Step | Hyp | Ref
| Expression |
1 | | elfznn 12370 |
. . . . . . . 8
⊢ (𝑥 ∈ (1...sup((◡𝐺 “ (𝑀...𝑁)), ℝ, < )) → 𝑥 ∈
ℕ) |
2 | 1 | a1i 11 |
. . . . . . 7
⊢ ((𝜑 ∧ 𝑁 ∈ (ℤ≥‘(𝐺‘1))) → (𝑥 ∈ (1...sup((◡𝐺 “ (𝑀...𝑁)), ℝ, < )) → 𝑥 ∈
ℕ)) |
3 | | cnvimass 5485 |
. . . . . . . . 9
⊢ (◡𝐺 “ (𝑀...𝑁)) ⊆ dom 𝐺 |
4 | | isercoll.g |
. . . . . . . . . . 11
⊢ (𝜑 → 𝐺:ℕ⟶𝑍) |
5 | 4 | adantr 481 |
. . . . . . . . . 10
⊢ ((𝜑 ∧ 𝑁 ∈ (ℤ≥‘(𝐺‘1))) → 𝐺:ℕ⟶𝑍) |
6 | | fdm 6051 |
. . . . . . . . . 10
⊢ (𝐺:ℕ⟶𝑍 → dom 𝐺 = ℕ) |
7 | 5, 6 | syl 17 |
. . . . . . . . 9
⊢ ((𝜑 ∧ 𝑁 ∈ (ℤ≥‘(𝐺‘1))) → dom 𝐺 = ℕ) |
8 | 3, 7 | syl5sseq 3653 |
. . . . . . . 8
⊢ ((𝜑 ∧ 𝑁 ∈ (ℤ≥‘(𝐺‘1))) → (◡𝐺 “ (𝑀...𝑁)) ⊆ ℕ) |
9 | 8 | sseld 3602 |
. . . . . . 7
⊢ ((𝜑 ∧ 𝑁 ∈ (ℤ≥‘(𝐺‘1))) → (𝑥 ∈ (◡𝐺 “ (𝑀...𝑁)) → 𝑥 ∈ ℕ)) |
10 | | id 22 |
. . . . . . . . . . 11
⊢ (𝑥 ∈ ℕ → 𝑥 ∈
ℕ) |
11 | | nnuz 11723 |
. . . . . . . . . . 11
⊢ ℕ =
(ℤ≥‘1) |
12 | 10, 11 | syl6eleq 2711 |
. . . . . . . . . 10
⊢ (𝑥 ∈ ℕ → 𝑥 ∈
(ℤ≥‘1)) |
13 | | ltso 10118 |
. . . . . . . . . . . . . 14
⊢ < Or
ℝ |
14 | 13 | a1i 11 |
. . . . . . . . . . . . 13
⊢ ((𝜑 ∧ 𝑁 ∈ (ℤ≥‘(𝐺‘1))) → < Or
ℝ) |
15 | | fzfid 12772 |
. . . . . . . . . . . . . . 15
⊢ ((𝜑 ∧ 𝑁 ∈ (ℤ≥‘(𝐺‘1))) → (𝑀...𝑁) ∈ Fin) |
16 | | ffun 6048 |
. . . . . . . . . . . . . . . . 17
⊢ (𝐺:ℕ⟶𝑍 → Fun 𝐺) |
17 | | funimacnv 5970 |
. . . . . . . . . . . . . . . . 17
⊢ (Fun
𝐺 → (𝐺 “ (◡𝐺 “ (𝑀...𝑁))) = ((𝑀...𝑁) ∩ ran 𝐺)) |
18 | 5, 16, 17 | 3syl 18 |
. . . . . . . . . . . . . . . 16
⊢ ((𝜑 ∧ 𝑁 ∈ (ℤ≥‘(𝐺‘1))) → (𝐺 “ (◡𝐺 “ (𝑀...𝑁))) = ((𝑀...𝑁) ∩ ran 𝐺)) |
19 | | inss1 3833 |
. . . . . . . . . . . . . . . 16
⊢ ((𝑀...𝑁) ∩ ran 𝐺) ⊆ (𝑀...𝑁) |
20 | 18, 19 | syl6eqss 3655 |
. . . . . . . . . . . . . . 15
⊢ ((𝜑 ∧ 𝑁 ∈ (ℤ≥‘(𝐺‘1))) → (𝐺 “ (◡𝐺 “ (𝑀...𝑁))) ⊆ (𝑀...𝑁)) |
21 | | ssfi 8180 |
. . . . . . . . . . . . . . 15
⊢ (((𝑀...𝑁) ∈ Fin ∧ (𝐺 “ (◡𝐺 “ (𝑀...𝑁))) ⊆ (𝑀...𝑁)) → (𝐺 “ (◡𝐺 “ (𝑀...𝑁))) ∈ Fin) |
22 | 15, 20, 21 | syl2anc 693 |
. . . . . . . . . . . . . 14
⊢ ((𝜑 ∧ 𝑁 ∈ (ℤ≥‘(𝐺‘1))) → (𝐺 “ (◡𝐺 “ (𝑀...𝑁))) ∈ Fin) |
