Step | Hyp | Ref
| Expression |
1 | | smuval.a |
. . . . . . 7
⊢ (𝜑 → 𝐴 ⊆
ℕ0) |
2 | | smuval.b |
. . . . . . 7
⊢ (𝜑 → 𝐵 ⊆
ℕ0) |
3 | | smuval.p |
. . . . . . 7
⊢ 𝑃 = seq0((𝑝 ∈ 𝒫 ℕ0, 𝑚 ∈ ℕ0
↦ (𝑝 sadd {𝑛 ∈ ℕ0
∣ (𝑚 ∈ 𝐴 ∧ (𝑛 − 𝑚) ∈ 𝐵)})), (𝑛 ∈ ℕ0 ↦ if(𝑛 = 0, ∅, (𝑛 − 1)))) |
4 | 1, 2, 3 | smupf 15821 |
. . . . . 6
⊢ (𝜑 → 𝑃:ℕ0⟶𝒫
ℕ0) |
5 | | smuval.n |
. . . . . . 7
⊢ (𝜑 → 𝑁 ∈
ℕ0) |
6 | | smupvallem.m |
. . . . . . 7
⊢ (𝜑 → 𝑀 ∈ (ℤ≥‘𝑁)) |
7 | | eluznn0 12311 |
. . . . . . 7
⊢ ((𝑁 ∈ ℕ0
∧ 𝑀 ∈
(ℤ≥‘𝑁)) → 𝑀 ∈
ℕ0) |
8 | 5, 6, 7 | syl2anc 586 |
. . . . . 6
⊢ (𝜑 → 𝑀 ∈
ℕ0) |
9 | 4, 8 | ffvelrnd 6847 |
. . . . 5
⊢ (𝜑 → (𝑃‘𝑀) ∈ 𝒫
ℕ0) |
10 | 9 | elpwid 4553 |
. . . 4
⊢ (𝜑 → (𝑃‘𝑀) ⊆
ℕ0) |
11 | 10 | sseld 3966 |
. . 3
⊢ (𝜑 → (𝑘 ∈ (𝑃‘𝑀) → 𝑘 ∈
ℕ0)) |
12 | 1, 2, 3 | smufval 15820 |
. . . . 5
⊢ (𝜑 → (𝐴 smul 𝐵) = {𝑘 ∈ ℕ0 ∣ 𝑘 ∈ (𝑃‘(𝑘 + 1))}) |
13 | | ssrab2 4056 |
. . . . 5
⊢ {𝑘 ∈ ℕ0
∣ 𝑘 ∈ (𝑃‘(𝑘 + 1))} ⊆
ℕ0 |
14 | 12, 13 | eqsstrdi 4021 |
. . . 4
⊢ (𝜑 → (𝐴 smul 𝐵) ⊆
ℕ0) |
15 | 14 | sseld 3966 |
. . 3
⊢ (𝜑 → (𝑘 ∈ (𝐴 smul 𝐵) → 𝑘 ∈
ℕ0)) |
16 | 1 | ad2antrr 724 |
. . . . . . 7
⊢ (((𝜑 ∧ 𝑘 ∈ ℕ0) ∧ 𝑁 ∈
(ℤ≥‘(𝑘 + 1))) → 𝐴 ⊆
ℕ0) |
17 | 2 | ad2antrr 724 |
. . . . . . 7
⊢ (((𝜑 ∧ 𝑘 ∈ ℕ0) ∧ 𝑁 ∈
(ℤ≥‘(𝑘 + 1))) → 𝐵 ⊆
ℕ0) |
18 | | simplr 767 |
. . . . . . 7
⊢ (((𝜑 ∧ 𝑘 ∈ ℕ0) ∧ 𝑁 ∈
(ℤ≥‘(𝑘 + 1))) → 𝑘 ∈ ℕ0) |
19 | 6 | adantr 483 |
. . . . . . . 8
⊢ ((𝜑 ∧ 𝑘 ∈ ℕ0) → 𝑀 ∈
(ℤ≥‘𝑁)) |
20 | | uztrn 12255 |
