MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  itg1addlem2 Unicode version

Theorem itg1addlem2 19068
Description: Lemma for itg1add 19072. The function  I represents the pieces into which we will break up the domain of the sum. Since it is infinite only when both  i and  j are zero, we arbitrarily define it to be zero there to simplify the sums that are manipulated in itg1addlem4 19070 and itg1addlem5 19071. (Contributed by Mario Carneiro, 26-Jun-2014.)
Hypotheses
Ref Expression
i1fadd.1  |-  ( ph  ->  F  e.  dom  S.1 )
i1fadd.2  |-  ( ph  ->  G  e.  dom  S.1 )
itg1add.3  |-  I  =  ( i  e.  RR ,  j  e.  RR  |->  if ( ( i  =  0  /\  j  =  0 ) ,  0 ,  ( vol `  (
( `' F " { i } )  i^i  ( `' G " { j } ) ) ) ) )
Assertion
Ref Expression
itg1addlem2  |-  ( ph  ->  I : ( RR 
X.  RR ) --> RR )
Distinct variable groups:    i, j, F    i, G, j    ph, i,
j
Allowed substitution hints:    I( i, j)

Proof of Theorem itg1addlem2
StepHypRef Expression
1 iffalse 3585 . . . . . . . 8  |-  ( -.  ( i  =  0  /\  j  =  0 )  ->  if (
( i  =  0  /\  j  =  0 ) ,  0 ,  ( vol `  (
( `' F " { i } )  i^i  ( `' G " { j } ) ) ) )  =  ( vol `  (
( `' F " { i } )  i^i  ( `' G " { j } ) ) ) )
21adantl 452 . . . . . . 7  |-  ( ( ( ph  /\  (
i  e.  RR  /\  j  e.  RR )
)  /\  -.  (
i  =  0  /\  j  =  0 ) )  ->  if (
( i  =  0  /\  j  =  0 ) ,  0 ,  ( vol `  (
( `' F " { i } )  i^i  ( `' G " { j } ) ) ) )  =  ( vol `  (
( `' F " { i } )  i^i  ( `' G " { j } ) ) ) )
3 i1fadd.1 . . . . . . . . . . 11  |-  ( ph  ->  F  e.  dom  S.1 )
4 i1fima 19049 . . . . . . . . . . 11  |-  ( F  e.  dom  S.1  ->  ( `' F " { i } )  e.  dom  vol )
53, 4syl 15 . . . . . . . . . 10  |-  ( ph  ->  ( `' F " { i } )  e.  dom  vol )
6 i1fadd.2 . . . . . . . . . . 11  |-  ( ph  ->  G  e.  dom  S.1 )
7 i1fima 19049 . . . . . . . . . . 11  |-  ( G  e.  dom  S.1  ->  ( `' G " { j } )  e.  dom  vol )
86, 7syl 15 . . . . . . . . . 10  |-  ( ph  ->  ( `' G " { j } )  e.  dom  vol )
9 inmbl 18915 . . . . . . . . . 10  |-  ( ( ( `' F " { i } )  e.  dom  vol  /\  ( `' G " { j } )  e.  dom  vol )  ->  ( ( `' F " { i } )  i^i  ( `' G " { j } ) )  e. 
dom  vol )
105, 8, 9syl2anc 642 . . . . . . . . 9  |-  ( ph  ->  ( ( `' F " { i } )  i^i  ( `' G " { j } ) )  e.  dom  vol )
1110ad2antrr 706 . . . . . . . 8  |-  ( ( ( ph  /\  (
i  e.  RR  /\  j  e.  RR )
)  /\  -.  (
i  =  0  /\  j  =  0 ) )  ->  ( ( `' F " { i } )  i^i  ( `' G " { j } ) )  e. 
