Step | Hyp | Ref
| Expression |
1 | | peano2nn 11842 |
. . . . 5
⊢ (𝑁 ∈ ℕ → (𝑁 + 1) ∈
ℕ) |
2 | 1 | nnnn0d 12150 |
. . . 4
⊢ (𝑁 ∈ ℕ → (𝑁 + 1) ∈
ℕ0) |
3 | | derang.d |
. . . . 5
⊢ 𝐷 = (𝑥 ∈ Fin ↦ (♯‘{𝑓 ∣ (𝑓:𝑥–1-1-onto→𝑥 ∧ ∀𝑦 ∈ 𝑥 (𝑓‘𝑦) ≠ 𝑦)})) |
4 | | subfac.n |
. . . . 5
⊢ 𝑆 = (𝑛 ∈ ℕ0 ↦ (𝐷‘(1...𝑛))) |
5 | 3, 4 | subfacval 32848 |
. . . 4
⊢ ((𝑁 + 1) ∈ ℕ0
→ (𝑆‘(𝑁 + 1)) = (𝐷‘(1...(𝑁 + 1)))) |
6 | 2, 5 | syl 17 |
. . 3
⊢ (𝑁 ∈ ℕ → (𝑆‘(𝑁 + 1)) = (𝐷‘(1...(𝑁 + 1)))) |
7 | | fzfid 13546 |
. . . . 5
⊢ (𝑁 ∈ ℕ →
(1...(𝑁 + 1)) ∈
Fin) |
8 | 3 | derangval 32842 |
. . . . 5
⊢
((1...(𝑁 + 1))
∈ Fin → (𝐷‘(1...(𝑁 + 1))) = (♯‘{𝑓 ∣ (𝑓:(1...(𝑁 + 1))–1-1-onto→(1...(𝑁 + 1)) ∧ ∀𝑦 ∈ (1...(𝑁 + 1))(𝑓‘𝑦) ≠ 𝑦)})) |
9 | 7, 8 | syl 17 |
. . . 4
⊢ (𝑁 ∈ ℕ → (𝐷‘(1...(𝑁 + 1))) = (♯‘{𝑓 ∣ (𝑓:(1...(𝑁 + 1))–1-1-onto→(1...(𝑁 + 1)) ∧ ∀𝑦 ∈ (1...(𝑁 + 1))(𝑓‘𝑦) ≠ 𝑦)})) |
10 | | subfacp1lem.a |
. . . . 5
⊢ 𝐴 = {𝑓 ∣ (𝑓:(1...(𝑁 + 1))–1-1-onto→(1...(𝑁 + 1)) ∧ ∀𝑦 ∈ (1...(𝑁 + 1))(𝑓‘𝑦) ≠ 𝑦)} |
11 | 10 | fveq2i 6720 |
. . . 4
⊢
(♯‘𝐴) =
(♯‘{𝑓 ∣
(𝑓:(1...(𝑁 + 1))–1-1-onto→(1...(𝑁 + 1)) ∧ ∀𝑦 ∈ (1...(𝑁 + 1))(𝑓‘𝑦) ≠ 𝑦)}) |
12 | 9, 11 | eqtr4di 2796 |
. . 3
⊢ (𝑁 ∈ ℕ → (𝐷‘(1...(𝑁 + 1))) = (♯‘𝐴)) |
13 | | nnuz 12477 |
. . . . . . . . . . 11
⊢ ℕ =
(ℤ≥‘1) |
14 | 1, 13 | eleqtrdi 2848 |
. . . . . . . . . 10
⊢ (𝑁 ∈ ℕ → (𝑁 + 1) ∈
(ℤ≥‘1)) |
15 | | eluzfz1 13119 |
. . . . . . . . . 10
⊢ ((𝑁 + 1) ∈
(ℤ≥‘1) → 1 ∈ (1...(𝑁 + 1))) |
16 | 14, 15 | syl 17 |
. . . . . . . . 9
⊢ (𝑁 ∈ ℕ → 1 ∈
(1...(𝑁 +
1))) |
17 | | f1of 6661 |
. . . . . . . . . 10
⊢ (𝑓:(1...(𝑁 + 1))–1-1-onto→(1...(𝑁 + 1)) → 𝑓:(1...(𝑁 + 1))⟶(1...(𝑁 + 1))) |
18 | 17 | adantr 484 |
. . . . . . . . 9
⊢ ((𝑓:(1...(𝑁 + 1))–1-1-onto→(1...(𝑁 + 1)) ∧ ∀𝑦 ∈ (1...(𝑁 + 1))(𝑓‘𝑦) ≠ 𝑦) → 𝑓:(1...(𝑁 + 1))⟶(1...(𝑁 + 1))) |
19 | | ffvelrn 6902 |
. . . . . . . . . 10
⊢ ((𝑓:(1...(𝑁 + 1))⟶(1...(𝑁 + 1)) ∧ 1 ∈ (1...(𝑁 + 1))) → (𝑓‘1) ∈ (1...(𝑁 + 1))) |
20 | 19 | expcom 417 |
. . . . . . . . 9
⊢ (1 ∈
(1...(𝑁 + 1)) → (𝑓:(1...(𝑁 + 1))⟶(1...(𝑁 + 1)) → (𝑓‘1) ∈ (1...(𝑁 + 1)))) |
21 | 16, 18, 20 | syl2im 40 |
. . . . . . . 8
⊢ (𝑁 ∈ ℕ → ((𝑓:(1...(𝑁 + 1))–1-1-onto→(1...(𝑁 + 1)) ∧ ∀𝑦 ∈ (1...(𝑁 + 1))(𝑓‘𝑦) ≠ 𝑦) → (𝑓‘1) ∈ (1...(𝑁 + 1)))) |
22 | 21 | ss2abdv 3977 |
. . . . . . 7
⊢ (𝑁 ∈ ℕ → {𝑓 ∣ (𝑓:(1...(𝑁 + 1))–1-1-onto→(1...(𝑁 + 1)) ∧ ∀𝑦 ∈ (1...(𝑁 + 1))(𝑓‘𝑦) ≠ 𝑦)} ⊆ {𝑓 ∣ (𝑓‘1) ∈ (1...(𝑁 + 1))}) |
23 | | fveq1 6716 |
. . . . . . . . 9
⊢ (𝑔 = 𝑓 → (𝑔‘1) = (𝑓‘1)) |
24 | 23 | eleq1d 2822 |
. . . . . . . 8
⊢ (𝑔 = 𝑓 → ((𝑔‘1) ∈ (1...(𝑁 + 1)) ↔ (𝑓‘1) ∈ (1...(𝑁 + 1)))) |
25 | 24 | cbvabv 2811 |
. . . . . . 7
⊢ {𝑔 ∣ (𝑔‘1) ∈ (1...(𝑁 + 1))} = {𝑓 ∣ (𝑓‘1) ∈ (1...(𝑁 + 1))} |
26 | 22, 10, 25 | 3sstr4g 3946 |
. . . . . 6
⊢ (𝑁 ∈ ℕ → 𝐴 ⊆ {𝑔 ∣ (𝑔‘1) ∈ (1...(𝑁 + 1))}) |
27 | | ssabral 3976 |
. . . . . 6
⊢ (𝐴 ⊆ {𝑔 ∣ (𝑔‘1) ∈ (1...(𝑁 + 1))} ↔ ∀𝑔 ∈ 𝐴 (𝑔‘1) ∈ (1...(𝑁 + 1))) |
28 | 26, 27 | sylib 221 |
. . . . 5
⊢ (𝑁 ∈ ℕ →
∀𝑔 ∈ 𝐴 (𝑔‘1) ∈ (1...(𝑁 + 1))) |
29 | | rabid2 3293 |
. . . . 5
⊢ (𝐴 = {𝑔 ∈ 𝐴 ∣ (𝑔‘1) ∈ (1...(𝑁 + 1))} ↔ ∀𝑔 ∈ 𝐴 (𝑔‘1) ∈ (1...(𝑁 + 1))) |
30 | 28, 29 | sylibr 237 |
. . . 4
⊢ (𝑁 ∈ ℕ → 𝐴 = {𝑔 ∈ 𝐴 ∣ (𝑔‘1) ∈ (1...(𝑁 + 1))}) |
31 | 30 | fveq2d 6721 |
. . 3
⊢ (𝑁 ∈ ℕ →
(♯‘𝐴) =
(♯‘{𝑔 ∈
𝐴 ∣ (𝑔‘1) ∈ (1...(𝑁 + 1))})) |
32 | 6, 12, 31 | 3eqtrd 2781 |
. 2
⊢ (𝑁 ∈ ℕ → (𝑆‘(𝑁 + 1)) = (♯‘{𝑔 ∈ 𝐴 ∣ (𝑔‘1) ∈ (1...(𝑁 + 1))})) |
33 | | elfz1end 13142 |
. . . 4
⊢ ((𝑁 + 1) ∈ ℕ ↔
(𝑁 + 1) ∈ (1...(𝑁 + 1))) |
34 | 1, 33 | sylib 221 |
. . 3
⊢ (𝑁 ∈ ℕ → (𝑁 + 1) ∈ (1...(𝑁 + 1))) |
35 | | eleq1 2825 |
. . . . . . 7
⊢ (𝑥 = 1 → (𝑥 ∈ (1...(𝑁 + 1)) ↔ 1 ∈ (1...(𝑁 + 1)))) |
36 | | oveq2 7221 |
. . . . . . . . . . . . 13
⊢ (𝑥 = 1 → (1...𝑥) = (1...1)) |
37 | | 1z 12207 |
. . . . . . . . . . . . . 14
⊢ 1 ∈
ℤ |
38 | | fzsn 13154 |
. . . . . . . . . . . . . 14
⊢ (1 ∈
ℤ → (1...1) = {1}) |
39 | 37, 38 | ax-mp 5 |
. . . . . . . . . . . . 13
⊢ (1...1) =
{1} |
40 | 36, 39 | eqtrdi 2794 |
. . . . . . . . . . . 12
⊢ (𝑥 = 1 → (1...𝑥) = {1}) |
41 | 40 | eleq2d 2823 |
. . . . . . . . . . 11
⊢ (𝑥 = 1 → ((𝑔‘1) ∈ (1...𝑥) ↔ (𝑔‘1) ∈ {1})) |
42 | | fvex 6730 |
. . . . . . . . . . . 12
⊢ (𝑔‘1) ∈
V |
43 | 42 | elsn 4556 |
. . . . . . . . . . 11
⊢ ((𝑔‘1) ∈ {1} ↔
(𝑔‘1) =
1) |
44 | 41, 43 | bitrdi 290 |
. . . . . . . . . 10
⊢ (𝑥 = 1 → ((𝑔‘1) ∈ (1...𝑥) ↔ (𝑔‘1) = 1)) |
45 | 44 | rabbidv 3390 |
. . . . . . . . 9
⊢ (𝑥 = 1 → {𝑔 ∈ 𝐴 ∣ (𝑔‘1) ∈ (1...𝑥)} = {𝑔 ∈ 𝐴 ∣ (𝑔‘1) = 1}) |
46 | 45 | fveq2d 6721 |
. . . . . . . 8
⊢ (𝑥 = 1 →
(♯‘{𝑔 ∈
𝐴 ∣ (𝑔‘1) ∈ (1...𝑥)}) = (♯‘{𝑔 ∈ 𝐴 ∣ (𝑔‘1) = 1})) |
47 | | oveq1 7220 |
. . . . . . . . . 10
⊢ (𝑥 = 1 → (𝑥 − 1) = (1 − 1)) |
48 | | 1m1e0 11902 |
. . . . . . . . . 10
⊢ (1
− 1) = 0 |
49 | 47, 48 | eqtrdi 2794 |
