Step |
Hyp |
Ref |
Expression |
1 |
|
lsslindf.u |
⊢ 𝑈 = ( LSubSp ‘ 𝑊 ) |
2 |
|
lsslindf.x |
⊢ 𝑋 = ( 𝑊 ↾s 𝑆 ) |
3 |
|
rellindf |
⊢ Rel LIndF |
4 |
3
|
brrelex1i |
⊢ ( 𝐹 LIndF 𝑋 → 𝐹 ∈ V ) |
5 |
4
|
a1i |
⊢ ( ( 𝑊 ∈ LMod ∧ 𝑆 ∈ 𝑈 ∧ ran 𝐹 ⊆ 𝑆 ) → ( 𝐹 LIndF 𝑋 → 𝐹 ∈ V ) ) |
6 |
3
|
brrelex1i |
⊢ ( 𝐹 LIndF 𝑊 → 𝐹 ∈ V ) |
7 |
6
|
a1i |
⊢ ( ( 𝑊 ∈ LMod ∧ 𝑆 ∈ 𝑈 ∧ ran 𝐹 ⊆ 𝑆 ) → ( 𝐹 LIndF 𝑊 → 𝐹 ∈ V ) ) |
8 |
|
simpr |
⊢ ( ( ( 𝑊 ∈ LMod ∧ 𝑆 ∈ 𝑈 ∧ ran 𝐹 ⊆ 𝑆 ) ∧ 𝐹 : dom 𝐹 ⟶ ( Base ‘ 𝑋 ) ) → 𝐹 : dom 𝐹 ⟶ ( Base ‘ 𝑋 ) ) |
9 |
|
eqid |
⊢ ( Base ‘ 𝑊 ) = ( Base ‘ 𝑊 ) |
10 |
2 9
|
ressbasss |
⊢ ( Base ‘ 𝑋 ) ⊆ ( Base ‘ 𝑊 ) |
11 |
|
fss |
⊢ ( ( 𝐹 : dom 𝐹 ⟶ ( Base ‘ 𝑋 ) ∧ ( Base ‘ 𝑋 ) ⊆ ( Base ‘ 𝑊 ) ) → 𝐹 : dom 𝐹 ⟶ ( Base ‘ 𝑊 ) ) |
12 |
8 10 11
|
sylancl |
⊢ ( ( ( 𝑊 ∈ LMod ∧ 𝑆 ∈ 𝑈 ∧ ran 𝐹 ⊆ 𝑆 ) ∧ 𝐹 : dom 𝐹 ⟶ ( Base ‘ 𝑋 ) ) → 𝐹 : dom 𝐹 ⟶ ( Base ‘ 𝑊 ) ) |
13 |
|
ffn |
⊢ ( 𝐹 : dom 𝐹 ⟶ ( Base ‘ 𝑊 ) → 𝐹 Fn dom 𝐹 ) |
14 |
13
|
adantl |
⊢ ( ( ( 𝑊 ∈ LMod ∧ 𝑆 ∈ 𝑈 ∧ ran 𝐹 ⊆ 𝑆 ) ∧ 𝐹 : dom 𝐹 ⟶ ( Base ‘ 𝑊 ) ) → 𝐹 Fn dom 𝐹 ) |
15 |
|
simp3 |
⊢ ( ( 𝑊 ∈ LMod ∧ 𝑆 ∈ 𝑈 ∧ ran 𝐹 ⊆ 𝑆 ) → ran 𝐹 ⊆ 𝑆 ) |
16 |
9 1
|
lssss |
⊢ ( 𝑆 ∈ 𝑈 → 𝑆 ⊆ ( Base ‘ 𝑊 ) ) |
17 |
16
|
3ad2ant2 |
⊢ ( ( 𝑊 ∈ LMod ∧ 𝑆 ∈ 𝑈 ∧ ran 𝐹 ⊆ 𝑆 ) → 𝑆 ⊆ ( Base ‘ 𝑊 ) ) |
18 |
2 9
|
ressbas2 |
⊢ ( 𝑆 ⊆ ( Base ‘ 𝑊 ) → 𝑆 = ( Base ‘ 𝑋 ) ) |
19 |
17 18
|
syl |
