![]() |
Mathbox for Alexander van der Vekens |
< Previous
Next >
Nearby theorems |
|
Mirrors > Home > MPE Home > Th. List > Mathboxes > lincvalsng | Structured version Visualization version GIF version |
Description: The linear combination over a singleton. (Contributed by AV, 25-May-2019.) |
Ref | Expression |
---|---|
lincvalsn.b | ⊢ 𝐵 = (Base‘𝑀) |
lincvalsn.s | ⊢ 𝑆 = (Scalar‘𝑀) |
lincvalsn.r | ⊢ 𝑅 = (Base‘𝑆) |
lincvalsn.t | ⊢ · = ( ·𝑠 ‘𝑀) |
Ref | Expression |
---|---|
lincvalsng | ⊢ ((𝑀 ∈ LMod ∧ 𝑉 ∈ 𝐵 ∧ 𝑌 ∈ 𝑅) → ({〈𝑉, 𝑌〉} ( linC ‘𝑀){𝑉}) = (𝑌 · 𝑉)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | simp1 1170 | . . 3 ⊢ ((𝑀 ∈ LMod ∧ 𝑉 ∈ 𝐵 ∧ 𝑌 ∈ 𝑅) → 𝑀 ∈ LMod) | |
2 | simp2 1171 | . . . 4 ⊢ ((𝑀 ∈ LMod ∧ 𝑉 ∈ 𝐵 ∧ 𝑌 ∈ 𝑅) → 𝑉 ∈ 𝐵) | |
3 | lincvalsn.r | . . . . . . . 8 ⊢ 𝑅 = (Base‘𝑆) | |
4 | lincvalsn.s | . . . . . . . . 9 ⊢ 𝑆 = (Scalar‘𝑀) | |
5 | 4 | fveq2i 6436 | . . . . . . . 8 ⊢ (Base‘𝑆) = (Base‘(Scalar‘𝑀)) |
6 | 3, 5 | eqtri 2849 | . . . . . . 7 ⊢ 𝑅 = (Base‘(Scalar‘𝑀)) |
7 | 6 | eleq2i 2898 | . . . . . 6 ⊢ (𝑌 ∈ 𝑅 ↔ 𝑌 ∈ (Base‘(Scalar‘𝑀))) |
8 | 7 | biimpi 208 | . . . . 5 ⊢ (𝑌 ∈ 𝑅 → 𝑌 ∈ (Base‘(Scalar‘𝑀))) |
9 | 8 | 3ad2ant3 1169 | . . . 4 ⊢ ((𝑀 ∈ LMod ∧ 𝑉 ∈ 𝐵 ∧ 𝑌 ∈ 𝑅) → 𝑌 ∈ (Base‘(Scalar‘𝑀))) |
10 | fvexd 6448 | . . . 4 ⊢ ((𝑀 ∈ LMod ∧ 𝑉 ∈ 𝐵 ∧ 𝑌 ∈ 𝑅) → (Base‘(Scalar‘𝑀)) ∈ V) | |
11 | eqid 2825 | . . . . 5 ⊢ {〈𝑉, 𝑌〉} = {〈𝑉, 𝑌〉} | |
12 | 11 | mapsnop 42963 | . . . 4 ⊢ ((𝑉 ∈ 𝐵 ∧ 𝑌 ∈ (Base‘(Scalar‘𝑀)) ∧ (Base‘(Scalar‘𝑀)) ∈ V) → {〈𝑉, 𝑌〉} ∈ ((Base‘(Scalar‘𝑀)) ↑𝑚 {𝑉})) |
13 | 2, 9, 10, 12 | syl3anc 1494 | . . 3 ⊢ ((𝑀 ∈ LMod ∧ 𝑉 ∈ 𝐵 ∧ 𝑌 ∈ 𝑅) → {〈𝑉, 𝑌〉} ∈ ((Base‘(Scalar‘𝑀)) ↑𝑚 {𝑉})) |