. . . . . . . . 9
⊢ (𝑥 = 1 → (𝑥 − 1) = 0) |
50 | 49 | oveq1d 7228 |
. . . . . . . 8
⊢ (𝑥 = 1 → ((𝑥 − 1) · ((𝑆‘𝑁) + (𝑆‘(𝑁 − 1)))) = (0 · ((𝑆‘𝑁) + (𝑆‘(𝑁 − 1))))) |
51 | 46, 50 | eqeq12d 2753 |
. . . . . . 7
⊢ (𝑥 = 1 →
((♯‘{𝑔 ∈
𝐴 ∣ (𝑔‘1) ∈ (1...𝑥)}) = ((𝑥 − 1) · ((𝑆‘𝑁) + (𝑆‘(𝑁 − 1)))) ↔ (♯‘{𝑔 ∈ 𝐴 ∣ (𝑔‘1) = 1}) = (0 · ((𝑆‘𝑁) + (𝑆‘(𝑁 − 1)))))) |
52 | 35, 51 | imbi12d 348 |
. . . . . 6
⊢ (𝑥 = 1 → ((𝑥 ∈ (1...(𝑁 + 1)) → (♯‘{𝑔 ∈ 𝐴 ∣ (𝑔‘1) ∈ (1...𝑥)}) = ((𝑥 − 1) · ((𝑆‘𝑁) + (𝑆‘(𝑁 − 1))))) ↔ (1 ∈ (1...(𝑁 + 1)) →
(♯‘{𝑔 ∈
𝐴 ∣ (𝑔‘1) = 1}) = (0 ·
((𝑆‘𝑁) + (𝑆‘(𝑁 − 1))))))) |
53 | 52 | imbi2d 344 |
. . . . 5
⊢ (𝑥 = 1 → ((𝑁 ∈ ℕ → (𝑥 ∈ (1...(𝑁 + 1)) → (♯‘{𝑔 ∈ 𝐴 ∣ (𝑔‘1) ∈ (1...𝑥)}) = ((𝑥 − 1) · ((𝑆‘𝑁) + (𝑆‘(𝑁 − 1)))))) ↔ (𝑁 ∈ ℕ → (1 ∈ (1...(𝑁 + 1)) →
(♯‘{𝑔 ∈
𝐴 ∣ (𝑔‘1) = 1}) = (0 ·
((𝑆‘𝑁) + (𝑆‘(𝑁 − 1)))))))) |
54 | | eleq1 2825 |
. . . . . . 7
⊢ (𝑥 = 𝑚 → (𝑥 ∈ (1...(𝑁 + 1)) ↔ 𝑚 ∈ (1...(𝑁 + 1)))) |
55 | | oveq2 7221 |
. . . . . . . . . . 11
⊢ (𝑥 = 𝑚 → (1...𝑥) = (1...𝑚)) |
56 | 55 | eleq2d 2823 |
. . . . . . . . . 10
⊢ (𝑥 = 𝑚 → ((𝑔‘1) ∈ (1...𝑥) ↔ (𝑔‘1) ∈ (1...𝑚))) |
57 | 56 | rabbidv 3390 |
. . . . . . . . 9
⊢ (𝑥 = 𝑚 → {𝑔 ∈ 𝐴 ∣ (𝑔‘1) ∈ (1...𝑥)} = {𝑔 ∈ 𝐴 ∣ (𝑔‘1) ∈ (1...𝑚)}) |
58 | 57 | fveq2d 6721 |
. . . . . . . 8
⊢ (𝑥 = 𝑚 → (♯‘{𝑔 ∈ 𝐴 ∣ (𝑔‘1) ∈ (1...𝑥)}) = (♯‘{𝑔 ∈ 𝐴 ∣ (𝑔‘1) ∈ (1...𝑚)})) |
59 | | oveq1 7220 |
. . . . . . . . 9
⊢ (𝑥 = 𝑚 → (𝑥 − 1) = (𝑚 − 1)) |
60 | 59 | oveq1d 7228 |
. . . . . . . 8
⊢ (𝑥 = 𝑚 → ((𝑥 − 1) · ((𝑆‘𝑁) + (𝑆‘(𝑁 − 1)))) = ((𝑚 − 1) · ((𝑆‘𝑁) + (𝑆‘(𝑁 − 1))))) |
61 | 58, 60 | eqeq12d 2753 |
. . . . . . 7
⊢ (𝑥 = 𝑚 → ((♯‘{𝑔 ∈ 𝐴 ∣ (𝑔‘1) ∈ (1...𝑥)}) = ((𝑥 − 1) · ((𝑆‘𝑁) + (𝑆‘(𝑁 − 1)))) ↔ (♯‘{𝑔 ∈ 𝐴 ∣ (𝑔‘1) ∈ (1...𝑚)}) = ((𝑚 − 1) · ((𝑆‘𝑁) + (𝑆‘(𝑁 − 1)))))) |
62 | 54, 61 | imbi12d 348 |
. . . . . 6
⊢ (𝑥 = 𝑚 → ((𝑥 ∈ (1...(𝑁 + 1)) → (♯‘{𝑔 ∈ 𝐴 ∣ (𝑔‘1) ∈ (1...𝑥)}) = ((𝑥 − 1) · ((𝑆‘𝑁) + (𝑆‘(𝑁 − 1))))) ↔ (𝑚 ∈ (1...(𝑁 + 1)) → (♯‘{𝑔 ∈ 𝐴 ∣ (𝑔‘1) ∈ (1...𝑚)}) = ((𝑚 − 1) · ((𝑆‘𝑁) + (𝑆‘(𝑁 − 1))))))) |
63 | 62 | imbi2d 344 |
. . . . 5
⊢ (𝑥 = 𝑚 → ((𝑁 ∈ ℕ → (𝑥 ∈ (1...(𝑁 + 1)) → (♯‘{𝑔 ∈ 𝐴 ∣ (𝑔‘1) ∈ (1...𝑥)}) = ((𝑥 − 1) · ((𝑆‘𝑁) + (𝑆‘(𝑁 − 1)))))) ↔ (𝑁 ∈ ℕ → (𝑚 ∈ (1...(𝑁 + 1)) → (♯‘{𝑔 ∈ 𝐴 ∣ (𝑔‘1) ∈ (1...𝑚)}) = ((𝑚 − 1) · ((𝑆‘𝑁) + (𝑆‘(𝑁 − 1)))))))) |
64 | | eleq1 2825 |
. . . . . . 7
⊢ (𝑥 = (𝑚 + 1) → (𝑥 ∈ (1...(𝑁 + 1)) ↔ (𝑚 + 1) ∈ (1...(𝑁 + 1)))) |
65 | | oveq2 7221 |
. . . . . . . . . . 11
⊢ (𝑥 = (𝑚 + 1) → (1...𝑥) = (1...(𝑚 + 1))) |
66 | 65 | eleq2d 2823 |
. . . . . . . . . 10
⊢ (𝑥 = (𝑚 + 1) → ((𝑔‘1) ∈ (1...𝑥) ↔ (𝑔‘1) ∈ (1...(𝑚 + 1)))) |
67 | 66 | rabbidv 3390 |
. . . . . . . . 9
⊢ (𝑥 = (𝑚 + 1) → {𝑔 ∈ 𝐴 ∣ (𝑔‘1) ∈ (1...𝑥)} = {𝑔 ∈ 𝐴 ∣ (𝑔‘1) ∈ (1...(𝑚 + 1))}) |
68 | 67 | fveq2d 6721 |
. . . . . . . 8
⊢ (𝑥 = (𝑚 + 1) → (♯‘{𝑔 ∈ 𝐴 ∣ (𝑔‘1) ∈ (1...𝑥)}) = (♯‘{𝑔 ∈ 𝐴 ∣ (𝑔‘1) ∈ (1...(𝑚 + 1))})) |
69 | | oveq1 7220 |
. . . . . . . . 9
⊢ (𝑥 = (𝑚 + 1) → (𝑥 − 1) = ((𝑚 + 1) − 1)) |
70 | 69 | oveq1d 7228 |
. . . . . . . 8
⊢ (𝑥 = (𝑚 + 1) → ((𝑥 − 1) · ((𝑆‘𝑁) + (𝑆‘(𝑁 − 1)))) = (((𝑚 + 1) − 1) · ((𝑆‘𝑁) + (𝑆‘(𝑁 − 1))))) |
71 | 68, 70 | eqeq12d 2753 |
. . . . . . 7
⊢ (𝑥 = (𝑚 + 1) → ((♯‘{𝑔 ∈ 𝐴 ∣ (𝑔‘1) ∈ (1...𝑥)}) = ((𝑥 − 1) · ((𝑆‘𝑁) + (𝑆‘(𝑁 − 1)))) ↔ (♯‘{𝑔 ∈ 𝐴 ∣ (𝑔‘1) ∈ (1...(𝑚 + 1))}) = (((𝑚 + 1) − 1) · ((𝑆‘𝑁) + (𝑆‘(𝑁 − 1)))))) |
72 | 64, 71 | imbi12d 348 |
. . . . . 6
⊢ (𝑥 = (𝑚 + 1) → ((𝑥 ∈ (1...(𝑁 + 1)) → (♯‘{𝑔 ∈ 𝐴 ∣ (𝑔‘1) ∈ (1...𝑥)}) = ((𝑥 − 1) · ((𝑆‘𝑁) + (𝑆‘(𝑁 − 1))))) ↔ ((𝑚 + 1) ∈ (1...(𝑁 + 1)) → (♯‘{𝑔 ∈ 𝐴 ∣ (𝑔‘1) ∈ (1...(𝑚 + 1))}) = (((𝑚 + 1) − 1) · ((𝑆‘𝑁) + (𝑆‘(𝑁 − 1))))))) |
73 | 72 | imbi2d 344 |
. . . . 5
⊢ (𝑥 = (𝑚 + 1) → ((𝑁 ∈ ℕ → (𝑥 ∈ (1...(𝑁 + 1)) → (♯‘{𝑔 ∈ 𝐴 ∣ (𝑔‘1) ∈ (1...𝑥)}) = ((𝑥 − 1) · ((𝑆‘𝑁) + (𝑆‘(𝑁 − 1)))))) ↔ (𝑁 ∈ ℕ → ((𝑚 + 1) ∈ (1...(𝑁 + 1)) → (♯‘{𝑔 ∈ 𝐴 ∣ (𝑔‘1) ∈ (1...(𝑚 + 1))}) = (((𝑚 + 1) − 1) · ((𝑆‘𝑁) + (𝑆‘(𝑁 − 1)))))))) |
74 | | eleq1 2825 |
. . . . . . 7
⊢ (𝑥 = (𝑁 + 1) → (𝑥 ∈ (1...(𝑁 + 1)) ↔ (𝑁 + 1) ∈ (1...(𝑁 + 1)))) |
75 | | oveq2 7221 |
. . . . . . . . . . 11
⊢ (𝑥 = (𝑁 + 1) → (1...𝑥) = (1...(𝑁 + 1))) |
76 | 75 | eleq2d 2823 |
. . . . . . . . . 10
⊢ (𝑥 = (𝑁 + 1) → ((𝑔‘1) ∈ (1...𝑥) ↔ (𝑔‘1) ∈ (1...(𝑁 + 1)))) |
77 | 76 | rabbidv 3390 |
. . . . . . . . 9
⊢ (𝑥 = (𝑁 + 1) → {𝑔 ∈ 𝐴 ∣ (𝑔‘1) ∈ (1...𝑥)} = {𝑔 ∈ 𝐴 ∣ (𝑔‘1) ∈ (1...(𝑁 + 1))}) |
78 | 77 | fveq2d 6721 |
. . . . . . . 8
⊢ (𝑥 = (𝑁 + 1) → (♯‘{𝑔 ∈ 𝐴 ∣ (𝑔‘1) ∈ (1...𝑥)}) = (♯‘{𝑔 ∈ 𝐴 ∣ (𝑔‘1) ∈ (1...(𝑁 + 1))})) |
79 | | oveq1 7220 |
. . . . . . . . 9
⊢ (𝑥 = (𝑁 + 1) → (𝑥 − 1) = ((𝑁 + 1) − 1)) |
80 | 79 | oveq1d 7228 |
. . . . . . . 8
⊢ (𝑥 = (𝑁 + 1) → ((𝑥 − 1) · ((𝑆‘𝑁) + (𝑆‘(𝑁 − 1)))) = (((𝑁 + 1) − 1) · ((𝑆‘𝑁) + (𝑆‘(𝑁 − 1))))) |
81 | 78, 80 | eqeq12d 2753 |
. . . . . . 7
