Theorem smfsuplem1 47416

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
smfsuplem1.m	⊢ (𝜑 → 𝑀 ∈ ℤ)
smfsuplem1.z	⊢ 𝑍 = (ℤ≥‘𝑀)
smfsuplem1.s	⊢ (𝜑 → 𝑆 ∈ SAlg)
smfsuplem1.f	⊢ (𝜑 → 𝐹:𝑍⟶(SMblFn‘𝑆))
smfsuplem1.d	⊢ 𝐷 = {𝑥 ∈ ∩ 𝑛 ∈ 𝑍 dom (𝐹‘𝑛) ∣ ∃𝑦 ∈ ℝ ∀𝑛 ∈ 𝑍 ((𝐹‘𝑛)‘𝑥) ≤ 𝑦}
smfsuplem1.g	⊢ 𝐺 = (𝑥 ∈ 𝐷 ↦ sup(ran (𝑛 ∈ 𝑍 ↦ ((𝐹‘𝑛)‘𝑥)), ℝ, < ))
smfsuplem1.a	⊢ (𝜑 → 𝐴 ∈ ℝ)
smfsuplem1.h	⊢ (𝜑 → 𝐻:𝑍⟶𝑆)
smfsuplem1.i	⊢ ((𝜑 ∧ 𝑛 ∈ 𝑍) → (◡(𝐹‘𝑛) “ (-∞(,]𝐴)) = ((𝐻‘𝑛) ∩ dom (𝐹‘𝑛)))

Assertion

Ref	Expression
smfsuplem1	⊢ (𝜑 → (◡𝐺 “ (-∞(,]𝐴)) ∈ (𝑆 ↾t 𝐷))

Distinct variable groups:   𝐴,𝑛,𝑥   𝐷,𝑛,𝑥,𝑦   𝑥,𝐹,𝑦   𝑛,𝐺,𝑥   𝑛,𝐻,𝑥,𝑦   𝑛,𝑀   𝑆,𝑛   𝑛,𝑍,𝑥,𝑦   𝜑,𝑛,𝑥,𝑦

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

Proof of Theorem smfsuplem1

Step	Hyp	Ref	Expression
1		smfsuplem1.s	. . . . . . . . . . . . 13 ⊢ (𝜑 → 𝑆 ∈ SAlg)
2	1	adantr 485	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑛 ∈ 𝑍) → 𝑆 ∈ SAlg)
3		smfsuplem1.f	. . . . . . . . . . . . 13 ⊢ (𝜑 → 𝐹:𝑍⟶(SMblFn‘𝑆))
4	3	ffvelcdmda 7080	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑛 ∈ 𝑍) → (𝐹‘𝑛) ∈ (SMblFn‘𝑆))
5		eqid 2769	. . . . . . . . . . . 12 ⊢ dom (𝐹‘𝑛) = dom (𝐹‘𝑛)
6	2, 4, 5	smff 47337	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑛 ∈ 𝑍) → (𝐹‘𝑛):dom (𝐹‘𝑛)⟶ℝ)
7	6	ffnd 6707	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑛 ∈ 𝑍) → (𝐹‘𝑛) Fn dom (𝐹‘𝑛))
8	7	adantr 485	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑛 ∈ 𝑍) ∧ 𝑥 ∈ (◡𝐺 “ (-∞(,]𝐴))) → (𝐹‘𝑛) Fn dom (𝐹‘𝑛))
9		smfsuplem1.d	. . . . . . . . . . . . 13 ⊢ 𝐷 = {𝑥 ∈ ∩ 𝑛 ∈ 𝑍 dom (𝐹‘𝑛) ∣ ∃𝑦 ∈ ℝ ∀𝑛 ∈ 𝑍 ((𝐹‘𝑛)‘𝑥) ≤ 𝑦}
10		ssrab2 4042	. . . . . . . . . . . . 13 ⊢ {𝑥 ∈ ∩ 𝑛 ∈ 𝑍 dom (𝐹‘𝑛) ∣ ∃𝑦 ∈ ℝ ∀𝑛 ∈ 𝑍 ((𝐹‘𝑛)‘𝑥) ≤ 𝑦} ⊆ ∩ 𝑛 ∈ 𝑍 dom (𝐹‘𝑛)
11	9, 10	eqsstri 3991	. . . . . . . . . . . 12 ⊢ 𝐷 ⊆ ∩ 𝑛 ∈ 𝑍 dom (𝐹‘𝑛)
12		iinss2 5026	. . . . . . . . . . . 12 ⊢ (𝑛 ∈ 𝑍 → ∩ 𝑛 ∈ 𝑍 dom (𝐹‘𝑛) ⊆ dom (𝐹‘𝑛))
13	11, 12	sstrid 3956	. . . . . . . . . . 11 ⊢ (𝑛 ∈ 𝑍 → 𝐷 ⊆ dom (𝐹‘𝑛))
14	13	ad2antlr 739	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝑛 ∈ 𝑍) ∧ 𝑥 ∈ (◡𝐺 “ (-∞(,]𝐴))) → 𝐷 ⊆ dom (𝐹‘𝑛))
15		cnvimass 6085	. . . . . . . . . . . . 13 ⊢ (◡𝐺 “ (-∞(,]𝐴)) ⊆ dom 𝐺
16	15	sseli 3941	. . . . . . . . . . . 12 ⊢ (𝑥 ∈ (◡𝐺 “ (-∞(,]𝐴)) → 𝑥 ∈ dom 𝐺)
17	16	adantl 486	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ 𝑛 ∈ 𝑍) ∧ 𝑥 ∈ (◡𝐺 “ (-∞(,]𝐴))) → 𝑥 ∈ dom 𝐺)
18		nfv 1941	. . . . . . . . . . . . . . 15 ⊢ Ⅎ𝑛(𝜑 ∧ 𝑥 ∈ 𝐷)
19		smfsuplem1.m	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝜑 → 𝑀 ∈ ℤ)
