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Theorem sincosq4sgn 22079
Description: The signs of the sine and cosine functions in the fourth quadrant. (Contributed by Paul Chapman, 24-Jan-2008.)
Assertion
Ref Expression
sincosq4sgn  |-  ( A  e.  ( ( 3  x.  ( pi  / 
2 ) ) (,) ( 2  x.  pi ) )  ->  (
( sin `  A
)  <  0  /\  0  <  ( cos `  A
) ) )

Proof of Theorem sincosq4sgn
StepHypRef Expression
1 3re 10496 . . . . 5  |-  3  e.  RR
2 halfpire 22042 . . . . 5  |-  ( pi 
/  2 )  e.  RR
31, 2remulcli 9501 . . . 4  |-  ( 3  x.  ( pi  / 
2 ) )  e.  RR
43rexri 9537 . . 3  |-  ( 3  x.  ( pi  / 
2 ) )  e. 
RR*
5 2re 10492 . . . . 5  |-  2  e.  RR
6 pire 22037 . . . . 5  |-  pi  e.  RR
75, 6remulcli 9501 . . . 4  |-  ( 2  x.  pi )  e.  RR
87rexri 9537 . . 3  |-  ( 2  x.  pi )  e. 
RR*
9 elioo2 11442 . . 3  |-  ( ( ( 3  x.  (
pi  /  2 ) )  e.  RR*  /\  (
2  x.  pi )  e.  RR* )  ->  ( A  e.  ( (
3  x.  ( pi 
/  2 ) ) (,) ( 2  x.  pi ) )  <->  ( A  e.  RR  /\  ( 3  x.  ( pi  / 
2 ) )  < 
A  /\  A  <  ( 2  x.  pi ) ) ) )
104, 8, 9mp2an 672 . 2  |-  ( A  e.  ( ( 3  x.  ( pi  / 
2 ) ) (,) ( 2  x.  pi ) )  <->  ( A  e.  RR  /\  ( 3  x.  ( pi  / 
2 ) )  < 
A  /\  A  <  ( 2  x.  pi ) ) )
11 df-3 10482 . . . . . . . . . . . 12  |-  3  =  ( 2  +  1 )
1211oveq1i 6200 . . . . . . . . . . 11  |-  ( 3  x.  ( pi  / 
2 ) )  =  ( ( 2  +  1 )  x.  (
pi  /  2 ) )
13 2cn 10493 . . . . . . . . . . . 12  |-  2  e.  CC
14 ax-1cn 9441 . . . . . . . . . . . 12  |-  1  e.  CC
152recni 9499 . . . . . . . . . . . 12  |-  ( pi 
/  2 )  e.  CC
1613, 14, 15adddiri 9498 . . . . . . . . . . 11  |-  ( ( 2  +  1 )  x.  ( pi  / 
2 ) )  =  ( ( 2  x.  ( pi  /  2
) )  +  ( 1  x.  ( pi 
/  2 ) ) )
176recni 9499 . . . . . . . . . . . . 13  |-  pi  e.  CC
18 2ne0 10515 . . . . . . . . . . . . 13  |-  2  =/=  0
1917, 13, 18divcan2i 10175 . . . . . . . . . . . 12  |-  ( 2  x.  ( pi  / 
2 ) )  =  pi
2015mulid2i 9490 . . . . . . . . . . . 12  |-  ( 1  x.  ( pi  / 
2 ) )  =  ( pi  /  2
)
2119, 20oveq12i 6202 . . . . . . . . . . 11  |-  ( ( 2  x.  ( pi 
/  2 ) )  +  ( 1  x.  ( pi  /  2
) ) )  =  ( pi  +  ( pi  /  2 ) )
2212, 16, 213eqtrri 2485 . . . . . . . . . 10  |-  ( pi  +  ( pi  / 
2 ) )  =  ( 3  x.  (
pi  /  2 ) )
2322breq1i 4397 . . . . . . . . 9  |-  ( ( pi  +  ( pi 
/  2 ) )  <  A  <->  ( 3  x.  ( pi  / 
2 ) )  < 
A )
24 ltaddsub 9914 . . . . . . . . . 10  |-  ( ( pi  e.  RR  /\  ( pi  /  2
)  e.  RR  /\  A  e.  RR )  ->  ( ( pi  +  ( pi  /  2
) )  <  A  <->  pi 
