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Theorem xltnegi 11191
Description: Forward direction of xltneg 11192. (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 11101 . . 3  |-  ( A  e.  RR*  <->  ( A  e.  RR  \/  A  = +oo  \/  A  = -oo ) )
2 elxr 11101 . . . . . 6  |-  ( B  e.  RR*  <->  ( B  e.  RR  \/  B  = +oo  \/  B  = -oo ) )
3 ltneg 9844 . . . . . . . . 9  |-  ( ( A  e.  RR  /\  B  e.  RR )  ->  ( A  <  B  <->  -u B  <  -u A
) )
4 rexneg 11186 . . . . . . . . . 10  |-  ( B  e.  RR  ->  -e
B  =  -u B
)
5 rexneg 11186 . . . . . . . . . 10  |-  ( A  e.  RR  ->  -e
A  =  -u A
)
64, 5breqan12rd 4313 . . . . . . . . 9  |-  ( ( A  e.  RR  /\  B  e.  RR )  ->  (  -e B  <  -e A  <->  -u B  <  -u A ) )
73, 6bitr4d 256 . . . . . . . 8  |-  ( ( A  e.  RR  /\  B  e.  RR )  ->  ( A  <  B  <->  -e B  <  -e
A ) )
87biimpd 207 . . . . . . 7  |-  ( ( A  e.  RR  /\  B  e.  RR )  ->  ( A  <  B  -> 
-e B  <  -e A ) )
9 xnegeq 11182 . . . . . . . . . . 11  |-  ( B  = +oo  ->  -e
B  =  -e +oo )
10 xnegpnf 11184 . . . . . . . . . . 11  |-  -e +oo  = -oo
119, 10syl6eq 2491 . . . . . . . . . 10  |-  ( B  = +oo  ->  -e
B  = -oo )
1211adantl 466 . . . . . . . . 9  |-  ( ( A  e.  RR  /\  B  = +oo )  -> 
-e B  = -oo )
13 renegcl 9677 . . . . . . . . . . . 12  |-  ( A  e.  RR  ->  -u A  e.  RR )
145, 13eqeltrd 2517 . . . . . . . . . . 11  |-  ( A  e.  RR  ->  -e
A  e.  RR )
15 mnflt 11109 . . . . . . . . . . 11  |-  (  -e A  e.  RR  -> -oo  <  -e A )
1614, 15syl 16 . . . . . . . . . 10  |-  ( A  e.  RR  -> -oo  <  -e A )
1716adantr 465 . . . . . . . . 9  |-  ( ( A  e.  RR  /\  B  = +oo )  -> -oo  <  -e A )
1812, 17eqbrtrd 4317 . . . . . . . 8  |-  ( ( A  e.  RR  /\  B  = +oo )  -> 
-e B  <  -e A )
1918a1d 25 . . . . . . 7  |-  ( ( A  e.  RR  /\  B  = +oo )  ->  ( A  <  B  -> 
-e B  <  -e A ) )
20 simpr 461 . . . . . . . . 9  |-  ( ( A  e.  RR  /\  B  = -oo )  ->  B  = -oo )
2120breq2d 4309 . . . . . . . 8  |-  ( ( A  e.  RR  /\  B  = -oo )  ->  ( A  <  B  <->  A  < -oo ) )
22 rexr 9434 . . . . . . . . . . 11  |-  ( A  e.  RR  ->  A  e.  RR* )
23 nltmnf 11114 . . . . . . . . . . 11  |-  ( A  e.  RR*  ->  -.  A  < -oo )
2422, 23syl 16 . . . . . . . . . 10  |-  ( A  e.  RR  ->  -.  A  < -oo )
2524adantr 465 . . . . . . . . 9  |-  ( ( A  e.  RR  /\  B  = -oo )  ->  -.  A  < -oo )
2625pm2.21d 106 . . . . . . . 8  |-  ( ( A  e.  RR  /\  B  = -oo )  ->  ( A  < -oo  -> 
-e B  <  -e A ) )
2721, 26sylbid 215 . . . . . . 7  |-  ( ( A  e.  RR  /\  B  = -oo )  ->  ( A  <  B  -> 
-e B  <  -e A ) )
288, 19, 273jaodan 1284 . . . . . 6  |-  ( ( A  e.  RR  /\  ( B  e.  RR  \/  B  = +oo  \/  B  = -oo ) )  ->  ( A  <  B  ->  -e
