Theorem naddcnfass 44197

Description: Component-wise addition of Cantor normal forms is associative. (Contributed by RP, 3-Jan-2025.)

Assertion

Ref	Expression
naddcnfass	⊢ (((𝑋 ∈ On ∧ 𝑆 = dom (ω CNF 𝑋)) ∧ (𝐹 ∈ 𝑆 ∧ 𝐺 ∈ 𝑆 ∧ 𝐻 ∈ 𝑆)) → ((𝐹 ∘f +o 𝐺) ∘f +o 𝐻) = (𝐹 ∘f +o (𝐺 ∘f +o 𝐻)))

Proof of Theorem naddcnfass

Dummy variable 𝑥 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		simpr 490	. . . . . . . 8 ⊢ ((𝑋 ∈ On ∧ 𝑆 = dom (ω CNF 𝑋)) → 𝑆 = dom (ω CNF 𝑋))
2	1	eleq2d 2848	. . . . . . 7 ⊢ ((𝑋 ∈ On ∧ 𝑆 = dom (ω CNF 𝑋)) → (𝐹 ∈ 𝑆 ↔ 𝐹 ∈ dom (ω CNF 𝑋)))
3		eqid 2762	. . . . . . . 8 ⊢ dom (ω CNF 𝑋) = dom (ω CNF 𝑋)
4		omelon 9628	. . . . . . . . 9 ⊢ ω ∈ On
5	4	a1i 11	. . . . . . . 8 ⊢ ((𝑋 ∈ On ∧ 𝑆 = dom (ω CNF 𝑋)) → ω ∈ On)
6		simpl 488	. . . . . . . 8 ⊢ ((𝑋 ∈ On ∧ 𝑆 = dom (ω CNF 𝑋)) → 𝑋 ∈ On)
7	3, 5, 6	cantnfs 9648	. . . . . . 7 ⊢ ((𝑋 ∈ On ∧ 𝑆 = dom (ω CNF 𝑋)) → (𝐹 ∈ dom (ω CNF 𝑋) ↔ (𝐹:𝑋⟶ω ∧ 𝐹 finSupp ∅)))
8	2, 7	bitrd 282	. . . . . 6 ⊢ ((𝑋 ∈ On ∧ 𝑆 = dom (ω CNF 𝑋)) → (𝐹 ∈ 𝑆 ↔ (𝐹:𝑋⟶ω ∧ 𝐹 finSupp ∅)))
9		simpl 488	. . . . . . 7 ⊢ ((𝐹:𝑋⟶ω ∧ 𝐹 finSupp ∅) → 𝐹:𝑋⟶ω)
10	9	ffnd 6707	. . . . . 6 ⊢ ((𝐹:𝑋⟶ω ∧ 𝐹 finSupp ∅) → 𝐹 Fn 𝑋)
11	8, 10	biimtrdi 256	. . . . 5 ⊢ ((𝑋 ∈ On ∧ 𝑆 = dom (ω CNF 𝑋)) → (𝐹 ∈ 𝑆 → 𝐹 Fn 𝑋))
12		simp1 1154	. . . . 5 ⊢ ((𝐹 ∈ 𝑆 ∧ 𝐺 ∈ 𝑆 ∧ 𝐻 ∈ 𝑆) → 𝐹 ∈ 𝑆)
13	11, 12	impel 515	. . . 4 ⊢ (((𝑋 ∈ On ∧ 𝑆 = dom (ω CNF 𝑋)) ∧ (𝐹 ∈ 𝑆 ∧ 𝐺 ∈ 𝑆 ∧ 𝐻 ∈ 𝑆)) → 𝐹 Fn 𝑋)
14	1	eleq2d 2848	. . . . . . 7 ⊢ ((𝑋 ∈ On ∧ 𝑆 = dom (ω CNF 𝑋)) → (𝐺 ∈ 𝑆 ↔ 𝐺 ∈ dom (ω CNF 𝑋)))
