Theorem smfsupmpt 44348

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

Assertion

Ref	Expression
smfsupmpt	⊢ (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 smfsupmpt

Step	Hyp	Ref	Expression
1		smfsupmpt.g	. . 3 ⊢ 𝐺 = (𝑥 ∈ 𝐷 ↦ sup(ran (𝑛 ∈ 𝑍 ↦ 𝐵), ℝ, < ))
2		smfsupmpt.x	. . . 4 ⊢ Ⅎ𝑥𝜑
3		smfsupmpt.d	. . . . 5 ⊢ 𝐷 = {𝑥 ∈ ∩ 𝑛 ∈ 𝑍 𝐴 ∣ ∃𝑦 ∈ ℝ ∀𝑛 ∈ 𝑍 𝐵 ≤ 𝑦}
4		smfsupmpt.n	. . . . . . . 8 ⊢ Ⅎ𝑛𝜑
5		eqidd 2739	. . . . . . . . . . 11 ⊢ (𝜑 → (𝑛 ∈ 𝑍 ↦ (𝑥 ∈ 𝐴 ↦ 𝐵)) = (𝑛 ∈ 𝑍 ↦ (𝑥 ∈ 𝐴 ↦ 𝐵)))
6		smfsupmpt.f	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑛 ∈ 𝑍) → (𝑥 ∈ 𝐴 ↦ 𝐵) ∈ (SMblFn‘𝑆))
7	5, 6	fvmpt2d 6888	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑛 ∈ 𝑍) → ((𝑛 ∈ 𝑍 ↦ (𝑥 ∈ 𝐴 ↦ 𝐵))‘𝑛) = (𝑥 ∈ 𝐴 ↦ 𝐵))
8	7	dmeqd 5814	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑛 ∈ 𝑍) → dom ((𝑛 ∈ 𝑍 ↦ (𝑥 ∈ 𝐴 ↦ 𝐵))‘𝑛) = dom (𝑥 ∈ 𝐴 ↦ 𝐵))
9		nfcv 2907	. . . . . . . . . . . 12 ⊢ Ⅎ𝑥𝑍
10	9	nfcri 2894	. . . . . . . . . . 11 ⊢ Ⅎ𝑥 𝑛 ∈ 𝑍
11	2, 10	nfan 1902	. . . . . . . . . 10 ⊢ Ⅎ𝑥(𝜑 ∧ 𝑛 ∈ 𝑍)
12		eqid 2738	. . . . . . . . . 10 ⊢ (𝑥 ∈ 𝐴 ↦ 𝐵) = (𝑥 ∈ 𝐴 ↦ 𝐵)
13		smfsupmpt.b	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑛 ∈ 𝑍 ∧ 𝑥 ∈ 𝐴) → 𝐵 ∈ 𝑉)
14	13	3expa 1117	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝑛 ∈ 𝑍) ∧ 𝑥 ∈ 𝐴) → 𝐵 ∈ 𝑉)
15	11, 12, 14	dmmptdf 42763	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑛 ∈ 𝑍) → dom (𝑥 ∈ 𝐴 ↦ 𝐵) = 𝐴)
