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Theorem uniioombllem2a 21021
Description: Lemma for uniioombl 21028. (Contributed by Mario Carneiro, 7-May-2015.)
Hypotheses
Ref Expression
uniioombl.1  |-  ( ph  ->  F : NN --> (  <_  i^i  ( RR  X.  RR ) ) )
uniioombl.2  |-  ( ph  -> Disj  x  e.  NN  ( (,) `  ( F `  x ) ) )
uniioombl.3  |-  S  =  seq 1 (  +  ,  ( ( abs 
o.  -  )  o.  F ) )
uniioombl.a  |-  A  = 
U. ran  ( (,)  o.  F )
uniioombl.e  |-  ( ph  ->  ( vol* `  E )  e.  RR )
uniioombl.c  |-  ( ph  ->  C  e.  RR+ )
uniioombl.g  |-  ( ph  ->  G : NN --> (  <_  i^i  ( RR  X.  RR ) ) )
uniioombl.s  |-  ( ph  ->  E  C_  U. ran  ( (,)  o.  G ) )
uniioombl.t  |-  T  =  seq 1 (  +  ,  ( ( abs 
o.  -  )  o.  G ) )
uniioombl.v  |-  ( ph  ->  sup ( ran  T ,  RR* ,  <  )  <_  ( ( vol* `  E )  +  C
) )
Assertion
Ref Expression
uniioombllem2a  |-  ( ( ( ph  /\  J  e.  NN )  /\  z  e.  NN )  ->  (
( (,) `  ( F `  z )
)  i^i  ( (,) `  ( G `  J
) ) )  e. 
ran  (,) )
Distinct variable groups:    x, z, F    x, G, z    x, A, z    x, C, z   
x, J, z    ph, x, z    x, T, z
Allowed substitution hints:    S( x, z)    E( x, z)

Proof of Theorem uniioombllem2a
StepHypRef Expression
1 inss2 3568 . . . . . . . 8  |-  (  <_  i^i  ( RR  X.  RR ) )  C_  ( RR  X.  RR )
2 uniioombl.1 . . . . . . . . . 10  |-  ( ph  ->  F : NN --> (  <_  i^i  ( RR  X.  RR ) ) )
32adantr 462 . . . . . . . . 9  |-  ( (
ph  /\  J  e.  NN )  ->  F : NN
--> (  <_  i^i  ( RR  X.  RR ) ) )
43ffvelrnda 5840 . . . . . . . 8  |-  ( ( ( ph  /\  J  e.  NN )  /\  z  e.  NN )  ->  ( F `  z )  e.  (  <_  i^i  ( RR  X.  RR ) ) )
51, 4sseldi 3351 . . . . . . 7  |-  ( ( ( ph  /\  J  e.  NN )  /\  z  e.  NN )  ->  ( F `  z )  e.  ( RR  X.  RR ) )
6 1st2nd2 6612 . . . . . . 7  |-  ( ( F `  z )  e.  ( RR  X.  RR )  ->  ( F `
 z )  = 
<. ( 1st `  ( F `  z )
) ,  ( 2nd `  ( F `  z
) ) >. )
75, 6syl 16 . . . . . 6  |-  ( ( ( ph  /\  J  e.  NN )  /\  z  e.  NN )  ->  ( F `  z )  =  <. ( 1st `  ( F `  z )
) ,  ( 2nd `  ( F `  z
) ) >. )
87fveq2d 5692 . . . . 5  |-  ( ( ( ph  /\  J  e.  NN )  /\  z  e.  NN )  ->  ( (,) `  ( F `  z ) )  =  ( (,) `  <. ( 1st `  ( F `
 z ) ) ,  ( 2nd `  ( F `  z )
) >. ) )
9 df-ov 6093 . . . . 5  |-  ( ( 1st `  ( F `
 z ) ) (,) ( 2nd `  ( F `  z )
) )  =  ( (,) `  <. ( 1st `  ( F `  z ) ) ,  ( 2nd `  ( F `  z )
) >. )
108, 9syl6eqr 2491 . . . 4  |-  ( ( ( ph  /\  J  e.  NN )  /\  z  e.  NN )  ->  ( (,) `  ( F `  z ) )  =  ( ( 1st `  ( F `  z )
