Step |
Hyp |
Ref |
Expression |
1 |
|
supadd.a1 |
⊢ ( 𝜑 → 𝐴 ⊆ ℝ ) |
2 |
|
supadd.a2 |
⊢ ( 𝜑 → 𝐴 ≠ ∅ ) |
3 |
|
supadd.a3 |
⊢ ( 𝜑 → ∃ 𝑥 ∈ ℝ ∀ 𝑦 ∈ 𝐴 𝑦 ≤ 𝑥 ) |
4 |
|
supadd.b1 |
⊢ ( 𝜑 → 𝐵 ⊆ ℝ ) |
5 |
|
supadd.b2 |
⊢ ( 𝜑 → 𝐵 ≠ ∅ ) |
6 |
|
supadd.b3 |
⊢ ( 𝜑 → ∃ 𝑥 ∈ ℝ ∀ 𝑦 ∈ 𝐵 𝑦 ≤ 𝑥 ) |
7 |
|
supadd.c |
⊢ 𝐶 = { 𝑧 ∣ ∃ 𝑣 ∈ 𝐴 ∃ 𝑏 ∈ 𝐵 𝑧 = ( 𝑣 + 𝑏 ) } |
8 |
4 5 6
|
suprcld |
⊢ ( 𝜑 → sup ( 𝐵 , ℝ , < ) ∈ ℝ ) |
9 |
|
eqid |
⊢ { 𝑧 ∣ ∃ 𝑎 ∈ 𝐴 𝑧 = ( 𝑎 + sup ( 𝐵 , ℝ , < ) ) } = { 𝑧 ∣ ∃ 𝑎 ∈ 𝐴 𝑧 = ( 𝑎 + sup ( 𝐵 , ℝ , < ) ) } |
10 |
1 2 3 8 9
|
supaddc |
⊢ ( 𝜑 → ( sup ( 𝐴 , ℝ , < ) + sup ( 𝐵 , ℝ , < ) ) = sup ( { 𝑧 ∣ ∃ 𝑎 ∈ 𝐴 𝑧 = ( 𝑎 + sup ( 𝐵 , ℝ , < ) ) } , ℝ , < ) ) |
11 |
1
|
sselda |
⊢ ( ( 𝜑 ∧ 𝑎 ∈ 𝐴 ) → 𝑎 ∈ ℝ ) |
12 |
11
|
recnd |
⊢ ( ( 𝜑 ∧ 𝑎 ∈ 𝐴 ) → 𝑎 ∈ ℂ ) |
13 |
8
|
adantr |
⊢ ( ( 𝜑 ∧ 𝑎 ∈ 𝐴 ) → sup ( 𝐵 , ℝ , < ) ∈ ℝ ) |
14 |
13
|
recnd |
⊢ ( ( 𝜑 ∧ 𝑎 ∈ 𝐴 ) → sup ( 𝐵 , ℝ , < ) ∈ ℂ ) |
15 |
12 14
|
addcomd |
⊢ ( ( 𝜑 ∧ 𝑎 ∈ 𝐴 ) → ( 𝑎 + sup ( 𝐵 , ℝ , < ) ) = ( sup ( 𝐵 , ℝ , < ) + 𝑎 ) ) |
16 |
15
|
eqeq2d |
⊢ ( ( 𝜑 ∧ 𝑎 ∈ 𝐴 ) → ( 𝑧 = ( 𝑎 + sup ( 𝐵 , ℝ , < ) ) ↔ 𝑧 = ( sup ( 𝐵 , ℝ , < ) + 𝑎 ) ) ) |
17 |
16
|
rexbidva |
⊢ ( 𝜑 → ( ∃ 𝑎 ∈ 𝐴 𝑧 = ( 𝑎 + sup ( 𝐵 , ℝ , < ) ) ↔ ∃ 𝑎 ∈ 𝐴 𝑧 = ( sup ( 𝐵 , ℝ , < ) + 𝑎 ) ) ) |
18 |
17
|
abbidv |
⊢ ( 𝜑 → { 𝑧 ∣ ∃ 𝑎 ∈ 𝐴 𝑧 = ( 𝑎 + sup ( 𝐵 , ℝ , < ) ) } = { 𝑧 ∣ ∃ 𝑎 ∈ 𝐴 𝑧 = ( sup ( 𝐵 , ℝ , < ) + 𝑎 ) } ) |
19 |
18
|
supeq1d |
⊢ ( 𝜑 → sup ( { 𝑧 ∣ ∃ 𝑎 ∈ 𝐴 𝑧 = ( 𝑎 + sup ( 𝐵 , ℝ , < ) ) } , ℝ , < ) = sup ( { 𝑧 ∣ ∃ 𝑎 ∈ 𝐴 𝑧 = ( sup ( 𝐵 , ℝ , < ) + 𝑎 ) } , ℝ , < ) ) |
20 |
10 19
|
eqtrd |
⊢ ( 𝜑 → ( sup ( 𝐴 , ℝ , < ) + sup ( 𝐵 , ℝ , < ) ) = sup ( { 𝑧 ∣ ∃ 𝑎 ∈ 𝐴 𝑧 = ( sup ( 𝐵 , ℝ , < ) + 𝑎 ) } , ℝ , < ) ) |
21 |
|
vex |
⊢ 𝑤 ∈ V |
22 |
|
eqeq1 |
⊢ ( 𝑧 = 𝑤 → ( 𝑧 = ( sup ( 𝐵 , ℝ , < ) + 𝑎 ) ↔ 𝑤 = ( sup ( 𝐵 , ℝ , < ) + 𝑎 ) ) ) |
23 |
22
|
rexbidv |
⊢ ( 𝑧 = 𝑤 → ( ∃ 𝑎 ∈ 𝐴 𝑧 = ( sup ( 𝐵 , ℝ , < ) + 𝑎 ) ↔ ∃ 𝑎 ∈ 𝐴 𝑤 = ( sup ( 𝐵 , ℝ , < ) + 𝑎 ) ) ) |
