Theorem infrpge 39567

Description: The infimum of a non empty, bounded subset of extended reals can be approximated from above by an element of the set. (Contributed by Glauco Siliprandi, 11-Oct-2020.)

Hypotheses

Ref	Expression
infrpge.xph	|- F/ x ph
infrpge.a	|- ( ph -> A C_ RR* )
infrpge.an0	|- ( ph -> A =/= (/) )
infrpge.bnd	|- ( ph -> E. x e. RR A. y e. A x <_ y )
infrpge.b	|- ( ph -> B e. RR+ )

Assertion

Ref	Expression
infrpge	|- ( ph -> E. z e. A z <_ (inf ( A , RR* , < ) +e B ) )

Distinct variable groups:   x, A, y   z, A   z, B   ph, z

Allowed substitution hints:   ph( x, y)   B( x, y)

Proof of Theorem infrpge

Step	Hyp	Ref	Expression
1		infrpge.an0	. . . . . 6 |- ( ph -> A =/= (/) )
2		n0 3931	. . . . . . 7 |- ( A =/= (/) <-> E. z z e. A )
3	2	biimpi 206	. . . . . 6 |- ( A =/= (/) -> E. z z e. A )
4	1, 3	syl 17	. . . . 5 |- ( ph -> E. z z e. A )
5	4	adantr 481	. . . 4 |- ( ( ph /\ inf ( A , RR* , < ) = +oo ) -> E. z z e. A )
6		nfv 1843	. . . . 5 |- F/ z ( ph /\ inf ( A , RR* , < ) = +oo )
7		simpr 477	. . . . . . 7 |- ( ( ( ph /\ inf ( A , RR* , < ) = +oo ) /\ z e. A ) -> z e. A )
8		infrpge.a	. . . . . . . . . . . 12 |- ( ph -> A C_ RR* )
9	8	adantr 481	. . . . . . . . . . 11 |- ( ( ph /\ z e. A ) -> A C_ RR* )
10		simpr 477	. . . . . . . . . . 11 |- ( ( ph /\ z e. A ) -> z e. A )
11	9, 10	sseldd 3604	. . . . . . . . . 10 |- ( ( ph /\ z e. A ) -> z e. RR* )
12		pnfge 11964	. . . . . . . . . 10 |- ( z e. RR* -> z <_ +oo )
13	11, 12	syl 17	. . . . . . . . 9 |- ( ( ph /\ z e. A ) -> z <_ +oo )
14	13	adantlr 751	. . . . . . . 8 |- ( ( ( ph /\ inf ( A , RR* , < ) = +oo ) /\ z e. A ) -> z <_ +oo )
15		oveq1 6657	. . . . . . . . . . 11 |- (inf ( A , RR* , < ) = +oo -> (inf ( A , RR* , < ) +e B ) = ( +oo +e B ) )
16	15	adantl 482	. . . . . . . . . 10 |- ( ( ph /\ inf ( A , RR* , < ) = +oo ) -> (inf ( A , RR* , < ) +e B ) = ( +oo +e B ) )
17		infrpge.b	. . . . . . . . . . . . 13 |- ( ph -> B e. RR+ )
18	17	rpxrd 11873	. . . . . . . . . . . 12 |- ( ph -> B e. RR* )
19	17	rpred 11872	. . . . . . . . . . . . 13 |- ( ph -> B e. RR )
20		renemnf 10088	. . . . . . . . . . . . 13 |- ( B e. RR -> B =/= -oo )
21	19, 20	syl 17	. . . . . . . . . . . 12 |- ( ph -> B =/= -oo )
22		xaddpnf2 12058	. . . . . . . . . . . 12 |- ( ( B e. RR* /\ B =/= -oo ) -> ( +oo +e B ) = +oo )
23	18, 21, 22	syl2anc 693	. . . . . . . . . . 11 |- ( ph -> ( +oo +e B ) = +oo )
24	23	adantr 481	. . . . . . . . . 10 |- ( ( ph /\ inf ( A , RR* , < ) = +oo ) -> ( +oo +e B ) = +oo )
25	16, 24	eqtr2d 2657	. . . . . . . . 9 |- ( ( ph /\ inf ( A , RR* , < ) = +oo ) -> +oo = (inf ( A , RR* , < ) +e B ) )
26	25	adantr 481	. . . . . . . 8 |- ( ( ( ph /\ inf ( A , RR* , < ) = +oo ) /\ z e. A ) -> +oo = (inf ( A , RR* , < ) +e B ) )
27	14, 26	breqtrd 4679	. . . . . . 7 |- ( ( ( ph /\ inf ( A , RR* , < ) = +oo ) /\ z e. A ) -> z <_ (inf ( A , RR* , < ) +e B ) )
28	7, 27	jca 554	. . . . . 6 |- ( ( ( ph /\ inf ( A , RR* , < ) = +oo ) /\ z e. A ) -> ( z e. A /\ z <_ (inf ( A , RR* , < ) +e B ) ) )
