Theorem smfinfmpt 47503

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

Hypotheses

Ref	Expression
smfinfmpt.n	⊢ Ⅎ𝑛𝜑
smfinfmpt.x	⊢ Ⅎ𝑥𝜑
smfinfmpt.y	⊢ Ⅎ𝑦𝜑
smfinfmpt.m	⊢ (𝜑 → 𝑀 ∈ ℤ)
smfinfmpt.z	⊢ 𝑍 = (ℤ≥‘𝑀)
smfinfmpt.s	⊢ (𝜑 → 𝑆 ∈ SAlg)
smfinfmpt.b	⊢ ((𝜑 ∧ 𝑛 ∈ 𝑍 ∧ 𝑥 ∈ 𝐴) → 𝐵 ∈ 𝑉)
smfinfmpt.f	⊢ ((𝜑 ∧ 𝑛 ∈ 𝑍) → (𝑥 ∈ 𝐴 ↦ 𝐵) ∈ (SMblFn‘𝑆))
smfinfmpt.d	⊢ 𝐷 = {𝑥 ∈ ∩ 𝑛 ∈ 𝑍 𝐴 ∣ ∃𝑦 ∈ ℝ ∀𝑛 ∈ 𝑍 𝑦 ≤ 𝐵}
smfinfmpt.g	⊢ 𝐺 = (𝑥 ∈ 𝐷 ↦ inf(ran (𝑛 ∈ 𝑍 ↦ 𝐵), ℝ, < ))

Assertion

Ref	Expression
smfinfmpt	⊢ (𝜑 → 𝐺 ∈ (SMblFn‘𝑆))

Distinct variable groups:   𝑥,𝐴,𝑦   𝑦,𝐵   𝑆,𝑛   𝑛,𝑍,𝑥,𝑦

Allowed substitution hints:   𝜑(𝑥,𝑦,𝑛)   𝐴(𝑛)   𝐵(𝑥,𝑛)   𝐷(𝑥,𝑦,𝑛)   𝑆(𝑥,𝑦)   𝐺(𝑥,𝑦,𝑛)   𝑀(𝑥,𝑦,𝑛)   𝑉(𝑥,𝑦,𝑛)

Proof of Theorem smfinfmpt

Step	Hyp	Ref	Expression
1		smfinfmpt.g	. . 3 ⊢ 𝐺 = (𝑥 ∈ 𝐷 ↦ inf(ran (𝑛 ∈ 𝑍 ↦ 𝐵), ℝ, < ))
2		smfinfmpt.x	. . . 4 ⊢ Ⅎ𝑥𝜑
3		smfinfmpt.d	. . . . 5 ⊢ 𝐷 = {𝑥 ∈ ∩ 𝑛 ∈ 𝑍 𝐴 ∣ ∃𝑦 ∈ ℝ ∀𝑛 ∈ 𝑍 𝑦 ≤ 𝐵}
4		smfinfmpt.n	. . . . . . . 8 ⊢ Ⅎ𝑛𝜑
5		eqidd 2762	. . . . . . . . . . 11 ⊢ (𝜑 → (𝑛 ∈ 𝑍 ↦ (𝑥 ∈ 𝐴 ↦ 𝐵)) = (𝑛 ∈ 𝑍 ↦ (𝑥 ∈ 𝐴 ↦ 𝐵)))
6		smfinfmpt.f	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑛 ∈ 𝑍) → (𝑥 ∈ 𝐴 ↦ 𝐵) ∈ (SMblFn‘𝑆))
7	5, 6	fvmpt2d 7003	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑛 ∈ 𝑍) → ((𝑛 ∈ 𝑍 ↦ (𝑥 ∈ 𝐴 ↦ 𝐵))‘𝑛) = (𝑥 ∈ 𝐴 ↦ 𝐵))
8	7	dmeqd 5895	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑛 ∈ 𝑍) → dom ((𝑛 ∈ 𝑍 ↦ (𝑥 ∈ 𝐴 ↦ 𝐵))‘𝑛) = dom (𝑥 ∈ 𝐴 ↦ 𝐵))
9		nfcv 2923	. . . . . . . . . . . 12 ⊢ Ⅎ𝑥𝑍
10	9	nfcri 2915	. . . . . . . . . . 11 ⊢ Ⅎ𝑥 𝑛 ∈ 𝑍
