Theorem naddcnff 43352

Description: Addition operator for Cantor normal forms is a function into Cantor normal forms. (Contributed by RP, 2-Jan-2025.)

Assertion

Ref	Expression
naddcnff	⊢ ((𝑋 ∈ On ∧ 𝑆 = dom (ω CNF 𝑋)) → ( ∘f +o ↾ (𝑆 × 𝑆)):(𝑆 × 𝑆)⟶𝑆)

Proof of Theorem naddcnff

Dummy variables 𝑓 𝑔 𝑥 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		simpr 484	. . . . . 6 ⊢ ((𝑋 ∈ On ∧ 𝑆 = dom (ω CNF 𝑋)) → 𝑆 = dom (ω CNF 𝑋))
2	1	eleq2d 2825	. . . . 5 ⊢ ((𝑋 ∈ On ∧ 𝑆 = dom (ω CNF 𝑋)) → (𝑓 ∈ 𝑆 ↔ 𝑓 ∈ dom (ω CNF 𝑋)))
3		eqid 2735	. . . . . 6 ⊢ dom (ω CNF 𝑋) = dom (ω CNF 𝑋)
4		omelon 9684	. . . . . . 7 ⊢ ω ∈ On
5	4	a1i 11	. . . . . 6 ⊢ ((𝑋 ∈ On ∧ 𝑆 = dom (ω CNF 𝑋)) → ω ∈ On)
6		simpl 482	. . . . . 6 ⊢ ((𝑋 ∈ On ∧ 𝑆 = dom (ω CNF 𝑋)) → 𝑋 ∈ On)
7	3, 5, 6	cantnfs 9704	. . . . 5 ⊢ ((𝑋 ∈ On ∧ 𝑆 = dom (ω CNF 𝑋)) → (𝑓 ∈ dom (ω CNF 𝑋) ↔ (𝑓:𝑋⟶ω ∧ 𝑓 finSupp ∅)))
8	2, 7	bitrd 279	. . . 4 ⊢ ((𝑋 ∈ On ∧ 𝑆 = dom (ω CNF 𝑋)) → (𝑓 ∈ 𝑆 ↔ (𝑓:𝑋⟶ω ∧ 𝑓 finSupp ∅)))
9	1	eleq2d 2825	. . . . . . . . 9 ⊢ ((𝑋 ∈ On ∧ 𝑆 = dom (ω CNF 𝑋)) → (𝑔 ∈ 𝑆 ↔ 𝑔 ∈ dom (ω CNF 𝑋)))
10	3, 5, 6	cantnfs 9704	. . . . . . . . 9 ⊢ ((𝑋 ∈ On ∧ 𝑆 = dom (ω CNF 𝑋)) → (𝑔 ∈ dom (ω CNF 𝑋) ↔ (𝑔:𝑋⟶ω ∧ 𝑔 finSupp ∅)))
11	9, 10	bitrd 279	. . . . . . . 8 ⊢ ((𝑋 ∈ On ∧ 𝑆 = dom (ω CNF 𝑋)) → (𝑔 ∈ 𝑆 ↔ (𝑔:𝑋⟶ω ∧ 𝑔 finSupp ∅)))
12	11	adantr 480	. . . . . . 7 ⊢ (((𝑋 ∈ On ∧ 𝑆 = dom (ω CNF 𝑋)) ∧ (𝑓:𝑋⟶ω ∧ 𝑓 finSupp ∅)) → (𝑔 ∈ 𝑆 ↔ (𝑔:𝑋⟶ω ∧ 𝑔 finSupp ∅)))
13		simpl 482	. . . . . . . . . . . . . 14 ⊢ ((𝑓:𝑋⟶ω ∧ 𝑓 finSupp ∅) → 𝑓:𝑋⟶ω)
14		simpl 482	. . . . . . . . . . . . . 14 ⊢ ((𝑔:𝑋⟶ω ∧ 𝑔 finSupp ∅) → 𝑔:𝑋⟶ω)
