Theorem nncdiv3lem2 6274

Description: Lemma for nncdiv3 6275. Set up stratification for induction. (Contributed by SF, 2-Mar-2015.)

Assertion

Ref	Expression
nncdiv3lem2	⊢ {a ∣ ∃n ∈ Nn (a = ((n +c n) +c n) ∨ a = (((n +c n) +c n) +c 1c) ∨ a = (((n +c n) +c n) +c 2c))} ∈ V

Distinct variable group:   n,a

Proof of Theorem nncdiv3lem2

Dummy variable b is distinct from all other variables.

Step	Hyp	Ref	Expression
1		elima 4754	. . . 4 ⊢ (a ∈ (((ran ( Ins3 ◡((ran (◡1st ⊗ (1st ∩ 2nd )) ⊗ 2nd ) “ AddC ) ∩ (((1st ∘ 1st ) ⊗ ((2nd ∘ 1st ) ⊗ 2nd )) “ AddC )) ∪ ran (◡ran ( Ins3 ◡((ran (◡1st ⊗ (1st ∩ 2nd )) ⊗ 2nd ) “ AddC ) ∩ (((1st ∘ 1st ) ⊗ ((2nd ∘ 1st ) ⊗ 2nd )) “ AddC )) ⊗ ran ((1st ∩ ((◡2nd “ {1c}) × V)) ⊗ AddC ))) ∪ ran (◡ran ( Ins3 ◡((ran (◡1st ⊗ (1st ∩ 2nd )) ⊗ 2nd ) “ AddC ) ∩ (((1st ∘ 1st ) ⊗ ((2nd ∘ 1st ) ⊗ 2nd )) “ AddC )) ⊗ ran ((1st ∩ ((◡2nd “ {2c}) × V)) ⊗ AddC ))) “ Nn ) ↔ ∃n ∈ Nn n((ran ( Ins3 ◡((ran (◡1st ⊗ (1st ∩ 2nd )) ⊗ 2nd ) “ AddC ) ∩ (((1st ∘ 1st ) ⊗ ((2nd ∘ 1st ) ⊗ 2nd )) “ AddC )) ∪ ran (◡ran ( Ins3 ◡((ran (◡1st ⊗ (1st ∩ 2nd )) ⊗ 2nd ) “ AddC ) ∩ (((1st ∘ 1st ) ⊗ ((2nd ∘ 1st ) ⊗ 2nd )) “ AddC )) ⊗ ran ((1st ∩ ((◡2nd “ {1c}) × V)) ⊗ AddC ))) ∪ ran (◡ran ( Ins3 ◡((ran (◡1st ⊗ (1st ∩ 2nd )) ⊗ 2nd ) “ AddC ) ∩ (((1st ∘ 1st ) ⊗ ((2nd ∘ 1st ) ⊗ 2nd )) “ AddC )) ⊗ ran ((1st ∩ ((◡2nd “ {2c}) × V)) ⊗ AddC )))a)
2		df-br 4640	. . . . . 6 ⊢ (n((ran ( Ins3 ◡((ran (◡1st ⊗ (1st ∩ 2nd )) ⊗ 2nd ) “ AddC ) ∩ (((1st ∘ 1st ) ⊗ ((2nd ∘ 1st ) ⊗ 2nd )) “ AddC )) ∪ ran (◡ran ( Ins3 ◡((ran (◡1st ⊗ (1st ∩ 2nd )) ⊗ 2nd ) “ AddC ) ∩ (((1st ∘ 1st ) ⊗ ((2nd ∘ 1st ) ⊗ 2nd )) “ AddC )) ⊗ ran ((1st ∩ ((◡2nd “ {1c}) × V)) ⊗ AddC ))) ∪ ran (◡ran ( Ins3 ◡((ran (◡1st ⊗ (1st ∩ 2nd )) ⊗ 2nd ) “ AddC ) ∩ (((1st ∘ 1st ) ⊗ ((2nd ∘ 1st ) ⊗ 2nd )) “ AddC )) ⊗ ran ((1st ∩ ((◡2nd “ {2c}) × V)) ⊗ AddC )))a ↔ ⟨n, a⟩ ∈ ((ran ( Ins3 ◡((ran (◡1st ⊗ (1st ∩ 2nd )) ⊗ 2nd ) “ AddC ) ∩ (((1st ∘ 1st ) ⊗ ((2nd ∘ 1st ) ⊗ 2nd )) “ AddC )) ∪ ran (◡ran ( Ins3 ◡((ran (◡1st ⊗ (1st ∩ 2nd )) ⊗ 2nd ) “ AddC ) ∩ (((1st ∘ 1st ) ⊗ ((2nd ∘ 1st ) ⊗ 2nd )) “ AddC )) ⊗ ran ((1st ∩ ((◡2nd “ {1c}) × V)) ⊗ AddC ))) ∪ ran (◡ran ( Ins3 ◡((ran (◡1st ⊗ (1st ∩ 2nd )) ⊗ 2nd ) “ AddC ) ∩ (((1st ∘ 1st ) ⊗ ((2nd ∘ 1st ) ⊗ 2nd )) “ AddC )) ⊗ ran ((1st ∩ ((◡2nd “ {2c}) × V)) ⊗ AddC ))))
3		elun 3220	. . . . . . . . 9 ⊢ (⟨n, a⟩ ∈ (ran ( Ins3 ◡((ran (◡1st ⊗ (1st ∩ 2nd )) ⊗ 2nd ) “ AddC ) ∩ (((1st ∘ 1st ) ⊗ ((2nd ∘ 1st ) ⊗ 2nd )) “ AddC )) ∪ ran (◡ran ( Ins3 ◡((ran (◡1st ⊗ (1st ∩ 2nd )) ⊗ 2nd ) “ AddC ) ∩ (((1st ∘ 1st ) ⊗ ((2nd ∘ 1st ) ⊗ 2nd )) “ AddC )) ⊗ ran ((1st ∩ ((◡2nd “ {1c}) × V)) ⊗ AddC ))) ↔ (⟨n, a⟩ ∈ ran ( Ins3 ◡((ran (◡1st ⊗ (1st ∩ 2nd )) ⊗ 2nd ) “ AddC ) ∩ (((1st ∘ 1st ) ⊗ ((2nd ∘ 1st ) ⊗ 2nd )) “ AddC )) ∨ ⟨n, a⟩ ∈ ran (◡ran ( Ins3 ◡((ran (◡1st ⊗ (1st ∩ 2nd )) ⊗ 2nd ) “ AddC ) ∩ (((1st ∘ 1st ) ⊗ ((2nd ∘ 1st ) ⊗ 2nd )) “ AddC )) ⊗ ran ((1st ∩ ((◡2nd “ {1c}) × V)) ⊗ AddC ))))
4		nncdiv3lem1 6273	. . . . . . . . . 10 ⊢ (⟨n, a⟩ ∈ ran ( Ins3 ◡((ran (◡1st ⊗ (1st ∩ 2nd )) ⊗ 2nd ) “ AddC ) ∩ (((1st ∘ 1st ) ⊗ ((2nd ∘ 1st ) ⊗ 2nd )) “ AddC )) ↔ a = ((n +c n) +c n))
5		elrn2 4897	. . . . . . . . . . 11 ⊢ (⟨n, a⟩ ∈ ran (◡ran ( Ins3 ◡((ran (◡1st ⊗ (1st ∩ 2nd )) ⊗ 2nd ) “ AddC ) ∩ (((1st ∘ 1st ) ⊗ ((2nd ∘ 1st ) ⊗ 2nd )) “ AddC )) ⊗ ran ((1st ∩ ((◡2nd “ {1c}) × V)) ⊗ AddC )) ↔ ∃b⟨b, ⟨n, a⟩⟩ ∈ (◡ran ( Ins3 ◡((ran (◡1st ⊗ (1st ∩ 2nd )) ⊗ 2nd ) “ AddC ) ∩ (((1st ∘ 1st ) ⊗ ((2nd ∘ 1st ) ⊗ 2nd )) “ AddC )) ⊗ ran ((1st ∩ ((◡2nd “ {1c}) × V)) ⊗ AddC )))
