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Theorem pilem2 21939
Description: Lemma for pire 21943, pigt2lt4 21941 and sinpi 21942. (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 13379 . . . 4  |-  pi  =  sup ( ( RR+  i^i  ( `' sin " { 0 } ) ) ,  RR ,  `'  <  )
2 inss1 3591 . . . . . . 7  |-  ( RR+  i^i  ( `' sin " {
0 } ) ) 
C_  RR+
3 rpssre 11022 . . . . . . 7  |-  RR+  C_  RR
42, 3sstri 3386 . . . . . 6  |-  ( RR+  i^i  ( `' sin " {
0 } ) ) 
C_  RR
54a1i 11 . . . . 5  |-  ( ph  ->  ( RR+  i^i  ( `' sin " { 0 } ) )  C_  RR )
6 0re 9407 . . . . . . 7  |-  0  e.  RR
72sseli 3373 . . . . . . . . 9  |-  ( y  e.  ( RR+  i^i  ( `' sin " { 0 } ) )  -> 
y  e.  RR+ )
87rpge0d 11052 . . . . . . . 8  |-  ( y  e.  ( RR+  i^i  ( `' sin " { 0 } ) )  -> 
0  <_  y )
98rgen 2802 . . . . . . 7  |-  A. y  e.  ( RR+  i^i  ( `' sin " { 0 } ) ) 0  <_  y
10 breq1 4316 . . . . . . . . 9  |-  ( x  =  0  ->  (
x  <_  y  <->  0  <_  y ) )
1110ralbidv 2756 . . . . . . . 8  |-  ( x  =  0  ->  ( A. y  e.  ( RR+  i^i  ( `' sin " { 0 } ) ) x  <_  y  <->  A. y  e.  ( RR+  i^i  ( `' sin " {
0 } ) ) 0  <_  y )
)
1211rspcev 3094 . . . . . . 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 672 . . . . . 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 10412 . . . . . . . . 9  |-  2  e.  RR
16 pilem.2 . . . . . . . . . 10  |-  ( ph  ->  B  e.  RR+ )
1716rpred 11048 . . . . . . . . 9  |-  ( ph  ->  B  e.  RR )
18 remulcl 9388 . . . . . . . . 9  |-  ( ( 2  e.  RR  /\  B  e.  RR )  ->  ( 2  x.  B
)  e.  RR )
1915, 17, 18sylancr 663 . . . . . . . 8  |-  ( ph  ->  ( 2  x.  B
)  e.  RR )
20 pilem.1 . . . . . . . . 9  |-  ( ph  ->  A  e.  ( 2 (,) 4 ) )
21 elioore 11351 . . . . . . . . 9  |-  ( A  e.  ( 2 (,) 4 )  ->  A  e.  RR )
2220, 21syl 16 . . . . . . . 8  |-  ( ph  ->  A  e.  RR )
2319, 22resubcld 9797 . . . . . . 7  |-  ( ph  ->  ( ( 2  x.  B )  -  A
)  e.  RR )
24 4re 10419 . . . . . . . . . 10  |-  4  e.  RR
2524a1i 11 . . . . . . . . 9  |-  ( ph  ->  4  e.  RR )
26 eliooord 11376 . . . . . . . . . . 11  |-  ( A  e.  ( 2 (,) 4 )  ->  (
2  <  A  /\  A  <  4 ) )
2720, 26syl 16 . . . . . . . . . 10  |-  ( ph  ->  ( 2  <  A  /\  A  <  4
) )
2827simprd 463 . . . . . . . . 9  |-  ( ph  ->  A  <  4 )
29 2t2e4 10492 . . . . . . . . . 10  |-  ( 2  x.  2 )  =  4
3015a1i 11 . . . . . . . . . . . 12  |-  ( ph  ->  2  e.  RR )
31 0red 9408 . . . . . . . . . . . . . . . . 17  |-  ( ph  ->  0  e.  RR )
32 2pos 10434 . . . . . . . . . . . . . . . . . 18  |-  0  <  2
3332a1i 11 . . . . . . . . . . . . . . . . 17  |-  ( ph  ->  0  <  2 )
3427simpld 459 . . . . . . . . . . . . . . . . 17  |-  ( ph  ->  2  <  A )