14 | snelpwi 5133 | . . . . 5 ⊢ (𝑉 ∈ (Base‘𝑀) → {𝑉} ∈ 𝒫 (Base‘𝑀)) | |
15 | lincvalsn.b | . . . . 5 ⊢ 𝐵 = (Base‘𝑀) | |
16 | 14, 15 | eleq2s 2924 | . . . 4 ⊢ (𝑉 ∈ 𝐵 → {𝑉} ∈ 𝒫 (Base‘𝑀)) |
17 | 16 | 3ad2ant2 1168 | . . 3 ⊢ ((𝑀 ∈ LMod ∧ 𝑉 ∈ 𝐵 ∧ 𝑌 ∈ 𝑅) → {𝑉} ∈ 𝒫 (Base‘𝑀)) |
18 | lincval 43038 | . . 3 ⊢ ((𝑀 ∈ LMod ∧ {〈𝑉, 𝑌〉} ∈ ((Base‘(Scalar‘𝑀)) ↑𝑚 {𝑉}) ∧ {𝑉} ∈ 𝒫 (Base‘𝑀)) → ({〈𝑉, 𝑌〉} ( linC ‘𝑀){𝑉}) = (𝑀 Σg (𝑣 ∈ {𝑉} ↦ (({〈𝑉, 𝑌〉}‘𝑣)( ·𝑠 ‘𝑀)𝑣)))) | |
19 | 1, 13, 17, 18 | syl3anc 1494 | . 2 ⊢ ((𝑀 ∈ LMod ∧ 𝑉 ∈ 𝐵 ∧ 𝑌 ∈ 𝑅) → ({〈𝑉, 𝑌〉} ( linC ‘𝑀){𝑉}) = (𝑀 Σg (𝑣 ∈ {𝑉} ↦ (({〈𝑉, 𝑌〉}‘𝑣)( ·𝑠 ‘𝑀)𝑣)))) |
20 | lmodgrp 19226 | . . . . 5 ⊢ (𝑀 ∈ LMod → 𝑀 ∈ Grp) | |
21 | grpmnd 17783 | . . . . 5 ⊢ (𝑀 ∈ Grp → 𝑀 ∈ Mnd) | |
22 | 20, 21 | syl 17 | . . . 4 ⊢ (𝑀 ∈ LMod → 𝑀 ∈ Mnd) |
23 | 22 | 3ad2ant1 1167 | . . 3 ⊢ ((𝑀 ∈ LMod ∧ 𝑉 ∈ 𝐵 ∧ 𝑌 ∈ 𝑅) → 𝑀 ∈ Mnd) |
24 | fvsng 6698 | . . . . . 6 ⊢ ((𝑉 ∈ 𝐵 ∧ 𝑌 ∈ 𝑅) → ({〈𝑉, 𝑌〉}‘𝑉) = 𝑌) | |
25 | 24 | 3adant1 1164 | . . . . 5 ⊢ ((𝑀 ∈ LMod ∧ 𝑉 ∈ 𝐵 ∧ 𝑌 ∈ 𝑅) → ({〈𝑉, 𝑌〉}‘𝑉) = 𝑌) |
26 | 25 | oveq1d 6920 | . . . 4 ⊢ ((𝑀 ∈ LMod ∧ 𝑉 ∈ 𝐵 ∧ 𝑌 ∈ 𝑅) → (({〈𝑉, 𝑌〉}‘𝑉)( ·𝑠 ‘𝑀)𝑉) = (𝑌( ·𝑠 ‘𝑀)𝑉)) |
27 | eqid 2825 | . . . . . 6 ⊢ ( ·𝑠 ‘𝑀) = ( ·𝑠 ‘𝑀) | |
28 | 15, 4, 27, 3 | lmodvscl 19236 | . . . . 5 ⊢ ((𝑀 ∈ LMod ∧ 𝑌 ∈ 𝑅 ∧ 𝑉 ∈ 𝐵) → (𝑌( ·𝑠 ‘𝑀)𝑉) ∈ 𝐵) |
29 | 28 | 3com23 1160 | . . . 4 ⊢ ((𝑀 ∈ LMod ∧ 𝑉 ∈ 𝐵 ∧ 𝑌 ∈ 𝑅) → (𝑌( ·𝑠 ‘𝑀)𝑉) ∈ 𝐵) |
30 | 26, 29 | eqeltrd 2906 | . . 3 ⊢ ((𝑀 ∈ LMod ∧ 𝑉 ∈ 𝐵 ∧ 𝑌 ∈ 𝑅) → (({〈𝑉, 𝑌〉}‘𝑉)( ·𝑠 ‘𝑀)𝑉) ∈ 𝐵) |
31 | fveq2 6433 | . . . . 5 ⊢ (𝑣 = 𝑉 → ({〈𝑉, 𝑌〉}‘𝑣) = ({〈𝑉, 𝑌〉}‘𝑉)) | |
32 | id 22 | . . . . 5 ⊢ (𝑣 = 𝑉 → 𝑣 = 𝑉) | |
33 | 31, 32 | oveq12d 6923 | . . . 4 ⊢ (𝑣 = 𝑉 → (({〈𝑉, 𝑌〉}‘𝑣)( ·𝑠 ‘𝑀)𝑣) = (({〈𝑉, 𝑌〉}‘𝑉)( ·𝑠 ‘𝑀)𝑉)) |
