Theorem xsubge0 10002

Description: Extended real version of subge0 8547. (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 9897	. 2 |- ( B e. RR* <-> ( B e. RR \/ B = +oo \/ B = -oo ) )
2		0xr 8118	. . . . 5 |- 0 e. RR*
3		rexr 8117	. . . . . 6 |- ( B e. RR -> B e. RR* )
4		xnegcl 9953	. . . . . . 7 |- ( B e. RR* -> -e B e. RR* )
5		xaddcl 9981	. . . . . . 7 |- ( ( A e. RR* /\ -e B e. RR* ) -> ( A +e -e B ) e. RR* )
6	4, 5	sylan2 286	. . . . . 6 |- ( ( A e. RR* /\ B e. RR* ) -> ( A +e -e B ) e. RR* )
7	3, 6	sylan2 286	. . . . 5 |- ( ( A e. RR* /\ B e. RR ) -> ( A +e -e B ) e. RR* )
8		simpr 110	. . . . 5 |- ( ( A e. RR* /\ B e. RR ) -> B e. RR )
9		xleadd1 9996	. . . . 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 1354	. . . 4 |- ( ( A e. RR* /\ B e. RR ) -> ( 0 <_ ( A +e -e B ) <-> ( 0 +e B ) <_ ( ( A +e -e B ) +e B ) ) )
11	3	adantl 277	. . . . . 6 |- ( ( A e. RR* /\ B e. RR ) -> B e. RR* )
12		xaddid2 9984	. . . . . 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 9993	. . . . 5 |- ( ( A e. RR* /\ B e. RR ) -> ( ( A +e -e B ) +e B ) = A )
15	13, 14	breq12d 4056	. . . 4 |- ( ( A e. RR* /\ B e. RR ) -> ( ( 0 +e B ) <_ ( ( A +e -e B ) +e B ) <-> B <_ A ) )
16	10, 15	bitrd 188	. . 3 |- ( ( A e. RR* /\ B e. RR ) -> ( 0 <_ ( A +e -e B ) <-> B <_ A ) )
17		pnfxr 8124	. . . . . . 7 |- +oo e. RR*
18		xrletri3 9925	. . . . . . 7 |- ( ( A e. RR* /\ +oo e. RR* ) -> ( A = +oo <-> ( A <_ +oo /\ +oo <_ A ) ) )
19	17, 18	mpan2 425	. . . . . 6 |- ( A e. RR* -> ( A = +oo <-> ( A <_ +oo /\ +oo <_ A ) ) )
20		rexr 8117	. . . . . . . . . . 11 |- ( A e. RR -> A e. RR* )
21		renepnf 8119	. . . . . . . . . . 11 |- ( A e. RR -> A =/= +oo )
22		xaddmnf1 9969	. . . . . . . . . . 11 |- ( ( A e. RR* /\ A =/= +oo ) -> ( A +e -oo ) = -oo )
23	20, 21, 22	syl2anc 411	. . . . . . . . . 10 |- ( A e. RR -> ( A +e -oo ) = -oo )
24		mnflt0 9905	. . . . . . . . . . . . 13 |- -oo < 0
25		mnfxr 8128	. . . . . . . . . . . . . . 15 |- -oo e. RR*
26		xrlenlt 8136	. . . . . . . . . . . . . . 15 |- ( ( 0 e. RR* /\ -oo e. RR* ) -> ( 0 <_ -oo <-> -. -oo < 0 ) )
27	2, 25, 26	mp2an 426	. . . . . . . . . . . . . 14 |- ( 0 <_ -oo <-> -. -oo < 0 )
28	27	biimpi 120	. . . . . . . . . . . . 13 |- ( 0 <_ -oo -> -. -oo < 0 )
29	24, 28	mt2 641	. . . . . . . . . . . 12 |- -. 0 <_ -oo
30		breq2 4047	. . . . . . . . . . . 12 |- ( ( A +e -oo ) = -oo -> ( 0 <_ ( A +e -oo ) <-> 0 <_ -oo ) )
31	29, 30	mtbiri 676	. . . . . . . . . . 11 |- ( ( A +e -oo ) = -oo -> -. 0 <_ ( A +e -oo ) )
32	31	pm2.21d 620	. . . . . . . . . 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 277	. . . . . . . 8 |- ( ( A e. RR* /\ A e. RR ) -> ( 0 <_ ( A +e -oo ) -> A = +oo ) )
35		simpr 110	. . . . . . . . 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 2267	. . . . . . . . . . . 12 |- ( A = -oo -> ( A e. RR* <-> -oo e. RR* ) )
38	25, 37	mpbiri 168	. . . . . . . . . . 11 |- ( A = -oo -> A e. RR* )
39		mnfnepnf 8127	. . . . . . . . . . . 12 |- -oo =/= +oo
40		neeq1 2388	. . . . . . . . . . . 12 |- ( A = -oo -> ( A =/= +oo <-> -oo =/= +oo ) )
41	39, 40	mpbiri 168	. . . . . . . . . . 11 |- ( A = -oo -> A =/= +oo )
42	38, 41, 22	syl2anc 411	. . . . . . . . . 10 |- ( A = -oo -> ( A +e -oo ) = -oo )
43	42, 32	syl 14	. . . . . . . . 9 |- ( A = -oo -> ( 0 <_ ( A +e -oo ) -> A = +oo ) )
44	43	adantl 277	. . . . . . . 8 |- ( ( A e. RR* /\ A = -oo ) -> ( 0 <_ ( A +e -oo ) -> A = +oo ) )
45		elxr 9897	. . . . . . . . 9 |- ( A e. RR* <-> ( A e. RR \/ A = +oo \/ A = -oo ) )
46	45	biimpi 120	. . . . . . . 8 |- ( A e. RR* -> ( A e. RR \/ A = +oo \/ A = -oo ) )
47	34, 36, 44, 46	mpjao3dan 1319	. . . . . . 7 |- ( A e. RR* -> ( 0 <_ ( A +e -oo ) -> A = +oo ) )