3531, 30, 22, 33, 34lttrd 9553 . . . . . . . . . . . . . . . 16  |-  ( ph  ->  0  <  A )
3622, 35elrpd 11046 . . . . . . . . . . . . . . 15  |-  ( ph  ->  A  e.  RR+ )
37 pilem.3 . . . . . . . . . . . . . . 15  |-  ( ph  ->  ( sin `  A
)  =  0 )
38 pilem1 21938 . . . . . . . . . . . . . . 15  |-  ( A  e.  ( RR+  i^i  ( `' sin " { 0 } ) )  <->  ( A  e.  RR+  /\  ( sin `  A )  =  0 ) )
3936, 37, 38sylanbrc 664 . . . . . . . . . . . . . 14  |-  ( ph  ->  A  e.  ( RR+  i^i  ( `' sin " {
0 } ) ) )
40 ne0i 3664 . . . . . . . . . . . . . 14  |-  ( A  e.  ( RR+  i^i  ( `' sin " { 0 } ) )  -> 
( RR+  i^i  ( `' sin " { 0 } ) )  =/=  (/) )
4139, 40syl 16 . . . . . . . . . . . . 13  |-  ( ph  ->  ( RR+  i^i  ( `' sin " { 0 } ) )  =/=  (/) )
42 infmrcl 10330 . . . . . . . . . . . . . 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 1305 . . . . . . . . . . . . 13  |-  ( (
RR+  i^i  ( `' sin " { 0 } ) )  =/=  (/)  ->  sup ( ( RR+  i^i  ( `' sin " { 0 } ) ) ,  RR ,  `'  <  )  e.  RR )
4441, 43syl 16 . . . . . . . . . . . 12  |-  ( ph  ->  sup ( ( RR+  i^i  ( `' sin " {
0 } ) ) ,  RR ,  `'  <  )  e.  RR )
45 pilem1 21938 . . . . . . . . . . . . . . 15  |-  ( x  e.  ( RR+  i^i  ( `' sin " { 0 } ) )  <->  ( x  e.  RR+  /\  ( sin `  x )  =  0 ) )
46 rpre 11018 . . . . . . . . . . . . . . . . . . . . 21  |-  ( x  e.  RR+  ->  x  e.  RR )
4746adantl 466 . . . . . . . . . . . . . . . . . . . 20  |-  ( (
ph  /\  x  e.  RR+ )  ->  x  e.  RR )
48 letric 9496 . . . . . . . . . . . . . . . . . . . 20  |-  ( ( 2  e.  RR  /\  x  e.  RR )  ->  ( 2  <_  x  \/  x  <_  2 ) )
4915, 47, 48sylancr 663 . . . . . . . . . . . . . . . . . . 19  |-  ( (
ph  /\  x  e.  RR+ )  ->  ( 2  <_  x  \/  x  <_  2 ) )
5049ord 377 . . . . . . . . . . . . . . . . . 18  |-  ( (
ph  /\  x  e.  RR+ )  ->  ( -.  2  <_  x  ->  x  <_  2 ) )
5146ad2antlr 726 . . . . . . . . . . . . . . . . . . . . . 22  |-  ( ( ( ph  /\  x  e.  RR+ )  /\  x  <_  2 )  ->  x  e.  RR )
52 rpgt0 11023 . . . . . . . . . . . . . . . . . . . . . . 23  |-  ( x  e.  RR+  ->  0  < 
x )
5352ad2antlr 726 . . . . . . . . . . . . . . . . . . . . . 22  |-  ( ( ( ph  /\  x  e.  RR+ )  /\  x  <_  2 )  ->  0  <  x )
54 simpr 461 . . . . . . . . . . . . . . . . . . . . . 22  |-  ( ( ( ph  /\  x  e.  RR+ )  /\  x  <_  2 )  ->  x  <_  2 )
55 0xr 9451 . . . . . . . . . . . . . . . . . . . . . . 23  |-  0  e.  RR*
56 elioc2 11379 . . . . . . . . . . . . . . . . . . . . . . 23  |-  ( ( 0  e.  RR*  /\  2  e.  RR )  ->  (
x  e.  ( 0 (,] 2 )  <->  ( x  e.  RR  /\  0  < 
x  /\  x  <_  2 ) ) )
