Step |
Hyp |
Ref |
Expression |
1 |
|
lbspropd.b1 |
⊢ ( 𝜑 → 𝐵 = ( Base ‘ 𝐾 ) ) |
2 |
|
lbspropd.b2 |
⊢ ( 𝜑 → 𝐵 = ( Base ‘ 𝐿 ) ) |
3 |
|
lbspropd.w |
⊢ ( 𝜑 → 𝐵 ⊆ 𝑊 ) |
4 |
|
lbspropd.p |
⊢ ( ( 𝜑 ∧ ( 𝑥 ∈ 𝑊 ∧ 𝑦 ∈ 𝑊 ) ) → ( 𝑥 ( +g ‘ 𝐾 ) 𝑦 ) = ( 𝑥 ( +g ‘ 𝐿 ) 𝑦 ) ) |
5 |
|
lbspropd.s1 |
⊢ ( ( 𝜑 ∧ ( 𝑥 ∈ 𝑃 ∧ 𝑦 ∈ 𝐵 ) ) → ( 𝑥 ( ·𝑠 ‘ 𝐾 ) 𝑦 ) ∈ 𝑊 ) |
6 |
|
lbspropd.s2 |
⊢ ( ( 𝜑 ∧ ( 𝑥 ∈ 𝑃 ∧ 𝑦 ∈ 𝐵 ) ) → ( 𝑥 ( ·𝑠 ‘ 𝐾 ) 𝑦 ) = ( 𝑥 ( ·𝑠 ‘ 𝐿 ) 𝑦 ) ) |
7 |
|
lbspropd.f |
⊢ 𝐹 = ( Scalar ‘ 𝐾 ) |
8 |
|
lbspropd.g |
⊢ 𝐺 = ( Scalar ‘ 𝐿 ) |
9 |
|
lbspropd.p1 |
⊢ ( 𝜑 → 𝑃 = ( Base ‘ 𝐹 ) ) |
10 |
|
lbspropd.p2 |
⊢ ( 𝜑 → 𝑃 = ( Base ‘ 𝐺 ) ) |
11 |
|
lbspropd.a |
⊢ ( ( 𝜑 ∧ ( 𝑥 ∈ 𝑃 ∧ 𝑦 ∈ 𝑃 ) ) → ( 𝑥 ( +g ‘ 𝐹 ) 𝑦 ) = ( 𝑥 ( +g ‘ 𝐺 ) 𝑦 ) ) |
12 |
|
lbspropd.v1 |
⊢ ( 𝜑 → 𝐾 ∈ 𝑋 ) |
13 |
|
lbspropd.v2 |
⊢ ( 𝜑 → 𝐿 ∈ 𝑌 ) |
14 |
|
simplll |
⊢ ( ( ( ( 𝜑 ∧ 𝑧 ⊆ 𝐵 ) ∧ 𝑢 ∈ 𝑧 ) ∧ 𝑣 ∈ ( 𝑃 ∖ { ( 0g ‘ 𝐹 ) } ) ) → 𝜑 ) |
15 |
|
eldifi |
⊢ ( 𝑣 ∈ ( 𝑃 ∖ { ( 0g ‘ 𝐹 ) } ) → 𝑣 ∈ 𝑃 ) |
16 |
15
|
adantl |
⊢ ( ( ( ( 𝜑 ∧ 𝑧 ⊆ 𝐵 ) ∧ 𝑢 ∈ 𝑧 ) ∧ 𝑣 ∈ ( 𝑃 ∖ { ( 0g ‘ 𝐹 ) } ) ) → 𝑣 ∈ 𝑃 ) |
17 |
|
simpr |
⊢ ( ( 𝜑 ∧ 𝑧 ⊆ 𝐵 ) → 𝑧 ⊆ 𝐵 ) |
18 |
17
|
sselda |
⊢ ( ( ( 𝜑 ∧ 𝑧 ⊆ 𝐵 ) ∧ 𝑢 ∈ 𝑧 ) → 𝑢 ∈ 𝐵 ) |
19 |
18
|
adantr |
⊢ ( ( ( ( 𝜑 ∧ 𝑧 ⊆ 𝐵 ) ∧ 𝑢 ∈ 𝑧 ) ∧ 𝑣 ∈ ( 𝑃 ∖ { ( 0g ‘ 𝐹 ) } ) ) → 𝑢 ∈ 𝐵 ) |
