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Theorem xltnegi 9792
Description: Forward direction of xltneg 9793. (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 9733 . . 3  |-  ( A  e.  RR*  <->  ( A  e.  RR  \/  A  = +oo  \/  A  = -oo ) )
2 elxr 9733 . . . . . 6  |-  ( B  e.  RR*  <->  ( B  e.  RR  \/  B  = +oo  \/  B  = -oo ) )
3 ltneg 8381 . . . . . . . . 9  |-  ( ( A  e.  RR  /\  B  e.  RR )  ->  ( A  <  B  <->  -u B  <  -u A
) )
4 rexneg 9787 . . . . . . . . . 10  |-  ( B  e.  RR  ->  -e
B  =  -u B
)
5 rexneg 9787 . . . . . . . . . 10  |-  ( A  e.  RR  ->  -e
A  =  -u A
)
64, 5breqan12rd 4006 . . . . . . . . 9  |-  ( ( A  e.  RR  /\  B  e.  RR )  ->  (  -e B  <  -e A  <->  -u B  <  -u A ) )
73, 6bitr4d 190 . . . . . . . 8  |-  ( ( A  e.  RR  /\  B  e.  RR )  ->  ( A  <  B  <->  -e B  <  -e
A ) )
87biimpd 143 . . . . . . 7  |-  ( ( A  e.  RR  /\  B  e.  RR )  ->  ( A  <  B  -> 
-e B  <  -e A ) )
9 xnegeq 9784 . . . . . . . . . . 11  |-  ( B  = +oo  ->  -e
B  =  -e +oo )
10 xnegpnf 9785 . . . . . . . . . . 11  |-  -e +oo  = -oo
119, 10eqtrdi 2219 . . . . . . . . . 10  |-  ( B  = +oo  ->  -e
B  = -oo )
1211adantl 275 . . . . . . . . 9  |-  ( ( A  e.  RR  /\  B  = +oo )  -> 
-e B  = -oo )
13 renegcl 8180 . . . . . . . . . . . 12  |-  ( A  e.  RR  ->  -u A  e.  RR )
145, 13eqeltrd 2247 . . . . . . . . . . 11  |-  ( A  e.  RR  ->  -e
A  e.  RR )
15 mnflt 9740 . . . . . . . . . . 11  |-  (  -e A  e.  RR  -> -oo  <  -e A )
1614, 15syl 14 . . . . . . . . . 10  |-  ( A  e.  RR  -> -oo  <  -e A )
1716adantr 274 . . . . . . . . 9  |-  ( ( A  e.  RR  /\  B  = +oo )  -> -oo  <  -e A )
1812, 17eqbrtrd 4011 . . . . . . . 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 109 . . . . . . . . 9  |-  ( ( A  e.  RR  /\  B  = -oo )  ->  B  = -oo )
2120breq2d 4001 . . . . . . . 8  |-  ( ( A  e.  RR  /\  B  = -oo )  ->  ( A  <  B  <->  A  < -oo ) )
22 rexr 7965 . . . . . . . . . . 11  |-  ( A  e.  RR  ->  A  e.  RR* )
23 nltmnf 9745 . . . . . . . . . . 11  |-  ( A  e.  RR*  ->  -.  A  < -oo )
2422, 23syl 14 . . . . . . . . . 10  |-  ( A  e.  RR  ->  -.  A  < -oo )
2524adantr 274 . . . . . . . . 9  |-  ( ( A  e.  RR  /\  B  = -oo )  ->  -.  A  < -oo )
2625pm2.21d 614 . . . . . . . 8  |-  ( ( A  e.  RR  /\  B  = -oo )  ->  ( A  < -oo  -> 
-e B  <  -e A ) )
2721, 26sylbid 149 . . . . . . 7  |-  ( ( A  e.  RR  /\  B  = -oo )  ->  ( A  <  B  -> 
-e B  <  -e A ) )
