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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 30962 | . 2 ⊢ ((𝐴 ∈ ℋ ∧ 𝐵 ∈ ℋ) → (𝐴 ·ih 𝐵) ∈ ℂ) | |
4 | 1, 2, 3 | mp2an 690 | 1 ⊢ (𝐴 ·ih 𝐵) ∈ ℂ |
Colors of variables: wff setvar class |
Syntax hints: ∈ wcel 2098 (class class class)co 7419 ℂcc 11138 ℋchba 30801 ·ih csp 30804 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1789 ax-4 1803 ax-5 1905 ax-6 1963 ax-7 2003 ax-8 2100 ax-9 2108 ax-10 2129 ax-11 2146 ax-12 2166 ax-ext 2696 ax-sep 5300 ax-nul 5307 ax-pr 5429 ax-hfi 30961 |
This theorem depends on definitions: df-bi 206 df-an 395 df-or 846 df-3an 1086 df-tru 1536 df-fal 1546 df-ex 1774 df-nf 1778 df-sb 2060 df-mo 2528 df-eu 2557 df-clab 2703 df-cleq 2717 df-clel 2802 df-nfc 2877 df-ne 2930 df-ral 3051 df-rex 3060 df-rab 3419 df-v 3463 df-sbc 3774 df-csb 3890 df-dif 3947 df-un 3949 df-ss 3961 df-nul 4323 df-if 4531 df-sn 4631 df-pr 4633 df-op 4637 df-uni 4910 df-iun 4999 df-br 5150 df-opab 5212 df-mpt 5233 df-id 5576 df-xp 5684 df-rel 5685 df-cnv 5686 df-co 5687 df-dm 5688 df-rn 5689 df-iota 6501 df-fun 6551 df-fn 6552 df-f 6553 df-fv 6557 df-ov 7422 |
This theorem is referenced by: hisubcomi 30986 normlem0 30991 normlem2 30993 normlem3 30994 normlem7 30998 normlem8 30999 normlem9 31000 bcseqi 31002 norm-ii-i 31019 normpythi 31024 normpari 31036 polid2i 31039 bcsiALT 31061 h1de2i 31435 h1de2bi 31436 h1de2ctlem 31437 eigrei 31716 eigorthi 31719 lnopunilem1 31892 lnopunilem2 31893 |
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