Step |
Hyp |
Ref |
Expression |
1 |
|
supcvg.1 |
⊢ 𝑋 ∈ V |
2 |
|
supcvg.2 |
⊢ 𝑆 = sup ( 𝐴 , ℝ , < ) |
3 |
|
supcvg.3 |
⊢ 𝑅 = ( 𝑛 ∈ ℕ ↦ ( 𝑆 − ( 1 / 𝑛 ) ) ) |
4 |
|
supcvg.4 |
⊢ ( 𝜑 → 𝑋 ≠ ∅ ) |
5 |
|
supcvg.5 |
⊢ ( 𝜑 → 𝐹 : 𝑋 –onto→ 𝐴 ) |
6 |
|
supcvg.6 |
⊢ ( 𝜑 → 𝐴 ⊆ ℝ ) |
7 |
|
supcvg.7 |
⊢ ( 𝜑 → ∃ 𝑥 ∈ ℝ ∀ 𝑦 ∈ 𝐴 𝑦 ≤ 𝑥 ) |
8 |
|
oveq2 |
⊢ ( 𝑛 = 𝑘 → ( 1 / 𝑛 ) = ( 1 / 𝑘 ) ) |
9 |
8
|
oveq2d |
⊢ ( 𝑛 = 𝑘 → ( 𝑆 − ( 1 / 𝑛 ) ) = ( 𝑆 − ( 1 / 𝑘 ) ) ) |
10 |
|
ovex |
⊢ ( 𝑆 − ( 1 / 𝑘 ) ) ∈ V |
11 |
9 3 10
|
fvmpt |
⊢ ( 𝑘 ∈ ℕ → ( 𝑅 ‘ 𝑘 ) = ( 𝑆 − ( 1 / 𝑘 ) ) ) |
12 |
11
|
adantl |
⊢ ( ( 𝜑 ∧ 𝑘 ∈ ℕ ) → ( 𝑅 ‘ 𝑘 ) = ( 𝑆 − ( 1 / 𝑘 ) ) ) |
13 |
|
fof |
⊢ ( 𝐹 : 𝑋 –onto→ 𝐴 → 𝐹 : 𝑋 ⟶ 𝐴 ) |
14 |
5 13
|
syl |
⊢ ( 𝜑 → 𝐹 : 𝑋 ⟶ 𝐴 ) |
15 |
|
feq3 |
⊢ ( 𝐴 = ∅ → ( 𝐹 : 𝑋 ⟶ 𝐴 ↔ 𝐹 : 𝑋 ⟶ ∅ ) ) |
16 |
14 15
|
syl5ibcom |
⊢ ( 𝜑 → ( 𝐴 = ∅ → 𝐹 : 𝑋 ⟶ ∅ ) ) |
17 |
|
f00 |
⊢ ( 𝐹 : 𝑋 ⟶ ∅ ↔ ( 𝐹 = ∅ ∧ 𝑋 = ∅ ) ) |
18 |
17
|
simprbi |
⊢ ( 𝐹 : 𝑋 ⟶ ∅ → 𝑋 = ∅ ) |
19 |
16 18
|
syl6 |
⊢ ( 𝜑 → ( 𝐴 = ∅ → 𝑋 = ∅ ) ) |
20 |
19
|
necon3d |
⊢ ( 𝜑 → ( 𝑋 ≠ ∅ → 𝐴 ≠ ∅ ) ) |
21 |
4 20
|
mpd |
⊢ ( 𝜑 → 𝐴 ≠ ∅ ) |
22 |
6 21 7
|
suprcld |
⊢ ( 𝜑 → sup ( 𝐴 , ℝ , < ) ∈ ℝ ) |
23 |
2 22
|
eqeltrid |
⊢ ( 𝜑 → 𝑆 ∈ ℝ ) |
24 |
|
nnrp |
⊢ ( 𝑘 ∈ ℕ → 𝑘 ∈ ℝ+ ) |
