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Theorem mendlmod 42510
Description: The module endomorphism algebra is a left module. (Contributed by Mario Carneiro, 22-Sep-2015.)
Hypotheses
Ref Expression
mendassa.a 𝐴 = (MEndoβ€˜π‘€)
mendassa.s 𝑆 = (Scalarβ€˜π‘€)
Assertion
Ref Expression
mendlmod ((𝑀 ∈ LMod ∧ 𝑆 ∈ CRing) β†’ 𝐴 ∈ LMod)

Proof of Theorem mendlmod
Dummy variables π‘₯ 𝑦 𝑧 𝑒 π‘˜ 𝑣 𝑀 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 mendassa.a . . . 4 𝐴 = (MEndoβ€˜π‘€)
21mendbas 42501 . . 3 (𝑀 LMHom 𝑀) = (Baseβ€˜π΄)
32a1i 11 . 2 ((𝑀 ∈ LMod ∧ 𝑆 ∈ CRing) β†’ (𝑀 LMHom 𝑀) = (Baseβ€˜π΄))
4 eqidd 2727 . 2 ((𝑀 ∈ LMod ∧ 𝑆 ∈ CRing) β†’ (+gβ€˜π΄) = (+gβ€˜π΄))
5 mendassa.s . . . 4 𝑆 = (Scalarβ€˜π‘€)
61, 5mendsca 42506 . . 3 𝑆 = (Scalarβ€˜π΄)
76a1i 11 . 2 ((𝑀 ∈ LMod ∧ 𝑆 ∈ CRing) β†’ 𝑆 = (Scalarβ€˜π΄))
8 eqidd 2727 . 2 ((𝑀 ∈ LMod ∧ 𝑆 ∈ CRing) β†’ ( ·𝑠 β€˜π΄) = ( ·𝑠 β€˜π΄))
9 eqidd 2727 . 2 ((𝑀 ∈ LMod ∧ 𝑆 ∈ CRing) β†’ (Baseβ€˜π‘†) = (Baseβ€˜π‘†))
10 eqidd 2727 . 2 ((𝑀 ∈ LMod ∧ 𝑆 ∈ CRing) β†’ (+gβ€˜π‘†) = (+gβ€˜π‘†))
11 eqidd 2727 . 2 ((𝑀 ∈ LMod ∧ 𝑆 ∈ CRing) β†’ (.rβ€˜π‘†) = (.rβ€˜π‘†))
12 eqidd 2727 . 2 ((𝑀 ∈ LMod ∧ 𝑆 ∈ CRing) β†’ (1rβ€˜π‘†) = (1rβ€˜π‘†))
13 crngring 20150 . . 3 (𝑆 ∈ CRing β†’ 𝑆 ∈ Ring)
1413adantl 481 . 2 ((𝑀 ∈ LMod ∧ 𝑆 ∈ CRing) β†’ 𝑆 ∈ Ring)
151mendring 42509 . . . 4 (𝑀 ∈ LMod β†’ 𝐴 ∈ Ring)
1615adantr 480 . . 3 ((𝑀 ∈ LMod ∧ 𝑆 ∈ CRing) β†’ 𝐴 ∈ Ring)
17 ringgrp 20143 . . 3 (𝐴 ∈ Ring β†’ 𝐴 ∈ Grp)
1816, 17syl 17 . 2 ((𝑀 ∈ LMod ∧ 𝑆 ∈ CRing) β†’ 𝐴 ∈ Grp)
19 eqid 2726 . . . . 5 ( ·𝑠 β€˜π‘€) = ( ·𝑠 β€˜π‘€)
20 eqid 2726 . . . . 5 (Baseβ€˜π‘†) = (Baseβ€˜π‘†)
21 eqid 2726 . . . . 5 (Baseβ€˜π‘€) = (Baseβ€˜π‘€)
22 eqid 2726 . . . . 5 ( ·𝑠 β€˜π΄) = ( ·𝑠 β€˜π΄)
231, 19, 2, 5, 20, 21, 22mendvsca 42508 . . . 4 ((π‘₯ ∈ (Baseβ€˜π‘†) ∧ 𝑦 ∈ (𝑀 LMHom 𝑀)) β†’ (π‘₯( ·𝑠 β€˜π΄)𝑦) = (((Baseβ€˜π‘€) Γ— {π‘₯}) ∘f ( ·𝑠 β€˜π‘€)𝑦))
24233adant1 1127 . . 3 (((𝑀 ∈ LMod ∧ 𝑆 ∈ CRing) ∧ π‘₯ ∈ (Baseβ€˜π‘†) ∧ 𝑦 ∈ (𝑀 LMHom 𝑀)) β†’ (π‘₯( ·𝑠 β€˜π΄)𝑦) = (((Baseβ€˜π‘€) Γ— {π‘₯}) ∘f ( ·𝑠 β€˜π‘€)𝑦))
2521, 19, 5, 20lmhmvsca 20893 . . . 4 ((𝑆 ∈ CRing ∧ π‘₯ ∈ (Baseβ€˜π‘†) ∧ 𝑦 ∈ (𝑀 LMHom 𝑀)) β†’ (((Baseβ€˜π‘€) Γ— {π‘₯}) ∘f ( ·𝑠 β€˜π‘€)𝑦) ∈ (𝑀 LMHom 𝑀))
26253adant1l 1173 . . 3 (((𝑀 ∈ LMod ∧ 𝑆 ∈ CRing) ∧ π‘₯ ∈ (Baseβ€˜π‘†) ∧ 𝑦 ∈ (𝑀 LMHom 𝑀)) β†’ (((Baseβ€˜π‘€) Γ— {π‘₯}) ∘f ( ·𝑠 β€˜π‘€)𝑦) ∈ (𝑀 LMHom 𝑀))
2724, 26eqeltrd 2827 . 2 (((𝑀 ∈ LMod ∧ 𝑆 ∈ CRing) ∧ π‘₯ ∈ (Baseβ€˜π‘†) ∧ 𝑦 ∈ (𝑀 LMHom 𝑀)) β†’ (π‘₯( ·𝑠 β€˜π΄)𝑦) ∈ (𝑀 LMHom 𝑀))
28 simpr2 1192 . . . . 5 (((𝑀 ∈ LMod ∧ 𝑆 ∈ CRing) ∧ (π‘₯ ∈ (Baseβ€˜π‘†) ∧ 𝑦 ∈ (𝑀 LMHom 𝑀) ∧ 𝑧 ∈ (𝑀 LMHom 𝑀))) β†’ 𝑦 ∈ (𝑀 LMHom 𝑀))
29 simpr3 1193 . . . . 5 (((𝑀 ∈ LMod ∧ 𝑆 ∈ CRing) ∧ (π‘₯ ∈ (Baseβ€˜π‘†) ∧ 𝑦 ∈ (𝑀 LMHom 𝑀) ∧ 𝑧 ∈ (𝑀 LMHom 𝑀))) β†’ 𝑧 ∈ (𝑀 LMHom 𝑀))
30 eqid 2726 . . . . . 6 (+gβ€˜π‘€) = (+gβ€˜π‘€)
31 eqid 2726 . . . . . 6 (+gβ€˜π΄) = (+gβ€˜π΄)
321, 2, 30, 31mendplusg 42503 . . . . 5 ((𝑦 ∈ (𝑀 LMHom 𝑀) ∧ 𝑧 ∈ (𝑀 LMHom 𝑀)) β†’ (𝑦(+gβ€˜π΄)𝑧) = (𝑦 ∘f (+gβ€˜π‘€)𝑧))
3328, 29, 32syl2anc 583 . . . 4 (((𝑀 ∈ LMod ∧ 𝑆 ∈ CRing) ∧ (π‘₯ ∈ (Baseβ€˜π‘†) ∧ 𝑦 ∈ (𝑀 LMHom 𝑀) ∧ 𝑧 ∈ (𝑀 LMHom 𝑀))) β†’ (𝑦(+gβ€˜π΄)𝑧) = (𝑦 ∘f (+gβ€˜π‘€)𝑧))