16	8, 15	eqtr2d 2779	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑛 ∈ 𝑍) → 𝐴 = dom ((𝑛 ∈ 𝑍 ↦ (𝑥 ∈ 𝐴 ↦ 𝐵))‘𝑛))
17	4, 16	iineq2d 4947	. . . . . . 7 ⊢ (𝜑 → ∩ 𝑛 ∈ 𝑍 𝐴 = ∩ 𝑛 ∈ 𝑍 dom ((𝑛 ∈ 𝑍 ↦ (𝑥 ∈ 𝐴 ↦ 𝐵))‘𝑛))
18		nfcv 2907	. . . . . . . 8 ⊢ Ⅎ𝑥∩ 𝑛 ∈ 𝑍 𝐴
19		nfmpt1 5182	. . . . . . . . . . . 12 ⊢ Ⅎ𝑥(𝑥 ∈ 𝐴 ↦ 𝐵)
20	9, 19	nfmpt 5181	. . . . . . . . . . 11 ⊢ Ⅎ𝑥(𝑛 ∈ 𝑍 ↦ (𝑥 ∈ 𝐴 ↦ 𝐵))
21		nfcv 2907	. . . . . . . . . . 11 ⊢ Ⅎ𝑥𝑛
22	20, 21	nffv 6784	. . . . . . . . . 10 ⊢ Ⅎ𝑥((𝑛 ∈ 𝑍 ↦ (𝑥 ∈ 𝐴 ↦ 𝐵))‘𝑛)
23	22	nfdm 5860	. . . . . . . . 9 ⊢ Ⅎ𝑥dom ((𝑛 ∈ 𝑍 ↦ (𝑥 ∈ 𝐴 ↦ 𝐵))‘𝑛)
24	9, 23	nfiin 4955	. . . . . . . 8 ⊢ Ⅎ𝑥∩ 𝑛 ∈ 𝑍 dom ((𝑛 ∈ 𝑍 ↦ (𝑥 ∈ 𝐴 ↦ 𝐵))‘𝑛)
25	18, 24	rabeqf 3415	. . . . . . 7 ⊢ (∩ 𝑛 ∈ 𝑍 𝐴 = ∩ 𝑛 ∈ 𝑍 dom ((𝑛 ∈ 𝑍 ↦ (𝑥 ∈ 𝐴 ↦ 𝐵))‘𝑛) → {𝑥 ∈ ∩ 𝑛 ∈ 𝑍 𝐴 ∣ ∃𝑦 ∈ ℝ ∀𝑛 ∈ 𝑍 𝐵 ≤ 𝑦} = {𝑥 ∈ ∩ 𝑛 ∈ 𝑍 dom ((𝑛 ∈ 𝑍 ↦ (𝑥 ∈ 𝐴 ↦ 𝐵))‘𝑛) ∣ ∃𝑦 ∈ ℝ ∀𝑛 ∈ 𝑍 𝐵 ≤ 𝑦})
26	17, 25	syl 17	. . . . . 6 ⊢ (𝜑 → {𝑥 ∈ ∩ 𝑛 ∈ 𝑍 𝐴 ∣ ∃𝑦 ∈ ℝ ∀𝑛 ∈ 𝑍 𝐵 ≤ 𝑦} = {𝑥 ∈ ∩ 𝑛 ∈ 𝑍 dom ((𝑛 ∈ 𝑍 ↦ (𝑥 ∈ 𝐴 ↦ 𝐵))‘𝑛) ∣ ∃𝑦 ∈ ℝ ∀𝑛 ∈ 𝑍 𝐵 ≤ 𝑦})
27		smfsupmpt.y	. . . . . . . . 9 ⊢ Ⅎ𝑦𝜑
28		nfv 1917	. . . . . . . . 9 ⊢ Ⅎ𝑦 𝑥 ∈ ∩ 𝑛 ∈ 𝑍 dom ((𝑛 ∈ 𝑍 ↦ (𝑥 ∈ 𝐴 ↦ 𝐵))‘𝑛)
29	27, 28	nfan 1902	. . . . . . . 8 ⊢ Ⅎ𝑦(𝜑 ∧ 𝑥 ∈ ∩ 𝑛 ∈ 𝑍 dom ((𝑛 ∈ 𝑍 ↦ (𝑥 ∈ 𝐴 ↦ 𝐵))‘𝑛))
30		nfii1 4959	. . . . . . . . . . 11 ⊢ Ⅎ𝑛∩ 𝑛 ∈ 𝑍 dom ((𝑛 ∈ 𝑍 ↦ (𝑥 ∈ 𝐴 ↦ 𝐵))‘𝑛)
31	30	nfcri 2894	. . . . . . . . . 10 ⊢ Ⅎ𝑛 𝑥 ∈ ∩ 𝑛 ∈ 𝑍 dom ((𝑛 ∈ 𝑍 ↦ (𝑥 ∈ 𝐴 ↦ 𝐵))‘𝑛)
32	4, 31	nfan 1902	. . . . . . . . 9 ⊢ Ⅎ𝑛(𝜑 ∧ 𝑥 ∈ ∩ 𝑛 ∈ 𝑍 dom ((𝑛 ∈ 𝑍 ↦ (𝑥 ∈ 𝐴 ↦ 𝐵))‘𝑛))
