Theorem smflimlem1 46692

Description: Lemma for the proof that the limit of a sequence of sigma-measurable functions is sigma-measurable, Proposition 121F (a) of [Fremlin1] p. 38 . This lemma proves that (𝐷 ∩ 𝐼) is in the subspace sigma-algebra induced by 𝐷. (Contributed by Glauco Siliprandi, 26-Jun-2021.)

Hypotheses

Ref	Expression
smflimlem1.1	⊢ 𝑍 = (ℤ≥‘𝑀)
smflimlem1.2	⊢ (𝜑 → 𝑆 ∈ SAlg)
smflimlem1.3	⊢ 𝐷 = {𝑥 ∈ ∪ 𝑛 ∈ 𝑍 ∩ 𝑚 ∈ (ℤ≥‘𝑛)dom (𝐹‘𝑚) ∣ (𝑚 ∈ 𝑍 ↦ ((𝐹‘𝑚)‘𝑥)) ∈ dom ⇝ }
smflimlem1.4	⊢ 𝑃 = (𝑚 ∈ 𝑍, 𝑘 ∈ ℕ ↦ {𝑠 ∈ 𝑆 ∣ {𝑥 ∈ dom (𝐹‘𝑚) ∣ ((𝐹‘𝑚)‘𝑥) < (𝐴 + (1 / 𝑘))} = (𝑠 ∩ dom (𝐹‘𝑚))})
smflimlem1.5	⊢ 𝐻 = (𝑚 ∈ 𝑍, 𝑘 ∈ ℕ ↦ (𝐶‘(𝑚𝑃𝑘)))
smflimlem1.6	⊢ 𝐼 = ∩ 𝑘 ∈ ℕ ∪ 𝑛 ∈ 𝑍 ∩ 𝑚 ∈ (ℤ≥‘𝑛)(𝑚𝐻𝑘)
smflimlem1.7	⊢ ((𝜑 ∧ 𝑟 ∈ ran 𝑃) → (𝐶‘𝑟) ∈ 𝑟)

Assertion

Ref	Expression
smflimlem1	⊢ (𝜑 → (𝐷 ∩ 𝐼) ∈ (𝑆 ↾t 𝐷))

Distinct variable groups:   𝐶,𝑟   𝑥,𝐹   𝑃,𝑟   𝑆,𝑘,𝑚,𝑛   𝑆,𝑠   𝑛,𝑍,𝑘,𝑚   𝑥,𝑍,𝑚,𝑛   𝜑,𝑘,𝑚,𝑛   𝑘,𝑟,𝑚,𝜑

Allowed substitution hints:   𝜑(𝑥,𝑠)   𝐴(𝑥,𝑘,𝑚,𝑛,𝑠,𝑟)   𝐶(𝑥,𝑘,𝑚,𝑛,𝑠)   𝐷(𝑥,𝑘,𝑚,𝑛,𝑠,𝑟)   𝑃(𝑥,𝑘,𝑚,𝑛,𝑠)   𝑆(𝑥,𝑟)   𝐹(𝑘,𝑚,𝑛,𝑠,𝑟)   𝐻(𝑥,𝑘,𝑚,𝑛,𝑠,𝑟)   𝐼(𝑥,𝑘,𝑚,𝑛,𝑠,𝑟)   𝑀(𝑥,𝑘,𝑚,𝑛,𝑠,𝑟)   𝑍(𝑠,𝑟)

