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Theorem mendassa 42018
Description: The module endomorphism algebra is an algebra. (Contributed by Mario Carneiro, 22-Sep-2015.)
Hypotheses
Ref Expression
mendassa.a 𝐴 = (MEndoβ€˜π‘€)
mendassa.s 𝑆 = (Scalarβ€˜π‘€)
Assertion
Ref Expression
mendassa ((𝑀 ∈ LMod ∧ 𝑆 ∈ CRing) β†’ 𝐴 ∈ AssAlg)

Proof of Theorem mendassa
Dummy variables π‘₯ 𝑦 𝑧 𝑣 𝑀 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 mendassa.a . . . 4 𝐴 = (MEndoβ€˜π‘€)
21mendbas 42008 . . 3 (𝑀 LMHom 𝑀) = (Baseβ€˜π΄)
32a1i 11 . 2 ((𝑀 ∈ LMod ∧ 𝑆 ∈ CRing) β†’ (𝑀 LMHom 𝑀) = (Baseβ€˜π΄))
4 mendassa.s . . . 4 𝑆 = (Scalarβ€˜π‘€)
51, 4mendsca 42013 . . 3 𝑆 = (Scalarβ€˜π΄)
65a1i 11 . 2 ((𝑀 ∈ LMod ∧ 𝑆 ∈ CRing) β†’ 𝑆 = (Scalarβ€˜π΄))
7 eqidd 2733 . 2 ((𝑀 ∈ LMod ∧ 𝑆 ∈ CRing) β†’ (Baseβ€˜π‘†) = (Baseβ€˜π‘†))
8 eqidd 2733 . 2 ((𝑀 ∈ LMod ∧ 𝑆 ∈ CRing) β†’ ( ·𝑠 β€˜π΄) = ( ·𝑠 β€˜π΄))
9 eqidd 2733 . 2 ((𝑀 ∈ LMod ∧ 𝑆 ∈ CRing) β†’ (.rβ€˜π΄) = (.rβ€˜π΄))
101, 4mendlmod 42017 . 2 ((𝑀 ∈ LMod ∧ 𝑆 ∈ CRing) β†’ 𝐴 ∈ LMod)
111mendring 42016 . . 3 (𝑀 ∈ LMod β†’ 𝐴 ∈ Ring)
1211adantr 481 . 2 ((𝑀 ∈ LMod ∧ 𝑆 ∈ CRing) β†’ 𝐴 ∈ Ring)
13 simpr3 1196 . . . . . . 7 (((𝑀 ∈ LMod ∧ 𝑆 ∈ CRing) ∧ (π‘₯ ∈ (Baseβ€˜π‘†) ∧ 𝑦 ∈ (𝑀 LMHom 𝑀) ∧ 𝑧 ∈ (𝑀 LMHom 𝑀))) β†’ 𝑧 ∈ (𝑀 LMHom 𝑀))
14 eqid 2732 . . . . . . . 8 (Baseβ€˜π‘€) = (Baseβ€˜π‘€)
1514, 14lmhmf 20650 . . . . . . 7 (𝑧 ∈ (𝑀 LMHom 𝑀) β†’ 𝑧:(Baseβ€˜π‘€)⟢(Baseβ€˜π‘€))
1613, 15syl 17 . . . . . 6 (((𝑀 ∈ LMod ∧ 𝑆 ∈ CRing) ∧ (π‘₯ ∈ (Baseβ€˜π‘†) ∧ 𝑦 ∈ (𝑀 LMHom 𝑀) ∧ 𝑧 ∈ (𝑀 LMHom 𝑀))) β†’ 𝑧:(Baseβ€˜π‘€)⟢(Baseβ€˜π‘€))
1716ffvelcdmda 7086 . . . . 5 ((((𝑀 ∈ LMod ∧ 𝑆 ∈ CRing) ∧ (π‘₯ ∈ (Baseβ€˜π‘†) ∧ 𝑦 ∈ (𝑀 LMHom 𝑀) ∧ 𝑧 ∈ (𝑀 LMHom 𝑀))) ∧ 𝑣 ∈ (Baseβ€˜π‘€)) β†’ (π‘§β€˜π‘£) ∈ (Baseβ€˜π‘€))
