Theorem smfsupmpt 42554

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	. . . 4 ⊢ 𝐺 = (𝑥 ∈ 𝐷 ↦ sup(ran (𝑛 ∈ 𝑍 ↦ 𝐵), ℝ, < ))
2	1	a1i 11	. . 3 ⊢ (𝜑 → 𝐺 = (𝑥 ∈ 𝐷 ↦ sup(ran (𝑛 ∈ 𝑍 ↦ 𝐵), ℝ, < )))
3		smfsupmpt.x	. . . . 5 ⊢ Ⅎ𝑥𝜑
4		smfsupmpt.d	. . . . . . 7 ⊢ 𝐷 = {𝑥 ∈ ∩ 𝑛 ∈ 𝑍 𝐴 ∣ ∃𝑦 ∈ ℝ ∀𝑛 ∈ 𝑍 𝐵 ≤ 𝑦}
5	4	a1i 11	. . . . . 6 ⊢ (𝜑 → 𝐷 = {𝑥 ∈ ∩ 𝑛 ∈ 𝑍 𝐴 ∣ ∃𝑦 ∈ ℝ ∀𝑛 ∈ 𝑍 𝐵 ≤ 𝑦})
6		smfsupmpt.n	. . . . . . . . 9 ⊢ Ⅎ𝑛𝜑
7		eqidd 2772	. . . . . . . . . . . 12 ⊢ (𝜑 → (𝑛 ∈ 𝑍 ↦ (𝑥 ∈ 𝐴 ↦ 𝐵)) = (𝑛 ∈ 𝑍 ↦ (𝑥 ∈ 𝐴 ↦ 𝐵)))
8		smfsupmpt.f	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑛 ∈ 𝑍) → (𝑥 ∈ 𝐴 ↦ 𝐵) ∈ (SMblFn‘𝑆))
9	7, 8	fvmpt2d 6605	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑛 ∈ 𝑍) → ((𝑛 ∈ 𝑍 ↦ (𝑥 ∈ 𝐴 ↦ 𝐵))‘𝑛) = (𝑥 ∈ 𝐴 ↦ 𝐵))
10	9	dmeqd 5620	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑛 ∈ 𝑍) → dom ((𝑛 ∈ 𝑍 ↦ (𝑥 ∈ 𝐴 ↦ 𝐵))‘𝑛) = dom (𝑥 ∈ 𝐴 ↦ 𝐵))
11		nfcv 2925	. . . . . . . . . . . . 13 ⊢ Ⅎ𝑥𝑛
12		nfcv 2925	. . . . . . . . . . . . 13 ⊢ Ⅎ𝑥𝑍
13	11, 12	nfel 2937	. . . . . . . . . . . 12 ⊢ Ⅎ𝑥 𝑛 ∈ 𝑍
14	3, 13	nfan 1863	. . . . . . . . . . 11 ⊢ Ⅎ𝑥(𝜑 ∧ 𝑛 ∈ 𝑍)
15		eqid 2771	. . . . . . . . . . 11 ⊢ (𝑥 ∈ 𝐴 ↦ 𝐵) = (𝑥 ∈ 𝐴 ↦ 𝐵)
16		smfsupmpt.s	. . . . . . . . . . . . . 14 ⊢ (𝜑 → 𝑆 ∈ SAlg)
17	16	adantr 473	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑛 ∈ 𝑍) → 𝑆 ∈ SAlg)
18		smfsupmpt.b	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ 𝑛 ∈ 𝑍 ∧ 𝑥 ∈ 𝐴) → 𝐵 ∈ 𝑉)
19	18	3expa 1099	. . . . . . . . . . . . 13 ⊢ (((𝜑 ∧ 𝑛 ∈ 𝑍) ∧ 𝑥 ∈ 𝐴) → 𝐵 ∈ 𝑉)
20	14, 17, 19, 8	smffmpt 42544	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑛 ∈ 𝑍) → (𝑥 ∈ 𝐴 ↦ 𝐵):𝐴⟶ℝ)
21	20	fvmptelrn 6698	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ 𝑛 ∈ 𝑍) ∧ 𝑥 ∈ 𝐴) → 𝐵 ∈ ℝ)
