Theorem fge0iccico 40587

Description: A range of nonnegative extended reals without plus infinity. (Contributed by Glauco Siliprandi, 17-Aug-2020.)

Hypotheses

Ref	Expression
fge0iccico.f	|- ( ph -> F : X --> ( 0 [,] +oo ) )
fge0iccico.re	|- ( ph -> -. +oo e. ran F )

Assertion

Ref	Expression
fge0iccico	|- ( ph -> F : X --> ( 0 [,) +oo ) )

Proof of Theorem fge0iccico

Dummy variable x is distinct from all other variables.

Step	Hyp	Ref	Expression
1		fge0iccico.f	. . . 4 |- ( ph -> F : X --> ( 0 [,] +oo ) )
2		ffn 6045	. . . 4 |- ( F : X --> ( 0 [,] +oo ) -> F Fn X )
3	1, 2	syl 17	. . 3 |- ( ph -> F Fn X )
4		0xr 10086	. . . . . 6 |- 0 e. RR*
5	4	a1i 11	. . . . 5 |- ( ( ph /\ x e. X ) -> 0 e. RR* )
6		pnfxr 10092	. . . . . 6 |- +oo e. RR*
7	6	a1i 11	. . . . 5 |- ( ( ph /\ x e. X ) -> +oo e. RR* )
8		iccssxr 12256	. . . . . 6 |- ( 0 [,] +oo ) C_ RR*
9	1	ffvelrnda 6359	. . . . . 6 |- ( ( ph /\ x e. X ) -> ( F ` x ) e. ( 0 [,] +oo ) )
10	8, 9	sseldi 3601	. . . . 5 |- ( ( ph /\ x e. X ) -> ( F ` x ) e. RR* )
11		iccgelb 12230	. . . . . 6 |- ( ( 0 e. RR* /\ +oo e. RR* /\ ( F ` x ) e. ( 0 [,] +oo ) ) -> 0 <_ ( F ` x ) )
12	5, 7, 9, 11	syl3anc 1326	. . . . 5 |- ( ( ph /\ x e. X ) -> 0 <_ ( F ` x ) )
13	10	adantr 481	. . . . . . . . 9 |- ( ( ( ph /\ x e. X ) /\ -. ( F ` x ) < +oo ) -> ( F ` x ) e. RR* )
14		simpr 477	. . . . . . . . . 10 |- ( ( ( ph /\ x e. X ) /\ -. ( F ` x ) < +oo ) -> -. ( F ` x ) < +oo )
15	6	a1i 11	. . . . . . . . . . 11 |- ( ( ( ph /\ x e. X ) /\ -. ( F ` x ) < +oo ) -> +oo e. RR* )
16	15, 13	xrlenltd 10104	. . . . . . . . . 10 |- ( ( ( ph /\ x e. X ) /\ -. ( F ` x ) < +oo ) -> ( +oo <_ ( F ` x ) <-> -. ( F ` x ) < +oo ) )
17	14, 16	mpbird 247	. . . . . . . . 9 |- ( ( ( ph /\ x e. X ) /\ -. ( F ` x ) < +oo ) -> +oo <_ ( F ` x ) )
18	13, 17	xrgepnfd 39547	. . . . . . . 8 |- ( ( ( ph /\ x e. X ) /\ -. ( F ` x ) < +oo ) -> ( F ` x ) = +oo )
19	18	eqcomd 2628	. . . . . . 7 |- ( ( ( ph /\ x e. X ) /\ -. ( F ` x ) < +oo ) -> +oo = ( F ` x ) )
20		ffun 6048	. . . . . . . . . . 11 |- ( F : X --> ( 0 [,] +oo ) -> Fun F )
21	1, 20	syl 17	. . . . . . . . . 10 |- ( ph -> Fun F )
22	21	adantr 481	. . . . . . . . 9 |- ( ( ph /\ x e. X ) -> Fun F )
23		simpr 477	. . . . . . . . . 10 |- ( ( ph /\ x e. X ) -> x e. X )
24		fdm 6051	. . . . . . . . . . . . 13 |- ( F : X --> ( 0 [,] +oo ) -> dom F = X )