24 |
21 23
|
elab |
⊢ ( 𝑤 ∈ { 𝑧 ∣ ∃ 𝑎 ∈ 𝐴 𝑧 = ( sup ( 𝐵 , ℝ , < ) + 𝑎 ) } ↔ ∃ 𝑎 ∈ 𝐴 𝑤 = ( sup ( 𝐵 , ℝ , < ) + 𝑎 ) ) |
25 |
4
|
adantr |
⊢ ( ( 𝜑 ∧ 𝑎 ∈ 𝐴 ) → 𝐵 ⊆ ℝ ) |
26 |
5
|
adantr |
⊢ ( ( 𝜑 ∧ 𝑎 ∈ 𝐴 ) → 𝐵 ≠ ∅ ) |
27 |
6
|
adantr |
⊢ ( ( 𝜑 ∧ 𝑎 ∈ 𝐴 ) → ∃ 𝑥 ∈ ℝ ∀ 𝑦 ∈ 𝐵 𝑦 ≤ 𝑥 ) |
28 |
|
eqid |
⊢ { 𝑧 ∣ ∃ 𝑏 ∈ 𝐵 𝑧 = ( 𝑏 + 𝑎 ) } = { 𝑧 ∣ ∃ 𝑏 ∈ 𝐵 𝑧 = ( 𝑏 + 𝑎 ) } |
29 |
25 26 27 11 28
|
supaddc |
⊢ ( ( 𝜑 ∧ 𝑎 ∈ 𝐴 ) → ( sup ( 𝐵 , ℝ , < ) + 𝑎 ) = sup ( { 𝑧 ∣ ∃ 𝑏 ∈ 𝐵 𝑧 = ( 𝑏 + 𝑎 ) } , ℝ , < ) ) |
30 |
4
|
sselda |
⊢ ( ( 𝜑 ∧ 𝑏 ∈ 𝐵 ) → 𝑏 ∈ ℝ ) |
31 |
30
|
adantlr |
⊢ ( ( ( 𝜑 ∧ 𝑎 ∈ 𝐴 ) ∧ 𝑏 ∈ 𝐵 ) → 𝑏 ∈ ℝ ) |
32 |
31
|
recnd |
⊢ ( ( ( 𝜑 ∧ 𝑎 ∈ 𝐴 ) ∧ 𝑏 ∈ 𝐵 ) → 𝑏 ∈ ℂ ) |
33 |
11
|
adantr |
⊢ ( ( ( 𝜑 ∧ 𝑎 ∈ 𝐴 ) ∧ 𝑏 ∈ 𝐵 ) → 𝑎 ∈ ℝ ) |
34 |
33
|
recnd |
⊢ ( ( ( 𝜑 ∧ 𝑎 ∈ 𝐴 ) ∧ 𝑏 ∈ 𝐵 ) → 𝑎 ∈ ℂ ) |
35 |
32 34
|
addcomd |
⊢ ( ( ( 𝜑 ∧ 𝑎 ∈ 𝐴 ) ∧ 𝑏 ∈ 𝐵 ) → ( 𝑏 + 𝑎 ) = ( 𝑎 + 𝑏 ) ) |
36 |
35
|
eqeq2d |
⊢ ( ( ( 𝜑 ∧ 𝑎 ∈ 𝐴 ) ∧ 𝑏 ∈ 𝐵 ) → ( 𝑧 = ( 𝑏 + 𝑎 ) ↔ 𝑧 = ( 𝑎 + 𝑏 ) ) ) |
37 |
36
|
rexbidva |
⊢ ( ( 𝜑 ∧ 𝑎 ∈ 𝐴 ) → ( ∃ 𝑏 ∈ 𝐵 𝑧 = ( 𝑏 + 𝑎 ) ↔ ∃ 𝑏 ∈ 𝐵 𝑧 = ( 𝑎 + 𝑏 ) ) ) |
38 |
37
|
abbidv |
⊢ ( ( 𝜑 ∧ 𝑎 ∈ 𝐴 ) → { 𝑧 ∣ ∃ 𝑏 ∈ 𝐵 𝑧 = ( 𝑏 + 𝑎 ) } = { 𝑧 ∣ ∃ 𝑏 ∈ 𝐵 𝑧 = ( 𝑎 + 𝑏 ) } ) |
39 |
38
|
supeq1d |
⊢ ( ( 𝜑 ∧ 𝑎 ∈ 𝐴 ) → sup ( { 𝑧 ∣ ∃ 𝑏 ∈ 𝐵 𝑧 = ( 𝑏 + 𝑎 ) } , ℝ , < ) = sup ( { 𝑧 ∣ ∃ 𝑏 ∈ 𝐵 𝑧 = ( 𝑎 + 𝑏 ) } , ℝ , < ) ) |
40 |
29 39
|
eqtrd |
⊢ ( ( 𝜑 ∧ 𝑎 ∈ 𝐴 ) → ( sup ( 𝐵 , ℝ , < ) + 𝑎 ) = sup ( { 𝑧 ∣ ∃ 𝑏 ∈ 𝐵 𝑧 = ( 𝑎 + 𝑏 ) } , ℝ , < ) ) |
41 |
|
eqeq1 |
⊢ ( 𝑧 = 𝑤 → ( 𝑧 = ( 𝑎 + 𝑏 ) ↔ 𝑤 = ( 𝑎 + 𝑏 ) ) ) |
42 |
41
|
rexbidv |
⊢ ( 𝑧 = 𝑤 → ( ∃ 𝑏 ∈ 𝐵 𝑧 = ( 𝑎 + 𝑏 ) ↔ ∃ 𝑏 ∈ 𝐵 𝑤 = ( 𝑎 + 𝑏 ) ) ) |
43 |
21 42
|
elab |
⊢ ( 𝑤 ∈ { 𝑧 ∣ ∃ 𝑏 ∈ 𝐵 𝑧 = ( 𝑎 + 𝑏 ) } ↔ ∃ 𝑏 ∈ 𝐵 𝑤 = ( 𝑎 + 𝑏 ) ) |
44 |
|
rspe |
⊢ ( ( 𝑎 ∈ 𝐴 ∧ ∃ 𝑏 ∈ 𝐵 𝑤 = ( 𝑎 + 𝑏 ) ) → ∃ 𝑎 ∈ 𝐴 ∃ 𝑏 ∈ 𝐵 𝑤 = ( 𝑎 + 𝑏 ) ) |
45 |
|
oveq1 |
⊢ ( 𝑣 = 𝑎 → ( 𝑣 + 𝑏 ) = ( 𝑎 + 𝑏 ) ) |
46 |
45
|
eqeq2d |