23 | | ssid 3624 |
. . . . . . . . . . . . . . . . . . . . 21
⊢ ℕ
⊆ ℕ |
24 | | isercoll.z |
. . . . . . . . . . . . . . . . . . . . . 22
⊢ 𝑍 =
(ℤ≥‘𝑀) |
25 | | isercoll.m |
. . . . . . . . . . . . . . . . . . . . . 22
⊢ (𝜑 → 𝑀 ∈ ℤ) |
26 | | isercoll.i |
. . . . . . . . . . . . . . . . . . . . . 22
⊢ ((𝜑 ∧ 𝑘 ∈ ℕ) → (𝐺‘𝑘) < (𝐺‘(𝑘 + 1))) |
27 | 24, 25, 4, 26 | isercolllem1 14395 |
. . . . . . . . . . . . . . . . . . . . 21
⊢ ((𝜑 ∧ ℕ ⊆ ℕ)
→ (𝐺 ↾ ℕ)
Isom < , < (ℕ, (𝐺 “ ℕ))) |
28 | 23, 27 | mpan2 707 |
. . . . . . . . . . . . . . . . . . . 20
⊢ (𝜑 → (𝐺 ↾ ℕ) Isom < , < (ℕ,
(𝐺 “
ℕ))) |
29 | | ffn 6045 |
. . . . . . . . . . . . . . . . . . . . 21
⊢ (𝐺:ℕ⟶𝑍 → 𝐺 Fn ℕ) |
30 | | fnresdm 6000 |
. . . . . . . . . . . . . . . . . . . . 21
⊢ (𝐺 Fn ℕ → (𝐺 ↾ ℕ) = 𝐺) |
31 | | isoeq1 6567 |
. . . . . . . . . . . . . . . . . . . . 21
⊢ ((𝐺 ↾ ℕ) = 𝐺 → ((𝐺 ↾ ℕ) Isom < , < (ℕ,
(𝐺 “ ℕ)) ↔
𝐺 Isom < , <
(ℕ, (𝐺 “
ℕ)))) |
32 | 4, 29, 30, 31 | 4syl 19 |
. . . . . . . . . . . . . . . . . . . 20
⊢ (𝜑 → ((𝐺 ↾ ℕ) Isom < , < (ℕ,
(𝐺 “ ℕ)) ↔
𝐺 Isom < , <
(ℕ, (𝐺 “
ℕ)))) |
33 | 28, 32 | mpbid 222 |
. . . . . . . . . . . . . . . . . . 19
⊢ (𝜑 → 𝐺 Isom < , < (ℕ, (𝐺 “
ℕ))) |
34 | | isof1o 6573 |
. . . . . . . . . . . . . . . . . . 19
⊢ (𝐺 Isom < , < (ℕ,
(𝐺 “ ℕ)) →
𝐺:ℕ–1-1-onto→(𝐺 “ ℕ)) |
35 | | f1ocnv 6149 |
. . . . . . . . . . . . . . . . . . 19
⊢ (𝐺:ℕ–1-1-onto→(𝐺 “ ℕ) → ◡𝐺:(𝐺 “ ℕ)–1-1-onto→ℕ) |
36 | | f1ofun 6139 |
. . . . . . . . . . . . . . . . . . 19
⊢ (◡𝐺:(𝐺 “ ℕ)–1-1-onto→ℕ → Fun ◡𝐺) |
37 | 33, 34, 35, 36 | 4syl 19 |
. . . . . . . . . . . . . . . . . 18
⊢ (𝜑 → Fun ◡𝐺) |
38 | | df-f1 5893 |
. . . . . . . . . . . . . . . . . 18
⊢ (𝐺:ℕ–1-1→𝑍 ↔ (𝐺:ℕ⟶𝑍 ∧ Fun ◡𝐺)) |
39 | 4, 37, 38 | sylanbrc 698 |
. . . . . . . . . . . . . . . . 17
⊢ (𝜑 → 𝐺:ℕ–1-1→𝑍) |
40 | 39 | adantr 481 |
. . . . . . . . . . . . . . . 16
⊢ ((𝜑 ∧ 𝑁 ∈ (ℤ≥‘(𝐺‘1))) → 𝐺:ℕ–1-1→𝑍) |
41 | | nnex 11026 |
. . . . . . . . . . . . . . . . 17
⊢ ℕ
∈ V |
42 | | ssexg 4804 |
. . . . . . . . . . . . . . . . 17
