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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 28792 | . 2 ⊢ ((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℋ) → (𝐴 ·ℎ 𝐵) ∈ ℋ) | |
4 | 1, 2, 3 | mp2an 690 | 1 ⊢ (𝐴 ·ℎ 𝐵) ∈ ℋ |
Colors of variables: wff setvar class |
Syntax hints: ∈ wcel 2114 (class class class)co 7158 ℂcc 10537 ℋchba 28698 ·ℎ csm 28700 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1796 ax-4 1810 ax-5 1911 ax-6 1970 ax-7 2015 ax-8 2116 ax-9 2124 ax-10 2145 ax-11 2161 ax-12 2177 ax-ext 2795 ax-sep 5205 ax-nul 5212 ax-pr 5332 ax-hfvmul 28784 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3an 1085 df-tru 1540 df-ex 1781 df-nf 1785 df-sb 2070 df-mo 2622 df-eu 2654 df-clab 2802 df-cleq 2816 df-clel 2895 df-nfc 2965 df-ral 3145 df-rex 3146 df-rab 3149 df-v 3498 df-sbc 3775 df-csb 3886 df-dif 3941 df-un 3943 df-in 3945 df-ss 3954 df-nul 4294 df-if 4470 df-sn 4570 df-pr 4572 df-op 4576 df-uni 4841 df-iun 4923 df-br 5069 df-opab 5131 df-mpt 5149 df-id 5462 df-xp 5563 df-rel 5564 df-cnv 5565 df-co 5566 df-dm 5567 df-rn 5568 df-iota 6316 df-fun 6359 df-fn 6360 df-f 6361 df-fv 6365 df-ov 7161 |
This theorem is referenced by: hvsubsub4i 28838 hvnegdii 28841 hvsubeq0i 28842 hvsubcan2i 28843 hvaddcani 28844 hvsubaddi 28845 normlem0 28888 normlem5 28893 normlem9 28897 bcseqi 28899 norm-iii-i 28918 norm3difi 28926 normpar2i 28935 polid2i 28936 polidi 28937 h1de2i 29332 pjsubii 29457 eigposi 29615 lnop0 29745 lnopunilem1 29789 lnophmlem2 29796 lnfn0i 29821 |
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