Theorem smfsuplem1 46732

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 480	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑛 ∈ 𝑍) → 𝑆 ∈ SAlg)
3		smfsuplem1.f	. . . . . . . . . . . . 13 ⊢ (𝜑 → 𝐹:𝑍⟶(SMblFn‘𝑆))
4	3	ffvelcdmda 7118	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑛 ∈ 𝑍) → (𝐹‘𝑛) ∈ (SMblFn‘𝑆))
5		eqid 2740	. . . . . . . . . . . 12 ⊢ dom (𝐹‘𝑛) = dom (𝐹‘𝑛)
6	2, 4, 5	smff 46653	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑛 ∈ 𝑍) → (𝐹‘𝑛):dom (𝐹‘𝑛)⟶ℝ)
7	6	ffnd 6748	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑛 ∈ 𝑍) → (𝐹‘𝑛) Fn dom (𝐹‘𝑛))
8	7	adantr 480	. . . . . . . . 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 4103	. . . . . . . . . . . . 13 ⊢ {𝑥 ∈ ∩ 𝑛 ∈ 𝑍 dom (𝐹‘𝑛) ∣ ∃𝑦 ∈ ℝ ∀𝑛 ∈ 𝑍 ((𝐹‘𝑛)‘𝑥) ≤ 𝑦} ⊆ ∩ 𝑛 ∈ 𝑍 dom (𝐹‘𝑛)
11	9, 10	eqsstri 4043	. . . . . . . . . . . 12 ⊢ 𝐷 ⊆ ∩ 𝑛 ∈ 𝑍 dom (𝐹‘𝑛)
12		iinss2 5080	. . . . . . . . . . . 12 ⊢ (𝑛 ∈ 𝑍 → ∩ 𝑛 ∈ 𝑍 dom (𝐹‘𝑛) ⊆ dom (𝐹‘𝑛))
13	11, 12	sstrid 4020	. . . . . . . . . . 11 ⊢ (𝑛 ∈ 𝑍 → 𝐷 ⊆ dom (𝐹‘𝑛))
14	13	ad2antlr 726	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝑛 ∈ 𝑍) ∧ 𝑥 ∈ (◡𝐺 “ (-∞(,]𝐴))) → 𝐷 ⊆ dom (𝐹‘𝑛))
15		cnvimass 6111	. . . . . . . . . . . . 13 ⊢ (◡𝐺 “ (-∞(,]𝐴)) ⊆ dom 𝐺
16	15	sseli 4004	. . . . . . . . . . . 12 ⊢ (𝑥 ∈ (◡𝐺 “ (-∞(,]𝐴)) → 𝑥 ∈ dom 𝐺)
17	16	adantl 481	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ 𝑛 ∈ 𝑍) ∧ 𝑥 ∈ (◡𝐺 “ (-∞(,]𝐴))) → 𝑥 ∈ dom 𝐺)
18		nfv 1913	. . . . . . . . . . . . . . 15 ⊢ Ⅎ𝑛(𝜑 ∧ 𝑥 ∈ 𝐷)
19		smfsuplem1.m	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝜑 → 𝑀 ∈ ℤ)
20		uzid 12918	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝑀 ∈ ℤ → 𝑀 ∈ (ℤ≥‘𝑀))
21	19, 20	syl 17	. . . . . . . . . . . . . . . . . 18 ⊢ (𝜑 → 𝑀 ∈ (ℤ≥‘𝑀))
22		smfsuplem1.z	. . . . . . . . . . . . . . . . . 18 ⊢ 𝑍 = (ℤ≥‘𝑀)
23	21, 22	eleqtrrdi 2855	. . . . . . . . . . . . . . . . 17 ⊢ (𝜑 → 𝑀 ∈ 𝑍)
24	23	ne0d 4365	. . . . . . . . . . . . . . . 16 ⊢ (𝜑 → 𝑍 ≠ ∅)
25	24	adantr 480	. . . . . . . . . . . . . . 15 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐷) → 𝑍 ≠ ∅)
26	6	adantlr 714	. . . . . . . . . . . . . . . 16 ⊢ (((𝜑 ∧ 𝑥 ∈ 𝐷) ∧ 𝑛 ∈ 𝑍) → (𝐹‘𝑛):dom (𝐹‘𝑛)⟶ℝ)
27	12	adantl 481	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝑥 ∈ 𝐷 ∧ 𝑛 ∈ 𝑍) → ∩ 𝑛 ∈ 𝑍 dom (𝐹‘𝑛) ⊆ dom (𝐹‘𝑛))
28	11	sseli 4004	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝑥 ∈ 𝐷 → 𝑥 ∈ ∩ 𝑛 ∈ 𝑍 dom (𝐹‘𝑛))
