Proof of Theorem seqf1oglem2
Step | Hyp | Ref
| Expression |
1 | | seqf1olem.6 |
. . . . . . . . . 10
⊢ (𝜑 → 𝐺:(𝑀...(𝑁 + 1))⟶𝐶) |
2 | 1 | ffnd 5396 |
. . . . . . . . 9
⊢ (𝜑 → 𝐺 Fn (𝑀...(𝑁 + 1))) |
3 | | fzssp1 10123 |
. . . . . . . . 9
⊢ (𝑀...𝑁) ⊆ (𝑀...(𝑁 + 1)) |
4 | | fnssres 5359 |
. . . . . . . . 9
⊢ ((𝐺 Fn (𝑀...(𝑁 + 1)) ∧ (𝑀...𝑁) ⊆ (𝑀...(𝑁 + 1))) → (𝐺 ↾ (𝑀...𝑁)) Fn (𝑀...𝑁)) |
5 | 2, 3, 4 | sylancl 413 |
. . . . . . . 8
⊢ (𝜑 → (𝐺 ↾ (𝑀...𝑁)) Fn (𝑀...𝑁)) |
6 | | seqf1o.4 |
. . . . . . . . . 10
⊢ (𝜑 → 𝑁 ∈ (ℤ≥‘𝑀)) |
7 | | eluzel2 9587 |
. . . . . . . . . 10
⊢ (𝑁 ∈
(ℤ≥‘𝑀) → 𝑀 ∈ ℤ) |
8 | 6, 7 | syl 14 |
. . . . . . . . 9
⊢ (𝜑 → 𝑀 ∈ ℤ) |
9 | | eluzelz 9591 |
. . . . . . . . . 10
⊢ (𝑁 ∈
(ℤ≥‘𝑀) → 𝑁 ∈ ℤ) |
10 | 6, 9 | syl 14 |
. . . . . . . . 9
⊢ (𝜑 → 𝑁 ∈ ℤ) |
11 | 8, 10 | fzfigd 10492 |
. . . . . . . 8
⊢ (𝜑 → (𝑀...𝑁) ∈ Fin) |
12 | | fnfi 6985 |
. . . . . . . 8
⊢ (((𝐺 ↾ (𝑀...𝑁)) Fn (𝑀...𝑁) ∧ (𝑀...𝑁) ∈ Fin) → (𝐺 ↾ (𝑀...𝑁)) ∈ Fin) |
13 | 5, 11, 12 | syl2anc 411 |
. . . . . . 7
⊢ (𝜑 → (𝐺 ↾ (𝑀...𝑁)) ∈ Fin) |
14 | 13 | elexd 2773 |
. . . . . 6
⊢ (𝜑 → (𝐺 ↾ (𝑀...𝑁)) ∈ V) |
15 | | seqf1o.1 |
. . . . . . . . 9
⊢ ((𝜑 ∧ (𝑥 ∈ 𝑆 ∧ 𝑦 ∈ 𝑆)) → (𝑥 + 𝑦) ∈ 𝑆) |
16 | | seqf1o.2 |
. . . . . . . . 9
⊢ ((𝜑 ∧ (𝑥 ∈ 𝐶 ∧ 𝑦 ∈ 𝐶)) → (𝑥 + 𝑦) = (𝑦 + 𝑥)) |
17 | | seqf1o.3 |
. . . . . . . . 9
⊢ ((𝜑 ∧ (𝑥 ∈ 𝑆 ∧ 𝑦 ∈ 𝑆 ∧ 𝑧 ∈ 𝑆)) → ((𝑥 + 𝑦) + 𝑧) = (𝑥 + (𝑦 + 𝑧))) |
18 | | seqf1o.5 |
. . . . . . . . 9
⊢ (𝜑 → 𝐶 ⊆ 𝑆) |
19 | | seqf1og.p |
. . . . . . . . 9
⊢ (𝜑 → + ∈ 𝑉) |
20 | | seqf1olem.5 |
. . . . . . . . 9
⊢ (𝜑 → 𝐹:(𝑀...(𝑁 + 1))–1-1-onto→(𝑀...(𝑁 + 1))) |
21 | | seqf1olem.7 |
. . . . . . . . 9
⊢ 𝐽 = (𝑘 ∈ (𝑀...𝑁) ↦ (𝐹‘if(𝑘 < 𝐾, 𝑘, (𝑘 + 1)))) |
22 | | seqf1olem.8 |
. . . . . . . . 9
⊢ 𝐾 = (◡𝐹‘(𝑁 + 1)) |
23 | 15, 16, 17, 6, 18, 19, 20, 1, 21, 22 | seqf1oglem1 10580 |
. . . . . . . 8
⊢ (𝜑 → 𝐽:(𝑀...𝑁)–1-1-onto→(𝑀...𝑁)) |
24 | | f1of 5492 |
. . . . . . . 8
⊢ (𝐽:(𝑀...𝑁)–1-1-onto→(𝑀...𝑁) → 𝐽:(𝑀...𝑁)⟶(𝑀...𝑁)) |
25 | 23, 24 | syl 14 |
. . . . . . 7
⊢ (𝜑 → 𝐽:(𝑀...𝑁)⟶(𝑀...𝑁)) |
26 | 25, 11 | fexd 5780 |
. . . . . 6
⊢ (𝜑 → 𝐽 ∈ V) |
27 | 14, 26 | jca 306 |
. . . . 5
⊢ (𝜑 → ((𝐺 ↾ (𝑀...𝑁)) ∈ V ∧ 𝐽 ∈ V)) |
28 | | 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))) |
29 | | fssres 5421 |
. . . . . . 7
⊢ ((𝐺:(𝑀...(𝑁 + 1))⟶𝐶 ∧ (𝑀...𝑁) ⊆ (𝑀...(𝑁 + 1))) → (𝐺 ↾ (𝑀...𝑁)):(𝑀...𝑁)⟶𝐶) |
30 | 1, 3, 29 | sylancl 413 |
. . . . . 6
⊢ (𝜑 → (𝐺 ↾ (𝑀...𝑁)):(𝑀...𝑁)⟶𝐶) |
31 | 23, 30 | jca 306 |
. . . . 5
⊢ (𝜑 → (𝐽:(𝑀...𝑁)–1-1-onto→(𝑀...𝑁) ∧ (𝐺 ↾ (𝑀...𝑁)):(𝑀...𝑁)⟶𝐶)) |
32 | | f1oeq1 5480 |
. . . . . . . 8
⊢ (𝑓 = 𝐽 → (𝑓:(𝑀...𝑁)–1-1-onto→(𝑀...𝑁) ↔ 𝐽:(𝑀...𝑁)–1-1-onto→(𝑀...𝑁))) |
33 | | feq1 5378 |
. . . . . . . 8
⊢ (𝑔 = (𝐺 ↾ (𝑀...𝑁)) → (𝑔:(𝑀...𝑁)⟶𝐶 ↔ (𝐺 ↾ (𝑀...𝑁)):(𝑀...𝑁)⟶𝐶)) |
34 | 32, 33 | bi2anan9r 607 |
. . . . . . 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))) |
35 | | coeq1 4813 |
. . . . . . . . . . 11
⊢ (𝑔 = (𝐺 ↾ (𝑀...𝑁)) → (𝑔 ∘ 𝑓) = ((𝐺 ↾ (𝑀...𝑁)) ∘ 𝑓)) |
36 | | coeq2 4814 |
. . . . . . . . . . 11
⊢ (𝑓 = 𝐽 → ((𝐺 ↾ (𝑀...𝑁)) ∘ 𝑓) = ((𝐺 ↾ (𝑀...𝑁)) ∘ 𝐽)) |
37 | 35, 36 | sylan9eq 2246 |
. . . . . . . . . 10
⊢ ((𝑔 = (𝐺 ↾ (𝑀...𝑁)) ∧ 𝑓 = 𝐽) → (𝑔 ∘ 𝑓) = ((𝐺 ↾ (𝑀...𝑁)) ∘ 𝐽)) |
38 | 37 | seqeq3d 10516 |
. . . . . . . . 9
⊢ ((𝑔 = (𝐺 ↾ (𝑀...𝑁)) ∧ 𝑓 = 𝐽) → seq𝑀( + , (𝑔 ∘ 𝑓)) = seq𝑀( + , ((𝐺 ↾ (𝑀...𝑁)) ∘ 𝐽))) |
39 | 38 | fveq1d 5548 |
. . . . . . . 8
⊢ ((𝑔 = (𝐺 ↾ (𝑀...𝑁)) ∧ 𝑓 = 𝐽) → (seq𝑀( + , (𝑔 ∘ 𝑓))‘𝑁) = (seq𝑀( + , ((𝐺 ↾ (𝑀...𝑁)) ∘ 𝐽))‘𝑁)) |
40 | | simpl 109 |
. . . . . . . . . 10
⊢ ((𝑔 = (𝐺 ↾ (𝑀...𝑁)) ∧ 𝑓 = 𝐽) → 𝑔 = (𝐺 ↾ (𝑀...𝑁))) |
41 | 40 | seqeq3d 10516 |
. . . . . . . . 9
⊢ ((𝑔 = (𝐺 ↾ (𝑀...𝑁)) ∧ 𝑓 = 𝐽) → seq𝑀( + , 𝑔) = seq𝑀( + , (𝐺 ↾ (𝑀...𝑁)))) |
42 | 41 | fveq1d 5548 |
. . . . . . . 8
⊢ ((𝑔 = (𝐺 ↾ (𝑀...𝑁)) ∧ 𝑓 = 𝐽) → (seq𝑀( + , 𝑔)‘𝑁) = (seq𝑀( + , (𝐺 ↾ (𝑀...𝑁)))‘𝑁)) |
43 | 39, 42 | eqeq12d 2208 |
. . . . . . 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))) |
44 | 34, 43 | imbi12d 234 |
. . . . . 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)))) |
45 | 44 | spc2gv 2851 |
. . . . 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)))) |
46 | 27, 28, 31, 45 | syl3c 63 |
. . . 4
⊢ (𝜑 → (seq𝑀( + , ((𝐺 ↾ (𝑀...𝑁)) ∘ 𝐽))‘𝑁) = (seq𝑀( + , (𝐺 ↾ (𝑀...𝑁)))‘𝑁)) |
47 | | fvres 5570 |
. . . . . 6
⊢ (𝑥 ∈ (𝑀...𝑁) → ((𝐺 ↾ (𝑀...𝑁))‘𝑥) = (𝐺‘𝑥)) |
48 | 47 | adantl 277 |
. . . . 5
⊢ ((𝜑 ∧ 𝑥 ∈ (𝑀...𝑁)) → ((𝐺 ↾ (𝑀...𝑁))‘𝑥) = (𝐺‘𝑥)) |
49 | 10 | peano2zd 9432 |
. . . . . . . 8
⊢ (𝜑 → (𝑁 + 1) ∈ ℤ) |
50 | 8, 49 | fzfigd 10492 |
. . . . . . 7
⊢ (𝜑 → (𝑀...(𝑁 + 1)) ∈ Fin) |
51 | 1, 50 | fexd 5780 |
. . . . . 6
⊢ (𝜑 → 𝐺 ∈ V) |
52 | | resexg 4976 |
. . . . . 6
⊢ (𝐺 ∈ V → (𝐺 ↾ (𝑀...𝑁)) ∈ V) |
53 | 51, 52 | syl 14 |
. . . . 5
⊢ (𝜑 → (𝐺 ↾ (𝑀...𝑁)) ∈ V) |
54 | 6, 48, 19, 53, 51 | seqfveqg 10539 |
. . . 4
⊢ (𝜑 → (seq𝑀( + , (𝐺 ↾ (𝑀...𝑁)))‘𝑁) = (seq𝑀( + , 𝐺)‘𝑁)) |
55 | 46, 54 | eqtrd 2226 |
. . 3
⊢ (𝜑 → (seq𝑀( + , ((𝐺 ↾ (𝑀...𝑁)) ∘ 𝐽))‘𝑁) = (seq𝑀( + , 𝐺)‘𝑁)) |
56 | 55 | oveq1d 5925 |
. 2
⊢ (𝜑 → ((seq𝑀( + , ((𝐺 ↾ (𝑀...𝑁)) ∘ 𝐽))‘𝑁) + (𝐺‘(𝑁 + 1))) = ((seq𝑀( + , 𝐺)‘𝑁) + (𝐺‘(𝑁 + 1)))) |
57 | 15 | adantlr 477 |
