Step |
Hyp |
Ref |
Expression |
1 |
|
infnsuprnmpt.x |
⊢ Ⅎ 𝑥 𝜑 |
2 |
|
infnsuprnmpt.a |
⊢ ( 𝜑 → 𝐴 ≠ ∅ ) |
3 |
|
infnsuprnmpt.b |
⊢ ( ( 𝜑 ∧ 𝑥 ∈ 𝐴 ) → 𝐵 ∈ ℝ ) |
4 |
|
infnsuprnmpt.l |
⊢ ( 𝜑 → ∃ 𝑦 ∈ ℝ ∀ 𝑥 ∈ 𝐴 𝑦 ≤ 𝐵 ) |
5 |
|
eqid |
⊢ ( 𝑥 ∈ 𝐴 ↦ 𝐵 ) = ( 𝑥 ∈ 𝐴 ↦ 𝐵 ) |
6 |
1 5 3
|
rnmptssd |
⊢ ( 𝜑 → ran ( 𝑥 ∈ 𝐴 ↦ 𝐵 ) ⊆ ℝ ) |
7 |
1 3 5 2
|
rnmptn0 |
⊢ ( 𝜑 → ran ( 𝑥 ∈ 𝐴 ↦ 𝐵 ) ≠ ∅ ) |
8 |
4
|
rnmptlb |
⊢ ( 𝜑 → ∃ 𝑦 ∈ ℝ ∀ 𝑧 ∈ ran ( 𝑥 ∈ 𝐴 ↦ 𝐵 ) 𝑦 ≤ 𝑧 ) |
9 |
|
infrenegsup |
⊢ ( ( ran ( 𝑥 ∈ 𝐴 ↦ 𝐵 ) ⊆ ℝ ∧ ran ( 𝑥 ∈ 𝐴 ↦ 𝐵 ) ≠ ∅ ∧ ∃ 𝑦 ∈ ℝ ∀ 𝑧 ∈ ran ( 𝑥 ∈ 𝐴 ↦ 𝐵 ) 𝑦 ≤ 𝑧 ) → inf ( ran ( 𝑥 ∈ 𝐴 ↦ 𝐵 ) , ℝ , < ) = - sup ( { 𝑤 ∈ ℝ ∣ - 𝑤 ∈ ran ( 𝑥 ∈ 𝐴 ↦ 𝐵 ) } , ℝ , < ) ) |
10 |
6 7 8 9
|
syl3anc |
⊢ ( 𝜑 → inf ( ran ( 𝑥 ∈ 𝐴 ↦ 𝐵 ) , ℝ , < ) = - sup ( { 𝑤 ∈ ℝ ∣ - 𝑤 ∈ ran ( 𝑥 ∈ 𝐴 ↦ 𝐵 ) } , ℝ , < ) ) |
11 |
|
eqid |
⊢ ( 𝑥 ∈ 𝐴 ↦ - 𝐵 ) = ( 𝑥 ∈ 𝐴 ↦ - 𝐵 ) |
12 |
|
rabidim2 |
⊢ ( 𝑤 ∈ { 𝑤 ∈ ℝ ∣ - 𝑤 ∈ ran ( 𝑥 ∈ 𝐴 ↦ 𝐵 ) } → - 𝑤 ∈ ran ( 𝑥 ∈ 𝐴 ↦ 𝐵 ) ) |
13 |
12
|
adantl |
⊢ ( ( 𝜑 ∧ 𝑤 ∈ { 𝑤 ∈ ℝ ∣ - 𝑤 ∈ ran ( 𝑥 ∈ 𝐴 ↦ 𝐵 ) } ) → - 𝑤 ∈ ran ( 𝑥 ∈ 𝐴 ↦ 𝐵 ) ) |
14 |
|
negex |
⊢ - 𝑤 ∈ V |
15 |
5
|
elrnmpt |
⊢ ( - 𝑤 ∈ V → ( - 𝑤 ∈ ran ( 𝑥 ∈ 𝐴 ↦ 𝐵 ) ↔ ∃ 𝑥 ∈ 𝐴 - 𝑤 = 𝐵 ) ) |
16 |
14 15
|
ax-mp |
⊢ ( - 𝑤 ∈ ran ( 𝑥 ∈ 𝐴 ↦ 𝐵 ) ↔ ∃ 𝑥 ∈ 𝐴 - 𝑤 = 𝐵 ) |
17 |
13 16
|
sylib |
