Step | Hyp | Ref
| Expression |
1 | | dchrisum0flb.a |
. . . 4
⊢ (𝜑 → 𝐴 ∈ ℕ) |
2 | | nnuz 11723 |
. . . 4
⊢ ℕ =
(ℤ≥‘1) |
3 | 1, 2 | syl6eleq 2711 |
. . 3
⊢ (𝜑 → 𝐴 ∈
(ℤ≥‘1)) |
4 | | eluzfz2 12349 |
. . 3
⊢ (𝐴 ∈
(ℤ≥‘1) → 𝐴 ∈ (1...𝐴)) |
5 | 3, 4 | syl 17 |
. 2
⊢ (𝜑 → 𝐴 ∈ (1...𝐴)) |
6 | | oveq2 6658 |
. . . . . 6
⊢ (𝑘 = 1 → (1...𝑘) = (1...1)) |
7 | 6 | raleqdv 3144 |
. . . . 5
⊢ (𝑘 = 1 → (∀𝑦 ∈ (1...𝑘)if((√‘𝑦) ∈ ℕ, 1, 0) ≤ (𝐹‘𝑦) ↔ ∀𝑦 ∈ (1...1)if((√‘𝑦) ∈ ℕ, 1, 0) ≤
(𝐹‘𝑦))) |
8 | 7 | imbi2d 330 |
. . . 4
⊢ (𝑘 = 1 → ((𝜑 → ∀𝑦 ∈ (1...𝑘)if((√‘𝑦) ∈ ℕ, 1, 0) ≤ (𝐹‘𝑦)) ↔ (𝜑 → ∀𝑦 ∈ (1...1)if((√‘𝑦) ∈ ℕ, 1, 0) ≤
(𝐹‘𝑦)))) |
9 | | oveq2 6658 |
. . . . . 6
⊢ (𝑘 = 𝑖 → (1...𝑘) = (1...𝑖)) |
10 | 9 | raleqdv 3144 |
. . . . 5
⊢ (𝑘 = 𝑖 → (∀𝑦 ∈ (1...𝑘)if((√‘𝑦) ∈ ℕ, 1, 0) ≤ (𝐹‘𝑦) ↔ ∀𝑦 ∈ (1...𝑖)if((√‘𝑦) ∈ ℕ, 1, 0) ≤ (𝐹‘𝑦))) |
11 | 10 | imbi2d 330 |
. . . 4
⊢ (𝑘 = 𝑖 → ((𝜑 → ∀𝑦 ∈ (1...𝑘)if((√‘𝑦) ∈ ℕ, 1, 0) ≤ (𝐹‘𝑦)) ↔ (𝜑 → ∀𝑦 ∈ (1...𝑖)if((√‘𝑦) ∈ ℕ, 1, 0) ≤ (𝐹‘𝑦)))) |
12 | | oveq2 6658 |
. . . . . 6
⊢ (𝑘 = (𝑖 + 1) → (1...𝑘) = (1...(𝑖 + 1))) |
13 | 12 | raleqdv 3144 |
. . . . 5
⊢ (𝑘 = (𝑖 + 1) → (∀𝑦 ∈ (1...𝑘)if((√‘𝑦) ∈ ℕ, 1, 0) ≤ (𝐹‘𝑦) ↔ ∀𝑦 ∈ (1...(𝑖 + 1))if((√‘𝑦) ∈ ℕ, 1, 0) ≤ (𝐹‘𝑦))) |
14 | 13 | imbi2d 330 |
. . . 4
⊢ (𝑘 = (𝑖 + 1) → ((𝜑 → ∀𝑦 ∈ (1...𝑘)if((√‘𝑦) ∈ ℕ, 1, 0) ≤ (𝐹‘𝑦)) ↔ (𝜑 → ∀𝑦 ∈ (1...(𝑖 + 1))if((√‘𝑦) ∈ ℕ, 1, 0) ≤ (𝐹‘𝑦)))) |
15 | | oveq2 6658 |
. . . . . 6
⊢ (𝑘 = 𝐴 → (1...𝑘) = (1...𝐴)) |
16 | 15 | raleqdv 3144 |
. . . . 5
