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Mirrors > Home > MPE Home > Th. List > Mathboxes > lcosn0 | Structured version Visualization version GIF version |
Description: Properties of a linear combination over a singleton mapping to 0. (Contributed by AV, 12-Apr-2019.) (Revised by AV, 28-Jul-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 |
---|---|
lcosn0 | ⊢ ((𝑀 ∈ LMod ∧ 𝑉 ∈ 𝐵) → (𝐹 ∈ (𝑅 ↑m {𝑉}) ∧ 𝐹 finSupp (0g‘𝑆) ∧ (𝐹( linC ‘𝑀){𝑉}) = (0g‘𝑀))) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | simpr 487 | . . 3 ⊢ ((𝑀 ∈ LMod ∧ 𝑉 ∈ 𝐵) → 𝑉 ∈ 𝐵) | |
2 | lincval1.s | . . . . 5 ⊢ 𝑆 = (Scalar‘𝑀) | |
3 | lincval1.r | . . . . 5 ⊢ 𝑅 = (Base‘𝑆) | |
4 | eqid 2821 | . . . . 5 ⊢ (0g‘𝑆) = (0g‘𝑆) | |
5 | 2, 3, 4 | lmod0cl 19654 | . . . 4 ⊢ (𝑀 ∈ LMod → (0g‘𝑆) ∈ 𝑅) |
6 | 5 | adantr 483 | . . 3 ⊢ ((𝑀 ∈ LMod ∧ 𝑉 ∈ 𝐵) → (0g‘𝑆) ∈ 𝑅) |
7 | 3 | fvexi 6678 | . . . 4 ⊢ 𝑅 ∈ V |
8 | 7 | a1i 11 | . . 3 ⊢ ((𝑀 ∈ LMod ∧ 𝑉 ∈ 𝐵) → 𝑅 ∈ V) |
9 | lincval1.f | . . . 4 ⊢ 𝐹 = {〈𝑉, (0g‘𝑆)〉} | |
10 | 9 | mapsnop 44387 | . . 3 ⊢ ((𝑉 ∈ 𝐵 ∧ (0g‘𝑆) ∈ 𝑅 ∧ 𝑅 ∈ V) → 𝐹 ∈ (𝑅 ↑m {𝑉})) |
11 | 1, 6, 8, 10 | syl3anc 1367 | . 2 ⊢ ((𝑀 ∈ LMod ∧ 𝑉 ∈ 𝐵) → 𝐹 ∈ (𝑅 ↑m {𝑉})) |
12 | elmapi 8422 | . . . 4 ⊢ (𝐹 ∈ (𝑅 ↑m {𝑉}) → 𝐹:{𝑉}⟶𝑅) | |
13 | 11, 12 | syl 17 | . . 3 ⊢ ((𝑀 ∈ LMod ∧ 𝑉 ∈ 𝐵) → 𝐹:{𝑉}⟶𝑅) |
14 | snfi 8588 | . . . 4 ⊢ {𝑉} ∈ Fin | |
15 | 14 | a1i 11 | . . 3 ⊢ ((𝑀 ∈ LMod ∧ 𝑉 ∈ 𝐵) → {𝑉} ∈ Fin) |
16 | fvex 6677 | . . . 4 ⊢ (0g‘𝑆) ∈ V | |
17 | 16 | a1i 11 | . . 3 ⊢ ((𝑀 ∈ LMod ∧ 𝑉 ∈ 𝐵) → (0g‘𝑆) ∈ V) |
18 | 13, 15, 17 | fdmfifsupp 8837 | . 2 ⊢ ((𝑀 ∈ LMod ∧ 𝑉 ∈ 𝐵) → 𝐹 finSupp (0g‘𝑆)) |
19 | lincval1.b | . . 3 ⊢ 𝐵 = (Base‘𝑀) | |
20 | 19, 2, 3, 9 | lincval1 44468 | . 2 ⊢ ((𝑀 ∈ LMod ∧ 𝑉 ∈ 𝐵) → (𝐹( linC ‘𝑀){𝑉}) = (0g‘𝑀)) |
21 | 11, 18, 20 | 3jca 1124 | 1 ⊢ ((𝑀 ∈ LMod ∧ 𝑉 ∈ 𝐵) → (𝐹 ∈ (𝑅 ↑m {𝑉}) ∧ 𝐹 finSupp (0g‘𝑆) ∧ (𝐹( linC ‘𝑀){𝑉}) = (0g‘𝑀))) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 398 ∧ w3a 1083 = wceq 1533 ∈ wcel 2110 Vcvv 3494 {csn 4560 〈cop 4566 class class class wbr 5058 ⟶wf 6345 ‘cfv 6349 (class class class)co 7150 ↑m cmap 8400 Fincfn 8503 finSupp cfsupp 8827 Basecbs 16477 Scalarcsca 16562 0gc0g 16707 LModclmod 19628 linC clinc 44453 