B  <  -e A ) )
292, 28sylan2b 475 . . . . 5  |-  ( ( A  e.  RR  /\  B  e.  RR* )  -> 
( A  <  B  -> 
-e B  <  -e A ) )
3029expimpd 603 . . . 4  |-  ( A  e.  RR  ->  (
( B  e.  RR*  /\  A  <  B )  ->  -e B  <  -e A ) )
31 simpl 457 . . . . . . 7  |-  ( ( A  = +oo  /\  B  e.  RR* )  ->  A  = +oo )
3231breq1d 4307 . . . . . 6  |-  ( ( A  = +oo  /\  B  e.  RR* )  -> 
( A  <  B  <-> +oo 
<  B ) )
33 pnfnlt 11113 . . . . . . . 8  |-  ( B  e.  RR*  ->  -. +oo  <  B )
3433adantl 466 . . . . . . 7  |-  ( ( A  = +oo  /\  B  e.  RR* )  ->  -. +oo  <  B )
3534pm2.21d 106 . . . . . 6  |-  ( ( A  = +oo  /\  B  e.  RR* )  -> 
( +oo  <  B  ->  -e B  <  -e
A ) )
3632, 35sylbid 215 . . . . 5  |-  ( ( A  = +oo  /\  B  e.  RR* )  -> 
( A  <  B  -> 
-e B  <  -e A ) )
3736expimpd 603 . . . 4  |-  ( A  = +oo  ->  (
( B  e.  RR*  /\  A  <  B )  ->  -e B  <  -e A ) )
38 breq1 4300 . . . . . 6  |-  ( A  = -oo  ->  ( A  <  B  <-> -oo  <  B
) )
3938anbi2d 703 . . . . 5  |-  ( A  = -oo  ->  (
( B  e.  RR*  /\  A  <  B )  <-> 
( B  e.  RR*  /\ -oo  <  B ) ) )
40 renegcl 9677 . . . . . . . . . . 11  |-  ( B  e.  RR  ->  -u B  e.  RR )
414, 40eqeltrd 2517 . . . . . . . . . 10  |-  ( B  e.  RR  ->  -e
B  e.  RR )
4241adantr 465 . . . . . . . . 9  |-  ( ( B  e.  RR  /\ -oo 
<  B )  ->  -e
B  e.  RR )
43 ltpnf 11107 . . . . . . . . 9  |-  (  -e B  e.  RR  -> 
-e B  < +oo )
4442, 43syl 16 . . . . . . . 8  |-  ( ( B  e.  RR  /\ -oo 
<  B )  ->  -e
B  < +oo )
4511adantr 465 . . . . . . . . 9  |-  ( ( B  = +oo  /\ -oo 
<  B )  ->  -e
B  = -oo )
46 mnfltpnf 11111 . . . . . . . . 9  |- -oo  < +oo
4745, 46syl6eqbr 4334 . . . . . . . 8  |-  ( ( B  = +oo  /\ -oo 
<  B )  ->  -e
B  < +oo )
48 breq2 4301 . . . . . . . . . 10  |-  ( B  = -oo  ->  ( -oo  <  B  <-> -oo  < -oo ) )
49 mnfxr 11099 . . . . . . . . . . . 12  |- -oo  e.  RR*
50 nltmnf 11114 . . . . . . . . . . . 12  |-  ( -oo  e.  RR*  ->  -. -oo  < -oo )
5149, 50ax-mp 5 . . . . . . . . . . 11  |-  -. -oo  < -oo
5251pm2.21i 131 . . . . . . . . . 10  |-  ( -oo  < -oo  ->  -e B  < +oo )
5348, 52syl6bi 228 . . . . . . . . 9  |-  ( B  = -oo  ->  ( -oo  <  B  ->  -e
B  < +oo )
)
5453imp 429 . . . . . . . 8  |-  ( ( B  = -oo  /\ -oo 
<  B )  ->  -e
B  < +oo )
5544, 47, 543jaoian 1283 . . . . . . 7  |-  ( ( ( B  e.  RR  \/  B  = +oo  \/  B  = -oo )  /\ -oo  <  B
)  ->  -e B  < +oo )
562, 55sylanb 472 . . . . . 6  |-  ( ( B  e.  RR*  /\ -oo  <  B )  ->  -e
B  < +oo )
57 xnegeq 11182 . . . . . . . 8  |-  ( A  = -oo  ->  -e
A  =  -e -oo )
58 xnegmnf 11185 . . . . . . . 8  |-  -e -oo  = +oo
