Theorem smflimsup 41034

Description: The superior limit of a sequence of sigma-measurable functions is sigma-measurable. Proposition 121F (d) of [Fremlin1] p. 39 . (Contributed by Glauco Siliprandi, 23-Oct-2021.)

Hypotheses

Ref	Expression
smflimsup.n	|- F/_ m F
smflimsup.x	|- F/_ x F
smflimsup.m	|- ( ph -> M e. ZZ )
smflimsup.z	|- Z = ( ZZ>= ` M )
smflimsup.s	|- ( ph -> S e. SAlg )
smflimsup.f	|- ( ph -> F : Z --> (SMblFn ` S ) )
smflimsup.d	|- D = { x e. U_ n e. Z |^|_ m e. ( ZZ>= ` n ) dom ( F ` m ) | ( limsup ` ( m e. Z |-> ( ( F ` m ) ` x ) ) ) e. RR }
smflimsup.g	|- G = ( x e. D |-> ( limsup ` ( m e. Z |-> ( ( F ` m ) ` x ) ) ) )

Assertion

Ref	Expression
smflimsup	|- ( ph -> G e. (SMblFn ` S ) )

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

Allowed substitution hints:   ph( x, m, n)   D( x, m, n)   S( x, m, n)   F( x, m)   G( x, m, n)   M( x, m, n)

Proof of Theorem smflimsup

Dummy variables k j q w i l p y are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		smflimsup.m	. 2 |- ( ph -> M e. ZZ )
2		smflimsup.z	. 2 |- Z = ( ZZ>= ` M )
3		smflimsup.s	. 2 |- ( ph -> S e. SAlg )
4		smflimsup.f	. 2 |- ( ph -> F : Z --> (SMblFn ` S ) )
5		smflimsup.d	. . 3 |- D = { x e. U_ n e. Z |^|_ m e. ( ZZ>= ` n ) dom ( F ` m ) | ( limsup ` ( m e. Z |-> ( ( F ` m ) ` x ) ) ) e. RR }
6		fveq2 6191	. . . . . . . . 9 |- ( n = j -> ( ZZ>= ` n ) = ( ZZ>= ` j ) )
7	6	iineq1d 39267	. . . . . . . 8 |- ( n = j -> |^|_ m e. ( ZZ>= ` n ) dom ( F ` m ) = |^|_ m e. ( ZZ>= ` j ) dom ( F ` m ) )
8		nfcv 2764	. . . . . . . . . 10 |- F/_ q dom ( F ` m )
9		smflimsup.n	. . . . . . . . . . . 12 |- F/_ m F
10		nfcv 2764	. . . . . . . . . . . 12 |- F/_ m q
11	9, 10	nffv 6198	. . . . . . . . . . 11 |- F/_ m ( F ` q )
12	11	nfdm 5367	. . . . . . . . . 10 |- F/_ m dom ( F ` q )
13		fveq2 6191	. . . . . . . . . . 11 |- ( m = q -> ( F ` m ) = ( F ` q ) )
14	13	dmeqd 5326	. . . . . . . . . 10 |- ( m = q -> dom ( F ` m ) = dom ( F ` q ) )
15	8, 12, 14	cbviin 4558	. . . . . . . . 9 |- |^|_ m e. ( ZZ>= ` j ) dom ( F ` m ) = |^|_ q e. ( ZZ>= ` j ) dom ( F ` q )
16	15	a1i 11	. . . . . . . 8 |- ( n = j -> |^|_ m e. ( ZZ>= ` j ) dom ( F ` m ) = |^|_ q e. ( ZZ>= ` j ) dom ( F ` q ) )
17	7, 16	eqtrd 2656	. . . . . . 7 |- ( n = j -> |^|_ m e. ( ZZ>= ` n ) dom ( F ` m ) = |^|_ q e. ( ZZ>= ` j ) dom ( F ` q ) )
18	17	cbviunv 4559	. . . . . 6 |- U_ n e. Z |^|_ m e. ( ZZ>= ` n ) dom ( F ` m ) = U_ j e. Z |^|_ q e. ( ZZ>= ` j ) dom ( F ` q )