25 |
24
|
rpreccld |
⊢ ( 𝑘 ∈ ℕ → ( 1 / 𝑘 ) ∈ ℝ+ ) |
26 |
|
ltsubrp |
⊢ ( ( 𝑆 ∈ ℝ ∧ ( 1 / 𝑘 ) ∈ ℝ+ ) → ( 𝑆 − ( 1 / 𝑘 ) ) < 𝑆 ) |
27 |
23 25 26
|
syl2an |
⊢ ( ( 𝜑 ∧ 𝑘 ∈ ℕ ) → ( 𝑆 − ( 1 / 𝑘 ) ) < 𝑆 ) |
28 |
12 27
|
eqbrtrd |
⊢ ( ( 𝜑 ∧ 𝑘 ∈ ℕ ) → ( 𝑅 ‘ 𝑘 ) < 𝑆 ) |
29 |
28 2
|
breqtrdi |
⊢ ( ( 𝜑 ∧ 𝑘 ∈ ℕ ) → ( 𝑅 ‘ 𝑘 ) < sup ( 𝐴 , ℝ , < ) ) |
30 |
6 21 7
|
3jca |
⊢ ( 𝜑 → ( 𝐴 ⊆ ℝ ∧ 𝐴 ≠ ∅ ∧ ∃ 𝑥 ∈ ℝ ∀ 𝑦 ∈ 𝐴 𝑦 ≤ 𝑥 ) ) |
31 |
|
nnrecre |
⊢ ( 𝑛 ∈ ℕ → ( 1 / 𝑛 ) ∈ ℝ ) |
32 |
|
resubcl |
⊢ ( ( 𝑆 ∈ ℝ ∧ ( 1 / 𝑛 ) ∈ ℝ ) → ( 𝑆 − ( 1 / 𝑛 ) ) ∈ ℝ ) |
33 |
23 31 32
|
syl2an |
⊢ ( ( 𝜑 ∧ 𝑛 ∈ ℕ ) → ( 𝑆 − ( 1 / 𝑛 ) ) ∈ ℝ ) |
34 |
33 3
|
fmptd |
⊢ ( 𝜑 → 𝑅 : ℕ ⟶ ℝ ) |
35 |
34
|
ffvelrnda |
⊢ ( ( 𝜑 ∧ 𝑘 ∈ ℕ ) → ( 𝑅 ‘ 𝑘 ) ∈ ℝ ) |
36 |
|
suprlub |
⊢ ( ( ( 𝐴 ⊆ ℝ ∧ 𝐴 ≠ ∅ ∧ ∃ 𝑥 ∈ ℝ ∀ 𝑦 ∈ 𝐴 𝑦 ≤ 𝑥 ) ∧ ( 𝑅 ‘ 𝑘 ) ∈ ℝ ) → ( ( 𝑅 ‘ 𝑘 ) < sup ( 𝐴 , ℝ , < ) ↔ ∃ 𝑧 ∈ 𝐴 ( 𝑅 ‘ 𝑘 ) < 𝑧 ) ) |
37 |
30 35 36
|
syl2an2r |
⊢ ( ( 𝜑 ∧ 𝑘 ∈ ℕ ) → ( ( 𝑅 ‘ 𝑘 ) < sup ( 𝐴 , ℝ , < ) ↔ ∃ 𝑧 ∈ 𝐴 ( 𝑅 ‘ 𝑘 ) < 𝑧 ) ) |
38 |
29 37
|
mpbid |
⊢ ( ( 𝜑 ∧ 𝑘 ∈ ℕ ) → ∃ 𝑧 ∈ 𝐴 ( 𝑅 ‘ 𝑘 ) < 𝑧 ) |
39 |
6
|
adantr |
⊢ ( ( 𝜑 ∧ 𝑘 ∈ ℕ ) → 𝐴 ⊆ ℝ ) |
40 |
39
|
sselda |
⊢ ( ( ( 𝜑 ∧ 𝑘 ∈ ℕ ) ∧ 𝑧 ∈ 𝐴 ) → 𝑧 ∈ ℝ ) |
41 |
|
ltle |
⊢ ( ( ( 𝑅 ‘ 𝑘 ) ∈ ℝ ∧ 𝑧 ∈ ℝ ) → ( ( 𝑅 ‘ 𝑘 ) < 𝑧 → ( 𝑅 ‘ 𝑘 ) ≤ 𝑧 ) ) |
