Users' Mathboxes Mathbox for Richard Penner < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >   Mathboxes  >  naddcnff Structured version   Visualization version   GIF version

Theorem naddcnff 42194
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.
StepHypRef Expression
1 simpr 485 . . . . . 6 ((𝑋 ∈ On ∧ 𝑆 = dom (ω CNF 𝑋)) → 𝑆 = dom (ω CNF 𝑋))
21eleq2d 2819 . . . . 5 ((𝑋 ∈ On ∧ 𝑆 = dom (ω CNF 𝑋)) → (𝑓𝑆𝑓 ∈ dom (ω CNF 𝑋)))
3 eqid 2732 . . . . . 6 dom (ω CNF 𝑋) = dom (ω CNF 𝑋)
4 omelon 9643 . . . . . . 7 ω ∈ On
54a1i 11 . . . . . 6 ((𝑋 ∈ On ∧ 𝑆 = dom (ω CNF 𝑋)) → ω ∈ On)
6 simpl 483 . . . . . 6 ((𝑋 ∈ On ∧ 𝑆 = dom (ω CNF 𝑋)) → 𝑋 ∈ On)
73, 5, 6cantnfs 9663 . . . . 5 ((𝑋 ∈ On ∧ 𝑆 = dom (ω CNF 𝑋)) → (𝑓 ∈ dom (ω CNF 𝑋) ↔ (𝑓:𝑋⟶ω ∧ 𝑓 finSupp ∅)))
82, 7bitrd 278 . . . 4 ((𝑋 ∈ On ∧ 𝑆 = dom (ω CNF 𝑋)) → (𝑓𝑆 ↔ (𝑓:𝑋⟶ω ∧ 𝑓 finSupp ∅)))
91eleq2d 2819 . . . . . . . . 9 ((𝑋 ∈ On ∧ 𝑆 = dom (ω CNF 𝑋)) → (𝑔𝑆𝑔 ∈ dom (ω CNF 𝑋)))
103, 5, 6cantnfs 9663 . . . . . . . . 9 ((𝑋 ∈ On ∧ 𝑆 = dom (ω CNF 𝑋)) → (𝑔 ∈ dom (ω CNF 𝑋) ↔ (𝑔:𝑋⟶ω ∧ 𝑔 finSupp ∅)))
119, 10bitrd 278 . . . . . . . 8 ((𝑋 ∈ On ∧ 𝑆 = dom (ω CNF 𝑋)) → (𝑔𝑆 ↔ (𝑔:𝑋⟶ω ∧ 𝑔 finSupp ∅)))
1211adantr 481 . . . . . . 7 (((𝑋 ∈ On ∧ 𝑆 = dom (ω CNF 𝑋)) ∧ (𝑓:𝑋⟶ω ∧ 𝑓 finSupp ∅)) → (𝑔𝑆 ↔ (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)
1513, 14anim12i 613 . . . . . . . . . . . . 13 (((𝑓:𝑋⟶ω ∧ 𝑓 finSupp ∅) ∧ (𝑔:𝑋⟶ω ∧ 𝑔 finSupp ∅)) → (𝑓:𝑋⟶ω ∧ 𝑔:𝑋⟶ω))
166, 15anim12i 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)))
1716anassrs 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)
1918ffnd 6718 . . . . . . . . . . . . . 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)
