Theorem nnacl 3172

Description: Closure of addition of natural numbers. Proposition 8.9 of [TakeutiZaring] p. 59.

Assertion

Ref	Expression
nnacl	⊢ ((A ∈ ω ∧ B ∈ ω) → (A +o B) ∈ ω)

Proof of Theorem nnacl

Step	Hyp	Ref	Expression
1		opreq2 3007	. . . . . 6 ⊢ (x = ∅ → (A +o x) = (A +o ∅))
2	1	eleq1d 1155	. . . . 5 ⊢ (x = ∅ → ((A +o x) ∈ ω ↔ (A +o ∅) ∈ ω))
3	2	imbi2d 464	. . . 4 ⊢ (x = ∅ → ((A ∈ ω → (A +o x) ∈ ω) ↔ (A ∈ ω → (A +o ∅) ∈ ω)))
4		opreq2 3007	. . . . . 6 ⊢ (x = y → (A +o x) = (A +o y))
5	4	eleq1d 1155	. . . . 5 ⊢ (x = y → ((A +o x) ∈ ω ↔ (A +o y) ∈ ω))
6	5	imbi2d 464	. . . 4 ⊢ (x = y → ((A ∈ ω → (A +o x) ∈ ω) ↔ (A ∈ ω → (A +o y) ∈ ω)))
7		opreq2 3007	. . . . . 6 ⊢ (x = suc y → (A +o x) = (A +o suc y))
8	7	eleq1d 1155	. . . . 5 ⊢ (x = suc y → ((A +o x) ∈ ω ↔ (A +o suc y) ∈ ω))
9	8	imbi2d 464	. . . 4 ⊢ (x = suc y → ((A ∈ ω → (A +o x) ∈ ω) ↔ (A ∈ ω → (A +o suc y) ∈ ω)))
10		opreq2 3007	. . . . . 6 ⊢ (x = B → (A +o x) = (A +o B))
11	10	eleq1d 1155	. . . . 5 ⊢ (x = B → ((A +o x) ∈ ω ↔ (A +o B) ∈ ω))
12	11	imbi2d 464	. . . 4 ⊢ (x = B → ((A ∈ ω → (A +o x) ∈ ω) ↔ (A ∈ ω → (A +o B) ∈ ω)))
13		nna0 3166	. . . . . . 7 ⊢ (A ∈ ω → (A +o ∅) = A)
14	13	eleq1d 1155	. . . . . 6 ⊢ (A ∈ ω → ((A +o ∅) ∈ ω ↔ A ∈ ω))
15	14	biimprd 136	. . . . 5 ⊢ (A ∈ ω → (A ∈ ω → (A +o ∅) ∈ ω))
16	15	pm2.43i 58	. . . 4 ⊢ (A ∈ ω → (A +o ∅) ∈ ω)
17		nnasuc 3168	. . . . . . . . 9 ⊢ ((A ∈ ω ∧ y ∈ ω) → (A +o suc y) = suc (A +o y))
18	17	eleq1d 1155	. . . . . . . 8 ⊢ ((A ∈ ω ∧ y ∈ ω) → ((A +o suc y) ∈ ω ↔ suc (A +o y) ∈ ω))
19		peano2 2391	. . . . . . . 8 ⊢ ((A +o y) ∈ ω → suc (A +o y) ∈ ω)
20	18, 19	syl5bir 184	. . . . . . 7 ⊢ ((A ∈ ω ∧ y ∈ ω) → ((A +o y) ∈ ω → (A +o suc y) ∈ ω))
21	20	exp 291	. . . . . 6 ⊢ (A ∈ ω → (y ∈ ω → ((A +o y) ∈ ω → (A +o suc y) ∈ ω)))
22	21	com12 13	. . . . 5 ⊢ (y ∈ ω → (A ∈ ω → ((A +o y) ∈ ω → (A +o suc y) ∈ ω)))
23	22	a2d 15	. . . 4 ⊢ (y ∈ ω → ((A ∈ ω → (A +o y) ∈ ω) → (A ∈ ω → (A +o suc y) ∈ ω)))
24	3, 6, 9, 12, 16, 23	finds 2397	. . 3 ⊢ (B ∈ ω → (A ∈ ω → (A +o B) ∈ ω))
25	24	com12 13	. 2 ⊢ (A ∈ ω → (B ∈ ω → (A +o B) ∈ ω))
26	25	imp 277	1 ⊢ ((A ∈ ω ∧ B ∈ ω) → (A +o B) ∈ ω)

Syntax hints:   → wi 2   ∧ wa 196   = weq 797   = wceq 1091   ∈ wcel 1092  ∅c0 1707  suc csuc 2201  ωcom 2372  (class class class)co 3001   +_o coa 3101

This theorem is referenced by:  nnmcl 3173  nnarcl 3174  nndi 3180  unfilem1 3438  unfi 3441  addclpi 3814

This theorem was proved from axioms:  ax-1 3  ax-2 4  ax-3 5  ax-mp 6  ax-4 673  ax-5 674  ax-6 675  ax-7 676  ax-gen 677  ax-8 798  ax-9 799  ax-10 800  ax-11 801  ax-12 802  ax-13 804  ax-14 805  ax-16 922  ax-17 925  ax-ext 1074  ax-rep 1075  ax-un 1076  ax-pow 1077

This theorem depends on definitions:  df-bi 128  df-or 197  df-an 198  df-3or 582  df-3an 583  df-ex 679  df-sb 853  df-eu 1009  df-mo 1010  df-clab 1093  df-cleq 1097  df-clel 1099  df-ral 1205  df-rex 1206  df-rab 1208  df-v 1349  df-sbc 1441  df-dif 1489  df-un 1490  df-in 1491  df-ss 1492  df-nul 1708  df-if 1777  df-pw 1799  df-sn 1811  df-pr 1812  df-tp 1814  df-op 1815  df-uni 1920  df-tr 2042  df-br 2063  df-opab 2098  df-eprel 2122  df-id 2125  df-po 2128  df-so 2138  df-fr 2169  df-we 2186  df-ord 2202  df-on 2203  df-lim 2204  df-suc 2205  df-om 2373  df-xp 2424  df-rel 2425  df-cnv 2426  df-co 2427  df-dm 2428  df-rn 2429  df-res 2430  df-ima 2431  df-fun 2432  df-fn 2433  df-fv 2438  df-rdg 2970  df-opr 3003  df-oprab 3004  df-oadd 3106

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