smfsupxr - Mathbox for Glauco Siliprandi

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Theorem smfsupxr 45518

Description: The supremum of a countable set of sigma-measurable functions is sigma-measurable. Proposition 121F (b) of [Fremlin1] p. 38 . (Contributed by Glauco Siliprandi, 23-Oct-2021.)

**Hypotheses**
Ref	Expression
smfsupxr.n	⊢ Ⅎ𝑛𝐹
smfsupxr.x	⊢ Ⅎ𝑥𝐹
smfsupxr.m	⊢ (𝜑 → 𝑀 ∈ ℤ)
smfsupxr.z	⊢ 𝑍 = (ℤ_≥‘𝑀)
smfsupxr.s	⊢ (𝜑 → 𝑆 ∈ SAlg)
smfsupxr.f	⊢ (𝜑 → 𝐹:𝑍⟶(SMblFn‘𝑆))
smfsupxr.d	⊢ 𝐷 = {𝑥 ∈ ∩ 𝑛 ∈ 𝑍 dom (𝐹‘𝑛) ∣ sup(ran (𝑛 ∈ 𝑍 ↦ ((𝐹‘𝑛)‘𝑥)), ℝ^*, < ) ∈ ℝ}
smfsupxr.g	⊢ 𝐺 = (𝑥 ∈ 𝐷 ↦ sup(ran (𝑛 ∈ 𝑍 ↦ ((𝐹‘𝑛)‘𝑥)), ℝ^*, < ))

**Assertion**
Ref	Expression
smfsupxr	⊢ (𝜑 → 𝐺 ∈ (SMblFn‘𝑆))

Distinct variable groups: 𝑛,𝑍,𝑥 𝜑,𝑛,𝑥 Allowed substitution hints: 𝐷(𝑥,𝑛) 𝑆(𝑥,𝑛) 𝐹(𝑥,𝑛) 𝐺(𝑥,𝑛) 𝑀(𝑥,𝑛)

