Step |
Hyp |
Ref |
Expression |
1 |
|
arearect.1 |
|- A e. RR |
2 |
|
arearect.2 |
|- B e. RR |
3 |
|
arearect.3 |
|- C e. RR |
4 |
|
arearect.4 |
|- D e. RR |
5 |
|
arearect.5 |
|- A <_ B |
6 |
|
arearect.6 |
|- C <_ D |
7 |
|
arearect.7 |
|- S = ( ( A [,] B ) X. ( C [,] D ) ) |
8 |
|
iccssre |
|- ( ( A e. RR /\ B e. RR ) -> ( A [,] B ) C_ RR ) |
9 |
1 2 8
|
mp2an |
|- ( A [,] B ) C_ RR |
10 |
|
iccssre |
|- ( ( C e. RR /\ D e. RR ) -> ( C [,] D ) C_ RR ) |
11 |
3 4 10
|
mp2an |
|- ( C [,] D ) C_ RR |
12 |
|
xpss12 |
|- ( ( ( A [,] B ) C_ RR /\ ( C [,] D ) C_ RR ) -> ( ( A [,] B ) X. ( C [,] D ) ) C_ ( RR X. RR ) ) |
13 |
9 11 12
|
mp2an |
|- ( ( A [,] B ) X. ( C [,] D ) ) C_ ( RR X. RR ) |
14 |
7 13
|
eqsstri |
|- S C_ ( RR X. RR ) |
15 |
|
iftrue |
|- ( x e. ( A [,] B ) -> if ( x e. ( A [,] B ) , ( D - C ) , 0 ) = ( D - C ) ) |
16 |
7
|
imaeq1i |
|- ( S " { x } ) = ( ( ( A [,] B ) X. ( C [,] D ) ) " { x } ) |
17 |
|
iftrue |
|- ( x e. ( A [,] B ) -> if ( x e. ( A [,] B ) , ( C [,] D ) , (/) ) = ( C [,] D ) ) |
18 |
|
xpimasn |
|- ( x e. ( A [,] B ) -> ( ( ( A [,] B ) X. ( C [,] D ) ) " { x } ) = ( C [,] D ) ) |
19 |
17 18
|
eqtr4d |
|- ( x e. ( A [,] B ) -> if ( x e. ( A [,] B ) , ( C [,] D ) , (/) ) = ( ( ( A [,] B ) X. ( C [,] D ) ) " { x } ) ) |
20 |
|
iffalse |
|- ( -. x e. ( A [,] B ) -> if ( x e. ( A [,] B ) , ( C [,] D ) , (/) ) = (/) ) |
21 |
|
disjsn |
|- ( ( ( A [,] B ) i^i { x } ) = (/) <-> -. x e. ( A [,] B ) ) |
22 |
|
xpima1 |
|- ( ( ( A [,] B ) i^i { x } ) = (/) -> ( ( ( A [,] B ) X. ( C [,] D ) ) " { x } ) = (/) ) |
23 |
21 22
|
sylbir |
|- ( -. x e. ( A [,] B ) -> ( ( ( A [,] B ) X. ( C [,] D ) ) " { x } ) = (/) ) |
24 |
20 23
|
eqtr4d |
|- ( -. x e. ( A [,] B ) -> if ( x e. ( A [,] B ) , ( C [,] D ) , (/) ) = ( ( ( A [,] B ) X. ( C [,] D ) ) " { x } ) ) |
25 |
19 24
|
pm2.61i |
|- if ( x e. ( A [,] B ) , ( C [,] D ) , (/) ) = ( ( ( A [,] B ) X. ( C [,] D ) ) " { x } ) |
26 |
16 25
|
eqtr4i |
|- ( S " { x } ) = if ( x e. ( A [,] B ) , ( C [,] D ) , (/) ) |
27 |
26
|
fveq2i |
|- ( vol ` ( S " { x } ) ) = ( vol ` if ( x e. ( A [,] B ) , ( C [,] D ) , (/) ) ) |
28 |
17
|
fveq2d |
