Proof of Theorem dochdmj1
Step | Hyp | Ref
| Expression |
1 | | simp1 1135 |
. . . 4
⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ 𝑋 ⊆ 𝑉 ∧ 𝑌 ⊆ 𝑉) → (𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻)) |
2 | | simp2 1136 |
. . . . 5
⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ 𝑋 ⊆ 𝑉 ∧ 𝑌 ⊆ 𝑉) → 𝑋 ⊆ 𝑉) |
3 | | simp3 1137 |
. . . . 5
⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ 𝑋 ⊆ 𝑉 ∧ 𝑌 ⊆ 𝑉) → 𝑌 ⊆ 𝑉) |
4 | 2, 3 | unssd 4125 |
. . . 4
⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ 𝑋 ⊆ 𝑉 ∧ 𝑌 ⊆ 𝑉) → (𝑋 ∪ 𝑌) ⊆ 𝑉) |
5 | | ssun1 4111 |
. . . . 5
⊢ 𝑋 ⊆ (𝑋 ∪ 𝑌) |
6 | 5 | a1i 11 |
. . . 4
⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ 𝑋 ⊆ 𝑉 ∧ 𝑌 ⊆ 𝑉) → 𝑋 ⊆ (𝑋 ∪ 𝑌)) |
7 | | dochdmj1.h |
. . . . 5
⊢ 𝐻 = (LHyp‘𝐾) |
8 | | dochdmj1.u |
. . . . 5
⊢ 𝑈 = ((DVecH‘𝐾)‘𝑊) |
9 | | dochdmj1.v |
. . . . 5
⊢ 𝑉 = (Base‘𝑈) |
10 | | dochdmj1.o |
. . . . 5
⊢ ⊥ =
((ocH‘𝐾)‘𝑊) |
11 | 7, 8, 9, 10 | dochss 39375 |
. . . 4
⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑋 ∪ 𝑌) ⊆ 𝑉 ∧ 𝑋 ⊆ (𝑋 ∪ 𝑌)) → ( ⊥ ‘(𝑋 ∪ 𝑌)) ⊆ ( ⊥ ‘𝑋)) |
12 | 1, 4, 6, 11 | syl3anc 1370 |
. . 3
⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ 𝑋 ⊆ 𝑉 ∧ 𝑌 ⊆ 𝑉) → ( ⊥ ‘(𝑋 ∪ 𝑌)) ⊆ ( ⊥ ‘𝑋)) |
13 | | ssun2 4112 |
. . . . 5
⊢ 𝑌 ⊆ (𝑋 ∪ 𝑌) |
14 | 13 | a1i 11 |
. . . 4
⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ 𝑋 ⊆ 𝑉 ∧ 𝑌 ⊆ 𝑉) → 𝑌 ⊆ (𝑋 ∪ 𝑌)) |
15 | 7, 8, 9, 10 | dochss 39375 |
. . . 4
⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑋 ∪ 𝑌) ⊆ 𝑉 ∧ 𝑌 ⊆ (𝑋 ∪ 𝑌)) → ( ⊥ ‘(𝑋 ∪ 𝑌)) ⊆ ( ⊥ ‘𝑌)) |
16 | 1, 4, 14, 15 | syl3anc 1370 |
. . 3
⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ 𝑋 ⊆ 𝑉 ∧ 𝑌 ⊆ 𝑉) → ( ⊥ ‘(𝑋 ∪ 𝑌)) ⊆ ( ⊥ ‘𝑌)) |
17 | 12, 16 | ssind 4172 |
. 2
⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ 𝑋 ⊆ 𝑉 ∧ 𝑌 ⊆ 𝑉) → ( ⊥ ‘(𝑋 ∪ 𝑌)) ⊆ (( ⊥ ‘𝑋) ∩ ( ⊥ ‘𝑌))) |
