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Theorem arearect 31351
Description: The area of a rectangle whose sides are parallel to the coordinate axes in  ( RR  X.  RR ) is its width multiplied by its height. (Contributed by Jon Pennant, 19-Mar-2019.)
Hypotheses
Ref Expression
arearect.1  |-  A  e.  RR
arearect.2  |-  B  e.  RR
arearect.3  |-  C  e.  RR
arearect.4  |-  D  e.  RR
arearect.5  |-  A  <_  B
arearect.6  |-  C  <_  D
arearect.7  |-  S  =  ( ( A [,] B )  X.  ( C [,] D ) )
Assertion
Ref Expression
arearect  |-  (area `  S )  =  ( ( B  -  A
)  x.  ( D  -  C ) )

Proof of Theorem arearect
Dummy variable  x is distinct from all other variables.
StepHypRef Expression
1 arearect.7 . . . . 5  |-  S  =  ( ( A [,] B )  X.  ( C [,] D ) )
2 arearect.1 . . . . . . 7  |-  A  e.  RR
3 arearect.2 . . . . . . 7  |-  B  e.  RR
4 iccssre 11527 . . . . . . 7  |-  ( ( A  e.  RR  /\  B  e.  RR )  ->  ( A [,] B
)  C_  RR )
52, 3, 4mp2an 670 . . . . . 6  |-  ( A [,] B )  C_  RR
6 arearect.3 . . . . . . 7  |-  C  e.  RR
7 arearect.4 . . . . . . 7  |-  D  e.  RR
8 iccssre 11527 . . . . . . 7  |-  ( ( C  e.  RR  /\  D  e.  RR )  ->  ( C [,] D
)  C_  RR )
96, 7, 8mp2an 670 . . . . . 6  |-  ( C [,] D )  C_  RR
10 xpss12 5021 . . . . . 6  |-  ( ( ( A [,] B
)  C_  RR  /\  ( C [,] D )  C_  RR )  ->  ( ( A [,] B )  X.  ( C [,] D ) )  C_  ( RR  X.  RR ) )
115, 9, 10mp2an 670 . . . . 5  |-  ( ( A [,] B )  X.  ( C [,] D ) )  C_  ( RR  X.  RR )
121, 11eqsstri 3447 . . . 4  |-  S  C_  ( RR  X.  RR )
13 iftrue 3863 . . . . . . . . . . 11  |-  ( x  e.  ( A [,] B )  ->  if ( x  e.  ( A [,] B ) ,  ( D  -  C
) ,  0 )  =  ( D  -  C ) )
141imaeq1i 5246 . . . . . . . . . . . . . . 15  |-  ( S
" { x }
)  =  ( ( ( A [,] B
)  X.  ( C [,] D ) )
" { x }
)
15 iftrue 3863 . . . . . . . . . . . . . . . . 17  |-  ( x  e.  ( A [,] B )  ->  if ( x  e.  ( A [,] B ) ,  ( C [,] D
) ,  (/) )  =  ( C [,] D
) )
16 xpimasn 5362 . . . . . . . . . . . . . . . . 17  |-  ( x  e.  ( A [,] B )  ->  (
( ( A [,] B )  X.  ( C [,] D ) )
" { x }
)  =  ( C [,] D ) )
1715, 16eqtr4d 2426 . . . . . . . . . . . . . . . 16  |-  ( x  e.  ( A [,] B )  ->  if ( x  e.  ( A [,] B ) ,  ( C [,] D
) ,  (/) )  =  ( ( ( A [,] B )  X.  ( C [,] D
) ) " {
x } ) )
