Proof of Theorem seqf1olem2
Step | Hyp | Ref
| Expression |
1 | | seqf1olem.6 |
. . . . . . . . . 10
⊢ (𝜑 → 𝐺:(𝑀...(𝑁 + 1))⟶𝐶) |
2 | 1 | ffnd 6280 |
. . . . . . . . 9
⊢ (𝜑 → 𝐺 Fn (𝑀...(𝑁 + 1))) |
3 | | fzssp1 12678 |
. . . . . . . . 9
⊢ (𝑀...𝑁) ⊆ (𝑀...(𝑁 + 1)) |
4 | | fnssres 6238 |
. . . . . . . . 9
⊢ ((𝐺 Fn (𝑀...(𝑁 + 1)) ∧ (𝑀...𝑁) ⊆ (𝑀...(𝑁 + 1))) → (𝐺 ↾ (𝑀...𝑁)) Fn (𝑀...𝑁)) |
5 | 2, 3, 4 | sylancl 582 |
. . . . . . . 8
⊢ (𝜑 → (𝐺 ↾ (𝑀...𝑁)) Fn (𝑀...𝑁)) |
6 | | fzfid 13068 |
. . . . . . . 8
⊢ (𝜑 → (𝑀...𝑁) ∈ Fin) |
7 | | fnfi 8508 |
. . . . . . . 8
⊢ (((𝐺 ↾ (𝑀...𝑁)) Fn (𝑀...𝑁) ∧ (𝑀...𝑁) ∈ Fin) → (𝐺 ↾ (𝑀...𝑁)) ∈ Fin) |
8 | 5, 6, 7 | syl2anc 581 |
. . . . . . 7
⊢ (𝜑 → (𝐺 ↾ (𝑀...𝑁)) ∈ Fin) |
9 | | elex 3430 |
. . . . . . 7
⊢ ((𝐺 ↾ (𝑀...𝑁)) ∈ Fin → (𝐺 ↾ (𝑀...𝑁)) ∈ V) |
10 | 8, 9 | syl 17 |
. . . . . 6
⊢ (𝜑 → (𝐺 ↾ (𝑀...𝑁)) ∈ V) |
11 | | seqf1o.1 |
. . . . . . . . 9
⊢ ((𝜑 ∧ (𝑥 ∈ 𝑆 ∧ 𝑦 ∈ 𝑆)) → (𝑥 + 𝑦) ∈ 𝑆) |
12 | | seqf1o.2 |
. . . . . . . . 9
⊢ ((𝜑 ∧ (𝑥 ∈ 𝐶 ∧ 𝑦 ∈ 𝐶)) → (𝑥 + 𝑦) = (𝑦 + 𝑥)) |
13 | | seqf1o.3 |
. . . . . . . . 9
⊢ ((𝜑 ∧ (𝑥 ∈ 𝑆 ∧ 𝑦 ∈ 𝑆 ∧ 𝑧 ∈ 𝑆)) → ((𝑥 + 𝑦) + 𝑧) = (𝑥 + (𝑦 + 𝑧))) |
14 | | seqf1o.4 |
. . . . . . . . 9
⊢ (𝜑 → 𝑁 ∈ (ℤ≥‘𝑀)) |
15 | | seqf1o.5 |
. . . . . . . . 9
⊢ (𝜑 → 𝐶 ⊆ 𝑆) |
16 | | seqf1olem.5 |
. . . . . . . . 9
⊢ (𝜑 → 𝐹:(𝑀...(𝑁 + 1))–1-1-onto→(𝑀...(𝑁 + 1))) |
17 | | seqf1olem.7 |
. . . . . . . . 9
⊢ 𝐽 = (𝑘 ∈ (𝑀...𝑁) ↦ (𝐹‘if(𝑘 < 𝐾, 𝑘, (𝑘 + 1)))) |
18 | | seqf1olem.8 |
. . . . . . . . 9
⊢ 𝐾 = (◡𝐹‘(𝑁 + 1)) |
19 | 11, 12, 13, 14, 15, 16, 1, 17, 18 | seqf1olem1 13135 |
. . . . . . . 8
⊢ (𝜑 → 𝐽:(𝑀...𝑁)–1-1-onto→(𝑀...𝑁)) |
20 | | f1of 6379 |
. . . . . . . 8
⊢ (𝐽:(𝑀...𝑁)–1-1-onto→(𝑀...𝑁) → 𝐽:(𝑀...𝑁)⟶(𝑀...𝑁)) |
21 | 19, 20 | syl 17 |
. . . . . . 7
⊢ (𝜑 → 𝐽:(𝑀...𝑁)⟶(𝑀...𝑁)) |
22 | | fex2 7384 |
. . . . . . 7
⊢ ((𝐽:(𝑀...𝑁)⟶(𝑀...𝑁) ∧ (𝑀...𝑁) ∈ Fin ∧ (𝑀...𝑁) ∈ Fin) → 𝐽 ∈ V) |
23 | 21, 6, 6, 22 | syl3anc 1496 |
. . . . . 6
⊢ (𝜑 → 𝐽 ∈ V) |
24 | 10, 23 | jca 509 |
. . . . 5
⊢ (𝜑 → ((𝐺 ↾ (𝑀...𝑁)) ∈ V ∧ 𝐽 ∈ V)) |
25 | | seqf1olem.9 |
. . . . 5
⊢ (𝜑 → ∀𝑔∀𝑓((𝑓:(𝑀...𝑁)–1-1-onto→(𝑀...𝑁) ∧ 𝑔:(𝑀...𝑁)⟶𝐶) → (seq𝑀( + , (𝑔 ∘ 𝑓))‘𝑁) = (seq𝑀( + , 𝑔)‘𝑁))) |
26 | | fssres 6308 |
. . . . . . 7
⊢ ((𝐺:(𝑀...(𝑁 + 1))⟶𝐶 ∧ (𝑀...𝑁) ⊆ (𝑀...(𝑁 + 1))) → (𝐺 ↾ (𝑀...𝑁)):(𝑀...𝑁)⟶𝐶) |
27 | 1, 3, 26 | sylancl 582 |
. . . . . 6
⊢ (𝜑 → (𝐺 ↾ (𝑀...𝑁)):(𝑀...𝑁)⟶𝐶) |
28 | 19, 27 | jca 509 |
. . . . 5
⊢ (𝜑 → (𝐽:(𝑀...𝑁)–1-1-onto→(𝑀...𝑁) ∧ (𝐺 ↾ (𝑀...𝑁)):(𝑀...𝑁)⟶𝐶)) |
29 | | f1oeq1 6368 |
. . . . . . . 8
⊢ (𝑓 = 𝐽 → (𝑓:(𝑀...𝑁)–1-1-onto→(𝑀...𝑁) ↔ 𝐽:(𝑀...𝑁)–1-1-onto→(𝑀...𝑁))) |
30 | | feq1 6260 |
. . . . . . . 8
⊢ (𝑔 = (𝐺 ↾ (𝑀...𝑁)) → (𝑔:(𝑀...𝑁)⟶𝐶 ↔ (𝐺 ↾ (𝑀...𝑁)):(𝑀...𝑁)⟶𝐶)) |
31 | 29, 30 | bi2anan9r 632 |
. . . . . . 7
⊢ ((𝑔 = (𝐺 ↾ (𝑀...𝑁)) ∧ 𝑓 = 𝐽) → ((𝑓:(𝑀...𝑁)–1-1-onto→(𝑀...𝑁) ∧ 𝑔:(𝑀...𝑁)⟶𝐶) ↔ (𝐽:(𝑀...𝑁)–1-1-onto→(𝑀...𝑁) ∧ (𝐺 ↾ (𝑀...𝑁)):(𝑀...𝑁)⟶𝐶))) |
32 | | coeq1 5513 |
. . . . . . . . . . 11
⊢ (𝑔 = (𝐺 ↾ (𝑀...𝑁)) → (𝑔 ∘ 𝑓) = ((𝐺 ↾ (𝑀...𝑁)) ∘ 𝑓)) |
33 | | coeq2 5514 |
. . . . . . . . . . 11
⊢ (𝑓 = 𝐽 → ((𝐺 ↾ (𝑀...𝑁)) ∘ 𝑓) = ((𝐺 ↾ (𝑀...𝑁)) ∘ 𝐽)) |
34 | 32, 33 | sylan9eq 2882 |
. . . . . . . . . 10
⊢ ((𝑔 = (𝐺 ↾ (𝑀...𝑁)) ∧ 𝑓 = 𝐽) → (𝑔 ∘ 𝑓) = ((𝐺 ↾ (𝑀...𝑁)) ∘ 𝐽)) |
35 | 34 | seqeq3d 13104 |
. . . . . . . . 9
⊢ ((𝑔 = (𝐺 ↾ (𝑀...𝑁)) ∧ 𝑓 = 𝐽) → seq𝑀( + , (𝑔 ∘ 𝑓)) = seq𝑀( + , ((𝐺 ↾ (𝑀...𝑁)) ∘ 𝐽))) |
36 | 35 | fveq1d 6436 |
. . . . . . . 8
⊢ ((𝑔 = (𝐺 ↾ (𝑀...𝑁)) ∧ 𝑓 = 𝐽) → (seq𝑀( + , (𝑔 ∘ 𝑓))‘𝑁) = (seq𝑀( + , ((𝐺 ↾ (𝑀...𝑁)) ∘ 𝐽))‘𝑁)) |
37 | | simpl 476 |
. . . . . . . . . 10
⊢ ((𝑔 = (𝐺 ↾ (𝑀...𝑁)) ∧ 𝑓 = 𝐽) → 𝑔 = (𝐺 ↾ (𝑀...𝑁))) |
38 | 37 | seqeq3d 13104 |
. . . . . . . . 9
⊢ ((𝑔 = (𝐺 ↾ (𝑀...𝑁)) ∧ 𝑓 = 𝐽) → seq𝑀( + , 𝑔) = seq𝑀( + , (𝐺 ↾ (𝑀...𝑁)))) |
39 | 38 | fveq1d 6436 |
. . . . . . . 8
⊢ ((𝑔 = (𝐺 ↾ (𝑀...𝑁)) ∧ 𝑓 = 𝐽) → (seq𝑀( + , 𝑔)‘𝑁) = (seq𝑀( + , (𝐺 ↾ (𝑀...𝑁)))‘𝑁)) |
40 | 36, 39 | eqeq12d 2841 |
. . . . . . 7
⊢ ((𝑔 = (𝐺 ↾ (𝑀...𝑁)) ∧ 𝑓 = 𝐽) → ((seq𝑀( + , (𝑔 ∘ 𝑓))‘𝑁) = (seq𝑀( + , 𝑔)‘𝑁) ↔ (seq𝑀( + , ((𝐺 ↾ (𝑀...𝑁)) ∘ 𝐽))‘𝑁) = (seq𝑀( + , (𝐺 ↾ (𝑀...𝑁)))‘𝑁))) |
41 | 31, 40 | imbi12d 336 |
. . . . . 6
⊢ ((𝑔 = (𝐺 ↾ (𝑀...𝑁)) ∧ 𝑓 = 𝐽) → (((𝑓:(𝑀...𝑁)–1-1-onto→(𝑀...𝑁) ∧ 𝑔:(𝑀...𝑁)⟶𝐶) → (seq𝑀( + , (𝑔 ∘ 𝑓))‘𝑁) = (seq𝑀( + , 𝑔)‘𝑁)) ↔ ((𝐽:(𝑀...𝑁)–1-1-onto→(𝑀...𝑁) ∧ (𝐺 ↾ (𝑀...𝑁)):(𝑀...𝑁)⟶𝐶) → (seq𝑀( + , ((𝐺 ↾ (𝑀...𝑁)) ∘ 𝐽))‘𝑁) = (seq𝑀( + , (𝐺 ↾ (𝑀...𝑁)))‘𝑁)))) |
42 | 41 | spc2gv 3514 |
. . . . 5
⊢ (((𝐺 ↾ (𝑀...𝑁)) ∈ V ∧ 𝐽 ∈ V) → (∀𝑔∀𝑓((𝑓:(𝑀...𝑁)–1-1-onto→(𝑀...𝑁) ∧ 𝑔:(𝑀...𝑁)⟶𝐶) → (seq𝑀( + , (𝑔 ∘ 𝑓))‘𝑁) = (seq𝑀( + , 𝑔)‘𝑁)) → ((𝐽:(𝑀...𝑁)–1-1-onto→(𝑀...𝑁) ∧ (𝐺 ↾ (𝑀...𝑁)):(𝑀...𝑁)⟶𝐶) → (seq𝑀( + , ((𝐺 ↾ (𝑀...𝑁)) ∘ 𝐽))‘𝑁) = (seq𝑀( + , (𝐺 ↾ (𝑀...𝑁)))‘𝑁)))) |
43 | 24, 25, 28, 42 | syl3c 66 |
. . . 4
⊢ (𝜑 → (seq𝑀( + , ((𝐺 ↾ (𝑀...𝑁)) ∘ 𝐽))‘𝑁) = (seq𝑀( + , (𝐺 ↾ (𝑀...𝑁)))‘𝑁)) |
44 | | fvres 6453 |
. . . . . 6
⊢ (𝑥 ∈ (𝑀...𝑁) → ((𝐺 ↾ (𝑀...𝑁))‘𝑥) = (𝐺‘𝑥)) |
45 | 44 | adantl 475 |
. . . . 5
⊢ ((𝜑 ∧ 𝑥 ∈ (𝑀...𝑁)) → ((𝐺 ↾ (𝑀...𝑁))‘𝑥) = (𝐺‘𝑥)) |