33		simpll 764	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝑥 ∈ ∩ 𝑛 ∈ 𝑍 dom ((𝑛 ∈ 𝑍 ↦ (𝑥 ∈ 𝐴 ↦ 𝐵))‘𝑛)) ∧ 𝑛 ∈ 𝑍) → 𝜑)
34		simpr 485	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝑥 ∈ ∩ 𝑛 ∈ 𝑍 dom ((𝑛 ∈ 𝑍 ↦ (𝑥 ∈ 𝐴 ↦ 𝐵))‘𝑛)) ∧ 𝑛 ∈ 𝑍) → 𝑛 ∈ 𝑍)
35		eliinid 42661	. . . . . . . . . . . 12 ⊢ ((𝑥 ∈ ∩ 𝑛 ∈ 𝑍 dom ((𝑛 ∈ 𝑍 ↦ (𝑥 ∈ 𝐴 ↦ 𝐵))‘𝑛) ∧ 𝑛 ∈ 𝑍) → 𝑥 ∈ dom ((𝑛 ∈ 𝑍 ↦ (𝑥 ∈ 𝐴 ↦ 𝐵))‘𝑛))
36	35	adantll 711	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ 𝑥 ∈ ∩ 𝑛 ∈ 𝑍 dom ((𝑛 ∈ 𝑍 ↦ (𝑥 ∈ 𝐴 ↦ 𝐵))‘𝑛)) ∧ 𝑛 ∈ 𝑍) → 𝑥 ∈ dom ((𝑛 ∈ 𝑍 ↦ (𝑥 ∈ 𝐴 ↦ 𝐵))‘𝑛))
37	8, 15	eqtrd 2778	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑛 ∈ 𝑍) → dom ((𝑛 ∈ 𝑍 ↦ (𝑥 ∈ 𝐴 ↦ 𝐵))‘𝑛) = 𝐴)
38	37	adantlr 712	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ 𝑥 ∈ ∩ 𝑛 ∈ 𝑍 dom ((𝑛 ∈ 𝑍 ↦ (𝑥 ∈ 𝐴 ↦ 𝐵))‘𝑛)) ∧ 𝑛 ∈ 𝑍) → dom ((𝑛 ∈ 𝑍 ↦ (𝑥 ∈ 𝐴 ↦ 𝐵))‘𝑛) = 𝐴)
39	36, 38	eleqtrd 2841	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝑥 ∈ ∩ 𝑛 ∈ 𝑍 dom ((𝑛 ∈ 𝑍 ↦ (𝑥 ∈ 𝐴 ↦ 𝐵))‘𝑛)) ∧ 𝑛 ∈ 𝑍) → 𝑥 ∈ 𝐴)
40	7	fveq1d 6776	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑛 ∈ 𝑍) → (((𝑛 ∈ 𝑍 ↦ (𝑥 ∈ 𝐴 ↦ 𝐵))‘𝑛)‘𝑥) = ((𝑥 ∈ 𝐴 ↦ 𝐵)‘𝑥))
41	40	3adant3 1131	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑛 ∈ 𝑍 ∧ 𝑥 ∈ 𝐴) → (((𝑛 ∈ 𝑍 ↦ (𝑥 ∈ 𝐴 ↦ 𝐵))‘𝑛)‘𝑥) = ((𝑥 ∈ 𝐴 ↦ 𝐵)‘𝑥))
42		simp3 1137	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑛 ∈ 𝑍 ∧ 𝑥 ∈ 𝐴) → 𝑥 ∈ 𝐴)
43		fvmpt4 42782	. . . . . . . . . . . . 13 ⊢ ((𝑥 ∈ 𝐴 ∧ 𝐵 ∈ 𝑉) → ((𝑥 ∈ 𝐴 ↦ 𝐵)‘𝑥) = 𝐵)
