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Theorem lcoss 47205
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 4612 . . . . . . 7 ((𝑣 ∈ 𝑉 ∧ 𝑉 ∈ 𝒫 (Baseβ€˜π‘€)) β†’ 𝑣 ∈ (Baseβ€˜π‘€))
21expcom 414 . . . . . 6 (𝑉 ∈ 𝒫 (Baseβ€˜π‘€) β†’ (𝑣 ∈ 𝑉 β†’ 𝑣 ∈ (Baseβ€˜π‘€)))
32adantl 482 . . . . 5 ((𝑀 ∈ LMod ∧ 𝑉 ∈ 𝒫 (Baseβ€˜π‘€)) β†’ (𝑣 ∈ 𝑉 β†’ 𝑣 ∈ (Baseβ€˜π‘€)))
43imp 407 . . . 4 (((𝑀 ∈ LMod ∧ 𝑉 ∈ 𝒫 (Baseβ€˜π‘€)) ∧ 𝑣 ∈ 𝑉) β†’ 𝑣 ∈ (Baseβ€˜π‘€))
5 eqid 2732 . . . . . . 7 (Baseβ€˜π‘€) = (Baseβ€˜π‘€)
6 eqid 2732 . . . . . . 7 (Scalarβ€˜π‘€) = (Scalarβ€˜π‘€)
7 eqid 2732 . . . . . . 7 (0gβ€˜(Scalarβ€˜π‘€)) = (0gβ€˜(Scalarβ€˜π‘€))
8 eqid 2732 . . . . . . 7 (1rβ€˜(Scalarβ€˜π‘€)) = (1rβ€˜(Scalarβ€˜π‘€))
9 equequ1 2028 . . . . . . . . 9 (π‘₯ = 𝑦 β†’ (π‘₯ = 𝑣 ↔ 𝑦 = 𝑣))
109ifbid 4551 . . . . . . . 8 (π‘₯ = 𝑦 β†’ if(π‘₯ = 𝑣, (1rβ€˜(Scalarβ€˜π‘€)), (0gβ€˜(Scalarβ€˜π‘€))) = if(𝑦 = 𝑣, (1rβ€˜(Scalarβ€˜π‘€)), (0gβ€˜(Scalarβ€˜π‘€))))
1110cbvmptv 5261 . . . . . . 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 47145 . . . . . 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 1118 . . . . 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 2732 . . . . . . . 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 47194 . . . . . . 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 1118 . . . . . 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 2738 . . . . 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 2732 . . . . . . . . . . 11 (Baseβ€˜(Scalarβ€˜π‘€)) = (Baseβ€˜(Scalarβ€˜π‘€))
196, 18, 8lmod1cl 20643 . . . . . . . . . 10 (𝑀 ∈ LMod β†’ (1rβ€˜(Scalarβ€˜π‘€)) ∈ (Baseβ€˜(Scalarβ€˜π‘€)))
206, 18, 7lmod0cl 20642 . . . . . . . . . 10 (𝑀 ∈ LMod β†’ (0gβ€˜(Scalarβ€˜π‘€)) ∈ (Baseβ€˜(Scalarβ€˜π‘€)))
2119, 20ifcld 4574 . . . . . . . . 9 (𝑀 ∈ LMod β†’ if(π‘₯ = 𝑣, (1rβ€˜(Scalarβ€˜π‘€)), (0gβ€˜(Scalarβ€˜π‘€))) ∈ (Baseβ€˜(Scalarβ€˜π‘€)))
2221ad3antrrr 728 . . . . . . . 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 7116 . . . . . . 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 767 . . . . . . . 8 (((𝑀 ∈ LMod ∧ 𝑉 ∈ 𝒫 (Baseβ€˜π‘€)) ∧ 𝑣 ∈ 𝑉) β†’ 𝑉 ∈ 𝒫 (Baseβ€˜π‘€))
26 elmapg 8835 . . . . . . . 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 587 . . . . . . 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 256 . . . . . 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 5151 . . . . . . . 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 7418 . . . . . . . . 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 2743 . . . . . . . 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 631 . . . . . . 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 482 . . . . . 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 3605 . . . . 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 697 . . . 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 47181 . . . . 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 481 . . . 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 711 . . 3 (((𝑀 ∈ LMod ∧ 𝑉 ∈ 𝒫 (Baseβ€˜π‘€)) ∧ 𝑣 ∈ 𝑉) β†’ 𝑣 ∈ (𝑀 LinCo 𝑉))