⊢ (𝑘 = 𝐴 → (∀𝑦 ∈ (1...𝑘)if((√‘𝑦) ∈ ℕ, 1, 0) ≤ (𝐹‘𝑦) ↔ ∀𝑦 ∈ (1...𝐴)if((√‘𝑦) ∈ ℕ, 1, 0) ≤ (𝐹‘𝑦))) |
17 | 16 | imbi2d 330 |
. . . 4
⊢ (𝑘 = 𝐴 → ((𝜑 → ∀𝑦 ∈ (1...𝑘)if((√‘𝑦) ∈ ℕ, 1, 0) ≤ (𝐹‘𝑦)) ↔ (𝜑 → ∀𝑦 ∈ (1...𝐴)if((√‘𝑦) ∈ ℕ, 1, 0) ≤ (𝐹‘𝑦)))) |
18 | | rpvmasum.z |
. . . . . 6
⊢ 𝑍 =
(ℤ/nℤ‘𝑁) |
19 | | rpvmasum.l |
. . . . . 6
⊢ 𝐿 = (ℤRHom‘𝑍) |
20 | | rpvmasum.a |
. . . . . 6
⊢ (𝜑 → 𝑁 ∈ ℕ) |
21 | | rpvmasum2.g |
. . . . . 6
⊢ 𝐺 = (DChr‘𝑁) |
22 | | rpvmasum2.d |
. . . . . 6
⊢ 𝐷 = (Base‘𝐺) |
23 | | rpvmasum2.1 |
. . . . . 6
⊢ 1 =
(0g‘𝐺) |
24 | | dchrisum0f.f |
. . . . . 6
⊢ 𝐹 = (𝑏 ∈ ℕ ↦ Σ𝑣 ∈ {𝑞 ∈ ℕ ∣ 𝑞 ∥ 𝑏} (𝑋‘(𝐿‘𝑣))) |
25 | | dchrisum0f.x |
. . . . . 6
⊢ (𝜑 → 𝑋 ∈ 𝐷) |
26 | | dchrisum0flb.r |
. . . . . 6
⊢ (𝜑 → 𝑋:(Base‘𝑍)⟶ℝ) |
27 | | 2prm 15405 |
. . . . . . 7
⊢ 2 ∈
ℙ |
28 | 27 | a1i 11 |
. . . . . 6
⊢ (𝜑 → 2 ∈
ℙ) |
29 | | 0nn0 11307 |
. . . . . . 7
⊢ 0 ∈
ℕ0 |
30 | 29 | a1i 11 |
. . . . . 6
⊢ (𝜑 → 0 ∈
ℕ0) |
31 | 18, 19, 20, 21, 22, 23, 24, 25, 26, 28, 30 | dchrisum0flblem1 25197 |
. . . . 5
⊢ (𝜑 →
if((√‘(2↑0)) ∈ ℕ, 1, 0) ≤ (𝐹‘(2↑0))) |
32 | | elfz1eq 12352 |
. . . . . . . . . . . 12
⊢ (𝑦 ∈ (1...1) → 𝑦 = 1) |
33 | | 2nn0 11309 |
. . . . . . . . . . . . 13
⊢ 2 ∈
ℕ0 |
34 | 33 | numexp0 15780 |
. . . . . . . . . . . 12
⊢
(2↑0) = 1 |
35 | 32, 34 | syl6eqr 2674 |
. . . . . . . . . . 11
⊢ (𝑦 ∈ (1...1) → 𝑦 = (2↑0)) |
36 | 35 | fveq2d 6195 |
. . . . . . . . . 10
⊢ (𝑦 ∈ (1...1) →
(√‘𝑦) =
(√‘(2↑0))) |
37 | 36 | eleq1d 2686 |
. . . . . . . . 9
⊢ (𝑦 ∈ (1...1) →
((√‘𝑦) ∈
ℕ ↔ (√‘(2↑0)) ∈ ℕ)) |
