Step | Hyp | Ref
| Expression |
1 | | nclennlem1 6248 |
. . . . 5
⊢ {x ∣ ∀n ∈ NC (n ≤c x → n ∈ Nn )} ∈ V |
2 | | breq2 4643 |
. . . . . . 7
⊢ (x = 0c → (n ≤c x ↔ n
≤c 0c)) |
3 | 2 | imbi1d 308 |
. . . . . 6
⊢ (x = 0c → ((n ≤c x → n ∈ Nn ) ↔
(n ≤c 0c
→ n ∈ Nn
))) |
4 | 3 | ralbidv 2634 |
. . . . 5
⊢ (x = 0c → (∀n ∈ NC (n ≤c x → n ∈ Nn ) ↔ ∀n ∈ NC (n ≤c 0c →
n ∈ Nn ))) |
5 | | breq2 4643 |
. . . . . . 7
⊢ (x = m →
(n ≤c x ↔ n
≤c m)) |
6 | 5 | imbi1d 308 |
. . . . . 6
⊢ (x = m →
((n ≤c x → n ∈ Nn ) ↔
(n ≤c m → n ∈ Nn
))) |
7 | 6 | ralbidv 2634 |
. . . . 5
⊢ (x = m →
(∀n
∈ NC (n ≤c x → n ∈ Nn ) ↔ ∀n ∈ NC (n ≤c m → n ∈ Nn
))) |
8 | | breq2 4643 |
. . . . . . 7
⊢ (x = (m
+c 1c) → (n ≤c x ↔ n
≤c (m +c
1c))) |
9 | 8 | imbi1d 308 |
. . . . . 6
⊢ (x = (m
+c 1c) → ((n ≤c x → n ∈ Nn ) ↔
(n ≤c (m +c 1c) →
n ∈ Nn ))) |
10 | 9 | ralbidv 2634 |
. . . . 5
⊢ (x = (m
+c 1c) → (∀n ∈ NC (n ≤c x → n ∈ Nn ) ↔ ∀n ∈ NC (n ≤c (m +c 1c) →
n ∈ Nn ))) |
11 | | breq2 4643 |
. . . . . . 7
⊢ (x = N →
(n ≤c x ↔ n
≤c N)) |
12 | 11 | imbi1d 308 |
. . . . . 6
⊢ (x = N →
((n ≤c x → n ∈ Nn ) ↔
(n ≤c N → n ∈ Nn
))) |
13 | 12 | ralbidv 2634 |
. . . . 5
⊢ (x = N →
(∀n
∈ NC (n ≤c x → n ∈ Nn ) ↔ ∀n ∈ NC (n ≤c N → n ∈ Nn
))) |
14 | | le0nc 6200 |
. . . . . . 7
⊢ (n ∈ NC → 0c ≤c n) |
15 | | 0cnc 6138 |
. . . . . . . . . . 11
⊢
0c ∈ NC |
16 | | sbth 6206 |
. . . . . . . . . . 11
⊢ ((n ∈ NC ∧
0c ∈ NC ) → ((n
≤c 0c ∧
0c ≤c n)
→ n =
0c)) |
17 | 15, 16 | mpan2 652 |
. . . . . . . . . 10
⊢ (n ∈ NC → ((n
≤c 0c ∧
0c ≤c n)
→ n =
0c)) |
18 | 17 | imp 418 |
. . . . . . . . 9
⊢ ((n ∈ NC ∧ (n ≤c 0c ∧ 0c ≤c n)) → n =
0c) |
19 | | peano1 4402 |
. . . . . . . . 9
⊢
0c ∈ Nn |
20 | 18, 19 | syl6eqel 2441 |
. . . . . . . 8
⊢ ((n ∈ NC ∧ (n ≤c 0c ∧ 0c ≤c n)) → n
∈ Nn
) |
21 | 20 | ex 423 |
. . . . . . 7
⊢ (n ∈ NC → ((n
≤c 0c ∧
0c ≤c n)
→ n ∈ Nn
)) |
22 | 14, 21 | mpan2d 655 |
. . . . . 6
⊢ (n ∈ NC → (n
≤c 0c → n ∈ Nn )) |
23 | 22 | rgen 2679 |
. . . . 5
⊢ ∀n ∈ NC (n ≤c 0c →
n ∈ Nn ) |
24 | | peano2 4403 |
. . . . . . . . . . . 12
⊢ (m ∈ Nn → (m
+c 1c) ∈
Nn ) |
25 | | nnnc 6146 |
. . . . . . . . . . . 12
⊢ ((m +c 1c) ∈ Nn → (m +c 1c) ∈ NC
) |
26 | 24, 25 | syl 15 |
. . . . . . . . . . 11
⊢ (m ∈ Nn → (m
+c 1c) ∈
NC ) |
