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 1132 | . . 3 ⊢ ((𝑀 ∈ LMod ∧ 𝑉 ∈ 𝐵 ∧ 𝑌 ∈ 𝑅) → 𝑀 ∈ LMod) | |
2 | simp2 1133 | . . . 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 6675 | . . . . . . . 8 ⊢ (Base‘𝑆) = (Base‘(Scalar‘𝑀)) |
6 | 3, 5 | eqtri 2846 | . . . . . . 7 ⊢ 𝑅 = (Base‘(Scalar‘𝑀)) |
7 | 6 | eleq2i 2906 | . . . . . 6 ⊢ (𝑌 ∈ 𝑅 ↔ 𝑌 ∈ (Base‘(Scalar‘𝑀))) |
8 | 7 | biimpi 218 | . . . . 5 ⊢ (𝑌 ∈ 𝑅 → 𝑌 ∈ (Base‘(Scalar‘𝑀))) |
9 | 8 | 3ad2ant3 1131 | . . . 4 ⊢ ((𝑀 ∈ LMod ∧ 𝑉 ∈ 𝐵 ∧ 𝑌 ∈ 𝑅) → 𝑌 ∈ (Base‘(Scalar‘𝑀))) |
10 | fvexd 6687 | . . . 4 ⊢ ((𝑀 ∈ LMod ∧ 𝑉 ∈ 𝐵 ∧ 𝑌 ∈ 𝑅) → (Base‘(Scalar‘𝑀)) ∈ V) | |
11 | eqid 2823 | . . . . 5 ⊢ {〈𝑉, 𝑌〉} = {〈𝑉, 𝑌〉} | |
12 | 11 | mapsnop 44400 | . . . 4 ⊢ ((𝑉 ∈ 𝐵 ∧ 𝑌 ∈ (Base‘(Scalar‘𝑀)) ∧ (Base‘(Scalar‘𝑀)) ∈ V) → {〈𝑉, 𝑌〉} ∈ ((Base‘(Scalar‘𝑀)) ↑m {𝑉})) |
13 | 2, 9, 10, 12 | syl3anc 1367 | . . 3 ⊢ ((𝑀 ∈ LMod ∧ 𝑉 ∈ 𝐵 ∧ 𝑌 ∈ 𝑅) → {〈𝑉, 𝑌〉} ∈ ((Base‘(Scalar‘𝑀)) ↑m {𝑉})) |
14 | snelpwi 5339 | . . . . 5 ⊢ (𝑉 ∈ (Base‘𝑀) → {𝑉} ∈ 𝒫 (Base‘𝑀)) | |
15 | lincvalsn.b | . . . . 5 ⊢ 𝐵 = (Base‘𝑀) | |
16 | 14, 15 | eleq2s 2933 | . . . 4 ⊢ (𝑉 ∈ 𝐵 → {𝑉} ∈ 𝒫 (Base‘𝑀)) |
17 | 16 | 3ad2ant2 1130 | . . 3 ⊢ ((𝑀 ∈ LMod ∧ 𝑉 ∈ 𝐵 ∧ 𝑌 ∈ 𝑅) → {𝑉} ∈ 𝒫 (Base‘𝑀)) |
18 | lincval 44471 | . . 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 1367 | . 2 ⊢ ((𝑀 ∈ LMod ∧ 𝑉 ∈ 𝐵 ∧ 𝑌 ∈ 𝑅) → ({〈𝑉, 𝑌〉} ( linC ‘𝑀){𝑉}) = (𝑀 Σg (𝑣 ∈ {𝑉} ↦ (({〈𝑉, 𝑌〉}‘𝑣)( ·𝑠 ‘𝑀)𝑣)))) |
20 | lmodgrp 19643 | . . . . 5 ⊢ (𝑀 ∈ LMod → 𝑀 ∈ Grp) | |
21 | grpmnd 18112 | . . . . 5 ⊢ (𝑀 ∈ Grp → 𝑀 ∈ Mnd) | |
22 | 20, 21 | syl 17 | . . . 4 ⊢ (𝑀 ∈ LMod → 𝑀 ∈ Mnd) |
23 | 22 | 3ad2ant1 1129 | . . 3 ⊢ ((𝑀 ∈ LMod ∧ 𝑉 ∈ 𝐵 ∧ 𝑌 ∈ 𝑅) → 𝑀 ∈ Mnd) |
24 | fvsng 6944 | . . . . . 6 ⊢ ((𝑉 ∈ 𝐵 ∧ 𝑌 ∈ 𝑅) → ({〈𝑉, 𝑌〉}‘𝑉) = 𝑌) | |
25 | 24 | 3adant1 1126 | . . . . 5 ⊢ ((𝑀 ∈ LMod ∧ 𝑉 ∈ 𝐵 ∧ 𝑌 ∈ 𝑅) → ({〈𝑉, 𝑌〉}‘𝑉) = 𝑌) |
