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.

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 11610	. . . . . . 7 |- ( ( A e. RR /\ B e. RR ) -> ( A [,] B ) C_ RR )
5	2, 3, 4	mp2an 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 )
9	6, 7, 8	mp2an 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 ) )
11	5, 9, 10	mp2an 672	. . . . 5 |- ( ( A [,] B ) X. ( C [,] D ) ) C_ ( RR X. RR )
12	1, 11	eqsstri 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 ) )
14	1	imaeq1i 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 ) )
17	15, 16	eqtr4d 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 } ) = (/) )
21	19, 20	sylbir 213	. . . . . . . . . . . . . . . . 17 |- ( -. x e. ( A [,] B ) -> ( ( ( A [,] B ) X. ( C [,] D ) ) " { x } ) = (/) )
22	18, 21	eqtr4d 2485	. . . . . . . . . . . . . . . 16 |- ( -. x e. ( A [,] B ) -> if ( x e. ( A [,] B ) , ( C [,] D ) , (/) ) = ( ( ( A [,] B ) X. ( C [,] D ) ) " { x } ) )
23	17, 22	pm2.61i 164	. . . . . . . . . . . . . . 15 |- if ( x e. ( A [,] B ) , ( C [,] D ) , (/) ) = ( ( ( A [,] B ) X. ( C [,] D ) ) " { x } )
24	14, 23	eqtr4i 2473	. . . . . . . . . . . . . 14 |- ( S " { x } ) = if ( x e. ( A [,] B ) , ( C [,] D ) , (/) )
25	24	fveq2i 5855	. . . . . . . . . . . . 13 |- ( vol ` ( S " { x } ) ) = ( vol ` if ( x e. ( A [,] B ) , ( C [,] D ) , (/) ) )
26	15	fveq2d 5856	. . . . . . . . . . . . 13 |- ( x e. ( A [,] B ) -> ( vol ` if ( x e. ( A [,] B ) , ( C [,] D ) , (/) ) ) = ( vol ` ( C [,] D ) ) )
27	25, 26	syl5eq 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 )
29	6, 7, 28	mp2an 672	. . . . . . . . . . . . . 14 |- ( C [,] D ) e. dom vol
30		mblvol 21807	. . . . . . . . . . . . . 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 21800	. . . . . . . . . . . . . 14 |- ( ( C e. RR /\ D e. RR /\ C <_ D ) -> ( vol* ` ( C [,] D ) ) = ( D - C ) )
34	6, 7, 32, 33	mp3an 1323	. . . . . . . . . . . . 13 |- ( vol* ` ( C [,] D ) ) = ( D - C )
35	31, 34	eqtri 2470	. . . . . . . . . . . 12 |- ( vol ` ( C [,] D ) ) = ( D - C )
36	27, 35	syl6eq 2498	. . . . . . . . . . 11 |- ( x e. ( A [,] B ) -> ( vol ` ( S " { x } ) ) = ( D - C ) )
37	13, 36	eqtr4d 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 )
39	18	fveq2d 5856	. . . . . . . . . . . . 13 |- ( -. x e. ( A [,] B ) -> ( vol ` if ( x e. ( A [,] B ) , ( C [,] D ) , (/) ) ) = ( vol ` (/) ) )
40	25, 39	syl5eq 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* ` (/) ) )
43	41, 42	ax-mp 5	. . . . . . . . . . . . 13 |- ( vol ` (/) ) = ( vol* ` (/) )
44		ovol0 21770	. . . . . . . . . . . . 13 |- ( vol* ` (/) ) = 0
45	43, 44	eqtri 2470	. . . . . . . . . . . 12 |- ( vol ` (/) ) = 0
46	40, 45	syl6eq 2498	. . . . . . . . . . 11 |- ( -. x e. ( A [,] B ) -> ( vol ` ( S " { x } ) ) = 0 )
47	38, 46	eqtr4d 2485	. . . . . . . . . 10 |- ( -. x e. ( A [,] B ) -> if ( x e. ( A [,] B ) , ( D - C ) , 0 ) = ( vol ` ( S " { x } ) ) )
48	37, 47	pm2.61i 164	. . . . . . . . 9 |- if ( x e. ( A [,] B ) , ( D - C ) , 0 ) = ( vol ` ( S " { x } ) )
49	48	eqcomi 2454	. . . . . . . 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 9881	. . . . . . . 8 |- ( D - C ) e. RR
52		0re 9594	. . . . . . . 8 |- 0 e. RR
53	51, 52	keepel 3990	. . . . . . 7 |- if ( x e. ( A [,] B ) , ( D - C ) , 0 ) e. RR
