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.

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 11713	. . . . . . 7 |- ( ( A e. RR /\ B e. RR ) -> ( A [,] B ) C_ RR )
5	2, 3, 4	mp2an 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 )
9	6, 7, 8	mp2an 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 ) )
11	5, 9, 10	mp2an 677	. . . . 5 |- ( ( A [,] B ) X. ( C [,] D ) ) C_ ( RR X. RR )
12	1, 11	eqsstri 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 ) )
14	1	imaeq1i 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 ) )
17	15, 16	eqtr4d 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 } ) = (/) )
21	19, 20	sylbir 217	. . . . . . . . . . . . . . . . 17 |- ( -. x e. ( A [,] B ) -> ( ( ( A [,] B ) X. ( C [,] D ) ) " { x } ) = (/) )
22	18, 21	eqtr4d 2487	. . . . . . . . . . . . . . . 16 |- ( -. x e. ( A [,] B ) -> if ( x e. ( A [,] B ) , ( C [,] D ) , (/) ) = ( ( ( A [,] B ) X. ( C [,] D ) ) " { x } ) )
23	17, 22	pm2.61i 168	. . . . . . . . . . . . . . 15 |- if ( x e. ( A [,] B ) , ( C [,] D ) , (/) ) = ( ( ( A [,] B ) X. ( C [,] D ) ) " { x } )
24	14, 23	eqtr4i 2475	. . . . . . . . . . . . . 14 |- ( S " { x } ) = if ( x e. ( A [,] B ) , ( C [,] D ) , (/) )
25	24	fveq2i 5866	. . . . . . . . . . . . 13 |- ( vol ` ( S " { x } ) ) = ( vol ` if ( x e. ( A [,] B ) , ( C [,] D ) , (/) ) )
26	15	fveq2d 5867	. . . . . . . . . . . . 13 |- ( x e. ( A [,] B ) -> ( vol ` if ( x e. ( A [,] B ) , ( C [,] D ) , (/) ) ) = ( vol ` ( C [,] D ) ) )
27	25, 26	syl5eq 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 )
29	6, 7, 28	mp2an 677	. . . . . . . . . . . . . 14 |- ( C [,] D ) e. dom vol
30		mblvol 22477	. . . . . . . . . . . . . 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 22470	. . . . . . . . . . . . . 14 |- ( ( C e. RR /\ D e. RR /\ C <_ D ) -> ( vol* ` ( C [,] D ) ) = ( D - C ) )
34	6, 7, 32, 33	mp3an 1363	. . . . . . . . . . . . 13 |- ( vol* ` ( C [,] D ) ) = ( D - C )
35	31, 34	eqtri 2472	. . . . . . . . . . . 12 |- ( vol ` ( C [,] D ) ) = ( D - C )
36	27, 35	syl6eq 2500	. . . . . . . . . . 11 |- ( x e. ( A [,] B ) -> ( vol ` ( S " { x } ) ) = ( D - C ) )
37	13, 36	eqtr4d 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 )
39	18	fveq2d 5867	. . . . . . . . . . . . 13 |- ( -. x e. ( A [,] B ) -> ( vol ` if ( x e. ( A [,] B ) , ( C [,] D ) , (/) ) ) = ( vol ` (/) ) )
40	25, 39	syl5eq 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* ` (/) ) )
43	41, 42	ax-mp 5	. . . . . . . . . . . . 13 |- ( vol ` (/) ) = ( vol* ` (/) )
44		ovol0 22439	. . . . . . . . . . . . 13 |- ( vol* ` (/) ) = 0
45	43, 44	eqtri 2472	. . . . . . . . . . . 12 |- ( vol ` (/) ) = 0
46	40, 45	syl6eq 2500	. . . . . . . . . . 11 |- ( -. x e. ( A [,] B ) -> ( vol ` ( S " { x } ) ) = 0 )
47	38, 46	eqtr4d 2487	. . . . . . . . . 10 |- ( -. x e. ( A [,] B ) -> if ( x e. ( A [,] B ) , ( D - C ) , 0 ) = ( vol ` ( S " { x } ) ) )
48	37, 47	pm2.61i 168	. . . . . . . . 9 |- if ( x e. ( A [,] B ) , ( D - C ) , 0 ) = ( vol ` ( S " { x } ) )
49	48	eqcomi 2459	. . . . . . . 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 9933	. . . . . . . 8 |- ( D - C ) e. RR
52		0re 9640	. . . . . . . 8 |- 0 e. RR
53	51, 52	keepel 3947	. . . . . . 7 |- if ( x e. ( A [,] B ) , ( D - C ) , 0 ) e. RR