Proof of Theorem smflimlem1

Dummy variable 𝑗 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		smflimlem1.2	. 2 ⊢ (𝜑 → 𝑆 ∈ SAlg)
2		smflimlem1.3	. . . 4 ⊢ 𝐷 = {𝑥 ∈ ∪ 𝑛 ∈ 𝑍 ∩ 𝑚 ∈ (ℤ≥‘𝑛)dom (𝐹‘𝑚) ∣ (𝑚 ∈ 𝑍 ↦ ((𝐹‘𝑚)‘𝑥)) ∈ dom ⇝ }
3		smflimlem1.1	. . . . . . 7 ⊢ 𝑍 = (ℤ≥‘𝑀)
4		fvex 6933	. . . . . . 7 ⊢ (ℤ≥‘𝑀) ∈ V
5	3, 4	eqeltri 2840	. . . . . 6 ⊢ 𝑍 ∈ V
6		uzssz 12924	. . . . . . . . . . 11 ⊢ (ℤ≥‘𝑀) ⊆ ℤ
7	3	eleq2i 2836	. . . . . . . . . . . 12 ⊢ (𝑛 ∈ 𝑍 ↔ 𝑛 ∈ (ℤ≥‘𝑀))
8	7	biimpi 216	. . . . . . . . . . 11 ⊢ (𝑛 ∈ 𝑍 → 𝑛 ∈ (ℤ≥‘𝑀))
9	6, 8	sselid 4006	. . . . . . . . . 10 ⊢ (𝑛 ∈ 𝑍 → 𝑛 ∈ ℤ)
10		uzid 12918	. . . . . . . . . 10 ⊢ (𝑛 ∈ ℤ → 𝑛 ∈ (ℤ≥‘𝑛))
11	9, 10	syl 17	. . . . . . . . 9 ⊢ (𝑛 ∈ 𝑍 → 𝑛 ∈ (ℤ≥‘𝑛))
12	11	ne0d 4365	. . . . . . . 8 ⊢ (𝑛 ∈ 𝑍 → (ℤ≥‘𝑛) ≠ ∅)
13		fvex 6933	. . . . . . . . . . 11 ⊢ (𝐹‘𝑚) ∈ V
14	13	dmex 7949	. . . . . . . . . 10 ⊢ dom (𝐹‘𝑚) ∈ V
15	14	rgenw 3071	. . . . . . . . 9 ⊢ ∀𝑚 ∈ (ℤ≥‘𝑛)dom (𝐹‘𝑚) ∈ V
16	15	a1i 11	. . . . . . . 8 ⊢ (𝑛 ∈ 𝑍 → ∀𝑚 ∈ (ℤ≥‘𝑛)dom (𝐹‘𝑚) ∈ V)
17		iinexg 5366	. . . . . . . 8 ⊢ (((ℤ≥‘𝑛) ≠ ∅ ∧ ∀𝑚 ∈ (ℤ≥‘𝑛)dom (𝐹‘𝑚) ∈ V) → ∩ 𝑚 ∈ (ℤ≥‘𝑛)dom (𝐹‘𝑚) ∈ V)
18	12, 16, 17	syl2anc 583	. . . . . . 7 ⊢ (𝑛 ∈ 𝑍 → ∩ 𝑚 ∈ (ℤ≥‘𝑛)dom (𝐹‘𝑚) ∈ V)
19	18	rgen 3069	. . . . . 6 ⊢ ∀𝑛 ∈ 𝑍 ∩ 𝑚 ∈ (ℤ≥‘𝑛)dom (𝐹‘𝑚) ∈ V
20		iunexg 8004	. . . . . 6 ⊢ ((𝑍 ∈ V ∧ ∀𝑛 ∈ 𝑍 ∩ 𝑚 ∈ (ℤ≥‘𝑛)dom (𝐹‘𝑚) ∈ V) → ∪ 𝑛 ∈ 𝑍 ∩ 𝑚 ∈ (ℤ≥‘𝑛)dom (𝐹‘𝑚) ∈ V)
21	5, 19, 20	mp2an 691	. . . . 5 ⊢ ∪ 𝑛 ∈ 𝑍 ∩ 𝑚 ∈ (ℤ≥‘𝑛)dom (𝐹‘𝑚) ∈ V
22	21	rabex 5357	. . . 4 ⊢ {𝑥 ∈ ∪ 𝑛 ∈ 𝑍 ∩ 𝑚 ∈ (ℤ≥‘𝑛)dom (𝐹‘𝑚) ∣ (𝑚 ∈ 𝑍 ↦ ((𝐹‘𝑚)‘𝑥)) ∈ dom ⇝ } ∈ V
23	2, 22	eqeltri 2840	. . 3 ⊢ 𝐷 ∈ V
24	23	a1i 11	. 2 ⊢ (𝜑 → 𝐷 ∈ V)
