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Theorem itg1addlem2 22532
Description: Lemma for itg1add 22536. 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 22534 and itg1addlem5 22535. (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 3924 . . . . . . . 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 467 . . . . . . 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 22513 . . . . . . . . . . 11  |-  ( F  e.  dom  S.1  ->  ( `' F " { i } )  e.  dom  vol )
53, 4syl 17 . . . . . . . . . 10  |-  ( ph  ->  ( `' F " { i } )  e.  dom  vol )
6 i1fadd.2 . . . . . . . . . . 11  |-  ( ph  ->  G  e.  dom  S.1 )
7 i1fima 22513 . . . . . . . . . . 11  |-  ( G  e.  dom  S.1  ->  ( `' G " { j } )  e.  dom  vol )
86, 7syl 17 . . . . . . . . . 10  |-  ( ph  ->  ( `' G " { j } )  e.  dom  vol )
9 inmbl 22372 . . . . . . . . . 10  |-  ( ( ( `' F " { i } )  e.  dom  vol  /\  ( `' G " { j } )  e.  dom  vol )  ->  ( ( `' F " { i } )  i^i  ( `' G " { j } ) )  e. 
dom  vol )
105, 8, 9syl2anc 665 . . . . . . . . 9  |-  ( ph  ->  ( ( `' F " { i } )  i^i  ( `' G " { j } ) )  e.  dom  vol )
1110ad2antrr 730 . . . . . . . 8  |-  ( ( ( ph  /\  (
i  e.  RR  /\  j  e.  RR )
)  /\  -.  (
i  =  0  /\  j  =  0 ) )  ->  ( ( `' F " { i } )  i^i  ( `' G " { j } ) )  e. 
dom  vol )
12 mblvol 22361 . . . . . . . 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 17 . . . . . . 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 2758 . . . . . . 7  |-  ( ( i  =/=  0  \/  j  =/=  0 )  <->  -.  ( i  =  0  /\  j  =  0 ) )
16 inss1 3688 . . . . . . . . . 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 730 . . . . . . . . . 10  |-  ( ( ( ph  /\  (
i  e.  RR  /\  j  e.  RR )
)  /\  i  =/=  0 )  ->  ( `' F " { i } )  e.  dom  vol )
19 mblss 22362 . . . . . . . . . 10  |-  ( ( `' F " { i } )  e.  dom  vol 
->  ( `' F " { i } ) 
C_  RR )
2018, 19syl 17 . . . . . . . . 9  |-  ( ( ( ph  /\  (
i  e.  RR  /\  j  e.  RR )
)  /\  i  =/=  0 )  ->  ( `' F " { i } )  C_  RR )
21 mblvol 22361 . . . . . . . . . . 11  |-  ( ( `' F " { i } )  e.  dom  vol 
->  ( vol `  ( `' F " { i } ) )  =  ( vol* `  ( `' F " { i } ) ) )
2218, 21syl 17 . . . . . . . . . 10  |-  ( ( ( ph  /\  (
i  e.  RR  /\  j  e.  RR )
)  /\  i  =/=  0 )  ->  ( vol `  ( `' F " { i } ) )  =  ( vol* `  ( `' F " { i } ) ) )
233ad2antrr 730 . . . . . . . . . . 11  |-  ( ( ( ph  /\  (
i  e.  RR  /\  j  e.  RR )
)  /\  i  =/=  0 )  ->  F  e.  dom  S.1 )
24 simplrl 768 . . . . . . . . . . . 12  |-  ( ( ( ph  /\  (
i  e.  RR  /\  j  e.  RR )
)  /\  i  =/=  0 )  ->  i  e.  RR )
