Step |
Hyp |
Ref |
Expression |
1 |
|
lindslinind.r |
⊢ 𝑅 = ( Scalar ‘ 𝑀 ) |
2 |
|
lindslinind.b |
⊢ 𝐵 = ( Base ‘ 𝑅 ) |
3 |
|
lindslinind.0 |
⊢ 0 = ( 0g ‘ 𝑅 ) |
4 |
|
lindslinind.z |
⊢ 𝑍 = ( 0g ‘ 𝑀 ) |
5 |
|
simprl |
⊢ ( ( ( 𝑆 ∈ 𝑉 ∧ 𝑀 ∈ LMod ) ∧ ( 𝑆 ⊆ ( Base ‘ 𝑀 ) ∧ ∀ 𝑠 ∈ 𝑆 ∀ 𝑦 ∈ ( 𝐵 ∖ { 0 } ) ¬ ( 𝑦 ( ·𝑠 ‘ 𝑀 ) 𝑠 ) ∈ ( ( LSpan ‘ 𝑀 ) ‘ ( 𝑆 ∖ { 𝑠 } ) ) ) ) → 𝑆 ⊆ ( Base ‘ 𝑀 ) ) |
6 |
|
elpwg |
⊢ ( 𝑆 ∈ 𝑉 → ( 𝑆 ∈ 𝒫 ( Base ‘ 𝑀 ) ↔ 𝑆 ⊆ ( Base ‘ 𝑀 ) ) ) |
7 |
6
|
ad2antrr |
⊢ ( ( ( 𝑆 ∈ 𝑉 ∧ 𝑀 ∈ LMod ) ∧ ( 𝑆 ⊆ ( Base ‘ 𝑀 ) ∧ ∀ 𝑠 ∈ 𝑆 ∀ 𝑦 ∈ ( 𝐵 ∖ { 0 } ) ¬ ( 𝑦 ( ·𝑠 ‘ 𝑀 ) 𝑠 ) ∈ ( ( LSpan ‘ 𝑀 ) ‘ ( 𝑆 ∖ { 𝑠 } ) ) ) ) → ( 𝑆 ∈ 𝒫 ( Base ‘ 𝑀 ) ↔ 𝑆 ⊆ ( Base ‘ 𝑀 ) ) ) |
8 |
5 7
|
mpbird |
⊢ ( ( ( 𝑆 ∈ 𝑉 ∧ 𝑀 ∈ LMod ) ∧ ( 𝑆 ⊆ ( Base ‘ 𝑀 ) ∧ ∀ 𝑠 ∈ 𝑆 ∀ 𝑦 ∈ ( 𝐵 ∖ { 0 } ) ¬ ( 𝑦 ( ·𝑠 ‘ 𝑀 ) 𝑠 ) ∈ ( ( LSpan ‘ 𝑀 ) ‘ ( 𝑆 ∖ { 𝑠 } ) ) ) ) → 𝑆 ∈ 𝒫 ( Base ‘ 𝑀 ) ) |
9 |
|
simplr |
⊢ ( ( ( 𝑆 ∈ 𝑉 ∧ 𝑀 ∈ LMod ) ∧ 𝑆 ⊆ ( Base ‘ 𝑀 ) ) → 𝑀 ∈ LMod ) |
10 |
|
ssdifss |
⊢ ( 𝑆 ⊆ ( Base ‘ 𝑀 ) → ( 𝑆 ∖ { 𝑠 } ) ⊆ ( Base ‘ 𝑀 ) ) |
11 |
10
|
adantl |
⊢ ( ( ( 𝑆 ∈ 𝑉 ∧ 𝑀 ∈ LMod ) ∧ 𝑆 ⊆ ( Base ‘ 𝑀 ) ) → ( 𝑆 ∖ { 𝑠 } ) ⊆ ( Base ‘ 𝑀 ) ) |
12 |
|
difexg |
⊢ ( 𝑆 ∈ 𝑉 → ( 𝑆 ∖ { 𝑠 } ) ∈ V ) |
13 |
12
|
ad2antrr |
⊢ ( ( ( 𝑆 ∈ 𝑉 ∧ 𝑀 ∈ LMod ) ∧ 𝑆 ⊆ ( Base ‘ 𝑀 ) ) → ( 𝑆 ∖ { 𝑠 } ) ∈ V ) |
14 |
|
elpwg |
⊢ ( ( 𝑆 ∖ { 𝑠 } ) ∈ V → ( ( 𝑆 ∖ { 𝑠 } ) ∈ 𝒫 ( Base ‘ 𝑀 ) ↔ ( 𝑆 ∖ { 𝑠 } ) ⊆ ( Base ‘ 𝑀 ) ) ) |
15 |
13 14
|
syl |
⊢ ( ( ( 𝑆 ∈ 𝑉 ∧ 𝑀 ∈ LMod ) ∧ 𝑆 ⊆ ( Base ‘ 𝑀 ) ) → ( ( 𝑆 ∖ { 𝑠 } ) ∈ 𝒫 ( Base ‘ 𝑀 ) ↔ ( 𝑆 ∖ { 𝑠 } ) ⊆ ( Base ‘ 𝑀 ) ) ) |
16 |
11 15
|
mpbird |
⊢ ( ( ( 𝑆 ∈ 𝑉 ∧ 𝑀 ∈ LMod ) ∧ 𝑆 ⊆ ( Base ‘ 𝑀 ) ) → ( 𝑆 ∖ { 𝑠 } ) ∈ 𝒫 ( Base ‘ 𝑀 ) ) |
17 |
|
eqid |
⊢ ( Base ‘ 𝑀 ) = ( Base ‘ 𝑀 ) |
18 |
17
|
lspeqlco |
⊢ ( ( 𝑀 ∈ LMod ∧ ( 𝑆 ∖ { 𝑠 } ) ∈ 𝒫 ( Base ‘ 𝑀 ) ) → ( 𝑀 LinCo ( 𝑆 ∖ { 𝑠 } ) ) = ( ( LSpan ‘ 𝑀 ) ‘ ( 𝑆 ∖ { 𝑠 } ) ) ) |
19 |
18
|
eleq2d |
⊢ ( ( 𝑀 ∈ LMod ∧ ( 𝑆 ∖ { 𝑠 } ) ∈ 𝒫 ( Base ‘ 𝑀 ) ) → ( ( 𝑦 ( ·𝑠 ‘ 𝑀 ) 𝑠 ) ∈ ( 𝑀 LinCo ( 𝑆 ∖ { 𝑠 } ) ) ↔ ( 𝑦 ( ·𝑠 ‘ 𝑀 ) 𝑠 ) ∈ ( ( LSpan ‘ 𝑀 ) ‘ ( 𝑆 ∖ { 𝑠 } ) ) ) ) |
20 |
19
|
bicomd |
⊢ ( ( 𝑀 ∈ LMod ∧ ( 𝑆 ∖ { 𝑠 } ) ∈ 𝒫 ( Base ‘ 𝑀 ) ) → ( ( 𝑦 ( ·𝑠 ‘ 𝑀 ) 𝑠 ) ∈ ( ( LSpan ‘ 𝑀 ) ‘ ( 𝑆 ∖ { 𝑠 } ) ) ↔ ( 𝑦 ( ·𝑠 ‘ 𝑀 ) 𝑠 ) ∈ ( 𝑀 LinCo ( 𝑆 ∖ { 𝑠 } ) ) ) ) |
21 |
9 16 20
|
syl2anc |
⊢ ( ( ( 𝑆 ∈ 𝑉 ∧ 𝑀 ∈ LMod ) ∧ 𝑆 ⊆ ( Base ‘ 𝑀 ) ) → ( ( 𝑦 ( ·𝑠 ‘ 𝑀 ) 𝑠 ) ∈ ( ( LSpan ‘ 𝑀 ) ‘ ( 𝑆 ∖ { 𝑠 } ) ) ↔ ( 𝑦 ( ·𝑠 ‘ 𝑀 ) 𝑠 ) ∈ ( 𝑀 LinCo ( 𝑆 ∖ { 𝑠 } ) ) ) ) |
