Theorem nncdiv3lem2 6277

Description: Lemma for nncdiv3 6278. Set up stratification for induction. (Contributed by SF, 2-Mar-2015.)

Assertion

Ref	Expression
nncdiv3lem2	|- {a | E.n e. Nn (a = ((n +o n) +o n) \/ a = (((n +o n) +o n) +o 1c) \/ a = (((n +o n) +o n) +o 2c))} e. _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 e. (((ran ( Ins3 `'((ran (`'1st (x) (1st i^i 2nd)) (x) 2nd)" AddC ) i^i (((1st o. 1st) (x) ((2nd o. 1st) (x) 2nd))" AddC )) u. ran (`'ran ( Ins3 `'((ran (`'1st (x) (1st i^i 2nd)) (x) 2nd)" AddC ) i^i (((1st o. 1st) (x) ((2nd o. 1st) (x) 2nd))" AddC )) (x) ran ((1st i^i ((`'2nd"{1c}) X. _V)) (x) AddC ))) u. ran (`'ran ( Ins3 `'((ran (`'1st (x) (1st i^i 2nd)) (x) 2nd)" AddC ) i^i (((1st o. 1st) (x) ((2nd o. 1st) (x) 2nd))" AddC )) (x) ran ((1st i^i ((`'2nd"{2c}) X. _V)) (x) AddC )))" Nn ) <-> E.n e. Nn n((ran ( Ins3 `'((ran (`'1st (x) (1st i^i 2nd)) (x) 2nd)" AddC ) i^i (((1st o. 1st) (x) ((2nd o. 1st) (x) 2nd))" AddC )) u. ran (`'ran ( Ins3 `'((ran (`'1st (x) (1st i^i 2nd)) (x) 2nd)" AddC ) i^i (((1st o. 1st) (x) ((2nd o. 1st) (x) 2nd))" AddC )) (x) ran ((1st i^i ((`'2nd"{1c}) X. _V)) (x) AddC ))) u. ran (`'ran ( Ins3 `'((ran (`'1st (x) (1st i^i 2nd)) (x) 2nd)" AddC ) i^i (((1st o. 1st) (x) ((2nd o. 1st) (x) 2nd))" AddC )) (x) ran ((1st i^i ((`'2nd"{2c}) X. _V)) (x) AddC )))a)
2		df-br 4641	. . . . . 6 |- (n((ran ( Ins3 `'((ran (`'1st (x) (1st i^i 2nd)) (x) 2nd)" AddC ) i^i (((1st o. 1st) (x) ((2nd o. 1st) (x) 2nd))" AddC )) u. ran (`'ran ( Ins3 `'((ran (`'1st (x) (1st i^i 2nd)) (x) 2nd)" AddC ) i^i (((1st o. 1st) (x) ((2nd o. 1st) (x) 2nd))" AddC )) (x) ran ((1st i^i ((`'2nd"{1c}) X. _V)) (x) AddC ))) u. ran (`'ran ( Ins3 `'((ran (`'1st (x) (1st i^i 2nd)) (x) 2nd)" AddC ) i^i (((1st o. 1st) (x) ((2nd o. 1st) (x) 2nd))" AddC )) (x) ran ((1st i^i ((`'2nd"{2c}) X. _V)) (x) AddC )))a <-> <.n, a>. e. ((ran ( Ins3 `'((ran (`'1st (x) (1st i^i 2nd)) (x) 2nd)" AddC ) i^i (((1st o. 1st) (x) ((2nd o. 1st) (x) 2nd))" AddC )) u. ran (`'ran ( Ins3 `'((ran (`'1st (x) (1st i^i 2nd)) (x) 2nd)" AddC ) i^i (((1st o. 1st) (x) ((2nd o. 1st) (x) 2nd))" AddC )) (x) ran ((1st i^i ((`'2nd"{1c}) X. _V)) (x) AddC ))) u. ran (`'ran ( Ins3 `'((ran (`'1st (x) (1st i^i 2nd)) (x) 2nd)" AddC ) i^i (((1st o. 1st) (x) ((2nd o. 1st) (x) 2nd))" AddC )) (x) ran ((1st i^i ((`'2nd"{2c}) X. _V)) (x) AddC ))))
3		elun 3221	. . . . . . . . 9 |- (<.n, a>. e. (ran ( Ins3 `'((ran (`'1st (x) (1st i^i 2nd)) (x) 2nd)" AddC ) i^i (((1st o. 1st) (x) ((2nd o. 1st) (x) 2nd))" AddC )) u. ran (`'ran ( Ins3 `'((ran (`'1st (x) (1st i^i 2nd)) (x) 2nd)" AddC ) i^i (((1st o. 1st) (x) ((2nd o. 1st) (x) 2nd))" AddC )) (x) ran ((1st i^i ((`'2nd"{1c}) X. _V)) (x) AddC ))) <-> (<.n, a>. e. ran ( Ins3 `'((ran (`'1st (x) (1st i^i 2nd)) (x) 2nd)" AddC ) i^i (((1st o. 1st) (x) ((2nd o. 1st) (x) 2nd))" AddC )) \/ <.n, a>. e. ran (`'ran ( Ins3 `'((ran (`'1st (x) (1st i^i 2nd)) (x) 2nd)" AddC ) i^i (((1st o. 1st) (x) ((2nd o. 1st) (x) 2nd))" AddC )) (x) ran ((1st i^i ((`'2nd"{1c}) X. _V)) (x) AddC ))))
4		nncdiv3lem1 6276	. . . . . . . . . 10 |- (<.n, a>. e. ran ( Ins3 `'((ran (`'1st (x) (1st i^i 2nd)) (x) 2nd)" AddC ) i^i (((1st o. 1st) (x) ((2nd o. 1st) (x) 2nd))" AddC )) <-> a = ((n +o n) +o n))
