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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 2733 | . . . . 5 ⊢ (0g‘𝑆) = (0g‘𝑆) | |
4 | 1, 2, 3 | lmod0cl 20491 | . . . 4 ⊢ (𝑀 ∈ LMod → (0g‘𝑆) ∈ 𝑅) |
5 | 4 | adantr 482 | . . 3 ⊢ ((𝑀 ∈ LMod ∧ 𝑉 ∈ 𝐵) → (0g‘𝑆) ∈ 𝑅) |
6 | lincval1.b | . . . 4 ⊢ 𝐵 = (Base‘𝑀) | |
7 | eqid 2733 | . . . 4 ⊢ ( ·𝑠 ‘𝑀) = ( ·𝑠 ‘𝑀) | |
8 | lincval1.f | . . . 4 ⊢ 𝐹 = {〈𝑉, (0g‘𝑆)〉} | |
9 | 6, 1, 2, 7, 8 | lincvalsn 47052 | . . 3 ⊢ ((𝑀 ∈ LMod ∧ 𝑉 ∈ 𝐵 ∧ (0g‘𝑆) ∈ 𝑅) → (𝐹( linC ‘𝑀){𝑉}) = ((0g‘𝑆)( ·𝑠 ‘𝑀)𝑉)) |
10 | 5, 9 | mpd3an3 1463 | . 2 ⊢ ((𝑀 ∈ LMod ∧ 𝑉 ∈ 𝐵) → (𝐹( linC ‘𝑀){𝑉}) = ((0g‘𝑆)( ·𝑠 ‘𝑀)𝑉)) |
11 | eqid 2733 | . . 3 ⊢ (0g‘𝑀) = (0g‘𝑀) | |
12 | 6, 1, 7, 3, 11 | lmod0vs 20498 | . 2 ⊢ ((𝑀 ∈ LMod ∧ 𝑉 ∈ 𝐵) → ((0g‘𝑆)( ·𝑠 ‘𝑀)𝑉) = (0g‘𝑀)) |
13 | 10, 12 | eqtrd 2773 | 1 ⊢ ((𝑀 ∈ LMod ∧ 𝑉 ∈ 𝐵) → (𝐹( linC ‘𝑀){𝑉}) = (0g‘𝑀)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 397 = wceq 1542 ∈ wcel 2107 {csn 4628 〈cop 4634 ‘cfv 6541 (class class class)co 7406 Basecbs 17141 Scalarcsca 17197 ·𝑠 cvsca 17198 0gc0g 17382 LModclmod 20464 linC clinc 47039 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1798 ax-4 1812 ax-5 1914 ax-6 1972 ax-7 2012 ax-8 2109 ax-9 2117 ax-10 2138 ax-11 2155 ax-12 2172 ax-ext 2704 ax-rep 5285 ax-sep 5299 ax-nul 5306 ax-pow 5363 ax-pr 5427 ax-un 7722 ax-cnex 11163 ax-resscn 11164 ax-1cn 11165 ax-icn 11166 ax-addcl 11167 ax-addrcl 11168 ax-mulcl 11169 ax-mulrcl 11170 ax-mulcom 11171 ax-addass 11172 ax-mulass 11173 ax-distr 11174 ax-i2m1 11175 ax-1ne0 11176 ax-1rid 11177 ax-rnegex 11178 ax-rrecex 11179 ax-cnre 11180 ax-pre-lttri 11181 ax-pre-lttrn 11182 ax-pre-ltadd 11183 ax-pre-mulgt0 11184 |
This theorem depends on definitions: df-bi 206 df-an 398 df-or 847 df-3or 1089 df-3an 1090 df-tru 1545 df-fal 1555 df-ex 1783 df-nf 1787 df-sb 2069 df-mo 2535 df-eu 2564 df-clab 2711 df-cleq 2725 df-clel 2811 df-nfc 2886 df-ne 2942 df-nel 3048 df-ral 3063 df-rex 3072 df-rmo 3377 df-reu 3378 df-rab 3434 df-v 3477 df-sbc 3778 df-csb 3894 df-dif 3951 df-un 3953 df-in 3955 df-ss 3965 df-pss 3967 df-nul 4323 df-if 4529 df-pw 4604 df-sn 4629 df-pr 4631 df-op 4635 df-uni 4909 df-int 4951 df-iun 4999 df-br 5149 df-opab 5211 df-mpt 5232 df-tr 5266 df-id 5574 df-eprel 5580 df-po 5588 df-so 5589 df-fr 5631 df-se 5632 df-we 5633 df-xp 5682 df-rel 5683 df-cnv 5684 df-co 5685 df-dm 5686 df-rn 5687 df-res 5688 df-ima 5689 df-pred 6298 df-ord 6365 df-on 6366 df-lim 6367 df-suc 6368 df-iota 6493 df-fun 6543 df-fn 6544 df-f 6545 df-f1 6546 df-fo 6547 df-f1o 6548 df-fv 6549 df-isom 6550 df-riota 7362 df-ov 7409 df-oprab 7410 df-mpo 7411 df-om 7853 df-1st 7972 df-2nd 7973 df-supp 8144 df-frecs 8263 df-wrecs 8294 df-recs 8368 df-rdg 8407 df-1o 8463 df-er 8700 df-map 8819 df-en 8937 df-dom 8938 df-sdom 8939 df-fin 8940 df-oi 9502 df-card 9931 df-pnf 11247 df-mnf 11248 df-xr 11249 df-ltxr 11250 df-le 11251 df-sub 11443 df-neg 11444 df-nn 12210 df-n0 12470 df-z 12556 df-uz 12820 df-fz 13482 df-fzo 13625 df-seq 13964 df-hash 14288 df-0g 17384 df-gsum 17385 df-mgm 18558 df-sgrp 18607 df-mnd 18623 df-grp 18819 df-mulg 18946 df-cntz 19176 df-ring 20052 df-lmod 20466 df-linc 47041 |
This theorem is referenced by: lcosn0 47055 |
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