| Hilbert Space Explorer |
< Previous
Next >
Nearby theorems |
||
| Mirrors > Home > HSE Home > Th. List > hicl | Structured version Visualization version GIF version | ||
| Description: Closure of inner product. (Contributed by NM, 17-Nov-2007.) (New usage is discouraged.) |
| Ref | Expression |
|---|---|
| hicl | ⊢ ((𝐴 ∈ ℋ ∧ 𝐵 ∈ ℋ) → (𝐴 ·ih 𝐵) ∈ ℂ) |
| Step | Hyp | Ref | Expression |
|---|---|---|---|
| 1 | ax-hfi 31098 | . 2 ⊢ ·ih :( ℋ × ℋ)⟶ℂ | |
| 2 | 1 | fovcl 7561 | 1 ⊢ ((𝐴 ∈ ℋ ∧ 𝐵 ∈ ℋ) → (𝐴 ·ih 𝐵) ∈ ℂ) |
| Colors of variables: wff setvar class |
| Syntax hints: → wi 4 ∧ wa 395 ∈ wcel 2108 (class class class)co 7431 ℂcc 11153 ℋchba 30938 ·ih csp 30941 |
| This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1795 ax-4 1809 ax-5 1910 ax-6 1967 ax-7 2007 ax-8 2110 ax-9 2118 ax-10 2141 ax-11 2157 ax-12 2177 ax-ext 2708 ax-sep 5296 ax-nul 5306 ax-pr 5432 ax-hfi 31098 |
| This theorem depends on definitions: df-bi 207 df-an 396 df-or 849 df-3an 1089 df-tru 1543 df-fal 1553 df-ex 1780 df-nf 1784 df-sb 2065 df-mo 2540 df-eu 2569 df-clab 2715 df-cleq 2729 df-clel 2816 df-nfc 2892 df-ne 2941 df-ral 3062 df-rex 3071 df-rab 3437 df-v 3482 df-sbc 3789 df-csb 3900 df-dif 3954 df-un 3956 df-ss 3968 df-nul 4334 df-if 4526 df-sn 4627 df-pr 4629 df-op 4633 df-uni 4908 df-iun 4993 df-br 5144 df-opab 5206 df-mpt 5226 df-id 5578 df-xp 5691 df-rel 5692 df-cnv 5693 df-co 5694 df-dm 5695 df-rn 5696 df-iota 6514 df-fun 6563 df-fn 6564 df-f 6565 df-fv 6569 df-ov 7434 |
| This theorem is referenced by: hicli 31100 his5 31105 his35 31107 his7 31109 his2sub 31111 his2sub2 31112 hire 31113 hi01 31115 abshicom 31120 hi2eq 31124 hial2eq2 31126 bcs2 31201 pjhthlem1 31410 normcan 31595 pjspansn 31596 adjsym 31852 cnvadj 31911 adj2 31953 brafn 31966 kbop 31972 kbmul 31974 kbpj 31975 eigvalcl 31980 lnopeqi 32027 riesz3i 32081 cnlnadjlem2 32087 cnlnadjlem7 32092 nmopcoadji 32120 kbass2 32136 kbass5 32139 kbass6 32140 hmopidmpji 32171 pjclem4 32218 pj3si 32226 |
| Copyright terms: Public domain | W3C validator |