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Theorem scmsuppss 47001
Description: The support of a mapping of a scalar multiplication with a function of scalars is a subset of the support of the function of scalars. (Contributed by AV, 5-Apr-2019.)
Hypotheses
Ref Expression
scmsuppss.s 𝑆 = (Scalarβ€˜π‘€)
scmsuppss.r 𝑅 = (Baseβ€˜π‘†)
Assertion
Ref Expression
scmsuppss ((𝑀 ∈ LMod ∧ 𝑉 ∈ 𝒫 (Baseβ€˜π‘€) ∧ 𝐴 ∈ (𝑅 ↑m 𝑉)) β†’ ((𝑣 ∈ 𝑉 ↦ ((π΄β€˜π‘£)( ·𝑠 β€˜π‘€)𝑣)) supp (0gβ€˜π‘€)) βŠ† (𝐴 supp (0gβ€˜π‘†)))
Distinct variable groups:   𝑣,𝐴   𝑣,𝑀   𝑣,𝑅   𝑣,𝑉
Allowed substitution hint:   𝑆(𝑣)

Proof of Theorem scmsuppss
Dummy variable π‘₯ is distinct from all other variables.
StepHypRef Expression
1 elmapi 8839 . . . . 5 (𝐴 ∈ (𝑅 ↑m 𝑉) β†’ 𝐴:π‘‰βŸΆπ‘…)
2 fdm 6723 . . . . . 6 (𝐴:π‘‰βŸΆπ‘… β†’ dom 𝐴 = 𝑉)
3 eqidd 2733 . . . . . . . . . . . 12 ((((dom 𝐴 = 𝑉 ∧ 𝐴:π‘‰βŸΆπ‘…) ∧ (𝑉 ∈ 𝒫 (Baseβ€˜π‘€) ∧ 𝑀 ∈ LMod)) ∧ π‘₯ ∈ 𝑉) β†’ (𝑣 ∈ 𝑉 ↦ ((π΄β€˜π‘£)( ·𝑠 β€˜π‘€)𝑣)) = (𝑣 ∈ 𝑉 ↦ ((π΄β€˜π‘£)( ·𝑠 β€˜π‘€)𝑣)))
4 fveq2 6888 . . . . . . . . . . . . . 14 (𝑣 = π‘₯ β†’ (π΄β€˜π‘£) = (π΄β€˜π‘₯))
5 id 22 . . . . . . . . . . . . . 14 (𝑣 = π‘₯ β†’ 𝑣 = π‘₯)
64, 5oveq12d 7423 . . . . . . . . . . . . 13 (𝑣 = π‘₯ β†’ ((π΄β€˜π‘£)( ·𝑠 β€˜π‘€)𝑣) = ((π΄β€˜π‘₯)( ·𝑠 β€˜π‘€)π‘₯))
76adantl 482 . . . . . . . . . . . 12 (((((dom 𝐴 = 𝑉 ∧ 𝐴:π‘‰βŸΆπ‘…) ∧ (𝑉 ∈ 𝒫 (Baseβ€˜π‘€) ∧ 𝑀 ∈ LMod)) ∧ π‘₯ ∈ 𝑉) ∧ 𝑣 = π‘₯) β†’ ((π΄β€˜π‘£)( ·𝑠 β€˜π‘€)𝑣) = ((π΄β€˜π‘₯)( ·𝑠 β€˜π‘€)π‘₯))