⊢ ( ( 𝑊 ∈ LMod ∧ 𝑆 ∈ 𝑈 ∧ ran 𝐹 ⊆ 𝑆 ) → 𝑆 = ( Base ‘ 𝑋 ) ) |
20 |
15 19
|
sseqtrd |
⊢ ( ( 𝑊 ∈ LMod ∧ 𝑆 ∈ 𝑈 ∧ ran 𝐹 ⊆ 𝑆 ) → ran 𝐹 ⊆ ( Base ‘ 𝑋 ) ) |
21 |
20
|
adantr |
⊢ ( ( ( 𝑊 ∈ LMod ∧ 𝑆 ∈ 𝑈 ∧ ran 𝐹 ⊆ 𝑆 ) ∧ 𝐹 : dom 𝐹 ⟶ ( Base ‘ 𝑊 ) ) → ran 𝐹 ⊆ ( Base ‘ 𝑋 ) ) |
22 |
|
df-f |
⊢ ( 𝐹 : dom 𝐹 ⟶ ( Base ‘ 𝑋 ) ↔ ( 𝐹 Fn dom 𝐹 ∧ ran 𝐹 ⊆ ( Base ‘ 𝑋 ) ) ) |
23 |
14 21 22
|
sylanbrc |
⊢ ( ( ( 𝑊 ∈ LMod ∧ 𝑆 ∈ 𝑈 ∧ ran 𝐹 ⊆ 𝑆 ) ∧ 𝐹 : dom 𝐹 ⟶ ( Base ‘ 𝑊 ) ) → 𝐹 : dom 𝐹 ⟶ ( Base ‘ 𝑋 ) ) |
24 |
12 23
|
impbida |
⊢ ( ( 𝑊 ∈ LMod ∧ 𝑆 ∈ 𝑈 ∧ ran 𝐹 ⊆ 𝑆 ) → ( 𝐹 : dom 𝐹 ⟶ ( Base ‘ 𝑋 ) ↔ 𝐹 : dom 𝐹 ⟶ ( Base ‘ 𝑊 ) ) ) |
25 |
24
|
adantr |
⊢ ( ( ( 𝑊 ∈ LMod ∧ 𝑆 ∈ 𝑈 ∧ ran 𝐹 ⊆ 𝑆 ) ∧ 𝐹 ∈ V ) → ( 𝐹 : dom 𝐹 ⟶ ( Base ‘ 𝑋 ) ↔ 𝐹 : dom 𝐹 ⟶ ( Base ‘ 𝑊 ) ) ) |
26 |
|
simpl2 |
⊢ ( ( ( 𝑊 ∈ LMod ∧ 𝑆 ∈ 𝑈 ∧ ran 𝐹 ⊆ 𝑆 ) ∧ 𝐹 ∈ V ) → 𝑆 ∈ 𝑈 ) |
27 |
|
eqid |
⊢ ( Scalar ‘ 𝑊 ) = ( Scalar ‘ 𝑊 ) |
28 |
2 27
|
resssca |
⊢ ( 𝑆 ∈ 𝑈 → ( Scalar ‘ 𝑊 ) = ( Scalar ‘ 𝑋 ) ) |
29 |
28
|
eqcomd |
⊢ ( 𝑆 ∈ 𝑈 → ( Scalar ‘ 𝑋 ) = ( Scalar ‘ 𝑊 ) ) |
30 |
26 29
|
syl |
⊢ ( ( ( 𝑊 ∈ LMod ∧ 𝑆 ∈ 𝑈 ∧ ran 𝐹 ⊆ 𝑆 ) ∧ 𝐹 ∈ V ) → ( Scalar ‘ 𝑋 ) = ( Scalar ‘ 𝑊 ) ) |
31 |
30
|
fveq2d |
⊢ ( ( ( 𝑊 ∈ LMod ∧ 𝑆 ∈ 𝑈 ∧ ran 𝐹 ⊆ 𝑆 ) ∧ 𝐹 ∈ V ) → ( Base ‘ ( Scalar ‘ 𝑋 ) ) = ( Base ‘ ( Scalar ‘ 𝑊 ) ) ) |
32 |
30
|
fveq2d |
⊢ ( ( ( 𝑊 ∈ LMod ∧ 𝑆 ∈ 𝑈 ∧ ran 𝐹 ⊆ 𝑆 ) ∧ 𝐹 ∈ V ) → ( 0g ‘ ( Scalar ‘ 𝑋 ) ) = ( 0g ‘ ( Scalar ‘ 𝑊 ) ) ) |