44	42, 13, 43	syl2anc 584	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑛 ∈ 𝑍 ∧ 𝑥 ∈ 𝐴) → ((𝑥 ∈ 𝐴 ↦ 𝐵)‘𝑥) = 𝐵)
45	41, 44	eqtr2d 2779	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑛 ∈ 𝑍 ∧ 𝑥 ∈ 𝐴) → 𝐵 = (((𝑛 ∈ 𝑍 ↦ (𝑥 ∈ 𝐴 ↦ 𝐵))‘𝑛)‘𝑥))
46	45	breq1d 5084	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑛 ∈ 𝑍 ∧ 𝑥 ∈ 𝐴) → (𝐵 ≤ 𝑦 ↔ (((𝑛 ∈ 𝑍 ↦ (𝑥 ∈ 𝐴 ↦ 𝐵))‘𝑛)‘𝑥) ≤ 𝑦))
47	33, 34, 39, 46	syl3anc 1370	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑥 ∈ ∩ 𝑛 ∈ 𝑍 dom ((𝑛 ∈ 𝑍 ↦ (𝑥 ∈ 𝐴 ↦ 𝐵))‘𝑛)) ∧ 𝑛 ∈ 𝑍) → (𝐵 ≤ 𝑦 ↔ (((𝑛 ∈ 𝑍 ↦ (𝑥 ∈ 𝐴 ↦ 𝐵))‘𝑛)‘𝑥) ≤ 𝑦))
48	32, 47	ralbida 3159	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑥 ∈ ∩ 𝑛 ∈ 𝑍 dom ((𝑛 ∈ 𝑍 ↦ (𝑥 ∈ 𝐴 ↦ 𝐵))‘𝑛)) → (∀𝑛 ∈ 𝑍 𝐵 ≤ 𝑦 ↔ ∀𝑛 ∈ 𝑍 (((𝑛 ∈ 𝑍 ↦ (𝑥 ∈ 𝐴 ↦ 𝐵))‘𝑛)‘𝑥) ≤ 𝑦))
49	29, 48	rexbid 3253	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑥 ∈ ∩ 𝑛 ∈ 𝑍 dom ((𝑛 ∈ 𝑍 ↦ (𝑥 ∈ 𝐴 ↦ 𝐵))‘𝑛)) → (∃𝑦 ∈ ℝ ∀𝑛 ∈ 𝑍 𝐵 ≤ 𝑦 ↔ ∃𝑦 ∈ ℝ ∀𝑛 ∈ 𝑍 (((𝑛 ∈ 𝑍 ↦ (𝑥 ∈ 𝐴 ↦ 𝐵))‘𝑛)‘𝑥) ≤ 𝑦))
50	2, 49	rabbida 3409	. . . . . 6 ⊢ (𝜑 → {𝑥 ∈ ∩ 𝑛 ∈ 𝑍 dom ((𝑛 ∈ 𝑍 ↦ (𝑥 ∈ 𝐴 ↦ 𝐵))‘𝑛) ∣ ∃𝑦 ∈ ℝ ∀𝑛 ∈ 𝑍 𝐵 ≤ 𝑦} = {𝑥 ∈ ∩ 𝑛 ∈ 𝑍 dom ((𝑛 ∈ 𝑍 ↦ (𝑥 ∈ 𝐴 ↦ 𝐵))‘𝑛) ∣ ∃𝑦 ∈ ℝ ∀𝑛 ∈ 𝑍 (((𝑛 ∈ 𝑍 ↦ (𝑥 ∈ 𝐴 ↦ 𝐵))‘𝑛)‘𝑥) ≤ 𝑦})
51	26, 50	eqtrd 2778	. . . . 5 ⊢ (𝜑 → {𝑥 ∈ ∩ 𝑛 ∈ 𝑍 𝐴 ∣ ∃𝑦 ∈ ℝ ∀𝑛 ∈ 𝑍 𝐵 ≤ 𝑦} = {𝑥 ∈ ∩ 𝑛 ∈ 𝑍 dom ((𝑛 ∈ 𝑍 ↦ (𝑥 ∈ 𝐴 ↦ 𝐵))‘𝑛) ∣ ∃𝑦 ∈ ℝ ∀𝑛 ∈ 𝑍 (((𝑛 ∈ 𝑍 ↦ (𝑥 ∈ 𝐴 ↦ 𝐵))‘𝑛)‘𝑥) ≤ 𝑦})
52	3, 51	eqtrid 2790	. . . 4 ⊢ (𝜑 → 𝐷 = {𝑥 ∈ ∩ 𝑛 ∈ 𝑍 dom ((𝑛 ∈ 𝑍 ↦ (𝑥 ∈ 𝐴 ↦ 𝐵))‘𝑛) ∣ ∃𝑦 ∈ ℝ ∀𝑛 ∈ 𝑍 (((𝑛 ∈ 𝑍 ↦ (𝑥 ∈ 𝐴 ↦ 𝐵))‘𝑛)‘𝑥) ≤ 𝑦})
53		nfcv 2907	. . . . . . . . . . . 12 ⊢ Ⅎ𝑛ℝ
