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 15038 |
. . . . . 6
⊢ (𝜑 → 𝑃:ℕ0⟶𝒫
ℕ0) |
5 | | smuval.n |
. . . . . . 7
⊢ (𝜑 → 𝑁 ∈
ℕ0) |
6 | | smupvallem.m |
. . . . . . 7
⊢ (𝜑 → 𝑀 ∈ (ℤ≥‘𝑁)) |
7 | | eluznn0 11633 |
. . . . . . 7
⊢ ((𝑁 ∈ ℕ0
∧ 𝑀 ∈
(ℤ≥‘𝑁)) → 𝑀 ∈
ℕ0) |
8 | 5, 6, 7 | syl2anc 691 |
. . . . . 6
⊢ (𝜑 → 𝑀 ∈
ℕ0) |
9 | 4, 8 | ffvelrnd 6268 |
. . . . 5
⊢ (𝜑 → (𝑃‘𝑀) ∈ 𝒫
ℕ0) |
10 | 9 | elpwid 4118 |
. . . 4
⊢ (𝜑 → (𝑃‘𝑀) ⊆
ℕ0) |
11 | 10 | sseld 3567 |
. . 3
⊢ (𝜑 → (𝑘 ∈ (𝑃‘𝑀) → 𝑘 ∈
ℕ0)) |
12 | 1, 2, 3 | smufval 15037 |
. . . . 5
⊢ (𝜑 → (𝐴 smul 𝐵) = {𝑘 ∈ ℕ0 ∣ 𝑘 ∈ (𝑃‘(𝑘 + 1))}) |
13 | | ssrab2 3650 |
. . . . 5
⊢ {𝑘 ∈ ℕ0
∣ 𝑘 ∈ (𝑃‘(𝑘 + 1))} ⊆
ℕ0 |
14 | 12, 13 | syl6eqss 3618 |
. . . 4
⊢ (𝜑 → (𝐴 smul 𝐵) ⊆
ℕ0) |
15 | 14 | sseld 3567 |
. . 3
⊢ (𝜑 → (𝑘 ∈ (𝐴 smul 𝐵) → 𝑘 ∈
ℕ0)) |
16 | 1 | ad2antrr 758 |
. . . . . . 7
⊢ (((𝜑 ∧ 𝑘 ∈ ℕ0) ∧ 𝑁 ∈
(ℤ≥‘(𝑘 + 1))) → 𝐴 ⊆
ℕ0) |
17 | 2 | ad2antrr 758 |
. . . . . . 7
⊢ (((𝜑 ∧ 𝑘 ∈ ℕ0) ∧ 𝑁 ∈
(ℤ≥‘(𝑘 + 1))) → 𝐵 ⊆
ℕ0) |
18 | | simplr 788 |
. . . . . . 7
⊢ (((𝜑 ∧ 𝑘 ∈ ℕ0) ∧ 𝑁 ∈
(ℤ≥‘(𝑘 + 1))) → 𝑘 ∈ ℕ0) |
19 | 6 | adantr 480 |
. . . . . . . 8
⊢ ((𝜑 ∧ 𝑘 ∈ ℕ0) → 𝑀 ∈
(ℤ≥‘𝑁)) |
20 | | uztrn 11580 |
. . . . . . . 8
⊢ ((𝑀 ∈
(ℤ≥‘𝑁) ∧ 𝑁 ∈ (ℤ≥‘(𝑘 + 1))) → 𝑀 ∈ (ℤ≥‘(𝑘 + 1))) |
21 | 19, 20 | sylan 487 |
. . . . . . 7
⊢ (((𝜑 ∧ 𝑘 ∈ ℕ0) ∧ 𝑁 ∈
(ℤ≥‘(𝑘 + 1))) → 𝑀 ∈ (ℤ≥‘(𝑘 + 1))) |
22 | 16, 17, 3, 18, 21 | smuval2 15042 |
. . . . . 6
⊢ (((𝜑 ∧ 𝑘 ∈ ℕ0) ∧ 𝑁 ∈
(ℤ≥‘(𝑘 + 1))) → (𝑘 ∈ (𝐴 smul 𝐵) ↔ 𝑘 ∈ (𝑃‘𝑀))) |
