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Theorem arearect 36094
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 11713 . . . . . . 7  |-  ( ( A  e.  RR  /\  B  e.  RR )  ->  ( A [,] B
)  C_  RR )
52, 3, 4mp2an 677 . . . . . 6  |-  ( A [,] B )  C_  RR
6 arearect.3 . . . . . . 7  |-  C  e.  RR
7 arearect.4 . . . . . . 7  |-  D  e.  RR
8 iccssre 11713 . . . . . . 7  |-  ( ( C  e.  RR  /\  D  e.  RR )  ->  ( C [,] D
)  C_  RR )
96, 7, 8mp2an 677 . . . . . 6  |-  ( C [,] D )  C_  RR
10 xpss12 4939 . . . . . 6  |-  ( ( ( A [,] B
)  C_  RR  /\  ( C [,] D )  C_  RR )  ->  ( ( A [,] B )  X.  ( C [,] D ) )  C_  ( RR  X.  RR ) )
115, 9, 10mp2an 677 . . . . 5  |-  ( ( A [,] B )  X.  ( C [,] D ) )  C_  ( RR  X.  RR )
121, 11eqsstri 3461 . . . 4  |-  S  C_  ( RR  X.  RR )
13 iftrue 3886 . . . . . . . . . . 11  |-  ( x  e.  ( A [,] B )  ->  if ( x  e.  ( A [,] B ) ,  ( D  -  C
) ,  0 )  =  ( D  -  C ) )
141imaeq1i 5164 . . . . . . . . . . . . . . 15  |-  ( S
" { x }
)  =  ( ( ( A [,] B
)  X.  ( C [,] D ) )
" { x }
)
15 iftrue 3886 . . . . . . . . . . . . . . . . 17  |-  ( x  e.  ( A [,] B )  ->  if ( x  e.  ( A [,] B ) ,  ( C [,] D
) ,  (/) )  =  ( C [,] D
) )
16 xpimasn 5281 . . . . . . . . . . . . . . . . 17  |-  ( x  e.  ( A [,] B )  ->  (
( ( A [,] B )  X.  ( C [,] D ) )
" { x }
)  =  ( C [,] D ) )
1715, 16eqtr4d 2487 . . . . . . . . . . . . . . . 16  |-  ( x  e.  ( A [,] B )  ->  if ( x  e.  ( A [,] B ) ,  ( C [,] D
) ,  (/) )  =  ( ( ( A [,] B )  X.  ( C [,] D
) ) " {
x } ) )
18 iffalse 3889 . . . . . . . . . . . . . . . . 17  |-  ( -.  x  e.  ( A [,] B )  ->  if ( x  e.  ( A [,] B ) ,  ( C [,] D ) ,  (/) )  =  (/) )
19 disjsn 4031 . . . . . . . . . . . . . . . . . 18  |-  ( ( ( A [,] B
)  i^i  { x } )  =  (/)  <->  -.  x  e.  ( A [,] B ) )
20 xpima1 5279 . . . . . . . . . . . . . . . . . 18  |-  ( ( ( A [,] B
)  i^i  { x } )  =  (/)  ->  ( ( ( A [,] B )  X.  ( C [,] D
) ) " {
x } )  =  (/) )
2119, 20sylbir 217 . . . . . . . . . . . . . . . . 17  |-  ( -.  x  e.  ( A [,] B )  -> 
( ( ( A [,] B )  X.  ( C [,] D
) ) " {
x } )  =  (/) )
2218, 21eqtr4d 2487 . . . . . . . . . . . . . . . 16  |-  ( -.  x  e.  ( A [,] B )  ->  if ( x  e.  ( A [,] B ) ,  ( C [,] D ) ,  (/) )  =  ( (
( A [,] B
)  X.  ( C [,] D ) )
" { x }
) )
2317, 22pm2.61i 168 . . . . . . . . . . . . . . 15  |-  if ( x  e.  ( A [,] B ) ,  ( C [,] D
) ,  (/) )  =  ( ( ( A [,] B )  X.  ( C [,] D
) ) " {
x } )
2414, 23eqtr4i 2475 . . . . . . . . . . . . . 14  |-  ( S
" { x }
)  =  if ( x  e.  ( A [,] B ) ,  ( C [,] D
) ,  (/) )
2524fveq2i 5866 . . . . . . . . . . . . 13  |-  ( vol `  ( S " {
x } ) )  =  ( vol `  if ( x  e.  ( A [,] B ) ,  ( C [,] D
) ,  (/) ) )
2615fveq2d 5867 . . . . . . . . . . . . 13  |-  ( x  e.  ( A [,] B )  ->  ( vol `  if ( x  e.  ( A [,] B ) ,  ( C [,] D ) ,  (/) ) )  =  ( vol `  ( C [,] D ) ) )
2725, 26syl5eq 2496 . . . . . . . . . . . 12  |-  ( x  e.  ( A [,] B )  ->  ( vol `  ( S " { x } ) )  =  ( vol `  ( C [,] D
) ) )
28 iccmbl 22512 . . . . . . . . . . . . . . 15  |-  ( ( C  e.  RR  /\  D  e.  RR )  ->  ( C [,] D
)  e.  dom  vol )
296, 7, 28mp2an 677 . . . . . . . . . . . . . 14  |-  ( C [,] D )  e. 
