Proof of Theorem seqz
| Step | Hyp | Ref
| Expression |
| 1 | | seqz.5 |
. . . 4
⊢ (𝜑 → 𝐾 ∈ (𝑀...𝑁)) |
| 2 | | elfzuz 13560 |
. . . 4
⊢ (𝐾 ∈ (𝑀...𝑁) → 𝐾 ∈ (ℤ≥‘𝑀)) |
| 3 | 1, 2 | syl 17 |
. . 3
⊢ (𝜑 → 𝐾 ∈ (ℤ≥‘𝑀)) |
| 4 | 1 | elfzelzd 13565 |
. . . . . . . 8
⊢ (𝜑 → 𝐾 ∈ ℤ) |
| 5 | | seq1 14055 |
. . . . . . . 8
⊢ (𝐾 ∈ ℤ → (seq𝐾( + , 𝐹)‘𝐾) = (𝐹‘𝐾)) |
| 6 | 4, 5 | syl 17 |
. . . . . . 7
⊢ (𝜑 → (seq𝐾( + , 𝐹)‘𝐾) = (𝐹‘𝐾)) |
| 7 | | seqz.7 |
. . . . . . 7
⊢ (𝜑 → (𝐹‘𝐾) = 𝑍) |
| 8 | 6, 7 | eqtrd 2777 |
. . . . . 6
⊢ (𝜑 → (seq𝐾( + , 𝐹)‘𝐾) = 𝑍) |
| 9 | | seqeq1 14045 |
. . . . . . . 8
⊢ (𝐾 = 𝑀 → seq𝐾( + , 𝐹) = seq𝑀( + , 𝐹)) |
| 10 | 9 | fveq1d 6908 |
. . . . . . 7
⊢ (𝐾 = 𝑀 → (seq𝐾( + , 𝐹)‘𝐾) = (seq𝑀( + , 𝐹)‘𝐾)) |
| 11 | 10 | eqeq1d 2739 |
. . . . . 6
⊢ (𝐾 = 𝑀 → ((seq𝐾( + , 𝐹)‘𝐾) = 𝑍 ↔ (seq𝑀( + , 𝐹)‘𝐾) = 𝑍)) |
| 12 | 8, 11 | syl5ibcom 245 |
. . . . 5
⊢ (𝜑 → (𝐾 = 𝑀 → (seq𝑀( + , 𝐹)‘𝐾) = 𝑍)) |
| 13 | | eluzel2 12883 |
. . . . . . . . 9
⊢ (𝐾 ∈
(ℤ≥‘𝑀) → 𝑀 ∈ ℤ) |
| 14 | 3, 13 | syl 17 |
. . . . . . . 8
⊢ (𝜑 → 𝑀 ∈ ℤ) |
| 15 | | seqm1 14060 |
. . . . . . . 8
⊢ ((𝑀 ∈ ℤ ∧ 𝐾 ∈
(ℤ≥‘(𝑀 + 1))) → (seq𝑀( + , 𝐹)‘𝐾) = ((seq𝑀( + , 𝐹)‘(𝐾 − 1)) + (𝐹‘𝐾))) |
| 16 | 14, 15 | sylan 580 |
. . . . . . 7
⊢ ((𝜑 ∧ 𝐾 ∈ (ℤ≥‘(𝑀 + 1))) → (seq𝑀( + , 𝐹)‘𝐾) = ((seq𝑀( + , 𝐹)‘(𝐾 − 1)) + (𝐹‘𝐾))) |
| 17 | 7 | adantr 480 |
. . . . . . . . 9
⊢ ((𝜑 ∧ 𝐾 ∈ (ℤ≥‘(𝑀 + 1))) → (𝐹‘𝐾) = 𝑍) |
| 18 | 17 | oveq2d 7447 |
. . . . . . . 8