3433oveq2d 7421 . . 3 (((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(+gβ€˜π΄)𝑧)) = (((Baseβ€˜π‘€) Γ— {π‘₯}) ∘f ( ·𝑠 β€˜π‘€)(𝑦 ∘f (+gβ€˜π‘€)𝑧)))
35 simpr1 1191 . . . 4 (((𝑀 ∈ LMod ∧ 𝑆 ∈ CRing) ∧ (π‘₯ ∈ (Baseβ€˜π‘†) ∧ 𝑦 ∈ (𝑀 LMHom 𝑀) ∧ 𝑧 ∈ (𝑀 LMHom 𝑀))) β†’ π‘₯ ∈ (Baseβ€˜π‘†))
3618adantr 480 . . . . 5 (((𝑀 ∈ LMod ∧ 𝑆 ∈ CRing) ∧ (π‘₯ ∈ (Baseβ€˜π‘†) ∧ 𝑦 ∈ (𝑀 LMHom 𝑀) ∧ 𝑧 ∈ (𝑀 LMHom 𝑀))) β†’ 𝐴 ∈ Grp)
372, 31grpcl 18871 . . . . 5 ((𝐴 ∈ Grp ∧ 𝑦 ∈ (𝑀 LMHom 𝑀) ∧ 𝑧 ∈ (𝑀 LMHom 𝑀)) β†’ (𝑦(+gβ€˜π΄)𝑧) ∈ (𝑀 LMHom 𝑀))
3836, 28, 29, 37syl3anc 1368 . . . 4 (((𝑀 ∈ LMod ∧ 𝑆 ∈ CRing) ∧ (π‘₯ ∈ (Baseβ€˜π‘†) ∧ 𝑦 ∈ (𝑀 LMHom 𝑀) ∧ 𝑧 ∈ (𝑀 LMHom 𝑀))) β†’ (𝑦(+gβ€˜π΄)𝑧) ∈ (𝑀 LMHom 𝑀))
391, 19, 2, 5, 20, 21, 22mendvsca 42508 . . . 4 ((π‘₯ ∈ (Baseβ€˜π‘†) ∧ (𝑦(+gβ€˜π΄)𝑧) ∈ (𝑀 LMHom 𝑀)) β†’ (π‘₯( ·𝑠 β€˜π΄)(𝑦(+gβ€˜π΄)𝑧)) = (((Baseβ€˜π‘€) Γ— {π‘₯}) ∘f ( ·𝑠 β€˜π‘€)(𝑦(+gβ€˜π΄)𝑧)))
4035, 38, 39syl2anc 583 . . 3 (((𝑀 ∈ LMod ∧ 𝑆 ∈ CRing) ∧ (π‘₯ ∈ (Baseβ€˜π‘†) ∧ 𝑦 ∈ (𝑀 LMHom 𝑀) ∧ 𝑧 ∈ (𝑀 LMHom 𝑀))) β†’ (π‘₯( ·𝑠 β€˜π΄)(𝑦(+gβ€˜π΄)𝑧)) = (((Baseβ€˜π‘€) Γ— {π‘₯}) ∘f ( ·𝑠 β€˜π‘€)(𝑦(+gβ€˜π΄)𝑧)))
4135, 28, 23syl2anc 583 . . . . 5 (((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))
421, 19, 2, 5, 20, 21, 22mendvsca 42508 . . . . . 6 ((π‘₯ ∈ (Baseβ€˜π‘†) ∧ 𝑧 ∈ (𝑀 LMHom 𝑀)) β†’ (π‘₯( ·𝑠 β€˜π΄)𝑧) = (((Baseβ€˜π‘€) Γ— {π‘₯}) ∘f ( ·𝑠 β€˜π‘€)𝑧))
4335, 29, 42syl2anc 583 . . . . 5 (((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))
4441, 43oveq12d 7423 . . . 4 (((𝑀 ∈ LMod ∧ 𝑆 ∈ CRing) ∧ (π‘₯ ∈ (Baseβ€˜π‘†) ∧ 𝑦 ∈ (𝑀 LMHom 𝑀) ∧ 𝑧 ∈ (𝑀 LMHom 𝑀))) β†’ ((π‘₯( ·𝑠 β€˜π΄)𝑦) ∘f (+gβ€˜π‘€)(π‘₯( ·𝑠 β€˜π΄)𝑧)) = ((((Baseβ€˜π‘€) Γ— {π‘₯}) ∘f ( ·𝑠 β€˜π‘€)𝑦) ∘f (+gβ€˜π‘€)(((Baseβ€˜π‘€) Γ— {π‘₯}) ∘f ( ·𝑠 β€˜π‘€)𝑧)))
45273adant3r3 1181 . . . . 5 (((𝑀 ∈ LMod ∧ 𝑆 ∈ CRing) ∧ (π‘₯ ∈ (Baseβ€˜π‘†) ∧ 𝑦 ∈ (𝑀 LMHom 𝑀) ∧ 𝑧 ∈ (𝑀 LMHom 𝑀))) β†’ (π‘₯( ·𝑠 β€˜π΄)𝑦) ∈ (𝑀 LMHom 𝑀))
46 eleq1w 2810 . . . . . . . . 9 (𝑦 = 𝑧 β†’ (𝑦 ∈ (𝑀 LMHom 𝑀) ↔ 𝑧 ∈ (𝑀 LMHom 𝑀)))
47463anbi3d 1438 . . . . . . . 8 (𝑦 = 𝑧 β†’ (((𝑀 ∈ LMod ∧ 𝑆 ∈ CRing) ∧ π‘₯ ∈ (Baseβ€˜π‘†) ∧ 𝑦 ∈ (𝑀 LMHom 𝑀)) ↔ ((𝑀 ∈ LMod ∧ 𝑆 ∈ CRing) ∧ π‘₯ ∈ (Baseβ€˜π‘†) ∧ 𝑧 ∈ (𝑀 LMHom 𝑀))))
48 oveq2 7413 . . . . . . . . 9 (𝑦 = 𝑧 β†’ (π‘₯( ·𝑠 β€˜π΄)𝑦) = (π‘₯( ·𝑠 β€˜π΄)𝑧))
4948eleq1d 2812 . . . . . . . 8 (𝑦 = 𝑧 β†’ ((π‘₯( ·𝑠 β€˜π΄)𝑦) ∈ (𝑀 LMHom 𝑀) ↔ (π‘₯( ·𝑠 β€˜π΄)𝑧) ∈ (𝑀 LMHom 𝑀)))
5047, 49imbi12d 344 . . . . . . 7 (𝑦 = 𝑧 β†’ ((((𝑀 ∈ LMod ∧ 𝑆 ∈ CRing) ∧ π‘₯ ∈ (Baseβ€˜π‘†) ∧ 𝑦 ∈ (𝑀 LMHom 𝑀)) β†’ (π‘₯( ·𝑠 β€˜π΄)𝑦) ∈ (𝑀 LMHom 𝑀)) ↔ (((𝑀 ∈ LMod ∧ 𝑆 ∈ CRing) ∧ π‘₯ ∈ (Baseβ€˜π‘†) ∧ 𝑧 ∈ (𝑀 LMHom 𝑀)) β†’ (π‘₯( ·𝑠 β€˜π΄)𝑧) ∈ (𝑀 LMHom 𝑀))))
5150, 27chvarvv 1994 . . . . . 6 (((𝑀 ∈ LMod ∧ 𝑆 ∈ CRing) ∧ π‘₯ ∈ (Baseβ€˜π‘†) ∧ 𝑧 ∈ (𝑀 LMHom 𝑀)) β†’ (π‘₯( ·𝑠 β€˜π΄)𝑧) ∈ (𝑀 LMHom 𝑀))
52513adant3r2 1180 . . . . 5 (((𝑀 ∈ LMod ∧ 𝑆 ∈ CRing) ∧ (π‘₯ ∈ (Baseβ€˜π‘†) ∧ 𝑦 ∈ (𝑀 LMHom 𝑀) ∧ 𝑧 ∈ (𝑀 LMHom 𝑀))) β†’ (π‘₯( ·𝑠 β€˜π΄)𝑧) ∈ (𝑀 LMHom 𝑀))
531, 2, 30, 31mendplusg 42503 . . . . 5 (((π‘₯( ·𝑠 β€˜π΄)𝑦) ∈ (𝑀 LMHom 𝑀) ∧ (π‘₯( ·𝑠 β€˜π΄)𝑧) ∈ (𝑀 LMHom 𝑀)) β†’ ((π‘₯( ·𝑠 β€˜π΄)𝑦)(+gβ€˜π΄)(π‘₯( ·𝑠 β€˜π΄)𝑧)) = ((π‘₯( ·𝑠 β€˜π΄)𝑦) ∘f (+gβ€˜π‘€)(π‘₯( ·𝑠 β€˜π΄)𝑧)))