22 |
21
|
notbid |
⊢ ( ( ( 𝑆 ∈ 𝑉 ∧ 𝑀 ∈ LMod ) ∧ 𝑆 ⊆ ( Base ‘ 𝑀 ) ) → ( ¬ ( 𝑦 ( ·𝑠 ‘ 𝑀 ) 𝑠 ) ∈ ( ( LSpan ‘ 𝑀 ) ‘ ( 𝑆 ∖ { 𝑠 } ) ) ↔ ¬ ( 𝑦 ( ·𝑠 ‘ 𝑀 ) 𝑠 ) ∈ ( 𝑀 LinCo ( 𝑆 ∖ { 𝑠 } ) ) ) ) |
23 |
17 1 2
|
lcoval |
⊢ ( ( 𝑀 ∈ LMod ∧ ( 𝑆 ∖ { 𝑠 } ) ∈ 𝒫 ( Base ‘ 𝑀 ) ) → ( ( 𝑦 ( ·𝑠 ‘ 𝑀 ) 𝑠 ) ∈ ( 𝑀 LinCo ( 𝑆 ∖ { 𝑠 } ) ) ↔ ( ( 𝑦 ( ·𝑠 ‘ 𝑀 ) 𝑠 ) ∈ ( Base ‘ 𝑀 ) ∧ ∃ 𝑔 ∈ ( 𝐵 ↑m ( 𝑆 ∖ { 𝑠 } ) ) ( 𝑔 finSupp ( 0g ‘ 𝑅 ) ∧ ( 𝑦 ( ·𝑠 ‘ 𝑀 ) 𝑠 ) = ( 𝑔 ( linC ‘ 𝑀 ) ( 𝑆 ∖ { 𝑠 } ) ) ) ) ) ) |
24 |
3
|
eqcomi |
⊢ ( 0g ‘ 𝑅 ) = 0 |
25 |
24
|
breq2i |
⊢ ( 𝑔 finSupp ( 0g ‘ 𝑅 ) ↔ 𝑔 finSupp 0 ) |
26 |
25
|
anbi1i |
⊢ ( ( 𝑔 finSupp ( 0g ‘ 𝑅 ) ∧ ( 𝑦 ( ·𝑠 ‘ 𝑀 ) 𝑠 ) = ( 𝑔 ( linC ‘ 𝑀 ) ( 𝑆 ∖ { 𝑠 } ) ) ) ↔ ( 𝑔 finSupp 0 ∧ ( 𝑦 ( ·𝑠 ‘ 𝑀 ) 𝑠 ) = ( 𝑔 ( linC ‘ 𝑀 ) ( 𝑆 ∖ { 𝑠 } ) ) ) ) |
27 |
26
|
rexbii |
⊢ ( ∃ 𝑔 ∈ ( 𝐵 ↑m ( 𝑆 ∖ { 𝑠 } ) ) ( 𝑔 finSupp ( 0g ‘ 𝑅 ) ∧ ( 𝑦 ( ·𝑠 ‘ 𝑀 ) 𝑠 ) = ( 𝑔 ( linC ‘ 𝑀 ) ( 𝑆 ∖ { 𝑠 } ) ) ) ↔ ∃ 𝑔 ∈ ( 𝐵 ↑m ( 𝑆 ∖ { 𝑠 } ) ) ( 𝑔 finSupp 0 ∧ ( 𝑦 ( ·𝑠 ‘ 𝑀 ) 𝑠 ) = ( 𝑔 ( linC ‘ 𝑀 ) ( 𝑆 ∖ { 𝑠 } ) ) ) ) |
28 |
27
|
anbi2i |
⊢ ( ( ( 𝑦 ( ·𝑠 ‘ 𝑀 ) 𝑠 ) ∈ ( Base ‘ 𝑀 ) ∧ ∃ 𝑔 ∈ ( 𝐵 ↑m ( 𝑆 ∖ { 𝑠 } ) ) ( 𝑔 finSupp ( 0g ‘ 𝑅 ) ∧ ( 𝑦 ( ·𝑠 ‘ 𝑀 ) 𝑠 ) = ( 𝑔 ( linC ‘ 𝑀 ) ( 𝑆 ∖ { 𝑠 } ) ) ) ) ↔ ( ( 𝑦 ( ·𝑠 ‘ 𝑀 ) 𝑠 ) ∈ ( Base ‘ 𝑀 ) ∧ ∃ 𝑔 ∈ ( 𝐵 ↑m ( 𝑆 ∖ { 𝑠 } ) ) ( 𝑔 finSupp 0 ∧ ( 𝑦 ( ·𝑠 ‘ 𝑀 ) 𝑠 ) = ( 𝑔 ( linC ‘ 𝑀 ) ( 𝑆 ∖ { 𝑠 } ) ) ) ) ) |
29 |
23 28
|
bitrdi |
⊢ ( ( 𝑀 ∈ LMod ∧ ( 𝑆 ∖ { 𝑠 } ) ∈ 𝒫 ( Base ‘ 𝑀 ) ) → ( ( 𝑦 ( ·𝑠 ‘ 𝑀 ) 𝑠 ) ∈ ( 𝑀 LinCo ( 𝑆 ∖ { 𝑠 } ) ) ↔ ( ( 𝑦 ( ·𝑠 ‘ 𝑀 ) 𝑠 ) ∈ ( Base ‘ 𝑀 ) ∧ ∃ 𝑔 ∈ ( 𝐵 ↑m ( 𝑆 ∖ { 𝑠 } ) ) ( 𝑔 finSupp 0 ∧ ( 𝑦 ( ·𝑠 ‘ 𝑀 ) 𝑠 ) = ( 𝑔 ( linC ‘ 𝑀 ) ( 𝑆 ∖ { 𝑠 } ) ) ) ) ) ) |
30 |
9 16 29
|
syl2anc |
⊢ ( ( ( 𝑆 ∈ 𝑉 ∧ 𝑀 ∈ LMod ) ∧ 𝑆 ⊆ ( Base ‘ 𝑀 ) ) → ( ( 𝑦 ( ·𝑠 ‘ 𝑀 ) 𝑠 ) ∈ ( 𝑀 LinCo ( 𝑆 ∖ { 𝑠 } ) ) ↔ ( ( 𝑦 ( ·𝑠 ‘ 𝑀 ) 𝑠 ) ∈ ( Base ‘ 𝑀 ) ∧ ∃ 𝑔 ∈ ( 𝐵 ↑m ( 𝑆 ∖ { 𝑠 } ) ) ( 𝑔 finSupp 0 ∧ ( 𝑦 ( ·𝑠 ‘ 𝑀 ) 𝑠 ) = ( 𝑔 ( linC ‘ 𝑀 ) ( 𝑆 ∖ { 𝑠 } ) ) ) ) ) ) |
31 |
30
|
notbid |
⊢ ( ( ( 𝑆 ∈ 𝑉 ∧ 𝑀 ∈ LMod ) ∧ 𝑆 ⊆ ( Base ‘ 𝑀 ) ) → ( ¬ ( 𝑦 ( ·𝑠 ‘ 𝑀 ) 𝑠 ) ∈ ( 𝑀 LinCo ( 𝑆 ∖ { 𝑠 } ) ) ↔ ¬ ( ( 𝑦 ( ·𝑠 ‘ 𝑀 ) 𝑠 ) ∈ ( Base ‘ 𝑀 ) ∧ ∃ 𝑔 ∈ ( 𝐵 ↑m ( 𝑆 ∖ { 𝑠 } ) ) ( 𝑔 finSupp 0 ∧ ( 𝑦 ( ·𝑠 ‘ 𝑀 ) 𝑠 ) = ( 𝑔 ( linC ‘ 𝑀 ) ( 𝑆 ∖ { 𝑠 } ) ) ) ) ) ) |
32 |
|
ianor |
