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Theorem xltnegi 8748
Description: Forward direction of xltneg 8749. (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 8696 . . 3  |-  ( A  e.  RR*  <->  ( A  e.  RR  \/  A  = +oo  \/  A  = -oo ) )
2 elxr 8696 . . . . . 6  |-  ( B  e.  RR*  <->  ( B  e.  RR  \/  B  = +oo  \/  B  = -oo ) )
3 ltneg 7457 . . . . . . . . 9  |-  ( ( A  e.  RR  /\  B  e.  RR )  ->  ( A  <  B  <->  -u B  <  -u A
) )
4 rexneg 8743 . . . . . . . . . 10  |-  ( B  e.  RR  ->  -e
B  =  -u B
)
5 rexneg 8743 . . . . . . . . . 10  |-  ( A  e.  RR  ->  -e
A  =  -u A
)
64, 5breqan12rd 3780 . . . . . . . . 9  |-  ( ( A  e.  RR  /\  B  e.  RR )  ->  (  -e B  <  -e A  <->  -u B  <  -u A ) )
73, 6bitr4d 180 . . . . . . . 8  |-  ( ( A  e.  RR  /\  B  e.  RR )  ->  ( A  <  B  <->  -e B  <  -e
A ) )
87biimpd 132 . . . . . . 7  |-  ( ( A  e.  RR  /\  B  e.  RR )  ->  ( A  <  B  -> 
-e B  <  -e A ) )
9 xnegeq 8740 . . . . . . . . . . 11  |-  ( B  = +oo  ->  -e
B  =  -e +oo )
10 xnegpnf 8741 . . . . . . . . . . 11  |-  -e +oo  = -oo
119, 10syl6eq 2088 . . . . . . . . . 10  |-  ( B  = +oo  ->  -e
B  = -oo )
1211adantl 262 . . . . . . . . 9  |-  ( ( A  e.  RR  /\  B  = +oo )  -> 
-e B  = -oo )
13 renegcl 7272 . . . . . . . . . . . 12  |-  ( A  e.  RR  ->  -u A  e.  RR )
145, 13eqeltrd 2114 . . . . . . . . . . 11  |-  ( A  e.  RR  ->  -e
A  e.  RR )
15 mnflt 8704 . . . . . . . . . . 11  |-  (  -e A  e.  RR  -> -oo  <  -e A )
1614, 15syl 14 . . . . . . . . . 10  |-  ( A  e.  RR  -> -oo  <  -e A )
1716adantr 261 . . . . . . . . 9  |-  ( ( A  e.  RR  /\  B  = +oo )  -> -oo  <  -e A )
1812, 17eqbrtrd 3784 . . . . . . . 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 103 . . . . . . . . 9  |-  ( ( A  e.  RR  /\  B  = -oo )  ->  B  = -oo )
2120breq2d 3776 . . . . . . . 8  |-  ( ( A  e.  RR  /\  B  = -oo )  ->  ( A  <  B  <->  A  < -oo ) )
22 rexr 7071 . . . . . . . . . . 11  |-  ( A  e.  RR  ->  A  e.  RR* )
23 nltmnf 8709 . . . . . . . . . . 11  |-  ( A  e.  RR*  ->  -.  A  < -oo )
2422, 23syl 14 . . . . . . . . . 10  |-  ( A  e.  RR  ->  -.  A  < -oo )
2524adantr 261 . . . . . . . . 9  |-  ( ( A  e.  RR  /\  B  = -oo )  ->  -.  A  < -oo )
2625pm2.21d 549 . . . . . . . 8  |-  ( ( A  e.  RR  /\  B  = -oo )  ->  ( A  < -oo  -> 
-e B  <  -e A ) )
2721, 26sylbid 139 . . . . . . 7  |-  ( ( A  e.  RR  /\  B  = -oo )  ->  ( A  <  B  -> 
-e B  <  -e A ) )
