Step |
Hyp |
Ref |
Expression |
1 |
|
uzfissfz.m |
⊢ ( 𝜑 → 𝑀 ∈ ℤ ) |
2 |
|
uzfissfz.z |
⊢ 𝑍 = ( ℤ≥ ‘ 𝑀 ) |
3 |
|
uzfissfz.a |
⊢ ( 𝜑 → 𝐴 ⊆ 𝑍 ) |
4 |
|
uzfissfz.fi |
⊢ ( 𝜑 → 𝐴 ∈ Fin ) |
5 |
|
uzid |
⊢ ( 𝑀 ∈ ℤ → 𝑀 ∈ ( ℤ≥ ‘ 𝑀 ) ) |
6 |
1 5
|
syl |
⊢ ( 𝜑 → 𝑀 ∈ ( ℤ≥ ‘ 𝑀 ) ) |
7 |
2
|
a1i |
⊢ ( 𝜑 → 𝑍 = ( ℤ≥ ‘ 𝑀 ) ) |
8 |
7
|
eqcomd |
⊢ ( 𝜑 → ( ℤ≥ ‘ 𝑀 ) = 𝑍 ) |
9 |
6 8
|
eleqtrd |
⊢ ( 𝜑 → 𝑀 ∈ 𝑍 ) |
10 |
9
|
adantr |
⊢ ( ( 𝜑 ∧ 𝐴 = ∅ ) → 𝑀 ∈ 𝑍 ) |
11 |
|
id |
⊢ ( 𝐴 = ∅ → 𝐴 = ∅ ) |
12 |
|
0ss |
⊢ ∅ ⊆ ( 𝑀 ... 𝑀 ) |
13 |
12
|
a1i |
⊢ ( 𝐴 = ∅ → ∅ ⊆ ( 𝑀 ... 𝑀 ) ) |
14 |
11 13
|
eqsstrd |
⊢ ( 𝐴 = ∅ → 𝐴 ⊆ ( 𝑀 ... 𝑀 ) ) |
15 |
14
|
adantl |
⊢ ( ( 𝜑 ∧ 𝐴 = ∅ ) → 𝐴 ⊆ ( 𝑀 ... 𝑀 ) ) |
16 |
|
oveq2 |
⊢ ( 𝑘 = 𝑀 → ( 𝑀 ... 𝑘 ) = ( 𝑀 ... 𝑀 ) ) |
17 |
16
|
sseq2d |
⊢ ( 𝑘 = 𝑀 → ( 𝐴 ⊆ ( 𝑀 ... 𝑘 ) ↔ 𝐴 ⊆ ( 𝑀 ... 𝑀 ) ) ) |
18 |
17
|
rspcev |
⊢ ( ( 𝑀 ∈ 𝑍 ∧ 𝐴 ⊆ ( 𝑀 ... 𝑀 ) ) → ∃ 𝑘 ∈ 𝑍 𝐴 ⊆ ( 𝑀 ... 𝑘 ) ) |
19 |
10 15 18
|
syl2anc |
⊢ ( ( 𝜑 ∧ 𝐴 = ∅ ) → ∃ 𝑘 ∈ 𝑍 𝐴 ⊆ ( 𝑀 ... 𝑘 ) ) |
20 |
3
|
adantr |
⊢ ( ( 𝜑 ∧ ¬ 𝐴 = ∅ ) → 𝐴 ⊆ 𝑍 ) |
21 |
|
uzssz |
⊢ ( ℤ≥ ‘ 𝑀 ) ⊆ ℤ |
22 |
2 21
|
eqsstri |
⊢ 𝑍 ⊆ ℤ |
23 |
22
|
a1i |
⊢ ( 𝜑 → 𝑍 ⊆ ℤ ) |
24 |
3 23
|
sstrd |
⊢ ( 𝜑 → 𝐴 ⊆ ℤ ) |
25 |
24
|
adantr |
⊢ ( ( 𝜑 ∧ ¬ 𝐴 = ∅ ) → 𝐴 ⊆ ℤ ) |
26 |
11
|
necon3bi |