46 | 14, 45 | seqfveq 13120 |
. . . 4
⊢ (𝜑 → (seq𝑀( + , (𝐺 ↾ (𝑀...𝑁)))‘𝑁) = (seq𝑀( + , 𝐺)‘𝑁)) |
47 | 43, 46 | eqtrd 2862 |
. . 3
⊢ (𝜑 → (seq𝑀( + , ((𝐺 ↾ (𝑀...𝑁)) ∘ 𝐽))‘𝑁) = (seq𝑀( + , 𝐺)‘𝑁)) |
48 | 47 | oveq1d 6921 |
. 2
⊢ (𝜑 → ((seq𝑀( + , ((𝐺 ↾ (𝑀...𝑁)) ∘ 𝐽))‘𝑁) + (𝐺‘(𝑁 + 1))) = ((seq𝑀( + , 𝐺)‘𝑁) + (𝐺‘(𝑁 + 1)))) |
49 | 11 | adantlr 708 |
. . . . 5
⊢ (((𝜑 ∧ 𝐾 ∈ (𝑀...𝑁)) ∧ (𝑥 ∈ 𝑆 ∧ 𝑦 ∈ 𝑆)) → (𝑥 + 𝑦) ∈ 𝑆) |
50 | 13 | adantlr 708 |
. . . . 5
⊢ (((𝜑 ∧ 𝐾 ∈ (𝑀...𝑁)) ∧ (𝑥 ∈ 𝑆 ∧ 𝑦 ∈ 𝑆 ∧ 𝑧 ∈ 𝑆)) → ((𝑥 + 𝑦) + 𝑧) = (𝑥 + (𝑦 + 𝑧))) |
51 | | elfzuz3 12633 |
. . . . . . 7
⊢ (𝐾 ∈ (𝑀...𝑁) → 𝑁 ∈ (ℤ≥‘𝐾)) |
52 | 51 | adantl 475 |
. . . . . 6
⊢ ((𝜑 ∧ 𝐾 ∈ (𝑀...𝑁)) → 𝑁 ∈ (ℤ≥‘𝐾)) |
53 | | eluzp1p1 11995 |
. . . . . 6
⊢ (𝑁 ∈
(ℤ≥‘𝐾) → (𝑁 + 1) ∈
(ℤ≥‘(𝐾 + 1))) |
54 | 52, 53 | syl 17 |
. . . . 5
⊢ ((𝜑 ∧ 𝐾 ∈ (𝑀...𝑁)) → (𝑁 + 1) ∈
(ℤ≥‘(𝐾 + 1))) |
55 | | elfzuz 12632 |
. . . . . 6
⊢ (𝐾 ∈ (𝑀...𝑁) → 𝐾 ∈ (ℤ≥‘𝑀)) |
56 | 55 | adantl 475 |
. . . . 5
⊢ ((𝜑 ∧ 𝐾 ∈ (𝑀...𝑁)) → 𝐾 ∈ (ℤ≥‘𝑀)) |
57 | | f1of 6379 |
. . . . . . . . . 10
⊢ (𝐹:(𝑀...(𝑁 + 1))–1-1-onto→(𝑀...(𝑁 + 1)) → 𝐹:(𝑀...(𝑁 + 1))⟶(𝑀...(𝑁 + 1))) |
58 | 16, 57 | syl 17 |
. . . . . . . . 9
⊢ (𝜑 → 𝐹:(𝑀...(𝑁 + 1))⟶(𝑀...(𝑁 + 1))) |
59 | | fco 6296 |
. . . . . . . . 9
⊢ ((𝐺:(𝑀...(𝑁 + 1))⟶𝐶 ∧ 𝐹:(𝑀...(𝑁 + 1))⟶(𝑀...(𝑁 + 1))) → (𝐺 ∘ 𝐹):(𝑀...(𝑁 + 1))⟶𝐶) |
60 | 1, 58, 59 | syl2anc 581 |
. . . . . . . 8
⊢ (𝜑 → (𝐺 ∘ 𝐹):(𝑀...(𝑁 + 1))⟶𝐶) |
61 | 60, 15 | fssd 6293 |
. . . . . . 7
⊢ (𝜑 → (𝐺 ∘ 𝐹):(𝑀...(𝑁 + 1))⟶𝑆) |
62 | 61 | ffvelrnda 6609 |
. . . . . 6
⊢ ((𝜑 ∧ 𝑥 ∈ (𝑀...(𝑁 + 1))) → ((𝐺 ∘ 𝐹)‘𝑥) ∈ 𝑆) |
63 | 62 | adantlr 708 |
. . . . 5
⊢ (((𝜑 ∧ 𝐾 ∈ (𝑀...𝑁)) ∧ 𝑥 ∈ (𝑀...(𝑁 + 1))) → ((𝐺 ∘ 𝐹)‘𝑥) ∈ 𝑆) |
64 | 49, 50, 54, 56, 63 | seqsplit 13129 |
. . . 4
⊢ ((𝜑 ∧ 𝐾 ∈ (𝑀...𝑁)) → (seq𝑀( + , (𝐺 ∘ 𝐹))‘(𝑁 + 1)) = ((seq𝑀( + , (𝐺 ∘ 𝐹))‘𝐾) + (seq(𝐾 + 1)( + , (𝐺 ∘ 𝐹))‘(𝑁 + 1)))) |
65 | | elfzp12 12714 |
. . . . . . 7
⊢ (𝑁 ∈
(ℤ≥‘𝑀) → (𝐾 ∈ (𝑀...𝑁) ↔ (𝐾 = 𝑀 ∨ 𝐾 ∈ ((𝑀 + 1)...𝑁)))) |
66 | 65 | biimpa 470 |
. . . . . 6
⊢ ((𝑁 ∈
(ℤ≥‘𝑀) ∧ 𝐾 ∈ (𝑀...𝑁)) → (𝐾 = 𝑀 ∨ 𝐾 ∈ ((𝑀 + 1)...𝑁))) |
67 | 14, 66 | sylan 577 |
. . . . 5
⊢ ((𝜑 ∧ 𝐾 ∈ (𝑀...𝑁)) → (𝐾 = 𝑀 ∨ 𝐾 ∈ ((𝑀 + 1)...𝑁))) |
68 | | seqeq1 13099 |
. . . . . . . . . . 11
⊢ (𝐾 = 𝑀 → seq𝐾( + , (𝐺 ∘ 𝐹)) = seq𝑀( + , (𝐺 ∘ 𝐹))) |
69 | 68 | eqcomd 2832 |
. . . . . . . . . 10
⊢ (𝐾 = 𝑀 → seq𝑀( + , (𝐺 ∘ 𝐹)) = seq𝐾( + , (𝐺 ∘ 𝐹))) |
70 | 69 | fveq1d 6436 |
. . . . . . . . 9
⊢ (𝐾 = 𝑀 → (seq𝑀( + , (𝐺 ∘ 𝐹))‘𝐾) = (seq𝐾( + , (𝐺 ∘ 𝐹))‘𝐾)) |
71 | | f1ocnv 6391 |
. . . . . . . . . . . . 13
⊢ (𝐹:(𝑀...(𝑁 + 1))–1-1-onto→(𝑀...(𝑁 + 1)) → ◡𝐹:(𝑀...(𝑁 + 1))–1-1-onto→(𝑀...(𝑁 + 1))) |
72 | | f1of 6379 |
. . . . . . . . . . . . 13
⊢ (◡𝐹:(𝑀...(𝑁 + 1))–1-1-onto→(𝑀...(𝑁 + 1)) → ◡𝐹:(𝑀...(𝑁 + 1))⟶(𝑀...(𝑁 + 1))) |
73 | 16, 71, 72 | 3syl 18 |
. . . . . . . . . . . 12
⊢ (𝜑 → ◡𝐹:(𝑀...(𝑁 + 1))⟶(𝑀...(𝑁 + 1))) |
74 | | peano2uz 12024 |
. . . . . . . . . . . . 13
⊢ (𝑁 ∈
(ℤ≥‘𝑀) → (𝑁 + 1) ∈
(ℤ≥‘𝑀)) |
75 | | eluzfz2 12643 |
. . . . . . . . . . . . 13
⊢ ((𝑁 + 1) ∈
(ℤ≥‘𝑀) → (𝑁 + 1) ∈ (𝑀...(𝑁 + 1))) |
76 | 14, 74, 75 | 3syl 18 |
. . . . . . . . . . . 12
⊢ (𝜑 → (𝑁 + 1) ∈ (𝑀...(𝑁 + 1))) |
77 | 73, 76 | ffvelrnd 6610 |
. . . . . . . . . . 11
⊢ (𝜑 → (◡𝐹‘(𝑁 + 1)) ∈ (𝑀...(𝑁 + 1))) |
78 | 18, 77 | syl5eqel 2911 |
. . . . . . . . . 10
⊢ (𝜑 → 𝐾 ∈ (𝑀...(𝑁 + 1))) |
79 | | elfzelz 12636 |
. . . . . . . . . 10
⊢ (𝐾 ∈ (𝑀...(𝑁 + 1)) → 𝐾 ∈ ℤ) |
80 | | seq1 13109 |
. . . . . . . . . 10
⊢ (𝐾 ∈ ℤ → (seq𝐾( + , (𝐺 ∘ 𝐹))‘𝐾) = ((𝐺 ∘ 𝐹)‘𝐾)) |
81 | 78, 79, 80 | 3syl 18 |
. . . . . . . . 9
⊢ (𝜑 → (seq𝐾( + , (𝐺 ∘ 𝐹))‘𝐾) = ((𝐺 ∘ 𝐹)‘𝐾)) |
82 | 70, 81 | sylan9eqr 2884 |
. . . . . . . 8
⊢ ((𝜑 ∧ 𝐾 = 𝑀) → (seq𝑀( + , (𝐺 ∘ 𝐹))‘𝐾) = ((𝐺 ∘ 𝐹)‘𝐾)) |
83 | 82 | oveq1d 6921 |
. . . . . . 7
⊢ ((𝜑 ∧ 𝐾 = 𝑀) → ((seq𝑀( + , (𝐺 ∘ 𝐹))‘𝐾) + (seq(𝐾 + 1)( + , (𝐺 ∘ 𝐹))‘(𝑁 + 1))) = (((𝐺 ∘ 𝐹)‘𝐾) + (seq(𝐾 + 1)( + , (𝐺 ∘ 𝐹))‘(𝑁 + 1)))) |
84 | | simpr 479 |
. . . . . . . . 9
⊢ ((𝜑 ∧ 𝐾 = 𝑀) → 𝐾 = 𝑀) |
85 | | eluzfz1 12642 |
. . . . . . . . . . 11
⊢ (𝑁 ∈
(ℤ≥‘𝑀) → 𝑀 ∈ (𝑀...𝑁)) |
86 | 14, 85 | syl 17 |
. . . . . . . . . 10
⊢ (𝜑 → 𝑀 ∈ (𝑀...𝑁)) |
87 | 86 | adantr 474 |
. . . . . . . . 9
⊢ ((𝜑 ∧ 𝐾 = 𝑀) → 𝑀 ∈ (𝑀...𝑁)) |
88 | 84, 87 | eqeltrd 2907 |
. . . . . . . 8
⊢ ((𝜑 ∧ 𝐾 = 𝑀) → 𝐾 ∈ (𝑀...𝑁)) |
89 | 12 | adantlr 708 |
. . . . . . . . . 10
⊢ (((𝜑 ∧ 𝐾 ∈ (𝑀...𝑁)) ∧ (𝑥 ∈ 𝐶 ∧ 𝑦 ∈ 𝐶)) → (𝑥 + 𝑦) = (𝑦 + 𝑥)) |
90 | 15 | adantr 474 |
. . . . . . . . . 10
⊢ ((𝜑 ∧ 𝐾 ∈ (𝑀...𝑁)) → 𝐶 ⊆ 𝑆) |
91 | 60 | adantr 474 |
. . . . . . . . . 10
⊢ ((𝜑 ∧ 𝐾 ∈ (𝑀...𝑁)) → (𝐺 ∘ 𝐹):(𝑀...(𝑁 + 1))⟶𝐶) |
92 | 78 | adantr 474 |
. . . . . . . . . 10
⊢ ((𝜑 ∧ 𝐾 ∈ (𝑀...𝑁)) → 𝐾 ∈ (𝑀...(𝑁 + 1))) |
93 | | peano2uz 12024 |
. . . . . . . . . . 11
⊢ (𝐾 ∈
(ℤ≥‘𝑀) → (𝐾 + 1) ∈
(ℤ≥‘𝑀)) |
94 | | fzss1 12674 |
. . . . . . . . . . 11
⊢ ((𝐾 + 1) ∈
(ℤ≥‘𝑀) → ((𝐾 + 1)...(𝑁 + 1)) ⊆ (𝑀...(𝑁 + 1))) |
95 | 56, 93, 94 | 3syl 18 |
. . . . . . . . . 10
⊢ ((𝜑 ∧ 𝐾 ∈ (𝑀...𝑁)) → ((𝐾 + 1)...(𝑁 + 1)) ⊆ (𝑀...(𝑁 + 1))) |