23 | 22 | bicomd 212 |
. . . . 5
⊢ (((𝜑 ∧ 𝑘 ∈ ℕ0) ∧ 𝑁 ∈
(ℤ≥‘(𝑘 + 1))) → (𝑘 ∈ (𝑃‘𝑀) ↔ 𝑘 ∈ (𝐴 smul 𝐵))) |
24 | 6 | ad2antrr 758 |
. . . . . . . . 9
⊢ (((𝜑 ∧ 𝑘 ∈ ℕ0) ∧ (𝑘 + 1) ∈
(ℤ≥‘𝑁)) → 𝑀 ∈ (ℤ≥‘𝑁)) |
25 | | simpll 786 |
. . . . . . . . 9
⊢ (((𝜑 ∧ 𝑘 ∈ ℕ0) ∧ (𝑘 + 1) ∈
(ℤ≥‘𝑁)) → 𝜑) |
26 | | fveq2 6103 |
. . . . . . . . . . . 12
⊢ (𝑥 = 𝑁 → (𝑃‘𝑥) = (𝑃‘𝑁)) |
27 | 26 | eqeq1d 2612 |
. . . . . . . . . . 11
⊢ (𝑥 = 𝑁 → ((𝑃‘𝑥) = (𝑃‘𝑁) ↔ (𝑃‘𝑁) = (𝑃‘𝑁))) |
28 | 27 | imbi2d 329 |
. . . . . . . . . 10
⊢ (𝑥 = 𝑁 → ((𝜑 → (𝑃‘𝑥) = (𝑃‘𝑁)) ↔ (𝜑 → (𝑃‘𝑁) = (𝑃‘𝑁)))) |
29 | | fveq2 6103 |
. . . . . . . . . . . 12
⊢ (𝑥 = 𝑘 → (𝑃‘𝑥) = (𝑃‘𝑘)) |
30 | 29 | eqeq1d 2612 |
. . . . . . . . . . 11
⊢ (𝑥 = 𝑘 → ((𝑃‘𝑥) = (𝑃‘𝑁) ↔ (𝑃‘𝑘) = (𝑃‘𝑁))) |
31 | 30 | imbi2d 329 |
. . . . . . . . . 10
⊢ (𝑥 = 𝑘 → ((𝜑 → (𝑃‘𝑥) = (𝑃‘𝑁)) ↔ (𝜑 → (𝑃‘𝑘) = (𝑃‘𝑁)))) |
32 | | fveq2 6103 |
. . . . . . . . . . . 12
⊢ (𝑥 = (𝑘 + 1) → (𝑃‘𝑥) = (𝑃‘(𝑘 + 1))) |
33 | 32 | eqeq1d 2612 |
. . . . . . . . . . 11
⊢ (𝑥 = (𝑘 + 1) → ((𝑃‘𝑥) = (𝑃‘𝑁) ↔ (𝑃‘(𝑘 + 1)) = (𝑃‘𝑁))) |
34 | 33 | imbi2d 329 |
. . . . . . . . . 10
⊢ (𝑥 = (𝑘 + 1) → ((𝜑 → (𝑃‘𝑥) = (𝑃‘𝑁)) ↔ (𝜑 → (𝑃‘(𝑘 + 1)) = (𝑃‘𝑁)))) |
35 | | fveq2 6103 |
. . . . . . . . . . . 12
⊢ (𝑥 = 𝑀 → (𝑃‘𝑥) = (𝑃‘𝑀)) |
36 | 35 | eqeq1d 2612 |
. . . . . . . . . . 11
⊢ (𝑥 = 𝑀 → ((𝑃‘𝑥) = (𝑃‘𝑁) ↔ (𝑃‘𝑀) = (𝑃‘𝑁))) |
37 | 36 | imbi2d 329 |
. . . . . . . . . 10
⊢ (𝑥 = 𝑀 → ((𝜑 → (𝑃‘𝑥) = (𝑃‘𝑁)) ↔ (𝜑 → (𝑃‘𝑀) = (𝑃‘𝑁)))) |
