Theorem xsubge0 9883

Description: Extended real version of subge0 8434. (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 9778	. 2 |- ( B e. RR* <-> ( B e. RR \/ B = +oo \/ B = -oo ) )
2		0xr 8006	. . . . 5 |- 0 e. RR*
3		rexr 8005	. . . . . 6 |- ( B e. RR -> B e. RR* )
4		xnegcl 9834	. . . . . . 7 |- ( B e. RR* -> -e B e. RR* )
5		xaddcl 9862	. . . . . . 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 9877	. . . . 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 1342	. . . 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 9865	. . . . . 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 9874	. . . . 5 |- ( ( A e. RR* /\ B e. RR ) -> ( ( A +e -e B ) +e B ) = A )
15	13, 14	breq12d 4018	. . . 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 8012	. . . . . . 7 |- +oo e. RR*
18		xrletri3 9806	. . . . . . 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 8005	. . . . . . . . . . 11 |- ( A e. RR -> A e. RR* )
21		renepnf 8007	. . . . . . . . . . 11 |- ( A e. RR -> A =/= +oo )
22		xaddmnf1 9850	. . . . . . . . . . 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 9786	. . . . . . . . . . . . 13 |- -oo < 0
25		mnfxr 8016	. . . . . . . . . . . . . . 15 |- -oo e. RR*
26		xrlenlt 8024	. . . . . . . . . . . . . . 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 640	. . . . . . . . . . . 12 |- -. 0 <_ -oo
30		breq2 4009	. . . . . . . . . . . 12 |- ( ( A +e -oo ) = -oo -> ( 0 <_ ( A +e -oo ) <-> 0 <_ -oo ) )
31	29, 30	mtbiri 675	. . . . . . . . . . 11 |- ( ( A +e -oo ) = -oo -> -. 0 <_ ( A +e -oo ) )
32	31	pm2.21d 619	. . . . . . . . . 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 2240	. . . . . . . . . . . 12 |- ( A = -oo -> ( A e. RR* <-> -oo e. RR* ) )
38	25, 37	mpbiri 168	. . . . . . . . . . 11 |- ( A = -oo -> A e. RR* )
39		mnfnepnf 8015	. . . . . . . . . . . 12 |- -oo =/= +oo
40		neeq1 2360	. . . . . . . . . . . 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 9778	. . . . . . . . 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 1307	. . . . . . 7 |- ( A e. RR* -> ( 0 <_ ( A +e -oo ) -> A = +oo ) )
48		0le0 9010	. . . . . . . 8 |- 0 <_ 0
49		oveq1 5884	. . . . . . . . 9 |- ( A = +oo -> ( A +e -oo ) = ( +oo +e -oo ) )
50		pnfaddmnf 9852	. . . . . . . . 9 |- ( +oo +e -oo ) = 0
51	49, 50	eqtrdi 2226	. . . . . . . 8 |- ( A = +oo -> ( A +e -oo ) = 0 )
52	48, 51	breqtrrid 4043	. . . . . . 7 |- ( A = +oo -> 0 <_ ( A +e -oo ) )
53	47, 52	impbid1 142	. . . . . 6 |- ( A e. RR* -> ( 0 <_ ( A +e -oo ) <-> A = +oo ) )
54		pnfge 9791	. . . . . . 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 9829	. . . . . . . 8 |- ( B = +oo -> -e B = -e +oo )
59		xnegpnf 9830	. . . . . . . 8 |- -e +oo = -oo
60	58, 59	eqtrdi 2226	. . . . . . 7 |- ( B = +oo -> -e B = -oo )
61	60	adantl 277	. . . . . 6 |- ( ( A e. RR* /\ B = +oo ) -> -e B = -oo )
62	61	oveq2d 5893	. . . . 5 |- ( ( A e. RR* /\ B = +oo ) -> ( A +e -e B ) = ( A +e -oo ) )
63	62	breq2d 4017	. . . 4 |- ( ( A e. RR* /\ B = +oo ) -> ( 0 <_ ( A +e -e B ) <-> 0 <_ ( A +e -oo ) ) )
64		breq1 4008	. . . . 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 5884	. . . . . . . . . 10 |- ( A = -oo -> ( A +e +oo ) = ( -oo +e +oo ) )
68		mnfaddpnf 9853	. . . . . . . . . 10 |- ( -oo +e +oo ) = 0
69	67, 68	eqtrdi 2226	. . . . . . . . 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 4043	. . . . . . 7 |- ( ( A e. RR* /\ A = -oo ) -> 0 <_ ( A +e +oo ) )
72		df-ne 2348	. . . . . . . 8 |- ( A =/= -oo <-> -. A = -oo )
73		0lepnf 9792	. . . . . . . . 9 |- 0 <_ +oo
74		xaddpnf1 9848	. . . . . . . . 9 |- ( ( A e. RR* /\ A =/= -oo ) -> ( A +e +oo ) = +oo )
75	73, 74	breqtrrid 4043	. . . . . . . 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 9845	. . . . . . . 8 |- ( A e. RR* -> DECID A = -oo )
78		exmiddc 836	. . . . . . . 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 798	. . . . . 6 |- ( A e. RR* -> 0 <_ ( A +e +oo ) )
81		mnfle 9794	. . . . . 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 9829	. . . . . . . 8 |- ( B = -oo -> -e B = -e -oo )
85		xnegmnf 9831	. . . . . . . 8 |- -e -oo = +oo
86	84, 85	eqtrdi 2226	. . . . . . 7 |- ( B = -oo -> -e B = +oo )
87	86	adantl 277	. . . . . 6 |- ( ( A e. RR* /\ B = -oo ) -> -e B = +oo )
88	87	oveq2d 5893	. . . . 5 |- ( ( A e. RR* /\ B = -oo ) -> ( A +e -e B ) = ( A +e +oo ) )
89	88	breq2d 4017	. . . 4 |- ( ( A e. RR* /\ B = -oo ) -> ( 0 <_ ( A +e -e B ) <-> 0 <_ ( A +e +oo ) ) )
90		breq1 4008	. . . . 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 1306	. 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 708  DECID wdc 834   \/ w3o 977   = wceq 1353   e. wcel 2148   =/= wne 2347   class class class wbr 4005  (class class class)co 5877   RRcr 7812   0cc0 7813   +oocpnf 7991   -oocmnf 7992   RR*cxr 7993   < clt 7994   <_ cle 7995   -ecxne 9771   +ecxad 9772