⊢ ( 𝑣 = 𝑎 → ( 𝑧 = ( 𝑣 + 𝑏 ) ↔ 𝑧 = ( 𝑎 + 𝑏 ) ) ) |
47 |
46
|
rexbidv |
⊢ ( 𝑣 = 𝑎 → ( ∃ 𝑏 ∈ 𝐵 𝑧 = ( 𝑣 + 𝑏 ) ↔ ∃ 𝑏 ∈ 𝐵 𝑧 = ( 𝑎 + 𝑏 ) ) ) |
48 |
47
|
cbvrexvw |
⊢ ( ∃ 𝑣 ∈ 𝐴 ∃ 𝑏 ∈ 𝐵 𝑧 = ( 𝑣 + 𝑏 ) ↔ ∃ 𝑎 ∈ 𝐴 ∃ 𝑏 ∈ 𝐵 𝑧 = ( 𝑎 + 𝑏 ) ) |
49 |
41
|
2rexbidv |
⊢ ( 𝑧 = 𝑤 → ( ∃ 𝑎 ∈ 𝐴 ∃ 𝑏 ∈ 𝐵 𝑧 = ( 𝑎 + 𝑏 ) ↔ ∃ 𝑎 ∈ 𝐴 ∃ 𝑏 ∈ 𝐵 𝑤 = ( 𝑎 + 𝑏 ) ) ) |
50 |
48 49
|
syl5bb |
⊢ ( 𝑧 = 𝑤 → ( ∃ 𝑣 ∈ 𝐴 ∃ 𝑏 ∈ 𝐵 𝑧 = ( 𝑣 + 𝑏 ) ↔ ∃ 𝑎 ∈ 𝐴 ∃ 𝑏 ∈ 𝐵 𝑤 = ( 𝑎 + 𝑏 ) ) ) |
51 |
21 50 7
|
elab2 |
⊢ ( 𝑤 ∈ 𝐶 ↔ ∃ 𝑎 ∈ 𝐴 ∃ 𝑏 ∈ 𝐵 𝑤 = ( 𝑎 + 𝑏 ) ) |
52 |
44 51
|
sylibr |
⊢ ( ( 𝑎 ∈ 𝐴 ∧ ∃ 𝑏 ∈ 𝐵 𝑤 = ( 𝑎 + 𝑏 ) ) → 𝑤 ∈ 𝐶 ) |
53 |
52
|
ex |
⊢ ( 𝑎 ∈ 𝐴 → ( ∃ 𝑏 ∈ 𝐵 𝑤 = ( 𝑎 + 𝑏 ) → 𝑤 ∈ 𝐶 ) ) |
54 |
1
|
sseld |
⊢ ( 𝜑 → ( 𝑎 ∈ 𝐴 → 𝑎 ∈ ℝ ) ) |
55 |
4
|
sseld |
⊢ ( 𝜑 → ( 𝑏 ∈ 𝐵 → 𝑏 ∈ ℝ ) ) |
56 |
54 55
|
anim12d |
⊢ ( 𝜑 → ( ( 𝑎 ∈ 𝐴 ∧ 𝑏 ∈ 𝐵 ) → ( 𝑎 ∈ ℝ ∧ 𝑏 ∈ ℝ ) ) ) |
57 |
|
readdcl |
⊢ ( ( 𝑎 ∈ ℝ ∧ 𝑏 ∈ ℝ ) → ( 𝑎 + 𝑏 ) ∈ ℝ ) |
58 |
56 57
|
syl6 |
⊢ ( 𝜑 → ( ( 𝑎 ∈ 𝐴 ∧ 𝑏 ∈ 𝐵 ) → ( 𝑎 + 𝑏 ) ∈ ℝ ) ) |
59 |
|
eleq1a |
⊢ ( ( 𝑎 + 𝑏 ) ∈ ℝ → ( 𝑤 = ( 𝑎 + 𝑏 ) → 𝑤 ∈ ℝ ) ) |
60 |
58 59
|
syl6 |
⊢ ( 𝜑 → ( ( 𝑎 ∈ 𝐴 ∧ 𝑏 ∈ 𝐵 ) → ( 𝑤 = ( 𝑎 + 𝑏 ) → 𝑤 ∈ ℝ ) ) ) |
61 |
60
|
rexlimdvv |
⊢ ( 𝜑 → ( ∃ 𝑎 ∈ 𝐴 ∃ 𝑏 ∈ 𝐵 𝑤 = ( 𝑎 + 𝑏 ) → 𝑤 ∈ ℝ ) ) |
62 |
51 61
|
syl5bi |
⊢ ( 𝜑 → ( 𝑤 ∈ 𝐶 → 𝑤 ∈ ℝ ) ) |
63 |
62
|
ssrdv |
⊢ ( 𝜑 → 𝐶 ⊆ ℝ ) |
64 |
|
ovex |
⊢ ( 𝑎 + 𝑏 ) ∈ V |
65 |
64
|
isseti |
⊢ ∃ 𝑤 𝑤 = ( 𝑎 + 𝑏 ) |
66 |
65
|
rgenw |
⊢ ∀ 𝑏 ∈ 𝐵 ∃ 𝑤 𝑤 = ( 𝑎 + 𝑏 ) |
67 |
|
r19.2z |
⊢ ( ( 𝐵 ≠ ∅ ∧ ∀ 𝑏 ∈ 𝐵 ∃ 𝑤 𝑤 = ( 𝑎 + 𝑏 ) ) → ∃ 𝑏 ∈ 𝐵 ∃ 𝑤 𝑤 = ( 𝑎 + 𝑏 ) ) |
68 |
5 66 67
|
sylancl |
⊢ ( 𝜑 → ∃ 𝑏 ∈ 𝐵 ∃ 𝑤 𝑤 = ( 𝑎 + 𝑏 ) ) |
69 |
|
rexcom4 |
⊢ ( ∃ 𝑏 ∈ 𝐵 ∃ 𝑤 𝑤 = ( 𝑎 + 𝑏 ) ↔ ∃ 𝑤 ∃ 𝑏 ∈ 𝐵 𝑤 = ( 𝑎 + 𝑏 ) ) |
70 |
68 69
|
sylib |
⊢ ( 𝜑 → ∃ 𝑤 ∃ 𝑏 ∈ 𝐵 𝑤 = ( 𝑎 + 𝑏 ) ) |
71 |
70
|
ralrimivw |
⊢ ( 𝜑 → ∀ 𝑎 ∈ 𝐴 ∃ 𝑤 ∃ 𝑏 ∈ 𝐵 𝑤 = ( 𝑎 + 𝑏 ) ) |
72 |
|
r19.2z |
⊢ ( ( 𝐴 ≠ ∅ ∧ ∀ 𝑎 ∈ 𝐴 ∃ 𝑤 ∃ 𝑏 ∈ 𝐵 𝑤 = ( 𝑎 + 𝑏 ) ) → ∃ 𝑎 ∈ 𝐴 ∃ 𝑤 ∃ 𝑏 ∈ 𝐵 𝑤 = ( 𝑎 + 𝑏 ) ) |