dom  vol
30 mblvol 22477 . . . . . . . . . . . . . 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 22470 . . . . . . . . . . . . . 14  |-  ( ( C  e.  RR  /\  D  e.  RR  /\  C  <_  D )  ->  ( vol* `  ( C [,] D ) )  =  ( D  -  C ) )
346, 7, 32, 33mp3an 1363 . . . . . . . . . . . . 13  |-  ( vol* `  ( C [,] D ) )  =  ( D  -  C
)
3531, 34eqtri 2472 . . . . . . . . . . . 12  |-  ( vol `  ( C [,] D
) )  =  ( D  -  C )
3627, 35syl6eq 2500 . . . . . . . . . . 11  |-  ( x  e.  ( A [,] B )  ->  ( vol `  ( S " { x } ) )  =  ( D  -  C ) )
3713, 36eqtr4d 2487 . . . . . . . . . 10  |-  ( x  e.  ( A [,] B )  ->  if ( x  e.  ( A [,] B ) ,  ( D  -  C
) ,  0 )  =  ( vol `  ( S " { x }
) ) )
38 iffalse 3889 . . . . . . . . . . 11  |-  ( -.  x  e.  ( A [,] B )  ->  if ( x  e.  ( A [,] B ) ,  ( D  -  C ) ,  0 )  =  0 )
3918fveq2d 5867 . . . . . . . . . . . . 13  |-  ( -.  x  e.  ( A [,] B )  -> 
( vol `  if ( x  e.  ( A [,] B ) ,  ( C [,] D
) ,  (/) ) )  =  ( vol `  (/) ) )
4025, 39syl5eq 2496 . . . . . . . . . . . 12  |-  ( -.  x  e.  ( A [,] B )  -> 
( vol `  ( S " { x }
) )  =  ( vol `  (/) ) )
41 0mbl 22486 . . . . . . . . . . . . . 14  |-  (/)  e.  dom  vol
42 mblvol 22477 . . . . . . . . . . . . . 14  |-  ( (/)  e.  dom  vol  ->  ( vol `  (/) )  =  ( vol* `  (/) ) )
4341, 42ax-mp 5 . . . . . . . . . . . . 13  |-  ( vol `  (/) )  =  ( vol* `  (/) )
44 ovol0 22439 . . . . . . . . . . . . 13  |-  ( vol* `  (/) )  =  0
4543, 44eqtri 2472 . . . . . . . . . . . 12  |-  ( vol `  (/) )  =  0
4640, 45syl6eq 2500 . . . . . . . . . . 11  |-  ( -.  x  e.  ( A [,] B )  -> 
( vol `  ( S " { x }
) )  =  0 )
4738, 46eqtr4d 2487 . . . . . . . . . 10  |-  ( -.  x  e.  ( A [,] B )  ->  if ( x  e.  ( A [,] B ) ,  ( D  -  C ) ,  0 )  =  ( vol `  ( S " {
x } ) ) )
4837, 47pm2.61i 168 . . . . . . . . 9  |-  if ( x  e.  ( A [,] B ) ,  ( D  -  C
) ,  0 )  =  ( vol `  ( S " { x }
) )
4948eqcomi 2459 . . . . . . . 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 9933 . . . . . . . 8  |-  ( D  -  C )  e.  RR
52 0re 9640 . . . . . . . 8  |-  0  e.  RR
5351, 52keepel 3947 . . . . . . 7  |-  if ( x  e.  ( A [,] B ) ,  ( D  -  C
) ,  0 )  e.  RR
5450, 53syl6eqel 2536 . . . . . 6  |-  ( x  e.  RR  ->  ( vol `  ( S " { x } ) )  e.  RR )
55 volf 22476 . . . . . . . 8  |-  vol : dom  vol --> ( 0 [,] +oo )
56 ffun 5729 . . . . . . . 8  |-  ( vol
: dom  vol --> ( 0 [,] +oo )  ->  Fun  vol )
5755, 56ax-mp 5 . . . . . . 7  |-  Fun  vol
5829, 41keepel 3947 . . . . . . . 8  |-  if ( x  e.  ( A [,] B ) ,  ( C [,] D
) ,  (/) )  e. 