29	28	ex 450	. . . . 5 |- ( ( ph /\ inf ( A , RR* , < ) = +oo ) -> ( z e. A -> ( z e. A /\ z <_ (inf ( A , RR* , < ) +e B ) ) ) )
30	6, 29	eximd 2085	. . . 4 |- ( ( ph /\ inf ( A , RR* , < ) = +oo ) -> ( E. z z e. A -> E. z ( z e. A /\ z <_ (inf ( A , RR* , < ) +e B ) ) ) )
31	5, 30	mpd 15	. . 3 |- ( ( ph /\ inf ( A , RR* , < ) = +oo ) -> E. z ( z e. A /\ z <_ (inf ( A , RR* , < ) +e B ) ) )
32		df-rex 2918	. . 3 |- ( E. z e. A z <_ (inf ( A , RR* , < ) +e B ) <-> E. z ( z e. A /\ z <_ (inf ( A , RR* , < ) +e B ) ) )
33	31, 32	sylibr 224	. 2 |- ( ( ph /\ inf ( A , RR* , < ) = +oo ) -> E. z e. A z <_ (inf ( A , RR* , < ) +e B ) )
34		simpl 473	. . . 4 |- ( ( ph /\ -. inf ( A , RR* , < ) = +oo ) -> ph )
35		infrpge.bnd	. . . . . . . . 9 |- ( ph -> E. x e. RR A. y e. A x <_ y )
36		infrpge.xph	. . . . . . . . . 10 |- F/ x ph
37		nfv 1843	. . . . . . . . . 10 |- F/ x -oo < inf ( A , RR* , < )
38		mnfxr 10096	. . . . . . . . . . . . 13 |- -oo e. RR*
39	38	a1i 11	. . . . . . . . . . . 12 |- ( ( ph /\ x e. RR /\ A. y e. A x <_ y ) -> -oo e. RR* )
40		rexr 10085	. . . . . . . . . . . . 13 |- ( x e. RR -> x e. RR* )
41	40	3ad2ant2 1083	. . . . . . . . . . . 12 |- ( ( ph /\ x e. RR /\ A. y e. A x <_ y ) -> x e. RR* )
42		infxrcl 12163	. . . . . . . . . . . . . 14 |- ( A C_ RR* -> inf ( A , RR* , < ) e. RR* )
43	8, 42	syl 17	. . . . . . . . . . . . 13 |- ( ph -> inf ( A , RR* , < ) e. RR* )
44	43	3ad2ant1 1082	. . . . . . . . . . . 12 |- ( ( ph /\ x e. RR /\ A. y e. A x <_ y ) -> inf ( A , RR* , < ) e. RR* )
45		mnflt 11957	. . . . . . . . . . . . 13 |- ( x e. RR -> -oo < x )
46	45	3ad2ant2 1083	. . . . . . . . . . . 12 |- ( ( ph /\ x e. RR /\ A. y e. A x <_ y ) -> -oo < x )
47		simp3 1063	. . . . . . . . . . . . 13 |- ( ( ph /\ x e. RR /\ A. y e. A x <_ y ) -> A. y e. A x <_ y )
48	8	adantr 481	. . . . . . . . . . . . . . 15 |- ( ( ph /\ x e. RR ) -> A C_ RR* )
49	40	adantl 482	. . . . . . . . . . . . . . 15 |- ( ( ph /\ x e. RR ) -> x e. RR* )
50		infxrgelb 12165	. . . . . . . . . . . . . . 15 |- ( ( A C_ RR* /\ x e. RR* ) -> ( x <_ inf ( A , RR* , < ) <-> A. y e. A x <_ y ) )
51	48, 49, 50	syl2anc 693	. . . . . . . . . . . . . 14 |- ( ( ph /\ x e. RR ) -> ( x <_ inf ( A , RR* , < ) <-> A. y e. A x <_ y ) )
52	51	3adant3 1081	. . . . . . . . . . . . 13 |- ( ( ph /\ x e. RR /\ A. y e. A x <_ y ) -> ( x <_ inf ( A , RR* , < ) <-> A. y e. A x <_ y ) )
53	47, 52	mpbird 247	. . . . . . . . . . . 12 |- ( ( ph /\ x e. RR /\ A. y e. A x <_ y ) -> x <_ inf ( A , RR* , < ) )
54	39, 41, 44, 46, 53	xrltletrd 11992	. . . . . . . . . . 11 |- ( ( ph /\ x e. RR /\ A. y e. A x <_ y ) -> -oo < inf ( A , RR* , < ) )
55	54	3exp 1264	. . . . . . . . . 10 |- ( ph -> ( x e. RR -> ( A. y e. A x <_ y -> -oo < inf ( A , RR* , < ) ) ) )
56	36, 37, 55	rexlimd 3026	. . . . . . . . 9 |- ( ph -> ( E. x e. RR A. y e. A x <_ y -> -oo < inf ( A , RR* , < ) ) )
57	35, 56	mpd 15	. . . . . . . 8 |- ( ph -> -oo < inf ( A , RR* , < ) )
58	57	adantr 481	. . . . . . 7 |- ( ( ph /\ -. inf ( A , RR* , < ) = +oo ) -> -oo < inf ( A , RR* , < ) )
59		neqne 2802	. . . . . . . . 9 |- ( -. inf ( A , RR* , < ) = +oo -> inf ( A , RR* , < ) =/= +oo )
