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Theorem itg1addlem2 19591
Description: Lemma for itg1add 19595. 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 19593 and itg1addlem5 19594. (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 3748 . . . . . . . 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 454 . . . . . . 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 19572 . . . . . . . . . . 11  |-  ( F  e.  dom  S.1  ->  ( `' F " { i } )  e.  dom  vol )
53, 4syl 16 . . . . . . . . . 10  |-  ( ph  ->  ( `' F " { i } )  e.  dom  vol )
6 i1fadd.2 . . . . . . . . . . 11  |-  ( ph  ->  G  e.  dom  S.1 )
7 i1fima 19572 . . . . . . . . . . 11  |-  ( G  e.  dom  S.1  ->  ( `' G " { j } )  e.  dom  vol )
86, 7syl 16 . . . . . . . . . 10  |-  ( ph  ->  ( `' G " { j } )  e.  dom  vol )
9 inmbl 19438 . . . . . . . . . 10  |-  ( ( ( `' F " { i } )  e.  dom  vol  /\  ( `' G " { j } )  e.  dom  vol )  ->  ( ( `' F " { i } )  i^i  ( `' G " { j } ) )  e. 
dom  vol )
105, 8, 9syl2anc 644 . . . . . . . . 9  |-  ( ph  ->  ( ( `' F " { i } )  i^i  ( `' G " { j } ) )  e.  dom  vol )
1110ad2antrr 708 . . . . . . . 8  |-  ( ( ( ph  /\  (
i  e.  RR  /\  j  e.  RR )
)  /\  -.  (
i  =  0  /\  j  =  0 ) )  ->  ( ( `' F " { i } )  i^i  ( `' G " { j } ) )  e. 
dom  vol )
12 mblvol 19428 . . . . . . . 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 16 . . . . . . 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 2470 . . . . . 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 2693 . . . . . . 7  |-  ( ( i  =/=  0  \/  j  =/=  0 )  <->  -.  ( i  =  0  /\  j  =  0 ) )
16 inss1 3563 . . . . . . . . . 10  |-  ( ( `' F " { i } )  i^i  ( `' G " { j } ) )  C_  ( `' F " { i } )
1716a1i 11 . . . . . . . . 9  |-  ( ( ( ph  /\  (
i  e.  RR  /\  j  e.  RR )
)  /\  i  =/=  0 )  ->  (
( `' F " { i } )  i^i  ( `' G " { j } ) )  C_  ( `' F " { i } ) )
185ad2antrr 708 . . . . . . . . . 10  |-  ( ( ( ph  /\  (
i  e.  RR  /\  j  e.  RR )
)  /\  i  =/=  0 )  ->  ( `' F " { i } )  e.  dom  vol )
19 mblss 19429 . . . . . . . . . 10  |-  ( ( `' F " { i } )  e.  dom  vol 
->  ( `' F " { i } ) 
C_  RR )
2018, 19syl 16 . . . . . . . . 9  |-  ( ( ( ph  /\  (
i  e.  RR  /\  j  e.  RR )
)  /\  i  =/=  0 )  ->  ( `' F " { i } )  C_  RR )
21 mblvol 19428 . . . . . . . . . . 11  |-  ( ( `' F " { i } )  e.  dom  vol 
->  ( vol `  ( `' F " { i } ) )  =  ( vol * `  ( `' F " { i } ) ) )
2218, 21syl 16 . . . . . . . . . 10  |-  ( ( ( ph  /\  (
i  e.  RR  /\  j  e.  RR )
)  /\  i  =/=  0 )  ->  ( vol `  ( `' F " { i } ) )  =  ( vol
* `  ( `' F " { i } ) ) )
233ad2antrr 708 . . . . . . . . . . 11  |-  ( ( ( ph  /\  (
i  e.  RR  /\  j  e.  RR )
