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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 28863 | . 2 ⊢ ((𝐴 ∈ ℋ ∧ 𝐵 ∈ ℋ) → (𝐴 ·ih 𝐵) ∈ ℂ) | |
4 | 1, 2, 3 | mp2an 691 | 1 ⊢ (𝐴 ·ih 𝐵) ∈ ℂ |
Colors of variables: wff setvar class |
Syntax hints: ∈ wcel 2111 (class class class)co 7135 ℂcc 10524 ℋchba 28702 ·ih csp 28705 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1797 ax-4 1811 ax-5 1911 ax-6 1970 ax-7 2015 ax-8 2113 ax-9 2121 ax-10 2142 ax-11 2158 ax-12 2175 ax-ext 2770 ax-sep 5167 ax-nul 5174 ax-pr 5295 ax-hfi 28862 |
This theorem depends on definitions: df-bi 210 df-an 400 df-or 845 df-3an 1086 df-tru 1541 df-ex 1782 df-nf 1786 df-sb 2070 df-mo 2598 df-eu 2629 df-clab 2777 df-cleq 2791 df-clel 2870 df-nfc 2938 df-ral 3111 df-rex 3112 df-v 3443 df-sbc 3721 df-csb 3829 df-dif 3884 df-un 3886 df-in 3888 df-ss 3898 df-nul 4244 df-if 4426 df-sn 4526 df-pr 4528 df-op 4532 df-uni 4801 df-iun 4883 df-br 5031 df-opab 5093 df-mpt 5111 df-id 5425 df-xp 5525 df-rel 5526 df-cnv 5527 df-co 5528 df-dm 5529 df-rn 5530 df-iota 6283 df-fun 6326 df-fn 6327 df-f 6328 df-fv 6332 df-ov 7138 |
This theorem is referenced by: hisubcomi 28887 normlem0 28892 normlem2 28894 normlem3 28895 normlem7 28899 normlem8 28900 normlem9 28901 bcseqi 28903 norm-ii-i 28920 normpythi 28925 normpari 28937 polid2i 28940 bcsiALT 28962 h1de2i 29336 h1de2bi 29337 h1de2ctlem 29338 eigrei 29617 eigorthi 29620 lnopunilem1 29793 lnopunilem2 29794 |
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