. . . . . . . 8
⊢ ((𝑀 ∈
(ℤ≥‘𝑁) ∧ 𝑁 ∈ (ℤ≥‘(𝑘 + 1))) → 𝑀 ∈ (ℤ≥‘(𝑘 + 1))) |
21 | 19, 20 | sylan 582 |
. . . . . . 7
⊢ (((𝜑 ∧ 𝑘 ∈ ℕ0) ∧ 𝑁 ∈
(ℤ≥‘(𝑘 + 1))) → 𝑀 ∈ (ℤ≥‘(𝑘 + 1))) |
22 | 16, 17, 3, 18, 21 | smuval2 15825 |
. . . . . 6
⊢ (((𝜑 ∧ 𝑘 ∈ ℕ0) ∧ 𝑁 ∈
(ℤ≥‘(𝑘 + 1))) → (𝑘 ∈ (𝐴 smul 𝐵) ↔ 𝑘 ∈ (𝑃‘𝑀))) |
23 | 22 | bicomd 225 |
. . . . 5
⊢ (((𝜑 ∧ 𝑘 ∈ ℕ0) ∧ 𝑁 ∈
(ℤ≥‘(𝑘 + 1))) → (𝑘 ∈ (𝑃‘𝑀) ↔ 𝑘 ∈ (𝐴 smul 𝐵))) |
24 | 6 | ad2antrr 724 |
. . . . . . . . 9
⊢ (((𝜑 ∧ 𝑘 ∈ ℕ0) ∧ (𝑘 + 1) ∈
(ℤ≥‘𝑁)) → 𝑀 ∈ (ℤ≥‘𝑁)) |
25 | | simpll 765 |
. . . . . . . . 9
⊢ (((𝜑 ∧ 𝑘 ∈ ℕ0) ∧ (𝑘 + 1) ∈
(ℤ≥‘𝑁)) → 𝜑) |
26 | | fveqeq2 6674 |
. . . . . . . . . . 11
⊢ (𝑥 = 𝑁 → ((𝑃‘𝑥) = (𝑃‘𝑁) ↔ (𝑃‘𝑁) = (𝑃‘𝑁))) |
27 | 26 | imbi2d 343 |
. . . . . . . . . 10
⊢ (𝑥 = 𝑁 → ((𝜑 → (𝑃‘𝑥) = (𝑃‘𝑁)) ↔ (𝜑 → (𝑃‘𝑁) = (𝑃‘𝑁)))) |
28 | | fveqeq2 6674 |
. . . . . . . . . . 11
⊢ (𝑥 = 𝑘 → ((𝑃‘𝑥) = (𝑃‘𝑁) ↔ (𝑃‘𝑘) = (𝑃‘𝑁))) |
29 | 28 | imbi2d 343 |
. . . . . . . . . 10
⊢ (𝑥 = 𝑘 → ((𝜑 → (𝑃‘𝑥) = (𝑃‘𝑁)) ↔ (𝜑 → (𝑃‘𝑘) = (𝑃‘𝑁)))) |
30 | | fveqeq2 6674 |
. . . . . . . . . . 11
⊢ (𝑥 = (𝑘 + 1) → ((𝑃‘𝑥) = (𝑃‘𝑁) ↔ (𝑃‘(𝑘 + 1)) = (𝑃‘𝑁))) |
31 | 30 | imbi2d 343 |
. . . . . . . . . 10
⊢ (𝑥 = (𝑘 + 1) → ((𝜑 → (𝑃‘𝑥) = (𝑃‘𝑁)) ↔ (𝜑 → (𝑃‘(𝑘 + 1)) = (𝑃‘𝑁)))) |
32 | | fveqeq2 6674 |
. . . . . . . . . . 11
⊢ (𝑥 = 𝑀 → ((𝑃‘𝑥) = (𝑃‘𝑁) ↔ (𝑃‘𝑀) = (𝑃‘𝑁))) |
33 | 32 | imbi2d 343 |
. . . . . . . . . 10
⊢ (𝑥 = 𝑀 → ((𝜑 → (𝑃‘𝑥) = (𝑃‘𝑁)) ↔ (𝜑 → (𝑃‘𝑀) = (𝑃‘𝑁)))) |
