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Theorem uniioombllem2a 18953
Description: Lemma for uniioombl 18960. (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 3403 . . . . . . . 8  |-  (  <_  i^i  ( RR  X.  RR ) )  C_  ( RR  X.  RR )
2 uniioombl.1 . . . . . . . . . 10  |-  ( ph  ->  F : NN --> (  <_  i^i  ( RR  X.  RR ) ) )
32adantr 451 . . . . . . . . 9  |-  ( (
ph  /\  J  e.  NN )  ->  F : NN
--> (  <_  i^i  ( RR  X.  RR ) ) )
4 ffvelrn 5679 . . . . . . . . 9  |-  ( ( F : NN --> (  <_  i^i  ( RR  X.  RR ) )  /\  z  e.  NN )  ->  ( F `  z )  e.  (  <_  i^i  ( RR  X.  RR ) ) )
53, 4sylan 457 . . . . . . . 8  |-  ( ( ( ph  /\  J  e.  NN )  /\  z  e.  NN )  ->  ( F `  z )  e.  (  <_  i^i  ( RR  X.  RR ) ) )
61, 5sseldi 3191 . . . . . . 7  |-  ( ( ( ph  /\  J  e.  NN )  /\  z  e.  NN )  ->  ( F `  z )  e.  ( RR  X.  RR ) )
7 1st2nd2 6175 . . . . . . 7  |-  ( ( F `  z )  e.  ( RR  X.  RR )  ->  ( F `
 z )  = 
<. ( 1st `  ( F `  z )
) ,  ( 2nd `  ( F `  z
) ) >. )
86, 7syl 15 . . . . . 6  |-  ( ( ( ph  /\  J  e.  NN )  /\  z  e.  NN )  ->  ( F `  z )  =  <. ( 1st `  ( F `  z )
) ,  ( 2nd `  ( F `  z
) ) >. )
98fveq2d 5545 . . . . 5  |-  ( ( ( ph  /\  J  e.  NN )  /\  z  e.  NN )  ->  ( (,) `  ( F `  z ) )  =  ( (,) `  <. ( 1st `  ( F `
 z ) ) ,  ( 2nd `  ( F `  z )
) >. ) )
10 df-ov 5877 . . . . 5  |-  ( ( 1st `  ( F `
 z ) ) (,) ( 2nd `  ( F `  z )
) )  =  ( (,) `  <. ( 1st `  ( F `  z ) ) ,  ( 2nd `  ( F `  z )
) >. )
119, 10syl6eqr 2346 . . . 4  |-  ( ( ( ph  /\  J  e.  NN )  /\  z  e.  NN )  ->  ( (,) `  ( F `  z ) )  =  ( ( 1st `  ( F `  z )
) (,) ( 2nd `  ( F `  z
) ) ) )
12 uniioombl.g . . . . . . . . . 10  |-  ( ph  ->  G : NN --> (  <_  i^i  ( RR  X.  RR ) ) )
13 ffvelrn 5679 . . . . . . . . . 10  |-  ( ( G : NN --> (  <_  i^i  ( RR  X.  RR ) )  /\  J  e.  NN )  ->  ( G `  J )  e.  (  <_  i^i  ( RR  X.  RR ) ) )
1412, 13sylan 457 . . . . . . . . 9  |-  ( (
ph  /\  J  e.  NN )  ->  ( G `
 J )  e.  (  <_  i^i  ( RR  X.  RR ) ) )
151, 14sseldi 3191 . . . . . . . 8  |-  ( (
ph  /\  J  e.  NN )  ->  ( G `
 J )  e.  ( RR  X.  RR ) )
16 1st2nd2 6175 . . . . . . . 8  |-  ( ( G `  J )  e.  ( RR  X.  RR )  ->  ( G `
 J )  = 
<. ( 1st `  ( G `  J )
) ,  ( 2nd `  ( G `  J
) ) >. )
1715, 16syl 15 . . . . . . 7  |-  ( (
ph  /\  J  e.  NN )  ->  ( G `
 J )  = 
<. ( 1st `  ( G `  J )
