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Theorem nnc3n3p1 6278
 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 Nn B Nn ) → ¬ ((A +c A) +c A) = (((B +c B) +c B) +c 1c))

Proof of Theorem nnc3n3p1
Dummy variables a m n p q x are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 vex 2862 . . . . . . . 8 a V
21elcompl 3225 . . . . . . 7 (a ∼ (ran (ran (( AddC (1st (2nd “ {1c}))) ⊗ ran ( Ins3 ((ran (1st ⊗ (1st ∩ 2nd )) ⊗ 2nd ) “ AddC ) ∩ (((1st 1st ) ⊗ ((2nd 1st ) ⊗ 2nd )) “ AddC ))) ⊗ ran ( Ins3 ((ran (1st ⊗ (1st ∩ 2nd )) ⊗ 2nd ) “ AddC ) ∩ (((1st 1st ) ⊗ ((2nd 1st ) ⊗ 2nd )) “ AddC ))) “ Nn ) ↔ ¬ a (ran (ran (( AddC (1st (2nd “ {1c}))) ⊗ ran ( Ins3 ((ran (1st ⊗ (1st ∩ 2nd )) ⊗ 2nd ) “ AddC ) ∩ (((1st 1st ) ⊗ ((2nd 1st ) ⊗ 2nd )) “ AddC ))) ⊗ ran ( Ins3 ((ran (1st ⊗ (1st ∩ 2nd )) ⊗ 2nd ) “ AddC ) ∩ (((1st 1st ) ⊗ ((2nd 1st ) ⊗ 2nd )) “ AddC ))) “ Nn ))
3 elima 4754 . . . . . . . . 9 (a (ran (ran (( AddC (1st (2nd “ {1c}))) ⊗ ran ( Ins3 ((ran (1st ⊗ (1st ∩ 2nd )) ⊗ 2nd ) “ AddC ) ∩ (((1st 1st ) ⊗ ((2nd 1st ) ⊗ 2nd )) “ AddC ))) ⊗ ran ( Ins3 ((ran (1st ⊗ (1st ∩ 2nd )) ⊗ 2nd ) “ AddC ) ∩ (((1st 1st ) ⊗ ((2nd 1st ) ⊗ 2nd )) “ AddC ))) “ Nn ) ↔ n Nn nran (ran (( AddC (1st (2nd “ {1c}))) ⊗ ran ( Ins3 ((ran (1st ⊗ (1st ∩ 2nd )) ⊗ 2nd ) “ AddC ) ∩ (((1st 1st ) ⊗ ((2nd 1st ) ⊗ 2nd )) “ AddC ))) ⊗ ran ( Ins3 ((ran (1st ⊗ (1st ∩ 2nd )) ⊗ 2nd ) “ AddC ) ∩ (((1st 1st ) ⊗ ((2nd 1st ) ⊗ 2nd )) “ AddC )))a)
4 df-br 4640 . . . . . . . . . . 11 (nran (ran (( AddC (1st (2nd “ {1c}))) ⊗ ran ( Ins3 ((ran (1st ⊗ (1st ∩ 2nd )) ⊗ 2nd ) “ AddC ) ∩ (((1st 1st ) ⊗ ((2nd 1st ) ⊗ 2nd )) “ AddC ))) ⊗ ran ( Ins3 ((ran (1st ⊗ (1st ∩ 2nd )) ⊗ 2nd ) “ AddC ) ∩ (((1st 1st ) ⊗ ((2nd 1st ) ⊗ 2nd )) “ AddC )))an, a ran (ran (( AddC (1st (2nd “ {1c}))) ⊗ ran ( Ins3 ((ran (1st ⊗ (1st ∩ 2nd )) ⊗ 2nd ) “ AddC ) ∩ (((1st 1st ) ⊗ ((2nd 1st ) ⊗ 2nd )) “ AddC ))) ⊗ ran ( Ins3 ((ran (1st ⊗ (1st ∩ 2nd )) ⊗ 2nd ) “ AddC ) ∩ (((1st 1st ) ⊗ ((2nd 1st ) ⊗ 2nd )) “ AddC ))))
5 elrn 4896 . . . . . . . . . . . 12 (n, a ran (ran (( AddC (1st (2nd “ {1c}))) ⊗ ran ( Ins3 ((ran (1st ⊗ (1st ∩ 2nd )) ⊗ 2nd ) “ AddC ) ∩ (((1st 1st ) ⊗ ((2nd 1st ) ⊗ 2nd )) “ AddC ))) ⊗ ran ( Ins3 ((ran (1st ⊗ (1st ∩ 2nd )) ⊗ 2nd ) “ AddC ) ∩ (((1st 1st ) ⊗ ((2nd 1st ) ⊗ 2nd )) “ AddC ))) ↔ p p(ran (( AddC (1st (2nd “ {1c}))) ⊗ ran ( Ins3 ((ran (1st ⊗ (1st ∩ 2nd )) ⊗ 2nd ) “ AddC ) ∩ (((1st 1st ) ⊗ ((2nd 1st ) ⊗ 2nd )) “ AddC ))) ⊗ ran ( Ins3 ((ran (1st ⊗ (1st ∩ 2nd )) ⊗ 2nd ) “ AddC ) ∩ (((1st 1st ) ⊗ ((2nd 1st ) ⊗ 2nd )) “ AddC )))n, a)
6 df-br 4640 . . . . . . . . . . . . . . 15 (p(ran (( AddC (1st (2nd “ {1c}))) ⊗ ran ( Ins3 ((ran (1st ⊗ (1st ∩ 2nd )) ⊗ 2nd ) “ AddC ) ∩ (((1st 1st ) ⊗ ((2nd 1st ) ⊗ 2nd )) “ AddC ))) ⊗ ran ( Ins3 ((ran (1st ⊗ (1st ∩ 2nd )) ⊗ 2nd ) “ AddC ) ∩ (((1st 1st ) ⊗ ((2nd 1st ) ⊗ 2nd )) “ AddC )))n, ap, n, a (ran (( AddC (1st (2nd “ {1c}))) ⊗ ran ( Ins3 ((ran (1st ⊗ (1st ∩ 2nd )) ⊗ 2nd ) “ AddC ) ∩ (((1st 1st ) ⊗ ((2nd 1st ) ⊗ 2nd )) “ AddC ))) ⊗ ran ( Ins3 ((ran (1st ⊗ (1st ∩ 2nd )) ⊗ 2nd ) “ AddC ) ∩ (((1st 1st ) ⊗ ((2nd 1st ) ⊗ 2nd )) “ AddC ))))
7 oteltxp 5782 . . . . . . . . . . . . . . 15 (p, n, a (ran (( AddC (1st (2nd “ {1c}))) ⊗ ran ( Ins3 ((ran (1st ⊗ (1st ∩ 2nd )) ⊗ 2nd ) “ AddC ) ∩ (((1st 1st ) ⊗ ((2nd 1st ) ⊗ 2nd )) “ AddC ))) ⊗ ran ( Ins3 ((ran (1st ⊗ (1st ∩ 2nd )) ⊗ 2nd ) “ AddC ) ∩ (((1st 1st ) ⊗ ((2nd 1st ) ⊗ 2nd )) “ AddC ))) ↔ (p, n ran (( AddC (1st (2nd “ {1c}))) ⊗ ran ( Ins3 ((ran (1st ⊗ (1st ∩ 2nd )) ⊗ 2nd ) “ AddC ) ∩ (((1st 1st ) ⊗ ((2nd 1st ) ⊗ 2nd )) “ AddC ))) p, a ran ( Ins3 ((ran (1st ⊗ (1st ∩ 2nd )) ⊗ 2nd ) “ AddC ) ∩ (((1st 1st ) ⊗ ((2nd 1st ) ⊗ 2nd )) “ AddC ))))
86, 7bitri 240 . . . . . . . . . . . . . 14 (p(ran (( AddC (1st (2nd “ {1c}))) ⊗ ran ( Ins3 ((ran (1st ⊗ (1st ∩ 2nd )) ⊗ 2nd ) “ AddC ) ∩ (((1st 1st ) ⊗ ((2nd 1st ) ⊗ 2nd )) “ AddC ))) ⊗ ran ( Ins3 ((ran (1st ⊗ (1st ∩ 2nd )) ⊗ 2nd ) “ AddC ) ∩ (((1st 1st ) ⊗ ((2nd 1st ) ⊗ 2nd )) “ AddC )))n, a ↔ (p, n ran (( AddC (1st (2nd “ {1c}))) ⊗ ran ( Ins3 ((ran (1st ⊗ (1st ∩ 2nd )) ⊗ 2nd ) “ AddC ) ∩ (((1st 1st ) ⊗ ((2nd 1st ) ⊗ 2nd )) “ AddC ))) p, a ran ( Ins3 ((ran (1st ⊗ (1st ∩ 2nd )) ⊗ 2nd ) “ AddC ) ∩ (((1st 1st ) ⊗ ((2nd 1st ) ⊗ 2nd )) “ AddC ))))
9 elrn2 4897 . . . . . . . . . . . . . . . 16 (p, n ran (( AddC (1st (2nd “ {1c}))) ⊗ ran ( Ins3 ((ran (1st ⊗ (1st ∩ 2nd )) ⊗ 2nd ) “ AddC ) ∩ (((1st 1st ) ⊗ ((2nd 1st ) ⊗ 2nd )) “ AddC ))) ↔ qq, p, n (( AddC (1st (2nd “ {1c}))) ⊗ ran ( Ins3 ((ran (1st ⊗ (1st ∩ 2nd )) ⊗ 2nd ) “ AddC ) ∩ (((1st 1st ) ⊗ ((2nd 1st ) ⊗ 2nd )) “ AddC ))))