⊢ ( ¬ ( ( 𝑦 ( ·𝑠 ‘ 𝑀 ) 𝑠 ) ∈ ( Base ‘ 𝑀 ) ∧ ∃ 𝑔 ∈ ( 𝐵 ↑m ( 𝑆 ∖ { 𝑠 } ) ) ( 𝑔 finSupp 0 ∧ ( 𝑦 ( ·𝑠 ‘ 𝑀 ) 𝑠 ) = ( 𝑔 ( linC ‘ 𝑀 ) ( 𝑆 ∖ { 𝑠 } ) ) ) ) ↔ ( ¬ ( 𝑦 ( ·𝑠 ‘ 𝑀 ) 𝑠 ) ∈ ( Base ‘ 𝑀 ) ∨ ¬ ∃ 𝑔 ∈ ( 𝐵 ↑m ( 𝑆 ∖ { 𝑠 } ) ) ( 𝑔 finSupp 0 ∧ ( 𝑦 ( ·𝑠 ‘ 𝑀 ) 𝑠 ) = ( 𝑔 ( linC ‘ 𝑀 ) ( 𝑆 ∖ { 𝑠 } ) ) ) ) ) |
33 |
|
ralnex |
⊢ ( ∀ 𝑔 ∈ ( 𝐵 ↑m ( 𝑆 ∖ { 𝑠 } ) ) ¬ ( 𝑔 finSupp 0 ∧ ( 𝑦 ( ·𝑠 ‘ 𝑀 ) 𝑠 ) = ( 𝑔 ( linC ‘ 𝑀 ) ( 𝑆 ∖ { 𝑠 } ) ) ) ↔ ¬ ∃ 𝑔 ∈ ( 𝐵 ↑m ( 𝑆 ∖ { 𝑠 } ) ) ( 𝑔 finSupp 0 ∧ ( 𝑦 ( ·𝑠 ‘ 𝑀 ) 𝑠 ) = ( 𝑔 ( linC ‘ 𝑀 ) ( 𝑆 ∖ { 𝑠 } ) ) ) ) |
34 |
|
ianor |
⊢ ( ¬ ( 𝑔 finSupp 0 ∧ ( 𝑦 ( ·𝑠 ‘ 𝑀 ) 𝑠 ) = ( 𝑔 ( linC ‘ 𝑀 ) ( 𝑆 ∖ { 𝑠 } ) ) ) ↔ ( ¬ 𝑔 finSupp 0 ∨ ¬ ( 𝑦 ( ·𝑠 ‘ 𝑀 ) 𝑠 ) = ( 𝑔 ( linC ‘ 𝑀 ) ( 𝑆 ∖ { 𝑠 } ) ) ) ) |
35 |
34
|
ralbii |
⊢ ( ∀ 𝑔 ∈ ( 𝐵 ↑m ( 𝑆 ∖ { 𝑠 } ) ) ¬ ( 𝑔 finSupp 0 ∧ ( 𝑦 ( ·𝑠 ‘ 𝑀 ) 𝑠 ) = ( 𝑔 ( linC ‘ 𝑀 ) ( 𝑆 ∖ { 𝑠 } ) ) ) ↔ ∀ 𝑔 ∈ ( 𝐵 ↑m ( 𝑆 ∖ { 𝑠 } ) ) ( ¬ 𝑔 finSupp 0 ∨ ¬ ( 𝑦 ( ·𝑠 ‘ 𝑀 ) 𝑠 ) = ( 𝑔 ( linC ‘ 𝑀 ) ( 𝑆 ∖ { 𝑠 } ) ) ) ) |
36 |
33 35
|
bitr3i |
⊢ ( ¬ ∃ 𝑔 ∈ ( 𝐵 ↑m ( 𝑆 ∖ { 𝑠 } ) ) ( 𝑔 finSupp 0 ∧ ( 𝑦 ( ·𝑠 ‘ 𝑀 ) 𝑠 ) = ( 𝑔 ( linC ‘ 𝑀 ) ( 𝑆 ∖ { 𝑠 } ) ) ) ↔ ∀ 𝑔 ∈ ( 𝐵 ↑m ( 𝑆 ∖ { 𝑠 } ) ) ( ¬ 𝑔 finSupp 0 ∨ ¬ ( 𝑦 ( ·𝑠 ‘ 𝑀 ) 𝑠 ) = ( 𝑔 ( linC ‘ 𝑀 ) ( 𝑆 ∖ { 𝑠 } ) ) ) ) |
37 |
36
|
orbi2i |
⊢ ( ( ¬ ( 𝑦 ( ·𝑠 ‘ 𝑀 ) 𝑠 ) ∈ ( Base ‘ 𝑀 ) ∨ ¬ ∃ 𝑔 ∈ ( 𝐵 ↑m ( 𝑆 ∖ { 𝑠 } ) ) ( 𝑔 finSupp 0 ∧ ( 𝑦 ( ·𝑠 ‘ 𝑀 ) 𝑠 ) = ( 𝑔 ( linC ‘ 𝑀 ) ( 𝑆 ∖ { 𝑠 } ) ) ) ) ↔ ( ¬ ( 𝑦 ( ·𝑠 ‘ 𝑀 ) 𝑠 ) ∈ ( Base ‘ 𝑀 ) ∨ ∀ 𝑔 ∈ ( 𝐵 ↑m ( 𝑆 ∖ { 𝑠 } ) ) ( ¬ 𝑔 finSupp 0 ∨ ¬ ( 𝑦 ( ·𝑠 ‘ 𝑀 ) 𝑠 ) = ( 𝑔 ( linC ‘ 𝑀 ) ( 𝑆 ∖ { 𝑠 } ) ) ) ) ) |
38 |
32 37
|
bitri |
⊢ ( ¬ ( ( 𝑦 ( ·𝑠 ‘ 𝑀 ) 𝑠 ) ∈ ( Base ‘ 𝑀 ) ∧ ∃ 𝑔 ∈ ( 𝐵 ↑m ( 𝑆 ∖ { 𝑠 } ) ) ( 𝑔 finSupp 0 ∧ ( 𝑦 ( ·𝑠 ‘ 𝑀 ) 𝑠 ) = ( 𝑔 ( linC ‘ 𝑀 ) ( 𝑆 ∖ { 𝑠 } ) ) ) ) ↔ ( ¬ ( 𝑦 ( ·𝑠 ‘ 𝑀 ) 𝑠 ) ∈ ( Base ‘ 𝑀 ) ∨ ∀ 𝑔 ∈ ( 𝐵 ↑m ( 𝑆 ∖ { 𝑠 } ) ) ( ¬ 𝑔 finSupp 0 ∨ ¬ ( 𝑦 ( ·𝑠 ‘ 𝑀 ) 𝑠 ) = ( 𝑔 ( linC ‘ 𝑀 ) ( 𝑆 ∖ { 𝑠 } ) ) ) ) ) |
39 |
31 38
|
bitrdi |
⊢ ( ( ( 𝑆 ∈ 𝑉 ∧ 𝑀 ∈ LMod ) ∧ 𝑆 ⊆ ( Base ‘ 𝑀 ) ) → ( ¬ ( 𝑦 ( ·𝑠 ‘ 𝑀 ) 𝑠 ) ∈ ( 𝑀 LinCo ( 𝑆 ∖ { 𝑠 } ) ) ↔ ( ¬ ( 𝑦 ( ·𝑠 ‘ 𝑀 ) 𝑠 ) ∈ ( Base ‘ 𝑀 ) ∨ ∀ 𝑔 ∈ ( 𝐵 ↑m ( 𝑆 ∖ { 𝑠 } ) ) ( ¬ 𝑔 finSupp 0 ∨ ¬ ( 𝑦 ( ·𝑠 ‘ 𝑀 ) 𝑠 ) = ( 𝑔 ( linC ‘ 𝑀 ) ( 𝑆 ∖ { 𝑠 } ) ) ) ) ) ) |
40 |
22 39
|
bitrd |
⊢ ( ( ( 𝑆 ∈ 𝑉 ∧ 𝑀 ∈ LMod ) ∧ 𝑆 ⊆ ( Base ‘ 𝑀 ) ) → ( ¬ ( 𝑦 ( ·𝑠 ‘ 𝑀 ) 𝑠 ) ∈ ( ( LSpan ‘ 𝑀 ) ‘ ( 𝑆 ∖ { 𝑠 } ) ) ↔ ( ¬ ( 𝑦 ( ·𝑠 ‘ 𝑀 ) 𝑠 ) ∈ ( Base ‘ 𝑀 ) ∨ ∀ 𝑔 ∈ ( 𝐵 ↑m ( 𝑆 ∖ { 𝑠 } ) ) ( ¬ 𝑔 finSupp 0 ∨ ¬ ( 𝑦 ( ·𝑠 ‘ 𝑀 ) 𝑠 ) = ( 𝑔 ( linC ‘ 𝑀 ) ( 𝑆 ∖ { 𝑠 } ) ) ) ) ) ) |
41 |
40
|
2ralbidv |