|- ( x e. ( A [,] B ) -> ( vol ` if ( x e. ( A [,] B ) , ( C [,] D ) , (/) ) ) = ( vol ` ( C [,] D ) ) ) |
29 |
27 28
|
syl5eq |
|- ( x e. ( A [,] B ) -> ( vol ` ( S " { x } ) ) = ( vol ` ( C [,] D ) ) ) |
30 |
|
iccmbl |
|- ( ( C e. RR /\ D e. RR ) -> ( C [,] D ) e. dom vol ) |
31 |
3 4 30
|
mp2an |
|- ( C [,] D ) e. dom vol |
32 |
|
mblvol |
|- ( ( C [,] D ) e. dom vol -> ( vol ` ( C [,] D ) ) = ( vol* ` ( C [,] D ) ) ) |
33 |
31 32
|
ax-mp |
|- ( vol ` ( C [,] D ) ) = ( vol* ` ( C [,] D ) ) |
34 |
|
ovolicc |
|- ( ( C e. RR /\ D e. RR /\ C <_ D ) -> ( vol* ` ( C [,] D ) ) = ( D - C ) ) |
35 |
3 4 6 34
|
mp3an |
|- ( vol* ` ( C [,] D ) ) = ( D - C ) |
36 |
33 35
|
eqtri |
|- ( vol ` ( C [,] D ) ) = ( D - C ) |
37 |
29 36
|
eqtrdi |
|- ( x e. ( A [,] B ) -> ( vol ` ( S " { x } ) ) = ( D - C ) ) |
38 |
15 37
|
eqtr4d |
|- ( x e. ( A [,] B ) -> if ( x e. ( A [,] B ) , ( D - C ) , 0 ) = ( vol ` ( S " { x } ) ) ) |
39 |
|
iffalse |
|- ( -. x e. ( A [,] B ) -> if ( x e. ( A [,] B ) , ( D - C ) , 0 ) = 0 ) |
40 |
20
|
fveq2d |
|- ( -. x e. ( A [,] B ) -> ( vol ` if ( x e. ( A [,] B ) , ( C [,] D ) , (/) ) ) = ( vol ` (/) ) ) |
41 |
27 40
|
syl5eq |
|- ( -. x e. ( A [,] B ) -> ( vol ` ( S " { x } ) ) = ( vol ` (/) ) ) |
42 |
|
0mbl |
|- (/) e. dom vol |
43 |
|
mblvol |
|- ( (/) e. dom vol -> ( vol ` (/) ) = ( vol* ` (/) ) ) |
44 |
42 43
|
ax-mp |
|- ( vol ` (/) ) = ( vol* ` (/) ) |
45 |
|
ovol0 |
|- ( vol* ` (/) ) = 0 |
46 |
44 45
|
eqtri |
|- ( vol ` (/) ) = 0 |
47 |
41 46
|
eqtrdi |
|- ( -. x e. ( A [,] B ) -> ( vol ` ( S " { x } ) ) = 0 ) |
48 |
39 47
|
eqtr4d |
|- ( -. x e. ( A [,] B ) -> if ( x e. ( A [,] B ) , ( D - C ) , 0 ) = ( vol ` ( S " { x } ) ) ) |
49 |
38 48
|
pm2.61i |
|- if ( x e. ( A [,] B ) , ( D - C ) , 0 ) = ( vol ` ( S " { x } ) ) |
50 |
49
|
eqcomi |
|- ( vol ` ( S " { x } ) ) = if ( x e. ( A [,] B ) , ( D - C ) , 0 ) |
51 |
50
|
a1i |
|- ( x e. RR -> ( vol ` ( S " { x } ) ) = if ( x e. ( A [,] B ) , ( D - C ) , 0 ) ) |
52 |
4 3
|
resubcli |
|- ( D - C ) e. RR |
53 |
|
0re |
|- 0 e. RR |
54 |
52 53
|
ifcli |
|- if ( x e. ( A [,] B ) , ( D - C ) , 0 ) e. RR |
55 |
51 54
|
eqeltrdi |
