Theorem xsubge0 9664

Description: Extended real version of subge0 8237. (Contributed by Mario Carneiro, 24-Aug-2015.)

Assertion

Ref	Expression
xsubge0	|- ( ( A e. RR* /\ B e. RR* ) -> ( 0 <_ ( A +e -e B ) <-> B <_ A ) )

Proof of Theorem xsubge0

Step	Hyp	Ref	Expression
1		elxr 9563	. 2 |- ( B e. RR* <-> ( B e. RR \/ B = +oo \/ B = -oo ) )
2		0xr 7812	. . . . 5 |- 0 e. RR*
3		rexr 7811	. . . . . 6 |- ( B e. RR -> B e. RR* )
4		xnegcl 9615	. . . . . . 7 |- ( B e. RR* -> -e B e. RR* )
5		xaddcl 9643	. . . . . . 7 |- ( ( A e. RR* /\ -e B e. RR* ) -> ( A +e -e B ) e. RR* )
6	4, 5	sylan2 284	. . . . . 6 |- ( ( A e. RR* /\ B e. RR* ) -> ( A +e -e B ) e. RR* )
7	3, 6	sylan2 284	. . . . 5 |- ( ( A e. RR* /\ B e. RR ) -> ( A +e -e B ) e. RR* )
8		simpr 109	. . . . 5 |- ( ( A e. RR* /\ B e. RR ) -> B e. RR )
9		xleadd1 9658	. . . . 5 |- ( ( 0 e. RR* /\ ( A +e -e B ) e. RR* /\ B e. RR ) -> ( 0 <_ ( A +e -e B ) <-> ( 0 +e B ) <_ ( ( A +e -e B ) +e B ) ) )
10	2, 7, 8, 9	mp3an2i 1320	. . . 4 |- ( ( A e. RR* /\ B e. RR ) -> ( 0 <_ ( A +e -e B ) <-> ( 0 +e B ) <_ ( ( A +e -e B ) +e B ) ) )
11	3	adantl 275	. . . . . 6 |- ( ( A e. RR* /\ B e. RR ) -> B e. RR* )
12		xaddid2 9646	. . . . . 6 |- ( B e. RR* -> ( 0 +e B ) = B )
13	11, 12	syl 14	. . . . 5 |- ( ( A e. RR* /\ B e. RR ) -> ( 0 +e B ) = B )
14		xnpcan 9655	. . . . 5 |- ( ( A e. RR* /\ B e. RR ) -> ( ( A +e -e B ) +e B ) = A )
15	13, 14	breq12d 3942	. . . 4 |- ( ( A e. RR* /\ B e. RR ) -> ( ( 0 +e B ) <_ ( ( A +e -e B ) +e B ) <-> B <_ A ) )
16	10, 15	bitrd 187	. . 3 |- ( ( A e. RR* /\ B e. RR ) -> ( 0 <_ ( A +e -e B ) <-> B <_ A ) )
17		pnfxr 7818	. . . . . . 7 |- +oo e. RR*
18		xrletri3 9588	. . . . . . 7 |- ( ( A e. RR* /\ +oo e. RR* ) -> ( A = +oo <-> ( A <_ +oo /\ +oo <_ A ) ) )
19	17, 18	mpan2 421	. . . . . 6 |- ( A e. RR* -> ( A = +oo <-> ( A <_ +oo /\ +oo <_ A ) ) )
20		rexr 7811	. . . . . . . . . . 11 |- ( A e. RR -> A e. RR* )
21		renepnf 7813	. . . . . . . . . . 11 |- ( A e. RR -> A =/= +oo )
22		xaddmnf1 9631	. . . . . . . . . . 11 |- ( ( A e. RR* /\ A =/= +oo ) -> ( A +e -oo ) = -oo )
23	20, 21, 22	syl2anc 408	. . . . . . . . . 10 |- ( A e. RR -> ( A +e -oo ) = -oo )
24		mnflt0 9570	. . . . . . . . . . . . 13 |- -oo < 0
25		mnfxr 7822	. . . . . . . . . . . . . . 15 |- -oo e. RR*
26		xrlenlt 7829	. . . . . . . . . . . . . . 15 |- ( ( 0 e. RR* /\ -oo e. RR* ) -> ( 0 <_ -oo <-> -. -oo < 0 ) )
27	2, 25, 26	mp2an 422	. . . . . . . . . . . . . 14 |- ( 0 <_ -oo <-> -. -oo < 0 )
28	27	biimpi 119	. . . . . . . . . . . . 13 |- ( 0 <_ -oo -> -. -oo < 0 )
29	24, 28	mt2 629	. . . . . . . . . . . 12 |- -. 0 <_ -oo
