Theorem smflimsuplem1 41026

Description: If H converges, the limsup of F is real. (Contributed by Glauco Siliprandi, 23-Oct-2021.)

Hypotheses

Ref	Expression
smflimsuplem1.z	|- Z = ( ZZ>= ` M )
smflimsuplem1.e	|- E = ( n e. Z |-> { x e. |^|_ m e. ( ZZ>= ` n ) dom ( F ` m ) | sup ( ran ( m e. ( ZZ>= ` n ) |-> ( ( F ` m ) ` x ) ) , RR* , < ) e. RR } )
smflimsuplem1.h	|- H = ( n e. Z |-> ( x e. ( E ` n ) |-> sup ( ran ( m e. ( ZZ>= ` n ) |-> ( ( F ` m ) ` x ) ) , RR* , < ) ) )
smflimsuplem1.k	|- ( ph -> K e. Z )

Assertion

Ref	Expression
smflimsuplem1	|- ( ph -> dom ( H ` K ) C_ dom ( F ` K ) )

Distinct variable groups:   n, E, x   m, F, n, x   n, K, x   n, Z

Allowed substitution hints:   ph( x, m, n)   E( m)   H( x, m, n)   K( m)   M( x, m, n)   Z( x, m)

Proof of Theorem smflimsuplem1

Dummy variable j is distinct from all other variables.

