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Theorem itg1addlem2 22590
Description: Lemma for itg1add 22594. 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 22592 and itg1addlem5 22593. (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 3856 . . . . . . . 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 22571 . . . . . . . . . . 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 22571 . . . . . . . . . . 11  |-  ( G  e.  dom  S.1  ->  ( `' G " { j } )  e.  dom  vol )
86, 7syl 17 . . . . . . . . . 10  |-  ( ph  ->  ( `' G " { j } )  e.  dom  vol )
9 inmbl 22430 . . . . . . . . . 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 22419 . . . . . . . 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 2456 . . . . . 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 2689 . . . . . . 7  |-  ( ( i  =/=  0  \/  j  =/=  0 )  <->  -.  ( i  =  0  /\  j  =  0 ) )
16 inss1 3618 . . . . . . . . . 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 22420 . . . . . . . . . 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 22419 . . . . . . . . . . 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 4061 . . . . . . . . . . . 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 22573 . . . . . . . . . . 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 2501 . . . . . . . . 9  |-  ( ( ( ph  /\  (
i  e.  RR  /\  j  e.  RR )
)  /\  i  =/=  0 )  ->  ( vol* `  ( `' F " { i } ) )  e.  RR )
31 ovolsscl 22374 . . . . . . . . 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 3619 . . . . . . . . . 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 22420 . . . . . . . . . 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 22419 . . . . . . . . . . 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 4061 . . . . . . . . . . . 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 22573 . . . . . . . . . . 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 2501 . . . . . . . . 9  |-  ( ( ( ph  /\  (
i  e.  RR  /\  j  e.  RR )
)  /\  j  =/=  0 )  ->  ( vol* `  ( `' G " { j } ) )  e.  RR )
50 ovolsscl 22374 . . . . . . . . 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 2500 . . . . 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 3853 . . . . 5  |-  ( ( i  =  0  /\  j  =  0 )  ->  if ( ( i  =  0  /\  j  =  0 ) ,  0 ,  ( vol `  ( ( `' F " { i } )  i^i  ( `' G " { j } ) ) ) )  =  0 )
57 0re 9587 . . . . 5  |-  0  e.  RR
5856, 57syl6eqel 2508 . . . 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 2780 . 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 6811 . 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 1872    =/= wne 2593   A.wral 2708    \ cdif 3369    i^i cin 3371    C_ wss 3372   ifcif 3847   {csn 3934    X. cxp 4787   `'ccnv 4788   dom cdm 4789   "cima 4792   -->wf 5533   ` cfv 5537    |-> cmpt2 6244   RRcr 9482   0cc0 9483   vol*covol 22348   volcvol 22350   S.1citg1 22508
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1663  ax-4 1676  ax-5 1752  ax-6 1798  ax-7 1843  ax-8 1874  ax-9 1876  ax-10 1891  ax-11 1896  ax-12 1909  ax-13 2058  ax-ext 2402  ax-rep 4472  ax-sep 4482  ax-nul 4491  ax-pow 4538  ax-pr 4596  ax-un 6534  ax-inf2 8092  ax-cnex 9539  ax-resscn 9540  ax-1cn 9541  ax-icn 9542  ax-addcl 9543  ax-addrcl 9544  ax-mulcl 9545  ax-mulrcl 9546  ax-mulcom 9547  ax-addass 9548  ax-mulass 9549  ax-distr 9550  ax-i2m1 9551  ax-1ne0 9552  ax-1rid 9553  ax-rnegex 9554  ax-rrecex 9555  ax-cnre 9556  ax-pre-lttri 9557  ax-pre-lttrn 9558  ax-pre-ltadd 9559  ax-pre-mulgt0 9560  ax-pre-sup 9561
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 1658  df-nf 1662  df-sb 1791  df-eu 2274  df-mo 2275  df-clab 2409  df-cleq 2415  df-clel 2418  df-nfc 2552  df-ne 2595  df-nel 2596  df-ral 2713  df-rex 2714  df-reu 2715  df-rmo 2716  df-rab 2717  df-v 3018  df-sbc 3236  df-csb 3332  df-dif 3375  df-un 3377  df-in 3379  df-ss 3386  df-pss 3388  df-nul 3698  df-if 3848  df-pw 3919  df-sn 3935  df-pr 3937  df-tp 3939  df-op 3941  df-uni 4156  df-int 4192  df-iun 4237  df-br 4360  df-opab 4419  df-mpt 4420  df-tr 4455  df-eprel 4700  df-id 4704  df-po 4710  df-so 4711  df-fr 4748  df-se 4749  df-we 4750  df-xp 4795  df-rel 4796  df-cnv 4797  df-co 4798  df-dm 4799  df-rn 4800  df-res 4801  df-ima 4802  df-pred 5335  df-ord 5381  df-on 5382  df-lim 5383  df-suc 5384  df-iota 5501  df-fun 5539  df-fn 5540  df-f 5541  df-f1 5542  df-fo 5543  df-f1o 5544  df-fv 5545  df-isom 5546  df-riota 6204  df-ov 6245  df-oprab 6246  df-mpt2 6247  df-of 6482  df-om 6644  df-1st 6744  df-2nd 6745  df-wrecs 6976  df-recs 7038  df-rdg 7076  df-1o 7130  df-2o 7131  df-oadd 7134  df-er 7311  df-map 7422  df-pm 7423  df-en 7518  df-dom 7519  df-sdom 7520  df-fin 7521  df-sup 7902  df-inf 7903  df-oi 7971  df-card 8318  df-cda 8542  df-pnf 9621  df-mnf 9622  df-xr 9623  df-ltxr 9624  df-le 9625  df-sub 9806  df-neg 9807  df-div 10214  df-nn 10554  df-2 10612  df-3 10613  df-n0 10814  df-z 10882  df-uz 11104  df-q 11209  df-rp 11247  df-xadd 11354  df-ioo 11583  df-ico 11585  df-icc 11586  df-fz 11729  df-fzo 11860  df-fl 11971  df-seq 12157  df-exp 12216  df-hash 12459  df-cj 13099  df-re 13100  df-im 13101  df-sqrt 13235  df-abs 13236  df-clim 13488  df-sum 13689  df-xmet 18899  df-met 18900  df-ovol 22351  df-vol 22353  df-mbf 22512  df-itg1 22513
This theorem is referenced by:  itg1addlem4  22592  itg1addlem5  22593
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