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Theorem pilem2OLD 23401
Description: Lemma for pire 23406, pigt2lt4 23404 and sinpi 23405. (Contributed by Mario Carneiro, 12-Jun-2014.) Obsolete version of pilem2 23400 as of 14-Sep-2020. (New usage is discouraged.) (Proof modification is discouraged.)
Hypotheses
Ref Expression
pilemOLD.1  |-  ( ph  ->  A  e.  ( 2 (,) 4 ) )
pilemOLD.2  |-  ( ph  ->  B  e.  RR+ )
pilemOLD.3  |-  ( ph  ->  ( sin `  A
)  =  0 )
pilemOLD.4  |-  ( ph  ->  ( sin `  B
)  =  0 )
pilemOLD.5  |-  ( ph  ->  pi  <  A )
Assertion
Ref Expression
pilem2OLD  |-  ( ph  ->  ( ( pi  +  A )  /  2
)  <_  B )

Proof of Theorem pilem2OLD
Dummy variables  x  y are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 df-piOLD 14120 . . . 4  |-  pi  =  sup ( ( RR+  i^i  ( `' sin " { 0 } ) ) ,  RR ,  `'  <  )
2 inss1 3651 . . . . . . 7  |-  ( RR+  i^i  ( `' sin " {
0 } ) ) 
C_  RR+
3 rpssre 11309 . . . . . . 7  |-  RR+  C_  RR
42, 3sstri 3440 . . . . . 6  |-  ( RR+  i^i  ( `' sin " {
0 } ) ) 
C_  RR
54a1i 11 . . . . 5  |-  ( ph  ->  ( RR+  i^i  ( `' sin " { 0 } ) )  C_  RR )
6 0re 9640 . . . . . . 7  |-  0  e.  RR
72sseli 3427 . . . . . . . . 9  |-  ( y  e.  ( RR+  i^i  ( `' sin " { 0 } ) )  -> 
y  e.  RR+ )
87rpge0d 11342 . . . . . . . 8  |-  ( y  e.  ( RR+  i^i  ( `' sin " { 0 } ) )  -> 
0  <_  y )
98rgen 2746 . . . . . . 7  |-  A. y  e.  ( RR+  i^i  ( `' sin " { 0 } ) ) 0  <_  y
10 breq1 4404 . . . . . . . . 9  |-  ( x  =  0  ->  (
x  <_  y  <->  0  <_  y ) )
1110ralbidv 2826 . . . . . . . 8  |-  ( x  =  0  ->  ( A. y  e.  ( RR+  i^i  ( `' sin " { 0 } ) ) x  <_  y  <->  A. y  e.  ( RR+  i^i  ( `' sin " {
0 } ) ) 0  <_  y )
)
1211rspcev 3149 . . . . . . 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 677 . . . . . 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 10676 . . . . . . . . 9  |-  2  e.  RR
16 pilemOLD.2 . . . . . . . . . 10  |-  ( ph  ->  B  e.  RR+ )
1716rpred 11338 . . . . . . . . 9  |-  ( ph  ->  B  e.  RR )
18 remulcl 9621 . . . . . . . . 9  |-  ( ( 2  e.  RR  /\  B  e.  RR )  ->  ( 2  x.  B
)  e.  RR )
1915, 17, 18sylancr 668 . . . . . . . 8  |-  ( ph  ->  ( 2  x.  B
)  e.  RR )
20 pilemOLD.1 . . . . . . . . 9  |-  ( ph  ->  A  e.  ( 2 (,) 4 ) )
21 elioore 11663 . . . . . . . . 9  |-  ( A  e.  ( 2 (,) 4 )  ->  A  e.  RR )
2220, 21syl 17 . . . . . . . 8  |-  ( ph  ->  A  e.  RR )
2319, 22resubcld 10044 . . . . . . 7  |-  ( ph  ->  ( ( 2  x.  B )  -  A
)  e.  RR )
24 4re 10683 . . . . . . . . . 10  |-  4  e.  RR
2524a1i 11 . . . . . . . . 9  |-  ( ph  ->  4  e.  RR )
26 eliooord 11691 . . . . . . . . . . 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 10756 . . . . . . . . . 10  |-  ( 2  x.  2 )  =  4
3015a1i 11 . . . . . . . . . . . 12  |-  ( ph  ->  2  e.  RR )
31 0red 9641 . . . . . . . . . . . . . . . . 17  |-  ( ph  ->  0  e.  RR )
32 2pos 10698 . . . . . . . . . . . . . . . . . 18  |-  0  <  2
3332a1i 11 . . . . . . . . . . . . . . . . 17  |-  ( ph  ->  0  <  2 )