⊢ ((𝜑 ∧ 𝐾 ∈ (ℤ≥‘(𝑀 + 1))) → ((seq𝑀( + , 𝐹)‘(𝐾 − 1)) + (𝐹‘𝐾)) = ((seq𝑀( + , 𝐹)‘(𝐾 − 1)) + 𝑍)) |
| 19 | | oveq1 7438 |
. . . . . . . . . 10
⊢ (𝑥 = (seq𝑀( + , 𝐹)‘(𝐾 − 1)) → (𝑥 + 𝑍) = ((seq𝑀( + , 𝐹)‘(𝐾 − 1)) + 𝑍)) |
| 20 | 19 | eqeq1d 2739 |
. . . . . . . . 9
⊢ (𝑥 = (seq𝑀( + , 𝐹)‘(𝐾 − 1)) → ((𝑥 + 𝑍) = 𝑍 ↔ ((seq𝑀( + , 𝐹)‘(𝐾 − 1)) + 𝑍) = 𝑍)) |
| 21 | | seqz.4 |
. . . . . . . . . . 11
⊢ ((𝜑 ∧ 𝑥 ∈ 𝑆) → (𝑥 + 𝑍) = 𝑍) |
| 22 | 21 | ralrimiva 3146 |
. . . . . . . . . 10
⊢ (𝜑 → ∀𝑥 ∈ 𝑆 (𝑥 + 𝑍) = 𝑍) |
| 23 | 22 | adantr 480 |
. . . . . . . . 9
⊢ ((𝜑 ∧ 𝐾 ∈ (ℤ≥‘(𝑀 + 1))) → ∀𝑥 ∈ 𝑆 (𝑥 + 𝑍) = 𝑍) |
| 24 | | eluzp1m1 12904 |
. . . . . . . . . . 11
⊢ ((𝑀 ∈ ℤ ∧ 𝐾 ∈
(ℤ≥‘(𝑀 + 1))) → (𝐾 − 1) ∈
(ℤ≥‘𝑀)) |
| 25 | 14, 24 | sylan 580 |
. . . . . . . . . 10
⊢ ((𝜑 ∧ 𝐾 ∈ (ℤ≥‘(𝑀 + 1))) → (𝐾 − 1) ∈
(ℤ≥‘𝑀)) |
| 26 | | fzssp1 13607 |
. . . . . . . . . . . . . . 15
⊢ (𝑀...(𝐾 − 1)) ⊆ (𝑀...((𝐾 − 1) + 1)) |
| 27 | 4 | zcnd 12723 |
. . . . . . . . . . . . . . . . 17
⊢ (𝜑 → 𝐾 ∈ ℂ) |
| 28 | | ax-1cn 11213 |
. . . . . . . . . . . . . . . . 17
⊢ 1 ∈
ℂ |
| 29 | | npcan 11517 |
. . . . . . . . . . . . . . . . 17
⊢ ((𝐾 ∈ ℂ ∧ 1 ∈
ℂ) → ((𝐾 −
1) + 1) = 𝐾) |
| 30 | 27, 28, 29 | sylancl 586 |
. . . . . . . . . . . . . . . 16
⊢ (𝜑 → ((𝐾 − 1) + 1) = 𝐾) |
| 31 | 30 | oveq2d 7447 |
. . . . . . . . . . . . . . 15
⊢ (𝜑 → (𝑀...((𝐾 − 1) + 1)) = (𝑀...𝐾)) |
| 32 | 26, 31 | sseqtrid 4026 |
. . . . . . . . . . . . . 14
⊢ (𝜑 → (𝑀...(𝐾 − 1)) ⊆ (𝑀...𝐾)) |
