Theorem limsupresre 39928

Description: The supremum limit of a function only depends on the real part of its domain. (Contributed by Glauco Siliprandi, 23-Oct-2021.)

Hypothesis

Ref	Expression
limsupresre.1	|- ( ph -> F e. V )

Assertion

Ref	Expression
limsupresre	|- ( ph -> ( limsup ` ( F |` RR ) ) = ( limsup ` F ) )

Proof of Theorem limsupresre

Dummy variable k is distinct from all other variables.

Step	Hyp	Ref	Expression
1		id 22	. . . . . . . . . 10 |- ( k e. RR -> k e. RR )
2		pnfxr 10092	. . . . . . . . . . 11 |- +oo e. RR*
3	2	a1i 11	. . . . . . . . . 10 |- ( k e. RR -> +oo e. RR* )
4		icossre 12254	. . . . . . . . . 10 |- ( ( k e. RR /\ +oo e. RR* ) -> ( k [,) +oo ) C_ RR )
5	1, 3, 4	syl2anc 693	. . . . . . . . 9 |- ( k e. RR -> ( k [,) +oo ) C_ RR )
6		resima2 5432	. . . . . . . . 9 |- ( ( k [,) +oo ) C_ RR -> ( ( F |` RR ) " ( k [,) +oo ) ) = ( F " ( k [,) +oo ) ) )
7	5, 6	syl 17	. . . . . . . 8 |- ( k e. RR -> ( ( F |` RR ) " ( k [,) +oo ) ) = ( F " ( k [,) +oo ) ) )
8	7	ineq1d 3813	. . . . . . 7 |- ( k e. RR -> ( ( ( F |` RR ) " ( k [,) +oo ) ) i^i RR* ) = ( ( F " ( k [,) +oo ) ) i^i RR* ) )
9	8	supeq1d 8352	. . . . . 6 |- ( k e. RR -> sup ( ( ( ( F |` RR ) " ( k [,) +oo ) ) i^i RR* ) , RR* , < ) = sup ( ( ( F " ( k [,) +oo ) ) i^i RR* ) , RR* , < ) )
10	9	mpteq2ia 4740	. . . . 5 |- ( k e. RR |-> sup ( ( ( ( F |` RR ) " ( k [,) +oo ) ) i^i RR* ) , RR* , < ) ) = ( k e. RR |-> sup ( ( ( F " ( k [,) +oo ) ) i^i RR* ) , RR* , < ) )
11	10	a1i 11	. . . 4 |- ( ph -> ( k e. RR |-> sup ( ( ( ( F |` RR ) " ( k [,) +oo ) ) i^i RR* ) , RR* , < ) ) = ( k e. RR |-> sup ( ( ( F " ( k [,) +oo ) ) i^i RR* ) , RR* , < ) ) )
12	11	rneqd 5353	. . 3 |- ( ph -> ran ( k e. RR |-> sup ( ( ( ( F |` RR ) " ( k [,) +oo ) ) i^i RR* ) , RR* , < ) ) = ran ( k e. RR |-> sup ( ( ( F " ( k [,) +oo ) ) i^i RR* ) , RR* , < ) ) )
13	12	infeq1d 8383	. 2 |- ( ph -> inf ( ran ( k e. RR |-> sup ( ( ( ( F |` RR ) " ( k [,) +oo ) ) i^i RR* ) , RR* , < ) ) , RR* , < ) = inf ( ran ( k e. RR |-> sup ( ( ( F " ( k [,) +oo ) ) i^i RR* ) , RR* , < ) ) , RR* , < ) )
14		limsupresre.1	. . . 4 |- ( ph -> F e. V )
15	14	resexd 39321	. . 3 |- ( ph -> ( F |` RR ) e. _V )
16		eqid 2622	. . . 4 |- ( k e. RR |-> sup ( ( ( ( F |` RR ) " ( k [,) +oo ) ) i^i RR* ) , RR* , < ) ) = ( k e. RR |-> sup ( ( ( ( F |` RR ) " ( k [,) +oo ) ) i^i RR* ) , RR* , < ) )
17	16	limsupval 14205	. . 3 |- ( ( F |` RR ) e. _V -> ( limsup ` ( F |` RR ) ) = inf ( ran ( k e. RR |-> sup ( ( ( ( F |` RR ) " ( k [,) +oo ) ) i^i RR* ) , RR* , < ) ) , RR* , < ) )
18	15, 17	syl 17	. 2 |- ( ph -> ( limsup ` ( F |` RR ) ) = inf ( ran ( k e. RR |-> sup ( ( ( ( F |` RR ) " ( k [,) +oo ) ) i^i RR* ) , RR* , < ) ) , RR* , < ) )
19		eqid 2622	. . . 4 |- ( k e. RR |-> sup ( ( ( F " ( k [,) +oo ) ) i^i RR* ) , RR* , < ) ) = ( k e. RR |-> sup ( ( ( F " ( k [,) +oo ) ) i^i RR* ) , RR* , < ) )
20	19	limsupval 14205	. . 3 |- ( F e. V -> ( limsup ` F ) = inf ( ran ( k e. RR |-> sup ( ( ( F " ( k [,) +oo ) ) i^i RR* ) , RR* , < ) ) , RR* , < ) )
21	14, 20	syl 17	. 2 |- ( ph -> ( limsup ` F ) = inf ( ran ( k e. RR |-> sup ( ( ( F " ( k [,) +oo ) ) i^i RR* ) , RR* , < ) ) , RR* , < ) )
22	13, 18, 21	3eqtr4d 2666	1 |- ( ph -> ( limsup ` ( F |` RR ) ) = ( limsup ` F ) )

Syntax hints:   -> wi 4   = wceq 1483   e. wcel 1990   _Vcvv 3200   i^i cin 3573   C_ wss 3574   |-> cmpt 4729   ran crn 5115   |` cres 5116   "cima 5117   ` cfv 5888  (class class class)co 6650   supcsup 8346  infcinf 8347   RRcr 9935   +oocpnf 10071   RR*cxr 10073   < clt 10074   [,)cico 12177   limsupclsp 14201

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-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-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-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-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-ico 12181  df-limsup 14202

This theorem is referenced by:  limsupresuz  39935