3427simpld 461 . . . . . . . . . . . . . . . . 17  |-  ( ph  ->  2  <  A )
3531, 30, 22, 33, 34lttrd 9793 . . . . . . . . . . . . . . . 16  |-  ( ph  ->  0  <  A )
3622, 35elrpd 11335 . . . . . . . . . . . . . . 15  |-  ( ph  ->  A  e.  RR+ )
37 pilemOLD.3 . . . . . . . . . . . . . . 15  |-  ( ph  ->  ( sin `  A
)  =  0 )
38 pilem1 23399 . . . . . . . . . . . . . . 15  |-  ( A  e.  ( RR+  i^i  ( `' sin " { 0 } ) )  <->  ( A  e.  RR+  /\  ( sin `  A )  =  0 ) )
3936, 37, 38sylanbrc 669 . . . . . . . . . . . . . 14  |-  ( ph  ->  A  e.  ( RR+  i^i  ( `' sin " {
0 } ) ) )
40 ne0i 3736 . . . . . . . . . . . . . 14  |-  ( A  e.  ( RR+  i^i  ( `' sin " { 0 } ) )  -> 
( RR+  i^i  ( `' sin " { 0 } ) )  =/=  (/) )
4139, 40syl 17 . . . . . . . . . . . . 13  |-  ( ph  ->  ( RR+  i^i  ( `' sin " { 0 } ) )  =/=  (/) )
42 infmrclOLD 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 )  ->  sup ( ( RR+  i^i  ( `' sin " { 0 } ) ) ,  RR ,  `'  <  )  e.  RR )
434, 13, 42mp3an13 1354 . . . . . . . . . . . . 13  |-  ( (
RR+  i^i  ( `' sin " { 0 } ) )  =/=  (/)  ->  sup ( ( RR+  i^i  ( `' sin " { 0 } ) ) ,  RR ,  `'  <  )  e.  RR )
4441, 43syl 17 . . . . . . . . . . . 12  |-  ( ph  ->  sup ( ( RR+  i^i  ( `' sin " {
0 } ) ) ,  RR ,  `'  <  )  e.  RR )
45 pilem1 23399 . . . . . . . . . . . . . . 15  |-  ( x  e.  ( RR+  i^i  ( `' sin " { 0 } ) )  <->  ( x  e.  RR+  /\  ( sin `  x )  =  0 ) )
46 rpre 11305 . . . . . . . . . . . . . . . . . . . . 21  |-  ( x  e.  RR+  ->  x  e.  RR )
4746adantl 468 . . . . . . . . . . . . . . . . . . . 20  |-  ( (
ph  /\  x  e.  RR+ )  ->  x  e.  RR )
48 letric 9731 . . . . . . . . . . . . . . . . . . . 20  |-  ( ( 2  e.  RR  /\  x  e.  RR )  ->  ( 2  <_  x  \/  x  <_  2 ) )
4915, 47, 48sylancr 668 . . . . . . . . . . . . . . . . . . 19  |-  ( (
ph  /\  x  e.  RR+ )  ->  ( 2  <_  x  \/  x  <_  2 ) )
5049ord 379 . . . . . . . . . . . . . . . . . 18  |-  ( (
ph  /\  x  e.  RR+ )  ->  ( -.  2  <_  x  ->  x  <_  2 ) )
5146ad2antlr 732 . . . . . . . . . . . . . . . . . . . . . 22  |-  ( ( ( ph  /\  x  e.  RR+ )  /\  x  <_  2 )  ->  x  e.  RR )
52 rpgt0 11310 . . . . . . . . . . . . . . . . . . . . . . 23  |-  ( x  e.  RR+  ->  0  < 
x )
5352ad2antlr 732 . . . . . . . . . . . . . . . . . . . . . 22  |-  ( ( ( ph  /\  x  e.  RR+ )  /\  x  <_  2 )  ->  0  <  x )
54 simpr 463 . . . . . . . . . . . . . . . . . . . . . 22  |-  ( ( ( ph  /\  x  e.  RR+ )  /\  x  <_  2 )  ->  x  <_  2 )
55 0xr 9684 . . . . . . . . . . . . . . . . . . . . . . 23  |-  0  e.  RR*
56 elioc2 11694 . . . . . . . . . . . . . . . . . . . . . . 23  |-  ( ( 0  e.  RR*  /\  2  e.  RR )  ->  (
x  e.  ( 0 (,] 2 )  <->  ( x  e.  RR  /\  0  < 
x  /\  x  <_  2 ) ) )
5755, 15, 56mp2an 677 . . . . . . . . . . . . . . . . . . . . . 22  |-  ( x  e.  ( 0 (,] 2 )  <->  ( x  e.  RR  /\  0  < 
x  /\  x  <_  2 ) )
5851, 53, 54, 57syl3anbrc 1191 . . . . . . . . . . . . . . . . . . . . 21  |-  ( ( ( ph  /\  x  e.  RR+ )  /\  x  <_  2 )  ->  x  e.  ( 0 (,] 2
