Step |
Hyp |
Ref |
Expression |
1 |
|
spanun.1 |
⊢ 𝐴 ⊆ ℋ |
2 |
|
spanun.2 |
⊢ 𝐵 ⊆ ℋ |
3 |
|
spancl |
⊢ ( 𝐴 ⊆ ℋ → ( span ‘ 𝐴 ) ∈ Sℋ ) |
4 |
1 3
|
ax-mp |
⊢ ( span ‘ 𝐴 ) ∈ Sℋ |
5 |
|
spancl |
⊢ ( 𝐵 ⊆ ℋ → ( span ‘ 𝐵 ) ∈ Sℋ ) |
6 |
2 5
|
ax-mp |
⊢ ( span ‘ 𝐵 ) ∈ Sℋ |
7 |
4 6
|
shscli |
⊢ ( ( span ‘ 𝐴 ) +ℋ ( span ‘ 𝐵 ) ) ∈ Sℋ |
8 |
7
|
shssii |
⊢ ( ( span ‘ 𝐴 ) +ℋ ( span ‘ 𝐵 ) ) ⊆ ℋ |
9 |
|
spanss2 |
⊢ ( 𝐴 ⊆ ℋ → 𝐴 ⊆ ( span ‘ 𝐴 ) ) |
10 |
1 9
|
ax-mp |
⊢ 𝐴 ⊆ ( span ‘ 𝐴 ) |
11 |
|
spanss2 |
⊢ ( 𝐵 ⊆ ℋ → 𝐵 ⊆ ( span ‘ 𝐵 ) ) |
12 |
2 11
|
ax-mp |
⊢ 𝐵 ⊆ ( span ‘ 𝐵 ) |
13 |
|
unss12 |
⊢ ( ( 𝐴 ⊆ ( span ‘ 𝐴 ) ∧ 𝐵 ⊆ ( span ‘ 𝐵 ) ) → ( 𝐴 ∪ 𝐵 ) ⊆ ( ( span ‘ 𝐴 ) ∪ ( span ‘ 𝐵 ) ) ) |
14 |
10 12 13
|
mp2an |
⊢ ( 𝐴 ∪ 𝐵 ) ⊆ ( ( span ‘ 𝐴 ) ∪ ( span ‘ 𝐵 ) ) |
15 |
4 6
|
shunssi |
⊢ ( ( span ‘ 𝐴 ) ∪ ( span ‘ 𝐵 ) ) ⊆ ( ( span ‘ 𝐴 ) +ℋ ( span ‘ 𝐵 ) ) |
16 |
14 15
|
sstri |
⊢ ( 𝐴 ∪ 𝐵 ) ⊆ ( ( span ‘ 𝐴 ) +ℋ ( span ‘ 𝐵 ) ) |
17 |
|
spanss |
⊢ ( ( ( ( span ‘ 𝐴 ) +ℋ ( span ‘ 𝐵 ) ) ⊆ ℋ ∧ ( 𝐴 ∪ 𝐵 ) ⊆ ( ( span ‘ 𝐴 ) +ℋ ( span ‘ 𝐵 ) ) ) → ( span ‘ ( 𝐴 ∪ 𝐵 ) ) ⊆ ( span ‘ ( ( span ‘ 𝐴 ) +ℋ ( span ‘ 𝐵 ) ) ) ) |
18 |
8 16 17
|
mp2an |
⊢ ( span ‘ ( 𝐴 ∪ 𝐵 ) ) ⊆ ( span ‘ ( ( span ‘ 𝐴 ) +ℋ ( span ‘ 𝐵 ) ) ) |
19 |
|
spanid |