34 | | eqidd 2822 |
. . . . . . . . . 10
⊢ (𝜑 → (𝑃‘𝑁) = (𝑃‘𝑁)) |
35 | 1 | adantr 483 |
. . . . . . . . . . . . . . . 16
⊢ ((𝜑 ∧ 𝑘 ∈ (ℤ≥‘𝑁)) → 𝐴 ⊆
ℕ0) |
36 | 2 | adantr 483 |
. . . . . . . . . . . . . . . 16
⊢ ((𝜑 ∧ 𝑘 ∈ (ℤ≥‘𝑁)) → 𝐵 ⊆
ℕ0) |
37 | | eluznn0 12311 |
. . . . . . . . . . . . . . . . 17
⊢ ((𝑁 ∈ ℕ0
∧ 𝑘 ∈
(ℤ≥‘𝑁)) → 𝑘 ∈ ℕ0) |
38 | 5, 37 | sylan 582 |
. . . . . . . . . . . . . . . 16
⊢ ((𝜑 ∧ 𝑘 ∈ (ℤ≥‘𝑁)) → 𝑘 ∈ ℕ0) |
39 | 35, 36, 3, 38 | smupp1 15823 |
. . . . . . . . . . . . . . 15
⊢ ((𝜑 ∧ 𝑘 ∈ (ℤ≥‘𝑁)) → (𝑃‘(𝑘 + 1)) = ((𝑃‘𝑘) sadd {𝑛 ∈ ℕ0 ∣ (𝑘 ∈ 𝐴 ∧ (𝑛 − 𝑘) ∈ 𝐵)})) |
40 | 5 | nn0red 11950 |
. . . . . . . . . . . . . . . . . . . . 21
⊢ (𝜑 → 𝑁 ∈ ℝ) |
41 | 40 | adantr 483 |
. . . . . . . . . . . . . . . . . . . 20
⊢ ((𝜑 ∧ 𝑘 ∈ (ℤ≥‘𝑁)) → 𝑁 ∈ ℝ) |
42 | 38 | nn0red 11950 |
. . . . . . . . . . . . . . . . . . . 20
⊢ ((𝜑 ∧ 𝑘 ∈ (ℤ≥‘𝑁)) → 𝑘 ∈ ℝ) |
43 | | eluzle 12250 |
. . . . . . . . . . . . . . . . . . . . 21
⊢ (𝑘 ∈
(ℤ≥‘𝑁) → 𝑁 ≤ 𝑘) |
44 | 43 | adantl 484 |
. . . . . . . . . . . . . . . . . . . 20
⊢ ((𝜑 ∧ 𝑘 ∈ (ℤ≥‘𝑁)) → 𝑁 ≤ 𝑘) |
45 | 41, 42, 44 | lensymd 10785 |
. . . . . . . . . . . . . . . . . . 19
⊢ ((𝜑 ∧ 𝑘 ∈ (ℤ≥‘𝑁)) → ¬ 𝑘 < 𝑁) |
46 | | smupvallem.a |
. . . . . . . . . . . . . . . . . . . . . . 23
⊢ (𝜑 → 𝐴 ⊆ (0..^𝑁)) |
47 | 46 | adantr 483 |
. . . . . . . . . . . . . . . . . . . . . 22
⊢ ((𝜑 ∧ 𝑘 ∈ (ℤ≥‘𝑁)) → 𝐴 ⊆ (0..^𝑁)) |
48 | 47 | sseld 3966 |
