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Theorem arearect 30779
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 11597 . . . . . . 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 11597 . . . . . . 7  |-  ( ( C  e.  RR  /\  D  e.  RR )  ->  ( C [,] D
)  C_  RR )
96, 7, 8mp2an 672 . . . . . 6  |-  ( C [,] D )  C_  RR
10 xpss12 5101 . . . . . 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 3529 . . . 4  |-  S  C_  ( RR  X.  RR )
13 iftrue 3940 . . . . . . . . . . 11  |-  ( x  e.  ( A [,] B )  ->  if ( x  e.  ( A [,] B ) ,  ( D  -  C
) ,  0 )  =  ( D  -  C ) )
141imaeq1i 5327 . . . . . . . . . . . . . . 15  |-  ( S
" { x }
)  =  ( ( ( A [,] B
)  X.  ( C [,] D ) )
" { x }
)
15 iftrue 3940 . . . . . . . . . . . . . . . . 17  |-  ( x  e.  ( A [,] B )  ->  if ( x  e.  ( A [,] B ) ,  ( C [,] D
) ,  (/) )  =  ( C [,] D
) )
16 xpimasn 5445 . . . . . . . . . . . . . . . . 17  |-  ( x  e.  ( A [,] B )  ->  (
( ( A [,] B )  X.  ( C [,] D ) )
" { x }
)  =  ( C [,] D ) )
1715, 16eqtr4d 2506 . . . . . . . . . . . . . . . 16  |-  ( x  e.  ( A [,] B )  ->  if ( x  e.  ( A [,] B ) ,  ( C [,] D
) ,  (/) )  =  ( ( ( A [,] B )  X.  ( C [,] D
) ) " {
x } ) )
18 iffalse 3943 . . . . . . . . . . . . . . . . 17  |-  ( -.  x  e.  ( A [,] B )  ->  if ( x  e.  ( A [,] B ) ,  ( C [,] D ) ,  (/) )  =  (/) )
19 disjsn 4083 . . . . . . . . . . . . . . . . . 18  |-  ( ( ( A [,] B
)  i^i  { x } )  =  (/)  <->  -.  x  e.  ( A [,] B ) )
20 xpima1 5443 . . . . . . . . . . . . . . . . . 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 2506 . . . . . . . . . . . . . . . 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 2494 . . . . . . . . . . . . . 14  |-  ( S
" { x }
)  =  if ( x  e.  ( A [,] B ) ,  ( C [,] D
) ,  (/) )
2524fveq2i 5862 . . . . . . . . . . . . 13  |-  ( vol `  ( S " {
x } ) )  =  ( vol `  if ( x  e.  ( A [,] B ) ,  ( C [,] D
) ,  (/) ) )
2615fveq2d 5863 . . . . . . . . . . . . 13  |-  ( x  e.  ( A [,] B )  ->  ( vol `  if ( x  e.  ( A [,] B ) ,  ( C [,] D ) ,  (/) ) )  =  ( vol `  ( C [,] D ) ) )
2725, 26syl5eq 2515 . . . . . . . . . . . 12  |-  ( x  e.  ( A [,] B )  ->  ( vol `  ( S " { x } ) )  =  ( vol `  ( C [,] D
) ) )
28 iccmbl 21706 . . . . . . . . . . . . . . 15  |-  ( ( C  e.  RR  /\  D  e.  RR )  ->  ( C [,] D
)  e.  dom  vol )
296, 7, 28mp2an 672 . . . . . . . . . . . . . 14  |-  ( C [,] D )  e. 
