Theorem smfinflem 46473

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
smfinflem.m	⊢ (𝜑 → 𝑀 ∈ ℤ)
smfinflem.z	⊢ 𝑍 = (ℤ≥‘𝑀)
smfinflem.s	⊢ (𝜑 → 𝑆 ∈ SAlg)
smfinflem.f	⊢ (𝜑 → 𝐹:𝑍⟶(SMblFn‘𝑆))
smfinflem.d	⊢ 𝐷 = {𝑥 ∈ ∩ 𝑛 ∈ 𝑍 dom (𝐹‘𝑛) ∣ ∃𝑦 ∈ ℝ ∀𝑛 ∈ 𝑍 𝑦 ≤ ((𝐹‘𝑛)‘𝑥)}
smfinflem.g	⊢ 𝐺 = (𝑥 ∈ 𝐷 ↦ inf(ran (𝑛 ∈ 𝑍 ↦ ((𝐹‘𝑛)‘𝑥)), ℝ, < ))

Assertion

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

Distinct variable groups:   𝐷,𝑛,𝑥,𝑦   𝑛,𝐹,𝑥,𝑦   𝑆,𝑛   𝑛,𝑍,𝑥,𝑦   𝜑,𝑛,𝑥,𝑦

Allowed substitution hints:   𝑆(𝑥,𝑦)   𝐺(𝑥,𝑦,𝑛)   𝑀(𝑥,𝑦,𝑛)

Proof of Theorem smfinflem

Dummy variables 𝑚 𝑤 𝑧 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		smfinflem.g	. . . 4 ⊢ 𝐺 = (𝑥 ∈ 𝐷 ↦ inf(ran (𝑛 ∈ 𝑍 ↦ ((𝐹‘𝑛)‘𝑥)), ℝ, < ))
2	1	a1i 11	. . 3 ⊢ (𝜑 → 𝐺 = (𝑥 ∈ 𝐷 ↦ inf(ran (𝑛 ∈ 𝑍 ↦ ((𝐹‘𝑛)‘𝑥)), ℝ, < )))
3		nfv 1910	. . . . 5 ⊢ Ⅎ𝑛(𝜑 ∧ 𝑥 ∈ 𝐷)
4		smfinflem.m	. . . . . . 7 ⊢ (𝜑 → 𝑀 ∈ ℤ)
5		smfinflem.z	. . . . . . 7 ⊢ 𝑍 = (ℤ≥‘𝑀)
6	4, 5	uzn0d 45075	. . . . . 6 ⊢ (𝜑 → 𝑍 ≠ ∅)
7	6	adantr 479	. . . . 5 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐷) → 𝑍 ≠ ∅)
8		smfinflem.s	. . . . . . . . 9 ⊢ (𝜑 → 𝑆 ∈ SAlg)
9	8	adantr 479	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑛 ∈ 𝑍) → 𝑆 ∈ SAlg)
10		smfinflem.f	. . . . . . . . 9 ⊢ (𝜑 → 𝐹:𝑍⟶(SMblFn‘𝑆))
11	10	ffvelcdmda 7089	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑛 ∈ 𝑍) → (𝐹‘𝑛) ∈ (SMblFn‘𝑆))
12		eqid 2726	. . . . . . . 8 ⊢ dom (𝐹‘𝑛) = dom (𝐹‘𝑛)
13	9, 11, 12	smff 46388	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑛 ∈ 𝑍) → (𝐹‘𝑛):dom (𝐹‘𝑛)⟶ℝ)
14	13	adantlr 713	. . . . . 6 ⊢ (((𝜑 ∧ 𝑥 ∈ 𝐷) ∧ 𝑛 ∈ 𝑍) → (𝐹‘𝑛):dom (𝐹‘𝑛)⟶ℝ)
15		ssrab2 4075	. . . . . . . . . 10 ⊢ {𝑥 ∈ ∩ 𝑛 ∈ 𝑍 dom (𝐹‘𝑛) ∣ ∃𝑦 ∈ ℝ ∀𝑛 ∈ 𝑍 𝑦 ≤ ((𝐹‘𝑛)‘𝑥)} ⊆ ∩ 𝑛 ∈ 𝑍 dom (𝐹‘𝑛)
16		smfinflem.d	. . . . . . . . . . . 12 ⊢ 𝐷 = {𝑥 ∈ ∩ 𝑛 ∈ 𝑍 dom (𝐹‘𝑛) ∣ ∃𝑦 ∈ ℝ ∀𝑛 ∈ 𝑍 𝑦 ≤ ((𝐹‘𝑛)‘𝑥)}
17	16	eleq2i 2818	. . . . . . . . . . 11 ⊢ (𝑥 ∈ 𝐷 ↔ 𝑥 ∈ {𝑥 ∈ ∩ 𝑛 ∈ 𝑍 dom (𝐹‘𝑛) ∣ ∃𝑦 ∈ ℝ ∀𝑛 ∈ 𝑍 𝑦 ≤ ((𝐹‘𝑛)‘𝑥)})
18	17	biimpi 215	. . . . . . . . . 10 ⊢ (𝑥 ∈ 𝐷 → 𝑥 ∈ {𝑥 ∈ ∩ 𝑛 ∈ 𝑍 dom (𝐹‘𝑛) ∣ ∃𝑦 ∈ ℝ ∀𝑛 ∈ 𝑍 𝑦 ≤ ((𝐹‘𝑛)‘𝑥)})
19	15, 18	sselid 3978	. . . . . . . . 9 ⊢ (𝑥 ∈ 𝐷 → 𝑥 ∈ ∩ 𝑛 ∈ 𝑍 dom (𝐹‘𝑛))
20	19	adantr 479	. . . . . . . 8 ⊢ ((𝑥 ∈ 𝐷 ∧ 𝑛 ∈ 𝑍) → 𝑥 ∈ ∩ 𝑛 ∈ 𝑍 dom (𝐹‘𝑛))
21		simpr 483	. . . . . . . 8 ⊢ ((𝑥 ∈ 𝐷 ∧ 𝑛 ∈ 𝑍) → 𝑛 ∈ 𝑍)
22		eliinid 44748	. . . . . . . 8 ⊢ ((𝑥 ∈ ∩ 𝑛 ∈ 𝑍 dom (𝐹‘𝑛) ∧ 𝑛 ∈ 𝑍) → 𝑥 ∈ dom (𝐹‘𝑛))
23	20, 21, 22	syl2anc 582	. . . . . . 7 ⊢ ((𝑥 ∈ 𝐷 ∧ 𝑛 ∈ 𝑍) → 𝑥 ∈ dom (𝐹‘𝑛))
24	23	adantll 712	. . . . . 6 ⊢ (((𝜑 ∧ 𝑥 ∈ 𝐷) ∧ 𝑛 ∈ 𝑍) → 𝑥 ∈ dom (𝐹‘𝑛))
25	14, 24	ffvelcdmd 7090	. . . . 5 ⊢ (((𝜑 ∧ 𝑥 ∈ 𝐷) ∧ 𝑛 ∈ 𝑍) → ((𝐹‘𝑛)‘𝑥) ∈ ℝ)
26		rabidim2 44739	. . . . . . 7 ⊢ (𝑥 ∈ {𝑥 ∈ ∩ 𝑛 ∈ 𝑍 dom (𝐹‘𝑛) ∣ ∃𝑦 ∈ ℝ ∀𝑛 ∈ 𝑍 𝑦 ≤ ((𝐹‘𝑛)‘𝑥)} → ∃𝑦 ∈ ℝ ∀𝑛 ∈ 𝑍 𝑦 ≤ ((𝐹‘𝑛)‘𝑥))
27	18, 26	syl 17	. . . . . 6 ⊢ (𝑥 ∈ 𝐷 → ∃𝑦 ∈ ℝ ∀𝑛 ∈ 𝑍 𝑦 ≤ ((𝐹‘𝑛)‘𝑥))
28	27	adantl 480	. . . . 5 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐷) → ∃𝑦 ∈ ℝ ∀𝑛 ∈ 𝑍 𝑦 ≤ ((𝐹‘𝑛)‘𝑥))
