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Theorem itg1addlem2 19052
Description: Lemma for itg1add 19056. 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 19054 and itg1addlem5 19055. (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 3572 . . . . . . . 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 19033 . . . . . . . . . . 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 19033 . . . . . . . . . . 11  |-  ( G  e.  dom  S.1  ->  ( `' G " { j } )  e.  dom  vol )
86, 7syl 15 . . . . . . . . . 10  |-  ( ph  ->  ( `' G " { j } )  e.  dom  vol )
9 inmbl 18899 . . . . . . . . . 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 18889 . . . . . . . 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 2315 . . . . . 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 2533 . . . . . . 7  |-  ( ( i  =/=  0  \/  j  =/=  0 )  <->  -.  ( i  =  0  /\  j  =  0 ) )
16 inss1 3389 . . . . . . . . . 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 18890 . . . . . . . . . 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 18889 . . . . . . . . . . 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 3749 . . . . . . . . . . . 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 19035 . . . . . . . . . . 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 2358 . . . . . . . . 9  |-  ( ( ( ph  /\  (
i  e.  RR  /\  j  e.  RR )
)  /\  i  =/=  0 )  ->  ( vol * `  ( `' F " { i } ) )  e.  RR )
31 ovolsscl 18845 . . . . . . . . 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 3390 . . . . . . . . . 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 18890 . . . . . . . . . 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 18889 . . . . . . . . . . 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 3749 . . . . . . . . . . . 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 19035 . . . . . . . . . . 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 2358 . . . . . . . . 9  |-  ( ( ( ph  /\  (
i  e.  RR  /\  j  e.  RR )
)  /\  j  =/=  0 )  ->  ( vol * `  ( `' G " { j } ) )  e.  RR )
50 ovolsscl 18845 . . . . . . . . 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 2357 . . . . 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 3571 . . . . 5  |-  ( ( i  =  0  /\  j  =  0 )  ->  if ( ( i  =  0  /\  j  =  0 ) ,  0 ,  ( vol `  ( ( `' F " { i } )  i^i  ( `' G " { j } ) ) ) )  =  0 )
57 0re 8838 . . . . 5  |-  0  e.  RR
5856, 57syl6eqel 2371 . . . 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 2635 . 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 6191 . 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 1623    e. wcel 1684    =/= wne 2446   A.wral 2543    \ cdif 3149    i^i cin 3151    C_ wss 3152   ifcif 3565   {csn 3640    X. cxp 4687   `'ccnv 4688   dom cdm 4689   "cima 4692   -->wf 5251   ` cfv 5255    e. cmpt2 5860   RRcr 8736   0cc0 8737   vol *covol 18822   volcvol 18823   S.1citg1 18970
This theorem is referenced by:  itg1addlem4  19054  itg1addlem5  19055
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1533  ax-5 1544  ax-17 1603  ax-9 1635  ax-8 1643  ax-13 1686  ax-14 1688  ax-6 1703  ax-7 1708  ax-11 1715  ax-12 1866  ax-ext 2264  ax-rep 4131  ax-sep 4141  ax-nul 4149  ax-pow 4188  ax-pr 4214  ax-un 4512  ax-inf2 7342  ax-cnex 8793  ax-resscn 8794  ax-1cn 8795  ax-icn 8796  ax-addcl 8797  ax-addrcl 8798  ax-mulcl 8799  ax-mulrcl 8800  ax-mulcom 8801  ax-addass 8802  ax-mulass 8803  ax-distr 8804  ax-i2m1 8805  ax-1ne0 8806  ax-1rid 8807  ax-rnegex 8808  ax-rrecex 8809  ax-cnre 8810  ax-pre-lttri 8811  ax-pre-lttrn 8812  ax-pre-ltadd 8813  ax-pre-mulgt0 8814  ax-pre-sup 8815
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 1529  df-nf 1532  df-sb 1630  df-eu 2147  df-mo 2148  df-clab 2270  df-cleq 2276  df-clel 2279  df-nfc 2408  df-ne 2448  df-nel 2449  df-ral 2548  df-rex 2549  df-reu 2550  df-rmo 2551  df-rab 2552  df-v 2790  df-sbc 2992  df-csb 3082  df-dif 3155  df-un 3157  df-in 3159  df-ss 3166  df-pss 3168  df-nul 3456  df-if 3566  df-pw 3627  df-sn 3646  df-pr 3647  df-tp 3648  df-op 3649  df-uni 3828  df-int 3863  df-iun 3907  df-br 4024  df-opab 4078  df-mpt 4079  df-tr 4114  df-eprel 4305  df-id 4309  df-po 4314  df-so 4315  df-fr 4352  df-se 4353  df-we 4354  df-ord 4395  df-on 4396  df-lim 4397  df-suc 4398  df-om 4657  df-xp 4695  df-rel 4696  df-cnv 4697  df-co 4698  df-dm 4699  df-rn 4700  df-res 4701  df-ima 4702  df-iota 5219  df-fun 5257  df-fn 5258  df-f 5259  df-f1 5260  df-fo 5261  df-f1o 5262  df-fv 5263  df-isom 5264  df-ov 5861  df-oprab 5862  df-mpt2 5863  df-of 6078  df-1st 6122  df-2nd 6123  df-riota 6304  df-recs 6388  df-rdg 6423  df-1o 6479  df-2o 6480  df-oadd 6483  df-er 6660  df-map 6774  df-pm 6775  df-en 6864  df-dom 6865  df-sdom 6866  df-fin 6867  df-sup 7194  df-oi 7225  df-card 7572  df-cda 7794  df-pnf 8869  df-mnf 8870  df-xr 8871  df-ltxr 8872  df-le 8873  df-sub 9039  df-neg 9040  df-div 9424  df-nn 9747  df-2 9804  df-3 9805  df-n0 9966  df-z 10025  df-uz 10231  df-q 10317  df-rp 10355  df-xadd 10453  df-ioo 10660  df-ico 10662  df-icc 10663  df-fz 10783  df-fzo 10871  df-fl 10925  df-seq 11047  df-exp 11105  df-hash 11338  df-cj 11584  df-re 11585  df-im 11586  df-sqr 11720  df-abs 11721  df-clim 11962  df-sum 12159  df-xmet 16373  df-met 16374  df-ovol 18824  df-vol 18825  df-mbf 18975  df-itg1 18976
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