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Theorem uniioombllem2a 21077
Description: Lemma for uniioombl 21084. (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 3586 . . . . . . . 8  |-  (  <_  i^i  ( RR  X.  RR ) )  C_  ( RR  X.  RR )
2 uniioombl.1 . . . . . . . . . 10  |-  ( ph  ->  F : NN --> (  <_  i^i  ( RR  X.  RR ) ) )
32adantr 465 . . . . . . . . 9  |-  ( (
ph  /\  J  e.  NN )  ->  F : NN
--> (  <_  i^i  ( RR  X.  RR ) ) )
43ffvelrnda 5858 . . . . . . . 8  |-  ( ( ( ph  /\  J  e.  NN )  /\  z  e.  NN )  ->  ( F `  z )  e.  (  <_  i^i  ( RR  X.  RR ) ) )
51, 4sseldi 3369 . . . . . . 7  |-  ( ( ( ph  /\  J  e.  NN )  /\  z  e.  NN )  ->  ( F `  z )  e.  ( RR  X.  RR ) )
6 1st2nd2 6628 . . . . . . 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 5710 . . . . 5  |-  ( ( ( ph  /\  J  e.  NN )  /\  z  e.  NN )  ->  ( (,) `  ( F `  z ) )  =  ( (,) `  <. ( 1st `  ( F `
 z ) ) ,  ( 2nd `  ( F `  z )
) >. ) )
9 df-ov 6109 . . . . 5  |-  ( ( 1st `  ( F `
 z ) ) (,) ( 2nd `  ( F `  z )
) )  =  ( (,) `  <. ( 1st `  ( F `  z ) ) ,  ( 2nd `  ( F `  z )
) >. )
108, 9syl6eqr 2493 . . . 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 5858 . . . . . . . . 9  |-  ( (
ph  /\  J  e.  NN )  ->  ( G `
 J )  e.  (  <_  i^i  ( RR  X.  RR ) ) )
131, 12sseldi 3369 . . . . . . . 8  |-  ( (
ph  /\  J  e.  NN )  ->  ( G `
 J )  e.  ( RR  X.  RR ) )
14 1st2nd2 6628 . . . . . . . 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 5710 . . . . . 6  |-  ( (
ph  /\  J  e.  NN )  ->  ( (,) `  ( G `  J
) )  =  ( (,) `  <. ( 1st `  ( G `  J ) ) ,  ( 2nd `  ( G `  J )
) >. ) )
17 df-ov 6109 . . . . . 6  |-  ( ( 1st `  ( G `
 J ) ) (,) ( 2nd `  ( G `  J )
) )  =  ( (,) `  <. ( 1st `  ( G `  J ) ) ,  ( 2nd `  ( G `  J )
) >. )
1816, 17syl6eqr 2493 . . . . 5  |-  ( (
ph  /\  J  e.  NN )  ->  ( (,) `  ( G `  J
) )  =  ( ( 1st `  ( G `  J )
) (,) ( 2nd `  ( G `  J
) ) ) )
1918adantr 465 . . . 4  |-  ( ( ( ph  /\  J  e.  NN )  /\  z  e.  NN )  ->  ( (,) `  ( G `  J ) )  =  ( ( 1st `  ( G `  J )
) (,) ( 2nd `  ( G `  J
) ) ) )
2010, 19ineq12d 3568 . . 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 20965 . . . . . . 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 471 . . . . . 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 1000 . . . . 5  |-  ( ( ( ph  /\  J  e.  NN )  /\  z  e.  NN )  ->  ( 1st `  ( F `  z ) )  e.  RR )
2423rexrd 9448 . . . 4  |-  ( ( ( ph  /\  J  e.  NN )  /\  z  e.  NN )  ->  ( 1st `  ( F `  z ) )  e. 
RR* )
2522simp2d 1001 . . . . 5  |-  ( ( ( ph  /\  J  e.  NN )  /\  z  e.  NN )  ->  ( 2nd `  ( F `  z ) )  e.  RR )
2625rexrd 9448 . . . 4  |-  ( ( ( ph  /\  J  e.  NN )  /\  z  e.  NN )  ->  ( 2nd `  ( F `  z ) )  e. 
RR* )
27 ovolfcl 20965 . . . . . . . 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 471 . . . . . . 7  |-  ( (
ph  /\  J  e.  NN )  ->  ( ( 1st `  ( G `
 J ) )  e.  RR  /\  ( 2nd `  ( G `  J ) )  e.  RR  /\  ( 1st `  ( G `  J
) )  <_  ( 2nd `  ( G `  J ) ) ) )
2928simp1d 1000 . . . . . 6  |-  ( (
ph  /\  J  e.  NN )  ->  ( 1st `  ( G `  J
) )  e.  RR )
3029rexrd 9448 . . . . 5  |-  ( (
ph  /\  J  e.  NN )  ->  ( 1st `  ( G `  J
) )  e.  RR* )
3130adantr 465 . . . 4  |-  ( ( ( ph  /\  J  e.  NN )  /\  z  e.  NN )  ->  ( 1st `  ( G `  J ) )  e. 