54	50, 53	syl6eqel 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 )
57	55, 56	ax-mp 5	. . . . . . 7 |- Fun vol
58	29, 41	keepel 3990	. . . . . . . 8 |- if ( x e. ( A [,] B ) , ( C [,] D ) , (/) ) e. dom vol
59	24, 58	eqeltri 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 ) ) )
61	57, 59, 60	mp2an 672	. . . . . 6 |- ( ( vol ` ( S " { x } ) ) e. RR <-> ( S " { x } ) e. ( `' vol " RR ) )
62	54, 61	sylib 196	. . . . 5 |- ( x e. RR -> ( S " { x } ) e. ( `' vol " RR ) )
63	62	rgen 2801	. . . 4 |- A. x e. RR ( S " { x } ) e. ( `' vol " RR )
64	5	a1i 11	. . . . . 6 |- ( 0 e. RR -> ( A [,] B ) C_ RR )
65		rembl 21817	. . . . . . 7 |- RR e. dom vol
66	65	a1i 11	. . . . . 6 |- ( 0 e. RR -> RR e. dom vol )
67	36, 51	syl6eqel 2537	. . . . . . 7 |- ( x e. ( A [,] B ) -> ( vol ` ( S " { x } ) ) e. RR )
68	67	adantl 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 ) )
70	69, 46	syl 16	. . . . . . 7 |- ( x e. ( RR \ ( A [,] B ) ) -> ( vol ` ( S " { x } ) ) = 0 )
71	70	adantl 466	. . . . . 6 |- ( ( 0 e. RR /\ x e. ( RR \ ( A [,] B ) ) ) -> ( vol ` ( S " { x } ) ) = 0 )
72	36	mpteq2ia 4515	. . . . . . . 8 |- ( x e. ( A [,] B ) |-> ( vol ` ( S " { x } ) ) ) = ( x e. ( A [,] B ) |-> ( D - C ) )
73	51	recni 9606	. . . . . . . . . 10 |- ( D - C ) e. CC
74		ax-resscn 9547	. . . . . . . . . . 11 |- RR C_ CC
75	5, 74	sstri 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 ) )
78	73, 75, 76, 77	mp3an 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 )
80	2, 3, 78, 79	mp3an 1323	. . . . . . . 8 |- ( x e. ( A [,] B ) |-> ( D - C ) ) e. L^1
81	72, 80	eqeltri 2525	. . . . . . 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 22078	. . . . 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 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 ) )
86	12, 63, 84, 85	mpbir3an 1177	. . 3 |- S e. dom area
87		areaval 23159	. . 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 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 )
90	89, 50	mprg 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 )
92	2, 3, 91	mp2an 672	. . . . 5 |- ( A [,] B ) e. dom vol
93		mblvol 21807	. . . . . . . 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 21800	. . . . . . . 8 |- ( ( A e. RR /\ B e. RR /\ A <_ B ) -> ( vol* ` ( A [,] B ) ) = ( B - A ) )
97	2, 3, 95, 96	mp3an 1323	. . . . . . 7 |- ( vol* ` ( A [,] B ) ) = ( B - A )
98	94, 97	eqtri 2470	. . . . . 6 |- ( vol ` ( A [,] B ) ) = ( B - A )
99	3, 2	resubcli 9881	. . . . . 6 |- ( B - A ) e. RR
100	98, 99	eqeltri 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 ) ) ) )
102	92, 100, 73, 101	mp3an 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 )
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 6288	. . . 4 |- ( ( D - C ) x. ( vol ` ( A [,] B ) ) ) = ( ( D - C ) x. ( B - A ) )
106	102, 104, 105	3eqtr3i 2478	. . 3 |- S. RR if ( x e. ( A [,] B ) , ( D - C ) , 0 ) _d x = ( ( D - C ) x. ( B - A ) )
107	90, 106	eqtri 2470	. 2 |- S. RR ( vol ` ( S " { x } ) ) _d x = ( ( D - C ) x. ( B - A ) )
108	99	recni 9606	. . 3 |- ( B - A ) e. CC
109	73, 108	mulcomi 9600	. 2 |- ( ( D - C ) x. ( B - A ) ) = ( ( B - A ) x. ( D - C ) )
110	88, 107, 109	3eqtri 2474	1 |- (area ` S ) = ( ( B - A ) x. ( D - C ) )

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)