. . . . 5
⊢ (((𝜑 ∧ 𝐾 ∈ (𝑀...𝑁)) ∧ (𝑥 ∈ 𝑆 ∧ 𝑦 ∈ 𝑆)) → (𝑥 + 𝑦) ∈ 𝑆) |
58 | 17 | adantlr 477 |
. . . . 5
⊢ (((𝜑 ∧ 𝐾 ∈ (𝑀...𝑁)) ∧ (𝑥 ∈ 𝑆 ∧ 𝑦 ∈ 𝑆 ∧ 𝑧 ∈ 𝑆)) → ((𝑥 + 𝑦) + 𝑧) = (𝑥 + (𝑦 + 𝑧))) |
59 | | elfzuz3 10078 |
. . . . . . 7
⊢ (𝐾 ∈ (𝑀...𝑁) → 𝑁 ∈ (ℤ≥‘𝐾)) |
60 | 59 | adantl 277 |
. . . . . 6
⊢ ((𝜑 ∧ 𝐾 ∈ (𝑀...𝑁)) → 𝑁 ∈ (ℤ≥‘𝐾)) |
61 | | eluzp1p1 9608 |
. . . . . 6
⊢ (𝑁 ∈
(ℤ≥‘𝐾) → (𝑁 + 1) ∈
(ℤ≥‘(𝐾 + 1))) |
62 | 60, 61 | syl 14 |
. . . . 5
⊢ ((𝜑 ∧ 𝐾 ∈ (𝑀...𝑁)) → (𝑁 + 1) ∈
(ℤ≥‘(𝐾 + 1))) |
63 | 19 | adantr 276 |
. . . . 5
⊢ ((𝜑 ∧ 𝐾 ∈ (𝑀...𝑁)) → + ∈ 𝑉) |
64 | 51 | adantr 276 |
. . . . . 6
⊢ ((𝜑 ∧ 𝐾 ∈ (𝑀...𝑁)) → 𝐺 ∈ V) |
65 | | f1of 5492 |
. . . . . . . . 9
⊢ (𝐹:(𝑀...(𝑁 + 1))–1-1-onto→(𝑀...(𝑁 + 1)) → 𝐹:(𝑀...(𝑁 + 1))⟶(𝑀...(𝑁 + 1))) |
66 | 20, 65 | syl 14 |
. . . . . . . 8
⊢ (𝜑 → 𝐹:(𝑀...(𝑁 + 1))⟶(𝑀...(𝑁 + 1))) |
67 | 66, 50 | fexd 5780 |
. . . . . . 7
⊢ (𝜑 → 𝐹 ∈ V) |
68 | 67 | adantr 276 |
. . . . . 6
⊢ ((𝜑 ∧ 𝐾 ∈ (𝑀...𝑁)) → 𝐹 ∈ V) |
69 | | coexg 5202 |
. . . . . 6
⊢ ((𝐺 ∈ V ∧ 𝐹 ∈ V) → (𝐺 ∘ 𝐹) ∈ V) |
70 | 64, 68, 69 | syl2anc 411 |
. . . . 5
⊢ ((𝜑 ∧ 𝐾 ∈ (𝑀...𝑁)) → (𝐺 ∘ 𝐹) ∈ V) |
71 | | elfzuz 10077 |
. . . . . 6
⊢ (𝐾 ∈ (𝑀...𝑁) → 𝐾 ∈ (ℤ≥‘𝑀)) |
72 | 71 | adantl 277 |
. . . . 5
⊢ ((𝜑 ∧ 𝐾 ∈ (𝑀...𝑁)) → 𝐾 ∈ (ℤ≥‘𝑀)) |
73 | | fco 5411 |
. . . . . . . . 9
⊢ ((𝐺:(𝑀...(𝑁 + 1))⟶𝐶 ∧ 𝐹:(𝑀...(𝑁 + 1))⟶(𝑀...(𝑁 + 1))) → (𝐺 ∘ 𝐹):(𝑀...(𝑁 + 1))⟶𝐶) |
74 | 1, 66, 73 | syl2anc 411 |
. . . . . . . 8
⊢ (𝜑 → (𝐺 ∘ 𝐹):(𝑀...(𝑁 + 1))⟶𝐶) |
75 | 74, 18 | fssd 5408 |
. . . . . . 7
⊢ (𝜑 → (𝐺 ∘ 𝐹):(𝑀...(𝑁 + 1))⟶𝑆) |
76 | 75 | ffvelcdmda 5685 |
. . . . . 6
⊢ ((𝜑 ∧ 𝑥 ∈ (𝑀...(𝑁 + 1))) → ((𝐺 ∘ 𝐹)‘𝑥) ∈ 𝑆) |
77 | 76 | adantlr 477 |
. . . . 5
⊢ (((𝜑 ∧ 𝐾 ∈ (𝑀...𝑁)) ∧ 𝑥 ∈ (𝑀...(𝑁 + 1))) → ((𝐺 ∘ 𝐹)‘𝑥) ∈ 𝑆) |
78 | 57, 58, 62, 63, 70, 72, 77 | seqsplitg 10550 |
. . . 4
⊢ ((𝜑 ∧ 𝐾 ∈ (𝑀...𝑁)) → (seq𝑀( + , (𝐺 ∘ 𝐹))‘(𝑁 + 1)) = ((seq𝑀( + , (𝐺 ∘ 𝐹))‘𝐾) + (seq(𝐾 + 1)( + , (𝐺 ∘ 𝐹))‘(𝑁 + 1)))) |
79 | | elfzp12 10155 |
. . . . . . 7
⊢ (𝑁 ∈
(ℤ≥‘𝑀) → (𝐾 ∈ (𝑀...𝑁) ↔ (𝐾 = 𝑀 ∨ 𝐾 ∈ ((𝑀 + 1)...𝑁)))) |
80 | 79 | biimpa 296 |
. . . . . 6
⊢ ((𝑁 ∈
(ℤ≥‘𝑀) ∧ 𝐾 ∈ (𝑀...𝑁)) → (𝐾 = 𝑀 ∨ 𝐾 ∈ ((𝑀 + 1)...𝑁))) |
81 | 6, 80 | sylan 283 |
. . . . 5
⊢ ((𝜑 ∧ 𝐾 ∈ (𝑀...𝑁)) → (𝐾 = 𝑀 ∨ 𝐾 ∈ ((𝑀 + 1)...𝑁))) |
82 | | seqeq1 10511 |
. . . . . . . . . . 11
⊢ (𝐾 = 𝑀 → seq𝐾( + , (𝐺 ∘ 𝐹)) = seq𝑀( + , (𝐺 ∘ 𝐹))) |
83 | 82 | eqcomd 2199 |
. . . . . . . . . 10
⊢ (𝐾 = 𝑀 → seq𝑀( + , (𝐺 ∘ 𝐹)) = seq𝐾( + , (𝐺 ∘ 𝐹))) |
84 | 83 | fveq1d 5548 |
. . . . . . . . 9
⊢ (𝐾 = 𝑀 → (seq𝑀( + , (𝐺 ∘ 𝐹))‘𝐾) = (seq𝐾( + , (𝐺 ∘ 𝐹))‘𝐾)) |
85 | | f1ocnv 5505 |
. . . . . . . . . . . . . 14
⊢ (𝐹:(𝑀...(𝑁 + 1))–1-1-onto→(𝑀...(𝑁 + 1)) → ◡𝐹:(𝑀...(𝑁 + 1))–1-1-onto→(𝑀...(𝑁 + 1))) |
86 | | f1of 5492 |
. . . . . . . . . . . . . 14
⊢ (◡𝐹:(𝑀...(𝑁 + 1))–1-1-onto→(𝑀...(𝑁 + 1)) → ◡𝐹:(𝑀...(𝑁 + 1))⟶(𝑀...(𝑁 + 1))) |
87 | 20, 85, 86 | 3syl 17 |
. . . . . . . . . . . . 13
⊢ (𝜑 → ◡𝐹:(𝑀...(𝑁 + 1))⟶(𝑀...(𝑁 + 1))) |
88 | | peano2uz 9638 |
. . . . . . . . . . . . . 14
⊢ (𝑁 ∈
(ℤ≥‘𝑀) → (𝑁 + 1) ∈
(ℤ≥‘𝑀)) |
89 | | eluzfz2 10088 |
. . . . . . . . . . . . . 14
⊢ ((𝑁 + 1) ∈
(ℤ≥‘𝑀) → (𝑁 + 1) ∈ (𝑀...(𝑁 + 1))) |
90 | 6, 88, 89 | 3syl 17 |
. . . . . . . . . . . . 13
⊢ (𝜑 → (𝑁 + 1) ∈ (𝑀...(𝑁 + 1))) |
91 | 87, 90 | ffvelcdmd 5686 |
. . . . . . . . . . . 12
⊢ (𝜑 → (◡𝐹‘(𝑁 + 1)) ∈ (𝑀...(𝑁 + 1))) |
92 | 22, 91 | eqeltrid 2280 |
. . . . . . . . . . 11
⊢ (𝜑 → 𝐾 ∈ (𝑀...(𝑁 + 1))) |
93 | 92 | elfzelzd 10082 |
. . . . . . . . . 10
⊢ (𝜑 → 𝐾 ∈ ℤ) |
94 | 51, 67, 69 | syl2anc 411 |
. . . . . . . . . 10
⊢ (𝜑 → (𝐺 ∘ 𝐹) ∈ V) |
95 | | seq1g 10524 |
. . . . . . . . . 10
⊢ ((𝐾 ∈ ℤ ∧ (𝐺 ∘ 𝐹) ∈ V ∧ + ∈ 𝑉) → (seq𝐾( + , (𝐺 ∘ 𝐹))‘𝐾) = ((𝐺 ∘ 𝐹)‘𝐾)) |
96 | 93, 94, 19, 95 | syl3anc 1249 |
. . . . . . . . 9
⊢ (𝜑 → (seq𝐾( + , (𝐺 ∘ 𝐹))‘𝐾) = ((𝐺 ∘ 𝐹)‘𝐾)) |
97 | 84, 96 | sylan9eqr 2248 |
. . . . . . . 8
⊢ ((𝜑 ∧ 𝐾 = 𝑀) → (seq𝑀( + , (𝐺 ∘ 𝐹))‘𝐾) = ((𝐺 ∘ 𝐹)‘𝐾)) |
98 | 97 | oveq1d 5925 |
. . . . . . 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)))) |
99 | | simpr 110 |
. . . . . . . . 9
⊢ ((𝜑 ∧ 𝐾 = 𝑀) → 𝐾 = 𝑀) |
100 | | eluzfz1 10087 |
. . . . . . . . . . 11
⊢ (𝑁 ∈
(ℤ≥‘𝑀) → 𝑀 ∈ (𝑀...𝑁)) |
101 | 6, 100 | syl 14 |
. . . . . . . . . 10
⊢ (𝜑 → 𝑀 ∈ (𝑀...𝑁)) |
102 | 101 | adantr 276 |
. . . . . . . . 9
⊢ ((𝜑 ∧ 𝐾 = 𝑀) → 𝑀 ∈ (𝑀...𝑁)) |
103 | 99, 102 | eqeltrd 2270 |
. . . . . . . 8
⊢ ((𝜑 ∧ 𝐾 = 𝑀) → 𝐾 ∈ (𝑀...𝑁)) |
104 | 16 | adantlr 477 |
. . . . . . . . . 10
⊢ (((𝜑 ∧ 𝐾 ∈ (𝑀...𝑁)) ∧ (𝑥 ∈ 𝐶 ∧ 𝑦 ∈ 𝐶)) → (𝑥 + 𝑦) = (𝑦 + 𝑥)) |
105 | 18 | adantr 276 |
. . . . . . . . . 10
⊢ ((𝜑 ∧ 𝐾 ∈ (𝑀...𝑁)) → 𝐶 ⊆ 𝑆) |
106 | 74 | adantr 276 |
. . . . . . . . . 10
⊢ ((𝜑 ∧ 𝐾 ∈ (𝑀...𝑁)) → (𝐺 ∘ 𝐹):(𝑀...(𝑁 + 1))⟶𝐶) |
107 | 92 | adantr 276 |
. . . . . . . . . 10
⊢ ((𝜑 ∧ 𝐾 ∈ (𝑀...𝑁)) → 𝐾 ∈ (𝑀...(𝑁 + 1))) |
108 | | peano2uz 9638 |
. . . . . . . . . . 11
⊢ (𝐾 ∈
(ℤ≥‘𝑀) → (𝐾 + 1) ∈
(ℤ≥‘𝑀)) |
109 | | fzss1 10119 |
. . . . . . . . . . 11
⊢ ((𝐾 + 1) ∈
(ℤ≥‘𝑀) → ((𝐾 + 1)...(𝑁 + 1)) ⊆ (𝑀...(𝑁 + 1))) |
110 | 72, 108, 109 | 3syl 17 |
. . . . . . . . . 10
⊢ ((𝜑 ∧ 𝐾 ∈ (𝑀...𝑁)) → ((𝐾 + 1)...(𝑁 + 1)) ⊆ (𝑀...(𝑁 + 1))) |
111 | 50 | adantr 276 |