6		oteltxp 5782	. . . . . . . . . . . . 13 ⊢ (⟨b, ⟨n, a⟩⟩ ∈ (◡ran ( Ins3 ◡((ran (◡1st ⊗ (1st ∩ 2nd )) ⊗ 2nd ) “ AddC ) ∩ (((1st ∘ 1st ) ⊗ ((2nd ∘ 1st ) ⊗ 2nd )) “ AddC )) ⊗ ran ((1st ∩ ((◡2nd “ {1c}) × V)) ⊗ AddC )) ↔ (⟨b, n⟩ ∈ ◡ran ( Ins3 ◡((ran (◡1st ⊗ (1st ∩ 2nd )) ⊗ 2nd ) “ AddC ) ∩ (((1st ∘ 1st ) ⊗ ((2nd ∘ 1st ) ⊗ 2nd )) “ AddC )) ∧ ⟨b, a⟩ ∈ ran ((1st ∩ ((◡2nd “ {1c}) × V)) ⊗ AddC )))
7		opelcnv 4893	. . . . . . . . . . . . . . 15 ⊢ (⟨b, n⟩ ∈ ◡ran ( Ins3 ◡((ran (◡1st ⊗ (1st ∩ 2nd )) ⊗ 2nd ) “ AddC ) ∩ (((1st ∘ 1st ) ⊗ ((2nd ∘ 1st ) ⊗ 2nd )) “ AddC )) ↔ ⟨n, b⟩ ∈ ran ( Ins3 ◡((ran (◡1st ⊗ (1st ∩ 2nd )) ⊗ 2nd ) “ AddC ) ∩ (((1st ∘ 1st ) ⊗ ((2nd ∘ 1st ) ⊗ 2nd )) “ AddC )))
8		nncdiv3lem1 6273	. . . . . . . . . . . . . . 15 ⊢ (⟨n, b⟩ ∈ ran ( Ins3 ◡((ran (◡1st ⊗ (1st ∩ 2nd )) ⊗ 2nd ) “ AddC ) ∩ (((1st ∘ 1st ) ⊗ ((2nd ∘ 1st ) ⊗ 2nd )) “ AddC )) ↔ b = ((n +c n) +c n))
9	7, 8	bitri 240	. . . . . . . . . . . . . 14 ⊢ (⟨b, n⟩ ∈ ◡ran ( Ins3 ◡((ran (◡1st ⊗ (1st ∩ 2nd )) ⊗ 2nd ) “ AddC ) ∩ (((1st ∘ 1st ) ⊗ ((2nd ∘ 1st ) ⊗ 2nd )) “ AddC )) ↔ b = ((n +c n) +c n))
10		elrn2 4897	. . . . . . . . . . . . . . . 16 ⊢ (⟨b, a⟩ ∈ ran ((1st ∩ ((◡2nd “ {1c}) × V)) ⊗ AddC ) ↔ ∃n⟨n, ⟨b, a⟩⟩ ∈ ((1st ∩ ((◡2nd “ {1c}) × V)) ⊗ AddC ))
11		oteltxp 5782	. . . . . . . . . . . . . . . . . 18 ⊢ (⟨n, ⟨b, a⟩⟩ ∈ ((1st ∩ ((◡2nd “ {1c}) × V)) ⊗ AddC ) ↔ (⟨n, b⟩ ∈ (1st ∩ ((◡2nd “ {1c}) × V)) ∧ ⟨n, a⟩ ∈ AddC ))
12		elin 3219	. . . . . . . . . . . . . . . . . . . 20 ⊢ (⟨n, b⟩ ∈ (1st ∩ ((◡2nd “ {1c}) × V)) ↔ (⟨n, b⟩ ∈ 1st ∧ ⟨n, b⟩ ∈ ((◡2nd “ {1c}) × V)))
13		df-br 4640	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (n1st b ↔ ⟨n, b⟩ ∈ 1st )
14	13	bicomi 193	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (⟨n, b⟩ ∈ 1st ↔ n1st b)
15		vex 2862	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ b ∈ V
16		opelxp 4811	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ (⟨n, b⟩ ∈ ((◡2nd “ {1c}) × V) ↔ (n ∈ (◡2nd “ {1c}) ∧ b ∈ V))
17	15, 16	mpbiran2 885	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (⟨n, b⟩ ∈ ((◡2nd “ {1c}) × V) ↔ n ∈ (◡2nd “ {1c}))
18		eliniseg 5020	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (n ∈ (◡2nd “ {1c}) ↔ n2nd 1c)
19	17, 18	bitri 240	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (⟨n, b⟩ ∈ ((◡2nd “ {1c}) × V) ↔ n2nd 1c)
20	14, 19	anbi12i 678	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((⟨n, b⟩ ∈ 1st ∧ ⟨n, b⟩ ∈ ((◡2nd “ {1c}) × V)) ↔ (n1st b ∧ n2nd 1c))
21		1cex 4142	. . . . . . . . . . . . . . . . . . . . 21 ⊢ 1c ∈ V
22	15, 21	op1st2nd 5790	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((n1st b ∧ n2nd 1c) ↔ n = ⟨b, 1c⟩)
23	12, 20, 22	3bitri 262	. . . . . . . . . . . . . . . . . . 19 ⊢ (⟨n, b⟩ ∈ (1st ∩ ((◡2nd “ {1c}) × V)) ↔ n = ⟨b, 1c⟩)
24		df-br 4640	. . . . . . . . . . . . . . . . . . . 20 ⊢ (n AddC a ↔ ⟨n, a⟩ ∈ AddC )
25	24	bicomi 193	. . . . . . . . . . . . . . . . . . 19 ⊢ (⟨n, a⟩ ∈ AddC ↔ n AddC a)
26	23, 25	anbi12i 678	. . . . . . . . . . . . . . . . . 18 ⊢ ((⟨n, b⟩ ∈ (1st ∩ ((◡2nd “ {1c}) × V)) ∧ ⟨n, a⟩ ∈ AddC ) ↔ (n = ⟨b, 1c⟩ ∧ n AddC a))
