Step |
Hyp |
Ref |
Expression |
1 |
|
icccmp.1 |
⊢ 𝐽 = ( topGen ‘ ran (,) ) |
2 |
|
icccmp.2 |
⊢ 𝑇 = ( 𝐽 ↾t ( 𝐴 [,] 𝐵 ) ) |
3 |
|
icccmp.3 |
⊢ 𝐷 = ( ( abs ∘ − ) ↾ ( ℝ × ℝ ) ) |
4 |
|
icccmp.4 |
⊢ 𝑆 = { 𝑥 ∈ ( 𝐴 [,] 𝐵 ) ∣ ∃ 𝑧 ∈ ( 𝒫 𝑈 ∩ Fin ) ( 𝐴 [,] 𝑥 ) ⊆ ∪ 𝑧 } |
5 |
|
icccmp.5 |
⊢ ( 𝜑 → 𝐴 ∈ ℝ ) |
6 |
|
icccmp.6 |
⊢ ( 𝜑 → 𝐵 ∈ ℝ ) |
7 |
|
icccmp.7 |
⊢ ( 𝜑 → 𝐴 ≤ 𝐵 ) |
8 |
|
icccmp.8 |
⊢ ( 𝜑 → 𝑈 ⊆ 𝐽 ) |
9 |
|
icccmp.9 |
⊢ ( 𝜑 → ( 𝐴 [,] 𝐵 ) ⊆ ∪ 𝑈 ) |
10 |
5
|
rexrd |
⊢ ( 𝜑 → 𝐴 ∈ ℝ* ) |
11 |
6
|
rexrd |
⊢ ( 𝜑 → 𝐵 ∈ ℝ* ) |
12 |
|
lbicc2 |
⊢ ( ( 𝐴 ∈ ℝ* ∧ 𝐵 ∈ ℝ* ∧ 𝐴 ≤ 𝐵 ) → 𝐴 ∈ ( 𝐴 [,] 𝐵 ) ) |
13 |
10 11 7 12
|
syl3anc |
⊢ ( 𝜑 → 𝐴 ∈ ( 𝐴 [,] 𝐵 ) ) |
14 |
9 13
|
sseldd |
⊢ ( 𝜑 → 𝐴 ∈ ∪ 𝑈 ) |
15 |
|
eluni2 |
⊢ ( 𝐴 ∈ ∪ 𝑈 ↔ ∃ 𝑢 ∈ 𝑈 𝐴 ∈ 𝑢 ) |
16 |
14 15
|
sylib |
⊢ ( 𝜑 → ∃ 𝑢 ∈ 𝑈 𝐴 ∈ 𝑢 ) |
17 |
|
snssi |
⊢ ( 𝑢 ∈ 𝑈 → { 𝑢 } ⊆ 𝑈 ) |
18 |
17
|
ad2antrl |
⊢ ( ( 𝜑 ∧ ( 𝑢 ∈ 𝑈 ∧ 𝐴 ∈ 𝑢 ) ) → { 𝑢 } ⊆ 𝑈 ) |
19 |
|
snex |
⊢ { 𝑢 } ∈ V |
20 |
19
|
elpw |
⊢ ( { 𝑢 } ∈ 𝒫 𝑈 ↔ { 𝑢 } ⊆ 𝑈 ) |
21 |
18 20
|
sylibr |
⊢ ( ( 𝜑 ∧ ( 𝑢 ∈ 𝑈 ∧ 𝐴 ∈ 𝑢 ) ) → { 𝑢 } ∈ 𝒫 𝑈 ) |
22 |
|
snfi |
⊢ { 𝑢 } ∈ Fin |
23 |
22
|
a1i |
⊢ ( ( 𝜑 ∧ ( 𝑢 ∈ 𝑈 ∧ 𝐴 ∈ 𝑢 ) ) → { 𝑢 } ∈ Fin ) |
24 |
21 23
|
elind |
⊢ ( ( 𝜑 ∧ ( 𝑢 ∈ 𝑈 ∧ 𝐴 ∈ 𝑢 ) ) → { 𝑢 } ∈ ( 𝒫 𝑈 ∩ Fin ) ) |
25 |
10
|
adantr |
⊢ ( ( 𝜑 ∧ ( 𝑢 ∈ 𝑈 ∧ 𝐴 ∈ 𝑢 ) ) → 𝐴 ∈ ℝ* ) |
26 |
|
iccid |
⊢ ( 𝐴 ∈ ℝ* → ( 𝐴 [,] 𝐴 ) = { 𝐴 } ) |
27 |
25 26
|
syl |
⊢ ( ( 𝜑 ∧ ( 𝑢 ∈ 𝑈 ∧ 𝐴 ∈ 𝑢 ) ) → ( 𝐴 [,] 𝐴 ) = { 𝐴 } ) |
28 |
|
snssi |
⊢ ( 𝐴 ∈ 𝑢 → { 𝐴 } ⊆ 𝑢 ) |
29 |
28
|
ad2antll |
⊢ ( ( 𝜑 ∧ ( 𝑢 ∈ 𝑈 ∧ 𝐴 ∈ 𝑢 ) ) → { 𝐴 } ⊆ 𝑢 ) |
30 |
27 29
|
eqsstrd |