96 | 49, 89, 50, 54, 90, 91, 92, 95 | seqf1olem2a 13134 |
. . . . . . . . 9
⊢ ((𝜑 ∧ 𝐾 ∈ (𝑀...𝑁)) → (((𝐺 ∘ 𝐹)‘𝐾) + (seq(𝐾 + 1)( + , (𝐺 ∘ 𝐹))‘(𝑁 + 1))) = ((seq(𝐾 + 1)( + , (𝐺 ∘ 𝐹))‘(𝑁 + 1)) + ((𝐺 ∘ 𝐹)‘𝐾))) |
97 | | 1zzd 11737 |
. . . . . . . . . . 11
⊢ ((𝜑 ∧ 𝐾 ∈ (𝑀...𝑁)) → 1 ∈ ℤ) |
98 | | elfzuz 12632 |
. . . . . . . . . . . . . . . . . 18
⊢ (𝐾 ∈ (𝑀...(𝑁 + 1)) → 𝐾 ∈ (ℤ≥‘𝑀)) |
99 | | fzss1 12674 |
. . . . . . . . . . . . . . . . . 18
⊢ (𝐾 ∈
(ℤ≥‘𝑀) → (𝐾...𝑁) ⊆ (𝑀...𝑁)) |
100 | 78, 98, 99 | 3syl 18 |
. . . . . . . . . . . . . . . . 17
⊢ (𝜑 → (𝐾...𝑁) ⊆ (𝑀...𝑁)) |
101 | 100 | sselda 3828 |
. . . . . . . . . . . . . . . 16
⊢ ((𝜑 ∧ 𝑥 ∈ (𝐾...𝑁)) → 𝑥 ∈ (𝑀...𝑁)) |
102 | 21 | ffvelrnda 6609 |
. . . . . . . . . . . . . . . 16
⊢ ((𝜑 ∧ 𝑥 ∈ (𝑀...𝑁)) → (𝐽‘𝑥) ∈ (𝑀...𝑁)) |
103 | 101, 102 | syldan 587 |
. . . . . . . . . . . . . . 15
⊢ ((𝜑 ∧ 𝑥 ∈ (𝐾...𝑁)) → (𝐽‘𝑥) ∈ (𝑀...𝑁)) |
104 | | fvres 6453 |
. . . . . . . . . . . . . . 15
⊢ ((𝐽‘𝑥) ∈ (𝑀...𝑁) → ((𝐺 ↾ (𝑀...𝑁))‘(𝐽‘𝑥)) = (𝐺‘(𝐽‘𝑥))) |
105 | 103, 104 | syl 17 |
. . . . . . . . . . . . . 14
⊢ ((𝜑 ∧ 𝑥 ∈ (𝐾...𝑁)) → ((𝐺 ↾ (𝑀...𝑁))‘(𝐽‘𝑥)) = (𝐺‘(𝐽‘𝑥))) |
106 | | breq1 4877 |
. . . . . . . . . . . . . . . . . . . 20
⊢ (𝑘 = 𝑥 → (𝑘 < 𝐾 ↔ 𝑥 < 𝐾)) |
107 | | id 22 |
. . . . . . . . . . . . . . . . . . . 20
⊢ (𝑘 = 𝑥 → 𝑘 = 𝑥) |
108 | | oveq1 6913 |
. . . . . . . . . . . . . . . . . . . 20
⊢ (𝑘 = 𝑥 → (𝑘 + 1) = (𝑥 + 1)) |
109 | 106, 107,
108 | ifbieq12d 4334 |
. . . . . . . . . . . . . . . . . . 19
⊢ (𝑘 = 𝑥 → if(𝑘 < 𝐾, 𝑘, (𝑘 + 1)) = if(𝑥 < 𝐾, 𝑥, (𝑥 + 1))) |
110 | 109 | fveq2d 6438 |
. . . . . . . . . . . . . . . . . 18
⊢ (𝑘 = 𝑥 → (𝐹‘if(𝑘 < 𝐾, 𝑘, (𝑘 + 1))) = (𝐹‘if(𝑥 < 𝐾, 𝑥, (𝑥 + 1)))) |
111 | | fvex 6447 |
. . . . . . . . . . . . . . . . . 18
⊢ (𝐹‘if(𝑥 < 𝐾, 𝑥, (𝑥 + 1))) ∈ V |
112 | 110, 17, 111 | fvmpt 6530 |
. . . . . . . . . . . . . . . . 17
⊢ (𝑥 ∈ (𝑀...𝑁) → (𝐽‘𝑥) = (𝐹‘if(𝑥 < 𝐾, 𝑥, (𝑥 + 1)))) |
113 | 101, 112 | syl 17 |
. . . . . . . . . . . . . . . 16
⊢ ((𝜑 ∧ 𝑥 ∈ (𝐾...𝑁)) → (𝐽‘𝑥) = (𝐹‘if(𝑥 < 𝐾, 𝑥, (𝑥 + 1)))) |
114 | | elfzle1 12638 |
. . . . . . . . . . . . . . . . . . 19
⊢ (𝑥 ∈ (𝐾...𝑁) → 𝐾 ≤ 𝑥) |
115 | 114 | adantl 475 |
. . . . . . . . . . . . . . . . . 18
⊢ ((𝜑 ∧ 𝑥 ∈ (𝐾...𝑁)) → 𝐾 ≤ 𝑥) |
116 | 78, 79 | syl 17 |
. . . . . . . . . . . . . . . . . . . . 21
⊢ (𝜑 → 𝐾 ∈ ℤ) |
117 | 116 | zred 11811 |
. . . . . . . . . . . . . . . . . . . 20
⊢ (𝜑 → 𝐾 ∈ ℝ) |
118 | 117 | adantr 474 |
. . . . . . . . . . . . . . . . . . 19
⊢ ((𝜑 ∧ 𝑥 ∈ (𝐾...𝑁)) → 𝐾 ∈ ℝ) |
119 | | elfzelz 12636 |
. . . . . . . . . . . . . . . . . . . . 21
⊢ (𝑥 ∈ (𝐾...𝑁) → 𝑥 ∈ ℤ) |
120 | 119 | adantl 475 |
. . . . . . . . . . . . . . . . . . . 20
⊢ ((𝜑 ∧ 𝑥 ∈ (𝐾...𝑁)) → 𝑥 ∈ ℤ) |
121 | 120 | zred 11811 |
. . . . . . . . . . . . . . . . . . 19
⊢ ((𝜑 ∧ 𝑥 ∈ (𝐾...𝑁)) → 𝑥 ∈ ℝ) |
122 | 118, 121 | lenltd 10503 |
. . . . . . . . . . . . . . . . . 18
⊢ ((𝜑 ∧ 𝑥 ∈ (𝐾...𝑁)) → (𝐾 ≤ 𝑥 ↔ ¬ 𝑥 < 𝐾)) |
123 | 115, 122 | mpbid 224 |
. . . . . . . . . . . . . . . . 17
⊢ ((𝜑 ∧ 𝑥 ∈ (𝐾...𝑁)) → ¬ 𝑥 < 𝐾) |
124 | | iffalse 4316 |
. . . . . . . . . . . . . . . . . 18
⊢ (¬
𝑥 < 𝐾 → if(𝑥 < 𝐾, 𝑥, (𝑥 + 1)) = (𝑥 + 1)) |
125 | 124 | fveq2d 6438 |
. . . . . . . . . . . . . . . . 17
⊢ (¬
𝑥 < 𝐾 → (𝐹‘if(𝑥 < 𝐾, 𝑥, (𝑥 + 1))) = (𝐹‘(𝑥 + 1))) |
126 | 123, 125 | syl 17 |
. . . . . . . . . . . . . . . 16
⊢ ((𝜑 ∧ 𝑥 ∈ (𝐾...𝑁)) → (𝐹‘if(𝑥 < 𝐾, 𝑥, (𝑥 + 1))) = (𝐹‘(𝑥 + 1))) |
127 | 113, 126 | eqtrd 2862 |
. . . . . . . . . . . . . . 15
⊢ ((𝜑 ∧ 𝑥 ∈ (𝐾...𝑁)) → (𝐽‘𝑥) = (𝐹‘(𝑥 + 1))) |
128 | 127 | fveq2d 6438 |
. . . . . . . . . . . . . 14
⊢ ((𝜑 ∧ 𝑥 ∈ (𝐾...𝑁)) → (𝐺‘(𝐽‘𝑥)) = (𝐺‘(𝐹‘(𝑥 + 1)))) |
129 | 105, 128 | eqtrd 2862 |
. . . . . . . . . . . . 13
⊢ ((𝜑 ∧ 𝑥 ∈ (𝐾...𝑁)) → ((𝐺 ↾ (𝑀...𝑁))‘(𝐽‘𝑥)) = (𝐺‘(𝐹‘(𝑥 + 1)))) |
130 | | fvco3 6523 |
. . . . . . . . . . . . . . 15
⊢ ((𝐽:(𝑀...𝑁)⟶(𝑀...𝑁) ∧ 𝑥 ∈ (𝑀...𝑁)) → (((𝐺 ↾ (𝑀...𝑁)) ∘ 𝐽)‘𝑥) = ((𝐺 ↾ (𝑀...𝑁))‘(𝐽‘𝑥))) |
131 | 21, 130 | sylan 577 |
. . . . . . . . . . . . . 14
⊢ ((𝜑 ∧ 𝑥 ∈ (𝑀...𝑁)) → (((𝐺 ↾ (𝑀...𝑁)) ∘ 𝐽)‘𝑥) = ((𝐺 ↾ (𝑀...𝑁))‘(𝐽‘𝑥))) |
132 | 101, 131 | syldan 587 |
. . . . . . . . . . . . 13
⊢ ((𝜑 ∧ 𝑥 ∈ (𝐾...𝑁)) → (((𝐺 ↾ (𝑀...𝑁)) ∘ 𝐽)‘𝑥) = ((𝐺 ↾ (𝑀...𝑁))‘(𝐽‘𝑥))) |
133 | | fzp1elp1 12688 |
. . . . . . . . . . . . . . 15
⊢ (𝑥 ∈ (𝑀...𝑁) → (𝑥 + 1) ∈ (𝑀...(𝑁 + 1))) |
134 | 101, 133 | syl 17 |
. . . . . . . . . . . . . 14
⊢ ((𝜑 ∧ 𝑥 ∈ (𝐾...𝑁)) → (𝑥 + 1) ∈ (𝑀...(𝑁 + 1))) |
135 | | fvco3 6523 |
. . . . . . . . . . . . . . 15
⊢ ((𝐹:(𝑀...(𝑁 + 1))⟶(𝑀...(𝑁 + 1)) ∧ (𝑥 + 1) ∈ (𝑀...(𝑁 + 1))) → ((𝐺 ∘ 𝐹)‘(𝑥 + 1)) = (𝐺‘(𝐹‘(𝑥 + 1)))) |
136 | 58, 135 | sylan 577 |
. . . . . . . . . . . . . 14
⊢ ((𝜑 ∧ (𝑥 + 1) ∈ (𝑀...(𝑁 + 1))) → ((𝐺 ∘ 𝐹)‘(𝑥 + 1)) = (𝐺‘(𝐹‘(𝑥 + 1)))) |
137 | 134, 136 | syldan 587 |
. . . . . . . . . . . . 13
⊢ ((𝜑 ∧ 𝑥 ∈ (𝐾...𝑁)) → ((𝐺 ∘ 𝐹)‘(𝑥 + 1)) = (𝐺‘(𝐹‘(𝑥 + 1)))) |
138 | 129, 132,
137 | 3eqtr4d 2872 |
. . . . . . . . . . . 12
⊢ ((𝜑 ∧ 𝑥 ∈ (𝐾...𝑁)) → (((𝐺 ↾ (𝑀...𝑁)) ∘ 𝐽)‘𝑥) = ((𝐺 ∘ 𝐹)‘(𝑥 + 1))) |
139 | 138 | adantlr 708 |
. . . . . . . . . . 11