5445, 52, 53syl2anc 583 . . . 4 (((𝑀 ∈ LMod ∧ 𝑆 ∈ CRing) ∧ (π‘₯ ∈ (Baseβ€˜π‘†) ∧ 𝑦 ∈ (𝑀 LMHom 𝑀) ∧ 𝑧 ∈ (𝑀 LMHom 𝑀))) β†’ ((π‘₯( ·𝑠 β€˜π΄)𝑦)(+gβ€˜π΄)(π‘₯( ·𝑠 β€˜π΄)𝑧)) = ((π‘₯( ·𝑠 β€˜π΄)𝑦) ∘f (+gβ€˜π‘€)(π‘₯( ·𝑠 β€˜π΄)𝑧)))
55 fvexd 6900 . . . . 5 (((𝑀 ∈ LMod ∧ 𝑆 ∈ CRing) ∧ (π‘₯ ∈ (Baseβ€˜π‘†) ∧ 𝑦 ∈ (𝑀 LMHom 𝑀) ∧ 𝑧 ∈ (𝑀 LMHom 𝑀))) β†’ (Baseβ€˜π‘€) ∈ V)
56 fconst6g 6774 . . . . . 6 (π‘₯ ∈ (Baseβ€˜π‘†) β†’ ((Baseβ€˜π‘€) Γ— {π‘₯}):(Baseβ€˜π‘€)⟢(Baseβ€˜π‘†))
5735, 56syl 17 . . . . 5 (((𝑀 ∈ LMod ∧ 𝑆 ∈ CRing) ∧ (π‘₯ ∈ (Baseβ€˜π‘†) ∧ 𝑦 ∈ (𝑀 LMHom 𝑀) ∧ 𝑧 ∈ (𝑀 LMHom 𝑀))) β†’ ((Baseβ€˜π‘€) Γ— {π‘₯}):(Baseβ€˜π‘€)⟢(Baseβ€˜π‘†))
5821, 21lmhmf 20882 . . . . . 6 (𝑦 ∈ (𝑀 LMHom 𝑀) β†’ 𝑦:(Baseβ€˜π‘€)⟢(Baseβ€˜π‘€))
5928, 58syl 17 . . . . 5 (((𝑀 ∈ LMod ∧ 𝑆 ∈ CRing) ∧ (π‘₯ ∈ (Baseβ€˜π‘†) ∧ 𝑦 ∈ (𝑀 LMHom 𝑀) ∧ 𝑧 ∈ (𝑀 LMHom 𝑀))) β†’ 𝑦:(Baseβ€˜π‘€)⟢(Baseβ€˜π‘€))
6021, 21lmhmf 20882 . . . . . 6 (𝑧 ∈ (𝑀 LMHom 𝑀) β†’ 𝑧:(Baseβ€˜π‘€)⟢(Baseβ€˜π‘€))
6129, 60syl 17 . . . . 5 (((𝑀 ∈ LMod ∧ 𝑆 ∈ CRing) ∧ (π‘₯ ∈ (Baseβ€˜π‘†) ∧ 𝑦 ∈ (𝑀 LMHom 𝑀) ∧ 𝑧 ∈ (𝑀 LMHom 𝑀))) β†’ 𝑧:(Baseβ€˜π‘€)⟢(Baseβ€˜π‘€))
62 simpll 764 . . . . . 6 (((𝑀 ∈ LMod ∧ 𝑆 ∈ CRing) ∧ (π‘₯ ∈ (Baseβ€˜π‘†) ∧ 𝑦 ∈ (𝑀 LMHom 𝑀) ∧ 𝑧 ∈ (𝑀 LMHom 𝑀))) β†’ 𝑀 ∈ LMod)
6321, 30, 5, 19, 20lmodvsdi 20731 . . . . . 6 ((𝑀 ∈ LMod ∧ (𝑀 ∈ (Baseβ€˜π‘†) ∧ 𝑣 ∈ (Baseβ€˜π‘€) ∧ 𝑒 ∈ (Baseβ€˜π‘€))) β†’ (𝑀( ·𝑠 β€˜π‘€)(𝑣(+gβ€˜π‘€)𝑒)) = ((𝑀( ·𝑠 β€˜π‘€)𝑣)(+gβ€˜π‘€)(𝑀( ·𝑠 β€˜π‘€)𝑒)))
6462, 63sylan 579 . . . . 5 ((((𝑀 ∈ LMod ∧ 𝑆 ∈ CRing) ∧ (π‘₯ ∈ (Baseβ€˜π‘†) ∧ 𝑦 ∈ (𝑀 LMHom 𝑀) ∧ 𝑧 ∈ (𝑀 LMHom 𝑀))) ∧ (𝑀 ∈ (Baseβ€˜π‘†) ∧ 𝑣 ∈ (Baseβ€˜π‘€) ∧ 𝑒 ∈ (Baseβ€˜π‘€))) β†’ (𝑀( ·𝑠 β€˜π‘€)(𝑣(+gβ€˜π‘€)𝑒)) = ((𝑀( ·𝑠 β€˜π‘€)𝑣)(+gβ€˜π‘€)(𝑀( ·𝑠 β€˜π‘€)𝑒)))
6555, 57, 59, 61, 64caofdi 7706 . . . 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 ∘f (+gβ€˜π‘€)𝑧)) = ((((Baseβ€˜π‘€) Γ— {π‘₯}) ∘f ( ·𝑠 β€˜π‘€)𝑦) ∘f (+gβ€˜π‘€)(((Baseβ€˜π‘€) Γ— {π‘₯}) ∘f ( ·𝑠 β€˜π‘€)𝑧)))
6644, 54, 653eqtr4d 2776 . . 3 (((𝑀 ∈ LMod ∧ 𝑆 ∈ CRing) ∧ (π‘₯ ∈ (Baseβ€˜π‘†) ∧ 𝑦 ∈ (𝑀 LMHom 𝑀) ∧ 𝑧 ∈ (𝑀 LMHom 𝑀))) β†’ ((π‘₯( ·𝑠 β€˜π΄)𝑦)(+gβ€˜π΄)(π‘₯( ·𝑠 β€˜π΄)𝑧)) = (((Baseβ€˜π‘€) Γ— {π‘₯}) ∘f ( ·𝑠 β€˜π‘€)(𝑦 ∘f (+gβ€˜π‘€)𝑧)))
6734, 40, 663eqtr4d 2776 . 2 (((𝑀 ∈ LMod ∧ 𝑆 ∈ CRing) ∧ (π‘₯ ∈ (Baseβ€˜π‘†) ∧ 𝑦 ∈ (𝑀 LMHom 𝑀) ∧ 𝑧 ∈ (𝑀 LMHom 𝑀))) β†’ (π‘₯( ·𝑠 β€˜π΄)(𝑦(+gβ€˜π΄)𝑧)) = ((π‘₯( ·𝑠 β€˜π΄)𝑦)(+gβ€˜π΄)(π‘₯( ·𝑠 β€˜π΄)𝑧)))
68 fvexd 6900 . . . 4 (((𝑀 ∈ LMod ∧ 𝑆 ∈ CRing) ∧ (π‘₯ ∈ (Baseβ€˜π‘†) ∧ 𝑦 ∈ (Baseβ€˜π‘†) ∧ 𝑧 ∈ (𝑀 LMHom 𝑀))) β†’ (Baseβ€˜π‘€) ∈ V)
69 simpr3 1193 . . . . 5 (((𝑀 ∈ LMod ∧ 𝑆 ∈ CRing) ∧ (π‘₯ ∈ (Baseβ€˜π‘†) ∧ 𝑦 ∈ (Baseβ€˜π‘†) ∧ 𝑧 ∈ (𝑀 LMHom 𝑀))) β†’ 𝑧 ∈ (𝑀 LMHom 𝑀))
7069, 60syl 17 . . . 4 (((𝑀 ∈ LMod ∧ 𝑆 ∈ CRing) ∧ (π‘₯ ∈ (Baseβ€˜π‘†) ∧ 𝑦 ∈ (Baseβ€˜π‘†) ∧ 𝑧 ∈ (𝑀 LMHom 𝑀))) β†’ 𝑧:(Baseβ€˜π‘€)⟢(Baseβ€˜π‘€))
71 simpr1 1191 . . . . 5 (((𝑀 ∈ LMod ∧ 𝑆 ∈ CRing) ∧ (π‘₯ ∈ (Baseβ€˜π‘†) ∧ 𝑦 ∈ (Baseβ€˜π‘†) ∧ 𝑧 ∈ (𝑀 LMHom 𝑀))) β†’ π‘₯ ∈ (Baseβ€˜π‘†))
7271, 56syl 17 . . . 4 (((𝑀 ∈ LMod ∧ 𝑆 ∈ CRing) ∧ (π‘₯ ∈ (Baseβ€˜π‘†) ∧ 𝑦 ∈ (Baseβ€˜π‘†) ∧ 𝑧 ∈ (𝑀 LMHom 𝑀))) β†’ ((Baseβ€˜π‘€) Γ— {π‘₯}):(Baseβ€˜π‘€)⟢(Baseβ€˜π‘†))