20		uzid 12876	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝑀 ∈ ℤ → 𝑀 ∈ (ℤ≥‘𝑀))
21	19, 20	syl 18	. . . . . . . . . . . . . . . . . 18 ⊢ (𝜑 → 𝑀 ∈ (ℤ≥‘𝑀))
22		smfsuplem1.z	. . . . . . . . . . . . . . . . . 18 ⊢ 𝑍 = (ℤ≥‘𝑀)
23	21, 22	eleqtrrdi 2880	. . . . . . . . . . . . . . . . 17 ⊢ (𝜑 → 𝑀 ∈ 𝑍)
24	23	ne0d 4303	. . . . . . . . . . . . . . . 16 ⊢ (𝜑 → 𝑍 ≠ ∅)
25	24	adantr 485	. . . . . . . . . . . . . . 15 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐷) → 𝑍 ≠ ∅)
26	6	adantlr 727	. . . . . . . . . . . . . . . 16 ⊢ (((𝜑 ∧ 𝑥 ∈ 𝐷) ∧ 𝑛 ∈ 𝑍) → (𝐹‘𝑛):dom (𝐹‘𝑛)⟶ℝ)
27	12	adantl 486	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝑥 ∈ 𝐷 ∧ 𝑛 ∈ 𝑍) → ∩ 𝑛 ∈ 𝑍 dom (𝐹‘𝑛) ⊆ dom (𝐹‘𝑛))
28	11	sseli 3941	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝑥 ∈ 𝐷 → 𝑥 ∈ ∩ 𝑛 ∈ 𝑍 dom (𝐹‘𝑛))
29	28	adantr 485	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝑥 ∈ 𝐷 ∧ 𝑛 ∈ 𝑍) → 𝑥 ∈ ∩ 𝑛 ∈ 𝑍 dom (𝐹‘𝑛))
30	27, 29	sseldd 3946	. . . . . . . . . . . . . . . . 17 ⊢ ((𝑥 ∈ 𝐷 ∧ 𝑛 ∈ 𝑍) → 𝑥 ∈ dom (𝐹‘𝑛))
31	30	adantll 726	. . . . . . . . . . . . . . . 16 ⊢ (((𝜑 ∧ 𝑥 ∈ 𝐷) ∧ 𝑛 ∈ 𝑍) → 𝑥 ∈ dom (𝐹‘𝑛))
32	26, 31	ffvelcdmd 7081	. . . . . . . . . . . . . . 15 ⊢ (((𝜑 ∧ 𝑥 ∈ 𝐷) ∧ 𝑛 ∈ 𝑍) → ((𝐹‘𝑛)‘𝑥) ∈ ℝ)
33	9	reqabi 3446	. . . . . . . . . . . . . . . . 17 ⊢ (𝑥 ∈ 𝐷 ↔ (𝑥 ∈ ∩ 𝑛 ∈ 𝑍 dom (𝐹‘𝑛) ∧ ∃𝑦 ∈ ℝ ∀𝑛 ∈ 𝑍 ((𝐹‘𝑛)‘𝑥) ≤ 𝑦))
34	33	simprbi 502	. . . . . . . . . . . . . . . 16 ⊢ (𝑥 ∈ 𝐷 → ∃𝑦 ∈ ℝ ∀𝑛 ∈ 𝑍 ((𝐹‘𝑛)‘𝑥) ≤ 𝑦)
35	34	adantl 486	. . . . . . . . . . . . . . 15 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐷) → ∃𝑦 ∈ ℝ ∀𝑛 ∈ 𝑍 ((𝐹‘𝑛)‘𝑥) ≤ 𝑦)
36	18, 25, 32, 35	suprclrnmpt 45857	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐷) → sup(ran (𝑛 ∈ 𝑍 ↦ ((𝐹‘𝑛)‘𝑥)), ℝ, < ) ∈ ℝ)
37		smfsuplem1.g	. . . . . . . . . . . . . 14 ⊢ 𝐺 = (𝑥 ∈ 𝐷 ↦ sup(ran (𝑛 ∈ 𝑍 ↦ ((𝐹‘𝑛)‘𝑥)), ℝ, < ))
38	36, 37	fmptd 7110	. . . . . . . . . . . . 13 ⊢ (𝜑 → 𝐺:𝐷⟶ℝ)
39	38	fdmd 6717	. . . . . . . . . . . 12 ⊢ (𝜑 → dom 𝐺 = 𝐷)
40	39	ad2antrr 738	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ 𝑛 ∈ 𝑍) ∧ 𝑥 ∈ (◡𝐺 “ (-∞(,]𝐴))) → dom 𝐺 = 𝐷)
41	17, 40	eleqtrd 2871	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝑛 ∈ 𝑍) ∧ 𝑥 ∈ (◡𝐺 “ (-∞(,]𝐴))) → 𝑥 ∈ 𝐷)
42	14, 41	sseldd 3946	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑛 ∈ 𝑍) ∧ 𝑥 ∈ (◡𝐺 “ (-∞(,]𝐴))) → 𝑥 ∈ dom (𝐹‘𝑛))
43		mnfxr 11265	. . . . . . . . . . 11 ⊢ -∞ ∈ ℝ*
44	43	a1i 11	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝑛 ∈ 𝑍) ∧ 𝑥 ∈ (◡𝐺 “ (-∞(,]𝐴))) → -∞ ∈ ℝ*)
45		smfsuplem1.a	. . . . . . . . . . . 12 ⊢ (𝜑 → 𝐴 ∈ ℝ)
46	45	rexrd 11258	. . . . . . . . . . 11 ⊢ (𝜑 → 𝐴 ∈ ℝ*)