29	3, 7, 25, 28	infnsuprnmpt 44894	. . . 4 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐷) → inf(ran (𝑛 ∈ 𝑍 ↦ ((𝐹‘𝑛)‘𝑥)), ℝ, < ) = -sup(ran (𝑛 ∈ 𝑍 ↦ -((𝐹‘𝑛)‘𝑥)), ℝ, < ))
30	29	mpteq2dva 5245	. . 3 ⊢ (𝜑 → (𝑥 ∈ 𝐷 ↦ inf(ran (𝑛 ∈ 𝑍 ↦ ((𝐹‘𝑛)‘𝑥)), ℝ, < )) = (𝑥 ∈ 𝐷 ↦ -sup(ran (𝑛 ∈ 𝑍 ↦ -((𝐹‘𝑛)‘𝑥)), ℝ, < )))
31	2, 30	eqtrd 2766	. 2 ⊢ (𝜑 → 𝐺 = (𝑥 ∈ 𝐷 ↦ -sup(ran (𝑛 ∈ 𝑍 ↦ -((𝐹‘𝑛)‘𝑥)), ℝ, < )))
32		nfv 1910	. . 3 ⊢ Ⅎ𝑥𝜑
33		fvex 6905	. . . . . . . 8 ⊢ (𝐹‘𝑛) ∈ V
34	33	dmex 7913	. . . . . . 7 ⊢ dom (𝐹‘𝑛) ∈ V
35	34	rgenw 3055	. . . . . 6 ⊢ ∀𝑛 ∈ 𝑍 dom (𝐹‘𝑛) ∈ V
36	35	a1i 11	. . . . 5 ⊢ (𝜑 → ∀𝑛 ∈ 𝑍 dom (𝐹‘𝑛) ∈ V)
37	6, 36	iinexd 44770	. . . 4 ⊢ (𝜑 → ∩ 𝑛 ∈ 𝑍 dom (𝐹‘𝑛) ∈ V)
38	16, 37	rabexd 5332	. . 3 ⊢ (𝜑 → 𝐷 ∈ V)
39	25	renegcld 11681	. . . 4 ⊢ (((𝜑 ∧ 𝑥 ∈ 𝐷) ∧ 𝑛 ∈ 𝑍) → -((𝐹‘𝑛)‘𝑥) ∈ ℝ)
40		fveq2 6892	. . . . . . . . . . . 12 ⊢ (𝑤 = 𝑥 → ((𝐹‘𝑚)‘𝑤) = ((𝐹‘𝑚)‘𝑥))
41	40	breq2d 5157	. . . . . . . . . . 11 ⊢ (𝑤 = 𝑥 → (𝑧 ≤ ((𝐹‘𝑚)‘𝑤) ↔ 𝑧 ≤ ((𝐹‘𝑚)‘𝑥)))
42	41	ralbidv 3168	. . . . . . . . . 10 ⊢ (𝑤 = 𝑥 → (∀𝑚 ∈ 𝑍 𝑧 ≤ ((𝐹‘𝑚)‘𝑤) ↔ ∀𝑚 ∈ 𝑍 𝑧 ≤ ((𝐹‘𝑚)‘𝑥)))
43	42	rexbidv 3169	. . . . . . . . 9 ⊢ (𝑤 = 𝑥 → (∃𝑧 ∈ ℝ ∀𝑚 ∈ 𝑍 𝑧 ≤ ((𝐹‘𝑚)‘𝑤) ↔ ∃𝑧 ∈ ℝ ∀𝑚 ∈ 𝑍 𝑧 ≤ ((𝐹‘𝑚)‘𝑥)))
44		nfcv 2892	. . . . . . . . . . 11 ⊢ Ⅎ𝑤∩ 𝑛 ∈ 𝑍 dom (𝐹‘𝑛)
45		nfcv 2892	. . . . . . . . . . . 12 ⊢ Ⅎ𝑥𝑍
46		nfcv 2892	. . . . . . . . . . . . 13 ⊢ Ⅎ𝑥(𝐹‘𝑚)
47	46	nfdm 5949	. . . . . . . . . . . 12 ⊢ Ⅎ𝑥dom (𝐹‘𝑚)
48	45, 47	nfiin 5026	. . . . . . . . . . 11 ⊢ Ⅎ𝑥∩ 𝑚 ∈ 𝑍 dom (𝐹‘𝑚)
49		nfv 1910	. . . . . . . . . . 11 ⊢ Ⅎ𝑤∃𝑦 ∈ ℝ ∀𝑛 ∈ 𝑍 𝑦 ≤ ((𝐹‘𝑛)‘𝑥)
50		nfv 1910	. . . . . . . . . . 11 ⊢ Ⅎ𝑥∃𝑧 ∈ ℝ ∀𝑚 ∈ 𝑍 𝑧 ≤ ((𝐹‘𝑚)‘𝑤)
51		nfcv 2892	. . . . . . . . . . . . 13 ⊢ Ⅎ𝑚dom (𝐹‘𝑛)
52		nfcv 2892	. . . . . . . . . . . . . 14 ⊢ Ⅎ𝑛(𝐹‘𝑚)
53	52	nfdm 5949	. . . . . . . . . . . . 13 ⊢ Ⅎ𝑛dom (𝐹‘𝑚)
54		fveq2 6892	. . . . . . . . . . . . . 14 ⊢ (𝑛 = 𝑚 → (𝐹‘𝑛) = (𝐹‘𝑚))
55	54	dmeqd 5904	. . . . . . . . . . . . 13 ⊢ (𝑛 = 𝑚 → dom (𝐹‘𝑛) = dom (𝐹‘𝑚))
56	51, 53, 55	cbviin 5039	. . . . . . . . . . . 12 ⊢ ∩ 𝑛 ∈ 𝑍 dom (𝐹‘𝑛) = ∩ 𝑚 ∈ 𝑍 dom (𝐹‘𝑚)
57	56	a1i 11	. . . . . . . . . . 11 ⊢ (𝑥 = 𝑤 → ∩ 𝑛 ∈ 𝑍 dom (𝐹‘𝑛) = ∩ 𝑚 ∈ 𝑍 dom (𝐹‘𝑚))
58		fveq2 6892	. . . . . . . . . . . . . . . 16 ⊢ (𝑥 = 𝑤 → ((𝐹‘𝑛)‘𝑥) = ((𝐹‘𝑛)‘𝑤))
59	58	breq2d 5157	. . . . . . . . . . . . . . 15 ⊢ (𝑥 = 𝑤 → (𝑦 ≤ ((𝐹‘𝑛)‘𝑥) ↔ 𝑦 ≤ ((𝐹‘𝑛)‘𝑤)))
60	59	ralbidv 3168	. . . . . . . . . . . . . 14 ⊢ (𝑥 = 𝑤 → (∀𝑛 ∈ 𝑍 𝑦 ≤ ((𝐹‘𝑛)‘𝑥) ↔ ∀𝑛 ∈ 𝑍 𝑦 ≤ ((𝐹‘𝑛)‘𝑤)))
61		nfv 1910	. . . . . . . . . . . . . . . 16 ⊢ Ⅎ𝑚 𝑦 ≤ ((𝐹‘𝑛)‘𝑤)
62		nfcv 2892	. . . . . . . . . . . . . . . . 17 ⊢ Ⅎ𝑛𝑦
63		nfcv 2892	. . . . . . . . . . . . . . . . 17 ⊢ Ⅎ𝑛 ≤
64		nfcv 2892	. . . . . . . . . . . . . . . . . 18 ⊢ Ⅎ𝑛𝑤
65	52, 64	nffv 6902	. . . . . . . . . . . . . . . . 17 ⊢ Ⅎ𝑛((𝐹‘𝑚)‘𝑤)
66	62, 63, 65	nfbr 5192	. . . . . . . . . . . . . . . 16 ⊢ Ⅎ𝑛 𝑦 ≤ ((𝐹‘𝑚)‘𝑤)
67	54	fveq1d 6894	. . . . . . . . . . . . . . . . 17 ⊢ (𝑛 = 𝑚 → ((𝐹‘𝑛)‘𝑤) = ((𝐹‘𝑚)‘𝑤))
