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Theorem ovolscalem2 22217
Description: Lemma for ovolshft 22214. (Contributed by Mario Carneiro, 22-Mar-2014.)
Hypotheses
Ref Expression
ovolsca.1  |-  ( ph  ->  A  C_  RR )
ovolsca.2  |-  ( ph  ->  C  e.  RR+ )
ovolsca.3  |-  ( ph  ->  B  =  { x  e.  RR  |  ( C  x.  x )  e.  A } )
ovolsca.4  |-  ( ph  ->  ( vol* `  A )  e.  RR )
Assertion
Ref Expression
ovolscalem2  |-  ( ph  ->  ( vol* `  B )  <_  (
( vol* `  A )  /  C
) )
Distinct variable groups:    x, A    x, C
Allowed substitution hints:    ph( x)    B( x)

Proof of Theorem ovolscalem2
Dummy variables  f  n  y  m are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 ovolsca.1 . . . . . 6  |-  ( ph  ->  A  C_  RR )
21adantr 463 . . . . 5  |-  ( (
ph  /\  y  e.  RR+ )  ->  A  C_  RR )
3 ovolsca.4 . . . . . 6  |-  ( ph  ->  ( vol* `  A )  e.  RR )
43adantr 463 . . . . 5  |-  ( (
ph  /\  y  e.  RR+ )  ->  ( vol* `  A )  e.  RR )
5 ovolsca.2 . . . . . 6  |-  ( ph  ->  C  e.  RR+ )
6 rpmulcl 11287 . . . . . 6  |-  ( ( C  e.  RR+  /\  y  e.  RR+ )  ->  ( C  x.  y )  e.  RR+ )
75, 6sylan 469 . . . . 5  |-  ( (
ph  /\  y  e.  RR+ )  ->  ( C  x.  y )  e.  RR+ )
8 eqid 2402 . . . . . 6  |-  seq 1
(  +  ,  ( ( abs  o.  -  )  o.  f )
)  =  seq 1
(  +  ,  ( ( abs  o.  -  )  o.  f )
)
98ovolgelb 22183 . . . . 5  |-  ( ( A  C_  RR  /\  ( vol* `  A )  e.  RR  /\  ( C  x.  y )  e.  RR+ )  ->  E. f  e.  ( (  <_  i^i  ( RR  X.  RR ) )  ^m  NN ) ( A  C_  U.
ran  ( (,)  o.  f )  /\  sup ( ran  seq 1 (  +  ,  ( ( abs  o.  -  )  o.  f ) ) , 
RR* ,  <  )  <_ 
( ( vol* `  A )  +  ( C  x.  y ) ) ) )
102, 4, 7, 9syl3anc 1230 . . . 4  |-  ( (
ph  /\  y  e.  RR+ )  ->  E. f  e.  ( (  <_  i^i  ( RR  X.  RR ) )  ^m  NN ) ( A  C_  U.
ran  ( (,)  o.  f )  /\  sup ( ran  seq 1 (  +  ,  ( ( abs  o.  -  )  o.  f ) ) , 
RR* ,  <  )  <_ 
( ( vol* `  A )  +  ( C  x.  y ) ) ) )
111ad2antrr 724 . . . . 5  |-  ( ( ( ph  /\  y  e.  RR+ )  /\  (
f  e.  ( (  <_  i^i  ( RR  X.  RR ) )  ^m  NN )  /\  ( A  C_  U. ran  ( (,)  o.  f )  /\  sup ( ran  seq 1
(  +  ,  ( ( abs  o.  -  )  o.  f )
) ,  RR* ,  <  )  <_  ( ( vol* `  A )  +  ( C  x.  y ) ) ) ) )  ->  A  C_  RR )
125ad2antrr 724 . . . . 5  |-  ( ( ( ph  /\  y  e.  RR+ )  /\  (
f  e.  ( (  <_  i^i  ( RR  X.  RR ) )  ^m  NN )  /\  ( A  C_  U. ran  ( (,)  o.  f )  /\  sup ( ran  seq 1
(  +  ,  ( ( abs  o.  -  )  o.  f )
) ,  RR* ,  <  )  <_  ( ( vol* `  A )  +  ( C  x.  y ) ) ) ) )  ->  C  e.  RR+ )
