Theorem smfsuplem1 46919

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 7017	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑛 ∈ 𝑍) → (𝐹‘𝑛) ∈ (SMblFn‘𝑆))
5		eqid 2731	. . . . . . . . . . . 12 ⊢ dom (𝐹‘𝑛) = dom (𝐹‘𝑛)
6	2, 4, 5	smff 46840	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑛 ∈ 𝑍) → (𝐹‘𝑛):dom (𝐹‘𝑛)⟶ℝ)
7	6	ffnd 6652	. . . . . . . . . 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 4027	. . . . . . . . . . . . 13 ⊢ {𝑥 ∈ ∩ 𝑛 ∈ 𝑍 dom (𝐹‘𝑛) ∣ ∃𝑦 ∈ ℝ ∀𝑛 ∈ 𝑍 ((𝐹‘𝑛)‘𝑥) ≤ 𝑦} ⊆ ∩ 𝑛 ∈ 𝑍 dom (𝐹‘𝑛)
11	9, 10	eqsstri 3976	. . . . . . . . . . . 12 ⊢ 𝐷 ⊆ ∩ 𝑛 ∈ 𝑍 dom (𝐹‘𝑛)
12		iinss2 5004	. . . . . . . . . . . 12 ⊢ (𝑛 ∈ 𝑍 → ∩ 𝑛 ∈ 𝑍 dom (𝐹‘𝑛) ⊆ dom (𝐹‘𝑛))
13	11, 12	sstrid 3941	. . . . . . . . . . 11 ⊢ (𝑛 ∈ 𝑍 → 𝐷 ⊆ dom (𝐹‘𝑛))
14	13	ad2antlr 727	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝑛 ∈ 𝑍) ∧ 𝑥 ∈ (◡𝐺 “ (-∞(,]𝐴))) → 𝐷 ⊆ dom (𝐹‘𝑛))
15		cnvimass 6030	. . . . . . . . . . . . 13 ⊢ (◡𝐺 “ (-∞(,]𝐴)) ⊆ dom 𝐺
16	15	sseli 3925	. . . . . . . . . . . 12 ⊢ (𝑥 ∈ (◡𝐺 “ (-∞(,]𝐴)) → 𝑥 ∈ dom 𝐺)
17	16	adantl 481	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ 𝑛 ∈ 𝑍) ∧ 𝑥 ∈ (◡𝐺 “ (-∞(,]𝐴))) → 𝑥 ∈ dom 𝐺)
18		nfv 1915	. . . . . . . . . . . . . . 15 ⊢ Ⅎ𝑛(𝜑 ∧ 𝑥 ∈ 𝐷)
19		smfsuplem1.m	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝜑 → 𝑀 ∈ ℤ)
20		uzid 12747	. . . . . . . . . . . . . . . . . . 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 2842	. . . . . . . . . . . . . . . . 17 ⊢ (𝜑 → 𝑀 ∈ 𝑍)
24	23	ne0d 4289	. . . . . . . . . . . . . . . 16 ⊢ (𝜑 → 𝑍 ≠ ∅)
25	24	adantr 480	. . . . . . . . . . . . . . 15 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐷) → 𝑍 ≠ ∅)
26	6	adantlr 715	. . . . . . . . . . . . . . . 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 3925	. . . . . . . . . . . . . . . . . . 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 3930	. . . . . . . . . . . . . . . . 17 ⊢ ((𝑥 ∈ 𝐷 ∧ 𝑛 ∈ 𝑍) → 𝑥 ∈ dom (𝐹‘𝑛))
31	30	adantll 714	. . . . . . . . . . . . . . . 16 ⊢ (((𝜑 ∧ 𝑥 ∈ 𝐷) ∧ 𝑛 ∈ 𝑍) → 𝑥 ∈ dom (𝐹‘𝑛))
32	26, 31	ffvelcdmd 7018	. . . . . . . . . . . . . . 15 ⊢ (((𝜑 ∧ 𝑥 ∈ 𝐷) ∧ 𝑛 ∈ 𝑍) → ((𝐹‘𝑛)‘𝑥) ∈ ℝ)
