![]() |
Hilbert Space Explorer |
< Previous
Next >
Nearby theorems |
|
Mirrors > Home > HSE Home > Th. List > hilablo | Structured version Visualization version GIF version |
Description: Hilbert space vector addition is an Abelian group operation. (Contributed by NM, 15-Apr-2007.) (New usage is discouraged.) |
Ref | Expression |
---|---|
hilablo | ⊢ +ℎ ∈ AbelOp |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | ax-hilex 31028 | . . 3 ⊢ ℋ ∈ V | |
2 | ax-hfvadd 31029 | . . 3 ⊢ +ℎ :( ℋ × ℋ)⟶ ℋ | |
3 | ax-hvass 31031 | . . 3 ⊢ ((𝑥 ∈ ℋ ∧ 𝑦 ∈ ℋ ∧ 𝑧 ∈ ℋ) → ((𝑥 +ℎ 𝑦) +ℎ 𝑧) = (𝑥 +ℎ (𝑦 +ℎ 𝑧))) | |
4 | ax-hv0cl 31032 | . . 3 ⊢ 0ℎ ∈ ℋ | |
5 | hvaddlid 31052 | . . 3 ⊢ (𝑥 ∈ ℋ → (0ℎ +ℎ 𝑥) = 𝑥) | |
6 | neg1cn 12378 | . . . 4 ⊢ -1 ∈ ℂ | |
7 | hvmulcl 31042 | . . . 4 ⊢ ((-1 ∈ ℂ ∧ 𝑥 ∈ ℋ) → (-1 ·ℎ 𝑥) ∈ ℋ) | |
8 | 6, 7 | mpan 690 | . . 3 ⊢ (𝑥 ∈ ℋ → (-1 ·ℎ 𝑥) ∈ ℋ) |
9 | ax-hvcom 31030 | . . . . 5 ⊢ (((-1 ·ℎ 𝑥) ∈ ℋ ∧ 𝑥 ∈ ℋ) → ((-1 ·ℎ 𝑥) +ℎ 𝑥) = (𝑥 +ℎ (-1 ·ℎ 𝑥))) | |
10 | 8, 9 | mpancom 688 | . . . 4 ⊢ (𝑥 ∈ ℋ → ((-1 ·ℎ 𝑥) +ℎ 𝑥) = (𝑥 +ℎ (-1 ·ℎ 𝑥))) |
11 | hvnegid 31056 | . . . 4 ⊢ (𝑥 ∈ ℋ → (𝑥 +ℎ (-1 ·ℎ 𝑥)) = 0ℎ) | |
12 | 10, 11 | eqtrd 2775 | . . 3 ⊢ (𝑥 ∈ ℋ → ((-1 ·ℎ 𝑥) +ℎ 𝑥) = 0ℎ) |
13 | 1, 2, 3, 4, 5, 8, 12 | isgrpoi 30527 | . 2 ⊢ +ℎ ∈ GrpOp |
14 | 2 | fdmi 6748 | . 2 ⊢ dom +ℎ = ( ℋ × ℋ) |
15 | ax-hvcom 31030 | . 2 ⊢ ((𝑥 ∈ ℋ ∧ 𝑦 ∈ ℋ) → (𝑥 +ℎ 𝑦) = (𝑦 +ℎ 𝑥)) | |
16 | 13, 14, 15 | isabloi 30580 | 1 ⊢ +ℎ ∈ AbelOp |
Colors of variables: wff setvar class |
Syntax hints: = wceq 1537 ∈ wcel 2106 × cxp 5687 (class class class)co 7431 ℂcc 11151 1c1 11154 -cneg 11491 AbelOpcablo 30573 ℋchba 30948 +ℎ cva 30949 ·ℎ csm 30950 0ℎc0v 30953 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1792 ax-4 1806 ax-5 1908 ax-6 1965 ax-7 2005 ax-8 2108 ax-9 2116 ax-10 2139 ax-11 2155 ax-12 2175 ax-ext 2706 ax-rep 5285 ax-sep 5302 ax-nul 5312 ax-pow 5371 ax-pr 5438 ax-un 7754 ax-resscn 11210 ax-1cn 11211 ax-icn 11212 ax-addcl 11213 ax-addrcl 11214 ax-mulcl 11215 ax-mulrcl 11216 ax-mulcom 11217 ax-addass 11218 ax-mulass 11219 ax-distr 11220 ax-i2m1 11221 ax-1ne0 11222 ax-1rid 11223 ax-rnegex 11224 ax-rrecex 11225 ax-cnre 11226 ax-pre-lttri 11227 ax-pre-lttrn 11228 ax-pre-ltadd 11229 ax-hilex 31028 ax-hfvadd 31029 ax-hvcom 31030 ax-hvass 31031 ax-hv0cl 31032 ax-hvaddid 31033 ax-hfvmul 31034 ax-hvmulid 31035 ax-hvdistr2 31038 ax-hvmul0 31039 |
This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3or 1087 df-3an 1088 df-tru 1540 df-fal 1550 df-ex 1777 df-nf 1781 df-sb 2063 df-mo 2538 df-eu 2567 df-clab 2713 df-cleq 2727 df-clel 2814 df-nfc 2890 df-ne 2939 df-nel 3045 df-ral 3060 df-rex 3069 df-reu 3379 df-rab 3434 df-v 3480 df-sbc 3792 df-csb 3909 df-dif 3966 df-un 3968 df-in 3970 df-ss 3980 df-nul 4340 df-if 4532 df-pw 4607 df-sn 4632 df-pr 4634 df-op 4638 df-uni 4913 df-iun 4998 df-br 5149 df-opab 5211 df-mpt 5232 df-id 5583 df-po 5597 df-so 5598 df-xp 5695 df-rel 5696 df-cnv 5697 df-co 5698 df-dm 5699 df-rn 5700 df-res 5701 df-ima 5702 df-iota 6516 df-fun 6565 df-fn 6566 df-f 6567 df-f1 6568 df-fo 6569 df-f1o 6570 df-fv 6571 df-riota 7388 df-ov 7434 df-oprab 7435 df-mpo 7436 df-er 8744 df-en 8985 df-dom 8986 df-sdom 8987 df-pnf 11295 df-mnf 11296 df-ltxr 11298 df-sub 11492 df-neg 11493 df-grpo 30522 df-ablo 30574 df-hvsub 31000 |
This theorem is referenced by: hilid 31190 hilvc 31191 hhnv 31194 hhba 31196 hhph 31207 hhssva 31286 hhsssm 31287 hhssabloilem 31290 hhshsslem1 31296 shsval 31341 |
Copyright terms: Public domain | W3C validator |