Theorem nncdiv3lem2 6277

Description: Lemma for nncdiv3 6278. 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 4755	. . . 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 4641	. . . . . 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 3221	. . . . . . . . 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 6276	. . . . . . . . . 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 4898	. . . . . . . . . . 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 5783	. . . . . . . . . . . . 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 4894	. . . . . . . . . . . . . . 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 6276	. . . . . . . . . . . . . . 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 4898	. . . . . . . . . . . . . . . 16 ⊢ (⟨b, a⟩ ∈ ran ((1st ∩ ((◡2nd “ {1c}) × V)) ⊗ AddC ) ↔ ∃n⟨n, ⟨b, a⟩⟩ ∈ ((1st ∩ ((◡2nd “ {1c}) × V)) ⊗ AddC ))
11		oteltxp 5783	. . . . . . . . . . . . . . . . . 18 ⊢ (⟨n, ⟨b, a⟩⟩ ∈ ((1st ∩ ((◡2nd “ {1c}) × V)) ⊗ AddC ) ↔ (⟨n, b⟩ ∈ (1st ∩ ((◡2nd “ {1c}) × V)) ∧ ⟨n, a⟩ ∈ AddC ))
12		elin 3220	. . . . . . . . . . . . . . . . . . . 20 ⊢ (⟨n, b⟩ ∈ (1st ∩ ((◡2nd “ {1c}) × V)) ↔ (⟨n, b⟩ ∈ 1st ∧ ⟨n, b⟩ ∈ ((◡2nd “ {1c}) × V)))
13		df-br 4641	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (n1st b ↔ ⟨n, b⟩ ∈ 1st )
14	13	bicomi 193	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (⟨n, b⟩ ∈ 1st ↔ n1st b)
15		vex 2863	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ b ∈ V
16		opelxp 4812	. . . . . . . . . . . . . . . . . . . . . . 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 5021	. . . . . . . . . . . . . . . . . . . . . 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 4143	. . . . . . . . . . . . . . . . . . . . 21 ⊢ 1c ∈ V
22	15, 21	op1st2nd 5791	. . . . . . . . . . . . . . . . . . . 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 4641	. . . . . . . . . . . . . . . . . . . 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 4589	. . . . . . . . . . . . . . . 16 ⊢ ⟨b, 1c⟩ ∈ V
31		breq1 4643	. . . . . . . . . . . . . . . 16 ⊢ (n = ⟨b, 1c⟩ → (n AddC a ↔ ⟨b, 1c⟩ AddC a))
32	30, 31	ceqsexv 2895	. . . . . . . . . . . . . . 15 ⊢ (∃n(n = ⟨b, 1c⟩ ∧ n AddC a) ↔ ⟨b, 1c⟩ AddC a)
33	15, 21	braddcfn 5827	. . . . . . . . . . . . . . . 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 2863	. . . . . . . . . . . . . 14 ⊢ n ∈ V
41	40, 40	addcex 4395	. . . . . . . . . . . . 13 ⊢ (n +c n) ∈ V
42	41, 40	addcex 4395	. . . . . . . . . . . 12 ⊢ ((n +c n) +c n) ∈ V
43		addceq1 4384	. . . . . . . . . . . . 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 2895	. . . . . . . . . . 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 4898	. . . . . . . . 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 5783	. . . . . . . . . . 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 4898	. . . . . . . . . . . . . 14 ⊢ (⟨b, a⟩ ∈ ran ((1st ∩ ((◡2nd “ {2c}) × V)) ⊗ AddC ) ↔ ∃n⟨n, ⟨b, a⟩⟩ ∈ ((1st ∩ ((◡2nd “ {2c}) × V)) ⊗ AddC ))
52		oteltxp 5783	. . . . . . . . . . . . . . . 16 ⊢ (⟨n, ⟨b, a⟩⟩ ∈ ((1st ∩ ((◡2nd “ {2c}) × V)) ⊗ AddC ) ↔ (⟨n, b⟩ ∈ (1st ∩ ((◡2nd “ {2c}) × V)) ∧ ⟨n, a⟩ ∈ AddC ))
53		elin 3220	. . . . . . . . . . . . . . . . . 18 ⊢ (⟨n, b⟩ ∈ (1st ∩ ((◡2nd “ {2c}) × V)) ↔ (⟨n, b⟩ ∈ 1st ∧ ⟨n, b⟩ ∈ ((◡2nd “ {2c}) × V)))
54		opelxp 4812	. . . . . . . . . . . . . . . . . . . . 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 5021	. . . . . . . . . . . . . . . . . . . 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 6105	. . . . . . . . . . . . . . . . . . . 20 ⊢ 2c = Nc {∅, V}
60		ncex 6118	. . . . . . . . . . . . . . . . . . . 20 ⊢ Nc {∅, V} ∈ V
61	59, 60	eqeltri 2423	. . . . . . . . . . . . . . . . . . 19 ⊢ 2c ∈ V