20 |
6
|
oveqrspc2v |
⊢ ( ( 𝜑 ∧ ( 𝑣 ∈ 𝑃 ∧ 𝑢 ∈ 𝐵 ) ) → ( 𝑣 ( ·𝑠 ‘ 𝐾 ) 𝑢 ) = ( 𝑣 ( ·𝑠 ‘ 𝐿 ) 𝑢 ) ) |
21 |
14 16 19 20
|
syl12anc |
⊢ ( ( ( ( 𝜑 ∧ 𝑧 ⊆ 𝐵 ) ∧ 𝑢 ∈ 𝑧 ) ∧ 𝑣 ∈ ( 𝑃 ∖ { ( 0g ‘ 𝐹 ) } ) ) → ( 𝑣 ( ·𝑠 ‘ 𝐾 ) 𝑢 ) = ( 𝑣 ( ·𝑠 ‘ 𝐿 ) 𝑢 ) ) |
22 |
7
|
fveq2i |
⊢ ( Base ‘ 𝐹 ) = ( Base ‘ ( Scalar ‘ 𝐾 ) ) |
23 |
9 22
|
eqtrdi |
⊢ ( 𝜑 → 𝑃 = ( Base ‘ ( Scalar ‘ 𝐾 ) ) ) |
24 |
8
|
fveq2i |
⊢ ( Base ‘ 𝐺 ) = ( Base ‘ ( Scalar ‘ 𝐿 ) ) |
25 |
10 24
|
eqtrdi |
⊢ ( 𝜑 → 𝑃 = ( Base ‘ ( Scalar ‘ 𝐿 ) ) ) |
26 |
1 2 3 4 5 6 23 25 12 13
|
lsppropd |
⊢ ( 𝜑 → ( LSpan ‘ 𝐾 ) = ( LSpan ‘ 𝐿 ) ) |
27 |
14 26
|
syl |
⊢ ( ( ( ( 𝜑 ∧ 𝑧 ⊆ 𝐵 ) ∧ 𝑢 ∈ 𝑧 ) ∧ 𝑣 ∈ ( 𝑃 ∖ { ( 0g ‘ 𝐹 ) } ) ) → ( LSpan ‘ 𝐾 ) = ( LSpan ‘ 𝐿 ) ) |
28 |
27
|
fveq1d |
⊢ ( ( ( ( 𝜑 ∧ 𝑧 ⊆ 𝐵 ) ∧ 𝑢 ∈ 𝑧 ) ∧ 𝑣 ∈ ( 𝑃 ∖ { ( 0g ‘ 𝐹 ) } ) ) → ( ( LSpan ‘ 𝐾 ) ‘ ( 𝑧 ∖ { 𝑢 } ) ) = ( ( LSpan ‘ 𝐿 ) ‘ ( 𝑧 ∖ { 𝑢 } ) ) ) |
29 |
21 28
|
eleq12d |
⊢ ( ( ( ( 𝜑 ∧ 𝑧 ⊆ 𝐵 ) ∧ 𝑢 ∈ 𝑧 ) ∧ 𝑣 ∈ ( 𝑃 ∖ { ( 0g ‘ 𝐹 ) } ) ) → ( ( 𝑣 ( ·𝑠 ‘ 𝐾 ) 𝑢 ) ∈ ( ( LSpan ‘ 𝐾 ) ‘ ( 𝑧 ∖ { 𝑢 } ) ) ↔ ( 𝑣 ( ·𝑠 ‘ 𝐿 ) 𝑢 ) ∈ ( ( LSpan ‘ 𝐿 ) ‘ ( 𝑧 ∖ { 𝑢 } ) ) ) ) |
30 |
29
|
notbid |