18 iffalse 3866 . . . . . . . . . . . . . . . . 17  |-  ( -.  x  e.  ( A [,] B )  ->  if ( x  e.  ( A [,] B ) ,  ( C [,] D ) ,  (/) )  =  (/) )
19 disjsn 4004 . . . . . . . . . . . . . . . . . 18  |-  ( ( ( A [,] B
)  i^i  { x } )  =  (/)  <->  -.  x  e.  ( A [,] B ) )
20 xpima1 5360 . . . . . . . . . . . . . . . . . 18  |-  ( ( ( A [,] B
)  i^i  { x } )  =  (/)  ->  ( ( ( A [,] B )  X.  ( C [,] D
) ) " {
x } )  =  (/) )
2119, 20sylbir 213 . . . . . . . . . . . . . . . . 17  |-  ( -.  x  e.  ( A [,] B )  -> 
( ( ( A [,] B )  X.  ( C [,] D
) ) " {
x } )  =  (/) )
2218, 21eqtr4d 2426 . . . . . . . . . . . . . . . 16  |-  ( -.  x  e.  ( A [,] B )  ->  if ( x  e.  ( A [,] B ) ,  ( C [,] D ) ,  (/) )  =  ( (
( A [,] B
)  X.  ( C [,] D ) )
" { x }
) )
2317, 22pm2.61i 164 . . . . . . . . . . . . . . 15  |-  if ( x  e.  ( A [,] B ) ,  ( C [,] D
) ,  (/) )  =  ( ( ( A [,] B )  X.  ( C [,] D
) ) " {
x } )
2414, 23eqtr4i 2414 . . . . . . . . . . . . . 14  |-  ( S
" { x }
)  =  if ( x  e.  ( A [,] B ) ,  ( C [,] D
) ,  (/) )
2524fveq2i 5777 . . . . . . . . . . . . 13  |-  ( vol `  ( S " {
x } ) )  =  ( vol `  if ( x  e.  ( A [,] B ) ,  ( C [,] D
) ,  (/) ) )
2615fveq2d 5778 . . . . . . . . . . . . 13  |-  ( x  e.  ( A [,] B )  ->  ( vol `  if ( x  e.  ( A [,] B ) ,  ( C [,] D ) ,  (/) ) )  =  ( vol `  ( C [,] D ) ) )
2725, 26syl5eq 2435 . . . . . . . . . . . 12  |-  ( x  e.  ( A [,] B )  ->  ( vol `  ( S " { x } ) )  =  ( vol `  ( C [,] D
) ) )
28 iccmbl 22061 . . . . . . . . . . . . . . 15  |-  ( ( C  e.  RR  /\  D  e.  RR )  ->  ( C [,] D
)  e.  dom  vol )
296, 7, 28mp2an 670 . . . . . . . . . . . . . 14  |-  ( C [,] D )  e. 
dom  vol
30 mblvol 22026 . . . . . . . . . . . . . 14  |-  ( ( C [,] D )  e.  dom  vol  ->  ( vol `  ( C [,] D ) )  =  ( vol* `  ( C [,] D
) ) )
3129, 30ax-mp 5 . . . . . . . . . . . . 13  |-  ( vol `  ( C [,] D
) )  =  ( vol* `  ( C [,] D ) )
32 arearect.6 . . . . . . . . . . . . . 14  |-  C  <_  D
33 ovolicc 22019 . . . . . . . . . . . . . 14  |-  ( ( C  e.  RR  /\  D  e.  RR  /\  C  <_  D )  ->  ( vol* `  ( C [,] D ) )  =  ( D  -  C ) )
346, 7, 32, 33mp3an 1322 . . . . . . . . . . . . 13  |-  ( vol* `  ( C [,] D ) )  =  ( D  -  C
)
3531, 34eqtri 2411 . . . . . . . . . . . 12  |-  ( vol `  ( C [,] D
) )  =  ( D  -  C )
