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Mirrors > Home > MPE Home > Th. List > Mathboxes > slmd0vs | Structured version Visualization version GIF version |
Description: Zero times a vector is the zero vector. Equation 1a of [Kreyszig] p. 51. (ax-hvmul0 28789 analog.) (Contributed by NM, 12-Jan-2014.) (Revised by Mario Carneiro, 19-Jun-2014.) (Revised by Thierry Arnoux, 1-Apr-2018.) |
Ref | Expression |
---|---|
slmd0vs.v | ⊢ 𝑉 = (Base‘𝑊) |
slmd0vs.f | ⊢ 𝐹 = (Scalar‘𝑊) |
slmd0vs.s | ⊢ · = ( ·𝑠 ‘𝑊) |
slmd0vs.o | ⊢ 𝑂 = (0g‘𝐹) |
slmd0vs.z | ⊢ 0 = (0g‘𝑊) |
Ref | Expression |
---|---|
slmd0vs | ⊢ ((𝑊 ∈ SLMod ∧ 𝑋 ∈ 𝑉) → (𝑂 · 𝑋) = 0 ) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | simpl 485 | . . . 4 ⊢ ((𝑊 ∈ SLMod ∧ 𝑋 ∈ 𝑉) → 𝑊 ∈ SLMod) | |
2 | slmd0vs.f | . . . . . 6 ⊢ 𝐹 = (Scalar‘𝑊) | |
3 | eqid 2823 | . . . . . 6 ⊢ (Base‘𝐹) = (Base‘𝐹) | |
4 | slmd0vs.o | . . . . . 6 ⊢ 𝑂 = (0g‘𝐹) | |
5 | 2, 3, 4 | slmd0cl 30848 | . . . . 5 ⊢ (𝑊 ∈ SLMod → 𝑂 ∈ (Base‘𝐹)) |
6 | 5 | adantr 483 | . . . 4 ⊢ ((𝑊 ∈ SLMod ∧ 𝑋 ∈ 𝑉) → 𝑂 ∈ (Base‘𝐹)) |
7 | simpr 487 | . . . 4 ⊢ ((𝑊 ∈ SLMod ∧ 𝑋 ∈ 𝑉) → 𝑋 ∈ 𝑉) | |
8 | slmd0vs.v | . . . . 5 ⊢ 𝑉 = (Base‘𝑊) | |
9 | eqid 2823 | . . . . 5 ⊢ (+g‘𝑊) = (+g‘𝑊) | |
10 | slmd0vs.s | . . . . 5 ⊢ · = ( ·𝑠 ‘𝑊) | |
11 | slmd0vs.z | . . . . 5 ⊢ 0 = (0g‘𝑊) | |
12 | eqid 2823 | . . . . 5 ⊢ (+g‘𝐹) = (+g‘𝐹) | |
13 | eqid 2823 | . . . . 5 ⊢ (.r‘𝐹) = (.r‘𝐹) | |
14 | eqid 2823 | . . . . 5 ⊢ (1r‘𝐹) = (1r‘𝐹) | |
15 | 8, 9, 10, 11, 2, 3, 12, 13, 14, 4 | slmdlema 30833 | . . . 4 ⊢ ((𝑊 ∈ SLMod ∧ (𝑂 ∈ (Base‘𝐹) ∧ 𝑂 ∈ (Base‘𝐹)) ∧ (𝑋 ∈ 𝑉 ∧ 𝑋 ∈ 𝑉)) → (((𝑂 · 𝑋) ∈ 𝑉 ∧ (𝑂 · (𝑋(+g‘𝑊)𝑋)) = ((𝑂 · 𝑋)(+g‘𝑊)(𝑂 · 𝑋)) ∧ ((𝑂(+g‘𝐹)𝑂) · 𝑋) = ((𝑂 · 𝑋)(+g‘𝑊)(𝑂 · 𝑋))) ∧ (((𝑂(.r‘𝐹)𝑂) · 𝑋) = (𝑂 · (𝑂 · 𝑋)) ∧ ((1r‘𝐹) · 𝑋) = 𝑋 ∧ (𝑂 · 𝑋) = 0 ))) |
16 | 1, 6, 6, 7, 7, 15 | syl122anc 1375 | . . 3 ⊢ ((𝑊 ∈ SLMod ∧ 𝑋 ∈ 𝑉) → (((𝑂 · 𝑋) ∈ 𝑉 ∧ (𝑂 · (𝑋(+g‘𝑊)𝑋)) = ((𝑂 · 𝑋)(+g‘𝑊)(𝑂 · 𝑋)) ∧ ((𝑂(+g‘𝐹)𝑂) · 𝑋) = ((𝑂 · 𝑋)(+g‘𝑊)(𝑂 · 𝑋))) ∧ (((𝑂(.r‘𝐹)𝑂) · 𝑋) = (𝑂 · (𝑂 · 𝑋)) ∧ ((1r‘𝐹) · 𝑋) = 𝑋 ∧ (𝑂 · 𝑋) = 0 ))) |
17 | 16 | simprd 498 | . 2 ⊢ ((𝑊 ∈ SLMod ∧ 𝑋 ∈ 𝑉) → (((𝑂(.r‘𝐹)𝑂) · 𝑋) = (𝑂 · (𝑂 · 𝑋)) ∧ ((1r‘𝐹) · 𝑋) = 𝑋 ∧ (𝑂 · 𝑋) = 0 )) |
18 | 17 | simp3d 1140 | 1 ⊢ ((𝑊 ∈ SLMod ∧ 𝑋 ∈ 𝑉) → (𝑂 · 𝑋) = 0 ) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 398 ∧ w3a 1083 = wceq 1537 ∈ wcel 2114 ‘cfv 6357 (class class class)co 7158 Basecbs 16485 +gcplusg 16567 .rcmulr 16568 Scalarcsca 16570 ·𝑠 cvsca 16571 0gc0g 16715 1rcur 19253 SLModcslmd 30830 |
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-pow 5268 ax-pr 5332 |
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-ne 3019 df-ral 3145 df-rex 3146 df-reu 3147 df-rmo 3148 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-opab 5131 df-mpt 5149 df-id 5462 df-xp 5563 df-rel 5564 df-cnv 5565 df-co 5566 df-dm 5567 df-iota 6316 df-fun 6359 df-fv 6365 df-riota 7116 df-ov 7161 df-0g 16717 df-mgm 17854 df-sgrp 17903 df-mnd 17914 df-cmn 18910 df-srg 19258 df-slmd 30831 |
This theorem is referenced by: slmdvs0 30855 gsumvsca2 30857 |
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