Theorem arearect 31351

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 11527	. . . . . . 7 |- ( ( A e. RR /\ B e. RR ) -> ( A [,] B ) C_ RR )
5	2, 3, 4	mp2an 670	. . . . . 6 |- ( A [,] B ) C_ RR
6		arearect.3	. . . . . . 7 |- C e. RR
7		arearect.4	. . . . . . 7 |- D e. RR
8		iccssre 11527	. . . . . . 7 |- ( ( C e. RR /\ D e. RR ) -> ( C [,] D ) C_ RR )
9	6, 7, 8	mp2an 670	. . . . . 6 |- ( C [,] D ) C_ RR
10		xpss12 5021	. . . . . 6 |- ( ( ( A [,] B ) C_ RR /\ ( C [,] D ) C_ RR ) -> ( ( A [,] B ) X. ( C [,] D ) ) C_ ( RR X. RR ) )
11	5, 9, 10	mp2an 670	. . . . 5 |- ( ( A [,] B ) X. ( C [,] D ) ) C_ ( RR X. RR )
12	1, 11	eqsstri 3447	. . . 4 |- S C_ ( RR X. RR )
13		iftrue 3863	. . . . . . . . . . 11 |- ( x e. ( A [,] B ) -> if ( x e. ( A [,] B ) , ( D - C ) , 0 ) = ( D - C ) )
14	1	imaeq1i 5246	. . . . . . . . . . . . . . 15 |- ( S " { x } ) = ( ( ( A [,] B ) X. ( C [,] D ) ) " { x } )
15		iftrue 3863	. . . . . . . . . . . . . . . . 17 |- ( x e. ( A [,] B ) -> if ( x e. ( A [,] B ) , ( C [,] D ) , (/) ) = ( C [,] D ) )
16		xpimasn 5362	. . . . . . . . . . . . . . . . 17 |- ( x e. ( A [,] B ) -> ( ( ( A [,] B ) X. ( C [,] D ) ) " { x } ) = ( C [,] D ) )
17	15, 16	eqtr4d 2426	. . . . . . . . . . . . . . . 16 |- ( x e. ( A [,] B ) -> if ( x e. ( A [,] B ) , ( C [,] D ) , (/) ) = ( ( ( A [,] B ) X. ( C [,] D ) ) " { x } ) )
18		iffalse 3866	. . . . . . . . . . . . . . . . 17 |- ( -. x e. ( A [,] B ) -> if ( x e. ( A [,] B ) , ( C [,] D ) , (/) ) = (/) )
19		disjsn 4004	. . . . . . . . . . . . . . . . . 18 |- ( ( ( A [,] B ) i^i { x } ) = (/) <-> -. x e. ( A [,] B ) )
20		xpima1 5360	. . . . . . . . . . . . . . . . . 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 2426	. . . . . . . . . . . . . . . 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 2414	. . . . . . . . . . . . . 14 |- ( S " { x } ) = if ( x e. ( A [,] B ) , ( C [,] D ) , (/) )
25	24	fveq2i 5777	. . . . . . . . . . . . 13 |- ( vol ` ( S " { x } ) ) = ( vol ` if ( x e. ( A [,] B ) , ( C [,] D ) , (/) ) )
26	15	fveq2d 5778	. . . . . . . . . . . . 13 |- ( x e. ( A [,] B ) -> ( vol ` if ( x e. ( A [,] B ) , ( C [,] D ) , (/) ) ) = ( vol ` ( C [,] D ) ) )
27	25, 26	syl5eq 2435	. . . . . . . . . . . 12 |- ( x e. ( A [,] B ) -> ( vol ` ( S " { x } ) ) = ( vol ` ( C [,] D ) ) )
28		iccmbl 22061	. . . . . . . . . . . . . . 15 |- ( ( C e. RR /\ D e. RR ) -> ( C [,] D ) e. dom vol )
29	6, 7, 28	mp2an 670	. . . . . . . . . . . . . 14 |- ( C [,] D ) e. dom vol
30		mblvol 22026	. . . . . . . . . . . . . 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 22019	. . . . . . . . . . . . . 14 |- ( ( C e. RR /\ D e. RR /\ C <_ D ) -> ( vol* ` ( C [,] D ) ) = ( D - C ) )
34	6, 7, 32, 33	mp3an 1322	. . . . . . . . . . . . 13 |- ( vol* ` ( C [,] D ) ) = ( D - C )
35	31, 34	eqtri 2411	. . . . . . . . . . . 12 |- ( vol ` ( C [,] D ) ) = ( D - C )
36	27, 35	syl6eq 2439	. . . . . . . . . . 11 |- ( x e. ( A [,] B ) -> ( vol ` ( S " { x } ) ) = ( D - C ) )
37	13, 36	eqtr4d 2426	. . . . . . . . . 10 |- ( x e. ( A [,] B ) -> if ( x e. ( A [,] B ) , ( D - C ) , 0 ) = ( vol ` ( S " { x } ) ) )