8 simpr 485 . . . . . . . . . . . 12 ((((dom 𝐴 = 𝑉 ∧ 𝐴:π‘‰βŸΆπ‘…) ∧ (𝑉 ∈ 𝒫 (Baseβ€˜π‘€) ∧ 𝑀 ∈ LMod)) ∧ π‘₯ ∈ 𝑉) β†’ π‘₯ ∈ 𝑉)
9 ovex 7438 . . . . . . . . . . . . 13 ((π΄β€˜π‘₯)( ·𝑠 β€˜π‘€)π‘₯) ∈ V
109a1i 11 . . . . . . . . . . . 12 ((((dom 𝐴 = 𝑉 ∧ 𝐴:π‘‰βŸΆπ‘…) ∧ (𝑉 ∈ 𝒫 (Baseβ€˜π‘€) ∧ 𝑀 ∈ LMod)) ∧ π‘₯ ∈ 𝑉) β†’ ((π΄β€˜π‘₯)( ·𝑠 β€˜π‘€)π‘₯) ∈ V)
113, 7, 8, 10fvmptd 7002 . . . . . . . . . . 11 ((((dom 𝐴 = 𝑉 ∧ 𝐴:π‘‰βŸΆπ‘…) ∧ (𝑉 ∈ 𝒫 (Baseβ€˜π‘€) ∧ 𝑀 ∈ LMod)) ∧ π‘₯ ∈ 𝑉) β†’ ((𝑣 ∈ 𝑉 ↦ ((π΄β€˜π‘£)( ·𝑠 β€˜π‘€)𝑣))β€˜π‘₯) = ((π΄β€˜π‘₯)( ·𝑠 β€˜π‘€)π‘₯))
1211neeq1d 3000 . . . . . . . . . 10 ((((dom 𝐴 = 𝑉 ∧ 𝐴:π‘‰βŸΆπ‘…) ∧ (𝑉 ∈ 𝒫 (Baseβ€˜π‘€) ∧ 𝑀 ∈ LMod)) ∧ π‘₯ ∈ 𝑉) β†’ (((𝑣 ∈ 𝑉 ↦ ((π΄β€˜π‘£)( ·𝑠 β€˜π‘€)𝑣))β€˜π‘₯) β‰  (0gβ€˜π‘€) ↔ ((π΄β€˜π‘₯)( ·𝑠 β€˜π‘€)π‘₯) β‰  (0gβ€˜π‘€)))
13 oveq1 7412 . . . . . . . . . . . . 13 ((π΄β€˜π‘₯) = (0gβ€˜π‘†) β†’ ((π΄β€˜π‘₯)( ·𝑠 β€˜π‘€)π‘₯) = ((0gβ€˜π‘†)( ·𝑠 β€˜π‘€)π‘₯))
14 simplrr 776 . . . . . . . . . . . . . 14 ((((dom 𝐴 = 𝑉 ∧ 𝐴:π‘‰βŸΆπ‘…) ∧ (𝑉 ∈ 𝒫 (Baseβ€˜π‘€) ∧ 𝑀 ∈ LMod)) ∧ π‘₯ ∈ 𝑉) β†’ 𝑀 ∈ LMod)
15 elelpwi 4611 . . . . . . . . . . . . . . . . . 18 ((π‘₯ ∈ 𝑉 ∧ 𝑉 ∈ 𝒫 (Baseβ€˜π‘€)) β†’ π‘₯ ∈ (Baseβ€˜π‘€))
1615expcom 414 . . . . . . . . . . . . . . . . 17 (𝑉 ∈ 𝒫 (Baseβ€˜π‘€) β†’ (π‘₯ ∈ 𝑉 β†’ π‘₯ ∈ (Baseβ€˜π‘€)))
1716adantr 481 . . . . . . . . . . . . . . . 16 ((𝑉 ∈ 𝒫 (Baseβ€˜π‘€) ∧ 𝑀 ∈ LMod) β†’ (π‘₯ ∈ 𝑉 β†’ π‘₯ ∈ (Baseβ€˜π‘€)))