5755, 15, 56mp2an 672 . . . . . . . . . . . . . . . . . . . . . 22  |-  ( x  e.  ( 0 (,] 2 )  <->  ( x  e.  RR  /\  0  < 
x  /\  x  <_  2 ) )
5851, 53, 54, 57syl3anbrc 1172 . . . . . . . . . . . . . . . . . . . . 21  |-  ( ( ( ph  /\  x  e.  RR+ )  /\  x  <_  2 )  ->  x  e.  ( 0 (,] 2
) )
59 sin02gt0 13497 . . . . . . . . . . . . . . . . . . . . 21  |-  ( x  e.  ( 0 (,] 2 )  ->  0  <  ( sin `  x
) )
6058, 59syl 16 . . . . . . . . . . . . . . . . . . . 20  |-  ( ( ( ph  /\  x  e.  RR+ )  /\  x  <_  2 )  ->  0  <  ( sin `  x
) )
6160gt0ne0d 9925 . . . . . . . . . . . . . . . . . . 19  |-  ( ( ( ph  /\  x  e.  RR+ )  /\  x  <_  2 )  ->  ( sin `  x )  =/=  0 )
6261ex 434 . . . . . . . . . . . . . . . . . 18  |-  ( (
ph  /\  x  e.  RR+ )  ->  ( x  <_  2  ->  ( sin `  x )  =/=  0
) )
6350, 62syld 44 . . . . . . . . . . . . . . . . 17  |-  ( (
ph  /\  x  e.  RR+ )  ->  ( -.  2  <_  x  ->  ( sin `  x )  =/=  0 ) )
6463necon4bd 2697 . . . . . . . . . . . . . . . 16  |-  ( (
ph  /\  x  e.  RR+ )  ->  ( ( sin `  x )  =  0  ->  2  <_  x ) )
6564expimpd 603 . . . . . . . . . . . . . . 15  |-  ( ph  ->  ( ( x  e.  RR+  /\  ( sin `  x
)  =  0 )  ->  2  <_  x
) )
6645, 65syl5bi 217 . . . . . . . . . . . . . 14  |-  ( ph  ->  ( x  e.  (
RR+  i^i  ( `' sin " { 0 } ) )  ->  2  <_  x ) )
6766ralrimiv 2819 . . . . . . . . . . . . 13  |-  ( ph  ->  A. x  e.  (
RR+  i^i  ( `' sin " { 0 } ) ) 2  <_  x )
68 infmrgelb 10331 . . . . . . . . . . . . . 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 1221 . . . . . . . . . . . . 13  |-  ( ph  ->  ( 2  <_  sup ( ( RR+  i^i  ( `' sin " { 0 } ) ) ,  RR ,  `'  <  )  <->  A. x  e.  ( RR+  i^i  ( `' sin " { 0 } ) ) 2  <_  x
) )
7067, 69mpbird 232 . . . . . . . . . . . 12  |-  ( ph  ->  2  <_  sup (
( RR+  i^i  ( `' sin " { 0 } ) ) ,  RR ,  `'  <  ) )
71 pilem.4 . . . . . . . . . . . . . 14  |-  ( ph  ->  ( sin `  B
)  =  0 )
72 pilem1 21938 . . . . . . . . . . . . . 14  |-  ( B  e.  ( RR+  i^i  ( `' sin " { 0 } ) )  <->  ( B  e.  RR+  /\  ( sin `  B )  =  0 ) )
7316, 71, 72sylanbrc 664 . . . . . . . . . . . . 13  |-  ( ph  ->  B  e.  ( RR+  i^i  ( `' sin " {
0 } ) ) )
74 infmrlb 10332 . . . . . . . . . . . . 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 1218 . . . . . . . . . . . 12  |-  ( ph  ->  sup ( ( RR+  i^i  ( `' sin " {
0 } ) ) ,  RR ,  `'  <  )  <_  B )
7630, 44, 17, 70, 75letrd 9549 . . . . . . . . . . 11  |-  ( ph  ->  2  <_  B )
7715, 32pm3.2i 455 . . . . . . . . . . . . 13  |-  ( 2  e.  RR  /\  0  <  2 )
7877a1i 11 . . . . . . . . . . . 12  |-  ( ph  ->  ( 2  e.  RR  /\  0  <  2 ) )
79 lemul2 10203 . . . . . . . . . . . 12  |-  ( ( 2  e.  RR  /\  B  e.  RR  /\  (