33 |
32
|
sneqd |
⊢ ( ( ( 𝑊 ∈ LMod ∧ 𝑆 ∈ 𝑈 ∧ ran 𝐹 ⊆ 𝑆 ) ∧ 𝐹 ∈ V ) → { ( 0g ‘ ( Scalar ‘ 𝑋 ) ) } = { ( 0g ‘ ( Scalar ‘ 𝑊 ) ) } ) |
34 |
31 33
|
difeq12d |
⊢ ( ( ( 𝑊 ∈ LMod ∧ 𝑆 ∈ 𝑈 ∧ ran 𝐹 ⊆ 𝑆 ) ∧ 𝐹 ∈ V ) → ( ( Base ‘ ( Scalar ‘ 𝑋 ) ) ∖ { ( 0g ‘ ( Scalar ‘ 𝑋 ) ) } ) = ( ( Base ‘ ( Scalar ‘ 𝑊 ) ) ∖ { ( 0g ‘ ( Scalar ‘ 𝑊 ) ) } ) ) |
35 |
|
eqid |
⊢ ( ·𝑠 ‘ 𝑊 ) = ( ·𝑠 ‘ 𝑊 ) |
36 |
2 35
|
ressvsca |
⊢ ( 𝑆 ∈ 𝑈 → ( ·𝑠 ‘ 𝑊 ) = ( ·𝑠 ‘ 𝑋 ) ) |
37 |
36
|
eqcomd |
⊢ ( 𝑆 ∈ 𝑈 → ( ·𝑠 ‘ 𝑋 ) = ( ·𝑠 ‘ 𝑊 ) ) |
38 |
26 37
|
syl |
⊢ ( ( ( 𝑊 ∈ LMod ∧ 𝑆 ∈ 𝑈 ∧ ran 𝐹 ⊆ 𝑆 ) ∧ 𝐹 ∈ V ) → ( ·𝑠 ‘ 𝑋 ) = ( ·𝑠 ‘ 𝑊 ) ) |
39 |
38
|
oveqd |
⊢ ( ( ( 𝑊 ∈ LMod ∧ 𝑆 ∈ 𝑈 ∧ ran 𝐹 ⊆ 𝑆 ) ∧ 𝐹 ∈ V ) → ( 𝑘 ( ·𝑠 ‘ 𝑋 ) ( 𝐹 ‘ 𝑥 ) ) = ( 𝑘 ( ·𝑠 ‘ 𝑊 ) ( 𝐹 ‘ 𝑥 ) ) ) |
40 |
|
simpl1 |
⊢ ( ( ( 𝑊 ∈ LMod ∧ 𝑆 ∈ 𝑈 ∧ ran 𝐹 ⊆ 𝑆 ) ∧ 𝐹 ∈ V ) → 𝑊 ∈ LMod ) |
41 |
|
imassrn |
⊢ ( 𝐹 “ ( dom 𝐹 ∖ { 𝑥 } ) ) ⊆ ran 𝐹 |
42 |
|
simpl3 |
⊢ ( ( ( 𝑊 ∈ LMod ∧ 𝑆 ∈ 𝑈 ∧ ran 𝐹 ⊆ 𝑆 ) ∧ 𝐹 ∈ V ) → ran 𝐹 ⊆ 𝑆 ) |
43 |
41 42
|
sstrid |
⊢ ( ( ( 𝑊 ∈ LMod ∧ 𝑆 ∈ 𝑈 ∧ ran 𝐹 ⊆ 𝑆 ) ∧ 𝐹 ∈ V ) → ( 𝐹 “ ( dom 𝐹 ∖ { 𝑥 } ) ) ⊆ 𝑆 ) |
44 |
|
eqid |
⊢ ( LSpan ‘ 𝑊 ) = ( LSpan ‘ 𝑊 ) |
45 |
|
eqid |
⊢ ( LSpan ‘ 𝑋 ) = ( LSpan ‘ 𝑋 ) |
46 |
2 44 45 1
|
lsslsp |
⊢ ( ( 𝑊 ∈ LMod ∧ 𝑆 ∈ 𝑈 ∧ ( 𝐹 “ ( dom 𝐹 ∖ { 𝑥 } ) ) ⊆ 𝑆 ) → ( ( LSpan ‘ 𝑊 ) ‘ ( 𝐹 “ ( dom 𝐹 ∖ { 𝑥 } ) ) ) = ( ( LSpan ‘ 𝑋 ) ‘ ( 𝐹 “ ( dom 𝐹 ∖ { 𝑥 } ) ) ) ) |