54	50, 53	syl6eqel 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 )
57	55, 56	ax-mp 5	. . . . . . 7 |- Fun vol
58	29, 41	keepel 3947	. . . . . . . 8 |- if ( x e. ( A [,] B ) , ( C [,] D ) , (/) ) e. dom vol
59	24, 58	eqeltri 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 ) ) )
61	57, 59, 60	mp2an 677	. . . . . 6 |- ( ( vol ` ( S " { x } ) ) e. RR <-> ( S " { x } ) e. ( `' vol " RR ) )
62	54, 61	sylib 200	. . . . 5 |- ( x e. RR -> ( S " { x } ) e. ( `' vol " RR ) )
63	62	rgen 2746	. . . 4 |- A. x e. RR ( S " { x } ) e. ( `' vol " RR )
64	5	a1i 11	. . . . . 6 |- ( 0 e. RR -> ( A [,] B ) C_ RR )
65		rembl 22487	. . . . . . 7 |- RR e. dom vol
66	65	a1i 11	. . . . . 6 |- ( 0 e. RR -> RR e. dom vol )
67	36, 51	syl6eqel 2536	. . . . . . 7 |- ( x e. ( A [,] B ) -> ( vol ` ( S " { x } ) ) e. RR )
68	67	adantl 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 ) )
70	69, 46	syl 17	. . . . . . 7 |- ( x e. ( RR \ ( A [,] B ) ) -> ( vol ` ( S " { x } ) ) = 0 )
71	70	adantl 468	. . . . . 6 |- ( ( 0 e. RR /\ x e. ( RR \ ( A [,] B ) ) ) -> ( vol ` ( S " { x } ) ) = 0 )
72	36	mpteq2ia 4484	. . . . . . . 8 |- ( x e. ( A [,] B ) |-> ( vol ` ( S " { x } ) ) ) = ( x e. ( A [,] B ) |-> ( D - C ) )
73	51	recni 9652	. . . . . . . . . 10 |- ( D - C ) e. CC
74		ax-resscn 9593	. . . . . . . . . . 11 |- RR C_ CC
75	5, 74	sstri 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 ) )
78	73, 75, 76, 77	mp3an 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 )
80	2, 3, 78, 79	mp3an 1363	. . . . . . . 8 |- ( x e. ( A [,] B ) |-> ( D - C ) ) e. L^1
81	72, 80	eqeltri 2524	. . . . . . 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 22756	. . . . 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 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 ) )
86	12, 63, 84, 85	mpbir3an 1189	. . 3 |- S e. dom area
87		areaval 23883	. . 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 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 )
90	89, 50	mprg 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 )
92	2, 3, 91	mp2an 677	. . . . 5 |- ( A [,] B ) e. dom vol
93		mblvol 22477	. . . . . . . 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 22470	. . . . . . . 8 |- ( ( A e. RR /\ B e. RR /\ A <_ B ) -> ( vol* ` ( A [,] B ) ) = ( B - A ) )
97	2, 3, 95, 96	mp3an 1363	. . . . . . 7 |- ( vol* ` ( A [,] B ) ) = ( B - A )
98	94, 97	eqtri 2472	. . . . . 6 |- ( vol ` ( A [,] B ) ) = ( B - A )
99	3, 2	resubcli 9933	. . . . . 6 |- ( B - A ) e. RR
100	98, 99	eqeltri 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 ) ) ) )
102	92, 100, 73, 101	mp3an 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 )
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 6299	. . . 4 |- ( ( D - C ) x. ( vol ` ( A [,] B ) ) ) = ( ( D - C ) x. ( B - A ) )
106	102, 104, 105	3eqtr3i 2480	. . 3 |- S. RR if ( x e. ( A [,] B ) , ( D - C ) , 0 ) _d x = ( ( D - C ) x. ( B - A ) )
107	90, 106	eqtri 2472	. 2 |- S. RR ( vol ` ( S " { x } ) ) _d x = ( ( D - C ) x. ( B - A ) )
108	99	recni 9652	. . 3 |- ( B - A ) e. CC
109	73, 108	mulcomi 9646	. 2 |- ( ( D - C ) x. ( B - A ) ) = ( ( B - A ) x. ( D - C ) )
110	88, 107, 109	3eqtri 2476	1 |- (area ` S ) = ( ( B - A ) x. ( D - C ) )

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)