| Step | Hyp | Ref
| Expression |
| 1 | | isercoll2.z |
. . 3
⊢ 𝑍 =
(ℤ≥‘𝑀) |
| 2 | | isercoll2.m |
. . 3
⊢ (𝜑 → 𝑀 ∈ ℤ) |
| 3 | | 1z 12647 |
. . . 4
⊢ 1 ∈
ℤ |
| 4 | | zsubcl 12659 |
. . . 4
⊢ ((1
∈ ℤ ∧ 𝑀
∈ ℤ) → (1 − 𝑀) ∈ ℤ) |
| 5 | 3, 2, 4 | sylancr 587 |
. . 3
⊢ (𝜑 → (1 − 𝑀) ∈
ℤ) |
| 6 | | seqex 14044 |
. . . 4
⊢ seq𝑀( + , 𝐻) ∈ V |
| 7 | 6 | a1i 11 |
. . 3
⊢ (𝜑 → seq𝑀( + , 𝐻) ∈ V) |
| 8 | | seqex 14044 |
. . . 4
⊢ seq1( + ,
(𝑥 ∈ ℕ ↦
(𝐻‘(𝑀 + (𝑥 − 1))))) ∈ V |
| 9 | 8 | a1i 11 |
. . 3
⊢ (𝜑 → seq1( + , (𝑥 ∈ ℕ ↦ (𝐻‘(𝑀 + (𝑥 − 1))))) ∈ V) |
| 10 | | simpr 484 |
. . . . . 6
⊢ ((𝜑 ∧ 𝑘 ∈ 𝑍) → 𝑘 ∈ 𝑍) |
| 11 | 10, 1 | eleqtrdi 2851 |
. . . . 5
⊢ ((𝜑 ∧ 𝑘 ∈ 𝑍) → 𝑘 ∈ (ℤ≥‘𝑀)) |
| 12 | 5 | adantr 480 |
. . . . 5
⊢ ((𝜑 ∧ 𝑘 ∈ 𝑍) → (1 − 𝑀) ∈ ℤ) |
| 13 | | simpl 482 |
. . . . . 6
⊢ ((𝜑 ∧ 𝑘 ∈ 𝑍) → 𝜑) |
| 14 | | elfzuz 13560 |
. . . . . . 7
⊢ (𝑗 ∈ (𝑀...𝑘) → 𝑗 ∈ (ℤ≥‘𝑀)) |
| 15 | 14, 1 | eleqtrrdi 2852 |
. . . . . 6
⊢ (𝑗 ∈ (𝑀...𝑘) → 𝑗 ∈ 𝑍) |
| 16 | | simpr 484 |
. . . . . . . . . . . . 13
⊢ ((𝜑 ∧ 𝑗 ∈ 𝑍) → 𝑗 ∈ 𝑍) |
| 17 | 16, 1 | eleqtrdi 2851 |
. . . . . . . . . . . 12
⊢ ((𝜑 ∧ 𝑗 ∈ 𝑍) → 𝑗 ∈ (ℤ≥‘𝑀)) |
| 18 | | eluzelz 12888 |
. . . . . . . . . . . 12
⊢ (𝑗 ∈
(ℤ≥‘𝑀) → 𝑗 ∈ ℤ) |
| 19 | 17, 18 | syl 17 |
. . . . . . . . . . 11
⊢ ((𝜑 ∧ 𝑗 ∈ 𝑍) → 𝑗 ∈ ℤ) |
| 20 | 19 | zcnd 12723 |
. . . . . . . . . 10
⊢ ((𝜑 ∧ 𝑗 ∈ 𝑍) → 𝑗 ∈ ℂ) |
| 21 | 2 | zcnd 12723 |
. . . . . . . . . . 11
⊢ (𝜑 → 𝑀 ∈ ℂ) |
| 22 | 21 | adantr 480 |
. . . . . . . . . 10
⊢ ((𝜑 ∧ 𝑗 ∈ 𝑍) → 𝑀 ∈ ℂ) |
| 23 | | 1cnd 11256 |
. . . . . . . . . 10
⊢ ((𝜑 ∧ 𝑗 ∈ 𝑍) → 1 ∈ ℂ) |
| 24 | 20, 22, 23 | subadd23d 11642 |
. . . . . . . . 9
⊢ ((𝜑 ∧ 𝑗 ∈ 𝑍) → ((𝑗 − 𝑀) + 1) = (𝑗 + (1 − 𝑀))) |
| 25 | | uznn0sub 12917 |
. . . . . . . . . . 11
⊢ (𝑗 ∈
(ℤ≥‘𝑀) → (𝑗 − 𝑀) ∈
ℕ0) |
| 26 | 17, 25 | syl 17 |
. . . . . . . . . 10
⊢ ((𝜑 ∧ 𝑗 ∈ 𝑍) → (𝑗 − 𝑀) ∈
ℕ0) |
