Theorem smflimlem1 47199

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 6853	. . . . . . 7 ⊢ (ℤ≥‘𝑀) ∈ V
5	3, 4	eqeltri 2832	. . . . . 6 ⊢ 𝑍 ∈ V
6		uzssz 12809	. . . . . . . . . . 11 ⊢ (ℤ≥‘𝑀) ⊆ ℤ
7	3	eleq2i 2828	. . . . . . . . . . . 12 ⊢ (𝑛 ∈ 𝑍 ↔ 𝑛 ∈ (ℤ≥‘𝑀))
8	7	biimpi 216	. . . . . . . . . . 11 ⊢ (𝑛 ∈ 𝑍 → 𝑛 ∈ (ℤ≥‘𝑀))
9	6, 8	sselid 3919	. . . . . . . . . 10 ⊢ (𝑛 ∈ 𝑍 → 𝑛 ∈ ℤ)
10		uzid 12803	. . . . . . . . . 10 ⊢ (𝑛 ∈ ℤ → 𝑛 ∈ (ℤ≥‘𝑛))
11	9, 10	syl 17	. . . . . . . . 9 ⊢ (𝑛 ∈ 𝑍 → 𝑛 ∈ (ℤ≥‘𝑛))
12	11	ne0d 4282	. . . . . . . 8 ⊢ (𝑛 ∈ 𝑍 → (ℤ≥‘𝑛) ≠ ∅)
13		fvex 6853	. . . . . . . . . . 11 ⊢ (𝐹‘𝑚) ∈ V
14	13	dmex 7860	. . . . . . . . . 10 ⊢ dom (𝐹‘𝑚) ∈ V
15	14	rgenw 3055	. . . . . . . . 9 ⊢ ∀𝑚 ∈ (ℤ≥‘𝑛)dom (𝐹‘𝑚) ∈ V
16	15	a1i 11	. . . . . . . 8 ⊢ (𝑛 ∈ 𝑍 → ∀𝑚 ∈ (ℤ≥‘𝑛)dom (𝐹‘𝑚) ∈ V)
17		iinexg 5289	. . . . . . . 8 ⊢ (((ℤ≥‘𝑛) ≠ ∅ ∧ ∀𝑚 ∈ (ℤ≥‘𝑛)dom (𝐹‘𝑚) ∈ V) → ∩ 𝑚 ∈ (ℤ≥‘𝑛)dom (𝐹‘𝑚) ∈ V)
18	12, 16, 17	syl2anc 585	. . . . . . 7 ⊢ (𝑛 ∈ 𝑍 → ∩ 𝑚 ∈ (ℤ≥‘𝑛)dom (𝐹‘𝑚) ∈ V)
19	18	rgen 3053	. . . . . 6 ⊢ ∀𝑛 ∈ 𝑍 ∩ 𝑚 ∈ (ℤ≥‘𝑛)dom (𝐹‘𝑚) ∈ V
20		iunexg 7916	. . . . . 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 693	. . . . 5 ⊢ ∪ 𝑛 ∈ 𝑍 ∩ 𝑚 ∈ (ℤ≥‘𝑛)dom (𝐹‘𝑚) ∈ V
22	21	rabex 5280	. . . 4 ⊢ {𝑥 ∈ ∪ 𝑛 ∈ 𝑍 ∩ 𝑚 ∈ (ℤ≥‘𝑛)dom (𝐹‘𝑚) ∣ (𝑚 ∈ 𝑍 ↦ ((𝐹‘𝑚)‘𝑥)) ∈ dom ⇝ } ∈ V
23	2, 22	eqeltri 2832	. . 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 13943	. . . . 5 ⊢ ℕ ≼ ω
27	26	a1i 11	. . . 4 ⊢ (𝜑 → ℕ ≼ ω)
28		nnn0 45807	. . . . 5 ⊢ ℕ ≠ ∅
29	28	a1i 11	. . . 4 ⊢ (𝜑 → ℕ ≠ ∅)
30	1	adantr 480	. . . . 5 ⊢ ((𝜑 ∧ 𝑘 ∈ ℕ) → 𝑆 ∈ SAlg)
31	3	uzct 45494	. . . . . 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 2736	. . . . . . . 8 ⊢ (ℤ≥‘𝑛) = (ℤ≥‘𝑛)
35	34	uzct 45494	. . . . . . 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 767	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝑛 ∈ 𝑍) ∧ 𝑚 ∈ (ℤ≥‘𝑛)) → 𝜑)
39	38	adantllr 720	. . . . . . 7 ⊢ ((((𝜑 ∧ 𝑘 ∈ ℕ) ∧ 𝑛 ∈ 𝑍) ∧ 𝑚 ∈ (ℤ≥‘𝑛)) → 𝜑)
40		simpll 767	. . . . . . . 8 ⊢ (((𝑘 ∈ ℕ ∧ 𝑛 ∈ 𝑍) ∧ 𝑚 ∈ (ℤ≥‘𝑛)) → 𝑘 ∈ ℕ)