⊢ (((𝜑 ∧ 𝐾 ∈ (𝑀...𝑁)) ∧ 𝑥 ∈ (𝐾...𝑁)) → (((𝐺 ↾ (𝑀...𝑁)) ∘ 𝐽)‘𝑥) = ((𝐺 ∘ 𝐹)‘(𝑥 + 1))) |
140 | 52, 97, 139 | seqshft2 13122 |
. . . . . . . . . 10
⊢ ((𝜑 ∧ 𝐾 ∈ (𝑀...𝑁)) → (seq𝐾( + , ((𝐺 ↾ (𝑀...𝑁)) ∘ 𝐽))‘𝑁) = (seq(𝐾 + 1)( + , (𝐺 ∘ 𝐹))‘(𝑁 + 1))) |
141 | | fvco3 6523 |
. . . . . . . . . . . . 13
⊢ ((𝐹:(𝑀...(𝑁 + 1))⟶(𝑀...(𝑁 + 1)) ∧ 𝐾 ∈ (𝑀...(𝑁 + 1))) → ((𝐺 ∘ 𝐹)‘𝐾) = (𝐺‘(𝐹‘𝐾))) |
142 | 58, 78, 141 | syl2anc 581 |
. . . . . . . . . . . 12
⊢ (𝜑 → ((𝐺 ∘ 𝐹)‘𝐾) = (𝐺‘(𝐹‘𝐾))) |
143 | 18 | fveq2i 6437 |
. . . . . . . . . . . . . 14
⊢ (𝐹‘𝐾) = (𝐹‘(◡𝐹‘(𝑁 + 1))) |
144 | | f1ocnvfv2 6789 |
. . . . . . . . . . . . . . 15
⊢ ((𝐹:(𝑀...(𝑁 + 1))–1-1-onto→(𝑀...(𝑁 + 1)) ∧ (𝑁 + 1) ∈ (𝑀...(𝑁 + 1))) → (𝐹‘(◡𝐹‘(𝑁 + 1))) = (𝑁 + 1)) |
145 | 16, 76, 144 | syl2anc 581 |
. . . . . . . . . . . . . 14
⊢ (𝜑 → (𝐹‘(◡𝐹‘(𝑁 + 1))) = (𝑁 + 1)) |
146 | 143, 145 | syl5eq 2874 |
. . . . . . . . . . . . 13
⊢ (𝜑 → (𝐹‘𝐾) = (𝑁 + 1)) |
147 | 146 | fveq2d 6438 |
. . . . . . . . . . . 12
⊢ (𝜑 → (𝐺‘(𝐹‘𝐾)) = (𝐺‘(𝑁 + 1))) |
148 | 142, 147 | eqtr2d 2863 |
. . . . . . . . . . 11
⊢ (𝜑 → (𝐺‘(𝑁 + 1)) = ((𝐺 ∘ 𝐹)‘𝐾)) |
149 | 148 | adantr 474 |
. . . . . . . . . 10
⊢ ((𝜑 ∧ 𝐾 ∈ (𝑀...𝑁)) → (𝐺‘(𝑁 + 1)) = ((𝐺 ∘ 𝐹)‘𝐾)) |
150 | 140, 149 | oveq12d 6924 |
. . . . . . . . 9
⊢ ((𝜑 ∧ 𝐾 ∈ (𝑀...𝑁)) → ((seq𝐾( + , ((𝐺 ↾ (𝑀...𝑁)) ∘ 𝐽))‘𝑁) + (𝐺‘(𝑁 + 1))) = ((seq(𝐾 + 1)( + , (𝐺 ∘ 𝐹))‘(𝑁 + 1)) + ((𝐺 ∘ 𝐹)‘𝐾))) |
151 | 96, 150 | eqtr4d 2865 |
. . . . . . . 8
⊢ ((𝜑 ∧ 𝐾 ∈ (𝑀...𝑁)) → (((𝐺 ∘ 𝐹)‘𝐾) + (seq(𝐾 + 1)( + , (𝐺 ∘ 𝐹))‘(𝑁 + 1))) = ((seq𝐾( + , ((𝐺 ↾ (𝑀...𝑁)) ∘ 𝐽))‘𝑁) + (𝐺‘(𝑁 + 1)))) |
152 | 88, 151 | syldan 587 |
. . . . . . 7
⊢ ((𝜑 ∧ 𝐾 = 𝑀) → (((𝐺 ∘ 𝐹)‘𝐾) + (seq(𝐾 + 1)( + , (𝐺 ∘ 𝐹))‘(𝑁 + 1))) = ((seq𝐾( + , ((𝐺 ↾ (𝑀...𝑁)) ∘ 𝐽))‘𝑁) + (𝐺‘(𝑁 + 1)))) |
153 | 84 | seqeq1d 13102 |
. . . . . . . . 9
⊢ ((𝜑 ∧ 𝐾 = 𝑀) → seq𝐾( + , ((𝐺 ↾ (𝑀...𝑁)) ∘ 𝐽)) = seq𝑀( + , ((𝐺 ↾ (𝑀...𝑁)) ∘ 𝐽))) |
154 | 153 | fveq1d 6436 |
. . . . . . . 8
⊢ ((𝜑 ∧ 𝐾 = 𝑀) → (seq𝐾( + , ((𝐺 ↾ (𝑀...𝑁)) ∘ 𝐽))‘𝑁) = (seq𝑀( + , ((𝐺 ↾ (𝑀...𝑁)) ∘ 𝐽))‘𝑁)) |
155 | 154 | oveq1d 6921 |
. . . . . . 7
⊢ ((𝜑 ∧ 𝐾 = 𝑀) → ((seq𝐾( + , ((𝐺 ↾ (𝑀...𝑁)) ∘ 𝐽))‘𝑁) + (𝐺‘(𝑁 + 1))) = ((seq𝑀( + , ((𝐺 ↾ (𝑀...𝑁)) ∘ 𝐽))‘𝑁) + (𝐺‘(𝑁 + 1)))) |
156 | 83, 152, 155 | 3eqtrd 2866 |
. . . . . 6
⊢ ((𝜑 ∧ 𝐾 = 𝑀) → ((seq𝑀( + , (𝐺 ∘ 𝐹))‘𝐾) + (seq(𝐾 + 1)( + , (𝐺 ∘ 𝐹))‘(𝑁 + 1))) = ((seq𝑀( + , ((𝐺 ↾ (𝑀...𝑁)) ∘ 𝐽))‘𝑁) + (𝐺‘(𝑁 + 1)))) |
157 | | eluzel2 11974 |
. . . . . . . . . . . 12
⊢ (𝑁 ∈
(ℤ≥‘𝑀) → 𝑀 ∈ ℤ) |
158 | 14, 157 | syl 17 |
. . . . . . . . . . 11
⊢ (𝜑 → 𝑀 ∈ ℤ) |
159 | | elfzuz 12632 |
. . . . . . . . . . 11
⊢ (𝐾 ∈ ((𝑀 + 1)...𝑁) → 𝐾 ∈ (ℤ≥‘(𝑀 + 1))) |
160 | | eluzp1m1 11993 |
. . . . . . . . . . 11
⊢ ((𝑀 ∈ ℤ ∧ 𝐾 ∈
(ℤ≥‘(𝑀 + 1))) → (𝐾 − 1) ∈
(ℤ≥‘𝑀)) |
161 | 158, 159,
160 | syl2an 591 |
. . . . . . . . . 10
⊢ ((𝜑 ∧ 𝐾 ∈ ((𝑀 + 1)...𝑁)) → (𝐾 − 1) ∈
(ℤ≥‘𝑀)) |
162 | | eluzelz 11979 |
. . . . . . . . . . . . . . . . . . . . 21
⊢ (𝑁 ∈
(ℤ≥‘𝑀) → 𝑁 ∈ ℤ) |
163 | 14, 162 | syl 17 |
. . . . . . . . . . . . . . . . . . . 20
⊢ (𝜑 → 𝑁 ∈ ℤ) |
164 | 163 | zcnd 11812 |
. . . . . . . . . . . . . . . . . . 19
⊢ (𝜑 → 𝑁 ∈ ℂ) |
165 | | ax-1cn 10311 |
. . . . . . . . . . . . . . . . . . 19
⊢ 1 ∈
ℂ |
166 | | pncan 10608 |
. . . . . . . . . . . . . . . . . . 19
⊢ ((𝑁 ∈ ℂ ∧ 1 ∈
ℂ) → ((𝑁 + 1)
− 1) = 𝑁) |
167 | 164, 165,
166 | sylancl 582 |
. . . . . . . . . . . . . . . . . 18
⊢ (𝜑 → ((𝑁 + 1) − 1) = 𝑁) |
168 | | peano2zm 11749 |
. . . . . . . . . . . . . . . . . . . 20
⊢ (𝐾 ∈ ℤ → (𝐾 − 1) ∈
ℤ) |
169 | 78, 79, 168 | 3syl 18 |
. . . . . . . . . . . . . . . . . . 19
⊢ (𝜑 → (𝐾 − 1) ∈ ℤ) |
170 | | elfzuz3 12633 |
. . . . . . . . . . . . . . . . . . . . 21
⊢ (𝐾 ∈ (𝑀...(𝑁 + 1)) → (𝑁 + 1) ∈
(ℤ≥‘𝐾)) |
171 | 78, 170 | syl 17 |
. . . . . . . . . . . . . . . . . . . 20
⊢ (𝜑 → (𝑁 + 1) ∈
(ℤ≥‘𝐾)) |
172 | 116 | zcnd 11812 |
. . . . . . . . . . . . . . . . . . . . . 22
⊢ (𝜑 → 𝐾 ∈ ℂ) |
173 | | npcan 10612 |
. . . . . . . . . . . . . . . . . . . . . 22
⊢ ((𝐾 ∈ ℂ ∧ 1 ∈
ℂ) → ((𝐾 −
1) + 1) = 𝐾) |
174 | 172, 165,
173 | sylancl 582 |
. . . . . . . . . . . . . . . . . . . . 21
⊢ (𝜑 → ((𝐾 − 1) + 1) = 𝐾) |
175 | 174 | fveq2d 6438 |
. . . . . . . . . . . . . . . . . . . 20
⊢ (𝜑 →
(ℤ≥‘((𝐾 − 1) + 1)) =
(ℤ≥‘𝐾)) |
176 | 171, 175 | eleqtrrd 2910 |
. . . . . . . . . . . . . . . . . . 19
⊢ (𝜑 → (𝑁 + 1) ∈
(ℤ≥‘((𝐾 − 1) + 1))) |
177 | | eluzp1m1 11993 |
. . . . . . . . . . . . . . . . . . 19
⊢ (((𝐾 − 1) ∈ ℤ ∧
(𝑁 + 1) ∈
(ℤ≥‘((𝐾 − 1) + 1))) → ((𝑁 + 1) − 1) ∈
(ℤ≥‘(𝐾 − 1))) |
178 | 169, 176,
177 | syl2anc 581 |
. . . . . . . . . . . . . . . . . 18
⊢ (𝜑 → ((𝑁 + 1) − 1) ∈
(ℤ≥‘(𝐾 − 1))) |
179 | 167, 178 | eqeltrrd 2908 |
. . . . . . . . . . . . . . . . 17
⊢ (𝜑 → 𝑁 ∈ (ℤ≥‘(𝐾 − 1))) |
180 | | fzss2 12675 |
. . . . . . . . . . . . . . . . 17
⊢ (𝑁 ∈
(ℤ≥‘(𝐾 − 1)) → (𝑀...(𝐾 − 1)) ⊆ (𝑀...𝑁)) |
181 | 179, 180 | syl 17 |
. . . . . . . . . . . . . . . 16
⊢ (𝜑 → (𝑀...(𝐾 − 1)) ⊆ (𝑀...𝑁)) |
182 | 181 | sselda 3828 |