)  /\  i  =/=  0 )  ->  F  e.  dom  S.1 )
24 simplrl 738 . . . . . . . . . . . 12  |-  ( ( ( ph  /\  (
i  e.  RR  /\  j  e.  RR )
)  /\  i  =/=  0 )  ->  i  e.  RR )
25 simpr 449 . . . . . . . . . . . 12  |-  ( ( ( ph  /\  (
i  e.  RR  /\  j  e.  RR )
)  /\  i  =/=  0 )  ->  i  =/=  0 )
26 eldifsn 3929 . . . . . . . . . . . 12  |-  ( i  e.  ( RR  \  { 0 } )  <-> 
( i  e.  RR  /\  i  =/=  0 ) )
2724, 25, 26sylanbrc 647 . . . . . . . . . . 11  |-  ( ( ( ph  /\  (
i  e.  RR  /\  j  e.  RR )
)  /\  i  =/=  0 )  ->  i  e.  ( RR  \  {
0 } ) )
28 i1fima2sn 19574 . . . . . . . . . . 11  |-  ( ( F  e.  dom  S.1  /\  i  e.  ( RR 
\  { 0 } ) )  ->  ( vol `  ( `' F " { i } ) )  e.  RR )
2923, 27, 28syl2anc 644 . . . . . . . . . 10  |-  ( ( ( ph  /\  (
i  e.  RR  /\  j  e.  RR )
)  /\  i  =/=  0 )  ->  ( vol `  ( `' F " { i } ) )  e.  RR )
3022, 29eqeltrrd 2513 . . . . . . . . 9  |-  ( ( ( ph  /\  (
i  e.  RR  /\  j  e.  RR )
)  /\  i  =/=  0 )  ->  ( vol * `  ( `' F " { i } ) )  e.  RR )
31 ovolsscl 19384 . . . . . . . . 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 1185 . . . . . . . 8  |-  ( ( ( ph  /\  (
i  e.  RR  /\  j  e.  RR )
)  /\  i  =/=  0 )  ->  ( vol * `  ( ( `' F " { i } )  i^i  ( `' G " { j } ) ) )  e.  RR )
33 inss2 3564 . . . . . . . . . 10  |-  ( ( `' F " { i } )  i^i  ( `' G " { j } ) )  C_  ( `' G " { j } )
3433a1i 11 . . . . . . . . 9  |-  ( ( ( ph  /\  (
i  e.  RR  /\  j  e.  RR )
)  /\  j  =/=  0 )  ->  (
( `' F " { i } )  i^i  ( `' G " { j } ) )  C_  ( `' G " { j } ) )
356adantr 453 . . . . . . . . . . . 12  |-  ( (
ph  /\  ( i  e.  RR  /\  j  e.  RR ) )  ->  G  e.  dom  S.1 )
3635, 7syl 16 . . . . . . . . . . 11  |-  ( (
ph  /\  ( i  e.  RR  /\  j  e.  RR ) )  -> 
( `' G " { j } )  e.  dom  vol )
3736adantr 453 . . . . . . . . . 10  |-  ( ( ( ph  /\  (
i  e.  RR  /\  j  e.  RR )
)  /\  j  =/=  0 )  ->  ( `' G " { j } )  e.  dom  vol )
38 mblss 19429 . . . . . . . . . 10  |-  ( ( `' G " { j } )  e.  dom  vol 
->  ( `' G " { j } ) 
C_  RR )
3937, 38syl 16 . . . . . . . . 9  |-  ( ( ( ph  /\  (
i  e.  RR  /\  j  e.  RR )
)  /\  j  =/=  0 )  ->  ( `' G " { j } )  C_  RR )
40 mblvol 19428 . . . . . . . . . . 11  |-  ( ( `' G " { j } )  e.  dom  vol 
->  ( vol `  ( `' G " { j } ) )  =  ( vol * `  ( `' G " { j } ) ) )
4137, 40syl 16 . . . . . . . . . 10  |-  ( ( ( ph  /\  (
i  e.  RR  /\  j  e.  RR )
)  /\  j  =/=  0 )  ->  ( vol `  ( `' G " { j } ) )  =  ( vol
* `  ( `' G " { j } ) ) )
426ad2antrr 708 . . . . . . . . . . 11  |-  ( ( ( ph  /\  (
i  e.  RR  /\  j  e.  RR )
)  /\  j  =/=  0 )  ->  G  e.  dom  S.1 )
43 simplrr 739 . . . . . . . . . . . 12  |-  ( ( ( ph  /\  (
i  e.  RR  /\  j  e.  RR )