. . . . . . . . . . . . . . . . . . . . 21
⊢ ((𝜑 ∧ 𝑘 ∈ (ℤ≥‘𝑁)) → (𝑘 ∈ 𝐴 → 𝑘 ∈ (0..^𝑁))) |
49 | | elfzolt2 13041 |
. . . . . . . . . . . . . . . . . . . . 21
⊢ (𝑘 ∈ (0..^𝑁) → 𝑘 < 𝑁) |
50 | 48, 49 | syl6 35 |
. . . . . . . . . . . . . . . . . . . 20
⊢ ((𝜑 ∧ 𝑘 ∈ (ℤ≥‘𝑁)) → (𝑘 ∈ 𝐴 → 𝑘 < 𝑁)) |
51 | 50 | adantrd 494 |
. . . . . . . . . . . . . . . . . . 19
⊢ ((𝜑 ∧ 𝑘 ∈ (ℤ≥‘𝑁)) → ((𝑘 ∈ 𝐴 ∧ (𝑛 − 𝑘) ∈ 𝐵) → 𝑘 < 𝑁)) |
52 | 45, 51 | mtod 200 |
. . . . . . . . . . . . . . . . . 18
⊢ ((𝜑 ∧ 𝑘 ∈ (ℤ≥‘𝑁)) → ¬ (𝑘 ∈ 𝐴 ∧ (𝑛 − 𝑘) ∈ 𝐵)) |
53 | 52 | ralrimivw 3183 |
. . . . . . . . . . . . . . . . 17
⊢ ((𝜑 ∧ 𝑘 ∈ (ℤ≥‘𝑁)) → ∀𝑛 ∈ ℕ0
¬ (𝑘 ∈ 𝐴 ∧ (𝑛 − 𝑘) ∈ 𝐵)) |
54 | | rabeq0 4338 |
. . . . . . . . . . . . . . . . 17
⊢ ({𝑛 ∈ ℕ0
∣ (𝑘 ∈ 𝐴 ∧ (𝑛 − 𝑘) ∈ 𝐵)} = ∅ ↔ ∀𝑛 ∈ ℕ0
¬ (𝑘 ∈ 𝐴 ∧ (𝑛 − 𝑘) ∈ 𝐵)) |
55 | 53, 54 | sylibr 236 |
. . . . . . . . . . . . . . . 16
⊢ ((𝜑 ∧ 𝑘 ∈ (ℤ≥‘𝑁)) → {𝑛 ∈ ℕ0 ∣ (𝑘 ∈ 𝐴 ∧ (𝑛 − 𝑘) ∈ 𝐵)} = ∅) |
56 | 55 | oveq2d 7166 |
. . . . . . . . . . . . . . 15
⊢ ((𝜑 ∧ 𝑘 ∈ (ℤ≥‘𝑁)) → ((𝑃‘𝑘) sadd {𝑛 ∈ ℕ0 ∣ (𝑘 ∈ 𝐴 ∧ (𝑛 − 𝑘) ∈ 𝐵)}) = ((𝑃‘𝑘) sadd ∅)) |
57 | 4 | adantr 483 |
. . . . . . . . . . . . . . . . . 18
⊢ ((𝜑 ∧ 𝑘 ∈ (ℤ≥‘𝑁)) → 𝑃:ℕ0⟶𝒫
ℕ0) |
58 | 57, 38 | ffvelrnd 6847 |
. . . . . . . . . . . . . . . . 17
⊢ ((𝜑 ∧ 𝑘 ∈ (ℤ≥‘𝑁)) → (𝑃‘𝑘) ∈ 𝒫
ℕ0) |
59 | 58 | elpwid 4553 |
. . . . . . . . . . . . . . . 16
⊢ ((𝜑 ∧ 𝑘 ∈ (ℤ≥‘𝑁)) → (𝑃‘𝑘) ⊆
ℕ0) |
60 | | sadid1 15811 |
. . . . . . . . . . . . . . . 16
