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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 1054 | . . 3 ⊢ ((𝑀 ∈ LMod ∧ 𝑉 ∈ 𝐵 ∧ 𝑌 ∈ 𝑅) → 𝑀 ∈ LMod) | |
2 | simp2 1055 | . . . 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 6106 | . . . . . . . 8 ⊢ (Base‘𝑆) = (Base‘(Scalar‘𝑀)) |
6 | 3, 5 | eqtri 2632 | . . . . . . 7 ⊢ 𝑅 = (Base‘(Scalar‘𝑀)) |
7 | 6 | eleq2i 2680 | . . . . . 6 ⊢ (𝑌 ∈ 𝑅 ↔ 𝑌 ∈ (Base‘(Scalar‘𝑀))) |
8 | 7 | biimpi 205 | . . . . 5 ⊢ (𝑌 ∈ 𝑅 → 𝑌 ∈ (Base‘(Scalar‘𝑀))) |
9 | 8 | 3ad2ant3 1077 | . . . 4 ⊢ ((𝑀 ∈ LMod ∧ 𝑉 ∈ 𝐵 ∧ 𝑌 ∈ 𝑅) → 𝑌 ∈ (Base‘(Scalar‘𝑀))) |
10 | fvex 6113 | . . . . 5 ⊢ (Base‘(Scalar‘𝑀)) ∈ V | |
11 | 10 | a1i 11 | . . . 4 ⊢ ((𝑀 ∈ LMod ∧ 𝑉 ∈ 𝐵 ∧ 𝑌 ∈ 𝑅) → (Base‘(Scalar‘𝑀)) ∈ V) |
12 | eqid 2610 | . . . . 5 ⊢ {〈𝑉, 𝑌〉} = {〈𝑉, 𝑌〉} | |
13 | 12 | mapsnop 41916 | . . . 4 ⊢ ((𝑉 ∈ 𝐵 ∧ 𝑌 ∈ (Base‘(Scalar‘𝑀)) ∧ (Base‘(Scalar‘𝑀)) ∈ V) → {〈𝑉, 𝑌〉} ∈ ((Base‘(Scalar‘𝑀)) ↑𝑚 {𝑉})) |
14 | 2, 9, 11, 13 | syl3anc 1318 | . . 3 ⊢ ((𝑀 ∈ LMod ∧ 𝑉 ∈ 𝐵 ∧ 𝑌 ∈ 𝑅) → {〈𝑉, 𝑌〉} ∈ ((Base‘(Scalar‘𝑀)) ↑𝑚 {𝑉})) |
15 | snelpwi 4839 | . . . . 5 ⊢ (𝑉 ∈ (Base‘𝑀) → {𝑉} ∈ 𝒫 (Base‘𝑀)) | |
16 | lincvalsn.b | . . . . 5 ⊢ 𝐵 = (Base‘𝑀) | |
17 | 15, 16 | eleq2s 2706 | . . . 4 ⊢ (𝑉 ∈ 𝐵 → {𝑉} ∈ 𝒫 (Base‘𝑀)) |
18 | 17 | 3ad2ant2 1076 | . . 3 ⊢ ((𝑀 ∈ LMod ∧ 𝑉 ∈ 𝐵 ∧ 𝑌 ∈ 𝑅) → {𝑉} ∈ 𝒫 (Base‘𝑀)) |
19 | lincval 41992 | . . 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)))) | |
20 | 1, 14, 18, 19 | syl3anc 1318 | . 2 ⊢ ((𝑀 ∈ LMod ∧ 𝑉 ∈ 𝐵 ∧ 𝑌 ∈ 𝑅) → ({〈𝑉, 𝑌〉} ( linC ‘𝑀){𝑉}) = (𝑀 Σg (𝑣 ∈ {𝑉} ↦ (({〈𝑉, 𝑌〉}‘𝑣)( ·𝑠 ‘𝑀)𝑣)))) |
21 | lmodgrp 18693 | . . . . 5 ⊢ (𝑀 ∈ LMod → 𝑀 ∈ Grp) | |
22 | grpmnd 17252 | . . . . 5 ⊢ (𝑀 ∈ Grp → 𝑀 ∈ Mnd) | |
23 | 21, 22 | syl 17 | . . . 4 ⊢ (𝑀 ∈ LMod → 𝑀 ∈ Mnd) |
24 | 23 | 3ad2ant1 1075 | . . 3 ⊢ ((𝑀 ∈ LMod ∧ 𝑉 ∈ 𝐵 ∧ 𝑌 ∈ 𝑅) → 𝑀 ∈ Mnd) |
25 | fvsng 6352 | . . . . . 6 ⊢ ((𝑉 ∈ 𝐵 ∧ 𝑌 ∈ 𝑅) → ({〈𝑉, 𝑌〉}‘𝑉) = 𝑌) | |
