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Theorem rpnnen2lem7 12515
Description: Lemma for rpnnen2 12520. (Contributed by Mario Carneiro, 13-May-2013.) (Revised by Mario Carneiro, 30-Apr-2014.)
Hypothesis
Ref Expression
rpnnen2.1  |-  F  =  ( x  e.  ~P NN  |->  ( n  e.  NN  |->  if ( n  e.  x ,  ( ( 1  /  3
) ^ n ) ,  0 ) ) )
Assertion
Ref Expression
rpnnen2lem7  |-  ( ( A  C_  B  /\  B  C_  NN  /\  M  e.  NN )  ->  sum_ k  e.  ( ZZ>= `  M )
( ( F `  A ) `  k
)  <_  sum_ k  e.  ( ZZ>= `  M )
( ( F `  B ) `  k
) )
Distinct variable groups:    x, n, k, A    B, k, n, x    k, F    k, M, n, x
Allowed substitution hints:    F( x, n)

Proof of Theorem rpnnen2lem7
StepHypRef Expression
1 eqid 2296 . 2  |-  ( ZZ>= `  M )  =  (
ZZ>= `  M )
2 simp3 957 . . 3  |-  ( ( A  C_  B  /\  B  C_  NN  /\  M  e.  NN )  ->  M  e.  NN )
32nnzd 10132 . 2  |-  ( ( A  C_  B  /\  B  C_  NN  /\  M  e.  NN )  ->  M  e.  ZZ )
4 eqidd 2297 . 2  |-  ( ( ( A  C_  B  /\  B  C_  NN  /\  M  e.  NN )  /\  k  e.  ( ZZ>=
`  M ) )  ->  ( ( F `
 A ) `  k )  =  ( ( F `  A
) `  k )
)
5 nnuz 10279 . . . . 5  |-  NN  =  ( ZZ>= `  1 )
65uztrn2 10261 . . . 4  |-  ( ( M  e.  NN  /\  k  e.  ( ZZ>= `  M ) )  -> 
k  e.  NN )
72, 6sylan 457 . . 3  |-  ( ( ( A  C_  B  /\  B  C_  NN  /\  M  e.  NN )  /\  k  e.  ( ZZ>=
`  M ) )  ->  k  e.  NN )
8 sstr 3200 . . . . . 6  |-  ( ( A  C_  B  /\  B  C_  NN )  ->  A  C_  NN )
983adant3 975 . . . . 5  |-  ( ( A  C_  B  /\  B  C_  NN  /\  M  e.  NN )  ->  A  C_  NN )
10 rpnnen2.1 . . . . . 6  |-  F  =  ( x  e.  ~P NN  |->  ( n  e.  NN  |->  if ( n  e.  x ,  ( ( 1  /  3
) ^ n ) ,  0 ) ) )
1110rpnnen2lem2 12510 . . . . 5  |-  ( A 
C_  NN  ->  ( F `
 A ) : NN --> RR )
129, 11syl 15 . . . 4  |-  ( ( A  C_  B  /\  B  C_  NN  /\  M  e.  NN )  ->  ( F `  A ) : NN --> RR )
13 ffvelrn 5679 . . . 4  |-  ( ( ( F `  A
) : NN --> RR  /\  k  e.  NN )  ->  ( ( F `  A ) `  k
)  e.  RR )
1412, 13sylan 457 . . 3  |-  ( ( ( A  C_  B  /\  B  C_  NN  /\  M  e.  NN )  /\  k  e.  NN )  ->  ( ( F `
 A ) `  k )  e.  RR )
157, 14syldan 456 . 2  |-  ( ( ( A  C_  B  /\  B  C_  NN  /\  M  e.  NN )  /\  k  e.  ( ZZ>=
`  M ) )  ->  ( ( F `
 A ) `  k )  e.  RR )
16 eqidd 2297 . 2  |-  ( ( ( A  C_  B  /\  B  C_  NN  /\  M  e.  NN )  /\  k  e.  ( ZZ>=
`  M ) )  ->  ( ( F `
 B ) `  k )  =  ( ( F `  B
) `  k )
)
1710rpnnen2lem2 12510 . . . . 5  |-  ( B 
C_  NN  ->  ( F `
 B ) : NN --> RR )
18173ad2ant2 977 . . . 4  |-  ( ( A  C_  B  /\  B  C_  NN  /\  M  e.  NN )  ->  ( F `  B ) : NN --> RR )
19 ffvelrn 5679 . . . 4  |-  ( ( ( F `  B
) : NN --> RR  /\  k  e.  NN )  ->  ( ( F `  B ) `  k
)  e.  RR )
2018, 19sylan 457 . . 3  |-  ( ( ( A  C_  B  /\  B  C_  NN  /\  M  e.  NN )  /\  k  e.  NN )  ->  ( ( F `
 B ) `  k )  e.  RR )
217, 20syldan 456 . 2  |-  ( ( ( A  C_  B  /\  B  C_  NN  /\  M  e.  NN )  /\  k  e.  ( ZZ>=
`  M ) )  ->  ( ( F `
 B ) `  k )  e.  RR )
2210rpnnen2lem4 12512 . . . . . 6  |-  ( ( A  C_  B  /\  B  C_  NN  /\  k  e.  NN )  ->  (