1816feqmptd 6960 . . . . 5 (((𝑀 ∈ LMod ∧ 𝑆 ∈ CRing) ∧ (π‘₯ ∈ (Baseβ€˜π‘†) ∧ 𝑦 ∈ (𝑀 LMHom 𝑀) ∧ 𝑧 ∈ (𝑀 LMHom 𝑀))) β†’ 𝑧 = (𝑣 ∈ (Baseβ€˜π‘€) ↦ (π‘§β€˜π‘£)))
19 simpr1 1194 . . . . . . 7 (((𝑀 ∈ LMod ∧ 𝑆 ∈ CRing) ∧ (π‘₯ ∈ (Baseβ€˜π‘†) ∧ 𝑦 ∈ (𝑀 LMHom 𝑀) ∧ 𝑧 ∈ (𝑀 LMHom 𝑀))) β†’ π‘₯ ∈ (Baseβ€˜π‘†))
20 simpr2 1195 . . . . . . 7 (((𝑀 ∈ LMod ∧ 𝑆 ∈ CRing) ∧ (π‘₯ ∈ (Baseβ€˜π‘†) ∧ 𝑦 ∈ (𝑀 LMHom 𝑀) ∧ 𝑧 ∈ (𝑀 LMHom 𝑀))) β†’ 𝑦 ∈ (𝑀 LMHom 𝑀))
21 eqid 2732 . . . . . . . 8 ( ·𝑠 β€˜π‘€) = ( ·𝑠 β€˜π‘€)
22 eqid 2732 . . . . . . . 8 (Baseβ€˜π‘†) = (Baseβ€˜π‘†)
23 eqid 2732 . . . . . . . 8 ( ·𝑠 β€˜π΄) = ( ·𝑠 β€˜π΄)
241, 21, 2, 4, 22, 14, 23mendvsca 42015 . . . . . . 7 ((π‘₯ ∈ (Baseβ€˜π‘†) ∧ 𝑦 ∈ (𝑀 LMHom 𝑀)) β†’ (π‘₯( ·𝑠 β€˜π΄)𝑦) = (((Baseβ€˜π‘€) Γ— {π‘₯}) ∘f ( ·𝑠 β€˜π‘€)𝑦))
2519, 20, 24syl2anc 584 . . . . . 6 (((𝑀 ∈ LMod ∧ 𝑆 ∈ CRing) ∧ (π‘₯ ∈ (Baseβ€˜π‘†) ∧ 𝑦 ∈ (𝑀 LMHom 𝑀) ∧ 𝑧 ∈ (𝑀 LMHom 𝑀))) β†’ (π‘₯( ·𝑠 β€˜π΄)𝑦) = (((Baseβ€˜π‘€) Γ— {π‘₯}) ∘f ( ·𝑠 β€˜π‘€)𝑦))
26 fvexd 6906 . . . . . . 7 (((𝑀 ∈ LMod ∧ 𝑆 ∈ CRing) ∧ (π‘₯ ∈ (Baseβ€˜π‘†) ∧ 𝑦 ∈ (𝑀 LMHom 𝑀) ∧ 𝑧 ∈ (𝑀 LMHom 𝑀))) β†’ (Baseβ€˜π‘€) ∈ V)
27 simplr1 1215 . . . . . . 7 ((((𝑀 ∈ LMod ∧ 𝑆 ∈ CRing) ∧ (π‘₯ ∈ (Baseβ€˜π‘†) ∧ 𝑦 ∈ (𝑀 LMHom 𝑀) ∧ 𝑧 ∈ (𝑀 LMHom 𝑀))) ∧ 𝑀 ∈ (Baseβ€˜π‘€)) β†’ π‘₯ ∈ (Baseβ€˜π‘†))
28 fvexd 6906 . . . . . . 7 ((((𝑀 ∈ LMod ∧ 𝑆 ∈ CRing) ∧ (π‘₯ ∈ (Baseβ€˜π‘†) ∧ 𝑦 ∈ (𝑀 LMHom 𝑀) ∧ 𝑧 ∈ (𝑀 LMHom 𝑀))) ∧ 𝑀 ∈ (Baseβ€˜π‘€)) β†’ (π‘¦β€˜π‘€) ∈ V)
29 fconstmpt 5738 . . . . . . . 8 ((Baseβ€˜π‘€) Γ— {π‘₯}) = (𝑀 ∈ (Baseβ€˜π‘€) ↦ π‘₯)
3029a1i 11 . . . . . . 7 (((𝑀 ∈ LMod ∧ 𝑆 ∈ CRing) ∧ (π‘₯ ∈ (Baseβ€˜π‘†) ∧ 𝑦 ∈ (𝑀 LMHom 𝑀) ∧ 𝑧 ∈ (𝑀 LMHom 𝑀))) β†’ ((Baseβ€˜π‘€) Γ— {π‘₯}) = (𝑀 ∈ (Baseβ€˜π‘€) ↦ π‘₯))