68	67	breq2d 5157	. . . . . . . . . . . . . . . 16 ⊢ (𝑛 = 𝑚 → (𝑦 ≤ ((𝐹‘𝑛)‘𝑤) ↔ 𝑦 ≤ ((𝐹‘𝑚)‘𝑤)))
69	61, 66, 68	cbvralw 3294	. . . . . . . . . . . . . . 15 ⊢ (∀𝑛 ∈ 𝑍 𝑦 ≤ ((𝐹‘𝑛)‘𝑤) ↔ ∀𝑚 ∈ 𝑍 𝑦 ≤ ((𝐹‘𝑚)‘𝑤))
70	69	a1i 11	. . . . . . . . . . . . . 14 ⊢ (𝑥 = 𝑤 → (∀𝑛 ∈ 𝑍 𝑦 ≤ ((𝐹‘𝑛)‘𝑤) ↔ ∀𝑚 ∈ 𝑍 𝑦 ≤ ((𝐹‘𝑚)‘𝑤)))
71	60, 70	bitrd 278	. . . . . . . . . . . . 13 ⊢ (𝑥 = 𝑤 → (∀𝑛 ∈ 𝑍 𝑦 ≤ ((𝐹‘𝑛)‘𝑥) ↔ ∀𝑚 ∈ 𝑍 𝑦 ≤ ((𝐹‘𝑚)‘𝑤)))
72	71	rexbidv 3169	. . . . . . . . . . . 12 ⊢ (𝑥 = 𝑤 → (∃𝑦 ∈ ℝ ∀𝑛 ∈ 𝑍 𝑦 ≤ ((𝐹‘𝑛)‘𝑥) ↔ ∃𝑦 ∈ ℝ ∀𝑚 ∈ 𝑍 𝑦 ≤ ((𝐹‘𝑚)‘𝑤)))
73		breq1 5148	. . . . . . . . . . . . . . 15 ⊢ (𝑦 = 𝑧 → (𝑦 ≤ ((𝐹‘𝑚)‘𝑤) ↔ 𝑧 ≤ ((𝐹‘𝑚)‘𝑤)))
74	73	ralbidv 3168	. . . . . . . . . . . . . 14 ⊢ (𝑦 = 𝑧 → (∀𝑚 ∈ 𝑍 𝑦 ≤ ((𝐹‘𝑚)‘𝑤) ↔ ∀𝑚 ∈ 𝑍 𝑧 ≤ ((𝐹‘𝑚)‘𝑤)))
75	74	cbvrexvw 3226	. . . . . . . . . . . . 13 ⊢ (∃𝑦 ∈ ℝ ∀𝑚 ∈ 𝑍 𝑦 ≤ ((𝐹‘𝑚)‘𝑤) ↔ ∃𝑧 ∈ ℝ ∀𝑚 ∈ 𝑍 𝑧 ≤ ((𝐹‘𝑚)‘𝑤))
76	75	a1i 11	. . . . . . . . . . . 12 ⊢ (𝑥 = 𝑤 → (∃𝑦 ∈ ℝ ∀𝑚 ∈ 𝑍 𝑦 ≤ ((𝐹‘𝑚)‘𝑤) ↔ ∃𝑧 ∈ ℝ ∀𝑚 ∈ 𝑍 𝑧 ≤ ((𝐹‘𝑚)‘𝑤)))
77	72, 76	bitrd 278	. . . . . . . . . . 11 ⊢ (𝑥 = 𝑤 → (∃𝑦 ∈ ℝ ∀𝑛 ∈ 𝑍 𝑦 ≤ ((𝐹‘𝑛)‘𝑥) ↔ ∃𝑧 ∈ ℝ ∀𝑚 ∈ 𝑍 𝑧 ≤ ((𝐹‘𝑚)‘𝑤)))
78	44, 48, 49, 50, 57, 77	cbvrabcsfw 3937	. . . . . . . . . 10 ⊢ {𝑥 ∈ ∩ 𝑛 ∈ 𝑍 dom (𝐹‘𝑛) ∣ ∃𝑦 ∈ ℝ ∀𝑛 ∈ 𝑍 𝑦 ≤ ((𝐹‘𝑛)‘𝑥)} = {𝑤 ∈ ∩ 𝑚 ∈ 𝑍 dom (𝐹‘𝑚) ∣ ∃𝑧 ∈ ℝ ∀𝑚 ∈ 𝑍 𝑧 ≤ ((𝐹‘𝑚)‘𝑤)}
79	16, 78	eqtri 2754	. . . . . . . . 9 ⊢ 𝐷 = {𝑤 ∈ ∩ 𝑚 ∈ 𝑍 dom (𝐹‘𝑚) ∣ ∃𝑧 ∈ ℝ ∀𝑚 ∈ 𝑍 𝑧 ≤ ((𝐹‘𝑚)‘𝑤)}
80	43, 79	elrab2 3685	. . . . . . . 8 ⊢ (𝑥 ∈ 𝐷 ↔ (𝑥 ∈ ∩ 𝑚 ∈ 𝑍 dom (𝐹‘𝑚) ∧ ∃𝑧 ∈ ℝ ∀𝑚 ∈ 𝑍 𝑧 ≤ ((𝐹‘𝑚)‘𝑥)))
81	80	biimpi 215	. . . . . . 7 ⊢ (𝑥 ∈ 𝐷 → (𝑥 ∈ ∩ 𝑚 ∈ 𝑍 dom (𝐹‘𝑚) ∧ ∃𝑧 ∈ ℝ ∀𝑚 ∈ 𝑍 𝑧 ≤ ((𝐹‘𝑚)‘𝑥)))
82	81	simprd 494	. . . . . 6 ⊢ (𝑥 ∈ 𝐷 → ∃𝑧 ∈ ℝ ∀𝑚 ∈ 𝑍 𝑧 ≤ ((𝐹‘𝑚)‘𝑥))
83	82	adantl 480	. . . . 5 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐷) → ∃𝑧 ∈ ℝ ∀𝑚 ∈ 𝑍 𝑧 ≤ ((𝐹‘𝑚)‘𝑥))
84		renegcl 11563	. . . . . . . 8 ⊢ (𝑧 ∈ ℝ → -𝑧 ∈ ℝ)
85	84	ad2antlr 725	. . . . . . 7 ⊢ ((((𝜑 ∧ 𝑥 ∈ 𝐷) ∧ 𝑧 ∈ ℝ) ∧ ∀𝑚 ∈ 𝑍 𝑧 ≤ ((𝐹‘𝑚)‘𝑥)) → -𝑧 ∈ ℝ)
86		fveq2 6892	. . . . . . . . . . . . . 14 ⊢ (𝑚 = 𝑛 → (𝐹‘𝑚) = (𝐹‘𝑛))
87	86	fveq1d 6894	. . . . . . . . . . . . 13 ⊢ (𝑚 = 𝑛 → ((𝐹‘𝑚)‘𝑥) = ((𝐹‘𝑛)‘𝑥))
88	87	breq2d 5157	. . . . . . . . . . . 12 ⊢ (𝑚 = 𝑛 → (𝑧 ≤ ((𝐹‘𝑚)‘𝑥) ↔ 𝑧 ≤ ((𝐹‘𝑛)‘𝑥)))
89	88	rspcva 3607	. . . . . . . . . . 11 ⊢ ((𝑛 ∈ 𝑍 ∧ ∀𝑚 ∈ 𝑍 𝑧 ≤ ((𝐹‘𝑚)‘𝑥)) → 𝑧 ≤ ((𝐹‘𝑛)‘𝑥))
90	89	ancoms 457	. . . . . . . . . 10 ⊢ ((∀𝑚 ∈ 𝑍 𝑧 ≤ ((𝐹‘𝑚)‘𝑥) ∧ 𝑛 ∈ 𝑍) → 𝑧 ≤ ((𝐹‘𝑛)‘𝑥))
91	90	adantll 712	. . . . . . . . 9 ⊢ (((((𝜑 ∧ 𝑥 ∈ 𝐷) ∧ 𝑧 ∈ ℝ) ∧ ∀𝑚 ∈ 𝑍 𝑧 ≤ ((𝐹‘𝑚)‘𝑥)) ∧ 𝑛 ∈ 𝑍) → 𝑧 ≤ ((𝐹‘𝑛)‘𝑥))
92		simpllr 774	. . . . . . . . . 10 ⊢ (((((𝜑 ∧ 𝑥 ∈ 𝐷) ∧ 𝑧 ∈ ℝ) ∧ ∀𝑚 ∈ 𝑍 𝑧 ≤ ((𝐹‘𝑚)‘𝑥)) ∧ 𝑛 ∈ 𝑍) → 𝑧 ∈ ℝ)
93	25	ad4ant14 750	. . . . . . . . . 10 ⊢ (((((𝜑 ∧ 𝑥 ∈ 𝐷) ∧ 𝑧 ∈ ℝ) ∧ ∀𝑚 ∈ 𝑍 𝑧 ≤ ((𝐹‘𝑚)‘𝑥)) ∧ 𝑛 ∈ 𝑍) → ((𝐹‘𝑛)‘𝑥) ∈ ℝ)
94	92, 93	lenegd 11833	. . . . . . . . 9 ⊢ (((((𝜑 ∧ 𝑥 ∈ 𝐷) ∧ 𝑧 ∈ ℝ) ∧ ∀𝑚 ∈ 𝑍 𝑧 ≤ ((𝐹‘𝑚)‘𝑥)) ∧ 𝑛 ∈ 𝑍) → (𝑧 ≤ ((𝐹‘𝑛)‘𝑥) ↔ -((𝐹‘𝑛)‘𝑥) ≤ -𝑧))