38 | | eqidd 2611 |
. . . . . . . . . . 11
⊢ (𝜑 → (𝑃‘𝑁) = (𝑃‘𝑁)) |
39 | 38 | a1i 11 |
. . . . . . . . . 10
⊢ (𝑁 ∈ ℤ → (𝜑 → (𝑃‘𝑁) = (𝑃‘𝑁))) |
40 | 1 | adantr 480 |
. . . . . . . . . . . . . . . 16
⊢ ((𝜑 ∧ 𝑘 ∈ (ℤ≥‘𝑁)) → 𝐴 ⊆
ℕ0) |
41 | 2 | adantr 480 |
. . . . . . . . . . . . . . . 16
⊢ ((𝜑 ∧ 𝑘 ∈ (ℤ≥‘𝑁)) → 𝐵 ⊆
ℕ0) |
42 | | eluznn0 11633 |
. . . . . . . . . . . . . . . . 17
⊢ ((𝑁 ∈ ℕ0
∧ 𝑘 ∈
(ℤ≥‘𝑁)) → 𝑘 ∈ ℕ0) |
43 | 5, 42 | sylan 487 |
. . . . . . . . . . . . . . . 16
⊢ ((𝜑 ∧ 𝑘 ∈ (ℤ≥‘𝑁)) → 𝑘 ∈ ℕ0) |
44 | 40, 41, 3, 43 | smupp1 15040 |
. . . . . . . . . . . . . . 15
⊢ ((𝜑 ∧ 𝑘 ∈ (ℤ≥‘𝑁)) → (𝑃‘(𝑘 + 1)) = ((𝑃‘𝑘) sadd {𝑛 ∈ ℕ0 ∣ (𝑘 ∈ 𝐴 ∧ (𝑛 − 𝑘) ∈ 𝐵)})) |
45 | | eluzle 11576 |
. . . . . . . . . . . . . . . . . . . . 21
⊢ (𝑘 ∈
(ℤ≥‘𝑁) → 𝑁 ≤ 𝑘) |
46 | 45 | adantl 481 |
. . . . . . . . . . . . . . . . . . . 20
⊢ ((𝜑 ∧ 𝑘 ∈ (ℤ≥‘𝑁)) → 𝑁 ≤ 𝑘) |
47 | 5 | nn0red 11229 |
. . . . . . . . . . . . . . . . . . . . . 22
⊢ (𝜑 → 𝑁 ∈ ℝ) |
48 | 47 | adantr 480 |
. . . . . . . . . . . . . . . . . . . . 21
⊢ ((𝜑 ∧ 𝑘 ∈ (ℤ≥‘𝑁)) → 𝑁 ∈ ℝ) |
49 | 43 | nn0red 11229 |
. . . . . . . . . . . . . . . . . . . . 21
⊢ ((𝜑 ∧ 𝑘 ∈ (ℤ≥‘𝑁)) → 𝑘 ∈ ℝ) |
50 | 48, 49 | lenltd 10062 |
. . . . . . . . . . . . . . . . . . . 20
⊢ ((𝜑 ∧ 𝑘 ∈ (ℤ≥‘𝑁)) → (𝑁 ≤ 𝑘 ↔ ¬ 𝑘 < 𝑁)) |
51 | 46, 50 | mpbid 221 |
. . . . . . . . . . . . . . . . . . 19
⊢ ((𝜑 ∧ 𝑘 ∈ (ℤ≥‘𝑁)) → ¬ 𝑘 < 𝑁) |
52 | | smupvallem.a |
. . . . . . . . . . . . . . . . . . . . . . 23
⊢ (𝜑 → 𝐴 ⊆ (0..^𝑁)) |
