Theorem naddcnff 41947

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 485	. . . . . 6 ⊢ ((𝑋 ∈ On ∧ 𝑆 = dom (ω CNF 𝑋)) → 𝑆 = dom (ω CNF 𝑋))
2	1	eleq2d 2819	. . . . 5 ⊢ ((𝑋 ∈ On ∧ 𝑆 = dom (ω CNF 𝑋)) → (𝑓 ∈ 𝑆 ↔ 𝑓 ∈ dom (ω CNF 𝑋)))
3		eqid 2732	. . . . . 6 ⊢ dom (ω CNF 𝑋) = dom (ω CNF 𝑋)
4		omelon 9625	. . . . . . 7 ⊢ ω ∈ On
5	4	a1i 11	. . . . . 6 ⊢ ((𝑋 ∈ On ∧ 𝑆 = dom (ω CNF 𝑋)) → ω ∈ On)
6		simpl 483	. . . . . 6 ⊢ ((𝑋 ∈ On ∧ 𝑆 = dom (ω CNF 𝑋)) → 𝑋 ∈ On)
7	3, 5, 6	cantnfs 9645	. . . . 5 ⊢ ((𝑋 ∈ On ∧ 𝑆 = dom (ω CNF 𝑋)) → (𝑓 ∈ dom (ω CNF 𝑋) ↔ (𝑓:𝑋⟶ω ∧ 𝑓 finSupp ∅)))
8	2, 7	bitrd 278	. . . 4 ⊢ ((𝑋 ∈ On ∧ 𝑆 = dom (ω CNF 𝑋)) → (𝑓 ∈ 𝑆 ↔ (𝑓:𝑋⟶ω ∧ 𝑓 finSupp ∅)))
9	1	eleq2d 2819	. . . . . . . . 9 ⊢ ((𝑋 ∈ On ∧ 𝑆 = dom (ω CNF 𝑋)) → (𝑔 ∈ 𝑆 ↔ 𝑔 ∈ dom (ω CNF 𝑋)))
10	3, 5, 6	cantnfs 9645	. . . . . . . . 9 ⊢ ((𝑋 ∈ On ∧ 𝑆 = dom (ω CNF 𝑋)) → (𝑔 ∈ dom (ω CNF 𝑋) ↔ (𝑔:𝑋⟶ω ∧ 𝑔 finSupp ∅)))
11	9, 10	bitrd 278	. . . . . . . 8 ⊢ ((𝑋 ∈ On ∧ 𝑆 = dom (ω CNF 𝑋)) → (𝑔 ∈ 𝑆 ↔ (𝑔:𝑋⟶ω ∧ 𝑔 finSupp ∅)))
12	11	adantr 481	. . . . . . 7 ⊢ (((𝑋 ∈ On ∧ 𝑆 = dom (ω CNF 𝑋)) ∧ (𝑓:𝑋⟶ω ∧ 𝑓 finSupp ∅)) → (𝑔 ∈ 𝑆 ↔ (𝑔:𝑋⟶ω ∧ 𝑔 finSupp ∅)))
13		simpl 483	. . . . . . . . . . . . . 14 ⊢ ((𝑓:𝑋⟶ω ∧ 𝑓 finSupp ∅) → 𝑓:𝑋⟶ω)
14		simpl 483	. . . . . . . . . . . . . 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 468	. . . . . . . . . . 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 769	. . . . . . . . . . . . . . 15 ⊢ ((𝑋 ∈ On ∧ (𝑓:𝑋⟶ω ∧ 𝑔:𝑋⟶ω)) → 𝑓:𝑋⟶ω)
19	18	ffnd 6706	. . . . . . . . . . . . . 14 ⊢ ((𝑋 ∈ On ∧ (𝑓:𝑋⟶ω ∧ 𝑔:𝑋⟶ω)) → 𝑓 Fn 𝑋)
20		simprr 771	. . . . . . . . . . . . . . 15 ⊢ ((𝑋 ∈ On ∧ (𝑓:𝑋⟶ω ∧ 𝑔:𝑋⟶ω)) → 𝑔:𝑋⟶ω)
21	20	ffnd 6706	. . . . . . . . . . . . . 14 ⊢ ((𝑋 ∈ On ∧ (𝑓:𝑋⟶ω ∧ 𝑔:𝑋⟶ω)) → 𝑔 Fn 𝑋)
22		simpl 483	. . . . . . . . . . . . . 14 ⊢ ((𝑋 ∈ On ∧ (𝑓:𝑋⟶ω ∧ 𝑔:𝑋⟶ω)) → 𝑋 ∈ On)
23		inidm 4215	. . . . . . . . . . . . . 14 ⊢ (𝑋 ∩ 𝑋) = 𝑋
24	19, 21, 22, 22, 23	offn 7667	. . . . . . . . . . . . 13 ⊢ ((𝑋 ∈ On ∧ (𝑓:𝑋⟶ω ∧ 𝑔:𝑋⟶ω)) → (𝑓 ∘f +o 𝑔) Fn 𝑋)
25		simpr 485	. . . . . . . . . . . . . . 15 ⊢ (((𝑋 ∈ On ∧ (𝑓:𝑋⟶ω ∧ 𝑔:𝑋⟶ω)) ∧ (𝑓 ∘f +o 𝑔) Fn 𝑋) → (𝑓 ∘f +o 𝑔) Fn 𝑋)
26		simplrl 775	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (((𝑋 ∈ On ∧ (𝑓:𝑋⟶ω ∧ 𝑔:𝑋⟶ω)) ∧ 𝑥 ∈ 𝑋) → 𝑓:𝑋⟶ω)