54		nfra1 3144	. . . . . . . . . . . 12 ⊢ Ⅎ𝑛∀𝑛 ∈ 𝑍 𝐵 ≤ 𝑦
55	53, 54	nfrex 3242	. . . . . . . . . . 11 ⊢ Ⅎ𝑛∃𝑦 ∈ ℝ ∀𝑛 ∈ 𝑍 𝐵 ≤ 𝑦
56		nfii1 4959	. . . . . . . . . . 11 ⊢ Ⅎ𝑛∩ 𝑛 ∈ 𝑍 𝐴
57	55, 56	nfrabw 3318	. . . . . . . . . 10 ⊢ Ⅎ𝑛{𝑥 ∈ ∩ 𝑛 ∈ 𝑍 𝐴 ∣ ∃𝑦 ∈ ℝ ∀𝑛 ∈ 𝑍 𝐵 ≤ 𝑦}
58	3, 57	nfcxfr 2905	. . . . . . . . 9 ⊢ Ⅎ𝑛𝐷
59	58	nfcri 2894	. . . . . . . 8 ⊢ Ⅎ𝑛 𝑥 ∈ 𝐷
60	4, 59	nfan 1902	. . . . . . 7 ⊢ Ⅎ𝑛(𝜑 ∧ 𝑥 ∈ 𝐷)
61		simpll 764	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝑥 ∈ 𝐷) ∧ 𝑛 ∈ 𝑍) → 𝜑)
62		simpr 485	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝑥 ∈ 𝐷) ∧ 𝑛 ∈ 𝑍) → 𝑛 ∈ 𝑍)
63		rabidim1 3312	. . . . . . . . . . 11 ⊢ (𝑥 ∈ {𝑥 ∈ ∩ 𝑛 ∈ 𝑍 𝐴 ∣ ∃𝑦 ∈ ℝ ∀𝑛 ∈ 𝑍 𝐵 ≤ 𝑦} → 𝑥 ∈ ∩ 𝑛 ∈ 𝑍 𝐴)
64	63, 3	eleq2s 2857	. . . . . . . . . 10 ⊢ (𝑥 ∈ 𝐷 → 𝑥 ∈ ∩ 𝑛 ∈ 𝑍 𝐴)
65		eliinid 42661	. . . . . . . . . 10 ⊢ ((𝑥 ∈ ∩ 𝑛 ∈ 𝑍 𝐴 ∧ 𝑛 ∈ 𝑍) → 𝑥 ∈ 𝐴)
66	64, 65	sylan 580	. . . . . . . . 9 ⊢ ((𝑥 ∈ 𝐷 ∧ 𝑛 ∈ 𝑍) → 𝑥 ∈ 𝐴)
67	66	adantll 711	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝑥 ∈ 𝐷) ∧ 𝑛 ∈ 𝑍) → 𝑥 ∈ 𝐴)
68	61, 62, 67, 45	syl3anc 1370	. . . . . . 7 ⊢ (((𝜑 ∧ 𝑥 ∈ 𝐷) ∧ 𝑛 ∈ 𝑍) → 𝐵 = (((𝑛 ∈ 𝑍 ↦ (𝑥 ∈ 𝐴 ↦ 𝐵))‘𝑛)‘𝑥))
69	60, 68	mpteq2da 5172	. . . . . 6 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐷) → (𝑛 ∈ 𝑍 ↦ 𝐵) = (𝑛 ∈ 𝑍 ↦ (((𝑛 ∈ 𝑍 ↦ (𝑥 ∈ 𝐴 ↦ 𝐵))‘𝑛)‘𝑥)))
70	69	rneqd 5847	. . . . 5 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐷) → ran (𝑛 ∈ 𝑍 ↦ 𝐵) = ran (𝑛 ∈ 𝑍 ↦ (((𝑛 ∈ 𝑍 ↦ (𝑥 ∈ 𝐴 ↦ 𝐵))‘𝑛)‘𝑥)))