38 | 37 | ifbid 4108 |
. . . . . . . 8
⊢ (𝑦 ∈ (1...1) →
if((√‘𝑦) ∈
ℕ, 1, 0) = if((√‘(2↑0)) ∈ ℕ, 1,
0)) |
39 | 35 | fveq2d 6195 |
. . . . . . . 8
⊢ (𝑦 ∈ (1...1) → (𝐹‘𝑦) = (𝐹‘(2↑0))) |
40 | 38, 39 | breq12d 4666 |
. . . . . . 7
⊢ (𝑦 ∈ (1...1) →
(if((√‘𝑦)
∈ ℕ, 1, 0) ≤ (𝐹‘𝑦) ↔ if((√‘(2↑0)) ∈
ℕ, 1, 0) ≤ (𝐹‘(2↑0)))) |
41 | 40 | biimprcd 240 |
. . . . . 6
⊢
(if((√‘(2↑0)) ∈ ℕ, 1, 0) ≤ (𝐹‘(2↑0)) → (𝑦 ∈ (1...1) →
if((√‘𝑦) ∈
ℕ, 1, 0) ≤ (𝐹‘𝑦))) |
42 | 41 | ralrimiv 2965 |
. . . . 5
⊢
(if((√‘(2↑0)) ∈ ℕ, 1, 0) ≤ (𝐹‘(2↑0)) →
∀𝑦 ∈
(1...1)if((√‘𝑦)
∈ ℕ, 1, 0) ≤ (𝐹‘𝑦)) |
43 | 31, 42 | syl 17 |
. . . 4
⊢ (𝜑 → ∀𝑦 ∈ (1...1)if((√‘𝑦) ∈ ℕ, 1, 0) ≤
(𝐹‘𝑦)) |
44 | | simpr 477 |
. . . . . . . . . . . . . . . 16
⊢ ((𝜑 ∧ 𝑖 ∈ ℕ) → 𝑖 ∈ ℕ) |
45 | 44, 2 | syl6eleq 2711 |
. . . . . . . . . . . . . . 15
⊢ ((𝜑 ∧ 𝑖 ∈ ℕ) → 𝑖 ∈
(ℤ≥‘1)) |
46 | 45 | adantrr 753 |
. . . . . . . . . . . . . 14
⊢ ((𝜑 ∧ (𝑖 ∈ ℕ ∧ ∀𝑦 ∈ (1...𝑖)if((√‘𝑦) ∈ ℕ, 1, 0) ≤ (𝐹‘𝑦))) → 𝑖 ∈
(ℤ≥‘1)) |
47 | | eluzp1p1 11713 |
. . . . . . . . . . . . . 14
⊢ (𝑖 ∈
(ℤ≥‘1) → (𝑖 + 1) ∈ (ℤ≥‘(1
+ 1))) |
48 | 46, 47 | syl 17 |
. . . . . . . . . . . . 13
⊢ ((𝜑 ∧ (𝑖 ∈ ℕ ∧ ∀𝑦 ∈ (1...𝑖)if((√‘𝑦) ∈ ℕ, 1, 0) ≤ (𝐹‘𝑦))) → (𝑖 + 1) ∈ (ℤ≥‘(1
+ 1))) |
49 | | df-2 11079 |
. . . . . . . . . . . . . 14
⊢ 2 = (1 +
1) |
50 | 49 | fveq2i 6194 |
. . . . . . . . . . . . 13
⊢
(ℤ≥‘2) = (ℤ≥‘(1 +
1)) |
51 | 48, 50 | syl6eleqr 2712 |
. . . . . . . . . . . 12
⊢ ((𝜑 ∧ (𝑖 ∈ ℕ ∧ ∀𝑦 ∈ (1...𝑖)if((√‘𝑦) ∈ ℕ, 1, 0) ≤ (𝐹‘𝑦))) → (𝑖 + 1) ∈
