Theorem nnc3n3p1 6279

Description: Three times a natural is not one more than three times a natural. Another part of Theorem 3.4 of [Specker] p. 973. (Contributed by SF, 13-Mar-2015.)

Assertion

Ref	Expression
nnc3n3p1	|- ((A e. Nn /\ B e. Nn ) -> -. ((A +o A) +o A) = (((B +o B) +o B) +o 1c))

Proof of Theorem nnc3n3p1

Dummy variables a m n p q x are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		vex 2863	. . . . . . . 8 |- a e. _V
2	1	elcompl 3226	. . . . . . 7 |- (a e. ∼ (ran (ran (( AddC o. `'(1st |` (`'2nd"{1c}))) (x) `'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 ( Ins3 `'((ran (`'1st (x) (1st i^i 2nd)) (x) 2nd)" AddC ) i^i (((1st o. 1st) (x) ((2nd o. 1st) (x) 2nd))" AddC )))" Nn ) <-> -. a e. (ran (ran (( AddC o. `'(1st |` (`'2nd"{1c}))) (x) `'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 ( Ins3 `'((ran (`'1st (x) (1st i^i 2nd)) (x) 2nd)" AddC ) i^i (((1st o. 1st) (x) ((2nd o. 1st) (x) 2nd))" AddC )))" Nn ))
3		elima 4755	. . . . . . . . 9 |- (a e. (ran (ran (( AddC o. `'(1st |` (`'2nd"{1c}))) (x) `'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 ( Ins3 `'((ran (`'1st (x) (1st i^i 2nd)) (x) 2nd)" AddC ) i^i (((1st o. 1st) (x) ((2nd o. 1st) (x) 2nd))" AddC )))" Nn ) <-> E.n e. Nn nran (ran (( AddC o. `'(1st |` (`'2nd"{1c}))) (x) `'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 ( Ins3 `'((ran (`'1st (x) (1st i^i 2nd)) (x) 2nd)" AddC ) i^i (((1st o. 1st) (x) ((2nd o. 1st) (x) 2nd))" AddC )))a)
4		df-br 4641	. . . . . . . . . . 11 |- (nran (ran (( AddC o. `'(1st |` (`'2nd"{1c}))) (x) `'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 ( 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, a>. e. ran (ran (( AddC o. `'(1st |` (`'2nd"{1c}))) (x) `'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 ( Ins3 `'((ran (`'1st (x) (1st i^i 2nd)) (x) 2nd)" AddC ) i^i (((1st o. 1st) (x) ((2nd o. 1st) (x) 2nd))" AddC ))))
5		elrn 4897	. . . . . . . . . . . 12 |- (<.n, a>. e. ran (ran (( AddC o. `'(1st |` (`'2nd"{1c}))) (x) `'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 ( Ins3 `'((ran (`'1st (x) (1st i^i 2nd)) (x) 2nd)" AddC ) i^i (((1st o. 1st) (x) ((2nd o. 1st) (x) 2nd))" AddC ))) <-> E.p p(ran (( AddC o. `'(1st |` (`'2nd"{1c}))) (x) `'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 ( 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>.)
6		df-br 4641	. . . . . . . . . . . . . . 15 |- (p(ran (( AddC o. `'(1st |` (`'2nd"{1c}))) (x) `'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 ( 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>. <-> <.p, <.n, a>.>. e. (ran (( AddC o. `'(1st |` (`'2nd"{1c}))) (x) `'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 ( Ins3 `'((ran (`'1st (x) (1st i^i 2nd)) (x) 2nd)" AddC ) i^i (((1st o. 1st) (x) ((2nd o. 1st) (x) 2nd))" AddC ))))
7		oteltxp 5783	. . . . . . . . . . . . . . 15 |- (<.p, <.n, a>.>. e. (ran (( AddC o. `'(1st |` (`'2nd"{1c}))) (x) `'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 ( Ins3 `'((ran (`'1st (x) (1st i^i 2nd)) (x) 2nd)" AddC ) i^i (((1st o. 1st) (x) ((2nd o. 1st) (x) 2nd))" AddC ))) <-> (<.p, n>. e. ran (( AddC o. `'(1st |` (`'2nd"{1c}))) (x) `'ran ( Ins3 `'((ran (`'1st (x) (1st i^i 2nd)) (x) 2nd)" AddC ) i^i (((1st o. 1st) (x) ((2nd o. 1st) (x) 2nd))" AddC ))) /\ <.p, 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 ))))
8	6, 7	bitri 240	. . . . . . . . . . . . . 14 |- (p(ran (( AddC o. `'(1st |` (`'2nd"{1c}))) (x) `'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 ( 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>. <-> (<.p, n>. e. ran (( AddC o. `'(1st |` (`'2nd"{1c}))) (x) `'ran ( Ins3 `'((ran (`'1st (x) (1st i^i 2nd)) (x) 2nd)" AddC ) i^i (((1st o. 1st) (x) ((2nd o. 1st) (x) 2nd))" AddC ))) /\ <.p, 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 ))))
9		elrn2 4898	. . . . . . . . . . . . . . . 16 |- (<.p, n>. e. ran (( AddC o. `'(1st |` (`'2nd"{1c}))) (x) `'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.q<.q, <.p, n>.>. e. (( AddC o. `'(1st |` (`'2nd"{1c}))) (x) `'ran ( Ins3 `'((ran (`'1st (x) (1st i^i 2nd)) (x) 2nd)" AddC ) i^i (((1st o. 1st) (x) ((2nd o. 1st) (x) 2nd))" AddC ))))
10		oteltxp 5783	. . . . . . . . . . . . . . . . . 18 |- (<.q, <.p, n>.>. e. (( AddC o. `'(1st |` (`'2nd"{1c}))) (x) `'ran ( Ins3 `'((ran (`'1st (x) (1st i^i 2nd)) (x) 2nd)" AddC ) i^i (((1st o. 1st) (x) ((2nd o. 1st) (x) 2nd))" AddC ))) <-> (<.q, p>. e. ( AddC o. `'(1st |` (`'2nd"{1c}))) /\ <.q, 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 ))))
11		opelco 4885	. . . . . . . . . . . . . . . . . . . 20 |- (<.q, p>. e. ( AddC o. `'(1st |` (`'2nd"{1c}))) <-> E.x(q`'(1st |` (`'2nd"{1c}))x /\ x AddC p))
12		brcnv 4893	. . . . . . . . . . . . . . . . . . . . . . . . 25 |- (q`'(1st |` (`'2nd"{1c}))x <-> x(1st |` (`'2nd"{1c}))q)
13		brres 4950	. . . . . . . . . . . . . . . . . . . . . . . . 25 |- (x(1st |` (`'2nd"{1c}))q <-> (x1stq /\ x e. (`'2nd"{1c})))
14	12, 13	bitri 240	. . . . . . . . . . . . . . . . . . . . . . . 24 |- (q`'(1st |` (`'2nd"{1c}))x <-> (x1stq /\ x e. (`'2nd"{1c})))
15		eliniseg 5021	. . . . . . . . . . . . . . . . . . . . . . . . 25 |- (x e. (`'2nd"{1c}) <-> x2nd1c)
16	15	anbi2i 675	. . . . . . . . . . . . . . . . . . . . . . . 24 |- ((x1stq /\ x e. (`'2nd"{1c})) <-> (x1stq /\ x2nd1c))
17		vex 2863	. . . . . . . . . . . . . . . . . . . . . . . . 25 |- q e. _V
18		1cex 4143	. . . . . . . . . . . . . . . . . . . . . . . . 25 |- 1c e. _V
19	17, 18	op1st2nd 5791	. . . . . . . . . . . . . . . . . . . . . . . 24 |- ((x1stq /\ x2nd1c) <-> x = <.q, 1c>.)
