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Theorem fmul01lt1lem2 27729
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
fmul01lt1lem2.1  |-  F/_ i B
fmul01lt1lem2.2  |-  F/ i
ph
fmul01lt1lem2.3  |-  A  =  seq  L (  x.  ,  B )
fmul01lt1lem2.4  |-  ( ph  ->  L  e.  ZZ )
fmul01lt1lem2.5  |-  ( ph  ->  M  e.  ( ZZ>= `  L ) )
fmul01lt1lem2.6  |-  ( (
ph  /\  i  e.  ( L ... M ) )  ->  ( B `  i )  e.  RR )
fmul01lt1lem2.7  |-  ( (
ph  /\  i  e.  ( L ... M ) )  ->  0  <_  ( B `  i ) )
fmul01lt1lem2.8  |-  ( (
ph  /\  i  e.  ( L ... M ) )  ->  ( B `  i )  <_  1
)
fmul01lt1lem2.9  |-  ( ph  ->  E  e.  RR+ )
fmul01lt1lem2.10  |-  ( ph  ->  J  e.  ( L ... M ) )
fmul01lt1lem2.11  |-  ( ph  ->  ( B `  J
)  <  E )
Assertion
Ref Expression
fmul01lt1lem2  |-  ( ph  ->  ( A `  M
)  <  E )
Distinct variable groups:    i, J    i, L    i, M
Allowed substitution hints:    ph( i)    A( i)    B( i)    E( i)

Proof of Theorem fmul01lt1lem2
Dummy variables  a 
b  c  j are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 fmul01lt1lem2.1 . . 3  |-  F/_ i B
2 fmul01lt1lem2.2 . . . 4  |-  F/ i
ph
3 nfv 1630 . . . 4  |-  F/ i  J  =  L
42, 3nfan 1848 . . 3  |-  F/ i ( ph  /\  J  =  L )
5 fmul01lt1lem2.3 . . 3  |-  A  =  seq  L (  x.  ,  B )
6 fmul01lt1lem2.4 . . . 4  |-  ( ph  ->  L  e.  ZZ )
76adantr 453 . . 3  |-  ( (
ph  /\  J  =  L )  ->  L  e.  ZZ )
8 fmul01lt1lem2.5 . . . 4  |-  ( ph  ->  M  e.  ( ZZ>= `  L ) )
98adantr 453 . . 3  |-  ( (
ph  /\  J  =  L )  ->  M  e.  ( ZZ>= `  L )
)
10 fmul01lt1lem2.6 . . . 4  |-  ( (
ph  /\  i  e.  ( L ... M ) )  ->  ( B `  i )  e.  RR )
1110adantlr 697 . . 3  |-  ( ( ( ph  /\  J  =  L )  /\  i  e.  ( L ... M
) )  ->  ( B `  i )  e.  RR )
12 fmul01lt1lem2.7 . . . 4  |-  ( (
ph  /\  i  e.  ( L ... M ) )  ->  0  <_  ( B `  i ) )
1312adantlr 697 . . 3  |-  ( ( ( ph  /\  J  =  L )  /\  i  e.  ( L ... M
) )  ->  0  <_  ( B `  i
) )
14 fmul01lt1lem2.8 . . . 4  |-  ( (
ph  /\  i  e.  ( L ... M ) )  ->  ( B `  i )  <_  1
)
1514adantlr 697 . . 3  |-  ( ( ( ph  /\  J  =  L )  /\  i  e.  ( L ... M
) )  ->  ( B `  i )  <_  1 )
16 fmul01lt1lem2.9 . . . 4  |-  ( ph  ->  E  e.  RR+ )
1716adantr 453 . . 3  |-  ( (
ph  /\  J  =  L )  ->  E  e.  RR+ )
18 simpr 449 . . . . 5  |-  ( (
ph  /\  J  =  L )  ->  J  =  L )
1918fveq2d 5761 . . . 4  |-  ( (
ph  /\  J  =  L )  ->  ( B `  J )  =  ( B `  L ) )
20 fmul01lt1lem2.11 . . . . 5  |-  ( ph  ->  ( B `  J
)  <  E )
2120adantr 453 . . . 4  |-  ( (
ph  /\  J  =  L )  ->  ( B `  J )  <  E )
2219, 21eqbrtrrd 4259 . . 3  |-  ( (
ph  /\  J  =  L )  ->  ( B `  L )  <  E )
231, 4, 5, 7, 9, 11, 13, 15, 17, 22fmul01lt1lem1 27728 . 2  |-  ( (
ph  /\  J  =  L )  ->  ( A `  M )  <  E )
245fveq1i 5758 . . 3  |-  ( A `
 M )  =  (  seq  L (  x.  ,  B ) `
 M )
25 nfv 1630 . . . . . . . . 9  |-  F/ i  a  e.  ( L ... M )
262, 25nfan 1848 . . . . . . . 8  |-  F/ i ( ph  /\  a  e.  ( L ... M
) )
27 nfcv 2578 . . . . . . . . . 10  |-  F/_ i
a
281, 27nffv 5764 . . . . . . . . 9  |-  F/_ i
( B `  a
)
2928nfel1 2588 . . . . . . . 8  |-  F/ i ( B `  a
)  e.  RR
3026, 29nfim 1834 . . . . . . 7  |-  F/ i ( ( ph  /\  a  e.  ( L ... M ) )  -> 
( B `  a
)  e.  RR )
31 eleq1 2502 . . . . . . . . 9  |-  ( i  =  a  ->  (
