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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 29442 | . 2 ⊢ ((𝐴 ∈ ℋ ∧ 𝐵 ∈ ℋ) → (𝐴 ·ih 𝐵) ∈ ℂ) | |
4 | 1, 2, 3 | mp2an 689 | 1 ⊢ (𝐴 ·ih 𝐵) ∈ ℂ |
Colors of variables: wff setvar class |
Syntax hints: ∈ wcel 2106 (class class class)co 7275 ℂcc 10869 ℋchba 29281 ·ih csp 29284 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1798 ax-4 1812 ax-5 1913 ax-6 1971 ax-7 2011 ax-8 2108 ax-9 2116 ax-10 2137 ax-11 2154 ax-12 2171 ax-ext 2709 ax-sep 5223 ax-nul 5230 ax-pr 5352 ax-hfi 29441 |
This theorem depends on definitions: df-bi 206 df-an 397 df-or 845 df-3an 1088 df-tru 1542 df-fal 1552 df-ex 1783 df-nf 1787 df-sb 2068 df-mo 2540 df-eu 2569 df-clab 2716 df-cleq 2730 df-clel 2816 df-nfc 2889 df-ral 3069 df-rex 3070 df-rab 3073 df-v 3434 df-sbc 3717 df-csb 3833 df-dif 3890 df-un 3892 df-in 3894 df-ss 3904 df-nul 4257 df-if 4460 df-sn 4562 df-pr 4564 df-op 4568 df-uni 4840 df-iun 4926 df-br 5075 df-opab 5137 df-mpt 5158 df-id 5489 df-xp 5595 df-rel 5596 df-cnv 5597 df-co 5598 df-dm 5599 df-rn 5600 df-iota 6391 df-fun 6435 df-fn 6436 df-f 6437 df-fv 6441 df-ov 7278 |
This theorem is referenced by: hisubcomi 29466 normlem0 29471 normlem2 29473 normlem3 29474 normlem7 29478 normlem8 29479 normlem9 29480 bcseqi 29482 norm-ii-i 29499 normpythi 29504 normpari 29516 polid2i 29519 bcsiALT 29541 h1de2i 29915 h1de2bi 29916 h1de2ctlem 29917 eigrei 30196 eigorthi 30199 lnopunilem1 30372 lnopunilem2 30373 |
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