⊢ ( ( 𝜑 ∧ 𝑤 ∈ { 𝑤 ∈ ℝ ∣ - 𝑤 ∈ ran ( 𝑥 ∈ 𝐴 ↦ 𝐵 ) } ) → ∃ 𝑥 ∈ 𝐴 - 𝑤 = 𝐵 ) |
18 |
|
nfcv |
⊢ Ⅎ 𝑥 𝑤 |
19 |
18
|
nfneg |
⊢ Ⅎ 𝑥 - 𝑤 |
20 |
|
nfmpt1 |
⊢ Ⅎ 𝑥 ( 𝑥 ∈ 𝐴 ↦ 𝐵 ) |
21 |
20
|
nfrn |
⊢ Ⅎ 𝑥 ran ( 𝑥 ∈ 𝐴 ↦ 𝐵 ) |
22 |
19 21
|
nfel |
⊢ Ⅎ 𝑥 - 𝑤 ∈ ran ( 𝑥 ∈ 𝐴 ↦ 𝐵 ) |
23 |
|
nfcv |
⊢ Ⅎ 𝑥 ℝ |
24 |
22 23
|
nfrabw |
⊢ Ⅎ 𝑥 { 𝑤 ∈ ℝ ∣ - 𝑤 ∈ ran ( 𝑥 ∈ 𝐴 ↦ 𝐵 ) } |
25 |
18 24
|
nfel |
⊢ Ⅎ 𝑥 𝑤 ∈ { 𝑤 ∈ ℝ ∣ - 𝑤 ∈ ran ( 𝑥 ∈ 𝐴 ↦ 𝐵 ) } |
26 |
1 25
|
nfan |
⊢ Ⅎ 𝑥 ( 𝜑 ∧ 𝑤 ∈ { 𝑤 ∈ ℝ ∣ - 𝑤 ∈ ran ( 𝑥 ∈ 𝐴 ↦ 𝐵 ) } ) |
27 |
|
rabidim1 |
⊢ ( 𝑤 ∈ { 𝑤 ∈ ℝ ∣ - 𝑤 ∈ ran ( 𝑥 ∈ 𝐴 ↦ 𝐵 ) } → 𝑤 ∈ ℝ ) |
28 |
27
|
adantl |
⊢ ( ( 𝜑 ∧ 𝑤 ∈ { 𝑤 ∈ ℝ ∣ - 𝑤 ∈ ran ( 𝑥 ∈ 𝐴 ↦ 𝐵 ) } ) → 𝑤 ∈ ℝ ) |
29 |
|
negeq |
⊢ ( - 𝑤 = 𝐵 → - - 𝑤 = - 𝐵 ) |
30 |
29
|
eqcomd |
⊢ ( - 𝑤 = 𝐵 → - 𝐵 = - - 𝑤 ) |
31 |
30
|
3ad2ant3 |
⊢ ( ( ( 𝜑 ∧ 𝑤 ∈ ℝ ) ∧ 𝑥 ∈ 𝐴 ∧ - 𝑤 = 𝐵 ) → - 𝐵 = - - 𝑤 ) |
32 |
|
simp1r |
⊢ ( ( ( 𝜑 ∧ 𝑤 ∈ ℝ ) ∧ 𝑥 ∈ 𝐴 ∧ - 𝑤 = 𝐵 ) → 𝑤 ∈ ℝ ) |
33 |
|
recn |
⊢ ( 𝑤 ∈ ℝ → 𝑤 ∈ ℂ ) |
34 |
33
|
negnegd |
⊢ ( 𝑤 ∈ ℝ → - - 𝑤 = 𝑤 ) |
35 |
32 34
|
syl |
⊢ ( ( ( 𝜑 ∧ 𝑤 ∈ ℝ ) ∧ 𝑥 ∈ 𝐴 ∧ - 𝑤 = 𝐵 ) → - - 𝑤 = 𝑤 ) |
36 |
31 35
|
eqtr2d |
⊢ ( ( ( 𝜑 ∧ 𝑤 ∈ ℝ ) ∧ 𝑥 ∈ 𝐴 ∧ - 𝑤 = 𝐵 ) → 𝑤 = - 𝐵 ) |
37 |
36
|
3exp |
⊢ ( ( 𝜑 ∧ 𝑤 ∈ ℝ ) → ( 𝑥 ∈ 𝐴 → ( - 𝑤 = 𝐵 → 𝑤 = - 𝐵 ) ) ) |
38 |
28 37
|
syldan |