) )
59 sin02gt0 14239 . . . . . . . . . . . . . . . . . . . . 21  |-  ( x  e.  ( 0 (,] 2 )  ->  0  <  ( sin `  x
) )
6058, 59syl 17 . . . . . . . . . . . . . . . . . . . 20  |-  ( ( ( ph  /\  x  e.  RR+ )  /\  x  <_  2 )  ->  0  <  ( sin `  x
) )
6160gt0ne0d 10175 . . . . . . . . . . . . . . . . . . 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 2643 . . . . . . . . . . . . . . . 16  |-  ( (
ph  /\  x  e.  RR+ )  ->  ( ( sin `  x )  =  0  ->  2  <_  x ) )
6564expimpd 607 . . . . . . . . . . . . . . 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 2799 . . . . . . . . . . . . 13  |-  ( ph  ->  A. x  e.  (
RR+  i^i  ( `' sin " { 0 } ) ) 2  <_  x )
68 infmrgelbOLD 10592 . . . . . . . . . . . . . 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 1270 . . . . . . . . . . . . 13  |-  ( ph  ->  ( 2  <_  sup ( ( RR+  i^i  ( `' sin " { 0 } ) ) ,  RR ,  `'  <  )  <->  A. x  e.  ( RR+  i^i  ( `' sin " { 0 } ) ) 2  <_  x
) )
7067, 69mpbird 236 . . . . . . . . . . . 12  |-  ( ph  ->  2  <_  sup (
( RR+  i^i  ( `' sin " { 0 } ) ) ,  RR ,  `'  <  ) )
71 pilemOLD.4 . . . . . . . . . . . . . 14  |-  ( ph  ->  ( sin `  B
)  =  0 )
72 pilem1 23399 . . . . . . . . . . . . . 14  |-  ( B  e.  ( RR+  i^i  ( `' sin " { 0 } ) )  <->  ( B  e.  RR+  /\  ( sin `  B )  =  0 ) )
7316, 71, 72sylanbrc 669 . . . . . . . . . . . . 13  |-  ( ph  ->  B  e.  ( RR+  i^i  ( `' sin " {
0 } ) ) )
74 infmrlbOLD 10594 . . . . . . . . . . . . 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 1267 . . . . . . . . . . . 12  |-  ( ph  ->  sup ( ( RR+  i^i  ( `' sin " {
0 } ) ) ,  RR ,  `'  <  )  <_  B )
7630, 44, 17, 70, 75letrd 9789 . . . . . . . . . . 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 10455 . . . . . . . . . . . 12  |-  ( ( 2  e.  RR  /\  B  e.  RR  /\  (
2  e.  RR  /\  0  <  2 ) )  ->  ( 2  <_  B 
<->  ( 2  x.  2 )  <_  ( 2  x.  B ) ) )
8030, 17, 78, 79syl3anc 1267 . . . . . . . . . . 11  |-  ( ph  ->  ( 2  <_  B  <->  ( 2  x.  2 )  <_  ( 2  x.  B ) ) )
8176, 80mpbid 214 . . . . . . . . . 10  |-  ( ph  ->  ( 2  x.  2 )  <_  ( 2  x.  B ) )
8229, 81syl5eqbrr 4436 . . . . . . . . 9  |-  ( ph  ->  4  <_  ( 2  x.  B ) )
8322, 25, 19, 28, 82ltletrd 9792 . . . . . . . 8  |-  ( ph  ->  A  <  ( 2  x.  B ) )
8422, 19posdifd 10197 . . . . . . . 8  |-  ( ph  ->  ( A  <  (
2  x.  B )  <->  0  <  ( ( 2  x.  B )  -  A ) ) )
8583, 84mpbid 214 . . . . . . 7  |-  ( ph  ->  0  <  ( ( 2  x.  B )  -  A ) )
8623, 85elrpd 11335 . . . . . 6  |-  ( ph  ->  ( ( 2  x.  B )  -  A
)  e.  RR+ )
8719recnd 9666 . . . . . . . 8  |-  ( ph  ->  ( 2  x.  B
)  e.  CC )
8822recnd 9666 . . . . . . . 8  |-  ( ph  ->  A  e.  CC )
89 sinsub 14215 . . . . . . . 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 666 . . . . . . 7  |-  ( ph  ->  ( sin `  (
( 2  x.  B
)  -  A ) )  =  ( ( ( sin `  (
2  x.  B ) )  x.  ( cos `  A ) )  -  ( ( cos `  (
2  x.  B ) )  x.  ( sin `  A ) ) ) )
9117recnd 9666 . . . . . . . . . . . . 13  |-  ( ph  ->  B  e.  CC )