71	70	supeq1d 9205	. . . 4 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐷) → sup(ran (𝑛 ∈ 𝑍 ↦ 𝐵), ℝ, < ) = sup(ran (𝑛 ∈ 𝑍 ↦ (((𝑛 ∈ 𝑍 ↦ (𝑥 ∈ 𝐴 ↦ 𝐵))‘𝑛)‘𝑥)), ℝ, < ))
72	2, 52, 71	mpteq12da 5159	. . 3 ⊢ (𝜑 → (𝑥 ∈ 𝐷 ↦ sup(ran (𝑛 ∈ 𝑍 ↦ 𝐵), ℝ, < )) = (𝑥 ∈ {𝑥 ∈ ∩ 𝑛 ∈ 𝑍 dom ((𝑛 ∈ 𝑍 ↦ (𝑥 ∈ 𝐴 ↦ 𝐵))‘𝑛) ∣ ∃𝑦 ∈ ℝ ∀𝑛 ∈ 𝑍 (((𝑛 ∈ 𝑍 ↦ (𝑥 ∈ 𝐴 ↦ 𝐵))‘𝑛)‘𝑥) ≤ 𝑦} ↦ sup(ran (𝑛 ∈ 𝑍 ↦ (((𝑛 ∈ 𝑍 ↦ (𝑥 ∈ 𝐴 ↦ 𝐵))‘𝑛)‘𝑥)), ℝ, < )))
73	1, 72	eqtrid 2790	. 2 ⊢ (𝜑 → 𝐺 = (𝑥 ∈ {𝑥 ∈ ∩ 𝑛 ∈ 𝑍 dom ((𝑛 ∈ 𝑍 ↦ (𝑥 ∈ 𝐴 ↦ 𝐵))‘𝑛) ∣ ∃𝑦 ∈ ℝ ∀𝑛 ∈ 𝑍 (((𝑛 ∈ 𝑍 ↦ (𝑥 ∈ 𝐴 ↦ 𝐵))‘𝑛)‘𝑥) ≤ 𝑦} ↦ sup(ran (𝑛 ∈ 𝑍 ↦ (((𝑛 ∈ 𝑍 ↦ (𝑥 ∈ 𝐴 ↦ 𝐵))‘𝑛)‘𝑥)), ℝ, < )))
74		nfmpt1 5182	. . 3 ⊢ Ⅎ𝑛(𝑛 ∈ 𝑍 ↦ (𝑥 ∈ 𝐴 ↦ 𝐵))
75		smfsupmpt.m	. . 3 ⊢ (𝜑 → 𝑀 ∈ ℤ)
76		smfsupmpt.z	. . 3 ⊢ 𝑍 = (ℤ≥‘𝑀)
77		smfsupmpt.s	. . 3 ⊢ (𝜑 → 𝑆 ∈ SAlg)
78	4, 6	fmptd2f 42778	. . 3 ⊢ (𝜑 → (𝑛 ∈ 𝑍 ↦ (𝑥 ∈ 𝐴 ↦ 𝐵)):𝑍⟶(SMblFn‘𝑆))
79		eqid 2738	. . 3 ⊢ {𝑥 ∈ ∩ 𝑛 ∈ 𝑍 dom ((𝑛 ∈ 𝑍 ↦ (𝑥 ∈ 𝐴 ↦ 𝐵))‘𝑛) ∣ ∃𝑦 ∈ ℝ ∀𝑛 ∈ 𝑍 (((𝑛 ∈ 𝑍 ↦ (𝑥 ∈ 𝐴 ↦ 𝐵))‘𝑛)‘𝑥) ≤ 𝑦} = {𝑥 ∈ ∩ 𝑛 ∈ 𝑍 dom ((𝑛 ∈ 𝑍 ↦ (𝑥 ∈ 𝐴 ↦ 𝐵))‘𝑛) ∣ ∃𝑦 ∈ ℝ ∀𝑛 ∈ 𝑍 (((𝑛 ∈ 𝑍 ↦ (𝑥 ∈ 𝐴 ↦ 𝐵))‘𝑛)‘𝑥) ≤ 𝑦}
80		eqid 2738	. . 3 ⊢ (𝑥 ∈ {𝑥 ∈ ∩ 𝑛 ∈ 𝑍 dom ((𝑛 ∈ 𝑍 ↦ (𝑥 ∈ 𝐴 ↦ 𝐵))‘𝑛) ∣ ∃𝑦 ∈ ℝ ∀𝑛 ∈ 𝑍 (((𝑛 ∈ 𝑍 ↦ (𝑥 ∈ 𝐴 ↦ 𝐵))‘𝑛)‘𝑥) ≤ 𝑦} ↦ sup(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 (((𝑛 ∈ 𝑍 ↦ (𝑥 ∈ 𝐴 ↦ 𝐵))‘𝑛)‘𝑥) ≤ 𝑦} ↦ sup(ran (𝑛 ∈ 𝑍 ↦ (((𝑛 ∈ 𝑍 ↦ (𝑥 ∈ 𝐴 ↦ 𝐵))‘𝑛)‘𝑥)), ℝ, < ))