3114, 14lmhmf 20650 . . . . . . . . 9 (𝑦 ∈ (𝑀 LMHom 𝑀) β†’ 𝑦:(Baseβ€˜π‘€)⟢(Baseβ€˜π‘€))
3220, 31syl 17 . . . . . . . 8 (((𝑀 ∈ LMod ∧ 𝑆 ∈ CRing) ∧ (π‘₯ ∈ (Baseβ€˜π‘†) ∧ 𝑦 ∈ (𝑀 LMHom 𝑀) ∧ 𝑧 ∈ (𝑀 LMHom 𝑀))) β†’ 𝑦:(Baseβ€˜π‘€)⟢(Baseβ€˜π‘€))
3332feqmptd 6960 . . . . . . 7 (((𝑀 ∈ LMod ∧ 𝑆 ∈ CRing) ∧ (π‘₯ ∈ (Baseβ€˜π‘†) ∧ 𝑦 ∈ (𝑀 LMHom 𝑀) ∧ 𝑧 ∈ (𝑀 LMHom 𝑀))) β†’ 𝑦 = (𝑀 ∈ (Baseβ€˜π‘€) ↦ (π‘¦β€˜π‘€)))
3426, 27, 28, 30, 33offval2 7692 . . . . . 6 (((𝑀 ∈ LMod ∧ 𝑆 ∈ CRing) ∧ (π‘₯ ∈ (Baseβ€˜π‘†) ∧ 𝑦 ∈ (𝑀 LMHom 𝑀) ∧ 𝑧 ∈ (𝑀 LMHom 𝑀))) β†’ (((Baseβ€˜π‘€) Γ— {π‘₯}) ∘f ( ·𝑠 β€˜π‘€)𝑦) = (𝑀 ∈ (Baseβ€˜π‘€) ↦ (π‘₯( ·𝑠 β€˜π‘€)(π‘¦β€˜π‘€))))
3525, 34eqtrd 2772 . . . . 5 (((𝑀 ∈ LMod ∧ 𝑆 ∈ CRing) ∧ (π‘₯ ∈ (Baseβ€˜π‘†) ∧ 𝑦 ∈ (𝑀 LMHom 𝑀) ∧ 𝑧 ∈ (𝑀 LMHom 𝑀))) β†’ (π‘₯( ·𝑠 β€˜π΄)𝑦) = (𝑀 ∈ (Baseβ€˜π‘€) ↦ (π‘₯( ·𝑠 β€˜π‘€)(π‘¦β€˜π‘€))))
36 fveq2 6891 . . . . . 6 (𝑀 = (π‘§β€˜π‘£) β†’ (π‘¦β€˜π‘€) = (π‘¦β€˜(π‘§β€˜π‘£)))
3736oveq2d 7427 . . . . 5 (𝑀 = (π‘§β€˜π‘£) β†’ (π‘₯( ·𝑠 β€˜π‘€)(π‘¦β€˜π‘€)) = (π‘₯( ·𝑠 β€˜π‘€)(π‘¦β€˜(π‘§β€˜π‘£))))
3817, 18, 35, 37fmptco 7129 . . . 4 (((𝑀 ∈ LMod ∧ 𝑆 ∈ CRing) ∧ (π‘₯ ∈ (Baseβ€˜π‘†) ∧ 𝑦 ∈ (𝑀 LMHom 𝑀) ∧ 𝑧 ∈ (𝑀 LMHom 𝑀))) β†’ ((π‘₯( ·𝑠 β€˜π΄)𝑦) ∘ 𝑧) = (𝑣 ∈ (Baseβ€˜π‘€) ↦ (π‘₯( ·𝑠 β€˜π‘€)(π‘¦β€˜(π‘§β€˜π‘£)))))
39 simplr1 1215 . . . . 5 ((((𝑀 ∈ LMod ∧ 𝑆 ∈ CRing) ∧ (π‘₯ ∈ (Baseβ€˜π‘†) ∧ 𝑦 ∈ (𝑀 LMHom 𝑀) ∧ 𝑧 ∈ (𝑀 LMHom 𝑀))) ∧ 𝑣 ∈ (Baseβ€˜π‘€)) β†’ π‘₯ ∈ (Baseβ€˜π‘†))
40 fvexd 6906 . . . . 5 ((((𝑀 ∈ LMod ∧ 𝑆 ∈ CRing) ∧ (π‘₯ ∈ (Baseβ€˜π‘†) ∧ 𝑦 ∈ (𝑀 LMHom 𝑀) ∧ 𝑧 ∈ (𝑀 LMHom 𝑀))) ∧ 𝑣 ∈ (Baseβ€˜π‘€)) β†’ (π‘¦β€˜(π‘§β€˜π‘£)) ∈ V)
41 fconstmpt 5738 . . . . . 6 ((Baseβ€˜π‘€) Γ— {π‘₯}) = (𝑣 ∈ (Baseβ€˜π‘€) ↦ π‘₯)
