Step | Hyp | Ref
| Expression |
1 | | zssre 12326 |
. . . . . 6
⊢ ℤ
⊆ ℝ |
2 | | sstr 3934 |
. . . . . 6
⊢ ((𝐴 ⊆ ℤ ∧ ℤ
⊆ ℝ) → 𝐴
⊆ ℝ) |
3 | 1, 2 | mpan2 688 |
. . . . 5
⊢ (𝐴 ⊆ ℤ → 𝐴 ⊆
ℝ) |
4 | | suprcl 11935 |
. . . . 5
⊢ ((𝐴 ⊆ ℝ ∧ 𝐴 ≠ ∅ ∧ ∃𝑥 ∈ ℝ ∀𝑦 ∈ 𝐴 𝑦 ≤ 𝑥) → sup(𝐴, ℝ, < ) ∈
ℝ) |
5 | 3, 4 | syl3an1 1162 |
. . . 4
⊢ ((𝐴 ⊆ ℤ ∧ 𝐴 ≠ ∅ ∧ ∃𝑥 ∈ ℝ ∀𝑦 ∈ 𝐴 𝑦 ≤ 𝑥) → sup(𝐴, ℝ, < ) ∈
ℝ) |
6 | 5 | ltm1d 11907 |
. . 3
⊢ ((𝐴 ⊆ ℤ ∧ 𝐴 ≠ ∅ ∧ ∃𝑥 ∈ ℝ ∀𝑦 ∈ 𝐴 𝑦 ≤ 𝑥) → (sup(𝐴, ℝ, < ) − 1) < sup(𝐴, ℝ, <
)) |
7 | | peano2rem 11288 |
. . . . . 6
⊢
(sup(𝐴, ℝ,
< ) ∈ ℝ → (sup(𝐴, ℝ, < ) − 1) ∈
ℝ) |
8 | 4, 7 | syl 17 |
. . . . 5
⊢ ((𝐴 ⊆ ℝ ∧ 𝐴 ≠ ∅ ∧ ∃𝑥 ∈ ℝ ∀𝑦 ∈ 𝐴 𝑦 ≤ 𝑥) → (sup(𝐴, ℝ, < ) − 1) ∈
ℝ) |
9 | | suprlub 11939 |
. . . . 5
⊢ (((𝐴 ⊆ ℝ ∧ 𝐴 ≠ ∅ ∧ ∃𝑥 ∈ ℝ ∀𝑦 ∈ 𝐴 𝑦 ≤ 𝑥) ∧ (sup(𝐴, ℝ, < ) − 1) ∈ ℝ)
→ ((sup(𝐴, ℝ,
< ) − 1) < sup(𝐴, ℝ, < ) ↔ ∃𝑧 ∈ 𝐴 (sup(𝐴, ℝ, < ) − 1) < 𝑧)) |
10 | 8, 9 | mpdan 684 |
. . . 4
⊢ ((𝐴 ⊆ ℝ ∧ 𝐴 ≠ ∅ ∧ ∃𝑥 ∈ ℝ ∀𝑦 ∈ 𝐴 𝑦 ≤ 𝑥) → ((sup(𝐴, ℝ, < ) − 1) < sup(𝐴, ℝ, < ) ↔
∃𝑧 ∈ 𝐴 (sup(𝐴, ℝ, < ) − 1) < 𝑧)) |
11 | 3, 10 | syl3an1 1162 |
. . 3
⊢ ((𝐴 ⊆ ℤ ∧ 𝐴 ≠ ∅ ∧ ∃𝑥 ∈ ℝ ∀𝑦 ∈ 𝐴 𝑦 ≤ 𝑥) → ((sup(𝐴, ℝ, < ) − 1) < sup(𝐴, ℝ, < ) ↔
∃𝑧 ∈ 𝐴 (sup(𝐴, ℝ, < ) − 1) < 𝑧)) |
12 | 6, 11 | mpbid 231 |
. 2
⊢ ((𝐴 ⊆ ℤ ∧ 𝐴 ≠ ∅ ∧ ∃𝑥 ∈ ℝ ∀𝑦 ∈ 𝐴 𝑦 ≤ 𝑥) → ∃𝑧 ∈ 𝐴 (sup(𝐴, ℝ, < ) − 1) < 𝑧) |