15	13, 14	anim12i 613	. . . . . . . . . . . . 13 ⊢ (((𝑓:𝑋⟶ω ∧ 𝑓 finSupp ∅) ∧ (𝑔:𝑋⟶ω ∧ 𝑔 finSupp ∅)) → (𝑓:𝑋⟶ω ∧ 𝑔:𝑋⟶ω))
16	6, 15	anim12i 613	. . . . . . . . . . . 12 ⊢ (((𝑋 ∈ On ∧ 𝑆 = dom (ω CNF 𝑋)) ∧ ((𝑓:𝑋⟶ω ∧ 𝑓 finSupp ∅) ∧ (𝑔:𝑋⟶ω ∧ 𝑔 finSupp ∅))) → (𝑋 ∈ On ∧ (𝑓:𝑋⟶ω ∧ 𝑔:𝑋⟶ω)))
17	16	anassrs 467	. . . . . . . . . . 11 ⊢ ((((𝑋 ∈ On ∧ 𝑆 = dom (ω CNF 𝑋)) ∧ (𝑓:𝑋⟶ω ∧ 𝑓 finSupp ∅)) ∧ (𝑔:𝑋⟶ω ∧ 𝑔 finSupp ∅)) → (𝑋 ∈ On ∧ (𝑓:𝑋⟶ω ∧ 𝑔:𝑋⟶ω)))
18		simprl 771	. . . . . . . . . . . . . . 15 ⊢ ((𝑋 ∈ On ∧ (𝑓:𝑋⟶ω ∧ 𝑔:𝑋⟶ω)) → 𝑓:𝑋⟶ω)
19	18	ffnd 6738	. . . . . . . . . . . . . 14 ⊢ ((𝑋 ∈ On ∧ (𝑓:𝑋⟶ω ∧ 𝑔:𝑋⟶ω)) → 𝑓 Fn 𝑋)
20		simprr 773	. . . . . . . . . . . . . . 15 ⊢ ((𝑋 ∈ On ∧ (𝑓:𝑋⟶ω ∧ 𝑔:𝑋⟶ω)) → 𝑔:𝑋⟶ω)
21	20	ffnd 6738	. . . . . . . . . . . . . 14 ⊢ ((𝑋 ∈ On ∧ (𝑓:𝑋⟶ω ∧ 𝑔:𝑋⟶ω)) → 𝑔 Fn 𝑋)
22		simpl 482	. . . . . . . . . . . . . 14 ⊢ ((𝑋 ∈ On ∧ (𝑓:𝑋⟶ω ∧ 𝑔:𝑋⟶ω)) → 𝑋 ∈ On)
23		inidm 4235	. . . . . . . . . . . . . 14 ⊢ (𝑋 ∩ 𝑋) = 𝑋
24	19, 21, 22, 22, 23	offn 7710	. . . . . . . . . . . . 13 ⊢ ((𝑋 ∈ On ∧ (𝑓:𝑋⟶ω ∧ 𝑔:𝑋⟶ω)) → (𝑓 ∘f +o 𝑔) Fn 𝑋)
25		simpr 484	. . . . . . . . . . . . . . 15 ⊢ (((𝑋 ∈ On ∧ (𝑓:𝑋⟶ω ∧ 𝑔:𝑋⟶ω)) ∧ (𝑓 ∘f +o 𝑔) Fn 𝑋) → (𝑓 ∘f +o 𝑔) Fn 𝑋)
26		simplrl 777	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (((𝑋 ∈ On ∧ (𝑓:𝑋⟶ω ∧ 𝑔:𝑋⟶ω)) ∧ 𝑥 ∈ 𝑋) → 𝑓:𝑋⟶ω)
27	26	ffnd 6738	. . . . . . . . . . . . . . . . . . . 20 ⊢ (((𝑋 ∈ On ∧ (𝑓:𝑋⟶ω ∧ 𝑔:𝑋⟶ω)) ∧ 𝑥 ∈ 𝑋) → 𝑓 Fn 𝑋)
28		simplrr 778	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (((𝑋 ∈ On ∧ (𝑓:𝑋⟶ω ∧ 𝑔:𝑋⟶ω)) ∧ 𝑥 ∈ 𝑋) → 𝑔:𝑋⟶ω)
29	28	ffnd 6738	. . . . . . . . . . . . . . . . . . . 20 ⊢ (((𝑋 ∈ On ∧ (𝑓:𝑋⟶ω ∧ 𝑔:𝑋⟶ω)) ∧ 𝑥 ∈ 𝑋) → 𝑔 Fn 𝑋)