5		elrn2 4898	. . . . . . . . . . 11 |- (<.n, a>. e. ran (`'ran ( Ins3 `'((ran (`'1st (x) (1st i^i 2nd)) (x) 2nd)" AddC ) i^i (((1st o. 1st) (x) ((2nd o. 1st) (x) 2nd))" AddC )) (x) ran ((1st i^i ((`'2nd"{1c}) X. _V)) (x) AddC )) <-> E.b<.b, <.n, a>.>. e. (`'ran ( Ins3 `'((ran (`'1st (x) (1st i^i 2nd)) (x) 2nd)" AddC ) i^i (((1st o. 1st) (x) ((2nd o. 1st) (x) 2nd))" AddC )) (x) ran ((1st i^i ((`'2nd"{1c}) X. _V)) (x) AddC )))
6		oteltxp 5783	. . . . . . . . . . . . 13 |- (<.b, <.n, a>.>. e. (`'ran ( Ins3 `'((ran (`'1st (x) (1st i^i 2nd)) (x) 2nd)" AddC ) i^i (((1st o. 1st) (x) ((2nd o. 1st) (x) 2nd))" AddC )) (x) ran ((1st i^i ((`'2nd"{1c}) X. _V)) (x) AddC )) <-> (<.b, n>. e. `'ran ( Ins3 `'((ran (`'1st (x) (1st i^i 2nd)) (x) 2nd)" AddC ) i^i (((1st o. 1st) (x) ((2nd o. 1st) (x) 2nd))" AddC )) /\ <.b, a>. e. ran ((1st i^i ((`'2nd"{1c}) X. _V)) (x) AddC )))
7		opelcnv 4894	. . . . . . . . . . . . . . 15 |- (<.b, n>. e. `'ran ( Ins3 `'((ran (`'1st (x) (1st i^i 2nd)) (x) 2nd)" AddC ) i^i (((1st o. 1st) (x) ((2nd o. 1st) (x) 2nd))" AddC )) <-> <.n, b>. e. ran ( Ins3 `'((ran (`'1st (x) (1st i^i 2nd)) (x) 2nd)" AddC ) i^i (((1st o. 1st) (x) ((2nd o. 1st) (x) 2nd))" AddC )))
8		nncdiv3lem1 6276	. . . . . . . . . . . . . . 15 |- (<.n, b>. e. ran ( Ins3 `'((ran (`'1st (x) (1st i^i 2nd)) (x) 2nd)" AddC ) i^i (((1st o. 1st) (x) ((2nd o. 1st) (x) 2nd))" AddC )) <-> b = ((n +o n) +o n))
9	7, 8	bitri 240	. . . . . . . . . . . . . 14 |- (<.b, n>. e. `'ran ( Ins3 `'((ran (`'1st (x) (1st i^i 2nd)) (x) 2nd)" AddC ) i^i (((1st o. 1st) (x) ((2nd o. 1st) (x) 2nd))" AddC )) <-> b = ((n +o n) +o n))
10		elrn2 4898	. . . . . . . . . . . . . . . 16 |- (<.b, a>. e. ran ((1st i^i ((`'2nd"{1c}) X. _V)) (x) AddC ) <-> E.n<.n, <.b, a>.>. e. ((1st i^i ((`'2nd"{1c}) X. _V)) (x) AddC ))
11		oteltxp 5783	. . . . . . . . . . . . . . . . . 18 |- (<.n, <.b, a>.>. e. ((1st i^i ((`'2nd"{1c}) X. _V)) (x) AddC ) <-> (<.n, b>. e. (1st i^i ((`'2nd"{1c}) X. _V)) /\ <.n, a>. e. AddC ))
12		elin 3220	. . . . . . . . . . . . . . . . . . . 20 |- (<.n, b>. e. (1st i^i ((`'2nd"{1c}) X. _V)) <-> (<.n, b>. e. 1st /\ <.n, b>. e. ((`'2nd"{1c}) X. _V)))
13		df-br 4641	. . . . . . . . . . . . . . . . . . . . . 22 |- (n1stb <-> <.n, b>. e. 1st)
14	13	bicomi 193	. . . . . . . . . . . . . . . . . . . . 21 |- (<.n, b>. e. 1st <-> n1stb)
15		vex 2863	. . . . . . . . . . . . . . . . . . . . . . 23 |- b e. _V
16		opelxp 4812	. . . . . . . . . . . . . . . . . . . . . . 23 |- (<.n, b>. e. ((`'2nd"{1c}) X. _V) <-> (n e. (`'2nd"{1c}) /\ b e. _V))
17	15, 16	mpbiran2 885	. . . . . . . . . . . . . . . . . . . . . 22 |- (<.n, b>. e. ((`'2nd"{1c}) X. _V) <-> n e. (`'2nd"{1c}))
18		eliniseg 5021	. . . . . . . . . . . . . . . . . . . . . 22 |- (n e. (`'2nd"{1c}) <-> n2nd1c)
19	17, 18	bitri 240	. . . . . . . . . . . . . . . . . . . . 21 |- (<.n, b>. e. ((`'2nd"{1c}) X. _V) <-> n2nd1c)
20	14, 19	anbi12i 678	. . . . . . . . . . . . . . . . . . . 20 |- ((<.n, b>. e. 1st /\ <.n, b>. e. ((`'2nd"{1c}) X. _V)) <-> (n1stb /\ n2nd1c))
21		1cex 4143	. . . . . . . . . . . . . . . . . . . . 21 |- 1c e. _V
22	15, 21	op1st2nd 5791	. . . . . . . . . . . . . . . . . . . 20 |- ((n1stb /\ n2nd1c) <-> n = <.b, 1c>.)
23	12, 20, 22	3bitri 262	. . . . . . . . . . . . . . . . . . 19 |- (<.n, b>. e. (1st i^i ((`'2nd"{1c}) X. _V)) <-> n = <.b, 1c>.)