. . . . . . . . . 10
⊢ ((𝜑 ∧ 𝐾 ∈ (𝑀...𝑁)) → (𝑀...(𝑁 + 1)) ∈ Fin) |
112 | 57, 104, 58, 62, 105, 63, 106, 107, 110, 111 | seqf1oglem2a 10579 |
. . . . . . . . 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))) |
113 | | 1zzd 9334 |
. . . . . . . . . . 11
⊢ ((𝜑 ∧ 𝐾 ∈ (𝑀...𝑁)) → 1 ∈ ℤ) |
114 | | elfzuz 10077 |
. . . . . . . . . . . . . . . . . 18
⊢ (𝐾 ∈ (𝑀...(𝑁 + 1)) → 𝐾 ∈ (ℤ≥‘𝑀)) |
115 | | fzss1 10119 |
. . . . . . . . . . . . . . . . . 18
⊢ (𝐾 ∈
(ℤ≥‘𝑀) → (𝐾...𝑁) ⊆ (𝑀...𝑁)) |
116 | 92, 114, 115 | 3syl 17 |
. . . . . . . . . . . . . . . . 17
⊢ (𝜑 → (𝐾...𝑁) ⊆ (𝑀...𝑁)) |
117 | 116 | sselda 3179 |
. . . . . . . . . . . . . . . 16
⊢ ((𝜑 ∧ 𝑥 ∈ (𝐾...𝑁)) → 𝑥 ∈ (𝑀...𝑁)) |
118 | 25 | ffvelcdmda 5685 |
. . . . . . . . . . . . . . . 16
⊢ ((𝜑 ∧ 𝑥 ∈ (𝑀...𝑁)) → (𝐽‘𝑥) ∈ (𝑀...𝑁)) |
119 | 117, 118 | syldan 282 |
. . . . . . . . . . . . . . 15
⊢ ((𝜑 ∧ 𝑥 ∈ (𝐾...𝑁)) → (𝐽‘𝑥) ∈ (𝑀...𝑁)) |
120 | 119 | fvresd 5571 |
. . . . . . . . . . . . . 14
⊢ ((𝜑 ∧ 𝑥 ∈ (𝐾...𝑁)) → ((𝐺 ↾ (𝑀...𝑁))‘(𝐽‘𝑥)) = (𝐺‘(𝐽‘𝑥))) |
121 | | breq1 4032 |
. . . . . . . . . . . . . . . . . . 19
⊢ (𝑘 = 𝑥 → (𝑘 < 𝐾 ↔ 𝑥 < 𝐾)) |
122 | | id 19 |
. . . . . . . . . . . . . . . . . . 19
⊢ (𝑘 = 𝑥 → 𝑘 = 𝑥) |
123 | | oveq1 5917 |
. . . . . . . . . . . . . . . . . . 19
⊢ (𝑘 = 𝑥 → (𝑘 + 1) = (𝑥 + 1)) |
124 | 121, 122,
123 | ifbieq12d 3583 |
. . . . . . . . . . . . . . . . . 18
⊢ (𝑘 = 𝑥 → if(𝑘 < 𝐾, 𝑘, (𝑘 + 1)) = if(𝑥 < 𝐾, 𝑥, (𝑥 + 1))) |
125 | 124 | fveq2d 5550 |
. . . . . . . . . . . . . . . . 17
⊢ (𝑘 = 𝑥 → (𝐹‘if(𝑘 < 𝐾, 𝑘, (𝑘 + 1))) = (𝐹‘if(𝑥 < 𝐾, 𝑥, (𝑥 + 1)))) |
126 | 66 | adantr 276 |
. . . . . . . . . . . . . . . . . 18
⊢ ((𝜑 ∧ 𝑥 ∈ (𝐾...𝑁)) → 𝐹:(𝑀...(𝑁 + 1))⟶(𝑀...(𝑁 + 1))) |
127 | 3, 117 | sselid 3177 |
. . . . . . . . . . . . . . . . . . 19
⊢ ((𝜑 ∧ 𝑥 ∈ (𝐾...𝑁)) → 𝑥 ∈ (𝑀...(𝑁 + 1))) |
128 | | fzp1elp1 10131 |
. . . . . . . . . . . . . . . . . . . 20
⊢ (𝑥 ∈ (𝑀...𝑁) → (𝑥 + 1) ∈ (𝑀...(𝑁 + 1))) |
129 | 117, 128 | syl 14 |
. . . . . . . . . . . . . . . . . . 19
⊢ ((𝜑 ∧ 𝑥 ∈ (𝐾...𝑁)) → (𝑥 + 1) ∈ (𝑀...(𝑁 + 1))) |
130 | 117 | elfzelzd 10082 |
. . . . . . . . . . . . . . . . . . . 20
⊢ ((𝜑 ∧ 𝑥 ∈ (𝐾...𝑁)) → 𝑥 ∈ ℤ) |
131 | 93 | adantr 276 |
. . . . . . . . . . . . . . . . . . . 20
⊢ ((𝜑 ∧ 𝑥 ∈ (𝐾...𝑁)) → 𝐾 ∈ ℤ) |
132 | | zdclt 9384 |
. . . . . . . . . . . . . . . . . . . 20
⊢ ((𝑥 ∈ ℤ ∧ 𝐾 ∈ ℤ) →
DECID 𝑥 <
𝐾) |
133 | 130, 131,
132 | syl2anc 411 |
. . . . . . . . . . . . . . . . . . 19
⊢ ((𝜑 ∧ 𝑥 ∈ (𝐾...𝑁)) → DECID 𝑥 < 𝐾) |
134 | 127, 129,
133 | ifcldcd 3593 |
. . . . . . . . . . . . . . . . . 18
⊢ ((𝜑 ∧ 𝑥 ∈ (𝐾...𝑁)) → if(𝑥 < 𝐾, 𝑥, (𝑥 + 1)) ∈ (𝑀...(𝑁 + 1))) |
135 | 126, 134 | ffvelcdmd 5686 |
. . . . . . . . . . . . . . . . 17
⊢ ((𝜑 ∧ 𝑥 ∈ (𝐾...𝑁)) → (𝐹‘if(𝑥 < 𝐾, 𝑥, (𝑥 + 1))) ∈ (𝑀...(𝑁 + 1))) |
136 | 21, 125, 117, 135 | fvmptd3 5643 |
. . . . . . . . . . . . . . . 16
⊢ ((𝜑 ∧ 𝑥 ∈ (𝐾...𝑁)) → (𝐽‘𝑥) = (𝐹‘if(𝑥 < 𝐾, 𝑥, (𝑥 + 1)))) |
137 | 93 | zred 9429 |
. . . . . . . . . . . . . . . . . . 19
⊢ (𝜑 → 𝐾 ∈ ℝ) |
138 | 137 | adantr 276 |
. . . . . . . . . . . . . . . . . 18
⊢ ((𝜑 ∧ 𝑥 ∈ (𝐾...𝑁)) → 𝐾 ∈ ℝ) |
139 | | elfzelz 10081 |
. . . . . . . . . . . . . . . . . . . 20
⊢ (𝑥 ∈ (𝐾...𝑁) → 𝑥 ∈ ℤ) |
140 | 139 | adantl 277 |
. . . . . . . . . . . . . . . . . . 19
⊢ ((𝜑 ∧ 𝑥 ∈ (𝐾...𝑁)) → 𝑥 ∈ ℤ) |
141 | 140 | zred 9429 |
. . . . . . . . . . . . . . . . . 18
⊢ ((𝜑 ∧ 𝑥 ∈ (𝐾...𝑁)) → 𝑥 ∈ ℝ) |
142 | | elfzle1 10083 |
. . . . . . . . . . . . . . . . . . 19
⊢ (𝑥 ∈ (𝐾...𝑁) → 𝐾 ≤ 𝑥) |
143 | 142 | adantl 277 |
. . . . . . . . . . . . . . . . . 18
⊢ ((𝜑 ∧ 𝑥 ∈ (𝐾...𝑁)) → 𝐾 ≤ 𝑥) |
144 | 138, 141,
143 | lensymd 8131 |
. . . . . . . . . . . . . . . . 17
⊢ ((𝜑 ∧ 𝑥 ∈ (𝐾...𝑁)) → ¬ 𝑥 < 𝐾) |
145 | | iffalse 3565 |
. . . . . . . . . . . . . . . . . 18
⊢ (¬
𝑥 < 𝐾 → if(𝑥 < 𝐾, 𝑥, (𝑥 + 1)) = (𝑥 + 1)) |
146 | 145 | fveq2d 5550 |
. . . . . . . . . . . . . . . . 17
⊢ (¬
𝑥 < 𝐾 → (𝐹‘if(𝑥 < 𝐾, 𝑥, (𝑥 + 1))) = (𝐹‘(𝑥 + 1))) |
147 | 144, 146 | syl 14 |
. . . . . . . . . . . . . . . 16
⊢ ((𝜑 ∧ 𝑥 ∈ (𝐾...𝑁)) → (𝐹‘if(𝑥 < 𝐾, 𝑥, (𝑥 + 1))) = (𝐹‘(𝑥 + 1))) |
148 | 136, 147 | eqtrd 2226 |
. . . . . . . . . . . . . . 15
⊢ ((𝜑 ∧ 𝑥 ∈ (𝐾...𝑁)) → (𝐽‘𝑥) = (𝐹‘(𝑥 + 1))) |
149 | 148 | fveq2d 5550 |
. . . . . . . . . . . . . 14
⊢ ((𝜑 ∧ 𝑥 ∈ (𝐾...𝑁)) → (𝐺‘(𝐽‘𝑥)) = (𝐺‘(𝐹‘(𝑥 + 1)))) |
150 | 120, 149 | eqtrd 2226 |
. . . . . . . . . . . . 13
⊢ ((𝜑 ∧ 𝑥 ∈ (𝐾...𝑁)) → ((𝐺 ↾ (𝑀...𝑁))‘(𝐽‘𝑥)) = (𝐺‘(𝐹‘(𝑥 + 1)))) |
151 | | fvco3 5620 |
. . . . . . . . . . . . . . 15
⊢ ((𝐽:(𝑀...𝑁)⟶(𝑀...𝑁) ∧ 𝑥 ∈ (𝑀...𝑁)) → (((𝐺 ↾ (𝑀...𝑁)) ∘ 𝐽)‘𝑥) = ((𝐺 ↾ (𝑀...𝑁))‘(𝐽‘𝑥))) |
152 | 25, 151 | sylan 283 |
. . . . . . . . . . . . . 14
⊢ ((𝜑 ∧ 𝑥 ∈ (𝑀...𝑁)) → (((𝐺 ↾ (𝑀...𝑁)) ∘ 𝐽)‘𝑥) = ((𝐺 ↾ (𝑀...𝑁))‘(𝐽‘𝑥))) |
153 | 117, 152 | syldan 282 |
. . . . . . . . . . . . 13
⊢ ((𝜑 ∧ 𝑥 ∈ (𝐾...𝑁)) → (((𝐺 ↾ (𝑀...𝑁)) ∘ 𝐽)‘𝑥) = ((𝐺 ↾ (𝑀...𝑁))‘(𝐽‘𝑥))) |
154 | | fvco3 5620 |
. . . . . . . . . . . . . . 15
⊢ ((𝐹:(𝑀...(𝑁 + 1))⟶(𝑀...(𝑁 + 1)) ∧ (𝑥 + 1) ∈ (𝑀...(𝑁 + 1))) → ((𝐺 ∘ 𝐹)‘(𝑥 + 1)) = (𝐺‘(𝐹‘(𝑥 + 1)))) |
155 | 66, 154 | sylan 283 |
. . . . . . . . . . . . . 14
⊢ ((𝜑 ∧ (𝑥 + 1) ∈ (𝑀...(𝑁 + 1))) → ((𝐺 ∘ 𝐹)‘(𝑥 + 1)) = (𝐺‘(𝐹‘(𝑥 + 1)))) |
