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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 6648 | . . . . . . . 8 ⊢ (Base‘𝑆) = (Base‘(Scalar‘𝑀)) |
6 | 3, 5 | eqtri 2821 | . . . . . . 7 ⊢ 𝑅 = (Base‘(Scalar‘𝑀)) |
7 | 6 | eleq2i 2881 | . . . . . 6 ⊢ (𝑌 ∈ 𝑅 ↔ 𝑌 ∈ (Base‘(Scalar‘𝑀))) |
8 | 7 | biimpi 219 | . . . . 5 ⊢ (𝑌 ∈ 𝑅 → 𝑌 ∈ (Base‘(Scalar‘𝑀))) |
9 | 8 | 3ad2ant3 1132 | . . . 4 ⊢ ((𝑀 ∈ LMod ∧ 𝑉 ∈ 𝐵 ∧ 𝑌 ∈ 𝑅) → 𝑌 ∈ (Base‘(Scalar‘𝑀))) |
10 | fvexd 6660 | . . . 4 ⊢ ((𝑀 ∈ LMod ∧ 𝑉 ∈ 𝐵 ∧ 𝑌 ∈ 𝑅) → (Base‘(Scalar‘𝑀)) ∈ V) | |
11 | eqid 2798 | . . . . 5 ⊢ {〈𝑉, 𝑌〉} = {〈𝑉, 𝑌〉} | |
12 | 11 | mapsnop 44746 | . . . 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 5302 | . . . . 5 ⊢ (𝑉 ∈ (Base‘𝑀) → {𝑉} ∈ 𝒫 (Base‘𝑀)) | |
15 | lincvalsn.b | . . . . 5 ⊢ 𝐵 = (Base‘𝑀) | |
16 | 14, 15 | eleq2s 2908 | . . . 4 ⊢ (𝑉 ∈ 𝐵 → {𝑉} ∈ 𝒫 (Base‘𝑀)) |
17 | 16 | 3ad2ant2 1131 | . . 3 ⊢ ((𝑀 ∈ LMod ∧ 𝑉 ∈ 𝐵 ∧ 𝑌 ∈ 𝑅) → {𝑉} ∈ 𝒫 (Base‘𝑀)) |
18 | lincval 44818 | . . 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 19634 | . . . . 5 ⊢ (𝑀 ∈ LMod → 𝑀 ∈ Grp) | |
21 | grpmnd 18102 | . . . . 5 ⊢ (𝑀 ∈ Grp → 𝑀 ∈ Mnd) | |
22 | 20, 21 | syl 17 | . . . 4 ⊢ (𝑀 ∈ LMod → 𝑀 ∈ Mnd) |
23 | 22 | 3ad2ant1 1130 | . . 3 ⊢ ((𝑀 ∈ LMod ∧ 𝑉 ∈ 𝐵 ∧ 𝑌 ∈ 𝑅) → 𝑀 ∈ Mnd) |
24 | fvsng 6919 | . . . . . 6 ⊢ ((𝑉 ∈ 𝐵 ∧ 𝑌 ∈ 𝑅) → ({〈𝑉, 𝑌〉}‘𝑉) = 𝑌) | |
25 | 24 | 3adant1 1127 | . . . . 5 ⊢ ((𝑀 ∈ LMod ∧ 𝑉 ∈ 𝐵 ∧ 𝑌 ∈ 𝑅) → ({〈𝑉, 𝑌〉}‘𝑉) = 𝑌) |
26 | 25 | oveq1d 7150 | . . . 4 ⊢ ((𝑀 ∈ LMod ∧ 𝑉 ∈ 𝐵 ∧ 𝑌 ∈ 𝑅) → (({〈𝑉, 𝑌〉}‘𝑉)( ·𝑠 ‘𝑀)𝑉) = (𝑌( ·𝑠 ‘𝑀)𝑉)) |
27 | eqid 2798 | . . . . . 6 ⊢ ( ·𝑠 ‘𝑀) = ( ·𝑠 ‘𝑀) | |
28 | 15, 4, 27, 3 | lmodvscl 19644 | . . . . 5 ⊢ ((𝑀 ∈ LMod ∧ 𝑌 ∈ 𝑅 ∧ 𝑉 ∈ 𝐵) → (𝑌( ·𝑠 ‘𝑀)𝑉) ∈ 𝐵) |
29 | 28 | 3com23 1123 | . . . 4 ⊢ ((𝑀 ∈ LMod ∧ 𝑉 ∈ 𝐵 ∧ 𝑌 ∈ 𝑅) → (𝑌( ·𝑠 ‘𝑀)𝑉) ∈ 𝐵) |
30 | 26, 29 | eqeltrd 2890 | . . 3 ⊢ ((𝑀 ∈ LMod ∧ 𝑉 ∈ 𝐵 ∧ 𝑌 ∈ 𝑅) → (({〈𝑉, 𝑌〉}‘𝑉)( ·𝑠 ‘𝑀)𝑉) ∈ 𝐵) |
31 | fveq2 6645 | . . . . 5 ⊢ (𝑣 = 𝑉 → ({〈𝑉, 𝑌〉}‘𝑣) = ({〈𝑉, 𝑌〉}‘𝑉)) | |
32 | id 22 | . . . . 5 ⊢ (𝑣 = 𝑉 → 𝑣 = 𝑉) | |
33 | 31, 32 | oveq12d 7153 | . . . 4 ⊢ (𝑣 = 𝑉 → (({〈𝑉, 𝑌〉}‘𝑣)( ·𝑠 ‘𝑀)𝑣) = (({〈𝑉, 𝑌〉}‘𝑉)( ·𝑠 ‘𝑀)𝑉)) |
34 | 15, 33 | gsumsn 19067 | . . 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 1368 | . 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 2807 | . . . 4 ⊢ ( ·𝑠 ‘𝑀) = · |
38 | 37 | a1i 11 | . . 3 ⊢ ((𝑀 ∈ LMod ∧ 𝑉 ∈ 𝐵 ∧ 𝑌 ∈ 𝑅) → ( ·𝑠 ‘𝑀) = · ) |
39 | eqidd 2799 | . . 3 ⊢ ((𝑀 ∈ LMod ∧ 𝑉 ∈ 𝐵 ∧ 𝑌 ∈ 𝑅) → 𝑉 = 𝑉) | |
40 | 38, 25, 39 | oveq123d 7156 | . 2 ⊢ ((𝑀 ∈ LMod ∧ 𝑉 ∈ 𝐵 ∧ 𝑌 ∈ 𝑅) → (({〈𝑉, 𝑌〉}‘𝑉)( ·𝑠 ‘𝑀)𝑉) = (𝑌 · 𝑉)) |
