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Theorem arearect 36171
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 11741 . . . . . . 7  |-  ( ( A  e.  RR  /\  B  e.  RR )  ->  ( A [,] B
)  C_  RR )
52, 3, 4mp2an 686 . . . . . 6  |-  ( A [,] B )  C_  RR
6 arearect.3 . . . . . . 7  |-  C  e.  RR
7 arearect.4 . . . . . . 7  |-  D  e.  RR
8 iccssre 11741 . . . . . . 7  |-  ( ( C  e.  RR  /\  D  e.  RR )  ->  ( C [,] D
)  C_  RR )
96, 7, 8mp2an 686 . . . . . 6  |-  ( C [,] D )  C_  RR
10 xpss12 4945 . . . . . 6  |-  ( ( ( A [,] B
)  C_  RR  /\  ( C [,] D )  C_  RR )  ->  ( ( A [,] B )  X.  ( C [,] D ) )  C_  ( RR  X.  RR ) )
115, 9, 10mp2an 686 . . . . 5  |-  ( ( A [,] B )  X.  ( C [,] D ) )  C_  ( RR  X.  RR )
121, 11eqsstri 3448 . . . 4  |-  S  C_  ( RR  X.  RR )
13 iftrue 3878 . . . . . . . . . . 11  |-  ( x  e.  ( A [,] B )  ->  if ( x  e.  ( A [,] B ) ,  ( D  -  C
) ,  0 )  =  ( D  -  C ) )
141imaeq1i 5171 . . . . . . . . . . . . . . 15  |-  ( S
" { x }
)  =  ( ( ( A [,] B
)  X.  ( C [,] D ) )
" { x }
)
15 iftrue 3878 . . . . . . . . . . . . . . . . 17  |-  ( x  e.  ( A [,] B )  ->  if ( x  e.  ( A [,] B ) ,  ( C [,] D
) ,  (/) )  =  ( C [,] D
) )
16 xpimasn 5288 . . . . . . . . . . . . . . . . 17  |-  ( x  e.  ( A [,] B )  ->  (
( ( A [,] B )  X.  ( C [,] D ) )
" { x }
)  =  ( C [,] D ) )
1715, 16eqtr4d 2508 . . . . . . . . . . . . . . . 16  |-  ( x  e.  ( A [,] B )  ->  if ( x  e.  ( A [,] B ) ,  ( C [,] D
) ,  (/) )  =  ( ( ( A [,] B )  X.  ( C [,] D
) ) " {
x } ) )
18 iffalse 3881 . . . . . . . . . . . . . . . . 17  |-  ( -.  x  e.  ( A [,] B )  ->  if ( x  e.  ( A [,] B ) ,  ( C [,] D ) ,  (/) )  =  (/) )
19 disjsn 4023 . . . . . . . . . . . . . . . . . 18  |-  ( ( ( A [,] B
)  i^i  { x } )  =  (/)  <->  -.  x  e.  ( A [,] B ) )
20 xpima1 5286 . . . . . . . . . . . . . . . . . 18  |-  ( ( ( A [,] B
)  i^i  { x } )  =  (/)  ->  ( ( ( A [,] B )  X.  ( C [,] D
) ) " {
x } )  =  (/) )
2119, 20sylbir 218 . . . . . . . . . . . . . . . . 17  |-  ( -.  x  e.  ( A [,] B )  -> 
( ( ( A [,] B )  X.  ( C [,] D
) ) " {
x } )  =  (/) )
2218, 21eqtr4d 2508 . . . . . . . . . . . . . . . 16  |-  ( -.  x  e.  ( A [,] B )  ->  if ( x  e.  ( A [,] B ) ,  ( C [,] D ) ,  (/) )  =  ( (
( A [,] B
)  X.  ( C [,] D ) )
" { x }
) )
2317, 22pm2.61i 169 . . . . . . . . . . . . . . 15  |-  if ( x  e.  ( A [,] B ) ,  ( C [,] D
) ,  (/) )  =  ( ( ( A [,] B )  X.  ( C [,] D
) ) " {
x } )
2414, 23eqtr4i 2496 . . . . . . . . . . . . . 14  |-  ( S
" { x }
)  =  if ( x  e.  ( A [,] B ) ,  ( C [,] D
) ,  (/) )
2524fveq2i 5882 . . . . . . . . . . . . 13  |-  ( vol `  ( S " {
x } ) )  =  ( vol `  if ( x  e.  ( A [,] B ) ,  ( C [,] D
) ,  (/) ) )
2615fveq2d 5883 . . . . . . . . . . . . 13  |-  ( x  e.  ( A [,] B )  ->  ( vol `  if ( x  e.  ( A [,] B ) ,  ( C [,] D ) ,  (/) ) )  =  ( vol `  ( C [,] D ) ) )
2725, 26syl5eq 2517 . . . . . . . . . . . 12  |-  ( x  e.  ( A [,] B )  ->  ( vol `  ( S " { x } ) )  =  ( vol `  ( C [,] D
) ) )
28 iccmbl 22598 . . . . . . . . . . . . . . 15  |-  ( ( C  e.  RR  /\  D  e.  RR )  ->  ( C [,] D
)  e.  dom  vol )
296, 7, 28mp2an 686 . . . . . . . . . . . . . 14  |-  ( C [,] D )  e. 
