Step |
Hyp |
Ref |
Expression |
1 |
|
hlhillvec.h |
⊢ 𝐻 = ( LHyp ‘ 𝐾 ) |
2 |
|
hlhillvec.u |
⊢ 𝑈 = ( ( HLHil ‘ 𝐾 ) ‘ 𝑊 ) |
3 |
|
hlhillvec.k |
⊢ ( 𝜑 → ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ) |
4 |
|
hlhildrng.r |
⊢ 𝑅 = ( Scalar ‘ 𝑈 ) |
5 |
|
hlhilsrng.l |
⊢ 𝐿 = ( ( DVecH ‘ 𝐾 ) ‘ 𝑊 ) |
6 |
|
hlhilsrng.s |
⊢ 𝑆 = ( Scalar ‘ 𝐿 ) |
7 |
|
hlhilsrng.b |
⊢ 𝐵 = ( Base ‘ 𝑆 ) |
8 |
|
hlhilsrng.p |
⊢ + = ( +g ‘ 𝑆 ) |
9 |
|
hlhilsrng.t |
⊢ · = ( .r ‘ 𝑆 ) |
10 |
|
hlhilsrng.g |
⊢ 𝐺 = ( ( HGMap ‘ 𝐾 ) ‘ 𝑊 ) |
11 |
1 5 6 2 4 3 7
|
hlhilsbase2 |
⊢ ( 𝜑 → 𝐵 = ( Base ‘ 𝑅 ) ) |
12 |
1 5 6 2 4 3 8
|
hlhilsplus2 |
⊢ ( 𝜑 → + = ( +g ‘ 𝑅 ) ) |
13 |
1 5 6 2 4 3 9
|
hlhilsmul2 |
⊢ ( 𝜑 → · = ( .r ‘ 𝑅 ) ) |
14 |
1 2 4 10 3
|
hlhilnvl |
⊢ ( 𝜑 → 𝐺 = ( *𝑟 ‘ 𝑅 ) ) |
15 |
1 2 3 4
|
hlhildrng |
⊢ ( 𝜑 → 𝑅 ∈ DivRing ) |
16 |
|
drngring |
⊢ ( 𝑅 ∈ DivRing → 𝑅 ∈ Ring ) |
17 |
15 16
|
syl |
⊢ ( 𝜑 → 𝑅 ∈ Ring ) |
18 |
3
|
adantr |
⊢ ( ( 𝜑 ∧ 𝑥 ∈ 𝐵 ) → ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ) |
19 |
|
simpr |
⊢ ( ( 𝜑 ∧ 𝑥 ∈ 𝐵 ) → 𝑥 ∈ 𝐵 ) |
20 |
1 5 6 7 10 18 19
|
hgmapcl |
⊢ ( ( 𝜑 ∧ 𝑥 ∈ 𝐵 ) → ( 𝐺 ‘ 𝑥 ) ∈ 𝐵 ) |
21 |
3
|
3ad2ant1 |
⊢ ( ( 𝜑 ∧ 𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵 ) → ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ) |
22 |
|
simp2 |
⊢ ( ( 𝜑 ∧ 𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵 ) → 𝑥 ∈ 𝐵 ) |
23 |
|
simp3 |
⊢ ( ( 𝜑 ∧ 𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵 ) → 𝑦 ∈ 𝐵 ) |
24 |
1 5 6 7 8 10 21 22 23
|
hgmapadd |
⊢ ( ( 𝜑 ∧ 𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵 ) → ( 𝐺 ‘ ( 𝑥 + 𝑦 ) ) = ( ( 𝐺 ‘ 𝑥 ) + ( 𝐺 ‘ 𝑦 ) ) ) |
25 |
1 5 6 7 9 10 21 22 23
|
hgmapmul |
⊢ ( ( 𝜑 ∧ 𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵 ) → ( 𝐺 ‘ ( 𝑥 · 𝑦 ) ) = ( ( 𝐺 ‘ 𝑦 ) · ( 𝐺 ‘ 𝑥 ) ) ) |
26 |
1 5 6 7 10 18 19
|
hgmapvv |
⊢ ( ( 𝜑 ∧ 𝑥 ∈ 𝐵 ) → ( 𝐺 ‘ ( 𝐺 ‘ 𝑥 ) ) = 𝑥 ) |
27 |
11 12 13 14 17 20 24 25 26
|
issrngd |
⊢ ( 𝜑 → 𝑅 ∈ *-Ring ) |