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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 28784 | . 2 ⊢ ((𝐴 ∈ ℋ ∧ 𝐵 ∈ ℋ) → (𝐴 ·ih 𝐵) ∈ ℂ) | |
4 | 1, 2, 3 | mp2an 688 | 1 ⊢ (𝐴 ·ih 𝐵) ∈ ℂ |
Colors of variables: wff setvar class |
Syntax hints: ∈ 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: hisubcomi 28808 normlem0 28813 normlem2 28815 normlem3 28816 normlem7 28820 normlem8 28821 normlem9 28822 bcseqi 28824 norm-ii-i 28841 normpythi 28846 normpari 28858 polid2i 28861 bcsiALT 28883 h1de2i 29257 h1de2bi 29258 h1de2ctlem 29259 eigrei 29538 eigorthi 29541 lnopunilem1 29714 lnopunilem2 29715 |
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