Step |
Hyp |
Ref |
Expression |
1 |
|
elat2 |
⊢ ( 𝐴 ∈ HAtoms ↔ ( 𝐴 ∈ Cℋ ∧ ( 𝐴 ≠ 0ℋ ∧ ∀ 𝑦 ∈ Cℋ ( 𝑦 ⊆ 𝐴 → ( 𝑦 = 𝐴 ∨ 𝑦 = 0ℋ ) ) ) ) ) |
2 |
|
chne0 |
⊢ ( 𝐴 ∈ Cℋ → ( 𝐴 ≠ 0ℋ ↔ ∃ 𝑥 ∈ 𝐴 𝑥 ≠ 0ℎ ) ) |
3 |
|
nfv |
⊢ Ⅎ 𝑥 𝐴 ∈ Cℋ |
4 |
|
nfv |
⊢ Ⅎ 𝑥 ∀ 𝑦 ∈ Cℋ ( 𝑦 ⊆ 𝐴 → ( 𝑦 = 𝐴 ∨ 𝑦 = 0ℋ ) ) |
5 |
|
nfre1 |
⊢ Ⅎ 𝑥 ∃ 𝑥 ∈ ℋ ( 𝑥 ≠ 0ℎ ∧ 𝐴 = ( ⊥ ‘ ( ⊥ ‘ { 𝑥 } ) ) ) |
6 |
4 5
|
nfim |
⊢ Ⅎ 𝑥 ( ∀ 𝑦 ∈ Cℋ ( 𝑦 ⊆ 𝐴 → ( 𝑦 = 𝐴 ∨ 𝑦 = 0ℋ ) ) → ∃ 𝑥 ∈ ℋ ( 𝑥 ≠ 0ℎ ∧ 𝐴 = ( ⊥ ‘ ( ⊥ ‘ { 𝑥 } ) ) ) ) |
7 |
|
chel |
⊢ ( ( 𝐴 ∈ Cℋ ∧ 𝑥 ∈ 𝐴 ) → 𝑥 ∈ ℋ ) |
8 |
7
|
adantrr |
⊢ ( ( 𝐴 ∈ Cℋ ∧ ( 𝑥 ∈ 𝐴 ∧ 𝑥 ≠ 0ℎ ) ) → 𝑥 ∈ ℋ ) |
9 |
8
|
adantrr |
⊢ ( ( 𝐴 ∈ Cℋ ∧ ( ( 𝑥 ∈ 𝐴 ∧ 𝑥 ≠ 0ℎ ) ∧ ∀ 𝑦 ∈ Cℋ ( 𝑦 ⊆ 𝐴 → ( 𝑦 = 𝐴 ∨ 𝑦 = 0ℋ ) ) ) ) → 𝑥 ∈ ℋ ) |
10 |
|
simprlr |
⊢ ( ( 𝐴 ∈ Cℋ ∧ ( ( 𝑥 ∈ 𝐴 ∧ 𝑥 ≠ 0ℎ ) ∧ ∀ 𝑦 ∈ Cℋ ( 𝑦 ⊆ 𝐴 → ( 𝑦 = 𝐴 ∨ 𝑦 = 0ℋ ) ) ) ) → 𝑥 ≠ 0ℎ ) |
11 |
|
h1dn0 |
⊢ ( ( 𝑥 ∈ ℋ ∧ 𝑥 ≠ 0ℎ ) → ( ⊥ ‘ ( ⊥ ‘ { 𝑥 } ) ) ≠ 0ℋ ) |
12 |
7 11
|
sylan |
⊢ ( ( ( 𝐴 ∈ Cℋ ∧ 𝑥 ∈ 𝐴 ) ∧ 𝑥 ≠ 0ℎ ) → ( ⊥ ‘ ( ⊥ ‘ { 𝑥 } ) ) ≠ 0ℋ ) |
13 |
12
|
anasss |
⊢ ( ( 𝐴 ∈ Cℋ ∧ ( 𝑥 ∈ 𝐴 ∧ 𝑥 ≠ 0ℎ ) ) → ( ⊥ ‘ ( ⊥ ‘ { 𝑥 } ) ) ≠ 0ℋ ) |
14 |
13
|
adantrr |
⊢ ( ( 𝐴 ∈ Cℋ ∧ ( ( 𝑥 ∈ 𝐴 ∧ 𝑥 ≠ 0ℎ ) ∧ ∀ 𝑦 ∈ Cℋ ( 𝑦 ⊆ 𝐴 → ( 𝑦 = 𝐴 ∨ 𝑦 = 0ℋ ) ) ) ) → ( ⊥ ‘ ( ⊥ ‘ { 𝑥 } ) ) ≠ 0ℋ ) |
