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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 28783 | . 2 ⊢ ·ih :( ℋ × ℋ)⟶ℂ | |
2 | 1 | fovcl 7268 | 1 ⊢ ((𝐴 ∈ ℋ ∧ 𝐵 ∈ ℋ) → (𝐴 ·ih 𝐵) ∈ ℂ) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 396 ∈ wcel 2105 (class class class)co 7145 ℂcc 10523 ℋchba 28623 ·ih csp 28626 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1787 ax-4 1801 ax-5 1902 ax-6 1961 ax-7 2006 ax-8 2107 ax-9 2115 ax-10 2136 ax-11 2151 ax-12 2167 ax-ext 2790 ax-sep 5194 ax-nul 5201 ax-pr 5320 ax-hfi 28783 |
This theorem depends on definitions: df-bi 208 df-an 397 df-or 842 df-3an 1081 df-tru 1531 df-ex 1772 df-nf 1776 df-sb 2061 df-mo 2615 df-eu 2647 df-clab 2797 df-cleq 2811 df-clel 2890 df-nfc 2960 df-ral 3140 df-rex 3141 df-rab 3144 df-v 3494 df-sbc 3770 df-csb 3881 df-dif 3936 df-un 3938 df-in 3940 df-ss 3949 df-nul 4289 df-if 4464 df-sn 4558 df-pr 4560 df-op 4564 df-uni 4831 df-iun 4912 df-br 5058 df-opab 5120 df-mpt 5138 df-id 5453 df-xp 5554 df-rel 5555 df-cnv 5556 df-co 5557 df-dm 5558 df-rn 5559 df-iota 6307 df-fun 6350 df-fn 6351 df-f 6352 df-fv 6356 df-ov 7148 |
This theorem is referenced by: hicli 28785 his5 28790 his35 28792 his7 28794 his2sub 28796 his2sub2 28797 hire 28798 hi01 28800 abshicom 28805 hi2eq 28809 hial2eq2 28811 bcs2 28886 pjhthlem1 29095 normcan 29280 pjspansn 29281 adjsym 29537 cnvadj 29596 adj2 29638 brafn 29651 kbop 29657 kbmul 29659 kbpj 29660 eigvalcl 29665 lnopeqi 29712 riesz3i 29766 cnlnadjlem2 29772 cnlnadjlem7 29777 nmopcoadji 29805 kbass2 29821 kbass5 29824 kbass6 29825 hmopidmpji 29856 pjclem4 29903 pj3si 29911 |
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