47	46	ad2antrr 738	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝑛 ∈ 𝑍) ∧ 𝑥 ∈ (◡𝐺 “ (-∞(,]𝐴))) → 𝐴 ∈ ℝ*)
48	32	an32s 664	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ 𝑛 ∈ 𝑍) ∧ 𝑥 ∈ 𝐷) → ((𝐹‘𝑛)‘𝑥) ∈ ℝ)
49	41, 48	syldan 602	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ 𝑛 ∈ 𝑍) ∧ 𝑥 ∈ (◡𝐺 “ (-∞(,]𝐴))) → ((𝐹‘𝑛)‘𝑥) ∈ ℝ)
50	49	rexrd 11258	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝑛 ∈ 𝑍) ∧ 𝑥 ∈ (◡𝐺 “ (-∞(,]𝐴))) → ((𝐹‘𝑛)‘𝑥) ∈ ℝ*)
51	49	mnfltd 13148	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝑛 ∈ 𝑍) ∧ 𝑥 ∈ (◡𝐺 “ (-∞(,]𝐴))) → -∞ < ((𝐹‘𝑛)‘𝑥))
52	16	adantl 486	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑥 ∈ (◡𝐺 “ (-∞(,]𝐴))) → 𝑥 ∈ dom 𝐺)
53	38	ffdmd 6737	. . . . . . . . . . . . . 14 ⊢ (𝜑 → 𝐺:dom 𝐺⟶ℝ)
54	53	ffvelcdmda 7080	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑥 ∈ dom 𝐺) → (𝐺‘𝑥) ∈ ℝ)
55	52, 54	syldan 602	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑥 ∈ (◡𝐺 “ (-∞(,]𝐴))) → (𝐺‘𝑥) ∈ ℝ)
56	55	adantlr 727	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ 𝑛 ∈ 𝑍) ∧ 𝑥 ∈ (◡𝐺 “ (-∞(,]𝐴))) → (𝐺‘𝑥) ∈ ℝ)
57	45	ad2antrr 738	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ 𝑛 ∈ 𝑍) ∧ 𝑥 ∈ (◡𝐺 “ (-∞(,]𝐴))) → 𝐴 ∈ ℝ)
58		an32 658	. . . . . . . . . . . . . . 15 ⊢ (((𝜑 ∧ 𝑛 ∈ 𝑍) ∧ 𝑥 ∈ 𝐷) ↔ ((𝜑 ∧ 𝑥 ∈ 𝐷) ∧ 𝑛 ∈ 𝑍))
59	58	biimpi 219	. . . . . . . . . . . . . 14 ⊢ (((𝜑 ∧ 𝑛 ∈ 𝑍) ∧ 𝑥 ∈ 𝐷) → ((𝜑 ∧ 𝑥 ∈ 𝐷) ∧ 𝑛 ∈ 𝑍))
60	18, 32, 35	suprubrnmpt 45859	. . . . . . . . . . . . . 14 ⊢ (((𝜑 ∧ 𝑥 ∈ 𝐷) ∧ 𝑛 ∈ 𝑍) → ((𝐹‘𝑛)‘𝑥) ≤ sup(ran (𝑛 ∈ 𝑍 ↦ ((𝐹‘𝑛)‘𝑥)), ℝ, < ))
61	59, 60	syl 18	. . . . . . . . . . . . 13 ⊢ (((𝜑 ∧ 𝑛 ∈ 𝑍) ∧ 𝑥 ∈ 𝐷) → ((𝐹‘𝑛)‘𝑥) ≤ sup(ran (𝑛 ∈ 𝑍 ↦ ((𝐹‘𝑛)‘𝑥)), ℝ, < ))
62	37	a1i 11	. . . . . . . . . . . . . . 15 ⊢ (𝜑 → 𝐺 = (𝑥 ∈ 𝐷 ↦ sup(ran (𝑛 ∈ 𝑍 ↦ ((𝐹‘𝑛)‘𝑥)), ℝ, < )))
63	62, 36	fvmpt2d 7004	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐷) → (𝐺‘𝑥) = sup(ran (𝑛 ∈ 𝑍 ↦ ((𝐹‘𝑛)‘𝑥)), ℝ, < ))
64	63	adantlr 727	. . . . . . . . . . . . 13 ⊢ (((𝜑 ∧ 𝑛 ∈ 𝑍) ∧ 𝑥 ∈ 𝐷) → (𝐺‘𝑥) = sup(ran (𝑛 ∈ 𝑍 ↦ ((𝐹‘𝑛)‘𝑥)), ℝ, < ))
65	61, 64	breqtrrd 5143	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ 𝑛 ∈ 𝑍) ∧ 𝑥 ∈ 𝐷) → ((𝐹‘𝑛)‘𝑥) ≤ (𝐺‘𝑥))
66	41, 65	syldan 602	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ 𝑛 ∈ 𝑍) ∧ 𝑥 ∈ (◡𝐺 “ (-∞(,]𝐴))) → ((𝐹‘𝑛)‘𝑥) ≤ (𝐺‘𝑥))
67	43	a1i 11	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑥 ∈ (◡𝐺 “ (-∞(,]𝐴))) → -∞ ∈ ℝ*)
68	46	adantr 485	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑥 ∈ (◡𝐺 “ (-∞(,]𝐴))) → 𝐴 ∈ ℝ*)
