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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 27881 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 29760 | . . . 4 ⊢ (𝑊 ∈ SLMod → 𝐹 ∈ SRing) |
3 | slmdvs0.k | . . . . 5 ⊢ 𝐾 = (Base‘𝐹) | |
4 | eqid 2622 | . . . . 5 ⊢ (.r‘𝐹) = (.r‘𝐹) | |
5 | eqid 2622 | . . . . 5 ⊢ (0g‘𝐹) = (0g‘𝐹) | |
6 | 3, 4, 5 | srgrz 18526 | . . . 4 ⊢ ((𝐹 ∈ SRing ∧ 𝑋 ∈ 𝐾) → (𝑋(.r‘𝐹)(0g‘𝐹)) = (0g‘𝐹)) |
7 | 2, 6 | sylan 488 | . . 3 ⊢ ((𝑊 ∈ SLMod ∧ 𝑋 ∈ 𝐾) → (𝑋(.r‘𝐹)(0g‘𝐹)) = (0g‘𝐹)) |
8 | 7 | oveq1d 6665 | . 2 ⊢ ((𝑊 ∈ SLMod ∧ 𝑋 ∈ 𝐾) → ((𝑋(.r‘𝐹)(0g‘𝐹)) · 0 ) = ((0g‘𝐹) · 0 )) |
9 | simpl 473 | . . . 4 ⊢ ((𝑊 ∈ SLMod ∧ 𝑋 ∈ 𝐾) → 𝑊 ∈ SLMod) | |
10 | simpr 477 | . . . 4 ⊢ ((𝑊 ∈ SLMod ∧ 𝑋 ∈ 𝐾) → 𝑋 ∈ 𝐾) | |
11 | 2 | adantr 481 | . . . . 5 ⊢ ((𝑊 ∈ SLMod ∧ 𝑋 ∈ 𝐾) → 𝐹 ∈ SRing) |
12 | 3, 5 | srg0cl 18519 | . . . . 5 ⊢ (𝐹 ∈ SRing → (0g‘𝐹) ∈ 𝐾) |
13 | 11, 12 | syl 17 | . . . 4 ⊢ ((𝑊 ∈ SLMod ∧ 𝑋 ∈ 𝐾) → (0g‘𝐹) ∈ 𝐾) |
14 | eqid 2622 | . . . . . 6 ⊢ (Base‘𝑊) = (Base‘𝑊) | |
15 | slmdvs0.z | . . . . . 6 ⊢ 0 = (0g‘𝑊) | |
16 | 14, 15 | slmd0vcl 29774 | . . . . 5 ⊢ (𝑊 ∈ SLMod → 0 ∈ (Base‘𝑊)) |
17 | 16 | adantr 481 | . . . 4 ⊢ ((𝑊 ∈ SLMod ∧ 𝑋 ∈ 𝐾) → 0 ∈ (Base‘𝑊)) |
18 | slmdvs0.s | . . . . 5 ⊢ · = ( ·𝑠 ‘𝑊) | |
19 | 14, 1, 18, 3, 4 | slmdvsass 29770 | . . . 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 1328 | . . 3 ⊢ ((𝑊 ∈ SLMod ∧ 𝑋 ∈ 𝐾) → ((𝑋(.r‘𝐹)(0g‘𝐹)) · 0 ) = (𝑋 · ((0g‘𝐹) · 0 ))) |
21 | 14, 1, 18, 5, 15 | slmd0vs 29777 | . . . . 5 ⊢ ((𝑊 ∈ SLMod ∧ 0 ∈ (Base‘𝑊)) → ((0g‘𝐹) · 0 ) = 0 ) |
22 | 17, 21 | syldan 487 | . . . 4 ⊢ ((𝑊 ∈ SLMod ∧ 𝑋 ∈ 𝐾) → ((0g‘𝐹) · 0 ) = 0 ) |
23 | 22 | oveq2d 6666 | . . 3 ⊢ ((𝑊 ∈ SLMod ∧ 𝑋 ∈ 𝐾) → (𝑋 · ((0g‘𝐹) · 0 )) = (𝑋 · 0 )) |
24 | 20, 23 | eqtrd 2656 | . 2 ⊢ ((𝑊 ∈ SLMod ∧ 𝑋 ∈ 𝐾) → ((𝑋(.r‘𝐹)(0g‘𝐹)) · 0 ) = (𝑋 · 0 )) |
25 | 8, 24, 22 | 3eqtr3d 2664 | 1 ⊢ ((𝑊 ∈ SLMod ∧ 𝑋 ∈ 𝐾) → (𝑋 · 0 ) = 0 ) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 384 = wceq 1483 ∈ wcel 1990 ‘cfv 5888 (class class class)co 6650 Basecbs 15857 .rcmulr 15942 Scalarcsca 15944 ·𝑠 cvsca 15945 0gc0g 16100 SRingcsrg 18505 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: gsumvsca1 29782 |
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