20	14, 16, 19	3bitri 262	. . . . . . . . . . . . . . . . . . . . . . 23 |- (q`'(1st |` (`'2nd"{1c}))x <-> x = <.q, 1c>.)
21	20	anbi1i 676	. . . . . . . . . . . . . . . . . . . . . 22 |- ((q`'(1st |` (`'2nd"{1c}))x /\ x AddC p) <-> (x = <.q, 1c>. /\ x AddC p))
22	21	exbii 1582	. . . . . . . . . . . . . . . . . . . . 21 |- (E.x(q`'(1st |` (`'2nd"{1c}))x /\ x AddC p) <-> E.x(x = <.q, 1c>. /\ x AddC p))
23	17, 18	opex 4589	. . . . . . . . . . . . . . . . . . . . . 22 |- <.q, 1c>. e. _V
24		breq1 4643	. . . . . . . . . . . . . . . . . . . . . 22 |- (x = <.q, 1c>. -> (x AddC p <-> <.q, 1c>. AddC p))
25	23, 24	ceqsexv 2895	. . . . . . . . . . . . . . . . . . . . 21 |- (E.x(x = <.q, 1c>. /\ x AddC p) <-> <.q, 1c>. AddC p)
26	22, 25	bitri 240	. . . . . . . . . . . . . . . . . . . 20 |- (E.x(q`'(1st |` (`'2nd"{1c}))x /\ x AddC p) <-> <.q, 1c>. AddC p)
27	17, 18	braddcfn 5827	. . . . . . . . . . . . . . . . . . . . 21 |- (<.q, 1c>. AddC p <-> (q +o 1c) = p)
28		eqcom 2355	. . . . . . . . . . . . . . . . . . . . 21 |- ((q +o 1c) = p <-> p = (q +o 1c))
29	27, 28	bitri 240	. . . . . . . . . . . . . . . . . . . 20 |- (<.q, 1c>. AddC p <-> p = (q +o 1c))
30	11, 26, 29	3bitri 262	. . . . . . . . . . . . . . . . . . 19 |- (<.q, p>. e. ( AddC o. `'(1st |` (`'2nd"{1c}))) <-> p = (q +o 1c))
31		opelcnv 4894	. . . . . . . . . . . . . . . . . . . 20 |- (<.q, 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, q>. 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 )))
32		nncdiv3lem1 6276	. . . . . . . . . . . . . . . . . . . 20 |- (<.n, q>. 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 )) <-> q = ((n +o n) +o n))
33	31, 32	bitri 240	. . . . . . . . . . . . . . . . . . 19 |- (<.q, 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 )) <-> q = ((n +o n) +o n))
34	30, 33	anbi12i 678	. . . . . . . . . . . . . . . . . 18 |- ((<.q, p>. e. ( AddC o. `'(1st |` (`'2nd"{1c}))) /\ <.q, 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 ))) <-> (p = (q +o 1c) /\ q = ((n +o n) +o n)))
35		ancom 437	. . . . . . . . . . . . . . . . . 18 |- ((p = (q +o 1c) /\ q = ((n +o n) +o n)) <-> (q = ((n +o n) +o n) /\ p = (q +o 1c)))
36	10, 34, 35	3bitri 262	. . . . . . . . . . . . . . . . 17 |- (<.q, <.p, n>.>. e. (( AddC o. `'(1st |` (`'2nd"{1c}))) (x) `'ran ( Ins3 `'((ran (`'1st (x) (1st i^i 2nd)) (x) 2nd)" AddC ) i^i (((1st o. 1st) (x) ((2nd o. 1st) (x) 2nd))" AddC ))) <-> (q = ((n +o n) +o n) /\ p = (q +o 1c)))
37	36	exbii 1582	. . . . . . . . . . . . . . . 16 |- (E.q<.q, <.p, n>.>. e. (( AddC o. `'(1st |` (`'2nd"{1c}))) (x) `'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.q(q = ((n +o n) +o n) /\ p = (q +o 1c)))
38		vex 2863	. . . . . . . . . . . . . . . . . . 19 |- n e. _V
39	38, 38	addcex 4395	. . . . . . . . . . . . . . . . . 18 |- (n +o n) e. _V
40	39, 38	addcex 4395	. . . . . . . . . . . . . . . . 17 |- ((n +o n) +o n) e. _V
41		addceq1 4384	. . . . . . . . . . . . . . . . . 18 |- (q = ((n +o n) +o n) -> (q +o 1c) = (((n +o n) +o n) +o 1c))
42	41	eqeq2d 2364	. . . . . . . . . . . . . . . . 17 |- (q = ((n +o n) +o n) -> (p = (q +o 1c) <-> p = (((n +o n) +o n) +o 1c)))
43	40, 42	ceqsexv 2895	. . . . . . . . . . . . . . . 16 |- (E.q(q = ((n +o n) +o n) /\ p = (q +o 1c)) <-> p = (((n +o n) +o n) +o 1c))
44	9, 37, 43	3bitri 262	. . . . . . . . . . . . . . 15 |- (<.p, n>. e. ran (( AddC o. `'(1st |` (`'2nd"{1c}))) (x) `'ran ( Ins3 `'((ran (`'1st (x) (1st i^i 2nd)) (x) 2nd)" AddC ) i^i (((1st o. 1st) (x) ((2nd o. 1st) (x) 2nd))" AddC ))) <-> p = (((n +o n) +o n) +o 1c))
45		opelcnv 4894	. . . . . . . . . . . . . . . 16 |- (<.p, 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, p>. 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 )))
46		nncdiv3lem1 6276	. . . . . . . . . . . . . . . 16 |- (<.a, p>. 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 )) <-> p = ((a +o a) +o a))
47	45, 46	bitri 240	. . . . . . . . . . . . . . 15 |- (<.p, 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 )) <-> p = ((a +o a) +o a))
48	44, 47	anbi12i 678	. . . . . . . . . . . . . 14 |- ((<.p, n>. e. ran (( AddC o. `'(1st |` (`'2nd"{1c}))) (x) `'ran ( Ins3 `'((ran (`'1st (x) (1st i^i 2nd)) (x) 2nd)" AddC ) i^i (((1st o. 1st) (x) ((2nd o. 1st) (x) 2nd))" AddC ))) /\ <.p, 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 ))) <-> (p = (((n +o n) +o n) +o 1c) /\ p = ((a +o a) +o a)))
49		ancom 437	. . . . . . . . . . . . . 14 |- ((p = (((n +o n) +o n) +o 1c) /\ p = ((a +o a) +o a)) <-> (p = ((a +o a) +o a) /\ p = (((n +o n) +o n) +o 1c)))