<  ( A  -  ( pi  /  2
) ) ) )
256, 2, 24mp3an12 1305 . . . . . . . . 9  |-  ( A  e.  RR  ->  (
( pi  +  ( pi  /  2 ) )  <  A  <->  pi  <  ( A  -  ( pi 
/  2 ) ) ) )
2623, 25syl5bbr 259 . . . . . . . 8  |-  ( A  e.  RR  ->  (
( 3  x.  (
pi  /  2 ) )  <  A  <->  pi  <  ( A  -  ( pi 
/  2 ) ) ) )
27 ltsubadd 9910 . . . . . . . . . 10  |-  ( ( A  e.  RR  /\  ( pi  /  2
)  e.  RR  /\  ( 3  x.  (
pi  /  2 ) )  e.  RR )  ->  ( ( A  -  ( pi  / 
2 ) )  < 
( 3  x.  (
pi  /  2 ) )  <->  A  <  ( ( 3  x.  ( pi 
/  2 ) )  +  ( pi  / 
2 ) ) ) )
282, 3, 27mp3an23 1307 . . . . . . . . 9  |-  ( A  e.  RR  ->  (
( A  -  (
pi  /  2 ) )  <  ( 3  x.  ( pi  / 
2 ) )  <->  A  <  ( ( 3  x.  (
pi  /  2 ) )  +  ( pi 
/  2 ) ) ) )
29 df-4 10483 . . . . . . . . . . . . 13  |-  4  =  ( 3  +  1 )
3029oveq1i 6200 . . . . . . . . . . . 12  |-  ( 4  x.  ( pi  / 
2 ) )  =  ( ( 3  +  1 )  x.  (
pi  /  2 ) )
311recni 9499 . . . . . . . . . . . . 13  |-  3  e.  CC
3231, 14, 15adddiri 9498 . . . . . . . . . . . 12  |-  ( ( 3  +  1 )  x.  ( pi  / 
2 ) )  =  ( ( 3  x.  ( pi  /  2
) )  +  ( 1  x.  ( pi 
/  2 ) ) )
3320oveq2i 6201 . . . . . . . . . . . 12  |-  ( ( 3  x.  ( pi 
/  2 ) )  +  ( 1  x.  ( pi  /  2
) ) )  =  ( ( 3  x.  ( pi  /  2
) )  +  ( pi  /  2 ) )
3430, 32, 333eqtrri 2485 . . . . . . . . . . 11  |-  ( ( 3  x.  ( pi 
/  2 ) )  +  ( pi  / 
2 ) )  =  ( 4  x.  (
pi  /  2 ) )
35 4cn 10500 . . . . . . . . . . . . 13  |-  4  e.  CC
36 2cnne0 10637 . . . . . . . . . . . . 13  |-  ( 2  e.  CC  /\  2  =/=  0 )
37 div12 10117 . . . . . . . . . . . . 13  |-  ( ( 4  e.  CC  /\  pi  e.  CC  /\  (
2  e.  CC  /\  2  =/=  0 ) )  ->  ( 4  x.  ( pi  /  2
) )  =  ( pi  x.  ( 4  /  2 ) ) )
3835, 17, 36, 37mp3an 1315 . . . . . . . . . . . 12  |-  ( 4  x.  ( pi  / 
2 ) )  =  ( pi  x.  (
4  /  2 ) )
39 4d2e2 10579 . . . . . . . . . . . . . 14  |-  ( 4  /  2 )  =  2
4039oveq2i 6201 . . . . . . . . . . . . 13  |-  ( pi  x.  ( 4  / 
2 ) )  =  ( pi  x.  2 )
4117, 13mulcomi 9493 . . . . . . . . . . . . 13  |-  ( pi  x.  2 )  =  ( 2  x.  pi )
4240, 41eqtri 2480 . . . . . . . . . . . 12  |-  ( pi  x.  ( 4  / 
2 ) )  =  ( 2  x.  pi )
4338, 42eqtri 2480 . . . . . . . . . . 11  |-  ( 4  x.  ( pi  / 
2 ) )  =  ( 2  x.  pi )
4434, 43eqtri 2480 . . . . . . . . . 10  |-  ( ( 3  x.  ( pi 
/  2 ) )  +  ( pi  / 
2 ) )  =  ( 2  x.  pi )
4544breq2i 4398 . . . . . . . . 9  |-  ( A  <  ( ( 3  x.  ( pi  / 
2 ) )  +  ( pi  /  2
) )  <->  A  <  ( 2  x.  pi ) )
