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Theorem xltnegi 8978
Description: Forward direction of xltneg 8979. (Contributed by Mario Carneiro, 20-Aug-2015.)
Assertion
Ref Expression
xltnegi  |-  ( ( A  e.  RR*  /\  B  e.  RR*  /\  A  < 
B )  ->  -e
B  <  -e A )

Proof of Theorem xltnegi
StepHypRef Expression
1 elxr 8928 . . 3  |-  ( A  e.  RR*  <->  ( A  e.  RR  \/  A  = +oo  \/  A  = -oo ) )
2 elxr 8928 . . . . . 6  |-  ( B  e.  RR*  <->  ( B  e.  RR  \/  B  = +oo  \/  B  = -oo ) )
3 ltneg 7633 . . . . . . . . 9  |-  ( ( A  e.  RR  /\  B  e.  RR )  ->  ( A  <  B  <->  -u B  <  -u A
) )
4 rexneg 8973 . . . . . . . . . 10  |-  ( B  e.  RR  ->  -e
B  =  -u B
)
5 rexneg 8973 . . . . . . . . . 10  |-  ( A  e.  RR  ->  -e
A  =  -u A
)
64, 5breqan12rd 3809 . . . . . . . . 9  |-  ( ( A  e.  RR  /\  B  e.  RR )  ->  (  -e B  <  -e A  <->  -u B  <  -u A ) )
73, 6bitr4d 189 . . . . . . . 8  |-  ( ( A  e.  RR  /\  B  e.  RR )  ->  ( A  <  B  <->  -e B  <  -e
A ) )
87biimpd 142 . . . . . . 7  |-  ( ( A  e.  RR  /\  B  e.  RR )  ->  ( A  <  B  -> 
-e B  <  -e A ) )
9 xnegeq 8970 . . . . . . . . . . 11  |-  ( B  = +oo  ->  -e
B  =  -e +oo )
10 xnegpnf 8971 . . . . . . . . . . 11  |-  -e +oo  = -oo
119, 10syl6eq 2130 . . . . . . . . . 10  |-  ( B  = +oo  ->  -e
B  = -oo )
1211adantl 271 . . . . . . . . 9  |-  ( ( A  e.  RR  /\  B  = +oo )  -> 
-e B  = -oo )
13 renegcl 7436 . . . . . . . . . . . 12  |-  ( A  e.  RR  ->  -u A  e.  RR )
145, 13eqeltrd 2156 . . . . . . . . . . 11  |-  ( A  e.  RR  ->  -e
A  e.  RR )
15 mnflt 8934 . . . . . . . . . . 11  |-  (  -e A  e.  RR  -> -oo  <  -e A )
1614, 15syl 14 . . . . . . . . . 10  |-  ( A  e.  RR  -> -oo  <  -e A )
1716adantr 270 . . . . . . . . 9  |-  ( ( A  e.  RR  /\  B  = +oo )  -> -oo  <  -e A )
1812, 17eqbrtrd 3813 . . . . . . . 8  |-  ( ( A  e.  RR  /\  B  = +oo )  -> 
-e B  <  -e A )
1918a1d 22 . . . . . . 7  |-  ( ( A  e.  RR  /\  B  = +oo )  ->  ( A  <  B  -> 
-e B  <  -e A ) )
20 simpr 108 . . . . . . . . 9  |-  ( ( A  e.  RR  /\  B  = -oo )  ->  B  = -oo )
2120breq2d 3805 . . . . . . . 8  |-  ( ( A  e.  RR  /\  B  = -oo )  ->  ( A  <  B  <->  A  < -oo ) )
22 rexr 7226 . . . . . . . . . . 11  |-  ( A  e.  RR  ->  A  e.  RR* )
23 nltmnf 8939 . . . . . . . . . . 11  |-  ( A  e.  RR*  ->  -.  A  < -oo )
2422, 23syl 14 . . . . . . . . . 10  |-  ( A  e.  RR  ->  -.  A  < -oo )
2524adantr 270 . . . . . . . . 9  |-  ( ( A  e.  RR  /\  B  = -oo )  ->  -.  A  < -oo )
2625pm2.21d 582 . . . . . . . 8  |-  ( ( A  e.  RR  /\  B  = -oo )  ->  ( A  < -oo  -> 
-e B  <  -e A ) )