⊢ ( ( ( 𝑆 ∈ 𝑉 ∧ 𝑀 ∈ LMod ) ∧ 𝑆 ⊆ ( Base ‘ 𝑀 ) ) → ( ∀ 𝑠 ∈ 𝑆 ∀ 𝑦 ∈ ( 𝐵 ∖ { 0 } ) ¬ ( 𝑦 ( ·𝑠 ‘ 𝑀 ) 𝑠 ) ∈ ( ( LSpan ‘ 𝑀 ) ‘ ( 𝑆 ∖ { 𝑠 } ) ) ↔ ∀ 𝑠 ∈ 𝑆 ∀ 𝑦 ∈ ( 𝐵 ∖ { 0 } ) ( ¬ ( 𝑦 ( ·𝑠 ‘ 𝑀 ) 𝑠 ) ∈ ( Base ‘ 𝑀 ) ∨ ∀ 𝑔 ∈ ( 𝐵 ↑m ( 𝑆 ∖ { 𝑠 } ) ) ( ¬ 𝑔 finSupp 0 ∨ ¬ ( 𝑦 ( ·𝑠 ‘ 𝑀 ) 𝑠 ) = ( 𝑔 ( linC ‘ 𝑀 ) ( 𝑆 ∖ { 𝑠 } ) ) ) ) ) ) |
42 |
|
simpllr |
⊢ ( ( ( ( 𝑆 ∈ 𝑉 ∧ 𝑀 ∈ LMod ) ∧ 𝑆 ⊆ ( Base ‘ 𝑀 ) ) ∧ ( 𝑠 ∈ 𝑆 ∧ 𝑦 ∈ ( 𝐵 ∖ { 0 } ) ) ) → 𝑀 ∈ LMod ) |
43 |
|
eldifi |
⊢ ( 𝑦 ∈ ( 𝐵 ∖ { 0 } ) → 𝑦 ∈ 𝐵 ) |
44 |
43
|
adantl |
⊢ ( ( 𝑠 ∈ 𝑆 ∧ 𝑦 ∈ ( 𝐵 ∖ { 0 } ) ) → 𝑦 ∈ 𝐵 ) |
45 |
44
|
adantl |
⊢ ( ( ( ( 𝑆 ∈ 𝑉 ∧ 𝑀 ∈ LMod ) ∧ 𝑆 ⊆ ( Base ‘ 𝑀 ) ) ∧ ( 𝑠 ∈ 𝑆 ∧ 𝑦 ∈ ( 𝐵 ∖ { 0 } ) ) ) → 𝑦 ∈ 𝐵 ) |
46 |
|
ssel2 |
⊢ ( ( 𝑆 ⊆ ( Base ‘ 𝑀 ) ∧ 𝑠 ∈ 𝑆 ) → 𝑠 ∈ ( Base ‘ 𝑀 ) ) |
47 |
46
|
ad2ant2lr |
⊢ ( ( ( ( 𝑆 ∈ 𝑉 ∧ 𝑀 ∈ LMod ) ∧ 𝑆 ⊆ ( Base ‘ 𝑀 ) ) ∧ ( 𝑠 ∈ 𝑆 ∧ 𝑦 ∈ ( 𝐵 ∖ { 0 } ) ) ) → 𝑠 ∈ ( Base ‘ 𝑀 ) ) |
48 |
|
eqid |
⊢ ( ·𝑠 ‘ 𝑀 ) = ( ·𝑠 ‘ 𝑀 ) |
49 |
17 1 48 2
|
lmodvscl |
⊢ ( ( 𝑀 ∈ LMod ∧ 𝑦 ∈ 𝐵 ∧ 𝑠 ∈ ( Base ‘ 𝑀 ) ) → ( 𝑦 ( ·𝑠 ‘ 𝑀 ) 𝑠 ) ∈ ( Base ‘ 𝑀 ) ) |
50 |
42 45 47 49
|
syl3anc |
⊢ ( ( ( ( 𝑆 ∈ 𝑉 ∧ 𝑀 ∈ LMod ) ∧ 𝑆 ⊆ ( Base ‘ 𝑀 ) ) ∧ ( 𝑠 ∈ 𝑆 ∧ 𝑦 ∈ ( 𝐵 ∖ { 0 } ) ) ) → ( 𝑦 ( ·𝑠 ‘ 𝑀 ) 𝑠 ) ∈ ( Base ‘ 𝑀 ) ) |
51 |
50
|
notnotd |
⊢ ( ( ( ( 𝑆 ∈ 𝑉 ∧ 𝑀 ∈ LMod ) ∧ 𝑆 ⊆ ( Base ‘ 𝑀 ) ) ∧ ( 𝑠 ∈ 𝑆 ∧ 𝑦 ∈ ( 𝐵 ∖ { 0 } ) ) ) → ¬ ¬ ( 𝑦 ( ·𝑠 ‘ 𝑀 ) 𝑠 ) ∈ ( Base ‘ 𝑀 ) ) |
52 |
|
nbfal |
⊢ ( ¬ ¬ ( 𝑦 ( ·𝑠 ‘ 𝑀 ) 𝑠 ) ∈ ( Base ‘ 𝑀 ) ↔ ( ¬ ( 𝑦 ( ·𝑠 ‘ 𝑀 ) 𝑠 ) ∈ ( Base ‘ 𝑀 ) ↔ ⊥ ) ) |
53 |
51 52
|
sylib |
⊢ ( ( ( ( 𝑆 ∈ 𝑉 ∧ 𝑀 ∈ LMod ) ∧ 𝑆 ⊆ ( Base ‘ 𝑀 ) ) ∧ ( 𝑠 ∈ 𝑆 ∧ 𝑦 ∈ ( 𝐵 ∖ { 0 } ) ) ) → ( ¬ ( 𝑦 ( ·𝑠 ‘ 𝑀 ) 𝑠 ) ∈ ( Base ‘ 𝑀 ) ↔ ⊥ ) ) |
54 |
53
|
orbi1d |
⊢ ( ( ( ( 𝑆 ∈ 𝑉 ∧ 𝑀 ∈ LMod ) ∧ 𝑆 ⊆ ( Base ‘ 𝑀 ) ) ∧ ( 𝑠 ∈ 𝑆 ∧ 𝑦 ∈ ( 𝐵 ∖ { 0 } ) ) ) → ( ( ¬ ( 𝑦 ( ·𝑠 ‘ 𝑀 ) 𝑠 ) ∈ ( Base ‘ 𝑀 ) ∨ ∀ 𝑔 ∈ ( 𝐵 ↑m ( 𝑆 ∖ { 𝑠 } ) ) ( ¬ 𝑔 finSupp 0 ∨ ¬ ( 𝑦 ( ·𝑠 ‘ 𝑀 ) 𝑠 ) = ( 𝑔 ( linC ‘ 𝑀 ) ( 𝑆 ∖ { 𝑠 } ) ) ) ) ↔ ( ⊥ ∨ ∀ 𝑔 ∈ ( 𝐵 ↑m ( 𝑆 ∖ { 𝑠 } ) ) ( ¬ 𝑔 finSupp 0 ∨ ¬ ( 𝑦 ( ·𝑠 ‘ 𝑀 ) 𝑠 ) = ( 𝑔 ( linC ‘ 𝑀 ) ( 𝑆 ∖ { 𝑠 } ) ) ) ) ) ) |
55 |
54
|
2ralbidva |
⊢ ( ( ( 𝑆 ∈ 𝑉 ∧ 𝑀 ∈ LMod ) ∧ 𝑆 ⊆ ( Base ‘ 𝑀 ) ) → ( ∀ 𝑠 ∈ 𝑆 ∀ 𝑦 ∈ ( 𝐵 ∖ { 0 } ) ( ¬ ( 𝑦 ( ·𝑠 ‘ 𝑀 ) 𝑠 ) ∈ ( Base ‘ 𝑀 ) ∨ ∀ 𝑔 ∈ ( 𝐵 ↑m ( 𝑆 ∖ { 𝑠 } ) ) ( ¬ 𝑔 finSupp 0 ∨ ¬ ( 𝑦 ( ·𝑠 ‘ 𝑀 ) 𝑠 ) = ( 𝑔 ( linC ‘ 𝑀 ) ( 𝑆 ∖ { 𝑠 } ) ) ) ) ↔ ∀ 𝑠 ∈ 𝑆 ∀ 𝑦 ∈ ( 𝐵 ∖ { 0 } ) ( ⊥ ∨ ∀ 𝑔 ∈ ( 𝐵 ↑m ( 𝑆 ∖ { 𝑠 } ) ) ( ¬ 𝑔 finSupp 0 ∨ ¬ ( 𝑦 ( ·𝑠 ‘ 𝑀 ) 𝑠 ) = ( 𝑔 ( linC ‘ 𝑀 ) ( 𝑆 ∖ { 𝑠 } ) ) ) ) ) ) |