10 oteltxp 5782 . . . . . . . . . . . . . . . . . 18 (q, p, n (( AddC (1st (2nd “ {1c}))) ⊗ ran ( Ins3 ((ran (1st ⊗ (1st ∩ 2nd )) ⊗ 2nd ) “ AddC ) ∩ (((1st 1st ) ⊗ ((2nd 1st ) ⊗ 2nd )) “ AddC ))) ↔ (q, p ( AddC (1st (2nd “ {1c}))) q, n ran ( Ins3 ((ran (1st ⊗ (1st ∩ 2nd )) ⊗ 2nd ) “ AddC ) ∩ (((1st 1st ) ⊗ ((2nd 1st ) ⊗ 2nd )) “ AddC ))))
11 opelco 4884 . . . . . . . . . . . . . . . . . . . 20 (q, p ( AddC (1st (2nd “ {1c}))) ↔ x(q(1st (2nd “ {1c}))x x AddC p))
12 brcnv 4892 . . . . . . . . . . . . . . . . . . . . . . . . 25 (q(1st (2nd “ {1c}))xx(1st (2nd “ {1c}))q)
13 brres 4949 . . . . . . . . . . . . . . . . . . . . . . . . 25 (x(1st (2nd “ {1c}))q ↔ (x1st q x (2nd “ {1c})))
1412, 13bitri 240 . . . . . . . . . . . . . . . . . . . . . . . 24 (q(1st (2nd “ {1c}))x ↔ (x1st q x (2nd “ {1c})))
15 eliniseg 5020 . . . . . . . . . . . . . . . . . . . . . . . . 25 (x (2nd “ {1c}) ↔ x2nd 1c)
1615anbi2i 675 . . . . . . . . . . . . . . . . . . . . . . . 24 ((x1st q x (2nd “ {1c})) ↔ (x1st q x2nd 1c))
17 vex 2862 . . . . . . . . . . . . . . . . . . . . . . . . 25 q V
18 1cex 4142 . . . . . . . . . . . . . . . . . . . . . . . . 25 1c V
1917, 18op1st2nd 5790 . . . . . . . . . . . . . . . . . . . . . . . 24 ((x1st q x2nd 1c) ↔ x = q, 1c)
2014, 16, 193bitri 262 . . . . . . . . . . . . . . . . . . . . . . 23 (q(1st (2nd “ {1c}))xx = q, 1c)
2120anbi1i 676 . . . . . . . . . . . . . . . . . . . . . 22 ((q(1st (2nd “ {1c}))x x AddC p) ↔ (x = q, 1c x AddC p))
2221exbii 1582 . . . . . . . . . . . . . . . . . . . . 21 (x(q(1st (2nd “ {1c}))x x AddC p) ↔ x(x = q, 1c x AddC p))
2317, 18opex 4588 . . . . . . . . . . . . . . . . . . . . . 22 q, 1c V
24 breq1 4642 . . . . . . . . . . . . . . . . . . . . . 22 (x = q, 1c → (x AddC pq, 1c AddC p))
2523, 24ceqsexv 2894 . . . . . . . . . . . . . . . . . . . . 21 (x(x = q, 1c x AddC p) ↔ q, 1c AddC p)
2622, 25bitri 240 . . . . . . . . . . . . . . . . . . . 20 (x(q(1st (2nd “ {1c}))x x AddC p) ↔ q, 1c AddC p)
2717, 18braddcfn 5826 . . . . . . . . . . . . . . . . . . . . 21 (q, 1c AddC p ↔ (q +c 1c) = p)
28 eqcom 2355 . . . . . . . . . . . . . . . . . . . . 21 ((q +c 1c) = pp = (q +c 1c))
2927, 28bitri 240 . . . . . . . . . . . . . . . . . . . 20 (q, 1c AddC pp = (q +c 1c))
3011, 26, 293bitri 262 . . . . . . . . . . . . . . . . . . 19 (q, p ( AddC (1st (2nd “ {1c}))) ↔ p = (q +c 1c))
31 opelcnv 4893 . . . . . . . . . . . . . . . . . . . 20 (q, n ran ( Ins3 ((ran (1st ⊗ (1st ∩ 2nd )) ⊗ 2nd ) “ AddC ) ∩ (((1st 1st ) ⊗ ((2nd 1st ) ⊗ 2nd )) “ AddC )) ↔ n, q ran ( Ins3 ((ran (1st ⊗ (1st ∩ 2nd )) ⊗ 2nd ) “ AddC ) ∩ (((1st 1st ) ⊗ ((2nd 1st ) ⊗ 2nd )) “ AddC )))
32 nncdiv3lem1 6275 . . . . . . . . . . . . . . . . . . . 20 (n, q ran ( Ins3 ((ran (1st ⊗ (1st ∩ 2nd )) ⊗ 2nd ) “ AddC ) ∩ (((1st 1st ) ⊗ ((2nd 1st ) ⊗ 2nd )) “ AddC )) ↔ q = ((n +c n) +c n))
3331, 32bitri 240 . . . . . . . . . . . . . . . . . . 19 (q, n ran ( Ins3 ((ran (1st ⊗ (1st ∩ 2nd )) ⊗ 2nd ) “ AddC ) ∩ (((1st 1st ) ⊗ ((2nd 1st ) ⊗ 2nd )) “ AddC )) ↔ q = ((n +c n) +c n))
3430, 33anbi12i 678 . . . . . . . . . . . . . . . . . 18 ((q, p ( AddC (1st (2nd “ {1c}))) q, n ran ( Ins3 ((ran (1st ⊗ (1st ∩ 2nd )) ⊗ 2nd ) “ AddC ) ∩ (((1st 1st ) ⊗ ((2nd 1st ) ⊗ 2nd )) “ AddC ))) ↔ (p = (q +c 1c) q = ((n +c n) +c n)))
35 ancom 437 . . . . . . . . . . . . . . . . . 18 ((p = (q +c 1c) q = ((n +c n) +c n)) ↔ (q = ((n +c n) +c n) p = (q +c 1c)))
3610, 34, 353bitri 262 . . . . . . . . . . . . . . . . 17 (q, p, n (( AddC (1st (2nd “ {1c}))) ⊗ ran ( Ins3 ((ran (1st ⊗ (1st ∩ 2nd )) ⊗ 2nd ) “ AddC ) ∩ (((1st 1st ) ⊗ ((2nd 1st ) ⊗ 2nd )) “ AddC ))) ↔ (q = ((n +c n) +c n) p = (q +c 1c)))
3736exbii 1582 . . . . . . . . . . . . . . . 16 (qq, p, n (( AddC (1st (2nd “ {1c}))) ⊗ ran ( Ins3 ((ran (1st ⊗ (1st ∩ 2nd )) ⊗ 2nd ) “ AddC ) ∩ (((1st 1st ) ⊗ ((2nd 1st ) ⊗ 2nd )) “ AddC ))) ↔ q(q = ((n +c n) +c n) p = (q +c 1c)))
38 vex 2862 . . . . . . . . . . . . . . . . . . 19 n V
3938, 38addcex 4394 . . . . . . . . . . . . . . . . . 18 (n +c n) V
4039, 38addcex 4394 . . . . . . . . . . . . . . . . 17 ((n +c n) +c n) V
41 addceq1 4383 . . . . . . . . . . . . . . . . . 18 (q = ((n +c n) +c n) → (q +c 1c) = (((n +c n) +c n) +c 1c))
4241eqeq2d 2364 . . . . . . . . . . . . . . . . 17 (q = ((n +c n) +c n) → (p = (q +c 1c) ↔ p = (((n +c n) +c n) +c 1c)))
4340, 42ceqsexv 2894 . . . . . . . . . . . . . . . 16 (q(q = ((n +c n) +c n) p = (q +c 1c)) ↔ p = (((n +c n) +c n) +c 1c))
449, 37, 433bitri 262 . . . . . . . . . . . . . . 15 (p, n ran (( AddC (1st (2nd “ {1c}))) ⊗ ran ( Ins3 ((ran (1st ⊗ (1st ∩ 2nd )) ⊗ 2nd ) “ AddC ) ∩ (((1st 1st ) ⊗ ((2nd 1st ) ⊗ 2nd )) “ AddC ))) ↔ p = (((n +c n) +c n) +c 1c))
45 opelcnv 4893 . . . . . . . . . . . . . . . 16 (p, a ran ( Ins3 ((ran (1st ⊗ (1st ∩ 2nd )) ⊗ 2nd ) “ AddC ) ∩ (((1st 1st ) ⊗ ((2nd 1st ) ⊗ 2nd )) “ AddC )) ↔ a, p ran ( Ins3 ((ran (1st ⊗ (1st ∩ 2nd )) ⊗ 2nd ) “ AddC ) ∩ (((1st 1st ) ⊗ ((2nd 1st ) ⊗ 2nd )) “ AddC )))