29	28	adantr 480	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝑥 ∈ 𝐷 ∧ 𝑛 ∈ 𝑍) → 𝑥 ∈ ∩ 𝑛 ∈ 𝑍 dom (𝐹‘𝑛))
30	27, 29	sseldd 4009	. . . . . . . . . . . . . . . . 17 ⊢ ((𝑥 ∈ 𝐷 ∧ 𝑛 ∈ 𝑍) → 𝑥 ∈ dom (𝐹‘𝑛))
31	30	adantll 713	. . . . . . . . . . . . . . . 16 ⊢ (((𝜑 ∧ 𝑥 ∈ 𝐷) ∧ 𝑛 ∈ 𝑍) → 𝑥 ∈ dom (𝐹‘𝑛))
32	26, 31	ffvelcdmd 7119	. . . . . . . . . . . . . . 15 ⊢ (((𝜑 ∧ 𝑥 ∈ 𝐷) ∧ 𝑛 ∈ 𝑍) → ((𝐹‘𝑛)‘𝑥) ∈ ℝ)
33	9	reqabi 3467	. . . . . . . . . . . . . . . . 17 ⊢ (𝑥 ∈ 𝐷 ↔ (𝑥 ∈ ∩ 𝑛 ∈ 𝑍 dom (𝐹‘𝑛) ∧ ∃𝑦 ∈ ℝ ∀𝑛 ∈ 𝑍 ((𝐹‘𝑛)‘𝑥) ≤ 𝑦))
34	33	simprbi 496	. . . . . . . . . . . . . . . 16 ⊢ (𝑥 ∈ 𝐷 → ∃𝑦 ∈ ℝ ∀𝑛 ∈ 𝑍 ((𝐹‘𝑛)‘𝑥) ≤ 𝑦)
35	34	adantl 481	. . . . . . . . . . . . . . 15 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐷) → ∃𝑦 ∈ ℝ ∀𝑛 ∈ 𝑍 ((𝐹‘𝑛)‘𝑥) ≤ 𝑦)
36	18, 25, 32, 35	suprclrnmpt 45160	. . . . . . . . . . . . . 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 7148	. . . . . . . . . . . . 13 ⊢ (𝜑 → 𝐺:𝐷⟶ℝ)
39	38	fdmd 6757	. . . . . . . . . . . 12 ⊢ (𝜑 → dom 𝐺 = 𝐷)
40	39	ad2antrr 725	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ 𝑛 ∈ 𝑍) ∧ 𝑥 ∈ (◡𝐺 “ (-∞(,]𝐴))) → dom 𝐺 = 𝐷)
41	17, 40	eleqtrd 2846	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝑛 ∈ 𝑍) ∧ 𝑥 ∈ (◡𝐺 “ (-∞(,]𝐴))) → 𝑥 ∈ 𝐷)
42	14, 41	sseldd 4009	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑛 ∈ 𝑍) ∧ 𝑥 ∈ (◡𝐺 “ (-∞(,]𝐴))) → 𝑥 ∈ dom (𝐹‘𝑛))
43		mnfxr 11347	. . . . . . . . . . 11 ⊢ -∞ ∈ ℝ*
44	43	a1i 11	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝑛 ∈ 𝑍) ∧ 𝑥 ∈ (◡𝐺 “ (-∞(,]𝐴))) → -∞ ∈ ℝ*)
45		smfsuplem1.a	. . . . . . . . . . . 12 ⊢ (𝜑 → 𝐴 ∈ ℝ)
46	45	rexrd 11340	. . . . . . . . . . 11 ⊢ (𝜑 → 𝐴 ∈ ℝ*)
47	46	ad2antrr 725	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝑛 ∈ 𝑍) ∧ 𝑥 ∈ (◡𝐺 “ (-∞(,]𝐴))) → 𝐴 ∈ ℝ*)
48	32	an32s 651	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ 𝑛 ∈ 𝑍) ∧ 𝑥 ∈ 𝐷) → ((𝐹‘𝑛)‘𝑥) ∈ ℝ)
49	41, 48	syldan 590	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ 𝑛 ∈ 𝑍) ∧ 𝑥 ∈ (◡𝐺 “ (-∞(,]𝐴))) → ((𝐹‘𝑛)‘𝑥) ∈ ℝ)
50	49	rexrd 11340	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝑛 ∈ 𝑍) ∧ 𝑥 ∈ (◡𝐺 “ (-∞(,]𝐴))) → ((𝐹‘𝑛)‘𝑥) ∈ ℝ*)
51	49	mnfltd 13187	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝑛 ∈ 𝑍) ∧ 𝑥 ∈ (◡𝐺 “ (-∞(,]𝐴))) → -∞ < ((𝐹‘𝑛)‘𝑥))
52	16	adantl 481	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑥 ∈ (◡𝐺 “ (-∞(,]𝐴))) → 𝑥 ∈ dom 𝐺)