⊢ ( ( ( ( 𝜑 ∧ 𝑧 ⊆ 𝐵 ) ∧ 𝑢 ∈ 𝑧 ) ∧ 𝑣 ∈ ( 𝑃 ∖ { ( 0g ‘ 𝐹 ) } ) ) → ( ¬ ( 𝑣 ( ·𝑠 ‘ 𝐾 ) 𝑢 ) ∈ ( ( LSpan ‘ 𝐾 ) ‘ ( 𝑧 ∖ { 𝑢 } ) ) ↔ ¬ ( 𝑣 ( ·𝑠 ‘ 𝐿 ) 𝑢 ) ∈ ( ( LSpan ‘ 𝐿 ) ‘ ( 𝑧 ∖ { 𝑢 } ) ) ) ) |
31 |
30
|
ralbidva |
⊢ ( ( ( 𝜑 ∧ 𝑧 ⊆ 𝐵 ) ∧ 𝑢 ∈ 𝑧 ) → ( ∀ 𝑣 ∈ ( 𝑃 ∖ { ( 0g ‘ 𝐹 ) } ) ¬ ( 𝑣 ( ·𝑠 ‘ 𝐾 ) 𝑢 ) ∈ ( ( LSpan ‘ 𝐾 ) ‘ ( 𝑧 ∖ { 𝑢 } ) ) ↔ ∀ 𝑣 ∈ ( 𝑃 ∖ { ( 0g ‘ 𝐹 ) } ) ¬ ( 𝑣 ( ·𝑠 ‘ 𝐿 ) 𝑢 ) ∈ ( ( LSpan ‘ 𝐿 ) ‘ ( 𝑧 ∖ { 𝑢 } ) ) ) ) |
32 |
9
|
ad2antrr |
⊢ ( ( ( 𝜑 ∧ 𝑧 ⊆ 𝐵 ) ∧ 𝑢 ∈ 𝑧 ) → 𝑃 = ( Base ‘ 𝐹 ) ) |
33 |
32
|
difeq1d |
⊢ ( ( ( 𝜑 ∧ 𝑧 ⊆ 𝐵 ) ∧ 𝑢 ∈ 𝑧 ) → ( 𝑃 ∖ { ( 0g ‘ 𝐹 ) } ) = ( ( Base ‘ 𝐹 ) ∖ { ( 0g ‘ 𝐹 ) } ) ) |
34 |
33
|
raleqdv |
⊢ ( ( ( 𝜑 ∧ 𝑧 ⊆ 𝐵 ) ∧ 𝑢 ∈ 𝑧 ) → ( ∀ 𝑣 ∈ ( 𝑃 ∖ { ( 0g ‘ 𝐹 ) } ) ¬ ( 𝑣 ( ·𝑠 ‘ 𝐾 ) 𝑢 ) ∈ ( ( LSpan ‘ 𝐾 ) ‘ ( 𝑧 ∖ { 𝑢 } ) ) ↔ ∀ 𝑣 ∈ ( ( Base ‘ 𝐹 ) ∖ { ( 0g ‘ 𝐹 ) } ) ¬ ( 𝑣 ( ·𝑠 ‘ 𝐾 ) 𝑢 ) ∈ ( ( LSpan ‘ 𝐾 ) ‘ ( 𝑧 ∖ { 𝑢 } ) ) ) ) |
35 |
10
|
ad2antrr |
⊢ ( ( ( 𝜑 ∧ 𝑧 ⊆ 𝐵 ) ∧ 𝑢 ∈ 𝑧 ) → 𝑃 = ( Base ‘ 𝐺 ) ) |
36 |
9 10 11
|
grpidpropd |
⊢ ( 𝜑 → ( 0g ‘ 𝐹 ) = ( 0g ‘ 𝐺 ) ) |
37 |
36
|
ad2antrr |
⊢ ( ( ( 𝜑 ∧ 𝑧 ⊆ 𝐵 ) ∧ 𝑢 ∈ 𝑧 ) → ( 0g ‘ 𝐹 ) = ( 0g ‘ 𝐺 ) ) |
38 |
37
|
sneqd |
⊢ ( ( ( 𝜑 ∧ 𝑧 ⊆ 𝐵 ) ∧ 𝑢 ∈ 𝑧 ) → { ( 0g ‘ 𝐹 ) } = { ( 0g ‘ 𝐺 ) } ) |
39 |
35 38
|
difeq12d |
⊢ ( ( ( 𝜑 ∧ 𝑧 ⊆ 𝐵 ) ∧ 𝑢 ∈ 𝑧 ) → ( 𝑃 ∖ { ( 0g ‘ 𝐹 ) } ) = ( ( Base ‘ 𝐺 ) ∖ { ( 0g ‘ 𝐺 ) } ) ) |
40 |
39
|
raleqdv |