⊢ (((◡𝐺 “ (𝑀...𝑁)) ⊆ ℕ ∧ ℕ ∈ V)
→ (◡𝐺 “ (𝑀...𝑁)) ∈ V) |
43 | 8, 41, 42 | sylancl 694 |
. . . . . . . . . . . . . . . 16
⊢ ((𝜑 ∧ 𝑁 ∈ (ℤ≥‘(𝐺‘1))) → (◡𝐺 “ (𝑀...𝑁)) ∈ V) |
44 | | f1imaeng 8016 |
. . . . . . . . . . . . . . . 16
⊢ ((𝐺:ℕ–1-1→𝑍 ∧ (◡𝐺 “ (𝑀...𝑁)) ⊆ ℕ ∧ (◡𝐺 “ (𝑀...𝑁)) ∈ V) → (𝐺 “ (◡𝐺 “ (𝑀...𝑁))) ≈ (◡𝐺 “ (𝑀...𝑁))) |
45 | 40, 8, 43, 44 | syl3anc 1326 |
. . . . . . . . . . . . . . 15
⊢ ((𝜑 ∧ 𝑁 ∈ (ℤ≥‘(𝐺‘1))) → (𝐺 “ (◡𝐺 “ (𝑀...𝑁))) ≈ (◡𝐺 “ (𝑀...𝑁))) |
46 | 45 | ensymd 8007 |
. . . . . . . . . . . . . 14
⊢ ((𝜑 ∧ 𝑁 ∈ (ℤ≥‘(𝐺‘1))) → (◡𝐺 “ (𝑀...𝑁)) ≈ (𝐺 “ (◡𝐺 “ (𝑀...𝑁)))) |
47 | | enfii 8177 |
. . . . . . . . . . . . . 14
⊢ (((𝐺 “ (◡𝐺 “ (𝑀...𝑁))) ∈ Fin ∧ (◡𝐺 “ (𝑀...𝑁)) ≈ (𝐺 “ (◡𝐺 “ (𝑀...𝑁)))) → (◡𝐺 “ (𝑀...𝑁)) ∈ Fin) |
48 | 22, 46, 47 | syl2anc 693 |
. . . . . . . . . . . . 13
⊢ ((𝜑 ∧ 𝑁 ∈ (ℤ≥‘(𝐺‘1))) → (◡𝐺 “ (𝑀...𝑁)) ∈ Fin) |
49 | | 1nn 11031 |
. . . . . . . . . . . . . . . 16
⊢ 1 ∈
ℕ |
50 | 49 | a1i 11 |
. . . . . . . . . . . . . . 15
⊢ ((𝜑 ∧ 𝑁 ∈ (ℤ≥‘(𝐺‘1))) → 1 ∈
ℕ) |
51 | | ffvelrn 6357 |
. . . . . . . . . . . . . . . . . . 19
⊢ ((𝐺:ℕ⟶𝑍 ∧ 1 ∈ ℕ) →
(𝐺‘1) ∈ 𝑍) |
52 | 4, 49, 51 | sylancl 694 |
. . . . . . . . . . . . . . . . . 18
⊢ (𝜑 → (𝐺‘1) ∈ 𝑍) |
53 | 52, 24 | syl6eleq 2711 |
. . . . . . . . . . . . . . . . 17
⊢ (𝜑 → (𝐺‘1) ∈
(ℤ≥‘𝑀)) |
54 | 53 | adantr 481 |
. . . . . . . . . . . . . . . 16
⊢ ((𝜑 ∧ 𝑁 ∈ (ℤ≥‘(𝐺‘1))) → (𝐺‘1) ∈
(ℤ≥‘𝑀)) |
55 | | simpr 477 |
. . . . . . . . . . . . . . . 16
⊢ ((𝜑 ∧ 𝑁 ∈ (ℤ≥‘(𝐺‘1))) → 𝑁 ∈
(ℤ≥‘(𝐺‘1))) |
56 | | elfzuzb 12336 |
. . . . . . . . . . . . . . . 16
⊢ ((𝐺‘1) ∈ (𝑀...𝑁) ↔ ((𝐺‘1) ∈
(ℤ≥‘𝑀) ∧ 𝑁 ∈ (ℤ≥‘(𝐺‘1)))) |
57 | 54, 55, 56 | sylanbrc 698 |
. . . . . . . . . . . . . . 15
⊢ ((𝜑 ∧ 𝑁 ∈ (ℤ≥‘(𝐺‘1))) → (𝐺‘1) ∈ (𝑀...𝑁)) |
58 | 5, 29 | syl 17 |
. . . . . . . . . . . . . . . 16
⊢ ((𝜑 ∧ 𝑁 ∈ (ℤ≥‘(𝐺‘1))) → 𝐺 Fn ℕ) |
59 | | elpreima 6337 |
. . . . . . . . . . . . . . . 16
⊢ (𝐺 Fn ℕ → (1 ∈
(◡𝐺 “ (𝑀...𝑁)) ↔ (1 ∈ ℕ ∧ (𝐺‘1) ∈ (𝑀...𝑁)))) |
60 | 58, 59 | syl 17 |
. . . . . . . . . . . . . . 15