156 | 129, 155 | syldan 282 |
. . . . . . . . . . . . 13
⊢ ((𝜑 ∧ 𝑥 ∈ (𝐾...𝑁)) → ((𝐺 ∘ 𝐹)‘(𝑥 + 1)) = (𝐺‘(𝐹‘(𝑥 + 1)))) |
157 | 150, 153,
156 | 3eqtr4d 2236 |
. . . . . . . . . . . 12
⊢ ((𝜑 ∧ 𝑥 ∈ (𝐾...𝑁)) → (((𝐺 ↾ (𝑀...𝑁)) ∘ 𝐽)‘𝑥) = ((𝐺 ∘ 𝐹)‘(𝑥 + 1))) |
158 | 157 | adantlr 477 |
. . . . . . . . . . 11
⊢ (((𝜑 ∧ 𝐾 ∈ (𝑀...𝑁)) ∧ 𝑥 ∈ (𝐾...𝑁)) → (((𝐺 ↾ (𝑀...𝑁)) ∘ 𝐽)‘𝑥) = ((𝐺 ∘ 𝐹)‘(𝑥 + 1))) |
159 | 64, 52 | syl 14 |
. . . . . . . . . . . 12
⊢ ((𝜑 ∧ 𝐾 ∈ (𝑀...𝑁)) → (𝐺 ↾ (𝑀...𝑁)) ∈ V) |
160 | 26 | adantr 276 |
. . . . . . . . . . . 12
⊢ ((𝜑 ∧ 𝐾 ∈ (𝑀...𝑁)) → 𝐽 ∈ V) |
161 | | coexg 5202 |
. . . . . . . . . . . 12
⊢ (((𝐺 ↾ (𝑀...𝑁)) ∈ V ∧ 𝐽 ∈ V) → ((𝐺 ↾ (𝑀...𝑁)) ∘ 𝐽) ∈ V) |
162 | 159, 160,
161 | syl2anc 411 |
. . . . . . . . . . 11
⊢ ((𝜑 ∧ 𝐾 ∈ (𝑀...𝑁)) → ((𝐺 ↾ (𝑀...𝑁)) ∘ 𝐽) ∈ V) |
163 | 60, 113, 158, 63, 162, 70 | seqshft2g 10543 |
. . . . . . . . . 10
⊢ ((𝜑 ∧ 𝐾 ∈ (𝑀...𝑁)) → (seq𝐾( + , ((𝐺 ↾ (𝑀...𝑁)) ∘ 𝐽))‘𝑁) = (seq(𝐾 + 1)( + , (𝐺 ∘ 𝐹))‘(𝑁 + 1))) |
164 | | fvco3 5620 |
. . . . . . . . . . . . 13
⊢ ((𝐹:(𝑀...(𝑁 + 1))⟶(𝑀...(𝑁 + 1)) ∧ 𝐾 ∈ (𝑀...(𝑁 + 1))) → ((𝐺 ∘ 𝐹)‘𝐾) = (𝐺‘(𝐹‘𝐾))) |
165 | 66, 92, 164 | syl2anc 411 |
. . . . . . . . . . . 12
⊢ (𝜑 → ((𝐺 ∘ 𝐹)‘𝐾) = (𝐺‘(𝐹‘𝐾))) |
166 | 22 | fveq2i 5549 |
. . . . . . . . . . . . . 14
⊢ (𝐹‘𝐾) = (𝐹‘(◡𝐹‘(𝑁 + 1))) |
167 | | f1ocnvfv2 5813 |
. . . . . . . . . . . . . . 15
⊢ ((𝐹:(𝑀...(𝑁 + 1))–1-1-onto→(𝑀...(𝑁 + 1)) ∧ (𝑁 + 1) ∈ (𝑀...(𝑁 + 1))) → (𝐹‘(◡𝐹‘(𝑁 + 1))) = (𝑁 + 1)) |
168 | 20, 90, 167 | syl2anc 411 |
. . . . . . . . . . . . . 14
⊢ (𝜑 → (𝐹‘(◡𝐹‘(𝑁 + 1))) = (𝑁 + 1)) |
169 | 166, 168 | eqtrid 2238 |
. . . . . . . . . . . . 13
⊢ (𝜑 → (𝐹‘𝐾) = (𝑁 + 1)) |
170 | 169 | fveq2d 5550 |
. . . . . . . . . . . 12
⊢ (𝜑 → (𝐺‘(𝐹‘𝐾)) = (𝐺‘(𝑁 + 1))) |
171 | 165, 170 | eqtr2d 2227 |
. . . . . . . . . . 11
⊢ (𝜑 → (𝐺‘(𝑁 + 1)) = ((𝐺 ∘ 𝐹)‘𝐾)) |
172 | 171 | adantr 276 |
. . . . . . . . . 10
⊢ ((𝜑 ∧ 𝐾 ∈ (𝑀...𝑁)) → (𝐺‘(𝑁 + 1)) = ((𝐺 ∘ 𝐹)‘𝐾)) |
173 | 163, 172 | oveq12d 5928 |
. . . . . . . . 9
⊢ ((𝜑 ∧ 𝐾 ∈ (𝑀...𝑁)) → ((seq𝐾( + , ((𝐺 ↾ (𝑀...𝑁)) ∘ 𝐽))‘𝑁) + (𝐺‘(𝑁 + 1))) = ((seq(𝐾 + 1)( + , (𝐺 ∘ 𝐹))‘(𝑁 + 1)) + ((𝐺 ∘ 𝐹)‘𝐾))) |
174 | 112, 173 | eqtr4d 2229 |
. . . . . . . 8
⊢ ((𝜑 ∧ 𝐾 ∈ (𝑀...𝑁)) → (((𝐺 ∘ 𝐹)‘𝐾) + (seq(𝐾 + 1)( + , (𝐺 ∘ 𝐹))‘(𝑁 + 1))) = ((seq𝐾( + , ((𝐺 ↾ (𝑀...𝑁)) ∘ 𝐽))‘𝑁) + (𝐺‘(𝑁 + 1)))) |
175 | 103, 174 | syldan 282 |
. . . . . . 7
⊢ ((𝜑 ∧ 𝐾 = 𝑀) → (((𝐺 ∘ 𝐹)‘𝐾) + (seq(𝐾 + 1)( + , (𝐺 ∘ 𝐹))‘(𝑁 + 1))) = ((seq𝐾( + , ((𝐺 ↾ (𝑀...𝑁)) ∘ 𝐽))‘𝑁) + (𝐺‘(𝑁 + 1)))) |
176 | 99 | seqeq1d 10514 |
. . . . . . . . 9
⊢ ((𝜑 ∧ 𝐾 = 𝑀) → seq𝐾( + , ((𝐺 ↾ (𝑀...𝑁)) ∘ 𝐽)) = seq𝑀( + , ((𝐺 ↾ (𝑀...𝑁)) ∘ 𝐽))) |
177 | 176 | fveq1d 5548 |
. . . . . . . 8
⊢ ((𝜑 ∧ 𝐾 = 𝑀) → (seq𝐾( + , ((𝐺 ↾ (𝑀...𝑁)) ∘ 𝐽))‘𝑁) = (seq𝑀( + , ((𝐺 ↾ (𝑀...𝑁)) ∘ 𝐽))‘𝑁)) |
178 | 177 | oveq1d 5925 |
. . . . . . 7
⊢ ((𝜑 ∧ 𝐾 = 𝑀) → ((seq𝐾( + , ((𝐺 ↾ (𝑀...𝑁)) ∘ 𝐽))‘𝑁) + (𝐺‘(𝑁 + 1))) = ((seq𝑀( + , ((𝐺 ↾ (𝑀...𝑁)) ∘ 𝐽))‘𝑁) + (𝐺‘(𝑁 + 1)))) |
179 | 98, 175, 178 | 3eqtrd 2230 |
. . . . . 6
⊢ ((𝜑 ∧ 𝐾 = 𝑀) → ((seq𝑀( + , (𝐺 ∘ 𝐹))‘𝐾) + (seq(𝐾 + 1)( + , (𝐺 ∘ 𝐹))‘(𝑁 + 1))) = ((seq𝑀( + , ((𝐺 ↾ (𝑀...𝑁)) ∘ 𝐽))‘𝑁) + (𝐺‘(𝑁 + 1)))) |
180 | | elfzuz 10077 |
. . . . . . . . . . 11
⊢ (𝐾 ∈ ((𝑀 + 1)...𝑁) → 𝐾 ∈ (ℤ≥‘(𝑀 + 1))) |
181 | | eluzp1m1 9606 |
. . . . . . . . . . 11
⊢ ((𝑀 ∈ ℤ ∧ 𝐾 ∈
(ℤ≥‘(𝑀 + 1))) → (𝐾 − 1) ∈
(ℤ≥‘𝑀)) |
182 | 8, 180, 181 | syl2an 289 |
. . . . . . . . . 10
⊢ ((𝜑 ∧ 𝐾 ∈ ((𝑀 + 1)...𝑁)) → (𝐾 − 1) ∈
(ℤ≥‘𝑀)) |
183 | 10 | zcnd 9430 |
. . . . . . . . . . . . . . . . . . 19
⊢ (𝜑 → 𝑁 ∈ ℂ) |
184 | | ax-1cn 7955 |
. . . . . . . . . . . . . . . . . . 19
⊢ 1 ∈
ℂ |
185 | | pncan 8215 |
. . . . . . . . . . . . . . . . . . 19
⊢ ((𝑁 ∈ ℂ ∧ 1 ∈
ℂ) → ((𝑁 + 1)
− 1) = 𝑁) |
186 | 183, 184,
185 | sylancl 413 |
. . . . . . . . . . . . . . . . . 18
⊢ (𝜑 → ((𝑁 + 1) − 1) = 𝑁) |
187 | | elfzelz 10081 |
. . . . . . . . . . . . . . . . . . . 20
⊢ (𝐾 ∈ (𝑀...(𝑁 + 1)) → 𝐾 ∈ ℤ) |
188 | | peano2zm 9345 |
. . . . . . . . . . . . . . . . . . . 20
⊢ (𝐾 ∈ ℤ → (𝐾 − 1) ∈
ℤ) |
189 | 92, 187, 188 | 3syl 17 |
. . . . . . . . . . . . . . . . . . 19
⊢ (𝜑 → (𝐾 − 1) ∈ ℤ) |
190 | | elfzuz3 10078 |
. . . . . . . . . . . . . . . . . . . . 21
⊢ (𝐾 ∈ (𝑀...(𝑁 + 1)) → (𝑁 + 1) ∈
(ℤ≥‘𝐾)) |
191 | 92, 190 | syl 14 |
. . . . . . . . . . . . . . . . . . . 20
⊢ (𝜑 → (𝑁 + 1) ∈
(ℤ≥‘𝐾)) |
192 | 93 | zcnd 9430 |
. . . . . . . . . . . . . . . . . . . . . 22
⊢ (𝜑 → 𝐾 ∈ ℂ) |
193 | | npcan 8218 |
. . . . . . . . . . . . . . . . . . . . . 22
⊢ ((𝐾 ∈ ℂ ∧ 1 ∈
ℂ) → ((𝐾 −
1) + 1) = 𝐾) |
194 | 192, 184,
193 | sylancl 413 |
. . . . . . . . . . . . . . . . . . . . 21
⊢ (𝜑 → ((𝐾 − 1) + 1) = 𝐾) |
195 | 194 | fveq2d 5550 |
. . . . . . . . . . . . . . . . . . . 20
⊢ (𝜑 →
(ℤ≥‘((𝐾 − 1) + 1)) =
(ℤ≥‘𝐾)) |
196 | 191, 195 | eleqtrrd 2273 |
. . . . . . . . . . . . . . . . . . 19
⊢ (𝜑 → (𝑁 + 1) ∈
(ℤ≥‘((𝐾 − 1) + 1))) |
197 | | eluzp1m1 9606 |
. . . . . . . . . . . . . . . . . . 19
⊢ (((𝐾 − 1) ∈ ℤ ∧
(𝑁 + 1) ∈
(ℤ≥‘((𝐾 − 1) + 1))) → ((𝑁 + 1) − 1) ∈
(ℤ≥‘(𝐾 − 1))) |
198 | 189, 196,
197 | syl2anc 411 |
. . . . . . . . . . . . . . . . . 18
⊢ (𝜑 → ((𝑁 + 1) − 1) ∈