73 |
2 71 72
|
syl2anc |
⊢ ( 𝜑 → ∃ 𝑎 ∈ 𝐴 ∃ 𝑤 ∃ 𝑏 ∈ 𝐵 𝑤 = ( 𝑎 + 𝑏 ) ) |
74 |
|
rexcom4 |
⊢ ( ∃ 𝑎 ∈ 𝐴 ∃ 𝑤 ∃ 𝑏 ∈ 𝐵 𝑤 = ( 𝑎 + 𝑏 ) ↔ ∃ 𝑤 ∃ 𝑎 ∈ 𝐴 ∃ 𝑏 ∈ 𝐵 𝑤 = ( 𝑎 + 𝑏 ) ) |
75 |
73 74
|
sylib |
⊢ ( 𝜑 → ∃ 𝑤 ∃ 𝑎 ∈ 𝐴 ∃ 𝑏 ∈ 𝐵 𝑤 = ( 𝑎 + 𝑏 ) ) |
76 |
|
n0 |
⊢ ( 𝐶 ≠ ∅ ↔ ∃ 𝑤 𝑤 ∈ 𝐶 ) |
77 |
51
|
exbii |
⊢ ( ∃ 𝑤 𝑤 ∈ 𝐶 ↔ ∃ 𝑤 ∃ 𝑎 ∈ 𝐴 ∃ 𝑏 ∈ 𝐵 𝑤 = ( 𝑎 + 𝑏 ) ) |
78 |
76 77
|
bitri |
⊢ ( 𝐶 ≠ ∅ ↔ ∃ 𝑤 ∃ 𝑎 ∈ 𝐴 ∃ 𝑏 ∈ 𝐵 𝑤 = ( 𝑎 + 𝑏 ) ) |
79 |
75 78
|
sylibr |
⊢ ( 𝜑 → 𝐶 ≠ ∅ ) |
80 |
1 2 3
|
suprcld |
⊢ ( 𝜑 → sup ( 𝐴 , ℝ , < ) ∈ ℝ ) |
81 |
80 8
|
readdcld |
⊢ ( 𝜑 → ( sup ( 𝐴 , ℝ , < ) + sup ( 𝐵 , ℝ , < ) ) ∈ ℝ ) |
82 |
11
|
adantrr |
⊢ ( ( 𝜑 ∧ ( 𝑎 ∈ 𝐴 ∧ 𝑏 ∈ 𝐵 ) ) → 𝑎 ∈ ℝ ) |
83 |
30
|
adantrl |
⊢ ( ( 𝜑 ∧ ( 𝑎 ∈ 𝐴 ∧ 𝑏 ∈ 𝐵 ) ) → 𝑏 ∈ ℝ ) |
84 |
80
|
adantr |
⊢ ( ( 𝜑 ∧ ( 𝑎 ∈ 𝐴 ∧ 𝑏 ∈ 𝐵 ) ) → sup ( 𝐴 , ℝ , < ) ∈ ℝ ) |
85 |
8
|
adantr |
⊢ ( ( 𝜑 ∧ ( 𝑎 ∈ 𝐴 ∧ 𝑏 ∈ 𝐵 ) ) → sup ( 𝐵 , ℝ , < ) ∈ ℝ ) |
86 |
1 2 3
|
3jca |
⊢ ( 𝜑 → ( 𝐴 ⊆ ℝ ∧ 𝐴 ≠ ∅ ∧ ∃ 𝑥 ∈ ℝ ∀ 𝑦 ∈ 𝐴 𝑦 ≤ 𝑥 ) ) |
87 |
|
suprub |
⊢ ( ( ( 𝐴 ⊆ ℝ ∧ 𝐴 ≠ ∅ ∧ ∃ 𝑥 ∈ ℝ ∀ 𝑦 ∈ 𝐴 𝑦 ≤ 𝑥 ) ∧ 𝑎 ∈ 𝐴 ) → 𝑎 ≤ sup ( 𝐴 , ℝ , < ) ) |
88 |
86 87
|
sylan |
⊢ ( ( 𝜑 ∧ 𝑎 ∈ 𝐴 ) → 𝑎 ≤ sup ( 𝐴 , ℝ , < ) ) |
89 |
88
|
adantrr |
⊢ ( ( 𝜑 ∧ ( 𝑎 ∈ 𝐴 ∧ 𝑏 ∈ 𝐵 ) ) → 𝑎 ≤ sup ( 𝐴 , ℝ , < ) ) |
90 |
4 5 6
|
3jca |
⊢ ( 𝜑 → ( 𝐵 ⊆ ℝ ∧ 𝐵 ≠ ∅ ∧ ∃ 𝑥 ∈ ℝ ∀ 𝑦 ∈ 𝐵 𝑦 ≤ 𝑥 ) ) |
91 |
|
suprub |
⊢ ( ( ( 𝐵 ⊆ ℝ ∧ 𝐵 ≠ ∅ ∧ ∃ 𝑥 ∈ ℝ ∀ 𝑦 ∈ 𝐵 𝑦 ≤ 𝑥 ) ∧ 𝑏 ∈ 𝐵 ) → 𝑏 ≤ sup ( 𝐵 , ℝ , < ) ) |
92 |
90 91
|
sylan |
⊢ ( ( 𝜑 ∧ 𝑏 ∈ 𝐵 ) → 𝑏 ≤ sup ( 𝐵 , ℝ , < ) ) |
93 |
92
|
adantrl |
⊢ ( ( 𝜑 ∧ ( 𝑎 ∈ 𝐴 ∧ 𝑏 ∈ 𝐵 ) ) → 𝑏 ≤ sup ( 𝐵 , ℝ , < ) ) |
94 |
82 83 84 85 89 93
|
le2addd |
⊢ ( ( 𝜑 ∧ ( 𝑎 ∈ 𝐴 ∧ 𝑏 ∈ 𝐵 ) ) → ( 𝑎 + 𝑏 ) ≤ ( sup ( 𝐴 , ℝ , < ) + sup ( 𝐵 , ℝ , < ) ) ) |
95 |
94
|
ex |
⊢ ( 𝜑 → ( ( 𝑎 ∈ 𝐴 ∧ 𝑏 ∈ 𝐵 ) → ( 𝑎 + 𝑏 ) ≤ ( sup ( 𝐴 , ℝ , < ) + sup ( 𝐵 , ℝ , < ) ) ) ) |
96 |
|
breq1 |
⊢ ( 𝑤 = ( 𝑎 + 𝑏 ) → ( 𝑤 ≤ ( sup ( 𝐴 , ℝ , < ) + sup ( 𝐵 , ℝ , < ) ) ↔ ( 𝑎 + 𝑏 ) ≤ ( sup ( 𝐴 , ℝ , < ) + sup ( 𝐵 , ℝ , < ) ) ) ) |