19	18	eleq2i 2693	. . . . 5 |- ( x e. U_ n e. Z |^|_ m e. ( ZZ>= ` n ) dom ( F ` m ) <-> x e. U_ j e. Z |^|_ q e. ( ZZ>= ` j ) dom ( F ` q ) )
20		nfcv 2764	. . . . . . . 8 |- F/_ q ( ( F ` m ) ` x )
21		nfcv 2764	. . . . . . . . 9 |- F/_ m x
22	11, 21	nffv 6198	. . . . . . . 8 |- F/_ m ( ( F ` q ) ` x )
23	13	fveq1d 6193	. . . . . . . 8 |- ( m = q -> ( ( F ` m ) ` x ) = ( ( F ` q ) ` x ) )
24	20, 22, 23	cbvmpt 4749	. . . . . . 7 |- ( m e. Z |-> ( ( F ` m ) ` x ) ) = ( q e. Z |-> ( ( F ` q ) ` x ) )
25	24	fveq2i 6194	. . . . . 6 |- ( limsup ` ( m e. Z |-> ( ( F ` m ) ` x ) ) ) = ( limsup ` ( q e. Z |-> ( ( F ` q ) ` x ) ) )
26	25	eleq1i 2692	. . . . 5 |- ( ( limsup ` ( m e. Z |-> ( ( F ` m ) ` x ) ) ) e. RR <-> ( limsup ` ( q e. Z |-> ( ( F ` q ) ` x ) ) ) e. RR )
27	19, 26	anbi12i 733	. . . 4 |- ( ( x e. U_ n e. Z |^|_ m e. ( ZZ>= ` n ) dom ( F ` m ) /\ ( limsup ` ( m e. Z |-> ( ( F ` m ) ` x ) ) ) e. RR ) <-> ( x e. U_ j e. Z |^|_ q e. ( ZZ>= ` j ) dom ( F ` q ) /\ ( limsup ` ( q e. Z |-> ( ( F ` q ) ` x ) ) ) e. RR ) )
28	27	rabbia2 3187	. . 3 |- { x e. U_ n e. Z |^|_ m e. ( ZZ>= ` n ) dom ( F ` m ) | ( limsup ` ( m e. Z |-> ( ( F ` m ) ` x ) ) ) e. RR } = { x e. U_ j e. Z |^|_ q e. ( ZZ>= ` j ) dom ( F ` q ) | ( limsup ` ( q e. Z |-> ( ( F ` q ) ` x ) ) ) e. RR }
29		nfcv 2764	. . . . 5 |- F/_ x Z
30		nfcv 2764	. . . . . 6 |- F/_ x ( ZZ>= ` j )
31		smflimsup.x	. . . . . . . 8 |- F/_ x F
32		nfcv 2764	. . . . . . . 8 |- F/_ x q
33	31, 32	nffv 6198	. . . . . . 7 |- F/_ x ( F ` q )
34	33	nfdm 5367	. . . . . 6 |- F/_ x dom ( F ` q )
35	30, 34	nfiin 4549	. . . . 5 |- F/_ x |^|_ q e. ( ZZ>= ` j ) dom ( F ` q )
36	29, 35	nfiun 4548	. . . 4 |- F/_ x U_ j e. Z |^|_ q e. ( ZZ>= ` j ) dom ( F ` q )
37		nfcv 2764	. . . 4 |- F/_ w U_ j e. Z |^|_ q e. ( ZZ>= ` j ) dom ( F ` q )
38		nfv 1843	. . . 4 |- F/ w ( limsup ` ( q e. Z |-> ( ( F ` q ) ` x ) ) ) e. RR
39		nfcv 2764	. . . . . 6 |- F/_ x limsup
40		nfcv 2764	. . . . . . . 8 |- F/_ x w
41	33, 40	nffv 6198	. . . . . . 7 |- F/_ x ( ( F ` q ) ` w )
42	29, 41	nfmpt 4746	. . . . . 6 |- F/_ x ( q e. Z |-> ( ( F ` q ) ` w ) )
43	39, 42	nffv 6198	. . . . 5 |- F/_ x ( limsup ` ( q e. Z |-> ( ( F ` q ) ` w ) ) )
44		nfcv 2764	. . . . 5 |- F/_ x RR
45	43, 44	nfel 2777	. . . 4 |- F/ x ( limsup ` ( q e. Z |-> ( ( F ` q ) ` w ) ) ) e. RR
46		fveq2 6191	. . . . . . 7 |- ( x = w -> ( ( F ` q ) ` x ) = ( ( F ` q ) ` w ) )
47	46	mpteq2dv 4745	. . . . . 6 |- ( x = w -> ( q e. Z |-> ( ( F ` q ) ` x ) ) = ( q e. Z |-> ( ( F ` q ) ` w ) ) )