i  e.  ( L ... M )  <->  a  e.  ( L ... M ) ) )
3231anbi2d 686 . . . . . . . 8  |-  ( i  =  a  ->  (
( ph  /\  i  e.  ( L ... M
) )  <->  ( ph  /\  a  e.  ( L ... M ) ) ) )
33 fveq2 5757 . . . . . . . . 9  |-  ( i  =  a  ->  ( B `  i )  =  ( B `  a ) )
3433eleq1d 2508 . . . . . . . 8  |-  ( i  =  a  ->  (
( B `  i
)  e.  RR  <->  ( B `  a )  e.  RR ) )
3532, 34imbi12d 313 . . . . . . 7  |-  ( i  =  a  ->  (
( ( ph  /\  i  e.  ( L ... M ) )  -> 
( B `  i
)  e.  RR )  <-> 
( ( ph  /\  a  e.  ( L ... M ) )  -> 
( B `  a
)  e.  RR ) ) )
3630, 35, 10chvar 1971 . . . . . 6  |-  ( (
ph  /\  a  e.  ( L ... M ) )  ->  ( B `  a )  e.  RR )
37 remulcl 9106 . . . . . . 7  |-  ( ( a  e.  RR  /\  j  e.  RR )  ->  ( a  x.  j
)  e.  RR )
3837adantl 454 . . . . . 6  |-  ( (
ph  /\  ( a  e.  RR  /\  j  e.  RR ) )  -> 
( a  x.  j
)  e.  RR )
398, 36, 38seqcl 11374 . . . . 5  |-  ( ph  ->  (  seq  L (  x.  ,  B ) `
 M )  e.  RR )
4039adantr 453 . . . 4  |-  ( (
ph  /\  -.  J  =  L )  ->  (  seq  L (  x.  ,  B ) `  M
)  e.  RR )
41 fmul01lt1lem2.10 . . . . . . 7  |-  ( ph  ->  J  e.  ( L ... M ) )
42 elfzuz3 11087 . . . . . . 7  |-  ( J  e.  ( L ... M )  ->  M  e.  ( ZZ>= `  J )
)
4341, 42syl 16 . . . . . 6  |-  ( ph  ->  M  e.  ( ZZ>= `  J ) )
44 nfv 1630 . . . . . . . . 9  |-  F/ i  a  e.  ( J ... M )
452, 44nfan 1848 . . . . . . . 8  |-  F/ i ( ph  /\  a  e.  ( J ... M
) )
4645, 29nfim 1834 . . . . . . 7  |-  F/ i ( ( ph  /\  a  e.  ( J ... M ) )  -> 
( B `  a
)  e.  RR )
47 eleq1 2502 . . . . . . . . 9  |-  ( i  =  a  ->  (
i  e.  ( J ... M )  <->  a  e.  ( J ... M ) ) )
4847anbi2d 686 . . . . . . . 8  |-  ( i  =  a  ->  (
( ph  /\  i  e.  ( J ... M
) )  <->  ( ph  /\  a  e.  ( J ... M ) ) ) )
4948, 34imbi12d 313 . . . . . . 7  |-  ( i  =  a  ->  (
( ( ph  /\  i  e.  ( J ... M ) )  -> 
( B `  i
)  e.  RR )  <-> 
( ( ph  /\  a  e.  ( J ... M ) )  -> 
( B `  a
)  e.  RR ) ) )
506adantr 453 . . . . . . . . . 10  |-  ( (
ph  /\  i  e.  ( J ... M ) )  ->  L  e.  ZZ )
51 eluzelz 10527 . . . . . . . . . . . 12  |-  ( M  e.  ( ZZ>= `  L
)  ->  M  e.  ZZ )
528, 51syl 16 . . . . . . . . . . 11  |-  ( ph  ->  M  e.  ZZ )
5352adantr 453 . . . . . . . . . 10  |-  ( (
ph  /\  i  e.  ( J ... M ) )  ->  M  e.  ZZ )
54 elfzelz 11090 . . . . . . . . . . 11  |-  ( i  e.  ( J ... M )  ->  i  e.  ZZ )
5554adantl 454 . . . . . . . . . 10  |-  ( (
ph  /\  i  e.  ( J ... M ) )  ->  i  e.  ZZ )
5650, 53, 553jca 1135 . . . . . . . . 9  |-  ( (
ph  /\  i  e.  ( J ... M ) )  ->  ( L  e.  ZZ  /\  M  e.  ZZ  /\  i  e.  ZZ ) )
576zred 10406 . . . . . . . . . . . 12  |-  ( ph  ->  L  e.  RR )
5857adantr 453 . . . . . . . . . . 11  |-  ( (
ph  /\  i  e.  ( J ... M ) )  ->  L  e.  RR )
59 elfzelz 11090 . . . . . . . . . . . . . 14  |-  ( J  e.  ( L ... M )  ->  J  e.  ZZ )
6041, 59syl 16 . . . . . . . . . . . . 13  |-  ( ph  ->  J  e.  ZZ )
6160zred 10406 . . . . . . . . . . . 12  |-  ( ph  ->  J  e.  RR )
6261adantr 453 . . . . . . . . . . 11  |-  ( (
ph  /\  i  e.  ( J ... M ) )  ->  J  e.  RR )
6354zred 10406 . . . . . . . . . . . 12  |-  ( i  e.  ( J ... M )  ->  i  e.  RR )
6463adantl 454 . . . . . . . . . . 11  |-  ( (
ph  /\  i  e.  ( J ... M ) )  ->  i  e.  RR )