46 nncdiv3lem1 6275 . . . . . . . . . . . . . . . 16 (a, p ran ( Ins3 ((ran (1st ⊗ (1st ∩ 2nd )) ⊗ 2nd ) “ AddC ) ∩ (((1st 1st ) ⊗ ((2nd 1st ) ⊗ 2nd )) “ AddC )) ↔ p = ((a +c a) +c a))
4745, 46bitri 240 . . . . . . . . . . . . . . 15 (p, a ran ( Ins3 ((ran (1st ⊗ (1st ∩ 2nd )) ⊗ 2nd ) “ AddC ) ∩ (((1st 1st ) ⊗ ((2nd 1st ) ⊗ 2nd )) “ AddC )) ↔ p = ((a +c a) +c a))
4844, 47anbi12i 678 . . . . . . . . . . . . . 14 ((p, n ran (( AddC (1st (2nd “ {1c}))) ⊗ ran ( Ins3 ((ran (1st ⊗ (1st ∩ 2nd )) ⊗ 2nd ) “ AddC ) ∩ (((1st 1st ) ⊗ ((2nd 1st ) ⊗ 2nd )) “ AddC ))) p, a ran ( Ins3 ((ran (1st ⊗ (1st ∩ 2nd )) ⊗ 2nd ) “ AddC ) ∩ (((1st 1st ) ⊗ ((2nd 1st ) ⊗ 2nd )) “ AddC ))) ↔ (p = (((n +c n) +c n) +c 1c) p = ((a +c a) +c a)))
49 ancom 437 . . . . . . . . . . . . . 14 ((p = (((n +c n) +c n) +c 1c) p = ((a +c a) +c a)) ↔ (p = ((a +c a) +c a) p = (((n +c n) +c n) +c 1c)))
508, 48, 493bitri 262 . . . . . . . . . . . . 13 (p(ran (( AddC (1st (2nd “ {1c}))) ⊗ ran ( Ins3 ((ran (1st ⊗ (1st ∩ 2nd )) ⊗ 2nd ) “ AddC ) ∩ (((1st 1st ) ⊗ ((2nd 1st ) ⊗ 2nd )) “ AddC ))) ⊗ ran ( Ins3 ((ran (1st ⊗ (1st ∩ 2nd )) ⊗ 2nd ) “ AddC ) ∩ (((1st 1st ) ⊗ ((2nd 1st ) ⊗ 2nd )) “ AddC )))n, a ↔ (p = ((a +c a) +c a) p = (((n +c n) +c n) +c 1c)))
5150exbii 1582 . . . . . . . . . . . 12 (p p(ran (( AddC (1st (2nd “ {1c}))) ⊗ ran ( Ins3 ((ran (1st ⊗ (1st ∩ 2nd )) ⊗ 2nd ) “ AddC ) ∩ (((1st 1st ) ⊗ ((2nd 1st ) ⊗ 2nd )) “ AddC ))) ⊗ ran ( Ins3 ((ran (1st ⊗ (1st ∩ 2nd )) ⊗ 2nd ) “ AddC ) ∩ (((1st 1st ) ⊗ ((2nd 1st ) ⊗ 2nd )) “ AddC )))n, ap(p = ((a +c a) +c a) p = (((n +c n) +c n) +c 1c)))
521, 1addcex 4394 . . . . . . . . . . . . . 14 (a +c a) V
5352, 1addcex 4394 . . . . . . . . . . . . 13 ((a +c a) +c a) V
54 eqeq1 2359 . . . . . . . . . . . . 13 (p = ((a +c a) +c a) → (p = (((n +c n) +c n) +c 1c) ↔ ((a +c a) +c a) = (((n +c n) +c n) +c 1c)))
5553, 54ceqsexv 2894 . . . . . . . . . . . 12 (p(p = ((a +c a) +c a) p = (((n +c n) +c n) +c 1c)) ↔ ((a +c a) +c a) = (((n +c n) +c n) +c 1c))
565, 51, 553bitri 262 . . . . . . . . . . 11 (n, a ran (ran (( AddC (1st (2nd “ {1c}))) ⊗ ran ( Ins3 ((ran (1st ⊗ (1st ∩ 2nd )) ⊗ 2nd ) “ AddC ) ∩ (((1st 1st ) ⊗ ((2nd 1st ) ⊗ 2nd )) “ AddC ))) ⊗ ran ( Ins3 ((ran (1st ⊗ (1st ∩ 2nd )) ⊗ 2nd ) “ AddC ) ∩ (((1st 1st ) ⊗ ((2nd 1st ) ⊗ 2nd )) “ AddC ))) ↔ ((a +c a) +c a) = (((n +c n) +c n) +c 1c))
574, 56bitri 240 . . . . . . . . . 10 (nran (ran (( AddC (1st (2nd “ {1c}))) ⊗ ran ( Ins3 ((ran (1st ⊗ (1st ∩ 2nd )) ⊗ 2nd ) “ AddC ) ∩ (((1st 1st ) ⊗ ((2nd 1st ) ⊗ 2nd )) “ AddC ))) ⊗ ran ( Ins3 ((ran (1st ⊗ (1st ∩ 2nd )) ⊗ 2nd ) “ AddC ) ∩ (((1st 1st ) ⊗ ((2nd 1st ) ⊗ 2nd )) “ AddC )))a ↔ ((a +c a) +c a) = (((n +c n) +c n) +c 1c))
5857rexbii 2639 . . . . . . . . 9 (n Nn nran (ran (( AddC (1st (2nd “ {1c}))) ⊗ ran ( Ins3 ((ran (1st ⊗ (1st ∩ 2nd )) ⊗ 2nd ) “ AddC ) ∩ (((1st 1st ) ⊗ ((2nd 1st ) ⊗ 2nd )) “ AddC ))) ⊗ ran ( Ins3 ((ran (1st ⊗ (1st ∩ 2nd )) ⊗ 2nd ) “ AddC ) ∩ (((1st 1st ) ⊗ ((2nd 1st ) ⊗ 2nd )) “ AddC )))an Nn ((a +c a) +c a) = (((n +c n) +c n) +c 1c))
59 dfrex2 2627 . . . . . . . . 9 (n Nn ((a +c a) +c a) = (((n +c n) +c n) +c 1c) ↔ ¬ n Nn ¬ ((a +c a) +c a) = (((n +c n) +c n) +c 1c))
603, 58, 593bitrri 263 . . . . . . . 8 n Nn ¬ ((a +c a) +c a) = (((n +c n) +c n) +c 1c) ↔ a (ran (ran (( AddC (1st (2nd “ {1c}))) ⊗ ran ( Ins3 ((ran (1st ⊗ (1st ∩ 2nd )) ⊗ 2nd ) “ AddC ) ∩ (((1st 1st ) ⊗ ((2nd 1st ) ⊗ 2nd )) “ AddC ))) ⊗ ran ( Ins3 ((ran (1st ⊗ (1st ∩ 2nd )) ⊗ 2nd ) “ AddC ) ∩ (((1st 1st ) ⊗ ((2nd 1st ) ⊗ 2nd )) “ AddC ))) “ Nn ))
6160con1bii 321 . . . . . . 7 a (ran (ran (( AddC (1st (2nd “ {1c}))) ⊗ ran ( Ins3 ((ran (1st ⊗ (1st ∩ 2nd )) ⊗ 2nd ) “ AddC ) ∩ (((1st 1st ) ⊗ ((2nd 1st ) ⊗ 2nd )) “ AddC ))) ⊗ ran ( Ins3 ((ran (1st ⊗ (1st ∩ 2nd )) ⊗ 2nd ) “ AddC ) ∩ (((1st 1st ) ⊗ ((2nd 1st ) ⊗ 2nd )) “ AddC ))) “ Nn ) ↔ n Nn ¬ ((a +c a) +c a) = (((n +c n) +c n) +c 1c))
622, 61bitri 240 . . . . . 6 (a ∼ (ran (ran (( AddC (1st (2nd “ {1c}))) ⊗ ran ( Ins3 ((ran (1st ⊗ (1st ∩ 2nd )) ⊗ 2nd ) “ AddC ) ∩ (((1st 1st ) ⊗ ((2nd 1st ) ⊗ 2nd )) “ AddC ))) ⊗ ran ( Ins3 ((ran (1st ⊗ (1st ∩ 2nd )) ⊗ 2nd ) “ AddC ) ∩ (((1st 1st ) ⊗ ((2nd 1st ) ⊗ 2nd )) “ AddC ))) “ Nn ) ↔ n Nn ¬ ((a +c a) +c a) = (((n +c n) +c n) +c 1c))
6362abbi2i 2464 . . . . 5 ∼ (ran (ran (( AddC (1st (2nd “ {1c}))) ⊗ ran ( Ins3 ((ran (1st ⊗ (1st ∩ 2nd )) ⊗ 2nd ) “ AddC ) ∩ (((1st 1st ) ⊗ ((2nd 1st ) ⊗ 2nd )) “ AddC ))) ⊗ ran ( Ins3 ((ran (1st ⊗ (1st ∩ 2nd )) ⊗ 2nd ) “ AddC ) ∩ (((1st 1st ) ⊗ ((2nd 1st ) ⊗ 2nd )) “ AddC ))) “ Nn ) = {a n Nn ¬ ((a +c a) +c a) = (((n +c n) +c n) +c 1c)}
64 addcfnex 5824 . . . . . . . . . . . 12 AddC V