47 |
40 26 43 46
|
syl3anc |
⊢ ( ( ( 𝑊 ∈ LMod ∧ 𝑆 ∈ 𝑈 ∧ ran 𝐹 ⊆ 𝑆 ) ∧ 𝐹 ∈ V ) → ( ( LSpan ‘ 𝑊 ) ‘ ( 𝐹 “ ( dom 𝐹 ∖ { 𝑥 } ) ) ) = ( ( LSpan ‘ 𝑋 ) ‘ ( 𝐹 “ ( dom 𝐹 ∖ { 𝑥 } ) ) ) ) |
48 |
47
|
eqcomd |
⊢ ( ( ( 𝑊 ∈ LMod ∧ 𝑆 ∈ 𝑈 ∧ ran 𝐹 ⊆ 𝑆 ) ∧ 𝐹 ∈ V ) → ( ( LSpan ‘ 𝑋 ) ‘ ( 𝐹 “ ( dom 𝐹 ∖ { 𝑥 } ) ) ) = ( ( LSpan ‘ 𝑊 ) ‘ ( 𝐹 “ ( dom 𝐹 ∖ { 𝑥 } ) ) ) ) |
49 |
39 48
|
eleq12d |
⊢ ( ( ( 𝑊 ∈ LMod ∧ 𝑆 ∈ 𝑈 ∧ ran 𝐹 ⊆ 𝑆 ) ∧ 𝐹 ∈ V ) → ( ( 𝑘 ( ·𝑠 ‘ 𝑋 ) ( 𝐹 ‘ 𝑥 ) ) ∈ ( ( LSpan ‘ 𝑋 ) ‘ ( 𝐹 “ ( dom 𝐹 ∖ { 𝑥 } ) ) ) ↔ ( 𝑘 ( ·𝑠 ‘ 𝑊 ) ( 𝐹 ‘ 𝑥 ) ) ∈ ( ( LSpan ‘ 𝑊 ) ‘ ( 𝐹 “ ( dom 𝐹 ∖ { 𝑥 } ) ) ) ) ) |
50 |
49
|
notbid |
⊢ ( ( ( 𝑊 ∈ LMod ∧ 𝑆 ∈ 𝑈 ∧ ran 𝐹 ⊆ 𝑆 ) ∧ 𝐹 ∈ V ) → ( ¬ ( 𝑘 ( ·𝑠 ‘ 𝑋 ) ( 𝐹 ‘ 𝑥 ) ) ∈ ( ( LSpan ‘ 𝑋 ) ‘ ( 𝐹 “ ( dom 𝐹 ∖ { 𝑥 } ) ) ) ↔ ¬ ( 𝑘 ( ·𝑠 ‘ 𝑊 ) ( 𝐹 ‘ 𝑥 ) ) ∈ ( ( LSpan ‘ 𝑊 ) ‘ ( 𝐹 “ ( dom 𝐹 ∖ { 𝑥 } ) ) ) ) ) |
51 |
34 50
|
raleqbidv |
⊢ ( ( ( 𝑊 ∈ LMod ∧ 𝑆 ∈ 𝑈 ∧ ran 𝐹 ⊆ 𝑆 ) ∧ 𝐹 ∈ V ) → ( ∀ 𝑘 ∈ ( ( Base ‘ ( Scalar ‘ 𝑋 ) ) ∖ { ( 0g ‘ ( Scalar ‘ 𝑋 ) ) } ) ¬ ( 𝑘 ( ·𝑠 ‘ 𝑋 ) ( 𝐹 ‘ 𝑥 ) ) ∈ ( ( LSpan ‘ 𝑋 ) ‘ ( 𝐹 “ ( dom 𝐹 ∖ { 𝑥 } ) ) ) ↔ ∀ 𝑘 ∈ ( ( Base ‘ ( Scalar ‘ 𝑊 ) ) ∖ { ( 0g ‘ ( Scalar ‘ 𝑊 ) ) } ) ¬ ( 𝑘 ( ·𝑠 ‘ 𝑊 ) ( 𝐹 ‘ 𝑥 ) ) ∈ ( ( LSpan ‘ 𝑊 ) ‘ ( 𝐹 “ ( dom 𝐹 ∖ { 𝑥 } ) ) ) ) ) |
52 |
51
|
ralbidv |