1817adantl 482 . . . . . . . . . . . . . . 15 (((dom 𝐴 = 𝑉 ∧ 𝐴:π‘‰βŸΆπ‘…) ∧ (𝑉 ∈ 𝒫 (Baseβ€˜π‘€) ∧ 𝑀 ∈ LMod)) β†’ (π‘₯ ∈ 𝑉 β†’ π‘₯ ∈ (Baseβ€˜π‘€)))
1918imp 407 . . . . . . . . . . . . . 14 ((((dom 𝐴 = 𝑉 ∧ 𝐴:π‘‰βŸΆπ‘…) ∧ (𝑉 ∈ 𝒫 (Baseβ€˜π‘€) ∧ 𝑀 ∈ LMod)) ∧ π‘₯ ∈ 𝑉) β†’ π‘₯ ∈ (Baseβ€˜π‘€))
20 eqid 2732 . . . . . . . . . . . . . . 15 (Baseβ€˜π‘€) = (Baseβ€˜π‘€)
21 scmsuppss.s . . . . . . . . . . . . . . 15 𝑆 = (Scalarβ€˜π‘€)
22 eqid 2732 . . . . . . . . . . . . . . 15 ( ·𝑠 β€˜π‘€) = ( ·𝑠 β€˜π‘€)
23 eqid 2732 . . . . . . . . . . . . . . 15 (0gβ€˜π‘†) = (0gβ€˜π‘†)
24 eqid 2732 . . . . . . . . . . . . . . 15 (0gβ€˜π‘€) = (0gβ€˜π‘€)
2520, 21, 22, 23, 24lmod0vs 20497 . . . . . . . . . . . . . 14 ((𝑀 ∈ LMod ∧ π‘₯ ∈ (Baseβ€˜π‘€)) β†’ ((0gβ€˜π‘†)( ·𝑠 β€˜π‘€)π‘₯) = (0gβ€˜π‘€))
2614, 19, 25syl2anc 584 . . . . . . . . . . . . 13 ((((dom 𝐴 = 𝑉 ∧ 𝐴:π‘‰βŸΆπ‘…) ∧ (𝑉 ∈ 𝒫 (Baseβ€˜π‘€) ∧ 𝑀 ∈ LMod)) ∧ π‘₯ ∈ 𝑉) β†’ ((0gβ€˜π‘†)( ·𝑠 β€˜π‘€)π‘₯) = (0gβ€˜π‘€))
2713, 26sylan9eqr 2794 . . . . . . . . . . . 12 (((((dom 𝐴 = 𝑉 ∧ 𝐴:π‘‰βŸΆπ‘…) ∧ (𝑉 ∈ 𝒫 (Baseβ€˜π‘€) ∧ 𝑀 ∈ LMod)) ∧ π‘₯ ∈ 𝑉) ∧ (π΄β€˜π‘₯) = (0gβ€˜π‘†)) β†’ ((π΄β€˜π‘₯)( ·𝑠 β€˜π‘€)π‘₯) = (0gβ€˜π‘€))
2827ex 413 . . . . . . . . . . 11 ((((dom 𝐴 = 𝑉 ∧ 𝐴:π‘‰βŸΆπ‘…) ∧ (𝑉 ∈ 𝒫 (Baseβ€˜π‘€) ∧ 𝑀 ∈ LMod)) ∧ π‘₯ ∈ 𝑉) β†’ ((π΄β€˜π‘₯) = (0gβ€˜π‘†) β†’ ((π΄β€˜π‘₯)( ·𝑠 β€˜π‘€)π‘₯) = (0gβ€˜π‘€)))