13 | | simpl1 1190 |
. . . . . . . . . 10
⊢ (((𝐴 ⊆ ℤ ∧ 𝐴 ≠ ∅ ∧ ∃𝑥 ∈ ℝ ∀𝑦 ∈ 𝐴 𝑦 ≤ 𝑥) ∧ (𝑧 ∈ 𝐴 ∧ (sup(𝐴, ℝ, < ) − 1) < 𝑧)) → 𝐴 ⊆ ℤ) |
14 | 13 | sselda 3926 |
. . . . . . . . 9
⊢ ((((𝐴 ⊆ ℤ ∧ 𝐴 ≠ ∅ ∧ ∃𝑥 ∈ ℝ ∀𝑦 ∈ 𝐴 𝑦 ≤ 𝑥) ∧ (𝑧 ∈ 𝐴 ∧ (sup(𝐴, ℝ, < ) − 1) < 𝑧)) ∧ 𝑤 ∈ 𝐴) → 𝑤 ∈ ℤ) |
15 | 1, 14 | sselid 3924 |
. . . . . . . 8
⊢ ((((𝐴 ⊆ ℤ ∧ 𝐴 ≠ ∅ ∧ ∃𝑥 ∈ ℝ ∀𝑦 ∈ 𝐴 𝑦 ≤ 𝑥) ∧ (𝑧 ∈ 𝐴 ∧ (sup(𝐴, ℝ, < ) − 1) < 𝑧)) ∧ 𝑤 ∈ 𝐴) → 𝑤 ∈ ℝ) |
16 | 5 | adantr 481 |
. . . . . . . . 9
⊢ (((𝐴 ⊆ ℤ ∧ 𝐴 ≠ ∅ ∧ ∃𝑥 ∈ ℝ ∀𝑦 ∈ 𝐴 𝑦 ≤ 𝑥) ∧ (𝑧 ∈ 𝐴 ∧ (sup(𝐴, ℝ, < ) − 1) < 𝑧)) → sup(𝐴, ℝ, < ) ∈
ℝ) |
17 | 16 | adantr 481 |
. . . . . . . 8
⊢ ((((𝐴 ⊆ ℤ ∧ 𝐴 ≠ ∅ ∧ ∃𝑥 ∈ ℝ ∀𝑦 ∈ 𝐴 𝑦 ≤ 𝑥) ∧ (𝑧 ∈ 𝐴 ∧ (sup(𝐴, ℝ, < ) − 1) < 𝑧)) ∧ 𝑤 ∈ 𝐴) → sup(𝐴, ℝ, < ) ∈
ℝ) |
18 | | simprl 768 |
. . . . . . . . . . . 12
⊢ (((𝐴 ⊆ ℤ ∧ 𝐴 ≠ ∅ ∧ ∃𝑥 ∈ ℝ ∀𝑦 ∈ 𝐴 𝑦 ≤ 𝑥) ∧ (𝑧 ∈ 𝐴 ∧ (sup(𝐴, ℝ, < ) − 1) < 𝑧)) → 𝑧 ∈ 𝐴) |
19 | 13, 18 | sseldd 3927 |
. . . . . . . . . . 11
⊢ (((𝐴 ⊆ ℤ ∧ 𝐴 ≠ ∅ ∧ ∃𝑥 ∈ ℝ ∀𝑦 ∈ 𝐴 𝑦 ≤ 𝑥) ∧ (𝑧 ∈ 𝐴 ∧ (sup(𝐴, ℝ, < ) − 1) < 𝑧)) → 𝑧 ∈ ℤ) |
20 | | zre 12323 |
. . . . . . . . . . 11
⊢ (𝑧 ∈ ℤ → 𝑧 ∈
ℝ) |
21 | 19, 20 | syl 17 |
. . . . . . . . . 10
⊢ (((𝐴 ⊆ ℤ ∧ 𝐴 ≠ ∅ ∧ ∃𝑥 ∈ ℝ ∀𝑦 ∈ 𝐴 𝑦 ≤ 𝑥) ∧ (𝑧 ∈ 𝐴 ∧ (sup(𝐴, ℝ, < ) − 1) < 𝑧)) → 𝑧 ∈ ℝ) |
22 | | peano2re 11148 |
. . . . . . . . . 10
⊢ (𝑧 ∈ ℝ → (𝑧 + 1) ∈
