Users' Mathboxes Mathbox for Jon Pennant < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >   Mathboxes  >  arearect Structured version   Unicode version

Theorem arearect 31152
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 11610 . . . . . . 7  |-  ( ( A  e.  RR  /\  B  e.  RR )  ->  ( A [,] B
)  C_  RR )
52, 3, 4mp2an 672 . . . . . 6  |-  ( A [,] B )  C_  RR
6 arearect.3 . . . . . . 7  |-  C  e.  RR
7 arearect.4 . . . . . . 7  |-  D  e.  RR
8 iccssre 11610 . . . . . . 7  |-  ( ( C  e.  RR  /\  D  e.  RR )  ->  ( C [,] D
)  C_  RR )
96, 7, 8mp2an 672 . . . . . 6  |-  ( C [,] D )  C_  RR
10 xpss12 5094 . . . . . 6  |-  ( ( ( A [,] B
)  C_  RR  /\  ( C [,] D )  C_  RR )  ->  ( ( A [,] B )  X.  ( C [,] D ) )  C_  ( RR  X.  RR ) )
115, 9, 10mp2an 672 . . . . 5  |-  ( ( A [,] B )  X.  ( C [,] D ) )  C_  ( RR  X.  RR )
121, 11eqsstri 3516 . . . 4  |-  S  C_  ( RR  X.  RR )
13 iftrue 3928 . . . . . . . . . . 11  |-  ( x  e.  ( A [,] B )  ->  if ( x  e.  ( A [,] B ) ,  ( D  -  C
) ,  0 )  =  ( D  -  C ) )
141imaeq1i 5320 . . . . . . . . . . . . . . 15  |-  ( S
" { x }
)  =  ( ( ( A [,] B
)  X.  ( C [,] D ) )
" { x }
)
15 iftrue 3928 . . . . . . . . . . . . . . . . 17  |-  ( x  e.  ( A [,] B )  ->  if ( x  e.  ( A [,] B ) ,  ( C [,] D
) ,  (/) )  =  ( C [,] D
) )
16 xpimasn 5438 . . . . . . . . . . . . . . . . 17  |-  ( x  e.  ( A [,] B )  ->  (
( ( A [,] B )  X.  ( C [,] D ) )
" { x }
)  =  ( C [,] D ) )
1715, 16eqtr4d 2485 . . . . . . . . . . . . . . . 16  |-  ( x  e.  ( A [,] B )  ->  if ( x  e.  ( A [,] B ) ,  ( C [,] D
) ,  (/) )  =  ( ( ( A [,] B )  X.  ( C [,] D
) ) " {
x } ) )
18 iffalse 3931 . . . . . . . . . . . . . . . . 17  |-  ( -.  x  e.  ( A [,] B )  ->  if ( x  e.  ( A [,] B ) ,  ( C [,] D ) ,  (/) )  =  (/) )
19 disjsn 4071 . . . . . . . . . . . . . . . . . 18  |-  ( ( ( A [,] B
)  i^i  { x } )  =  (/)  <->  -.  x  e.  ( A [,] B ) )
20 xpima1 5436 . . . . . . . . . . . . . . . . . 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 2485 . . . . . . . . . . . . . . . 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 2473 . . . . . . . . . . . . . 14  |-  ( S
" { x }
)  =  if ( x  e.  ( A [,] B ) ,  ( C [,] D
) ,  (/) )
2524fveq2i 5855 . . . . . . . . . . . . 13  |-  ( vol `  ( S " {
x } ) )  =  ( vol `  if ( x  e.  ( A [,] B ) ,  ( C [,] D
) ,  (/) ) )
2615fveq2d 5856 . . . . . . . . . . . . 13  |-  ( x  e.  ( A [,] B )  ->  ( vol `  if ( x  e.  ( A [,] B ) ,  ( C [,] D ) ,  (/) ) )  =  ( vol `  ( C [,] D ) ) )
2725, 26syl5eq 2494 . . . . . . . . . . . 12  |-  ( x  e.  ( A [,] B )  ->  ( vol `  ( S " { x } ) )  =  ( vol `  ( C [,] D
) ) )
28 iccmbl 21842 . . . . . . . . . . . . . . 15  |-  ( ( C  e.  RR  /\  D  e.  RR )  ->  ( C [,] D
)  e.  dom  vol )
296, 7, 28mp2an 672 . . . . . . . . . . . . . 14  |-  ( C [,] D )  e. 