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 614  ax-in2 615  ax-io 709  ax-5 1447  ax-7 1448  ax-gen 1449  ax-ie1 1493  ax-ie2 1494  ax-8 1504  ax-10 1505  ax-11 1506  ax-i12 1507  ax-bndl 1509  ax-4 1510  ax-17 1526  ax-i9 1530  ax-ial 1534  ax-i5r 1535  ax-13 2150  ax-14 2151  ax-ext 2159  ax-sep 4123  ax-pow 4176  ax-pr 4211  ax-un 4435  ax-setind 4538  ax-cnex 7904  ax-resscn 7905  ax-1cn 7906  ax-1re 7907  ax-icn 7908  ax-addcl 7909  ax-addrcl 7910  ax-mulcl 7911  ax-addcom 7913  ax-addass 7915  ax-distr 7917  ax-i2m1 7918  ax-0id 7921  ax-rnegex 7922  ax-cnre 7924  ax-pre-ltirr 7925  ax-pre-apti 7928  ax-pre-ltadd 7929

This theorem depends on definitions:  df-bi 117  df-dc 835  df-3or 979  df-3an 980  df-tru 1356  df-fal 1359  df-nf 1461  df-sb 1763  df-eu 2029  df-mo 2030  df-clab 2164  df-cleq 2170  df-clel 2173  df-nfc 2308  df-ne 2348  df-nel 2443  df-ral 2460  df-rex 2461  df-reu 2462  df-rab 2464  df-v 2741  df-sbc 2965  df-csb 3060  df-dif 3133  df-un 3135  df-in 3137  df-ss 3144  df-if 3537  df-pw 3579  df-sn 3600  df-pr 3601  df-op 3603  df-uni 3812  df-iun 3890  df-br 4006  df-opab 4067  df-mpt 4068  df-id 4295  df-xp 4634  df-rel 4635  df-cnv 4636  df-co 4637  df-dm 4638  df-rn 4639  df-res 4640  df-ima 4641  df-iota 5180  df-fun 5220  df-fn 5221  df-f 5222  df-fv 5226  df-riota 5833  df-ov 5880  df-oprab 5881  df-mpo 5882  df-1st 6143  df-2nd 6144  df-pnf 7996  df-mnf 7997  df-xr 7998  df-ltxr 7999  df-le 8000  df-sub 8132  df-neg 8133  df-xneg 9774  df-xadd 9775

This theorem is referenced by:  ssblps  13964  ssbl  13965