60	59	adantl 482	. . . . . . . 8 |- ( ( ph /\ -. inf ( A , RR* , < ) = +oo ) -> inf ( A , RR* , < ) =/= +oo )
61	43	adantr 481	. . . . . . . 8 |- ( ( ph /\ -. inf ( A , RR* , < ) = +oo ) -> inf ( A , RR* , < ) e. RR* )
62	60, 61	nepnfltpnf 39558	. . . . . . 7 |- ( ( ph /\ -. inf ( A , RR* , < ) = +oo ) -> inf ( A , RR* , < ) < +oo )
63	58, 62	jca 554	. . . . . 6 |- ( ( ph /\ -. inf ( A , RR* , < ) = +oo ) -> ( -oo < inf ( A , RR* , < ) /\ inf ( A , RR* , < ) < +oo ) )
64		xrrebnd 11999	. . . . . . . 8 |- (inf ( A , RR* , < ) e. RR* -> (inf ( A , RR* , < ) e. RR <-> ( -oo < inf ( A , RR* , < ) /\ inf ( A , RR* , < ) < +oo ) ) )
65	43, 64	syl 17	. . . . . . 7 |- ( ph -> (inf ( A , RR* , < ) e. RR <-> ( -oo < inf ( A , RR* , < ) /\ inf ( A , RR* , < ) < +oo ) ) )
66	65	adantr 481	. . . . . 6 |- ( ( ph /\ -. inf ( A , RR* , < ) = +oo ) -> (inf ( A , RR* , < ) e. RR <-> ( -oo < inf ( A , RR* , < ) /\ inf ( A , RR* , < ) < +oo ) ) )
67	63, 66	mpbird 247	. . . . 5 |- ( ( ph /\ -. inf ( A , RR* , < ) = +oo ) -> inf ( A , RR* , < ) e. RR )
68		simpr 477	. . . . . . . 8 |- ( ( ph /\ inf ( A , RR* , < ) e. RR ) -> inf ( A , RR* , < ) e. RR )
69	17	adantr 481	. . . . . . . 8 |- ( ( ph /\ inf ( A , RR* , < ) e. RR ) -> B e. RR+ )
70	68, 69	ltaddrpd 11905	. . . . . . 7 |- ( ( ph /\ inf ( A , RR* , < ) e. RR ) -> inf ( A , RR* , < ) < (inf ( A , RR* , < ) + B ) )
71	19	adantr 481	. . . . . . . . 9 |- ( ( ph /\ inf ( A , RR* , < ) e. RR ) -> B e. RR )
72		rexadd 12063	. . . . . . . . 9 |- ( (inf ( A , RR* , < ) e. RR /\ B e. RR ) -> (inf ( A , RR* , < ) +e B ) = (inf ( A , RR* , < ) + B ) )
73	68, 71, 72	syl2anc 693	. . . . . . . 8 |- ( ( ph /\ inf ( A , RR* , < ) e. RR ) -> (inf ( A , RR* , < ) +e B ) = (inf ( A , RR* , < ) + B ) )
74	73	eqcomd 2628	. . . . . . 7 |- ( ( ph /\ inf ( A , RR* , < ) e. RR ) -> (inf ( A , RR* , < ) + B ) = (inf ( A , RR* , < ) +e B ) )
75	70, 74	breqtrd 4679	. . . . . 6 |- ( ( ph /\ inf ( A , RR* , < ) e. RR ) -> inf ( A , RR* , < ) < (inf ( A , RR* , < ) +e B ) )
76	43	adantr 481	. . . . . . 7 |- ( ( ph /\ inf ( A , RR* , < ) e. RR ) -> inf ( A , RR* , < ) e. RR* )
77	43, 18	xaddcld 12131	. . . . . . . 8 |- ( ph -> (inf ( A , RR* , < ) +e B ) e. RR* )
78	77	adantr 481	. . . . . . 7 |- ( ( ph /\ inf ( A , RR* , < ) e. RR ) -> (inf ( A , RR* , < ) +e B ) e. RR* )
79		xrltnle 10105	. . . . . . 7 |- ( (inf ( A , RR* , < ) e. RR* /\ (inf ( A , RR* , < ) +e B ) e. RR* ) -> (inf ( A , RR* , < ) < (inf ( A , RR* , < ) +e B ) <-> -. (inf ( A , RR* , < ) +e B ) <_ inf ( A , RR* , < ) ) )
80	76, 78, 79	syl2anc 693	. . . . . 6 |- ( ( ph /\ inf ( A , RR* , < ) e. RR ) -> (inf ( A , RR* , < ) < (inf ( A , RR* , < ) +e B ) <-> -. (inf ( A , RR* , < ) +e B ) <_ inf ( A , RR* , < ) ) )
81	75, 80	mpbid 222	. . . . 5 |- ( ( ph /\ inf ( A , RR* , < ) e. RR ) -> -. (inf ( A , RR* , < ) +e B ) <_ inf ( A , RR* , < ) )
82	34, 67, 81	syl2anc 693	. . . 4 |- ( ( ph /\ -. inf ( A , RR* , < ) = +oo ) -> -. (inf ( A , RR* , < ) +e B ) <_ inf ( A , RR* , < ) )