4241a1i 11 . . . . 5 (((𝑀 ∈ LMod ∧ 𝑆 ∈ CRing) ∧ (π‘₯ ∈ (Baseβ€˜π‘†) ∧ 𝑦 ∈ (𝑀 LMHom 𝑀) ∧ 𝑧 ∈ (𝑀 LMHom 𝑀))) β†’ ((Baseβ€˜π‘€) Γ— {π‘₯}) = (𝑣 ∈ (Baseβ€˜π‘€) ↦ π‘₯))
43 eqid 2732 . . . . . . . 8 (.rβ€˜π΄) = (.rβ€˜π΄)
441, 2, 43mendmulr 42012 . . . . . . 7 ((𝑦 ∈ (𝑀 LMHom 𝑀) ∧ 𝑧 ∈ (𝑀 LMHom 𝑀)) β†’ (𝑦(.rβ€˜π΄)𝑧) = (𝑦 ∘ 𝑧))
4520, 13, 44syl2anc 584 . . . . . 6 (((𝑀 ∈ LMod ∧ 𝑆 ∈ CRing) ∧ (π‘₯ ∈ (Baseβ€˜π‘†) ∧ 𝑦 ∈ (𝑀 LMHom 𝑀) ∧ 𝑧 ∈ (𝑀 LMHom 𝑀))) β†’ (𝑦(.rβ€˜π΄)𝑧) = (𝑦 ∘ 𝑧))
46 fcompt 7133 . . . . . . 7 ((𝑦:(Baseβ€˜π‘€)⟢(Baseβ€˜π‘€) ∧ 𝑧:(Baseβ€˜π‘€)⟢(Baseβ€˜π‘€)) β†’ (𝑦 ∘ 𝑧) = (𝑣 ∈ (Baseβ€˜π‘€) ↦ (π‘¦β€˜(π‘§β€˜π‘£))))
4732, 16, 46syl2anc 584 . . . . . 6 (((𝑀 ∈ LMod ∧ 𝑆 ∈ CRing) ∧ (π‘₯ ∈ (Baseβ€˜π‘†) ∧ 𝑦 ∈ (𝑀 LMHom 𝑀) ∧ 𝑧 ∈ (𝑀 LMHom 𝑀))) β†’ (𝑦 ∘ 𝑧) = (𝑣 ∈ (Baseβ€˜π‘€) ↦ (π‘¦β€˜(π‘§β€˜π‘£))))
4845, 47eqtrd 2772 . . . . 5 (((𝑀 ∈ LMod ∧ 𝑆 ∈ CRing) ∧ (π‘₯ ∈ (Baseβ€˜π‘†) ∧ 𝑦 ∈ (𝑀 LMHom 𝑀) ∧ 𝑧 ∈ (𝑀 LMHom 𝑀))) β†’ (𝑦(.rβ€˜π΄)𝑧) = (𝑣 ∈ (Baseβ€˜π‘€) ↦ (π‘¦β€˜(π‘§β€˜π‘£))))
4926, 39, 40, 42, 48offval2 7692 . . . 4 (((𝑀 ∈ LMod ∧ 𝑆 ∈ CRing) ∧ (π‘₯ ∈ (Baseβ€˜π‘†) ∧ 𝑦 ∈ (𝑀 LMHom 𝑀) ∧ 𝑧 ∈ (𝑀 LMHom 𝑀))) β†’ (((Baseβ€˜π‘€) Γ— {π‘₯}) ∘f ( ·𝑠 β€˜π‘€)(𝑦(.rβ€˜π΄)𝑧)) = (𝑣 ∈ (Baseβ€˜π‘€) ↦ (π‘₯( ·𝑠 β€˜π‘€)(π‘¦β€˜(π‘§β€˜π‘£)))))
5038, 49eqtr4d 2775 . . 3 (((𝑀 ∈ LMod ∧ 𝑆 ∈ CRing) ∧ (π‘₯ ∈ (Baseβ€˜π‘†) ∧ 𝑦 ∈ (𝑀 LMHom 𝑀) ∧ 𝑧 ∈ (𝑀 LMHom 𝑀))) β†’ ((π‘₯( ·𝑠 β€˜π΄)𝑦) ∘ 𝑧) = (((Baseβ€˜π‘€) Γ— {π‘₯}) ∘f ( ·𝑠 β€˜π‘€)(𝑦(.rβ€˜π΄)𝑧)))
5110adantr 481 . . . . 5 (((𝑀 ∈ LMod ∧ 𝑆 ∈ CRing) ∧ (π‘₯ ∈ (Baseβ€˜π‘†) ∧ 𝑦 ∈ (𝑀 LMHom 𝑀) ∧ 𝑧 ∈ (𝑀 LMHom 𝑀))) β†’ 𝐴 ∈ LMod)
522, 5, 23, 22lmodvscl 20493 . . . . 5 ((𝐴 ∈ LMod ∧ π‘₯ ∈ (Baseβ€˜π‘†) ∧ 𝑦 ∈ (𝑀 LMHom 𝑀)) β†’ (π‘₯( ·𝑠 β€˜π΄)𝑦) ∈ (𝑀 LMHom 𝑀))