⊢ ( ( 𝜑 ∧ 𝑤 ∈ { 𝑤 ∈ ℝ ∣ - 𝑤 ∈ ran ( 𝑥 ∈ 𝐴 ↦ 𝐵 ) } ) → ( 𝑥 ∈ 𝐴 → ( - 𝑤 = 𝐵 → 𝑤 = - 𝐵 ) ) ) |
39 |
26 38
|
reximdai |
⊢ ( ( 𝜑 ∧ 𝑤 ∈ { 𝑤 ∈ ℝ ∣ - 𝑤 ∈ ran ( 𝑥 ∈ 𝐴 ↦ 𝐵 ) } ) → ( ∃ 𝑥 ∈ 𝐴 - 𝑤 = 𝐵 → ∃ 𝑥 ∈ 𝐴 𝑤 = - 𝐵 ) ) |
40 |
17 39
|
mpd |
⊢ ( ( 𝜑 ∧ 𝑤 ∈ { 𝑤 ∈ ℝ ∣ - 𝑤 ∈ ran ( 𝑥 ∈ 𝐴 ↦ 𝐵 ) } ) → ∃ 𝑥 ∈ 𝐴 𝑤 = - 𝐵 ) |
41 |
|
simpr |
⊢ ( ( 𝜑 ∧ 𝑤 ∈ { 𝑤 ∈ ℝ ∣ - 𝑤 ∈ ran ( 𝑥 ∈ 𝐴 ↦ 𝐵 ) } ) → 𝑤 ∈ { 𝑤 ∈ ℝ ∣ - 𝑤 ∈ ran ( 𝑥 ∈ 𝐴 ↦ 𝐵 ) } ) |
42 |
11 40 41
|
elrnmptd |
⊢ ( ( 𝜑 ∧ 𝑤 ∈ { 𝑤 ∈ ℝ ∣ - 𝑤 ∈ ran ( 𝑥 ∈ 𝐴 ↦ 𝐵 ) } ) → 𝑤 ∈ ran ( 𝑥 ∈ 𝐴 ↦ - 𝐵 ) ) |
43 |
42
|
ex |
⊢ ( 𝜑 → ( 𝑤 ∈ { 𝑤 ∈ ℝ ∣ - 𝑤 ∈ ran ( 𝑥 ∈ 𝐴 ↦ 𝐵 ) } → 𝑤 ∈ ran ( 𝑥 ∈ 𝐴 ↦ - 𝐵 ) ) ) |
44 |
|
vex |
⊢ 𝑤 ∈ V |
45 |
11
|
elrnmpt |
⊢ ( 𝑤 ∈ V → ( 𝑤 ∈ ran ( 𝑥 ∈ 𝐴 ↦ - 𝐵 ) ↔ ∃ 𝑥 ∈ 𝐴 𝑤 = - 𝐵 ) ) |
46 |
44 45
|
ax-mp |
⊢ ( 𝑤 ∈ ran ( 𝑥 ∈ 𝐴 ↦ - 𝐵 ) ↔ ∃ 𝑥 ∈ 𝐴 𝑤 = - 𝐵 ) |
47 |
46
|
biimpi |
⊢ ( 𝑤 ∈ ran ( 𝑥 ∈ 𝐴 ↦ - 𝐵 ) → ∃ 𝑥 ∈ 𝐴 𝑤 = - 𝐵 ) |
48 |
47
|
adantl |
⊢ ( ( 𝜑 ∧ 𝑤 ∈ ran ( 𝑥 ∈ 𝐴 ↦ - 𝐵 ) ) → ∃ 𝑥 ∈ 𝐴 𝑤 = - 𝐵 ) |
49 |
18 23
|
nfel |
⊢ Ⅎ 𝑥 𝑤 ∈ ℝ |
50 |
49 22
|
nfan |
⊢ Ⅎ 𝑥 ( 𝑤 ∈ ℝ ∧ - 𝑤 ∈ ran ( 𝑥 ∈ 𝐴 ↦ 𝐵 ) ) |
51 |
|
simp3 |
⊢ ( ( 𝜑 ∧ 𝑥 ∈ 𝐴 ∧ 𝑤 = - 𝐵 ) → 𝑤 = - 𝐵 ) |
52 |
3
|
renegcld |
⊢ ( ( 𝜑 ∧ 𝑥 ∈ 𝐴 ) → - 𝐵 ∈ ℝ ) |
53 |
52
|
3adant3 |
⊢ ( ( 𝜑 ∧ 𝑥 ∈ 𝐴 ∧ 𝑤 = - 𝐵 ) → - 𝐵 ∈ ℝ ) |
54 |
51 53
|
eqeltrd |