95	91, 94	mpbid 231	. . . . . . . 8 ⊢ (((((𝜑 ∧ 𝑥 ∈ 𝐷) ∧ 𝑧 ∈ ℝ) ∧ ∀𝑚 ∈ 𝑍 𝑧 ≤ ((𝐹‘𝑚)‘𝑥)) ∧ 𝑛 ∈ 𝑍) → -((𝐹‘𝑛)‘𝑥) ≤ -𝑧)
96	95	ralrimiva 3136	. . . . . . 7 ⊢ ((((𝜑 ∧ 𝑥 ∈ 𝐷) ∧ 𝑧 ∈ ℝ) ∧ ∀𝑚 ∈ 𝑍 𝑧 ≤ ((𝐹‘𝑚)‘𝑥)) → ∀𝑛 ∈ 𝑍 -((𝐹‘𝑛)‘𝑥) ≤ -𝑧)
97		brralrspcev 5205	. . . . . . 7 ⊢ ((-𝑧 ∈ ℝ ∧ ∀𝑛 ∈ 𝑍 -((𝐹‘𝑛)‘𝑥) ≤ -𝑧) → ∃𝑦 ∈ ℝ ∀𝑛 ∈ 𝑍 -((𝐹‘𝑛)‘𝑥) ≤ 𝑦)
98	85, 96, 97	syl2anc 582	. . . . . 6 ⊢ ((((𝜑 ∧ 𝑥 ∈ 𝐷) ∧ 𝑧 ∈ ℝ) ∧ ∀𝑚 ∈ 𝑍 𝑧 ≤ ((𝐹‘𝑚)‘𝑥)) → ∃𝑦 ∈ ℝ ∀𝑛 ∈ 𝑍 -((𝐹‘𝑛)‘𝑥) ≤ 𝑦)
99	98	rexlimdva2 3147	. . . . 5 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐷) → (∃𝑧 ∈ ℝ ∀𝑚 ∈ 𝑍 𝑧 ≤ ((𝐹‘𝑚)‘𝑥) → ∃𝑦 ∈ ℝ ∀𝑛 ∈ 𝑍 -((𝐹‘𝑛)‘𝑥) ≤ 𝑦))
100	83, 99	mpd 15	. . . 4 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐷) → ∃𝑦 ∈ ℝ ∀𝑛 ∈ 𝑍 -((𝐹‘𝑛)‘𝑥) ≤ 𝑦)
101	3, 7, 39, 100	suprclrnmpt 44895	. . 3 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐷) → sup(ran (𝑛 ∈ 𝑍 ↦ -((𝐹‘𝑛)‘𝑥)), ℝ, < ) ∈ ℝ)
102	16	a1i 11	. . . . . . 7 ⊢ (𝜑 → 𝐷 = {𝑥 ∈ ∩ 𝑛 ∈ 𝑍 dom (𝐹‘𝑛) ∣ ∃𝑦 ∈ ℝ ∀𝑛 ∈ 𝑍 𝑦 ≤ ((𝐹‘𝑛)‘𝑥)})
103		nfv 1910	. . . . . . . . . 10 ⊢ Ⅎ𝑦(𝜑 ∧ 𝑥 ∈ ∩ 𝑛 ∈ 𝑍 dom (𝐹‘𝑛))
104		nfv 1910	. . . . . . . . . 10 ⊢ Ⅎ𝑦∃𝑧 ∈ ℝ ∀𝑛 ∈ 𝑍 -((𝐹‘𝑛)‘𝑥) ≤ 𝑧
105		renegcl 11563	. . . . . . . . . . . . 13 ⊢ (𝑦 ∈ ℝ → -𝑦 ∈ ℝ)
106	105	3ad2ant2 1131	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ 𝑥 ∈ ∩ 𝑛 ∈ 𝑍 dom (𝐹‘𝑛)) ∧ 𝑦 ∈ ℝ ∧ ∀𝑛 ∈ 𝑍 𝑦 ≤ ((𝐹‘𝑛)‘𝑥)) → -𝑦 ∈ ℝ)
107		nfv 1910	. . . . . . . . . . . . . . 15 ⊢ Ⅎ𝑛𝜑
108		nfcv 2892	. . . . . . . . . . . . . . . 16 ⊢ Ⅎ𝑛𝑥
109		nfii1 5031	. . . . . . . . . . . . . . . 16 ⊢ Ⅎ𝑛∩ 𝑛 ∈ 𝑍 dom (𝐹‘𝑛)
110	108, 109	nfel 2907	. . . . . . . . . . . . . . 15 ⊢ Ⅎ𝑛 𝑥 ∈ ∩ 𝑛 ∈ 𝑍 dom (𝐹‘𝑛)
111	107, 110	nfan 1895	. . . . . . . . . . . . . 14 ⊢ Ⅎ𝑛(𝜑 ∧ 𝑥 ∈ ∩ 𝑛 ∈ 𝑍 dom (𝐹‘𝑛))
112	62	nfel1 2909	. . . . . . . . . . . . . 14 ⊢ Ⅎ𝑛 𝑦 ∈ ℝ
113		nfra1 3272	. . . . . . . . . . . . . 14 ⊢ Ⅎ𝑛∀𝑛 ∈ 𝑍 𝑦 ≤ ((𝐹‘𝑛)‘𝑥)
114	111, 112, 113	nf3an 1897	. . . . . . . . . . . . 13 ⊢ Ⅎ𝑛((𝜑 ∧ 𝑥 ∈ ∩ 𝑛 ∈ 𝑍 dom (𝐹‘𝑛)) ∧ 𝑦 ∈ ℝ ∧ ∀𝑛 ∈ 𝑍 𝑦 ≤ ((𝐹‘𝑛)‘𝑥))
115		simpl2 1189	. . . . . . . . . . . . . . 15 ⊢ ((((𝜑 ∧ 𝑥 ∈ ∩ 𝑛 ∈ 𝑍 dom (𝐹‘𝑛)) ∧ 𝑦 ∈ ℝ ∧ ∀𝑛 ∈ 𝑍 𝑦 ≤ ((𝐹‘𝑛)‘𝑥)) ∧ 𝑛 ∈ 𝑍) → 𝑦 ∈ ℝ)
116		simpll 765	. . . . . . . . . . . . . . . . 17 ⊢ (((𝜑 ∧ 𝑥 ∈ ∩ 𝑛 ∈ 𝑍 dom (𝐹‘𝑛)) ∧ 𝑛 ∈ 𝑍) → 𝜑)
117		simpr 483	. . . . . . . . . . . . . . . . 17 ⊢ (((𝜑 ∧ 𝑥 ∈ ∩ 𝑛 ∈ 𝑍 dom (𝐹‘𝑛)) ∧ 𝑛 ∈ 𝑍) → 𝑛 ∈ 𝑍)
118	22	adantll 712	. . . . . . . . . . . . . . . . 17 ⊢ (((𝜑 ∧ 𝑥 ∈ ∩ 𝑛 ∈ 𝑍 dom (𝐹‘𝑛)) ∧ 𝑛 ∈ 𝑍) → 𝑥 ∈ dom (𝐹‘𝑛))
119	13	3adant3 1129	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝜑 ∧ 𝑛 ∈ 𝑍 ∧ 𝑥 ∈ dom (𝐹‘𝑛)) → (𝐹‘𝑛):dom (𝐹‘𝑛)⟶ℝ)
120		simp3 1135	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝜑 ∧ 𝑛 ∈ 𝑍 ∧ 𝑥 ∈ dom (𝐹‘𝑛)) → 𝑥 ∈ dom (𝐹‘𝑛))
121	119, 120	ffvelcdmd 7090	. . . . . . . . . . . . . . . . 17 ⊢ ((𝜑 ∧ 𝑛 ∈ 𝑍 ∧ 𝑥 ∈ dom (𝐹‘𝑛)) → ((𝐹‘𝑛)‘𝑥) ∈ ℝ)
122	116, 117, 118, 121	syl3anc 1368	. . . . . . . . . . . . . . . 16 ⊢ (((𝜑 ∧ 𝑥 ∈ ∩ 𝑛 ∈ 𝑍 dom (𝐹‘𝑛)) ∧ 𝑛 ∈ 𝑍) → ((𝐹‘𝑛)‘𝑥) ∈ ℝ)