⊢ ((𝜑 ∧ 𝑁 ∈ (ℤ≥‘(𝐺‘1))) → (1 ∈
(◡𝐺 “ (𝑀...𝑁)) ↔ (1 ∈ ℕ ∧ (𝐺‘1) ∈ (𝑀...𝑁)))) |
61 | 50, 57, 60 | mpbir2and 957 |
. . . . . . . . . . . . . 14
⊢ ((𝜑 ∧ 𝑁 ∈ (ℤ≥‘(𝐺‘1))) → 1 ∈
(◡𝐺 “ (𝑀...𝑁))) |
62 | | ne0i 3921 |
. . . . . . . . . . . . . 14
⊢ (1 ∈
(◡𝐺 “ (𝑀...𝑁)) → (◡𝐺 “ (𝑀...𝑁)) ≠ ∅) |
63 | 61, 62 | syl 17 |
. . . . . . . . . . . . 13
⊢ ((𝜑 ∧ 𝑁 ∈ (ℤ≥‘(𝐺‘1))) → (◡𝐺 “ (𝑀...𝑁)) ≠ ∅) |
64 | | nnssre 11024 |
. . . . . . . . . . . . . 14
⊢ ℕ
⊆ ℝ |
65 | 8, 64 | syl6ss 3615 |
. . . . . . . . . . . . 13
⊢ ((𝜑 ∧ 𝑁 ∈ (ℤ≥‘(𝐺‘1))) → (◡𝐺 “ (𝑀...𝑁)) ⊆ ℝ) |
66 | | fisupcl 8375 |
. . . . . . . . . . . . 13
⊢ (( <
Or ℝ ∧ ((◡𝐺 “ (𝑀...𝑁)) ∈ Fin ∧ (◡𝐺 “ (𝑀...𝑁)) ≠ ∅ ∧ (◡𝐺 “ (𝑀...𝑁)) ⊆ ℝ)) → sup((◡𝐺 “ (𝑀...𝑁)), ℝ, < ) ∈ (◡𝐺 “ (𝑀...𝑁))) |
67 | 14, 48, 63, 65, 66 | syl13anc 1328 |
. . . . . . . . . . . 12
⊢ ((𝜑 ∧ 𝑁 ∈ (ℤ≥‘(𝐺‘1))) → sup((◡𝐺 “ (𝑀...𝑁)), ℝ, < ) ∈ (◡𝐺 “ (𝑀...𝑁))) |
68 | 8, 67 | sseldd 3604 |
. . . . . . . . . . 11
⊢ ((𝜑 ∧ 𝑁 ∈ (ℤ≥‘(𝐺‘1))) → sup((◡𝐺 “ (𝑀...𝑁)), ℝ, < ) ∈
ℕ) |
69 | 68 | nnzd 11481 |
. . . . . . . . . 10
⊢ ((𝜑 ∧ 𝑁 ∈ (ℤ≥‘(𝐺‘1))) → sup((◡𝐺 “ (𝑀...𝑁)), ℝ, < ) ∈
ℤ) |
70 | | elfz5 12334 |
. . . . . . . . . 10
⊢ ((𝑥 ∈
(ℤ≥‘1) ∧ sup((◡𝐺 “ (𝑀...𝑁)), ℝ, < ) ∈ ℤ) →
(𝑥 ∈ (1...sup((◡𝐺 “ (𝑀...𝑁)), ℝ, < )) ↔ 𝑥 ≤ sup((◡𝐺 “ (𝑀...𝑁)), ℝ, < ))) |
71 | 12, 69, 70 | syl2anr 495 |
. . . . . . . . 9
⊢ (((𝜑 ∧ 𝑁 ∈ (ℤ≥‘(𝐺‘1))) ∧ 𝑥 ∈ ℕ) → (𝑥 ∈ (1...sup((◡𝐺 “ (𝑀...𝑁)), ℝ, < )) ↔ 𝑥 ≤ sup((◡𝐺 “ (𝑀...𝑁)), ℝ, < ))) |
72 | | elpreima 6337 |
. . . . . . . . . . . . . . . . . 18
⊢ (𝐺 Fn ℕ → (sup((◡𝐺 “ (𝑀...𝑁)), ℝ, < ) ∈ (◡𝐺 “ (𝑀...𝑁)) ↔ (sup((◡𝐺 “ (𝑀...𝑁)), ℝ, < ) ∈ ℕ ∧
(𝐺‘sup((◡𝐺 “ (𝑀...𝑁)), ℝ, < )) ∈ (𝑀...𝑁)))) |
73 | 58, 72 | syl 17 |
. . . . . . . . . . . . . . . . 17
⊢ ((𝜑 ∧ 𝑁 ∈ (ℤ≥‘(𝐺‘1))) → (sup((◡𝐺 “ (𝑀...𝑁)), ℝ, < ) ∈ (◡𝐺 “ (𝑀...𝑁)) ↔ (sup((◡𝐺 “ (𝑀...𝑁)), ℝ, < ) ∈ ℕ ∧
(𝐺‘sup((◡𝐺 “ (𝑀...𝑁)), ℝ, < )) ∈ (𝑀...𝑁)))) |
74 | 67, 73 | mpbid 222 |
. . . . . . . . . . . . . . . 16
⊢ ((𝜑 ∧ 𝑁 ∈ (ℤ≥‘(𝐺‘1))) → (sup((◡𝐺 “ (𝑀...𝑁)), ℝ, < ) ∈ ℕ ∧
(𝐺‘sup((◡𝐺 “ (𝑀...𝑁)), ℝ, < )) ∈ (𝑀...𝑁))) |
