Users' Mathboxes Mathbox for Alexander van der Vekens < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >   Mathboxes  >  lincvalsc0 Structured version   Visualization version   GIF version

Theorem lincvalsc0 47092
Description: The linear combination where all scalars are 0. (Contributed by AV, 12-Apr-2019.)
Hypotheses
Ref Expression
lincvalsc0.b 𝐡 = (Baseβ€˜π‘€)
lincvalsc0.s 𝑆 = (Scalarβ€˜π‘€)
lincvalsc0.0 0 = (0gβ€˜π‘†)
lincvalsc0.z 𝑍 = (0gβ€˜π‘€)
lincvalsc0.f 𝐹 = (π‘₯ ∈ 𝑉 ↦ 0 )
Assertion
Ref Expression
lincvalsc0 ((𝑀 ∈ LMod ∧ 𝑉 ∈ 𝒫 𝐡) β†’ (𝐹( linC β€˜π‘€)𝑉) = 𝑍)
Distinct variable groups:   π‘₯,𝐡   π‘₯,𝑀   π‘₯,𝑉   π‘₯, 0
Allowed substitution hints:   𝑆(π‘₯)   𝐹(π‘₯)   𝑍(π‘₯)

Proof of Theorem lincvalsc0
Dummy variable 𝑣 is distinct from all other variables.
StepHypRef Expression
1 simpl 483 . . 3 ((𝑀 ∈ LMod ∧ 𝑉 ∈ 𝒫 𝐡) β†’ 𝑀 ∈ LMod)
2 lincvalsc0.s . . . . . . . 8 𝑆 = (Scalarβ€˜π‘€)
32eqcomi 2741 . . . . . . . . 9 (Scalarβ€˜π‘€) = 𝑆
43fveq2i 6894 . . . . . . . 8 (Baseβ€˜(Scalarβ€˜π‘€)) = (Baseβ€˜π‘†)
5 lincvalsc0.0 . . . . . . . 8 0 = (0gβ€˜π‘†)
62, 4, 5lmod0cl 20497 . . . . . . 7 (𝑀 ∈ LMod β†’ 0 ∈ (Baseβ€˜(Scalarβ€˜π‘€)))
76adantr 481 . . . . . 6 ((𝑀 ∈ LMod ∧ 𝑉 ∈ 𝒫 𝐡) β†’ 0 ∈ (Baseβ€˜(Scalarβ€˜π‘€)))
87adantr 481 . . . . 5 (((𝑀 ∈ LMod ∧ 𝑉 ∈ 𝒫 𝐡) ∧ π‘₯ ∈ 𝑉) β†’ 0 ∈ (Baseβ€˜(Scalarβ€˜π‘€)))
9 lincvalsc0.f . . . . 5 𝐹 = (π‘₯ ∈ 𝑉 ↦ 0 )
108, 9fmptd 7113 . . . 4 ((𝑀 ∈ LMod ∧ 𝑉 ∈ 𝒫 𝐡) β†’ 𝐹:π‘‰βŸΆ(Baseβ€˜(Scalarβ€˜π‘€)))
11 fvexd 6906 . . . . 5 (𝑀 ∈ LMod β†’ (Baseβ€˜(Scalarβ€˜π‘€)) ∈ V)
12 elmapg 8832 . . . . 5 (((Baseβ€˜(Scalarβ€˜π‘€)) ∈ V ∧ 𝑉 ∈ 𝒫 𝐡) β†’ (𝐹 ∈ ((Baseβ€˜(Scalarβ€˜π‘€)) ↑m 𝑉) ↔ 𝐹:π‘‰βŸΆ(Baseβ€˜(Scalarβ€˜π‘€))))
1311, 12sylan 580 . . . 4 ((𝑀 ∈ LMod ∧ 𝑉 ∈ 𝒫 𝐡) β†’ (𝐹 ∈ ((Baseβ€˜(Scalarβ€˜π‘€)) ↑m 𝑉) ↔ 𝐹:π‘‰βŸΆ(Baseβ€˜(Scalarβ€˜π‘€))))