11	2, 10	nfan 1927	. . . . . . . . . 10 ⊢ Ⅎ𝑥(𝜑 ∧ 𝑛 ∈ 𝑍)
12		eqid 2761	. . . . . . . . . 10 ⊢ (𝑥 ∈ 𝐴 ↦ 𝐵) = (𝑥 ∈ 𝐴 ↦ 𝐵)
13		smfinfmpt.b	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑛 ∈ 𝑍 ∧ 𝑥 ∈ 𝐴) → 𝐵 ∈ 𝑉)
14	13	3expa 1134	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝑛 ∈ 𝑍) ∧ 𝑥 ∈ 𝐴) → 𝐵 ∈ 𝑉)
15	11, 12, 14	dmmptdf 45910	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑛 ∈ 𝑍) → dom (𝑥 ∈ 𝐴 ↦ 𝐵) = 𝐴)
16	8, 15	eqtr2d 2797	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑛 ∈ 𝑍) → 𝐴 = dom ((𝑛 ∈ 𝑍 ↦ (𝑥 ∈ 𝐴 ↦ 𝐵))‘𝑛))
17	4, 16	iineq2d 4979	. . . . . . 7 ⊢ (𝜑 → ∩ 𝑛 ∈ 𝑍 𝐴 = ∩ 𝑛 ∈ 𝑍 dom ((𝑛 ∈ 𝑍 ↦ (𝑥 ∈ 𝐴 ↦ 𝐵))‘𝑛))
18	2, 17	rabeqd 3442	. . . . . 6 ⊢ (𝜑 → {𝑥 ∈ ∩ 𝑛 ∈ 𝑍 𝐴 ∣ ∃𝑦 ∈ ℝ ∀𝑛 ∈ 𝑍 𝑦 ≤ 𝐵} = {𝑥 ∈ ∩ 𝑛 ∈ 𝑍 dom ((𝑛 ∈ 𝑍 ↦ (𝑥 ∈ 𝐴 ↦ 𝐵))‘𝑛) ∣ ∃𝑦 ∈ ℝ ∀𝑛 ∈ 𝑍 𝑦 ≤ 𝐵})
19		smfinfmpt.y	. . . . . . . . 9 ⊢ Ⅎ𝑦𝜑
20		nfv 1942	. . . . . . . . 9 ⊢ Ⅎ𝑦 𝑥 ∈ ∩ 𝑛 ∈ 𝑍 dom ((𝑛 ∈ 𝑍 ↦ (𝑥 ∈ 𝐴 ↦ 𝐵))‘𝑛)
21	19, 20	nfan 1927	. . . . . . . 8 ⊢ Ⅎ𝑦(𝜑 ∧ 𝑥 ∈ ∩ 𝑛 ∈ 𝑍 dom ((𝑛 ∈ 𝑍 ↦ (𝑥 ∈ 𝐴 ↦ 𝐵))‘𝑛))
22		nfii1 4992	. . . . . . . . . . 11 ⊢ Ⅎ𝑛∩ 𝑛 ∈ 𝑍 dom ((𝑛 ∈ 𝑍 ↦ (𝑥 ∈ 𝐴 ↦ 𝐵))‘𝑛)
23	22	nfcri 2915	. . . . . . . . . 10 ⊢ Ⅎ𝑛 𝑥 ∈ ∩ 𝑛 ∈ 𝑍 dom ((𝑛 ∈ 𝑍 ↦ (𝑥 ∈ 𝐴 ↦ 𝐵))‘𝑛)
24	4, 23	nfan 1927	. . . . . . . . 9 ⊢ Ⅎ𝑛(𝜑 ∧ 𝑥 ∈ ∩ 𝑛 ∈ 𝑍 dom ((𝑛 ∈ 𝑍 ↦ (𝑥 ∈ 𝐴 ↦ 𝐵))‘𝑛))
25		simpll 778	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝑥 ∈ ∩ 𝑛 ∈ 𝑍 dom ((𝑛 ∈ 𝑍 ↦ (𝑥 ∈ 𝐴 ↦ 𝐵))‘𝑛)) ∧ 𝑛 ∈ 𝑍) → 𝜑)
26		simpr 489	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝑥 ∈ ∩ 𝑛 ∈ 𝑍 dom ((𝑛 ∈ 𝑍 ↦ (𝑥 ∈ 𝐴 ↦ 𝐵))‘𝑛)) ∧ 𝑛 ∈ 𝑍) → 𝑛 ∈ 𝑍)
