![]() |
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 30309 | . 2 ⊢ ·ih :( ℋ × ℋ)⟶ℂ | |
2 | 1 | fovcl 7531 | 1 ⊢ ((𝐴 ∈ ℋ ∧ 𝐵 ∈ ℋ) → (𝐴 ·ih 𝐵) ∈ ℂ) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 397 ∈ wcel 2107 (class class class)co 7403 ℂcc 11103 ℋchba 30149 ·ih csp 30152 |
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 2704 ax-sep 5297 ax-nul 5304 ax-pr 5425 ax-hfi 30309 |
This theorem depends on definitions: df-bi 206 df-an 398 df-or 847 df-3an 1090 df-tru 1545 df-fal 1555 df-ex 1783 df-nf 1787 df-sb 2069 df-mo 2535 df-eu 2564 df-clab 2711 df-cleq 2725 df-clel 2811 df-nfc 2886 df-ne 2942 df-ral 3063 df-rex 3072 df-rab 3434 df-v 3477 df-sbc 3776 df-csb 3892 df-dif 3949 df-un 3951 df-in 3953 df-ss 3963 df-nul 4321 df-if 4527 df-sn 4627 df-pr 4629 df-op 4633 df-uni 4907 df-iun 4997 df-br 5147 df-opab 5209 df-mpt 5230 df-id 5572 df-xp 5680 df-rel 5681 df-cnv 5682 df-co 5683 df-dm 5684 df-rn 5685 df-iota 6491 df-fun 6541 df-fn 6542 df-f 6543 df-fv 6547 df-ov 7406 |
This theorem is referenced by: hicli 30311 his5 30316 his35 30318 his7 30320 his2sub 30322 his2sub2 30323 hire 30324 hi01 30326 abshicom 30331 hi2eq 30335 hial2eq2 30337 bcs2 30412 pjhthlem1 30621 normcan 30806 pjspansn 30807 adjsym 31063 cnvadj 31122 adj2 31164 brafn 31177 kbop 31183 kbmul 31185 kbpj 31186 eigvalcl 31191 lnopeqi 31238 riesz3i 31292 cnlnadjlem2 31298 cnlnadjlem7 31303 nmopcoadji 31331 kbass2 31347 kbass5 31350 kbass6 31351 hmopidmpji 31382 pjclem4 31429 pj3si 31437 |
Copyright terms: Public domain | W3C validator |