27	26	ffnd 6706	. . . . . . . . . . . . . . . . . . . 20 ⊢ (((𝑋 ∈ On ∧ (𝑓:𝑋⟶ω ∧ 𝑔:𝑋⟶ω)) ∧ 𝑥 ∈ 𝑋) → 𝑓 Fn 𝑋)
28		simplrr 776	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (((𝑋 ∈ On ∧ (𝑓:𝑋⟶ω ∧ 𝑔:𝑋⟶ω)) ∧ 𝑥 ∈ 𝑋) → 𝑔:𝑋⟶ω)
29	28	ffnd 6706	. . . . . . . . . . . . . . . . . . . 20 ⊢ (((𝑋 ∈ On ∧ (𝑓:𝑋⟶ω ∧ 𝑔:𝑋⟶ω)) ∧ 𝑥 ∈ 𝑋) → 𝑔 Fn 𝑋)
30		simpll 765	. . . . . . . . . . . . . . . . . . . 20 ⊢ (((𝑋 ∈ On ∧ (𝑓:𝑋⟶ω ∧ 𝑔:𝑋⟶ω)) ∧ 𝑥 ∈ 𝑋) → 𝑋 ∈ On)
31		simpr 485	. . . . . . . . . . . . . . . . . . . 20 ⊢ (((𝑋 ∈ On ∧ (𝑓:𝑋⟶ω ∧ 𝑔:𝑋⟶ω)) ∧ 𝑥 ∈ 𝑋) → 𝑥 ∈ 𝑋)
32		fnfvof 7671	. . . . . . . . . . . . . . . . . . . 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 837	. . . . . . . . . . . . . . . . . . 19 ⊢ (((𝑋 ∈ On ∧ (𝑓:𝑋⟶ω ∧ 𝑔:𝑋⟶ω)) ∧ 𝑥 ∈ 𝑋) → ((𝑓 ∘f +o 𝑔)‘𝑥) = ((𝑓‘𝑥) +o (𝑔‘𝑥)))
34	18	ffvelcdmda 7072	. . . . . . . . . . . . . . . . . . . 20 ⊢ (((𝑋 ∈ On ∧ (𝑓:𝑋⟶ω ∧ 𝑔:𝑋⟶ω)) ∧ 𝑥 ∈ 𝑋) → (𝑓‘𝑥) ∈ ω)
35	20	ffvelcdmda 7072	. . . . . . . . . . . . . . . . . . . 20 ⊢ (((𝑋 ∈ On ∧ (𝑓:𝑋⟶ω ∧ 𝑔:𝑋⟶ω)) ∧ 𝑥 ∈ 𝑋) → (𝑔‘𝑥) ∈ ω)
36		nnacl 8596	. . . . . . . . . . . . . . . . . . . 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 2833	. . . . . . . . . . . . . . . . . 18 ⊢ (((𝑋 ∈ On ∧ (𝑓:𝑋⟶ω ∧ 𝑔:𝑋⟶ω)) ∧ 𝑥 ∈ 𝑋) → ((𝑓 ∘f +o 𝑔)‘𝑥) ∈ ω)
39	38	ex 413	. . . . . . . . . . . . . . . . 17 ⊢ ((𝑋 ∈ On ∧ (𝑓:𝑋⟶ω ∧ 𝑔:𝑋⟶ω)) → (𝑥 ∈ 𝑋 → ((𝑓 ∘f +o 𝑔)‘𝑥) ∈ ω))
40	39	ralrimiv 3145	. . . . . . . . . . . . . . . 16 ⊢ ((𝑋 ∈ On ∧ (𝑓:𝑋⟶ω ∧ 𝑔:𝑋⟶ω)) → ∀𝑥 ∈ 𝑋 ((𝑓 ∘f +o 𝑔)‘𝑥) ∈ ω)
41	40	adantr 481	. . . . . . . . . . . . . . 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 7105	. . . . . . . . . . . . . . 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 413	. . . . . . . . . . . . 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 517	. . . . . . . . . . . 12 ⊢ ((𝑋 ∈ On ∧ (𝑓:𝑋⟶ω ∧ 𝑔:𝑋⟶ω)) → ((𝑓 ∘f +o 𝑔) Fn 𝑋 ∧ ran (𝑓 ∘f +o 𝑔) ⊆ ω))
46		df-f 6537	. . . . . . . . . . . 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 6708	. . . . . . . . . . . . 13 ⊢ ((𝑓 ∘f +o 𝑔):𝑋⟶ω → Fun (𝑓 ∘f +o 𝑔))