73 simpr2 1192 . . . . 5 (((𝑀 ∈ LMod ∧ 𝑆 ∈ CRing) ∧ (π‘₯ ∈ (Baseβ€˜π‘†) ∧ 𝑦 ∈ (Baseβ€˜π‘†) ∧ 𝑧 ∈ (𝑀 LMHom 𝑀))) β†’ 𝑦 ∈ (Baseβ€˜π‘†))
74 fconst6g 6774 . . . . 5 (𝑦 ∈ (Baseβ€˜π‘†) β†’ ((Baseβ€˜π‘€) Γ— {𝑦}):(Baseβ€˜π‘€)⟢(Baseβ€˜π‘†))
7573, 74syl 17 . . . 4 (((𝑀 ∈ LMod ∧ 𝑆 ∈ CRing) ∧ (π‘₯ ∈ (Baseβ€˜π‘†) ∧ 𝑦 ∈ (Baseβ€˜π‘†) ∧ 𝑧 ∈ (𝑀 LMHom 𝑀))) β†’ ((Baseβ€˜π‘€) Γ— {𝑦}):(Baseβ€˜π‘€)⟢(Baseβ€˜π‘†))
76 simpll 764 . . . . 5 (((𝑀 ∈ LMod ∧ 𝑆 ∈ CRing) ∧ (π‘₯ ∈ (Baseβ€˜π‘†) ∧ 𝑦 ∈ (Baseβ€˜π‘†) ∧ 𝑧 ∈ (𝑀 LMHom 𝑀))) β†’ 𝑀 ∈ LMod)
77 eqid 2726 . . . . . 6 (+gβ€˜π‘†) = (+gβ€˜π‘†)
7821, 30, 5, 19, 20, 77lmodvsdir 20732 . . . . 5 ((𝑀 ∈ LMod ∧ (𝑀 ∈ (Baseβ€˜π‘†) ∧ 𝑣 ∈ (Baseβ€˜π‘†) ∧ 𝑒 ∈ (Baseβ€˜π‘€))) β†’ ((𝑀(+gβ€˜π‘†)𝑣)( ·𝑠 β€˜π‘€)𝑒) = ((𝑀( ·𝑠 β€˜π‘€)𝑒)(+gβ€˜π‘€)(𝑣( ·𝑠 β€˜π‘€)𝑒)))
7976, 78sylan 579 . . . 4 ((((𝑀 ∈ LMod ∧ 𝑆 ∈ CRing) ∧ (π‘₯ ∈ (Baseβ€˜π‘†) ∧ 𝑦 ∈ (Baseβ€˜π‘†) ∧ 𝑧 ∈ (𝑀 LMHom 𝑀))) ∧ (𝑀 ∈ (Baseβ€˜π‘†) ∧ 𝑣 ∈ (Baseβ€˜π‘†) ∧ 𝑒 ∈ (Baseβ€˜π‘€))) β†’ ((𝑀(+gβ€˜π‘†)𝑣)( ·𝑠 β€˜π‘€)𝑒) = ((𝑀( ·𝑠 β€˜π‘€)𝑒)(+gβ€˜π‘€)(𝑣( ·𝑠 β€˜π‘€)𝑒)))
8068, 70, 72, 75, 79caofdir 7707 . . 3 (((𝑀 ∈ LMod ∧ 𝑆 ∈ CRing) ∧ (π‘₯ ∈ (Baseβ€˜π‘†) ∧ 𝑦 ∈ (Baseβ€˜π‘†) ∧ 𝑧 ∈ (𝑀 LMHom 𝑀))) β†’ ((((Baseβ€˜π‘€) Γ— {π‘₯}) ∘f (+gβ€˜π‘†)((Baseβ€˜π‘€) Γ— {𝑦})) ∘f ( ·𝑠 β€˜π‘€)𝑧) = ((((Baseβ€˜π‘€) Γ— {π‘₯}) ∘f ( ·𝑠 β€˜π‘€)𝑧) ∘f (+gβ€˜π‘€)(((Baseβ€˜π‘€) Γ— {𝑦}) ∘f ( ·𝑠 β€˜π‘€)𝑧)))
8114adantr 480 . . . . . 6 (((𝑀 ∈ LMod ∧ 𝑆 ∈ CRing) ∧ (π‘₯ ∈ (Baseβ€˜π‘†) ∧ 𝑦 ∈ (Baseβ€˜π‘†) ∧ 𝑧 ∈ (𝑀 LMHom 𝑀))) β†’ 𝑆 ∈ Ring)
8220, 77ringacl 20177 . . . . . 6 ((𝑆 ∈ Ring ∧ π‘₯ ∈ (Baseβ€˜π‘†) ∧ 𝑦 ∈ (Baseβ€˜π‘†)) β†’ (π‘₯(+gβ€˜π‘†)𝑦) ∈ (Baseβ€˜π‘†))
8381, 71, 73, 82syl3anc 1368 . . . . 5 (((𝑀 ∈ LMod ∧ 𝑆 ∈ CRing) ∧ (π‘₯ ∈ (Baseβ€˜π‘†) ∧ 𝑦 ∈ (Baseβ€˜π‘†) ∧ 𝑧 ∈ (𝑀 LMHom 𝑀))) β†’ (π‘₯(+gβ€˜π‘†)𝑦) ∈ (Baseβ€˜π‘†))
841, 19, 2, 5, 20, 21, 22mendvsca 42508 . . . . 5 (((π‘₯(+gβ€˜π‘†)𝑦) ∈ (Baseβ€˜π‘†) ∧ 𝑧 ∈ (𝑀 LMHom 𝑀)) β†’ ((π‘₯(+gβ€˜π‘†)𝑦)( ·𝑠 β€˜π΄)𝑧) = (((Baseβ€˜π‘€) Γ— {(π‘₯(+gβ€˜π‘†)𝑦)}) ∘f ( ·𝑠 β€˜π‘€)𝑧))
8583, 69, 84syl2anc 583 . . . 4 (((𝑀 ∈ LMod ∧ 𝑆 ∈ CRing) ∧ (π‘₯ ∈ (Baseβ€˜π‘†) ∧ 𝑦 ∈ (Baseβ€˜π‘†) ∧ 𝑧 ∈ (𝑀 LMHom 𝑀))) β†’ ((π‘₯(+gβ€˜π‘†)𝑦)( ·𝑠 β€˜π΄)𝑧) = (((Baseβ€˜π‘€) Γ— {(π‘₯(+gβ€˜π‘†)𝑦)}) ∘f ( ·𝑠 β€˜π‘€)𝑧))
8668, 71, 73ofc12 7695 . . . . 5 (((𝑀 ∈ LMod ∧ 𝑆 ∈ CRing) ∧ (π‘₯ ∈ (Baseβ€˜π‘†) ∧ 𝑦 ∈ (Baseβ€˜π‘†) ∧ 𝑧 ∈ (𝑀 LMHom 𝑀))) β†’ (((Baseβ€˜π‘€) Γ— {π‘₯}) ∘f (+gβ€˜π‘†)((Baseβ€˜π‘€) Γ— {𝑦})) = ((Baseβ€˜π‘€) Γ— {(π‘₯(+gβ€˜π‘†)𝑦)}))
8786oveq1d 7420 . . . 4 (((𝑀 ∈ LMod ∧ 𝑆 ∈ CRing) ∧ (π‘₯ ∈ (Baseβ€˜π‘†) ∧ 𝑦 ∈ (Baseβ€˜π‘†) ∧ 𝑧 ∈ (𝑀 LMHom 𝑀))) β†’ ((((Baseβ€˜π‘€) Γ— {π‘₯}) ∘f (+gβ€˜π‘†)((Baseβ€˜π‘€) Γ— {𝑦})) ∘f ( ·𝑠 β€˜π‘€)𝑧) = (((Baseβ€˜π‘€) Γ— {(π‘₯(+gβ€˜π‘†)𝑦)}) ∘f ( ·𝑠 β€˜π‘€)𝑧))
8885, 87eqtr4d 2769 . . 3 (((𝑀 ∈ LMod ∧ 𝑆 ∈ CRing) ∧ (π‘₯ ∈ (Baseβ€˜π‘†) ∧ 𝑦 ∈ (Baseβ€˜π‘†) ∧ 𝑧 ∈ (𝑀 LMHom 𝑀))) β†’ ((π‘₯(+gβ€˜π‘†)𝑦)( ·𝑠 β€˜π΄)𝑧) = ((((Baseβ€˜π‘€) Γ— {π‘₯}) ∘f (+gβ€˜π‘†)((Baseβ€˜π‘€) Γ— {𝑦})) ∘f ( ·𝑠 β€˜π‘€)𝑧))
89513adant3r2 1180 . . . . 5 (((𝑀 ∈ LMod ∧ 𝑆 ∈ CRing) ∧ (π‘₯ ∈ (Baseβ€˜π‘†) ∧ 𝑦 ∈ (Baseβ€˜π‘†) ∧ 𝑧 ∈ (𝑀 LMHom 𝑀))) β†’ (π‘₯( ·𝑠 β€˜π΄)𝑧) ∈ (𝑀 LMHom 𝑀))