) ,  ( 2nd `  ( G `  J
) ) >. )
1817fveq2d 5545 . . . . . 6  |-  ( (
ph  /\  J  e.  NN )  ->  ( (,) `  ( G `  J
) )  =  ( (,) `  <. ( 1st `  ( G `  J ) ) ,  ( 2nd `  ( G `  J )
) >. ) )
19 df-ov 5877 . . . . . 6  |-  ( ( 1st `  ( G `
 J ) ) (,) ( 2nd `  ( G `  J )
) )  =  ( (,) `  <. ( 1st `  ( G `  J ) ) ,  ( 2nd `  ( G `  J )
) >. )
2018, 19syl6eqr 2346 . . . . 5  |-  ( (
ph  /\  J  e.  NN )  ->  ( (,) `  ( G `  J
) )  =  ( ( 1st `  ( G `  J )
) (,) ( 2nd `  ( G `  J
) ) ) )
2120adantr 451 . . . 4  |-  ( ( ( ph  /\  J  e.  NN )  /\  z  e.  NN )  ->  ( (,) `  ( G `  J ) )  =  ( ( 1st `  ( G `  J )
) (,) ( 2nd `  ( G `  J
) ) ) )
2211, 21ineq12d 3384 . . 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 ) ) ) ) )
23 ovolfcl 18842 . . . . . . 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 )
) ) )
243, 23sylan 457 . . . . . 6  |-  ( ( ( ph  /\  J  e.  NN )  /\  z  e.  NN )  ->  (
( 1st `  ( F `  z )
)  e.  RR  /\  ( 2nd `  ( F `
 z ) )  e.  RR  /\  ( 1st `  ( F `  z ) )  <_ 
( 2nd `  ( F `  z )
) ) )
2524simp1d 967 . . . . 5  |-  ( ( ( ph  /\  J  e.  NN )  /\  z  e.  NN )  ->  ( 1st `  ( F `  z ) )  e.  RR )
2625rexrd 8897 . . . 4  |-  ( ( ( ph  /\  J  e.  NN )  /\  z  e.  NN )  ->  ( 1st `  ( F `  z ) )  e. 
RR* )
2724simp2d 968 . . . . 5  |-  ( ( ( ph  /\  J  e.  NN )  /\  z  e.  NN )  ->  ( 2nd `  ( F `  z ) )  e.  RR )
2827rexrd 8897 . . . 4  |-  ( ( ( ph  /\  J  e.  NN )  /\  z  e.  NN )  ->  ( 2nd `  ( F `  z ) )  e. 
RR* )
29 ovolfcl 18842 . . . . . . . 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 )
) ) )
3012, 29sylan 457 . . . . . . 7  |-  ( (
ph  /\  J  e.  NN )  ->  ( ( 1st `  ( G `
 J ) )  e.  RR  /\  ( 2nd `  ( G `  J ) )  e.  RR  /\  ( 1st `  ( G `  J
) )  <_  ( 2nd `  ( G `  J ) ) ) )
3130simp1d 967 . . . . . 6  |-  ( (
ph  /\  J  e.  NN )  ->  ( 1st `  ( G `  J
) )  e.  RR )
3231rexrd 8897 . . . . 5  |-  ( (
ph  /\  J  e.  NN )  ->  ( 1st `  ( G `  J
) )  e.  RR* )
3332adantr 451 . . . 4  |-  ( ( ( ph  /\  J  e.  NN )  /\  z  e.  NN )  ->  ( 1st `  ( G `  J ) )  e. 
RR* )
3430simp2d 968 . . . . . 6  |-  ( (
ph  /\  J  e.  NN )  ->  ( 2nd `  ( G `  J
) )  e.  RR )
3534rexrd 8897 . . . . 5  |-  ( (
ph  /\  J  e.  NN )  ->  ( 2nd `  ( G `  J
) )  e.  RR* )
3635adantr 451 . . . 4  |-  ( ( ( ph  /\  J  e.  NN )  /\  z  e.  NN )  ->  ( 2nd `  ( G `  J ) )  e. 
