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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 27320 | . 2 ⊢ ·ih :( ℋ × ℋ)⟶ℂ | |
2 | 1 | fovcl 6663 | 1 ⊢ ((𝐴 ∈ ℋ ∧ 𝐵 ∈ ℋ) → (𝐴 ·ih 𝐵) ∈ ℂ) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 383 ∈ wcel 1977 (class class class)co 6549 ℂcc 9813 ℋchil 27160 ·ih csp 27163 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1713 ax-4 1728 ax-5 1827 ax-6 1875 ax-7 1922 ax-9 1986 ax-10 2006 ax-11 2021 ax-12 2034 ax-13 2234 ax-ext 2590 ax-sep 4709 ax-nul 4717 ax-pr 4833 ax-hfi 27320 |
This theorem depends on definitions: df-bi 196 df-or 384 df-an 385 df-3an 1033 df-tru 1478 df-ex 1696 df-nf 1701 df-sb 1868 df-eu 2462 df-mo 2463 df-clab 2597 df-cleq 2603 df-clel 2606 df-nfc 2740 df-ral 2901 df-rex 2902 df-rab 2905 df-v 3175 df-sbc 3403 df-csb 3500 df-dif 3543 df-un 3545 df-in 3547 df-ss 3554 df-nul 3875 df-if 4037 df-sn 4126 df-pr 4128 df-op 4132 df-uni 4373 df-iun 4457 df-br 4584 df-opab 4644 df-mpt 4645 df-id 4953 df-xp 5044 df-rel 5045 df-cnv 5046 df-co 5047 df-dm 5048 df-rn 5049 df-iota 5768 df-fun 5806 df-fn 5807 df-f 5808 df-fv 5812 df-ov 6552 |
This theorem is referenced by: hicli 27322 his5 27327 his35 27329 his7 27331 his2sub 27333 his2sub2 27334 hire 27335 hi01 27337 abshicom 27342 hi2eq 27346 hial2eq2 27348 bcs2 27423 pjhthlem1 27634 normcan 27819 pjspansn 27820 adjsym 28076 cnvadj 28135 adj2 28177 brafn 28190 kbop 28196 kbmul 28198 kbpj 28199 eigvalcl 28204 lnopeqi 28251 riesz3i 28305 cnlnadjlem2 28311 cnlnadjlem7 28316 nmopcoadji 28344 kbass2 28360 kbass5 28363 kbass6 28364 hmopidmpji 28395 pjclem4 28442 pj3si 28450 |
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