⊢ ((𝑃‘𝑘) ⊆ ℕ0 → ((𝑃‘𝑘) sadd ∅) = (𝑃‘𝑘)) |
61 | 59, 60 | syl 17 |
. . . . . . . . . . . . . . 15
⊢ ((𝜑 ∧ 𝑘 ∈ (ℤ≥‘𝑁)) → ((𝑃‘𝑘) sadd ∅) = (𝑃‘𝑘)) |
62 | 39, 56, 61 | 3eqtrd 2860 |
. . . . . . . . . . . . . 14
⊢ ((𝜑 ∧ 𝑘 ∈ (ℤ≥‘𝑁)) → (𝑃‘(𝑘 + 1)) = (𝑃‘𝑘)) |
63 | 62 | eqeq1d 2823 |
. . . . . . . . . . . . 13
⊢ ((𝜑 ∧ 𝑘 ∈ (ℤ≥‘𝑁)) → ((𝑃‘(𝑘 + 1)) = (𝑃‘𝑁) ↔ (𝑃‘𝑘) = (𝑃‘𝑁))) |
64 | 63 | biimprd 250 |
. . . . . . . . . . . 12
⊢ ((𝜑 ∧ 𝑘 ∈ (ℤ≥‘𝑁)) → ((𝑃‘𝑘) = (𝑃‘𝑁) → (𝑃‘(𝑘 + 1)) = (𝑃‘𝑁))) |
65 | 64 | expcom 416 |
. . . . . . . . . . 11
⊢ (𝑘 ∈
(ℤ≥‘𝑁) → (𝜑 → ((𝑃‘𝑘) = (𝑃‘𝑁) → (𝑃‘(𝑘 + 1)) = (𝑃‘𝑁)))) |
66 | 65 | a2d 29 |
. . . . . . . . . 10
⊢ (𝑘 ∈
(ℤ≥‘𝑁) → ((𝜑 → (𝑃‘𝑘) = (𝑃‘𝑁)) → (𝜑 → (𝑃‘(𝑘 + 1)) = (𝑃‘𝑁)))) |
67 | 27, 29, 31, 33, 34, 66 | uzind4i 12304 |
. . . . . . . . 9
⊢ (𝑀 ∈
(ℤ≥‘𝑁) → (𝜑 → (𝑃‘𝑀) = (𝑃‘𝑁))) |
68 | 24, 25, 67 | sylc 65 |
. . . . . . . 8
⊢ (((𝜑 ∧ 𝑘 ∈ ℕ0) ∧ (𝑘 + 1) ∈
(ℤ≥‘𝑁)) → (𝑃‘𝑀) = (𝑃‘𝑁)) |
69 | | simpr 487 |
. . . . . . . . 9
⊢ (((𝜑 ∧ 𝑘 ∈ ℕ0) ∧ (𝑘 + 1) ∈
(ℤ≥‘𝑁)) → (𝑘 + 1) ∈
(ℤ≥‘𝑁)) |
70 | 27, 29, 31, 31, 34, 66 | uzind4i 12304 |
. . . . . . . . 9
⊢ ((𝑘 + 1) ∈
(ℤ≥‘𝑁) → (𝜑 → (𝑃‘(𝑘 + 1)) = (𝑃‘𝑁))) |
71 | 69, 25, 70 | sylc 65 |
. . . . . . . 8
⊢ (((𝜑 ∧ 𝑘 ∈ ℕ0) ∧ (𝑘 + 1) ∈
(ℤ≥‘𝑁)) → (𝑃‘(𝑘 + 1)) = (𝑃‘𝑁)) |
72 | 68, 71 | eqtr4d 2859 |
. . . . . . 7
⊢ (((𝜑 ∧ 𝑘 ∈ ℕ0) ∧ (𝑘 + 1) ∈
(ℤ≥‘𝑁)) → (𝑃‘𝑀) = (𝑃‘(𝑘 + 1))) |
73 | 72 | eleq2d 2898 |
. . . . . 6
