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Theorem fmul01lt1lem2 27383
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 1626 . . . 4  |-  F/ i  J  =  L
42, 3nfan 1836 . . 3  |-  F/ i ( ph  /\  J  =  L )
5 fmul01lt1lem2.3 . . 3  |-  A  =  seq  L (  x.  ,  B )
6 fmul01lt1lem2.4 . . . 4  |-  ( ph  ->  L  e.  ZZ )
76adantr 452 . . 3  |-  ( (
ph  /\  J  =  L )  ->  L  e.  ZZ )
8 fmul01lt1lem2.5 . . . 4  |-  ( ph  ->  M  e.  ( ZZ>= `  L ) )
98adantr 452 . . 3  |-  ( (
ph  /\  J  =  L )  ->  M  e.  ( ZZ>= `  L )
)
10 fmul01lt1lem2.6 . . . 4  |-  ( (
ph  /\  i  e.  ( L ... M ) )  ->  ( B `  i )  e.  RR )
1110adantlr 696 . . 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 696 . . 3  |-  ( ( ( ph  /\  J  =  L )  /\  i  e.  ( L ... M
) )  ->  0  <_  ( B `  i
) )
14 fmul01lt1lem2.8 . . . 4  |-  ( (
ph  /\  i  e.  ( L ... M ) )  ->  ( B `  i )  <_  1
)
1514adantlr 696 . . 3  |-  ( ( ( ph  /\  J  =  L )  /\  i  e.  ( L ... M
) )  ->  ( B `  i )  <_  1 )
16 fmul01lt1lem2.9 . . . 4  |-  ( ph  ->  E  e.  RR+ )
1716adantr 452 . . 3  |-  ( (
ph  /\  J  =  L )  ->  E  e.  RR+ )
18 simpr 448 . . . . 5  |-  ( (
ph  /\  J  =  L )  ->  J  =  L )
1918fveq2d 5672 . . . 4  |-  ( (
ph  /\  J  =  L )  ->  ( B `  J )  =  ( B `  L ) )
20 fmul01lt1lem2.11 . . . . 5  |-  ( ph  ->  ( B `  J
)  <  E )
2120adantr 452 . . . 4  |-  ( (
ph  /\  J  =  L )  ->  ( B `  J )  <  E )
2219, 21eqbrtrrd 4175 . . 3  |-  ( (
ph  /\  J  =  L )  ->  ( B `  L )  <  E )
231, 4, 5, 7, 9, 11, 13, 15, 17, 22fmul01lt1lem1 27382 . 2  |-  ( (
ph  /\  J  =  L )  ->  ( A `  M )  <  E )
245fveq1i 5669 . . 3  |-  ( A `
 M )  =  (  seq  L (  x.  ,  B ) `
 M )
25 nfv 1626 . . . . . . . . 9  |-  F/ i  a  e.  ( L ... M )
262, 25nfan 1836 . . . . . . . 8  |-  F/ i ( ph  /\  a  e.  ( L ... M
) )
27 nfcv 2523 . . . . . . . . . 10  |-  F/_ i
a
281, 27nffv 5675 . . . . . . . . 9  |-  F/_ i
( B `  a
)
2928nfel1 2533 . . . . . . . 8  |-  F/ i ( B `  a
)  e.  RR
3026, 29nfim 1822 . . . . . . 7  |-  F/ i ( ( ph  /\  a  e.  ( L ... M ) )  -> 
( B `  a
)  e.  RR )
31 eleq1 2447 . . . . . . . . 9  |-  ( i  =  a  ->  (
i  e.  ( L ... M )  <->  a  e.  ( L ... M ) ) )
3231anbi2d 685 . . . . . . . 8  |-  ( i  =  a  ->  (
( ph  /\  i  e.  ( L ... M
) )  <->  ( ph  /\  a  e.  ( L ... M ) ) ) )
33 fveq2 5668 . . . . . . . . 9  |-  ( i  =  a  ->  ( B `  i )  =  ( B `  a ) )
3433eleq1d 2453 . . . . . . . 8  |-  ( i  =  a  ->  (
( B `  i
)  e.  RR  <->  ( B `  a )  e.  RR ) )
3532, 34imbi12d 312 . . . . . . 7  |-  ( i  =  a  ->  (
( ( ph  /\  i  e.  ( L ... M ) )  -> 
( B `  i
)  e.  RR )  <-> 
( ( ph  /\  a  e.  ( L ... M ) )  -> 
( B `  a
)  e.  RR ) ) )
3630, 35, 10chvar 2021 . . . . . 6  |-  ( (
ph  /\  a  e.  ( L ... M ) )  ->  ( B `  a )  e.  RR )