42 |
35 40 41
|
syl2an2r |
⊢ ( ( ( 𝜑 ∧ 𝑘 ∈ ℕ ) ∧ 𝑧 ∈ 𝐴 ) → ( ( 𝑅 ‘ 𝑘 ) < 𝑧 → ( 𝑅 ‘ 𝑘 ) ≤ 𝑧 ) ) |
43 |
42
|
reximdva |
⊢ ( ( 𝜑 ∧ 𝑘 ∈ ℕ ) → ( ∃ 𝑧 ∈ 𝐴 ( 𝑅 ‘ 𝑘 ) < 𝑧 → ∃ 𝑧 ∈ 𝐴 ( 𝑅 ‘ 𝑘 ) ≤ 𝑧 ) ) |
44 |
38 43
|
mpd |
⊢ ( ( 𝜑 ∧ 𝑘 ∈ ℕ ) → ∃ 𝑧 ∈ 𝐴 ( 𝑅 ‘ 𝑘 ) ≤ 𝑧 ) |
45 |
|
forn |
⊢ ( 𝐹 : 𝑋 –onto→ 𝐴 → ran 𝐹 = 𝐴 ) |
46 |
5 45
|
syl |
⊢ ( 𝜑 → ran 𝐹 = 𝐴 ) |
47 |
46
|
rexeqdv |
⊢ ( 𝜑 → ( ∃ 𝑧 ∈ ran 𝐹 ( 𝑅 ‘ 𝑘 ) ≤ 𝑧 ↔ ∃ 𝑧 ∈ 𝐴 ( 𝑅 ‘ 𝑘 ) ≤ 𝑧 ) ) |
48 |
|
ffn |
⊢ ( 𝐹 : 𝑋 ⟶ 𝐴 → 𝐹 Fn 𝑋 ) |
49 |
|
breq2 |
⊢ ( 𝑧 = ( 𝐹 ‘ 𝑥 ) → ( ( 𝑅 ‘ 𝑘 ) ≤ 𝑧 ↔ ( 𝑅 ‘ 𝑘 ) ≤ ( 𝐹 ‘ 𝑥 ) ) ) |
50 |
49
|
rexrn |
⊢ ( 𝐹 Fn 𝑋 → ( ∃ 𝑧 ∈ ran 𝐹 ( 𝑅 ‘ 𝑘 ) ≤ 𝑧 ↔ ∃ 𝑥 ∈ 𝑋 ( 𝑅 ‘ 𝑘 ) ≤ ( 𝐹 ‘ 𝑥 ) ) ) |
51 |
14 48 50
|
3syl |
⊢ ( 𝜑 → ( ∃ 𝑧 ∈ ran 𝐹 ( 𝑅 ‘ 𝑘 ) ≤ 𝑧 ↔ ∃ 𝑥 ∈ 𝑋 ( 𝑅 ‘ 𝑘 ) ≤ ( 𝐹 ‘ 𝑥 ) ) ) |
52 |
47 51
|
bitr3d |
⊢ ( 𝜑 → ( ∃ 𝑧 ∈ 𝐴 ( 𝑅 ‘ 𝑘 ) ≤ 𝑧 ↔ ∃ 𝑥 ∈ 𝑋 ( 𝑅 ‘ 𝑘 ) ≤ ( 𝐹 ‘ 𝑥 ) ) ) |
53 |
52
|
adantr |
⊢ ( ( 𝜑 ∧ 𝑘 ∈ ℕ ) → ( ∃ 𝑧 ∈ 𝐴 ( 𝑅 ‘ 𝑘 ) ≤ 𝑧 ↔ ∃ 𝑥 ∈ 𝑋 ( 𝑅 ‘ 𝑘 ) ≤ ( 𝐹 ‘ 𝑥 ) ) ) |
54 |
44 53
|
mpbid |
⊢ ( ( 𝜑 ∧ 𝑘 ∈ ℕ ) → ∃ 𝑥 ∈ 𝑋 ( 𝑅 ‘ 𝑘 ) ≤ ( 𝐹 ‘ 𝑥 ) ) |
55 |
54
|
ralrimiva |
⊢ ( 𝜑 → ∀ 𝑘 ∈ ℕ ∃ 𝑥 ∈ 𝑋 ( 𝑅 ‘ 𝑘 ) ≤ ( 𝐹 ‘ 𝑥 ) ) |
56 |
|
nnenom |
⊢ ℕ ≈ ω |
57 |
|
fveq2 |
⊢ ( 𝑥 = ( 𝑓 ‘ 𝑘 ) → ( 𝐹 ‘ 𝑥 ) = ( 𝐹 ‘ ( 𝑓 ‘ 𝑘 ) ) ) |
