Theorem xsubge0 10283

Description: Extended real version of subge0 8803. (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 10178	. 2 |- ( B e. RR* <-> ( B e. RR \/ B = +oo \/ B = -oo ) )
2		0xr 8372	. . . . 5 |- 0 e. RR*
3		rexr 8371	. . . . . 6 |- ( B e. RR -> B e. RR* )
4		xnegcl 10234	. . . . . . 7 |- ( B e. RR* -> -e B e. RR* )
5		xaddcl 10262	. . . . . . 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 10277	. . . . 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 1383	. . . 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 10265	. . . . . 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 10274	. . . . 5 |- ( ( A e. RR* /\ B e. RR ) -> ( ( A +e -e B ) +e B ) = A )
15	13, 14	breq12d 4143	. . . 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 8378	. . . . . . 7 |- +oo e. RR*
18		xrletri3 10206	. . . . . . 7 |- ( ( A e. RR* /\ +oo e. RR* ) -> ( A = +oo <-> ( A <_ +oo /\ +oo <_ A ) ) )
19	17, 18	mpan2 429	. . . . . 6 |- ( A e. RR* -> ( A = +oo <-> ( A <_ +oo /\ +oo <_ A ) ) )
20		rexr 8371	. . . . . . . . . . 11 |- ( A e. RR -> A e. RR* )
21		renepnf 8373	. . . . . . . . . . 11 |- ( A e. RR -> A =/= +oo )
22		xaddmnf1 10250	. . . . . . . . . . 11 |- ( ( A e. RR* /\ A =/= +oo ) -> ( A +e -oo ) = -oo )
23	20, 21, 22	syl2anc 415	. . . . . . . . . 10 |- ( A e. RR -> ( A +e -oo ) = -oo )
24		mnflt0 10186	. . . . . . . . . . . . 13 |- -oo < 0
25		mnfxr 8382	. . . . . . . . . . . . . . 15 |- -oo e. RR*
26		xrlenlt 8390	. . . . . . . . . . . . . . 15 |- ( ( 0 e. RR* /\ -oo e. RR* ) -> ( 0 <_ -oo <-> -. -oo < 0 ) )
27	2, 25, 26	mp2an 430	. . . . . . . . . . . . . 14 |- ( 0 <_ -oo <-> -. -oo < 0 )
28	27	biimpi 120	. . . . . . . . . . . . 13 |- ( 0 <_ -oo -> -. -oo < 0 )
29	24, 28	mt2 649	. . . . . . . . . . . 12 |- -. 0 <_ -oo
30		breq2 4134	. . . . . . . . . . . 12 |- ( ( A +e -oo ) = -oo -> ( 0 <_ ( A +e -oo ) <-> 0 <_ -oo ) )
31	29, 30	mtbiri 686	. . . . . . . . . . 11 |- ( ( A +e -oo ) = -oo -> -. 0 <_ ( A +e -oo ) )
32	31	pm2.21d 628	. . . . . . . . . 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 2301	. . . . . . . . . . . 12 |- ( A = -oo -> ( A e. RR* <-> -oo e. RR* ) )
38	25, 37	mpbiri 168	. . . . . . . . . . 11 |- ( A = -oo -> A e. RR* )
39		mnfnepnf 8381	. . . . . . . . . . . 12 |- -oo =/= +oo
40		neeq1 2433	. . . . . . . . . . . 12 |- ( A = -oo -> ( A =/= +oo <-> -oo =/= +oo ) )
41	39, 40	mpbiri 168	. . . . . . . . . . 11 |- ( A = -oo -> A =/= +oo )
42	38, 41, 22	syl2anc 415	. . . . . . . . . 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 10178	. . . . . . . . 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 1348	. . . . . . 7 |- ( A e. RR* -> ( 0 <_ ( A +e -oo ) -> A = +oo ) )
48		0le0 9393	. . . . . . . 8 |- 0 <_ 0
49		oveq1 6092	. . . . . . . . 9 |- ( A = +oo -> ( A +e -oo ) = ( +oo +e -oo ) )
50		pnfaddmnf 10252	. . . . . . . . 9 |- ( +oo +e -oo ) = 0
51	49, 50	eqtrdi 2287	. . . . . . . 8 |- ( A = +oo -> ( A +e -oo ) = 0 )
52	48, 51	breqtrrid 4168	. . . . . . 7 |- ( A = +oo -> 0 <_ ( A +e -oo ) )
53	47, 52	impbid1 142	. . . . . 6 |- ( A e. RR* -> ( 0 <_ ( A +e -oo ) <-> A = +oo ) )
54		pnfge 10191	. . . . . . 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 10229	. . . . . . . 8 |- ( B = +oo -> -e B = -e +oo )
59		xnegpnf 10230	. . . . . . . 8 |- -e +oo = -oo
60	58, 59	eqtrdi 2287	. . . . . . 7 |- ( B = +oo -> -e B = -oo )
61	60	adantl 277	. . . . . 6 |- ( ( A e. RR* /\ B = +oo ) -> -e B = -oo )
62	61	oveq2d 6101	. . . . 5 |- ( ( A e. RR* /\ B = +oo ) -> ( A +e -e B ) = ( A +e -oo ) )
63	62	breq2d 4142	. . . 4 |- ( ( A e. RR* /\ B = +oo ) -> ( 0 <_ ( A +e -e B ) <-> 0 <_ ( A +e -oo ) ) )
64		breq1 4133	. . . . 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 6092	. . . . . . . . . 10 |- ( A = -oo -> ( A +e +oo ) = ( -oo +e +oo ) )
68		mnfaddpnf 10253	. . . . . . . . . 10 |- ( -oo +e +oo ) = 0
69	67, 68	eqtrdi 2287	. . . . . . . . 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 4168	. . . . . . 7 |- ( ( A e. RR* /\ A = -oo ) -> 0 <_ ( A +e +oo ) )
72		df-ne 2421	. . . . . . . 8 |- ( A =/= -oo <-> -. A = -oo )
73		0lepnf 10192	. . . . . . . . 9 |- 0 <_ +oo
74		xaddpnf1 10248	. . . . . . . . 9 |- ( ( A e. RR* /\ A =/= -oo ) -> ( A +e +oo ) = +oo )
75	73, 74	breqtrrid 4168	. . . . . . . 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 10245	. . . . . . . 8 |- ( A e. RR* -> DECID A = -oo )
78		exmiddc 848	. . . . . . . 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 810	. . . . . 6 |- ( A e. RR* -> 0 <_ ( A +e +oo ) )
81		mnfle 10194	. . . . . 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 10229	. . . . . . . 8 |- ( B = -oo -> -e B = -e -oo )
85		xnegmnf 10231	. . . . . . . 8 |- -e -oo = +oo
86	84, 85	eqtrdi 2287	. . . . . . 7 |- ( B = -oo -> -e B = +oo )
87	86	adantl 277	. . . . . 6 |- ( ( A e. RR* /\ B = -oo ) -> -e B = +oo )
88	87	oveq2d 6101	. . . . 5 |- ( ( A e. RR* /\ B = -oo ) -> ( A +e -e B ) = ( A +e +oo ) )
89	88	breq2d 4142	. . . 4 |- ( ( A e. RR* /\ B = -oo ) -> ( 0 <_ ( A +e -e B ) <-> 0 <_ ( A +e +oo ) ) )
90		breq1 4133	. . . . 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 1347	. 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 ) )