15 |
|
ch1dle |
⊢ ( ( 𝐴 ∈ Cℋ ∧ 𝑥 ∈ 𝐴 ) → ( ⊥ ‘ ( ⊥ ‘ { 𝑥 } ) ) ⊆ 𝐴 ) |
16 |
|
snssi |
⊢ ( 𝑥 ∈ ℋ → { 𝑥 } ⊆ ℋ ) |
17 |
|
occl |
⊢ ( { 𝑥 } ⊆ ℋ → ( ⊥ ‘ { 𝑥 } ) ∈ Cℋ ) |
18 |
7 16 17
|
3syl |
⊢ ( ( 𝐴 ∈ Cℋ ∧ 𝑥 ∈ 𝐴 ) → ( ⊥ ‘ { 𝑥 } ) ∈ Cℋ ) |
19 |
|
choccl |
⊢ ( ( ⊥ ‘ { 𝑥 } ) ∈ Cℋ → ( ⊥ ‘ ( ⊥ ‘ { 𝑥 } ) ) ∈ Cℋ ) |
20 |
|
sseq1 |
⊢ ( 𝑦 = ( ⊥ ‘ ( ⊥ ‘ { 𝑥 } ) ) → ( 𝑦 ⊆ 𝐴 ↔ ( ⊥ ‘ ( ⊥ ‘ { 𝑥 } ) ) ⊆ 𝐴 ) ) |
21 |
|
eqeq1 |
⊢ ( 𝑦 = ( ⊥ ‘ ( ⊥ ‘ { 𝑥 } ) ) → ( 𝑦 = 𝐴 ↔ ( ⊥ ‘ ( ⊥ ‘ { 𝑥 } ) ) = 𝐴 ) ) |
22 |
|
eqeq1 |
⊢ ( 𝑦 = ( ⊥ ‘ ( ⊥ ‘ { 𝑥 } ) ) → ( 𝑦 = 0ℋ ↔ ( ⊥ ‘ ( ⊥ ‘ { 𝑥 } ) ) = 0ℋ ) ) |
23 |
21 22
|
orbi12d |
⊢ ( 𝑦 = ( ⊥ ‘ ( ⊥ ‘ { 𝑥 } ) ) → ( ( 𝑦 = 𝐴 ∨ 𝑦 = 0ℋ ) ↔ ( ( ⊥ ‘ ( ⊥ ‘ { 𝑥 } ) ) = 𝐴 ∨ ( ⊥ ‘ ( ⊥ ‘ { 𝑥 } ) ) = 0ℋ ) ) ) |
24 |
20 23
|
imbi12d |
⊢ ( 𝑦 = ( ⊥ ‘ ( ⊥ ‘ { 𝑥 } ) ) → ( ( 𝑦 ⊆ 𝐴 → ( 𝑦 = 𝐴 ∨ 𝑦 = 0ℋ ) ) ↔ ( ( ⊥ ‘ ( ⊥ ‘ { 𝑥 } ) ) ⊆ 𝐴 → ( ( ⊥ ‘ ( ⊥ ‘ { 𝑥 } ) ) = 𝐴 ∨ ( ⊥ ‘ ( ⊥ ‘ { 𝑥 } ) ) = 0ℋ ) ) ) ) |
25 |
24
|
rspcv |
⊢ ( ( ⊥ ‘ ( ⊥ ‘ { 𝑥 } ) ) ∈ Cℋ → ( ∀ 𝑦 ∈ Cℋ ( 𝑦 ⊆ 𝐴 → ( 𝑦 = 𝐴 ∨ 𝑦 = 0ℋ ) ) → ( ( ⊥ ‘ ( ⊥ ‘ { 𝑥 } ) ) ⊆ 𝐴 → ( ( ⊥ ‘ ( ⊥ ‘ { 𝑥 } ) ) = 𝐴 ∨ ( ⊥ ‘ ( ⊥ ‘ { 𝑥 } ) ) = 0ℋ ) ) ) ) |