(ℤ≥‘(𝐾 − 1))) |
199 | 186, 198 | eqeltrrd 2271 |
. . . . . . . . . . . . . . . . 17
⊢ (𝜑 → 𝑁 ∈ (ℤ≥‘(𝐾 − 1))) |
200 | | fzss2 10120 |
. . . . . . . . . . . . . . . . 17
⊢ (𝑁 ∈
(ℤ≥‘(𝐾 − 1)) → (𝑀...(𝐾 − 1)) ⊆ (𝑀...𝑁)) |
201 | 199, 200 | syl 14 |
. . . . . . . . . . . . . . . 16
⊢ (𝜑 → (𝑀...(𝐾 − 1)) ⊆ (𝑀...𝑁)) |
202 | 201 | sselda 3179 |
. . . . . . . . . . . . . . 15
⊢ ((𝜑 ∧ 𝑥 ∈ (𝑀...(𝐾 − 1))) → 𝑥 ∈ (𝑀...𝑁)) |
203 | 202, 118 | syldan 282 |
. . . . . . . . . . . . . 14
⊢ ((𝜑 ∧ 𝑥 ∈ (𝑀...(𝐾 − 1))) → (𝐽‘𝑥) ∈ (𝑀...𝑁)) |
204 | 203 | fvresd 5571 |
. . . . . . . . . . . . 13
⊢ ((𝜑 ∧ 𝑥 ∈ (𝑀...(𝐾 − 1))) → ((𝐺 ↾ (𝑀...𝑁))‘(𝐽‘𝑥)) = (𝐺‘(𝐽‘𝑥))) |
205 | | simpr 110 |
. . . . . . . . . . . . . . . . 17
⊢ ((𝜑 ∧ 𝑥 ∈ (𝑀...𝑁)) → 𝑥 ∈ (𝑀...𝑁)) |
206 | 66 | adantr 276 |
. . . . . . . . . . . . . . . . . 18
⊢ ((𝜑 ∧ 𝑥 ∈ (𝑀...𝑁)) → 𝐹:(𝑀...(𝑁 + 1))⟶(𝑀...(𝑁 + 1))) |
207 | | fzelp1 10130 |
. . . . . . . . . . . . . . . . . . . 20
⊢ (𝑥 ∈ (𝑀...𝑁) → 𝑥 ∈ (𝑀...(𝑁 + 1))) |
208 | 207 | adantl 277 |
. . . . . . . . . . . . . . . . . . 19
⊢ ((𝜑 ∧ 𝑥 ∈ (𝑀...𝑁)) → 𝑥 ∈ (𝑀...(𝑁 + 1))) |
209 | 128 | adantl 277 |
. . . . . . . . . . . . . . . . . . 19
⊢ ((𝜑 ∧ 𝑥 ∈ (𝑀...𝑁)) → (𝑥 + 1) ∈ (𝑀...(𝑁 + 1))) |
210 | | elfzelz 10081 |
. . . . . . . . . . . . . . . . . . . 20
⊢ (𝑥 ∈ (𝑀...𝑁) → 𝑥 ∈ ℤ) |
211 | 210, 93, 132 | syl2anr 290 |
. . . . . . . . . . . . . . . . . . 19
⊢ ((𝜑 ∧ 𝑥 ∈ (𝑀...𝑁)) → DECID 𝑥 < 𝐾) |
212 | 208, 209,
211 | ifcldcd 3593 |
. . . . . . . . . . . . . . . . . 18
⊢ ((𝜑 ∧ 𝑥 ∈ (𝑀...𝑁)) → if(𝑥 < 𝐾, 𝑥, (𝑥 + 1)) ∈ (𝑀...(𝑁 + 1))) |
213 | 206, 212 | ffvelcdmd 5686 |
. . . . . . . . . . . . . . . . 17
⊢ ((𝜑 ∧ 𝑥 ∈ (𝑀...𝑁)) → (𝐹‘if(𝑥 < 𝐾, 𝑥, (𝑥 + 1))) ∈ (𝑀...(𝑁 + 1))) |
214 | 21, 125, 205, 213 | fvmptd3 5643 |
. . . . . . . . . . . . . . . 16
⊢ ((𝜑 ∧ 𝑥 ∈ (𝑀...𝑁)) → (𝐽‘𝑥) = (𝐹‘if(𝑥 < 𝐾, 𝑥, (𝑥 + 1)))) |
215 | 202, 214 | syldan 282 |
. . . . . . . . . . . . . . 15
⊢ ((𝜑 ∧ 𝑥 ∈ (𝑀...(𝐾 − 1))) → (𝐽‘𝑥) = (𝐹‘if(𝑥 < 𝐾, 𝑥, (𝑥 + 1)))) |
216 | | elfzm11 10147 |
. . . . . . . . . . . . . . . . . . 19
⊢ ((𝑀 ∈ ℤ ∧ 𝐾 ∈ ℤ) → (𝑥 ∈ (𝑀...(𝐾 − 1)) ↔ (𝑥 ∈ ℤ ∧ 𝑀 ≤ 𝑥 ∧ 𝑥 < 𝐾))) |
217 | 8, 93, 216 | syl2anc 411 |
. . . . . . . . . . . . . . . . . 18
⊢ (𝜑 → (𝑥 ∈ (𝑀...(𝐾 − 1)) ↔ (𝑥 ∈ ℤ ∧ 𝑀 ≤ 𝑥 ∧ 𝑥 < 𝐾))) |
218 | 217 | biimpa 296 |
. . . . . . . . . . . . . . . . 17
⊢ ((𝜑 ∧ 𝑥 ∈ (𝑀...(𝐾 − 1))) → (𝑥 ∈ ℤ ∧ 𝑀 ≤ 𝑥 ∧ 𝑥 < 𝐾)) |
219 | 218 | simp3d 1013 |
. . . . . . . . . . . . . . . 16
⊢ ((𝜑 ∧ 𝑥 ∈ (𝑀...(𝐾 − 1))) → 𝑥 < 𝐾) |
220 | | iftrue 3562 |
. . . . . . . . . . . . . . . . 17
⊢ (𝑥 < 𝐾 → if(𝑥 < 𝐾, 𝑥, (𝑥 + 1)) = 𝑥) |
221 | 220 | fveq2d 5550 |
. . . . . . . . . . . . . . . 16
⊢ (𝑥 < 𝐾 → (𝐹‘if(𝑥 < 𝐾, 𝑥, (𝑥 + 1))) = (𝐹‘𝑥)) |
222 | 219, 221 | syl 14 |
. . . . . . . . . . . . . . 15
⊢ ((𝜑 ∧ 𝑥 ∈ (𝑀...(𝐾 − 1))) → (𝐹‘if(𝑥 < 𝐾, 𝑥, (𝑥 + 1))) = (𝐹‘𝑥)) |
223 | 215, 222 | eqtrd 2226 |
. . . . . . . . . . . . . 14
⊢ ((𝜑 ∧ 𝑥 ∈ (𝑀...(𝐾 − 1))) → (𝐽‘𝑥) = (𝐹‘𝑥)) |
224 | 223 | fveq2d 5550 |
. . . . . . . . . . . . 13
⊢ ((𝜑 ∧ 𝑥 ∈ (𝑀...(𝐾 − 1))) → (𝐺‘(𝐽‘𝑥)) = (𝐺‘(𝐹‘𝑥))) |
225 | 204, 224 | eqtr2d 2227 |
. . . . . . . . . . . 12
⊢ ((𝜑 ∧ 𝑥 ∈ (𝑀...(𝐾 − 1))) → (𝐺‘(𝐹‘𝑥)) = ((𝐺 ↾ (𝑀...𝑁))‘(𝐽‘𝑥))) |
226 | | peano2uz 9638 |
. . . . . . . . . . . . . . 15
⊢ (𝑁 ∈
(ℤ≥‘(𝐾 − 1)) → (𝑁 + 1) ∈
(ℤ≥‘(𝐾 − 1))) |
227 | | fzss2 10120 |
. . . . . . . . . . . . . . 15
⊢ ((𝑁 + 1) ∈
(ℤ≥‘(𝐾 − 1)) → (𝑀...(𝐾 − 1)) ⊆ (𝑀...(𝑁 + 1))) |
228 | 199, 226,
227 | 3syl 17 |
. . . . . . . . . . . . . 14
⊢ (𝜑 → (𝑀...(𝐾 − 1)) ⊆ (𝑀...(𝑁 + 1))) |
229 | 228 | sselda 3179 |
. . . . . . . . . . . . 13
⊢ ((𝜑 ∧ 𝑥 ∈ (𝑀...(𝐾 − 1))) → 𝑥 ∈ (𝑀...(𝑁 + 1))) |
230 | | fvco3 5620 |
. . . . . . . . . . . . . 14
⊢ ((𝐹:(𝑀...(𝑁 + 1))⟶(𝑀...(𝑁 + 1)) ∧ 𝑥 ∈ (𝑀...(𝑁 + 1))) → ((𝐺 ∘ 𝐹)‘𝑥) = (𝐺‘(𝐹‘𝑥))) |
231 | 66, 230 | sylan 283 |
. . . . . . . . . . . . 13
⊢ ((𝜑 ∧ 𝑥 ∈ (𝑀...(𝑁 + 1))) → ((𝐺 ∘ 𝐹)‘𝑥) = (𝐺‘(𝐹‘𝑥))) |
232 | 229, 231 | syldan 282 |
. . . . . . . . . . . 12
⊢ ((𝜑 ∧ 𝑥 ∈ (𝑀...(𝐾 − 1))) → ((𝐺 ∘ 𝐹)‘𝑥) = (𝐺‘(𝐹‘𝑥))) |
233 | 202, 152 | syldan 282 |
. . . . . . . . . . . 12
⊢ ((𝜑 ∧ 𝑥 ∈ (𝑀...(𝐾 − 1))) → (((𝐺 ↾ (𝑀...𝑁)) ∘ 𝐽)‘𝑥) = ((𝐺 ↾ (𝑀...𝑁))‘(𝐽‘𝑥))) |
234 | 225, 232,
233 | 3eqtr4d 2236 |
. . . . . . . . . . 11
⊢ ((𝜑 ∧ 𝑥 ∈ (𝑀...(𝐾 − 1))) → ((𝐺 ∘ 𝐹)‘𝑥) = (((𝐺 ↾ (𝑀...𝑁)) ∘ 𝐽)‘𝑥)) |
235 | 234 | adantlr 477 |
. . . . . . . . . 10
⊢ (((𝜑 ∧ 𝐾 ∈ ((𝑀 + 1)...𝑁)) ∧ 𝑥 ∈ (𝑀...(𝐾 − 1))) → ((𝐺 ∘ 𝐹)‘𝑥) = (((𝐺 ↾ (𝑀...𝑁)) ∘ 𝐽)‘𝑥)) |
236 | 19 | adantr 276 |
. . . . . . . . . 10
⊢ ((𝜑 ∧ 𝐾 ∈ ((𝑀 + 1)...𝑁)) → + ∈ 𝑉) |
237 | 94 | adantr 276 |
. . . . . . . . . 10
⊢ ((𝜑 ∧ 𝐾 ∈ ((𝑀 + 1)...𝑁)) → (𝐺 ∘ 𝐹) ∈ V) |
238 | 27, 161 | syl 14 |
. . . . . . . . . . 11
⊢ (𝜑 → ((𝐺 ↾ (𝑀...𝑁)) ∘ 𝐽) ∈ V) |
239 | 238 | adantr 276 |
. . . . . . . . . 10
⊢ ((𝜑 ∧ 𝐾 ∈ ((𝑀 + 1)...𝑁)) → ((𝐺 ↾ (𝑀...𝑁)) ∘ 𝐽) ∈ V) |
240 | 182, 235,
236, 237, 239 | seqfveqg 10539 |
. . . . . . . . 9
⊢ ((𝜑 ∧ 𝐾 ∈ ((𝑀 + 1)...𝑁)) → (seq𝑀( + , (𝐺 ∘ 𝐹))‘(𝐾 − 1)) = (seq𝑀( + , ((𝐺 ↾ (𝑀...𝑁)) ∘ 𝐽))‘(𝐾 − 1))) |
241 | | fzp1ss 10129 |
. . . . . . . . . . . 12
⊢ (𝑀 ∈ ℤ → ((𝑀 + 1)...𝑁) ⊆ (𝑀...𝑁)) |
242 | 6, 7, 241 | 3syl 17 |
. . . . . . . . . . 11
