Theorem smfsuplem1 44231

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	ffvelrnda 6943	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑛 ∈ 𝑍) → (𝐹‘𝑛) ∈ (SMblFn‘𝑆))
5		eqid 2738	. . . . . . . . . . . 12 ⊢ dom (𝐹‘𝑛) = dom (𝐹‘𝑛)
6	2, 4, 5	smff 44155	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑛 ∈ 𝑍) → (𝐹‘𝑛):dom (𝐹‘𝑛)⟶ℝ)
7	6	ffnd 6585	. . . . . . . . . 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 4009	. . . . . . . . . . . . 13 ⊢ {𝑥 ∈ ∩ 𝑛 ∈ 𝑍 dom (𝐹‘𝑛) ∣ ∃𝑦 ∈ ℝ ∀𝑛 ∈ 𝑍 ((𝐹‘𝑛)‘𝑥) ≤ 𝑦} ⊆ ∩ 𝑛 ∈ 𝑍 dom (𝐹‘𝑛)
11	9, 10	eqsstri 3951	. . . . . . . . . . . 12 ⊢ 𝐷 ⊆ ∩ 𝑛 ∈ 𝑍 dom (𝐹‘𝑛)
12		iinss2 4983	. . . . . . . . . . . 12 ⊢ (𝑛 ∈ 𝑍 → ∩ 𝑛 ∈ 𝑍 dom (𝐹‘𝑛) ⊆ dom (𝐹‘𝑛))
13	11, 12	sstrid 3928	. . . . . . . . . . 11 ⊢ (𝑛 ∈ 𝑍 → 𝐷 ⊆ dom (𝐹‘𝑛))
14	13	ad2antlr 723	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝑛 ∈ 𝑍) ∧ 𝑥 ∈ (◡𝐺 “ (-∞(,]𝐴))) → 𝐷 ⊆ dom (𝐹‘𝑛))
15		cnvimass 5978	. . . . . . . . . . . . 13 ⊢ (◡𝐺 “ (-∞(,]𝐴)) ⊆ dom 𝐺
16	15	sseli 3913	. . . . . . . . . . . 12 ⊢ (𝑥 ∈ (◡𝐺 “ (-∞(,]𝐴)) → 𝑥 ∈ dom 𝐺)
17	16	adantl 481	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ 𝑛 ∈ 𝑍) ∧ 𝑥 ∈ (◡𝐺 “ (-∞(,]𝐴))) → 𝑥 ∈ dom 𝐺)
18		nfv 1918	. . . . . . . . . . . . . . 15 ⊢ Ⅎ𝑛(𝜑 ∧ 𝑥 ∈ 𝐷)
19		smfsuplem1.m	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝜑 → 𝑀 ∈ ℤ)
20		uzid 12526	. . . . . . . . . . . . . . . . . . 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 2850	. . . . . . . . . . . . . . . . 17 ⊢ (𝜑 → 𝑀 ∈ 𝑍)
24	23	ne0d 4266	. . . . . . . . . . . . . . . 16 ⊢ (𝜑 → 𝑍 ≠ ∅)
25	24	adantr 480	. . . . . . . . . . . . . . 15 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐷) → 𝑍 ≠ ∅)
26	6	adantlr 711	. . . . . . . . . . . . . . . 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 3913	. . . . . . . . . . . . . . . . . . 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 3918	. . . . . . . . . . . . . . . . 17 ⊢ ((𝑥 ∈ 𝐷 ∧ 𝑛 ∈ 𝑍) → 𝑥 ∈ dom (𝐹‘𝑛))
31	30	adantll 710	. . . . . . . . . . . . . . . 16 ⊢ (((𝜑 ∧ 𝑥 ∈ 𝐷) ∧ 𝑛 ∈ 𝑍) → 𝑥 ∈ dom (𝐹‘𝑛))
