Theorem smfsuplem1 47545

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 7079	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑛 ∈ 𝑍) → (𝐹‘𝑛) ∈ (SMblFn‘𝑆))
5		eqid 2763	. . . . . . . . . . . 12 ⊢ dom (𝐹‘𝑛) = dom (𝐹‘𝑛)
6	2, 4, 5	smff 47466	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑛 ∈ 𝑍) → (𝐹‘𝑛):dom (𝐹‘𝑛)⟶ℝ)
7	6	ffnd 6706	. . . . . . . . . 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 4034	. . . . . . . . . . . . 13 ⊢ {𝑥 ∈ ∩ 𝑛 ∈ 𝑍 dom (𝐹‘𝑛) ∣ ∃𝑦 ∈ ℝ ∀𝑛 ∈ 𝑍 ((𝐹‘𝑛)‘𝑥) ≤ 𝑦} ⊆ ∩ 𝑛 ∈ 𝑍 dom (𝐹‘𝑛)
11	9, 10	eqsstri 3983	. . . . . . . . . . . 12 ⊢ 𝐷 ⊆ ∩ 𝑛 ∈ 𝑍 dom (𝐹‘𝑛)
12		iinss2 5022	. . . . . . . . . . . 12 ⊢ (𝑛 ∈ 𝑍 → ∩ 𝑛 ∈ 𝑍 dom (𝐹‘𝑛) ⊆ dom (𝐹‘𝑛))
13	11, 12	sstrid 3948	. . . . . . . . . . 11 ⊢ (𝑛 ∈ 𝑍 → 𝐷 ⊆ dom (𝐹‘𝑛))
14	13	ad2antlr 739	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝑛 ∈ 𝑍) ∧ 𝑥 ∈ (◡𝐺 “ (-∞(,]𝐴))) → 𝐷 ⊆ dom (𝐹‘𝑛))
15		cnvimass 6084	. . . . . . . . . . . . 13 ⊢ (◡𝐺 “ (-∞(,]𝐴)) ⊆ dom 𝐺
16	15	sseli 3933	. . . . . . . . . . . 12 ⊢ (𝑥 ∈ (◡𝐺 “ (-∞(,]𝐴)) → 𝑥 ∈ dom 𝐺)
17	16	adantl 486	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ 𝑛 ∈ 𝑍) ∧ 𝑥 ∈ (◡𝐺 “ (-∞(,]𝐴))) → 𝑥 ∈ dom 𝐺)
18		nfv 1944	. . . . . . . . . . . . . . 15 ⊢ Ⅎ𝑛(𝜑 ∧ 𝑥 ∈ 𝐷)
19		smfsuplem1.m	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝜑 → 𝑀 ∈ ℤ)
20		uzid 12872	. . . . . . . . . . . . . . . . . . 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 2874	. . . . . . . . . . . . . . . . 17 ⊢ (𝜑 → 𝑀 ∈ 𝑍)
24	23	ne0d 4295	. . . . . . . . . . . . . . . 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 3933	. . . . . . . . . . . . . . . . . . 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 3938	. . . . . . . . . . . . . . . . 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 7080	. . . . . . . . . . . . . . 15 ⊢ (((𝜑 ∧ 𝑥 ∈ 𝐷) ∧ 𝑛 ∈ 𝑍) → ((𝐹‘𝑛)‘𝑥) ∈ ℝ)
33	9	reqabi 3439	. . . . . . . . . . . . . . . . 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 45986	. . . . . . . . . . . . . 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 7109	. . . . . . . . . . . . 13 ⊢ (𝜑 → 𝐺:𝐷⟶ℝ)
39	38	fdmd 6716	. . . . . . . . . . . 12 ⊢ (𝜑 → dom 𝐺 = 𝐷)
40	39	ad2antrr 738	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ 𝑛 ∈ 𝑍) ∧ 𝑥 ∈ (◡𝐺 “ (-∞(,]𝐴))) → dom 𝐺 = 𝐷)
