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Theorem fmul01lt1 27716
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 1605 . . 3  |-  F/ j
ph
3 fmul01lt1.3 . . . . 5  |-  F/_ j A
4 nfcv 2419 . . . . 5  |-  F/_ j M
53, 4nffv 5532 . . . 4  |-  F/_ j
( A `  M
)
6 nfcv 2419 . . . 4  |-  F/_ j  <
7 nfcv 2419 . . . 4  |-  F/_ j E
85, 6, 7nfbr 4067 . . 3  |-  F/ j ( A `  M
)  <  E
9 fmul01lt1.1 . . . . 5  |-  F/_ i B
10 fmul01lt1.2 . . . . . 6  |-  F/ i
ph
11 nfv 1605 . . . . . 6  |-  F/ i  j  e.  ( 1 ... M )
12 nfcv 2419 . . . . . . . 8  |-  F/_ i
j
139, 12nffv 5532 . . . . . . 7  |-  F/_ i
( B `  j
)
14 nfcv 2419 . . . . . . 7  |-  F/_ i  <
15 nfcv 2419 . . . . . . 7  |-  F/_ i E
1613, 14, 15nfbr 4067 . . . . . 6  |-  F/ i ( B `  j
)  <  E
1710, 11, 16nf3an 1774 . . . . 5  |-  F/ i ( ph  /\  j  e.  ( 1 ... M
)  /\  ( B `  j )  <  E
)
18 fmul01lt1.4 . . . . 5  |-  A  =  seq  1 (  x.  ,  B )
19 1z 10053 . . . . . 6  |-  1  e.  ZZ
2019a1i 10 . . . . 5  |-  ( (
ph  /\  j  e.  ( 1 ... M
)  /\  ( B `  j )  <  E
)  ->  1  e.  ZZ )
21 fmul01lt1.5 . . . . . . 7  |-  ( ph  ->  M  e.  NN )
22 elnnuz 10264 . . . . . . 7  |-  ( M  e.  NN  <->  M  e.  ( ZZ>= `  1 )
)
2321, 22sylib 188 . . . . . 6  |-  ( ph  ->  M  e.  ( ZZ>= ` 
1 ) )
24233ad2ant1 976 . . . . 5  |-  ( (
ph  /\  j  e.  ( 1 ... M
)  /\  ( B `  j )  <  E
)  ->  M  e.  ( ZZ>= `  1 )
)
25 fmul01lt1.6 . . . . . . . . 9  |-  ( ph  ->  B : ( 1 ... M ) --> RR )
2625adantr 451 . . . . . . . 8  |-  ( (
ph  /\  i  e.  ( 1 ... M
) )  ->  B : ( 1 ... M ) --> RR )
27 id 19 . . . . . . . . 9  |-  ( i  e.  ( 1 ... M )  ->  i  e.  ( 1 ... M
) )
2827adantl 452 . . . . . . . 8  |-  ( (
ph  /\  i  e.  ( 1 ... M
) )  ->  i  e.  ( 1 ... M
) )
2926, 28jca 518 . . . . . . 7  |-  ( (
ph  /\  i  e.  ( 1 ... M
) )  ->  ( B : ( 1 ... M ) --> RR  /\  i  e.  ( 1 ... M ) ) )
30 ffvelrn 5663 . . . . . . 7  |-  ( ( B : ( 1 ... M ) --> RR 
/\  i  e.  ( 1 ... M ) )  ->  ( B `  i )  e.  RR )
3129, 30syl 15 . . . . . 6  |-  ( (
ph  /\  i  e.  ( 1 ... M
) )  ->  ( B `  i )  e.  RR )
32313ad2antl1 1117 . . . . 5  |-  ( ( ( ph  /\  j  e.  ( 1 ... M
)  /\  ( B `  j )  <  E
)  /\  i  e.  ( 1 ... M
) )  ->  ( B `  i )  e.  RR )
33 fmul01lt1.7 . . . . . 6  |-  ( (
ph  /\  i  e.  ( 1 ... M
) )  ->  0  <_  ( B `  i
) )
34333ad2antl1 1117 . . . . 5  |-  ( ( ( ph  /\  j  e.  ( 1 ... M
)  /\  ( B `  j )  <  E
)  /\  i  e.  ( 1 ... M
) )  ->  0  <_  ( B `  i
) )
35 fmul01lt1.8 . . . . . 6  |-  ( (
ph  /\  i  e.  ( 1 ... M
) )  ->  ( B `  i )  <_  1 )
36353ad2antl1 1117 . . . . 5  |-  ( ( ( ph  /\  j  e.  ( 1 ... M
)  /\  ( B `  j )  <  E
)  /\  i  e.  ( 1 ... M
) )  ->  ( B `  i )  <_  1 )
37 fmul01lt1.9 . . . . . 6  |-  ( ph  ->  E  e.  RR+ )
38373ad2ant1 976 . . . . 5  |-  ( (
ph  /\  j  e.  ( 1 ... M
)  /\  ( B `  j )  <  E
)  ->  E  e.  RR+ )
39 simp2 956 . . . . 5  |-  ( (
ph  /\  j  e.  ( 1 ... M
)  /\  ( B `  j )  <  E
)  ->  j  e.  ( 1 ... M
) )
40 simp3 957 . . . . 5  |-  ( (
ph  /\  j  e.  ( 1 ... M
)  /\  ( B `  j )  <  E
)  ->  ( B `  j )  <  E
)
419, 17, 18, 20, 24, 32, 34, 36, 38, 39, 40fmul01lt1lem2 27715 . . . 4  |-  ( (
ph  /\  j  e.  ( 1 ... M
)  /\  ( B `  j )  <  E
)  ->  ( A `  M )  <  E
)
42413exp 1150 . . 3  |-  ( ph  ->  ( j  e.  ( 1 ... M )  ->  ( ( B `
 j )  < 
E  ->  ( A `  M )  <  E
) ) )
432, 8, 42rexlimd 2664 . 2  |-  ( ph  ->  ( E. j  e.  ( 1 ... M
) ( B `  j )  <  E  ->  ( A `  M
)  <  E )
)
441, 43mpd 14 1  |-  ( ph  ->  ( A `  M
)  <  E )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 358    /\ w3a 934   F/wnf 1531    = wceq 1623    e. wcel 1684   F/_wnfc 2406   E.wrex 2544   class class class wbr 4023   -->wf 5251   ` cfv 5255  (class class class)co 5858   RRcr 8736   0cc0 8737   1c1 8738    x. cmul 8742    < clt 8867    <_ cle 8868   NNcn 9746   ZZcz 10024   ZZ>=cuz 10230   RR+crp 10354   ...cfz 10782    seq cseq 11046
This theorem is referenced by:  stoweidlem48  27797
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
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-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-1st 6122  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-pnf 8869  df-mnf 8870  df-xr 8871  df-ltxr 8872  df-le 8873  df-sub 9039  df-neg 9040  df-nn 9747  df-n0 9966  df-z 10025  df-uz 10231  df-rp 10355  df-fz 10783  df-fzo 10871  df-seq 11047
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