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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 28479 | . . 3 ⊢ (𝐴 +ℎ 𝐵) = (𝐵 +ℎ 𝐴) |
4 | 3 | oveq1i 7029 | . 2 ⊢ ((𝐴 +ℎ 𝐵) +ℎ 𝐶) = ((𝐵 +ℎ 𝐴) +ℎ 𝐶) |
5 | hvass.3 | . . 3 ⊢ 𝐶 ∈ ℋ | |
6 | 1, 2, 5 | hvassi 28513 | . 2 ⊢ ((𝐴 +ℎ 𝐵) +ℎ 𝐶) = (𝐴 +ℎ (𝐵 +ℎ 𝐶)) |
7 | 2, 1, 5 | hvassi 28513 | . 2 ⊢ ((𝐵 +ℎ 𝐴) +ℎ 𝐶) = (𝐵 +ℎ (𝐴 +ℎ 𝐶)) |
8 | 4, 6, 7 | 3eqtr3i 2826 | 1 ⊢ (𝐴 +ℎ (𝐵 +ℎ 𝐶)) = (𝐵 +ℎ (𝐴 +ℎ 𝐶)) |
Colors of variables: wff setvar class |
Syntax hints: = wceq 1522 ∈ wcel 2080 (class class class)co 7019 ℋchba 28379 +ℎ cva 28380 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1778 ax-4 1792 ax-5 1889 ax-6 1948 ax-7 1993 ax-8 2082 ax-9 2090 ax-10 2111 ax-11 2125 ax-12 2140 ax-ext 2768 ax-hvcom 28461 ax-hvass 28462 |
This theorem depends on definitions: df-bi 208 df-an 397 df-or 843 df-3an 1082 df-tru 1525 df-ex 1763 df-nf 1767 df-sb 2042 df-clab 2775 df-cleq 2787 df-clel 2862 df-nfc 2934 df-rex 3110 df-rab 3113 df-v 3438 df-dif 3864 df-un 3866 df-in 3868 df-ss 3876 df-nul 4214 df-if 4384 df-sn 4475 df-pr 4477 df-op 4481 df-uni 4748 df-br 4965 df-iota 6192 df-fv 6236 df-ov 7022 |
This theorem is referenced by: hvsubaddi 28526 |
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