32	26, 31	ffvelrnd 6944	. . . . . . . . . . . . . . 15 ⊢ (((𝜑 ∧ 𝑥 ∈ 𝐷) ∧ 𝑛 ∈ 𝑍) → ((𝐹‘𝑛)‘𝑥) ∈ ℝ)
33	9	rabeq2i 3412	. . . . . . . . . . . . . . . . 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 42686	. . . . . . . . . . . . . 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 6970	. . . . . . . . . . . . 13 ⊢ (𝜑 → 𝐺:𝐷⟶ℝ)
39	38	fdmd 6595	. . . . . . . . . . . 12 ⊢ (𝜑 → dom 𝐺 = 𝐷)
40	39	ad2antrr 722	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ 𝑛 ∈ 𝑍) ∧ 𝑥 ∈ (◡𝐺 “ (-∞(,]𝐴))) → dom 𝐺 = 𝐷)
41	17, 40	eleqtrd 2841	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝑛 ∈ 𝑍) ∧ 𝑥 ∈ (◡𝐺 “ (-∞(,]𝐴))) → 𝑥 ∈ 𝐷)
42	14, 41	sseldd 3918	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑛 ∈ 𝑍) ∧ 𝑥 ∈ (◡𝐺 “ (-∞(,]𝐴))) → 𝑥 ∈ dom (𝐹‘𝑛))
43		mnfxr 10963	. . . . . . . . . . 11 ⊢ -∞ ∈ ℝ*
44	43	a1i 11	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝑛 ∈ 𝑍) ∧ 𝑥 ∈ (◡𝐺 “ (-∞(,]𝐴))) → -∞ ∈ ℝ*)
45		smfsuplem1.a	. . . . . . . . . . . 12 ⊢ (𝜑 → 𝐴 ∈ ℝ)
46	45	rexrd 10956	. . . . . . . . . . 11 ⊢ (𝜑 → 𝐴 ∈ ℝ*)
47	46	ad2antrr 722	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝑛 ∈ 𝑍) ∧ 𝑥 ∈ (◡𝐺 “ (-∞(,]𝐴))) → 𝐴 ∈ ℝ*)
48	32	an32s 648	. . . . . . . . . . . 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 10956	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝑛 ∈ 𝑍) ∧ 𝑥 ∈ (◡𝐺 “ (-∞(,]𝐴))) → ((𝐹‘𝑛)‘𝑥) ∈ ℝ*)
51	49	mnfltd 12789	. . . . . . . . . 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 6615	. . . . . . . . . . . . . 14 ⊢ (𝜑 → 𝐺:dom 𝐺⟶ℝ)
54	53	ffvelrnda 6943	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑥 ∈ dom 𝐺) → (𝐺‘𝑥) ∈ ℝ)
55	52, 54	syldan 590	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑥 ∈ (◡𝐺 “ (-∞(,]𝐴))) → (𝐺‘𝑥) ∈ ℝ)
56	55	adantlr 711	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ 𝑛 ∈ 𝑍) ∧ 𝑥 ∈ (◡𝐺 “ (-∞(,]𝐴))) → (𝐺‘𝑥) ∈ ℝ)
57	45	ad2antrr 722	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ 𝑛 ∈ 𝑍) ∧ 𝑥 ∈ (◡𝐺 “ (-∞(,]𝐴))) → 𝐴 ∈ ℝ)