58 |
57
|
breq2d |
⊢ ( 𝑥 = ( 𝑓 ‘ 𝑘 ) → ( ( 𝑅 ‘ 𝑘 ) ≤ ( 𝐹 ‘ 𝑥 ) ↔ ( 𝑅 ‘ 𝑘 ) ≤ ( 𝐹 ‘ ( 𝑓 ‘ 𝑘 ) ) ) ) |
59 |
1 56 58
|
axcc4 |
⊢ ( ∀ 𝑘 ∈ ℕ ∃ 𝑥 ∈ 𝑋 ( 𝑅 ‘ 𝑘 ) ≤ ( 𝐹 ‘ 𝑥 ) → ∃ 𝑓 ( 𝑓 : ℕ ⟶ 𝑋 ∧ ∀ 𝑘 ∈ ℕ ( 𝑅 ‘ 𝑘 ) ≤ ( 𝐹 ‘ ( 𝑓 ‘ 𝑘 ) ) ) ) |
60 |
55 59
|
syl |
⊢ ( 𝜑 → ∃ 𝑓 ( 𝑓 : ℕ ⟶ 𝑋 ∧ ∀ 𝑘 ∈ ℕ ( 𝑅 ‘ 𝑘 ) ≤ ( 𝐹 ‘ ( 𝑓 ‘ 𝑘 ) ) ) ) |
61 |
|
nnuz |
⊢ ℕ = ( ℤ≥ ‘ 1 ) |
62 |
|
1zzd |
⊢ ( ( ( 𝜑 ∧ 𝑓 : ℕ ⟶ 𝑋 ) ∧ ∀ 𝑘 ∈ ℕ ( 𝑅 ‘ 𝑘 ) ≤ ( 𝐹 ‘ ( 𝑓 ‘ 𝑘 ) ) ) → 1 ∈ ℤ ) |
63 |
|
1zzd |
⊢ ( 𝜑 → 1 ∈ ℤ ) |
64 |
23
|
recnd |
⊢ ( 𝜑 → 𝑆 ∈ ℂ ) |
65 |
|
1z |
⊢ 1 ∈ ℤ |
66 |
61
|
eqimss2i |
⊢ ( ℤ≥ ‘ 1 ) ⊆ ℕ |
67 |
|
nnex |
⊢ ℕ ∈ V |
68 |
66 67
|
climconst2 |
⊢ ( ( 𝑆 ∈ ℂ ∧ 1 ∈ ℤ ) → ( ℕ × { 𝑆 } ) ⇝ 𝑆 ) |
69 |
64 65 68
|
sylancl |
⊢ ( 𝜑 → ( ℕ × { 𝑆 } ) ⇝ 𝑆 ) |
70 |
67
|
mptex |
⊢ ( 𝑛 ∈ ℕ ↦ ( 𝑆 − ( 1 / 𝑛 ) ) ) ∈ V |
71 |
3 70
|
eqeltri |
⊢ 𝑅 ∈ V |
72 |
71
|
a1i |
⊢ ( 𝜑 → 𝑅 ∈ V ) |
73 |
|
ax-1cn |
⊢ 1 ∈ ℂ |
74 |
|
divcnv |
⊢ ( 1 ∈ ℂ → ( 𝑛 ∈ ℕ ↦ ( 1 / 𝑛 ) ) ⇝ 0 ) |
75 |
73 74
|
mp1i |
⊢ ( 𝜑 → ( 𝑛 ∈ ℕ ↦ ( 1 / 𝑛 ) ) ⇝ 0 ) |
76 |
|
fvconst2g |
⊢ ( ( 𝑆 ∈ ℝ ∧ 𝑘 ∈ ℕ ) → ( ( ℕ × { 𝑆 } ) ‘ 𝑘 ) = 𝑆 ) |
77 |
23 76
|
sylan |
⊢ ( ( 𝜑 ∧ 𝑘 ∈ ℕ ) → ( ( ℕ × { 𝑆 } ) ‘ 𝑘 ) = 𝑆 ) |
78 |
64
|
adantr |
⊢ ( ( 𝜑 ∧ 𝑘 ∈ ℕ ) → 𝑆 ∈ ℂ ) |
