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Mirrors > Home > MPE Home > Th. List > Mathboxes > lincval1 | Structured version Visualization version GIF version |
Description: The linear combination over a singleton mapping to 0. (Contributed by AV, 12-Apr-2019.) |
Ref | Expression |
---|---|
lincval1.b | ⊢ 𝐵 = (Base‘𝑀) |
lincval1.s | ⊢ 𝑆 = (Scalar‘𝑀) |
lincval1.r | ⊢ 𝑅 = (Base‘𝑆) |
lincval1.f | ⊢ 𝐹 = {〈𝑉, (0g‘𝑆)〉} |
Ref | Expression |
---|---|
lincval1 | ⊢ ((𝑀 ∈ LMod ∧ 𝑉 ∈ 𝐵) → (𝐹( linC ‘𝑀){𝑉}) = (0g‘𝑀)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | lincval1.s | . . . . 5 ⊢ 𝑆 = (Scalar‘𝑀) | |
2 | lincval1.r | . . . . 5 ⊢ 𝑅 = (Base‘𝑆) | |
3 | eqid 2736 | . . . . 5 ⊢ (0g‘𝑆) = (0g‘𝑆) | |
4 | 1, 2, 3 | lmod0cl 19879 | . . . 4 ⊢ (𝑀 ∈ LMod → (0g‘𝑆) ∈ 𝑅) |
5 | 4 | adantr 484 | . . 3 ⊢ ((𝑀 ∈ LMod ∧ 𝑉 ∈ 𝐵) → (0g‘𝑆) ∈ 𝑅) |
6 | lincval1.b | . . . 4 ⊢ 𝐵 = (Base‘𝑀) | |
7 | eqid 2736 | . . . 4 ⊢ ( ·𝑠 ‘𝑀) = ( ·𝑠 ‘𝑀) | |
8 | lincval1.f | . . . 4 ⊢ 𝐹 = {〈𝑉, (0g‘𝑆)〉} | |
9 | 6, 1, 2, 7, 8 | lincvalsn 45374 | . . 3 ⊢ ((𝑀 ∈ LMod ∧ 𝑉 ∈ 𝐵 ∧ (0g‘𝑆) ∈ 𝑅) → (𝐹( linC ‘𝑀){𝑉}) = ((0g‘𝑆)( ·𝑠 ‘𝑀)𝑉)) |
10 | 5, 9 | mpd3an3 1464 | . 2 ⊢ ((𝑀 ∈ LMod ∧ 𝑉 ∈ 𝐵) → (𝐹( linC ‘𝑀){𝑉}) = ((0g‘𝑆)( ·𝑠 ‘𝑀)𝑉)) |
11 | eqid 2736 | . . 3 ⊢ (0g‘𝑀) = (0g‘𝑀) | |
12 | 6, 1, 7, 3, 11 | lmod0vs 19886 | . 2 ⊢ ((𝑀 ∈ LMod ∧ 𝑉 ∈ 𝐵) → ((0g‘𝑆)( ·𝑠 ‘𝑀)𝑉) = (0g‘𝑀)) |
13 | 10, 12 | eqtrd 2771 | 1 ⊢ ((𝑀 ∈ LMod ∧ 𝑉 ∈ 𝐵) → (𝐹( linC ‘𝑀){𝑉}) = (0g‘𝑀)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 399 = wceq 1543 ∈ wcel 2112 {csn 4527 〈cop 4533 ‘cfv 6358 (class class class)co 7191 Basecbs 16666 Scalarcsca 16752 ·𝑠 cvsca 16753 0gc0g 16898 LModclmod 19853 linC clinc 45361 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1803 ax-4 1817 ax-5 1918 ax-6 1976 ax-7 2018 ax-8 2114 ax-9 2122 ax-10 2143 ax-11 2160 ax-12 2177 ax-ext 2708 ax-rep 5164 ax-sep 5177 ax-nul 5184 ax-pow 5243 ax-pr 5307 ax-un 7501 ax-cnex 10750 ax-resscn 10751 ax-1cn 10752 ax-icn 10753 ax-addcl 10754 ax-addrcl 10755 ax-mulcl 10756 ax-mulrcl 10757 ax-mulcom 10758 ax-addass 10759 ax-mulass 10760 ax-distr 10761 ax-i2m1 10762 ax-1ne0 10763 ax-1rid 10764 ax-rnegex 10765 ax-rrecex 10766 ax-cnre 10767 ax-pre-lttri 10768 ax-pre-lttrn 10769 ax-pre-ltadd 10770 ax-pre-mulgt0 10771 |
This theorem depends on definitions: df-bi 210 df-an 400 df-or 848 df-3or 1090 df-3an 1091 df-tru 1546 df-fal 1556 df-ex 1788 df-nf 1792 df-sb 2073 df-mo 2539 df-eu 2568 df-clab 2715 df-cleq 2728 df-clel 2809 df-nfc 2879 df-ne 2933 df-nel 3037 df-ral 3056 df-rex 3057 df-reu 3058 df-rmo 3059 df-rab 3060 df-v 3400 df-sbc 3684 df-csb 3799 df-dif 3856 df-un 3858 df-in 3860 df-ss 3870 df-pss 3872 df-nul 4224 df-if 4426 df-pw 4501 df-sn 4528 df-pr 4530 df-tp 4532 df-op 4534 df-uni 4806 df-int 4846 df-iun 4892 df-br 5040 df-opab 5102 df-mpt 5121 df-tr 5147 df-id 5440 df-eprel 5445 df-po 5453 df-so 5454 df-fr 5494 df-se 5495 df-we 5496 df-xp 5542 df-rel 5543 df-cnv 5544 df-co 5545 df-dm 5546 df-rn 5547 df-res 5548 df-ima 5549 df-pred 6140 df-ord 6194 df-on 6195 df-lim 6196 df-suc 6197 df-iota 6316 df-fun 6360 df-fn 6361 df-f 6362 df-f1 6363 df-fo 6364 df-f1o 6365 df-fv 6366 df-isom 6367 df-riota 7148 df-ov 7194 df-oprab 7195 df-mpo 7196 df-om 7623 df-1st 7739 df-2nd 7740 df-supp 7882 df-wrecs 8025 df-recs 8086 df-rdg 8124 df-1o 8180 df-er 8369 df-map 8488 df-en 8605 df-dom 8606 df-sdom 8607 df-fin 8608 df-oi 9104 df-card 9520 df-pnf 10834 df-mnf 10835 df-xr 10836 df-ltxr 10837 df-le 10838 df-sub 11029 df-neg 11030 df-nn 11796 df-n0 12056 df-z 12142 df-uz 12404 df-fz 13061 df-fzo 13204 df-seq 13540 df-hash 13862 df-0g 16900 df-gsum 16901 df-mgm 18068 df-sgrp 18117 df-mnd 18128 df-grp 18322 df-mulg 18443 df-cntz 18665 df-ring 19518 df-lmod 19855 df-linc 45363 |
This theorem is referenced by: lcosn0 45377 |
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