50	49	adantl 482	. . . . . . . . . . . 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 776	. . . . . . . . . . . . . . 15 ⊢ ((((𝑋 ∈ On ∧ 𝑆 = dom (ω CNF 𝑋)) ∧ (𝑓:𝑋⟶ω ∧ 𝑓 finSupp ∅)) ∧ (𝑔:𝑋⟶ω ∧ 𝑔 finSupp ∅)) → 𝑓 finSupp ∅)
52	51	adantr 481	. . . . . . . . . . . . . 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 776	. . . . . . . . . . . . . 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 9368	. . . . . . . . . . . . 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 781	. . . . . . . . . . . . . 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 7863	. . . . . . . . . . . . . . 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 775	. . . . . . . . . . . . . . 15 ⊢ ((((𝑋 ∈ On ∧ 𝑆 = dom (ω CNF 𝑋)) ∧ (𝑓:𝑋⟶ω ∧ 𝑓 finSupp ∅)) ∧ (𝑔:𝑋⟶ω ∧ 𝑔 finSupp ∅)) → 𝑓:𝑋⟶ω)
59	58	adantr 481	. . . . . . . . . . . . . 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 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:𝑋⟶ω)
61		0elon 6408	. . . . . . . . . . . . . . 15 ⊢ ∅ ∈ On
62		oa0 8500	. . . . . . . . . . . . . . 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 8172	. . . . . . . . . . . . 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 9252	. . . . . . . . . . . 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 7429	. . . . . . . . . . . . 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 9350	. . . . . . . . . . . . 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 711	. . . . . . . . . . 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 413	. . . . . . . . . 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 517	. . . . . . . . 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 2819	. . . . . . . . . . 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 9645	. . . . . . . . . . 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 278	. . . . . . . . . 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 724	. . . . . . . . 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 256	. . . . . . . 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 413	. . . . . . 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 3145	. . . . 5 ⊢ (((𝑋 ∈ On ∧ 𝑆 = dom (ω CNF 𝑋)) ∧ (𝑓:𝑋⟶ω ∧ 𝑓 finSupp ∅)) → ∀𝑔 ∈ 𝑆 (𝑓 ∘f +o 𝑔) ∈ 𝑆)
