Theorem smflimlem1 42611

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 6558	. . . . . . 7 ⊢ (ℤ≥‘𝑀) ∈ V
5	3, 4	eqeltri 2881	. . . . . 6 ⊢ 𝑍 ∈ V
6		uzssz 12117	. . . . . . . . . . 11 ⊢ (ℤ≥‘𝑀) ⊆ ℤ
7	3	eleq2i 2876	. . . . . . . . . . . 12 ⊢ (𝑛 ∈ 𝑍 ↔ 𝑛 ∈ (ℤ≥‘𝑀))
8	7	biimpi 217	. . . . . . . . . . 11 ⊢ (𝑛 ∈ 𝑍 → 𝑛 ∈ (ℤ≥‘𝑀))
9	6, 8	sseldi 3893	. . . . . . . . . 10 ⊢ (𝑛 ∈ 𝑍 → 𝑛 ∈ ℤ)
10		uzid 12112	. . . . . . . . . 10 ⊢ (𝑛 ∈ ℤ → 𝑛 ∈ (ℤ≥‘𝑛))
11	9, 10	syl 17	. . . . . . . . 9 ⊢ (𝑛 ∈ 𝑍 → 𝑛 ∈ (ℤ≥‘𝑛))
12	11	ne0d 4227	. . . . . . . 8 ⊢ (𝑛 ∈ 𝑍 → (ℤ≥‘𝑛) ≠ ∅)
13		fvex 6558	. . . . . . . . . . 11 ⊢ (𝐹‘𝑚) ∈ V
14	13	dmex 7479	. . . . . . . . . 10 ⊢ dom (𝐹‘𝑚) ∈ V
15	14	rgenw 3119	. . . . . . . . 9 ⊢ ∀𝑚 ∈ (ℤ≥‘𝑛)dom (𝐹‘𝑚) ∈ V
16	15	a1i 11	. . . . . . . 8 ⊢ (𝑛 ∈ 𝑍 → ∀𝑚 ∈ (ℤ≥‘𝑛)dom (𝐹‘𝑚) ∈ V)
17		iinexg 5142	. . . . . . . 8 ⊢ (((ℤ≥‘𝑛) ≠ ∅ ∧ ∀𝑚 ∈ (ℤ≥‘𝑛)dom (𝐹‘𝑚) ∈ V) → ∩ 𝑚 ∈ (ℤ≥‘𝑛)dom (𝐹‘𝑚) ∈ V)
18	12, 16, 17	syl2anc 584	. . . . . . 7 ⊢ (𝑛 ∈ 𝑍 → ∩ 𝑚 ∈ (ℤ≥‘𝑛)dom (𝐹‘𝑚) ∈ V)
19	18	rgen 3117	. . . . . 6 ⊢ ∀𝑛 ∈ 𝑍 ∩ 𝑚 ∈ (ℤ≥‘𝑛)dom (𝐹‘𝑚) ∈ V
20		iunexg 7527	. . . . . 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 688	. . . . 5 ⊢ ∪ 𝑛 ∈ 𝑍 ∩ 𝑚 ∈ (ℤ≥‘𝑛)dom (𝐹‘𝑚) ∈ V
22	21	rabex 5133	. . . 4 ⊢ {𝑥 ∈ ∪ 𝑛 ∈ 𝑍 ∩ 𝑚 ∈ (ℤ≥‘𝑛)dom (𝐹‘𝑚) ∣ (𝑚 ∈ 𝑍 ↦ ((𝐹‘𝑚)‘𝑥)) ∈ dom ⇝ } ∈ V
23	2, 22	eqeltri 2881	. . 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 13203	. . . . 5 ⊢ ℕ ≼ ω
27	26	a1i 11	. . . 4 ⊢ (𝜑 → ℕ ≼ ω)
28		nnn0 41209	. . . . 5 ⊢ ℕ ≠ ∅
29	28	a1i 11	. . . 4 ⊢ (𝜑 → ℕ ≠ ∅)
30	1	adantr 481	. . . . 5 ⊢ ((𝜑 ∧ 𝑘 ∈ ℕ) → 𝑆 ∈ SAlg)
31	3	uzct 40885	. . . . . 6 ⊢ 𝑍 ≼ ω
32	31	a1i 11	. . . . 5 ⊢ ((𝜑 ∧ 𝑘 ∈ ℕ) → 𝑍 ≼ ω)
33	30	adantr 481	. . . . . 6 ⊢ (((𝜑 ∧ 𝑘 ∈ ℕ) ∧ 𝑛 ∈ 𝑍) → 𝑆 ∈ SAlg)
34		eqid 2797	. . . . . . . 8 ⊢ (ℤ≥‘𝑛) = (ℤ≥‘𝑛)
35	34	uzct 40885	. . . . . . 7 ⊢ (ℤ≥‘𝑛) ≼ ω
36	35	a1i 11	. . . . . 6 ⊢ (((𝜑 ∧ 𝑘 ∈ ℕ) ∧ 𝑛 ∈ 𝑍) → (ℤ≥‘𝑛) ≼ ω)
37	12	adantl 482	. . . . . 6 ⊢ (((𝜑 ∧ 𝑘 ∈ ℕ) ∧ 𝑛 ∈ 𝑍) → (ℤ≥‘𝑛) ≠ ∅)
38		simpll 763	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝑛 ∈ 𝑍) ∧ 𝑚 ∈ (ℤ≥‘𝑛)) → 𝜑)
39	38	adantllr 715	. . . . . . 7 ⊢ ((((𝜑 ∧ 𝑘 ∈ ℕ) ∧ 𝑛 ∈ 𝑍) ∧ 𝑚 ∈ (ℤ≥‘𝑛)) → 𝜑)