81	74, 20, 75, 76, 77, 78, 79, 80	smfsup 44347	. 2 ⊢ (𝜑 → (𝑥 ∈ {𝑥 ∈ ∩ 𝑛 ∈ 𝑍 dom ((𝑛 ∈ 𝑍 ↦ (𝑥 ∈ 𝐴 ↦ 𝐵))‘𝑛) ∣ ∃𝑦 ∈ ℝ ∀𝑛 ∈ 𝑍 (((𝑛 ∈ 𝑍 ↦ (𝑥 ∈ 𝐴 ↦ 𝐵))‘𝑛)‘𝑥) ≤ 𝑦} ↦ sup(ran (𝑛 ∈ 𝑍 ↦ (((𝑛 ∈ 𝑍 ↦ (𝑥 ∈ 𝐴 ↦ 𝐵))‘𝑛)‘𝑥)), ℝ, < )) ∈ (SMblFn‘𝑆))
82	73, 81	eqeltrd 2839	1 ⊢ (𝜑 → 𝐺 ∈ (SMblFn‘𝑆))

Syntax hints:   → wi 4   ↔ wb 205   ∧ wa 396   ∧ w3a 1086   = wceq 1539  Ⅎwnf 1786   ∈ wcel 2106  ∀wral 3064  ∃wrex 3065  {crab 3068  ∩ ciin 4925   class class class wbr 5074   ↦ cmpt 5157  dom cdm 5589  ran crn 5590  ‘cfv 6433  supcsup 9199  ℝcr 10870   < clt 11009   ≤ cle 11010  ℤcz 12319  ℤ_≥cuz 12582  SAlgcsalg 43849  SMblFncsmblfn 44233

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  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 2709  ax-rep 5209  ax-sep 5223  ax-nul 5230  ax-pow 5288  ax-pr 5352  ax-un 7588  ax-inf2 9399  ax-cc 10191  ax-ac2 10219  ax-cnex 10927  ax-resscn 10928  ax-1cn 10929  ax-icn 10930  ax-addcl 10931  ax-addrcl 10932  ax-mulcl 10933  ax-mulrcl 10934  ax-mulcom 10935  ax-addass 10936  ax-mulass 10937  ax-distr 10938  ax-i2m1 10939  ax-1ne0 10940  ax-1rid 10941  ax-rnegex 10942  ax-rrecex 10943  ax-cnre 10944  ax-pre-lttri 10945  ax-pre-lttrn 10946  ax-pre-ltadd 10947  ax-pre-mulgt0 10948  ax-pre-sup 10949

This theorem depends on definitions:  df-bi 206  df-an 397  df-or 845  df-3or 1087  df-3an 1088  df-tru 1542  df-fal 1552  df-ex 1783  df-nf 1787  df-sb 2068  df-mo 2540  df-eu 2569  df-clab 2716  df-cleq 2730  df-clel 2816  df-nfc 2889  df-ne 2944  df-nel 3050  df-ral 3069  df-rex 3070  df-rmo 