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Theorem lincvalsc0 47102
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 484 . . 3 ((𝑀 ∈ LMod ∧ 𝑉 ∈ 𝒫 𝐡) β†’ 𝑀 ∈ LMod)
2 lincvalsc0.s . . . . . . . 8 𝑆 = (Scalarβ€˜π‘€)
32eqcomi 2742 . . . . . . . . 9 (Scalarβ€˜π‘€) = 𝑆
43fveq2i 6895 . . . . . . . 8 (Baseβ€˜(Scalarβ€˜π‘€)) = (Baseβ€˜π‘†)
5 lincvalsc0.0 . . . . . . . 8 0 = (0gβ€˜π‘†)
62, 4, 5lmod0cl 20498 . . . . . . 7 (𝑀 ∈ LMod β†’ 0 ∈ (Baseβ€˜(Scalarβ€˜π‘€)))
76adantr 482 . . . . . 6 ((𝑀 ∈ LMod ∧ 𝑉 ∈ 𝒫 𝐡) β†’ 0 ∈ (Baseβ€˜(Scalarβ€˜π‘€)))
87adantr 482 . . . . 5 (((𝑀 ∈ LMod ∧ 𝑉 ∈ 𝒫 𝐡) ∧ π‘₯ ∈ 𝑉) β†’ 0 ∈ (Baseβ€˜(Scalarβ€˜π‘€)))
9 lincvalsc0.f . . . . 5 𝐹 = (π‘₯ ∈ 𝑉 ↦ 0 )
108, 9fmptd 7114 . . . 4 ((𝑀 ∈ LMod ∧ 𝑉 ∈ 𝒫 𝐡) β†’ 𝐹:π‘‰βŸΆ(Baseβ€˜(Scalarβ€˜π‘€)))
11 fvexd 6907 . . . . 5 (𝑀 ∈ LMod β†’ (Baseβ€˜(Scalarβ€˜π‘€)) ∈ V)
12 elmapg 8833 . . . . 5 (((Baseβ€˜(Scalarβ€˜π‘€)) ∈ V ∧ 𝑉 ∈ 𝒫 𝐡) β†’ (𝐹 ∈ ((Baseβ€˜(Scalarβ€˜π‘€)) ↑m 𝑉) ↔ 𝐹:π‘‰βŸΆ(Baseβ€˜(Scalarβ€˜π‘€))))
1311, 12sylan 581 . . . 4 ((𝑀 ∈ LMod ∧ 𝑉 ∈ 𝒫 𝐡) β†’ (𝐹 ∈ ((Baseβ€˜(Scalarβ€˜π‘€)) ↑m 𝑉) ↔ 𝐹:π‘‰βŸΆ(Baseβ€˜(Scalarβ€˜π‘€))))
1410, 13mpbird 257 . . 3 ((𝑀 ∈ LMod ∧ 𝑉 ∈ 𝒫 𝐡) β†’ 𝐹 ∈ ((Baseβ€˜(Scalarβ€˜π‘€)) ↑m 𝑉))
15 lincvalsc0.b . . . . . . 7 𝐡 = (Baseβ€˜π‘€)
1615pweqi 4619 . . . . . 6 𝒫 𝐡 = 𝒫 (Baseβ€˜π‘€)
1716eleq2i 2826 . . . . 5 (𝑉 ∈ 𝒫 𝐡 ↔ 𝑉 ∈ 𝒫 (Baseβ€˜π‘€))
1817biimpi 215 . . . 4 (𝑉 ∈ 𝒫 𝐡 β†’ 𝑉 ∈ 𝒫 (Baseβ€˜π‘€))
1918adantl 483 . . 3 ((𝑀 ∈ LMod ∧ 𝑉 ∈ 𝒫 𝐡) β†’ 𝑉 ∈ 𝒫 (Baseβ€˜π‘€))
20 lincval 47090 . . 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 1372 . 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 486 . . . . . . 7 (((𝑀 ∈ LMod ∧ 𝑉 ∈ 𝒫 𝐡) ∧ 𝑣 ∈ 𝑉) β†’ 𝑣 ∈ 𝑉)
235fvexi 6906 . . . . . . 7 0 ∈ V
24 eqidd 2734 . . . . . . . 8 (π‘₯ = 𝑣 β†’ 0 = 0 )
2524, 9fvmptg 6997 . . . . . . 7 ((𝑣 ∈ 𝑉 ∧ 0 ∈ V) β†’ (πΉβ€˜π‘£) = 0 )
2622, 23, 25sylancl 587 . . . . . 6 (((𝑀 ∈ LMod ∧ 𝑉 ∈ 𝒫 𝐡) ∧ 𝑣 ∈ 𝑉) β†’ (πΉβ€˜π‘£) = 0 )
2726oveq1d 7424 . . . . 5 (((𝑀 ∈ LMod ∧ 𝑉 ∈ 𝒫 𝐡) ∧ 𝑣 ∈ 𝑉) β†’ ((πΉβ€˜π‘£)( ·𝑠 β€˜π‘€)𝑣) = ( 0 ( ·𝑠 β€˜π‘€)𝑣))
281adantr 482 . . . . . 6 (((𝑀 ∈ LMod ∧ 𝑉 ∈ 𝒫 𝐡) ∧ 𝑣 ∈ 𝑉) β†’ 𝑀 ∈ LMod)
29 elelpwi 4613 . . . . . . . . 9 ((𝑣 ∈ 𝑉 ∧ 𝑉 ∈ 𝒫 𝐡) β†’ 𝑣 ∈ 𝐡)
3029expcom 415 . . . . . . . 8 (𝑉 ∈ 𝒫 𝐡 β†’ (𝑣 ∈ 𝑉 β†’ 𝑣 ∈ 𝐡))
3130adantl 483 . . . . . . 7 ((𝑀 ∈ LMod ∧ 𝑉 ∈ 𝒫 𝐡) β†’ (𝑣 ∈ 𝑉 β†’ 𝑣 ∈ 𝐡))
3231imp 408 . . . . . 6 (((𝑀 ∈ LMod ∧ 𝑉 ∈ 𝒫 𝐡) ∧ 𝑣 ∈ 𝑉) β†’ 𝑣 ∈ 𝐡)
33 eqid 2733 . . . . . . 7 ( ·𝑠 β€˜π‘€) = ( ·𝑠 β€˜π‘€)