Step	Hyp	Ref	Expression
1		smflimsuplem1.h	. . . . 5 |- H = ( n e. Z |-> ( x e. ( E ` n ) |-> sup ( ran ( m e. ( ZZ>= ` n ) |-> ( ( F ` m ) ` x ) ) , RR* , < ) ) )
2		fveq2 6191	. . . . . . . . . . . 12 |- ( m = j -> ( F ` m ) = ( F ` j ) )
3	2	fveq1d 6193	. . . . . . . . . . 11 |- ( m = j -> ( ( F ` m ) ` x ) = ( ( F ` j ) ` x ) )
4	3	cbvmptv 4750	. . . . . . . . . 10 |- ( m e. ( ZZ>= ` n ) |-> ( ( F ` m ) ` x ) ) = ( j e. ( ZZ>= ` n ) |-> ( ( F ` j ) ` x ) )
5	4	rneqi 5352	. . . . . . . . 9 |- ran ( m e. ( ZZ>= ` n ) |-> ( ( F ` m ) ` x ) ) = ran ( j e. ( ZZ>= ` n ) |-> ( ( F ` j ) ` x ) )
6	5	supeq1i 8353	. . . . . . . 8 |- sup ( ran ( m e. ( ZZ>= ` n ) |-> ( ( F ` m ) ` x ) ) , RR* , < ) = sup ( ran ( j e. ( ZZ>= ` n ) |-> ( ( F ` j ) ` x ) ) , RR* , < )
7	6	mpteq2i 4741	. . . . . . 7 |- ( x e. ( E ` n ) |-> sup ( ran ( m e. ( ZZ>= ` n ) |-> ( ( F ` m ) ` x ) ) , RR* , < ) ) = ( x e. ( E ` n ) |-> sup ( ran ( j e. ( ZZ>= ` n ) |-> ( ( F ` j ) ` x ) ) , RR* , < ) )
8	7	a1i 11	. . . . . 6 |- ( n = K -> ( x e. ( E ` n ) |-> sup ( ran ( m e. ( ZZ>= ` n ) |-> ( ( F ` m ) ` x ) ) , RR* , < ) ) = ( x e. ( E ` n ) |-> sup ( ran ( j e. ( ZZ>= ` n ) |-> ( ( F ` j ) ` x ) ) , RR* , < ) ) )
9		fveq2 6191	. . . . . . 7 |- ( n = K -> ( E ` n ) = ( E ` K ) )
10		fveq2 6191	. . . . . . . . . 10 |- ( n = K -> ( ZZ>= ` n ) = ( ZZ>= ` K ) )
11	10	mpteq1d 4738	. . . . . . . . 9 |- ( n = K -> ( j e. ( ZZ>= ` n ) |-> ( ( F ` j ) ` x ) ) = ( j e. ( ZZ>= ` K ) |-> ( ( F ` j ) ` x ) ) )
12	11	rneqd 5353	. . . . . . . 8 |- ( n = K -> ran ( j e. ( ZZ>= ` n ) |-> ( ( F ` j ) ` x ) ) = ran ( j e. ( ZZ>= ` K ) |-> ( ( F ` j ) ` x ) ) )
13	12	supeq1d 8352	. . . . . . 7 |- ( n = K -> sup ( ran ( j e. ( ZZ>= ` n ) |-> ( ( F ` j ) ` x ) ) , RR* , < ) = sup ( ran ( j e. ( ZZ>= ` K ) |-> ( ( F ` j ) ` x ) ) , RR* , < ) )
14	9, 13	mpteq12dv 4733	. . . . . 6 |- ( n = K -> ( x e. ( E ` n ) |-> sup ( ran ( j e. ( ZZ>= ` n ) |-> ( ( F ` j ) ` x ) ) , RR* , < ) ) = ( x e. ( E ` K ) |-> sup ( ran ( j e. ( ZZ>= ` K ) |-> ( ( F ` j ) ` x ) ) , RR* , < ) ) )
15	8, 14	eqtrd 2656	. . . . 5 |- ( n = K -> ( x e. ( E ` n ) |-> sup ( ran ( m e. ( ZZ>= ` n ) |-> ( ( F ` m ) ` x ) ) , RR* , < ) ) = ( x e. ( E ` K ) |-> sup ( ran ( j e. ( ZZ>= ` K ) |-> ( ( F ` j ) ` x ) ) , RR* , < ) ) )
16		smflimsuplem1.k	. . . . 5 |- ( ph -> K e. Z )
17		fvex 6201	. . . . . . 7 |- ( E ` K ) e. _V