18 | | eqid 2740 |
. . . . . . 7
⊢
((DIsoH‘𝐾)‘𝑊) = ((DIsoH‘𝐾)‘𝑊) |
19 | 7, 18, 8, 9, 10 | dochcl 39363 |
. . . . . 6
⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ 𝑋 ⊆ 𝑉) → ( ⊥ ‘𝑋) ∈ ran ((DIsoH‘𝐾)‘𝑊)) |
20 | 19 | 3adant3 1131 |
. . . . 5
⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ 𝑋 ⊆ 𝑉 ∧ 𝑌 ⊆ 𝑉) → ( ⊥ ‘𝑋) ∈ ran ((DIsoH‘𝐾)‘𝑊)) |
21 | 7, 18, 8, 9, 10 | dochcl 39363 |
. . . . . 6
⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ 𝑌 ⊆ 𝑉) → ( ⊥ ‘𝑌) ∈ ran ((DIsoH‘𝐾)‘𝑊)) |
22 | 21 | 3adant2 1130 |
. . . . 5
⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ 𝑋 ⊆ 𝑉 ∧ 𝑌 ⊆ 𝑉) → ( ⊥ ‘𝑌) ∈ ran ((DIsoH‘𝐾)‘𝑊)) |
23 | 7, 18 | dihmeetcl 39355 |
. . . . 5
⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (( ⊥ ‘𝑋) ∈ ran ((DIsoH‘𝐾)‘𝑊) ∧ ( ⊥ ‘𝑌) ∈ ran ((DIsoH‘𝐾)‘𝑊))) → (( ⊥ ‘𝑋) ∩ ( ⊥ ‘𝑌)) ∈ ran
((DIsoH‘𝐾)‘𝑊)) |
24 | 1, 20, 22, 23 | syl12anc 834 |
. . . 4
⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ 𝑋 ⊆ 𝑉 ∧ 𝑌 ⊆ 𝑉) → (( ⊥ ‘𝑋) ∩ ( ⊥ ‘𝑌)) ∈ ran
((DIsoH‘𝐾)‘𝑊)) |
25 | 7, 18, 10 | dochoc 39377 |
. . . 4
⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (( ⊥ ‘𝑋) ∩ ( ⊥ ‘𝑌)) ∈ ran
((DIsoH‘𝐾)‘𝑊)) → ( ⊥ ‘( ⊥
‘(( ⊥ ‘𝑋) ∩ ( ⊥ ‘𝑌)))) = (( ⊥ ‘𝑋) ∩ ( ⊥ ‘𝑌))) |
26 | 1, 24, 25 | syl2anc 584 |
. . 3
⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ 𝑋 ⊆ 𝑉 ∧ 𝑌 ⊆ 𝑉) → ( ⊥ ‘( ⊥
‘(( ⊥ ‘𝑋) ∩ ( ⊥ ‘𝑌)))) = (( ⊥ ‘𝑋) ∩ ( ⊥ ‘𝑌))) |
27 | 7, 8, 9, 10 | dochssv 39365 |
. . . . . . 7
⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ 𝑋 ⊆ 𝑉) → ( ⊥ ‘𝑋) ⊆ 𝑉) |
28 | 27 | 3adant3 1131 |
. . . . . 6
⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ 𝑋 ⊆ 𝑉 ∧ 𝑌 ⊆ 𝑉) → ( ⊥ ‘𝑋) ⊆ 𝑉) |
29 | | ssinss1 4177 |
. . . . . 6
⊢ (( ⊥
‘𝑋) ⊆ 𝑉 → (( ⊥ ‘𝑋) ∩ ( ⊥ ‘𝑌)) ⊆ 𝑉) |
30 | 28, 29 | syl 17 |
. . . . 5
⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ 𝑋 ⊆ 𝑉 ∧ 𝑌 ⊆ 𝑉) → (( ⊥ ‘𝑋) ∩ ( ⊥ ‘𝑌)) ⊆ 𝑉) |
31 | 7, 8, 9, 10 | dochssv 39365 |
. . . . 5
⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (( ⊥ ‘𝑋) ∩ ( ⊥ ‘𝑌)) ⊆ 𝑉) → ( ⊥ ‘(( ⊥
‘𝑋) ∩ ( ⊥
‘𝑌))) ⊆ 𝑉) |
32 | 1, 30, 31 | syl2anc 584 |
. . . 4
⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ 𝑋 ⊆ 𝑉 ∧ 𝑌 ⊆ 𝑉) → ( ⊥ ‘(( ⊥
‘𝑋) ∩ ( ⊥
‘𝑌))) ⊆ 𝑉) |
33 | 7, 8, 9, 10 | dochocss 39376 |
. . . . . . 7
⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ 𝑋 ⊆ 𝑉) → 𝑋 ⊆ ( ⊥ ‘( ⊥
‘𝑋))) |
34 | 33 | 3adant3 1131 |
. . . . . 6
⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ 𝑋 ⊆ 𝑉 ∧ 𝑌 ⊆ 𝑉) → 𝑋 ⊆ ( ⊥ ‘( ⊥
‘𝑋))) |
35 | 7, 8, 9, 10 | dochocss 39376 |
. . . . . . 7
⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ 𝑌 ⊆ 𝑉) → 𝑌 ⊆ ( ⊥ ‘( ⊥
‘𝑌))) |
36 | 35 | 3adant2 1130 |
. . . . . 6
⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ 𝑋 ⊆ 𝑉 ∧ 𝑌 ⊆ 𝑉) → 𝑌 ⊆ ( ⊥ ‘( ⊥
‘𝑌))) |
37 | | unss12 4121 |
. . . . . 6
⊢ ((𝑋 ⊆ ( ⊥ ‘( ⊥
‘𝑋)) ∧ 𝑌 ⊆ ( ⊥ ‘( ⊥
‘𝑌))) → (𝑋 ∪ 𝑌) ⊆ (( ⊥ ‘( ⊥
‘𝑋)) ∪ ( ⊥
‘( ⊥ ‘𝑌)))) |
38 | 34, 36, 37 | syl2anc 584 |
. . . . 5
⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ 𝑋 ⊆ 𝑉 ∧ 𝑌 ⊆ 𝑉) → (𝑋 ∪ 𝑌) ⊆ (( ⊥ ‘( ⊥
‘𝑋)) ∪ ( ⊥
‘( ⊥ ‘𝑌)))) |
39 | | inss1 4168 |
. . . . . . . 8
⊢ (( ⊥
‘𝑋) ∩ ( ⊥
‘𝑌)) ⊆ ( ⊥
‘𝑋) |
40 | 39 | a1i 11 |
. . . . . . 7
⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ 𝑋 ⊆ 𝑉 ∧ 𝑌 ⊆ 𝑉) → (( ⊥ ‘𝑋) ∩ ( ⊥ ‘𝑌)) ⊆ ( ⊥ ‘𝑋)) |
41 | 7, 8, 9, 10 | dochss 39375 |
. . . . . . 7
⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ ( ⊥ ‘𝑋) ⊆ 𝑉 ∧ (( ⊥ ‘𝑋) ∩ ( ⊥ ‘𝑌)) ⊆ ( ⊥ ‘𝑋)) → ( ⊥ ‘( ⊥
‘𝑋)) ⊆ ( ⊥
‘(( ⊥ ‘𝑋) ∩ ( ⊥ ‘𝑌)))) |
42 | 1, 28, 40, 41 | syl3anc 1370 |
. . . . . 6
⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ 𝑋 ⊆ 𝑉 ∧ 𝑌 ⊆ 𝑉) → ( ⊥ ‘( ⊥
‘𝑋)) ⊆ ( ⊥
‘(( ⊥ ‘𝑋) ∩ ( ⊥ ‘𝑌)))) |
43 | 7, 8, 9, 10 | dochssv 39365 |