⊢ ( ( ( 𝑊 ∈ LMod ∧ 𝑆 ∈ 𝑈 ∧ ran 𝐹 ⊆ 𝑆 ) ∧ 𝐹 ∈ V ) → ( ∀ 𝑥 ∈ dom 𝐹 ∀ 𝑘 ∈ ( ( Base ‘ ( Scalar ‘ 𝑋 ) ) ∖ { ( 0g ‘ ( Scalar ‘ 𝑋 ) ) } ) ¬ ( 𝑘 ( ·𝑠 ‘ 𝑋 ) ( 𝐹 ‘ 𝑥 ) ) ∈ ( ( LSpan ‘ 𝑋 ) ‘ ( 𝐹 “ ( dom 𝐹 ∖ { 𝑥 } ) ) ) ↔ ∀ 𝑥 ∈ dom 𝐹 ∀ 𝑘 ∈ ( ( Base ‘ ( Scalar ‘ 𝑊 ) ) ∖ { ( 0g ‘ ( Scalar ‘ 𝑊 ) ) } ) ¬ ( 𝑘 ( ·𝑠 ‘ 𝑊 ) ( 𝐹 ‘ 𝑥 ) ) ∈ ( ( LSpan ‘ 𝑊 ) ‘ ( 𝐹 “ ( dom 𝐹 ∖ { 𝑥 } ) ) ) ) ) |
53 |
25 52
|
anbi12d |
⊢ ( ( ( 𝑊 ∈ LMod ∧ 𝑆 ∈ 𝑈 ∧ ran 𝐹 ⊆ 𝑆 ) ∧ 𝐹 ∈ V ) → ( ( 𝐹 : dom 𝐹 ⟶ ( Base ‘ 𝑋 ) ∧ ∀ 𝑥 ∈ dom 𝐹 ∀ 𝑘 ∈ ( ( Base ‘ ( Scalar ‘ 𝑋 ) ) ∖ { ( 0g ‘ ( Scalar ‘ 𝑋 ) ) } ) ¬ ( 𝑘 ( ·𝑠 ‘ 𝑋 ) ( 𝐹 ‘ 𝑥 ) ) ∈ ( ( LSpan ‘ 𝑋 ) ‘ ( 𝐹 “ ( dom 𝐹 ∖ { 𝑥 } ) ) ) ) ↔ ( 𝐹 : dom 𝐹 ⟶ ( Base ‘ 𝑊 ) ∧ ∀ 𝑥 ∈ dom 𝐹 ∀ 𝑘 ∈ ( ( Base ‘ ( Scalar ‘ 𝑊 ) ) ∖ { ( 0g ‘ ( Scalar ‘ 𝑊 ) ) } ) ¬ ( 𝑘 ( ·𝑠 ‘ 𝑊 ) ( 𝐹 ‘ 𝑥 ) ) ∈ ( ( LSpan ‘ 𝑊 ) ‘ ( 𝐹 “ ( dom 𝐹 ∖ { 𝑥 } ) ) ) ) ) ) |
54 |
2
|
ovexi |
⊢ 𝑋 ∈ V |
55 |
54
|
a1i |
⊢ ( ( 𝑊 ∈ LMod ∧ 𝑆 ∈ 𝑈 ∧ ran 𝐹 ⊆ 𝑆 ) → 𝑋 ∈ V ) |
56 |
|
eqid |
⊢ ( Base ‘ 𝑋 ) = ( Base ‘ 𝑋 ) |
57 |
|
eqid |
⊢ ( ·𝑠 ‘ 𝑋 ) = ( ·𝑠 ‘ 𝑋 ) |
58 |
|
eqid |
⊢ ( Scalar ‘ 𝑋 ) = ( Scalar ‘ 𝑋 ) |
59 |
|
eqid |
⊢ ( Base ‘ ( Scalar ‘ 𝑋 ) ) = ( Base ‘ ( Scalar ‘ 𝑋 ) ) |
60 |
|
eqid |
⊢ ( 0g ‘ ( Scalar ‘ 𝑋 ) ) = ( 0g ‘ ( Scalar ‘ 𝑋 ) ) |
61 |
56 57 45 58 59 60
|
islindf |
⊢ ( ( 𝑋 ∈ V ∧ 𝐹 ∈ V ) → ( 𝐹 LIndF 𝑋 ↔ ( 𝐹 : dom 𝐹 ⟶ ( Base ‘ 𝑋 ) ∧ ∀ 𝑥 ∈ dom 𝐹 ∀ 𝑘 ∈ ( ( Base ‘ ( Scalar ‘ 𝑋 ) ) ∖ { ( 0g ‘ ( Scalar ‘ 𝑋 ) ) } ) ¬ ( 𝑘 ( ·𝑠 ‘ 𝑋 ) ( 𝐹 ‘ 𝑥 ) ) ∈ ( ( LSpan ‘ 𝑋 ) ‘ ( 𝐹 “ ( dom 𝐹 ∖ { 𝑥 } ) ) ) ) ) ) |
