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Theorem pilem2 19828
Description: Lemma for pire 19832, pigt2lt4 19830 and sinpi 19831. (Contributed by Mario Carneiro, 12-Jun-2014.)
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 12354 . . . 4  |-  pi  =  sup ( ( RR+  i^i  ( `' sin " { 0 } ) ) ,  RR ,  `'  <  )
2 inss1 3389 . . . . . . 7  |-  ( RR+  i^i  ( `' sin " {
0 } ) ) 
C_  RR+
3 rpssre 10364 . . . . . . 7  |-  RR+  C_  RR
42, 3sstri 3188 . . . . . 6  |-  ( RR+  i^i  ( `' sin " {
0 } ) ) 
C_  RR
54a1i 10 . . . . 5  |-  ( ph  ->  ( RR+  i^i  ( `' sin " { 0 } ) )  C_  RR )
6 0re 8838 . . . . . . 7  |-  0  e.  RR
72sseli 3176 . . . . . . . . 9  |-  ( y  e.  ( RR+  i^i  ( `' sin " { 0 } ) )  -> 
y  e.  RR+ )
87rpge0d 10394 . . . . . . . 8  |-  ( y  e.  ( RR+  i^i  ( `' sin " { 0 } ) )  -> 
0  <_  y )
98rgen 2608 . . . . . . 7  |-  A. y  e.  ( RR+  i^i  ( `' sin " { 0 } ) ) 0  <_  y
10 breq1 4026 . . . . . . . . 9  |-  ( x  =  0  ->  (
x  <_  y  <->  0  <_  y ) )
1110ralbidv 2563 . . . . . . . 8  |-  ( x  =  0  ->  ( A. y  e.  ( RR+  i^i  ( `' sin " { 0 } ) ) x  <_  y  <->  A. y  e.  ( RR+  i^i  ( `' sin " {
0 } ) ) 0  <_  y )
)
1211rspcev 2884 . . . . . . 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 653 . . . . . 6  |-  E. x  e.  RR  A. y  e.  ( RR+  i^i  ( `' sin " { 0 } ) ) x  <_  y
1413a1i 10 . . . . 5  |-  ( ph  ->  E. x  e.  RR  A. y  e.  ( RR+  i^i  ( `' sin " {
0 } ) ) x  <_  y )
15 2re 9815 . . . . . . . . 9  |-  2  e.  RR
16 pilem.2 . . . . . . . . . 10  |-  ( ph  ->  B  e.  RR+ )
1716rpred 10390 . . . . . . . . 9  |-  ( ph  ->  B  e.  RR )
18 remulcl 8822 . . . . . . . . 9  |-  ( ( 2  e.  RR  /\  B  e.  RR )  ->  ( 2  x.  B
)  e.  RR )
1915, 17, 18sylancr 644 . . . . . . . 8  |-  ( ph  ->  ( 2  x.  B
)  e.  RR )
20 pilem.1 . . . . . . . . 9  |-  ( ph  ->  A  e.  ( 2 (,) 4 ) )
21 elioore 10686 . . . . . . . . 9  |-  ( A  e.  ( 2 (,) 4 )  ->  A  e.  RR )
2220, 21syl 15 . . . . . . . 8  |-  ( ph  ->  A  e.  RR )
2319, 22resubcld 9211 . . . . . . 7  |-  ( ph  ->  ( ( 2  x.  B )  -  A
)  e.  RR )
24 4re 9819 . . . . . . . . . 10  |-  4  e.  RR
2524a1i 10 . . . . . . . . 9  |-  ( ph  ->  4  e.  RR )
26 eliooord 10710 . . . . . . . . . . 11  |-  ( A  e.  ( 2 (,) 4 )  ->  (
2  <  A  /\  A  <  4 ) )
2720, 26syl 15 . . . . . . . . . 10  |-  ( ph  ->  ( 2  <  A  /\  A  <  4
) )
2827simprd 449 . . . . . . . . 9  |-  ( ph  ->  A  <  4 )