dom  vol
30 mblvol 21807 . . . . . . . . . . . . . 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 21800 . . . . . . . . . . . . . 14  |-  ( ( C  e.  RR  /\  D  e.  RR  /\  C  <_  D )  ->  ( vol* `  ( C [,] D ) )  =  ( D  -  C ) )
346, 7, 32, 33mp3an 1323 . . . . . . . . . . . . 13  |-  ( vol* `  ( C [,] D ) )  =  ( D  -  C
)
3531, 34eqtri 2470 . . . . . . . . . . . 12  |-  ( vol `  ( C [,] D
) )  =  ( D  -  C )
3627, 35syl6eq 2498 . . . . . . . . . . 11  |-  ( x  e.  ( A [,] B )  ->  ( vol `  ( S " { x } ) )  =  ( D  -  C ) )
3713, 36eqtr4d 2485 . . . . . . . . . 10  |-  ( x  e.  ( A [,] B )  ->  if ( x  e.  ( A [,] B ) ,  ( D  -  C
) ,  0 )  =  ( vol `  ( S " { x }
) ) )
38 iffalse 3931 . . . . . . . . . . 11  |-  ( -.  x  e.  ( A [,] B )  ->  if ( x  e.  ( A [,] B ) ,  ( D  -  C ) ,  0 )  =  0 )
3918fveq2d 5856 . . . . . . . . . . . . 13  |-  ( -.  x  e.  ( A [,] B )  -> 
( vol `  if ( x  e.  ( A [,] B ) ,  ( C [,] D
) ,  (/) ) )  =  ( vol `  (/) ) )
4025, 39syl5eq 2494 . . . . . . . . . . . 12  |-  ( -.  x  e.  ( A [,] B )  -> 
( vol `  ( S " { x }
) )  =  ( vol `  (/) ) )
41 0mbl 21816 . . . . . . . . . . . . . 14  |-  (/)  e.  dom  vol
42 mblvol 21807 . . . . . . . . . . . . . 14  |-  ( (/)  e.  dom  vol  ->  ( vol `  (/) )  =  ( vol* `  (/) ) )
4341, 42ax-mp 5 . . . . . . . . . . . . 13  |-  ( vol `  (/) )  =  ( vol* `  (/) )
44 ovol0 21770 . . . . . . . . . . . . 13  |-  ( vol* `  (/) )  =  0
4543, 44eqtri 2470 . . . . . . . . . . . 12  |-  ( vol `  (/) )  =  0
4640, 45syl6eq 2498 . . . . . . . . . . 11  |-  ( -.  x  e.  ( A [,] B )  -> 
( vol `  ( S " { x }
) )  =  0 )
4738, 46eqtr4d 2485 . . . . . . . . . 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 2454 . . . . . . . 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 9881 . . . . . . . 8  |-  ( D  -  C )  e.  RR
52 0re 9594 . . . . . . . 8  |-  0  e.  RR
5351, 52keepel 3990 . . . . . . 7  |-  if ( x  e.  ( A [,] B ) ,  ( D  -  C
) ,  0 )  e.  RR
5450, 53syl6eqel 2537 . . . . . 6  |-  ( x  e.  RR  ->  ( vol `  ( S " { x } ) )  e.  RR )
55 volf 21806 . . . . . . . 8  |-  vol : dom  vol --> ( 0 [,] +oo )
56 ffun 5719 . . . . . . . 8  |-  ( vol
: dom  vol --> ( 0 [,] +oo )  ->  Fun  vol )
5755, 56ax-mp 5 . . . . . . 7  |-  Fun  vol
5829, 41keepel 3990 . . . . . . . 8  |-  if ( x  e.  ( A [,] B ) ,  ( C [,] D
) ,  (/) )  e. 