2721, 26sylbid 148 . . . . . . 7  |-  ( ( A  e.  RR  /\  B  = -oo )  ->  ( A  <  B  -> 
-e B  <  -e A ) )
288, 19, 273jaodan 1238 . . . . . 6  |-  ( ( A  e.  RR  /\  ( B  e.  RR  \/  B  = +oo  \/  B  = -oo ) )  ->  ( A  <  B  ->  -e
B  <  -e A ) )
292, 28sylan2b 281 . . . . 5  |-  ( ( A  e.  RR  /\  B  e.  RR* )  -> 
( A  <  B  -> 
-e B  <  -e A ) )
3029expimpd 355 . . . 4  |-  ( A  e.  RR  ->  (
( B  e.  RR*  /\  A  <  B )  ->  -e B  <  -e A ) )
31 simpl 107 . . . . . . 7  |-  ( ( A  = +oo  /\  B  e.  RR* )  ->  A  = +oo )
3231breq1d 3803 . . . . . 6  |-  ( ( A  = +oo  /\  B  e.  RR* )  -> 
( A  <  B  <-> +oo 
<  B ) )
33 pnfnlt 8938 . . . . . . . 8  |-  ( B  e.  RR*  ->  -. +oo  <  B )
3433adantl 271 . . . . . . 7  |-  ( ( A  = +oo  /\  B  e.  RR* )  ->  -. +oo  <  B )
3534pm2.21d 582 . . . . . 6  |-  ( ( A  = +oo  /\  B  e.  RR* )  -> 
( +oo  <  B  ->  -e B  <  -e
A ) )
3632, 35sylbid 148 . . . . 5  |-  ( ( A  = +oo  /\  B  e.  RR* )  -> 
( A  <  B  -> 
-e B  <  -e A ) )
3736expimpd 355 . . . 4  |-  ( A  = +oo  ->  (
( B  e.  RR*  /\  A  <  B )  ->  -e B  <  -e A ) )
38 breq1 3796 . . . . . 6  |-  ( A  = -oo  ->  ( A  <  B  <-> -oo  <  B
) )
3938anbi2d 452 . . . . 5  |-  ( A  = -oo  ->  (
( B  e.  RR*  /\  A  <  B )  <-> 
( B  e.  RR*  /\ -oo  <  B ) ) )
40 renegcl 7436 . . . . . . . . . . 11  |-  ( B  e.  RR  ->  -u B  e.  RR )
414, 40eqeltrd 2156 . . . . . . . . . 10  |-  ( B  e.  RR  ->  -e
B  e.  RR )
4241adantr 270 . . . . . . . . 9  |-  ( ( B  e.  RR  /\ -oo 
<  B )  ->  -e
B  e.  RR )
43 ltpnf 8932 . . . . . . . . 9  |-  (  -e B  e.  RR  -> 
-e B  < +oo )
4442, 43syl 14 . . . . . . . 8  |-  ( ( B  e.  RR  /\ -oo 
<  B )  ->  -e
B  < +oo )
4511adantr 270 . . . . . . . . 9  |-  ( ( B  = +oo  /\ -oo 
<  B )  ->  -e
B  = -oo )
46 mnfltpnf 8936 . . . . . . . . 9  |- -oo  < +oo
4745, 46syl6eqbr 3830 . . . . . . . 8  |-  ( ( B  = +oo  /\ -oo 
<  B )  ->  -e
B  < +oo )
48 breq2 3797 . . . . . . . . . 10  |-  ( B  = -oo  ->  ( -oo  <  B  <-> -oo  < -oo ) )
49 mnfxr 7237 . . . . . . . . . . . 12  |- -oo  e.  RR*
50 nltmnf 8939 . . . . . . . . . . . 12  |-  ( -oo  e.  RR*  ->  -. -oo  < -oo )
5149, 50ax-mp 7 . . . . . . . . . . 11  |-  -. -oo  < -oo
5251pm2.21i 608 . . . . . . . . . 10  |-  ( -oo  < -oo  ->  -e B  < +oo )
5348, 52syl6bi 161 . . . . . . . . 9  |-  ( B  = -oo  ->  ( -oo  <  B  ->  -e
B  < +oo )
)
5453imp 122 . . . . . . . 8  |-  ( ( B  = -oo  /\ -oo 
<  B )  ->  -e
B  < +oo )
5544, 47, 543jaoian 1237 . . . . . . 7  |-  ( ( ( B  e.  RR  \/  B  = +oo  \/  B  = -oo )  /\ -oo  <  B
)  ->  -e B  < +oo )