| 33 | | elfzuz3 13561 |
. . . . . . . . . . . . . . . 16
⊢ (𝐾 ∈ (𝑀...𝑁) → 𝑁 ∈ (ℤ≥‘𝐾)) |
| 34 | 1, 33 | syl 17 |
. . . . . . . . . . . . . . 15
⊢ (𝜑 → 𝑁 ∈ (ℤ≥‘𝐾)) |
| 35 | | fzss2 13604 |
. . . . . . . . . . . . . . 15
⊢ (𝑁 ∈
(ℤ≥‘𝐾) → (𝑀...𝐾) ⊆ (𝑀...𝑁)) |
| 36 | 34, 35 | syl 17 |
. . . . . . . . . . . . . 14
⊢ (𝜑 → (𝑀...𝐾) ⊆ (𝑀...𝑁)) |
| 37 | 32, 36 | sstrd 3994 |
. . . . . . . . . . . . 13
⊢ (𝜑 → (𝑀...(𝐾 − 1)) ⊆ (𝑀...𝑁)) |
| 38 | 37 | adantr 480 |
. . . . . . . . . . . 12
⊢ ((𝜑 ∧ 𝐾 ∈ (ℤ≥‘(𝑀 + 1))) → (𝑀...(𝐾 − 1)) ⊆ (𝑀...𝑁)) |
| 39 | 38 | sselda 3983 |
. . . . . . . . . . 11
⊢ (((𝜑 ∧ 𝐾 ∈ (ℤ≥‘(𝑀 + 1))) ∧ 𝑥 ∈ (𝑀...(𝐾 − 1))) → 𝑥 ∈ (𝑀...𝑁)) |
| 40 | | seqhomo.2 |
. . . . . . . . . . . 12
⊢ ((𝜑 ∧ 𝑥 ∈ (𝑀...𝑁)) → (𝐹‘𝑥) ∈ 𝑆) |
| 41 | 40 | adantlr 715 |
. . . . . . . . . . 11
⊢ (((𝜑 ∧ 𝐾 ∈ (ℤ≥‘(𝑀 + 1))) ∧ 𝑥 ∈ (𝑀...𝑁)) → (𝐹‘𝑥) ∈ 𝑆) |
| 42 | 39, 41 | syldan 591 |
. . . . . . . . . 10
⊢ (((𝜑 ∧ 𝐾 ∈ (ℤ≥‘(𝑀 + 1))) ∧ 𝑥 ∈ (𝑀...(𝐾 − 1))) → (𝐹‘𝑥) ∈ 𝑆) |
| 43 | | seqhomo.1 |
. . . . . . . . . . 11
⊢ ((𝜑 ∧ (𝑥 ∈ 𝑆 ∧ 𝑦 ∈ 𝑆)) → (𝑥 + 𝑦) ∈ 𝑆) |
| 44 | 43 | adantlr 715 |
. . . . . . . . . 10
⊢ (((𝜑 ∧ 𝐾 ∈ (ℤ≥‘(𝑀 + 1))) ∧ (𝑥 ∈ 𝑆 ∧ 𝑦 ∈ 𝑆)) → (𝑥 + 𝑦) ∈ 𝑆) |
| 45 | 25, 42, 44 | seqcl 14063 |
. . . . . . . . 9
⊢ ((𝜑 ∧ 𝐾 ∈ (ℤ≥‘(𝑀 + 1))) → (seq𝑀( + , 𝐹)‘(𝐾 − 1)) ∈ 𝑆) |
| 46 | 20, 23, 45 | rspcdva 3623 |
. . . . . . . 8
⊢ ((𝜑 ∧ 𝐾 ∈ (ℤ≥‘(𝑀 + 1))) → ((seq𝑀( + , 𝐹)‘(𝐾 − 1)) + 𝑍) = 𝑍) |