123	122	3ad2antl1 1182	. . . . . . . . . . . . . . 15 ⊢ ((((𝜑 ∧ 𝑥 ∈ ∩ 𝑛 ∈ 𝑍 dom (𝐹‘𝑛)) ∧ 𝑦 ∈ ℝ ∧ ∀𝑛 ∈ 𝑍 𝑦 ≤ ((𝐹‘𝑛)‘𝑥)) ∧ 𝑛 ∈ 𝑍) → ((𝐹‘𝑛)‘𝑥) ∈ ℝ)
124		rspa 3236	. . . . . . . . . . . . . . . 16 ⊢ ((∀𝑛 ∈ 𝑍 𝑦 ≤ ((𝐹‘𝑛)‘𝑥) ∧ 𝑛 ∈ 𝑍) → 𝑦 ≤ ((𝐹‘𝑛)‘𝑥))
125	124	3ad2antl3 1184	. . . . . . . . . . . . . . 15 ⊢ ((((𝜑 ∧ 𝑥 ∈ ∩ 𝑛 ∈ 𝑍 dom (𝐹‘𝑛)) ∧ 𝑦 ∈ ℝ ∧ ∀𝑛 ∈ 𝑍 𝑦 ≤ ((𝐹‘𝑛)‘𝑥)) ∧ 𝑛 ∈ 𝑍) → 𝑦 ≤ ((𝐹‘𝑛)‘𝑥))
126		leneg 11757	. . . . . . . . . . . . . . . 16 ⊢ ((𝑦 ∈ ℝ ∧ ((𝐹‘𝑛)‘𝑥) ∈ ℝ) → (𝑦 ≤ ((𝐹‘𝑛)‘𝑥) ↔ -((𝐹‘𝑛)‘𝑥) ≤ -𝑦))
127	126	biimp3a 1466	. . . . . . . . . . . . . . 15 ⊢ ((𝑦 ∈ ℝ ∧ ((𝐹‘𝑛)‘𝑥) ∈ ℝ ∧ 𝑦 ≤ ((𝐹‘𝑛)‘𝑥)) → -((𝐹‘𝑛)‘𝑥) ≤ -𝑦)
128	115, 123, 125, 127	syl3anc 1368	. . . . . . . . . . . . . 14 ⊢ ((((𝜑 ∧ 𝑥 ∈ ∩ 𝑛 ∈ 𝑍 dom (𝐹‘𝑛)) ∧ 𝑦 ∈ ℝ ∧ ∀𝑛 ∈ 𝑍 𝑦 ≤ ((𝐹‘𝑛)‘𝑥)) ∧ 𝑛 ∈ 𝑍) → -((𝐹‘𝑛)‘𝑥) ≤ -𝑦)
129	128	ex 411	. . . . . . . . . . . . 13 ⊢ (((𝜑 ∧ 𝑥 ∈ ∩ 𝑛 ∈ 𝑍 dom (𝐹‘𝑛)) ∧ 𝑦 ∈ ℝ ∧ ∀𝑛 ∈ 𝑍 𝑦 ≤ ((𝐹‘𝑛)‘𝑥)) → (𝑛 ∈ 𝑍 → -((𝐹‘𝑛)‘𝑥) ≤ -𝑦))
130	114, 129	ralrimi 3245	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ 𝑥 ∈ ∩ 𝑛 ∈ 𝑍 dom (𝐹‘𝑛)) ∧ 𝑦 ∈ ℝ ∧ ∀𝑛 ∈ 𝑍 𝑦 ≤ ((𝐹‘𝑛)‘𝑥)) → ∀𝑛 ∈ 𝑍 -((𝐹‘𝑛)‘𝑥) ≤ -𝑦)
131		brralrspcev 5205	. . . . . . . . . . . 12 ⊢ ((-𝑦 ∈ ℝ ∧ ∀𝑛 ∈ 𝑍 -((𝐹‘𝑛)‘𝑥) ≤ -𝑦) → ∃𝑧 ∈ ℝ ∀𝑛 ∈ 𝑍 -((𝐹‘𝑛)‘𝑥) ≤ 𝑧)
132	106, 130, 131	syl2anc 582	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ 𝑥 ∈ ∩ 𝑛 ∈ 𝑍 dom (𝐹‘𝑛)) ∧ 𝑦 ∈ ℝ ∧ ∀𝑛 ∈ 𝑍 𝑦 ≤ ((𝐹‘𝑛)‘𝑥)) → ∃𝑧 ∈ ℝ ∀𝑛 ∈ 𝑍 -((𝐹‘𝑛)‘𝑥) ≤ 𝑧)
133	132	3exp 1116	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑥 ∈ ∩ 𝑛 ∈ 𝑍 dom (𝐹‘𝑛)) → (𝑦 ∈ ℝ → (∀𝑛 ∈ 𝑍 𝑦 ≤ ((𝐹‘𝑛)‘𝑥) → ∃𝑧 ∈ ℝ ∀𝑛 ∈ 𝑍 -((𝐹‘𝑛)‘𝑥) ≤ 𝑧)))
134	103, 104, 133	rexlimd 3254	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑥 ∈ ∩ 𝑛 ∈ 𝑍 dom (𝐹‘𝑛)) → (∃𝑦 ∈ ℝ ∀𝑛 ∈ 𝑍 𝑦 ≤ ((𝐹‘𝑛)‘𝑥) → ∃𝑧 ∈ ℝ ∀𝑛 ∈ 𝑍 -((𝐹‘𝑛)‘𝑥) ≤ 𝑧))
135	84	3ad2ant2 1131	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ 𝑥 ∈ ∩ 𝑛 ∈ 𝑍 dom (𝐹‘𝑛)) ∧ 𝑧 ∈ ℝ ∧ ∀𝑛 ∈ 𝑍 -((𝐹‘𝑛)‘𝑥) ≤ 𝑧) → -𝑧 ∈ ℝ)
136		nfv 1910	. . . . . . . . . . . . . 14 ⊢ Ⅎ𝑛 𝑧 ∈ ℝ
137		nfra1 3272	. . . . . . . . . . . . . 14 ⊢ Ⅎ𝑛∀𝑛 ∈ 𝑍 -((𝐹‘𝑛)‘𝑥) ≤ 𝑧
138	111, 136, 137	nf3an 1897	. . . . . . . . . . . . 13 ⊢ Ⅎ𝑛((𝜑 ∧ 𝑥 ∈ ∩ 𝑛 ∈ 𝑍 dom (𝐹‘𝑛)) ∧ 𝑧 ∈ ℝ ∧ ∀𝑛 ∈ 𝑍 -((𝐹‘𝑛)‘𝑥) ≤ 𝑧)
139	122	3ad2antl1 1182	. . . . . . . . . . . . . . 15 ⊢ ((((𝜑 ∧ 𝑥 ∈ ∩ 𝑛 ∈ 𝑍 dom (𝐹‘𝑛)) ∧ 𝑧 ∈ ℝ ∧ ∀𝑛 ∈ 𝑍 -((𝐹‘𝑛)‘𝑥) ≤ 𝑧) ∧ 𝑛 ∈ 𝑍) → ((𝐹‘𝑛)‘𝑥) ∈ ℝ)
140		simpl2 1189	. . . . . . . . . . . . . . 15 ⊢ ((((𝜑 ∧ 𝑥 ∈ ∩ 𝑛 ∈ 𝑍 dom (𝐹‘𝑛)) ∧ 𝑧 ∈ ℝ ∧ ∀𝑛 ∈ 𝑍 -((𝐹‘𝑛)‘𝑥) ≤ 𝑧) ∧ 𝑛 ∈ 𝑍) → 𝑧 ∈ ℝ)
141		rspa 3236	. . . . . . . . . . . . . . . 16 ⊢ ((∀𝑛 ∈ 𝑍 -((𝐹‘𝑛)‘𝑥) ≤ 𝑧 ∧ 𝑛 ∈ 𝑍) → -((𝐹‘𝑛)‘𝑥) ≤ 𝑧)
142	141	3ad2antl3 1184	. . . . . . . . . . . . . . 15 ⊢ ((((𝜑 ∧ 𝑥 ∈ ∩ 𝑛 ∈ 𝑍 dom (𝐹‘𝑛)) ∧ 𝑧 ∈ ℝ ∧ ∀𝑛 ∈ 𝑍 -((𝐹‘𝑛)‘𝑥) ≤ 𝑧) ∧ 𝑛 ∈ 𝑍) → -((𝐹‘𝑛)‘𝑥) ≤ 𝑧)
143		simp3 1135	. . . . . . . . . . . . . . . . 17 ⊢ ((((𝐹‘𝑛)‘𝑥) ∈ ℝ ∧ 𝑧 ∈ ℝ ∧ -((𝐹‘𝑛)‘𝑥) ≤ 𝑧) → -((𝐹‘𝑛)‘𝑥) ≤ 𝑧)
144		renegcl 11563	. . . . . . . . . . . . . . . . . . . 20 ⊢ (((𝐹‘𝑛)‘𝑥) ∈ ℝ → -((𝐹‘𝑛)‘𝑥) ∈ ℝ)
