Theorem xsubge0 9817

Description: Extended real version of subge0 8373. (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 9712	. 2 |- ( B e. RR* <-> ( B e. RR \/ B = +oo \/ B = -oo ) )
2		0xr 7945	. . . . 5 |- 0 e. RR*
3		rexr 7944	. . . . . 6 |- ( B e. RR -> B e. RR* )
4		xnegcl 9768	. . . . . . 7 |- ( B e. RR* -> -e B e. RR* )
5		xaddcl 9796	. . . . . . 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 9811	. . . . 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 1332	. . . 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 9799	. . . . . 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 9808	. . . . 5 |- ( ( A e. RR* /\ B e. RR ) -> ( ( A +e -e B ) +e B ) = A )
15	13, 14	breq12d 3995	. . . 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 7951	. . . . . . 7 |- +oo e. RR*
18		xrletri3 9740	. . . . . . 7 |- ( ( A e. RR* /\ +oo e. RR* ) -> ( A = +oo <-> ( A <_ +oo /\ +oo <_ A ) ) )
19	17, 18	mpan2 422	. . . . . 6 |- ( A e. RR* -> ( A = +oo <-> ( A <_ +oo /\ +oo <_ A ) ) )
20		rexr 7944	. . . . . . . . . . 11 |- ( A e. RR -> A e. RR* )
21		renepnf 7946	. . . . . . . . . . 11 |- ( A e. RR -> A =/= +oo )
22		xaddmnf1 9784	. . . . . . . . . . 11 |- ( ( A e. RR* /\ A =/= +oo ) -> ( A +e -oo ) = -oo )
23	20, 21, 22	syl2anc 409	. . . . . . . . . 10 |- ( A e. RR -> ( A +e -oo ) = -oo )
24		mnflt0 9720	. . . . . . . . . . . . 13 |- -oo < 0
25		mnfxr 7955	. . . . . . . . . . . . . . 15 |- -oo e. RR*
26		xrlenlt 7963	. . . . . . . . . . . . . . 15 |- ( ( 0 e. RR* /\ -oo e. RR* ) -> ( 0 <_ -oo <-> -. -oo < 0 ) )
27	2, 25, 26	mp2an 423	. . . . . . . . . . . . . 14 |- ( 0 <_ -oo <-> -. -oo < 0 )
28	27	biimpi 119	. . . . . . . . . . . . 13 |- ( 0 <_ -oo -> -. -oo < 0 )
29	24, 28	mt2 630	. . . . . . . . . . . 12 |- -. 0 <_ -oo
30		breq2 3986	. . . . . . . . . . . 12 |- ( ( A +e -oo ) = -oo -> ( 0 <_ ( A +e -oo ) <-> 0 <_ -oo ) )
31	29, 30	mtbiri 665	. . . . . . . . . . 11 |- ( ( A +e -oo ) = -oo -> -. 0 <_ ( A +e -oo ) )
32	31	pm2.21d 609	. . . . . . . . . 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 2229	. . . . . . . . . . . 12 |- ( A = -oo -> ( A e. RR* <-> -oo e. RR* ) )
38	25, 37	mpbiri 167	. . . . . . . . . . 11 |- ( A = -oo -> A e. RR* )
39		mnfnepnf 7954	. . . . . . . . . . . 12 |- -oo =/= +oo
40		neeq1 2349	. . . . . . . . . . . 12 |- ( A = -oo -> ( A =/= +oo <-> -oo =/= +oo ) )
41	39, 40	mpbiri 167	. . . . . . . . . . 11 |- ( A = -oo -> A =/= +oo )
42	38, 41, 22	syl2anc 409	. . . . . . . . . 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 9712	. . . . . . . . 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 1297	. . . . . . 7 |- ( A e. RR* -> ( 0 <_ ( A +e -oo ) -> A = +oo ) )
48		0le0 8946	. . . . . . . 8 |- 0 <_ 0
49		oveq1 5849	. . . . . . . . 9 |- ( A = +oo -> ( A +e -oo ) = ( +oo +e -oo ) )
50		pnfaddmnf 9786	. . . . . . . . 9 |- ( +oo +e -oo ) = 0
51	49, 50	eqtrdi 2215	. . . . . . . 8 |- ( A = +oo -> ( A +e -oo ) = 0 )
52	48, 51	breqtrrid 4020	. . . . . . 7 |- ( A = +oo -> 0 <_ ( A +e -oo ) )
53	47, 52	impbid1 141	. . . . . 6 |- ( A e. RR* -> ( 0 <_ ( A +e -oo ) <-> A = +oo ) )
54		pnfge 9725	. . . . . . 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 9763	. . . . . . . 8 |- ( B = +oo -> -e B = -e +oo )
59		xnegpnf 9764	. . . . . . . 8 |- -e +oo = -oo
60	58, 59	eqtrdi 2215	. . . . . . 7 |- ( B = +oo -> -e B = -oo )
61	60	adantl 275	. . . . . 6 |- ( ( A e. RR* /\ B = +oo ) -> -e B = -oo )
62	61	oveq2d 5858	. . . . 5 |- ( ( A e. RR* /\ B = +oo ) -> ( A +e -e B ) = ( A +e -oo ) )
63	62	breq2d 3994	. . . 4 |- ( ( A e. RR* /\ B = +oo ) -> ( 0 <_ ( A +e -e B ) <-> 0 <_ ( A +e -oo ) ) )
64		breq1 3985	. . . . 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 5849	. . . . . . . . . 10 |- ( A = -oo -> ( A +e +oo ) = ( -oo +e +oo ) )
68		mnfaddpnf 9787	. . . . . . . . . 10 |- ( -oo +e +oo ) = 0
69	67, 68	eqtrdi 2215	. . . . . . . . 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 4020	. . . . . . 7 |- ( ( A e. RR* /\ A = -oo ) -> 0 <_ ( A +e +oo ) )
72		df-ne 2337	. . . . . . . 8 |- ( A =/= -oo <-> -. A = -oo )
73		0lepnf 9726	. . . . . . . . 9 |- 0 <_ +oo
74		xaddpnf1 9782	. . . . . . . . 9 |- ( ( A e. RR* /\ A =/= -oo ) -> ( A +e +oo ) = +oo )
75	73, 74	breqtrrid 4020	. . . . . . . 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 9779	. . . . . . . 8 |- ( A e. RR* -> DECID A = -oo )
78		exmiddc 826	. . . . . . . 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 788	. . . . . 6 |- ( A e. RR* -> 0 <_ ( A +e +oo ) )
81		mnfle 9728	. . . . . 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 9763	. . . . . . . 8 |- ( B = -oo -> -e B = -e -oo )
85		xnegmnf 9765	. . . . . . . 8 |- -e -oo = +oo
86	84, 85	eqtrdi 2215	. . . . . . 7 |- ( B = -oo -> -e B = +oo )
87	86	adantl 275	. . . . . 6 |- ( ( A e. RR* /\ B = -oo ) -> -e B = +oo )
88	87	oveq2d 5858	. . . . 5 |- ( ( A e. RR* /\ B = -oo ) -> ( A +e -e B ) = ( A +e +oo ) )
89	88	breq2d 3994	. . . 4 |- ( ( A e. RR* /\ B = -oo ) -> ( 0 <_ ( A +e -e B ) <-> 0 <_ ( A +e +oo ) ) )
90		breq1 3985	. . . . 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 1296	. 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 698  DECID wdc 824   \/ w3o 967   = wceq 1343   e. wcel 2136   =/= wne 2336   class class class wbr 3982  (class class class)co 5842   RRcr 7752   0cc0 7753   +oocpnf 7930   -oocmnf 7931   RR*cxr 7932   < clt 7933   <_ cle 7934   -ecxne 9705   +ecxad 9706