. . . . . . . . . . . . . . 15
⊢ ((𝜑 ∧ 𝑥 ∈ (𝑀...(𝐾 − 1))) → 𝑥 ∈ (𝑀...𝑁)) |
183 | 182, 102 | syldan 587 |
. . . . . . . . . . . . . 14
⊢ ((𝜑 ∧ 𝑥 ∈ (𝑀...(𝐾 − 1))) → (𝐽‘𝑥) ∈ (𝑀...𝑁)) |
184 | 183, 104 | syl 17 |
. . . . . . . . . . . . 13
⊢ ((𝜑 ∧ 𝑥 ∈ (𝑀...(𝐾 − 1))) → ((𝐺 ↾ (𝑀...𝑁))‘(𝐽‘𝑥)) = (𝐺‘(𝐽‘𝑥))) |
185 | 182, 112 | syl 17 |
. . . . . . . . . . . . . . 15
⊢ ((𝜑 ∧ 𝑥 ∈ (𝑀...(𝐾 − 1))) → (𝐽‘𝑥) = (𝐹‘if(𝑥 < 𝐾, 𝑥, (𝑥 + 1)))) |
186 | | elfzm11 12706 |
. . . . . . . . . . . . . . . . . . 19
⊢ ((𝑀 ∈ ℤ ∧ 𝐾 ∈ ℤ) → (𝑥 ∈ (𝑀...(𝐾 − 1)) ↔ (𝑥 ∈ ℤ ∧ 𝑀 ≤ 𝑥 ∧ 𝑥 < 𝐾))) |
187 | 158, 116,
186 | syl2anc 581 |
. . . . . . . . . . . . . . . . . 18
⊢ (𝜑 → (𝑥 ∈ (𝑀...(𝐾 − 1)) ↔ (𝑥 ∈ ℤ ∧ 𝑀 ≤ 𝑥 ∧ 𝑥 < 𝐾))) |
188 | 187 | biimpa 470 |
. . . . . . . . . . . . . . . . 17
⊢ ((𝜑 ∧ 𝑥 ∈ (𝑀...(𝐾 − 1))) → (𝑥 ∈ ℤ ∧ 𝑀 ≤ 𝑥 ∧ 𝑥 < 𝐾)) |
189 | 188 | simp3d 1180 |
. . . . . . . . . . . . . . . 16
⊢ ((𝜑 ∧ 𝑥 ∈ (𝑀...(𝐾 − 1))) → 𝑥 < 𝐾) |
190 | | iftrue 4313 |
. . . . . . . . . . . . . . . . 17
⊢ (𝑥 < 𝐾 → if(𝑥 < 𝐾, 𝑥, (𝑥 + 1)) = 𝑥) |
191 | 190 | fveq2d 6438 |
. . . . . . . . . . . . . . . 16
⊢ (𝑥 < 𝐾 → (𝐹‘if(𝑥 < 𝐾, 𝑥, (𝑥 + 1))) = (𝐹‘𝑥)) |
192 | 189, 191 | syl 17 |
. . . . . . . . . . . . . . 15
⊢ ((𝜑 ∧ 𝑥 ∈ (𝑀...(𝐾 − 1))) → (𝐹‘if(𝑥 < 𝐾, 𝑥, (𝑥 + 1))) = (𝐹‘𝑥)) |
193 | 185, 192 | eqtrd 2862 |
. . . . . . . . . . . . . 14
⊢ ((𝜑 ∧ 𝑥 ∈ (𝑀...(𝐾 − 1))) → (𝐽‘𝑥) = (𝐹‘𝑥)) |
194 | 193 | fveq2d 6438 |
. . . . . . . . . . . . 13
⊢ ((𝜑 ∧ 𝑥 ∈ (𝑀...(𝐾 − 1))) → (𝐺‘(𝐽‘𝑥)) = (𝐺‘(𝐹‘𝑥))) |
195 | 184, 194 | eqtr2d 2863 |
. . . . . . . . . . . 12
⊢ ((𝜑 ∧ 𝑥 ∈ (𝑀...(𝐾 − 1))) → (𝐺‘(𝐹‘𝑥)) = ((𝐺 ↾ (𝑀...𝑁))‘(𝐽‘𝑥))) |
196 | | peano2uz 12024 |
. . . . . . . . . . . . . . 15
⊢ (𝑁 ∈
(ℤ≥‘(𝐾 − 1)) → (𝑁 + 1) ∈
(ℤ≥‘(𝐾 − 1))) |
197 | | fzss2 12675 |
. . . . . . . . . . . . . . 15
⊢ ((𝑁 + 1) ∈
(ℤ≥‘(𝐾 − 1)) → (𝑀...(𝐾 − 1)) ⊆ (𝑀...(𝑁 + 1))) |
198 | 179, 196,
197 | 3syl 18 |
. . . . . . . . . . . . . 14
⊢ (𝜑 → (𝑀...(𝐾 − 1)) ⊆ (𝑀...(𝑁 + 1))) |
199 | 198 | sselda 3828 |
. . . . . . . . . . . . 13
⊢ ((𝜑 ∧ 𝑥 ∈ (𝑀...(𝐾 − 1))) → 𝑥 ∈ (𝑀...(𝑁 + 1))) |
200 | | fvco3 6523 |
. . . . . . . . . . . . . 14
⊢ ((𝐹:(𝑀...(𝑁 + 1))⟶(𝑀...(𝑁 + 1)) ∧ 𝑥 ∈ (𝑀...(𝑁 + 1))) → ((𝐺 ∘ 𝐹)‘𝑥) = (𝐺‘(𝐹‘𝑥))) |
201 | 58, 200 | sylan 577 |
. . . . . . . . . . . . 13
⊢ ((𝜑 ∧ 𝑥 ∈ (𝑀...(𝑁 + 1))) → ((𝐺 ∘ 𝐹)‘𝑥) = (𝐺‘(𝐹‘𝑥))) |
202 | 199, 201 | syldan 587 |
. . . . . . . . . . . 12
⊢ ((𝜑 ∧ 𝑥 ∈ (𝑀...(𝐾 − 1))) → ((𝐺 ∘ 𝐹)‘𝑥) = (𝐺‘(𝐹‘𝑥))) |
203 | 182, 131 | syldan 587 |
. . . . . . . . . . . 12
⊢ ((𝜑 ∧ 𝑥 ∈ (𝑀...(𝐾 − 1))) → (((𝐺 ↾ (𝑀...𝑁)) ∘ 𝐽)‘𝑥) = ((𝐺 ↾ (𝑀...𝑁))‘(𝐽‘𝑥))) |
204 | 195, 202,
203 | 3eqtr4d 2872 |
. . . . . . . . . . 11
⊢ ((𝜑 ∧ 𝑥 ∈ (𝑀...(𝐾 − 1))) → ((𝐺 ∘ 𝐹)‘𝑥) = (((𝐺 ↾ (𝑀...𝑁)) ∘ 𝐽)‘𝑥)) |
205 | 204 | adantlr 708 |
. . . . . . . . . 10
⊢ (((𝜑 ∧ 𝐾 ∈ ((𝑀 + 1)...𝑁)) ∧ 𝑥 ∈ (𝑀...(𝐾 − 1))) → ((𝐺 ∘ 𝐹)‘𝑥) = (((𝐺 ↾ (𝑀...𝑁)) ∘ 𝐽)‘𝑥)) |
206 | 161, 205 | seqfveq 13120 |
. . . . . . . . 9
⊢ ((𝜑 ∧ 𝐾 ∈ ((𝑀 + 1)...𝑁)) → (seq𝑀( + , (𝐺 ∘ 𝐹))‘(𝐾 − 1)) = (seq𝑀( + , ((𝐺 ↾ (𝑀...𝑁)) ∘ 𝐽))‘(𝐾 − 1))) |
207 | | fzp1ss 12686 |
. . . . . . . . . . . 12
⊢ (𝑀 ∈ ℤ → ((𝑀 + 1)...𝑁) ⊆ (𝑀...𝑁)) |
208 | 14, 157, 207 | 3syl 18 |
. . . . . . . . . . 11
⊢ (𝜑 → ((𝑀 + 1)...𝑁) ⊆ (𝑀...𝑁)) |
209 | 208 | sselda 3828 |
. . . . . . . . . 10
⊢ ((𝜑 ∧ 𝐾 ∈ ((𝑀 + 1)...𝑁)) → 𝐾 ∈ (𝑀...𝑁)) |
210 | 209, 151 | syldan 587 |
. . . . . . . . 9
⊢ ((𝜑 ∧ 𝐾 ∈ ((𝑀 + 1)...𝑁)) → (((𝐺 ∘ 𝐹)‘𝐾) + (seq(𝐾 + 1)( + , (𝐺 ∘ 𝐹))‘(𝑁 + 1))) = ((seq𝐾( + , ((𝐺 ↾ (𝑀...𝑁)) ∘ 𝐽))‘𝑁) + (𝐺‘(𝑁 + 1)))) |
211 | 206, 210 | oveq12d 6924 |
. . . . . . . 8
⊢ ((𝜑 ∧ 𝐾 ∈ ((𝑀 + 1)...𝑁)) → ((seq𝑀( + , (𝐺 ∘ 𝐹))‘(𝐾 − 1)) + (((𝐺 ∘ 𝐹)‘𝐾) + (seq(𝐾 + 1)( + , (𝐺 ∘ 𝐹))‘(𝑁 + 1)))) = ((seq𝑀( + , ((𝐺 ↾ (𝑀...𝑁)) ∘ 𝐽))‘(𝐾 − 1)) + ((seq𝐾( + , ((𝐺 ↾ (𝑀...𝑁)) ∘ 𝐽))‘𝑁) + (𝐺‘(𝑁 + 1))))) |
212 | 199, 62 | syldan 587 |
. . . . . . . . . . . 12
⊢ ((𝜑 ∧ 𝑥 ∈ (𝑀...(𝐾 − 1))) → ((𝐺 ∘ 𝐹)‘𝑥) ∈ 𝑆) |
213 | 212 | adantlr 708 |
. . . . . . . . . . 11
⊢ (((𝜑 ∧ 𝐾 ∈ ((𝑀 + 1)...𝑁)) ∧ 𝑥 ∈ (𝑀...(𝐾 − 1))) → ((𝐺 ∘ 𝐹)‘𝑥) ∈ 𝑆) |
214 | 11 | adantlr 708 |
. . . . . . . . . . 11
⊢ (((𝜑 ∧ 𝐾 ∈ ((𝑀 + 1)...𝑁)) ∧ (𝑥 ∈ 𝑆 ∧ 𝑦 ∈ 𝑆)) → (𝑥 + 𝑦) ∈ 𝑆) |
215 | 161, 213,
214 | seqcl 13116 |
. . . . . . . . . 10
⊢ ((𝜑 ∧ 𝐾 ∈ ((𝑀 + 1)...𝑁)) → (seq𝑀( + , (𝐺 ∘ 𝐹))‘(𝐾 − 1)) ∈ 𝑆) |
216 | 60, 78 | ffvelrnd 6610 |
. . . . . . . . . . . 12
⊢ (𝜑 → ((𝐺 ∘ 𝐹)‘𝐾) ∈ 𝐶) |
217 | 15, 216 | sseldd 3829 |
. . . . . . . . . . 11
⊢ (𝜑 → ((𝐺 ∘ 𝐹)‘𝐾) ∈ 𝑆) |
218 | 217 | adantr 474 |
. . . . . . . . . 10
⊢ ((𝜑 ∧ 𝐾 ∈ ((𝑀 + 1)...𝑁)) → ((𝐺 ∘ 𝐹)‘𝐾) ∈ 𝑆) |
219 | 95 | sselda 3828 |
. . . . . . . . . . . . 13
⊢ (((𝜑 ∧ 𝐾 ∈ (𝑀...𝑁)) ∧ 𝑥 ∈ ((𝐾 + 1)...(𝑁 + 1))) → 𝑥 ∈ (𝑀...(𝑁 + 1))) |
220 | 219, 63 | syldan 587 |
. . . . . . . . . . . 12
⊢ (((𝜑 ∧ 𝐾 ∈ (𝑀...𝑁)) ∧ 𝑥 ∈ ((𝐾 + 1)...(𝑁 + 1))) → ((𝐺 ∘ 𝐹)‘𝑥) ∈ 𝑆) |
221 | 54, 220, 49 | seqcl 13116 |
. . . . . . . . . . 11
⊢ ((𝜑 ∧ 𝐾 ∈ (𝑀...𝑁)) → (seq(𝐾 + 1)( + , (𝐺 ∘ 𝐹))‘(𝑁 + 1)) ∈ 𝑆) |
222 | 209, 221 | syldan 587 |
. . . . . . . . . 10