|- ( x e. RR -> ( vol ` ( S " { x } ) ) e. RR ) |
56 |
|
volf |
|- vol : dom vol --> ( 0 [,] +oo ) |
57 |
|
ffun |
|- ( vol : dom vol --> ( 0 [,] +oo ) -> Fun vol ) |
58 |
56 57
|
ax-mp |
|- Fun vol |
59 |
31 42
|
ifcli |
|- if ( x e. ( A [,] B ) , ( C [,] D ) , (/) ) e. dom vol |
60 |
26 59
|
eqeltri |
|- ( S " { x } ) e. dom vol |
61 |
|
fvimacnv |
|- ( ( Fun vol /\ ( S " { x } ) e. dom vol ) -> ( ( vol ` ( S " { x } ) ) e. RR <-> ( S " { x } ) e. ( `' vol " RR ) ) ) |
62 |
58 60 61
|
mp2an |
|- ( ( vol ` ( S " { x } ) ) e. RR <-> ( S " { x } ) e. ( `' vol " RR ) ) |
63 |
55 62
|
sylib |
|- ( x e. RR -> ( S " { x } ) e. ( `' vol " RR ) ) |
64 |
63
|
rgen |
|- A. x e. RR ( S " { x } ) e. ( `' vol " RR ) |
65 |
9
|
a1i |
|- ( 0 e. RR -> ( A [,] B ) C_ RR ) |
66 |
|
rembl |
|- RR e. dom vol |
67 |
66
|
a1i |
|- ( 0 e. RR -> RR e. dom vol ) |
68 |
37 52
|
eqeltrdi |
|- ( x e. ( A [,] B ) -> ( vol ` ( S " { x } ) ) e. RR ) |
69 |
68
|
adantl |
|- ( ( 0 e. RR /\ x e. ( A [,] B ) ) -> ( vol ` ( S " { x } ) ) e. RR ) |
70 |
|
eldifn |
|- ( x e. ( RR \ ( A [,] B ) ) -> -. x e. ( A [,] B ) ) |
71 |
70 47
|
syl |
|- ( x e. ( RR \ ( A [,] B ) ) -> ( vol ` ( S " { x } ) ) = 0 ) |
72 |
71
|
adantl |
|- ( ( 0 e. RR /\ x e. ( RR \ ( A [,] B ) ) ) -> ( vol ` ( S " { x } ) ) = 0 ) |
73 |
37
|
mpteq2ia |
|- ( x e. ( A [,] B ) |-> ( vol ` ( S " { x } ) ) ) = ( x e. ( A [,] B ) |-> ( D - C ) ) |
74 |
52
|
recni |
|- ( D - C ) e. CC |
75 |
|
ax-resscn |
|- RR C_ CC |
76 |
9 75
|
sstri |
|- ( A [,] B ) C_ CC |
77 |
|
ssid |
|- CC C_ CC |
78 |
|
cncfmptc |
|- ( ( ( D - C ) e. CC /\ ( A [,] B ) C_ CC /\ CC C_ CC ) -> ( x e. ( A [,] B ) |-> ( D - C ) ) e. ( ( A [,] B ) -cn-> CC ) ) |
79 |
74 76 77 78
|
mp3an |
|- ( x e. ( A [,] B ) |-> ( D - C ) ) e. ( ( A [,] B ) -cn-> CC ) |
80 |
|
cniccibl |
|- ( ( 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 ) |
81 |
1 2 79 80
|
mp3an |
|- ( x e. ( A [,] B ) |-> ( D - C ) ) e. L^1 |
82 |
73 81
|
eqeltri |
|- ( x e. ( A [,] B ) |-> ( vol ` ( S " { x } ) ) ) e. L^1 |
83 |
82
|
a1i |
|- ( 0 e. RR -> ( x e. ( A [,] B ) |-> ( vol ` ( S " { x } ) ) ) e. L^1 ) |
84 |
65 67 69 72 83
|
iblss2 |
