![]() |
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 28710 | . 2 ⊢ +ℎ ∈ AbelOp | |
2 | ax-hfvadd 28550 | . . 3 ⊢ +ℎ :( ℋ × ℋ)⟶ ℋ | |
3 | 2 | fdmi 6348 | . 2 ⊢ dom +ℎ = ( ℋ × ℋ) |
4 | ax-hfvmul 28555 | . 2 ⊢ ·ℎ :(ℂ × ℋ)⟶ ℋ | |
5 | ax-hvmulid 28556 | . 2 ⊢ (𝑥 ∈ ℋ → (1 ·ℎ 𝑥) = 𝑥) | |
6 | ax-hvdistr1 28558 | . 2 ⊢ ((𝑦 ∈ ℂ ∧ 𝑥 ∈ ℋ ∧ 𝑧 ∈ ℋ) → (𝑦 ·ℎ (𝑥 +ℎ 𝑧)) = ((𝑦 ·ℎ 𝑥) +ℎ (𝑦 ·ℎ 𝑧))) | |
7 | ax-hvdistr2 28559 | . 2 ⊢ ((𝑦 ∈ ℂ ∧ 𝑧 ∈ ℂ ∧ 𝑥 ∈ ℋ) → ((𝑦 + 𝑧) ·ℎ 𝑥) = ((𝑦 ·ℎ 𝑥) +ℎ (𝑧 ·ℎ 𝑥))) | |
8 | ax-hvmulass 28557 | . 2 ⊢ ((𝑦 ∈ ℂ ∧ 𝑧 ∈ ℂ ∧ 𝑥 ∈ ℋ) → ((𝑦 · 𝑧) ·ℎ 𝑥) = (𝑦 ·ℎ (𝑧 ·ℎ 𝑥))) | |
9 | eqid 2772 | . 2 ⊢ 〈 +ℎ , ·ℎ 〉 = 〈 +ℎ , ·ℎ 〉 | |
10 | 1, 3, 4, 5, 6, 7, 8, 9 | isvciOLD 28128 | 1 ⊢ 〈 +ℎ , ·ℎ 〉 ∈ CVecOLD |
Colors of variables: wff setvar class |
Syntax hints: ∈ wcel 2050 〈cop 4441 × cxp 5399 CVecOLDcvc 28106 ℋchba 28469 +ℎ cva 28470 ·ℎ csm 28471 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1758 ax-4 1772 ax-5 1869 ax-6 1928 ax-7 1965 ax-8 2052 ax-9 2059 ax-10 2079 ax-11 2093 ax-12 2106 ax-13 2301 ax-ext 2744 ax-rep 5043 ax-sep 5054 ax-nul 5061 ax-pow 5113 ax-pr 5180 ax-un 7273 ax-cnex 10385 ax-resscn 10386 ax-1cn 10387 ax-icn 10388 ax-addcl 10389 ax-addrcl 10390 ax-mulcl 10391 ax-mulrcl 10392 ax-mulcom 10393 ax-addass 10394 ax-mulass 10395 ax-distr 10396 ax-i2m1 10397 ax-1ne0 10398 ax-1rid 10399 ax-rnegex 10400 ax-rrecex 10401 ax-cnre 10402 ax-pre-lttri 10403 ax-pre-lttrn 10404 ax-pre-ltadd 10405 ax-hilex 28549 ax-hfvadd 28550 ax-hvcom 28551 ax-hvass 28552 ax-hv0cl 28553 ax-hvaddid 28554 ax-hfvmul 28555 ax-hvmulid 28556 ax-hvmulass 28557 ax-hvdistr1 28558 ax-hvdistr2 28559 ax-hvmul0 28560 |
This theorem depends on definitions: df-bi 199 df-an 388 df-or 834 df-3or 1069 df-3an 1070 df-tru 1510 df-ex 1743 df-nf 1747 df-sb 2016 df-mo 2547 df-eu 2584 df-clab 2753 df-cleq 2765 df-clel 2840 df-nfc 2912 df-ne 2962 df-nel 3068 df-ral 3087 df-rex 3088 df-reu 3089 df-rab 3091 df-v 3411 df-sbc 3676 df-csb 3781 df-dif 3826 df-un 3828 df-in 3830 df-ss 3837 df-nul 4173 df-if 4345 df-pw 4418 df-sn 4436 df-pr 4438 df-op 4442 df-uni 4707 df-iun 4788 df-br 4924 df-opab 4986 df-mpt 5003 df-id 5306 df-po 5320 df-so 5321 df-xp 5407 df-rel 5408 df-cnv 5409 df-co 5410 df-dm 5411 df-rn 5412 df-res 5413 df-ima 5414 df-iota 6146 df-fun 6184 df-fn 6185 df-f 6186 df-f1 6187 df-fo 6188 df-f1o 6189 df-fv 6190 df-riota 6931 df-ov 6973 df-oprab 6974 df-mpo 6975 df-er 8083 df-en 8301 df-dom 8302 df-sdom 8303 df-pnf 10470 df-mnf 10471 df-ltxr 10473 df-sub 10666 df-neg 10667 df-grpo 28041 df-ablo 28093 df-vc 28107 df-hvsub 28521 |
This theorem is referenced by: hhnv 28715 |
Copyright terms: Public domain | W3C validator |