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.

Step	Hyp	Ref	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 )
5	2, 3, 4	mp2an 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 )
9	6, 7, 8	mp2an 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 ) )
11	5, 9, 10	mp2an 686	. . . . 5 |- ( ( A [,] B ) X. ( C [,] D ) ) C_ ( RR X. RR )
12	1, 11	eqsstri 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 ) )
14	1	imaeq1i 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 ) )
17	15, 16	eqtr4d 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 } ) = (/) )
21	19, 20	sylbir 218	. . . . . . . . . . . . . . . . 17 |- ( -. x e. ( A [,] B ) -> ( ( ( A [,] B ) X. ( C [,] D ) ) " { x } ) = (/) )
22	18, 21	eqtr4d 2508	. . . . . . . . . . . . . . . 16 |- ( -. x e. ( A [,] B ) -> if ( x e. ( A [,] B ) , ( C [,] D ) , (/) ) = ( ( ( A [,] B ) X. ( C [,] D ) ) " { x } ) )
23	17, 22	pm2.61i 169	. . . . . . . . . . . . . . 15 |- if ( x e. ( A [,] B ) , ( C [,] D ) , (/) ) = ( ( ( A [,] B ) X. ( C [,] D ) ) " { x } )
24	14, 23	eqtr4i 2496	. . . . . . . . . . . . . 14 |- ( S " { x } ) = if ( x e. ( A [,] B ) , ( C [,] D ) , (/) )
25	24	fveq2i 5882	. . . . . . . . . . . . 13 |- ( vol ` ( S " { x } ) ) = ( vol ` if ( x e. ( A [,] B ) , ( C [,] D ) , (/) ) )
26	15	fveq2d 5883	. . . . . . . . . . . . 13 |- ( x e. ( A [,] B ) -> ( vol ` if ( x e. ( A [,] B ) , ( C [,] D ) , (/) ) ) = ( vol ` ( C [,] D ) ) )
27	25, 26	syl5eq 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 )
29	6, 7, 28	mp2an 686	. . . . . . . . . . . . . 14 |- ( C [,] D ) e. dom vol
30		mblvol 22562	. . . . . . . . . . . . . 14 |- ( ( C [,] D ) e. dom vol -> ( vol ` ( C [,] D ) ) = ( vol* ` ( C [,] D ) ) )
31	29, 30	ax-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 ) )
34	6, 7, 32, 33	mp3an 1390	. . . . . . . . . . . . 13 |- ( vol* ` ( C [,] D ) ) = ( D - C )
35	31, 34	eqtri 2493	. . . . . . . . . . . 12 |- ( vol ` ( C [,] D ) ) = ( D - C )
36	27, 35	syl6eq 2521	. . . . . . . . . . 11 |- ( x e. ( A [,] B ) -> ( vol ` ( S " { x } ) ) = ( D - C ) )
37	13, 36	eqtr4d 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 )
39	18	fveq2d 5883	. . . . . . . . . . . . 13 |- ( -. x e. ( A [,] B ) -> ( vol ` if ( x e. ( A [,] B ) , ( C [,] D ) , (/) ) ) = ( vol ` (/) ) )
40	25, 39	syl5eq 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* ` (/) ) )
43	41, 42	ax-mp 5	. . . . . . . . . . . . 13 |- ( vol ` (/) ) = ( vol* ` (/) )
44		ovol0 22524	. . . . . . . . . . . . 13 |- ( vol* ` (/) ) = 0
45	43, 44	eqtri 2493	. . . . . . . . . . . 12 |- ( vol ` (/) ) = 0
46	40, 45	syl6eq 2521	. . . . . . . . . . 11 |- ( -. x e. ( A [,] B ) -> ( vol ` ( S " { x } ) ) = 0 )
47	38, 46	eqtr4d 2508	. . . . . . . . . 10 |- ( -. x e. ( A [,] B ) -> if ( x e. ( A [,] B ) , ( D - C ) , 0 ) = ( vol ` ( S " { x } ) ) )
48	37, 47	pm2.61i 169	. . . . . . . . 9 |- if ( x e. ( A [,] B ) , ( D - C ) , 0 ) = ( vol ` ( S " { x } ) )
49	48	eqcomi 2480	. . . . . . . 8 |- ( vol ` ( S " { x } ) ) = if ( x e. ( A [,] B ) , ( D - C ) , 0 )
50	49	a1i 11	. . . . . . 7 |- ( x e. RR -> ( vol ` ( S " { x } ) ) = if ( x e. ( A [,] B ) , ( D - C ) , 0 ) )
51	7, 6	resubcli 9956	. . . . . . . 8 |- ( D - C ) e. RR
52		0re 9661	. . . . . . . 8 |- 0 e. RR
53	51, 52	keepel 3939	. . . . . . 7 |- if ( x e. ( A [,] B ) , ( D - C ) , 0 ) e. RR
54	50, 53	syl6eqel 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 )
57	55, 56	ax-mp 5	. . . . . . 7 |- Fun vol