33	9	reqabi 3418	. . . . . . . . . . . . . . . . 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 45358	. . . . . . . . . . . . . 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 7047	. . . . . . . . . . . . 13 ⊢ (𝜑 → 𝐺:𝐷⟶ℝ)
39	38	fdmd 6661	. . . . . . . . . . . 12 ⊢ (𝜑 → dom 𝐺 = 𝐷)
40	39	ad2antrr 726	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ 𝑛 ∈ 𝑍) ∧ 𝑥 ∈ (◡𝐺 “ (-∞(,]𝐴))) → dom 𝐺 = 𝐷)
41	17, 40	eleqtrd 2833	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝑛 ∈ 𝑍) ∧ 𝑥 ∈ (◡𝐺 “ (-∞(,]𝐴))) → 𝑥 ∈ 𝐷)
42	14, 41	sseldd 3930	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑛 ∈ 𝑍) ∧ 𝑥 ∈ (◡𝐺 “ (-∞(,]𝐴))) → 𝑥 ∈ dom (𝐹‘𝑛))
43		mnfxr 11169	. . . . . . . . . . 11 ⊢ -∞ ∈ ℝ*
44	43	a1i 11	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝑛 ∈ 𝑍) ∧ 𝑥 ∈ (◡𝐺 “ (-∞(,]𝐴))) → -∞ ∈ ℝ*)
45		smfsuplem1.a	. . . . . . . . . . . 12 ⊢ (𝜑 → 𝐴 ∈ ℝ)
46	45	rexrd 11162	. . . . . . . . . . 11 ⊢ (𝜑 → 𝐴 ∈ ℝ*)
47	46	ad2antrr 726	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝑛 ∈ 𝑍) ∧ 𝑥 ∈ (◡𝐺 “ (-∞(,]𝐴))) → 𝐴 ∈ ℝ*)
48	32	an32s 652	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ 𝑛 ∈ 𝑍) ∧ 𝑥 ∈ 𝐷) → ((𝐹‘𝑛)‘𝑥) ∈ ℝ)
49	41, 48	syldan 591	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ 𝑛 ∈ 𝑍) ∧ 𝑥 ∈ (◡𝐺 “ (-∞(,]𝐴))) → ((𝐹‘𝑛)‘𝑥) ∈ ℝ)
50	49	rexrd 11162	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝑛 ∈ 𝑍) ∧ 𝑥 ∈ (◡𝐺 “ (-∞(,]𝐴))) → ((𝐹‘𝑛)‘𝑥) ∈ ℝ*)
51	49	mnfltd 13023	. . . . . . . . . 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 6681	. . . . . . . . . . . . . 14 ⊢ (𝜑 → 𝐺:dom 𝐺⟶ℝ)
54	53	ffvelcdmda 7017	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑥 ∈ dom 𝐺) → (𝐺‘𝑥) ∈ ℝ)
55	52, 54	syldan 591	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑥 ∈ (◡𝐺 “ (-∞(,]𝐴))) → (𝐺‘𝑥) ∈ ℝ)
56	55	adantlr 715	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ 𝑛 ∈ 𝑍) ∧ 𝑥 ∈ (◡𝐺 “ (-∞(,]𝐴))) → (𝐺‘𝑥) ∈ ℝ)
57	45	ad2antrr 726	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ 𝑛 ∈ 𝑍) ∧ 𝑥 ∈ (◡𝐺 “ (-∞(,]𝐴))) → 𝐴 ∈ ℝ)
58		an32 646	. . . . . . . . . . . . . . 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 45360	. . . . . . . . . . . . . 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 6942	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐷) → (𝐺‘𝑥) = sup(ran (𝑛 ∈ 𝑍 ↦ ((𝐹‘𝑛)‘𝑥)), ℝ, < ))