27	11, 26	bitri 240	. . . . . . . . . . . . . . . . 17 ⊢ (⟨n, ⟨b, a⟩⟩ ∈ ((1st ∩ ((◡2nd “ {1c}) × V)) ⊗ AddC ) ↔ (n = ⟨b, 1c⟩ ∧ n AddC a))
28	27	exbii 1582	. . . . . . . . . . . . . . . 16 ⊢ (∃n⟨n, ⟨b, a⟩⟩ ∈ ((1st ∩ ((◡2nd “ {1c}) × V)) ⊗ AddC ) ↔ ∃n(n = ⟨b, 1c⟩ ∧ n AddC a))
29	10, 28	bitri 240	. . . . . . . . . . . . . . 15 ⊢ (⟨b, a⟩ ∈ ran ((1st ∩ ((◡2nd “ {1c}) × V)) ⊗ AddC ) ↔ ∃n(n = ⟨b, 1c⟩ ∧ n AddC a))
30	15, 21	opex 4588	. . . . . . . . . . . . . . . 16 ⊢ ⟨b, 1c⟩ ∈ V
31		breq1 4642	. . . . . . . . . . . . . . . 16 ⊢ (n = ⟨b, 1c⟩ → (n AddC a ↔ ⟨b, 1c⟩ AddC a))
32	30, 31	ceqsexv 2894	. . . . . . . . . . . . . . 15 ⊢ (∃n(n = ⟨b, 1c⟩ ∧ n AddC a) ↔ ⟨b, 1c⟩ AddC a)
33	15, 21	braddcfn 5826	. . . . . . . . . . . . . . . 16 ⊢ (⟨b, 1c⟩ AddC a ↔ (b +c 1c) = a)
34		eqcom 2355	. . . . . . . . . . . . . . . 16 ⊢ ((b +c 1c) = a ↔ a = (b +c 1c))
35	33, 34	bitri 240	. . . . . . . . . . . . . . 15 ⊢ (⟨b, 1c⟩ AddC a ↔ a = (b +c 1c))
36	29, 32, 35	3bitri 262	. . . . . . . . . . . . . 14 ⊢ (⟨b, a⟩ ∈ ran ((1st ∩ ((◡2nd “ {1c}) × V)) ⊗ AddC ) ↔ a = (b +c 1c))
37	9, 36	anbi12i 678	. . . . . . . . . . . . 13 ⊢ ((⟨b, n⟩ ∈ ◡ran ( Ins3 ◡((ran (◡1st ⊗ (1st ∩ 2nd )) ⊗ 2nd ) “ AddC ) ∩ (((1st ∘ 1st ) ⊗ ((2nd ∘ 1st ) ⊗ 2nd )) “ AddC )) ∧ ⟨b, a⟩ ∈ ran ((1st ∩ ((◡2nd “ {1c}) × V)) ⊗ AddC )) ↔ (b = ((n +c n) +c n) ∧ a = (b +c 1c)))
38	6, 37	bitri 240	. . . . . . . . . . . 12 ⊢ (⟨b, ⟨n, a⟩⟩ ∈ (◡ran ( Ins3 ◡((ran (◡1st ⊗ (1st ∩ 2nd )) ⊗ 2nd ) “ AddC ) ∩ (((1st ∘ 1st ) ⊗ ((2nd ∘ 1st ) ⊗ 2nd )) “ AddC )) ⊗ ran ((1st ∩ ((◡2nd “ {1c}) × V)) ⊗ AddC )) ↔ (b = ((n +c n) +c n) ∧ a = (b +c 1c)))
39	38	exbii 1582	. . . . . . . . . . 11 ⊢ (∃b⟨b, ⟨n, a⟩⟩ ∈ (◡ran ( Ins3 ◡((ran (◡1st ⊗ (1st ∩ 2nd )) ⊗ 2nd ) “ AddC ) ∩ (((1st ∘ 1st ) ⊗ ((2nd ∘ 1st ) ⊗ 2nd )) “ AddC )) ⊗ ran ((1st ∩ ((◡2nd “ {1c}) × V)) ⊗ AddC )) ↔ ∃b(b = ((n +c n) +c n) ∧ a = (b +c 1c)))
40		vex 2862	. . . . . . . . . . . . . 14 ⊢ n ∈ V
41	40, 40	addcex 4394	. . . . . . . . . . . . 13 ⊢ (n +c n) ∈ V
42	41, 40	addcex 4394	. . . . . . . . . . . 12 ⊢ ((n +c n) +c n) ∈ V
43		addceq1 4383	. . . . . . . . . . . . 13 ⊢ (b = ((n +c n) +c n) → (b +c 1c) = (((n +c n) +c n) +c 1c))
44	43	eqeq2d 2364	. . . . . . . . . . . 12 ⊢ (b = ((n +c n) +c n) → (a = (b +c 1c) ↔ a = (((n +c n) +c n) +c 1c)))
45	42, 44	ceqsexv 2894	. . . . . . . . . . 11 ⊢ (∃b(b = ((n +c n) +c n) ∧ a = (b +c 1c)) ↔ a = (((n +c n) +c n) +c 1c))
46	5, 39, 45	3bitri 262	. . . . . . . . . 10 ⊢ (⟨n, a⟩ ∈ ran (◡ran ( Ins3 ◡((ran (◡1st ⊗ (1st ∩ 2nd )) ⊗ 2nd ) “ AddC ) ∩ (((1st ∘ 1st ) ⊗ ((2nd ∘ 1st ) ⊗ 2nd )) “ AddC )) ⊗ ran ((1st ∩ ((◡2nd “ {1c}) × V)) ⊗ AddC )) ↔ a = (((n +c n) +c n) +c 1c))
47	4, 46	orbi12i 507	. . . . . . . . 9 ⊢ ((⟨n, a⟩ ∈ ran ( Ins3 ◡((ran (◡1st ⊗ (1st ∩ 2nd )) ⊗ 2nd ) “ AddC ) ∩ (((1st ∘ 1st ) ⊗ ((2nd ∘ 1st ) ⊗ 2nd )) “ AddC )) ∨ ⟨n, a⟩ ∈ ran (◡ran ( Ins3 ◡((ran (◡1st ⊗ (1st ∩ 2nd )) ⊗ 2nd ) “ AddC ) ∩ (((1st ∘ 1st ) ⊗ ((2nd ∘ 1st ) ⊗ 2nd )) “ AddC )) ⊗ ran ((1st ∩ ((◡2nd “ {1c}) × V)) ⊗ AddC ))) ↔ (a = ((n +c n) +c n) ∨ a = (((n +c n) +c n) +c 1c)))
48	3, 47	bitri 240	. . . . . . . 8 ⊢ (⟨n, a⟩ ∈ (ran ( Ins3 ◡((ran (◡1st ⊗ (1st ∩ 2nd )) ⊗ 2nd ) “ AddC ) ∩ (((1st ∘ 1st ) ⊗ ((2nd ∘ 1st ) ⊗ 2nd )) “ AddC )) ∪ ran (◡ran ( Ins3 ◡((ran (◡1st ⊗ (1st ∩ 2nd )) ⊗ 2nd ) “ AddC ) ∩ (((1st ∘ 1st ) ⊗ ((2nd ∘ 1st ) ⊗ 2nd )) “ AddC )) ⊗ ran ((1st ∩ ((◡2nd “ {1c}) × V)) ⊗ AddC ))) ↔ (a = ((n +c n) +c n) ∨ a = (((n +c n) +c n) +c 1c)))
