Step | Hyp | Ref
| Expression |
1 | | vex 3424 |
. . . . . . 7
⊢ 𝑤 ∈ V |
2 | | oveq1 7238 |
. . . . . . . . . 10
⊢ (𝑣 = 𝑎 → (𝑣 + 𝐵) = (𝑎 + 𝐵)) |
3 | 2 | eqeq2d 2749 |
. . . . . . . . 9
⊢ (𝑣 = 𝑎 → (𝑧 = (𝑣 + 𝐵) ↔ 𝑧 = (𝑎 + 𝐵))) |
4 | 3 | cbvrexvw 3371 |
. . . . . . . 8
⊢
(∃𝑣 ∈
𝐴 𝑧 = (𝑣 + 𝐵) ↔ ∃𝑎 ∈ 𝐴 𝑧 = (𝑎 + 𝐵)) |
5 | | eqeq1 2742 |
. . . . . . . . 9
⊢ (𝑧 = 𝑤 → (𝑧 = (𝑎 + 𝐵) ↔ 𝑤 = (𝑎 + 𝐵))) |
6 | 5 | rexbidv 3224 |
. . . . . . . 8
⊢ (𝑧 = 𝑤 → (∃𝑎 ∈ 𝐴 𝑧 = (𝑎 + 𝐵) ↔ ∃𝑎 ∈ 𝐴 𝑤 = (𝑎 + 𝐵))) |
7 | 4, 6 | syl5bb 286 |
. . . . . . 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 3603 |
. . . . . 6
⊢ (𝑤 ∈ 𝐶 ↔ ∃𝑎 ∈ 𝐴 𝑤 = (𝑎 + 𝐵)) |
10 | | supadd.a1 |
. . . . . . . . . 10
⊢ (𝜑 → 𝐴 ⊆ ℝ) |
11 | 10 | sselda 3915 |
. . . . . . . . 9
⊢ ((𝜑 ∧ 𝑎 ∈ 𝐴) → 𝑎 ∈ ℝ) |
12 | | supadd.a2 |
. . . . . . . . . . 11
⊢ (𝜑 → 𝐴 ≠ ∅) |
13 | | supadd.a3 |
. . . . . . . . . . 11
⊢ (𝜑 → ∃𝑥 ∈ ℝ ∀𝑦 ∈ 𝐴 𝑦 ≤ 𝑥) |
14 | 10, 12, 13 | suprcld 11819 |
. . . . . . . . . 10
⊢ (𝜑 → sup(𝐴, ℝ, < ) ∈
ℝ) |
15 | 14 | adantr 484 |
. . . . . . . . 9
⊢ ((𝜑 ∧ 𝑎 ∈ 𝐴) → sup(𝐴, ℝ, < ) ∈
ℝ) |
16 | | supaddc.b |
. . . . . . . . . 10
⊢ (𝜑 → 𝐵 ∈ ℝ) |
17 | 16 | adantr 484 |
. . . . . . . . 9
⊢ ((𝜑 ∧ 𝑎 ∈ 𝐴) → 𝐵 ∈ ℝ) |
18 | 10, 12, 13 | 3jca 1130 |
. . . . . . . . . 10
⊢ (𝜑 → (𝐴 ⊆ ℝ ∧ 𝐴 ≠ ∅ ∧ ∃𝑥 ∈ ℝ ∀𝑦 ∈ 𝐴 𝑦 ≤ 𝑥)) |
19 | | suprub 11817 |
. . . . . . . . . 10
⊢ (((𝐴 ⊆ ℝ ∧ 𝐴 ≠ ∅ ∧ ∃𝑥 ∈ ℝ ∀𝑦 ∈ 𝐴 𝑦 ≤ 𝑥) ∧ 𝑎 ∈ 𝐴) → 𝑎 ≤ sup(𝐴, ℝ, < )) |
20 | 18, 19 | sylan 583 |
. . . . . . . . 9
⊢ ((𝜑 ∧ 𝑎 ∈ 𝐴) → 𝑎 ≤ sup(𝐴, ℝ, < )) |
21 | 11, 15, 17, 20 | leadd1dd 11470 |