69		simpr 489	. . . . . . . . . . . . . . 15 ⊢ ((𝜑 ∧ 𝑥 ∈ (◡𝐺 “ (-∞(,]𝐴))) → 𝑥 ∈ (◡𝐺 “ (-∞(,]𝐴)))
70	38	ffnd 6707	. . . . . . . . . . . . . . . . 17 ⊢ (𝜑 → 𝐺 Fn 𝐷)
71		elpreima 7054	. . . . . . . . . . . . . . . . 17 ⊢ (𝐺 Fn 𝐷 → (𝑥 ∈ (◡𝐺 “ (-∞(,]𝐴)) ↔ (𝑥 ∈ 𝐷 ∧ (𝐺‘𝑥) ∈ (-∞(,]𝐴))))
72	70, 71	syl 18	. . . . . . . . . . . . . . . 16 ⊢ (𝜑 → (𝑥 ∈ (◡𝐺 “ (-∞(,]𝐴)) ↔ (𝑥 ∈ 𝐷 ∧ (𝐺‘𝑥) ∈ (-∞(,]𝐴))))
73	72	adantr 485	. . . . . . . . . . . . . . 15 ⊢ ((𝜑 ∧ 𝑥 ∈ (◡𝐺 “ (-∞(,]𝐴))) → (𝑥 ∈ (◡𝐺 “ (-∞(,]𝐴)) ↔ (𝑥 ∈ 𝐷 ∧ (𝐺‘𝑥) ∈ (-∞(,]𝐴))))
74	69, 73	mpbid 235	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ 𝑥 ∈ (◡𝐺 “ (-∞(,]𝐴))) → (𝑥 ∈ 𝐷 ∧ (𝐺‘𝑥) ∈ (-∞(,]𝐴)))
75	74	simprd 500	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑥 ∈ (◡𝐺 “ (-∞(,]𝐴))) → (𝐺‘𝑥) ∈ (-∞(,]𝐴))
76	67, 68, 75	iocleubd 46165	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑥 ∈ (◡𝐺 “ (-∞(,]𝐴))) → (𝐺‘𝑥) ≤ 𝐴)
77	76	adantlr 727	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ 𝑛 ∈ 𝑍) ∧ 𝑥 ∈ (◡𝐺 “ (-∞(,]𝐴))) → (𝐺‘𝑥) ≤ 𝐴)
78	49, 56, 57, 66, 77	letrd 11366	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝑛 ∈ 𝑍) ∧ 𝑥 ∈ (◡𝐺 “ (-∞(,]𝐴))) → ((𝐹‘𝑛)‘𝑥) ≤ 𝐴)
79	44, 47, 50, 51, 78	eliocd 46114	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑛 ∈ 𝑍) ∧ 𝑥 ∈ (◡𝐺 “ (-∞(,]𝐴))) → ((𝐹‘𝑛)‘𝑥) ∈ (-∞(,]𝐴))
80	8, 42, 79	elpreimad 7055	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝑛 ∈ 𝑍) ∧ 𝑥 ∈ (◡𝐺 “ (-∞(,]𝐴))) → 𝑥 ∈ (◡(𝐹‘𝑛) “ (-∞(,]𝐴)))
81	80	ssd 45691	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑛 ∈ 𝑍) → (◡𝐺 “ (-∞(,]𝐴)) ⊆ (◡(𝐹‘𝑛) “ (-∞(,]𝐴)))
82		smfsuplem1.i	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑛 ∈ 𝑍) → (◡(𝐹‘𝑛) “ (-∞(,]𝐴)) = ((𝐻‘𝑛) ∩ dom (𝐹‘𝑛)))
83		inss1 4197	. . . . . . . 8 ⊢ ((𝐻‘𝑛) ∩ dom (𝐹‘𝑛)) ⊆ (𝐻‘𝑛)
84	82, 83	eqsstrdi 3989	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑛 ∈ 𝑍) → (◡(𝐹‘𝑛) “ (-∞(,]𝐴)) ⊆ (𝐻‘𝑛))
85	81, 84	sstrd 3955	. . . . . 6 ⊢ ((𝜑 ∧ 𝑛 ∈ 𝑍) → (◡𝐺 “ (-∞(,]𝐴)) ⊆ (𝐻‘𝑛))
86	85	ralrimiva 3163	. . . . 5 ⊢ (𝜑 → ∀𝑛 ∈ 𝑍 (◡𝐺 “ (-∞(,]𝐴)) ⊆ (𝐻‘𝑛))
87		ssiin 5024	. . . . 5 ⊢ ((◡𝐺 “ (-∞(,]𝐴)) ⊆ ∩ 𝑛 ∈ 𝑍 (𝐻‘𝑛) ↔ ∀𝑛 ∈ 𝑍 (◡𝐺 “ (-∞(,]𝐴)) ⊆ (𝐻‘𝑛))
88	86, 87	sylibr 237	. . . 4 ⊢ (𝜑 → (◡𝐺 “ (-∞(,]𝐴)) ⊆ ∩ 𝑛 ∈ 𝑍 (𝐻‘𝑛))
89	15, 38	fssdm 6726	. . . 4 ⊢ (𝜑 → (◡𝐺 “ (-∞(,]𝐴)) ⊆ 𝐷)
90	88, 89	ssind 4201	. . 3 ⊢ (𝜑 → (◡𝐺 “ (-∞(,]𝐴)) ⊆ (∩ 𝑛 ∈ 𝑍 (𝐻‘𝑛) ∩ 𝐷))
91		iniin1 45734	. . . . 5 ⊢ (𝑍 ≠ ∅ → (∩ 𝑛 ∈ 𝑍 (𝐻‘𝑛) ∩ 𝐷) = ∩ 𝑛 ∈ 𝑍 ((𝐻‘𝑛) ∩ 𝐷))
92	24, 91	syl 18	. . . 4 ⊢ (𝜑 → (∩ 𝑛 ∈ 𝑍 (𝐻‘𝑛) ∩ 𝐷) = ∩ 𝑛 ∈ 𝑍 ((𝐻‘𝑛) ∩ 𝐷))
93	70	adantr 485	. . . . . 6 ⊢ ((𝜑 ∧ 𝑥 ∈ ∩ 𝑛 ∈ 𝑍 ((𝐻‘𝑛) ∩ 𝐷)) → 𝐺 Fn 𝐷)