50	8, 48, 49	3bitri 262	. . . . . . . . . . . . 13 |- (p(ran (( AddC o. `'(1st |` (`'2nd"{1c}))) (x) `'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 ( 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>. <-> (p = ((a +o a) +o a) /\ p = (((n +o n) +o n) +o 1c)))
51	50	exbii 1582	. . . . . . . . . . . 12 |- (E.p p(ran (( AddC o. `'(1st |` (`'2nd"{1c}))) (x) `'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 ( 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.p(p = ((a +o a) +o a) /\ p = (((n +o n) +o n) +o 1c)))
52	1, 1	addcex 4395	. . . . . . . . . . . . . 14 |- (a +o a) e. _V
53	52, 1	addcex 4395	. . . . . . . . . . . . 13 |- ((a +o a) +o a) e. _V
54		eqeq1 2359	. . . . . . . . . . . . 13 |- (p = ((a +o a) +o a) -> (p = (((n +o n) +o n) +o 1c) <-> ((a +o a) +o a) = (((n +o n) +o n) +o 1c)))
55	53, 54	ceqsexv 2895	. . . . . . . . . . . 12 |- (E.p(p = ((a +o a) +o a) /\ p = (((n +o n) +o n) +o 1c)) <-> ((a +o a) +o a) = (((n +o n) +o n) +o 1c))
56	5, 51, 55	3bitri 262	. . . . . . . . . . 11 |- (<.n, a>. e. ran (ran (( AddC o. `'(1st |` (`'2nd"{1c}))) (x) `'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 ( Ins3 `'((ran (`'1st (x) (1st i^i 2nd)) (x) 2nd)" AddC ) i^i (((1st o. 1st) (x) ((2nd o. 1st) (x) 2nd))" AddC ))) <-> ((a +o a) +o a) = (((n +o n) +o n) +o 1c))
57	4, 56	bitri 240	. . . . . . . . . 10 |- (nran (ran (( AddC o. `'(1st |` (`'2nd"{1c}))) (x) `'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 ( Ins3 `'((ran (`'1st (x) (1st i^i 2nd)) (x) 2nd)" AddC ) i^i (((1st o. 1st) (x) ((2nd o. 1st) (x) 2nd))" AddC )))a <-> ((a +o a) +o a) = (((n +o n) +o n) +o 1c))
58	57	rexbii 2640	. . . . . . . . 9 |- (E.n e. Nn nran (ran (( AddC o. `'(1st |` (`'2nd"{1c}))) (x) `'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 ( Ins3 `'((ran (`'1st (x) (1st i^i 2nd)) (x) 2nd)" AddC ) i^i (((1st o. 1st) (x) ((2nd o. 1st) (x) 2nd))" AddC )))a <-> E.n e. Nn ((a +o a) +o a) = (((n +o n) +o n) +o 1c))
59		dfrex2 2628	. . . . . . . . 9 |- (E.n e. Nn ((a +o a) +o a) = (((n +o n) +o n) +o 1c) <-> -. A.n e. Nn -. ((a +o a) +o a) = (((n +o n) +o n) +o 1c))
60	3, 58, 59	3bitrri 263	. . . . . . . 8 |- (-. A.n e. Nn -. ((a +o a) +o a) = (((n +o n) +o n) +o 1c) <-> a e. (ran (ran (( AddC o. `'(1st |` (`'2nd"{1c}))) (x) `'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 ( Ins3 `'((ran (`'1st (x) (1st i^i 2nd)) (x) 2nd)" AddC ) i^i (((1st o. 1st) (x) ((2nd o. 1st) (x) 2nd))" AddC )))" Nn ))
61	60	con1bii 321	. . . . . . 7 |- (-. a e. (ran (ran (( AddC o. `'(1st |` (`'2nd"{1c}))) (x) `'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 ( Ins3 `'((ran (`'1st (x) (1st i^i 2nd)) (x) 2nd)" AddC ) i^i (((1st o. 1st) (x) ((2nd o. 1st) (x) 2nd))" AddC )))" Nn ) <-> A.n e. Nn -. ((a +o a) +o a) = (((n +o n) +o n) +o 1c))
62	2, 61	bitri 240	. . . . . 6 |- (a e. ∼ (ran (ran (( AddC o. `'(1st |` (`'2nd"{1c}))) (x) `'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 ( Ins3 `'((ran (`'1st (x) (1st i^i 2nd)) (x) 2nd)" AddC ) i^i (((1st o. 1st) (x) ((2nd o. 1st) (x) 2nd))" AddC )))" Nn ) <-> A.n e. Nn -. ((a +o a) +o a) = (((n +o n) +o n) +o 1c))
63	62	abbi2i 2465	. . . . 5 |- ∼ (ran (ran (( AddC o. `'(1st |` (`'2nd"{1c}))) (x) `'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 ( Ins3 `'((ran (`'1st (x) (1st i^i 2nd)) (x) 2nd)" AddC ) i^i (((1st o. 1st) (x) ((2nd o. 1st) (x) 2nd))" AddC )))" Nn ) = {a | A.n e. Nn -. ((a +o a) +o a) = (((n +o n) +o n) +o 1c)}
64		addcfnex 5825	. . . . . . . . . . . 12 |- AddC e. _V
65		1stex 4740	. . . . . . . . . . . . . 14 |- 1st e. _V
66		2ndex 5113	. . . . . . . . . . . . . . . 16 |- 2nd e. _V
67	66	cnvex 5103	. . . . . . . . . . . . . . 15 |- `'2nd e. _V
68		snex 4112	. . . . . . . . . . . . . . 15 |- {1c} e. _V
69	67, 68	imaex 4748	. . . . . . . . . . . . . 14 |- (`'2nd"{1c}) e. _V
70	65, 69	resex 5118	. . . . . . . . . . . . 13 |- (1st |` (`'2nd"{1c})) e. _V
71	70	cnvex 5103	. . . . . . . . . . . 12 |- `'(1st |` (`'2nd"{1c})) e. _V
72	64, 71	coex 4751	. . . . . . . . . . 11 |- ( AddC o. `'(1st |` (`'2nd"{1c}))) e. _V
73	65	cnvex 5103	. . . . . . . . . . . . . . . . . . . 20 |- `'1st e. _V
74	65, 66	inex 4106	. . . . . . . . . . . . . . . . . . . 20 |- (1st i^i 2nd) e. _V
75	73, 74	txpex 5786	. . . . . . . . . . . . . . . . . . 19 |- (`'1st (x) (1st i^i 2nd)) e. _V
76	75	rnex 5108	. . . . . . . . . . . . . . . . . 18 |- ran (`'1st (x) (1st i^i 2nd)) e. _V