79 |
77 78
|
eqeltrd |
⊢ ( ( 𝜑 ∧ 𝑘 ∈ ℕ ) → ( ( ℕ × { 𝑆 } ) ‘ 𝑘 ) ∈ ℂ ) |
80 |
|
eqid |
⊢ ( 𝑛 ∈ ℕ ↦ ( 1 / 𝑛 ) ) = ( 𝑛 ∈ ℕ ↦ ( 1 / 𝑛 ) ) |
81 |
|
ovex |
⊢ ( 1 / 𝑘 ) ∈ V |
82 |
8 80 81
|
fvmpt |
⊢ ( 𝑘 ∈ ℕ → ( ( 𝑛 ∈ ℕ ↦ ( 1 / 𝑛 ) ) ‘ 𝑘 ) = ( 1 / 𝑘 ) ) |
83 |
82
|
adantl |
⊢ ( ( 𝜑 ∧ 𝑘 ∈ ℕ ) → ( ( 𝑛 ∈ ℕ ↦ ( 1 / 𝑛 ) ) ‘ 𝑘 ) = ( 1 / 𝑘 ) ) |
84 |
|
nnrecre |
⊢ ( 𝑘 ∈ ℕ → ( 1 / 𝑘 ) ∈ ℝ ) |
85 |
84
|
recnd |
⊢ ( 𝑘 ∈ ℕ → ( 1 / 𝑘 ) ∈ ℂ ) |
86 |
85
|
adantl |
⊢ ( ( 𝜑 ∧ 𝑘 ∈ ℕ ) → ( 1 / 𝑘 ) ∈ ℂ ) |
87 |
83 86
|
eqeltrd |
⊢ ( ( 𝜑 ∧ 𝑘 ∈ ℕ ) → ( ( 𝑛 ∈ ℕ ↦ ( 1 / 𝑛 ) ) ‘ 𝑘 ) ∈ ℂ ) |
88 |
77 83
|
oveq12d |
⊢ ( ( 𝜑 ∧ 𝑘 ∈ ℕ ) → ( ( ( ℕ × { 𝑆 } ) ‘ 𝑘 ) − ( ( 𝑛 ∈ ℕ ↦ ( 1 / 𝑛 ) ) ‘ 𝑘 ) ) = ( 𝑆 − ( 1 / 𝑘 ) ) ) |
89 |
12 88
|
eqtr4d |
⊢ ( ( 𝜑 ∧ 𝑘 ∈ ℕ ) → ( 𝑅 ‘ 𝑘 ) = ( ( ( ℕ × { 𝑆 } ) ‘ 𝑘 ) − ( ( 𝑛 ∈ ℕ ↦ ( 1 / 𝑛 ) ) ‘ 𝑘 ) ) ) |
90 |
61 63 69 72 75 79 87 89
|
climsub |
⊢ ( 𝜑 → 𝑅 ⇝ ( 𝑆 − 0 ) ) |
91 |
64
|
subid1d |
⊢ ( 𝜑 → ( 𝑆 − 0 ) = 𝑆 ) |
92 |
90 91
|
breqtrd |
⊢ ( 𝜑 → 𝑅 ⇝ 𝑆 ) |
93 |
92
|
ad2antrr |
⊢ ( ( ( 𝜑 ∧ 𝑓 : ℕ ⟶ 𝑋 ) ∧ ∀ 𝑘 ∈ ℕ ( 𝑅 ‘ 𝑘 ) ≤ ( 𝐹 ‘ ( 𝑓 ‘ 𝑘 ) ) ) → 𝑅 ⇝ 𝑆 ) |
94 |
14
|
ad2antrr |
⊢ ( ( ( 𝜑 ∧ 𝑓 : ℕ ⟶ 𝑋 ) ∧ ∀ 𝑘 ∈ ℕ ( 𝑅 ‘ 𝑘 ) ≤ ( 𝐹 ‘ ( 𝑓 ‘ 𝑘 ) ) ) → 𝐹 : 𝑋 ⟶ 𝐴 ) |
95 |
|
fex |