97 |
96
|
biimprcd |
⊢ ( ( 𝑎 + 𝑏 ) ≤ ( sup ( 𝐴 , ℝ , < ) + sup ( 𝐵 , ℝ , < ) ) → ( 𝑤 = ( 𝑎 + 𝑏 ) → 𝑤 ≤ ( sup ( 𝐴 , ℝ , < ) + sup ( 𝐵 , ℝ , < ) ) ) ) |
98 |
95 97
|
syl6 |
⊢ ( 𝜑 → ( ( 𝑎 ∈ 𝐴 ∧ 𝑏 ∈ 𝐵 ) → ( 𝑤 = ( 𝑎 + 𝑏 ) → 𝑤 ≤ ( sup ( 𝐴 , ℝ , < ) + sup ( 𝐵 , ℝ , < ) ) ) ) ) |
99 |
98
|
rexlimdvv |
⊢ ( 𝜑 → ( ∃ 𝑎 ∈ 𝐴 ∃ 𝑏 ∈ 𝐵 𝑤 = ( 𝑎 + 𝑏 ) → 𝑤 ≤ ( sup ( 𝐴 , ℝ , < ) + sup ( 𝐵 , ℝ , < ) ) ) ) |
100 |
51 99
|
syl5bi |
⊢ ( 𝜑 → ( 𝑤 ∈ 𝐶 → 𝑤 ≤ ( sup ( 𝐴 , ℝ , < ) + sup ( 𝐵 , ℝ , < ) ) ) ) |
101 |
100
|
ralrimiv |
⊢ ( 𝜑 → ∀ 𝑤 ∈ 𝐶 𝑤 ≤ ( sup ( 𝐴 , ℝ , < ) + sup ( 𝐵 , ℝ , < ) ) ) |
102 |
|
brralrspcev |
⊢ ( ( ( sup ( 𝐴 , ℝ , < ) + sup ( 𝐵 , ℝ , < ) ) ∈ ℝ ∧ ∀ 𝑤 ∈ 𝐶 𝑤 ≤ ( sup ( 𝐴 , ℝ , < ) + sup ( 𝐵 , ℝ , < ) ) ) → ∃ 𝑥 ∈ ℝ ∀ 𝑤 ∈ 𝐶 𝑤 ≤ 𝑥 ) |
103 |
81 101 102
|
syl2anc |
⊢ ( 𝜑 → ∃ 𝑥 ∈ ℝ ∀ 𝑤 ∈ 𝐶 𝑤 ≤ 𝑥 ) |
104 |
|
suprub |
⊢ ( ( ( 𝐶 ⊆ ℝ ∧ 𝐶 ≠ ∅ ∧ ∃ 𝑥 ∈ ℝ ∀ 𝑤 ∈ 𝐶 𝑤 ≤ 𝑥 ) ∧ 𝑤 ∈ 𝐶 ) → 𝑤 ≤ sup ( 𝐶 , ℝ , < ) ) |
105 |
104
|
ex |
⊢ ( ( 𝐶 ⊆ ℝ ∧ 𝐶 ≠ ∅ ∧ ∃ 𝑥 ∈ ℝ ∀ 𝑤 ∈ 𝐶 𝑤 ≤ 𝑥 ) → ( 𝑤 ∈ 𝐶 → 𝑤 ≤ sup ( 𝐶 , ℝ , < ) ) ) |
106 |
63 79 103 105
|
syl3anc |
⊢ ( 𝜑 → ( 𝑤 ∈ 𝐶 → 𝑤 ≤ sup ( 𝐶 , ℝ , < ) ) ) |
107 |
53 106
|
sylan9r |
⊢ ( ( 𝜑 ∧ 𝑎 ∈ 𝐴 ) → ( ∃ 𝑏 ∈ 𝐵 𝑤 = ( 𝑎 + 𝑏 ) → 𝑤 ≤ sup ( 𝐶 , ℝ , < ) ) ) |
108 |
43 107
|
syl5bi |
⊢ ( ( 𝜑 ∧ 𝑎 ∈ 𝐴 ) → ( 𝑤 ∈ { 𝑧 ∣ ∃ 𝑏 ∈ 𝐵 𝑧 = ( 𝑎 + 𝑏 ) } → 𝑤 ≤ sup ( 𝐶 , ℝ , < ) ) ) |
109 |
108
|
ralrimiv |
⊢ ( ( 𝜑 ∧ 𝑎 ∈ 𝐴 ) → ∀ 𝑤 ∈ { 𝑧 ∣ ∃ 𝑏 ∈ 𝐵 𝑧 = ( 𝑎 + 𝑏 ) } 𝑤 ≤ sup ( 𝐶 , ℝ , < ) ) |
110 |
33 31
|
readdcld |
⊢ ( ( ( 𝜑 ∧ 𝑎 ∈ 𝐴 ) ∧ 𝑏 ∈ 𝐵 ) → ( 𝑎 + 𝑏 ) ∈ ℝ ) |
111 |
|
eleq1a |
⊢ ( ( 𝑎 + 𝑏 ) ∈ ℝ → ( 𝑧 = ( 𝑎 + 𝑏 ) → 𝑧 ∈ ℝ ) ) |
112 |
110 111
|
syl |
⊢ ( ( ( 𝜑 ∧ 𝑎 ∈ 𝐴 ) ∧ 𝑏 ∈ 𝐵 ) → ( 𝑧 = ( 𝑎 + 𝑏 ) → 𝑧 ∈ ℝ ) ) |
113 |
112
|
rexlimdva |
⊢ ( ( 𝜑 ∧ 𝑎 ∈ 𝐴 ) → ( ∃ 𝑏 ∈ 𝐵 𝑧 = ( 𝑎 + 𝑏 ) → 𝑧 ∈ ℝ ) ) |
114 |
113
|
abssdv |
⊢ ( ( 𝜑 ∧ 𝑎 ∈ 𝐴 ) → { 𝑧 ∣ ∃ 𝑏 ∈ 𝐵 𝑧 = ( 𝑎 + 𝑏 ) } ⊆ ℝ ) |