49		elrn2 4897	. . . . . . . . 9 ⊢ (⟨n, a⟩ ∈ ran (◡ran ( Ins3 ◡((ran (◡1st ⊗ (1st ∩ 2nd )) ⊗ 2nd ) “ AddC ) ∩ (((1st ∘ 1st ) ⊗ ((2nd ∘ 1st ) ⊗ 2nd )) “ AddC )) ⊗ ran ((1st ∩ ((◡2nd “ {2c}) × V)) ⊗ AddC )) ↔ ∃b⟨b, ⟨n, a⟩⟩ ∈ (◡ran ( Ins3 ◡((ran (◡1st ⊗ (1st ∩ 2nd )) ⊗ 2nd ) “ AddC ) ∩ (((1st ∘ 1st ) ⊗ ((2nd ∘ 1st ) ⊗ 2nd )) “ AddC )) ⊗ ran ((1st ∩ ((◡2nd “ {2c}) × V)) ⊗ AddC )))
50		oteltxp 5782	. . . . . . . . . . 11 ⊢ (⟨b, ⟨n, a⟩⟩ ∈ (◡ran ( Ins3 ◡((ran (◡1st ⊗ (1st ∩ 2nd )) ⊗ 2nd ) “ AddC ) ∩ (((1st ∘ 1st ) ⊗ ((2nd ∘ 1st ) ⊗ 2nd )) “ AddC )) ⊗ ran ((1st ∩ ((◡2nd “ {2c}) × V)) ⊗ AddC )) ↔ (⟨b, n⟩ ∈ ◡ran ( Ins3 ◡((ran (◡1st ⊗ (1st ∩ 2nd )) ⊗ 2nd ) “ AddC ) ∩ (((1st ∘ 1st ) ⊗ ((2nd ∘ 1st ) ⊗ 2nd )) “ AddC )) ∧ ⟨b, a⟩ ∈ ran ((1st ∩ ((◡2nd “ {2c}) × V)) ⊗ AddC )))
51		elrn2 4897	. . . . . . . . . . . . . 14 ⊢ (⟨b, a⟩ ∈ ran ((1st ∩ ((◡2nd “ {2c}) × V)) ⊗ AddC ) ↔ ∃n⟨n, ⟨b, a⟩⟩ ∈ ((1st ∩ ((◡2nd “ {2c}) × V)) ⊗ AddC ))
52		oteltxp 5782	. . . . . . . . . . . . . . . 16 ⊢ (⟨n, ⟨b, a⟩⟩ ∈ ((1st ∩ ((◡2nd “ {2c}) × V)) ⊗ AddC ) ↔ (⟨n, b⟩ ∈ (1st ∩ ((◡2nd “ {2c}) × V)) ∧ ⟨n, a⟩ ∈ AddC ))
53		elin 3219	. . . . . . . . . . . . . . . . . 18 ⊢ (⟨n, b⟩ ∈ (1st ∩ ((◡2nd “ {2c}) × V)) ↔ (⟨n, b⟩ ∈ 1st ∧ ⟨n, b⟩ ∈ ((◡2nd “ {2c}) × V)))
54		opelxp 4811	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (⟨n, b⟩ ∈ ((◡2nd “ {2c}) × V) ↔ (n ∈ (◡2nd “ {2c}) ∧ b ∈ V))
55	15, 54	mpbiran2 885	. . . . . . . . . . . . . . . . . . . 20 ⊢ (⟨n, b⟩ ∈ ((◡2nd “ {2c}) × V) ↔ n ∈ (◡2nd “ {2c}))
56		eliniseg 5020	. . . . . . . . . . . . . . . . . . . 20 ⊢ (n ∈ (◡2nd “ {2c}) ↔ n2nd 2c)
57	55, 56	bitri 240	. . . . . . . . . . . . . . . . . . 19 ⊢ (⟨n, b⟩ ∈ ((◡2nd “ {2c}) × V) ↔ n2nd 2c)
58	14, 57	anbi12i 678	. . . . . . . . . . . . . . . . . 18 ⊢ ((⟨n, b⟩ ∈ 1st ∧ ⟨n, b⟩ ∈ ((◡2nd “ {2c}) × V)) ↔ (n1st b ∧ n2nd 2c))
59		df-2c 6104	. . . . . . . . . . . . . . . . . . . 20 ⊢ 2c = Nc {∅, V}
60		ncex 6117	. . . . . . . . . . . . . . . . . . . 20 ⊢ Nc {∅, V} ∈ V
61	59, 60	eqeltri 2423	. . . . . . . . . . . . . . . . . . 19 ⊢ 2c ∈ V
62	15, 61	op1st2nd 5790	. . . . . . . . . . . . . . . . . 18 ⊢ ((n1st b ∧ n2nd 2c) ↔ n = ⟨b, 2c⟩)
63	53, 58, 62	3bitri 262	. . . . . . . . . . . . . . . . 17 ⊢ (⟨n, b⟩ ∈ (1st ∩ ((◡2nd “ {2c}) × V)) ↔ n = ⟨b, 2c⟩)
64	63, 25	anbi12i 678	. . . . . . . . . . . . . . . 16 ⊢ ((⟨n, b⟩ ∈ (1st ∩ ((◡2nd “ {2c}) × V)) ∧ ⟨n, a⟩ ∈ AddC ) ↔ (n = ⟨b, 2c⟩ ∧ n AddC a))
65	52, 64	bitri 240	. . . . . . . . . . . . . . 15 ⊢ (⟨n, ⟨b, a⟩⟩ ∈ ((1st ∩ ((◡2nd “ {2c}) × V)) ⊗ AddC ) ↔ (n = ⟨b, 2c⟩ ∧ n AddC a))
66	65	exbii 1582	. . . . . . . . . . . . . 14 ⊢ (∃n⟨n, ⟨b, a⟩⟩ ∈ ((1st ∩ ((◡2nd “ {2c}) × V)) ⊗ AddC ) ↔ ∃n(n = ⟨b, 2c⟩ ∧ n AddC a))
67	15, 61	opex 4588	. . . . . . . . . . . . . . 15 ⊢ ⟨b, 2c⟩ ∈ V
68		breq1 4642	. . . . . . . . . . . . . . 15 ⊢ (n = ⟨b, 2c⟩ → (n AddC a ↔ ⟨b, 2c⟩ AddC a))
69	67, 68	ceqsexv 2894	. . . . . . . . . . . . . 14 ⊢ (∃n(n = ⟨b, 2c⟩ ∧ n AddC a) ↔ ⟨b, 2c⟩ AddC a)
70	51, 66, 69	3bitri 262	. . . . . . . . . . . . 13 ⊢ (⟨b, a⟩ ∈ ran ((1st ∩ ((◡2nd “ {2c}) × V)) ⊗ AddC ) ↔ ⟨b, 2c⟩ AddC a)
71	15, 61	braddcfn 5826	. . . . . . . . . . . . 13 ⊢ (⟨b, 2c⟩ AddC a ↔ (b +c 2c) = a)
72		eqcom 2355	. . . . . . . . . . . . 13 ⊢ ((b +c 2c) = a ↔ a = (b +c 2c))