145	144	adantr 479	. . . . . . . . . . . . . . . . . . 19 ⊢ ((((𝐹‘𝑛)‘𝑥) ∈ ℝ ∧ 𝑧 ∈ ℝ) → -((𝐹‘𝑛)‘𝑥) ∈ ℝ)
146		simpr 483	. . . . . . . . . . . . . . . . . . 19 ⊢ ((((𝐹‘𝑛)‘𝑥) ∈ ℝ ∧ 𝑧 ∈ ℝ) → 𝑧 ∈ ℝ)
147		leneg 11757	. . . . . . . . . . . . . . . . . . 19 ⊢ ((-((𝐹‘𝑛)‘𝑥) ∈ ℝ ∧ 𝑧 ∈ ℝ) → (-((𝐹‘𝑛)‘𝑥) ≤ 𝑧 ↔ -𝑧 ≤ --((𝐹‘𝑛)‘𝑥)))
148	145, 146, 147	syl2anc 582	. . . . . . . . . . . . . . . . . 18 ⊢ ((((𝐹‘𝑛)‘𝑥) ∈ ℝ ∧ 𝑧 ∈ ℝ) → (-((𝐹‘𝑛)‘𝑥) ≤ 𝑧 ↔ -𝑧 ≤ --((𝐹‘𝑛)‘𝑥)))
149	148	3adant3 1129	. . . . . . . . . . . . . . . . 17 ⊢ ((((𝐹‘𝑛)‘𝑥) ∈ ℝ ∧ 𝑧 ∈ ℝ ∧ -((𝐹‘𝑛)‘𝑥) ≤ 𝑧) → (-((𝐹‘𝑛)‘𝑥) ≤ 𝑧 ↔ -𝑧 ≤ --((𝐹‘𝑛)‘𝑥)))
150	143, 149	mpbid 231	. . . . . . . . . . . . . . . 16 ⊢ ((((𝐹‘𝑛)‘𝑥) ∈ ℝ ∧ 𝑧 ∈ ℝ ∧ -((𝐹‘𝑛)‘𝑥) ≤ 𝑧) → -𝑧 ≤ --((𝐹‘𝑛)‘𝑥))
151		recn 11238	. . . . . . . . . . . . . . . . . 18 ⊢ (((𝐹‘𝑛)‘𝑥) ∈ ℝ → ((𝐹‘𝑛)‘𝑥) ∈ ℂ)
152	151	negnegd 11602	. . . . . . . . . . . . . . . . 17 ⊢ (((𝐹‘𝑛)‘𝑥) ∈ ℝ → --((𝐹‘𝑛)‘𝑥) = ((𝐹‘𝑛)‘𝑥))
153	152	3ad2ant1 1130	. . . . . . . . . . . . . . . 16 ⊢ ((((𝐹‘𝑛)‘𝑥) ∈ ℝ ∧ 𝑧 ∈ ℝ ∧ -((𝐹‘𝑛)‘𝑥) ≤ 𝑧) → --((𝐹‘𝑛)‘𝑥) = ((𝐹‘𝑛)‘𝑥))
154	150, 153	breqtrd 5171	. . . . . . . . . . . . . . 15 ⊢ ((((𝐹‘𝑛)‘𝑥) ∈ ℝ ∧ 𝑧 ∈ ℝ ∧ -((𝐹‘𝑛)‘𝑥) ≤ 𝑧) → -𝑧 ≤ ((𝐹‘𝑛)‘𝑥))
155	139, 140, 142, 154	syl3anc 1368	. . . . . . . . . . . . . 14 ⊢ ((((𝜑 ∧ 𝑥 ∈ ∩ 𝑛 ∈ 𝑍 dom (𝐹‘𝑛)) ∧ 𝑧 ∈ ℝ ∧ ∀𝑛 ∈ 𝑍 -((𝐹‘𝑛)‘𝑥) ≤ 𝑧) ∧ 𝑛 ∈ 𝑍) → -𝑧 ≤ ((𝐹‘𝑛)‘𝑥))
156	155	ex 411	. . . . . . . . . . . . 13 ⊢ (((𝜑 ∧ 𝑥 ∈ ∩ 𝑛 ∈ 𝑍 dom (𝐹‘𝑛)) ∧ 𝑧 ∈ ℝ ∧ ∀𝑛 ∈ 𝑍 -((𝐹‘𝑛)‘𝑥) ≤ 𝑧) → (𝑛 ∈ 𝑍 → -𝑧 ≤ ((𝐹‘𝑛)‘𝑥)))
157	138, 156	ralrimi 3245	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ 𝑥 ∈ ∩ 𝑛 ∈ 𝑍 dom (𝐹‘𝑛)) ∧ 𝑧 ∈ ℝ ∧ ∀𝑛 ∈ 𝑍 -((𝐹‘𝑛)‘𝑥) ≤ 𝑧) → ∀𝑛 ∈ 𝑍 -𝑧 ≤ ((𝐹‘𝑛)‘𝑥))
158		breq1 5148	. . . . . . . . . . . . . 14 ⊢ (𝑦 = -𝑧 → (𝑦 ≤ ((𝐹‘𝑛)‘𝑥) ↔ -𝑧 ≤ ((𝐹‘𝑛)‘𝑥)))
159	158	ralbidv 3168	. . . . . . . . . . . . 13 ⊢ (𝑦 = -𝑧 → (∀𝑛 ∈ 𝑍 𝑦 ≤ ((𝐹‘𝑛)‘𝑥) ↔ ∀𝑛 ∈ 𝑍 -𝑧 ≤ ((𝐹‘𝑛)‘𝑥)))
160	159	rspcev 3609	. . . . . . . . . . . 12 ⊢ ((-𝑧 ∈ ℝ ∧ ∀𝑛 ∈ 𝑍 -𝑧 ≤ ((𝐹‘𝑛)‘𝑥)) → ∃𝑦 ∈ ℝ ∀𝑛 ∈ 𝑍 𝑦 ≤ ((𝐹‘𝑛)‘𝑥))
161	135, 157, 160	syl2anc 582	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ 𝑥 ∈ ∩ 𝑛 ∈ 𝑍 dom (𝐹‘𝑛)) ∧ 𝑧 ∈ ℝ ∧ ∀𝑛 ∈ 𝑍 -((𝐹‘𝑛)‘𝑥) ≤ 𝑧) → ∃𝑦 ∈ ℝ ∀𝑛 ∈ 𝑍 𝑦 ≤ ((𝐹‘𝑛)‘𝑥))
162	161	3exp 1116	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑥 ∈ ∩ 𝑛 ∈ 𝑍 dom (𝐹‘𝑛)) → (𝑧 ∈ ℝ → (∀𝑛 ∈ 𝑍 -((𝐹‘𝑛)‘𝑥) ≤ 𝑧 → ∃𝑦 ∈ ℝ ∀𝑛 ∈ 𝑍 𝑦 ≤ ((𝐹‘𝑛)‘𝑥))))
163	162	rexlimdv 3143	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑥 ∈ ∩ 𝑛 ∈ 𝑍 dom (𝐹‘𝑛)) → (∃𝑧 ∈ ℝ ∀𝑛 ∈ 𝑍 -((𝐹‘𝑛)‘𝑥) ≤ 𝑧 → ∃𝑦 ∈ ℝ ∀𝑛 ∈ 𝑍 𝑦 ≤ ((𝐹‘𝑛)‘𝑥)))
164	134, 163	impbid 211	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑥 ∈ ∩ 𝑛 ∈ 𝑍 dom (𝐹‘𝑛)) → (∃𝑦 ∈ ℝ ∀𝑛 ∈ 𝑍 𝑦 ≤ ((𝐹‘𝑛)‘𝑥) ↔ ∃𝑧 ∈ ℝ ∀𝑛 ∈ 𝑍 -((𝐹‘𝑛)‘𝑥) ≤ 𝑧))
165	32, 164	rabbida 3447	. . . . . . 7 ⊢ (𝜑 → {𝑥 ∈ ∩ 𝑛 ∈ 𝑍 dom (𝐹‘𝑛) ∣ ∃𝑦 ∈ ℝ ∀𝑛 ∈ 𝑍 𝑦 ≤ ((𝐹‘𝑛)‘𝑥)} = {𝑥 ∈ ∩ 𝑛 ∈ 𝑍 dom (𝐹‘𝑛) ∣ ∃𝑧 ∈ ℝ ∀𝑛 ∈ 𝑍 -((𝐹‘𝑛)‘𝑥) ≤ 𝑧})
166	102, 165	eqtrd 2766	. . . . . 6 ⊢ (𝜑 → 𝐷 = {𝑥 ∈ ∩ 𝑛 ∈ 𝑍 dom (𝐹‘𝑛) ∣ ∃𝑧 ∈ ℝ ∀𝑛 ∈ 𝑍 -((𝐹‘𝑛)‘𝑥) ≤ 𝑧})