⊢ (𝑥 = (𝑁 + 1) → ((♯‘{𝑔 ∈ 𝐴 ∣ (𝑔‘1) ∈ (1...𝑥)}) = ((𝑥 − 1) · ((𝑆‘𝑁) + (𝑆‘(𝑁 − 1)))) ↔ (♯‘{𝑔 ∈ 𝐴 ∣ (𝑔‘1) ∈ (1...(𝑁 + 1))}) = (((𝑁 + 1) − 1) · ((𝑆‘𝑁) + (𝑆‘(𝑁 − 1)))))) |
82 | 74, 81 | imbi12d 348 |
. . . . . 6
⊢ (𝑥 = (𝑁 + 1) → ((𝑥 ∈ (1...(𝑁 + 1)) → (♯‘{𝑔 ∈ 𝐴 ∣ (𝑔‘1) ∈ (1...𝑥)}) = ((𝑥 − 1) · ((𝑆‘𝑁) + (𝑆‘(𝑁 − 1))))) ↔ ((𝑁 + 1) ∈ (1...(𝑁 + 1)) → (♯‘{𝑔 ∈ 𝐴 ∣ (𝑔‘1) ∈ (1...(𝑁 + 1))}) = (((𝑁 + 1) − 1) · ((𝑆‘𝑁) + (𝑆‘(𝑁 − 1))))))) |
83 | 82 | imbi2d 344 |
. . . . 5
⊢ (𝑥 = (𝑁 + 1) → ((𝑁 ∈ ℕ → (𝑥 ∈ (1...(𝑁 + 1)) → (♯‘{𝑔 ∈ 𝐴 ∣ (𝑔‘1) ∈ (1...𝑥)}) = ((𝑥 − 1) · ((𝑆‘𝑁) + (𝑆‘(𝑁 − 1)))))) ↔ (𝑁 ∈ ℕ → ((𝑁 + 1) ∈ (1...(𝑁 + 1)) → (♯‘{𝑔 ∈ 𝐴 ∣ (𝑔‘1) ∈ (1...(𝑁 + 1))}) = (((𝑁 + 1) − 1) · ((𝑆‘𝑁) + (𝑆‘(𝑁 − 1)))))))) |
84 | | hash0 13934 |
. . . . . . 7
⊢
(♯‘∅) = 0 |
85 | | fveq2 6717 |
. . . . . . . . . . . . . . . 16
⊢ (𝑦 = 1 → (𝑓‘𝑦) = (𝑓‘1)) |
86 | | id 22 |
. . . . . . . . . . . . . . . 16
⊢ (𝑦 = 1 → 𝑦 = 1) |
87 | 85, 86 | neeq12d 3002 |
. . . . . . . . . . . . . . 15
⊢ (𝑦 = 1 → ((𝑓‘𝑦) ≠ 𝑦 ↔ (𝑓‘1) ≠ 1)) |
88 | 87 | rspcv 3532 |
. . . . . . . . . . . . . 14
⊢ (1 ∈
(1...(𝑁 + 1)) →
(∀𝑦 ∈
(1...(𝑁 + 1))(𝑓‘𝑦) ≠ 𝑦 → (𝑓‘1) ≠ 1)) |
89 | 16, 88 | syl 17 |
. . . . . . . . . . . . 13
⊢ (𝑁 ∈ ℕ →
(∀𝑦 ∈
(1...(𝑁 + 1))(𝑓‘𝑦) ≠ 𝑦 → (𝑓‘1) ≠ 1)) |
90 | 89 | adantld 494 |
. . . . . . . . . . . 12
⊢ (𝑁 ∈ ℕ → ((𝑓:(1...(𝑁 + 1))–1-1-onto→(1...(𝑁 + 1)) ∧ ∀𝑦 ∈ (1...(𝑁 + 1))(𝑓‘𝑦) ≠ 𝑦) → (𝑓‘1) ≠ 1)) |
91 | 90 | ss2abdv 3977 |
. . . . . . . . . . 11
⊢ (𝑁 ∈ ℕ → {𝑓 ∣ (𝑓:(1...(𝑁 + 1))–1-1-onto→(1...(𝑁 + 1)) ∧ ∀𝑦 ∈ (1...(𝑁 + 1))(𝑓‘𝑦) ≠ 𝑦)} ⊆ {𝑓 ∣ (𝑓‘1) ≠ 1}) |
92 | | df-ne 2941 |
. . . . . . . . . . . . 13
⊢ ((𝑔‘1) ≠ 1 ↔ ¬
(𝑔‘1) =
1) |
93 | 23 | neeq1d 3000 |
. . . . . . . . . . . . 13
⊢ (𝑔 = 𝑓 → ((𝑔‘1) ≠ 1 ↔ (𝑓‘1) ≠ 1)) |
94 | 92, 93 | bitr3id 288 |
. . . . . . . . . . . 12
⊢ (𝑔 = 𝑓 → (¬ (𝑔‘1) = 1 ↔ (𝑓‘1) ≠ 1)) |
95 | 94 | cbvabv 2811 |
. . . . . . . . . . 11
⊢ {𝑔 ∣ ¬ (𝑔‘1) = 1} = {𝑓 ∣ (𝑓‘1) ≠ 1} |
96 | 91, 10, 95 | 3sstr4g 3946 |
. . . . . . . . . 10
⊢ (𝑁 ∈ ℕ → 𝐴 ⊆ {𝑔 ∣ ¬ (𝑔‘1) = 1}) |
97 | | ssabral 3976 |
. . . . . . . . . 10
⊢ (𝐴 ⊆ {𝑔 ∣ ¬ (𝑔‘1) = 1} ↔ ∀𝑔 ∈ 𝐴 ¬ (𝑔‘1) = 1) |
98 | 96, 97 | sylib 221 |
. . . . . . . . 9
⊢ (𝑁 ∈ ℕ →
∀𝑔 ∈ 𝐴 ¬ (𝑔‘1) = 1) |
99 | | rabeq0 4299 |
. . . . . . . . 9
⊢ ({𝑔 ∈ 𝐴 ∣ (𝑔‘1) = 1} = ∅ ↔ ∀𝑔 ∈ 𝐴 ¬ (𝑔‘1) = 1) |
100 | 98, 99 | sylibr 237 |
. . . . . . . 8
⊢ (𝑁 ∈ ℕ → {𝑔 ∈ 𝐴 ∣ (𝑔‘1) = 1} = ∅) |
101 | 100 | fveq2d 6721 |
. . . . . . 7
⊢ (𝑁 ∈ ℕ →
(♯‘{𝑔 ∈
𝐴 ∣ (𝑔‘1) = 1}) =
(♯‘∅)) |
102 | | nnnn0 12097 |
. . . . . . . . . . 11
⊢ (𝑁 ∈ ℕ → 𝑁 ∈
ℕ0) |
103 | 3, 4 | subfacf 32850 |
. . . . . . . . . . . 12
⊢ 𝑆:ℕ0⟶ℕ0 |
104 | 103 | ffvelrni 6903 |
. . . . . . . . . . 11
⊢ (𝑁 ∈ ℕ0
→ (𝑆‘𝑁) ∈
ℕ0) |
105 | 102, 104 | syl 17 |
. . . . . . . . . 10
⊢ (𝑁 ∈ ℕ → (𝑆‘𝑁) ∈
ℕ0) |
106 | | nnm1nn0 12131 |
. . . . . . . . . . 11
⊢ (𝑁 ∈ ℕ → (𝑁 − 1) ∈
ℕ0) |
107 | 103 | ffvelrni 6903 |
. . . . . . . . . . 11
⊢ ((𝑁 − 1) ∈
ℕ0 → (𝑆‘(𝑁 − 1)) ∈
ℕ0) |
108 | 106, 107 | syl 17 |
. . . . . . . . . 10
⊢ (𝑁 ∈ ℕ → (𝑆‘(𝑁 − 1)) ∈
ℕ0) |
109 | 105, 108 | nn0addcld 12154 |
. . . . . . . . 9
⊢ (𝑁 ∈ ℕ → ((𝑆‘𝑁) + (𝑆‘(𝑁 − 1))) ∈
ℕ0) |
110 | 109 | nn0cnd 12152 |
. . . . . . . 8
⊢ (𝑁 ∈ ℕ → ((𝑆‘𝑁) + (𝑆‘(𝑁 − 1))) ∈
ℂ) |
111 | 110 | mul02d 11030 |
. . . . . . 7
⊢ (𝑁 ∈ ℕ → (0
· ((𝑆‘𝑁) + (𝑆‘(𝑁 − 1)))) = 0) |
112 | 84, 101, 111 | 3eqtr4a 2804 |
. . . . . 6
⊢ (𝑁 ∈ ℕ →
(♯‘{𝑔 ∈
𝐴 ∣ (𝑔‘1) = 1}) = (0 ·
((𝑆‘𝑁) + (𝑆‘(𝑁 − 1))))) |
113 | 112 | a1d 25 |
. . . . 5
⊢ (𝑁 ∈ ℕ → (1 ∈
(1...(𝑁 + 1)) →
(♯‘{𝑔 ∈
𝐴 ∣ (𝑔‘1) = 1}) = (0 ·
((𝑆‘𝑁) + (𝑆‘(𝑁 − 1)))))) |
114 | | simplr 769 |
. . . . . . . . . . . 12
⊢ (((𝑁 ∈ ℕ ∧ 𝑚 ∈ ℕ) ∧ (𝑚 + 1) ∈ (1...(𝑁 + 1))) → 𝑚 ∈
ℕ) |
115 | 114, 13 | eleqtrdi 2848 |
. . . . . . . . . . 11
⊢ (((𝑁 ∈ ℕ ∧ 𝑚 ∈ ℕ) ∧ (𝑚 + 1) ∈ (1...(𝑁 + 1))) → 𝑚 ∈
(ℤ≥‘1)) |
116 | | peano2fzr 13125 |
. . . . . . . . . . 11
⊢ ((𝑚 ∈
(ℤ≥‘1) ∧ (𝑚 + 1) ∈ (1...(𝑁 + 1))) → 𝑚 ∈ (1...(𝑁 + 1))) |
117 | 115, 116 | sylancom 591 |
. . . . . . . . . 10
⊢ (((𝑁 ∈ ℕ ∧ 𝑚 ∈ ℕ) ∧ (𝑚 + 1) ∈ (1...(𝑁 + 1))) → 𝑚 ∈ (1...(𝑁 + 1))) |
118 | 117 | ex 416 |
. . . . . . . . 9
⊢ ((𝑁 ∈ ℕ ∧ 𝑚 ∈ ℕ) → ((𝑚 + 1) ∈ (1...(𝑁 + 1)) → 𝑚 ∈ (1...(𝑁 + 1)))) |
119 | 118 | imim1d 82 |
. . . . . . . 8
⊢ ((𝑁 ∈ ℕ ∧ 𝑚 ∈ ℕ) → ((𝑚 ∈ (1...(𝑁 + 1)) → (♯‘{𝑔 ∈ 𝐴 ∣ (𝑔‘1) ∈ (1...𝑚)}) = ((𝑚 − 1) · ((𝑆‘𝑁) + (𝑆‘(𝑁 − 1))))) → ((𝑚 + 1) ∈ (1...(𝑁 + 1)) → (♯‘{𝑔 ∈ 𝐴 ∣ (𝑔‘1) ∈ (1...𝑚)}) = ((𝑚 − 1) · ((𝑆‘𝑁) + (𝑆‘(𝑁 − 1))))))) |
120 | | oveq1 7220 |
. . . . . . . . . . 11
⊢
((♯‘{𝑔
∈ 𝐴 ∣ (𝑔‘1) ∈ (1...𝑚)}) = ((𝑚 − 1) · ((𝑆‘𝑁) + (𝑆‘(𝑁 − 1)))) → ((♯‘{𝑔 ∈ 𝐴 ∣ (𝑔‘1) ∈ (1...𝑚)}) + (♯‘{𝑔 ∈ 𝐴 ∣ (𝑔‘1) = (𝑚 + 1)})) = (((𝑚 − 1) · ((𝑆‘𝑁) + (𝑆‘(𝑁 − 1)))) + (♯‘{𝑔 ∈ 𝐴 ∣ (𝑔‘1) = (𝑚 + 1)}))) |
121 | | elfzp1 13162 |