⊢ ( ( ( span ‘ 𝐴 ) +ℋ ( span ‘ 𝐵 ) ) ∈ Sℋ → ( span ‘ ( ( span ‘ 𝐴 ) +ℋ ( span ‘ 𝐵 ) ) ) = ( ( span ‘ 𝐴 ) +ℋ ( span ‘ 𝐵 ) ) ) |
20 |
7 19
|
ax-mp |
⊢ ( span ‘ ( ( span ‘ 𝐴 ) +ℋ ( span ‘ 𝐵 ) ) ) = ( ( span ‘ 𝐴 ) +ℋ ( span ‘ 𝐵 ) ) |
21 |
18 20
|
sseqtri |
⊢ ( span ‘ ( 𝐴 ∪ 𝐵 ) ) ⊆ ( ( span ‘ 𝐴 ) +ℋ ( span ‘ 𝐵 ) ) |
22 |
4 6
|
shseli |
⊢ ( 𝑥 ∈ ( ( span ‘ 𝐴 ) +ℋ ( span ‘ 𝐵 ) ) ↔ ∃ 𝑧 ∈ ( span ‘ 𝐴 ) ∃ 𝑤 ∈ ( span ‘ 𝐵 ) 𝑥 = ( 𝑧 +ℎ 𝑤 ) ) |
23 |
|
r2ex |
⊢ ( ∃ 𝑧 ∈ ( span ‘ 𝐴 ) ∃ 𝑤 ∈ ( span ‘ 𝐵 ) 𝑥 = ( 𝑧 +ℎ 𝑤 ) ↔ ∃ 𝑧 ∃ 𝑤 ( ( 𝑧 ∈ ( span ‘ 𝐴 ) ∧ 𝑤 ∈ ( span ‘ 𝐵 ) ) ∧ 𝑥 = ( 𝑧 +ℎ 𝑤 ) ) ) |
24 |
22 23
|
bitri |
⊢ ( 𝑥 ∈ ( ( span ‘ 𝐴 ) +ℋ ( span ‘ 𝐵 ) ) ↔ ∃ 𝑧 ∃ 𝑤 ( ( 𝑧 ∈ ( span ‘ 𝐴 ) ∧ 𝑤 ∈ ( span ‘ 𝐵 ) ) ∧ 𝑥 = ( 𝑧 +ℎ 𝑤 ) ) ) |
25 |
|
vex |
⊢ 𝑧 ∈ V |
26 |
25
|
elspani |
⊢ ( 𝐴 ⊆ ℋ → ( 𝑧 ∈ ( span ‘ 𝐴 ) ↔ ∀ 𝑦 ∈ Sℋ ( 𝐴 ⊆ 𝑦 → 𝑧 ∈ 𝑦 ) ) ) |
27 |
1 26
|
ax-mp |
⊢ ( 𝑧 ∈ ( span ‘ 𝐴 ) ↔ ∀ 𝑦 ∈ Sℋ ( 𝐴 ⊆ 𝑦 → 𝑧 ∈ 𝑦 ) ) |
28 |
|
vex |
⊢ 𝑤 ∈ V |
29 |
28
|
elspani |
⊢ ( 𝐵 ⊆ ℋ → ( 𝑤 ∈ ( span ‘ 𝐵 ) ↔ ∀ 𝑦 ∈ Sℋ ( 𝐵 ⊆ 𝑦 → 𝑤 ∈ 𝑦 ) ) ) |
30 |
2 29
|
ax-mp |
⊢ ( 𝑤 ∈ ( span ‘ 𝐵 ) ↔ ∀ 𝑦 ∈ Sℋ ( 𝐵 ⊆ 𝑦 → 𝑤 ∈ 𝑦 ) ) |
31 |
27 30
|
anbi12i |
⊢ ( ( 𝑧 ∈ ( span ‘ 𝐴 ) ∧ 𝑤 ∈ ( span ‘ 𝐵 ) ) ↔ ( ∀ 𝑦 ∈ Sℋ ( 𝐴 ⊆ 𝑦 → 𝑧 ∈ 𝑦 ) ∧ ∀ 𝑦 ∈ Sℋ ( 𝐵 ⊆ 𝑦 → 𝑤 ∈ 𝑦 ) ) ) |
32 |
|
r19.26 |
