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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 27260 | . . 3 ⊢ (𝐴 +ℎ 𝐵) = (𝐵 +ℎ 𝐴) |
4 | 3 | oveq1i 6559 | . 2 ⊢ ((𝐴 +ℎ 𝐵) +ℎ 𝐶) = ((𝐵 +ℎ 𝐴) +ℎ 𝐶) |
5 | hvass.3 | . . 3 ⊢ 𝐶 ∈ ℋ | |
6 | 1, 2, 5 | hvassi 27294 | . 2 ⊢ ((𝐴 +ℎ 𝐵) +ℎ 𝐶) = (𝐴 +ℎ (𝐵 +ℎ 𝐶)) |
7 | 2, 1, 5 | hvassi 27294 | . 2 ⊢ ((𝐵 +ℎ 𝐴) +ℎ 𝐶) = (𝐵 +ℎ (𝐴 +ℎ 𝐶)) |
8 | 4, 6, 7 | 3eqtr3i 2640 | 1 ⊢ (𝐴 +ℎ (𝐵 +ℎ 𝐶)) = (𝐵 +ℎ (𝐴 +ℎ 𝐶)) |
Colors of variables: wff setvar class |
Syntax hints: = wceq 1475 ∈ wcel 1977 (class class class)co 6549 ℋchil 27160 +ℎ cva 27161 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1713 ax-4 1728 ax-5 1827 ax-6 1875 ax-7 1922 ax-10 2006 ax-11 2021 ax-12 2034 ax-13 2234 ax-ext 2590 ax-hvcom 27242 ax-hvass 27243 |
This theorem depends on definitions: df-bi 196 df-or 384 df-an 385 df-3an 1033 df-tru 1478 df-ex 1696 df-nf 1701 df-sb 1868 df-clab 2597 df-cleq 2603 df-clel 2606 df-nfc 2740 df-rex 2902 df-rab 2905 df-v 3175 df-dif 3543 df-un 3545 df-in 3547 df-ss 3554 df-nul 3875 df-if 4037 df-sn 4126 df-pr 4128 df-op 4132 df-uni 4373 df-br 4584 df-iota 5768 df-fv 5812 df-ov 6552 |
This theorem is referenced by: hvsubaddi 27307 |
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