| 47 | 18, 46 | eqtrd 2777 |
. . . . . . 7
⊢ ((𝜑 ∧ 𝐾 ∈ (ℤ≥‘(𝑀 + 1))) → ((seq𝑀( + , 𝐹)‘(𝐾 − 1)) + (𝐹‘𝐾)) = 𝑍) |
| 48 | 16, 47 | eqtrd 2777 |
. . . . . 6
⊢ ((𝜑 ∧ 𝐾 ∈ (ℤ≥‘(𝑀 + 1))) → (seq𝑀( + , 𝐹)‘𝐾) = 𝑍) |
| 49 | 48 | ex 412 |
. . . . 5
⊢ (𝜑 → (𝐾 ∈ (ℤ≥‘(𝑀 + 1)) → (seq𝑀( + , 𝐹)‘𝐾) = 𝑍)) |
| 50 | | uzp1 12919 |
. . . . . 6
⊢ (𝐾 ∈
(ℤ≥‘𝑀) → (𝐾 = 𝑀 ∨ 𝐾 ∈ (ℤ≥‘(𝑀 + 1)))) |
| 51 | 3, 50 | syl 17 |
. . . . 5
⊢ (𝜑 → (𝐾 = 𝑀 ∨ 𝐾 ∈ (ℤ≥‘(𝑀 + 1)))) |
| 52 | 12, 49, 51 | mpjaod 861 |
. . . 4
⊢ (𝜑 → (seq𝑀( + , 𝐹)‘𝐾) = 𝑍) |
| 53 | 52, 7 | eqtr4d 2780 |
. . 3
⊢ (𝜑 → (seq𝑀( + , 𝐹)‘𝐾) = (𝐹‘𝐾)) |
| 54 | | eqidd 2738 |
. . 3
⊢ ((𝜑 ∧ 𝑥 ∈ ((𝐾 + 1)...𝑁)) → (𝐹‘𝑥) = (𝐹‘𝑥)) |
| 55 | 3, 53, 34, 54 | seqfveq2 14065 |
. 2
⊢ (𝜑 → (seq𝑀( + , 𝐹)‘𝑁) = (seq𝐾( + , 𝐹)‘𝑁)) |
| 56 | | fvex 6919 |
. . . . . 6
⊢ (𝐹‘𝐾) ∈ V |
| 57 | 56 | elsn 4641 |
. . . . 5
⊢ ((𝐹‘𝐾) ∈ {𝑍} ↔ (𝐹‘𝐾) = 𝑍) |
| 58 | 7, 57 | sylibr 234 |
. . . 4
⊢ (𝜑 → (𝐹‘𝐾) ∈ {𝑍}) |
| 59 | | simprl 771 |
. . . . . . . 8
⊢ ((𝜑 ∧ (𝑥 ∈ {𝑍} ∧ 𝑦 ∈ 𝑆)) → 𝑥 ∈ {𝑍}) |
| 60 | | velsn 4642 |
. . . . . . . 8
⊢ (𝑥 ∈ {𝑍} ↔ 𝑥 = 𝑍) |
| 61 | 59, 60 | sylib 218 |
. . . . . . 7
⊢ ((𝜑 ∧ (𝑥 ∈ {𝑍} ∧ 𝑦 ∈ 𝑆)) → 𝑥 = 𝑍) |
| 62 | 61 | oveq1d 7446 |
. . . . . 6
⊢ ((𝜑 ∧ (𝑥 ∈ {𝑍} ∧ 𝑦 ∈ 𝑆)) → (𝑥 + 𝑦) = (𝑍 + 𝑦)) |
| 63 | | oveq2 7439 |
. . . . . . . 8
⊢ (𝑥 = 𝑦 → (𝑍 + 𝑥) = (𝑍 + 𝑦)) |
| 64 | 63 | eqeq1d 2739 |
. . . . . . 7
⊢ (𝑥 = 𝑦 → ((𝑍 + 𝑥) = 𝑍 ↔ (𝑍 + 𝑦) = 𝑍)) |
| 65 | | seqz.3 |