40		simpll 763	. . . . . . . 8 ⊢ (((𝑘 ∈ ℕ ∧ 𝑛 ∈ 𝑍) ∧ 𝑚 ∈ (ℤ≥‘𝑛)) → 𝑘 ∈ ℕ)
41	40	adantlll 714	. . . . . . 7 ⊢ ((((𝜑 ∧ 𝑘 ∈ ℕ) ∧ 𝑛 ∈ 𝑍) ∧ 𝑚 ∈ (ℤ≥‘𝑛)) → 𝑘 ∈ ℕ)
42	3	uztrn2 12115	. . . . . . . . . 10 ⊢ ((𝑛 ∈ 𝑍 ∧ 𝑗 ∈ (ℤ≥‘𝑛)) → 𝑗 ∈ 𝑍)
43	42	ssd 40904	. . . . . . . . 9 ⊢ (𝑛 ∈ 𝑍 → (ℤ≥‘𝑛) ⊆ 𝑍)
44	43	sselda 3895	. . . . . . . 8 ⊢ ((𝑛 ∈ 𝑍 ∧ 𝑚 ∈ (ℤ≥‘𝑛)) → 𝑚 ∈ 𝑍)
45	44	adantll 710	. . . . . . 7 ⊢ ((((𝜑 ∧ 𝑘 ∈ ℕ) ∧ 𝑛 ∈ 𝑍) ∧ 𝑚 ∈ (ℤ≥‘𝑛)) → 𝑚 ∈ 𝑍)
46		simp3 1131	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑘 ∈ ℕ ∧ 𝑚 ∈ 𝑍) → 𝑚 ∈ 𝑍)
47		simp2 1130	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑘 ∈ ℕ ∧ 𝑚 ∈ 𝑍) → 𝑘 ∈ ℕ)
48		fvex 6558	. . . . . . . . . 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 7160	. . . . . . . . 9 ⊢ ((𝑚 ∈ 𝑍 ∧ 𝑘 ∈ ℕ ∧ (𝐶‘(𝑚𝑃𝑘)) ∈ V) → (𝑚𝐻𝑘) = (𝐶‘(𝑚𝑃𝑘)))
52	46, 47, 49, 51	syl3anc 1364	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑘 ∈ ℕ ∧ 𝑚 ∈ 𝑍) → (𝑚𝐻𝑘) = (𝐶‘(𝑚𝑃𝑘)))
53		simp1 1129	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑘 ∈ ℕ ∧ 𝑚 ∈ 𝑍) → 𝜑)
54		eqid 2797	. . . . . . . . . . . . 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 5134	. . . . . . . . . . . 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 7160	. . . . . . . . . . 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 1364	. . . . . . . . . 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 3983	. . . . . . . . . 10 ⊢ {𝑠 ∈ 𝑆 ∣ {𝑥 ∈ dom (𝐹‘𝑚) ∣ ((𝐹‘𝑚)‘𝑥) < (𝐴 + (1 / 𝑘))} = (𝑠 ∩ dom (𝐹‘𝑚))} ⊆ 𝑆
61	59, 60	syl6eqss 3948	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑘 ∈ ℕ ∧ 𝑚 ∈ 𝑍) → (𝑚𝑃𝑘) ⊆ 𝑆)
62	55	ralrimivw 3152	. . . . . . . . . . . . 13 ⊢ (𝜑 → ∀𝑘 ∈ ℕ {𝑠 ∈ 𝑆 ∣ {𝑥 ∈ dom (𝐹‘𝑚) ∣ ((𝐹‘𝑚)‘𝑥) < (𝐴 + (1 / 𝑘))} = (𝑠 ∩ dom (𝐹‘𝑚))} ∈ V)
63	62	ralrimivw 3152	. . . . . . . . . . . 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 1126	. . . . . . . . . . 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 41055	. . . . . . . . . . 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 1364	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑘 ∈ ℕ ∧ 𝑚 ∈ 𝑍) → (𝑚𝑃𝑘) ∈ ran 𝑃)