2  e.  RR  /\  0  <  2 ) )  ->  ( 2  <_  B 
<->  ( 2  x.  2 )  <_  ( 2  x.  B ) ) )
8030, 17, 78, 79syl3anc 1218 . . . . . . . . . . 11  |-  ( ph  ->  ( 2  <_  B  <->  ( 2  x.  2 )  <_  ( 2  x.  B ) ) )
8176, 80mpbid 210 . . . . . . . . . 10  |-  ( ph  ->  ( 2  x.  2 )  <_  ( 2  x.  B ) )
8229, 81syl5eqbrr 4347 . . . . . . . . 9  |-  ( ph  ->  4  <_  ( 2  x.  B ) )
8322, 25, 19, 28, 82ltletrd 9552 . . . . . . . 8  |-  ( ph  ->  A  <  ( 2  x.  B ) )
8422, 19posdifd 9947 . . . . . . . 8  |-  ( ph  ->  ( A  <  (
2  x.  B )  <->  0  <  ( ( 2  x.  B )  -  A ) ) )
8583, 84mpbid 210 . . . . . . 7  |-  ( ph  ->  0  <  ( ( 2  x.  B )  -  A ) )
8623, 85elrpd 11046 . . . . . 6  |-  ( ph  ->  ( ( 2  x.  B )  -  A
)  e.  RR+ )
8719recnd 9433 . . . . . . . 8  |-  ( ph  ->  ( 2  x.  B
)  e.  CC )
8822recnd 9433 . . . . . . . 8  |-  ( ph  ->  A  e.  CC )
89 sinsub 13473 . . . . . . . 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 661 . . . . . . 7  |-  ( ph  ->  ( sin `  (
( 2  x.  B
)  -  A ) )  =  ( ( ( sin `  (
2  x.  B ) )  x.  ( cos `  A ) )  -  ( ( cos `  (
2  x.  B ) )  x.  ( sin `  A ) ) ) )
9117recnd 9433 . . . . . . . . . . . . 13  |-  ( ph  ->  B  e.  CC )
92 sin2t 13482 . . . . . . . . . . . . 13  |-  ( B  e.  CC  ->  ( sin `  ( 2  x.  B ) )  =  ( 2  x.  (
( sin `  B
)  x.  ( cos `  B ) ) ) )
9391, 92syl 16 . . . . . . . . . . . 12  |-  ( ph  ->  ( sin `  (
2  x.  B ) )  =  ( 2  x.  ( ( sin `  B )  x.  ( cos `  B ) ) ) )
9471oveq1d 6127 . . . . . . . . . . . . . . 15  |-  ( ph  ->  ( ( sin `  B
)  x.  ( cos `  B ) )  =  ( 0  x.  ( cos `  B ) ) )
9591coscld 13436 . . . . . . . . . . . . . . . 16  |-  ( ph  ->  ( cos `  B
)  e.  CC )
9695mul02d 9588 . . . . . . . . . . . . . . 15  |-  ( ph  ->  ( 0  x.  ( cos `  B ) )  =  0 )
9794, 96eqtrd 2475 . . . . . . . . . . . . . 14  |-  ( ph  ->  ( ( sin `  B
)  x.  ( cos `  B ) )  =  0 )
9897oveq2d 6128 . . . . . . . . . . . . 13  |-  ( ph  ->  ( 2  x.  (
( sin `  B
)  x.  ( cos `  B ) ) )  =  ( 2  x.  0 ) )
99 2t0e0 10498 . . . . . . . . . . . . 13  |-  ( 2  x.  0 )  =  0
10098, 99syl6eq 2491 . . . . . . . . . . . 12  |-  ( ph  ->  ( 2  x.  (
( sin `  B
)  x.  ( cos `  B ) ) )  =  0 )
10193, 100eqtrd 2475 . . . . . . . . . . 11  |-  ( ph  ->  ( sin `  (
2  x.  B ) )  =  0 )
102101oveq1d 6127 . . . . . . . . . 10  |-  ( ph  ->  ( ( sin `  (
2  x.  B ) )  x.  ( cos `  A ) )  =  ( 0  x.  ( cos `  A ) ) )
10388coscld 13436 . . . . . . . . . . 11  |-  ( ph  ->  ( cos `  A
)  e.  CC )
104103mul02d 9588 . . . . . . . . . 10  |-  ( ph  ->  ( 0  x.  ( cos `  A ) )  =  0 )
105102, 104eqtrd 2475 . . . . . . . . 9  |-  ( ph  ->  ( ( sin `  (