37 remulcl 9008 . . . . . . 7  |-  ( ( a  e.  RR  /\  j  e.  RR )  ->  ( a  x.  j
)  e.  RR )
3837adantl 453 . . . . . 6  |-  ( (
ph  /\  ( a  e.  RR  /\  j  e.  RR ) )  -> 
( a  x.  j
)  e.  RR )
398, 36, 38seqcl 11270 . . . . 5  |-  ( ph  ->  (  seq  L (  x.  ,  B ) `
 M )  e.  RR )
4039adantr 452 . . . 4  |-  ( (
ph  /\  -.  J  =  L )  ->  (  seq  L (  x.  ,  B ) `  M
)  e.  RR )
41 fmul01lt1lem2.10 . . . . . . 7  |-  ( ph  ->  J  e.  ( L ... M ) )
42 elfzuz3 10988 . . . . . . 7  |-  ( J  e.  ( L ... M )  ->  M  e.  ( ZZ>= `  J )
)
4341, 42syl 16 . . . . . 6  |-  ( ph  ->  M  e.  ( ZZ>= `  J ) )
44 nfv 1626 . . . . . . . . 9  |-  F/ i  a  e.  ( J ... M )
452, 44nfan 1836 . . . . . . . 8  |-  F/ i ( ph  /\  a  e.  ( J ... M
) )
4645, 29nfim 1822 . . . . . . 7  |-  F/ i ( ( ph  /\  a  e.  ( J ... M ) )  -> 
( B `  a
)  e.  RR )
47 eleq1 2447 . . . . . . . . 9  |-  ( i  =  a  ->  (
i  e.  ( J ... M )  <->  a  e.  ( J ... M ) ) )
4847anbi2d 685 . . . . . . . 8  |-  ( i  =  a  ->  (
( ph  /\  i  e.  ( J ... M
) )  <->  ( ph  /\  a  e.  ( J ... M ) ) ) )
4948, 34imbi12d 312 . . . . . . 7  |-  ( i  =  a  ->  (
( ( ph  /\  i  e.  ( J ... M ) )  -> 
( B `  i
)  e.  RR )  <-> 
( ( ph  /\  a  e.  ( J ... M ) )  -> 
( B `  a
)  e.  RR ) ) )
506adantr 452 . . . . . . . . . 10  |-  ( (
ph  /\  i  e.  ( J ... M ) )  ->  L  e.  ZZ )
51 eluzelz 10428 . . . . . . . . . . . 12  |-  ( M  e.  ( ZZ>= `  L
)  ->  M  e.  ZZ )
528, 51syl 16 . . . . . . . . . . 11  |-  ( ph  ->  M  e.  ZZ )
5352adantr 452 . . . . . . . . . 10  |-  ( (
ph  /\  i  e.  ( J ... M ) )  ->  M  e.  ZZ )
54 elfzelz 10991 . . . . . . . . . . 11  |-  ( i  e.  ( J ... M )  ->  i  e.  ZZ )
5554adantl 453 . . . . . . . . . 10  |-  ( (
ph  /\  i  e.  ( J ... M ) )  ->  i  e.  ZZ )
5650, 53, 553jca 1134 . . . . . . . . 9  |-  ( (
ph  /\  i  e.  ( J ... M ) )  ->  ( L  e.  ZZ  /\  M  e.  ZZ  /\  i  e.  ZZ ) )
576zred 10307 . . . . . . . . . . . 12  |-  ( ph  ->  L  e.  RR )
5857adantr 452 . . . . . . . . . . 11  |-  ( (
ph  /\  i  e.  ( J ... M ) )  ->  L  e.  RR )
59 elfzelz 10991 . . . . . . . . . . . . . 14  |-  ( J  e.  ( L ... M )  ->  J  e.  ZZ )
6041, 59syl 16 . . . . . . . . . . . . 13  |-  ( ph  ->  J  e.  ZZ )
6160zred 10307 . . . . . . . . . . . 12  |-  ( ph  ->  J  e.  RR )
6261adantr 452 . . . . . . . . . . 11  |-  ( (
ph  /\  i  e.  ( J ... M ) )  ->  J  e.  RR )
6354zred 10307 . . . . . . . . . . . 12  |-  ( i  e.  ( J ... M )  ->  i  e.  RR )
6463adantl 453 . . . . . . . . . . 11  |-  ( (
ph  /\  i  e.  ( J ... M ) )  ->  i  e.  RR )
65 elfzle1 10992 . . . . . . . . . . . . 13  |-  ( J  e.  ( L ... M )  ->  L  <_  J )
6641, 65syl 16 . . . . . . . . . . . 12  |-  ( ph  ->  L  <_  J )
6766adantr 452 . . . . . . . . . . 11  |-  ( (
