![]() |
Hilbert Space Explorer |
< Previous
Next >
Nearby theorems |
|
Mirrors > Home > HSE Home > Th. List > hilvc | Structured version Visualization version GIF version |
Description: Hilbert space is a complex vector space. Vector addition is +ℎ, and scalar product is ·ℎ. (Contributed by NM, 15-Apr-2007.) (New usage is discouraged.) |
Ref | Expression |
---|---|
hilvc | ⊢ 〈 +ℎ , ·ℎ 〉 ∈ CVecOLD |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | hilablo 31085 | . 2 ⊢ +ℎ ∈ AbelOp | |
2 | ax-hfvadd 30925 | . . 3 ⊢ +ℎ :( ℋ × ℋ)⟶ ℋ | |
3 | 2 | fdmi 6738 | . 2 ⊢ dom +ℎ = ( ℋ × ℋ) |
4 | ax-hfvmul 30930 | . 2 ⊢ ·ℎ :(ℂ × ℋ)⟶ ℋ | |
5 | ax-hvmulid 30931 | . 2 ⊢ (𝑥 ∈ ℋ → (1 ·ℎ 𝑥) = 𝑥) | |
6 | ax-hvdistr1 30933 | . 2 ⊢ ((𝑦 ∈ ℂ ∧ 𝑥 ∈ ℋ ∧ 𝑧 ∈ ℋ) → (𝑦 ·ℎ (𝑥 +ℎ 𝑧)) = ((𝑦 ·ℎ 𝑥) +ℎ (𝑦 ·ℎ 𝑧))) | |
7 | ax-hvdistr2 30934 | . 2 ⊢ ((𝑦 ∈ ℂ ∧ 𝑧 ∈ ℂ ∧ 𝑥 ∈ ℋ) → ((𝑦 + 𝑧) ·ℎ 𝑥) = ((𝑦 ·ℎ 𝑥) +ℎ (𝑧 ·ℎ 𝑥))) | |
8 | ax-hvmulass 30932 | . 2 ⊢ ((𝑦 ∈ ℂ ∧ 𝑧 ∈ ℂ ∧ 𝑥 ∈ ℋ) → ((𝑦 · 𝑧) ·ℎ 𝑥) = (𝑦 ·ℎ (𝑧 ·ℎ 𝑥))) | |
9 | eqid 2725 | . 2 ⊢ 〈 +ℎ , ·ℎ 〉 = 〈 +ℎ , ·ℎ 〉 | |
10 | 1, 3, 4, 5, 6, 7, 8, 9 | isvciOLD 30505 | 1 ⊢ 〈 +ℎ , ·ℎ 〉 ∈ CVecOLD |
Colors of variables: wff setvar class |
Syntax hints: ∈ wcel 2098 〈cop 4638 × cxp 5679 CVecOLDcvc 30483 ℋchba 30844 +ℎ cva 30845 ·ℎ csm 30846 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1789 ax-4 1803 ax-5 1905 ax-6 1963 ax-7 2003 ax-8 2100 ax-9 2108 ax-10 2129 ax-11 2146 ax-12 2166 ax-ext 2696 ax-rep 5289 ax-sep 5303 ax-nul 5310 ax-pow 5368 ax-pr 5432 ax-un 7745 ax-cnex 11210 ax-resscn 11211 ax-1cn 11212 ax-icn 11213 ax-addcl 11214 ax-addrcl 11215 ax-mulcl 11216 ax-mulrcl 11217 ax-mulcom 11218 ax-addass 11219 ax-mulass 11220 ax-distr 11221 ax-i2m1 11222 ax-1ne0 11223 ax-1rid 11224 ax-rnegex 11225 ax-rrecex 11226 ax-cnre 11227 ax-pre-lttri 11228 ax-pre-lttrn 11229 ax-pre-ltadd 11230 ax-hilex 30924 ax-hfvadd 30925 ax-hvcom 30926 ax-hvass 30927 ax-hv0cl 30928 ax-hvaddid 30929 ax-hfvmul 30930 ax-hvmulid 30931 ax-hvmulass 30932 ax-hvdistr1 30933 ax-hvdistr2 30934 ax-hvmul0 30935 |
This theorem depends on definitions: df-bi 206 df-an 395 df-or 846 df-3or 1085 df-3an 1086 df-tru 1536 df-fal 1546 df-ex 1774 df-nf 1778 df-sb 2060 df-mo 2528 df-eu 2557 df-clab 2703 df-cleq 2717 df-clel 2802 df-nfc 2877 df-ne 2930 df-nel 3036 df-ral 3051 df-rex 3060 df-reu 3364 df-rab 3419 df-v 3463 df-sbc 3776 df-csb 3892 df-dif 3949 df-un 3951 df-in 3953 df-ss 3963 df-nul 4325 df-if 4533 df-pw 4608 df-sn 4633 df-pr 4635 df-op 4639 df-uni 4913 df-iun 5002 df-br 5153 df-opab 5215 df-mpt 5236 df-id 5579 df-po 5593 df-so 5594 df-xp 5687 df-rel 5688 df-cnv 5689 df-co 5690 df-dm 5691 df-rn 5692 df-res 5693 df-ima 5694 df-iota 6505 df-fun 6555 df-fn 6556 df-f 6557 df-f1 6558 df-fo 6559 df-f1o 6560 df-fv 6561 df-riota 7379 df-ov 7426 df-oprab 7427 df-mpo 7428 df-er 8733 df-en 8974 df-dom 8975 df-sdom 8976 df-pnf 11296 df-mnf 11297 df-ltxr 11299 df-sub 11492 df-neg 11493 df-grpo 30418 df-ablo 30470 df-vc 30484 df-hvsub 30896 |
This theorem is referenced by: hhnv 31090 |
Copyright terms: Public domain | W3C validator |