56 |
|
r19.32v |
⊢ ( ∀ 𝑦 ∈ ( 𝐵 ∖ { 0 } ) ( ⊥ ∨ ∀ 𝑔 ∈ ( 𝐵 ↑m ( 𝑆 ∖ { 𝑠 } ) ) ( ¬ 𝑔 finSupp 0 ∨ ¬ ( 𝑦 ( ·𝑠 ‘ 𝑀 ) 𝑠 ) = ( 𝑔 ( linC ‘ 𝑀 ) ( 𝑆 ∖ { 𝑠 } ) ) ) ) ↔ ( ⊥ ∨ ∀ 𝑦 ∈ ( 𝐵 ∖ { 0 } ) ∀ 𝑔 ∈ ( 𝐵 ↑m ( 𝑆 ∖ { 𝑠 } ) ) ( ¬ 𝑔 finSupp 0 ∨ ¬ ( 𝑦 ( ·𝑠 ‘ 𝑀 ) 𝑠 ) = ( 𝑔 ( linC ‘ 𝑀 ) ( 𝑆 ∖ { 𝑠 } ) ) ) ) ) |
57 |
56
|
ralbii |
⊢ ( ∀ 𝑠 ∈ 𝑆 ∀ 𝑦 ∈ ( 𝐵 ∖ { 0 } ) ( ⊥ ∨ ∀ 𝑔 ∈ ( 𝐵 ↑m ( 𝑆 ∖ { 𝑠 } ) ) ( ¬ 𝑔 finSupp 0 ∨ ¬ ( 𝑦 ( ·𝑠 ‘ 𝑀 ) 𝑠 ) = ( 𝑔 ( linC ‘ 𝑀 ) ( 𝑆 ∖ { 𝑠 } ) ) ) ) ↔ ∀ 𝑠 ∈ 𝑆 ( ⊥ ∨ ∀ 𝑦 ∈ ( 𝐵 ∖ { 0 } ) ∀ 𝑔 ∈ ( 𝐵 ↑m ( 𝑆 ∖ { 𝑠 } ) ) ( ¬ 𝑔 finSupp 0 ∨ ¬ ( 𝑦 ( ·𝑠 ‘ 𝑀 ) 𝑠 ) = ( 𝑔 ( linC ‘ 𝑀 ) ( 𝑆 ∖ { 𝑠 } ) ) ) ) ) |
58 |
|
r19.32v |
⊢ ( ∀ 𝑠 ∈ 𝑆 ( ⊥ ∨ ∀ 𝑦 ∈ ( 𝐵 ∖ { 0 } ) ∀ 𝑔 ∈ ( 𝐵 ↑m ( 𝑆 ∖ { 𝑠 } ) ) ( ¬ 𝑔 finSupp 0 ∨ ¬ ( 𝑦 ( ·𝑠 ‘ 𝑀 ) 𝑠 ) = ( 𝑔 ( linC ‘ 𝑀 ) ( 𝑆 ∖ { 𝑠 } ) ) ) ) ↔ ( ⊥ ∨ ∀ 𝑠 ∈ 𝑆 ∀ 𝑦 ∈ ( 𝐵 ∖ { 0 } ) ∀ 𝑔 ∈ ( 𝐵 ↑m ( 𝑆 ∖ { 𝑠 } ) ) ( ¬ 𝑔 finSupp 0 ∨ ¬ ( 𝑦 ( ·𝑠 ‘ 𝑀 ) 𝑠 ) = ( 𝑔 ( linC ‘ 𝑀 ) ( 𝑆 ∖ { 𝑠 } ) ) ) ) ) |
59 |
57 58
|
bitri |
⊢ ( ∀ 𝑠 ∈ 𝑆 ∀ 𝑦 ∈ ( 𝐵 ∖ { 0 } ) ( ⊥ ∨ ∀ 𝑔 ∈ ( 𝐵 ↑m ( 𝑆 ∖ { 𝑠 } ) ) ( ¬ 𝑔 finSupp 0 ∨ ¬ ( 𝑦 ( ·𝑠 ‘ 𝑀 ) 𝑠 ) = ( 𝑔 ( linC ‘ 𝑀 ) ( 𝑆 ∖ { 𝑠 } ) ) ) ) ↔ ( ⊥ ∨ ∀ 𝑠 ∈ 𝑆 ∀ 𝑦 ∈ ( 𝐵 ∖ { 0 } ) ∀ 𝑔 ∈ ( 𝐵 ↑m ( 𝑆 ∖ { 𝑠 } ) ) ( ¬ 𝑔 finSupp 0 ∨ ¬ ( 𝑦 ( ·𝑠 ‘ 𝑀 ) 𝑠 ) = ( 𝑔 ( linC ‘ 𝑀 ) ( 𝑆 ∖ { 𝑠 } ) ) ) ) ) |
60 |
|
falim |
⊢ ( ⊥ → ( ( ( 𝑆 ∈ 𝑉 ∧ 𝑀 ∈ LMod ) ∧ 𝑆 ⊆ ( Base ‘ 𝑀 ) ) → ∀ 𝑓 ∈ ( 𝐵 ↑m 𝑆 ) ( ( 𝑓 finSupp 0 ∧ ( 𝑓 ( linC ‘ 𝑀 ) 𝑆 ) = 𝑍 ) → ∀ 𝑥 ∈ 𝑆 ( 𝑓 ‘ 𝑥 ) = 0 ) ) ) |
61 |
|
sneq |
⊢ ( 𝑠 = 𝑥 → { 𝑠 } = { 𝑥 } ) |
62 |
61
|
difeq2d |
⊢ ( 𝑠 = 𝑥 → ( 𝑆 ∖ { 𝑠 } ) = ( 𝑆 ∖ { 𝑥 } ) ) |
63 |
62
|
oveq2d |
⊢ ( 𝑠 = 𝑥 → ( 𝐵 ↑m ( 𝑆 ∖ { 𝑠 } ) ) = ( 𝐵 ↑m ( 𝑆 ∖ { 𝑥 } ) ) ) |
64 |
|
oveq2 |
⊢ ( 𝑠 = 𝑥 → ( 𝑦 ( ·𝑠 ‘ 𝑀 ) 𝑠 ) = ( 𝑦 ( ·𝑠 ‘ 𝑀 ) 𝑥 ) ) |
65 |
62
|
oveq2d |
⊢ ( 𝑠 = 𝑥 → ( 𝑔 ( linC ‘ 𝑀 ) ( 𝑆 ∖ { 𝑠 } ) ) = ( 𝑔 ( linC ‘ 𝑀 ) ( 𝑆 ∖ { 𝑥 } ) ) ) |
66 |
64 65
|
eqeq12d |
⊢ ( 𝑠 = 𝑥 → ( ( 𝑦 ( ·𝑠 ‘ 𝑀 ) 𝑠 ) = ( 𝑔 ( linC ‘ 𝑀 ) ( 𝑆 ∖ { 𝑠 } ) ) ↔ ( 𝑦 ( ·𝑠 ‘ 𝑀 ) 𝑥 ) = ( 𝑔 ( linC ‘ 𝑀 ) ( 𝑆 ∖ { 𝑥 } ) ) ) ) |
67 |
66
|
notbid |
⊢ ( 𝑠 = 𝑥 → ( ¬ ( 𝑦 ( ·𝑠 ‘ 𝑀 ) 𝑠 ) = ( 𝑔 ( linC ‘ 𝑀 ) ( 𝑆 ∖ { 𝑠 } ) ) ↔ ¬ ( 𝑦 ( ·𝑠 ‘ 𝑀 ) 𝑥 ) = ( 𝑔 ( linC ‘ 𝑀 ) ( 𝑆 ∖ { 𝑥 } ) ) ) ) |
68 |
67
|
orbi2d |
⊢ ( 𝑠 = 𝑥 → ( ( ¬ 𝑔 finSupp 0 ∨ ¬ ( 𝑦 ( ·𝑠 ‘ 𝑀 ) 𝑠 ) = ( 𝑔 ( linC ‘ 𝑀 ) ( 𝑆 ∖ { 𝑠 } ) ) ) ↔ ( ¬ 𝑔 finSupp 0 ∨ ¬ ( 𝑦 ( ·𝑠 ‘ 𝑀 ) 𝑥 ) = ( 𝑔 ( linC ‘ 𝑀 ) ( 𝑆 ∖ { 𝑥 } ) ) ) ) ) |
