MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  pilem2 Structured version   Unicode version

Theorem pilem2 22719
Description: Lemma for pire 22723, pigt2lt4 22721 and sinpi 22722. (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 13686 . . . 4  |-  pi  =  sup ( ( RR+  i^i  ( `' sin " { 0 } ) ) ,  RR ,  `'  <  )
2 inss1 3703 . . . . . . 7  |-  ( RR+  i^i  ( `' sin " {
0 } ) ) 
C_  RR+
3 rpssre 11239 . . . . . . 7  |-  RR+  C_  RR
42, 3sstri 3498 . . . . . 6  |-  ( RR+  i^i  ( `' sin " {
0 } ) ) 
C_  RR
54a1i 11 . . . . 5  |-  ( ph  ->  ( RR+  i^i  ( `' sin " { 0 } ) )  C_  RR )
6 0re 9599 . . . . . . 7  |-  0  e.  RR
72sseli 3485 . . . . . . . . 9  |-  ( y  e.  ( RR+  i^i  ( `' sin " { 0 } ) )  -> 
y  e.  RR+ )
87rpge0d 11269 . . . . . . . 8  |-  ( y  e.  ( RR+  i^i  ( `' sin " { 0 } ) )  -> 
0  <_  y )
98rgen 2803 . . . . . . 7  |-  A. y  e.  ( RR+  i^i  ( `' sin " { 0 } ) ) 0  <_  y
10 breq1 4440 . . . . . . . . 9  |-  ( x  =  0  ->  (
x  <_  y  <->  0  <_  y ) )
1110ralbidv 2882 . . . . . . . 8  |-  ( x  =  0  ->  ( A. y  e.  ( RR+  i^i  ( `' sin " { 0 } ) ) x  <_  y  <->  A. y  e.  ( RR+  i^i  ( `' sin " {
0 } ) ) 0  <_  y )
)
1211rspcev 3196 . . . . . . 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 10611 . . . . . . . . 9  |-  2  e.  RR
16 pilem.2 . . . . . . . . . 10  |-  ( ph  ->  B  e.  RR+ )
1716rpred 11265 . . . . . . . . 9  |-  ( ph  ->  B  e.  RR )
18 remulcl 9580 . . . . . . . . 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 11568 . . . . . . . . 9  |-  ( A  e.  ( 2 (,) 4 )  ->  A  e.  RR )
2220, 21syl 16 . . . . . . . 8  |-  ( ph  ->  A  e.  RR )
2319, 22resubcld 9993 . . . . . . 7  |-  ( ph  ->  ( ( 2  x.  B )  -  A
)  e.  RR )
24 4re 10618 . . . . . . . . . 10  |-  4  e.  RR
2524a1i 11 . . . . . . . . 9  |-  ( ph  ->  4  e.  RR )
26 eliooord 11593 . . . . . . . . . . 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 10691 . . . . . . . . . 10  |-  ( 2  x.  2 )  =  4
3015a1i 11 . . . . . . . . . . . 12  |-  ( ph  ->  2  e.  RR )
31 0red 9600 . . . . . . . . . . . . . . . . 17  |-  ( ph  ->  0  e.  RR )
32 2pos 10633 . . . . . . . . . . . . . . . . . 18  |-  0  <  2
3332a1i 11 . . . . . . . . . . . . . . . . 17  |-  ( ph  ->  0  <  2 )
3427simpld 459 . . . . . . . . . . . . . . . . 17  |-  ( ph  ->  2  <  A )
3531, 30, 22, 33, 34lttrd 9746 . . . . . . . . . . . . . . . 16  |-  ( ph  ->  0  <  A )
3622, 35elrpd 11263 . . . . . . . . . . . . . . 15  |-  ( ph  ->  A  e.  RR+ )
37 pilem.3 . . . . . . . . . . . . . . 15  |-  ( ph  ->  ( sin `  A
)  =  0 )
38 pilem1 22718 . . . . . . . . . . . . . . 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 3776 . . . . . . . . . . . . . 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 10528 . . . . . . . . . . . . . 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 1316 . . . . . . . . . . . . 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 22718 . . . . . . . . . . . . . . 15  |-  ( x  e.  ( RR+  i^i  ( `' sin " { 0 } ) )  <->  ( x  e.  RR+  /\  ( sin `  x )  =  0 ) )