)  /\  j  =/=  0 )  ->  j  e.  RR )
44 simpr 449 . . . . . . . . . . . 12  |-  ( ( ( ph  /\  (
i  e.  RR  /\  j  e.  RR )
)  /\  j  =/=  0 )  ->  j  =/=  0 )
45 eldifsn 3929 . . . . . . . . . . . 12  |-  ( j  e.  ( RR  \  { 0 } )  <-> 
( j  e.  RR  /\  j  =/=  0 ) )
4643, 44, 45sylanbrc 647 . . . . . . . . . . 11  |-  ( ( ( ph  /\  (
i  e.  RR  /\  j  e.  RR )
)  /\  j  =/=  0 )  ->  j  e.  ( RR  \  {
0 } ) )
47 i1fima2sn 19574 . . . . . . . . . . 11  |-  ( ( G  e.  dom  S.1  /\  j  e.  ( RR 
\  { 0 } ) )  ->  ( vol `  ( `' G " { j } ) )  e.  RR )
4842, 46, 47syl2anc 644 . . . . . . . . . 10  |-  ( ( ( ph  /\  (
i  e.  RR  /\  j  e.  RR )
)  /\  j  =/=  0 )  ->  ( vol `  ( `' G " { j } ) )  e.  RR )
4941, 48eqeltrrd 2513 . . . . . . . . 9  |-  ( ( ( ph  /\  (
i  e.  RR  /\  j  e.  RR )
)  /\  j  =/=  0 )  ->  ( vol * `  ( `' G " { j } ) )  e.  RR )
50 ovolsscl 19384 . . . . . . . . 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 1185 . . . . . . . 8  |-  ( ( ( ph  /\  (
i  e.  RR  /\  j  e.  RR )
)  /\  j  =/=  0 )  ->  ( vol * `  ( ( `' F " { i } )  i^i  ( `' G " { j } ) ) )  e.  RR )
5232, 51jaodan 762 . . . . . . 7  |-  ( ( ( ph  /\  (
i  e.  RR  /\  j  e.  RR )
)  /\  ( i  =/=  0  \/  j  =/=  0 ) )  -> 
( vol * `  ( ( `' F " { i } )  i^i  ( `' G " { j } ) ) )  e.  RR )
5315, 52sylan2br 464 . . . . . 6  |-  ( ( ( ph  /\  (
i  e.  RR  /\  j  e.  RR )
)  /\  -.  (
i  =  0  /\  j  =  0 ) )  ->  ( vol * `
 ( ( `' F " { i } )  i^i  ( `' G " { j } ) ) )  e.  RR )
5414, 53eqeltrd 2512 . . . . 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 425 . . . 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 3747 . . . . 5  |-  ( ( i  =  0  /\  j  =  0 )  ->  if ( ( i  =  0  /\  j  =  0 ) ,  0 ,  ( vol `  ( ( `' F " { i } )  i^i  ( `' G " { j } ) ) ) )  =  0 )
57 0re 9093 . . . . 5  |-  0  e.  RR
5856, 57syl6eqel 2526 . . . 4  |-  ( ( i  =  0  /\  j  =  0 )  ->  if ( ( i  =  0  /\  j  =  0 ) ,  0 ,  ( vol `  ( ( `' F " { i } )  i^i  ( `' G " { j } ) ) ) )  e.  RR )
5955, 58pm2.61d2 155 . . 3  |-  ( (
ph  /\  ( i  e.  RR  /\  j  e.  RR ) )  ->  if ( ( i  =  0  /\  j  =  0 ) ,  0 ,  ( vol `  (
( `' F " { i } )  i^i  ( `' G " { j } ) ) ) )  e.  RR )
6059ralrimivva 2800 . 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 6420 . 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 190 1  |-  ( ph  ->  I : ( RR 
X.  RR ) --> RR )
Colors of variables: wff set class
Syntax hints:   -. wn 3    -> wi 4    \/ wo 359    /\ wa 360    = wceq 1653    e. wcel 1726    =/= wne 2601   A.wral 2707    \ cdif 3319    i^i cin 3321    C_ wss 3322   ifcif 3741   {csn 3816    X. cxp 4878   `'ccnv 4879   dom cdm 4880   "cima 4883   -->wf 5452   ` cfv 5456    e. cmpt2 6085   RRcr 8991   0cc0 8992   vol *covol 19361   volcvol 19362   S.1citg1 19509