2928necon3d 2961 . . . . . . . . . 10 ((((dom 𝐴 = 𝑉 ∧ 𝐴:π‘‰βŸΆπ‘…) ∧ (𝑉 ∈ 𝒫 (Baseβ€˜π‘€) ∧ 𝑀 ∈ LMod)) ∧ π‘₯ ∈ 𝑉) β†’ (((π΄β€˜π‘₯)( ·𝑠 β€˜π‘€)π‘₯) β‰  (0gβ€˜π‘€) β†’ (π΄β€˜π‘₯) β‰  (0gβ€˜π‘†)))
3012, 29sylbid 239 . . . . . . . . 9 ((((dom 𝐴 = 𝑉 ∧ 𝐴:π‘‰βŸΆπ‘…) ∧ (𝑉 ∈ 𝒫 (Baseβ€˜π‘€) ∧ 𝑀 ∈ LMod)) ∧ π‘₯ ∈ 𝑉) β†’ (((𝑣 ∈ 𝑉 ↦ ((π΄β€˜π‘£)( ·𝑠 β€˜π‘€)𝑣))β€˜π‘₯) β‰  (0gβ€˜π‘€) β†’ (π΄β€˜π‘₯) β‰  (0gβ€˜π‘†)))
3130ss2rabdv 4072 . . . . . . . 8 (((dom 𝐴 = 𝑉 ∧ 𝐴:π‘‰βŸΆπ‘…) ∧ (𝑉 ∈ 𝒫 (Baseβ€˜π‘€) ∧ 𝑀 ∈ LMod)) β†’ {π‘₯ ∈ 𝑉 ∣ ((𝑣 ∈ 𝑉 ↦ ((π΄β€˜π‘£)( ·𝑠 β€˜π‘€)𝑣))β€˜π‘₯) β‰  (0gβ€˜π‘€)} βŠ† {π‘₯ ∈ 𝑉 ∣ (π΄β€˜π‘₯) β‰  (0gβ€˜π‘†)})
32 ovex 7438 . . . . . . . . . . . . 13 ((π΄β€˜π‘£)( ·𝑠 β€˜π‘€)𝑣) ∈ V
33 eqid 2732 . . . . . . . . . . . . 13 (𝑣 ∈ 𝑉 ↦ ((π΄β€˜π‘£)( ·𝑠 β€˜π‘€)𝑣)) = (𝑣 ∈ 𝑉 ↦ ((π΄β€˜π‘£)( ·𝑠 β€˜π‘€)𝑣))
3432, 33dmmpti 6691 . . . . . . . . . . . 12 dom (𝑣 ∈ 𝑉 ↦ ((π΄β€˜π‘£)( ·𝑠 β€˜π‘€)𝑣)) = 𝑉
35 rabeq 3446 . . . . . . . . . . . 12 (dom (𝑣 ∈ 𝑉 ↦ ((π΄β€˜π‘£)( ·𝑠 β€˜π‘€)𝑣)) = 𝑉 β†’ {π‘₯ ∈ dom (𝑣 ∈ 𝑉 ↦ ((π΄β€˜π‘£)( ·𝑠 β€˜π‘€)𝑣)) ∣ ((𝑣 ∈ 𝑉 ↦ ((π΄β€˜π‘£)( ·𝑠 β€˜π‘€)𝑣))β€˜π‘₯) β‰  (0gβ€˜π‘€)} = {π‘₯ ∈ 𝑉 ∣ ((𝑣 ∈ 𝑉 ↦ ((π΄β€˜π‘£)( ·𝑠 β€˜π‘€)𝑣))β€˜π‘₯) β‰  (0gβ€˜π‘€)})
3634, 35mp1i 13 . . . . . . . . . . 11 (dom 𝐴 = 𝑉 β†’ {π‘₯ ∈ dom (𝑣 ∈ 𝑉 ↦ ((π΄β€˜π‘£)( ·𝑠 β€˜π‘€)𝑣)) ∣ ((𝑣 ∈ 𝑉 ↦ ((π΄β€˜π‘£)( ·𝑠 β€˜π‘€)𝑣))β€˜π‘₯) β‰  (0gβ€˜π‘€)} = {π‘₯ ∈ 𝑉 ∣ ((𝑣 ∈ 𝑉 ↦ ((π΄β€˜π‘£)( ·𝑠 β€˜π‘€)𝑣))β€˜π‘₯) β‰  (0gβ€˜π‘€)})
37 rabeq 3446 . . . . . . . . . . 11 (dom 𝐴 = 𝑉 β†’ {π‘₯ ∈ dom 𝐴 ∣ (π΄β€˜π‘₯) β‰  (0gβ€˜π‘†)} = {π‘₯ ∈ 𝑉 ∣ (π΄β€˜π‘₯) β‰  (0gβ€˜π‘†)})