46 rpre 11235 . . . . . . . . . . . . . . . . . . . . 21  |-  ( x  e.  RR+  ->  x  e.  RR )
4746adantl 466 . . . . . . . . . . . . . . . . . . . 20  |-  ( (
ph  /\  x  e.  RR+ )  ->  x  e.  RR )
48 letric 9688 . . . . . . . . . . . . . . . . . . . 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 11240 . . . . . . . . . . . . . . . . . . . . . . 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 9643 . . . . . . . . . . . . . . . . . . . . . . 23  |-  0  e.  RR*
56 elioc2 11596 . . . . . . . . . . . . . . . . . . . . . . 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 1181 . . . . . . . . . . . . . . . . . . . . 21  |-  ( ( ( ph  /\  x  e.  RR+ )  /\  x  <_  2 )  ->  x  e.  ( 0 (,] 2
) )
59 sin02gt0 13804 . . . . . . . . . . . . . . . . . . . . 21  |-  ( x  e.  ( 0 (,] 2 )  ->  0  <  ( sin `  x
) )
6058, 59syl 16 . . . . . . . . . . . . . . . . . . . 20  |-  ( ( ( ph  /\  x  e.  RR+ )  /\  x  <_  2 )  ->  0  <  ( sin `  x
) )
6160gt0ne0d 10123 . . . . . . . . . . . . . . . . . . 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 2665 . . . . . . . . . . . . . . . 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 2855 . . . . . . . . . . . . 13  |-  ( ph  ->  A. x  e.  (
RR+  i^i  ( `' sin " { 0 } ) ) 2  <_  x )
68 infmrgelb 10529 . . . . . . . . . . . . . 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 1232 . . . . . . . . . . . . 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 22718 . . . . . . . . . . . . . 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 10530 . . . . . . . . . . . . 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 1229 . . . . . . . . . . . 12  |-  ( ph  ->  sup ( ( RR+  i^i  ( `' sin " {
0 } ) ) ,  RR ,  `'  <  )  <_  B )
7630, 44, 17, 70, 75letrd 9742 . . . . . . . . . . 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 10401 . . . . . . . . . . . 12  |-  ( ( 2  e.  RR  /\  B  e.  RR  /\  (
2  e.  RR  /\  0  <  2 ) )  ->  ( 2  <_  B 
<->  ( 2  x.  2 )  <_  ( 2  x.  B ) ) )
8030, 17, 78, 79syl3anc 1229 . . . . . . . . . . 11  |-  ( ph  ->  ( 2  <_  B  <->  ( 2  x.  2 )  <_  ( 2  x.  B ) ) )
8176, 80mpbid 210 . . . . . . . . . 10  |-  ( ph  ->  ( 2  x.  2 )  <_  ( 2  x.  B ) )
8229, 81syl5eqbrr 4471 . . . . . . . . 9  |-  ( ph  ->  4  <_  ( 2  x.  B ) )
8322, 25, 19, 28, 82ltletrd 9745 . . . . . . . 8  |-  ( ph  ->  A  <  ( 2  x.  B ) )
8422, 19posdifd 10145 . . . . . . . 8  |-  ( ph  ->  ( A  <  (
2  x.  B )  <->  0  <  ( ( 2  x.  B )  -  A ) ) )
8583, 84mpbid 210 . . . . . . 7  |-  ( ph  ->  0  <  ( ( 2  x.  B )  -  A ) )
8623, 85elrpd 11263 . . . . . 6  |-  ( ph  ->  ( ( 2  x.  B )  -  A
)  e.  RR+ )
8719recnd 9625 . . . . . . . 8  |-  ( ph  ->  ( 2  x.  B
)  e.  CC )
8822recnd 9625 . . . . . . . 8  |-  ( ph  ->  A  e.  CC )
89 sinsub 13780 . . . . . . . 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 9625 . . . . . . . . . . . . 13  |-  ( ph  ->  B  e.  CC )
92 sin2t 13789 . . . . . . . . . . . . 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 6296 . . . . . . . . . . . . . . 15  |-  ( ph  ->  ( ( sin `  B