53 | 52 | adantr 480 |
. . . . . . . . . . . . . . . . . . . . . 22
⊢ ((𝜑 ∧ 𝑘 ∈ (ℤ≥‘𝑁)) → 𝐴 ⊆ (0..^𝑁)) |
54 | 53 | sseld 3567 |
. . . . . . . . . . . . . . . . . . . . 21
⊢ ((𝜑 ∧ 𝑘 ∈ (ℤ≥‘𝑁)) → (𝑘 ∈ 𝐴 → 𝑘 ∈ (0..^𝑁))) |
55 | | elfzolt2 12348 |
. . . . . . . . . . . . . . . . . . . . 21
⊢ (𝑘 ∈ (0..^𝑁) → 𝑘 < 𝑁) |
56 | 54, 55 | syl6 34 |
. . . . . . . . . . . . . . . . . . . 20
⊢ ((𝜑 ∧ 𝑘 ∈ (ℤ≥‘𝑁)) → (𝑘 ∈ 𝐴 → 𝑘 < 𝑁)) |
57 | 56 | adantrd 483 |
. . . . . . . . . . . . . . . . . . 19
⊢ ((𝜑 ∧ 𝑘 ∈ (ℤ≥‘𝑁)) → ((𝑘 ∈ 𝐴 ∧ (𝑛 − 𝑘) ∈ 𝐵) → 𝑘 < 𝑁)) |
58 | 51, 57 | mtod 188 |
. . . . . . . . . . . . . . . . . 18
⊢ ((𝜑 ∧ 𝑘 ∈ (ℤ≥‘𝑁)) → ¬ (𝑘 ∈ 𝐴 ∧ (𝑛 − 𝑘) ∈ 𝐵)) |
59 | 58 | ralrimivw 2950 |
. . . . . . . . . . . . . . . . 17
⊢ ((𝜑 ∧ 𝑘 ∈ (ℤ≥‘𝑁)) → ∀𝑛 ∈ ℕ0
¬ (𝑘 ∈ 𝐴 ∧ (𝑛 − 𝑘) ∈ 𝐵)) |
60 | | rabeq0 3911 |
. . . . . . . . . . . . . . . . 17
⊢ ({𝑛 ∈ ℕ0
∣ (𝑘 ∈ 𝐴 ∧ (𝑛 − 𝑘) ∈ 𝐵)} = ∅ ↔ ∀𝑛 ∈ ℕ0
¬ (𝑘 ∈ 𝐴 ∧ (𝑛 − 𝑘) ∈ 𝐵)) |
61 | 59, 60 | sylibr 223 |
. . . . . . . . . . . . . . . 16
⊢ ((𝜑 ∧ 𝑘 ∈ (ℤ≥‘𝑁)) → {𝑛 ∈ ℕ0 ∣ (𝑘 ∈ 𝐴 ∧ (𝑛 − 𝑘) ∈ 𝐵)} = ∅) |
62 | 61 | oveq2d 6565 |
. . . . . . . . . . . . . . 15
⊢ ((𝜑 ∧ 𝑘 ∈ (ℤ≥‘𝑁)) → ((𝑃‘𝑘) sadd {𝑛 ∈ ℕ0 ∣ (𝑘 ∈ 𝐴 ∧ (𝑛 − 𝑘) ∈ 𝐵)}) = ((𝑃‘𝑘) sadd ∅)) |
63 | 4 | adantr 480 |
. . . . . . . . . . . . . . . . . 18
⊢ ((𝜑 ∧ 𝑘 ∈ (ℤ≥‘𝑁)) → 𝑃:ℕ0⟶𝒫
ℕ0) |
64 | 63, 43 | ffvelrnd 6268 |
. . . . . . . . . . . . . . . . 17
⊢ ((𝜑 ∧ 𝑘 ∈ (ℤ≥‘𝑁)) → (𝑃‘𝑘) ∈ 𝒫
ℕ0) |
65 | 64 | elpwid 4118 |
. . . . . . . . . . . . . . . 16