34 | 15, 33 | gsumsn 18707 | . . 3 ⊢ ((𝑀 ∈ Mnd ∧ 𝑉 ∈ 𝐵 ∧ (({〈𝑉, 𝑌〉}‘𝑉)( ·𝑠 ‘𝑀)𝑉) ∈ 𝐵) → (𝑀 Σg (𝑣 ∈ {𝑉} ↦ (({〈𝑉, 𝑌〉}‘𝑣)( ·𝑠 ‘𝑀)𝑣))) = (({〈𝑉, 𝑌〉}‘𝑉)( ·𝑠 ‘𝑀)𝑉)) |
35 | 23, 2, 30, 34 | syl3anc 1494 | . 2 ⊢ ((𝑀 ∈ LMod ∧ 𝑉 ∈ 𝐵 ∧ 𝑌 ∈ 𝑅) → (𝑀 Σg (𝑣 ∈ {𝑉} ↦ (({〈𝑉, 𝑌〉}‘𝑣)( ·𝑠 ‘𝑀)𝑣))) = (({〈𝑉, 𝑌〉}‘𝑉)( ·𝑠 ‘𝑀)𝑉)) |
36 | lincvalsn.t | . . . . 5 ⊢ · = ( ·𝑠 ‘𝑀) | |
37 | 36 | eqcomi 2834 | . . . 4 ⊢ ( ·𝑠 ‘𝑀) = · |
38 | 37 | a1i 11 | . . 3 ⊢ ((𝑀 ∈ LMod ∧ 𝑉 ∈ 𝐵 ∧ 𝑌 ∈ 𝑅) → ( ·𝑠 ‘𝑀) = · ) |
39 | eqidd 2826 | . . 3 ⊢ ((𝑀 ∈ LMod ∧ 𝑉 ∈ 𝐵 ∧ 𝑌 ∈ 𝑅) → 𝑉 = 𝑉) | |
40 | 38, 25, 39 | oveq123d 6926 | . 2 ⊢ ((𝑀 ∈ LMod ∧ 𝑉 ∈ 𝐵 ∧ 𝑌 ∈ 𝑅) → (({〈𝑉, 𝑌〉}‘𝑉)( ·𝑠 ‘𝑀)𝑉) = (𝑌 · 𝑉)) |
41 | 19, 35, 40 | 3eqtrd 2865 | 1 ⊢ ((𝑀 ∈ LMod ∧ 𝑉 ∈ 𝐵 ∧ 𝑌 ∈ 𝑅) → ({〈𝑉, 𝑌〉} ( linC ‘𝑀){𝑉}) = (𝑌 · 𝑉)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ w3a 1111 = wceq 1656 ∈ wcel 2164 Vcvv 3414 𝒫 cpw 4378 {csn 4397 〈cop 4403 ↦ cmpt 4952 ‘cfv 6123 (class class class)co 6905 ↑𝑚 cmap 8122 Basecbs 16222 Scalarcsca 16308 ·𝑠 cvsca 16309 Σg cgsu 16454 Mndcmnd 17647 Grpcgrp 17776 LModclmod 19219 linC clinc 43033 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1894 ax-4 1908 ax-5 2009 ax-6 2075 ax-7 2112 ax-8 2166 ax-9 2173 ax-10 2192 ax-11 2207 ax-12 2220 ax-13 2389 ax-ext 2803 ax-rep 4994 ax-sep 5005 ax-nul 5013 ax-pow 5065 ax-pr 5127 ax-un 7209 ax-inf2 8815 ax-cnex 10308 ax-resscn 10309 ax-1cn 10310 ax-icn 10311 ax-addcl 10312 ax-addrcl 10313 ax-mulcl 10314 ax-mulrcl 10315 ax-mulcom 10316 ax-addass 10317 ax-mulass 10318 ax-distr 10319 ax-i2m1 10320 ax-1ne0 10321 ax-1rid 10322 ax-rnegex 10323 ax-rrecex 10324 ax-cnre 10325 ax-pre-lttri 10326 ax-pre-lttrn 10327 ax-pre-ltadd 10328 ax-pre-mulgt0 10329 |