. . . . . . . 8
⊢ ((𝜑 ∧ 𝑎 ∈ 𝐴) → (𝑎 + 𝐵) ≤ (sup(𝐴, ℝ, < ) + 𝐵)) |
22 | | breq1 5070 |
. . . . . . . 8
⊢ (𝑤 = (𝑎 + 𝐵) → (𝑤 ≤ (sup(𝐴, ℝ, < ) + 𝐵) ↔ (𝑎 + 𝐵) ≤ (sup(𝐴, ℝ, < ) + 𝐵))) |
23 | 21, 22 | syl5ibrcom 250 |
. . . . . . 7
⊢ ((𝜑 ∧ 𝑎 ∈ 𝐴) → (𝑤 = (𝑎 + 𝐵) → 𝑤 ≤ (sup(𝐴, ℝ, < ) + 𝐵))) |
24 | 23 | rexlimdva 3211 |
. . . . . 6
⊢ (𝜑 → (∃𝑎 ∈ 𝐴 𝑤 = (𝑎 + 𝐵) → 𝑤 ≤ (sup(𝐴, ℝ, < ) + 𝐵))) |
25 | 9, 24 | syl5bi 245 |
. . . . 5
⊢ (𝜑 → (𝑤 ∈ 𝐶 → 𝑤 ≤ (sup(𝐴, ℝ, < ) + 𝐵))) |
26 | 25 | ralrimiv 3105 |
. . . 4
⊢ (𝜑 → ∀𝑤 ∈ 𝐶 𝑤 ≤ (sup(𝐴, ℝ, < ) + 𝐵)) |
27 | 11, 17 | readdcld 10886 |
. . . . . . . . 9
⊢ ((𝜑 ∧ 𝑎 ∈ 𝐴) → (𝑎 + 𝐵) ∈ ℝ) |
28 | | eleq1a 2834 |
. . . . . . . . 9
⊢ ((𝑎 + 𝐵) ∈ ℝ → (𝑤 = (𝑎 + 𝐵) → 𝑤 ∈ ℝ)) |
29 | 27, 28 | syl 17 |
. . . . . . . 8
⊢ ((𝜑 ∧ 𝑎 ∈ 𝐴) → (𝑤 = (𝑎 + 𝐵) → 𝑤 ∈ ℝ)) |
30 | 29 | rexlimdva 3211 |
. . . . . . 7
⊢ (𝜑 → (∃𝑎 ∈ 𝐴 𝑤 = (𝑎 + 𝐵) → 𝑤 ∈ ℝ)) |
31 | 9, 30 | syl5bi 245 |
. . . . . 6
⊢ (𝜑 → (𝑤 ∈ 𝐶 → 𝑤 ∈ ℝ)) |
32 | 31 | ssrdv 3921 |
. . . . 5
⊢ (𝜑 → 𝐶 ⊆ ℝ) |
33 | | ovex 7264 |
. . . . . . . . 9
⊢ (𝑎 + 𝐵) ∈ V |
34 | 33 | isseti 3435 |
. . . . . . . 8
⊢
∃𝑤 𝑤 = (𝑎 + 𝐵) |
35 | 34 | rgenw 3074 |
. . . . . . 7
⊢
∀𝑎 ∈
𝐴 ∃𝑤 𝑤 = (𝑎 + 𝐵) |
36 | | r19.2z 4420 |
. . . . . . 7
⊢ ((𝐴 ≠ ∅ ∧
∀𝑎 ∈ 𝐴 ∃𝑤 𝑤 = (𝑎 + 𝐵)) → ∃𝑎 ∈ 𝐴 ∃𝑤 𝑤 = (𝑎 + 𝐵)) |
37 | 12, 35, 36 | sylancl 589 |
. . . . . 6
⊢ (𝜑 → ∃𝑎 ∈ 𝐴 ∃𝑤 𝑤 = (𝑎 + 𝐵)) |
38 | 9 | exbii 1855 |
. . . . . . 7
⊢
(∃𝑤 𝑤 ∈ 𝐶 ↔ ∃𝑤∃𝑎 ∈ 𝐴 𝑤 = (𝑎 + 𝐵)) |
39 | | n0 4275 |
. . . . . . 7
⊢ (𝐶 ≠ ∅ ↔
∃𝑤 𝑤 ∈ 𝐶) |
40 | | rexcom4 3178 |
. . . . . . 7
⊢
(∃𝑎 ∈