24		df-br 4641	. . . . . . . . . . . . . . . . . . . 20 |- (n AddC a <-> <.n, a>. e. AddC )
25	24	bicomi 193	. . . . . . . . . . . . . . . . . . 19 |- (<.n, a>. e. AddC <-> n AddC a)
26	23, 25	anbi12i 678	. . . . . . . . . . . . . . . . . 18 |- ((<.n, b>. e. (1st i^i ((`'2nd"{1c}) X. _V)) /\ <.n, a>. e. AddC ) <-> (n = <.b, 1c>. /\ n AddC a))
27	11, 26	bitri 240	. . . . . . . . . . . . . . . . 17 |- (<.n, <.b, a>.>. e. ((1st i^i ((`'2nd"{1c}) X. _V)) (x) AddC ) <-> (n = <.b, 1c>. /\ n AddC a))
28	27	exbii 1582	. . . . . . . . . . . . . . . 16 |- (E.n<.n, <.b, a>.>. e. ((1st i^i ((`'2nd"{1c}) X. _V)) (x) AddC ) <-> E.n(n = <.b, 1c>. /\ n AddC a))
29	10, 28	bitri 240	. . . . . . . . . . . . . . 15 |- (<.b, a>. e. ran ((1st i^i ((`'2nd"{1c}) X. _V)) (x) AddC ) <-> E.n(n = <.b, 1c>. /\ n AddC a))
30	15, 21	opex 4589	. . . . . . . . . . . . . . . 16 |- <.b, 1c>. e. _V
31		breq1 4643	. . . . . . . . . . . . . . . 16 |- (n = <.b, 1c>. -> (n AddC a <-> <.b, 1c>. AddC a))
32	30, 31	ceqsexv 2895	. . . . . . . . . . . . . . 15 |- (E.n(n = <.b, 1c>. /\ n AddC a) <-> <.b, 1c>. AddC a)
33	15, 21	braddcfn 5827	. . . . . . . . . . . . . . . 16 |- (<.b, 1c>. AddC a <-> (b +o 1c) = a)
34		eqcom 2355	. . . . . . . . . . . . . . . 16 |- ((b +o 1c) = a <-> a = (b +o 1c))
35	33, 34	bitri 240	. . . . . . . . . . . . . . 15 |- (<.b, 1c>. AddC a <-> a = (b +o 1c))
36	29, 32, 35	3bitri 262	. . . . . . . . . . . . . 14 |- (<.b, a>. e. ran ((1st i^i ((`'2nd"{1c}) X. _V)) (x) AddC ) <-> a = (b +o 1c))
37	9, 36	anbi12i 678	. . . . . . . . . . . . 13 |- ((<.b, n>. e. `'ran ( Ins3 `'((ran (`'1st (x) (1st i^i 2nd)) (x) 2nd)" AddC ) i^i (((1st o. 1st) (x) ((2nd o. 1st) (x) 2nd))" AddC )) /\ <.b, a>. e. ran ((1st i^i ((`'2nd"{1c}) X. _V)) (x) AddC )) <-> (b = ((n +o n) +o n) /\ a = (b +o 1c)))
38	6, 37	bitri 240	. . . . . . . . . . . 12 |- (<.b, <.n, a>.>. e. (`'ran ( Ins3 `'((ran (`'1st (x) (1st i^i 2nd)) (x) 2nd)" AddC ) i^i (((1st o. 1st) (x) ((2nd o. 1st) (x) 2nd))" AddC )) (x) ran ((1st i^i ((`'2nd"{1c}) X. _V)) (x) AddC )) <-> (b = ((n +o n) +o n) /\ a = (b +o 1c)))
39	38	exbii 1582	. . . . . . . . . . 11 |- (E.b<.b, <.n, a>.>. e. (`'ran ( Ins3 `'((ran (`'1st (x) (1st i^i 2nd)) (x) 2nd)" AddC ) i^i (((1st o. 1st) (x) ((2nd o. 1st) (x) 2nd))" AddC )) (x) ran ((1st i^i ((`'2nd"{1c}) X. _V)) (x) AddC )) <-> E.b(b = ((n +o n) +o n) /\ a = (b +o 1c)))
40		vex 2863	. . . . . . . . . . . . . 14 |- n e. _V
41	40, 40	addcex 4395	. . . . . . . . . . . . 13 |- (n +o n) e. _V
42	41, 40	addcex 4395	. . . . . . . . . . . 12 |- ((n +o n) +o n) e. _V
43		addceq1 4384	. . . . . . . . . . . . 13 |- (b = ((n +o n) +o n) -> (b +o 1c) = (((n +o n) +o n) +o 1c))
44	43	eqeq2d 2364	. . . . . . . . . . . 12 |- (b = ((n +o n) +o n) -> (a = (b +o 1c) <-> a = (((n +o n) +o n) +o 1c)))
45	42, 44	ceqsexv 2895	. . . . . . . . . . 11 |- (E.b(b = ((n +o n) +o n) /\ a = (b +o 1c)) <-> a = (((n +o n) +o n) +o 1c))
46	5, 39, 45	3bitri 262	. . . . . . . . . 10 |- (<.n, a>. e. ran (`'ran ( Ins3 `'((ran (`'1st (x) (1st i^i 2nd)) (x) 2nd)" AddC ) i^i (((1st o. 1st) (x) ((2nd o. 1st) (x) 2nd))" AddC )) (x) ran ((1st i^i ((`'2nd"{1c}) X. _V)) (x) AddC )) <-> a = (((n +o n) +o n) +o 1c))