65 elfzle1 11091 . . . . . . . . . . . . 13  |-  ( J  e.  ( L ... M )  ->  L  <_  J )
6641, 65syl 16 . . . . . . . . . . . 12  |-  ( ph  ->  L  <_  J )
6766adantr 453 . . . . . . . . . . 11  |-  ( (
ph  /\  i  e.  ( J ... M ) )  ->  L  <_  J )
68 elfzle1 11091 . . . . . . . . . . . 12  |-  ( i  e.  ( J ... M )  ->  J  <_  i )
6968adantl 454 . . . . . . . . . . 11  |-  ( (
ph  /\  i  e.  ( J ... M ) )  ->  J  <_  i )
7058, 62, 64, 67, 69letrd 9258 . . . . . . . . . 10  |-  ( (
ph  /\  i  e.  ( J ... M ) )  ->  L  <_  i )
71 elfzle2 11092 . . . . . . . . . . 11  |-  ( i  e.  ( J ... M )  ->  i  <_  M )
7271adantl 454 . . . . . . . . . 10  |-  ( (
ph  /\  i  e.  ( J ... M ) )  ->  i  <_  M )
7370, 72jca 520 . . . . . . . . 9  |-  ( (
ph  /\  i  e.  ( J ... M ) )  ->  ( L  <_  i  /\  i  <_  M ) )
74 elfz2 11081 . . . . . . . . 9  |-  ( i  e.  ( L ... M )  <->  ( ( L  e.  ZZ  /\  M  e.  ZZ  /\  i  e.  ZZ )  /\  ( L  <_  i  /\  i  <_  M ) ) )
7556, 73, 74sylanbrc 647 . . . . . . . 8  |-  ( (
ph  /\  i  e.  ( J ... M ) )  ->  i  e.  ( L ... M ) )
7675, 10syldan 458 . . . . . . 7  |-  ( (
ph  /\  i  e.  ( J ... M ) )  ->  ( B `  i )  e.  RR )
7746, 49, 76chvar 1971 . . . . . 6  |-  ( (
ph  /\  a  e.  ( J ... M ) )  ->  ( B `  a )  e.  RR )
7843, 77, 38seqcl 11374 . . . . 5  |-  ( ph  ->  (  seq  J (  x.  ,  B ) `
 M )  e.  RR )
7978adantr 453 . . . 4  |-  ( (
ph  /\  -.  J  =  L )  ->  (  seq  J (  x.  ,  B ) `  M
)  e.  RR )
8016rpred 10679 . . . . 5  |-  ( ph  ->  E  e.  RR )
8180adantr 453 . . . 4  |-  ( (
ph  /\  -.  J  =  L )  ->  E  e.  RR )
82 remulcl 9106 . . . . . . . . 9  |-  ( ( a  e.  RR  /\  b  e.  RR )  ->  ( a  x.  b
)  e.  RR )
8382adantl 454 . . . . . . . 8  |-  ( ( ( ph  /\  -.  J  =  L )  /\  ( a  e.  RR  /\  b  e.  RR ) )  ->  ( a  x.  b )  e.  RR )
84 simp1 958 . . . . . . . . . . 11  |-  ( ( a  e.  RR  /\  b  e.  RR  /\  c  e.  RR )  ->  a  e.  RR )
8584recnd 9145 . . . . . . . . . 10  |-  ( ( a  e.  RR  /\  b  e.  RR  /\  c  e.  RR )  ->  a  e.  CC )
86 simp2 959 . . . . . . . . . . 11  |-  ( ( a  e.  RR  /\  b  e.  RR  /\  c  e.  RR )  ->  b  e.  RR )
8786recnd 9145 . . . . . . . . . 10  |-  ( ( a  e.  RR  /\  b  e.  RR  /\  c  e.  RR )  ->  b  e.  CC )
88 simp3 960 . . . . . . . . . . 11  |-  ( ( a  e.  RR  /\  b  e.  RR  /\  c  e.  RR )  ->  c  e.  RR )
8988recnd 9145 . . . . . . . . . 10  |-  ( ( a  e.  RR  /\  b  e.  RR  /\  c  e.  RR )  ->  c  e.  CC )
9085, 87, 89mulassd 9142 . . . . . . . . 9  |-  ( ( a  e.  RR  /\  b  e.  RR  /\  c  e.  RR )  ->  (
( a  x.  b
)  x.  c )  =  ( a  x.  ( b  x.  c
) ) )
9190adantl 454 . . . . . . . 8  |-  ( ( ( ph  /\  -.  J  =  L )  /\  ( a  e.  RR  /\  b  e.  RR  /\  c  e.  RR )
)  ->  ( (
a  x.  b )  x.  c )  =  ( a  x.  (
b  x.  c ) ) )
9260zcnd 10407 . . . . . . . . . . . 12  |-  ( ph  ->  J  e.  CC )
93 ax-1cn 9079 . . . . . . . . . . . . 13  |-  1  e.  CC
9493a1i 11 . . . . . . . . . . . 12  |-  ( ph  ->  1  e.  CC )
9592, 94npcand 9446 . . . . . . . . . . 11  |-  ( ph  ->  ( ( J  - 
1 )  +  1 )  =  J )
9695fveq2d 5761 . . . . . . . . . 10  |-  ( ph  ->  ( ZZ>= `  ( ( J  -  1 )  +  1 ) )  =  ( ZZ>= `  J
) )
9743, 96eleqtrrd 2519 . . . . . . . . 9  |-  ( ph  ->  M  e.  ( ZZ>= `  ( ( J  - 
1 )  +  1 ) ) )
