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Theorem pilem2 23407
Description: Lemma for pire 23413, pigt2lt4 23411 and sinpi 23412. (Contributed by Mario Carneiro, 12-Jun-2014.) (Revised by AV, 14-Sep-2020.)
Hypotheses
Ref Expression
pilem.1  |-  ( ph  ->  A  e.  ( 2 (,) 4 ) )
pilem.2  |-  ( ph  ->  B  e.  RR+ )
pilem.3  |-  ( ph  ->  ( sin `  A
)  =  0 )
pilem.4  |-  ( ph  ->  ( sin `  B
)  =  0 )
pilem.5  |-  ( ph  ->  pi  <  A )
Assertion
Ref Expression
pilem2  |-  ( ph  ->  ( ( pi  +  A )  /  2
)  <_  B )

Proof of Theorem pilem2
Dummy variables  x  y are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 df-pi 14126 . . . 4  |-  pi  = inf ( ( RR+  i^i  ( `' sin " { 0 } ) ) ,  RR ,  <  )
2 inss1 3652 . . . . . . 7  |-  ( RR+  i^i  ( `' sin " {
0 } ) ) 
C_  RR+
3 rpssre 11312 . . . . . . 7  |-  RR+  C_  RR
42, 3sstri 3441 . . . . . 6  |-  ( RR+  i^i  ( `' sin " {
0 } ) ) 
C_  RR
54a1i 11 . . . . 5  |-  ( ph  ->  ( RR+  i^i  ( `' sin " { 0 } ) )  C_  RR )
6 0re 9643 . . . . . . 7  |-  0  e.  RR
72sseli 3428 . . . . . . . . 9  |-  ( y  e.  ( RR+  i^i  ( `' sin " { 0 } ) )  -> 
y  e.  RR+ )
87rpge0d 11345 . . . . . . . 8  |-  ( y  e.  ( RR+  i^i  ( `' sin " { 0 } ) )  -> 
0  <_  y )
98rgen 2747 . . . . . . 7  |-  A. y  e.  ( RR+  i^i  ( `' sin " { 0 } ) ) 0  <_  y
10 breq1 4405 . . . . . . . . 9  |-  ( x  =  0  ->  (
x  <_  y  <->  0  <_  y ) )
1110ralbidv 2827 . . . . . . . 8  |-  ( x  =  0  ->  ( A. y  e.  ( RR+  i^i  ( `' sin " { 0 } ) ) x  <_  y  <->  A. y  e.  ( RR+  i^i  ( `' sin " {
0 } ) ) 0  <_  y )
)
1211rspcev 3150 . . . . . . 7  |-  ( ( 0  e.  RR  /\  A. y  e.  ( RR+  i^i  ( `' sin " {
0 } ) ) 0  <_  y )  ->  E. x  e.  RR  A. y  e.  ( RR+  i^i  ( `' sin " {
0 } ) ) x  <_  y )
136, 9, 12mp2an 678 . . . . . 6  |-  E. x  e.  RR  A. y  e.  ( RR+  i^i  ( `' sin " { 0 } ) ) x  <_  y
1413a1i 11 . . . . 5  |-  ( ph  ->  E. x  e.  RR  A. y  e.  ( RR+  i^i  ( `' sin " {
0 } ) ) x  <_  y )
15 2re 10679 . . . . . . . . 9  |-  2  e.  RR
16 pilem.2 . . . . . . . . . 10  |-  ( ph  ->  B  e.  RR+ )
1716rpred 11341 . . . . . . . . 9  |-  ( ph  ->  B  e.  RR )
18 remulcl 9624 . . . . . . . . 9  |-  ( ( 2  e.  RR  /\  B  e.  RR )  ->  ( 2  x.  B
)  e.  RR )
1915, 17, 18sylancr 669 . . . . . . . 8  |-  ( ph  ->  ( 2  x.  B
)  e.  RR )
20 pilem.1 . . . . . . . . 9  |-  ( ph  ->  A  e.  ( 2 (,) 4 ) )
21 elioore 11666 . . . . . . . . 9  |-  ( A  e.  ( 2 (,) 4 )  ->  A  e.  RR )
2220, 21syl 17 . . . . . . . 8  |-  ( ph  ->  A  e.  RR )
2319, 22resubcld 10047 . . . . . . 7  |-  ( ph  ->  ( ( 2  x.  B )  -  A
