Theorem climlimsupcex 40001

Description: Counterexample for climlimsup 39992, showing that the first hypothesis is needed, if the empty set is a complex number (see 0ncn 9954 and its comment) (Contributed by Glauco Siliprandi, 2-Jan-2022.)

Hypotheses

Ref	Expression
climlimsupcex.1	|- -. M e. ZZ
climlimsupcex.2	|- Z = ( ZZ>= ` M )
climlimsupcex.3	|- F = (/)

Assertion

Ref	Expression
climlimsupcex	|- ( ( (/) e. CC /\ -. -oo e. CC ) -> ( F : Z --> RR /\ F e. dom ~~> /\ -. F ~~> ( limsup ` F ) ) )

Proof of Theorem climlimsupcex

Step	Hyp	Ref	Expression
1		f0 6086	. . . 4 |- (/) : (/) --> RR
2		climlimsupcex.3	. . . . 5 |- F = (/)
3		climlimsupcex.2	. . . . . 6 |- Z = ( ZZ>= ` M )
4		climlimsupcex.1	. . . . . . 7 |- -. M e. ZZ
5		uz0 39639	. . . . . . 7 |- ( -. M e. ZZ -> ( ZZ>= ` M ) = (/) )
6	4, 5	ax-mp 5	. . . . . 6 |- ( ZZ>= ` M ) = (/)
7	3, 6	eqtri 2644	. . . . 5 |- Z = (/)
8	2, 7	feq12i 6038	. . . 4 |- ( F : Z --> RR <-> (/) : (/) --> RR )
9	1, 8	mpbir 221	. . 3 |- F : Z --> RR
10	9	a1i 11	. 2 |- ( ( (/) e. CC /\ -. -oo e. CC ) -> F : Z --> RR )
11		climrel 14223	. . . . 5 |- Rel ~~>
12	11	a1i 11	. . . 4 |- ( (/) e. CC -> Rel ~~> )
13		0cnv 39974	. . . . 5 |- ( (/) e. CC -> (/) ~~> (/) )
14	2, 13	syl5eqbr 4688	. . . 4 |- ( (/) e. CC -> F ~~> (/) )
15		releldm 5358	. . . 4 |- ( ( Rel ~~> /\ F ~~> (/) ) -> F e. dom ~~> )
16	12, 14, 15	syl2anc 693	. . 3 |- ( (/) e. CC -> F e. dom ~~> )
17	16	adantr 481	. 2 |- ( ( (/) e. CC /\ -. -oo e. CC ) -> F e. dom ~~> )
18	13	adantr 481	. . . 4 |- ( ( (/) e. CC /\ F ~~> ( limsup ` F ) ) -> (/) ~~> (/) )
19	18	adantlr 751	. . 3 |- ( ( ( (/) e. CC /\ -. -oo e. CC ) /\ F ~~> ( limsup ` F ) ) -> (/) ~~> (/) )
20		simpr 477	. . . . . 6 |- ( ( F ~~> ( limsup ` F ) /\ (/) ~~> (/) ) -> (/) ~~> (/) )
21	2	fveq2i 6194	. . . . . . . . . 10 |- ( limsup ` F ) = ( limsup ` (/) )
22		limsup0 39926	. . . . . . . . . 10 |- ( limsup ` (/) ) = -oo
23	21, 22	eqtri 2644	. . . . . . . . 9 |- ( limsup ` F ) = -oo
24	2, 23	breq12i 4662	. . . . . . . 8 |- ( F ~~> ( limsup ` F ) <-> (/) ~~> -oo )
25	24	biimpi 206	. . . . . . 7 |- ( F ~~> ( limsup ` F ) -> (/) ~~> -oo )
26	25	adantr 481	. . . . . 6 |- ( ( F ~~> ( limsup ` F ) /\ (/) ~~> (/) ) -> (/) ~~> -oo )
27		climuni 14283	. . . . . 6 |- ( ( (/) ~~> (/) /\ (/) ~~> -oo ) -> (/) = -oo )
28	20, 26, 27	syl2anc 693	. . . . 5 |- ( ( F ~~> ( limsup ` F ) /\ (/) ~~> (/) ) -> (/) = -oo )
29	28	adantll 750	. . . 4 |- ( ( ( ( (/) e. CC /\ -. -oo e. CC ) /\ F ~~> ( limsup ` F ) ) /\ (/) ~~> (/) ) -> (/) = -oo )
30		nelneq 2725	. . . . 5 |- ( ( (/) e. CC /\ -. -oo e. CC ) -> -. (/) = -oo )
31	30	ad2antrr 762	. . . 4 |- ( ( ( ( (/) e. CC /\ -. -oo e. CC ) /\ F ~~> ( limsup ` F ) ) /\ (/) ~~> (/) ) -> -. (/) = -oo )
32	29, 31	pm2.65da 600	. . 3 |- ( ( ( (/) e. CC /\ -. -oo e. CC ) /\ F ~~> ( limsup ` F ) ) -> -. (/) ~~> (/) )
33	19, 32	pm2.65da 600	. 2 |- ( ( (/) e. CC /\ -. -oo e. CC ) -> -. F ~~> ( limsup ` F ) )
34	10, 17, 33	3jca 1242	1 |- ( ( (/) e. CC /\ -. -oo e. CC ) -> ( F : Z --> RR /\ F e. dom ~~> /\ -. F ~~> ( limsup ` F ) ) )

Syntax hints:   -. wn 3   -> wi 4   /\ wa 384   /\ w3a 1037   = wceq 1483   e. wcel 1990   (/)c0 3915   class class class wbr 4653   dom cdm 5114   Rel wrel 5119   -->wf 5884   ` cfv 5888   CCcc 9934   RRcr 9935   -oocmnf 10072   ZZcz 11377   ZZ>=cuz 11687   limsupclsp 14201   ~~> cli 14215

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-pss 3590  df-nul 3916  df-if 4087  df-pw 4160  df-sn 4178  df-pr 4180  df-tp 4182  df-op 4184  df-uni 4437  df-iun 4522  df-br 4654  df-opab 4713  df-mpt 4730  df-tr 4753  df-id 5024  df-eprel 5029  df-po 5035  df-so 5036  df-fr 5073  df-we 5075  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-pred 5680  df-ord 5726  df-on 5727  df-lim 5728  df-suc 5729  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-om 7066  df-2nd 7169  df-wrecs 7407  df-recs 7468  df-rdg 7506  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-div 10685  df-nn 11021  df-2 11079  df-3 11080  df-n0 11293  df-z 11378  df-uz 11688  df-rp 11833  df-seq 12802  df-exp 12861  df-cj 13839  df-re 13840  df-im 13841  df-sqrt 13975  df-abs 13976  df-limsup 14202  df-clim 14219

This theorem is referenced by: (None)