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Theorem fmul01lt1 27384
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 1626 . . 3  |-  F/ j
ph
3 fmul01lt1.3 . . . . 5  |-  F/_ j A
4 nfcv 2523 . . . . 5  |-  F/_ j M
53, 4nffv 5675 . . . 4  |-  F/_ j
( A `  M
)
6 nfcv 2523 . . . 4  |-  F/_ j  <
7 nfcv 2523 . . . 4  |-  F/_ j E
85, 6, 7nfbr 4197 . . 3  |-  F/ j ( A `  M
)  <  E
9 fmul01lt1.1 . . . . 5  |-  F/_ i B
10 fmul01lt1.2 . . . . . 6  |-  F/ i
ph
11 nfv 1626 . . . . . 6  |-  F/ i  j  e.  ( 1 ... M )
12 nfcv 2523 . . . . . . . 8  |-  F/_ i
j
139, 12nffv 5675 . . . . . . 7  |-  F/_ i
( B `  j
)
14 nfcv 2523 . . . . . . 7  |-  F/_ i  <
15 nfcv 2523 . . . . . . 7  |-  F/_ i E
1613, 14, 15nfbr 4197 . . . . . 6  |-  F/ i ( B `  j
)  <  E
1710, 11, 16nf3an 1839 . . . . 5  |-  F/ i ( ph  /\  j  e.  ( 1 ... M
)  /\  ( B `  j )  <  E
)
18 fmul01lt1.4 . . . . 5  |-  A  =  seq  1 (  x.  ,  B )
19 1z 10243 . . . . . 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 10454 . . . . . . 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 27353 . . . . . 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 27383 . . . 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 2770 . 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 1550    = wceq 1649    e. wcel 1717   F/_wnfc 2510   E.wrex 2650   class class class wbr 4153   -->wf 5390   ` cfv 5394  (class class class)co 6020   RRcr 8922   0cc0 8923   1c1 8924    x. cmul 8928    < clt 9053    <_ cle 9054   NNcn 9932   ZZcz 10214   ZZ>=cuz 10420   RR+crp 10544   ...cfz 10975    seq cseq 11250
This theorem is referenced by:  stoweidlem48  27465
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1552  ax-5 1563  ax-17 1623  ax-9 1661  ax-8 1682  ax-13 1719  ax-14 1721  ax-6 1736  ax-7 1741  ax-11 1753  ax-12 1939  ax-ext 2368  ax-sep 4271  ax-nul 4279  ax-pow 4318  ax-pr 4344  ax-un 4641  ax-cnex 8979  ax-resscn 8980  ax-1cn 8981  ax-icn 8982  ax-addcl 8983  ax-addrcl 8984  ax-mulcl 8985  ax-mulrcl 8986  ax-mulcom 8987  ax-addass 8988  ax-mulass 8989  ax-distr 8990  ax-i2m1 8991  ax-1ne0 8992  ax-1rid 8993  ax-rnegex 8994  ax-rrecex 8995  ax-cnre 8996  ax-pre-lttri 8997  ax-pre-lttrn 8998  ax-pre-ltadd 8999  ax-pre-mulgt0 9000
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3or 937  df-3an 938  df-tru 1325  df-ex 1548  df-nf 1551  df-sb 1656  df-eu 2242  df-mo 2243  df-clab 2374  df-cleq 2380  df-clel 2383  df-nfc 2512  df-ne 2552  df-nel 2553  df-ral 2654  df-rex 2655  df-reu 2656  df-rab 2658  df-v 2901  df-sbc 3105  df-csb 3195  df-dif 3266  df-un 3268  df-in 3270  df-ss 3277  df-pss 3279  df-nul 3572  df-if 3683  df-pw 3744  df-sn 3763  df-pr 3764  df-tp 3765  df-op 3766  df-uni 3958  df-iun 4037  df-br 4154  df-opab 4208  df-mpt 4209  df-tr 4244  df-eprel 4435  df-id 4439  df-po 4444  df-so 4445  df-fr 4482  df-we 4484  df-ord 4525  df-on 4526  df-lim 4527  df-suc 4528  df-om 4786  df-xp 4824  df-rel 4825  df-cnv 4826  df-co 4827  df-dm 4828  df-rn 4829  df-res 4830  df-ima 4831  df-iota 5358  df-fun 5396  df-fn 5397  df-f 5398  df-f1 5399  df-fo 5400  df-f1o 5401  df-fv 5402  df-ov 6023  df-oprab 6024  df-mpt2 6025  df-1st 6288  df-2nd 6289  df-riota 6485  df-recs 6569  df-rdg 6604  df-er 6841  df-en 7046  df-dom 7047  df-sdom 7048  df-pnf 9055  df-mnf 9056  df-xr 9057  df-ltxr 9058  df-le 9059  df-sub 9225  df-neg 9226  df-nn 9933  df-n0 10154  df-z 10215  df-uz 10421  df-rp 10545  df-fz 10976  df-fzo 11066  df-seq 11251
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