Theorem xltnegi 9618

Description: Forward direction of xltneg 9619. (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

Step	Hyp	Ref	Expression
1		elxr 9563	. . 3 |- ( A e. RR* <-> ( A e. RR \/ A = +oo \/ A = -oo ) )
2		elxr 9563	. . . . . 6 |- ( B e. RR* <-> ( B e. RR \/ B = +oo \/ B = -oo ) )
3		ltneg 8224	. . . . . . . . 9 |- ( ( A e. RR /\ B e. RR ) -> ( A < B <-> -u B < -u A ) )
4		rexneg 9613	. . . . . . . . . 10 |- ( B e. RR -> -e B = -u B )
5		rexneg 9613	. . . . . . . . . 10 |- ( A e. RR -> -e A = -u A )
6	4, 5	breqan12rd 3946	. . . . . . . . 9 |- ( ( A e. RR /\ B e. RR ) -> ( -e B < -e A <-> -u B < -u A ) )
7	3, 6	bitr4d 190	. . . . . . . 8 |- ( ( A e. RR /\ B e. RR ) -> ( A < B <-> -e B < -e A ) )
8	7	biimpd 143	. . . . . . 7 |- ( ( A e. RR /\ B e. RR ) -> ( A < B -> -e B < -e A ) )
9		xnegeq 9610	. . . . . . . . . . 11 |- ( B = +oo -> -e B = -e +oo )
10		xnegpnf 9611	. . . . . . . . . . 11 |- -e +oo = -oo
11	9, 10	syl6eq 2188	. . . . . . . . . 10 |- ( B = +oo -> -e B = -oo )
12	11	adantl 275	. . . . . . . . 9 |- ( ( A e. RR /\ B = +oo ) -> -e B = -oo )
13		renegcl 8023	. . . . . . . . . . . 12 |- ( A e. RR -> -u A e. RR )
14	5, 13	eqeltrd 2216	. . . . . . . . . . 11 |- ( A e. RR -> -e A e. RR )
15		mnflt 9569	. . . . . . . . . . 11 |- ( -e A e. RR -> -oo < -e A )
16	14, 15	syl 14	. . . . . . . . . 10 |- ( A e. RR -> -oo < -e A )
17	16	adantr 274	. . . . . . . . 9 |- ( ( A e. RR /\ B = +oo ) -> -oo < -e A )
18	12, 17	eqbrtrd 3950	. . . . . . . 8 |- ( ( A e. RR /\ B = +oo ) -> -e B < -e A )
19	18	a1d 22	. . . . . . 7 |- ( ( A e. RR /\ B = +oo ) -> ( A < B -> -e B < -e A ) )
20		simpr 109	. . . . . . . . 9 |- ( ( A e. RR /\ B = -oo ) -> B = -oo )
21	20	breq2d 3941	. . . . . . . 8 |- ( ( A e. RR /\ B = -oo ) -> ( A < B <-> A < -oo ) )
22		rexr 7811	. . . . . . . . . . 11 |- ( A e. RR -> A e. RR* )
23		nltmnf 9574	. . . . . . . . . . 11 |- ( A e. RR* -> -. A < -oo )
24	22, 23	syl 14	. . . . . . . . . 10 |- ( A e. RR -> -. A < -oo )
25	24	adantr 274	. . . . . . . . 9 |- ( ( A e. RR /\ B = -oo ) -> -. A < -oo )
26	25	pm2.21d 608	. . . . . . . 8 |- ( ( A e. RR /\ B = -oo ) -> ( A < -oo -> -e B < -e A ) )
27	21, 26	sylbid 149	. . . . . . 7 |- ( ( A e. RR /\ B = -oo ) -> ( A < B -> -e B < -e A ) )
28	8, 19, 27	3jaodan 1284	. . . . . 6 |- ( ( A e. RR /\ ( B e. RR \/ B = +oo \/ B = -oo ) ) -> ( A < B -> -e B < -e A ) )
29	2, 28	sylan2b 285	. . . . 5 |- ( ( A e. RR /\ B e. RR* ) -> ( A < B -> -e B < -e A ) )
30	29	expimpd 360	. . . 4 |- ( A e. RR -> ( ( B e. RR* /\ A < B ) -> -e B < -e A ) )
31		simpl 108	. . . . . . 7 |- ( ( A = +oo /\ B e. RR* ) -> A = +oo )
32	31	breq1d 3939	. . . . . 6 |- ( ( A = +oo /\ B e. RR* ) -> ( A < B <-> +oo < B ) )
33		pnfnlt 9573	. . . . . . . 8 |- ( B e. RR* -> -. +oo < B )
34	33	adantl 275	. . . . . . 7 |- ( ( A = +oo /\ B e. RR* ) -> -. +oo < B )
35	34	pm2.21d 608	. . . . . 6 |- ( ( A = +oo /\ B e. RR* ) -> ( +oo < B -> -e B < -e A ) )
36	32, 35	sylbid 149	. . . . 5 |- ( ( A = +oo /\ B e. RR* ) -> ( A < B -> -e B < -e A ) )
37	36	expimpd 360	. . . 4 |- ( A = +oo -> ( ( B e. RR* /\ A < B ) -> -e B < -e A ) )
38		breq1 3932	. . . . . 6 |- ( A = -oo -> ( A < B <-> -oo < B ) )
39	38	anbi2d 459	. . . . 5 |- ( A = -oo -> ( ( B e. RR* /\ A < B ) <-> ( B e. RR* /\ -oo < B ) ) )
40		renegcl 8023	. . . . . . . . . . 11 |- ( B e. RR -> -u B e. RR )
41	4, 40	eqeltrd 2216	. . . . . . . . . 10 |- ( B e. RR -> -e B e. RR )
42	41	adantr 274	. . . . . . . . 9 |- ( ( B e. RR /\ -oo < B ) -> -e B e. RR )
43		ltpnf 9567	. . . . . . . . 9 |- ( -e B e. RR -> -e B < +oo )
44	42, 43	syl 14	. . . . . . . 8 |- ( ( B e. RR /\ -oo < B ) -> -e B < +oo )
45	11	adantr 274	. . . . . . . . 9 |- ( ( B = +oo /\ -oo < B ) -> -e B = -oo )
46		mnfltpnf 9571	. . . . . . . . 9 |- -oo < +oo
47	45, 46	eqbrtrdi 3967	. . . . . . . 8 |- ( ( B = +oo /\ -oo < B ) -> -e B < +oo )
48		breq2 3933	. . . . . . . . . 10 |- ( B = -oo -> ( -oo < B <-> -oo < -oo ) )
49		mnfxr 7822	. . . . . . . . . . . 12 |- -oo e. RR*
50		nltmnf 9574	. . . . . . . . . . . 12 |- ( -oo e. RR* -> -. -oo < -oo )
51	49, 50	ax-mp 5	. . . . . . . . . . 11 |- -. -oo < -oo
52	51	pm2.21i 635	. . . . . . . . . 10 |- ( -oo < -oo -> -e B < +oo )
53	48, 52	syl6bi 162	. . . . . . . . 9 |- ( B = -oo -> ( -oo < B -> -e B < +oo ) )
54	53	imp 123	. . . . . . . 8 |- ( ( B = -oo /\ -oo < B ) -> -e B < +oo )
55	44, 47, 54	3jaoian 1283	. . . . . . 7 |- ( ( ( B e. RR \/ B = +oo \/ B = -oo ) /\ -oo < B ) -> -e B < +oo )
56	2, 55	sylanb 282	. . . . . 6 |- ( ( B e. RR* /\ -oo < B ) -> -e B < +oo )
57		xnegeq 9610	. . . . . . . 8 |- ( A = -oo -> -e A = -e -oo )
58		xnegmnf 9612	. . . . . . . 8 |- -e -oo = +oo
59	57, 58	syl6eq 2188	. . . . . . 7 |- ( A = -oo -> -e A = +oo )
60	59	breq2d 3941	. . . . . 6 |- ( A = -oo -> ( -e B < -e A <-> -e B < +oo ) )
61	56, 60	syl5ibr 155	. . . . 5 |- ( A = -oo -> ( ( B e. RR* /\ -oo < B ) -> -e B < -e A ) )
62	39, 61	sylbid 149	. . . 4 |- ( A = -oo -> ( ( B e. RR* /\ A < B ) -> -e B < -e A ) )
63	30, 37, 62	3jaoi 1281	. . 3 |- ( ( A e. RR \/ A = +oo \/ A = -oo ) -> ( ( B e. RR* /\ A < B ) -> -e B < -e A ) )
64	1, 63	sylbi 120	. 2 |- ( A e. RR* -> ( ( B e. RR* /\ A < B ) -> -e B < -e A ) )
65	64	3impib 1179	1 |- ( ( A e. RR* /\ B e. RR* /\ A < B ) -> -e B < -e A )