26 | 25 | oveq1d 7173 | . . . 4 ⊢ ((𝑀 ∈ LMod ∧ 𝑉 ∈ 𝐵 ∧ 𝑌 ∈ 𝑅) → (({〈𝑉, 𝑌〉}‘𝑉)( ·𝑠 ‘𝑀)𝑉) = (𝑌( ·𝑠 ‘𝑀)𝑉)) |
27 | eqid 2823 | . . . . . 6 ⊢ ( ·𝑠 ‘𝑀) = ( ·𝑠 ‘𝑀) | |
28 | 15, 4, 27, 3 | lmodvscl 19653 | . . . . 5 ⊢ ((𝑀 ∈ LMod ∧ 𝑌 ∈ 𝑅 ∧ 𝑉 ∈ 𝐵) → (𝑌( ·𝑠 ‘𝑀)𝑉) ∈ 𝐵) |
29 | 28 | 3com23 1122 | . . . 4 ⊢ ((𝑀 ∈ LMod ∧ 𝑉 ∈ 𝐵 ∧ 𝑌 ∈ 𝑅) → (𝑌( ·𝑠 ‘𝑀)𝑉) ∈ 𝐵) |
30 | 26, 29 | eqeltrd 2915 | . . 3 ⊢ ((𝑀 ∈ LMod ∧ 𝑉 ∈ 𝐵 ∧ 𝑌 ∈ 𝑅) → (({〈𝑉, 𝑌〉}‘𝑉)( ·𝑠 ‘𝑀)𝑉) ∈ 𝐵) |
31 | fveq2 6672 | . . . . 5 ⊢ (𝑣 = 𝑉 → ({〈𝑉, 𝑌〉}‘𝑣) = ({〈𝑉, 𝑌〉}‘𝑉)) | |
32 | id 22 | . . . . 5 ⊢ (𝑣 = 𝑉 → 𝑣 = 𝑉) | |
33 | 31, 32 | oveq12d 7176 | . . . 4 ⊢ (𝑣 = 𝑉 → (({〈𝑉, 𝑌〉}‘𝑣)( ·𝑠 ‘𝑀)𝑣) = (({〈𝑉, 𝑌〉}‘𝑉)( ·𝑠 ‘𝑀)𝑉)) |
34 | 15, 33 | gsumsn 19076 | . . 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 1367 | . 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 2832 | . . . 4 ⊢ ( ·𝑠 ‘𝑀) = · |
38 | 37 | a1i 11 | . . 3 ⊢ ((𝑀 ∈ LMod ∧ 𝑉 ∈ 𝐵 ∧ 𝑌 ∈ 𝑅) → ( ·𝑠 ‘𝑀) = · ) |
39 | eqidd 2824 | . . 3 ⊢ ((𝑀 ∈ LMod ∧ 𝑉 ∈ 𝐵 ∧ 𝑌 ∈ 𝑅) → 𝑉 = 𝑉) | |
40 | 38, 25, 39 | oveq123d 7179 | . 2 ⊢ ((𝑀 ∈ LMod ∧ 𝑉 ∈ 𝐵 ∧ 𝑌 ∈ 𝑅) → (({〈𝑉, 𝑌〉}‘𝑉)( ·𝑠 ‘𝑀)𝑉) = (𝑌 · 𝑉)) |
41 | 19, 35, 40 | 3eqtrd 2862 | 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 1083 = wceq 1537 ∈ wcel 2114 Vcvv 3496 𝒫 cpw 4541 {csn 4569 〈cop 4575 ↦ cmpt 5148 ‘cfv 6357 (class class class)co 7158 ↑m cmap 8408 Basecbs 16485 Scalarcsca 16570 ·𝑠 cvsca 16571 Σg cgsu 16716 Mndcmnd 17913 Grpcgrp 18105 LModclmod 19636 linC clinc 44466 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1796 ax-4 1810 ax-5 1911 ax-6 1970 ax-7 2015 ax-8 2116 ax-9 2124 ax-10 2145 ax-11 2161 ax-12 2177 ax-ext 2795 ax-rep 5192 ax-sep 5205 ax-nul 5212 ax-pow 5268 ax-pr 5332 ax-un 7463 ax-cnex 10595 ax-resscn 10596 ax-1cn 10597 ax-icn 10598 ax-addcl 10599 ax-addrcl 10600 ax-mulcl 10601 ax-mulrcl 10602 ax-mulcom 10603 ax-addass 10604 