41	17, 40	eleqtrd 2865	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝑛 ∈ 𝑍) ∧ 𝑥 ∈ (◡𝐺 “ (-∞(,]𝐴))) → 𝑥 ∈ 𝐷)
42	14, 41	sseldd 3938	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑛 ∈ 𝑍) ∧ 𝑥 ∈ (◡𝐺 “ (-∞(,]𝐴))) → 𝑥 ∈ dom (𝐹‘𝑛))
43		mnfxr 11261	. . . . . . . . . . 11 ⊢ -∞ ∈ ℝ*
44	43	a1i 11	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝑛 ∈ 𝑍) ∧ 𝑥 ∈ (◡𝐺 “ (-∞(,]𝐴))) → -∞ ∈ ℝ*)
45		smfsuplem1.a	. . . . . . . . . . . 12 ⊢ (𝜑 → 𝐴 ∈ ℝ)
46	45	rexrd 11254	. . . . . . . . . . 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 11254	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝑛 ∈ 𝑍) ∧ 𝑥 ∈ (◡𝐺 “ (-∞(,]𝐴))) → ((𝐹‘𝑛)‘𝑥) ∈ ℝ*)
51	49	mnfltd 13144	. . . . . . . . . 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 6736	. . . . . . . . . . . . . 14 ⊢ (𝜑 → 𝐺:dom 𝐺⟶ℝ)
54	53	ffvelcdmda 7079	. . . . . . . . . . . . 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 45988	. . . . . . . . . . . . . 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 7003	. . . . . . . . . . . . . 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 5139	. . . . . . . . . . . 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 6706	. . . . . . . . . . . . . . . . 17 ⊢ (𝜑 → 𝐺 Fn 𝐷)
71		elpreima 7053	. . . . . . . . . . . . . . . . 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 46294	. . . . . . . . . . . 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 11362	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝑛 ∈ 𝑍) ∧ 𝑥 ∈ (◡𝐺 “ (-∞(,]𝐴))) → ((𝐹‘𝑛)‘𝑥) ≤ 𝐴)
79	44, 47, 50, 51, 78	eliocd 46243	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑛 ∈ 𝑍) ∧ 𝑥 ∈ (◡𝐺 “ (-∞(,]𝐴))) → ((𝐹‘𝑛)‘𝑥) ∈ (-∞(,]𝐴))
80	8, 42, 79	elpreimad 7054	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝑛 ∈ 𝑍) ∧ 𝑥 ∈ (◡𝐺 “ (-∞(,]𝐴))) → 𝑥 ∈ (◡(𝐹‘𝑛) “ (-∞(,]𝐴)))
81	80	ssd 45820	. . . . . . 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 4189	. . . . . . . 8 ⊢ ((𝐻‘𝑛) ∩ dom (𝐹‘𝑛)) ⊆ (𝐻‘𝑛)
84	82, 83	eqsstrdi 3981	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑛 ∈ 𝑍) → (◡(𝐹‘𝑛) “ (-∞(,]𝐴)) ⊆ (𝐻‘𝑛))
85	81, 84	sstrd 3947	. . . . . 6 ⊢ ((𝜑 ∧ 𝑛 ∈ 𝑍) → (◡𝐺 “ (-∞(,]𝐴)) ⊆ (𝐻‘𝑛))
86	85	ralrimiva 3157	. . . . 5 ⊢ (𝜑 → ∀𝑛 ∈ 𝑍 (◡𝐺 “ (-∞(,]𝐴)) ⊆ (𝐻‘𝑛))
87		ssiin 5020	. . . . 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 6725	. . . 4 ⊢ (𝜑 → (◡𝐺 “ (-∞(,]𝐴)) ⊆ 𝐷)