4628, 45syl6rbb 262 . . . . . . . 8  |-  ( A  e.  RR  ->  ( A  <  ( 2  x.  pi )  <->  ( A  -  ( pi  / 
2 ) )  < 
( 3  x.  (
pi  /  2 ) ) ) )
4726, 46anbi12d 710 . . . . . . 7  |-  ( A  e.  RR  ->  (
( ( 3  x.  ( pi  /  2
) )  <  A  /\  A  <  ( 2  x.  pi ) )  <-> 
( pi  <  ( A  -  ( pi  /  2 ) )  /\  ( A  -  (
pi  /  2 ) )  <  ( 3  x.  ( pi  / 
2 ) ) ) ) )
48 resubcl 9774 . . . . . . . . . 10  |-  ( ( A  e.  RR  /\  ( pi  /  2
)  e.  RR )  ->  ( A  -  ( pi  /  2
) )  e.  RR )
492, 48mpan2 671 . . . . . . . . 9  |-  ( A  e.  RR  ->  ( A  -  ( pi  /  2 ) )  e.  RR )
506rexri 9537 . . . . . . . . . . 11  |-  pi  e.  RR*
51 elioo2 11442 . . . . . . . . . . 11  |-  ( ( pi  e.  RR*  /\  (
3  x.  ( pi 
/  2 ) )  e.  RR* )  ->  (
( A  -  (
pi  /  2 ) )  e.  ( pi
(,) ( 3  x.  ( pi  /  2
) ) )  <->  ( ( A  -  ( pi  /  2 ) )  e.  RR  /\  pi  <  ( A  -  ( pi 
/  2 ) )  /\  ( A  -  ( pi  /  2
) )  <  (
3  x.  ( pi 
/  2 ) ) ) ) )
5250, 4, 51mp2an 672 . . . . . . . . . 10  |-  ( ( A  -  ( pi 
/  2 ) )  e.  ( pi (,) ( 3  x.  (
pi  /  2 ) ) )  <->  ( ( A  -  ( pi  /  2 ) )  e.  RR  /\  pi  <  ( A  -  ( pi 
/  2 ) )  /\  ( A  -  ( pi  /  2
) )  <  (
3  x.  ( pi 
/  2 ) ) ) )
53 sincosq3sgn 22078 . . . . . . . . . 10  |-  ( ( A  -  ( pi 
/  2 ) )  e.  ( pi (,) ( 3  x.  (
pi  /  2 ) ) )  ->  (
( sin `  ( A  -  ( pi  /  2 ) ) )  <  0  /\  ( cos `  ( A  -  ( pi  /  2
) ) )  <  0 ) )
5452, 53sylbir 213 . . . . . . . . 9  |-  ( ( ( A  -  (
pi  /  2 ) )  e.  RR  /\  pi  <  ( A  -  ( pi  /  2
) )  /\  ( A  -  ( pi  /  2 ) )  < 
( 3  x.  (
pi  /  2 ) ) )  ->  (
( sin `  ( A  -  ( pi  /  2 ) ) )  <  0  /\  ( cos `  ( A  -  ( pi  /  2
) ) )  <  0 ) )
5549, 54syl3an1 1252 . . . . . . . 8  |-  ( ( A  e.  RR  /\  pi  <  ( A  -  ( pi  /  2
) )  /\  ( A  -  ( pi  /  2 ) )  < 
( 3  x.  (
pi  /  2 ) ) )  ->  (
( sin `  ( A  -  ( pi  /  2 ) ) )  <  0  /\  ( cos `  ( A  -  ( pi  /  2
) ) )  <  0 ) )
56553expib 1191 . . . . . . 7  |-  ( A  e.  RR  ->  (
( pi  <  ( A  -  ( pi  /  2 ) )  /\  ( A  -  (
pi  /  2 ) )  <  ( 3  x.  ( pi  / 
2 ) ) )  ->  ( ( sin `  ( A  -  (
pi  /  2 ) ) )  <  0  /\  ( cos `  ( A  -  ( pi  /  2 ) ) )  <  0 ) ) )
5747, 56sylbid 215 . . . . . 6  |-  ( A  e.  RR  ->  (
( ( 3  x.  ( pi  /  2
) )  <  A  /\  A  <  ( 2  x.  pi ) )  ->  ( ( sin `  ( A  -  (
pi  /  2 ) ) )  <  0  /\  ( cos `  ( A  -  ( pi  /  2 ) ) )  <  0 ) ) )
5849resincld 13529 . . . . . . . 8  |-  ( A  e.  RR  ->  ( sin `  ( A  -  ( pi  /  2
) ) )  e.  RR )
5958lt0neg1d 10010 . . . . . . 7  |-  ( A  e.  RR  ->  (