115 |
64
|
isseti |
⊢ ∃ 𝑧 𝑧 = ( 𝑎 + 𝑏 ) |
116 |
115
|
rgenw |
⊢ ∀ 𝑏 ∈ 𝐵 ∃ 𝑧 𝑧 = ( 𝑎 + 𝑏 ) |
117 |
|
r19.2z |
⊢ ( ( 𝐵 ≠ ∅ ∧ ∀ 𝑏 ∈ 𝐵 ∃ 𝑧 𝑧 = ( 𝑎 + 𝑏 ) ) → ∃ 𝑏 ∈ 𝐵 ∃ 𝑧 𝑧 = ( 𝑎 + 𝑏 ) ) |
118 |
5 116 117
|
sylancl |
⊢ ( 𝜑 → ∃ 𝑏 ∈ 𝐵 ∃ 𝑧 𝑧 = ( 𝑎 + 𝑏 ) ) |
119 |
|
rexcom4 |
⊢ ( ∃ 𝑏 ∈ 𝐵 ∃ 𝑧 𝑧 = ( 𝑎 + 𝑏 ) ↔ ∃ 𝑧 ∃ 𝑏 ∈ 𝐵 𝑧 = ( 𝑎 + 𝑏 ) ) |
120 |
118 119
|
sylib |
⊢ ( 𝜑 → ∃ 𝑧 ∃ 𝑏 ∈ 𝐵 𝑧 = ( 𝑎 + 𝑏 ) ) |
121 |
|
abn0 |
⊢ ( { 𝑧 ∣ ∃ 𝑏 ∈ 𝐵 𝑧 = ( 𝑎 + 𝑏 ) } ≠ ∅ ↔ ∃ 𝑧 ∃ 𝑏 ∈ 𝐵 𝑧 = ( 𝑎 + 𝑏 ) ) |
122 |
120 121
|
sylibr |
⊢ ( 𝜑 → { 𝑧 ∣ ∃ 𝑏 ∈ 𝐵 𝑧 = ( 𝑎 + 𝑏 ) } ≠ ∅ ) |
123 |
122
|
adantr |
⊢ ( ( 𝜑 ∧ 𝑎 ∈ 𝐴 ) → { 𝑧 ∣ ∃ 𝑏 ∈ 𝐵 𝑧 = ( 𝑎 + 𝑏 ) } ≠ ∅ ) |
124 |
63 79 103
|
suprcld |
⊢ ( 𝜑 → sup ( 𝐶 , ℝ , < ) ∈ ℝ ) |
125 |
124
|
adantr |
⊢ ( ( 𝜑 ∧ 𝑎 ∈ 𝐴 ) → sup ( 𝐶 , ℝ , < ) ∈ ℝ ) |
126 |
|
brralrspcev |
⊢ ( ( sup ( 𝐶 , ℝ , < ) ∈ ℝ ∧ ∀ 𝑤 ∈ { 𝑧 ∣ ∃ 𝑏 ∈ 𝐵 𝑧 = ( 𝑎 + 𝑏 ) } 𝑤 ≤ sup ( 𝐶 , ℝ , < ) ) → ∃ 𝑥 ∈ ℝ ∀ 𝑤 ∈ { 𝑧 ∣ ∃ 𝑏 ∈ 𝐵 𝑧 = ( 𝑎 + 𝑏 ) } 𝑤 ≤ 𝑥 ) |
127 |
125 109 126
|
syl2anc |
⊢ ( ( 𝜑 ∧ 𝑎 ∈ 𝐴 ) → ∃ 𝑥 ∈ ℝ ∀ 𝑤 ∈ { 𝑧 ∣ ∃ 𝑏 ∈ 𝐵 𝑧 = ( 𝑎 + 𝑏 ) } 𝑤 ≤ 𝑥 ) |
128 |
|
suprleub |
⊢ ( ( ( { 𝑧 ∣ ∃ 𝑏 ∈ 𝐵 𝑧 = ( 𝑎 + 𝑏 ) } ⊆ ℝ ∧ { 𝑧 ∣ ∃ 𝑏 ∈ 𝐵 𝑧 = ( 𝑎 + 𝑏 ) } ≠ ∅ ∧ ∃ 𝑥 ∈ ℝ ∀ 𝑤 ∈ { 𝑧 ∣ ∃ 𝑏 ∈ 𝐵 𝑧 = ( 𝑎 + 𝑏 ) } 𝑤 ≤ 𝑥 ) ∧ sup ( 𝐶 , ℝ , < ) ∈ ℝ ) → ( sup ( { 𝑧 ∣ ∃ 𝑏 ∈ 𝐵 𝑧 = ( 𝑎 + 𝑏 ) } , ℝ , < ) ≤ sup ( 𝐶 , ℝ , < ) ↔ ∀ 𝑤 ∈ { 𝑧 ∣ ∃ 𝑏 ∈ 𝐵 𝑧 = ( 𝑎 + 𝑏 ) } 𝑤 ≤ sup ( 𝐶 , ℝ , < ) ) ) |
129 |
114 123 127 125 128
|
syl31anc |
⊢ ( ( 𝜑 ∧ 𝑎 ∈ 𝐴 ) → ( sup ( { 𝑧 ∣ ∃ 𝑏 ∈ 𝐵 𝑧 = ( 𝑎 + 𝑏 ) } , ℝ , < ) ≤ sup ( 𝐶 , ℝ , < ) ↔ ∀ 𝑤 ∈ { 𝑧 ∣ ∃ 𝑏 ∈ 𝐵 𝑧 = ( 𝑎 + 𝑏 ) } 𝑤 ≤ sup ( 𝐶 , ℝ , < ) ) ) |
130 |
109 129
|
mpbird |
⊢ ( ( 𝜑 ∧ 𝑎 ∈ 𝐴 ) → sup ( { 𝑧 ∣ ∃ 𝑏 ∈ 𝐵 𝑧 = ( 𝑎 + 𝑏 ) } , ℝ , < ) ≤ sup ( 𝐶 , ℝ , < ) ) |
131 |
40 130
|
eqbrtrd |
⊢ ( ( 𝜑 ∧ 𝑎 ∈ 𝐴 ) → ( sup ( 𝐵 , ℝ , < ) + 𝑎 ) ≤ sup ( 𝐶 , ℝ , < ) ) |
132 |
|
breq1 |
⊢ ( 𝑤 = ( sup ( 𝐵 , ℝ , < ) + 𝑎 ) → ( 𝑤 ≤ sup ( 𝐶 , ℝ , < ) ↔ ( sup ( 𝐵 , ℝ , < ) + 𝑎 ) ≤ sup ( 𝐶 , ℝ , < ) ) ) |