This theorem depends on definitions: df-bi 199 df-an 387 df-or 879 df-3or 1112 df-3an 1113 df-tru 1660 df-ex 1879 df-nf 1883 df-sb 2068 df-mo 2605 df-eu 2640 df-clab 2812 df-cleq 2818 df-clel 2821 df-nfc 2958 df-ne 3000 df-nel 3103 df-ral 3122 df-rex 3123 df-reu 3124 df-rmo 3125 df-rab 3126 df-v 3416 df-sbc 3663 df-csb 3758 df-dif 3801 df-un 3803 df-in 3805 df-ss 3812 df-pss 3814 df-nul 4145 df-if 4307 df-pw 4380 df-sn 4398 df-pr 4400 df-tp 4402 df-op 4404 df-uni 4659 df-int 4698 df-iun 4742 df-br 4874 df-opab 4936 df-mpt 4953 df-tr 4976 df-id 5250 df-eprel 5255 df-po 5263 df-so 5264 df-fr 5301 df-se 5302 df-we 5303 df-xp 5348 df-rel 5349 df-cnv 5350 df-co 5351 df-dm 5352 df-rn 5353 df-res 5354 df-ima 5355 df-pred 5920 df-ord 5966 df-on 5967 df-lim 5968 df-suc 5969 df-iota 6086 df-fun 6125 df-fn 6126 df-f 6127 df-f1 6128 df-fo 6129 df-f1o 6130 df-fv 6131 df-isom 6132 df-riota 6866 df-ov 6908 df-oprab 6909 df-mpt2 6910 df-om 7327 df-1st 7428 df-2nd 7429 df-supp 7560 df-wrecs 7672 df-recs 7734 df-rdg 7772 df-1o 7826 df-oadd 7830 df-er 8009 df-map 8124 df-en 8223 df-dom 8224 df-sdom 8225 df-fin 8226 df-oi 8684 df-card 9078 df-pnf 10393 df-mnf 10394 df-xr 10395 df-ltxr 10396 df-le 10397 df-sub 10587 df-neg 10588 df-nn 11351 df-n0 11619 df-z 11705 df-uz 11969 df-fz 12620 df-fzo 12761 df-seq 13096 df-hash 13411 df-0g 16455 df-gsum 16456 df-mgm 17595 df-sgrp 17637 df-mnd 17648 df-grp 17779 df-mulg 17895 df-cntz 18100 df-lmod 19221 df-linc 43035 |
This theorem is referenced by: lincvalsn 43046 snlindsntorlem 43099 ldepsnlinclem1 43134 ldepsnlinclem2 43135 |
Copyright terms: Public domain | W3C validator |