77	76, 66	txpex 5786	. . . . . . . . . . . . . . . . 17 |- (ran (`'1st (x) (1st i^i 2nd)) (x) 2nd) e. _V
78	77, 64	imaex 4748	. . . . . . . . . . . . . . . 16 |- ((ran (`'1st (x) (1st i^i 2nd)) (x) 2nd)" AddC ) e. _V
79	78	cnvex 5103	. . . . . . . . . . . . . . 15 |- `'((ran (`'1st (x) (1st i^i 2nd)) (x) 2nd)" AddC ) e. _V
80	79	ins3ex 5799	. . . . . . . . . . . . . 14 |- Ins3 `'((ran (`'1st (x) (1st i^i 2nd)) (x) 2nd)" AddC ) e. _V
81	65, 65	coex 4751	. . . . . . . . . . . . . . . 16 |- (1st o. 1st) e. _V
82	66, 65	coex 4751	. . . . . . . . . . . . . . . . 17 |- (2nd o. 1st) e. _V
83	82, 66	txpex 5786	. . . . . . . . . . . . . . . 16 |- ((2nd o. 1st) (x) 2nd) e. _V
84	81, 83	txpex 5786	. . . . . . . . . . . . . . 15 |- ((1st o. 1st) (x) ((2nd o. 1st) (x) 2nd)) e. _V
85	84, 64	imaex 4748	. . . . . . . . . . . . . 14 |- (((1st o. 1st) (x) ((2nd o. 1st) (x) 2nd))" AddC ) e. _V
86	80, 85	inex 4106	. . . . . . . . . . . . 13 |- ( 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
87	86	rnex 5108	. . . . . . . . . . . 12 |- 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
88	87	cnvex 5103	. . . . . . . . . . 11 |- `'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
89	72, 88	txpex 5786	. . . . . . . . . 10 |- (( AddC o. `'(1st |` (`'2nd"{1c}))) (x) `'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
90	89	rnex 5108	. . . . . . . . 9 |- ran (( AddC o. `'(1st |` (`'2nd"{1c}))) (x) `'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
91	90, 88	txpex 5786	. . . . . . . 8 |- (ran (( AddC o. `'(1st |` (`'2nd"{1c}))) (x) `'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 ( 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
92	91	rnex 5108	. . . . . . 7 |- ran (ran (( AddC o. `'(1st |` (`'2nd"{1c}))) (x) `'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 ( 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
93		nncex 4397	. . . . . . 7 |- Nn e. _V
94	92, 93	imaex 4748	. . . . . 6 |- (ran (ran (( AddC o. `'(1st |` (`'2nd"{1c}))) (x) `'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 ( Ins3 `'((ran (`'1st (x) (1st i^i 2nd)) (x) 2nd)" AddC ) i^i (((1st o. 1st) (x) ((2nd o. 1st) (x) 2nd))" AddC )))" Nn ) e. _V
95	94	complex 4105	. . . . 5 |- ∼ (ran (ran (( AddC o. `'(1st |` (`'2nd"{1c}))) (x) `'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 ( Ins3 `'((ran (`'1st (x) (1st i^i 2nd)) (x) 2nd)" AddC ) i^i (((1st o. 1st) (x) ((2nd o. 1st) (x) 2nd))" AddC )))" Nn ) e. _V
96	63, 95	eqeltrri 2424	. . . 4 |- {a | A.n e. Nn -. ((a +o a) +o a) = (((n +o n) +o n) +o 1c)} e. _V
97		addceq12 4386	. . . . . . . . . 10 |- ((a = 0c /\ a = 0c) -> (a +o a) = (0c +o 0c))
98	97	anidms 626	. . . . . . . . 9 |- (a = 0c -> (a +o a) = (0c +o 0c))
99		id 19	. . . . . . . . 9 |- (a = 0c -> a = 0c)
100	98, 99	addceq12d 4392	. . . . . . . 8 |- (a = 0c -> ((a +o a) +o a) = ((0c +o 0c) +o 0c))
101		addcid1 4406	. . . . . . . . 9 |- ((0c +o 0c) +o 0c) = (0c +o 0c)
102		addcid2 4408	. . . . . . . . 9 |- (0c +o 0c) = 0c
103	101, 102	eqtri 2373	. . . . . . . 8 |- ((0c +o 0c) +o 0c) = 0c
104	100, 103	syl6eq 2401	. . . . . . 7 |- (a = 0c -> ((a +o a) +o a) = 0c)
105	104	eqeq1d 2361	. . . . . 6 |- (a = 0c -> (((a +o a) +o a) = (((n +o n) +o n) +o 1c) <-> 0c = (((n +o n) +o n) +o 1c)))
106	105	notbid 285	. . . . 5 |- (a = 0c -> (-. ((a +o a) +o a) = (((n +o n) +o n) +o 1c) <-> -. 0c = (((n +o n) +o n) +o 1c)))
107	106	ralbidv 2635	. . . 4 |- (a = 0c -> (A.n e. Nn -. ((a +o a) +o a) = (((n +o n) +o n) +o 1c) <-> A.n e. Nn -. 0c = (((n +o n) +o n) +o 1c)))
108		addceq12 4386	. . . . . . . . 9 |- ((a = m /\ a = m) -> (a +o a) = (m +o m))
109	108	anidms 626	. . . . . . . 8 |- (a = m -> (a +o a) = (m +o m))
110		id 19	. . . . . . . 8 |- (a = m -> a = m)
111	109, 110	addceq12d 4392	. . . . . . 7 |- (a = m -> ((a +o a) +o a) = ((m +o m) +o m))
112	111	eqeq1d 2361	. . . . . 6 |- (a = m -> (((a +o a) +o a) = (((n +o n) +o n) +o 1c) <-> ((m +o m) +o m) = (((n +o n) +o n) +o 1c)))
113	112	notbid 285	. . . . 5 |- (a = m -> (-. ((a +o a) +o a) = (((n +o n) +o n) +o 1c) <-> -. ((m +o m) +o m) = (((n +o n) +o n) +o 1c)))
114	113	ralbidv 2635	. . . 4 |- (a = m -> (A.n e. Nn -. ((a +o a) +o a) = (((n +o n) +o n) +o 1c) <-> A.n e. Nn -. ((m +o m) +o m) = (((n +o n) +o n) +o 1c)))
115		addceq12 4386	. . . . . . . . . 10 |- ((a = (m +o 1c) /\ a = (m +o 1c)) -> (a +o a) = ((m +o 1c) +o (m +o 1c)))
116	115	anidms 626	. . . . . . . . 9 |- (a = (m +o 1c) -> (a +o a) = ((m +o 1c) +o (m +o 1c)))