48		0le0 9124	. . . . . . . 8 |- 0 <_ 0
49		oveq1 5950	. . . . . . . . 9 |- ( A = +oo -> ( A +e -oo ) = ( +oo +e -oo ) )
50		pnfaddmnf 9971	. . . . . . . . 9 |- ( +oo +e -oo ) = 0
51	49, 50	eqtrdi 2253	. . . . . . . 8 |- ( A = +oo -> ( A +e -oo ) = 0 )
52	48, 51	breqtrrid 4081	. . . . . . 7 |- ( A = +oo -> 0 <_ ( A +e -oo ) )
53	47, 52	impbid1 142	. . . . . 6 |- ( A e. RR* -> ( 0 <_ ( A +e -oo ) <-> A = +oo ) )
54		pnfge 9910	. . . . . . 7 |- ( A e. RR* -> A <_ +oo )
55	54	biantrurd 305	. . . . . 6 |- ( A e. RR* -> ( +oo <_ A <-> ( A <_ +oo /\ +oo <_ A ) ) )
56	19, 53, 55	3bitr4d 220	. . . . 5 |- ( A e. RR* -> ( 0 <_ ( A +e -oo ) <-> +oo <_ A ) )
57	56	adantr 276	. . . 4 |- ( ( A e. RR* /\ B = +oo ) -> ( 0 <_ ( A +e -oo ) <-> +oo <_ A ) )
58		xnegeq 9948	. . . . . . . 8 |- ( B = +oo -> -e B = -e +oo )
59		xnegpnf 9949	. . . . . . . 8 |- -e +oo = -oo
60	58, 59	eqtrdi 2253	. . . . . . 7 |- ( B = +oo -> -e B = -oo )
61	60	adantl 277	. . . . . 6 |- ( ( A e. RR* /\ B = +oo ) -> -e B = -oo )
62	61	oveq2d 5959	. . . . 5 |- ( ( A e. RR* /\ B = +oo ) -> ( A +e -e B ) = ( A +e -oo ) )
63	62	breq2d 4055	. . . 4 |- ( ( A e. RR* /\ B = +oo ) -> ( 0 <_ ( A +e -e B ) <-> 0 <_ ( A +e -oo ) ) )
64		breq1 4046	. . . . 5 |- ( B = +oo -> ( B <_ A <-> +oo <_ A ) )
65	64	adantl 277	. . . 4 |- ( ( A e. RR* /\ B = +oo ) -> ( B <_ A <-> +oo <_ A ) )
66	57, 63, 65	3bitr4d 220	. . 3 |- ( ( A e. RR* /\ B = +oo ) -> ( 0 <_ ( A +e -e B ) <-> B <_ A ) )
67		oveq1 5950	. . . . . . . . . 10 |- ( A = -oo -> ( A +e +oo ) = ( -oo +e +oo ) )
68		mnfaddpnf 9972	. . . . . . . . . 10 |- ( -oo +e +oo ) = 0
69	67, 68	eqtrdi 2253	. . . . . . . . 9 |- ( A = -oo -> ( A +e +oo ) = 0 )
70	69	adantl 277	. . . . . . . 8 |- ( ( A e. RR* /\ A = -oo ) -> ( A +e +oo ) = 0 )
71	48, 70	breqtrrid 4081	. . . . . . 7 |- ( ( A e. RR* /\ A = -oo ) -> 0 <_ ( A +e +oo ) )
72		df-ne 2376	. . . . . . . 8 |- ( A =/= -oo <-> -. A = -oo )
73		0lepnf 9911	. . . . . . . . 9 |- 0 <_ +oo
74		xaddpnf1 9967	. . . . . . . . 9 |- ( ( A e. RR* /\ A =/= -oo ) -> ( A +e +oo ) = +oo )
75	73, 74	breqtrrid 4081	. . . . . . . 8 |- ( ( A e. RR* /\ A =/= -oo ) -> 0 <_ ( A +e +oo ) )
76	72, 75	sylan2br 288	. . . . . . 7 |- ( ( A e. RR* /\ -. A = -oo ) -> 0 <_ ( A +e +oo ) )
77		xrmnfdc 9964	. . . . . . . 8 |- ( A e. RR* -> DECID A = -oo )
78		exmiddc 837	. . . . . . . 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 799	. . . . . 6 |- ( A e. RR* -> 0 <_ ( A +e +oo ) )
81		mnfle 9913	. . . . . 6 |- ( A e. RR* -> -oo <_ A )
82	80, 81	2thd 175	. . . . 5 |- ( A e. RR* -> ( 0 <_ ( A +e +oo ) <-> -oo <_ A ) )
83	82	adantr 276	. . . 4 |- ( ( A e. RR* /\ B = -oo ) -> ( 0 <_ ( A +e +oo ) <-> -oo <_ A ) )
84		xnegeq 9948	. . . . . . . 8 |- ( B = -oo -> -e B = -e -oo )
85		xnegmnf 9950	. . . . . . . 8 |- -e -oo = +oo
86	84, 85	eqtrdi 2253	. . . . . . 7 |- ( B = -oo -> -e B = +oo )
87	86	adantl 277	. . . . . 6 |- ( ( A e. RR* /\ B = -oo ) -> -e B = +oo )
88	87	oveq2d 5959	. . . . 5 |- ( ( A e. RR* /\ B = -oo ) -> ( A +e -e B ) = ( A +e +oo ) )
89	88	breq2d 4055	. . . 4 |- ( ( A e. RR* /\ B = -oo ) -> ( 0 <_ ( A +e -e B ) <-> 0 <_ ( A +e +oo ) ) )
90		breq1 4046	. . . . 5 |- ( B = -oo -> ( B <_ A <-> -oo <_ A ) )
91	90	adantl 277	. . . 4 |- ( ( A e. RR* /\ B = -oo ) -> ( B <_ A <-> -oo <_ A ) )
92	83, 89, 91	3bitr4d 220	. . 3 |- ( ( A e. RR* /\ B = -oo ) -> ( 0 <_ ( A +e -e B ) <-> B <_ A ) )
93	16, 66, 92	3jaodan 1318	. 2 |- ( ( A e. RR* /\ ( B e. RR \/ B = +oo \/ B = -oo ) ) -> ( 0 <_ ( A +e -e B ) <-> B <_ A ) )
94	1, 93	sylan2b 287	1 |- ( ( A e. RR* /\ B e. RR* ) -> ( 0 <_ ( A +e -e B ) <-> B <_ A ) )

