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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 29358 | . . 3 ⊢ ℋ ∈ V | |
2 | ax-hfvadd 29359 | . . 3 ⊢ +ℎ :( ℋ × ℋ)⟶ ℋ | |
3 | ax-hvass 29361 | . . 3 ⊢ ((𝑥 ∈ ℋ ∧ 𝑦 ∈ ℋ ∧ 𝑧 ∈ ℋ) → ((𝑥 +ℎ 𝑦) +ℎ 𝑧) = (𝑥 +ℎ (𝑦 +ℎ 𝑧))) | |
4 | ax-hv0cl 29362 | . . 3 ⊢ 0ℎ ∈ ℋ | |
5 | hvaddid2 29382 | . . 3 ⊢ (𝑥 ∈ ℋ → (0ℎ +ℎ 𝑥) = 𝑥) | |
6 | neg1cn 12085 | . . . 4 ⊢ -1 ∈ ℂ | |
7 | hvmulcl 29372 | . . . 4 ⊢ ((-1 ∈ ℂ ∧ 𝑥 ∈ ℋ) → (-1 ·ℎ 𝑥) ∈ ℋ) | |
8 | 6, 7 | mpan 687 | . . 3 ⊢ (𝑥 ∈ ℋ → (-1 ·ℎ 𝑥) ∈ ℋ) |
9 | ax-hvcom 29360 | . . . . 5 ⊢ (((-1 ·ℎ 𝑥) ∈ ℋ ∧ 𝑥 ∈ ℋ) → ((-1 ·ℎ 𝑥) +ℎ 𝑥) = (𝑥 +ℎ (-1 ·ℎ 𝑥))) | |
10 | 8, 9 | mpancom 685 | . . . 4 ⊢ (𝑥 ∈ ℋ → ((-1 ·ℎ 𝑥) +ℎ 𝑥) = (𝑥 +ℎ (-1 ·ℎ 𝑥))) |
11 | hvnegid 29386 | . . . 4 ⊢ (𝑥 ∈ ℋ → (𝑥 +ℎ (-1 ·ℎ 𝑥)) = 0ℎ) | |
12 | 10, 11 | eqtrd 2778 | . . 3 ⊢ (𝑥 ∈ ℋ → ((-1 ·ℎ 𝑥) +ℎ 𝑥) = 0ℎ) |
13 | 1, 2, 3, 4, 5, 8, 12 | isgrpoi 28857 | . 2 ⊢ +ℎ ∈ GrpOp |
14 | 2 | fdmi 6614 | . 2 ⊢ dom +ℎ = ( ℋ × ℋ) |
15 | ax-hvcom 29360 | . 2 ⊢ ((𝑥 ∈ ℋ ∧ 𝑦 ∈ ℋ) → (𝑥 +ℎ 𝑦) = (𝑦 +ℎ 𝑥)) | |
16 | 13, 14, 15 | isabloi 28910 | 1 ⊢ +ℎ ∈ AbelOp |
Colors of variables: wff setvar class |
Syntax hints: = wceq 1539 ∈ wcel 2106 × cxp 5589 (class class class)co 7277 ℂcc 10867 1c1 10870 -cneg 11204 AbelOpcablo 28903 ℋchba 29278 +ℎ cva 29279 ·ℎ csm 29280 0ℎc0v 29283 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1798 ax-4 1812 ax-5 1913 ax-6 1971 ax-7 2011 ax-8 2108 ax-9 2116 ax-10 2137 ax-11 2154 ax-12 2171 ax-ext 2709 ax-rep 5211 ax-sep 5225 ax-nul 5232 ax-pow 5290 ax-pr 5354 ax-un 7588 ax-resscn 10926 ax-1cn 10927 ax-icn 10928 ax-addcl 10929 ax-addrcl 10930 ax-mulcl 10931 ax-mulrcl 10932 ax-mulcom 10933 ax-addass 10934 ax-mulass 10935 ax-distr 10936 ax-i2m1 10937 ax-1ne0 10938 ax-1rid 10939 ax-rnegex 10940 ax-rrecex 10941 ax-cnre 10942 ax-pre-lttri 10943 ax-pre-lttrn 10944 ax-pre-ltadd 10945 ax-hilex 29358 ax-hfvadd 29359 ax-hvcom 29360 ax-hvass 29361 ax-hv0cl 29362 ax-hvaddid 29363 ax-hfvmul 29364 ax-hvmulid 29365 ax-hvdistr2 29368 ax-hvmul0 29369 |
This theorem depends on definitions: df-bi 206 df-an 397 df-or 845 df-3or 1087 df-3an 1088 df-tru 1542 df-fal 1552 df-ex 1783 df-nf 1787 df-sb 2068 df-mo 2540 df-eu 2569 df-clab 2716 df-cleq 2730 df-clel 2816 df-nfc 2889 df-ne 2944 df-nel 3050 df-ral 3069 df-rex 3070 df-reu 3072 df-rab 3073 df-v 3433 df-sbc 3718 df-csb 3834 df-dif 3891 df-un 3893 df-in 3895 df-ss 3905 df-nul 4259 df-if 4462 df-pw 4537 df-sn 4564 df-pr 4566 df-op 4570 df-uni 4842 df-iun 4928 df-br 5077 df-opab 5139 df-mpt 5160 df-id 5491 df-po 5505 df-so 5506 df-xp 5597 df-rel 5598 df-cnv 5599 df-co 5600 df-dm 5601 df-rn 5602 df-res 5603 df-ima 5604 df-iota 6393 df-fun 6437 df-fn 6438 df-f 6439 df-f1 6440 df-fo 6441 df-f1o 6442 df-fv 6443 df-riota 7234 df-ov 7280 df-oprab 7281 df-mpo 7282 df-er 8496 df-en 8732 df-dom 8733 df-sdom 8734 df-pnf 11009 df-mnf 11010 df-ltxr 11012 df-sub 11205 df-neg 11206 df-grpo 28852 df-ablo 28904 df-hvsub 29330 |
This theorem is referenced by: hilid 29520 hilvc 29521 hhnv 29524 hhba 29526 hhph 29537 hhssva 29616 hhsssm 29617 hhssabloilem 29620 hhshsslem1 29626 shsval 29671 |
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