58		an32 642	. . . . . . . . . . . . . . 15 ⊢ (((𝜑 ∧ 𝑛 ∈ 𝑍) ∧ 𝑥 ∈ 𝐷) ↔ ((𝜑 ∧ 𝑥 ∈ 𝐷) ∧ 𝑛 ∈ 𝑍))
59	58	biimpi 215	. . . . . . . . . . . . . 14 ⊢ (((𝜑 ∧ 𝑛 ∈ 𝑍) ∧ 𝑥 ∈ 𝐷) → ((𝜑 ∧ 𝑥 ∈ 𝐷) ∧ 𝑛 ∈ 𝑍))
60	18, 32, 35	suprubrnmpt 42688	. . . . . . . . . . . . . 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 6870	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐷) → (𝐺‘𝑥) = sup(ran (𝑛 ∈ 𝑍 ↦ ((𝐹‘𝑛)‘𝑥)), ℝ, < ))
64	63	adantlr 711	. . . . . . . . . . . . 13 ⊢ (((𝜑 ∧ 𝑛 ∈ 𝑍) ∧ 𝑥 ∈ 𝐷) → (𝐺‘𝑥) = sup(ran (𝑛 ∈ 𝑍 ↦ ((𝐹‘𝑛)‘𝑥)), ℝ, < ))
65	61, 64	breqtrrd 5098	. . . . . . . . . . . 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 6585	. . . . . . . . . . . . . . . . 17 ⊢ (𝜑 → 𝐺 Fn 𝐷)
71		elpreima 6917	. . . . . . . . . . . . . . . . 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 231	. . . . . . . . . . . . . 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 42987	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑥 ∈ (◡𝐺 “ (-∞(,]𝐴))) → (𝐺‘𝑥) ≤ 𝐴)
77	76	adantlr 711	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ 𝑛 ∈ 𝑍) ∧ 𝑥 ∈ (◡𝐺 “ (-∞(,]𝐴))) → (𝐺‘𝑥) ≤ 𝐴)
78	49, 56, 57, 66, 77	letrd 11062	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝑛 ∈ 𝑍) ∧ 𝑥 ∈ (◡𝐺 “ (-∞(,]𝐴))) → ((𝐹‘𝑛)‘𝑥) ≤ 𝐴)
79	44, 47, 50, 51, 78	eliocd 42935	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑛 ∈ 𝑍) ∧ 𝑥 ∈ (◡𝐺 “ (-∞(,]𝐴))) → ((𝐹‘𝑛)‘𝑥) ∈ (-∞(,]𝐴))
80	8, 42, 79	elpreimad 6918	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝑛 ∈ 𝑍) ∧ 𝑥 ∈ (◡𝐺 “ (-∞(,]𝐴))) → 𝑥 ∈ (◡(𝐹‘𝑛) “ (-∞(,]𝐴)))
81	80	ssd 42519	. . . . . . 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 4159	. . . . . . . 8 ⊢ ((𝐻‘𝑛) ∩ dom (𝐹‘𝑛)) ⊆ (𝐻‘𝑛)
84	82, 83	eqsstrdi 3971	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑛 ∈ 𝑍) → (◡(𝐹‘𝑛) “ (-∞(,]𝐴)) ⊆ (𝐻‘𝑛))
85	81, 84	sstrd 3927	. . . . . 6 ⊢ ((𝜑 ∧ 𝑛 ∈ 𝑍) → (◡𝐺 “ (-∞(,]𝐴)) ⊆ (𝐻‘𝑛))
86	85	ralrimiva 3107	. . . . 5 ⊢ (𝜑 → ∀𝑛 ∈ 𝑍 (◡𝐺 “ (-∞(,]𝐴)) ⊆ (𝐻‘𝑛))
87		ssiin 4981	. . . . 5 ⊢ ((◡𝐺 “ (-∞(,]𝐴)) ⊆ ∩ 𝑛 ∈ 𝑍 (𝐻‘𝑛) ↔ ∀𝑛 ∈ 𝑍 (◡𝐺 “ (-∞(,]𝐴)) ⊆ (𝐻‘𝑛))