dom  vol
30 mblvol 22562 . . . . . . . . . . . . . 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 22555 . . . . . . . . . . . . . 14  |-  ( ( C  e.  RR  /\  D  e.  RR  /\  C  <_  D )  ->  ( vol* `  ( C [,] D ) )  =  ( D  -  C ) )
346, 7, 32, 33mp3an 1390 . . . . . . . . . . . . 13  |-  ( vol* `  ( C [,] D ) )  =  ( D  -  C
)
3531, 34eqtri 2493 . . . . . . . . . . . 12  |-  ( vol `  ( C [,] D
) )  =  ( D  -  C )
3627, 35syl6eq 2521 . . . . . . . . . . 11  |-  ( x  e.  ( A [,] B )  ->  ( vol `  ( S " { x } ) )  =  ( D  -  C ) )
3713, 36eqtr4d 2508 . . . . . . . . . 10  |-  ( x  e.  ( A [,] B )  ->  if ( x  e.  ( A [,] B ) ,  ( D  -  C
) ,  0 )  =  ( vol `  ( S " { x }
) ) )
38 iffalse 3881 . . . . . . . . . . 11  |-  ( -.  x  e.  ( A [,] B )  ->  if ( x  e.  ( A [,] B ) ,  ( D  -  C ) ,  0 )  =  0 )
3918fveq2d 5883 . . . . . . . . . . . . 13  |-  ( -.  x  e.  ( A [,] B )  -> 
( vol `  if ( x  e.  ( A [,] B ) ,  ( C [,] D
) ,  (/) ) )  =  ( vol `  (/) ) )
4025, 39syl5eq 2517 . . . . . . . . . . . 12  |-  ( -.  x  e.  ( A [,] B )  -> 
( vol `  ( S " { x }
) )  =  ( vol `  (/) ) )
41 0mbl 22571 . . . . . . . . . . . . . 14  |-  (/)  e.  dom  vol
42 mblvol 22562 . . . . . . . . . . . . . 14  |-  ( (/)  e.  dom  vol  ->  ( vol `  (/) )  =  ( vol* `  (/) ) )
4341, 42ax-mp 5 . . . . . . . . . . . . 13  |-  ( vol `  (/) )  =  ( vol* `  (/) )
44 ovol0 22524 . . . . . . . . . . . . 13  |-  ( vol* `  (/) )  =  0
4543, 44eqtri 2493 . . . . . . . . . . . 12  |-  ( vol `  (/) )  =  0
4640, 45syl6eq 2521 . . . . . . . . . . 11  |-  ( -.  x  e.  ( A [,] B )  -> 
( vol `  ( S " { x }
) )  =  0 )
4738, 46eqtr4d 2508 . . . . . . . . . 10  |-  ( -.  x  e.  ( A [,] B )  ->  if ( x  e.  ( A [,] B ) ,  ( D  -  C ) ,  0 )  =  ( vol `  ( S " {
x } ) ) )
4837, 47pm2.61i 169 . . . . . . . . 9  |-  if ( x  e.  ( A [,] B ) ,  ( D  -  C
) ,  0 )  =  ( vol `  ( S " { x }
) )
4948eqcomi 2480 . . . . . . . 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 9956 . . . . . . . 8  |-  ( D  -  C )  e.  RR
52 0re 9661 . . . . . . . 8  |-  0  e.  RR
5351, 52keepel 3939 . . . . . . 7  |-  if ( x  e.  ( A [,] B ) ,  ( D  -  C
) ,  0 )  e.  RR
5450, 53syl6eqel 2557 . . . . . 6  |-  ( x  e.  RR  ->  ( vol `  ( S " { x } ) )  e.  RR )
55 volf 22561 . . . . . . . 8  |-  vol : dom  vol --> ( 0 [,] +oo )
56 ffun 5742 . . . . . . . 8  |-  ( vol
: dom  vol --> ( 0 [,] +oo )  ->  Fun  vol )
5755, 56ax-mp 5 . . . . . . 7  |-  Fun  vol
5829, 41keepel 3939 . . . . . . . 8  |-  if ( x  e.  ( A [,] B ) ,  ( C [,] D
) ,  (/) )  e. 