288, 19, 273jaodan 1301 . . . . . 6  |-  ( ( A  e.  RR  /\  ( B  e.  RR  \/  B  = +oo  \/  B  = -oo ) )  ->  ( A  <  B  ->  -e
B  <  -e A ) )
292, 28sylan2b 285 . . . . 5  |-  ( ( A  e.  RR  /\  B  e.  RR* )  -> 
( A  <  B  -> 
-e B  <  -e A ) )
3029expimpd 361 . . . 4  |-  ( A  e.  RR  ->  (
( B  e.  RR*  /\  A  <  B )  ->  -e B  <  -e A ) )
31 simpl 108 . . . . . . 7  |-  ( ( A  = +oo  /\  B  e.  RR* )  ->  A  = +oo )
3231breq1d 3999 . . . . . 6  |-  ( ( A  = +oo  /\  B  e.  RR* )  -> 
( A  <  B  <-> +oo 
<  B ) )
33 pnfnlt 9744 . . . . . . . 8  |-  ( B  e.  RR*  ->  -. +oo  <  B )
3433adantl 275 . . . . . . 7  |-  ( ( A  = +oo  /\  B  e.  RR* )  ->  -. +oo  <  B )
3534pm2.21d 614 . . . . . 6  |-  ( ( A  = +oo  /\  B  e.  RR* )  -> 
( +oo  <  B  ->  -e B  <  -e
A ) )
3632, 35sylbid 149 . . . . 5  |-  ( ( A  = +oo  /\  B  e.  RR* )  -> 
( A  <  B  -> 
-e B  <  -e A ) )
3736expimpd 361 . . . 4  |-  ( A  = +oo  ->  (
( B  e.  RR*  /\  A  <  B )  ->  -e B  <  -e A ) )
38 breq1 3992 . . . . . 6  |-  ( A  = -oo  ->  ( A  <  B  <-> -oo  <  B
) )
3938anbi2d 461 . . . . 5  |-  ( A  = -oo  ->  (
( B  e.  RR*  /\  A  <  B )  <-> 
( B  e.  RR*  /\ -oo  <  B ) ) )
40 renegcl 8180 . . . . . . . . . . 11  |-  ( B  e.  RR  ->  -u B  e.  RR )
414, 40eqeltrd 2247 . . . . . . . . . 10  |-  ( B  e.  RR  ->  -e
B  e.  RR )
4241adantr 274 . . . . . . . . 9  |-  ( ( B  e.  RR  /\ -oo 
<  B )  ->  -e
B  e.  RR )
43 ltpnf 9737 . . . . . . . . 9  |-  (  -e B  e.  RR  -> 
-e B  < +oo )
4442, 43syl 14 . . . . . . . 8  |-  ( ( B  e.  RR  /\ -oo 
<  B )  ->  -e
B  < +oo )
4511adantr 274 . . . . . . . . 9  |-  ( ( B  = +oo  /\ -oo 
<  B )  ->  -e
B  = -oo )
46 mnfltpnf 9742 . . . . . . . . 9  |- -oo  < +oo
4745, 46eqbrtrdi 4028 . . . . . . . 8  |-  ( ( B  = +oo  /\ -oo 
<  B )  ->  -e
B  < +oo )
48 breq2 3993 . . . . . . . . . 10  |-  ( B  = -oo  ->  ( -oo  <  B  <-> -oo  < -oo ) )
49 mnfxr 7976 . . . . . . . . . . . 12  |- -oo  e.  RR*
50 nltmnf 9745 . . . . . . . . . . . 12  |-  ( -oo  e.  RR*  ->  -. -oo  < -oo )
5149, 50ax-mp 5 . . . . . . . . . . 11  |-  -. -oo  < -oo
5251pm2.21i 641 . . . . . . . . . 10  |-  ( -oo  < -oo  ->  -e B  < +oo )
5348, 52syl6bi 162 . . . . . . . . 9  |-  ( B  = -oo  ->  ( -oo  <  B  ->  -e
B  < +oo )
)
5453imp 123 . . . . . . . 8  |-  ( ( B  = -oo  /\ -oo 
<  B )  ->  -e
B  < +oo )
5544, 47, 543jaoian 1300 . . . . . . 7  |-  ( ( ( B  e.  RR  \/  B  = +oo  \/  B  = -oo )  /\ -oo  <  B
)  ->  -e B  < +oo )