15	3, 5, 6	cantnfs 9648	. . . . . . 7 ⊢ ((𝑋 ∈ On ∧ 𝑆 = dom (ω CNF 𝑋)) → (𝐺 ∈ dom (ω CNF 𝑋) ↔ (𝐺:𝑋⟶ω ∧ 𝐺 finSupp ∅)))
16	14, 15	bitrd 282	. . . . . 6 ⊢ ((𝑋 ∈ On ∧ 𝑆 = dom (ω CNF 𝑋)) → (𝐺 ∈ 𝑆 ↔ (𝐺:𝑋⟶ω ∧ 𝐺 finSupp ∅)))
17		simpl 488	. . . . . . 7 ⊢ ((𝐺:𝑋⟶ω ∧ 𝐺 finSupp ∅) → 𝐺:𝑋⟶ω)
18	17	ffnd 6707	. . . . . 6 ⊢ ((𝐺:𝑋⟶ω ∧ 𝐺 finSupp ∅) → 𝐺 Fn 𝑋)
19	16, 18	biimtrdi 256	. . . . 5 ⊢ ((𝑋 ∈ On ∧ 𝑆 = dom (ω CNF 𝑋)) → (𝐺 ∈ 𝑆 → 𝐺 Fn 𝑋))
20		simp2 1155	. . . . 5 ⊢ ((𝐹 ∈ 𝑆 ∧ 𝐺 ∈ 𝑆 ∧ 𝐻 ∈ 𝑆) → 𝐺 ∈ 𝑆)
21	19, 20	impel 515	. . . 4 ⊢ (((𝑋 ∈ On ∧ 𝑆 = dom (ω CNF 𝑋)) ∧ (𝐹 ∈ 𝑆 ∧ 𝐺 ∈ 𝑆 ∧ 𝐻 ∈ 𝑆)) → 𝐺 Fn 𝑋)
22	6	adantr 486	. . . 4 ⊢ (((𝑋 ∈ On ∧ 𝑆 = dom (ω CNF 𝑋)) ∧ (𝐹 ∈ 𝑆 ∧ 𝐺 ∈ 𝑆 ∧ 𝐻 ∈ 𝑆)) → 𝑋 ∈ On)
23		inidm 4175	. . . 4 ⊢ (𝑋 ∩ 𝑋) = 𝑋
24	13, 21, 22, 22, 23	offn 7694	. . 3 ⊢ (((𝑋 ∈ On ∧ 𝑆 = dom (ω CNF 𝑋)) ∧ (𝐹 ∈ 𝑆 ∧ 𝐺 ∈ 𝑆 ∧ 𝐻 ∈ 𝑆)) → (𝐹 ∘f +o 𝐺) Fn 𝑋)
25	1	eleq2d 2848	. . . . . 6 ⊢ ((𝑋 ∈ On ∧ 𝑆 = dom (ω CNF 𝑋)) → (𝐻 ∈ 𝑆 ↔ 𝐻 ∈ dom (ω CNF 𝑋)))
26	3, 5, 6	cantnfs 9648	. . . . . 6 ⊢ ((𝑋 ∈ On ∧ 𝑆 = dom (ω CNF 𝑋)) → (𝐻 ∈ dom (ω CNF 𝑋) ↔ (𝐻:𝑋⟶ω ∧ 𝐻 finSupp ∅)))
27	25, 26	bitrd 282	. . . . 5 ⊢ ((𝑋 ∈ On ∧ 𝑆 = dom (ω CNF 𝑋)) → (𝐻 ∈ 𝑆 ↔ (𝐻:𝑋⟶ω ∧ 𝐻 finSupp ∅)))
28		simpl 488	. . . . . 6 ⊢ ((𝐻:𝑋⟶ω ∧ 𝐻 finSupp ∅) → 𝐻:𝑋⟶ω)
29	28	ffnd 6707	. . . . 5 ⊢ ((𝐻:𝑋⟶ω ∧ 𝐻 finSupp ∅) → 𝐻 Fn 𝑋)
30	27, 29	biimtrdi 256	. . . 4 ⊢ ((𝑋 ∈ On ∧ 𝑆 = dom (ω CNF 𝑋)) → (𝐻 ∈ 𝑆 → 𝐻 Fn 𝑋))
31		simp3 1156	. . . 4 ⊢ ((𝐹 ∈ 𝑆 ∧ 𝐺 ∈ 𝑆 ∧ 𝐻 ∈ 𝑆) → 𝐻 ∈ 𝑆)
32	30, 31	impel 515	. . 3 ⊢ (((𝑋 ∈ On ∧ 𝑆 = dom (ω CNF 𝑋)) ∧ (𝐹 ∈ 𝑆 ∧ 𝐺 ∈ 𝑆 ∧ 𝐻 ∈ 𝑆)) → 𝐻 Fn 𝑋)