| 27 | | nn0p1nn 12565 |
. . . . . . . . . 10
⊢ ((𝑗 − 𝑀) ∈ ℕ0 → ((𝑗 − 𝑀) + 1) ∈ ℕ) |
| 28 | 26, 27 | syl 17 |
. . . . . . . . 9
⊢ ((𝜑 ∧ 𝑗 ∈ 𝑍) → ((𝑗 − 𝑀) + 1) ∈ ℕ) |
| 29 | 24, 28 | eqeltrrd 2842 |
. . . . . . . 8
⊢ ((𝜑 ∧ 𝑗 ∈ 𝑍) → (𝑗 + (1 − 𝑀)) ∈ ℕ) |
| 30 | | oveq1 7438 |
. . . . . . . . . . 11
⊢ (𝑥 = (𝑗 + (1 − 𝑀)) → (𝑥 − 1) = ((𝑗 + (1 − 𝑀)) − 1)) |
| 31 | 30 | oveq2d 7447 |
. . . . . . . . . 10
⊢ (𝑥 = (𝑗 + (1 − 𝑀)) → (𝑀 + (𝑥 − 1)) = (𝑀 + ((𝑗 + (1 − 𝑀)) − 1))) |
| 32 | 31 | fveq2d 6910 |
. . . . . . . . 9
⊢ (𝑥 = (𝑗 + (1 − 𝑀)) → (𝐻‘(𝑀 + (𝑥 − 1))) = (𝐻‘(𝑀 + ((𝑗 + (1 − 𝑀)) − 1)))) |
| 33 | | eqid 2737 |
. . . . . . . . 9
⊢ (𝑥 ∈ ℕ ↦ (𝐻‘(𝑀 + (𝑥 − 1)))) = (𝑥 ∈ ℕ ↦ (𝐻‘(𝑀 + (𝑥 − 1)))) |
| 34 | | fvex 6919 |
. . . . . . . . 9
⊢ (𝐻‘(𝑀 + ((𝑗 + (1 − 𝑀)) − 1))) ∈ V |
| 35 | 32, 33, 34 | fvmpt 7016 |
. . . . . . . 8
⊢ ((𝑗 + (1 − 𝑀)) ∈ ℕ → ((𝑥 ∈ ℕ ↦ (𝐻‘(𝑀 + (𝑥 − 1))))‘(𝑗 + (1 − 𝑀))) = (𝐻‘(𝑀 + ((𝑗 + (1 − 𝑀)) − 1)))) |
| 36 | 29, 35 | syl 17 |
. . . . . . 7
⊢ ((𝜑 ∧ 𝑗 ∈ 𝑍) → ((𝑥 ∈ ℕ ↦ (𝐻‘(𝑀 + (𝑥 − 1))))‘(𝑗 + (1 − 𝑀))) = (𝐻‘(𝑀 + ((𝑗 + (1 − 𝑀)) − 1)))) |
| 37 | 24 | oveq1d 7446 |
. . . . . . . . . . 11
⊢ ((𝜑 ∧ 𝑗 ∈ 𝑍) → (((𝑗 − 𝑀) + 1) − 1) = ((𝑗 + (1 − 𝑀)) − 1)) |
| 38 | 26 | nn0cnd 12589 |
. . . . . . . . . . . 12
⊢ ((𝜑 ∧ 𝑗 ∈ 𝑍) → (𝑗 − 𝑀) ∈ ℂ) |
| 39 | | ax-1cn 11213 |
. . . . . . . . . . . 12
⊢ 1 ∈
ℂ |
| 40 | | pncan 11514 |
. . . . . . . . . . . 12
⊢ (((𝑗 − 𝑀) ∈ ℂ ∧ 1 ∈ ℂ)
→ (((𝑗 − 𝑀) + 1) − 1) = (𝑗 − 𝑀)) |
| 41 | 38, 39, 40 | sylancl 586 |
. . . . . . . . . . 11
⊢ ((𝜑 ∧ 𝑗 ∈ 𝑍) → (((𝑗 − 𝑀) + 1) − 1) = (𝑗 − 𝑀)) |
| 42 | 37, 41 | eqtr3d 2779 |
. . . . . . . . . 10
⊢ ((𝜑 ∧ 𝑗 ∈ 𝑍) → ((𝑗 + (1 − 𝑀)) − 1) = (𝑗 − 𝑀)) |
| 43 | 42 | oveq2d 7447 |
. . . . . . . . 9
⊢ ((𝜑 ∧ 𝑗 ∈ 𝑍) → (𝑀 + ((𝑗 + (1 − 𝑀)) − 1)) = (𝑀 + (𝑗 − 𝑀))) |
| 44 | 22, 20 | pncan3d 11623 |
. . . . . . . . 9