ph  /\  i  e.  ( J ... M ) )  ->  L  <_  J )
68 elfzle1 10992 . . . . . . . . . . . 12  |-  ( i  e.  ( J ... M )  ->  J  <_  i )
6968adantl 453 . . . . . . . . . . 11  |-  ( (
ph  /\  i  e.  ( J ... M ) )  ->  J  <_  i )
7058, 62, 64, 67, 69letrd 9159 . . . . . . . . . 10  |-  ( (
ph  /\  i  e.  ( J ... M ) )  ->  L  <_  i )
71 elfzle2 10993 . . . . . . . . . . 11  |-  ( i  e.  ( J ... M )  ->  i  <_  M )
7271adantl 453 . . . . . . . . . 10  |-  ( (
ph  /\  i  e.  ( J ... M ) )  ->  i  <_  M )
7370, 72jca 519 . . . . . . . . 9  |-  ( (
ph  /\  i  e.  ( J ... M ) )  ->  ( L  <_  i  /\  i  <_  M ) )
74 elfz2 10982 . . . . . . . . 9  |-  ( i  e.  ( L ... M )  <->  ( ( L  e.  ZZ  /\  M  e.  ZZ  /\  i  e.  ZZ )  /\  ( L  <_  i  /\  i  <_  M ) ) )
7556, 73, 74sylanbrc 646 . . . . . . . 8  |-  ( (
ph  /\  i  e.  ( J ... M ) )  ->  i  e.  ( L ... M ) )
7675, 10syldan 457 . . . . . . 7  |-  ( (
ph  /\  i  e.  ( J ... M ) )  ->  ( B `  i )  e.  RR )
7746, 49, 76chvar 2021 . . . . . 6  |-  ( (
ph  /\  a  e.  ( J ... M ) )  ->  ( B `  a )  e.  RR )
7843, 77, 38seqcl 11270 . . . . 5  |-  ( ph  ->  (  seq  J (  x.  ,  B ) `
 M )  e.  RR )
7978adantr 452 . . . 4  |-  ( (
ph  /\  -.  J  =  L )  ->  (  seq  J (  x.  ,  B ) `  M
)  e.  RR )
8016rpred 10580 . . . . 5  |-  ( ph  ->  E  e.  RR )
8180adantr 452 . . . 4  |-  ( (
ph  /\  -.  J  =  L )  ->  E  e.  RR )
82 remulcl 9008 . . . . . . . . 9  |-  ( ( a  e.  RR  /\  b  e.  RR )  ->  ( a  x.  b
)  e.  RR )
8382adantl 453 . . . . . . . 8  |-  ( ( ( ph  /\  -.  J  =  L )  /\  ( a  e.  RR  /\  b  e.  RR ) )  ->  ( a  x.  b )  e.  RR )
84 simp1 957 . . . . . . . . . . 11  |-  ( ( a  e.  RR  /\  b  e.  RR  /\  c  e.  RR )  ->  a  e.  RR )
8584recnd 9047 . . . . . . . . . 10  |-  ( ( a  e.  RR  /\  b  e.  RR  /\  c  e.  RR )  ->  a  e.  CC )
86 simp2 958 . . . . . . . . . . 11  |-  ( ( a  e.  RR  /\  b  e.  RR  /\  c  e.  RR )  ->  b  e.  RR )
8786recnd 9047 . . . . . . . . . 10  |-  ( ( a  e.  RR  /\  b  e.  RR  /\  c  e.  RR )  ->  b  e.  CC )
88 simp3 959 . . . . . . . . . . 11  |-  ( ( a  e.  RR  /\  b  e.  RR  /\  c  e.  RR )  ->  c  e.  RR )
8988recnd 9047 . . . . . . . . . 10  |-  ( ( a  e.  RR  /\  b  e.  RR  /\  c  e.  RR )  ->  c  e.  CC )
9085, 87, 89mulassd 9044 . . . . . . . . 9  |-  ( ( a  e.  RR  /\  b  e.  RR  /\  c  e.  RR )  ->  (
( a  x.  b
)  x.  c )  =  ( a  x.  ( b  x.  c
) ) )
9190adantl 453 . . . . . . . 8  |-  ( ( ( ph  /\  -.  J  =  L )  /\  ( a  e.  RR  /\  b  e.  RR  /\  c  e.  RR )
)  ->  ( (
a  x.  b )  x.  c )  =  ( a  x.  (
b  x.  c ) ) )
9260zcnd 10308 . . . . . . . . . . . 12  |-  ( ph  ->  J  e.  CC )
93 ax-1cn 8981 . . . . . . . . . . . . 13  |-  1  e.  CC
9493a1i 11 . . . . . . . . . . . 12  |-  ( ph  ->  1  e.  CC )