dom  vol
30 mblvol 21671 . . . . . . . . . . . . . 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 21664 . . . . . . . . . . . . . 14  |-  ( ( C  e.  RR  /\  D  e.  RR  /\  C  <_  D )  ->  ( vol* `  ( C [,] D ) )  =  ( D  -  C ) )
346, 7, 32, 33mp3an 1319 . . . . . . . . . . . . 13  |-  ( vol* `  ( C [,] D ) )  =  ( D  -  C
)
3531, 34eqtri 2491 . . . . . . . . . . . 12  |-  ( vol `  ( C [,] D
) )  =  ( D  -  C )
3627, 35syl6eq 2519 . . . . . . . . . . 11  |-  ( x  e.  ( A [,] B )  ->  ( vol `  ( S " { x } ) )  =  ( D  -  C ) )
3713, 36eqtr4d 2506 . . . . . . . . . 10  |-  ( x  e.  ( A [,] B )  ->  if ( x  e.  ( A [,] B ) ,  ( D  -  C
) ,  0 )  =  ( vol `  ( S " { x }
) ) )
38 iffalse 3943 . . . . . . . . . . 11  |-  ( -.  x  e.  ( A [,] B )  ->  if ( x  e.  ( A [,] B ) ,  ( D  -  C ) ,  0 )  =  0 )
3918fveq2d 5863 . . . . . . . . . . . . 13  |-  ( -.  x  e.  ( A [,] B )  -> 
( vol `  if ( x  e.  ( A [,] B ) ,  ( C [,] D
) ,  (/) ) )  =  ( vol `  (/) ) )
4025, 39syl5eq 2515 . . . . . . . . . . . 12  |-  ( -.  x  e.  ( A [,] B )  -> 
( vol `  ( S " { x }
) )  =  ( vol `  (/) ) )
41 0mbl 21680 . . . . . . . . . . . . . 14  |-  (/)  e.  dom  vol
42 mblvol 21671 . . . . . . . . . . . . . 14  |-  ( (/)  e.  dom  vol  ->  ( vol `  (/) )  =  ( vol* `  (/) ) )
4341, 42ax-mp 5 . . . . . . . . . . . . 13  |-  ( vol `  (/) )  =  ( vol* `  (/) )
44 ovol0 21634 . . . . . . . . . . . . 13  |-  ( vol* `  (/) )  =  0
4543, 44eqtri 2491 . . . . . . . . . . . 12  |-  ( vol `  (/) )  =  0
4640, 45syl6eq 2519 . . . . . . . . . . 11  |-  ( -.  x  e.  ( A [,] B )  -> 
( vol `  ( S " { x }
) )  =  0 )
4738, 46eqtr4d 2506 . . . . . . . . . 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 2475 . . . . . . . 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 9872 . . . . . . . 8  |-  ( D  -  C )  e.  RR
52 0re 9587 . . . . . . . 8  |-  0  e.  RR
5351, 52keepel 4002 . . . . . . 7  |-  if ( x  e.  ( A [,] B ) ,  ( D  -  C
) ,  0 )  e.  RR
5450, 53syl6eqel 2558 . . . . . 6  |-  ( x  e.  RR  ->  ( vol `  ( S " { x } ) )  e.  RR )
55 volf 21670 . . . . . . . 8  |-  vol : dom  vol --> ( 0 [,] +oo )
56 ffun 5726 . . . . . . . 8  |-  ( vol
: dom  vol --> ( 0 [,] +oo )  ->  Fun  vol )
5755, 56ax-mp 5 . . . . . . 7  |-  Fun  vol
5829, 41keepel 4002 . . . . . . . 8  |-  if ( x  e.  ( A [,] B ) ,  ( C [,] D
) ,  (/) )  e. 
dom  vol
5924, 58eqeltri 2546 . . . . . . 7  |-  ( S
" { x }
)  e.  dom  vol
60 fvimacnv 5989 . . . . . . 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 2819 . . . 4  |-  A. x  e.  RR  ( S " { x } )  e.  ( `' vol " RR )
645a1i 11 . . . . . 6  |-  ( 0  e.  RR  ->  ( A [,] B )  C_  RR )
65 rembl 21681 . . . . . . 7  |-  RR  e.  dom  vol
6665a1i 11 . . . . . 6  |-  ( 0  e.  RR  ->  RR  e.  dom  vol )
6736, 51syl6eqel 2558 . . . . . . 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 3622 . . . . . . . 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 4524 . . . . . . . 8  |-  ( x  e.  ( A [,] B )  |->  ( vol `  ( S " {
x } ) ) )  =  ( x  e.  ( A [,] B )  |->  ( D  -  C ) )
7351recni 9599 . . . . . . . . . 10  |-  ( D  -  C )  e.  CC
74 ax-resscn 9540 . . . . . . . . . . 11  |-  RR  C_  CC
755, 74sstri 3508 . . . . . . . . . 10  |-  ( A [,] B )  C_  CC
76 ssid 3518 . . . . . . . . . 10  |-  CC  C_  CC
77 cncfmptc 21145 . . . . . . . . . 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 1319 . . . . . . . . 9  |-  ( x  e.  ( A [,] B )  |->  ( D  -  C ) )  e.  ( ( A [,] B ) -cn-> CC )
79 cniccibl 21977 . . . . . . . . 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 1319 . . . . . . . 8  |-  ( x  e.  ( A [,] B )  |->  ( D  -  C ) )  e.  L^1
8172, 80eqeltri 2546 . . . . . . 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 21942 . . . . 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 23010 . . . 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 1173 . . 3  |-  S  e. 