41	40	adantlll 719	. . . . . . 7 ⊢ ((((𝜑 ∧ 𝑘 ∈ ℕ) ∧ 𝑛 ∈ 𝑍) ∧ 𝑚 ∈ (ℤ≥‘𝑛)) → 𝑘 ∈ ℕ)
42	3	uztrn2 12807	. . . . . . . . . 10 ⊢ ((𝑛 ∈ 𝑍 ∧ 𝑗 ∈ (ℤ≥‘𝑛)) → 𝑗 ∈ 𝑍)
43	42	ssd 45511	. . . . . . . . 9 ⊢ (𝑛 ∈ 𝑍 → (ℤ≥‘𝑛) ⊆ 𝑍)
44	43	sselda 3921	. . . . . . . 8 ⊢ ((𝑛 ∈ 𝑍 ∧ 𝑚 ∈ (ℤ≥‘𝑛)) → 𝑚 ∈ 𝑍)
45	44	adantll 715	. . . . . . 7 ⊢ ((((𝜑 ∧ 𝑘 ∈ ℕ) ∧ 𝑛 ∈ 𝑍) ∧ 𝑚 ∈ (ℤ≥‘𝑛)) → 𝑚 ∈ 𝑍)
46		simp3 1139	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑘 ∈ ℕ ∧ 𝑚 ∈ 𝑍) → 𝑚 ∈ 𝑍)
47		simp2 1138	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑘 ∈ ℕ ∧ 𝑚 ∈ 𝑍) → 𝑘 ∈ ℕ)
48		fvex 6853	. . . . . . . . . 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 7514	. . . . . . . . 9 ⊢ ((𝑚 ∈ 𝑍 ∧ 𝑘 ∈ ℕ ∧ (𝐶‘(𝑚𝑃𝑘)) ∈ V) → (𝑚𝐻𝑘) = (𝐶‘(𝑚𝑃𝑘)))
52	46, 47, 49, 51	syl3anc 1374	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑘 ∈ ℕ ∧ 𝑚 ∈ 𝑍) → (𝑚𝐻𝑘) = (𝐶‘(𝑚𝑃𝑘)))
53		simp1 1137	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑘 ∈ ℕ ∧ 𝑚 ∈ 𝑍) → 𝜑)
54		eqid 2736	. . . . . . . . . . . . 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 5281	. . . . . . . . . . . 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 7514	. . . . . . . . . . 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 1374	. . . . . . . . . 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 4020	. . . . . . . . . 10 ⊢ {𝑠 ∈ 𝑆 ∣ {𝑥 ∈ dom (𝐹‘𝑚) ∣ ((𝐹‘𝑚)‘𝑥) < (𝐴 + (1 / 𝑘))} = (𝑠 ∩ dom (𝐹‘𝑚))} ⊆ 𝑆
61	59, 60	eqsstrdi 3966	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑘 ∈ ℕ ∧ 𝑚 ∈ 𝑍) → (𝑚𝑃𝑘) ⊆ 𝑆)
62	55	ralrimivw 3133	. . . . . . . . . . . . 13 ⊢ (𝜑 → ∀𝑘 ∈ ℕ {𝑠 ∈ 𝑆 ∣ {𝑥 ∈ dom (𝐹‘𝑚) ∣ ((𝐹‘𝑚)‘𝑥) < (𝐴 + (1 / 𝑘))} = (𝑠 ∩ dom (𝐹‘𝑚))} ∈ V)
63	62	ralrimivw 3133	. . . . . . . . . . . 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 1134	. . . . . . . . . . 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 45657	. . . . . . . . . . 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 1374	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑘 ∈ ℕ ∧ 𝑚 ∈ 𝑍) → (𝑚𝑃𝑘) ∈ ran 𝑃)