3627, 35syl6eq 2439 . . . . . . . . . . 11  |-  ( x  e.  ( A [,] B )  ->  ( vol `  ( S " { x } ) )  =  ( D  -  C ) )
3713, 36eqtr4d 2426 . . . . . . . . . 10  |-  ( x  e.  ( A [,] B )  ->  if ( x  e.  ( A [,] B ) ,  ( D  -  C
) ,  0 )  =  ( vol `  ( S " { x }
) ) )
38 iffalse 3866 . . . . . . . . . . 11  |-  ( -.  x  e.  ( A [,] B )  ->  if ( x  e.  ( A [,] B ) ,  ( D  -  C ) ,  0 )  =  0 )
3918fveq2d 5778 . . . . . . . . . . . . 13  |-  ( -.  x  e.  ( A [,] B )  -> 
( vol `  if ( x  e.  ( A [,] B ) ,  ( C [,] D
) ,  (/) ) )  =  ( vol `  (/) ) )
4025, 39syl5eq 2435 . . . . . . . . . . . 12  |-  ( -.  x  e.  ( A [,] B )  -> 
( vol `  ( S " { x }
) )  =  ( vol `  (/) ) )
41 0mbl 22035 . . . . . . . . . . . . . 14  |-  (/)  e.  dom  vol
42 mblvol 22026 . . . . . . . . . . . . . 14  |-  ( (/)  e.  dom  vol  ->  ( vol `  (/) )  =  ( vol* `  (/) ) )
4341, 42ax-mp 5 . . . . . . . . . . . . 13  |-  ( vol `  (/) )  =  ( vol* `  (/) )
44 ovol0 21989 . . . . . . . . . . . . 13  |-  ( vol* `  (/) )  =  0
4543, 44eqtri 2411 . . . . . . . . . . . 12  |-  ( vol `  (/) )  =  0
4640, 45syl6eq 2439 . . . . . . . . . . 11  |-  ( -.  x  e.  ( A [,] B )  -> 
( vol `  ( S " { x }
) )  =  0 )
4738, 46eqtr4d 2426 . . . . . . . . . 10  |-  ( -.  x  e.  ( A [,] B )  ->  if ( x  e.  ( A [,] B ) ,  ( D  -  C ) ,  0 )  =  ( vol `  ( S " {
x } ) ) )
4837, 47pm2.61i 164 . . . . . . . . 9  |-  if ( x  e.  ( A [,] B ) ,  ( D  -  C
) ,  0 )  =  ( vol `  ( S " { x }
) )
4948eqcomi 2395 . . . . . . . 8  |-  ( vol `  ( S " {
x } ) )  =  if ( x  e.  ( A [,] B ) ,  ( D  -  C ) ,  0 )
5049a1i 11 . . . . . . 7  |-  ( x  e.  RR  ->  ( vol `  ( S " { x } ) )  =  if ( x  e.  ( A [,] B ) ,  ( D  -  C
) ,  0 ) )
517, 6resubcli 9794 . . . . . . . 8  |-  ( D  -  C )  e.  RR
52 0re 9507 . . . . . . . 8  |-  0  e.  RR
5351, 52keepel 3924 . . . . . . 7  |-  if ( x  e.  ( A [,] B ) ,  ( D  -  C
) ,  0 )  e.  RR
5450, 53syl6eqel 2478 . . . . . 6  |-  ( x  e.  RR  ->  ( vol `  ( S " { x } ) )  e.  RR )
55 volf 22025 . . . . . . . 8  |-  vol : dom  vol --> ( 0 [,] +oo )
56 ffun 5641 . . . . . . . 8  |-  ( vol
: dom  vol --> ( 0 [,] +oo )  ->  Fun  vol )
5755, 56ax-mp 5 . . . . . . 7  |-  Fun  vol
5829, 41keepel 3924 . . . . . . . 8  |-  if ( x  e.  ( A [,] B ) ,  ( C [,] D
) ,  (/) )  e. 