83		simpr 477	. . . . . 6 |- ( ( ph /\ -. (inf ( A , RR* , < ) +e B ) <_ inf ( A , RR* , < ) ) -> -. (inf ( A , RR* , < ) +e B ) <_ inf ( A , RR* , < ) )
84		simpl 473	. . . . . . 7 |- ( ( ph /\ -. (inf ( A , RR* , < ) +e B ) <_ inf ( A , RR* , < ) ) -> ph )
85		infxrgelb 12165	. . . . . . . 8 |- ( ( A C_ RR* /\ (inf ( A , RR* , < ) +e B ) e. RR* ) -> ( (inf ( A , RR* , < ) +e B ) <_ inf ( A , RR* , < ) <-> A. z e. A (inf ( A , RR* , < ) +e B ) <_ z ) )
86	8, 77, 85	syl2anc 693	. . . . . . 7 |- ( ph -> ( (inf ( A , RR* , < ) +e B ) <_ inf ( A , RR* , < ) <-> A. z e. A (inf ( A , RR* , < ) +e B ) <_ z ) )
87	84, 86	syl 17	. . . . . 6 |- ( ( ph /\ -. (inf ( A , RR* , < ) +e B ) <_ inf ( A , RR* , < ) ) -> ( (inf ( A , RR* , < ) +e B ) <_ inf ( A , RR* , < ) <-> A. z e. A (inf ( A , RR* , < ) +e B ) <_ z ) )
88	83, 87	mtbid 314	. . . . 5 |- ( ( ph /\ -. (inf ( A , RR* , < ) +e B ) <_ inf ( A , RR* , < ) ) -> -. A. z e. A (inf ( A , RR* , < ) +e B ) <_ z )
89		rexnal 2995	. . . . 5 |- ( E. z e. A -. (inf ( A , RR* , < ) +e B ) <_ z <-> -. A. z e. A (inf ( A , RR* , < ) +e B ) <_ z )
90	88, 89	sylibr 224	. . . 4 |- ( ( ph /\ -. (inf ( A , RR* , < ) +e B ) <_ inf ( A , RR* , < ) ) -> E. z e. A -. (inf ( A , RR* , < ) +e B ) <_ z )
91	34, 82, 90	syl2anc 693	. . 3 |- ( ( ph /\ -. inf ( A , RR* , < ) = +oo ) -> E. z e. A -. (inf ( A , RR* , < ) +e B ) <_ z )
92	11	adantr 481	. . . . . . 7 |- ( ( ( ph /\ z e. A ) /\ -. (inf ( A , RR* , < ) +e B ) <_ z ) -> z e. RR* )
93	77	ad2antrr 762	. . . . . . 7 |- ( ( ( ph /\ z e. A ) /\ -. (inf ( A , RR* , < ) +e B ) <_ z ) -> (inf ( A , RR* , < ) +e B ) e. RR* )
94		simpr 477	. . . . . . . 8 |- ( ( ( ph /\ z e. A ) /\ -. (inf ( A , RR* , < ) +e B ) <_ z ) -> -. (inf ( A , RR* , < ) +e B ) <_ z )
95		xrltnle 10105	. . . . . . . . 9 |- ( ( z e. RR* /\ (inf ( A , RR* , < ) +e B ) e. RR* ) -> ( z < (inf ( A , RR* , < ) +e B ) <-> -. (inf ( A , RR* , < ) +e B ) <_ z ) )
96	92, 93, 95	syl2anc 693	. . . . . . . 8 |- ( ( ( ph /\ z e. A ) /\ -. (inf ( A , RR* , < ) +e B ) <_ z ) -> ( z < (inf ( A , RR* , < ) +e B ) <-> -. (inf ( A , RR* , < ) +e B ) <_ z ) )
97	94, 96	mpbird 247	. . . . . . 7 |- ( ( ( ph /\ z e. A ) /\ -. (inf ( A , RR* , < ) +e B ) <_ z ) -> z < (inf ( A , RR* , < ) +e B ) )
98	92, 93, 97	xrltled 39486	. . . . . 6 |- ( ( ( ph /\ z e. A ) /\ -. (inf ( A , RR* , < ) +e B ) <_ z ) -> z <_ (inf ( A , RR* , < ) +e B ) )
99	98	ex 450	. . . . 5 |- ( ( ph /\ z e. A ) -> ( -. (inf ( A , RR* , < ) +e B ) <_ z -> z <_ (inf ( A , RR* , < ) +e B ) ) )
100	99	adantlr 751	. . . 4 |- ( ( ( ph /\ -. inf ( A , RR* , < ) = +oo ) /\ z e. A ) -> ( -. (inf ( A , RR* , < ) +e B ) <_ z -> z <_ (inf ( A , RR* , < ) +e B ) ) )
101	100	reximdva 3017	. . 3 |- ( ( ph /\ -. inf ( A , RR* , < ) = +oo ) -> ( E. z e. A -. (inf ( A , RR* , < ) +e B ) <_ z -> E. z e. A z <_ (inf ( A , RR* , < ) +e B ) ) )
102	91, 101	mpd 15	. 2 |- ( ( ph /\ -. inf ( A , RR* , < ) = +oo ) -> E. z e. A z <_ (inf ( A , RR* , < ) +e B ) )
103	33, 102	pm2.61dan 832	1 |- ( ph -> E. z e. A z <_ (inf ( A , RR* , < ) +e B ) )

