Users' Mathboxes Mathbox for Alexander van der Vekens < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >   Mathboxes  >  lcoss Structured version   Visualization version   GIF version

Theorem lcoss 47427
Description: A set of vectors of a module is a subset of the set of all linear combinations of the set. (Contributed by AV, 18-Apr-2019.) (Proof shortened by AV, 30-Jul-2019.)
Assertion
Ref Expression
lcoss ((𝑀 ∈ LMod ∧ 𝑉 ∈ 𝒫 (Baseβ€˜π‘€)) β†’ 𝑉 βŠ† (𝑀 LinCo 𝑉))

Proof of Theorem lcoss
Dummy variables 𝑓 𝑣 π‘₯ 𝑦 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 elelpwi 4608 . . . . . . 7 ((𝑣 ∈ 𝑉 ∧ 𝑉 ∈ 𝒫 (Baseβ€˜π‘€)) β†’ 𝑣 ∈ (Baseβ€˜π‘€))
21expcom 413 . . . . . 6 (𝑉 ∈ 𝒫 (Baseβ€˜π‘€) β†’ (𝑣 ∈ 𝑉 β†’ 𝑣 ∈ (Baseβ€˜π‘€)))
32adantl 481 . . . . 5 ((𝑀 ∈ LMod ∧ 𝑉 ∈ 𝒫 (Baseβ€˜π‘€)) β†’ (𝑣 ∈ 𝑉 β†’ 𝑣 ∈ (Baseβ€˜π‘€)))
43imp 406 . . . 4 (((𝑀 ∈ LMod ∧ 𝑉 ∈ 𝒫 (Baseβ€˜π‘€)) ∧ 𝑣 ∈ 𝑉) β†’ 𝑣 ∈ (Baseβ€˜π‘€))
5 eqid 2727 . . . . . . 7 (Baseβ€˜π‘€) = (Baseβ€˜π‘€)
6 eqid 2727 . . . . . . 7 (Scalarβ€˜π‘€) = (Scalarβ€˜π‘€)
7 eqid 2727 . . . . . . 7 (0gβ€˜(Scalarβ€˜π‘€)) = (0gβ€˜(Scalarβ€˜π‘€))
8 eqid 2727 . . . . . . 7 (1rβ€˜(Scalarβ€˜π‘€)) = (1rβ€˜(Scalarβ€˜π‘€))
9 equequ1 2021 . . . . . . . . 9 (π‘₯ = 𝑦 β†’ (π‘₯ = 𝑣 ↔ 𝑦 = 𝑣))
109ifbid 4547 . . . . . . . 8 (π‘₯ = 𝑦 β†’ if(π‘₯ = 𝑣, (1rβ€˜(Scalarβ€˜π‘€)), (0gβ€˜(Scalarβ€˜π‘€))) = if(𝑦 = 𝑣, (1rβ€˜(Scalarβ€˜π‘€)), (0gβ€˜(Scalarβ€˜π‘€))))
1110cbvmptv 5255 . . . . . . 7 (π‘₯ ∈ 𝑉 ↦ if(π‘₯ = 𝑣, (1rβ€˜(Scalarβ€˜π‘€)), (0gβ€˜(Scalarβ€˜π‘€)))) = (𝑦 ∈ 𝑉 ↦ if(𝑦 = 𝑣, (1rβ€˜(Scalarβ€˜π‘€)), (0gβ€˜(Scalarβ€˜π‘€))))