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 604  ax-in2 605  ax-io 699  ax-5 1435  ax-7 1436  ax-gen 1437  ax-ie1 1481  ax-ie2 1482  ax-8 1492  ax-10 1493  ax-11 1494  ax-i12 1495  ax-bndl 1497  ax-4 1498  ax-17 1514  ax-i9 1518  ax-ial 1522  ax-i5r 1523  ax-13 2138  ax-14 2139  ax-ext 2147  ax-sep 4100  ax-pow 4153  ax-pr 4187  ax-un 4411  ax-setind 4514  ax-cnex 7844  ax-resscn 7845  ax-1cn 7846  ax-1re 7847  ax-icn 7848  ax-addcl 7849  ax-addrcl 7850  ax-mulcl 7851  ax-addcom 7853  ax-addass 7855  ax-distr 7857  ax-i2m1 7858  ax-0id 7861  ax-rnegex 7862  ax-cnre 7864  ax-pre-ltirr 7865  ax-pre-apti 7868  ax-pre-ltadd 7869

This theorem depends on definitions:  df-bi 116  df-dc 825  df-3or 969  df-3an 970  df-tru 1346  df-fal 1349  df-nf 1449  df-sb 1751  df-eu 2017  df-mo 2018  df-clab 2152  df-cleq 2158  df-clel 2161  df-nfc 2297  df-ne 2337  df-nel 2432  df-ral 2449  df-rex 2450  df-reu 2451  df-rab 2453  df-v 2728  df-sbc 2952  df-csb 3046  df-dif 3118  df-un 3120  df-in 3122  df-ss 3129  df-if 3521  df-pw 3561  df-sn 3582  df-pr 3583  df-op 3585  df-uni 3790  df-iun 3868  df-br 3983  df-opab 4044  df-mpt 4045  df-id 4271  df-xp 4610  df-rel 4611  df-cnv 4612  df-co 4613  df-dm 4614  df-rn 4615  df-res 4616  df-ima 4617  df-iota 5153  df-fun 5190  df-fn 5191  df-f 5192  df-fv 5196  df-riota 5798  df-ov 5845  df-oprab 5846  df-mpo 5847  df-1st 6108  df-2nd 6109  df-pnf 7935  df-mnf 7936  df-xr 7937  df-ltxr 7938  df-le 7939  df-sub 8071  df-neg 8072  df-xneg 9708  df-xadd 9709

This theorem is referenced by:  ssblps  13065  ssbl  13066