Metamath Proof Explorer
Description: Scalar multiplication for the final constructed Hilbert space.
(Contributed by NM, 22-Jun-2015) (Revised by Mario Carneiro, 28-Jun-2015)
|
|
Ref |
Expression |
|
Hypotheses |
hlhilslem.h |
⊢ 𝐻 = ( LHyp ‘ 𝐾 ) |
|
|
hlhilslem.e |
⊢ 𝐸 = ( ( EDRing ‘ 𝐾 ) ‘ 𝑊 ) |
|
|
hlhilslem.u |
⊢ 𝑈 = ( ( HLHil ‘ 𝐾 ) ‘ 𝑊 ) |
|
|
hlhilslem.r |
⊢ 𝑅 = ( Scalar ‘ 𝑈 ) |
|
|
hlhilslem.k |
⊢ ( 𝜑 → ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ) |
|
|
hlhilsmul.m |
⊢ · = ( .r ‘ 𝐸 ) |
|
Assertion |
hlhilsmul |
⊢ ( 𝜑 → · = ( .r ‘ 𝑅 ) ) |
Proof
Step |
Hyp |
Ref |
Expression |
1 |
|
hlhilslem.h |
⊢ 𝐻 = ( LHyp ‘ 𝐾 ) |
2 |
|
hlhilslem.e |
⊢ 𝐸 = ( ( EDRing ‘ 𝐾 ) ‘ 𝑊 ) |
3 |
|
hlhilslem.u |
⊢ 𝑈 = ( ( HLHil ‘ 𝐾 ) ‘ 𝑊 ) |
4 |
|
hlhilslem.r |
⊢ 𝑅 = ( Scalar ‘ 𝑈 ) |
5 |
|
hlhilslem.k |
⊢ ( 𝜑 → ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ) |
6 |
|
hlhilsmul.m |
⊢ · = ( .r ‘ 𝐸 ) |
7 |
|
df-mulr |
⊢ .r = Slot 3 |
8 |
|
3nn |
⊢ 3 ∈ ℕ |
9 |
|
3lt4 |
⊢ 3 < 4 |
10 |
1 2 3 4 5 7 8 9 6
|
hlhilslem |
⊢ ( 𝜑 → · = ( .r ‘ 𝑅 ) ) |