29 2t2e4 9871 . . . . . . . . . 10  |-  ( 2  x.  2 )  =  4
3015a1i 10 . . . . . . . . . . . 12  |-  ( ph  ->  2  e.  RR )
316a1i 10 . . . . . . . . . . . . . . . . 17  |-  ( ph  ->  0  e.  RR )
32 2pos 9828 . . . . . . . . . . . . . . . . . 18  |-  0  <  2
3332a1i 10 . . . . . . . . . . . . . . . . 17  |-  ( ph  ->  0  <  2 )
3427simpld 445 . . . . . . . . . . . . . . . . 17  |-  ( ph  ->  2  <  A )
3531, 30, 22, 33, 34lttrd 8977 . . . . . . . . . . . . . . . 16  |-  ( ph  ->  0  <  A )
3622, 35elrpd 10388 . . . . . . . . . . . . . . 15  |-  ( ph  ->  A  e.  RR+ )
37 pilem.3 . . . . . . . . . . . . . . 15  |-  ( ph  ->  ( sin `  A
)  =  0 )
38 pilem1 19827 . . . . . . . . . . . . . . 15  |-  ( A  e.  ( RR+  i^i  ( `' sin " { 0 } ) )  <->  ( A  e.  RR+  /\  ( sin `  A )  =  0 ) )
3936, 37, 38sylanbrc 645 . . . . . . . . . . . . . 14  |-  ( ph  ->  A  e.  ( RR+  i^i  ( `' sin " {
0 } ) ) )
40 ne0i 3461 . . . . . . . . . . . . . 14  |-  ( A  e.  ( RR+  i^i  ( `' sin " { 0 } ) )  -> 
( RR+  i^i  ( `' sin " { 0 } ) )  =/=  (/) )
4139, 40syl 15 . . . . . . . . . . . . 13  |-  ( ph  ->  ( RR+  i^i  ( `' sin " { 0 } ) )  =/=  (/) )
42 infmrcl 9733 . . . . . . . . . . . . . 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 )  ->  sup ( ( RR+  i^i  ( `' sin " { 0 } ) ) ,  RR ,  `'  <  )  e.  RR )
434, 13, 42mp3an13 1268 . . . . . . . . . . . . 13  |-  ( (
RR+  i^i  ( `' sin " { 0 } ) )  =/=  (/)  ->  sup ( ( RR+  i^i  ( `' sin " { 0 } ) ) ,  RR ,  `'  <  )  e.  RR )
4441, 43syl 15 . . . . . . . . . . . 12  |-  ( ph  ->  sup ( ( RR+  i^i  ( `' sin " {
0 } ) ) ,  RR ,  `'  <  )  e.  RR )
45 pilem1 19827 . . . . . . . . . . . . . . 15  |-  ( x  e.  ( RR+  i^i  ( `' sin " { 0 } ) )  <->  ( x  e.  RR+  /\  ( sin `  x )  =  0 ) )
46 rpre 10360 . . . . . . . . . . . . . . . . . . . . 21  |-  ( x  e.  RR+  ->  x  e.  RR )
4746adantl 452 . . . . . . . . . . . . . . . . . . . 20  |-  ( (
ph  /\  x  e.  RR+ )  ->  x  e.  RR )
48 letric 8921 . . . . . . . . . . . . . . . . . . . 20  |-  ( ( 2  e.  RR  /\  x  e.  RR )  ->  ( 2  <_  x  \/  x  <_  2 ) )
4915, 47, 48sylancr 644 . . . . . . . . . . . . . . . . . . 19  |-  ( (
ph  /\  x  e.  RR+ )  ->  ( 2  <_  x  \/  x  <_  2 ) )
5049ord 366 . . . . . . . . . . . . . . . . . 18  |-  ( (
ph  /\  x  e.  RR+ )  ->  ( -.  2  <_  x  ->  x  <_  2 ) )
5146ad2antlr 707 . . . . . . . . . . . . . . . . . . . . . 22  |-  ( ( ( ph  /\  x  e.  RR+ )  /\  x  <_  2 )  ->  x  e.  RR )