⊢ ((𝜑 ∧ 𝑘 ∈ (ℤ≥‘𝑁)) → (𝑃‘𝑘) ⊆
ℕ0) |
66 | | sadid1 15028 |
. . . . . . . . . . . . . . . 16
⊢ ((𝑃‘𝑘) ⊆ ℕ0 → ((𝑃‘𝑘) sadd ∅) = (𝑃‘𝑘)) |
67 | 65, 66 | syl 17 |
. . . . . . . . . . . . . . 15
⊢ ((𝜑 ∧ 𝑘 ∈ (ℤ≥‘𝑁)) → ((𝑃‘𝑘) sadd ∅) = (𝑃‘𝑘)) |
68 | 44, 62, 67 | 3eqtrd 2648 |
. . . . . . . . . . . . . 14
⊢ ((𝜑 ∧ 𝑘 ∈ (ℤ≥‘𝑁)) → (𝑃‘(𝑘 + 1)) = (𝑃‘𝑘)) |
69 | 68 | eqeq1d 2612 |
. . . . . . . . . . . . 13
⊢ ((𝜑 ∧ 𝑘 ∈ (ℤ≥‘𝑁)) → ((𝑃‘(𝑘 + 1)) = (𝑃‘𝑁) ↔ (𝑃‘𝑘) = (𝑃‘𝑁))) |
70 | 69 | biimprd 237 |
. . . . . . . . . . . 12
⊢ ((𝜑 ∧ 𝑘 ∈ (ℤ≥‘𝑁)) → ((𝑃‘𝑘) = (𝑃‘𝑁) → (𝑃‘(𝑘 + 1)) = (𝑃‘𝑁))) |
71 | 70 | expcom 450 |
. . . . . . . . . . 11
⊢ (𝑘 ∈
(ℤ≥‘𝑁) → (𝜑 → ((𝑃‘𝑘) = (𝑃‘𝑁) → (𝑃‘(𝑘 + 1)) = (𝑃‘𝑁)))) |
72 | 71 | a2d 29 |
. . . . . . . . . 10
⊢ (𝑘 ∈
(ℤ≥‘𝑁) → ((𝜑 → (𝑃‘𝑘) = (𝑃‘𝑁)) → (𝜑 → (𝑃‘(𝑘 + 1)) = (𝑃‘𝑁)))) |
73 | 28, 31, 34, 37, 39, 72 | uzind4 11622 |
. . . . . . . . 9
⊢ (𝑀 ∈
(ℤ≥‘𝑁) → (𝜑 → (𝑃‘𝑀) = (𝑃‘𝑁))) |
74 | 24, 25, 73 | sylc 63 |
. . . . . . . 8
⊢ (((𝜑 ∧ 𝑘 ∈ ℕ0) ∧ (𝑘 + 1) ∈
(ℤ≥‘𝑁)) → (𝑃‘𝑀) = (𝑃‘𝑁)) |
75 | | simpr 476 |
. . . . . . . . 9
⊢ (((𝜑 ∧ 𝑘 ∈ ℕ0) ∧ (𝑘 + 1) ∈
(ℤ≥‘𝑁)) → (𝑘 + 1) ∈
(ℤ≥‘𝑁)) |
76 | 28, 31, 34, 34, 39, 72 | uzind4 11622 |
. . . . . . . . 9
⊢ ((𝑘 + 1) ∈
(ℤ≥‘𝑁) → (𝜑 → (𝑃‘(𝑘 + 1)) = (𝑃‘𝑁))) |
77 | 75, 25, 76 | sylc 63 |
. . . . . . . 8
⊢ (((𝜑 ∧ 𝑘 ∈ ℕ0) ∧ (𝑘 + 1) ∈
(ℤ≥‘𝑁)) → (𝑃‘(𝑘 + 1)) = (𝑃‘𝑁)) |
78 | 74, 77 | eqtr4d 2647 |
. . . . . . 7
⊢ (((𝜑 ∧ 𝑘 ∈ ℕ0) ∧ (𝑘 + 1) ∈
(ℤ≥‘𝑁)) → (𝑃‘𝑀) = (𝑃‘(𝑘 + 1))) |
