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Mathbox for Alexander van der Vekens |
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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 1133 | . . 3 ⊢ ((𝑀 ∈ LMod ∧ 𝑉 ∈ 𝐵 ∧ 𝑌 ∈ 𝑅) → 𝑀 ∈ LMod) | |
2 | simp2 1134 | . . . 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 6894 | . . . . . . . 8 ⊢ (Base‘𝑆) = (Base‘(Scalar‘𝑀)) |
6 | 3, 5 | eqtri 2754 | . . . . . . 7 ⊢ 𝑅 = (Base‘(Scalar‘𝑀)) |
7 | 6 | eleq2i 2818 | . . . . . 6 ⊢ (𝑌 ∈ 𝑅 ↔ 𝑌 ∈ (Base‘(Scalar‘𝑀))) |
8 | 7 | biimpi 215 | . . . . 5 ⊢ (𝑌 ∈ 𝑅 → 𝑌 ∈ (Base‘(Scalar‘𝑀))) |
9 | 8 | 3ad2ant3 1132 | . . . 4 ⊢ ((𝑀 ∈ LMod ∧ 𝑉 ∈ 𝐵 ∧ 𝑌 ∈ 𝑅) → 𝑌 ∈ (Base‘(Scalar‘𝑀))) |
10 | fvexd 6906 | . . . 4 ⊢ ((𝑀 ∈ LMod ∧ 𝑉 ∈ 𝐵 ∧ 𝑌 ∈ 𝑅) → (Base‘(Scalar‘𝑀)) ∈ V) | |
11 | eqid 2726 | . . . . 5 ⊢ {〈𝑉, 𝑌〉} = {〈𝑉, 𝑌〉} | |
12 | 11 | mapsnop 47757 | . . . 4 ⊢ ((𝑉 ∈ 𝐵 ∧ 𝑌 ∈ (Base‘(Scalar‘𝑀)) ∧ (Base‘(Scalar‘𝑀)) ∈ V) → {〈𝑉, 𝑌〉} ∈ ((Base‘(Scalar‘𝑀)) ↑m {𝑉})) |
13 | 2, 9, 10, 12 | syl3anc 1368 | . . 3 ⊢ ((𝑀 ∈ LMod ∧ 𝑉 ∈ 𝐵 ∧ 𝑌 ∈ 𝑅) → {〈𝑉, 𝑌〉} ∈ ((Base‘(Scalar‘𝑀)) ↑m {𝑉})) |
14 | snelpwi 5440 | . . . . 5 ⊢ (𝑉 ∈ (Base‘𝑀) → {𝑉} ∈ 𝒫 (Base‘𝑀)) | |
15 | lincvalsn.b | . . . . 5 ⊢ 𝐵 = (Base‘𝑀) | |
16 | 14, 15 | eleq2s 2844 | . . . 4 ⊢ (𝑉 ∈ 𝐵 → {𝑉} ∈ 𝒫 (Base‘𝑀)) |
17 | 16 | 3ad2ant2 1131 | . . 3 ⊢ ((𝑀 ∈ LMod ∧ 𝑉 ∈ 𝐵 ∧ 𝑌 ∈ 𝑅) → {𝑉} ∈ 𝒫 (Base‘𝑀)) |
18 | lincval 47826 | . . 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 1368 | . 2 ⊢ ((𝑀 ∈ LMod ∧ 𝑉 ∈ 𝐵 ∧ 𝑌 ∈ 𝑅) → ({〈𝑉, 𝑌〉} ( linC ‘𝑀){𝑉}) = (𝑀 Σg (𝑣 ∈ {𝑉} ↦ (({〈𝑉, 𝑌〉}‘𝑣)( ·𝑠 ‘𝑀)𝑣)))) |
20 | lmodgrp 20837 | . . . . 5 ⊢ (𝑀 ∈ LMod → 𝑀 ∈ Grp) | |
21 | 20 | grpmndd 18934 | . . . 4 ⊢ (𝑀 ∈ LMod → 𝑀 ∈ Mnd) |
22 | 21 | 3ad2ant1 1130 | . . 3 ⊢ ((𝑀 ∈ LMod ∧ 𝑉 ∈ 𝐵 ∧ 𝑌 ∈ 𝑅) → 𝑀 ∈ Mnd) |
23 | fvsng 7184 | . . . . . 6 ⊢ ((𝑉 ∈ 𝐵 ∧ 𝑌 ∈ 𝑅) → ({〈𝑉, 𝑌〉}‘𝑉) = 𝑌) | |
24 | 23 | 3adant1 1127 | . . . . 5 ⊢ ((𝑀 ∈ LMod ∧ 𝑉 ∈ 𝐵 ∧ 𝑌 ∈ 𝑅) → ({〈𝑉, 𝑌〉}‘𝑉) = 𝑌) |