)  x.  ( cos `  B ) )  =  ( 0  x.  ( cos `  B ) ) )
9591coscld 13743 . . . . . . . . . . . . . . . 16  |-  ( ph  ->  ( cos `  B
)  e.  CC )
9695mul02d 9781 . . . . . . . . . . . . . . 15  |-  ( ph  ->  ( 0  x.  ( cos `  B ) )  =  0 )
9794, 96eqtrd 2484 . . . . . . . . . . . . . 14  |-  ( ph  ->  ( ( sin `  B
)  x.  ( cos `  B ) )  =  0 )
9897oveq2d 6297 . . . . . . . . . . . . 13  |-  ( ph  ->  ( 2  x.  (
( sin `  B
)  x.  ( cos `  B ) ) )  =  ( 2  x.  0 ) )
99 2t0e0 10697 . . . . . . . . . . . . 13  |-  ( 2  x.  0 )  =  0
10098, 99syl6eq 2500 . . . . . . . . . . . 12  |-  ( ph  ->  ( 2  x.  (
( sin `  B
)  x.  ( cos `  B ) ) )  =  0 )
10193, 100eqtrd 2484 . . . . . . . . . . 11  |-  ( ph  ->  ( sin `  (
2  x.  B ) )  =  0 )
102101oveq1d 6296 . . . . . . . . . 10  |-  ( ph  ->  ( ( sin `  (
2  x.  B ) )  x.  ( cos `  A ) )  =  ( 0  x.  ( cos `  A ) ) )
10388coscld 13743 . . . . . . . . . . 11  |-  ( ph  ->  ( cos `  A
)  e.  CC )
104103mul02d 9781 . . . . . . . . . 10  |-  ( ph  ->  ( 0  x.  ( cos `  A ) )  =  0 )
105102, 104eqtrd 2484 . . . . . . . . 9  |-  ( ph  ->  ( ( sin `  (
2  x.  B ) )  x.  ( cos `  A ) )  =  0 )
10637oveq2d 6297 . . . . . . . . . 10  |-  ( ph  ->  ( ( cos `  (
2  x.  B ) )  x.  ( sin `  A ) )  =  ( ( cos `  (
2  x.  B ) )  x.  0 ) )
10787coscld 13743 . . . . . . . . . . 11  |-  ( ph  ->  ( cos `  (
2  x.  B ) )  e.  CC )
108107mul01d 9782 . . . . . . . . . 10  |-  ( ph  ->  ( ( cos `  (
2  x.  B ) )  x.  0 )  =  0 )
109106, 108eqtrd 2484 . . . . . . . . 9  |-  ( ph  ->  ( ( cos `  (
2  x.  B ) )  x.  ( sin `  A ) )  =  0 )
110105, 109oveq12d 6299 . . . . . . . 8  |-  ( ph  ->  ( ( ( sin `  ( 2  x.  B
) )  x.  ( cos `  A ) )  -  ( ( cos `  ( 2  x.  B
) )  x.  ( sin `  A ) ) )  =  ( 0  -  0 ) )
111 0m0e0 10651 . . . . . . . 8  |-  ( 0  -  0 )  =  0
112110, 111syl6eq 2500 . . . . . . 7  |-  ( ph  ->  ( ( ( sin `  ( 2  x.  B
) )  x.  ( cos `  A ) )  -  ( ( cos `  ( 2  x.  B
) )  x.  ( sin `  A ) ) )  =  0 )
11390, 112eqtrd 2484 . . . . . 6  |-  ( ph  ->  ( sin `  (
( 2  x.  B
)  -  A ) )  =  0 )
114 pilem1 22718 . . . . . 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 10530 . . . . 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 1229 . . . 4  |-  ( ph  ->  sup ( ( RR+  i^i  ( `' sin " {
0 } ) ) ,  RR ,  `'  <  )  <_  ( (
2  x.  B )  -  A ) )
1181, 117syl5eqbr 4470 . . 3  |-  ( ph  ->  pi  <_  ( (
2  x.  B )  -  A ) )
1191, 44syl5eqel 2535 . . . 4  |-  ( ph  ->  pi  e.  RR )
120 leaddsub 10034 . . . 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 1229 . . 3  |-  ( ph  ->  ( ( pi  +  A )  <_  (
2  x.  B )  <-> 
pi  <_  ( ( 2  x.  B )  -  A ) ) )
122118, 121mpbird 232 . 2  |-  ( ph  ->  ( pi  +  A
)  <_  ( 2  x.  B ) )
123119, 22readdcld 9626 . . 3  |-  ( ph  ->  ( pi  +  A
)  e.  RR )
