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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 27251 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 472 | . . . 4 ⊢ ((𝑊 ∈ SLMod ∧ 𝑋 ∈ 𝑉) → 𝑊 ∈ SLMod) | |
2 | slmd0vs.f | . . . . . 6 ⊢ 𝐹 = (Scalar‘𝑊) | |
3 | eqid 2610 | . . . . . 6 ⊢ (Base‘𝐹) = (Base‘𝐹) | |
4 | slmd0vs.o | . . . . . 6 ⊢ 𝑂 = (0g‘𝐹) | |
5 | 2, 3, 4 | slmd0cl 29102 | . . . . 5 ⊢ (𝑊 ∈ SLMod → 𝑂 ∈ (Base‘𝐹)) |
6 | 5 | adantr 480 | . . . 4 ⊢ ((𝑊 ∈ SLMod ∧ 𝑋 ∈ 𝑉) → 𝑂 ∈ (Base‘𝐹)) |
7 | simpr 476 | . . . 4 ⊢ ((𝑊 ∈ SLMod ∧ 𝑋 ∈ 𝑉) → 𝑋 ∈ 𝑉) | |
8 | slmd0vs.v | . . . . 5 ⊢ 𝑉 = (Base‘𝑊) | |
9 | eqid 2610 | . . . . 5 ⊢ (+g‘𝑊) = (+g‘𝑊) | |
10 | slmd0vs.s | . . . . 5 ⊢ · = ( ·𝑠 ‘𝑊) | |
11 | slmd0vs.z | . . . . 5 ⊢ 0 = (0g‘𝑊) | |
12 | eqid 2610 | . . . . 5 ⊢ (+g‘𝐹) = (+g‘𝐹) | |
13 | eqid 2610 | . . . . 5 ⊢ (.r‘𝐹) = (.r‘𝐹) | |
14 | eqid 2610 | . . . . 5 ⊢ (1r‘𝐹) = (1r‘𝐹) | |
15 | 8, 9, 10, 11, 2, 3, 12, 13, 14, 4 | slmdlema 29087 | . . . 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 1327 | . . 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 478 | . 2 ⊢ ((𝑊 ∈ SLMod ∧ 𝑋 ∈ 𝑉) → (((𝑂(.r‘𝐹)𝑂) · 𝑋) = (𝑂 · (𝑂 · 𝑋)) ∧ ((1r‘𝐹) · 𝑋) = 𝑋 ∧ (𝑂 · 𝑋) = 0 )) |
18 | 17 | simp3d 1068 | 1 ⊢ ((𝑊 ∈ SLMod ∧ 𝑋 ∈ 𝑉) → (𝑂 · 𝑋) = 0 ) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 383 ∧ w3a 1031 = wceq 1475 ∈ wcel 1977 ‘cfv 5804 (class class class)co 6549 Basecbs 15695 +gcplusg 15768 .rcmulr 15769 Scalarcsca 15771 ·𝑠 cvsca 15772 0gc0g 15923 1rcur 18324 SLModcslmd 29084 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1713 ax-4 1728 ax-5 1827 ax-6 1875 ax-7 1922 ax-8 1979 ax-9 1986 ax-10 2006 ax-11 2021 ax-12 2034 ax-13 2234 ax-ext 2590 ax-sep 4709 ax-nul 4717 ax-pow 4769 ax-pr 4833 |
This theorem depends on definitions: df-bi 196 df-or 384 df-an 385 df-3an 1033 df-tru 1478 df-ex 1696 df-nf 1701 df-sb 1868 df-eu 2462 df-mo 2463 df-clab 2597 df-cleq 2603 df-clel 2606 df-nfc 2740 df-ne 2782 df-ral 2901 df-rex 2902 df-reu 2903 df-rmo 2904 df-rab 2905 df-v 3175 df-sbc 3403 df-dif 3543 df-un 3545 df-in 3547 df-ss 3554 df-nul 3875 df-if 4037 df-sn 4126 df-pr 4128 df-op 4132 df-uni 4373 df-br 4584 df-opab 4644 df-mpt 4645 df-id 4953 df-xp 5044 df-rel 5045 df-cnv 5046 df-co 5047 df-dm 5048 df-iota 5768 df-fun 5806 df-fv 5812 df-riota 6511 df-ov 6552 df-0g 15925 df-mgm 17065 df-sgrp 17107 df-mnd 17118 df-cmn 18018 df-srg 18329 df-slmd 29085 |
This theorem is referenced by: slmdvs0 29109 gsumvsca2 29114 |
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