26 |
18 19 25
|
3syl |
⊢ ( ( 𝐴 ∈ Cℋ ∧ 𝑥 ∈ 𝐴 ) → ( ∀ 𝑦 ∈ Cℋ ( 𝑦 ⊆ 𝐴 → ( 𝑦 = 𝐴 ∨ 𝑦 = 0ℋ ) ) → ( ( ⊥ ‘ ( ⊥ ‘ { 𝑥 } ) ) ⊆ 𝐴 → ( ( ⊥ ‘ ( ⊥ ‘ { 𝑥 } ) ) = 𝐴 ∨ ( ⊥ ‘ ( ⊥ ‘ { 𝑥 } ) ) = 0ℋ ) ) ) ) |
27 |
15 26
|
mpid |
⊢ ( ( 𝐴 ∈ Cℋ ∧ 𝑥 ∈ 𝐴 ) → ( ∀ 𝑦 ∈ Cℋ ( 𝑦 ⊆ 𝐴 → ( 𝑦 = 𝐴 ∨ 𝑦 = 0ℋ ) ) → ( ( ⊥ ‘ ( ⊥ ‘ { 𝑥 } ) ) = 𝐴 ∨ ( ⊥ ‘ ( ⊥ ‘ { 𝑥 } ) ) = 0ℋ ) ) ) |
28 |
27
|
impr |
⊢ ( ( 𝐴 ∈ Cℋ ∧ ( 𝑥 ∈ 𝐴 ∧ ∀ 𝑦 ∈ Cℋ ( 𝑦 ⊆ 𝐴 → ( 𝑦 = 𝐴 ∨ 𝑦 = 0ℋ ) ) ) ) → ( ( ⊥ ‘ ( ⊥ ‘ { 𝑥 } ) ) = 𝐴 ∨ ( ⊥ ‘ ( ⊥ ‘ { 𝑥 } ) ) = 0ℋ ) ) |
29 |
28
|
adantrlr |
⊢ ( ( 𝐴 ∈ Cℋ ∧ ( ( 𝑥 ∈ 𝐴 ∧ 𝑥 ≠ 0ℎ ) ∧ ∀ 𝑦 ∈ Cℋ ( 𝑦 ⊆ 𝐴 → ( 𝑦 = 𝐴 ∨ 𝑦 = 0ℋ ) ) ) ) → ( ( ⊥ ‘ ( ⊥ ‘ { 𝑥 } ) ) = 𝐴 ∨ ( ⊥ ‘ ( ⊥ ‘ { 𝑥 } ) ) = 0ℋ ) ) |
30 |
29
|
ord |
⊢ ( ( 𝐴 ∈ Cℋ ∧ ( ( 𝑥 ∈ 𝐴 ∧ 𝑥 ≠ 0ℎ ) ∧ ∀ 𝑦 ∈ Cℋ ( 𝑦 ⊆ 𝐴 → ( 𝑦 = 𝐴 ∨ 𝑦 = 0ℋ ) ) ) ) → ( ¬ ( ⊥ ‘ ( ⊥ ‘ { 𝑥 } ) ) = 𝐴 → ( ⊥ ‘ ( ⊥ ‘ { 𝑥 } ) ) = 0ℋ ) ) |
31 |
|
nne |
⊢ ( ¬ ( ⊥ ‘ ( ⊥ ‘ { 𝑥 } ) ) ≠ 0ℋ ↔ ( ⊥ ‘ ( ⊥ ‘ { 𝑥 } ) ) = 0ℋ ) |
32 |
30 31
|
syl6ibr |
⊢ ( ( 𝐴 ∈ Cℋ ∧ ( ( 𝑥 ∈ 𝐴 ∧ 𝑥 ≠ 0ℎ ) ∧ ∀ 𝑦 ∈ Cℋ ( 𝑦 ⊆ 𝐴 → ( 𝑦 = 𝐴 ∨ 𝑦 = 0ℋ ) ) ) ) → ( ¬ ( ⊥ ‘ ( ⊥ ‘ { 𝑥 } ) ) = 𝐴 → ¬ ( ⊥ ‘ ( ⊥ ‘ { 𝑥 } ) ) ≠ 0ℋ ) ) |
33 |
14 32
|
mt4d |