26 | 25 | 3adant1 1072 | . . . . 5 ⊢ ((𝑀 ∈ LMod ∧ 𝑉 ∈ 𝐵 ∧ 𝑌 ∈ 𝑅) → ({〈𝑉, 𝑌〉}‘𝑉) = 𝑌) |
27 | 26 | oveq1d 6564 | . . . 4 ⊢ ((𝑀 ∈ LMod ∧ 𝑉 ∈ 𝐵 ∧ 𝑌 ∈ 𝑅) → (({〈𝑉, 𝑌〉}‘𝑉)( ·𝑠 ‘𝑀)𝑉) = (𝑌( ·𝑠 ‘𝑀)𝑉)) |
28 | eqid 2610 | . . . . . 6 ⊢ ( ·𝑠 ‘𝑀) = ( ·𝑠 ‘𝑀) | |
29 | 16, 4, 28, 3 | lmodvscl 18703 | . . . . 5 ⊢ ((𝑀 ∈ LMod ∧ 𝑌 ∈ 𝑅 ∧ 𝑉 ∈ 𝐵) → (𝑌( ·𝑠 ‘𝑀)𝑉) ∈ 𝐵) |
30 | 29 | 3com23 1263 | . . . 4 ⊢ ((𝑀 ∈ LMod ∧ 𝑉 ∈ 𝐵 ∧ 𝑌 ∈ 𝑅) → (𝑌( ·𝑠 ‘𝑀)𝑉) ∈ 𝐵) |
31 | 27, 30 | eqeltrd 2688 | . . 3 ⊢ ((𝑀 ∈ LMod ∧ 𝑉 ∈ 𝐵 ∧ 𝑌 ∈ 𝑅) → (({〈𝑉, 𝑌〉}‘𝑉)( ·𝑠 ‘𝑀)𝑉) ∈ 𝐵) |
32 | fveq2 6103 | . . . . 5 ⊢ (𝑣 = 𝑉 → ({〈𝑉, 𝑌〉}‘𝑣) = ({〈𝑉, 𝑌〉}‘𝑉)) | |
33 | id 22 | . . . . 5 ⊢ (𝑣 = 𝑉 → 𝑣 = 𝑉) | |
34 | 32, 33 | oveq12d 6567 | . . . 4 ⊢ (𝑣 = 𝑉 → (({〈𝑉, 𝑌〉}‘𝑣)( ·𝑠 ‘𝑀)𝑣) = (({〈𝑉, 𝑌〉}‘𝑉)( ·𝑠 ‘𝑀)𝑉)) |
35 | 16, 34 | gsumsn 18177 | . . 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)) |
36 | 24, 2, 31, 35 | syl3anc 1318 | . 2 ⊢ ((𝑀 ∈ LMod ∧ 𝑉 ∈ 𝐵 ∧ 𝑌 ∈ 𝑅) → (𝑀 Σg (𝑣 ∈ {𝑉} ↦ (({〈𝑉, 𝑌〉}‘𝑣)( ·𝑠 ‘𝑀)𝑣))) = (({〈𝑉, 𝑌〉}‘𝑉)( ·𝑠 ‘𝑀)𝑉)) |
37 | lincvalsn.t | . . . . 5 ⊢ · = ( ·𝑠 ‘𝑀) | |
38 | 37 | eqcomi 2619 | . . . 4 ⊢ ( ·𝑠 ‘𝑀) = · |
39 | 38 | a1i 11 | . . 3 ⊢ ((𝑀 ∈ LMod ∧ 𝑉 ∈ 𝐵 ∧ 𝑌 ∈ 𝑅) → ( ·𝑠 ‘𝑀) = · ) |
40 | eqidd 2611 | . . 3 ⊢ ((𝑀 ∈ LMod ∧ 𝑉 ∈ 𝐵 ∧ 𝑌 ∈ 𝑅) → 𝑉 = 𝑉) | |
41 | 39, 26, 40 | oveq123d 6570 | . 2 ⊢ ((𝑀 ∈ LMod ∧ 𝑉 ∈ 𝐵 ∧ 𝑌 ∈ 𝑅) → (({〈𝑉, 𝑌〉}‘𝑉)( ·𝑠 ‘𝑀)𝑉) = (𝑌 · 𝑉)) |
42 | 20, 36, 41 | 3eqtrd 2648 | 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 1031 = wceq 1475 ∈ wcel 1977 Vcvv 3173 𝒫 cpw 4108 {csn 4125 〈cop 4131 ↦ cmpt 4643 ‘cfv 5804 (class class class)co 6549 ↑𝑚 cmap 7744 Basecbs 15695 Scalarcsca 15771 ·𝑠 cvsca 15772 Σg cgsu 15924 Mndcmnd 17117 Grpcgrp 17245 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-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-lmod 18688 df-linc 41989 |
This theorem is referenced by: lincvalsn 42000 snlindsntorlem 42053 ldepsnlinclem1 42088 ldepsnlinclem2 42089 |
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