) (,) ( 2nd `  ( F `  z
) ) ) )
11 uniioombl.g . . . . . . . . . 10  |-  ( ph  ->  G : NN --> (  <_  i^i  ( RR  X.  RR ) ) )
1211ffvelrnda 5840 . . . . . . . . 9  |-  ( (
ph  /\  J  e.  NN )  ->  ( G `
 J )  e.  (  <_  i^i  ( RR  X.  RR ) ) )
131, 12sseldi 3351 . . . . . . . 8  |-  ( (
ph  /\  J  e.  NN )  ->  ( G `
 J )  e.  ( RR  X.  RR ) )
14 1st2nd2 6612 . . . . . . . 8  |-  ( ( G `  J )  e.  ( RR  X.  RR )  ->  ( G `
 J )  = 
<. ( 1st `  ( G `  J )
) ,  ( 2nd `  ( G `  J
) ) >. )
1513, 14syl 16 . . . . . . 7  |-  ( (
ph  /\  J  e.  NN )  ->  ( G `
 J )  = 
<. ( 1st `  ( G `  J )
) ,  ( 2nd `  ( G `  J
) ) >. )
1615fveq2d 5692 . . . . . 6  |-  ( (
ph  /\  J  e.  NN )  ->  ( (,) `  ( G `  J
) )  =  ( (,) `  <. ( 1st `  ( G `  J ) ) ,  ( 2nd `  ( G `  J )
) >. ) )
17 df-ov 6093 . . . . . 6  |-  ( ( 1st `  ( G `
 J ) ) (,) ( 2nd `  ( G `  J )
) )  =  ( (,) `  <. ( 1st `  ( G `  J ) ) ,  ( 2nd `  ( G `  J )
) >. )
1816, 17syl6eqr 2491 . . . . 5  |-  ( (
ph  /\  J  e.  NN )  ->  ( (,) `  ( G `  J
) )  =  ( ( 1st `  ( G `  J )
) (,) ( 2nd `  ( G `  J
) ) ) )
1918adantr 462 . . . 4  |-  ( ( ( ph  /\  J  e.  NN )  /\  z  e.  NN )  ->  ( (,) `  ( G `  J ) )  =  ( ( 1st `  ( G `  J )
) (,) ( 2nd `  ( G `  J
) ) ) )
2010, 19ineq12d 3550 . . 3  |-  ( ( ( ph  /\  J  e.  NN )  /\  z  e.  NN )  ->  (
( (,) `  ( F `  z )
)  i^i  ( (,) `  ( G `  J
) ) )  =  ( ( ( 1st `  ( F `  z
) ) (,) ( 2nd `  ( F `  z ) ) )  i^i  ( ( 1st `  ( G `  J
) ) (,) ( 2nd `  ( G `  J ) ) ) ) )
21 ovolfcl 20909 . . . . . . 7  |-  ( ( F : NN --> (  <_  i^i  ( RR  X.  RR ) )  /\  z  e.  NN )  ->  (
( 1st `  ( F `  z )
)  e.  RR  /\  ( 2nd `  ( F `
 z ) )  e.  RR  /\  ( 1st `  ( F `  z ) )  <_ 
( 2nd `  ( F `  z )
) ) )
223, 21sylan 468 . . . . . 6  |-  ( ( ( ph  /\  J  e.  NN )  /\  z  e.  NN )  ->  (
( 1st `  ( F `  z )
)  e.  RR  /\  ( 2nd `  ( F `
 z ) )  e.  RR  /\  ( 1st `  ( F `  z ) )  <_ 
( 2nd `  ( F `  z )
) ) )
2322simp1d 995 . . . . 5  |-  ( ( ( ph  /\  J  e.  NN )  /\  z  e.  NN )  ->  ( 1st `  ( F `  z ) )  e.  RR )
2423rexrd 9429 . . . 4  |-  ( ( ( ph  /\  J  e.  NN )  /\  z  e.  NN )  ->  ( 1st `  ( F `  z ) )  e. 
RR* )
2522simp2d 996 . . . . 5  |-  ( ( ( ph  /\  J  e.  NN )  /\  z  e.  NN )  ->  ( 2nd `  ( F `  z ) )  e.  RR )
2625rexrd 9429 . . . 4  |-  ( ( ( ph  /\  J  e.  NN )  /\  z  e.  NN )  ->  ( 2nd `  ( F `  z ) )  e. 