27 | | dflec2 6210 |
. . . . . . . . . . 11
⊢ ((n ∈ NC ∧ (m +c 1c) ∈ NC ) →
(n ≤c (m +c 1c) ↔
∃p ∈ NC (m +c 1c) =
(n +c p))) |
28 | 26, 27 | sylan2 460 |
. . . . . . . . . 10
⊢ ((n ∈ NC ∧ m ∈ Nn ) → (n
≤c (m +c
1c) ↔ ∃p ∈ NC (m
+c 1c) = (n +c p))) |
29 | 28 | ancoms 439 |
. . . . . . . . 9
⊢ ((m ∈ Nn ∧ n ∈ NC ) → (n
≤c (m +c
1c) ↔ ∃p ∈ NC (m
+c 1c) = (n +c p))) |
30 | 29 | 3adant3 975 |
. . . . . . . 8
⊢ ((m ∈ Nn ∧ n ∈ NC ∧ (n ≤c m → n ∈ Nn )) →
(n ≤c (m +c 1c) ↔
∃p ∈ NC (m +c 1c) =
(n +c p))) |
31 | | nc0suc 6217 |
. . . . . . . . . 10
⊢ (p ∈ NC → (p =
0c ∨ ∃q ∈ NC p = (q
+c 1c))) |
32 | | addceq2 4384 |
. . . . . . . . . . . . . . . . . 18
⊢ (p = 0c → (n +c p) = (n
+c 0c)) |
33 | | addcid1 4405 |
. . . . . . . . . . . . . . . . . 18
⊢ (n +c 0c) =
n |
34 | 32, 33 | syl6eq 2401 |
. . . . . . . . . . . . . . . . 17
⊢ (p = 0c → (n +c p) = n) |
35 | 34 | eqeq2d 2364 |
. . . . . . . . . . . . . . . 16
⊢ (p = 0c → ((m +c 1c) =
(n +c p) ↔ (m
+c 1c) = n)) |
36 | 35 | biimpa 470 |
. . . . . . . . . . . . . . 15
⊢ ((p = 0c ∧ (m
+c 1c) = (n +c p)) → (m
+c 1c) = n) |
37 | | eleq1 2413 |
. . . . . . . . . . . . . . . 16
⊢ ((m +c 1c) =
n → ((m +c 1c) ∈ Nn ↔ n ∈ Nn )) |
38 | 37 | biimpcd 215 |
. . . . . . . . . . . . . . 15
⊢ ((m +c 1c) ∈ Nn → ((m +c 1c) =
n → n ∈ Nn )) |
39 | 36, 38 | syl5 28 |
. . . . . . . . . . . . . 14
⊢ ((m +c 1c) ∈ Nn → ((p = 0c ∧ (m
+c 1c) = (n +c p)) → n
∈ Nn
)) |
40 | 39 | exp3a 425 |
. . . . . . . . . . . . 13
⊢ ((m +c 1c) ∈ Nn → (p = 0c → ((m +c 1c) =
(n +c p) → n
∈ Nn
))) |
41 | 24, 40 | syl 15 |
. . . . . . . . . . . 12
⊢ (m ∈ Nn → (p =
0c → ((m
+c 1c) = (n +c p) → n
∈ Nn
))) |
42 | 41 | 3ad2ant1 976 |
. . . . . . . . . . 11
⊢ ((m ∈ Nn ∧ n ∈ NC ∧ (n ≤c m → n ∈ Nn )) →
(p = 0c → ((m +c 1c) =
(n +c p) → n
∈ Nn
))) |
43 | | addceq2 4384 |
. . . . . . . . . . . . . . . . 17
⊢ (p = (q
+c 1c) → (n +c p) = (n
+c (q
+c 1c))) |
44 | | addcass 4415 |
. . . . . . . . . . . . . . . . 17
⊢ ((n +c q) +c 1c) =
(n +c (q +c
1c)) |
45 | 43, 44 | syl6eqr 2403 |
. . . . . . . . . . . . . . . 16
⊢ (p = (q
+c 1c) → (n +c p) = ((n
+c q)
+c 1c)) |
46 | 45 | eqeq2d 2364 |
. . . . . . . . . . . . . . 15
⊢ (p = (q
+c 1c) → ((m +c 1c) =
(n +c p) ↔ (m
+c 1c) = ((n +c q) +c
