Metamath Proof Explorer
Description: Equation 6.47 of Ponnusamy p. 362. (Contributed by NM, 27-Apr-2007) (New usage is discouraged.)
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|
Ref |
Expression |
|
Hypotheses |
ip1i.1 |
⊢ 𝑋 = ( BaseSet ‘ 𝑈 ) |
|
|
ip1i.2 |
⊢ 𝐺 = ( +𝑣 ‘ 𝑈 ) |
|
|
ip1i.4 |
⊢ 𝑆 = ( ·𝑠OLD ‘ 𝑈 ) |
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|
ip1i.7 |
⊢ 𝑃 = ( ·𝑖OLD ‘ 𝑈 ) |
|
|
ip1i.9 |
⊢ 𝑈 ∈ CPreHilOLD |
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|
ip1i.a |
⊢ 𝐴 ∈ 𝑋 |
|
|
ip1i.b |
⊢ 𝐵 ∈ 𝑋 |
|
|
ip1i.c |
⊢ 𝐶 ∈ 𝑋 |
|
Assertion |
ip1i |
⊢ ( ( ( 𝐴 𝐺 𝐵 ) 𝑃 𝐶 ) + ( ( 𝐴 𝐺 ( - 1 𝑆 𝐵 ) ) 𝑃 𝐶 ) ) = ( 2 · ( 𝐴 𝑃 𝐶 ) ) |
Proof
Step |
Hyp |
Ref |
Expression |
1 |
|
ip1i.1 |
⊢ 𝑋 = ( BaseSet ‘ 𝑈 ) |
2 |
|
ip1i.2 |
⊢ 𝐺 = ( +𝑣 ‘ 𝑈 ) |
3 |
|
ip1i.4 |
⊢ 𝑆 = ( ·𝑠OLD ‘ 𝑈 ) |
4 |
|
ip1i.7 |
⊢ 𝑃 = ( ·𝑖OLD ‘ 𝑈 ) |
5 |
|
ip1i.9 |
⊢ 𝑈 ∈ CPreHilOLD |
6 |
|
ip1i.a |
⊢ 𝐴 ∈ 𝑋 |
7 |
|
ip1i.b |
⊢ 𝐵 ∈ 𝑋 |
8 |
|
ip1i.c |
⊢ 𝐶 ∈ 𝑋 |
9 |
|
eqid |
⊢ ( normCV ‘ 𝑈 ) = ( normCV ‘ 𝑈 ) |
10 |
|
ax-1cn |
⊢ 1 ∈ ℂ |
11 |
1 2 3 4 5 6 7 8 9 10
|
ip1ilem |
⊢ ( ( ( 𝐴 𝐺 𝐵 ) 𝑃 𝐶 ) + ( ( 𝐴 𝐺 ( - 1 𝑆 𝐵 ) ) 𝑃 𝐶 ) ) = ( 2 · ( 𝐴 𝑃 𝐶 ) ) |