48	47	fveq2d 6195	. . . . 5 |- ( x = w -> ( limsup ` ( q e. Z |-> ( ( F ` q ) ` x ) ) ) = ( limsup ` ( q e. Z |-> ( ( F ` q ) ` w ) ) ) )
49	48	eleq1d 2686	. . . 4 |- ( x = w -> ( ( limsup ` ( q e. Z |-> ( ( F ` q ) ` x ) ) ) e. RR <-> ( limsup ` ( q e. Z |-> ( ( F ` q ) ` w ) ) ) e. RR ) )
50	36, 37, 38, 45, 49	cbvrab 3198	. . 3 |- { x e. U_ j e. Z |^|_ q e. ( ZZ>= ` j ) dom ( F ` q ) | ( limsup ` ( q e. Z |-> ( ( F ` q ) ` x ) ) ) e. RR } = { w e. U_ j e. Z |^|_ q e. ( ZZ>= ` j ) dom ( F ` q ) | ( limsup ` ( q e. Z |-> ( ( F ` q ) ` w ) ) ) e. RR }
51	5, 28, 50	3eqtri 2648	. 2 |- D = { w e. U_ j e. Z |^|_ q e. ( ZZ>= ` j ) dom ( F ` q ) | ( limsup ` ( q e. Z |-> ( ( F ` q ) ` w ) ) ) e. RR }
52		smflimsup.g	. . 3 |- G = ( x e. D |-> ( limsup ` ( m e. Z |-> ( ( F ` m ) ` x ) ) ) )
53	25	mpteq2i 4741	. . 3 |- ( x e. D |-> ( limsup ` ( m e. Z |-> ( ( F ` m ) ` x ) ) ) ) = ( x e. D |-> ( limsup ` ( q e. Z |-> ( ( F ` q ) ` x ) ) ) )
54		nfrab1 3122	. . . . 5 |- F/_ x { x e. U_ n e. Z |^|_ m e. ( ZZ>= ` n ) dom ( F ` m ) | ( limsup ` ( m e. Z |-> ( ( F ` m ) ` x ) ) ) e. RR }
55	5, 54	nfcxfr 2762	. . . 4 |- F/_ x D
56		nfcv 2764	. . . 4 |- F/_ w D
57		nfcv 2764	. . . 4 |- F/_ w ( limsup ` ( q e. Z |-> ( ( F ` q ) ` x ) ) )
58	55, 56, 57, 43, 48	cbvmptf 4748	. . 3 |- ( x e. D |-> ( limsup ` ( q e. Z |-> ( ( F ` q ) ` x ) ) ) ) = ( w e. D |-> ( limsup ` ( q e. Z |-> ( ( F ` q ) ` w ) ) ) )
59	52, 53, 58	3eqtri 2648	. 2 |- G = ( w e. D |-> ( limsup ` ( q e. Z |-> ( ( F ` q ) ` w ) ) ) )
60		nfcv 2764	. . . . . . 7 |- F/_ x ( ZZ>= ` i )
61	60, 34	nfiin 4549	. . . . . 6 |- F/_ x |^|_ q e. ( ZZ>= ` i ) dom ( F ` q )
62		nfcv 2764	. . . . . 6 |- F/_ w |^|_ q e. ( ZZ>= ` i ) dom ( F ` q )
63		nfv 1843	. . . . . 6 |- F/ w sup ( ran ( q e. ( ZZ>= ` i ) |-> ( ( F ` q ) ` x ) ) , RR* , < ) e. RR
64	60, 41	nfmpt 4746	. . . . . . . . 9 |- F/_ x ( q e. ( ZZ>= ` i ) |-> ( ( F ` q ) ` w ) )
65	64	nfrn 5368	. . . . . . . 8 |- F/_ x ran ( q e. ( ZZ>= ` i ) |-> ( ( F ` q ) ` w ) )
66		nfcv 2764	. . . . . . . 8 |- F/_ x RR*
67		nfcv 2764	. . . . . . . 8 |- F/_ x <
68	65, 66, 67	nfsup 8357	. . . . . . 7 |- F/_ x sup ( ran ( q e. ( ZZ>= ` i ) |-> ( ( F ` q ) ` w ) ) , RR* , < )
69	68, 44	nfel 2777	. . . . . 6 |- F/ x sup ( ran ( q e. ( ZZ>= ` i ) |-> ( ( F ` q ) ` w ) ) , RR* , < ) e. RR
70	46	mpteq2dv 4745	. . . . . . . . 9 |- ( x = w -> ( q e. ( ZZ>= ` i ) |-> ( ( F ` q ) ` x ) ) = ( q e. ( ZZ>= ` i ) |-> ( ( F ` q ) ` w ) ) )
71	70	rneqd 5353	. . . . . . . 8 |- ( x = w -> ran ( q e. ( ZZ>= ` i ) |-> ( ( F ` q ) ` x ) ) = ran ( q e. ( ZZ>= ` i ) |-> ( ( F ` q ) ` w ) ) )