94		simpr 489	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑥 ∈ ∩ 𝑛 ∈ 𝑍 ((𝐻‘𝑛) ∩ 𝐷)) → 𝑥 ∈ ∩ 𝑛 ∈ 𝑍 ((𝐻‘𝑛) ∩ 𝐷))
95	23	adantr 485	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑥 ∈ ∩ 𝑛 ∈ 𝑍 ((𝐻‘𝑛) ∩ 𝐷)) → 𝑀 ∈ 𝑍)
96		fveq2 6882	. . . . . . . . . 10 ⊢ (𝑛 = 𝑀 → (𝐻‘𝑛) = (𝐻‘𝑀))
97	96	ineq1d 4180	. . . . . . . . 9 ⊢ (𝑛 = 𝑀 → ((𝐻‘𝑛) ∩ 𝐷) = ((𝐻‘𝑀) ∩ 𝐷))
98	97	eleq2d 2855	. . . . . . . 8 ⊢ (𝑛 = 𝑀 → (𝑥 ∈ ((𝐻‘𝑛) ∩ 𝐷) ↔ 𝑥 ∈ ((𝐻‘𝑀) ∩ 𝐷)))
99	94, 95, 98	eliind 45682	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑥 ∈ ∩ 𝑛 ∈ 𝑍 ((𝐻‘𝑛) ∩ 𝐷)) → 𝑥 ∈ ((𝐻‘𝑀) ∩ 𝐷))
100		elinel2 4163	. . . . . . 7 ⊢ (𝑥 ∈ ((𝐻‘𝑀) ∩ 𝐷) → 𝑥 ∈ 𝐷)
101	99, 100	syl 18	. . . . . 6 ⊢ ((𝜑 ∧ 𝑥 ∈ ∩ 𝑛 ∈ 𝑍 ((𝐻‘𝑛) ∩ 𝐷)) → 𝑥 ∈ 𝐷)
102	43	a1i 11	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑥 ∈ ∩ 𝑛 ∈ 𝑍 ((𝐻‘𝑛) ∩ 𝐷)) → -∞ ∈ ℝ*)
103	46	adantr 485	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑥 ∈ ∩ 𝑛 ∈ 𝑍 ((𝐻‘𝑛) ∩ 𝐷)) → 𝐴 ∈ ℝ*)
104	63, 36	eqeltrd 2869	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐷) → (𝐺‘𝑥) ∈ ℝ)
105	104	rexrd 11258	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐷) → (𝐺‘𝑥) ∈ ℝ*)
106	101, 105	syldan 602	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑥 ∈ ∩ 𝑛 ∈ 𝑍 ((𝐻‘𝑛) ∩ 𝐷)) → (𝐺‘𝑥) ∈ ℝ*)
107	100	adantl 486	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑥 ∈ ((𝐻‘𝑀) ∩ 𝐷)) → 𝑥 ∈ 𝐷)
108	107, 104	syldan 602	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑥 ∈ ((𝐻‘𝑀) ∩ 𝐷)) → (𝐺‘𝑥) ∈ ℝ)
109	108	mnfltd 13148	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑥 ∈ ((𝐻‘𝑀) ∩ 𝐷)) → -∞ < (𝐺‘𝑥))
110	99, 109	syldan 602	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑥 ∈ ∩ 𝑛 ∈ 𝑍 ((𝐻‘𝑛) ∩ 𝐷)) → -∞ < (𝐺‘𝑥))
111	101, 63	syldan 602	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑥 ∈ ∩ 𝑛 ∈ 𝑍 ((𝐻‘𝑛) ∩ 𝐷)) → (𝐺‘𝑥) = sup(ran (𝑛 ∈ 𝑍 ↦ ((𝐹‘𝑛)‘𝑥)), ℝ, < ))
112		nfv 1941	. . . . . . . . . . 11 ⊢ Ⅎ𝑛𝜑
113		nfcv 2931	. . . . . . . . . . . 12 ⊢ Ⅎ𝑛𝑥
114		nfii1 4997	. . . . . . . . . . . 12 ⊢ Ⅎ𝑛∩ 𝑛 ∈ 𝑍 ((𝐻‘𝑛) ∩ 𝐷)
115	113, 114	nfel 2945	. . . . . . . . . . 11 ⊢ Ⅎ𝑛 𝑥 ∈ ∩ 𝑛 ∈ 𝑍 ((𝐻‘𝑛) ∩ 𝐷)
116	112, 115	nfan 1926	. . . . . . . . . 10 ⊢ Ⅎ𝑛(𝜑 ∧ 𝑥 ∈ ∩ 𝑛 ∈ 𝑍 ((𝐻‘𝑛) ∩ 𝐷))
117		simpll 778	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ 𝑥 ∈ ∩ 𝑛 ∈ 𝑍 ((𝐻‘𝑛) ∩ 𝐷)) ∧ 𝑛 ∈ 𝑍) → 𝜑)
118		simpr 489	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ 𝑥 ∈ ∩ 𝑛 ∈ 𝑍 ((𝐻‘𝑛) ∩ 𝐷)) ∧ 𝑛 ∈ 𝑍) → 𝑛 ∈ 𝑍)
119		eliinid 45720	. . . . . . . . . . . . 13 ⊢ ((𝑥 ∈ ∩ 𝑛 ∈ 𝑍 ((𝐻‘𝑛) ∩ 𝐷) ∧ 𝑛 ∈ 𝑍) → 𝑥 ∈ ((𝐻‘𝑛) ∩ 𝐷))