52 rpgt0 10365 . . . . . . . . . . . . . . . . . . . . . . 23  |-  ( x  e.  RR+  ->  0  < 
x )
5352ad2antlr 707 . . . . . . . . . . . . . . . . . . . . . 22  |-  ( ( ( ph  /\  x  e.  RR+ )  /\  x  <_  2 )  ->  0  <  x )
54 simpr 447 . . . . . . . . . . . . . . . . . . . . . 22  |-  ( ( ( ph  /\  x  e.  RR+ )  /\  x  <_  2 )  ->  x  <_  2 )
55 0xr 8878 . . . . . . . . . . . . . . . . . . . . . . 23  |-  0  e.  RR*
56 elioc2 10713 . . . . . . . . . . . . . . . . . . . . . . 23  |-  ( ( 0  e.  RR*  /\  2  e.  RR )  ->  (
x  e.  ( 0 (,] 2 )  <->  ( x  e.  RR  /\  0  < 
x  /\  x  <_  2 ) ) )
5755, 15, 56mp2an 653 . . . . . . . . . . . . . . . . . . . . . 22  |-  ( x  e.  ( 0 (,] 2 )  <->  ( x  e.  RR  /\  0  < 
x  /\  x  <_  2 ) )
5851, 53, 54, 57syl3anbrc 1136 . . . . . . . . . . . . . . . . . . . . 21  |-  ( ( ( ph  /\  x  e.  RR+ )  /\  x  <_  2 )  ->  x  e.  ( 0 (,] 2
) )
59 sin02gt0 12472 . . . . . . . . . . . . . . . . . . . . 21  |-  ( x  e.  ( 0 (,] 2 )  ->  0  <  ( sin `  x
) )
6058, 59syl 15 . . . . . . . . . . . . . . . . . . . 20  |-  ( ( ( ph  /\  x  e.  RR+ )  /\  x  <_  2 )  ->  0  <  ( sin `  x
) )
6160gt0ne0d 9337 . . . . . . . . . . . . . . . . . . 19  |-  ( ( ( ph  /\  x  e.  RR+ )  /\  x  <_  2 )  ->  ( sin `  x )  =/=  0 )
6261ex 423 . . . . . . . . . . . . . . . . . 18  |-  ( (
ph  /\  x  e.  RR+ )  ->  ( x  <_  2  ->  ( sin `  x )  =/=  0
) )
6350, 62syld 40 . . . . . . . . . . . . . . . . 17  |-  ( (
ph  /\  x  e.  RR+ )  ->  ( -.  2  <_  x  ->  ( sin `  x )  =/=  0 ) )
6463necon4bd 2508 . . . . . . . . . . . . . . . 16  |-  ( (
ph  /\  x  e.  RR+ )  ->  ( ( sin `  x )  =  0  ->  2  <_  x ) )
6564expimpd 586 . . . . . . . . . . . . . . 15  |-  ( ph  ->  ( ( x  e.  RR+  /\  ( sin `  x
)  =  0 )  ->  2  <_  x
) )
6645, 65syl5bi 208 . . . . . . . . . . . . . 14  |-  ( ph  ->  ( x  e.  (
RR+  i^i  ( `' sin " { 0 } ) )  ->  2  <_  x ) )
6766ralrimiv 2625 . . . . . . . . . . . . 13  |-  ( ph  ->  A. x  e.  (
RR+  i^i  ( `' sin " { 0 } ) ) 2  <_  x )
68 infmrgelb 9734 . . . . . . . . . . . . . 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  <_  sup ( ( RR+  i^i  ( `' sin " { 0 } ) ) ,  RR ,  `'  <  )  <->  A. x  e.  ( RR+  i^i  ( `' sin " { 0 } ) ) 2  <_  x
) )
695, 41, 14, 30, 68syl31anc 1185 . . . . . . . . . . . . 13  |-  ( ph  ->  ( 2  <_  sup ( ( RR+  i^i  ( `' sin " { 0 } ) ) ,  RR ,  `'  <  )  <->  A. x  e.  ( RR+  i^i  ( `' sin " { 0 } ) ) 2  <_  x