ℝ) |
23 | 21, 22 | syl 17 |
. . . . . . . . 9
⊢ (((𝐴 ⊆ ℤ ∧ 𝐴 ≠ ∅ ∧ ∃𝑥 ∈ ℝ ∀𝑦 ∈ 𝐴 𝑦 ≤ 𝑥) ∧ (𝑧 ∈ 𝐴 ∧ (sup(𝐴, ℝ, < ) − 1) < 𝑧)) → (𝑧 + 1) ∈ ℝ) |
24 | 23 | adantr 481 |
. . . . . . . 8
⊢ ((((𝐴 ⊆ ℤ ∧ 𝐴 ≠ ∅ ∧ ∃𝑥 ∈ ℝ ∀𝑦 ∈ 𝐴 𝑦 ≤ 𝑥) ∧ (𝑧 ∈ 𝐴 ∧ (sup(𝐴, ℝ, < ) − 1) < 𝑧)) ∧ 𝑤 ∈ 𝐴) → (𝑧 + 1) ∈ ℝ) |
25 | | suprub 11936 |
. . . . . . . . . 10
⊢ (((𝐴 ⊆ ℝ ∧ 𝐴 ≠ ∅ ∧ ∃𝑥 ∈ ℝ ∀𝑦 ∈ 𝐴 𝑦 ≤ 𝑥) ∧ 𝑤 ∈ 𝐴) → 𝑤 ≤ sup(𝐴, ℝ, < )) |
26 | 3, 25 | syl3anl1 1411 |
. . . . . . . . 9
⊢ (((𝐴 ⊆ ℤ ∧ 𝐴 ≠ ∅ ∧ ∃𝑥 ∈ ℝ ∀𝑦 ∈ 𝐴 𝑦 ≤ 𝑥) ∧ 𝑤 ∈ 𝐴) → 𝑤 ≤ sup(𝐴, ℝ, < )) |
27 | 26 | adantlr 712 |
. . . . . . . 8
⊢ ((((𝐴 ⊆ ℤ ∧ 𝐴 ≠ ∅ ∧ ∃𝑥 ∈ ℝ ∀𝑦 ∈ 𝐴 𝑦 ≤ 𝑥) ∧ (𝑧 ∈ 𝐴 ∧ (sup(𝐴, ℝ, < ) − 1) < 𝑧)) ∧ 𝑤 ∈ 𝐴) → 𝑤 ≤ sup(𝐴, ℝ, < )) |
28 | | simprr 770 |
. . . . . . . . . 10
⊢ (((𝐴 ⊆ ℤ ∧ 𝐴 ≠ ∅ ∧ ∃𝑥 ∈ ℝ ∀𝑦 ∈ 𝐴 𝑦 ≤ 𝑥) ∧ (𝑧 ∈ 𝐴 ∧ (sup(𝐴, ℝ, < ) − 1) < 𝑧)) → (sup(𝐴, ℝ, < ) − 1) < 𝑧) |
29 | | 1red 10977 |
. . . . . . . . . . 11
⊢ (((𝐴 ⊆ ℤ ∧ 𝐴 ≠ ∅ ∧ ∃𝑥 ∈ ℝ ∀𝑦 ∈ 𝐴 𝑦 ≤ 𝑥) ∧ (𝑧 ∈ 𝐴 ∧ (sup(𝐴, ℝ, < ) − 1) < 𝑧)) → 1 ∈
ℝ) |
30 | 16, 29, 21 | ltsubaddd 11571 |
. . . . . . . . . 10
⊢ (((𝐴 ⊆ ℤ ∧ 𝐴 ≠ ∅ ∧ ∃𝑥 ∈ ℝ ∀𝑦 ∈ 𝐴 𝑦 ≤ 𝑥) ∧ (𝑧 ∈ 𝐴 ∧ (sup(𝐴, ℝ, < ) − 1) < 𝑧)) → ((sup(𝐴, ℝ, < ) − 1)
< 𝑧 ↔ sup(𝐴, ℝ, < ) < (𝑧 + 1))) |
31 | 28, 30 | mpbid 231 |
. . . . . . . . 9
