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Theorem fmul01lt1 31746
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 1715 . . 3  |-  F/ j
ph
3 fmul01lt1.3 . . . . 5  |-  F/_ j A
4 nfcv 2544 . . . . 5  |-  F/_ j M
53, 4nffv 5781 . . . 4  |-  F/_ j
( A `  M
)
6 nfcv 2544 . . . 4  |-  F/_ j  <
7 nfcv 2544 . . . 4  |-  F/_ j E
85, 6, 7nfbr 4411 . . 3  |-  F/ j ( A `  M
)  <  E
9 fmul01lt1.1 . . . . 5  |-  F/_ i B
10 fmul01lt1.2 . . . . . 6  |-  F/ i
ph
11 nfv 1715 . . . . . 6  |-  F/ i  j  e.  ( 1 ... M )
12 nfcv 2544 . . . . . . . 8  |-  F/_ i
j
139, 12nffv 5781 . . . . . . 7  |-  F/_ i
( B `  j
)
14 nfcv 2544 . . . . . . 7  |-  F/_ i  <
15 nfcv 2544 . . . . . . 7  |-  F/_ i E
1613, 14, 15nfbr 4411 . . . . . 6  |-  F/ i ( B `  j
)  <  E
1710, 11, 16nf3an 1938 . . . . 5  |-  F/ i ( ph  /\  j  e.  ( 1 ... M
)  /\  ( B `  j )  <  E
)
18 fmul01lt1.4 . . . . 5  |-  A  =  seq 1 (  x.  ,  B )
19 1zzd 10812 . . . . 5  |-  ( (
ph  /\  j  e.  ( 1 ... M
)  /\  ( B `  j )  <  E
)  ->  1  e.  ZZ )
20 fmul01lt1.5 . . . . . . 7  |-  ( ph  ->  M  e.  NN )
21 elnnuz 11037 . . . . . . 7  |-  ( M  e.  NN  <->  M  e.  ( ZZ>= `  1 )
)
2220, 21sylib 196 . . . . . 6  |-  ( ph  ->  M  e.  ( ZZ>= ` 
1 ) )
23223ad2ant1 1015 . . . . 5  |-  ( (
ph  /\  j  e.  ( 1 ... M
)  /\  ( B `  j )  <  E
)  ->  M  e.  ( ZZ>= `  1 )
)
24 fmul01lt1.6 . . . . . . 7  |-  ( ph  ->  B : ( 1 ... M ) --> RR )
2524fnvinran 31556 . . . . . 6  |-  ( (
ph  /\  i  e.  ( 1 ... M
) )  ->  ( B `  i )  e.  RR )
26253ad2antl1 1156 . . . . 5  |-  ( ( ( ph  /\  j  e.  ( 1 ... M
)  /\  ( B `  j )  <  E
)  /\  i  e.  ( 1 ... M
) )  ->  ( B `  i )  e.  RR )
27 fmul01lt1.7 . . . . . 6  |-  ( (
ph  /\  i  e.  ( 1 ... M
) )  ->  0  <_  ( B `  i
) )
28273ad2antl1 1156 . . . . 5  |-  ( ( ( ph  /\  j  e.  ( 1 ... M
)  /\  ( B `  j )  <  E
)  /\  i  e.  ( 1 ... M
) )  ->  0  <_  ( B `  i
) )
29 fmul01lt1.8 . . . . . 6  |-  ( (
ph  /\  i  e.  ( 1 ... M
) )  ->  ( B `  i )  <_  1 )
30293ad2antl1 1156 . . . . 5  |-  ( ( ( ph  /\  j  e.  ( 1 ... M
)  /\  ( B `  j )  <  E
)  /\  i  e.  ( 1 ... M
) )  ->  ( B `  i )  <_  1 )
31 fmul01lt1.9 . . . . . 6  |-  ( ph  ->  E  e.  RR+ )
32313ad2ant1 1015 . . . . 5  |-  ( (
ph  /\  j  e.  ( 1 ... M
)  /\  ( B `  j )  <  E
)  ->  E  e.  RR+ )
33 simp2 995 . . . . 5  |-  ( (
ph  /\  j  e.  ( 1 ... M
)  /\  ( B `  j )  <  E
)  ->  j  e.  ( 1 ... M
) )
34 simp3 996 . . . . 5  |-  ( (
ph  /\  j  e.  ( 1 ... M
)  /\  ( B `  j )  <  E
)  ->  ( B `  j )  <  E
)
359, 17, 18, 19, 23, 26, 28, 30, 32, 33, 34fmul01lt1lem2 31745 . . . 4  |-  ( (
ph  /\  j  e.  ( 1 ... M
)  /\  ( B `  j )  <  E
)  ->  ( A `  M )  <  E