5351, 19, 20, 52syl3anc 1371 . . . 4 (((𝑀 ∈ LMod ∧ 𝑆 ∈ CRing) ∧ (π‘₯ ∈ (Baseβ€˜π‘†) ∧ 𝑦 ∈ (𝑀 LMHom 𝑀) ∧ 𝑧 ∈ (𝑀 LMHom 𝑀))) β†’ (π‘₯( ·𝑠 β€˜π΄)𝑦) ∈ (𝑀 LMHom 𝑀))
541, 2, 43mendmulr 42012 . . . 4 (((π‘₯( ·𝑠 β€˜π΄)𝑦) ∈ (𝑀 LMHom 𝑀) ∧ 𝑧 ∈ (𝑀 LMHom 𝑀)) β†’ ((π‘₯( ·𝑠 β€˜π΄)𝑦)(.rβ€˜π΄)𝑧) = ((π‘₯( ·𝑠 β€˜π΄)𝑦) ∘ 𝑧))
5553, 13, 54syl2anc 584 . . 3 (((𝑀 ∈ LMod ∧ 𝑆 ∈ CRing) ∧ (π‘₯ ∈ (Baseβ€˜π‘†) ∧ 𝑦 ∈ (𝑀 LMHom 𝑀) ∧ 𝑧 ∈ (𝑀 LMHom 𝑀))) β†’ ((π‘₯( ·𝑠 β€˜π΄)𝑦)(.rβ€˜π΄)𝑧) = ((π‘₯( ·𝑠 β€˜π΄)𝑦) ∘ 𝑧))
5612adantr 481 . . . . 5 (((𝑀 ∈ LMod ∧ 𝑆 ∈ CRing) ∧ (π‘₯ ∈ (Baseβ€˜π‘†) ∧ 𝑦 ∈ (𝑀 LMHom 𝑀) ∧ 𝑧 ∈ (𝑀 LMHom 𝑀))) β†’ 𝐴 ∈ Ring)
572, 43ringcl 20075 . . . . 5 ((𝐴 ∈ Ring ∧ 𝑦 ∈ (𝑀 LMHom 𝑀) ∧ 𝑧 ∈ (𝑀 LMHom 𝑀)) β†’ (𝑦(.rβ€˜π΄)𝑧) ∈ (𝑀 LMHom 𝑀))
5856, 20, 13, 57syl3anc 1371 . . . 4 (((𝑀 ∈ LMod ∧ 𝑆 ∈ CRing) ∧ (π‘₯ ∈ (Baseβ€˜π‘†) ∧ 𝑦 ∈ (𝑀 LMHom 𝑀) ∧ 𝑧 ∈ (𝑀 LMHom 𝑀))) β†’ (𝑦(.rβ€˜π΄)𝑧) ∈ (𝑀 LMHom 𝑀))
591, 21, 2, 4, 22, 14, 23mendvsca 42015 . . . 4 ((π‘₯ ∈ (Baseβ€˜π‘†) ∧ (𝑦(.rβ€˜π΄)𝑧) ∈ (𝑀 LMHom 𝑀)) β†’ (π‘₯( ·𝑠 β€˜π΄)(𝑦(.rβ€˜π΄)𝑧)) = (((Baseβ€˜π‘€) Γ— {π‘₯}) ∘f ( ·𝑠 β€˜π‘€)(𝑦(.rβ€˜π΄)𝑧)))
6019, 58, 59syl2anc 584 . . 3 (((𝑀 ∈ LMod ∧ 𝑆 ∈ CRing) ∧ (π‘₯ ∈ (Baseβ€˜π‘†) ∧ 𝑦 ∈ (𝑀 LMHom 𝑀) ∧ 𝑧 ∈ (𝑀 LMHom 𝑀))) β†’ (π‘₯( ·𝑠 β€˜π΄)(𝑦(.rβ€˜π΄)𝑧)) = (((Baseβ€˜π‘€) Γ— {π‘₯}) ∘f ( ·𝑠 β€˜π‘€)(𝑦(.rβ€˜π΄)𝑧)))
6150, 55, 603eqtr4d 2782 . 2 (((𝑀 ∈ LMod ∧ 𝑆 ∈ CRing) ∧ (π‘₯ ∈ (Baseβ€˜π‘†) ∧ 𝑦 ∈ (𝑀 LMHom 𝑀) ∧ 𝑧 ∈ (𝑀 LMHom 𝑀))) β†’ ((π‘₯( ·𝑠 β€˜π΄)𝑦)(.rβ€˜π΄)𝑧) = (π‘₯( ·𝑠 β€˜π΄)(𝑦(.rβ€˜π΄)𝑧)))
62 simplr2 1216 . . . . . 6 ((((𝑀 ∈ LMod ∧ 𝑆 ∈ CRing) ∧ (π‘₯ ∈ (Baseβ€˜π‘†) ∧ 𝑦 ∈ (𝑀 LMHom 𝑀) ∧ 𝑧 ∈ (𝑀 LMHom 𝑀))) ∧ 𝑣 ∈ (Baseβ€˜π‘€)) β†’ 𝑦 ∈ (𝑀 LMHom 𝑀))