dom  vol
5924, 58eqeltri 2525 . . . . . . 7  |-  ( S
" { x }
)  e.  dom  vol
60 fvimacnv 5983 . . . . . . 7  |-  ( ( Fun  vol  /\  ( S " { x }
)  e.  dom  vol )  ->  ( ( vol `  ( S " {
x } ) )  e.  RR  <->  ( S " { x } )  e.  ( `' vol " RR ) ) )
6157, 59, 60mp2an 672 . . . . . 6  |-  ( ( vol `  ( S
" { x }
) )  e.  RR  <->  ( S " { x } )  e.  ( `' vol " RR ) )
6254, 61sylib 196 . . . . 5  |-  ( x  e.  RR  ->  ( S " { x }
)  e.  ( `' vol " RR ) )
6362rgen 2801 . . . 4  |-  A. x  e.  RR  ( S " { x } )  e.  ( `' vol " RR )
645a1i 11 . . . . . 6  |-  ( 0  e.  RR  ->  ( A [,] B )  C_  RR )
65 rembl 21817 . . . . . . 7  |-  RR  e.  dom  vol
6665a1i 11 . . . . . 6  |-  ( 0  e.  RR  ->  RR  e.  dom  vol )
6736, 51syl6eqel 2537 . . . . . . 7  |-  ( x  e.  ( A [,] B )  ->  ( vol `  ( S " { x } ) )  e.  RR )
6867adantl 466 . . . . . 6  |-  ( ( 0  e.  RR  /\  x  e.  ( A [,] B ) )  -> 
( vol `  ( S " { x }
) )  e.  RR )
69 eldifn 3609 . . . . . . . 8  |-  ( x  e.  ( RR  \ 
( A [,] B
) )  ->  -.  x  e.  ( A [,] B ) )
7069, 46syl 16 . . . . . . 7  |-  ( x  e.  ( RR  \ 
( A [,] B
) )  ->  ( vol `  ( S " { x } ) )  =  0 )
7170adantl 466 . . . . . 6  |-  ( ( 0  e.  RR  /\  x  e.  ( RR  \  ( A [,] B
) ) )  -> 
( vol `  ( S " { x }
) )  =  0 )
7236mpteq2ia 4515 . . . . . . . 8  |-  ( x  e.  ( A [,] B )  |->  ( vol `  ( S " {
x } ) ) )  =  ( x  e.  ( A [,] B )  |->  ( D  -  C ) )
7351recni 9606 . . . . . . . . . 10  |-  ( D  -  C )  e.  CC
74 ax-resscn 9547 . . . . . . . . . . 11  |-  RR  C_  CC
755, 74sstri 3495 . . . . . . . . . 10  |-  ( A [,] B )  C_  CC
76 ssid 3505 . . . . . . . . . 10  |-  CC  C_  CC
77 cncfmptc 21281 . . . . . . . . . 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 1323 . . . . . . . . 9  |-  ( x  e.  ( A [,] B )  |->  ( D  -  C ) )  e.  ( ( A [,] B ) -cn-> CC )
79 cniccibl 22113 . . . . . . . . 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 1323 . . . . . . . 8  |-  ( x  e.  ( A [,] B )  |->  ( D  -  C ) )  e.  L^1
8172, 80eqeltri 2525 . . . . . . 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 22078 . . . . 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 23152 . . . 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 1177 . . 3  |-  S  e. 
dom area
87 areaval 23159 . . 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 22050 . . . 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 2804 . . 3  |-  S. RR ( vol `  ( S
" { x }
) )  _d x  =  S. RR if ( x  e.  ( A [,] B ) ,  ( D  -  C
) ,  0 )  _d x
91 iccmbl 21842 . . . . . 6  |-  ( ( A  e.  RR  /\  B  e.  RR )  ->  ( A [,] B
)  e.  dom  vol )
922, 3, 91mp2an 672 . . . . 5  |-  ( A [,] B )  e. 
