Theorem xltnegi 8518

Description: Forward direction of xltneg 8519. (Contributed by Mario Carneiro, 20-Aug-2015.)

Assertion

Ref	Expression
xltnegi	|- ((A e. RR* /\ B e. RR* /\ A < B) -> -eB < -eA)

Proof of Theorem xltnegi

Step	Hyp	Ref	Expression
1		elxr 8466	. . 3 |- (A e. RR* <-> (A e. RR \/ A = +oo \/ A = -oo))
2		elxr 8466	. . . . . 6 |- (B e. RR* <-> (B e. RR \/ B = +oo \/ B = -oo))
3		ltneg 7252	. . . . . . . . 9 |- ((A e. RR /\ B e. RR) -> (A < B <-> -uB < -uA))
4		rexneg 8513	. . . . . . . . . 10 |- (B e. RR -> -eB = -uB)
5		rexneg 8513	. . . . . . . . . 10 |- (A e. RR -> -eA = -uA)
6	4, 5	breqan12rd 3771	. . . . . . . . 9 |- ((A e. RR /\ B e. RR) -> ( -eB < -eA <-> -uB < -uA))
7	3, 6	bitr4d 180	. . . . . . . 8 |- ((A e. RR /\ B e. RR) -> (A < B <-> -eB < -eA))
8	7	biimpd 132	. . . . . . 7 |- ((A e. RR /\ B e. RR) -> (A < B -> -eB < -eA))
9		xnegeq 8510	. . . . . . . . . . 11 |- (B = +oo -> -eB = -e +oo)
10		xnegpnf 8511	. . . . . . . . . . 11 |- -e +oo = -oo
11	9, 10	syl6eq 2085	. . . . . . . . . 10 |- (B = +oo -> -eB = -oo)
12	11	adantl 262	. . . . . . . . 9 |- ((A e. RR /\ B = +oo) -> -eB = -oo)
13		renegcl 7068	. . . . . . . . . . . 12 |- (A e. RR -> -uA e. RR)
14	5, 13	eqeltrd 2111	. . . . . . . . . . 11 |- (A e. RR -> -eA e. RR)
15		mnflt 8474	. . . . . . . . . . 11 |- ( -eA e. RR -> -oo < -eA)
16	14, 15	syl 14	. . . . . . . . . 10 |- (A e. RR -> -oo < -eA)
17	16	adantr 261	. . . . . . . . 9 |- ((A e. RR /\ B = +oo) -> -oo < -eA)
18	12, 17	eqbrtrd 3775	. . . . . . . 8 |- ((A e. RR /\ B = +oo) -> -eB < -eA)
19	18	a1d 22	. . . . . . 7 |- ((A e. RR /\ B = +oo) -> (A < B -> -eB < -eA))
20		simpr 103	. . . . . . . . 9 |- ((A e. RR /\ B = -oo) -> B = -oo)
21	20	breq2d 3767	. . . . . . . 8 |- ((A e. RR /\ B = -oo) -> (A < B <-> A < -oo))
22		rexr 6868	. . . . . . . . . . 11 |- (A e. RR -> A e. RR*)
23		nltmnf 8479	. . . . . . . . . . 11 |- (A e. RR* -> -. A < -oo)
24	22, 23	syl 14	. . . . . . . . . 10 |- (A e. RR -> -. A < -oo)
25	24	adantr 261	. . . . . . . . 9 |- ((A e. RR /\ B = -oo) -> -. A < -oo)
26	25	pm2.21d 549	. . . . . . . 8 |- ((A e. RR /\ B = -oo) -> (A < -oo -> -eB < -eA))
27	21, 26	sylbid 139	. . . . . . 7 |- ((A e. RR /\ B = -oo) -> (A < B -> -eB < -eA))
28	8, 19, 27	3jaodan 1200	. . . . . 6 |- ((A e. RR /\ (B e. RR \/ B = +oo \/ B = -oo)) -> (A < B -> -eB < -eA))
29	2, 28	sylan2b 271	. . . . 5 |- ((A e. RR /\ B e. RR*) -> (A < B -> -eB < -eA))
30	29	expimpd 345	. . . 4 |- (A e. RR -> ((B e. RR* /\ A < B) -> -eB < -eA))
31		simpl 102	. . . . . . 7 |- ((A = +oo /\ B e. RR*) -> A = +oo)
32	31	breq1d 3765	. . . . . 6 |- ((A = +oo /\ B e. RR*) -> (A < B <-> +oo < B))
33		pnfnlt 8478	. . . . . . . 8 |- (B e. RR* -> -. +oo < B)
34	33	adantl 262	. . . . . . 7 |- ((A = +oo /\ B e. RR*) -> -. +oo < B)
35	34	pm2.21d 549	. . . . . 6 |- ((A = +oo /\ B e. RR*) -> ( +oo < B -> -eB < -eA))
36	32, 35	sylbid 139	. . . . 5 |- ((A = +oo /\ B e. RR*) -> (A < B -> -eB < -eA))
37	36	expimpd 345	. . . 4 |- (A = +oo -> ((B e. RR* /\ A < B) -> -eB < -eA))
38		breq1 3758	. . . . . 6 |- (A = -oo -> (A < B <-> -oo < B))
39	38	anbi2d 437	. . . . 5 |- (A = -oo -> ((B e. RR* /\ A < B) <-> (B e. RR* /\ -oo < B)))
40		renegcl 7068	. . . . . . . . . . 11 |- (B e. RR -> -uB e. RR)
41	4, 40	eqeltrd 2111	. . . . . . . . . 10 |- (B e. RR -> -eB e. RR)
42	41	adantr 261	. . . . . . . . 9 |- ((B e. RR /\ -oo < B) -> -eB e. RR)
43		ltpnf 8472	. . . . . . . . 9 |- ( -eB e. RR -> -eB < +oo)
44	42, 43	syl 14	. . . . . . . 8 |- ((B e. RR /\ -oo < B) -> -eB < +oo)
45	11	adantr 261	. . . . . . . . 9 |- ((B = +oo /\ -oo < B) -> -eB = -oo)
46		mnfltpnf 8476	. . . . . . . . 9 |- -oo < +oo
47	45, 46	syl6eqbr 3792	. . . . . . . 8 |- ((B = +oo /\ -oo < B) -> -eB < +oo)
48		breq2 3759	. . . . . . . . . 10 |- (B = -oo -> ( -oo < B <-> -oo < -oo))
49		mnfxr 8464	. . . . . . . . . . . 12 |- -oo e. RR*
50		nltmnf 8479	. . . . . . . . . . . 12 |- ( -oo e. RR* -> -. -oo < -oo)
51	49, 50	ax-mp 7	. . . . . . . . . . 11 |- -. -oo < -oo
52	51	pm2.21i 574	. . . . . . . . . 10 |- ( -oo < -oo -> -eB < +oo)
53	48, 52	syl6bi 152	. . . . . . . . 9 |- (B = -oo -> ( -oo < B -> -eB < +oo))
54	53	imp 115	. . . . . . . 8 |- ((B = -oo /\ -oo < B) -> -eB < +oo)
55	44, 47, 54	3jaoian 1199	. . . . . . 7 |- (((B e. RR \/ B = +oo \/ B = -oo) /\ -oo < B) -> -eB < +oo)
56	2, 55	sylanb 268	. . . . . 6 |- ((B e. RR* /\ -oo < B) -> -eB < +oo)
57		xnegeq 8510	. . . . . . . 8 |- (A = -oo -> -eA = -e -oo)
58		xnegmnf 8512	. . . . . . . 8 |- -e -oo = +oo
59	57, 58	syl6eq 2085	. . . . . . 7 |- (A = -oo -> -eA = +oo)
60	59	breq2d 3767	. . . . . 6 |- (A = -oo -> ( -eB < -eA <-> -eB < +oo))
61	56, 60	syl5ibr 145	. . . . 5 |- (A = -oo -> ((B e. RR* /\ -oo < B) -> -eB < -eA))
62	39, 61	sylbid 139	. . . 4 |- (A = -oo -> ((B e. RR* /\ A < B) -> -eB < -eA))
63	30, 37, 62	3jaoi 1197	. . 3 |- ((A e. RR \/ A = +oo \/ A = -oo) -> ((B e. RR* /\ A < B) -> -eB < -eA))
64	1, 63	sylbi 114	. 2 |- (A e. RR* -> ((B e. RR* /\ A < B) -> -eB < -eA))
65	64	3impib 1101	1 |- ((A e. RR* /\ B e. RR* /\ A < B) -> -eB < -eA)