62	15, 61	op1st2nd 5791	. . . . . . . . . . . . . . . . . 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 4589	. . . . . . . . . . . . . . 15 ⊢ ⟨b, 2c⟩ ∈ V
68		breq1 4643	. . . . . . . . . . . . . . 15 ⊢ (n = ⟨b, 2c⟩ → (n AddC a ↔ ⟨b, 2c⟩ AddC a))
69	67, 68	ceqsexv 2895	. . . . . . . . . . . . . 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 5827	. . . . . . . . . . . . 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 4384	. . . . . . . . . . 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 2895	. . . . . . . . 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 3221	. . . . . . 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 2640	. . . 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	eqabi 2465	. 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 4740	. . . . . . . . . . . . . 14 ⊢ 1st ∈ V
90	89	cnvex 5103	. . . . . . . . . . . . 13 ⊢ ◡1st ∈ V
91		2ndex 5113	. . . . . . . . . . . . . 14 ⊢ 2nd ∈ V
92	89, 91	inex 4106	. . . . . . . . . . . . 13 ⊢ (1st ∩ 2nd ) ∈ V
93	90, 92	txpex 5786	. . . . . . . . . . . 12 ⊢ (◡1st ⊗ (1st ∩ 2nd )) ∈ V
94	93	rnex 5108	. . . . . . . . . . 11 ⊢ ran (◡1st ⊗ (1st ∩ 2nd )) ∈ V
95	94, 91	txpex 5786	. . . . . . . . . 10 ⊢ (ran (◡1st ⊗ (1st ∩ 2nd )) ⊗ 2nd ) ∈ V
96		addcfnex 5825	. . . . . . . . . 10 ⊢ AddC ∈ V
97	95, 96	imaex 4748	. . . . . . . . 9 ⊢ ((ran (◡1st ⊗ (1st ∩ 2nd )) ⊗ 2nd ) “ AddC ) ∈ V
98	97	cnvex 5103	. . . . . . . 8 ⊢ ◡((ran (◡1st ⊗ (1st ∩ 2nd )) ⊗ 2nd ) “ AddC ) ∈ V
99	98	ins3ex 5799	. . . . . . 7 ⊢ Ins3 ◡((ran (◡1st ⊗ (1st ∩ 2nd )) ⊗ 2nd ) “ AddC ) ∈ V
100	89, 89	coex 4751	. . . . . . . . 9 ⊢ (1st ∘ 1st ) ∈ V
101	91, 89	coex 4751	. . . . . . . . . 10 ⊢ (2nd ∘ 1st ) ∈ V
102	101, 91	txpex 5786	. . . . . . . . 9 ⊢ ((2nd ∘ 1st ) ⊗ 2nd ) ∈ V
103	100, 102	txpex 5786	. . . . . . . 8 ⊢ ((1st ∘ 1st ) ⊗ ((2nd ∘ 1st ) ⊗ 2nd )) ∈ V
104	103, 96	imaex 4748	. . . . . . 7 ⊢ (((1st ∘ 1st ) ⊗ ((2nd ∘ 1st ) ⊗ 2nd )) “ AddC ) ∈ V
105	99, 104	inex 4106	. . . . . 6 ⊢ ( Ins3 ◡((ran (◡1st ⊗ (1st ∩ 2nd )) ⊗ 2nd ) “ AddC ) ∩ (((1st ∘ 1st ) ⊗ ((2nd ∘ 1st ) ⊗ 2nd )) “ AddC )) ∈ V
106	105	rnex 5108	. . . . 5 ⊢ ran ( Ins3 ◡((ran (◡1st ⊗ (1st ∩ 2nd )) ⊗ 2nd ) “ AddC ) ∩ (((1st ∘ 1st ) ⊗ ((2nd ∘ 1st ) ⊗ 2nd )) “ AddC )) ∈ V
107	106	cnvex 5103	. . . . . . 7 ⊢ ◡ran ( Ins3 ◡((ran (◡1st ⊗ (1st ∩ 2nd )) ⊗ 2nd ) “ AddC ) ∩ (((1st ∘ 1st ) ⊗ ((2nd ∘ 1st ) ⊗ 2nd )) “ AddC )) ∈ V
108	91	cnvex 5103	. . . . . . . . . . . 12 ⊢ ◡2nd ∈ V
109		snex 4112	. . . . . . . . . . . 12 ⊢ {1c} ∈ V
110	108, 109	imaex 4748	. . . . . . . . . . 11 ⊢ (◡2nd “ {1c}) ∈ V
111		vvex 4110	. . . . . . . . . . 11 ⊢ V ∈ V
112	110, 111	xpex 5116	. . . . . . . . . 10 ⊢ ((◡2nd “ {1c}) × V) ∈ V
113	89, 112	inex 4106	. . . . . . . . 9 ⊢ (1st ∩ ((◡2nd “ {1c}) × V)) ∈ V
114	113, 96	txpex 5786	. . . . . . . 8 ⊢ ((1st ∩ ((◡2nd “ {1c}) × V)) ⊗ AddC ) ∈ V
115	114	rnex 5108	. . . . . . 7 ⊢ ran ((1st ∩ ((◡2nd “ {1c}) × V)) ⊗ AddC ) ∈ V
116	107, 115	txpex 5786	. . . . . 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 5108	. . . . 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 4107	. . . 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 4112	. . . . . . . . . . 11 ⊢ {2c} ∈ V
120	108, 119	imaex 4748	. . . . . . . . . 10 ⊢ (◡2nd “ {2c}) ∈ V
121	120, 111	xpex 5116	. . . . . . . . 9 ⊢ ((◡2nd “ {2c}) × V) ∈ V
122	89, 121	inex 4106	. . . . . . . 8 ⊢ (1st ∩ ((◡2nd “ {2c}) × V)) ∈ V
123	122, 96	txpex 5786	. . . . . . 7 ⊢ ((1st ∩ ((◡2nd “ {2c}) × V)) ⊗ AddC ) ∈ V
124	123	rnex 5108	. . . . . 6 ⊢ ran ((1st ∩ ((◡2nd “ {2c}) × V)) ⊗ AddC ) ∈ V
125	107, 124	txpex 5786	. . . . 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 5108	. . . 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 4107	. . 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 4397	. . 3 ⊢ Nn ∈ V
129	127, 128	imaex 4748	. 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