⊢ ( ∀ 𝑦 ∈ Sℋ ( ( 𝐴 ⊆ 𝑦 → 𝑧 ∈ 𝑦 ) ∧ ( 𝐵 ⊆ 𝑦 → 𝑤 ∈ 𝑦 ) ) ↔ ( ∀ 𝑦 ∈ Sℋ ( 𝐴 ⊆ 𝑦 → 𝑧 ∈ 𝑦 ) ∧ ∀ 𝑦 ∈ Sℋ ( 𝐵 ⊆ 𝑦 → 𝑤 ∈ 𝑦 ) ) ) |
33 |
31 32
|
bitr4i |
⊢ ( ( 𝑧 ∈ ( span ‘ 𝐴 ) ∧ 𝑤 ∈ ( span ‘ 𝐵 ) ) ↔ ∀ 𝑦 ∈ Sℋ ( ( 𝐴 ⊆ 𝑦 → 𝑧 ∈ 𝑦 ) ∧ ( 𝐵 ⊆ 𝑦 → 𝑤 ∈ 𝑦 ) ) ) |
34 |
|
r19.27v |
⊢ ( ( ∀ 𝑦 ∈ Sℋ ( ( 𝐴 ⊆ 𝑦 → 𝑧 ∈ 𝑦 ) ∧ ( 𝐵 ⊆ 𝑦 → 𝑤 ∈ 𝑦 ) ) ∧ 𝑥 = ( 𝑧 +ℎ 𝑤 ) ) → ∀ 𝑦 ∈ Sℋ ( ( ( 𝐴 ⊆ 𝑦 → 𝑧 ∈ 𝑦 ) ∧ ( 𝐵 ⊆ 𝑦 → 𝑤 ∈ 𝑦 ) ) ∧ 𝑥 = ( 𝑧 +ℎ 𝑤 ) ) ) |
35 |
33 34
|
sylanb |
⊢ ( ( ( 𝑧 ∈ ( span ‘ 𝐴 ) ∧ 𝑤 ∈ ( span ‘ 𝐵 ) ) ∧ 𝑥 = ( 𝑧 +ℎ 𝑤 ) ) → ∀ 𝑦 ∈ Sℋ ( ( ( 𝐴 ⊆ 𝑦 → 𝑧 ∈ 𝑦 ) ∧ ( 𝐵 ⊆ 𝑦 → 𝑤 ∈ 𝑦 ) ) ∧ 𝑥 = ( 𝑧 +ℎ 𝑤 ) ) ) |
36 |
|
unss |
⊢ ( ( 𝐴 ⊆ 𝑦 ∧ 𝐵 ⊆ 𝑦 ) ↔ ( 𝐴 ∪ 𝐵 ) ⊆ 𝑦 ) |
37 |
|
anim12 |
⊢ ( ( ( 𝐴 ⊆ 𝑦 → 𝑧 ∈ 𝑦 ) ∧ ( 𝐵 ⊆ 𝑦 → 𝑤 ∈ 𝑦 ) ) → ( ( 𝐴 ⊆ 𝑦 ∧ 𝐵 ⊆ 𝑦 ) → ( 𝑧 ∈ 𝑦 ∧ 𝑤 ∈ 𝑦 ) ) ) |
38 |
36 37
|
syl5bir |
⊢ ( ( ( 𝐴 ⊆ 𝑦 → 𝑧 ∈ 𝑦 ) ∧ ( 𝐵 ⊆ 𝑦 → 𝑤 ∈ 𝑦 ) ) → ( ( 𝐴 ∪ 𝐵 ) ⊆ 𝑦 → ( 𝑧 ∈ 𝑦 ∧ 𝑤 ∈ 𝑦 ) ) ) |
39 |
|
shaddcl |
⊢ ( ( 𝑦 ∈ Sℋ ∧ 𝑧 ∈ 𝑦 ∧ 𝑤 ∈ 𝑦 ) → ( 𝑧 +ℎ 𝑤 ) ∈ 𝑦 ) |
40 |
39
|
3expib |
⊢ ( 𝑦 ∈ Sℋ → ( ( 𝑧 ∈ 𝑦 ∧ 𝑤 ∈ 𝑦 ) → ( 𝑧 +ℎ 𝑤 ) ∈ 𝑦 ) ) |
41 |
38 40
|
sylan9r |
⊢ ( ( 𝑦 ∈ Sℋ ∧ ( ( 𝐴 ⊆ 𝑦 → 𝑧 ∈ 𝑦 ) ∧ ( 𝐵 ⊆ 𝑦 → 𝑤 ∈ 𝑦 ) ) ) → ( ( 𝐴 ∪ 𝐵 ) ⊆ 𝑦 → ( 𝑧 +ℎ 𝑤 ) ∈ 𝑦 ) ) |
42 |
|
eleq1 |
⊢ ( 𝑥 = ( 𝑧 +ℎ 𝑤 ) → ( 𝑥 ∈ 𝑦 ↔ ( 𝑧 +ℎ 𝑤 ) ∈ 𝑦 ) ) |
43 |
42
|
biimprd |
⊢ ( 𝑥 = ( 𝑧 +ℎ 𝑤 ) → ( ( 𝑧 +ℎ 𝑤 ) ∈ 𝑦 → 𝑥 ∈ 𝑦 ) ) |
44 |
41 43
|
sylan9 |