73	70, 71, 72	3bitri 262	. . . . . . . . . . . 12 ⊢ (⟨b, a⟩ ∈ ran ((1st ∩ ((◡2nd “ {2c}) × V)) ⊗ AddC ) ↔ a = (b +c 2c))
74	9, 73	anbi12i 678	. . . . . . . . . . 11 ⊢ ((⟨b, n⟩ ∈ ◡ran ( Ins3 ◡((ran (◡1st ⊗ (1st ∩ 2nd )) ⊗ 2nd ) “ AddC ) ∩ (((1st ∘ 1st ) ⊗ ((2nd ∘ 1st ) ⊗ 2nd )) “ AddC )) ∧ ⟨b, a⟩ ∈ ran ((1st ∩ ((◡2nd “ {2c}) × V)) ⊗ AddC )) ↔ (b = ((n +c n) +c n) ∧ a = (b +c 2c)))
75	50, 74	bitri 240	. . . . . . . . . 10 ⊢ (⟨b, ⟨n, a⟩⟩ ∈ (◡ran ( Ins3 ◡((ran (◡1st ⊗ (1st ∩ 2nd )) ⊗ 2nd ) “ AddC ) ∩ (((1st ∘ 1st ) ⊗ ((2nd ∘ 1st ) ⊗ 2nd )) “ AddC )) ⊗ ran ((1st ∩ ((◡2nd “ {2c}) × V)) ⊗ AddC )) ↔ (b = ((n +c n) +c n) ∧ a = (b +c 2c)))
76	75	exbii 1582	. . . . . . . . 9 ⊢ (∃b⟨b, ⟨n, a⟩⟩ ∈ (◡ran ( Ins3 ◡((ran (◡1st ⊗ (1st ∩ 2nd )) ⊗ 2nd ) “ AddC ) ∩ (((1st ∘ 1st ) ⊗ ((2nd ∘ 1st ) ⊗ 2nd )) “ AddC )) ⊗ ran ((1st ∩ ((◡2nd “ {2c}) × V)) ⊗ AddC )) ↔ ∃b(b = ((n +c n) +c n) ∧ a = (b +c 2c)))
77		addceq1 4383	. . . . . . . . . . 11 ⊢ (b = ((n +c n) +c n) → (b +c 2c) = (((n +c n) +c n) +c 2c))
78	77	eqeq2d 2364	. . . . . . . . . 10 ⊢ (b = ((n +c n) +c n) → (a = (b +c 2c) ↔ a = (((n +c n) +c n) +c 2c)))
79	42, 78	ceqsexv 2894	. . . . . . . . 9 ⊢ (∃b(b = ((n +c n) +c n) ∧ a = (b +c 2c)) ↔ a = (((n +c n) +c n) +c 2c))
80	49, 76, 79	3bitri 262	. . . . . . . 8 ⊢ (⟨n, a⟩ ∈ ran (◡ran ( Ins3 ◡((ran (◡1st ⊗ (1st ∩ 2nd )) ⊗ 2nd ) “ AddC ) ∩ (((1st ∘ 1st ) ⊗ ((2nd ∘ 1st ) ⊗ 2nd )) “ AddC )) ⊗ ran ((1st ∩ ((◡2nd “ {2c}) × V)) ⊗ AddC )) ↔ a = (((n +c n) +c n) +c 2c))
81	48, 80	orbi12i 507	. . . . . . 7 ⊢ ((⟨n, a⟩ ∈ (ran ( Ins3 ◡((ran (◡1st ⊗ (1st ∩ 2nd )) ⊗ 2nd ) “ AddC ) ∩ (((1st ∘ 1st ) ⊗ ((2nd ∘ 1st ) ⊗ 2nd )) “ AddC )) ∪ ran (◡ran ( Ins3 ◡((ran (◡1st ⊗ (1st ∩ 2nd )) ⊗ 2nd ) “ AddC ) ∩ (((1st ∘ 1st ) ⊗ ((2nd ∘ 1st ) ⊗ 2nd )) “ AddC )) ⊗ ran ((1st ∩ ((◡2nd “ {1c}) × V)) ⊗ AddC ))) ∨ ⟨n, a⟩ ∈ ran (◡ran ( Ins3 ◡((ran (◡1st ⊗ (1st ∩ 2nd )) ⊗ 2nd ) “ AddC ) ∩ (((1st ∘ 1st ) ⊗ ((2nd ∘ 1st ) ⊗ 2nd )) “ AddC )) ⊗ ran ((1st ∩ ((◡2nd “ {2c}) × V)) ⊗ AddC ))) ↔ ((a = ((n +c n) +c n) ∨ a = (((n +c n) +c n) +c 1c)) ∨ a = (((n +c n) +c n) +c 2c)))
82		elun 3220	. . . . . . 7 ⊢ (⟨n, a⟩ ∈ ((ran ( Ins3 ◡((ran (◡1st ⊗ (1st ∩ 2nd )) ⊗ 2nd ) “ AddC ) ∩ (((1st ∘ 1st ) ⊗ ((2nd ∘ 1st ) ⊗ 2nd )) “ AddC )) ∪ ran (◡ran ( Ins3 ◡((ran (◡1st ⊗ (1st ∩ 2nd )) ⊗ 2nd ) “ AddC ) ∩ (((1st ∘ 1st ) ⊗ ((2nd ∘ 1st ) ⊗ 2nd )) “ AddC )) ⊗ ran ((1st ∩ ((◡2nd “ {1c}) × V)) ⊗ AddC ))) ∪ ran (◡ran ( Ins3 ◡((ran (◡1st ⊗ (1st ∩ 2nd )) ⊗ 2nd ) “ AddC ) ∩ (((1st ∘ 1st ) ⊗ ((2nd ∘ 1st ) ⊗ 2nd )) “ AddC )) ⊗ ran ((1st ∩ ((◡2nd “ {2c}) × V)) ⊗ AddC ))) ↔ (⟨n, a⟩ ∈ (ran ( Ins3 ◡((ran (◡1st ⊗ (1st ∩ 2nd )) ⊗ 2nd ) “ AddC ) ∩ (((1st ∘ 1st ) ⊗ ((2nd ∘ 1st ) ⊗ 2nd )) “ AddC )) ∪ ran (◡ran ( Ins3 ◡((ran (◡1st ⊗ (1st ∩ 2nd )) ⊗ 2nd ) “ AddC ) ∩ (((1st ∘ 1st ) ⊗ ((2nd ∘ 1st ) ⊗ 2nd )) “ AddC )) ⊗ ran ((1st ∩ ((◡2nd “ {1c}) × V)) ⊗ AddC ))) ∨ ⟨n, a⟩ ∈ ran (◡ran ( Ins3 ◡((ran (◡1st ⊗ (1st ∩ 2nd )) ⊗ 2nd ) “ AddC ) ∩ (((1st ∘ 1st ) ⊗ ((2nd ∘ 1st ) ⊗ 2nd )) “ AddC )) ⊗ ran ((1st ∩ ((◡2nd “ {2c}) × V)) ⊗ AddC ))))
83		df-3or 935	. . . . . . 7 ⊢ ((a = ((n +c n) +c n) ∨ a = (((n +c n) +c n) +c 1c) ∨ a = (((n +c n) +c n) +c 2c)) ↔ ((a = ((n +c n) +c n) ∨ a = (((n +c n) +c n) +c 1c)) ∨ a = (((n +c n) +c n) +c 2c)))