3836, 37sseq12d 4014 . . . . . . . . . 10 (dom 𝐴 = 𝑉 β†’ ({π‘₯ ∈ dom (𝑣 ∈ 𝑉 ↦ ((π΄β€˜π‘£)( ·𝑠 β€˜π‘€)𝑣)) ∣ ((𝑣 ∈ 𝑉 ↦ ((π΄β€˜π‘£)( ·𝑠 β€˜π‘€)𝑣))β€˜π‘₯) β‰  (0gβ€˜π‘€)} βŠ† {π‘₯ ∈ dom 𝐴 ∣ (π΄β€˜π‘₯) β‰  (0gβ€˜π‘†)} ↔ {π‘₯ ∈ 𝑉 ∣ ((𝑣 ∈ 𝑉 ↦ ((π΄β€˜π‘£)( ·𝑠 β€˜π‘€)𝑣))β€˜π‘₯) β‰  (0gβ€˜π‘€)} βŠ† {π‘₯ ∈ 𝑉 ∣ (π΄β€˜π‘₯) β‰  (0gβ€˜π‘†)}))
3938adantr 481 . . . . . . . . 9 ((dom 𝐴 = 𝑉 ∧ 𝐴:π‘‰βŸΆπ‘…) β†’ ({π‘₯ ∈ dom (𝑣 ∈ 𝑉 ↦ ((π΄β€˜π‘£)( ·𝑠 β€˜π‘€)𝑣)) ∣ ((𝑣 ∈ 𝑉 ↦ ((π΄β€˜π‘£)( ·𝑠 β€˜π‘€)𝑣))β€˜π‘₯) β‰  (0gβ€˜π‘€)} βŠ† {π‘₯ ∈ dom 𝐴 ∣ (π΄β€˜π‘₯) β‰  (0gβ€˜π‘†)} ↔ {π‘₯ ∈ 𝑉 ∣ ((𝑣 ∈ 𝑉 ↦ ((π΄β€˜π‘£)( ·𝑠 β€˜π‘€)𝑣))β€˜π‘₯) β‰  (0gβ€˜π‘€)} βŠ† {π‘₯ ∈ 𝑉 ∣ (π΄β€˜π‘₯) β‰  (0gβ€˜π‘†)}))
4039adantr 481 . . . . . . . 8 (((dom 𝐴 = 𝑉 ∧ 𝐴:π‘‰βŸΆπ‘…) ∧ (𝑉 ∈ 𝒫 (Baseβ€˜π‘€) ∧ 𝑀 ∈ LMod)) β†’ ({π‘₯ ∈ dom (𝑣 ∈ 𝑉 ↦ ((π΄β€˜π‘£)( ·𝑠 β€˜π‘€)𝑣)) ∣ ((𝑣 ∈ 𝑉 ↦ ((π΄β€˜π‘£)( ·𝑠 β€˜π‘€)𝑣))β€˜π‘₯) β‰  (0gβ€˜π‘€)} βŠ† {π‘₯ ∈ dom 𝐴 ∣ (π΄β€˜π‘₯) β‰  (0gβ€˜π‘†)} ↔ {π‘₯ ∈ 𝑉 ∣ ((𝑣 ∈ 𝑉 ↦ ((π΄β€˜π‘£)( ·𝑠 β€˜π‘€)𝑣))β€˜π‘₯) β‰  (0gβ€˜π‘€)} βŠ† {π‘₯ ∈ 𝑉 ∣ (π΄β€˜π‘₯) β‰  (0gβ€˜π‘†)}))
4131, 40mpbird 256 . . . . . . 7 (((dom 𝐴 = 𝑉 ∧ 𝐴:π‘‰βŸΆπ‘…) ∧ (𝑉 ∈ 𝒫 (Baseβ€˜π‘€) ∧ 𝑀 ∈ LMod)) β†’ {π‘₯ ∈ dom (𝑣 ∈ 𝑉 ↦ ((π΄β€˜π‘£)( ·𝑠 β€˜π‘€)𝑣)) ∣ ((𝑣 ∈ 𝑉 ↦ ((π΄β€˜π‘£)( ·𝑠 β€˜π‘€)𝑣))β€˜π‘₯) β‰  (0gβ€˜π‘€)} βŠ† {π‘₯ ∈ dom 𝐴 ∣ (π΄β€˜π‘₯) β‰  (0gβ€˜π‘†)})
