Theorem naddcnff 42415

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 2818	. . . . 5 ⊢ ((𝑋 ∈ On ∧ 𝑆 = dom (ω CNF 𝑋)) → (𝑓 ∈ 𝑆 ↔ 𝑓 ∈ dom (ω CNF 𝑋)))
3		eqid 2731	. . . . . 6 ⊢ dom (ω CNF 𝑋) = dom (ω CNF 𝑋)
4		omelon 9644	. . . . . . 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 9664	. . . . 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 2818	. . . . . . . . 9 ⊢ ((𝑋 ∈ On ∧ 𝑆 = dom (ω CNF 𝑋)) → (𝑔 ∈ 𝑆 ↔ 𝑔 ∈ dom (ω CNF 𝑋)))
10	3, 5, 6	cantnfs 9664	. . . . . . . . 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 612	. . . . . . . . . . . . 13 ⊢ (((𝑓:𝑋⟶ω ∧ 𝑓 finSupp ∅) ∧ (𝑔:𝑋⟶ω ∧ 𝑔 finSupp ∅)) → (𝑓:𝑋⟶ω ∧ 𝑔:𝑋⟶ω))
16	6, 15	anim12i 612	. . . . . . . . . . . 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 768	. . . . . . . . . . . . . . 15 ⊢ ((𝑋 ∈ On ∧ (𝑓:𝑋⟶ω ∧ 𝑔:𝑋⟶ω)) → 𝑓:𝑋⟶ω)
19	18	ffnd 6718	. . . . . . . . . . . . . 14 ⊢ ((𝑋 ∈ On ∧ (𝑓:𝑋⟶ω ∧ 𝑔:𝑋⟶ω)) → 𝑓 Fn 𝑋)
20		simprr 770	. . . . . . . . . . . . . . 15 ⊢ ((𝑋 ∈ On ∧ (𝑓:𝑋⟶ω ∧ 𝑔:𝑋⟶ω)) → 𝑔:𝑋⟶ω)
21	20	ffnd 6718	. . . . . . . . . . . . . 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 4218	. . . . . . . . . . . . . 14 ⊢ (𝑋 ∩ 𝑋) = 𝑋
24	19, 21, 22, 22, 23	offn 7686	. . . . . . . . . . . . 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 774	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (((𝑋 ∈ On ∧ (𝑓:𝑋⟶ω ∧ 𝑔:𝑋⟶ω)) ∧ 𝑥 ∈ 𝑋) → 𝑓:𝑋⟶ω)
27	26	ffnd 6718	. . . . . . . . . . . . . . . . . . . 20 ⊢ (((𝑋 ∈ On ∧ (𝑓:𝑋⟶ω ∧ 𝑔:𝑋⟶ω)) ∧ 𝑥 ∈ 𝑋) → 𝑓 Fn 𝑋)
28		simplrr 775	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (((𝑋 ∈ On ∧ (𝑓:𝑋⟶ω ∧ 𝑔:𝑋⟶ω)) ∧ 𝑥 ∈ 𝑋) → 𝑔:𝑋⟶ω)
29	28	ffnd 6718	. . . . . . . . . . . . . . . . . . . 20 ⊢ (((𝑋 ∈ On ∧ (𝑓:𝑋⟶ω ∧ 𝑔:𝑋⟶ω)) ∧ 𝑥 ∈ 𝑋) → 𝑔 Fn 𝑋)
30		simpll 764	. . . . . . . . . . . . . . . . . . . 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 7690	. . . . . . . . . . . . . . . . . . . 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 836	. . . . . . . . . . . . . . . . . . 19 ⊢ (((𝑋 ∈ On ∧ (𝑓:𝑋⟶ω ∧ 𝑔:𝑋⟶ω)) ∧ 𝑥 ∈ 𝑋) → ((𝑓 ∘f +o 𝑔)‘𝑥) = ((𝑓‘𝑥) +o (𝑔‘𝑥)))
34	18	ffvelcdmda 7086	. . . . . . . . . . . . . . . . . . . 20 ⊢ (((𝑋 ∈ On ∧ (𝑓:𝑋⟶ω ∧ 𝑔:𝑋⟶ω)) ∧ 𝑥 ∈ 𝑋) → (𝑓‘𝑥) ∈ ω)
35	20	ffvelcdmda 7086	. . . . . . . . . . . . . . . . . . . 20 ⊢ (((𝑋 ∈ On ∧ (𝑓:𝑋⟶ω ∧ 𝑔:𝑋⟶ω)) ∧ 𝑥 ∈ 𝑋) → (𝑔‘𝑥) ∈ ω)
36		nnacl 8614	. . . . . . . . . . . . . . . . . . . 20 ⊢ (((𝑓‘𝑥) ∈ ω ∧ (𝑔‘𝑥) ∈ ω) → ((𝑓‘𝑥) +o (𝑔‘𝑥)) ∈ ω)
37	34, 35, 36	syl2anc 583	. . . . . . . . . . . . . . . . . . 19 ⊢ (((𝑋 ∈ On ∧ (𝑓:𝑋⟶ω ∧ 𝑔:𝑋⟶ω)) ∧ 𝑥 ∈ 𝑋) → ((𝑓‘𝑥) +o (𝑔‘𝑥)) ∈ ω)
38	33, 37	eqeltrd 2832	. . . . . . . . . . . . . . . . . 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 3144	. . . . . . . . . . . . . . . 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 7122	. . . . . . . . . . . . . . 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 583	. . . . . . . . . . . . . 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 6547	. . . . . . . . . . . 12 ⊢ ((𝑓 ∘f +o 𝑔):𝑋⟶ω ↔ ((𝑓 ∘f +o 𝑔) Fn 𝑋 ∧ ran (𝑓 ∘f +o 𝑔) ⊆ ω))
47	45, 46	sylibr 233	. . . . . . . . . . 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 6720	. . . . . . . . . . . . 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 775	. . . . . . . . . . . . . . 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 775	. . . . . . . . . . . . . 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 9386	. . . . . . . . . . . . 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 780	. . . . . . . . . . . . . 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 7882	. . . . . . . . . . . . . . 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 774	. . . . . . . . . . . . . . 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 774	. . . . . . . . . . . . . 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 6418	. . . . . . . . . . . . . . 15 ⊢ ∅ ∈ On
62		oa0 8519	. . . . . . . . . . . . . . 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 8191	. . . . . . . . . . . . 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 9270	. . . . . . . . . . . 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 7447	. . . . . . . . . . . . 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 9368	. . . . . . . . . . . . 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 585	. . . . . . . . . . . 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 710	. . . . . . . . . . 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 2818	. . . . . . . . . . 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 9664	. . . . . . . . . . 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 723	. . . . . . . . 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 239	. . . . . 6 ⊢ (((𝑋 ∈ On ∧ 𝑆 = dom (ω CNF 𝑋)) ∧ (𝑓:𝑋⟶ω ∧ 𝑓 finSupp ∅)) → (𝑔 ∈ 𝑆 → (𝑓 ∘f +o 𝑔) ∈ 𝑆))
79	78	ralrimiv 3144	. . . . 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 239	. . 3 ⊢ ((𝑋 ∈ On ∧ 𝑆 = dom (ω CNF 𝑋)) → (𝑓 ∈ 𝑆 → ∀𝑔 ∈ 𝑆 (𝑓 ∘f +o 𝑔) ∈ 𝑆))
82	81	ralrimiv 3144	. 2 ⊢ ((𝑋 ∈ On ∧ 𝑆 = dom (ω CNF 𝑋)) → ∀𝑓 ∈ 𝑆 ∀𝑔 ∈ 𝑆 (𝑓 ∘f +o 𝑔) ∈ 𝑆)
83		ofmres 7974	. . 3 ⊢ ( ∘f +o ↾ (𝑆 × 𝑆)) = (𝑓 ∈ 𝑆, 𝑔 ∈ 𝑆 ↦ (𝑓 ∘f +o 𝑔))
84	83	fmpo 8057	. 2 ⊢ (∀𝑓 ∈ 𝑆 ∀𝑔 ∈ 𝑆 (𝑓 ∘f +o 𝑔) ∈ 𝑆 ↔ ( ∘f +o ↾ (𝑆 × 𝑆)):(𝑆 × 𝑆)⟶𝑆)
85	82, 84	sylib 217	1 ⊢ ((𝑋 ∈ On ∧ 𝑆 = dom (ω CNF 𝑋)) → ( ∘f +o ↾ (𝑆 × 𝑆)):(𝑆 × 𝑆)⟶𝑆)