90	88, 89	ssind 4193	. . 3 ⊢ (𝜑 → (◡𝐺 “ (-∞(,]𝐴)) ⊆ (∩ 𝑛 ∈ 𝑍 (𝐻‘𝑛) ∩ 𝐷))
91		iniin1 45863	. . . . 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 6881	. . . . . . . . . 10 ⊢ (𝑛 = 𝑀 → (𝐻‘𝑛) = (𝐻‘𝑀))
97	96	ineq1d 4172	. . . . . . . . 9 ⊢ (𝑛 = 𝑀 → ((𝐻‘𝑛) ∩ 𝐷) = ((𝐻‘𝑀) ∩ 𝐷))
98	97	eleq2d 2849	. . . . . . . 8 ⊢ (𝑛 = 𝑀 → (𝑥 ∈ ((𝐻‘𝑛) ∩ 𝐷) ↔ 𝑥 ∈ ((𝐻‘𝑀) ∩ 𝐷)))
99	94, 95, 98	eliind 45811	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑥 ∈ ∩ 𝑛 ∈ 𝑍 ((𝐻‘𝑛) ∩ 𝐷)) → 𝑥 ∈ ((𝐻‘𝑀) ∩ 𝐷))
100		elinel2 4155	. . . . . . 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 2863	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐷) → (𝐺‘𝑥) ∈ ℝ)
105	104	rexrd 11254	. . . . . . . 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 13144	. . . . . . . 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 1944	. . . . . . . . . . 11 ⊢ Ⅎ𝑛𝜑
113		nfcv 2925	. . . . . . . . . . . 12 ⊢ Ⅎ𝑛𝑥
114		nfii1 4993	. . . . . . . . . . . 12 ⊢ Ⅎ𝑛∩ 𝑛 ∈ 𝑍 ((𝐻‘𝑛) ∩ 𝐷)
115	113, 114	nfel 2939	. . . . . . . . . . 11 ⊢ Ⅎ𝑛 𝑥 ∈ ∩ 𝑛 ∈ 𝑍 ((𝐻‘𝑛) ∩ 𝐷)
116	112, 115	nfan 1929	. . . . . . . . . 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 45849	. . . . . . . . . . . . 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 4154	. . . . . . . . . . . . . . . 16 ⊢ (𝑥 ∈ ((𝐻‘𝑛) ∩ 𝐷) → 𝑥 ∈ (𝐻‘𝑛))
122	121	3ad2ant3 1153	. . . . . . . . . . . . . . 15 ⊢ ((𝜑 ∧ 𝑛 ∈ 𝑍 ∧ 𝑥 ∈ ((𝐻‘𝑛) ∩ 𝐷)) → 𝑥 ∈ (𝐻‘𝑛))
123		elinel2 4155	. . . . . . . . . . . . . . . . . 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 1148	. . . . . . . . . . . . . . 15 ⊢ ((𝜑 ∧ 𝑛 ∈ 𝑍 ∧ 𝑥 ∈ ((𝐻‘𝑛) ∩ 𝐷)) → 𝑥 ∈ dom (𝐹‘𝑛))
128	122, 127	elind 4153	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ 𝑛 ∈ 𝑍 ∧ 𝑥 ∈ ((𝐻‘𝑛) ∩ 𝐷)) → 𝑥 ∈ ((𝐻‘𝑛) ∩ dom (𝐹‘𝑛)))
129	82	3adant3 1150	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ 𝑛 ∈ 𝑍 ∧ 𝑥 ∈ ((𝐻‘𝑛) ∩ 𝐷)) → (◡(𝐹‘𝑛) “ (-∞(,]𝐴)) = ((𝐻‘𝑛) ∩ dom (𝐹‘𝑛)))
130	128, 129	eleqtrrd 2866	. . . . . . . . . . . . 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 1151	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ 𝑛 ∈ 𝑍 ∧ 𝑥 ∈ (◡(𝐹‘𝑛) “ (-∞(,]𝐴))) → 𝐴 ∈ ℝ*)
133		simp3 1156	. . . . . . . . . . . . . . . 16 ⊢ ((𝜑 ∧ 𝑛 ∈ 𝑍 ∧ 𝑥 ∈ (◡(𝐹‘𝑛) “ (-∞(,]𝐴))) → 𝑥 ∈ (◡(𝐹‘𝑛) “ (-∞(,]𝐴)))
