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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 30105 | . 2 ⊢ +ℎ ∈ AbelOp | |
2 | ax-hfvadd 29945 | . . 3 ⊢ +ℎ :( ℋ × ℋ)⟶ ℋ | |
3 | 2 | fdmi 6681 | . 2 ⊢ dom +ℎ = ( ℋ × ℋ) |
4 | ax-hfvmul 29950 | . 2 ⊢ ·ℎ :(ℂ × ℋ)⟶ ℋ | |
5 | ax-hvmulid 29951 | . 2 ⊢ (𝑥 ∈ ℋ → (1 ·ℎ 𝑥) = 𝑥) | |
6 | ax-hvdistr1 29953 | . 2 ⊢ ((𝑦 ∈ ℂ ∧ 𝑥 ∈ ℋ ∧ 𝑧 ∈ ℋ) → (𝑦 ·ℎ (𝑥 +ℎ 𝑧)) = ((𝑦 ·ℎ 𝑥) +ℎ (𝑦 ·ℎ 𝑧))) | |
7 | ax-hvdistr2 29954 | . 2 ⊢ ((𝑦 ∈ ℂ ∧ 𝑧 ∈ ℂ ∧ 𝑥 ∈ ℋ) → ((𝑦 + 𝑧) ·ℎ 𝑥) = ((𝑦 ·ℎ 𝑥) +ℎ (𝑧 ·ℎ 𝑥))) | |
8 | ax-hvmulass 29952 | . 2 ⊢ ((𝑦 ∈ ℂ ∧ 𝑧 ∈ ℂ ∧ 𝑥 ∈ ℋ) → ((𝑦 · 𝑧) ·ℎ 𝑥) = (𝑦 ·ℎ (𝑧 ·ℎ 𝑥))) | |
9 | eqid 2737 | . 2 ⊢ ⟨ +ℎ , ·ℎ ⟩ = ⟨ +ℎ , ·ℎ ⟩ | |
10 | 1, 3, 4, 5, 6, 7, 8, 9 | isvciOLD 29525 | 1 ⊢ ⟨ +ℎ , ·ℎ ⟩ ∈ CVecOLD |
Colors of variables: wff setvar class |
Syntax hints: ∈ wcel 2107 ⟨cop 4593 × cxp 5632 CVecOLDcvc 29503 ℋchba 29864 +ℎ cva 29865 ·ℎ csm 29866 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1798 ax-4 1812 ax-5 1914 ax-6 1972 ax-7 2012 ax-8 2109 ax-9 2117 ax-10 2138 ax-11 2155 ax-12 2172 ax-ext 2708 ax-rep 5243 ax-sep 5257 ax-nul 5264 ax-pow 5321 ax-pr 5385 ax-un 7673 ax-cnex 11108 ax-resscn 11109 ax-1cn 11110 ax-icn 11111 ax-addcl 11112 ax-addrcl 11113 ax-mulcl 11114 ax-mulrcl 11115 ax-mulcom 11116 ax-addass 11117 ax-mulass 11118 ax-distr 11119 ax-i2m1 11120 ax-1ne0 11121 ax-1rid 11122 ax-rnegex 11123 ax-rrecex 11124 ax-cnre 11125 ax-pre-lttri 11126 ax-pre-lttrn 11127 ax-pre-ltadd 11128 ax-hilex 29944 ax-hfvadd 29945 ax-hvcom 29946 ax-hvass 29947 ax-hv0cl 29948 ax-hvaddid 29949 ax-hfvmul 29950 ax-hvmulid 29951 ax-hvmulass 29952 ax-hvdistr1 29953 ax-hvdistr2 29954 ax-hvmul0 29955 |
This theorem depends on definitions: df-bi 206 df-an 398 df-or 847 df-3or 1089 df-3an 1090 df-tru 1545 df-fal 1555 df-ex 1783 df-nf 1787 df-sb 2069 df-mo 2539 df-eu 2568 df-clab 2715 df-cleq 2729 df-clel 2815 df-nfc 2890 df-ne 2945 df-nel 3051 df-ral 3066 df-rex 3075 df-reu 3355 df-rab 3409 df-v 3448 df-sbc 3741 df-csb 3857 df-dif 3914 df-un 3916 df-in 3918 df-ss 3928 df-nul 4284 df-if 4488 df-pw 4563 df-sn 4588 df-pr 4590 df-op 4594 df-uni 4867 df-iun 4957 df-br 5107 df-opab 5169 df-mpt 5190 df-id 5532 df-po 5546 df-so 5547 df-xp 5640 df-rel 5641 df-cnv 5642 df-co 5643 df-dm 5644 df-rn 5645 df-res 5646 df-ima 5647 df-iota 6449 df-fun 6499 df-fn 6500 df-f 6501 df-f1 6502 df-fo 6503 df-f1o 6504 df-fv 6505 df-riota 7314 df-ov 7361 df-oprab 7362 df-mpo 7363 df-er 8649 df-en 8885 df-dom 8886 df-sdom 8887 df-pnf 11192 df-mnf 11193 df-ltxr 11195 df-sub 11388 df-neg 11389 df-grpo 29438 df-ablo 29490 df-vc 29504 df-hvsub 29916 |
This theorem is referenced by: hhnv 30110 |
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