Step | Hyp | Ref
| Expression |
1 | | vex 3417 |
. . . . . . 7
⊢ 𝑤 ∈ V |
2 | | oveq1 6917 |
. . . . . . . . . 10
⊢ (𝑣 = 𝑎 → (𝑣 + 𝐵) = (𝑎 + 𝐵)) |
3 | 2 | eqeq2d 2835 |
. . . . . . . . 9
⊢ (𝑣 = 𝑎 → (𝑧 = (𝑣 + 𝐵) ↔ 𝑧 = (𝑎 + 𝐵))) |
4 | 3 | cbvrexv 3384 |
. . . . . . . 8
⊢
(∃𝑣 ∈
𝐴 𝑧 = (𝑣 + 𝐵) ↔ ∃𝑎 ∈ 𝐴 𝑧 = (𝑎 + 𝐵)) |
5 | | eqeq1 2829 |
. . . . . . . . 9
⊢ (𝑧 = 𝑤 → (𝑧 = (𝑎 + 𝐵) ↔ 𝑤 = (𝑎 + 𝐵))) |
6 | 5 | rexbidv 3262 |
. . . . . . . 8
⊢ (𝑧 = 𝑤 → (∃𝑎 ∈ 𝐴 𝑧 = (𝑎 + 𝐵) ↔ ∃𝑎 ∈ 𝐴 𝑤 = (𝑎 + 𝐵))) |
7 | 4, 6 | syl5bb 275 |
. . . . . . 7
⊢ (𝑧 = 𝑤 → (∃𝑣 ∈ 𝐴 𝑧 = (𝑣 + 𝐵) ↔ ∃𝑎 ∈ 𝐴 𝑤 = (𝑎 + 𝐵))) |
8 | | supaddc.c |
. . . . . . 7
⊢ 𝐶 = {𝑧 ∣ ∃𝑣 ∈ 𝐴 𝑧 = (𝑣 + 𝐵)} |
9 | 1, 7, 8 | elab2 3575 |
. . . . . 6
⊢ (𝑤 ∈ 𝐶 ↔ ∃𝑎 ∈ 𝐴 𝑤 = (𝑎 + 𝐵)) |
10 | | supadd.a1 |
. . . . . . . . . 10
⊢ (𝜑 → 𝐴 ⊆ ℝ) |
11 | 10 | sselda 3827 |
. . . . . . . . 9
⊢ ((𝜑 ∧ 𝑎 ∈ 𝐴) → 𝑎 ∈ ℝ) |
12 | | supadd.a2 |
. . . . . . . . . . 11
⊢ (𝜑 → 𝐴 ≠ ∅) |
13 | | supadd.a3 |
. . . . . . . . . . 11
⊢ (𝜑 → ∃𝑥 ∈ ℝ ∀𝑦 ∈ 𝐴 𝑦 ≤ 𝑥) |
14 | | suprcl 11320 |
. . . . . . . . . . 11
⊢ ((𝐴 ⊆ ℝ ∧ 𝐴 ≠ ∅ ∧ ∃𝑥 ∈ ℝ ∀𝑦 ∈ 𝐴 𝑦 ≤ 𝑥) → sup(𝐴, ℝ, < ) ∈
ℝ) |
15 | 10, 12, 13, 14 | syl3anc 1494 |
. . . . . . . . . 10
⊢ (𝜑 → sup(𝐴, ℝ, < ) ∈
ℝ) |
16 | 15 | adantr 474 |
. . . . . . . . 9
⊢ ((𝜑 ∧ 𝑎 ∈ 𝐴) → sup(𝐴, ℝ, < ) ∈
ℝ) |
17 | | supaddc.b |
. . . . . . . . . 10
⊢ (𝜑 → 𝐵 ∈ ℝ) |
18 | 17 | adantr 474 |
. . . . . . . . 9