13 ovolsca.3 . . . . . 6  |-  ( ph  ->  B  =  { x  e.  RR  |  ( C  x.  x )  e.  A } )
1413ad2antrr 724 . . . . 5  |-  ( ( ( ph  /\  y  e.  RR+ )  /\  (
f  e.  ( (  <_  i^i  ( RR  X.  RR ) )  ^m  NN )  /\  ( A  C_  U. ran  ( (,)  o.  f )  /\  sup ( ran  seq 1
(  +  ,  ( ( abs  o.  -  )  o.  f )
) ,  RR* ,  <  )  <_  ( ( vol* `  A )  +  ( C  x.  y ) ) ) ) )  ->  B  =  { x  e.  RR  |  ( C  x.  x )  e.  A } )
153ad2antrr 724 . . . . 5  |-  ( ( ( ph  /\  y  e.  RR+ )  /\  (
f  e.  ( (  <_  i^i  ( RR  X.  RR ) )  ^m  NN )  /\  ( A  C_  U. ran  ( (,)  o.  f )  /\  sup ( ran  seq 1
(  +  ,  ( ( abs  o.  -  )  o.  f )
) ,  RR* ,  <  )  <_  ( ( vol* `  A )  +  ( C  x.  y ) ) ) ) )  ->  ( vol* `  A )  e.  RR )
16 fveq2 5849 . . . . . . . . 9  |-  ( m  =  n  ->  (
f `  m )  =  ( f `  n ) )
1716fveq2d 5853 . . . . . . . 8  |-  ( m  =  n  ->  ( 1st `  ( f `  m ) )  =  ( 1st `  (
f `  n )
) )
1817oveq1d 6293 . . . . . . 7  |-  ( m  =  n  ->  (
( 1st `  (
f `  m )
)  /  C )  =  ( ( 1st `  ( f `  n
) )  /  C
) )
1916fveq2d 5853 . . . . . . . 8  |-  ( m  =  n  ->  ( 2nd `  ( f `  m ) )  =  ( 2nd `  (
f `  n )
) )
2019oveq1d 6293 . . . . . . 7  |-  ( m  =  n  ->  (
( 2nd `  (
f `  m )
)  /  C )  =  ( ( 2nd `  ( f `  n
) )  /  C
) )
2118, 20opeq12d 4167 . . . . . 6  |-  ( m  =  n  ->  <. (
( 1st `  (
f `  m )
)  /  C ) ,  ( ( 2nd `  ( f `  m
) )  /  C
) >.  =  <. (
( 1st `  (
f `  n )
)  /  C ) ,  ( ( 2nd `  ( f `  n
) )  /  C
) >. )
2221cbvmptv 4487 . . . . 5  |-  ( m  e.  NN  |->  <. (
( 1st `  (
f `  m )
)  /  C ) ,  ( ( 2nd `  ( f `  m
) )  /  C
) >. )  =  ( n  e.  NN  |->  <.
( ( 1st `  (
f `  n )
)  /  C ) ,  ( ( 2nd `  ( f `  n
) )  /  C
) >. )
23 elmapi 7478 . . . . . 6  |-  ( f  e.  ( (  <_  i^i  ( RR  X.  RR ) )  ^m  NN )  ->  f : NN --> (  <_  i^i  ( RR  X.  RR ) ) )
2423ad2antrl 726 . . . . 5  |-  ( ( ( ph  /\  y  e.  RR+ )  /\  (
f  e.  ( (  <_  i^i  ( RR  X.  RR ) )  ^m  NN )  /\  ( A  C_  U. ran  ( (,)  o.  f )  /\  sup ( ran  seq 1
(  +  ,  ( ( abs  o.  -  )  o.  f )
) ,  RR* ,  <  )  <_  ( ( vol* `  A )  +  ( C  x.  y ) ) ) ) )  ->  f : NN --> (  <_  i^i  ( RR  X.  RR ) ) )
25 simprrl 766 . . . . 5  |-  ( ( ( ph  /\  y  e.  RR+ )  /\  (
f  e.  ( (  <_  i^i  ( RR  X.  RR ) )  ^m  NN )  /\  ( A  C_  U. ran  ( (,)  o.  f )  /\  sup ( ran  seq 1
(  +  ,  ( ( abs  o.  -  )  o.  f )
) ,  RR* ,  <  )  <_  ( ( vol* `  A )  +  ( C  x.  y ) ) ) ) )  ->  A  C_ 
U. ran  ( (,)  o.  f ) )
26 simplr 754 . . . . 5  |-  ( ( ( ph  /\  y  e.  RR+ )  /\  (
f  e.  ( (  <_  i^i  ( RR  X.  RR ) )  ^m  NN )  /\  ( A  C_  U. ran  ( (,)  o.  f )  /\  sup ( ran  seq 1
