Step |
Hyp |
Ref |
Expression |
1 |
|
suprnmpt.a |
⊢ ( 𝜑 → 𝐴 ≠ ∅ ) |
2 |
|
suprnmpt.b |
⊢ ( ( 𝜑 ∧ 𝑥 ∈ 𝐴 ) → 𝐵 ∈ ℝ ) |
3 |
|
suprnmpt.bnd |
⊢ ( 𝜑 → ∃ 𝑦 ∈ ℝ ∀ 𝑥 ∈ 𝐴 𝐵 ≤ 𝑦 ) |
4 |
|
suprnmpt.f |
⊢ 𝐹 = ( 𝑥 ∈ 𝐴 ↦ 𝐵 ) |
5 |
|
suprnmpt.c |
⊢ 𝐶 = sup ( ran 𝐹 , ℝ , < ) |
6 |
2
|
ralrimiva |
⊢ ( 𝜑 → ∀ 𝑥 ∈ 𝐴 𝐵 ∈ ℝ ) |
7 |
4
|
rnmptss |
⊢ ( ∀ 𝑥 ∈ 𝐴 𝐵 ∈ ℝ → ran 𝐹 ⊆ ℝ ) |
8 |
6 7
|
syl |
⊢ ( 𝜑 → ran 𝐹 ⊆ ℝ ) |
9 |
|
nfv |
⊢ Ⅎ 𝑥 𝜑 |
10 |
9 2 4 1
|
rnmptn0 |
⊢ ( 𝜑 → ran 𝐹 ≠ ∅ ) |
11 |
|
nfv |
⊢ Ⅎ 𝑦 𝜑 |
12 |
|
nfre1 |
⊢ Ⅎ 𝑦 ∃ 𝑦 ∈ ℝ ∀ 𝑧 ∈ ran 𝐹 𝑧 ≤ 𝑦 |
13 |
|
simp2 |
⊢ ( ( 𝜑 ∧ 𝑦 ∈ ℝ ∧ ∀ 𝑥 ∈ 𝐴 𝐵 ≤ 𝑦 ) → 𝑦 ∈ ℝ ) |
14 |
|
simpl1 |
⊢ ( ( ( 𝜑 ∧ 𝑦 ∈ ℝ ∧ ∀ 𝑥 ∈ 𝐴 𝐵 ≤ 𝑦 ) ∧ 𝑧 ∈ ran 𝐹 ) → 𝜑 ) |
15 |
|
simpl3 |
⊢ ( ( ( 𝜑 ∧ 𝑦 ∈ ℝ ∧ ∀ 𝑥 ∈ 𝐴 𝐵 ≤ 𝑦 ) ∧ 𝑧 ∈ ran 𝐹 ) → ∀ 𝑥 ∈ 𝐴 𝐵 ≤ 𝑦 ) |
16 |
|
vex |
⊢ 𝑧 ∈ V |
17 |
4
|
elrnmpt |
⊢ ( 𝑧 ∈ V → ( 𝑧 ∈ ran 𝐹 ↔ ∃ 𝑥 ∈ 𝐴 𝑧 = 𝐵 ) ) |
18 |
16 17
|
ax-mp |
⊢ ( 𝑧 ∈ ran 𝐹 ↔ ∃ 𝑥 ∈ 𝐴 𝑧 = 𝐵 ) |
19 |
18
|
biimpi |
⊢ ( 𝑧 ∈ ran 𝐹 → ∃ 𝑥 ∈ 𝐴 𝑧 = 𝐵 ) |
20 |
19
|
adantl |
⊢ ( ( ( 𝜑 ∧ 𝑦 ∈ ℝ ∧ ∀ 𝑥 ∈ 𝐴 𝐵 ≤ 𝑦 ) ∧ 𝑧 ∈ ran 𝐹 ) → ∃ 𝑥 ∈ 𝐴 𝑧 = 𝐵 ) |
21 |
|
simp3 |
⊢ ( ( 𝜑 ∧ ∀ 𝑥 ∈ 𝐴 𝐵 ≤ 𝑦 ∧ ∃ 𝑥 ∈ 𝐴 𝑧 = 𝐵 ) → ∃ 𝑥 ∈ 𝐴 𝑧 = 𝐵 ) |
22 |
|
nfra1 |
⊢ Ⅎ 𝑥 ∀ 𝑥 ∈ 𝐴 𝐵 ≤ 𝑦 |
23 |
|
nfre1 |
⊢ Ⅎ 𝑥 ∃ 𝑥 ∈ 𝐴 𝑧 = 𝐵 |
