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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 28796 | . . 3 ⊢ (𝐴 +ℎ 𝐵) = (𝐵 +ℎ 𝐴) |
4 | 3 | oveq1i 7166 | . 2 ⊢ ((𝐴 +ℎ 𝐵) +ℎ 𝐶) = ((𝐵 +ℎ 𝐴) +ℎ 𝐶) |
5 | hvass.3 | . . 3 ⊢ 𝐶 ∈ ℋ | |
6 | 1, 2, 5 | hvassi 28830 | . 2 ⊢ ((𝐴 +ℎ 𝐵) +ℎ 𝐶) = (𝐴 +ℎ (𝐵 +ℎ 𝐶)) |
7 | 2, 1, 5 | hvassi 28830 | . 2 ⊢ ((𝐵 +ℎ 𝐴) +ℎ 𝐶) = (𝐵 +ℎ (𝐴 +ℎ 𝐶)) |
8 | 4, 6, 7 | 3eqtr3i 2852 | 1 ⊢ (𝐴 +ℎ (𝐵 +ℎ 𝐶)) = (𝐵 +ℎ (𝐴 +ℎ 𝐶)) |
Colors of variables: wff setvar class |
Syntax hints: = wceq 1537 ∈ wcel 2114 (class class class)co 7156 ℋchba 28696 +ℎ cva 28697 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1796 ax-4 1810 ax-5 1911 ax-6 1970 ax-7 2015 ax-8 2116 ax-9 2124 ax-10 2145 ax-11 2161 ax-12 2177 ax-ext 2793 ax-hvcom 28778 ax-hvass 28779 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3an 1085 df-tru 1540 df-ex 1781 df-nf 1785 df-sb 2070 df-clab 2800 df-cleq 2814 df-clel 2893 df-nfc 2963 df-rab 3147 df-v 3496 df-dif 3939 df-un 3941 df-in 3943 df-ss 3952 df-nul 4292 df-if 4468 df-sn 4568 df-pr 4570 df-op 4574 df-uni 4839 df-br 5067 df-iota 6314 df-fv 6363 df-ov 7159 |
This theorem is referenced by: hvsubaddi 28843 |
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