(ℤ≥‘2)) |
52 | | exprmfct 15416 |
. . . . . . . . . . . 12
⊢ ((𝑖 + 1) ∈
(ℤ≥‘2) → ∃𝑝 ∈ ℙ 𝑝 ∥ (𝑖 + 1)) |
53 | 51, 52 | syl 17 |
. . . . . . . . . . 11
⊢ ((𝜑 ∧ (𝑖 ∈ ℕ ∧ ∀𝑦 ∈ (1...𝑖)if((√‘𝑦) ∈ ℕ, 1, 0) ≤ (𝐹‘𝑦))) → ∃𝑝 ∈ ℙ 𝑝 ∥ (𝑖 + 1)) |
54 | 20 | ad2antrr 762 |
. . . . . . . . . . . 12
⊢ (((𝜑 ∧ (𝑖 ∈ ℕ ∧ ∀𝑦 ∈ (1...𝑖)if((√‘𝑦) ∈ ℕ, 1, 0) ≤ (𝐹‘𝑦))) ∧ (𝑝 ∈ ℙ ∧ 𝑝 ∥ (𝑖 + 1))) → 𝑁 ∈ ℕ) |
55 | 25 | ad2antrr 762 |
. . . . . . . . . . . 12
⊢ (((𝜑 ∧ (𝑖 ∈ ℕ ∧ ∀𝑦 ∈ (1...𝑖)if((√‘𝑦) ∈ ℕ, 1, 0) ≤ (𝐹‘𝑦))) ∧ (𝑝 ∈ ℙ ∧ 𝑝 ∥ (𝑖 + 1))) → 𝑋 ∈ 𝐷) |
56 | 26 | ad2antrr 762 |
. . . . . . . . . . . 12
⊢ (((𝜑 ∧ (𝑖 ∈ ℕ ∧ ∀𝑦 ∈ (1...𝑖)if((√‘𝑦) ∈ ℕ, 1, 0) ≤ (𝐹‘𝑦))) ∧ (𝑝 ∈ ℙ ∧ 𝑝 ∥ (𝑖 + 1))) → 𝑋:(Base‘𝑍)⟶ℝ) |
57 | 51 | adantr 481 |
. . . . . . . . . . . 12
⊢ (((𝜑 ∧ (𝑖 ∈ ℕ ∧ ∀𝑦 ∈ (1...𝑖)if((√‘𝑦) ∈ ℕ, 1, 0) ≤ (𝐹‘𝑦))) ∧ (𝑝 ∈ ℙ ∧ 𝑝 ∥ (𝑖 + 1))) → (𝑖 + 1) ∈
(ℤ≥‘2)) |
58 | | simprl 794 |
. . . . . . . . . . . 12
⊢ (((𝜑 ∧ (𝑖 ∈ ℕ ∧ ∀𝑦 ∈ (1...𝑖)if((√‘𝑦) ∈ ℕ, 1, 0) ≤ (𝐹‘𝑦))) ∧ (𝑝 ∈ ℙ ∧ 𝑝 ∥ (𝑖 + 1))) → 𝑝 ∈ ℙ) |
59 | | simprr 796 |
. . . . . . . . . . . 12
⊢ (((𝜑 ∧ (𝑖 ∈ ℕ ∧ ∀𝑦 ∈ (1...𝑖)if((√‘𝑦) ∈ ℕ, 1, 0) ≤ (𝐹‘𝑦))) ∧ (𝑝 ∈ ℙ ∧ 𝑝 ∥ (𝑖 + 1))) → 𝑝 ∥ (𝑖 + 1)) |
60 | | simplrr 801 |
. . . . . . . . . . . . 13
⊢ (((𝜑 ∧ (𝑖 ∈ ℕ ∧ ∀𝑦 ∈ (1...𝑖)if((√‘𝑦) ∈ ℕ, 1, 0) ≤ (𝐹‘𝑦))) ∧ (𝑝 ∈ ℙ ∧ 𝑝 ∥ (𝑖 + 1))) → ∀𝑦 ∈ (1...𝑖)if((√‘𝑦) ∈ ℕ, 1, 0) ≤ (𝐹‘𝑦)) |
61 | | simplrl 800 |
. . . . . . . . . . . . . . . 16
⊢ (((𝜑 ∧ (𝑖 ∈ ℕ ∧ ∀𝑦 ∈ (1...𝑖)if((√‘𝑦) ∈ ℕ, 1, 0) ≤ (𝐹‘𝑦))) ∧ (𝑝 ∈ ℙ ∧ 𝑝 ∥ (𝑖 + 1))) → 𝑖 ∈ ℕ) |