⊢ (((𝜑 ∧ 𝑘 ∈ ℕ0) ∧ (𝑘 + 1) ∈
(ℤ≥‘𝑁)) → (𝑘 ∈ (𝑃‘𝑀) ↔ 𝑘 ∈ (𝑃‘(𝑘 + 1)))) |
74 | 1 | ad2antrr 724 |
. . . . . . 7
⊢ (((𝜑 ∧ 𝑘 ∈ ℕ0) ∧ (𝑘 + 1) ∈
(ℤ≥‘𝑁)) → 𝐴 ⊆
ℕ0) |
75 | 2 | ad2antrr 724 |
. . . . . . 7
⊢ (((𝜑 ∧ 𝑘 ∈ ℕ0) ∧ (𝑘 + 1) ∈
(ℤ≥‘𝑁)) → 𝐵 ⊆
ℕ0) |
76 | | simplr 767 |
. . . . . . 7
⊢ (((𝜑 ∧ 𝑘 ∈ ℕ0) ∧ (𝑘 + 1) ∈
(ℤ≥‘𝑁)) → 𝑘 ∈ ℕ0) |
77 | 74, 75, 3, 76 | smuval 15824 |
. . . . . 6
⊢ (((𝜑 ∧ 𝑘 ∈ ℕ0) ∧ (𝑘 + 1) ∈
(ℤ≥‘𝑁)) → (𝑘 ∈ (𝐴 smul 𝐵) ↔ 𝑘 ∈ (𝑃‘(𝑘 + 1)))) |
78 | 73, 77 | bitr4d 284 |
. . . . 5
⊢ (((𝜑 ∧ 𝑘 ∈ ℕ0) ∧ (𝑘 + 1) ∈
(ℤ≥‘𝑁)) → (𝑘 ∈ (𝑃‘𝑀) ↔ 𝑘 ∈ (𝐴 smul 𝐵))) |
79 | | simpr 487 |
. . . . . . . 8
⊢ ((𝜑 ∧ 𝑘 ∈ ℕ0) → 𝑘 ∈
ℕ0) |
80 | 79 | nn0zd 12079 |
. . . . . . 7
⊢ ((𝜑 ∧ 𝑘 ∈ ℕ0) → 𝑘 ∈
ℤ) |
81 | 80 | peano2zd 12084 |
. . . . . 6
⊢ ((𝜑 ∧ 𝑘 ∈ ℕ0) → (𝑘 + 1) ∈
ℤ) |
82 | 5 | nn0zd 12079 |
. . . . . . 7
⊢ (𝜑 → 𝑁 ∈ ℤ) |
83 | 82 | adantr 483 |
. . . . . 6
⊢ ((𝜑 ∧ 𝑘 ∈ ℕ0) → 𝑁 ∈
ℤ) |
84 | | uztric 12260 |
. . . . . 6
⊢ (((𝑘 + 1) ∈ ℤ ∧ 𝑁 ∈ ℤ) → (𝑁 ∈
(ℤ≥‘(𝑘 + 1)) ∨ (𝑘 + 1) ∈
(ℤ≥‘𝑁))) |
85 | 81, 83, 84 | syl2anc 586 |
. . . . 5
⊢ ((𝜑 ∧ 𝑘 ∈ ℕ0) → (𝑁 ∈
(ℤ≥‘(𝑘 + 1)) ∨ (𝑘 + 1) ∈
(ℤ≥‘𝑁))) |
86 | 23, 78, 85 | mpjaodan 955 |
. . . 4
⊢ ((𝜑 ∧ 𝑘 ∈ ℕ0) → (𝑘 ∈ (𝑃‘𝑀) ↔ 𝑘 ∈ (𝐴 smul 𝐵))) |
87 | 86 | ex 415 |
. . 3
⊢ (𝜑 → (𝑘 ∈ ℕ0 → (𝑘 ∈ (𝑃‘𝑀) ↔ 𝑘 ∈ (𝐴 smul 𝐵)))) |
88 | 11, 15, 87 | pm5.21ndd 383 |
. 2
⊢ (𝜑 → (𝑘 ∈ (𝑃‘𝑀) ↔ 𝑘 ∈ (𝐴 smul 𝐵))) |
89 | 88 | eqrdv 2819 |
1
⊢ (𝜑 → (𝑃‘𝑀) = (𝐴 smul 𝐵)) |