Syntax hints:   -. wn 3   -> wi 4   <-> wb 196   /\ wa 384   /\ w3a 1037   = wceq 1483   E.wex 1704   F/wnf 1708   e. wcel 1990   =/= wne 2794   A.wral 2912   E.wrex 2913   C_ wss 3574   (/)c0 3915   class class class wbr 4653  (class class class)co 6650  infcinf 8347   RRcr 9935   + caddc 9939   +oocpnf 10071   -oocmnf 10072   RR*cxr 10073   < clt 10074   <_ cle 10075   RR+crp 11832   +ecxad 11944

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-8 1992  ax-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246  ax-ext 2602  ax-sep 4781  ax-nul 4789  ax-pow 4843  ax-pr 4906  ax-un 6949  ax-cnex 9992  ax-resscn 9993  ax-1cn 9994  ax-icn 9995  ax-addcl 9996  ax-addrcl 9997  ax-mulcl 9998  ax-mulrcl 9999  ax-mulcom 10000  ax-addass 10001  ax-mulass 10002  ax-distr 10003  ax-i2m1 10004  ax-1ne0 10005  ax-1rid 10006  ax-rnegex 10007  ax-rrecex 10008  ax-cnre 10009  ax-pre-lttri 10010  ax-pre-lttrn 10011  ax-pre-ltadd 10012  ax-pre-mulgt0 10013  ax-pre-sup 10014