25		smflimlem1.6	. . 3 ⊢ 𝐼 = ∩ 𝑘 ∈ ℕ ∪ 𝑛 ∈ 𝑍 ∩ 𝑚 ∈ (ℤ≥‘𝑛)(𝑚𝐻𝑘)
26		nnct 14032	. . . . 5 ⊢ ℕ ≼ ω
27	26	a1i 11	. . . 4 ⊢ (𝜑 → ℕ ≼ ω)
28		nnn0 45293	. . . . 5 ⊢ ℕ ≠ ∅
29	28	a1i 11	. . . 4 ⊢ (𝜑 → ℕ ≠ ∅)
30	1	adantr 480	. . . . 5 ⊢ ((𝜑 ∧ 𝑘 ∈ ℕ) → 𝑆 ∈ SAlg)
31	3	uzct 44965	. . . . . 6 ⊢ 𝑍 ≼ ω
32	31	a1i 11	. . . . 5 ⊢ ((𝜑 ∧ 𝑘 ∈ ℕ) → 𝑍 ≼ ω)
33	30	adantr 480	. . . . . 6 ⊢ (((𝜑 ∧ 𝑘 ∈ ℕ) ∧ 𝑛 ∈ 𝑍) → 𝑆 ∈ SAlg)
34		eqid 2740	. . . . . . . 8 ⊢ (ℤ≥‘𝑛) = (ℤ≥‘𝑛)
35	34	uzct 44965	. . . . . . 7 ⊢ (ℤ≥‘𝑛) ≼ ω
36	35	a1i 11	. . . . . 6 ⊢ (((𝜑 ∧ 𝑘 ∈ ℕ) ∧ 𝑛 ∈ 𝑍) → (ℤ≥‘𝑛) ≼ ω)
37	12	adantl 481	. . . . . 6 ⊢ (((𝜑 ∧ 𝑘 ∈ ℕ) ∧ 𝑛 ∈ 𝑍) → (ℤ≥‘𝑛) ≠ ∅)
38		simpll 766	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝑛 ∈ 𝑍) ∧ 𝑚 ∈ (ℤ≥‘𝑛)) → 𝜑)
39	38	adantllr 718	. . . . . . 7 ⊢ ((((𝜑 ∧ 𝑘 ∈ ℕ) ∧ 𝑛 ∈ 𝑍) ∧ 𝑚 ∈ (ℤ≥‘𝑛)) → 𝜑)
40		simpll 766	. . . . . . . 8 ⊢ (((𝑘 ∈ ℕ ∧ 𝑛 ∈ 𝑍) ∧ 𝑚 ∈ (ℤ≥‘𝑛)) → 𝑘 ∈ ℕ)
41	40	adantlll 717	. . . . . . 7 ⊢ ((((𝜑 ∧ 𝑘 ∈ ℕ) ∧ 𝑛 ∈ 𝑍) ∧ 𝑚 ∈ (ℤ≥‘𝑛)) → 𝑘 ∈ ℕ)
42	3	uztrn2 12922	. . . . . . . . . 10 ⊢ ((𝑛 ∈ 𝑍 ∧ 𝑗 ∈ (ℤ≥‘𝑛)) → 𝑗 ∈ 𝑍)
43	42	ssd 44982	. . . . . . . . 9 ⊢ (𝑛 ∈ 𝑍 → (ℤ≥‘𝑛) ⊆ 𝑍)
44	43	sselda 4008	. . . . . . . 8 ⊢ ((𝑛 ∈ 𝑍 ∧ 𝑚 ∈ (ℤ≥‘𝑛)) → 𝑚 ∈ 𝑍)
45	44	adantll 713	. . . . . . 7 ⊢ ((((𝜑 ∧ 𝑘 ∈ ℕ) ∧ 𝑛 ∈ 𝑍) ∧ 𝑚 ∈ (ℤ≥‘𝑛)) → 𝑚 ∈ 𝑍)
46		simp3 1138	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑘 ∈ ℕ ∧ 𝑚 ∈ 𝑍) → 𝑚 ∈ 𝑍)
47		simp2 1137	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑘 ∈ ℕ ∧ 𝑚 ∈ 𝑍) → 𝑘 ∈ ℕ)
48		fvex 6933	. . . . . . . . . 10 ⊢ (𝐶‘(𝑚𝑃𝑘)) ∈ V
49	48	a1i 11	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑘 ∈ ℕ ∧ 𝑚 ∈ 𝑍) → (𝐶‘(𝑚𝑃𝑘)) ∈ V)
