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Theorem fmul01lt1 27683
Description: Given a finite multiplication of values betweeen 0 and 1, a value E larger than any multiplicand, is larger than the whole multiplication. (Contributed by Glauco Siliprandi, 20-Apr-2017.)
Hypotheses
Ref Expression
fmul01lt1.1  |-  F/_ i B
fmul01lt1.2  |-  F/ i
ph
fmul01lt1.3  |-  F/_ j A
fmul01lt1.4  |-  A  =  seq  1 (  x.  ,  B )
fmul01lt1.5  |-  ( ph  ->  M  e.  NN )
fmul01lt1.6  |-  ( ph  ->  B : ( 1 ... M ) --> RR )
fmul01lt1.7  |-  ( (
ph  /\  i  e.  ( 1 ... M
) )  ->  0  <_  ( B `  i
) )
fmul01lt1.8  |-  ( (
ph  /\  i  e.  ( 1 ... M
) )  ->  ( B `  i )  <_  1 )
fmul01lt1.9  |-  ( ph  ->  E  e.  RR+ )
fmul01lt1.10  |-  ( ph  ->  E. j  e.  ( 1 ... M ) ( B `  j
)  <  E )
Assertion
Ref Expression
fmul01lt1  |-  ( ph  ->  ( A `  M
)  <  E )
Distinct variable groups:    i, j, E    i, M, j    ph, j
Allowed substitution hints:    ph( i)    A( i, j)    B( i, j)

Proof of Theorem fmul01lt1
StepHypRef Expression
1 fmul01lt1.10 . 2  |-  ( ph  ->  E. j  e.  ( 1 ... M ) ( B `  j
)  <  E )
2 nfv 1629 . . 3  |-  F/ j
ph
3 fmul01lt1.3 . . . . 5  |-  F/_ j A
4 nfcv 2571 . . . . 5  |-  F/_ j M
53, 4nffv 5727 . . . 4  |-  F/_ j
( A `  M
)
6 nfcv 2571 . . . 4  |-  F/_ j  <
7 nfcv 2571 . . . 4  |-  F/_ j E
85, 6, 7nfbr 4248 . . 3  |-  F/ j ( A `  M
)  <  E
9 fmul01lt1.1 . . . . 5  |-  F/_ i B
10 fmul01lt1.2 . . . . . 6  |-  F/ i
ph
11 nfv 1629 . . . . . 6  |-  F/ i  j  e.  ( 1 ... M )
12 nfcv 2571 . . . . . . . 8  |-  F/_ i
j
139, 12nffv 5727 . . . . . . 7  |-  F/_ i
( B `  j
)
14 nfcv 2571 . . . . . . 7  |-  F/_ i  <
15 nfcv 2571 . . . . . . 7  |-  F/_ i E
1613, 14, 15nfbr 4248 . . . . . 6  |-  F/ i ( B `  j
)  <  E
1710, 11, 16nf3an 1849 . . . . 5  |-  F/ i ( ph  /\  j  e.  ( 1 ... M
)  /\  ( B `  j )  <  E
)
18 fmul01lt1.4 . . . . 5  |-  A  =  seq  1 (  x.  ,  B )
19 1z 10303 . . . . . 6  |-  1  e.  ZZ
2019a1i 11 . . . . 5  |-  ( (
ph  /\  j  e.  ( 1 ... M
)  /\  ( B `  j )  <  E
)  ->  1  e.  ZZ )
21 fmul01lt1.5 . . . . . . 7  |-  ( ph  ->  M  e.  NN )
22 elnnuz 10514 . . . . . . 7  |-  ( M  e.  NN  <->  M  e.  ( ZZ>= `  1 )
)
2321, 22sylib 189 . . . . . 6  |-  ( ph  ->  M  e.  ( ZZ>= ` 
1 ) )
24233ad2ant1 978 . . . . 5  |-  ( (
ph  /\  j  e.  ( 1 ... M
)  /\  ( B `  j )  <  E
)  ->  M  e.  ( ZZ>= `  1 )
)
25 fmul01lt1.6 . . . . . . 7  |-  ( ph  ->  B : ( 1 ... M ) --> RR )
2625fnvinran 27652 . . . . . 6  |-  ( (
ph  /\  i  e.  ( 1 ... M
) )  ->  ( B `  i )  e.  RR )
27263ad2antl1 1119 . . . . 5  |-  ( ( ( ph  /\  j  e.  ( 1 ... M
)  /\  ( B `  j )  <  E
)  /\  i  e.  ( 1 ... M
) )  ->  ( B `  i )  e.  RR )
28 fmul01lt1.7 . . . . . 6  |-  ( (
ph  /\  i  e.  ( 1 ... M
) )  ->  0  <_  ( B `  i
) )
29283ad2antl1 1119 . . . . 5  |-  ( ( ( ph  /\  j  e.  ( 1 ... M
)  /\  ( B `  j )  <  E
)  /\  i  e.  ( 1 ... M
) )  ->  0  <_  ( B `  i
) )
30 fmul01lt1.8 . . . . . 6  |-  ( (
ph  /\  i  e.  ( 1 ... M
) )  ->  ( B `  i )  <_  1 )
31303ad2antl1 1119 . . . . 5  |-  ( ( ( ph  /\  j  e.  ( 1 ... M