125, 6, 7, 8, 11mptcfsupp 47367 . . . . . 6 ((𝑀 ∈ LMod ∧ 𝑉 ∈ 𝒫 (Baseβ€˜π‘€) ∧ 𝑣 ∈ 𝑉) β†’ (π‘₯ ∈ 𝑉 ↦ if(π‘₯ = 𝑣, (1rβ€˜(Scalarβ€˜π‘€)), (0gβ€˜(Scalarβ€˜π‘€)))) finSupp (0gβ€˜(Scalarβ€˜π‘€)))
13123expa 1116 . . . . 5 (((𝑀 ∈ LMod ∧ 𝑉 ∈ 𝒫 (Baseβ€˜π‘€)) ∧ 𝑣 ∈ 𝑉) β†’ (π‘₯ ∈ 𝑉 ↦ if(π‘₯ = 𝑣, (1rβ€˜(Scalarβ€˜π‘€)), (0gβ€˜(Scalarβ€˜π‘€)))) finSupp (0gβ€˜(Scalarβ€˜π‘€)))
14 eqid 2727 . . . . . . . 8 (π‘₯ ∈ 𝑉 ↦ if(π‘₯ = 𝑣, (1rβ€˜(Scalarβ€˜π‘€)), (0gβ€˜(Scalarβ€˜π‘€)))) = (π‘₯ ∈ 𝑉 ↦ if(π‘₯ = 𝑣, (1rβ€˜(Scalarβ€˜π‘€)), (0gβ€˜(Scalarβ€˜π‘€))))
155, 6, 7, 8, 14linc1 47416 . . . . . . 7 ((𝑀 ∈ LMod ∧ 𝑉 ∈ 𝒫 (Baseβ€˜π‘€) ∧ 𝑣 ∈ 𝑉) β†’ ((π‘₯ ∈ 𝑉 ↦ if(π‘₯ = 𝑣, (1rβ€˜(Scalarβ€˜π‘€)), (0gβ€˜(Scalarβ€˜π‘€))))( linC β€˜π‘€)𝑉) = 𝑣)
16153expa 1116 . . . . . 6 (((𝑀 ∈ LMod ∧ 𝑉 ∈ 𝒫 (Baseβ€˜π‘€)) ∧ 𝑣 ∈ 𝑉) β†’ ((π‘₯ ∈ 𝑉 ↦ if(π‘₯ = 𝑣, (1rβ€˜(Scalarβ€˜π‘€)), (0gβ€˜(Scalarβ€˜π‘€))))( linC β€˜π‘€)𝑉) = 𝑣)
1716eqcomd 2733 . . . . 5 (((𝑀 ∈ LMod ∧ 𝑉 ∈ 𝒫 (Baseβ€˜π‘€)) ∧ 𝑣 ∈ 𝑉) β†’ 𝑣 = ((π‘₯ ∈ 𝑉 ↦ if(π‘₯ = 𝑣, (1rβ€˜(Scalarβ€˜π‘€)), (0gβ€˜(Scalarβ€˜π‘€))))( linC β€˜π‘€)𝑉))
18 eqid 2727 . . . . . . . . . . 11 (Baseβ€˜(Scalarβ€˜π‘€)) = (Baseβ€˜(Scalarβ€˜π‘€))
196, 18, 8lmod1cl 20761 . . . . . . . . . 10 (𝑀 ∈ LMod β†’ (1rβ€˜(Scalarβ€˜π‘€)) ∈ (Baseβ€˜(Scalarβ€˜π‘€)))
206, 18, 7lmod0cl 20760 . . . . . . . . . 10 (𝑀 ∈ LMod β†’ (0gβ€˜(Scalarβ€˜π‘€)) ∈ (Baseβ€˜(Scalarβ€˜π‘€)))
2119, 20ifcld 4570 . . . . . . . . 9 (𝑀 ∈ LMod β†’ if(π‘₯ = 𝑣, (1rβ€˜(Scalarβ€˜π‘€)), (0gβ€˜(Scalarβ€˜π‘€))) ∈ (Baseβ€˜(Scalarβ€˜π‘€)))
2221ad3antrrr 729 . . . . . . . 8 ((((𝑀 ∈ LMod ∧ 𝑉 ∈ 𝒫 (Baseβ€˜π‘€)) ∧ 𝑣 ∈ 𝑉) ∧ π‘₯ ∈ 𝑉) β†’ if(π‘₯ = 𝑣, (1rβ€˜(Scalarβ€˜π‘€)), (0gβ€˜(Scalarβ€˜π‘€))) ∈ (Baseβ€˜(Scalarβ€˜π‘€)))