) )
7067, 69mpbird 223 . . . . . . . . . . . 12  |-  ( ph  ->  2  <_  sup (
( RR+  i^i  ( `' sin " { 0 } ) ) ,  RR ,  `'  <  ) )
71 pilem.4 . . . . . . . . . . . . . 14  |-  ( ph  ->  ( sin `  B
)  =  0 )
72 pilem1 19827 . . . . . . . . . . . . . 14  |-  ( B  e.  ( RR+  i^i  ( `' sin " { 0 } ) )  <->  ( B  e.  RR+  /\  ( sin `  B )  =  0 ) )
7316, 71, 72sylanbrc 645 . . . . . . . . . . . . 13  |-  ( ph  ->  B  e.  ( RR+  i^i  ( `' sin " {
0 } ) ) )
74 infmrlb 9735 . . . . . . . . . . . . 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 } ) ) )  ->  sup ( ( RR+  i^i  ( `' sin " {
0 } ) ) ,  RR ,  `'  <  )  <_  B )
755, 14, 73, 74syl3anc 1182 . . . . . . . . . . . 12  |-  ( ph  ->  sup ( ( RR+  i^i  ( `' sin " {
0 } ) ) ,  RR ,  `'  <  )  <_  B )
7630, 44, 17, 70, 75letrd 8973 . . . . . . . . . . 11  |-  ( ph  ->  2  <_  B )
7715, 32pm3.2i 441 . . . . . . . . . . . . 13  |-  ( 2  e.  RR  /\  0  <  2 )
7877a1i 10 . . . . . . . . . . . 12  |-  ( ph  ->  ( 2  e.  RR  /\  0  <  2 ) )
79 lemul2 9609 . . . . . . . . . . . 12  |-  ( ( 2  e.  RR  /\  B  e.  RR  /\  (
2  e.  RR  /\  0  <  2 ) )  ->  ( 2  <_  B 
<->  ( 2  x.  2 )  <_  ( 2  x.  B ) ) )
8030, 17, 78, 79syl3anc 1182 . . . . . . . . . . 11  |-  ( ph  ->  ( 2  <_  B  <->  ( 2  x.  2 )  <_  ( 2  x.  B ) ) )
8176, 80mpbid 201 . . . . . . . . . 10  |-  ( ph  ->  ( 2  x.  2 )  <_  ( 2  x.  B ) )
8229, 81syl5eqbrr 4057 . . . . . . . . 9  |-  ( ph  ->  4  <_  ( 2  x.  B ) )
8322, 25, 19, 28, 82ltletrd 8976 . . . . . . . 8  |-  ( ph  ->  A  <  ( 2  x.  B ) )
8422, 19posdifd 9359 . . . . . . . 8  |-  ( ph  ->  ( A  <  (
2  x.  B )  <->  0  <  ( ( 2  x.  B )  -  A ) ) )
8583, 84mpbid 201 . . . . . . 7  |-  ( ph  ->  0  <  ( ( 2  x.  B )  -  A ) )
8623, 85elrpd 10388 . . . . . 6  |-  ( ph  ->  ( ( 2  x.  B )  -  A
)  e.  RR+ )
8719recnd 8861 . . . . . . . 8  |-  ( ph  ->  ( 2  x.  B
)  e.  CC )
8822recnd 8861 . . . . . . . 8  |-  ( ph  ->  A  e.  CC )
89 sinsub 12448 . . . . . . . 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 642 . . . . . . 7  |-  ( ph  ->  ( sin `  (
( 2  x.  B
)  -  A ) )  =  ( ( ( sin `  (
2  x.  B ) )  x.  ( cos `  A ) )  -  ( ( cos `  (
2  x.  B ) )  x.  ( sin `  A ) ) ) )
9117recnd 8861 . . . . . . . . . . . . 13  |-  ( ph  ->  B  e.  CC )
92 sin2t 12457 . . . . . . . . . . . . 13  |-  ( B  e.  CC  ->  ( sin `  ( 2  x.  B ) )  =  ( 2  x.  (