2120ffnd 6718 . . . . . . . . . . . . . 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 4218 . . . . . . . . . . . . . 14 (𝑋𝑋) = 𝑋
2419, 21, 22, 22, 23offn 7685 . . . . . . . . . . . . 13 ((𝑋 ∈ On ∧ (𝑓:𝑋⟶ω ∧ 𝑔:𝑋⟶ω)) → (𝑓f +o 𝑔) Fn 𝑋)
25 simpr 485 . . . . . . . . . . . . . . 15 (((𝑋 ∈ On ∧ (𝑓:𝑋⟶ω ∧ 𝑔:𝑋⟶ω)) ∧ (𝑓f +o 𝑔) Fn 𝑋) → (𝑓f +o 𝑔) Fn 𝑋)
26 simplrl 775 . . . . . . . . . . . . . . . . . . . . 21 (((𝑋 ∈ On ∧ (𝑓:𝑋⟶ω ∧ 𝑔:𝑋⟶ω)) ∧ 𝑥𝑋) → 𝑓:𝑋⟶ω)
2726ffnd 6718 . . . . . . . . . . . . . . . . . . . 20 (((𝑋 ∈ On ∧ (𝑓:𝑋⟶ω ∧ 𝑔:𝑋⟶ω)) ∧ 𝑥𝑋) → 𝑓 Fn 𝑋)
28 simplrr 776 . . . . . . . . . . . . . . . . . . . . 21 (((𝑋 ∈ On ∧ (𝑓:𝑋⟶ω ∧ 𝑔:𝑋⟶ω)) ∧ 𝑥𝑋) → 𝑔:𝑋⟶ω)
2928ffnd 6718 . . . . . . . . . . . . . . . . . . . 20 (((𝑋 ∈ On ∧ (𝑓:𝑋⟶ω ∧ 𝑔:𝑋⟶ω)) ∧ 𝑥𝑋) → 𝑔 Fn 𝑋)
30 simpll 765 . . . . . . . . . . . . . . . . . . . 20 (((𝑋 ∈ On ∧ (𝑓:𝑋⟶ω ∧ 𝑔:𝑋⟶ω)) ∧ 𝑥𝑋) → 𝑋 ∈ On)
31 simpr 485 . . . . . . . . . . . . . . . . . . . 20 (((𝑋 ∈ On ∧ (𝑓:𝑋⟶ω ∧ 𝑔:𝑋⟶ω)) ∧ 𝑥𝑋) → 𝑥𝑋)
32 fnfvof 7689 . . . . . . . . . . . . . . . . . . . 20 (((𝑓 Fn 𝑋𝑔 Fn 𝑋) ∧ (𝑋 ∈ On ∧ 𝑥𝑋)) → ((𝑓f +o 𝑔)‘𝑥) = ((𝑓𝑥) +o (𝑔𝑥)))
3327, 29, 30, 31, 32syl22anc 837 . . . . . . . . . . . . . . . . . . 19 (((𝑋 ∈ On ∧ (𝑓:𝑋⟶ω ∧ 𝑔:𝑋⟶ω)) ∧ 𝑥𝑋) → ((𝑓f +o 𝑔)‘𝑥) = ((𝑓𝑥) +o (𝑔𝑥)))
3418ffvelcdmda 7086 . . . . . . . . . . . . . . . . . . . 20 (((𝑋 ∈ On ∧ (𝑓:𝑋⟶ω ∧ 𝑔:𝑋⟶ω)) ∧ 𝑥𝑋) → (𝑓𝑥) ∈ ω)
3520ffvelcdmda 7086 . . . . . . . . . . . . . . . . . . . 20 (((𝑋 ∈ On ∧ (𝑓:𝑋⟶ω ∧ 𝑔:𝑋⟶ω)) ∧ 𝑥𝑋) → (𝑔𝑥) ∈ ω)
36 nnacl 8613 . . . . . . . . . . . . . . . . . . . 20 (((𝑓𝑥) ∈ ω ∧ (𝑔𝑥) ∈ ω) → ((𝑓𝑥) +o (𝑔𝑥)) ∈ ω)
3734, 35, 36syl2anc 584 . . . . . . . . . . . . . . . . . . 19 (((𝑋 ∈ On ∧ (𝑓:𝑋⟶ω ∧ 𝑔:𝑋⟶ω)) ∧ 𝑥𝑋) → ((𝑓𝑥) +o (𝑔𝑥)) ∈ ω)