)  e.  RR )
24 4re 10686 . . . . . . . . . 10  |-  4  e.  RR
2524a1i 11 . . . . . . . . 9  |-  ( ph  ->  4  e.  RR )
26 eliooord 11694 . . . . . . . . . . 11  |-  ( A  e.  ( 2 (,) 4 )  ->  (
2  <  A  /\  A  <  4 ) )
2720, 26syl 17 . . . . . . . . . 10  |-  ( ph  ->  ( 2  <  A  /\  A  <  4
) )
2827simprd 465 . . . . . . . . 9  |-  ( ph  ->  A  <  4 )
29 2t2e4 10759 . . . . . . . . . 10  |-  ( 2  x.  2 )  =  4
3015a1i 11 . . . . . . . . . . . 12  |-  ( ph  ->  2  e.  RR )
31 0red 9644 . . . . . . . . . . . . . . . . 17  |-  ( ph  ->  0  e.  RR )
32 2pos 10701 . . . . . . . . . . . . . . . . . 18  |-  0  <  2
3332a1i 11 . . . . . . . . . . . . . . . . 17  |-  ( ph  ->  0  <  2 )
3427simpld 461 . . . . . . . . . . . . . . . . 17  |-  ( ph  ->  2  <  A )
3531, 30, 22, 33, 34lttrd 9796 . . . . . . . . . . . . . . . 16  |-  ( ph  ->  0  <  A )
3622, 35elrpd 11338 . . . . . . . . . . . . . . 15  |-  ( ph  ->  A  e.  RR+ )
37 pilem.3 . . . . . . . . . . . . . . 15  |-  ( ph  ->  ( sin `  A
)  =  0 )
38 pilem1 23406 . . . . . . . . . . . . . . 15  |-  ( A  e.  ( RR+  i^i  ( `' sin " { 0 } ) )  <->  ( A  e.  RR+  /\  ( sin `  A )  =  0 ) )
3936, 37, 38sylanbrc 670 . . . . . . . . . . . . . 14  |-  ( ph  ->  A  e.  ( RR+  i^i  ( `' sin " {
0 } ) ) )
40 ne0i 3737 . . . . . . . . . . . . . 14  |-  ( A  e.  ( RR+  i^i  ( `' sin " { 0 } ) )  -> 
( RR+  i^i  ( `' sin " { 0 } ) )  =/=  (/) )
4139, 40syl 17 . . . . . . . . . . . . 13  |-  ( ph  ->  ( RR+  i^i  ( `' sin " { 0 } ) )  =/=  (/) )
42 infrecl 10590 . . . . . . . . . . . . . 14  |-  ( ( ( RR+  i^i  ( `' sin " { 0 } ) )  C_  RR  /\  ( RR+  i^i  ( `' sin " { 0 } ) )  =/=  (/)  /\  E. x  e.  RR  A. y  e.  ( RR+  i^i  ( `' sin " { 0 } ) ) x  <_  y )  -> inf ( ( RR+  i^i  ( `' sin " { 0 } ) ) ,  RR ,  <  )  e.  RR )
434, 13, 42mp3an13 1355 . . . . . . . . . . . . 13  |-  ( (
RR+  i^i  ( `' sin " { 0 } ) )  =/=  (/)  -> inf ( (
RR+  i^i  ( `' sin " { 0 } ) ) ,  RR ,  <  )  e.  RR )
4441, 43syl 17 . . . . . . . . . . . 12  |-  ( ph  -> inf ( ( RR+  i^i  ( `' sin " { 0 } ) ) ,  RR ,  <  )  e.  RR )
45 pilem1 23406 . . . . . . . . . . . . . . 15  |-  ( x  e.  ( RR+  i^i  ( `' sin " { 0 } ) )  <->  ( x  e.  RR+  /\  ( sin `  x )  =  0 ) )
46 rpre 11308 . . . . . . . . . . . . . . . . . . . . 21  |-  ( x  e.  RR+  ->  x  e.  RR )
4746adantl 468 . . . . . . . . . . . . . . . . . . . 20  |-  ( (
ph  /\  x  e.  RR+ )  ->  x  e.  RR )
48 letric 9734 . . . . . . . . . . . . . . . . . . . 20  |-  ( ( 2  e.  RR  /\  x  e.  RR )  ->  ( 2  <_  x  \/  x  <_  2 ) )