)  /\  ( B `  j )  <  E
)  /\  i  e.  ( 1 ... M
) )  ->  ( B `  i )  <_  1 )
32 fmul01lt1.9 . . . . . 6  |-  ( ph  ->  E  e.  RR+ )
33323ad2ant1 978 . . . . 5  |-  ( (
ph  /\  j  e.  ( 1 ... M
)  /\  ( B `  j )  <  E
)  ->  E  e.  RR+ )
34 simp2 958 . . . . 5  |-  ( (
ph  /\  j  e.  ( 1 ... M
)  /\  ( B `  j )  <  E
)  ->  j  e.  ( 1 ... M
) )
35 simp3 959 . . . . 5  |-  ( (
ph  /\  j  e.  ( 1 ... M
)  /\  ( B `  j )  <  E
)  ->  ( B `  j )  <  E
)
369, 17, 18, 20, 24, 27, 29, 31, 33, 34, 35fmul01lt1lem2 27682 . . . 4  |-  ( (
ph  /\  j  e.  ( 1 ... M
)  /\  ( B `  j )  <  E
)  ->  ( A `  M )  <  E
)
37363exp 1152 . . 3  |-  ( ph  ->  ( j  e.  ( 1 ... M )  ->  ( ( B `
 j )  < 
E  ->  ( A `  M )  <  E
) ) )
382, 8, 37rexlimd 2819 . 2  |-  ( ph  ->  ( E. j  e.  ( 1 ... M
) ( B `  j )  <  E  ->  ( A `  M
)  <  E )
)
391, 38mpd 15 1  |-  ( ph  ->  ( A `  M
)  <  E )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 359    /\ w3a 936   F/wnf 1553    = wceq 1652    e. wcel 1725   F/_wnfc 2558   E.wrex 2698   class class class wbr 4204   -->wf 5442   ` cfv 5446  (class class class)co 6073   RRcr 8981   0cc0 8982   1c1 8983    x. cmul 8987    < clt 9112    <_ cle 9113   NNcn 9992   ZZcz 10274   ZZ>=cuz 10480   RR+crp 10604   ...cfz 11035    seq cseq 11315
This theorem is referenced by:  stoweidlem48  27764
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 2416  ax-sep 4322  ax-nul 4330  ax-pow 4369  ax-pr 4395  ax-un 4693  ax-cnex 9038  ax-resscn 9039  ax-1cn 9040  ax-icn 9041  ax-addcl 9042  ax-addrcl 9043  ax-mulcl 9044  ax-mulrcl 9045  ax-mulcom 9046  ax-addass 9047  ax-mulass 9048  ax-distr 9049  ax-i2m1 9050  ax-1ne0 9051  ax-1rid 9052  ax-rnegex 9053  ax-rrecex 9054  ax-cnre 9055  ax-pre-lttri 9056  ax-pre-lttrn 9057  ax-pre-ltadd 9058  ax-pre-mulgt0 9059
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 2284  df-mo 2285  df-clab 2422  df-cleq 2428  df-clel 2431  df-nfc 2560  df-ne 2600  df-nel 2601  df-ral 2702  df-rex 2703  df-reu 2704  df-rab 2706  df-v 2950  df-sbc 3154  df-csb 3244  df-dif 3315  df-un 3317  df-in 3319  df-ss 3326  df-pss 3328  df-nul 3621  df-if 3732  df-pw 3793  df-sn 3812  df-pr 3813  df-tp 3814  df-op 3815  df-uni 4008  df-iun 4087  df-br 4205  df-opab 4259  df-mpt 4260  df-tr 4295  df-eprel 4486  df-id 4490  df-po 4495  df-so 4496  df-fr 4533  df-we 4535  df-ord 4576  df-on 4577  df-lim 4578  df-suc 4579  df-om 4838  df-xp 4876  df-rel 4877  df-cnv 4878  df-co 4879  df-dm 4880  df-rn 4881  df-res 4882  df-ima 4883  df-iota 5410  df-fun 5448  df-fn 5449  df-f 5450  df-f1 5451  df-fo 5452  df-f1o 5453  df-fv 5454  df-ov 6076  df-oprab 6077  df-mpt2 6078  df-1st 6341  df-2nd 6342  df-riota 6541  df-recs 6625  df-rdg 6660  df-er 6897  df-en 7102  df-dom 7103  df-sdom 7104  df-pnf 9114  df-mnf 9115  df-xr 9116  df-ltxr 9117  df-le 9118  df-sub 9285  df-neg 9286  df-nn 9993  df-n0 10214  df-z 10275  df-uz 10481  df-rp 10605  df-fz 11036  df-fzo 11128  df-seq 11316
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