dom  vol
5924, 58eqeltri 2466 . . . . . . 7  |-  ( S
" { x }
)  e.  dom  vol
60 fvimacnv 5904 . . . . . . 7  |-  ( ( Fun  vol  /\  ( S " { x }
)  e.  dom  vol )  ->  ( ( vol `  ( S " {
x } ) )  e.  RR  <->  ( S " { x } )  e.  ( `' vol " RR ) ) )
6157, 59, 60mp2an 670 . . . . . 6  |-  ( ( vol `  ( S
" { x }
) )  e.  RR  <->  ( S " { x } )  e.  ( `' vol " RR ) )
6254, 61sylib 196 . . . . 5  |-  ( x  e.  RR  ->  ( S " { x }
)  e.  ( `' vol " RR ) )
6362rgen 2742 . . . 4  |-  A. x  e.  RR  ( S " { x } )  e.  ( `' vol " RR )
645a1i 11 . . . . . 6  |-  ( 0  e.  RR  ->  ( A [,] B )  C_  RR )
65 rembl 22036 . . . . . . 7  |-  RR  e.  dom  vol
6665a1i 11 . . . . . 6  |-  ( 0  e.  RR  ->  RR  e.  dom  vol )
6736, 51syl6eqel 2478 . . . . . . 7  |-  ( x  e.  ( A [,] B )  ->  ( vol `  ( S " { x } ) )  e.  RR )
6867adantl 464 . . . . . 6  |-  ( ( 0  e.  RR  /\  x  e.  ( A [,] B ) )  -> 
( vol `  ( S " { x }
) )  e.  RR )
69 eldifn 3541 . . . . . . . 8  |-  ( x  e.  ( RR  \ 
( A [,] B
) )  ->  -.  x  e.  ( A [,] B ) )
7069, 46syl 16 . . . . . . 7  |-  ( x  e.  ( RR  \ 
( A [,] B
) )  ->  ( vol `  ( S " { x } ) )  =  0 )
7170adantl 464 . . . . . 6  |-  ( ( 0  e.  RR  /\  x  e.  ( RR  \  ( A [,] B
) ) )  -> 
( vol `  ( S " { x }
) )  =  0 )
7236mpteq2ia 4449 . . . . . . . 8  |-  ( x  e.  ( A [,] B )  |->  ( vol `  ( S " {
x } ) ) )  =  ( x  e.  ( A [,] B )  |->  ( D  -  C ) )
7351recni 9519 . . . . . . . . . 10  |-  ( D  -  C )  e.  CC
74 ax-resscn 9460 . . . . . . . . . . 11  |-  RR  C_  CC
755, 74sstri 3426 . . . . . . . . . 10  |-  ( A [,] B )  C_  CC
76 ssid 3436 . . . . . . . . . 10  |-  CC  C_  CC
77 cncfmptc 21500 . . . . . . . . . 10  |-  ( ( ( D  -  C
)  e.  CC  /\  ( A [,] B ) 
C_  CC  /\  CC  C_  CC )  ->  ( x  e.  ( A [,] B )  |->  ( D  -  C ) )  e.  ( ( A [,] B ) -cn-> CC ) )
7873, 75, 76, 77mp3an 1322 . . . . . . . . 9  |-  ( x  e.  ( A [,] B )  |->  ( D  -  C ) )  e.  ( ( A [,] B ) -cn-> CC )
79 cniccibl 22332 . . . . . . . . 9  |-  ( ( A  e.  RR  /\  B  e.  RR  /\  (
x  e.  ( A [,] B )  |->  ( D  -  C ) )  e.  ( ( A [,] B )
-cn-> CC ) )  -> 
( x  e.  ( A [,] B ) 
|->  ( D  -  C
) )  e.  L^1 )
802, 3, 78, 79mp3an 1322 . . . . . . . 8  |-  ( x  e.  ( A [,] B )  |->  ( D  -  C ) )  e.  L^1
8172, 80eqeltri 2466 . . . . . . 7  |-  ( x  e.  ( A [,] B )  |->  ( vol `  ( S " {
x } ) ) )  e.  L^1
8281a1i 11 . . . . . 6  |-  ( 0  e.  RR  ->  (
x  e.  ( A [,] B )  |->  ( vol `  ( S
" { x }
) ) )  e.  L^1 )
8364, 66, 68, 71, 82iblss2 22297 . . . . 5  |-  ( 0  e.  RR  ->  (
x  e.  RR  |->  ( vol `  ( S
" { x }
) ) )  e.  L^1 )
8452, 83ax-mp 5 . . . 4  |-  ( x  e.  RR  |->  ( vol `  ( S " {
x } ) ) )  e.  L^1
85 dmarea 23404 . . . 4  |-  ( S  e.  dom area  <->  ( S  C_  ( RR  X.  RR )  /\  A. x  e.  RR  ( S " { x } )  e.  ( `' vol " RR )  /\  (
x  e.  RR  |->  ( vol `  ( S
" { x }
) ) )  e.  L^1 ) )
8612, 63, 84, 85mpbir3an 1176 . . 3  |-  S  e. 