53	38	ffdmd 6778	. . . . . . . . . . . . . 14 ⊢ (𝜑 → 𝐺:dom 𝐺⟶ℝ)
54	53	ffvelcdmda 7118	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑥 ∈ dom 𝐺) → (𝐺‘𝑥) ∈ ℝ)
55	52, 54	syldan 590	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑥 ∈ (◡𝐺 “ (-∞(,]𝐴))) → (𝐺‘𝑥) ∈ ℝ)
56	55	adantlr 714	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ 𝑛 ∈ 𝑍) ∧ 𝑥 ∈ (◡𝐺 “ (-∞(,]𝐴))) → (𝐺‘𝑥) ∈ ℝ)
57	45	ad2antrr 725	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ 𝑛 ∈ 𝑍) ∧ 𝑥 ∈ (◡𝐺 “ (-∞(,]𝐴))) → 𝐴 ∈ ℝ)
58		an32 645	. . . . . . . . . . . . . . 15 ⊢ (((𝜑 ∧ 𝑛 ∈ 𝑍) ∧ 𝑥 ∈ 𝐷) ↔ ((𝜑 ∧ 𝑥 ∈ 𝐷) ∧ 𝑛 ∈ 𝑍))
59	58	biimpi 216	. . . . . . . . . . . . . 14 ⊢ (((𝜑 ∧ 𝑛 ∈ 𝑍) ∧ 𝑥 ∈ 𝐷) → ((𝜑 ∧ 𝑥 ∈ 𝐷) ∧ 𝑛 ∈ 𝑍))
60	18, 32, 35	suprubrnmpt 45162	. . . . . . . . . . . . . 14 ⊢ (((𝜑 ∧ 𝑥 ∈ 𝐷) ∧ 𝑛 ∈ 𝑍) → ((𝐹‘𝑛)‘𝑥) ≤ sup(ran (𝑛 ∈ 𝑍 ↦ ((𝐹‘𝑛)‘𝑥)), ℝ, < ))
61	59, 60	syl 17	. . . . . . . . . . . . 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 7042	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐷) → (𝐺‘𝑥) = sup(ran (𝑛 ∈ 𝑍 ↦ ((𝐹‘𝑛)‘𝑥)), ℝ, < ))
64	63	adantlr 714	. . . . . . . . . . . . 13 ⊢ (((𝜑 ∧ 𝑛 ∈ 𝑍) ∧ 𝑥 ∈ 𝐷) → (𝐺‘𝑥) = sup(ran (𝑛 ∈ 𝑍 ↦ ((𝐹‘𝑛)‘𝑥)), ℝ, < ))
65	61, 64	breqtrrd 5194	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ 𝑛 ∈ 𝑍) ∧ 𝑥 ∈ 𝐷) → ((𝐹‘𝑛)‘𝑥) ≤ (𝐺‘𝑥))
66	41, 65	syldan 590	. . . . . . . . . . 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 480	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑥 ∈ (◡𝐺 “ (-∞(,]𝐴))) → 𝐴 ∈ ℝ*)
69		simpr 484	. . . . . . . . . . . . . . 15 ⊢ ((𝜑 ∧ 𝑥 ∈ (◡𝐺 “ (-∞(,]𝐴))) → 𝑥 ∈ (◡𝐺 “ (-∞(,]𝐴)))
70	38	ffnd 6748	. . . . . . . . . . . . . . . . 17 ⊢ (𝜑 → 𝐺 Fn 𝐷)
71		elpreima 7091	. . . . . . . . . . . . . . . . 17 ⊢ (𝐺 Fn 𝐷 → (𝑥 ∈ (◡𝐺 “ (-∞(,]𝐴)) ↔ (𝑥 ∈ 𝐷 ∧ (𝐺‘𝑥) ∈ (-∞(,]𝐴))))
72	70, 71	syl 17	. . . . . . . . . . . . . . . 16 ⊢ (𝜑 → (𝑥 ∈ (◡𝐺 “ (-∞(,]𝐴)) ↔ (𝑥 ∈ 𝐷 ∧ (𝐺‘𝑥) ∈ (-∞(,]𝐴))))
73	72	adantr 480	. . . . . . . . . . . . . . 15 ⊢ ((𝜑 ∧ 𝑥 ∈ (◡𝐺 “ (-∞(,]𝐴))) → (𝑥 ∈ (◡𝐺 “ (-∞(,]𝐴)) ↔ (𝑥 ∈ 𝐷 ∧ (𝐺‘𝑥) ∈ (-∞(,]𝐴))))
74	69, 73	mpbid 232	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ 𝑥 ∈ (◡𝐺 “ (-∞(,]𝐴))) → (𝑥 ∈ 𝐷 ∧ (𝐺‘𝑥) ∈ (-∞(,]𝐴)))
75	74	simprd 495	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑥 ∈ (◡𝐺 “ (-∞(,]𝐴))) → (𝐺‘𝑥) ∈ (-∞(,]𝐴))