22	14, 15, 21	dmmptdf 40948	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑛 ∈ 𝑍) → dom (𝑥 ∈ 𝐴 ↦ 𝐵) = 𝐴)
23		eqidd 2772	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑛 ∈ 𝑍) → 𝐴 = 𝐴)
24	10, 22, 23	3eqtrrd 2812	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑛 ∈ 𝑍) → 𝐴 = dom ((𝑛 ∈ 𝑍 ↦ (𝑥 ∈ 𝐴 ↦ 𝐵))‘𝑛))
25	6, 24	iineq2d 4810	. . . . . . . 8 ⊢ (𝜑 → ∩ 𝑛 ∈ 𝑍 𝐴 = ∩ 𝑛 ∈ 𝑍 dom ((𝑛 ∈ 𝑍 ↦ (𝑥 ∈ 𝐴 ↦ 𝐵))‘𝑛))
26		nfcv 2925	. . . . . . . . 9 ⊢ Ⅎ𝑥∩ 𝑛 ∈ 𝑍 𝐴
27		nfmpt1 5021	. . . . . . . . . . . . 13 ⊢ Ⅎ𝑥(𝑥 ∈ 𝐴 ↦ 𝐵)
28	12, 27	nfmpt 5020	. . . . . . . . . . . 12 ⊢ Ⅎ𝑥(𝑛 ∈ 𝑍 ↦ (𝑥 ∈ 𝐴 ↦ 𝐵))
29	28, 11	nffv 6506	. . . . . . . . . . 11 ⊢ Ⅎ𝑥((𝑛 ∈ 𝑍 ↦ (𝑥 ∈ 𝐴 ↦ 𝐵))‘𝑛)
30	29	nfdm 5663	. . . . . . . . . 10 ⊢ Ⅎ𝑥dom ((𝑛 ∈ 𝑍 ↦ (𝑥 ∈ 𝐴 ↦ 𝐵))‘𝑛)
31	12, 30	nfiin 4818	. . . . . . . . 9 ⊢ Ⅎ𝑥∩ 𝑛 ∈ 𝑍 dom ((𝑛 ∈ 𝑍 ↦ (𝑥 ∈ 𝐴 ↦ 𝐵))‘𝑛)
32	26, 31	rabeqf 3397	. . . . . . . 8 ⊢ (∩ 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})
33	25, 32	syl 17	. . . . . . 7 ⊢ (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})
34		smfsupmpt.y	. . . . . . . . . 10 ⊢ Ⅎ𝑦𝜑
35		nfv 1874	. . . . . . . . . 10 ⊢ Ⅎ𝑦 𝑥 ∈ ∩ 𝑛 ∈ 𝑍 dom ((𝑛 ∈ 𝑍 ↦ (𝑥 ∈ 𝐴 ↦ 𝐵))‘𝑛)
36	34, 35	nfan 1863	. . . . . . . . 9 ⊢ Ⅎ𝑦(𝜑 ∧ 𝑥 ∈ ∩ 𝑛 ∈ 𝑍 dom ((𝑛 ∈ 𝑍 ↦ (𝑥 ∈ 𝐴 ↦ 𝐵))‘𝑛))
37		nfcv 2925	. . . . . . . . . . . 12 ⊢ Ⅎ𝑛𝑥
38		nfii1 4820	. . . . . . . . . . . 12 ⊢ Ⅎ𝑛∩ 𝑛 ∈ 𝑍 dom ((𝑛 ∈ 𝑍 ↦ (𝑥 ∈ 𝐴 ↦ 𝐵))‘𝑛)
39	37, 38	nfel 2937	. . . . . . . . . . 11 ⊢ Ⅎ𝑛 𝑥 ∈ ∩ 𝑛 ∈ 𝑍 dom ((𝑛 ∈ 𝑍 ↦ (𝑥 ∈ 𝐴 ↦ 𝐵))‘𝑛)
40	6, 39	nfan 1863	. . . . . . . . . 10 ⊢ Ⅎ𝑛(𝜑 ∧ 𝑥 ∈ ∩ 𝑛 ∈ 𝑍 dom ((𝑛 ∈ 𝑍 ↦ (𝑥 ∈ 𝐴 ↦ 𝐵))‘𝑛))
41		simpll 755	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ 𝑥 ∈ ∩ 𝑛 ∈ 𝑍 dom ((𝑛 ∈ 𝑍 ↦ (𝑥 ∈ 𝐴 ↦ 𝐵))‘𝑛)) ∧ 𝑛 ∈ 𝑍) → 𝜑)
42		simpr 477	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ 𝑥 ∈ ∩ 𝑛 ∈ 𝑍 dom ((𝑛 ∈ 𝑍 ↦ (𝑥 ∈ 𝐴 ↦ 𝐵))‘𝑛)) ∧ 𝑛 ∈ 𝑍) → 𝑛 ∈ 𝑍)