⊢ (𝜑 → ((𝑀 + 1)...𝑁) ⊆ (𝑀...𝑁)) |
243 | 242 | sselda 3179 |
. . . . . . . . . 10
⊢ ((𝜑 ∧ 𝐾 ∈ ((𝑀 + 1)...𝑁)) → 𝐾 ∈ (𝑀...𝑁)) |
244 | 243, 174 | syldan 282 |
. . . . . . . . 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)))) |
245 | 240, 244 | oveq12d 5928 |
. . . . . . . 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))))) |
246 | 229, 76 | syldan 282 |
. . . . . . . . . . . 12
⊢ ((𝜑 ∧ 𝑥 ∈ (𝑀...(𝐾 − 1))) → ((𝐺 ∘ 𝐹)‘𝑥) ∈ 𝑆) |
247 | 246 | adantlr 477 |
. . . . . . . . . . 11
⊢ (((𝜑 ∧ 𝐾 ∈ ((𝑀 + 1)...𝑁)) ∧ 𝑥 ∈ (𝑀...(𝐾 − 1))) → ((𝐺 ∘ 𝐹)‘𝑥) ∈ 𝑆) |
248 | 15 | adantlr 477 |
. . . . . . . . . . 11
⊢ (((𝜑 ∧ 𝐾 ∈ ((𝑀 + 1)...𝑁)) ∧ (𝑥 ∈ 𝑆 ∧ 𝑦 ∈ 𝑆)) → (𝑥 + 𝑦) ∈ 𝑆) |
249 | 182, 247,
248, 237, 236 | seqclg 10533 |
. . . . . . . . . 10
⊢ ((𝜑 ∧ 𝐾 ∈ ((𝑀 + 1)...𝑁)) → (seq𝑀( + , (𝐺 ∘ 𝐹))‘(𝐾 − 1)) ∈ 𝑆) |
250 | 74, 92 | ffvelcdmd 5686 |
. . . . . . . . . . . 12
⊢ (𝜑 → ((𝐺 ∘ 𝐹)‘𝐾) ∈ 𝐶) |
251 | 18, 250 | sseldd 3180 |
. . . . . . . . . . 11
⊢ (𝜑 → ((𝐺 ∘ 𝐹)‘𝐾) ∈ 𝑆) |
252 | 251 | adantr 276 |
. . . . . . . . . 10
⊢ ((𝜑 ∧ 𝐾 ∈ ((𝑀 + 1)...𝑁)) → ((𝐺 ∘ 𝐹)‘𝐾) ∈ 𝑆) |
253 | 110 | sselda 3179 |
. . . . . . . . . . . . 13
⊢ (((𝜑 ∧ 𝐾 ∈ (𝑀...𝑁)) ∧ 𝑥 ∈ ((𝐾 + 1)...(𝑁 + 1))) → 𝑥 ∈ (𝑀...(𝑁 + 1))) |
254 | 253, 77 | syldan 282 |
. . . . . . . . . . . 12
⊢ (((𝜑 ∧ 𝐾 ∈ (𝑀...𝑁)) ∧ 𝑥 ∈ ((𝐾 + 1)...(𝑁 + 1))) → ((𝐺 ∘ 𝐹)‘𝑥) ∈ 𝑆) |
255 | 62, 254, 57, 70, 63 | seqclg 10533 |
. . . . . . . . . . 11
⊢ ((𝜑 ∧ 𝐾 ∈ (𝑀...𝑁)) → (seq(𝐾 + 1)( + , (𝐺 ∘ 𝐹))‘(𝑁 + 1)) ∈ 𝑆) |
256 | 243, 255 | syldan 282 |
. . . . . . . . . 10
⊢ ((𝜑 ∧ 𝐾 ∈ ((𝑀 + 1)...𝑁)) → (seq(𝐾 + 1)( + , (𝐺 ∘ 𝐹))‘(𝑁 + 1)) ∈ 𝑆) |
257 | 249, 252,
256 | 3jca 1179 |
. . . . . . . . 9
⊢ ((𝜑 ∧ 𝐾 ∈ ((𝑀 + 1)...𝑁)) → ((seq𝑀( + , (𝐺 ∘ 𝐹))‘(𝐾 − 1)) ∈ 𝑆 ∧ ((𝐺 ∘ 𝐹)‘𝐾) ∈ 𝑆 ∧ (seq(𝐾 + 1)( + , (𝐺 ∘ 𝐹))‘(𝑁 + 1)) ∈ 𝑆)) |
258 | 17 | caovassg 6069 |
. . . . . . . . 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))))) |
259 | 257, 258 | syldan 282 |
. . . . . . . 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))))) |
260 | 1, 18 | fssd 5408 |
. . . . . . . . . . . . . . . 16
⊢ (𝜑 → 𝐺:(𝑀...(𝑁 + 1))⟶𝑆) |
261 | | fssres 5421 |
. . . . . . . . . . . . . . . 16
⊢ ((𝐺:(𝑀...(𝑁 + 1))⟶𝑆 ∧ (𝑀...𝑁) ⊆ (𝑀...(𝑁 + 1))) → (𝐺 ↾ (𝑀...𝑁)):(𝑀...𝑁)⟶𝑆) |
262 | 260, 3, 261 | sylancl 413 |
. . . . . . . . . . . . . . 15
⊢ (𝜑 → (𝐺 ↾ (𝑀...𝑁)):(𝑀...𝑁)⟶𝑆) |
263 | | fco 5411 |
. . . . . . . . . . . . . . 15
⊢ (((𝐺 ↾ (𝑀...𝑁)):(𝑀...𝑁)⟶𝑆 ∧ 𝐽:(𝑀...𝑁)⟶(𝑀...𝑁)) → ((𝐺 ↾ (𝑀...𝑁)) ∘ 𝐽):(𝑀...𝑁)⟶𝑆) |
264 | 262, 25, 263 | syl2anc 411 |
. . . . . . . . . . . . . 14
⊢ (𝜑 → ((𝐺 ↾ (𝑀...𝑁)) ∘ 𝐽):(𝑀...𝑁)⟶𝑆) |
265 | 264 | ffvelcdmda 5685 |
. . . . . . . . . . . . 13
⊢ ((𝜑 ∧ 𝑥 ∈ (𝑀...𝑁)) → (((𝐺 ↾ (𝑀...𝑁)) ∘ 𝐽)‘𝑥) ∈ 𝑆) |
266 | 202, 265 | syldan 282 |
. . . . . . . . . . . 12
⊢ ((𝜑 ∧ 𝑥 ∈ (𝑀...(𝐾 − 1))) → (((𝐺 ↾ (𝑀...𝑁)) ∘ 𝐽)‘𝑥) ∈ 𝑆) |
267 | 266 | adantlr 477 |
. . . . . . . . . . 11
⊢ (((𝜑 ∧ 𝐾 ∈ ((𝑀 + 1)...𝑁)) ∧ 𝑥 ∈ (𝑀...(𝐾 − 1))) → (((𝐺 ↾ (𝑀...𝑁)) ∘ 𝐽)‘𝑥) ∈ 𝑆) |
268 | 182, 267,
248, 239, 236 | seqclg 10533 |
. . . . . . . . . 10
⊢ ((𝜑 ∧ 𝐾 ∈ ((𝑀 + 1)...𝑁)) → (seq𝑀( + , ((𝐺 ↾ (𝑀...𝑁)) ∘ 𝐽))‘(𝐾 − 1)) ∈ 𝑆) |
269 | | elfzuz3 10078 |
. . . . . . . . . . . 12
⊢ (𝐾 ∈ ((𝑀 + 1)...𝑁) → 𝑁 ∈ (ℤ≥‘𝐾)) |
270 | 269 | adantl 277 |
. . . . . . . . . . 11
⊢ ((𝜑 ∧ 𝐾 ∈ ((𝑀 + 1)...𝑁)) → 𝑁 ∈ (ℤ≥‘𝐾)) |
271 | 117, 265 | syldan 282 |
. . . . . . . . . . . 12
⊢ ((𝜑 ∧ 𝑥 ∈ (𝐾...𝑁)) → (((𝐺 ↾ (𝑀...𝑁)) ∘ 𝐽)‘𝑥) ∈ 𝑆) |
272 | 271 | adantlr 477 |
. . . . . . . . . . 11
⊢ (((𝜑 ∧ 𝐾 ∈ ((𝑀 + 1)...𝑁)) ∧ 𝑥 ∈ (𝐾...𝑁)) → (((𝐺 ↾ (𝑀...𝑁)) ∘ 𝐽)‘𝑥) ∈ 𝑆) |
273 | 270, 272,
248, 239, 236 | seqclg 10533 |
. . . . . . . . . 10
⊢ ((𝜑 ∧ 𝐾 ∈ ((𝑀 + 1)...𝑁)) → (seq𝐾( + , ((𝐺 ↾ (𝑀...𝑁)) ∘ 𝐽))‘𝑁) ∈ 𝑆) |
274 | 260, 90 | ffvelcdmd 5686 |
. . . . . . . . . . 11
⊢ (𝜑 → (𝐺‘(𝑁 + 1)) ∈ 𝑆) |
275 | 274 | adantr 276 |
. . . . . . . . . 10
⊢ ((𝜑 ∧ 𝐾 ∈ ((𝑀 + 1)...𝑁)) → (𝐺‘(𝑁 + 1)) ∈ 𝑆) |
276 | 268, 273,
275 | 3jca 1179 |
. . . . . . . . 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)) |
277 | 17 | caovassg 6069 |
. . . . . . . . 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))))) |
278 | 276, 277 | syldan 282 |
. . . . . . . 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))))) |
279 | 245, 259,
278 | 3eqtr4d 2236 |
. . . . . . 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)))) |
280 | 8 | adantr 276 |
. . . . . . . . 9
⊢ ((𝜑 ∧ 𝐾 ∈ ((𝑀 + 1)...𝑁)) → 𝑀 ∈ ℤ) |
281 | 180 | adantl 277 |
. . . . . . . . 9
⊢ ((𝜑 ∧ 𝐾 ∈ ((𝑀 + 1)...𝑁)) → 𝐾 ∈ (ℤ≥‘(𝑀 + 1))) |
282 | 280, 281,
236, 237 | seqm1g 10535 |
. . . . . . . 8
⊢ ((𝜑 ∧ 𝐾 ∈ ((𝑀 + 1)...𝑁)) → (seq𝑀( + , (𝐺 ∘ 𝐹))‘𝐾) = ((seq𝑀( + , (𝐺 ∘ 𝐹))‘(𝐾 − 1)) + ((𝐺 ∘ 𝐹)‘𝐾))) |
283 | 282 | oveq1d 5925 |
. . . . . . 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)))) |
284 | 17 | adantlr 477 |
. . . . . . . . . 10
⊢ (((𝜑 ∧ 𝐾 ∈ ((𝑀 + 1)...𝑁)) ∧ (𝑥 ∈ 𝑆 ∧ 𝑦 ∈ 𝑆 ∧ 𝑧 ∈ 𝑆)) → ((𝑥 + 𝑦) + 𝑧) = (𝑥 + (𝑦 + 𝑧))) |
285 | | elfzelz 10081 |
. . . . . . . . . . . . . . 15
⊢ (𝐾 ∈ ((𝑀 + 1)...𝑁) → 𝐾 ∈ ℤ) |
286 | 285 | adantl 277 |
. . . . . . . . . . . . . 14
⊢ ((𝜑 ∧ 𝐾 ∈ ((𝑀 + 1)...𝑁)) → 𝐾 ∈ ℤ) |