2322fmpttd 7119 . . . . . . 7 (((𝑀 ∈ LMod ∧ 𝑉 ∈ 𝒫 (Baseβ€˜π‘€)) ∧ 𝑣 ∈ 𝑉) β†’ (π‘₯ ∈ 𝑉 ↦ if(π‘₯ = 𝑣, (1rβ€˜(Scalarβ€˜π‘€)), (0gβ€˜(Scalarβ€˜π‘€)))):π‘‰βŸΆ(Baseβ€˜(Scalarβ€˜π‘€)))
24 fvex 6904 . . . . . . . 8 (Baseβ€˜(Scalarβ€˜π‘€)) ∈ V
25 simplr 768 . . . . . . . 8 (((𝑀 ∈ LMod ∧ 𝑉 ∈ 𝒫 (Baseβ€˜π‘€)) ∧ 𝑣 ∈ 𝑉) β†’ 𝑉 ∈ 𝒫 (Baseβ€˜π‘€))
26 elmapg 8849 . . . . . . . 8 (((Baseβ€˜(Scalarβ€˜π‘€)) ∈ V ∧ 𝑉 ∈ 𝒫 (Baseβ€˜π‘€)) β†’ ((π‘₯ ∈ 𝑉 ↦ if(π‘₯ = 𝑣, (1rβ€˜(Scalarβ€˜π‘€)), (0gβ€˜(Scalarβ€˜π‘€)))) ∈ ((Baseβ€˜(Scalarβ€˜π‘€)) ↑m 𝑉) ↔ (π‘₯ ∈ 𝑉 ↦ if(π‘₯ = 𝑣, (1rβ€˜(Scalarβ€˜π‘€)), (0gβ€˜(Scalarβ€˜π‘€)))):π‘‰βŸΆ(Baseβ€˜(Scalarβ€˜π‘€))))
2724, 25, 26sylancr 586 . . . . . . 7 (((𝑀 ∈ LMod ∧ 𝑉 ∈ 𝒫 (Baseβ€˜π‘€)) ∧ 𝑣 ∈ 𝑉) β†’ ((π‘₯ ∈ 𝑉 ↦ if(π‘₯ = 𝑣, (1rβ€˜(Scalarβ€˜π‘€)), (0gβ€˜(Scalarβ€˜π‘€)))) ∈ ((Baseβ€˜(Scalarβ€˜π‘€)) ↑m 𝑉) ↔ (π‘₯ ∈ 𝑉 ↦ if(π‘₯ = 𝑣, (1rβ€˜(Scalarβ€˜π‘€)), (0gβ€˜(Scalarβ€˜π‘€)))):π‘‰βŸΆ(Baseβ€˜(Scalarβ€˜π‘€))))
2823, 27mpbird 257 . . . . . 6 (((𝑀 ∈ LMod ∧ 𝑉 ∈ 𝒫 (Baseβ€˜π‘€)) ∧ 𝑣 ∈ 𝑉) β†’ (π‘₯ ∈ 𝑉 ↦ if(π‘₯ = 𝑣, (1rβ€˜(Scalarβ€˜π‘€)), (0gβ€˜(Scalarβ€˜π‘€)))) ∈ ((Baseβ€˜(Scalarβ€˜π‘€)) ↑m 𝑉))
29 breq1 5145 . . . . . . . 8 (𝑓 = (π‘₯ ∈ 𝑉 ↦ if(π‘₯ = 𝑣, (1rβ€˜(Scalarβ€˜π‘€)), (0gβ€˜(Scalarβ€˜π‘€)))) β†’ (𝑓 finSupp (0gβ€˜(Scalarβ€˜π‘€)) ↔ (π‘₯ ∈ 𝑉 ↦ if(π‘₯ = 𝑣, (1rβ€˜(Scalarβ€˜π‘€)), (0gβ€˜(Scalarβ€˜π‘€)))) finSupp (0gβ€˜(Scalarβ€˜π‘€))))
30 oveq1 7421 . . . . . . . . 9 (𝑓 = (π‘₯ ∈ 𝑉 ↦ if(π‘₯ = 𝑣, (1rβ€˜(Scalarβ€˜π‘€)), (0gβ€˜(Scalarβ€˜π‘€)))) β†’ (𝑓( linC β€˜π‘€)𝑉) = ((π‘₯ ∈ 𝑉 ↦ if(π‘₯ = 𝑣, (1rβ€˜(Scalarβ€˜π‘€)), (0gβ€˜(Scalarβ€˜π‘€))))( linC β€˜π‘€)𝑉))
3130eqeq2d 2738 . . . . . . . 8 (𝑓 = (π‘₯ ∈ 𝑉 ↦ if(π‘₯ = 𝑣, (1rβ€˜(Scalarβ€˜π‘€)), (0gβ€˜(Scalarβ€˜π‘€)))) β†’ (𝑣 = (𝑓( linC β€˜π‘€)𝑉) ↔ 𝑣 = ((π‘₯ ∈ 𝑉 ↦ if(π‘₯ = 𝑣, (1rβ€˜(Scalarβ€˜π‘€)), (0gβ€˜(Scalarβ€˜π‘€))))( linC β€˜π‘€)𝑉)))