⊢ ((𝜑 ∧ 𝑎 ∈ 𝐴) → 𝐵 ∈ ℝ) |
19 | 10, 12, 13 | 3jca 1162 |
. . . . . . . . . 10
⊢ (𝜑 → (𝐴 ⊆ ℝ ∧ 𝐴 ≠ ∅ ∧ ∃𝑥 ∈ ℝ ∀𝑦 ∈ 𝐴 𝑦 ≤ 𝑥)) |
20 | | suprub 11321 |
. . . . . . . . . 10
⊢ (((𝐴 ⊆ ℝ ∧ 𝐴 ≠ ∅ ∧ ∃𝑥 ∈ ℝ ∀𝑦 ∈ 𝐴 𝑦 ≤ 𝑥) ∧ 𝑎 ∈ 𝐴) → 𝑎 ≤ sup(𝐴, ℝ, < )) |
21 | 19, 20 | sylan 575 |
. . . . . . . . 9
⊢ ((𝜑 ∧ 𝑎 ∈ 𝐴) → 𝑎 ≤ sup(𝐴, ℝ, < )) |
22 | 11, 16, 18, 21 | leadd1dd 10973 |
. . . . . . . 8
⊢ ((𝜑 ∧ 𝑎 ∈ 𝐴) → (𝑎 + 𝐵) ≤ (sup(𝐴, ℝ, < ) + 𝐵)) |
23 | | breq1 4878 |
. . . . . . . 8
⊢ (𝑤 = (𝑎 + 𝐵) → (𝑤 ≤ (sup(𝐴, ℝ, < ) + 𝐵) ↔ (𝑎 + 𝐵) ≤ (sup(𝐴, ℝ, < ) + 𝐵))) |
24 | 22, 23 | syl5ibrcom 239 |
. . . . . . 7
⊢ ((𝜑 ∧ 𝑎 ∈ 𝐴) → (𝑤 = (𝑎 + 𝐵) → 𝑤 ≤ (sup(𝐴, ℝ, < ) + 𝐵))) |
25 | 24 | rexlimdva 3240 |
. . . . . 6
⊢ (𝜑 → (∃𝑎 ∈ 𝐴 𝑤 = (𝑎 + 𝐵) → 𝑤 ≤ (sup(𝐴, ℝ, < ) + 𝐵))) |
26 | 9, 25 | syl5bi 234 |
. . . . 5
⊢ (𝜑 → (𝑤 ∈ 𝐶 → 𝑤 ≤ (sup(𝐴, ℝ, < ) + 𝐵))) |
27 | 26 | ralrimiv 3174 |
. . . 4
⊢ (𝜑 → ∀𝑤 ∈ 𝐶 𝑤 ≤ (sup(𝐴, ℝ, < ) + 𝐵)) |
28 | 11, 18 | readdcld 10393 |
. . . . . . . . 9
⊢ ((𝜑 ∧ 𝑎 ∈ 𝐴) → (𝑎 + 𝐵) ∈ ℝ) |
29 | | eleq1a 2901 |
. . . . . . . . 9
⊢ ((𝑎 + 𝐵) ∈ ℝ → (𝑤 = (𝑎 + 𝐵) → 𝑤 ∈ ℝ)) |
30 | 28, 29 | syl 17 |
. . . . . . . 8
⊢ ((𝜑 ∧ 𝑎 ∈ 𝐴) → (𝑤 = (𝑎 + 𝐵) → 𝑤 ∈ ℝ)) |
31 | 30 | rexlimdva 3240 |
. . . . . . 7
⊢ (𝜑 → (∃𝑎 ∈ 𝐴 𝑤 = (𝑎 + 𝐵) → 𝑤 ∈ ℝ)) |
32 | 9, 31 | syl5bi 234 |
. . . . . 6
⊢ (𝜑 → (𝑤 ∈ 𝐶 → 𝑤 ∈ ℝ)) |
33 | 32 | ssrdv 3833 |
. . . . 5
⊢ (𝜑 → 𝐶 ⊆ ℝ) |
34 | | ovex 6942 |
. . . . . . . . 9