62 |
55 61
|
sylan |
⊢ ( ( ( 𝑊 ∈ LMod ∧ 𝑆 ∈ 𝑈 ∧ ran 𝐹 ⊆ 𝑆 ) ∧ 𝐹 ∈ V ) → ( 𝐹 LIndF 𝑋 ↔ ( 𝐹 : dom 𝐹 ⟶ ( Base ‘ 𝑋 ) ∧ ∀ 𝑥 ∈ dom 𝐹 ∀ 𝑘 ∈ ( ( Base ‘ ( Scalar ‘ 𝑋 ) ) ∖ { ( 0g ‘ ( Scalar ‘ 𝑋 ) ) } ) ¬ ( 𝑘 ( ·𝑠 ‘ 𝑋 ) ( 𝐹 ‘ 𝑥 ) ) ∈ ( ( LSpan ‘ 𝑋 ) ‘ ( 𝐹 “ ( dom 𝐹 ∖ { 𝑥 } ) ) ) ) ) ) |
63 |
|
eqid |
⊢ ( Base ‘ ( Scalar ‘ 𝑊 ) ) = ( Base ‘ ( Scalar ‘ 𝑊 ) ) |
64 |
|
eqid |
⊢ ( 0g ‘ ( Scalar ‘ 𝑊 ) ) = ( 0g ‘ ( Scalar ‘ 𝑊 ) ) |
65 |
9 35 44 27 63 64
|
islindf |
⊢ ( ( 𝑊 ∈ LMod ∧ 𝐹 ∈ V ) → ( 𝐹 LIndF 𝑊 ↔ ( 𝐹 : dom 𝐹 ⟶ ( Base ‘ 𝑊 ) ∧ ∀ 𝑥 ∈ dom 𝐹 ∀ 𝑘 ∈ ( ( Base ‘ ( Scalar ‘ 𝑊 ) ) ∖ { ( 0g ‘ ( Scalar ‘ 𝑊 ) ) } ) ¬ ( 𝑘 ( ·𝑠 ‘ 𝑊 ) ( 𝐹 ‘ 𝑥 ) ) ∈ ( ( LSpan ‘ 𝑊 ) ‘ ( 𝐹 “ ( dom 𝐹 ∖ { 𝑥 } ) ) ) ) ) ) |
66 |
65
|
3ad2antl1 |
⊢ ( ( ( 𝑊 ∈ LMod ∧ 𝑆 ∈ 𝑈 ∧ ran 𝐹 ⊆ 𝑆 ) ∧ 𝐹 ∈ V ) → ( 𝐹 LIndF 𝑊 ↔ ( 𝐹 : dom 𝐹 ⟶ ( Base ‘ 𝑊 ) ∧ ∀ 𝑥 ∈ dom 𝐹 ∀ 𝑘 ∈ ( ( Base ‘ ( Scalar ‘ 𝑊 ) ) ∖ { ( 0g ‘ ( Scalar ‘ 𝑊 ) ) } ) ¬ ( 𝑘 ( ·𝑠 ‘ 𝑊 ) ( 𝐹 ‘ 𝑥 ) ) ∈ ( ( LSpan ‘ 𝑊 ) ‘ ( 𝐹 “ ( dom 𝐹 ∖ { 𝑥 } ) ) ) ) ) ) |
67 |
53 62 66
|
3bitr4d |
⊢ ( ( ( 𝑊 ∈ LMod ∧ 𝑆 ∈ 𝑈 ∧ ran 𝐹 ⊆ 𝑆 ) ∧ 𝐹 ∈ V ) → ( 𝐹 LIndF 𝑋 ↔ 𝐹 LIndF 𝑊 ) ) |
68 |
67
|
ex |
⊢ ( ( 𝑊 ∈ LMod ∧ 𝑆 ∈ 𝑈 ∧ ran 𝐹 ⊆ 𝑆 ) → ( 𝐹 ∈ V → ( 𝐹 LIndF 𝑋 ↔ 𝐹 LIndF 𝑊 ) ) ) |
69 |
5 7 68
|
pm5.21ndd |
⊢ ( ( 𝑊 ∈ LMod ∧ 𝑆 ∈ 𝑈 ∧ ran 𝐹 ⊆ 𝑆 ) → ( 𝐹 LIndF 𝑋 ↔ 𝐹 LIndF 𝑊 ) ) |