Proof of Theorem smfsupxr Dummy variable 𝑦 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		smfsupxr.g	. . . 4 ⊢ 𝐺 = (𝑥 ∈ 𝐷 ↦ sup(ran (𝑛 ∈ 𝑍 ↦ ((𝐹‘𝑛)‘𝑥)), ℝ^*, < ))
2	1	a1i 11	. . 3 ⊢ (𝜑 → 𝐺 = (𝑥 ∈ 𝐷 ↦ sup(ran (𝑛 ∈ 𝑍 ↦ ((𝐹‘𝑛)‘𝑥)), ℝ^*, < )))
3		smfsupxr.d	. . . . . 6 ⊢ 𝐷 = {𝑥 ∈ ∩ 𝑛 ∈ 𝑍 dom (𝐹‘𝑛) ∣ sup(ran (𝑛 ∈ 𝑍 ↦ ((𝐹‘𝑛)‘𝑥)), ℝ^*, < ) ∈ ℝ}
4	3	a1i 11	. . . . 5 ⊢ (𝜑 → 𝐷 = {𝑥 ∈ ∩ 𝑛 ∈ 𝑍 dom (𝐹‘𝑛) ∣ sup(ran (𝑛 ∈ 𝑍 ↦ ((𝐹‘𝑛)‘𝑥)), ℝ^*, < ) ∈ ℝ})
5		nfv 1917	. . . . . . . 8 ⊢ Ⅎ𝑛𝜑
6		nfcv 2903	. . . . . . . . 9 ⊢ Ⅎ𝑛𝑥
7		nfii1 5031	. . . . . . . . 9 ⊢ Ⅎ𝑛∩ 𝑛 ∈ 𝑍 dom (𝐹‘𝑛)
8	6, 7	nfel 2917	. . . . . . . 8 ⊢ Ⅎ𝑛 𝑥 ∈ ∩ 𝑛 ∈ 𝑍 dom (𝐹‘𝑛)
9	5, 8	nfan 1902	. . . . . . 7 ⊢ Ⅎ𝑛(𝜑 ∧ 𝑥 ∈ ∩ 𝑛 ∈ 𝑍 dom (𝐹‘𝑛))
10		smfsupxr.m	. . . . . . . . 9 ⊢ (𝜑 → 𝑀 ∈ ℤ)
11		smfsupxr.z	. . . . . . . . 9 ⊢ 𝑍 = (ℤ_≥‘𝑀)
12	10, 11	uzn0d 44121	. . . . . . . 8 ⊢ (𝜑 → 𝑍 ≠ ∅)
13	12	adantr 481	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑥 ∈ ∩ 𝑛 ∈ 𝑍 dom (𝐹‘𝑛)) → 𝑍 ≠ ∅)
14		smfsupxr.s	. . . . . . . . . . 11 ⊢ (𝜑 → 𝑆 ∈ SAlg)
15	14	adantr 481	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑛 ∈ 𝑍) → 𝑆 ∈ SAlg)
16		smfsupxr.f	. . . . . . . . . . 11 ⊢ (𝜑 → 𝐹:𝑍⟶(SMblFn‘𝑆))
17	16	ffvelcdmda 7083	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑛 ∈ 𝑍) → (𝐹‘𝑛) ∈ (SMblFn‘𝑆))
18		eqid 2732	. . . . . . . . . 10 ⊢ dom (𝐹‘𝑛) = dom (𝐹‘𝑛)
19	15, 17, 18	smff 45434	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑛 ∈ 𝑍) → (𝐹‘𝑛):dom (𝐹‘𝑛)⟶ℝ)
