Step |
Hyp |
Ref |
Expression |
1 |
|
lincresunit.b |
⊢ 𝐵 = ( Base ‘ 𝑀 ) |
2 |
|
lincresunit.r |
⊢ 𝑅 = ( Scalar ‘ 𝑀 ) |
3 |
|
lincresunit.e |
⊢ 𝐸 = ( Base ‘ 𝑅 ) |
4 |
|
lincresunit.u |
⊢ 𝑈 = ( Unit ‘ 𝑅 ) |
5 |
|
lincresunit.0 |
⊢ 0 = ( 0g ‘ 𝑅 ) |
6 |
|
lincresunit.z |
⊢ 𝑍 = ( 0g ‘ 𝑀 ) |
7 |
|
lincresunit.n |
⊢ 𝑁 = ( invg ‘ 𝑅 ) |
8 |
|
lincresunit.i |
⊢ 𝐼 = ( invr ‘ 𝑅 ) |
9 |
|
lincresunit.t |
⊢ · = ( .r ‘ 𝑅 ) |
10 |
|
lincresunit.g |
⊢ 𝐺 = ( 𝑠 ∈ ( 𝑆 ∖ { 𝑋 } ) ↦ ( ( 𝐼 ‘ ( 𝑁 ‘ ( 𝐹 ‘ 𝑋 ) ) ) · ( 𝐹 ‘ 𝑠 ) ) ) |
11 |
2
|
lmodring |
⊢ ( 𝑀 ∈ LMod → 𝑅 ∈ Ring ) |
12 |
11
|
3ad2ant2 |
⊢ ( ( 𝑆 ∈ 𝒫 𝐵 ∧ 𝑀 ∈ LMod ∧ 𝑋 ∈ 𝑆 ) → 𝑅 ∈ Ring ) |
13 |
12
|
adantr |
⊢ ( ( ( 𝑆 ∈ 𝒫 𝐵 ∧ 𝑀 ∈ LMod ∧ 𝑋 ∈ 𝑆 ) ∧ ( 𝐹 ∈ ( 𝐸 ↑m 𝑆 ) ∧ ( 𝐹 ‘ 𝑋 ) ∈ 𝑈 ) ) → 𝑅 ∈ Ring ) |
14 |
|
simpr |
⊢ ( ( 𝐹 ∈ ( 𝐸 ↑m 𝑆 ) ∧ ( 𝐹 ‘ 𝑋 ) ∈ 𝑈 ) → ( 𝐹 ‘ 𝑋 ) ∈ 𝑈 ) |
15 |
4 7
|
unitnegcl |
⊢ ( ( 𝑅 ∈ Ring ∧ ( 𝐹 ‘ 𝑋 ) ∈ 𝑈 ) → ( 𝑁 ‘ ( 𝐹 ‘ 𝑋 ) ) ∈ 𝑈 ) |
16 |
12 14 15
|
syl2an |
⊢ ( ( ( 𝑆 ∈ 𝒫 𝐵 ∧ 𝑀 ∈ LMod ∧ 𝑋 ∈ 𝑆 ) ∧ ( 𝐹 ∈ ( 𝐸 ↑m 𝑆 ) ∧ ( 𝐹 ‘ 𝑋 ) ∈ 𝑈 ) ) → ( 𝑁 ‘ ( 𝐹 ‘ 𝑋 ) ) ∈ 𝑈 ) |
17 |
4 8 3
|
ringinvcl |
⊢ ( ( 𝑅 ∈ Ring ∧ ( 𝑁 ‘ ( 𝐹 ‘ 𝑋 ) ) ∈ 𝑈 ) → ( 𝐼 ‘ ( 𝑁 ‘ ( 𝐹 ‘ 𝑋 ) ) ) ∈ 𝐸 ) |
18 |
13 16 17
|
syl2anc |
⊢ ( ( ( 𝑆 ∈ 𝒫 𝐵 ∧ 𝑀 ∈ LMod ∧ 𝑋 ∈ 𝑆 ) ∧ ( 𝐹 ∈ ( 𝐸 ↑m 𝑆 ) ∧ ( 𝐹 ‘ 𝑋 ) ∈ 𝑈 ) ) → ( 𝐼 ‘ ( 𝑁 ‘ ( 𝐹 ‘ 𝑋 ) ) ) ∈ 𝐸 ) |