90 eleq1w 2810 . . . . . . . . 9 (π‘₯ = 𝑦 β†’ (π‘₯ ∈ (Baseβ€˜π‘†) ↔ 𝑦 ∈ (Baseβ€˜π‘†)))
91903anbi2d 1437 . . . . . . . 8 (π‘₯ = 𝑦 β†’ (((𝑀 ∈ LMod ∧ 𝑆 ∈ CRing) ∧ π‘₯ ∈ (Baseβ€˜π‘†) ∧ 𝑧 ∈ (𝑀 LMHom 𝑀)) ↔ ((𝑀 ∈ LMod ∧ 𝑆 ∈ CRing) ∧ 𝑦 ∈ (Baseβ€˜π‘†) ∧ 𝑧 ∈ (𝑀 LMHom 𝑀))))
92 oveq1 7412 . . . . . . . . 9 (π‘₯ = 𝑦 β†’ (π‘₯( ·𝑠 β€˜π΄)𝑧) = (𝑦( ·𝑠 β€˜π΄)𝑧))
9392eleq1d 2812 . . . . . . . 8 (π‘₯ = 𝑦 β†’ ((π‘₯( ·𝑠 β€˜π΄)𝑧) ∈ (𝑀 LMHom 𝑀) ↔ (𝑦( ·𝑠 β€˜π΄)𝑧) ∈ (𝑀 LMHom 𝑀)))
9491, 93imbi12d 344 . . . . . . 7 (π‘₯ = 𝑦 β†’ ((((𝑀 ∈ LMod ∧ 𝑆 ∈ CRing) ∧ π‘₯ ∈ (Baseβ€˜π‘†) ∧ 𝑧 ∈ (𝑀 LMHom 𝑀)) β†’ (π‘₯( ·𝑠 β€˜π΄)𝑧) ∈ (𝑀 LMHom 𝑀)) ↔ (((𝑀 ∈ LMod ∧ 𝑆 ∈ CRing) ∧ 𝑦 ∈ (Baseβ€˜π‘†) ∧ 𝑧 ∈ (𝑀 LMHom 𝑀)) β†’ (𝑦( ·𝑠 β€˜π΄)𝑧) ∈ (𝑀 LMHom 𝑀))))
9594, 51chvarvv 1994 . . . . . 6 (((𝑀 ∈ LMod ∧ 𝑆 ∈ CRing) ∧ 𝑦 ∈ (Baseβ€˜π‘†) ∧ 𝑧 ∈ (𝑀 LMHom 𝑀)) β†’ (𝑦( ·𝑠 β€˜π΄)𝑧) ∈ (𝑀 LMHom 𝑀))
96953adant3r1 1179 . . . . 5 (((𝑀 ∈ LMod ∧ 𝑆 ∈ CRing) ∧ (π‘₯ ∈ (Baseβ€˜π‘†) ∧ 𝑦 ∈ (Baseβ€˜π‘†) ∧ 𝑧 ∈ (𝑀 LMHom 𝑀))) β†’ (𝑦( ·𝑠 β€˜π΄)𝑧) ∈ (𝑀 LMHom 𝑀))
971, 2, 30, 31mendplusg 42503 . . . . 5 (((π‘₯( ·𝑠 β€˜π΄)𝑧) ∈ (𝑀 LMHom 𝑀) ∧ (𝑦( ·𝑠 β€˜π΄)𝑧) ∈ (𝑀 LMHom 𝑀)) β†’ ((π‘₯( ·𝑠 β€˜π΄)𝑧)(+gβ€˜π΄)(𝑦( ·𝑠 β€˜π΄)𝑧)) = ((π‘₯( ·𝑠 β€˜π΄)𝑧) ∘f (+gβ€˜π‘€)(𝑦( ·𝑠 β€˜π΄)𝑧)))
9889, 96, 97syl2anc 583 . . . 4 (((𝑀 ∈ LMod ∧ 𝑆 ∈ CRing) ∧ (π‘₯ ∈ (Baseβ€˜π‘†) ∧ 𝑦 ∈ (Baseβ€˜π‘†) ∧ 𝑧 ∈ (𝑀 LMHom 𝑀))) β†’ ((π‘₯( ·𝑠 β€˜π΄)𝑧)(+gβ€˜π΄)(𝑦( ·𝑠 β€˜π΄)𝑧)) = ((π‘₯( ·𝑠 β€˜π΄)𝑧) ∘f (+gβ€˜π‘€)(𝑦( ·𝑠 β€˜π΄)𝑧)))
9971, 69, 42syl2anc 583 . . . . 5 (((𝑀 ∈ LMod ∧ 𝑆 ∈ CRing) ∧ (π‘₯ ∈ (Baseβ€˜π‘†) ∧ 𝑦 ∈ (Baseβ€˜π‘†) ∧ 𝑧 ∈ (𝑀 LMHom 𝑀))) β†’ (π‘₯( ·𝑠 β€˜π΄)𝑧) = (((Baseβ€˜π‘€) Γ— {π‘₯}) ∘f ( ·𝑠 β€˜π‘€)𝑧))
1001, 19, 2, 5, 20, 21, 22mendvsca 42508 . . . . . 6 ((𝑦 ∈ (Baseβ€˜π‘†) ∧ 𝑧 ∈ (𝑀 LMHom 𝑀)) β†’ (𝑦( ·𝑠 β€˜π΄)𝑧) = (((Baseβ€˜π‘€) Γ— {𝑦}) ∘f ( ·𝑠 β€˜π‘€)𝑧))
10173, 69, 100syl2anc 583 . . . . 5 (((𝑀 ∈ LMod ∧ 𝑆 ∈ CRing) ∧ (π‘₯ ∈ (Baseβ€˜π‘†) ∧ 𝑦 ∈ (Baseβ€˜π‘†) ∧ 𝑧 ∈ (𝑀 LMHom 𝑀))) β†’ (𝑦( ·𝑠 β€˜π΄)𝑧) = (((Baseβ€˜π‘€) Γ— {𝑦}) ∘f ( ·𝑠 β€˜π‘€)𝑧))
10299, 101oveq12d 7423 . . . 4 (((𝑀 ∈ LMod ∧ 𝑆 ∈ CRing) ∧ (π‘₯ ∈ (Baseβ€˜π‘†) ∧ 𝑦 ∈ (Baseβ€˜π‘†) ∧ 𝑧 ∈ (𝑀 LMHom 𝑀))) β†’ ((π‘₯( ·𝑠 β€˜π΄)𝑧) ∘f (+gβ€˜π‘€)(𝑦( ·𝑠 β€˜π΄)𝑧)) = ((((Baseβ€˜π‘€) Γ— {π‘₯}) ∘f ( ·𝑠 β€˜π‘€)𝑧) ∘f (+gβ€˜π‘€)(((Baseβ€˜π‘€) Γ— {𝑦}) ∘f ( ·𝑠 β€˜π‘€)𝑧)))
10398, 102eqtrd 2766 . . 3 (((𝑀 ∈ LMod ∧ 𝑆 ∈ CRing) ∧ (π‘₯ ∈ (Baseβ€˜π‘†) ∧ 𝑦 ∈ (Baseβ€˜π‘†) ∧ 𝑧 ∈ (𝑀 LMHom 𝑀))) β†’ ((π‘₯( ·𝑠 β€˜π΄)𝑧)(+gβ€˜π΄)(𝑦( ·𝑠 β€˜π΄)𝑧)) = ((((Baseβ€˜π‘€) Γ— {π‘₯}) ∘f ( ·𝑠 β€˜π‘€)𝑧) ∘f (+gβ€˜π‘€)(((Baseβ€˜π‘€) Γ— {𝑦}) ∘f ( ·𝑠 β€˜π‘€)𝑧)))
10480, 88, 1033eqtr4d 2776 . 2 (((𝑀 ∈ LMod ∧ 𝑆 ∈ CRing) ∧ (π‘₯ ∈ (Baseβ€˜π‘†) ∧ 𝑦 ∈ (Baseβ€˜π‘†) ∧ 𝑧 ∈ (𝑀 LMHom 𝑀))) β†’ ((π‘₯(+gβ€˜π‘†)𝑦)( ·𝑠 β€˜π΄)𝑧) = ((π‘₯( ·𝑠 β€˜π΄)𝑧)(+gβ€˜π΄)(𝑦( ·𝑠 β€˜π΄)𝑧)))
105 ovexd 7440 . . . 4 ((((𝑀 ∈ LMod ∧ 𝑆 ∈ CRing) ∧ (π‘₯ ∈ (Baseβ€˜π‘†) ∧ 𝑦 ∈ (Baseβ€˜π‘†) ∧ 𝑧 ∈ (𝑀 LMHom 𝑀))) ∧ π‘˜ ∈ (Baseβ€˜π‘€)) β†’ (π‘₯(.rβ€˜π‘†)𝑦) ∈ V)
10670ffvelcdmda 7080 . . . 4 ((((𝑀 ∈ LMod ∧ 𝑆 ∈ CRing) ∧ (π‘₯ ∈ (Baseβ€˜π‘†) ∧ 𝑦 ∈ (Baseβ€˜π‘†) ∧ 𝑧 ∈ (𝑀 LMHom 𝑀))) ∧ π‘˜ ∈ (Baseβ€˜π‘€)) β†’ (π‘§β€˜π‘˜) ∈ (Baseβ€˜π‘€))
