Proof of Theorem vdwlem5
| Step | Hyp | Ref
| Expression |
| 1 | | vdwlem6.t |
. 2
⊢ 𝑇 = (𝐵 + (𝑊 · ((𝐴 + (𝑉 − 𝐷)) − 1))) |
| 2 | | vdwlem6.b |
. . 3
⊢ (𝜑 → 𝐵 ∈ ℕ) |
| 3 | | vdwlem3.w |
. . . . 5
⊢ (𝜑 → 𝑊 ∈ ℕ) |
| 4 | 3 | nnnn0d 12587 |
. . . 4
⊢ (𝜑 → 𝑊 ∈
ℕ0) |
| 5 | | vdwlem7.a |
. . . . . 6
⊢ (𝜑 → 𝐴 ∈ ℕ) |
| 6 | | vdwlem3.v |
. . . . . . . . . 10
⊢ (𝜑 → 𝑉 ∈ ℕ) |
| 7 | 6 | nncnd 12282 |
. . . . . . . . 9
⊢ (𝜑 → 𝑉 ∈ ℂ) |
| 8 | | vdwlem7.d |
. . . . . . . . . 10
⊢ (𝜑 → 𝐷 ∈ ℕ) |
| 9 | 8 | nncnd 12282 |
. . . . . . . . 9
⊢ (𝜑 → 𝐷 ∈ ℂ) |
| 10 | 7, 9 | subcld 11620 |
. . . . . . . 8
⊢ (𝜑 → (𝑉 − 𝐷) ∈ ℂ) |
| 11 | 5 | nncnd 12282 |
. . . . . . . 8
⊢ (𝜑 → 𝐴 ∈ ℂ) |
| 12 | 10, 11 | npcand 11624 |
. . . . . . 7
⊢ (𝜑 → (((𝑉 − 𝐷) − 𝐴) + 𝐴) = (𝑉 − 𝐷)) |
| 13 | 7, 9, 11 | subsub4d 11651 |
. . . . . . . . . 10
⊢ (𝜑 → ((𝑉 − 𝐷) − 𝐴) = (𝑉 − (𝐷 + 𝐴))) |
| 14 | 9, 11 | addcomd 11463 |
. . . . . . . . . . 11
⊢ (𝜑 → (𝐷 + 𝐴) = (𝐴 + 𝐷)) |
| 15 | 14 | oveq2d 7447 |
. . . . . . . . . 10
⊢ (𝜑 → (𝑉 − (𝐷 + 𝐴)) = (𝑉 − (𝐴 + 𝐷))) |
| 16 | 13, 15 | eqtrd 2777 |
. . . . . . . . 9
⊢ (𝜑 → ((𝑉 − 𝐷) − 𝐴) = (𝑉 − (𝐴 + 𝐷))) |
| 17 | | cnvimass 6100 |
. . . . . . . . . . . . 13
⊢ (◡𝐹 “ {𝐺}) ⊆ dom 𝐹 |
| 18 | | vdwlem4.r |
. . . . . . . . . . . . . 14
⊢ (𝜑 → 𝑅 ∈ Fin) |
| 19 | | vdwlem4.h |
. . . . . . . . . . . . . 14
⊢ (𝜑 → 𝐻:(1...(𝑊 · (2 · 𝑉)))⟶𝑅) |
| 20 | | vdwlem4.f |
. . . . . . . . . . . . . 14
⊢ 𝐹 = (𝑥 ∈ (1...𝑉) ↦ (𝑦 ∈ (1...𝑊) ↦ (𝐻‘(𝑦 + (𝑊 · ((𝑥 − 1) + 𝑉)))))) |
| 21 | 6, 3, 18, 19, 20 | vdwlem4 17022 |
. . . . . . . . . . . . 13
⊢ (𝜑 → 𝐹:(1...𝑉)⟶(𝑅 ↑m (1...𝑊))) |
| 22 | 17, 21 | fssdm 6755 |
. . . . . . . . . . . 12
⊢ (𝜑 → (◡𝐹 “ {𝐺}) ⊆ (1...𝑉)) |
| 23 | | vdwlem7.s |
. . . . . . . . . . . . 13
⊢ (𝜑 → (𝐴(AP‘𝐾)𝐷) ⊆ (◡𝐹 “ {𝐺})) |
| 24 | | ssun2 4179 |
. . . . . . . . . . . . . . 15