43		eliinid 40836	. . . . . . . . . . . . 13 ⊢ ((𝑥 ∈ ∩ 𝑛 ∈ 𝑍 dom ((𝑛 ∈ 𝑍 ↦ (𝑥 ∈ 𝐴 ↦ 𝐵))‘𝑛) ∧ 𝑛 ∈ 𝑍) → 𝑥 ∈ dom ((𝑛 ∈ 𝑍 ↦ (𝑥 ∈ 𝐴 ↦ 𝐵))‘𝑛))
44	43	adantll 702	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ 𝑥 ∈ ∩ 𝑛 ∈ 𝑍 dom ((𝑛 ∈ 𝑍 ↦ (𝑥 ∈ 𝐴 ↦ 𝐵))‘𝑛)) ∧ 𝑛 ∈ 𝑍) → 𝑥 ∈ dom ((𝑛 ∈ 𝑍 ↦ (𝑥 ∈ 𝐴 ↦ 𝐵))‘𝑛))
45	24	eqcomd 2777	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑛 ∈ 𝑍) → dom ((𝑛 ∈ 𝑍 ↦ (𝑥 ∈ 𝐴 ↦ 𝐵))‘𝑛) = 𝐴)
46	45	adantlr 703	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ 𝑥 ∈ ∩ 𝑛 ∈ 𝑍 dom ((𝑛 ∈ 𝑍 ↦ (𝑥 ∈ 𝐴 ↦ 𝐵))‘𝑛)) ∧ 𝑛 ∈ 𝑍) → dom ((𝑛 ∈ 𝑍 ↦ (𝑥 ∈ 𝐴 ↦ 𝐵))‘𝑛) = 𝐴)
47	44, 46	eleqtrd 2861	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ 𝑥 ∈ ∩ 𝑛 ∈ 𝑍 dom ((𝑛 ∈ 𝑍 ↦ (𝑥 ∈ 𝐴 ↦ 𝐵))‘𝑛)) ∧ 𝑛 ∈ 𝑍) → 𝑥 ∈ 𝐴)
48	9	fveq1d 6498	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ 𝑛 ∈ 𝑍) → (((𝑛 ∈ 𝑍 ↦ (𝑥 ∈ 𝐴 ↦ 𝐵))‘𝑛)‘𝑥) = ((𝑥 ∈ 𝐴 ↦ 𝐵)‘𝑥))
49	48	3adant3 1113	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑛 ∈ 𝑍 ∧ 𝑥 ∈ 𝐴) → (((𝑛 ∈ 𝑍 ↦ (𝑥 ∈ 𝐴 ↦ 𝐵))‘𝑛)‘𝑥) = ((𝑥 ∈ 𝐴 ↦ 𝐵)‘𝑥))
50		simp3 1119	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ 𝑛 ∈ 𝑍 ∧ 𝑥 ∈ 𝐴) → 𝑥 ∈ 𝐴)
51	15	fvmpt2 6603	. . . . . . . . . . . . . 14 ⊢ ((𝑥 ∈ 𝐴 ∧ 𝐵 ∈ 𝑉) → ((𝑥 ∈ 𝐴 ↦ 𝐵)‘𝑥) = 𝐵)
52	50, 18, 51	syl2anc 576	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑛 ∈ 𝑍 ∧ 𝑥 ∈ 𝐴) → ((𝑥 ∈ 𝐴 ↦ 𝐵)‘𝑥) = 𝐵)
53	49, 52	eqtr2d 2808	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑛 ∈ 𝑍 ∧ 𝑥 ∈ 𝐴) → 𝐵 = (((𝑛 ∈ 𝑍 ↦ (𝑥 ∈ 𝐴 ↦ 𝐵))‘𝑛)‘𝑥))
54	53	breq1d 4935	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑛 ∈ 𝑍 ∧ 𝑥 ∈ 𝐴) → (𝐵 ≤ 𝑦 ↔ (((𝑛 ∈ 𝑍 ↦ (𝑥 ∈ 𝐴 ↦ 𝐵))‘𝑛)‘𝑥) ≤ 𝑦))
55	41, 42, 47, 54	syl3anc 1352	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝑥 ∈ ∩ 𝑛 ∈ 𝑍 dom ((𝑛 ∈ 𝑍 ↦ (𝑥 ∈ 𝐴 ↦ 𝐵))‘𝑛)) ∧ 𝑛 ∈ 𝑍) → (𝐵 ≤ 𝑦 ↔ (((𝑛 ∈ 𝑍 ↦ (𝑥 ∈ 𝐴 ↦ 𝐵))‘𝑛)‘𝑥) ≤ 𝑦))