47	4, 46	orbi12i 507	. . . . . . . . 9 |- ((<.n, a>. e. ran ( Ins3 `'((ran (`'1st (x) (1st i^i 2nd)) (x) 2nd)" AddC ) i^i (((1st o. 1st) (x) ((2nd o. 1st) (x) 2nd))" AddC )) \/ <.n, a>. e. ran (`'ran ( Ins3 `'((ran (`'1st (x) (1st i^i 2nd)) (x) 2nd)" AddC ) i^i (((1st o. 1st) (x) ((2nd o. 1st) (x) 2nd))" AddC )) (x) ran ((1st i^i ((`'2nd"{1c}) X. _V)) (x) AddC ))) <-> (a = ((n +o n) +o n) \/ a = (((n +o n) +o n) +o 1c)))
48	3, 47	bitri 240	. . . . . . . 8 |- (<.n, a>. e. (ran ( Ins3 `'((ran (`'1st (x) (1st i^i 2nd)) (x) 2nd)" AddC ) i^i (((1st o. 1st) (x) ((2nd o. 1st) (x) 2nd))" AddC )) u. ran (`'ran ( Ins3 `'((ran (`'1st (x) (1st i^i 2nd)) (x) 2nd)" AddC ) i^i (((1st o. 1st) (x) ((2nd o. 1st) (x) 2nd))" AddC )) (x) ran ((1st i^i ((`'2nd"{1c}) X. _V)) (x) AddC ))) <-> (a = ((n +o n) +o n) \/ a = (((n +o n) +o n) +o 1c)))
49		elrn2 4898	. . . . . . . . 9 |- (<.n, a>. e. ran (`'ran ( Ins3 `'((ran (`'1st (x) (1st i^i 2nd)) (x) 2nd)" AddC ) i^i (((1st o. 1st) (x) ((2nd o. 1st) (x) 2nd))" AddC )) (x) ran ((1st i^i ((`'2nd"{2c}) X. _V)) (x) AddC )) <-> E.b<.b, <.n, a>.>. e. (`'ran ( Ins3 `'((ran (`'1st (x) (1st i^i 2nd)) (x) 2nd)" AddC ) i^i (((1st o. 1st) (x) ((2nd o. 1st) (x) 2nd))" AddC )) (x) ran ((1st i^i ((`'2nd"{2c}) X. _V)) (x) AddC )))
50		oteltxp 5783	. . . . . . . . . . 11 |- (<.b, <.n, a>.>. e. (`'ran ( Ins3 `'((ran (`'1st (x) (1st i^i 2nd)) (x) 2nd)" AddC ) i^i (((1st o. 1st) (x) ((2nd o. 1st) (x) 2nd))" AddC )) (x) ran ((1st i^i ((`'2nd"{2c}) X. _V)) (x) AddC )) <-> (<.b, n>. e. `'ran ( Ins3 `'((ran (`'1st (x) (1st i^i 2nd)) (x) 2nd)" AddC ) i^i (((1st o. 1st) (x) ((2nd o. 1st) (x) 2nd))" AddC )) /\ <.b, a>. e. ran ((1st i^i ((`'2nd"{2c}) X. _V)) (x) AddC )))
51		elrn2 4898	. . . . . . . . . . . . . 14 |- (<.b, a>. e. ran ((1st i^i ((`'2nd"{2c}) X. _V)) (x) AddC ) <-> E.n<.n, <.b, a>.>. e. ((1st i^i ((`'2nd"{2c}) X. _V)) (x) AddC ))
52		oteltxp 5783	. . . . . . . . . . . . . . . 16 |- (<.n, <.b, a>.>. e. ((1st i^i ((`'2nd"{2c}) X. _V)) (x) AddC ) <-> (<.n, b>. e. (1st i^i ((`'2nd"{2c}) X. _V)) /\ <.n, a>. e. AddC ))
53		elin 3220	. . . . . . . . . . . . . . . . . 18 |- (<.n, b>. e. (1st i^i ((`'2nd"{2c}) X. _V)) <-> (<.n, b>. e. 1st /\ <.n, b>. e. ((`'2nd"{2c}) X. _V)))
54		opelxp 4812	. . . . . . . . . . . . . . . . . . . . 21 |- (<.n, b>. e. ((`'2nd"{2c}) X. _V) <-> (n e. (`'2nd"{2c}) /\ b e. _V))
55	15, 54	mpbiran2 885	. . . . . . . . . . . . . . . . . . . 20 |- (<.n, b>. e. ((`'2nd"{2c}) X. _V) <-> n e. (`'2nd"{2c}))
56		eliniseg 5021	. . . . . . . . . . . . . . . . . . . 20 |- (n e. (`'2nd"{2c}) <-> n2nd2c)
57	55, 56	bitri 240	. . . . . . . . . . . . . . . . . . 19 |- (<.n, b>. e. ((`'2nd"{2c}) X. _V) <-> n2nd2c)
58	14, 57	anbi12i 678	. . . . . . . . . . . . . . . . . 18 |- ((<.n, b>. e. 1st /\ <.n, b>. e. ((`'2nd"{2c}) X. _V)) <-> (n1stb /\ n2nd2c))
59		df-2c 6105	. . . . . . . . . . . . . . . . . . . 20 |- 2c = Nc {(/), _V}
60		ncex 6118	. . . . . . . . . . . . . . . . . . . 20 |- Nc {(/), _V} e. _V
61	59, 60	eqeltri 2423	. . . . . . . . . . . . . . . . . . 19 |- 2c e. _V
62	15, 61	op1st2nd 5791	. . . . . . . . . . . . . . . . . 18 |- ((n1stb /\ n2nd2c) <-> n = <.b, 2c>.)
63	53, 58, 62	3bitri 262	. . . . . . . . . . . . . . . . 17 |- (<.n, b>. e. (1st i^i ((`'2nd"{2c}) X. _V)) <-> n = <.b, 2c>.)
