Theorem limsupvaluz2 39970

Description: The superior limit, when the domain of a real-valued function is a set of upper integers, and the superior limit is real. (Contributed by Glauco Siliprandi, 23-Oct-2021.)

Hypotheses

Ref	Expression
limsupvaluz2.m	|- ( ph -> M e. ZZ )
limsupvaluz2.z	|- Z = ( ZZ>= ` M )
limsupvaluz2.f	|- ( ph -> F : Z --> RR )
limsupvaluz2.r	|- ( ph -> ( limsup ` F ) e. RR )

Assertion

Ref	Expression
limsupvaluz2	|- ( ph -> ( limsup ` F ) = inf ( ran ( k e. Z |-> sup ( ran ( F |` ( ZZ>= ` k ) ) , RR* , < ) ) , RR , < ) )

Distinct variable groups:   k, F   k, Z

Allowed substitution hints:   ph( k)   M( k)

Proof of Theorem limsupvaluz2

Dummy variables i j x n m y are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		limsupvaluz2.m	. . 3 |- ( ph -> M e. ZZ )
2		limsupvaluz2.z	. . 3 |- Z = ( ZZ>= ` M )
3		limsupvaluz2.f	. . . 4 |- ( ph -> F : Z --> RR )
4	3	frexr 39604	. . 3 |- ( ph -> F : Z --> RR* )
5	1, 2, 4	limsupvaluz 39940	. 2 |- ( ph -> ( limsup ` F ) = inf ( ran ( n e. Z |-> sup ( ran ( F |` ( ZZ>= ` n ) ) , RR* , < ) ) , RR* , < ) )
6	3	adantr 481	. . . . . . . . 9 |- ( ( ph /\ n e. Z ) -> F : Z --> RR )
7		id 22	. . . . . . . . . . 11 |- ( n e. Z -> n e. Z )
8	2, 7	uzssd2 39644	. . . . . . . . . 10 |- ( n e. Z -> ( ZZ>= ` n ) C_ Z )
9	8	adantl 482	. . . . . . . . 9 |- ( ( ph /\ n e. Z ) -> ( ZZ>= ` n ) C_ Z )
10	6, 9	feqresmpt 6250	. . . . . . . 8 |- ( ( ph /\ n e. Z ) -> ( F |` ( ZZ>= ` n ) ) = ( m e. ( ZZ>= ` n ) |-> ( F ` m ) ) )
11	10	rneqd 5353	. . . . . . 7 |- ( ( ph /\ n e. Z ) -> ran ( F |` ( ZZ>= ` n ) ) = ran ( m e. ( ZZ>= ` n ) |-> ( F ` m ) ) )
12	11	supeq1d 8352	. . . . . 6 |- ( ( ph /\ n e. Z ) -> sup ( ran ( F |` ( ZZ>= ` n ) ) , RR* , < ) = sup ( ran ( m e. ( ZZ>= ` n ) |-> ( F ` m ) ) , RR* , < ) )
13		nfcv 2764	. . . . . . . . . 10 |- F/_ m F
14		limsupvaluz2.r	. . . . . . . . . . 11 |- ( ph -> ( limsup ` F ) e. RR )
15	14	renepnfd 10090	. . . . . . . . . 10 |- ( ph -> ( limsup ` F ) =/= +oo )
16	13, 2, 3, 15	limsupubuz 39945	. . . . . . . . 9 |- ( ph -> E. x e. RR A. m e. Z ( F ` m ) <_ x )
17	16	adantr 481	. . . . . . . 8 |- ( ( ph /\ n e. Z ) -> E. x e. RR A. m e. Z ( F ` m ) <_ x )
18		ssralv 3666	. . . . . . . . . . 11 |- ( ( ZZ>= ` n ) C_ Z -> ( A. m e. Z ( F ` m ) <_ x -> A. m e. ( ZZ>= ` n ) ( F ` m ) <_ x ) )
