Theorem xltnegi 8748

Description: Forward direction of xltneg 8749. (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 8696	. . 3 |- ( A e. RR* <-> ( A e. RR \/ A = +oo \/ A = -oo ) )
2		elxr 8696	. . . . . 6 |- ( B e. RR* <-> ( B e. RR \/ B = +oo \/ B = -oo ) )
3		ltneg 7457	. . . . . . . . 9 |- ( ( A e. RR /\ B e. RR ) -> ( A < B <-> -u B < -u A ) )
4		rexneg 8743	. . . . . . . . . 10 |- ( B e. RR -> -e B = -u B )
5		rexneg 8743	. . . . . . . . . 10 |- ( A e. RR -> -e A = -u A )
6	4, 5	breqan12rd 3780	. . . . . . . . 9 |- ( ( A e. RR /\ B e. RR ) -> ( -e B < -e A <-> -u B < -u A ) )
7	3, 6	bitr4d 180	. . . . . . . 8 |- ( ( A e. RR /\ B e. RR ) -> ( A < B <-> -e B < -e A ) )
8	7	biimpd 132	. . . . . . 7 |- ( ( A e. RR /\ B e. RR ) -> ( A < B -> -e B < -e A ) )
9		xnegeq 8740	. . . . . . . . . . 11 |- ( B = +oo -> -e B = -e +oo )
10		xnegpnf 8741	. . . . . . . . . . 11 |- -e +oo = -oo
11	9, 10	syl6eq 2088	. . . . . . . . . 10 |- ( B = +oo -> -e B = -oo )
12	11	adantl 262	. . . . . . . . 9 |- ( ( A e. RR /\ B = +oo ) -> -e B = -oo )
13		renegcl 7272	. . . . . . . . . . . 12 |- ( A e. RR -> -u A e. RR )
14	5, 13	eqeltrd 2114	. . . . . . . . . . 11 |- ( A e. RR -> -e A e. RR )
15		mnflt 8704	. . . . . . . . . . 11 |- ( -e A e. RR -> -oo < -e A )
16	14, 15	syl 14	. . . . . . . . . 10 |- ( A e. RR -> -oo < -e A )
17	16	adantr 261	. . . . . . . . 9 |- ( ( A e. RR /\ B = +oo ) -> -oo < -e A )
18	12, 17	eqbrtrd 3784	. . . . . . . 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 103	. . . . . . . . 9 |- ( ( A e. RR /\ B = -oo ) -> B = -oo )
21	20	breq2d 3776	. . . . . . . 8 |- ( ( A e. RR /\ B = -oo ) -> ( A < B <-> A < -oo ) )
22		rexr 7071	. . . . . . . . . . 11 |- ( A e. RR -> A e. RR* )
23		nltmnf 8709	. . . . . . . . . . 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 -> -e B < -e A ) )
27	21, 26	sylbid 139	. . . . . . 7 |- ( ( A e. RR /\ B = -oo ) -> ( A < B -> -e B < -e A ) )
28	8, 19, 27	3jaodan 1201	. . . . . 6 |- ( ( A e. RR /\ ( B e. RR \/ B = +oo \/ B = -oo ) ) -> ( A < B -> -e B < -e A ) )
29	2, 28	sylan2b 271	. . . . 5 |- ( ( A e. RR /\ B e. RR* ) -> ( A < B -> -e B < -e A ) )
30	29	expimpd 345	. . . 4 |- ( A e. RR -> ( ( B e. RR* /\ A < B ) -> -e B < -e A ) )
31		simpl 102	. . . . . . 7 |- ( ( A = +oo /\ B e. RR* ) -> A = +oo )
32	31	breq1d 3774	. . . . . 6 |- ( ( A = +oo /\ B e. RR* ) -> ( A < B <-> +oo < B ) )
33		pnfnlt 8708	. . . . . . . 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 -> -e B < -e A ) )
36	32, 35	sylbid 139	. . . . 5 |- ( ( A = +oo /\ B e. RR* ) -> ( A < B -> -e B < -e A ) )
37	36	expimpd 345	. . . 4 |- ( A = +oo -> ( ( B e. RR* /\ A < B ) -> -e B < -e A ) )
38		breq1 3767	. . . . . 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 7272	. . . . . . . . . . 11 |- ( B e. RR -> -u B e. RR )
41	4, 40	eqeltrd 2114	. . . . . . . . . 10 |- ( B e. RR -> -e B e. RR )
42	41	adantr 261	. . . . . . . . 9 |- ( ( B e. RR /\ -oo < B ) -> -e B e. RR )
43		ltpnf 8702	. . . . . . . . 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 261	. . . . . . . . 9 |- ( ( B = +oo /\ -oo < B ) -> -e B = -oo )
46		mnfltpnf 8706	. . . . . . . . 9 |- -oo < +oo
47	45, 46	syl6eqbr 3801	. . . . . . . 8 |- ( ( B = +oo /\ -oo < B ) -> -e B < +oo )
48		breq2 3768	. . . . . . . . . 10 |- ( B = -oo -> ( -oo < B <-> -oo < -oo ) )
49		mnfxr 8694	. . . . . . . . . . . 12 |- -oo e. RR*
50		nltmnf 8709	. . . . . . . . . . . 12 |- ( -oo e. RR* -> -. -oo < -oo )
51	49, 50	ax-mp 7	. . . . . . . . . . 11 |- -. -oo < -oo
52	51	pm2.21i 575	. . . . . . . . . 10 |- ( -oo < -oo -> -e B < +oo )
53	48, 52	syl6bi 152	. . . . . . . . 9 |- ( B = -oo -> ( -oo < B -> -e B < +oo ) )
54	53	imp 115	. . . . . . . 8 |- ( ( B = -oo /\ -oo < B ) -> -e B < +oo )
55	44, 47, 54	3jaoian 1200	. . . . . . 7 |- ( ( ( B e. RR \/ B = +oo \/ B = -oo ) /\ -oo < B ) -> -e B < +oo )
56	2, 55	sylanb 268	. . . . . 6 |- ( ( B e. RR* /\ -oo < B ) -> -e B < +oo )
57		xnegeq 8740	. . . . . . . 8 |- ( A = -oo -> -e A = -e -oo )
58		xnegmnf 8742	. . . . . . . 8 |- -e -oo = +oo
59	57, 58	syl6eq 2088	. . . . . . 7 |- ( A = -oo -> -e A = +oo )
60	59	breq2d 3776	. . . . . 6 |- ( A = -oo -> ( -e B < -e A <-> -e B < +oo ) )
61	56, 60	syl5ibr 145	. . . . 5 |- ( A = -oo -> ( ( B e. RR* /\ -oo < B ) -> -e B < -e A ) )
62	39, 61	sylbid 139	. . . 4 |- ( A = -oo -> ( ( B e. RR* /\ A < B ) -> -e B < -e A ) )
63	30, 37, 62	3jaoi 1198	. . 3 |- ( ( A e. RR \/ A = +oo \/ A = -oo ) -> ( ( B e. RR* /\ A < B ) -> -e B < -e A ) )
64	1, 63	sylbi 114	. 2 |- ( A e. RR* -> ( ( B e. RR* /\ A < B ) -> -e B < -e A ) )
65	64	3impib 1102	1 |- ( ( A e. RR* /\ B e. RR* /\ A < B ) -> -e B < -e A )