⊢ (((𝐴 ⊆ ℤ ∧ 𝐴 ≠ ∅ ∧ ∃𝑥 ∈ ℝ ∀𝑦 ∈ 𝐴 𝑦 ≤ 𝑥) ∧ (𝑧 ∈ 𝐴 ∧ (sup(𝐴, ℝ, < ) − 1) < 𝑧)) → sup(𝐴, ℝ, < ) < (𝑧 + 1)) |
32 | 31 | adantr 481 |
. . . . . . . 8
⊢ ((((𝐴 ⊆ ℤ ∧ 𝐴 ≠ ∅ ∧ ∃𝑥 ∈ ℝ ∀𝑦 ∈ 𝐴 𝑦 ≤ 𝑥) ∧ (𝑧 ∈ 𝐴 ∧ (sup(𝐴, ℝ, < ) − 1) < 𝑧)) ∧ 𝑤 ∈ 𝐴) → sup(𝐴, ℝ, < ) < (𝑧 + 1)) |
33 | 15, 17, 24, 27, 32 | lelttrd 11133 |
. . . . . . 7
⊢ ((((𝐴 ⊆ ℤ ∧ 𝐴 ≠ ∅ ∧ ∃𝑥 ∈ ℝ ∀𝑦 ∈ 𝐴 𝑦 ≤ 𝑥) ∧ (𝑧 ∈ 𝐴 ∧ (sup(𝐴, ℝ, < ) − 1) < 𝑧)) ∧ 𝑤 ∈ 𝐴) → 𝑤 < (𝑧 + 1)) |
34 | 19 | adantr 481 |
. . . . . . . 8
⊢ ((((𝐴 ⊆ ℤ ∧ 𝐴 ≠ ∅ ∧ ∃𝑥 ∈ ℝ ∀𝑦 ∈ 𝐴 𝑦 ≤ 𝑥) ∧ (𝑧 ∈ 𝐴 ∧ (sup(𝐴, ℝ, < ) − 1) < 𝑧)) ∧ 𝑤 ∈ 𝐴) → 𝑧 ∈ ℤ) |
35 | | zleltp1 12371 |
. . . . . . . 8
⊢ ((𝑤 ∈ ℤ ∧ 𝑧 ∈ ℤ) → (𝑤 ≤ 𝑧 ↔ 𝑤 < (𝑧 + 1))) |
36 | 14, 34, 35 | syl2anc 584 |
. . . . . . 7
⊢ ((((𝐴 ⊆ ℤ ∧ 𝐴 ≠ ∅ ∧ ∃𝑥 ∈ ℝ ∀𝑦 ∈ 𝐴 𝑦 ≤ 𝑥) ∧ (𝑧 ∈ 𝐴 ∧ (sup(𝐴, ℝ, < ) − 1) < 𝑧)) ∧ 𝑤 ∈ 𝐴) → (𝑤 ≤ 𝑧 ↔ 𝑤 < (𝑧 + 1))) |
37 | 33, 36 | mpbird 256 |
. . . . . 6
⊢ ((((𝐴 ⊆ ℤ ∧ 𝐴 ≠ ∅ ∧ ∃𝑥 ∈ ℝ ∀𝑦 ∈ 𝐴 𝑦 ≤ 𝑥) ∧ (𝑧 ∈ 𝐴 ∧ (sup(𝐴, ℝ, < ) − 1) < 𝑧)) ∧ 𝑤 ∈ 𝐴) → 𝑤 ≤ 𝑧) |
38 | 37 | ralrimiva 3110 |
. . . . 5
⊢ (((𝐴 ⊆ ℤ ∧ 𝐴 ≠ ∅ ∧ ∃𝑥 ∈ ℝ ∀𝑦 ∈ 𝐴 𝑦 ≤ 𝑥) ∧ (𝑧 ∈ 𝐴 ∧ (sup(𝐴, ℝ, < ) − 1) < 𝑧)) → ∀𝑤 ∈ 𝐴 𝑤 ≤ 𝑧) |
39 | | suprleub 11941 |
. . . . . . 7
⊢ (((𝐴 ⊆ ℝ ∧ 𝐴 ≠ ∅ ∧ ∃𝑥 ∈ ℝ ∀𝑦 ∈ 𝐴 𝑦 ≤ 𝑥) ∧ 𝑧 ∈ ℝ) → (sup(𝐴, ℝ, < ) ≤ 𝑧 ↔ ∀𝑤 ∈ 𝐴 𝑤 ≤ 𝑧)) |
40 | 3, 39 | syl3anl1 1411 |
. . . . . 6