⊢ ( ¬ 𝐴 = ∅ → 𝐴 ≠ ∅ ) |
27 |
26
|
adantl |
⊢ ( ( 𝜑 ∧ ¬ 𝐴 = ∅ ) → 𝐴 ≠ ∅ ) |
28 |
4
|
adantr |
⊢ ( ( 𝜑 ∧ ¬ 𝐴 = ∅ ) → 𝐴 ∈ Fin ) |
29 |
|
suprfinzcl |
⊢ ( ( 𝐴 ⊆ ℤ ∧ 𝐴 ≠ ∅ ∧ 𝐴 ∈ Fin ) → sup ( 𝐴 , ℝ , < ) ∈ 𝐴 ) |
30 |
25 27 28 29
|
syl3anc |
⊢ ( ( 𝜑 ∧ ¬ 𝐴 = ∅ ) → sup ( 𝐴 , ℝ , < ) ∈ 𝐴 ) |
31 |
20 30
|
sseldd |
⊢ ( ( 𝜑 ∧ ¬ 𝐴 = ∅ ) → sup ( 𝐴 , ℝ , < ) ∈ 𝑍 ) |
32 |
1
|
ad2antrr |
⊢ ( ( ( 𝜑 ∧ ¬ 𝐴 = ∅ ) ∧ 𝑗 ∈ 𝐴 ) → 𝑀 ∈ ℤ ) |
33 |
22 31
|
sseldi |
⊢ ( ( 𝜑 ∧ ¬ 𝐴 = ∅ ) → sup ( 𝐴 , ℝ , < ) ∈ ℤ ) |
34 |
33
|
adantr |
⊢ ( ( ( 𝜑 ∧ ¬ 𝐴 = ∅ ) ∧ 𝑗 ∈ 𝐴 ) → sup ( 𝐴 , ℝ , < ) ∈ ℤ ) |
35 |
25
|
sselda |
⊢ ( ( ( 𝜑 ∧ ¬ 𝐴 = ∅ ) ∧ 𝑗 ∈ 𝐴 ) → 𝑗 ∈ ℤ ) |
36 |
3
|
sselda |
⊢ ( ( 𝜑 ∧ 𝑗 ∈ 𝐴 ) → 𝑗 ∈ 𝑍 ) |
37 |
2
|
a1i |
⊢ ( ( 𝜑 ∧ 𝑗 ∈ 𝐴 ) → 𝑍 = ( ℤ≥ ‘ 𝑀 ) ) |
38 |
36 37
|
eleqtrd |
⊢ ( ( 𝜑 ∧ 𝑗 ∈ 𝐴 ) → 𝑗 ∈ ( ℤ≥ ‘ 𝑀 ) ) |
39 |
|
eluzle |
⊢ ( 𝑗 ∈ ( ℤ≥ ‘ 𝑀 ) → 𝑀 ≤ 𝑗 ) |
40 |
38 39
|
syl |
⊢ ( ( 𝜑 ∧ 𝑗 ∈ 𝐴 ) → 𝑀 ≤ 𝑗 ) |
41 |
40
|
adantlr |
⊢ ( ( ( 𝜑 ∧ ¬ 𝐴 = ∅ ) ∧ 𝑗 ∈ 𝐴 ) → 𝑀 ≤ 𝑗 ) |
42 |
|
zssre |
⊢ ℤ ⊆ ℝ |
43 |
24 42
|
sstrdi |
⊢ ( 𝜑 → 𝐴 ⊆ ℝ ) |
44 |
43
|
ad2antrr |
⊢ ( ( ( 𝜑 ∧ ¬ 𝐴 = ∅ ) ∧ 𝑗 ∈ 𝐴 ) → 𝐴 ⊆ ℝ ) |
45 |
27
|
adantr |
⊢ ( ( ( 𝜑 ∧ ¬ 𝐴 = ∅ ) ∧ 𝑗 ∈ 𝐴 ) → 𝐴 ≠ ∅ ) |
46 |
|
fimaxre2 |
⊢ ( ( 𝐴 ⊆ ℝ ∧ 𝐴 ∈ Fin ) → ∃ 𝑥 ∈ ℝ ∀ 𝑦 ∈ 𝐴 𝑦 ≤ 𝑥 ) |