75 | 74 | simprd 479 |
. . . . . . . . . . . . . . 15
⊢ ((𝜑 ∧ 𝑁 ∈ (ℤ≥‘(𝐺‘1))) → (𝐺‘sup((◡𝐺 “ (𝑀...𝑁)), ℝ, < )) ∈ (𝑀...𝑁)) |
76 | | elfzle2 12345 |
. . . . . . . . . . . . . . 15
⊢ ((𝐺‘sup((◡𝐺 “ (𝑀...𝑁)), ℝ, < )) ∈ (𝑀...𝑁) → (𝐺‘sup((◡𝐺 “ (𝑀...𝑁)), ℝ, < )) ≤ 𝑁) |
77 | 75, 76 | syl 17 |
. . . . . . . . . . . . . 14
⊢ ((𝜑 ∧ 𝑁 ∈ (ℤ≥‘(𝐺‘1))) → (𝐺‘sup((◡𝐺 “ (𝑀...𝑁)), ℝ, < )) ≤ 𝑁) |
78 | 77 | adantr 481 |
. . . . . . . . . . . . 13
⊢ (((𝜑 ∧ 𝑁 ∈ (ℤ≥‘(𝐺‘1))) ∧ 𝑥 ∈ ℕ) → (𝐺‘sup((◡𝐺 “ (𝑀...𝑁)), ℝ, < )) ≤ 𝑁) |
79 | | uzssz 11707 |
. . . . . . . . . . . . . . . . 17
⊢
(ℤ≥‘𝑀) ⊆ ℤ |
80 | 24, 79 | eqsstri 3635 |
. . . . . . . . . . . . . . . 16
⊢ 𝑍 ⊆
ℤ |
81 | | zssre 11384 |
. . . . . . . . . . . . . . . 16
⊢ ℤ
⊆ ℝ |
82 | 80, 81 | sstri 3612 |
. . . . . . . . . . . . . . 15
⊢ 𝑍 ⊆
ℝ |
83 | 5 | ffvelrnda 6359 |
. . . . . . . . . . . . . . 15
⊢ (((𝜑 ∧ 𝑁 ∈ (ℤ≥‘(𝐺‘1))) ∧ 𝑥 ∈ ℕ) → (𝐺‘𝑥) ∈ 𝑍) |
84 | 82, 83 | sseldi 3601 |
. . . . . . . . . . . . . 14
⊢ (((𝜑 ∧ 𝑁 ∈ (ℤ≥‘(𝐺‘1))) ∧ 𝑥 ∈ ℕ) → (𝐺‘𝑥) ∈ ℝ) |
85 | 5, 68 | ffvelrnd 6360 |
. . . . . . . . . . . . . . . 16
⊢ ((𝜑 ∧ 𝑁 ∈ (ℤ≥‘(𝐺‘1))) → (𝐺‘sup((◡𝐺 “ (𝑀...𝑁)), ℝ, < )) ∈ 𝑍) |
86 | 85 | adantr 481 |
. . . . . . . . . . . . . . 15
⊢ (((𝜑 ∧ 𝑁 ∈ (ℤ≥‘(𝐺‘1))) ∧ 𝑥 ∈ ℕ) → (𝐺‘sup((◡𝐺 “ (𝑀...𝑁)), ℝ, < )) ∈ 𝑍) |
87 | 82, 86 | sseldi 3601 |
. . . . . . . . . . . . . 14
⊢ (((𝜑 ∧ 𝑁 ∈ (ℤ≥‘(𝐺‘1))) ∧ 𝑥 ∈ ℕ) → (𝐺‘sup((◡𝐺 “ (𝑀...𝑁)), ℝ, < )) ∈
ℝ) |
88 | | eluzelz 11697 |
. . . . . . . . . . . . . . . 16
⊢ (𝑁 ∈
(ℤ≥‘(𝐺‘1)) → 𝑁 ∈ ℤ) |
89 | 88 | ad2antlr 763 |
. . . . . . . . . . . . . . 15
⊢ (((𝜑 ∧ 𝑁 ∈ (ℤ≥‘(𝐺‘1))) ∧ 𝑥 ∈ ℕ) → 𝑁 ∈
ℤ) |
90 | 81, 89 | sseldi 3601 |
. . . . . . . . . . . . . 14
⊢ (((𝜑 ∧ 𝑁 ∈ (ℤ≥‘(𝐺‘1))) ∧ 𝑥 ∈ ℕ) → 𝑁 ∈
ℝ) |
91 | | letr 10131 |
. . . . . . . . . . . . . 14
⊢ (((𝐺‘𝑥) ∈ ℝ ∧ (𝐺‘sup((◡𝐺 “ (𝑀...𝑁)), ℝ, < )) ∈ ℝ ∧
𝑁 ∈ ℝ) →
(((𝐺‘𝑥) ≤ (𝐺‘sup((◡𝐺 “ (𝑀...𝑁)), ℝ, < )) ∧ (𝐺‘sup((◡𝐺 “ (𝑀...𝑁)), ℝ, < )) ≤ 𝑁) → (𝐺‘𝑥) ≤ 𝑁)) |
92 | 84, 87, 90, 91 | syl3anc 1326 |
. . . . . . . . . . . . 13