5957, 58syl6eq 2491 . . . . . . 7  |-  ( A  = -oo  ->  -e
A  = +oo )
6059breq2d 4309 . . . . . 6  |-  ( A  = -oo  ->  (  -e B  <  -e
A  <->  -e B  < +oo ) )
6156, 60syl5ibr 221 . . . . 5  |-  ( A  = -oo  ->  (
( B  e.  RR*  /\ -oo  <  B )  ->  -e B  <  -e
A ) )
6239, 61sylbid 215 . . . 4  |-  ( A  = -oo  ->  (
( B  e.  RR*  /\  A  <  B )  ->  -e B  <  -e A ) )
6330, 37, 623jaoi 1281 . . 3  |-  ( ( A  e.  RR  \/  A  = +oo  \/  A  = -oo )  ->  (
( B  e.  RR*  /\  A  <  B )  ->  -e B  <  -e A ) )
641, 63sylbi 195 . 2  |-  ( A  e.  RR*  ->  ( ( B  e.  RR*  /\  A  <  B )  ->  -e
B  <  -e A ) )
65643impib 1185 1  |-  ( ( A  e.  RR*  /\  B  e.  RR*  /\  A  < 
B )  ->  -e
B  <  -e A )
Colors of variables: wff setvar class
Syntax hints:   -. wn 3    -> wi 4    /\ wa 369    \/ w3o 964    /\ w3a 965    = wceq 1369    e. wcel 1756   class class class wbr 4297   RRcr 9286   +oocpnf 9420   -oocmnf 9421   RR*cxr 9422    < clt 9423   -ucneg 9601    -ecxne 11091
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1591  ax-4 1602  ax-5 1670  ax-6 1708  ax-7 1728  ax-8 1758  ax-9 1760  ax-10 1775  ax-11 1780  ax-12 1792  ax-13 1943  ax-ext 2423  ax-sep 4418  ax-nul 4426  ax-pow 4475  ax-pr 4536  ax-un 6377  ax-cnex 9343  ax-resscn 9344  ax-1cn 9345  ax-icn 9346  ax-addcl 9347  ax-addrcl 9348  ax-mulcl 9349  ax-mulrcl 9350  ax-mulcom 9351  ax-addass 9352  ax-mulass 9353  ax-distr 9354  ax-i2m1 9355  ax-1ne0 9356  ax-1rid 9357  ax-rnegex 9358  ax-rrecex 9359  ax-cnre 9360  ax-pre-lttri 9361  ax-pre-lttrn 9362  ax-pre-ltadd 9363
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 966  df-3an 967  df-tru 1372  df-ex 1587  df-nf 1590  df-sb 1701  df-eu 2257  df-mo 2258  df-clab 2430  df-cleq 2436  df-clel 2439  df-nfc 2573  df-ne 2613  df-nel 2614  df-ral 2725  df-rex 2726  df-reu 2727  df-rab 2729  df-v 2979  df-sbc 3192  df-csb 3294  df-dif 3336  df-un 3338  df-in 3340  df-ss 3347  df-nul 3643  df-if 3797  df-pw 3867  df-sn 3883  df-pr 3885  df-op 3889  df-uni 4097  df-br 4298  df-opab 4356  df-mpt 4357  df-id 4641  df-po 4646  df-so 4647  df-xp 4851  df-rel 4852  df-cnv 4853  df-co 4854  df-dm 4855  df-rn 4856  df-res 4857  df-ima 4858  df-iota 5386  df-fun 5425  df-fn 5426  df-f 5427  df-f1 5428  df-fo 5429  df-f1o 5430  df-fv 5431  df-riota 6057  df-ov 6099  df-oprab 6100  df-mpt2 6101  df-er 7106  df-en 7316  df-dom 7317  df-sdom 7318  df-pnf 9425  df-mnf 9426  df-xr 9427  df-ltxr 9428  df-le 9429  df-sub 9602  df-neg 9603  df-xneg 11094
This theorem is referenced by:  xltneg  11192  xrsdsreclblem  17864
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