RR* )
37 iooin 10706 . . . 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 )
) ) ) )
3826, 28, 33, 36, 37syl22anc 1183 . . 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 )
) ) ) )
3922, 38eqtrd 2328 . 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 )
) ) ) )
40 ioorebas 10761 . 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  (,)
4139, 40syl6eqel 2384 1  |-  ( ( ( ph  /\  J  e.  NN )  /\  z  e.  NN )  ->  (
( (,) `  ( F `  z )
)  i^i  ( (,) `  ( G `  J
) ) )  e. 
ran  (,) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 358    /\ w3a 934    = wceq 1632    e. wcel 1696    i^i cin 3164    C_ wss 3165   ifcif 3578   <.cop 3656   U.cuni 3843  Disj wdisj 4009   class class class wbr 4039    X. cxp 4703   ran crn 4706    o. ccom 4709   -->wf 5267   ` cfv 5271  (class class class)co 5874   1stc1st 6136   2ndc2nd 6137   supcsup 7209   RRcr 8752   1c1 8754    + caddc 8756   RR*cxr 8882    < clt 8883    <_ cle 8884    - cmin 9053   NNcn 9762   RR+crp 10370   (,)cioo 10672    seq cseq 11062   abscabs 11735   vol
*covol 18838
This theorem is referenced by:  uniioombllem2  18954
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1536  ax-5 1547  ax-17 1606  ax-9 1644  ax-8 1661  ax-13 1698  ax-14 1700  ax-6 1715  ax-7 1720  ax-11 1727  ax-12 1878  ax-ext 2277  ax-sep 4157  ax-nul 4165  ax-pow 4204  ax-pr 4230  ax-un 4528  ax-cnex 8809  ax-resscn 8810  ax-1cn 8811  ax-icn 8812  ax-addcl 8813  ax-addrcl 8814  ax-mulcl 8815  ax-mulrcl 8816  ax-mulcom 8817  ax-addass 8818  ax-mulass 8819  ax-distr 8820  ax-i2m1 8821  ax-1ne0 8822  ax-1rid 8823  ax-rnegex 8824  ax-rrecex 8825  ax-cnre 8826  ax-pre-lttri 8827  ax-pre-lttrn 8828  ax-pre-ltadd 8829  ax-pre-mulgt0 8830  ax-pre-sup 8831
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3or 935  df-3an 936  df-tru 1310  df-ex 1532  df-nf 1535  df-sb 1639  df-eu 2160  df-mo 2161  df-clab 2283  df-cleq 2289  df-clel 2292  df-nfc 2421  df-ne 2461  df-nel 2462  df-ral 2561  df-rex 2562  df-reu 2563  df-rmo 2564  df-rab 2565  df-v 2803  df-sbc 3005  df-csb 3095  df-dif 3168  df-un 3170  df-in 3172  df-ss 3179  df-pss 3181  df-nul 3469  df-if 3579  df-pw 3640  df-sn 3659  df-pr 3660  df-tp 3661  df-op 3662  df-uni 3844  df-iun 3923  df-br 4040  df-opab 4094  df-mpt 4095  df-tr 4130  df-eprel 4321  df-id 4325  df-po 4330  df-so 4331  df-fr 4368  df-we 4370  df-ord 4411  df-on 4412  df-lim 4413  df-suc 4414  df-om 4673  df-xp 4711  df-rel 4712  df-cnv 4713  df-co 4714  df-dm 4715  df-rn 4716  df-res 4717  df-ima 4718  df-iota 5235  df-fun 5273  df-fn 5274  df-f 5275  df-f1 5276  df-fo 5277  df-f1o 5278  df-fv 5279  df-ov 5877  df-oprab 5878  df-mpt2 5879  df-1st 6138  df-2nd 6139  df-riota 6320  df-recs 6404  df-rdg 6439  df-er 6676  df-en 6880  df-dom 6881  df-sdom 6882  df-sup 7210  df-pnf 8885  df-mnf 8886  df-xr 8887  df-ltxr 8888  df-le 8889  df-sub 9055  df-neg 9056  df-div 9440  df-nn 9763  df-n0 9982  df-z 10041  df-uz 10247  df-q 10333  df-ioo 10676
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