RR* )
27 ovolfcl 20909 . . . . . . . 8  |-  ( ( G : NN --> (  <_  i^i  ( RR  X.  RR ) )  /\  J  e.  NN )  ->  (
( 1st `  ( G `  J )
)  e.  RR  /\  ( 2nd `  ( G `
 J ) )  e.  RR  /\  ( 1st `  ( G `  J ) )  <_ 
( 2nd `  ( G `  J )
) ) )
2811, 27sylan 468 . . . . . . 7  |-  ( (
ph  /\  J  e.  NN )  ->  ( ( 1st `  ( G `
 J ) )  e.  RR  /\  ( 2nd `  ( G `  J ) )  e.  RR  /\  ( 1st `  ( G `  J
) )  <_  ( 2nd `  ( G `  J ) ) ) )
2928simp1d 995 . . . . . 6  |-  ( (
ph  /\  J  e.  NN )  ->  ( 1st `  ( G `  J
) )  e.  RR )
3029rexrd 9429 . . . . 5  |-  ( (
ph  /\  J  e.  NN )  ->  ( 1st `  ( G `  J
) )  e.  RR* )
3130adantr 462 . . . 4  |-  ( ( ( ph  /\  J  e.  NN )  /\  z  e.  NN )  ->  ( 1st `  ( G `  J ) )  e. 
RR* )
3228simp2d 996 . . . . . 6  |-  ( (
ph  /\  J  e.  NN )  ->  ( 2nd `  ( G `  J
) )  e.  RR )
3332rexrd 9429 . . . . 5  |-  ( (
ph  /\  J  e.  NN )  ->  ( 2nd `  ( G `  J
) )  e.  RR* )
3433adantr 462 . . . 4  |-  ( ( ( ph  /\  J  e.  NN )  /\  z  e.  NN )  ->  ( 2nd `  ( G `  J ) )  e. 
RR* )
35 iooin 11330 . . . 4  |-  ( ( ( ( 1st `  ( F `  z )
)  e.  RR*  /\  ( 2nd `  ( F `  z ) )  e. 
RR* )  /\  (
( 1st `  ( G `  J )
)  e.  RR*  /\  ( 2nd `  ( G `  J ) )  e. 
RR* ) )  -> 
( ( ( 1st `  ( F `  z
) ) (,) ( 2nd `  ( F `  z ) ) )  i^i  ( ( 1st `  ( G `  J
) ) (,) ( 2nd `  ( G `  J ) ) ) )  =  ( if ( ( 1st `  ( F `  z )
)  <_  ( 1st `  ( G `  J
) ) ,  ( 1st `  ( G `
 J ) ) ,  ( 1st `  ( F `  z )
) ) (,) if ( ( 2nd `  ( F `  z )
)  <_  ( 2nd `  ( G `  J
) ) ,  ( 2nd `  ( F `
 z ) ) ,  ( 2nd `  ( G `  J )
) ) ) )
3624, 26, 31, 34, 35syl22anc 1214 . . 3  |-  ( ( ( ph  /\  J  e.  NN )  /\  z  e.  NN )  ->  (
( ( 1st `  ( F `  z )
) (,) ( 2nd `  ( F `  z
) ) )  i^i  ( ( 1st `  ( G `  J )
) (,) ( 2nd `  ( G `  J
) ) ) )  =  ( if ( ( 1st `  ( F `  z )
)  <_  ( 1st `  ( G `  J
) ) ,  ( 1st `  ( G `
 J ) ) ,  ( 1st `  ( F `  z )
) ) (,) if ( ( 2nd `  ( F `  z )
)  <_  ( 2nd `  ( G `  J
) ) ,  ( 2nd `  ( F `
 z ) ) ,  ( 2nd `  ( G `  J )
) ) ) )
3720, 36eqtrd 2473 . 2  |-  ( ( ( ph  /\  J  e.  NN )  /\  z  e.  NN )  ->  (
( (,) `  ( F `  z )
)  i^i  ( (,) `  ( G `  J
) ) )  =  ( if ( ( 1st `  ( F `
 z ) )  <_  ( 1st `  ( G `  J )
) ,  ( 1st `  ( G `  J
) ) ,  ( 1st `  ( F `
 z ) ) ) (,) if ( ( 2nd `  ( F `  z )
)  <_  ( 2nd `  ( G `  J
) ) ,  ( 2nd `  ( F `
 z ) ) ,  ( 2nd `  ( G `  J )
) ) ) )
38 ioorebas 11387 . 2  |-  ( if ( ( 1st `  ( F `  z )
)  <_  ( 1st `  ( G `  J
) ) ,  ( 1st `  ( G `
 J ) ) ,  ( 1st `  ( F `  z )
) ) (,) if ( ( 2nd `  ( F `  z )
)  <_  ( 2nd `  ( G `  J
) ) ,  ( 2nd `  ( F `
 z ) ) ,  ( 2nd `  ( G `  J )
) ) )  e. 