58	29, 41	keepel 3939	. . . . . . . 8 |- if ( x e. ( A [,] B ) , ( C [,] D ) , (/) ) e. dom vol
59	24, 58	eqeltri 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 ) ) )
61	57, 59, 60	mp2an 686	. . . . . 6 |- ( ( vol ` ( S " { x } ) ) e. RR <-> ( S " { x } ) e. ( `' vol " RR ) )
62	54, 61	sylib 201	. . . . 5 |- ( x e. RR -> ( S " { x } ) e. ( `' vol " RR ) )
63	62	rgen 2766	. . . 4 |- A. x e. RR ( S " { x } ) e. ( `' vol " RR )
64	5	a1i 11	. . . . . 6 |- ( 0 e. RR -> ( A [,] B ) C_ RR )
65		rembl 22572	. . . . . . 7 |- RR e. dom vol
66	65	a1i 11	. . . . . 6 |- ( 0 e. RR -> RR e. dom vol )
67	36, 51	syl6eqel 2557	. . . . . . 7 |- ( x e. ( A [,] B ) -> ( vol ` ( S " { x } ) ) e. RR )
68	67	adantl 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 ) )
70	69, 46	syl 17	. . . . . . 7 |- ( x e. ( RR \ ( A [,] B ) ) -> ( vol ` ( S " { x } ) ) = 0 )
71	70	adantl 473	. . . . . 6 |- ( ( 0 e. RR /\ x e. ( RR \ ( A [,] B ) ) ) -> ( vol ` ( S " { x } ) ) = 0 )
72	36	mpteq2ia 4478	. . . . . . . 8 |- ( x e. ( A [,] B ) |-> ( vol ` ( S " { x } ) ) ) = ( x e. ( A [,] B ) |-> ( D - C ) )
73	51	recni 9673	. . . . . . . . . 10 |- ( D - C ) e. CC
74		ax-resscn 9614	. . . . . . . . . . 11 |- RR C_ CC
75	5, 74	sstri 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 ) )
78	73, 75, 76, 77	mp3an 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 )
80	2, 3, 78, 79	mp3an 1390	. . . . . . . 8 |- ( x e. ( A [,] B ) |-> ( D - C ) ) e. L^1
81	72, 80	eqeltri 2545	. . . . . . 7 |- ( x e. ( A [,] B ) |-> ( vol ` ( S " { x } ) ) ) e. L^1
82	81	a1i 11	. . . . . 6 |- ( 0 e. RR -> ( x e. ( A [,] B ) |-> ( vol ` ( S " { x } ) ) ) e. L^1 )
83	64, 66, 68, 71, 82	iblss2 22842	. . . . 5 |- ( 0 e. RR -> ( x e. RR |-> ( vol ` ( S " { x } ) ) ) e. L^1 )
84	52, 83	ax-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 ) )
86	12, 63, 84, 85	mpbir3an 1212	. . 3 |- S e. dom area
87		areaval 23969	. . 3 |- ( S e. dom area -> (area ` S ) = S. RR ( vol ` ( S " { x } ) ) _d x )
88	86, 87	ax-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 )
90	89, 50	mprg 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 )
92	2, 3, 91	mp2an 686	. . . . 5 |- ( A [,] B ) e. dom vol
93		mblvol 22562	. . . . . . . 8 |- ( ( A [,] B ) e. dom vol -> ( vol ` ( A [,] B ) ) = ( vol* ` ( A [,] B ) ) )
94	92, 93	ax-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 ) )
97	2, 3, 95, 96	mp3an 1390	. . . . . . 7 |- ( vol* ` ( A [,] B ) ) = ( B - A )
98	94, 97	eqtri 2493	. . . . . 6 |- ( vol ` ( A [,] B ) ) = ( B - A )
99	3, 2	resubcli 9956	. . . . . 6 |- ( B - A ) e. RR
100	98, 99	eqeltri 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 ) ) ) )
102	92, 100, 73, 101	mp3an 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 )
104	5, 103	ax-mp 5	. . . 4 |- S. ( A [,] B ) ( D - C ) _d x = S. RR if ( x e. ( A [,] B ) , ( D - C ) , 0 ) _d x
105	98	oveq2i 6319	. . . 4 |- ( ( D - C ) x. ( vol ` ( A [,] B ) ) ) = ( ( D - C ) x. ( B - A ) )
106	102, 104, 105	3eqtr3i 2501	. . 3 |- S. RR if ( x e. ( A [,] B ) , ( D - C ) , 0 ) _d x = ( ( D - C ) x. ( B - A ) )
107	90, 106	eqtri 2493	. 2 |- S. RR ( vol ` ( S " { x } ) ) _d x = ( ( D - C ) x. ( B - A ) )
108	99	recni 9673	. . 3 |- ( B - A ) e. CC
109	73, 108	mulcomi 9667	. 2 |- ( ( D - C ) x. ( B - A ) ) = ( ( B - A ) x. ( D - C ) )
110	88, 107, 109	3eqtri 2497	1 |- (area ` S ) = ( ( B - A ) x. ( D - C ) )

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)