Syntax hints:   → wi 4   ↔ wb 205   ∧ wa 395   = wceq 1540   ∈ wcel 2105  ∀wral 3060  Vcvv 3473   ∪ cun 3946   ⊆ wss 3948  ∅c0 4322   class class class wbr 5148   × cxp 5674  dom cdm 5676  ran crn 5677   ↾ cres 5678  Oncon0 6364  Fun wfun 6537   Fn wfn 6538  ⟶wf 6539  ‘cfv 6543  (class class class)co 7412   ∘_f cof 7671  ωcom 7858   supp csupp 8149   +_o coa 8466  Fincfn 8942   finSupp cfsupp 9364   CNF ccnf 9659

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1912  ax-6 1970  ax-7 2010  ax-8 2107  ax-9 2115  ax-10 2136  ax-11 2153  ax-12 2170  ax-ext 2702  ax-rep 5285  ax-sep 5299  ax-nul 5306  ax-pow 5363  ax-pr 5427  ax-un 7728  ax-inf2 9639

This theorem depends on definitions:  df-bi 206  df-an 396  df-or 845  df-3or 1087  df-3an 1088  df-tru 1543  df-fal 1553  df-ex 1781  df-nf 1785  df-sb 2067  df-mo 2533  df-eu 2562  df-clab 2709  df-cleq 2723  df-clel 2809  df-nfc 2884  df-ne 2940  df-ral 3061  df-rex 3070  df-reu 3376  df-rab 3432  df-v 3475  df-sbc 3778  df-csb 3894  df-dif 3951  df-un 3953  df-in 3955  df-ss 3965  df-pss 3967  df-nul 4323  df-if 4529  df-pw 4604  df-sn 4629  df-pr 4631  df-op 4635  df-uni 4909  df-iun 4999  df-br 5149  df-opab 5211  df-mpt 5232  df-tr 5266  df-id 5574  df-eprel 5580  df-po 5588  df-so 5589  df-fr 5631  df-we 5633  df-xp 5682  df-rel 5683  df-cnv 5684  df-co 5685  df-dm 5686  df-rn 5687  df-res 5688  df-ima 5689  df-pred 6300  df-ord 6367  df-on 6368  df-lim 6369  df-suc 6370  df-iota 6495  df-fun 6545  df-fn 6546  df-f 6547  df-f1 6548  df-fo 6549  df-f1o 6550  df-fv 6551  df-ov 7415  df-oprab 7416  df-mpo 7417  df-of 7673  df-om 7859  df-1st 7978  df-2nd 7979  df-supp 8150  df-frecs 8269  df-wrecs 8300  df-recs 8374  df-rdg 8413  df-seqom 8451  df-1o 8469  df-oadd 8473  df-map 8825  df-en 8943  df-fin 8946  df-fsupp 9365  df-cnf 9660

This theorem is referenced by:  naddcnffn  42416  naddcnffo  42417  naddcnfcl  42418