⊢ ((𝜑 ∧ 𝑗 ∈ 𝑍) → (𝑀 + (𝑗 − 𝑀)) = 𝑗) |
| 45 | 43, 44 | eqtrd 2777 |
. . . . . . . 8
⊢ ((𝜑 ∧ 𝑗 ∈ 𝑍) → (𝑀 + ((𝑗 + (1 − 𝑀)) − 1)) = 𝑗) |
| 46 | 45 | fveq2d 6910 |
. . . . . . 7
⊢ ((𝜑 ∧ 𝑗 ∈ 𝑍) → (𝐻‘(𝑀 + ((𝑗 + (1 − 𝑀)) − 1))) = (𝐻‘𝑗)) |
| 47 | 36, 46 | eqtr2d 2778 |
. . . . . 6
⊢ ((𝜑 ∧ 𝑗 ∈ 𝑍) → (𝐻‘𝑗) = ((𝑥 ∈ ℕ ↦ (𝐻‘(𝑀 + (𝑥 − 1))))‘(𝑗 + (1 − 𝑀)))) |
| 48 | 13, 15, 47 | syl2an 596 |
. . . . 5
⊢ (((𝜑 ∧ 𝑘 ∈ 𝑍) ∧ 𝑗 ∈ (𝑀...𝑘)) → (𝐻‘𝑗) = ((𝑥 ∈ ℕ ↦ (𝐻‘(𝑀 + (𝑥 − 1))))‘(𝑗 + (1 − 𝑀)))) |
| 49 | 11, 12, 48 | seqshft2 14069 |
. . . 4
⊢ ((𝜑 ∧ 𝑘 ∈ 𝑍) → (seq𝑀( + , 𝐻)‘𝑘) = (seq(𝑀 + (1 − 𝑀))( + , (𝑥 ∈ ℕ ↦ (𝐻‘(𝑀 + (𝑥 − 1)))))‘(𝑘 + (1 − 𝑀)))) |
| 50 | 21 | adantr 480 |
. . . . . . 7
⊢ ((𝜑 ∧ 𝑘 ∈ 𝑍) → 𝑀 ∈ ℂ) |
| 51 | | pncan3 11516 |
. . . . . . 7
⊢ ((𝑀 ∈ ℂ ∧ 1 ∈
ℂ) → (𝑀 + (1
− 𝑀)) =
1) |
| 52 | 50, 39, 51 | sylancl 586 |
. . . . . 6
⊢ ((𝜑 ∧ 𝑘 ∈ 𝑍) → (𝑀 + (1 − 𝑀)) = 1) |
| 53 | 52 | seqeq1d 14048 |
. . . . 5
⊢ ((𝜑 ∧ 𝑘 ∈ 𝑍) → seq(𝑀 + (1 − 𝑀))( + , (𝑥 ∈ ℕ ↦ (𝐻‘(𝑀 + (𝑥 − 1))))) = seq1( + , (𝑥 ∈ ℕ ↦ (𝐻‘(𝑀 + (𝑥 − 1)))))) |
| 54 | 53 | fveq1d 6908 |
. . . 4
⊢ ((𝜑 ∧ 𝑘 ∈ 𝑍) → (seq(𝑀 + (1 − 𝑀))( + , (𝑥 ∈ ℕ ↦ (𝐻‘(𝑀 + (𝑥 − 1)))))‘(𝑘 + (1 − 𝑀))) = (seq1( + , (𝑥 ∈ ℕ ↦ (𝐻‘(𝑀 + (𝑥 − 1)))))‘(𝑘 + (1 − 𝑀)))) |
| 55 | 49, 54 | eqtr2d 2778 |
. . 3
⊢ ((𝜑 ∧ 𝑘 ∈ 𝑍) → (seq1( + , (𝑥 ∈ ℕ ↦ (𝐻‘(𝑀 + (𝑥 − 1)))))‘(𝑘 + (1 − 𝑀))) = (seq𝑀( + , 𝐻)‘𝑘)) |
| 56 | 1, 2, 5, 7, 9, 55 | climshft2 15618 |
. 2
⊢ (𝜑 → (seq𝑀( + , 𝐻) ⇝ 𝐴 ↔ seq1( + , (𝑥 ∈ ℕ ↦ (𝐻‘(𝑀 + (𝑥 − 1))))) ⇝ 𝐴)) |
| 57 | | isercoll2.w |
. . 3
⊢ 𝑊 =
(ℤ≥‘𝑁) |
| 58 | | isercoll2.n |
. . 3
⊢ (𝜑 → 𝑁 ∈ ℤ) |
| 59 | | isercoll2.g |
. . . . . 6
⊢ (𝜑 → 𝐺:𝑍⟶𝑊) |
| 60 | 59 | adantr 480 |
. . . . 5
⊢ ((𝜑 ∧ 𝑥 ∈ ℕ) → 𝐺:𝑍⟶𝑊) |
| 61 | | uzid 12893 |
. . . . . . . 8
⊢ (𝑀 ∈ ℤ → 𝑀 ∈
(ℤ≥‘𝑀)) |
| 62 | 2, 61 | syl 17 |
. . . . . . 7
⊢ (𝜑 → 𝑀 ∈ (ℤ≥‘𝑀)) |
| 63 | | nnm1nn0 12567 |
. . . . . . 7
