|   | Mathbox for Thierry Arnoux | < Previous  
      Next > Nearby theorems | |
| Mirrors > Home > MPE Home > Th. List > Mathboxes > slmdvs0 | Structured version Visualization version GIF version | ||
| Description: Anything times the zero vector is the zero vector. Equation 1b of [Kreyszig] p. 51. (hvmul0 31043 analog.) (Contributed by NM, 12-Jan-2014.) (Revised by Mario Carneiro, 19-Jun-2014.) (Revised by Thierry Arnoux, 1-Apr-2018.) | 
| Ref | Expression | 
|---|---|
| slmdvs0.f | ⊢ 𝐹 = (Scalar‘𝑊) | 
| slmdvs0.s | ⊢ · = ( ·𝑠 ‘𝑊) | 
| slmdvs0.k | ⊢ 𝐾 = (Base‘𝐹) | 
| slmdvs0.z | ⊢ 0 = (0g‘𝑊) | 
| Ref | Expression | 
|---|---|
| slmdvs0 | ⊢ ((𝑊 ∈ SLMod ∧ 𝑋 ∈ 𝐾) → (𝑋 · 0 ) = 0 ) | 
| Step | Hyp | Ref | Expression | 
|---|---|---|---|
| 1 | slmdvs0.f | . . . . 5 ⊢ 𝐹 = (Scalar‘𝑊) | |
| 2 | 1 | slmdsrg 33213 | . . . 4 ⊢ (𝑊 ∈ SLMod → 𝐹 ∈ SRing) | 
| 3 | slmdvs0.k | . . . . 5 ⊢ 𝐾 = (Base‘𝐹) | |
| 4 | eqid 2737 | . . . . 5 ⊢ (.r‘𝐹) = (.r‘𝐹) | |
| 5 | eqid 2737 | . . . . 5 ⊢ (0g‘𝐹) = (0g‘𝐹) | |
| 6 | 3, 4, 5 | srgrz 20204 | . . . 4 ⊢ ((𝐹 ∈ SRing ∧ 𝑋 ∈ 𝐾) → (𝑋(.r‘𝐹)(0g‘𝐹)) = (0g‘𝐹)) | 
| 7 | 2, 6 | sylan 580 | . . 3 ⊢ ((𝑊 ∈ SLMod ∧ 𝑋 ∈ 𝐾) → (𝑋(.r‘𝐹)(0g‘𝐹)) = (0g‘𝐹)) | 
| 8 | 7 | oveq1d 7446 | . 2 ⊢ ((𝑊 ∈ SLMod ∧ 𝑋 ∈ 𝐾) → ((𝑋(.r‘𝐹)(0g‘𝐹)) · 0 ) = ((0g‘𝐹) · 0 )) | 
| 9 | simpl 482 | . . . 4 ⊢ ((𝑊 ∈ SLMod ∧ 𝑋 ∈ 𝐾) → 𝑊 ∈ SLMod) | |
| 10 | simpr 484 | . . . 4 ⊢ ((𝑊 ∈ SLMod ∧ 𝑋 ∈ 𝐾) → 𝑋 ∈ 𝐾) | |
| 11 | 2 | adantr 480 | . . . . 5 ⊢ ((𝑊 ∈ SLMod ∧ 𝑋 ∈ 𝐾) → 𝐹 ∈ SRing) | 
| 12 | 3, 5 | srg0cl 20197 | . . . . 5 ⊢ (𝐹 ∈ SRing → (0g‘𝐹) ∈ 𝐾) | 
| 13 | 11, 12 | syl 17 | . . . 4 ⊢ ((𝑊 ∈ SLMod ∧ 𝑋 ∈ 𝐾) → (0g‘𝐹) ∈ 𝐾) | 
| 14 | eqid 2737 | . . . . . 6 ⊢ (Base‘𝑊) = (Base‘𝑊) | |
| 15 | slmdvs0.z | . . . . . 6 ⊢ 0 = (0g‘𝑊) | |
| 16 | 14, 15 | slmd0vcl 33227 | . . . . 5 ⊢ (𝑊 ∈ SLMod → 0 ∈ (Base‘𝑊)) | 
| 17 | 16 | adantr 480 | . . . 4 ⊢ ((𝑊 ∈ SLMod ∧ 𝑋 ∈ 𝐾) → 0 ∈ (Base‘𝑊)) | 
| 18 | slmdvs0.s | . . . . 5 ⊢ · = ( ·𝑠 ‘𝑊) | |