76	67, 68, 75	iocleubd 45477	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑥 ∈ (◡𝐺 “ (-∞(,]𝐴))) → (𝐺‘𝑥) ≤ 𝐴)
77	76	adantlr 714	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ 𝑛 ∈ 𝑍) ∧ 𝑥 ∈ (◡𝐺 “ (-∞(,]𝐴))) → (𝐺‘𝑥) ≤ 𝐴)
78	49, 56, 57, 66, 77	letrd 11447	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝑛 ∈ 𝑍) ∧ 𝑥 ∈ (◡𝐺 “ (-∞(,]𝐴))) → ((𝐹‘𝑛)‘𝑥) ≤ 𝐴)
79	44, 47, 50, 51, 78	eliocd 45425	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑛 ∈ 𝑍) ∧ 𝑥 ∈ (◡𝐺 “ (-∞(,]𝐴))) → ((𝐹‘𝑛)‘𝑥) ∈ (-∞(,]𝐴))
80	8, 42, 79	elpreimad 7092	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝑛 ∈ 𝑍) ∧ 𝑥 ∈ (◡𝐺 “ (-∞(,]𝐴))) → 𝑥 ∈ (◡(𝐹‘𝑛) “ (-∞(,]𝐴)))
81	80	ssd 44982	. . . . . . 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 4258	. . . . . . . 8 ⊢ ((𝐻‘𝑛) ∩ dom (𝐹‘𝑛)) ⊆ (𝐻‘𝑛)
84	82, 83	eqsstrdi 4063	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑛 ∈ 𝑍) → (◡(𝐹‘𝑛) “ (-∞(,]𝐴)) ⊆ (𝐻‘𝑛))
85	81, 84	sstrd 4019	. . . . . 6 ⊢ ((𝜑 ∧ 𝑛 ∈ 𝑍) → (◡𝐺 “ (-∞(,]𝐴)) ⊆ (𝐻‘𝑛))
86	85	ralrimiva 3152	. . . . 5 ⊢ (𝜑 → ∀𝑛 ∈ 𝑍 (◡𝐺 “ (-∞(,]𝐴)) ⊆ (𝐻‘𝑛))
87		ssiin 5078	. . . . 5 ⊢ ((◡𝐺 “ (-∞(,]𝐴)) ⊆ ∩ 𝑛 ∈ 𝑍 (𝐻‘𝑛) ↔ ∀𝑛 ∈ 𝑍 (◡𝐺 “ (-∞(,]𝐴)) ⊆ (𝐻‘𝑛))
88	86, 87	sylibr 234	. . . 4 ⊢ (𝜑 → (◡𝐺 “ (-∞(,]𝐴)) ⊆ ∩ 𝑛 ∈ 𝑍 (𝐻‘𝑛))
89	15, 38	fssdm 6766	. . . 4 ⊢ (𝜑 → (◡𝐺 “ (-∞(,]𝐴)) ⊆ 𝐷)
90	88, 89	ssind 4262	. . 3 ⊢ (𝜑 → (◡𝐺 “ (-∞(,]𝐴)) ⊆ (∩ 𝑛 ∈ 𝑍 (𝐻‘𝑛) ∩ 𝐷))
91		iniin1 45027	. . . . 5 ⊢ (𝑍 ≠ ∅ → (∩ 𝑛 ∈ 𝑍 (𝐻‘𝑛) ∩ 𝐷) = ∩ 𝑛 ∈ 𝑍 ((𝐻‘𝑛) ∩ 𝐷))
92	24, 91	syl 17	. . . 4 ⊢ (𝜑 → (∩ 𝑛 ∈ 𝑍 (𝐻‘𝑛) ∩ 𝐷) = ∩ 𝑛 ∈ 𝑍 ((𝐻‘𝑛) ∩ 𝐷))
93	70	adantr 480	. . . . . 6 ⊢ ((𝜑 ∧ 𝑥 ∈ ∩ 𝑛 ∈ 𝑍 ((𝐻‘𝑛) ∩ 𝐷)) → 𝐺 Fn 𝐷)
94		simpr 484	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑥 ∈ ∩ 𝑛 ∈ 𝑍 ((𝐻‘𝑛) ∩ 𝐷)) → 𝑥 ∈ ∩ 𝑛 ∈ 𝑍 ((𝐻‘𝑛) ∩ 𝐷))
95	23	adantr 480	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑥 ∈ ∩ 𝑛 ∈ 𝑍 ((𝐻‘𝑛) ∩ 𝐷)) → 𝑀 ∈ 𝑍)
96		fveq2 6920	. . . . . . . . . 10 ⊢ (𝑛 = 𝑀 → (𝐻‘𝑛) = (𝐻‘𝑀))
97	96	ineq1d 4240	. . . . . . . . 9 ⊢ (𝑛 = 𝑀 → ((𝐻‘𝑛) ∩ 𝐷) = ((𝐻‘𝑀) ∩ 𝐷))
98	97	eleq2d 2830	. . . . . . . 8 ⊢ (𝑛 = 𝑀 → (𝑥 ∈ ((𝐻‘𝑛) ∩ 𝐷) ↔ 𝑥 ∈ ((𝐻‘𝑀) ∩ 𝐷)))
99	94, 95, 98	eliind 44973	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑥 ∈ ∩ 𝑛 ∈ 𝑍 ((𝐻‘𝑛) ∩ 𝐷)) → 𝑥 ∈ ((𝐻‘𝑀) ∩ 𝐷))
100		elinel2 4225	. . . . . . 7 ⊢ (𝑥 ∈ ((𝐻‘𝑀) ∩ 𝐷) → 𝑥 ∈ 𝐷)
101	99, 100	syl 17	. . . . . 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 480	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑥 ∈ ∩ 𝑛 ∈ 𝑍 ((𝐻‘𝑛) ∩ 𝐷)) → 𝐴 ∈ ℝ*)