1410, 13mpbird 256 . . 3 ((𝑀 ∈ LMod ∧ 𝑉 ∈ 𝒫 𝐡) β†’ 𝐹 ∈ ((Baseβ€˜(Scalarβ€˜π‘€)) ↑m 𝑉))
15 lincvalsc0.b . . . . . . 7 𝐡 = (Baseβ€˜π‘€)
1615pweqi 4618 . . . . . 6 𝒫 𝐡 = 𝒫 (Baseβ€˜π‘€)
1716eleq2i 2825 . . . . 5 (𝑉 ∈ 𝒫 𝐡 ↔ 𝑉 ∈ 𝒫 (Baseβ€˜π‘€))
1817biimpi 215 . . . 4 (𝑉 ∈ 𝒫 𝐡 β†’ 𝑉 ∈ 𝒫 (Baseβ€˜π‘€))
1918adantl 482 . . 3 ((𝑀 ∈ LMod ∧ 𝑉 ∈ 𝒫 𝐡) β†’ 𝑉 ∈ 𝒫 (Baseβ€˜π‘€))
20 lincval 47080 . . 3 ((𝑀 ∈ LMod ∧ 𝐹 ∈ ((Baseβ€˜(Scalarβ€˜π‘€)) ↑m 𝑉) ∧ 𝑉 ∈ 𝒫 (Baseβ€˜π‘€)) β†’ (𝐹( linC β€˜π‘€)𝑉) = (𝑀 Ξ£g (𝑣 ∈ 𝑉 ↦ ((πΉβ€˜π‘£)( ·𝑠 β€˜π‘€)𝑣))))
211, 14, 19, 20syl3anc 1371 . 2 ((𝑀 ∈ LMod ∧ 𝑉 ∈ 𝒫 𝐡) β†’ (𝐹( linC β€˜π‘€)𝑉) = (𝑀 Ξ£g (𝑣 ∈ 𝑉 ↦ ((πΉβ€˜π‘£)( ·𝑠 β€˜π‘€)𝑣))))
22 simpr 485 . . . . . . 7 (((𝑀 ∈ LMod ∧ 𝑉 ∈ 𝒫 𝐡) ∧ 𝑣 ∈ 𝑉) β†’ 𝑣 ∈ 𝑉)
235fvexi 6905 . . . . . . 7 0 ∈ V
24 eqidd 2733 . . . . . . . 8 (π‘₯ = 𝑣 β†’ 0 = 0 )
2524, 9fvmptg 6996 . . . . . . 7 ((𝑣 ∈ 𝑉 ∧ 0 ∈ V) β†’ (πΉβ€˜π‘£) = 0 )
2622, 23, 25sylancl 586 . . . . . 6 (((𝑀 ∈ LMod ∧ 𝑉 ∈ 𝒫 𝐡) ∧ 𝑣 ∈ 𝑉) β†’ (πΉβ€˜π‘£) = 0 )
2726oveq1d 7423 . . . . 5 (((𝑀 ∈ LMod ∧ 𝑉 ∈ 𝒫 𝐡) ∧ 𝑣 ∈ 𝑉) β†’ ((πΉβ€˜π‘£)( ·𝑠 β€˜π‘€)𝑣) = ( 0 ( ·𝑠 β€˜π‘€)𝑣))
281adantr 481 . . . . . 6 (((𝑀 ∈ LMod ∧ 𝑉 ∈ 𝒫 𝐡) ∧ 𝑣 ∈ 𝑉) β†’ 𝑀 ∈ LMod)
29 elelpwi 4612 . . . . . . . . 9 ((𝑣 ∈ 𝑉 ∧ 𝑉 ∈ 𝒫 𝐡) β†’ 𝑣 ∈ 𝐡)
3029expcom 414 . . . . . . . 8 (𝑉 ∈ 𝒫 𝐡 β†’ (𝑣 ∈ 𝑉 β†’ 𝑣 ∈ 𝐡))
3130adantl 482 . . . . . . 7 ((𝑀 ∈ LMod ∧ 𝑉 ∈ 𝒫 𝐡) β†’ (𝑣 ∈ 𝑉 β†’ 𝑣 ∈ 𝐡))
3231imp 407 . . . . . 6 (((𝑀 ∈ LMod ∧ 𝑉 ∈ 𝒫 𝐡) ∧ 𝑣 ∈ 𝑉) β†’ 𝑣 ∈ 𝐡)
33 eqid 2732 . . . . . . 7 ( ·𝑠 β€˜π‘€) = ( ·𝑠 β€˜π‘€)
34 lincvalsc0.z . . . . . . 7 𝑍 = (0gβ€˜π‘€)
3515, 2, 33, 5, 34lmod0vs 20504 . . . . . 6 ((𝑀 ∈ LMod ∧ 𝑣 ∈ 𝐡) β†’ ( 0 ( ·𝑠 β€˜π‘€)𝑣) = 𝑍)
3628, 32, 35syl2anc 584 . . . . 5 (((𝑀 ∈ LMod ∧ 𝑉 ∈ 𝒫 𝐡) ∧ 𝑣 ∈ 𝑉) β†’ ( 0 ( ·𝑠 β€˜π‘€)𝑣) = 𝑍)
3727, 36eqtrd 2772 . . . 4 (((𝑀 ∈ LMod ∧ 𝑉 ∈ 𝒫 𝐡) ∧ 𝑣 ∈ 𝑉) β†’ ((πΉβ€˜π‘£)( ·𝑠 β€˜π‘€)𝑣) = 𝑍)