⊢ ( ( 𝐴 ∈ Cℋ ∧ ( ( 𝑥 ∈ 𝐴 ∧ 𝑥 ≠ 0ℎ ) ∧ ∀ 𝑦 ∈ Cℋ ( 𝑦 ⊆ 𝐴 → ( 𝑦 = 𝐴 ∨ 𝑦 = 0ℋ ) ) ) ) → ( ⊥ ‘ ( ⊥ ‘ { 𝑥 } ) ) = 𝐴 ) |
34 |
33
|
eqcomd |
⊢ ( ( 𝐴 ∈ Cℋ ∧ ( ( 𝑥 ∈ 𝐴 ∧ 𝑥 ≠ 0ℎ ) ∧ ∀ 𝑦 ∈ Cℋ ( 𝑦 ⊆ 𝐴 → ( 𝑦 = 𝐴 ∨ 𝑦 = 0ℋ ) ) ) ) → 𝐴 = ( ⊥ ‘ ( ⊥ ‘ { 𝑥 } ) ) ) |
35 |
|
rspe |
⊢ ( ( 𝑥 ∈ ℋ ∧ ( 𝑥 ≠ 0ℎ ∧ 𝐴 = ( ⊥ ‘ ( ⊥ ‘ { 𝑥 } ) ) ) ) → ∃ 𝑥 ∈ ℋ ( 𝑥 ≠ 0ℎ ∧ 𝐴 = ( ⊥ ‘ ( ⊥ ‘ { 𝑥 } ) ) ) ) |
36 |
9 10 34 35
|
syl12anc |
⊢ ( ( 𝐴 ∈ Cℋ ∧ ( ( 𝑥 ∈ 𝐴 ∧ 𝑥 ≠ 0ℎ ) ∧ ∀ 𝑦 ∈ Cℋ ( 𝑦 ⊆ 𝐴 → ( 𝑦 = 𝐴 ∨ 𝑦 = 0ℋ ) ) ) ) → ∃ 𝑥 ∈ ℋ ( 𝑥 ≠ 0ℎ ∧ 𝐴 = ( ⊥ ‘ ( ⊥ ‘ { 𝑥 } ) ) ) ) |
37 |
36
|
exp44 |
⊢ ( 𝐴 ∈ Cℋ → ( 𝑥 ∈ 𝐴 → ( 𝑥 ≠ 0ℎ → ( ∀ 𝑦 ∈ Cℋ ( 𝑦 ⊆ 𝐴 → ( 𝑦 = 𝐴 ∨ 𝑦 = 0ℋ ) ) → ∃ 𝑥 ∈ ℋ ( 𝑥 ≠ 0ℎ ∧ 𝐴 = ( ⊥ ‘ ( ⊥ ‘ { 𝑥 } ) ) ) ) ) ) ) |
38 |
3 6 37
|
rexlimd |
⊢ ( 𝐴 ∈ Cℋ → ( ∃ 𝑥 ∈ 𝐴 𝑥 ≠ 0ℎ → ( ∀ 𝑦 ∈ Cℋ ( 𝑦 ⊆ 𝐴 → ( 𝑦 = 𝐴 ∨ 𝑦 = 0ℋ ) ) → ∃ 𝑥 ∈ ℋ ( 𝑥 ≠ 0ℎ ∧ 𝐴 = ( ⊥ ‘ ( ⊥ ‘ { 𝑥 } ) ) ) ) ) ) |
39 |
2 38
|
sylbid |
⊢ ( 𝐴 ∈ Cℋ → ( 𝐴 ≠ 0ℋ → ( ∀ 𝑦 ∈ Cℋ ( 𝑦 ⊆ 𝐴 → ( 𝑦 = 𝐴 ∨ 𝑦 = 0ℋ ) ) → ∃ 𝑥 ∈ ℋ ( 𝑥 ≠ 0ℎ ∧ 𝐴 = ( ⊥ ‘ ( ⊥ ‘ { 𝑥 } ) ) ) ) ) ) |
40 |
39
|
imp32 |
⊢ ( ( 𝐴 ∈ Cℋ ∧ ( 𝐴 ≠ 0ℋ ∧ ∀ 𝑦 ∈ Cℋ ( 𝑦 ⊆ 𝐴 → ( 𝑦 = 𝐴 ∨ 𝑦 = 0ℋ ) ) ) ) → ∃ 𝑥 ∈ ℋ ( 𝑥 ≠ 0ℎ ∧ 𝐴 = ( ⊥ ‘ ( ⊥ ‘ { 𝑥 } ) ) ) ) |
41 |
1 40
|
sylbi |