⊢ (𝑥 ∈ ℕ → (𝑥 − 1) ∈
ℕ0) |
| 64 | | uzaddcl 12946 |
. . . . . . 7
⊢ ((𝑀 ∈
(ℤ≥‘𝑀) ∧ (𝑥 − 1) ∈ ℕ0)
→ (𝑀 + (𝑥 − 1)) ∈
(ℤ≥‘𝑀)) |
| 65 | 62, 63, 64 | syl2an 596 |
. . . . . 6
⊢ ((𝜑 ∧ 𝑥 ∈ ℕ) → (𝑀 + (𝑥 − 1)) ∈
(ℤ≥‘𝑀)) |
| 66 | 65, 1 | eleqtrrdi 2852 |
. . . . 5
⊢ ((𝜑 ∧ 𝑥 ∈ ℕ) → (𝑀 + (𝑥 − 1)) ∈ 𝑍) |
| 67 | 60, 66 | ffvelcdmd 7105 |
. . . 4
⊢ ((𝜑 ∧ 𝑥 ∈ ℕ) → (𝐺‘(𝑀 + (𝑥 − 1))) ∈ 𝑊) |
| 68 | 67 | fmpttd 7135 |
. . 3
⊢ (𝜑 → (𝑥 ∈ ℕ ↦ (𝐺‘(𝑀 + (𝑥 − 1)))):ℕ⟶𝑊) |
| 69 | | fveq2 6906 |
. . . . . . 7
⊢ (𝑘 = (𝑀 + (𝑗 − 1)) → (𝐺‘𝑘) = (𝐺‘(𝑀 + (𝑗 − 1)))) |
| 70 | | fvoveq1 7454 |
. . . . . . 7
⊢ (𝑘 = (𝑀 + (𝑗 − 1)) → (𝐺‘(𝑘 + 1)) = (𝐺‘((𝑀 + (𝑗 − 1)) + 1))) |
| 71 | 69, 70 | breq12d 5156 |
. . . . . 6
⊢ (𝑘 = (𝑀 + (𝑗 − 1)) → ((𝐺‘𝑘) < (𝐺‘(𝑘 + 1)) ↔ (𝐺‘(𝑀 + (𝑗 − 1))) < (𝐺‘((𝑀 + (𝑗 − 1)) + 1)))) |
| 72 | | isercoll2.i |
. . . . . . . 8
⊢ ((𝜑 ∧ 𝑘 ∈ 𝑍) → (𝐺‘𝑘) < (𝐺‘(𝑘 + 1))) |
| 73 | 72 | ralrimiva 3146 |
. . . . . . 7
⊢ (𝜑 → ∀𝑘 ∈ 𝑍 (𝐺‘𝑘) < (𝐺‘(𝑘 + 1))) |
| 74 | 73 | adantr 480 |
. . . . . 6
⊢ ((𝜑 ∧ 𝑗 ∈ ℕ) → ∀𝑘 ∈ 𝑍 (𝐺‘𝑘) < (𝐺‘(𝑘 + 1))) |
| 75 | | nnm1nn0 12567 |
. . . . . . . 8
⊢ (𝑗 ∈ ℕ → (𝑗 − 1) ∈
ℕ0) |
| 76 | | uzaddcl 12946 |
. . . . . . . 8
⊢ ((𝑀 ∈
(ℤ≥‘𝑀) ∧ (𝑗 − 1) ∈ ℕ0)
→ (𝑀 + (𝑗 − 1)) ∈
(ℤ≥‘𝑀)) |
| 77 | 62, 75, 76 | syl2an 596 |
. . . . . . 7
⊢ ((𝜑 ∧ 𝑗 ∈ ℕ) → (𝑀 + (𝑗 − 1)) ∈
(ℤ≥‘𝑀)) |
| 78 | 77, 1 | eleqtrrdi 2852 |
. . . . . 6
⊢ ((𝜑 ∧ 𝑗 ∈ ℕ) → (𝑀 + (𝑗 − 1)) ∈ 𝑍) |
| 79 | 71, 74, 78 | rspcdva 3623 |
. . . . 5
⊢ ((𝜑 ∧ 𝑗 ∈ ℕ) → (𝐺‘(𝑀 + (𝑗 − 1))) < (𝐺‘((𝑀 + (𝑗 − 1)) + 1))) |
| 80 | | nncn 12274 |
. . . . . . . . . 10
⊢ (𝑗 ∈ ℕ → 𝑗 ∈
ℂ) |
| 81 | 80 | adantl 481 |
. . . . . . . . 9
⊢ ((𝜑 ∧ 𝑗 ∈ ℕ) → 𝑗 ∈ ℂ) |
| 82 | | 1cnd 11256 |
. . . . . . . . 9
⊢ ((𝜑 ∧ 𝑗 ∈ ℕ) → 1 ∈
ℂ) |
| 83 | 81, 82, 82 | addsubd 11641 |
. . . . . . . 8
⊢ ((𝜑 ∧ 𝑗 ∈ ℕ) → ((𝑗 + 1) − 1) = ((𝑗 − 1) + 1)) |
| 84 | 83 | oveq2d 7447 |