⊢ ( ( 𝜑 ∧ 𝑥 ∈ 𝐴 ∧ 𝑤 = - 𝐵 ) → 𝑤 ∈ ℝ ) |
55 |
|
simp2 |
⊢ ( ( 𝜑 ∧ 𝑥 ∈ 𝐴 ∧ 𝑤 = - 𝐵 ) → 𝑥 ∈ 𝐴 ) |
56 |
51
|
negeqd |
⊢ ( ( 𝜑 ∧ 𝑥 ∈ 𝐴 ∧ 𝑤 = - 𝐵 ) → - 𝑤 = - - 𝐵 ) |
57 |
3
|
recnd |
⊢ ( ( 𝜑 ∧ 𝑥 ∈ 𝐴 ) → 𝐵 ∈ ℂ ) |
58 |
57
|
negnegd |
⊢ ( ( 𝜑 ∧ 𝑥 ∈ 𝐴 ) → - - 𝐵 = 𝐵 ) |
59 |
58
|
3adant3 |
⊢ ( ( 𝜑 ∧ 𝑥 ∈ 𝐴 ∧ 𝑤 = - 𝐵 ) → - - 𝐵 = 𝐵 ) |
60 |
56 59
|
eqtrd |
⊢ ( ( 𝜑 ∧ 𝑥 ∈ 𝐴 ∧ 𝑤 = - 𝐵 ) → - 𝑤 = 𝐵 ) |
61 |
|
rspe |
⊢ ( ( 𝑥 ∈ 𝐴 ∧ - 𝑤 = 𝐵 ) → ∃ 𝑥 ∈ 𝐴 - 𝑤 = 𝐵 ) |
62 |
55 60 61
|
syl2anc |
⊢ ( ( 𝜑 ∧ 𝑥 ∈ 𝐴 ∧ 𝑤 = - 𝐵 ) → ∃ 𝑥 ∈ 𝐴 - 𝑤 = 𝐵 ) |
63 |
14
|
a1i |
⊢ ( ( 𝜑 ∧ 𝑥 ∈ 𝐴 ∧ 𝑤 = - 𝐵 ) → - 𝑤 ∈ V ) |
64 |
5 62 63
|
elrnmptd |
⊢ ( ( 𝜑 ∧ 𝑥 ∈ 𝐴 ∧ 𝑤 = - 𝐵 ) → - 𝑤 ∈ ran ( 𝑥 ∈ 𝐴 ↦ 𝐵 ) ) |
65 |
54 64
|
jca |
⊢ ( ( 𝜑 ∧ 𝑥 ∈ 𝐴 ∧ 𝑤 = - 𝐵 ) → ( 𝑤 ∈ ℝ ∧ - 𝑤 ∈ ran ( 𝑥 ∈ 𝐴 ↦ 𝐵 ) ) ) |
66 |
65
|
3exp |
⊢ ( 𝜑 → ( 𝑥 ∈ 𝐴 → ( 𝑤 = - 𝐵 → ( 𝑤 ∈ ℝ ∧ - 𝑤 ∈ ran ( 𝑥 ∈ 𝐴 ↦ 𝐵 ) ) ) ) ) |
67 |
1 50 66
|
rexlimd |
⊢ ( 𝜑 → ( ∃ 𝑥 ∈ 𝐴 𝑤 = - 𝐵 → ( 𝑤 ∈ ℝ ∧ - 𝑤 ∈ ran ( 𝑥 ∈ 𝐴 ↦ 𝐵 ) ) ) ) |
68 |
67
|
imp |
⊢ ( ( 𝜑 ∧ ∃ 𝑥 ∈ 𝐴 𝑤 = - 𝐵 ) → ( 𝑤 ∈ ℝ ∧ - 𝑤 ∈ ran ( 𝑥 ∈ 𝐴 ↦ 𝐵 ) ) ) |
69 |
48 68
|
syldan |
⊢ ( ( 𝜑 ∧ 𝑤 ∈ ran ( 𝑥 ∈ 𝐴 ↦ - 𝐵 ) ) → ( 𝑤 ∈ ℝ ∧ - 𝑤 ∈ ran ( 𝑥 ∈ 𝐴 ↦ 𝐵 ) ) ) |
70 |
|
rabid |
⊢ ( 𝑤 ∈ { 𝑤 ∈ ℝ ∣ - 𝑤 ∈ ran ( 𝑥 ∈ 𝐴 ↦ 𝐵 ) } ↔ ( 𝑤 ∈ ℝ ∧ - 𝑤 ∈ ran ( 𝑥 ∈ 𝐴 ↦ 𝐵 ) ) ) |