(  +  ,  ( ( abs  o.  -  )  o.  f )
) ,  RR* ,  <  )  <_  ( ( vol* `  A )  +  ( C  x.  y ) ) ) ) )  ->  y  e.  RR+ )
27 simprrr 767 . . . . 5  |-  ( ( ( ph  /\  y  e.  RR+ )  /\  (
f  e.  ( (  <_  i^i  ( RR  X.  RR ) )  ^m  NN )  /\  ( A  C_  U. ran  ( (,)  o.  f )  /\  sup ( ran  seq 1
(  +  ,  ( ( abs  o.  -  )  o.  f )
) ,  RR* ,  <  )  <_  ( ( vol* `  A )  +  ( C  x.  y ) ) ) ) )  ->  sup ( ran  seq 1 (  +  ,  ( ( abs  o.  -  )  o.  f ) ) , 
RR* ,  <  )  <_ 
( ( vol* `  A )  +  ( C  x.  y ) ) )
2811, 12, 14, 15, 8, 22, 24, 25, 26, 27ovolscalem1 22216 . . . 4  |-  ( ( ( ph  /\  y  e.  RR+ )  /\  (
f  e.  ( (  <_  i^i  ( RR  X.  RR ) )  ^m  NN )  /\  ( A  C_  U. ran  ( (,)  o.  f )  /\  sup ( ran  seq 1
(  +  ,  ( ( abs  o.  -  )  o.  f )
) ,  RR* ,  <  )  <_  ( ( vol* `  A )  +  ( C  x.  y ) ) ) ) )  ->  ( vol* `  B )  <_  ( ( ( vol* `  A
)  /  C )  +  y ) )
2910, 28rexlimddv 2900 . . 3  |-  ( (
ph  /\  y  e.  RR+ )  ->  ( vol* `  B )  <_ 
( ( ( vol* `  A )  /  C )  +  y ) )
3029ralrimiva 2818 . 2  |-  ( ph  ->  A. y  e.  RR+  ( vol* `  B
)  <_  ( (
( vol* `  A )  /  C
)  +  y ) )
31 ssrab2 3524 . . . . 5  |-  { x  e.  RR  |  ( C  x.  x )  e.  A }  C_  RR
3213, 31syl6eqss 3492 . . . 4  |-  ( ph  ->  B  C_  RR )
33 ovolcl 22181 . . . 4  |-  ( B 
C_  RR  ->  ( vol* `  B )  e.  RR* )
3432, 33syl 17 . . 3  |-  ( ph  ->  ( vol* `  B )  e.  RR* )
353, 5rerpdivcld 11331 . . 3  |-  ( ph  ->  ( ( vol* `  A )  /  C
)  e.  RR )
36 xralrple 11457 . . 3  |-  ( ( ( vol* `  B )  e.  RR*  /\  ( ( vol* `  A )  /  C
)  e.  RR )  ->  ( ( vol* `  B )  <_  ( ( vol* `  A )  /  C
)  <->  A. y  e.  RR+  ( vol* `  B
)  <_  ( (
( vol* `  A )  /  C
)  +  y ) ) )
3734, 35, 36syl2anc 659 . 2  |-  ( ph  ->  ( ( vol* `  B )  <_  (
( vol* `  A )  /  C
)  <->  A. y  e.  RR+  ( vol* `  B
)  <_  ( (
( vol* `  A )  /  C
)  +  y ) ) )
3830, 37mpbird 232 1  |-  ( ph  ->  ( vol* `  B )  <_  (
( vol* `  A )  /  C
) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    <-> wb 184    /\ wa 367    = wceq 1405    e. wcel 1842   A.wral 2754   E.wrex 2755   {crab 2758    i^i cin 3413    C_ wss 3414   <.cop 3978   U.cuni 4191   class class class wbr 4395    |-> cmpt 4453    X. cxp 4821   ran crn 4824    o. ccom 4827   -->wf 5565   ` cfv 5569  (class class class)co 6278   1stc1st 6782   2ndc2nd 6783    ^m cmap 7457   supcsup 7934   RRcr 9521   1c1 9523    + caddc 9525    x. cmul 9527   RR*cxr 9657    < clt 9658    <_ cle 9659    - cmin 9841    / cdiv 10247   NNcn 10576   RR+crp 11265   (,)cioo 11582    seqcseq 12151   abscabs 13216   vol*covol 22166