25 | 24 | oveq1d 7429 | . . . 4 ⊢ ((𝑀 ∈ LMod ∧ 𝑉 ∈ 𝐵 ∧ 𝑌 ∈ 𝑅) → (({〈𝑉, 𝑌〉}‘𝑉)( ·𝑠 ‘𝑀)𝑉) = (𝑌( ·𝑠 ‘𝑀)𝑉)) |
26 | eqid 2726 | . . . . . 6 ⊢ ( ·𝑠 ‘𝑀) = ( ·𝑠 ‘𝑀) | |
27 | 15, 4, 26, 3 | lmodvscl 20848 | . . . . 5 ⊢ ((𝑀 ∈ LMod ∧ 𝑌 ∈ 𝑅 ∧ 𝑉 ∈ 𝐵) → (𝑌( ·𝑠 ‘𝑀)𝑉) ∈ 𝐵) |
28 | 27 | 3com23 1123 | . . . 4 ⊢ ((𝑀 ∈ LMod ∧ 𝑉 ∈ 𝐵 ∧ 𝑌 ∈ 𝑅) → (𝑌( ·𝑠 ‘𝑀)𝑉) ∈ 𝐵) |
29 | 25, 28 | eqeltrd 2826 | . . 3 ⊢ ((𝑀 ∈ LMod ∧ 𝑉 ∈ 𝐵 ∧ 𝑌 ∈ 𝑅) → (({〈𝑉, 𝑌〉}‘𝑉)( ·𝑠 ‘𝑀)𝑉) ∈ 𝐵) |
30 | fveq2 6891 | . . . . 5 ⊢ (𝑣 = 𝑉 → ({〈𝑉, 𝑌〉}‘𝑣) = ({〈𝑉, 𝑌〉}‘𝑉)) | |
31 | id 22 | . . . . 5 ⊢ (𝑣 = 𝑉 → 𝑣 = 𝑉) | |
32 | 30, 31 | oveq12d 7432 | . . . 4 ⊢ (𝑣 = 𝑉 → (({〈𝑉, 𝑌〉}‘𝑣)( ·𝑠 ‘𝑀)𝑣) = (({〈𝑉, 𝑌〉}‘𝑉)( ·𝑠 ‘𝑀)𝑉)) |
33 | 15, 32 | gsumsn 19946 | . . 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)) |
34 | 22, 2, 29, 33 | syl3anc 1368 | . 2 ⊢ ((𝑀 ∈ LMod ∧ 𝑉 ∈ 𝐵 ∧ 𝑌 ∈ 𝑅) → (𝑀 Σg (𝑣 ∈ {𝑉} ↦ (({〈𝑉, 𝑌〉}‘𝑣)( ·𝑠 ‘𝑀)𝑣))) = (({〈𝑉, 𝑌〉}‘𝑉)( ·𝑠 ‘𝑀)𝑉)) |
35 | lincvalsn.t | . . . . 5 ⊢ · = ( ·𝑠 ‘𝑀) | |
36 | 35 | eqcomi 2735 | . . . 4 ⊢ ( ·𝑠 ‘𝑀) = · |
37 | 36 | a1i 11 | . . 3 ⊢ ((𝑀 ∈ LMod ∧ 𝑉 ∈ 𝐵 ∧ 𝑌 ∈ 𝑅) → ( ·𝑠 ‘𝑀) = · ) |
38 | eqidd 2727 | . . 3 ⊢ ((𝑀 ∈ LMod ∧ 𝑉 ∈ 𝐵 ∧ 𝑌 ∈ 𝑅) → 𝑉 = 𝑉) | |
39 | 37, 24, 38 | oveq123d 7435 | . 2 ⊢ ((𝑀 ∈ LMod ∧ 𝑉 ∈ 𝐵 ∧ 𝑌 ∈ 𝑅) → (({〈𝑉, 𝑌〉}‘𝑉)( ·𝑠 ‘𝑀)𝑉) = (𝑌 · 𝑉)) |
40 | 19, 34, 39 | 3eqtrd 2770 | 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 1084 = wceq 1534 ∈ wcel 2099 Vcvv 3463 𝒫 cpw 4598 {csn 4624 〈cop 4630 ↦ cmpt 5227 ‘cfv 6544 (class class class)co 7414 ↑m cmap 8845 Basecbs 17206 Scalarcsca 17262 ·𝑠 cvsca 17263 Σg cgsu 17448 Mndcmnd 18720 LModclmod 20830 linC clinc 47821 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1790 ax-4 1804 ax-5 1906 ax-6 1964 ax-7 2004 ax-8 2101 ax-9 2109 ax-10 2130 ax-11 2147 ax-12 2167 ax-ext 2697 ax-rep 5281 ax-sep 5295 ax-nul 5302 ax-pow 5360 ax-pr 5424 ax-un 7736 ax-cnex 11203 ax-resscn 11204 ax-1cn 11205 ax-icn 11206 ax-addcl 11207 ax-addrcl 11208 ax-mulcl 11209 ax-mulrcl 11210 ax-mulcom 11211 ax-addass 11212 ax-mulass 11213 ax-distr 11214 ax-i2m1 11215 ax-1ne0 11216 ax-1rid 11217 ax-rnegex 11218 ax-rrecex 11219 ax-cnre 11220 ax-pre-lttri 11221 ax-pre-lttrn 11222 ax-pre-ltadd 11223 ax-pre-mulgt0 11224 |