⊢ ( ( 𝜑 ∧ ( 𝑢 ∈ 𝑈 ∧ 𝐴 ∈ 𝑢 ) ) → ( 𝐴 [,] 𝐴 ) ⊆ 𝑢 ) |
31 |
|
unieq |
⊢ ( 𝑧 = { 𝑢 } → ∪ 𝑧 = ∪ { 𝑢 } ) |
32 |
|
vex |
⊢ 𝑢 ∈ V |
33 |
32
|
unisn |
⊢ ∪ { 𝑢 } = 𝑢 |
34 |
31 33
|
eqtrdi |
⊢ ( 𝑧 = { 𝑢 } → ∪ 𝑧 = 𝑢 ) |
35 |
34
|
sseq2d |
⊢ ( 𝑧 = { 𝑢 } → ( ( 𝐴 [,] 𝐴 ) ⊆ ∪ 𝑧 ↔ ( 𝐴 [,] 𝐴 ) ⊆ 𝑢 ) ) |
36 |
35
|
rspcev |
⊢ ( ( { 𝑢 } ∈ ( 𝒫 𝑈 ∩ Fin ) ∧ ( 𝐴 [,] 𝐴 ) ⊆ 𝑢 ) → ∃ 𝑧 ∈ ( 𝒫 𝑈 ∩ Fin ) ( 𝐴 [,] 𝐴 ) ⊆ ∪ 𝑧 ) |
37 |
24 30 36
|
syl2anc |
⊢ ( ( 𝜑 ∧ ( 𝑢 ∈ 𝑈 ∧ 𝐴 ∈ 𝑢 ) ) → ∃ 𝑧 ∈ ( 𝒫 𝑈 ∩ Fin ) ( 𝐴 [,] 𝐴 ) ⊆ ∪ 𝑧 ) |
38 |
16 37
|
rexlimddv |
⊢ ( 𝜑 → ∃ 𝑧 ∈ ( 𝒫 𝑈 ∩ Fin ) ( 𝐴 [,] 𝐴 ) ⊆ ∪ 𝑧 ) |
39 |
|
oveq2 |
⊢ ( 𝑥 = 𝐴 → ( 𝐴 [,] 𝑥 ) = ( 𝐴 [,] 𝐴 ) ) |
40 |
39
|
sseq1d |
⊢ ( 𝑥 = 𝐴 → ( ( 𝐴 [,] 𝑥 ) ⊆ ∪ 𝑧 ↔ ( 𝐴 [,] 𝐴 ) ⊆ ∪ 𝑧 ) ) |
41 |
40
|
rexbidv |
⊢ ( 𝑥 = 𝐴 → ( ∃ 𝑧 ∈ ( 𝒫 𝑈 ∩ Fin ) ( 𝐴 [,] 𝑥 ) ⊆ ∪ 𝑧 ↔ ∃ 𝑧 ∈ ( 𝒫 𝑈 ∩ Fin ) ( 𝐴 [,] 𝐴 ) ⊆ ∪ 𝑧 ) ) |
42 |
41 4
|
elrab2 |
⊢ ( 𝐴 ∈ 𝑆 ↔ ( 𝐴 ∈ ( 𝐴 [,] 𝐵 ) ∧ ∃ 𝑧 ∈ ( 𝒫 𝑈 ∩ Fin ) ( 𝐴 [,] 𝐴 ) ⊆ ∪ 𝑧 ) ) |
43 |
13 38 42
|
sylanbrc |
⊢ ( 𝜑 → 𝐴 ∈ 𝑆 ) |
44 |
4
|
ssrab3 |
⊢ 𝑆 ⊆ ( 𝐴 [,] 𝐵 ) |
45 |
44
|
sseli |
⊢ ( 𝑦 ∈ 𝑆 → 𝑦 ∈ ( 𝐴 [,] 𝐵 ) ) |
46 |
|
elicc2 |
⊢ ( ( 𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ ) → ( 𝑦 ∈ ( 𝐴 [,] 𝐵 ) ↔ ( 𝑦 ∈ ℝ ∧ 𝐴 ≤ 𝑦 ∧ 𝑦 ≤ 𝐵 ) ) ) |
47 |
5 6 46
|
syl2anc |
⊢ ( 𝜑 → ( 𝑦 ∈ ( 𝐴 [,] 𝐵 ) ↔ ( 𝑦 ∈ ℝ ∧ 𝐴 ≤ 𝑦 ∧ 𝑦 ≤ 𝐵 ) ) ) |
48 |
47
|
biimpa |
⊢ ( ( 𝜑 ∧ 𝑦 ∈ ( 𝐴 [,] 𝐵 ) ) → ( 𝑦 ∈ ℝ ∧ 𝐴 ≤ 𝑦 ∧ 𝑦 ≤ 𝐵 ) ) |
49 |
48
|
simp3d |
⊢ ( ( 𝜑 ∧ 𝑦 ∈ ( 𝐴 [,] 𝐵 ) ) → 𝑦 ≤ 𝐵 ) |
50 |
45 49
|
sylan2 |
⊢ ( ( 𝜑 ∧ 𝑦 ∈ 𝑆 ) → 𝑦 ≤ 𝐵 ) |
51 |
50
|
ralrimiva |
⊢ ( 𝜑 → ∀ 𝑦 ∈ 𝑆 𝑦 ≤ 𝐵 ) |
52 |
43 51
|
jca |
⊢ ( 𝜑 → ( 𝐴 ∈ 𝑆 ∧ ∀ 𝑦 ∈ 𝑆 𝑦 ≤ 𝐵 ) ) |