⊢ ( ( ( 𝜑 ∧ 𝑧 ⊆ 𝐵 ) ∧ 𝑢 ∈ 𝑧 ) → ( ∀ 𝑣 ∈ ( 𝑃 ∖ { ( 0g ‘ 𝐹 ) } ) ¬ ( 𝑣 ( ·𝑠 ‘ 𝐿 ) 𝑢 ) ∈ ( ( LSpan ‘ 𝐿 ) ‘ ( 𝑧 ∖ { 𝑢 } ) ) ↔ ∀ 𝑣 ∈ ( ( Base ‘ 𝐺 ) ∖ { ( 0g ‘ 𝐺 ) } ) ¬ ( 𝑣 ( ·𝑠 ‘ 𝐿 ) 𝑢 ) ∈ ( ( LSpan ‘ 𝐿 ) ‘ ( 𝑧 ∖ { 𝑢 } ) ) ) ) |
41 |
31 34 40
|
3bitr3d |
⊢ ( ( ( 𝜑 ∧ 𝑧 ⊆ 𝐵 ) ∧ 𝑢 ∈ 𝑧 ) → ( ∀ 𝑣 ∈ ( ( Base ‘ 𝐹 ) ∖ { ( 0g ‘ 𝐹 ) } ) ¬ ( 𝑣 ( ·𝑠 ‘ 𝐾 ) 𝑢 ) ∈ ( ( LSpan ‘ 𝐾 ) ‘ ( 𝑧 ∖ { 𝑢 } ) ) ↔ ∀ 𝑣 ∈ ( ( Base ‘ 𝐺 ) ∖ { ( 0g ‘ 𝐺 ) } ) ¬ ( 𝑣 ( ·𝑠 ‘ 𝐿 ) 𝑢 ) ∈ ( ( LSpan ‘ 𝐿 ) ‘ ( 𝑧 ∖ { 𝑢 } ) ) ) ) |
42 |
41
|
ralbidva |
⊢ ( ( 𝜑 ∧ 𝑧 ⊆ 𝐵 ) → ( ∀ 𝑢 ∈ 𝑧 ∀ 𝑣 ∈ ( ( Base ‘ 𝐹 ) ∖ { ( 0g ‘ 𝐹 ) } ) ¬ ( 𝑣 ( ·𝑠 ‘ 𝐾 ) 𝑢 ) ∈ ( ( LSpan ‘ 𝐾 ) ‘ ( 𝑧 ∖ { 𝑢 } ) ) ↔ ∀ 𝑢 ∈ 𝑧 ∀ 𝑣 ∈ ( ( Base ‘ 𝐺 ) ∖ { ( 0g ‘ 𝐺 ) } ) ¬ ( 𝑣 ( ·𝑠 ‘ 𝐿 ) 𝑢 ) ∈ ( ( LSpan ‘ 𝐿 ) ‘ ( 𝑧 ∖ { 𝑢 } ) ) ) ) |
43 |
42
|
anbi2d |
⊢ ( ( 𝜑 ∧ 𝑧 ⊆ 𝐵 ) → ( ( ( ( LSpan ‘ 𝐾 ) ‘ 𝑧 ) = ( Base ‘ 𝐾 ) ∧ ∀ 𝑢 ∈ 𝑧 ∀ 𝑣 ∈ ( ( Base ‘ 𝐹 ) ∖ { ( 0g ‘ 𝐹 ) } ) ¬ ( 𝑣 ( ·𝑠 ‘ 𝐾 ) 𝑢 ) ∈ ( ( LSpan ‘ 𝐾 ) ‘ ( 𝑧 ∖ { 𝑢 } ) ) ) ↔ ( ( ( LSpan ‘ 𝐾 ) ‘ 𝑧 ) = ( Base ‘ 𝐾 ) ∧ ∀ 𝑢 ∈ 𝑧 ∀ 𝑣 ∈ ( ( Base ‘ 𝐺 ) ∖ { ( 0g ‘ 𝐺 ) } ) ¬ ( 𝑣 ( ·𝑠 ‘ 𝐿 ) 𝑢 ) ∈ ( ( LSpan ‘ 𝐿 ) ‘ ( 𝑧 ∖ { 𝑢 } ) ) ) ) ) |
44 |
43
|
pm5.32da |