25 simpr 462 . . . . . . . . . . . 12  |-  ( ( ( ph  /\  (
i  e.  RR  /\  j  e.  RR )
)  /\  i  =/=  0 )  ->  i  =/=  0 )
26 eldifsn 4128 . . . . . . . . . . . 12  |-  ( i  e.  ( RR  \  { 0 } )  <-> 
( i  e.  RR  /\  i  =/=  0 ) )
2724, 25, 26sylanbrc 668 . . . . . . . . . . 11  |-  ( ( ( ph  /\  (
i  e.  RR  /\  j  e.  RR )
)  /\  i  =/=  0 )  ->  i  e.  ( RR  \  {
0 } ) )
28 i1fima2sn 22515 . . . . . . . . . . 11  |-  ( ( F  e.  dom  S.1  /\  i  e.  ( RR 
\  { 0 } ) )  ->  ( vol `  ( `' F " { i } ) )  e.  RR )
2923, 27, 28syl2anc 665 . . . . . . . . . 10  |-  ( ( ( ph  /\  (
i  e.  RR  /\  j  e.  RR )
)  /\  i  =/=  0 )  ->  ( vol `  ( `' F " { i } ) )  e.  RR )
3022, 29eqeltrrd 2518 . . . . . . . . 9  |-  ( ( ( ph  /\  (
i  e.  RR  /\  j  e.  RR )
)  /\  i  =/=  0 )  ->  ( vol* `  ( `' F " { i } ) )  e.  RR )
31 ovolsscl 22317 . . . . . . . . 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 1264 . . . . . . . 8  |-  ( ( ( ph  /\  (
i  e.  RR  /\  j  e.  RR )
)  /\  i  =/=  0 )  ->  ( vol* `  ( ( `' F " { i } )  i^i  ( `' G " { j } ) ) )  e.  RR )
33 inss2 3689 . . . . . . . . . 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 466 . . . . . . . . . . . 12  |-  ( (
ph  /\  ( i  e.  RR  /\  j  e.  RR ) )  ->  G  e.  dom  S.1 )
3635, 7syl 17 . . . . . . . . . . 11  |-  ( (
ph  /\  ( i  e.  RR  /\  j  e.  RR ) )  -> 
( `' G " { j } )  e.  dom  vol )
3736adantr 466 . . . . . . . . . 10  |-  ( ( ( ph  /\  (
i  e.  RR  /\  j  e.  RR )
)  /\  j  =/=  0 )  ->  ( `' G " { j } )  e.  dom  vol )
38 mblss 22362 . . . . . . . . . 10  |-  ( ( `' G " { j } )  e.  dom  vol 
->  ( `' G " { j } ) 
C_  RR )
3937, 38syl 17 . . . . . . . . 9  |-  ( ( ( ph  /\  (
i  e.  RR  /\  j  e.  RR )
)  /\  j  =/=  0 )  ->  ( `' G " { j } )  C_  RR )
40 mblvol 22361 . . . . . . . . . . 11  |-  ( ( `' G " { j } )  e.  dom  vol 
->  ( vol `  ( `' G " { j } ) )  =  ( vol* `  ( `' G " { j } ) ) )
4137, 40syl 17 . . . . . . . . . 10  |-  ( ( ( ph  /\  (
i  e.  RR  /\  j  e.  RR )
)  /\  j  =/=  0 )  ->  ( vol `  ( `' G " { j } ) )  =  ( vol* `  ( `' G " { j } ) ) )
426ad2antrr 730 . . . . . . . . . . 11  |-  ( ( ( ph  /\  (
i  e.  RR  /\  j  e.  RR )
)  /\  j  =/=  0 )  ->  G  e.  dom  S.1 )
43 simplrr 769 . . . . . . . . . . . 12  |-  ( ( ( ph  /\  (
i  e.  RR  /\  j  e.  RR )
)  /\  j  =/=  0 )  ->  j  e.  RR )
44 simpr 462 . . . . . . . . . . . 12  |-  ( ( ( ph  /\  (
i  e.  RR  /\  j  e.  RR )
)  /\  j  =/=  0 )  ->  j  =/=  0 )
45 eldifsn 4128 . . . . . . . . . . . 12  |-  ( j  e.  ( RR  \  { 0 } )  <-> 
( j  e.  RR  /\  j  =/=  0 ) )