𝐴 ∃𝑤 𝑤 = (𝑎 + 𝐵) ↔ ∃𝑤∃𝑎 ∈ 𝐴 𝑤 = (𝑎 + 𝐵)) |
41 | 38, 39, 40 | 3bitr4i 306 |
. . . . . 6
⊢ (𝐶 ≠ ∅ ↔
∃𝑎 ∈ 𝐴 ∃𝑤 𝑤 = (𝑎 + 𝐵)) |
42 | 37, 41 | sylibr 237 |
. . . . 5
⊢ (𝜑 → 𝐶 ≠ ∅) |
43 | 14, 16 | readdcld 10886 |
. . . . . 6
⊢ (𝜑 → (sup(𝐴, ℝ, < ) + 𝐵) ∈ ℝ) |
44 | | brralrspcev 5127 |
. . . . . 6
⊢
(((sup(𝐴, ℝ,
< ) + 𝐵) ∈ ℝ
∧ ∀𝑤 ∈
𝐶 𝑤 ≤ (sup(𝐴, ℝ, < ) + 𝐵)) → ∃𝑥 ∈ ℝ ∀𝑤 ∈ 𝐶 𝑤 ≤ 𝑥) |
45 | 43, 26, 44 | syl2anc 587 |
. . . . 5
⊢ (𝜑 → ∃𝑥 ∈ ℝ ∀𝑤 ∈ 𝐶 𝑤 ≤ 𝑥) |
46 | | suprleub 11822 |
. . . . 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))) |
47 | 32, 42, 45, 43, 46 | syl31anc 1375 |
. . . 4
⊢ (𝜑 → (sup(𝐶, ℝ, < ) ≤ (sup(𝐴, ℝ, < ) + 𝐵) ↔ ∀𝑤 ∈ 𝐶 𝑤 ≤ (sup(𝐴, ℝ, < ) + 𝐵))) |
48 | 26, 47 | mpbird 260 |
. . 3
⊢ (𝜑 → sup(𝐶, ℝ, < ) ≤ (sup(𝐴, ℝ, < ) + 𝐵)) |
49 | 32, 42, 45 | suprcld 11819 |
. . . . . . 7
⊢ (𝜑 → sup(𝐶, ℝ, < ) ∈
ℝ) |
50 | 49, 16, 14 | ltsubaddd 11452 |
. . . . . 6
⊢ (𝜑 → ((sup(𝐶, ℝ, < ) − 𝐵) < sup(𝐴, ℝ, < ) ↔ sup(𝐶, ℝ, < ) <
(sup(𝐴, ℝ, < ) +
𝐵))) |
51 | 50 | biimpar 481 |
. . . . 5
⊢ ((𝜑 ∧ sup(𝐶, ℝ, < ) < (sup(𝐴, ℝ, < ) + 𝐵)) → (sup(𝐶, ℝ, < ) − 𝐵) < sup(𝐴, ℝ, < )) |
52 | 49, 16 | resubcld 11284 |
. . . . . . 7
⊢ (𝜑 → (sup(𝐶, ℝ, < ) − 𝐵) ∈ ℝ) |
53 | | suprlub 11820 |
. . . . . . 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)) |
54 | 10, 12, 13, 52, 53 | syl31anc 1375 |
. . . . . 6
⊢ (𝜑 → ((sup(𝐶, ℝ, < ) − 𝐵) < sup(𝐴, ℝ, < ) ↔ ∃𝑎 ∈ 𝐴 (sup(𝐶, ℝ, < ) − 𝐵) < 𝑎)) |
55 | 54 | adantr 484 |
. . . . 5
⊢ ((𝜑 ∧ sup(𝐶, ℝ, < ) < (sup(𝐴, ℝ, < ) + 𝐵)) → ((sup(𝐶, ℝ, < ) − 𝐵) < sup(𝐴, ℝ, < ) ↔ ∃𝑎 ∈ 𝐴 (sup(𝐶, ℝ, < ) − 𝐵) < 𝑎)) |