56	40, 55	ralbida 3170	. . . . . . . . 9 ⊢ ((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))
57	36, 56	rexbid 3256	. . . . . . . 8 ⊢ ((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))
58	3, 57	rabbida 40819	. . . . . . 7 ⊢ (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})
59	33, 58	eqtrd 2807	. . . . . 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 (((𝑛 ∈ 𝑍 ↦ (𝑥 ∈ 𝐴 ↦ 𝐵))‘𝑛)‘𝑥) ≤ 𝑦})
60	5, 59	eqtrd 2807	. . . . 5 ⊢ (𝜑 → 𝐷 = {𝑥 ∈ ∩ 𝑛 ∈ 𝑍 dom ((𝑛 ∈ 𝑍 ↦ (𝑥 ∈ 𝐴 ↦ 𝐵))‘𝑛) ∣ ∃𝑦 ∈ ℝ ∀𝑛 ∈ 𝑍 (((𝑛 ∈ 𝑍 ↦ (𝑥 ∈ 𝐴 ↦ 𝐵))‘𝑛)‘𝑥) ≤ 𝑦})
61	3, 60	alrimi 2144	. . . 4 ⊢ (𝜑 → ∀𝑥 𝐷 = {𝑥 ∈ ∩ 𝑛 ∈ 𝑍 dom ((𝑛 ∈ 𝑍 ↦ (𝑥 ∈ 𝐴 ↦ 𝐵))‘𝑛) ∣ ∃𝑦 ∈ ℝ ∀𝑛 ∈ 𝑍 (((𝑛 ∈ 𝑍 ↦ (𝑥 ∈ 𝐴 ↦ 𝐵))‘𝑛)‘𝑥) ≤ 𝑦})
62		nfcv 2925	. . . . . . . . . . . . . 14 ⊢ Ⅎ𝑛ℝ
63		nfra1 3162	. . . . . . . . . . . . . 14 ⊢ Ⅎ𝑛∀𝑛 ∈ 𝑍 𝐵 ≤ 𝑦
64	62, 63	nfrex 3246	. . . . . . . . . . . . 13 ⊢ Ⅎ𝑛∃𝑦 ∈ ℝ ∀𝑛 ∈ 𝑍 𝐵 ≤ 𝑦
65		nfii1 4820	. . . . . . . . . . . . 13 ⊢ Ⅎ𝑛∩ 𝑛 ∈ 𝑍 𝐴
66	64, 65	nfrab 3318	. . . . . . . . . . . 12 ⊢ Ⅎ𝑛{𝑥 ∈ ∩ 𝑛 ∈ 𝑍 𝐴 ∣ ∃𝑦 ∈ ℝ ∀𝑛 ∈ 𝑍 𝐵 ≤ 𝑦}
67	4, 66	nfcxfr 2923	. . . . . . . . . . 11 ⊢ Ⅎ𝑛𝐷
68	37, 67	nfel 2937	. . . . . . . . . 10 ⊢ Ⅎ𝑛 𝑥 ∈ 𝐷
69	6, 68	nfan 1863	. . . . . . . . 9 ⊢ Ⅎ𝑛(𝜑 ∧ 𝑥 ∈ 𝐷)
70		simpll 755	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝑥 ∈ 𝐷) ∧ 𝑛 ∈ 𝑍) → 𝜑)
71		simpr 477	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝑥 ∈ 𝐷) ∧ 𝑛 ∈ 𝑍) → 𝑛 ∈ 𝑍)
72	4	eleq2i 2850	. . . . . . . . . . . . . . 15 ⊢ (𝑥 ∈ 𝐷 ↔ 𝑥 ∈ {𝑥 ∈ ∩ 𝑛 ∈ 𝑍 𝐴 ∣ ∃𝑦 ∈ ℝ ∀𝑛 ∈ 𝑍 𝐵 ≤ 𝑦})
73	72	biimpi 208	. . . . . . . . . . . . . 14 ⊢ (𝑥 ∈ 𝐷 → 𝑥 ∈ {𝑥 ∈ ∩ 𝑛 ∈ 𝑍 𝐴 ∣ ∃𝑦 ∈ ℝ ∀𝑛 ∈ 𝑍 𝐵 ≤ 𝑦})
74		rabidim1 3312	. . . . . . . . . . . . . 14 ⊢ (𝑥 ∈ {𝑥 ∈ ∩ 𝑛 ∈ 𝑍 𝐴 ∣ ∃𝑦 ∈ ℝ ∀𝑛 ∈ 𝑍 𝐵 ≤ 𝑦} → 𝑥 ∈ ∩ 𝑛 ∈ 𝑍 𝐴)
75	73, 74	syl 17	. . . . . . . . . . . . 13 ⊢ (𝑥 ∈ 𝐷 → 𝑥 ∈ ∩ 𝑛 ∈ 𝑍 𝐴)