288, 19, 273jaodan 1201 . . . . . 6  |-  ( ( A  e.  RR  /\  ( B  e.  RR  \/  B  = +oo  \/  B  = -oo ) )  ->  ( A  <  B  ->  -e
B  <  -e A ) )
292, 28sylan2b 271 . . . . 5  |-  ( ( A  e.  RR  /\  B  e.  RR* )  -> 
( A  <  B  -> 
-e B  <  -e A ) )
3029expimpd 345 . . . 4  |-  ( A  e.  RR  ->  (
( B  e.  RR*  /\  A  <  B )  ->  -e B  <  -e A ) )
31 simpl 102 . . . . . . 7  |-  ( ( A  = +oo  /\  B  e.  RR* )  ->  A  = +oo )
3231breq1d 3774 . . . . . 6  |-  ( ( A  = +oo  /\  B  e.  RR* )  -> 
( A  <  B  <-> +oo 
<  B ) )
33 pnfnlt 8708 . . . . . . . 8  |-  ( B  e.  RR*  ->  -. +oo  <  B )
3433adantl 262 . . . . . . 7  |-  ( ( A  = +oo  /\  B  e.  RR* )  ->  -. +oo  <  B )
3534pm2.21d 549 . . . . . 6  |-  ( ( A  = +oo  /\  B  e.  RR* )  -> 
( +oo  <  B  ->  -e B  <  -e
A ) )
3632, 35sylbid 139 . . . . 5  |-  ( ( A  = +oo  /\  B  e.  RR* )  -> 
( A  <  B  -> 
-e B  <  -e A ) )
3736expimpd 345 . . . 4  |-  ( A  = +oo  ->  (
( B  e.  RR*  /\  A  <  B )  ->  -e B  <  -e A ) )
38 breq1 3767 . . . . . 6  |-  ( A  = -oo  ->  ( A  <  B  <-> -oo  <  B
) )
3938anbi2d 437 . . . . 5  |-  ( A  = -oo  ->  (
( B  e.  RR*  /\  A  <  B )  <-> 
( B  e.  RR*  /\ -oo  <  B ) ) )
40 renegcl 7272 . . . . . . . . . . 11  |-  ( B  e.  RR  ->  -u B  e.  RR )
414, 40eqeltrd 2114 . . . . . . . . . 10  |-  ( B  e.  RR  ->  -e
B  e.  RR )
4241adantr 261 . . . . . . . . 9  |-  ( ( B  e.  RR  /\ -oo 
<  B )  ->  -e
B  e.  RR )
43 ltpnf 8702 . . . . . . . . 9  |-  (  -e B  e.  RR  -> 
-e B  < +oo )
4442, 43syl 14 . . . . . . . 8  |-  ( ( B  e.  RR  /\ -oo 
<  B )  ->  -e
B  < +oo )
4511adantr 261 . . . . . . . . 9  |-  ( ( B  = +oo  /\ -oo 
<  B )  ->  -e
B  = -oo )
46 mnfltpnf 8706 . . . . . . . . 9  |- -oo  < +oo
4745, 46syl6eqbr 3801 . . . . . . . 8  |-  ( ( B  = +oo  /\ -oo 
<  B )  ->  -e
B  < +oo )
48 breq2 3768 . . . . . . . . . 10  |-  ( B  = -oo  ->  ( -oo  <  B  <-> -oo  < -oo ) )
49 mnfxr 8694 . . . . . . . . . . . 12  |- -oo  e.  RR*
50 nltmnf 8709 . . . . . . . . . . . 12  |-  ( -oo  e.  RR*  ->  -. -oo  < -oo )
5149, 50ax-mp 7 . . . . . . . . . . 11  |-  -. -oo  < -oo
5251pm2.21i 575 . . . . . . . . . 10  |-  ( -oo  < -oo  ->  -e B  < +oo )
5348, 52syl6bi 152 . . . . . . . . 9  |-  ( B  = -oo  ->  ( -oo  <  B  ->  -e
B  < +oo )
)
5453imp 115 . . . . . . . 8  |-  ( ( B  = -oo  /\ -oo 
<  B )  ->  -e
B  < +oo )
5544, 47, 543jaoian 1200 . . . . . . 7  |-  ( ( ( B  e.  RR  \/  B  = +oo  \/  B  = -oo )  /\ -oo  <  B
)  ->  -e B  < +oo )
562, 55sylanb 268 . . . . . 6  |-  ( ( B  e.  RR*  /\ -oo  <  B )  ->  -e
