| 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 16515 |
. . . . . 6
⊢ (𝜑 → 𝑃:ℕ0⟶𝒫
ℕ0) |
| 5 | | smuval.n |
. . . . . . 7
⊢ (𝜑 → 𝑁 ∈
ℕ0) |
| 6 | | smupvallem.m |
. . . . . . 7
⊢ (𝜑 → 𝑀 ∈ (ℤ≥‘𝑁)) |
| 7 | | eluznn0 12959 |
. . . . . . 7
⊢ ((𝑁 ∈ ℕ0
∧ 𝑀 ∈
(ℤ≥‘𝑁)) → 𝑀 ∈
ℕ0) |
| 8 | 5, 6, 7 | syl2anc 584 |
. . . . . 6
⊢ (𝜑 → 𝑀 ∈
ℕ0) |
| 9 | 4, 8 | ffvelcdmd 7105 |
. . . . 5
⊢ (𝜑 → (𝑃‘𝑀) ∈ 𝒫
ℕ0) |
| 10 | 9 | elpwid 4609 |
. . . 4
⊢ (𝜑 → (𝑃‘𝑀) ⊆
ℕ0) |
| 11 | 10 | sseld 3982 |
. . 3
⊢ (𝜑 → (𝑘 ∈ (𝑃‘𝑀) → 𝑘 ∈
ℕ0)) |
| 12 | 1, 2, 3 | smufval 16514 |
. . . . 5
⊢ (𝜑 → (𝐴 smul 𝐵) = {𝑘 ∈ ℕ0 ∣ 𝑘 ∈ (𝑃‘(𝑘 + 1))}) |
| 13 | | ssrab2 4080 |
. . . . 5
⊢ {𝑘 ∈ ℕ0
∣ 𝑘 ∈ (𝑃‘(𝑘 + 1))} ⊆
ℕ0 |
| 14 | 12, 13 | eqsstrdi 4028 |
. . . 4
⊢ (𝜑 → (𝐴 smul 𝐵) ⊆
ℕ0) |
| 15 | 14 | sseld 3982 |
. . 3
⊢ (𝜑 → (𝑘 ∈ (𝐴 smul 𝐵) → 𝑘 ∈
ℕ0)) |
| 16 | 1 | ad2antrr 726 |
. . . . . . 7
⊢ (((𝜑 ∧ 𝑘 ∈ ℕ0) ∧ 𝑁 ∈
(ℤ≥‘(𝑘 + 1))) → 𝐴 ⊆
ℕ0) |
| 17 | 2 | ad2antrr 726 |
. . . . . . 7
⊢ (((𝜑 ∧ 𝑘 ∈ ℕ0) ∧ 𝑁 ∈
(ℤ≥‘(𝑘 + 1))) → 𝐵 ⊆
ℕ0) |
| 18 | | simplr 769 |
. . . . . . 7
⊢ (((𝜑 ∧ 𝑘 ∈ ℕ0) ∧ 𝑁 ∈
(ℤ≥‘(𝑘 + 1))) → 𝑘 ∈ ℕ0) |
| 19 | 6 | adantr 480 |
. . . . . . . 8
⊢ ((𝜑 ∧ 𝑘 ∈ ℕ0) → 𝑀 ∈
(ℤ≥‘𝑁)) |
| 20 | | uztrn 12896 |
. . . . . . . 8
⊢ ((𝑀 ∈
(ℤ≥‘𝑁) ∧ 𝑁 ∈ (ℤ≥‘(𝑘 + 1))) → 𝑀 ∈ (ℤ≥‘(𝑘 + 1))) |
| 21 | 19, 20 | sylan 580 |
. . . . . . 7
⊢ (((𝜑 ∧ 𝑘 ∈ ℕ0) ∧ 𝑁 ∈
(ℤ≥‘(𝑘 + 1))) → 𝑀 ∈ (ℤ≥‘(𝑘 + 1))) |
| 22 | 16, 17, 3, 18, 21 | smuval2 16519 |
. . . . . 6
⊢ (((𝜑 ∧ 𝑘 ∈ ℕ0) ∧ 𝑁 ∈
(ℤ≥‘(𝑘 + 1))) → (𝑘 ∈ (𝐴 smul 𝐵) ↔ 𝑘 ∈ (𝑃‘𝑀))) |
| 23 | 22 | bicomd 223 |
. . . . 5
⊢ (((𝜑 ∧ 𝑘 ∈ ℕ0) ∧ 𝑁 ∈