69 |
63 68
|
raleqbidv |
⊢ ( 𝑠 = 𝑥 → ( ∀ 𝑔 ∈ ( 𝐵 ↑m ( 𝑆 ∖ { 𝑠 } ) ) ( ¬ 𝑔 finSupp 0 ∨ ¬ ( 𝑦 ( ·𝑠 ‘ 𝑀 ) 𝑠 ) = ( 𝑔 ( linC ‘ 𝑀 ) ( 𝑆 ∖ { 𝑠 } ) ) ) ↔ ∀ 𝑔 ∈ ( 𝐵 ↑m ( 𝑆 ∖ { 𝑥 } ) ) ( ¬ 𝑔 finSupp 0 ∨ ¬ ( 𝑦 ( ·𝑠 ‘ 𝑀 ) 𝑥 ) = ( 𝑔 ( linC ‘ 𝑀 ) ( 𝑆 ∖ { 𝑥 } ) ) ) ) ) |
70 |
69
|
ralbidv |
⊢ ( 𝑠 = 𝑥 → ( ∀ 𝑦 ∈ ( 𝐵 ∖ { 0 } ) ∀ 𝑔 ∈ ( 𝐵 ↑m ( 𝑆 ∖ { 𝑠 } ) ) ( ¬ 𝑔 finSupp 0 ∨ ¬ ( 𝑦 ( ·𝑠 ‘ 𝑀 ) 𝑠 ) = ( 𝑔 ( linC ‘ 𝑀 ) ( 𝑆 ∖ { 𝑠 } ) ) ) ↔ ∀ 𝑦 ∈ ( 𝐵 ∖ { 0 } ) ∀ 𝑔 ∈ ( 𝐵 ↑m ( 𝑆 ∖ { 𝑥 } ) ) ( ¬ 𝑔 finSupp 0 ∨ ¬ ( 𝑦 ( ·𝑠 ‘ 𝑀 ) 𝑥 ) = ( 𝑔 ( linC ‘ 𝑀 ) ( 𝑆 ∖ { 𝑥 } ) ) ) ) ) |
71 |
70
|
rspcva |
⊢ ( ( 𝑥 ∈ 𝑆 ∧ ∀ 𝑠 ∈ 𝑆 ∀ 𝑦 ∈ ( 𝐵 ∖ { 0 } ) ∀ 𝑔 ∈ ( 𝐵 ↑m ( 𝑆 ∖ { 𝑠 } ) ) ( ¬ 𝑔 finSupp 0 ∨ ¬ ( 𝑦 ( ·𝑠 ‘ 𝑀 ) 𝑠 ) = ( 𝑔 ( linC ‘ 𝑀 ) ( 𝑆 ∖ { 𝑠 } ) ) ) ) → ∀ 𝑦 ∈ ( 𝐵 ∖ { 0 } ) ∀ 𝑔 ∈ ( 𝐵 ↑m ( 𝑆 ∖ { 𝑥 } ) ) ( ¬ 𝑔 finSupp 0 ∨ ¬ ( 𝑦 ( ·𝑠 ‘ 𝑀 ) 𝑥 ) = ( 𝑔 ( linC ‘ 𝑀 ) ( 𝑆 ∖ { 𝑥 } ) ) ) ) |
72 |
1 2 3 4
|
lindslinindsimp2lem5 |
⊢ ( ( ( 𝑆 ∈ 𝑉 ∧ 𝑀 ∈ LMod ) ∧ ( 𝑆 ⊆ ( Base ‘ 𝑀 ) ∧ 𝑥 ∈ 𝑆 ) ) → ( ( 𝑓 ∈ ( 𝐵 ↑m 𝑆 ) ∧ ( 𝑓 finSupp 0 ∧ ( 𝑓 ( linC ‘ 𝑀 ) 𝑆 ) = 𝑍 ) ) → ( ∀ 𝑦 ∈ ( 𝐵 ∖ { 0 } ) ∀ 𝑔 ∈ ( 𝐵 ↑m ( 𝑆 ∖ { 𝑥 } ) ) ( ¬ 𝑔 finSupp 0 ∨ ¬ ( 𝑦 ( ·𝑠 ‘ 𝑀 ) 𝑥 ) = ( 𝑔 ( linC ‘ 𝑀 ) ( 𝑆 ∖ { 𝑥 } ) ) ) → ( 𝑓 ‘ 𝑥 ) = 0 ) ) ) |
73 |
72
|
expr |
⊢ ( ( ( 𝑆 ∈ 𝑉 ∧ 𝑀 ∈ LMod ) ∧ 𝑆 ⊆ ( Base ‘ 𝑀 ) ) → ( 𝑥 ∈ 𝑆 → ( ( 𝑓 ∈ ( 𝐵 ↑m 𝑆 ) ∧ ( 𝑓 finSupp 0 ∧ ( 𝑓 ( linC ‘ 𝑀 ) 𝑆 ) = 𝑍 ) ) → ( ∀ 𝑦 ∈ ( 𝐵 ∖ { 0 } ) ∀ 𝑔 ∈ ( 𝐵 ↑m ( 𝑆 ∖ { 𝑥 } ) ) ( ¬ 𝑔 finSupp 0 ∨ ¬ ( 𝑦 ( ·𝑠 ‘ 𝑀 ) 𝑥 ) = ( 𝑔 ( linC ‘ 𝑀 ) ( 𝑆 ∖ { 𝑥 } ) ) ) → ( 𝑓 ‘ 𝑥 ) = 0 ) ) ) ) |
74 |
73
|
com14 |
⊢ ( ∀ 𝑦 ∈ ( 𝐵 ∖ { 0 } ) ∀ 𝑔 ∈ ( 𝐵 ↑m ( 𝑆 ∖ { 𝑥 } ) ) ( ¬ 𝑔 finSupp 0 ∨ ¬ ( 𝑦 ( ·𝑠 ‘ 𝑀 ) 𝑥 ) = ( 𝑔 ( linC ‘ 𝑀 ) ( 𝑆 ∖ { 𝑥 } ) ) ) → ( 𝑥 ∈ 𝑆 → ( ( 𝑓 ∈ ( 𝐵 ↑m 𝑆 ) ∧ ( 𝑓 finSupp 0 ∧ ( 𝑓 ( linC ‘ 𝑀 ) 𝑆 ) = 𝑍 ) ) → ( ( ( 𝑆 ∈ 𝑉 ∧ 𝑀 ∈ LMod ) ∧ 𝑆 ⊆ ( Base ‘ 𝑀 ) ) → ( 𝑓 ‘ 𝑥 ) = 0 ) ) ) ) |
75 |
71 74
|
syl |
⊢ ( ( 𝑥 ∈ 𝑆 ∧ ∀ 𝑠 ∈ 𝑆 ∀ 𝑦 ∈ ( 𝐵 ∖ { 0 } ) ∀ 𝑔 ∈ ( 𝐵 ↑m ( 𝑆 ∖ { 𝑠 } ) ) ( ¬ 𝑔 finSupp 0 ∨ ¬ ( 𝑦 ( ·𝑠 ‘ 𝑀 ) 𝑠 ) = ( 𝑔 ( linC ‘ 𝑀 ) ( 𝑆 ∖ { 𝑠 } ) ) ) ) → ( 𝑥 ∈ 𝑆 → ( ( 𝑓 ∈ ( 𝐵 ↑m 𝑆 ) ∧ ( 𝑓 finSupp 0 ∧ ( 𝑓 ( linC ‘ 𝑀 ) 𝑆 ) = 𝑍 ) ) → ( ( ( 𝑆 ∈ 𝑉 ∧ 𝑀 ∈ LMod ) ∧ 𝑆 ⊆ ( Base ‘ 𝑀 ) ) → ( 𝑓 ‘ 𝑥 ) = 0 ) ) ) ) |
76 |
75
|
ex |
⊢ ( 𝑥 ∈ 𝑆 → ( ∀ 𝑠 ∈ 𝑆 ∀ 𝑦 ∈ ( 𝐵 ∖ { 0 } ) ∀ 𝑔 ∈ ( 𝐵 ↑m ( 𝑆 ∖ { 𝑠 } ) ) ( ¬ 𝑔 finSupp 0 ∨ ¬ ( 𝑦 ( ·𝑠 ‘ 𝑀 ) 𝑠 ) = ( 𝑔 ( linC ‘ 𝑀 ) ( 𝑆 ∖ { 𝑠 } ) ) ) → ( 𝑥 ∈ 𝑆 → ( ( 𝑓 ∈ ( 𝐵 ↑m 𝑆 ) ∧ ( 𝑓 finSupp 0 ∧ ( 𝑓 ( linC ‘ 𝑀 ) 𝑆 ) = 𝑍 ) ) → ( ( ( 𝑆 ∈ 𝑉 ∧ 𝑀 ∈ LMod ) ∧ 𝑆 ⊆ ( Base ‘ 𝑀 ) ) → ( 𝑓 ‘ 𝑥 ) = 0 ) ) ) ) ) |
