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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 476 | . . 3 ⊢ ((𝑀 ∈ LMod ∧ 𝑉 ∈ 𝐵) → 𝑉 ∈ 𝐵) | |
2 | lincval1.s | . . . . 5 ⊢ 𝑆 = (Scalar‘𝑀) | |
3 | lincval1.r | . . . . 5 ⊢ 𝑅 = (Base‘𝑆) | |
4 | eqid 2610 | . . . . 5 ⊢ (0g‘𝑆) = (0g‘𝑆) | |
5 | 2, 3, 4 | lmod0cl 18712 | . . . 4 ⊢ (𝑀 ∈ LMod → (0g‘𝑆) ∈ 𝑅) |
6 | 5 | adantr 480 | . . 3 ⊢ ((𝑀 ∈ LMod ∧ 𝑉 ∈ 𝐵) → (0g‘𝑆) ∈ 𝑅) |
7 | fvex 6113 | . . . . 5 ⊢ (Base‘𝑆) ∈ V | |
8 | 3, 7 | eqeltri 2684 | . . . 4 ⊢ 𝑅 ∈ V |
9 | 8 | a1i 11 | . . 3 ⊢ ((𝑀 ∈ LMod ∧ 𝑉 ∈ 𝐵) → 𝑅 ∈ V) |
10 | lincval1.f | . . . 4 ⊢ 𝐹 = {〈𝑉, (0g‘𝑆)〉} | |
11 | 10 | mapsnop 41916 | . . 3 ⊢ ((𝑉 ∈ 𝐵 ∧ (0g‘𝑆) ∈ 𝑅 ∧ 𝑅 ∈ V) → 𝐹 ∈ (𝑅 ↑𝑚 {𝑉})) |
12 | 1, 6, 9, 11 | syl3anc 1318 | . 2 ⊢ ((𝑀 ∈ LMod ∧ 𝑉 ∈ 𝐵) → 𝐹 ∈ (𝑅 ↑𝑚 {𝑉})) |
13 | elmapi 7765 | . . . 4 ⊢ (𝐹 ∈ (𝑅 ↑𝑚 {𝑉}) → 𝐹:{𝑉}⟶𝑅) | |
14 | 12, 13 | syl 17 | . . 3 ⊢ ((𝑀 ∈ LMod ∧ 𝑉 ∈ 𝐵) → 𝐹:{𝑉}⟶𝑅) |
15 | snfi 7923 | . . . 4 ⊢ {𝑉} ∈ Fin | |
16 | 15 | a1i 11 | . . 3 ⊢ ((𝑀 ∈ LMod ∧ 𝑉 ∈ 𝐵) → {𝑉} ∈ Fin) |
17 | fvex 6113 | . . . 4 ⊢ (0g‘𝑆) ∈ V | |
18 | 17 | a1i 11 | . . 3 ⊢ ((𝑀 ∈ LMod ∧ 𝑉 ∈ 𝐵) → (0g‘𝑆) ∈ V) |
19 | 14, 16, 18 | fdmfifsupp 8168 | . 2 ⊢ ((𝑀 ∈ LMod ∧ 𝑉 ∈ 𝐵) → 𝐹 finSupp (0g‘𝑆)) |
20 | lincval1.b | . . 3 ⊢ 𝐵 = (Base‘𝑀) | |
21 | 20, 2, 3, 10 | lincval1 42002 | . 2 ⊢ ((𝑀 ∈ LMod ∧ 𝑉 ∈ 𝐵) → (𝐹( linC ‘𝑀){𝑉}) = (0g‘𝑀)) |
22 | 12, 19, 21 | 3jca 1235 | 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 383 ∧ w3a 1031 = wceq 1475 ∈ wcel 1977 Vcvv 3173 {csn 4125 〈cop 4131 class class class wbr 4583 ⟶wf 5800 ‘cfv 5804 (class class class)co 6549 ↑𝑚 cmap 7744 Fincfn 7841 finSupp cfsupp 8158 Basecbs 15695 Scalarcsca 15771 0gc0g 15923 LModclmod 18686 linC clinc 41987 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1713 ax-4 1728 ax-5 1827 ax-6 1875 ax-7 1922 ax-8 1979 ax-9 1986 ax-10 2006 ax-11 2021 ax-12 2034 ax-13 2234 ax-ext 2590 ax-rep 4699 ax-sep 4709 ax-nul 4717 ax-pow 4769 ax-pr 4833 ax-un 6847 ax-inf2 8421 ax-cnex 9871 ax-resscn 9872 ax-1cn 9873 ax-icn 9874 ax-addcl 9875 ax-addrcl 9876 ax-mulcl 9877 ax-mulrcl 9878 ax-mulcom 9879 ax-addass 9880 ax-mulass 9881 ax-distr 9882 ax-i2m1 9883 ax-1ne0 9884 ax-1rid 9885 ax-rnegex 9886 ax-rrecex 9887 ax-cnre 9888 ax-pre-lttri 9889 ax-pre-lttrn 9890 ax-pre-ltadd 9891 ax-pre-mulgt0 9892 |
This theorem depends on definitions: df-bi 196 df-or 384 df-an 385 df-3or 1032 df-3an 1033 df-tru 1478 df-ex 1696 df-nf 1701 df-sb 1868 df-eu 2462 df-mo 2463 df-clab 2597 df-cleq 2603 df-clel 2606 df-nfc 2740 df-ne 2782 df-nel 2783 df-ral 2901 df-rex 2902 df-reu 2903 df-rmo 2904 df-rab 2905 df-v 3175 df-sbc 3403 df-csb 3500 df-dif 3543 df-un 3545 df-in 3547 df-ss 3554 df-pss 3556 df-nul 3875 df-if 4037 df-pw 4110 df-sn 4126 df-pr 4128 df-tp 4130 df-op 4132 df-uni 4373 df-int 4411 df-iun 4457 df-br 4584 df-opab 4644 df-mpt 4645 df-tr 4681 df-eprel 4949 df-id 4953 df-po 4959 df-so 4960 df-fr 4997 df-se 4998 df-we 4999 df-xp 5044 df-rel 5045 df-cnv 5046 df-co 5047 df-dm 5048 df-rn 5049 df-res 5050 df-ima 5051 df-pred 5597 df-ord 5643 df-on 5644 df-lim 5645 df-suc 5646 df-iota 5768 df-fun 5806 df-fn 5807 df-f 5808 df-f1 5809 df-fo 5810 df-f1o 5811 df-fv 5812 df-isom 5813 df-riota 6511 df-ov 6552 df-oprab 6553 df-mpt2 6554 df-om 6958 df-1st 7059 df-2nd 7060 df-supp 7183 df-wrecs 7294 df-recs 7355 df-rdg 7393 df-1o 7447 df-oadd 7451 df-er 7629 df-map 7746 df-en 7842 df-dom 7843 df-sdom 7844 df-fin 7845 df-fsupp 8159 df-oi 8298 df-card 8648 df-pnf 9955 df-mnf 9956 df-xr 9957 df-ltxr 9958 df-le 9959 df-sub 10147 df-neg 10148 df-nn 10898 df-n0 11170 df-z 11255 df-uz 11564 df-fz 12198 df-fzo 12335 df-seq 12664 df-hash 12980 df-0g 15925 df-gsum 15926 df-mgm 17065 df-sgrp 17107 df-mnd 17118 df-grp 17248 df-mulg 17364 df-cntz 17573 df-ring 18372 df-lmod 18688 df-linc 41989 |
This theorem is referenced by: (None) |
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