⊢ ( 𝜑 → ( ( 𝑧 ⊆ 𝐵 ∧ ( ( ( LSpan ‘ 𝐾 ) ‘ 𝑧 ) = ( Base ‘ 𝐾 ) ∧ ∀ 𝑢 ∈ 𝑧 ∀ 𝑣 ∈ ( ( Base ‘ 𝐹 ) ∖ { ( 0g ‘ 𝐹 ) } ) ¬ ( 𝑣 ( ·𝑠 ‘ 𝐾 ) 𝑢 ) ∈ ( ( LSpan ‘ 𝐾 ) ‘ ( 𝑧 ∖ { 𝑢 } ) ) ) ) ↔ ( 𝑧 ⊆ 𝐵 ∧ ( ( ( LSpan ‘ 𝐾 ) ‘ 𝑧 ) = ( Base ‘ 𝐾 ) ∧ ∀ 𝑢 ∈ 𝑧 ∀ 𝑣 ∈ ( ( Base ‘ 𝐺 ) ∖ { ( 0g ‘ 𝐺 ) } ) ¬ ( 𝑣 ( ·𝑠 ‘ 𝐿 ) 𝑢 ) ∈ ( ( LSpan ‘ 𝐿 ) ‘ ( 𝑧 ∖ { 𝑢 } ) ) ) ) ) ) |
45 |
1
|
sseq2d |
⊢ ( 𝜑 → ( 𝑧 ⊆ 𝐵 ↔ 𝑧 ⊆ ( Base ‘ 𝐾 ) ) ) |
46 |
45
|
anbi1d |
⊢ ( 𝜑 → ( ( 𝑧 ⊆ 𝐵 ∧ ( ( ( LSpan ‘ 𝐾 ) ‘ 𝑧 ) = ( Base ‘ 𝐾 ) ∧ ∀ 𝑢 ∈ 𝑧 ∀ 𝑣 ∈ ( ( Base ‘ 𝐹 ) ∖ { ( 0g ‘ 𝐹 ) } ) ¬ ( 𝑣 ( ·𝑠 ‘ 𝐾 ) 𝑢 ) ∈ ( ( LSpan ‘ 𝐾 ) ‘ ( 𝑧 ∖ { 𝑢 } ) ) ) ) ↔ ( 𝑧 ⊆ ( Base ‘ 𝐾 ) ∧ ( ( ( LSpan ‘ 𝐾 ) ‘ 𝑧 ) = ( Base ‘ 𝐾 ) ∧ ∀ 𝑢 ∈ 𝑧 ∀ 𝑣 ∈ ( ( Base ‘ 𝐹 ) ∖ { ( 0g ‘ 𝐹 ) } ) ¬ ( 𝑣 ( ·𝑠 ‘ 𝐾 ) 𝑢 ) ∈ ( ( LSpan ‘ 𝐾 ) ‘ ( 𝑧 ∖ { 𝑢 } ) ) ) ) ) ) |
47 |
2
|
sseq2d |
⊢ ( 𝜑 → ( 𝑧 ⊆ 𝐵 ↔ 𝑧 ⊆ ( Base ‘ 𝐿 ) ) ) |
48 |
26
|
fveq1d |
⊢ ( 𝜑 → ( ( LSpan ‘ 𝐾 ) ‘ 𝑧 ) = ( ( LSpan ‘ 𝐿 ) ‘ 𝑧 ) ) |
49 |
1 2
|
eqtr3d |
⊢ ( 𝜑 → ( Base ‘ 𝐾 ) = ( Base ‘ 𝐿 ) ) |
50 |
48 49
|
eqeq12d |
⊢ ( 𝜑 → ( ( ( LSpan ‘ 𝐾 ) ‘ 𝑧 ) = ( Base ‘ 𝐾 ) ↔ ( ( LSpan ‘ 𝐿 ) ‘ 𝑧 ) = ( Base ‘ 𝐿 ) ) ) |
51 |
50
|
anbi1d |
⊢ ( 𝜑 → ( ( ( ( LSpan ‘ 𝐾 ) ‘ 𝑧 ) = ( Base ‘ 𝐾 ) ∧ ∀ 𝑢 ∈ 𝑧 ∀ 𝑣 ∈ ( ( Base ‘ 𝐺 ) ∖ { ( 0g ‘ 𝐺 ) } ) ¬ ( 𝑣 ( ·𝑠 ‘ 𝐿 ) 𝑢 ) ∈ ( ( LSpan ‘ 𝐿 ) ‘ ( 𝑧 ∖ { 𝑢 } ) ) ) ↔ ( ( ( LSpan ‘ 𝐿 ) ‘ 𝑧 ) = ( Base ‘ 𝐿 ) ∧ ∀ 𝑢 ∈ 𝑧 ∀ 𝑣 ∈ ( ( Base ‘ 𝐺 ) ∖ { ( 0g ‘ 𝐺 ) } ) ¬ ( 𝑣 ( ·𝑠 ‘ 𝐿 ) 𝑢 ) ∈ ( ( LSpan ‘ 𝐿 ) ‘ ( 𝑧 ∖ { 𝑢 } ) ) ) ) ) |
