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Mirrors > Home > MPE Home > Th. List > Mathboxes > slmdvscl | Structured version Visualization version GIF version |
Description: Closure of scalar product for a semiring left module. (hvmulcl 28784 analog.) (Contributed by NM, 8-Dec-2013.) (Revised by Mario Carneiro, 19-Jun-2014.) (Revised by Thierry Arnoux, 1-Apr-2018.) |
Ref | Expression |
---|---|
slmdvscl.v | ⊢ 𝑉 = (Base‘𝑊) |
slmdvscl.f | ⊢ 𝐹 = (Scalar‘𝑊) |
slmdvscl.s | ⊢ · = ( ·𝑠 ‘𝑊) |
slmdvscl.k | ⊢ 𝐾 = (Base‘𝐹) |
Ref | Expression |
---|---|
slmdvscl | ⊢ ((𝑊 ∈ SLMod ∧ 𝑅 ∈ 𝐾 ∧ 𝑋 ∈ 𝑉) → (𝑅 · 𝑋) ∈ 𝑉) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | biid 263 | . 2 ⊢ (𝑊 ∈ SLMod ↔ 𝑊 ∈ SLMod) | |
2 | pm4.24 566 | . 2 ⊢ (𝑅 ∈ 𝐾 ↔ (𝑅 ∈ 𝐾 ∧ 𝑅 ∈ 𝐾)) | |
3 | pm4.24 566 | . 2 ⊢ (𝑋 ∈ 𝑉 ↔ (𝑋 ∈ 𝑉 ∧ 𝑋 ∈ 𝑉)) | |
4 | slmdvscl.v | . . . . 5 ⊢ 𝑉 = (Base‘𝑊) | |
5 | eqid 2821 | . . . . 5 ⊢ (+g‘𝑊) = (+g‘𝑊) | |
6 | slmdvscl.s | . . . . 5 ⊢ · = ( ·𝑠 ‘𝑊) | |
7 | eqid 2821 | . . . . 5 ⊢ (0g‘𝑊) = (0g‘𝑊) | |
8 | slmdvscl.f | . . . . 5 ⊢ 𝐹 = (Scalar‘𝑊) | |
9 | slmdvscl.k | . . . . 5 ⊢ 𝐾 = (Base‘𝐹) | |
10 | eqid 2821 | . . . . 5 ⊢ (+g‘𝐹) = (+g‘𝐹) | |
11 | eqid 2821 | . . . . 5 ⊢ (.r‘𝐹) = (.r‘𝐹) | |
12 | eqid 2821 | . . . . 5 ⊢ (1r‘𝐹) = (1r‘𝐹) | |
13 | eqid 2821 | . . . . 5 ⊢ (0g‘𝐹) = (0g‘𝐹) | |
14 | 4, 5, 6, 7, 8, 9, 10, 11, 12, 13 | slmdlema 30826 | . . . 4 ⊢ ((𝑊 ∈ SLMod ∧ (𝑅 ∈ 𝐾 ∧ 𝑅 ∈ 𝐾) ∧ (𝑋 ∈ 𝑉 ∧ 𝑋 ∈ 𝑉)) → (((𝑅 · 𝑋) ∈ 𝑉 ∧ (𝑅 · (𝑋(+g‘𝑊)𝑋)) = ((𝑅 · 𝑋)(+g‘𝑊)(𝑅 · 𝑋)) ∧ ((𝑅(+g‘𝐹)𝑅) · 𝑋) = ((𝑅 · 𝑋)(+g‘𝑊)(𝑅 · 𝑋))) ∧ (((𝑅(.r‘𝐹)𝑅) · 𝑋) = (𝑅 · (𝑅 · 𝑋)) ∧ ((1r‘𝐹) · 𝑋) = 𝑋 ∧ ((0g‘𝐹) · 𝑋) = (0g‘𝑊)))) |
15 | 14 | simpld 497 | . . 3 ⊢ ((𝑊 ∈ SLMod ∧ (𝑅 ∈ 𝐾 ∧ 𝑅 ∈ 𝐾) ∧ (𝑋 ∈ 𝑉 ∧ 𝑋 ∈ 𝑉)) → ((𝑅 · 𝑋) ∈ 𝑉 ∧ (𝑅 · (𝑋(+g‘𝑊)𝑋)) = ((𝑅 · 𝑋)(+g‘𝑊)(𝑅 · 𝑋)) ∧ ((𝑅(+g‘𝐹)𝑅) · 𝑋) = ((𝑅 · 𝑋)(+g‘𝑊)(𝑅 · 𝑋)))) |
16 | 15 | simp1d 1138 | . 2 ⊢ ((𝑊 ∈ SLMod ∧ (𝑅 ∈ 𝐾 ∧ 𝑅 ∈ 𝐾) ∧ (𝑋 ∈ 𝑉 ∧ 𝑋 ∈ 𝑉)) → (𝑅 · 𝑋) ∈ 𝑉) |
17 | 1, 2, 3, 16 | syl3anb 1157 | 1 ⊢ ((𝑊 ∈ SLMod ∧ 𝑅 ∈ 𝐾 ∧ 𝑋 ∈ 𝑉) → (𝑅 · 𝑋) ∈ 𝑉) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 398 ∧ w3a 1083 = wceq 1533 ∈ wcel 2110 ‘cfv 6349 (class class class)co 7150 Basecbs 16477 +gcplusg 16559 .rcmulr 16560 Scalarcsca 16562 ·𝑠 cvsca 16563 0gc0g 16707 1rcur 19245 SLModcslmd 30823 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1792 ax-4 1806 ax-5 1907 ax-6 1966 ax-7 2011 ax-8 2112 ax-9 2120 ax-10 2141 ax-11 2157 ax-12 2173 ax-ext 2793 ax-nul 5202 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3an 1085 df-tru 1536 df-ex 1777 df-nf 1781 df-sb 2066 df-mo 2618 df-eu 2650 df-clab 2800 df-cleq 2814 df-clel 2893 df-nfc 2963 df-ral 3143 df-rex 3144 df-rab 3147 df-v 3496 df-sbc 3772 df-dif 3938 df-un 3940 df-in 3942 df-ss 3951 df-nul 4291 df-if 4467 df-sn 4561 df-pr 4563 df-op 4567 df-uni 4832 df-br 5059 df-iota 6308 df-fv 6357 df-ov 7153 df-slmd 30824 |
This theorem is referenced by: gsumvsca1 30849 gsumvsca2 30850 sitgaddlemb 31601 |
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