9592, 94npcand 9347 . . . . . . . . . . 11  |-  ( ph  ->  ( ( J  - 
1 )  +  1 )  =  J )
9695fveq2d 5672 . . . . . . . . . 10  |-  ( ph  ->  ( ZZ>= `  ( ( J  -  1 )  +  1 ) )  =  ( ZZ>= `  J
) )
9743, 96eleqtrrd 2464 . . . . . . . . 9  |-  ( ph  ->  M  e.  ( ZZ>= `  ( ( J  - 
1 )  +  1 ) ) )
9897adantr 452 . . . . . . . 8  |-  ( (
ph  /\  -.  J  =  L )  ->  M  e.  ( ZZ>= `  ( ( J  -  1 )  +  1 ) ) )
996adantr 452 . . . . . . . . 9  |-  ( (
ph  /\  -.  J  =  L )  ->  L  e.  ZZ )
10060adantr 452 . . . . . . . . . 10  |-  ( (
ph  /\  -.  J  =  L )  ->  J  e.  ZZ )
101 1z 10243 . . . . . . . . . . 11  |-  1  e.  ZZ
102101a1i 11 . . . . . . . . . 10  |-  ( (
ph  /\  -.  J  =  L )  ->  1  e.  ZZ )
103100, 102zsubcld 10312 . . . . . . . . 9  |-  ( (
ph  /\  -.  J  =  L )  ->  ( J  -  1 )  e.  ZZ )
104 simpr 448 . . . . . . . . . . . 12  |-  ( (
ph  /\  -.  J  =  L )  ->  -.  J  =  L )
105 eqcom 2389 . . . . . . . . . . . 12  |-  ( J  =  L  <->  L  =  J )
106104, 105sylnib 296 . . . . . . . . . . 11  |-  ( (
ph  /\  -.  J  =  L )  ->  -.  L  =  J )
10757, 61leloed 9148 . . . . . . . . . . . . 13  |-  ( ph  ->  ( L  <_  J  <->  ( L  <  J  \/  L  =  J )
) )
10866, 107mpbid 202 . . . . . . . . . . . 12  |-  ( ph  ->  ( L  <  J  \/  L  =  J
) )
109108adantr 452 . . . . . . . . . . 11  |-  ( (
ph  /\  -.  J  =  L )  ->  ( L  <  J  \/  L  =  J ) )
110 orel2 373 . . . . . . . . . . 11  |-  ( -.  L  =  J  -> 
( ( L  < 
J  \/  L  =  J )  ->  L  <  J ) )
111106, 109, 110sylc 58 . . . . . . . . . 10  |-  ( (
ph  /\  -.  J  =  L )  ->  L  <  J )
112 zltlem1 10260 . . . . . . . . . . . 12  |-  ( ( L  e.  ZZ  /\  J  e.  ZZ )  ->  ( L  <  J  <->  L  <_  ( J  - 
1 ) ) )
1136, 60, 112syl2anc 643 . . . . . . . . . . 11  |-  ( ph  ->  ( L  <  J  <->  L  <_  ( J  - 
1 ) ) )
114113adantr 452 . . . . . . . . . 10  |-  ( (
ph  /\  -.  J  =  L )  ->  ( L  <  J  <->  L  <_  ( J  -  1 ) ) )
115111, 114mpbid 202 . . . . . . . . 9  |-  ( (
ph  /\  -.  J  =  L )  ->  L  <_  ( J  -  1 ) )
116 eluz2 10426 . . . . . . . . 9  |-  ( ( J  -  1 )  e.  ( ZZ>= `  L
)  <->  ( L  e.  ZZ  /\  ( J  -  1 )  e.  ZZ  /\  L  <_ 
( J  -  1 ) ) )
11799, 103, 115, 116syl3anbrc 1138 . . . . . . . 8  |-  ( (
ph  /\  -.  J  =  L )  ->  ( J  -  1 )  e.  ( ZZ>= `  L
) )
118 nfv 1626 . . . . . . . . . . . 12  |-  F/ i  -.  J  =  L
1192, 118nfan 1836 . . . . . . . . . . 11  |-  F/ i ( ph  /\  -.  J  =  L )
120119, 25nfan 1836 . . . . . . . . . 10  |-  F/ i ( ( ph  /\  -.  J  =  L
)  /\  a  e.  ( L ... M ) )
121120, 29nfim 1822 . . . . . . . . 9  |-  F/ i ( ( ( ph  /\ 
-.  J  =  L )  /\  a  e.  ( L ... M
) )  ->  ( B `  a )  e.  RR )
12231anbi2d 685 . . . . . . . . . 10  |-  ( i  =  a  ->  (
( ( ph  /\  -.  J  =  L
)  /\  i  e.  ( L ... M ) )  <->  ( ( ph  /\ 