⊢ ((𝜑 ∧ 𝐾 ∈ ((𝑀 + 1)...𝑁)) → (seq(𝐾 + 1)( + , (𝐺 ∘ 𝐹))‘(𝑁 + 1)) ∈ 𝑆) |
223 | 215, 218,
222 | 3jca 1164 |
. . . . . . . . 9
⊢ ((𝜑 ∧ 𝐾 ∈ ((𝑀 + 1)...𝑁)) → ((seq𝑀( + , (𝐺 ∘ 𝐹))‘(𝐾 − 1)) ∈ 𝑆 ∧ ((𝐺 ∘ 𝐹)‘𝐾) ∈ 𝑆 ∧ (seq(𝐾 + 1)( + , (𝐺 ∘ 𝐹))‘(𝑁 + 1)) ∈ 𝑆)) |
224 | 13 | caovassg 7093 |
. . . . . . . . 9
⊢ ((𝜑 ∧ ((seq𝑀( + , (𝐺 ∘ 𝐹))‘(𝐾 − 1)) ∈ 𝑆 ∧ ((𝐺 ∘ 𝐹)‘𝐾) ∈ 𝑆 ∧ (seq(𝐾 + 1)( + , (𝐺 ∘ 𝐹))‘(𝑁 + 1)) ∈ 𝑆)) → (((seq𝑀( + , (𝐺 ∘ 𝐹))‘(𝐾 − 1)) + ((𝐺 ∘ 𝐹)‘𝐾)) + (seq(𝐾 + 1)( + , (𝐺 ∘ 𝐹))‘(𝑁 + 1))) = ((seq𝑀( + , (𝐺 ∘ 𝐹))‘(𝐾 − 1)) + (((𝐺 ∘ 𝐹)‘𝐾) + (seq(𝐾 + 1)( + , (𝐺 ∘ 𝐹))‘(𝑁 + 1))))) |
225 | 223, 224 | syldan 587 |
. . . . . . . 8
⊢ ((𝜑 ∧ 𝐾 ∈ ((𝑀 + 1)...𝑁)) → (((seq𝑀( + , (𝐺 ∘ 𝐹))‘(𝐾 − 1)) + ((𝐺 ∘ 𝐹)‘𝐾)) + (seq(𝐾 + 1)( + , (𝐺 ∘ 𝐹))‘(𝑁 + 1))) = ((seq𝑀( + , (𝐺 ∘ 𝐹))‘(𝐾 − 1)) + (((𝐺 ∘ 𝐹)‘𝐾) + (seq(𝐾 + 1)( + , (𝐺 ∘ 𝐹))‘(𝑁 + 1))))) |
226 | 1, 15 | fssd 6293 |
. . . . . . . . . . . . . . . 16
⊢ (𝜑 → 𝐺:(𝑀...(𝑁 + 1))⟶𝑆) |
227 | | fssres 6308 |
. . . . . . . . . . . . . . . 16
⊢ ((𝐺:(𝑀...(𝑁 + 1))⟶𝑆 ∧ (𝑀...𝑁) ⊆ (𝑀...(𝑁 + 1))) → (𝐺 ↾ (𝑀...𝑁)):(𝑀...𝑁)⟶𝑆) |
228 | 226, 3, 227 | sylancl 582 |
. . . . . . . . . . . . . . 15
⊢ (𝜑 → (𝐺 ↾ (𝑀...𝑁)):(𝑀...𝑁)⟶𝑆) |
229 | | fco 6296 |
. . . . . . . . . . . . . . 15
⊢ (((𝐺 ↾ (𝑀...𝑁)):(𝑀...𝑁)⟶𝑆 ∧ 𝐽:(𝑀...𝑁)⟶(𝑀...𝑁)) → ((𝐺 ↾ (𝑀...𝑁)) ∘ 𝐽):(𝑀...𝑁)⟶𝑆) |
230 | 228, 21, 229 | syl2anc 581 |
. . . . . . . . . . . . . 14
⊢ (𝜑 → ((𝐺 ↾ (𝑀...𝑁)) ∘ 𝐽):(𝑀...𝑁)⟶𝑆) |
231 | 230 | ffvelrnda 6609 |
. . . . . . . . . . . . 13
⊢ ((𝜑 ∧ 𝑥 ∈ (𝑀...𝑁)) → (((𝐺 ↾ (𝑀...𝑁)) ∘ 𝐽)‘𝑥) ∈ 𝑆) |
232 | 182, 231 | syldan 587 |
. . . . . . . . . . . 12
⊢ ((𝜑 ∧ 𝑥 ∈ (𝑀...(𝐾 − 1))) → (((𝐺 ↾ (𝑀...𝑁)) ∘ 𝐽)‘𝑥) ∈ 𝑆) |
233 | 232 | adantlr 708 |
. . . . . . . . . . 11
⊢ (((𝜑 ∧ 𝐾 ∈ ((𝑀 + 1)...𝑁)) ∧ 𝑥 ∈ (𝑀...(𝐾 − 1))) → (((𝐺 ↾ (𝑀...𝑁)) ∘ 𝐽)‘𝑥) ∈ 𝑆) |
234 | 161, 233,
214 | seqcl 13116 |
. . . . . . . . . 10
⊢ ((𝜑 ∧ 𝐾 ∈ ((𝑀 + 1)...𝑁)) → (seq𝑀( + , ((𝐺 ↾ (𝑀...𝑁)) ∘ 𝐽))‘(𝐾 − 1)) ∈ 𝑆) |
235 | | elfzuz3 12633 |
. . . . . . . . . . . 12
⊢ (𝐾 ∈ ((𝑀 + 1)...𝑁) → 𝑁 ∈ (ℤ≥‘𝐾)) |
236 | 235 | adantl 475 |
. . . . . . . . . . 11
⊢ ((𝜑 ∧ 𝐾 ∈ ((𝑀 + 1)...𝑁)) → 𝑁 ∈ (ℤ≥‘𝐾)) |
237 | 101, 231 | syldan 587 |
. . . . . . . . . . . 12
⊢ ((𝜑 ∧ 𝑥 ∈ (𝐾...𝑁)) → (((𝐺 ↾ (𝑀...𝑁)) ∘ 𝐽)‘𝑥) ∈ 𝑆) |
238 | 237 | adantlr 708 |
. . . . . . . . . . 11
⊢ (((𝜑 ∧ 𝐾 ∈ ((𝑀 + 1)...𝑁)) ∧ 𝑥 ∈ (𝐾...𝑁)) → (((𝐺 ↾ (𝑀...𝑁)) ∘ 𝐽)‘𝑥) ∈ 𝑆) |
239 | 236, 238,
214 | seqcl 13116 |
. . . . . . . . . 10
⊢ ((𝜑 ∧ 𝐾 ∈ ((𝑀 + 1)...𝑁)) → (seq𝐾( + , ((𝐺 ↾ (𝑀...𝑁)) ∘ 𝐽))‘𝑁) ∈ 𝑆) |
240 | 226, 76 | ffvelrnd 6610 |
. . . . . . . . . . 11
⊢ (𝜑 → (𝐺‘(𝑁 + 1)) ∈ 𝑆) |
241 | 240 | adantr 474 |
. . . . . . . . . 10
⊢ ((𝜑 ∧ 𝐾 ∈ ((𝑀 + 1)...𝑁)) → (𝐺‘(𝑁 + 1)) ∈ 𝑆) |
242 | 234, 239,
241 | 3jca 1164 |
. . . . . . . . 9
⊢ ((𝜑 ∧ 𝐾 ∈ ((𝑀 + 1)...𝑁)) → ((seq𝑀( + , ((𝐺 ↾ (𝑀...𝑁)) ∘ 𝐽))‘(𝐾 − 1)) ∈ 𝑆 ∧ (seq𝐾( + , ((𝐺 ↾ (𝑀...𝑁)) ∘ 𝐽))‘𝑁) ∈ 𝑆 ∧ (𝐺‘(𝑁 + 1)) ∈ 𝑆)) |
243 | 13 | caovassg 7093 |
. . . . . . . . 9
⊢ ((𝜑 ∧ ((seq𝑀( + , ((𝐺 ↾ (𝑀...𝑁)) ∘ 𝐽))‘(𝐾 − 1)) ∈ 𝑆 ∧ (seq𝐾( + , ((𝐺 ↾ (𝑀...𝑁)) ∘ 𝐽))‘𝑁) ∈ 𝑆 ∧ (𝐺‘(𝑁 + 1)) ∈ 𝑆)) → (((seq𝑀( + , ((𝐺 ↾ (𝑀...𝑁)) ∘ 𝐽))‘(𝐾 − 1)) + (seq𝐾( + , ((𝐺 ↾ (𝑀...𝑁)) ∘ 𝐽))‘𝑁)) + (𝐺‘(𝑁 + 1))) = ((seq𝑀( + , ((𝐺 ↾ (𝑀...𝑁)) ∘ 𝐽))‘(𝐾 − 1)) + ((seq𝐾( + , ((𝐺 ↾ (𝑀...𝑁)) ∘ 𝐽))‘𝑁) + (𝐺‘(𝑁 + 1))))) |
244 | 242, 243 | syldan 587 |
. . . . . . . 8
⊢ ((𝜑 ∧ 𝐾 ∈ ((𝑀 + 1)...𝑁)) → (((seq𝑀( + , ((𝐺 ↾ (𝑀...𝑁)) ∘ 𝐽))‘(𝐾 − 1)) + (seq𝐾( + , ((𝐺 ↾ (𝑀...𝑁)) ∘ 𝐽))‘𝑁)) + (𝐺‘(𝑁 + 1))) = ((seq𝑀( + , ((𝐺 ↾ (𝑀...𝑁)) ∘ 𝐽))‘(𝐾 − 1)) + ((seq𝐾( + , ((𝐺 ↾ (𝑀...𝑁)) ∘ 𝐽))‘𝑁) + (𝐺‘(𝑁 + 1))))) |
245 | 211, 225,
244 | 3eqtr4d 2872 |
. . . . . . 7
⊢ ((𝜑 ∧ 𝐾 ∈ ((𝑀 + 1)...𝑁)) → (((seq𝑀( + , (𝐺 ∘ 𝐹))‘(𝐾 − 1)) + ((𝐺 ∘ 𝐹)‘𝐾)) + (seq(𝐾 + 1)( + , (𝐺 ∘ 𝐹))‘(𝑁 + 1))) = (((seq𝑀( + , ((𝐺 ↾ (𝑀...𝑁)) ∘ 𝐽))‘(𝐾 − 1)) + (seq𝐾( + , ((𝐺 ↾ (𝑀...𝑁)) ∘ 𝐽))‘𝑁)) + (𝐺‘(𝑁 + 1)))) |
246 | | seqm1 13113 |
. . . . . . . . 9
⊢ ((𝑀 ∈ ℤ ∧ 𝐾 ∈
(ℤ≥‘(𝑀 + 1))) → (seq𝑀( + , (𝐺 ∘ 𝐹))‘𝐾) = ((seq𝑀( + , (𝐺 ∘ 𝐹))‘(𝐾 − 1)) + ((𝐺 ∘ 𝐹)‘𝐾))) |
247 | 158, 159,
246 | syl2an 591 |
. . . . . . . 8
⊢ ((𝜑 ∧ 𝐾 ∈ ((𝑀 + 1)...𝑁)) → (seq𝑀( + , (𝐺 ∘ 𝐹))‘𝐾) = ((seq𝑀( + , (𝐺 ∘ 𝐹))‘(𝐾 − 1)) + ((𝐺 ∘ 𝐹)‘𝐾))) |
248 | 247 | oveq1d 6921 |
. . . . . . 7
⊢ ((𝜑 ∧ 𝐾 ∈ ((𝑀 + 1)...𝑁)) → ((seq𝑀( + , (𝐺 ∘ 𝐹))‘𝐾) + (seq(𝐾 + 1)( + , (𝐺 ∘ 𝐹))‘(𝑁 + 1))) = (((seq𝑀( + , (𝐺 ∘ 𝐹))‘(𝐾 − 1)) + ((𝐺 ∘ 𝐹)‘𝐾)) + (seq(𝐾 + 1)( + , (𝐺 ∘ 𝐹))‘(𝑁 + 1)))) |
249 | 13 | adantlr 708 |
. . . . . . . . . 10
⊢ (((𝜑 ∧ 𝐾 ∈ ((𝑀 + 1)...𝑁)) ∧ (𝑥 ∈ 𝑆 ∧ 𝑦 ∈ 𝑆 ∧ 𝑧 ∈ 𝑆)) → ((𝑥 + 𝑦) + 𝑧) = (𝑥 + (𝑦 + 𝑧))) |
250 | | elfzelz 12636 |
. . . . . . . . . . . . . . 15
⊢ (𝐾 ∈ ((𝑀 + 1)...𝑁) → 𝐾 ∈ ℤ) |
251 | 250 | adantl 475 |
. . . . . . . . . . . . . 14
⊢ ((𝜑 ∧ 𝐾 ∈ ((𝑀 + 1)...𝑁)) → 𝐾 ∈ ℤ) |
252 | 251 | zcnd 11812 |
. . . . . . . . . . . . 13
⊢ ((𝜑 ∧ 𝐾 ∈ ((𝑀 + 1)...𝑁)) → 𝐾 ∈ ℂ) |
253 | 252, 165,
173 | sylancl 582 |
. . . . . . . . . . . 12
⊢ ((𝜑 ∧ 𝐾 ∈ ((𝑀 + 1)...𝑁)) → ((𝐾 − 1) + 1) = 𝐾) |
254 | 253 | fveq2d 6438 |
. . . . . . . . . . 11
⊢ ((𝜑 ∧ 𝐾 ∈ ((𝑀 + 1)...𝑁)) →