. . . . . . 7
⊢ ((𝜑 ∧ 𝑗 ∈ ℕ) → (𝑀 + ((𝑗 + 1) − 1)) = (𝑀 + ((𝑗 − 1) + 1))) |
| 85 | 21 | adantr 480 |
. . . . . . . 8
⊢ ((𝜑 ∧ 𝑗 ∈ ℕ) → 𝑀 ∈ ℂ) |
| 86 | 75 | adantl 481 |
. . . . . . . . 9
⊢ ((𝜑 ∧ 𝑗 ∈ ℕ) → (𝑗 − 1) ∈
ℕ0) |
| 87 | 86 | nn0cnd 12589 |
. . . . . . . 8
⊢ ((𝜑 ∧ 𝑗 ∈ ℕ) → (𝑗 − 1) ∈ ℂ) |
| 88 | 85, 87, 82 | addassd 11283 |
. . . . . . 7
⊢ ((𝜑 ∧ 𝑗 ∈ ℕ) → ((𝑀 + (𝑗 − 1)) + 1) = (𝑀 + ((𝑗 − 1) + 1))) |
| 89 | 84, 88 | eqtr4d 2780 |
. . . . . 6
⊢ ((𝜑 ∧ 𝑗 ∈ ℕ) → (𝑀 + ((𝑗 + 1) − 1)) = ((𝑀 + (𝑗 − 1)) + 1)) |
| 90 | 89 | fveq2d 6910 |
. . . . 5
⊢ ((𝜑 ∧ 𝑗 ∈ ℕ) → (𝐺‘(𝑀 + ((𝑗 + 1) − 1))) = (𝐺‘((𝑀 + (𝑗 − 1)) + 1))) |
| 91 | 79, 90 | breqtrrd 5171 |
. . . 4
⊢ ((𝜑 ∧ 𝑗 ∈ ℕ) → (𝐺‘(𝑀 + (𝑗 − 1))) < (𝐺‘(𝑀 + ((𝑗 + 1) − 1)))) |
| 92 | | oveq1 7438 |
. . . . . . . 8
⊢ (𝑥 = 𝑗 → (𝑥 − 1) = (𝑗 − 1)) |
| 93 | 92 | oveq2d 7447 |
. . . . . . 7
⊢ (𝑥 = 𝑗 → (𝑀 + (𝑥 − 1)) = (𝑀 + (𝑗 − 1))) |
| 94 | 93 | fveq2d 6910 |
. . . . . 6
⊢ (𝑥 = 𝑗 → (𝐺‘(𝑀 + (𝑥 − 1))) = (𝐺‘(𝑀 + (𝑗 − 1)))) |
| 95 | | eqid 2737 |
. . . . . 6
⊢ (𝑥 ∈ ℕ ↦ (𝐺‘(𝑀 + (𝑥 − 1)))) = (𝑥 ∈ ℕ ↦ (𝐺‘(𝑀 + (𝑥 − 1)))) |
| 96 | | fvex 6919 |
. . . . . 6
⊢ (𝐺‘(𝑀 + (𝑗 − 1))) ∈ V |
| 97 | 94, 95, 96 | fvmpt 7016 |
. . . . 5
⊢ (𝑗 ∈ ℕ → ((𝑥 ∈ ℕ ↦ (𝐺‘(𝑀 + (𝑥 − 1))))‘𝑗) = (𝐺‘(𝑀 + (𝑗 − 1)))) |
| 98 | 97 | adantl 481 |
. . . 4
⊢ ((𝜑 ∧ 𝑗 ∈ ℕ) → ((𝑥 ∈ ℕ ↦ (𝐺‘(𝑀 + (𝑥 − 1))))‘𝑗) = (𝐺‘(𝑀 + (𝑗 − 1)))) |
| 99 | | peano2nn 12278 |
. . . . . 6
⊢ (𝑗 ∈ ℕ → (𝑗 + 1) ∈
ℕ) |
| 100 | 99 | adantl 481 |
. . . . 5
⊢ ((𝜑 ∧ 𝑗 ∈ ℕ) → (𝑗 + 1) ∈ ℕ) |
| 101 | | oveq1 7438 |
. . . . . . . 8
⊢ (𝑥 = (𝑗 + 1) → (𝑥 − 1) = ((𝑗 + 1) − 1)) |
| 102 | 101 | oveq2d 7447 |
. . . . . . 7
⊢ (𝑥 = (𝑗 + 1) → (𝑀 + (𝑥 − 1)) = (𝑀 + ((𝑗 + 1) − 1))) |
| 103 | 102 | fveq2d 6910 |
. . . . . 6
⊢ (𝑥 = (𝑗 + 1) → (𝐺‘(𝑀 + (𝑥 − 1))) = (𝐺‘(𝑀 + ((𝑗 + 1) − 1)))) |
| 104 | | fvex 6919 |
. . . . . 6
