Theorem xltnegi 10031

Description: Forward direction of xltneg 10032. (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 9972	. . 3 |- ( A e. RR* <-> ( A e. RR \/ A = +oo \/ A = -oo ) )
2		elxr 9972	. . . . . 6 |- ( B e. RR* <-> ( B e. RR \/ B = +oo \/ B = -oo ) )
3		ltneg 8609	. . . . . . . . 9 |- ( ( A e. RR /\ B e. RR ) -> ( A < B <-> -u B < -u A ) )
4		rexneg 10026	. . . . . . . . . 10 |- ( B e. RR -> -e B = -u B )
5		rexneg 10026	. . . . . . . . . 10 |- ( A e. RR -> -e A = -u A )
6	4, 5	breqan12rd 4100	. . . . . . . . 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 10023	. . . . . . . . . . 11 |- ( B = +oo -> -e B = -e +oo )
10		xnegpnf 10024	. . . . . . . . . . 11 |- -e +oo = -oo
11	9, 10	eqtrdi 2278	. . . . . . . . . 10 |- ( B = +oo -> -e B = -oo )
12	11	adantl 277	. . . . . . . . 9 |- ( ( A e. RR /\ B = +oo ) -> -e B = -oo )
13		renegcl 8407	. . . . . . . . . . . 12 |- ( A e. RR -> -u A e. RR )
14	5, 13	eqeltrd 2306	. . . . . . . . . . 11 |- ( A e. RR -> -e A e. RR )
15		mnflt 9979	. . . . . . . . . . 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 4105	. . . . . . . 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 4095	. . . . . . . 8 |- ( ( A e. RR /\ B = -oo ) -> ( A < B <-> A < -oo ) )
22		rexr 8192	. . . . . . . . . . 11 |- ( A e. RR -> A e. RR* )
23		nltmnf 9984	. . . . . . . . . . 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 622	. . . . . . . 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 1340	. . . . . 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 4093	. . . . . 6 |- ( ( A = +oo /\ B e. RR* ) -> ( A < B <-> +oo < B ) )
33		pnfnlt 9983	. . . . . . . 8 |- ( B e. RR* -> -. +oo < B )
34	33	adantl 277	. . . . . . 7 |- ( ( A = +oo /\ B e. RR* ) -> -. +oo < B )
35	34	pm2.21d 622	. . . . . 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 4086	. . . . . 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 8407	. . . . . . . . . . 11 |- ( B e. RR -> -u B e. RR )
41	4, 40	eqeltrd 2306	. . . . . . . . . 10 |- ( B e. RR -> -e B e. RR )
42	41	adantr 276	. . . . . . . . 9 |- ( ( B e. RR /\ -oo < B ) -> -e B e. RR )
43		ltpnf 9976	. . . . . . . . 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 9981	. . . . . . . . 9 |- -oo < +oo
47	45, 46	eqbrtrdi 4122	. . . . . . . 8 |- ( ( B = +oo /\ -oo < B ) -> -e B < +oo )
48		breq2 4087	. . . . . . . . . 10 |- ( B = -oo -> ( -oo < B <-> -oo < -oo ) )
49		mnfxr 8203	. . . . . . . . . . . 12 |- -oo e. RR*
50		nltmnf 9984	. . . . . . . . . . . 12 |- ( -oo e. RR* -> -. -oo < -oo )
51	49, 50	ax-mp 5	. . . . . . . . . . 11 |- -. -oo < -oo
52	51	pm2.21i 649	. . . . . . . . . 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 1339	. . . . . . 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 10023	. . . . . . . 8 |- ( A = -oo -> -e A = -e -oo )
58		xnegmnf 10025	. . . . . . . 8 |- -e -oo = +oo
59	57, 58	eqtrdi 2278	. . . . . . 7 |- ( A = -oo -> -e A = +oo )
60	59	breq2d 4095	. . . . . 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 1337	. . 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 1225	1 |- ( ( A e. RR* /\ B e. RR* /\ A < B ) -> -e B < -e A )

Syntax hints:   -. wn 3   -> wi 4   /\ wa 104   \/ w3o 1001   /\ w3a 1002   = wceq 1395   e. wcel 2200   class class class wbr 4083   RRcr 7998   +oocpnf 8178   -oocmnf 8179   RR*cxr 8180   < clt 8181   -ucneg 8318   -ecxne 9965

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 617  ax-in2 618  ax-io 714  ax-5 1493  ax-7 1494  ax-gen 1495  ax-ie1 1539  ax-ie2 1540  ax-8 1550  ax-10 1551  ax-11 1552  ax-i12 1553  ax-bndl 1555  ax-4 1556  ax-17 1572  ax-i9 1576  ax-ial 1580  ax-i5r 1581  ax-13 2202  ax-14 2203  ax-ext 2211  ax-sep 4202  ax-pow 4258  ax-pr 4293  ax-un 4524  ax-setind 4629  ax-cnex 8090  ax-resscn 8091  ax-1cn 8092  ax-1re 8093  ax-icn 8094  ax-addcl 8095  ax-addrcl 8096  ax-mulcl 8097  ax-addcom 8099  ax-addass 8101  ax-distr 8103  ax-i2m1 8104  ax-0id 8107  ax-rnegex 8108  ax-cnre 8110  ax-pre-ltadd 8115

This theorem depends on definitions:  df-bi 117  df-3or 1003  df-3an 1004  df-tru 1398  df-fal 1401  df-nf 1507  df-sb 1809  df-eu 2080  df-mo 2081  df-clab 2216  df-cleq 2222  df-clel 2225  df-nfc 2361  df-ne 2401  df-nel 2496  df-ral 2513  df-rex 2514  df-reu 2515  df-rab 2517  df-v 2801  df-sbc 3029  df-dif 3199  df-un 3201  df-in 3203  df-ss 3210  df-if 3603  df-pw 3651  df-sn 3672  df-pr 3673  df-op 3675  df-uni 3889  df-br 4084  df-opab 4146  df-id 4384  df-xp 4725  df-rel 4726  df-cnv 4727  df-co 4728  df-dm 4729  df-iota 5278  df-fun 5320  df-fv 5326  df-riota 5954  df-ov 6004  df-oprab 6005  df-mpo 6006  df-pnf 8183  df-mnf 8184  df-xr 8185  df-ltxr 8186  df-sub 8319  df-neg 8320  df-xneg 9968

This theorem is referenced by:  xltneg  10032