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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 31018 | . . 3 ⊢ ℋ ∈ V | |
| 2 | ax-hfvadd 31019 | . . 3 ⊢ +ℎ :( ℋ × ℋ)⟶ ℋ | |
| 3 | ax-hvass 31021 | . . 3 ⊢ ((𝑥 ∈ ℋ ∧ 𝑦 ∈ ℋ ∧ 𝑧 ∈ ℋ) → ((𝑥 +ℎ 𝑦) +ℎ 𝑧) = (𝑥 +ℎ (𝑦 +ℎ 𝑧))) | |
| 4 | ax-hv0cl 31022 | . . 3 ⊢ 0ℎ ∈ ℋ | |
| 5 | hvaddlid 31042 | . . 3 ⊢ (𝑥 ∈ ℋ → (0ℎ +ℎ 𝑥) = 𝑥) | |
| 6 | neg1cn 12380 | . . . 4 ⊢ -1 ∈ ℂ | |
| 7 | hvmulcl 31032 | . . . 4 ⊢ ((-1 ∈ ℂ ∧ 𝑥 ∈ ℋ) → (-1 ·ℎ 𝑥) ∈ ℋ) | |
| 8 | 6, 7 | mpan 690 | . . 3 ⊢ (𝑥 ∈ ℋ → (-1 ·ℎ 𝑥) ∈ ℋ) | 
| 9 | ax-hvcom 31020 | . . . . 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 31046 | . . . 4 ⊢ (𝑥 ∈ ℋ → (𝑥 +ℎ (-1 ·ℎ 𝑥)) = 0ℎ) | |
| 12 | 10, 11 | eqtrd 2777 | . . 3 ⊢ (𝑥 ∈ ℋ → ((-1 ·ℎ 𝑥) +ℎ 𝑥) = 0ℎ) | 
| 13 | 1, 2, 3, 4, 5, 8, 12 | isgrpoi 30517 | . 2 ⊢ +ℎ ∈ GrpOp | 
| 14 | 2 | fdmi 6747 | . 2 ⊢ dom +ℎ = ( ℋ × ℋ) | 
| 15 | ax-hvcom 31020 | . 2 ⊢ ((𝑥 ∈ ℋ ∧ 𝑦 ∈ ℋ) → (𝑥 +ℎ 𝑦) = (𝑦 +ℎ 𝑥)) | |
| 16 | 13, 14, 15 | isabloi 30570 | 1 ⊢ +ℎ ∈ AbelOp | 
| Colors of variables: wff setvar class | 
| Syntax hints: = wceq 1540 ∈ wcel 2108 × cxp 5683 (class class class)co 7431 ℂcc 11153 1c1 11156 -cneg 11493 AbelOpcablo 30563 ℋchba 30938 +ℎ cva 30939 ·ℎ csm 30940 0ℎc0v 30943 | 
| This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1795 ax-4 1809 ax-5 1910 ax-6 1967 ax-7 2007 ax-8 2110 ax-9 2118 ax-10 2141 ax-11 2157 ax-12 2177 ax-ext 2708 ax-rep 5279 ax-sep 5296 ax-nul 5306 ax-pow 5365 ax-pr 5432 ax-un 7755 ax-resscn 11212 ax-1cn 11213 ax-icn 11214 ax-addcl 11215 ax-addrcl 11216 ax-mulcl 11217 ax-mulrcl 11218 ax-mulcom 11219 ax-addass 11220 ax-mulass 11221 ax-distr 11222 ax-i2m1 11223 ax-1ne0 11224 ax-1rid 11225 ax-rnegex 11226 ax-rrecex 11227 ax-cnre 11228 ax-pre-lttri 11229 ax-pre-lttrn 11230 ax-pre-ltadd 11231 ax-hilex 31018 ax-hfvadd 31019 ax-hvcom 31020 ax-hvass 31021 ax-hv0cl 31022 ax-hvaddid 31023 ax-hfvmul 31024 ax-hvmulid 31025 ax-hvdistr2 31028 ax-hvmul0 31029 | 
| This theorem depends on definitions: df-bi 207 df-an 396 df-or 849 df-3or 1088 df-3an 1089 df-tru 1543 df-fal 1553 df-ex 1780 df-nf 1784 df-sb 2065 df-mo 2540 df-eu 2569 df-clab 2715 df-cleq 2729 df-clel 2816 df-nfc 2892 df-ne 2941 df-nel 3047 df-ral 3062 df-rex 3071 df-reu 3381 df-rab 3437 df-v 3482 df-sbc 3789 df-csb 3900 df-dif 3954 df-un 3956 df-in 3958 df-ss 3968 df-nul 4334 df-if 4526 df-pw 4602 df-sn 4627 df-pr 4629 df-op 4633 df-uni 4908 df-iun 4993 df-br 5144 df-opab 5206 df-mpt 5226 df-id 5578 df-po 5592 df-so 5593 df-xp 5691 df-rel 5692 df-cnv 5693 df-co 5694 df-dm 5695 df-rn 5696 df-res 5697 df-ima 5698 df-iota 6514 df-fun 6563 df-fn 6564 df-f 6565 df-f1 6566 df-fo 6567 df-f1o 6568 df-fv 6569 df-riota 7388 df-ov 7434 df-oprab 7435 df-mpo 7436 df-er 8745 df-en 8986 df-dom 8987 df-sdom 8988 df-pnf 11297 df-mnf 11298 df-ltxr 11300 df-sub 11494 df-neg 11495 df-grpo 30512 df-ablo 30564 df-hvsub 30990 | 
| This theorem is referenced by: hilid 31180 hilvc 31181 hhnv 31184 hhba 31186 hhph 31197 hhssva 31276 hhsssm 31277 hhssabloilem 31280 hhshsslem1 31286 shsval 31331 | 
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