1c))) |
47 | 46 | biimpa 470 |
. . . . . . . . . . . . . 14
⊢ ((p = (q
+c 1c) ∧
(m +c
1c) = (n
+c p)) → (m +c 1c) =
((n +c q) +c
1c)) |
48 | | nnnc 6146 |
. . . . . . . . . . . . . . . . . 18
⊢ (m ∈ Nn → m ∈ NC
) |
49 | 48 | 3ad2ant1 976 |
. . . . . . . . . . . . . . . . 17
⊢ ((m ∈ Nn ∧ n ∈ NC ∧ (n ≤c m → n ∈ Nn )) →
m ∈ NC ) |
50 | 49 | adantr 451 |
. . . . . . . . . . . . . . . 16
⊢ (((m ∈ Nn ∧ n ∈ NC ∧ (n ≤c m → n ∈ Nn )) ∧ q ∈ NC ) → m ∈ NC ) |
51 | | ncaddccl 6144 |
. . . . . . . . . . . . . . . . 17
⊢ ((n ∈ NC ∧ q ∈ NC ) → (n
+c q) ∈ NC
) |
52 | 51 | 3ad2antl2 1118 |
. . . . . . . . . . . . . . . 16
⊢ (((m ∈ Nn ∧ n ∈ NC ∧ (n ≤c m → n ∈ Nn )) ∧ q ∈ NC ) →
(n +c q) ∈ NC ) |
53 | | peano4nc 6150 |
. . . . . . . . . . . . . . . 16
⊢ ((m ∈ NC ∧ (n +c q) ∈ NC ) → ((m
+c 1c) = ((n +c q) +c 1c) ↔
m = (n
+c q))) |
54 | 50, 52, 53 | syl2anc 642 |
. . . . . . . . . . . . . . 15
⊢ (((m ∈ Nn ∧ n ∈ NC ∧ (n ≤c m → n ∈ Nn )) ∧ q ∈ NC ) →
((m +c
1c) = ((n
+c q)
+c 1c) ↔ m = (n
+c q))) |
55 | | addlecncs 6209 |
. . . . . . . . . . . . . . . . . . . . . . 23
⊢ ((n ∈ NC ∧ q ∈ NC ) → n
≤c (n +c
q)) |
56 | | breq2 4643 |
. . . . . . . . . . . . . . . . . . . . . . 23
⊢ (m = (n
+c q) → (n ≤c m ↔ n
≤c (n +c
q))) |
57 | 55, 56 | syl5ibrcom 213 |
. . . . . . . . . . . . . . . . . . . . . 22
⊢ ((n ∈ NC ∧ q ∈ NC ) → (m =
(n +c q) → n
≤c m)) |
58 | 57 | ex 423 |
. . . . . . . . . . . . . . . . . . . . 21
⊢ (n ∈ NC → (q ∈ NC → (m = (n
+c q) → n ≤c m))) |
59 | 58 | com23 72 |
. . . . . . . . . . . . . . . . . . . 20
⊢ (n ∈ NC → (m =
(n +c q) → (q
∈ NC →
n ≤c m))) |
60 | 59 | adantl 452 |
. . . . . . . . . . . . . . . . . . 19
⊢ ((m ∈ Nn ∧ n ∈ NC ) → (m =
(n +c q) → (q
∈ NC →
n ≤c m))) |
61 | | pm2.27 35 |
. . . . . . . . . . . . . . . . . . 19
⊢ (n ≤c m → ((n
≤c m → n ∈ Nn ) → n ∈ Nn
)) |
62 | 60, 61 | syl8 65 |
. . . . . . . . . . . . . . . . . 18
⊢ ((m ∈ Nn ∧ n ∈ NC ) → (m =
(n +c q) → (q
∈ NC →
((n ≤c m → n ∈ Nn ) → n ∈ Nn )))) |
63 | 62 | com24 81 |
. . . . . . . . . . . . . . . . 17
⊢ ((m ∈ Nn ∧ n ∈ NC ) → ((n
≤c m → n ∈ Nn ) → (q ∈ NC → (m = (n
+c q) → n ∈ Nn )))) |
64 | 63 | 3impia 1148 |
. . . . . . . . . . . . . . . 16
⊢ ((m ∈ Nn ∧ n ∈ NC ∧ (n ≤c m → n ∈ Nn )) →
(q ∈