dom  vol )
12 mblvol 18905 . . . . . . . 8  |-  ( ( ( `' F " { i } )  i^i  ( `' G " { j } ) )  e.  dom  vol  ->  ( vol `  (
( `' F " { i } )  i^i  ( `' G " { j } ) ) )  =  ( vol * `  (
( `' F " { i } )  i^i  ( `' G " { j } ) ) ) )
1311, 12syl 15 . . . . . . 7  |-  ( ( ( ph  /\  (
i  e.  RR  /\  j  e.  RR )
)  /\  -.  (
i  =  0  /\  j  =  0 ) )  ->  ( vol `  ( ( `' F " { i } )  i^i  ( `' G " { j } ) ) )  =  ( vol * `  (
( `' F " { i } )  i^i  ( `' G " { j } ) ) ) )
142, 13eqtrd 2328 . . . . . 6  |-  ( ( ( ph  /\  (
i  e.  RR  /\  j  e.  RR )
)  /\  -.  (
i  =  0  /\  j  =  0 ) )  ->  if (
( i  =  0  /\  j  =  0 ) ,  0 ,  ( vol `  (
( `' F " { i } )  i^i  ( `' G " { j } ) ) ) )  =  ( vol * `  ( ( `' F " { i } )  i^i  ( `' G " { j } ) ) ) )
15 neorian 2546 . . . . . . 7  |-  ( ( i  =/=  0  \/  j  =/=  0 )  <->  -.  ( i  =  0  /\  j  =  0 ) )
16 inss1 3402 . . . . . . . . . 10  |-  ( ( `' F " { i } )  i^i  ( `' G " { j } ) )  C_  ( `' F " { i } )
1716a1i 10 . . . . . . . . 9  |-  ( ( ( ph  /\  (
i  e.  RR  /\  j  e.  RR )
)  /\  i  =/=  0 )  ->  (
( `' F " { i } )  i^i  ( `' G " { j } ) )  C_  ( `' F " { i } ) )
185ad2antrr 706 . . . . . . . . . 10  |-  ( ( ( ph  /\  (
i  e.  RR  /\  j  e.  RR )
)  /\  i  =/=  0 )  ->  ( `' F " { i } )  e.  dom  vol )
19 mblss 18906 . . . . . . . . . 10  |-  ( ( `' F " { i } )  e.  dom  vol 
->  ( `' F " { i } ) 
C_  RR )
2018, 19syl 15 . . . . . . . . 9  |-  ( ( ( ph  /\  (
i  e.  RR  /\  j  e.  RR )
)  /\  i  =/=  0 )  ->  ( `' F " { i } )  C_  RR )
21 mblvol 18905 . . . . . . . . . . 11  |-  ( ( `' F " { i } )  e.  dom  vol 
->  ( vol `  ( `' F " { i } ) )  =  ( vol * `  ( `' F " { i } ) ) )
2218, 21syl 15 . . . . . . . . . 10  |-  ( ( ( ph  /\  (
i  e.  RR  /\  j  e.  RR )
)  /\  i  =/=  0 )  ->  ( vol `  ( `' F " { i } ) )  =  ( vol
* `  ( `' F " { i } ) ) )
233ad2antrr 706 . . . . . . . . . . 11  |-  ( ( ( ph  /\  (
i  e.  RR  /\  j  e.  RR )
)  /\  i  =/=  0 )  ->  F  e.  dom  S.1 )
24 simplrl 736 . . . . . . . . . . . 12  |-  ( ( ( ph  /\  (
i  e.  RR  /\  j  e.  RR )
)  /\  i  =/=  0 )  ->  i  e.  RR )
25 simpr 447 . . . . . . . . . . . 12  |-  ( ( ( ph  /\  (
i  e.  RR  /\  j  e.  RR )
)  /\  i  =/=  0 )  ->  i  =/=  0 )
26 eldifsn 3762 . . . . . . . . . . . 12  |-  ( i  e.  ( RR  \  { 0 } )  <-> 
( i  e.  RR  /\  i  =/=  0 ) )
2724, 25, 26sylanbrc 645 . . . . . . . . . . 11  |-  ( ( ( ph  /\  (
i  e.  RR  /\  j  e.  RR )