20	19	adantlr 713	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝑥 ∈ ∩ 𝑛 ∈ 𝑍 dom (𝐹‘𝑛)) ∧ 𝑛 ∈ 𝑍) → (𝐹‘𝑛):dom (𝐹‘𝑛)⟶ℝ)
21		eliinid 43785	. . . . . . . . 9 ⊢ ((𝑥 ∈ ∩ 𝑛 ∈ 𝑍 dom (𝐹‘𝑛) ∧ 𝑛 ∈ 𝑍) → 𝑥 ∈ dom (𝐹‘𝑛))
22	21	adantll 712	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝑥 ∈ ∩ 𝑛 ∈ 𝑍 dom (𝐹‘𝑛)) ∧ 𝑛 ∈ 𝑍) → 𝑥 ∈ dom (𝐹‘𝑛))
23	20, 22	ffvelcdmd 7084	. . . . . . 7 ⊢ (((𝜑 ∧ 𝑥 ∈ ∩ 𝑛 ∈ 𝑍 dom (𝐹‘𝑛)) ∧ 𝑛 ∈ 𝑍) → ((𝐹‘𝑛)‘𝑥) ∈ ℝ)
24	9, 13, 23	supxrre3rnmpt 44125	. . . . . 6 ⊢ ((𝜑 ∧ 𝑥 ∈ ∩ 𝑛 ∈ 𝑍 dom (𝐹‘𝑛)) → (sup(ran (𝑛 ∈ 𝑍 ↦ ((𝐹‘𝑛)‘𝑥)), ℝ^*, < ) ∈ ℝ ↔ ∃𝑦 ∈ ℝ ∀𝑛 ∈ 𝑍 ((𝐹‘𝑛)‘𝑥) ≤ 𝑦))
25	24	rabbidva 3439	. . . . 5 ⊢ (𝜑 → {𝑥 ∈ ∩ 𝑛 ∈ 𝑍 dom (𝐹‘𝑛) ∣ sup(ran (𝑛 ∈ 𝑍 ↦ ((𝐹‘𝑛)‘𝑥)), ℝ^*, < ) ∈ ℝ} = {𝑥 ∈ ∩ 𝑛 ∈ 𝑍 dom (𝐹‘𝑛) ∣ ∃𝑦 ∈ ℝ ∀𝑛 ∈ 𝑍 ((𝐹‘𝑛)‘𝑥) ≤ 𝑦})
26	4, 25	eqtrd 2772	. . . 4 ⊢ (𝜑 → 𝐷 = {𝑥 ∈ ∩ 𝑛 ∈ 𝑍 dom (𝐹‘𝑛) ∣ ∃𝑦 ∈ ℝ ∀𝑛 ∈ 𝑍 ((𝐹‘𝑛)‘𝑥) ≤ 𝑦})
27		nfmpt1 5255	. . . . . . . . . . . 12 ⊢ Ⅎ𝑛(𝑛 ∈ 𝑍 ↦ ((𝐹‘𝑛)‘𝑥))
28	27	nfrn 5949	. . . . . . . . . . 11 ⊢ Ⅎ𝑛ran (𝑛 ∈ 𝑍 ↦ ((𝐹‘𝑛)‘𝑥))
29		nfcv 2903	. . . . . . . . . . 11 ⊢ Ⅎ𝑛ℝ^*
30		nfcv 2903	. . . . . . . . . . 11 ⊢ Ⅎ𝑛 <
31	28, 29, 30	nfsup 9442	. . . . . . . . . 10 ⊢ Ⅎ𝑛sup(ran (𝑛 ∈ 𝑍 ↦ ((𝐹‘𝑛)‘𝑥)), ℝ^*, < )
32		nfcv 2903	. . . . . . . . . 10 ⊢ Ⅎ𝑛ℝ
33	31, 32	nfel 2917	. . . . . . . . 9 ⊢ Ⅎ𝑛sup(ran (𝑛 ∈ 𝑍 ↦ ((𝐹‘𝑛)‘𝑥)), ℝ^*, < ) ∈ ℝ
34	33, 7	nfrabw 3468	. . . . . . . 8 ⊢ Ⅎ𝑛{𝑥 ∈ ∩ 𝑛 ∈ 𝑍 dom (𝐹‘𝑛) ∣ sup(ran (𝑛 ∈ 𝑍 ↦ ((𝐹‘𝑛)‘𝑥)), ℝ^*, < ) ∈ ℝ}
35	3, 34	nfcxfr 2901	. . . . . . 7 ⊢ Ⅎ𝑛𝐷
36	6, 35	nfel 2917	. . . . . 6 ⊢ Ⅎ𝑛 𝑥 ∈ 𝐷
37	5, 36	nfan 1902	. . . . 5 ⊢ Ⅎ𝑛(𝜑 ∧ 𝑥 ∈ 𝐷)