(ℤ≥‘(𝑘 + 1))) → (𝑘 ∈ (𝑃‘𝑀) ↔ 𝑘 ∈ (𝐴 smul 𝐵))) |
| 24 | 6 | ad2antrr 726 |
. . . . . . . . 9
⊢ (((𝜑 ∧ 𝑘 ∈ ℕ0) ∧ (𝑘 + 1) ∈
(ℤ≥‘𝑁)) → 𝑀 ∈ (ℤ≥‘𝑁)) |
| 25 | | simpll 767 |
. . . . . . . . 9
⊢ (((𝜑 ∧ 𝑘 ∈ ℕ0) ∧ (𝑘 + 1) ∈
(ℤ≥‘𝑁)) → 𝜑) |
| 26 | | fveqeq2 6915 |
. . . . . . . . . . 11
⊢ (𝑥 = 𝑁 → ((𝑃‘𝑥) = (𝑃‘𝑁) ↔ (𝑃‘𝑁) = (𝑃‘𝑁))) |
| 27 | 26 | imbi2d 340 |
. . . . . . . . . 10
⊢ (𝑥 = 𝑁 → ((𝜑 → (𝑃‘𝑥) = (𝑃‘𝑁)) ↔ (𝜑 → (𝑃‘𝑁) = (𝑃‘𝑁)))) |
| 28 | | fveqeq2 6915 |
. . . . . . . . . . 11
⊢ (𝑥 = 𝑘 → ((𝑃‘𝑥) = (𝑃‘𝑁) ↔ (𝑃‘𝑘) = (𝑃‘𝑁))) |
| 29 | 28 | imbi2d 340 |
. . . . . . . . . 10
⊢ (𝑥 = 𝑘 → ((𝜑 → (𝑃‘𝑥) = (𝑃‘𝑁)) ↔ (𝜑 → (𝑃‘𝑘) = (𝑃‘𝑁)))) |
| 30 | | fveqeq2 6915 |
. . . . . . . . . . 11
⊢ (𝑥 = (𝑘 + 1) → ((𝑃‘𝑥) = (𝑃‘𝑁) ↔ (𝑃‘(𝑘 + 1)) = (𝑃‘𝑁))) |
| 31 | 30 | imbi2d 340 |
. . . . . . . . . 10
⊢ (𝑥 = (𝑘 + 1) → ((𝜑 → (𝑃‘𝑥) = (𝑃‘𝑁)) ↔ (𝜑 → (𝑃‘(𝑘 + 1)) = (𝑃‘𝑁)))) |
| 32 | | fveqeq2 6915 |
. . . . . . . . . . 11
⊢ (𝑥 = 𝑀 → ((𝑃‘𝑥) = (𝑃‘𝑁) ↔ (𝑃‘𝑀) = (𝑃‘𝑁))) |
| 33 | 32 | imbi2d 340 |
. . . . . . . . . 10
⊢ (𝑥 = 𝑀 → ((𝜑 → (𝑃‘𝑥) = (𝑃‘𝑁)) ↔ (𝜑 → (𝑃‘𝑀) = (𝑃‘𝑁)))) |
| 34 | | eqidd 2738 |
. . . . . . . . . 10
⊢ (𝜑 → (𝑃‘𝑁) = (𝑃‘𝑁)) |
| 35 | 1 | adantr 480 |
. . . . . . . . . . . . . . . 16
⊢ ((𝜑 ∧ 𝑘 ∈ (ℤ≥‘𝑁)) → 𝐴 ⊆
ℕ0) |
| 36 | 2 | adantr 480 |
. . . . . . . . . . . . . . . 16
⊢ ((𝜑 ∧ 𝑘 ∈ (ℤ≥‘𝑁)) → 𝐵 ⊆
ℕ0) |
| 37 | | eluznn0 12959 |
. . . . . . . . . . . . . . . . 17
⊢ ((𝑁 ∈ ℕ0
∧ 𝑘 ∈
(ℤ≥‘𝑁)) → 𝑘 ∈ ℕ0) |
| 38 | 5, 37 | sylan 580 |
. . . . . . . . . . . . . . . 16
⊢ ((𝜑 ∧ 𝑘 ∈ (ℤ≥‘𝑁)) → 𝑘 ∈ ℕ0) |
| 39 | 35, 36, 3, 38 | smupp1 16517 |