76	75	adantr 473	. . . . . . . . . . . 12 ⊢ ((𝑥 ∈ 𝐷 ∧ 𝑛 ∈ 𝑍) → 𝑥 ∈ ∩ 𝑛 ∈ 𝑍 𝐴)
77		simpr 477	. . . . . . . . . . . 12 ⊢ ((𝑥 ∈ 𝐷 ∧ 𝑛 ∈ 𝑍) → 𝑛 ∈ 𝑍)
78		eliinid 40836	. . . . . . . . . . . 12 ⊢ ((𝑥 ∈ ∩ 𝑛 ∈ 𝑍 𝐴 ∧ 𝑛 ∈ 𝑍) → 𝑥 ∈ 𝐴)
79	76, 77, 78	syl2anc 576	. . . . . . . . . . 11 ⊢ ((𝑥 ∈ 𝐷 ∧ 𝑛 ∈ 𝑍) → 𝑥 ∈ 𝐴)
80	79	adantll 702	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝑥 ∈ 𝐷) ∧ 𝑛 ∈ 𝑍) → 𝑥 ∈ 𝐴)
81	53	idi 1	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑛 ∈ 𝑍 ∧ 𝑥 ∈ 𝐴) → 𝐵 = (((𝑛 ∈ 𝑍 ↦ (𝑥 ∈ 𝐴 ↦ 𝐵))‘𝑛)‘𝑥))
82	70, 71, 80, 81	syl3anc 1352	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑥 ∈ 𝐷) ∧ 𝑛 ∈ 𝑍) → 𝐵 = (((𝑛 ∈ 𝑍 ↦ (𝑥 ∈ 𝐴 ↦ 𝐵))‘𝑛)‘𝑥))
83	69, 82	mpteq2da 5017	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐷) → (𝑛 ∈ 𝑍 ↦ 𝐵) = (𝑛 ∈ 𝑍 ↦ (((𝑛 ∈ 𝑍 ↦ (𝑥 ∈ 𝐴 ↦ 𝐵))‘𝑛)‘𝑥)))
84	83	rneqd 5648	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐷) → ran (𝑛 ∈ 𝑍 ↦ 𝐵) = ran (𝑛 ∈ 𝑍 ↦ (((𝑛 ∈ 𝑍 ↦ (𝑥 ∈ 𝐴 ↦ 𝐵))‘𝑛)‘𝑥)))
85	84	supeq1d 8703	. . . . . 6 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐷) → sup(ran (𝑛 ∈ 𝑍 ↦ 𝐵), ℝ, < ) = sup(ran (𝑛 ∈ 𝑍 ↦ (((𝑛 ∈ 𝑍 ↦ (𝑥 ∈ 𝐴 ↦ 𝐵))‘𝑛)‘𝑥)), ℝ, < ))
86	85	ex 405	. . . . 5 ⊢ (𝜑 → (𝑥 ∈ 𝐷 → sup(ran (𝑛 ∈ 𝑍 ↦ 𝐵), ℝ, < ) = sup(ran (𝑛 ∈ 𝑍 ↦ (((𝑛 ∈ 𝑍 ↦ (𝑥 ∈ 𝐴 ↦ 𝐵))‘𝑛)‘𝑥)), ℝ, < )))
87	3, 86	ralrimi 3159	. . . 4 ⊢ (𝜑 → ∀𝑥 ∈ 𝐷 sup(ran (𝑛 ∈ 𝑍 ↦ 𝐵), ℝ, < ) = sup(ran (𝑛 ∈ 𝑍 ↦ (((𝑛 ∈ 𝑍 ↦ (𝑥 ∈ 𝐴 ↦ 𝐵))‘𝑛)‘𝑥)), ℝ, < ))
88		mpteq12f 5006	. . . 4 ⊢ ((∀𝑥 𝐷 = {𝑥 ∈ ∩ 𝑛 ∈ 𝑍 dom ((𝑛 ∈ 𝑍 ↦ (𝑥 ∈ 𝐴 ↦ 𝐵))‘𝑛) ∣ ∃𝑦 ∈ ℝ ∀𝑛 ∈ 𝑍 (((𝑛 ∈ 𝑍 ↦ (𝑥 ∈ 𝐴 ↦ 𝐵))‘𝑛)‘𝑥) ≤ 𝑦} ∧ ∀𝑥 ∈ 𝐷 sup(ran (𝑛 ∈ 𝑍 ↦ 𝐵), ℝ, < ) = sup(ran (𝑛 ∈ 𝑍 ↦ (((𝑛 ∈ 𝑍 ↦ (𝑥 ∈ 𝐴 ↦ 𝐵))‘𝑛)‘𝑥)), ℝ, < )) → (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, < )))