18	17	mptex 6486	. . . . . 6 |- ( x e. ( E ` K ) |-> sup ( ran ( j e. ( ZZ>= ` K ) |-> ( ( F ` j ) ` x ) ) , RR* , < ) ) e. _V
19	18	a1i 11	. . . . 5 |- ( ph -> ( x e. ( E ` K ) |-> sup ( ran ( j e. ( ZZ>= ` K ) |-> ( ( F ` j ) ` x ) ) , RR* , < ) ) e. _V )
20	1, 15, 16, 19	fvmptd3 39447	. . . 4 |- ( ph -> ( H ` K ) = ( x e. ( E ` K ) |-> sup ( ran ( j e. ( ZZ>= ` K ) |-> ( ( F ` j ) ` x ) ) , RR* , < ) ) )
21	20	dmeqd 5326	. . 3 |- ( ph -> dom ( H ` K ) = dom ( x e. ( E ` K ) |-> sup ( ran ( j e. ( ZZ>= ` K ) |-> ( ( F ` j ) ` x ) ) , RR* , < ) ) )
22		xrltso 11974	. . . . . 6 |- < Or RR*
23	22	supex 8369	. . . . 5 |- sup ( ran ( j e. ( ZZ>= ` K ) |-> ( ( F ` j ) ` x ) ) , RR* , < ) e. _V
24		eqid 2622	. . . . 5 |- ( x e. ( E ` K ) |-> sup ( ran ( j e. ( ZZ>= ` K ) |-> ( ( F ` j ) ` x ) ) , RR* , < ) ) = ( x e. ( E ` K ) |-> sup ( ran ( j e. ( ZZ>= ` K ) |-> ( ( F ` j ) ` x ) ) , RR* , < ) )
25	23, 24	dmmpti 6023	. . . 4 |- dom ( x e. ( E ` K ) |-> sup ( ran ( j e. ( ZZ>= ` K ) |-> ( ( F ` j ) ` x ) ) , RR* , < ) ) = ( E ` K )
26	25	a1i 11	. . 3 |- ( ph -> dom ( x e. ( E ` K ) |-> sup ( ran ( j e. ( ZZ>= ` K ) |-> ( ( F ` j ) ` x ) ) , RR* , < ) ) = ( E ` K ) )
27		smflimsuplem1.e	. . . 4 |- E = ( n e. Z |-> { x e. |^|_ m e. ( ZZ>= ` n ) dom ( F ` m ) | sup ( ran ( m e. ( ZZ>= ` n ) |-> ( ( F ` m ) ` x ) ) , RR* , < ) e. RR } )
28	2	dmeqd 5326	. . . . . . . . . 10 |- ( m = j -> dom ( F ` m ) = dom ( F ` j ) )
29	28	cbviinv 4560	. . . . . . . . 9 |- |^|_ m e. ( ZZ>= ` n ) dom ( F ` m ) = |^|_ j e. ( ZZ>= ` n ) dom ( F ` j )
30	29	eleq2i 2693	. . . . . . . 8 |- ( x e. |^|_ m e. ( ZZ>= ` n ) dom ( F ` m ) <-> x e. |^|_ j e. ( ZZ>= ` n ) dom ( F ` j ) )
31	6	eleq1i 2692	. . . . . . . 8 |- ( sup ( ran ( m e. ( ZZ>= ` n ) |-> ( ( F ` m ) ` x ) ) , RR* , < ) e. RR <-> sup ( ran ( j e. ( ZZ>= ` n ) |-> ( ( F ` j ) ` x ) ) , RR* , < ) e. RR )
32	30, 31	anbi12i 733	. . . . . . 7 |- ( ( x e. |^|_ m e. ( ZZ>= ` n ) dom ( F ` m ) /\ sup ( ran ( m e. ( ZZ>= ` n ) |-> ( ( F ` m ) ` x ) ) , RR* , < ) e. RR ) <-> ( x e. |^|_ j e. ( ZZ>= ` n ) dom ( F ` j ) /\ sup ( ran ( j e. ( ZZ>= ` n ) |-> ( ( F ` j ) ` x ) ) , RR* , < ) e. RR ) )
33	32	rabbia2 3187	. . . . . 6 |- { x e. |^|_ m e. ( ZZ>= ` n ) dom ( F ` m ) | sup ( ran ( m e. ( ZZ>= ` n ) |-> ( ( F ` m ) ` x ) ) , RR* , < ) e. RR } = { x e. |^|_ j e. ( ZZ>= ` n ) dom ( F ` j ) | sup ( ran ( j e. ( ZZ>= ` n ) |-> ( ( F ` j ) ` x ) ) , RR* , < ) e. RR }