56 | 51, 55 | mpbid 235 |
. . . 4
⊢ ((𝜑 ∧ sup(𝐶, ℝ, < ) < (sup(𝐴, ℝ, < ) + 𝐵)) → ∃𝑎 ∈ 𝐴 (sup(𝐶, ℝ, < ) − 𝐵) < 𝑎) |
57 | 27 | adantlr 715 |
. . . . . . 7
⊢ (((𝜑 ∧ sup(𝐶, ℝ, < ) < (sup(𝐴, ℝ, < ) + 𝐵)) ∧ 𝑎 ∈ 𝐴) → (𝑎 + 𝐵) ∈ ℝ) |
58 | 49 | ad2antrr 726 |
. . . . . . 7
⊢ (((𝜑 ∧ sup(𝐶, ℝ, < ) < (sup(𝐴, ℝ, < ) + 𝐵)) ∧ 𝑎 ∈ 𝐴) → sup(𝐶, ℝ, < ) ∈
ℝ) |
59 | | rspe 3231 |
. . . . . . . . . . . . . 14
⊢ ((𝑎 ∈ 𝐴 ∧ 𝑤 = (𝑎 + 𝐵)) → ∃𝑎 ∈ 𝐴 𝑤 = (𝑎 + 𝐵)) |
60 | 59, 9 | sylibr 237 |
. . . . . . . . . . . . 13
⊢ ((𝑎 ∈ 𝐴 ∧ 𝑤 = (𝑎 + 𝐵)) → 𝑤 ∈ 𝐶) |
61 | 60 | adantl 485 |
. . . . . . . . . . . 12
⊢ ((𝜑 ∧ (𝑎 ∈ 𝐴 ∧ 𝑤 = (𝑎 + 𝐵))) → 𝑤 ∈ 𝐶) |
62 | | simplrr 778 |
. . . . . . . . . . . . 13
⊢ (((𝜑 ∧ (𝑎 ∈ 𝐴 ∧ 𝑤 = (𝑎 + 𝐵))) ∧ 𝑤 ∈ 𝐶) → 𝑤 = (𝑎 + 𝐵)) |
63 | 32, 42, 45 | 3jca 1130 |
. . . . . . . . . . . . . . 15
⊢ (𝜑 → (𝐶 ⊆ ℝ ∧ 𝐶 ≠ ∅ ∧ ∃𝑥 ∈ ℝ ∀𝑤 ∈ 𝐶 𝑤 ≤ 𝑥)) |
64 | | suprub 11817 |
. . . . . . . . . . . . . . 15
⊢ (((𝐶 ⊆ ℝ ∧ 𝐶 ≠ ∅ ∧ ∃𝑥 ∈ ℝ ∀𝑤 ∈ 𝐶 𝑤 ≤ 𝑥) ∧ 𝑤 ∈ 𝐶) → 𝑤 ≤ sup(𝐶, ℝ, < )) |
65 | 63, 64 | sylan 583 |
. . . . . . . . . . . . . 14
⊢ ((𝜑 ∧ 𝑤 ∈ 𝐶) → 𝑤 ≤ sup(𝐶, ℝ, < )) |
66 | 65 | adantlr 715 |
. . . . . . . . . . . . 13
⊢ (((𝜑 ∧ (𝑎 ∈ 𝐴 ∧ 𝑤 = (𝑎 + 𝐵))) ∧ 𝑤 ∈ 𝐶) → 𝑤 ≤ sup(𝐶, ℝ, < )) |
67 | 62, 66 | eqbrtrrd 5091 |
. . . . . . . . . . . 12
⊢ (((𝜑 ∧ (𝑎 ∈ 𝐴 ∧ 𝑤 = (𝑎 + 𝐵))) ∧ 𝑤 ∈ 𝐶) → (𝑎 + 𝐵) ≤ sup(𝐶, ℝ, < )) |
68 | 61, 67 | mpdan 687 |
. . . . . . . . . . 11
⊢ ((𝜑 ∧ (𝑎 ∈ 𝐴 ∧ 𝑤 = (𝑎 + 𝐵))) → (𝑎 + 𝐵) ≤ sup(𝐶, ℝ, < )) |
69 | 68 | expr 460 |
. . . . . . . . . 10