88	86, 87	sylibr 233	. . . 4 ⊢ (𝜑 → (◡𝐺 “ (-∞(,]𝐴)) ⊆ ∩ 𝑛 ∈ 𝑍 (𝐻‘𝑛))
89	15, 38	fssdm 6604	. . . 4 ⊢ (𝜑 → (◡𝐺 “ (-∞(,]𝐴)) ⊆ 𝐷)
90	88, 89	ssind 4163	. . 3 ⊢ (𝜑 → (◡𝐺 “ (-∞(,]𝐴)) ⊆ (∩ 𝑛 ∈ 𝑍 (𝐻‘𝑛) ∩ 𝐷))
91		iniin1 42563	. . . . 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 6756	. . . . . . . . . 10 ⊢ (𝑛 = 𝑀 → (𝐻‘𝑛) = (𝐻‘𝑀))
97	96	ineq1d 4142	. . . . . . . . 9 ⊢ (𝑛 = 𝑀 → ((𝐻‘𝑛) ∩ 𝐷) = ((𝐻‘𝑀) ∩ 𝐷))
98	97	eleq2d 2824	. . . . . . . 8 ⊢ (𝑛 = 𝑀 → (𝑥 ∈ ((𝐻‘𝑛) ∩ 𝐷) ↔ 𝑥 ∈ ((𝐻‘𝑀) ∩ 𝐷)))
99	94, 95, 98	eliind 42508	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑥 ∈ ∩ 𝑛 ∈ 𝑍 ((𝐻‘𝑛) ∩ 𝐷)) → 𝑥 ∈ ((𝐻‘𝑀) ∩ 𝐷))
100		elinel2 4126	. . . . . . 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 2839	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐷) → (𝐺‘𝑥) ∈ ℝ)
105	104	rexrd 10956	. . . . . . . 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 12789	. . . . . . . 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 1918	. . . . . . . . . . 11 ⊢ Ⅎ𝑛𝜑
113		nfcv 2906	. . . . . . . . . . . 12 ⊢ Ⅎ𝑛𝑥
114		nfii1 4956	. . . . . . . . . . . 12 ⊢ Ⅎ𝑛∩ 𝑛 ∈ 𝑍 ((𝐻‘𝑛) ∩ 𝐷)
115	113, 114	nfel 2920	. . . . . . . . . . 11 ⊢ Ⅎ𝑛 𝑥 ∈ ∩ 𝑛 ∈ 𝑍 ((𝐻‘𝑛) ∩ 𝐷)
116	112, 115	nfan 1903	. . . . . . . . . 10 ⊢ Ⅎ𝑛(𝜑 ∧ 𝑥 ∈ ∩ 𝑛 ∈ 𝑍 ((𝐻‘𝑛) ∩ 𝐷))
117		simpll 763	. . . . . . . . . . . 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 42550	. . . . . . . . . . . . 13 ⊢ ((𝑥 ∈ ∩ 𝑛 ∈ 𝑍 ((𝐻‘𝑛) ∩ 𝐷) ∧ 𝑛 ∈ 𝑍) → 𝑥 ∈ ((𝐻‘𝑛) ∩ 𝐷))
120	119	adantll 710	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ 𝑥 ∈ ∩ 𝑛 ∈ 𝑍 ((𝐻‘𝑛) ∩ 𝐷)) ∧ 𝑛 ∈ 𝑍) → 𝑥 ∈ ((𝐻‘𝑛) ∩ 𝐷))
121		elinel1 4125	. . . . . . . . . . . . . . . 16 ⊢ (𝑥 ∈ ((𝐻‘𝑛) ∩ 𝐷) → 𝑥 ∈ (𝐻‘𝑛))
122	121	3ad2ant3 1133	. . . . . . . . . . . . . . 15 ⊢ ((𝜑 ∧ 𝑛 ∈ 𝑍 ∧ 𝑥 ∈ ((𝐻‘𝑛) ∩ 𝐷)) → 𝑥 ∈ (𝐻‘𝑛))
123		elinel2 4126	. . . . . . . . . . . . . . . . . 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 1128	. . . . . . . . . . . . . . 15 ⊢ ((𝜑 ∧ 𝑛 ∈ 𝑍 ∧ 𝑥 ∈ ((𝐻‘𝑛) ∩ 𝐷)) → 𝑥 ∈ dom (𝐹‘𝑛))