50		smflimlem1.5	. . . . . . . . . 10 ⊢ 𝐻 = (𝑚 ∈ 𝑍, 𝑘 ∈ ℕ ↦ (𝐶‘(𝑚𝑃𝑘)))
51	50	ovmpt4g 7597	. . . . . . . . 9 ⊢ ((𝑚 ∈ 𝑍 ∧ 𝑘 ∈ ℕ ∧ (𝐶‘(𝑚𝑃𝑘)) ∈ V) → (𝑚𝐻𝑘) = (𝐶‘(𝑚𝑃𝑘)))
52	46, 47, 49, 51	syl3anc 1371	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑘 ∈ ℕ ∧ 𝑚 ∈ 𝑍) → (𝑚𝐻𝑘) = (𝐶‘(𝑚𝑃𝑘)))
53		simp1 1136	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑘 ∈ ℕ ∧ 𝑚 ∈ 𝑍) → 𝜑)
54		eqid 2740	. . . . . . . . . . . . 13 ⊢ {𝑠 ∈ 𝑆 ∣ {𝑥 ∈ dom (𝐹‘𝑚) ∣ ((𝐹‘𝑚)‘𝑥) < (𝐴 + (1 / 𝑘))} = (𝑠 ∩ dom (𝐹‘𝑚))} = {𝑠 ∈ 𝑆 ∣ {𝑥 ∈ dom (𝐹‘𝑚) ∣ ((𝐹‘𝑚)‘𝑥) < (𝐴 + (1 / 𝑘))} = (𝑠 ∩ dom (𝐹‘𝑚))}
55	54, 1	rabexd 5358	. . . . . . . . . . . 12 ⊢ (𝜑 → {𝑠 ∈ 𝑆 ∣ {𝑥 ∈ dom (𝐹‘𝑚) ∣ ((𝐹‘𝑚)‘𝑥) < (𝐴 + (1 / 𝑘))} = (𝑠 ∩ dom (𝐹‘𝑚))} ∈ V)
56	53, 55	syl 17	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑘 ∈ ℕ ∧ 𝑚 ∈ 𝑍) → {𝑠 ∈ 𝑆 ∣ {𝑥 ∈ dom (𝐹‘𝑚) ∣ ((𝐹‘𝑚)‘𝑥) < (𝐴 + (1 / 𝑘))} = (𝑠 ∩ dom (𝐹‘𝑚))} ∈ V)
57		smflimlem1.4	. . . . . . . . . . . 12 ⊢ 𝑃 = (𝑚 ∈ 𝑍, 𝑘 ∈ ℕ ↦ {𝑠 ∈ 𝑆 ∣ {𝑥 ∈ dom (𝐹‘𝑚) ∣ ((𝐹‘𝑚)‘𝑥) < (𝐴 + (1 / 𝑘))} = (𝑠 ∩ dom (𝐹‘𝑚))})
58	57	ovmpt4g 7597	. . . . . . . . . . 11 ⊢ ((𝑚 ∈ 𝑍 ∧ 𝑘 ∈ ℕ ∧ {𝑠 ∈ 𝑆 ∣ {𝑥 ∈ dom (𝐹‘𝑚) ∣ ((𝐹‘𝑚)‘𝑥) < (𝐴 + (1 / 𝑘))} = (𝑠 ∩ dom (𝐹‘𝑚))} ∈ V) → (𝑚𝑃𝑘) = {𝑠 ∈ 𝑆 ∣ {𝑥 ∈ dom (𝐹‘𝑚) ∣ ((𝐹‘𝑚)‘𝑥) < (𝐴 + (1 / 𝑘))} = (𝑠 ∩ dom (𝐹‘𝑚))})
59	46, 47, 56, 58	syl3anc 1371	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑘 ∈ ℕ ∧ 𝑚 ∈ 𝑍) → (𝑚𝑃𝑘) = {𝑠 ∈ 𝑆 ∣ {𝑥 ∈ dom (𝐹‘𝑚) ∣ ((𝐹‘𝑚)‘𝑥) < (𝐴 + (1 / 𝑘))} = (𝑠 ∩ dom (𝐹‘𝑚))})
60		ssrab2 4103	. . . . . . . . . 10 ⊢ {𝑠 ∈ 𝑆 ∣ {𝑥 ∈ dom (𝐹‘𝑚) ∣ ((𝐹‘𝑚)‘𝑥) < (𝐴 + (1 / 𝑘))} = (𝑠 ∩ dom (𝐹‘𝑚))} ⊆ 𝑆
61	59, 60	eqsstrdi 4063	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑘 ∈ ℕ ∧ 𝑚 ∈ 𝑍) → (𝑚𝑃𝑘) ⊆ 𝑆)
