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.

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 11597	. . . . . . 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 11597	. . . . . . 7 |- ( ( C e. RR /\ D e. RR ) -> ( C [,] D ) C_ RR )
9	6, 7, 8	mp2an 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 ) )
11	5, 9, 10	mp2an 672	. . . . 5 |- ( ( A [,] B ) X. ( C [,] D ) ) C_ ( RR X. RR )
12	1, 11	eqsstri 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 ) )
14	1	imaeq1i 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 ) )
17	15, 16	eqtr4d 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 } ) = (/) )
21	19, 20	sylbir 213	. . . . . . . . . . . . . . . . 17 |- ( -. x e. ( A [,] B ) -> ( ( ( A [,] B ) X. ( C [,] D ) ) " { x } ) = (/) )
22	18, 21	eqtr4d 2506	. . . . . . . . . . . . . . . 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 2494	. . . . . . . . . . . . . 14 |- ( S " { x } ) = if ( x e. ( A [,] B ) , ( C [,] D ) , (/) )
25	24	fveq2i 5862	. . . . . . . . . . . . 13 |- ( vol ` ( S " { x } ) ) = ( vol ` if ( x e. ( A [,] B ) , ( C [,] D ) , (/) ) )
26	15	fveq2d 5863	. . . . . . . . . . . . 13 |- ( x e. ( A [,] B ) -> ( vol ` if ( x e. ( A [,] B ) , ( C [,] D ) , (/) ) ) = ( vol ` ( C [,] D ) ) )
27	25, 26	syl5eq 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 )
29	6, 7, 28	mp2an 672	. . . . . . . . . . . . . 14 |- ( C [,] D ) e. dom vol
30		mblvol 21671	. . . . . . . . . . . . . 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 21664	. . . . . . . . . . . . . 14 |- ( ( C e. RR /\ D e. RR /\ C <_ D ) -> ( vol* ` ( C [,] D ) ) = ( D - C ) )
34	6, 7, 32, 33	mp3an 1319	. . . . . . . . . . . . 13 |- ( vol* ` ( C [,] D ) ) = ( D - C )
35	31, 34	eqtri 2491	. . . . . . . . . . . 12 |- ( vol ` ( C [,] D ) ) = ( D - C )
36	27, 35	syl6eq 2519	. . . . . . . . . . 11 |- ( x e. ( A [,] B ) -> ( vol ` ( S " { x } ) ) = ( D - C ) )
37	13, 36	eqtr4d 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 )
39	18	fveq2d 5863	. . . . . . . . . . . . 13 |- ( -. x e. ( A [,] B ) -> ( vol ` if ( x e. ( A [,] B ) , ( C [,] D ) , (/) ) ) = ( vol ` (/) ) )
40	25, 39	syl5eq 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* ` (/) ) )
43	41, 42	ax-mp 5	. . . . . . . . . . . . 13 |- ( vol ` (/) ) = ( vol* ` (/) )
44		ovol0 21634	. . . . . . . . . . . . 13 |- ( vol* ` (/) ) = 0
45	43, 44	eqtri 2491	. . . . . . . . . . . 12 |- ( vol ` (/) ) = 0
46	40, 45	syl6eq 2519	. . . . . . . . . . 11 |- ( -. x e. ( A [,] B ) -> ( vol ` ( S " { x } ) ) = 0 )
47	38, 46	eqtr4d 2506	. . . . . . . . . 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 2475	. . . . . . . 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 9872	. . . . . . . 8 |- ( D - C ) e. RR
52		0re 9587	. . . . . . . 8 |- 0 e. RR
53	51, 52	keepel 4002	. . . . . . 7 |- if ( x e. ( A [,] B ) , ( D - C ) , 0 ) e. RR
54	50, 53	syl6eqel 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 )
57	55, 56	ax-mp 5	. . . . . . 7 |- Fun vol
58	29, 41	keepel 4002	. . . . . . . 8 |- if ( x e. ( A [,] B ) , ( C [,] D ) , (/) ) e. dom vol
59	24, 58	eqeltri 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 ) ) )
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 2819	. . . 4 |- A. x e. RR ( S " { x } ) e. ( `' vol " RR )
64	5	a1i 11	. . . . . 6 |- ( 0 e. RR -> ( A [,] B ) C_ RR )
65		rembl 21681	. . . . . . 7 |- RR e. dom vol
66	65	a1i 11	. . . . . 6 |- ( 0 e. RR -> RR e. dom vol )
67	36, 51	syl6eqel 2558	. . . . . . 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 3622	. . . . . . . 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 4524	. . . . . . . 8 |- ( x e. ( A [,] B ) |-> ( vol ` ( S " { x } ) ) ) = ( x e. ( A [,] B ) |-> ( D - C ) )
73	51	recni 9599	. . . . . . . . . 10 |- ( D - C ) e. CC
74		ax-resscn 9540	. . . . . . . . . . 11 |- RR C_ CC
75	5, 74	sstri 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 ) )
78	73, 75, 76, 77	mp3an 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 )
80	2, 3, 78, 79	mp3an 1319	. . . . . . . 8 |- ( x e. ( A [,] B ) |-> ( D - C ) ) e. L^1
81	72, 80	eqeltri 2546	. . . . . . 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 21942	. . . . 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 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 ) )
86	12, 63, 84, 85	mpbir3an 1173	. . 3 |- S e. dom area
87		areaval 23017	. . 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 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 )
90	89, 50	mprg 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 )
92	2, 3, 91	mp2an 672	. . . . 5 |- ( A [,] B ) e. dom vol
93		mblvol 21671	. . . . . . . 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 21664	. . . . . . . 8 |- ( ( A e. RR /\ B e. RR /\ A <_ B ) -> ( vol* ` ( A [,] B ) ) = ( B - A ) )
97	2, 3, 95, 96	mp3an 1319	. . . . . . 7 |- ( vol* ` ( A [,] B ) ) = ( B - A )
98	94, 97	eqtri 2491	. . . . . 6 |- ( vol ` ( A [,] B ) ) = ( B - A )
99	3, 2	resubcli 9872	. . . . . 6 |- ( B - A ) e. RR
100	98, 99	eqeltri 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 ) ) ) )
102	92, 100, 73, 101	mp3an 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 )
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 2499	. . 3 |- S. RR if ( x e. ( A [,] B ) , ( D - C ) , 0 ) _d x = ( ( D - C ) x. ( B - A ) )
107	90, 106	eqtri 2491	. 2 |- S. RR ( vol ` ( S " { x } ) ) _d x = ( ( D - C ) x. ( B - A ) )
108	99	recni 9599	. . 3 |- ( B - A ) e. CC
109	73, 108	mulcomi 9593	. 2 |- ( ( D - C ) x. ( B - A ) ) = ( ( B - A ) x. ( D - C ) )
110	88, 107, 109	3eqtri 2495	1 |- (area ` S ) = ( ( B - A ) x. ( D - C ) )

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)