19	8, 18	syl 17	. . . . . . . . . 10 |- ( n e. Z -> ( A. m e. Z ( F ` m ) <_ x -> A. m e. ( ZZ>= ` n ) ( F ` m ) <_ x ) )
20	19	adantl 482	. . . . . . . . 9 |- ( ( ph /\ n e. Z ) -> ( A. m e. Z ( F ` m ) <_ x -> A. m e. ( ZZ>= ` n ) ( F ` m ) <_ x ) )
21	20	reximdv 3016	. . . . . . . 8 |- ( ( ph /\ n e. Z ) -> ( E. x e. RR A. m e. Z ( F ` m ) <_ x -> E. x e. RR A. m e. ( ZZ>= ` n ) ( F ` m ) <_ x ) )
22	17, 21	mpd 15	. . . . . . 7 |- ( ( ph /\ n e. Z ) -> E. x e. RR A. m e. ( ZZ>= ` n ) ( F ` m ) <_ x )
23		nfv 1843	. . . . . . . 8 |- F/ m ( ph /\ n e. Z )
24	2	eluzelz2 39627	. . . . . . . . . 10 |- ( n e. Z -> n e. ZZ )
25		uzid 11702	. . . . . . . . . 10 |- ( n e. ZZ -> n e. ( ZZ>= ` n ) )
26		ne0i 3921	. . . . . . . . . 10 |- ( n e. ( ZZ>= ` n ) -> ( ZZ>= ` n ) =/= (/) )
27	24, 25, 26	3syl 18	. . . . . . . . 9 |- ( n e. Z -> ( ZZ>= ` n ) =/= (/) )
28	27	adantl 482	. . . . . . . 8 |- ( ( ph /\ n e. Z ) -> ( ZZ>= ` n ) =/= (/) )
29	6	adantr 481	. . . . . . . . 9 |- ( ( ( ph /\ n e. Z ) /\ m e. ( ZZ>= ` n ) ) -> F : Z --> RR )
30	9	sselda 3603	. . . . . . . . 9 |- ( ( ( ph /\ n e. Z ) /\ m e. ( ZZ>= ` n ) ) -> m e. Z )
31	29, 30	ffvelrnd 6360	. . . . . . . 8 |- ( ( ( ph /\ n e. Z ) /\ m e. ( ZZ>= ` n ) ) -> ( F ` m ) e. RR )
32	23, 28, 31	supxrre3rnmpt 39656	. . . . . . 7 |- ( ( ph /\ n e. Z ) -> ( sup ( ran ( m e. ( ZZ>= ` n ) |-> ( F ` m ) ) , RR* , < ) e. RR <-> E. x e. RR A. m e. ( ZZ>= ` n ) ( F ` m ) <_ x ) )
33	22, 32	mpbird 247	. . . . . 6 |- ( ( ph /\ n e. Z ) -> sup ( ran ( m e. ( ZZ>= ` n ) |-> ( F ` m ) ) , RR* , < ) e. RR )
34	12, 33	eqeltrd 2701	. . . . 5 |- ( ( ph /\ n e. Z ) -> sup ( ran ( F |` ( ZZ>= ` n ) ) , RR* , < ) e. RR )
35		eqid 2622	. . . . 5 |- ( n e. Z |-> sup ( ran ( F |` ( ZZ>= ` n ) ) , RR* , < ) ) = ( n e. Z |-> sup ( ran ( F |` ( ZZ>= ` n ) ) , RR* , < ) )
36	34, 35	fmptd 6385	. . . 4 |- ( ph -> ( n e. Z |-> sup ( ran ( F |` ( ZZ>= ` n ) ) , RR* , < ) ) : Z --> RR )
37	36	frnd 39426	. . 3 |- ( ph -> ran ( n e. Z |-> sup ( ran ( F |` ( ZZ>= ` n ) ) , RR* , < ) ) C_ RR )
38		nfv 1843	. . . 4 |- F/ n ph
39	34	elexd 3214	. . . 4 |- ( ( ph /\ n e. Z ) -> sup ( ran ( F |` ( ZZ>= ` n ) ) , RR* , < ) e. _V )
40	1, 2	uzn0d 39652	. . . 4 |- ( ph -> Z =/= (/) )
41	38, 39, 35, 40	rnmptn0 39413	. . 3 |- ( ph -> ran ( n e. Z |-> sup ( ran ( F |` ( ZZ>= ` n ) ) , RR* , < ) ) =/= (/) )