89	61, 87, 88	syl2anc 576	. . 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, < )))
90	2, 89	eqtrd 2807	. 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, < )))
91		nfmpt1 5021	. . 3 ⊢ Ⅎ𝑛(𝑛 ∈ 𝑍 ↦ (𝑥 ∈ 𝐴 ↦ 𝐵))
92		smfsupmpt.m	. . 3 ⊢ (𝜑 → 𝑀 ∈ ℤ)
93		smfsupmpt.z	. . 3 ⊢ 𝑍 = (ℤ≥‘𝑀)
94		eqid 2771	. . . 4 ⊢ (𝑛 ∈ 𝑍 ↦ (𝑥 ∈ 𝐴 ↦ 𝐵)) = (𝑛 ∈ 𝑍 ↦ (𝑥 ∈ 𝐴 ↦ 𝐵))
95	6, 8, 94	fmptdf 6702	. . 3 ⊢ (𝜑 → (𝑛 ∈ 𝑍 ↦ (𝑥 ∈ 𝐴 ↦ 𝐵)):𝑍⟶(SMblFn‘𝑆))
96		eqid 2771	. . 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}
97		eqid 2771	. . 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, < ))
98	91, 28, 92, 93, 16, 95, 96, 97	smfsup 42553	. 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))
99	90, 98	eqeltrd 2859	1 ⊢ (𝜑 → 𝐺 ∈ (SMblFn‘𝑆))

Syntax hints:   → wi 4   ↔ wb 198   ∧ wa 387   ∧ w3a 1069  ∀wal 1506   = wceq 1508  Ⅎwnf 1747   ∈ wcel 2051  ∀wral 3081  ∃wrex 3082  {crab 3085  ∩ ciin 4789   class class class wbr 4925   ↦ cmpt 5004  dom cdm 5403  ran crn 5404  ‘cfv 6185  supcsup 8697  ℝcr 10332   < clt 10472   ≤ cle 10473  ℤcz 11791  ℤ_≥cuz 12056  SAlgcsalg 42058  SMblFncsmblfn 42442

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1759  ax-4 1773  ax-5 1870  ax-6 1929  ax-7 1966  ax-8 2053  ax-9 2060  ax-10 2080  ax-11 2094  ax-12 2107  ax-13 2302  ax-ext 2743  ax-rep 5045  ax-sep 5056  ax-nul 5063  ax-pow 5115  ax-pr 5182  ax-un 7277  ax-inf2 8896  ax-cc 9653  ax-ac2 9681  ax-cnex 10389  ax-resscn 10390  ax-1cn 10391  ax-icn 10392  ax-addcl 10393  ax-addrcl 10394  ax-mulcl 10395  ax-mulrcl 10396  ax-mulcom 10397  ax-addass 10398  ax-mulass 10399  ax-distr 10400  ax-i2m1 10401  ax-1ne0 10402  ax-1rid 10403  ax-rnegex 10404  ax-rrecex 10405  ax-cnre 10406  ax-pre-lttri 10407  ax-pre-lttrn 10408  ax-pre-ltadd 10409  ax-pre-mulgt0 10410  ax-pre-sup 10411