133 |
131 132
|
syl5ibrcom |
⊢ ( ( 𝜑 ∧ 𝑎 ∈ 𝐴 ) → ( 𝑤 = ( sup ( 𝐵 , ℝ , < ) + 𝑎 ) → 𝑤 ≤ sup ( 𝐶 , ℝ , < ) ) ) |
134 |
133
|
rexlimdva |
⊢ ( 𝜑 → ( ∃ 𝑎 ∈ 𝐴 𝑤 = ( sup ( 𝐵 , ℝ , < ) + 𝑎 ) → 𝑤 ≤ sup ( 𝐶 , ℝ , < ) ) ) |
135 |
24 134
|
syl5bi |
⊢ ( 𝜑 → ( 𝑤 ∈ { 𝑧 ∣ ∃ 𝑎 ∈ 𝐴 𝑧 = ( sup ( 𝐵 , ℝ , < ) + 𝑎 ) } → 𝑤 ≤ sup ( 𝐶 , ℝ , < ) ) ) |
136 |
135
|
ralrimiv |
⊢ ( 𝜑 → ∀ 𝑤 ∈ { 𝑧 ∣ ∃ 𝑎 ∈ 𝐴 𝑧 = ( sup ( 𝐵 , ℝ , < ) + 𝑎 ) } 𝑤 ≤ sup ( 𝐶 , ℝ , < ) ) |
137 |
13 11
|
readdcld |
⊢ ( ( 𝜑 ∧ 𝑎 ∈ 𝐴 ) → ( sup ( 𝐵 , ℝ , < ) + 𝑎 ) ∈ ℝ ) |
138 |
|
eleq1a |
⊢ ( ( sup ( 𝐵 , ℝ , < ) + 𝑎 ) ∈ ℝ → ( 𝑧 = ( sup ( 𝐵 , ℝ , < ) + 𝑎 ) → 𝑧 ∈ ℝ ) ) |
139 |
137 138
|
syl |
⊢ ( ( 𝜑 ∧ 𝑎 ∈ 𝐴 ) → ( 𝑧 = ( sup ( 𝐵 , ℝ , < ) + 𝑎 ) → 𝑧 ∈ ℝ ) ) |
140 |
139
|
rexlimdva |
⊢ ( 𝜑 → ( ∃ 𝑎 ∈ 𝐴 𝑧 = ( sup ( 𝐵 , ℝ , < ) + 𝑎 ) → 𝑧 ∈ ℝ ) ) |
141 |
140
|
abssdv |
⊢ ( 𝜑 → { 𝑧 ∣ ∃ 𝑎 ∈ 𝐴 𝑧 = ( sup ( 𝐵 , ℝ , < ) + 𝑎 ) } ⊆ ℝ ) |
142 |
|
ovex |
⊢ ( sup ( 𝐵 , ℝ , < ) + 𝑎 ) ∈ V |
143 |
142
|
isseti |
⊢ ∃ 𝑧 𝑧 = ( sup ( 𝐵 , ℝ , < ) + 𝑎 ) |
144 |
143
|
rgenw |
⊢ ∀ 𝑎 ∈ 𝐴 ∃ 𝑧 𝑧 = ( sup ( 𝐵 , ℝ , < ) + 𝑎 ) |
145 |
|
r19.2z |
⊢ ( ( 𝐴 ≠ ∅ ∧ ∀ 𝑎 ∈ 𝐴 ∃ 𝑧 𝑧 = ( sup ( 𝐵 , ℝ , < ) + 𝑎 ) ) → ∃ 𝑎 ∈ 𝐴 ∃ 𝑧 𝑧 = ( sup ( 𝐵 , ℝ , < ) + 𝑎 ) ) |
146 |
2 144 145
|
sylancl |
⊢ ( 𝜑 → ∃ 𝑎 ∈ 𝐴 ∃ 𝑧 𝑧 = ( sup ( 𝐵 , ℝ , < ) + 𝑎 ) ) |
147 |
|
rexcom4 |
⊢ ( ∃ 𝑎 ∈ 𝐴 ∃ 𝑧 𝑧 = ( sup ( 𝐵 , ℝ , < ) + 𝑎 ) ↔ ∃ 𝑧 ∃ 𝑎 ∈ 𝐴 𝑧 = ( sup ( 𝐵 , ℝ , < ) + 𝑎 ) ) |
148 |
146 147
|
sylib |
⊢ ( 𝜑 → ∃ 𝑧 ∃ 𝑎 ∈ 𝐴 𝑧 = ( sup ( 𝐵 , ℝ , < ) + 𝑎 ) ) |
149 |
|
abn0 |
⊢ ( { 𝑧 ∣ ∃ 𝑎 ∈ 𝐴 𝑧 = ( sup ( 𝐵 , ℝ , < ) + 𝑎 ) } ≠ ∅ ↔ ∃ 𝑧 ∃ 𝑎 ∈ 𝐴 𝑧 = ( sup ( 𝐵 , ℝ , < ) + 𝑎 ) ) |
150 |
148 149
|
sylibr |
⊢ ( 𝜑 → { 𝑧 ∣ ∃ 𝑎 ∈ 𝐴 𝑧 = ( sup ( 𝐵 , ℝ , < ) + 𝑎 ) } ≠ ∅ ) |
151 |
|
brralrspcev |
⊢ ( ( sup ( 𝐶 , ℝ , < ) ∈ ℝ ∧ ∀ 𝑤 ∈ { 𝑧 ∣ ∃ 𝑎 ∈ 𝐴 𝑧 = ( sup ( 𝐵 , ℝ , < ) + 𝑎 ) } 𝑤 ≤ sup ( 𝐶 , ℝ , < ) ) → ∃ 𝑥 ∈ ℝ ∀ 𝑤 ∈ { 𝑧 ∣ ∃ 𝑎 ∈ 𝐴 𝑧 = ( sup ( 𝐵 , ℝ , < ) + 𝑎 ) } 𝑤 ≤ 𝑥 ) |