⊢ ((𝐴 + 𝐷)(AP‘(𝐾 − 1))𝐷) ⊆ ({𝐴} ∪ ((𝐴 + 𝐷)(AP‘(𝐾 − 1))𝐷)) |
| 25 | | vdwlem7.k |
. . . . . . . . . . . . . . . . 17
⊢ (𝜑 → 𝐾 ∈
(ℤ≥‘2)) |
| 26 | | uz2m1nn 12965 |
. . . . . . . . . . . . . . . . 17
⊢ (𝐾 ∈
(ℤ≥‘2) → (𝐾 − 1) ∈ ℕ) |
| 27 | 25, 26 | syl 17 |
. . . . . . . . . . . . . . . 16
⊢ (𝜑 → (𝐾 − 1) ∈ ℕ) |
| 28 | 5, 8 | nnaddcld 12318 |
. . . . . . . . . . . . . . . 16
⊢ (𝜑 → (𝐴 + 𝐷) ∈ ℕ) |
| 29 | | vdwapid1 17013 |
. . . . . . . . . . . . . . . 16
⊢ (((𝐾 − 1) ∈ ℕ ∧
(𝐴 + 𝐷) ∈ ℕ ∧ 𝐷 ∈ ℕ) → (𝐴 + 𝐷) ∈ ((𝐴 + 𝐷)(AP‘(𝐾 − 1))𝐷)) |
| 30 | 27, 28, 8, 29 | syl3anc 1373 |
. . . . . . . . . . . . . . 15
⊢ (𝜑 → (𝐴 + 𝐷) ∈ ((𝐴 + 𝐷)(AP‘(𝐾 − 1))𝐷)) |
| 31 | 24, 30 | sselid 3981 |
. . . . . . . . . . . . . 14
⊢ (𝜑 → (𝐴 + 𝐷) ∈ ({𝐴} ∪ ((𝐴 + 𝐷)(AP‘(𝐾 − 1))𝐷))) |
| 32 | | eluz2nn 12924 |
. . . . . . . . . . . . . . . . . . . 20
⊢ (𝐾 ∈
(ℤ≥‘2) → 𝐾 ∈ ℕ) |
| 33 | 25, 32 | syl 17 |
. . . . . . . . . . . . . . . . . . 19
⊢ (𝜑 → 𝐾 ∈ ℕ) |
| 34 | 33 | nncnd 12282 |
. . . . . . . . . . . . . . . . . 18
⊢ (𝜑 → 𝐾 ∈ ℂ) |
| 35 | | ax-1cn 11213 |
. . . . . . . . . . . . . . . . . 18
⊢ 1 ∈
ℂ |
| 36 | | npcan 11517 |
. . . . . . . . . . . . . . . . . 18
⊢ ((𝐾 ∈ ℂ ∧ 1 ∈
ℂ) → ((𝐾 −
1) + 1) = 𝐾) |
| 37 | 34, 35, 36 | sylancl 586 |
. . . . . . . . . . . . . . . . 17
⊢ (𝜑 → ((𝐾 − 1) + 1) = 𝐾) |
| 38 | 37 | fveq2d 6910 |
. . . . . . . . . . . . . . . 16
⊢ (𝜑 → (AP‘((𝐾 − 1) + 1)) =
(AP‘𝐾)) |
| 39 | 38 | oveqd 7448 |
. . . . . . . . . . . . . . 15
⊢ (𝜑 → (𝐴(AP‘((𝐾 − 1) + 1))𝐷) = (𝐴(AP‘𝐾)𝐷)) |
| 40 | | nnm1nn0 12567 |
. . . . . . . . . . . . . . . . 17
⊢ (𝐾 ∈ ℕ → (𝐾 − 1) ∈
ℕ0) |
| 41 | 33, 40 | syl 17 |
. . . . . . . . . . . . . . . 16
⊢ (𝜑 → (𝐾 − 1) ∈
ℕ0) |
| 42 | | vdwapun 17012 |
. . . . . . . . . . . . . . . 16
⊢ (((𝐾 − 1) ∈
ℕ0 ∧ 𝐴
∈ ℕ ∧ 𝐷