3071  df-reu 3072  df-rab 3073  df-v 3434  df-sbc 3717  df-csb 3833  df-dif 3890  df-un 3892  df-in 3894  df-ss 3904  df-pss 3906  df-nul 4257  df-if 4460  df-pw 4535  df-sn 4562  df-pr 4564  df-op 4568  df-uni 4840  df-int 4880  df-iun 4926  df-iin 4927  df-br 5075  df-opab 5137  df-mpt 5158  df-tr 5192  df-id 5489  df-eprel 5495  df-po 5503  df-so 5504  df-fr 5544  df-se 5545  df-we 5546  df-xp 5595  df-rel 5596  df-cnv 5597  df-co 5598  df-dm 5599  df-rn 5600  df-res 5601  df-ima 5602  df-pred 6202  df-ord 6269  df-on 6270  df-lim 6271  df-suc 6272  df-iota 6391  df-fun 6435  df-fn 6436  df-f 6437  df-f1 6438  df-fo 6439  df-f1o 6440  df-fv 6441  df-isom 6442  df-riota 7232  df-ov 7278  df-oprab 7279  df-mpo 7280  df-om 7713  df-1st 7831  df-2nd 7832  df-frecs 8097  df-wrecs 8128  df-recs 8202  df-rdg 8241  df-1o 8297  df-oadd 8301  df-omul 8302  df-er 8498  df-map 8617  df-pm 8618  df-en 8734  df-dom 8735  df-sdom 8736  df-fin 8737  df-sup 9201  df-inf 9202  df-oi 9269  df-card 9697  df-acn 9700  df-ac 9872  df-pnf 11011  df-mnf 11012  df-xr 11013  df-ltxr 11014  df-le 11015  df-sub 11207  df-neg 11208  df-div 11633  df-nn 11974  df-n0 12234  df-z 12320  df-uz 12583  df-q 12689  df-rp 12731  df-ioo 13083  df-ioc 13084  df-ico 13085  df-fl 13512  df-rest 17133  df-topgen 17154  df-top 22043  df-bases 22096  df-salg 43850  df-salgen 43854  df-smblfn 44234

This theorem is referenced by:  smfinflem  44350