65 1stex 4739 . . . . . . . . . . . . . 14 1st V
66 2ndex 5112 . . . . . . . . . . . . . . . 16 2nd V
6766cnvex 5102 . . . . . . . . . . . . . . 15 2nd V
68 snex 4111 . . . . . . . . . . . . . . 15 {1c} V
6967, 68imaex 4747 . . . . . . . . . . . . . 14 (2nd “ {1c}) V
7065, 69resex 5117 . . . . . . . . . . . . 13 (1st (2nd “ {1c})) V
7170cnvex 5102 . . . . . . . . . . . 12 (1st (2nd “ {1c})) V
7264, 71coex 4750 . . . . . . . . . . 11 ( AddC (1st (2nd “ {1c}))) V
7365cnvex 5102 . . . . . . . . . . . . . . . . . . . 20 1st V
7465, 66inex 4105 . . . . . . . . . . . . . . . . . . . 20 (1st ∩ 2nd ) V
7573, 74txpex 5785 . . . . . . . . . . . . . . . . . . 19 (1st ⊗ (1st ∩ 2nd )) V
7675rnex 5107 . . . . . . . . . . . . . . . . . 18 ran (1st ⊗ (1st ∩ 2nd )) V
7776, 66txpex 5785 . . . . . . . . . . . . . . . . 17 (ran (1st ⊗ (1st ∩ 2nd )) ⊗ 2nd ) V
7877, 64imaex 4747 . . . . . . . . . . . . . . . 16 ((ran (1st ⊗ (1st ∩ 2nd )) ⊗ 2nd ) “ AddC ) V
7978cnvex 5102 . . . . . . . . . . . . . . 15 ((ran (1st ⊗ (1st ∩ 2nd )) ⊗ 2nd ) “ AddC ) V
8079ins3ex 5798 . . . . . . . . . . . . . 14 Ins3 ((ran (1st ⊗ (1st ∩ 2nd )) ⊗ 2nd ) “ AddC ) V
8165, 65coex 4750 . . . . . . . . . . . . . . . 16 (1st 1st ) V
8266, 65coex 4750 . . . . . . . . . . . . . . . . 17 (2nd 1st ) V
8382, 66txpex 5785 . . . . . . . . . . . . . . . 16 ((2nd 1st ) ⊗ 2nd ) V
8481, 83txpex 5785 . . . . . . . . . . . . . . 15 ((1st 1st ) ⊗ ((2nd 1st ) ⊗ 2nd )) V
8584, 64imaex 4747 . . . . . . . . . . . . . 14 (((1st 1st ) ⊗ ((2nd 1st ) ⊗ 2nd )) “ AddC ) V
8680, 85inex 4105 . . . . . . . . . . . . 13 ( Ins3 ((ran (1st ⊗ (1st ∩ 2nd )) ⊗ 2nd ) “ AddC ) ∩ (((1st 1st ) ⊗ ((2nd 1st ) ⊗ 2nd )) “ AddC )) V
8786rnex 5107 . . . . . . . . . . . 12 ran ( Ins3 ((ran (1st ⊗ (1st ∩ 2nd )) ⊗ 2nd ) “ AddC ) ∩ (((1st 1st ) ⊗ ((2nd 1st ) ⊗ 2nd )) “ AddC )) V
8887cnvex 5102 . . . . . . . . . . 11 ran ( Ins3 ((ran (1st ⊗ (1st ∩ 2nd )) ⊗ 2nd ) “ AddC ) ∩ (((1st 1st ) ⊗ ((2nd 1st ) ⊗ 2nd )) “ AddC )) V
8972, 88txpex 5785 . . . . . . . . . 10 (( AddC (1st (2nd “ {1c}))) ⊗ ran ( Ins3 ((ran (1st ⊗ (1st ∩ 2nd )) ⊗ 2nd ) “ AddC ) ∩ (((1st 1st ) ⊗ ((2nd 1st ) ⊗ 2nd )) “ AddC ))) V
9089rnex 5107 . . . . . . . . 9 ran (( AddC (1st (2nd “ {1c}))) ⊗ ran ( Ins3 ((ran (1st ⊗ (1st ∩ 2nd )) ⊗ 2nd ) “ AddC ) ∩ (((1st 1st ) ⊗ ((2nd 1st ) ⊗ 2nd )) “ AddC ))) V
9190, 88txpex 5785 . . . . . . . 8 (ran (( AddC (1st (2nd “ {1c}))) ⊗ ran ( Ins3 ((ran (1st ⊗ (1st ∩ 2nd )) ⊗ 2nd ) “ AddC ) ∩ (((1st 1st ) ⊗ ((2nd 1st ) ⊗ 2nd )) “ AddC ))) ⊗ ran ( Ins3 ((ran (1st ⊗ (1st ∩ 2nd )) ⊗ 2nd ) “ AddC ) ∩ (((1st 1st ) ⊗ ((2nd 1st ) ⊗ 2nd )) “ AddC ))) V
9291rnex 5107 . . . . . . 7 ran (ran (( AddC (1st (2nd “ {1c}))) ⊗ ran ( Ins3 ((ran (1st ⊗ (1st ∩ 2nd )) ⊗ 2nd ) “ AddC ) ∩ (((1st 1st ) ⊗ ((2nd 1st ) ⊗ 2nd )) “ AddC ))) ⊗ ran ( Ins3 ((ran (1st ⊗ (1st ∩ 2nd )) ⊗ 2nd ) “ AddC ) ∩ (((1st 1st ) ⊗ ((2nd 1st ) ⊗ 2nd )) “ AddC ))) V
93 nncex 4396 . . . . . . 7 Nn V
9492, 93imaex 4747 . . . . . 6 (ran (ran (( AddC (1st (2nd “ {1c}))) ⊗ ran ( Ins3 ((ran (1st ⊗ (1st ∩ 2nd )) ⊗ 2nd ) “ AddC ) ∩ (((1st 1st ) ⊗ ((2nd 1st ) ⊗ 2nd )) “ AddC ))) ⊗ ran ( Ins3 ((ran (1st ⊗ (1st ∩ 2nd )) ⊗ 2nd ) “ AddC ) ∩ (((1st 1st ) ⊗ ((2nd 1st ) ⊗ 2nd )) “ AddC ))) “ Nn ) V
9594complex 4104 . . . . 5 ∼ (ran (ran (( AddC (1st (2nd “ {1c}))) ⊗ ran ( Ins3 ((ran (1st ⊗ (1st ∩ 2nd )) ⊗ 2nd ) “ AddC ) ∩ (((1st 1st ) ⊗ ((2nd 1st ) ⊗ 2nd )) “ AddC ))) ⊗ ran ( Ins3 ((ran (1st ⊗ (1st ∩ 2nd )) ⊗ 2nd ) “ AddC ) ∩ (((1st 1st ) ⊗ ((2nd 1st ) ⊗ 2nd )) “ AddC ))) “ Nn ) V
9663, 95eqeltrri 2424 . . . 4 {a n Nn ¬ ((a +c a) +c a) = (((n +c n) +c n) +c 1c)} V
97 addceq12 4385 . . . . . . . . . 10 ((a = 0c a = 0c) → (a +c a) = (0c +c 0c))
9897anidms 626 . . . . . . . . 9 (a = 0c → (a +c a) = (0c +c 0c))
99 id 19 . . . . . . . . 9 (a = 0ca = 0c)
10098, 99addceq12d 4391 . . . . . . . 8 (a = 0c → ((a +c a) +c a) = ((0c +c 0c) +c 0c))
101 addcid1 4405 . . . . . . . . 9 ((0c +c 0c) +c 0c) = (0c +c 0c)
102 addcid2 4407 . . . . . . . . 9 (0c +c 0c) = 0c
103101, 102eqtri 2373 . . . . . . . 8 ((0c +c 0c) +c 0c) = 0c
104100, 103syl6eq 2401 . . . . . . 7 (a = 0c → ((a +c a) +c a) = 0c)
105104eqeq1d 2361 . . . . . 6 (a = 0c → (((a +c a) +c a) = (((n +c n) +c n) +c 1c) ↔ 0c = (((n +c n) +c n) +c 1c)))
106105notbid 285 . . . . 5 (a = 0c → (¬ ((a +c a) +c a) = (((n +c n) +c n) +c 1c) ↔ ¬ 0c = (((n +c n) +c n) +c 1c)))
107106ralbidv 2634 . . . 4 (a = 0c → (n Nn ¬ ((a +c a) +c a) = (((n +c n) +c n) +c 1c) ↔ n Nn ¬ 0c = (((n +c n) +c n) +c 1c)))
108 addceq12 4385 . . . . . . . . 9 ((a = m a = m) → (a +c a) = (m +c m))
109108anidms 626 . . . . . . . 8 (a = m → (a +c a) = (m +c m))
110 id 19 . . . . . . . 8 (a = ma = m)