104	63, 36	eqeltrd 2844	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐷) → (𝐺‘𝑥) ∈ ℝ)
105	104	rexrd 11340	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐷) → (𝐺‘𝑥) ∈ ℝ*)
106	101, 105	syldan 590	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑥 ∈ ∩ 𝑛 ∈ 𝑍 ((𝐻‘𝑛) ∩ 𝐷)) → (𝐺‘𝑥) ∈ ℝ*)
107	100	adantl 481	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑥 ∈ ((𝐻‘𝑀) ∩ 𝐷)) → 𝑥 ∈ 𝐷)
108	107, 104	syldan 590	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑥 ∈ ((𝐻‘𝑀) ∩ 𝐷)) → (𝐺‘𝑥) ∈ ℝ)
109	108	mnfltd 13187	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑥 ∈ ((𝐻‘𝑀) ∩ 𝐷)) → -∞ < (𝐺‘𝑥))
110	99, 109	syldan 590	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑥 ∈ ∩ 𝑛 ∈ 𝑍 ((𝐻‘𝑛) ∩ 𝐷)) → -∞ < (𝐺‘𝑥))
111	101, 63	syldan 590	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑥 ∈ ∩ 𝑛 ∈ 𝑍 ((𝐻‘𝑛) ∩ 𝐷)) → (𝐺‘𝑥) = sup(ran (𝑛 ∈ 𝑍 ↦ ((𝐹‘𝑛)‘𝑥)), ℝ, < ))
112		nfv 1913	. . . . . . . . . . 11 ⊢ Ⅎ𝑛𝜑
113		nfcv 2908	. . . . . . . . . . . 12 ⊢ Ⅎ𝑛𝑥
114		nfii1 5052	. . . . . . . . . . . 12 ⊢ Ⅎ𝑛∩ 𝑛 ∈ 𝑍 ((𝐻‘𝑛) ∩ 𝐷)
115	113, 114	nfel 2923	. . . . . . . . . . 11 ⊢ Ⅎ𝑛 𝑥 ∈ ∩ 𝑛 ∈ 𝑍 ((𝐻‘𝑛) ∩ 𝐷)
116	112, 115	nfan 1898	. . . . . . . . . 10 ⊢ Ⅎ𝑛(𝜑 ∧ 𝑥 ∈ ∩ 𝑛 ∈ 𝑍 ((𝐻‘𝑛) ∩ 𝐷))
117		simpll 766	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ 𝑥 ∈ ∩ 𝑛 ∈ 𝑍 ((𝐻‘𝑛) ∩ 𝐷)) ∧ 𝑛 ∈ 𝑍) → 𝜑)
118		simpr 484	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ 𝑥 ∈ ∩ 𝑛 ∈ 𝑍 ((𝐻‘𝑛) ∩ 𝐷)) ∧ 𝑛 ∈ 𝑍) → 𝑛 ∈ 𝑍)
119		eliinid 45013	. . . . . . . . . . . . 13 ⊢ ((𝑥 ∈ ∩ 𝑛 ∈ 𝑍 ((𝐻‘𝑛) ∩ 𝐷) ∧ 𝑛 ∈ 𝑍) → 𝑥 ∈ ((𝐻‘𝑛) ∩ 𝐷))
120	119	adantll 713	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ 𝑥 ∈ ∩ 𝑛 ∈ 𝑍 ((𝐻‘𝑛) ∩ 𝐷)) ∧ 𝑛 ∈ 𝑍) → 𝑥 ∈ ((𝐻‘𝑛) ∩ 𝐷))
121		elinel1 4224	. . . . . . . . . . . . . . . 16 ⊢ (𝑥 ∈ ((𝐻‘𝑛) ∩ 𝐷) → 𝑥 ∈ (𝐻‘𝑛))
122	121	3ad2ant3 1135	. . . . . . . . . . . . . . 15 ⊢ ((𝜑 ∧ 𝑛 ∈ 𝑍 ∧ 𝑥 ∈ ((𝐻‘𝑛) ∩ 𝐷)) → 𝑥 ∈ (𝐻‘𝑛))
123		elinel2 4225	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑥 ∈ ((𝐻‘𝑛) ∩ 𝐷) → 𝑥 ∈ 𝐷)
124	123	adantl 481	. . . . . . . . . . . . . . . . 17 ⊢ ((𝑛 ∈ 𝑍 ∧ 𝑥 ∈ ((𝐻‘𝑛) ∩ 𝐷)) → 𝑥 ∈ 𝐷)
125	30	ancoms 458	. . . . . . . . . . . . . . . . 17 ⊢ ((𝑛 ∈ 𝑍 ∧ 𝑥 ∈ 𝐷) → 𝑥 ∈ dom (𝐹‘𝑛))
126	124, 125	syldan 590	. . . . . . . . . . . . . . . 16 ⊢ ((𝑛 ∈ 𝑍 ∧ 𝑥 ∈ ((𝐻‘𝑛) ∩ 𝐷)) → 𝑥 ∈ dom (𝐹‘𝑛))