77 |
76
|
pm2.43a |
⊢ ( 𝑥 ∈ 𝑆 → ( ∀ 𝑠 ∈ 𝑆 ∀ 𝑦 ∈ ( 𝐵 ∖ { 0 } ) ∀ 𝑔 ∈ ( 𝐵 ↑m ( 𝑆 ∖ { 𝑠 } ) ) ( ¬ 𝑔 finSupp 0 ∨ ¬ ( 𝑦 ( ·𝑠 ‘ 𝑀 ) 𝑠 ) = ( 𝑔 ( linC ‘ 𝑀 ) ( 𝑆 ∖ { 𝑠 } ) ) ) → ( ( 𝑓 ∈ ( 𝐵 ↑m 𝑆 ) ∧ ( 𝑓 finSupp 0 ∧ ( 𝑓 ( linC ‘ 𝑀 ) 𝑆 ) = 𝑍 ) ) → ( ( ( 𝑆 ∈ 𝑉 ∧ 𝑀 ∈ LMod ) ∧ 𝑆 ⊆ ( Base ‘ 𝑀 ) ) → ( 𝑓 ‘ 𝑥 ) = 0 ) ) ) ) |
78 |
77
|
com14 |
⊢ ( ( ( 𝑆 ∈ 𝑉 ∧ 𝑀 ∈ LMod ) ∧ 𝑆 ⊆ ( Base ‘ 𝑀 ) ) → ( ∀ 𝑠 ∈ 𝑆 ∀ 𝑦 ∈ ( 𝐵 ∖ { 0 } ) ∀ 𝑔 ∈ ( 𝐵 ↑m ( 𝑆 ∖ { 𝑠 } ) ) ( ¬ 𝑔 finSupp 0 ∨ ¬ ( 𝑦 ( ·𝑠 ‘ 𝑀 ) 𝑠 ) = ( 𝑔 ( linC ‘ 𝑀 ) ( 𝑆 ∖ { 𝑠 } ) ) ) → ( ( 𝑓 ∈ ( 𝐵 ↑m 𝑆 ) ∧ ( 𝑓 finSupp 0 ∧ ( 𝑓 ( linC ‘ 𝑀 ) 𝑆 ) = 𝑍 ) ) → ( 𝑥 ∈ 𝑆 → ( 𝑓 ‘ 𝑥 ) = 0 ) ) ) ) |
79 |
78
|
imp |
⊢ ( ( ( ( 𝑆 ∈ 𝑉 ∧ 𝑀 ∈ LMod ) ∧ 𝑆 ⊆ ( Base ‘ 𝑀 ) ) ∧ ∀ 𝑠 ∈ 𝑆 ∀ 𝑦 ∈ ( 𝐵 ∖ { 0 } ) ∀ 𝑔 ∈ ( 𝐵 ↑m ( 𝑆 ∖ { 𝑠 } ) ) ( ¬ 𝑔 finSupp 0 ∨ ¬ ( 𝑦 ( ·𝑠 ‘ 𝑀 ) 𝑠 ) = ( 𝑔 ( linC ‘ 𝑀 ) ( 𝑆 ∖ { 𝑠 } ) ) ) ) → ( ( 𝑓 ∈ ( 𝐵 ↑m 𝑆 ) ∧ ( 𝑓 finSupp 0 ∧ ( 𝑓 ( linC ‘ 𝑀 ) 𝑆 ) = 𝑍 ) ) → ( 𝑥 ∈ 𝑆 → ( 𝑓 ‘ 𝑥 ) = 0 ) ) ) |
80 |
79
|
expdimp |
⊢ ( ( ( ( ( 𝑆 ∈ 𝑉 ∧ 𝑀 ∈ LMod ) ∧ 𝑆 ⊆ ( Base ‘ 𝑀 ) ) ∧ ∀ 𝑠 ∈ 𝑆 ∀ 𝑦 ∈ ( 𝐵 ∖ { 0 } ) ∀ 𝑔 ∈ ( 𝐵 ↑m ( 𝑆 ∖ { 𝑠 } ) ) ( ¬ 𝑔 finSupp 0 ∨ ¬ ( 𝑦 ( ·𝑠 ‘ 𝑀 ) 𝑠 ) = ( 𝑔 ( linC ‘ 𝑀 ) ( 𝑆 ∖ { 𝑠 } ) ) ) ) ∧ 𝑓 ∈ ( 𝐵 ↑m 𝑆 ) ) → ( ( 𝑓 finSupp 0 ∧ ( 𝑓 ( linC ‘ 𝑀 ) 𝑆 ) = 𝑍 ) → ( 𝑥 ∈ 𝑆 → ( 𝑓 ‘ 𝑥 ) = 0 ) ) ) |
81 |
80
|
ralrimdv |
⊢ ( ( ( ( ( 𝑆 ∈ 𝑉 ∧ 𝑀 ∈ LMod ) ∧ 𝑆 ⊆ ( Base ‘ 𝑀 ) ) ∧ ∀ 𝑠 ∈ 𝑆 ∀ 𝑦 ∈ ( 𝐵 ∖ { 0 } ) ∀ 𝑔 ∈ ( 𝐵 ↑m ( 𝑆 ∖ { 𝑠 } ) ) ( ¬ 𝑔 finSupp 0 ∨ ¬ ( 𝑦 ( ·𝑠 ‘ 𝑀 ) 𝑠 ) = ( 𝑔 ( linC ‘ 𝑀 ) ( 𝑆 ∖ { 𝑠 } ) ) ) ) ∧ 𝑓 ∈ ( 𝐵 ↑m 𝑆 ) ) → ( ( 𝑓 finSupp 0 ∧ ( 𝑓 ( linC ‘ 𝑀 ) 𝑆 ) = 𝑍 ) → ∀ 𝑥 ∈ 𝑆 ( 𝑓 ‘ 𝑥 ) = 0 ) ) |
82 |
81
|
ralrimiva |
⊢ ( ( ( ( 𝑆 ∈ 𝑉 ∧ 𝑀 ∈ LMod ) ∧ 𝑆 ⊆ ( Base ‘ 𝑀 ) ) ∧ ∀ 𝑠 ∈ 𝑆 ∀ 𝑦 ∈ ( 𝐵 ∖ { 0 } ) ∀ 𝑔 ∈ ( 𝐵 ↑m ( 𝑆 ∖ { 𝑠 } ) ) ( ¬ 𝑔 finSupp 0 ∨ ¬ ( 𝑦 ( ·𝑠 ‘ 𝑀 ) 𝑠 ) = ( 𝑔 ( linC ‘ 𝑀 ) ( 𝑆 ∖ { 𝑠 } ) ) ) ) → ∀ 𝑓 ∈ ( 𝐵 ↑m 𝑆 ) ( ( 𝑓 finSupp 0 ∧ ( 𝑓 ( linC ‘ 𝑀 ) 𝑆 ) = 𝑍 ) → ∀ 𝑥 ∈ 𝑆 ( 𝑓 ‘ 𝑥 ) = 0 ) ) |
83 |
82
|
expcom |
⊢ ( ∀ 𝑠 ∈ 𝑆 ∀ 𝑦 ∈ ( 𝐵 ∖ { 0 } ) ∀ 𝑔 ∈ ( 𝐵 ↑m ( 𝑆 ∖ { 𝑠 } ) ) ( ¬ 𝑔 finSupp 0 ∨ ¬ ( 𝑦 ( ·𝑠 ‘ 𝑀 ) 𝑠 ) = ( 𝑔 ( linC ‘ 𝑀 ) ( 𝑆 ∖ { 𝑠 } ) ) ) → ( ( ( 𝑆 ∈ 𝑉 ∧ 𝑀 ∈ LMod ) ∧ 𝑆 ⊆ ( Base ‘ 𝑀 ) ) → ∀ 𝑓 ∈ ( 𝐵 ↑m 𝑆 ) ( ( 𝑓 finSupp 0 ∧ ( 𝑓 ( linC ‘ 𝑀 ) 𝑆 ) = 𝑍 ) → ∀ 𝑥 ∈ 𝑆 ( 𝑓 ‘ 𝑥 ) = 0 ) ) ) |
84 |
60 83
|
jaoi |