⊢ ( 𝐴 ∈ HAtoms → ∃ 𝑥 ∈ ℋ ( 𝑥 ≠ 0ℎ ∧ 𝐴 = ( ⊥ ‘ ( ⊥ ‘ { 𝑥 } ) ) ) ) |
42 |
|
h1da |
⊢ ( ( 𝑥 ∈ ℋ ∧ 𝑥 ≠ 0ℎ ) → ( ⊥ ‘ ( ⊥ ‘ { 𝑥 } ) ) ∈ HAtoms ) |
43 |
|
eleq1 |
⊢ ( 𝐴 = ( ⊥ ‘ ( ⊥ ‘ { 𝑥 } ) ) → ( 𝐴 ∈ HAtoms ↔ ( ⊥ ‘ ( ⊥ ‘ { 𝑥 } ) ) ∈ HAtoms ) ) |
44 |
42 43
|
syl5ibr |
⊢ ( 𝐴 = ( ⊥ ‘ ( ⊥ ‘ { 𝑥 } ) ) → ( ( 𝑥 ∈ ℋ ∧ 𝑥 ≠ 0ℎ ) → 𝐴 ∈ HAtoms ) ) |
45 |
44
|
expdcom |
⊢ ( 𝑥 ∈ ℋ → ( 𝑥 ≠ 0ℎ → ( 𝐴 = ( ⊥ ‘ ( ⊥ ‘ { 𝑥 } ) ) → 𝐴 ∈ HAtoms ) ) ) |
46 |
45
|
impd |
⊢ ( 𝑥 ∈ ℋ → ( ( 𝑥 ≠ 0ℎ ∧ 𝐴 = ( ⊥ ‘ ( ⊥ ‘ { 𝑥 } ) ) ) → 𝐴 ∈ HAtoms ) ) |
47 |
46
|
rexlimiv |
⊢ ( ∃ 𝑥 ∈ ℋ ( 𝑥 ≠ 0ℎ ∧ 𝐴 = ( ⊥ ‘ ( ⊥ ‘ { 𝑥 } ) ) ) → 𝐴 ∈ HAtoms ) |
48 |
41 47
|
impbii |
⊢ ( 𝐴 ∈ HAtoms ↔ ∃ 𝑥 ∈ ℋ ( 𝑥 ≠ 0ℎ ∧ 𝐴 = ( ⊥ ‘ ( ⊥ ‘ { 𝑥 } ) ) ) ) |
49 |
|
spansn |
⊢ ( 𝑥 ∈ ℋ → ( span ‘ { 𝑥 } ) = ( ⊥ ‘ ( ⊥ ‘ { 𝑥 } ) ) ) |
50 |
49
|
eqeq2d |
⊢ ( 𝑥 ∈ ℋ → ( 𝐴 = ( span ‘ { 𝑥 } ) ↔ 𝐴 = ( ⊥ ‘ ( ⊥ ‘ { 𝑥 } ) ) ) ) |
51 |
50
|
anbi2d |
⊢ ( 𝑥 ∈ ℋ → ( ( 𝑥 ≠ 0ℎ ∧ 𝐴 = ( span ‘ { 𝑥 } ) ) ↔ ( 𝑥 ≠ 0ℎ ∧ 𝐴 = ( ⊥ ‘ ( ⊥ ‘ { 𝑥 } ) ) ) ) ) |
52 |
51
|
rexbiia |
⊢ ( ∃ 𝑥 ∈ ℋ ( 𝑥 ≠ 0ℎ ∧ 𝐴 = ( span ‘ { 𝑥 } ) ) ↔ ∃ 𝑥 ∈ ℋ ( 𝑥 ≠ 0ℎ ∧ 𝐴 = ( ⊥ ‘ ( ⊥ ‘ { 𝑥 } ) ) ) ) |
53 |
48 52
|
bitr4i |
⊢ ( 𝐴 ∈ HAtoms ↔ ∃ 𝑥 ∈ ℋ ( 𝑥 ≠ 0ℎ ∧ 𝐴 = ( span ‘ { 𝑥 } ) ) ) |