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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 29195 | . 2 ⊢ +ℎ ∈ AbelOp | |
2 | ax-hfvadd 29035 | . . 3 ⊢ +ℎ :( ℋ × ℋ)⟶ ℋ | |
3 | 2 | fdmi 6535 | . 2 ⊢ dom +ℎ = ( ℋ × ℋ) |
4 | ax-hfvmul 29040 | . 2 ⊢ ·ℎ :(ℂ × ℋ)⟶ ℋ | |
5 | ax-hvmulid 29041 | . 2 ⊢ (𝑥 ∈ ℋ → (1 ·ℎ 𝑥) = 𝑥) | |
6 | ax-hvdistr1 29043 | . 2 ⊢ ((𝑦 ∈ ℂ ∧ 𝑥 ∈ ℋ ∧ 𝑧 ∈ ℋ) → (𝑦 ·ℎ (𝑥 +ℎ 𝑧)) = ((𝑦 ·ℎ 𝑥) +ℎ (𝑦 ·ℎ 𝑧))) | |
7 | ax-hvdistr2 29044 | . 2 ⊢ ((𝑦 ∈ ℂ ∧ 𝑧 ∈ ℂ ∧ 𝑥 ∈ ℋ) → ((𝑦 + 𝑧) ·ℎ 𝑥) = ((𝑦 ·ℎ 𝑥) +ℎ (𝑧 ·ℎ 𝑥))) | |
8 | ax-hvmulass 29042 | . 2 ⊢ ((𝑦 ∈ ℂ ∧ 𝑧 ∈ ℂ ∧ 𝑥 ∈ ℋ) → ((𝑦 · 𝑧) ·ℎ 𝑥) = (𝑦 ·ℎ (𝑧 ·ℎ 𝑥))) | |
9 | eqid 2736 | . 2 ⊢ 〈 +ℎ , ·ℎ 〉 = 〈 +ℎ , ·ℎ 〉 | |
10 | 1, 3, 4, 5, 6, 7, 8, 9 | isvciOLD 28615 | 1 ⊢ 〈 +ℎ , ·ℎ 〉 ∈ CVecOLD |
Colors of variables: wff setvar class |
Syntax hints: ∈ wcel 2112 〈cop 4533 × cxp 5534 CVecOLDcvc 28593 ℋchba 28954 +ℎ cva 28955 ·ℎ csm 28956 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1803 ax-4 1817 ax-5 1918 ax-6 1976 ax-7 2018 ax-8 2114 ax-9 2122 ax-10 2143 ax-11 2160 ax-12 2177 ax-ext 2708 ax-rep 5164 ax-sep 5177 ax-nul 5184 ax-pow 5243 ax-pr 5307 ax-un 7501 ax-cnex 10750 ax-resscn 10751 ax-1cn 10752 ax-icn 10753 ax-addcl 10754 ax-addrcl 10755 ax-mulcl 10756 ax-mulrcl 10757 ax-mulcom 10758 ax-addass 10759 ax-mulass 10760 ax-distr 10761 ax-i2m1 10762 ax-1ne0 10763 ax-1rid 10764 ax-rnegex 10765 ax-rrecex 10766 ax-cnre 10767 ax-pre-lttri 10768 ax-pre-lttrn 10769 ax-pre-ltadd 10770 ax-hilex 29034 ax-hfvadd 29035 ax-hvcom 29036 ax-hvass 29037 ax-hv0cl 29038 ax-hvaddid 29039 ax-hfvmul 29040 ax-hvmulid 29041 ax-hvmulass 29042 ax-hvdistr1 29043 ax-hvdistr2 29044 ax-hvmul0 29045 |
This theorem depends on definitions: df-bi 210 df-an 400 df-or 848 df-3or 1090 df-3an 1091 df-tru 1546 df-fal 1556 df-ex 1788 df-nf 1792 df-sb 2073 df-mo 2539 df-eu 2568 df-clab 2715 df-cleq 2728 df-clel 2809 df-nfc 2879 df-ne 2933 df-nel 3037 df-ral 3056 df-rex 3057 df-reu 3058 df-rab 3060 df-v 3400 df-sbc 3684 df-csb 3799 df-dif 3856 df-un 3858 df-in 3860 df-ss 3870 df-nul 4224 df-if 4426 df-pw 4501 df-sn 4528 df-pr 4530 df-op 4534 df-uni 4806 df-iun 4892 df-br 5040 df-opab 5102 df-mpt 5121 df-id 5440 df-po 5453 df-so 5454 df-xp 5542 df-rel 5543 df-cnv 5544 df-co 5545 df-dm 5546 df-rn 5547 df-res 5548 df-ima 5549 df-iota 6316 df-fun 6360 df-fn 6361 df-f 6362 df-f1 6363 df-fo 6364 df-f1o 6365 df-fv 6366 df-riota 7148 df-ov 7194 df-oprab 7195 df-mpo 7196 df-er 8369 df-en 8605 df-dom 8606 df-sdom 8607 df-pnf 10834 df-mnf 10835 df-ltxr 10837 df-sub 11029 df-neg 11030 df-grpo 28528 df-ablo 28580 df-vc 28594 df-hvsub 29006 |
This theorem is referenced by: hhnv 29200 |
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