![]() |
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 30816 | . 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 2099 (class class class)co 7414 ℂcc 11130 ℋchba 30722 ·ℎ csm 30724 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1790 ax-4 1804 ax-5 1906 ax-6 1964 ax-7 2004 ax-8 2101 ax-9 2109 ax-10 2130 ax-11 2147 ax-12 2167 ax-ext 2699 ax-sep 5293 ax-nul 5300 ax-pr 5423 ax-hfvmul 30808 |
This theorem depends on definitions: df-bi 206 df-an 396 df-or 847 df-3an 1087 df-tru 1537 df-fal 1547 df-ex 1775 df-nf 1779 df-sb 2061 df-mo 2530 df-eu 2559 df-clab 2706 df-cleq 2720 df-clel 2806 df-nfc 2881 df-ne 2937 df-ral 3058 df-rex 3067 df-rab 3429 df-v 3472 df-sbc 3776 df-csb 3891 df-dif 3948 df-un 3950 df-in 3952 df-ss 3962 df-nul 4319 df-if 4525 df-sn 4625 df-pr 4627 df-op 4631 df-uni 4904 df-iun 4993 df-br 5143 df-opab 5205 df-mpt 5226 df-id 5570 df-xp 5678 df-rel 5679 df-cnv 5680 df-co 5681 df-dm 5682 df-rn 5683 df-iota 6494 df-fun 6544 df-fn 6545 df-f 6546 df-fv 6550 df-ov 7417 |
This theorem is referenced by: hvsubsub4i 30862 hvnegdii 30865 hvsubeq0i 30866 hvsubcan2i 30867 hvaddcani 30868 hvsubaddi 30869 normlem0 30912 normlem5 30917 normlem9 30921 bcseqi 30923 norm-iii-i 30942 norm3difi 30950 normpar2i 30959 polid2i 30960 polidi 30961 h1de2i 31356 pjsubii 31481 eigposi 31639 lnop0 31769 lnopunilem1 31813 lnophmlem2 31820 lnfn0i 31845 |
Copyright terms: Public domain | W3C validator |