27		eliinid 45799	. . . . . . . . . . . 12 ⊢ ((𝑥 ∈ ∩ 𝑛 ∈ 𝑍 dom ((𝑛 ∈ 𝑍 ↦ (𝑥 ∈ 𝐴 ↦ 𝐵))‘𝑛) ∧ 𝑛 ∈ 𝑍) → 𝑥 ∈ dom ((𝑛 ∈ 𝑍 ↦ (𝑥 ∈ 𝐴 ↦ 𝐵))‘𝑛))
28	27	adantll 726	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ 𝑥 ∈ ∩ 𝑛 ∈ 𝑍 dom ((𝑛 ∈ 𝑍 ↦ (𝑥 ∈ 𝐴 ↦ 𝐵))‘𝑛)) ∧ 𝑛 ∈ 𝑍) → 𝑥 ∈ dom ((𝑛 ∈ 𝑍 ↦ (𝑥 ∈ 𝐴 ↦ 𝐵))‘𝑛))
29	8, 15	eqtrd 2796	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑛 ∈ 𝑍) → dom ((𝑛 ∈ 𝑍 ↦ (𝑥 ∈ 𝐴 ↦ 𝐵))‘𝑛) = 𝐴)
30	29	adantlr 727	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ 𝑥 ∈ ∩ 𝑛 ∈ 𝑍 dom ((𝑛 ∈ 𝑍 ↦ (𝑥 ∈ 𝐴 ↦ 𝐵))‘𝑛)) ∧ 𝑛 ∈ 𝑍) → dom ((𝑛 ∈ 𝑍 ↦ (𝑥 ∈ 𝐴 ↦ 𝐵))‘𝑛) = 𝐴)
31	28, 30	eleqtrd 2863	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝑥 ∈ ∩ 𝑛 ∈ 𝑍 dom ((𝑛 ∈ 𝑍 ↦ (𝑥 ∈ 𝐴 ↦ 𝐵))‘𝑛)) ∧ 𝑛 ∈ 𝑍) → 𝑥 ∈ 𝐴)
32	7	fveq1d 6883	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑛 ∈ 𝑍) → (((𝑛 ∈ 𝑍 ↦ (𝑥 ∈ 𝐴 ↦ 𝐵))‘𝑛)‘𝑥) = ((𝑥 ∈ 𝐴 ↦ 𝐵)‘𝑥))
33	32	3adant3 1148	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑛 ∈ 𝑍 ∧ 𝑥 ∈ 𝐴) → (((𝑛 ∈ 𝑍 ↦ (𝑥 ∈ 𝐴 ↦ 𝐵))‘𝑛)‘𝑥) = ((𝑥 ∈ 𝐴 ↦ 𝐵)‘𝑥))
34		simp3 1154	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑛 ∈ 𝑍 ∧ 𝑥 ∈ 𝐴) → 𝑥 ∈ 𝐴)
35		fvmpt4 45923	. . . . . . . . . . . . 13 ⊢ ((𝑥 ∈ 𝐴 ∧ 𝐵 ∈ 𝑉) → ((𝑥 ∈ 𝐴 ↦ 𝐵)‘𝑥) = 𝐵)
36	34, 13, 35	syl2anc 595	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑛 ∈ 𝑍 ∧ 𝑥 ∈ 𝐴) → ((𝑥 ∈ 𝐴 ↦ 𝐵)‘𝑥) = 𝐵)
37	33, 36	eqtr2d 2797	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑛 ∈ 𝑍 ∧ 𝑥 ∈ 𝐴) → 𝐵 = (((𝑛 ∈ 𝑍 ↦ (𝑥 ∈ 𝐴 ↦ 𝐵))‘𝑛)‘𝑥))