30		breq2 3933	. . . . . . . . . . . 12 |- ( ( A +e -oo ) = -oo -> ( 0 <_ ( A +e -oo ) <-> 0 <_ -oo ) )
31	29, 30	mtbiri 664	. . . . . . . . . . 11 |- ( ( A +e -oo ) = -oo -> -. 0 <_ ( A +e -oo ) )
32	31	pm2.21d 608	. . . . . . . . . 10 |- ( ( A +e -oo ) = -oo -> ( 0 <_ ( A +e -oo ) -> A = +oo ) )
33	23, 32	syl 14	. . . . . . . . 9 |- ( A e. RR -> ( 0 <_ ( A +e -oo ) -> A = +oo ) )
34	33	adantl 275	. . . . . . . 8 |- ( ( A e. RR* /\ A e. RR ) -> ( 0 <_ ( A +e -oo ) -> A = +oo ) )
35		simpr 109	. . . . . . . . 9 |- ( ( A e. RR* /\ A = +oo ) -> A = +oo )
36	35	a1d 22	. . . . . . . 8 |- ( ( A e. RR* /\ A = +oo ) -> ( 0 <_ ( A +e -oo ) -> A = +oo ) )
37		eleq1 2202	. . . . . . . . . . . 12 |- ( A = -oo -> ( A e. RR* <-> -oo e. RR* ) )
38	25, 37	mpbiri 167	. . . . . . . . . . 11 |- ( A = -oo -> A e. RR* )
39		mnfnepnf 7821	. . . . . . . . . . . 12 |- -oo =/= +oo
40		neeq1 2321	. . . . . . . . . . . 12 |- ( A = -oo -> ( A =/= +oo <-> -oo =/= +oo ) )
41	39, 40	mpbiri 167	. . . . . . . . . . 11 |- ( A = -oo -> A =/= +oo )
42	38, 41, 22	syl2anc 408	. . . . . . . . . 10 |- ( A = -oo -> ( A +e -oo ) = -oo )
43	42, 32	syl 14	. . . . . . . . 9 |- ( A = -oo -> ( 0 <_ ( A +e -oo ) -> A = +oo ) )
44	43	adantl 275	. . . . . . . 8 |- ( ( A e. RR* /\ A = -oo ) -> ( 0 <_ ( A +e -oo ) -> A = +oo ) )
45		elxr 9563	. . . . . . . . 9 |- ( A e. RR* <-> ( A e. RR \/ A = +oo \/ A = -oo ) )
46	45	biimpi 119	. . . . . . . 8 |- ( A e. RR* -> ( A e. RR \/ A = +oo \/ A = -oo ) )
47	34, 36, 44, 46	mpjao3dan 1285	. . . . . . 7 |- ( A e. RR* -> ( 0 <_ ( A +e -oo ) -> A = +oo ) )
48		0le0 8809	. . . . . . . 8 |- 0 <_ 0
49		oveq1 5781	. . . . . . . . 9 |- ( A = +oo -> ( A +e -oo ) = ( +oo +e -oo ) )
50		pnfaddmnf 9633	. . . . . . . . 9 |- ( +oo +e -oo ) = 0
51	49, 50	syl6eq 2188	. . . . . . . 8 |- ( A = +oo -> ( A +e -oo ) = 0 )
52	48, 51	breqtrrid 3966	. . . . . . 7 |- ( A = +oo -> 0 <_ ( A +e -oo ) )
53	47, 52	impbid1 141	. . . . . 6 |- ( A e. RR* -> ( 0 <_ ( A +e -oo ) <-> A = +oo ) )
54		pnfge 9575	. . . . . . 7 |- ( A e. RR* -> A <_ +oo )
55	54	biantrurd 303	. . . . . 6 |- ( A e. RR* -> ( +oo <_ A <-> ( A <_ +oo /\ +oo <_ A ) ) )
56	19, 53, 55	3bitr4d 219	. . . . 5 |- ( A e. RR* -> ( 0 <_ ( A +e -oo ) <-> +oo <_ A ) )
57	56	adantr 274	. . . 4 |- ( ( A e. RR* /\ B = +oo ) -> ( 0 <_ ( A +e -oo ) <-> +oo <_ A ) )
58		xnegeq 9610	. . . . . . . 8 |- ( B = +oo -> -e B = -e +oo )
59		xnegpnf 9611	. . . . . . . 8 |- -e +oo = -oo
60	58, 59	syl6eq 2188	. . . . . . 7 |- ( B = +oo -> -e B = -oo )
61	60	adantl 275	. . . . . 6 |- ( ( A e. RR* /\ B = +oo ) -> -e B = -oo )
62	61	oveq2d 5790	. . . . 5 |- ( ( A e. RR* /\ B = +oo ) -> ( A +e -e B ) = ( A +e -oo ) )
