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Theorem uniioombllem2a 21726
Description: Lemma for uniioombl 21733. (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 3719 . . . . . . . 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 6019 . . . . . . . 8  |-  ( ( ( ph  /\  J  e.  NN )  /\  z  e.  NN )  ->  ( F `  z )  e.  (  <_  i^i  ( RR  X.  RR ) ) )
51, 4sseldi 3502 . . . . . . 7  |-  ( ( ( ph  /\  J  e.  NN )  /\  z  e.  NN )  ->  ( F `  z )  e.  ( RR  X.  RR ) )
6 1st2nd2 6818 . . . . . . 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 5868 . . . . 5  |-  ( ( ( ph  /\  J  e.  NN )  /\  z  e.  NN )  ->  ( (,) `  ( F `  z ) )  =  ( (,) `  <. ( 1st `  ( F `
 z ) ) ,  ( 2nd `  ( F `  z )
) >. ) )
9 df-ov 6285 . . . . 5  |-  ( ( 1st `  ( F `
 z ) ) (,) ( 2nd `  ( F `  z )
) )  =  ( (,) `  <. ( 1st `  ( F `  z ) ) ,  ( 2nd `  ( F `  z )
) >. )
108, 9syl6eqr 2526 . . . 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 6019 . . . . . . . . 9  |-  ( (
ph  /\  J  e.  NN )  ->  ( G `
 J )  e.  (  <_  i^i  ( RR  X.  RR ) ) )
131, 12sseldi 3502 . . . . . . . 8  |-  ( (
ph  /\  J  e.  NN )  ->  ( G `
 J )  e.  ( RR  X.  RR ) )
14 1st2nd2 6818 . . . . . . . 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 5868 . . . . . 6  |-  ( (
ph  /\  J  e.  NN )  ->  ( (,) `  ( G `  J
) )  =  ( (,) `  <. ( 1st `  ( G `  J ) ) ,  ( 2nd `  ( G `  J )
) >. ) )
17 df-ov 6285 . . . . . 6  |-  ( ( 1st `  ( G `
 J ) ) (,) ( 2nd `  ( G `  J )
) )  =  ( (,) `  <. ( 1st `  ( G `  J ) ) ,  ( 2nd `  ( G `  J )
) >. )
1816, 17syl6eqr 2526 . . . . 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 3701 . . 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 21613 . . . . . . 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 1008 . . . . 5  |-  ( ( ( ph  /\  J  e.  NN )  /\  z  e.  NN )  ->  ( 1st `  ( F `  z ) )  e.  RR )
2423rexrd 9639 . . . 4  |-  ( ( ( ph  /\  J  e.  NN )  /\  z  e.  NN )  ->  ( 1st `  ( F `  z ) )  e. 
RR* )
2522simp2d 1009 . . . . 5  |-  ( ( ( ph  /\  J  e.  NN )  /\  z  e.  NN )  ->  ( 2nd `  ( F `  z ) )  e.  RR )
2625rexrd 9639 . . . 4  |-  ( ( ( ph  /\  J  e.  NN )  /\  z  e.  NN )  ->  ( 2nd `  ( F `  z ) )  e. 
RR* )
27 ovolfcl 21613 . . . . . . . 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 1008 . . . . . 6  |-  ( (
ph  /\  J  e.  NN )  ->  ( 1st `  ( G `  J
) )  e.  RR )
3029rexrd 9639 . . . . 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 1009 . . . . . 6  |-  ( (
ph  /\  J  e.  NN )  ->  ( 2nd `  ( G `  J
) )  e.  RR )
3332rexrd 9639 . . . . 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 11559 . . . 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 1229 . . 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 2508 . 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 11622 . 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 2563 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 973    = wceq 1379    e. wcel 1767    i^i cin 3475    C_ wss 3476   ifcif 3939   <.cop 4033   U.cuni 4245  Disj wdisj 4417   class class class wbr 4447    X. cxp 4997   ran crn 5000    o. ccom 5003   -->wf 5582   ` cfv 5586  (class class class)co 6282   1stc1st 6779   2ndc2nd 6780   supcsup 7896   RRcr 9487   1c1 9489    + caddc 9491   RR*cxr 9623    < clt 9624    <_ cle 9625    - cmin 9801   NNcn 10532   RR+crp 11216   (,)cioo 11525    seqcseq 12071   abscabs 13026   vol*covol 21609
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1601  ax-4 1612  ax-5 1680  ax-6 1719  ax-7 1739  ax-8 1769  ax-9 1771  ax-10 1786  ax-11 1791  ax-12 1803  ax-13 1968  ax-ext 2445  ax-sep 4568  ax-nul 4576  ax-pow 4625  ax-pr 4686  ax-un 6574  ax-cnex 9544  ax-resscn 9545  ax-1cn 9546  ax-icn 9547  ax-addcl 9548  ax-addrcl 9549  ax-mulcl 9550  ax-mulrcl 9551  ax-mulcom 9552  ax-addass 9553  ax-mulass 9554  ax-distr 9555  ax-i2m1 9556  ax-1ne0 9557  ax-1rid 9558  ax-rnegex 9559  ax-rrecex 9560  ax-cnre 9561  ax-pre-lttri 9562  ax-pre-lttrn 9563  ax-pre-ltadd 9564  ax-pre-mulgt0 9565  ax-pre-sup 9566
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 974  df-3an 975  df-tru 1382  df-ex 1597  df-nf 1600  df-sb 1712  df-eu 2279  df-mo 2280  df-clab 2453  df-cleq 2459  df-clel 2462  df-nfc 2617  df-ne 2664  df-nel 2665  df-ral 2819  df-rex 2820  df-reu 2821  df-rmo 2822  df-rab 2823  df-v 3115  df-sbc 3332  df-csb 3436  df-dif 3479  df-un 3481  df-in 3483  df-ss 3490  df-pss 3492  df-nul 3786  df-if 3940  df-pw 4012  df-sn 4028  df-pr 4030  df-tp 4032  df-op 4034  df-uni 4246  df-iun 4327  df-br 4448  df-opab 4506  df-mpt 4507  df-tr 4541  df-eprel 4791  df-id 4795  df-po 4800  df-so 4801  df-fr 4838  df-we 4840  df-ord 4881  df-on 4882  df-lim 4883  df-suc 4884  df-xp 5005  df-rel 5006  df-cnv 5007  df-co 5008  df-dm 5009  df-rn 5010  df-res 5011  df-ima 5012  df-iota 5549  df-fun 5588  df-fn 5589  df-f 5590  df-f1 5591  df-fo 5592  df-f1o 5593  df-fv 5594  df-riota 6243  df-ov 6285  df-oprab 6286  df-mpt2 6287  df-om 6679  df-1st 6781  df-2nd 6782  df-recs 7039  df-rdg 7073  df-er 7308  df-en 7514  df-dom 7515  df-sdom 7516  df-sup 7897  df-pnf 9626  df-mnf 9627  df-xr 9628  df-ltxr 9629  df-le 9630  df-sub 9803  df-neg 9804  df-div 10203  df-nn 10533  df-n0 10792  df-z 10861  df-uz 11079  df-q 11179  df-ioo 11529
This theorem is referenced by:  uniioombllem2  21727
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