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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 27856 | . . 3 ⊢ ℋ ∈ V | |
2 | ax-hfvadd 27857 | . . 3 ⊢ +ℎ :( ℋ × ℋ)⟶ ℋ | |
3 | ax-hvass 27859 | . . 3 ⊢ ((𝑥 ∈ ℋ ∧ 𝑦 ∈ ℋ ∧ 𝑧 ∈ ℋ) → ((𝑥 +ℎ 𝑦) +ℎ 𝑧) = (𝑥 +ℎ (𝑦 +ℎ 𝑧))) | |
4 | ax-hv0cl 27860 | . . 3 ⊢ 0ℎ ∈ ℋ | |
5 | hvaddid2 27880 | . . 3 ⊢ (𝑥 ∈ ℋ → (0ℎ +ℎ 𝑥) = 𝑥) | |
6 | neg1cn 11124 | . . . 4 ⊢ -1 ∈ ℂ | |
7 | hvmulcl 27870 | . . . 4 ⊢ ((-1 ∈ ℂ ∧ 𝑥 ∈ ℋ) → (-1 ·ℎ 𝑥) ∈ ℋ) | |
8 | 6, 7 | mpan 706 | . . 3 ⊢ (𝑥 ∈ ℋ → (-1 ·ℎ 𝑥) ∈ ℋ) |
9 | ax-hvcom 27858 | . . . . 5 ⊢ (((-1 ·ℎ 𝑥) ∈ ℋ ∧ 𝑥 ∈ ℋ) → ((-1 ·ℎ 𝑥) +ℎ 𝑥) = (𝑥 +ℎ (-1 ·ℎ 𝑥))) | |
10 | 8, 9 | mpancom 703 | . . . 4 ⊢ (𝑥 ∈ ℋ → ((-1 ·ℎ 𝑥) +ℎ 𝑥) = (𝑥 +ℎ (-1 ·ℎ 𝑥))) |
11 | hvnegid 27884 | . . . 4 ⊢ (𝑥 ∈ ℋ → (𝑥 +ℎ (-1 ·ℎ 𝑥)) = 0ℎ) | |
12 | 10, 11 | eqtrd 2656 | . . 3 ⊢ (𝑥 ∈ ℋ → ((-1 ·ℎ 𝑥) +ℎ 𝑥) = 0ℎ) |
13 | 1, 2, 3, 4, 5, 8, 12 | isgrpoi 27352 | . 2 ⊢ +ℎ ∈ GrpOp |
14 | 2 | fdmi 6052 | . 2 ⊢ dom +ℎ = ( ℋ × ℋ) |
15 | ax-hvcom 27858 | . 2 ⊢ ((𝑥 ∈ ℋ ∧ 𝑦 ∈ ℋ) → (𝑥 +ℎ 𝑦) = (𝑦 +ℎ 𝑥)) | |
16 | 13, 14, 15 | isabloi 27405 | 1 ⊢ +ℎ ∈ AbelOp |
Colors of variables: wff setvar class |
Syntax hints: = wceq 1483 ∈ wcel 1990 × cxp 5112 (class class class)co 6650 ℂcc 9934 1c1 9937 -cneg 10267 AbelOpcablo 27398 ℋchil 27776 +ℎ cva 27777 ·ℎ csm 27778 0ℎc0v 27781 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1722 ax-4 1737 ax-5 1839 ax-6 1888 ax-7 1935 ax-8 1992 ax-9 1999 ax-10 2019 ax-11 2034 ax-12 2047 ax-13 2246 ax-ext 2602 ax-rep 4771 ax-sep 4781 ax-nul 4789 ax-pow 4843 ax-pr 4906 ax-un 6949 ax-resscn 9993 ax-1cn 9994 ax-icn 9995 ax-addcl 9996 ax-addrcl 9997 ax-mulcl 9998 ax-mulrcl 9999 ax-mulcom 10000 ax-addass 10001 ax-mulass 10002 ax-distr 10003 ax-i2m1 10004 ax-1ne0 10005 ax-1rid 10006 ax-rnegex 10007 ax-rrecex 10008 ax-cnre 10009 ax-pre-lttri 10010 ax-pre-lttrn 10011 ax-pre-ltadd 10012 ax-hilex 27856 ax-hfvadd 27857 ax-hvcom 27858 ax-hvass 27859 ax-hv0cl 27860 ax-hvaddid 27861 ax-hfvmul 27862 ax-hvmulid 27863 ax-hvdistr2 27866 ax-hvmul0 27867 |
This theorem depends on definitions: df-bi 197 df-or 385 df-an 386 df-3or 1038 df-3an 1039 df-tru 1486 df-ex 1705 df-nf 1710 df-sb 1881 df-eu 2474 df-mo 2475 df-clab 2609 df-cleq 2615 df-clel 2618 df-nfc 2753 df-ne 2795 df-nel 2898 df-ral 2917 df-rex 2918 df-reu 2919 df-rab 2921 df-v 3202 df-sbc 3436 df-csb 3534 df-dif 3577 df-un 3579 df-in 3581 df-ss 3588 df-nul 3916 df-if 4087 df-pw 4160 df-sn 4178 df-pr 4180 df-op 4184 df-uni 4437 df-iun 4522 df-br 4654 df-opab 4713 df-mpt 4730 df-id 5024 df-po 5035 df-so 5036 df-xp 5120 df-rel 5121 df-cnv 5122 df-co 5123 df-dm 5124 df-rn 5125 df-res 5126 df-ima 5127 df-iota 5851 df-fun 5890 df-fn 5891 df-f 5892 df-f1 5893 df-fo 5894 df-f1o 5895 df-fv 5896 df-riota 6611 df-ov 6653 df-oprab 6654 df-mpt2 6655 df-er 7742 df-en 7956 df-dom 7957 df-sdom 7958 df-pnf 10076 df-mnf 10077 df-ltxr 10079 df-sub 10268 df-neg 10269 df-grpo 27347 df-ablo 27399 df-hvsub 27828 |
This theorem is referenced by: hilid 28018 hilvc 28019 hhnv 28022 hhba 28024 hhph 28035 hhssva 28114 hhsssm 28115 hhssabloilem 28118 hhshsslem1 28124 shsval 28171 |
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