287 | 286 | zcnd 9430 |
. . . . . . . . . . . . 13
⊢ ((𝜑 ∧ 𝐾 ∈ ((𝑀 + 1)...𝑁)) → 𝐾 ∈ ℂ) |
288 | 287, 184,
193 | sylancl 413 |
. . . . . . . . . . . 12
⊢ ((𝜑 ∧ 𝐾 ∈ ((𝑀 + 1)...𝑁)) → ((𝐾 − 1) + 1) = 𝐾) |
289 | 288 | fveq2d 5550 |
. . . . . . . . . . 11
⊢ ((𝜑 ∧ 𝐾 ∈ ((𝑀 + 1)...𝑁)) →
(ℤ≥‘((𝐾 − 1) + 1)) =
(ℤ≥‘𝐾)) |
290 | 270, 289 | eleqtrrd 2273 |
. . . . . . . . . 10
⊢ ((𝜑 ∧ 𝐾 ∈ ((𝑀 + 1)...𝑁)) → 𝑁 ∈
(ℤ≥‘((𝐾 − 1) + 1))) |
291 | 265 | adantlr 477 |
. . . . . . . . . 10
⊢ (((𝜑 ∧ 𝐾 ∈ ((𝑀 + 1)...𝑁)) ∧ 𝑥 ∈ (𝑀...𝑁)) → (((𝐺 ↾ (𝑀...𝑁)) ∘ 𝐽)‘𝑥) ∈ 𝑆) |
292 | 248, 284,
290, 236, 239, 182, 291 | seqsplitg 10550 |
. . . . . . . . 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))) |
293 | 288 | seqeq1d 10514 |
. . . . . . . . . . 11
⊢ ((𝜑 ∧ 𝐾 ∈ ((𝑀 + 1)...𝑁)) → seq((𝐾 − 1) + 1)( + , ((𝐺 ↾ (𝑀...𝑁)) ∘ 𝐽)) = seq𝐾( + , ((𝐺 ↾ (𝑀...𝑁)) ∘ 𝐽))) |
294 | 293 | fveq1d 5548 |
. . . . . . . . . 10
⊢ ((𝜑 ∧ 𝐾 ∈ ((𝑀 + 1)...𝑁)) → (seq((𝐾 − 1) + 1)( + , ((𝐺 ↾ (𝑀...𝑁)) ∘ 𝐽))‘𝑁) = (seq𝐾( + , ((𝐺 ↾ (𝑀...𝑁)) ∘ 𝐽))‘𝑁)) |
295 | 294 | oveq2d 5926 |
. . . . . . . . 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))) |
296 | 292, 295 | eqtrd 2226 |
. . . . . . . 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))) |
297 | 296 | oveq1d 5925 |
. . . . . . 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)))) |
298 | 279, 283,
297 | 3eqtr4d 2236 |
. . . . . 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)))) |
299 | 179, 298 | jaodan 798 |
. . . . 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)))) |
300 | 81, 299 | syldan 282 |
. . . 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)))) |
301 | 78, 300 | eqtrd 2226 |
. . 3
⊢ ((𝜑 ∧ 𝐾 ∈ (𝑀...𝑁)) → (seq𝑀( + , (𝐺 ∘ 𝐹))‘(𝑁 + 1)) = ((seq𝑀( + , ((𝐺 ↾ (𝑀...𝑁)) ∘ 𝐽))‘𝑁) + (𝐺‘(𝑁 + 1)))) |
302 | 6 | adantr 276 |
. . . . 5
⊢ ((𝜑 ∧ 𝐾 = (𝑁 + 1)) → 𝑁 ∈ (ℤ≥‘𝑀)) |
303 | 94 | adantr 276 |
. . . . 5
⊢ ((𝜑 ∧ 𝐾 = (𝑁 + 1)) → (𝐺 ∘ 𝐹) ∈ V) |
304 | 19 | adantr 276 |
. . . . 5
⊢ ((𝜑 ∧ 𝐾 = (𝑁 + 1)) → + ∈ 𝑉) |
305 | | seqp1g 10527 |
. . . . 5
⊢ ((𝑁 ∈
(ℤ≥‘𝑀) ∧ (𝐺 ∘ 𝐹) ∈ V ∧ + ∈ 𝑉) → (seq𝑀( + , (𝐺 ∘ 𝐹))‘(𝑁 + 1)) = ((seq𝑀( + , (𝐺 ∘ 𝐹))‘𝑁) + ((𝐺 ∘ 𝐹)‘(𝑁 + 1)))) |
306 | 302, 303,
304, 305 | syl3anc 1249 |
. . . 4
⊢ ((𝜑 ∧ 𝐾 = (𝑁 + 1)) → (seq𝑀( + , (𝐺 ∘ 𝐹))‘(𝑁 + 1)) = ((seq𝑀( + , (𝐺 ∘ 𝐹))‘𝑁) + ((𝐺 ∘ 𝐹)‘(𝑁 + 1)))) |
307 | 214 | adantlr 477 |
. . . . . . . . . 10
⊢ (((𝜑 ∧ 𝐾 = (𝑁 + 1)) ∧ 𝑥 ∈ (𝑀...𝑁)) → (𝐽‘𝑥) = (𝐹‘if(𝑥 < 𝐾, 𝑥, (𝑥 + 1)))) |
308 | 210 | zred 9429 |
. . . . . . . . . . . . . . 15
⊢ (𝑥 ∈ (𝑀...𝑁) → 𝑥 ∈ ℝ) |
309 | 308 | adantl 277 |
. . . . . . . . . . . . . 14
⊢ ((𝜑 ∧ 𝑥 ∈ (𝑀...𝑁)) → 𝑥 ∈ ℝ) |
310 | 10 | zred 9429 |
. . . . . . . . . . . . . . 15
⊢ (𝜑 → 𝑁 ∈ ℝ) |
311 | 310 | adantr 276 |
. . . . . . . . . . . . . 14
⊢ ((𝜑 ∧ 𝑥 ∈ (𝑀...𝑁)) → 𝑁 ∈ ℝ) |
312 | | peano2re 8145 |
. . . . . . . . . . . . . . 15
⊢ (𝑁 ∈ ℝ → (𝑁 + 1) ∈
ℝ) |
313 | 311, 312 | syl 14 |
. . . . . . . . . . . . . 14
⊢ ((𝜑 ∧ 𝑥 ∈ (𝑀...𝑁)) → (𝑁 + 1) ∈ ℝ) |
314 | | elfzle2 10084 |
. . . . . . . . . . . . . . 15
⊢ (𝑥 ∈ (𝑀...𝑁) → 𝑥 ≤ 𝑁) |
315 | 314 | adantl 277 |
. . . . . . . . . . . . . 14
⊢ ((𝜑 ∧ 𝑥 ∈ (𝑀...𝑁)) → 𝑥 ≤ 𝑁) |
316 | 311 | ltp1d 8939 |
. . . . . . . . . . . . . 14
⊢ ((𝜑 ∧ 𝑥 ∈ (𝑀...𝑁)) → 𝑁 < (𝑁 + 1)) |
317 | 309, 311,
313, 315, 316 | lelttrd 8134 |
. . . . . . . . . . . . 13
⊢ ((𝜑 ∧ 𝑥 ∈ (𝑀...𝑁)) → 𝑥 < (𝑁 + 1)) |
318 | 317 | adantlr 477 |
. . . . . . . . . . . 12
⊢ (((𝜑 ∧ 𝐾 = (𝑁 + 1)) ∧ 𝑥 ∈ (𝑀...𝑁)) → 𝑥 < (𝑁 + 1)) |
319 | | simplr 528 |
. . . . . . . . . . . 12
⊢ (((𝜑 ∧ 𝐾 = (𝑁 + 1)) ∧ 𝑥 ∈ (𝑀...𝑁)) → 𝐾 = (𝑁 + 1)) |
320 | 318, 319 | breqtrrd 4057 |
. . . . . . . . . . 11
⊢ (((𝜑 ∧ 𝐾 = (𝑁 + 1)) ∧ 𝑥 ∈ (𝑀...𝑁)) → 𝑥 < 𝐾) |
321 | 320, 221 | syl 14 |
. . . . . . . . . 10
⊢ (((𝜑 ∧ 𝐾 = (𝑁 + 1)) ∧ 𝑥 ∈ (𝑀...𝑁)) → (𝐹‘if(𝑥 < 𝐾, 𝑥, (𝑥 + 1))) = (𝐹‘𝑥)) |
322 | 307, 321 | eqtrd 2226 |
. . . . . . . . 9
⊢ (((𝜑 ∧ 𝐾 = (𝑁 + 1)) ∧ 𝑥 ∈ (𝑀...𝑁)) → (𝐽‘𝑥) = (𝐹‘𝑥)) |
323 | 322 | fveq2d 5550 |
. . . . . . . 8
⊢ (((𝜑 ∧ 𝐾 = (𝑁 + 1)) ∧ 𝑥 ∈ (𝑀...𝑁)) → ((𝐺 ↾ (𝑀...𝑁))‘(𝐽‘𝑥)) = ((𝐺 ↾ (𝑀...𝑁))‘(𝐹‘𝑥))) |
324 | 66 | ad2antrr 488 |
. . . . . . . . . . . . 13
⊢ (((𝜑 ∧ 𝐾 = (𝑁 + 1)) ∧ 𝑥 ∈ (𝑀...𝑁)) → 𝐹:(𝑀...(𝑁 + 1))⟶(𝑀...(𝑁 + 1))) |
325 | 207 | adantl 277 |
. . . . . . . . . . . . 13
⊢ (((𝜑 ∧ 𝐾 = (𝑁 + 1)) ∧ 𝑥 ∈ (𝑀...𝑁)) → 𝑥 ∈ (𝑀...(𝑁 + 1))) |
326 | 324, 325 | ffvelcdmd 5686 |
. . . . . . . . . . . 12
⊢ (((𝜑 ∧ 𝐾 = (𝑁 + 1)) ∧ 𝑥 ∈ (𝑀...𝑁)) → (𝐹‘𝑥) ∈ (𝑀...(𝑁 + 1))) |
327 | 326 | elfzelzd 10082 |
. . . . . . . . . . 11
⊢ (((𝜑 ∧ 𝐾 = (𝑁 + 1)) ∧ 𝑥 ∈ (𝑀...𝑁)) → (𝐹‘𝑥) ∈ ℤ) |
328 | 6 | ad2antrr 488 |
. . . . . . . . . . . 12
⊢ (((𝜑 ∧ 𝐾 = (𝑁 + 1)) ∧ 𝑥 ∈ (𝑀...𝑁)) → 𝑁 ∈ (ℤ≥‘𝑀)) |
329 | 328, 7 | syl 14 |
. . . . . . . . . . 11
⊢ (((𝜑 ∧ 𝐾 = (𝑁 + 1)) ∧ 𝑥 ∈ (𝑀...𝑁)) → 𝑀 ∈ ℤ) |
330 | 328, 9 | syl 14 |
. . . . . . . . . . 11
⊢ (((𝜑 ∧ 𝐾 = (𝑁 + 1)) ∧ 𝑥 ∈ (𝑀...𝑁)) → 𝑁 ∈ ℤ) |
331 | | fzdcel 10096 |
. . . . . . . . . . 11
⊢ (((𝐹‘𝑥) ∈ ℤ ∧ 𝑀 ∈ ℤ ∧ 𝑁 ∈ ℤ) → DECID
(𝐹‘𝑥) ∈ (𝑀...𝑁)) |
332 | 327, 329,
330, 331 | syl3anc 1249 |
. . . . . . . . . 10
⊢ (((𝜑 ∧ 𝐾 = (𝑁 + 1)) ∧ 𝑥 ∈ (𝑀...𝑁)) → DECID (𝐹‘𝑥) ∈ (𝑀...𝑁)) |