| 19 | 14, 1, 18, 3, 4 | slmdvsass 33223 | . . . 4 ⊢ ((𝑊 ∈ SLMod ∧ (𝑋 ∈ 𝐾 ∧ (0g‘𝐹) ∈ 𝐾 ∧ 0 ∈ (Base‘𝑊))) → ((𝑋(.r‘𝐹)(0g‘𝐹)) · 0 ) = (𝑋 · ((0g‘𝐹) · 0 ))) | 
| 20 | 9, 10, 13, 17, 19 | syl13anc 1374 | . . 3 ⊢ ((𝑊 ∈ SLMod ∧ 𝑋 ∈ 𝐾) → ((𝑋(.r‘𝐹)(0g‘𝐹)) · 0 ) = (𝑋 · ((0g‘𝐹) · 0 ))) | 
| 21 | 14, 1, 18, 5, 15 | slmd0vs 33230 | . . . . 5 ⊢ ((𝑊 ∈ SLMod ∧ 0 ∈ (Base‘𝑊)) → ((0g‘𝐹) · 0 ) = 0 ) | 
| 22 | 17, 21 | syldan 591 | . . . 4 ⊢ ((𝑊 ∈ SLMod ∧ 𝑋 ∈ 𝐾) → ((0g‘𝐹) · 0 ) = 0 ) | 
| 23 | 22 | oveq2d 7447 | . . 3 ⊢ ((𝑊 ∈ SLMod ∧ 𝑋 ∈ 𝐾) → (𝑋 · ((0g‘𝐹) · 0 )) = (𝑋 · 0 )) | 
| 24 | 20, 23 | eqtrd 2777 | . 2 ⊢ ((𝑊 ∈ SLMod ∧ 𝑋 ∈ 𝐾) → ((𝑋(.r‘𝐹)(0g‘𝐹)) · 0 ) = (𝑋 · 0 )) | 
| 25 | 8, 24, 22 | 3eqtr3d 2785 | 1 ⊢ ((𝑊 ∈ SLMod ∧ 𝑋 ∈ 𝐾) → (𝑋 · 0 ) = 0 ) | 
| Colors of variables: wff setvar class | 
| Syntax hints: → wi 4 ∧ wa 395 = wceq 1540 ∈ wcel 2108 ‘cfv 6561 (class class class)co 7431 Basecbs 17247 .rcmulr 17298 Scalarcsca 17300 ·𝑠 cvsca 17301 0gc0g 17484 SRingcsrg 20183 SLModcslmd 33206 | 
| This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1795 ax-4 1809 ax-5 1910 ax-6 1967 ax-7 2007 ax-8 2110 ax-9 2118 ax-10 2141 ax-11 2157 ax-12 2177 ax-ext 2708 ax-sep 5296 ax-nul 5306 ax-pr 5432 | 
| This theorem depends on definitions: df-bi 207 df-an 396 df-or 849 df-3an 1089 df-tru 1543 df-fal 1553 df-ex 1780 df-nf 1784 df-sb 2065 df-mo 2540 df-eu 2569 df-clab 2715 df-cleq 2729 df-clel 2816 df-nfc 2892 df-ne 2941 df-ral 3062 df-rex 3071 df-rmo 3380 df-reu 3381 df-rab 3437 df-v 3482 df-sbc 3789 df-dif 3954 df-un 3956 df-ss 3968 df-nul 4334 df-if 4526 df-sn 4627 df-pr 4629 df-op 4633 df-uni 4908 df-br 5144 df-opab 5206 df-mpt 5226 df-id 5578 df-xp 5691 df-rel 5692 df-cnv 5693 df-co 5694 df-dm 5695 df-iota 6514 df-fun 6563 df-fv 6569 df-riota 7388 df-ov 7434 df-0g 17486 df-mgm 18653 df-sgrp 18732 df-mnd 18748 df-cmn 19800 df-srg 20184 df-slmd 33207 | 
| This theorem is referenced by: gsumvsca1 33232 | 
| Copyright terms: Public domain | W3C validator |