3938ex 413 . 2 ((𝑀 ∈ LMod ∧ 𝑉 ∈ 𝒫 (Baseβ€˜π‘€)) β†’ (𝑣 ∈ 𝑉 β†’ 𝑣 ∈ (𝑀 LinCo 𝑉)))
4039ssrdv 3988 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 396   = wceq 1541   ∈ wcel 2106  βˆƒwrex 3070  Vcvv 3474   βŠ† wss 3948  ifcif 4528  π’« cpw 4602   class class class wbr 5148   ↦ cmpt 5231  βŸΆwf 6539  β€˜cfv 6543  (class class class)co 7411   ↑m cmap 8822   finSupp cfsupp 9363  Basecbs 17148  Scalarcsca 17204  0gc0g 17389  1rcur 20075  LModclmod 20614   linC clinc 47173   LinCo clinco 47174
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 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-10 2137  ax-11 2154  ax-12 2171  ax-ext 2703  ax-rep 5285  ax-sep 5299  ax-nul 5306  ax-pow 5363  ax-pr 5427  ax-un 7727  ax-cnex 11168  ax-resscn 11169  ax-1cn 11170  ax-icn 11171  ax-addcl 11172  ax-addrcl 11173  ax-mulcl 11174  ax-mulrcl 11175  ax-mulcom 11176  ax-addass 11177  ax-mulass 11178  ax-distr 11179  ax-i2m1 11180  ax-1ne0 11181  ax-1rid 11182  ax-rnegex 11183  ax-rrecex 11184  ax-cnre 11185  ax-pre-lttri 11186  ax-pre-lttrn 11187  ax-pre-ltadd 11188  ax-pre-mulgt0 11189
This theorem depends on definitions:  df-bi 206  df-an 397  df-or 846  df-3or 1088  df-3an 1089  df-tru 1544  df-fal 1554  df-ex 1782  df-nf 1786  df-sb 2068  df-mo 2534  df-eu 2563  df-clab 2710  df-cleq 2724  df-clel 2810  df-nfc 2885  df-ne 2941  df-nel 3047  df-ral 3062  df-rex 3071  df-rmo 3376  df-reu 3377  df-rab 3433  df-v 3476  df-sbc 3778  df-csb 3894  df-dif 3951  df-un 3953  df-in 3955  df-ss 3965  df-pss 3967  df-nul 4323  df-if 4529  df-pw 4604  df-sn 4629  df-pr 4631  df-op 4635  df-uni 4909  df-int 4951  df-iun 4999  df-iin 5000  df-br 5149  df-opab 5211  df-mpt 5232  df-tr 5266  df-id 5574  df-eprel 5580  df-po 5588  df-so 5589  df-fr 5631  df-se 5632  df-we 5633  df-xp 5682  df-rel 5683  df-cnv 5684  df-co 5685  df-dm 5686  df-rn 5687  df-res 5688  df-ima 5689  df-pred 6300  df-ord 6367  df-on 6368  df-lim 6369  df-suc 6370  df-iota 6495  df-fun 6545  df-fn 6546  df-f 6547  df-f1 6548  df-fo 6549  df-f1o 6550  df-fv 6551  df-isom 6552  df-riota 7367  df-ov 7414  df-oprab 7415  df-mpo 7416  df-om 7858  df-1st 7977  df-2nd 7978  df-supp 8149  df-frecs 8268  df-wrecs 8299  df-recs 8373  df-rdg 8412  df-1o 8468  df-er 8705  df-map 8824  df-en 8942  df-dom 8943  df-sdom 8944  df-fin 8945  df-fsupp 9364  df-oi 9507  df-card 9936  df-pnf 11254  df-mnf 11255  df-xr 11256  df-ltxr 11257  df-le 11258  df-sub 11450  df-neg 11451  df-nn 12217  df-2 12279  df-n0 12477  df-z 12563  df-uz 12827  df-fz 13489  df-fzo 13632  df-seq 13971  df-hash 14295  df-sets 17101  df-slot 17119  df-ndx 17131  df-base 17149  df-ress 17178  df-plusg 17214  df-0g 17391  df-gsum 17392  df-mre 17534  df-mrc 17535  df-acs 17537  df-mgm 18565  df-sgrp 18644  df-mnd 18660  df-submnd 18706  df-grp 18858  df-mulg 18987  df-cntz 19222  df-cmn 19691  df-mgp 20029  df-ur 20076  df-ring 20129  df-lmod 20616  df-linc 47175  df-lco 47176
This theorem is referenced by:  lspsslco  47206
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