562, 55sylanb 278 . . . . . 6  |-  ( ( B  e.  RR*  /\ -oo  <  B )  ->  -e
B  < +oo )
57 xnegeq 8970 . . . . . . . 8  |-  ( A  = -oo  ->  -e
A  =  -e -oo )
58 xnegmnf 8972 . . . . . . . 8  |-  -e -oo  = +oo
5957, 58syl6eq 2130 . . . . . . 7  |-  ( A  = -oo  ->  -e
A  = +oo )
6059breq2d 3805 . . . . . 6  |-  ( A  = -oo  ->  (  -e B  <  -e
A  <->  -e B  < +oo ) )
6156, 60syl5ibr 154 . . . . 5  |-  ( A  = -oo  ->  (
( B  e.  RR*  /\ -oo  <  B )  ->  -e B  <  -e
A ) )
6239, 61sylbid 148 . . . 4  |-  ( A  = -oo  ->  (
( B  e.  RR*  /\  A  <  B )  ->  -e B  <  -e A ) )
6330, 37, 623jaoi 1235 . . 3  |-  ( ( A  e.  RR  \/  A  = +oo  \/  A  = -oo )  ->  (
( B  e.  RR*  /\  A  <  B )  ->  -e B  <  -e A ) )
641, 63sylbi 119 . 2  |-  ( A  e.  RR*  ->  ( ( B  e.  RR*  /\  A  <  B )  ->  -e
B  <  -e A ) )
65643impib 1137 1  |-  ( ( A  e.  RR*  /\  B  e.  RR*  /\  A  < 
B )  ->  -e
B  <  -e A )
Colors of variables: wff set class
Syntax hints:   -. wn 3    -> wi 4    /\ wa 102    \/ w3o 919    /\ w3a 920    = wceq 1285    e. wcel 1434   class class class wbr 3793   RRcr 7042   +oocpnf 7212   -oocmnf 7213   RR*cxr 7214    < clt 7215   -ucneg 7347    -ecxne 8921
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-in1 577  ax-in2 578  ax-io 663  ax-5 1377  ax-7 1378  ax-gen 1379  ax-ie1 1423  ax-ie2 1424  ax-8 1436  ax-10 1437  ax-11 1438  ax-i12 1439  ax-bndl 1440  ax-4 1441  ax-13 1445  ax-14 1446  ax-17 1460  ax-i9 1464  ax-ial 1468  ax-i5r 1469  ax-ext 2064  ax-sep 3904  ax-pow 3956  ax-pr 3972  ax-un 4196  ax-setind 4288  ax-cnex 7129  ax-resscn 7130  ax-1cn 7131  ax-1re 7132  ax-icn 7133  ax-addcl 7134  ax-addrcl 7135  ax-mulcl 7136  ax-addcom 7138  ax-addass 7140  ax-distr 7142  ax-i2m1 7143  ax-0id 7146  ax-rnegex 7147  ax-cnre 7149  ax-pre-ltadd 7154
This theorem depends on definitions:  df-bi 115  df-3or 921  df-3an 922  df-tru 1288  df-fal 1291  df-nf 1391  df-sb 1687  df-eu 1945  df-mo 1946  df-clab 2069  df-cleq 2075  df-clel 2078  df-nfc 2209  df-ne 2247  df-nel 2341  df-ral 2354  df-rex 2355  df-reu 2356  df-rab 2358  df-v 2604  df-sbc 2817  df-dif 2976  df-un 2978  df-in 2980  df-ss 2987  df-if 3360  df-pw 3392  df-sn 3412  df-pr 3413  df-op 3415  df-uni 3610  df-br 3794  df-opab 3848  df-id 4056  df-xp 4377  df-rel 4378  df-cnv 4379  df-co 4380  df-dm 4381  df-iota 4897  df-fun 4934  df-fv 4940  df-riota 5499  df-ov 5546  df-oprab 5547  df-mpt2 5548  df-pnf 7217  df-mnf 7218  df-xr 7219  df-ltxr 7220  df-sub 7348  df-neg 7349  df-xneg 8924
This theorem is referenced by:  xltneg  8979
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