120	119	adantll 726	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ 𝑥 ∈ ∩ 𝑛 ∈ 𝑍 ((𝐻‘𝑛) ∩ 𝐷)) ∧ 𝑛 ∈ 𝑍) → 𝑥 ∈ ((𝐻‘𝑛) ∩ 𝐷))
121		elinel1 4162	. . . . . . . . . . . . . . . 16 ⊢ (𝑥 ∈ ((𝐻‘𝑛) ∩ 𝐷) → 𝑥 ∈ (𝐻‘𝑛))
122	121	3ad2ant3 1151	. . . . . . . . . . . . . . 15 ⊢ ((𝜑 ∧ 𝑛 ∈ 𝑍 ∧ 𝑥 ∈ ((𝐻‘𝑛) ∩ 𝐷)) → 𝑥 ∈ (𝐻‘𝑛))
123		elinel2 4163	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑥 ∈ ((𝐻‘𝑛) ∩ 𝐷) → 𝑥 ∈ 𝐷)
124	123	adantl 486	. . . . . . . . . . . . . . . . 17 ⊢ ((𝑛 ∈ 𝑍 ∧ 𝑥 ∈ ((𝐻‘𝑛) ∩ 𝐷)) → 𝑥 ∈ 𝐷)
125	30	ancoms 463	. . . . . . . . . . . . . . . . 17 ⊢ ((𝑛 ∈ 𝑍 ∧ 𝑥 ∈ 𝐷) → 𝑥 ∈ dom (𝐹‘𝑛))
126	124, 125	syldan 602	. . . . . . . . . . . . . . . 16 ⊢ ((𝑛 ∈ 𝑍 ∧ 𝑥 ∈ ((𝐻‘𝑛) ∩ 𝐷)) → 𝑥 ∈ dom (𝐹‘𝑛))
127	126	3adant1 1146	. . . . . . . . . . . . . . 15 ⊢ ((𝜑 ∧ 𝑛 ∈ 𝑍 ∧ 𝑥 ∈ ((𝐻‘𝑛) ∩ 𝐷)) → 𝑥 ∈ dom (𝐹‘𝑛))
128	122, 127	elind 4161	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ 𝑛 ∈ 𝑍 ∧ 𝑥 ∈ ((𝐻‘𝑛) ∩ 𝐷)) → 𝑥 ∈ ((𝐻‘𝑛) ∩ dom (𝐹‘𝑛)))
129	82	3adant3 1148	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ 𝑛 ∈ 𝑍 ∧ 𝑥 ∈ ((𝐻‘𝑛) ∩ 𝐷)) → (◡(𝐹‘𝑛) “ (-∞(,]𝐴)) = ((𝐻‘𝑛) ∩ dom (𝐹‘𝑛)))
130	128, 129	eleqtrrd 2872	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑛 ∈ 𝑍 ∧ 𝑥 ∈ ((𝐻‘𝑛) ∩ 𝐷)) → 𝑥 ∈ (◡(𝐹‘𝑛) “ (-∞(,]𝐴)))
131	43	a1i 11	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ 𝑛 ∈ 𝑍 ∧ 𝑥 ∈ (◡(𝐹‘𝑛) “ (-∞(,]𝐴))) → -∞ ∈ ℝ*)
132	46	3ad2ant1 1149	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ 𝑛 ∈ 𝑍 ∧ 𝑥 ∈ (◡(𝐹‘𝑛) “ (-∞(,]𝐴))) → 𝐴 ∈ ℝ*)
133		simp3 1154	. . . . . . . . . . . . . . . 16 ⊢ ((𝜑 ∧ 𝑛 ∈ 𝑍 ∧ 𝑥 ∈ (◡(𝐹‘𝑛) “ (-∞(,]𝐴))) → 𝑥 ∈ (◡(𝐹‘𝑛) “ (-∞(,]𝐴)))
134		elpreima 7054	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝐹‘𝑛) Fn dom (𝐹‘𝑛) → (𝑥 ∈ (◡(𝐹‘𝑛) “ (-∞(,]𝐴)) ↔ (𝑥 ∈ dom (𝐹‘𝑛) ∧ ((𝐹‘𝑛)‘𝑥) ∈ (-∞(,]𝐴))))
135	7, 134	syl 18	. . . . . . . . . . . . . . . . 17 ⊢ ((𝜑 ∧ 𝑛 ∈ 𝑍) → (𝑥 ∈ (◡(𝐹‘𝑛) “ (-∞(,]𝐴)) ↔ (𝑥 ∈ dom (𝐹‘𝑛) ∧ ((𝐹‘𝑛)‘𝑥) ∈ (-∞(,]𝐴))))
136	135	3adant3 1148	. . . . . . . . . . . . . . . 16 ⊢ ((𝜑 ∧ 𝑛 ∈ 𝑍 ∧ 𝑥 ∈ (◡(𝐹‘𝑛) “ (-∞(,]𝐴))) → (𝑥 ∈ (◡(𝐹‘𝑛) “ (-∞(,]𝐴)) ↔ (𝑥 ∈ dom (𝐹‘𝑛) ∧ ((𝐹‘𝑛)‘𝑥) ∈ (-∞(,]𝐴))))
137	133, 136	mpbid 235	. . . . . . . . . . . . . . 15 ⊢ ((𝜑 ∧ 𝑛 ∈ 𝑍 ∧ 𝑥 ∈ (◡(𝐹‘𝑛) “ (-∞(,]𝐴))) → (𝑥 ∈ dom (𝐹‘𝑛) ∧ ((𝐹‘𝑛)‘𝑥) ∈ (-∞(,]𝐴)))
138	137	simprd 500	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ 𝑛 ∈ 𝑍 ∧ 𝑥 ∈ (◡(𝐹‘𝑛) “ (-∞(,]𝐴))) → ((𝐹‘𝑛)‘𝑥) ∈ (-∞(,]𝐴))
139	131, 132, 138	iocleubd 46165	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑛 ∈ 𝑍 ∧ 𝑥 ∈ (◡(𝐹‘𝑛) “ (-∞(,]𝐴))) → ((𝐹‘𝑛)‘𝑥) ≤ 𝐴)
140	130, 139	syld3an3 1434	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑛 ∈ 𝑍 ∧ 𝑥 ∈ ((𝐻‘𝑛) ∩ 𝐷)) → ((𝐹‘𝑛)‘𝑥) ≤ 𝐴)