⊢ (𝑎 + 𝐵) ∈ V |
35 | 34 | isseti 3426 |
. . . . . . . 8
⊢
∃𝑤 𝑤 = (𝑎 + 𝐵) |
36 | 35 | rgenw 3133 |
. . . . . . 7
⊢
∀𝑎 ∈
𝐴 ∃𝑤 𝑤 = (𝑎 + 𝐵) |
37 | | r19.2z 4284 |
. . . . . . 7
⊢ ((𝐴 ≠ ∅ ∧
∀𝑎 ∈ 𝐴 ∃𝑤 𝑤 = (𝑎 + 𝐵)) → ∃𝑎 ∈ 𝐴 ∃𝑤 𝑤 = (𝑎 + 𝐵)) |
38 | 12, 36, 37 | sylancl 580 |
. . . . . 6
⊢ (𝜑 → ∃𝑎 ∈ 𝐴 ∃𝑤 𝑤 = (𝑎 + 𝐵)) |
39 | 9 | exbii 1947 |
. . . . . . 7
⊢
(∃𝑤 𝑤 ∈ 𝐶 ↔ ∃𝑤∃𝑎 ∈ 𝐴 𝑤 = (𝑎 + 𝐵)) |
40 | | n0 4162 |
. . . . . . 7
⊢ (𝐶 ≠ ∅ ↔
∃𝑤 𝑤 ∈ 𝐶) |
41 | | rexcom4 3442 |
. . . . . . 7
⊢
(∃𝑎 ∈
𝐴 ∃𝑤 𝑤 = (𝑎 + 𝐵) ↔ ∃𝑤∃𝑎 ∈ 𝐴 𝑤 = (𝑎 + 𝐵)) |
42 | 39, 40, 41 | 3bitr4i 295 |
. . . . . 6
⊢ (𝐶 ≠ ∅ ↔
∃𝑎 ∈ 𝐴 ∃𝑤 𝑤 = (𝑎 + 𝐵)) |
43 | 38, 42 | sylibr 226 |
. . . . 5
⊢ (𝜑 → 𝐶 ≠ ∅) |
44 | 15, 17 | readdcld 10393 |
. . . . . 6
⊢ (𝜑 → (sup(𝐴, ℝ, < ) + 𝐵) ∈ ℝ) |
45 | | brralrspcev 4935 |
. . . . . 6
⊢
(((sup(𝐴, ℝ,
< ) + 𝐵) ∈ ℝ
∧ ∀𝑤 ∈
𝐶 𝑤 ≤ (sup(𝐴, ℝ, < ) + 𝐵)) → ∃𝑥 ∈ ℝ ∀𝑤 ∈ 𝐶 𝑤 ≤ 𝑥) |
46 | 44, 27, 45 | syl2anc 579 |
. . . . 5
⊢ (𝜑 → ∃𝑥 ∈ ℝ ∀𝑤 ∈ 𝐶 𝑤 ≤ 𝑥) |
47 | | suprleub 11326 |
. . . . 5
⊢ (((𝐶 ⊆ ℝ ∧ 𝐶 ≠ ∅ ∧ ∃𝑥 ∈ ℝ ∀𝑤 ∈ 𝐶 𝑤 ≤ 𝑥) ∧ (sup(𝐴, ℝ, < ) + 𝐵) ∈ ℝ) → (sup(𝐶, ℝ, < ) ≤
(sup(𝐴, ℝ, < ) +
𝐵) ↔ ∀𝑤 ∈ 𝐶 𝑤 ≤ (sup(𝐴, ℝ, < ) + 𝐵))) |
48 | 33, 43, 46, 44, 47 | syl31anc 1496 |
. . . 4
⊢ (𝜑 → (sup(𝐶, ℝ, < ) ≤ (sup(𝐴, ℝ, < ) + 𝐵) ↔ ∀𝑤 ∈ 𝐶 𝑤 ≤ (sup(𝐴, ℝ, < ) + 𝐵))) |
49 | 27, 48 | mpbird 249 |
. . 3
⊢ (𝜑 → sup(𝐶, ℝ, < ) ≤ (sup(𝐴, ℝ, < ) + 𝐵)) |
50 | | suprcl 11320 |