117		id 19	. . . . . . . . 9 |- (a = (m +o 1c) -> a = (m +o 1c))
118	116, 117	addceq12d 4392	. . . . . . . 8 |- (a = (m +o 1c) -> ((a +o a) +o a) = (((m +o 1c) +o (m +o 1c)) +o (m +o 1c)))
119	118	eqeq1d 2361	. . . . . . 7 |- (a = (m +o 1c) -> (((a +o a) +o a) = (((n +o n) +o n) +o 1c) <-> (((m +o 1c) +o (m +o 1c)) +o (m +o 1c)) = (((n +o n) +o n) +o 1c)))
120	119	notbid 285	. . . . . 6 |- (a = (m +o 1c) -> (-. ((a +o a) +o a) = (((n +o n) +o n) +o 1c) <-> -. (((m +o 1c) +o (m +o 1c)) +o (m +o 1c)) = (((n +o n) +o n) +o 1c)))
121	120	ralbidv 2635	. . . . 5 |- (a = (m +o 1c) -> (A.n e. Nn -. ((a +o a) +o a) = (((n +o n) +o n) +o 1c) <-> A.n e. Nn -. (((m +o 1c) +o (m +o 1c)) +o (m +o 1c)) = (((n +o n) +o n) +o 1c)))
122		addceq12 4386	. . . . . . . . . . 11 |- ((n = p /\ n = p) -> (n +o n) = (p +o p))
123	122	anidms 626	. . . . . . . . . 10 |- (n = p -> (n +o n) = (p +o p))
124		id 19	. . . . . . . . . 10 |- (n = p -> n = p)
125	123, 124	addceq12d 4392	. . . . . . . . 9 |- (n = p -> ((n +o n) +o n) = ((p +o p) +o p))
126	125	addceq1d 4390	. . . . . . . 8 |- (n = p -> (((n +o n) +o n) +o 1c) = (((p +o p) +o p) +o 1c))
127	126	eqeq2d 2364	. . . . . . 7 |- (n = p -> ((((m +o 1c) +o (m +o 1c)) +o (m +o 1c)) = (((n +o n) +o n) +o 1c) <-> (((m +o 1c) +o (m +o 1c)) +o (m +o 1c)) = (((p +o p) +o p) +o 1c)))
128	127	notbid 285	. . . . . 6 |- (n = p -> (-. (((m +o 1c) +o (m +o 1c)) +o (m +o 1c)) = (((n +o n) +o n) +o 1c) <-> -. (((m +o 1c) +o (m +o 1c)) +o (m +o 1c)) = (((p +o p) +o p) +o 1c)))
129	128	cbvralv 2836	. . . . 5 |- (A.n e. Nn -. (((m +o 1c) +o (m +o 1c)) +o (m +o 1c)) = (((n +o n) +o n) +o 1c) <-> A.p e. Nn -. (((m +o 1c) +o (m +o 1c)) +o (m +o 1c)) = (((p +o p) +o p) +o 1c))
130	121, 129	syl6bb 252	. . . 4 |- (a = (m +o 1c) -> (A.n e. Nn -. ((a +o a) +o a) = (((n +o n) +o n) +o 1c) <-> A.p e. Nn -. (((m +o 1c) +o (m +o 1c)) +o (m +o 1c)) = (((p +o p) +o p) +o 1c)))
131		addceq12 4386	. . . . . . . . 9 |- ((a = A /\ a = A) -> (a +o a) = (A +o A))
132	131	anidms 626	. . . . . . . 8 |- (a = A -> (a +o a) = (A +o A))
133		id 19	. . . . . . . 8 |- (a = A -> a = A)
134	132, 133	addceq12d 4392	. . . . . . 7 |- (a = A -> ((a +o a) +o a) = ((A +o A) +o A))
135	134	eqeq1d 2361	. . . . . 6 |- (a = A -> (((a +o a) +o a) = (((n +o n) +o n) +o 1c) <-> ((A +o A) +o A) = (((n +o n) +o n) +o 1c)))
136	135	notbid 285	. . . . 5 |- (a = A -> (-. ((a +o a) +o a) = (((n +o n) +o n) +o 1c) <-> -. ((A +o A) +o A) = (((n +o n) +o n) +o 1c)))
137	136	ralbidv 2635	. . . 4 |- (a = A -> (A.n e. Nn -. ((a +o a) +o a) = (((n +o n) +o n) +o 1c) <-> A.n e. Nn -. ((A +o A) +o A) = (((n +o n) +o n) +o 1c)))
138		1ne0c 6242	. . . . . . . 8 |- 1c =/= 0c
139		df-ne 2519	. . . . . . . 8 |- (1c =/= 0c <-> -. 1c = 0c)
140	138, 139	mpbi 199	. . . . . . 7 |- -. 1c = 0c
141	140	intnan 880	. . . . . 6 |- -. (((n +o n) +o n) = 0c /\ 1c = 0c)
142		eqcom 2355	. . . . . . 7 |- (0c = (((n +o n) +o n) +o 1c) <-> (((n +o n) +o n) +o 1c) = 0c)
143		nncaddccl 4420	. . . . . . . . . . 11 |- ((n e. Nn /\ n e. Nn ) -> (n +o n) e. Nn )
144	143	anidms 626	. . . . . . . . . 10 |- (n e. Nn -> (n +o n) e. Nn )
145		nncaddccl 4420	. . . . . . . . . 10 |- (((n +o n) e. Nn /\ n e. Nn ) -> ((n +o n) +o n) e. Nn )
146	144, 145	mpancom 650	. . . . . . . . 9 |- (n e. Nn -> ((n +o n) +o n) e. Nn )
147		nnnc 6147	. . . . . . . . 9 |- (((n +o n) +o n) e. Nn -> ((n +o n) +o n) e. NC )
148	146, 147	syl 15	. . . . . . . 8 |- (n e. Nn -> ((n +o n) +o n) e. NC )
149		1cnc 6140	. . . . . . . 8 |- 1c e. NC
150		addceq0 6220	. . . . . . . 8 |- ((((n +o n) +o n) e. NC /\ 1c e. NC ) -> ((((n +o n) +o n) +o 1c) = 0c <-> (((n +o n) +o n) = 0c /\ 1c = 0c)))
151	148, 149, 150	sylancl 643	. . . . . . 7 |- (n e. Nn -> ((((n +o n) +o n) +o 1c) = 0c <-> (((n +o n) +o n) = 0c /\ 1c = 0c)))
152	142, 151	syl5bb 248	. . . . . 6 |- (n e. Nn -> (0c = (((n +o n) +o n) +o 1c) <-> (((n +o n) +o n) = 0c /\ 1c = 0c)))
153	141, 152	mtbiri 294	. . . . 5 |- (n e. Nn -> -. 0c = (((n +o n) +o n) +o 1c))
154	153	rgen 2680	. . . 4 |- A.n e. Nn -. 0c = (((n +o n) +o n) +o 1c)
155		nnc0suc 4413	. . . . . . 7 |- (p e. Nn <-> (p = 0c \/ E.q e. Nn p = (q +o 1c)))
156		0cnsuc 4402	. . . . . . . . . . . . . . 15 |- ((((m +o 1c) +o m) +o m) +o 1c) =/= 0c
157		df-ne 2519	. . . . . . . . . . . . . . 15 |- (((((m +o 1c) +o m) +o m) +o 1c) =/= 0c <-> -. ((((m +o 1c) +o m) +o m) +o 1c) = 0c)
158	156, 157	mpbi 199	. . . . . . . . . . . . . 14 |- -. ((((m +o 1c) +o m) +o m) +o 1c) = 0c