( sin `  ( A  -  ( pi  /  2 ) ) )  <  0  <->  0  <  -u ( sin `  ( A  -  ( pi  /  2 ) ) ) ) )
6059anbi1d 704 . . . . . 6  |-  ( A  e.  RR  ->  (
( ( sin `  ( A  -  ( pi  /  2 ) ) )  <  0  /\  ( cos `  ( A  -  ( pi  /  2
) ) )  <  0 )  <->  ( 0  <  -u ( sin `  ( A  -  ( pi  /  2 ) ) )  /\  ( cos `  ( A  -  ( pi  /  2 ) ) )  <  0 ) ) )
6157, 60sylibd 214 . . . . 5  |-  ( A  e.  RR  ->  (
( ( 3  x.  ( pi  /  2
) )  <  A  /\  A  <  ( 2  x.  pi ) )  ->  ( 0  <  -u ( sin `  ( A  -  ( pi  /  2 ) ) )  /\  ( cos `  ( A  -  ( pi  /  2 ) ) )  <  0 ) ) )
62 recn 9473 . . . . . . . . . 10  |-  ( A  e.  RR  ->  A  e.  CC )
63 pncan3 9719 . . . . . . . . . 10  |-  ( ( ( pi  /  2
)  e.  CC  /\  A  e.  CC )  ->  ( ( pi  / 
2 )  +  ( A  -  ( pi 
/  2 ) ) )  =  A )
6415, 62, 63sylancr 663 . . . . . . . . 9  |-  ( A  e.  RR  ->  (
( pi  /  2
)  +  ( A  -  ( pi  / 
2 ) ) )  =  A )
6564fveq2d 5793 . . . . . . . 8  |-  ( A  e.  RR  ->  ( cos `  ( ( pi 
/  2 )  +  ( A  -  (
pi  /  2 ) ) ) )  =  ( cos `  A
) )
6649recnd 9513 . . . . . . . . 9  |-  ( A  e.  RR  ->  ( A  -  ( pi  /  2 ) )  e.  CC )
67 coshalfpip 22072 . . . . . . . . 9  |-  ( ( A  -  ( pi 
/  2 ) )  e.  CC  ->  ( cos `  ( ( pi 
/  2 )  +  ( A  -  (
pi  /  2 ) ) ) )  = 
-u ( sin `  ( A  -  ( pi  /  2 ) ) ) )
6866, 67syl 16 . . . . . . . 8  |-  ( A  e.  RR  ->  ( cos `  ( ( pi 
/  2 )  +  ( A  -  (
pi  /  2 ) ) ) )  = 
-u ( sin `  ( A  -  ( pi  /  2 ) ) ) )
6965, 68eqtr3d 2494 . . . . . . 7  |-  ( A  e.  RR  ->  ( cos `  A )  = 
-u ( sin `  ( A  -  ( pi  /  2 ) ) ) )
7069breq2d 4402 . . . . . 6  |-  ( A  e.  RR  ->  (
0  <  ( cos `  A )  <->  0  <  -u ( sin `  ( A  -  ( pi  /  2 ) ) ) ) )
7164fveq2d 5793 . . . . . . . 8  |-  ( A  e.  RR  ->  ( sin `  ( ( pi 
/  2 )  +  ( A  -  (
pi  /  2 ) ) ) )  =  ( sin `  A
) )
72 sinhalfpip 22070 . . . . . . . . 9  |-  ( ( A  -  ( pi 
/  2 ) )  e.  CC  ->  ( sin `  ( ( pi 
/  2 )  +  ( A  -  (
pi  /  2 ) ) ) )  =  ( cos `  ( A  -  ( pi  /  2 ) ) ) )
7366, 72syl 16 . . . . . . . 8  |-  ( A  e.  RR  ->  ( sin `  ( ( pi 
/  2 )  +  ( A  -  (
pi  /  2 ) ) ) )  =  ( cos `  ( A  -  ( pi  /  2 ) ) ) )
7471, 73eqtr3d 2494 . . . . . . 7  |-  ( A  e.  RR  ->  ( sin `  A )  =  ( cos `  ( A  -  ( pi  /  2 ) ) ) )
7574breq1d 4400 . . . . . 6  |-  ( A  e.  RR  ->  (
( sin `  A
)  <  0  <->  ( cos `  ( A  -  (
pi  /  2 ) ) )  <  0
) )
7670, 75anbi12d 710 . . . . 5  |-  ( A  e.  RR  ->  (
( 0  <  ( cos `  A )  /\  ( sin `  A )  <  0 )  <->  ( 0  <  -u ( sin `  ( A  -  ( pi  /  2 ) ) )  /\  ( cos `  ( A  -  ( pi  /  2 ) ) )  <  0 ) ) )