167	32, 166	alrimi 2202	. . . . 5 ⊢ (𝜑 → ∀𝑥 𝐷 = {𝑥 ∈ ∩ 𝑛 ∈ 𝑍 dom (𝐹‘𝑛) ∣ ∃𝑧 ∈ ℝ ∀𝑛 ∈ 𝑍 -((𝐹‘𝑛)‘𝑥) ≤ 𝑧})
168		eqid 2726	. . . . . . 7 ⊢ sup(ran (𝑛 ∈ 𝑍 ↦ -((𝐹‘𝑛)‘𝑥)), ℝ, < ) = sup(ran (𝑛 ∈ 𝑍 ↦ -((𝐹‘𝑛)‘𝑥)), ℝ, < )
169	168	rgenw 3055	. . . . . 6 ⊢ ∀𝑥 ∈ 𝐷 sup(ran (𝑛 ∈ 𝑍 ↦ -((𝐹‘𝑛)‘𝑥)), ℝ, < ) = sup(ran (𝑛 ∈ 𝑍 ↦ -((𝐹‘𝑛)‘𝑥)), ℝ, < )
170	169	a1i 11	. . . . 5 ⊢ (𝜑 → ∀𝑥 ∈ 𝐷 sup(ran (𝑛 ∈ 𝑍 ↦ -((𝐹‘𝑛)‘𝑥)), ℝ, < ) = sup(ran (𝑛 ∈ 𝑍 ↦ -((𝐹‘𝑛)‘𝑥)), ℝ, < ))
171		mpteq12f 5233	. . . . 5 ⊢ ((∀𝑥 𝐷 = {𝑥 ∈ ∩ 𝑛 ∈ 𝑍 dom (𝐹‘𝑛) ∣ ∃𝑧 ∈ ℝ ∀𝑛 ∈ 𝑍 -((𝐹‘𝑛)‘𝑥) ≤ 𝑧} ∧ ∀𝑥 ∈ 𝐷 sup(ran (𝑛 ∈ 𝑍 ↦ -((𝐹‘𝑛)‘𝑥)), ℝ, < ) = sup(ran (𝑛 ∈ 𝑍 ↦ -((𝐹‘𝑛)‘𝑥)), ℝ, < )) → (𝑥 ∈ 𝐷 ↦ sup(ran (𝑛 ∈ 𝑍 ↦ -((𝐹‘𝑛)‘𝑥)), ℝ, < )) = (𝑥 ∈ {𝑥 ∈ ∩ 𝑛 ∈ 𝑍 dom (𝐹‘𝑛) ∣ ∃𝑧 ∈ ℝ ∀𝑛 ∈ 𝑍 -((𝐹‘𝑛)‘𝑥) ≤ 𝑧} ↦ sup(ran (𝑛 ∈ 𝑍 ↦ -((𝐹‘𝑛)‘𝑥)), ℝ, < )))
172	167, 170, 171	syl2anc 582	. . . 4 ⊢ (𝜑 → (𝑥 ∈ 𝐷 ↦ sup(ran (𝑛 ∈ 𝑍 ↦ -((𝐹‘𝑛)‘𝑥)), ℝ, < )) = (𝑥 ∈ {𝑥 ∈ ∩ 𝑛 ∈ 𝑍 dom (𝐹‘𝑛) ∣ ∃𝑧 ∈ ℝ ∀𝑛 ∈ 𝑍 -((𝐹‘𝑛)‘𝑥) ≤ 𝑧} ↦ sup(ran (𝑛 ∈ 𝑍 ↦ -((𝐹‘𝑛)‘𝑥)), ℝ, < )))
173		nfv 1910	. . . . 5 ⊢ Ⅎ𝑧𝜑
174	121	renegcld 11681	. . . . 5 ⊢ ((𝜑 ∧ 𝑛 ∈ 𝑍 ∧ 𝑥 ∈ dom (𝐹‘𝑛)) → -((𝐹‘𝑛)‘𝑥) ∈ ℝ)
175		nfv 1910	. . . . . 6 ⊢ Ⅎ𝑥(𝜑 ∧ 𝑛 ∈ 𝑍)
176	34	a1i 11	. . . . . 6 ⊢ ((𝜑 ∧ 𝑛 ∈ 𝑍) → dom (𝐹‘𝑛) ∈ V)
177	121	3expa 1115	. . . . . 6 ⊢ (((𝜑 ∧ 𝑛 ∈ 𝑍) ∧ 𝑥 ∈ dom (𝐹‘𝑛)) → ((𝐹‘𝑛)‘𝑥) ∈ ℝ)
178	13	feqmptd 6962	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑛 ∈ 𝑍) → (𝐹‘𝑛) = (𝑥 ∈ dom (𝐹‘𝑛) ↦ ((𝐹‘𝑛)‘𝑥)))
179	178	eqcomd 2732	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑛 ∈ 𝑍) → (𝑥 ∈ dom (𝐹‘𝑛) ↦ ((𝐹‘𝑛)‘𝑥)) = (𝐹‘𝑛))
180	179, 11	eqeltrd 2826	. . . . . 6 ⊢ ((𝜑 ∧ 𝑛 ∈ 𝑍) → (𝑥 ∈ dom (𝐹‘𝑛) ↦ ((𝐹‘𝑛)‘𝑥)) ∈ (SMblFn‘𝑆))
181	175, 9, 176, 177, 180	smfneg 46459	. . . . 5 ⊢ ((𝜑 ∧ 𝑛 ∈ 𝑍) → (𝑥 ∈ dom (𝐹‘𝑛) ↦ -((𝐹‘𝑛)‘𝑥)) ∈ (SMblFn‘𝑆))
182		eqid 2726	. . . . 5 ⊢ {𝑥 ∈ ∩ 𝑛 ∈ 𝑍 dom (𝐹‘𝑛) ∣ ∃𝑧 ∈ ℝ ∀𝑛 ∈ 𝑍 -((𝐹‘𝑛)‘𝑥) ≤ 𝑧} = {𝑥 ∈ ∩ 𝑛 ∈ 𝑍 dom (𝐹‘𝑛) ∣ ∃𝑧 ∈ ℝ ∀𝑛 ∈ 𝑍 -((𝐹‘𝑛)‘𝑥) ≤ 𝑧}
183		eqid 2726	. . . . 5 ⊢ (𝑥 ∈ {𝑥 ∈ ∩ 𝑛 ∈ 𝑍 dom (𝐹‘𝑛) ∣ ∃𝑧 ∈ ℝ ∀𝑛 ∈ 𝑍 -((𝐹‘𝑛)‘𝑥) ≤ 𝑧} ↦ sup(ran (𝑛 ∈ 𝑍 ↦ -((𝐹‘𝑛)‘𝑥)), ℝ, < )) = (𝑥 ∈ {𝑥 ∈ ∩ 𝑛 ∈ 𝑍 dom (𝐹‘𝑛) ∣ ∃𝑧 ∈ ℝ ∀𝑛 ∈ 𝑍 -((𝐹‘𝑛)‘𝑥) ≤ 𝑧} ↦ sup(ran (𝑛 ∈ 𝑍 ↦ -((𝐹‘𝑛)‘𝑥)), ℝ, < ))
184	107, 32, 173, 4, 5, 8, 174, 181, 182, 183	smfsupmpt 46471	. . . 4 ⊢ (𝜑 → (𝑥 ∈ {𝑥 ∈ ∩ 𝑛 ∈ 𝑍 dom (𝐹‘𝑛) ∣ ∃𝑧 ∈ ℝ ∀𝑛 ∈ 𝑍 -((𝐹‘𝑛)‘𝑥) ≤ 𝑧} ↦ sup(ran (𝑛 ∈ 𝑍 ↦ -((𝐹‘𝑛)‘𝑥)), ℝ, < )) ∈ (SMblFn‘𝑆))
185	172, 184	eqeltrd 2826	. . 3 ⊢ (𝜑 → (𝑥 ∈ 𝐷 ↦ sup(ran (𝑛 ∈ 𝑍 ↦ -((𝐹‘𝑛)‘𝑥)), ℝ, < )) ∈ (SMblFn‘𝑆))
186	32, 8, 38, 101, 185	smfneg 46459	. 2 ⊢ (𝜑 → (𝑥 ∈ 𝐷 ↦ -sup(ran (𝑛 ∈ 𝑍 ↦ -((𝐹‘𝑛)‘𝑥)), ℝ, < )) ∈ (SMblFn‘𝑆))
187	31, 186	eqeltrd 2826	1 ⊢ (𝜑 → 𝐺 ∈ (SMblFn‘𝑆))

Syntax hints:   → wi 4   ↔ wb 205   ∧ wa 394   ∧ w3a 1084  ∀wal 1532   = wceq 1534   ∈ wcel 2099   ≠ wne 2930  ∀wral 3051  ∃wrex 3060  {crab 3420  Vcvv 3464  ∅c0 4324  ∩ ciin 4996   class class class wbr 5145   ↦ cmpt 5228  dom cdm 5674  ran crn 5675  ⟶wf 6541  ‘cfv 6545  supcsup 9475  infcinf 9476  ℝcr 11147   < clt 11288   ≤ cle 11289  -cneg 11485  ℤcz 12603  ℤ_≥cuz 12867  SAlgcsalg 45964  SMblFncsmblfn 46351