NC → (m =
(n +c q) → n
∈ Nn
))) |
65 | 64 | imp 418 |
. . . . . . . . . . . . . . 15
⊢ (((m ∈ Nn ∧ n ∈ NC ∧ (n ≤c m → n ∈ Nn )) ∧ q ∈ NC ) →
(m = (n
+c q) → n ∈ Nn )) |
66 | 54, 65 | sylbid 206 |
. . . . . . . . . . . . . 14
⊢ (((m ∈ Nn ∧ n ∈ NC ∧ (n ≤c m → n ∈ Nn )) ∧ q ∈ NC ) →
((m +c
1c) = ((n
+c q)
+c 1c) → n ∈ Nn )) |
67 | 47, 66 | syl5 28 |
. . . . . . . . . . . . 13
⊢ (((m ∈ Nn ∧ n ∈ NC ∧ (n ≤c m → n ∈ Nn )) ∧ q ∈ NC ) →
((p = (q +c 1c) ∧ (m
+c 1c) = (n +c p)) → n
∈ Nn
)) |
68 | 67 | exp3a 425 |
. . . . . . . . . . . 12
⊢ (((m ∈ Nn ∧ n ∈ NC ∧ (n ≤c m → n ∈ Nn )) ∧ q ∈ NC ) →
(p = (q
+c 1c) → ((m +c 1c) =
(n +c p) → n
∈ Nn
))) |
69 | 68 | rexlimdva 2738 |
. . . . . . . . . . 11
⊢ ((m ∈ Nn ∧ n ∈ NC ∧ (n ≤c m → n ∈ Nn )) → (∃q ∈ NC p = (q
+c 1c) → ((m +c 1c) =
(n +c p) → n
∈ Nn
))) |
70 | 42, 69 | jaod 369 |
. . . . . . . . . 10
⊢ ((m ∈ Nn ∧ n ∈ NC ∧ (n ≤c m → n ∈ Nn )) →
((p = 0c ∨ ∃q ∈ NC p = (q +c 1c)) →
((m +c
1c) = (n
+c p) → n ∈ Nn ))) |
71 | 31, 70 | syl5 28 |
. . . . . . . . 9
⊢ ((m ∈ Nn ∧ n ∈ NC ∧ (n ≤c m → n ∈ Nn )) →
(p ∈
NC → ((m
+c 1c) = (n +c p) → n
∈ Nn
))) |
72 | 71 | rexlimdv 2737 |
. . . . . . . 8
⊢ ((m ∈ Nn ∧ n ∈ NC ∧ (n ≤c m → n ∈ Nn )) → (∃p ∈ NC (m +c 1c) =
(n +c p) → n
∈ Nn
)) |
73 | 30, 72 | sylbid 206 |
. . . . . . 7
⊢ ((m ∈ Nn ∧ n ∈ NC ∧ (n ≤c m → n ∈ Nn )) →
(n ≤c (m +c 1c) →
n ∈ Nn )) |
74 | 73 | 3expia 1153 |
. . . . . 6
⊢ ((m ∈ Nn ∧ n ∈ NC ) → ((n
≤c m → n ∈ Nn ) → (n
≤c (m +c
1c) → n ∈ Nn
))) |
75 | 74 | ralimdva 2692 |
. . . . 5
⊢ (m ∈ Nn → (∀n ∈ NC (n
≤c m → n ∈ Nn ) → ∀n ∈ NC (n ≤c (m +c 1c) →
n ∈ Nn ))) |
76 | 1, 4, 7, 10, 13, 23, 75 | finds 4411 |
. . . 4
⊢ (N ∈ Nn → ∀n ∈ NC (n
≤c N → n ∈ Nn )) |
77 | | breq1 4642 |
. . . . . 6
⊢ (n = M →
(n ≤c N ↔ M
≤c N)) |
78 | | eleq1 2413 |
. . . . . 6
⊢ (n = M →
(n ∈
Nn ↔ M
∈ Nn
)) |
79 | 77, 78 | imbi12d 311 |
. . . . 5
⊢ (n = M →
((n ≤c N → n ∈ Nn ) ↔
(M ≤c N → M ∈ Nn
))) |
80 | 79 | rspccv 2952 |
. . . 4
⊢ (∀n ∈ NC (n ≤c N → n ∈ Nn ) →
(M ∈
NC → (M
≤c N → M ∈ Nn ))) |
81 | 76, 80 | syl 15 |
. . 3
⊢ (N ∈ Nn → (M ∈ NC → (M ≤c N → M ∈ Nn
))) |
82 | 81 | com12 27 |
. 2
⊢ (M ∈ NC → (N ∈ Nn → (M ≤c N → M ∈ Nn
))) |
83 | 82 | 3imp 1145 |
1
⊢ ((M ∈ NC ∧ N ∈ Nn ∧ M ≤c N) → M
∈ Nn
) |