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Theorem fmul01lt1 27819
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 1609 . . 3  |-  F/ j
ph
3 fmul01lt1.3 . . . . 5  |-  F/_ j A
4 nfcv 2432 . . . . 5  |-  F/_ j M
53, 4nffv 5548 . . . 4  |-  F/_ j
( A `  M
)
6 nfcv 2432 . . . 4  |-  F/_ j  <
7 nfcv 2432 . . . 4  |-  F/_ j E
85, 6, 7nfbr 4083 . . 3  |-  F/ j ( A `  M
)  <  E
9 fmul01lt1.1 . . . . 5  |-  F/_ i B
10 fmul01lt1.2 . . . . . 6  |-  F/ i
ph
11 nfv 1609 . . . . . 6  |-  F/ i  j  e.  ( 1 ... M )
12 nfcv 2432 . . . . . . . 8  |-  F/_ i
j
139, 12nffv 5548 . . . . . . 7  |-  F/_ i
( B `  j
)
14 nfcv 2432 . . . . . . 7  |-  F/_ i  <
15 nfcv 2432 . . . . . . 7  |-  F/_ i E
1613, 14, 15nfbr 4083 . . . . . 6  |-  F/ i ( B `  j
)  <  E
1710, 11, 16nf3an 1786 . . . . 5  |-  F/ i ( ph  /\  j  e.  ( 1 ... M
)  /\  ( B `  j )  <  E
)
18 fmul01lt1.4 . . . . 5  |-  A  =  seq  1 (  x.  ,  B )
19 1z 10069 . . . . . 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 10280 . . . . . . 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 5679 . . . . . . 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 27818 . . . 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 2677 . 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 1534    = wceq 1632    e. wcel 1696   F/_wnfc 2419   E.wrex 2557   class class class wbr 4039   -->wf 5267   ` cfv 5271  (class class class)co 5874   RRcr 8752   0cc0 8753   1c1 8754    x. cmul 8758    < clt 8883    <_ cle 8884   NNcn 9762   ZZcz 10040   ZZ>=cuz 10246   RR+crp 10370   ...cfz 10798    seq cseq 11062
This theorem is referenced by:  stoweidlem48  27900
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
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-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-1st 6138  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-pnf 8885  df-mnf 8886  df-xr 8887  df-ltxr 8888  df-le 8889  df-sub 9055  df-neg 9056  df-nn 9763  df-n0 9982  df-z 10041  df-uz 10247  df-rp 10371  df-fz 10799  df-fzo 10887  df-seq 11063
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