3837mpteq2dva 5248 . . 3 ((𝑀 ∈ LMod ∧ 𝑉 ∈ 𝒫 𝐡) β†’ (𝑣 ∈ 𝑉 ↦ ((πΉβ€˜π‘£)( ·𝑠 β€˜π‘€)𝑣)) = (𝑣 ∈ 𝑉 ↦ 𝑍))
3938oveq2d 7424 . 2 ((𝑀 ∈ LMod ∧ 𝑉 ∈ 𝒫 𝐡) β†’ (𝑀 Ξ£g (𝑣 ∈ 𝑉 ↦ ((πΉβ€˜π‘£)( ·𝑠 β€˜π‘€)𝑣))) = (𝑀 Ξ£g (𝑣 ∈ 𝑉 ↦ 𝑍)))
40 lmodgrp 20477 . . . 4 (𝑀 ∈ LMod β†’ 𝑀 ∈ Grp)
4140grpmndd 18831 . . 3 (𝑀 ∈ LMod β†’ 𝑀 ∈ Mnd)
4234gsumz 18716 . . 3 ((𝑀 ∈ Mnd ∧ 𝑉 ∈ 𝒫 𝐡) β†’ (𝑀 Ξ£g (𝑣 ∈ 𝑉 ↦ 𝑍)) = 𝑍)
4341, 42sylan 580 . 2 ((𝑀 ∈ LMod ∧ 𝑉 ∈ 𝒫 𝐡) β†’ (𝑀 Ξ£g (𝑣 ∈ 𝑉 ↦ 𝑍)) = 𝑍)
4421, 39, 433eqtrd 2776 1 ((𝑀 ∈ LMod ∧ 𝑉 ∈ 𝒫 𝐡) β†’ (𝐹( linC β€˜π‘€)𝑉) = 𝑍)
Colors of variables: wff setvar class
Syntax hints:   β†’ wi 4   ↔ wb 205   ∧ wa 396   = wceq 1541   ∈ wcel 2106  Vcvv 3474  π’« cpw 4602   ↦ cmpt 5231  βŸΆwf 6539  β€˜cfv 6543  (class class class)co 7408   ↑m cmap 8819  Basecbs 17143  Scalarcsca 17199   ·𝑠 cvsca 17200  0gc0g 17384   Ξ£g cgsu 17385  Mndcmnd 18624  LModclmod 20470   linC clinc 47075
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 7724
This theorem depends on definitions:  df-bi 206  df-an 397  df-or 846  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-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-nul 4323  df-if 4529  df-pw 4604  df-sn 4629  df-pr 4631  df-op 4635  df-uni 4909  df-iun 4999  df-br 5149  df-opab 5211  df-mpt 5232  df-id 5574  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-iota 6495  df-fun 6545  df-fn 6546  df-f 6547  df-f1 6548  df-fo 6549  df-f1o 6550  df-fv 6551  df-riota 7364  df-ov 7411  df-oprab 7412  df-mpo 7413  df-1st 7974  df-2nd 7975  df-frecs 8265  df-wrecs 8296  df-recs 8370  df-rdg 8409  df-map 8821  df-seq 13966  df-0g 17386  df-gsum 17387  df-mgm 18560  df-sgrp 18609  df-mnd 18625  df-grp 18821  df-ring 20057  df-lmod 20472  df-linc 47077
This theorem is referenced by:  lcoc0  47093
  Copyright terms: Public domain W3C validator