47 |
43 4 46
|
syl2anc |
⊢ ( 𝜑 → ∃ 𝑥 ∈ ℝ ∀ 𝑦 ∈ 𝐴 𝑦 ≤ 𝑥 ) |
48 |
47
|
ad2antrr |
⊢ ( ( ( 𝜑 ∧ ¬ 𝐴 = ∅ ) ∧ 𝑗 ∈ 𝐴 ) → ∃ 𝑥 ∈ ℝ ∀ 𝑦 ∈ 𝐴 𝑦 ≤ 𝑥 ) |
49 |
|
simpr |
⊢ ( ( ( 𝜑 ∧ ¬ 𝐴 = ∅ ) ∧ 𝑗 ∈ 𝐴 ) → 𝑗 ∈ 𝐴 ) |
50 |
|
suprub |
⊢ ( ( ( 𝐴 ⊆ ℝ ∧ 𝐴 ≠ ∅ ∧ ∃ 𝑥 ∈ ℝ ∀ 𝑦 ∈ 𝐴 𝑦 ≤ 𝑥 ) ∧ 𝑗 ∈ 𝐴 ) → 𝑗 ≤ sup ( 𝐴 , ℝ , < ) ) |
51 |
44 45 48 49 50
|
syl31anc |
⊢ ( ( ( 𝜑 ∧ ¬ 𝐴 = ∅ ) ∧ 𝑗 ∈ 𝐴 ) → 𝑗 ≤ sup ( 𝐴 , ℝ , < ) ) |
52 |
32 34 35 41 51
|
elfzd |
⊢ ( ( ( 𝜑 ∧ ¬ 𝐴 = ∅ ) ∧ 𝑗 ∈ 𝐴 ) → 𝑗 ∈ ( 𝑀 ... sup ( 𝐴 , ℝ , < ) ) ) |
53 |
52
|
ralrimiva |
⊢ ( ( 𝜑 ∧ ¬ 𝐴 = ∅ ) → ∀ 𝑗 ∈ 𝐴 𝑗 ∈ ( 𝑀 ... sup ( 𝐴 , ℝ , < ) ) ) |
54 |
|
dfss3 |
⊢ ( 𝐴 ⊆ ( 𝑀 ... sup ( 𝐴 , ℝ , < ) ) ↔ ∀ 𝑗 ∈ 𝐴 𝑗 ∈ ( 𝑀 ... sup ( 𝐴 , ℝ , < ) ) ) |
55 |
53 54
|
sylibr |
⊢ ( ( 𝜑 ∧ ¬ 𝐴 = ∅ ) → 𝐴 ⊆ ( 𝑀 ... sup ( 𝐴 , ℝ , < ) ) ) |
56 |
|
oveq2 |
⊢ ( 𝑘 = sup ( 𝐴 , ℝ , < ) → ( 𝑀 ... 𝑘 ) = ( 𝑀 ... sup ( 𝐴 , ℝ , < ) ) ) |
57 |
56
|
sseq2d |
⊢ ( 𝑘 = sup ( 𝐴 , ℝ , < ) → ( 𝐴 ⊆ ( 𝑀 ... 𝑘 ) ↔ 𝐴 ⊆ ( 𝑀 ... sup ( 𝐴 , ℝ , < ) ) ) ) |
58 |
57
|
rspcev |
⊢ ( ( sup ( 𝐴 , ℝ , < ) ∈ 𝑍 ∧ 𝐴 ⊆ ( 𝑀 ... sup ( 𝐴 , ℝ , < ) ) ) → ∃ 𝑘 ∈ 𝑍 𝐴 ⊆ ( 𝑀 ... 𝑘 ) ) |
59 |
31 55 58
|
syl2anc |
⊢ ( ( 𝜑 ∧ ¬ 𝐴 = ∅ ) → ∃ 𝑘 ∈ 𝑍 𝐴 ⊆ ( 𝑀 ... 𝑘 ) ) |
60 |
19 59
|
pm2.61dan |
⊢ ( 𝜑 → ∃ 𝑘 ∈ 𝑍 𝐴 ⊆ ( 𝑀 ... 𝑘 ) ) |