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1639  ax-4 1652  ax-5 1725  ax-6 1771  ax-7 1814  ax-8 1844  ax-9 1846  ax-10 1861  ax-11 1866  ax-12 1878  ax-13 2026  ax-ext 2380  ax-rep 4507  ax-sep 4517  ax-nul 4525  ax-pow 4572  ax-pr 4630  ax-un 6574  ax-inf2 8091  ax-cnex 9578  ax-resscn 9579  ax-1cn 9580  ax-icn 9581  ax-addcl 9582  ax-addrcl 9583  ax-mulcl 9584  ax-mulrcl 9585  ax-mulcom 9586  ax-addass 9587  ax-mulass 9588  ax-distr 9589  ax-i2m1 9590  ax-1ne0 9591  ax-1rid 9592  ax-rnegex 9593  ax-rrecex 9594  ax-cnre 9595  ax-pre-lttri 9596  ax-pre-lttrn 9597  ax-pre-ltadd 9598  ax-pre-mulgt0 9599  ax-pre-sup 9600
This theorem depends on definitions:  df-bi 185  df-or 368  df-an 369  df-3or 975  df-3an 976  df-tru 1408  df-fal 1411  df-ex 1634  df-nf 1638  df-sb 1764  df-eu 2242  df-mo 2243  df-clab 2388  df-cleq 2394  df-clel 2397  df-nfc 2552  df-ne 2600  df-nel 2601  df-ral 2759  df-rex 2760  df-reu 2761  df-rmo 2762  df-rab 2763  df-v 3061  df-sbc 3278  df-csb 3374  df-dif 3417  df-un 3419  df-in 3421  df-ss 3428  df-pss 3430  df-nul 3739  df-if 3886  df-pw 3957  df-sn 3973  df-pr 3975  df-tp 3977  df-op 3979  df-uni 4192  df-int 4228  df-iun 4273  df-br 4396  df-opab 4454  df-mpt 4455  df-tr 4490  df-eprel 4734  df-id 4738  df-po 4744  df-so 4745  df-fr 4782  df-se 4783  df-we 4784  df-xp 4829  df-rel 4830  df-cnv 4831  df-co 4832  df-dm 4833  df-rn 4834  df-res 4835  df-ima 4836  df-pred 5367  df-ord 5413  df-on 5414  df-lim 5415  df-suc 5416  df-iota 5533  df-fun 5571  df-fn 5572  df-f 5573  df-f1 5574  df-fo 5575  df-f1o 5576  df-fv 5577  df-isom 5578  df-riota 6240  df-ov 6281  df-oprab 6282  df-mpt2 6283  df-om 6684  df-1st 6784  df-2nd 6785  df-wrecs 7013  df-recs 7075  df-rdg 7113  df-1o 7167  df-oadd 7171  df-er 7348  df-map 7459  df-en 7555  df-dom 7556  df-sdom 7557  df-fin 7558  df-sup 7935  df-oi 7969  df-card 8352  df-pnf 9660  df-mnf 9661  df-xr 9662  df-ltxr 9663  df-le 9664  df-sub 9843  df-neg 9844  df-div 10248  df-nn 10577  df-2 10635  df-3 10636  df-n0 10837  df-z 10906  df-uz 11128  df-q 11228  df-rp 11266  df-ioo 11586  df-ico 11588  df-fz 11727  df-fzo 11855  df-seq 12152  df-exp 12211  df-hash 12453  df-cj 13081  df-re 13082  df-im 13083  df-sqrt 13217  df-abs 13218  df-clim 13460  df-sum 13658  df-ovol 22168
This theorem is referenced by:  ovolsca  22218
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