562, 55sylanb 282 . . . . . 6  |-  ( ( B  e.  RR*  /\ -oo  <  B )  ->  -e
B  < +oo )
57 xnegeq 9784 . . . . . . . 8  |-  ( A  = -oo  ->  -e
A  =  -e -oo )
58 xnegmnf 9786 . . . . . . . 8  |-  -e -oo  = +oo
5957, 58eqtrdi 2219 . . . . . . 7  |-  ( A  = -oo  ->  -e
A  = +oo )
6059breq2d 4001 . . . . . 6  |-  ( A  = -oo  ->  (  -e B  <  -e
A  <->  -e B  < +oo ) )
6156, 60syl5ibr 155 . . . . 5  |-  ( A  = -oo  ->  (
( B  e.  RR*  /\ -oo  <  B )  ->  -e B  <  -e
A ) )
6239, 61sylbid 149 . . . 4  |-  ( A  = -oo  ->  (
( B  e.  RR*  /\  A  <  B )  ->  -e B  <  -e A ) )
6330, 37, 623jaoi 1298 . . 3  |-  ( ( A  e.  RR  \/  A  = +oo  \/  A  = -oo )  ->  (
( B  e.  RR*  /\  A  <  B )  ->  -e B  <  -e A ) )
641, 63sylbi 120 . 2  |-  ( A  e.  RR*  ->  ( ( B  e.  RR*  /\  A  <  B )  ->  -e
B  <  -e A ) )
65643impib 1196 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 103    \/ w3o 972    /\ w3a 973    = wceq 1348    e. wcel 2141   class class class wbr 3989   RRcr 7773   +oocpnf 7951   -oocmnf 7952   RR*cxr 7953    < clt 7954   -ucneg 8091    -ecxne 9726
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-in1 609  ax-in2 610  ax-io 704  ax-5 1440  ax-7 1441  ax-gen 1442  ax-ie1 1486  ax-ie2 1487  ax-8 1497  ax-10 1498  ax-11 1499  ax-i12 1500  ax-bndl 1502  ax-4 1503  ax-17 1519  ax-i9 1523  ax-ial 1527  ax-i5r 1528  ax-13 2143  ax-14 2144  ax-ext 2152  ax-sep 4107  ax-pow 4160  ax-pr 4194  ax-un 4418  ax-setind 4521  ax-cnex 7865  ax-resscn 7866  ax-1cn 7867  ax-1re 7868  ax-icn 7869  ax-addcl 7870  ax-addrcl 7871  ax-mulcl 7872  ax-addcom 7874  ax-addass 7876  ax-distr 7878  ax-i2m1 7879  ax-0id 7882  ax-rnegex 7883  ax-cnre 7885  ax-pre-ltadd 7890
This theorem depends on definitions:  df-bi 116  df-3or 974  df-3an 975  df-tru 1351  df-fal 1354  df-nf 1454  df-sb 1756  df-eu 2022  df-mo 2023  df-clab 2157  df-cleq 2163  df-clel 2166  df-nfc 2301  df-ne 2341  df-nel 2436  df-ral 2453  df-rex 2454  df-reu 2455  df-rab 2457  df-v 2732  df-sbc 2956  df-dif 3123  df-un 3125  df-in 3127  df-ss 3134  df-if 3527  df-pw 3568  df-sn 3589  df-pr 3590  df-op 3592  df-uni 3797  df-br 3990  df-opab 4051  df-id 4278  df-xp 4617  df-rel 4618  df-cnv 4619  df-co 4620  df-dm 4621  df-iota 5160  df-fun 5200  df-fv 5206  df-riota 5809  df-ov 5856  df-oprab 5857  df-mpo 5858  df-pnf 7956  df-mnf 7957  df-xr 7958  df-ltxr 7959  df-sub 8092  df-neg 8093  df-xneg 9729
This theorem is referenced by:  xltneg  9793
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