3833, 37eqeltrd 2833 . . . . . . . . . . . . . . . . . 18 (((𝑋 ∈ On ∧ (𝑓:𝑋⟶ω ∧ 𝑔:𝑋⟶ω)) ∧ 𝑥𝑋) → ((𝑓f +o 𝑔)‘𝑥) ∈ ω)
3938ex 413 . . . . . . . . . . . . . . . . 17 ((𝑋 ∈ On ∧ (𝑓:𝑋⟶ω ∧ 𝑔:𝑋⟶ω)) → (𝑥𝑋 → ((𝑓f +o 𝑔)‘𝑥) ∈ ω))
4039ralrimiv 3145 . . . . . . . . . . . . . . . 16 ((𝑋 ∈ On ∧ (𝑓:𝑋⟶ω ∧ 𝑔:𝑋⟶ω)) → ∀𝑥𝑋 ((𝑓f +o 𝑔)‘𝑥) ∈ ω)
4140adantr 481 . . . . . . . . . . . . . . 15 (((𝑋 ∈ On ∧ (𝑓:𝑋⟶ω ∧ 𝑔:𝑋⟶ω)) ∧ (𝑓f +o 𝑔) Fn 𝑋) → ∀𝑥𝑋 ((𝑓f +o 𝑔)‘𝑥) ∈ ω)
42 fnfvrnss 7122 . . . . . . . . . . . . . . 15 (((𝑓f +o 𝑔) Fn 𝑋 ∧ ∀𝑥𝑋 ((𝑓f +o 𝑔)‘𝑥) ∈ ω) → ran (𝑓f +o 𝑔) ⊆ ω)
4325, 41, 42syl2anc 584 . . . . . . . . . . . . . 14 (((𝑋 ∈ On ∧ (𝑓:𝑋⟶ω ∧ 𝑔:𝑋⟶ω)) ∧ (𝑓f +o 𝑔) Fn 𝑋) → ran (𝑓f +o 𝑔) ⊆ ω)
4443ex 413 . . . . . . . . . . . . 13 ((𝑋 ∈ On ∧ (𝑓:𝑋⟶ω ∧ 𝑔:𝑋⟶ω)) → ((𝑓f +o 𝑔) Fn 𝑋 → ran (𝑓f +o 𝑔) ⊆ ω))
4524, 44jcai 517 . . . . . . . . . . . 12 ((𝑋 ∈ On ∧ (𝑓:𝑋⟶ω ∧ 𝑔:𝑋⟶ω)) → ((𝑓f +o 𝑔) Fn 𝑋 ∧ ran (𝑓f +o 𝑔) ⊆ ω))
46 df-f 6547 . . . . . . . . . . . 12 ((𝑓f +o 𝑔):𝑋⟶ω ↔ ((𝑓f +o 𝑔) Fn 𝑋 ∧ ran (𝑓f +o 𝑔) ⊆ ω))
4745, 46sylibr 233 . . . . . . . . . . 11 ((𝑋 ∈ On ∧ (𝑓:𝑋⟶ω ∧ 𝑔:𝑋⟶ω)) → (𝑓f +o 𝑔):𝑋⟶ω)
4817, 47syl 17 . . . . . . . . . 10 ((((𝑋 ∈ On ∧ 𝑆 = dom (ω CNF 𝑋)) ∧ (𝑓:𝑋⟶ω ∧ 𝑓 finSupp ∅)) ∧ (𝑔:𝑋⟶ω ∧ 𝑔 finSupp ∅)) → (𝑓f +o 𝑔):𝑋⟶ω)
49 ffun 6720 . . . . . . . . . . . . 13 ((𝑓f +o 𝑔):𝑋⟶ω → Fun (𝑓f +o 𝑔))
5049adantl 482 . . . . . . . . . . . 12 (((((𝑋 ∈ On ∧ 𝑆 = dom (ω CNF 𝑋)) ∧ (𝑓:𝑋⟶ω ∧ 𝑓 finSupp ∅)) ∧ (𝑔:𝑋⟶ω ∧ 𝑔 finSupp ∅)) ∧ (𝑓f +o 𝑔):𝑋⟶ω) → Fun (𝑓f +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 ∅)