64	63	adantlr 715	. . . . . . . . . . . . 13 ⊢ (((𝜑 ∧ 𝑛 ∈ 𝑍) ∧ 𝑥 ∈ 𝐷) → (𝐺‘𝑥) = sup(ran (𝑛 ∈ 𝑍 ↦ ((𝐹‘𝑛)‘𝑥)), ℝ, < ))
65	61, 64	breqtrrd 5117	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ 𝑛 ∈ 𝑍) ∧ 𝑥 ∈ 𝐷) → ((𝐹‘𝑛)‘𝑥) ≤ (𝐺‘𝑥))
66	41, 65	syldan 591	. . . . . . . . . . 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 6652	. . . . . . . . . . . . . . . . 17 ⊢ (𝜑 → 𝐺 Fn 𝐷)
71		elpreima 6991	. . . . . . . . . . . . . . . . 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 45668	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑥 ∈ (◡𝐺 “ (-∞(,]𝐴))) → (𝐺‘𝑥) ≤ 𝐴)
77	76	adantlr 715	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ 𝑛 ∈ 𝑍) ∧ 𝑥 ∈ (◡𝐺 “ (-∞(,]𝐴))) → (𝐺‘𝑥) ≤ 𝐴)
78	49, 56, 57, 66, 77	letrd 11270	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝑛 ∈ 𝑍) ∧ 𝑥 ∈ (◡𝐺 “ (-∞(,]𝐴))) → ((𝐹‘𝑛)‘𝑥) ≤ 𝐴)
79	44, 47, 50, 51, 78	eliocd 45617	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑛 ∈ 𝑍) ∧ 𝑥 ∈ (◡𝐺 “ (-∞(,]𝐴))) → ((𝐹‘𝑛)‘𝑥) ∈ (-∞(,]𝐴))
80	8, 42, 79	elpreimad 6992	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝑛 ∈ 𝑍) ∧ 𝑥 ∈ (◡𝐺 “ (-∞(,]𝐴))) → 𝑥 ∈ (◡(𝐹‘𝑛) “ (-∞(,]𝐴)))
81	80	ssd 45187	. . . . . . 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 4184	. . . . . . . 8 ⊢ ((𝐻‘𝑛) ∩ dom (𝐹‘𝑛)) ⊆ (𝐻‘𝑛)
84	82, 83	eqsstrdi 3974	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑛 ∈ 𝑍) → (◡(𝐹‘𝑛) “ (-∞(,]𝐴)) ⊆ (𝐻‘𝑛))
85	81, 84	sstrd 3940	. . . . . 6 ⊢ ((𝜑 ∧ 𝑛 ∈ 𝑍) → (◡𝐺 “ (-∞(,]𝐴)) ⊆ (𝐻‘𝑛))
86	85	ralrimiva 3124	. . . . 5 ⊢ (𝜑 → ∀𝑛 ∈ 𝑍 (◡𝐺 “ (-∞(,]𝐴)) ⊆ (𝐻‘𝑛))
87		ssiin 5002	. . . . 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 6670	. . . 4 ⊢ (𝜑 → (◡𝐺 “ (-∞(,]𝐴)) ⊆ 𝐷)
90	88, 89	ssind 4188	. . 3 ⊢ (𝜑 → (◡𝐺 “ (-∞(,]𝐴)) ⊆ (∩ 𝑛 ∈ 𝑍 (𝐻‘𝑛) ∩ 𝐷))
91		iniin1 45232	. . . . 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 6822	. . . . . . . . . 10 ⊢ (𝑛 = 𝑀 → (𝐻‘𝑛) = (𝐻‘𝑀))
97	96	ineq1d 4166	. . . . . . . . 9 ⊢ (𝑛 = 𝑀 → ((𝐻‘𝑛) ∩ 𝐷) = ((𝐻‘𝑀) ∩ 𝐷))
98	97	eleq2d 2817	. . . . . . . 8 ⊢ (𝑛 = 𝑀 → (𝑥 ∈ ((𝐻‘𝑛) ∩ 𝐷) ↔ 𝑥 ∈ ((𝐻‘𝑀) ∩ 𝐷)))
99	94, 95, 98	eliind 45178	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑥 ∈ ∩ 𝑛 ∈ 𝑍 ((𝐻‘𝑛) ∩ 𝐷)) → 𝑥 ∈ ((𝐻‘𝑀) ∩ 𝐷))
100		elinel2 4149	. . . . . . 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 2831	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐷) → (𝐺‘𝑥) ∈ ℝ)