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1790  ax-4 1804  ax-5 1906  ax-6 1964  ax-7 2004  ax-8 2101  ax-9 2109  ax-10 2130  ax-11 2147  ax-12 2167  ax-ext 2697  ax-rep 5282  ax-sep 5296  ax-nul 5303  ax-pow 5361  ax-pr 5425  ax-un 7737  ax-inf2 9676  ax-cc 10468  ax-ac2 10496  ax-cnex 11204  ax-resscn 11205  ax-1cn 11206  ax-icn 11207  ax-addcl 11208  ax-addrcl 11209  ax-mulcl 11210  ax-mulrcl 11211  ax-mulcom 11212  ax-addass 11213  ax-mulass 11214  ax-distr 11215  ax-i2m1 11216  ax-1ne0 11217  ax-1rid 11218  ax-rnegex 11219  ax-rrecex 11220  ax-cnre 11221  ax-pre-lttri 11222  ax-pre-lttrn 11223  ax-pre-ltadd 11224  ax-pre-mulgt0 11225  ax-pre-sup 11226

This theorem depends on definitions:  df-bi 206  df-an 395  df-or 846  df-3or 1085  df-3an 1086  df-tru 1537  df-fal 1547  df-ex 1775  df-nf 1779  df-sb 2061  df-mo 2529  df-eu 2558  df-clab 2704  df-cleq 2718  df-clel 2803  df-nfc 2878  df-ne 2931  df-nel 3037  df-ral 3052  df-rex 3061  df-rmo 3365  df-reu 3366  df-rab 3421  df-v 3466  df-sbc 3778  df-csb 3894  df-dif 3951  df-un 3953  df-in 3955  df-ss 3965  df-pss 3968  df-nul 4325  df-if 4526  df-pw 4601  df-sn 4626  df-pr 4628  df-op 4632  df-uni 4908  df-int 4949  df-iun 4997  df-iin 4998  df-br 5146  df-opab 5208  df-mpt 5229  df-tr 5263  df-id 5572  df-eprel 5578  df-po 5586  df-so 5587  df-fr 5629  df-se 5630  df-we 5631  df-xp 5680  df-rel 5681  df-cnv 5682  df-co 5683  df-dm 5684  df-rn 5685  df-res 5686  df-ima 5687  df-pred 6304  df-ord 6370  df-on 6371  df-lim 6372  df-suc 6373  df-iota 6497  df-fun 6547  df-fn 6548  df-f 6549  df-f1 6550  df-fo 6551  df-f1o 6552  df-fv 6553  df-isom 6554  df-riota 7371  df-ov 7418  df-oprab 7419  df-mpo 7420  df-om 7868  df-1st 7994  df-2nd 7995  df-frecs 8287  df-wrecs 8318  df-recs 8392  df-rdg 8431  df-1o 8487  df-2o 8488  df-oadd 8491  df-omul 8492  df-er 8725  df-map 8848  df-pm 8849  df-en 8966  df-dom 8967  df-sdom 8968  df-fin 8969  df-sup 9477  df-inf 9478  df-oi 9545  df-card 9974  df-acn 9977  df-ac 10151  df-pnf 11290  df-mnf 11291  df-xr 11292  df-ltxr 11293  df-le 11294  df-sub 11486  df-neg 11487  df-div 11912  df-nn 12258  df-2 12320  df-3 12321  df-4 12322  df-n0 12518  df-z 12604  df-uz 12868  df-q 12978  df-rp 13022  df-ioo 13375  df-ioc 13376  df-ico 13377  df-icc 13378  df-fz 13532  df-fzo 13675  df-fl 13805  df-seq 14015  df-exp 14075  df-hash 14342  df-word 14517  df-concat 14573  df-s1 14598  df-s2 14851  df-s3 14852  df-s4 14853  df-cj 15098  df-re 15099  df-im 15100  df-sqrt 15234  df-abs 15235  df-rest 17431  df-topgen 17452  df-top 22883  df-bases 22936  df-salg 45965  df-salgen 45969  df-smblfn 46352

This theorem is referenced by:  smfinf  46474