62 | 61 | nnzd 11481 |
. . . . . . . . . . . . . . 15
⊢ (((𝜑 ∧ (𝑖 ∈ ℕ ∧ ∀𝑦 ∈ (1...𝑖)if((√‘𝑦) ∈ ℕ, 1, 0) ≤ (𝐹‘𝑦))) ∧ (𝑝 ∈ ℙ ∧ 𝑝 ∥ (𝑖 + 1))) → 𝑖 ∈ ℤ) |
63 | | fzval3 12536 |
. . . . . . . . . . . . . . 15
⊢ (𝑖 ∈ ℤ →
(1...𝑖) = (1..^(𝑖 + 1))) |
64 | 62, 63 | syl 17 |
. . . . . . . . . . . . . 14
⊢ (((𝜑 ∧ (𝑖 ∈ ℕ ∧ ∀𝑦 ∈ (1...𝑖)if((√‘𝑦) ∈ ℕ, 1, 0) ≤ (𝐹‘𝑦))) ∧ (𝑝 ∈ ℙ ∧ 𝑝 ∥ (𝑖 + 1))) → (1...𝑖) = (1..^(𝑖 + 1))) |
65 | 64 | raleqdv 3144 |
. . . . . . . . . . . . 13
⊢ (((𝜑 ∧ (𝑖 ∈ ℕ ∧ ∀𝑦 ∈ (1...𝑖)if((√‘𝑦) ∈ ℕ, 1, 0) ≤ (𝐹‘𝑦))) ∧ (𝑝 ∈ ℙ ∧ 𝑝 ∥ (𝑖 + 1))) → (∀𝑦 ∈ (1...𝑖)if((√‘𝑦) ∈ ℕ, 1, 0) ≤ (𝐹‘𝑦) ↔ ∀𝑦 ∈ (1..^(𝑖 + 1))if((√‘𝑦) ∈ ℕ, 1, 0) ≤ (𝐹‘𝑦))) |
66 | 60, 65 | mpbid 222 |
. . . . . . . . . . . 12
⊢ (((𝜑 ∧ (𝑖 ∈ ℕ ∧ ∀𝑦 ∈ (1...𝑖)if((√‘𝑦) ∈ ℕ, 1, 0) ≤ (𝐹‘𝑦))) ∧ (𝑝 ∈ ℙ ∧ 𝑝 ∥ (𝑖 + 1))) → ∀𝑦 ∈ (1..^(𝑖 + 1))if((√‘𝑦) ∈ ℕ, 1, 0) ≤ (𝐹‘𝑦)) |
67 | 18, 19, 54, 21, 22, 23, 24, 55, 56, 57, 58, 59, 66 | dchrisum0flblem2 25198 |
. . . . . . . . . . 11
⊢ (((𝜑 ∧ (𝑖 ∈ ℕ ∧ ∀𝑦 ∈ (1...𝑖)if((√‘𝑦) ∈ ℕ, 1, 0) ≤ (𝐹‘𝑦))) ∧ (𝑝 ∈ ℙ ∧ 𝑝 ∥ (𝑖 + 1))) → if((√‘(𝑖 + 1)) ∈ ℕ, 1, 0)
≤ (𝐹‘(𝑖 + 1))) |
68 | 53, 67 | rexlimddv 3035 |
. . . . . . . . . 10
⊢ ((𝜑 ∧ (𝑖 ∈ ℕ ∧ ∀𝑦 ∈ (1...𝑖)if((√‘𝑦) ∈ ℕ, 1, 0) ≤ (𝐹‘𝑦))) → if((√‘(𝑖 + 1)) ∈ ℕ, 1, 0)
≤ (𝐹‘(𝑖 + 1))) |
69 | | ovex 6678 |
. . . . . . . . . . 11
⊢ (𝑖 + 1) ∈ V |
70 | | fveq2 6191 |
. . . . . . . . . . . . . 14
⊢ (𝑦 = (𝑖 + 1) → (√‘𝑦) = (√‘(𝑖 + 1))) |
71 | 70 | eleq1d 2686 |
. . . . . . . . . . . . 13
⊢ (𝑦 = (𝑖 + 1) → ((√‘𝑦) ∈ ℕ ↔
(√‘(𝑖 + 1))