dom  vol
5924, 58eqeltri 2545 . . . . . . 7  |-  ( S
" { x }
)  e.  dom  vol
60 fvimacnv 6012 . . . . . . 7  |-  ( ( Fun  vol  /\  ( S " { x }
)  e.  dom  vol )  ->  ( ( vol `  ( S " {
x } ) )  e.  RR  <->  ( S " { x } )  e.  ( `' vol " RR ) ) )
6157, 59, 60mp2an 686 . . . . . 6  |-  ( ( vol `  ( S
" { x }
) )  e.  RR  <->  ( S " { x } )  e.  ( `' vol " RR ) )
6254, 61sylib 201 . . . . 5  |-  ( x  e.  RR  ->  ( S " { x }
)  e.  ( `' vol " RR ) )
6362rgen 2766 . . . 4  |-  A. x  e.  RR  ( S " { x } )  e.  ( `' vol " RR )
645a1i 11 . . . . . 6  |-  ( 0  e.  RR  ->  ( A [,] B )  C_  RR )
65 rembl 22572 . . . . . . 7  |-  RR  e.  dom  vol
6665a1i 11 . . . . . 6  |-  ( 0  e.  RR  ->  RR  e.  dom  vol )
6736, 51syl6eqel 2557 . . . . . . 7  |-  ( x  e.  ( A [,] B )  ->  ( vol `  ( S " { x } ) )  e.  RR )
6867adantl 473 . . . . . 6  |-  ( ( 0  e.  RR  /\  x  e.  ( A [,] B ) )  -> 
( vol `  ( S " { x }
) )  e.  RR )
69 eldifn 3545 . . . . . . . 8  |-  ( x  e.  ( RR  \ 
( A [,] B
) )  ->  -.  x  e.  ( A [,] B ) )
7069, 46syl 17 . . . . . . 7  |-  ( x  e.  ( RR  \ 
( A [,] B
) )  ->  ( vol `  ( S " { x } ) )  =  0 )
7170adantl 473 . . . . . 6  |-  ( ( 0  e.  RR  /\  x  e.  ( RR  \  ( A [,] B
) ) )  -> 
( vol `  ( S " { x }
) )  =  0 )
7236mpteq2ia 4478 . . . . . . . 8  |-  ( x  e.  ( A [,] B )  |->  ( vol `  ( S " {
x } ) ) )  =  ( x  e.  ( A [,] B )  |->  ( D  -  C ) )
7351recni 9673 . . . . . . . . . 10  |-  ( D  -  C )  e.  CC
74 ax-resscn 9614 . . . . . . . . . . 11  |-  RR  C_  CC
755, 74sstri 3427 . . . . . . . . . 10  |-  ( A [,] B )  C_  CC
76 ssid 3437 . . . . . . . . . 10  |-  CC  C_  CC
77 cncfmptc 22021 . . . . . . . . . 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 1390 . . . . . . . . 9  |-  ( x  e.  ( A [,] B )  |->  ( D  -  C ) )  e.  ( ( A [,] B ) -cn-> CC )
79 cniccibl 22877 . . . . . . . . 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 1390 . . . . . . . 8  |-  ( x  e.  ( A [,] B )  |->  ( D  -  C ) )  e.  L^1
8172, 80eqeltri 2545 . . . . . . 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 22842 . . . . 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 23962 . . . 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 1212 . . 3  |-  S  e. 
dom area
87 areaval 23969 . . 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 22814 . . . 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 2770 . . 3  |-  S. RR ( vol `  ( S
" { x }
) )  _d x  =  S. RR if ( x  e.  ( A [,] B ) ,  ( D  -  C
) ,  0 )  _d x
91 iccmbl 22598 . . . . . 6  |-  ( ( A  e.  RR  /\  B  e.  RR )  ->  ( A [,] B
)  e.  dom  vol )
922, 3, 91mp2an 686 . . . . 5  |-  ( A [,] B )  e. 