5251adantr 481 . . . . . . . . . . . . . 14 (((((𝑋 ∈ On ∧ 𝑆 = dom (ω CNF 𝑋)) ∧ (𝑓:𝑋⟶ω ∧ 𝑓 finSupp ∅)) ∧ (𝑔:𝑋⟶ω ∧ 𝑔 finSupp ∅)) ∧ (𝑓f +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 ∅)) ∧ (𝑓f +o 𝑔):𝑋⟶ω) → 𝑔 finSupp ∅)
5452, 53fsuppunfi 9385 . . . . . . . . . . . . 13 (((((𝑋 ∈ On ∧ 𝑆 = dom (ω CNF 𝑋)) ∧ (𝑓:𝑋⟶ω ∧ 𝑓 finSupp ∅)) ∧ (𝑔:𝑋⟶ω ∧ 𝑔 finSupp ∅)) ∧ (𝑓f +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 ∅)) ∧ (𝑓f +o 𝑔):𝑋⟶ω) → 𝑋 ∈ On)
56 peano1 7881 . . . . . . . . . . . . . . 15 ∅ ∈ ω
5756a1i 11 . . . . . . . . . . . . . 14 (((((𝑋 ∈ On ∧ 𝑆 = dom (ω CNF 𝑋)) ∧ (𝑓:𝑋⟶ω ∧ 𝑓 finSupp ∅)) ∧ (𝑔:𝑋⟶ω ∧ 𝑔 finSupp ∅)) ∧ (𝑓f +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)
5958adantr 481 . . . . . . . . . . . . . 14 (((((𝑋 ∈ On ∧ 𝑆 = dom (ω CNF 𝑋)) ∧ (𝑓:𝑋⟶ω ∧ 𝑓 finSupp ∅)) ∧ (𝑔:𝑋⟶ω ∧ 𝑔 finSupp ∅)) ∧ (𝑓f +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 ∅)) ∧ (𝑓f +o 𝑔):𝑋⟶ω) → 𝑔:𝑋⟶ω)
61 0elon 6418 . . . . . . . . . . . . . . 15 ∅ ∈ On
62 oa0 8518 . . . . . . . . . . . . . . 15 (∅ ∈ On → (∅ +o ∅) = ∅)
6361, 62mp1i 13 . . . . . . . . . . . . . 14 (((((𝑋 ∈ On ∧ 𝑆 = dom (ω CNF 𝑋)) ∧ (𝑓:𝑋⟶ω ∧ 𝑓 finSupp ∅)) ∧ (𝑔:𝑋⟶ω ∧ 𝑔 finSupp ∅)) ∧ (𝑓f +o 𝑔):𝑋⟶ω) → (∅ +o ∅) = ∅)
6455, 57, 59, 60, 63suppofssd 8190 . . . . . . . . . . . . 13 (((((𝑋 ∈ On ∧ 𝑆 = dom (ω CNF 𝑋)) ∧ (𝑓:𝑋⟶ω ∧ 𝑓 finSupp ∅)) ∧ (𝑔:𝑋⟶ω ∧ 𝑔 finSupp ∅)) ∧ (𝑓f +o 𝑔):𝑋⟶ω) → ((𝑓f +o 𝑔) supp ∅) ⊆ ((𝑓 supp ∅) ∪ (𝑔 supp ∅)))
6554, 64ssfid 9269 . . . . . . . . . . . 12 (((((𝑋 ∈ On ∧ 𝑆 = dom (ω CNF 𝑋)) ∧ (𝑓:𝑋⟶ω ∧ 𝑓 finSupp ∅)) ∧ (𝑔:𝑋⟶ω ∧ 𝑔 finSupp ∅)) ∧ (𝑓f +o 𝑔):𝑋⟶ω) → ((𝑓f +o 𝑔) supp ∅) ∈ Fin)
66 ovexd 7446 . . . . . . . . . . . . 13 (((((𝑋 ∈ On ∧ 𝑆 = dom (ω CNF 𝑋)) ∧ (𝑓:𝑋⟶ω ∧ 𝑓 finSupp ∅)) ∧ (𝑔:𝑋⟶ω ∧ 𝑔 finSupp ∅)) ∧ (𝑓f +o 𝑔):𝑋⟶ω) → (𝑓f +o 𝑔) ∈ V)