80	79	ex 413	. . . 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 3145	. 2 ⊢ ((𝑋 ∈ On ∧ 𝑆 = dom (ω CNF 𝑋)) → ∀𝑓 ∈ 𝑆 ∀𝑔 ∈ 𝑆 (𝑓 ∘f +o 𝑔) ∈ 𝑆)
83		ofmres 7955	. . 3 ⊢ ( ∘f +o ↾ (𝑆 × 𝑆)) = (𝑓 ∈ 𝑆, 𝑔 ∈ 𝑆 ↦ (𝑓 ∘f +o 𝑔))
84	83	fmpo 8038	. 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 396   = wceq 1541   ∈ wcel 2106  ∀wral 3061  Vcvv 3474   ∪ cun 3943   ⊆ wss 3945  ∅c0 4319   class class class wbr 5142   × cxp 5668  dom cdm 5670  ran crn 5671   ↾ cres 5672  Oncon0 6354  Fun wfun 6527   Fn wfn 6528  ⟶wf 6529  ‘cfv 6533  (class class class)co 7394   ∘_f cof 7652  ωcom 7839   supp csupp 8130   +_o coa 8447  Fincfn 8924   finSupp cfsupp 9346   CNF ccnf 9640

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 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-10 2137  ax-11 2154  ax-12 2171  ax-ext 2703  ax-rep 5279  ax-sep 5293  ax-nul 5300  ax-pow 5357  ax-pr 5421  ax-un 7709  ax-inf2 9620

This theorem depends on definitions:  df-bi 206  df-an 397  df-or 846  df-3or 1088  df-3an 1089  df-tru 1544  df-fal 1554  df-ex 1782  df-nf 1786  df-sb 2068  df-mo 2534  df-eu 2563  df-clab 2710  df-cleq 2724  df-clel 2810  df-nfc 2885  df-ne 2941  df-ral 3062  df-rex 3071  df-reu 3377  df-rab 3433  df-v 3476  df-sbc 3775  df-csb 3891  df-dif 3948  df-un 3950  df-in 3952  df-ss 3962  df-pss 3964  df-nul 4320  df-if 4524  df-pw 4599  df-sn 4624  df-pr 4626  df-op 4630  df-uni 4903  df-iun 4993  df-br 5143  df-opab 5205  df-mpt 5226  df-tr 5260  df-id 5568  df-eprel 5574  df-po 5582  df-so 5583  df-fr 5625  df-we 5627  df-xp 5676  df-rel 5677  df-cnv 5678  df-co 5679  df-dm 5680  df-rn 5681  df-res 5682  df-ima 5683  df-pred 6290  df-ord 6357  df-on 6358  df-lim 6359  df-suc 6360  df-iota 6485  df-fun 6535  df-fn 6536  df-f 6537  df-f1 6538  df-fo 6539  df-f1o 6540  df-fv 6541  df-ov 7397  df-oprab 7398  df-mpo 7399  df-of 7654  df-om 7840  df-1st 7959  df-2nd 7960  df-supp 8131  df-frecs 8250  df-wrecs 8281  df-recs 8355  df-rdg 8394  df-seqom 8432  df-1o 8450  df-oadd 8454  df-map 8807  df-en 8925  df-fin 8928  df-fsupp 9347  df-cnf 9641

This theorem is referenced by:  naddcnffn  41948  naddcnffo  41949  naddcnfcl  41950