Theorem naddcnfass 43647

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 484	. . . . . . . 8 ⊢ ((𝑋 ∈ On ∧ 𝑆 = dom (ω CNF 𝑋)) → 𝑆 = dom (ω CNF 𝑋))
2	1	eleq2d 2823	. . . . . . 7 ⊢ ((𝑋 ∈ On ∧ 𝑆 = dom (ω CNF 𝑋)) → (𝐹 ∈ 𝑆 ↔ 𝐹 ∈ dom (ω CNF 𝑋)))
3		eqid 2737	. . . . . . . 8 ⊢ dom (ω CNF 𝑋) = dom (ω CNF 𝑋)
4		omelon 9559	. . . . . . . . 9 ⊢ ω ∈ On
5	4	a1i 11	. . . . . . . 8 ⊢ ((𝑋 ∈ On ∧ 𝑆 = dom (ω CNF 𝑋)) → ω ∈ On)
6		simpl 482	. . . . . . . 8 ⊢ ((𝑋 ∈ On ∧ 𝑆 = dom (ω CNF 𝑋)) → 𝑋 ∈ On)
7	3, 5, 6	cantnfs 9579	. . . . . . 7 ⊢ ((𝑋 ∈ On ∧ 𝑆 = dom (ω CNF 𝑋)) → (𝐹 ∈ dom (ω CNF 𝑋) ↔ (𝐹:𝑋⟶ω ∧ 𝐹 finSupp ∅)))
8	2, 7	bitrd 279	. . . . . 6 ⊢ ((𝑋 ∈ On ∧ 𝑆 = dom (ω CNF 𝑋)) → (𝐹 ∈ 𝑆 ↔ (𝐹:𝑋⟶ω ∧ 𝐹 finSupp ∅)))
9		simpl 482	. . . . . . 7 ⊢ ((𝐹:𝑋⟶ω ∧ 𝐹 finSupp ∅) → 𝐹:𝑋⟶ω)
10	9	ffnd 6664	. . . . . 6 ⊢ ((𝐹:𝑋⟶ω ∧ 𝐹 finSupp ∅) → 𝐹 Fn 𝑋)
11	8, 10	biimtrdi 253	. . . . 5 ⊢ ((𝑋 ∈ On ∧ 𝑆 = dom (ω CNF 𝑋)) → (𝐹 ∈ 𝑆 → 𝐹 Fn 𝑋))
12		simp1 1137	. . . . 5 ⊢ ((𝐹 ∈ 𝑆 ∧ 𝐺 ∈ 𝑆 ∧ 𝐻 ∈ 𝑆) → 𝐹 ∈ 𝑆)
13	11, 12	impel 505	. . . 4 ⊢ (((𝑋 ∈ On ∧ 𝑆 = dom (ω CNF 𝑋)) ∧ (𝐹 ∈ 𝑆 ∧ 𝐺 ∈ 𝑆 ∧ 𝐻 ∈ 𝑆)) → 𝐹 Fn 𝑋)
14	1	eleq2d 2823	. . . . . . 7 ⊢ ((𝑋 ∈ On ∧ 𝑆 = dom (ω CNF 𝑋)) → (𝐺 ∈ 𝑆 ↔ 𝐺 ∈ dom (ω CNF 𝑋)))
15	3, 5, 6	cantnfs 9579	. . . . . . 7 ⊢ ((𝑋 ∈ On ∧ 𝑆 = dom (ω CNF 𝑋)) → (𝐺 ∈ dom (ω CNF 𝑋) ↔ (𝐺:𝑋⟶ω ∧ 𝐺 finSupp ∅)))
16	14, 15	bitrd 279	. . . . . 6 ⊢ ((𝑋 ∈ On ∧ 𝑆 = dom (ω CNF 𝑋)) → (𝐺 ∈ 𝑆 ↔ (𝐺:𝑋⟶ω ∧ 𝐺 finSupp ∅)))
17		simpl 482	. . . . . . 7 ⊢ ((𝐺:𝑋⟶ω ∧ 𝐺 finSupp ∅) → 𝐺:𝑋⟶ω)
18	17	ffnd 6664	. . . . . 6 ⊢ ((𝐺:𝑋⟶ω ∧ 𝐺 finSupp ∅) → 𝐺 Fn 𝑋)
19	16, 18	biimtrdi 253	. . . . 5 ⊢ ((𝑋 ∈ On ∧ 𝑆 = dom (ω CNF 𝑋)) → (𝐺 ∈ 𝑆 → 𝐺 Fn 𝑋))
20		simp2 1138	. . . . 5 ⊢ ((𝐹 ∈ 𝑆 ∧ 𝐺 ∈ 𝑆 ∧ 𝐻 ∈ 𝑆) → 𝐺 ∈ 𝑆)
21	19, 20	impel 505	. . . 4 ⊢ (((𝑋 ∈ On ∧ 𝑆 = dom (ω CNF 𝑋)) ∧ (𝐹 ∈ 𝑆 ∧ 𝐺 ∈ 𝑆 ∧ 𝐻 ∈ 𝑆)) → 𝐺 Fn 𝑋)
