Theorem xltnegi 8902

Description: Forward direction of xltneg 8903. (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 8850	. . 3 |- ( A e. RR* <-> ( A e. RR \/ A = +oo \/ A = -oo ) )
2		elxr 8850	. . . . . 6 |- ( B e. RR* <-> ( B e. RR \/ B = +oo \/ B = -oo ) )
3		ltneg 7566	. . . . . . . . 9 |- ( ( A e. RR /\ B e. RR ) -> ( A < B <-> -u B < -u A ) )
4		rexneg 8897	. . . . . . . . . 10 |- ( B e. RR -> -e B = -u B )
5		rexneg 8897	. . . . . . . . . 10 |- ( A e. RR -> -e A = -u A )
6	4, 5	breqan12rd 3801	. . . . . . . . 9 |- ( ( A e. RR /\ B e. RR ) -> ( -e B < -e A <-> -u B < -u A ) )
7	3, 6	bitr4d 189	. . . . . . . 8 |- ( ( A e. RR /\ B e. RR ) -> ( A < B <-> -e B < -e A ) )
8	7	biimpd 142	. . . . . . 7 |- ( ( A e. RR /\ B e. RR ) -> ( A < B -> -e B < -e A ) )
9		xnegeq 8894	. . . . . . . . . . 11 |- ( B = +oo -> -e B = -e +oo )
10		xnegpnf 8895	. . . . . . . . . . 11 |- -e +oo = -oo
11	9, 10	syl6eq 2129	. . . . . . . . . 10 |- ( B = +oo -> -e B = -oo )
12	11	adantl 271	. . . . . . . . 9 |- ( ( A e. RR /\ B = +oo ) -> -e B = -oo )
13		renegcl 7369	. . . . . . . . . . . 12 |- ( A e. RR -> -u A e. RR )
14	5, 13	eqeltrd 2155	. . . . . . . . . . 11 |- ( A e. RR -> -e A e. RR )
15		mnflt 8858	. . . . . . . . . . 11 |- ( -e A e. RR -> -oo < -e A )
16	14, 15	syl 14	. . . . . . . . . 10 |- ( A e. RR -> -oo < -e A )
17	16	adantr 270	. . . . . . . . 9 |- ( ( A e. RR /\ B = +oo ) -> -oo < -e A )
18	12, 17	eqbrtrd 3805	. . . . . . . 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 108	. . . . . . . . 9 |- ( ( A e. RR /\ B = -oo ) -> B = -oo )
21	20	breq2d 3797	. . . . . . . 8 |- ( ( A e. RR /\ B = -oo ) -> ( A < B <-> A < -oo ) )
22		rexr 7164	. . . . . . . . . . 11 |- ( A e. RR -> A e. RR* )
23		nltmnf 8863	. . . . . . . . . . 11 |- ( A e. RR* -> -. A < -oo )
24	22, 23	syl 14	. . . . . . . . . 10 |- ( A e. RR -> -. A < -oo )
25	24	adantr 270	. . . . . . . . 9 |- ( ( A e. RR /\ B = -oo ) -> -. A < -oo )
26	25	pm2.21d 581	. . . . . . . 8 |- ( ( A e. RR /\ B = -oo ) -> ( A < -oo -> -e B < -e A ) )
27	21, 26	sylbid 148	. . . . . . 7 |- ( ( A e. RR /\ B = -oo ) -> ( A < B -> -e B < -e A ) )
28	8, 19, 27	3jaodan 1237	. . . . . 6 |- ( ( A e. RR /\ ( B e. RR \/ B = +oo \/ B = -oo ) ) -> ( A < B -> -e B < -e A ) )
29	2, 28	sylan2b 281	. . . . 5 |- ( ( A e. RR /\ B e. RR* ) -> ( A < B -> -e B < -e A ) )
30	29	expimpd 355	. . . 4 |- ( A e. RR -> ( ( B e. RR* /\ A < B ) -> -e B < -e A ) )
31		simpl 107	. . . . . . 7 |- ( ( A = +oo /\ B e. RR* ) -> A = +oo )
32	31	breq1d 3795	. . . . . 6 |- ( ( A = +oo /\ B e. RR* ) -> ( A < B <-> +oo < B ) )
33		pnfnlt 8862	. . . . . . . 8 |- ( B e. RR* -> -. +oo < B )
34	33	adantl 271	. . . . . . 7 |- ( ( A = +oo /\ B e. RR* ) -> -. +oo < B )
35	34	pm2.21d 581	. . . . . 6 |- ( ( A = +oo /\ B e. RR* ) -> ( +oo < B -> -e B < -e A ) )
36	32, 35	sylbid 148	. . . . 5 |- ( ( A = +oo /\ B e. RR* ) -> ( A < B -> -e B < -e A ) )
37	36	expimpd 355	. . . 4 |- ( A = +oo -> ( ( B e. RR* /\ A < B ) -> -e B < -e A ) )
38		breq1 3788	. . . . . 6 |- ( A = -oo -> ( A < B <-> -oo < B ) )
39	38	anbi2d 451	. . . . 5 |- ( A = -oo -> ( ( B e. RR* /\ A < B ) <-> ( B e. RR* /\ -oo < B ) ) )
40		renegcl 7369	. . . . . . . . . . 11 |- ( B e. RR -> -u B e. RR )
41	4, 40	eqeltrd 2155	. . . . . . . . . 10 |- ( B e. RR -> -e B e. RR )
42	41	adantr 270	. . . . . . . . 9 |- ( ( B e. RR /\ -oo < B ) -> -e B e. RR )
43		ltpnf 8856	. . . . . . . . 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 270	. . . . . . . . 9 |- ( ( B = +oo /\ -oo < B ) -> -e B = -oo )
46		mnfltpnf 8860	. . . . . . . . 9 |- -oo < +oo
47	45, 46	syl6eqbr 3822	. . . . . . . 8 |- ( ( B = +oo /\ -oo < B ) -> -e B < +oo )
48		breq2 3789	. . . . . . . . . 10 |- ( B = -oo -> ( -oo < B <-> -oo < -oo ) )
49		mnfxr 8848	. . . . . . . . . . . 12 |- -oo e. RR*
50		nltmnf 8863	. . . . . . . . . . . 12 |- ( -oo e. RR* -> -. -oo < -oo )
51	49, 50	ax-mp 7	. . . . . . . . . . 11 |- -. -oo < -oo
52	51	pm2.21i 607	. . . . . . . . . 10 |- ( -oo < -oo -> -e B < +oo )
53	48, 52	syl6bi 161	. . . . . . . . 9 |- ( B = -oo -> ( -oo < B -> -e B < +oo ) )
54	53	imp 122	. . . . . . . 8 |- ( ( B = -oo /\ -oo < B ) -> -e B < +oo )
55	44, 47, 54	3jaoian 1236	. . . . . . 7 |- ( ( ( B e. RR \/ B = +oo \/ B = -oo ) /\ -oo < B ) -> -e B < +oo )
56	2, 55	sylanb 278	. . . . . 6 |- ( ( B e. RR* /\ -oo < B ) -> -e B < +oo )
57		xnegeq 8894	. . . . . . . 8 |- ( A = -oo -> -e A = -e -oo )
58		xnegmnf 8896	. . . . . . . 8 |- -e -oo = +oo
59	57, 58	syl6eq 2129	. . . . . . 7 |- ( A = -oo -> -e A = +oo )
60	59	breq2d 3797	. . . . . 6 |- ( A = -oo -> ( -e B < -e A <-> -e B < +oo ) )
61	56, 60	syl5ibr 154	. . . . 5 |- ( A = -oo -> ( ( B e. RR* /\ -oo < B ) -> -e B < -e A ) )
62	39, 61	sylbid 148	. . . 4 |- ( A = -oo -> ( ( B e. RR* /\ A < B ) -> -e B < -e A ) )
63	30, 37, 62	3jaoi 1234	. . 3 |- ( ( A e. RR \/ A = +oo \/ A = -oo ) -> ( ( B e. RR* /\ A < B ) -> -e B < -e A ) )
64	1, 63	sylbi 119	. 2 |- ( A e. RR* -> ( ( B e. RR* /\ A < B ) -> -e B < -e A ) )
65	64	3impib 1136	1 |- ( ( A e. RR* /\ B e. RR* /\ A < B ) -> -e B < -e A )