-.  J  =  L )  /\  a  e.  ( L ... M
) ) ) )
123122, 34imbi12d 312 . . . . . . . . 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 696 . . . . . . . . 9  |-  ( ( ( ph  /\  -.  J  =  L )  /\  i  e.  ( L ... M ) )  ->  ( B `  i )  e.  RR )
125121, 123, 124chvar 2021 . . . . . . . 8  |-  ( ( ( ph  /\  -.  J  =  L )  /\  a  e.  ( L ... M ) )  ->  ( B `  a )  e.  RR )
12683, 91, 98, 117, 125seqsplit 11283 . . . . . . 7  |-  ( (
ph  /\  -.  J  =  L )  ->  (  seq  L (  x.  ,  B ) `  M
)  =  ( (  seq  L (  x.  ,  B ) `  ( J  -  1
) )  x.  (  seq  ( ( J  - 
1 )  +  1 ) (  x.  ,  B ) `  M
) ) )
12795adantr 452 . . . . . . . . . 10  |-  ( (
ph  /\  -.  J  =  L )  ->  (
( J  -  1 )  +  1 )  =  J )
128127seqeq1d 11256 . . . . . . . . 9  |-  ( (
ph  /\  -.  J  =  L )  ->  seq  ( ( J  - 
1 )  +  1 ) (  x.  ,  B )  =  seq  J (  x.  ,  B
) )
129128fveq1d 5670 . . . . . . . 8  |-  ( (
ph  /\  -.  J  =  L )  ->  (  seq  ( ( J  - 
1 )  +  1 ) (  x.  ,  B ) `  M
)  =  (  seq 
J (  x.  ,  B ) `  M
) )
130129oveq2d 6036 . . . . . . 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 2419 . . . . . 6  |-  ( (
ph  /\  -.  J  =  L )  ->  (  seq  L (  x.  ,  B ) `  M
)  =  ( (  seq  L (  x.  ,  B ) `  ( J  -  1
) )  x.  (  seq  J (  x.  ,  B ) `  M
) ) )
132 nfv 1626 . . . . . . . . . . 11  |-  F/ i  a  e.  ( L ... ( J  - 
1 ) )
133119, 132nfan 1836 . . . . . . . . . 10  |-  F/ i ( ( ph  /\  -.  J  =  L
)  /\  a  e.  ( L ... ( J  -  1 ) ) )
134133, 29nfim 1822 . . . . . . . . 9  |-  F/ i ( ( ( ph  /\ 
-.  J  =  L )  /\  a  e.  ( L ... ( J  -  1 ) ) )  ->  ( B `  a )  e.  RR )
135 eleq1 2447 . . . . . . . . . . 11  |-  ( i  =  a  ->  (
i  e.  ( L ... ( J  - 
1 ) )  <->  a  e.  ( L ... ( J  -  1 ) ) ) )
136135anbi2d 685 . . . . . . . . . 10  |-  ( i  =  a  ->  (
( ( ph  /\  -.  J  =  L
)  /\  i  e.  ( L ... ( J  -  1 ) ) )  <->  ( ( ph  /\ 
-.  J  =  L )  /\  a  e.  ( L ... ( J  -  1 ) ) ) ) )
137136, 34imbi12d 312 . . . . . . . . 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 452 . . . . . . . . . . . . 13  |-  ( (
ph  /\  i  e.  ( L ... ( J  -  1 ) ) )  ->  L  e.  ZZ )
13952adantr 452 . . . . . . . . . . . . 13  |-  ( (
ph  /\  i  e.  ( L ... ( J  -  1 ) ) )  ->  M  e.  ZZ )
140 elfzelz 10991 . . . . . . . . . . . . . 14  |-  ( i  e.  ( L ... ( J  -  1
) )  ->  i  e.  ZZ )
141140adantl 453 . . . . . . . . . . . . 13  |-  ( (
ph  /\  i  e.  ( L ... ( J  -  1 ) ) )  ->  i  e.  ZZ )
142138, 139, 1413jca 1134 . . . . . . . . . . . 12  |-  ( (
ph  /\  i  e.  ( L ... ( J  -  1 ) ) )  ->  ( L  e.  ZZ  /\  M  e.  ZZ  /\  i  e.  ZZ ) )
143 elfzle1 10992 . . . . . . . . . . . . . 14  |-  ( i  e.  ( L ... ( J  -  1
) )  ->  L  <_  i )
144143adantl 453 . . . . . . . . . . . . 13  |-  ( (
ph  /\  i  e.  ( L ... ( J  -  1 ) ) )  ->  L  <_  i )