111109, 110addceq12d 4391 . . . . . . 7 (a = m → ((a +c a) +c a) = ((m +c m) +c m))
112111eqeq1d 2361 . . . . . 6 (a = m → (((a +c a) +c a) = (((n +c n) +c n) +c 1c) ↔ ((m +c m) +c m) = (((n +c n) +c n) +c 1c)))
113112notbid 285 . . . . 5 (a = m → (¬ ((a +c a) +c a) = (((n +c n) +c n) +c 1c) ↔ ¬ ((m +c m) +c m) = (((n +c n) +c n) +c 1c)))
114113ralbidv 2634 . . . 4 (a = m → (n Nn ¬ ((a +c a) +c a) = (((n +c n) +c n) +c 1c) ↔ n Nn ¬ ((m +c m) +c m) = (((n +c n) +c n) +c 1c)))
115 addceq12 4385 . . . . . . . . . 10 ((a = (m +c 1c) a = (m +c 1c)) → (a +c a) = ((m +c 1c) +c (m +c 1c)))
116115anidms 626 . . . . . . . . 9 (a = (m +c 1c) → (a +c a) = ((m +c 1c) +c (m +c 1c)))
117 id 19 . . . . . . . . 9 (a = (m +c 1c) → a = (m +c 1c))
118116, 117addceq12d 4391 . . . . . . . 8 (a = (m +c 1c) → ((a +c a) +c a) = (((m +c 1c) +c (m +c 1c)) +c (m +c 1c)))
119118eqeq1d 2361 . . . . . . 7 (a = (m +c 1c) → (((a +c a) +c a) = (((n +c n) +c n) +c 1c) ↔ (((m +c 1c) +c (m +c 1c)) +c (m +c 1c)) = (((n +c n) +c n) +c 1c)))
120119notbid 285 . . . . . 6 (a = (m +c 1c) → (¬ ((a +c a) +c a) = (((n +c n) +c n) +c 1c) ↔ ¬ (((m +c 1c) +c (m +c 1c)) +c (m +c 1c)) = (((n +c n) +c n) +c 1c)))
121120ralbidv 2634 . . . . 5 (a = (m +c 1c) → (n Nn ¬ ((a +c a) +c a) = (((n +c n) +c n) +c 1c) ↔ n Nn ¬ (((m +c 1c) +c (m +c 1c)) +c (m +c 1c)) = (((n +c n) +c n) +c 1c)))
122 addceq12 4385 . . . . . . . . . . 11 ((n = p n = p) → (n +c n) = (p +c p))
123122anidms 626 . . . . . . . . . 10 (n = p → (n +c n) = (p +c p))
124 id 19 . . . . . . . . . 10 (n = pn = p)
125123, 124addceq12d 4391 . . . . . . . . 9 (n = p → ((n +c n) +c n) = ((p +c p) +c p))
126125addceq1d 4389 . . . . . . . 8 (n = p → (((n +c n) +c n) +c 1c) = (((p +c p) +c p) +c 1c))
127126eqeq2d 2364 . . . . . . 7 (n = p → ((((m +c 1c) +c (m +c 1c)) +c (m +c 1c)) = (((n +c n) +c n) +c 1c) ↔ (((m +c 1c) +c (m +c 1c)) +c (m +c 1c)) = (((p +c p) +c p) +c 1c)))
128127notbid 285 . . . . . 6 (n = p → (¬ (((m +c 1c) +c (m +c 1c)) +c (m +c 1c)) = (((n +c n) +c n) +c 1c) ↔ ¬ (((m +c 1c) +c (m +c 1c)) +c (m +c 1c)) = (((p +c p) +c p) +c 1c)))
129128cbvralv 2835 . . . . 5 (n Nn ¬ (((m +c 1c) +c (m +c 1c)) +c (m +c 1c)) = (((n +c n) +c n) +c 1c) ↔ p Nn ¬ (((m +c 1c) +c (m +c 1c)) +c (m +c 1c)) = (((p +c p) +c p) +c 1c))
130121, 129syl6bb 252 . . . 4 (a = (m +c 1c) → (n Nn ¬ ((a +c a) +c a) = (((n +c n) +c n) +c 1c) ↔ p Nn ¬ (((m +c 1c) +c (m +c 1c)) +c (m +c 1c)) = (((p +c p) +c p) +c 1c)))
131 addceq12 4385 . . . . . . . . 9 ((a = A a = A) → (a +c a) = (A +c A))
132131anidms 626 . . . . . . . 8 (a = A → (a +c a) = (A +c A))
133 id 19 . . . . . . . 8 (a = Aa = A)
134132, 133addceq12d 4391 . . . . . . 7 (a = A → ((a +c a) +c a) = ((A +c A) +c A))
135134eqeq1d 2361 . . . . . 6 (a = A → (((a +c a) +c a) = (((n +c n) +c n) +c 1c) ↔ ((A +c A) +c A) = (((n +c n) +c n) +c 1c)))
136135notbid 285 . . . . 5 (a = A → (¬ ((a +c a) +c a) = (((n +c n) +c n) +c 1c) ↔ ¬ ((A +c A) +c A) = (((n +c n) +c n) +c 1c)))
137136ralbidv 2634 . . . 4 (a = A → (n Nn ¬ ((a +c a) +c a) = (((n +c n) +c n) +c 1c) ↔ n Nn ¬ ((A +c A) +c A) = (((n +c n) +c n) +c 1c)))
138 1ne0c 6241 . . . . . . . 8 1c ≠ 0c
139 df-ne 2518 . . . . . . . 8 (1c ≠ 0c ↔ ¬ 1c = 0c)
140138, 139mpbi 199 . . . . . . 7 ¬ 1c = 0c
141140intnan 880 . . . . . 6 ¬ (((n +c n) +c n) = 0c 1c = 0c)
142 eqcom 2355 . . . . . . 7 (0c = (((n +c n) +c n) +c 1c) ↔ (((n +c n) +c n) +c 1c) = 0c)
143 nncaddccl 4419 . . . . . . . . . . 11 ((n Nn n Nn ) → (n +c n) Nn )
144143anidms 626 . . . . . . . . . 10 (n Nn → (n +c n) Nn )
145 nncaddccl 4419 . . . . . . . . . 10 (((n +c n) Nn n Nn ) → ((n +c n) +c n) Nn )
146144, 145mpancom 650 . . . . . . . . 9 (n Nn → ((n +c n) +c n) Nn )
147 nnnc 6146 . . . . . . . . 9 (((n +c n) +c n) Nn → ((n +c n) +c n) NC )
148146, 147syl 15 . . . . . . . 8 (n Nn → ((n +c n) +c n) NC )
149 1cnc 6139 . . . . . . . 8 1c NC
150 addceq0 6219 . . . . . . . 8 ((((n +c n) +c n) NC 1c NC ) → ((((n +c n) +c n) +c 1c) = 0c ↔ (((n +c n) +c n) = 0c 1c = 0c)))
151148, 149, 150sylancl 643 . . . . . . 7 (n Nn → ((((n +c n) +c n) +c 1c) = 0c ↔ (((n +c n) +c n) = 0c 1c = 0c)))
152142, 151syl5bb 248 . . . . . 6 (n Nn → (0c = (((n +c n) +c n) +c 1c) ↔ (((n +c n) +c n) = 0c 1c = 0c)))
153141, 152mtbiri 294 . . . . 5 (n Nn → ¬ 0c = (((n +c n) +c n) +c 1c))
154153rgen 2679 . . . 4 n Nn ¬ 0c = (((n +c n) +c n) +c 1c)
155 nnc0suc 4412 . . . . . . 7 (p Nn ↔ (p = 0c q Nn p = (q +c 1c)))
156 0cnsuc 4401 . . . . . . . . . . . . . . 15 ((((m +c 1c) +c m) +c m) +c 1c) ≠ 0c
157 df-ne 2518 . . . . . . . . . . . . . . 15 (((((m +c 1c) +c m) +c m) +c 1c) ≠ 0c ↔ ¬ ((((m +c 1c) +c m) +c m) +c 1c) = 0c)
158156, 157mpbi 199 . . . . . . . . . . . . . 14 ¬ ((((m +c 1c) +c m) +c m) +c 1c) = 0c