Syntax hints:   -. wn 3   -> wi 4   /\ wa 104   <-> wb 105   \/ wo 709  DECID wdc 835   \/ w3o 979   = wceq 1372   e. wcel 2175   =/= wne 2375   class class class wbr 4043  (class class class)co 5943   RRcr 7923   0cc0 7924   +oocpnf 8103   -oocmnf 8104   RR*cxr 8105   < clt 8106   <_ cle 8107   -ecxne 9890   +ecxad 9891

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 1469  ax-7 1470  ax-gen 1471  ax-ie1 1515  ax-ie2 1516  ax-8 1526  ax-10 1527  ax-11 1528  ax-i12 1529  ax-bndl 1531  ax-4 1532  ax-17 1548  ax-i9 1552  ax-ial 1556  ax-i5r 1557  ax-13 2177  ax-14 2178  ax-ext 2186  ax-sep 4161  ax-pow 4217  ax-pr 4252  ax-un 4479  ax-setind 4584  ax-cnex 8015  ax-resscn 8016  ax-1cn 8017  ax-1re 8018  ax-icn 8019  ax-addcl 8020  ax-addrcl 8021  ax-mulcl 8022  ax-addcom 8024  ax-addass 8026  ax-distr 8028  ax-i2m1 8029  ax-0id 8032  ax-rnegex 8033  ax-cnre 8035  ax-pre-ltirr 8036  ax-pre-apti 8039  ax-pre-ltadd 8040

This theorem depends on definitions:  df-bi 117  df-dc 836  df-3or 981  df-3an 982  df-tru 1375  df-fal 1378  df-nf 1483  df-sb 1785  df-eu 2056  df-mo 2057  df-clab 2191  df-cleq 2197  df-clel 2200  df-nfc 2336  df-ne 2376  df-nel 2471  df-ral 2488  df-rex 2489  df-reu 2490  df-rab 2492  df-v 2773  df-sbc 2998  df-csb 3093  df-dif 3167  df-un 3169  df-in 3171  df-ss 3178  df-if 3571  df-pw 3617  df-sn 3638  df-pr 3639  df-op 3641  df-uni 3850  df-iun 3928  df-br 4044  df-opab 4105  df-mpt 4106  df-id 4339  df-xp 4680  df-rel 4681  df-cnv 4682  df-co 4683  df-dm 4684  df-rn 4685  df-res 4686  df-ima 4687  df-iota 5231  df-fun 5272  df-fn 5273  df-f 5274  df-fv 5278  df-riota 5898  df-ov 5946  df-oprab 5947  df-mpo 5948  df-1st 6225  df-2nd 6226  df-pnf 8108  df-mnf 8109  df-xr 8110  df-ltxr 8111  df-le 8112  df-sub 8244  df-neg 8245  df-xneg 9893  df-xadd 9894

This theorem is referenced by:  ssblps  14868  ssbl  14869