2  x.  B ) )  x.  ( cos `  A ) )  =  0 )
10637oveq2d 6128 . . . . . . . . . 10  |-  ( ph  ->  ( ( cos `  (
2  x.  B ) )  x.  ( sin `  A ) )  =  ( ( cos `  (
2  x.  B ) )  x.  0 ) )
10787coscld 13436 . . . . . . . . . . 11  |-  ( ph  ->  ( cos `  (
2  x.  B ) )  e.  CC )
108107mul01d 9589 . . . . . . . . . 10  |-  ( ph  ->  ( ( cos `  (
2  x.  B ) )  x.  0 )  =  0 )
109106, 108eqtrd 2475 . . . . . . . . 9  |-  ( ph  ->  ( ( cos `  (
2  x.  B ) )  x.  ( sin `  A ) )  =  0 )
110105, 109oveq12d 6130 . . . . . . . 8  |-  ( ph  ->  ( ( ( sin `  ( 2  x.  B
) )  x.  ( cos `  A ) )  -  ( ( cos `  ( 2  x.  B
) )  x.  ( sin `  A ) ) )  =  ( 0  -  0 ) )
111 0m0e0 10452 . . . . . . . 8  |-  ( 0  -  0 )  =  0
112110, 111syl6eq 2491 . . . . . . 7  |-  ( ph  ->  ( ( ( sin `  ( 2  x.  B
) )  x.  ( cos `  A ) )  -  ( ( cos `  ( 2  x.  B
) )  x.  ( sin `  A ) ) )  =  0 )
11390, 112eqtrd 2475 . . . . . 6  |-  ( ph  ->  ( sin `  (
( 2  x.  B
)  -  A ) )  =  0 )
114 pilem1 21938 . . . . . 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 664 . . . . 5  |-  ( ph  ->  ( ( 2  x.  B )  -  A
)  e.  ( RR+  i^i  ( `' sin " {
0 } ) ) )
116 infmrlb 10332 . . . . 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 ) )
1175, 14, 115, 116syl3anc 1218 . . . 4  |-  ( ph  ->  sup ( ( RR+  i^i  ( `' sin " {
0 } ) ) ,  RR ,  `'  <  )  <_  ( (
2  x.  B )  -  A ) )
1181, 117syl5eqbr 4346 . . 3  |-  ( ph  ->  pi  <_  ( (
2  x.  B )  -  A ) )
1191, 44syl5eqel 2527 . . . 4  |-  ( ph  ->  pi  e.  RR )
120 leaddsub 9836 . . . 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 1218 . . 3  |-  ( ph  ->  ( ( pi  +  A )  <_  (
2  x.  B )  <-> 
pi  <_  ( ( 2  x.  B )  -  A ) ) )
122118, 121mpbird 232 . 2  |-  ( ph  ->  ( pi  +  A
)  <_  ( 2  x.  B ) )
123119, 22readdcld 9434 . . 3  |-  ( ph  ->  ( pi  +  A
)  e.  RR )
124 ledivmul 10226 . . 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 1218 . 2  |-  ( ph  ->  ( ( ( pi  +  A )  / 
2 )  <_  B  <->  ( pi  +  A )  <_  ( 2  x.  B ) ) )
126122, 125mpbird 232 1  |-  ( ph  ->  ( ( pi  +  A )  /  2
)  <_  B )
Colors of variables: wff setvar class
Syntax hints:   -. wn 3    -> wi 4    <-> wb 184    \/ wo 368    /\ wa 369    /\ w3a 965    = wceq 1369    e. wcel 1756    =/= wne 2620   A.wral 2736   E.wrex 2737    i^i cin 3348    C_ wss 3349   (/)c0 3658   {csn 3898   class class class wbr 4313   `'ccnv 4860   "cima 4864   ` cfv 5439  (class class class)co 6112   supcsup 7711   CCcc 9301   RRcr 9302   0cc0 9303    + caddc 9306    x. cmul 9308   RR*cxr 9438    < clt 9439    <_ cle 9440    - cmin 9616    / cdiv 10014   2c2 10392   4c4 10394   RR+crp 11012   (,)cioo 11321   (,]cioc 11322   sincsin 13370   cosccos 13371   picpi 13373