dom area
87 areaval 23411 . . 3  |-  ( S  e.  dom area  ->  (area `  S )  =  S. RR ( vol `  ( S " { x }
) )  _d x )
8886, 87ax-mp 5 . 2  |-  (area `  S )  =  S. RR ( vol `  ( S " { x }
) )  _d x
89 itgeq2 22269 . . . 4  |-  ( A. x  e.  RR  ( vol `  ( S " { x } ) )  =  if ( x  e.  ( A [,] B ) ,  ( D  -  C
) ,  0 )  ->  S. RR ( vol `  ( S
" { x }
) )  _d x  =  S. RR if ( x  e.  ( A [,] B ) ,  ( D  -  C
) ,  0 )  _d x )
9089, 50mprg 2745 . . 3  |-  S. RR ( vol `  ( S
" { x }
) )  _d x  =  S. RR if ( x  e.  ( A [,] B ) ,  ( D  -  C
) ,  0 )  _d x
91 iccmbl 22061 . . . . . 6  |-  ( ( A  e.  RR  /\  B  e.  RR )  ->  ( A [,] B
)  e.  dom  vol )
922, 3, 91mp2an 670 . . . . 5  |-  ( A [,] B )  e. 
dom  vol
93 mblvol 22026 . . . . . . . 8  |-  ( ( A [,] B )  e.  dom  vol  ->  ( vol `  ( A [,] B ) )  =  ( vol* `  ( A [,] B
) ) )
9492, 93ax-mp 5 . . . . . . 7  |-  ( vol `  ( A [,] B
) )  =  ( vol* `  ( A [,] B ) )
95 arearect.5 . . . . . . . 8  |-  A  <_  B
96 ovolicc 22019 . . . . . . . 8  |-  ( ( A  e.  RR  /\  B  e.  RR  /\  A  <_  B )  ->  ( vol* `  ( A [,] B ) )  =  ( B  -  A ) )
972, 3, 95, 96mp3an 1322 . . . . . . 7  |-  ( vol* `  ( A [,] B ) )  =  ( B  -  A
)
9894, 97eqtri 2411 . . . . . 6  |-  ( vol `  ( A [,] B
) )  =  ( B  -  A )
993, 2resubcli 9794 . . . . . 6  |-  ( B  -  A )  e.  RR
10098, 99eqeltri 2466 . . . . 5  |-  ( vol `  ( A [,] B
) )  e.  RR
101 itgconst 22310 . . . . 5  |-  ( ( ( A [,] B
)  e.  dom  vol  /\  ( vol `  ( A [,] B ) )  e.  RR  /\  ( D  -  C )  e.  CC )  ->  S. ( A [,] B ) ( D  -  C
)  _d x  =  ( ( D  -  C )  x.  ( vol `  ( A [,] B ) ) ) )
10292, 100, 73, 101mp3an 1322 . . . 4  |-  S. ( A [,] B ) ( D  -  C
)  _d x  =  ( ( D  -  C )  x.  ( vol `  ( A [,] B ) ) )
103 itgss2 22304 . . . . 5  |-  ( ( A [,] B ) 
C_  RR  ->  S. ( A [,] B ) ( D  -  C
)  _d x  =  S. RR if ( x  e.  ( A [,] B ) ,  ( D  -  C
) ,  0 )  _d x )
1045, 103ax-mp 5 . . . 4  |-  S. ( A [,] B ) ( D  -  C
)  _d x  =  S. RR if ( x  e.  ( A [,] B ) ,  ( D  -  C
) ,  0 )  _d x
10598oveq2i 6207 . . . 4  |-  ( ( D  -  C )  x.  ( vol `  ( A [,] B ) ) )  =  ( ( D  -  C )  x.  ( B  -  A ) )
106102, 104, 1053eqtr3i 2419 . . 3  |-  S. RR if ( x  e.  ( A [,] B ) ,  ( D  -  C ) ,  0 )  _d x  =  ( ( D  -  C )  x.  ( B  -  A )
)
10790, 106eqtri 2411 . 2  |-  S. RR ( vol `  ( S
" { x }