22	6	adantr 480	. . . 4 ⊢ (((𝑋 ∈ On ∧ 𝑆 = dom (ω CNF 𝑋)) ∧ (𝐹 ∈ 𝑆 ∧ 𝐺 ∈ 𝑆 ∧ 𝐻 ∈ 𝑆)) → 𝑋 ∈ On)
23		inidm 4180	. . . 4 ⊢ (𝑋 ∩ 𝑋) = 𝑋
24	13, 21, 22, 22, 23	offn 7637	. . 3 ⊢ (((𝑋 ∈ On ∧ 𝑆 = dom (ω CNF 𝑋)) ∧ (𝐹 ∈ 𝑆 ∧ 𝐺 ∈ 𝑆 ∧ 𝐻 ∈ 𝑆)) → (𝐹 ∘f +o 𝐺) Fn 𝑋)
25	1	eleq2d 2823	. . . . . 6 ⊢ ((𝑋 ∈ On ∧ 𝑆 = dom (ω CNF 𝑋)) → (𝐻 ∈ 𝑆 ↔ 𝐻 ∈ dom (ω CNF 𝑋)))
26	3, 5, 6	cantnfs 9579	. . . . . 6 ⊢ ((𝑋 ∈ On ∧ 𝑆 = dom (ω CNF 𝑋)) → (𝐻 ∈ dom (ω CNF 𝑋) ↔ (𝐻:𝑋⟶ω ∧ 𝐻 finSupp ∅)))
27	25, 26	bitrd 279	. . . . 5 ⊢ ((𝑋 ∈ On ∧ 𝑆 = dom (ω CNF 𝑋)) → (𝐻 ∈ 𝑆 ↔ (𝐻:𝑋⟶ω ∧ 𝐻 finSupp ∅)))
28		simpl 482	. . . . . 6 ⊢ ((𝐻:𝑋⟶ω ∧ 𝐻 finSupp ∅) → 𝐻:𝑋⟶ω)
29	28	ffnd 6664	. . . . 5 ⊢ ((𝐻:𝑋⟶ω ∧ 𝐻 finSupp ∅) → 𝐻 Fn 𝑋)
30	27, 29	biimtrdi 253	. . . 4 ⊢ ((𝑋 ∈ On ∧ 𝑆 = dom (ω CNF 𝑋)) → (𝐻 ∈ 𝑆 → 𝐻 Fn 𝑋))
31		simp3 1139	. . . 4 ⊢ ((𝐹 ∈ 𝑆 ∧ 𝐺 ∈ 𝑆 ∧ 𝐻 ∈ 𝑆) → 𝐻 ∈ 𝑆)
32	30, 31	impel 505	. . 3 ⊢ (((𝑋 ∈ On ∧ 𝑆 = dom (ω CNF 𝑋)) ∧ (𝐹 ∈ 𝑆 ∧ 𝐺 ∈ 𝑆 ∧ 𝐻 ∈ 𝑆)) → 𝐻 Fn 𝑋)
33	24, 32, 22, 22, 23	offn 7637	. 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 7637	. . 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 7637	. 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 253	. . . . . . 7 ⊢ ((𝑋 ∈ On ∧ 𝑆 = dom (ω CNF 𝑋)) → (𝐹 ∈ 𝑆 → 𝐹:𝑋⟶ω))
37	36, 12	impel 505	. . . . . 6 ⊢ (((𝑋 ∈ On ∧ 𝑆 = dom (ω CNF 𝑋)) ∧ (𝐹 ∈ 𝑆 ∧ 𝐺 ∈ 𝑆 ∧ 𝐻 ∈ 𝑆)) → 𝐹:𝑋⟶ω)
38	37	ffvelcdmda 7031	. . . . 5 ⊢ ((((𝑋 ∈ On ∧ 𝑆 = dom (ω CNF 𝑋)) ∧ (𝐹 ∈ 𝑆 ∧ 𝐺 ∈ 𝑆 ∧ 𝐻 ∈ 𝑆)) ∧ 𝑥 ∈ 𝑋) → (𝐹‘𝑥) ∈ ω)
39	16, 17	biimtrdi 253	. . . . . . 7 ⊢ ((𝑋 ∈ On ∧ 𝑆 = dom (ω CNF 𝑋)) → (𝐺 ∈ 𝑆 → 𝐺:𝑋⟶ω))
40	39, 20	impel 505	. . . . . 6 ⊢ (((𝑋 ∈ On ∧ 𝑆 = dom (ω CNF 𝑋)) ∧ (𝐹 ∈ 𝑆 ∧ 𝐺 ∈ 𝑆 ∧ 𝐻 ∈ 𝑆)) → 𝐺:𝑋⟶ω)
41	40	ffvelcdmda 7031	. . . . 5 ⊢ ((((𝑋 ∈ On ∧ 𝑆 = dom (ω CNF 𝑋)) ∧ (𝐹 ∈ 𝑆 ∧ 𝐺 ∈ 𝑆 ∧ 𝐻 ∈ 𝑆)) ∧ 𝑥 ∈ 𝑋) → (𝐺‘𝑥) ∈ ω)