42		nfcv 2764	. . . . . . . . . 10 |- F/_ j F
43	42, 1, 2, 4	limsupre3uz 39968	. . . . . . . . 9 |- ( ph -> ( ( limsup ` F ) e. RR <-> ( E. x e. RR A. i e. Z E. j e. ( ZZ>= ` i ) x <_ ( F ` j ) /\ E. x e. RR E. i e. Z A. j e. ( ZZ>= ` i ) ( F ` j ) <_ x ) ) )
44	14, 43	mpbid 222	. . . . . . . 8 |- ( ph -> ( E. x e. RR A. i e. Z E. j e. ( ZZ>= ` i ) x <_ ( F ` j ) /\ E. x e. RR E. i e. Z A. j e. ( ZZ>= ` i ) ( F ` j ) <_ x ) )
45	44	simpld 475	. . . . . . 7 |- ( ph -> E. x e. RR A. i e. Z E. j e. ( ZZ>= ` i ) x <_ ( F ` j ) )
46		simp-4r 807	. . . . . . . . . . . . 13 |- ( ( ( ( ( ph /\ x e. RR ) /\ i e. Z ) /\ j e. ( ZZ>= ` i ) ) /\ x <_ ( F ` j ) ) -> x e. RR )
47	46	rexrd 10089	. . . . . . . . . . . 12 |- ( ( ( ( ( ph /\ x e. RR ) /\ i e. Z ) /\ j e. ( ZZ>= ` i ) ) /\ x <_ ( F ` j ) ) -> x e. RR* )
48	4	3ad2ant1 1082	. . . . . . . . . . . . . 14 |- ( ( ph /\ i e. Z /\ j e. ( ZZ>= ` i ) ) -> F : Z --> RR* )
49	2	uztrn2 11705	. . . . . . . . . . . . . . 15 |- ( ( i e. Z /\ j e. ( ZZ>= ` i ) ) -> j e. Z )
50	49	3adant1 1079	. . . . . . . . . . . . . 14 |- ( ( ph /\ i e. Z /\ j e. ( ZZ>= ` i ) ) -> j e. Z )
51	48, 50	ffvelrnd 6360	. . . . . . . . . . . . 13 |- ( ( ph /\ i e. Z /\ j e. ( ZZ>= ` i ) ) -> ( F ` j ) e. RR* )
52	51	ad5ant134 1313	. . . . . . . . . . . 12 |- ( ( ( ( ( ph /\ x e. RR ) /\ i e. Z ) /\ j e. ( ZZ>= ` i ) ) /\ x <_ ( F ` j ) ) -> ( F ` j ) e. RR* )
53		rnresss 39365	. . . . . . . . . . . . . . . . 17 |- ran ( F |` ( ZZ>= ` i ) ) C_ ran F
54	53	a1i 11	. . . . . . . . . . . . . . . 16 |- ( ( ph /\ i e. Z ) -> ran ( F |` ( ZZ>= ` i ) ) C_ ran F )
55	3	frnd 39426	. . . . . . . . . . . . . . . . 17 |- ( ph -> ran F C_ RR )
56	55	adantr 481	. . . . . . . . . . . . . . . 16 |- ( ( ph /\ i e. Z ) -> ran F C_ RR )
57	54, 56	sstrd 3613	. . . . . . . . . . . . . . 15 |- ( ( ph /\ i e. Z ) -> ran ( F |` ( ZZ>= ` i ) ) C_ RR )
58		ressxr 10083	. . . . . . . . . . . . . . . 16 |- RR C_ RR*
59	58	a1i 11	. . . . . . . . . . . . . . 15 |- ( ( ph /\ i e. Z ) -> RR C_ RR* )
60	57, 59	sstrd 3613	. . . . . . . . . . . . . 14 |- ( ( ph /\ i e. Z ) -> ran ( F |` ( ZZ>= ` i ) ) C_ RR* )
61	60	supxrcld 39290	. . . . . . . . . . . . 13 |- ( ( ph /\ i e. Z ) -> sup ( ran ( F |` ( ZZ>= ` i ) ) , RR* , < ) e. RR* )