⊢ (𝐺‘(𝑀 + ((𝑗 + 1) − 1))) ∈ V |
| 105 | 103, 95, 104 | fvmpt 7016 |
. . . . 5
⊢ ((𝑗 + 1) ∈ ℕ →
((𝑥 ∈ ℕ ↦
(𝐺‘(𝑀 + (𝑥 − 1))))‘(𝑗 + 1)) = (𝐺‘(𝑀 + ((𝑗 + 1) − 1)))) |
| 106 | 100, 105 | syl 17 |
. . . 4
⊢ ((𝜑 ∧ 𝑗 ∈ ℕ) → ((𝑥 ∈ ℕ ↦ (𝐺‘(𝑀 + (𝑥 − 1))))‘(𝑗 + 1)) = (𝐺‘(𝑀 + ((𝑗 + 1) − 1)))) |
| 107 | 91, 98, 106 | 3brtr4d 5175 |
. . 3
⊢ ((𝜑 ∧ 𝑗 ∈ ℕ) → ((𝑥 ∈ ℕ ↦ (𝐺‘(𝑀 + (𝑥 − 1))))‘𝑗) < ((𝑥 ∈ ℕ ↦ (𝐺‘(𝑀 + (𝑥 − 1))))‘(𝑗 + 1))) |
| 108 | 59 | ffnd 6737 |
. . . . . . . 8
⊢ (𝜑 → 𝐺 Fn 𝑍) |
| 109 | | uznn0sub 12917 |
. . . . . . . . . . . . 13
⊢ (𝑘 ∈
(ℤ≥‘𝑀) → (𝑘 − 𝑀) ∈
ℕ0) |
| 110 | 11, 109 | syl 17 |
. . . . . . . . . . . 12
⊢ ((𝜑 ∧ 𝑘 ∈ 𝑍) → (𝑘 − 𝑀) ∈
ℕ0) |
| 111 | | nn0p1nn 12565 |
. . . . . . . . . . . 12
⊢ ((𝑘 − 𝑀) ∈ ℕ0 → ((𝑘 − 𝑀) + 1) ∈ ℕ) |
| 112 | 110, 111 | syl 17 |
. . . . . . . . . . 11
⊢ ((𝜑 ∧ 𝑘 ∈ 𝑍) → ((𝑘 − 𝑀) + 1) ∈ ℕ) |
| 113 | 110 | nn0cnd 12589 |
. . . . . . . . . . . . . . 15
⊢ ((𝜑 ∧ 𝑘 ∈ 𝑍) → (𝑘 − 𝑀) ∈ ℂ) |
| 114 | | pncan 11514 |
. . . . . . . . . . . . . . 15
⊢ (((𝑘 − 𝑀) ∈ ℂ ∧ 1 ∈ ℂ)
→ (((𝑘 − 𝑀) + 1) − 1) = (𝑘 − 𝑀)) |
| 115 | 113, 39, 114 | sylancl 586 |
. . . . . . . . . . . . . 14
⊢ ((𝜑 ∧ 𝑘 ∈ 𝑍) → (((𝑘 − 𝑀) + 1) − 1) = (𝑘 − 𝑀)) |
| 116 | 115 | oveq2d 7447 |
. . . . . . . . . . . . 13
⊢ ((𝜑 ∧ 𝑘 ∈ 𝑍) → (𝑀 + (((𝑘 − 𝑀) + 1) − 1)) = (𝑀 + (𝑘 − 𝑀))) |
| 117 | | eluzelz 12888 |
. . . . . . . . . . . . . . . 16
⊢ (𝑘 ∈
(ℤ≥‘𝑀) → 𝑘 ∈ ℤ) |
| 118 | 117, 1 | eleq2s 2859 |
. . . . . . . . . . . . . . 15
⊢ (𝑘 ∈ 𝑍 → 𝑘 ∈ ℤ) |
| 119 | 118 | zcnd 12723 |
. . . . . . . . . . . . . 14
⊢ (𝑘 ∈ 𝑍 → 𝑘 ∈ ℂ) |
| 120 | | pncan3 11516 |
. . . . . . . . . . . . . 14
⊢ ((𝑀 ∈ ℂ ∧ 𝑘 ∈ ℂ) → (𝑀 + (𝑘 − 𝑀)) = 𝑘) |
| 121 | 21, 119, 120 | syl2an 596 |
. . . . . . . . . . . . 13
⊢ ((𝜑 ∧ 𝑘 ∈ 𝑍) → (𝑀 + (𝑘 − 𝑀)) = 𝑘) |
| 122 | 116, 121 | eqtr2d 2778 |
. . . . . . . . . . . 12
⊢ ((𝜑 ∧ 𝑘 ∈ 𝑍) → 𝑘 = (𝑀 + (((𝑘 − 𝑀) + 1) − 1))) |
