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Mirrors > Home > HSE Home > Th. List > hicli | Structured version Visualization version GIF version |
Description: Closure inference for inner product. (Contributed by NM, 1-Aug-1999.) (New usage is discouraged.) |
Ref | Expression |
---|---|
hicl.1 | ⊢ 𝐴 ∈ ℋ |
hicl.2 | ⊢ 𝐵 ∈ ℋ |
Ref | Expression |
---|---|
hicli | ⊢ (𝐴 ·ih 𝐵) ∈ ℂ |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | hicl.1 | . 2 ⊢ 𝐴 ∈ ℋ | |
2 | hicl.2 | . 2 ⊢ 𝐵 ∈ ℋ | |
3 | hicl 29007 | . 2 ⊢ ((𝐴 ∈ ℋ ∧ 𝐵 ∈ ℋ) → (𝐴 ·ih 𝐵) ∈ ℂ) | |
4 | 1, 2, 3 | mp2an 692 | 1 ⊢ (𝐴 ·ih 𝐵) ∈ ℂ |
Colors of variables: wff setvar class |
Syntax hints: ∈ wcel 2113 (class class class)co 7164 ℂcc 10606 ℋchba 28846 ·ih csp 28849 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1802 ax-4 1816 ax-5 1916 ax-6 1974 ax-7 2019 ax-8 2115 ax-9 2123 ax-10 2144 ax-11 2161 ax-12 2178 ax-ext 2710 ax-sep 5164 ax-nul 5171 ax-pr 5293 ax-hfi 29006 |
This theorem depends on definitions: df-bi 210 df-an 400 df-or 847 df-3an 1090 df-tru 1545 df-fal 1555 df-ex 1787 df-nf 1791 df-sb 2074 df-mo 2540 df-eu 2570 df-clab 2717 df-cleq 2730 df-clel 2811 df-nfc 2881 df-ral 3058 df-rex 3059 df-v 3399 df-sbc 3680 df-csb 3789 df-dif 3844 df-un 3846 df-in 3848 df-ss 3858 df-nul 4210 df-if 4412 df-sn 4514 df-pr 4516 df-op 4520 df-uni 4794 df-iun 4880 df-br 5028 df-opab 5090 df-mpt 5108 df-id 5425 df-xp 5525 df-rel 5526 df-cnv 5527 df-co 5528 df-dm 5529 df-rn 5530 df-iota 6291 df-fun 6335 df-fn 6336 df-f 6337 df-fv 6341 df-ov 7167 |
This theorem is referenced by: hisubcomi 29031 normlem0 29036 normlem2 29038 normlem3 29039 normlem7 29043 normlem8 29044 normlem9 29045 bcseqi 29047 norm-ii-i 29064 normpythi 29069 normpari 29081 polid2i 29084 bcsiALT 29106 h1de2i 29480 h1de2bi 29481 h1de2ctlem 29482 eigrei 29761 eigorthi 29764 lnopunilem1 29937 lnopunilem2 29938 |
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