64	63, 25	anbi12i 678	. . . . . . . . . . . . . . . 16 |- ((<.n, b>. e. (1st i^i ((`'2nd"{2c}) X. _V)) /\ <.n, a>. e. AddC ) <-> (n = <.b, 2c>. /\ n AddC a))
65	52, 64	bitri 240	. . . . . . . . . . . . . . 15 |- (<.n, <.b, a>.>. e. ((1st i^i ((`'2nd"{2c}) X. _V)) (x) AddC ) <-> (n = <.b, 2c>. /\ n AddC a))
66	65	exbii 1582	. . . . . . . . . . . . . 14 |- (E.n<.n, <.b, a>.>. e. ((1st i^i ((`'2nd"{2c}) X. _V)) (x) AddC ) <-> E.n(n = <.b, 2c>. /\ n AddC a))
67	15, 61	opex 4589	. . . . . . . . . . . . . . 15 |- <.b, 2c>. e. _V
68		breq1 4643	. . . . . . . . . . . . . . 15 |- (n = <.b, 2c>. -> (n AddC a <-> <.b, 2c>. AddC a))
69	67, 68	ceqsexv 2895	. . . . . . . . . . . . . 14 |- (E.n(n = <.b, 2c>. /\ n AddC a) <-> <.b, 2c>. AddC a)
70	51, 66, 69	3bitri 262	. . . . . . . . . . . . 13 |- (<.b, a>. e. ran ((1st i^i ((`'2nd"{2c}) X. _V)) (x) AddC ) <-> <.b, 2c>. AddC a)
71	15, 61	braddcfn 5827	. . . . . . . . . . . . 13 |- (<.b, 2c>. AddC a <-> (b +o 2c) = a)
72		eqcom 2355	. . . . . . . . . . . . 13 |- ((b +o 2c) = a <-> a = (b +o 2c))
73	70, 71, 72	3bitri 262	. . . . . . . . . . . 12 |- (<.b, a>. e. ran ((1st i^i ((`'2nd"{2c}) X. _V)) (x) AddC ) <-> a = (b +o 2c))
74	9, 73	anbi12i 678	. . . . . . . . . . 11 |- ((<.b, n>. e. `'ran ( Ins3 `'((ran (`'1st (x) (1st i^i 2nd)) (x) 2nd)" AddC ) i^i (((1st o. 1st) (x) ((2nd o. 1st) (x) 2nd))" AddC )) /\ <.b, a>. e. ran ((1st i^i ((`'2nd"{2c}) X. _V)) (x) AddC )) <-> (b = ((n +o n) +o n) /\ a = (b +o 2c)))
75	50, 74	bitri 240	. . . . . . . . . 10 |- (<.b, <.n, a>.>. e. (`'ran ( Ins3 `'((ran (`'1st (x) (1st i^i 2nd)) (x) 2nd)" AddC ) i^i (((1st o. 1st) (x) ((2nd o. 1st) (x) 2nd))" AddC )) (x) ran ((1st i^i ((`'2nd"{2c}) X. _V)) (x) AddC )) <-> (b = ((n +o n) +o n) /\ a = (b +o 2c)))
76	75	exbii 1582	. . . . . . . . 9 |- (E.b<.b, <.n, a>.>. e. (`'ran ( Ins3 `'((ran (`'1st (x) (1st i^i 2nd)) (x) 2nd)" AddC ) i^i (((1st o. 1st) (x) ((2nd o. 1st) (x) 2nd))" AddC )) (x) ran ((1st i^i ((`'2nd"{2c}) X. _V)) (x) AddC )) <-> E.b(b = ((n +o n) +o n) /\ a = (b +o 2c)))
77		addceq1 4384	. . . . . . . . . . 11 |- (b = ((n +o n) +o n) -> (b +o 2c) = (((n +o n) +o n) +o 2c))
78	77	eqeq2d 2364	. . . . . . . . . 10 |- (b = ((n +o n) +o n) -> (a = (b +o 2c) <-> a = (((n +o n) +o n) +o 2c)))
79	42, 78	ceqsexv 2895	. . . . . . . . 9 |- (E.b(b = ((n +o n) +o n) /\ a = (b +o 2c)) <-> a = (((n +o n) +o n) +o 2c))
80	49, 76, 79	3bitri 262	. . . . . . . 8 |- (<.n, a>. e. ran (`'ran ( Ins3 `'((ran (`'1st (x) (1st i^i 2nd)) (x) 2nd)" AddC ) i^i (((1st o. 1st) (x) ((2nd o. 1st) (x) 2nd))" AddC )) (x) ran ((1st i^i ((`'2nd"{2c}) X. _V)) (x) AddC )) <-> a = (((n +o n) +o n) +o 2c))
81	48, 80	orbi12i 507	. . . . . . 7 |- ((<.n, a>. e. (ran ( Ins3 `'((ran (`'1st (x) (1st i^i 2nd)) (x) 2nd)" AddC ) i^i (((1st o. 1st) (x) ((2nd o. 1st) (x) 2nd))" AddC )) u. ran (`'ran ( Ins3 `'((ran (`'1st (x) (1st i^i 2nd)) (x) 2nd)" AddC ) i^i (((1st o. 1st) (x) ((2nd o. 1st) (x) 2nd))" AddC )) (x) ran ((1st i^i ((`'2nd"{1c}) X. _V)) (x) AddC ))) \/ <.n, a>. e. ran (`'ran ( Ins3 `'((ran (`'1st (x) (1st i^i 2nd)) (x) 2nd)" AddC ) i^i (((1st o. 1st) (x) ((2nd o. 1st) (x) 2nd))" AddC )) (x) ran ((1st i^i ((`'2nd"{2c}) X. _V)) (x) AddC ))) <-> ((a = ((n +o n) +o n) \/ a = (((n +o n) +o n) +o 1c)) \/ a = (((n +o n) +o n) +o 2c)))