62	61	ad5ant13 1301	. . . . . . . . . . . 12 |- ( ( ( ( ( ph /\ x e. RR ) /\ i e. Z ) /\ j e. ( ZZ>= ` i ) ) /\ x <_ ( F ` j ) ) -> sup ( ran ( F |` ( ZZ>= ` i ) ) , RR* , < ) e. RR* )
63		simpr 477	. . . . . . . . . . . 12 |- ( ( ( ( ( ph /\ x e. RR ) /\ i e. Z ) /\ j e. ( ZZ>= ` i ) ) /\ x <_ ( F ` j ) ) -> x <_ ( F ` j ) )
64	60	3adant3 1081	. . . . . . . . . . . . . 14 |- ( ( ph /\ i e. Z /\ j e. ( ZZ>= ` i ) ) -> ran ( F |` ( ZZ>= ` i ) ) C_ RR* )
65		fvres 6207	. . . . . . . . . . . . . . . . 17 |- ( j e. ( ZZ>= ` i ) -> ( ( F |` ( ZZ>= ` i ) ) ` j ) = ( F ` j ) )
66	65	eqcomd 2628	. . . . . . . . . . . . . . . 16 |- ( j e. ( ZZ>= ` i ) -> ( F ` j ) = ( ( F |` ( ZZ>= ` i ) ) ` j ) )
67	66	3ad2ant3 1084	. . . . . . . . . . . . . . 15 |- ( ( ph /\ i e. Z /\ j e. ( ZZ>= ` i ) ) -> ( F ` j ) = ( ( F |` ( ZZ>= ` i ) ) ` j ) )
68	3	ffnd 6046	. . . . . . . . . . . . . . . . . . 19 |- ( ph -> F Fn Z )
69	68	adantr 481	. . . . . . . . . . . . . . . . . 18 |- ( ( ph /\ i e. Z ) -> F Fn Z )
70		id 22	. . . . . . . . . . . . . . . . . . . 20 |- ( i e. Z -> i e. Z )
71	2, 70	uzssd2 39644	. . . . . . . . . . . . . . . . . . 19 |- ( i e. Z -> ( ZZ>= ` i ) C_ Z )
72	71	adantl 482	. . . . . . . . . . . . . . . . . 18 |- ( ( ph /\ i e. Z ) -> ( ZZ>= ` i ) C_ Z )
73		fnssres 6004	. . . . . . . . . . . . . . . . . 18 |- ( ( F Fn Z /\ ( ZZ>= ` i ) C_ Z ) -> ( F |` ( ZZ>= ` i ) ) Fn ( ZZ>= ` i ) )
74	69, 72, 73	syl2anc 693	. . . . . . . . . . . . . . . . 17 |- ( ( ph /\ i e. Z ) -> ( F |` ( ZZ>= ` i ) ) Fn ( ZZ>= ` i ) )
75	74	3adant3 1081	. . . . . . . . . . . . . . . 16 |- ( ( ph /\ i e. Z /\ j e. ( ZZ>= ` i ) ) -> ( F |` ( ZZ>= ` i ) ) Fn ( ZZ>= ` i ) )
76		simp3 1063	. . . . . . . . . . . . . . . 16 |- ( ( ph /\ i e. Z /\ j e. ( ZZ>= ` i ) ) -> j e. ( ZZ>= ` i ) )
77		fnfvelrn 6356	. . . . . . . . . . . . . . . 16 |- ( ( ( F |` ( ZZ>= ` i ) ) Fn ( ZZ>= ` i ) /\ j e. ( ZZ>= ` i ) ) -> ( ( F |` ( ZZ>= ` i ) ) ` j ) e. ran ( F |` ( ZZ>= ` i ) ) )
78	75, 76, 77	syl2anc 693	. . . . . . . . . . . . . . 15 |- ( ( ph /\ i e. Z /\ j e. ( ZZ>= ` i ) ) -> ( ( F |` ( ZZ>= ` i ) ) ` j ) e. ran ( F |` ( ZZ>= ` i ) ) )
79	67, 78	eqeltrd 2701	. . . . . . . . . . . . . 14 |- ( ( ph /\ i e. Z /\ j e. ( ZZ>= ` i ) ) -> ( F ` j ) e. ran ( F |` ( ZZ>= ` i ) ) )
80		eqid 2622	. . . . . . . . . . . . . 14 |- sup ( ran ( F |` ( ZZ>= ` i ) ) , RR* , < ) = sup ( ran ( F |` ( ZZ>= ` i ) ) , RR* , < )
