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| Mirrors > Home > HSE Home > Th. List > hvadd12i | Structured version Visualization version GIF version | ||
| Description: Hilbert vector space commutative/associative law. (Contributed by NM, 11-Sep-1999.) (New usage is discouraged.) | 
| Ref | Expression | 
|---|---|
| hvass.1 | ⊢ 𝐴 ∈ ℋ | 
| hvass.2 | ⊢ 𝐵 ∈ ℋ | 
| hvass.3 | ⊢ 𝐶 ∈ ℋ | 
| Ref | Expression | 
|---|---|
| hvadd12i | ⊢ (𝐴 +ℎ (𝐵 +ℎ 𝐶)) = (𝐵 +ℎ (𝐴 +ℎ 𝐶)) | 
| Step | Hyp | Ref | Expression | 
|---|---|---|---|
| 1 | hvass.1 | . . . 4 ⊢ 𝐴 ∈ ℋ | |
| 2 | hvass.2 | . . . 4 ⊢ 𝐵 ∈ ℋ | |
| 3 | 1, 2 | hvcomi 31039 | . . 3 ⊢ (𝐴 +ℎ 𝐵) = (𝐵 +ℎ 𝐴) | 
| 4 | 3 | oveq1i 7442 | . 2 ⊢ ((𝐴 +ℎ 𝐵) +ℎ 𝐶) = ((𝐵 +ℎ 𝐴) +ℎ 𝐶) | 
| 5 | hvass.3 | . . 3 ⊢ 𝐶 ∈ ℋ | |
| 6 | 1, 2, 5 | hvassi 31073 | . 2 ⊢ ((𝐴 +ℎ 𝐵) +ℎ 𝐶) = (𝐴 +ℎ (𝐵 +ℎ 𝐶)) | 
| 7 | 2, 1, 5 | hvassi 31073 | . 2 ⊢ ((𝐵 +ℎ 𝐴) +ℎ 𝐶) = (𝐵 +ℎ (𝐴 +ℎ 𝐶)) | 
| 8 | 4, 6, 7 | 3eqtr3i 2772 | 1 ⊢ (𝐴 +ℎ (𝐵 +ℎ 𝐶)) = (𝐵 +ℎ (𝐴 +ℎ 𝐶)) | 
| Colors of variables: wff setvar class | 
| Syntax hints: = wceq 1539 ∈ wcel 2107 (class class class)co 7432 ℋchba 30939 +ℎ cva 30940 | 
| This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1794 ax-4 1808 ax-5 1909 ax-6 1966 ax-7 2006 ax-8 2109 ax-9 2117 ax-ext 2707 ax-hvcom 31021 ax-hvass 31022 | 
| This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3an 1088 df-tru 1542 df-fal 1552 df-ex 1779 df-sb 2064 df-clab 2714 df-cleq 2728 df-clel 2815 df-rab 3436 df-v 3481 df-dif 3953 df-un 3955 df-ss 3967 df-nul 4333 df-if 4525 df-sn 4626 df-pr 4628 df-op 4632 df-uni 4907 df-br 5143 df-iota 6513 df-fv 6568 df-ov 7435 | 
| This theorem is referenced by: hvsubaddi 31086 | 
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