82		elun 3221	. . . . . . 7 |- (<.n, a>. e. ((ran ( Ins3 `'((ran (`'1st (x) (1st i^i 2nd)) (x) 2nd)" AddC ) i^i (((1st o. 1st) (x) ((2nd o. 1st) (x) 2nd))" AddC )) u. ran (`'ran ( Ins3 `'((ran (`'1st (x) (1st i^i 2nd)) (x) 2nd)" AddC ) i^i (((1st o. 1st) (x) ((2nd o. 1st) (x) 2nd))" AddC )) (x) ran ((1st i^i ((`'2nd"{1c}) X. _V)) (x) AddC ))) u. ran (`'ran ( Ins3 `'((ran (`'1st (x) (1st i^i 2nd)) (x) 2nd)" AddC ) i^i (((1st o. 1st) (x) ((2nd o. 1st) (x) 2nd))" AddC )) (x) ran ((1st i^i ((`'2nd"{2c}) X. _V)) (x) AddC ))) <-> (<.n, a>. e. (ran ( Ins3 `'((ran (`'1st (x) (1st i^i 2nd)) (x) 2nd)" AddC ) i^i (((1st o. 1st) (x) ((2nd o. 1st) (x) 2nd))" AddC )) u. ran (`'ran ( Ins3 `'((ran (`'1st (x) (1st i^i 2nd)) (x) 2nd)" AddC ) i^i (((1st o. 1st) (x) ((2nd o. 1st) (x) 2nd))" AddC )) (x) ran ((1st i^i ((`'2nd"{1c}) X. _V)) (x) AddC ))) \/ <.n, a>. e. ran (`'ran ( Ins3 `'((ran (`'1st (x) (1st i^i 2nd)) (x) 2nd)" AddC ) i^i (((1st o. 1st) (x) ((2nd o. 1st) (x) 2nd))" AddC )) (x) ran ((1st i^i ((`'2nd"{2c}) X. _V)) (x) AddC ))))
83		df-3or 935	. . . . . . 7 |- ((a = ((n +o n) +o n) \/ a = (((n +o n) +o n) +o 1c) \/ a = (((n +o n) +o n) +o 2c)) <-> ((a = ((n +o n) +o n) \/ a = (((n +o n) +o n) +o 1c)) \/ a = (((n +o n) +o n) +o 2c)))
84	81, 82, 83	3bitr4i 268	. . . . . 6 |- (<.n, a>. e. ((ran ( Ins3 `'((ran (`'1st (x) (1st i^i 2nd)) (x) 2nd)" AddC ) i^i (((1st o. 1st) (x) ((2nd o. 1st) (x) 2nd))" AddC )) u. ran (`'ran ( Ins3 `'((ran (`'1st (x) (1st i^i 2nd)) (x) 2nd)" AddC ) i^i (((1st o. 1st) (x) ((2nd o. 1st) (x) 2nd))" AddC )) (x) ran ((1st i^i ((`'2nd"{1c}) X. _V)) (x) AddC ))) u. ran (`'ran ( Ins3 `'((ran (`'1st (x) (1st i^i 2nd)) (x) 2nd)" AddC ) i^i (((1st o. 1st) (x) ((2nd o. 1st) (x) 2nd))" AddC )) (x) ran ((1st i^i ((`'2nd"{2c}) X. _V)) (x) AddC ))) <-> (a = ((n +o n) +o n) \/ a = (((n +o n) +o n) +o 1c) \/ a = (((n +o n) +o n) +o 2c)))
85	2, 84	bitri 240	. . . . 5 |- (n((ran ( Ins3 `'((ran (`'1st (x) (1st i^i 2nd)) (x) 2nd)" AddC ) i^i (((1st o. 1st) (x) ((2nd o. 1st) (x) 2nd))" AddC )) u. ran (`'ran ( Ins3 `'((ran (`'1st (x) (1st i^i 2nd)) (x) 2nd)" AddC ) i^i (((1st o. 1st) (x) ((2nd o. 1st) (x) 2nd))" AddC )) (x) ran ((1st i^i ((`'2nd"{1c}) X. _V)) (x) AddC ))) u. ran (`'ran ( Ins3 `'((ran (`'1st (x) (1st i^i 2nd)) (x) 2nd)" AddC ) i^i (((1st o. 1st) (x) ((2nd o. 1st) (x) 2nd))" AddC )) (x) ran ((1st i^i ((`'2nd"{2c}) X. _V)) (x) AddC )))a <-> (a = ((n +o n) +o n) \/ a = (((n +o n) +o n) +o 1c) \/ a = (((n +o n) +o n) +o 2c)))
86	85	rexbii 2640	. . . 4 |- (E.n e. Nn n((ran ( Ins3 `'((ran (`'1st (x) (1st i^i 2nd)) (x) 2nd)" AddC ) i^i (((1st o. 1st) (x) ((2nd o. 1st) (x) 2nd))" AddC )) u. ran (`'ran ( Ins3 `'((ran (`'1st (x) (1st i^i 2nd)) (x) 2nd)" AddC ) i^i (((1st o. 1st) (x) ((2nd o. 1st) (x) 2nd))" AddC )) (x) ran ((1st i^i ((`'2nd"{1c}) X. _V)) (x) AddC ))) u. ran (`'ran ( Ins3 `'((ran (`'1st (x) (1st i^i 2nd)) (x) 2nd)" AddC ) i^i (((1st o. 1st) (x) ((2nd o. 1st) (x) 2nd))" AddC )) (x) ran ((1st i^i ((`'2nd"{2c}) X. _V)) (x) AddC )))a <-> E.n e. Nn (a = ((n +o n) +o n) \/ a = (((n +o n) +o n) +o 1c) \/ a = (((n +o n) +o n) +o 2c)))
87	1, 86	bitri 240	. . 3 |- (a e. (((ran ( Ins3 `'((ran (`'1st (x) (1st i^i 2nd)) (x) 2nd)" AddC ) i^i (((1st o. 1st) (x) ((2nd o. 1st) (x) 2nd))" AddC )) u. ran (`'ran ( Ins3 `'((ran (`'1st (x) (1st i^i 2nd)) (x) 2nd)" AddC ) i^i (((1st o. 1st) (x) ((2nd o. 1st) (x) 2nd))" AddC )) (x) ran ((1st i^i ((`'2nd"{1c}) X. _V)) (x) AddC ))) u. ran (`'ran ( Ins3 `'((ran (`'1st (x) (1st i^i 2nd)) (x) 2nd)" AddC ) i^i (((1st o. 1st) (x) ((2nd o. 1st) (x) 2nd))" AddC )) (x) ran ((1st i^i ((`'2nd"{2c}) X. _V)) (x) AddC )))" Nn ) <-> E.n e. Nn (a = ((n +o n) +o n) \/ a = (((n +o n) +o n) +o 1c) \/ a = (((n +o n) +o n) +o 2c)))