. . . . . . . 8
⊢ ((𝐶 ⊆ ℝ ∧ 𝐶 ≠ ∅ ∧ ∃𝑥 ∈ ℝ ∀𝑤 ∈ 𝐶 𝑤 ≤ 𝑥) → sup(𝐶, ℝ, < ) ∈
ℝ) |
51 | 33, 43, 46, 50 | syl3anc 1494 |
. . . . . . 7
⊢ (𝜑 → sup(𝐶, ℝ, < ) ∈
ℝ) |
52 | 51, 17, 15 | ltsubaddd 10955 |
. . . . . 6
⊢ (𝜑 → ((sup(𝐶, ℝ, < ) − 𝐵) < sup(𝐴, ℝ, < ) ↔ sup(𝐶, ℝ, < ) <
(sup(𝐴, ℝ, < ) +
𝐵))) |
53 | 52 | biimpar 471 |
. . . . 5
⊢ ((𝜑 ∧ sup(𝐶, ℝ, < ) < (sup(𝐴, ℝ, < ) + 𝐵)) → (sup(𝐶, ℝ, < ) − 𝐵) < sup(𝐴, ℝ, < )) |
54 | 51, 17 | resubcld 10789 |
. . . . . . 7
⊢ (𝜑 → (sup(𝐶, ℝ, < ) − 𝐵) ∈ ℝ) |
55 | | suprlub 11324 |
. . . . . . 7
⊢ (((𝐴 ⊆ ℝ ∧ 𝐴 ≠ ∅ ∧ ∃𝑥 ∈ ℝ ∀𝑦 ∈ 𝐴 𝑦 ≤ 𝑥) ∧ (sup(𝐶, ℝ, < ) − 𝐵) ∈ ℝ) → ((sup(𝐶, ℝ, < ) − 𝐵) < sup(𝐴, ℝ, < ) ↔ ∃𝑎 ∈ 𝐴 (sup(𝐶, ℝ, < ) − 𝐵) < 𝑎)) |
56 | 10, 12, 13, 54, 55 | syl31anc 1496 |
. . . . . 6
⊢ (𝜑 → ((sup(𝐶, ℝ, < ) − 𝐵) < sup(𝐴, ℝ, < ) ↔ ∃𝑎 ∈ 𝐴 (sup(𝐶, ℝ, < ) − 𝐵) < 𝑎)) |
57 | 56 | adantr 474 |
. . . . 5
⊢ ((𝜑 ∧ sup(𝐶, ℝ, < ) < (sup(𝐴, ℝ, < ) + 𝐵)) → ((sup(𝐶, ℝ, < ) − 𝐵) < sup(𝐴, ℝ, < ) ↔ ∃𝑎 ∈ 𝐴 (sup(𝐶, ℝ, < ) − 𝐵) < 𝑎)) |
58 | 53, 57 | mpbid 224 |
. . . 4
⊢ ((𝜑 ∧ sup(𝐶, ℝ, < ) < (sup(𝐴, ℝ, < ) + 𝐵)) → ∃𝑎 ∈ 𝐴 (sup(𝐶, ℝ, < ) − 𝐵) < 𝑎) |
59 | | rspe 3211 |
. . . . . . . . . . . . . 14
⊢ ((𝑎 ∈ 𝐴 ∧ 𝑤 = (𝑎 + 𝐵)) → ∃𝑎 ∈ 𝐴 𝑤 = (𝑎 + 𝐵)) |
60 | 59, 9 | sylibr 226 |
. . . . . . . . . . . . 13
⊢ ((𝑎 ∈ 𝐴 ∧ 𝑤 = (𝑎 + 𝐵)) → 𝑤 ∈ 𝐶) |
61 | 60 | adantl 475 |
. . . . . . . . . . . 12
⊢ ((𝜑 ∧ (𝑎 ∈ 𝐴 ∧ 𝑤 = (𝑎 + 𝐵))) → 𝑤 ∈ 𝐶) |