105	104	rexrd 11162	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐷) → (𝐺‘𝑥) ∈ ℝ*)
106	101, 105	syldan 591	. . . . . . 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 591	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑥 ∈ ((𝐻‘𝑀) ∩ 𝐷)) → (𝐺‘𝑥) ∈ ℝ)
109	108	mnfltd 13023	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑥 ∈ ((𝐻‘𝑀) ∩ 𝐷)) → -∞ < (𝐺‘𝑥))
110	99, 109	syldan 591	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑥 ∈ ∩ 𝑛 ∈ 𝑍 ((𝐻‘𝑛) ∩ 𝐷)) → -∞ < (𝐺‘𝑥))
111	101, 63	syldan 591	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑥 ∈ ∩ 𝑛 ∈ 𝑍 ((𝐻‘𝑛) ∩ 𝐷)) → (𝐺‘𝑥) = sup(ran (𝑛 ∈ 𝑍 ↦ ((𝐹‘𝑛)‘𝑥)), ℝ, < ))
112		nfv 1915	. . . . . . . . . . 11 ⊢ Ⅎ𝑛𝜑
113		nfcv 2894	. . . . . . . . . . . 12 ⊢ Ⅎ𝑛𝑥
114		nfii1 4977	. . . . . . . . . . . 12 ⊢ Ⅎ𝑛∩ 𝑛 ∈ 𝑍 ((𝐻‘𝑛) ∩ 𝐷)
115	113, 114	nfel 2909	. . . . . . . . . . 11 ⊢ Ⅎ𝑛 𝑥 ∈ ∩ 𝑛 ∈ 𝑍 ((𝐻‘𝑛) ∩ 𝐷)
116	112, 115	nfan 1900	. . . . . . . . . 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 45218	. . . . . . . . . . . . 13 ⊢ ((𝑥 ∈ ∩ 𝑛 ∈ 𝑍 ((𝐻‘𝑛) ∩ 𝐷) ∧ 𝑛 ∈ 𝑍) → 𝑥 ∈ ((𝐻‘𝑛) ∩ 𝐷))
120	119	adantll 714	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ 𝑥 ∈ ∩ 𝑛 ∈ 𝑍 ((𝐻‘𝑛) ∩ 𝐷)) ∧ 𝑛 ∈ 𝑍) → 𝑥 ∈ ((𝐻‘𝑛) ∩ 𝐷))
121		elinel1 4148	. . . . . . . . . . . . . . . 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 4149	. . . . . . . . . . . . . . . . . 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 591	. . . . . . . . . . . . . . . 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 4147	. . . . . . . . . . . . . 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 2834	. . . . . . . . . . . . 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 6991	. . . . . . . . . . . . . . . . . 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 45668	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑛 ∈ 𝑍 ∧ 𝑥 ∈ (◡(𝐹‘𝑛) “ (-∞(,]𝐴))) → ((𝐹‘𝑛)‘𝑥) ≤ 𝐴)
140	130, 139	syld3an3 1411	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑛 ∈ 𝑍 ∧ 𝑥 ∈ ((𝐻‘𝑛) ∩ 𝐷)) → ((𝐹‘𝑛)‘𝑥) ≤ 𝐴)