dom  vol
5924, 58eqeltri 2524 . . . . . . 7  |-  ( S
" { x }
)  e.  dom  vol
60 fvimacnv 5995 . . . . . . 7  |-  ( ( Fun  vol  /\  ( S " { x }
)  e.  dom  vol )  ->  ( ( vol `  ( S " {
x } ) )  e.  RR  <->  ( S " { x } )  e.  ( `' vol " RR ) ) )
6157, 59, 60mp2an 677 . . . . . 6  |-  ( ( vol `  ( S
" { x }
) )  e.  RR  <->  ( S " { x } )  e.  ( `' vol " RR ) )
6254, 61sylib 200 . . . . 5  |-  ( x  e.  RR  ->  ( S " { x }
)  e.  ( `' vol " RR ) )
6362rgen 2746 . . . 4  |-  A. x  e.  RR  ( S " { x } )  e.  ( `' vol " RR )
645a1i 11 . . . . . 6  |-  ( 0  e.  RR  ->  ( A [,] B )  C_  RR )
65 rembl 22487 . . . . . . 7  |-  RR  e.  dom  vol
6665a1i 11 . . . . . 6  |-  ( 0  e.  RR  ->  RR  e.  dom  vol )
6736, 51syl6eqel 2536 . . . . . . 7  |-  ( x  e.  ( A [,] B )  ->  ( vol `  ( S " { x } ) )  e.  RR )
6867adantl 468 . . . . . 6  |-  ( ( 0  e.  RR  /\  x  e.  ( A [,] B ) )  -> 
( vol `  ( S " { x }
) )  e.  RR )
69 eldifn 3555 . . . . . . . 8  |-  ( x  e.  ( RR  \ 
( A [,] B
) )  ->  -.  x  e.  ( A [,] B ) )
7069, 46syl 17 . . . . . . 7  |-  ( x  e.  ( RR  \ 
( A [,] B
) )  ->  ( vol `  ( S " { x } ) )  =  0 )
7170adantl 468 . . . . . 6  |-  ( ( 0  e.  RR  /\  x  e.  ( RR  \  ( A [,] B
) ) )  -> 
( vol `  ( S " { x }
) )  =  0 )
7236mpteq2ia 4484 . . . . . . . 8  |-  ( x  e.  ( A [,] B )  |->  ( vol `  ( S " {
x } ) ) )  =  ( x  e.  ( A [,] B )  |->  ( D  -  C ) )
7351recni 9652 . . . . . . . . . 10  |-  ( D  -  C )  e.  CC
74 ax-resscn 9593 . . . . . . . . . . 11  |-  RR  C_  CC
755, 74sstri 3440 . . . . . . . . . 10  |-  ( A [,] B )  C_  CC
76 ssid 3450 . . . . . . . . . 10  |-  CC  C_  CC
77 cncfmptc 21936 . . . . . . . . . 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 1363 . . . . . . . . 9  |-  ( x  e.  ( A [,] B )  |->  ( D  -  C ) )  e.  ( ( A [,] B ) -cn-> CC )
79 cniccibl 22791 . . . . . . . . 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 1363 . . . . . . . 8  |-  ( x  e.  ( A [,] B )  |->  ( D  -  C ) )  e.  L^1
8172, 80eqeltri 2524 . . . . . . 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 22756 . . . . 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 23876 . . . 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 1189 . . 3  |-  S  e. 
dom area
87 areaval 23883 . . 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 22728 . . . 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 2750 . . 3  |-  S. RR ( vol `  ( S
" { x }
) )  _d x  =  S. RR if ( x  e.  ( A [,] B ) ,  ( D  -  C
) ,  0 )  _d x
91 iccmbl 22512 . . . . . 6  |-  ( ( A  e.  RR  /\  B  e.  RR )  ->  ( A [,] B
)  e.  dom  vol )
922, 3, 91mp2an 677 . . . . 5  |-  ( A [,] B )  e. 