62	55	ralrimivw 3156	. . . . . . . . . . . . 13 ⊢ (𝜑 → ∀𝑘 ∈ ℕ {𝑠 ∈ 𝑆 ∣ {𝑥 ∈ dom (𝐹‘𝑚) ∣ ((𝐹‘𝑚)‘𝑥) < (𝐴 + (1 / 𝑘))} = (𝑠 ∩ dom (𝐹‘𝑚))} ∈ V)
63	62	ralrimivw 3156	. . . . . . . . . . . 12 ⊢ (𝜑 → ∀𝑚 ∈ 𝑍 ∀𝑘 ∈ ℕ {𝑠 ∈ 𝑆 ∣ {𝑥 ∈ dom (𝐹‘𝑚) ∣ ((𝐹‘𝑚)‘𝑥) < (𝐴 + (1 / 𝑘))} = (𝑠 ∩ dom (𝐹‘𝑚))} ∈ V)
64	63	3ad2ant1 1133	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑘 ∈ ℕ ∧ 𝑚 ∈ 𝑍) → ∀𝑚 ∈ 𝑍 ∀𝑘 ∈ ℕ {𝑠 ∈ 𝑆 ∣ {𝑥 ∈ dom (𝐹‘𝑚) ∣ ((𝐹‘𝑚)‘𝑥) < (𝐴 + (1 / 𝑘))} = (𝑠 ∩ dom (𝐹‘𝑚))} ∈ V)
65	57	elrnmpoid 45135	. . . . . . . . . . 11 ⊢ ((𝑚 ∈ 𝑍 ∧ 𝑘 ∈ ℕ ∧ ∀𝑚 ∈ 𝑍 ∀𝑘 ∈ ℕ {𝑠 ∈ 𝑆 ∣ {𝑥 ∈ dom (𝐹‘𝑚) ∣ ((𝐹‘𝑚)‘𝑥) < (𝐴 + (1 / 𝑘))} = (𝑠 ∩ dom (𝐹‘𝑚))} ∈ V) → (𝑚𝑃𝑘) ∈ ran 𝑃)
66	46, 47, 64, 65	syl3anc 1371	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑘 ∈ ℕ ∧ 𝑚 ∈ 𝑍) → (𝑚𝑃𝑘) ∈ ran 𝑃)
67		ovex 7481	. . . . . . . . . . 11 ⊢ (𝑚𝑃𝑘) ∈ V
68		eleq1 2832	. . . . . . . . . . . . 13 ⊢ (𝑟 = (𝑚𝑃𝑘) → (𝑟 ∈ ran 𝑃 ↔ (𝑚𝑃𝑘) ∈ ran 𝑃))
69	68	anbi2d 629	. . . . . . . . . . . 12 ⊢ (𝑟 = (𝑚𝑃𝑘) → ((𝜑 ∧ 𝑟 ∈ ran 𝑃) ↔ (𝜑 ∧ (𝑚𝑃𝑘) ∈ ran 𝑃)))
70		fveq2 6920	. . . . . . . . . . . . 13 ⊢ (𝑟 = (𝑚𝑃𝑘) → (𝐶‘𝑟) = (𝐶‘(𝑚𝑃𝑘)))
71		id 22	. . . . . . . . . . . . 13 ⊢ (𝑟 = (𝑚𝑃𝑘) → 𝑟 = (𝑚𝑃𝑘))
72	70, 71	eleq12d 2838	. . . . . . . . . . . 12 ⊢ (𝑟 = (𝑚𝑃𝑘) → ((𝐶‘𝑟) ∈ 𝑟 ↔ (𝐶‘(𝑚𝑃𝑘)) ∈ (𝑚𝑃𝑘)))
73	69, 72	imbi12d 344	. . . . . . . . . . 11 ⊢ (𝑟 = (𝑚𝑃𝑘) → (((𝜑 ∧ 𝑟 ∈ ran 𝑃) → (𝐶‘𝑟) ∈ 𝑟) ↔ ((𝜑 ∧ (𝑚𝑃𝑘) ∈ ran 𝑃) → (𝐶‘(𝑚𝑃𝑘)) ∈ (𝑚𝑃𝑘))))
74		smflimlem1.7	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑟 ∈ ran 𝑃) → (𝐶‘𝑟) ∈ 𝑟)
75	67, 73, 74	vtocl 3570	. . . . . . . . . 10 ⊢ ((𝜑 ∧ (𝑚𝑃𝑘) ∈ ran 𝑃) → (𝐶‘(𝑚𝑃𝑘)) ∈ (𝑚𝑃𝑘))