4643, 44, 45sylanbrc 668 . . . . . . . . . . 11  |-  ( ( ( ph  /\  (
i  e.  RR  /\  j  e.  RR )
)  /\  j  =/=  0 )  ->  j  e.  ( RR  \  {
0 } ) )
47 i1fima2sn 22515 . . . . . . . . . . 11  |-  ( ( G  e.  dom  S.1  /\  j  e.  ( RR 
\  { 0 } ) )  ->  ( vol `  ( `' G " { j } ) )  e.  RR )
4842, 46, 47syl2anc 665 . . . . . . . . . 10  |-  ( ( ( ph  /\  (
i  e.  RR  /\  j  e.  RR )
)  /\  j  =/=  0 )  ->  ( vol `  ( `' G " { j } ) )  e.  RR )
4941, 48eqeltrrd 2518 . . . . . . . . 9  |-  ( ( ( ph  /\  (
i  e.  RR  /\  j  e.  RR )
)  /\  j  =/=  0 )  ->  ( vol* `  ( `' G " { j } ) )  e.  RR )
50 ovolsscl 22317 . . . . . . . . 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 1264 . . . . . . . 8  |-  ( ( ( ph  /\  (
i  e.  RR  /\  j  e.  RR )
)  /\  j  =/=  0 )  ->  ( vol* `  ( ( `' F " { i } )  i^i  ( `' G " { j } ) ) )  e.  RR )
5232, 51jaodan 792 . . . . . . 7  |-  ( ( ( ph  /\  (
i  e.  RR  /\  j  e.  RR )
)  /\  ( i  =/=  0  \/  j  =/=  0 ) )  -> 
( vol* `  ( ( `' F " { i } )  i^i  ( `' G " { j } ) ) )  e.  RR )
5315, 52sylan2br 478 . . . . . 6  |-  ( ( ( ph  /\  (
i  e.  RR  /\  j  e.  RR )
)  /\  -.  (
i  =  0  /\  j  =  0 ) )  ->  ( vol* `  ( ( `' F " { i } )  i^i  ( `' G " { j } ) ) )  e.  RR )
5414, 53eqeltrd 2517 . . . . 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 435 . . . 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 3921 . . . . 5  |-  ( ( i  =  0  /\  j  =  0 )  ->  if ( ( i  =  0  /\  j  =  0 ) ,  0 ,  ( vol `  ( ( `' F " { i } )  i^i  ( `' G " { j } ) ) ) )  =  0 )
57 0re 9642 . . . . 5  |-  0  e.  RR
5856, 57syl6eqel 2525 . . . 4  |-  ( ( i  =  0  /\  j  =  0 )  ->  if ( ( i  =  0  /\  j  =  0 ) ,  0 ,  ( vol `  ( ( `' F " { i } )  i^i  ( `' G " { j } ) ) ) )  e.  RR )
5955, 58pm2.61d2 163 . . 3  |-  ( (
ph  /\  ( i  e.  RR  /\  j  e.  RR ) )  ->  if ( ( i  =  0  /\  j  =  0 ) ,  0 ,  ( vol `  (
( `' F " { i } )  i^i  ( `' G " { j } ) ) ) )  e.  RR )
6059ralrimivva 2853 . 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 6874 . 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 199 1  |-  ( ph  ->  I : ( RR 
X.  RR ) --> RR )
Colors of variables: wff setvar class
Syntax hints:   -. wn 3    -> wi 4    \/ wo 369    /\ wa 370    = wceq 1437    e. wcel 1870    =/= wne 2625   A.wral 2782    \ cdif 3439    i^i cin 3441    C_ wss 3442   ifcif 3915   {csn 4002    X. cxp 4852   `'ccnv 4853   dom cdm 4854   "cima 4857   -->wf 5597   ` cfv 5601    |-> cmpt2 6307   RRcr 9537   0cc0 9538   vol*covol 22294   volcvol 22295   S.1citg1 22450