333 | 308 | adantl 277 |
. . . . . . . . . . 11
⊢ (((𝜑 ∧ 𝐾 = (𝑁 + 1)) ∧ 𝑥 ∈ (𝑀...𝑁)) → 𝑥 ∈ ℝ) |
334 | 333, 320 | gtned 8122 |
. . . . . . . . . 10
⊢ (((𝜑 ∧ 𝐾 = (𝑁 + 1)) ∧ 𝑥 ∈ (𝑀...𝑁)) → 𝐾 ≠ 𝑥) |
335 | | elfzp1 10128 |
. . . . . . . . . . . . . . . 16
⊢ (𝑁 ∈
(ℤ≥‘𝑀) → ((𝐹‘𝑥) ∈ (𝑀...(𝑁 + 1)) ↔ ((𝐹‘𝑥) ∈ (𝑀...𝑁) ∨ (𝐹‘𝑥) = (𝑁 + 1)))) |
336 | 328, 335 | syl 14 |
. . . . . . . . . . . . . . 15
⊢ (((𝜑 ∧ 𝐾 = (𝑁 + 1)) ∧ 𝑥 ∈ (𝑀...𝑁)) → ((𝐹‘𝑥) ∈ (𝑀...(𝑁 + 1)) ↔ ((𝐹‘𝑥) ∈ (𝑀...𝑁) ∨ (𝐹‘𝑥) = (𝑁 + 1)))) |
337 | 326, 336 | mpbid 147 |
. . . . . . . . . . . . . 14
⊢ (((𝜑 ∧ 𝐾 = (𝑁 + 1)) ∧ 𝑥 ∈ (𝑀...𝑁)) → ((𝐹‘𝑥) ∈ (𝑀...𝑁) ∨ (𝐹‘𝑥) = (𝑁 + 1))) |
338 | 337 | ord 725 |
. . . . . . . . . . . . 13
⊢ (((𝜑 ∧ 𝐾 = (𝑁 + 1)) ∧ 𝑥 ∈ (𝑀...𝑁)) → (¬ (𝐹‘𝑥) ∈ (𝑀...𝑁) → (𝐹‘𝑥) = (𝑁 + 1))) |
339 | 20 | ad2antrr 488 |
. . . . . . . . . . . . . . 15
⊢ (((𝜑 ∧ 𝐾 = (𝑁 + 1)) ∧ 𝑥 ∈ (𝑀...𝑁)) → 𝐹:(𝑀...(𝑁 + 1))–1-1-onto→(𝑀...(𝑁 + 1))) |
340 | | f1ocnvfv 5814 |
. . . . . . . . . . . . . . 15
⊢ ((𝐹:(𝑀...(𝑁 + 1))–1-1-onto→(𝑀...(𝑁 + 1)) ∧ 𝑥 ∈ (𝑀...(𝑁 + 1))) → ((𝐹‘𝑥) = (𝑁 + 1) → (◡𝐹‘(𝑁 + 1)) = 𝑥)) |
341 | 339, 325,
340 | syl2anc 411 |
. . . . . . . . . . . . . 14
⊢ (((𝜑 ∧ 𝐾 = (𝑁 + 1)) ∧ 𝑥 ∈ (𝑀...𝑁)) → ((𝐹‘𝑥) = (𝑁 + 1) → (◡𝐹‘(𝑁 + 1)) = 𝑥)) |
342 | 22 | eqeq1i 2201 |
. . . . . . . . . . . . . 14
⊢ (𝐾 = 𝑥 ↔ (◡𝐹‘(𝑁 + 1)) = 𝑥) |
343 | 341, 342 | imbitrrdi 162 |
. . . . . . . . . . . . 13
⊢ (((𝜑 ∧ 𝐾 = (𝑁 + 1)) ∧ 𝑥 ∈ (𝑀...𝑁)) → ((𝐹‘𝑥) = (𝑁 + 1) → 𝐾 = 𝑥)) |
344 | 338, 343 | syld 45 |
. . . . . . . . . . . 12
⊢ (((𝜑 ∧ 𝐾 = (𝑁 + 1)) ∧ 𝑥 ∈ (𝑀...𝑁)) → (¬ (𝐹‘𝑥) ∈ (𝑀...𝑁) → 𝐾 = 𝑥)) |
345 | 344 | a1d 22 |
. . . . . . . . . . 11
⊢ (((𝜑 ∧ 𝐾 = (𝑁 + 1)) ∧ 𝑥 ∈ (𝑀...𝑁)) → (DECID (𝐹‘𝑥) ∈ (𝑀...𝑁) → (¬ (𝐹‘𝑥) ∈ (𝑀...𝑁) → 𝐾 = 𝑥))) |
346 | 345 | necon1addc 2440 |
. . . . . . . . . 10
⊢ (((𝜑 ∧ 𝐾 = (𝑁 + 1)) ∧ 𝑥 ∈ (𝑀...𝑁)) → (DECID (𝐹‘𝑥) ∈ (𝑀...𝑁) → (𝐾 ≠ 𝑥 → (𝐹‘𝑥) ∈ (𝑀...𝑁)))) |
347 | 332, 334,
346 | mp2d 47 |
. . . . . . . . 9
⊢ (((𝜑 ∧ 𝐾 = (𝑁 + 1)) ∧ 𝑥 ∈ (𝑀...𝑁)) → (𝐹‘𝑥) ∈ (𝑀...𝑁)) |
348 | 347 | fvresd 5571 |
. . . . . . . 8
⊢ (((𝜑 ∧ 𝐾 = (𝑁 + 1)) ∧ 𝑥 ∈ (𝑀...𝑁)) → ((𝐺 ↾ (𝑀...𝑁))‘(𝐹‘𝑥)) = (𝐺‘(𝐹‘𝑥))) |
349 | 323, 348 | eqtr2d 2227 |
. . . . . . 7
⊢ (((𝜑 ∧ 𝐾 = (𝑁 + 1)) ∧ 𝑥 ∈ (𝑀...𝑁)) → (𝐺‘(𝐹‘𝑥)) = ((𝐺 ↾ (𝑀...𝑁))‘(𝐽‘𝑥))) |
350 | 66, 207, 230 | syl2an 289 |
. . . . . . . 8
⊢ ((𝜑 ∧ 𝑥 ∈ (𝑀...𝑁)) → ((𝐺 ∘ 𝐹)‘𝑥) = (𝐺‘(𝐹‘𝑥))) |
351 | 350 | adantlr 477 |
. . . . . . 7
⊢ (((𝜑 ∧ 𝐾 = (𝑁 + 1)) ∧ 𝑥 ∈ (𝑀...𝑁)) → ((𝐺 ∘ 𝐹)‘𝑥) = (𝐺‘(𝐹‘𝑥))) |
352 | 152 | adantlr 477 |
. . . . . . 7
⊢ (((𝜑 ∧ 𝐾 = (𝑁 + 1)) ∧ 𝑥 ∈ (𝑀...𝑁)) → (((𝐺 ↾ (𝑀...𝑁)) ∘ 𝐽)‘𝑥) = ((𝐺 ↾ (𝑀...𝑁))‘(𝐽‘𝑥))) |
353 | 349, 351,
352 | 3eqtr4d 2236 |
. . . . . 6
⊢ (((𝜑 ∧ 𝐾 = (𝑁 + 1)) ∧ 𝑥 ∈ (𝑀...𝑁)) → ((𝐺 ∘ 𝐹)‘𝑥) = (((𝐺 ↾ (𝑀...𝑁)) ∘ 𝐽)‘𝑥)) |
354 | 238 | adantr 276 |
. . . . . 6
⊢ ((𝜑 ∧ 𝐾 = (𝑁 + 1)) → ((𝐺 ↾ (𝑀...𝑁)) ∘ 𝐽) ∈ V) |
355 | 302, 353,
304, 303, 354 | seqfveqg 10539 |
. . . . 5
⊢ ((𝜑 ∧ 𝐾 = (𝑁 + 1)) → (seq𝑀( + , (𝐺 ∘ 𝐹))‘𝑁) = (seq𝑀( + , ((𝐺 ↾ (𝑀...𝑁)) ∘ 𝐽))‘𝑁)) |
356 | | fvco3 5620 |
. . . . . . . 8
⊢ ((𝐹:(𝑀...(𝑁 + 1))⟶(𝑀...(𝑁 + 1)) ∧ (𝑁 + 1) ∈ (𝑀...(𝑁 + 1))) → ((𝐺 ∘ 𝐹)‘(𝑁 + 1)) = (𝐺‘(𝐹‘(𝑁 + 1)))) |
357 | 66, 90, 356 | syl2anc 411 |
. . . . . . 7
⊢ (𝜑 → ((𝐺 ∘ 𝐹)‘(𝑁 + 1)) = (𝐺‘(𝐹‘(𝑁 + 1)))) |
358 | 357 | adantr 276 |
. . . . . 6
⊢ ((𝜑 ∧ 𝐾 = (𝑁 + 1)) → ((𝐺 ∘ 𝐹)‘(𝑁 + 1)) = (𝐺‘(𝐹‘(𝑁 + 1)))) |
359 | | simpr 110 |
. . . . . . . . . 10
⊢ ((𝜑 ∧ 𝐾 = (𝑁 + 1)) → 𝐾 = (𝑁 + 1)) |
360 | 22, 359 | eqtr3id 2240 |
. . . . . . . . 9
⊢ ((𝜑 ∧ 𝐾 = (𝑁 + 1)) → (◡𝐹‘(𝑁 + 1)) = (𝑁 + 1)) |
361 | 360 | fveq2d 5550 |
. . . . . . . 8
⊢ ((𝜑 ∧ 𝐾 = (𝑁 + 1)) → (𝐹‘(◡𝐹‘(𝑁 + 1))) = (𝐹‘(𝑁 + 1))) |
362 | 168 | adantr 276 |
. . . . . . . 8
⊢ ((𝜑 ∧ 𝐾 = (𝑁 + 1)) → (𝐹‘(◡𝐹‘(𝑁 + 1))) = (𝑁 + 1)) |
363 | 361, 362 | eqtr3d 2228 |
. . . . . . 7
⊢ ((𝜑 ∧ 𝐾 = (𝑁 + 1)) → (𝐹‘(𝑁 + 1)) = (𝑁 + 1)) |
364 | 363 | fveq2d 5550 |
. . . . . 6
⊢ ((𝜑 ∧ 𝐾 = (𝑁 + 1)) → (𝐺‘(𝐹‘(𝑁 + 1))) = (𝐺‘(𝑁 + 1))) |
365 | 358, 364 | eqtrd 2226 |
. . . . 5
⊢ ((𝜑 ∧ 𝐾 = (𝑁 + 1)) → ((𝐺 ∘ 𝐹)‘(𝑁 + 1)) = (𝐺‘(𝑁 + 1))) |
366 | 355, 365 | oveq12d 5928 |
. . . 4
⊢ ((𝜑 ∧ 𝐾 = (𝑁 + 1)) → ((seq𝑀( + , (𝐺 ∘ 𝐹))‘𝑁) + ((𝐺 ∘ 𝐹)‘(𝑁 + 1))) = ((seq𝑀( + , ((𝐺 ↾ (𝑀...𝑁)) ∘ 𝐽))‘𝑁) + (𝐺‘(𝑁 + 1)))) |
367 | 306, 366 | eqtrd 2226 |
. . 3
⊢ ((𝜑 ∧ 𝐾 = (𝑁 + 1)) → (seq𝑀( + , (𝐺 ∘ 𝐹))‘(𝑁 + 1)) = ((seq𝑀( + , ((𝐺 ↾ (𝑀...𝑁)) ∘ 𝐽))‘𝑁) + (𝐺‘(𝑁 + 1)))) |
368 | | elfzp1 10128 |
. . . . 5
⊢ (𝑁 ∈
(ℤ≥‘𝑀) → (𝐾 ∈ (𝑀...(𝑁 + 1)) ↔ (𝐾 ∈ (𝑀...𝑁) ∨ 𝐾 = (𝑁 + 1)))) |
369 | 6, 368 | syl 14 |
. . . 4
⊢ (𝜑 → (𝐾 ∈ (𝑀...(𝑁 + 1)) ↔ (𝐾 ∈ (𝑀...𝑁) ∨ 𝐾 = (𝑁 + 1)))) |
370 | 92, 369 | mpbid 147 |
. . 3
⊢ (𝜑 → (𝐾 ∈ (𝑀...𝑁) ∨ 𝐾 = (𝑁 + 1))) |
371 | 301, 367,
370 | mpjaodan 799 |
. 2
⊢ (𝜑 → (seq𝑀( + , (𝐺 ∘ 𝐹))‘(𝑁 + 1)) = ((seq𝑀( + , ((𝐺 ↾ (𝑀...𝑁)) ∘ 𝐽))‘𝑁) + (𝐺‘(𝑁 + 1)))) |
372 | | seqp1g 10527 |
. . 3
⊢ ((𝑁 ∈
(ℤ≥‘𝑀) ∧ 𝐺 ∈ V ∧ + ∈ 𝑉) → (seq𝑀( + , 𝐺)‘(𝑁 + 1)) = ((seq𝑀( + , 𝐺)‘𝑁) + (𝐺‘(𝑁 + 1)))) |
373 | 6, 51, 19, 372 | syl3anc 1249 |
. 2
⊢ (𝜑 → (seq𝑀( + , 𝐺)‘(𝑁 + 1)) = ((seq𝑀( + , 𝐺)‘𝑁) + (𝐺‘(𝑁 + 1)))) |
374 | 56, 371, 373 | 3eqtr4d 2236 |
1
⊢ (𝜑 → (seq𝑀( + , (𝐺 ∘ 𝐹))‘(𝑁 + 1)) = (seq𝑀( + , 𝐺)‘(𝑁 + 1))) |