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Mirrors > Home > HSE Home > Th. List > hvcomi | Structured version Visualization version GIF version |
Description: Commutation of vector addition. (Contributed by NM, 3-Sep-1999.) (New usage is discouraged.) |
Ref | Expression |
---|---|
hvaddcl.1 | ⊢ 𝐴 ∈ ℋ |
hvaddcl.2 | ⊢ 𝐵 ∈ ℋ |
Ref | Expression |
---|---|
hvcomi | ⊢ (𝐴 +ℎ 𝐵) = (𝐵 +ℎ 𝐴) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | hvaddcl.1 | . 2 ⊢ 𝐴 ∈ ℋ | |
2 | hvaddcl.2 | . 2 ⊢ 𝐵 ∈ ℋ | |
3 | ax-hvcom 29992 | . 2 ⊢ ((𝐴 ∈ ℋ ∧ 𝐵 ∈ ℋ) → (𝐴 +ℎ 𝐵) = (𝐵 +ℎ 𝐴)) | |
4 | 1, 2, 3 | mp2an 691 | 1 ⊢ (𝐴 +ℎ 𝐵) = (𝐵 +ℎ 𝐴) |
Colors of variables: wff setvar class |
Syntax hints: = wceq 1542 ∈ wcel 2107 (class class class)co 7361 ℋchba 29910 +ℎ cva 29911 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-hvcom 29992 |
This theorem depends on definitions: df-bi 206 df-an 398 |
This theorem is referenced by: hvadd12i 30048 hvnegdii 30053 norm3difi 30138 normpar2i 30147 nonbooli 30642 lnophmlem2 31008 |
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