This theorem depends on definitions: df-bi 206 df-an 395 df-or 846 df-3or 1085 df-3an 1086 df-tru 1537 df-fal 1547 df-ex 1775 df-nf 1779 df-sb 2061 df-mo 2529 df-eu 2558 df-clab 2704 df-cleq 2718 df-clel 2803 df-nfc 2878 df-ne 2931 df-nel 3037 df-ral 3052 df-rex 3061 df-rmo 3365 df-reu 3366 df-rab 3421 df-v 3465 df-sbc 3777 df-csb 3893 df-dif 3950 df-un 3952 df-in 3954 df-ss 3964 df-pss 3967 df-nul 4324 df-if 4525 df-pw 4600 df-sn 4625 df-pr 4627 df-op 4631 df-uni 4907 df-int 4948 df-iun 4996 df-br 5145 df-opab 5207 df-mpt 5228 df-tr 5262 df-id 5571 df-eprel 5577 df-po 5585 df-so 5586 df-fr 5628 df-se 5629 df-we 5630 df-xp 5679 df-rel 5680 df-cnv 5681 df-co 5682 df-dm 5683 df-rn 5684 df-res 5685 df-ima 5686 df-pred 6303 df-ord 6369 df-on 6370 df-lim 6371 df-suc 6372 df-iota 6496 df-fun 6546 df-fn 6547 df-f 6548 df-f1 6549 df-fo 6550 df-f1o 6551 df-fv 6552 df-isom 6553 df-riota 7370 df-ov 7417 df-oprab 7418 df-mpo 7419 df-om 7867 df-1st 7993 df-2nd 7994 df-supp 8165 df-frecs 8286 df-wrecs 8317 df-recs 8391 df-rdg 8430 df-1o 8486 df-er 8724 df-map 8847 df-en 8965 df-dom 8966 df-sdom 8967 df-fin 8968 df-oi 9544 df-card 9973 df-pnf 11289 df-mnf 11290 df-xr 11291 df-ltxr 11292 df-le 11293 df-sub 11485 df-neg 11486 df-nn 12257 df-n0 12517 df-z 12603 df-uz 12867 df-fz 13531 df-fzo 13674 df-seq 14014 df-hash 14341 df-0g 17449 df-gsum 17450 df-mgm 18626 df-sgrp 18705 df-mnd 18721 df-grp 18924 df-mulg 19056 df-cntz 19305 df-lmod 20832 df-linc 47823 |
This theorem is referenced by: lincvalsn 47834 snlindsntorlem 47887 ldepsnlinclem1 47922 ldepsnlinclem2 47923 |
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