⊢ (((𝜑 ∧ 𝑁 ∈ (ℤ≥‘(𝐺‘1))) ∧ 𝑥 ∈ ℕ) → (((𝐺‘𝑥) ≤ (𝐺‘sup((◡𝐺 “ (𝑀...𝑁)), ℝ, < )) ∧ (𝐺‘sup((◡𝐺 “ (𝑀...𝑁)), ℝ, < )) ≤ 𝑁) → (𝐺‘𝑥) ≤ 𝑁)) |
93 | 78, 92 | mpan2d 710 |
. . . . . . . . . . . 12
⊢ (((𝜑 ∧ 𝑁 ∈ (ℤ≥‘(𝐺‘1))) ∧ 𝑥 ∈ ℕ) → ((𝐺‘𝑥) ≤ (𝐺‘sup((◡𝐺 “ (𝑀...𝑁)), ℝ, < )) → (𝐺‘𝑥) ≤ 𝑁)) |
94 | 33 | ad2antrr 762 |
. . . . . . . . . . . . 13
⊢ (((𝜑 ∧ 𝑁 ∈ (ℤ≥‘(𝐺‘1))) ∧ 𝑥 ∈ ℕ) → 𝐺 Isom < , < (ℕ,
(𝐺 “
ℕ))) |
95 | 64 | a1i 11 |
. . . . . . . . . . . . . 14
⊢ (((𝜑 ∧ 𝑁 ∈ (ℤ≥‘(𝐺‘1))) ∧ 𝑥 ∈ ℕ) → ℕ
⊆ ℝ) |
96 | | ressxr 10083 |
. . . . . . . . . . . . . 14
⊢ ℝ
⊆ ℝ* |
97 | 95, 96 | syl6ss 3615 |
. . . . . . . . . . . . 13
⊢ (((𝜑 ∧ 𝑁 ∈ (ℤ≥‘(𝐺‘1))) ∧ 𝑥 ∈ ℕ) → ℕ
⊆ ℝ*) |
98 | | imassrn 5477 |
. . . . . . . . . . . . . . . 16
⊢ (𝐺 “ ℕ) ⊆ ran
𝐺 |
99 | 4 | ad2antrr 762 |
. . . . . . . . . . . . . . . . 17
⊢ (((𝜑 ∧ 𝑁 ∈ (ℤ≥‘(𝐺‘1))) ∧ 𝑥 ∈ ℕ) → 𝐺:ℕ⟶𝑍) |
100 | | frn 6053 |
. . . . . . . . . . . . . . . . 17
⊢ (𝐺:ℕ⟶𝑍 → ran 𝐺 ⊆ 𝑍) |
101 | 99, 100 | syl 17 |
. . . . . . . . . . . . . . . 16
⊢ (((𝜑 ∧ 𝑁 ∈ (ℤ≥‘(𝐺‘1))) ∧ 𝑥 ∈ ℕ) → ran
𝐺 ⊆ 𝑍) |
102 | 98, 101 | syl5ss 3614 |
. . . . . . . . . . . . . . 15
⊢ (((𝜑 ∧ 𝑁 ∈ (ℤ≥‘(𝐺‘1))) ∧ 𝑥 ∈ ℕ) → (𝐺 “ ℕ) ⊆ 𝑍) |
103 | 102, 82 | syl6ss 3615 |
. . . . . . . . . . . . . 14
⊢ (((𝜑 ∧ 𝑁 ∈ (ℤ≥‘(𝐺‘1))) ∧ 𝑥 ∈ ℕ) → (𝐺 “ ℕ) ⊆
ℝ) |
104 | 103, 96 | syl6ss 3615 |
. . . . . . . . . . . . 13
⊢ (((𝜑 ∧ 𝑁 ∈ (ℤ≥‘(𝐺‘1))) ∧ 𝑥 ∈ ℕ) → (𝐺 “ ℕ) ⊆
ℝ*) |
105 | | simpr 477 |
. . . . . . . . . . . . 13
⊢ (((𝜑 ∧ 𝑁 ∈ (ℤ≥‘(𝐺‘1))) ∧ 𝑥 ∈ ℕ) → 𝑥 ∈
ℕ) |
106 | 68 | adantr 481 |
. . . . . . . . . . . . 13
⊢ (((𝜑 ∧ 𝑁 ∈ (ℤ≥‘(𝐺‘1))) ∧ 𝑥 ∈ ℕ) →
sup((◡𝐺 “ (𝑀...𝑁)), ℝ, < ) ∈
ℕ) |
107 | | leisorel 13244 |
. . . . . . . . . . . . 13
⊢ ((𝐺 Isom < , < (ℕ,
(𝐺 “ ℕ)) ∧
(ℕ ⊆ ℝ* ∧ (𝐺 “ ℕ) ⊆
ℝ*) ∧ (𝑥 ∈ ℕ ∧ sup((◡𝐺 “ (𝑀...𝑁)), ℝ, < ) ∈ ℕ)) →
(𝑥 ≤ sup((◡𝐺 “ (𝑀...𝑁)), ℝ, < ) ↔ (𝐺‘𝑥) ≤ (𝐺‘sup((◡𝐺 “ (𝑀...𝑁)), ℝ, < )))) |
108 | 94, 97, 104, 105, 106, 107 | syl122anc 1335 |
. . . . . . . . . . . 12
⊢ (((𝜑 ∧ 𝑁 ∈ (ℤ≥‘(𝐺‘1))) ∧ 𝑥 ∈ ℕ) → (𝑥 ≤ sup((◡𝐺 “ (𝑀...𝑁)), ℝ, < ) ↔ (𝐺‘𝑥) ≤ (𝐺‘sup((◡𝐺 “ (𝑀...𝑁)), ℝ, < )))) |