159	158	a1i 10	. . . . . . . . . . . . 13 |- (m e. Nn -> -. ((((m +o 1c) +o m) +o m) +o 1c) = 0c)
160		addcass 4416	. . . . . . . . . . . . . . . 16 |- (((m +o 1c) +o m) +o 1c) = ((m +o 1c) +o (m +o 1c))
161	160	addceq1i 4387	. . . . . . . . . . . . . . 15 |- ((((m +o 1c) +o m) +o 1c) +o m) = (((m +o 1c) +o (m +o 1c)) +o m)
162		addc32 4417	. . . . . . . . . . . . . . 15 |- ((((m +o 1c) +o m) +o 1c) +o m) = ((((m +o 1c) +o m) +o m) +o 1c)
163	161, 162	eqtr3i 2375	. . . . . . . . . . . . . 14 |- (((m +o 1c) +o (m +o 1c)) +o m) = ((((m +o 1c) +o m) +o m) +o 1c)
164	163	eqeq1i 2360	. . . . . . . . . . . . 13 |- ((((m +o 1c) +o (m +o 1c)) +o m) = 0c <-> ((((m +o 1c) +o m) +o m) +o 1c) = 0c)
165	159, 164	sylnibr 296	. . . . . . . . . . . 12 |- (m e. Nn -> -. (((m +o 1c) +o (m +o 1c)) +o m) = 0c)
166		peano2 4404	. . . . . . . . . . . . . . 15 |- (m e. Nn -> (m +o 1c) e. Nn )
167		nncaddccl 4420	. . . . . . . . . . . . . . . 16 |- (((m +o 1c) e. Nn /\ (m +o 1c) e. Nn ) -> ((m +o 1c) +o (m +o 1c)) e. Nn )
168	167	anidms 626	. . . . . . . . . . . . . . 15 |- ((m +o 1c) e. Nn -> ((m +o 1c) +o (m +o 1c)) e. Nn )
169	166, 168	syl 15	. . . . . . . . . . . . . 14 |- (m e. Nn -> ((m +o 1c) +o (m +o 1c)) e. Nn )
170		nncaddccl 4420	. . . . . . . . . . . . . 14 |- ((((m +o 1c) +o (m +o 1c)) e. Nn /\ m e. Nn ) -> (((m +o 1c) +o (m +o 1c)) +o m) e. Nn )
171	169, 170	mpancom 650	. . . . . . . . . . . . 13 |- (m e. Nn -> (((m +o 1c) +o (m +o 1c)) +o m) e. Nn )
172		peano1 4403	. . . . . . . . . . . . 13 |- 0c e. Nn
173		suc11nnc 4559	. . . . . . . . . . . . 13 |- (((((m +o 1c) +o (m +o 1c)) +o m) e. Nn /\ 0c e. Nn ) -> (((((m +o 1c) +o (m +o 1c)) +o m) +o 1c) = (0c +o 1c) <-> (((m +o 1c) +o (m +o 1c)) +o m) = 0c))
174	171, 172, 173	sylancl 643	. . . . . . . . . . . 12 |- (m e. Nn -> (((((m +o 1c) +o (m +o 1c)) +o m) +o 1c) = (0c +o 1c) <-> (((m +o 1c) +o (m +o 1c)) +o m) = 0c))
175	165, 174	mtbird 292	. . . . . . . . . . 11 |- (m e. Nn -> -. ((((m +o 1c) +o (m +o 1c)) +o m) +o 1c) = (0c +o 1c))
176		addcass 4416	. . . . . . . . . . . 12 |- ((((m +o 1c) +o (m +o 1c)) +o m) +o 1c) = (((m +o 1c) +o (m +o 1c)) +o (m +o 1c))
177	176	eqeq1i 2360	. . . . . . . . . . 11 |- (((((m +o 1c) +o (m +o 1c)) +o m) +o 1c) = (0c +o 1c) <-> (((m +o 1c) +o (m +o 1c)) +o (m +o 1c)) = (0c +o 1c))
178	175, 177	sylnib 295	. . . . . . . . . 10 |- (m e. Nn -> -. (((m +o 1c) +o (m +o 1c)) +o (m +o 1c)) = (0c +o 1c))
179		addceq12 4386	. . . . . . . . . . . . . . . 16 |- ((p = 0c /\ p = 0c) -> (p +o p) = (0c +o 0c))
180	179	anidms 626	. . . . . . . . . . . . . . 15 |- (p = 0c -> (p +o p) = (0c +o 0c))
181		id 19	. . . . . . . . . . . . . . 15 |- (p = 0c -> p = 0c)
182	180, 181	addceq12d 4392	. . . . . . . . . . . . . 14 |- (p = 0c -> ((p +o p) +o p) = ((0c +o 0c) +o 0c))
183	182, 103	syl6eq 2401	. . . . . . . . . . . . 13 |- (p = 0c -> ((p +o p) +o p) = 0c)
184	183	addceq1d 4390	. . . . . . . . . . . 12 |- (p = 0c -> (((p +o p) +o p) +o 1c) = (0c +o 1c))
185	184	eqeq2d 2364	. . . . . . . . . . 11 |- (p = 0c -> ((((m +o 1c) +o (m +o 1c)) +o (m +o 1c)) = (((p +o p) +o p) +o 1c) <-> (((m +o 1c) +o (m +o 1c)) +o (m +o 1c)) = (0c +o 1c)))
186	185	notbid 285	. . . . . . . . . 10 |- (p = 0c -> (-. (((m +o 1c) +o (m +o 1c)) +o (m +o 1c)) = (((p +o p) +o p) +o 1c) <-> -. (((m +o 1c) +o (m +o 1c)) +o (m +o 1c)) = (0c +o 1c)))
187	178, 186	syl5ibrcom 213	. . . . . . . . 9 |- (m e. Nn -> (p = 0c -> -. (((m +o 1c) +o (m +o 1c)) +o (m +o 1c)) = (((p +o p) +o p) +o 1c)))
188	187	adantr 451	. . . . . . . 8 |- ((m e. Nn /\ A.n e. Nn -. ((m +o m) +o m) = (((n +o n) +o n) +o 1c)) -> (p = 0c -> -. (((m +o 1c) +o (m +o 1c)) +o (m +o 1c)) = (((p +o p) +o p) +o 1c)))
189		addceq12 4386	. . . . . . . . . . . . . . . . . . . 20 |- ((n = q /\ n = q) -> (n +o n) = (q +o q))
190	189	anidms 626	. . . . . . . . . . . . . . . . . . 19 |- (n = q -> (n +o n) = (q +o q))
191		id 19	. . . . . . . . . . . . . . . . . . 19 |- (n = q -> n = q)
192	190, 191	addceq12d 4392	. . . . . . . . . . . . . . . . . 18 |- (n = q -> ((n +o n) +o n) = ((q +o q) +o q))
193	192	addceq1d 4390	. . . . . . . . . . . . . . . . 17 |- (n = q -> (((n +o n) +o n) +o 1c) = (((q +o q) +o q) +o 1c))
194	193	eqeq2d 2364	. . . . . . . . . . . . . . . 16 |- (n = q -> (((m +o m) +o m) = (((n +o n) +o n) +o 1c) <-> ((m +o m) +o m) = (((q +o q) +o q) +o 1c)))
195	194	notbid 285	. . . . . . . . . . . . . . 15 |- (n = q -> (-. ((m +o m) +o m) = (((n +o n) +o n) +o 1c) <-> -. ((m +o m) +o m) = (((q +o q) +o q) +o 1c)))