41 | 19, 35, 40 | 3eqtrd 2837 | 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 1538 ∈ wcel 2111 Vcvv 3441 𝒫 cpw 4497 {csn 4525 〈cop 4531 ↦ cmpt 5110 ‘cfv 6324 (class class class)co 7135 ↑m cmap 8389 Basecbs 16475 Scalarcsca 16560 ·𝑠 cvsca 16561 Σg cgsu 16706 Mndcmnd 17903 Grpcgrp 18095 LModclmod 19627 linC clinc 44813 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1797 ax-4 1811 ax-5 1911 ax-6 1970 ax-7 2015 ax-8 2113 ax-9 2121 ax-10 2142 ax-11 2158 ax-12 2175 ax-ext 2770 ax-rep 5154 ax-sep 5167 ax-nul 5174 ax-pow 5231 ax-pr 5295 ax-un 7441 ax-cnex 10582 ax-resscn 10583 ax-1cn 10584 ax-icn 10585 ax-addcl 10586 ax-addrcl 10587 ax-mulcl 10588 ax-mulrcl 10589 ax-mulcom 10590 ax-addass 10591 ax-mulass 10592 ax-distr 10593 ax-i2m1 10594 ax-1ne0 10595 ax-1rid 10596 ax-rnegex 10597 ax-rrecex 10598 ax-cnre 10599 ax-pre-lttri 10600 ax-pre-lttrn 10601 ax-pre-ltadd 10602 ax-pre-mulgt0 10603 |
This theorem depends on definitions: df-bi 210 df-an 400 df-or 845 df-3or 1085 df-3an 1086 df-tru 1541 df-ex 1782 df-nf 1786 df-sb 2070 df-mo 2598 df-eu 2629 df-clab 2777 df-cleq 2791 df-clel 2870 df-nfc 2938 df-ne 2988 df-nel 3092 df-ral 3111 df-rex 3112 df-reu 3113 df-rmo 3114 df-rab 3115 df-v 3443 df-sbc 3721 df-csb 3829 df-dif 3884 df-un 3886 df-in 3888 df-ss 3898 df-pss 3900 df-nul 4244 df-if 4426 df-pw 4499 df-sn 4526 df-pr 4528 df-tp 4530 df-op 4532 df-uni 4801 df-int 4839 df-iun 4883 df-br 5031 df-opab 5093 df-mpt 5111 df-tr 5137 df-id 5425 df-eprel 5430 df-po 5438 df-so 5439 df-fr 5478 df-se 5479 df-we 5480 df-xp 5525 df-rel 5526 df-cnv 5527 df-co 5528 df-dm 5529 df-rn 5530 df-res 5531 df-ima 5532 df-pred 6116 df-ord 6162 df-on 6163 df-lim 6164 df-suc 6165 df-iota 6283 df-fun 6326 df-fn 6327 df-f 6328 df-f1 6329 df-fo 6330 df-f1o 6331 df-fv 6332 df-isom 6333 df-riota 7093 df-ov 7138 df-oprab 7139 df-mpo 7140 df-om 7561 df-1st 7671 df-2nd 7672 df-supp 7814 df-wrecs 7930 df-recs 7991 df-rdg 8029 df-1o 8085 df-oadd 8089 df-er 8272 df-map 8391 df-en 8493 df-dom 8494 df-sdom 8495 df-fin 8496 df-oi 8958 df-card 9352 df-pnf 10666 df-mnf 10667 df-xr 10668 df-ltxr 10669 df-le 10670 df-sub 10861 df-neg 10862 df-nn 11626 df-n0 11886 df-z 11970 df-uz 12232 df-fz 12886 df-fzo 13029 df-seq 13365 df-hash 13687 df-0g 16707 df-gsum 16708 df-mgm 17844 df-sgrp 17893 df-mnd 17904 df-grp 18098 df-mulg 18217 df-cntz 18439 df-lmod 19629 df-linc 44815 |
This theorem is referenced by: lincvalsn 44826 snlindsntorlem 44879 ldepsnlinclem1 44914 ldepsnlinclem2 44915 |
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