634, 22, 14, 21, 21lmhmlin 20651 . . . . . 6 ((𝑦 ∈ (𝑀 LMHom 𝑀) ∧ π‘₯ ∈ (Baseβ€˜π‘†) ∧ (π‘§β€˜π‘£) ∈ (Baseβ€˜π‘€)) β†’ (π‘¦β€˜(π‘₯( ·𝑠 β€˜π‘€)(π‘§β€˜π‘£))) = (π‘₯( ·𝑠 β€˜π‘€)(π‘¦β€˜(π‘§β€˜π‘£))))
6462, 39, 17, 63syl3anc 1371 . . . . 5 ((((𝑀 ∈ LMod ∧ 𝑆 ∈ CRing) ∧ (π‘₯ ∈ (Baseβ€˜π‘†) ∧ 𝑦 ∈ (𝑀 LMHom 𝑀) ∧ 𝑧 ∈ (𝑀 LMHom 𝑀))) ∧ 𝑣 ∈ (Baseβ€˜π‘€)) β†’ (π‘¦β€˜(π‘₯( ·𝑠 β€˜π‘€)(π‘§β€˜π‘£))) = (π‘₯( ·𝑠 β€˜π‘€)(π‘¦β€˜(π‘§β€˜π‘£))))
6564mpteq2dva 5248 . . . 4 (((𝑀 ∈ LMod ∧ 𝑆 ∈ CRing) ∧ (π‘₯ ∈ (Baseβ€˜π‘†) ∧ 𝑦 ∈ (𝑀 LMHom 𝑀) ∧ 𝑧 ∈ (𝑀 LMHom 𝑀))) β†’ (𝑣 ∈ (Baseβ€˜π‘€) ↦ (π‘¦β€˜(π‘₯( ·𝑠 β€˜π‘€)(π‘§β€˜π‘£)))) = (𝑣 ∈ (Baseβ€˜π‘€) ↦ (π‘₯( ·𝑠 β€˜π‘€)(π‘¦β€˜(π‘§β€˜π‘£)))))
66 simplll 773 . . . . . 6 ((((𝑀 ∈ LMod ∧ 𝑆 ∈ CRing) ∧ (π‘₯ ∈ (Baseβ€˜π‘†) ∧ 𝑦 ∈ (𝑀 LMHom 𝑀) ∧ 𝑧 ∈ (𝑀 LMHom 𝑀))) ∧ 𝑣 ∈ (Baseβ€˜π‘€)) β†’ 𝑀 ∈ LMod)
6714, 4, 21, 22lmodvscl 20493 . . . . . 6 ((𝑀 ∈ LMod ∧ π‘₯ ∈ (Baseβ€˜π‘†) ∧ (π‘§β€˜π‘£) ∈ (Baseβ€˜π‘€)) β†’ (π‘₯( ·𝑠 β€˜π‘€)(π‘§β€˜π‘£)) ∈ (Baseβ€˜π‘€))
6866, 39, 17, 67syl3anc 1371 . . . . 5 ((((𝑀 ∈ LMod ∧ 𝑆 ∈ CRing) ∧ (π‘₯ ∈ (Baseβ€˜π‘†) ∧ 𝑦 ∈ (𝑀 LMHom 𝑀) ∧ 𝑧 ∈ (𝑀 LMHom 𝑀))) ∧ 𝑣 ∈ (Baseβ€˜π‘€)) β†’ (π‘₯( ·𝑠 β€˜π‘€)(π‘§β€˜π‘£)) ∈ (Baseβ€˜π‘€))
691, 21, 2, 4, 22, 14, 23mendvsca 42015 . . . . . . 7 ((π‘₯ ∈ (Baseβ€˜π‘†) ∧ 𝑧 ∈ (𝑀 LMHom 𝑀)) β†’ (π‘₯( ·𝑠 β€˜π΄)𝑧) = (((Baseβ€˜π‘€) Γ— {π‘₯}) ∘f ( ·𝑠 β€˜π‘€)𝑧))
7019, 13, 69syl2anc 584 . . . . . 6 (((𝑀 ∈ LMod ∧ 𝑆 ∈ CRing) ∧ (π‘₯ ∈ (Baseβ€˜π‘†) ∧ 𝑦 ∈ (𝑀 LMHom 𝑀) ∧ 𝑧 ∈ (𝑀 LMHom 𝑀))) β†’ (π‘₯( ·𝑠 β€˜π΄)𝑧) = (((Baseβ€˜π‘€) Γ— {π‘₯}) ∘f ( ·𝑠 β€˜π‘€)𝑧))
71 fvexd 6906 . . . . . . 7 ((((𝑀 ∈ LMod ∧ 𝑆 ∈ CRing) ∧ (π‘₯ ∈ (Baseβ€˜π‘†) ∧ 𝑦 ∈ (𝑀 LMHom 𝑀) ∧ 𝑧 ∈ (𝑀 LMHom 𝑀))) ∧ 𝑣 ∈ (Baseβ€˜π‘€)) β†’ (π‘§β€˜π‘£) ∈ V)