42	27, 28	biimtrdi 253	. . . . . . 7 ⊢ ((𝑋 ∈ On ∧ 𝑆 = dom (ω CNF 𝑋)) → (𝐻 ∈ 𝑆 → 𝐻:𝑋⟶ω))
43	42, 31	impel 505	. . . . . 6 ⊢ (((𝑋 ∈ On ∧ 𝑆 = dom (ω CNF 𝑋)) ∧ (𝐹 ∈ 𝑆 ∧ 𝐺 ∈ 𝑆 ∧ 𝐻 ∈ 𝑆)) → 𝐻:𝑋⟶ω)
44	43	ffvelcdmda 7031	. . . . 5 ⊢ ((((𝑋 ∈ On ∧ 𝑆 = dom (ω CNF 𝑋)) ∧ (𝐹 ∈ 𝑆 ∧ 𝐺 ∈ 𝑆 ∧ 𝐻 ∈ 𝑆)) ∧ 𝑥 ∈ 𝑋) → (𝐻‘𝑥) ∈ ω)
45		nnaass 8552	. . . . 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 1374	. . . 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 480	. . . . . 6 ⊢ ((((𝑋 ∈ On ∧ 𝑆 = dom (ω CNF 𝑋)) ∧ (𝐹 ∈ 𝑆 ∧ 𝐺 ∈ 𝑆 ∧ 𝐻 ∈ 𝑆)) ∧ 𝑥 ∈ 𝑋) → 𝐹 Fn 𝑋)
48	21	adantr 480	. . . . . 6 ⊢ ((((𝑋 ∈ On ∧ 𝑆 = dom (ω CNF 𝑋)) ∧ (𝐹 ∈ 𝑆 ∧ 𝐺 ∈ 𝑆 ∧ 𝐻 ∈ 𝑆)) ∧ 𝑥 ∈ 𝑋) → 𝐺 Fn 𝑋)
49	22	anim1i 616	. . . . . 6 ⊢ ((((𝑋 ∈ On ∧ 𝑆 = dom (ω CNF 𝑋)) ∧ (𝐹 ∈ 𝑆 ∧ 𝐺 ∈ 𝑆 ∧ 𝐻 ∈ 𝑆)) ∧ 𝑥 ∈ 𝑋) → (𝑋 ∈ On ∧ 𝑥 ∈ 𝑋))
50		fnfvof 7641	. . . . . 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 838	. . . . 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 7375	. . . 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 480	. . . . . 6 ⊢ ((((𝑋 ∈ On ∧ 𝑆 = dom (ω CNF 𝑋)) ∧ (𝐹 ∈ 𝑆 ∧ 𝐺 ∈ 𝑆 ∧ 𝐻 ∈ 𝑆)) ∧ 𝑥 ∈ 𝑋) → 𝐻 Fn 𝑋)
54		fnfvof 7641	. . . . . 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 838	. . . . 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 7376	. . . 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 2782	. . 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 480	. . . 4 ⊢ ((((𝑋 ∈ On ∧ 𝑆 = dom (ω CNF 𝑋)) ∧ (𝐹 ∈ 𝑆 ∧ 𝐺 ∈ 𝑆 ∧ 𝐻 ∈ 𝑆)) ∧ 𝑥 ∈ 𝑋) → (𝐹 ∘f +o 𝐺) Fn 𝑋)
59		fnfvof 7641	. . . 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 838	. . 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 480	. . . 4 ⊢ ((((𝑋 ∈ On ∧ 𝑆 = dom (ω CNF 𝑋)) ∧ (𝐹 ∈ 𝑆 ∧ 𝐺 ∈ 𝑆 ∧ 𝐻 ∈ 𝑆)) ∧ 𝑥 ∈ 𝑋) → (𝐺 ∘f +o 𝐻) Fn 𝑋)