134		elpreima 7053	. . . . . . . . . . . . . . . . . 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 1150	. . . . . . . . . . . . . . . 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 46294	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑛 ∈ 𝑍 ∧ 𝑥 ∈ (◡(𝐹‘𝑛) “ (-∞(,]𝐴))) → ((𝐹‘𝑛)‘𝑥) ≤ 𝐴)
140	130, 139	syld3an3 1436	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑛 ∈ 𝑍 ∧ 𝑥 ∈ ((𝐻‘𝑛) ∩ 𝐷)) → ((𝐹‘𝑛)‘𝑥) ≤ 𝐴)
141	117, 118, 120, 140	syl3anc 1398	. . . . . . . . . . 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 3263	. . . . . . . . 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 46156	. . . . . . . . 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 5133	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑥 ∈ ∩ 𝑛 ∈ 𝑍 ((𝐻‘𝑛) ∩ 𝐷)) → (𝐺‘𝑥) ≤ 𝐴)
151	102, 103, 106, 110, 150	eliocd 46243	. . . . . 6 ⊢ ((𝜑 ∧ 𝑥 ∈ ∩ 𝑛 ∈ 𝑍 ((𝐻‘𝑛) ∩ 𝐷)) → (𝐺‘𝑥) ∈ (-∞(,]𝐴))
152	93, 101, 151	elpreimad 7054	. . . . 5 ⊢ ((𝜑 ∧ 𝑥 ∈ ∩ 𝑛 ∈ 𝑍 ((𝐻‘𝑛) ∩ 𝐷)) → 𝑥 ∈ (◡𝐺 “ (-∞(,]𝐴)))
153	152	ssd 45820	. . . 4 ⊢ (𝜑 → ∩ 𝑛 ∈ 𝑍 ((𝐻‘𝑛) ∩ 𝐷) ⊆ (◡𝐺 “ (-∞(,]𝐴)))
154	92, 153	eqsstrd 3971	. . 3 ⊢ (𝜑 → (∩ 𝑛 ∈ 𝑍 (𝐻‘𝑛) ∩ 𝐷) ⊆ (◡𝐺 “ (-∞(,]𝐴)))
155	90, 154	eqssd 3954	. 2 ⊢ (𝜑 → (◡𝐺 “ (-∞(,]𝐴)) = (∩ 𝑛 ∈ 𝑍 (𝐻‘𝑛) ∩ 𝐷))
156		eqid 2763	. . . . 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 6894	. . . . . . . . 9 ⊢ (𝐹‘𝑛) ∈ V
158	157	dmex 7902	. . . . . . . 8 ⊢ dom (𝐹‘𝑛) ∈ V
159	158	rgenw 3083	. . . . . . 7 ⊢ ∀𝑛 ∈ 𝑍 dom (𝐹‘𝑛) ∈ V
160	159	a1i 11	. . . . . 6 ⊢ (𝜑 → ∀𝑛 ∈ 𝑍 dom (𝐹‘𝑛) ∈ V)
161	24, 160	iinexd 45871	. . . . 5 ⊢ (𝜑 → ∩ 𝑛 ∈ 𝑍 dom (𝐹‘𝑛) ∈ V)
162	156, 161	rabexd 5310	. . . 4 ⊢ (𝜑 → {𝑥 ∈ ∩ 𝑛 ∈ 𝑍 dom (𝐹‘𝑛) ∣ ∃𝑦 ∈ ℝ ∀𝑛 ∈ 𝑍 ((𝐹‘𝑛)‘𝑥) ≤ 𝑦} ∈ V)
163	9, 162	eqeltrid 2867	. . 3 ⊢ (𝜑 → 𝐷 ∈ V)
164	22	uzct 45803	. . . . 5 ⊢ 𝑍 ≼ ω
165	164	a1i 11	. . . 4 ⊢ (𝜑 → 𝑍 ≼ ω)
166		smfsuplem1.h	. . . . 5 ⊢ (𝜑 → 𝐻:𝑍⟶𝑆)
167	166	ffvelcdmda 7079	. . . 4 ⊢ ((𝜑 ∧ 𝑛 ∈ 𝑍) → (𝐻‘𝑛) ∈ 𝑆)
168	1, 165, 24, 167	saliincl 47061	. . 3 ⊢ (𝜑 → ∩ 𝑛 ∈ 𝑍 (𝐻‘𝑛) ∈ 𝑆)
169		eqid 2763	. . 3 ⊢ (∩ 𝑛 ∈ 𝑍 (𝐻‘𝑛) ∩ 𝐷) = (∩ 𝑛 ∈ 𝑍 (𝐻‘𝑛) ∩ 𝐷)
170	1, 163, 168, 169	elrestd 45846	. 2 ⊢ (𝜑 → (∩ 𝑛 ∈ 𝑍 (𝐻‘𝑛) ∩ 𝐷) ∈ (𝑆 ↾t 𝐷))
171	155, 170	eqeltrd 2863	1 ⊢ (𝜑 → (◡𝐺 “ (-∞(,]𝐴)) ∈ (𝑆 ↾t 𝐷))

