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Mirrors > Home > HSE Home > Th. List > hvadd32i | Structured version Visualization version GIF version |
Description: Hilbert vector space commutative/associative law. (Contributed by NM, 18-Aug-1999.) (New usage is discouraged.) |
Ref | Expression |
---|---|
hvass.1 | ⊢ 𝐴 ∈ ℋ |
hvass.2 | ⊢ 𝐵 ∈ ℋ |
hvass.3 | ⊢ 𝐶 ∈ ℋ |
Ref | Expression |
---|---|
hvadd32i | ⊢ ((𝐴 +ℎ 𝐵) +ℎ 𝐶) = ((𝐴 +ℎ 𝐶) +ℎ 𝐵) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | hvass.1 | . 2 ⊢ 𝐴 ∈ ℋ | |
2 | hvass.2 | . 2 ⊢ 𝐵 ∈ ℋ | |
3 | hvass.3 | . 2 ⊢ 𝐶 ∈ ℋ | |
4 | hvadd32 28811 | . 2 ⊢ ((𝐴 ∈ ℋ ∧ 𝐵 ∈ ℋ ∧ 𝐶 ∈ ℋ) → ((𝐴 +ℎ 𝐵) +ℎ 𝐶) = ((𝐴 +ℎ 𝐶) +ℎ 𝐵)) | |
5 | 1, 2, 3, 4 | mp3an 1457 | 1 ⊢ ((𝐴 +ℎ 𝐵) +ℎ 𝐶) = ((𝐴 +ℎ 𝐶) +ℎ 𝐵) |
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: hvsubeq0i 28840 hvaddcani 28842 normpar2i 28933 |
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