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1665  ax-4 1678  ax-5 1751  ax-6 1797  ax-7 1841  ax-8 1872  ax-9 1874  ax-10 1889  ax-11 1894  ax-12 1907  ax-13 2055  ax-ext 2407  ax-rep 4538  ax-sep 4548  ax-nul 4556  ax-pow 4603  ax-pr 4661  ax-un 6597  ax-inf2 8146  ax-cnex 9594  ax-resscn 9595  ax-1cn 9596  ax-icn 9597  ax-addcl 9598  ax-addrcl 9599  ax-mulcl 9600  ax-mulrcl 9601  ax-mulcom 9602  ax-addass 9603  ax-mulass 9604  ax-distr 9605  ax-i2m1 9606  ax-1ne0 9607  ax-1rid 9608  ax-rnegex 9609  ax-rrecex 9610  ax-cnre 9611  ax-pre-lttri 9612  ax-pre-lttrn 9613  ax-pre-ltadd 9614  ax-pre-mulgt0 9615  ax-pre-sup 9616
This theorem depends on definitions:  df-bi 188  df-or 371  df-an 372  df-3or 983  df-3an 984  df-tru 1440  df-fal 1443  df-ex 1660  df-nf 1664  df-sb 1790  df-eu 2270  df-mo 2271  df-clab 2415  df-cleq 2421  df-clel 2424  df-nfc 2579  df-ne 2627  df-nel 2628  df-ral 2787  df-rex 2788  df-reu 2789  df-rmo 2790  df-rab 2791  df-v 3089  df-sbc 3306  df-csb 3402  df-dif 3445  df-un 3447  df-in 3449  df-ss 3456  df-pss 3458  df-nul 3768  df-if 3916  df-pw 3987  df-sn 4003  df-pr 4005  df-tp 4007  df-op 4009  df-uni 4223  df-int 4259  df-iun 4304  df-br 4427  df-opab 4485  df-mpt 4486  df-tr 4521  df-eprel 4765  df-id 4769  df-po 4775  df-so 4776  df-fr 4813  df-se 4814  df-we 4815  df-xp 4860  df-rel 4861  df-cnv 4862  df-co 4863  df-dm 4864  df-rn 4865  df-res 4866  df-ima 4867  df-pred 5399  df-ord 5445  df-on 5446  df-lim 5447  df-suc 5448  df-iota 5565  df-fun 5603  df-fn 5604  df-f 5605  df-f1 5606  df-fo 5607  df-f1o 5608  df-fv 5609  df-isom 5610  df-riota 6267  df-ov 6308  df-oprab 6309  df-mpt2 6310  df-of 6545  df-om 6707  df-1st 6807  df-2nd 6808  df-wrecs 7036  df-recs 7098  df-rdg 7136  df-1o 7190  df-2o 7191  df-oadd 7194  df-er 7371  df-map 7482  df-pm 7483  df-en 7578  df-dom 7579  df-sdom 7580  df-fin 7581  df-sup 7962  df-oi 8025  df-card 8372  df-cda 8596  df-pnf 9676  df-mnf 9677  df-xr 9678  df-ltxr 9679  df-le 9680  df-sub 9861  df-neg 9862  df-div 10269  df-nn 10610  df-2 10668  df-3 10669  df-n0 10870  df-z 10938  df-uz 11160  df-q 11265  df-rp 11303  df-xadd 11410  df-ioo 11639  df-ico 11641  df-icc 11642  df-fz 11783  df-fzo 11914  df-fl 12025  df-seq 12211  df-exp 12270  df-hash 12513  df-cj 13141  df-re 13142  df-im 13143  df-sqrt 13277  df-abs 13278  df-clim 13530  df-sum 13731  df-xmet 18898  df-met 18899  df-ovol 22296  df-vol 22297  df-mbf 22454  df-itg1 22455
This theorem is referenced by:  itg1addlem4  22534  itg1addlem5  22535
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