7226, 39, 71, 42, 18offval2 7692 . . . . . 6 (((𝑀 ∈ LMod ∧ 𝑆 ∈ CRing) ∧ (π‘₯ ∈ (Baseβ€˜π‘†) ∧ 𝑦 ∈ (𝑀 LMHom 𝑀) ∧ 𝑧 ∈ (𝑀 LMHom 𝑀))) β†’ (((Baseβ€˜π‘€) Γ— {π‘₯}) ∘f ( ·𝑠 β€˜π‘€)𝑧) = (𝑣 ∈ (Baseβ€˜π‘€) ↦ (π‘₯( ·𝑠 β€˜π‘€)(π‘§β€˜π‘£))))
7370, 72eqtrd 2772 . . . . 5 (((𝑀 ∈ LMod ∧ 𝑆 ∈ CRing) ∧ (π‘₯ ∈ (Baseβ€˜π‘†) ∧ 𝑦 ∈ (𝑀 LMHom 𝑀) ∧ 𝑧 ∈ (𝑀 LMHom 𝑀))) β†’ (π‘₯( ·𝑠 β€˜π΄)𝑧) = (𝑣 ∈ (Baseβ€˜π‘€) ↦ (π‘₯( ·𝑠 β€˜π‘€)(π‘§β€˜π‘£))))
74 fveq2 6891 . . . . 5 (𝑀 = (π‘₯( ·𝑠 β€˜π‘€)(π‘§β€˜π‘£)) β†’ (π‘¦β€˜π‘€) = (π‘¦β€˜(π‘₯( ·𝑠 β€˜π‘€)(π‘§β€˜π‘£))))
7568, 73, 33, 74fmptco 7129 . . . 4 (((𝑀 ∈ LMod ∧ 𝑆 ∈ CRing) ∧ (π‘₯ ∈ (Baseβ€˜π‘†) ∧ 𝑦 ∈ (𝑀 LMHom 𝑀) ∧ 𝑧 ∈ (𝑀 LMHom 𝑀))) β†’ (𝑦 ∘ (π‘₯( ·𝑠 β€˜π΄)𝑧)) = (𝑣 ∈ (Baseβ€˜π‘€) ↦ (π‘¦β€˜(π‘₯( ·𝑠 β€˜π‘€)(π‘§β€˜π‘£)))))
7665, 75, 493eqtr4d 2782 . . 3 (((𝑀 ∈ LMod ∧ 𝑆 ∈ CRing) ∧ (π‘₯ ∈ (Baseβ€˜π‘†) ∧ 𝑦 ∈ (𝑀 LMHom 𝑀) ∧ 𝑧 ∈ (𝑀 LMHom 𝑀))) β†’ (𝑦 ∘ (π‘₯( ·𝑠 β€˜π΄)𝑧)) = (((Baseβ€˜π‘€) Γ— {π‘₯}) ∘f ( ·𝑠 β€˜π‘€)(𝑦(.rβ€˜π΄)𝑧)))
772, 5, 23, 22lmodvscl 20493 . . . . 5 ((𝐴 ∈ LMod ∧ π‘₯ ∈ (Baseβ€˜π‘†) ∧ 𝑧 ∈ (𝑀 LMHom 𝑀)) β†’ (π‘₯( ·𝑠 β€˜π΄)𝑧) ∈ (𝑀 LMHom 𝑀))
7851, 19, 13, 77syl3anc 1371 . . . 4 (((𝑀 ∈ LMod ∧ 𝑆 ∈ CRing) ∧ (π‘₯ ∈ (Baseβ€˜π‘†) ∧ 𝑦 ∈ (𝑀 LMHom 𝑀) ∧ 𝑧 ∈ (𝑀 LMHom 𝑀))) β†’ (π‘₯( ·𝑠 β€˜π΄)𝑧) ∈ (𝑀 LMHom 𝑀))
791, 2, 43mendmulr 42012 . . . 4 ((𝑦 ∈ (𝑀 LMHom 𝑀) ∧ (π‘₯( ·𝑠 β€˜π΄)𝑧) ∈ (𝑀 LMHom 𝑀)) β†’ (𝑦(.rβ€˜π΄)(π‘₯( ·𝑠 β€˜π΄)𝑧)) = (𝑦 ∘ (π‘₯( ·𝑠 β€˜π΄)𝑧)))
8020, 78, 79syl2anc 584 . . 3 (((𝑀 ∈ LMod ∧ 𝑆 ∈ CRing) ∧ (π‘₯ ∈ (Baseβ€˜π‘†) ∧ 𝑦 ∈ (𝑀 LMHom 𝑀) ∧ 𝑧 ∈ (𝑀 LMHom 𝑀))) β†’ (𝑦(.rβ€˜π΄)(π‘₯( ·𝑠 β€˜π΄)𝑧)) = (𝑦 ∘ (π‘₯( ·𝑠 β€˜π΄)𝑧)))
8176, 80, 603eqtr4d 2782 . 2 (((𝑀 ∈ LMod ∧ 𝑆 ∈ CRing) ∧ (π‘₯ ∈ (Baseβ€˜π‘†) ∧ 𝑦 ∈ (𝑀 LMHom 𝑀) ∧ 𝑧 ∈ (𝑀 LMHom 𝑀))) β†’ (𝑦(.rβ€˜π΄)(π‘₯( ·𝑠 β€˜π΄)𝑧)) = (π‘₯( ·𝑠 β€˜π΄)(𝑦(.rβ€˜π΄)𝑧)))
