Description: Rearrangement of 4 terms in a vector sum. (Contributed by NM, 8-Feb-2008) (New usage is discouraged.)
| Ref | Expression | ||
|---|---|---|---|
| Hypotheses | nvgcl.1 | ⊢ 𝑋 = ( BaseSet ‘ 𝑈 ) | |
| nvgcl.2 | ⊢ 𝐺 = ( +𝑣 ‘ 𝑈 ) | ||
| Assertion | nvadd4 | ⊢ ( ( 𝑈 ∈ NrmCVec ∧ ( 𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋 ) ∧ ( 𝐶 ∈ 𝑋 ∧ 𝐷 ∈ 𝑋 ) ) → ( ( 𝐴 𝐺 𝐵 ) 𝐺 ( 𝐶 𝐺 𝐷 ) ) = ( ( 𝐴 𝐺 𝐶 ) 𝐺 ( 𝐵 𝐺 𝐷 ) ) ) |
| Step | Hyp | Ref | Expression |
|---|---|---|---|
| 1 | nvgcl.1 | ⊢ 𝑋 = ( BaseSet ‘ 𝑈 ) | |
| 2 | nvgcl.2 | ⊢ 𝐺 = ( +𝑣 ‘ 𝑈 ) | |
| 3 | 2 | nvablo | ⊢ ( 𝑈 ∈ NrmCVec → 𝐺 ∈ AbelOp ) |
| 4 | 1 2 | bafval | ⊢ 𝑋 = ran 𝐺 |
| 5 | 4 | ablo4 | ⊢ ( ( 𝐺 ∈ AbelOp ∧ ( 𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋 ) ∧ ( 𝐶 ∈ 𝑋 ∧ 𝐷 ∈ 𝑋 ) ) → ( ( 𝐴 𝐺 𝐵 ) 𝐺 ( 𝐶 𝐺 𝐷 ) ) = ( ( 𝐴 𝐺 𝐶 ) 𝐺 ( 𝐵 𝐺 𝐷 ) ) ) |
| 6 | 3 5 | syl3an1 | ⊢ ( ( 𝑈 ∈ NrmCVec ∧ ( 𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋 ) ∧ ( 𝐶 ∈ 𝑋 ∧ 𝐷 ∈ 𝑋 ) ) → ( ( 𝐴 𝐺 𝐵 ) 𝐺 ( 𝐶 𝐺 𝐷 ) ) = ( ( 𝐴 𝐺 𝐶 ) 𝐺 ( 𝐵 𝐺 𝐷 ) ) ) |