⊢ ( ( ( 𝑦 ∈ Sℋ ∧ ( ( 𝐴 ⊆ 𝑦 → 𝑧 ∈ 𝑦 ) ∧ ( 𝐵 ⊆ 𝑦 → 𝑤 ∈ 𝑦 ) ) ) ∧ 𝑥 = ( 𝑧 +ℎ 𝑤 ) ) → ( ( 𝐴 ∪ 𝐵 ) ⊆ 𝑦 → 𝑥 ∈ 𝑦 ) ) |
45 |
44
|
expl |
⊢ ( 𝑦 ∈ Sℋ → ( ( ( ( 𝐴 ⊆ 𝑦 → 𝑧 ∈ 𝑦 ) ∧ ( 𝐵 ⊆ 𝑦 → 𝑤 ∈ 𝑦 ) ) ∧ 𝑥 = ( 𝑧 +ℎ 𝑤 ) ) → ( ( 𝐴 ∪ 𝐵 ) ⊆ 𝑦 → 𝑥 ∈ 𝑦 ) ) ) |
46 |
45
|
ralimia |
⊢ ( ∀ 𝑦 ∈ Sℋ ( ( ( 𝐴 ⊆ 𝑦 → 𝑧 ∈ 𝑦 ) ∧ ( 𝐵 ⊆ 𝑦 → 𝑤 ∈ 𝑦 ) ) ∧ 𝑥 = ( 𝑧 +ℎ 𝑤 ) ) → ∀ 𝑦 ∈ Sℋ ( ( 𝐴 ∪ 𝐵 ) ⊆ 𝑦 → 𝑥 ∈ 𝑦 ) ) |
47 |
1 2
|
unssi |
⊢ ( 𝐴 ∪ 𝐵 ) ⊆ ℋ |
48 |
|
vex |
⊢ 𝑥 ∈ V |
49 |
48
|
elspani |
⊢ ( ( 𝐴 ∪ 𝐵 ) ⊆ ℋ → ( 𝑥 ∈ ( span ‘ ( 𝐴 ∪ 𝐵 ) ) ↔ ∀ 𝑦 ∈ Sℋ ( ( 𝐴 ∪ 𝐵 ) ⊆ 𝑦 → 𝑥 ∈ 𝑦 ) ) ) |
50 |
47 49
|
ax-mp |
⊢ ( 𝑥 ∈ ( span ‘ ( 𝐴 ∪ 𝐵 ) ) ↔ ∀ 𝑦 ∈ Sℋ ( ( 𝐴 ∪ 𝐵 ) ⊆ 𝑦 → 𝑥 ∈ 𝑦 ) ) |
51 |
46 50
|
sylibr |
⊢ ( ∀ 𝑦 ∈ Sℋ ( ( ( 𝐴 ⊆ 𝑦 → 𝑧 ∈ 𝑦 ) ∧ ( 𝐵 ⊆ 𝑦 → 𝑤 ∈ 𝑦 ) ) ∧ 𝑥 = ( 𝑧 +ℎ 𝑤 ) ) → 𝑥 ∈ ( span ‘ ( 𝐴 ∪ 𝐵 ) ) ) |
52 |
35 51
|
syl |
⊢ ( ( ( 𝑧 ∈ ( span ‘ 𝐴 ) ∧ 𝑤 ∈ ( span ‘ 𝐵 ) ) ∧ 𝑥 = ( 𝑧 +ℎ 𝑤 ) ) → 𝑥 ∈ ( span ‘ ( 𝐴 ∪ 𝐵 ) ) ) |
53 |
52
|
exlimivv |
⊢ ( ∃ 𝑧 ∃ 𝑤 ( ( 𝑧 ∈ ( span ‘ 𝐴 ) ∧ 𝑤 ∈ ( span ‘ 𝐵 ) ) ∧ 𝑥 = ( 𝑧 +ℎ 𝑤 ) ) → 𝑥 ∈ ( span ‘ ( 𝐴 ∪ 𝐵 ) ) ) |
54 |
24 53
|
sylbi |
⊢ ( 𝑥 ∈ ( ( span ‘ 𝐴 ) +ℋ ( span ‘ 𝐵 ) ) → 𝑥 ∈ ( span ‘ ( 𝐴 ∪ 𝐵 ) ) ) |
55 |
54
|
ssriv |
⊢ ( ( span ‘ 𝐴 ) +ℋ ( span ‘ 𝐵 ) ) ⊆ ( span ‘ ( 𝐴 ∪ 𝐵 ) ) |
56 |
21 55
|
eqssi |
⊢ ( span ‘ ( 𝐴 ∪ 𝐵 ) ) = ( ( span ‘ 𝐴 ) +ℋ ( span ‘ 𝐵 ) ) |