Theorem xltnegi 9910

Description: Forward direction of xltneg 9911. (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 9851	. . 3 |- ( A e. RR* <-> ( A e. RR \/ A = +oo \/ A = -oo ) )
2		elxr 9851	. . . . . 6 |- ( B e. RR* <-> ( B e. RR \/ B = +oo \/ B = -oo ) )
3		ltneg 8489	. . . . . . . . 9 |- ( ( A e. RR /\ B e. RR ) -> ( A < B <-> -u B < -u A ) )
4		rexneg 9905	. . . . . . . . . 10 |- ( B e. RR -> -e B = -u B )
5		rexneg 9905	. . . . . . . . . 10 |- ( A e. RR -> -e A = -u A )
6	4, 5	breqan12rd 4050	. . . . . . . . 9 |- ( ( A e. RR /\ B e. RR ) -> ( -e B < -e A <-> -u B < -u A ) )
7	3, 6	bitr4d 191	. . . . . . . 8 |- ( ( A e. RR /\ B e. RR ) -> ( A < B <-> -e B < -e A ) )
8	7	biimpd 144	. . . . . . 7 |- ( ( A e. RR /\ B e. RR ) -> ( A < B -> -e B < -e A ) )
9		xnegeq 9902	. . . . . . . . . . 11 |- ( B = +oo -> -e B = -e +oo )
10		xnegpnf 9903	. . . . . . . . . . 11 |- -e +oo = -oo
11	9, 10	eqtrdi 2245	. . . . . . . . . 10 |- ( B = +oo -> -e B = -oo )
12	11	adantl 277	. . . . . . . . 9 |- ( ( A e. RR /\ B = +oo ) -> -e B = -oo )
13		renegcl 8287	. . . . . . . . . . . 12 |- ( A e. RR -> -u A e. RR )
14	5, 13	eqeltrd 2273	. . . . . . . . . . 11 |- ( A e. RR -> -e A e. RR )
15		mnflt 9858	. . . . . . . . . . 11 |- ( -e A e. RR -> -oo < -e A )
16	14, 15	syl 14	. . . . . . . . . 10 |- ( A e. RR -> -oo < -e A )
17	16	adantr 276	. . . . . . . . 9 |- ( ( A e. RR /\ B = +oo ) -> -oo < -e A )
18	12, 17	eqbrtrd 4055	. . . . . . . 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 110	. . . . . . . . 9 |- ( ( A e. RR /\ B = -oo ) -> B = -oo )
21	20	breq2d 4045	. . . . . . . 8 |- ( ( A e. RR /\ B = -oo ) -> ( A < B <-> A < -oo ) )
22		rexr 8072	. . . . . . . . . . 11 |- ( A e. RR -> A e. RR* )
23		nltmnf 9863	. . . . . . . . . . 11 |- ( A e. RR* -> -. A < -oo )
24	22, 23	syl 14	. . . . . . . . . 10 |- ( A e. RR -> -. A < -oo )
25	24	adantr 276	. . . . . . . . 9 |- ( ( A e. RR /\ B = -oo ) -> -. A < -oo )
26	25	pm2.21d 620	. . . . . . . 8 |- ( ( A e. RR /\ B = -oo ) -> ( A < -oo -> -e B < -e A ) )
27	21, 26	sylbid 150	. . . . . . 7 |- ( ( A e. RR /\ B = -oo ) -> ( A < B -> -e B < -e A ) )
28	8, 19, 27	3jaodan 1317	. . . . . 6 |- ( ( A e. RR /\ ( B e. RR \/ B = +oo \/ B = -oo ) ) -> ( A < B -> -e B < -e A ) )
29	2, 28	sylan2b 287	. . . . 5 |- ( ( A e. RR /\ B e. RR* ) -> ( A < B -> -e B < -e A ) )
30	29	expimpd 363	. . . 4 |- ( A e. RR -> ( ( B e. RR* /\ A < B ) -> -e B < -e A ) )
31		simpl 109	. . . . . . 7 |- ( ( A = +oo /\ B e. RR* ) -> A = +oo )
32	31	breq1d 4043	. . . . . 6 |- ( ( A = +oo /\ B e. RR* ) -> ( A < B <-> +oo < B ) )
33		pnfnlt 9862	. . . . . . . 8 |- ( B e. RR* -> -. +oo < B )
34	33	adantl 277	. . . . . . 7 |- ( ( A = +oo /\ B e. RR* ) -> -. +oo < B )
35	34	pm2.21d 620	. . . . . 6 |- ( ( A = +oo /\ B e. RR* ) -> ( +oo < B -> -e B < -e A ) )
36	32, 35	sylbid 150	. . . . 5 |- ( ( A = +oo /\ B e. RR* ) -> ( A < B -> -e B < -e A ) )
37	36	expimpd 363	. . . 4 |- ( A = +oo -> ( ( B e. RR* /\ A < B ) -> -e B < -e A ) )
38		breq1 4036	. . . . . 6 |- ( A = -oo -> ( A < B <-> -oo < B ) )
39	38	anbi2d 464	. . . . 5 |- ( A = -oo -> ( ( B e. RR* /\ A < B ) <-> ( B e. RR* /\ -oo < B ) ) )
40		renegcl 8287	. . . . . . . . . . 11 |- ( B e. RR -> -u B e. RR )
41	4, 40	eqeltrd 2273	. . . . . . . . . 10 |- ( B e. RR -> -e B e. RR )
42	41	adantr 276	. . . . . . . . 9 |- ( ( B e. RR /\ -oo < B ) -> -e B e. RR )
43		ltpnf 9855	. . . . . . . . 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 276	. . . . . . . . 9 |- ( ( B = +oo /\ -oo < B ) -> -e B = -oo )
46		mnfltpnf 9860	. . . . . . . . 9 |- -oo < +oo
47	45, 46	eqbrtrdi 4072	. . . . . . . 8 |- ( ( B = +oo /\ -oo < B ) -> -e B < +oo )
48		breq2 4037	. . . . . . . . . 10 |- ( B = -oo -> ( -oo < B <-> -oo < -oo ) )
49		mnfxr 8083	. . . . . . . . . . . 12 |- -oo e. RR*
50		nltmnf 9863	. . . . . . . . . . . 12 |- ( -oo e. RR* -> -. -oo < -oo )
51	49, 50	ax-mp 5	. . . . . . . . . . 11 |- -. -oo < -oo
52	51	pm2.21i 647	. . . . . . . . . 10 |- ( -oo < -oo -> -e B < +oo )
53	48, 52	biimtrdi 163	. . . . . . . . 9 |- ( B = -oo -> ( -oo < B -> -e B < +oo ) )
54	53	imp 124	. . . . . . . 8 |- ( ( B = -oo /\ -oo < B ) -> -e B < +oo )
55	44, 47, 54	3jaoian 1316	. . . . . . 7 |- ( ( ( B e. RR \/ B = +oo \/ B = -oo ) /\ -oo < B ) -> -e B < +oo )
56	2, 55	sylanb 284	. . . . . 6 |- ( ( B e. RR* /\ -oo < B ) -> -e B < +oo )
57		xnegeq 9902	. . . . . . . 8 |- ( A = -oo -> -e A = -e -oo )
58		xnegmnf 9904	. . . . . . . 8 |- -e -oo = +oo
59	57, 58	eqtrdi 2245	. . . . . . 7 |- ( A = -oo -> -e A = +oo )
60	59	breq2d 4045	. . . . . 6 |- ( A = -oo -> ( -e B < -e A <-> -e B < +oo ) )
61	56, 60	imbitrrid 156	. . . . 5 |- ( A = -oo -> ( ( B e. RR* /\ -oo < B ) -> -e B < -e A ) )
62	39, 61	sylbid 150	. . . 4 |- ( A = -oo -> ( ( B e. RR* /\ A < B ) -> -e B < -e A ) )
63	30, 37, 62	3jaoi 1314	. . 3 |- ( ( A e. RR \/ A = +oo \/ A = -oo ) -> ( ( B e. RR* /\ A < B ) -> -e B < -e A ) )
64	1, 63	sylbi 121	. 2 |- ( A e. RR* -> ( ( B e. RR* /\ A < B ) -> -e B < -e A ) )
65	64	3impib 1203	1 |- ( ( A e. RR* /\ B e. RR* /\ A < B ) -> -e B < -e A )