9897adantr 453 . . . . . . . 8  |-  ( (
ph  /\  -.  J  =  L )  ->  M  e.  ( ZZ>= `  ( ( J  -  1 )  +  1 ) ) )
996adantr 453 . . . . . . . . 9  |-  ( (
ph  /\  -.  J  =  L )  ->  L  e.  ZZ )
10060adantr 453 . . . . . . . . . 10  |-  ( (
ph  /\  -.  J  =  L )  ->  J  e.  ZZ )
101 1z 10342 . . . . . . . . . . 11  |-  1  e.  ZZ
102101a1i 11 . . . . . . . . . 10  |-  ( (
ph  /\  -.  J  =  L )  ->  1  e.  ZZ )
103100, 102zsubcld 10411 . . . . . . . . 9  |-  ( (
ph  /\  -.  J  =  L )  ->  ( J  -  1 )  e.  ZZ )
104 simpr 449 . . . . . . . . . . . 12  |-  ( (
ph  /\  -.  J  =  L )  ->  -.  J  =  L )
105 eqcom 2444 . . . . . . . . . . . 12  |-  ( J  =  L  <->  L  =  J )
106104, 105sylnib 297 . . . . . . . . . . 11  |-  ( (
ph  /\  -.  J  =  L )  ->  -.  L  =  J )
10757, 61leloed 9247 . . . . . . . . . . . . 13  |-  ( ph  ->  ( L  <_  J  <->  ( L  <  J  \/  L  =  J )
) )
10866, 107mpbid 203 . . . . . . . . . . . 12  |-  ( ph  ->  ( L  <  J  \/  L  =  J
) )
109108adantr 453 . . . . . . . . . . 11  |-  ( (
ph  /\  -.  J  =  L )  ->  ( L  <  J  \/  L  =  J ) )
110 orel2 374 . . . . . . . . . . 11  |-  ( -.  L  =  J  -> 
( ( L  < 
J  \/  L  =  J )  ->  L  <  J ) )
111106, 109, 110sylc 59 . . . . . . . . . 10  |-  ( (
ph  /\  -.  J  =  L )  ->  L  <  J )
112 zltlem1 10359 . . . . . . . . . . . 12  |-  ( ( L  e.  ZZ  /\  J  e.  ZZ )  ->  ( L  <  J  <->  L  <_  ( J  - 
1 ) ) )
1136, 60, 112syl2anc 644 . . . . . . . . . . 11  |-  ( ph  ->  ( L  <  J  <->  L  <_  ( J  - 
1 ) ) )
114113adantr 453 . . . . . . . . . 10  |-  ( (
ph  /\  -.  J  =  L )  ->  ( L  <  J  <->  L  <_  ( J  -  1 ) ) )
115111, 114mpbid 203 . . . . . . . . 9  |-  ( (
ph  /\  -.  J  =  L )  ->  L  <_  ( J  -  1 ) )
116 eluz2 10525 . . . . . . . . 9  |-  ( ( J  -  1 )  e.  ( ZZ>= `  L
)  <->  ( L  e.  ZZ  /\  ( J  -  1 )  e.  ZZ  /\  L  <_ 
( J  -  1 ) ) )
11799, 103, 115, 116syl3anbrc 1139 . . . . . . . 8  |-  ( (
ph  /\  -.  J  =  L )  ->  ( J  -  1 )  e.  ( ZZ>= `  L
) )
118 nfv 1630 . . . . . . . . . . . 12  |-  F/ i  -.  J  =  L
1192, 118nfan 1848 . . . . . . . . . . 11  |-  F/ i ( ph  /\  -.  J  =  L )
120119, 25nfan 1848 . . . . . . . . . 10  |-  F/ i ( ( ph  /\  -.  J  =  L
)  /\  a  e.  ( L ... M ) )
121120, 29nfim 1834 . . . . . . . . 9  |-  F/ i ( ( ( ph  /\ 
-.  J  =  L )  /\  a  e.  ( L ... M
) )  ->  ( B `  a )  e.  RR )
12231anbi2d 686 . . . . . . . . . 10  |-  ( i  =  a  ->  (
( ( ph  /\  -.  J  =  L
)  /\  i  e.  ( L ... M ) )  <->  ( ( ph  /\ 
-.  J  =  L )  /\  a  e.  ( L ... M
) ) ) )
123122, 34imbi12d 313 . . . . . . . . 9  |-  ( i  =  a  ->  (
( ( ( ph  /\ 
-.  J  =  L )  /\  i  e.  ( L ... M
) )  ->  ( B `  i )  e.  RR )  <->  ( (
( ph  /\  -.  J  =  L )  /\  a  e.  ( L ... M
) )  ->  ( B `  a )  e.  RR ) ) )
12410adantlr 697 . . . . . . . . 9  |-  ( ( ( ph  /\  -.  J  =  L )  /\  i  e.  ( L ... M ) )  ->  ( B `  i )  e.  RR )
125121, 123, 124chvar 1971 . . . . . . . 8  |-  ( ( ( ph  /\  -.  J  =  L )  /\  a  e.  ( L ... M ) )  ->  ( B `  a )  e.  RR )
12683, 91, 98, 117, 125seqsplit 11387 . . . . . . 7  |-  ( (
ph  /\  -.  J  =  L )  ->  (  seq  L (  x.  ,  B ) `  M
)  =  ( (  seq  L (  x.  ,  B ) `  ( J  -  1
) )  x.  (  seq  ( ( J  - 
1 )  +  1 ) (  x.  ,  B ) `  M
) ) )