88	87	abbi2i 2465	. 2 |- (((ran ( Ins3 `'((ran (`'1st (x) (1st i^i 2nd)) (x) 2nd)" AddC ) i^i (((1st o. 1st) (x) ((2nd o. 1st) (x) 2nd))" AddC )) u. ran (`'ran ( Ins3 `'((ran (`'1st (x) (1st i^i 2nd)) (x) 2nd)" AddC ) i^i (((1st o. 1st) (x) ((2nd o. 1st) (x) 2nd))" AddC )) (x) ran ((1st i^i ((`'2nd"{1c}) X. _V)) (x) AddC ))) u. ran (`'ran ( Ins3 `'((ran (`'1st (x) (1st i^i 2nd)) (x) 2nd)" AddC ) i^i (((1st o. 1st) (x) ((2nd o. 1st) (x) 2nd))" AddC )) (x) ran ((1st i^i ((`'2nd"{2c}) X. _V)) (x) AddC )))" Nn ) = {a | E.n e. Nn (a = ((n +o n) +o n) \/ a = (((n +o n) +o n) +o 1c) \/ a = (((n +o n) +o n) +o 2c))}
89		1stex 4740	. . . . . . . . . . . . . 14 |- 1st e. _V
90	89	cnvex 5103	. . . . . . . . . . . . 13 |- `'1st e. _V
91		2ndex 5113	. . . . . . . . . . . . . 14 |- 2nd e. _V
92	89, 91	inex 4106	. . . . . . . . . . . . 13 |- (1st i^i 2nd) e. _V
93	90, 92	txpex 5786	. . . . . . . . . . . 12 |- (`'1st (x) (1st i^i 2nd)) e. _V
94	93	rnex 5108	. . . . . . . . . . 11 |- ran (`'1st (x) (1st i^i 2nd)) e. _V
95	94, 91	txpex 5786	. . . . . . . . . 10 |- (ran (`'1st (x) (1st i^i 2nd)) (x) 2nd) e. _V
96		addcfnex 5825	. . . . . . . . . 10 |- AddC e. _V
97	95, 96	imaex 4748	. . . . . . . . 9 |- ((ran (`'1st (x) (1st i^i 2nd)) (x) 2nd)" AddC ) e. _V
98	97	cnvex 5103	. . . . . . . 8 |- `'((ran (`'1st (x) (1st i^i 2nd)) (x) 2nd)" AddC ) e. _V
99	98	ins3ex 5799	. . . . . . 7 |- Ins3 `'((ran (`'1st (x) (1st i^i 2nd)) (x) 2nd)" AddC ) e. _V
100	89, 89	coex 4751	. . . . . . . . 9 |- (1st o. 1st) e. _V
101	91, 89	coex 4751	. . . . . . . . . 10 |- (2nd o. 1st) e. _V
102	101, 91	txpex 5786	. . . . . . . . 9 |- ((2nd o. 1st) (x) 2nd) e. _V
103	100, 102	txpex 5786	. . . . . . . 8 |- ((1st o. 1st) (x) ((2nd o. 1st) (x) 2nd)) e. _V
104	103, 96	imaex 4748	. . . . . . 7 |- (((1st o. 1st) (x) ((2nd o. 1st) (x) 2nd))" AddC ) e. _V
105	99, 104	inex 4106	. . . . . 6 |- ( Ins3 `'((ran (`'1st (x) (1st i^i 2nd)) (x) 2nd)" AddC ) i^i (((1st o. 1st) (x) ((2nd o. 1st) (x) 2nd))" AddC )) e. _V
106	105	rnex 5108	. . . . 5 |- ran ( Ins3 `'((ran (`'1st (x) (1st i^i 2nd)) (x) 2nd)" AddC ) i^i (((1st o. 1st) (x) ((2nd o. 1st) (x) 2nd))" AddC )) e. _V
107	106	cnvex 5103	. . . . . . 7 |- `'ran ( Ins3 `'((ran (`'1st (x) (1st i^i 2nd)) (x) 2nd)" AddC ) i^i (((1st o. 1st) (x) ((2nd o. 1st) (x) 2nd))" AddC )) e. _V
108	91	cnvex 5103	. . . . . . . . . . . 12 |- `'2nd e. _V
109		snex 4112	. . . . . . . . . . . 12 |- {1c} e. _V
110	108, 109	imaex 4748	. . . . . . . . . . 11 |- (`'2nd"{1c}) e. _V
111		vvex 4110	. . . . . . . . . . 11 |- _V e. _V
112	110, 111	xpex 5116	. . . . . . . . . 10 |- ((`'2nd"{1c}) X. _V) e. _V
113	89, 112	inex 4106	. . . . . . . . 9 |- (1st i^i ((`'2nd"{1c}) X. _V)) e. _V
114	113, 96	txpex 5786	. . . . . . . 8 |- ((1st i^i ((`'2nd"{1c}) X. _V)) (x) AddC ) e. _V
115	114	rnex 5108	. . . . . . 7 |- ran ((1st i^i ((`'2nd"{1c}) X. _V)) (x) AddC ) e. _V
116	107, 115	txpex 5786	. . . . . 6 |- (`'ran ( Ins3 `'((ran (`'1st (x) (1st i^i 2nd)) (x) 2nd)" AddC ) i^i (((1st o. 1st) (x) ((2nd o. 1st) (x) 2nd))" AddC )) (x) ran ((1st i^i ((`'2nd"{1c}) X. _V)) (x) AddC )) e. _V