(ℤ≥‘((𝐾 − 1) + 1)) =
(ℤ≥‘𝐾)) |
255 | 236, 254 | eleqtrrd 2910 |
. . . . . . . . . 10
⊢ ((𝜑 ∧ 𝐾 ∈ ((𝑀 + 1)...𝑁)) → 𝑁 ∈
(ℤ≥‘((𝐾 − 1) + 1))) |
256 | 231 | adantlr 708 |
. . . . . . . . . 10
⊢ (((𝜑 ∧ 𝐾 ∈ ((𝑀 + 1)...𝑁)) ∧ 𝑥 ∈ (𝑀...𝑁)) → (((𝐺 ↾ (𝑀...𝑁)) ∘ 𝐽)‘𝑥) ∈ 𝑆) |
257 | 214, 249,
255, 161, 256 | seqsplit 13129 |
. . . . . . . . 9
⊢ ((𝜑 ∧ 𝐾 ∈ ((𝑀 + 1)...𝑁)) → (seq𝑀( + , ((𝐺 ↾ (𝑀...𝑁)) ∘ 𝐽))‘𝑁) = ((seq𝑀( + , ((𝐺 ↾ (𝑀...𝑁)) ∘ 𝐽))‘(𝐾 − 1)) + (seq((𝐾 − 1) + 1)( + , ((𝐺 ↾ (𝑀...𝑁)) ∘ 𝐽))‘𝑁))) |
258 | 253 | seqeq1d 13102 |
. . . . . . . . . . 11
⊢ ((𝜑 ∧ 𝐾 ∈ ((𝑀 + 1)...𝑁)) → seq((𝐾 − 1) + 1)( + , ((𝐺 ↾ (𝑀...𝑁)) ∘ 𝐽)) = seq𝐾( + , ((𝐺 ↾ (𝑀...𝑁)) ∘ 𝐽))) |
259 | 258 | fveq1d 6436 |
. . . . . . . . . 10
⊢ ((𝜑 ∧ 𝐾 ∈ ((𝑀 + 1)...𝑁)) → (seq((𝐾 − 1) + 1)( + , ((𝐺 ↾ (𝑀...𝑁)) ∘ 𝐽))‘𝑁) = (seq𝐾( + , ((𝐺 ↾ (𝑀...𝑁)) ∘ 𝐽))‘𝑁)) |
260 | 259 | oveq2d 6922 |
. . . . . . . . 9
⊢ ((𝜑 ∧ 𝐾 ∈ ((𝑀 + 1)...𝑁)) → ((seq𝑀( + , ((𝐺 ↾ (𝑀...𝑁)) ∘ 𝐽))‘(𝐾 − 1)) + (seq((𝐾 − 1) + 1)( + , ((𝐺 ↾ (𝑀...𝑁)) ∘ 𝐽))‘𝑁)) = ((seq𝑀( + , ((𝐺 ↾ (𝑀...𝑁)) ∘ 𝐽))‘(𝐾 − 1)) + (seq𝐾( + , ((𝐺 ↾ (𝑀...𝑁)) ∘ 𝐽))‘𝑁))) |
261 | 257, 260 | eqtrd 2862 |
. . . . . . . 8
⊢ ((𝜑 ∧ 𝐾 ∈ ((𝑀 + 1)...𝑁)) → (seq𝑀( + , ((𝐺 ↾ (𝑀...𝑁)) ∘ 𝐽))‘𝑁) = ((seq𝑀( + , ((𝐺 ↾ (𝑀...𝑁)) ∘ 𝐽))‘(𝐾 − 1)) + (seq𝐾( + , ((𝐺 ↾ (𝑀...𝑁)) ∘ 𝐽))‘𝑁))) |
262 | 261 | oveq1d 6921 |
. . . . . . 7
⊢ ((𝜑 ∧ 𝐾 ∈ ((𝑀 + 1)...𝑁)) → ((seq𝑀( + , ((𝐺 ↾ (𝑀...𝑁)) ∘ 𝐽))‘𝑁) + (𝐺‘(𝑁 + 1))) = (((seq𝑀( + , ((𝐺 ↾ (𝑀...𝑁)) ∘ 𝐽))‘(𝐾 − 1)) + (seq𝐾( + , ((𝐺 ↾ (𝑀...𝑁)) ∘ 𝐽))‘𝑁)) + (𝐺‘(𝑁 + 1)))) |
263 | 245, 248,
262 | 3eqtr4d 2872 |
. . . . . 6
⊢ ((𝜑 ∧ 𝐾 ∈ ((𝑀 + 1)...𝑁)) → ((seq𝑀( + , (𝐺 ∘ 𝐹))‘𝐾) + (seq(𝐾 + 1)( + , (𝐺 ∘ 𝐹))‘(𝑁 + 1))) = ((seq𝑀( + , ((𝐺 ↾ (𝑀...𝑁)) ∘ 𝐽))‘𝑁) + (𝐺‘(𝑁 + 1)))) |
264 | 156, 263 | jaodan 987 |
. . . . 5
⊢ ((𝜑 ∧ (𝐾 = 𝑀 ∨ 𝐾 ∈ ((𝑀 + 1)...𝑁))) → ((seq𝑀( + , (𝐺 ∘ 𝐹))‘𝐾) + (seq(𝐾 + 1)( + , (𝐺 ∘ 𝐹))‘(𝑁 + 1))) = ((seq𝑀( + , ((𝐺 ↾ (𝑀...𝑁)) ∘ 𝐽))‘𝑁) + (𝐺‘(𝑁 + 1)))) |
265 | 67, 264 | syldan 587 |
. . . 4
⊢ ((𝜑 ∧ 𝐾 ∈ (𝑀...𝑁)) → ((seq𝑀( + , (𝐺 ∘ 𝐹))‘𝐾) + (seq(𝐾 + 1)( + , (𝐺 ∘ 𝐹))‘(𝑁 + 1))) = ((seq𝑀( + , ((𝐺 ↾ (𝑀...𝑁)) ∘ 𝐽))‘𝑁) + (𝐺‘(𝑁 + 1)))) |
266 | 64, 265 | eqtrd 2862 |
. . 3
⊢ ((𝜑 ∧ 𝐾 ∈ (𝑀...𝑁)) → (seq𝑀( + , (𝐺 ∘ 𝐹))‘(𝑁 + 1)) = ((seq𝑀( + , ((𝐺 ↾ (𝑀...𝑁)) ∘ 𝐽))‘𝑁) + (𝐺‘(𝑁 + 1)))) |
267 | 14 | adantr 474 |
. . . . 5
⊢ ((𝜑 ∧ 𝐾 = (𝑁 + 1)) → 𝑁 ∈ (ℤ≥‘𝑀)) |
268 | | seqp1 13111 |
. . . . 5
⊢ (𝑁 ∈
(ℤ≥‘𝑀) → (seq𝑀( + , (𝐺 ∘ 𝐹))‘(𝑁 + 1)) = ((seq𝑀( + , (𝐺 ∘ 𝐹))‘𝑁) + ((𝐺 ∘ 𝐹)‘(𝑁 + 1)))) |
269 | 267, 268 | syl 17 |
. . . 4
⊢ ((𝜑 ∧ 𝐾 = (𝑁 + 1)) → (seq𝑀( + , (𝐺 ∘ 𝐹))‘(𝑁 + 1)) = ((seq𝑀( + , (𝐺 ∘ 𝐹))‘𝑁) + ((𝐺 ∘ 𝐹)‘(𝑁 + 1)))) |
270 | 112 | adantl 475 |
. . . . . . . . . 10
⊢ (((𝜑 ∧ 𝐾 = (𝑁 + 1)) ∧ 𝑥 ∈ (𝑀...𝑁)) → (𝐽‘𝑥) = (𝐹‘if(𝑥 < 𝐾, 𝑥, (𝑥 + 1)))) |
271 | | elfzelz 12636 |
. . . . . . . . . . . . . . . 16
⊢ (𝑥 ∈ (𝑀...𝑁) → 𝑥 ∈ ℤ) |
272 | 271 | zred 11811 |
. . . . . . . . . . . . . . 15
⊢ (𝑥 ∈ (𝑀...𝑁) → 𝑥 ∈ ℝ) |
273 | 272 | adantl 475 |
. . . . . . . . . . . . . 14
⊢ ((𝜑 ∧ 𝑥 ∈ (𝑀...𝑁)) → 𝑥 ∈ ℝ) |
274 | 163 | zred 11811 |
. . . . . . . . . . . . . . 15
⊢ (𝜑 → 𝑁 ∈ ℝ) |
275 | 274 | adantr 474 |
. . . . . . . . . . . . . 14
⊢ ((𝜑 ∧ 𝑥 ∈ (𝑀...𝑁)) → 𝑁 ∈ ℝ) |
276 | | peano2re 10529 |
. . . . . . . . . . . . . . 15
⊢ (𝑁 ∈ ℝ → (𝑁 + 1) ∈
ℝ) |
277 | 275, 276 | syl 17 |
. . . . . . . . . . . . . 14
⊢ ((𝜑 ∧ 𝑥 ∈ (𝑀...𝑁)) → (𝑁 + 1) ∈ ℝ) |
278 | | elfzle2 12639 |
. . . . . . . . . . . . . . 15
⊢ (𝑥 ∈ (𝑀...𝑁) → 𝑥 ≤ 𝑁) |
279 | 278 | adantl 475 |
. . . . . . . . . . . . . 14
⊢ ((𝜑 ∧ 𝑥 ∈ (𝑀...𝑁)) → 𝑥 ≤ 𝑁) |
280 | 275 | ltp1d 11285 |
. . . . . . . . . . . . . 14
⊢ ((𝜑 ∧ 𝑥 ∈ (𝑀...𝑁)) → 𝑁 < (𝑁 + 1)) |
281 | 273, 275,
277, 279, 280 | lelttrd 10515 |
. . . . . . . . . . . . 13
⊢ ((𝜑 ∧ 𝑥 ∈ (𝑀...𝑁)) → 𝑥 < (𝑁 + 1)) |
282 | 281 | adantlr 708 |
. . . . . . . . . . . 12
⊢ (((𝜑 ∧ 𝐾 = (𝑁 + 1)) ∧ 𝑥 ∈ (𝑀...𝑁)) → 𝑥 < (𝑁 + 1)) |
283 | | simplr 787 |
. . . . . . . . . . . 12
⊢ (((𝜑 ∧ 𝐾 = (𝑁 + 1)) ∧ 𝑥 ∈ (𝑀...𝑁)) → 𝐾 = (𝑁 + 1)) |
284 | 282, 283 | breqtrrd 4902 |
. . . . . . . . . . 11
⊢ (((𝜑 ∧ 𝐾 = (𝑁 + 1)) ∧ 𝑥 ∈ (𝑀...𝑁)) → 𝑥 < 𝐾) |
285 | 284, 191 | syl 17 |
. . . . . . . . . 10
⊢ (((𝜑 ∧ 𝐾 = (𝑁 + 1)) ∧ 𝑥 ∈ (𝑀...𝑁)) → (𝐹‘if(𝑥 < 𝐾, 𝑥, (𝑥 + 1))) = (𝐹‘𝑥)) |
286 | 270, 285 | eqtrd 2862 |
. . . . . . . . 9
⊢ (((𝜑 ∧ 𝐾 = (𝑁 + 1)) ∧ 𝑥 ∈ (𝑀...𝑁)) → (𝐽‘𝑥) = (𝐹‘𝑥)) |
287 | 286 | fveq2d 6438 |
. . . . . . . 8
⊢ (((𝜑 ∧ 𝐾 = (𝑁 + 1)) ∧ 𝑥 ∈ (𝑀...𝑁)) → ((𝐺 ↾ (𝑀...𝑁))‘(𝐽‘𝑥)) = ((𝐺 ↾ (𝑀...𝑁))‘(𝐹‘𝑥))) |
288 | 272 | adantl 475 |
. . . . . . . . . . 11
⊢ (((𝜑 ∧ 𝐾 = (𝑁 + 1)) ∧ 𝑥 ∈ (𝑀...𝑁)) → 𝑥 ∈ ℝ) |
289 | 288, 284 | gtned 10492 |
. . . . . . . . . 10
⊢ (((𝜑 ∧ 𝐾 = (𝑁 + 1)) ∧ 𝑥 ∈ (𝑀...𝑁)) → 𝐾 ≠ 𝑥) |
290 | 58 | ad2antrr 719 |
. . . . . . . . . . . . . . 15
⊢ (((𝜑 ∧ 𝐾 = (𝑁 + 1)) ∧ 𝑥 ∈ (𝑀...𝑁)) → 𝐹:(𝑀...(𝑁 + 1))⟶(𝑀...(𝑁 + 1))) |
291 | | fzelp1 12687 |
. . . . . . . . . . . . . . . 16
⊢ (𝑥 ∈ (𝑀...𝑁) → 𝑥 ∈ (𝑀...(𝑁 + 1))) |
292 | 291 | adantl 475 |
. . . . . . . . . . . . . . 15