30		simpll 767	. . . . . . . . . . . . . . . . . . . 20 ⊢ (((𝑋 ∈ On ∧ (𝑓:𝑋⟶ω ∧ 𝑔:𝑋⟶ω)) ∧ 𝑥 ∈ 𝑋) → 𝑋 ∈ On)
31		simpr 484	. . . . . . . . . . . . . . . . . . . 20 ⊢ (((𝑋 ∈ On ∧ (𝑓:𝑋⟶ω ∧ 𝑔:𝑋⟶ω)) ∧ 𝑥 ∈ 𝑋) → 𝑥 ∈ 𝑋)
32		fnfvof 7714	. . . . . . . . . . . . . . . . . . . 20 ⊢ (((𝑓 Fn 𝑋 ∧ 𝑔 Fn 𝑋) ∧ (𝑋 ∈ On ∧ 𝑥 ∈ 𝑋)) → ((𝑓 ∘f +o 𝑔)‘𝑥) = ((𝑓‘𝑥) +o (𝑔‘𝑥)))
33	27, 29, 30, 31, 32	syl22anc 839	. . . . . . . . . . . . . . . . . . 19 ⊢ (((𝑋 ∈ On ∧ (𝑓:𝑋⟶ω ∧ 𝑔:𝑋⟶ω)) ∧ 𝑥 ∈ 𝑋) → ((𝑓 ∘f +o 𝑔)‘𝑥) = ((𝑓‘𝑥) +o (𝑔‘𝑥)))
34	18	ffvelcdmda 7104	. . . . . . . . . . . . . . . . . . . 20 ⊢ (((𝑋 ∈ On ∧ (𝑓:𝑋⟶ω ∧ 𝑔:𝑋⟶ω)) ∧ 𝑥 ∈ 𝑋) → (𝑓‘𝑥) ∈ ω)
35	20	ffvelcdmda 7104	. . . . . . . . . . . . . . . . . . . 20 ⊢ (((𝑋 ∈ On ∧ (𝑓:𝑋⟶ω ∧ 𝑔:𝑋⟶ω)) ∧ 𝑥 ∈ 𝑋) → (𝑔‘𝑥) ∈ ω)
36		nnacl 8648	. . . . . . . . . . . . . . . . . . . 20 ⊢ (((𝑓‘𝑥) ∈ ω ∧ (𝑔‘𝑥) ∈ ω) → ((𝑓‘𝑥) +o (𝑔‘𝑥)) ∈ ω)
37	34, 35, 36	syl2anc 584	. . . . . . . . . . . . . . . . . . 19 ⊢ (((𝑋 ∈ On ∧ (𝑓:𝑋⟶ω ∧ 𝑔:𝑋⟶ω)) ∧ 𝑥 ∈ 𝑋) → ((𝑓‘𝑥) +o (𝑔‘𝑥)) ∈ ω)
38	33, 37	eqeltrd 2839	. . . . . . . . . . . . . . . . . 18 ⊢ (((𝑋 ∈ On ∧ (𝑓:𝑋⟶ω ∧ 𝑔:𝑋⟶ω)) ∧ 𝑥 ∈ 𝑋) → ((𝑓 ∘f +o 𝑔)‘𝑥) ∈ ω)
39	38	ex 412	. . . . . . . . . . . . . . . . 17 ⊢ ((𝑋 ∈ On ∧ (𝑓:𝑋⟶ω ∧ 𝑔:𝑋⟶ω)) → (𝑥 ∈ 𝑋 → ((𝑓 ∘f +o 𝑔)‘𝑥) ∈ ω))
40	39	ralrimiv 3143	. . . . . . . . . . . . . . . 16 ⊢ ((𝑋 ∈ On ∧ (𝑓:𝑋⟶ω ∧ 𝑔:𝑋⟶ω)) → ∀𝑥 ∈ 𝑋 ((𝑓 ∘f +o 𝑔)‘𝑥) ∈ ω)
41	40	adantr 480	. . . . . . . . . . . . . . 15 ⊢ (((𝑋 ∈ On ∧ (𝑓:𝑋⟶ω ∧ 𝑔:𝑋⟶ω)) ∧ (𝑓 ∘f +o 𝑔) Fn 𝑋) → ∀𝑥 ∈ 𝑋 ((𝑓 ∘f +o 𝑔)‘𝑥) ∈ ω)
42		fnfvrnss 7141	. . . . . . . . . . . . . . 15 ⊢ (((𝑓 ∘f +o 𝑔) Fn 𝑋 ∧ ∀𝑥 ∈ 𝑋 ((𝑓 ∘f +o 𝑔)‘𝑥) ∈ ω) → ran (𝑓 ∘f +o 𝑔) ⊆ ω)