7761, 76sylibrd 234 . . . 4  |-  ( A  e.  RR  ->  (
( ( 3  x.  ( pi  /  2
) )  <  A  /\  A  <  ( 2  x.  pi ) )  ->  ( 0  < 
( cos `  A
)  /\  ( sin `  A )  <  0
) ) )
78773impib 1186 . . 3  |-  ( ( A  e.  RR  /\  ( 3  x.  (
pi  /  2 ) )  <  A  /\  A  <  ( 2  x.  pi ) )  -> 
( 0  <  ( cos `  A )  /\  ( sin `  A )  <  0 ) )
7978ancomd 451 . 2  |-  ( ( A  e.  RR  /\  ( 3  x.  (
pi  /  2 ) )  <  A  /\  A  <  ( 2  x.  pi ) )  -> 
( ( sin `  A
)  <  0  /\  0  <  ( cos `  A
) ) )
8010, 79sylbi 195 1  |-  ( A  e.  ( ( 3  x.  ( pi  / 
2 ) ) (,) ( 2  x.  pi ) )  ->  (
( sin `  A
)  <  0  /\  0  <  ( cos `  A
) ) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    <-> wb 184    /\ wa 369    /\ w3a 965    = wceq 1370    e. wcel 1758    =/= wne 2644   class class class wbr 4390   ` cfv 5516  (class class class)co 6190   CCcc 9381   RRcr 9382   0cc0 9383   1c1 9384    + caddc 9386    x. cmul 9388   RR*cxr 9518    < clt 9519    - cmin 9696   -ucneg 9697    / cdiv 10094   2c2 10472   3c3 10473   4c4 10474   (,)cioo 11401   sincsin 13451   cosccos 13452   picpi 13454
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1592  ax-4 1603  ax-5 1671  ax-6 1710  ax-7 1730  ax-8 1760  ax-9 1762  ax-10 1777  ax-11 1782  ax-12 1794  ax-13 1952  ax-ext 2430  ax-rep 4501  ax-sep 4511  ax-nul 4519  ax-pow 4568  ax-pr 4629  ax-un 6472  ax-inf2 7948  ax-cnex 9439  ax-resscn 9440  ax-1cn 9441  ax-icn 9442  ax-addcl 9443  ax-addrcl 9444  ax-mulcl 9445  ax-mulrcl 9446  ax-mulcom 9447  ax-addass 9448  ax-mulass 9449  ax-distr 9450  ax-i2m1 9451  ax-1ne0 9452  ax-1rid 9453  ax-rnegex 9454  ax-rrecex 9455  ax-cnre 9456  ax-pre-lttri 9457  ax-pre-lttrn 9458  ax-pre-ltadd 9459  ax-pre-mulgt0 9460  ax-pre-sup 9461  ax-addf 9462  ax-mulf 9463
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 966  df-3an 967  df-tru 1373  df-fal 1376  df-ex 1588  df-nf 1591  df-sb 1703  df-eu 2264  df-mo 2265  df-clab 2437  df-cleq 2443  df-clel 2446  df-nfc 2601  df-ne 2646  df-nel 2647  df-ral 2800  df-rex 2801  df-reu 2802  df-rmo 2803  df-rab 2804  df-v 3070  df-sbc 3285  df-csb 3387  df-dif 3429  df-un 3431  df-in 3433  df-ss 3440  df-pss 3442  df-nul 3736  df-if 3890  df-pw 3960  df-sn 3976  df-pr 3978  df-tp 3980  df-op 3982  df-uni 4190  df-int 4227  df-iun 4271  df-iin 4272  df-br 4391  df-opab 4449  df-mpt 4450  df-tr 4484  df-eprel 4730  df-id 4734  df-po 4739  df-so 4740  df-fr 4777  df-se 