⊢ ((𝜑 ∧ 𝑎 ∈ 𝐴) → (𝑤 = (𝑎 + 𝐵) → (𝑎 + 𝐵) ≤ sup(𝐶, ℝ, < ))) |
70 | 69 | exlimdv 1941 |
. . . . . . . . 9
⊢ ((𝜑 ∧ 𝑎 ∈ 𝐴) → (∃𝑤 𝑤 = (𝑎 + 𝐵) → (𝑎 + 𝐵) ≤ sup(𝐶, ℝ, < ))) |
71 | 34, 70 | mpi 20 |
. . . . . . . 8
⊢ ((𝜑 ∧ 𝑎 ∈ 𝐴) → (𝑎 + 𝐵) ≤ sup(𝐶, ℝ, < )) |
72 | 71 | adantlr 715 |
. . . . . . 7
⊢ (((𝜑 ∧ sup(𝐶, ℝ, < ) < (sup(𝐴, ℝ, < ) + 𝐵)) ∧ 𝑎 ∈ 𝐴) → (𝑎 + 𝐵) ≤ sup(𝐶, ℝ, < )) |
73 | 57, 58, 72 | lensymd 11007 |
. . . . . 6
⊢ (((𝜑 ∧ sup(𝐶, ℝ, < ) < (sup(𝐴, ℝ, < ) + 𝐵)) ∧ 𝑎 ∈ 𝐴) → ¬ sup(𝐶, ℝ, < ) < (𝑎 + 𝐵)) |
74 | 16 | ad2antrr 726 |
. . . . . . 7
⊢ (((𝜑 ∧ sup(𝐶, ℝ, < ) < (sup(𝐴, ℝ, < ) + 𝐵)) ∧ 𝑎 ∈ 𝐴) → 𝐵 ∈ ℝ) |
75 | 11 | adantlr 715 |
. . . . . . 7
⊢ (((𝜑 ∧ sup(𝐶, ℝ, < ) < (sup(𝐴, ℝ, < ) + 𝐵)) ∧ 𝑎 ∈ 𝐴) → 𝑎 ∈ ℝ) |
76 | 58, 74, 75 | ltsubaddd 11452 |
. . . . . 6
⊢ (((𝜑 ∧ sup(𝐶, ℝ, < ) < (sup(𝐴, ℝ, < ) + 𝐵)) ∧ 𝑎 ∈ 𝐴) → ((sup(𝐶, ℝ, < ) − 𝐵) < 𝑎 ↔ sup(𝐶, ℝ, < ) < (𝑎 + 𝐵))) |
77 | 73, 76 | mtbird 328 |
. . . . 5
⊢ (((𝜑 ∧ sup(𝐶, ℝ, < ) < (sup(𝐴, ℝ, < ) + 𝐵)) ∧ 𝑎 ∈ 𝐴) → ¬ (sup(𝐶, ℝ, < ) − 𝐵) < 𝑎) |
78 | 77 | nrexdv 3196 |
. . . 4
⊢ ((𝜑 ∧ sup(𝐶, ℝ, < ) < (sup(𝐴, ℝ, < ) + 𝐵)) → ¬ ∃𝑎 ∈ 𝐴 (sup(𝐶, ℝ, < ) − 𝐵) < 𝑎) |
79 | 56, 78 | pm2.65da 817 |
. . 3
⊢ (𝜑 → ¬ sup(𝐶, ℝ, < ) <
(sup(𝐴, ℝ, < ) +
𝐵)) |
80 | 49, 43 | eqleltd 11000 |
. . 3
⊢ (𝜑 → (sup(𝐶, ℝ, < ) = (sup(𝐴, ℝ, < ) + 𝐵) ↔ (sup(𝐶, ℝ, < ) ≤ (sup(𝐴, ℝ, < ) + 𝐵) ∧ ¬ sup(𝐶, ℝ, < ) <
(sup(𝐴, ℝ, < ) +
𝐵)))) |
81 | 48, 79, 80 | mpbir2and 713 |
. 2
⊢ (𝜑 → sup(𝐶, ℝ, < ) = (sup(𝐴, ℝ, < ) + 𝐵)) |
82 | 81 | eqcomd 2744 |
1
⊢ (𝜑 → (sup(𝐴, ℝ, < ) + 𝐵) = sup(𝐶, ℝ, < )) |