| 123 | 122 | fveq2d 6910 |
. . . . . . . . . . 11
⊢ ((𝜑 ∧ 𝑘 ∈ 𝑍) → (𝐺‘𝑘) = (𝐺‘(𝑀 + (((𝑘 − 𝑀) + 1) − 1)))) |
| 124 | | oveq1 7438 |
. . . . . . . . . . . . . 14
⊢ (𝑥 = ((𝑘 − 𝑀) + 1) → (𝑥 − 1) = (((𝑘 − 𝑀) + 1) − 1)) |
| 125 | 124 | oveq2d 7447 |
. . . . . . . . . . . . 13
⊢ (𝑥 = ((𝑘 − 𝑀) + 1) → (𝑀 + (𝑥 − 1)) = (𝑀 + (((𝑘 − 𝑀) + 1) − 1))) |
| 126 | 125 | fveq2d 6910 |
. . . . . . . . . . . 12
⊢ (𝑥 = ((𝑘 − 𝑀) + 1) → (𝐺‘(𝑀 + (𝑥 − 1))) = (𝐺‘(𝑀 + (((𝑘 − 𝑀) + 1) − 1)))) |
| 127 | 126 | rspceeqv 3645 |
. . . . . . . . . . 11
⊢ ((((𝑘 − 𝑀) + 1) ∈ ℕ ∧ (𝐺‘𝑘) = (𝐺‘(𝑀 + (((𝑘 − 𝑀) + 1) − 1)))) → ∃𝑥 ∈ ℕ (𝐺‘𝑘) = (𝐺‘(𝑀 + (𝑥 − 1)))) |
| 128 | 112, 123,
127 | syl2anc 584 |
. . . . . . . . . 10
⊢ ((𝜑 ∧ 𝑘 ∈ 𝑍) → ∃𝑥 ∈ ℕ (𝐺‘𝑘) = (𝐺‘(𝑀 + (𝑥 − 1)))) |
| 129 | | fvex 6919 |
. . . . . . . . . . 11
⊢ (𝐺‘𝑘) ∈ V |
| 130 | 95 | elrnmpt 5969 |
. . . . . . . . . . 11
⊢ ((𝐺‘𝑘) ∈ V → ((𝐺‘𝑘) ∈ ran (𝑥 ∈ ℕ ↦ (𝐺‘(𝑀 + (𝑥 − 1)))) ↔ ∃𝑥 ∈ ℕ (𝐺‘𝑘) = (𝐺‘(𝑀 + (𝑥 − 1))))) |
| 131 | 129, 130 | ax-mp 5 |
. . . . . . . . . 10
⊢ ((𝐺‘𝑘) ∈ ran (𝑥 ∈ ℕ ↦ (𝐺‘(𝑀 + (𝑥 − 1)))) ↔ ∃𝑥 ∈ ℕ (𝐺‘𝑘) = (𝐺‘(𝑀 + (𝑥 − 1)))) |
| 132 | 128, 131 | sylibr 234 |
. . . . . . . . 9
⊢ ((𝜑 ∧ 𝑘 ∈ 𝑍) → (𝐺‘𝑘) ∈ ran (𝑥 ∈ ℕ ↦ (𝐺‘(𝑀 + (𝑥 − 1))))) |
| 133 | 132 | ralrimiva 3146 |
. . . . . . . 8
⊢ (𝜑 → ∀𝑘 ∈ 𝑍 (𝐺‘𝑘) ∈ ran (𝑥 ∈ ℕ ↦ (𝐺‘(𝑀 + (𝑥 − 1))))) |
| 134 | | ffnfv 7139 |
. . . . . . . 8
⊢ (𝐺:𝑍⟶ran (𝑥 ∈ ℕ ↦ (𝐺‘(𝑀 + (𝑥 − 1)))) ↔ (𝐺 Fn 𝑍 ∧ ∀𝑘 ∈ 𝑍 (𝐺‘𝑘) ∈ ran (𝑥 ∈ ℕ ↦ (𝐺‘(𝑀 + (𝑥 − 1)))))) |
| 135 | 108, 133,
134 | sylanbrc 583 |
. . . . . . 7
⊢ (𝜑 → 𝐺:𝑍⟶ran (𝑥 ∈ ℕ ↦ (𝐺‘(𝑀 + (𝑥 − 1))))) |
| 136 | 135 | frnd 6744 |
. . . . . 6
⊢ (𝜑 → ran 𝐺 ⊆ ran (𝑥 ∈ ℕ ↦ (𝐺‘(𝑀 + (𝑥 − 1))))) |
| 137 | 136 | sscond 4146 |
. . . . 5
⊢ (𝜑 → (𝑊 ∖ ran (𝑥 ∈ ℕ ↦ (𝐺‘(𝑀 + (𝑥 − 1))))) ⊆ (𝑊 ∖ ran 𝐺)) |
| 138 | 137 | sselda 3983 |
. . . 4