92 sin2t 14224 . . . . . . . . . . . . 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 6303 . . . . . . . . . . . . . . 15  |-  ( ph  ->  ( ( sin `  B
)  x.  ( cos `  B ) )  =  ( 0  x.  ( cos `  B ) ) )
9591coscld 14178 . . . . . . . . . . . . . . . 16  |-  ( ph  ->  ( cos `  B
)  e.  CC )
9695mul02d 9828 . . . . . . . . . . . . . . 15  |-  ( ph  ->  ( 0  x.  ( cos `  B ) )  =  0 )
9794, 96eqtrd 2484 . . . . . . . . . . . . . 14  |-  ( ph  ->  ( ( sin `  B
)  x.  ( cos `  B ) )  =  0 )
9897oveq2d 6304 . . . . . . . . . . . . 13  |-  ( ph  ->  ( 2  x.  (
( sin `  B
)  x.  ( cos `  B ) ) )  =  ( 2  x.  0 ) )
99 2t0e0 10762 . . . . . . . . . . . . 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 6303 . . . . . . . . . 10  |-  ( ph  ->  ( ( sin `  (
2  x.  B ) )  x.  ( cos `  A ) )  =  ( 0  x.  ( cos `  A ) ) )
10388coscld 14178 . . . . . . . . . . 11  |-  ( ph  ->  ( cos `  A
)  e.  CC )
104103mul02d 9828 . . . . . . . . . 10  |-  ( ph  ->  ( 0  x.  ( cos `  A ) )  =  0 )
105102, 104eqtrd 2484 . . . . . . . . 9  |-  ( ph  ->  ( ( sin `  (
2  x.  B ) )  x.  ( cos `  A ) )  =  0 )
10637oveq2d 6304 . . . . . . . . . 10  |-  ( ph  ->  ( ( cos `  (
2  x.  B ) )  x.  ( sin `  A ) )  =  ( ( cos `  (
2  x.  B ) )  x.  0 ) )
10787coscld 14178 . . . . . . . . . . 11  |-  ( ph  ->  ( cos `  (
2  x.  B ) )  e.  CC )
108107mul01d 9829 . . . . . . . . . 10  |-  ( ph  ->  ( ( cos `  (
2  x.  B ) )  x.  0 )  =  0 )
109106, 108eqtrd 2484 . . . . . . . . 9  |-  ( ph  ->  ( ( cos `  (
2  x.  B ) )  x.  ( sin `  A ) )  =  0 )
110105, 109oveq12d 6306 . . . . . . . 8  |-  ( ph  ->  ( ( ( sin `  ( 2  x.  B
) )  x.  ( cos `  A ) )  -  ( ( cos `  ( 2  x.  B
) )  x.  ( sin `  A ) ) )  =  ( 0  -  0 ) )
111 0m0e0 10716 . . . . . . . 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 23399 . . . . . 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 669 . . . . 5  |-  ( ph  ->  ( ( 2  x.  B )  -  A
)  e.  ( RR+  i^i  ( `' sin " {
0 } ) ) )
116 infmrlbOLD 10594 . . . . 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 1267 . . . 4  |-  ( ph  ->  sup ( ( RR+  i^i  ( `' sin " {
0 } ) ) ,  RR ,  `'  <  )  <_  ( (
2  x.  B )  -  A ) )
1181, 117syl5eqbr 4435 . . 3  |-  ( ph  ->  pi  <_  ( (
2  x.  B )  -  A ) )
1191, 44syl5eqel 2532 . . . 4  |-  ( ph  ->  pi  e.  RR )
120 leaddsub 10087 . . . 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 1267 . . 3  |-  ( ph  ->  ( ( pi  +  A )  <_  (
2  x.  B )  <-> 
pi  <_  ( ( 2  x.  B )  -  A ) ) )
122118, 121mpbird 236 . 2  |-  ( ph  ->  ( pi  +  A
)  <_  ( 2  x.  B ) )
123119, 22readdcld 9667 . . 3  |-  ( ph  ->  ( pi  +  A
)  e.  RR )