38	37	breq2d 5120	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑛 ∈ 𝑍 ∧ 𝑥 ∈ 𝐴) → (𝑦 ≤ 𝐵 ↔ 𝑦 ≤ (((𝑛 ∈ 𝑍 ↦ (𝑥 ∈ 𝐴 ↦ 𝐵))‘𝑛)‘𝑥)))
39	25, 26, 31, 38	syl3anc 1396	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑥 ∈ ∩ 𝑛 ∈ 𝑍 dom ((𝑛 ∈ 𝑍 ↦ (𝑥 ∈ 𝐴 ↦ 𝐵))‘𝑛)) ∧ 𝑛 ∈ 𝑍) → (𝑦 ≤ 𝐵 ↔ 𝑦 ≤ (((𝑛 ∈ 𝑍 ↦ (𝑥 ∈ 𝐴 ↦ 𝐵))‘𝑛)‘𝑥)))
40	24, 39	ralbida 3274	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑥 ∈ ∩ 𝑛 ∈ 𝑍 dom ((𝑛 ∈ 𝑍 ↦ (𝑥 ∈ 𝐴 ↦ 𝐵))‘𝑛)) → (∀𝑛 ∈ 𝑍 𝑦 ≤ 𝐵 ↔ ∀𝑛 ∈ 𝑍 𝑦 ≤ (((𝑛 ∈ 𝑍 ↦ (𝑥 ∈ 𝐴 ↦ 𝐵))‘𝑛)‘𝑥)))
41	21, 40	rexbid 3277	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑥 ∈ ∩ 𝑛 ∈ 𝑍 dom ((𝑛 ∈ 𝑍 ↦ (𝑥 ∈ 𝐴 ↦ 𝐵))‘𝑛)) → (∃𝑦 ∈ ℝ ∀𝑛 ∈ 𝑍 𝑦 ≤ 𝐵 ↔ ∃𝑦 ∈ ℝ ∀𝑛 ∈ 𝑍 𝑦 ≤ (((𝑛 ∈ 𝑍 ↦ (𝑥 ∈ 𝐴 ↦ 𝐵))‘𝑛)‘𝑥)))
42	2, 41	rabbida 3440	. . . . . 6 ⊢ (𝜑 → {𝑥 ∈ ∩ 𝑛 ∈ 𝑍 dom ((𝑛 ∈ 𝑍 ↦ (𝑥 ∈ 𝐴 ↦ 𝐵))‘𝑛) ∣ ∃𝑦 ∈ ℝ ∀𝑛 ∈ 𝑍 𝑦 ≤ 𝐵} = {𝑥 ∈ ∩ 𝑛 ∈ 𝑍 dom ((𝑛 ∈ 𝑍 ↦ (𝑥 ∈ 𝐴 ↦ 𝐵))‘𝑛) ∣ ∃𝑦 ∈ ℝ ∀𝑛 ∈ 𝑍 𝑦 ≤ (((𝑛 ∈ 𝑍 ↦ (𝑥 ∈ 𝐴 ↦ 𝐵))‘𝑛)‘𝑥)})
43	18, 42	eqtrd 2796	. . . . 5 ⊢ (𝜑 → {𝑥 ∈ ∩ 𝑛 ∈ 𝑍 𝐴 ∣ ∃𝑦 ∈ ℝ ∀𝑛 ∈ 𝑍 𝑦 ≤ 𝐵} = {𝑥 ∈ ∩ 𝑛 ∈ 𝑍 dom ((𝑛 ∈ 𝑍 ↦ (𝑥 ∈ 𝐴 ↦ 𝐵))‘𝑛) ∣ ∃𝑦 ∈ ℝ ∀𝑛 ∈ 𝑍 𝑦 ≤ (((𝑛 ∈ 𝑍 ↦ (𝑥 ∈ 𝐴 ↦ 𝐵))‘𝑛)‘𝑥)})
44	3, 43	eqtrid 2808	. . . 4 ⊢ (𝜑 → 𝐷 = {𝑥 ∈ ∩ 𝑛 ∈ 𝑍 dom ((𝑛 ∈ 𝑍 ↦ (𝑥 ∈ 𝐴 ↦ 𝐵))‘𝑛) ∣ ∃𝑦 ∈ ℝ ∀𝑛 ∈ 𝑍 𝑦 ≤ (((𝑛 ∈ 𝑍 ↦ (𝑥 ∈ 𝐴 ↦ 𝐵))‘𝑛)‘𝑥)})
45		nfcv 2923	. . . . . . . . . . . 12 ⊢ Ⅎ𝑛ℝ
46		nfra1 3287	. . . . . . . . . . . 12 ⊢ Ⅎ𝑛∀𝑛 ∈ 𝑍 𝑦 ≤ 𝐵
47	45, 46	nfrexw 3311	. . . . . . . . . . 11 ⊢ Ⅎ𝑛∃𝑦 ∈ ℝ ∀𝑛 ∈ 𝑍 𝑦 ≤ 𝐵
48		nfii1 4992	. . . . . . . . . . 11 ⊢ Ⅎ𝑛∩ 𝑛 ∈ 𝑍 𝐴
49	47, 48	nfrabw 3450	. . . . . . . . . 10 ⊢ Ⅎ𝑛{𝑥 ∈ ∩ 𝑛 ∈ 𝑍 𝐴 ∣ ∃𝑦 ∈ ℝ ∀𝑛 ∈ 𝑍 𝑦 ≤ 𝐵}