52 |
47 51
|
anbi12d |
⊢ ( 𝜑 → ( ( 𝑧 ⊆ 𝐵 ∧ ( ( ( LSpan ‘ 𝐾 ) ‘ 𝑧 ) = ( Base ‘ 𝐾 ) ∧ ∀ 𝑢 ∈ 𝑧 ∀ 𝑣 ∈ ( ( Base ‘ 𝐺 ) ∖ { ( 0g ‘ 𝐺 ) } ) ¬ ( 𝑣 ( ·𝑠 ‘ 𝐿 ) 𝑢 ) ∈ ( ( LSpan ‘ 𝐿 ) ‘ ( 𝑧 ∖ { 𝑢 } ) ) ) ) ↔ ( 𝑧 ⊆ ( Base ‘ 𝐿 ) ∧ ( ( ( LSpan ‘ 𝐿 ) ‘ 𝑧 ) = ( Base ‘ 𝐿 ) ∧ ∀ 𝑢 ∈ 𝑧 ∀ 𝑣 ∈ ( ( Base ‘ 𝐺 ) ∖ { ( 0g ‘ 𝐺 ) } ) ¬ ( 𝑣 ( ·𝑠 ‘ 𝐿 ) 𝑢 ) ∈ ( ( LSpan ‘ 𝐿 ) ‘ ( 𝑧 ∖ { 𝑢 } ) ) ) ) ) ) |
53 |
44 46 52
|
3bitr3d |
⊢ ( 𝜑 → ( ( 𝑧 ⊆ ( Base ‘ 𝐾 ) ∧ ( ( ( LSpan ‘ 𝐾 ) ‘ 𝑧 ) = ( Base ‘ 𝐾 ) ∧ ∀ 𝑢 ∈ 𝑧 ∀ 𝑣 ∈ ( ( Base ‘ 𝐹 ) ∖ { ( 0g ‘ 𝐹 ) } ) ¬ ( 𝑣 ( ·𝑠 ‘ 𝐾 ) 𝑢 ) ∈ ( ( LSpan ‘ 𝐾 ) ‘ ( 𝑧 ∖ { 𝑢 } ) ) ) ) ↔ ( 𝑧 ⊆ ( Base ‘ 𝐿 ) ∧ ( ( ( LSpan ‘ 𝐿 ) ‘ 𝑧 ) = ( Base ‘ 𝐿 ) ∧ ∀ 𝑢 ∈ 𝑧 ∀ 𝑣 ∈ ( ( Base ‘ 𝐺 ) ∖ { ( 0g ‘ 𝐺 ) } ) ¬ ( 𝑣 ( ·𝑠 ‘ 𝐿 ) 𝑢 ) ∈ ( ( LSpan ‘ 𝐿 ) ‘ ( 𝑧 ∖ { 𝑢 } ) ) ) ) ) ) |
54 |
|
3anass |
⊢ ( ( 𝑧 ⊆ ( Base ‘ 𝐾 ) ∧ ( ( LSpan ‘ 𝐾 ) ‘ 𝑧 ) = ( Base ‘ 𝐾 ) ∧ ∀ 𝑢 ∈ 𝑧 ∀ 𝑣 ∈ ( ( Base ‘ 𝐹 ) ∖ { ( 0g ‘ 𝐹 ) } ) ¬ ( 𝑣 ( ·𝑠 ‘ 𝐾 ) 𝑢 ) ∈ ( ( LSpan ‘ 𝐾 ) ‘ ( 𝑧 ∖ { 𝑢 } ) ) ) ↔ ( 𝑧 ⊆ ( Base ‘ 𝐾 ) ∧ ( ( ( LSpan ‘ 𝐾 ) ‘ 𝑧 ) = ( Base ‘ 𝐾 ) ∧ ∀ 𝑢 ∈ 𝑧 ∀ 𝑣 ∈ ( ( Base ‘ 𝐹 ) ∖ { ( 0g ‘ 𝐹 ) } ) ¬ ( 𝑣 ( ·𝑠 ‘ 𝐾 ) 𝑢 ) ∈ ( ( LSpan ‘ 𝐾 ) ‘ ( 𝑧 ∖ { 𝑢 } ) ) ) ) ) |