128	122, 127	elind 4124	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ 𝑛 ∈ 𝑍 ∧ 𝑥 ∈ ((𝐻‘𝑛) ∩ 𝐷)) → 𝑥 ∈ ((𝐻‘𝑛) ∩ dom (𝐹‘𝑛)))
129	82	3adant3 1130	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ 𝑛 ∈ 𝑍 ∧ 𝑥 ∈ ((𝐻‘𝑛) ∩ 𝐷)) → (◡(𝐹‘𝑛) “ (-∞(,]𝐴)) = ((𝐻‘𝑛) ∩ dom (𝐹‘𝑛)))
130	128, 129	eleqtrrd 2842	. . . . . . . . . . . . 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 1131	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ 𝑛 ∈ 𝑍 ∧ 𝑥 ∈ (◡(𝐹‘𝑛) “ (-∞(,]𝐴))) → 𝐴 ∈ ℝ*)
133		simp3 1136	. . . . . . . . . . . . . . . 16 ⊢ ((𝜑 ∧ 𝑛 ∈ 𝑍 ∧ 𝑥 ∈ (◡(𝐹‘𝑛) “ (-∞(,]𝐴))) → 𝑥 ∈ (◡(𝐹‘𝑛) “ (-∞(,]𝐴)))
134		elpreima 6917	. . . . . . . . . . . . . . . . . 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 1130	. . . . . . . . . . . . . . . 16 ⊢ ((𝜑 ∧ 𝑛 ∈ 𝑍 ∧ 𝑥 ∈ (◡(𝐹‘𝑛) “ (-∞(,]𝐴))) → (𝑥 ∈ (◡(𝐹‘𝑛) “ (-∞(,]𝐴)) ↔ (𝑥 ∈ dom (𝐹‘𝑛) ∧ ((𝐹‘𝑛)‘𝑥) ∈ (-∞(,]𝐴))))
137	133, 136	mpbid 231	. . . . . . . . . . . . . . 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 42987	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑛 ∈ 𝑍 ∧ 𝑥 ∈ (◡(𝐹‘𝑛) “ (-∞(,]𝐴))) → ((𝐹‘𝑛)‘𝑥) ≤ 𝐴)
140	130, 139	syld3an3 1407	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑛 ∈ 𝑍 ∧ 𝑥 ∈ ((𝐻‘𝑛) ∩ 𝐷)) → ((𝐹‘𝑛)‘𝑥) ≤ 𝐴)
141	117, 118, 120, 140	syl3anc 1369	. . . . . . . . . . 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 3139	. . . . . . . . 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 42852	. . . . . . . . 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 256	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑥 ∈ ∩ 𝑛 ∈ 𝑍 ((𝐻‘𝑛) ∩ 𝐷)) → sup(ran (𝑛 ∈ 𝑍 ↦ ((𝐹‘𝑛)‘𝑥)), ℝ, < ) ≤ 𝐴)
150	111, 149	eqbrtrd 5092	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑥 ∈ ∩ 𝑛 ∈ 𝑍 ((𝐻‘𝑛) ∩ 𝐷)) → (𝐺‘𝑥) ≤ 𝐴)
151	102, 103, 106, 110, 150	eliocd 42935	. . . . . 6 ⊢ ((𝜑 ∧ 𝑥 ∈ ∩ 𝑛 ∈ 𝑍 ((𝐻‘𝑛) ∩ 𝐷)) → (𝐺‘𝑥) ∈ (-∞(,]𝐴))
152	93, 101, 151	elpreimad 6918	. . . . 5 ⊢ ((𝜑 ∧ 𝑥 ∈ ∩ 𝑛 ∈ 𝑍 ((𝐻‘𝑛) ∩ 𝐷)) → 𝑥 ∈ (◡𝐺 “ (-∞(,]𝐴)))
153	152	ssd 42519	. . . 4 ⊢ (𝜑 → ∩ 𝑛 ∈ 𝑍 ((𝐻‘𝑛) ∩ 𝐷) ⊆ (◡𝐺 “ (-∞(,]𝐴)))