72	71	supeq1d 8352	. . . . . . 7 |- ( x = w -> sup ( ran ( q e. ( ZZ>= ` i ) |-> ( ( F ` q ) ` x ) ) , RR* , < ) = sup ( ran ( q e. ( ZZ>= ` i ) |-> ( ( F ` q ) ` w ) ) , RR* , < ) )
73	72	eleq1d 2686	. . . . . 6 |- ( x = w -> ( sup ( ran ( q e. ( ZZ>= ` i ) |-> ( ( F ` q ) ` x ) ) , RR* , < ) e. RR <-> sup ( ran ( q e. ( ZZ>= ` i ) |-> ( ( F ` q ) ` w ) ) , RR* , < ) e. RR ) )
74	61, 62, 63, 69, 73	cbvrab 3198	. . . . 5 |- { x e. |^|_ q e. ( ZZ>= ` i ) dom ( F ` q ) | sup ( ran ( q e. ( ZZ>= ` i ) |-> ( ( F ` q ) ` x ) ) , RR* , < ) e. RR } = { w e. |^|_ q e. ( ZZ>= ` i ) dom ( F ` q ) | sup ( ran ( q e. ( ZZ>= ` i ) |-> ( ( F ` q ) ` w ) ) , RR* , < ) e. RR }
75	74	a1i 11	. . . 4 |- ( i = k -> { x e. |^|_ q e. ( ZZ>= ` i ) dom ( F ` q ) | sup ( ran ( q e. ( ZZ>= ` i ) |-> ( ( F ` q ) ` x ) ) , RR* , < ) e. RR } = { w e. |^|_ q e. ( ZZ>= ` i ) dom ( F ` q ) | sup ( ran ( q e. ( ZZ>= ` i ) |-> ( ( F ` q ) ` w ) ) , RR* , < ) e. RR } )
76		fveq2 6191	. . . . . . . 8 |- ( i = k -> ( ZZ>= ` i ) = ( ZZ>= ` k ) )
77	76	iineq1d 39267	. . . . . . 7 |- ( i = k -> |^|_ q e. ( ZZ>= ` i ) dom ( F ` q ) = |^|_ q e. ( ZZ>= ` k ) dom ( F ` q ) )
78	77	eleq2d 2687	. . . . . 6 |- ( i = k -> ( w e. |^|_ q e. ( ZZ>= ` i ) dom ( F ` q ) <-> w e. |^|_ q e. ( ZZ>= ` k ) dom ( F ` q ) ) )
79	76	mpteq1d 4738	. . . . . . . . 9 |- ( i = k -> ( q e. ( ZZ>= ` i ) |-> ( ( F ` q ) ` w ) ) = ( q e. ( ZZ>= ` k ) |-> ( ( F ` q ) ` w ) ) )
80	79	rneqd 5353	. . . . . . . 8 |- ( i = k -> ran ( q e. ( ZZ>= ` i ) |-> ( ( F ` q ) ` w ) ) = ran ( q e. ( ZZ>= ` k ) |-> ( ( F ` q ) ` w ) ) )
81	80	supeq1d 8352	. . . . . . 7 |- ( i = k -> sup ( ran ( q e. ( ZZ>= ` i ) |-> ( ( F ` q ) ` w ) ) , RR* , < ) = sup ( ran ( q e. ( ZZ>= ` k ) |-> ( ( F ` q ) ` w ) ) , RR* , < ) )
82	81	eleq1d 2686	. . . . . 6 |- ( i = k -> ( sup ( ran ( q e. ( ZZ>= ` i ) |-> ( ( F ` q ) ` w ) ) , RR* , < ) e. RR <-> sup ( ran ( q e. ( ZZ>= ` k ) |-> ( ( F ` q ) ` w ) ) , RR* , < ) e. RR ) )
83	78, 82	anbi12d 747	. . . . 5 |- ( i = k -> ( ( w e. |^|_ q e. ( ZZ>= ` i ) dom ( F ` q ) /\ sup ( ran ( q e. ( ZZ>= ` i ) |-> ( ( F ` q ) ` w ) ) , RR* , < ) e. RR ) <-> ( w e. |^|_ q e. ( ZZ>= ` k ) dom ( F ` q ) /\ sup ( ran ( q e. ( ZZ>= ` k ) |-> ( ( F ` q ) ` w ) ) , RR* , < ) e. RR ) ) )
84	83	rabbidva2 3186	. . . 4 |- ( i = k -> { w e. |^|_ q e. ( ZZ>= ` i ) dom ( F ` q ) | sup ( ran ( q e. ( ZZ>= ` i ) |-> ( ( F ` q ) ` w ) ) , RR* , < ) e. RR } = { w e. |^|_ q e. ( ZZ>= ` k ) dom ( F ` q ) | sup ( ran ( q e. ( ZZ>= ` k ) |-> ( ( F ` q ) ` w ) ) , RR* , < ) e. RR } )