38		iffalse 3866	. . . . . . . . . . 11 |- ( -. x e. ( A [,] B ) -> if ( x e. ( A [,] B ) , ( D - C ) , 0 ) = 0 )
39	18	fveq2d 5778	. . . . . . . . . . . . 13 |- ( -. x e. ( A [,] B ) -> ( vol ` if ( x e. ( A [,] B ) , ( C [,] D ) , (/) ) ) = ( vol ` (/) ) )
40	25, 39	syl5eq 2435	. . . . . . . . . . . 12 |- ( -. x e. ( A [,] B ) -> ( vol ` ( S " { x } ) ) = ( vol ` (/) ) )
41		0mbl 22035	. . . . . . . . . . . . . 14 |- (/) e. dom vol
42		mblvol 22026	. . . . . . . . . . . . . 14 |- ( (/) e. dom vol -> ( vol ` (/) ) = ( vol* ` (/) ) )
43	41, 42	ax-mp 5	. . . . . . . . . . . . 13 |- ( vol ` (/) ) = ( vol* ` (/) )
44		ovol0 21989	. . . . . . . . . . . . 13 |- ( vol* ` (/) ) = 0
45	43, 44	eqtri 2411	. . . . . . . . . . . 12 |- ( vol ` (/) ) = 0
46	40, 45	syl6eq 2439	. . . . . . . . . . 11 |- ( -. x e. ( A [,] B ) -> ( vol ` ( S " { x } ) ) = 0 )
47	38, 46	eqtr4d 2426	. . . . . . . . . 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 2395	. . . . . . . 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 9794	. . . . . . . 8 |- ( D - C ) e. RR
52		0re 9507	. . . . . . . 8 |- 0 e. RR
53	51, 52	keepel 3924	. . . . . . 7 |- if ( x e. ( A [,] B ) , ( D - C ) , 0 ) e. RR
54	50, 53	syl6eqel 2478	. . . . . 6 |- ( x e. RR -> ( vol ` ( S " { x } ) ) e. RR )
55		volf 22025	. . . . . . . 8 |- vol : dom vol --> ( 0 [,] +oo )
56		ffun 5641	. . . . . . . 8 |- ( vol : dom vol --> ( 0 [,] +oo ) -> Fun vol )
57	55, 56	ax-mp 5	. . . . . . 7 |- Fun vol
58	29, 41	keepel 3924	. . . . . . . 8 |- if ( x e. ( A [,] B ) , ( C [,] D ) , (/) ) e. dom vol
59	24, 58	eqeltri 2466	. . . . . . 7 |- ( S " { x } ) e. dom vol
60		fvimacnv 5904	. . . . . . 7 |- ( ( Fun vol /\ ( S " { x } ) e. dom vol ) -> ( ( vol ` ( S " { x } ) ) e. RR <-> ( S " { x } ) e. ( `' vol " RR ) ) )
61	57, 59, 60	mp2an 670	. . . . . 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 2742	. . . 4 |- A. x e. RR ( S " { x } ) e. ( `' vol " RR )
64	5	a1i 11	. . . . . 6 |- ( 0 e. RR -> ( A [,] B ) C_ RR )
65		rembl 22036	. . . . . . 7 |- RR e. dom vol
66	65	a1i 11	. . . . . 6 |- ( 0 e. RR -> RR e. dom vol )
67	36, 51	syl6eqel 2478	. . . . . . 7 |- ( x e. ( A [,] B ) -> ( vol ` ( S " { x } ) ) e. RR )
68	67	adantl 464	. . . . . 6 |- ( ( 0 e. RR /\ x e. ( A [,] B ) ) -> ( vol ` ( S " { x } ) ) e. RR )
69		eldifn 3541	. . . . . . . 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 464	. . . . . 6 |- ( ( 0 e. RR /\ x e. ( RR \ ( A [,] B ) ) ) -> ( vol ` ( S " { x } ) ) = 0 )
72	36	mpteq2ia 4449	. . . . . . . 8 |- ( x e. ( A [,] B ) |-> ( vol ` ( S " { x } ) ) ) = ( x e. ( A [,] B ) |-> ( D - C ) )
73	51	recni 9519	. . . . . . . . . 10 |- ( D - C ) e. CC
74		ax-resscn 9460	. . . . . . . . . . 11 |- RR C_ CC
75	5, 74	sstri 3426	. . . . . . . . . 10 |- ( A [,] B ) C_ CC
76		ssid 3436	. . . . . . . . . 10 |- CC C_ CC
77		cncfmptc 21500	. . . . . . . . . 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 1322	. . . . . . . . 9 |- ( x e. ( A [,] B ) |-> ( D - C ) ) e. ( ( A [,] B ) -cn-> CC )
79		cniccibl 22332	. . . . . . . . 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 1322	. . . . . . . 8 |- ( x e. ( A [,] B ) |-> ( D - C ) ) e. L^1