79 | 78 | eleq2d 2673 |
. . . . . 6
⊢ (((𝜑 ∧ 𝑘 ∈ ℕ0) ∧ (𝑘 + 1) ∈
(ℤ≥‘𝑁)) → (𝑘 ∈ (𝑃‘𝑀) ↔ 𝑘 ∈ (𝑃‘(𝑘 + 1)))) |
80 | 1 | ad2antrr 758 |
. . . . . . 7
⊢ (((𝜑 ∧ 𝑘 ∈ ℕ0) ∧ (𝑘 + 1) ∈
(ℤ≥‘𝑁)) → 𝐴 ⊆
ℕ0) |
81 | 2 | ad2antrr 758 |
. . . . . . 7
⊢ (((𝜑 ∧ 𝑘 ∈ ℕ0) ∧ (𝑘 + 1) ∈
(ℤ≥‘𝑁)) → 𝐵 ⊆
ℕ0) |
82 | | simplr 788 |
. . . . . . 7
⊢ (((𝜑 ∧ 𝑘 ∈ ℕ0) ∧ (𝑘 + 1) ∈
(ℤ≥‘𝑁)) → 𝑘 ∈ ℕ0) |
83 | 80, 81, 3, 82 | smuval 15041 |
. . . . . 6
⊢ (((𝜑 ∧ 𝑘 ∈ ℕ0) ∧ (𝑘 + 1) ∈
(ℤ≥‘𝑁)) → (𝑘 ∈ (𝐴 smul 𝐵) ↔ 𝑘 ∈ (𝑃‘(𝑘 + 1)))) |
84 | 79, 83 | bitr4d 270 |
. . . . 5
⊢ (((𝜑 ∧ 𝑘 ∈ ℕ0) ∧ (𝑘 + 1) ∈
(ℤ≥‘𝑁)) → (𝑘 ∈ (𝑃‘𝑀) ↔ 𝑘 ∈ (𝐴 smul 𝐵))) |
85 | | simpr 476 |
. . . . . . . 8
⊢ ((𝜑 ∧ 𝑘 ∈ ℕ0) → 𝑘 ∈
ℕ0) |
86 | 85 | nn0zd 11356 |
. . . . . . 7
⊢ ((𝜑 ∧ 𝑘 ∈ ℕ0) → 𝑘 ∈
ℤ) |
87 | 86 | peano2zd 11361 |
. . . . . 6
⊢ ((𝜑 ∧ 𝑘 ∈ ℕ0) → (𝑘 + 1) ∈
ℤ) |
88 | 5 | nn0zd 11356 |
. . . . . . 7
⊢ (𝜑 → 𝑁 ∈ ℤ) |
89 | 88 | adantr 480 |
. . . . . 6
⊢ ((𝜑 ∧ 𝑘 ∈ ℕ0) → 𝑁 ∈
ℤ) |
90 | | uztric 11585 |
. . . . . 6
⊢ (((𝑘 + 1) ∈ ℤ ∧ 𝑁 ∈ ℤ) → (𝑁 ∈
(ℤ≥‘(𝑘 + 1)) ∨ (𝑘 + 1) ∈
(ℤ≥‘𝑁))) |
91 | 87, 89, 90 | syl2anc 691 |
. . . . 5
⊢ ((𝜑 ∧ 𝑘 ∈ ℕ0) → (𝑁 ∈
(ℤ≥‘(𝑘 + 1)) ∨ (𝑘 + 1) ∈
(ℤ≥‘𝑁))) |
92 | 23, 84, 91 | mpjaodan 823 |
. . . 4
⊢ ((𝜑 ∧ 𝑘 ∈ ℕ0) → (𝑘 ∈ (𝑃‘𝑀) ↔ 𝑘 ∈ (𝐴 smul 𝐵))) |
93 | 92 | ex 449 |
. . 3
⊢ (𝜑 → (𝑘 ∈ ℕ0 → (𝑘 ∈ (𝑃‘𝑀) ↔ 𝑘 ∈ (𝐴 smul 𝐵)))) |
94 | 11, 15, 93 | pm5.21ndd 368 |
. 2
⊢ (𝜑 → (𝑘 ∈ (𝑃‘𝑀) ↔ 𝑘 ∈ (𝐴 smul 𝐵))) |
95 | 94 | eqrdv 2608 |
1
⊢ (𝜑 → (𝑃‘𝑀) = (𝐴 smul 𝐵)) |