152 |
124 136 151
|
syl2anc |
⊢ ( 𝜑 → ∃ 𝑥 ∈ ℝ ∀ 𝑤 ∈ { 𝑧 ∣ ∃ 𝑎 ∈ 𝐴 𝑧 = ( sup ( 𝐵 , ℝ , < ) + 𝑎 ) } 𝑤 ≤ 𝑥 ) |
153 |
|
suprleub |
⊢ ( ( ( { 𝑧 ∣ ∃ 𝑎 ∈ 𝐴 𝑧 = ( sup ( 𝐵 , ℝ , < ) + 𝑎 ) } ⊆ ℝ ∧ { 𝑧 ∣ ∃ 𝑎 ∈ 𝐴 𝑧 = ( sup ( 𝐵 , ℝ , < ) + 𝑎 ) } ≠ ∅ ∧ ∃ 𝑥 ∈ ℝ ∀ 𝑤 ∈ { 𝑧 ∣ ∃ 𝑎 ∈ 𝐴 𝑧 = ( sup ( 𝐵 , ℝ , < ) + 𝑎 ) } 𝑤 ≤ 𝑥 ) ∧ sup ( 𝐶 , ℝ , < ) ∈ ℝ ) → ( sup ( { 𝑧 ∣ ∃ 𝑎 ∈ 𝐴 𝑧 = ( sup ( 𝐵 , ℝ , < ) + 𝑎 ) } , ℝ , < ) ≤ sup ( 𝐶 , ℝ , < ) ↔ ∀ 𝑤 ∈ { 𝑧 ∣ ∃ 𝑎 ∈ 𝐴 𝑧 = ( sup ( 𝐵 , ℝ , < ) + 𝑎 ) } 𝑤 ≤ sup ( 𝐶 , ℝ , < ) ) ) |
154 |
141 150 152 124 153
|
syl31anc |
⊢ ( 𝜑 → ( sup ( { 𝑧 ∣ ∃ 𝑎 ∈ 𝐴 𝑧 = ( sup ( 𝐵 , ℝ , < ) + 𝑎 ) } , ℝ , < ) ≤ sup ( 𝐶 , ℝ , < ) ↔ ∀ 𝑤 ∈ { 𝑧 ∣ ∃ 𝑎 ∈ 𝐴 𝑧 = ( sup ( 𝐵 , ℝ , < ) + 𝑎 ) } 𝑤 ≤ sup ( 𝐶 , ℝ , < ) ) ) |
155 |
136 154
|
mpbird |
⊢ ( 𝜑 → sup ( { 𝑧 ∣ ∃ 𝑎 ∈ 𝐴 𝑧 = ( sup ( 𝐵 , ℝ , < ) + 𝑎 ) } , ℝ , < ) ≤ sup ( 𝐶 , ℝ , < ) ) |
156 |
20 155
|
eqbrtrd |
⊢ ( 𝜑 → ( sup ( 𝐴 , ℝ , < ) + sup ( 𝐵 , ℝ , < ) ) ≤ sup ( 𝐶 , ℝ , < ) ) |
157 |
|
suprleub |
⊢ ( ( ( 𝐶 ⊆ ℝ ∧ 𝐶 ≠ ∅ ∧ ∃ 𝑥 ∈ ℝ ∀ 𝑤 ∈ 𝐶 𝑤 ≤ 𝑥 ) ∧ ( sup ( 𝐴 , ℝ , < ) + sup ( 𝐵 , ℝ , < ) ) ∈ ℝ ) → ( sup ( 𝐶 , ℝ , < ) ≤ ( sup ( 𝐴 , ℝ , < ) + sup ( 𝐵 , ℝ , < ) ) ↔ ∀ 𝑤 ∈ 𝐶 𝑤 ≤ ( sup ( 𝐴 , ℝ , < ) + sup ( 𝐵 , ℝ , < ) ) ) ) |
158 |
63 79 103 81 157
|
syl31anc |
⊢ ( 𝜑 → ( sup ( 𝐶 , ℝ , < ) ≤ ( sup ( 𝐴 , ℝ , < ) + sup ( 𝐵 , ℝ , < ) ) ↔ ∀ 𝑤 ∈ 𝐶 𝑤 ≤ ( sup ( 𝐴 , ℝ , < ) + sup ( 𝐵 , ℝ , < ) ) ) ) |
159 |
101 158
|
mpbird |
⊢ ( 𝜑 → sup ( 𝐶 , ℝ , < ) ≤ ( sup ( 𝐴 , ℝ , < ) + sup ( 𝐵 , ℝ , < ) ) ) |
160 |
81 124
|
letri3d |
⊢ ( 𝜑 → ( ( sup ( 𝐴 , ℝ , < ) + sup ( 𝐵 , ℝ , < ) ) = sup ( 𝐶 , ℝ , < ) ↔ ( ( sup ( 𝐴 , ℝ , < ) + sup ( 𝐵 , ℝ , < ) ) ≤ sup ( 𝐶 , ℝ , < ) ∧ sup ( 𝐶 , ℝ , < ) ≤ ( sup ( 𝐴 , ℝ , < ) + sup ( 𝐵 , ℝ , < ) ) ) ) ) |
161 |
156 159 160
|
mpbir2and |
⊢ ( 𝜑 → ( sup ( 𝐴 , ℝ , < ) + sup ( 𝐵 , ℝ , < ) ) = sup ( 𝐶 , ℝ , < ) ) |