)
36353exp 1193 . . 3  |-  ( ph  ->  ( j  e.  ( 1 ... M )  ->  ( ( B `
 j )  < 
E  ->  ( A `  M )  <  E
) ) )
372, 8, 36rexlimd 2866 . 2  |-  ( ph  ->  ( E. j  e.  ( 1 ... M
) ( B `  j )  <  E  ->  ( A `  M
)  <  E )
)
381, 37mpd 15 1  |-  ( ph  ->  ( A `  M
)  <  E )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    /\ wa 367    /\ w3a 971    = wceq 1399   F/wnf 1624    e. wcel 1826   F/_wnfc 2530   E.wrex 2733   class class class wbr 4367   -->wf 5492   ` cfv 5496  (class class class)co 6196   RRcr 9402   0cc0 9403   1c1 9404    x. cmul 9408    < clt 9539    <_ cle 9540   NNcn 10452   ZZ>=cuz 11001   RR+crp 11139   ...cfz 11593    seqcseq 12010
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1626  ax-4 1639  ax-5 1712  ax-6 1755  ax-7 1798  ax-8 1828  ax-9 1830  ax-10 1845  ax-11 1850  ax-12 1862  ax-13 2006  ax-ext 2360  ax-sep 4488  ax-nul 4496  ax-pow 4543  ax-pr 4601  ax-un 6491  ax-cnex 9459  ax-resscn 9460  ax-1cn 9461  ax-icn 9462  ax-addcl 9463  ax-addrcl 9464  ax-mulcl 9465  ax-mulrcl 9466  ax-mulcom 9467  ax-addass 9468  ax-mulass 9469  ax-distr 9470  ax-i2m1 9471  ax-1ne0 9472  ax-1rid 9473  ax-rnegex 9474  ax-rrecex 9475  ax-cnre 9476  ax-pre-lttri 9477  ax-pre-lttrn 9478  ax-pre-ltadd 9479  ax-pre-mulgt0 9480
This theorem depends on definitions:  df-bi 185  df-or 368  df-an 369  df-3or 972  df-3an 973  df-tru 1402  df-ex 1621  df-nf 1625  df-sb 1748  df-eu 2222  df-mo 2223  df-clab 2368  df-cleq 2374  df-clel 2377  df-nfc 2532  df-ne 2579  df-nel 2580  df-ral 2737  df-rex 2738  df-reu 2739  df-rab 2741  df-v 3036  df-sbc 3253  df-csb 3349  df-dif 3392  df-un 3394  df-in 3396  df-ss 3403  df-pss 3405  df-nul 3712  df-if 3858  df-pw 3929  df-sn 3945  df-pr 3947  df-tp 3949  df-op 3951  df-uni 4164  df-iun 4245  df-br 4368  df-opab 4426  df-mpt 4427  df-tr 4461  df-eprel 4705  df-id 4709  df-po 4714  df-so 4715  df-fr 4752  df-we 4754  df-ord 4795  df-on 4796  df-lim 4797  df-suc 4798  df-xp 4919  df-rel 4920  df-cnv 4921  df-co 4922  df-dm 4923  df-rn 4924  df-res 4925  df-ima 4926  df-iota 5460  df-fun 5498  df-fn 5499  df-f 5500  df-f1 5501  df-fo 5502  df-f1o 5503  df-fv 5504  df-riota 6158  df-ov 6199  df-oprab 6200  df-mpt2 6201  df-om 6600  df-1st 6699  df-2nd 6700  df-recs 6960  df-rdg 6994  df-er 7229  df-en 7436  df-dom 7437  df-sdom 7438  df-pnf 9541  df-mnf 9542  df-xr 9543  df-ltxr 9544  df-le 9545  df-sub 9720  df-neg 9721  df-nn 10453  df-n0 10713  df-z 10782  df-uz 11002  df-rp 11140  df-fz 11594  df-fzo 11718  df-seq 12011
This theorem is referenced by:  stoweidlem48  31996
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