. . . . . . . . . . . . . . . . 17
⊢ (𝑚 ∈
(ℤ≥‘1) → ((𝑔‘1) ∈ (1...(𝑚 + 1)) ↔ ((𝑔‘1) ∈ (1...𝑚) ∨ (𝑔‘1) = (𝑚 + 1)))) |
122 | 115, 121 | syl 17 |
. . . . . . . . . . . . . . . 16
⊢ (((𝑁 ∈ ℕ ∧ 𝑚 ∈ ℕ) ∧ (𝑚 + 1) ∈ (1...(𝑁 + 1))) → ((𝑔‘1) ∈ (1...(𝑚 + 1)) ↔ ((𝑔‘1) ∈ (1...𝑚) ∨ (𝑔‘1) = (𝑚 + 1)))) |
123 | 122 | rabbidv 3390 |
. . . . . . . . . . . . . . 15
⊢ (((𝑁 ∈ ℕ ∧ 𝑚 ∈ ℕ) ∧ (𝑚 + 1) ∈ (1...(𝑁 + 1))) → {𝑔 ∈ 𝐴 ∣ (𝑔‘1) ∈ (1...(𝑚 + 1))} = {𝑔 ∈ 𝐴 ∣ ((𝑔‘1) ∈ (1...𝑚) ∨ (𝑔‘1) = (𝑚 + 1))}) |
124 | | unrab 4220 |
. . . . . . . . . . . . . . 15
⊢ ({𝑔 ∈ 𝐴 ∣ (𝑔‘1) ∈ (1...𝑚)} ∪ {𝑔 ∈ 𝐴 ∣ (𝑔‘1) = (𝑚 + 1)}) = {𝑔 ∈ 𝐴 ∣ ((𝑔‘1) ∈ (1...𝑚) ∨ (𝑔‘1) = (𝑚 + 1))} |
125 | 123, 124 | eqtr4di 2796 |
. . . . . . . . . . . . . 14
⊢ (((𝑁 ∈ ℕ ∧ 𝑚 ∈ ℕ) ∧ (𝑚 + 1) ∈ (1...(𝑁 + 1))) → {𝑔 ∈ 𝐴 ∣ (𝑔‘1) ∈ (1...(𝑚 + 1))} = ({𝑔 ∈ 𝐴 ∣ (𝑔‘1) ∈ (1...𝑚)} ∪ {𝑔 ∈ 𝐴 ∣ (𝑔‘1) = (𝑚 + 1)})) |
126 | 125 | fveq2d 6721 |
. . . . . . . . . . . . 13
⊢ (((𝑁 ∈ ℕ ∧ 𝑚 ∈ ℕ) ∧ (𝑚 + 1) ∈ (1...(𝑁 + 1))) →
(♯‘{𝑔 ∈
𝐴 ∣ (𝑔‘1) ∈ (1...(𝑚 + 1))}) =
(♯‘({𝑔 ∈
𝐴 ∣ (𝑔‘1) ∈ (1...𝑚)} ∪ {𝑔 ∈ 𝐴 ∣ (𝑔‘1) = (𝑚 + 1)}))) |
127 | | fzfi 13545 |
. . . . . . . . . . . . . . . . 17
⊢
(1...(𝑁 + 1)) ∈
Fin |
128 | | deranglem 32841 |
. . . . . . . . . . . . . . . . 17
⊢
((1...(𝑁 + 1))
∈ Fin → {𝑓
∣ (𝑓:(1...(𝑁 + 1))–1-1-onto→(1...(𝑁 + 1)) ∧ ∀𝑦 ∈ (1...(𝑁 + 1))(𝑓‘𝑦) ≠ 𝑦)} ∈ Fin) |
129 | 127, 128 | ax-mp 5 |
. . . . . . . . . . . . . . . 16
⊢ {𝑓 ∣ (𝑓:(1...(𝑁 + 1))–1-1-onto→(1...(𝑁 + 1)) ∧ ∀𝑦 ∈ (1...(𝑁 + 1))(𝑓‘𝑦) ≠ 𝑦)} ∈ Fin |
130 | 10, 129 | eqeltri 2834 |
. . . . . . . . . . . . . . 15
⊢ 𝐴 ∈ Fin |
131 | | ssrab2 3993 |
. . . . . . . . . . . . . . 15
⊢ {𝑔 ∈ 𝐴 ∣ (𝑔‘1) ∈ (1...𝑚)} ⊆ 𝐴 |
132 | | ssfi 8851 |
. . . . . . . . . . . . . . 15
⊢ ((𝐴 ∈ Fin ∧ {𝑔 ∈ 𝐴 ∣ (𝑔‘1) ∈ (1...𝑚)} ⊆ 𝐴) → {𝑔 ∈ 𝐴 ∣ (𝑔‘1) ∈ (1...𝑚)} ∈ Fin) |
133 | 130, 131,
132 | mp2an 692 |
. . . . . . . . . . . . . 14
⊢ {𝑔 ∈ 𝐴 ∣ (𝑔‘1) ∈ (1...𝑚)} ∈ Fin |
134 | | ssrab2 3993 |
. . . . . . . . . . . . . . 15
⊢ {𝑔 ∈ 𝐴 ∣ (𝑔‘1) = (𝑚 + 1)} ⊆ 𝐴 |
135 | | ssfi 8851 |
. . . . . . . . . . . . . . 15
⊢ ((𝐴 ∈ Fin ∧ {𝑔 ∈ 𝐴 ∣ (𝑔‘1) = (𝑚 + 1)} ⊆ 𝐴) → {𝑔 ∈ 𝐴 ∣ (𝑔‘1) = (𝑚 + 1)} ∈ Fin) |
136 | 130, 134,
135 | mp2an 692 |
. . . . . . . . . . . . . 14
⊢ {𝑔 ∈ 𝐴 ∣ (𝑔‘1) = (𝑚 + 1)} ∈ Fin |
137 | | inrab 4221 |
. . . . . . . . . . . . . . 15
⊢ ({𝑔 ∈ 𝐴 ∣ (𝑔‘1) ∈ (1...𝑚)} ∩ {𝑔 ∈ 𝐴 ∣ (𝑔‘1) = (𝑚 + 1)}) = {𝑔 ∈ 𝐴 ∣ ((𝑔‘1) ∈ (1...𝑚) ∧ (𝑔‘1) = (𝑚 + 1))} |
138 | | fzp1disj 13171 |
. . . . . . . . . . . . . . . . . 18
⊢
((1...𝑚) ∩
{(𝑚 + 1)}) =
∅ |
139 | 42 | elsn 4556 |
. . . . . . . . . . . . . . . . . . . 20
⊢ ((𝑔‘1) ∈ {(𝑚 + 1)} ↔ (𝑔‘1) = (𝑚 + 1)) |
140 | | inelcm 4379 |
. . . . . . . . . . . . . . . . . . . 20
⊢ (((𝑔‘1) ∈ (1...𝑚) ∧ (𝑔‘1) ∈ {(𝑚 + 1)}) → ((1...𝑚) ∩ {(𝑚 + 1)}) ≠ ∅) |
141 | 139, 140 | sylan2br 598 |
. . . . . . . . . . . . . . . . . . 19
⊢ (((𝑔‘1) ∈ (1...𝑚) ∧ (𝑔‘1) = (𝑚 + 1)) → ((1...𝑚) ∩ {(𝑚 + 1)}) ≠ ∅) |
142 | 141 | necon2bi 2971 |
. . . . . . . . . . . . . . . . . 18
⊢
(((1...𝑚) ∩
{(𝑚 + 1)}) = ∅ →
¬ ((𝑔‘1) ∈
(1...𝑚) ∧ (𝑔‘1) = (𝑚 + 1))) |
143 | 138, 142 | ax-mp 5 |
. . . . . . . . . . . . . . . . 17
⊢ ¬
((𝑔‘1) ∈
(1...𝑚) ∧ (𝑔‘1) = (𝑚 + 1)) |
144 | 143 | rgenw 3073 |
. . . . . . . . . . . . . . . 16
⊢
∀𝑔 ∈
𝐴 ¬ ((𝑔‘1) ∈ (1...𝑚) ∧ (𝑔‘1) = (𝑚 + 1)) |
145 | | rabeq0 4299 |
. . . . . . . . . . . . . . . 16
⊢ ({𝑔 ∈ 𝐴 ∣ ((𝑔‘1) ∈ (1...𝑚) ∧ (𝑔‘1) = (𝑚 + 1))} = ∅ ↔ ∀𝑔 ∈ 𝐴 ¬ ((𝑔‘1) ∈ (1...𝑚) ∧ (𝑔‘1) = (𝑚 + 1))) |
146 | 144, 145 | mpbir 234 |
. . . . . . . . . . . . . . 15
⊢ {𝑔 ∈ 𝐴 ∣ ((𝑔‘1) ∈ (1...𝑚) ∧ (𝑔‘1) = (𝑚 + 1))} = ∅ |
147 | 137, 146 | eqtri 2765 |
. . . . . . . . . . . . . 14
⊢ ({𝑔 ∈ 𝐴 ∣ (𝑔‘1) ∈ (1...𝑚)} ∩ {𝑔 ∈ 𝐴 ∣ (𝑔‘1) = (𝑚 + 1)}) = ∅ |
148 | | hashun 13949 |
. . . . . . . . . . . . . 14
⊢ (({𝑔 ∈ 𝐴 ∣ (𝑔‘1) ∈ (1...𝑚)} ∈ Fin ∧ {𝑔 ∈ 𝐴 ∣ (𝑔‘1) = (𝑚 + 1)} ∈ Fin ∧ ({𝑔 ∈ 𝐴 ∣ (𝑔‘1) ∈ (1...𝑚)} ∩ {𝑔 ∈ 𝐴 ∣ (𝑔‘1) = (𝑚 + 1)}) = ∅) →
(♯‘({𝑔 ∈
𝐴 ∣ (𝑔‘1) ∈ (1...𝑚)} ∪ {𝑔 ∈ 𝐴 ∣ (𝑔‘1) = (𝑚 + 1)})) = ((♯‘{𝑔 ∈ 𝐴 ∣ (𝑔‘1) ∈ (1...𝑚)}) + (♯‘{𝑔 ∈ 𝐴 ∣ (𝑔‘1) = (𝑚 + 1)}))) |
149 | 133, 136,
147, 148 | mp3an 1463 |
. . . . . . . . . . . . 13
⊢
(♯‘({𝑔
∈ 𝐴 ∣ (𝑔‘1) ∈ (1...𝑚)} ∪ {𝑔 ∈ 𝐴 ∣ (𝑔‘1) = (𝑚 + 1)})) = ((♯‘{𝑔 ∈ 𝐴 ∣ (𝑔‘1) ∈ (1...𝑚)}) + (♯‘{𝑔 ∈ 𝐴 ∣ (𝑔‘1) = (𝑚 + 1)})) |
150 | 126, 149 | eqtrdi 2794 |
. . . . . . . . . . . 12
⊢ (((𝑁 ∈ ℕ ∧ 𝑚 ∈ ℕ) ∧ (𝑚 + 1) ∈ (1...(𝑁 + 1))) →
(♯‘{𝑔 ∈
𝐴 ∣ (𝑔‘1) ∈ (1...(𝑚 + 1))}) =
((♯‘{𝑔 ∈
𝐴 ∣ (𝑔‘1) ∈ (1...𝑚)}) + (♯‘{𝑔 ∈ 𝐴 ∣ (𝑔‘1) = (𝑚 + 1)}))) |
151 | | nncn 11838 |
. . . . . . . . . . . . . . . 16
⊢ (𝑚 ∈ ℕ → 𝑚 ∈
ℂ) |
152 | 151 | ad2antlr 727 |
. . . . . . . . . . . . . . 15
⊢ (((𝑁 ∈ ℕ ∧ 𝑚 ∈ ℕ) ∧ (𝑚 + 1) ∈ (1...(𝑁 + 1))) → 𝑚 ∈
ℂ) |
153 | | ax-1cn 10787 |