43	25, 41, 42	syl2anc 584	. . . . . . . . . . . . . 14 ⊢ (((𝑋 ∈ On ∧ (𝑓:𝑋⟶ω ∧ 𝑔:𝑋⟶ω)) ∧ (𝑓 ∘f +o 𝑔) Fn 𝑋) → ran (𝑓 ∘f +o 𝑔) ⊆ ω)
44	43	ex 412	. . . . . . . . . . . . 13 ⊢ ((𝑋 ∈ On ∧ (𝑓:𝑋⟶ω ∧ 𝑔:𝑋⟶ω)) → ((𝑓 ∘f +o 𝑔) Fn 𝑋 → ran (𝑓 ∘f +o 𝑔) ⊆ ω))
45	24, 44	jcai 516	. . . . . . . . . . . 12 ⊢ ((𝑋 ∈ On ∧ (𝑓:𝑋⟶ω ∧ 𝑔:𝑋⟶ω)) → ((𝑓 ∘f +o 𝑔) Fn 𝑋 ∧ ran (𝑓 ∘f +o 𝑔) ⊆ ω))
46		df-f 6567	. . . . . . . . . . . 12 ⊢ ((𝑓 ∘f +o 𝑔):𝑋⟶ω ↔ ((𝑓 ∘f +o 𝑔) Fn 𝑋 ∧ ran (𝑓 ∘f +o 𝑔) ⊆ ω))
47	45, 46	sylibr 234	. . . . . . . . . . 11 ⊢ ((𝑋 ∈ On ∧ (𝑓:𝑋⟶ω ∧ 𝑔:𝑋⟶ω)) → (𝑓 ∘f +o 𝑔):𝑋⟶ω)
48	17, 47	syl 17	. . . . . . . . . 10 ⊢ ((((𝑋 ∈ On ∧ 𝑆 = dom (ω CNF 𝑋)) ∧ (𝑓:𝑋⟶ω ∧ 𝑓 finSupp ∅)) ∧ (𝑔:𝑋⟶ω ∧ 𝑔 finSupp ∅)) → (𝑓 ∘f +o 𝑔):𝑋⟶ω)
49		ffun 6740	. . . . . . . . . . . . 13 ⊢ ((𝑓 ∘f +o 𝑔):𝑋⟶ω → Fun (𝑓 ∘f +o 𝑔))
50	49	adantl 481	. . . . . . . . . . . 12 ⊢ (((((𝑋 ∈ On ∧ 𝑆 = dom (ω CNF 𝑋)) ∧ (𝑓:𝑋⟶ω ∧ 𝑓 finSupp ∅)) ∧ (𝑔:𝑋⟶ω ∧ 𝑔 finSupp ∅)) ∧ (𝑓 ∘f +o 𝑔):𝑋⟶ω) → Fun (𝑓 ∘f +o 𝑔))
51		simplrr 778	. . . . . . . . . . . . . . 15 ⊢ ((((𝑋 ∈ On ∧ 𝑆 = dom (ω CNF 𝑋)) ∧ (𝑓:𝑋⟶ω ∧ 𝑓 finSupp ∅)) ∧ (𝑔:𝑋⟶ω ∧ 𝑔 finSupp ∅)) → 𝑓 finSupp ∅)
52	51	adantr 480	. . . . . . . . . . . . . 14 ⊢ (((((𝑋 ∈ On ∧ 𝑆 = dom (ω CNF 𝑋)) ∧ (𝑓:𝑋⟶ω ∧ 𝑓 finSupp ∅)) ∧ (𝑔:𝑋⟶ω ∧ 𝑔 finSupp ∅)) ∧ (𝑓 ∘f +o 𝑔):𝑋⟶ω) → 𝑓 finSupp ∅)
53		simplrr 778	. . . . . . . . . . . . . 14 ⊢ (((((𝑋 ∈ On ∧ 𝑆 = dom (ω CNF 𝑋)) ∧ (𝑓:𝑋⟶ω ∧ 𝑓 finSupp ∅)) ∧ (𝑔:𝑋⟶ω ∧ 𝑔 finSupp ∅)) ∧ (𝑓 ∘f +o 𝑔):𝑋⟶ω) → 𝑔 finSupp ∅)