76	53, 66, 75	syl2anc 583	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑘 ∈ ℕ ∧ 𝑚 ∈ 𝑍) → (𝐶‘(𝑚𝑃𝑘)) ∈ (𝑚𝑃𝑘))
77	61, 76	sseldd 4009	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑘 ∈ ℕ ∧ 𝑚 ∈ 𝑍) → (𝐶‘(𝑚𝑃𝑘)) ∈ 𝑆)
78	52, 77	eqeltrd 2844	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑘 ∈ ℕ ∧ 𝑚 ∈ 𝑍) → (𝑚𝐻𝑘) ∈ 𝑆)
79	39, 41, 45, 78	syl3anc 1371	. . . . . 6 ⊢ ((((𝜑 ∧ 𝑘 ∈ ℕ) ∧ 𝑛 ∈ 𝑍) ∧ 𝑚 ∈ (ℤ≥‘𝑛)) → (𝑚𝐻𝑘) ∈ 𝑆)
80	33, 36, 37, 79	saliincl 46248	. . . . 5 ⊢ (((𝜑 ∧ 𝑘 ∈ ℕ) ∧ 𝑛 ∈ 𝑍) → ∩ 𝑚 ∈ (ℤ≥‘𝑛)(𝑚𝐻𝑘) ∈ 𝑆)
81	30, 32, 80	saliuncl 46244	. . . 4 ⊢ ((𝜑 ∧ 𝑘 ∈ ℕ) → ∪ 𝑛 ∈ 𝑍 ∩ 𝑚 ∈ (ℤ≥‘𝑛)(𝑚𝐻𝑘) ∈ 𝑆)
82	1, 27, 29, 81	saliincl 46248	. . 3 ⊢ (𝜑 → ∩ 𝑘 ∈ ℕ ∪ 𝑛 ∈ 𝑍 ∩ 𝑚 ∈ (ℤ≥‘𝑛)(𝑚𝐻𝑘) ∈ 𝑆)
83	25, 82	eqeltrid 2848	. 2 ⊢ (𝜑 → 𝐼 ∈ 𝑆)
84		incom 4230	. 2 ⊢ (𝐷 ∩ 𝐼) = (𝐼 ∩ 𝐷)
85	1, 24, 83, 84	elrestd 45010	1 ⊢ (𝜑 → (𝐷 ∩ 𝐼) ∈ (𝑆 ↾t 𝐷))

Syntax hints:   → wi 4   ∧ wa 395   ∧ w3a 1087   = wceq 1537   ∈ wcel 2108   ≠ wne 2946  ∀wral 3067  {crab 3443  Vcvv 3488   ∩ cin 3975  ∅c0 4352  ∪ ciun 5015  ∩ ciin 5016   class class class wbr 5166   ↦ cmpt 5249  dom cdm 5700  ran crn 5701  ‘cfv 6573  (class class class)co 7448   ∈ cmpo 7450  ωcom 7903   ≼ cdom 9001  1c1 11185   + caddc 11187   < clt 11324   / cdiv 11947  ℕcn 12293  ℤcz 12639  ℤ_≥cuz 12903   ⇝ cli 15530   ↾_t crest 17480  SAlgcsalg 46229

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

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-en 9004  df-dom 9005  df-sdom 9006  df-fin 9007  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-rest 17482  df-salg 46230

This theorem is referenced by:  smflimlem5  46696