67		ovex 7055	. . . . . . . . . . 11 ⊢ (𝑚𝑃𝑘) ∈ V
68		eleq1 2872	. . . . . . . . . . . . 13 ⊢ (𝑟 = (𝑚𝑃𝑘) → (𝑟 ∈ ran 𝑃 ↔ (𝑚𝑃𝑘) ∈ ran 𝑃))
69	68	anbi2d 628	. . . . . . . . . . . 12 ⊢ (𝑟 = (𝑚𝑃𝑘) → ((𝜑 ∧ 𝑟 ∈ ran 𝑃) ↔ (𝜑 ∧ (𝑚𝑃𝑘) ∈ ran 𝑃)))
70		fveq2 6545	. . . . . . . . . . . . 13 ⊢ (𝑟 = (𝑚𝑃𝑘) → (𝐶‘𝑟) = (𝐶‘(𝑚𝑃𝑘)))
71		id 22	. . . . . . . . . . . . 13 ⊢ (𝑟 = (𝑚𝑃𝑘) → 𝑟 = (𝑚𝑃𝑘))
72	70, 71	eleq12d 2879	. . . . . . . . . . . 12 ⊢ (𝑟 = (𝑚𝑃𝑘) → ((𝐶‘𝑟) ∈ 𝑟 ↔ (𝐶‘(𝑚𝑃𝑘)) ∈ (𝑚𝑃𝑘)))
73	69, 72	imbi12d 346	. . . . . . . . . . 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 3505	. . . . . . . . . 10 ⊢ ((𝜑 ∧ (𝑚𝑃𝑘) ∈ ran 𝑃) → (𝐶‘(𝑚𝑃𝑘)) ∈ (𝑚𝑃𝑘))
76	53, 66, 75	syl2anc 584	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑘 ∈ ℕ ∧ 𝑚 ∈ 𝑍) → (𝐶‘(𝑚𝑃𝑘)) ∈ (𝑚𝑃𝑘))
77	61, 76	sseldd 3896	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑘 ∈ ℕ ∧ 𝑚 ∈ 𝑍) → (𝐶‘(𝑚𝑃𝑘)) ∈ 𝑆)
78	52, 77	eqeltrd 2885	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑘 ∈ ℕ ∧ 𝑚 ∈ 𝑍) → (𝑚𝐻𝑘) ∈ 𝑆)
79	39, 41, 45, 78	syl3anc 1364	. . . . . 6 ⊢ ((((𝜑 ∧ 𝑘 ∈ ℕ) ∧ 𝑛 ∈ 𝑍) ∧ 𝑚 ∈ (ℤ≥‘𝑛)) → (𝑚𝐻𝑘) ∈ 𝑆)
80	33, 36, 37, 79	saliincl 42174	. . . . 5 ⊢ (((𝜑 ∧ 𝑘 ∈ ℕ) ∧ 𝑛 ∈ 𝑍) → ∩ 𝑚 ∈ (ℤ≥‘𝑛)(𝑚𝐻𝑘) ∈ 𝑆)
81	30, 32, 80	saliuncl 42171	. . . 4 ⊢ ((𝜑 ∧ 𝑘 ∈ ℕ) → ∪ 𝑛 ∈ 𝑍 ∩ 𝑚 ∈ (ℤ≥‘𝑛)(𝑚𝐻𝑘) ∈ 𝑆)
82	1, 27, 29, 81	saliincl 42174	. . 3 ⊢ (𝜑 → ∩ 𝑘 ∈ ℕ ∪ 𝑛 ∈ 𝑍 ∩ 𝑚 ∈ (ℤ≥‘𝑛)(𝑚𝐻𝑘) ∈ 𝑆)
83	25, 82	syl5eqel 2889	. 2 ⊢ (𝜑 → 𝐼 ∈ 𝑆)
84		incom 4105	. 2 ⊢ (𝐷 ∩ 𝐼) = (𝐼 ∩ 𝐷)
85	1, 24, 83, 84	elrestd 40935	1 ⊢ (𝜑 → (𝐷 ∩ 𝐼) ∈ (𝑆 ↾t 𝐷))

Syntax hints:   → wi 4   ∧ wa 396   ∧ w3a 1080   = wceq 1525   ∈ wcel 2083   ≠ wne 2986  ∀wral 3107  {crab 3111  Vcvv 3440   ∩ cin 3864  ∅c0 4217  ∪ ciun 4831  ∩ ciin 4832   class class class wbr 4968   ↦ cmpt 5047  dom cdm 5450  ran crn 5451  ‘cfv 6232  (class class class)co 7023   ∈ cmpo 7025  ωcom 7443   ≼ cdom 8362  1c1 10391   + caddc 10393   < clt 10528   / cdiv 11151  ℕcn 11492  ℤcz 11835  ℤ_≥cuz 12097   ⇝ cli 14679   ↾_t crest 16527  SAlgcsalg 42157