⊢ (((𝐴 ⊆ ℤ ∧ 𝐴 ≠ ∅ ∧ ∃𝑥 ∈ ℝ ∀𝑦 ∈ 𝐴 𝑦 ≤ 𝑥) ∧ 𝑧 ∈ ℝ) → (sup(𝐴, ℝ, < ) ≤ 𝑧 ↔ ∀𝑤 ∈ 𝐴 𝑤 ≤ 𝑧)) |
41 | 21, 40 | syldan 591 |
. . . . 5
⊢ (((𝐴 ⊆ ℤ ∧ 𝐴 ≠ ∅ ∧ ∃𝑥 ∈ ℝ ∀𝑦 ∈ 𝐴 𝑦 ≤ 𝑥) ∧ (𝑧 ∈ 𝐴 ∧ (sup(𝐴, ℝ, < ) − 1) < 𝑧)) → (sup(𝐴, ℝ, < ) ≤ 𝑧 ↔ ∀𝑤 ∈ 𝐴 𝑤 ≤ 𝑧)) |
42 | 38, 41 | mpbird 256 |
. . . 4
⊢ (((𝐴 ⊆ ℤ ∧ 𝐴 ≠ ∅ ∧ ∃𝑥 ∈ ℝ ∀𝑦 ∈ 𝐴 𝑦 ≤ 𝑥) ∧ (𝑧 ∈ 𝐴 ∧ (sup(𝐴, ℝ, < ) − 1) < 𝑧)) → sup(𝐴, ℝ, < ) ≤ 𝑧) |
43 | | suprub 11936 |
. . . . . 6
⊢ (((𝐴 ⊆ ℝ ∧ 𝐴 ≠ ∅ ∧ ∃𝑥 ∈ ℝ ∀𝑦 ∈ 𝐴 𝑦 ≤ 𝑥) ∧ 𝑧 ∈ 𝐴) → 𝑧 ≤ sup(𝐴, ℝ, < )) |
44 | 3, 43 | syl3anl1 1411 |
. . . . 5
⊢ (((𝐴 ⊆ ℤ ∧ 𝐴 ≠ ∅ ∧ ∃𝑥 ∈ ℝ ∀𝑦 ∈ 𝐴 𝑦 ≤ 𝑥) ∧ 𝑧 ∈ 𝐴) → 𝑧 ≤ sup(𝐴, ℝ, < )) |
45 | 44 | adantrr 714 |
. . . 4
⊢ (((𝐴 ⊆ ℤ ∧ 𝐴 ≠ ∅ ∧ ∃𝑥 ∈ ℝ ∀𝑦 ∈ 𝐴 𝑦 ≤ 𝑥) ∧ (𝑧 ∈ 𝐴 ∧ (sup(𝐴, ℝ, < ) − 1) < 𝑧)) → 𝑧 ≤ sup(𝐴, ℝ, < )) |
46 | 16, 21 | letri3d 11117 |
. . . 4
⊢ (((𝐴 ⊆ ℤ ∧ 𝐴 ≠ ∅ ∧ ∃𝑥 ∈ ℝ ∀𝑦 ∈ 𝐴 𝑦 ≤ 𝑥) ∧ (𝑧 ∈ 𝐴 ∧ (sup(𝐴, ℝ, < ) − 1) < 𝑧)) → (sup(𝐴, ℝ, < ) = 𝑧 ↔ (sup(𝐴, ℝ, < ) ≤ 𝑧 ∧ 𝑧 ≤ sup(𝐴, ℝ, < )))) |
47 | 42, 45, 46 | mpbir2and 710 |
. . 3
⊢ (((𝐴 ⊆ ℤ ∧ 𝐴 ≠ ∅ ∧ ∃𝑥 ∈ ℝ ∀𝑦 ∈ 𝐴 𝑦 ≤ 𝑥) ∧ (𝑧 ∈ 𝐴 ∧ (sup(𝐴, ℝ, < ) − 1) < 𝑧)) → sup(𝐴, ℝ, < ) = 𝑧) |
48 | 47, 18 | eqeltrd 2841 |
. 2
⊢ (((𝐴 ⊆ ℤ ∧ 𝐴 ≠ ∅ ∧ ∃𝑥 ∈ ℝ ∀𝑦 ∈ 𝐴 𝑦 ≤ 𝑥) ∧ (𝑧 ∈ 𝐴 ∧ (sup(𝐴, ℝ, < ) − 1) < 𝑧)) → sup(𝐴, ℝ, < ) ∈ 𝐴) |
49 | 12, 48 | rexlimddv 3222 |
1
⊢ ((𝐴 ⊆ ℤ ∧ 𝐴 ≠ ∅ ∧ ∃𝑥 ∈ ℝ ∀𝑦 ∈ 𝐴 𝑦 ≤ 𝑥) → sup(𝐴, ℝ, < ) ∈ 𝐴) |