This proof depends on syntax axioms:   -. wn 3   -> wi 4   /\ wa 104   <-> wb 105   \/ wo 720  DECID wdc 846   \/ w3o 1008   = wceq 1402   e. wcel 2209   =/= wne 2420   class class class wbr 4130  (class class class)co 6085   RRcr 8178   0cc0 8179   +oocpnf 8357   -oocmnf 8358   RR*cxr 8359   < clt 8360   <_ cle 8361   -ecxne 10171   +ecxad 10172

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-in1 623  ax-in2 624  ax-io 721  ax-5 1500  ax-7 1501  ax-gen 1502  ax-ie1 1546  ax-ie2 1547  ax-8 1557  ax-10 1558  ax-11 1559  ax-i12 1560  ax-bndl 1562  ax-4 1563  ax-17 1579  ax-i9 1583  ax-ial 1587  ax-i5r 1588  ax-14 2212  ax-ext 2220  ax-sep 4249  ax-pow 4311  ax-pr 4346  ax-un 4578  ax-setind 4684  ax-cnex 8270  ax-resscn 8271  ax-1cn 8272  ax-1re 8273  ax-icn 8274  ax-addcl 8275  ax-addrcl 8276  ax-mulcl 8277  ax-addcom 8279  ax-addass 8281  ax-distr 8283  ax-i2m1 8284  ax-0id 8287  ax-rnegex 8288  ax-cnre 8290  ax-pre-ltirr 8291  ax-pre-apti 8294  ax-pre-ltadd 8295

This proof depends on definitions:  df-bi 117  df-dc 847  df-3or 1010  df-3an 1011  df-tru 1405  df-fal 1408  df-nf 1514  df-sb 1816  df-eu 2089  df-mo 2090  df-clab 2225  df-cleq 2231  df-clel 2234  df-nfc 2381  df-ne 2421  df-nel 2516  df-ral 2533  df-rex 2534  df-reu 2535  df-rab 2537  df-v 2823  df-sbc 3052  df-csb 3148  df-dif 3222  df-un 3224  df-in 3226  df-ss 3233  df-if 3639  df-pw 3690  df-sn 3715  df-pr 3716  df-op 3718  df-uni 3936  df-iun 4014  df-br 4131  df-opab 4193  df-mpt 4194  df-id 4438  df-xp 4780  df-rel 4781  df-cnv 4782  df-co 4783  df-dm 4784  df-rn 4785  df-res 4786  df-ima 4787  df-iota 5337  df-fun 5379  df-fn 5380  df-f 5381  df-fv 5385  df-riota 6038  df-ov 6088  df-oprab 6089  df-mpo 6090  df-1st 6374  df-2nd 6375  df-pnf 8362  df-mnf 8363  df-xr 8364  df-ltxr 8365  df-le 8366  df-sub 8499  df-neg 8500  df-xneg 10174  df-xadd 10175

This theorem is used by:  ssblps  15526  ssbl  15527