159158a1i 10 . . . . . . . . . . . . 13 (m Nn → ¬ ((((m +c 1c) +c m) +c m) +c 1c) = 0c)
160 addcass 4415 . . . . . . . . . . . . . . . 16 (((m +c 1c) +c m) +c 1c) = ((m +c 1c) +c (m +c 1c))
161160addceq1i 4386 . . . . . . . . . . . . . . 15 ((((m +c 1c) +c m) +c 1c) +c m) = (((m +c 1c) +c (m +c 1c)) +c m)
162 addc32 4416 . . . . . . . . . . . . . . 15 ((((m +c 1c) +c m) +c 1c) +c m) = ((((m +c 1c) +c m) +c m) +c 1c)
163161, 162eqtr3i 2375 . . . . . . . . . . . . . 14 (((m +c 1c) +c (m +c 1c)) +c m) = ((((m +c 1c) +c m) +c m) +c 1c)
164163eqeq1i 2360 . . . . . . . . . . . . 13 ((((m +c 1c) +c (m +c 1c)) +c m) = 0c ↔ ((((m +c 1c) +c m) +c m) +c 1c) = 0c)
165159, 164sylnibr 296 . . . . . . . . . . . 12 (m Nn → ¬ (((m +c 1c) +c (m +c 1c)) +c m) = 0c)
166 peano2 4403 . . . . . . . . . . . . . . 15 (m Nn → (m +c 1c) Nn )
167 nncaddccl 4419 . . . . . . . . . . . . . . . 16 (((m +c 1c) Nn (m +c 1c) Nn ) → ((m +c 1c) +c (m +c 1c)) Nn )
168167anidms 626 . . . . . . . . . . . . . . 15 ((m +c 1c) Nn → ((m +c 1c) +c (m +c 1c)) Nn )
169166, 168syl 15 . . . . . . . . . . . . . 14 (m Nn → ((m +c 1c) +c (m +c 1c)) Nn )
170 nncaddccl 4419 . . . . . . . . . . . . . 14 ((((m +c 1c) +c (m +c 1c)) Nn m Nn ) → (((m +c 1c) +c (m +c 1c)) +c m) Nn )
171169, 170mpancom 650 . . . . . . . . . . . . 13 (m Nn → (((m +c 1c) +c (m +c 1c)) +c m) Nn )
172 peano1 4402 . . . . . . . . . . . . 13 0c Nn
173 suc11nnc 4558 . . . . . . . . . . . . 13 (((((m +c 1c) +c (m +c 1c)) +c m) Nn 0c Nn ) → (((((m +c 1c) +c (m +c 1c)) +c m) +c 1c) = (0c +c 1c) ↔ (((m +c 1c) +c (m +c 1c)) +c m) = 0c))
174171, 172, 173sylancl 643 . . . . . . . . . . . 12 (m Nn → (((((m +c 1c) +c (m +c 1c)) +c m) +c 1c) = (0c +c 1c) ↔ (((m +c 1c) +c (m +c 1c)) +c m) = 0c))
175165, 174mtbird 292 . . . . . . . . . . 11 (m Nn → ¬ ((((m +c 1c) +c (m +c 1c)) +c m) +c 1c) = (0c +c 1c))
176 addcass 4415 . . . . . . . . . . . 12 ((((m +c 1c) +c (m +c 1c)) +c m) +c 1c) = (((m +c 1c) +c (m +c 1c)) +c (m +c 1c))
177176eqeq1i 2360 . . . . . . . . . . 11 (((((m +c 1c) +c (m +c 1c)) +c m) +c 1c) = (0c +c 1c) ↔ (((m +c 1c) +c (m +c 1c)) +c (m +c 1c)) = (0c +c 1c))
178175, 177sylnib 295 . . . . . . . . . 10 (m Nn → ¬ (((m +c 1c) +c (m +c 1c)) +c (m +c 1c)) = (0c +c 1c))
179 addceq12 4385 . . . . . . . . . . . . . . . 16 ((p = 0c p = 0c) → (p +c p) = (0c +c 0c))
180179anidms 626 . . . . . . . . . . . . . . 15 (p = 0c → (p +c p) = (0c +c 0c))
181 id 19 . . . . . . . . . . . . . . 15 (p = 0cp = 0c)
182180, 181addceq12d 4391 . . . . . . . . . . . . . 14 (p = 0c → ((p +c p) +c p) = ((0c +c 0c) +c 0c))
183182, 103syl6eq 2401 . . . . . . . . . . . . 13 (p = 0c → ((p +c p) +c p) = 0c)
184183addceq1d 4389 . . . . . . . . . . . 12 (p = 0c → (((p +c p) +c p) +c 1c) = (0c +c 1c))
185184eqeq2d 2364 . . . . . . . . . . 11 (p = 0c → ((((m +c 1c) +c (m +c 1c)) +c (m +c 1c)) = (((p +c p) +c p) +c 1c) ↔ (((m +c 1c) +c (m +c 1c)) +c (m +c 1c)) = (0c +c 1c)))
186185notbid 285 . . . . . . . . . 10 (p = 0c → (¬ (((m +c 1c) +c (m +c 1c)) +c (m +c 1c)) = (((p +c p) +c p) +c 1c) ↔ ¬ (((m +c 1c) +c (m +c 1c)) +c (m +c 1c)) = (0c +c 1c)))
187178, 186syl5ibrcom 213 . . . . . . . . 9 (m Nn → (p = 0c → ¬ (((m +c 1c) +c (m +c 1c)) +c (m +c 1c)) = (((p +c p) +c p) +c 1c)))
188187adantr 451 . . . . . . . 8 ((m Nn n Nn ¬ ((m +c m) +c m) = (((n +c n) +c n) +c 1c)) → (p = 0c → ¬ (((m +c 1c) +c (m +c 1c)) +c (m +c 1c)) = (((p +c p) +c p) +c 1c)))
189 addceq12 4385 . . . . . . . . . . . . . . . . . . . 20 ((n = q n = q) → (n +c n) = (q +c q))
190189anidms 626 . . . . . . . . . . . . . . . . . . 19 (n = q → (n +c n) = (q +c q))
191 id 19 . . . . . . . . . . . . . . . . . . 19 (n = qn = q)
192190, 191addceq12d 4391 . . . . . . . . . . . . . . . . . 18 (n = q → ((n +c n) +c n) = ((q +c q) +c q))
193192addceq1d 4389 . . . . . . . . . . . . . . . . 17 (n = q → (((n +c n) +c n) +c 1c) = (((q +c q) +c q) +c 1c))
194193eqeq2d 2364 . . . . . . . . . . . . . . . 16 (n = q → (((m +c m) +c m) = (((n +c n) +c n) +c 1c) ↔ ((m +c m) +c m) = (((q +c q) +c q) +c 1c)))
195194notbid 285 . . . . . . . . . . . . . . 15 (n = q → (¬ ((m +c m) +c m) = (((n +c n) +c n) +c 1c) ↔ ¬ ((m +c m) +c m) = (((q +c q) +c q) +c 1c)))
196195rspcv 2951 . . . . . . . . . . . . . 14 (q Nn → (n Nn ¬ ((m +c m) +c m) = (((n +c n) +c n) +c 1c) → ¬ ((m +c m) +c m) = (((q +c q) +c q) +c 1c)))
197196adantl 452 . . . . . . . . . . . . 13 ((m Nn q Nn ) → (n Nn ¬ ((m +c m) +c m) = (((n +c n) +c n) +c 1c) → ¬ ((m +c m) +c m) = (((q +c q) +c q) +c 1c)))
198 addc6 4418 . . . . . . . . . . . . . . . 16 (((m +c 1c) +c (m +c 1c)) +c (m +c 1c)) = (((m +c m) +c m) +c ((1c +c 1c) +c 1c))