This theorem depends on definitions:  df-bi 199  df-an 388  df-or 835  df-3or 1070  df-3an 1071  df-tru 1511  df-ex 1744  df-nf 1748  df-sb 2017  df-mo 2548  df-eu 2585  df-clab 2752  df-cleq 2764  df-clel 2839  df-nfc 2911  df-ne 2961  df-nel 3067  df-ral 3086  df-rex 3087  df-reu 3088  df-rmo 3089  df-rab 3090  df-v 3410  df-sbc 3675  df-csb 3780  df-dif 3825  df-un 3827  df-in 3829  df-ss 3836  df-pss 3838  df-nul 4173  df-if 4345  df-pw 4418  df-sn 4436  df-pr 4438  df-tp 4440  df-op 4442  df-uni 4709  df-int 4746  df-iun 4790  df-iin 4791  df-br 4926  df-opab 4988  df-mpt 5005  df-tr 5027  df-id 5308  df-eprel 5313  df-po 5322  df-so 5323  df-fr 5362  df-se 5363  df-we 5364  df-xp 5409  df-rel 5410  df-cnv 5411  df-co 5412  df-dm 5413  df-rn 5414  df-res 5415  df-ima 5416  df-pred 5983  df-ord 6029  df-on 6030  df-lim 6031  df-suc 6032  df-iota 6149  df-fun 6187  df-fn 6188  df-f 6189  df-f1 6190  df-fo 6191  df-f1o 6192  df-fv 6193  df-isom 6194  df-riota 6935  df-ov 6977  df-oprab 6978  df-mpo 6979  df-om 7395  df-1st 7499  df-2nd 7500  df-wrecs 7748  df-recs 7810  df-rdg 7848  df-1o 7903  df-oadd 7907  df-omul 7908  df-er 8087  df-map 8206  df-pm 8207  df-en 8305  df-dom 8306  df-sdom 8307  df-fin 8308  df-sup 8699  df-inf 8700  df-oi 8767  df-card 9160  df-acn 9163  df-ac 9334  df-pnf 10474  df-mnf 10475  df-xr 10476  df-ltxr 10477  df-le 10478  df-sub 10670  df-neg 10671  df-div 11097  df-nn 11438  df-n0 11706  df-z 11792  df-uz 12057  df-q 12161  df-rp 12203  df-ioo 12556  df-ioc 12557  df-ico 12558  df-fl 12975  df-rest 16550  df-topgen 16571  df-top 21221  df-bases 21273  df-salg 42059  df-salgen 42063  df-smblfn 42443

This theorem is referenced by:  smfinflem  42556