dom  vol
93 mblvol 22562 . . . . . . . 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 22555 . . . . . . . 8  |-  ( ( A  e.  RR  /\  B  e.  RR  /\  A  <_  B )  ->  ( vol* `  ( A [,] B ) )  =  ( B  -  A ) )
972, 3, 95, 96mp3an 1390 . . . . . . 7  |-  ( vol* `  ( A [,] B ) )  =  ( B  -  A
)
9894, 97eqtri 2493 . . . . . 6  |-  ( vol `  ( A [,] B
) )  =  ( B  -  A )
993, 2resubcli 9956 . . . . . 6  |-  ( B  -  A )  e.  RR
10098, 99eqeltri 2545 . . . . 5  |-  ( vol `  ( A [,] B
) )  e.  RR
101 itgconst 22855 . . . . 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 1390 . . . 4  |-  S. ( A [,] B ) ( D  -  C
)  _d x  =  ( ( D  -  C )  x.  ( vol `  ( A [,] B ) ) )
103 itgss2 22849 . . . . 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 6319 . . . 4  |-  ( ( D  -  C )  x.  ( vol `  ( A [,] B ) ) )  =  ( ( D  -  C )  x.  ( B  -  A ) )
106102, 104, 1053eqtr3i 2501 . . 3  |-  S. RR if ( x  e.  ( A [,] B ) ,  ( D  -  C ) ,  0 )  _d x  =  ( ( D  -  C )  x.  ( B  -  A )
)
10790, 106eqtri 2493 . 2  |-  S. RR ( vol `  ( S
" { x }
) )  _d x  =  ( ( D  -  C )  x.  ( B  -  A
) )
10899recni 9673 . . 3  |-  ( B  -  A )  e.  CC
10973, 108mulcomi 9667 . 2  |-  ( ( D  -  C )  x.  ( B  -  A ) )  =  ( ( B  -  A )  x.  ( D  -  C )
)
11088, 107, 1093eqtri 2497 1  |-  (area `  S )  =  ( ( B  -  A
)  x.  ( D  -  C ) )
Colors of variables: wff setvar class
Syntax hints:   -. wn 3    <-> wb 189    = wceq 1452    e. wcel 1904   A.wral 2756    \ cdif 3387    i^i cin 3389    C_ wss 3390   (/)c0 3722   ifcif 3872   {csn 3959   class class class wbr 4395    |-> cmpt 4454    X. cxp 4837   `'ccnv 4838   dom cdm 4839   "cima 4842   Fun wfun 5583   -->wf 5585   ` cfv 5589  (class class class)co 6308   CCcc 9555   RRcr 9556   0cc0 9557    x. cmul 9562   +oocpnf 9690    <_ cle 9694    - cmin 9880   [,]cicc 11663   -cn->ccncf 21986   vol*covol 22491   volcvol 22493   L^1cibl 22654   S.citg 22655  areacarea 23960
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1677  ax-4 1690  ax-5 1766  ax-6 1813  ax-7 1859  ax-8 1906  ax-9 1913  ax-10 1932  ax-11 1937  ax-12 1950  ax-13 2104  ax-ext 2451  ax-rep 4508  ax-sep 4518  ax-nul 4527  ax-pow 4579  ax-pr 4639  ax-un 6602  ax-inf2 8164  ax-cc 8883  ax-cnex 9613  ax-resscn 9614  ax-1cn 9615  ax-icn 9616  ax-addcl 9617  ax-addrcl 9618  ax-mulcl 9619  ax-mulrcl 9620  ax-mulcom 9621  ax-addass 9622  ax-mulass 9623  ax-distr 9624  ax-i2m1 9625  ax-1ne0 9626  ax-1rid 9627  ax-rnegex 9628  ax-rrecex 9629  ax-cnre 9630  ax-pre-lttri 9631  ax-pre-lttrn 9632  ax-pre-ltadd 9633  ax-pre-mulgt0 9634  ax-pre-sup 9635  ax-addf 9636  ax-mulf 9637