54	52, 53	fsuppunfi 9426	. . . . . . . . . . . . 13 ⊢ (((((𝑋 ∈ On ∧ 𝑆 = dom (ω CNF 𝑋)) ∧ (𝑓:𝑋⟶ω ∧ 𝑓 finSupp ∅)) ∧ (𝑔:𝑋⟶ω ∧ 𝑔 finSupp ∅)) ∧ (𝑓 ∘f +o 𝑔):𝑋⟶ω) → ((𝑓 supp ∅) ∪ (𝑔 supp ∅)) ∈ Fin)
55		simp-4l 783	. . . . . . . . . . . . . 14 ⊢ (((((𝑋 ∈ On ∧ 𝑆 = dom (ω CNF 𝑋)) ∧ (𝑓:𝑋⟶ω ∧ 𝑓 finSupp ∅)) ∧ (𝑔:𝑋⟶ω ∧ 𝑔 finSupp ∅)) ∧ (𝑓 ∘f +o 𝑔):𝑋⟶ω) → 𝑋 ∈ On)
56		peano1 7911	. . . . . . . . . . . . . . 15 ⊢ ∅ ∈ ω
57	56	a1i 11	. . . . . . . . . . . . . 14 ⊢ (((((𝑋 ∈ On ∧ 𝑆 = dom (ω CNF 𝑋)) ∧ (𝑓:𝑋⟶ω ∧ 𝑓 finSupp ∅)) ∧ (𝑔:𝑋⟶ω ∧ 𝑔 finSupp ∅)) ∧ (𝑓 ∘f +o 𝑔):𝑋⟶ω) → ∅ ∈ ω)
58		simplrl 777	. . . . . . . . . . . . . . 15 ⊢ ((((𝑋 ∈ On ∧ 𝑆 = dom (ω CNF 𝑋)) ∧ (𝑓:𝑋⟶ω ∧ 𝑓 finSupp ∅)) ∧ (𝑔:𝑋⟶ω ∧ 𝑔 finSupp ∅)) → 𝑓:𝑋⟶ω)
59	58	adantr 480	. . . . . . . . . . . . . 14 ⊢ (((((𝑋 ∈ On ∧ 𝑆 = dom (ω CNF 𝑋)) ∧ (𝑓:𝑋⟶ω ∧ 𝑓 finSupp ∅)) ∧ (𝑔:𝑋⟶ω ∧ 𝑔 finSupp ∅)) ∧ (𝑓 ∘f +o 𝑔):𝑋⟶ω) → 𝑓:𝑋⟶ω)
60		simplrl 777	. . . . . . . . . . . . . 14 ⊢ (((((𝑋 ∈ On ∧ 𝑆 = dom (ω CNF 𝑋)) ∧ (𝑓:𝑋⟶ω ∧ 𝑓 finSupp ∅)) ∧ (𝑔:𝑋⟶ω ∧ 𝑔 finSupp ∅)) ∧ (𝑓 ∘f +o 𝑔):𝑋⟶ω) → 𝑔:𝑋⟶ω)
61		0elon 6440	. . . . . . . . . . . . . . 15 ⊢ ∅ ∈ On
62		oa0 8553	. . . . . . . . . . . . . . 15 ⊢ (∅ ∈ On → (∅ +o ∅) = ∅)
63	61, 62	mp1i 13	. . . . . . . . . . . . . 14 ⊢ (((((𝑋 ∈ On ∧ 𝑆 = dom (ω CNF 𝑋)) ∧ (𝑓:𝑋⟶ω ∧ 𝑓 finSupp ∅)) ∧ (𝑔:𝑋⟶ω ∧ 𝑔 finSupp ∅)) ∧ (𝑓 ∘f +o 𝑔):𝑋⟶ω) → (∅ +o ∅) = ∅)
64	55, 57, 59, 60, 63	suppofssd 8227	. . . . . . . . . . . . 13 ⊢ (((((𝑋 ∈ On ∧ 𝑆 = dom (ω CNF 𝑋)) ∧ (𝑓:𝑋⟶ω ∧ 𝑓 finSupp ∅)) ∧ (𝑔:𝑋⟶ω ∧ 𝑔 finSupp ∅)) ∧ (𝑓 ∘f +o 𝑔):𝑋⟶ω) → ((𝑓 ∘f +o 𝑔) supp ∅) ⊆ ((𝑓 supp ∅) ∪ (𝑔 supp ∅)))
65	54, 64	ssfid 9299	. . . . . . . . . . . 12 ⊢ (((((𝑋 ∈ On ∧ 𝑆 = dom (ω CNF 𝑋)) ∧ (𝑓:𝑋⟶ω ∧ 𝑓 finSupp ∅)) ∧ (𝑔:𝑋⟶ω ∧ 𝑔 finSupp ∅)) ∧ (𝑓 ∘f +o 𝑔):𝑋⟶ω) → ((𝑓 ∘f +o 𝑔) supp ∅) ∈ Fin)