. . . . . . . . . . . . . . 15
⊢ ((𝜑 ∧ 𝑘 ∈ (ℤ≥‘𝑁)) → (𝑃‘(𝑘 + 1)) = ((𝑃‘𝑘) sadd {𝑛 ∈ ℕ0 ∣ (𝑘 ∈ 𝐴 ∧ (𝑛 − 𝑘) ∈ 𝐵)})) |
| 40 | 5 | nn0red 12588 |
. . . . . . . . . . . . . . . . . . . . 21
⊢ (𝜑 → 𝑁 ∈ ℝ) |
| 41 | 40 | adantr 480 |
. . . . . . . . . . . . . . . . . . . 20
⊢ ((𝜑 ∧ 𝑘 ∈ (ℤ≥‘𝑁)) → 𝑁 ∈ ℝ) |
| 42 | 38 | nn0red 12588 |
. . . . . . . . . . . . . . . . . . . 20
⊢ ((𝜑 ∧ 𝑘 ∈ (ℤ≥‘𝑁)) → 𝑘 ∈ ℝ) |
| 43 | | eluzle 12891 |
. . . . . . . . . . . . . . . . . . . . 21
⊢ (𝑘 ∈
(ℤ≥‘𝑁) → 𝑁 ≤ 𝑘) |
| 44 | 43 | adantl 481 |
. . . . . . . . . . . . . . . . . . . 20
⊢ ((𝜑 ∧ 𝑘 ∈ (ℤ≥‘𝑁)) → 𝑁 ≤ 𝑘) |
| 45 | 41, 42, 44 | lensymd 11412 |
. . . . . . . . . . . . . . . . . . 19
⊢ ((𝜑 ∧ 𝑘 ∈ (ℤ≥‘𝑁)) → ¬ 𝑘 < 𝑁) |
| 46 | | smupvallem.a |
. . . . . . . . . . . . . . . . . . . . . . 23
⊢ (𝜑 → 𝐴 ⊆ (0..^𝑁)) |
| 47 | 46 | adantr 480 |
. . . . . . . . . . . . . . . . . . . . . 22
⊢ ((𝜑 ∧ 𝑘 ∈ (ℤ≥‘𝑁)) → 𝐴 ⊆ (0..^𝑁)) |
| 48 | 47 | sseld 3982 |
. . . . . . . . . . . . . . . . . . . . 21
⊢ ((𝜑 ∧ 𝑘 ∈ (ℤ≥‘𝑁)) → (𝑘 ∈ 𝐴 → 𝑘 ∈ (0..^𝑁))) |
| 49 | | elfzolt2 13708 |
. . . . . . . . . . . . . . . . . . . . 21
⊢ (𝑘 ∈ (0..^𝑁) → 𝑘 < 𝑁) |
| 50 | 48, 49 | syl6 35 |
. . . . . . . . . . . . . . . . . . . 20
⊢ ((𝜑 ∧ 𝑘 ∈ (ℤ≥‘𝑁)) → (𝑘 ∈ 𝐴 → 𝑘 < 𝑁)) |
| 51 | 50 | adantrd 491 |
. . . . . . . . . . . . . . . . . . 19
⊢ ((𝜑 ∧ 𝑘 ∈ (ℤ≥‘𝑁)) → ((𝑘 ∈ 𝐴 ∧ (𝑛 − 𝑘) ∈ 𝐵) → 𝑘 < 𝑁)) |
| 52 | 45, 51 | mtod 198 |
. . . . . . . . . . . . . . . . . 18
⊢ ((𝜑 ∧ 𝑘 ∈ (ℤ≥‘𝑁)) → ¬ (𝑘 ∈ 𝐴 ∧ (𝑛 − 𝑘) ∈ 𝐵)) |
| 53 | 52 | ralrimivw 3150 |
. . . . . . . . . . . . . . . . 17
⊢ ((𝜑 ∧ 𝑘 ∈ (ℤ≥‘𝑁)) → ∀𝑛 ∈ ℕ0