63	62	breq2d 3941	. . . 4 |- ( ( A e. RR* /\ B = +oo ) -> ( 0 <_ ( A +e -e B ) <-> 0 <_ ( A +e -oo ) ) )
64		breq1 3932	. . . . 5 |- ( B = +oo -> ( B <_ A <-> +oo <_ A ) )
65	64	adantl 275	. . . 4 |- ( ( A e. RR* /\ B = +oo ) -> ( B <_ A <-> +oo <_ A ) )
66	57, 63, 65	3bitr4d 219	. . 3 |- ( ( A e. RR* /\ B = +oo ) -> ( 0 <_ ( A +e -e B ) <-> B <_ A ) )
67		oveq1 5781	. . . . . . . . . 10 |- ( A = -oo -> ( A +e +oo ) = ( -oo +e +oo ) )
68		mnfaddpnf 9634	. . . . . . . . . 10 |- ( -oo +e +oo ) = 0
69	67, 68	syl6eq 2188	. . . . . . . . 9 |- ( A = -oo -> ( A +e +oo ) = 0 )
70	69	adantl 275	. . . . . . . 8 |- ( ( A e. RR* /\ A = -oo ) -> ( A +e +oo ) = 0 )
71	48, 70	breqtrrid 3966	. . . . . . 7 |- ( ( A e. RR* /\ A = -oo ) -> 0 <_ ( A +e +oo ) )
72		df-ne 2309	. . . . . . . 8 |- ( A =/= -oo <-> -. A = -oo )
73		0lepnf 9576	. . . . . . . . 9 |- 0 <_ +oo
74		xaddpnf1 9629	. . . . . . . . 9 |- ( ( A e. RR* /\ A =/= -oo ) -> ( A +e +oo ) = +oo )
75	73, 74	breqtrrid 3966	. . . . . . . 8 |- ( ( A e. RR* /\ A =/= -oo ) -> 0 <_ ( A +e +oo ) )
76	72, 75	sylan2br 286	. . . . . . 7 |- ( ( A e. RR* /\ -. A = -oo ) -> 0 <_ ( A +e +oo ) )
77		xrmnfdc 9626	. . . . . . . 8 |- ( A e. RR* -> DECID A = -oo )
78		exmiddc 821	. . . . . . . 8 |- (DECID A = -oo -> ( A = -oo \/ -. A = -oo ) )
79	77, 78	syl 14	. . . . . . 7 |- ( A e. RR* -> ( A = -oo \/ -. A = -oo ) )
80	71, 76, 79	mpjaodan 787	. . . . . 6 |- ( A e. RR* -> 0 <_ ( A +e +oo ) )
81		mnfle 9578	. . . . . 6 |- ( A e. RR* -> -oo <_ A )
82	80, 81	2thd 174	. . . . 5 |- ( A e. RR* -> ( 0 <_ ( A +e +oo ) <-> -oo <_ A ) )
83	82	adantr 274	. . . 4 |- ( ( A e. RR* /\ B = -oo ) -> ( 0 <_ ( A +e +oo ) <-> -oo <_ A ) )
84		xnegeq 9610	. . . . . . . 8 |- ( B = -oo -> -e B = -e -oo )
85		xnegmnf 9612	. . . . . . . 8 |- -e -oo = +oo
86	84, 85	syl6eq 2188	. . . . . . 7 |- ( B = -oo -> -e B = +oo )
87	86	adantl 275	. . . . . 6 |- ( ( A e. RR* /\ B = -oo ) -> -e B = +oo )
88	87	oveq2d 5790	. . . . 5 |- ( ( A e. RR* /\ B = -oo ) -> ( A +e -e B ) = ( A +e +oo ) )
89	88	breq2d 3941	. . . 4 |- ( ( A e. RR* /\ B = -oo ) -> ( 0 <_ ( A +e -e B ) <-> 0 <_ ( A +e +oo ) ) )
90		breq1 3932	. . . . 5 |- ( B = -oo -> ( B <_ A <-> -oo <_ A ) )
91	90	adantl 275	. . . 4 |- ( ( A e. RR* /\ B = -oo ) -> ( B <_ A <-> -oo <_ A ) )
92	83, 89, 91	3bitr4d 219	. . 3 |- ( ( A e. RR* /\ B = -oo ) -> ( 0 <_ ( A +e -e B ) <-> B <_ A ) )
93	16, 66, 92	3jaodan 1284	. 2 |- ( ( A e. RR* /\ ( B e. RR \/ B = +oo \/ B = -oo ) ) -> ( 0 <_ ( A +e -e B ) <-> B <_ A ) )
94	1, 93	sylan2b 285	1 |- ( ( A e. RR* /\ B e. RR* ) -> ( 0 <_ ( A +e -e B ) <-> B <_ A ) )