67 isfsupp 9367 . . . . . . . . . . . . 13 (((𝑓f +o 𝑔) ∈ V ∧ ∅ ∈ On) → ((𝑓f +o 𝑔) finSupp ∅ ↔ (Fun (𝑓f +o 𝑔) ∧ ((𝑓f +o 𝑔) supp ∅) ∈ Fin)))
6866, 61, 67sylancl 586 . . . . . . . . . . . 12 (((((𝑋 ∈ On ∧ 𝑆 = dom (ω CNF 𝑋)) ∧ (𝑓:𝑋⟶ω ∧ 𝑓 finSupp ∅)) ∧ (𝑔:𝑋⟶ω ∧ 𝑔 finSupp ∅)) ∧ (𝑓f +o 𝑔):𝑋⟶ω) → ((𝑓f +o 𝑔) finSupp ∅ ↔ (Fun (𝑓f +o 𝑔) ∧ ((𝑓f +o 𝑔) supp ∅) ∈ Fin)))
6950, 65, 68mpbir2and 711 . . . . . . . . . . 11 (((((𝑋 ∈ On ∧ 𝑆 = dom (ω CNF 𝑋)) ∧ (𝑓:𝑋⟶ω ∧ 𝑓 finSupp ∅)) ∧ (𝑔:𝑋⟶ω ∧ 𝑔 finSupp ∅)) ∧ (𝑓f +o 𝑔):𝑋⟶ω) → (𝑓f +o 𝑔) finSupp ∅)
7069ex 413 . . . . . . . . . 10 ((((𝑋 ∈ On ∧ 𝑆 = dom (ω CNF 𝑋)) ∧ (𝑓:𝑋⟶ω ∧ 𝑓 finSupp ∅)) ∧ (𝑔:𝑋⟶ω ∧ 𝑔 finSupp ∅)) → ((𝑓f +o 𝑔):𝑋⟶ω → (𝑓f +o 𝑔) finSupp ∅))
7148, 70jcai 517 . . . . . . . . 9 ((((𝑋 ∈ On ∧ 𝑆 = dom (ω CNF 𝑋)) ∧ (𝑓:𝑋⟶ω ∧ 𝑓 finSupp ∅)) ∧ (𝑔:𝑋⟶ω ∧ 𝑔 finSupp ∅)) → ((𝑓f +o 𝑔):𝑋⟶ω ∧ (𝑓f +o 𝑔) finSupp ∅))
721eleq2d 2819 . . . . . . . . . . 11 ((𝑋 ∈ On ∧ 𝑆 = dom (ω CNF 𝑋)) → ((𝑓f +o 𝑔) ∈ 𝑆 ↔ (𝑓f +o 𝑔) ∈ dom (ω CNF 𝑋)))
733, 5, 6cantnfs 9663 . . . . . . . . . . 11 ((𝑋 ∈ On ∧ 𝑆 = dom (ω CNF 𝑋)) → ((𝑓f +o 𝑔) ∈ dom (ω CNF 𝑋) ↔ ((𝑓f +o 𝑔):𝑋⟶ω ∧ (𝑓f +o 𝑔) finSupp ∅)))
7472, 73bitrd 278 . . . . . . . . . 10 ((𝑋 ∈ On ∧ 𝑆 = dom (ω CNF 𝑋)) → ((𝑓f +o 𝑔) ∈ 𝑆 ↔ ((𝑓f +o 𝑔):𝑋⟶ω ∧ (𝑓f +o 𝑔) finSupp ∅)))
7574ad2antrr 724 . . . . . . . . 9 ((((𝑋 ∈ On ∧ 𝑆 = dom (ω CNF 𝑋)) ∧ (𝑓:𝑋⟶ω ∧ 𝑓 finSupp ∅)) ∧ (𝑔:𝑋⟶ω ∧ 𝑔 finSupp ∅)) → ((𝑓f +o 𝑔) ∈ 𝑆 ↔ ((𝑓f +o 𝑔):𝑋⟶ω ∧ (𝑓f +o 𝑔) finSupp ∅)))
7671, 75mpbird 256 . . . . . . . 8 ((((𝑋 ∈ On ∧ 𝑆 = dom (ω CNF 𝑋)) ∧ (𝑓:𝑋⟶ω ∧ 𝑓 finSupp ∅)) ∧ (𝑔:𝑋⟶ω ∧ 𝑔 finSupp ∅)) → (𝑓f +o 𝑔) ∈ 𝑆)
7776ex 413 . . . . . . 7 (((𝑋 ∈ On ∧ 𝑆 = dom (ω CNF 𝑋)) ∧ (𝑓:𝑋⟶ω ∧ 𝑓 finSupp ∅)) → ((𝑔:𝑋⟶ω ∧ 𝑔 finSupp ∅) → (𝑓f +o 𝑔) ∈ 𝑆))