Syntax hints:   -. wn 3   -> wi 4   /\ wa 104   \/ w3o 979   /\ w3a 980   = wceq 1364   e. wcel 2167   class class class wbr 4033   RRcr 7878   +oocpnf 8058   -oocmnf 8059   RR*cxr 8060   < clt 8061   -ucneg 8198   -ecxne 9844

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-in1 615  ax-in2 616  ax-io 710  ax-5 1461  ax-7 1462  ax-gen 1463  ax-ie1 1507  ax-ie2 1508  ax-8 1518  ax-10 1519  ax-11 1520  ax-i12 1521  ax-bndl 1523  ax-4 1524  ax-17 1540  ax-i9 1544  ax-ial 1548  ax-i5r 1549  ax-13 2169  ax-14 2170  ax-ext 2178  ax-sep 4151  ax-pow 4207  ax-pr 4242  ax-un 4468  ax-setind 4573  ax-cnex 7970  ax-resscn 7971  ax-1cn 7972  ax-1re 7973  ax-icn 7974  ax-addcl 7975  ax-addrcl 7976  ax-mulcl 7977  ax-addcom 7979  ax-addass 7981  ax-distr 7983  ax-i2m1 7984  ax-0id 7987  ax-rnegex 7988  ax-cnre 7990  ax-pre-ltadd 7995

This theorem depends on definitions:  df-bi 117  df-3or 981  df-3an 982  df-tru 1367  df-fal 1370  df-nf 1475  df-sb 1777  df-eu 2048  df-mo 2049  df-clab 2183  df-cleq 2189  df-clel 2192  df-nfc 2328  df-ne 2368  df-nel 2463  df-ral 2480  df-rex 2481  df-reu 2482  df-rab 2484  df-v 2765  df-sbc 2990  df-dif 3159  df-un 3161  df-in 3163  df-ss 3170  df-if 3562  df-pw 3607  df-sn 3628  df-pr 3629  df-op 3631  df-uni 3840  df-br 4034  df-opab 4095  df-id 4328  df-xp 4669  df-rel 4670  df-cnv 4671  df-co 4672  df-dm 4673  df-iota 5219  df-fun 5260  df-fv 5266  df-riota 5877  df-ov 5925  df-oprab 5926  df-mpo 5927  df-pnf 8063  df-mnf 8064  df-xr 8065  df-ltxr 8066  df-sub 8199  df-neg 8200  df-xneg 9847

This theorem is referenced by:  xltneg  9911