∈ ℕ)) |
72 | 71 | ifbid 4108 |
. . . . . . . . . . . 12
⊢ (𝑦 = (𝑖 + 1) → if((√‘𝑦) ∈ ℕ, 1, 0) =
if((√‘(𝑖 + 1))
∈ ℕ, 1, 0)) |
73 | | fveq2 6191 |
. . . . . . . . . . . 12
⊢ (𝑦 = (𝑖 + 1) → (𝐹‘𝑦) = (𝐹‘(𝑖 + 1))) |
74 | 72, 73 | breq12d 4666 |
. . . . . . . . . . 11
⊢ (𝑦 = (𝑖 + 1) → (if((√‘𝑦) ∈ ℕ, 1, 0) ≤
(𝐹‘𝑦) ↔ if((√‘(𝑖 + 1)) ∈ ℕ, 1, 0)
≤ (𝐹‘(𝑖 + 1)))) |
75 | 69, 74 | ralsn 4222 |
. . . . . . . . . 10
⊢
(∀𝑦 ∈
{(𝑖 +
1)}if((√‘𝑦)
∈ ℕ, 1, 0) ≤ (𝐹‘𝑦) ↔ if((√‘(𝑖 + 1)) ∈ ℕ, 1, 0)
≤ (𝐹‘(𝑖 + 1))) |
76 | 68, 75 | sylibr 224 |
. . . . . . . . 9
⊢ ((𝜑 ∧ (𝑖 ∈ ℕ ∧ ∀𝑦 ∈ (1...𝑖)if((√‘𝑦) ∈ ℕ, 1, 0) ≤ (𝐹‘𝑦))) → ∀𝑦 ∈ {(𝑖 + 1)}if((√‘𝑦) ∈ ℕ, 1, 0) ≤ (𝐹‘𝑦)) |
77 | 76 | expr 643 |
. . . . . . . 8
⊢ ((𝜑 ∧ 𝑖 ∈ ℕ) → (∀𝑦 ∈ (1...𝑖)if((√‘𝑦) ∈ ℕ, 1, 0) ≤ (𝐹‘𝑦) → ∀𝑦 ∈ {(𝑖 + 1)}if((√‘𝑦) ∈ ℕ, 1, 0) ≤ (𝐹‘𝑦))) |
78 | 77 | ancld 576 |
. . . . . . 7
⊢ ((𝜑 ∧ 𝑖 ∈ ℕ) → (∀𝑦 ∈ (1...𝑖)if((√‘𝑦) ∈ ℕ, 1, 0) ≤ (𝐹‘𝑦) → (∀𝑦 ∈ (1...𝑖)if((√‘𝑦) ∈ ℕ, 1, 0) ≤ (𝐹‘𝑦) ∧ ∀𝑦 ∈ {(𝑖 + 1)}if((√‘𝑦) ∈ ℕ, 1, 0) ≤ (𝐹‘𝑦)))) |
79 | | fzsuc 12388 |
. . . . . . . . . 10
⊢ (𝑖 ∈
(ℤ≥‘1) → (1...(𝑖 + 1)) = ((1...𝑖) ∪ {(𝑖 + 1)})) |
80 | 45, 79 | syl 17 |
. . . . . . . . 9
⊢ ((𝜑 ∧ 𝑖 ∈ ℕ) → (1...(𝑖 + 1)) = ((1...𝑖) ∪ {(𝑖 + 1)})) |
81 | 80 | raleqdv 3144 |
. . . . . . . 8
⊢ ((𝜑 ∧ 𝑖 ∈ ℕ) → (∀𝑦 ∈ (1...(𝑖 + 1))if((√‘𝑦) ∈ ℕ, 1, 0) ≤ (𝐹‘𝑦) ↔ ∀𝑦 ∈ ((1...𝑖) ∪ {(𝑖 + 1)})if((√‘𝑦) ∈ ℕ, 1, 0) ≤ (𝐹‘𝑦))) |
82 | | ralunb 3794 |
. . . . . . . 8
⊢
(∀𝑦 ∈
((1...𝑖) ∪ {(𝑖 + 1)})if((√‘𝑦) ∈ ℕ, 1, 0) ≤