50	3, 49	nfcxfr 2921	. . . . . . . . 9 ⊢ Ⅎ𝑛𝐷
51	50	nfcri 2915	. . . . . . . 8 ⊢ Ⅎ𝑛 𝑥 ∈ 𝐷
52	4, 51	nfan 1927	. . . . . . 7 ⊢ Ⅎ𝑛(𝜑 ∧ 𝑥 ∈ 𝐷)
53		simpll 778	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝑥 ∈ 𝐷) ∧ 𝑛 ∈ 𝑍) → 𝜑)
54		simpr 489	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝑥 ∈ 𝐷) ∧ 𝑛 ∈ 𝑍) → 𝑛 ∈ 𝑍)
55		rabidim1 3436	. . . . . . . . . . 11 ⊢ (𝑥 ∈ {𝑥 ∈ ∩ 𝑛 ∈ 𝑍 𝐴 ∣ ∃𝑦 ∈ ℝ ∀𝑛 ∈ 𝑍 𝑦 ≤ 𝐵} → 𝑥 ∈ ∩ 𝑛 ∈ 𝑍 𝐴)
56	55, 3	eleq2s 2879	. . . . . . . . . 10 ⊢ (𝑥 ∈ 𝐷 → 𝑥 ∈ ∩ 𝑛 ∈ 𝑍 𝐴)
57		eliinid 45799	. . . . . . . . . 10 ⊢ ((𝑥 ∈ ∩ 𝑛 ∈ 𝑍 𝐴 ∧ 𝑛 ∈ 𝑍) → 𝑥 ∈ 𝐴)
58	56, 57	sylan 591	. . . . . . . . 9 ⊢ ((𝑥 ∈ 𝐷 ∧ 𝑛 ∈ 𝑍) → 𝑥 ∈ 𝐴)
59	58	adantll 726	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝑥 ∈ 𝐷) ∧ 𝑛 ∈ 𝑍) → 𝑥 ∈ 𝐴)
60	53, 54, 59, 37	syl3anc 1396	. . . . . . 7 ⊢ (((𝜑 ∧ 𝑥 ∈ 𝐷) ∧ 𝑛 ∈ 𝑍) → 𝐵 = (((𝑛 ∈ 𝑍 ↦ (𝑥 ∈ 𝐴 ↦ 𝐵))‘𝑛)‘𝑥))
61	52, 60	mpteq2da 5202	. . . . . 6 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐷) → (𝑛 ∈ 𝑍 ↦ 𝐵) = (𝑛 ∈ 𝑍 ↦ (((𝑛 ∈ 𝑍 ↦ (𝑥 ∈ 𝐴 ↦ 𝐵))‘𝑛)‘𝑥)))
62	61	rneqd 5928	. . . . 5 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐷) → ran (𝑛 ∈ 𝑍 ↦ 𝐵) = ran (𝑛 ∈ 𝑍 ↦ (((𝑛 ∈ 𝑍 ↦ (𝑥 ∈ 𝐴 ↦ 𝐵))‘𝑛)‘𝑥)))
63	62	infeq1d 9437	. . . 4 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐷) → inf(ran (𝑛 ∈ 𝑍 ↦ 𝐵), ℝ, < ) = inf(ran (𝑛 ∈ 𝑍 ↦ (((𝑛 ∈ 𝑍 ↦ (𝑥 ∈ 𝐴 ↦ 𝐵))‘𝑛)‘𝑥)), ℝ, < ))
64	2, 44, 63	mpteq12da 5193	. . 3 ⊢ (𝜑 → (𝑥 ∈ 𝐷 ↦ inf(ran (𝑛 ∈ 𝑍 ↦ 𝐵), ℝ, < )) = (𝑥 ∈ {𝑥 ∈ ∩ 𝑛 ∈ 𝑍 dom ((𝑛 ∈ 𝑍 ↦ (𝑥 ∈ 𝐴 ↦ 𝐵))‘𝑛) ∣ ∃𝑦 ∈ ℝ ∀𝑛 ∈ 𝑍 𝑦 ≤ (((𝑛 ∈ 𝑍 ↦ (𝑥 ∈ 𝐴 ↦ 𝐵))‘𝑛)‘𝑥)} ↦ inf(ran (𝑛 ∈ 𝑍 ↦ (((𝑛 ∈ 𝑍 ↦ (𝑥 ∈ 𝐴 ↦ 𝐵))‘𝑛)‘𝑥)), ℝ, < )))