This theorem is referenced by:  itg1addlem4  19593  itg1addlem5  19594
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1556  ax-5 1567  ax-17 1627  ax-9 1667  ax-8 1688  ax-13 1728  ax-14 1730  ax-6 1745  ax-7 1750  ax-11 1762  ax-12 1951  ax-ext 2419  ax-rep 4322  ax-sep 4332  ax-nul 4340  ax-pow 4379  ax-pr 4405  ax-un 4703  ax-inf2 7598  ax-cnex 9048  ax-resscn 9049  ax-1cn 9050  ax-icn 9051  ax-addcl 9052  ax-addrcl 9053  ax-mulcl 9054  ax-mulrcl 9055  ax-mulcom 9056  ax-addass 9057  ax-mulass 9058  ax-distr 9059  ax-i2m1 9060  ax-1ne0 9061  ax-1rid 9062  ax-rnegex 9063  ax-rrecex 9064  ax-cnre 9065  ax-pre-lttri 9066  ax-pre-lttrn 9067  ax-pre-ltadd 9068  ax-pre-mulgt0 9069  ax-pre-sup 9070
This theorem depends on definitions:  df-bi 179  df-or 361  df-an 362  df-3or 938  df-3an 939  df-tru 1329  df-ex 1552  df-nf 1555  df-sb 1660  df-eu 2287  df-mo 2288  df-clab 2425  df-cleq 2431  df-clel 2434  df-nfc 2563  df-ne 2603  df-nel 2604  df-ral 2712  df-rex 2713  df-reu 2714  df-rmo 2715  df-rab 2716  df-v 2960  df-sbc 3164  df-csb 3254  df-dif 3325  df-un 3327  df-in 3329  df-ss 3336  df-pss 3338  df-nul 3631  df-if 3742  df-pw 3803  df-sn 3822  df-pr 3823  df-tp 3824  df-op 3825  df-uni 4018  df-int 4053  df-iun 4097  df-br 4215  df-opab 4269  df-mpt 4270  df-tr 4305  df-eprel 4496  df-id 4500  df-po 4505  df-so 4506  df-fr 4543  df-se 4544  df-we 4545  df-ord 4586  df-on 4587  df-lim 4588  df-suc 4589  df-om 4848  df-xp 4886  df-rel 4887  df-cnv 4888  df-co 4889  df-dm 4890  df-rn 4891  df-res 4892  df-ima 4893  df-iota 5420  df-fun 5458  df-fn 5459  df-f 5460  df-f1 5461  df-fo 5462  df-f1o 5463  df-fv 5464  df-isom 5465  df-ov 6086  df-oprab 6087  df-mpt2 6088  df-of 6307  df-1st 6351  df-2nd 6352  df-riota 6551  df-recs 6635  df-rdg 6670  df-1o 6726  df-2o 6727  df-oadd 6730  df-er 6907  df-map 7022  df-pm 7023  df-en 7112  df-dom 7113  df-sdom 7114  df-fin 7115  df-sup 7448  df-oi 7481  df-card 7828  df-cda 8050  df-pnf 9124  df-mnf 9125  df-xr 9126  df-ltxr 9127  df-le 9128  df-sub 9295  df-neg 9296  df-div 9680  df-nn 10003  df-2 10060  df-3 10061  df-n0 10224  df-z 10285  df-uz 10491  df-q 10577  df-rp 10615  df-xadd 10713  df-ioo 10922  df-ico 10924  df-icc 10925  df-fz 11046  df-fzo 11138  df-fl 11204  df-seq 11326  df-exp 11385  df-hash 11621  df-cj 11906  df-re 11907  df-im 11908  df-sqr 12042  df-abs 12043  df-clim 12284  df-sum 12482  df-xmet 16697  df-met 16698  df-ovol 19363  df-vol 19364  df-mbf 19514  df-itg1 19515
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