Syntax hints:   -. wn 3   -> wi 4   /\ wa 97   \/ w3o 884   /\ w3a 885   = wceq 1243   e. wcel 1393   class class class wbr 3764   RRcr 6888   +oocpnf 7057   -oocmnf 7058   RR*cxr 7059   < clt 7060   -ucneg 7183   -ecxne 8686

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 630  ax-5 1336  ax-7 1337  ax-gen 1338  ax-ie1 1382  ax-ie2 1383  ax-8 1395  ax-10 1396  ax-11 1397  ax-i12 1398  ax-bndl 1399  ax-4 1400  ax-13 1404  ax-14 1405  ax-17 1419  ax-i9 1423  ax-ial 1427  ax-i5r 1428  ax-ext 2022  ax-sep 3875  ax-pow 3927  ax-pr 3944  ax-un 4170  ax-setind 4262  ax-cnex 6975  ax-resscn 6976  ax-1cn 6977  ax-1re 6978  ax-icn 6979  ax-addcl 6980  ax-addrcl 6981  ax-mulcl 6982  ax-addcom 6984  ax-addass 6986  ax-distr 6988  ax-i2m1 6989  ax-0id 6992  ax-rnegex 6993  ax-cnre 6995  ax-pre-ltadd 7000

This theorem depends on definitions:  df-bi 110  df-3or 886  df-3an 887  df-tru 1246  df-fal 1249  df-nf 1350  df-sb 1646  df-eu 1903  df-mo 1904  df-clab 2027  df-cleq 2033  df-clel 2036  df-nfc 2167  df-ne 2206  df-nel 2207  df-ral 2311  df-rex 2312  df-reu 2313  df-rab 2315  df-v 2559  df-sbc 2765  df-dif 2920  df-un 2922  df-in 2924  df-ss 2931  df-if 3332  df-pw 3361  df-sn 3381  df-pr 3382  df-op 3384  df-uni 3581  df-br 3765  df-opab 3819  df-id 4030  df-xp 4351  df-rel 4352  df-cnv 4353  df-co 4354  df-dm 4355  df-iota 4867  df-fun 4904  df-fv 4910  df-riota 5468  df-ov 5515  df-oprab 5516  df-mpt2 5517  df-pnf 7062  df-mnf 7063  df-xr 7064  df-ltxr 7065  df-sub 7184  df-neg 7185  df-xneg 8689

This theorem is referenced by:  xltneg  8749