33	24, 32, 22, 22, 23	offn 7694	. 2 ⊢ (((𝑋 ∈ On ∧ 𝑆 = dom (ω CNF 𝑋)) ∧ (𝐹 ∈ 𝑆 ∧ 𝐺 ∈ 𝑆 ∧ 𝐻 ∈ 𝑆)) → ((𝐹 ∘f +o 𝐺) ∘f +o 𝐻) Fn 𝑋)
34	21, 32, 22, 22, 23	offn 7694	. . 3 ⊢ (((𝑋 ∈ On ∧ 𝑆 = dom (ω CNF 𝑋)) ∧ (𝐹 ∈ 𝑆 ∧ 𝐺 ∈ 𝑆 ∧ 𝐻 ∈ 𝑆)) → (𝐺 ∘f +o 𝐻) Fn 𝑋)
35	13, 34, 22, 22, 23	offn 7694	. 2 ⊢ (((𝑋 ∈ On ∧ 𝑆 = dom (ω CNF 𝑋)) ∧ (𝐹 ∈ 𝑆 ∧ 𝐺 ∈ 𝑆 ∧ 𝐻 ∈ 𝑆)) → (𝐹 ∘f +o (𝐺 ∘f +o 𝐻)) Fn 𝑋)
36	8, 9	biimtrdi 256	. . . . . . 7 ⊢ ((𝑋 ∈ On ∧ 𝑆 = dom (ω CNF 𝑋)) → (𝐹 ∈ 𝑆 → 𝐹:𝑋⟶ω))
37	36, 12	impel 515	. . . . . 6 ⊢ (((𝑋 ∈ On ∧ 𝑆 = dom (ω CNF 𝑋)) ∧ (𝐹 ∈ 𝑆 ∧ 𝐺 ∈ 𝑆 ∧ 𝐻 ∈ 𝑆)) → 𝐹:𝑋⟶ω)
38	37	ffvelcdmda 7080	. . . . 5 ⊢ ((((𝑋 ∈ On ∧ 𝑆 = dom (ω CNF 𝑋)) ∧ (𝐹 ∈ 𝑆 ∧ 𝐺 ∈ 𝑆 ∧ 𝐻 ∈ 𝑆)) ∧ 𝑥 ∈ 𝑋) → (𝐹‘𝑥) ∈ ω)
39	16, 17	biimtrdi 256	. . . . . . 7 ⊢ ((𝑋 ∈ On ∧ 𝑆 = dom (ω CNF 𝑋)) → (𝐺 ∈ 𝑆 → 𝐺:𝑋⟶ω))
40	39, 20	impel 515	. . . . . 6 ⊢ (((𝑋 ∈ On ∧ 𝑆 = dom (ω CNF 𝑋)) ∧ (𝐹 ∈ 𝑆 ∧ 𝐺 ∈ 𝑆 ∧ 𝐻 ∈ 𝑆)) → 𝐺:𝑋⟶ω)
41	40	ffvelcdmda 7080	. . . . 5 ⊢ ((((𝑋 ∈ On ∧ 𝑆 = dom (ω CNF 𝑋)) ∧ (𝐹 ∈ 𝑆 ∧ 𝐺 ∈ 𝑆 ∧ 𝐻 ∈ 𝑆)) ∧ 𝑥 ∈ 𝑋) → (𝐺‘𝑥) ∈ ω)
42	27, 28	biimtrdi 256	. . . . . . 7 ⊢ ((𝑋 ∈ On ∧ 𝑆 = dom (ω CNF 𝑋)) → (𝐻 ∈ 𝑆 → 𝐻:𝑋⟶ω))
43	42, 31	impel 515	. . . . . 6 ⊢ (((𝑋 ∈ On ∧ 𝑆 = dom (ω CNF 𝑋)) ∧ (𝐹 ∈ 𝑆 ∧ 𝐺 ∈ 𝑆 ∧ 𝐻 ∈ 𝑆)) → 𝐻:𝑋⟶ω)
44	43	ffvelcdmda 7080	. . . . 5 ⊢ ((((𝑋 ∈ On ∧ 𝑆 = dom (ω CNF 𝑋)) ∧ (𝐹 ∈ 𝑆 ∧ 𝐺 ∈ 𝑆 ∧ 𝐻 ∈ 𝑆)) ∧ 𝑥 ∈ 𝑋) → (𝐻‘𝑥) ∈ ω)
45		nnaass 8613	. . . . 5 ⊢ (((𝐹‘𝑥) ∈ ω ∧ (𝐺‘𝑥) ∈ ω ∧ (𝐻‘𝑥) ∈ ω) → (((𝐹‘𝑥) +o (𝐺‘𝑥)) +o (𝐻‘𝑥)) = ((𝐹‘𝑥) +o ((𝐺‘𝑥) +o (𝐻‘𝑥))))