62 | | simplrr 796 |
. . . . . . . . . . . . 13
⊢ (((𝜑 ∧ (𝑎 ∈ 𝐴 ∧ 𝑤 = (𝑎 + 𝐵))) ∧ 𝑤 ∈ 𝐶) → 𝑤 = (𝑎 + 𝐵)) |
63 | 33, 43, 46 | 3jca 1162 |
. . . . . . . . . . . . . . 15
⊢ (𝜑 → (𝐶 ⊆ ℝ ∧ 𝐶 ≠ ∅ ∧ ∃𝑥 ∈ ℝ ∀𝑤 ∈ 𝐶 𝑤 ≤ 𝑥)) |
64 | | suprub 11321 |
. . . . . . . . . . . . . . 15
⊢ (((𝐶 ⊆ ℝ ∧ 𝐶 ≠ ∅ ∧ ∃𝑥 ∈ ℝ ∀𝑤 ∈ 𝐶 𝑤 ≤ 𝑥) ∧ 𝑤 ∈ 𝐶) → 𝑤 ≤ sup(𝐶, ℝ, < )) |
65 | 63, 64 | sylan 575 |
. . . . . . . . . . . . . 14
⊢ ((𝜑 ∧ 𝑤 ∈ 𝐶) → 𝑤 ≤ sup(𝐶, ℝ, < )) |
66 | 65 | adantlr 706 |
. . . . . . . . . . . . 13
⊢ (((𝜑 ∧ (𝑎 ∈ 𝐴 ∧ 𝑤 = (𝑎 + 𝐵))) ∧ 𝑤 ∈ 𝐶) → 𝑤 ≤ sup(𝐶, ℝ, < )) |
67 | 62, 66 | eqbrtrrd 4899 |
. . . . . . . . . . . 12
⊢ (((𝜑 ∧ (𝑎 ∈ 𝐴 ∧ 𝑤 = (𝑎 + 𝐵))) ∧ 𝑤 ∈ 𝐶) → (𝑎 + 𝐵) ≤ sup(𝐶, ℝ, < )) |
68 | 61, 67 | mpdan 678 |
. . . . . . . . . . 11
⊢ ((𝜑 ∧ (𝑎 ∈ 𝐴 ∧ 𝑤 = (𝑎 + 𝐵))) → (𝑎 + 𝐵) ≤ sup(𝐶, ℝ, < )) |
69 | 68 | expr 450 |
. . . . . . . . . 10
⊢ ((𝜑 ∧ 𝑎 ∈ 𝐴) → (𝑤 = (𝑎 + 𝐵) → (𝑎 + 𝐵) ≤ sup(𝐶, ℝ, < ))) |
70 | 69 | exlimdv 2032 |
. . . . . . . . 9
⊢ ((𝜑 ∧ 𝑎 ∈ 𝐴) → (∃𝑤 𝑤 = (𝑎 + 𝐵) → (𝑎 + 𝐵) ≤ sup(𝐶, ℝ, < ))) |
71 | 35, 70 | mpi 20 |
. . . . . . . 8
⊢ ((𝜑 ∧ 𝑎 ∈ 𝐴) → (𝑎 + 𝐵) ≤ sup(𝐶, ℝ, < )) |
72 | 71 | adantlr 706 |
. . . . . . 7
⊢ (((𝜑 ∧ sup(𝐶, ℝ, < ) < (sup(𝐴, ℝ, < ) + 𝐵)) ∧ 𝑎 ∈ 𝐴) → (𝑎 + 𝐵) ≤ sup(𝐶, ℝ, < )) |
73 | 28 | adantlr 706 |
. . . . . . . 8
⊢ (((𝜑 ∧ sup(𝐶, ℝ, < ) < (sup(𝐴, ℝ, < ) + 𝐵)) ∧ 𝑎 ∈ 𝐴) → (𝑎 + 𝐵) ∈ ℝ) |
74 | 51 | ad2antrr 717 |
. . . . . . . 8