⊢ (((𝜑 ∧ 𝐾 = (𝑁 + 1)) ∧ 𝑥 ∈ (𝑀...𝑁)) → 𝑥 ∈ (𝑀...(𝑁 + 1))) |
293 | 290, 292 | ffvelrnd 6610 |
. . . . . . . . . . . . . 14
⊢ (((𝜑 ∧ 𝐾 = (𝑁 + 1)) ∧ 𝑥 ∈ (𝑀...𝑁)) → (𝐹‘𝑥) ∈ (𝑀...(𝑁 + 1))) |
294 | 14 | ad2antrr 719 |
. . . . . . . . . . . . . . 15
⊢ (((𝜑 ∧ 𝐾 = (𝑁 + 1)) ∧ 𝑥 ∈ (𝑀...𝑁)) → 𝑁 ∈ (ℤ≥‘𝑀)) |
295 | | elfzp1 12685 |
. . . . . . . . . . . . . . 15
⊢ (𝑁 ∈
(ℤ≥‘𝑀) → ((𝐹‘𝑥) ∈ (𝑀...(𝑁 + 1)) ↔ ((𝐹‘𝑥) ∈ (𝑀...𝑁) ∨ (𝐹‘𝑥) = (𝑁 + 1)))) |
296 | 294, 295 | syl 17 |
. . . . . . . . . . . . . 14
⊢ (((𝜑 ∧ 𝐾 = (𝑁 + 1)) ∧ 𝑥 ∈ (𝑀...𝑁)) → ((𝐹‘𝑥) ∈ (𝑀...(𝑁 + 1)) ↔ ((𝐹‘𝑥) ∈ (𝑀...𝑁) ∨ (𝐹‘𝑥) = (𝑁 + 1)))) |
297 | 293, 296 | mpbid 224 |
. . . . . . . . . . . . 13
⊢ (((𝜑 ∧ 𝐾 = (𝑁 + 1)) ∧ 𝑥 ∈ (𝑀...𝑁)) → ((𝐹‘𝑥) ∈ (𝑀...𝑁) ∨ (𝐹‘𝑥) = (𝑁 + 1))) |
298 | 297 | ord 897 |
. . . . . . . . . . . 12
⊢ (((𝜑 ∧ 𝐾 = (𝑁 + 1)) ∧ 𝑥 ∈ (𝑀...𝑁)) → (¬ (𝐹‘𝑥) ∈ (𝑀...𝑁) → (𝐹‘𝑥) = (𝑁 + 1))) |
299 | 16 | ad2antrr 719 |
. . . . . . . . . . . . . 14
⊢ (((𝜑 ∧ 𝐾 = (𝑁 + 1)) ∧ 𝑥 ∈ (𝑀...𝑁)) → 𝐹:(𝑀...(𝑁 + 1))–1-1-onto→(𝑀...(𝑁 + 1))) |
300 | | f1ocnvfv 6790 |
. . . . . . . . . . . . . 14
⊢ ((𝐹:(𝑀...(𝑁 + 1))–1-1-onto→(𝑀...(𝑁 + 1)) ∧ 𝑥 ∈ (𝑀...(𝑁 + 1))) → ((𝐹‘𝑥) = (𝑁 + 1) → (◡𝐹‘(𝑁 + 1)) = 𝑥)) |
301 | 299, 292,
300 | syl2anc 581 |
. . . . . . . . . . . . 13
⊢ (((𝜑 ∧ 𝐾 = (𝑁 + 1)) ∧ 𝑥 ∈ (𝑀...𝑁)) → ((𝐹‘𝑥) = (𝑁 + 1) → (◡𝐹‘(𝑁 + 1)) = 𝑥)) |
302 | 18 | eqeq1i 2831 |
. . . . . . . . . . . . 13
⊢ (𝐾 = 𝑥 ↔ (◡𝐹‘(𝑁 + 1)) = 𝑥) |
303 | 301, 302 | syl6ibr 244 |
. . . . . . . . . . . 12
⊢ (((𝜑 ∧ 𝐾 = (𝑁 + 1)) ∧ 𝑥 ∈ (𝑀...𝑁)) → ((𝐹‘𝑥) = (𝑁 + 1) → 𝐾 = 𝑥)) |
304 | 298, 303 | syld 47 |
. . . . . . . . . . 11
⊢ (((𝜑 ∧ 𝐾 = (𝑁 + 1)) ∧ 𝑥 ∈ (𝑀...𝑁)) → (¬ (𝐹‘𝑥) ∈ (𝑀...𝑁) → 𝐾 = 𝑥)) |
305 | 304 | necon1ad 3017 |
. . . . . . . . . 10
⊢ (((𝜑 ∧ 𝐾 = (𝑁 + 1)) ∧ 𝑥 ∈ (𝑀...𝑁)) → (𝐾 ≠ 𝑥 → (𝐹‘𝑥) ∈ (𝑀...𝑁))) |
306 | 289, 305 | mpd 15 |
. . . . . . . . 9
⊢ (((𝜑 ∧ 𝐾 = (𝑁 + 1)) ∧ 𝑥 ∈ (𝑀...𝑁)) → (𝐹‘𝑥) ∈ (𝑀...𝑁)) |
307 | | fvres 6453 |
. . . . . . . . 9
⊢ ((𝐹‘𝑥) ∈ (𝑀...𝑁) → ((𝐺 ↾ (𝑀...𝑁))‘(𝐹‘𝑥)) = (𝐺‘(𝐹‘𝑥))) |
308 | 306, 307 | syl 17 |
. . . . . . . 8
⊢ (((𝜑 ∧ 𝐾 = (𝑁 + 1)) ∧ 𝑥 ∈ (𝑀...𝑁)) → ((𝐺 ↾ (𝑀...𝑁))‘(𝐹‘𝑥)) = (𝐺‘(𝐹‘𝑥))) |
309 | 287, 308 | eqtr2d 2863 |
. . . . . . 7
⊢ (((𝜑 ∧ 𝐾 = (𝑁 + 1)) ∧ 𝑥 ∈ (𝑀...𝑁)) → (𝐺‘(𝐹‘𝑥)) = ((𝐺 ↾ (𝑀...𝑁))‘(𝐽‘𝑥))) |
310 | 58, 291, 200 | syl2an 591 |
. . . . . . . 8
⊢ ((𝜑 ∧ 𝑥 ∈ (𝑀...𝑁)) → ((𝐺 ∘ 𝐹)‘𝑥) = (𝐺‘(𝐹‘𝑥))) |
311 | 310 | adantlr 708 |
. . . . . . 7
⊢ (((𝜑 ∧ 𝐾 = (𝑁 + 1)) ∧ 𝑥 ∈ (𝑀...𝑁)) → ((𝐺 ∘ 𝐹)‘𝑥) = (𝐺‘(𝐹‘𝑥))) |
312 | 131 | adantlr 708 |
. . . . . . 7
⊢ (((𝜑 ∧ 𝐾 = (𝑁 + 1)) ∧ 𝑥 ∈ (𝑀...𝑁)) → (((𝐺 ↾ (𝑀...𝑁)) ∘ 𝐽)‘𝑥) = ((𝐺 ↾ (𝑀...𝑁))‘(𝐽‘𝑥))) |
313 | 309, 311,
312 | 3eqtr4d 2872 |
. . . . . 6
⊢ (((𝜑 ∧ 𝐾 = (𝑁 + 1)) ∧ 𝑥 ∈ (𝑀...𝑁)) → ((𝐺 ∘ 𝐹)‘𝑥) = (((𝐺 ↾ (𝑀...𝑁)) ∘ 𝐽)‘𝑥)) |
314 | 267, 313 | seqfveq 13120 |
. . . . 5
⊢ ((𝜑 ∧ 𝐾 = (𝑁 + 1)) → (seq𝑀( + , (𝐺 ∘ 𝐹))‘𝑁) = (seq𝑀( + , ((𝐺 ↾ (𝑀...𝑁)) ∘ 𝐽))‘𝑁)) |
315 | | fvco3 6523 |
. . . . . . . 8
⊢ ((𝐹:(𝑀...(𝑁 + 1))⟶(𝑀...(𝑁 + 1)) ∧ (𝑁 + 1) ∈ (𝑀...(𝑁 + 1))) → ((𝐺 ∘ 𝐹)‘(𝑁 + 1)) = (𝐺‘(𝐹‘(𝑁 + 1)))) |
316 | 58, 76, 315 | syl2anc 581 |
. . . . . . 7
⊢ (𝜑 → ((𝐺 ∘ 𝐹)‘(𝑁 + 1)) = (𝐺‘(𝐹‘(𝑁 + 1)))) |
317 | 316 | adantr 474 |
. . . . . 6
⊢ ((𝜑 ∧ 𝐾 = (𝑁 + 1)) → ((𝐺 ∘ 𝐹)‘(𝑁 + 1)) = (𝐺‘(𝐹‘(𝑁 + 1)))) |
318 | | simpr 479 |
. . . . . . . . . 10
⊢ ((𝜑 ∧ 𝐾 = (𝑁 + 1)) → 𝐾 = (𝑁 + 1)) |
319 | 18, 318 | syl5eqr 2876 |
. . . . . . . . 9
⊢ ((𝜑 ∧ 𝐾 = (𝑁 + 1)) → (◡𝐹‘(𝑁 + 1)) = (𝑁 + 1)) |
320 | 319 | fveq2d 6438 |
. . . . . . . 8
⊢ ((𝜑 ∧ 𝐾 = (𝑁 + 1)) → (𝐹‘(◡𝐹‘(𝑁 + 1))) = (𝐹‘(𝑁 + 1))) |
321 | 145 | adantr 474 |
. . . . . . . 8
⊢ ((𝜑 ∧ 𝐾 = (𝑁 + 1)) → (𝐹‘(◡𝐹‘(𝑁 + 1))) = (𝑁 + 1)) |
322 | 320, 321 | eqtr3d 2864 |
. . . . . . 7
⊢ ((𝜑 ∧ 𝐾 = (𝑁 + 1)) → (𝐹‘(𝑁 + 1)) = (𝑁 + 1)) |
323 | 322 | fveq2d 6438 |
. . . . . 6
⊢ ((𝜑 ∧ 𝐾 = (𝑁 + 1)) → (𝐺‘(𝐹‘(𝑁 + 1))) = (𝐺‘(𝑁 + 1))) |
324 | 317, 323 | eqtrd 2862 |
. . . . 5
⊢ ((𝜑 ∧ 𝐾 = (𝑁 + 1)) → ((𝐺 ∘ 𝐹)‘(𝑁 + 1)) = (𝐺‘(𝑁 + 1))) |
325 | 314, 324 | oveq12d 6924 |
. . . 4
⊢ ((𝜑 ∧ 𝐾 = (𝑁 + 1)) → ((seq𝑀( + , (𝐺 ∘ 𝐹))‘𝑁) + ((𝐺 ∘ 𝐹)‘(𝑁 + 1))) = ((seq𝑀( + , ((𝐺 ↾ (𝑀...𝑁)) ∘ 𝐽))‘𝑁) + (𝐺‘(𝑁 + 1)))) |
326 | 269, 325 | eqtrd 2862 |
. . 3
⊢ ((𝜑 ∧ 𝐾 = (𝑁 + 1)) → (seq𝑀( + , (𝐺 ∘ 𝐹))‘(𝑁 + 1)) = ((seq𝑀( + , ((𝐺 ↾ (𝑀...𝑁)) ∘ 𝐽))‘𝑁) + (𝐺‘(𝑁 + 1)))) |
327 | | elfzp1 12685 |
. . . . 5
⊢ (𝑁 ∈
(ℤ≥‘𝑀) → (𝐾 ∈ (𝑀...(𝑁 + 1)) ↔ (𝐾 ∈ (𝑀...𝑁) ∨ 𝐾 = (𝑁 + 1)))) |
328 | 14, 327 | syl 17 |
. . . 4
⊢ (𝜑 → (𝐾 ∈ (𝑀...(𝑁 + 1)) ↔ (𝐾 ∈ (𝑀...𝑁) ∨ 𝐾 = (𝑁 + 1)))) |
329 | 78, 328 | mpbid 224 |
. . 3
⊢ (𝜑 → (𝐾 ∈ (𝑀...𝑁) ∨ 𝐾 = (𝑁 + 1))) |
330 | 266, 326,
329 | mpjaodan 988 |
. 2
⊢ (𝜑 → (seq𝑀( + , (𝐺 ∘ 𝐹))‘(𝑁 + 1)) = ((seq𝑀( + , ((𝐺 ↾ (𝑀...𝑁)) ∘ 𝐽))‘𝑁) + (𝐺‘(𝑁 + 1)))) |
331 | | seqp1 13111 |
. . 3
⊢ (𝑁 ∈
(ℤ≥‘𝑀) → (seq𝑀( + , 𝐺)‘(𝑁 + 1)) = ((seq𝑀( + , 𝐺)‘𝑁) + (𝐺‘(𝑁 + 1)))) |
332 | 14, 331 | syl 17 |
. 2
⊢ (𝜑 → (seq𝑀( + , 𝐺)‘(𝑁 + 1)) = ((seq𝑀( + , 𝐺)‘𝑁) + (𝐺‘(𝑁 + 1)))) |
333 | 48, 330, 332 | 3eqtr4d 2872 |
1
⊢ (𝜑 → (seq𝑀( + , (𝐺 ∘ 𝐹))‘(𝑁 + 1)) = (seq𝑀( + , 𝐺)‘(𝑁 + 1))) |