12795adantr 453 . . . . . . . . . 10  |-  ( (
ph  /\  -.  J  =  L )  ->  (
( J  -  1 )  +  1 )  =  J )
128127seqeq1d 11360 . . . . . . . . 9  |-  ( (
ph  /\  -.  J  =  L )  ->  seq  ( ( J  - 
1 )  +  1 ) (  x.  ,  B )  =  seq  J (  x.  ,  B
) )
129128fveq1d 5759 . . . . . . . 8  |-  ( (
ph  /\  -.  J  =  L )  ->  (  seq  ( ( J  - 
1 )  +  1 ) (  x.  ,  B ) `  M
)  =  (  seq 
J (  x.  ,  B ) `  M
) )
130129oveq2d 6126 . . . . . . 7  |-  ( (
ph  /\  -.  J  =  L )  ->  (
(  seq  L (  x.  ,  B ) `  ( J  -  1
) )  x.  (  seq  ( ( J  - 
1 )  +  1 ) (  x.  ,  B ) `  M
) )  =  ( (  seq  L (  x.  ,  B ) `
 ( J  - 
1 ) )  x.  (  seq  J (  x.  ,  B ) `
 M ) ) )
131126, 130eqtrd 2474 . . . . . 6  |-  ( (
ph  /\  -.  J  =  L )  ->  (  seq  L (  x.  ,  B ) `  M
)  =  ( (  seq  L (  x.  ,  B ) `  ( J  -  1
) )  x.  (  seq  J (  x.  ,  B ) `  M
) ) )
132 nfv 1630 . . . . . . . . . . 11  |-  F/ i  a  e.  ( L ... ( J  - 
1 ) )
133119, 132nfan 1848 . . . . . . . . . 10  |-  F/ i ( ( ph  /\  -.  J  =  L
)  /\  a  e.  ( L ... ( J  -  1 ) ) )
134133, 29nfim 1834 . . . . . . . . 9  |-  F/ i ( ( ( ph  /\ 
-.  J  =  L )  /\  a  e.  ( L ... ( J  -  1 ) ) )  ->  ( B `  a )  e.  RR )
135 eleq1 2502 . . . . . . . . . . 11  |-  ( i  =  a  ->  (
i  e.  ( L ... ( J  - 
1 ) )  <->  a  e.  ( L ... ( J  -  1 ) ) ) )
136135anbi2d 686 . . . . . . . . . 10  |-  ( i  =  a  ->  (
( ( ph  /\  -.  J  =  L
)  /\  i  e.  ( L ... ( J  -  1 ) ) )  <->  ( ( ph  /\ 
-.  J  =  L )  /\  a  e.  ( L ... ( J  -  1 ) ) ) ) )
137136, 34imbi12d 313 . . . . . . . . 9  |-  ( i  =  a  ->  (
( ( ( ph  /\ 
-.  J  =  L )  /\  i  e.  ( L ... ( J  -  1 ) ) )  ->  ( B `  i )  e.  RR )  <->  ( (
( ph  /\  -.  J  =  L )  /\  a  e.  ( L ... ( J  -  1 ) ) )  ->  ( B `  a )  e.  RR ) ) )
1386adantr 453 . . . . . . . . . . . . 13  |-  ( (
ph  /\  i  e.  ( L ... ( J  -  1 ) ) )  ->  L  e.  ZZ )
13952adantr 453 . . . . . . . . . . . . 13  |-  ( (
ph  /\  i  e.  ( L ... ( J  -  1 ) ) )  ->  M  e.  ZZ )
140 elfzelz 11090 . . . . . . . . . . . . . 14  |-  ( i  e.  ( L ... ( J  -  1
) )  ->  i  e.  ZZ )
141140adantl 454 . . . . . . . . . . . . 13  |-  ( (
ph  /\  i  e.  ( L ... ( J  -  1 ) ) )  ->  i  e.  ZZ )
142138, 139, 1413jca 1135 . . . . . . . . . . . 12  |-  ( (
ph  /\  i  e.  ( L ... ( J  -  1 ) ) )  ->  ( L  e.  ZZ  /\  M  e.  ZZ  /\  i  e.  ZZ ) )
143 elfzle1 11091 . . . . . . . . . . . . . 14  |-  ( i  e.  ( L ... ( J  -  1
) )  ->  L  <_  i )
144143adantl 454 . . . . . . . . . . . . 13  |-  ( (
ph  /\  i  e.  ( L ... ( J  -  1 ) ) )  ->  L  <_  i )
145140zred 10406 . . . . . . . . . . . . . . 15  |-  ( i  e.  ( L ... ( J  -  1
) )  ->  i  e.  RR )
146145adantl 454 . . . . . . . . . . . . . 14  |-  ( (
ph  /\  i  e.  ( L ... ( J  -  1 ) ) )  ->  i  e.  RR )
14761adantr 453 . . . . . . . . . . . . . 14  |-  ( (
ph  /\  i  e.  ( L ... ( J  -  1 ) ) )  ->  J  e.  RR )
14852zred 10406 . . . . . . . . . . . . . . 15  |-  ( ph  ->  M  e.  RR )
149148adantr 453 . . . . . . . . . . . . . 14  |-  ( (
ph  /\  i  e.  ( L ... ( J  -  1 ) ) )  ->  M  e.  RR )
150 1re 9121 . . . . . . . . . . . . . . . . . 18  |-  1  e.  RR
151150a1i 11 . . . . . . . . . . . . . . . . 17  |-  ( ph  ->  1  e.  RR )