¬ (𝑘 ∈ 𝐴 ∧ (𝑛 − 𝑘) ∈ 𝐵)) |
| 54 | | rabeq0 4388 |
. . . . . . . . . . . . . . . . 17
⊢ ({𝑛 ∈ ℕ0
∣ (𝑘 ∈ 𝐴 ∧ (𝑛 − 𝑘) ∈ 𝐵)} = ∅ ↔ ∀𝑛 ∈ ℕ0
¬ (𝑘 ∈ 𝐴 ∧ (𝑛 − 𝑘) ∈ 𝐵)) |
| 55 | 53, 54 | sylibr 234 |
. . . . . . . . . . . . . . . 16
⊢ ((𝜑 ∧ 𝑘 ∈ (ℤ≥‘𝑁)) → {𝑛 ∈ ℕ0 ∣ (𝑘 ∈ 𝐴 ∧ (𝑛 − 𝑘) ∈ 𝐵)} = ∅) |
| 56 | 55 | oveq2d 7447 |
. . . . . . . . . . . . . . 15
⊢ ((𝜑 ∧ 𝑘 ∈ (ℤ≥‘𝑁)) → ((𝑃‘𝑘) sadd {𝑛 ∈ ℕ0 ∣ (𝑘 ∈ 𝐴 ∧ (𝑛 − 𝑘) ∈ 𝐵)}) = ((𝑃‘𝑘) sadd ∅)) |
| 57 | 4 | adantr 480 |
. . . . . . . . . . . . . . . . . 18
⊢ ((𝜑 ∧ 𝑘 ∈ (ℤ≥‘𝑁)) → 𝑃:ℕ0⟶𝒫
ℕ0) |
| 58 | 57, 38 | ffvelcdmd 7105 |
. . . . . . . . . . . . . . . . 17
⊢ ((𝜑 ∧ 𝑘 ∈ (ℤ≥‘𝑁)) → (𝑃‘𝑘) ∈ 𝒫
ℕ0) |
| 59 | 58 | elpwid 4609 |
. . . . . . . . . . . . . . . 16
⊢ ((𝜑 ∧ 𝑘 ∈ (ℤ≥‘𝑁)) → (𝑃‘𝑘) ⊆
ℕ0) |
| 60 | | sadid1 16505 |
. . . . . . . . . . . . . . . 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 2781 |
. . . . . . . . . . . . . 14
⊢ ((𝜑 ∧ 𝑘 ∈ (ℤ≥‘𝑁)) → (𝑃‘(𝑘 + 1)) = (𝑃‘𝑘)) |
| 63 | 62 | eqeq1d 2739 |
. . . . . . . . . . . . 13
⊢ ((𝜑 ∧ 𝑘 ∈ (ℤ≥‘𝑁)) → ((𝑃‘(𝑘 + 1)) = (𝑃‘𝑁) ↔ (𝑃‘𝑘) = (𝑃‘𝑁))) |
| 64 | 63 | biimprd 248 |
. . . . . . . . . . . 12
⊢ ((𝜑 ∧ 𝑘 ∈ (ℤ≥‘𝑁)) → ((𝑃‘𝑘) = (𝑃‘𝑁) → (𝑃‘(𝑘 + 1)) = (𝑃‘𝑁))) |
| 65 | 64 | expcom 413 |
. . . . . . . . . . 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 12952 |
. . . . . . . . 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 484 |
. . . . . . . . 9
⊢ (((𝜑 ∧ 𝑘 ∈ ℕ0) ∧ (𝑘 + 1) ∈
(ℤ≥‘𝑁)) → (𝑘 + 1) ∈
(ℤ≥‘𝑁)) |
| 70 | 27, 29, 31, 31, 34, 66 | uzind4i 12952 |
. . . . . . . . 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 2780 |