67		ovex 7400	. . . . . . . . . . 11 ⊢ (𝑚𝑃𝑘) ∈ V
68		eleq1 2824	. . . . . . . . . . . . 13 ⊢ (𝑟 = (𝑚𝑃𝑘) → (𝑟 ∈ ran 𝑃 ↔ (𝑚𝑃𝑘) ∈ ran 𝑃))
69	68	anbi2d 631	. . . . . . . . . . . 12 ⊢ (𝑟 = (𝑚𝑃𝑘) → ((𝜑 ∧ 𝑟 ∈ ran 𝑃) ↔ (𝜑 ∧ (𝑚𝑃𝑘) ∈ ran 𝑃)))
70		fveq2 6840	. . . . . . . . . . . . 13 ⊢ (𝑟 = (𝑚𝑃𝑘) → (𝐶‘𝑟) = (𝐶‘(𝑚𝑃𝑘)))
71		id 22	. . . . . . . . . . . . 13 ⊢ (𝑟 = (𝑚𝑃𝑘) → 𝑟 = (𝑚𝑃𝑘))
72	70, 71	eleq12d 2830	. . . . . . . . . . . 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 3503	. . . . . . . . . 10 ⊢ ((𝜑 ∧ (𝑚𝑃𝑘) ∈ ran 𝑃) → (𝐶‘(𝑚𝑃𝑘)) ∈ (𝑚𝑃𝑘))
76	53, 66, 75	syl2anc 585	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑘 ∈ ℕ ∧ 𝑚 ∈ 𝑍) → (𝐶‘(𝑚𝑃𝑘)) ∈ (𝑚𝑃𝑘))
77	61, 76	sseldd 3922	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑘 ∈ ℕ ∧ 𝑚 ∈ 𝑍) → (𝐶‘(𝑚𝑃𝑘)) ∈ 𝑆)
78	52, 77	eqeltrd 2836	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑘 ∈ ℕ ∧ 𝑚 ∈ 𝑍) → (𝑚𝐻𝑘) ∈ 𝑆)
79	39, 41, 45, 78	syl3anc 1374	. . . . . 6 ⊢ ((((𝜑 ∧ 𝑘 ∈ ℕ) ∧ 𝑛 ∈ 𝑍) ∧ 𝑚 ∈ (ℤ≥‘𝑛)) → (𝑚𝐻𝑘) ∈ 𝑆)
80	33, 36, 37, 79	saliincl 46755	. . . . 5 ⊢ (((𝜑 ∧ 𝑘 ∈ ℕ) ∧ 𝑛 ∈ 𝑍) → ∩ 𝑚 ∈ (ℤ≥‘𝑛)(𝑚𝐻𝑘) ∈ 𝑆)
81	30, 32, 80	saliuncl 46751	. . . 4 ⊢ ((𝜑 ∧ 𝑘 ∈ ℕ) → ∪ 𝑛 ∈ 𝑍 ∩ 𝑚 ∈ (ℤ≥‘𝑛)(𝑚𝐻𝑘) ∈ 𝑆)
82	1, 27, 29, 81	saliincl 46755	. . 3 ⊢ (𝜑 → ∩ 𝑘 ∈ ℕ ∪ 𝑛 ∈ 𝑍 ∩ 𝑚 ∈ (ℤ≥‘𝑛)(𝑚𝐻𝑘) ∈ 𝑆)
83	25, 82	eqeltrid 2840	. 2 ⊢ (𝜑 → 𝐼 ∈ 𝑆)
84		incom 4149	. 2 ⊢ (𝐷 ∩ 𝐼) = (𝐼 ∩ 𝐷)
85	1, 24, 83, 84	elrestd 45538	1 ⊢ (𝜑 → (𝐷 ∩ 𝐼) ∈ (𝑆 ↾t 𝐷))

Syntax hints:   → wi 4   ∧ wa 395   ∧ w3a 1087   = wceq 1542   ∈ wcel 2114   ≠ wne 2932  ∀wral 3051  {crab 3389  Vcvv 3429   ∩ cin 3888  ∅c0 4273  ∪ ciun 4933  ∩ ciin 4934   class class class wbr 5085   ↦ cmpt 5166  dom cdm 5631  ran crn 5632  ‘cfv 6498  (class class class)co 7367   ∈ cmpo 7369  ωcom 7817   ≼ cdom 8891  1c1 11039   + caddc 11041   < clt 11179   / cdiv 11807  ℕcn 12174  ℤcz 12524  ℤ_≥cuz 12788   ⇝ cli 15446   ↾_t crest 17383  SAlgcsalg 46736