. . . . . . . . . . . . . . . 16
⊢ 1 ∈
ℂ |
154 | 153 | a1i 11 |
. . . . . . . . . . . . . . 15
⊢ (((𝑁 ∈ ℕ ∧ 𝑚 ∈ ℕ) ∧ (𝑚 + 1) ∈ (1...(𝑁 + 1))) → 1 ∈
ℂ) |
155 | 152, 154,
154 | addsubd 11210 |
. . . . . . . . . . . . . 14
⊢ (((𝑁 ∈ ℕ ∧ 𝑚 ∈ ℕ) ∧ (𝑚 + 1) ∈ (1...(𝑁 + 1))) → ((𝑚 + 1) − 1) = ((𝑚 − 1) +
1)) |
156 | 155 | oveq1d 7228 |
. . . . . . . . . . . . 13
⊢ (((𝑁 ∈ ℕ ∧ 𝑚 ∈ ℕ) ∧ (𝑚 + 1) ∈ (1...(𝑁 + 1))) → (((𝑚 + 1) − 1) ·
((𝑆‘𝑁) + (𝑆‘(𝑁 − 1)))) = (((𝑚 − 1) + 1) · ((𝑆‘𝑁) + (𝑆‘(𝑁 − 1))))) |
157 | | subcl 11077 |
. . . . . . . . . . . . . . 15
⊢ ((𝑚 ∈ ℂ ∧ 1 ∈
ℂ) → (𝑚 −
1) ∈ ℂ) |
158 | 152, 153,
157 | sylancl 589 |
. . . . . . . . . . . . . 14
⊢ (((𝑁 ∈ ℕ ∧ 𝑚 ∈ ℕ) ∧ (𝑚 + 1) ∈ (1...(𝑁 + 1))) → (𝑚 − 1) ∈
ℂ) |
159 | 109 | ad2antrr 726 |
. . . . . . . . . . . . . . 15
⊢ (((𝑁 ∈ ℕ ∧ 𝑚 ∈ ℕ) ∧ (𝑚 + 1) ∈ (1...(𝑁 + 1))) → ((𝑆‘𝑁) + (𝑆‘(𝑁 − 1))) ∈
ℕ0) |
160 | 159 | nn0cnd 12152 |
. . . . . . . . . . . . . 14
⊢ (((𝑁 ∈ ℕ ∧ 𝑚 ∈ ℕ) ∧ (𝑚 + 1) ∈ (1...(𝑁 + 1))) → ((𝑆‘𝑁) + (𝑆‘(𝑁 − 1))) ∈
ℂ) |
161 | 158, 154,
160 | adddird 10858 |
. . . . . . . . . . . . 13
⊢ (((𝑁 ∈ ℕ ∧ 𝑚 ∈ ℕ) ∧ (𝑚 + 1) ∈ (1...(𝑁 + 1))) → (((𝑚 − 1) + 1) ·
((𝑆‘𝑁) + (𝑆‘(𝑁 − 1)))) = (((𝑚 − 1) · ((𝑆‘𝑁) + (𝑆‘(𝑁 − 1)))) + (1 · ((𝑆‘𝑁) + (𝑆‘(𝑁 − 1)))))) |
162 | 160 | mulid2d 10851 |
. . . . . . . . . . . . . . 15
⊢ (((𝑁 ∈ ℕ ∧ 𝑚 ∈ ℕ) ∧ (𝑚 + 1) ∈ (1...(𝑁 + 1))) → (1 ·
((𝑆‘𝑁) + (𝑆‘(𝑁 − 1)))) = ((𝑆‘𝑁) + (𝑆‘(𝑁 − 1)))) |
163 | | exmidne 2950 |
. . . . . . . . . . . . . . . . . . . . . . 23
⊢ ((𝑔‘(𝑚 + 1)) = 1 ∨ (𝑔‘(𝑚 + 1)) ≠ 1) |
164 | | orcom 870 |
. . . . . . . . . . . . . . . . . . . . . . 23
⊢ (((𝑔‘(𝑚 + 1)) = 1 ∨ (𝑔‘(𝑚 + 1)) ≠ 1) ↔ ((𝑔‘(𝑚 + 1)) ≠ 1 ∨ (𝑔‘(𝑚 + 1)) = 1)) |
165 | 163, 164 | mpbi 233 |
. . . . . . . . . . . . . . . . . . . . . 22
⊢ ((𝑔‘(𝑚 + 1)) ≠ 1 ∨ (𝑔‘(𝑚 + 1)) = 1) |
166 | 165 | biantru 533 |
. . . . . . . . . . . . . . . . . . . . 21
⊢ ((𝑔‘1) = (𝑚 + 1) ↔ ((𝑔‘1) = (𝑚 + 1) ∧ ((𝑔‘(𝑚 + 1)) ≠ 1 ∨ (𝑔‘(𝑚 + 1)) = 1))) |
167 | | andi 1008 |
. . . . . . . . . . . . . . . . . . . . 21
⊢ (((𝑔‘1) = (𝑚 + 1) ∧ ((𝑔‘(𝑚 + 1)) ≠ 1 ∨ (𝑔‘(𝑚 + 1)) = 1)) ↔ (((𝑔‘1) = (𝑚 + 1) ∧ (𝑔‘(𝑚 + 1)) ≠ 1) ∨ ((𝑔‘1) = (𝑚 + 1) ∧ (𝑔‘(𝑚 + 1)) = 1))) |
168 | 166, 167 | bitri 278 |
. . . . . . . . . . . . . . . . . . . 20
⊢ ((𝑔‘1) = (𝑚 + 1) ↔ (((𝑔‘1) = (𝑚 + 1) ∧ (𝑔‘(𝑚 + 1)) ≠ 1) ∨ ((𝑔‘1) = (𝑚 + 1) ∧ (𝑔‘(𝑚 + 1)) = 1))) |
169 | 168 | rabbii 3383 |
. . . . . . . . . . . . . . . . . . 19
⊢ {𝑔 ∈ 𝐴 ∣ (𝑔‘1) = (𝑚 + 1)} = {𝑔 ∈ 𝐴 ∣ (((𝑔‘1) = (𝑚 + 1) ∧ (𝑔‘(𝑚 + 1)) ≠ 1) ∨ ((𝑔‘1) = (𝑚 + 1) ∧ (𝑔‘(𝑚 + 1)) = 1))} |
170 | | unrab 4220 |
. . . . . . . . . . . . . . . . . . 19
⊢ ({𝑔 ∈ 𝐴 ∣ ((𝑔‘1) = (𝑚 + 1) ∧ (𝑔‘(𝑚 + 1)) ≠ 1)} ∪ {𝑔 ∈ 𝐴 ∣ ((𝑔‘1) = (𝑚 + 1) ∧ (𝑔‘(𝑚 + 1)) = 1)}) = {𝑔 ∈ 𝐴 ∣ (((𝑔‘1) = (𝑚 + 1) ∧ (𝑔‘(𝑚 + 1)) ≠ 1) ∨ ((𝑔‘1) = (𝑚 + 1) ∧ (𝑔‘(𝑚 + 1)) = 1))} |
171 | 169, 170 | eqtr4i 2768 |
. . . . . . . . . . . . . . . . . 18
⊢ {𝑔 ∈ 𝐴 ∣ (𝑔‘1) = (𝑚 + 1)} = ({𝑔 ∈ 𝐴 ∣ ((𝑔‘1) = (𝑚 + 1) ∧ (𝑔‘(𝑚 + 1)) ≠ 1)} ∪ {𝑔 ∈ 𝐴 ∣ ((𝑔‘1) = (𝑚 + 1) ∧ (𝑔‘(𝑚 + 1)) = 1)}) |
172 | 171 | fveq2i 6720 |
. . . . . . . . . . . . . . . . 17
⊢
(♯‘{𝑔
∈ 𝐴 ∣ (𝑔‘1) = (𝑚 + 1)}) = (♯‘({𝑔 ∈ 𝐴 ∣ ((𝑔‘1) = (𝑚 + 1) ∧ (𝑔‘(𝑚 + 1)) ≠ 1)} ∪ {𝑔 ∈ 𝐴 ∣ ((𝑔‘1) = (𝑚 + 1) ∧ (𝑔‘(𝑚 + 1)) = 1)})) |
173 | | ssrab2 3993 |
. . . . . . . . . . . . . . . . . . 19
⊢ {𝑔 ∈ 𝐴 ∣ ((𝑔‘1) = (𝑚 + 1) ∧ (𝑔‘(𝑚 + 1)) ≠ 1)} ⊆ 𝐴 |
174 | | ssfi 8851 |
. . . . . . . . . . . . . . . . . . 19
⊢ ((𝐴 ∈ Fin ∧ {𝑔 ∈ 𝐴 ∣ ((𝑔‘1) = (𝑚 + 1) ∧ (𝑔‘(𝑚 + 1)) ≠ 1)} ⊆ 𝐴) → {𝑔 ∈ 𝐴 ∣ ((𝑔‘1) = (𝑚 + 1) ∧ (𝑔‘(𝑚 + 1)) ≠ 1)} ∈ Fin) |
175 | 130, 173,
174 | mp2an 692 |
. . . . . . . . . . . . . . . . . 18
⊢ {𝑔 ∈ 𝐴 ∣ ((𝑔‘1) = (𝑚 + 1) ∧ (𝑔‘(𝑚 + 1)) ≠ 1)} ∈ Fin |
176 | | ssrab2 3993 |
. . . . . . . . . . . . . . . . . . 19
⊢ {𝑔 ∈ 𝐴 ∣ ((𝑔‘1) = (𝑚 + 1) ∧ (𝑔‘(𝑚 + 1)) = 1)} ⊆ 𝐴 |
177 | | ssfi 8851 |
. . . . . . . . . . . . . . . . . . 19
⊢ ((𝐴 ∈ Fin ∧ {𝑔 ∈ 𝐴 ∣ ((𝑔‘1) = (𝑚 + 1) ∧ (𝑔‘(𝑚 + 1)) = 1)} ⊆ 𝐴) → {𝑔 ∈ 𝐴 ∣ ((𝑔‘1) = (𝑚 + 1) ∧ (𝑔‘(𝑚 + 1)) = 1)} ∈ Fin) |
178 | 130, 176,
177 | mp2an 692 |
. . . . . . . . . . . . . . . . . 18
⊢ {𝑔 ∈ 𝐴 ∣ ((𝑔‘1) = (𝑚 + 1) ∧ (𝑔‘(𝑚 + 1)) = 1)} ∈ Fin |
179 | | inrab 4221 |
. . . . . . . . . . . . . . . . . . 19
⊢ ({𝑔 ∈ 𝐴 ∣ ((𝑔‘1) = (𝑚 + 1) ∧ (𝑔‘(𝑚 + 1)) ≠ 1)} ∩ {𝑔 ∈ 𝐴 ∣ ((𝑔‘1) = (𝑚 + 1) ∧ (𝑔‘(𝑚 + 1)) = 1)}) = {𝑔 ∈ 𝐴 ∣ (((𝑔‘1) = (𝑚 + 1) ∧ (𝑔‘(𝑚 + 1)) ≠ 1) ∧ ((𝑔‘1) = (𝑚 + 1) ∧ (𝑔‘(𝑚 + 1)) = 1))} |
180 | | simpr 488 |
. . . . . . . . . . . . . . . . . . . . . . . 24
⊢ (((𝑔‘1) = (𝑚 + 1) ∧ (𝑔‘(𝑚 + 1)) = 1) → (𝑔‘(𝑚 + 1)) = 1) |
181 | 180 | necon3ai 2965 |
. . . . . . . . . . . . . . . . . . . . . . 23
⊢ ((𝑔‘(𝑚 + 1)) ≠ 1 → ¬ ((𝑔‘1) = (𝑚 + 1) ∧ (𝑔‘(𝑚 + 1)) = 1)) |
182 | 181 | adantl 485 |
. . . . . . . . . . . . . . . . . . . . . 22