⊢ ( ( 𝐹 : 𝑋 ⟶ 𝐴 ∧ 𝑋 ∈ V ) → 𝐹 ∈ V ) |
96 |
94 1 95
|
sylancl |
⊢ ( ( ( 𝜑 ∧ 𝑓 : ℕ ⟶ 𝑋 ) ∧ ∀ 𝑘 ∈ ℕ ( 𝑅 ‘ 𝑘 ) ≤ ( 𝐹 ‘ ( 𝑓 ‘ 𝑘 ) ) ) → 𝐹 ∈ V ) |
97 |
|
vex |
⊢ 𝑓 ∈ V |
98 |
|
coexg |
⊢ ( ( 𝐹 ∈ V ∧ 𝑓 ∈ V ) → ( 𝐹 ∘ 𝑓 ) ∈ V ) |
99 |
96 97 98
|
sylancl |
⊢ ( ( ( 𝜑 ∧ 𝑓 : ℕ ⟶ 𝑋 ) ∧ ∀ 𝑘 ∈ ℕ ( 𝑅 ‘ 𝑘 ) ≤ ( 𝐹 ‘ ( 𝑓 ‘ 𝑘 ) ) ) → ( 𝐹 ∘ 𝑓 ) ∈ V ) |
100 |
34
|
ad2antrr |
⊢ ( ( ( 𝜑 ∧ 𝑓 : ℕ ⟶ 𝑋 ) ∧ ∀ 𝑘 ∈ ℕ ( 𝑅 ‘ 𝑘 ) ≤ ( 𝐹 ‘ ( 𝑓 ‘ 𝑘 ) ) ) → 𝑅 : ℕ ⟶ ℝ ) |
101 |
100
|
ffvelrnda |
⊢ ( ( ( ( 𝜑 ∧ 𝑓 : ℕ ⟶ 𝑋 ) ∧ ∀ 𝑘 ∈ ℕ ( 𝑅 ‘ 𝑘 ) ≤ ( 𝐹 ‘ ( 𝑓 ‘ 𝑘 ) ) ) ∧ 𝑚 ∈ ℕ ) → ( 𝑅 ‘ 𝑚 ) ∈ ℝ ) |
102 |
14 6
|
fssd |
⊢ ( 𝜑 → 𝐹 : 𝑋 ⟶ ℝ ) |
103 |
|
fco |
⊢ ( ( 𝐹 : 𝑋 ⟶ ℝ ∧ 𝑓 : ℕ ⟶ 𝑋 ) → ( 𝐹 ∘ 𝑓 ) : ℕ ⟶ ℝ ) |
104 |
102 103
|
sylan |
⊢ ( ( 𝜑 ∧ 𝑓 : ℕ ⟶ 𝑋 ) → ( 𝐹 ∘ 𝑓 ) : ℕ ⟶ ℝ ) |
105 |
104
|
adantr |
⊢ ( ( ( 𝜑 ∧ 𝑓 : ℕ ⟶ 𝑋 ) ∧ ∀ 𝑘 ∈ ℕ ( 𝑅 ‘ 𝑘 ) ≤ ( 𝐹 ‘ ( 𝑓 ‘ 𝑘 ) ) ) → ( 𝐹 ∘ 𝑓 ) : ℕ ⟶ ℝ ) |
106 |
105
|
ffvelrnda |
⊢ ( ( ( ( 𝜑 ∧ 𝑓 : ℕ ⟶ 𝑋 ) ∧ ∀ 𝑘 ∈ ℕ ( 𝑅 ‘ 𝑘 ) ≤ ( 𝐹 ‘ ( 𝑓 ‘ 𝑘 ) ) ) ∧ 𝑚 ∈ ℕ ) → ( ( 𝐹 ∘ 𝑓 ) ‘ 𝑚 ) ∈ ℝ ) |
107 |
|
fveq2 |
⊢ ( 𝑘 = 𝑚 → ( 𝑅 ‘ 𝑘 ) = ( 𝑅 ‘ 𝑚 ) ) |
108 |
|
2fveq3 |
⊢ ( 𝑘 = 𝑚 → ( 𝐹 ‘ ( 𝑓 ‘ 𝑘 ) ) = ( 𝐹 ‘ ( 𝑓 ‘ 𝑚 ) ) ) |
109 |
107 108
|
breq12d |
⊢ ( 𝑘 = 𝑚 → ( ( 𝑅 ‘ 𝑘 ) ≤ ( 𝐹 ‘ ( 𝑓 ‘ 𝑘 ) ) ↔ ( 𝑅 ‘ 𝑚 ) ≤ ( 𝐹 ‘ ( 𝑓 ‘ 𝑚 ) ) ) ) |