154	92, 153	eqsstrd 3955	. . 3 ⊢ (𝜑 → (∩ 𝑛 ∈ 𝑍 (𝐻‘𝑛) ∩ 𝐷) ⊆ (◡𝐺 “ (-∞(,]𝐴)))
155	90, 154	eqssd 3934	. 2 ⊢ (𝜑 → (◡𝐺 “ (-∞(,]𝐴)) = (∩ 𝑛 ∈ 𝑍 (𝐻‘𝑛) ∩ 𝐷))
156		eqid 2738	. . . . 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 6769	. . . . . . . . 9 ⊢ (𝐹‘𝑛) ∈ V
158	157	dmex 7732	. . . . . . . 8 ⊢ dom (𝐹‘𝑛) ∈ V
159	158	rgenw 3075	. . . . . . 7 ⊢ ∀𝑛 ∈ 𝑍 dom (𝐹‘𝑛) ∈ V
160	159	a1i 11	. . . . . 6 ⊢ (𝜑 → ∀𝑛 ∈ 𝑍 dom (𝐹‘𝑛) ∈ V)
161	24, 160	iinexd 42571	. . . . 5 ⊢ (𝜑 → ∩ 𝑛 ∈ 𝑍 dom (𝐹‘𝑛) ∈ V)
162	156, 161	rabexd 5252	. . . 4 ⊢ (𝜑 → {𝑥 ∈ ∩ 𝑛 ∈ 𝑍 dom (𝐹‘𝑛) ∣ ∃𝑦 ∈ ℝ ∀𝑛 ∈ 𝑍 ((𝐹‘𝑛)‘𝑥) ≤ 𝑦} ∈ V)
163	9, 162	eqeltrid 2843	. . 3 ⊢ (𝜑 → 𝐷 ∈ V)
164	22	uzct 42500	. . . . 5 ⊢ 𝑍 ≼ ω
165	164	a1i 11	. . . 4 ⊢ (𝜑 → 𝑍 ≼ ω)
166		smfsuplem1.h	. . . . 5 ⊢ (𝜑 → 𝐻:𝑍⟶𝑆)
167	166	ffvelrnda 6943	. . . 4 ⊢ ((𝜑 ∧ 𝑛 ∈ 𝑍) → (𝐻‘𝑛) ∈ 𝑆)
168	1, 165, 24, 167	saliincl 43756	. . 3 ⊢ (𝜑 → ∩ 𝑛 ∈ 𝑍 (𝐻‘𝑛) ∈ 𝑆)
169		eqid 2738	. . 3 ⊢ (∩ 𝑛 ∈ 𝑍 (𝐻‘𝑛) ∩ 𝐷) = (∩ 𝑛 ∈ 𝑍 (𝐻‘𝑛) ∩ 𝐷)
170	1, 163, 168, 169	elrestd 42547	. 2 ⊢ (𝜑 → (∩ 𝑛 ∈ 𝑍 (𝐻‘𝑛) ∩ 𝐷) ∈ (𝑆 ↾t 𝐷))
171	155, 170	eqeltrd 2839	1 ⊢ (𝜑 → (◡𝐺 “ (-∞(,]𝐴)) ∈ (𝑆 ↾t 𝐷))

Syntax hints:   → wi 4   ↔ wb 205   ∧ wa 395   ∧ w3a 1085   = wceq 1539   ∈ wcel 2108   ≠ wne 2942  ∀wral 3063  ∃wrex 3064  {crab 3067  Vcvv 3422   ∩ cin 3882   ⊆ wss 3883  ∅c0 4253  ∩ ciin 4922   class class class wbr 5070   ↦ cmpt 5153  ^◡ccnv 5579  dom cdm 5580  ran crn 5581   “ cima 5583   Fn wfn 6413  ⟶wf 6414  ‘cfv 6418  (class class class)co 7255  ωcom 7687   ≼ cdom 8689  supcsup 9129  ℝcr 10801  -∞cmnf 10938  ℝ^*cxr 10939   < clt 10940   ≤ cle 10941  ℤcz 12249  ℤ_≥cuz 12511  (,]cioc 13009   ↾_t crest 17048  SAlgcsalg 43739  SMblFncsmblfn 44123

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1799  ax-4 1813  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2110  ax-9 2118  ax-10 2139  ax-11 2156  ax-12 2173  ax-ext 2709  ax-rep 5205  ax-sep 5218  ax-nul 5225  ax-pow 5283  ax-pr 5347  ax-un 7566  ax-inf2 9329  ax-cnex 10858  ax-resscn 10859  