0  <_  ( ( F `  A ) `  k )  /\  (
( F `  A
) `  k )  <_  ( ( F `  B ) `  k
) ) )
2322simprd 449 . . . . 5  |-  ( ( A  C_  B  /\  B  C_  NN  /\  k  e.  NN )  ->  (
( F `  A
) `  k )  <_  ( ( F `  B ) `  k
) )
24233expa 1151 . . . 4  |-  ( ( ( A  C_  B  /\  B  C_  NN )  /\  k  e.  NN )  ->  ( ( F `
 A ) `  k )  <_  (
( F `  B
) `  k )
)
25243adantl3 1113 . . 3  |-  ( ( ( A  C_  B  /\  B  C_  NN  /\  M  e.  NN )  /\  k  e.  NN )  ->  ( ( F `
 A ) `  k )  <_  (
( F `  B
) `  k )
)
267, 25syldan 456 . 2  |-  ( ( ( A  C_  B  /\  B  C_  NN  /\  M  e.  NN )  /\  k  e.  ( ZZ>=
`  M ) )  ->  ( ( F `
 A ) `  k )  <_  (
( F `  B
) `  k )
)
2710rpnnen2lem5 12513 . . 3  |-  ( ( A  C_  NN  /\  M  e.  NN )  ->  seq  M (  +  ,  ( F `  A ) )  e.  dom  ~~>  )
289, 2, 27syl2anc 642 . 2  |-  ( ( A  C_  B  /\  B  C_  NN  /\  M  e.  NN )  ->  seq  M (  +  ,  ( F `  A ) )  e.  dom  ~~>  )
2910rpnnen2lem5 12513 . . 3  |-  ( ( B  C_  NN  /\  M  e.  NN )  ->  seq  M (  +  ,  ( F `  B ) )  e.  dom  ~~>  )
30293adant1 973 . 2  |-  ( ( A  C_  B  /\  B  C_  NN  /\  M  e.  NN )  ->  seq  M (  +  ,  ( F `  B ) )  e.  dom  ~~>  )
311, 3, 4, 15, 16, 21, 26, 28, 30isumle 12319 1  |-  ( ( A  C_  B  /\  B  C_  NN  /\  M  e.  NN )  ->  sum_ k  e.  ( ZZ>= `  M )
( ( F `  A ) `  k
)  <_  sum_ k  e.  ( ZZ>= `  M )
( ( F `  B ) `  k
) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 358    /\ w3a 934    = wceq 1632    e. wcel 1696    C_ wss 3165   ifcif 3578   ~Pcpw 3638   class class class wbr 4039    e. cmpt 4093   dom cdm 4705   -->wf 5267   ` cfv 5271  (class class class)co 5874   RRcr 8752   0cc0 8753   1c1 8754    + caddc 8756    <_ cle 8884    / cdiv 9439   NNcn 9762   3c3 9812   ZZ>=cuz 10246    seq cseq 11062   ^cexp 11120    ~~> cli 11974   sum_csu 12174
This theorem is referenced by:  rpnnen2lem11  12519  rpnnen2  12520
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-rep 4147  ax-sep 4157  ax-nul 4165  ax-pow 4204  ax-pr 4230  ax-un 4528  ax-inf2 7358  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  ax-pre-sup 8831
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-rmo 2564  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-int 3879  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-se 4369  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-isom 5280  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-1o 6495  df-oadd 6499  df-er 6676  df-pm 6791  df-en 6880  df-dom 6881  df-sdom 6882  df-fin 6883  df-sup 7210  df-oi 7241  df-card 7588  df-pnf 8885  df-mnf 8886  df-xr 8887  df-ltxr 8888  df-le 8889  df-sub 9055  df-neg 9056  df-div 9440  df-nn 9763  df-2 9820  df-3 9821  df-n0 9982  df-z 10041  df-uz 10247  df-rp 10371  df-ico 10678  df-fz 10799  df-fzo 10887  df-fl 10941  df-seq 11063  df-exp 11121  df-hash 11354  df-cj 11600  df-re 11601  df-im 11602  df-sqr 11736  df-abs 11737  df-limsup 11961  df-clim 11978  df-rlim 11979  df-sum 12175
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