55 |
|
3anass |
⊢ ( ( 𝑧 ⊆ ( Base ‘ 𝐿 ) ∧ ( ( LSpan ‘ 𝐿 ) ‘ 𝑧 ) = ( Base ‘ 𝐿 ) ∧ ∀ 𝑢 ∈ 𝑧 ∀ 𝑣 ∈ ( ( Base ‘ 𝐺 ) ∖ { ( 0g ‘ 𝐺 ) } ) ¬ ( 𝑣 ( ·𝑠 ‘ 𝐿 ) 𝑢 ) ∈ ( ( LSpan ‘ 𝐿 ) ‘ ( 𝑧 ∖ { 𝑢 } ) ) ) ↔ ( 𝑧 ⊆ ( Base ‘ 𝐿 ) ∧ ( ( ( LSpan ‘ 𝐿 ) ‘ 𝑧 ) = ( Base ‘ 𝐿 ) ∧ ∀ 𝑢 ∈ 𝑧 ∀ 𝑣 ∈ ( ( Base ‘ 𝐺 ) ∖ { ( 0g ‘ 𝐺 ) } ) ¬ ( 𝑣 ( ·𝑠 ‘ 𝐿 ) 𝑢 ) ∈ ( ( LSpan ‘ 𝐿 ) ‘ ( 𝑧 ∖ { 𝑢 } ) ) ) ) ) |
56 |
53 54 55
|
3bitr4g |
⊢ ( 𝜑 → ( ( 𝑧 ⊆ ( Base ‘ 𝐾 ) ∧ ( ( LSpan ‘ 𝐾 ) ‘ 𝑧 ) = ( Base ‘ 𝐾 ) ∧ ∀ 𝑢 ∈ 𝑧 ∀ 𝑣 ∈ ( ( Base ‘ 𝐹 ) ∖ { ( 0g ‘ 𝐹 ) } ) ¬ ( 𝑣 ( ·𝑠 ‘ 𝐾 ) 𝑢 ) ∈ ( ( LSpan ‘ 𝐾 ) ‘ ( 𝑧 ∖ { 𝑢 } ) ) ) ↔ ( 𝑧 ⊆ ( Base ‘ 𝐿 ) ∧ ( ( LSpan ‘ 𝐿 ) ‘ 𝑧 ) = ( Base ‘ 𝐿 ) ∧ ∀ 𝑢 ∈ 𝑧 ∀ 𝑣 ∈ ( ( Base ‘ 𝐺 ) ∖ { ( 0g ‘ 𝐺 ) } ) ¬ ( 𝑣 ( ·𝑠 ‘ 𝐿 ) 𝑢 ) ∈ ( ( LSpan ‘ 𝐿 ) ‘ ( 𝑧 ∖ { 𝑢 } ) ) ) ) ) |
57 |
|
eqid |
⊢ ( Base ‘ 𝐾 ) = ( Base ‘ 𝐾 ) |
58 |
|
eqid |
⊢ ( ·𝑠 ‘ 𝐾 ) = ( ·𝑠 ‘ 𝐾 ) |
59 |
|
eqid |
⊢ ( Base ‘ 𝐹 ) = ( Base ‘ 𝐹 ) |
60 |
|
eqid |
⊢ ( LBasis ‘ 𝐾 ) = ( LBasis ‘ 𝐾 ) |
61 |
|
eqid |
⊢ ( LSpan ‘ 𝐾 ) = ( LSpan ‘ 𝐾 ) |
62 |
|
eqid |
⊢ ( 0g ‘ 𝐹 ) = ( 0g ‘ 𝐹 ) |
63 |
57 7 58 59 60 61 62
|
islbs |
⊢ ( 𝐾 ∈ 𝑋 → ( 𝑧 ∈ ( LBasis ‘ 𝐾 ) ↔ ( 𝑧 ⊆ ( Base ‘ 𝐾 ) ∧ ( ( LSpan ‘ 𝐾 ) ‘ 𝑧 ) = ( Base ‘ 𝐾 ) ∧ ∀ 𝑢 ∈ 𝑧 ∀ 𝑣 ∈ ( ( Base ‘ 𝐹 ) ∖ { ( 0g ‘ 𝐹 ) } ) ¬ ( 𝑣 ( ·𝑠 ‘ 𝐾 ) 𝑢 ) ∈ ( ( LSpan ‘ 𝐾 ) ‘ ( 𝑧 ∖ { 𝑢 } ) ) ) ) ) |