ran  (,)
3937, 38syl6eqel 2529 1  |-  ( ( ( ph  /\  J  e.  NN )  /\  z  e.  NN )  ->  (
( (,) `  ( F `  z )
)  i^i  ( (,) `  ( G `  J
) ) )  e. 
ran  (,) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    /\ wa 369    /\ w3a 960    = wceq 1364    e. wcel 1761    i^i cin 3324    C_ wss 3325   ifcif 3788   <.cop 3880   U.cuni 4088  Disj wdisj 4259   class class class wbr 4289    X. cxp 4834   ran crn 4837    o. ccom 4840   -->wf 5411   ` cfv 5415  (class class class)co 6090   1stc1st 6574   2ndc2nd 6575   supcsup 7686   RRcr 9277   1c1 9279    + caddc 9281   RR*cxr 9413    < clt 9414    <_ cle 9415    - cmin 9591   NNcn 10318   RR+crp 10987   (,)cioo 11296    seqcseq 11802   abscabs 12719   vol*covol 20905
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1596  ax-4 1607  ax-5 1675  ax-6 1713  ax-7 1733  ax-8 1763  ax-9 1765  ax-10 1780  ax-11 1785  ax-12 1797  ax-13 1948  ax-ext 2422  ax-sep 4410  ax-nul 4418  ax-pow 4467  ax-pr 4528  ax-un 6371  ax-cnex 9334  ax-resscn 9335  ax-1cn 9336  ax-icn 9337  ax-addcl 9338  ax-addrcl 9339  ax-mulcl 9340  ax-mulrcl 9341  ax-mulcom 9342  ax-addass 9343  ax-mulass 9344  ax-distr 9345  ax-i2m1 9346  ax-1ne0 9347  ax-1rid 9348  ax-rnegex 9349  ax-rrecex 9350  ax-cnre 9351  ax-pre-lttri 9352  ax-pre-lttrn 9353  ax-pre-ltadd 9354  ax-pre-mulgt0 9355  ax-pre-sup 9356
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 961  df-3an 962  df-tru 1367  df-ex 1592  df-nf 1595  df-sb 1706  df-eu 2261  df-mo 2262  df-clab 2428  df-cleq 2434  df-clel 2437  df-nfc 2566  df-ne 2606  df-nel 2607  df-ral 2718  df-rex 2719  df-reu 2720  df-rmo 2721  df-rab 2722  df-v 2972  df-sbc 3184  df-csb 3286  df-dif 3328  df-un 3330  df-in 3332  df-ss 3339  df-pss 3341  df-nul 3635  df-if 3789  df-pw 3859  df-sn 3875  df-pr 3877  df-tp 3879  df-op 3881  df-uni 4089  df-iun 4170  df-br 4290  df-opab 4348  df-mpt 4349  df-tr 4383  df-eprel 4628  df-id 4632  df-po 4637  df-so 4638  df-fr 4675  df-we 4677  df-ord 4718  df-on 4719  df-lim 4720  df-suc 4721  df-xp 4842  df-rel 4843  df-cnv 4844  df-co 4845  df-dm 4846  df-rn 4847  df-res 4848  df-ima 4849  df-iota 5378  df-fun 5417  df-fn 5418  df-f 5419  df-f1 5420  df-fo 5421  df-f1o 5422  df-fv 5423  df-riota 6049  df-ov 6093  df-oprab 6094  df-mpt2 6095  df-om 6476  df-1st 6576  df-2nd 6577  df-recs 6828  df-rdg 6862  df-er 7097  df-en 7307  df-dom 7308  df-sdom 7309  df-sup 7687  df-pnf 9416  df-mnf 9417  df-xr 9418  df-ltxr 9419  df-le 9420  df-sub 9593  df-neg 9594  df-div 9990  df-nn 10319  df-n0 10576  df-z 10643  df-uz 10858  df-q 10950  df-ioo 11300
This theorem is referenced by:  uniioombllem2  21022
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