This proof depends on syntax axioms:   ∨ wo 357   ∧ wa 358   ∨ w3o 933  ∃wex 1541   = wceq 1642   ∈ wcel 1710  {cab 2339  ∃wrex 2616  Vcvv 2860   ∪ cun 3208   ∩ cin 3209  ∅c0 3551  {csn 3738  {cpr 3739  1_cc1c 4135   Nn cnnc 4374   +_c cplc 4376  ⟨cop 4562   class class class wbr 4640  1^st c1st 4718   ∘ ccom 4722   “ cima 4723   × cxp 4771  ^◡ccnv 4772  ran crn 4774  2^nd c2nd 4784   ⊗ ctxp 5736   AddC caddcfn 5746   Ins3 cins3 5752   Nc cnc 6092  2_cc2c 6095

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 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 4079  ax-xp 4080  ax-cnv 4081  ax-1c 4082  ax-sset 4083  ax-si 4084  ax-ins2 4085  ax-ins3 4086  ax-typlower 4087  ax-sn 4088

This proof 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 2479  df-ne 2519  df-ral 2620  df-rex 2621  df-reu 2622  df-rmo 2623  df-rab 2624  df-v 2862  df-sbc 3048  df-csb 3138  df-nin 3212  df-compl 3213  df-in 3214  df-un 3215  df-dif 3216  df-symdif 3217  df-ss 3260  df-pss 3262  df-nul 3552  df-if 3664  df-pw 3725  df-sn 3742  df-pr 3743  df-uni 3893  df-int 3928  df-iun 3972  df-opk 4059  df-1c 4137  df-pw1 4138  df-uni1 4139  df-xpk 4186  df-cnvk 4187  df-ins2k 4188  df-ins3k 4189  df-imak 4190  df-cok 4191  df-p6 4192  df-sik 4193  df-ssetk 4194  df-imagek 4195  df-idk 4196  df-iota 4340  df-0c 4378  df-addc 4379  df-nnc 4380  df-fin 4381  df-lefin 4441  df-ltfin 4442  df-ncfin 4443  df-tfin 4444  df-evenfin 4445  df-oddfin 4446  df-sfin 4447  df-spfin 4448  df-phi 4566  df-op 4567  df-proj1 4568  df-proj2 4569  df-opab 4624  df-br 4641  df-1st 4724  df-swap 4725  df-sset 4726  df-co 4727  df-ima 4728  df-si 4729  df-id 4768  df-xp 4785  df-cnv 4786  df-rn 4787  df-dm 4788  df-res 4789  df-fun 4790  df-fn 4791  df-f 4792  df-f1 4793  df-fo 4794  df-f1o 4795  df-fv 4796  df-2nd 4798  df-ov 5527  df-oprab 5529  df-mpt 5653  df-mpt2 5655  df-txp 5737  df-cup 5743  df-disj 5745  df-addcfn 5747  df-ins2 5751  df-ins3 5753  df-image 5755  df-ins4 5757  df-si3 5759  df-funs 5761  df-fns 5763  df-ec 5948  df-en 6030  df-nc 6102  df-2c 6105

This theorem is used by:  nncdiv3  6278