64 |
12 63
|
syl |
⊢ ( 𝜑 → ( 𝑧 ∈ ( LBasis ‘ 𝐾 ) ↔ ( 𝑧 ⊆ ( Base ‘ 𝐾 ) ∧ ( ( LSpan ‘ 𝐾 ) ‘ 𝑧 ) = ( Base ‘ 𝐾 ) ∧ ∀ 𝑢 ∈ 𝑧 ∀ 𝑣 ∈ ( ( Base ‘ 𝐹 ) ∖ { ( 0g ‘ 𝐹 ) } ) ¬ ( 𝑣 ( ·𝑠 ‘ 𝐾 ) 𝑢 ) ∈ ( ( LSpan ‘ 𝐾 ) ‘ ( 𝑧 ∖ { 𝑢 } ) ) ) ) ) |
65 |
|
eqid |
⊢ ( Base ‘ 𝐿 ) = ( Base ‘ 𝐿 ) |
66 |
|
eqid |
⊢ ( ·𝑠 ‘ 𝐿 ) = ( ·𝑠 ‘ 𝐿 ) |
67 |
|
eqid |
⊢ ( Base ‘ 𝐺 ) = ( Base ‘ 𝐺 ) |
68 |
|
eqid |
⊢ ( LBasis ‘ 𝐿 ) = ( LBasis ‘ 𝐿 ) |
69 |
|
eqid |
⊢ ( LSpan ‘ 𝐿 ) = ( LSpan ‘ 𝐿 ) |
70 |
|
eqid |
⊢ ( 0g ‘ 𝐺 ) = ( 0g ‘ 𝐺 ) |
71 |
65 8 66 67 68 69 70
|
islbs |
⊢ ( 𝐿 ∈ 𝑌 → ( 𝑧 ∈ ( LBasis ‘ 𝐿 ) ↔ ( 𝑧 ⊆ ( Base ‘ 𝐿 ) ∧ ( ( LSpan ‘ 𝐿 ) ‘ 𝑧 ) = ( Base ‘ 𝐿 ) ∧ ∀ 𝑢 ∈ 𝑧 ∀ 𝑣 ∈ ( ( Base ‘ 𝐺 ) ∖ { ( 0g ‘ 𝐺 ) } ) ¬ ( 𝑣 ( ·𝑠 ‘ 𝐿 ) 𝑢 ) ∈ ( ( LSpan ‘ 𝐿 ) ‘ ( 𝑧 ∖ { 𝑢 } ) ) ) ) ) |
72 |
13 71
|
syl |
⊢ ( 𝜑 → ( 𝑧 ∈ ( LBasis ‘ 𝐿 ) ↔ ( 𝑧 ⊆ ( Base ‘ 𝐿 ) ∧ ( ( LSpan ‘ 𝐿 ) ‘ 𝑧 ) = ( Base ‘ 𝐿 ) ∧ ∀ 𝑢 ∈ 𝑧 ∀ 𝑣 ∈ ( ( Base ‘ 𝐺 ) ∖ { ( 0g ‘ 𝐺 ) } ) ¬ ( 𝑣 ( ·𝑠 ‘ 𝐿 ) 𝑢 ) ∈ ( ( LSpan ‘ 𝐿 ) ‘ ( 𝑧 ∖ { 𝑢 } ) ) ) ) ) |
73 |
56 64 72
|
3bitr4d |
⊢ ( 𝜑 → ( 𝑧 ∈ ( LBasis ‘ 𝐾 ) ↔ 𝑧 ∈ ( LBasis ‘ 𝐿 ) ) ) |
74 |
73
|
eqrdv |
⊢ ( 𝜑 → ( LBasis ‘ 𝐾 ) = ( LBasis ‘ 𝐿 ) ) |