196	195	rspcv 2952	. . . . . . . . . . . . . 14 |- (q e. Nn -> (A.n e. Nn -. ((m +o m) +o m) = (((n +o n) +o n) +o 1c) -> -. ((m +o m) +o m) = (((q +o q) +o q) +o 1c)))
197	196	adantl 452	. . . . . . . . . . . . 13 |- ((m e. Nn /\ q e. Nn ) -> (A.n e. Nn -. ((m +o m) +o m) = (((n +o n) +o n) +o 1c) -> -. ((m +o m) +o m) = (((q +o q) +o q) +o 1c)))
198		addc6 4419	. . . . . . . . . . . . . . . 16 |- (((m +o 1c) +o (m +o 1c)) +o (m +o 1c)) = (((m +o m) +o m) +o ((1c +o 1c) +o 1c))
199		addc6 4419	. . . . . . . . . . . . . . . . . 18 |- (((q +o 1c) +o (q +o 1c)) +o (q +o 1c)) = (((q +o q) +o q) +o ((1c +o 1c) +o 1c))
200	199	addceq1i 4387	. . . . . . . . . . . . . . . . 17 |- ((((q +o 1c) +o (q +o 1c)) +o (q +o 1c)) +o 1c) = ((((q +o q) +o q) +o ((1c +o 1c) +o 1c)) +o 1c)
201		addc32 4417	. . . . . . . . . . . . . . . . 17 |- ((((q +o q) +o q) +o ((1c +o 1c) +o 1c)) +o 1c) = ((((q +o q) +o q) +o 1c) +o ((1c +o 1c) +o 1c))
202	200, 201	eqtri 2373	. . . . . . . . . . . . . . . 16 |- ((((q +o 1c) +o (q +o 1c)) +o (q +o 1c)) +o 1c) = ((((q +o q) +o q) +o 1c) +o ((1c +o 1c) +o 1c))
203	198, 202	eqeq12i 2366	. . . . . . . . . . . . . . 15 |- ((((m +o 1c) +o (m +o 1c)) +o (m +o 1c)) = ((((q +o 1c) +o (q +o 1c)) +o (q +o 1c)) +o 1c) <-> (((m +o m) +o m) +o ((1c +o 1c) +o 1c)) = ((((q +o q) +o q) +o 1c) +o ((1c +o 1c) +o 1c)))
204		nncaddccl 4420	. . . . . . . . . . . . . . . . . 18 |- ((m e. Nn /\ m e. Nn ) -> (m +o m) e. Nn )
205	204	anidms 626	. . . . . . . . . . . . . . . . 17 |- (m e. Nn -> (m +o m) e. Nn )
206		nncaddccl 4420	. . . . . . . . . . . . . . . . 17 |- (((m +o m) e. Nn /\ m e. Nn ) -> ((m +o m) +o m) e. Nn )
207	205, 206	mpancom 650	. . . . . . . . . . . . . . . 16 |- (m e. Nn -> ((m +o m) +o m) e. Nn )
208		nncaddccl 4420	. . . . . . . . . . . . . . . . . . 19 |- ((q e. Nn /\ q e. Nn ) -> (q +o q) e. Nn )
209	208	anidms 626	. . . . . . . . . . . . . . . . . 18 |- (q e. Nn -> (q +o q) e. Nn )
210		nncaddccl 4420	. . . . . . . . . . . . . . . . . 18 |- (((q +o q) e. Nn /\ q e. Nn ) -> ((q +o q) +o q) e. Nn )
211	209, 210	mpancom 650	. . . . . . . . . . . . . . . . 17 |- (q e. Nn -> ((q +o q) +o q) e. Nn )
212		peano2 4404	. . . . . . . . . . . . . . . . 17 |- (((q +o q) +o q) e. Nn -> (((q +o q) +o q) +o 1c) e. Nn )
213	211, 212	syl 15	. . . . . . . . . . . . . . . 16 |- (q e. Nn -> (((q +o q) +o q) +o 1c) e. Nn )
214		1cnnc 4409	. . . . . . . . . . . . . . . . . . 19 |- 1c e. Nn
215		nncaddccl 4420	. . . . . . . . . . . . . . . . . . 19 |- ((1c e. Nn /\ 1c e. Nn ) -> (1c +o 1c) e. Nn )
216	214, 214, 215	mp2an 653	. . . . . . . . . . . . . . . . . 18 |- (1c +o 1c) e. Nn
217		nncaddccl 4420	. . . . . . . . . . . . . . . . . 18 |- (((1c +o 1c) e. Nn /\ 1c e. Nn ) -> ((1c +o 1c) +o 1c) e. Nn )
218	216, 214, 217	mp2an 653	. . . . . . . . . . . . . . . . 17 |- ((1c +o 1c) +o 1c) e. Nn
219		addccan1 4561	. . . . . . . . . . . . . . . . 17 |- ((((m +o m) +o m) e. Nn /\ (((q +o q) +o q) +o 1c) e. Nn /\ ((1c +o 1c) +o 1c) e. Nn ) -> ((((m +o m) +o m) +o ((1c +o 1c) +o 1c)) = ((((q +o q) +o q) +o 1c) +o ((1c +o 1c) +o 1c)) <-> ((m +o m) +o m) = (((q +o q) +o q) +o 1c)))
220	218, 219	mp3an3 1266	. . . . . . . . . . . . . . . 16 |- ((((m +o m) +o m) e. Nn /\ (((q +o q) +o q) +o 1c) e. Nn ) -> ((((m +o m) +o m) +o ((1c +o 1c) +o 1c)) = ((((q +o q) +o q) +o 1c) +o ((1c +o 1c) +o 1c)) <-> ((m +o m) +o m) = (((q +o q) +o q) +o 1c)))
221	207, 213, 220	syl2an 463	. . . . . . . . . . . . . . 15 |- ((m e. Nn /\ q e. Nn ) -> ((((m +o m) +o m) +o ((1c +o 1c) +o 1c)) = ((((q +o q) +o q) +o 1c) +o ((1c +o 1c) +o 1c)) <-> ((m +o m) +o m) = (((q +o q) +o q) +o 1c)))
222	203, 221	syl5bb 248	. . . . . . . . . . . . . 14 |- ((m e. Nn /\ q e. Nn ) -> ((((m +o 1c) +o (m +o 1c)) +o (m +o 1c)) = ((((q +o 1c) +o (q +o 1c)) +o (q +o 1c)) +o 1c) <-> ((m +o m) +o m) = (((q +o q) +o q) +o 1c)))
223	222	biimpd 198	. . . . . . . . . . . . 13 |- ((m e. Nn /\ q e. Nn ) -> ((((m +o 1c) +o (m +o 1c)) +o (m +o 1c)) = ((((q +o 1c) +o (q +o 1c)) +o (q +o 1c)) +o 1c) -> ((m +o m) +o m) = (((q +o q) +o q) +o 1c)))
224	197, 223	nsyld 132	. . . . . . . . . . . 12 |- ((m e. Nn /\ q e. Nn ) -> (A.n e. Nn -. ((m +o m) +o m) = (((n +o n) +o n) +o 1c) -> -. (((m +o 1c) +o (m +o 1c)) +o (m +o 1c)) = ((((q +o 1c) +o (q +o 1c)) +o (q +o 1c)) +o 1c)))
225	224	imp 418	. . . . . . . . . . 11 |- (((m e. Nn /\ q e. Nn ) /\ A.n e. Nn -. ((m +o m) +o m) = (((n +o n) +o n) +o 1c)) -> -. (((m +o 1c) +o (m +o 1c)) +o (m +o 1c)) = ((((q +o 1c) +o (q +o 1c)) +o (q +o 1c)) +o 1c))