dom  vol
93 mblvol 22477 . . . . . . . 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 22470 . . . . . . . 8  |-  ( ( A  e.  RR  /\  B  e.  RR  /\  A  <_  B )  ->  ( vol* `  ( A [,] B ) )  =  ( B  -  A ) )
972, 3, 95, 96mp3an 1363 . . . . . . 7  |-  ( vol* `  ( A [,] B ) )  =  ( B  -  A
)
9894, 97eqtri 2472 . . . . . 6  |-  ( vol `  ( A [,] B
) )  =  ( B  -  A )
993, 2resubcli 9933 . . . . . 6  |-  ( B  -  A )  e.  RR
10098, 99eqeltri 2524 . . . . 5  |-  ( vol `  ( A [,] B
) )  e.  RR
101 itgconst 22769 . . . . 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 1363 . . . 4  |-  S. ( A [,] B ) ( D  -  C
)  _d x  =  ( ( D  -  C )  x.  ( vol `  ( A [,] B ) ) )
103 itgss2 22763 . . . . 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 6299 . . . 4  |-  ( ( D  -  C )  x.  ( vol `  ( A [,] B ) ) )  =  ( ( D  -  C )  x.  ( B  -  A ) )
106102, 104, 1053eqtr3i 2480 . . 3  |-  S. RR if ( x  e.  ( A [,] B ) ,  ( D  -  C ) ,  0 )  _d x  =  ( ( D  -  C )  x.  ( B  -  A )
)
10790, 106eqtri 2472 . 2  |-  S. RR ( vol `  ( S
" { x }
) )  _d x  =  ( ( D  -  C )  x.  ( B  -  A
) )
10899recni 9652 . . 3  |-  ( B  -  A )  e.  CC
10973, 108mulcomi 9646 . 2  |-  ( ( D  -  C )  x.  ( B  -  A ) )  =  ( ( B  -  A )  x.  ( D  -  C )
)
11088, 107, 1093eqtri 2476 1  |-  (area `  S )  =  ( ( B  -  A
)  x.  ( D  -  C ) )
Colors of variables: wff setvar class
Syntax hints:   -. wn 3    <-> wb 188    = wceq 1443    e. wcel 1886   A.wral 2736    \ cdif 3400    i^i cin 3402    C_ wss 3403   (/)c0 3730   ifcif 3880   {csn 3967   class class class wbr 4401    |-> cmpt 4460    X. cxp 4831   `'ccnv 4832   dom cdm 4833   "cima 4836   Fun wfun 5575   -->wf 5577   ` cfv 5581  (class class class)co 6288   CCcc 9534   RRcr 9535   0cc0 9536    x. cmul 9541   +oocpnf 9669    <_ cle 9673    - cmin 9857   [,]cicc 11635   -cn->ccncf 21901   vol*covol 22406   volcvol 22408   L^1cibl 22568   S.citg 22569  areacarea 23874
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1668  ax-4 1681  ax-5 1757  ax-6 1804  ax-7 1850  ax-8 1888  ax-9 1895  ax-10 1914  ax-11 1919  ax-12 1932  ax-13 2090  ax-ext 2430  ax-rep 4514  ax-sep 4524  ax-nul 4533  ax-pow 4580  ax-pr 4638  ax-un 6580  ax-inf2 8143  ax-cc 8862  ax-cnex 9592  ax-resscn 9593  ax-1cn 9594  ax-icn 9595  ax-addcl 9596  ax-addrcl 9597  ax-mulcl 9598  ax-mulrcl 9599  ax-mulcom 9600  ax-addass 9601  ax-mulass 9602  ax-distr 9603  ax-i2m1 9604  ax-1ne0 9605  ax-1rid 9606  ax-rnegex 9607  ax-rrecex 9608  ax-cnre 9609  ax-pre-lttri 9610  ax-pre-lttrn 9611  ax-pre-ltadd 9612  ax-pre-mulgt0 9613  ax-pre-sup 9614  ax-addf 9615  ax-mulf 9616