85	75, 84	eqtrd 2656	. . 3 |- ( i = k -> { x e. |^|_ q e. ( ZZ>= ` i ) dom ( F ` q ) | sup ( ran ( q e. ( ZZ>= ` i ) |-> ( ( F ` q ) ` x ) ) , RR* , < ) e. RR } = { w e. |^|_ q e. ( ZZ>= ` k ) dom ( F ` q ) | sup ( ran ( q e. ( ZZ>= ` k ) |-> ( ( F ` q ) ` w ) ) , RR* , < ) e. RR } )
86	85	cbvmptv 4750	. 2 |- ( i e. Z |-> { x e. |^|_ q e. ( ZZ>= ` i ) dom ( F ` q ) | sup ( ran ( q e. ( ZZ>= ` i ) |-> ( ( F ` q ) ` x ) ) , RR* , < ) e. RR } ) = ( k e. Z |-> { w e. |^|_ q e. ( ZZ>= ` k ) dom ( F ` q ) | sup ( ran ( q e. ( ZZ>= ` k ) |-> ( ( F ` q ) ` w ) ) , RR* , < ) e. RR } )
87		fveq2 6191	. . . . . . . . . 10 |- ( y = w -> ( ( F ` p ) ` y ) = ( ( F ` p ) ` w ) )
88	87	mpteq2dv 4745	. . . . . . . . 9 |- ( y = w -> ( p e. ( ZZ>= ` l ) |-> ( ( F ` p ) ` y ) ) = ( p e. ( ZZ>= ` l ) |-> ( ( F ` p ) ` w ) ) )
89	88	rneqd 5353	. . . . . . . 8 |- ( y = w -> ran ( p e. ( ZZ>= ` l ) |-> ( ( F ` p ) ` y ) ) = ran ( p e. ( ZZ>= ` l ) |-> ( ( F ` p ) ` w ) ) )
90	89	supeq1d 8352	. . . . . . 7 |- ( y = w -> sup ( ran ( p e. ( ZZ>= ` l ) |-> ( ( F ` p ) ` y ) ) , RR* , < ) = sup ( ran ( p e. ( ZZ>= ` l ) |-> ( ( F ` p ) ` w ) ) , RR* , < ) )
91	90	cbvmptv 4750	. . . . . 6 |- ( y e. ( ( i e. Z |-> { x e. |^|_ p e. ( ZZ>= ` i ) dom ( F ` p ) | sup ( ran ( p e. ( ZZ>= ` i ) |-> ( ( F ` p ) ` x ) ) , RR* , < ) e. RR } ) ` l ) |-> sup ( ran ( p e. ( ZZ>= ` l ) |-> ( ( F ` p ) ` y ) ) , RR* , < ) ) = ( w e. ( ( i e. Z |-> { x e. |^|_ p e. ( ZZ>= ` i ) dom ( F ` p ) | sup ( ran ( p e. ( ZZ>= ` i ) |-> ( ( F ` p ) ` x ) ) , RR* , < ) e. RR } ) ` l ) |-> sup ( ran ( p e. ( ZZ>= ` l ) |-> ( ( F ` p ) ` w ) ) , RR* , < ) )
92		fveq2 6191	. . . . . . . . . . . . . 14 |- ( p = q -> ( F ` p ) = ( F ` q ) )
93	92	dmeqd 5326	. . . . . . . . . . . . 13 |- ( p = q -> dom ( F ` p ) = dom ( F ` q ) )
94	93	cbviinv 4560	. . . . . . . . . . . 12 |- |^|_ p e. ( ZZ>= ` i ) dom ( F ` p ) = |^|_ q e. ( ZZ>= ` i ) dom ( F ` q )
95	94	eleq2i 2693	. . . . . . . . . . 11 |- ( x e. |^|_ p e. ( ZZ>= ` i ) dom ( F ` p ) <-> x e. |^|_ q e. ( ZZ>= ` i ) dom ( F ` q ) )
96		nfcv 2764	. . . . . . . . . . . . . . 15 |- F/_ q ( ( F ` p ) ` x )
97		nfcv 2764	. . . . . . . . . . . . . . . 16 |- F/_ p ( F ` q )
98		nfcv 2764	. . . . . . . . . . . . . . . 16 |- F/_ p x
99	97, 98	nffv 6198	. . . . . . . . . . . . . . 15 |- F/_ p ( ( F ` q ) ` x )
100	92	fveq1d 6193	. . . . . . . . . . . . . . 15 |- ( p = q -> ( ( F ` p ) ` x ) = ( ( F ` q ) ` x ) )
