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Theorem climmulf 27833
Description: A version of climmul 12122 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 2432 . . . 4  |-  F/_ k
j
9 climmulf.1 . . . . . 6  |-  F/ k
ph
10 nfcv 2432 . . . . . . 7  |-  F/_ k Z
118, 10nfel 2440 . . . . . 6  |-  F/ k  j  e.  Z
129, 11nfan 1783 . . . . 5  |-  F/ k ( ph  /\  j  e.  Z )
13 climmulf.2 . . . . . . 7  |-  F/_ k F
1413, 8nffv 5548 . . . . . 6  |-  F/_ k
( F `  j
)
15 nfcv 2432 . . . . . 6  |-  F/_ k CC
1614, 15nfel 2440 . . . . 5  |-  F/ k ( F `  j
)  e.  CC
1712, 16nfim 1781 . . . 4  |-  F/ k ( ( ph  /\  j  e.  Z )  ->  ( F `  j
)  e.  CC )
18 eleq1 2356 . . . . . 6  |-  ( k  =  j  ->  (
k  e.  Z  <->  j  e.  Z ) )
1918anbi2d 684 . . . . 5  |-  ( k  =  j  ->  (
( ph  /\  k  e.  Z )  <->  ( ph  /\  j  e.  Z ) ) )
20 fveq2 5541 . . . . . 6  |-  ( k  =  j  ->  ( F `  k )  =  ( F `  j ) )
2120eleq1d 2362 . . . . 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 2861 . . 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 5548 . . . . . 6  |-  F/_ k
( G `  j
)
2928, 15nfel 2440 . . . . 5  |-  F/ k ( G `  j
)  e.  CC
3012, 29nfim 1781 . . . 4  |-  F/ k ( ( ph  /\  j  e.  Z )  ->  ( G `  j
)  e.  CC )
31 fveq2 5541 . . . . . 6  |-  ( k  =  j  ->  ( G `  k )  =  ( G `  j ) )
3231eleq1d 2362 . . . . 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 2861 . . 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 5548 . . . . . 6  |-  F/_ k
( H `  j
)
40 nfcv 2432 . . . . . . 7  |-  F/_ k  x.
4114, 40, 28nfov 5897 . . . . . 6  |-  F/_ k
( ( F `  j )  x.  ( G `  j )
)
4239, 41nfeq 2439 . . . . 5  |-  F/ k ( H `  j
)  =  ( ( F `  j )  x.  ( G `  j ) )
4312, 42nfim 1781 . . . 4  |-  F/ k ( ( ph  /\  j  e.  Z )  ->  ( H `  j
)  =  ( ( F `  j )  x.  ( G `  j ) ) )
44 fveq2 5541 . . . . . 6  |-  ( k  =  j  ->  ( H `  k )  =  ( H `  j ) )
4520, 31oveq12d 5892 . . . . . 6  |-  ( k  =  j  ->  (
( F `  k
)  x.  ( G `
 k ) )  =  ( ( F `
 j )  x.  ( G `  j
) ) )
4644, 45eqeq12d 2310 . . . . 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 2861 . . 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 12122 1  |-  ( ph  ->  H  ~~>  ( A  x.  B ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 358   F/wnf 1534    = wceq 1632    e. wcel 1696   F/_wnfc 2419   class class class wbr 4039   ` cfv 5271  (class class class)co 5874   CCcc 8751    x. cmul 8758   ZZcz 10040   ZZ>=cuz 10246    ~~> cli 11974
This theorem is referenced by:  climneg  27839  climdivf  27841  stirlinglem15  27940
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1536  ax-5 1547  ax-17 1606  ax-9 1644  ax-8 1661  ax-13 1698  ax-14 1700  ax-6 1715  ax-7 1720  ax-11 1727  ax-12 1878  ax-ext 2277  ax-sep 4157  ax-nul 4165  ax-pow 4204  ax-pr 4230  ax-un 4528  ax-cnex 8809  ax-resscn 8810  ax-1cn 8811  ax-icn 8812  ax-addcl 8813  ax-addrcl 8814  ax-mulcl 8815  ax-mulrcl 8816  ax-mulcom 8817  ax-addass 8818  ax-mulass 8819  ax-distr 8820  ax-i2m1 8821  ax-1ne0 8822  ax-1rid 8823  ax-rnegex 8824  ax-rrecex 8825  ax-cnre 8826  ax-pre-lttri 8827  ax-pre-lttrn 8828  ax-pre-ltadd 8829  ax-pre-mulgt0 8830  ax-pre-sup 8831
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 1532  df-nf 1535  df-sb 1639  df-eu 2160  df-mo 2161  df-clab 2283  df-cleq 2289  df-clel 2292  df-nfc 2421  df-ne 2461  df-nel 2462  df-ral 2561  df-rex 2562  df-reu 2563  df-rmo 2564  df-rab 2565  df-v 2803  df-sbc 3005  df-csb 3095  df-dif 3168  df-un 3170  df-in 3172  df-ss 3179  df-pss 3181  df-nul 3469  df-if 3579  df-pw 3640  df-sn 3659  df-pr 3660  df-tp 3661  df-op 3662  df-uni 3844  df-iun 3923  df-br 4040  df-opab 4094  df-mpt 4095  df-tr 4130  df-eprel 4321  df-id 4325  df-po 4330  df-so 4331  df-fr 4368  df-we 4370  df-ord 4411  df-on 4412  df-lim 4413  df-suc 4414  df-om 4673  df-xp 4711  df-rel 4712  df-cnv 4713  df-co 4714  df-dm 4715  df-rn 4716  df-res 4717  df-ima 4718  df-iota 5235  df-fun 5273  df-fn 5274  df-f 5275  df-f1 5276  df-fo 5277  df-f1o 5278  df-fv 5279  df-ov 5877  df-oprab 5878  df-mpt2 5879  df-2nd 6139  df-riota 6320  df-recs 6404  df-rdg 6439  df-er 6676  df-en 6880  df-dom 6881  df-sdom 6882  df-sup 7210  df-pnf 8885  df-mnf 8886  df-xr 8887  df-ltxr 8888  df-le 8889  df-sub 9055  df-neg 9056  df-div 9440  df-nn 9763  df-2 9820  df-3 9821  df-n0 9982  df-z 10041  df-uz 10247  df-rp 10371  df-seq 11063  df-exp 11121  df-cj 11600  df-re 11601  df-im 11602  df-sqr 11736  df-abs 11737  df-clim 11978
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