84	81, 82, 83	3bitr4i 268	. . . . . 6 ⊢ (⟨n, a⟩ ∈ ((ran ( Ins3 ◡((ran (◡1st ⊗ (1st ∩ 2nd )) ⊗ 2nd ) “ AddC ) ∩ (((1st ∘ 1st ) ⊗ ((2nd ∘ 1st ) ⊗ 2nd )) “ AddC )) ∪ ran (◡ran ( Ins3 ◡((ran (◡1st ⊗ (1st ∩ 2nd )) ⊗ 2nd ) “ AddC ) ∩ (((1st ∘ 1st ) ⊗ ((2nd ∘ 1st ) ⊗ 2nd )) “ AddC )) ⊗ ran ((1st ∩ ((◡2nd “ {1c}) × V)) ⊗ AddC ))) ∪ ran (◡ran ( Ins3 ◡((ran (◡1st ⊗ (1st ∩ 2nd )) ⊗ 2nd ) “ AddC ) ∩ (((1st ∘ 1st ) ⊗ ((2nd ∘ 1st ) ⊗ 2nd )) “ AddC )) ⊗ ran ((1st ∩ ((◡2nd “ {2c}) × V)) ⊗ AddC ))) ↔ (a = ((n +c n) +c n) ∨ a = (((n +c n) +c n) +c 1c) ∨ a = (((n +c n) +c n) +c 2c)))
85	2, 84	bitri 240	. . . . 5 ⊢ (n((ran ( Ins3 ◡((ran (◡1st ⊗ (1st ∩ 2nd )) ⊗ 2nd ) “ AddC ) ∩ (((1st ∘ 1st ) ⊗ ((2nd ∘ 1st ) ⊗ 2nd )) “ AddC )) ∪ ran (◡ran ( Ins3 ◡((ran (◡1st ⊗ (1st ∩ 2nd )) ⊗ 2nd ) “ AddC ) ∩ (((1st ∘ 1st ) ⊗ ((2nd ∘ 1st ) ⊗ 2nd )) “ AddC )) ⊗ ran ((1st ∩ ((◡2nd “ {1c}) × V)) ⊗ AddC ))) ∪ ran (◡ran ( Ins3 ◡((ran (◡1st ⊗ (1st ∩ 2nd )) ⊗ 2nd ) “ AddC ) ∩ (((1st ∘ 1st ) ⊗ ((2nd ∘ 1st ) ⊗ 2nd )) “ AddC )) ⊗ ran ((1st ∩ ((◡2nd “ {2c}) × V)) ⊗ AddC )))a ↔ (a = ((n +c n) +c n) ∨ a = (((n +c n) +c n) +c 1c) ∨ a = (((n +c n) +c n) +c 2c)))
86	85	rexbii 2639	. . . 4 ⊢ (∃n ∈ Nn n((ran ( Ins3 ◡((ran (◡1st ⊗ (1st ∩ 2nd )) ⊗ 2nd ) “ AddC ) ∩ (((1st ∘ 1st ) ⊗ ((2nd ∘ 1st ) ⊗ 2nd )) “ AddC )) ∪ ran (◡ran ( Ins3 ◡((ran (◡1st ⊗ (1st ∩ 2nd )) ⊗ 2nd ) “ AddC ) ∩ (((1st ∘ 1st ) ⊗ ((2nd ∘ 1st ) ⊗ 2nd )) “ AddC )) ⊗ ran ((1st ∩ ((◡2nd “ {1c}) × V)) ⊗ AddC ))) ∪ ran (◡ran ( Ins3 ◡((ran (◡1st ⊗ (1st ∩ 2nd )) ⊗ 2nd ) “ AddC ) ∩ (((1st ∘ 1st ) ⊗ ((2nd ∘ 1st ) ⊗ 2nd )) “ AddC )) ⊗ ran ((1st ∩ ((◡2nd “ {2c}) × V)) ⊗ AddC )))a ↔ ∃n ∈ Nn (a = ((n +c n) +c n) ∨ a = (((n +c n) +c n) +c 1c) ∨ a = (((n +c n) +c n) +c 2c)))
87	1, 86	bitri 240	. . 3 ⊢ (a ∈ (((ran ( Ins3 ◡((ran (◡1st ⊗ (1st ∩ 2nd )) ⊗ 2nd ) “ AddC ) ∩ (((1st ∘ 1st ) ⊗ ((2nd ∘ 1st ) ⊗ 2nd )) “ AddC )) ∪ ran (◡ran ( Ins3 ◡((ran (◡1st ⊗ (1st ∩ 2nd )) ⊗ 2nd ) “ AddC ) ∩ (((1st ∘ 1st ) ⊗ ((2nd ∘ 1st ) ⊗ 2nd )) “ AddC )) ⊗ ran ((1st ∩ ((◡2nd “ {1c}) × V)) ⊗ AddC ))) ∪ ran (◡ran ( Ins3 ◡((ran (◡1st ⊗ (1st ∩ 2nd )) ⊗ 2nd ) “ AddC ) ∩ (((1st ∘ 1st ) ⊗ ((2nd ∘ 1st ) ⊗ 2nd )) “ AddC )) ⊗ ran ((1st ∩ ((◡2nd “ {2c}) × V)) ⊗ AddC ))) “ Nn ) ↔ ∃n ∈ Nn (a = ((n +c n) +c n) ∨ a = (((n +c n) +c n) +c 1c) ∨ a = (((n +c n) +c n) +c 2c)))
88	87	abbi2i 2464	. 2 ⊢ (((ran ( Ins3 ◡((ran (◡1st ⊗ (1st ∩ 2nd )) ⊗ 2nd ) “ AddC ) ∩ (((1st ∘ 1st ) ⊗ ((2nd ∘ 1st ) ⊗ 2nd )) “ AddC )) ∪ ran (◡ran ( Ins3 ◡((ran (◡1st ⊗ (1st ∩ 2nd )) ⊗ 2nd ) “ AddC ) ∩ (((1st ∘ 1st ) ⊗ ((2nd ∘ 1st ) ⊗ 2nd )) “ AddC )) ⊗ ran ((1st ∩ ((◡2nd “ {1c}) × V)) ⊗ AddC ))) ∪ ran (◡ran ( Ins3 ◡((ran (◡1st ⊗ (1st ∩ 2nd )) ⊗ 2nd ) “ AddC ) ∩ (((1st ∘ 1st ) ⊗ ((2nd ∘ 1st ) ⊗ 2nd )) “ AddC )) ⊗ ran ((1st ∩ ((◡2nd “ {2c}) × V)) ⊗ AddC ))) “ Nn ) = {a ∣ ∃n ∈ Nn (a = ((n +c n) +c n) ∨ a = (((n +c n) +c n) +c 1c) ∨ a = (((n +c n) +c n) +c 2c))}
89		1stex 4739	. . . . . . . . . . . . . 14 ⊢ 1st ∈ V
90	89	cnvex 5102	. . . . . . . . . . . . 13 ⊢ ◡1st ∈ V
91		2ndex 5112	. . . . . . . . . . . . . 14 ⊢ 2nd ∈ V
92	89, 91	inex 4105	. . . . . . . . . . . . 13 ⊢ (1st ∩ 2nd ) ∈ V
93	90, 92	txpex 5785	. . . . . . . . . . . 12 ⊢ (◡1st ⊗ (1st ∩ 2nd )) ∈ V
94	93	rnex 5107	. . . . . . . . . . 11 ⊢ ran (◡1st ⊗ (1st ∩ 2nd )) ∈ V
95	94, 91	txpex 5785	. . . . . . . . . 10 ⊢ (ran (◡1st ⊗ (1st ∩ 2nd )) ⊗ 2nd ) ∈ V
96		addcfnex 5824	. . . . . . . . . 10 ⊢ AddC ∈ V
