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Mirrors > Home > MPE Home > Th. List > clmvscl | Structured version Visualization version GIF version |
Description: Closure of scalar product for a subcomplex module. Analogue of lmodvscl 19653. (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 23673 | . 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 19653 | . 2 ⊢ ((𝑊 ∈ LMod ∧ 𝑄 ∈ 𝐾 ∧ 𝑋 ∈ 𝑉) → (𝑄 · 𝑋) ∈ 𝑉) |
7 | 1, 6 | syl3an1 1159 | 1 ⊢ ((𝑊 ∈ ℂMod ∧ 𝑄 ∈ 𝐾 ∧ 𝑋 ∈ 𝑉) → (𝑄 · 𝑋) ∈ 𝑉) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ w3a 1083 = wceq 1537 ∈ wcel 2114 ‘cfv 6357 (class class class)co 7158 Basecbs 16485 Scalarcsca 16570 ·𝑠 cvsca 16571 LModclmod 19636 ℂModcclm 23668 |
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-nul 5212 |
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-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-br 5069 df-iota 6316 df-fv 6365 df-ov 7161 df-lmod 19638 df-clm 23669 |
This theorem is referenced by: clmpm1dir 23709 clmnegsubdi2 23711 clmsub4 23712 clmvsubval2 23716 clmvz 23717 nmoleub2lem3 23721 nmoleub3 23725 ncvspi 23762 |
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