117	116	rnex 5108	. . . . 5 |- ran (`'ran ( Ins3 `'((ran (`'1st (x) (1st i^i 2nd)) (x) 2nd)" AddC ) i^i (((1st o. 1st) (x) ((2nd o. 1st) (x) 2nd))" AddC )) (x) ran ((1st i^i ((`'2nd"{1c}) X. _V)) (x) AddC )) e. _V
118	106, 117	unex 4107	. . . 4 |- (ran ( Ins3 `'((ran (`'1st (x) (1st i^i 2nd)) (x) 2nd)" AddC ) i^i (((1st o. 1st) (x) ((2nd o. 1st) (x) 2nd))" AddC )) u. ran (`'ran ( Ins3 `'((ran (`'1st (x) (1st i^i 2nd)) (x) 2nd)" AddC ) i^i (((1st o. 1st) (x) ((2nd o. 1st) (x) 2nd))" AddC )) (x) ran ((1st i^i ((`'2nd"{1c}) X. _V)) (x) AddC ))) e. _V
119		snex 4112	. . . . . . . . . . 11 |- {2c} e. _V
120	108, 119	imaex 4748	. . . . . . . . . 10 |- (`'2nd"{2c}) e. _V
121	120, 111	xpex 5116	. . . . . . . . 9 |- ((`'2nd"{2c}) X. _V) e. _V
122	89, 121	inex 4106	. . . . . . . 8 |- (1st i^i ((`'2nd"{2c}) X. _V)) e. _V
123	122, 96	txpex 5786	. . . . . . 7 |- ((1st i^i ((`'2nd"{2c}) X. _V)) (x) AddC ) e. _V
124	123	rnex 5108	. . . . . 6 |- ran ((1st i^i ((`'2nd"{2c}) X. _V)) (x) AddC ) e. _V
125	107, 124	txpex 5786	. . . . 5 |- (`'ran ( Ins3 `'((ran (`'1st (x) (1st i^i 2nd)) (x) 2nd)" AddC ) i^i (((1st o. 1st) (x) ((2nd o. 1st) (x) 2nd))" AddC )) (x) ran ((1st i^i ((`'2nd"{2c}) X. _V)) (x) AddC )) e. _V
126	125	rnex 5108	. . . 4 |- ran (`'ran ( Ins3 `'((ran (`'1st (x) (1st i^i 2nd)) (x) 2nd)" AddC ) i^i (((1st o. 1st) (x) ((2nd o. 1st) (x) 2nd))" AddC )) (x) ran ((1st i^i ((`'2nd"{2c}) X. _V)) (x) AddC )) e. _V
127	118, 126	unex 4107	. . 3 |- ((ran ( Ins3 `'((ran (`'1st (x) (1st i^i 2nd)) (x) 2nd)" AddC ) i^i (((1st o. 1st) (x) ((2nd o. 1st) (x) 2nd))" AddC )) u. ran (`'ran ( Ins3 `'((ran (`'1st (x) (1st i^i 2nd)) (x) 2nd)" AddC ) i^i (((1st o. 1st) (x) ((2nd o. 1st) (x) 2nd))" AddC )) (x) ran ((1st i^i ((`'2nd"{1c}) X. _V)) (x) AddC ))) u. ran (`'ran ( Ins3 `'((ran (`'1st (x) (1st i^i 2nd)) (x) 2nd)" AddC ) i^i (((1st o. 1st) (x) ((2nd o. 1st) (x) 2nd))" AddC )) (x) ran ((1st i^i ((`'2nd"{2c}) X. _V)) (x) AddC ))) e. _V
128		nncex 4397	. . 3 |- Nn e. _V
129	127, 128	imaex 4748	. 2 |- (((ran ( Ins3 `'((ran (`'1st (x) (1st i^i 2nd)) (x) 2nd)" AddC ) i^i (((1st o. 1st) (x) ((2nd o. 1st) (x) 2nd))" AddC )) u. ran (`'ran ( Ins3 `'((ran (`'1st (x) (1st i^i 2nd)) (x) 2nd)" AddC ) i^i (((1st o. 1st) (x) ((2nd o. 1st) (x) 2nd))" AddC )) (x) ran ((1st i^i ((`'2nd"{1c}) X. _V)) (x) AddC ))) u. ran (`'ran ( Ins3 `'((ran (`'1st (x) (1st i^i 2nd)) (x) 2nd)" AddC ) i^i (((1st o. 1st) (x) ((2nd o. 1st) (x) 2nd))" AddC )) (x) ran ((1st i^i ((`'2nd"{2c}) X. _V)) (x) AddC )))" Nn ) e. _V
130	88, 129	eqeltrri 2424	1 |- {a | E.n e. Nn (a = ((n +o n) +o n) \/ a = (((n +o n) +o n) +o 1c) \/ a = (((n +o n) +o n) +o 2c))} e. _V

Syntax hints:   \/ wo 357   /\ wa 358   \/ w3o 933  E.wex 1541   = wceq 1642   e. wcel 1710  {cab 2339  E.wrex 2616  _Vcvv 2860   u. cun 3208   i^i cin 3209  (/)c0 3551  {csn 3738  {cpr 3739  1_cc1c 4135   Nn cnnc 4374   +o cplc 4376  <.cop 4562   class class class wbr 4640  1stc1st 4718   o. ccom 4722  "cima 4723   X. cxp 4771  `'ccnv 4772  ran crn 4774  2ndc2nd 4784   (x) ctxp 5736   AddC caddcfn 5746   Ins3 cins3 5752   Nc cnc 6092  2_cc2c 6095

This theorem was proved from 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 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 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 referenced by:  nncdiv3  6278