⊢ (((𝜑 ∧ sup(𝐶, ℝ, < ) < (sup(𝐴, ℝ, < ) + 𝐵)) ∧ 𝑎 ∈ 𝐴) → sup(𝐶, ℝ, < ) ∈
ℝ) |
75 | 73, 74 | lenltd 10509 |
. . . . . . 7
⊢ (((𝜑 ∧ sup(𝐶, ℝ, < ) < (sup(𝐴, ℝ, < ) + 𝐵)) ∧ 𝑎 ∈ 𝐴) → ((𝑎 + 𝐵) ≤ sup(𝐶, ℝ, < ) ↔ ¬ sup(𝐶, ℝ, < ) < (𝑎 + 𝐵))) |
76 | 72, 75 | mpbid 224 |
. . . . . 6
⊢ (((𝜑 ∧ sup(𝐶, ℝ, < ) < (sup(𝐴, ℝ, < ) + 𝐵)) ∧ 𝑎 ∈ 𝐴) → ¬ sup(𝐶, ℝ, < ) < (𝑎 + 𝐵)) |
77 | 17 | ad2antrr 717 |
. . . . . . 7
⊢ (((𝜑 ∧ sup(𝐶, ℝ, < ) < (sup(𝐴, ℝ, < ) + 𝐵)) ∧ 𝑎 ∈ 𝐴) → 𝐵 ∈ ℝ) |
78 | 11 | adantlr 706 |
. . . . . . 7
⊢ (((𝜑 ∧ sup(𝐶, ℝ, < ) < (sup(𝐴, ℝ, < ) + 𝐵)) ∧ 𝑎 ∈ 𝐴) → 𝑎 ∈ ℝ) |
79 | 74, 77, 78 | ltsubaddd 10955 |
. . . . . 6
⊢ (((𝜑 ∧ sup(𝐶, ℝ, < ) < (sup(𝐴, ℝ, < ) + 𝐵)) ∧ 𝑎 ∈ 𝐴) → ((sup(𝐶, ℝ, < ) − 𝐵) < 𝑎 ↔ sup(𝐶, ℝ, < ) < (𝑎 + 𝐵))) |
80 | 76, 79 | mtbird 317 |
. . . . 5
⊢ (((𝜑 ∧ sup(𝐶, ℝ, < ) < (sup(𝐴, ℝ, < ) + 𝐵)) ∧ 𝑎 ∈ 𝐴) → ¬ (sup(𝐶, ℝ, < ) − 𝐵) < 𝑎) |
81 | 80 | nrexdv 3209 |
. . . 4
⊢ ((𝜑 ∧ sup(𝐶, ℝ, < ) < (sup(𝐴, ℝ, < ) + 𝐵)) → ¬ ∃𝑎 ∈ 𝐴 (sup(𝐶, ℝ, < ) − 𝐵) < 𝑎) |
82 | 58, 81 | pm2.65da 851 |
. . 3
⊢ (𝜑 → ¬ sup(𝐶, ℝ, < ) <
(sup(𝐴, ℝ, < ) +
𝐵)) |
83 | 51, 44 | eqleltd 10507 |
. . 3
⊢ (𝜑 → (sup(𝐶, ℝ, < ) = (sup(𝐴, ℝ, < ) + 𝐵) ↔ (sup(𝐶, ℝ, < ) ≤ (sup(𝐴, ℝ, < ) + 𝐵) ∧ ¬ sup(𝐶, ℝ, < ) <
(sup(𝐴, ℝ, < ) +
𝐵)))) |
84 | 49, 82, 83 | mpbir2and 704 |
. 2
⊢ (𝜑 → sup(𝐶, ℝ, < ) = (sup(𝐴, ℝ, < ) + 𝐵)) |
85 | 84 | eqcomd 2831 |
1
⊢ (𝜑 → (sup(𝐴, ℝ, < ) + 𝐵) = sup(𝐶, ℝ, < )) |