71 |
69 70
|
sylibr |
⊢ ( ( 𝜑 ∧ 𝑤 ∈ ran ( 𝑥 ∈ 𝐴 ↦ - 𝐵 ) ) → 𝑤 ∈ { 𝑤 ∈ ℝ ∣ - 𝑤 ∈ ran ( 𝑥 ∈ 𝐴 ↦ 𝐵 ) } ) |
72 |
71
|
ex |
⊢ ( 𝜑 → ( 𝑤 ∈ ran ( 𝑥 ∈ 𝐴 ↦ - 𝐵 ) → 𝑤 ∈ { 𝑤 ∈ ℝ ∣ - 𝑤 ∈ ran ( 𝑥 ∈ 𝐴 ↦ 𝐵 ) } ) ) |
73 |
43 72
|
impbid |
⊢ ( 𝜑 → ( 𝑤 ∈ { 𝑤 ∈ ℝ ∣ - 𝑤 ∈ ran ( 𝑥 ∈ 𝐴 ↦ 𝐵 ) } ↔ 𝑤 ∈ ran ( 𝑥 ∈ 𝐴 ↦ - 𝐵 ) ) ) |
74 |
73
|
alrimiv |
⊢ ( 𝜑 → ∀ 𝑤 ( 𝑤 ∈ { 𝑤 ∈ ℝ ∣ - 𝑤 ∈ ran ( 𝑥 ∈ 𝐴 ↦ 𝐵 ) } ↔ 𝑤 ∈ ran ( 𝑥 ∈ 𝐴 ↦ - 𝐵 ) ) ) |
75 |
|
nfrab1 |
⊢ Ⅎ 𝑤 { 𝑤 ∈ ℝ ∣ - 𝑤 ∈ ran ( 𝑥 ∈ 𝐴 ↦ 𝐵 ) } |
76 |
|
nfcv |
⊢ Ⅎ 𝑤 ran ( 𝑥 ∈ 𝐴 ↦ - 𝐵 ) |
77 |
75 76
|
cleqf |
⊢ ( { 𝑤 ∈ ℝ ∣ - 𝑤 ∈ ran ( 𝑥 ∈ 𝐴 ↦ 𝐵 ) } = ran ( 𝑥 ∈ 𝐴 ↦ - 𝐵 ) ↔ ∀ 𝑤 ( 𝑤 ∈ { 𝑤 ∈ ℝ ∣ - 𝑤 ∈ ran ( 𝑥 ∈ 𝐴 ↦ 𝐵 ) } ↔ 𝑤 ∈ ran ( 𝑥 ∈ 𝐴 ↦ - 𝐵 ) ) ) |
78 |
74 77
|
sylibr |
⊢ ( 𝜑 → { 𝑤 ∈ ℝ ∣ - 𝑤 ∈ ran ( 𝑥 ∈ 𝐴 ↦ 𝐵 ) } = ran ( 𝑥 ∈ 𝐴 ↦ - 𝐵 ) ) |
79 |
78
|
supeq1d |
⊢ ( 𝜑 → sup ( { 𝑤 ∈ ℝ ∣ - 𝑤 ∈ ran ( 𝑥 ∈ 𝐴 ↦ 𝐵 ) } , ℝ , < ) = sup ( ran ( 𝑥 ∈ 𝐴 ↦ - 𝐵 ) , ℝ , < ) ) |
80 |
79
|
negeqd |
⊢ ( 𝜑 → - sup ( { 𝑤 ∈ ℝ ∣ - 𝑤 ∈ ran ( 𝑥 ∈ 𝐴 ↦ 𝐵 ) } , ℝ , < ) = - sup ( ran ( 𝑥 ∈ 𝐴 ↦ - 𝐵 ) , ℝ , < ) ) |
81 |
|
eqidd |
⊢ ( 𝜑 → - sup ( ran ( 𝑥 ∈ 𝐴 ↦ - 𝐵 ) , ℝ , < ) = - sup ( ran ( 𝑥 ∈ 𝐴 ↦ - 𝐵 ) , ℝ , < ) ) |
82 |
10 80 81
|
3eqtrd |
⊢ ( 𝜑 → inf ( ran ( 𝑥 ∈ 𝐴 ↦ 𝐵 ) , ℝ , < ) = - sup ( ran ( 𝑥 ∈ 𝐴 ↦ - 𝐵 ) , ℝ , < ) ) |