34	33	a1i 11	. . . . 5 |- ( n = K -> { x e. |^|_ m e. ( ZZ>= ` n ) dom ( F ` m ) | sup ( ran ( m e. ( ZZ>= ` n ) |-> ( ( F ` m ) ` x ) ) , RR* , < ) e. RR } = { x e. |^|_ j e. ( ZZ>= ` n ) dom ( F ` j ) | sup ( ran ( j e. ( ZZ>= ` n ) |-> ( ( F ` j ) ` x ) ) , RR* , < ) e. RR } )
35	10	iineq1d 39267	. . . . . . . 8 |- ( n = K -> |^|_ j e. ( ZZ>= ` n ) dom ( F ` j ) = |^|_ j e. ( ZZ>= ` K ) dom ( F ` j ) )
36	35	eleq2d 2687	. . . . . . 7 |- ( n = K -> ( x e. |^|_ j e. ( ZZ>= ` n ) dom ( F ` j ) <-> x e. |^|_ j e. ( ZZ>= ` K ) dom ( F ` j ) ) )
37	13	eleq1d 2686	. . . . . . 7 |- ( n = K -> ( sup ( ran ( j e. ( ZZ>= ` n ) |-> ( ( F ` j ) ` x ) ) , RR* , < ) e. RR <-> sup ( ran ( j e. ( ZZ>= ` K ) |-> ( ( F ` j ) ` x ) ) , RR* , < ) e. RR ) )
38	36, 37	anbi12d 747	. . . . . 6 |- ( n = K -> ( ( x e. |^|_ j e. ( ZZ>= ` n ) dom ( F ` j ) /\ sup ( ran ( j e. ( ZZ>= ` n ) |-> ( ( F ` j ) ` x ) ) , RR* , < ) e. RR ) <-> ( x e. |^|_ j e. ( ZZ>= ` K ) dom ( F ` j ) /\ sup ( ran ( j e. ( ZZ>= ` K ) |-> ( ( F ` j ) ` x ) ) , RR* , < ) e. RR ) ) )
39	38	rabbidva2 3186	. . . . 5 |- ( n = K -> { x e. |^|_ j e. ( ZZ>= ` n ) dom ( F ` j ) | sup ( ran ( j e. ( ZZ>= ` n ) |-> ( ( F ` j ) ` x ) ) , RR* , < ) e. RR } = { x e. |^|_ j e. ( ZZ>= ` K ) dom ( F ` j ) | sup ( ran ( j e. ( ZZ>= ` K ) |-> ( ( F ` j ) ` x ) ) , RR* , < ) e. RR } )
40	34, 39	eqtrd 2656	. . . 4 |- ( n = K -> { x e. |^|_ m e. ( ZZ>= ` n ) dom ( F ` m ) | sup ( ran ( m e. ( ZZ>= ` n ) |-> ( ( F ` m ) ` x ) ) , RR* , < ) e. RR } = { x e. |^|_ j e. ( ZZ>= ` K ) dom ( F ` j ) | sup ( ran ( j e. ( ZZ>= ` K ) |-> ( ( F ` j ) ` x ) ) , RR* , < ) e. RR } )
41		eqid 2622	. . . . 5 |- { x e. |^|_ j e. ( ZZ>= ` K ) dom ( F ` j ) | sup ( ran ( j e. ( ZZ>= ` K ) |-> ( ( F ` j ) ` x ) ) , RR* , < ) e. RR } = { x e. |^|_ j e. ( ZZ>= ` K ) dom ( F ` j ) | sup ( ran ( j e. ( ZZ>= ` K ) |-> ( ( F ` j ) ` x ) ) , RR* , < ) e. RR }
42		smflimsuplem1.z	. . . . . . . 8 |- Z = ( ZZ>= ` M )
43	42, 16	eluzelz2d 39640	. . . . . . 7 |- ( ph -> K e. ZZ )
44		uzid 11702	. . . . . . 7 |- ( K e. ZZ -> K e. ( ZZ>= ` K ) )
45		ne0i 3921	. . . . . . 7 |- ( K e. ( ZZ>= ` K ) -> ( ZZ>= ` K ) =/= (/) )
46	43, 44, 45	3syl 18	. . . . . 6 |- ( ph -> ( ZZ>= ` K ) =/= (/) )
47		fvex 6201	. . . . . . . . 9 |- ( F ` j ) e. _V
48	47	dmex 7099	. . . . . . . 8 |- dom ( F ` j ) e. _V
49	48	rgenw 2924	. . . . . . 7 |- A. j e. ( ZZ>= ` K ) dom ( F ` j ) e. _V
50	49	a1i 11	. . . . . 6 |- ( ph -> A. j e. ( ZZ>= ` K ) dom ( F ` j ) e. _V )