101	96, 99, 100	cbvmpt 4749	. . . . . . . . . . . . . 14 |- ( p e. ( ZZ>= ` i ) |-> ( ( F ` p ) ` x ) ) = ( q e. ( ZZ>= ` i ) |-> ( ( F ` q ) ` x ) )
102	101	rneqi 5352	. . . . . . . . . . . . 13 |- ran ( p e. ( ZZ>= ` i ) |-> ( ( F ` p ) ` x ) ) = ran ( q e. ( ZZ>= ` i ) |-> ( ( F ` q ) ` x ) )
103	102	supeq1i 8353	. . . . . . . . . . . 12 |- sup ( ran ( p e. ( ZZ>= ` i ) |-> ( ( F ` p ) ` x ) ) , RR* , < ) = sup ( ran ( q e. ( ZZ>= ` i ) |-> ( ( F ` q ) ` x ) ) , RR* , < )
104	103	eleq1i 2692	. . . . . . . . . . 11 |- ( sup ( ran ( p e. ( ZZ>= ` i ) |-> ( ( F ` p ) ` x ) ) , RR* , < ) e. RR <-> sup ( ran ( q e. ( ZZ>= ` i ) |-> ( ( F ` q ) ` x ) ) , RR* , < ) e. RR )
105	95, 104	anbi12i 733	. . . . . . . . . 10 |- ( ( x e. |^|_ p e. ( ZZ>= ` i ) dom ( F ` p ) /\ sup ( ran ( p e. ( ZZ>= ` i ) |-> ( ( F ` p ) ` x ) ) , RR* , < ) e. RR ) <-> ( x e. |^|_ q e. ( ZZ>= ` i ) dom ( F ` q ) /\ sup ( ran ( q e. ( ZZ>= ` i ) |-> ( ( F ` q ) ` x ) ) , RR* , < ) e. RR ) )
106	105	rabbia2 3187	. . . . . . . . 9 |- { x e. |^|_ p e. ( ZZ>= ` i ) dom ( F ` p ) | sup ( ran ( p e. ( ZZ>= ` i ) |-> ( ( F ` p ) ` x ) ) , RR* , < ) e. RR } = { x e. |^|_ q e. ( ZZ>= ` i ) dom ( F ` q ) | sup ( ran ( q e. ( ZZ>= ` i ) |-> ( ( F ` q ) ` x ) ) , RR* , < ) e. RR }
107	106	mpteq2i 4741	. . . . . . . 8 |- ( i e. Z |-> { x e. |^|_ p e. ( ZZ>= ` i ) dom ( F ` p ) | sup ( ran ( p e. ( ZZ>= ` i ) |-> ( ( F ` p ) ` x ) ) , RR* , < ) e. RR } ) = ( i e. Z |-> { x e. |^|_ q e. ( ZZ>= ` i ) dom ( F ` q ) | sup ( ran ( q e. ( ZZ>= ` i ) |-> ( ( F ` q ) ` x ) ) , RR* , < ) e. RR } )
108	107	fveq1i 6192	. . . . . . 7 |- ( ( i e. Z |-> { x e. |^|_ p e. ( ZZ>= ` i ) dom ( F ` p ) | sup ( ran ( p e. ( ZZ>= ` i ) |-> ( ( F ` p ) ` x ) ) , RR* , < ) e. RR } ) ` l ) = ( ( i e. Z |-> { x e. |^|_ q e. ( ZZ>= ` i ) dom ( F ` q ) | sup ( ran ( q e. ( ZZ>= ` i ) |-> ( ( F ` q ) ` x ) ) , RR* , < ) e. RR } ) ` l )
109	92	fveq1d 6193	. . . . . . . . . 10 |- ( p = q -> ( ( F ` p ) ` w ) = ( ( F ` q ) ` w ) )
110	109	cbvmptv 4750	. . . . . . . . 9 |- ( p e. ( ZZ>= ` l ) |-> ( ( F ` p ) ` w ) ) = ( q e. ( ZZ>= ` l ) |-> ( ( F ` q ) ` w ) )
111	110	rneqi 5352	. . . . . . . 8 |- ran ( p e. ( ZZ>= ` l ) |-> ( ( F ` p ) ` w ) ) = ran ( q e. ( ZZ>= ` l ) |-> ( ( F ` q ) ` w ) )
112	111	supeq1i 8353	. . . . . . 7 |- sup ( ran ( p e. ( ZZ>= ` l ) |-> ( ( F ` p ) ` w ) ) , RR* , < ) = sup ( ran ( q e. ( ZZ>= ` l ) |-> ( ( F ` q ) ` w ) ) , RR* , < )
