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| Description: Closure of scalar product for a subcomplex module. Analogue of lmodvscl 20876. (Contributed by NM, 3-Nov-2006.) (Revised by AV, 28-Sep-2021.) | 
| Ref | Expression | 
|---|---|
| clmvscl.v | ⊢ 𝑉 = (Base‘𝑊) | 
| clmvscl.f | ⊢ 𝐹 = (Scalar‘𝑊) | 
| clmvscl.s | ⊢ · = ( ·𝑠 ‘𝑊) | 
| clmvscl.k | ⊢ 𝐾 = (Base‘𝐹) | 
| Ref | Expression | 
|---|---|
| clmvscl | ⊢ ((𝑊 ∈ ℂMod ∧ 𝑄 ∈ 𝐾 ∧ 𝑋 ∈ 𝑉) → (𝑄 · 𝑋) ∈ 𝑉) | 
| Step | Hyp | Ref | Expression | 
|---|---|---|---|
| 1 | clmlmod 25100 | . 2 ⊢ (𝑊 ∈ ℂMod → 𝑊 ∈ LMod) | |
| 2 | clmvscl.v | . . 3 ⊢ 𝑉 = (Base‘𝑊) | |
| 3 | clmvscl.f | . . 3 ⊢ 𝐹 = (Scalar‘𝑊) | |
| 4 | clmvscl.s | . . 3 ⊢ · = ( ·𝑠 ‘𝑊) | |
| 5 | clmvscl.k | . . 3 ⊢ 𝐾 = (Base‘𝐹) | |
| 6 | 2, 3, 4, 5 | lmodvscl 20876 | . 2 ⊢ ((𝑊 ∈ LMod ∧ 𝑄 ∈ 𝐾 ∧ 𝑋 ∈ 𝑉) → (𝑄 · 𝑋) ∈ 𝑉) | 
| 7 | 1, 6 | syl3an1 1164 | 1 ⊢ ((𝑊 ∈ ℂMod ∧ 𝑄 ∈ 𝐾 ∧ 𝑋 ∈ 𝑉) → (𝑄 · 𝑋) ∈ 𝑉) | 
| Colors of variables: wff setvar class | 
| Syntax hints: → wi 4 ∧ w3a 1087 = wceq 1540 ∈ wcel 2108 ‘cfv 6561 (class class class)co 7431 Basecbs 17247 Scalarcsca 17300 ·𝑠 cvsca 17301 LModclmod 20858 ℂModcclm 25095 | 
| This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1795 ax-4 1809 ax-5 1910 ax-6 1967 ax-7 2007 ax-8 2110 ax-9 2118 ax-ext 2708 ax-nul 5306 | 
| This theorem depends on definitions: df-bi 207 df-an 396 df-or 849 df-3an 1089 df-tru 1543 df-fal 1553 df-ex 1780 df-sb 2065 df-clab 2715 df-cleq 2729 df-clel 2816 df-ne 2941 df-ral 3062 df-rab 3437 df-v 3482 df-sbc 3789 df-dif 3954 df-un 3956 df-ss 3968 df-nul 4334 df-if 4526 df-sn 4627 df-pr 4629 df-op 4633 df-uni 4908 df-br 5144 df-iota 6514 df-fv 6569 df-ov 7434 df-lmod 20860 df-clm 25096 | 
| This theorem is referenced by: clmpm1dir 25136 clmnegsubdi2 25138 clmsub4 25139 clmvsubval2 25143 clmvz 25144 nmoleub2lem3 25148 nmoleub3 25152 ncvspi 25190 | 
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