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Theorem climmulf 37244
Description: A version of climmul 13674 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 2591 . . . . . 6  |-  F/_ k
j
87nfel1 2607 . . . . 5  |-  F/ k  j  e.  Z
96, 8nfan 1986 . . . 4  |-  F/ k ( ph  /\  j  e.  Z )
10 climmulf.2 . . . . . 6  |-  F/_ k F
1110, 7nffv 5888 . . . . 5  |-  F/_ k
( F `  j
)
1211nfel1 2607 . . . 4  |-  F/ k ( F `  j
)  e.  CC
139, 12nfim 1978 . . 3  |-  F/ k ( ( ph  /\  j  e.  Z )  ->  ( F `  j
)  e.  CC )
14 eleq1 2501 . . . . 5  |-  ( k  =  j  ->  (
k  e.  Z  <->  j  e.  Z ) )
1514anbi2d 708 . . . 4  |-  ( k  =  j  ->  (
( ph  /\  k  e.  Z )  <->  ( ph  /\  j  e.  Z ) ) )
16 fveq2 5881 . . . . 5  |-  ( k  =  j  ->  ( F `  k )  =  ( F `  j ) )
1716eleq1d 2498 . . . 4  |-  ( k  =  j  ->  (
( F `  k
)  e.  CC  <->  ( F `  j )  e.  CC ) )
1815, 17imbi12d 321 . . 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 2069 . 2  |-  ( (
ph  /\  j  e.  Z )  ->  ( F `  j )  e.  CC )
21 climmulf.3 . . . . . 6  |-  F/_ k G
2221, 7nffv 5888 . . . . 5  |-  F/_ k
( G `  j
)
2322nfel1 2607 . . . 4  |-  F/ k ( G `  j
)  e.  CC
249, 23nfim 1978 . . 3  |-  F/ k ( ( ph  /\  j  e.  Z )  ->  ( G `  j
)  e.  CC )
25 fveq2 5881 . . . . 5  |-  ( k  =  j  ->  ( G `  k )  =  ( G `  j ) )
2625eleq1d 2498 . . . 4  |-  ( k  =  j  ->  (
( G `  k
)  e.  CC  <->  ( G `  j )  e.  CC ) )
2715, 26imbi12d 321 . . 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 2069 . 2  |-  ( (
ph  /\  j  e.  Z )  ->  ( G `  j )  e.  CC )
30 climmulf.4 . . . . . 6  |-  F/_ k H
3130, 7nffv 5888 . . . . 5  |-  F/_ k
( H `  j
)
32 nfcv 2591 . . . . . 6  |-  F/_ k  x.
3311, 32, 22nfov 6331 . . . . 5  |-  F/_ k
( ( F `  j )  x.  ( G `  j )
)
3431, 33nfeq 2602 . . . 4  |-  F/ k ( H `  j
)  =  ( ( F `  j )  x.  ( G `  j ) )
359, 34nfim 1978 . . 3  |-  F/ k ( ( ph  /\  j  e.  Z )  ->  ( H `  j
)  =  ( ( F `  j )  x.  ( G `  j ) ) )
36 fveq2 5881 . . . . 5  |-  ( k  =  j  ->  ( H `  k )  =  ( H `  j ) )
3716, 25oveq12d 6323 . . . . 5  |-  ( k  =  j  ->  (
( F `  k
)  x.  ( G `
 k ) )  =  ( ( F `
 j )  x.  ( G `  j
) ) )
3836, 37eqeq12d 2451 . . . 4  |-  ( k  =  j  ->  (
( H `  k
)  =  ( ( F `  k )  x.  ( G `  k ) )  <->  ( H `  j )  =  ( ( F `  j
)  x.  ( G `
 j ) ) ) )
3915, 38imbi12d 321 . . 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 2069 . 2  |-  ( (
ph  /\  j  e.  Z )  ->  ( H `  j )  =  ( ( F `
 j )  x.  ( G `  j
) ) )
421, 2, 3, 4, 5, 20, 29, 41climmul 13674 1  |-  ( ph  ->  H  ~~>  ( A  x.  B ) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    /\ wa 370    = wceq 1437   F/wnf 1663    e. wcel 1870   F/_wnfc 2577   class class class wbr 4426   ` cfv 5601  (class class class)co 6305   CCcc 9536    x. cmul 9543   ZZcz 10937   ZZ>=cuz 11159    ~~> cli 13526
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1665  ax-4 1678  ax-5 1751  ax-6 1797  ax-7 1841  ax-8 1872  ax-9 1874  ax-10 1889  ax-11 1894  ax-12 1907  ax-13 2055  ax-ext 2407  ax-sep 4548  ax-nul 4556  ax-pow 4603  ax-pr 4661  ax-un 6597  ax-cnex 9594  ax-resscn 9595  ax-1cn 9596  ax-icn 9597  ax-addcl 9598  ax-addrcl 9599  ax-mulcl 9600  ax-mulrcl 9601  ax-mulcom 9602  ax-addass 9603  ax-mulass 9604  ax-distr 9605  ax-i2m1 9606  ax-1ne0 9607  ax-1rid 9608  ax-rnegex 9609  ax-rrecex 9610  ax-cnre 9611  ax-pre-lttri 9612  ax-pre-lttrn 9613  ax-pre-ltadd 9614  ax-pre-mulgt0 9615  ax-pre-sup 9616
This theorem depends on definitions:  df-bi 188  df-or 371  df-an 372  df-3or 983  df-3an 984  df-tru 1440  df-ex 1660  df-nf 1664  df-sb 1790  df-eu 2270  df-mo 2271  df-clab 2415  df-cleq 2421  df-clel 2424  df-nfc 2579  df-ne 2627  df-nel 2628  df-ral 2787  df-rex 2788  df-reu 2789  df-rmo 2790  df-rab 2791  df-v 3089  df-sbc 3306  df-csb 3402  df-dif 3445  df-un 3447  df-in 3449  df-ss 3456  df-pss 3458  df-nul 3768  df-if 3916  df-pw 3987  df-sn 4003  df-pr 4005  df-tp 4007  df-op 4009  df-uni 4223  df-iun 4304  df-br 4427  df-opab 4485  df-mpt 4486  df-tr 4521  df-eprel 4765  df-id 4769  df-po 4775  df-so 4776  df-fr 4813  df-we 4815  df-xp 4860  df-rel 4861  df-cnv 4862  df-co 4863  df-dm 4864  df-rn 4865  df-res 4866  df-ima 4867  df-pred 5399  df-ord 5445  df-on 5446  df-lim 5447  df-suc 5448  df-iota 5565  df-fun 5603  df-fn 5604  df-f 5605  df-f1 5606  df-fo 5607  df-f1o 5608  df-fv 5609  df-riota 6267  df-ov 6308  df-oprab 6309  df-mpt2 6310  df-om 6707  df-2nd 6808  df-wrecs 7036  df-recs 7098  df-rdg 7136  df-er 7371  df-en 7578  df-dom 7579  df-sdom 7580  df-sup 7962  df-pnf 9676  df-mnf 9677  df-xr 9678  df-ltxr 9679  df-le 9680  df-sub 9861  df-neg 9862  df-div 10269  df-nn 10610  df-2 10668  df-3 10669  df-n0 10870  df-z 10938  df-uz 11160  df-rp 11303  df-seq 12211  df-exp 12270  df-cj 13141  df-re 13142  df-im 13143  df-sqrt 13277  df-abs 13278  df-clim 13530
This theorem is referenced by:  climneg  37251  climdivf  37254  stirlinglem15  37509  etransclem48  37704
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