141	117, 118, 120, 140	syl3anc 1396	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ 𝑥 ∈ ∩ 𝑛 ∈ 𝑍 ((𝐻‘𝑛) ∩ 𝐷)) ∧ 𝑛 ∈ 𝑍) → ((𝐹‘𝑛)‘𝑥) ≤ 𝐴)
142	141	ex 417	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑥 ∈ ∩ 𝑛 ∈ 𝑍 ((𝐻‘𝑛) ∩ 𝐷)) → (𝑛 ∈ 𝑍 → ((𝐹‘𝑛)‘𝑥) ≤ 𝐴))
143	116, 142	ralrimi 3269	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑥 ∈ ∩ 𝑛 ∈ 𝑍 ((𝐻‘𝑛) ∩ 𝐷)) → ∀𝑛 ∈ 𝑍 ((𝐹‘𝑛)‘𝑥) ≤ 𝐴)
144	24	adantr 485	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑥 ∈ ∩ 𝑛 ∈ 𝑍 ((𝐻‘𝑛) ∩ 𝐷)) → 𝑍 ≠ ∅)
145	101, 32	syldanl 613	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝑥 ∈ ∩ 𝑛 ∈ 𝑍 ((𝐻‘𝑛) ∩ 𝐷)) ∧ 𝑛 ∈ 𝑍) → ((𝐹‘𝑛)‘𝑥) ∈ ℝ)
146	101, 34	syl 18	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑥 ∈ ∩ 𝑛 ∈ 𝑍 ((𝐻‘𝑛) ∩ 𝐷)) → ∃𝑦 ∈ ℝ ∀𝑛 ∈ 𝑍 ((𝐹‘𝑛)‘𝑥) ≤ 𝑦)
147	45	adantr 485	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑥 ∈ ∩ 𝑛 ∈ 𝑍 ((𝐻‘𝑛) ∩ 𝐷)) → 𝐴 ∈ ℝ)
148	116, 144, 145, 146, 147	suprleubrnmpt 46027	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑥 ∈ ∩ 𝑛 ∈ 𝑍 ((𝐻‘𝑛) ∩ 𝐷)) → (sup(ran (𝑛 ∈ 𝑍 ↦ ((𝐹‘𝑛)‘𝑥)), ℝ, < ) ≤ 𝐴 ↔ ∀𝑛 ∈ 𝑍 ((𝐹‘𝑛)‘𝑥) ≤ 𝐴))
149	143, 148	mpbird 260	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑥 ∈ ∩ 𝑛 ∈ 𝑍 ((𝐻‘𝑛) ∩ 𝐷)) → sup(ran (𝑛 ∈ 𝑍 ↦ ((𝐹‘𝑛)‘𝑥)), ℝ, < ) ≤ 𝐴)
150	111, 149	eqbrtrd 5137	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑥 ∈ ∩ 𝑛 ∈ 𝑍 ((𝐻‘𝑛) ∩ 𝐷)) → (𝐺‘𝑥) ≤ 𝐴)
151	102, 103, 106, 110, 150	eliocd 46114	. . . . . 6 ⊢ ((𝜑 ∧ 𝑥 ∈ ∩ 𝑛 ∈ 𝑍 ((𝐻‘𝑛) ∩ 𝐷)) → (𝐺‘𝑥) ∈ (-∞(,]𝐴))
152	93, 101, 151	elpreimad 7055	. . . . 5 ⊢ ((𝜑 ∧ 𝑥 ∈ ∩ 𝑛 ∈ 𝑍 ((𝐻‘𝑛) ∩ 𝐷)) → 𝑥 ∈ (◡𝐺 “ (-∞(,]𝐴)))
153	152	ssd 45691	. . . 4 ⊢ (𝜑 → ∩ 𝑛 ∈ 𝑍 ((𝐻‘𝑛) ∩ 𝐷) ⊆ (◡𝐺 “ (-∞(,]𝐴)))
154	92, 153	eqsstrd 3979	. . 3 ⊢ (𝜑 → (∩ 𝑛 ∈ 𝑍 (𝐻‘𝑛) ∩ 𝐷) ⊆ (◡𝐺 “ (-∞(,]𝐴)))
155	90, 154	eqssd 3962	. 2 ⊢ (𝜑 → (◡𝐺 “ (-∞(,]𝐴)) = (∩ 𝑛 ∈ 𝑍 (𝐻‘𝑛) ∩ 𝐷))
156		eqid 2769	. . . . 5 ⊢ {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}
157		fvex 6895	. . . . . . . . 9 ⊢ (𝐹‘𝑛) ∈ V
158	157	dmex 7905	. . . . . . . 8 ⊢ dom (𝐹‘𝑛) ∈ V
159	158	rgenw 3089	. . . . . . 7 ⊢ ∀𝑛 ∈ 𝑍 dom (𝐹‘𝑛) ∈ V
160	159	a1i 11	. . . . . 6 ⊢ (𝜑 → ∀𝑛 ∈ 𝑍 dom (𝐹‘𝑛) ∈ V)
161	24, 160	iinexd 45742	. . . . 5 ⊢ (𝜑 → ∩ 𝑛 ∈ 𝑍 dom (𝐹‘𝑛) ∈ V)
162	156, 161	rabexd 5311	. . . 4 ⊢ (𝜑 → {𝑥 ∈ ∩ 𝑛 ∈ 𝑍 dom (𝐹‘𝑛) ∣ ∃𝑦 ∈ ℝ ∀𝑛 ∈ 𝑍 ((𝐹‘𝑛)‘𝑥) ≤ 𝑦} ∈ V)
163	9, 162	eqeltrid 2873	. . 3 ⊢ (𝜑 → 𝐷 ∈ V)
164	22	uzct 45674	. . . . 5 ⊢ 𝑍 ≼ ω
165	164	a1i 11	. . . 4 ⊢ (𝜑 → 𝑍 ≼ ω)
166		smfsuplem1.h	. . . . 5 ⊢ (𝜑 → 𝐻:𝑍⟶𝑆)
167	166	ffvelcdmda 7080	. . . 4 ⊢ ((𝜑 ∧ 𝑛 ∈ 𝑍) → (𝐻‘𝑛) ∈ 𝑆)