81	64, 79, 80	supxrubd 39297	. . . . . . . . . . . . 13 |- ( ( ph /\ i e. Z /\ j e. ( ZZ>= ` i ) ) -> ( F ` j ) <_ sup ( ran ( F |` ( ZZ>= ` i ) ) , RR* , < ) )
82	81	ad5ant134 1313	. . . . . . . . . . . 12 |- ( ( ( ( ( ph /\ x e. RR ) /\ i e. Z ) /\ j e. ( ZZ>= ` i ) ) /\ x <_ ( F ` j ) ) -> ( F ` j ) <_ sup ( ran ( F |` ( ZZ>= ` i ) ) , RR* , < ) )
83	47, 52, 62, 63, 82	xrletrd 11993	. . . . . . . . . . 11 |- ( ( ( ( ( ph /\ x e. RR ) /\ i e. Z ) /\ j e. ( ZZ>= ` i ) ) /\ x <_ ( F ` j ) ) -> x <_ sup ( ran ( F |` ( ZZ>= ` i ) ) , RR* , < ) )
84	83	ex 450	. . . . . . . . . 10 |- ( ( ( ( ph /\ x e. RR ) /\ i e. Z ) /\ j e. ( ZZ>= ` i ) ) -> ( x <_ ( F ` j ) -> x <_ sup ( ran ( F |` ( ZZ>= ` i ) ) , RR* , < ) ) )
85	84	rexlimdva 3031	. . . . . . . . 9 |- ( ( ( ph /\ x e. RR ) /\ i e. Z ) -> ( E. j e. ( ZZ>= ` i ) x <_ ( F ` j ) -> x <_ sup ( ran ( F |` ( ZZ>= ` i ) ) , RR* , < ) ) )
86	85	ralimdva 2962	. . . . . . . 8 |- ( ( ph /\ x e. RR ) -> ( A. i e. Z E. j e. ( ZZ>= ` i ) x <_ ( F ` j ) -> A. i e. Z x <_ sup ( ran ( F |` ( ZZ>= ` i ) ) , RR* , < ) ) )
87	86	reximdva 3017	. . . . . . 7 |- ( ph -> ( E. x e. RR A. i e. Z E. j e. ( ZZ>= ` i ) x <_ ( F ` j ) -> E. x e. RR A. i e. Z x <_ sup ( ran ( F |` ( ZZ>= ` i ) ) , RR* , < ) ) )
88	45, 87	mpd 15	. . . . . 6 |- ( ph -> E. x e. RR A. i e. Z x <_ sup ( ran ( F |` ( ZZ>= ` i ) ) , RR* , < ) )
89	88	idi 2	. . . . 5 |- ( ph -> E. x e. RR A. i e. Z x <_ sup ( ran ( F |` ( ZZ>= ` i ) ) , RR* , < ) )
90		fveq2 6191	. . . . . . . . . . . 12 |- ( n = i -> ( ZZ>= ` n ) = ( ZZ>= ` i ) )
91	90	reseq2d 5396	. . . . . . . . . . 11 |- ( n = i -> ( F |` ( ZZ>= ` n ) ) = ( F |` ( ZZ>= ` i ) ) )
92	91	rneqd 5353	. . . . . . . . . 10 |- ( n = i -> ran ( F |` ( ZZ>= ` n ) ) = ran ( F |` ( ZZ>= ` i ) ) )
93	92	supeq1d 8352	. . . . . . . . 9 |- ( n = i -> sup ( ran ( F |` ( ZZ>= ` n ) ) , RR* , < ) = sup ( ran ( F |` ( ZZ>= ` i ) ) , RR* , < ) )
94		eqcom 2629	. . . . . . . . . . 11 |- ( n = i <-> i = n )
95	94	imbi1i 339	. . . . . . . . . 10 |- ( ( n = i -> sup ( ran ( F |` ( ZZ>= ` n ) ) , RR* , < ) = sup ( ran ( F |` ( ZZ>= ` i ) ) , RR* , < ) ) <-> ( i = n -> sup ( ran ( F |` ( ZZ>= ` n ) ) , RR* , < ) = sup ( ran ( F |` ( ZZ>= ` i ) ) , RR* , < ) ) )