127	126	3adant1 1130	. . . . . . . . . . . . . . 15 ⊢ ((𝜑 ∧ 𝑛 ∈ 𝑍 ∧ 𝑥 ∈ ((𝐻‘𝑛) ∩ 𝐷)) → 𝑥 ∈ dom (𝐹‘𝑛))
128	122, 127	elind 4223	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ 𝑛 ∈ 𝑍 ∧ 𝑥 ∈ ((𝐻‘𝑛) ∩ 𝐷)) → 𝑥 ∈ ((𝐻‘𝑛) ∩ dom (𝐹‘𝑛)))
129	82	3adant3 1132	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ 𝑛 ∈ 𝑍 ∧ 𝑥 ∈ ((𝐻‘𝑛) ∩ 𝐷)) → (◡(𝐹‘𝑛) “ (-∞(,]𝐴)) = ((𝐻‘𝑛) ∩ dom (𝐹‘𝑛)))
130	128, 129	eleqtrrd 2847	. . . . . . . . . . . . 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 1133	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ 𝑛 ∈ 𝑍 ∧ 𝑥 ∈ (◡(𝐹‘𝑛) “ (-∞(,]𝐴))) → 𝐴 ∈ ℝ*)
133		simp3 1138	. . . . . . . . . . . . . . . 16 ⊢ ((𝜑 ∧ 𝑛 ∈ 𝑍 ∧ 𝑥 ∈ (◡(𝐹‘𝑛) “ (-∞(,]𝐴))) → 𝑥 ∈ (◡(𝐹‘𝑛) “ (-∞(,]𝐴)))
134		elpreima 7091	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝐹‘𝑛) Fn dom (𝐹‘𝑛) → (𝑥 ∈ (◡(𝐹‘𝑛) “ (-∞(,]𝐴)) ↔ (𝑥 ∈ dom (𝐹‘𝑛) ∧ ((𝐹‘𝑛)‘𝑥) ∈ (-∞(,]𝐴))))
135	7, 134	syl 17	. . . . . . . . . . . . . . . . 17 ⊢ ((𝜑 ∧ 𝑛 ∈ 𝑍) → (𝑥 ∈ (◡(𝐹‘𝑛) “ (-∞(,]𝐴)) ↔ (𝑥 ∈ dom (𝐹‘𝑛) ∧ ((𝐹‘𝑛)‘𝑥) ∈ (-∞(,]𝐴))))
136	135	3adant3 1132	. . . . . . . . . . . . . . . 16 ⊢ ((𝜑 ∧ 𝑛 ∈ 𝑍 ∧ 𝑥 ∈ (◡(𝐹‘𝑛) “ (-∞(,]𝐴))) → (𝑥 ∈ (◡(𝐹‘𝑛) “ (-∞(,]𝐴)) ↔ (𝑥 ∈ dom (𝐹‘𝑛) ∧ ((𝐹‘𝑛)‘𝑥) ∈ (-∞(,]𝐴))))
137	133, 136	mpbid 232	. . . . . . . . . . . . . . 15 ⊢ ((𝜑 ∧ 𝑛 ∈ 𝑍 ∧ 𝑥 ∈ (◡(𝐹‘𝑛) “ (-∞(,]𝐴))) → (𝑥 ∈ dom (𝐹‘𝑛) ∧ ((𝐹‘𝑛)‘𝑥) ∈ (-∞(,]𝐴)))
138	137	simprd 495	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ 𝑛 ∈ 𝑍 ∧ 𝑥 ∈ (◡(𝐹‘𝑛) “ (-∞(,]𝐴))) → ((𝐹‘𝑛)‘𝑥) ∈ (-∞(,]𝐴))
139	131, 132, 138	iocleubd 45477	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑛 ∈ 𝑍 ∧ 𝑥 ∈ (◡(𝐹‘𝑛) “ (-∞(,]𝐴))) → ((𝐹‘𝑛)‘𝑥) ≤ 𝐴)
140	130, 139	syld3an3 1409	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑛 ∈ 𝑍 ∧ 𝑥 ∈ ((𝐻‘𝑛) ∩ 𝐷)) → ((𝐹‘𝑛)‘𝑥) ≤ 𝐴)
141	117, 118, 120, 140	syl3anc 1371	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ 𝑥 ∈ ∩ 𝑛 ∈ 𝑍 ((𝐻‘𝑛) ∩ 𝐷)) ∧ 𝑛 ∈ 𝑍) → ((𝐹‘𝑛)‘𝑥) ≤ 𝐴)
142	141	ex 412	. . . . . . . . . 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 480	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑥 ∈ ∩ 𝑛 ∈ 𝑍 ((𝐻‘𝑛) ∩ 𝐷)) → 𝑍 ≠ ∅)
145	101, 32	syldanl 601	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝑥 ∈ ∩ 𝑛 ∈ 𝑍 ((𝐻‘𝑛) ∩ 𝐷)) ∧ 𝑛 ∈ 𝑍) → ((𝐹‘𝑛)‘𝑥) ∈ ℝ)
146	101, 34	syl 17	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑥 ∈ ∩ 𝑛 ∈ 𝑍 ((𝐻‘𝑛) ∩ 𝐷)) → ∃𝑦 ∈ ℝ ∀𝑛 ∈ 𝑍 ((𝐹‘𝑛)‘𝑥) ≤ 𝑦)