4241exp43 437 . . . . . 6 (dom 𝐴 = 𝑉 β†’ (𝐴:π‘‰βŸΆπ‘… β†’ (𝑉 ∈ 𝒫 (Baseβ€˜π‘€) β†’ (𝑀 ∈ LMod β†’ {π‘₯ ∈ dom (𝑣 ∈ 𝑉 ↦ ((π΄β€˜π‘£)( ·𝑠 β€˜π‘€)𝑣)) ∣ ((𝑣 ∈ 𝑉 ↦ ((π΄β€˜π‘£)( ·𝑠 β€˜π‘€)𝑣))β€˜π‘₯) β‰  (0gβ€˜π‘€)} βŠ† {π‘₯ ∈ dom 𝐴 ∣ (π΄β€˜π‘₯) β‰  (0gβ€˜π‘†)}))))
432, 42mpcom 38 . . . . 5 (𝐴:π‘‰βŸΆπ‘… β†’ (𝑉 ∈ 𝒫 (Baseβ€˜π‘€) β†’ (𝑀 ∈ LMod β†’ {π‘₯ ∈ dom (𝑣 ∈ 𝑉 ↦ ((π΄β€˜π‘£)( ·𝑠 β€˜π‘€)𝑣)) ∣ ((𝑣 ∈ 𝑉 ↦ ((π΄β€˜π‘£)( ·𝑠 β€˜π‘€)𝑣))β€˜π‘₯) β‰  (0gβ€˜π‘€)} βŠ† {π‘₯ ∈ dom 𝐴 ∣ (π΄β€˜π‘₯) β‰  (0gβ€˜π‘†)})))
441, 43syl 17 . . . 4 (𝐴 ∈ (𝑅 ↑m 𝑉) β†’ (𝑉 ∈ 𝒫 (Baseβ€˜π‘€) β†’ (𝑀 ∈ LMod β†’ {π‘₯ ∈ dom (𝑣 ∈ 𝑉 ↦ ((π΄β€˜π‘£)( ·𝑠 β€˜π‘€)𝑣)) ∣ ((𝑣 ∈ 𝑉 ↦ ((π΄β€˜π‘£)( ·𝑠 β€˜π‘€)𝑣))β€˜π‘₯) β‰  (0gβ€˜π‘€)} βŠ† {π‘₯ ∈ dom 𝐴 ∣ (π΄β€˜π‘₯) β‰  (0gβ€˜π‘†)})))
4544com13 88 . . 3 (𝑀 ∈ LMod β†’ (𝑉 ∈ 𝒫 (Baseβ€˜π‘€) β†’ (𝐴 ∈ (𝑅 ↑m 𝑉) β†’ {π‘₯ ∈ dom (𝑣 ∈ 𝑉 ↦ ((π΄β€˜π‘£)( ·𝑠 β€˜π‘€)𝑣)) ∣ ((𝑣 ∈ 𝑉 ↦ ((π΄β€˜π‘£)( ·𝑠 β€˜π‘€)𝑣))β€˜π‘₯) β‰  (0gβ€˜π‘€)} βŠ† {π‘₯ ∈ dom 𝐴 ∣ (π΄β€˜π‘₯) β‰  (0gβ€˜π‘†)})))
46453imp 1111 . 2 ((𝑀 ∈ LMod ∧ 𝑉 ∈ 𝒫 (Baseβ€˜π‘€) ∧ 𝐴 ∈ (𝑅 ↑m 𝑉)) β†’ {π‘₯ ∈ dom (𝑣 ∈ 𝑉 ↦ ((π΄β€˜π‘£)( ·𝑠 β€˜π‘€)𝑣)) ∣ ((𝑣 ∈ 𝑉 ↦ ((π΄β€˜π‘£)( ·𝑠 β€˜π‘€)𝑣))β€˜π‘₯) β‰  (0gβ€˜π‘€)} βŠ† {π‘₯ ∈ dom 𝐴 ∣ (π΄β€˜π‘₯) β‰  (0gβ€˜π‘†)})
47 funmpt 6583 . . . 4 Fun (𝑣 ∈ 𝑉 ↦ ((π΄β€˜π‘£)( ·𝑠 β€˜π‘€)𝑣))
4847a1i 11 . . 3 ((𝑀 ∈ LMod ∧ 𝑉 ∈ 𝒫 (Baseβ€˜π‘€) ∧ 𝐴 ∈ (𝑅 ↑m 𝑉)) β†’ Fun (𝑣 ∈ 𝑉 ↦ ((π΄β€˜π‘£)( ·𝑠 β€˜π‘€)𝑣)))
49 mptexg 7219 . . . 4 (𝑉 ∈ 𝒫 (Baseβ€˜π‘€) β†’ (𝑣 ∈ 𝑉 ↦ ((π΄β€˜π‘£)( ·𝑠 β€˜π‘€)𝑣)) ∈ V)