34 lincvalsc0.z . . . . . . 7 𝑍 = (0gβ€˜π‘€)
3515, 2, 33, 5, 34lmod0vs 20505 . . . . . 6 ((𝑀 ∈ LMod ∧ 𝑣 ∈ 𝐡) β†’ ( 0 ( ·𝑠 β€˜π‘€)𝑣) = 𝑍)
3628, 32, 35syl2anc 585 . . . . 5 (((𝑀 ∈ LMod ∧ 𝑉 ∈ 𝒫 𝐡) ∧ 𝑣 ∈ 𝑉) β†’ ( 0 ( ·𝑠 β€˜π‘€)𝑣) = 𝑍)
3727, 36eqtrd 2773 . . . 4 (((𝑀 ∈ LMod ∧ 𝑉 ∈ 𝒫 𝐡) ∧ 𝑣 ∈ 𝑉) β†’ ((πΉβ€˜π‘£)( ·𝑠 β€˜π‘€)𝑣) = 𝑍)
3837mpteq2dva 5249 . . 3 ((𝑀 ∈ LMod ∧ 𝑉 ∈ 𝒫 𝐡) β†’ (𝑣 ∈ 𝑉 ↦ ((πΉβ€˜π‘£)( ·𝑠 β€˜π‘€)𝑣)) = (𝑣 ∈ 𝑉 ↦ 𝑍))
3938oveq2d 7425 . 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 20478 . . . 4 (𝑀 ∈ LMod β†’ 𝑀 ∈ Grp)
4140grpmndd 18832 . . 3 (𝑀 ∈ LMod β†’ 𝑀 ∈ Mnd)
4234gsumz 18717 . . 3 ((𝑀 ∈ Mnd ∧ 𝑉 ∈ 𝒫 𝐡) β†’ (𝑀 Ξ£g (𝑣 ∈ 𝑉 ↦ 𝑍)) = 𝑍)
4341, 42sylan 581 . 2 ((𝑀 ∈ LMod ∧ 𝑉 ∈ 𝒫 𝐡) β†’ (𝑀 Ξ£g (𝑣 ∈ 𝑉 ↦ 𝑍)) = 𝑍)
4421, 39, 433eqtrd 2777 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 397   = wceq 1542   ∈ wcel 2107  Vcvv 3475  π’« cpw 4603   ↦ cmpt 5232  βŸΆwf 6540  β€˜cfv 6544  (class class class)co 7409   ↑m cmap 8820  Basecbs 17144  Scalarcsca 17200   ·𝑠 cvsca 17201  0gc0g 17385   Ξ£g cgsu 17386  Mndcmnd 18625  LModclmod 20471   linC clinc 47085
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2109  ax-9 2117  ax-10 2138  ax-11 2155  ax-12 2172  ax-ext 2704  ax-rep 5286  ax-sep 5300  ax-nul 5307  ax-pow 5364  ax-pr 5428  ax-un 7725
This theorem depends on definitions:  df-bi 206  df-an 398  df-or 847  df-3an 1090  df-tru 1545  df-fal 1555  df-ex 1783  df-nf 1787  df-sb 2069  df-mo 2535  df-eu 2564  df-clab 2711  df-cleq 2725  df-clel 2811  df-nfc 2886  df-ne 2942  df-ral 3063  df-rex 3072  df-rmo 3377  df-reu 3378  df-rab 3434  df-v 3477  df-sbc 3779  df-csb 3895  df-dif 3952  df-un 3954  df-in 3956  df-ss 3966  df-nul 4324  df-if 4530  df-pw 4605  df-sn 4630  df-pr 4632  df-op 4636  df-uni 4910  df-iun 5000  df-br 5150  df-opab 5212  df-mpt 5233  df-id 5575  df-xp 5683  df-rel 5684  df-cnv 5685  df-co 5686  df-dm 5687  df-rn 5688  df-res 5689  df-ima 5690  df-pred 6301  df-iota 6496  df-fun 6546  df-fn 6547  df-f 6548  df-f1 6549  df-fo 6550  df-f1o 6551  df-fv 6552  df-riota 7365  df-ov 7412  df-oprab 7413  df-mpo 7414  df-1st 7975  df-2nd 7976  df-frecs 8266  df-wrecs 8297  df-recs 8371  df-rdg 8410  df-map 8822  df-seq 13967  df-0g 17387  df-gsum 17388  df-mgm 18561  df-sgrp 18610  df-mnd 18626  df-grp 18822  df-ring 20058  df-lmod 20473  df-linc 47087
This theorem is referenced by:  lcoc0  47103
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