51	46, 50	iinexd 39318	. . . . 5 |- ( ph -> |^|_ j e. ( ZZ>= ` K ) dom ( F ` j ) e. _V )
52	41, 51	rabexd 4814	. . . 4 |- ( ph -> { x e. |^|_ j e. ( ZZ>= ` K ) dom ( F ` j ) | sup ( ran ( j e. ( ZZ>= ` K ) |-> ( ( F ` j ) ` x ) ) , RR* , < ) e. RR } e. _V )
53	27, 40, 16, 52	fvmptd3 39447	. . 3 |- ( ph -> ( E ` K ) = { x e. |^|_ j e. ( ZZ>= ` K ) dom ( F ` j ) | sup ( ran ( j e. ( ZZ>= ` K ) |-> ( ( F ` j ) ` x ) ) , RR* , < ) e. RR } )
54	21, 26, 53	3eqtrd 2660	. 2 |- ( ph -> dom ( H ` K ) = { x e. |^|_ j e. ( ZZ>= ` K ) dom ( F ` j ) | sup ( ran ( j e. ( ZZ>= ` K ) |-> ( ( F ` j ) ` x ) ) , RR* , < ) e. RR } )
55		ssrab2 3687	. . . 4 |- { x e. |^|_ j e. ( ZZ>= ` K ) dom ( F ` j ) | sup ( ran ( j e. ( ZZ>= ` K ) |-> ( ( F ` j ) ` x ) ) , RR* , < ) e. RR } C_ |^|_ j e. ( ZZ>= ` K ) dom ( F ` j )
56	55	a1i 11	. . 3 |- ( ph -> { x e. |^|_ j e. ( ZZ>= ` K ) dom ( F ` j ) | sup ( ran ( j e. ( ZZ>= ` K ) |-> ( ( F ` j ) ` x ) ) , RR* , < ) e. RR } C_ |^|_ j e. ( ZZ>= ` K ) dom ( F ` j ) )
57	43, 44	syl 17	. . . 4 |- ( ph -> K e. ( ZZ>= ` K ) )
58		fveq2 6191	. . . . 5 |- ( j = K -> ( F ` j ) = ( F ` K ) )
59	58	dmeqd 5326	. . . 4 |- ( j = K -> dom ( F ` j ) = dom ( F ` K ) )
60		ssid 3624	. . . . 5 |- dom ( F ` K ) C_ dom ( F ` K )
61	60	a1i 11	. . . 4 |- ( ph -> dom ( F ` K ) C_ dom ( F ` K ) )
62	57, 59, 61	iinssd 39314	. . 3 |- ( ph -> |^|_ j e. ( ZZ>= ` K ) dom ( F ` j ) C_ dom ( F ` K ) )
63	56, 62	sstrd 3613	. 2 |- ( ph -> { x e. |^|_ j e. ( ZZ>= ` K ) dom ( F ` j ) | sup ( ran ( j e. ( ZZ>= ` K ) |-> ( ( F ` j ) ` x ) ) , RR* , < ) e. RR } C_ dom ( F ` K ) )
64	54, 63	eqsstrd 3639	1 |- ( ph -> dom ( H ` K ) C_ dom ( F ` K ) )

Syntax hints:   -> wi 4   = wceq 1483   e. wcel 1990   =/= wne 2794   A.wral 2912   {crab 2916   _Vcvv 3200   C_ wss 3574   (/)c0 3915   |^|_ciin 4521   |-> cmpt 4729   dom cdm 5114   ran crn 5115   ` cfv 5888   supcsup 8346   RRcr 9935   RR*cxr 10073   < clt 10074   ZZcz 11377   ZZ>=cuz 11687

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-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-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-int 4476  df-iun 4522  df-iin 4523  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-er 7742  df-en 7956  df-dom 7957  df-sdom 7958  df-sup 8348  df-pnf 10076  df-mnf 10077  df-xr 10078  df-ltxr 10079  df-le 10080  df-neg 10269  df-z 11378  df-uz 11688

This theorem is referenced by:  smflimsuplem4  41029