147	45	adantr 480	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑥 ∈ ∩ 𝑛 ∈ 𝑍 ((𝐻‘𝑛) ∩ 𝐷)) → 𝐴 ∈ ℝ)
148	116, 144, 145, 146, 147	suprleubrnmpt 45337	. . . . . . . . 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 257	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑥 ∈ ∩ 𝑛 ∈ 𝑍 ((𝐻‘𝑛) ∩ 𝐷)) → sup(ran (𝑛 ∈ 𝑍 ↦ ((𝐹‘𝑛)‘𝑥)), ℝ, < ) ≤ 𝐴)
150	111, 149	eqbrtrd 5188	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑥 ∈ ∩ 𝑛 ∈ 𝑍 ((𝐻‘𝑛) ∩ 𝐷)) → (𝐺‘𝑥) ≤ 𝐴)
151	102, 103, 106, 110, 150	eliocd 45425	. . . . . 6 ⊢ ((𝜑 ∧ 𝑥 ∈ ∩ 𝑛 ∈ 𝑍 ((𝐻‘𝑛) ∩ 𝐷)) → (𝐺‘𝑥) ∈ (-∞(,]𝐴))
152	93, 101, 151	elpreimad 7092	. . . . 5 ⊢ ((𝜑 ∧ 𝑥 ∈ ∩ 𝑛 ∈ 𝑍 ((𝐻‘𝑛) ∩ 𝐷)) → 𝑥 ∈ (◡𝐺 “ (-∞(,]𝐴)))
153	152	ssd 44982	. . . 4 ⊢ (𝜑 → ∩ 𝑛 ∈ 𝑍 ((𝐻‘𝑛) ∩ 𝐷) ⊆ (◡𝐺 “ (-∞(,]𝐴)))
154	92, 153	eqsstrd 4047	. . 3 ⊢ (𝜑 → (∩ 𝑛 ∈ 𝑍 (𝐻‘𝑛) ∩ 𝐷) ⊆ (◡𝐺 “ (-∞(,]𝐴)))
155	90, 154	eqssd 4026	. 2 ⊢ (𝜑 → (◡𝐺 “ (-∞(,]𝐴)) = (∩ 𝑛 ∈ 𝑍 (𝐻‘𝑛) ∩ 𝐷))
156		eqid 2740	. . . . 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 6933	. . . . . . . . 9 ⊢ (𝐹‘𝑛) ∈ V
158	157	dmex 7949	. . . . . . . 8 ⊢ dom (𝐹‘𝑛) ∈ V
159	158	rgenw 3071	. . . . . . 7 ⊢ ∀𝑛 ∈ 𝑍 dom (𝐹‘𝑛) ∈ V
160	159	a1i 11	. . . . . 6 ⊢ (𝜑 → ∀𝑛 ∈ 𝑍 dom (𝐹‘𝑛) ∈ V)
161	24, 160	iinexd 45035	. . . . 5 ⊢ (𝜑 → ∩ 𝑛 ∈ 𝑍 dom (𝐹‘𝑛) ∈ V)
162	156, 161	rabexd 5358	. . . 4 ⊢ (𝜑 → {𝑥 ∈ ∩ 𝑛 ∈ 𝑍 dom (𝐹‘𝑛) ∣ ∃𝑦 ∈ ℝ ∀𝑛 ∈ 𝑍 ((𝐹‘𝑛)‘𝑥) ≤ 𝑦} ∈ V)
163	9, 162	eqeltrid 2848	. . 3 ⊢ (𝜑 → 𝐷 ∈ V)
164	22	uzct 44965	. . . . 5 ⊢ 𝑍 ≼ ω
165	164	a1i 11	. . . 4 ⊢ (𝜑 → 𝑍 ≼ ω)
166		smfsuplem1.h	. . . . 5 ⊢ (𝜑 → 𝐻:𝑍⟶𝑆)
167	166	ffvelcdmda 7118	. . . 4 ⊢ ((𝜑 ∧ 𝑛 ∈ 𝑍) → (𝐻‘𝑛) ∈ 𝑆)
168	1, 165, 24, 167	saliincl 46248	. . 3 ⊢ (𝜑 → ∩ 𝑛 ∈ 𝑍 (𝐻‘𝑛) ∈ 𝑆)
169		eqid 2740	. . 3 ⊢ (∩ 𝑛 ∈ 𝑍 (𝐻‘𝑛) ∩ 𝐷) = (∩ 𝑛 ∈ 𝑍 (𝐻‘𝑛) ∩ 𝐷)
170	1, 163, 168, 169	elrestd 45010	. 2 ⊢ (𝜑 → (∩ 𝑛 ∈ 𝑍 (𝐻‘𝑛) ∩ 𝐷) ∈ (𝑆 ↾t 𝐷))
171	155, 170	eqeltrd 2844	1 ⊢ (𝜑 → (◡𝐺 “ (-∞(,]𝐴)) ∈ (𝑆 ↾t 𝐷))

Syntax hints:   → wi 4   ↔ wb 206   ∧ wa 395   ∧ w3a 1087   = wceq 1537   ∈ wcel 2108   ≠ wne 2946  ∀wral 3067  ∃wrex 3076  {crab 3443  Vcvv 3488   ∩ cin 3975   ⊆ wss 3976  ∅c0 4352  ∩ ciin 5016   class class class wbr 5166   ↦ cmpt 5249  ^◡ccnv 5699  dom cdm 5700  ran crn 5701   “ cima 5703   Fn wfn 6568  ⟶wf 6569  ‘cfv 6573  (class class class)co 7448  ωcom 7903   ≼ cdom 9001  supcsup 9509  ℝcr 11183  -∞cmnf 11322  ℝ^*cxr 11323   < clt 11324   ≤ cle 11325  ℤcz 12639  ℤ_≥cuz 12903  (,]cioc 13408   ↾_t crest 17480  SAlgcsalg 46229  SMblFncsmblfn 46616