( sin `  B
)  x.  ( cos `  B ) ) ) )
9391, 92syl 15 . . . . . . . . . . . 12  |-  ( ph  ->  ( sin `  (
2  x.  B ) )  =  ( 2  x.  ( ( sin `  B )  x.  ( cos `  B ) ) ) )
9471oveq1d 5873 . . . . . . . . . . . . . . 15  |-  ( ph  ->  ( ( sin `  B
)  x.  ( cos `  B ) )  =  ( 0  x.  ( cos `  B ) ) )
9591coscld 12411 . . . . . . . . . . . . . . . 16  |-  ( ph  ->  ( cos `  B
)  e.  CC )
9695mul02d 9010 . . . . . . . . . . . . . . 15  |-  ( ph  ->  ( 0  x.  ( cos `  B ) )  =  0 )
9794, 96eqtrd 2315 . . . . . . . . . . . . . 14  |-  ( ph  ->  ( ( sin `  B
)  x.  ( cos `  B ) )  =  0 )
9897oveq2d 5874 . . . . . . . . . . . . 13  |-  ( ph  ->  ( 2  x.  (
( sin `  B
)  x.  ( cos `  B ) ) )  =  ( 2  x.  0 ) )
99 2cn 9816 . . . . . . . . . . . . . 14  |-  2  e.  CC
10099mul01i 9002 . . . . . . . . . . . . 13  |-  ( 2  x.  0 )  =  0
10198, 100syl6eq 2331 . . . . . . . . . . . 12  |-  ( ph  ->  ( 2  x.  (
( sin `  B
)  x.  ( cos `  B ) ) )  =  0 )
10293, 101eqtrd 2315 . . . . . . . . . . 11  |-  ( ph  ->  ( sin `  (
2  x.  B ) )  =  0 )
103102oveq1d 5873 . . . . . . . . . 10  |-  ( ph  ->  ( ( sin `  (
2  x.  B ) )  x.  ( cos `  A ) )  =  ( 0  x.  ( cos `  A ) ) )
10488coscld 12411 . . . . . . . . . . 11  |-  ( ph  ->  ( cos `  A
)  e.  CC )
105104mul02d 9010 . . . . . . . . . 10  |-  ( ph  ->  ( 0  x.  ( cos `  A ) )  =  0 )
106103, 105eqtrd 2315 . . . . . . . . 9  |-  ( ph  ->  ( ( sin `  (
2  x.  B ) )  x.  ( cos `  A ) )  =  0 )
10737oveq2d 5874 . . . . . . . . . 10  |-  ( ph  ->  ( ( cos `  (
2  x.  B ) )  x.  ( sin `  A ) )  =  ( ( cos `  (
2  x.  B ) )  x.  0 ) )
10887coscld 12411 . . . . . . . . . . 11  |-  ( ph  ->  ( cos `  (
2  x.  B ) )  e.  CC )
109108mul01d 9011 . . . . . . . . . 10  |-  ( ph  ->  ( ( cos `  (
2  x.  B ) )  x.  0 )  =  0 )
110107, 109eqtrd 2315 . . . . . . . . 9  |-  ( ph  ->  ( ( cos `  (
2  x.  B ) )  x.  ( sin `  A ) )  =  0 )
111106, 110oveq12d 5876 . . . . . . . 8  |-  ( ph  ->  ( ( ( sin `  ( 2  x.  B
) )  x.  ( cos `  A ) )  -  ( ( cos `  ( 2  x.  B
) )  x.  ( sin `  A ) ) )  =  ( 0  -  0 ) )
112 0cn 8831 . . . . . . . . 9  |-  0  e.  CC
113112subidi 9117 . . . . . . . 8  |-  ( 0  -  0 )  =  0
114111, 113syl6eq 2331 . . . . . . 7  |-  ( ph  ->  ( ( ( sin `  ( 2  x.  B
) )  x.  ( cos `  A ) )  -  ( ( cos `  ( 2  x.  B
) )  x.  ( sin `  A ) ) )  =  0 )
11590, 114eqtrd 2315 . . . . . 6  |-  ( ph  ->  ( sin `  (