dom area
87 areaval 23017 . . 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 21914 . . . 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 2822 . . 3  |-  S. RR ( vol `  ( S
" { x }
) )  _d x  =  S. RR if ( x  e.  ( A [,] B ) ,  ( D  -  C
) ,  0 )  _d x
91 iccmbl 21706 . . . . . 6  |-  ( ( A  e.  RR  /\  B  e.  RR )  ->  ( A [,] B
)  e.  dom  vol )
922, 3, 91mp2an 672 . . . . 5  |-  ( A [,] B )  e. 
dom  vol
93 mblvol 21671 . . . . . . . 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 21664 . . . . . . . 8  |-  ( ( A  e.  RR  /\  B  e.  RR  /\  A  <_  B )  ->  ( vol* `  ( A [,] B ) )  =  ( B  -  A ) )
972, 3, 95, 96mp3an 1319 . . . . . . 7  |-  ( vol* `  ( A [,] B ) )  =  ( B  -  A
)
9894, 97eqtri 2491 . . . . . 6  |-  ( vol `  ( A [,] B
) )  =  ( B  -  A )
993, 2resubcli 9872 . . . . . 6  |-  ( B  -  A )  e.  RR
10098, 99eqeltri 2546 . . . . 5  |-  ( vol `  ( A [,] B
) )  e.  RR
101 itgconst 21955 . . . . 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 1319 . . . 4  |-  S. ( A [,] B ) ( D  -  C
)  _d x  =  ( ( D  -  C )  x.  ( vol `  ( A [,] B ) ) )
103 itgss2 21949 . . . . 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 2499 . . 3  |-  S. RR if ( x  e.  ( A [,] B ) ,  ( D  -  C ) ,  0 )  _d x  =  ( ( D  -  C )  x.  ( B  -  A )
)
10790, 106eqtri 2491 . 2  |-  S. RR ( vol `  ( S
" { x }
) )  _d x  =  ( ( D  -  C )  x.  ( B  -  A
) )
10899recni 9599 . . 3  |-  ( B  -  A )  e.  CC
10973, 108mulcomi 9593 . 2  |-  ( ( D  -  C )  x.  ( B  -  A ) )  =  ( ( B  -  A )  x.  ( D  -  C )
)
11088, 107, 1093eqtri 2495 1  |-  (area `  S )  =  ( ( B  -  A
)  x.  ( D  -  C ) )
Colors of variables: wff setvar class
Syntax hints:   -. wn 3    <-> wb 184    = wceq 1374    e. wcel 1762   A.wral 2809    \ cdif 3468    i^i cin 3470    C_ wss 3471   (/)c0 3780   ifcif 3934   {csn 4022   class class class wbr 4442    |-> cmpt 4500    X. cxp 4992   `'ccnv 4993   dom cdm 4994   "cima 4997   Fun wfun 5575   -->wf 5577   ` cfv 5581  (class class class)co 6277   CCcc 9481   RRcr 9482   0cc0 9483    x. cmul 9488   +oocpnf 9616    <_ cle 9620    - cmin 9796   [,]cicc 11523   -cn->ccncf 21110   vol*covol 21604   volcvol 21605   L^1cibl 21756   S.citg 21757  areacarea 23008
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1596  ax-4 1607  ax-5 1675  ax-6 1714  ax-7 1734  ax-8 1764  ax-9 1766  ax-10 1781  ax-11 1786  ax-12 1798  ax-13 1963  ax-ext 2440  ax-rep 4553  ax-sep 4563  ax-nul 4571  ax-pow 4620  ax-pr 4681  ax-un 6569  ax-inf2 8049  ax-cc 8806  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  ax-addf 9562  ax-mulf 9563