Syntax hints:  -. wn 3   -> wi 4   /\ wa 97   \/ w3o 883   /\ w3a 884   = wceq 1242   e. wcel 1390   class class class wbr 3755   RRcr 6710   +oocpnf 6854   -oocmnf 6855   RR*cxr 6856   < clt 6857   -ucneg 6980   -ecxne 8456

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 629  ax-5 1333  ax-7 1334  ax-gen 1335  ax-ie1 1379  ax-ie2 1380  ax-8 1392  ax-10 1393  ax-11 1394  ax-i12 1395  ax-bndl 1396  ax-4 1397  ax-13 1401  ax-14 1402  ax-17 1416  ax-i9 1420  ax-ial 1424  ax-i5r 1425  ax-ext 2019  ax-sep 3866  ax-pow 3918  ax-pr 3935  ax-un 4136  ax-setind 4220  ax-cnex 6774  ax-resscn 6775  ax-1cn 6776  ax-1re 6777  ax-icn 6778  ax-addcl 6779  ax-addrcl 6780  ax-mulcl 6781  ax-addcom 6783  ax-addass 6785  ax-distr 6787  ax-i2m1 6788  ax-0id 6791  ax-rnegex 6792  ax-cnre 6794  ax-pre-ltadd 6799

This theorem depends on definitions:  df-bi 110  df-3or 885  df-3an 886  df-tru 1245  df-fal 1248  df-nf 1347  df-sb 1643  df-eu 1900  df-mo 1901  df-clab 2024  df-cleq 2030  df-clel 2033  df-nfc 2164  df-ne 2203  df-nel 2204  df-ral 2305  df-rex 2306  df-reu 2307  df-rab 2309  df-v 2553  df-sbc 2759  df-dif 2914  df-un 2916  df-in 2918  df-ss 2925  df-if 3326  df-pw 3353  df-sn 3373  df-pr 3374  df-op 3376  df-uni 3572  df-br 3756  df-opab 3810  df-id 4021  df-xp 4294  df-rel 4295  df-cnv 4296  df-co 4297  df-dm 4298  df-iota 4810  df-fun 4847  df-fv 4853  df-riota 5411  df-ov 5458  df-oprab 5459  df-mpt2 5460  df-pnf 6859  df-mnf 6860  df-xr 6861  df-ltxr 6862  df-sub 6981  df-neg 6982  df-xneg 8459

This theorem is referenced by:  xltneg  8519