66		ovexd 7466	. . . . . . . . . . . . 13 ⊢ (((((𝑋 ∈ On ∧ 𝑆 = dom (ω CNF 𝑋)) ∧ (𝑓:𝑋⟶ω ∧ 𝑓 finSupp ∅)) ∧ (𝑔:𝑋⟶ω ∧ 𝑔 finSupp ∅)) ∧ (𝑓 ∘f +o 𝑔):𝑋⟶ω) → (𝑓 ∘f +o 𝑔) ∈ V)
67		isfsupp 9403	. . . . . . . . . . . . 13 ⊢ (((𝑓 ∘f +o 𝑔) ∈ V ∧ ∅ ∈ On) → ((𝑓 ∘f +o 𝑔) finSupp ∅ ↔ (Fun (𝑓 ∘f +o 𝑔) ∧ ((𝑓 ∘f +o 𝑔) supp ∅) ∈ Fin)))
68	66, 61, 67	sylancl 586	. . . . . . . . . . . 12 ⊢ (((((𝑋 ∈ On ∧ 𝑆 = dom (ω CNF 𝑋)) ∧ (𝑓:𝑋⟶ω ∧ 𝑓 finSupp ∅)) ∧ (𝑔:𝑋⟶ω ∧ 𝑔 finSupp ∅)) ∧ (𝑓 ∘f +o 𝑔):𝑋⟶ω) → ((𝑓 ∘f +o 𝑔) finSupp ∅ ↔ (Fun (𝑓 ∘f +o 𝑔) ∧ ((𝑓 ∘f +o 𝑔) supp ∅) ∈ Fin)))
69	50, 65, 68	mpbir2and 713	. . . . . . . . . . 11 ⊢ (((((𝑋 ∈ On ∧ 𝑆 = dom (ω CNF 𝑋)) ∧ (𝑓:𝑋⟶ω ∧ 𝑓 finSupp ∅)) ∧ (𝑔:𝑋⟶ω ∧ 𝑔 finSupp ∅)) ∧ (𝑓 ∘f +o 𝑔):𝑋⟶ω) → (𝑓 ∘f +o 𝑔) finSupp ∅)
70	69	ex 412	. . . . . . . . . 10 ⊢ ((((𝑋 ∈ On ∧ 𝑆 = dom (ω CNF 𝑋)) ∧ (𝑓:𝑋⟶ω ∧ 𝑓 finSupp ∅)) ∧ (𝑔:𝑋⟶ω ∧ 𝑔 finSupp ∅)) → ((𝑓 ∘f +o 𝑔):𝑋⟶ω → (𝑓 ∘f +o 𝑔) finSupp ∅))
71	48, 70	jcai 516	. . . . . . . . 9 ⊢ ((((𝑋 ∈ On ∧ 𝑆 = dom (ω CNF 𝑋)) ∧ (𝑓:𝑋⟶ω ∧ 𝑓 finSupp ∅)) ∧ (𝑔:𝑋⟶ω ∧ 𝑔 finSupp ∅)) → ((𝑓 ∘f +o 𝑔):𝑋⟶ω ∧ (𝑓 ∘f +o 𝑔) finSupp ∅))
72	1	eleq2d 2825	. . . . . . . . . . 11 ⊢ ((𝑋 ∈ On ∧ 𝑆 = dom (ω CNF 𝑋)) → ((𝑓 ∘f +o 𝑔) ∈ 𝑆 ↔ (𝑓 ∘f +o 𝑔) ∈ dom (ω CNF 𝑋)))
73	3, 5, 6	cantnfs 9704	. . . . . . . . . . 11 ⊢ ((𝑋 ∈ On ∧ 𝑆 = dom (ω CNF 𝑋)) → ((𝑓 ∘f +o 𝑔) ∈ dom (ω CNF 𝑋) ↔ ((𝑓 ∘f +o 𝑔):𝑋⟶ω ∧ (𝑓 ∘f +o 𝑔) finSupp ∅)))
74	72, 73	bitrd 279	. . . . . . . . . 10 ⊢ ((𝑋 ∈ On ∧ 𝑆 = dom (ω CNF 𝑋)) → ((𝑓 ∘f +o 𝑔) ∈ 𝑆 ↔ ((𝑓 ∘f +o 𝑔):𝑋⟶ω ∧ (𝑓 ∘f +o 𝑔) finSupp ∅)))