Syntax hints:   -. wn 3   -> wi 4   /\ wa 103   \/ w3o 961   /\ w3a 962   = wceq 1331   e. wcel 1480   class class class wbr 3929   RRcr 7619   +oocpnf 7797   -oocmnf 7798   RR*cxr 7799   < clt 7800   -ucneg 7934   -ecxne 9556

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 603  ax-in2 604  ax-io 698  ax-5 1423  ax-7 1424  ax-gen 1425  ax-ie1 1469  ax-ie2 1470  ax-8 1482  ax-10 1483  ax-11 1484  ax-i12 1485  ax-bndl 1486  ax-4 1487  ax-13 1491  ax-14 1492  ax-17 1506  ax-i9 1510  ax-ial 1514  ax-i5r 1515  ax-ext 2121  ax-sep 4046  ax-pow 4098  ax-pr 4131  ax-un 4355  ax-setind 4452  ax-cnex 7711  ax-resscn 7712  ax-1cn 7713  ax-1re 7714  ax-icn 7715  ax-addcl 7716  ax-addrcl 7717  ax-mulcl 7718  ax-addcom 7720  ax-addass 7722  ax-distr 7724  ax-i2m1 7725  ax-0id 7728  ax-rnegex 7729  ax-cnre 7731  ax-pre-ltadd 7736

This theorem depends on definitions:  df-bi 116  df-3or 963  df-3an 964  df-tru 1334  df-fal 1337  df-nf 1437  df-sb 1736  df-eu 2002  df-mo 2003  df-clab 2126  df-cleq 2132  df-clel 2135  df-nfc 2270  df-ne 2309  df-nel 2404  df-ral 2421  df-rex 2422  df-reu 2423  df-rab 2425  df-v 2688  df-sbc 2910  df-dif 3073  df-un 3075  df-in 3077  df-ss 3084  df-if 3475  df-pw 3512  df-sn 3533  df-pr 3534  df-op 3536  df-uni 3737  df-br 3930  df-opab 3990  df-id 4215  df-xp 4545  df-rel 4546  df-cnv 4547  df-co 4548  df-dm 4549  df-iota 5088  df-fun 5125  df-fv 5131  df-riota 5730  df-ov 5777  df-oprab 5778  df-mpo 5779  df-pnf 7802  df-mnf 7803  df-xr 7804  df-ltxr 7805  df-sub 7935  df-neg 7936  df-xneg 9559

This theorem is referenced by:  xltneg  9619