7812, 77sylbid 239 . . . . . 6 (((𝑋 ∈ On ∧ 𝑆 = dom (ω CNF 𝑋)) ∧ (𝑓:𝑋⟶ω ∧ 𝑓 finSupp ∅)) → (𝑔𝑆 → (𝑓f +o 𝑔) ∈ 𝑆))
7978ralrimiv 3145 . . . . 5 (((𝑋 ∈ On ∧ 𝑆 = dom (ω CNF 𝑋)) ∧ (𝑓:𝑋⟶ω ∧ 𝑓 finSupp ∅)) → ∀𝑔𝑆 (𝑓f +o 𝑔) ∈ 𝑆)
8079ex 413 . . . 4 ((𝑋 ∈ On ∧ 𝑆 = dom (ω CNF 𝑋)) → ((𝑓:𝑋⟶ω ∧ 𝑓 finSupp ∅) → ∀𝑔𝑆 (𝑓f +o 𝑔) ∈ 𝑆))
818, 80sylbid 239 . . 3 ((𝑋 ∈ On ∧ 𝑆 = dom (ω CNF 𝑋)) → (𝑓𝑆 → ∀𝑔𝑆 (𝑓f +o 𝑔) ∈ 𝑆))
8281ralrimiv 3145 . 2 ((𝑋 ∈ On ∧ 𝑆 = dom (ω CNF 𝑋)) → ∀𝑓𝑆𝑔𝑆 (𝑓f +o 𝑔) ∈ 𝑆)
83 ofmres 7973 . . 3 ( ∘f +o ↾ (𝑆 × 𝑆)) = (𝑓𝑆, 𝑔𝑆 ↦ (𝑓f +o 𝑔))
8483fmpo 8056 . 2 (∀𝑓𝑆𝑔𝑆 (𝑓f +o 𝑔) ∈ 𝑆 ↔ ( ∘f +o ↾ (𝑆 × 𝑆)):(𝑆 × 𝑆)⟶𝑆)
8582, 84sylib 217 1 ((𝑋 ∈ On ∧ 𝑆 = dom (ω CNF 𝑋)) → ( ∘f +o ↾ (𝑆 × 𝑆)):(𝑆 × 𝑆)⟶𝑆)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 205  wa 396   = wceq 1541  wcel 2106  wral 3061  Vcvv 3474  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 7411  f cof 7670  ωcom 7857   supp csupp 8148   +o coa 8465  Fincfn 8941   finSupp cfsupp 9363   CNF ccnf 9658
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 5285  ax-sep 5299  ax-nul 5306  ax-pow 5363  ax-pr 5427  ax-un 7727  ax-inf2 9638
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 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 7414  df-oprab 7415  df-mpo 7416  df-of 7672  df-om 7858  df-1st 7977  df-2nd 7978  df-supp 8149  df-frecs 8268  df-wrecs 8299  df-recs 8373  df-rdg 8412  df-seqom 8450  df-1o 8468  df-oadd 8472  df-map 8824  df-en 8942  df-fin 8945  df-fsupp 9364  df-cnf 9659
This theorem is referenced by:  naddcnffn  42195  naddcnffo  42196  naddcnfcl  42197
  Copyright terms: Public domain W3C validator