15261, 151resubcld 9496 . . . . . . . . . . . . . . . 16  |-  ( ph  ->  ( J  -  1 )  e.  RR )
153152adantr 453 . . . . . . . . . . . . . . 15  |-  ( (
ph  /\  i  e.  ( L ... ( J  -  1 ) ) )  ->  ( J  -  1 )  e.  RR )
154 elfzle2 11092 . . . . . . . . . . . . . . . 16  |-  ( i  e.  ( L ... ( J  -  1
) )  ->  i  <_  ( J  -  1 ) )
155154adantl 454 . . . . . . . . . . . . . . 15  |-  ( (
ph  /\  i  e.  ( L ... ( J  -  1 ) ) )  ->  i  <_  ( J  -  1 ) )
15661lem1d 9975 . . . . . . . . . . . . . . . 16  |-  ( ph  ->  ( J  -  1 )  <_  J )
157156adantr 453 . . . . . . . . . . . . . . 15  |-  ( (
ph  /\  i  e.  ( L ... ( J  -  1 ) ) )  ->  ( J  -  1 )  <_  J )
158146, 153, 147, 155, 157letrd 9258 . . . . . . . . . . . . . 14  |-  ( (
ph  /\  i  e.  ( L ... ( J  -  1 ) ) )  ->  i  <_  J )
159 elfzle2 11092 . . . . . . . . . . . . . . . 16  |-  ( J  e.  ( L ... M )  ->  J  <_  M )
16041, 159syl 16 . . . . . . . . . . . . . . 15  |-  ( ph  ->  J  <_  M )
161160adantr 453 . . . . . . . . . . . . . 14  |-  ( (
ph  /\  i  e.  ( L ... ( J  -  1 ) ) )  ->  J  <_  M )
162146, 147, 149, 158, 161letrd 9258 . . . . . . . . . . . . 13  |-  ( (
ph  /\  i  e.  ( L ... ( J  -  1 ) ) )  ->  i  <_  M )
163144, 162jca 520 . . . . . . . . . . . 12  |-  ( (
ph  /\  i  e.  ( L ... ( J  -  1 ) ) )  ->  ( L  <_  i  /\  i  <_  M ) )
164142, 163, 74sylanbrc 647 . . . . . . . . . . 11  |-  ( (
ph  /\  i  e.  ( L ... ( J  -  1 ) ) )  ->  i  e.  ( L ... M ) )
165164, 10syldan 458 . . . . . . . . . 10  |-  ( (
ph  /\  i  e.  ( L ... ( J  -  1 ) ) )  ->  ( B `  i )  e.  RR )
166165adantlr 697 . . . . . . . . 9  |-  ( ( ( ph  /\  -.  J  =  L )  /\  i  e.  ( L ... ( J  - 
1 ) ) )  ->  ( B `  i )  e.  RR )
167134, 137, 166chvar 1971 . . . . . . . 8  |-  ( ( ( ph  /\  -.  J  =  L )  /\  a  e.  ( L ... ( J  - 
1 ) ) )  ->  ( B `  a )  e.  RR )
16837adantl 454 . . . . . . . 8  |-  ( ( ( ph  /\  -.  J  =  L )  /\  ( a  e.  RR  /\  j  e.  RR ) )  ->  ( a  x.  j )  e.  RR )
169117, 167, 168seqcl 11374 . . . . . . 7  |-  ( (
ph  /\  -.  J  =  L )  ->  (  seq  L (  x.  ,  B ) `  ( J  -  1 ) )  e.  RR )
170150a1i 11 . . . . . . 7  |-  ( (
ph  /\  -.  J  =  L )  ->  1  e.  RR )
171 eqid 2442 . . . . . . . . 9  |-  seq  J
(  x.  ,  B
)  =  seq  J
(  x.  ,  B
)
17243adantr 453 . . . . . . . . 9  |-  ( (
ph  /\  -.  J  =  L )  ->  M  e.  ( ZZ>= `  J )
)
173 eluzfz2 11096 . . . . . . . . . . 11  |-  ( M  e.  ( ZZ>= `  J
)  ->  M  e.  ( J ... M ) )
17443, 173syl 16 . . . . . . . . . 10  |-  ( ph  ->  M  e.  ( J ... M ) )
175174adantr 453 . . . . . . . . 9  |-  ( (
ph  /\  -.  J  =  L )  ->  M  e.  ( J ... M
) )
17676adantlr 697 . . . . . . . . 9  |-  ( ( ( ph  /\  -.  J  =  L )  /\  i  e.  ( J ... M ) )  ->  ( B `  i )  e.  RR )
17775, 12syldan 458 . . . . . . . . . 10  |-  ( (
ph  /\  i  e.  ( J ... M ) )  ->  0  <_  ( B `  i ) )
178177adantlr 697 . . . . . . . . 9  |-  ( ( ( ph  /\  -.  J  =  L )  /\  i  e.  ( J ... M ) )  ->  0  <_  ( B `  i )
)
17975, 14syldan 458 . . . . . . . . . 10  |-  ( (
ph  /\  i  e.  ( J ... M ) )  ->  ( B `  i )  <_  1
)
180179adantlr 697 . . . . . . . . 9  |-  ( ( ( ph  /\  -.  J  =  L )  /\  i  e.  ( J ... M ) )  ->  ( B `  i )  <_  1
)
1811, 119, 171, 100, 172, 175, 176, 178, 180fmul01 27724 . . . . . . . 8  |-  ( (
