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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 28803 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 30837 | . . . 4 ⊢ (𝑊 ∈ SLMod → 𝐹 ∈ SRing) |
3 | slmdvs0.k | . . . . 5 ⊢ 𝐾 = (Base‘𝐹) | |
4 | eqid 2823 | . . . . 5 ⊢ (.r‘𝐹) = (.r‘𝐹) | |
5 | eqid 2823 | . . . . 5 ⊢ (0g‘𝐹) = (0g‘𝐹) | |
6 | 3, 4, 5 | srgrz 19278 | . . . 4 ⊢ ((𝐹 ∈ SRing ∧ 𝑋 ∈ 𝐾) → (𝑋(.r‘𝐹)(0g‘𝐹)) = (0g‘𝐹)) |
7 | 2, 6 | sylan 582 | . . 3 ⊢ ((𝑊 ∈ SLMod ∧ 𝑋 ∈ 𝐾) → (𝑋(.r‘𝐹)(0g‘𝐹)) = (0g‘𝐹)) |
8 | 7 | oveq1d 7173 | . 2 ⊢ ((𝑊 ∈ SLMod ∧ 𝑋 ∈ 𝐾) → ((𝑋(.r‘𝐹)(0g‘𝐹)) · 0 ) = ((0g‘𝐹) · 0 )) |
9 | simpl 485 | . . . 4 ⊢ ((𝑊 ∈ SLMod ∧ 𝑋 ∈ 𝐾) → 𝑊 ∈ SLMod) | |
10 | simpr 487 | . . . 4 ⊢ ((𝑊 ∈ SLMod ∧ 𝑋 ∈ 𝐾) → 𝑋 ∈ 𝐾) | |
11 | 2 | adantr 483 | . . . . 5 ⊢ ((𝑊 ∈ SLMod ∧ 𝑋 ∈ 𝐾) → 𝐹 ∈ SRing) |
12 | 3, 5 | srg0cl 19271 | . . . . 5 ⊢ (𝐹 ∈ SRing → (0g‘𝐹) ∈ 𝐾) |
13 | 11, 12 | syl 17 | . . . 4 ⊢ ((𝑊 ∈ SLMod ∧ 𝑋 ∈ 𝐾) → (0g‘𝐹) ∈ 𝐾) |
14 | eqid 2823 | . . . . . 6 ⊢ (Base‘𝑊) = (Base‘𝑊) | |
15 | slmdvs0.z | . . . . . 6 ⊢ 0 = (0g‘𝑊) | |
16 | 14, 15 | slmd0vcl 30851 | . . . . 5 ⊢ (𝑊 ∈ SLMod → 0 ∈ (Base‘𝑊)) |
17 | 16 | adantr 483 | . . . 4 ⊢ ((𝑊 ∈ SLMod ∧ 𝑋 ∈ 𝐾) → 0 ∈ (Base‘𝑊)) |
18 | slmdvs0.s | . . . . 5 ⊢ · = ( ·𝑠 ‘𝑊) | |
19 | 14, 1, 18, 3, 4 | slmdvsass 30847 | . . . 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 1368 | . . 3 ⊢ ((𝑊 ∈ SLMod ∧ 𝑋 ∈ 𝐾) → ((𝑋(.r‘𝐹)(0g‘𝐹)) · 0 ) = (𝑋 · ((0g‘𝐹) · 0 ))) |
21 | 14, 1, 18, 5, 15 | slmd0vs 30854 | . . . . 5 ⊢ ((𝑊 ∈ SLMod ∧ 0 ∈ (Base‘𝑊)) → ((0g‘𝐹) · 0 ) = 0 ) |
22 | 17, 21 | syldan 593 | . . . 4 ⊢ ((𝑊 ∈ SLMod ∧ 𝑋 ∈ 𝐾) → ((0g‘𝐹) · 0 ) = 0 ) |
23 | 22 | oveq2d 7174 | . . 3 ⊢ ((𝑊 ∈ SLMod ∧ 𝑋 ∈ 𝐾) → (𝑋 · ((0g‘𝐹) · 0 )) = (𝑋 · 0 )) |
24 | 20, 23 | eqtrd 2858 | . 2 ⊢ ((𝑊 ∈ SLMod ∧ 𝑋 ∈ 𝐾) → ((𝑋(.r‘𝐹)(0g‘𝐹)) · 0 ) = (𝑋 · 0 )) |
25 | 8, 24, 22 | 3eqtr3d 2866 | 1 ⊢ ((𝑊 ∈ SLMod ∧ 𝑋 ∈ 𝐾) → (𝑋 · 0 ) = 0 ) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 398 = wceq 1537 ∈ wcel 2114 ‘cfv 6357 (class class class)co 7158 Basecbs 16485 .rcmulr 16568 Scalarcsca 16570 ·𝑠 cvsca 16571 0gc0g 16715 SRingcsrg 19257 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: gsumvsca1 30856 |
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