226	225	an32s 779	. . . . . . . . . 10 |- (((m e. Nn /\ A.n e. Nn -. ((m +o m) +o m) = (((n +o n) +o n) +o 1c)) /\ q e. Nn ) -> -. (((m +o 1c) +o (m +o 1c)) +o (m +o 1c)) = ((((q +o 1c) +o (q +o 1c)) +o (q +o 1c)) +o 1c))
227		addceq12 4386	. . . . . . . . . . . . . . 15 |- ((p = (q +o 1c) /\ p = (q +o 1c)) -> (p +o p) = ((q +o 1c) +o (q +o 1c)))
228	227	anidms 626	. . . . . . . . . . . . . 14 |- (p = (q +o 1c) -> (p +o p) = ((q +o 1c) +o (q +o 1c)))
229		id 19	. . . . . . . . . . . . . 14 |- (p = (q +o 1c) -> p = (q +o 1c))
230	228, 229	addceq12d 4392	. . . . . . . . . . . . 13 |- (p = (q +o 1c) -> ((p +o p) +o p) = (((q +o 1c) +o (q +o 1c)) +o (q +o 1c)))
231	230	addceq1d 4390	. . . . . . . . . . . 12 |- (p = (q +o 1c) -> (((p +o p) +o p) +o 1c) = ((((q +o 1c) +o (q +o 1c)) +o (q +o 1c)) +o 1c))
232	231	eqeq2d 2364	. . . . . . . . . . 11 |- (p = (q +o 1c) -> ((((m +o 1c) +o (m +o 1c)) +o (m +o 1c)) = (((p +o p) +o p) +o 1c) <-> (((m +o 1c) +o (m +o 1c)) +o (m +o 1c)) = ((((q +o 1c) +o (q +o 1c)) +o (q +o 1c)) +o 1c)))
233	232	notbid 285	. . . . . . . . . 10 |- (p = (q +o 1c) -> (-. (((m +o 1c) +o (m +o 1c)) +o (m +o 1c)) = (((p +o p) +o p) +o 1c) <-> -. (((m +o 1c) +o (m +o 1c)) +o (m +o 1c)) = ((((q +o 1c) +o (q +o 1c)) +o (q +o 1c)) +o 1c)))
234	226, 233	syl5ibrcom 213	. . . . . . . . 9 |- (((m e. Nn /\ A.n e. Nn -. ((m +o m) +o m) = (((n +o n) +o n) +o 1c)) /\ q e. Nn ) -> (p = (q +o 1c) -> -. (((m +o 1c) +o (m +o 1c)) +o (m +o 1c)) = (((p +o p) +o p) +o 1c)))
235	234	rexlimdva 2739	. . . . . . . 8 |- ((m e. Nn /\ A.n e. Nn -. ((m +o m) +o m) = (((n +o n) +o n) +o 1c)) -> (E.q e. Nn p = (q +o 1c) -> -. (((m +o 1c) +o (m +o 1c)) +o (m +o 1c)) = (((p +o p) +o p) +o 1c)))
236	188, 235	jaod 369	. . . . . . 7 |- ((m e. Nn /\ A.n e. Nn -. ((m +o m) +o m) = (((n +o n) +o n) +o 1c)) -> ((p = 0c \/ E.q e. Nn p = (q +o 1c)) -> -. (((m +o 1c) +o (m +o 1c)) +o (m +o 1c)) = (((p +o p) +o p) +o 1c)))
237	155, 236	syl5bi 208	. . . . . 6 |- ((m e. Nn /\ A.n e. Nn -. ((m +o m) +o m) = (((n +o n) +o n) +o 1c)) -> (p e. Nn -> -. (((m +o 1c) +o (m +o 1c)) +o (m +o 1c)) = (((p +o p) +o p) +o 1c)))
238	237	ralrimiv 2697	. . . . 5 |- ((m e. Nn /\ A.n e. Nn -. ((m +o m) +o m) = (((n +o n) +o n) +o 1c)) -> A.p e. Nn -. (((m +o 1c) +o (m +o 1c)) +o (m +o 1c)) = (((p +o p) +o p) +o 1c))
239	238	ex 423	. . . 4 |- (m e. Nn -> (A.n e. Nn -. ((m +o m) +o m) = (((n +o n) +o n) +o 1c) -> A.p e. Nn -. (((m +o 1c) +o (m +o 1c)) +o (m +o 1c)) = (((p +o p) +o p) +o 1c)))
240	96, 107, 114, 130, 137, 154, 239	finds 4412	. . 3 |- (A e. Nn -> A.n e. Nn -. ((A +o A) +o A) = (((n +o n) +o n) +o 1c))
241		addceq12 4386	. . . . . . . . 9 |- ((n = B /\ n = B) -> (n +o n) = (B +o B))
242	241	anidms 626	. . . . . . . 8 |- (n = B -> (n +o n) = (B +o B))
243		id 19	. . . . . . . 8 |- (n = B -> n = B)
244	242, 243	addceq12d 4392	. . . . . . 7 |- (n = B -> ((n +o n) +o n) = ((B +o B) +o B))
245	244	addceq1d 4390	. . . . . 6 |- (n = B -> (((n +o n) +o n) +o 1c) = (((B +o B) +o B) +o 1c))
246	245	eqeq2d 2364	. . . . 5 |- (n = B -> (((A +o A) +o A) = (((n +o n) +o n) +o 1c) <-> ((A +o A) +o A) = (((B +o B) +o B) +o 1c)))
247	246	notbid 285	. . . 4 |- (n = B -> (-. ((A +o A) +o A) = (((n +o n) +o n) +o 1c) <-> -. ((A +o A) +o A) = (((B +o B) +o B) +o 1c)))
248	247	rspccv 2953	. . 3 |- (A.n e. Nn -. ((A +o A) +o A) = (((n +o n) +o n) +o 1c) -> (B e. Nn -> -. ((A +o A) +o A) = (((B +o B) +o B) +o 1c)))
249	240, 248	syl 15	. 2 |- (A e. Nn -> (B e. Nn -> -. ((A +o A) +o A) = (((B +o B) +o B) +o 1c)))
250	249	imp 418	1 |- ((A e. Nn /\ B e. Nn ) -> -. ((A +o A) +o A) = (((B +o B) +o B) +o 1c))

Syntax hints:  -. wn 3   -> wi 4   <-> wb 176   \/ wo 357   /\ wa 358  E.wex 1541   = wceq 1642   e. wcel 1710  {cab 2339   =/= wne 2517  A.wral 2615  E.wrex 2616  _Vcvv 2860   ∼ ccompl 3206   i^i cin 3209  {csn 3738  1_cc1c 4135   Nn cnnc 4374  0_cc0c 4375   +o cplc 4376  <.cop 4562   class class class wbr 4640  1stc1st 4718   o. ccom 4722  "cima 4723  `'ccnv 4772  ran crn 4774   |` cres 4775  2ndc2nd 4784   (x) ctxp 5736   AddC caddcfn 5746   Ins3 cins3 5752   NC cncs 6089

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-trans 5900  df-sym 5909  df-er 5910  df-ec 5948  df-qs 5952  df-en 6030  df-ncs 6099  df-lec 6100  df-nc 6102

This theorem is referenced by:  nnc3n3p2  6280  nnc3p1n3p2  6281  nchoicelem1  6290