This theorem depends on definitions:  df-bi 190  df-or 377  df-an 378  df-3or 1008  df-3an 1009  df-tru 1455  df-fal 1458  df-ex 1672  df-nf 1676  df-sb 1806  df-eu 2323  df-mo 2324  df-clab 2458  df-cleq 2464  df-clel 2467  df-nfc 2601  df-ne 2643  df-nel 2644  df-ral 2761  df-rex 2762  df-reu 2763  df-rmo 2764  df-rab 2765  df-v 3033  df-sbc 3256  df-csb 3350  df-dif 3393  df-un 3395  df-in 3397  df-ss 3404  df-pss 3406  df-nul 3723  df-if 3873  df-pw 3944  df-sn 3960  df-pr 3962  df-tp 3964  df-op 3966  df-uni 4191  df-int 4227  df-iun 4271  df-iin 4272  df-disj 4367  df-br 4396  df-opab 4455  df-mpt 4456  df-tr 4491  df-eprel 4750  df-id 4754  df-po 4760  df-so 4761  df-fr 4798  df-se 4799  df-we 4800  df-xp 4845  df-rel 4846  df-cnv 4847  df-co 4848  df-dm 4849  df-rn 4850  df-res 4851  df-ima 4852  df-pred 5387  df-ord 5433  df-on 5434  df-lim 5435  df-suc 5436  df-iota 5553  df-fun 5591  df-fn 5592  df-f 5593  df-f1 5594  df-fo 5595  df-f1o 5596  df-fv 5597  df-isom 5598  df-riota 6270  df-ov 6311  df-oprab 6312  df-mpt2 6313  df-of 6550  df-ofr 6551  df-om 6712  df-1st 6812  df-2nd 6813  df-supp 6934  df-wrecs 7046  df-recs 7108  df-rdg 7146  df-1o 7200  df-2o 7201  df-oadd 7204  df-omul 7205  df-er 7381  df-map 7492  df-pm 7493  df-ixp 7541  df-en 7588  df-dom 7589  df-sdom 7590  df-fin 7591  df-fsupp 7902  df-fi 7943  df-sup 7974  df-inf 7975  df-oi 8043  df-card 8391  df-acn 8394  df-cda 8616  df-pnf 9695  df-mnf 9696  df-xr 9697  df-ltxr 9698  df-le 9699  df-sub 9882  df-neg 9883  df-div 10292  df-nn 10632  df-2 10690  df-3 10691  df-4 10692  df-5 10693  df-6 10694  df-7 10695  df-8 10696  df-9 10697  df-10 10698  df-n0 10894  df-z 10962  df-dec 11075  df-uz 11183  df-q 11288  df-rp 11326  df-xneg 11432  df-xadd 11433  df-xmul 11434  df-ioo 11664  df-ioc 11665  df-ico 11666  df-icc 11667  df-fz 11811  df-fzo 11943  df-fl 12061  df-mod 12130  df-seq 12252  df-exp 12311  df-hash 12554  df-cj 13239  df-re 13240  df-im 13241  df-sqrt 13375  df-abs 13376  df-limsup 13603  df-clim 13629  df-rlim 13630  df-sum 13830  df-struct 15201  df-ndx 15202  df-slot 15203  df-base 15204  df-sets 15205  df-ress 15206  df-plusg 15281  df-mulr 15282  df-starv 15283  df-sca 15284  df-vsca 15285  df-ip 15286  df-tset 15287  df-ple 15288  df-ds 15290  df-unif 15291  df-hom 15292  df-cco 15293  df-rest 15399  df-topn 15400  df-0g 15418  df-gsum 15419  df-topgen 15420  df-pt 15421  df-prds 15424  df-xrs 15478  df-qtop 15484  df-imas 15485  df-xps 15488  df-mre 15570  df-mrc 15571  df-acs 15573  df-mgm 16566  df-sgrp 16605  df-mnd 16615  df-submnd 16661  df-mulg 16754  df-cntz 17049  df-cmn 17510  df-psmet 19039  df-xmet 19040  df-met 19041  df-bl 19042  df-mopn 19043  df-cnfld 19048  df-top 19998  df-bases 19999  df-topon 20000  df-topsp 20001  df-cn 20320  df-cnp 20321  df-cmp 20479  df-tx 20654  df-hmeo 20847  df-xms 21413  df-ms 21414  df-tms 21415  df-cncf 21988  df-ovol 22494  df-vol 22496  df-mbf 22656  df-itg1 22657  df-itg2 22658  df-ibl 22659  df-itg 22660  df-0p 22707  df-area 23961
This theorem is referenced by: (None)
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