24 |
9 22 23
|
nf3an |
⊢ Ⅎ 𝑥 ( 𝜑 ∧ ∀ 𝑥 ∈ 𝐴 𝐵 ≤ 𝑦 ∧ ∃ 𝑥 ∈ 𝐴 𝑧 = 𝐵 ) |
25 |
|
nfv |
⊢ Ⅎ 𝑥 𝑧 ≤ 𝑦 |
26 |
|
simp3 |
⊢ ( ( ∀ 𝑥 ∈ 𝐴 𝐵 ≤ 𝑦 ∧ 𝑥 ∈ 𝐴 ∧ 𝑧 = 𝐵 ) → 𝑧 = 𝐵 ) |
27 |
|
rspa |
⊢ ( ( ∀ 𝑥 ∈ 𝐴 𝐵 ≤ 𝑦 ∧ 𝑥 ∈ 𝐴 ) → 𝐵 ≤ 𝑦 ) |
28 |
27
|
3adant3 |
⊢ ( ( ∀ 𝑥 ∈ 𝐴 𝐵 ≤ 𝑦 ∧ 𝑥 ∈ 𝐴 ∧ 𝑧 = 𝐵 ) → 𝐵 ≤ 𝑦 ) |
29 |
26 28
|
eqbrtrd |
⊢ ( ( ∀ 𝑥 ∈ 𝐴 𝐵 ≤ 𝑦 ∧ 𝑥 ∈ 𝐴 ∧ 𝑧 = 𝐵 ) → 𝑧 ≤ 𝑦 ) |
30 |
29
|
3exp |
⊢ ( ∀ 𝑥 ∈ 𝐴 𝐵 ≤ 𝑦 → ( 𝑥 ∈ 𝐴 → ( 𝑧 = 𝐵 → 𝑧 ≤ 𝑦 ) ) ) |
31 |
30
|
3ad2ant2 |
⊢ ( ( 𝜑 ∧ ∀ 𝑥 ∈ 𝐴 𝐵 ≤ 𝑦 ∧ ∃ 𝑥 ∈ 𝐴 𝑧 = 𝐵 ) → ( 𝑥 ∈ 𝐴 → ( 𝑧 = 𝐵 → 𝑧 ≤ 𝑦 ) ) ) |
32 |
24 25 31
|
rexlimd |
⊢ ( ( 𝜑 ∧ ∀ 𝑥 ∈ 𝐴 𝐵 ≤ 𝑦 ∧ ∃ 𝑥 ∈ 𝐴 𝑧 = 𝐵 ) → ( ∃ 𝑥 ∈ 𝐴 𝑧 = 𝐵 → 𝑧 ≤ 𝑦 ) ) |
33 |
21 32
|
mpd |
⊢ ( ( 𝜑 ∧ ∀ 𝑥 ∈ 𝐴 𝐵 ≤ 𝑦 ∧ ∃ 𝑥 ∈ 𝐴 𝑧 = 𝐵 ) → 𝑧 ≤ 𝑦 ) |
34 |
14 15 20 33
|
syl3anc |
⊢ ( ( ( 𝜑 ∧ 𝑦 ∈ ℝ ∧ ∀ 𝑥 ∈ 𝐴 𝐵 ≤ 𝑦 ) ∧ 𝑧 ∈ ran 𝐹 ) → 𝑧 ≤ 𝑦 ) |
35 |
34
|
ralrimiva |
⊢ ( ( 𝜑 ∧ 𝑦 ∈ ℝ ∧ ∀ 𝑥 ∈ 𝐴 𝐵 ≤ 𝑦 ) → ∀ 𝑧 ∈ ran 𝐹 𝑧 ≤ 𝑦 ) |
36 |
|
19.8a |
⊢ ( ( 𝑦 ∈ ℝ ∧ ∀ 𝑧 ∈ ran 𝐹 𝑧 ≤ 𝑦 ) → ∃ 𝑦 ( 𝑦 ∈ ℝ ∧ ∀ 𝑧 ∈ ran 𝐹 𝑧 ≤ 𝑦 ) ) |
37 |
13 35 36
|
syl2anc |
⊢ ( ( 𝜑 ∧ 𝑦 ∈ ℝ ∧ ∀ 𝑥 ∈ 𝐴 𝐵 ≤ 𝑦 ) → ∃ 𝑦 ( 𝑦 ∈ ℝ ∧ ∀ 𝑧 ∈ ran 𝐹 𝑧 ≤ 𝑦 ) ) |
38 |
|
df-rex |
⊢ ( ∃ 𝑦 ∈ ℝ ∀ 𝑧 ∈ ran 𝐹 𝑧 ≤ 𝑦 ↔ ∃ 𝑦 ( 𝑦 ∈ ℝ ∧ ∀ 𝑧 ∈ ran 𝐹 𝑧 ≤ 𝑦 ) ) |
39 |
37 38
|
sylibr |