75	74	ad2antrr 726	. . . . . . . . 9 ⊢ ((((𝑋 ∈ On ∧ 𝑆 = dom (ω CNF 𝑋)) ∧ (𝑓:𝑋⟶ω ∧ 𝑓 finSupp ∅)) ∧ (𝑔:𝑋⟶ω ∧ 𝑔 finSupp ∅)) → ((𝑓 ∘f +o 𝑔) ∈ 𝑆 ↔ ((𝑓 ∘f +o 𝑔):𝑋⟶ω ∧ (𝑓 ∘f +o 𝑔) finSupp ∅)))
76	71, 75	mpbird 257	. . . . . . . 8 ⊢ ((((𝑋 ∈ On ∧ 𝑆 = dom (ω CNF 𝑋)) ∧ (𝑓:𝑋⟶ω ∧ 𝑓 finSupp ∅)) ∧ (𝑔:𝑋⟶ω ∧ 𝑔 finSupp ∅)) → (𝑓 ∘f +o 𝑔) ∈ 𝑆)
77	76	ex 412	. . . . . . 7 ⊢ (((𝑋 ∈ On ∧ 𝑆 = dom (ω CNF 𝑋)) ∧ (𝑓:𝑋⟶ω ∧ 𝑓 finSupp ∅)) → ((𝑔:𝑋⟶ω ∧ 𝑔 finSupp ∅) → (𝑓 ∘f +o 𝑔) ∈ 𝑆))
78	12, 77	sylbid 240	. . . . . 6 ⊢ (((𝑋 ∈ On ∧ 𝑆 = dom (ω CNF 𝑋)) ∧ (𝑓:𝑋⟶ω ∧ 𝑓 finSupp ∅)) → (𝑔 ∈ 𝑆 → (𝑓 ∘f +o 𝑔) ∈ 𝑆))
79	78	ralrimiv 3143	. . . . 5 ⊢ (((𝑋 ∈ On ∧ 𝑆 = dom (ω CNF 𝑋)) ∧ (𝑓:𝑋⟶ω ∧ 𝑓 finSupp ∅)) → ∀𝑔 ∈ 𝑆 (𝑓 ∘f +o 𝑔) ∈ 𝑆)
80	79	ex 412	. . . 4 ⊢ ((𝑋 ∈ On ∧ 𝑆 = dom (ω CNF 𝑋)) → ((𝑓:𝑋⟶ω ∧ 𝑓 finSupp ∅) → ∀𝑔 ∈ 𝑆 (𝑓 ∘f +o 𝑔) ∈ 𝑆))
81	8, 80	sylbid 240	. . 3 ⊢ ((𝑋 ∈ On ∧ 𝑆 = dom (ω CNF 𝑋)) → (𝑓 ∈ 𝑆 → ∀𝑔 ∈ 𝑆 (𝑓 ∘f +o 𝑔) ∈ 𝑆))
82	81	ralrimiv 3143	. 2 ⊢ ((𝑋 ∈ On ∧ 𝑆 = dom (ω CNF 𝑋)) → ∀𝑓 ∈ 𝑆 ∀𝑔 ∈ 𝑆 (𝑓 ∘f +o 𝑔) ∈ 𝑆)
83		ofmres 8008	. . 3 ⊢ ( ∘f +o ↾ (𝑆 × 𝑆)) = (𝑓 ∈ 𝑆, 𝑔 ∈ 𝑆 ↦ (𝑓 ∘f +o 𝑔))
84	83	fmpo 8092	. 2 ⊢ (∀𝑓 ∈ 𝑆 ∀𝑔 ∈ 𝑆 (𝑓 ∘f +o 𝑔) ∈ 𝑆 ↔ ( ∘f +o ↾ (𝑆 × 𝑆)):(𝑆 × 𝑆)⟶𝑆)
85	82, 84	sylib 218	1 ⊢ ((𝑋 ∈ On ∧ 𝑆 = dom (ω CNF 𝑋)) → ( ∘f +o ↾ (𝑆 × 𝑆)):(𝑆 × 𝑆)⟶𝑆)

Syntax hints:   → wi 4   ↔ wb 206   ∧ wa 395   = wceq 1537   ∈ wcel 2106  ∀wral 3059  Vcvv 3478   ∪ cun 3961   ⊆ wss 3963  ∅c0 4339   class class class wbr 5148   × cxp 5687  dom cdm 5689  ran crn 5690   ↾ cres 5691  Oncon0 6386  Fun wfun 6557   Fn wfn 6558  ⟶wf 6559  ‘cfv 6563  (class class class)co 7431   ∘_f cof 7695  ωcom 7887   supp csupp 8184   +_o coa 8502  Fincfn 8984   finSupp cfsupp 9399   CNF ccnf 9699