109 | 83, 24 | syl6eleq 2711 |
. . . . . . . . . . . . 13
⊢ (((𝜑 ∧ 𝑁 ∈ (ℤ≥‘(𝐺‘1))) ∧ 𝑥 ∈ ℕ) → (𝐺‘𝑥) ∈ (ℤ≥‘𝑀)) |
110 | | elfz5 12334 |
. . . . . . . . . . . . 13
⊢ (((𝐺‘𝑥) ∈ (ℤ≥‘𝑀) ∧ 𝑁 ∈ ℤ) → ((𝐺‘𝑥) ∈ (𝑀...𝑁) ↔ (𝐺‘𝑥) ≤ 𝑁)) |
111 | 109, 89, 110 | syl2anc 693 |
. . . . . . . . . . . 12
⊢ (((𝜑 ∧ 𝑁 ∈ (ℤ≥‘(𝐺‘1))) ∧ 𝑥 ∈ ℕ) → ((𝐺‘𝑥) ∈ (𝑀...𝑁) ↔ (𝐺‘𝑥) ≤ 𝑁)) |
112 | 93, 108, 111 | 3imtr4d 283 |
. . . . . . . . . . 11
⊢ (((𝜑 ∧ 𝑁 ∈ (ℤ≥‘(𝐺‘1))) ∧ 𝑥 ∈ ℕ) → (𝑥 ≤ sup((◡𝐺 “ (𝑀...𝑁)), ℝ, < ) → (𝐺‘𝑥) ∈ (𝑀...𝑁))) |
113 | | elpreima 6337 |
. . . . . . . . . . . . 13
⊢ (𝐺 Fn ℕ → (𝑥 ∈ (◡𝐺 “ (𝑀...𝑁)) ↔ (𝑥 ∈ ℕ ∧ (𝐺‘𝑥) ∈ (𝑀...𝑁)))) |
114 | 113 | baibd 948 |
. . . . . . . . . . . 12
⊢ ((𝐺 Fn ℕ ∧ 𝑥 ∈ ℕ) → (𝑥 ∈ (◡𝐺 “ (𝑀...𝑁)) ↔ (𝐺‘𝑥) ∈ (𝑀...𝑁))) |
115 | 58, 114 | sylan 488 |
. . . . . . . . . . 11
⊢ (((𝜑 ∧ 𝑁 ∈ (ℤ≥‘(𝐺‘1))) ∧ 𝑥 ∈ ℕ) → (𝑥 ∈ (◡𝐺 “ (𝑀...𝑁)) ↔ (𝐺‘𝑥) ∈ (𝑀...𝑁))) |
116 | 112, 115 | sylibrd 249 |
. . . . . . . . . 10
⊢ (((𝜑 ∧ 𝑁 ∈ (ℤ≥‘(𝐺‘1))) ∧ 𝑥 ∈ ℕ) → (𝑥 ≤ sup((◡𝐺 “ (𝑀...𝑁)), ℝ, < ) → 𝑥 ∈ (◡𝐺 “ (𝑀...𝑁)))) |
117 | | fimaxre2 10969 |
. . . . . . . . . . . . 13
⊢ (((◡𝐺 “ (𝑀...𝑁)) ⊆ ℝ ∧ (◡𝐺 “ (𝑀...𝑁)) ∈ Fin) → ∃𝑥 ∈ ℝ ∀𝑦 ∈ (◡𝐺 “ (𝑀...𝑁))𝑦 ≤ 𝑥) |
118 | 65, 48, 117 | syl2anc 693 |
. . . . . . . . . . . 12
⊢ ((𝜑 ∧ 𝑁 ∈ (ℤ≥‘(𝐺‘1))) → ∃𝑥 ∈ ℝ ∀𝑦 ∈ (◡𝐺 “ (𝑀...𝑁))𝑦 ≤ 𝑥) |
119 | | suprub 10984 |
. . . . . . . . . . . . 13
⊢ ((((◡𝐺 “ (𝑀...𝑁)) ⊆ ℝ ∧ (◡𝐺 “ (𝑀...𝑁)) ≠ ∅ ∧ ∃𝑥 ∈ ℝ ∀𝑦 ∈ (◡𝐺 “ (𝑀...𝑁))𝑦 ≤ 𝑥) ∧ 𝑥 ∈ (◡𝐺 “ (𝑀...𝑁))) → 𝑥 ≤ sup((◡𝐺 “ (𝑀...𝑁)), ℝ, < )) |
120 | 119 | ex 450 |
. . . . . . . . . . . 12
⊢ (((◡𝐺 “ (𝑀...𝑁)) ⊆ ℝ ∧ (◡𝐺 “ (𝑀...𝑁)) ≠ ∅ ∧ ∃𝑥 ∈ ℝ ∀𝑦 ∈ (◡𝐺 “ (𝑀...𝑁))𝑦 ≤ 𝑥) → (𝑥 ∈ (◡𝐺 “ (𝑀...𝑁)) → 𝑥 ≤ sup((◡𝐺 “ (𝑀...𝑁)), ℝ, < ))) |
121 | 65, 63, 118, 120 | syl3anc 1326 |
. . . . . . . . . . 11
⊢ ((𝜑 ∧ 𝑁 ∈ (ℤ≥‘(𝐺‘1))) → (𝑥 ∈ (◡𝐺 “ (𝑀...𝑁)) → 𝑥 ≤ sup((◡𝐺 “ (𝑀...𝑁)), ℝ, < ))) |
122 | 121 | adantr 481 |
. . . . . . . . . 10
⊢ (((𝜑 ∧ 𝑁 ∈ (ℤ≥‘(𝐺‘1))) ∧ 𝑥 ∈ ℕ) → (𝑥 ∈ (◡𝐺 “ (𝑀...𝑁)) → 𝑥 ≤ sup((◡𝐺 “ (𝑀...𝑁)), ℝ, < ))) |
123 | 116, 122 | impbid 202 |