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1591  ax-4 1602  ax-5 1670  ax-6 1708  ax-7 1728  ax-8 1758  ax-9 1760  ax-10 1775  ax-11 1780  ax-12 1792  ax-13 1943  ax-ext 2423  ax-rep 4424  ax-sep 4434  ax-nul 4442  ax-pow 4491  ax-pr 4552  ax-un 6393  ax-inf2 7868  ax-cnex 9359  ax-resscn 9360  ax-1cn 9361  ax-icn 9362  ax-addcl 9363  ax-addrcl 9364  ax-mulcl 9365  ax-mulrcl 9366  ax-mulcom 9367  ax-addass 9368  ax-mulass 9369  ax-distr 9370  ax-i2m1 9371  ax-1ne0 9372  ax-1rid 9373  ax-rnegex 9374  ax-rrecex 9375  ax-cnre 9376  ax-pre-lttri 9377  ax-pre-lttrn 9378  ax-pre-ltadd 9379  ax-pre-mulgt0 9380  ax-pre-sup 9381  ax-addf 9382  ax-mulf 9383
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 966  df-3an 967  df-tru 1372  df-fal 1375  df-ex 1587  df-nf 1590  df-sb 1701  df-eu 2257  df-mo 2258  df-clab 2430  df-cleq 2436  df-clel 2439  df-nfc 2577  df-ne 2622  df-nel 2623  df-ral 2741  df-rex 2742  df-reu 2743  df-rmo 2744  df-rab 2745  df-v 2995  df-sbc 3208  df-csb 3310  df-dif 3352  df-un 3354  df-in 3356  df-ss 3363  df-pss 3365  df-nul 3659  df-if 3813  df-pw 3883  df-sn 3899  df-pr 3901  df-tp 3903  df-op 3905  df-uni 4113  df-int 4150  df-iun 4194  df-br 4314  df-opab 4372  df-mpt 4373  df-tr 4407  df-eprel 4653  df-id 4657  df-po 4662  df-so 4663  df-fr 4700  df-se 4701  df-we 4702  df-ord 4743  df-on 4744  df-lim 4745  df-suc 4746  df-xp 4867  df-rel 4868  df-cnv 4869  df-co 4870  df-dm 4871  df-rn 4872  df-res 4873  df-ima 4874  df-iota 5402  df-fun 5441  df-fn 5442  df-f 5443  df-f1 5444  df-fo 5445  df-f1o 5446  df-fv 5447  df-isom 5448  df-riota 6073  df-ov 6115  df-oprab 6116  df-mpt2 6117  df-om 6498  df-1st 6598  df-2nd 6599  df-recs 6853  df-rdg 6887  df-1o 6941  df-oadd 6945  df-er 7122  df-pm 7238  df-en 7332  df-dom 7333  df-sdom 7334  df-fin 7335  df-sup 7712  df-oi 7745  df-card 8130  df-pnf 9441  df-mnf 9442  df-xr 9443  df-ltxr 9444  df-le 9445  df-sub 9618  df-neg 9619  df-div 10015  df-nn 10344  df-2 10401  df-3 10402  df-4 10403  df-5 10404  df-6 10405  df-7 10406  df-8 10407  df-n0 10601  df-z 10668  df-uz 10883  df-rp 11013  df-ioo 11325  df-ioc 11326  df-ico 11327  df-fz 11459  df-fzo 11570  df-fl 11663  df-seq 11828  df-exp 11887  df-fac 12073  df-bc 12100  df-hash 12125  df-shft 12577  df-cj 12609  df-re 12610  df-im 12611  df-sqr 12745  df-abs 12746  df-limsup 12970  df-clim 12987  df-rlim 12988  df-sum 13185  df-ef 13374  df-sin 13376  df-cos 13377  df-pi 13379
This theorem is referenced by:  pilem3  21940
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