65	1, 64	eqtrid 2808	. 2 ⊢ (𝜑 → 𝐺 = (𝑥 ∈ {𝑥 ∈ ∩ 𝑛 ∈ 𝑍 dom ((𝑛 ∈ 𝑍 ↦ (𝑥 ∈ 𝐴 ↦ 𝐵))‘𝑛) ∣ ∃𝑦 ∈ ℝ ∀𝑛 ∈ 𝑍 𝑦 ≤ (((𝑛 ∈ 𝑍 ↦ (𝑥 ∈ 𝐴 ↦ 𝐵))‘𝑛)‘𝑥)} ↦ inf(ran (𝑛 ∈ 𝑍 ↦ (((𝑛 ∈ 𝑍 ↦ (𝑥 ∈ 𝐴 ↦ 𝐵))‘𝑛)‘𝑥)), ℝ, < )))
66		nfmpt1 5209	. . 3 ⊢ Ⅎ𝑛(𝑛 ∈ 𝑍 ↦ (𝑥 ∈ 𝐴 ↦ 𝐵))
67		nfmpt1 5209	. . . 4 ⊢ Ⅎ𝑥(𝑥 ∈ 𝐴 ↦ 𝐵)
68	9, 67	nfmpt 5208	. . 3 ⊢ Ⅎ𝑥(𝑛 ∈ 𝑍 ↦ (𝑥 ∈ 𝐴 ↦ 𝐵))
69		smfinfmpt.m	. . 3 ⊢ (𝜑 → 𝑀 ∈ ℤ)
70		smfinfmpt.z	. . 3 ⊢ 𝑍 = (ℤ≥‘𝑀)
71		smfinfmpt.s	. . 3 ⊢ (𝜑 → 𝑆 ∈ SAlg)
72	4, 6	fmptd2f 45920	. . 3 ⊢ (𝜑 → (𝑛 ∈ 𝑍 ↦ (𝑥 ∈ 𝐴 ↦ 𝐵)):𝑍⟶(SMblFn‘𝑆))
73		eqid 2761	. . 3 ⊢ {𝑥 ∈ ∩ 𝑛 ∈ 𝑍 dom ((𝑛 ∈ 𝑍 ↦ (𝑥 ∈ 𝐴 ↦ 𝐵))‘𝑛) ∣ ∃𝑦 ∈ ℝ ∀𝑛 ∈ 𝑍 𝑦 ≤ (((𝑛 ∈ 𝑍 ↦ (𝑥 ∈ 𝐴 ↦ 𝐵))‘𝑛)‘𝑥)} = {𝑥 ∈ ∩ 𝑛 ∈ 𝑍 dom ((𝑛 ∈ 𝑍 ↦ (𝑥 ∈ 𝐴 ↦ 𝐵))‘𝑛) ∣ ∃𝑦 ∈ ℝ ∀𝑛 ∈ 𝑍 𝑦 ≤ (((𝑛 ∈ 𝑍 ↦ (𝑥 ∈ 𝐴 ↦ 𝐵))‘𝑛)‘𝑥)}
74		eqid 2761	. . 3 ⊢ (𝑥 ∈ {𝑥 ∈ ∩ 𝑛 ∈ 𝑍 dom ((𝑛 ∈ 𝑍 ↦ (𝑥 ∈ 𝐴 ↦ 𝐵))‘𝑛) ∣ ∃𝑦 ∈ ℝ ∀𝑛 ∈ 𝑍 𝑦 ≤ (((𝑛 ∈ 𝑍 ↦ (𝑥 ∈ 𝐴 ↦ 𝐵))‘𝑛)‘𝑥)} ↦ inf(ran (𝑛 ∈ 𝑍 ↦ (((𝑛 ∈ 𝑍 ↦ (𝑥 ∈ 𝐴 ↦ 𝐵))‘𝑛)‘𝑥)), ℝ, < )) = (𝑥 ∈ {𝑥 ∈ ∩ 𝑛 ∈ 𝑍 dom ((𝑛 ∈ 𝑍 ↦ (𝑥 ∈ 𝐴 ↦ 𝐵))‘𝑛) ∣ ∃𝑦 ∈ ℝ ∀𝑛 ∈ 𝑍 𝑦 ≤ (((𝑛 ∈ 𝑍 ↦ (𝑥 ∈ 𝐴 ↦ 𝐵))‘𝑛)‘𝑥)} ↦ inf(ran (𝑛 ∈ 𝑍 ↦ (((𝑛 ∈ 𝑍 ↦ (𝑥 ∈ 𝐴 ↦ 𝐵))‘𝑛)‘𝑥)), ℝ, < ))
75	66, 68, 69, 70, 71, 72, 73, 74	smfinf 47502	. 2 ⊢ (𝜑 → (𝑥 ∈ {𝑥 ∈ ∩ 𝑛 ∈ 𝑍 dom ((𝑛 ∈ 𝑍 ↦ (𝑥 ∈ 𝐴 ↦ 𝐵))‘𝑛) ∣ ∃𝑦 ∈ ℝ ∀𝑛 ∈ 𝑍 𝑦 ≤ (((𝑛 ∈ 𝑍 ↦ (𝑥 ∈ 𝐴 ↦ 𝐵))‘𝑛)‘𝑥)} ↦ inf(ran (𝑛 ∈ 𝑍 ↦ (((𝑛 ∈ 𝑍 ↦ (𝑥 ∈ 𝐴 ↦ 𝐵))‘𝑛)‘𝑥)), ℝ, < )) ∈ (SMblFn‘𝑆))
76	65, 75	eqeltrd 2861	1 ⊢ (𝜑 → 𝐺 ∈ (SMblFn‘𝑆))