)  /\  i  =/=  0 )  ->  i  e.  ( RR  \  {
0 } ) )
28 i1fima2sn 19051 . . . . . . . . . . 11  |-  ( ( F  e.  dom  S.1  /\  i  e.  ( RR 
\  { 0 } ) )  ->  ( vol `  ( `' F " { i } ) )  e.  RR )
2923, 27, 28syl2anc 642 . . . . . . . . . 10  |-  ( ( ( ph  /\  (
i  e.  RR  /\  j  e.  RR )
)  /\  i  =/=  0 )  ->  ( vol `  ( `' F " { i } ) )  e.  RR )
3022, 29eqeltrrd 2371 . . . . . . . . 9  |-  ( ( ( ph  /\  (
i  e.  RR  /\  j  e.  RR )
)  /\  i  =/=  0 )  ->  ( vol * `  ( `' F " { i } ) )  e.  RR )
31 ovolsscl 18861 . . . . . . . . 9  |-  ( ( ( ( `' F " { i } )  i^i  ( `' G " { j } ) )  C_  ( `' F " { i } )  /\  ( `' F " { i } )  C_  RR  /\  ( vol * `  ( `' F " { i } ) )  e.  RR )  ->  ( vol * `  ( ( `' F " { i } )  i^i  ( `' G " { j } ) ) )  e.  RR )
3217, 20, 30, 31syl3anc 1182 . . . . . . . 8  |-  ( ( ( ph  /\  (
i  e.  RR  /\  j  e.  RR )
)  /\  i  =/=  0 )  ->  ( vol * `  ( ( `' F " { i } )  i^i  ( `' G " { j } ) ) )  e.  RR )
33 inss2 3403 . . . . . . . . . 10  |-  ( ( `' F " { i } )  i^i  ( `' G " { j } ) )  C_  ( `' G " { j } )
3433a1i 10 . . . . . . . . 9  |-  ( ( ( ph  /\  (
i  e.  RR  /\  j  e.  RR )
)  /\  j  =/=  0 )  ->  (
( `' F " { i } )  i^i  ( `' G " { j } ) )  C_  ( `' G " { j } ) )
356adantr 451 . . . . . . . . . . . 12  |-  ( (
ph  /\  ( i  e.  RR  /\  j  e.  RR ) )  ->  G  e.  dom  S.1 )
3635, 7syl 15 . . . . . . . . . . 11  |-  ( (
ph  /\  ( i  e.  RR  /\  j  e.  RR ) )  -> 
( `' G " { j } )  e.  dom  vol )
3736adantr 451 . . . . . . . . . 10  |-  ( ( ( ph  /\  (
i  e.  RR  /\  j  e.  RR )
)  /\  j  =/=  0 )  ->  ( `' G " { j } )  e.  dom  vol )
38 mblss 18906 . . . . . . . . . 10  |-  ( ( `' G " { j } )  e.  dom  vol 
->  ( `' G " { j } ) 
C_  RR )
3937, 38syl 15 . . . . . . . . 9  |-  ( ( ( ph  /\  (
i  e.  RR  /\  j  e.  RR )
)  /\  j  =/=  0 )  ->  ( `' G " { j } )  C_  RR )
40 mblvol 18905 . . . . . . . . . . 11  |-  ( ( `' G " { j } )  e.  dom  vol 
->  ( vol `  ( `' G " { j } ) )  =  ( vol * `  ( `' G " { j } ) ) )
4137, 40syl 15 . . . . . . . . . 10  |-  ( ( ( ph  /\  (
i  e.  RR  /\  j  e.  RR )
)  /\  j  =/=  0 )  ->  ( vol `  ( `' G " { j } ) )  =  ( vol
* `  ( `' G " { j } ) ) )
426ad2antrr 706 . . . . . . . . . . 11  |-  ( ( ( ph  /\  (
i  e.  RR  /\  j  e.  RR )
)  /\  j  =/=  0 )  ->  G  e.  dom  S.1 )
43 simplrr 737 . . . . . . . . . . . 12  |-  ( ( ( ph  /\  (
i  e.  RR  /\  j  e.  RR )
)  /\  j  =/=  0 )  ->  j  e.  RR )