823, 6, 7, 8, 9, 10, 12, 61, 81isassad 21425 1 ((𝑀 ∈ LMod ∧ 𝑆 ∈ CRing) β†’ 𝐴 ∈ AssAlg)
Colors of variables: wff setvar class
Syntax hints:   β†’ wi 4   ∧ wa 396   ∧ w3a 1087   = wceq 1541   ∈ wcel 2106  Vcvv 3474  {csn 4628   ↦ cmpt 5231   Γ— cxp 5674   ∘ ccom 5680  βŸΆwf 6539  β€˜cfv 6543  (class class class)co 7411   ∘f cof 7670  Basecbs 17146  .rcmulr 17200  Scalarcsca 17202   ·𝑠 cvsca 17203  Ringcrg 20058  CRingccrg 20059  LModclmod 20475   LMHom clmhm 20635  AssAlgcasa 21411  MEndocmend 41999
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-tp 4633  df-op 4635  df-uni 4909  df-iun 4999  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-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-riota 7367  df-ov 7414  df-oprab 7415  df-mpo 7416  df-of 7672  df-om 7858  df-1st 7977  df-2nd 7978  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-pnf 11252  df-mnf 11253  df-xr 11254  df-ltxr 11255  df-le 11256  df-sub 11448  df-neg 11449  df-nn 12215  df-2 12277  df-3 12278  df-4 12279  df-5 12280  df-6 12281  df-n0 12475  df-z 12561  df-uz 12825  df-fz 13487  df-struct 17082  df-sets 17099  df-slot 17117  df-ndx 17129  df-base 17147  df-plusg 17212  df-mulr 17213  df-sca 17215  df-vsca 17216  df-0g 17389  df-mgm 18563  df-sgrp 18612  df-mnd 18628  df-mhm 18673  df-grp 18824  df-minusg 18825  df-ghm 19092  df-cmn 19652  df-abl 19653  df-mgp 19990  df-ur 20007  df-ring 20060  df-cring 20061  df-lmod 20477  df-lmhm 20638  df-assa 21414  df-mend 42000
This theorem is referenced by: (None)
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