46	38, 41, 44, 45	syl3anc 1398	. . . 4 ⊢ ((((𝑋 ∈ On ∧ 𝑆 = dom (ω CNF 𝑋)) ∧ (𝐹 ∈ 𝑆 ∧ 𝐺 ∈ 𝑆 ∧ 𝐻 ∈ 𝑆)) ∧ 𝑥 ∈ 𝑋) → (((𝐹‘𝑥) +o (𝐺‘𝑥)) +o (𝐻‘𝑥)) = ((𝐹‘𝑥) +o ((𝐺‘𝑥) +o (𝐻‘𝑥))))
47	13	adantr 486	. . . . . 6 ⊢ ((((𝑋 ∈ On ∧ 𝑆 = dom (ω CNF 𝑋)) ∧ (𝐹 ∈ 𝑆 ∧ 𝐺 ∈ 𝑆 ∧ 𝐻 ∈ 𝑆)) ∧ 𝑥 ∈ 𝑋) → 𝐹 Fn 𝑋)
48	21	adantr 486	. . . . . 6 ⊢ ((((𝑋 ∈ On ∧ 𝑆 = dom (ω CNF 𝑋)) ∧ (𝐹 ∈ 𝑆 ∧ 𝐺 ∈ 𝑆 ∧ 𝐻 ∈ 𝑆)) ∧ 𝑥 ∈ 𝑋) → 𝐺 Fn 𝑋)
49	22	anim1i 627	. . . . . 6 ⊢ ((((𝑋 ∈ On ∧ 𝑆 = dom (ω CNF 𝑋)) ∧ (𝐹 ∈ 𝑆 ∧ 𝐺 ∈ 𝑆 ∧ 𝐻 ∈ 𝑆)) ∧ 𝑥 ∈ 𝑋) → (𝑋 ∈ On ∧ 𝑥 ∈ 𝑋))
50		fnfvof 7698	. . . . . 6 ⊢ (((𝐹 Fn 𝑋 ∧ 𝐺 Fn 𝑋) ∧ (𝑋 ∈ On ∧ 𝑥 ∈ 𝑋)) → ((𝐹 ∘f +o 𝐺)‘𝑥) = ((𝐹‘𝑥) +o (𝐺‘𝑥)))
51	47, 48, 49, 50	syl21anc 851	. . . . 5 ⊢ ((((𝑋 ∈ On ∧ 𝑆 = dom (ω CNF 𝑋)) ∧ (𝐹 ∈ 𝑆 ∧ 𝐺 ∈ 𝑆 ∧ 𝐻 ∈ 𝑆)) ∧ 𝑥 ∈ 𝑋) → ((𝐹 ∘f +o 𝐺)‘𝑥) = ((𝐹‘𝑥) +o (𝐺‘𝑥)))
52	51	oveq1d 7431	. . . 4 ⊢ ((((𝑋 ∈ On ∧ 𝑆 = dom (ω CNF 𝑋)) ∧ (𝐹 ∈ 𝑆 ∧ 𝐺 ∈ 𝑆 ∧ 𝐻 ∈ 𝑆)) ∧ 𝑥 ∈ 𝑋) → (((𝐹 ∘f +o 𝐺)‘𝑥) +o (𝐻‘𝑥)) = (((𝐹‘𝑥) +o (𝐺‘𝑥)) +o (𝐻‘𝑥)))
53	32	adantr 486	. . . . . 6 ⊢ ((((𝑋 ∈ On ∧ 𝑆 = dom (ω CNF 𝑋)) ∧ (𝐹 ∈ 𝑆 ∧ 𝐺 ∈ 𝑆 ∧ 𝐻 ∈ 𝑆)) ∧ 𝑥 ∈ 𝑋) → 𝐻 Fn 𝑋)
54		fnfvof 7698	. . . . . 6 ⊢ (((𝐺 Fn 𝑋 ∧ 𝐻 Fn 𝑋) ∧ (𝑋 ∈ On ∧ 𝑥 ∈ 𝑋)) → ((𝐺 ∘f +o 𝐻)‘𝑥) = ((𝐺‘𝑥) +o (𝐻‘𝑥)))
55	48, 53, 49, 54	syl21anc 851	. . . . 5 ⊢ ((((𝑋 ∈ On ∧ 𝑆 = dom (ω CNF 𝑋)) ∧ (𝐹 ∈ 𝑆 ∧ 𝐺 ∈ 𝑆 ∧ 𝐻 ∈ 𝑆)) ∧ 𝑥 ∈ 𝑋) → ((𝐺 ∘f +o 𝐻)‘𝑥) = ((𝐺‘𝑥) +o (𝐻‘𝑥)))
56	55	oveq2d 7432	. . . 4 ⊢ ((((𝑋 ∈ On ∧ 𝑆 = dom (ω CNF 𝑋)) ∧ (𝐹 ∈ 𝑆 ∧ 𝐺 ∈ 𝑆 ∧ 𝐻 ∈ 𝑆)) ∧ 𝑥 ∈ 𝑋) → ((𝐹‘𝑥) +o ((𝐺 ∘f +o 𝐻)‘𝑥)) = ((𝐹‘𝑥) +o ((𝐺‘𝑥) +o (𝐻‘𝑥))))