44 simpr 447 . . . . . . . . . . . 12  |-  ( ( ( ph  /\  (
i  e.  RR  /\  j  e.  RR )
)  /\  j  =/=  0 )  ->  j  =/=  0 )
45 eldifsn 3762 . . . . . . . . . . . 12  |-  ( j  e.  ( RR  \  { 0 } )  <-> 
( j  e.  RR  /\  j  =/=  0 ) )
4643, 44, 45sylanbrc 645 . . . . . . . . . . 11  |-  ( ( ( ph  /\  (
i  e.  RR  /\  j  e.  RR )
)  /\  j  =/=  0 )  ->  j  e.  ( RR  \  {
0 } ) )
47 i1fima2sn 19051 . . . . . . . . . . 11  |-  ( ( G  e.  dom  S.1  /\  j  e.  ( RR 
\  { 0 } ) )  ->  ( vol `  ( `' G " { j } ) )  e.  RR )
4842, 46, 47syl2anc 642 . . . . . . . . . 10  |-  ( ( ( ph  /\  (
i  e.  RR  /\  j  e.  RR )
)  /\  j  =/=  0 )  ->  ( vol `  ( `' G " { j } ) )  e.  RR )
4941, 48eqeltrrd 2371 . . . . . . . . 9  |-  ( ( ( ph  /\  (
i  e.  RR  /\  j  e.  RR )
)  /\  j  =/=  0 )  ->  ( vol * `  ( `' G " { j } ) )  e.  RR )
50 ovolsscl 18861 . . . . . . . . 9  |-  ( ( ( ( `' F " { i } )  i^i  ( `' G " { j } ) )  C_  ( `' G " { j } )  /\  ( `' G " { j } )  C_  RR  /\  ( vol * `  ( `' G " { j } ) )  e.  RR )  ->  ( vol * `  ( ( `' F " { i } )  i^i  ( `' G " { j } ) ) )  e.  RR )
5134, 39, 49, 50syl3anc 1182 . . . . . . . 8  |-  ( ( ( ph  /\  (
i  e.  RR  /\  j  e.  RR )
)  /\  j  =/=  0 )  ->  ( vol * `  ( ( `' F " { i } )  i^i  ( `' G " { j } ) ) )  e.  RR )
5232, 51jaodan 760 . . . . . . 7  |-  ( ( ( ph  /\  (
i  e.  RR  /\  j  e.  RR )
)  /\  ( i  =/=  0  \/  j  =/=  0 ) )  -> 
( vol * `  ( ( `' F " { i } )  i^i  ( `' G " { j } ) ) )  e.  RR )
5315, 52sylan2br 462 . . . . . 6  |-  ( ( ( ph  /\  (
i  e.  RR  /\  j  e.  RR )
)  /\  -.  (
i  =  0  /\  j  =  0 ) )  ->  ( vol * `
 ( ( `' F " { i } )  i^i  ( `' G " { j } ) ) )  e.  RR )
5414, 53eqeltrd 2370 . . . . 5  |-  ( ( ( ph  /\  (
i  e.  RR  /\  j  e.  RR )
)  /\  -.  (
i  =  0  /\  j  =  0 ) )  ->  if (
( i  =  0  /\  j  =  0 ) ,  0 ,  ( vol `  (
( `' F " { i } )  i^i  ( `' G " { j } ) ) ) )  e.  RR )
5554ex 423 . . . 4  |-  ( (
ph  /\  ( i  e.  RR  /\  j  e.  RR ) )  -> 
( -.  ( i  =  0  /\  j  =  0 )  ->  if ( ( i  =  0  /\  j  =  0 ) ,  0 ,  ( vol `  (
( `' F " { i } )  i^i  ( `' G " { j } ) ) ) )  e.  RR ) )
56 iftrue 3584 . . . . 5  |-  ( ( i  =  0  /\  j  =  0 )  ->  if ( ( i  =  0  /\  j  =  0 ) ,  0 ,  ( vol `  ( ( `' F " { i } )  i^i  ( `' G " { j } ) ) ) )  =  0 )
57 0re 8854 . . . . 5  |-  0  e.  RR
5856, 57syl6eqel 2384 . . . 4  |-  ( ( i  =  0  /\  j  =  0 )  ->  if ( ( i  =  0  /\  j  =  0 ) ,  0 ,  ( vol `  ( ( `' F " { i } )  i^i  ( `' G " { j } ) ) ) )  e.  RR )