38	12	adantr 481	. . . . 5 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐷) → 𝑍 ≠ ∅)
39	19	adantlr 713	. . . . . 6 ⊢ (((𝜑 ∧ 𝑥 ∈ 𝐷) ∧ 𝑛 ∈ 𝑍) → (𝐹‘𝑛):dom (𝐹‘𝑛)⟶ℝ)
40		nfcv 2903	. . . . . . . . . . . 12 ⊢ Ⅎ𝑥𝑍
41		smfsupxr.x	. . . . . . . . . . . . . 14 ⊢ Ⅎ𝑥𝐹
42		nfcv 2903	. . . . . . . . . . . . . 14 ⊢ Ⅎ𝑥𝑛
43	41, 42	nffv 6898	. . . . . . . . . . . . 13 ⊢ Ⅎ𝑥(𝐹‘𝑛)
44	43	nfdm 5948	. . . . . . . . . . . 12 ⊢ Ⅎ𝑥dom (𝐹‘𝑛)
45	40, 44	nfiin 5027	. . . . . . . . . . 11 ⊢ Ⅎ𝑥∩ 𝑛 ∈ 𝑍 dom (𝐹‘𝑛)
46	45	ssrab2f 43791	. . . . . . . . . 10 ⊢ {𝑥 ∈ ∩ 𝑛 ∈ 𝑍 dom (𝐹‘𝑛) ∣ sup(ran (𝑛 ∈ 𝑍 ↦ ((𝐹‘𝑛)‘𝑥)), ℝ^*, < ) ∈ ℝ} ⊆ ∩ 𝑛 ∈ 𝑍 dom (𝐹‘𝑛)
47	3, 46	eqsstri 4015	. . . . . . . . 9 ⊢ 𝐷 ⊆ ∩ 𝑛 ∈ 𝑍 dom (𝐹‘𝑛)
48		id 22	. . . . . . . . 9 ⊢ (𝑥 ∈ 𝐷 → 𝑥 ∈ 𝐷)
49	47, 48	sselid 3979	. . . . . . . 8 ⊢ (𝑥 ∈ 𝐷 → 𝑥 ∈ ∩ 𝑛 ∈ 𝑍 dom (𝐹‘𝑛))
50	49, 21	sylan 580	. . . . . . 7 ⊢ ((𝑥 ∈ 𝐷 ∧ 𝑛 ∈ 𝑍) → 𝑥 ∈ dom (𝐹‘𝑛))
51	50	adantll 712	. . . . . 6 ⊢ (((𝜑 ∧ 𝑥 ∈ 𝐷) ∧ 𝑛 ∈ 𝑍) → 𝑥 ∈ dom (𝐹‘𝑛))
52	39, 51	ffvelcdmd 7084	. . . . 5 ⊢ (((𝜑 ∧ 𝑥 ∈ 𝐷) ∧ 𝑛 ∈ 𝑍) → ((𝐹‘𝑛)‘𝑥) ∈ ℝ)
53	48, 3	eleqtrdi 2843	. . . . . . . 8 ⊢ (𝑥 ∈ 𝐷 → 𝑥 ∈ {𝑥 ∈ ∩ 𝑛 ∈ 𝑍 dom (𝐹‘𝑛) ∣ sup(ran (𝑛 ∈ 𝑍 ↦ ((𝐹‘𝑛)‘𝑥)), ℝ^*, < ) ∈ ℝ})
54		rabidim2 43776	. . . . . . . 8 ⊢ (𝑥 ∈ {𝑥 ∈ ∩ 𝑛 ∈ 𝑍 dom (𝐹‘𝑛) ∣ sup(ran (𝑛 ∈ 𝑍 ↦ ((𝐹‘𝑛)‘𝑥)), ℝ^, < ) ∈ ℝ} → sup(ran (𝑛 ∈ 𝑍 ↦ ((𝐹‘𝑛)‘𝑥)), ℝ^, < ) ∈ ℝ)
55	53, 54	syl 17	. . . . . . 7 ⊢ (𝑥 ∈ 𝐷 → sup(ran (𝑛 ∈ 𝑍 ↦ ((𝐹‘𝑛)‘𝑥)), ℝ^*, < ) ∈ ℝ)
56	55	adantl 482	. . . . . 6 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐷) → sup(ran (𝑛 ∈ 𝑍 ↦ ((𝐹‘𝑛)‘𝑥)), ℝ^*, < ) ∈ ℝ)