107 fconstmpt 5731 . . . . 5 ((Baseβ€˜π‘€) Γ— {(π‘₯(.rβ€˜π‘†)𝑦)}) = (π‘˜ ∈ (Baseβ€˜π‘€) ↦ (π‘₯(.rβ€˜π‘†)𝑦))
108107a1i 11 . . . 4 (((𝑀 ∈ LMod ∧ 𝑆 ∈ CRing) ∧ (π‘₯ ∈ (Baseβ€˜π‘†) ∧ 𝑦 ∈ (Baseβ€˜π‘†) ∧ 𝑧 ∈ (𝑀 LMHom 𝑀))) β†’ ((Baseβ€˜π‘€) Γ— {(π‘₯(.rβ€˜π‘†)𝑦)}) = (π‘˜ ∈ (Baseβ€˜π‘€) ↦ (π‘₯(.rβ€˜π‘†)𝑦)))
10970feqmptd 6954 . . . 4 (((𝑀 ∈ LMod ∧ 𝑆 ∈ CRing) ∧ (π‘₯ ∈ (Baseβ€˜π‘†) ∧ 𝑦 ∈ (Baseβ€˜π‘†) ∧ 𝑧 ∈ (𝑀 LMHom 𝑀))) β†’ 𝑧 = (π‘˜ ∈ (Baseβ€˜π‘€) ↦ (π‘§β€˜π‘˜)))
11068, 105, 106, 108, 109offval2 7687 . . 3 (((𝑀 ∈ LMod ∧ 𝑆 ∈ CRing) ∧ (π‘₯ ∈ (Baseβ€˜π‘†) ∧ 𝑦 ∈ (Baseβ€˜π‘†) ∧ 𝑧 ∈ (𝑀 LMHom 𝑀))) β†’ (((Baseβ€˜π‘€) Γ— {(π‘₯(.rβ€˜π‘†)𝑦)}) ∘f ( ·𝑠 β€˜π‘€)𝑧) = (π‘˜ ∈ (Baseβ€˜π‘€) ↦ ((π‘₯(.rβ€˜π‘†)𝑦)( ·𝑠 β€˜π‘€)(π‘§β€˜π‘˜))))
111 eqid 2726 . . . . . 6 (.rβ€˜π‘†) = (.rβ€˜π‘†)
11220, 111ringcl 20155 . . . . 5 ((𝑆 ∈ Ring ∧ π‘₯ ∈ (Baseβ€˜π‘†) ∧ 𝑦 ∈ (Baseβ€˜π‘†)) β†’ (π‘₯(.rβ€˜π‘†)𝑦) ∈ (Baseβ€˜π‘†))
11381, 71, 73, 112syl3anc 1368 . . . 4 (((𝑀 ∈ LMod ∧ 𝑆 ∈ CRing) ∧ (π‘₯ ∈ (Baseβ€˜π‘†) ∧ 𝑦 ∈ (Baseβ€˜π‘†) ∧ 𝑧 ∈ (𝑀 LMHom 𝑀))) β†’ (π‘₯(.rβ€˜π‘†)𝑦) ∈ (Baseβ€˜π‘†))
1141, 19, 2, 5, 20, 21, 22mendvsca 42508 . . . 4 (((π‘₯(.rβ€˜π‘†)𝑦) ∈ (Baseβ€˜π‘†) ∧ 𝑧 ∈ (𝑀 LMHom 𝑀)) β†’ ((π‘₯(.rβ€˜π‘†)𝑦)( ·𝑠 β€˜π΄)𝑧) = (((Baseβ€˜π‘€) Γ— {(π‘₯(.rβ€˜π‘†)𝑦)}) ∘f ( ·𝑠 β€˜π‘€)𝑧))
115113, 69, 114syl2anc 583 . . 3 (((𝑀 ∈ LMod ∧ 𝑆 ∈ CRing) ∧ (π‘₯ ∈ (Baseβ€˜π‘†) ∧ 𝑦 ∈ (Baseβ€˜π‘†) ∧ 𝑧 ∈ (𝑀 LMHom 𝑀))) β†’ ((π‘₯(.rβ€˜π‘†)𝑦)( ·𝑠 β€˜π΄)𝑧) = (((Baseβ€˜π‘€) Γ— {(π‘₯(.rβ€˜π‘†)𝑦)}) ∘f ( ·𝑠 β€˜π‘€)𝑧))
11671adantr 480 . . . . 5 ((((𝑀 ∈ LMod ∧ 𝑆 ∈ CRing) ∧ (π‘₯ ∈ (Baseβ€˜π‘†) ∧ 𝑦 ∈ (Baseβ€˜π‘†) ∧ 𝑧 ∈ (𝑀 LMHom 𝑀))) ∧ π‘˜ ∈ (Baseβ€˜π‘€)) β†’ π‘₯ ∈ (Baseβ€˜π‘†))
117 ovexd 7440 . . . . 5 ((((𝑀 ∈ LMod ∧ 𝑆 ∈ CRing) ∧ (π‘₯ ∈ (Baseβ€˜π‘†) ∧ 𝑦 ∈ (Baseβ€˜π‘†) ∧ 𝑧 ∈ (𝑀 LMHom 𝑀))) ∧ π‘˜ ∈ (Baseβ€˜π‘€)) β†’ (𝑦( ·𝑠 β€˜π‘€)(π‘§β€˜π‘˜)) ∈ V)
118 fconstmpt 5731 . . . . . 6 ((Baseβ€˜π‘€) Γ— {π‘₯}) = (π‘˜ ∈ (Baseβ€˜π‘€) ↦ π‘₯)
119118a1i 11 . . . . 5 (((𝑀 ∈ LMod ∧ 𝑆 ∈ CRing) ∧ (π‘₯ ∈ (Baseβ€˜π‘†) ∧ 𝑦 ∈ (Baseβ€˜π‘†) ∧ 𝑧 ∈ (𝑀 LMHom 𝑀))) β†’ ((Baseβ€˜π‘€) Γ— {π‘₯}) = (π‘˜ ∈ (Baseβ€˜π‘€) ↦ π‘₯))
120 simplr2 1213 . . . . . . 7 ((((𝑀 ∈ LMod ∧ 𝑆 ∈ CRing) ∧ (π‘₯ ∈ (Baseβ€˜π‘†) ∧ 𝑦 ∈ (Baseβ€˜π‘†) ∧ 𝑧 ∈ (𝑀 LMHom 𝑀))) ∧ π‘˜ ∈ (Baseβ€˜π‘€)) β†’ 𝑦 ∈ (Baseβ€˜π‘†))
121 fconstmpt 5731 . . . . . . . 8 ((Baseβ€˜π‘€) Γ— {𝑦}) = (π‘˜ ∈ (Baseβ€˜π‘€) ↦ 𝑦)
122121a1i 11 . . . . . . 7 (((𝑀 ∈ LMod ∧ 𝑆 ∈ CRing) ∧ (π‘₯ ∈ (Baseβ€˜π‘†) ∧ 𝑦 ∈ (Baseβ€˜π‘†) ∧ 𝑧 ∈ (𝑀 LMHom 𝑀))) β†’ ((Baseβ€˜π‘€) Γ— {𝑦}) = (π‘˜ ∈ (Baseβ€˜π‘€) ↦ 𝑦))
12368, 120, 106, 122, 109offval2 7687 . . . . . 6 (((𝑀 ∈ LMod ∧ 𝑆 ∈ CRing) ∧ (π‘₯ ∈ (Baseβ€˜π‘†) ∧ 𝑦 ∈ (Baseβ€˜π‘†) ∧ 𝑧 ∈ (𝑀 LMHom 𝑀))) β†’ (((Baseβ€˜π‘€) Γ— {𝑦}) ∘f ( ·𝑠 β€˜π‘€)𝑧) = (π‘˜ ∈ (Baseβ€˜π‘€) ↦ (𝑦( ·𝑠 β€˜π‘€)(π‘§β€˜π‘˜))))
124101, 123eqtrd 2766 . . . . 5 (((𝑀 ∈ LMod ∧ 𝑆 ∈ CRing) ∧ (π‘₯ ∈ (Baseβ€˜π‘†) ∧ 𝑦 ∈ (Baseβ€˜π‘†) ∧ 𝑧 ∈ (𝑀 LMHom 𝑀))) β†’ (𝑦( ·𝑠 β€˜π΄)𝑧) = (π‘˜ ∈ (Baseβ€˜π‘€) ↦ (𝑦( ·𝑠 β€˜π‘€)(π‘§β€˜π‘˜))))