3229, 31anbi12d 630 . . . . . . 7 (𝑓 = (π‘₯ ∈ 𝑉 ↦ if(π‘₯ = 𝑣, (1rβ€˜(Scalarβ€˜π‘€)), (0gβ€˜(Scalarβ€˜π‘€)))) β†’ ((𝑓 finSupp (0gβ€˜(Scalarβ€˜π‘€)) ∧ 𝑣 = (𝑓( linC β€˜π‘€)𝑉)) ↔ ((π‘₯ ∈ 𝑉 ↦ if(π‘₯ = 𝑣, (1rβ€˜(Scalarβ€˜π‘€)), (0gβ€˜(Scalarβ€˜π‘€)))) finSupp (0gβ€˜(Scalarβ€˜π‘€)) ∧ 𝑣 = ((π‘₯ ∈ 𝑉 ↦ if(π‘₯ = 𝑣, (1rβ€˜(Scalarβ€˜π‘€)), (0gβ€˜(Scalarβ€˜π‘€))))( linC β€˜π‘€)𝑉))))
3332adantl 481 . . . . . 6 ((((𝑀 ∈ LMod ∧ 𝑉 ∈ 𝒫 (Baseβ€˜π‘€)) ∧ 𝑣 ∈ 𝑉) ∧ 𝑓 = (π‘₯ ∈ 𝑉 ↦ if(π‘₯ = 𝑣, (1rβ€˜(Scalarβ€˜π‘€)), (0gβ€˜(Scalarβ€˜π‘€))))) β†’ ((𝑓 finSupp (0gβ€˜(Scalarβ€˜π‘€)) ∧ 𝑣 = (𝑓( linC β€˜π‘€)𝑉)) ↔ ((π‘₯ ∈ 𝑉 ↦ if(π‘₯ = 𝑣, (1rβ€˜(Scalarβ€˜π‘€)), (0gβ€˜(Scalarβ€˜π‘€)))) finSupp (0gβ€˜(Scalarβ€˜π‘€)) ∧ 𝑣 = ((π‘₯ ∈ 𝑉 ↦ if(π‘₯ = 𝑣, (1rβ€˜(Scalarβ€˜π‘€)), (0gβ€˜(Scalarβ€˜π‘€))))( linC β€˜π‘€)𝑉))))
3428, 33rspcedv 3600 . . . . 5 (((𝑀 ∈ LMod ∧ 𝑉 ∈ 𝒫 (Baseβ€˜π‘€)) ∧ 𝑣 ∈ 𝑉) β†’ (((π‘₯ ∈ 𝑉 ↦ if(π‘₯ = 𝑣, (1rβ€˜(Scalarβ€˜π‘€)), (0gβ€˜(Scalarβ€˜π‘€)))) finSupp (0gβ€˜(Scalarβ€˜π‘€)) ∧ 𝑣 = ((π‘₯ ∈ 𝑉 ↦ if(π‘₯ = 𝑣, (1rβ€˜(Scalarβ€˜π‘€)), (0gβ€˜(Scalarβ€˜π‘€))))( linC β€˜π‘€)𝑉)) β†’ βˆƒπ‘“ ∈ ((Baseβ€˜(Scalarβ€˜π‘€)) ↑m 𝑉)(𝑓 finSupp (0gβ€˜(Scalarβ€˜π‘€)) ∧ 𝑣 = (𝑓( linC β€˜π‘€)𝑉))))
3513, 17, 34mp2and 698 . . . 4 (((𝑀 ∈ LMod ∧ 𝑉 ∈ 𝒫 (Baseβ€˜π‘€)) ∧ 𝑣 ∈ 𝑉) β†’ βˆƒπ‘“ ∈ ((Baseβ€˜(Scalarβ€˜π‘€)) ↑m 𝑉)(𝑓 finSupp (0gβ€˜(Scalarβ€˜π‘€)) ∧ 𝑣 = (𝑓( linC β€˜π‘€)𝑉)))
365, 6, 18lcoval 47403 . . . . 5 ((𝑀 ∈ LMod ∧ 𝑉 ∈ 𝒫 (Baseβ€˜π‘€)) β†’ (𝑣 ∈ (𝑀 LinCo 𝑉) ↔ (𝑣 ∈ (Baseβ€˜π‘€) ∧ βˆƒπ‘“ ∈ ((Baseβ€˜(Scalarβ€˜π‘€)) ↑m 𝑉)(𝑓 finSupp (0gβ€˜(Scalarβ€˜π‘€)) ∧ 𝑣 = (𝑓( linC β€˜π‘€)𝑉)))))