110 |
109
|
rspccva |
⊢ ( ( ∀ 𝑘 ∈ ℕ ( 𝑅 ‘ 𝑘 ) ≤ ( 𝐹 ‘ ( 𝑓 ‘ 𝑘 ) ) ∧ 𝑚 ∈ ℕ ) → ( 𝑅 ‘ 𝑚 ) ≤ ( 𝐹 ‘ ( 𝑓 ‘ 𝑚 ) ) ) |
111 |
110
|
adantll |
⊢ ( ( ( ( 𝜑 ∧ 𝑓 : ℕ ⟶ 𝑋 ) ∧ ∀ 𝑘 ∈ ℕ ( 𝑅 ‘ 𝑘 ) ≤ ( 𝐹 ‘ ( 𝑓 ‘ 𝑘 ) ) ) ∧ 𝑚 ∈ ℕ ) → ( 𝑅 ‘ 𝑚 ) ≤ ( 𝐹 ‘ ( 𝑓 ‘ 𝑚 ) ) ) |
112 |
|
simplr |
⊢ ( ( ( 𝜑 ∧ 𝑓 : ℕ ⟶ 𝑋 ) ∧ ∀ 𝑘 ∈ ℕ ( 𝑅 ‘ 𝑘 ) ≤ ( 𝐹 ‘ ( 𝑓 ‘ 𝑘 ) ) ) → 𝑓 : ℕ ⟶ 𝑋 ) |
113 |
|
fvco3 |
⊢ ( ( 𝑓 : ℕ ⟶ 𝑋 ∧ 𝑚 ∈ ℕ ) → ( ( 𝐹 ∘ 𝑓 ) ‘ 𝑚 ) = ( 𝐹 ‘ ( 𝑓 ‘ 𝑚 ) ) ) |
114 |
112 113
|
sylan |
⊢ ( ( ( ( 𝜑 ∧ 𝑓 : ℕ ⟶ 𝑋 ) ∧ ∀ 𝑘 ∈ ℕ ( 𝑅 ‘ 𝑘 ) ≤ ( 𝐹 ‘ ( 𝑓 ‘ 𝑘 ) ) ) ∧ 𝑚 ∈ ℕ ) → ( ( 𝐹 ∘ 𝑓 ) ‘ 𝑚 ) = ( 𝐹 ‘ ( 𝑓 ‘ 𝑚 ) ) ) |
115 |
111 114
|
breqtrrd |
⊢ ( ( ( ( 𝜑 ∧ 𝑓 : ℕ ⟶ 𝑋 ) ∧ ∀ 𝑘 ∈ ℕ ( 𝑅 ‘ 𝑘 ) ≤ ( 𝐹 ‘ ( 𝑓 ‘ 𝑘 ) ) ) ∧ 𝑚 ∈ ℕ ) → ( 𝑅 ‘ 𝑚 ) ≤ ( ( 𝐹 ∘ 𝑓 ) ‘ 𝑚 ) ) |
116 |
30
|
ad3antrrr |
⊢ ( ( ( ( 𝜑 ∧ 𝑓 : ℕ ⟶ 𝑋 ) ∧ ∀ 𝑘 ∈ ℕ ( 𝑅 ‘ 𝑘 ) ≤ ( 𝐹 ‘ ( 𝑓 ‘ 𝑘 ) ) ) ∧ 𝑚 ∈ ℕ ) → ( 𝐴 ⊆ ℝ ∧ 𝐴 ≠ ∅ ∧ ∃ 𝑥 ∈ ℝ ∀ 𝑦 ∈ 𝐴 𝑦 ≤ 𝑥 ) ) |
117 |
112
|
ffvelrnda |
⊢ ( ( ( ( 𝜑 ∧ 𝑓 : ℕ ⟶ 𝑋 ) ∧ ∀ 𝑘 ∈ ℕ ( 𝑅 ‘ 𝑘 ) ≤ ( 𝐹 ‘ ( 𝑓 ‘ 𝑘 ) ) ) ∧ 𝑚 ∈ ℕ ) → ( 𝑓 ‘ 𝑚 ) ∈ 𝑋 ) |
118 |
94
|
ffvelrnda |
⊢ ( ( ( ( 𝜑 ∧ 𝑓 : ℕ ⟶ 𝑋 ) ∧ ∀ 𝑘 ∈ ℕ ( 𝑅 ‘ 𝑘 ) ≤ ( 𝐹 ‘ ( 𝑓 ‘ 𝑘 ) ) ) ∧ ( 𝑓 ‘ 𝑚 ) ∈ 𝑋 ) → ( 𝐹 ‘ ( 𝑓 ‘ 𝑚 ) ) ∈ 𝐴 ) |
119 |
117 118
|
syldan |
