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Mirrors > Home > HSE Home > Th. List > hvmulcli | Structured version Visualization version GIF version |
Description: Closure inference for scalar multiplication. (Contributed by NM, 1-Aug-1999.) (New usage is discouraged.) |
Ref | Expression |
---|---|
hvmulcl.1 | ⊢ 𝐴 ∈ ℂ |
hvmulcl.2 | ⊢ 𝐵 ∈ ℋ |
Ref | Expression |
---|---|
hvmulcli | ⊢ (𝐴 ·ℎ 𝐵) ∈ ℋ |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | hvmulcl.1 | . 2 ⊢ 𝐴 ∈ ℂ | |
2 | hvmulcl.2 | . 2 ⊢ 𝐵 ∈ ℋ | |
3 | hvmulcl 27254 | . 2 ⊢ ((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℋ) → (𝐴 ·ℎ 𝐵) ∈ ℋ) | |
4 | 1, 2, 3 | mp2an 704 | 1 ⊢ (𝐴 ·ℎ 𝐵) ∈ ℋ |
Colors of variables: wff setvar class |
Syntax hints: ∈ wcel 1977 (class class class)co 6549 ℂcc 9813 ℋchil 27160 ·ℎ csm 27162 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1713 ax-4 1728 ax-5 1827 ax-6 1875 ax-7 1922 ax-9 1986 ax-10 2006 ax-11 2021 ax-12 2034 ax-13 2234 ax-ext 2590 ax-sep 4709 ax-nul 4717 ax-pr 4833 ax-hfvmul 27246 |
This theorem depends on definitions: df-bi 196 df-or 384 df-an 385 df-3an 1033 df-tru 1478 df-ex 1696 df-nf 1701 df-sb 1868 df-eu 2462 df-mo 2463 df-clab 2597 df-cleq 2603 df-clel 2606 df-nfc 2740 df-ral 2901 df-rex 2902 df-rab 2905 df-v 3175 df-sbc 3403 df-csb 3500 df-dif 3543 df-un 3545 df-in 3547 df-ss 3554 df-nul 3875 df-if 4037 df-sn 4126 df-pr 4128 df-op 4132 df-uni 4373 df-iun 4457 df-br 4584 df-opab 4644 df-mpt 4645 df-id 4953 df-xp 5044 df-rel 5045 df-cnv 5046 df-co 5047 df-dm 5048 df-rn 5049 df-iota 5768 df-fun 5806 df-fn 5807 df-f 5808 df-fv 5812 df-ov 6552 |
This theorem is referenced by: hvsubsub4i 27300 hvnegdii 27303 hvsubeq0i 27304 hvsubcan2i 27305 hvaddcani 27306 hvsubaddi 27307 normlem0 27350 normlem5 27355 normlem9 27359 bcseqi 27361 norm-iii-i 27380 norm3difi 27388 normpar2i 27397 polid2i 27398 polidi 27399 h1de2i 27796 pjsubii 27921 eigposi 28079 lnop0 28209 lnopunilem1 28253 lnophmlem2 28260 lnfn0i 28285 |
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