dom  vol
93 mblvol 21807 . . . . . . . 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 21800 . . . . . . . 8  |-  ( ( A  e.  RR  /\  B  e.  RR  /\  A  <_  B )  ->  ( vol* `  ( A [,] B ) )  =  ( B  -  A ) )
972, 3, 95, 96mp3an 1323 . . . . . . 7  |-  ( vol* `  ( A [,] B ) )  =  ( B  -  A
)
9894, 97eqtri 2470 . . . . . 6  |-  ( vol `  ( A [,] B
) )  =  ( B  -  A )
993, 2resubcli 9881 . . . . . 6  |-  ( B  -  A )  e.  RR
10098, 99eqeltri 2525 . . . . 5  |-  ( vol `  ( A [,] B
) )  e.  RR
101 itgconst 22091 . . . . 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 1323 . . . 4  |-  S. ( A [,] B ) ( D  -  C
)  _d x  =  ( ( D  -  C )  x.  ( vol `  ( A [,] B ) ) )
103 itgss2 22085 . . . . 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 6288 . . . 4  |-  ( ( D  -  C )  x.  ( vol `  ( A [,] B ) ) )  =  ( ( D  -  C )  x.  ( B  -  A ) )
106102, 104, 1053eqtr3i 2478 . . 3  |-  S. RR if ( x  e.  ( A [,] B ) ,  ( D  -  C ) ,  0 )  _d x  =  ( ( D  -  C )  x.  ( B  -  A )
)
10790, 106eqtri 2470 . 2  |-  S. RR ( vol `  ( S
" { x }
) )  _d x  =  ( ( D  -  C )  x.  ( B  -  A
) )
10899recni 9606 . . 3  |-  ( B  -  A )  e.  CC
10973, 108mulcomi 9600 . 2  |-  ( ( D  -  C )  x.  ( B  -  A ) )  =  ( ( B  -  A )  x.  ( D  -  C )
)
11088, 107, 1093eqtri 2474 1  |-  (area `  S )  =  ( ( B  -  A
)  x.  ( D  -  C ) )
Colors of variables: wff setvar class
Syntax hints:   -. wn 3    <-> wb 184    = wceq 1381    e. wcel 1802   A.wral 2791    \ cdif 3455    i^i cin 3457    C_ wss 3458   (/)c0 3767   ifcif 3922   {csn 4010   class class class wbr 4433    |-> cmpt 4491    X. cxp 4983   `'ccnv 4984   dom cdm 4985   "cima 4988   Fun wfun 5568   -->wf 5570   ` cfv 5574  (class class class)co 6277   CCcc 9488   RRcr 9489   0cc0 9490    x. cmul 9495   +oocpnf 9623    <_ cle 9627    - cmin 9805   [,]cicc 11536   -cn->ccncf 21246   vol*covol 21740   volcvol 21741   L^1cibl 21892   S.citg 21893  areacarea 23150
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1603  ax-4 1616  ax-5 1689  ax-6 1732  ax-7 1774  ax-8 1804  ax-9 1806  ax-10 1821  ax-11 1826  ax-12 1838  ax-13 1983  ax-ext 2419  ax-rep 4544  ax-sep 4554  ax-nul 4562  ax-pow 4611  ax-pr 4672  ax-un 6573  ax-inf2 8056  ax-cc 8813  ax-cnex 9546  ax-resscn 9547  ax-1cn 9548  ax-icn 9549  ax-addcl 9550  ax-addrcl 9551  ax-mulcl 9552  ax-mulrcl 9553  ax-mulcom 9554  ax-addass 9555  ax-mulass 9556  ax-distr 9557  ax-i2m1 9558  ax-1ne0 9559  ax-1rid 9560  ax-rnegex 9561  ax-rrecex 9562  ax-cnre 9563  ax-pre-lttri 9564  ax-pre-lttrn 9565  ax-pre-ltadd 9566  ax-pre-mulgt0 9567  ax-pre-sup 9568  ax-addf 9569  ax-mulf 9570