57	49	adantl 482	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐷) → 𝑥 ∈ ∩ 𝑛 ∈ 𝑍 dom (𝐹‘𝑛))
58	57, 24	syldan 591	. . . . . 6 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐷) → (sup(ran (𝑛 ∈ 𝑍 ↦ ((𝐹‘𝑛)‘𝑥)), ℝ^*, < ) ∈ ℝ ↔ ∃𝑦 ∈ ℝ ∀𝑛 ∈ 𝑍 ((𝐹‘𝑛)‘𝑥) ≤ 𝑦))
59	56, 58	mpbid 231	. . . . 5 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐷) → ∃𝑦 ∈ ℝ ∀𝑛 ∈ 𝑍 ((𝐹‘𝑛)‘𝑥) ≤ 𝑦)
60	37, 38, 52, 59	supxrrernmpt 44117	. . . 4 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐷) → sup(ran (𝑛 ∈ 𝑍 ↦ ((𝐹‘𝑛)‘𝑥)), ℝ^*, < ) = sup(ran (𝑛 ∈ 𝑍 ↦ ((𝐹‘𝑛)‘𝑥)), ℝ, < ))
61	26, 60	mpteq12dva 5236	. . 3 ⊢ (𝜑 → (𝑥 ∈ 𝐷 ↦ sup(ran (𝑛 ∈ 𝑍 ↦ ((𝐹‘𝑛)‘𝑥)), ℝ^*, < )) = (𝑥 ∈ {𝑥 ∈ ∩ 𝑛 ∈ 𝑍 dom (𝐹‘𝑛) ∣ ∃𝑦 ∈ ℝ ∀𝑛 ∈ 𝑍 ((𝐹‘𝑛)‘𝑥) ≤ 𝑦} ↦ sup(ran (𝑛 ∈ 𝑍 ↦ ((𝐹‘𝑛)‘𝑥)), ℝ, < )))
62	2, 61	eqtrd 2772	. 2 ⊢ (𝜑 → 𝐺 = (𝑥 ∈ {𝑥 ∈ ∩ 𝑛 ∈ 𝑍 dom (𝐹‘𝑛) ∣ ∃𝑦 ∈ ℝ ∀𝑛 ∈ 𝑍 ((𝐹‘𝑛)‘𝑥) ≤ 𝑦} ↦ sup(ran (𝑛 ∈ 𝑍 ↦ ((𝐹‘𝑛)‘𝑥)), ℝ, < )))
63		smfsupxr.n	. . 3 ⊢ Ⅎ𝑛𝐹
64		eqid 2732	. . 3 ⊢ {𝑥 ∈ ∩ 𝑛 ∈ 𝑍 dom (𝐹‘𝑛) ∣ ∃𝑦 ∈ ℝ ∀𝑛 ∈ 𝑍 ((𝐹‘𝑛)‘𝑥) ≤ 𝑦} = {𝑥 ∈ ∩ 𝑛 ∈ 𝑍 dom (𝐹‘𝑛) ∣ ∃𝑦 ∈ ℝ ∀𝑛 ∈ 𝑍 ((𝐹‘𝑛)‘𝑥) ≤ 𝑦}
65		eqid 2732	. . 3 ⊢ (𝑥 ∈ {𝑥 ∈ ∩ 𝑛 ∈ 𝑍 dom (𝐹‘𝑛) ∣ ∃𝑦 ∈ ℝ ∀𝑛 ∈ 𝑍 ((𝐹‘𝑛)‘𝑥) ≤ 𝑦} ↦ sup(ran (𝑛 ∈ 𝑍 ↦ ((𝐹‘𝑛)‘𝑥)), ℝ, < )) = (𝑥 ∈ {𝑥 ∈ ∩ 𝑛 ∈ 𝑍 dom (𝐹‘𝑛) ∣ ∃𝑦 ∈ ℝ ∀𝑛 ∈ 𝑍 ((𝐹‘𝑛)‘𝑥) ≤ 𝑦} ↦ sup(ran (𝑛 ∈ 𝑍 ↦ ((𝐹‘𝑛)‘𝑥)), ℝ, < ))
66	63, 41, 10, 11, 14, 16, 64, 65	smfsup 45516	. 2 ⊢ (𝜑 → (𝑥 ∈ {𝑥 ∈ ∩ 𝑛 ∈ 𝑍 dom (𝐹‘𝑛) ∣ ∃𝑦 ∈ ℝ ∀𝑛 ∈ 𝑍 ((𝐹‘𝑛)‘𝑥) ≤ 𝑦} ↦ sup(ran (𝑛 ∈ 𝑍 ↦ ((𝐹‘𝑛)‘𝑥)), ℝ, < )) ∈ (SMblFn‘𝑆))
67	62, 66	eqeltrd 2833	1 ⊢ (𝜑 → 𝐺 ∈ (SMblFn‘𝑆))