3736adantr 480 . . . 4 (((𝑀 ∈ LMod ∧ 𝑉 ∈ 𝒫 (Baseβ€˜π‘€)) ∧ 𝑣 ∈ 𝑉) β†’ (𝑣 ∈ (𝑀 LinCo 𝑉) ↔ (𝑣 ∈ (Baseβ€˜π‘€) ∧ βˆƒπ‘“ ∈ ((Baseβ€˜(Scalarβ€˜π‘€)) ↑m 𝑉)(𝑓 finSupp (0gβ€˜(Scalarβ€˜π‘€)) ∧ 𝑣 = (𝑓( linC β€˜π‘€)𝑉)))))
384, 35, 37mpbir2and 712 . . 3 (((𝑀 ∈ LMod ∧ 𝑉 ∈ 𝒫 (Baseβ€˜π‘€)) ∧ 𝑣 ∈ 𝑉) β†’ 𝑣 ∈ (𝑀 LinCo 𝑉))
3938ex 412 . 2 ((𝑀 ∈ LMod ∧ 𝑉 ∈ 𝒫 (Baseβ€˜π‘€)) β†’ (𝑣 ∈ 𝑉 β†’ 𝑣 ∈ (𝑀 LinCo 𝑉)))
4039ssrdv 3984 1 ((𝑀 ∈ LMod ∧ 𝑉 ∈ 𝒫 (Baseβ€˜π‘€)) β†’ 𝑉 βŠ† (𝑀 LinCo 𝑉))
Colors of variables: wff setvar class
Syntax hints:   β†’ wi 4   ↔ wb 205   ∧ wa 395   = wceq 1534   ∈ wcel 2099  βˆƒwrex 3065  Vcvv 3469   βŠ† wss 3944  ifcif 4524  π’« cpw 4598   class class class wbr 5142   ↦ cmpt 5225  βŸΆwf 6538  β€˜cfv 6542  (class class class)co 7414   ↑m cmap 8836   finSupp cfsupp 9377  Basecbs 17171  Scalarcsca 17227  0gc0g 17412  1rcur 20112  LModclmod 20732   linC clinc 47395   LinCo clinco 47396
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 2164  ax-ext 2698  ax-rep 5279  ax-sep 5293  ax-nul 5300  ax-pow 5359  ax-pr 5423  ax-un 7734  ax-cnex 11186  ax-resscn 11187  ax-1cn 11188  ax-icn 11189  ax-addcl 11190  ax-addrcl 11191  ax-mulcl 11192  ax-mulrcl 11193  ax-mulcom 11194  ax-addass 11195  ax-mulass 11196  ax-distr 11197  ax-i2m1 11198  ax-1ne0 11199  ax-1rid 11200  ax-rnegex 11201  ax-rrecex 11202  ax-cnre 11203  ax-pre-lttri 11204  ax-pre-lttrn 11205  ax-pre-ltadd 11206  ax-pre-mulgt0 11207
This theorem depends on definitions:  df-bi 206  df-an 396  df-or 847  df-3or 1086  df-3an 1087  df-tru 1537  df-fal 1547  df-ex 1775  df-nf 1779  df-sb 2061  df-mo 2529  df-eu 2558  df-clab 2705  df-cleq 2719  df-clel 2805  df-nfc 2880  df-ne 2936  df-nel 3042  df-ral 3057  df-rex 