50493ad2ant2 1134 . . 3 ((𝑀 ∈ LMod ∧ 𝑉 ∈ 𝒫 (Baseβ€˜π‘€) ∧ 𝐴 ∈ (𝑅 ↑m 𝑉)) β†’ (𝑣 ∈ 𝑉 ↦ ((π΄β€˜π‘£)( ·𝑠 β€˜π‘€)𝑣)) ∈ V)
51 fvexd 6903 . . 3 ((𝑀 ∈ LMod ∧ 𝑉 ∈ 𝒫 (Baseβ€˜π‘€) ∧ 𝐴 ∈ (𝑅 ↑m 𝑉)) β†’ (0gβ€˜π‘€) ∈ V)
52 suppval1 8148 . . 3 ((Fun (𝑣 ∈ 𝑉 ↦ ((π΄β€˜π‘£)( ·𝑠 β€˜π‘€)𝑣)) ∧ (𝑣 ∈ 𝑉 ↦ ((π΄β€˜π‘£)( ·𝑠 β€˜π‘€)𝑣)) ∈ V ∧ (0gβ€˜π‘€) ∈ V) β†’ ((𝑣 ∈ 𝑉 ↦ ((π΄β€˜π‘£)( ·𝑠 β€˜π‘€)𝑣)) supp (0gβ€˜π‘€)) = {π‘₯ ∈ dom (𝑣 ∈ 𝑉 ↦ ((π΄β€˜π‘£)( ·𝑠 β€˜π‘€)𝑣)) ∣ ((𝑣 ∈ 𝑉 ↦ ((π΄β€˜π‘£)( ·𝑠 β€˜π‘€)𝑣))β€˜π‘₯) β‰  (0gβ€˜π‘€)})
5348, 50, 51, 52syl3anc 1371 . 2 ((𝑀 ∈ LMod ∧ 𝑉 ∈ 𝒫 (Baseβ€˜π‘€) ∧ 𝐴 ∈ (𝑅 ↑m 𝑉)) β†’ ((𝑣 ∈ 𝑉 ↦ ((π΄β€˜π‘£)( ·𝑠 β€˜π‘€)𝑣)) supp (0gβ€˜π‘€)) = {π‘₯ ∈ dom (𝑣 ∈ 𝑉 ↦ ((π΄β€˜π‘£)( ·𝑠 β€˜π‘€)𝑣)) ∣ ((𝑣 ∈ 𝑉 ↦ ((π΄β€˜π‘£)( ·𝑠 β€˜π‘€)𝑣))β€˜π‘₯) β‰  (0gβ€˜π‘€)})
54 elmapfun 8856 . . . 4 (𝐴 ∈ (𝑅 ↑m 𝑉) β†’ Fun 𝐴)
55543ad2ant3 1135 . . 3 ((𝑀 ∈ LMod ∧ 𝑉 ∈ 𝒫 (Baseβ€˜π‘€) ∧ 𝐴 ∈ (𝑅 ↑m 𝑉)) β†’ Fun 𝐴)
56 simp3 1138 . . 3 ((𝑀 ∈ LMod ∧ 𝑉 ∈ 𝒫 (Baseβ€˜π‘€) ∧ 𝐴 ∈ (𝑅 ↑m 𝑉)) β†’ 𝐴 ∈ (𝑅 ↑m 𝑉))
57 fvexd 6903 . . 3 ((𝑀 ∈ LMod ∧ 𝑉 ∈ 𝒫 (Baseβ€˜π‘€) ∧ 𝐴 ∈ (𝑅 ↑m 𝑉)) β†’ (0gβ€˜π‘†) ∈ V)
58 suppval1 8148 . . 3 ((Fun 𝐴 ∧ 𝐴 ∈ (𝑅 ↑m 𝑉) ∧ (0gβ€˜π‘†) ∈ V) β†’ (𝐴 supp (0gβ€˜π‘†)) = {π‘₯ ∈ dom 𝐴 ∣ (π΄β€˜π‘₯) β‰  (0gβ€˜π‘†)})
5955, 56, 57, 58syl3anc 1371 . 2 ((𝑀 ∈ LMod ∧ 𝑉 ∈ 𝒫 (Baseβ€˜π‘€) ∧ 𝐴 ∈ (𝑅 ↑m 𝑉)) β†’ (𝐴 supp (0gβ€˜π‘†)) = {π‘₯ ∈ dom 𝐴 ∣ (π΄β€˜π‘₯) β‰  (0gβ€˜π‘†)})