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1792 ax-4 1806 ax-5 1907 ax-6 1966 ax-7 2011 ax-8 2112 ax-9 2120 ax-10 2141 ax-11 2157 ax-12 2173 ax-ext 2793 ax-rep 5182 ax-sep 5195 ax-nul 5202 ax-pow 5258 ax-pr 5321 ax-un 7455 ax-cnex 10587 ax-resscn 10588 ax-1cn 10589 ax-icn 10590 ax-addcl 10591 ax-addrcl 10592 ax-mulcl 10593 ax-mulrcl 10594 ax-mulcom 10595 ax-addass 10596 ax-mulass 10597 ax-distr 10598 ax-i2m1 10599 ax-1ne0 10600 ax-1rid 10601 ax-rnegex 10602 ax-rrecex 10603 ax-cnre 10604 ax-pre-lttri 10605 ax-pre-lttrn 10606 ax-pre-ltadd 10607 ax-pre-mulgt0 10608 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3or 1084 df-3an 1085 df-tru 1536 df-ex 1777 df-nf 1781 df-sb 2066 df-mo 2618 df-eu 2650 df-clab 2800 df-cleq 2814 df-clel 2893 df-nfc 2963 df-ne 3017 df-nel 3124 df-ral 3143 df-rex 3144 df-reu 3145 df-rmo 3146 df-rab 3147 df-v 3496 df-sbc 3772 df-csb 3883 df-dif 3938 df-un 3940 df-in 3942 df-ss 3951 df-pss 3953 df-nul 4291 df-if 4467 df-pw 4540 df-sn 4561 df-pr 4563 df-tp 4565 df-op 4567 df-uni 4832 df-int 4869 df-iun 4913 df-br 5059 df-opab 5121 df-mpt 5139 df-tr 5165 df-id 5454 df-eprel 5459 df-po 5468 df-so 5469 df-fr 5508 df-se 5509 df-we 5510 df-xp 5555 df-rel 5556 df-cnv 5557 df-co 5558 df-dm 5559 df-rn 5560 df-res 5561 df-ima 5562 df-pred 6142 df-ord 6188 df-on 6189 df-lim 6190 df-suc 6191 df-iota 6308 df-fun 6351 df-fn 6352 df-f 6353 df-f1 6354 df-fo 6355 df-f1o 6356 df-fv 6357 df-isom 6358 df-riota 7108 df-ov 7153 df-oprab 7154 df-mpo 7155 df-om 7575 df-1st 7683 df-2nd 7684 df-supp 7825 df-wrecs 7941 df-recs 8002 df-rdg 8040 df-1o 8096 df-oadd 8100 df-er 8283 df-map 8402 df-en 8504 df-dom 8505 df-sdom 8506 df-fin 8507 df-fsupp 8828 df-oi 8968 df-card 9362 df-pnf 10671 df-mnf 10672 df-xr 10673 df-ltxr 10674 df-le 10675 df-sub 10866 df-neg 10867 df-nn 11633 df-n0 11892 df-z 11976 df-uz 12238 df-fz 12887 df-fzo 13028 df-seq 13364 df-hash 13685 df-0g 16709 df-gsum 16710 df-mgm 17846 df-sgrp 17895 df-mnd 17906 df-grp 18100 df-mulg 18219 df-cntz 18441 df-ring 19293 df-lmod 19630 df-linc 44455 |
This theorem is referenced by: (None) |
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