Syntax hints:   -. wn 3   -> wi 4   /\ wa 102   \/ w3o 918   /\ w3a 919   = wceq 1284   e. wcel 1433   class class class wbr 3785   RRcr 6980   +oocpnf 7150   -oocmnf 7151   RR*cxr 7152   < clt 7153   -ucneg 7280   -ecxne 8840

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-in1 576  ax-in2 577  ax-io 662  ax-5 1376  ax-7 1377  ax-gen 1378  ax-ie1 1422  ax-ie2 1423  ax-8 1435  ax-10 1436  ax-11 1437  ax-i12 1438  ax-bndl 1439  ax-4 1440  ax-13 1444  ax-14 1445  ax-17 1459  ax-i9 1463  ax-ial 1467  ax-i5r 1468  ax-ext 2063  ax-sep 3896  ax-pow 3948  ax-pr 3964  ax-un 4188  ax-setind 4280  ax-cnex 7067  ax-resscn 7068  ax-1cn 7069  ax-1re 7070  ax-icn 7071  ax-addcl 7072  ax-addrcl 7073  ax-mulcl 7074  ax-addcom 7076  ax-addass 7078  ax-distr 7080  ax-i2m1 7081  ax-0id 7084  ax-rnegex 7085  ax-cnre 7087  ax-pre-ltadd 7092

This theorem depends on definitions:  df-bi 115  df-3or 920  df-3an 921  df-tru 1287  df-fal 1290  df-nf 1390  df-sb 1686  df-eu 1944  df-mo 1945  df-clab 2068  df-cleq 2074  df-clel 2077  df-nfc 2208  df-ne 2246  df-nel 2340  df-ral 2353  df-rex 2354  df-reu 2355  df-rab 2357  df-v 2603  df-sbc 2816  df-dif 2975  df-un 2977  df-in 2979  df-ss 2986  df-if 3352  df-pw 3384  df-sn 3404  df-pr 3405  df-op 3407  df-uni 3602  df-br 3786  df-opab 3840  df-id 4048  df-xp 4369  df-rel 4370  df-cnv 4371  df-co 4372  df-dm 4373  df-iota 4887  df-fun 4924  df-fv 4930  df-riota 5488  df-ov 5535  df-oprab 5536  df-mpt2 5537  df-pnf 7155  df-mnf 7156  df-xr 7157  df-ltxr 7158  df-sub 7281  df-neg 7282  df-xneg 8843

This theorem is referenced by:  xltneg  8903