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 969  df-3an 970  df-tru 1377  df-fal 1380  df-ex 1592  df-nf 1595  df-sb 1707  df-eu 2274  df-mo 2275  df-clab 2448  df-cleq 2454  df-clel 2457  df-nfc 2612  df-ne 2659  df-nel 2660  df-ral 2814  df-rex 2815  df-reu 2816  df-rmo 2817  df-rab 2818  df-v 3110  df-sbc 3327  df-csb 3431  df-dif 3474  df-un 3476  df-in 3478  df-ss 3485  df-pss 3487  df-nul 3781  df-if 3935  df-pw 4007  df-sn 4023  df-pr 4025  df-tp 4027  df-op 4029  df-uni 4241  df-int 4278  df-iun 4322  df-iin 4323  df-disj 4413  df-br 4443  df-opab 4501  df-mpt 4502  df-tr 4536  df-eprel 4786  df-id 4790  df-po 4795  df-so 4796  df-fr 4833  df-se 4834  df-we 4835  df-ord 4876  df-on 4877  df-lim 4878  df-suc 4879  df-xp 5000  df-rel 5001  df-cnv 5002  df-co 5003  df-dm 5004  df-rn 5005  df-res 5006  df-ima 5007  df-iota 5544  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 6238  df-ov 6280  df-oprab 6281  df-mpt2 6282  df-of 6517  df-ofr 6518  df-om 6674  df-1st 6776  df-2nd 6777  df-supp 6894  df-recs 7034  df-rdg 7068  df-1o 7122  df-2o 7123  df-oadd 7126  df-omul 7127  df-er 7303  df-map 7414  df-pm 7415  df-ixp 7462  df-en 7509  df-dom 7510  df-sdom 7511  df-fin 7512  df-fsupp 7821  df-fi 7862  df-sup 7892  df-oi 7926  df-card 8311  df-acn 8314  df-cda 8539  df-pnf 9621  df-mnf 9622  df-xr 9623  df-ltxr 9624  df-le 9625  df-sub 9798  df-neg 9799  df-div 10198  df-nn 10528  df-2 10585  df-3 10586  df-4 10587  df-5 10588  df-6 10589  df-7 10590  df-8 10591  df-9 10592  df-10 10593  df-n0 10787  df-z 10856  df-dec 10968  df-uz 11074  df-q 11174  df-rp 11212  df-xneg 11309  df-xadd 11310  df-xmul 11311  df-ioo 11524  df-ioc 11525  df-ico 11526  df-icc 11527  df-fz 11664  df-fzo 11784  df-fl 11888  df-mod 11955  df-seq 12066  df-exp 12125  df-hash 12363  df-cj 12884  df-re 12885  df-im 12886  df-sqr 13020  df-abs 13021  df-limsup 13245  df-clim 13262  df-rlim 13263  df-sum 13460  df-struct 14483  df-ndx 14484  df-slot 14485  df-base 14486  df-sets 14487  df-ress 14488  df-plusg 14559  df-mulr 14560  df-starv 14561  df-sca 14562  df-vsca 14563  df-ip 14564  df-tset 14565  df-ple 14566  df-ds 14568  df-unif 14569  df-hom 14570  df-cco 14571  df-rest 14669  df-topn 14670  df-0g 14688  df-gsum 14689  df-topgen 14690  df-pt 14691  df-prds 14694  df-xrs 14748  df-qtop 14753  df-imas 14754  df-xps 14756  df-mre 14832  df-mrc 14833  df-acs 14835  df-mnd 15723  df-submnd 15773  df-mulg 15856  df-cntz 16145  df-cmn 16591  df-psmet 18177  df-xmet 18178  df-met 18179  df-bl 18180  df-mopn 18181  df-cnfld 18187  df-top 19161  df-bases 19163  df-topon 19164  df-topsp 19165  df-cn 19489  df-cnp 19490  df-cmp 19648  df-tx 19793  df-hmeo 19986  df-xms 20553  df-ms 20554  df-tms 20555  df-cncf 21112  df-ovol 21606  df-vol 21607  df-mbf 21758  df-itg1 21759  df-itg2 21760  df-ibl 21761  df-itg 21762  df-0p 21807  df-area 23009
This theorem is referenced by: (None)
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