Theorem naddcnfass 44563

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 2850	. . . . . . 7 ⊢ ((𝑋 ∈ On ∧ 𝑆 = dom (ω CNF 𝑋)) → (𝐹 ∈ 𝑆 ↔ 𝐹 ∈ dom (ω CNF 𝑋)))
3		eqid 2763	. . . . . . . 8 ⊢ dom (ω CNF 𝑋) = dom (ω CNF 𝑋)
4		omelon 9674	. . . . . . . . 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 9694	. . . . . . 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 6732	. . . . . 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 2850	. . . . . . 7 ⊢ ((𝑋 ∈ On ∧ 𝑆 = dom (ω CNF 𝑋)) → (𝐺 ∈ 𝑆 ↔ 𝐺 ∈ dom (ω CNF 𝑋)))
15	3, 5, 6	cantnfs 9694	. . . . . . 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 6732	. . . . . 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 4126	. . . 4 ⊢ (𝑋 ∩ 𝑋) = 𝑋
24	13, 21, 22, 22, 23	offn 7730	. . 3 ⊢ (((𝑋 ∈ On ∧ 𝑆 = dom (ω CNF 𝑋)) ∧ (𝐹 ∈ 𝑆 ∧ 𝐺 ∈ 𝑆 ∧ 𝐻 ∈ 𝑆)) → (𝐹 ∘f +o 𝐺) Fn 𝑋)
25	1	eleq2d 2850	. . . . . 6 ⊢ ((𝑋 ∈ On ∧ 𝑆 = dom (ω CNF 𝑋)) → (𝐻 ∈ 𝑆 ↔ 𝐻 ∈ dom (ω CNF 𝑋)))
26	3, 5, 6	cantnfs 9694	. . . . . 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 6732	. . . . 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 7730	. 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 7730	. . 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 7730	. 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 7108	. . . . 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 7108	. . . . 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 7108	. . . . 5 ⊢ ((((𝑋 ∈ On ∧ 𝑆 = dom (ω CNF 𝑋)) ∧ (𝐹 ∈ 𝑆 ∧ 𝐺 ∈ 𝑆 ∧ 𝐻 ∈ 𝑆)) ∧ 𝑥 ∈ 𝑋) → (𝐻‘𝑥) ∈ ω)
45		nnaass 8659	. . . . 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 1399	. . . 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 7734	. . . . . 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 7458	. . . 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 7734	. . . . . 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 7459	. . . 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 2808	. . 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 7734	. . . 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 7734	. . . 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 2808	. 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 7055	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 2147  ∅c0 4281   class class class wbr 5106  dom cdm 5655  Oncon0 6380   Fn wfn 6553  ⟶wf 6554  ‘cfv 6558  (class class class)co 7443   ∘_f cof 7715  ωcom 7901   +_o coa 8501   finSupp cfsupp 9381   CNF ccnf 9689

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 1945  ax-6 2002  ax-7 2043  ax-8 2149  ax-9 2157  ax-10 2180  ax-11 2197  ax-12 2216  ax-ext 2735  ax-rep 5235  ax-sep 5252  ax-nul 5263  ax-pow 5331  ax-pr 5395  ax-un 7775  ax-inf2 9669

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 2102  df-mo 2567  df-eu 2597  df-clab 2742  df-cleq 2755  df-clel 2839  df-nfc 2913  df-ne 2960  df-ral 3081  df-rex 3091  df-reu 3370  df-rab 3417  df-v 3456  df-sbc 3743  df-csb 3852  df-ss 3901  df-pss 4017  df-dif 4050  df-in 4094  df-un 4154  df-nul 4282  df-if 4486  df-pw 4562  df-sn 4589  df-pr 4591  df-op 4839  df-uni 4871  df-iun 4956  df-br 5107  df-opab 5171  df-mpt 5190  df-tr 5216  df-id 5550  df-eprel 5555  df-po 5563  df-so 5564  df-fr 5608  df-we 5610  df-xp 5661  df-rel 5662  df-cnv 5663  df-co 5664  df-dm 5665  df-rn 5666  df-res 5667  df-ima 5668  df-pred 6322  df-ord 6383  df-on 6384  df-lim 6385  df-suc 6386  df-iota 6512  df-fun 6560  df-fn 6561  df-f 6562  df-f1 6563  df-fo 6564  df-f1o 6565  df-fv 6566  df-ov 7446  df-oprab 7447  df-mpo 7448  df-of 7717  df-om 7902  df-2nd 8026  df-frecs 8327  df-wrecs 8358  df-recs 8407  df-rdg 8446  df-seqom 8486  df-oadd 8508  df-map 8877  df-cnf 9690

This theorem is used by: (None)