Syntax hints:   -. wn 3   -> wi 4   /\ wa 103   <-> wb 104   \/ wo 697  DECID wdc 819   \/ w3o 961   = wceq 1331   e. wcel 1480   =/= wne 2308   class class class wbr 3929  (class class class)co 5774   RRcr 7619   0cc0 7620   +oocpnf 7797   -oocmnf 7798   RR*cxr 7799   < clt 7800   <_ cle 7801   -ecxne 9556   +ecxad 9557

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-ltirr 7732  ax-pre-apti 7735  ax-pre-ltadd 7736

This theorem depends on definitions:  df-bi 116  df-dc 820  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-csb 3004  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-iun 3815  df-br 3930  df-opab 3990  df-mpt 3991  df-id 4215  df-xp 4545  df-rel 4546  df-cnv 4547  df-co 4548  df-dm 4549  df-rn 4550  df-res 4551  df-ima 4552  df-iota 5088  df-fun 5125  df-fn 5126  df-f 5127  df-fv 5131  df-riota 5730  df-ov 5777  df-oprab 5778  df-mpo 5779  df-1st 6038  df-2nd 6039  df-pnf 7802  df-mnf 7803  df-xr 7804  df-ltxr 7805  df-le 7806  df-sub 7935  df-neg 7936  df-xneg 9559  df-xadd 9560

This theorem is referenced by:  ssblps  12594  ssbl  12595