3155  df-un 3157  df-in 3159  df-ss 3166  df-pss 3168  df-nul 3456  df-if 3566  df-pw 3627  df-sn 3646  df-pr 3647  df-tp 3648  df-op 3649  df-uni 3828  df-iun 3907  df-br 4024  df-opab 4078  df-mpt 4079  df-tr 4114  df-eprel 4305  df-id 4309  df-po 4314  df-so 4315  df-fr 4352  df-we 4354  df-ord 4395  df-on 4396  df-lim 4397  df-suc 4398  df-om 4657  df-xp 4695  df-rel 4696  df-cnv 4697  df-co 4698  df-dm 4699  df-rn 4700  df-res 4701  df-ima 4702  df-iota 5219  df-fun 5257  df-fn 5258  df-f 5259  df-f1 5260  df-fo 5261  df-f1o 5262  df-fv 5263  df-ov 5861  df-oprab 5862  df-mpt2 5863  df-2nd 6123  df-riota 6304  df-recs 6388  df-rdg 6423  df-er 6660  df-en 6864  df-dom 6865  df-sdom 6866  df-sup 7194  df-pnf 8869  df-mnf 8870  df-xr 8871  df-ltxr 8872  df-le 8873  df-sub 9039  df-neg 9040  df-div 9424  df-nn 9747  df-2 9804  df-3 9805  df-n0 9966  df-z 10025  df-uz 10231  df-rp 10355  df-seq 11047  df-exp 11105  df-cj 11584  df-re 11585  df-im 11586  df-sqr 11720  df-abs 11721  df-clim 11962
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