⊢ (((𝑔‘1) = (𝑚 + 1) ∧ (𝑔‘(𝑚 + 1)) ≠ 1) → ¬ ((𝑔‘1) = (𝑚 + 1) ∧ (𝑔‘(𝑚 + 1)) = 1)) |
183 | | imnan 403 |
. . . . . . . . . . . . . . . . . . . . . 22
⊢ ((((𝑔‘1) = (𝑚 + 1) ∧ (𝑔‘(𝑚 + 1)) ≠ 1) → ¬ ((𝑔‘1) = (𝑚 + 1) ∧ (𝑔‘(𝑚 + 1)) = 1)) ↔ ¬ (((𝑔‘1) = (𝑚 + 1) ∧ (𝑔‘(𝑚 + 1)) ≠ 1) ∧ ((𝑔‘1) = (𝑚 + 1) ∧ (𝑔‘(𝑚 + 1)) = 1))) |
184 | 182, 183 | mpbi 233 |
. . . . . . . . . . . . . . . . . . . . 21
⊢ ¬
(((𝑔‘1) = (𝑚 + 1) ∧ (𝑔‘(𝑚 + 1)) ≠ 1) ∧ ((𝑔‘1) = (𝑚 + 1) ∧ (𝑔‘(𝑚 + 1)) = 1)) |
185 | 184 | rgenw 3073 |
. . . . . . . . . . . . . . . . . . . 20
⊢
∀𝑔 ∈
𝐴 ¬ (((𝑔‘1) = (𝑚 + 1) ∧ (𝑔‘(𝑚 + 1)) ≠ 1) ∧ ((𝑔‘1) = (𝑚 + 1) ∧ (𝑔‘(𝑚 + 1)) = 1)) |
186 | | rabeq0 4299 |
. . . . . . . . . . . . . . . . . . . 20
⊢ ({𝑔 ∈ 𝐴 ∣ (((𝑔‘1) = (𝑚 + 1) ∧ (𝑔‘(𝑚 + 1)) ≠ 1) ∧ ((𝑔‘1) = (𝑚 + 1) ∧ (𝑔‘(𝑚 + 1)) = 1))} = ∅ ↔ ∀𝑔 ∈ 𝐴 ¬ (((𝑔‘1) = (𝑚 + 1) ∧ (𝑔‘(𝑚 + 1)) ≠ 1) ∧ ((𝑔‘1) = (𝑚 + 1) ∧ (𝑔‘(𝑚 + 1)) = 1))) |
187 | 185, 186 | mpbir 234 |
. . . . . . . . . . . . . . . . . . 19
⊢ {𝑔 ∈ 𝐴 ∣ (((𝑔‘1) = (𝑚 + 1) ∧ (𝑔‘(𝑚 + 1)) ≠ 1) ∧ ((𝑔‘1) = (𝑚 + 1) ∧ (𝑔‘(𝑚 + 1)) = 1))} = ∅ |
188 | 179, 187 | eqtri 2765 |
. . . . . . . . . . . . . . . . . 18
⊢ ({𝑔 ∈ 𝐴 ∣ ((𝑔‘1) = (𝑚 + 1) ∧ (𝑔‘(𝑚 + 1)) ≠ 1)} ∩ {𝑔 ∈ 𝐴 ∣ ((𝑔‘1) = (𝑚 + 1) ∧ (𝑔‘(𝑚 + 1)) = 1)}) = ∅ |
189 | | hashun 13949 |
. . . . . . . . . . . . . . . . . 18
⊢ (({𝑔 ∈ 𝐴 ∣ ((𝑔‘1) = (𝑚 + 1) ∧ (𝑔‘(𝑚 + 1)) ≠ 1)} ∈ Fin ∧ {𝑔 ∈ 𝐴 ∣ ((𝑔‘1) = (𝑚 + 1) ∧ (𝑔‘(𝑚 + 1)) = 1)} ∈ Fin ∧ ({𝑔 ∈ 𝐴 ∣ ((𝑔‘1) = (𝑚 + 1) ∧ (𝑔‘(𝑚 + 1)) ≠ 1)} ∩ {𝑔 ∈ 𝐴 ∣ ((𝑔‘1) = (𝑚 + 1) ∧ (𝑔‘(𝑚 + 1)) = 1)}) = ∅) →
(♯‘({𝑔 ∈
𝐴 ∣ ((𝑔‘1) = (𝑚 + 1) ∧ (𝑔‘(𝑚 + 1)) ≠ 1)} ∪ {𝑔 ∈ 𝐴 ∣ ((𝑔‘1) = (𝑚 + 1) ∧ (𝑔‘(𝑚 + 1)) = 1)})) = ((♯‘{𝑔 ∈ 𝐴 ∣ ((𝑔‘1) = (𝑚 + 1) ∧ (𝑔‘(𝑚 + 1)) ≠ 1)}) + (♯‘{𝑔 ∈ 𝐴 ∣ ((𝑔‘1) = (𝑚 + 1) ∧ (𝑔‘(𝑚 + 1)) = 1)}))) |
190 | 175, 178,
188, 189 | mp3an 1463 |
. . . . . . . . . . . . . . . . 17
⊢
(♯‘({𝑔
∈ 𝐴 ∣ ((𝑔‘1) = (𝑚 + 1) ∧ (𝑔‘(𝑚 + 1)) ≠ 1)} ∪ {𝑔 ∈ 𝐴 ∣ ((𝑔‘1) = (𝑚 + 1) ∧ (𝑔‘(𝑚 + 1)) = 1)})) = ((♯‘{𝑔 ∈ 𝐴 ∣ ((𝑔‘1) = (𝑚 + 1) ∧ (𝑔‘(𝑚 + 1)) ≠ 1)}) + (♯‘{𝑔 ∈ 𝐴 ∣ ((𝑔‘1) = (𝑚 + 1) ∧ (𝑔‘(𝑚 + 1)) = 1)})) |
191 | 172, 190 | eqtri 2765 |
. . . . . . . . . . . . . . . 16
⊢
(♯‘{𝑔
∈ 𝐴 ∣ (𝑔‘1) = (𝑚 + 1)}) = ((♯‘{𝑔 ∈ 𝐴 ∣ ((𝑔‘1) = (𝑚 + 1) ∧ (𝑔‘(𝑚 + 1)) ≠ 1)}) + (♯‘{𝑔 ∈ 𝐴 ∣ ((𝑔‘1) = (𝑚 + 1) ∧ (𝑔‘(𝑚 + 1)) = 1)})) |
192 | | simpll 767 |
. . . . . . . . . . . . . . . . . 18
⊢ (((𝑁 ∈ ℕ ∧ 𝑚 ∈ ℕ) ∧ (𝑚 + 1) ∈ (1...(𝑁 + 1))) → 𝑁 ∈ ℕ) |
193 | | nnne0 11864 |
. . . . . . . . . . . . . . . . . . . . . 22
⊢ (𝑚 ∈ ℕ → 𝑚 ≠ 0) |
194 | | 0p1e1 11952 |
. . . . . . . . . . . . . . . . . . . . . . . . 25
⊢ (0 + 1) =
1 |
195 | 194 | eqeq2i 2750 |
. . . . . . . . . . . . . . . . . . . . . . . 24
⊢ ((𝑚 + 1) = (0 + 1) ↔ (𝑚 + 1) = 1) |
196 | | 0cn 10825 |
. . . . . . . . . . . . . . . . . . . . . . . . . 26
⊢ 0 ∈
ℂ |
197 | | addcan2 11017 |
. . . . . . . . . . . . . . . . . . . . . . . . . 26
⊢ ((𝑚 ∈ ℂ ∧ 0 ∈
ℂ ∧ 1 ∈ ℂ) → ((𝑚 + 1) = (0 + 1) ↔ 𝑚 = 0)) |
198 | 196, 153,
197 | mp3an23 1455 |
. . . . . . . . . . . . . . . . . . . . . . . . 25
⊢ (𝑚 ∈ ℂ → ((𝑚 + 1) = (0 + 1) ↔ 𝑚 = 0)) |
199 | 151, 198 | syl 17 |
. . . . . . . . . . . . . . . . . . . . . . . 24
⊢ (𝑚 ∈ ℕ → ((𝑚 + 1) = (0 + 1) ↔ 𝑚 = 0)) |
200 | 195, 199 | bitr3id 288 |
. . . . . . . . . . . . . . . . . . . . . . 23
⊢ (𝑚 ∈ ℕ → ((𝑚 + 1) = 1 ↔ 𝑚 = 0)) |
201 | 200 | necon3bbid 2978 |
. . . . . . . . . . . . . . . . . . . . . 22
⊢ (𝑚 ∈ ℕ → (¬
(𝑚 + 1) = 1 ↔ 𝑚 ≠ 0)) |
202 | 193, 201 | mpbird 260 |
. . . . . . . . . . . . . . . . . . . . 21
⊢ (𝑚 ∈ ℕ → ¬
(𝑚 + 1) =
1) |
203 | 202 | ad2antlr 727 |
. . . . . . . . . . . . . . . . . . . 20
⊢ (((𝑁 ∈ ℕ ∧ 𝑚 ∈ ℕ) ∧ (𝑚 + 1) ∈ (1...(𝑁 + 1))) → ¬ (𝑚 + 1) = 1) |
204 | 14 | adantr 484 |
. . . . . . . . . . . . . . . . . . . . . . 23
⊢ ((𝑁 ∈ ℕ ∧ 𝑚 ∈ ℕ) → (𝑁 + 1) ∈
(ℤ≥‘1)) |
205 | | elfzp12 13191 |
. . . . . . . . . . . . . . . . . . . . . . 23
⊢ ((𝑁 + 1) ∈
(ℤ≥‘1) → ((𝑚 + 1) ∈ (1...(𝑁 + 1)) ↔ ((𝑚 + 1) = 1 ∨ (𝑚 + 1) ∈ ((1 + 1)...(𝑁 + 1))))) |
206 | 204, 205 | syl 17 |
. . . . . . . . . . . . . . . . . . . . . 22
⊢ ((𝑁 ∈ ℕ ∧ 𝑚 ∈ ℕ) → ((𝑚 + 1) ∈ (1...(𝑁 + 1)) ↔ ((𝑚 + 1) = 1 ∨ (𝑚 + 1) ∈ ((1 + 1)...(𝑁 + 1))))) |
207 | 206 | biimpa 480 |
. . . . . . . . . . . . . . . . . . . . 21
⊢ (((𝑁 ∈ ℕ ∧ 𝑚 ∈ ℕ) ∧ (𝑚 + 1) ∈ (1...(𝑁 + 1))) → ((𝑚 + 1) = 1 ∨ (𝑚 + 1) ∈ ((1 + 1)...(𝑁 + 1)))) |
208 | 207 | ord 864 |
. . . . . . . . . . . . . . . . . . . 20
⊢ (((𝑁 ∈ ℕ ∧ 𝑚 ∈ ℕ) ∧ (𝑚 + 1) ∈ (1...(𝑁 + 1))) → (¬ (𝑚 + 1) = 1 → (𝑚 + 1) ∈ ((1 + 1)...(𝑁 + 1)))) |
209 | 203, 208 | mpd 15 |
. . . . . . . . . . . . . . . . . . 19
⊢ (((𝑁 ∈ ℕ ∧ 𝑚 ∈ ℕ) ∧ (𝑚 + 1) ∈ (1...(𝑁 + 1))) → (𝑚 + 1) ∈ ((1 + 1)...(𝑁 + 1))) |
210 | | df-2 11893 |