) )  _d x  =  ( ( D  -  C )  x.  ( B  -  A
) )
10899recni 9519 . . 3  |-  ( B  -  A )  e.  CC
10973, 108mulcomi 9513 . 2  |-  ( ( D  -  C )  x.  ( B  -  A ) )  =  ( ( B  -  A )  x.  ( D  -  C )
)
11088, 107, 1093eqtri 2415 1  |-  (area `  S )  =  ( ( B  -  A
)  x.  ( D  -  C ) )
Colors of variables: wff setvar class
Syntax hints:   -. wn 3    <-> wb 184    = wceq 1399    e. wcel 1826   A.wral 2732    \ cdif 3386    i^i cin 3388    C_ wss 3389   (/)c0 3711   ifcif 3857   {csn 3944   class class class wbr 4367    |-> cmpt 4425    X. cxp 4911   `'ccnv 4912   dom cdm 4913   "cima 4916   Fun wfun 5490   -->wf 5492   ` cfv 5496  (class class class)co 6196   CCcc 9401   RRcr 9402   0cc0 9403    x. cmul 9408   +oocpnf 9536    <_ cle 9540    - cmin 9718   [,]cicc 11453   -cn->ccncf 21465   vol*covol 21959   volcvol 21960   L^1cibl 22111   S.citg 22112  areacarea 23402
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1626  ax-4 1639  ax-5 1712  ax-6 1755  ax-7 1798  ax-8 1828  ax-9 1830  ax-10 1845  ax-11 1850  ax-12 1862  ax-13 2006  ax-ext 2360  ax-rep 4478  ax-sep 4488  ax-nul 4496  ax-pow 4543  ax-pr 4601  ax-un 6491  ax-inf2 7972  ax-cc 8728  ax-cnex 9459  ax-resscn 9460  ax-1cn 9461  ax-icn 9462  ax-addcl 9463  ax-addrcl 9464  ax-mulcl 9465  ax-mulrcl 9466  ax-mulcom 9467  ax-addass 9468  ax-mulass 9469  ax-distr 9470  ax-i2m1 9471  ax-1ne0 9472  ax-1rid 9473  ax-rnegex 9474  ax-rrecex 9475  ax-cnre 9476  ax-pre-lttri 9477  ax-pre-lttrn 9478  ax-pre-ltadd 9479  ax-pre-mulgt0 9480  ax-pre-sup 9481  ax-addf 9482  ax-mulf 9483
This theorem depends on definitions:  df-bi 185  df-or 368  df-an 369  df-3or 972  df-3an 973  df-tru 1402  df-fal 1405  df-ex 1621  df-nf 1625  df-sb 1748  df-eu 2222  df-mo 2223  df-clab 2368  df-cleq 2374  df-clel 2377  df-nfc 2532  df-ne 2579  df-nel 2580  df-ral 2737  df-rex 2738  df-reu 2739  df-rmo 2740  df-rab 2741  df-v 3036  df-sbc 3253  df-csb 3349  df-dif 3392  df-un 3394  df-in 3396  df-ss 3403  df-pss 3405  df-nul 3712  df-if 3858  df-pw 3929  df-sn 3945  df-pr 3947  df-tp 3949  df-op 3951  df-uni 4164  df-int 4200  df-iun 4245  df-iin 4246  df-disj 4339  df-br 4368  df-opab 4426  df-mpt 4427  df-tr 4461  df-eprel 4705  df-id 4709  df-po 4714  df-so 4715  df-fr 4752  df-se 4753  df-we 4754  df-ord 4795  df-on 4796  df-lim 4797  df-suc 4798  df-xp 