( 2  x.  B
)  -  A ) )  =  0 )
116 pilem1 19827 . . . . . 6  |-  ( ( ( 2  x.  B
)  -  A )  e.  ( RR+  i^i  ( `' sin " { 0 } ) )  <->  ( (
( 2  x.  B
)  -  A )  e.  RR+  /\  ( sin `  ( ( 2  x.  B )  -  A ) )  =  0 ) )
11786, 115, 116sylanbrc 645 . . . . 5  |-  ( ph  ->  ( ( 2  x.  B )  -  A
)  e.  ( RR+  i^i  ( `' sin " {
0 } ) ) )
118 infmrlb 9735 . . . . 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 } ) ) )  ->  sup ( ( RR+  i^i  ( `' sin " {
0 } ) ) ,  RR ,  `'  <  )  <_  ( (
2  x.  B )  -  A ) )
1195, 14, 117, 118syl3anc 1182 . . . 4  |-  ( ph  ->  sup ( ( RR+  i^i  ( `' sin " {
0 } ) ) ,  RR ,  `'  <  )  <_  ( (
2  x.  B )  -  A ) )
1201, 119syl5eqbr 4056 . . 3  |-  ( ph  ->  pi  <_  ( (
2  x.  B )  -  A ) )
1211, 44syl5eqel 2367 . . . 4  |-  ( ph  ->  pi  e.  RR )
122 leaddsub 9250 . . . 4  |-  ( ( pi  e.  RR  /\  A  e.  RR  /\  (
2  x.  B )  e.  RR )  -> 
( ( pi  +  A )  <_  (
2  x.  B )  <-> 
pi  <_  ( ( 2  x.  B )  -  A ) ) )
123121, 22, 19, 122syl3anc 1182 . . 3  |-  ( ph  ->  ( ( pi  +  A )  <_  (
2  x.  B )  <-> 
pi  <_  ( ( 2  x.  B )  -  A ) ) )
124120, 123mpbird 223 . 2  |-  ( ph  ->  ( pi  +  A
)  <_  ( 2  x.  B ) )
125121, 22readdcld 8862 . . 3  |-  ( ph  ->  ( pi  +  A
)  e.  RR )
126 ledivmul 9629 . . 3  |-  ( ( ( pi  +  A
)  e.  RR  /\  B  e.  RR  /\  (
2  e.  RR  /\  0  <  2 ) )  ->  ( ( ( pi  +  A )  /  2 )  <_  B 
<->  ( pi  +  A
)  <_  ( 2  x.  B ) ) )
127125, 17, 78, 126syl3anc 1182 . 2  |-  ( ph  ->  ( ( ( pi  +  A )  / 
2 )  <_  B  <->  ( pi  +  A )  <_  ( 2  x.  B ) ) )
128124, 127mpbird 223 1  |-  ( ph  ->  ( ( pi  +  A )  /  2
)  <_  B )
Colors of variables: wff set class
Syntax hints:   -. wn 3    -> wi 4    <-> wb 176    \/ wo 357    /\ wa 358    /\ w3a 934    = wceq 1623    e. wcel 1684    =/= wne 2446   A.wral 2543   E.wrex 2544    i^i cin 3151    C_ wss 3152   (/)c0 3455   {csn 3640   class class class wbr 4023   `'ccnv 4688   "cima 4692   ` cfv 5255  (class class class)co 5858   supcsup 7193   CCcc 8735   RRcr 8736   0cc0 8737    + caddc 8740    x. cmul 8742   RR*cxr 8866    < clt 8867    <_ cle 8868    - cmin 9037    / cdiv 9423   2c2 9795   4c4 9797   RR+crp 10354   (,)cioo 10656   (,]cioc 10657   sincsin 12345   cosccos 12346   picpi 12348
This theorem is referenced by:  pilem3  19829
