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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 31036 | . 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 2103 (class class class)co 7445 ℂcc 11178 ℋchba 30942 ·ℎ csm 30944 |
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 2105 ax-9 2113 ax-10 2136 ax-11 2153 ax-12 2173 ax-ext 2705 ax-sep 5320 ax-nul 5327 ax-pr 5450 ax-hfvmul 31028 |
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 2537 df-eu 2566 df-clab 2712 df-cleq 2726 df-clel 2813 df-nfc 2890 df-ne 2943 df-ral 3064 df-rex 3073 df-rab 3439 df-v 3484 df-sbc 3799 df-csb 3916 df-dif 3973 df-un 3975 df-ss 3987 df-nul 4348 df-if 4549 df-sn 4649 df-pr 4651 df-op 4655 df-uni 4932 df-iun 5021 df-br 5170 df-opab 5232 df-mpt 5253 df-id 5597 df-xp 5705 df-rel 5706 df-cnv 5707 df-co 5708 df-dm 5709 df-rn 5710 df-iota 6524 df-fun 6574 df-fn 6575 df-f 6576 df-fv 6580 df-ov 7448 |
This theorem is referenced by: hvsubsub4i 31082 hvnegdii 31085 hvsubeq0i 31086 hvsubcan2i 31087 hvaddcani 31088 hvsubaddi 31089 normlem0 31132 normlem5 31137 normlem9 31141 bcseqi 31143 norm-iii-i 31162 norm3difi 31170 normpar2i 31179 polid2i 31180 polidi 31181 h1de2i 31576 pjsubii 31701 eigposi 31859 lnop0 31989 lnopunilem1 32033 lnophmlem2 32040 lnfn0i 32065 |
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