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Theorem uniioombllem2a 21864
Description: Lemma for uniioombl 21871. (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 3704 . . . . . . . 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 6016 . . . . . . . 8  |-  ( ( ( ph  /\  J  e.  NN )  /\  z  e.  NN )  ->  ( F `  z )  e.  (  <_  i^i  ( RR  X.  RR ) ) )
51, 4sseldi 3487 . . . . . . 7  |-  ( ( ( ph  /\  J  e.  NN )  /\  z  e.  NN )  ->  ( F `  z )  e.  ( RR  X.  RR ) )
6 1st2nd2 6822 . . . . . . 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 5860 . . . . 5  |-  ( ( ( ph  /\  J  e.  NN )  /\  z  e.  NN )  ->  ( (,) `  ( F `  z ) )  =  ( (,) `  <. ( 1st `  ( F `
 z ) ) ,  ( 2nd `  ( F `  z )
) >. ) )
9 df-ov 6284 . . . . 5  |-  ( ( 1st `  ( F `
 z ) ) (,) ( 2nd `  ( F `  z )
) )  =  ( (,) `  <. ( 1st `  ( F `  z ) ) ,  ( 2nd `  ( F `  z )
) >. )
108, 9syl6eqr 2502 . . . 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 6016 . . . . . . . . 9  |-  ( (
ph  /\  J  e.  NN )  ->  ( G `
 J )  e.  (  <_  i^i  ( RR  X.  RR ) ) )
131, 12sseldi 3487 . . . . . . . 8  |-  ( (
ph  /\  J  e.  NN )  ->  ( G `
 J )  e.  ( RR  X.  RR ) )
14 1st2nd2 6822 . . . . . . . 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 5860 . . . . . 6  |-  ( (
ph  /\  J  e.  NN )  ->  ( (,) `  ( G `  J
) )  =  ( (,) `  <. ( 1st `  ( G `  J ) ) ,  ( 2nd `  ( G `  J )
) >. ) )
17 df-ov 6284 . . . . . 6  |-  ( ( 1st `  ( G `
 J ) ) (,) ( 2nd `  ( G `  J )
) )  =  ( (,) `  <. ( 1st `  ( G `  J ) ) ,  ( 2nd `  ( G `  J )
) >. )
1816, 17syl6eqr 2502 . . . . 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 3686 . . 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 21751 . . . . . . 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 1009 . . . . 5  |-  ( ( ( ph  /\  J  e.  NN )  /\  z  e.  NN )  ->  ( 1st `  ( F `  z ) )  e.  RR )
2423rexrd 9646 . . . 4  |-  ( ( ( ph  /\  J  e.  NN )  /\  z  e.  NN )  ->  ( 1st `  ( F `  z ) )  e. 
RR* )
2522simp2d 1010 . . . . 5  |-  ( ( ( ph  /\  J  e.  NN )  /\  z  e.  NN )  ->  ( 2nd `  ( F `  z ) )  e.  RR )
2625rexrd 9646 . . . 4  |-  ( ( ( ph  /\  J  e.  NN )  /\  z  e.  NN )  ->  ( 2nd `  ( F `  z ) )  e. 
RR* )
27 ovolfcl 21751 . . . . . . . 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 1009 . . . . . 6  |-  ( (
ph  /\  J  e.  NN )  ->  ( 1st `  ( G `  J
) )  e.  RR )
3029rexrd 9646 . . . . 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 1010 . . . . . 6  |-  ( (
ph  /\  J  e.  NN )  ->  ( 2nd `  ( G `  J
) )  e.  RR )
3332rexrd 9646 . . . . 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 11572 . . . 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 1230 . . 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 2484 . 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 11635 . 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 2539 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 974    = wceq 1383    e. wcel 1804    i^i cin 3460    C_ wss 3461   ifcif 3926   <.cop 4020   U.cuni 4234  Disj wdisj 4407   class class class wbr 4437    X. cxp 4987   ran crn 4990    o. ccom 4993   -->wf 5574   ` cfv 5578  (class class class)co 6281   1stc1st 6783   2ndc2nd 6784   supcsup 7902   RRcr 9494   1c1 9496    + caddc 9498   RR*cxr 9630    < clt 9631    <_ cle 9632    - cmin 9810   NNcn 10542   RR+crp 11229   (,)cioo 11538    seqcseq 12086   abscabs 13046   vol*covol 21747
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1605  ax-4 1618  ax-5 1691  ax-6 1734  ax-7 1776  ax-8 1806  ax-9 1808  ax-10 1823  ax-11 1828  ax-12 1840  ax-13 1985  ax-ext 2421  ax-sep 4558  ax-nul 4566  ax-pow 4615  ax-pr 4676  ax-un 6577  ax-cnex 9551  ax-resscn 9552  ax-1cn 9553  ax-icn 9554  ax-addcl 9555  ax-addrcl 9556  ax-mulcl 9557  ax-mulrcl 9558  ax-mulcom 9559  ax-addass 9560  ax-mulass 9561  ax-distr 9562  ax-i2m1 9563  ax-1ne0 9564  ax-1rid 9565  ax-rnegex 9566  ax-rrecex 9567  ax-cnre 9568  ax-pre-lttri 9569  ax-pre-lttrn 9570  ax-pre-ltadd 9571  ax-pre-mulgt0 9572  ax-pre-sup 9573
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 975  df-3an 976  df-tru 1386  df-ex 1600  df-nf 1604  df-sb 1727  df-eu 2272  df-mo 2273  df-clab 2429  df-cleq 2435  df-clel 2438  df-nfc 2593  df-ne 2640  df-nel 2641  df-ral 2798  df-rex 2799  df-reu 2800  df-rmo 2801  df-rab 2802  df-v 3097  df-sbc 3314  df-csb 3421  df-dif 3464  df-un 3466  df-in 3468  df-ss 3475  df-pss 3477  df-nul 3771  df-if 3927  df-pw 3999  df-sn 4015  df-pr 4017  df-tp 4019  df-op 4021  df-uni 4235  df-iun 4317  df-br 4438  df-opab 4496  df-mpt 4497  df-tr 4531  df-eprel 4781  df-id 4785  df-po 4790  df-so 4791  df-fr 4828  df-we 4830  df-ord 4871  df-on 4872  df-lim 4873  df-suc 4874  df-xp 4995  df-rel 4996  df-cnv 4997  df-co 4998  df-dm 4999  df-rn 5000  df-res 5001  df-ima 5002  df-iota 5541  df-fun 5580  df-fn 5581  df-f 5582  df-f1 5583  df-fo 5584  df-f1o 5585  df-fv 5586  df-riota 6242  df-ov 6284  df-oprab 6285  df-mpt2 6286  df-om 6686  df-1st 6785  df-2nd 6786  df-recs 7044  df-rdg 7078  df-er 7313  df-en 7519  df-dom 7520  df-sdom 7521  df-sup 7903  df-pnf 9633  df-mnf 9634  df-xr 9635  df-ltxr 9636  df-le 9637  df-sub 9812  df-neg 9813  df-div 10213  df-nn 10543  df-n0 10802  df-z 10871  df-uz 11091  df-q 11192  df-ioo 11542
This theorem is referenced by:  uniioombllem2  21865
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