4915, 47, 48sylancr 669 . . . . . . . . . . . . . . . . . . 19  |-  ( (
ph  /\  x  e.  RR+ )  ->  ( 2  <_  x  \/  x  <_  2 ) )
5049ord 379 . . . . . . . . . . . . . . . . . 18  |-  ( (
ph  /\  x  e.  RR+ )  ->  ( -.  2  <_  x  ->  x  <_  2 ) )
5146ad2antlr 733 . . . . . . . . . . . . . . . . . . . . . 22  |-  ( ( ( ph  /\  x  e.  RR+ )  /\  x  <_  2 )  ->  x  e.  RR )
52 rpgt0 11313 . . . . . . . . . . . . . . . . . . . . . . 23  |-  ( x  e.  RR+  ->  0  < 
x )
5352ad2antlr 733 . . . . . . . . . . . . . . . . . . . . . 22  |-  ( ( ( ph  /\  x  e.  RR+ )  /\  x  <_  2 )  ->  0  <  x )
54 simpr 463 . . . . . . . . . . . . . . . . . . . . . 22  |-  ( ( ( ph  /\  x  e.  RR+ )  /\  x  <_  2 )  ->  x  <_  2 )
55 0xr 9687 . . . . . . . . . . . . . . . . . . . . . . 23  |-  0  e.  RR*
56 elioc2 11697 . . . . . . . . . . . . . . . . . . . . . . 23  |-  ( ( 0  e.  RR*  /\  2  e.  RR )  ->  (
x  e.  ( 0 (,] 2 )  <->  ( x  e.  RR  /\  0  < 
x  /\  x  <_  2 ) ) )
5755, 15, 56mp2an 678 . . . . . . . . . . . . . . . . . . . . . 22  |-  ( x  e.  ( 0 (,] 2 )  <->  ( x  e.  RR  /\  0  < 
x  /\  x  <_  2 ) )
5851, 53, 54, 57syl3anbrc 1192 . . . . . . . . . . . . . . . . . . . . 21  |-  ( ( ( ph  /\  x  e.  RR+ )  /\  x  <_  2 )  ->  x  e.  ( 0 (,] 2
) )
59 sin02gt0 14246 . . . . . . . . . . . . . . . . . . . . 21  |-  ( x  e.  ( 0 (,] 2 )  ->  0  <  ( sin `  x
) )
6058, 59syl 17 . . . . . . . . . . . . . . . . . . . 20  |-  ( ( ( ph  /\  x  e.  RR+ )  /\  x  <_  2 )  ->  0  <  ( sin `  x
) )
6160gt0ne0d 10178 . . . . . . . . . . . . . . . . . . 19  |-  ( ( ( ph  /\  x  e.  RR+ )  /\  x  <_  2 )  ->  ( sin `  x )  =/=  0 )
6261ex 436 . . . . . . . . . . . . . . . . . 18  |-  ( (
ph  /\  x  e.  RR+ )  ->  ( x  <_  2  ->  ( sin `  x )  =/=  0
) )
6350, 62syld 45 . . . . . . . . . . . . . . . . 17  |-  ( (
ph  /\  x  e.  RR+ )  ->  ( -.  2  <_  x  ->  ( sin `  x )  =/=  0 ) )
6463necon4bd 2644 . . . . . . . . . . . . . . . 16  |-  ( (
ph  /\  x  e.  RR+ )  ->  ( ( sin `  x )  =  0  ->  2  <_  x ) )
6564expimpd 608 . . . . . . . . . . . . . . 15  |-  ( ph  ->  ( ( x  e.  RR+  /\  ( sin `  x
)  =  0 )  ->  2  <_  x
) )
6645, 65syl5bi 221 . . . . . . . . . . . . . 14  |-  ( ph  ->  ( x  e.  (
RR+  i^i  ( `' sin " { 0 } ) )  ->  2  <_  x ) )
6766ralrimiv 2800 . . . . . . . . . . . . 13  |-  ( ph  ->  A. x  e.  (
RR+  i^i  ( `' sin " { 0 } ) ) 2  <_  x )
68 infregelb 10594 . . . . . . . . . . . . . 14  |-  ( ( ( ( RR+  i^i  ( `' sin " { 0 } ) )  C_  RR  /\  ( RR+  i^i  ( `' sin " { 0 } ) )  =/=  (/)  /\  E. x  e.  RR  A. y  e.  ( RR+  i^i  ( `' sin " { 0 } ) ) x  <_  y )  /\  2  e.  RR )  ->  ( 2  <_ inf ( (