. . . . . . . . 9
⊢ ((𝜑 ∧ 𝑥 ∈ 𝑆) → (𝑍 + 𝑥) = 𝑍) |
| 66 | 65 | ralrimiva 3146 |
. . . . . . . 8
⊢ (𝜑 → ∀𝑥 ∈ 𝑆 (𝑍 + 𝑥) = 𝑍) |
| 67 | 66 | adantr 480 |
. . . . . . 7
⊢ ((𝜑 ∧ (𝑥 ∈ {𝑍} ∧ 𝑦 ∈ 𝑆)) → ∀𝑥 ∈ 𝑆 (𝑍 + 𝑥) = 𝑍) |
| 68 | | simprr 773 |
. . . . . . 7
⊢ ((𝜑 ∧ (𝑥 ∈ {𝑍} ∧ 𝑦 ∈ 𝑆)) → 𝑦 ∈ 𝑆) |
| 69 | 64, 67, 68 | rspcdva 3623 |
. . . . . 6
⊢ ((𝜑 ∧ (𝑥 ∈ {𝑍} ∧ 𝑦 ∈ 𝑆)) → (𝑍 + 𝑦) = 𝑍) |
| 70 | 62, 69 | eqtrd 2777 |
. . . . 5
⊢ ((𝜑 ∧ (𝑥 ∈ {𝑍} ∧ 𝑦 ∈ 𝑆)) → (𝑥 + 𝑦) = 𝑍) |
| 71 | | ovex 7464 |
. . . . . 6
⊢ (𝑥 + 𝑦) ∈ V |
| 72 | 71 | elsn 4641 |
. . . . 5
⊢ ((𝑥 + 𝑦) ∈ {𝑍} ↔ (𝑥 + 𝑦) = 𝑍) |
| 73 | 70, 72 | sylibr 234 |
. . . 4
⊢ ((𝜑 ∧ (𝑥 ∈ {𝑍} ∧ 𝑦 ∈ 𝑆)) → (𝑥 + 𝑦) ∈ {𝑍}) |
| 74 | | peano2uz 12943 |
. . . . . . . 8
⊢ (𝐾 ∈
(ℤ≥‘𝑀) → (𝐾 + 1) ∈
(ℤ≥‘𝑀)) |
| 75 | 3, 74 | syl 17 |
. . . . . . 7
⊢ (𝜑 → (𝐾 + 1) ∈
(ℤ≥‘𝑀)) |
| 76 | | fzss1 13603 |
. . . . . . 7
⊢ ((𝐾 + 1) ∈
(ℤ≥‘𝑀) → ((𝐾 + 1)...𝑁) ⊆ (𝑀...𝑁)) |
| 77 | 75, 76 | syl 17 |
. . . . . 6
⊢ (𝜑 → ((𝐾 + 1)...𝑁) ⊆ (𝑀...𝑁)) |
| 78 | 77 | sselda 3983 |
. . . . 5
⊢ ((𝜑 ∧ 𝑥 ∈ ((𝐾 + 1)...𝑁)) → 𝑥 ∈ (𝑀...𝑁)) |
| 79 | 78, 40 | syldan 591 |
. . . 4
⊢ ((𝜑 ∧ 𝑥 ∈ ((𝐾 + 1)...𝑁)) → (𝐹‘𝑥) ∈ 𝑆) |
| 80 | 58, 73, 34, 79 | seqcl2 14061 |
. . 3
⊢ (𝜑 → (seq𝐾( + , 𝐹)‘𝑁) ∈ {𝑍}) |
| 81 | | elsni 4643 |
. . 3
⊢
((seq𝐾( + , 𝐹)‘𝑁) ∈ {𝑍} → (seq𝐾( + , 𝐹)‘𝑁) = 𝑍) |
| 82 | 80, 81 | syl 17 |
. 2
⊢ (𝜑 → (seq𝐾( + , 𝐹)‘𝑁) = 𝑍) |
| 83 | 55, 82 | eqtrd 2777 |
1
⊢ (𝜑 → (seq𝑀( + , 𝐹)‘𝑁) = 𝑍) |