Syntax hints:   → wi 4   ↔ wb 209   ∧ wa 400   ∧ w3a 1101   = wceq 1568  Ⅎwnf 1811   ∈ wcel 2141  ∀wral 3077  ∃wrex 3087  {crab 3414  ∩ ciin 4956   class class class wbr 5108   ↦ cmpt 5191  dom cdm 5661  ran crn 5662  ‘cfv 6536  infcinf 9400  ℝcr 11098   < clt 11242   ≤ cle 11243  ℤcz 12590  ℤ_≥cuz 12861  SAlgcsalg 46992  SMblFncsmblfn 47379

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1823  ax-4 1837  ax-5 1938  ax-6 1995  ax-7 2036  ax-8 2143  ax-9 2151  ax-10 2174  ax-11 2190  ax-12 2211  ax-ext 2733  ax-rep 5237  ax-sep 5256  ax-nul 5268  ax-pow 5336  ax-pr 5404  ax-un 7732  ax-inf2 9609  ax-cc 10418  ax-ac2 10446  ax-cnex 11155  ax-resscn 11156  ax-1cn 11157  ax-icn 11158  ax-addcl 11159  ax-addrcl 11160  ax-mulcl 11161  ax-mulrcl 11162  ax-mulcom 11163  ax-addass 11164  ax-mulass 11165  ax-distr 11166  ax-i2m1 11167  ax-1ne0 11168  ax-1rid 11169  ax-rnegex 11170  ax-rrecex 11171  ax-cnre 11172  ax-pre-lttri 11173  ax-pre-lttrn 11174  ax-pre-ltadd 11175  ax-pre-mulgt0 11176  ax-pre-sup 11177

This theorem depends on definitions:  df-bi 210  df-an 401  df-or 861  df-3or 1102  df-3an 1103  df-tru 1571  df-fal 1581  df-ex 1808  df-nf 1812  df-sb 2095  df-mo 2565  df-eu 2595  df-clab 2740  df-cleq 2753  df-clel 2836  df-nfc 2910  df-ne 2957  df-nel 3063  df-ral 3078  df-rex 3088  df-rmo 3367  df-reu 3368  df-rab 3415  df-v 3455  df-sbc 3744  df-csb 3853  df-dif 3907  df-un 3909  df-in 3911  df-ss 3921  df-pss 3924  df-nul 4286  df-if 4487  df-pw 4563  df-sn 4589  df-pr 4591  df-op 4595  df-uni 4872  df-int 