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 973  df-3an 974  df-tru 1384  df-fal 1387  df-ex 1598  df-nf 1602  df-sb 1725  df-eu 2270  df-mo 2271  df-clab 2427  df-cleq 2433  df-clel 2436  df-nfc 2591  df-ne 2638  df-nel 2639  df-ral 2796  df-rex 2797  df-reu 2798  df-rmo 2799  df-rab 2800  df-v 3095  df-sbc 3312  df-csb 3418  df-dif 3461  df-un 3463  df-in 3465  df-ss 3472  df-pss 3474  df-nul 3768  df-if 3923  df-pw 3995  df-sn 4011  df-pr 4013  df-tp 4015  df-op 4017  df-uni 4231  df-int 4268  df-iun 4313  df-iin 4314  df-disj 4404  df-br 4434  df-opab 4492  df-mpt 4493  df-tr 4527  df-eprel 4777  df-id 4781  df-po 4786  df-so 4787  df-fr 4824  df-se 4825  df-we 4826  df-ord 4867  df-on 4868  df-lim 4869  df-suc 4870  df-xp 4991  df-rel 4992  df-cnv 4993  df-co 4994  df-dm 4995  df-rn 4996  df-res 4997  df-ima 4998  df-iota 5537  df-fun 5576  df-fn 5577  df-f 5578  df-f1 5579  df-fo 5580  df-f1o 5581  df-fv 5582  df-isom 5583  df-riota 6238  df-ov 6280  df-oprab 6281  df-mpt2 6282  df-of 6521  df-ofr 6522  df-om 6682  df-1st 6781  df-2nd 6782  df-supp 6900  df-recs 7040  df-rdg 7074  df-1o 7128  df-2o 7129  df-oadd 7132  df-omul 7133  df-er 7309  df-map 7420  df-pm 7421  df-ixp 7468  df-en 7515  df-dom 7516  df-sdom 7517  df-fin 7518  df-fsupp 7828  df-fi 7869  df-sup 7899  df-oi 7933  df-card 8318  df-acn 8321  df-cda 8546  df-pnf 9628  df-mnf 9629  df-xr 9630  df-ltxr 9631  df-le 9632  df-sub 9807  df-neg 9808  df-div 10208  df-nn 10538  df-2 10595  df-3 10596  df-4 10597  df-5 10598  df-6 10599  df-7 10600  df-8 10601  df-9 10602  df-10 10603  df-n0 10797  df-z 10866  df-dec 10980  df-uz 11086  df-q 11187  df-rp 11225  df-xneg 11322  df-xadd 11323  df-xmul 11324  df-ioo 11537  df-ioc 11538  df-ico 11539  df-icc 11540  df-fz 11677  df-fzo 11799  df-fl 11903  df-mod 11971  df-seq 12082  df-exp 12141  df-hash 12380  df-cj 12906  df-re 12907  df-im 12908  df-sqrt 13042  df-abs 13043  df-limsup 13268  df-clim 13285  df-rlim 13286  df-sum 13483  df-struct 14506  df-ndx 14507  df-slot 14508  df-base 14509  df-sets 14510  df-ress 14511  df-plusg 14582  df-mulr 14583  df-starv 14584  df-sca 14585  df-vsca 14586  df-ip 14587  df-tset 14588  df-ple 14589  df-ds 14591  df-unif 14592  df-hom 14593  df-cco 14594  df-rest 14692  df-topn 14693  df-0g 14711  df-gsum 14712  df-topgen 14713  df-pt 14714  df-prds 14717  df-xrs 14771  df-qtop 14776  df-imas 14777  df-xps 14779  df-mre 14855  df-mrc 14856  df-acs 14858  df-mgm 15741  df-sgrp 15780  df-mnd 15790  df-submnd 15836  df-mulg 15929  df-cntz 16224  df-cmn 16669  df-psmet 18279  df-xmet 18280  df-met 18281  df-bl 18282  df-mopn 18283  df-cnfld 18289  df-top 19266  df-bases 19268  df-topon 19269  df-topsp 19270  df-cn 19594  df-cnp 19595  df-cmp 19753  df-tx 19929  df-hmeo 20122  df-xms 20689  df-ms 20690  df-tms 20691  df-cncf 21248  df-ovol 21742  df-vol 21743  df-mbf 21894  df-itg1 21895  df-itg2 21896  df-ibl 21897  df-itg 21898  df-0p 21943  df-area 23151
This theorem is referenced by: (None)
  Copyright terms: Public domain W3C validator