96		eqcom 2629	. . . . . . . . . . 11 |- ( sup ( ran ( F |` ( ZZ>= ` n ) ) , RR* , < ) = sup ( ran ( F |` ( ZZ>= ` i ) ) , RR* , < ) <-> sup ( ran ( F |` ( ZZ>= ` i ) ) , RR* , < ) = sup ( ran ( F |` ( ZZ>= ` n ) ) , RR* , < ) )
97	96	imbi2i 326	. . . . . . . . . 10 |- ( ( i = n -> sup ( ran ( F |` ( ZZ>= ` n ) ) , RR* , < ) = sup ( ran ( F |` ( ZZ>= ` i ) ) , RR* , < ) ) <-> ( i = n -> sup ( ran ( F |` ( ZZ>= ` i ) ) , RR* , < ) = sup ( ran ( F |` ( ZZ>= ` n ) ) , RR* , < ) ) )
98	95, 97	bitri 264	. . . . . . . . 9 |- ( ( n = i -> sup ( ran ( F |` ( ZZ>= ` n ) ) , RR* , < ) = sup ( ran ( F |` ( ZZ>= ` i ) ) , RR* , < ) ) <-> ( i = n -> sup ( ran ( F |` ( ZZ>= ` i ) ) , RR* , < ) = sup ( ran ( F |` ( ZZ>= ` n ) ) , RR* , < ) ) )
99	93, 98	mpbi 220	. . . . . . . 8 |- ( i = n -> sup ( ran ( F |` ( ZZ>= ` i ) ) , RR* , < ) = sup ( ran ( F |` ( ZZ>= ` n ) ) , RR* , < ) )
100	99	breq2d 4665	. . . . . . 7 |- ( i = n -> ( x <_ sup ( ran ( F |` ( ZZ>= ` i ) ) , RR* , < ) <-> x <_ sup ( ran ( F |` ( ZZ>= ` n ) ) , RR* , < ) ) )
101	100	cbvralv 3171	. . . . . 6 |- ( A. i e. Z x <_ sup ( ran ( F |` ( ZZ>= ` i ) ) , RR* , < ) <-> A. n e. Z x <_ sup ( ran ( F |` ( ZZ>= ` n ) ) , RR* , < ) )
102	101	rexbii 3041	. . . . 5 |- ( E. x e. RR A. i e. Z x <_ sup ( ran ( F |` ( ZZ>= ` i ) ) , RR* , < ) <-> E. x e. RR A. n e. Z x <_ sup ( ran ( F |` ( ZZ>= ` n ) ) , RR* , < ) )
103	89, 102	sylib 208	. . . 4 |- ( ph -> E. x e. RR A. n e. Z x <_ sup ( ran ( F |` ( ZZ>= ` n ) ) , RR* , < ) )
104	38, 39	rnmptbd2 39464	. . . 4 |- ( ph -> ( E. x e. RR A. n e. Z x <_ sup ( ran ( F |` ( ZZ>= ` n ) ) , RR* , < ) <-> E. x e. RR A. y e. ran ( n e. Z |-> sup ( ran ( F |` ( ZZ>= ` n ) ) , RR* , < ) ) x <_ y ) )
105	103, 104	mpbid 222	. . 3 |- ( ph -> E. x e. RR A. y e. ran ( n e. Z |-> sup ( ran ( F |` ( ZZ>= ` n ) ) , RR* , < ) ) x <_ y )
106		infxrre 12166	. . 3 |- ( ( ran ( n e. Z |-> sup ( ran ( F |` ( ZZ>= ` n ) ) , RR* , < ) ) C_ RR /\ ran ( n e. Z |-> sup ( ran ( F |` ( ZZ>= ` n ) ) , RR* , < ) ) =/= (/) /\ E. x e. RR A. y e. ran ( n e. Z |-> sup ( ran ( F |` ( ZZ>= ` n ) ) , RR* , < ) ) x <_ y ) -> inf ( ran ( n e. Z |-> sup ( ran ( F |` ( ZZ>= ` n ) ) , RR* , < ) ) , RR* , < ) = inf ( ran ( n e. Z |-> sup ( ran ( F |` ( ZZ>= ` n ) ) , RR* , < ) ) , RR , < ) )
107	37, 41, 105, 106	syl3anc 1326	. 2 |- ( ph -> inf ( ran ( n e. Z |-> sup ( ran ( F |` ( ZZ>= ` n ) ) , RR* , < ) ) , RR* , < ) = inf ( ran ( n e. Z |-> sup ( ran ( F |` ( ZZ>= ` n ) ) , RR* , < ) ) , RR , < ) )