. . . . . . . . . . . . . . . . . . . 20
⊢ 2 = (1 +
1) |
211 | 210 | oveq1i 7223 |
. . . . . . . . . . . . . . . . . . 19
⊢
(2...(𝑁 + 1)) = ((1
+ 1)...(𝑁 +
1)) |
212 | 209, 211 | eleqtrrdi 2849 |
. . . . . . . . . . . . . . . . . 18
⊢ (((𝑁 ∈ ℕ ∧ 𝑚 ∈ ℕ) ∧ (𝑚 + 1) ∈ (1...(𝑁 + 1))) → (𝑚 + 1) ∈ (2...(𝑁 + 1))) |
213 | | ovex 7246 |
. . . . . . . . . . . . . . . . . 18
⊢ (𝑚 + 1) ∈ V |
214 | | eqid 2737 |
. . . . . . . . . . . . . . . . . 18
⊢
((2...(𝑁 + 1))
∖ {(𝑚 + 1)}) =
((2...(𝑁 + 1)) ∖
{(𝑚 + 1)}) |
215 | | fveq1 6716 |
. . . . . . . . . . . . . . . . . . . . 21
⊢ (𝑔 = ℎ → (𝑔‘1) = (ℎ‘1)) |
216 | 215 | eqeq1d 2739 |
. . . . . . . . . . . . . . . . . . . 20
⊢ (𝑔 = ℎ → ((𝑔‘1) = (𝑚 + 1) ↔ (ℎ‘1) = (𝑚 + 1))) |
217 | | fveq1 6716 |
. . . . . . . . . . . . . . . . . . . . 21
⊢ (𝑔 = ℎ → (𝑔‘(𝑚 + 1)) = (ℎ‘(𝑚 + 1))) |
218 | 217 | neeq1d 3000 |
. . . . . . . . . . . . . . . . . . . 20
⊢ (𝑔 = ℎ → ((𝑔‘(𝑚 + 1)) ≠ 1 ↔ (ℎ‘(𝑚 + 1)) ≠ 1)) |
219 | 216, 218 | anbi12d 634 |
. . . . . . . . . . . . . . . . . . 19
⊢ (𝑔 = ℎ → (((𝑔‘1) = (𝑚 + 1) ∧ (𝑔‘(𝑚 + 1)) ≠ 1) ↔ ((ℎ‘1) = (𝑚 + 1) ∧ (ℎ‘(𝑚 + 1)) ≠ 1))) |
220 | 219 | cbvrabv 3402 |
. . . . . . . . . . . . . . . . . 18
⊢ {𝑔 ∈ 𝐴 ∣ ((𝑔‘1) = (𝑚 + 1) ∧ (𝑔‘(𝑚 + 1)) ≠ 1)} = {ℎ ∈ 𝐴 ∣ ((ℎ‘1) = (𝑚 + 1) ∧ (ℎ‘(𝑚 + 1)) ≠ 1)} |
221 | | eqid 2737 |
. . . . . . . . . . . . . . . . . 18
⊢ (( I
↾ ((2...(𝑁 + 1))
∖ {(𝑚 + 1)})) ∪
{〈1, (𝑚 + 1)〉,
〈(𝑚 + 1), 1〉}) =
(( I ↾ ((2...(𝑁 + 1))
∖ {(𝑚 + 1)})) ∪
{〈1, (𝑚 + 1)〉,
〈(𝑚 + 1),
1〉}) |
222 | | f1oeq1 6649 |
. . . . . . . . . . . . . . . . . . . 20
⊢ (𝑔 = 𝑓 → (𝑔:(2...(𝑁 + 1))–1-1-onto→(2...(𝑁 + 1)) ↔ 𝑓:(2...(𝑁 + 1))–1-1-onto→(2...(𝑁 + 1)))) |
223 | | fveq2 6717 |
. . . . . . . . . . . . . . . . . . . . . . 23
⊢ (𝑧 = 𝑦 → (𝑔‘𝑧) = (𝑔‘𝑦)) |
224 | | id 22 |
. . . . . . . . . . . . . . . . . . . . . . 23
⊢ (𝑧 = 𝑦 → 𝑧 = 𝑦) |
225 | 223, 224 | neeq12d 3002 |
. . . . . . . . . . . . . . . . . . . . . 22
⊢ (𝑧 = 𝑦 → ((𝑔‘𝑧) ≠ 𝑧 ↔ (𝑔‘𝑦) ≠ 𝑦)) |
226 | 225 | cbvralvw 3358 |
. . . . . . . . . . . . . . . . . . . . 21
⊢
(∀𝑧 ∈
(2...(𝑁 + 1))(𝑔‘𝑧) ≠ 𝑧 ↔ ∀𝑦 ∈ (2...(𝑁 + 1))(𝑔‘𝑦) ≠ 𝑦) |
227 | | fveq1 6716 |
. . . . . . . . . . . . . . . . . . . . . . 23
⊢ (𝑔 = 𝑓 → (𝑔‘𝑦) = (𝑓‘𝑦)) |
228 | 227 | neeq1d 3000 |
. . . . . . . . . . . . . . . . . . . . . 22
⊢ (𝑔 = 𝑓 → ((𝑔‘𝑦) ≠ 𝑦 ↔ (𝑓‘𝑦) ≠ 𝑦)) |
229 | 228 | ralbidv 3118 |
. . . . . . . . . . . . . . . . . . . . 21
⊢ (𝑔 = 𝑓 → (∀𝑦 ∈ (2...(𝑁 + 1))(𝑔‘𝑦) ≠ 𝑦 ↔ ∀𝑦 ∈ (2...(𝑁 + 1))(𝑓‘𝑦) ≠ 𝑦)) |
230 | 226, 229 | syl5bb 286 |
. . . . . . . . . . . . . . . . . . . 20
⊢ (𝑔 = 𝑓 → (∀𝑧 ∈ (2...(𝑁 + 1))(𝑔‘𝑧) ≠ 𝑧 ↔ ∀𝑦 ∈ (2...(𝑁 + 1))(𝑓‘𝑦) ≠ 𝑦)) |
231 | 222, 230 | anbi12d 634 |
. . . . . . . . . . . . . . . . . . 19
⊢ (𝑔 = 𝑓 → ((𝑔:(2...(𝑁 + 1))–1-1-onto→(2...(𝑁 + 1)) ∧ ∀𝑧 ∈ (2...(𝑁 + 1))(𝑔‘𝑧) ≠ 𝑧) ↔ (𝑓:(2...(𝑁 + 1))–1-1-onto→(2...(𝑁 + 1)) ∧ ∀𝑦 ∈ (2...(𝑁 + 1))(𝑓‘𝑦) ≠ 𝑦))) |
232 | 231 | cbvabv 2811 |
. . . . . . . . . . . . . . . . . 18
⊢ {𝑔 ∣ (𝑔:(2...(𝑁 + 1))–1-1-onto→(2...(𝑁 + 1)) ∧ ∀𝑧 ∈ (2...(𝑁 + 1))(𝑔‘𝑧) ≠ 𝑧)} = {𝑓 ∣ (𝑓:(2...(𝑁 + 1))–1-1-onto→(2...(𝑁 + 1)) ∧ ∀𝑦 ∈ (2...(𝑁 + 1))(𝑓‘𝑦) ≠ 𝑦)} |
233 | 3, 4, 10, 192, 212, 213, 214, 220, 221, 232 | subfacp1lem5 32859 |
. . . . . . . . . . . . . . . . 17
⊢ (((𝑁 ∈ ℕ ∧ 𝑚 ∈ ℕ) ∧ (𝑚 + 1) ∈ (1...(𝑁 + 1))) →
(♯‘{𝑔 ∈
𝐴 ∣ ((𝑔‘1) = (𝑚 + 1) ∧ (𝑔‘(𝑚 + 1)) ≠ 1)}) = (𝑆‘𝑁)) |
234 | 217 | eqeq1d 2739 |
. . . . . . . . . . . . . . . . . . . 20
⊢ (𝑔 = ℎ → ((𝑔‘(𝑚 + 1)) = 1 ↔ (ℎ‘(𝑚 + 1)) = 1)) |
235 | 216, 234 | anbi12d 634 |
. . . . . . . . . . . . . . . . . . 19
⊢ (𝑔 = ℎ → (((𝑔‘1) = (𝑚 + 1) ∧ (𝑔‘(𝑚 + 1)) = 1) ↔ ((ℎ‘1) = (𝑚 + 1) ∧ (ℎ‘(𝑚 + 1)) = 1))) |
236 | 235 | cbvrabv 3402 |
. . . . . . . . . . . . . . . . . 18
⊢ {𝑔 ∈ 𝐴 ∣ ((𝑔‘1) = (𝑚 + 1) ∧ (𝑔‘(𝑚 + 1)) = 1)} = {ℎ ∈ 𝐴 ∣ ((ℎ‘1) = (𝑚 + 1) ∧ (ℎ‘(𝑚 + 1)) = 1)} |
237 | | f1oeq1 6649 |
. . . . . . . . . . . . . . . . . . . 20
⊢ (𝑔 = 𝑓 → (𝑔:((2...(𝑁 + 1)) ∖ {(𝑚 + 1)})–1-1-onto→((2...(𝑁 + 1)) ∖ {(𝑚 + 1)}) ↔ 𝑓:((2...(𝑁 + 1)) ∖ {(𝑚 + 1)})–1-1-onto→((2...(𝑁 + 1)) ∖ {(𝑚 + 1)}))) |
238 | 225 | cbvralvw 3358 |
. . . . . . . . . . . . . . . . . . . . 21
⊢
(∀𝑧 ∈
((2...(𝑁 + 1)) ∖
{(𝑚 + 1)})(𝑔‘𝑧) ≠ 𝑧 ↔ ∀𝑦 ∈ ((2...(𝑁 + 1)) ∖ {(𝑚 + 1)})(𝑔‘𝑦) ≠ 𝑦) |
239 | 228 | ralbidv 3118 |
. . . . . . . . . . . . . . . . . . . . 21
⊢ (𝑔 = 𝑓 → (∀𝑦 ∈ ((2...(𝑁 + 1)) ∖ {(𝑚 + 1)})(𝑔‘𝑦) ≠ 𝑦 ↔ ∀𝑦 ∈ ((2...(𝑁 + 1)) ∖ {(𝑚 + 1)})(𝑓‘𝑦) ≠ 𝑦)) |
240 | 238, 239 | syl5bb 286 |
. . . . . . . . . . . . . . . . . . . 20
⊢ (𝑔 = 𝑓 → (∀𝑧 ∈ ((2...(𝑁 + 1)) ∖ {(𝑚 + 1)})(𝑔‘𝑧) ≠ 𝑧 ↔ ∀𝑦 ∈ ((2...(𝑁 + 1)) ∖ {(𝑚 + 1)})(𝑓‘𝑦) ≠ 𝑦)) |
241 | 237, 240 | anbi12d 634 |
. . . . . . . . . . . . . . . . . . 19