4778  df-we 4779  df-ord 4820  df-on 4821  df-lim 4822  df-suc 4823  df-xp 4944  df-rel 4945  df-cnv 4946  df-co 4947  df-dm 4948  df-rn 4949  df-res 4950  df-ima 4951  df-iota 5479  df-fun 5518  df-fn 5519  df-f 5520  df-f1 5521  df-fo 5522  df-f1o 5523  df-fv 5524  df-isom 5525  df-riota 6151  df-ov 6193  df-oprab 6194  df-mpt2 6195  df-of 6420  df-om 6577  df-1st 6677  df-2nd 6678  df-supp 6791  df-recs 6932  df-rdg 6966  df-1o 7020  df-2o 7021  df-oadd 7024  df-er 7201  df-map 7316  df-pm 7317  df-ixp 7364  df-en 7411  df-dom 7412  df-sdom 7413  df-fin 7414  df-fsupp 7722  df-fi 7762  df-sup 7792  df-oi 7825  df-card 8210  df-cda 8438  df-pnf 9521  df-mnf 9522  df-xr 9523  df-ltxr 9524  df-le 9525  df-sub 9698  df-neg 9699  df-div 10095  df-nn 10424  df-2 10481  df-3 10482  df-4 10483  df-5 10484  df-6 10485  df-7 10486  df-8 10487  df-9 10488  df-10 10489  df-n0 10681  df-z 10748  df-dec 10857  df-uz 10963  df-q 11055  df-rp 11093  df-xneg 11190  df-xadd 11191  df-xmul 11192  df-ioo 11405  df-ioc 11406  df-ico 11407  df-icc 11408  df-fz 11539  df-fzo 11650  df-fl 11743  df-seq 11908  df-exp 11967  df-fac 12153  df-bc 12180  df-hash 12205  df-shft 12658  df-cj 12690  df-re 12691  df-im 12692  df-sqr 12826  df-abs 12827  df-limsup 13051  df-clim 13068  df-rlim 13069  df-sum 13266  df-ef 13455  df-sin 13457  df-cos 13458  df-pi 13460  df-struct 14278  df-ndx 14279  df-slot 14280  df-base 14281  df-sets 14282  df-ress 14283  df-plusg 14353  df-mulr 14354  df-starv 14355  df-sca 14356  df-vsca 14357  df-ip 14358  df-tset 14359  df-ple 14360  df-ds 14362  df-unif 14363  df-hom 14364  df-cco 14365  df-rest 14463  df-topn 14464  df-0g 14482  df-gsum 14483  df-topgen 14484  df-pt 14485  df-prds 14488  df-xrs 14542  df-qtop 14547  df-imas 14548  df-xps 14550  df-mre 14626  df-mrc 14627  df-acs 14629  df-mnd 15517  df-submnd 15567  df-mulg 15650  df-cntz 15937  df-cmn 16383  df-psmet 17918  df-xmet 17919  df-met 17920  df-bl 17921  df-mopn 17922  df-fbas 17923  df-fg 17924  df-cnfld 17928  df-top 18619  df-bases 18621  df-topon 18622  df-topsp 18623  df-cld 18739  df-ntr 18740  df-cls 18741  df-nei 18818  df-lp 18856  df-perf 18857  df-cn 18947  df-cnp 18948  df-haus 19035  df-tx 19251  df-hmeo 19444  df-fil 19535  df-fm 19627  df-flim 19628  df-flf 19629  df-xms 20011  df-ms 20012  df-tms 20013  df-cncf 20570  df-limc 21457  df-dv 21458
This theorem is referenced by: (None)
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