This theorem depends on definitions:  df-bi 189  df-or 372  df-an 373  df-3or 985  df-3an 986  df-tru 1446  df-fal 1449  df-ex 1663  df-nf 1667  df-sb 1797  df-eu 2302  df-mo 2303  df-clab 2437  df-cleq 2443  df-clel 2446  df-nfc 2580  df-ne 2623  df-nel 2624  df-ral 2741  df-rex 2742  df-reu 2743  df-rmo 2744  df-rab 2745  df-v 3046  df-sbc 3267  df-csb 3363  df-dif 3406  df-un 3408  df-in 3410  df-ss 3417  df-pss 3419  df-nul 3731  df-if 3881  df-pw 3952  df-sn 3968  df-pr 3970  df-tp 3972  df-op 3974  df-uni 4198  df-int 4234  df-iun 4279  df-iin 4280  df-disj 4373  df-br 4402  df-opab 4461  df-mpt 4462  df-tr 4497  df-eprel 4744  df-id 4748  df-po 4754  df-so 4755  df-fr 4792  df-se 4793  df-we 4794  df-xp 4839  df-rel 4840  df-cnv 4841  df-co 4842  df-dm 4843  df-rn 4844  df-res 4845  df-ima 4846  df-pred 5379  df-ord 5425  df-on 5426  df-lim 5427  df-suc 5428  df-iota 5545  df-fun 5583  df-fn 5584  df-f 5585  df-f1 5586  df-fo 5587  df-f1o 5588  df-fv 5589  df-isom 5590  df-riota 6250  df-ov 6291  df-oprab 6292  df-mpt2 6293  df-of 6528  df-ofr 6529  df-om 6690  df-1st 6790  df-2nd 6791  df-supp 6912  df-wrecs 7025  df-recs 7087  df-rdg 7125  df-1o 7179  df-2o 7180  df-oadd 7183  df-omul 7184  df-er 7360  df-map 7471  df-pm 7472  df-ixp 7520  df-en 7567  df-dom 7568  df-sdom 7569  df-fin 7570  df-fsupp 7881  df-fi 7922  df-sup 7953  df-inf 7954  df-oi 8022  df-card 8370  df-acn 8373  df-cda 8595  df-pnf 9674  df-mnf 9675  df-xr 9676  df-ltxr 9677  df-le 9678  df-sub 9859  df-neg 9860  df-div 10267  df-nn 10607  df-2 10665  df-3 10666  df-4 10667  df-5 10668  df-6 10669  df-7 10670  df-8 10671  df-9 10672  df-10 10673  df-n0 10867  df-z 10935  df-dec 11049  df-uz 11157  df-q 11262  df-rp 11300  df-xneg 11406  df-xadd 11407  df-xmul 11408  df-ioo 11636  df-ioc 11637  df-ico 11638  df-icc 11639  df-fz 11782  df-fzo 11913  df-fl 12025  df-mod 12094  df-seq 12211  df-exp 12270  df-hash 12513  df-cj 13155  df-re 13156  df-im 13157  df-sqrt 13291  df-abs 13292  df-limsup 13519  df-clim 13545  df-rlim 13546  df-sum 13746  df-struct 15116  df-ndx 15117  df-slot 15118  df-base 15119  df-sets 15120  df-ress 15121  df-plusg 15196  df-mulr 15197  df-starv 15198  df-sca 15199  df-vsca 15200  df-ip 15201  df-tset 15202  df-ple 15203  df-ds 15205  df-unif 15206  df-hom 15207  df-cco 15208  df-rest 15314  df-topn 15315  df-0g 15333  df-gsum 15334  df-topgen 15335  df-pt 15336  df-prds 15339  df-xrs 15393  df-qtop 15399  df-imas 15400  df-xps 15403  df-mre 15485  df-mrc 15486  df-acs 15488  df-mgm 16481  df-sgrp 16520  df-mnd 16530  df-submnd 16576  df-mulg 16669  df-cntz 16964  df-cmn 17425  df-psmet 18955  df-xmet 18956  df-met 18957  df-bl 18958  df-mopn 18959  df-cnfld 18964  df-top 19914  df-bases 19915  df-topon 19916  df-topsp 19917  df-cn 20236  df-cnp 20237  df-cmp 20395  df-tx 20570  df-hmeo 20763  df-xms 21328  df-ms 21329  df-tms 21330  df-cncf 21903  df-ovol 22409  df-vol 22411  df-mbf 22570  df-itg1 22571  df-itg2 22572  df-ibl 22573  df-itg 22574  df-0p 22621  df-area 23875
This theorem is referenced by: (None)
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