113	108, 112	mpteq12i 4742	. . . . . 6 |- ( w e. ( ( i e. Z |-> { x e. |^|_ p e. ( ZZ>= ` i ) dom ( F ` p ) | sup ( ran ( p e. ( ZZ>= ` i ) |-> ( ( F ` p ) ` x ) ) , RR* , < ) e. RR } ) ` l ) |-> sup ( ran ( p e. ( ZZ>= ` l ) |-> ( ( F ` p ) ` w ) ) , RR* , < ) ) = ( w e. ( ( i e. Z |-> { x e. |^|_ q e. ( ZZ>= ` i ) dom ( F ` q ) | sup ( ran ( q e. ( ZZ>= ` i ) |-> ( ( F ` q ) ` x ) ) , RR* , < ) e. RR } ) ` l ) |-> sup ( ran ( q e. ( ZZ>= ` l ) |-> ( ( F ` q ) ` w ) ) , RR* , < ) )
114	91, 113	eqtri 2644	. . . . 5 |- ( y e. ( ( i e. Z |-> { x e. |^|_ p e. ( ZZ>= ` i ) dom ( F ` p ) | sup ( ran ( p e. ( ZZ>= ` i ) |-> ( ( F ` p ) ` x ) ) , RR* , < ) e. RR } ) ` l ) |-> sup ( ran ( p e. ( ZZ>= ` l ) |-> ( ( F ` p ) ` y ) ) , RR* , < ) ) = ( w e. ( ( i e. Z |-> { x e. |^|_ q e. ( ZZ>= ` i ) dom ( F ` q ) | sup ( ran ( q e. ( ZZ>= ` i ) |-> ( ( F ` q ) ` x ) ) , RR* , < ) e. RR } ) ` l ) |-> sup ( ran ( q e. ( ZZ>= ` l ) |-> ( ( F ` q ) ` w ) ) , RR* , < ) )
115	114	a1i 11	. . . 4 |- ( l = k -> ( y e. ( ( i e. Z |-> { x e. |^|_ p e. ( ZZ>= ` i ) dom ( F ` p ) | sup ( ran ( p e. ( ZZ>= ` i ) |-> ( ( F ` p ) ` x ) ) , RR* , < ) e. RR } ) ` l ) |-> sup ( ran ( p e. ( ZZ>= ` l ) |-> ( ( F ` p ) ` y ) ) , RR* , < ) ) = ( w e. ( ( i e. Z |-> { x e. |^|_ q e. ( ZZ>= ` i ) dom ( F ` q ) | sup ( ran ( q e. ( ZZ>= ` i ) |-> ( ( F ` q ) ` x ) ) , RR* , < ) e. RR } ) ` l ) |-> sup ( ran ( q e. ( ZZ>= ` l ) |-> ( ( F ` q ) ` w ) ) , RR* , < ) ) )
116		fveq2 6191	. . . . 5 |- ( l = k -> ( ( i e. Z |-> { x e. |^|_ q e. ( ZZ>= ` i ) dom ( F ` q ) | sup ( ran ( q e. ( ZZ>= ` i ) |-> ( ( F ` q ) ` x ) ) , RR* , < ) e. RR } ) ` l ) = ( ( i e. Z |-> { x e. |^|_ q e. ( ZZ>= ` i ) dom ( F ` q ) | sup ( ran ( q e. ( ZZ>= ` i ) |-> ( ( F ` q ) ` x ) ) , RR* , < ) e. RR } ) ` k ) )
117		fveq2 6191	. . . . . . . 8 |- ( l = k -> ( ZZ>= ` l ) = ( ZZ>= ` k ) )
118	117	mpteq1d 4738	. . . . . . 7 |- ( l = k -> ( q e. ( ZZ>= ` l ) |-> ( ( F ` q ) ` w ) ) = ( q e. ( ZZ>= ` k ) |-> ( ( F ` q ) ` w ) ) )
119	118	rneqd 5353	. . . . . 6 |- ( l = k -> ran ( q e. ( ZZ>= ` l ) |-> ( ( F ` q ) ` w ) ) = ran ( q e. ( ZZ>= ` k ) |-> ( ( F ` q ) ` w ) ) )
120	119	supeq1d 8352	. . . . 5 |- ( l = k -> sup ( ran ( q e. ( ZZ>= ` l ) |-> ( ( F ` q ) ` w ) ) , RR* , < ) = sup ( ran ( q e. ( ZZ>= ` k ) |-> ( ( F ` q ) ` w ) ) , RR* , < ) )
121	116, 120	mpteq12dv 4733	. . . 4 |- ( l = k -> ( w e. ( ( i e. Z |-> { x e. |^|_ q e. ( ZZ>= ` i ) dom ( F ` q ) | sup ( ran ( q e. ( ZZ>= ` i ) |-> ( ( F ` q ) ` x ) ) , RR* , < ) e. RR } ) ` l ) |-> sup ( ran ( q e. ( ZZ>= ` l ) |-> ( ( F ` q ) ` w ) ) , RR* , < ) ) = ( w e. ( ( i e. Z |-> { x e. |^|_ q e. ( ZZ>= ` i ) dom ( F ` q ) | sup ( ran ( q e. ( ZZ>= ` i ) |-> ( ( F ` q ) ` x ) ) , RR* , < ) e. RR } ) ` k ) |-> sup ( ran ( q e. ( ZZ>= ` k ) |-> ( ( F ` q ) ` w ) ) , RR* , < ) ) )