97	95, 96	imaex 4747	. . . . . . . . 9 ⊢ ((ran (◡1st ⊗ (1st ∩ 2nd )) ⊗ 2nd ) “ AddC ) ∈ V
98	97	cnvex 5102	. . . . . . . 8 ⊢ ◡((ran (◡1st ⊗ (1st ∩ 2nd )) ⊗ 2nd ) “ AddC ) ∈ V
99	98	ins3ex 5798	. . . . . . 7 ⊢ Ins3 ◡((ran (◡1st ⊗ (1st ∩ 2nd )) ⊗ 2nd ) “ AddC ) ∈ V
100	89, 89	coex 4750	. . . . . . . . 9 ⊢ (1st ∘ 1st ) ∈ V
101	91, 89	coex 4750	. . . . . . . . . 10 ⊢ (2nd ∘ 1st ) ∈ V
102	101, 91	txpex 5785	. . . . . . . . 9 ⊢ ((2nd ∘ 1st ) ⊗ 2nd ) ∈ V
103	100, 102	txpex 5785	. . . . . . . 8 ⊢ ((1st ∘ 1st ) ⊗ ((2nd ∘ 1st ) ⊗ 2nd )) ∈ V
104	103, 96	imaex 4747	. . . . . . 7 ⊢ (((1st ∘ 1st ) ⊗ ((2nd ∘ 1st ) ⊗ 2nd )) “ AddC ) ∈ V
105	99, 104	inex 4105	. . . . . 6 ⊢ ( Ins3 ◡((ran (◡1st ⊗ (1st ∩ 2nd )) ⊗ 2nd ) “ AddC ) ∩ (((1st ∘ 1st ) ⊗ ((2nd ∘ 1st ) ⊗ 2nd )) “ AddC )) ∈ V
106	105	rnex 5107	. . . . 5 ⊢ ran ( Ins3 ◡((ran (◡1st ⊗ (1st ∩ 2nd )) ⊗ 2nd ) “ AddC ) ∩ (((1st ∘ 1st ) ⊗ ((2nd ∘ 1st ) ⊗ 2nd )) “ AddC )) ∈ V
107	106	cnvex 5102	. . . . . . 7 ⊢ ◡ran ( Ins3 ◡((ran (◡1st ⊗ (1st ∩ 2nd )) ⊗ 2nd ) “ AddC ) ∩ (((1st ∘ 1st ) ⊗ ((2nd ∘ 1st ) ⊗ 2nd )) “ AddC )) ∈ V
108	91	cnvex 5102	. . . . . . . . . . . 12 ⊢ ◡2nd ∈ V
109		snex 4111	. . . . . . . . . . . 12 ⊢ {1c} ∈ V
110	108, 109	imaex 4747	. . . . . . . . . . 11 ⊢ (◡2nd “ {1c}) ∈ V
111		vvex 4109	. . . . . . . . . . 11 ⊢ V ∈ V
112	110, 111	xpex 5115	. . . . . . . . . 10 ⊢ ((◡2nd “ {1c}) × V) ∈ V
113	89, 112	inex 4105	. . . . . . . . 9 ⊢ (1st ∩ ((◡2nd “ {1c}) × V)) ∈ V
114	113, 96	txpex 5785	. . . . . . . 8 ⊢ ((1st ∩ ((◡2nd “ {1c}) × V)) ⊗ AddC ) ∈ V
115	114	rnex 5107	. . . . . . 7 ⊢ ran ((1st ∩ ((◡2nd “ {1c}) × V)) ⊗ AddC ) ∈ V
116	107, 115	txpex 5785	. . . . . 6 ⊢ (◡ran ( Ins3 ◡((ran (◡1st ⊗ (1st ∩ 2nd )) ⊗ 2nd ) “ AddC ) ∩ (((1st ∘ 1st ) ⊗ ((2nd ∘ 1st ) ⊗ 2nd )) “ AddC )) ⊗ ran ((1st ∩ ((◡2nd “ {1c}) × V)) ⊗ AddC )) ∈ V
117	116	rnex 5107	. . . . 5 ⊢ ran (◡ran ( Ins3 ◡((ran (◡1st ⊗ (1st ∩ 2nd )) ⊗ 2nd ) “ AddC ) ∩ (((1st ∘ 1st ) ⊗ ((2nd ∘ 1st ) ⊗ 2nd )) “ AddC )) ⊗ ran ((1st ∩ ((◡2nd “ {1c}) × V)) ⊗ AddC )) ∈ V
118	106, 117	unex 4106	. . . 4 ⊢ (ran ( Ins3 ◡((ran (◡1st ⊗ (1st ∩ 2nd )) ⊗ 2nd ) “ AddC ) ∩ (((1st ∘ 1st ) ⊗ ((2nd ∘ 1st ) ⊗ 2nd )) “ AddC )) ∪ ran (◡ran ( Ins3 ◡((ran (◡1st ⊗ (1st ∩ 2nd )) ⊗ 2nd ) “ AddC ) ∩ (((1st ∘ 1st ) ⊗ ((2nd ∘ 1st ) ⊗ 2nd )) “ AddC )) ⊗ ran ((1st ∩ ((◡2nd “ {1c}) × V)) ⊗ AddC ))) ∈ V
119		snex 4111	. . . . . . . . . . 11 ⊢ {2c} ∈ V
120	108, 119	imaex 4747	. . . . . . . . . 10 ⊢ (◡2nd “ {2c}) ∈ V
121	120, 111	xpex 5115	. . . . . . . . 9 ⊢ ((◡2nd “ {2c}) × V) ∈ V
122	89, 121	inex 4105	. . . . . . . 8 ⊢ (1st ∩ ((◡2nd “ {2c}) × V)) ∈ V
123	122, 96	txpex 5785	. . . . . . 7 ⊢ ((1st ∩ ((◡2nd “ {2c}) × V)) ⊗ AddC ) ∈ V
124	123	rnex 5107	. . . . . 6 ⊢ ran ((1st ∩ ((◡2nd “ {2c}) × V)) ⊗ AddC ) ∈ V
125	107, 124	txpex 5785	. . . . 5 ⊢ (◡ran ( Ins3 ◡((ran (◡1st ⊗ (1st ∩ 2nd )) ⊗ 2nd ) “ AddC ) ∩ (((1st ∘ 1st ) ⊗ ((2nd ∘ 1st ) ⊗ 2nd )) “ AddC )) ⊗ ran ((1st ∩ ((◡2nd “ {2c}) × V)) ⊗ AddC )) ∈ V
126	125	rnex 5107	. . . 4 ⊢ ran (◡ran ( Ins3 ◡((ran (◡1st ⊗ (1st ∩ 2nd )) ⊗ 2nd ) “ AddC ) ∩ (((1st ∘ 1st ) ⊗ ((2nd ∘ 1st ) ⊗ 2nd )) “ AddC )) ⊗ ran ((1st ∩ ((◡2nd “ {2c}) × V)) ⊗ AddC )) ∈ V