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1793  ax-4 1807  ax-5 1909  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-10 2141  ax-11 2158  ax-12 2178  ax-ext 2711  ax-rep 5303  ax-sep 5317  ax-nul 5324  ax-pow 5383  ax-pr 5447  ax-un 7770  ax-inf2 9710  ax-cnex 11240  ax-resscn 11241  ax-1cn 11242  ax-icn 11243  ax-addcl 11244  ax-addrcl 11245  ax-mulcl 11246  ax-mulrcl 11247  ax-mulcom 11248  ax-addass 11249  ax-mulass 11250  ax-distr 11251  ax-i2m1 11252  ax-1ne0 11253  ax-1rid 11254  ax-rnegex 11255  ax-rrecex 11256  ax-cnre 11257  ax-pre-lttri 11258  ax-pre-lttrn 11259  ax-pre-ltadd 11260  ax-pre-mulgt0 11261  ax-pre-sup 11262

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 847  df-3or 1088  df-3an 1089  df-tru 1540  df-fal 1550  df-ex 1778  df-nf 1782  df-sb 2065  df-mo 2543  df-eu 2572  df-clab 2718  df-cleq 2732  df-clel 2819  df-nfc 2895  df-ne 2947  df-nel 3053  df-ral 3068  df-rex 3077  df-rmo 3388  df-reu 3389  df-rab 3444  df-v 3490  df-sbc 3805  df-csb 3922  df-dif 3979  df-un 3981  df-in 3983  df-ss 3993  df-pss 3996  df-nul 4353  df-if 4549  df-pw 4624  df-sn 4649  df-pr 4651  df-op 4655  df-uni 4932  df-int 4971  df-iun 5017  df-iin 5018  df-br 5167  df-opab 5229  df-mpt 5250  df-tr 5284  df-id 5593  df-eprel 5599  df-po 5607  df-so 5608  df-fr 5652  df-se 5653  df-we 5654  df-xp 5706  df-rel 5707  df-cnv 5708  df-co 5709  df-dm 5710  df-rn 5711  df-res 5712  df-ima 5713  df-pred 6332  df-ord 6398  df-on 6399  df-lim 6400  df-suc 6401  df-iota 6525  df-fun 6575  df-fn 6576  df-f 6577  df-f1 6578  df-fo 6579  df-f1o 6580  df-fv 6581  df-isom 6582  df-riota 7404  df-ov 7451  df-oprab 7452  df-mpo 7453  df-om 7904  df-1st 8030  df-2nd 8031  df-frecs 8322  df-wrecs 8353  df-recs 8427  df-rdg 8466  df-1o 8522  df-oadd 8526  df-omul 8527  df-er 8763  df-map 8886  df-pm 8887  df-en 9004  df-dom 9005  df-sdom 9006  df-fin 9007  df-sup 9511  df-oi 9579  df-card 10008  df-acn 10011  df-pnf 11326  df-mnf 11327  df-xr 11328  df-ltxr 11329  df-le 11330  df-sub 11522  df-neg 11523  df-nn 12294  df-n0 12554  df-z 12640  df-uz 12904  df-ioo 13411  df-ioc 13412  df-ico 13413  df-rest 17482  df-salg 46230  df-smblfn 46617

This theorem is referenced by:  smfsuplem2  46733