12568, 116, 117, 119, 124offval2 7687 . . . 4 (((𝑀 ∈ LMod ∧ 𝑆 ∈ CRing) ∧ (π‘₯ ∈ (Baseβ€˜π‘†) ∧ 𝑦 ∈ (Baseβ€˜π‘†) ∧ 𝑧 ∈ (𝑀 LMHom 𝑀))) β†’ (((Baseβ€˜π‘€) Γ— {π‘₯}) ∘f ( ·𝑠 β€˜π‘€)(𝑦( ·𝑠 β€˜π΄)𝑧)) = (π‘˜ ∈ (Baseβ€˜π‘€) ↦ (π‘₯( ·𝑠 β€˜π‘€)(𝑦( ·𝑠 β€˜π‘€)(π‘§β€˜π‘˜)))))
1261, 19, 2, 5, 20, 21, 22mendvsca 42508 . . . . 5 ((π‘₯ ∈ (Baseβ€˜π‘†) ∧ (𝑦( ·𝑠 β€˜π΄)𝑧) ∈ (𝑀 LMHom 𝑀)) β†’ (π‘₯( ·𝑠 β€˜π΄)(𝑦( ·𝑠 β€˜π΄)𝑧)) = (((Baseβ€˜π‘€) Γ— {π‘₯}) ∘f ( ·𝑠 β€˜π‘€)(𝑦( ·𝑠 β€˜π΄)𝑧)))
12771, 96, 126syl2anc 583 . . . 4 (((𝑀 ∈ LMod ∧ 𝑆 ∈ CRing) ∧ (π‘₯ ∈ (Baseβ€˜π‘†) ∧ 𝑦 ∈ (Baseβ€˜π‘†) ∧ 𝑧 ∈ (𝑀 LMHom 𝑀))) β†’ (π‘₯( ·𝑠 β€˜π΄)(𝑦( ·𝑠 β€˜π΄)𝑧)) = (((Baseβ€˜π‘€) Γ— {π‘₯}) ∘f ( ·𝑠 β€˜π‘€)(𝑦( ·𝑠 β€˜π΄)𝑧)))
12876adantr 480 . . . . . 6 ((((𝑀 ∈ LMod ∧ 𝑆 ∈ CRing) ∧ (π‘₯ ∈ (Baseβ€˜π‘†) ∧ 𝑦 ∈ (Baseβ€˜π‘†) ∧ 𝑧 ∈ (𝑀 LMHom 𝑀))) ∧ π‘˜ ∈ (Baseβ€˜π‘€)) β†’ 𝑀 ∈ LMod)
12921, 5, 19, 20, 111lmodvsass 20733 . . . . . 6 ((𝑀 ∈ LMod ∧ (π‘₯ ∈ (Baseβ€˜π‘†) ∧ 𝑦 ∈ (Baseβ€˜π‘†) ∧ (π‘§β€˜π‘˜) ∈ (Baseβ€˜π‘€))) β†’ ((π‘₯(.rβ€˜π‘†)𝑦)( ·𝑠 β€˜π‘€)(π‘§β€˜π‘˜)) = (π‘₯( ·𝑠 β€˜π‘€)(𝑦( ·𝑠 β€˜π‘€)(π‘§β€˜π‘˜))))
130128, 116, 120, 106, 129syl13anc 1369 . . . . 5 ((((𝑀 ∈ LMod ∧ 𝑆 ∈ CRing) ∧ (π‘₯ ∈ (Baseβ€˜π‘†) ∧ 𝑦 ∈ (Baseβ€˜π‘†) ∧ 𝑧 ∈ (𝑀 LMHom 𝑀))) ∧ π‘˜ ∈ (Baseβ€˜π‘€)) β†’ ((π‘₯(.rβ€˜π‘†)𝑦)( ·𝑠 β€˜π‘€)(π‘§β€˜π‘˜)) = (π‘₯( ·𝑠 β€˜π‘€)(𝑦( ·𝑠 β€˜π‘€)(π‘§β€˜π‘˜))))
131130mpteq2dva 5241 . . . 4 (((𝑀 ∈ LMod ∧ 𝑆 ∈ CRing) ∧ (π‘₯ ∈ (Baseβ€˜π‘†) ∧ 𝑦 ∈ (Baseβ€˜π‘†) ∧ 𝑧 ∈ (𝑀 LMHom 𝑀))) β†’ (π‘˜ ∈ (Baseβ€˜π‘€) ↦ ((π‘₯(.rβ€˜π‘†)𝑦)( ·𝑠 β€˜π‘€)(π‘§β€˜π‘˜))) = (π‘˜ ∈ (Baseβ€˜π‘€) ↦ (π‘₯( ·𝑠 β€˜π‘€)(𝑦( ·𝑠 β€˜π‘€)(π‘§β€˜π‘˜)))))
132125, 127, 1313eqtr4d 2776 . . 3 (((𝑀 ∈ LMod ∧ 𝑆 ∈ CRing) ∧ (π‘₯ ∈ (Baseβ€˜π‘†) ∧ 𝑦 ∈ (Baseβ€˜π‘†) ∧ 𝑧 ∈ (𝑀 LMHom 𝑀))) β†’ (π‘₯( ·𝑠 β€˜π΄)(𝑦( ·𝑠 β€˜π΄)𝑧)) = (π‘˜ ∈ (Baseβ€˜π‘€) ↦ ((π‘₯(.rβ€˜π‘†)𝑦)( ·𝑠 β€˜π‘€)(π‘§β€˜π‘˜))))
133110, 115, 1323eqtr4d 2776 . 2 (((𝑀 ∈ LMod ∧ 𝑆 ∈ CRing) ∧ (π‘₯ ∈ (Baseβ€˜π‘†) ∧ 𝑦 ∈ (Baseβ€˜π‘†) ∧ 𝑧 ∈ (𝑀 LMHom 𝑀))) β†’ ((π‘₯(.rβ€˜π‘†)𝑦)( ·𝑠 β€˜π΄)𝑧) = (π‘₯( ·𝑠 β€˜π΄)(𝑦( ·𝑠 β€˜π΄)𝑧)))
13414adantr 480 . . . . 5 (((𝑀 ∈ LMod ∧ 𝑆 ∈ CRing) ∧ π‘₯ ∈ (𝑀 LMHom 𝑀)) β†’ 𝑆 ∈ Ring)
135 eqid 2726 . . . . . 6 (1rβ€˜π‘†) = (1rβ€˜π‘†)
13620, 135ringidcl 20165 . . . . 5 (𝑆 ∈ Ring β†’ (1rβ€˜π‘†) ∈ (Baseβ€˜π‘†))
137134, 136syl 17 . . . 4 (((𝑀 ∈ LMod ∧ 𝑆 ∈ CRing) ∧ π‘₯ ∈ (𝑀 LMHom 𝑀)) β†’ (1rβ€˜π‘†) ∈ (Baseβ€˜π‘†))
1381, 19, 2, 5, 20, 21, 22mendvsca 42508 . . . 4 (((1rβ€˜π‘†) ∈ (Baseβ€˜π‘†) ∧ π‘₯ ∈ (𝑀 LMHom 𝑀)) β†’ ((1rβ€˜π‘†)( ·𝑠 β€˜π΄)π‘₯) = (((Baseβ€˜π‘€) Γ— {(1rβ€˜π‘†)}) ∘f ( ·𝑠 β€˜π‘€)π‘₯))
139137, 138sylancom 587 . . 3 (((𝑀 ∈ LMod ∧ 𝑆 ∈ CRing) ∧ π‘₯ ∈ (𝑀 LMHom 𝑀)) β†’ ((1rβ€˜π‘†)( ·𝑠 β€˜π΄)π‘₯) = (((Baseβ€˜π‘€) Γ— {(1rβ€˜π‘†)}) ∘f ( ·𝑠 β€˜π‘€)π‘₯))
140 fvexd 6900 . . . 4 (((𝑀 ∈ LMod ∧ 𝑆 ∈ CRing) ∧ π‘₯ ∈ (𝑀 LMHom 𝑀)) β†’ (Baseβ€˜π‘€) ∈ V)
14121, 21lmhmf 20882 . . . . 5 (π‘₯ ∈ (𝑀 LMHom 𝑀) β†’ π‘₯:(Baseβ€˜π‘€)⟢(Baseβ€˜π‘€))
142141adantl 481 . . . 4 (((𝑀 ∈ LMod ∧ 𝑆 ∈ CRing) ∧ π‘₯ ∈ (𝑀 LMHom 𝑀)) β†’ π‘₯:(Baseβ€˜π‘€)⟢(Baseβ€˜π‘€))
143 simpll 764 . . . . 5 (((𝑀 ∈ LMod ∧ 𝑆 ∈ CRing) ∧ π‘₯ ∈ (𝑀 LMHom 𝑀)) β†’ 𝑀 ∈ LMod)
14421, 5, 19, 135lmodvs1 20736 . . . . 5 ((𝑀 ∈ LMod ∧ 𝑦 ∈ (Baseβ€˜π‘€)) β†’ ((1rβ€˜π‘†)( ·𝑠 β€˜π‘€)𝑦) = 𝑦)