124 ledivmul 10424 . . 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 1229 . 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 974    = wceq 1383    e. wcel 1804    =/= wne 2638   A.wral 2793   E.wrex 2794    i^i cin 3460    C_ wss 3461   (/)c0 3770   {csn 4014   class class class wbr 4437   `'ccnv 4988   "cima 4992   ` cfv 5578  (class class class)co 6281   supcsup 7902   CCcc 9493   RRcr 9494   0cc0 9495    + caddc 9498    x. cmul 9500   RR*cxr 9630    < clt 9631    <_ cle 9632    - cmin 9810    / cdiv 10212   2c2 10591   4c4 10593   RR+crp 11229   (,)cioo 11538   (,]cioc 11539   sincsin 13677   cosccos 13678   picpi 13680
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1605  ax-4 1618  ax-5 1691  ax-6 1734  ax-7 1776  ax-8 1806  ax-9 1808  ax-10 1823  ax-11 1828  ax-12 1840  ax-13 1985  ax-ext 2421  ax-rep 4548  ax-sep 4558  ax-nul 4566  ax-pow 4615  ax-pr 4676  ax-un 6577  ax-inf2 8061  ax-cnex 9551  ax-resscn 9552  ax-1cn 9553  ax-icn 9554  ax-addcl 9555  ax-addrcl 9556  ax-mulcl 9557  ax-mulrcl 9558  ax-mulcom 9559  ax-addass 9560  ax-mulass 9561  ax-distr 9562  ax-i2m1 9563  ax-1ne0 9564  ax-1rid 9565  ax-rnegex 9566  ax-rrecex 9567  ax-cnre 9568  ax-pre-lttri 9569  ax-pre-lttrn 9570  ax-pre-ltadd 9571  ax-pre-mulgt0 9572  ax-pre-sup 9573  ax-addf 9574  ax-mulf 9575
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 975  df-3an 976  df-tru 1386  df-fal 1389  df-ex 1600  df-nf 1604  df-sb 1727  df-eu 2272  df-mo 2273  df-clab 2429  df-cleq 2435  df-clel 2438  df-nfc 2593  df-ne 2640  df-nel 2641  df-ral 2798  df-rex 2799  df-reu 2800  df-rmo 2801  df-rab 2802  df-v 3097  df-sbc 3314  df-csb 3421  df-dif 3464  df-un 3466  df-in 3468  df-ss 3475  df-pss 3477  df-nul 3771  df-if 3927  df-pw 3999  df-sn 4015  df-pr 4017  df-tp 4019  df-op 4021  df-uni 4235  df-int 4272  df-iun 4317  df-br 4438  df-opab 4496  df-mpt 4497  df-tr 4531  df-eprel 4781  df-id 4785  df-po 4790  df-so 4791  df-fr 4828  df-se 4829  df-we 4830  df-ord 4871  df-on 4872  df-lim 4873  df-suc 4874  df-xp 4995  df-rel 4996  df-cnv 4997  df-co 4998  df-dm 4999  df-rn 5000  df-res 5001  df-ima 5002  df-iota 5541  df-fun 5580  df-fn 5581  df-f 5582  df-f1 5583  df-fo 5584  df-f1o 5585  df-fv 5586  df-isom 5587  df-riota 6242  df-ov 6284  df-oprab 6285  df-mpt2 6286  df-om 6686  df-1st 6785  df-2nd 6786  df-recs 7044  df-rdg 7078  df-1o 7132  df-oadd 7136  df-er 7313  df-pm 7425  df-en 7519  df-dom 7520  df-sdom 7521  df-fin 7522  df-sup 7903  df-oi 7938  df-card 8323  df-pnf 9633  df-mnf 9634  df-xr 9635  df-ltxr 9636  df-le 9637  df-sub 9812  df-neg 9813  df-div 10213  df-nn 10543  df-2 10600  df-3 10601  df-4 10602  df-5 10603  df-6 10604  df-7 10605  df-8 10606  df-n0 10802  df-z 10871  df-uz 11091  df-rp 11230  df-ioo 11542  df-ioc 11543  df-ico 11544  df-fz 11682  df-fzo 11804  df-fl 11908  df-seq 12087  df-exp 12146  df-fac 12333  df-bc 12360  df-hash 12385  df-shft 12879  df-cj 12911  df-re 12912  df-im 12913  df-sqrt 13047  df-abs 13048  df-limsup 13273  df-clim 13290  df-rlim 13291  df-sum 13488  df-ef 13681  df-sin 13683  df-cos 13684  df-pi 13686
This theorem is referenced by:  pilem3  22720
  Copyright terms: Public domain W3C validator