6046, 53, 593sstr4d 4028 1 ((𝑀 ∈ LMod ∧ 𝑉 ∈ 𝒫 (Baseβ€˜π‘€) ∧ 𝐴 ∈ (𝑅 ↑m 𝑉)) β†’ ((𝑣 ∈ 𝑉 ↦ ((π΄β€˜π‘£)( ·𝑠 β€˜π‘€)𝑣)) supp (0gβ€˜π‘€)) βŠ† (𝐴 supp (0gβ€˜π‘†)))
Colors of variables: wff setvar class
Syntax hints:   β†’ wi 4   ↔ wb 205   ∧ wa 396   ∧ w3a 1087   = wceq 1541   ∈ wcel 2106   β‰  wne 2940  {crab 3432  Vcvv 3474   βŠ† wss 3947  π’« cpw 4601   ↦ cmpt 5230  dom cdm 5675  Fun wfun 6534  βŸΆwf 6536  β€˜cfv 6540  (class class class)co 7405   supp csupp 8142   ↑m cmap 8816  Basecbs 17140  Scalarcsca 17196   ·𝑠 cvsca 17197  0gc0g 17381  LModclmod 20463
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 5284  ax-sep 5298  ax-nul 5305  ax-pow 5362  ax-pr 5426  ax-un 7721
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 3777  df-csb 3893  df-dif 3950  df-un 3952  df-in 3954  df-ss 3964  df-nul 4322  df-if 4528  df-pw 4603  df-sn 4628  df-pr 4630  df-op 4634  df-uni 4908  df-iun 4998  df-br 5148  df-opab 5210  df-mpt 5231  df-id 5573  df-xp 5681  df-rel 5682  df-cnv 5683  df-co 5684  df-dm 5685  df-rn 5686  df-res 5687  df-ima 5688  df-iota 6492  df-fun 6542  df-fn 6543  df-f 6544  df-f1 6545  df-fo 6546  df-f1o 6547  df-fv 6548  df-riota 7361  df-ov 7408  df-oprab 7409  df-mpo 7410  df-1st 7971  df-2nd 7972  df-supp 8143  df-map 8818  df-0g 17383  df-mgm 18557  df-sgrp 18606  df-mnd 18622  df-grp 18818  df-ring 20051  df-lmod 20465
This theorem is referenced by:  scmsuppfi  47006
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