ph  /\  -.  J  =  L )  ->  (
0  <_  (  seq  J (  x.  ,  B
) `  M )  /\  (  seq  J (  x.  ,  B ) `
 M )  <_ 
1 ) )
182181simpld 447 . . . . . . 7  |-  ( (
ph  /\  -.  J  =  L )  ->  0  <_  (  seq  J (  x.  ,  B ) `
 M ) )
183 eqid 2442 . . . . . . . . 9  |-  seq  L
(  x.  ,  B
)  =  seq  L
(  x.  ,  B
)
1848adantr 453 . . . . . . . . 9  |-  ( (
ph  /\  -.  J  =  L )  ->  M  e.  ( ZZ>= `  L )
)
185101a1i 11 . . . . . . . . . . . . 13  |-  ( ph  ->  1  e.  ZZ )
18660, 185zsubcld 10411 . . . . . . . . . . . 12  |-  ( ph  ->  ( J  -  1 )  e.  ZZ )
1876, 52, 1863jca 1135 . . . . . . . . . . 11  |-  ( ph  ->  ( L  e.  ZZ  /\  M  e.  ZZ  /\  ( J  -  1
)  e.  ZZ ) )
188187adantr 453 . . . . . . . . . 10  |-  ( (
ph  /\  -.  J  =  L )  ->  ( L  e.  ZZ  /\  M  e.  ZZ  /\  ( J  -  1 )  e.  ZZ ) )
189152, 61, 1483jca 1135 . . . . . . . . . . . . 13  |-  ( ph  ->  ( ( J  - 
1 )  e.  RR  /\  J  e.  RR  /\  M  e.  RR )
)
190189adantr 453 . . . . . . . . . . . 12  |-  ( (
ph  /\  -.  J  =  L )  ->  (
( J  -  1 )  e.  RR  /\  J  e.  RR  /\  M  e.  RR ) )
19161adantr 453 . . . . . . . . . . . . . 14  |-  ( (
ph  /\  -.  J  =  L )  ->  J  e.  RR )
192191lem1d 9975 . . . . . . . . . . . . 13  |-  ( (
ph  /\  -.  J  =  L )  ->  ( J  -  1 )  <_  J )
193160adantr 453 . . . . . . . . . . . . 13  |-  ( (
ph  /\  -.  J  =  L )  ->  J  <_  M )
194192, 193jca 520 . . . . . . . . . . . 12  |-  ( (
ph  /\  -.  J  =  L )  ->  (
( J  -  1 )  <_  J  /\  J  <_  M ) )
195 letr 9198 . . . . . . . . . . . 12  |-  ( ( ( J  -  1 )  e.  RR  /\  J  e.  RR  /\  M  e.  RR )  ->  (
( ( J  - 
1 )  <_  J  /\  J  <_  M )  ->  ( J  - 
1 )  <_  M
) )
196190, 194, 195sylc 59 . . . . . . . . . . 11  |-  ( (
ph  /\  -.  J  =  L )  ->  ( J  -  1 )  <_  M )
197115, 196jca 520 . . . . . . . . . 10  |-  ( (
ph  /\  -.  J  =  L )  ->  ( L  <_  ( J  - 
1 )  /\  ( J  -  1 )  <_  M ) )
198 elfz2 11081 . . . . . . . . . 10  |-  ( ( J  -  1 )  e.  ( L ... M )  <->  ( ( L  e.  ZZ  /\  M  e.  ZZ  /\  ( J  -  1 )  e.  ZZ )  /\  ( L  <_  ( J  - 
1 )  /\  ( J  -  1 )  <_  M ) ) )
199188, 197, 198sylanbrc 647 . . . . . . . . 9  |-  ( (
ph  /\  -.  J  =  L )  ->  ( J  -  1 )  e.  ( L ... M ) )
20012adantlr 697 . . . . . . . . 9  |-  ( ( ( ph  /\  -.  J  =  L )  /\  i  e.  ( L ... M ) )  ->  0  <_  ( B `  i )
)
20114adantlr 697 . . . . . . . . 9  |-  ( ( ( ph  /\  -.  J  =  L )  /\  i  e.  ( L ... M ) )  ->  ( B `  i )  <_  1
)
2021, 119, 183, 99, 184, 199, 124, 200, 201fmul01 27724 . . . . . . . 8  |-  ( (
ph  /\  -.  J  =  L )  ->  (
0  <_  (  seq  L (  x.  ,  B
) `  ( J  -  1 ) )  /\  (  seq  L
(  x.  ,  B
) `  ( J  -  1 ) )  <_  1 ) )
203202simprd 451 . . . . . . 7  |-  ( (
ph  /\  -.  J  =  L )  ->  (  seq  L (  x.  ,  B ) `  ( J  -  1 ) )  <_  1 )
204169, 170, 79, 182, 203lemul1ad 9981 . . . . . 6  |-  ( (
ph  /\  -.  J  =  L )  ->  (
(  seq  L (  x.  ,  B ) `  ( J  -  1
) )  x.  (  seq  J (  x.  ,  B ) `  M
) )  <_  (
1  x.  (  seq 
J (  x.  ,  B ) `  M
) ) )
205131, 204eqbrtrd 4257 . . . . 5  |-  ( (
ph  /\  -.  J  =  L )  ->  (  seq  L (  x.  ,  B ) `  M
)  <_  ( 1  x.  (  seq  J
(  x.  ,  B
) `  M )
) )
20679recnd 9145 . . . . . 6  |-  ( (
ph  /\  -.  J  =  L )  ->  (  seq  J (  x.  ,  B ) `  M