124 ledivmul 10478 . . 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 1267 . 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 984    = wceq 1443    e. wcel 1886    =/= wne 2621   A.wral 2736   E.wrex 2737    i^i cin 3402    C_ wss 3403   (/)c0 3730   {csn 3967   class class class wbr 4401   `'ccnv 4832   "cima 4836   ` cfv 5581  (class class class)co 6288   supcsup 7951   CCcc 9534   RRcr 9535   0cc0 9536    + caddc 9539    x. cmul 9541   RR*cxr 9671    < clt 9672    <_ cle 9673    - cmin 9857    / cdiv 10266   2c2 10656   4c4 10658   RR+crp 11299   (,)cioo 11632   (,]cioc 11633   sincsin 14109   cosccos 14110   picpiold 14113
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1668  ax-4 1681  ax-5 1757  ax-6 1804  ax-7 1850  ax-8 1888  ax-9 1895  ax-10 1914  ax-11 1919  ax-12 1932  ax-13 2090  ax-ext 2430  ax-rep 4514  ax-sep 4524  ax-nul 4533  ax-pow 4580  ax-pr 4638  ax-un 6580  ax-inf2 8143  ax-cnex 9592  ax-resscn 9593  ax-1cn 9594  ax-icn 9595  ax-addcl 9596  ax-addrcl 9597  ax-mulcl 9598  ax-mulrcl 9599  ax-mulcom 9600  ax-addass 9601  ax-mulass 9602  ax-distr 9603  ax-i2m1 9604  ax-1ne0 9605  ax-1rid 9606  ax-rnegex 9607  ax-rrecex 9608  ax-cnre 9609  ax-pre-lttri 9610  ax-pre-lttrn 9611  ax-pre-ltadd 9612  ax-pre-mulgt0 9613  ax-pre-sup 9614  ax-addf 9615  ax-mulf 9616
This theorem depends on definitions:  df-bi 189  df-or 372  df-an 373  df-3or 985  df-3an 986  df-tru 1446  df-fal 1449  df-ex 1663  df-nf 1667  df-sb 1797  df-eu 2302  df-mo 2303  df-clab 2437  df-cleq 2443  df-clel 2446  df-nfc 2580  df-ne 2623  df-nel 2624  df-ral 2741  df-rex 2742  df-reu 2743  df-rmo 2744  df-rab 2745  df-v 3046  df-sbc 3267  df-csb 3363  df-dif 3406  df-un 3408  df-in 3410  df-ss 3417  df-pss 3419  df-nul 3731  df-if 3881  df-pw 3952  df-sn 3968  df-pr 3970  df-tp 3972  df-op 3974  df-uni 4198  df-int 4234  df-iun 4279  df-br 4402  df-opab 4461  df-mpt 4462  df-tr 4497  df-eprel 4744  df-id 4748  df-po 4754  df-so 4755  df-fr 4792  df-se 4793  df-we 4794  df-xp 4839  df-rel 4840  df-cnv 4841  df-co 4842  df-dm 4843  df-rn 4844  df-res 4845  df-ima 4846  df-pred 5379  df-ord 5425  df-on 5426  df-lim 5427  df-suc 5428  df-iota 5545  df-fun 5583  df-fn 5584  df-f 5585  df-f1 5586  df-fo 5587  df-f1o 5588  df-fv 5589  df-isom 5590  df-riota 6250  df-ov 6291  df-oprab 6292  df-mpt2 6293  df-om 6690  df-1st 6790  df-2nd 6791  df-wrecs 7025  df-recs 7087  df-rdg 7125  df-1o 7179  df-oadd 7183  df-er 7360  df-pm 7472  df-en 7567  df-dom 7568  df-sdom 7569  df-fin 7570  df-sup 7953  df-inf 7954  df-oi 8022  df-card 8370  df-pnf 9674  df-mnf 9675  df-xr 9676  df-ltxr 9677  df-le 9678  df-sub 9859  df-neg 9860  df-div 10267  df-nn 10607  df-2 10665  df-3 10666  df-4 10667  df-5 10668  df-6 10669  df-7 10670  df-8 10671  df-n0 10867  df-z 10935  df-uz 11157  df-rp 11300  df-ioo 11636  df-ioc 11637  df-ico 11638  df-fz 11782  df-fzo 11913  df-fl 12025  df-seq 12211  df-exp 12270  df-fac 12457  df-bc 12485  df-hash 12513  df-shft 13123  df-cj 13155  df-re 13156  df-im 13157  df-sqrt 13291  df-abs 13292  df-limsup 13519  df-clim 13545  df-rlim 13546  df-sum 13746  df-ef 14114  df-sin 14116  df-cos 14117  df-piOLD 14120
This theorem is referenced by:  pilem3OLD  23403
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