RR+  i^i  ( `' sin " { 0 } ) ) ,  RR ,  <  )  <->  A. x  e.  ( RR+  i^i  ( `' sin " { 0 } ) ) 2  <_  x ) )
695, 41, 14, 30, 68syl31anc 1271 . . . . . . . . . . . . 13  |-  ( ph  ->  ( 2  <_ inf ( (
RR+  i^i  ( `' sin " { 0 } ) ) ,  RR ,  <  )  <->  A. x  e.  ( RR+  i^i  ( `' sin " { 0 } ) ) 2  <_  x ) )
7067, 69mpbird 236 . . . . . . . . . . . 12  |-  ( ph  ->  2  <_ inf ( ( RR+  i^i  ( `' sin " { 0 } ) ) ,  RR ,  <  ) )
71 pilem.4 . . . . . . . . . . . . . 14  |-  ( ph  ->  ( sin `  B
)  =  0 )
72 pilem1 23406 . . . . . . . . . . . . . 14  |-  ( B  e.  ( RR+  i^i  ( `' sin " { 0 } ) )  <->  ( B  e.  RR+  /\  ( sin `  B )  =  0 ) )
7316, 71, 72sylanbrc 670 . . . . . . . . . . . . 13  |-  ( ph  ->  B  e.  ( RR+  i^i  ( `' sin " {
0 } ) ) )
74 infrelb 10596 . . . . . . . . . . . . 13  |-  ( ( ( RR+  i^i  ( `' sin " { 0 } ) )  C_  RR  /\  E. x  e.  RR  A. y  e.  ( RR+  i^i  ( `' sin " { 0 } ) ) x  <_  y  /\  B  e.  ( RR+  i^i  ( `' sin " { 0 } ) ) )  -> inf ( ( RR+  i^i  ( `' sin " {
0 } ) ) ,  RR ,  <  )  <_  B )
755, 14, 73, 74syl3anc 1268 . . . . . . . . . . . 12  |-  ( ph  -> inf ( ( RR+  i^i  ( `' sin " { 0 } ) ) ,  RR ,  <  )  <_  B )
7630, 44, 17, 70, 75letrd 9792 . . . . . . . . . . 11  |-  ( ph  ->  2  <_  B )
7715, 32pm3.2i 457 . . . . . . . . . . . . 13  |-  ( 2  e.  RR  /\  0  <  2 )
7877a1i 11 . . . . . . . . . . . 12  |-  ( ph  ->  ( 2  e.  RR  /\  0  <  2 ) )
79 lemul2 10458 . . . . . . . . . . . 12  |-  ( ( 2  e.  RR  /\  B  e.  RR  /\  (
2  e.  RR  /\  0  <  2 ) )  ->  ( 2  <_  B 
<->  ( 2  x.  2 )  <_  ( 2  x.  B ) ) )
8030, 17, 78, 79syl3anc 1268 . . . . . . . . . . 11  |-  ( ph  ->  ( 2  <_  B  <->  ( 2  x.  2 )  <_  ( 2  x.  B ) ) )
8176, 80mpbid 214 . . . . . . . . . 10  |-  ( ph  ->  ( 2  x.  2 )  <_  ( 2  x.  B ) )
8229, 81syl5eqbrr 4437 . . . . . . . . 9  |-  ( ph  ->  4  <_  ( 2  x.  B ) )
8322, 25, 19, 28, 82ltletrd 9795 . . . . . . . 8  |-  ( ph  ->  A  <  ( 2  x.  B ) )
8422, 19posdifd 10200 . . . . . . . 8  |-  ( ph  ->  ( A  <  (
2  x.  B )  <->  0  <  ( ( 2  x.  B )  -  A ) ) )
8583, 84mpbid 214 . . . . . . 7  |-  ( ph  ->  0  <  ( ( 2  x.  B )  -  A ) )
8623, 85elrpd 11338 . . . . . 6  |-  ( ph  ->  ( ( 2  x.  B )  -  A
)  e.  RR+ )
8719recnd 9669 . . . . . . . 8  |-  ( ph  ->  ( 2  x.  B
)  e.  CC )
8822recnd 9669 . . . . . . . 8  |-  ( ph  ->  A  e.  CC )
89 sinsub 14222 . . . . . . . 8  |-  ( ( ( 2  x.  B
)  e.  CC  /\  A  e.  CC )  ->  ( sin `  (
( 2  x.  B
)  -  A ) )  =  ( ( ( sin `  (
2  x.  B ) )  x.  ( cos `  A ) )  -  ( ( cos `  (