57	46, 52, 56	3eqtr4d 2807	. . 3 ⊢ ((((𝑋 ∈ On ∧ 𝑆 = dom (ω CNF 𝑋)) ∧ (𝐹 ∈ 𝑆 ∧ 𝐺 ∈ 𝑆 ∧ 𝐻 ∈ 𝑆)) ∧ 𝑥 ∈ 𝑋) → (((𝐹 ∘f +o 𝐺)‘𝑥) +o (𝐻‘𝑥)) = ((𝐹‘𝑥) +o ((𝐺 ∘f +o 𝐻)‘𝑥)))
58	24	adantr 486	. . . 4 ⊢ ((((𝑋 ∈ On ∧ 𝑆 = dom (ω CNF 𝑋)) ∧ (𝐹 ∈ 𝑆 ∧ 𝐺 ∈ 𝑆 ∧ 𝐻 ∈ 𝑆)) ∧ 𝑥 ∈ 𝑋) → (𝐹 ∘f +o 𝐺) Fn 𝑋)
59		fnfvof 7698	. . . 4 ⊢ ((((𝐹 ∘f +o 𝐺) Fn 𝑋 ∧ 𝐻 Fn 𝑋) ∧ (𝑋 ∈ On ∧ 𝑥 ∈ 𝑋)) → (((𝐹 ∘f +o 𝐺) ∘f +o 𝐻)‘𝑥) = (((𝐹 ∘f +o 𝐺)‘𝑥) +o (𝐻‘𝑥)))
60	58, 53, 49, 59	syl21anc 851	. . 3 ⊢ ((((𝑋 ∈ On ∧ 𝑆 = dom (ω CNF 𝑋)) ∧ (𝐹 ∈ 𝑆 ∧ 𝐺 ∈ 𝑆 ∧ 𝐻 ∈ 𝑆)) ∧ 𝑥 ∈ 𝑋) → (((𝐹 ∘f +o 𝐺) ∘f +o 𝐻)‘𝑥) = (((𝐹 ∘f +o 𝐺)‘𝑥) +o (𝐻‘𝑥)))
61	34	adantr 486	. . . 4 ⊢ ((((𝑋 ∈ On ∧ 𝑆 = dom (ω CNF 𝑋)) ∧ (𝐹 ∈ 𝑆 ∧ 𝐺 ∈ 𝑆 ∧ 𝐻 ∈ 𝑆)) ∧ 𝑥 ∈ 𝑋) → (𝐺 ∘f +o 𝐻) Fn 𝑋)
62		fnfvof 7698	. . . 4 ⊢ (((𝐹 Fn 𝑋 ∧ (𝐺 ∘f +o 𝐻) Fn 𝑋) ∧ (𝑋 ∈ On ∧ 𝑥 ∈ 𝑋)) → ((𝐹 ∘f +o (𝐺 ∘f +o 𝐻))‘𝑥) = ((𝐹‘𝑥) +o ((𝐺 ∘f +o 𝐻)‘𝑥)))
63	47, 61, 49, 62	syl21anc 851	. . 3 ⊢ ((((𝑋 ∈ On ∧ 𝑆 = dom (ω CNF 𝑋)) ∧ (𝐹 ∈ 𝑆 ∧ 𝐺 ∈ 𝑆 ∧ 𝐻 ∈ 𝑆)) ∧ 𝑥 ∈ 𝑋) → ((𝐹 ∘f +o (𝐺 ∘f +o 𝐻))‘𝑥) = ((𝐹‘𝑥) +o ((𝐺 ∘f +o 𝐻)‘𝑥)))
64	57, 60, 63	3eqtr4d 2807	. 2 ⊢ ((((𝑋 ∈ On ∧ 𝑆 = dom (ω CNF 𝑋)) ∧ (𝐹 ∈ 𝑆 ∧ 𝐺 ∈ 𝑆 ∧ 𝐻 ∈ 𝑆)) ∧ 𝑥 ∈ 𝑋) → (((𝐹 ∘f +o 𝐺) ∘f +o 𝐻)‘𝑥) = ((𝐹 ∘f +o (𝐺 ∘f +o 𝐻))‘𝑥))
65	33, 35, 64	eqfnfvd 7029	1 ⊢ (((𝑋 ∈ On ∧ 𝑆 = dom (ω CNF 𝑋)) ∧ (𝐹 ∈ 𝑆 ∧ 𝐺 ∈ 𝑆 ∧ 𝐻 ∈ 𝑆)) → ((𝐹 ∘f +o 𝐺) ∘f +o 𝐻) = (𝐹 ∘f +o (𝐺 ∘f +o 𝐻)))

This proof depends on syntax axioms:   → wi 4   ∧ wa 401   ∧ w3a 1103   = wceq 1570   ∈ wcel 2145  ∅c0 4282   class class class wbr 5107  dom cdm 5659  Oncon0 6361   Fn wfn 6532  ⟶wf 6533  ‘cfv 6537  (class class class)co 7416   ∘_f cof 7679  ωcom 7865   +_o coa 8455   finSupp cfsupp 9334   CNF ccnf 9643

