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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 30272 | . . 3 ⊢ (𝐴 +ℎ 𝐵) = (𝐵 +ℎ 𝐴) |
4 | 3 | oveq1i 7419 | . 2 ⊢ ((𝐴 +ℎ 𝐵) +ℎ 𝐶) = ((𝐵 +ℎ 𝐴) +ℎ 𝐶) |
5 | hvass.3 | . . 3 ⊢ 𝐶 ∈ ℋ | |
6 | 1, 2, 5 | hvassi 30306 | . 2 ⊢ ((𝐴 +ℎ 𝐵) +ℎ 𝐶) = (𝐴 +ℎ (𝐵 +ℎ 𝐶)) |
7 | 2, 1, 5 | hvassi 30306 | . 2 ⊢ ((𝐵 +ℎ 𝐴) +ℎ 𝐶) = (𝐵 +ℎ (𝐴 +ℎ 𝐶)) |
8 | 4, 6, 7 | 3eqtr3i 2769 | 1 ⊢ (𝐴 +ℎ (𝐵 +ℎ 𝐶)) = (𝐵 +ℎ (𝐴 +ℎ 𝐶)) |
Colors of variables: wff setvar class |
Syntax hints: = wceq 1542 ∈ wcel 2107 (class class class)co 7409 ℋchba 30172 +ℎ cva 30173 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1798 ax-4 1812 ax-5 1914 ax-6 1972 ax-7 2012 ax-8 2109 ax-9 2117 ax-ext 2704 ax-hvcom 30254 ax-hvass 30255 |
This theorem depends on definitions: df-bi 206 df-an 398 df-or 847 df-3an 1090 df-tru 1545 df-fal 1555 df-ex 1783 df-sb 2069 df-clab 2711 df-cleq 2725 df-clel 2811 df-rab 3434 df-v 3477 df-dif 3952 df-un 3954 df-in 3956 df-ss 3966 df-nul 4324 df-if 4530 df-sn 4630 df-pr 4632 df-op 4636 df-uni 4910 df-br 5150 df-iota 6496 df-fv 6552 df-ov 7412 |
This theorem is referenced by: hvsubaddi 30319 |
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