ax-mulass 10605 ax-distr 10606 ax-i2m1 10607 ax-1ne0 10608 ax-1rid 10609 ax-rnegex 10610 ax-rrecex 10611 ax-cnre 10612 ax-pre-lttri 10613 ax-pre-lttrn 10614 ax-pre-ltadd 10615 ax-pre-mulgt0 10616 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3or 1084 df-3an 1085 df-tru 1540 df-ex 1781 df-nf 1785 df-sb 2070 df-mo 2622 df-eu 2654 df-clab 2802 df-cleq 2816 df-clel 2895 df-nfc 2965 df-ne 3019 df-nel 3126 df-ral 3145 df-rex 3146 df-reu 3147 df-rmo 3148 df-rab 3149 df-v 3498 df-sbc 3775 df-csb 3886 df-dif 3941 df-un 3943 df-in 3945 df-ss 3954 df-pss 3956 df-nul 4294 df-if 4470 df-pw 4543 df-sn 4570 df-pr 4572 df-tp 4574 df-op 4576 df-uni 4841 df-int 4879 df-iun 4923 df-br 5069 df-opab 5131 df-mpt 5149 df-tr 5175 df-id 5462 df-eprel 5467 df-po 5476 df-so 5477 df-fr 5516 df-se 5517 df-we 5518 df-xp 5563 df-rel 5564 df-cnv 5565 df-co 5566 df-dm 5567 df-rn 5568 df-res 5569 df-ima 5570 df-pred 6150 df-ord 6196 df-on 6197 df-lim 6198 df-suc 6199 df-iota 6316 df-fun 6359 df-fn 6360 df-f 6361 df-f1 6362 df-fo 6363 df-f1o 6364 df-fv 6365 df-isom 6366 df-riota 7116 df-ov 7161 df-oprab 7162 df-mpo 7163 df-om 7583 df-1st 7691 df-2nd 7692 df-supp 7833 df-wrecs 7949 df-recs 8010 df-rdg 8048 df-1o 8104 df-oadd 8108 df-er 8291 df-map 8410 df-en 8512 df-dom 8513 df-sdom 8514 df-fin 8515 df-oi 8976 df-card 9370 df-pnf 10679 df-mnf 10680 df-xr 10681 df-ltxr 10682 df-le 10683 df-sub 10874 df-neg 10875 df-nn 11641 df-n0 11901 df-z 11985 df-uz 12247 df-fz 12896 df-fzo 13037 df-seq 13373 df-hash 13694 df-0g 16717 df-gsum 16718 df-mgm 17854 df-sgrp 17903 df-mnd 17914 df-grp 18108 df-mulg 18227 df-cntz 18449 df-lmod 19638 df-linc 44468 |
This theorem is referenced by: lincvalsn 44479 snlindsntorlem 44532 ldepsnlinclem1 44567 ldepsnlinclem2 44568 |
Copyright terms: Public domain | W3C validator |