⊢ ( ( 𝜑 ∧ 𝑦 ∈ ℝ ∧ ∀ 𝑥 ∈ 𝐴 𝐵 ≤ 𝑦 ) → ∃ 𝑦 ∈ ℝ ∀ 𝑧 ∈ ran 𝐹 𝑧 ≤ 𝑦 ) |
40 |
39
|
3exp |
⊢ ( 𝜑 → ( 𝑦 ∈ ℝ → ( ∀ 𝑥 ∈ 𝐴 𝐵 ≤ 𝑦 → ∃ 𝑦 ∈ ℝ ∀ 𝑧 ∈ ran 𝐹 𝑧 ≤ 𝑦 ) ) ) |
41 |
11 12 40
|
rexlimd |
⊢ ( 𝜑 → ( ∃ 𝑦 ∈ ℝ ∀ 𝑥 ∈ 𝐴 𝐵 ≤ 𝑦 → ∃ 𝑦 ∈ ℝ ∀ 𝑧 ∈ ran 𝐹 𝑧 ≤ 𝑦 ) ) |
42 |
3 41
|
mpd |
⊢ ( 𝜑 → ∃ 𝑦 ∈ ℝ ∀ 𝑧 ∈ ran 𝐹 𝑧 ≤ 𝑦 ) |
43 |
|
suprcl |
⊢ ( ( ran 𝐹 ⊆ ℝ ∧ ran 𝐹 ≠ ∅ ∧ ∃ 𝑦 ∈ ℝ ∀ 𝑧 ∈ ran 𝐹 𝑧 ≤ 𝑦 ) → sup ( ran 𝐹 , ℝ , < ) ∈ ℝ ) |
44 |
8 10 42 43
|
syl3anc |
⊢ ( 𝜑 → sup ( ran 𝐹 , ℝ , < ) ∈ ℝ ) |
45 |
5 44
|
eqeltrid |
⊢ ( 𝜑 → 𝐶 ∈ ℝ ) |
46 |
8
|
adantr |
⊢ ( ( 𝜑 ∧ 𝑥 ∈ 𝐴 ) → ran 𝐹 ⊆ ℝ ) |
47 |
|
simpr |
⊢ ( ( 𝜑 ∧ 𝑥 ∈ 𝐴 ) → 𝑥 ∈ 𝐴 ) |
48 |
4
|
elrnmpt1 |
⊢ ( ( 𝑥 ∈ 𝐴 ∧ 𝐵 ∈ ℝ ) → 𝐵 ∈ ran 𝐹 ) |
49 |
47 2 48
|
syl2anc |
⊢ ( ( 𝜑 ∧ 𝑥 ∈ 𝐴 ) → 𝐵 ∈ ran 𝐹 ) |
50 |
49
|
ne0d |
⊢ ( ( 𝜑 ∧ 𝑥 ∈ 𝐴 ) → ran 𝐹 ≠ ∅ ) |
51 |
42
|
adantr |
⊢ ( ( 𝜑 ∧ 𝑥 ∈ 𝐴 ) → ∃ 𝑦 ∈ ℝ ∀ 𝑧 ∈ ran 𝐹 𝑧 ≤ 𝑦 ) |
52 |
|
suprub |
⊢ ( ( ( ran 𝐹 ⊆ ℝ ∧ ran 𝐹 ≠ ∅ ∧ ∃ 𝑦 ∈ ℝ ∀ 𝑧 ∈ ran 𝐹 𝑧 ≤ 𝑦 ) ∧ 𝐵 ∈ ran 𝐹 ) → 𝐵 ≤ sup ( ran 𝐹 , ℝ , < ) ) |
53 |
46 50 51 49 52
|
syl31anc |
⊢ ( ( 𝜑 ∧ 𝑥 ∈ 𝐴 ) → 𝐵 ≤ sup ( ran 𝐹 , ℝ , < ) ) |
54 |
53 5
|
breqtrrdi |
⊢ ( ( 𝜑 ∧ 𝑥 ∈ 𝐴 ) → 𝐵 ≤ 𝐶 ) |
55 |
54
|
ralrimiva |
⊢ ( 𝜑 → ∀ 𝑥 ∈ 𝐴 𝐵 ≤ 𝐶 ) |
56 |
45 55
|
jca |
⊢ ( 𝜑 → ( 𝐶 ∈ ℝ ∧ ∀ 𝑥 ∈ 𝐴 𝐵 ≤ 𝐶 ) ) |