127	118, 126	unex 4106	. . 3 ⊢ ((ran ( Ins3 ◡((ran (◡1st ⊗ (1st ∩ 2nd )) ⊗ 2nd ) “ AddC ) ∩ (((1st ∘ 1st ) ⊗ ((2nd ∘ 1st ) ⊗ 2nd )) “ AddC )) ∪ ran (◡ran ( Ins3 ◡((ran (◡1st ⊗ (1st ∩ 2nd )) ⊗ 2nd ) “ AddC ) ∩ (((1st ∘ 1st ) ⊗ ((2nd ∘ 1st ) ⊗ 2nd )) “ AddC )) ⊗ ran ((1st ∩ ((◡2nd “ {1c}) × V)) ⊗ AddC ))) ∪ ran (◡ran ( Ins3 ◡((ran (◡1st ⊗ (1st ∩ 2nd )) ⊗ 2nd ) “ AddC ) ∩ (((1st ∘ 1st ) ⊗ ((2nd ∘ 1st ) ⊗ 2nd )) “ AddC )) ⊗ ran ((1st ∩ ((◡2nd “ {2c}) × V)) ⊗ AddC ))) ∈ V
128		nncex 4396	. . 3 ⊢ Nn ∈ V
129	127, 128	imaex 4747	. 2 ⊢ (((ran ( Ins3 ◡((ran (◡1st ⊗ (1st ∩ 2nd )) ⊗ 2nd ) “ AddC ) ∩ (((1st ∘ 1st ) ⊗ ((2nd ∘ 1st ) ⊗ 2nd )) “ AddC )) ∪ ran (◡ran ( Ins3 ◡((ran (◡1st ⊗ (1st ∩ 2nd )) ⊗ 2nd ) “ AddC ) ∩ (((1st ∘ 1st ) ⊗ ((2nd ∘ 1st ) ⊗ 2nd )) “ AddC )) ⊗ ran ((1st ∩ ((◡2nd “ {1c}) × V)) ⊗ AddC ))) ∪ ran (◡ran ( Ins3 ◡((ran (◡1st ⊗ (1st ∩ 2nd )) ⊗ 2nd ) “ AddC ) ∩ (((1st ∘ 1st ) ⊗ ((2nd ∘ 1st ) ⊗ 2nd )) “ AddC )) ⊗ ran ((1st ∩ ((◡2nd “ {2c}) × V)) ⊗ AddC ))) “ Nn ) ∈ V
130	88, 129	eqeltrri 2424	1 ⊢ {a ∣ ∃n ∈ Nn (a = ((n +c n) +c n) ∨ a = (((n +c n) +c n) +c 1c) ∨ a = (((n +c n) +c n) +c 2c))} ∈ V

Syntax hints:   ∨ wo 357   ∧ wa 358   ∨ w3o 933  ∃wex 1541   = wceq 1642   ∈ wcel 1710  {cab 2339  ∃wrex 2615  Vcvv 2859   ∪ cun 3207   ∩ cin 3208  ∅c0 3550  {csn 3737  {cpr 3738  1_cc1c 4134   Nn cnnc 4373   +_c cplc 4375  ⟨cop 4561   class class class wbr 4639  1^st c1st 4717   ∘ ccom 4721   “ cima 4722   × cxp 4770  ^◡ccnv 4771  ran crn 4773  2^nd c2nd 4783   ⊗ ctxp 5735   AddC caddcfn 5745   Ins3 cins3 5751   Nc cnc 6091  2_cc2c 6094

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1546  ax-5 1557  ax-17 1616  ax-9 1654  ax-8 1675  ax-13 1712  ax-14 1714  ax-6 1729  ax-7 1734  ax-11 1746  ax-12 1925  ax-ext 2334  ax-nin 4078  ax-xp 4079  ax-cnv 4080  ax-1c 4081  ax-sset 4082  ax-si 4083  ax-ins2 4084  ax-ins3 4085  ax-typlower 4086  ax-sn 4087

This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3or 935  df-3an 936  df-nan 1288  df-tru 1319  df-ex 1542  df-nf 1545  df-sb 1649  df-eu 2208  df-mo 2209  df-clab 2340  df-cleq 2346  df-clel 2349  df-nfc 2478  df-ne 2518  df-ral 2619  df-rex 2620  df-reu 2621  df-rmo 2622  df-rab 2623  df-v 2861  df-sbc 3047  df-csb 3137  df-nin 3211  df-compl 3212  df-in 3213  df-un 3214  df-dif 3215  df-symdif 3216  df-ss 3259  df-pss 3261  df-nul 3551  df-if 3663  df-pw 3724  df-sn 3741  df-pr 3742  df-uni 3892  df-int 3927  df-iun 3971  df-opk 4058  df-1c 4136  df-pw1 4137  df-uni1 4138  df-xpk 4185  df-cnvk 4186  df-ins2k 4187  df-ins3k 4188  df-imak 4189  df-cok 4190  df-p6 4191  df-sik 4192  df-ssetk 4193  df-imagek 4194  df-idk 4195  df-iota 4339  df-0c 4377  df-addc 4378  df-nnc 4379  df-fin 4380  df-lefin 4440  df-ltfin 4441  df-ncfin 4442  df-tfin 4443  df-evenfin 4444  df-oddfin 4445  df-sfin 4446  df-spfin 4447  df-phi 4565  df-op 4566  df-proj1 4567  df-proj2 4568  df-opab 4623  df-br 4640  df-1st 4723  df-swap 4724  df-sset 4725  df-co 4726  df-ima 4727  df-si 4728  df-id 4767  df-xp 4784  df-cnv 4785  df-rn 4786  df-dm 4787  df-res 4788  df-fun 4789  df-fn 4790  df-f 4791  df-f1 4792  df-fo 4793  df-f1o 4794  df-fv 4795  df-2nd 4797  df-ov 5526  df-oprab 5528  df-mpt 5652  df-mpt2 5654  df-txp 5736  df-cup 5742  df-disj 5744  df-addcfn 5746  df-ins2 5750  df-ins3 5752  df-image 5754  df-ins4 5756  df-si3 5758  df-funs 5760  df-fns 5762  df-ec 5947  df-en 6029  df-nc 6101  df-2c 6104

This theorem is referenced by:  nncdiv3  6275