⊢ ((𝜑 ∧ 𝑛 ∈ (𝑊 ∖ ran (𝑥 ∈ ℕ ↦ (𝐺‘(𝑀 + (𝑥 − 1)))))) → 𝑛 ∈ (𝑊 ∖ ran 𝐺)) |
| 139 | | isercoll2.0 |
. . . 4
⊢ ((𝜑 ∧ 𝑛 ∈ (𝑊 ∖ ran 𝐺)) → (𝐹‘𝑛) = 0) |
| 140 | 138, 139 | syldan 591 |
. . 3
⊢ ((𝜑 ∧ 𝑛 ∈ (𝑊 ∖ ran (𝑥 ∈ ℕ ↦ (𝐺‘(𝑀 + (𝑥 − 1)))))) → (𝐹‘𝑛) = 0) |
| 141 | | isercoll2.f |
. . 3
⊢ ((𝜑 ∧ 𝑛 ∈ 𝑊) → (𝐹‘𝑛) ∈ ℂ) |
| 142 | | fveq2 6906 |
. . . . . 6
⊢ (𝑘 = (𝑀 + (𝑗 − 1)) → (𝐻‘𝑘) = (𝐻‘(𝑀 + (𝑗 − 1)))) |
| 143 | 69 | fveq2d 6910 |
. . . . . 6
⊢ (𝑘 = (𝑀 + (𝑗 − 1)) → (𝐹‘(𝐺‘𝑘)) = (𝐹‘(𝐺‘(𝑀 + (𝑗 − 1))))) |
| 144 | 142, 143 | eqeq12d 2753 |
. . . . 5
⊢ (𝑘 = (𝑀 + (𝑗 − 1)) → ((𝐻‘𝑘) = (𝐹‘(𝐺‘𝑘)) ↔ (𝐻‘(𝑀 + (𝑗 − 1))) = (𝐹‘(𝐺‘(𝑀 + (𝑗 − 1)))))) |
| 145 | | isercoll2.h |
. . . . . . 7
⊢ ((𝜑 ∧ 𝑘 ∈ 𝑍) → (𝐻‘𝑘) = (𝐹‘(𝐺‘𝑘))) |
| 146 | 145 | ralrimiva 3146 |
. . . . . 6
⊢ (𝜑 → ∀𝑘 ∈ 𝑍 (𝐻‘𝑘) = (𝐹‘(𝐺‘𝑘))) |
| 147 | 146 | adantr 480 |
. . . . 5
⊢ ((𝜑 ∧ 𝑗 ∈ ℕ) → ∀𝑘 ∈ 𝑍 (𝐻‘𝑘) = (𝐹‘(𝐺‘𝑘))) |
| 148 | 144, 147,
78 | rspcdva 3623 |
. . . 4
⊢ ((𝜑 ∧ 𝑗 ∈ ℕ) → (𝐻‘(𝑀 + (𝑗 − 1))) = (𝐹‘(𝐺‘(𝑀 + (𝑗 − 1))))) |
| 149 | 93 | fveq2d 6910 |
. . . . . 6
⊢ (𝑥 = 𝑗 → (𝐻‘(𝑀 + (𝑥 − 1))) = (𝐻‘(𝑀 + (𝑗 − 1)))) |
| 150 | | fvex 6919 |
. . . . . 6
⊢ (𝐻‘(𝑀 + (𝑗 − 1))) ∈ V |
| 151 | 149, 33, 150 | fvmpt 7016 |
. . . . 5
⊢ (𝑗 ∈ ℕ → ((𝑥 ∈ ℕ ↦ (𝐻‘(𝑀 + (𝑥 − 1))))‘𝑗) = (𝐻‘(𝑀 + (𝑗 − 1)))) |
| 152 | 151 | adantl 481 |
. . . 4
⊢ ((𝜑 ∧ 𝑗 ∈ ℕ) → ((𝑥 ∈ ℕ ↦ (𝐻‘(𝑀 + (𝑥 − 1))))‘𝑗) = (𝐻‘(𝑀 + (𝑗 − 1)))) |
| 153 | 98 | fveq2d 6910 |
. . . 4
⊢ ((𝜑 ∧ 𝑗 ∈ ℕ) → (𝐹‘((𝑥 ∈ ℕ ↦ (𝐺‘(𝑀 + (𝑥 − 1))))‘𝑗)) = (𝐹‘(𝐺‘(𝑀 + (𝑗 − 1))))) |
| 154 | 148, 152,
153 | 3eqtr4d 2787 |
. . 3
⊢ ((𝜑 ∧ 𝑗 ∈ ℕ) → ((𝑥 ∈ ℕ ↦ (𝐻‘(𝑀 + (𝑥 − 1))))‘𝑗) = (𝐹‘((𝑥 ∈ ℕ ↦ (𝐺‘(𝑀 + (𝑥 − 1))))‘𝑗))) |
| 155 | 57, 58, 68, 107, 140, 141, 154 | isercoll 15704 |
. 2
⊢ (𝜑 → (seq1( + , (𝑥 ∈ ℕ ↦ (𝐻‘(𝑀 + (𝑥 − 1))))) ⇝ 𝐴 ↔ seq𝑁( + , 𝐹) ⇝ 𝐴)) |
| 156 | 56, 155 | bitrd 279 |
1
⊢ (𝜑 → (seq𝑀( + , 𝐻) ⇝ 𝐴 ↔ seq𝑁( + , 𝐹) ⇝ 𝐴)) |