25	24	eqcomd 2628	. . . . . . . . . . . 12 |- ( F : X --> ( 0 [,] +oo ) -> X = dom F )
26	1, 25	syl 17	. . . . . . . . . . 11 |- ( ph -> X = dom F )
27	26	adantr 481	. . . . . . . . . 10 |- ( ( ph /\ x e. X ) -> X = dom F )
28	23, 27	eleqtrd 2703	. . . . . . . . 9 |- ( ( ph /\ x e. X ) -> x e. dom F )
29		fvelrn 6352	. . . . . . . . 9 |- ( ( Fun F /\ x e. dom F ) -> ( F ` x ) e. ran F )
30	22, 28, 29	syl2anc 693	. . . . . . . 8 |- ( ( ph /\ x e. X ) -> ( F ` x ) e. ran F )
31	30	adantr 481	. . . . . . 7 |- ( ( ( ph /\ x e. X ) /\ -. ( F ` x ) < +oo ) -> ( F ` x ) e. ran F )
32	19, 31	eqeltrd 2701	. . . . . 6 |- ( ( ( ph /\ x e. X ) /\ -. ( F ` x ) < +oo ) -> +oo e. ran F )
33		fge0iccico.re	. . . . . . 7 |- ( ph -> -. +oo e. ran F )
34	33	ad2antrr 762	. . . . . 6 |- ( ( ( ph /\ x e. X ) /\ -. ( F ` x ) < +oo ) -> -. +oo e. ran F )
35	32, 34	condan 835	. . . . 5 |- ( ( ph /\ x e. X ) -> ( F ` x ) < +oo )
36	5, 7, 10, 12, 35	elicod 12224	. . . 4 |- ( ( ph /\ x e. X ) -> ( F ` x ) e. ( 0 [,) +oo ) )
37	36	ralrimiva 2966	. . 3 |- ( ph -> A. x e. X ( F ` x ) e. ( 0 [,) +oo ) )
38	3, 37	jca 554	. 2 |- ( ph -> ( F Fn X /\ A. x e. X ( F ` x ) e. ( 0 [,) +oo ) ) )
39		ffnfv 6388	. 2 |- ( F : X --> ( 0 [,) +oo ) <-> ( F Fn X /\ A. x e. X ( F ` x ) e. ( 0 [,) +oo ) ) )
40	38, 39	sylibr 224	1 |- ( ph -> F : X --> ( 0 [,) +oo ) )

Syntax hints:   -. wn 3   -> wi 4   /\ wa 384   = wceq 1483   e. wcel 1990   A.wral 2912   class class class wbr 4653   dom cdm 5114   ran crn 5115   Fun wfun 5882   Fn wfn 5883   -->wf 5884   ` cfv 5888  (class class class)co 6650   0cc0 9936   +oocpnf 10071   RR*cxr 10073   < clt 10074   <_ cle 10075   [,)cico 12177   [,]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-i2m1 10004  ax-1ne0 10005  ax-rnegex 10007  ax-rrecex 10008  ax-cnre 10009  ax-pre-lttri 10010  ax-pre-lttrn 10011

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-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-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-pnf 10076  df-mnf 10077  df-xr 10078  df-ltxr 10079  df-le 10080  df-ico 12181  df-icc 12182

This theorem is referenced by:  fge0iccre  40591  sge00  40593  sge0sn  40596  sge0tsms  40597  sge0cl  40598  sge0supre  40606  sge0sup  40608  sge0less  40609  sge0rnbnd  40610  sge0ltfirp  40617  sge0resplit  40623  sge0le  40624  sge0split  40626  sge0iunmptlemre  40632