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1792  ax-4 1806  ax-5 1908  ax-6 1965  ax-7 2005  ax-8 2108  ax-9 2116  ax-10 2139  ax-11 2155  ax-12 2175  ax-ext 2706  ax-rep 5285  ax-sep 5302  ax-nul 5312  ax-pow 5371  ax-pr 5438  ax-un 7754  ax-inf2 9679

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3or 1087  df-3an 1088  df-tru 1540  df-fal 1550  df-ex 1777  df-nf 1781  df-sb 2063  df-mo 2538  df-eu 2567  df-clab 2713  df-cleq 2727  df-clel 2814  df-nfc 2890  df-ne 2939  df-ral 3060  df-rex 3069  df-reu 3379  df-rab 3434  df-v 3480  df-sbc 3792  df-csb 3909  df-dif 3966  df-un 3968  df-in 3970  df-ss 3980  df-pss 3983  df-nul 4340  df-if 4532  df-pw 4607  df-sn 4632  df-pr 4634  df-op 4638  df-uni 4913  df-iun 4998  df-br 5149  df-opab 5211  df-mpt 5232  df-tr 5266  df-id 5583  df-eprel 5589  df-po 5597  df-so 5598  df-fr 5641  df-we 5643  df-xp 5695  df-rel 5696  df-cnv 5697  df-co 5698  df-dm 5699  df-rn 5700  df-res 5701  df-ima 5702  df-pred 6323  df-ord 6389  df-on 6390  df-lim 6391  df-suc 6392  df-iota 6516  df-fun 6565  df-fn 6566  df-f 6567  df-f1 6568  df-fo 6569  df-f1o 6570  df-fv 6571  df-ov 7434  df-oprab 7435  df-mpo 7436  df-of 7697  df-om 7888  df-1st 8013  df-2nd 8014  df-supp 8185  df-frecs 8305  df-wrecs 8336  df-recs 8410  df-rdg 8449  df-seqom 8487  df-1o 8505  df-oadd 8509  df-map 8867  df-en 8985  df-fin 8988  df-fsupp 9400  df-cnf 9700

This theorem is referenced by:  naddcnffn  43353  naddcnffo  43354  naddcnfcl  43355