|- ( 0 e. RR -> ( x e. RR |-> ( vol ` ( S " { x } ) ) ) e. L^1 ) |
85 |
53 84
|
ax-mp |
|- ( x e. RR |-> ( vol ` ( S " { x } ) ) ) e. L^1 |
86 |
|
dmarea |
|- ( 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 ) ) |
87 |
14 64 85 86
|
mpbir3an |
|- S e. dom area |
88 |
|
areaval |
|- ( S e. dom area -> ( area ` S ) = S. RR ( vol ` ( S " { x } ) ) _d x ) |
89 |
87 88
|
ax-mp |
|- ( area ` S ) = S. RR ( vol ` ( S " { x } ) ) _d x |
90 |
|
itgeq2 |
|- ( 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 ) |
91 |
90 51
|
mprg |
|- S. RR ( vol ` ( S " { x } ) ) _d x = S. RR if ( x e. ( A [,] B ) , ( D - C ) , 0 ) _d x |
92 |
|
iccmbl |
|- ( ( A e. RR /\ B e. RR ) -> ( A [,] B ) e. dom vol ) |
93 |
1 2 92
|
mp2an |
|- ( A [,] B ) e. dom vol |
94 |
|
mblvol |
|- ( ( A [,] B ) e. dom vol -> ( vol ` ( A [,] B ) ) = ( vol* ` ( A [,] B ) ) ) |
95 |
93 94
|
ax-mp |
|- ( vol ` ( A [,] B ) ) = ( vol* ` ( A [,] B ) ) |
96 |
|
ovolicc |
|- ( ( A e. RR /\ B e. RR /\ A <_ B ) -> ( vol* ` ( A [,] B ) ) = ( B - A ) ) |
97 |
1 2 5 96
|
mp3an |
|- ( vol* ` ( A [,] B ) ) = ( B - A ) |
98 |
95 97
|
eqtri |
|- ( vol ` ( A [,] B ) ) = ( B - A ) |
99 |
2 1
|
resubcli |
|- ( B - A ) e. RR |
100 |
98 99
|
eqeltri |
|- ( vol ` ( A [,] B ) ) e. RR |
101 |
|
itgconst |
|- ( ( ( 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 |
93 100 74 101
|
mp3an |
|- S. ( A [,] B ) ( D - C ) _d x = ( ( D - C ) x. ( vol ` ( A [,] B ) ) ) |
103 |
|
itgss2 |
|- ( ( A [,] B ) C_ RR -> S. ( A [,] B ) ( D - C ) _d x = S. RR if ( x e. ( A [,] B ) , ( D - C ) , 0 ) _d x ) |
104 |
9 103
|
ax-mp |
|- S. ( A [,] B ) ( D - C ) _d x = S. RR if ( x e. ( A [,] B ) , ( D - C ) , 0 ) _d x |
105 |
98
|
oveq2i |
|- ( ( D - C ) x. ( vol ` ( A [,] B ) ) ) = ( ( D - C ) x. ( B - A ) ) |
106 |
102 104 105
|
3eqtr3i |
|- S. RR if ( x e. ( A [,] B ) , ( D - C ) , 0 ) _d x = ( ( D - C ) x. ( B - A ) ) |
107 |
91 106
|
eqtri |
|- S. RR ( vol ` ( S " { x } ) ) _d x = ( ( D - C ) x. ( B - A ) ) |
108 |
99
|
recni |
|- ( B - A ) e. CC |
109 |
74 108
|
mulcomi |
|- ( ( D - C ) x. ( B - A ) ) = ( ( B - A ) x. ( D - C ) ) |
110 |
89 107 109
|
3eqtri |
|- ( area ` S ) = ( ( B - A ) x. ( D - C ) ) |