145143, 144sylan 579 . . . 4 ((((𝑀 ∈ LMod ∧ 𝑆 ∈ CRing) ∧ π‘₯ ∈ (𝑀 LMHom 𝑀)) ∧ 𝑦 ∈ (Baseβ€˜π‘€)) β†’ ((1rβ€˜π‘†)( ·𝑠 β€˜π‘€)𝑦) = 𝑦)
146140, 142, 137, 145caofid0l 7698 . . 3 (((𝑀 ∈ LMod ∧ 𝑆 ∈ CRing) ∧ π‘₯ ∈ (𝑀 LMHom 𝑀)) β†’ (((Baseβ€˜π‘€) Γ— {(1rβ€˜π‘†)}) ∘f ( ·𝑠 β€˜π‘€)π‘₯) = π‘₯)
147139, 146eqtrd 2766 . 2 (((𝑀 ∈ LMod ∧ 𝑆 ∈ CRing) ∧ π‘₯ ∈ (𝑀 LMHom 𝑀)) β†’ ((1rβ€˜π‘†)( ·𝑠 β€˜π΄)π‘₯) = π‘₯)
1483, 4, 7, 8, 9, 10, 11, 12, 14, 18, 27, 67, 104, 133, 147islmodd 20712 1 ((𝑀 ∈ LMod ∧ 𝑆 ∈ CRing) β†’ 𝐴 ∈ LMod)
Colors of variables: wff setvar class
Syntax hints:   β†’ wi 4   ∧ wa 395   ∧ w3a 1084   = wceq 1533   ∈ wcel 2098  Vcvv 3468  {csn 4623   ↦ cmpt 5224   Γ— cxp 5667  βŸΆwf 6533  β€˜cfv 6537  (class class class)co 7405   ∘f cof 7665  Basecbs 17153  +gcplusg 17206  .rcmulr 17207  Scalarcsca 17209   ·𝑠 cvsca 17210  Grpcgrp 18863  1rcur 20086  Ringcrg 20138  CRingccrg 20139  LModclmod 20706   LMHom clmhm 20867  MEndocmend 42492
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1905  ax-6 1963  ax-7 2003  ax-8 2100  ax-9 2108  ax-10 2129  ax-11 2146  ax-12 2163  ax-ext 2697  ax-rep 5278  ax-sep 5292  ax-nul 5299  ax-pow 5356  ax-pr 5420  ax-un 7722  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 396  df-or 845  df-3or 1085  df-3an 1086  df-tru 1536  df-fal 1546  df-ex 1774  df-nf 1778  df-sb 2060  df-mo 2528  df-eu 2557  df-clab 2704  df-cleq 2718  df-clel 2804  df-nfc 2879  df-ne 2935  df-nel 3041  df-ral 3056  df-rex 3065  df-rmo 3370  df-reu 3371  df-rab 3427  df-v 3470  df-sbc 3773  df-csb 3889  df-dif 3946  df-un 3948  df-in 3950  df-ss 3960  df-pss 3962  df-nul 4318  df-if 4524  df-pw 4599  df-sn 4624  df-pr 4626  df-tp 4628  df-op 4630  df-uni 4903  df-iun 4992  df-br 5142  df-opab 5204  df-mpt 5225  df-tr 5259  df-id 5567  df-eprel 5573  df-po 5581  df-so 5582  df-fr 5624  df-we 5626  df-xp 5675  df-rel 5676  df-cnv 5677  df-co 5678  df-dm 5679  df-rn 5680  df-res 5681  df-ima 5682  df-pred 6294  df-ord 6361  df-on 6362  df-lim 6363  df-suc 6364  df-iota 6489  df-fun 6539  df-fn 6540  df-f 6541  df-f1 6542  df-fo 6543  df-f1o 6544  df-fv 6545  df-riota 7361  df-ov 7408  df-oprab 7409  df-mpo 7410  df-of 7667  df-om 7853  df-1st 7974  df-2nd 7975  df-frecs 8267  df-wrecs 8298  df-recs 8372  df-rdg 8411  df-1o 8467  df-er 8705  df-map 8824  df-en 8942  df-dom 8943  df-sdom 8944  df-fin 8945  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-3 12280  df-4 12281  df-5 12282  df-6 12283  df-n0 12477  df-z 12563  df-uz 12827  df-fz 13491  df-struct 17089  df-sets 17106  df-slot 17124  df-ndx 17136  df-base 17154  df-plusg 17219  df-mulr 17220  df-sca 17222  df-vsca 17223  df-0g 17396  df-mgm 18573  df-sgrp 18652  df-mnd 18668  df-mhm 18713  df-grp 18866  df-minusg 18867  df-ghm 19139  df-cmn 19702  df-abl 19703  df-mgp 20040  df-rng 20058  df-ur 20087  df-ring 20140  df-cring 20141  df-lmod 20708  df-lmhm 20870  df-mend 42493
This theorem is referenced by:  mendassa  42511
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