(𝐹‘𝑦) ↔ (∀𝑦 ∈ (1...𝑖)if((√‘𝑦) ∈ ℕ, 1, 0) ≤ (𝐹‘𝑦) ∧ ∀𝑦 ∈ {(𝑖 + 1)}if((√‘𝑦) ∈ ℕ, 1, 0) ≤ (𝐹‘𝑦))) |
83 | 81, 82 | syl6bb 276 |
. . . . . . 7
⊢ ((𝜑 ∧ 𝑖 ∈ ℕ) → (∀𝑦 ∈ (1...(𝑖 + 1))if((√‘𝑦) ∈ ℕ, 1, 0) ≤ (𝐹‘𝑦) ↔ (∀𝑦 ∈ (1...𝑖)if((√‘𝑦) ∈ ℕ, 1, 0) ≤ (𝐹‘𝑦) ∧ ∀𝑦 ∈ {(𝑖 + 1)}if((√‘𝑦) ∈ ℕ, 1, 0) ≤ (𝐹‘𝑦)))) |
84 | 78, 83 | sylibrd 249 |
. . . . . 6
⊢ ((𝜑 ∧ 𝑖 ∈ ℕ) → (∀𝑦 ∈ (1...𝑖)if((√‘𝑦) ∈ ℕ, 1, 0) ≤ (𝐹‘𝑦) → ∀𝑦 ∈ (1...(𝑖 + 1))if((√‘𝑦) ∈ ℕ, 1, 0) ≤ (𝐹‘𝑦))) |
85 | 84 | expcom 451 |
. . . . 5
⊢ (𝑖 ∈ ℕ → (𝜑 → (∀𝑦 ∈ (1...𝑖)if((√‘𝑦) ∈ ℕ, 1, 0) ≤ (𝐹‘𝑦) → ∀𝑦 ∈ (1...(𝑖 + 1))if((√‘𝑦) ∈ ℕ, 1, 0) ≤ (𝐹‘𝑦)))) |
86 | 85 | a2d 29 |
. . . 4
⊢ (𝑖 ∈ ℕ → ((𝜑 → ∀𝑦 ∈ (1...𝑖)if((√‘𝑦) ∈ ℕ, 1, 0) ≤ (𝐹‘𝑦)) → (𝜑 → ∀𝑦 ∈ (1...(𝑖 + 1))if((√‘𝑦) ∈ ℕ, 1, 0) ≤ (𝐹‘𝑦)))) |
87 | 8, 11, 14, 17, 43, 86 | nnind 11038 |
. . 3
⊢ (𝐴 ∈ ℕ → (𝜑 → ∀𝑦 ∈ (1...𝐴)if((√‘𝑦) ∈ ℕ, 1, 0) ≤ (𝐹‘𝑦))) |
88 | 1, 87 | mpcom 38 |
. 2
⊢ (𝜑 → ∀𝑦 ∈ (1...𝐴)if((√‘𝑦) ∈ ℕ, 1, 0) ≤ (𝐹‘𝑦)) |
89 | | fveq2 6191 |
. . . . . 6
⊢ (𝑦 = 𝐴 → (√‘𝑦) = (√‘𝐴)) |
90 | 89 | eleq1d 2686 |
. . . . 5
⊢ (𝑦 = 𝐴 → ((√‘𝑦) ∈ ℕ ↔ (√‘𝐴) ∈
ℕ)) |
91 | 90 | ifbid 4108 |
. . . 4
⊢ (𝑦 = 𝐴 → if((√‘𝑦) ∈ ℕ, 1, 0) =
if((√‘𝐴) ∈
ℕ, 1, 0)) |
92 | | fveq2 6191 |
. . . 4
⊢ (𝑦 = 𝐴 → (𝐹‘𝑦) = (𝐹‘𝐴)) |
93 | 91, 92 | breq12d 4666 |
. . 3
⊢ (𝑦 = 𝐴 → (if((√‘𝑦) ∈ ℕ, 1, 0) ≤
(𝐹‘𝑦) ↔ if((√‘𝐴) ∈ ℕ, 1, 0) ≤ (𝐹‘𝐴))) |
94 | 93 | rspcv 3305 |
. 2
⊢ (𝐴 ∈ (1...𝐴) → (∀𝑦 ∈ (1...𝐴)if((√‘𝑦) ∈ ℕ, 1, 0) ≤ (𝐹‘𝑦) → if((√‘𝐴) ∈ ℕ, 1, 0) ≤ (𝐹‘𝐴))) |
95 | 5, 88, 94 | sylc 65 |
1
⊢ (𝜑 → if((√‘𝐴) ∈ ℕ, 1, 0) ≤
(𝐹‘𝐴)) |