. . . . . . 7
⊢ (((𝜑 ∧ 𝑘 ∈ ℕ0) ∧ (𝑘 + 1) ∈
(ℤ≥‘𝑁)) → (𝑃‘𝑀) = (𝑃‘(𝑘 + 1))) |
| 73 | 72 | eleq2d 2827 |
. . . . . 6
⊢ (((𝜑 ∧ 𝑘 ∈ ℕ0) ∧ (𝑘 + 1) ∈
(ℤ≥‘𝑁)) → (𝑘 ∈ (𝑃‘𝑀) ↔ 𝑘 ∈ (𝑃‘(𝑘 + 1)))) |
| 74 | 1 | ad2antrr 726 |
. . . . . . 7
⊢ (((𝜑 ∧ 𝑘 ∈ ℕ0) ∧ (𝑘 + 1) ∈
(ℤ≥‘𝑁)) → 𝐴 ⊆
ℕ0) |
| 75 | 2 | ad2antrr 726 |
. . . . . . 7
⊢ (((𝜑 ∧ 𝑘 ∈ ℕ0) ∧ (𝑘 + 1) ∈
(ℤ≥‘𝑁)) → 𝐵 ⊆
ℕ0) |
| 76 | | simplr 769 |
. . . . . . 7
⊢ (((𝜑 ∧ 𝑘 ∈ ℕ0) ∧ (𝑘 + 1) ∈
(ℤ≥‘𝑁)) → 𝑘 ∈ ℕ0) |
| 77 | 74, 75, 3, 76 | smuval 16518 |
. . . . . 6
⊢ (((𝜑 ∧ 𝑘 ∈ ℕ0) ∧ (𝑘 + 1) ∈
(ℤ≥‘𝑁)) → (𝑘 ∈ (𝐴 smul 𝐵) ↔ 𝑘 ∈ (𝑃‘(𝑘 + 1)))) |
| 78 | 73, 77 | bitr4d 282 |
. . . . 5
⊢ (((𝜑 ∧ 𝑘 ∈ ℕ0) ∧ (𝑘 + 1) ∈
(ℤ≥‘𝑁)) → (𝑘 ∈ (𝑃‘𝑀) ↔ 𝑘 ∈ (𝐴 smul 𝐵))) |
| 79 | | simpr 484 |
. . . . . . . 8
⊢ ((𝜑 ∧ 𝑘 ∈ ℕ0) → 𝑘 ∈
ℕ0) |
| 80 | 79 | nn0zd 12639 |
. . . . . . 7
⊢ ((𝜑 ∧ 𝑘 ∈ ℕ0) → 𝑘 ∈
ℤ) |
| 81 | 80 | peano2zd 12725 |
. . . . . 6
⊢ ((𝜑 ∧ 𝑘 ∈ ℕ0) → (𝑘 + 1) ∈
ℤ) |
| 82 | 5 | nn0zd 12639 |
. . . . . . 7
⊢ (𝜑 → 𝑁 ∈ ℤ) |
| 83 | 82 | adantr 480 |
. . . . . 6
⊢ ((𝜑 ∧ 𝑘 ∈ ℕ0) → 𝑁 ∈
ℤ) |
| 84 | | uztric 12902 |
. . . . . 6
⊢ (((𝑘 + 1) ∈ ℤ ∧ 𝑁 ∈ ℤ) → (𝑁 ∈
(ℤ≥‘(𝑘 + 1)) ∨ (𝑘 + 1) ∈
(ℤ≥‘𝑁))) |
| 85 | 81, 83, 84 | syl2anc 584 |
. . . . 5
⊢ ((𝜑 ∧ 𝑘 ∈ ℕ0) → (𝑁 ∈
(ℤ≥‘(𝑘 + 1)) ∨ (𝑘 + 1) ∈
(ℤ≥‘𝑁))) |
| 86 | 23, 78, 85 | mpjaodan 961 |
. . . 4
⊢ ((𝜑 ∧ 𝑘 ∈ ℕ0) → (𝑘 ∈ (𝑃‘𝑀) ↔ 𝑘 ∈ (𝐴 smul 𝐵))) |
| 87 | 86 | ex 412 |
. . 3
⊢ (𝜑 → (𝑘 ∈ ℕ0 → (𝑘 ∈ (𝑃‘𝑀) ↔ 𝑘 ∈ (𝐴 smul 𝐵)))) |
| 88 | 11, 15, 87 | pm5.21ndd 379 |
. 2
⊢ (𝜑 → (𝑘 ∈ (𝑃‘𝑀) ↔ 𝑘 ∈ (𝐴 smul 𝐵))) |
| 89 | 88 | eqrdv 2735 |
1
⊢ (𝜑 → (𝑃‘𝑀) = (𝐴 smul 𝐵)) |