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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 27867 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 473 | . . . 4 ⊢ ((𝑊 ∈ SLMod ∧ 𝑋 ∈ 𝑉) → 𝑊 ∈ SLMod) | |
2 | slmd0vs.f | . . . . . 6 ⊢ 𝐹 = (Scalar‘𝑊) | |
3 | eqid 2622 | . . . . . 6 ⊢ (Base‘𝐹) = (Base‘𝐹) | |
4 | slmd0vs.o | . . . . . 6 ⊢ 𝑂 = (0g‘𝐹) | |
5 | 2, 3, 4 | slmd0cl 29771 | . . . . 5 ⊢ (𝑊 ∈ SLMod → 𝑂 ∈ (Base‘𝐹)) |
6 | 5 | adantr 481 | . . . 4 ⊢ ((𝑊 ∈ SLMod ∧ 𝑋 ∈ 𝑉) → 𝑂 ∈ (Base‘𝐹)) |
7 | simpr 477 | . . . 4 ⊢ ((𝑊 ∈ SLMod ∧ 𝑋 ∈ 𝑉) → 𝑋 ∈ 𝑉) | |
8 | slmd0vs.v | . . . . 5 ⊢ 𝑉 = (Base‘𝑊) | |
9 | eqid 2622 | . . . . 5 ⊢ (+g‘𝑊) = (+g‘𝑊) | |
10 | slmd0vs.s | . . . . 5 ⊢ · = ( ·𝑠 ‘𝑊) | |
11 | slmd0vs.z | . . . . 5 ⊢ 0 = (0g‘𝑊) | |
12 | eqid 2622 | . . . . 5 ⊢ (+g‘𝐹) = (+g‘𝐹) | |
13 | eqid 2622 | . . . . 5 ⊢ (.r‘𝐹) = (.r‘𝐹) | |
14 | eqid 2622 | . . . . 5 ⊢ (1r‘𝐹) = (1r‘𝐹) | |
15 | 8, 9, 10, 11, 2, 3, 12, 13, 14, 4 | slmdlema 29756 | . . . 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 1335 | . . 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 479 | . 2 ⊢ ((𝑊 ∈ SLMod ∧ 𝑋 ∈ 𝑉) → (((𝑂(.r‘𝐹)𝑂) · 𝑋) = (𝑂 · (𝑂 · 𝑋)) ∧ ((1r‘𝐹) · 𝑋) = 𝑋 ∧ (𝑂 · 𝑋) = 0 )) |
18 | 17 | simp3d 1075 | 1 ⊢ ((𝑊 ∈ SLMod ∧ 𝑋 ∈ 𝑉) → (𝑂 · 𝑋) = 0 ) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 384 ∧ w3a 1037 = wceq 1483 ∈ wcel 1990 ‘cfv 5888 (class class class)co 6650 Basecbs 15857 +gcplusg 15941 .rcmulr 15942 Scalarcsca 15944 ·𝑠 cvsca 15945 0gc0g 16100 1rcur 18501 SLModcslmd 29753 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1722 ax-4 1737 ax-5 1839 ax-6 1888 ax-7 1935 ax-8 1992 ax-9 1999 ax-10 2019 ax-11 2034 ax-12 2047 ax-13 2246 ax-ext 2602 ax-sep 4781 ax-nul 4789 ax-pow 4843 ax-pr 4906 |
This theorem depends on definitions: df-bi 197 df-or 385 df-an 386 df-3an 1039 df-tru 1486 df-ex 1705 df-nf 1710 df-sb 1881 df-eu 2474 df-mo 2475 df-clab 2609 df-cleq 2615 df-clel 2618 df-nfc 2753 df-ne 2795 df-ral 2917 df-rex 2918 df-reu 2919 df-rmo 2920 df-rab 2921 df-v 3202 df-sbc 3436 df-dif 3577 df-un 3579 df-in 3581 df-ss 3588 df-nul 3916 df-if 4087 df-sn 4178 df-pr 4180 df-op 4184 df-uni 4437 df-br 4654 df-opab 4713 df-mpt 4730 df-id 5024 df-xp 5120 df-rel 5121 df-cnv 5122 df-co 5123 df-dm 5124 df-iota 5851 df-fun 5890 df-fv 5896 df-riota 6611 df-ov 6653 df-0g 16102 df-mgm 17242 df-sgrp 17284 df-mnd 17295 df-cmn 18195 df-srg 18506 df-slmd 29754 |
This theorem is referenced by: slmdvs0 29778 gsumvsca2 29783 |
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