⊢ ( ( ( ( 𝜑 ∧ 𝑓 : ℕ ⟶ 𝑋 ) ∧ ∀ 𝑘 ∈ ℕ ( 𝑅 ‘ 𝑘 ) ≤ ( 𝐹 ‘ ( 𝑓 ‘ 𝑘 ) ) ) ∧ 𝑚 ∈ ℕ ) → ( 𝐹 ‘ ( 𝑓 ‘ 𝑚 ) ) ∈ 𝐴 ) |
120 |
|
suprub |
⊢ ( ( ( 𝐴 ⊆ ℝ ∧ 𝐴 ≠ ∅ ∧ ∃ 𝑥 ∈ ℝ ∀ 𝑦 ∈ 𝐴 𝑦 ≤ 𝑥 ) ∧ ( 𝐹 ‘ ( 𝑓 ‘ 𝑚 ) ) ∈ 𝐴 ) → ( 𝐹 ‘ ( 𝑓 ‘ 𝑚 ) ) ≤ sup ( 𝐴 , ℝ , < ) ) |
121 |
116 119 120
|
syl2anc |
⊢ ( ( ( ( 𝜑 ∧ 𝑓 : ℕ ⟶ 𝑋 ) ∧ ∀ 𝑘 ∈ ℕ ( 𝑅 ‘ 𝑘 ) ≤ ( 𝐹 ‘ ( 𝑓 ‘ 𝑘 ) ) ) ∧ 𝑚 ∈ ℕ ) → ( 𝐹 ‘ ( 𝑓 ‘ 𝑚 ) ) ≤ sup ( 𝐴 , ℝ , < ) ) |
122 |
121 2
|
breqtrrdi |
⊢ ( ( ( ( 𝜑 ∧ 𝑓 : ℕ ⟶ 𝑋 ) ∧ ∀ 𝑘 ∈ ℕ ( 𝑅 ‘ 𝑘 ) ≤ ( 𝐹 ‘ ( 𝑓 ‘ 𝑘 ) ) ) ∧ 𝑚 ∈ ℕ ) → ( 𝐹 ‘ ( 𝑓 ‘ 𝑚 ) ) ≤ 𝑆 ) |
123 |
114 122
|
eqbrtrd |
⊢ ( ( ( ( 𝜑 ∧ 𝑓 : ℕ ⟶ 𝑋 ) ∧ ∀ 𝑘 ∈ ℕ ( 𝑅 ‘ 𝑘 ) ≤ ( 𝐹 ‘ ( 𝑓 ‘ 𝑘 ) ) ) ∧ 𝑚 ∈ ℕ ) → ( ( 𝐹 ∘ 𝑓 ) ‘ 𝑚 ) ≤ 𝑆 ) |
124 |
61 62 93 99 101 106 115 123
|
climsqz |
⊢ ( ( ( 𝜑 ∧ 𝑓 : ℕ ⟶ 𝑋 ) ∧ ∀ 𝑘 ∈ ℕ ( 𝑅 ‘ 𝑘 ) ≤ ( 𝐹 ‘ ( 𝑓 ‘ 𝑘 ) ) ) → ( 𝐹 ∘ 𝑓 ) ⇝ 𝑆 ) |
125 |
124
|
ex |
⊢ ( ( 𝜑 ∧ 𝑓 : ℕ ⟶ 𝑋 ) → ( ∀ 𝑘 ∈ ℕ ( 𝑅 ‘ 𝑘 ) ≤ ( 𝐹 ‘ ( 𝑓 ‘ 𝑘 ) ) → ( 𝐹 ∘ 𝑓 ) ⇝ 𝑆 ) ) |
126 |
125
|
imdistanda |
⊢ ( 𝜑 → ( ( 𝑓 : ℕ ⟶ 𝑋 ∧ ∀ 𝑘 ∈ ℕ ( 𝑅 ‘ 𝑘 ) ≤ ( 𝐹 ‘ ( 𝑓 ‘ 𝑘 ) ) ) → ( 𝑓 : ℕ ⟶ 𝑋 ∧ ( 𝐹 ∘ 𝑓 ) ⇝ 𝑆 ) ) ) |
127 |
126
|
eximdv |
⊢ ( 𝜑 → ( ∃ 𝑓 ( 𝑓 : ℕ ⟶ 𝑋 ∧ ∀ 𝑘 ∈ ℕ ( 𝑅 ‘ 𝑘 ) ≤ ( 𝐹 ‘ ( 𝑓 ‘ 𝑘 ) ) ) → ∃ 𝑓 ( 𝑓 : ℕ ⟶ 𝑋 ∧ ( 𝐹 ∘ 𝑓 ) ⇝ 𝑆 ) ) ) |
128 |
60 127
|
mpd |
⊢ ( 𝜑 → ∃ 𝑓 ( 𝑓 : ℕ ⟶ 𝑋 ∧ ( 𝐹 ∘ 𝑓 ) ⇝ 𝑆 ) ) |