Theorem infxrge0gelb 29531

Description: The infimum of a set of nonnegative extended reals is greater than or equal to a lower bound. (Contributed by Thierry Arnoux, 19-Jul-2020.) (Revised by AV, 4-Oct-2020.)

Hypotheses

Ref	Expression
infxrge0glb.a	|- ( ph -> A C_ ( 0 [,] +oo ) )
infxrge0glb.b	|- ( ph -> B e. ( 0 [,] +oo ) )

Assertion

Ref	Expression
infxrge0gelb	|- ( ph -> ( B <_ inf ( A , ( 0 [,] +oo ) , < ) <-> A. x e. A B <_ x ) )

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

Proof of Theorem infxrge0gelb

Dummy variables y z are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		infxrge0glb.a	. . . 4 |- ( ph -> A C_ ( 0 [,] +oo ) )
2		infxrge0glb.b	. . . 4 |- ( ph -> B e. ( 0 [,] +oo ) )
3	1, 2	infxrge0glb 29530	. . 3 |- ( ph -> (inf ( A , ( 0 [,] +oo ) , < ) < B <-> E. x e. A x < B ) )
4	3	notbid 308	. 2 |- ( ph -> ( -. inf ( A , ( 0 [,] +oo ) , < ) < B <-> -. E. x e. A x < B ) )
5		iccssxr 12256	. . . 4 |- ( 0 [,] +oo ) C_ RR*
6	5, 2	sseldi 3601	. . 3 |- ( ph -> B e. RR* )
7		xrltso 11974	. . . . . . 7 |- < Or RR*
8		soss 5053	. . . . . . 7 |- ( ( 0 [,] +oo ) C_ RR* -> ( < Or RR* -> < Or ( 0 [,] +oo ) ) )
9	5, 7, 8	mp2 9	. . . . . 6 |- < Or ( 0 [,] +oo )
10	9	a1i 11	. . . . 5 |- ( ph -> < Or ( 0 [,] +oo ) )
11		xrge0infss 29525	. . . . . 6 |- ( A C_ ( 0 [,] +oo ) -> E. x e. ( 0 [,] +oo ) ( A. y e. A -. y < x /\ A. y e. ( 0 [,] +oo ) ( x < y -> E. z e. A z < y ) ) )
12	1, 11	syl 17	. . . . 5 |- ( ph -> E. x e. ( 0 [,] +oo ) ( A. y e. A -. y < x /\ A. y e. ( 0 [,] +oo ) ( x < y -> E. z e. A z < y ) ) )
13	10, 12	infcl 8394	. . . 4 |- ( ph -> inf ( A , ( 0 [,] +oo ) , < ) e. ( 0 [,] +oo ) )
14	5, 13	sseldi 3601	. . 3 |- ( ph -> inf ( A , ( 0 [,] +oo ) , < ) e. RR* )
15	6, 14	xrlenltd 10104	. 2 |- ( ph -> ( B <_ inf ( A , ( 0 [,] +oo ) , < ) <-> -. inf ( A , ( 0 [,] +oo ) , < ) < B ) )
16	6	adantr 481	. . . . 5 |- ( ( ph /\ x e. A ) -> B e. RR* )
17	1, 5	syl6ss 3615	. . . . . 6 |- ( ph -> A C_ RR* )
18	17	sselda 3603	. . . . 5 |- ( ( ph /\ x e. A ) -> x e. RR* )
19	16, 18	xrlenltd 10104	. . . 4 |- ( ( ph /\ x e. A ) -> ( B <_ x <-> -. x < B ) )
20	19	ralbidva 2985	. . 3 |- ( ph -> ( A. x e. A B <_ x <-> A. x e. A -. x < B ) )
21		ralnex 2992	. . 3 |- ( A. x e. A -. x < B <-> -. E. x e. A x < B )
22	20, 21	syl6bb 276	. 2 |- ( ph -> ( A. x e. A B <_ x <-> -. E. x e. A x < B ) )
23	4, 15, 22	3bitr4d 300	1 |- ( ph -> ( B <_ inf ( A , ( 0 [,] +oo ) , < ) <-> A. x e. A B <_ x ) )

Syntax hints:   -. wn 3   -> wi 4   <-> wb 196   /\ wa 384   e. wcel 1990   A.wral 2912   E.wrex 2913   C_ wss 3574   class class class wbr 4653   Or wor 5034  (class class class)co 6650  infcinf 8347   0cc0 9936   +oocpnf 10071   RR*cxr 10073   < clt 10074   <_ cle 10075   [,]cicc 12178

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-icc 12182

This theorem is referenced by: (None)