2  x.  B ) )  x.  ( sin `  A ) ) ) )
9087, 88, 89syl2anc 667 . . . . . . 7  |-  ( ph  ->  ( sin `  (
( 2  x.  B
)  -  A ) )  =  ( ( ( sin `  (
2  x.  B ) )  x.  ( cos `  A ) )  -  ( ( cos `  (
2  x.  B ) )  x.  ( sin `  A ) ) ) )
9117recnd 9669 . . . . . . . . . . . . 13  |-  ( ph  ->  B  e.  CC )
92 sin2t 14231 . . . . . . . . . . . . 13  |-  ( B  e.  CC  ->  ( sin `  ( 2  x.  B ) )  =  ( 2  x.  (
( sin `  B
)  x.  ( cos `  B ) ) ) )
9391, 92syl 17 . . . . . . . . . . . 12  |-  ( ph  ->  ( sin `  (
2  x.  B ) )  =  ( 2  x.  ( ( sin `  B )  x.  ( cos `  B ) ) ) )
9471oveq1d 6305 . . . . . . . . . . . . . . 15  |-  ( ph  ->  ( ( sin `  B
)  x.  ( cos `  B ) )  =  ( 0  x.  ( cos `  B ) ) )
9591coscld 14185 . . . . . . . . . . . . . . . 16  |-  ( ph  ->  ( cos `  B
)  e.  CC )
9695mul02d 9831 . . . . . . . . . . . . . . 15  |-  ( ph  ->  ( 0  x.  ( cos `  B ) )  =  0 )
9794, 96eqtrd 2485 . . . . . . . . . . . . . 14  |-  ( ph  ->  ( ( sin `  B
)  x.  ( cos `  B ) )  =  0 )
9897oveq2d 6306 . . . . . . . . . . . . 13  |-  ( ph  ->  ( 2  x.  (
( sin `  B
)  x.  ( cos `  B ) ) )  =  ( 2  x.  0 ) )
99 2t0e0 10765 . . . . . . . . . . . . 13  |-  ( 2  x.  0 )  =  0
10098, 99syl6eq 2501 . . . . . . . . . . . 12  |-  ( ph  ->  ( 2  x.  (
( sin `  B
)  x.  ( cos `  B ) ) )  =  0 )
10193, 100eqtrd 2485 . . . . . . . . . . 11  |-  ( ph  ->  ( sin `  (
2  x.  B ) )  =  0 )
102101oveq1d 6305 . . . . . . . . . 10  |-  ( ph  ->  ( ( sin `  (
2  x.  B ) )  x.  ( cos `  A ) )  =  ( 0  x.  ( cos `  A ) ) )
10388coscld 14185 . . . . . . . . . . 11  |-  ( ph  ->  ( cos `  A
)  e.  CC )
104103mul02d 9831 . . . . . . . . . 10  |-  ( ph  ->  ( 0  x.  ( cos `  A ) )  =  0 )
105102, 104eqtrd 2485 . . . . . . . . 9  |-  ( ph  ->  ( ( sin `  (
2  x.  B ) )  x.  ( cos `  A ) )  =  0 )
10637oveq2d 6306 . . . . . . . . . 10  |-  ( ph  ->  ( ( cos `  (
2  x.  B ) )  x.  ( sin `  A ) )  =  ( ( cos `  (
2  x.  B ) )  x.  0 ) )
10787coscld 14185 . . . . . . . . . . 11  |-  ( ph  ->  ( cos `  (
2  x.  B ) )  e.  CC )
108107mul01d 9832 . . . . . . . . . 10  |-  ( ph  ->  ( ( cos `  (
2  x.  B ) )  x.  0 )  =  0 )
109106, 108eqtrd 2485 . . . . . . . . 9  |-  ( ph  ->  ( ( cos `  (
2  x.  B ) )  x.  ( sin `  A ) )  =  0 )
110105, 109oveq12d 6308 . . . . . . . 8  |-  ( ph  ->  ( ( ( sin `  ( 2  x.  B
) )  x.  ( cos `  A ) )  -  ( ( cos `  ( 2  x.  B
) )  x.  ( sin `  A ) ) )  =  ( 0  -  0 ) )
111 0m0e0 10719 . . . . . . . 8  |-  ( 0  -  0 )  =  0
112110, 111syl6eq 2501 . . . . . . 7  |-  ( ph  ->  ( ( ( sin `  ( 2  x.  B