⊢ ( ( ⊥ ∨ ∀ 𝑠 ∈ 𝑆 ∀ 𝑦 ∈ ( 𝐵 ∖ { 0 } ) ∀ 𝑔 ∈ ( 𝐵 ↑m ( 𝑆 ∖ { 𝑠 } ) ) ( ¬ 𝑔 finSupp 0 ∨ ¬ ( 𝑦 ( ·𝑠 ‘ 𝑀 ) 𝑠 ) = ( 𝑔 ( linC ‘ 𝑀 ) ( 𝑆 ∖ { 𝑠 } ) ) ) ) → ( ( ( 𝑆 ∈ 𝑉 ∧ 𝑀 ∈ LMod ) ∧ 𝑆 ⊆ ( Base ‘ 𝑀 ) ) → ∀ 𝑓 ∈ ( 𝐵 ↑m 𝑆 ) ( ( 𝑓 finSupp 0 ∧ ( 𝑓 ( linC ‘ 𝑀 ) 𝑆 ) = 𝑍 ) → ∀ 𝑥 ∈ 𝑆 ( 𝑓 ‘ 𝑥 ) = 0 ) ) ) |
85 |
84
|
com12 |
⊢ ( ( ( 𝑆 ∈ 𝑉 ∧ 𝑀 ∈ LMod ) ∧ 𝑆 ⊆ ( Base ‘ 𝑀 ) ) → ( ( ⊥ ∨ ∀ 𝑠 ∈ 𝑆 ∀ 𝑦 ∈ ( 𝐵 ∖ { 0 } ) ∀ 𝑔 ∈ ( 𝐵 ↑m ( 𝑆 ∖ { 𝑠 } ) ) ( ¬ 𝑔 finSupp 0 ∨ ¬ ( 𝑦 ( ·𝑠 ‘ 𝑀 ) 𝑠 ) = ( 𝑔 ( linC ‘ 𝑀 ) ( 𝑆 ∖ { 𝑠 } ) ) ) ) → ∀ 𝑓 ∈ ( 𝐵 ↑m 𝑆 ) ( ( 𝑓 finSupp 0 ∧ ( 𝑓 ( linC ‘ 𝑀 ) 𝑆 ) = 𝑍 ) → ∀ 𝑥 ∈ 𝑆 ( 𝑓 ‘ 𝑥 ) = 0 ) ) ) |
86 |
59 85
|
syl5bi |
⊢ ( ( ( 𝑆 ∈ 𝑉 ∧ 𝑀 ∈ LMod ) ∧ 𝑆 ⊆ ( Base ‘ 𝑀 ) ) → ( ∀ 𝑠 ∈ 𝑆 ∀ 𝑦 ∈ ( 𝐵 ∖ { 0 } ) ( ⊥ ∨ ∀ 𝑔 ∈ ( 𝐵 ↑m ( 𝑆 ∖ { 𝑠 } ) ) ( ¬ 𝑔 finSupp 0 ∨ ¬ ( 𝑦 ( ·𝑠 ‘ 𝑀 ) 𝑠 ) = ( 𝑔 ( linC ‘ 𝑀 ) ( 𝑆 ∖ { 𝑠 } ) ) ) ) → ∀ 𝑓 ∈ ( 𝐵 ↑m 𝑆 ) ( ( 𝑓 finSupp 0 ∧ ( 𝑓 ( linC ‘ 𝑀 ) 𝑆 ) = 𝑍 ) → ∀ 𝑥 ∈ 𝑆 ( 𝑓 ‘ 𝑥 ) = 0 ) ) ) |
87 |
55 86
|
sylbid |
⊢ ( ( ( 𝑆 ∈ 𝑉 ∧ 𝑀 ∈ LMod ) ∧ 𝑆 ⊆ ( Base ‘ 𝑀 ) ) → ( ∀ 𝑠 ∈ 𝑆 ∀ 𝑦 ∈ ( 𝐵 ∖ { 0 } ) ( ¬ ( 𝑦 ( ·𝑠 ‘ 𝑀 ) 𝑠 ) ∈ ( Base ‘ 𝑀 ) ∨ ∀ 𝑔 ∈ ( 𝐵 ↑m ( 𝑆 ∖ { 𝑠 } ) ) ( ¬ 𝑔 finSupp 0 ∨ ¬ ( 𝑦 ( ·𝑠 ‘ 𝑀 ) 𝑠 ) = ( 𝑔 ( linC ‘ 𝑀 ) ( 𝑆 ∖ { 𝑠 } ) ) ) ) → ∀ 𝑓 ∈ ( 𝐵 ↑m 𝑆 ) ( ( 𝑓 finSupp 0 ∧ ( 𝑓 ( linC ‘ 𝑀 ) 𝑆 ) = 𝑍 ) → ∀ 𝑥 ∈ 𝑆 ( 𝑓 ‘ 𝑥 ) = 0 ) ) ) |
88 |
41 87
|
sylbid |
⊢ ( ( ( 𝑆 ∈ 𝑉 ∧ 𝑀 ∈ LMod ) ∧ 𝑆 ⊆ ( Base ‘ 𝑀 ) ) → ( ∀ 𝑠 ∈ 𝑆 ∀ 𝑦 ∈ ( 𝐵 ∖ { 0 } ) ¬ ( 𝑦 ( ·𝑠 ‘ 𝑀 ) 𝑠 ) ∈ ( ( LSpan ‘ 𝑀 ) ‘ ( 𝑆 ∖ { 𝑠 } ) ) → ∀ 𝑓 ∈ ( 𝐵 ↑m 𝑆 ) ( ( 𝑓 finSupp 0 ∧ ( 𝑓 ( linC ‘ 𝑀 ) 𝑆 ) = 𝑍 ) → ∀ 𝑥 ∈ 𝑆 ( 𝑓 ‘ 𝑥 ) = 0 ) ) ) |
89 |
88
|
impr |
⊢ ( ( ( 𝑆 ∈ 𝑉 ∧ 𝑀 ∈ LMod ) ∧ ( 𝑆 ⊆ ( Base ‘ 𝑀 ) ∧ ∀ 𝑠 ∈ 𝑆 ∀ 𝑦 ∈ ( 𝐵 ∖ { 0 } ) ¬ ( 𝑦 ( ·𝑠 ‘ 𝑀 ) 𝑠 ) ∈ ( ( LSpan ‘ 𝑀 ) ‘ ( 𝑆 ∖ { 𝑠 } ) ) ) ) → ∀ 𝑓 ∈ ( 𝐵 ↑m 𝑆 ) ( ( 𝑓 finSupp 0 ∧ ( 𝑓 ( linC ‘ 𝑀 ) 𝑆 ) = 𝑍 ) → ∀ 𝑥 ∈ 𝑆 ( 𝑓 ‘ 𝑥 ) = 0 ) ) |
90 |
8 89
|
jca |
⊢ ( ( ( 𝑆 ∈ 𝑉 ∧ 𝑀 ∈ LMod ) ∧ ( 𝑆 ⊆ ( Base ‘ 𝑀 ) ∧ ∀ 𝑠 ∈ 𝑆 ∀ 𝑦 ∈ ( 𝐵 ∖ { 0 } ) ¬ ( 𝑦 ( ·𝑠 ‘ 𝑀 ) 𝑠 ) ∈ ( ( LSpan ‘ 𝑀 ) ‘ ( 𝑆 ∖ { 𝑠 } ) ) ) ) → ( 𝑆 ∈ 𝒫 ( Base ‘ 𝑀 ) ∧ ∀ 𝑓 ∈ ( 𝐵 ↑m 𝑆 ) ( ( 𝑓 finSupp 0 ∧ ( 𝑓 ( linC ‘ 𝑀 ) 𝑆 ) = 𝑍 ) → ∀ 𝑥 ∈ 𝑆 ( 𝑓 ‘ 𝑥 ) = 0 ) ) ) |
91 |
90
|
ex |
⊢ ( ( 𝑆 ∈ 𝑉 ∧ 𝑀 ∈ LMod ) → ( ( 𝑆 ⊆ ( Base ‘ 𝑀 ) ∧ ∀ 𝑠 ∈ 𝑆 ∀ 𝑦 ∈ ( 𝐵 ∖ { 0 } ) ¬ ( 𝑦 ( ·𝑠 ‘ 𝑀 ) 𝑠 ) ∈ ( ( LSpan ‘ 𝑀 ) ‘ ( 𝑆 ∖ { 𝑠 } ) ) ) → ( 𝑆 ∈ 𝒫 ( Base ‘ 𝑀 ) ∧ ∀ 𝑓 ∈ ( 𝐵 ↑m 𝑆 ) ( ( 𝑓 finSupp 0 ∧ ( 𝑓 ( linC ‘ 𝑀 ) 𝑆 ) = 𝑍 ) → ∀ 𝑥 ∈ 𝑆 ( 𝑓 ‘ 𝑥 ) = 0 ) ) ) ) |