RR* )
3228simp2d 1001 . . . . . 6  |-  ( (
ph  /\  J  e.  NN )  ->  ( 2nd `  ( G `  J
) )  e.  RR )
3332rexrd 9448 . . . . 5  |-  ( (
ph  /\  J  e.  NN )  ->  ( 2nd `  ( G `  J
) )  e.  RR* )
3433adantr 465 . . . 4  |-  ( ( ( ph  /\  J  e.  NN )  /\  z  e.  NN )  ->  ( 2nd `  ( G `  J ) )  e. 
RR* )
35 iooin 11349 . . . 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 1219 . . 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 2475 . 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 11406 . 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 2531 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 965    = wceq 1369    e. wcel 1756    i^i cin 3342    C_ wss 3343   ifcif 3806   <.cop 3898   U.cuni 4106  Disj wdisj 4277   class class class wbr 4307    X. cxp 4853   ran crn 4856    o. ccom 4859   -->wf 5429   ` cfv 5433  (class class class)co 6106   1stc1st 6590   2ndc2nd 6591   supcsup 7705   RRcr 9296   1c1 9298    + caddc 9300   RR*cxr 9432    < clt 9433    <_ cle 9434    - cmin 9610   NNcn 10337   RR+crp 11006   (,)cioo 11315    seqcseq 11821   abscabs 12738   vol*covol 20961
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1591  ax-4 1602  ax-5 1670  ax-6 1708  ax-7 1728  ax-8 1758  ax-9 1760  ax-10 1775  ax-11 1780  ax-12 1792  ax-13 1943  ax-ext 2423  ax-sep 4428  ax-nul 4436  ax-pow 4485  ax-pr 4546  ax-un 6387  ax-cnex 9353  ax-resscn 9354  ax-1cn 9355  ax-icn 9356  ax-addcl 9357  ax-addrcl 9358  ax-mulcl 9359  ax-mulrcl 9360  ax-mulcom 9361  ax-addass 9362  ax-mulass 9363  ax-distr 9364  ax-i2m1 9365  ax-1ne0 9366  ax-1rid 9367  ax-rnegex 9368  ax-rrecex 9369  ax-cnre 9370  ax-pre-lttri 9371  ax-pre-lttrn 9372  ax-pre-ltadd 9373  ax-pre-mulgt0 9374  ax-pre-sup 9375
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 966  df-3an 967  df-tru 1372  df-ex 1587  df-nf 1590  df-sb 1701  df-eu 2257  df-mo 2258  df-clab 2430  df-cleq 2436  df-clel 2439  df-nfc 2577  df-ne 2622  df-nel 2623  df-ral 2735  df-rex 2736  df-reu 2737  df-rmo 2738  df-rab 2739  df-v 2989  df-sbc 3202  df-csb 3304  df-dif 3346  df-un 3348  df-in 3350  df-ss 3357  df-pss 3359  df-nul 3653  df-if 3807  df-pw 3877  df-sn 3893  df-pr 3895  df-tp 3897  df-op 3899  df-uni 4107  df-iun 4188  df-br 4308  df-opab 4366  df-mpt 4367  df-tr 4401  df-eprel 4647  df-id 4651  df-po 4656  df-so 4657  df-fr 4694  df-we 4696  df-ord 4737  df-on 4738  df-lim 4739  df-suc 4740  df-xp 4861  df-rel 4862  df-cnv 4863  df-co 4864  df-dm 4865  df-rn 4866  df-res 4867  df-ima 4868  df-iota 5396  df-fun 5435  df-fn 5436  df-f 5437  df-f1 5438  df-fo 5439  df-f1o 5440  df-fv 5441  df-riota 6067  df-ov 6109  df-oprab 6110  df-mpt2 6111  df-om 6492  df-1st 6592  df-2nd 6593  df-recs 6847  df-rdg 6881  df-er 7116  df-en 7326  df-dom 7327  df-sdom 7328  df-sup 7706  df-pnf 9435  df-mnf 9436  df-xr 9437  df-ltxr 9438  df-le 9439  df-sub 9612  df-neg 9613  df-div 10009  df-nn 10338  df-n0 10595  df-z 10662  df-uz 10877  df-q 10969  df-ioo 11319
This theorem is referenced by:  uniioombllem2  21078
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