141	117, 118, 120, 140	syl3anc 1373	. . . . . . . . . . 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 3230	. . . . . . . . 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 602	. . . . . . . . . 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 45530	. . . . . . . . 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 5111	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑥 ∈ ∩ 𝑛 ∈ 𝑍 ((𝐻‘𝑛) ∩ 𝐷)) → (𝐺‘𝑥) ≤ 𝐴)
151	102, 103, 106, 110, 150	eliocd 45617	. . . . . 6 ⊢ ((𝜑 ∧ 𝑥 ∈ ∩ 𝑛 ∈ 𝑍 ((𝐻‘𝑛) ∩ 𝐷)) → (𝐺‘𝑥) ∈ (-∞(,]𝐴))
152	93, 101, 151	elpreimad 6992	. . . . 5 ⊢ ((𝜑 ∧ 𝑥 ∈ ∩ 𝑛 ∈ 𝑍 ((𝐻‘𝑛) ∩ 𝐷)) → 𝑥 ∈ (◡𝐺 “ (-∞(,]𝐴)))
153	152	ssd 45187	. . . 4 ⊢ (𝜑 → ∩ 𝑛 ∈ 𝑍 ((𝐻‘𝑛) ∩ 𝐷) ⊆ (◡𝐺 “ (-∞(,]𝐴)))
154	92, 153	eqsstrd 3964	. . 3 ⊢ (𝜑 → (∩ 𝑛 ∈ 𝑍 (𝐻‘𝑛) ∩ 𝐷) ⊆ (◡𝐺 “ (-∞(,]𝐴)))
155	90, 154	eqssd 3947	. 2 ⊢ (𝜑 → (◡𝐺 “ (-∞(,]𝐴)) = (∩ 𝑛 ∈ 𝑍 (𝐻‘𝑛) ∩ 𝐷))
156		eqid 2731	. . . . 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 6835	. . . . . . . . 9 ⊢ (𝐹‘𝑛) ∈ V
158	157	dmex 7839	. . . . . . . 8 ⊢ dom (𝐹‘𝑛) ∈ V
159	158	rgenw 3051	. . . . . . 7 ⊢ ∀𝑛 ∈ 𝑍 dom (𝐹‘𝑛) ∈ V
160	159	a1i 11	. . . . . 6 ⊢ (𝜑 → ∀𝑛 ∈ 𝑍 dom (𝐹‘𝑛) ∈ V)
161	24, 160	iinexd 45240	. . . . 5 ⊢ (𝜑 → ∩ 𝑛 ∈ 𝑍 dom (𝐹‘𝑛) ∈ V)
162	156, 161	rabexd 5276	. . . 4 ⊢ (𝜑 → {𝑥 ∈ ∩ 𝑛 ∈ 𝑍 dom (𝐹‘𝑛) ∣ ∃𝑦 ∈ ℝ ∀𝑛 ∈ 𝑍 ((𝐹‘𝑛)‘𝑥) ≤ 𝑦} ∈ V)
163	9, 162	eqeltrid 2835	. . 3 ⊢ (𝜑 → 𝐷 ∈ V)
164	22	uzct 45170	. . . . 5 ⊢ 𝑍 ≼ ω
165	164	a1i 11	. . . 4 ⊢ (𝜑 → 𝑍 ≼ ω)
166		smfsuplem1.h	. . . . 5 ⊢ (𝜑 → 𝐻:𝑍⟶𝑆)
167	166	ffvelcdmda 7017	. . . 4 ⊢ ((𝜑 ∧ 𝑛 ∈ 𝑍) → (𝐻‘𝑛) ∈ 𝑆)
168	1, 165, 24, 167	saliincl 46435	. . 3 ⊢ (𝜑 → ∩ 𝑛 ∈ 𝑍 (𝐻‘𝑛) ∈ 𝑆)
169		eqid 2731	. . 3 ⊢ (∩ 𝑛 ∈ 𝑍 (𝐻‘𝑛) ∩ 𝐷) = (∩ 𝑛 ∈ 𝑍 (𝐻‘𝑛) ∩ 𝐷)
170	1, 163, 168, 169	elrestd 45215	. 2 ⊢ (𝜑 → (∩ 𝑛 ∈ 𝑍 (𝐻‘𝑛) ∩ 𝐷) ∈ (𝑆 ↾t 𝐷))
171	155, 170	eqeltrd 2831	1 ⊢ (𝜑 → (◡𝐺 “ (-∞(,]𝐴)) ∈ (𝑆 ↾t 𝐷))

Syntax hints:   → wi 4   ↔ wb 206   ∧ wa 395   ∧ w3a 1086   = wceq 1541   ∈ wcel 2111   ≠ wne 2928  ∀wral 3047  ∃wrex 3056  {crab 3395  Vcvv 3436   ∩ cin 3896   ⊆ wss 3897  ∅c0 4280  ∩ ciin 4940   class class class wbr 5089   ↦ cmpt 5170  ^◡ccnv 5613  dom cdm 5614  ran crn 5615   “ cima 5617   Fn wfn 6476  ⟶wf 6477  ‘cfv 6481  (class class class)co 7346  ωcom 7796   ≼ cdom 8867  supcsup 9324  ℝcr 11005  -∞cmnf 11144  ℝ^*cxr 11145   < clt 11146   ≤ cle 11147  ℤcz 12468  ℤ_≥cuz 12732  (,]cioc 13246   ↾_t crest 17324  SAlgcsalg 46416  SMblFncsmblfn 46803