)  e.  CC )
207206mulid2d 9137 . . . . 5  |-  ( (
ph  /\  -.  J  =  L )  ->  (
1  x.  (  seq 
J (  x.  ,  B ) `  M
) )  =  (  seq  J (  x.  ,  B ) `  M ) )
208205, 207breqtrd 4261 . . . 4  |-  ( (
ph  /\  -.  J  =  L )  ->  (  seq  L (  x.  ,  B ) `  M
)  <_  (  seq  J (  x.  ,  B
) `  M )
)
2091, 2, 171, 60, 43, 76, 177, 179, 16, 20fmul01lt1lem1 27728 . . . . 5  |-  ( ph  ->  (  seq  J (  x.  ,  B ) `
 M )  < 
E )
210209adantr 453 . . . 4  |-  ( (
ph  /\  -.  J  =  L )  ->  (  seq  J (  x.  ,  B ) `  M
)  <  E )
21140, 79, 81, 208, 210lelttrd 9259 . . 3  |-  ( (
ph  /\  -.  J  =  L )  ->  (  seq  L (  x.  ,  B ) `  M
)  <  E )
21224, 211syl5eqbr 4270 . 2  |-  ( (
ph  /\  -.  J  =  L )  ->  ( A `  M )  <  E )
21323, 212pm2.61dan 768 1  |-  ( ph  ->  ( A `  M
)  <  E )
Colors of variables: wff set class
Syntax hints:   -. wn 3    -> wi 4    <-> wb 178    \/ wo 359    /\ wa 360    /\ w3a 937   F/wnf 1554    = wceq 1653    e. wcel 1727   F/_wnfc 2565   class class class wbr 4237   ` cfv 5483  (class class class)co 6110   CCcc 9019   RRcr 9020   0cc0 9021   1c1 9022    + caddc 9024    x. cmul 9026    < clt 9151    <_ cle 9152    - cmin 9322   ZZcz 10313   ZZ>=cuz 10519   RR+crp 10643   ...cfz 11074    seq cseq 11354
This theorem is referenced by:  fmul01lt1  27730
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1556  ax-5 1567  ax-17 1627  ax-9 1668  ax-8 1689  ax-13 1729  ax-14 1731  ax-6 1746  ax-7 1751  ax-11 1763  ax-12 1953  ax-ext 2423  ax-sep 4355  ax-nul 4363  ax-pow 4406  ax-pr 4432  ax-un 4730  ax-cnex 9077  ax-resscn 9078  ax-1cn 9079  ax-icn 9080  ax-addcl 9081  ax-addrcl 9082  ax-mulcl 9083  ax-mulrcl 9084  ax-mulcom 9085  ax-addass 9086  ax-mulass 9087  ax-distr 9088  ax-i2m1 9089  ax-1ne0 9090  ax-1rid 9091  ax-rnegex 9092  ax-rrecex 9093  ax-cnre 9094  ax-pre-lttri 9095  ax-pre-lttrn 9096  ax-pre-ltadd 9097  ax-pre-mulgt0 9098
This theorem depends on definitions:  df-bi 179  df-or 361  df-an 362  df-3or 938  df-3an 939  df-tru 1329  df-ex 1552  df-nf 1555  df-sb 1660  df-eu 2291  df-mo 2292  df-clab 2429  df-cleq 2435  df-clel 2438  df-nfc 2567  df-ne 2607  df-nel 2608  df-ral 2716  df-rex 2717  df-reu 2718  df-rab 2720  df-v 2964  df-sbc 3168  df-csb 3268  df-dif 3309  df-un 3311  df-in 3313  df-ss 3320  df-pss 3322  df-nul 3614  df-if 3764  df-pw 3825  df-sn 3844  df-pr 3845  df-tp 3846  df-op 3847  df-uni 4040  df-iun 4119  df-br 4238  df-opab 4292  df-mpt 4293  df-tr 4328  df-eprel 4523  df-id 4527  df-po 4532  df-so 4533  df-fr 4570  df-we 4572  df-ord 4613  df-on 4614  df-lim 4615  df-suc 4616  df-om 4875  df-xp 4913  df-rel 4914  df-cnv 4915  df-co 4916  df-dm 4917  df-rn 4918  df-res 4919  df-ima 4920  df-iota 5447  df-fun 5485  df-fn 5486  df-f 5487  df-f1 5488  df-fo 5489  df-f1o 5490  df-fv 5491  df-ov 6113  df-oprab 6114  df-mpt2 6115  df-1st 6378  df-2nd 6379  df-riota 6578  df-recs 6662  df-rdg 6697  df-er 6934  df-en 7139  df-dom 7140  df-sdom 7141  df-pnf 9153  df-mnf 9154  df-xr 9155  df-ltxr 9156  df-le 9157  df-sub 9324  df-neg 9325  df-nn 10032  df-n0 10253  df-z 10314  df-uz 10520  df-rp 10644  df-fz 11075  df-fzo 11167  df-seq 11355
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