199 addc6 4418 . . . . . . . . . . . . . . . . . 18 (((q +c 1c) +c (q +c 1c)) +c (q +c 1c)) = (((q +c q) +c q) +c ((1c +c 1c) +c 1c))
200199addceq1i 4386 . . . . . . . . . . . . . . . . 17 ((((q +c 1c) +c (q +c 1c)) +c (q +c 1c)) +c 1c) = ((((q +c q) +c q) +c ((1c +c 1c) +c 1c)) +c 1c)
201 addc32 4416 . . . . . . . . . . . . . . . . 17 ((((q +c q) +c q) +c ((1c +c 1c) +c 1c)) +c 1c) = ((((q +c q) +c q) +c 1c) +c ((1c +c 1c) +c 1c))
202200, 201eqtri 2373 . . . . . . . . . . . . . . . 16 ((((q +c 1c) +c (q +c 1c)) +c (q +c 1c)) +c 1c) = ((((q +c q) +c q) +c 1c) +c ((1c +c 1c) +c 1c))
203198, 202eqeq12i 2366 . . . . . . . . . . . . . . 15 ((((m +c 1c) +c (m +c 1c)) +c (m +c 1c)) = ((((q +c 1c) +c (q +c 1c)) +c (q +c 1c)) +c 1c) ↔ (((m +c m) +c m) +c ((1c +c 1c) +c 1c)) = ((((q +c q) +c q) +c 1c) +c ((1c +c 1c) +c 1c)))
204 nncaddccl 4419 . . . . . . . . . . . . . . . . . 18 ((m Nn m Nn ) → (m +c m) Nn )
205204anidms 626 . . . . . . . . . . . . . . . . 17 (m Nn → (m +c m) Nn )
206 nncaddccl 4419 . . . . . . . . . . . . . . . . 17 (((m +c m) Nn m Nn ) → ((m +c m) +c m) Nn )
207205, 206mpancom 650 . . . . . . . . . . . . . . . 16 (m Nn → ((m +c m) +c m) Nn )
208 nncaddccl 4419 . . . . . . . . . . . . . . . . . . 19 ((q Nn q Nn ) → (q +c q) Nn )
209208anidms 626 . . . . . . . . . . . . . . . . . 18 (q Nn → (q +c q) Nn )
210 nncaddccl 4419 . . . . . . . . . . . . . . . . . 18 (((q +c q) Nn q Nn ) → ((q +c q) +c q) Nn )
211209, 210mpancom 650 . . . . . . . . . . . . . . . . 17 (q Nn → ((q +c q) +c q) Nn )
212 peano2 4403 . . . . . . . . . . . . . . . . 17 (((q +c q) +c q) Nn → (((q +c q) +c q) +c 1c) Nn )
213211, 212syl 15 . . . . . . . . . . . . . . . 16 (q Nn → (((q +c q) +c q) +c 1c) Nn )
214 1cnnc 4408 . . . . . . . . . . . . . . . . . . 19 1c Nn
215 nncaddccl 4419 . . . . . . . . . . . . . . . . . . 19 ((1c Nn 1c Nn ) → (1c +c 1c) Nn )
216214, 214, 215mp2an 653 . . . . . . . . . . . . . . . . . 18 (1c +c 1c) Nn
217 nncaddccl 4419 . . . . . . . . . . . . . . . . . 18 (((1c +c 1c) Nn 1c Nn ) → ((1c +c 1c) +c 1c) Nn )
218216, 214, 217mp2an 653 . . . . . . . . . . . . . . . . 17 ((1c +c 1c) +c 1c) Nn
219 addccan1 4560 . . . . . . . . . . . . . . . . 17 ((((m +c m) +c m) Nn (((q +c q) +c q) +c 1c) Nn ((1c +c 1c) +c 1c) Nn ) → ((((m +c m) +c m) +c ((1c +c 1c) +c 1c)) = ((((q +c q) +c q) +c 1c) +c ((1c +c 1c) +c 1c)) ↔ ((m +c m) +c m) = (((q +c q) +c q) +c 1c)))
220218, 219mp3an3 1266 . . . . . . . . . . . . . . . 16 ((((m +c m) +c m) Nn (((q +c q) +c q) +c 1c) Nn ) → ((((m +c m) +c m) +c ((1c +c 1c) +c 1c)) = ((((q +c q) +c q) +c 1c) +c ((1c +c 1c) +c 1c)) ↔ ((m +c m) +c m) = (((q +c q) +c q) +c 1c)))
221207, 213, 220syl2an 463 . . . . . . . . . . . . . . 15 ((m Nn q Nn ) → ((((m +c m) +c m) +c ((1c +c 1c) +c 1c)) = ((((q +c q) +c q) +c 1c) +c ((1c +c 1c) +c 1c)) ↔ ((m +c m) +c m) = (((q +c q) +c q) +c 1c)))
222203, 221syl5bb 248 . . . . . . . . . . . . . 14 ((m Nn q Nn ) → ((((m +c 1c) +c (m +c 1c)) +c (m +c 1c)) = ((((q +c 1c) +c (q +c 1c)) +c (q +c 1c)) +c 1c) ↔ ((m +c m) +c m) = (((q +c q) +c q) +c 1c)))
223222biimpd 198 . . . . . . . . . . . . 13 ((m Nn q Nn ) → ((((m +c 1c) +c (m +c 1c)) +c (m +c 1c)) = ((((q +c 1c) +c (q +c 1c)) +c (q +c 1c)) +c 1c) → ((m +c m) +c m) = (((q +c q) +c q) +c 1c)))
224197, 223nsyld 132 . . . . . . . . . . . 12 ((m Nn q Nn ) → (n Nn ¬ ((m +c m) +c m) = (((n +c n) +c n) +c 1c) → ¬ (((m +c 1c) +c (m +c 1c)) +c (m +c 1c)) = ((((q +c 1c) +c (q +c 1c)) +c (q +c 1c)) +c 1c)))
225224imp 418 . . . . . . . . . . 11 (((m Nn q Nn ) n Nn ¬ ((m +c m) +c m) = (((n +c n) +c n) +c 1c)) → ¬ (((m +c 1c) +c (m +c 1c)) +c (m +c 1c)) = ((((q +c 1c) +c (q +c 1c)) +c (q +c 1c)) +c 1c))
226225an32s 779 . . . . . . . . . 10 (((m Nn n Nn ¬ ((m +c m) +c m) = (((n +c n) +c n) +c 1c)) q Nn ) → ¬ (((m +c 1c) +c (m +c 1c)) +c (m +c 1c)) = ((((q +c 1c) +c (q +c 1c)) +c (q +c 1c)) +c 1c))
227 addceq12 4385 . . . . . . . . . . . . . . 15 ((p = (q +c 1c) p = (q +c 1c)) → (p +c p) = ((q +c 1c) +c (q +c 1c)))
228227anidms 626 . . . . . . . . . . . . . 14 (p = (q +c 1c) → (p +c p) = ((q +c 1c) +c (q +c 1c)))
229 id 19 . . . . . . . . . . . . . 14 (p = (q +c 1c) → p = (q +c 1c))
230228, 229addceq12d 4391 . . . . . . . . . . . . 13 (p = (q +c 1c) → ((p +c p) +c p) = (((q +c 1c) +c (q +c 1c)) +c (q +c 1c)))
231230addceq1d 4389 . . . . . . . . . . . 12 (p = (q +c 1c) → (((p +c p) +c p) +c 1c) = ((((q +c 1c) +c (q +c 1c)) +c (q +c 1c)) +c 1c))
232231eqeq2d 2364 . . . . . . . . . . 11 (p = (q +c 1c) → ((((m +c 1c) +c (m +c 1c)) +c (m +c 1c)) = (((p +c p) +c p) +c 1c) ↔ (((m +c 1c) +c (m +c 1c)) +c (m +c 1c)) = ((((q +c 1c) +c (q +c 1c)) +c (q +c 1c)) +c 1c)))