62		fnfvof 7641	. . . 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 838	. . 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 2782	. 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 6981	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)))

Syntax hints:   → wi 4   ∧ wa 395   ∧ w3a 1087   = wceq 1542   ∈ wcel 2114  ∅c0 4286   class class class wbr 5099  dom cdm 5625  Oncon0 6318   Fn wfn 6488  ⟶wf 6489  ‘cfv 6493  (class class class)co 7360   ∘_f cof 7622  ωcom 7810   +_o coa 8396   finSupp cfsupp 9268   CNF ccnf 9574

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 2709  ax-rep 5225  ax-sep 5242  ax-nul 5252  ax-pow 5311  ax-pr 5378  ax-un 7682  ax-inf2 9554

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 2540  df-eu 2570  df-clab 2716  df-cleq 2729  df-clel 2812  df-nfc 2886  df-ne 2934  df-ral 3053  df-rex 3062  df-reu 3352  df-rab 3401  df-v 3443  df-sbc 3742  df-csb 3851  df-dif 3905  df-un 3907  df-in 3909  df-ss 3919  df-pss 3922  df-nul 4287  df-if 4481  df-pw 4557  df-sn 4582  df-pr 4584  df-op 4588  df-uni 4865  df-iun 4949  df-br 5100  df-opab 5162  df-mpt 5181  df-tr 5207  df-id 5520  df-eprel 5525  df-po 5533  df-so 5534  df-fr 5578  df-we 5580  df-xp 5631  df-rel 5632  df-cnv 5633  df-co 5634  df-dm 5635  df-rn 5636  df-res 5637  df-ima 5638  df-pred 6260  df-ord 6321  df-on 6322  df-lim 6323  df-suc 6324  df-iota 6449  df-fun 6495  df-fn 6496  df-f 6497  df-f1 6498  df-fo 6499  df-f1o 6500  df-fv 6501  df-ov 7363  df-oprab 7364  df-mpo 7365  df-of 7624  df-om 7811  df-2nd 7936  df-frecs 8225  df-wrecs 8256  df-recs 8305  df-rdg 8343  df-seqom 8381  df-oadd 8403  df-map 8769  df-cnf 9575

This theorem is referenced by: (None)