. . . . . . . . 9
⊢ (((𝜑 ∧ 𝑁 ∈ (ℤ≥‘(𝐺‘1))) ∧ 𝑥 ∈ ℕ) → (𝑥 ≤ sup((◡𝐺 “ (𝑀...𝑁)), ℝ, < ) ↔ 𝑥 ∈ (◡𝐺 “ (𝑀...𝑁)))) |
124 | 71, 123 | bitrd 268 |
. . . . . . . 8
⊢ (((𝜑 ∧ 𝑁 ∈ (ℤ≥‘(𝐺‘1))) ∧ 𝑥 ∈ ℕ) → (𝑥 ∈ (1...sup((◡𝐺 “ (𝑀...𝑁)), ℝ, < )) ↔ 𝑥 ∈ (◡𝐺 “ (𝑀...𝑁)))) |
125 | 124 | ex 450 |
. . . . . . 7
⊢ ((𝜑 ∧ 𝑁 ∈ (ℤ≥‘(𝐺‘1))) → (𝑥 ∈ ℕ → (𝑥 ∈ (1...sup((◡𝐺 “ (𝑀...𝑁)), ℝ, < )) ↔ 𝑥 ∈ (◡𝐺 “ (𝑀...𝑁))))) |
126 | 2, 9, 125 | pm5.21ndd 369 |
. . . . . 6
⊢ ((𝜑 ∧ 𝑁 ∈ (ℤ≥‘(𝐺‘1))) → (𝑥 ∈ (1...sup((◡𝐺 “ (𝑀...𝑁)), ℝ, < )) ↔ 𝑥 ∈ (◡𝐺 “ (𝑀...𝑁)))) |
127 | 126 | eqrdv 2620 |
. . . . 5
⊢ ((𝜑 ∧ 𝑁 ∈ (ℤ≥‘(𝐺‘1))) →
(1...sup((◡𝐺 “ (𝑀...𝑁)), ℝ, < )) = (◡𝐺 “ (𝑀...𝑁))) |
128 | 127 | fveq2d 6195 |
. . . 4
⊢ ((𝜑 ∧ 𝑁 ∈ (ℤ≥‘(𝐺‘1))) →
(#‘(1...sup((◡𝐺 “ (𝑀...𝑁)), ℝ, < ))) = (#‘(◡𝐺 “ (𝑀...𝑁)))) |
129 | 68 | nnnn0d 11351 |
. . . . 5
⊢ ((𝜑 ∧ 𝑁 ∈ (ℤ≥‘(𝐺‘1))) → sup((◡𝐺 “ (𝑀...𝑁)), ℝ, < ) ∈
ℕ0) |
130 | | hashfz1 13134 |
. . . . 5
⊢
(sup((◡𝐺 “ (𝑀...𝑁)), ℝ, < ) ∈
ℕ0 → (#‘(1...sup((◡𝐺 “ (𝑀...𝑁)), ℝ, < ))) = sup((◡𝐺 “ (𝑀...𝑁)), ℝ, < )) |
131 | 129, 130 | syl 17 |
. . . 4
⊢ ((𝜑 ∧ 𝑁 ∈ (ℤ≥‘(𝐺‘1))) →
(#‘(1...sup((◡𝐺 “ (𝑀...𝑁)), ℝ, < ))) = sup((◡𝐺 “ (𝑀...𝑁)), ℝ, < )) |
132 | | hashen 13135 |
. . . . . 6
⊢ (((◡𝐺 “ (𝑀...𝑁)) ∈ Fin ∧ (𝐺 “ (◡𝐺 “ (𝑀...𝑁))) ∈ Fin) → ((#‘(◡𝐺 “ (𝑀...𝑁))) = (#‘(𝐺 “ (◡𝐺 “ (𝑀...𝑁)))) ↔ (◡𝐺 “ (𝑀...𝑁)) ≈ (𝐺 “ (◡𝐺 “ (𝑀...𝑁))))) |
133 | 48, 22, 132 | syl2anc 693 |
. . . . 5
⊢ ((𝜑 ∧ 𝑁 ∈ (ℤ≥‘(𝐺‘1))) →
((#‘(◡𝐺 “ (𝑀...𝑁))) = (#‘(𝐺 “ (◡𝐺 “ (𝑀...𝑁)))) ↔ (◡𝐺 “ (𝑀...𝑁)) ≈ (𝐺 “ (◡𝐺 “ (𝑀...𝑁))))) |
134 | 46, 133 | mpbird 247 |
. . . 4
⊢ ((𝜑 ∧ 𝑁 ∈ (ℤ≥‘(𝐺‘1))) →
(#‘(◡𝐺 “ (𝑀...𝑁))) = (#‘(𝐺 “ (◡𝐺 “ (𝑀...𝑁))))) |
135 | 128, 131,
134 | 3eqtr3d 2664 |
. . 3
⊢ ((𝜑 ∧ 𝑁 ∈ (ℤ≥‘(𝐺‘1))) → sup((◡𝐺 “ (𝑀...𝑁)), ℝ, < ) = (#‘(𝐺 “ (◡𝐺 “ (𝑀...𝑁))))) |
136 | 135 | oveq2d 6666 |
. 2
⊢ ((𝜑 ∧ 𝑁 ∈ (ℤ≥‘(𝐺‘1))) →
(1...sup((◡𝐺 “ (𝑀...𝑁)), ℝ, < )) = (1...(#‘(𝐺 “ (◡𝐺 “ (𝑀...𝑁)))))) |
137 | 136, 127 | eqtr3d 2658 |
1
⊢ ((𝜑 ∧ 𝑁 ∈ (ℤ≥‘(𝐺‘1))) →
(1...(#‘(𝐺 “
(◡𝐺 “ (𝑀...𝑁))))) = (◡𝐺 “ (𝑀...𝑁))) |