Colors of variables: wff setvar class

Syntax hints: → wi 4 ↔ wb 205 ∧ wa 396 = wceq 1541 ∈ wcel 2106 Ⅎwnfc 2883 ≠ wne 2940 ∀wral 3061 ∃wrex 3070 {crab 3432 ∅c0 4321 ∩ ciin 4997 class class class wbr 5147 ↦ cmpt 5230 dom cdm 5675 ran crn 5676 ⟶wf 6536 ‘cfv 6540 supcsup 9431 ℝcr 11105 ℝ^*cxr 11243 < clt 11244 ≤ cle 11245 ℤcz 12554 ℤ_≥cuz 12818 SAlgcsalg 45010 SMblFncsmblfn 45397

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1797 ax-4 1811 ax-5 1913 ax-6 1971 ax-7 2011 ax-8 2108 ax-9 2116 ax-10 2137 ax-11 2154 ax-12 2171 ax-ext 2703 ax-rep 5284 ax-sep 5298 ax-nul 5305 ax-pow 5362 ax-pr 5426 ax-un 7721 ax-inf2 9632 ax-cc 10426 ax-ac2 10454 ax-cnex 11162 ax-resscn 11163 ax-1cn 11164 ax-icn 11165 ax-addcl 11166 ax-addrcl 11167 ax-mulcl 11168 ax-mulrcl 11169 ax-mulcom 11170 ax-addass 11171 ax-mulass 11172 ax-distr 11173 ax-i2m1 11174 ax-1ne0 11175 ax-1rid 11176 ax-rnegex 11177 ax-rrecex 11178 ax-cnre 11179 ax-pre-lttri 11180 ax-pre-lttrn 11181 ax-pre-ltadd 11182 ax-pre-mulgt0 11183 ax-pre-sup 11184

This theorem depends on definitions: df-bi 206 df-an 397 df-or 846 df-3or 1088 df-3an 1089 df-tru 1544 df-fal 1554 df-ex 1782 df-nf 1786 df-sb 2068 df-mo 2534 df-eu 2563 df-clab 2710 df-cleq 2724 df-clel 2810 df-nfc 2885 df-ne 2941 df-nel 3047 df-ral 3062 df-rex 3071 df-rmo 3376 df-reu 3377 df-rab 3433 df-v 3476 df-sbc 3777 df-csb 3893 df-dif 3950 df-un 3952 df-in 3954 df-ss 3964 df-pss 3966 df-nul 4322 df-if 4528 df-pw 4603 df-sn 4628 df-pr 4630 df-op 4634 df-uni 4908 df-int 4950 df-iun 4998 df-iin 4999 df-br 5148 df-opab 5210 df-mpt 5231 df-tr 5265 df-id 5573 df-eprel 5579 df-po 5587 df-so 5588 df-fr 5630 df-se 5631 df-we 5632 df-xp 5681 df-rel 5682 df-cnv 5683 df-co 5684 df-dm 5685 df-rn 5686 df-res 5687 df-ima 5688 df-pred 6297 df-ord 6364 df-on 6365 df-lim 6366 df-suc 6367 df-iota 6492 df-fun 6542 df-fn 6543 df-f 6544 df-f1 6545 df-fo 6546 df-f1o 6547 df-fv 6548 df-isom 6549 df-riota 7361 df-ov 7408 df-oprab 7409 df-mpo 7410 df-om 7852 df-1st 7971 df-2nd 7972 df-frecs 8262 df-wrecs 8293 df-recs 8367 df-rdg 8406 df-1o 8462 df-oadd 8466 df-omul 8467 df-er 8699 df-map 8818 df-pm 8819 df-en 8936 df-dom 8937 df-sdom 8938 df-fin 8939 df-sup 9433 df-inf 9434 df-oi 9501 df-card 9930 df-acn 9933 df-ac 10107 df-pnf 11246 df-mnf 11247 df-xr 11248 df-ltxr 11249 df-le 11250 df-sub 11442 df-neg 11443 df-div 11868 df-nn 12209 df-n0 12469 df-z 12555 df-uz 12819 df-q 12929 df-rp 12971 df-ioo 13324 df-ioc 13325 df-ico 13326 df-fl 13753 df-rest 17364 df-topgen 17385 df-top 22387 df-bases 22440 df-salg 45011 df-salgen 45015 df-smblfn 45398

This theorem is referenced by: smflimsuplem3 45524

W3C validator