81	72, 80	eqeltri 2466	. . . . . . 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 22297	. . . . 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 23404	. . . 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 1176	. . 3 |- S e. dom area
87		areaval 23411	. . 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 22269	. . . 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 2745	. . 3 |- S. RR ( vol ` ( S " { x } ) ) _d x = S. RR if ( x e. ( A [,] B ) , ( D - C ) , 0 ) _d x
91		iccmbl 22061	. . . . . 6 |- ( ( A e. RR /\ B e. RR ) -> ( A [,] B ) e. dom vol )
92	2, 3, 91	mp2an 670	. . . . 5 |- ( A [,] B ) e. dom vol
93		mblvol 22026	. . . . . . . 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 22019	. . . . . . . 8 |- ( ( A e. RR /\ B e. RR /\ A <_ B ) -> ( vol* ` ( A [,] B ) ) = ( B - A ) )
97	2, 3, 95, 96	mp3an 1322	. . . . . . 7 |- ( vol* ` ( A [,] B ) ) = ( B - A )
98	94, 97	eqtri 2411	. . . . . 6 |- ( vol ` ( A [,] B ) ) = ( B - A )
99	3, 2	resubcli 9794	. . . . . 6 |- ( B - A ) e. RR
100	98, 99	eqeltri 2466	. . . . 5 |- ( vol ` ( A [,] B ) ) e. RR
101		itgconst 22310	. . . . 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 1322	. . . 4 |- S. ( A [,] B ) ( D - C ) _d x = ( ( D - C ) x. ( vol ` ( A [,] B ) ) )
103		itgss2 22304	. . . . 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 6207	. . . 4 |- ( ( D - C ) x. ( vol ` ( A [,] B ) ) ) = ( ( D - C ) x. ( B - A ) )
106	102, 104, 105	3eqtr3i 2419	. . 3 |- S. RR if ( x e. ( A [,] B ) , ( D - C ) , 0 ) _d x = ( ( D - C ) x. ( B - A ) )
107	90, 106	eqtri 2411	. 2 |- S. RR ( vol ` ( S " { x } ) ) _d x = ( ( D - C ) x. ( B - A ) )
108	99	recni 9519	. . 3 |- ( B - A ) e. CC
109	73, 108	mulcomi 9513	. 2 |- ( ( D - C ) x. ( B - A ) ) = ( ( B - A ) x. ( D - C ) )
110	88, 107, 109	3eqtri 2415	1 |- (area ` S ) = ( ( B - A ) x. ( D - C ) )

Syntax hints:   -. wn 3   <-> wb 184   = wceq 1399   e. wcel 1826   A.wral 2732   \ cdif 3386   i^i cin 3388   C_ wss 3389   (/)c0 3711   ifcif 3857   {csn 3944   class class class wbr 4367   |-> cmpt 4425   X. cxp 4911   `'ccnv 4912   dom cdm 4913   "cima 4916   Fun wfun 5490   -->wf 5492   ` cfv 5496  (class class class)co 6196   CCcc 9401   RRcr 9402   0cc0 9403   x. cmul 9408   +oocpnf 9536   <_ cle 9540   - cmin 9718   [,]cicc 11453   -cn->ccncf 21465   vol*covol 21959   volcvol 21960   L^1cibl 22111   S.citg 22112  areacarea 23402

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1626  ax-4 1639  ax-5 1712  ax-6 1755  ax-7 1798  ax-8 1828  ax-9 1830  ax-10 1845  ax-11 1850  ax-12 1862  ax-13 2006  ax-ext 2360  ax-rep 4478  ax-sep 4488  ax-nul 4496  ax-pow 4543  ax-pr 4601  ax-un 6491  ax-inf2 7972  ax-cc 8728  ax-cnex 9459  ax-resscn 9460  ax-1cn 9461  ax-icn 9462  ax-addcl 9463  ax-addrcl 9464  ax-mulcl 9465  ax-mulrcl 9466  ax-mulcom 9467  ax-addass 9468  ax-mulass 9469  ax-distr 9470  ax-i2m1 9471  ax-1ne0 9472  ax-1rid 9473  ax-rnegex 9474  ax-rrecex 9475  ax-cnre 9476  ax-pre-lttri 9477  ax-pre-lttrn 9478  ax-pre-ltadd 9479  ax-pre-mulgt0 9480  ax-pre-sup 9481  ax-addf 9482  ax-mulf 9483