3066  df-rmo 3371  df-reu 3372  df-rab 3428  df-v 3471  df-sbc 3775  df-csb 3890  df-dif 3947  df-un 3949  df-in 3951  df-ss 3961  df-pss 3963  df-nul 4319  df-if 4525  df-pw 4600  df-sn 4625  df-pr 4627  df-op 4631  df-uni 4904  df-int 4945  df-iun 4993  df-iin 4994  df-br 5143  df-opab 5205  df-mpt 5226  df-tr 5260  df-id 5570  df-eprel 5576  df-po 5584  df-so 5585  df-fr 5627  df-se 5628  df-we 5629  df-xp 5678  df-rel 5679  df-cnv 5680  df-co 5681  df-dm 5682  df-rn 5683  df-res 5684  df-ima 5685  df-pred 6299  df-ord 6366  df-on 6367  df-lim 6368  df-suc 6369  df-iota 6494  df-fun 6544  df-fn 6545  df-f 6546  df-f1 6547  df-fo 6548  df-f1o 6549  df-fv 6550  df-isom 6551  df-riota 7370  df-ov 7417  df-oprab 7418  df-mpo 7419  df-om 7865  df-1st 7987  df-2nd 7988  df-supp 8160  df-frecs 8280  df-wrecs 8311  df-recs 8385  df-rdg 8424  df-1o 8480  df-er 8718  df-map 8838  df-en 8956  df-dom 8957  df-sdom 8958  df-fin 8959  df-fsupp 9378  df-oi 9525  df-card 9954  df-pnf 11272  df-mnf 11273  df-xr 11274  df-ltxr 11275  df-le 11276  df-sub 11468  df-neg 11469  df-nn 12235  df-2 12297  df-n0 12495  df-z 12581  df-uz 12845  df-fz 13509  df-fzo 13652  df-seq 13991  df-hash 14314  df-sets 17124  df-slot 17142  df-ndx 17154  df-base 17172  df-ress 17201  df-plusg 17237  df-0g 17414  df-gsum 17415  df-mre 17557  df-mrc 17558  df-acs 17560  df-mgm 18591  df-sgrp 18670  df-mnd 18686  df-submnd 18732  df-grp 18884  df-mulg 19015  df-cntz 19259  df-cmn 19728  df-mgp 20066  df-ur 20113  df-ring 20166  df-lmod 20734  df-linc 47397  df-lco 47398
This theorem is referenced by:  lspsslco  47428
  Copyright terms: Public domain W3C validator