) )  x.  ( cos `  A ) )  -  ( ( cos `  ( 2  x.  B
) )  x.  ( sin `  A ) ) )  =  0 )
11390, 112eqtrd 2485 . . . . . 6  |-  ( ph  ->  ( sin `  (
( 2  x.  B
)  -  A ) )  =  0 )
114 pilem1 23406 . . . . . 6  |-  ( ( ( 2  x.  B
)  -  A )  e.  ( RR+  i^i  ( `' sin " { 0 } ) )  <->  ( (
( 2  x.  B
)  -  A )  e.  RR+  /\  ( sin `  ( ( 2  x.  B )  -  A ) )  =  0 ) )
11586, 113, 114sylanbrc 670 . . . . 5  |-  ( ph  ->  ( ( 2  x.  B )  -  A
)  e.  ( RR+  i^i  ( `' sin " {
0 } ) ) )
116 infrelb 10596 . . . . 5  |-  ( ( ( RR+  i^i  ( `' sin " { 0 } ) )  C_  RR  /\  E. x  e.  RR  A. y  e.  ( RR+  i^i  ( `' sin " { 0 } ) ) x  <_  y  /\  (
( 2  x.  B
)  -  A )  e.  ( RR+  i^i  ( `' sin " { 0 } ) ) )  -> inf ( ( RR+  i^i  ( `' sin " {
0 } ) ) ,  RR ,  <  )  <_  ( ( 2  x.  B )  -  A ) )
1175, 14, 115, 116syl3anc 1268 . . . 4  |-  ( ph  -> inf ( ( RR+  i^i  ( `' sin " { 0 } ) ) ,  RR ,  <  )  <_  ( ( 2  x.  B )  -  A
) )
1181, 117syl5eqbr 4436 . . 3  |-  ( ph  ->  pi  <_  ( (
2  x.  B )  -  A ) )
1191, 44syl5eqel 2533 . . . 4  |-  ( ph  ->  pi  e.  RR )
120 leaddsub 10090 . . . 4  |-  ( ( pi  e.  RR  /\  A  e.  RR  /\  (
2  x.  B )  e.  RR )  -> 
( ( pi  +  A )  <_  (
2  x.  B )  <-> 
pi  <_  ( ( 2  x.  B )  -  A ) ) )
121119, 22, 19, 120syl3anc 1268 . . 3  |-  ( ph  ->  ( ( pi  +  A )  <_  (
2  x.  B )  <-> 
pi  <_  ( ( 2  x.  B )  -  A ) ) )
122118, 121mpbird 236 . 2  |-  ( ph  ->  ( pi  +  A
)  <_  ( 2  x.  B ) )
123119, 22readdcld 9670 . . 3  |-  ( ph  ->  ( pi  +  A
)  e.  RR )
124 ledivmul 10481 . . 3  |-  ( ( ( pi  +  A
)  e.  RR  /\  B  e.  RR  /\  (
2  e.  RR  /\  0  <  2 ) )  ->  ( ( ( pi  +  A )  /  2 )  <_  B 
<->  ( pi  +  A
)  <_  ( 2  x.  B ) ) )
125123, 17, 78, 124syl3anc 1268 . 2  |-  ( ph  ->  ( ( ( pi  +  A )  / 
2 )  <_  B  <->  ( pi  +  A )  <_  ( 2  x.  B ) ) )
126122, 125mpbird 236 1  |-  ( ph  ->  ( ( pi  +  A )  /  2
)  <_  B )
Colors of variables: wff setvar class
Syntax hints:   -. wn 3    -> wi 4    <-> wb 188    \/ wo 370    /\ wa 371    /\ w3a 985    = wceq 1444    e. wcel 1887    =/= wne 2622   A.wral 2737   E.wrex 2738    i^i cin 3403    C_ wss 3404   (/)c0 3731   {csn 3968   class class class wbr 4402   `'ccnv 4833   "cima 4837   ` cfv 5582  (class class class)co 6290  infcinf 7955   CCcc 9537   RRcr 9538   0cc0 9539    + caddc 9542    x. cmul 9544   RR*cxr 9674    < clt 9675    <_ cle 9676    - cmin 9860    / cdiv 10269   2c2 10659   4c4 10661   RR+crp 11302   (,)cioo 11635   (,]cioc 11636   sincsin 14116   cosccos 14117   picpi 14119