B  < +oo )
57 xnegeq 8740 . . . . . . . 8  |-  ( A  = -oo  ->  -e
A  =  -e -oo )
58 xnegmnf 8742 . . . . . . . 8  |-  -e -oo  = +oo
5957, 58syl6eq 2088 . . . . . . 7  |-  ( A  = -oo  ->  -e
A  = +oo )
6059breq2d 3776 . . . . . 6  |-  ( A  = -oo  ->  (  -e B  <  -e
A  <->  -e B  < +oo ) )
6156, 60syl5ibr 145 . . . . 5  |-  ( A  = -oo  ->  (
( B  e.  RR*  /\ -oo  <  B )  ->  -e B  <  -e
A ) )
6239, 61sylbid 139 . . . 4  |-  ( A  = -oo  ->  (
( B  e.  RR*  /\  A  <  B )  ->  -e B  <  -e A ) )
6330, 37, 623jaoi 1198 . . 3  |-  ( ( A  e.  RR  \/  A  = +oo  \/  A  = -oo )  ->  (
( B  e.  RR*  /\  A  <  B )  ->  -e B  <  -e A ) )
641, 63sylbi 114 . 2  |-  ( A  e.  RR*  ->  ( ( B  e.  RR*  /\  A  <  B )  ->  -e
B  <  -e A ) )
65643impib 1102 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 97    \/ w3o 884    /\ w3a 885    = wceq 1243    e. wcel 1393   class class class wbr 3764   RRcr 6888   +oocpnf 7057   -oocmnf 7058   RR*cxr 7059    < clt 7060   -ucneg 7183    -ecxne 8686
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 99  ax-ia2 100  ax-ia3 101  ax-in1 544  ax-in2 545  ax-io 630  ax-5 1336  ax-7 1337  ax-gen 1338  ax-ie1 1382  ax-ie2 1383  ax-8 1395  ax-10 1396  ax-11 1397  ax-i12 1398  ax-bndl 1399  ax-4 1400  ax-13 1404  ax-14 1405  ax-17 1419  ax-i9 1423  ax-ial 1427  ax-i5r 1428  ax-ext 2022  ax-sep 3875  ax-pow 3927  ax-pr 3944  ax-un 4170  ax-setind 4262  ax-cnex 6975  ax-resscn 6976  ax-1cn 6977  ax-1re 6978  ax-icn 6979  ax-addcl 6980  ax-addrcl 6981  ax-mulcl 6982  ax-addcom 6984  ax-addass 6986  ax-distr 6988  ax-i2m1 6989  ax-0id 6992  ax-rnegex 6993  ax-cnre 6995  ax-pre-ltadd 7000
This theorem depends on definitions:  df-bi 110  df-3or 886  df-3an 887  df-tru 1246  df-fal 1249  df-nf 1350  df-sb 1646  df-eu 1903  df-mo 1904  df-clab 2027  df-cleq 2033  df-clel 2036  df-nfc 2167  df-ne 2206  df-nel 2207  df-ral 2311  df-rex 2312  df-reu 2313  df-rab 2315  df-v 2559  df-sbc 2765  df-dif 2920  df-un 2922  df-in 2924  df-ss 2931  df-if 3332  df-pw 3361  df-sn 3381  df-pr 3382  df-op 3384  df-uni 3581  df-br 3765  df-opab 3819  df-id 4030  df-xp 4351  df-rel 4352  df-cnv 4353  df-co 4354  df-dm 4355  df-iota 4867  df-fun 4904  df-fv 4910  df-riota 5468  df-ov 5515  df-oprab 5516  df-mpt2 5517  df-pnf 7062  df-mnf 7063  df-xr 7064  df-ltxr 7065  df-sub 7184  df-neg 7185  df-xneg 8689
This theorem is referenced by:  xltneg  8749
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