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1533  ax-5 1544  ax-17 1603  ax-9 1635  ax-8 1643  ax-13 1686  ax-14 1688  ax-6 1703  ax-7 1708  ax-11 1715  ax-12 1866  ax-ext 2264  ax-rep 4131  ax-sep 4141  ax-nul 4149  ax-pow 4188  ax-pr 4214  ax-un 4512  ax-inf2 7342  ax-cnex 8793  ax-resscn 8794  ax-1cn 8795  ax-icn 8796  ax-addcl 8797  ax-addrcl 8798  ax-mulcl 8799  ax-mulrcl 8800  ax-mulcom 8801  ax-addass 8802  ax-mulass 8803  ax-distr 8804  ax-i2m1 8805  ax-1ne0 8806  ax-1rid 8807  ax-rnegex 8808  ax-rrecex 8809  ax-cnre 8810  ax-pre-lttri 8811  ax-pre-lttrn 8812  ax-pre-ltadd 8813  ax-pre-mulgt0 8814  ax-pre-sup 8815  ax-addf 8816  ax-mulf 8817
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3or 935  df-3an 936  df-tru 1310  df-ex 1529  df-nf 1532  df-sb 1630  df-eu 2147  df-mo 2148  df-clab 2270  df-cleq 2276  df-clel 2279  df-nfc 2408  df-ne 2448  df-nel 2449  df-ral 2548  df-rex 2549  df-reu 2550  df-rmo 2551  df-rab 2552  df-v 2790  df-sbc 2992  df-csb 3082  df-dif 3155  df-un 3157  df-in 3159  df-ss 3166  df-pss 3168  df-nul 3456  df-if 3566  df-pw 3627  df-sn 3646  df-pr 3647  df-tp 3648  df-op 3649  df-uni 3828  df-int 3863  df-iun 3907  df-br 4024  df-opab 4078  df-mpt 4079  df-tr 4114  df-eprel 4305  df-id 4309  df-po 4314  df-so 4315  df-fr 4352  df-se 4353  df-we 4354  df-ord 4395  df-on 4396  df-lim 4397  df-suc 4398  df-om 4657  df-xp 4695  df-rel 4696  df-cnv 4697  df-co 4698  df-dm 4699  df-rn 4700  df-res 4701  df-ima 4702  df-iota 5219  df-fun 5257  df-fn 5258  df-f 5259  df-f1 5260  df-fo 5261  df-f1o 5262  df-fv 5263  df-isom 5264  df-ov 5861  df-oprab 5862  df-mpt2 5863  df-1st 6122  df-2nd 6123  df-riota 6304  df-recs 6388  df-rdg 6423  df-1o 6479  df-oadd 6483  df-er 6660  df-pm 6775  df-en 6864  df-dom 6865  df-sdom 6866  df-fin 6867  df-sup 7194  df-oi 7225  df-card 7572  df-pnf 8869  df-mnf 8870  df-xr 8871  df-ltxr 8872  df-le 8873  df-sub 9039  df-neg 9040  df-div 9424  df-nn 9747  df-2 9804  df-3 9805  df-4 9806  df-5 9807  df-6 9808  df-7 9809  df-8 9810  df-n0 9966  df-z 10025  df-uz 10231  df-rp 10355  df-ioo 10660  df-ioc 10661  df-ico 10662  df-fz 10783  df-fzo 10871  df-fl 10925  df-seq 11047  df-exp 11105  df-fac 11289  df-bc 11316  df-hash 11338  df-shft 11562  df-cj 11584  df-re 11585  df-im 11586  df-sqr 11720  df-abs 11721  df-limsup 11945  df-clim 11962  df-rlim 11963  df-sum 12159  df-ef 12349  df-sin 12351  df-cos 12352  df-pi 12354
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