![]() |
Hilbert Space Explorer |
< Previous
Next >
Nearby theorems |
|
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 31047 | . 2 ⊢ ((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℋ) → (𝐴 ·ℎ 𝐵) ∈ ℋ) | |
4 | 1, 2, 3 | mp2an 691 | 1 ⊢ (𝐴 ·ℎ 𝐵) ∈ ℋ |
Colors of variables: wff setvar class |
Syntax hints: ∈ wcel 2108 (class class class)co 7450 ℂcc 11184 ℋchba 30953 ·ℎ csm 30955 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1793 ax-4 1807 ax-5 1909 ax-6 1967 ax-7 2007 ax-8 2110 ax-9 2118 ax-10 2141 ax-11 2158 ax-12 2178 ax-ext 2711 ax-sep 5317 ax-nul 5324 ax-pr 5447 ax-hfvmul 31039 |
This theorem depends on definitions: df-bi 207 df-an 396 df-or 847 df-3an 1089 df-tru 1540 df-fal 1550 df-ex 1778 df-nf 1782 df-sb 2065 df-mo 2543 df-eu 2572 df-clab 2718 df-cleq 2732 df-clel 2819 df-nfc 2895 df-ne 2947 df-ral 3068 df-rex 3077 df-rab 3444 df-v 3490 df-sbc 3805 df-csb 3922 df-dif 3979 df-un 3981 df-ss 3993 df-nul 4353 df-if 4549 df-sn 4649 df-pr 4651 df-op 4655 df-uni 4932 df-iun 5017 df-br 5167 df-opab 5229 df-mpt 5250 df-id 5593 df-xp 5706 df-rel 5707 df-cnv 5708 df-co 5709 df-dm 5710 df-rn 5711 df-iota 6527 df-fun 6577 df-fn 6578 df-f 6579 df-fv 6583 df-ov 7453 |
This theorem is referenced by: hvsubsub4i 31093 hvnegdii 31096 hvsubeq0i 31097 hvsubcan2i 31098 hvaddcani 31099 hvsubaddi 31100 normlem0 31143 normlem5 31148 normlem9 31152 bcseqi 31154 norm-iii-i 31173 norm3difi 31181 normpar2i 31190 polid2i 31191 polidi 31192 h1de2i 31587 pjsubii 31712 eigposi 31870 lnop0 32000 lnopunilem1 32044 lnophmlem2 32051 lnfn0i 32076 |
Copyright terms: Public domain | W3C validator |