168	1, 165, 24, 167	saliincl 46932	. . 3 ⊢ (𝜑 → ∩ 𝑛 ∈ 𝑍 (𝐻‘𝑛) ∈ 𝑆)
169		eqid 2769	. . 3 ⊢ (∩ 𝑛 ∈ 𝑍 (𝐻‘𝑛) ∩ 𝐷) = (∩ 𝑛 ∈ 𝑍 (𝐻‘𝑛) ∩ 𝐷)
170	1, 163, 168, 169	elrestd 45717	. 2 ⊢ (𝜑 → (∩ 𝑛 ∈ 𝑍 (𝐻‘𝑛) ∩ 𝐷) ∈ (𝑆 ↾t 𝐷))
171	155, 170	eqeltrd 2869	1 ⊢ (𝜑 → (◡𝐺 “ (-∞(,]𝐴)) ∈ (𝑆 ↾t 𝐷))

Syntax hints:   → wi 4   ↔ wb 209   ∧ wa 400   ∧ w3a 1101   = wceq 1567   ∈ wcel 2149   ≠ wne 2964  ∀wral 3085  ∃wrex 3095  {crab 3423  Vcvv 3463   ∩ cin 3912   ⊆ wss 3913  ∅c0 4294  ∩ ciin 4961   class class class wbr 5113   ↦ cmpt 5196  ^◡ccnv 5661  dom cdm 5662  ran crn 5663   “ cima 5665   Fn wfn 6532  ⟶wf 6533  ‘cfv 6537  (class class class)co 7411  ωcom 7861   ≼ cdom 8940  supcsup 9399  ℝcr 11098  -∞cmnf 11240  ℝ^*cxr 11241   < clt 11242   ≤ cle 11243  ℤcz 12590  ℤ_≥cuz 12861  (,]cioc 13372   ↾_t crest 17472  SAlgcsalg 46913  SMblFncsmblfn 47300

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1822  ax-4 1836  ax-5 1937  ax-6 1994  ax-7 2035  ax-8 2151  ax-9 2159  ax-10 2182  ax-11 2198  ax-12 2219  ax-ext 2741  ax-rep 5242  ax-sep 5261  ax-nul 5271  ax-pow 5337  ax-pr 5405  ax-un 7733  ax-inf2 9609  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 1570  df-fal 1580  df-ex 1807  df-nf 1811  df-sb 2098  df-mo 2573  df-eu 2603  df-clab 2748  df-cleq 2761  df-clel 2844  df-nfc 2918  df-ne 2965  df-nel 3071  df-ral 3086  df-rex 3096  df-rmo 3376  df-reu 3377  df-rab 3424  df-v 3465  df-sbc 3754  df-csb 3862  df-dif 3916  df-un 3918  df-in 3920  df-ss 3930  df-pss 3933  df-nul 4295  df-if 4493  df-pw 4569  df-sn 4595  df-pr 4597  df-op 4601  df-uni 4877  df-int 4917  df-iun 4962  df-iin 4963  df-br 5114  df-opab 5178  df-mpt 5197  df-tr 5223  df-id 5557  df-eprel 5562  df-po 5570  df-so 5571  df-fr 5615  df-se 5616  df-we 5617  df-xp 5668  df-rel 5669  df-cnv 5670  df-co 5671  df-dm 5672  df-rn 5673  df-res 5674  df-ima 5675  df-pred 6303  df-ord 6364  df-on 6365  df-lim 6366  df-suc 6367  df-iota 6493  df-fun 6539  df-fn 6540  df-f 6541  df-f1 6542  df-fo 6543  df-f1o 6544  df-fv 6545  df-isom 6546  df-riota 7368  df-ov 7414  df-oprab 7415  df-mpo 7416  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-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-oi 9471  df-card 9924  df-acn 9927  df-pnf 11244  df-mnf 11245  df-xr 11246  df-ltxr 11247  df-le 11248  df-sub 11442  df-neg 11443  df-nn 12233  df-n0 12504  df-z 12591  df-uz 12862  df-ioo 13375  df-ioc 13376  df-ico 13377  df-rest 17474  df-salg 46914  df-smblfn 47301

This theorem is referenced by:  smfsuplem2  47417