233232notbid 285 . . . . . . . . . 10 (p = (q +c 1c) → (¬ (((m +c 1c) +c (m +c 1c)) +c (m +c 1c)) = (((p +c p) +c p) +c 1c) ↔ ¬ (((m +c 1c) +c (m +c 1c)) +c (m +c 1c)) = ((((q +c 1c) +c (q +c 1c)) +c (q +c 1c)) +c 1c)))
234226, 233syl5ibrcom 213 . . . . . . . . 9 (((m Nn n Nn ¬ ((m +c m) +c m) = (((n +c n) +c n) +c 1c)) q Nn ) → (p = (q +c 1c) → ¬ (((m +c 1c) +c (m +c 1c)) +c (m +c 1c)) = (((p +c p) +c p) +c 1c)))
235234rexlimdva 2738 . . . . . . . 8 ((m Nn n Nn ¬ ((m +c m) +c m) = (((n +c n) +c n) +c 1c)) → (q Nn p = (q +c 1c) → ¬ (((m +c 1c) +c (m +c 1c)) +c (m +c 1c)) = (((p +c p) +c p) +c 1c)))
236188, 235jaod 369 . . . . . . 7 ((m Nn n Nn ¬ ((m +c m) +c m) = (((n +c n) +c n) +c 1c)) → ((p = 0c q Nn p = (q +c 1c)) → ¬ (((m +c 1c) +c (m +c 1c)) +c (m +c 1c)) = (((p +c p) +c p) +c 1c)))
237155, 236syl5bi 208 . . . . . 6 ((m Nn n Nn ¬ ((m +c m) +c m) = (((n +c n) +c n) +c 1c)) → (p Nn → ¬ (((m +c 1c) +c (m +c 1c)) +c (m +c 1c)) = (((p +c p) +c p) +c 1c)))
238237ralrimiv 2696 . . . . 5 ((m Nn n Nn ¬ ((m +c m) +c m) = (((n +c n) +c n) +c 1c)) → p Nn ¬ (((m +c 1c) +c (m +c 1c)) +c (m +c 1c)) = (((p +c p) +c p) +c 1c))
239238ex 423 . . . 4 (m Nn → (n Nn ¬ ((m +c m) +c m) = (((n +c n) +c n) +c 1c) → p Nn ¬ (((m +c 1c) +c (m +c 1c)) +c (m +c 1c)) = (((p +c p) +c p) +c 1c)))
24096, 107, 114, 130, 137, 154, 239finds 4411 . . 3 (A Nnn Nn ¬ ((A +c A) +c A) = (((n +c n) +c n) +c 1c))
241 addceq12 4385 . . . . . . . . 9 ((n = B n = B) → (n +c n) = (B +c B))
242241anidms 626 . . . . . . . 8 (n = B → (n +c n) = (B +c B))
243 id 19 . . . . . . . 8 (n = Bn = B)
244242, 243addceq12d 4391 . . . . . . 7 (n = B → ((n +c n) +c n) = ((B +c B) +c B))
245244addceq1d 4389 . . . . . 6 (n = B → (((n +c n) +c n) +c 1c) = (((B +c B) +c B) +c 1c))
246245eqeq2d 2364 . . . . 5 (n = B → (((A +c A) +c A) = (((n +c n) +c n) +c 1c) ↔ ((A +c A) +c A) = (((B +c B) +c B) +c 1c)))
247246notbid 285 . . . 4 (n = B → (¬ ((A +c A) +c A) = (((n +c n) +c n) +c 1c) ↔ ¬ ((A +c A) +c A) = (((B +c B) +c B) +c 1c)))
248247rspccv 2952 . . 3 (n Nn ¬ ((A +c A) +c A) = (((n +c n) +c n) +c 1c) → (B Nn → ¬ ((A +c A) +c A) = (((B +c B) +c B) +c 1c)))
249240, 248syl 15 . 2 (A Nn → (B Nn → ¬ ((A +c A) +c A) = (((B +c B) +c B) +c 1c)))
250249imp 418 1 ((A Nn B Nn ) → ¬ ((A +c A) +c A) = (((B +c B) +c B) +c 1c))
 Colors of variables: wff setvar class Syntax hints:  ¬ wn 3   → wi 4   ↔ wb 176   ∨ wo 357   ∧ wa 358  ∃wex 1541   = wceq 1642   ∈ wcel 1710  {cab 2339   ≠ wne 2516  ∀wral 2614  ∃wrex 2615  Vcvv 2859   ∼ ccompl 3205   ∩ cin 3208  {csn 3737  1cc1c 4134   Nn cnnc 4373  0cc0c 4374   +c cplc 4375  ⟨cop 4561   class class class wbr 4639  1st c1st 4717   ∘ ccom 4721   “ cima 4722  ◡ccnv 4771  ran crn 4773   ↾ cres 4774  2nd c2nd 4783   ⊗ ctxp 5735   AddC caddcfn 5745   Ins3 cins3 5751   NC cncs 6088 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 4078  ax-xp 4079  ax-cnv 4080  ax-1c 4081  ax-sset 4082  ax-si 4083  ax-ins2 4084  ax-ins3 4085  ax-typlower 4086  ax-sn 4087 This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3or 935  df-3an 936  df-nan 1288  df-tru 1319  df-ex 1542  df-nf 1545  df-sb 1649  df-eu 2208  df-mo 2209  df-clab 2340  df-cleq 2346  df-clel 2349  df-nfc 2478  df-ne 2518  df-ral 2619  df-rex 2620  df-reu 2621  df-rmo 2622  df-rab 2623  df-v 2861  df-sbc 3047  df-csb 3137  df-nin 3211  df-compl 3212  df-in 3213  df-un 3214  df-dif 3215  df-symdif 3216  df-ss 3259  df-pss 3261  df-nul 3551  df-if 3663  df-pw 3724  df-sn 3741  df-pr 3742  df-uni 3892  df-int 3927  df-iun 3971  df-opk 4058  df-1c 4136  df-pw1 4137  df-uni1 4138  df-xpk 4185  df-cnvk 4186  df-ins2k 4187  df-ins3k 4188  df-imak 4189  df-cok 4190  df-p6 4191  df-sik 4192  df-ssetk 4193  df-imagek 4194  df-idk 4195  df-iota 4339  df-0c 4377  df-addc 4378  df-nnc 4379  df-fin 4380  df-lefin 4440  df-ltfin 4441  df-ncfin 4442  df-tfin 4443  df-evenfin 4444  df-oddfin 4445  df-sfin 4446  df-spfin 4447  df-phi 4565  df-op 4566  df-proj1 4567  df-proj2 4568  df-opab 4623  df-br 4640  df-1st 4723  df-swap 4724  df-sset 4725  df-co 4726  df-ima 4727  df-si 4728  df-id 4767  df-xp 4784  df-cnv 4785  df-rn 4786  df-dm 4787  df-res 4788  df-fun 4789  df-fn 4790  df-f 4791  df-f1 4792  df-fo 4793  df-f1o 4794  df-fv 4795  df-2nd 4797  df-ov 5526  df-oprab 5528  df-mpt 5652  df-mpt2 5654  df-txp 5736  df-cup 5742  df-disj 5744  df-addcfn 5746  df-ins2 5750  df-ins3 5752  df-image 5754  df-ins4 5756  df-si3 5758  df-funs 5760  df-fns 5762  df-trans 5899  df-sym 5908  df-er 5909  df-ec 5947  df-qs 5951  df-en 6029  df-ncs 6098  df-lec 6099  df-nc 6101 This theorem is referenced by:  nnc3n3p2  6279  nnc3p1n3p2  6280  nchoicelem1  6289
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