4912  df-iun 4957  df-iin 4958  df-br 5109  df-opab 5173  df-mpt 5192  df-tr 5218  df-id 5556  df-eprel 5561  df-po 5569  df-so 5570  df-fr 5614  df-se 5615  df-we 5616  df-xp 5667  df-rel 5668  df-cnv 5669  df-co 5670  df-dm 5671  df-rn 5672  df-res 5673  df-ima 5674  df-pred 6302  df-ord 6363  df-on 6364  df-lim 6365  df-suc 6366  df-iota 6492  df-fun 6538  df-fn 6539  df-f 6540  df-f1 6541  df-fo 6542  df-f1o 6543  df-fv 6544  df-isom 6545  df-riota 7367  df-ov 7413  df-oprab 7414  df-mpo 7415  df-om 7862  df-1st 7985  df-2nd 7986  df-frecs 8277  df-wrecs 8308  df-recs 8357  df-rdg 8396  df-1o 8452  df-2o 8453  df-oadd 8456  df-omul 8457  df-er 8693  df-map 8825  df-pm 8826  df-en 8943  df-dom 8944  df-sdom 8945  df-fin 8946  df-sup 9401  df-inf 9402  df-oi 9471  df-card 9924  df-acn 9927  df-ac 10099  df-pnf 11244  df-mnf 11245  df-xr 11246  df-ltxr 11247  df-le 11248  df-sub 11442  df-neg 11443  df-div 11871  df-nn 12233  df-2 12302  df-3 12303  df-4 12304  df-n0 12504  df-z 12591  df-uz 12862  df-q 12972  df-rp 13016  df-ioo 13375  df-ioc 13376  df-ico 13377  df-icc 13378  df-fz 13535  df-fzo 13683  df-fl 13825  df-seq 14038  df-exp 14098  df-hash 14367  df-word 14551  df-concat 14608  df-s1 14634  df-s2 14885  df-s3 14886  df-s4 14887  df-cj 15150  df-re 15151  df-im 15152  df-sqrt 15286  df-abs 15287  df-rest 17474  df-topgen 17495  df-top 23030  df-bases 23082  df-salg 46993  df-salgen 46997  df-smblfn 47380

This theorem is referenced by: (None)