This theorem depends on definitions:  df-bi 185  df-or 368  df-an 369  df-3or 972  df-3an 973  df-tru 1402  df-fal 1405  df-ex 1621  df-nf 1625  df-sb 1748  df-eu 2222  df-mo 2223  df-clab 2368  df-cleq 2374  df-clel 2377  df-nfc 2532  df-ne 2579  df-nel 2580  df-ral 2737  df-rex 2738  df-reu 2739  df-rmo 2740  df-rab 2741  df-v 3036  df-sbc 3253  df-csb 3349  df-dif 3392  df-un 3394  df-in 3396  df-ss 3403  df-pss 3405  df-nul 3712  df-if 3858  df-pw 3929  df-sn 3945  df-pr 3947  df-tp 3949  df-op 3951  df-uni 4164  df-int 4200  df-iun 4245  df-iin 4246  df-disj 4339  df-br 4368  df-opab 4426  df-mpt 4427  df-tr 4461  df-eprel 4705  df-id 4709  df-po 4714  df-so 4715  df-fr 4752  df-se 4753  df-we 4754  df-ord 4795  df-on 4796  df-lim 4797  df-suc 4798  df-xp 4919  df-rel 4920  df-cnv 4921  df-co 4922  df-dm 4923  df-rn 4924  df-res 4925  df-ima 4926  df-iota 5460  df-fun 5498  df-fn 5499  df-f 5500  df-f1 5501  df-fo 5502  df-f1o 5503  df-fv 5504  df-isom 5505  df-riota 6158  df-ov 6199  df-oprab 6200  df-mpt2 6201  df-of 6439  df-ofr 6440  df-om 6600  df-1st 6699  df-2nd 6700  df-supp 6818  df-recs 6960  df-rdg 6994  df-1o 7048  df-2o 7049  df-oadd 7052  df-omul 7053  df-er 7229  df-map 7340  df-pm 7341  df-ixp 7389  df-en 7436  df-dom 7437  df-sdom 7438  df-fin 7439  df-fsupp 7745  df-fi 7786  df-sup 7816  df-oi 7850  df-card 8233  df-acn 8236  df-cda 8461  df-pnf 9541  df-mnf 9542  df-xr 9543  df-ltxr 9544  df-le 9545  df-sub 9720  df-neg 9721  df-div 10124  df-nn 10453  df-2 10511  df-3 10512  df-4 10513  df-5 10514  df-6 10515  df-7 10516  df-8 10517  df-9 10518  df-10 10519  df-n0 10713  df-z 10782  df-dec 10896  df-uz 11002  df-q 11102  df-rp 11140  df-xneg 11239  df-xadd 11240  df-xmul 11241  df-ioo 11454  df-ioc 11455  df-ico 11456  df-icc 11457  df-fz 11594  df-fzo 11718  df-fl 11828  df-mod 11897  df-seq 12011  df-exp 12070  df-hash 12308  df-cj 12934  df-re 12935  df-im 12936  df-sqrt 13070  df-abs 13071  df-limsup 13296  df-clim 13313  df-rlim 13314  df-sum 13511  df-struct 14636  df-ndx 14637  df-slot 14638  df-base 14639  df-sets 14640  df-ress 14641  df-plusg 14715  df-mulr 14716  df-starv 14717  df-sca 14718  df-vsca 14719  df-ip 14720  df-tset 14721  df-ple 14722  df-ds 14724  df-unif 14725  df-hom 14726  df-cco 14727  df-rest 14830  df-topn 14831  df-0g 14849  df-gsum 14850  df-topgen 14851  df-pt 14852  df-prds 14855  df-xrs 14909  df-qtop 14914  df-imas 14915  df-xps 14917  df-mre 14993  df-mrc 14994  df-acs 14996  df-mgm 15989  df-sgrp 16028  df-mnd 16038  df-submnd 16084  df-mulg 16177  df-cntz 16472  df-cmn 16917  df-psmet 18524  df-xmet 18525  df-met 18526  df-bl 18527  df-mopn 18528  df-cnfld 18534  df-top 19484  df-bases 19486  df-topon 19487  df-topsp 19488  df-cn 19814  df-cnp 19815  df-cmp 19973  df-tx 20148  df-hmeo 20341  df-xms 20908  df-ms 20909  df-tms 20910  df-cncf 21467  df-ovol 21961  df-vol 21962  df-mbf 22113  df-itg1 22114  df-itg2 22115  df-ibl 22116  df-itg 22117  df-0p 22162  df-area 23403

This theorem is referenced by: (None)