145140zred 10307 . . . . . . . . . . . . . . 15  |-  ( i  e.  ( L ... ( J  -  1
) )  ->  i  e.  RR )
146145adantl 453 . . . . . . . . . . . . . 14  |-  ( (
ph  /\  i  e.  ( L ... ( J  -  1 ) ) )  ->  i  e.  RR )
14761adantr 452 . . . . . . . . . . . . . 14  |-  ( (
ph  /\  i  e.  ( L ... ( J  -  1 ) ) )  ->  J  e.  RR )
14852zred 10307 . . . . . . . . . . . . . . 15  |-  ( ph  ->  M  e.  RR )
149148adantr 452 . . . . . . . . . . . . . 14  |-  ( (
ph  /\  i  e.  ( L ... ( J  -  1 ) ) )  ->  M  e.  RR )
150 1re 9023 . . . . . . . . . . . . . . . . . 18  |-  1  e.  RR
151150a1i 11 . . . . . . . . . . . . . . . . 17  |-  ( ph  ->  1  e.  RR )
15261, 151resubcld 9397 . . . . . . . . . . . . . . . 16  |-  ( ph  ->  ( J  -  1 )  e.  RR )
153152adantr 452 . . . . . . . . . . . . . . 15  |-  ( (
ph  /\  i  e.  ( L ... ( J  -  1 ) ) )  ->  ( J  -  1 )  e.  RR )
154 elfzle2 10993 . . . . . . . . . . . . . . . 16  |-  ( i  e.  ( L ... ( J  -  1
) )  ->  i  <_  ( J  -  1 ) )
155154adantl 453 . . . . . . . . . . . . . . 15  |-  ( (
ph  /\  i  e.  ( L ... ( J  -  1 ) ) )  ->  i  <_  ( J  -  1 ) )
15661lem1d 9876 . . . . . . . . . . . . . . . 16  |-  ( ph  ->  ( J  -  1 )  <_  J )
157156adantr 452 . . . . . . . . . . . . . . 15  |-  ( (
ph  /\  i  e.  ( L ... ( J  -  1 ) ) )  ->  ( J  -  1 )  <_  J )
158146, 153, 147, 155, 157letrd 9159 . . . . . . . . . . . . . 14  |-  ( (
ph  /\  i  e.  ( L ... ( J  -  1 ) ) )  ->  i  <_  J )
159 elfzle2 10993 . . . . . . . . . . . . . . . 16  |-  ( J  e.  ( L ... M )  ->  J  <_  M )
16041, 159syl 16 . . . . . . . . . . . . . . 15  |-  ( ph  ->  J  <_  M )
161160adantr 452 . . . . . . . . . . . . . 14  |-  ( (
ph  /\  i  e.  ( L ... ( J  -  1 ) ) )  ->  J  <_  M )
162146, 147, 149, 158, 161letrd 9159 . . . . . . . . . . . . 13  |-  ( (
ph  /\  i  e.  ( L ... ( J  -  1 ) ) )  ->  i  <_  M )
163144, 162jca 519 . . . . . . . . . . . 12  |-  ( (
ph  /\  i  e.  ( L ... ( J  -  1 ) ) )  ->  ( L  <_  i  /\  i  <_  M ) )
164142, 163, 74sylanbrc 646 . . . . . . . . . . 11  |-  ( (
ph  /\  i  e.  ( L ... ( J  -  1 ) ) )  ->  i  e.  ( L ... M ) )
165164, 10syldan 457 . . . . . . . . . 10  |-  ( (
ph  /\  i  e.  ( L ... ( J  -  1 ) ) )  ->  ( B `  i )  e.  RR )
166165adantlr 696 . . . . . . . . 9  |-  ( ( ( ph  /\  -.  J  =  L )  /\  i  e.  ( L ... ( J  - 
1 ) ) )  ->  ( B `  i )  e.  RR )
167134, 137, 166chvar 2021 . . . . . . . 8  |-  ( ( ( ph  /\  -.  J  =  L )  /\  a  e.  ( L ... ( J  - 
1 ) ) )  ->  ( B `  a )  e.  RR )
16837adantl 453 . . . . . . . 8  |-  ( ( ( ph  /\  -.  J  =  L )  /\  ( a  e.  RR  /\  j  e.  RR ) )  ->  ( a  x.  j )  e.  RR )
169117, 167, 168seqcl 11270 . . . . . . 7  |-  ( (
ph  /\  -.  J  =  L )  ->  (  seq  L (  x.  ,  B ) `  ( J  -  1 ) )  e.  RR )