∈ ℕ) → (𝐴(AP‘((𝐾 − 1) + 1))𝐷) = ({𝐴} ∪ ((𝐴 + 𝐷)(AP‘(𝐾 − 1))𝐷))) |
| 43 | 41, 5, 8, 42 | syl3anc 1373 |
. . . . . . . . . . . . . . 15
⊢ (𝜑 → (𝐴(AP‘((𝐾 − 1) + 1))𝐷) = ({𝐴} ∪ ((𝐴 + 𝐷)(AP‘(𝐾 − 1))𝐷))) |
| 44 | 39, 43 | eqtr3d 2779 |
. . . . . . . . . . . . . 14
⊢ (𝜑 → (𝐴(AP‘𝐾)𝐷) = ({𝐴} ∪ ((𝐴 + 𝐷)(AP‘(𝐾 − 1))𝐷))) |
| 45 | 31, 44 | eleqtrrd 2844 |
. . . . . . . . . . . . 13
⊢ (𝜑 → (𝐴 + 𝐷) ∈ (𝐴(AP‘𝐾)𝐷)) |
| 46 | 23, 45 | sseldd 3984 |
. . . . . . . . . . . 12
⊢ (𝜑 → (𝐴 + 𝐷) ∈ (◡𝐹 “ {𝐺})) |
| 47 | 22, 46 | sseldd 3984 |
. . . . . . . . . . 11
⊢ (𝜑 → (𝐴 + 𝐷) ∈ (1...𝑉)) |
| 48 | | elfzuz3 13561 |
. . . . . . . . . . 11
⊢ ((𝐴 + 𝐷) ∈ (1...𝑉) → 𝑉 ∈ (ℤ≥‘(𝐴 + 𝐷))) |
| 49 | 47, 48 | syl 17 |
. . . . . . . . . 10
⊢ (𝜑 → 𝑉 ∈ (ℤ≥‘(𝐴 + 𝐷))) |
| 50 | | uznn0sub 12917 |
. . . . . . . . . 10
⊢ (𝑉 ∈
(ℤ≥‘(𝐴 + 𝐷)) → (𝑉 − (𝐴 + 𝐷)) ∈
ℕ0) |
| 51 | 49, 50 | syl 17 |
. . . . . . . . 9
⊢ (𝜑 → (𝑉 − (𝐴 + 𝐷)) ∈
ℕ0) |
| 52 | 16, 51 | eqeltrd 2841 |
. . . . . . . 8
⊢ (𝜑 → ((𝑉 − 𝐷) − 𝐴) ∈
ℕ0) |
| 53 | | nn0nnaddcl 12557 |
. . . . . . . 8
⊢ ((((𝑉 − 𝐷) − 𝐴) ∈ ℕ0 ∧ 𝐴 ∈ ℕ) → (((𝑉 − 𝐷) − 𝐴) + 𝐴) ∈ ℕ) |
| 54 | 52, 5, 53 | syl2anc 584 |
. . . . . . 7
⊢ (𝜑 → (((𝑉 − 𝐷) − 𝐴) + 𝐴) ∈ ℕ) |
| 55 | 12, 54 | eqeltrrd 2842 |
. . . . . 6
⊢ (𝜑 → (𝑉 − 𝐷) ∈ ℕ) |
| 56 | 5, 55 | nnaddcld 12318 |
. . . . 5
⊢ (𝜑 → (𝐴 + (𝑉 − 𝐷)) ∈ ℕ) |
| 57 | | nnm1nn0 12567 |
. . . . 5
⊢ ((𝐴 + (𝑉 − 𝐷)) ∈ ℕ → ((𝐴 + (𝑉 − 𝐷)) − 1) ∈
ℕ0) |
| 58 | 56, 57 | syl 17 |
. . . 4
⊢ (𝜑 → ((𝐴 + (𝑉 − 𝐷)) − 1) ∈
ℕ0) |
| 59 | 4, 58 | nn0mulcld 12592 |
. . 3
⊢ (𝜑 → (𝑊 · ((𝐴 + (𝑉 − 𝐷)) − 1)) ∈
ℕ0) |
| 60 | | nnnn0addcl 12556 |
. . 3
⊢ ((𝐵 ∈ ℕ ∧ (𝑊 · ((𝐴 + (𝑉 − 𝐷)) − 1)) ∈ ℕ0)
→ (𝐵 + (𝑊 · ((𝐴 + (𝑉 − 𝐷)) − 1))) ∈
ℕ) |
| 61 | 2, 59, 60 | syl2anc 584 |
. 2
⊢ (𝜑 → (𝐵 + (𝑊 · ((𝐴 + (𝑉 − 𝐷)) − 1))) ∈
ℕ) |
| 62 | 1, 61 | eqeltrid 2845 |
1
⊢ (𝜑 → 𝑇 ∈ ℕ) |