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1911  ax-6 1968  ax-7 2009  ax-8 2113  ax-9 2121  ax-10 2144  ax-11 2160  ax-12 2180  ax-ext 2703  ax-rep 5215  ax-sep 5232  ax-nul 5242  ax-pow 5301  ax-pr 5368  ax-un 7668  ax-inf2 9531  ax-cnex 11062  ax-resscn 11063  ax-1cn 11064  ax-icn 11065  ax-addcl 11066  ax-addrcl 11067  ax-mulcl 11068  ax-mulrcl 11069  ax-mulcom 11070  ax-addass 11071  ax-mulass 11072  ax-distr 11073  ax-i2m1 11074  ax-1ne0 11075  ax-1rid 11076  ax-rnegex 11077  ax-rrecex 11078  ax-cnre 11079  ax-pre-lttri 11080  ax-pre-lttrn 11081  ax-pre-ltadd 11082  ax-pre-mulgt0 11083  ax-pre-sup 11084

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3or 1087  df-3an 1088  df-tru 1544  df-fal 1554  df-ex 1781  df-nf 1785  df-sb 2068  df-mo 2535  df-eu 2564  df-clab 2710  df-cleq 2723  df-clel 2806  df-nfc 2881  df-ne 2929  df-nel 3033  df-ral 3048  df-rex 3057  df-rmo 3346  df-reu 3347  df-rab 3396  df-v 3438  df-sbc 3737  df-csb 3846  df-dif 3900  df-un 3902  df-in 3904  df-ss 3914  df-pss 3917  df-nul 4281  df-if 4473  df-pw 4549  df-sn 4574  df-pr 4576  df-op 4580  df-uni 4857  df-int 4896  df-iun 4941  df-iin 4942  df-br 5090  df-opab 5152  df-mpt 5171  df-tr 5197  df-id 5509  df-eprel 5514  df-po 5522  df-so 5523  df-fr 5567  df-se 5568  df-we 5569  df-xp 5620  df-rel 5621  df-cnv 5622  df-co 5623  df-dm 5624  df-rn 5625  df-res 5626  df-ima 5627  df-pred 6248  df-ord 6309  df-on 6310  df-lim 6311  df-suc 6312  df-iota 6437  df-fun 6483  df-fn 6484  df-f 6485  df-f1 6486  df-fo 6487  df-f1o 6488  df-fv 6489  df-isom 6490  df-riota 7303  df-ov 7349  df-oprab 7350  df-mpo 7351  df-om 7797  df-1st 7921  df-2nd 7922  df-frecs 8211  df-wrecs 8242  df-recs 8291  df-rdg 8329  df-1o 8385  df-oadd 8389  df-omul 8390  df-er 8622  df-map 8752  df-pm 8753  df-en 8870  df-dom 8871  df-sdom 8872  df-fin 8873  df-sup 9326  df-oi 9396  df-card 9832  df-acn 9835  df-pnf 11148  df-mnf 11149  df-xr 11150  df-ltxr 11151  df-le 11152  df-sub 11346  df-neg 11347  df-nn 12126  df-n0 12382  df-z 12469  df-uz 12733  df-ioo 13249  df-ioc 13250  df-ico 13251  df-rest 17326  df-salg 46417  df-smblfn 46804

This theorem is referenced by:  smfsuplem2  46920