. . . . . . . 8
⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ 𝑌 ⊆ 𝑉) → ( ⊥ ‘𝑌) ⊆ 𝑉) |
44 | 43 | 3adant2 1130 |
. . . . . . 7
⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ 𝑋 ⊆ 𝑉 ∧ 𝑌 ⊆ 𝑉) → ( ⊥ ‘𝑌) ⊆ 𝑉) |
45 | | inss2 4169 |
. . . . . . . 8
⊢ (( ⊥
‘𝑋) ∩ ( ⊥
‘𝑌)) ⊆ ( ⊥
‘𝑌) |
46 | 45 | a1i 11 |
. . . . . . 7
⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ 𝑋 ⊆ 𝑉 ∧ 𝑌 ⊆ 𝑉) → (( ⊥ ‘𝑋) ∩ ( ⊥ ‘𝑌)) ⊆ ( ⊥ ‘𝑌)) |
47 | 7, 8, 9, 10 | dochss 39375 |
. . . . . . 7
⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ ( ⊥ ‘𝑌) ⊆ 𝑉 ∧ (( ⊥ ‘𝑋) ∩ ( ⊥ ‘𝑌)) ⊆ ( ⊥ ‘𝑌)) → ( ⊥ ‘( ⊥
‘𝑌)) ⊆ ( ⊥
‘(( ⊥ ‘𝑋) ∩ ( ⊥ ‘𝑌)))) |
48 | 1, 44, 46, 47 | syl3anc 1370 |
. . . . . 6
⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ 𝑋 ⊆ 𝑉 ∧ 𝑌 ⊆ 𝑉) → ( ⊥ ‘( ⊥
‘𝑌)) ⊆ ( ⊥
‘(( ⊥ ‘𝑋) ∩ ( ⊥ ‘𝑌)))) |
49 | 42, 48 | unssd 4125 |
. . . . 5
⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ 𝑋 ⊆ 𝑉 ∧ 𝑌 ⊆ 𝑉) → (( ⊥ ‘( ⊥
‘𝑋)) ∪ ( ⊥
‘( ⊥ ‘𝑌))) ⊆ ( ⊥ ‘(( ⊥
‘𝑋) ∩ ( ⊥
‘𝑌)))) |
50 | 38, 49 | sstrd 3936 |
. . . 4
⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ 𝑋 ⊆ 𝑉 ∧ 𝑌 ⊆ 𝑉) → (𝑋 ∪ 𝑌) ⊆ ( ⊥ ‘(( ⊥
‘𝑋) ∩ ( ⊥
‘𝑌)))) |
51 | 7, 8, 9, 10 | dochss 39375 |
. . . 4
⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ ( ⊥ ‘(( ⊥
‘𝑋) ∩ ( ⊥
‘𝑌))) ⊆ 𝑉 ∧ (𝑋 ∪ 𝑌) ⊆ ( ⊥ ‘(( ⊥
‘𝑋) ∩ ( ⊥
‘𝑌)))) → ( ⊥
‘( ⊥ ‘(( ⊥
‘𝑋) ∩ ( ⊥
‘𝑌)))) ⊆ (
⊥
‘(𝑋 ∪ 𝑌))) |
52 | 1, 32, 50, 51 | syl3anc 1370 |
. . 3
⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ 𝑋 ⊆ 𝑉 ∧ 𝑌 ⊆ 𝑉) → ( ⊥ ‘( ⊥
‘(( ⊥ ‘𝑋) ∩ ( ⊥ ‘𝑌)))) ⊆ ( ⊥
‘(𝑋 ∪ 𝑌))) |
53 | 26, 52 | eqsstrrd 3965 |
. 2
⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ 𝑋 ⊆ 𝑉 ∧ 𝑌 ⊆ 𝑉) → (( ⊥ ‘𝑋) ∩ ( ⊥ ‘𝑌)) ⊆ ( ⊥ ‘(𝑋 ∪ 𝑌))) |
54 | 17, 53 | eqssd 3943 |
1
⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ 𝑋 ⊆ 𝑉 ∧ 𝑌 ⊆ 𝑉) → ( ⊥ ‘(𝑋 ∪ 𝑌)) = (( ⊥ ‘𝑋) ∩ ( ⊥ ‘𝑌))) |