122	115, 121	eqtrd 2656	. . 3 |- ( l = k -> ( y e. ( ( i e. Z |-> { x e. |^|_ p e. ( ZZ>= ` i ) dom ( F ` p ) | sup ( ran ( p e. ( ZZ>= ` i ) |-> ( ( F ` p ) ` x ) ) , RR* , < ) e. RR } ) ` l ) |-> sup ( ran ( p e. ( ZZ>= ` l ) |-> ( ( F ` p ) ` y ) ) , RR* , < ) ) = ( w e. ( ( i e. Z |-> { x e. |^|_ q e. ( ZZ>= ` i ) dom ( F ` q ) | sup ( ran ( q e. ( ZZ>= ` i ) |-> ( ( F ` q ) ` x ) ) , RR* , < ) e. RR } ) ` k ) |-> sup ( ran ( q e. ( ZZ>= ` k ) |-> ( ( F ` q ) ` w ) ) , RR* , < ) ) )
123	122	cbvmptv 4750	. 2 |- ( l e. Z |-> ( y e. ( ( i e. Z |-> { x e. |^|_ p e. ( ZZ>= ` i ) dom ( F ` p ) | sup ( ran ( p e. ( ZZ>= ` i ) |-> ( ( F ` p ) ` x ) ) , RR* , < ) e. RR } ) ` l ) |-> sup ( ran ( p e. ( ZZ>= ` l ) |-> ( ( F ` p ) ` y ) ) , RR* , < ) ) ) = ( k e. Z |-> ( w e. ( ( i e. Z |-> { x e. |^|_ q e. ( ZZ>= ` i ) dom ( F ` q ) | sup ( ran ( q e. ( ZZ>= ` i ) |-> ( ( F ` q ) ` x ) ) , RR* , < ) e. RR } ) ` k ) |-> sup ( ran ( q e. ( ZZ>= ` k ) |-> ( ( F ` q ) ` w ) ) , RR* , < ) ) )
124	1, 2, 3, 4, 51, 59, 86, 123	smflimsuplem8 41033	1 |- ( ph -> G e. (SMblFn ` S ) )

Syntax hints:   -> wi 4   = wceq 1483   e. wcel 1990   F/_wnfc 2751   {crab 2916   U_ciun 4520   |^|_ciin 4521   |-> cmpt 4729   dom cdm 5114   ran crn 5115   -->wf 5884   ` cfv 5888   supcsup 8346   RRcr 9935   RR*cxr 10073   < clt 10074   ZZcz 11377   ZZ>=cuz 11687   limsupclsp 14201  SAlgcsalg 40528  SMblFncsmblfn 40909

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-inf2 8538  ax-cc 9257  ax-ac2 9285  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-iin 4523  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-se 5074  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-isom 5897  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-omul 7565  df-er 7742  df-map 7859  df-pm 7860  df-en 7956  df-dom 7957  df-sdom 7958  df-fin 7959  df-sup 8348  df-inf 8349  df-oi 8415  df-card 8765  df-acn 8768  df-ac 8939  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-q 11789  df-rp 11833  df-ioo 12179  df-ioc 12180  df-ico 12181  df-fz 12327  df-fl 12593  df-ceil 12594  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  df-rlim 14220  df-rest 16083  df-topgen 16104  df-top 20699  df-bases 20750  df-salg 40529  df-salgen 40533  df-smblfn 40910

This theorem is referenced by:  smflimsupmpt  41035