⊢ (𝑔 = 𝑓 → ((𝑔:((2...(𝑁 + 1)) ∖ {(𝑚 + 1)})–1-1-onto→((2...(𝑁 + 1)) ∖ {(𝑚 + 1)}) ∧ ∀𝑧 ∈ ((2...(𝑁 + 1)) ∖ {(𝑚 + 1)})(𝑔‘𝑧) ≠ 𝑧) ↔ (𝑓:((2...(𝑁 + 1)) ∖ {(𝑚 + 1)})–1-1-onto→((2...(𝑁 + 1)) ∖ {(𝑚 + 1)}) ∧ ∀𝑦 ∈ ((2...(𝑁 + 1)) ∖ {(𝑚 + 1)})(𝑓‘𝑦) ≠ 𝑦))) |
242 | 241 | cbvabv 2811 |
. . . . . . . . . . . . . . . . . 18
⊢ {𝑔 ∣ (𝑔:((2...(𝑁 + 1)) ∖ {(𝑚 + 1)})–1-1-onto→((2...(𝑁 + 1)) ∖ {(𝑚 + 1)}) ∧ ∀𝑧 ∈ ((2...(𝑁 + 1)) ∖ {(𝑚 + 1)})(𝑔‘𝑧) ≠ 𝑧)} = {𝑓 ∣ (𝑓:((2...(𝑁 + 1)) ∖ {(𝑚 + 1)})–1-1-onto→((2...(𝑁 + 1)) ∖ {(𝑚 + 1)}) ∧ ∀𝑦 ∈ ((2...(𝑁 + 1)) ∖ {(𝑚 + 1)})(𝑓‘𝑦) ≠ 𝑦)} |
243 | 3, 4, 10, 192, 212, 213, 214, 236, 242 | subfacp1lem3 32857 |
. . . . . . . . . . . . . . . . 17
⊢ (((𝑁 ∈ ℕ ∧ 𝑚 ∈ ℕ) ∧ (𝑚 + 1) ∈ (1...(𝑁 + 1))) →
(♯‘{𝑔 ∈
𝐴 ∣ ((𝑔‘1) = (𝑚 + 1) ∧ (𝑔‘(𝑚 + 1)) = 1)}) = (𝑆‘(𝑁 − 1))) |
244 | 233, 243 | oveq12d 7231 |
. . . . . . . . . . . . . . . 16
⊢ (((𝑁 ∈ ℕ ∧ 𝑚 ∈ ℕ) ∧ (𝑚 + 1) ∈ (1...(𝑁 + 1))) →
((♯‘{𝑔 ∈
𝐴 ∣ ((𝑔‘1) = (𝑚 + 1) ∧ (𝑔‘(𝑚 + 1)) ≠ 1)}) + (♯‘{𝑔 ∈ 𝐴 ∣ ((𝑔‘1) = (𝑚 + 1) ∧ (𝑔‘(𝑚 + 1)) = 1)})) = ((𝑆‘𝑁) + (𝑆‘(𝑁 − 1)))) |
245 | 191, 244 | syl5eq 2790 |
. . . . . . . . . . . . . . 15
⊢ (((𝑁 ∈ ℕ ∧ 𝑚 ∈ ℕ) ∧ (𝑚 + 1) ∈ (1...(𝑁 + 1))) →
(♯‘{𝑔 ∈
𝐴 ∣ (𝑔‘1) = (𝑚 + 1)}) = ((𝑆‘𝑁) + (𝑆‘(𝑁 − 1)))) |
246 | 162, 245 | eqtr4d 2780 |
. . . . . . . . . . . . . 14
⊢ (((𝑁 ∈ ℕ ∧ 𝑚 ∈ ℕ) ∧ (𝑚 + 1) ∈ (1...(𝑁 + 1))) → (1 ·
((𝑆‘𝑁) + (𝑆‘(𝑁 − 1)))) = (♯‘{𝑔 ∈ 𝐴 ∣ (𝑔‘1) = (𝑚 + 1)})) |
247 | 246 | oveq2d 7229 |
. . . . . . . . . . . . 13
⊢ (((𝑁 ∈ ℕ ∧ 𝑚 ∈ ℕ) ∧ (𝑚 + 1) ∈ (1...(𝑁 + 1))) → (((𝑚 − 1) · ((𝑆‘𝑁) + (𝑆‘(𝑁 − 1)))) + (1 · ((𝑆‘𝑁) + (𝑆‘(𝑁 − 1))))) = (((𝑚 − 1) · ((𝑆‘𝑁) + (𝑆‘(𝑁 − 1)))) + (♯‘{𝑔 ∈ 𝐴 ∣ (𝑔‘1) = (𝑚 + 1)}))) |
248 | 156, 161,
247 | 3eqtrd 2781 |
. . . . . . . . . . . 12
⊢ (((𝑁 ∈ ℕ ∧ 𝑚 ∈ ℕ) ∧ (𝑚 + 1) ∈ (1...(𝑁 + 1))) → (((𝑚 + 1) − 1) ·
((𝑆‘𝑁) + (𝑆‘(𝑁 − 1)))) = (((𝑚 − 1) · ((𝑆‘𝑁) + (𝑆‘(𝑁 − 1)))) + (♯‘{𝑔 ∈ 𝐴 ∣ (𝑔‘1) = (𝑚 + 1)}))) |
249 | 150, 248 | eqeq12d 2753 |
. . . . . . . . . . 11
⊢ (((𝑁 ∈ ℕ ∧ 𝑚 ∈ ℕ) ∧ (𝑚 + 1) ∈ (1...(𝑁 + 1))) →
((♯‘{𝑔 ∈
𝐴 ∣ (𝑔‘1) ∈ (1...(𝑚 + 1))}) = (((𝑚 + 1) − 1) · ((𝑆‘𝑁) + (𝑆‘(𝑁 − 1)))) ↔ ((♯‘{𝑔 ∈ 𝐴 ∣ (𝑔‘1) ∈ (1...𝑚)}) + (♯‘{𝑔 ∈ 𝐴 ∣ (𝑔‘1) = (𝑚 + 1)})) = (((𝑚 − 1) · ((𝑆‘𝑁) + (𝑆‘(𝑁 − 1)))) + (♯‘{𝑔 ∈ 𝐴 ∣ (𝑔‘1) = (𝑚 + 1)})))) |
250 | 120, 249 | syl5ibr 249 |
. . . . . . . . . 10
⊢ (((𝑁 ∈ ℕ ∧ 𝑚 ∈ ℕ) ∧ (𝑚 + 1) ∈ (1...(𝑁 + 1))) →
((♯‘{𝑔 ∈
𝐴 ∣ (𝑔‘1) ∈ (1...𝑚)}) = ((𝑚 − 1) · ((𝑆‘𝑁) + (𝑆‘(𝑁 − 1)))) → (♯‘{𝑔 ∈ 𝐴 ∣ (𝑔‘1) ∈ (1...(𝑚 + 1))}) = (((𝑚 + 1) − 1) · ((𝑆‘𝑁) + (𝑆‘(𝑁 − 1)))))) |
251 | 250 | ex 416 |
. . . . . . . . 9
⊢ ((𝑁 ∈ ℕ ∧ 𝑚 ∈ ℕ) → ((𝑚 + 1) ∈ (1...(𝑁 + 1)) →
((♯‘{𝑔 ∈
𝐴 ∣ (𝑔‘1) ∈ (1...𝑚)}) = ((𝑚 − 1) · ((𝑆‘𝑁) + (𝑆‘(𝑁 − 1)))) → (♯‘{𝑔 ∈ 𝐴 ∣ (𝑔‘1) ∈ (1...(𝑚 + 1))}) = (((𝑚 + 1) − 1) · ((𝑆‘𝑁) + (𝑆‘(𝑁 − 1))))))) |
252 | 251 | a2d 29 |
. . . . . . . 8
⊢ ((𝑁 ∈ ℕ ∧ 𝑚 ∈ ℕ) → (((𝑚 + 1) ∈ (1...(𝑁 + 1)) →
(♯‘{𝑔 ∈
𝐴 ∣ (𝑔‘1) ∈ (1...𝑚)}) = ((𝑚 − 1) · ((𝑆‘𝑁) + (𝑆‘(𝑁 − 1))))) → ((𝑚 + 1) ∈ (1...(𝑁 + 1)) → (♯‘{𝑔 ∈ 𝐴 ∣ (𝑔‘1) ∈ (1...(𝑚 + 1))}) = (((𝑚 + 1) − 1) · ((𝑆‘𝑁) + (𝑆‘(𝑁 − 1))))))) |
253 | 119, 252 | syld 47 |
. . . . . . 7
⊢ ((𝑁 ∈ ℕ ∧ 𝑚 ∈ ℕ) → ((𝑚 ∈ (1...(𝑁 + 1)) → (♯‘{𝑔 ∈ 𝐴 ∣ (𝑔‘1) ∈ (1...𝑚)}) = ((𝑚 − 1) · ((𝑆‘𝑁) + (𝑆‘(𝑁 − 1))))) → ((𝑚 + 1) ∈ (1...(𝑁 + 1)) → (♯‘{𝑔 ∈ 𝐴 ∣ (𝑔‘1) ∈ (1...(𝑚 + 1))}) = (((𝑚 + 1) − 1) · ((𝑆‘𝑁) + (𝑆‘(𝑁 − 1))))))) |
254 | 253 | expcom 417 |
. . . . . 6
⊢ (𝑚 ∈ ℕ → (𝑁 ∈ ℕ → ((𝑚 ∈ (1...(𝑁 + 1)) → (♯‘{𝑔 ∈ 𝐴 ∣ (𝑔‘1) ∈ (1...𝑚)}) = ((𝑚 − 1) · ((𝑆‘𝑁) + (𝑆‘(𝑁 − 1))))) → ((𝑚 + 1) ∈ (1...(𝑁 + 1)) → (♯‘{𝑔 ∈ 𝐴 ∣ (𝑔‘1) ∈ (1...(𝑚 + 1))}) = (((𝑚 + 1) − 1) · ((𝑆‘𝑁) + (𝑆‘(𝑁 − 1)))))))) |
255 | 254 | a2d 29 |
. . . . 5
⊢ (𝑚 ∈ ℕ → ((𝑁 ∈ ℕ → (𝑚 ∈ (1...(𝑁 + 1)) → (♯‘{𝑔 ∈ 𝐴 ∣ (𝑔‘1) ∈ (1...𝑚)}) = ((𝑚 − 1) · ((𝑆‘𝑁) + (𝑆‘(𝑁 − 1)))))) → (𝑁 ∈ ℕ → ((𝑚 + 1) ∈ (1...(𝑁 + 1)) → (♯‘{𝑔 ∈ 𝐴 ∣ (𝑔‘1) ∈ (1...(𝑚 + 1))}) = (((𝑚 + 1) − 1) · ((𝑆‘𝑁) + (𝑆‘(𝑁 − 1)))))))) |
256 | 53, 63, 73, 83, 113, 255 | nnind 11848 |
. . . 4
⊢ ((𝑁 + 1) ∈ ℕ →
(𝑁 ∈ ℕ →
((𝑁 + 1) ∈ (1...(𝑁 + 1)) →
(♯‘{𝑔 ∈
𝐴 ∣ (𝑔‘1) ∈ (1...(𝑁 + 1))}) = (((𝑁 + 1) − 1) · ((𝑆‘𝑁) + (𝑆‘(𝑁 − 1))))))) |
257 | 1, 256 | mpcom 38 |
. . 3
⊢ (𝑁 ∈ ℕ → ((𝑁 + 1) ∈ (1...(𝑁 + 1)) →
(♯‘{𝑔 ∈
𝐴 ∣ (𝑔‘1) ∈ (1...(𝑁 + 1))}) = (((𝑁 + 1) − 1) · ((𝑆‘𝑁) + (𝑆‘(𝑁 − 1)))))) |
258 | 34, 257 | mpd 15 |
. 2
⊢ (𝑁 ∈ ℕ →
(♯‘{𝑔 ∈
𝐴 ∣ (𝑔‘1) ∈ (1...(𝑁 + 1))}) = (((𝑁 + 1) − 1) · ((𝑆‘𝑁) + (𝑆‘(𝑁 − 1))))) |
259 | | nncn 11838 |
. . . 4
⊢ (𝑁 ∈ ℕ → 𝑁 ∈
ℂ) |
260 | | pncan 11084 |
. . . 4
⊢ ((𝑁 ∈ ℂ ∧ 1 ∈
ℂ) → ((𝑁 + 1)
− 1) = 𝑁) |
261 | 259, 153,
260 | sylancl 589 |
. . 3
⊢ (𝑁 ∈ ℕ → ((𝑁 + 1) − 1) = 𝑁) |
262 | 261 | oveq1d 7228 |
. 2
⊢ (𝑁 ∈ ℕ → (((𝑁 + 1) − 1) ·
((𝑆‘𝑁) + (𝑆‘(𝑁 − 1)))) = (𝑁 · ((𝑆‘𝑁) + (𝑆‘(𝑁 − 1))))) |
263 | 32, 258, 262 | 3eqtrd 2781 |
1
⊢ (𝑁 ∈ ℕ → (𝑆‘(𝑁 + 1)) = (𝑁 · ((𝑆‘𝑁) + (𝑆‘(𝑁 − 1))))) |