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3or 1038  df-3an 1039  df-tru 1486  df-ex 1705  df-nf 1710  df-sb 1881  df-eu 2474  df-mo 2475  df-clab 2609  df-cleq 2615  df-clel 2618  df-nfc 2753  df-ne 2795  df-nel 2898  df-ral 2917  df-rex 2918  df-reu 2919  df-rmo 2920  df-rab 2921  df-v 3202  df-sbc 3436  df-csb 3534  df-dif 3577  df-un 3579  df-in 3581  df-ss 3588  df-nul 3916  df-if 4087  df-pw 4160  df-sn 4178  df-pr 4180  df-op 4184  df-uni 4437  df-iun 4522  df-br 4654  df-opab 4713  df-mpt 4730  df-id 5024  df-po 5035  df-so 5036  df-xp 5120  df-rel 5121  df-cnv 5122  df-co 5123  df-dm 5124  df-rn 5125  df-res 5126  df-ima 5127  df-iota 5851  df-fun 5890  df-fn 5891  df-f 5892  df-f1 5893  df-fo 5894  df-f1o 5895  df-fv 5896  df-riota 6611  df-ov 6653  df-oprab 6654  df-mpt2 6655  df-1st 7168  df-2nd 7169  df-er 7742  df-en 7956  df-dom 7957  df-sdom 7958  df-sup 8348  df-inf 8349  df-pnf 10076  df-mnf 10077  df-xr 10078  df-ltxr 10079  df-le 10080  df-sub 10268  df-neg 10269  df-rp 11833  df-xadd 11947

This theorem is referenced by:  infleinf  39588  infrpgernmpt  39695  ovnlerp  40776