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1912  ax-6 1969  ax-7 2010  ax-8 2116  ax-9 2124  ax-10 2147  ax-11 2163  ax-12 2185  ax-ext 2708  ax-rep 5212  ax-sep 5231  ax-nul 5241  ax-pow 5307  ax-pr 5375  ax-un 7689  ax-inf2 9562  ax-cnex 11094  ax-resscn 11095  ax-1cn 11096  ax-icn 11097  ax-addcl 11098  ax-addrcl 11099  ax-mulcl 11100  ax-mulrcl 11101  ax-mulcom 11102  ax-addass 11103  ax-mulass 11104  ax-distr 11105  ax-i2m1 11106  ax-1ne0 11107  ax-1rid 11108  ax-rnegex 11109  ax-rrecex 11110  ax-cnre 11111  ax-pre-lttri 11112  ax-pre-lttrn 11113  ax-pre-ltadd 11114  ax-pre-mulgt0 11115

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3or 1088  df-3an 1089  df-tru 1545  df-fal 1555  df-ex 1782  df-nf 1786  df-sb 2069  df-mo 2539  df-eu 2569  df-clab 2715  df-cleq 2728  df-clel 2811  df-nfc 2885  df-ne 2933  df-nel 3037  df-ral 3052  df-rex 3062  df-rmo 3342  df-reu 3343  df-rab 3390  df-v 3431  df-sbc 3729  df-csb 3838  df-dif 3892  df-un 3894  df-in 3896  df-ss 3906  df-pss 3909  df-nul 4274  df-if 4467  df-pw 4543  df-sn 4568  df-pr 4570  df-op 4574  df-uni 4851  df-int 4890  df-iun 4935  df-iin 4936  df-br 5086  df-opab 5148  df-mpt 5167  df-tr 5193  df-id 5526  df-eprel 5531  df-po 5539  df-so 5540  df-fr 5584  df-se 5585  df-we 5586  df-xp 5637  df-rel 5638  df-cnv 5639  df-co 5640  df-dm 5641  df-rn 5642  df-res 5643  df-ima 5644  df-pred 6265  df-ord 6326  df-on 6327  df-lim 6328  df-suc 6329  df-iota 6454  df-fun 6500  df-fn 6501  df-f 6502  df-f1 6503  df-fo 6504  df-f1o 6505  df-fv 6506  df-isom 6507  df-riota 7324  df-ov 7370  df-oprab 7371  df-mpo 7372  df-om 7818  df-1st 7942  df-2nd 7943  df-frecs 8231  df-wrecs 8262  df-recs 8311  df-rdg 8349  df-1o 8405  df-oadd 8409  df-omul 8410  df-er 8643  df-map 8775  df-en 8894  df-dom 8895  df-sdom 8896  df-fin 8897  df-oi 9425  df-card 9863  df-acn 9866  df-pnf 11181  df-mnf 11182  df-xr 11183  df-ltxr 11184  df-le 11185  df-sub 11379  df-neg 11380  df-nn 12175  df-n0 12438  df-z 12525  df-uz 12789  df-rest 17385  df-salg 46737

This theorem is referenced by:  smflimlem5  47203