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1669  ax-4 1682  ax-5 1758  ax-6 1805  ax-7 1851  ax-8 1889  ax-9 1896  ax-10 1915  ax-11 1920  ax-12 1933  ax-13 2091  ax-ext 2431  ax-rep 4515  ax-sep 4525  ax-nul 4534  ax-pow 4581  ax-pr 4639  ax-un 6583  ax-inf2 8146  ax-cnex 9595  ax-resscn 9596  ax-1cn 9597  ax-icn 9598  ax-addcl 9599  ax-addrcl 9600  ax-mulcl 9601  ax-mulrcl 9602  ax-mulcom 9603  ax-addass 9604  ax-mulass 9605  ax-distr 9606  ax-i2m1 9607  ax-1ne0 9608  ax-1rid 9609  ax-rnegex 9610  ax-rrecex 9611  ax-cnre 9612  ax-pre-lttri 9613  ax-pre-lttrn 9614  ax-pre-ltadd 9615  ax-pre-mulgt0 9616  ax-pre-sup 9617  ax-addf 9618  ax-mulf 9619
This theorem depends on definitions:  df-bi 189  df-or 372  df-an 373  df-3or 986  df-3an 987  df-tru 1447  df-fal 1450  df-ex 1664  df-nf 1668  df-sb 1798  df-eu 2303  df-mo 2304  df-clab 2438  df-cleq 2444  df-clel 2447  df-nfc 2581  df-ne 2624  df-nel 2625  df-ral 2742  df-rex 2743  df-reu 2744  df-rmo 2745  df-rab 2746  df-v 3047  df-sbc 3268  df-csb 3364  df-dif 3407  df-un 3409  df-in 3411  df-ss 3418  df-pss 3420  df-nul 3732  df-if 3882  df-pw 3953  df-sn 3969  df-pr 3971  df-tp 3973  df-op 3975  df-uni 4199  df-int 4235  df-iun 4280  df-br 4403  df-opab 4462  df-mpt 4463  df-tr 4498  df-eprel 4745  df-id 4749  df-po 4755  df-so 4756  df-fr 4793  df-se 4794  df-we 4795  df-xp 4840  df-rel 4841  df-cnv 4842  df-co 4843  df-dm 4844  df-rn 4845  df-res 4846  df-ima 4847  df-pred 5380  df-ord 5426  df-on 5427  df-lim 5428  df-suc 5429  df-iota 5546  df-fun 5584  df-fn 5585  df-f 5586  df-f1 5587  df-fo 5588  df-f1o 5589  df-fv 5590  df-isom 5591  df-riota 6252  df-ov 6293  df-oprab 6294  df-mpt2 6295  df-om 6693  df-1st 6793  df-2nd 6794  df-wrecs 7028  df-recs 7090  df-rdg 7128  df-1o 7182  df-oadd 7186  df-er 7363  df-pm 7475  df-en 7570  df-dom 7571  df-sdom 7572  df-fin 7573  df-sup 7956  df-inf 7957  df-oi 8025  df-card 8373  df-pnf 9677  df-mnf 9678  df-xr 9679  df-ltxr 9680  df-le 9681  df-sub 9862  df-neg 9863  df-div 10270  df-nn 10610  df-2 10668  df-3 10669  df-4 10670  df-5 10671  df-6 10672  df-7 10673  df-8 10674  df-n0 10870  df-z 10938  df-uz 11160  df-rp 11303  df-ioo 11639  df-ioc 11640  df-ico 11641  df-fz 11785  df-fzo 11916  df-fl 12028  df-seq 12214  df-exp 12273  df-fac 12460  df-bc 12488  df-hash 12516  df-shft 13130  df-cj 13162  df-re 13163  df-im 13164  df-sqrt 13298  df-abs 13299  df-limsup 13526  df-clim 13552  df-rlim 13553  df-sum 13753  df-ef 14121  df-sin 14123  df-cos 14124  df-pi 14126
This theorem is referenced by:  pilem3  23409
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