4919  df-rel 4920  df-cnv 4921  df-co 4922  df-dm 4923  df-rn 4924  df-res 4925  df-ima 4926  df-iota 5460  df-fun 5498  df-fn 5499  df-f 5500  df-f1 5501  df-fo 5502  df-f1o 5503  df-fv 5504  df-isom 5505  df-riota 6158  df-ov 6199  df-oprab 6200  df-mpt2 6201  df-of 6439  df-ofr 6440  df-om 6600  df-1st 6699  df-2nd 6700  df-supp 6818  df-recs 6960  df-rdg 6994  df-1o 7048  df-2o 7049  df-oadd 7052  df-omul 7053  df-er 7229  df-map 7340  df-pm 7341  df-ixp 7389  df-en 7436  df-dom 7437  df-sdom 7438  df-fin 7439  df-fsupp 7745  df-fi 7786  df-sup 7816  df-oi 7850  df-card 8233  df-acn 8236  df-cda 8461  df-pnf 9541  df-mnf 9542  df-xr 9543  df-ltxr 9544  df-le 9545  df-sub 9720  df-neg 9721  df-div 10124  df-nn 10453  df-2 10511  df-3 10512  df-4 10513  df-5 10514  df-6 10515  df-7 10516  df-8 10517  df-9 10518  df-10 10519  df-n0 10713  df-z 10782  df-dec 10896  df-uz 11002  df-q 11102  df-rp 11140  df-xneg 11239  df-xadd 11240  df-xmul 11241  df-ioo 11454  df-ioc 11455  df-ico 11456  df-icc 11457  df-fz 11594  df-fzo 11718  df-fl 11828  df-mod 11897  df-seq 12011  df-exp 12070  df-hash 12308  df-cj 12934  df-re 12935  df-im 12936  df-sqrt 13070  df-abs 13071  df-limsup 13296  df-clim 13313  df-rlim 13314  df-sum 13511  df-struct 14636  df-ndx 14637  df-slot 14638  df-base 14639  df-sets 14640  df-ress 14641  df-plusg 14715  df-mulr 14716  df-starv 14717  df-sca 14718  df-vsca 14719  df-ip 14720  df-tset 14721  df-ple 14722  df-ds 14724  df-unif 14725  df-hom 14726  df-cco 14727  df-rest 14830  df-topn 14831  df-0g 14849  df-gsum 14850  df-topgen 14851  df-pt 14852  df-prds 14855  df-xrs 14909  df-qtop 14914  df-imas 14915  df-xps 14917  df-mre 14993  df-mrc 14994  df-acs 14996  df-mgm 15989  df-sgrp 16028  df-mnd 16038  df-submnd 16084  df-mulg 16177  df-cntz 16472  df-cmn 16917  df-psmet 18524  df-xmet 18525  df-met 18526  df-bl 18527  df-mopn 18528  df-cnfld 18534  df-top 19484  df-bases 19486  df-topon 19487  df-topsp 19488  df-cn 19814  df-cnp 19815  df-cmp 19973  df-tx 20148  df-hmeo 20341  df-xms 20908  df-ms 20909  df-tms 20910  df-cncf 21467  df-ovol 21961  df-vol 21962  df-mbf 22113  df-itg1 22114  df-itg2 22115  df-ibl 22116  df-itg 22117  df-0p 22162  df-area 23403
This theorem is referenced by: (None)
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