ax-1cn 10860  ax-icn 10861  ax-addcl 10862  ax-addrcl 10863  ax-mulcl 10864  ax-mulrcl 10865  ax-mulcom 10866  ax-addass 10867  ax-mulass 10868  ax-distr 10869  ax-i2m1 10870  ax-1ne0 10871  ax-1rid 10872  ax-rnegex 10873  ax-rrecex 10874  ax-cnre 10875  ax-pre-lttri 10876  ax-pre-lttrn 10877  ax-pre-ltadd 10878  ax-pre-mulgt0 10879  ax-pre-sup 10880

This theorem depends on definitions:  df-bi 206  df-an 396  df-or 844  df-3or 1086  df-3an 1087  df-tru 1542  df-fal 1552  df-ex 1784  df-nf 1788  df-sb 2069  df-mo 2540  df-eu 2569  df-clab 2716  df-cleq 2730  df-clel 2817  df-nfc 2888  df-ne 2943  df-nel 3049  df-ral 3068  df-rex 3069  df-reu 3070  df-rmo 3071  df-rab 3072  df-v 3424  df-sbc 3712  df-csb 3829  df-dif 3886  df-un 3888  df-in 3890  df-ss 3900  df-pss 3902  df-nul 4254  df-if 4457  df-pw 4532  df-sn 4559  df-pr 4561  df-tp 4563  df-op 4565  df-uni 4837  df-int 4877  df-iun 4923  df-iin 4924  df-br 5071  df-opab 5133  df-mpt 5154  df-tr 5188  df-id 5480  df-eprel 5486  df-po 5494  df-so 5495  df-fr 5535  df-se 5536  df-we 5537  df-xp 5586  df-rel 5587  df-cnv 5588  df-co 5589  df-dm 5590  df-rn 5591  df-res 5592  df-ima 5593  df-pred 6191  df-ord 6254  df-on 6255  df-lim 6256  df-suc 6257  df-iota 6376  df-fun 6420  df-fn 6421  df-f 6422  df-f1 6423  df-fo 6424  df-f1o 6425  df-fv 6426  df-isom 6427  df-riota 7212  df-ov 7258  df-oprab 7259  df-mpo 7260  df-om 7688  df-1st 7804  df-2nd 7805  df-frecs 8068  df-wrecs 8099  df-recs 8173  df-rdg 8212  df-1o 8267  df-oadd 8271  df-omul 8272  df-er 8456  df-map 8575  df-pm 8576  df-en 8692  df-dom 8693  df-sdom 8694  df-fin 8695  df-sup 9131  df-oi 9199  df-card 9628  df-acn 9631  df-pnf 10942  df-mnf 10943  df-xr 10944  df-ltxr 10945  df-le 10946  df-sub 11137  df-neg 11138  df-nn 11904  df-n0 12164  df-z 12250  df-uz 12512  df-ioo 13012  df-ioc 13013  df-ico 13014  df-rest 17050  df-salg 43740  df-smblfn 44124

This theorem is referenced by:  smfsuplem2  44232