170150a1i 11 . . . . . . 7  |-  ( (
ph  /\  -.  J  =  L )  ->  1  e.  RR )
171 eqid 2387 . . . . . . . . 9  |-  seq  J
(  x.  ,  B
)  =  seq  J
(  x.  ,  B
)
17243adantr 452 . . . . . . . . 9  |-  ( (
ph  /\  -.  J  =  L )  ->  M  e.  ( ZZ>= `  J )
)
173 eluzfz2 10997 . . . . . . . . . . 11  |-  ( M  e.  ( ZZ>= `  J
)  ->  M  e.  ( J ... M ) )
17443, 173syl 16 . . . . . . . . . 10  |-  ( ph  ->  M  e.  ( J ... M ) )
175174adantr 452 . . . . . . . . 9  |-  ( (
ph  /\  -.  J  =  L )  ->  M  e.  ( J ... M
) )
17676adantlr 696 . . . . . . . . 9  |-  ( ( ( ph  /\  -.  J  =  L )  /\  i  e.  ( J ... M ) )  ->  ( B `  i )  e.  RR )
17775, 12syldan 457 . . . . . . . . . 10  |-  ( (
ph  /\  i  e.  ( J ... M ) )  ->  0  <_  ( B `  i ) )
178177adantlr 696 . . . . . . . . 9  |-  ( ( ( ph  /\  -.  J  =  L )  /\  i  e.  ( J ... M ) )  ->  0  <_  ( B `  i )
)
17975, 14syldan 457 . . . . . . . . . 10  |-  ( (
ph  /\  i  e.  ( J ... M ) )  ->  ( B `  i )  <_  1
)
180179adantlr 696 . . . . . . . . 9  |-  ( ( ( ph  /\  -.  J  =  L )  /\  i  e.  ( J ... M ) )  ->  ( B `  i )  <_  1
)
1811, 119, 171, 100, 172, 175, 176, 178, 180fmul01 27378 . . . . . . . 8  |-  ( (
ph  /\  -.  J  =  L )  ->  (
0  <_  (  seq  J (  x.  ,  B
) `  M )  /\  (  seq  J (  x.  ,  B ) `
 M )  <_ 
1 ) )
182181simpld 446 . . . . . . 7  |-  ( (
ph  /\  -.  J  =  L )  ->  0  <_  (  seq  J (  x.  ,  B ) `
 M ) )
183 eqid 2387 . . . . . . . . 9  |-  seq  L
(  x.  ,  B
)  =  seq  L
(  x.  ,  B
)
1848adantr 452 . . . . . . . . 9  |-  ( (
ph  /\  -.  J  =  L )  ->  M  e.  ( ZZ>= `  L )
)
185101a1i 11 . . . . . . . . . . . . 13  |-  ( ph  ->  1  e.  ZZ )
18660, 185zsubcld 10312 . . . . . . . . . . . 12  |-  ( ph  ->  ( J  -  1 )  e.  ZZ )
1876, 52, 1863jca 1134 . . . . . . . . . . 11  |-  ( ph  ->  ( L  e.  ZZ  /\  M  e.  ZZ  /\  ( J  -  1
)  e.  ZZ ) )
188187adantr 452 . . . . . . . . . 10  |-  ( (
ph  /\  -.  J  =  L )  ->  ( L  e.  ZZ  /\  M  e.  ZZ  /\  ( J  -  1 )  e.  ZZ ) )
189152, 61, 1483jca 1134 . . . . . . . . . . . . 13  |-  ( ph  ->  ( ( J  - 
1 )  e.  RR  /\  J  e.  RR  /\  M  e.  RR )
)
190189adantr 452 . . . . . . . . . . . 12  |-  ( (
ph  /\  -.  J  =  L )  ->  (
( J  -  1 )  e.  RR  /\  J  e.  RR  /\  M  e.  RR ) )
19161adantr 452 . . . . . . . . . . . . . 14  |-  ( (
ph  /\  -.  J  =  L )  ->  J  e.  RR )
192191lem1d 9876 . . . . . . . . . . . . 13  |-  ( (
ph  /\  -.  J  =  L )  ->  ( J  -  1 )  <_  J )
193160adantr 452 . . . . . . . . . . . . 13  |-  ( (
ph  /\  -.  J  =  L )  ->  J  <_  M )
194192, 193jca 519 . . . . . . . . . . . 12  |-  ( (
ph  /\  -.  J  =  L )  ->  (
( J  -  1 )  <_  J  /\  J  <_  M ) )
195 letr 9100 . . . . . . . . . . . 12  |-  ( ( ( J  -  1 )  e.  RR  /\  J  e.  RR  /\  M  e.  RR )  ->  (
( ( J  - 