5955, 58pm2.61d2 152 . . 3  |-  ( (
ph  /\  ( i  e.  RR  /\  j  e.  RR ) )  ->  if ( ( i  =  0  /\  j  =  0 ) ,  0 ,  ( vol `  (
( `' F " { i } )  i^i  ( `' G " { j } ) ) ) )  e.  RR )
6059ralrimivva 2648 . 2  |-  ( ph  ->  A. i  e.  RR  A. j  e.  RR  if ( ( i  =  0  /\  j  =  0 ) ,  0 ,  ( vol `  (
( `' F " { i } )  i^i  ( `' G " { j } ) ) ) )  e.  RR )
61 itg1add.3 . . 3  |-  I  =  ( i  e.  RR ,  j  e.  RR  |->  if ( ( i  =  0  /\  j  =  0 ) ,  0 ,  ( vol `  (
( `' F " { i } )  i^i  ( `' G " { j } ) ) ) ) )
6261fmpt2 6207 . 2  |-  ( A. i  e.  RR  A. j  e.  RR  if ( ( i  =  0  /\  j  =  0 ) ,  0 ,  ( vol `  ( ( `' F " { i } )  i^i  ( `' G " { j } ) ) ) )  e.  RR  <->  I :
( RR  X.  RR )
--> RR )
6360, 62sylib 188 1  |-  ( ph  ->  I : ( RR 
X.  RR ) --> RR )
Colors of variables: wff set class
Syntax hints:   -. wn 3    -> wi 4    \/ wo 357    /\ wa 358    = wceq 1632    e. wcel 1696    =/= wne 2459   A.wral 2556    \ cdif 3162    i^i cin 3164    C_ wss 3165   ifcif 3578   {csn 3653    X. cxp 4703   `'ccnv 4704   dom cdm 4705   "cima 4708   -->wf 5267   ` cfv 5271    e. cmpt2 5876   RRcr 8752   0cc0 8753   vol *covol 18838   volcvol 18839   S.1citg1 18986
This theorem is referenced by:  itg1addlem4  19070  itg1addlem5  19071
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-rep 4147  ax-sep 4157  ax-nul 4165  ax-pow 4204  ax-pr 4230  ax-un 4528  ax-inf2 7358  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-int 3879  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-se 4369  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-isom 5280  df-ov 5877  df-oprab 5878  df-mpt2 5879  df-of 6094  df-1st 6138  df-2nd 6139  df-riota 6320  df-recs 6404  df-rdg 6439  df-1o 6495  df-2o 6496  df-oadd 6499  df-er 6676  df-map 6790  df-pm 6791  df-en 6880  df-dom 6881  df-sdom 6882  df-fin 6883  df-sup 7210  df-oi 7241  df-card 7588  df-cda 7810  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-2 9820  df-3 9821  df-n0 9982  df-z 10041  df-uz 10247  df-q 10333  df-rp 10371  df-xadd 10469  df-ioo 10676  df-ico 10678  df-icc 10679  df-fz 10799  df-fzo 10887  df-fl 10941  df-seq 11063  df-exp 11121  df-hash 11354  df-cj 11600  df-re 11601  df-im 11602  df-sqr 11736  df-abs 11737  df-clim 11978  df-sum 12175  df-xmet 16389  df-met 16390  df-ovol 18840  df-vol 18841  df-mbf 18991  df-itg1 18992
  Copyright terms: Public domain W3C validator