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1828  ax-4 1842  ax-5 1943  ax-6 2000  ax-7 2041  ax-8 2147  ax-9 2155  ax-10 2178  ax-11 2194  ax-12 2215  ax-ext 2734  ax-rep 5236  ax-sep 5255  ax-nul 5267  ax-pow 5334  ax-pr 5402  ax-un 7739  ax-inf2 9623

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3or 1104  df-3an 1105  df-tru 1573  df-fal 1583  df-ex 1813  df-nf 1817  df-sb 2100  df-mo 2566  df-eu 2596  df-clab 2741  df-cleq 2754  df-clel 2837  df-nfc 2911  df-ne 2958  df-ral 3079  df-rex 3089  df-reu 3368  df-rab 3415  df-v 3455  df-sbc 3743  df-csb 3851  df-dif 3905  df-un 3907  df-in 3909  df-ss 3919  df-pss 3922  df-nul 4283  df-if 4486  df-pw 4562  df-sn 4588  df-pr 4590  df-op 4594  df-uni 4871  df-iun 4956  df-br 5108  df-opab 5172  df-mpt 5191  df-tr 5217  df-id 5554  df-eprel 5559  df-po 5567  df-so 5568  df-fr 5612  df-we 5614  df-xp 5665  df-rel 5666  df-cnv 5667  df-co 5668  df-dm 5669  df-rn 5670  df-res 5671  df-ima 5672  df-pred 6303  df-ord 6364  df-on 6365  df-lim 6366  df-suc 6367  df-iota 6493  df-fun 6539  df-fn 6540  df-f 6541  df-f1 6542  df-fo 6543  df-f1o 6544  df-fv 6545  df-ov 7419  df-oprab 7420  df-mpo 7421  df-of 7681  df-om 7866  df-2nd 7990  df-frecs 8283  df-wrecs 8314  df-recs 8363  df-rdg 8402  df-seqom 8440  df-oadd 8462  df-map 8831  df-cnf 9644

This theorem is used by: (None)