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1781  ax-4 1795  ax-5 1892  ax-6 1951  ax-7 1996  ax-8 2085  ax-9 2093  ax-10 2114  ax-11 2128  ax-12 2143  ax-13 2346  ax-ext 2771  ax-rep 5088  ax-sep 5101  ax-nul 5108  ax-pow 5164  ax-pr 5228  ax-un 7326  ax-inf2 8957  ax-cnex 10446  ax-resscn 10447  ax-1cn 10448  ax-icn 10449  ax-addcl 10450  ax-addrcl 10451  ax-mulcl 10452  ax-mulrcl 10453  ax-mulcom 10454  ax-addass 10455  ax-mulass 10456  ax-distr 10457  ax-i2m1 10458  ax-1ne0 10459  ax-1rid 10460  ax-rnegex 10461  ax-rrecex 10462  ax-cnre 10463  ax-pre-lttri 10464  ax-pre-lttrn 10465  ax-pre-ltadd 10466  ax-pre-mulgt0 10467

This theorem depends on definitions:  df-bi 208  df-an 397  df-or 843  df-3or 1081  df-3an 1082  df-tru 1528  df-ex 1766  df-nf 1770  df-sb 2045  df-mo 2578  df-eu 2614  df-clab 2778  df-cleq 2790  df-clel 2865  df-nfc 2937  df-ne 2987  df-nel 3093  df-ral 3112  df-rex 3113  df-reu 3114  df-rmo 3115  df-rab 3116  df-v 3442  df-sbc 3712  df-csb 3818  df-dif 3868  df-un 3870  df-in 3872  df-ss 3880  df-pss 3882  df-nul 4218  df-if 4388  df-pw 4461  df-sn 4479  df-pr 4481  df-tp 4483  df-op 4485  df-uni 4752  df-int 4789  df-iun 4833  df-iin 4834  df-br 4969  df-opab 5031  df-mpt 5048  df-tr 5071  df-id 5355  df-eprel 5360  df-po 5369  df-so 5370  df-fr 5409  df-se 5410  df-we 5411  df-xp 5456  df-rel 5457  df-cnv 5458  df-co 5459  df-dm 5460  df-rn 5461  df-res 5462  df-ima 5463  df-pred 6030  df-ord 6076  df-on 6077  df-lim 6078  df-suc 6079  df-iota 6196  df-fun 6234  df-fn 6235  df-f 6236  df-f1 6237  df-fo 6238  df-f1o 6239  df-fv 6240  df-isom 6241  df-riota 6984  df-ov 7026  df-oprab 7027  df-mpo 7028  df-om 7444  df-1st 7552  df-2nd 7553  df-wrecs 7805  df-recs 7867  df-rdg 7905  df-1o 7960  df-oadd 7964  df-omul 7965  df-er 8146  df-map 8265  df-en 8365  df-dom 8366  df-sdom 8367  df-fin 8368  df-oi 8827  df-card 9221  df-acn 9224  df-pnf 10530  df-mnf 10531  df-xr 10532  df-ltxr 10533  df-le 10534  df-sub 10725  df-neg 10726  df-nn 11493  df-n0 11752  df-z 11836  df-uz 12098  df-rest 16529  df-salg 42158

This theorem is referenced by:  smflimlem5  42615