1 )  <_  J  /\  J  <_  M )  ->  ( J  - 
1 )  <_  M
) )
196190, 194, 195sylc 58 . . . . . . . . . . 11  |-  ( (
ph  /\  -.  J  =  L )  ->  ( J  -  1 )  <_  M )
197115, 196jca 519 . . . . . . . . . 10  |-  ( (
ph  /\  -.  J  =  L )  ->  ( L  <_  ( J  - 
1 )  /\  ( J  -  1 )  <_  M ) )
198 elfz2 10982 . . . . . . . . . 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 646 . . . . . . . . 9  |-  ( (
ph  /\  -.  J  =  L )  ->  ( J  -  1 )  e.  ( L ... M ) )
20012adantlr 696 . . . . . . . . 9  |-  ( ( ( ph  /\  -.  J  =  L )  /\  i  e.  ( L ... M ) )  ->  0  <_  ( B `  i )
)
20114adantlr 696 . . . . . . . . 9  |-  ( ( ( ph  /\  -.  J  =  L )  /\  i  e.  ( L ... M ) )  ->  ( B `  i )  <_  1
)
2021, 119, 183, 99, 184, 199, 124, 200, 201fmul01 27378 . . . . . . . 8  |-  ( (
ph  /\  -.  J  =  L )  ->  (
0  <_  (  seq  L (  x.  ,  B
) `  ( J  -  1 ) )  /\  (  seq  L
(  x.  ,  B
) `  ( J  -  1 ) )  <_  1 ) )
203202simprd 450 . . . . . . 7  |-  ( (
ph  /\  -.  J  =  L )  ->  (  seq  L (  x.  ,  B ) `  ( J  -  1 ) )  <_  1 )
204169, 170, 79, 182, 203lemul1ad 9882 . . . . . 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 4173 . . . . 5  |-  ( (
ph  /\  -.  J  =  L )  ->  (  seq  L (  x.  ,  B ) `  M
)  <_  ( 1  x.  (  seq  J
(  x.  ,  B
) `  M )
) )
20679recnd 9047 . . . . . 6  |-  ( (
ph  /\  -.  J  =  L )  ->  (  seq  J (  x.  ,  B ) `  M
)  e.  CC )
207206mulid2d 9039 . . . . 5  |-  ( (
ph  /\  -.  J  =  L )  ->  (
1  x.  (  seq 
J (  x.  ,  B ) `  M
) )  =  (  seq  J (  x.  ,  B ) `  M ) )
208205, 207breqtrd 4177 . . . 4  |-  ( (
ph  /\  -.  J  =  L )  ->  (  seq  L (  x.  ,  B ) `  M
)  <_  (  seq  J (  x.  ,  B
) `  M )
)
2091, 2, 171, 60, 43, 76, 177, 179, 16, 20fmul01lt1lem1 27382 . . . . 5  |-  ( ph  ->  (  seq  J (  x.  ,  B ) `
 M )  < 
E )
210209adantr 452 . . . 4  |-  ( (
ph  /\  -.  J  =  L )  ->  (  seq  J (  x.  ,  B ) `  M
)  <  E )
21140, 79, 81, 208, 210lelttrd 9160 . . 3  |-  ( (
ph  /\  -.  J  =  L )  ->  (  seq  L (  x.  ,  B ) `  M
)  <  E )
21224, 211syl5eqbr 4186 . 2  |-  ( (
ph  /\  -.  J  =  L )  ->  ( A `  M )  <  E )
21323, 212pm2.61dan 767 1  |-  ( ph  ->  ( A `  M
)  <  E )
Colors of variables: wff set class
Syntax hints:   -. wn 3    -> wi 4    <-> wb 177    \/ wo 358    /\ wa 359    /\ w3a 936   F/wnf 1550    = wceq 1649    e. wcel 1717   F/_wnfc 2510   class class class wbr 4153   ` cfv 5394  (class class class)co 6020   CCcc 8921   RRcr 8922   0cc0 8923   1c1 8924    + caddc 8926    x. cmul 8928    < clt 9053    <_ cle 9054    - cmin 9223   ZZcz 10214   ZZ>=cuz 10420   RR+crp 10544   ...cfz 10975    seq cseq 11250
This theorem is referenced by:  fmul01lt1  27384
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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