Syntax hints:   → wi 4   ↔ wb 209   ∧ wa 400   ∧ w3a 1103   = wceq 1570   ∈ wcel 2143   ≠ wne 2958  ∀wral 3079  ∃wrex 3089  {crab 3416  Vcvv 3455   ∩ cin 3904   ⊆ wss 3905  ∅c0 4286  ∩ ciin 4957   class class class wbr 5109   ↦ cmpt 5192  ^◡ccnv 5660  dom cdm 5661  ran crn 5662   “ cima 5664   Fn wfn 6531  ⟶wf 6532  ‘cfv 6536  (class class class)co 7410  ωcom 7858   ≼ cdom 8937  supcsup 9396  ℝcr 11094  -∞cmnf 11236  ℝ^*cxr 11237   < clt 11238   ≤ cle 11239  ℤcz 12586  ℤ_≥cuz 12857  (,]cioc 13368   ↾_t crest 17468  SAlgcsalg 47042  SMblFncsmblfn 47429

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1825  ax-4 1839  ax-5 1940  ax-6 1997  ax-7 2038  ax-8 2145  ax-9 2153  ax-10 2176  ax-11 2192  ax-12 2213  ax-ext 2735  ax-rep 5238  ax-sep 5257  ax-nul 5269  ax-pow 5336  ax-pr 5404  ax-un 7732  ax-inf2 9606  ax-cnex 11151  ax-resscn 11152  ax-1cn 11153  ax-icn 11154  ax-addcl 11155  ax-addrcl 11156  ax-mulcl 11157  ax-mulrcl 11158  ax-mulcom 11159  ax-addass 11160  ax-mulass 11161  ax-distr 11162  ax-i2m1 11163  ax-1ne0 11164  ax-1rid 11165  ax-rnegex 11166  ax-rrecex 11167  ax-cnre 11168  ax-pre-lttri 11169  ax-pre-lttrn 11170  ax-pre-ltadd 11171  ax-pre-mulgt0 11172  ax-pre-sup 11173

This theorem depends on definitions:  df-bi 210  df-an 401  df-or 861  df-3or 1104  df-3an 1105  df-tru 1573  df-fal 1583  df-ex 1810  df-nf 1814  df-sb 2097  df-mo 2567  df-eu 2597  df-clab 2742  df-cleq 2755  df-clel 2838  df-nfc 2912  df-ne 2959  df-nel 3065  df-ral 3080  df-rex 3090  df-rmo 3369  df-reu 3370  df-rab 3417  df-v 3457  df-sbc 3745  df-csb 3854  df-dif 3908  df-un 3910  df-in 3912  df-ss 3922  df-pss 3925  df-nul 4287  df-if 4488  df-pw 4564  df-sn 4590  df-pr 4592  df-op 4596  df-uni 4873  df-int 4913  df-iun 4958  df-iin 4959  df-br 5110  df-opab 5174  df-mpt 5193  df-tr 5219  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 7859  df-1st 7982  df-2nd 7983  df-frecs 8274  df-wrecs 8305  df-recs 8354  df-rdg 8393  df-1o 8449  df-oadd 8453  df-omul 8454  df-er 8690  df-map 8822  df-pm 8823  df-en 8940  df-dom 8941  df-sdom 8942  df-fin 8943  df-sup 9398  df-oi 9468  df-card 9921  df-acn 9924  df-pnf 11240  df-mnf 11241  df-xr 11242  df-ltxr 11243  df-le 11244  df-sub 11438  df-neg 11439  df-nn 12229  df-n0 12500  df-z 12587  df-uz 12858  df-ioo 13371  df-ioc 13372  df-ico 13373  df-rest 17470  df-salg 47043  df-smblfn 47430

This theorem is referenced by:  smfsuplem2  47546