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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 28937 | . 2 ⊢ +ℎ ∈ AbelOp | |
2 | ax-hfvadd 28777 | . . 3 ⊢ +ℎ :( ℋ × ℋ)⟶ ℋ | |
3 | 2 | fdmi 6524 | . 2 ⊢ dom +ℎ = ( ℋ × ℋ) |
4 | ax-hfvmul 28782 | . 2 ⊢ ·ℎ :(ℂ × ℋ)⟶ ℋ | |
5 | ax-hvmulid 28783 | . 2 ⊢ (𝑥 ∈ ℋ → (1 ·ℎ 𝑥) = 𝑥) | |
6 | ax-hvdistr1 28785 | . 2 ⊢ ((𝑦 ∈ ℂ ∧ 𝑥 ∈ ℋ ∧ 𝑧 ∈ ℋ) → (𝑦 ·ℎ (𝑥 +ℎ 𝑧)) = ((𝑦 ·ℎ 𝑥) +ℎ (𝑦 ·ℎ 𝑧))) | |
7 | ax-hvdistr2 28786 | . 2 ⊢ ((𝑦 ∈ ℂ ∧ 𝑧 ∈ ℂ ∧ 𝑥 ∈ ℋ) → ((𝑦 + 𝑧) ·ℎ 𝑥) = ((𝑦 ·ℎ 𝑥) +ℎ (𝑧 ·ℎ 𝑥))) | |
8 | ax-hvmulass 28784 | . 2 ⊢ ((𝑦 ∈ ℂ ∧ 𝑧 ∈ ℂ ∧ 𝑥 ∈ ℋ) → ((𝑦 · 𝑧) ·ℎ 𝑥) = (𝑦 ·ℎ (𝑧 ·ℎ 𝑥))) | |
9 | eqid 2821 | . 2 ⊢ 〈 +ℎ , ·ℎ 〉 = 〈 +ℎ , ·ℎ 〉 | |
10 | 1, 3, 4, 5, 6, 7, 8, 9 | isvciOLD 28357 | 1 ⊢ 〈 +ℎ , ·ℎ 〉 ∈ CVecOLD |
Colors of variables: wff setvar class |
Syntax hints: ∈ wcel 2114 〈cop 4573 × cxp 5553 CVecOLDcvc 28335 ℋchba 28696 +ℎ cva 28697 ·ℎ csm 28698 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1796 ax-4 1810 ax-5 1911 ax-6 1970 ax-7 2015 ax-8 2116 ax-9 2124 ax-10 2145 ax-11 2161 ax-12 2177 ax-ext 2793 ax-rep 5190 ax-sep 5203 ax-nul 5210 ax-pow 5266 ax-pr 5330 ax-un 7461 ax-cnex 10593 ax-resscn 10594 ax-1cn 10595 ax-icn 10596 ax-addcl 10597 ax-addrcl 10598 ax-mulcl 10599 ax-mulrcl 10600 ax-mulcom 10601 ax-addass 10602 ax-mulass 10603 ax-distr 10604 ax-i2m1 10605 ax-1ne0 10606 ax-1rid 10607 ax-rnegex 10608 ax-rrecex 10609 ax-cnre 10610 ax-pre-lttri 10611 ax-pre-lttrn 10612 ax-pre-ltadd 10613 ax-hilex 28776 ax-hfvadd 28777 ax-hvcom 28778 ax-hvass 28779 ax-hv0cl 28780 ax-hvaddid 28781 ax-hfvmul 28782 ax-hvmulid 28783 ax-hvmulass 28784 ax-hvdistr1 28785 ax-hvdistr2 28786 ax-hvmul0 28787 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3or 1084 df-3an 1085 df-tru 1540 df-ex 1781 df-nf 1785 df-sb 2070 df-mo 2622 df-eu 2654 df-clab 2800 df-cleq 2814 df-clel 2893 df-nfc 2963 df-ne 3017 df-nel 3124 df-ral 3143 df-rex 3144 df-reu 3145 df-rab 3147 df-v 3496 df-sbc 3773 df-csb 3884 df-dif 3939 df-un 3941 df-in 3943 df-ss 3952 df-nul 4292 df-if 4468 df-pw 4541 df-sn 4568 df-pr 4570 df-op 4574 df-uni 4839 df-iun 4921 df-br 5067 df-opab 5129 df-mpt 5147 df-id 5460 df-po 5474 df-so 5475 df-xp 5561 df-rel 5562 df-cnv 5563 df-co 5564 df-dm 5565 df-rn 5566 df-res 5567 df-ima 5568 df-iota 6314 df-fun 6357 df-fn 6358 df-f 6359 df-f1 6360 df-fo 6361 df-f1o 6362 df-fv 6363 df-riota 7114 df-ov 7159 df-oprab 7160 df-mpo 7161 df-er 8289 df-en 8510 df-dom 8511 df-sdom 8512 df-pnf 10677 df-mnf 10678 df-ltxr 10680 df-sub 10872 df-neg 10873 df-grpo 28270 df-ablo 28322 df-vc 28336 df-hvsub 28748 |
This theorem is referenced by: hhnv 28942 |
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