108		fveq2 6191	. . . . . . . . 9 |- ( n = k -> ( ZZ>= ` n ) = ( ZZ>= ` k ) )
109	108	reseq2d 5396	. . . . . . . 8 |- ( n = k -> ( F |` ( ZZ>= ` n ) ) = ( F |` ( ZZ>= ` k ) ) )
110	109	rneqd 5353	. . . . . . 7 |- ( n = k -> ran ( F |` ( ZZ>= ` n ) ) = ran ( F |` ( ZZ>= ` k ) ) )
111	110	supeq1d 8352	. . . . . 6 |- ( n = k -> sup ( ran ( F |` ( ZZ>= ` n ) ) , RR* , < ) = sup ( ran ( F |` ( ZZ>= ` k ) ) , RR* , < ) )
112	111	cbvmptv 4750	. . . . 5 |- ( n e. Z |-> sup ( ran ( F |` ( ZZ>= ` n ) ) , RR* , < ) ) = ( k e. Z |-> sup ( ran ( F |` ( ZZ>= ` k ) ) , RR* , < ) )
113	112	rneqi 5352	. . . 4 |- ran ( n e. Z |-> sup ( ran ( F |` ( ZZ>= ` n ) ) , RR* , < ) ) = ran ( k e. Z |-> sup ( ran ( F |` ( ZZ>= ` k ) ) , RR* , < ) )
114	113	infeq1i 8384	. . 3 |- inf ( ran ( n e. Z |-> sup ( ran ( F |` ( ZZ>= ` n ) ) , RR* , < ) ) , RR , < ) = inf ( ran ( k e. Z |-> sup ( ran ( F |` ( ZZ>= ` k ) ) , RR* , < ) ) , RR , < )
115	114	a1i 11	. 2 |- ( ph -> inf ( ran ( n e. Z |-> sup ( ran ( F |` ( ZZ>= ` n ) ) , RR* , < ) ) , RR , < ) = inf ( ran ( k e. Z |-> sup ( ran ( F |` ( ZZ>= ` k ) ) , RR* , < ) ) , RR , < ) )
116	5, 107, 115	3eqtrd 2660	1 |- ( ph -> ( limsup ` F ) = inf ( ran ( k e. Z |-> sup ( ran ( F |` ( ZZ>= ` k ) ) , RR* , < ) ) , RR , < ) )

Syntax hints:   -> wi 4   /\ wa 384   /\ w3a 1037   = wceq 1483   e. wcel 1990   =/= wne 2794   A.wral 2912   E.wrex 2913   _Vcvv 3200   C_ wss 3574   (/)c0 3915   class class class wbr 4653   |-> cmpt 4729   ran crn 5115   |` cres 5116   Fn wfn 5883   -->wf 5884   ` cfv 5888   supcsup 8346  infcinf 8347   RRcr 9935   RR*cxr 10073   < clt 10074   <_ cle 10075   ZZcz 11377   ZZ>=cuz 11687   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-rep 4771  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-int 4476  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-1st 7168  df-2nd 7169  df-wrecs 7407  df-recs 7468  df-rdg 7506  df-1o 7560  df-oadd 7564  df-er 7742  df-en 7956  df-dom 7957  df-sdom 7958  df-fin 7959  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-nn 11021  df-n0 11293  df-z 11378  df-uz 11688  df-ico 12181  df-fz 12327  df-fl 12593  df-ceil 12594  df-limsup 14202

This theorem is referenced by:  supcnvlimsup  39972