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Theorem mendlmod 42648
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 42639 . . 3 (𝑀 LMHom 𝑀) = (Baseβ€˜π΄)
32a1i 11 . 2 ((𝑀 ∈ LMod ∧ 𝑆 ∈ CRing) β†’ (𝑀 LMHom 𝑀) = (Baseβ€˜π΄))
4 eqidd 2729 . 2 ((𝑀 ∈ LMod ∧ 𝑆 ∈ CRing) β†’ (+gβ€˜π΄) = (+gβ€˜π΄))
5 mendassa.s . . . 4 𝑆 = (Scalarβ€˜π‘€)
61, 5mendsca 42644 . . 3 𝑆 = (Scalarβ€˜π΄)
76a1i 11 . 2 ((𝑀 ∈ LMod ∧ 𝑆 ∈ CRing) β†’ 𝑆 = (Scalarβ€˜π΄))
8 eqidd 2729 . 2 ((𝑀 ∈ LMod ∧ 𝑆 ∈ CRing) β†’ ( ·𝑠 β€˜π΄) = ( ·𝑠 β€˜π΄))
9 eqidd 2729 . 2 ((𝑀 ∈ LMod ∧ 𝑆 ∈ CRing) β†’ (Baseβ€˜π‘†) = (Baseβ€˜π‘†))
10 eqidd 2729 . 2 ((𝑀 ∈ LMod ∧ 𝑆 ∈ CRing) β†’ (+gβ€˜π‘†) = (+gβ€˜π‘†))
11 eqidd 2729 . 2 ((𝑀 ∈ LMod ∧ 𝑆 ∈ CRing) β†’ (.rβ€˜π‘†) = (.rβ€˜π‘†))
12 eqidd 2729 . 2 ((𝑀 ∈ LMod ∧ 𝑆 ∈ CRing) β†’ (1rβ€˜π‘†) = (1rβ€˜π‘†))
13 crngring 20192 . . 3 (𝑆 ∈ CRing β†’ 𝑆 ∈ Ring)
1413adantl 480 . 2 ((𝑀 ∈ LMod ∧ 𝑆 ∈ CRing) β†’ 𝑆 ∈ Ring)
151mendring 42647 . . . 4 (𝑀 ∈ LMod β†’ 𝐴 ∈ Ring)
1615adantr 479 . . 3 ((𝑀 ∈ LMod ∧ 𝑆 ∈ CRing) β†’ 𝐴 ∈ Ring)
17 ringgrp 20185 . . 3 (𝐴 ∈ Ring β†’ 𝐴 ∈ Grp)
1816, 17syl 17 . 2 ((𝑀 ∈ LMod ∧ 𝑆 ∈ CRing) β†’ 𝐴 ∈ Grp)
19 eqid 2728 . . . . 5 ( ·𝑠 β€˜π‘€) = ( ·𝑠 β€˜π‘€)
20 eqid 2728 . . . . 5 (Baseβ€˜π‘†) = (Baseβ€˜π‘†)
21 eqid 2728 . . . . 5 (Baseβ€˜π‘€) = (Baseβ€˜π‘€)
22 eqid 2728 . . . . 5 ( ·𝑠 β€˜π΄) = ( ·𝑠 β€˜π΄)
231, 19, 2, 5, 20, 21, 22mendvsca 42646 . . . 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 20937 . . . 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 2829 . 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 2728 . . . . . 6 (+gβ€˜π‘€) = (+gβ€˜π‘€)
31 eqid 2728 . . . . . 6 (+gβ€˜π΄) = (+gβ€˜π΄)
321, 2, 30, 31mendplusg 42641 . . . . 5 ((𝑦 ∈ (𝑀 LMHom 𝑀) ∧ 𝑧 ∈ (𝑀 LMHom 𝑀)) β†’ (𝑦(+gβ€˜π΄)𝑧) = (𝑦 ∘f (+gβ€˜π‘€)𝑧))
3328, 29, 32syl2anc 582 . . . 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 7442 . . 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 479 . . . . 5 (((𝑀 ∈ LMod ∧ 𝑆 ∈ CRing) ∧ (π‘₯ ∈ (Baseβ€˜π‘†) ∧ 𝑦 ∈ (𝑀 LMHom 𝑀) ∧ 𝑧 ∈ (𝑀 LMHom 𝑀))) β†’ 𝐴 ∈ Grp)
372, 31grpcl 18905 . . . . 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 42646 . . . 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 582 . . 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 582 . . . . 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 42646 . . . . . 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 582 . . . . 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 7444 . . . 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 2812 . . . . . . . . 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 7434 . . . . . . . . 9 (𝑦 = 𝑧 β†’ (π‘₯( ·𝑠 β€˜π΄)𝑦) = (π‘₯( ·𝑠 β€˜π΄)𝑧))
4948eleq1d 2814 . . . . . . . 8 (𝑦 = 𝑧 β†’ ((π‘₯( ·𝑠 β€˜π΄)𝑦) ∈ (𝑀 LMHom 𝑀) ↔ (π‘₯( ·𝑠 β€˜π΄)𝑧) ∈ (𝑀 LMHom 𝑀)))
5047, 49imbi12d 343 . . . . . . 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 42641 . . . . 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 582 . . . 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 6917 . . . . 5 (((𝑀 ∈ LMod ∧ 𝑆 ∈ CRing) ∧ (π‘₯ ∈ (Baseβ€˜π‘†) ∧ 𝑦 ∈ (𝑀 LMHom 𝑀) ∧ 𝑧 ∈ (𝑀 LMHom 𝑀))) β†’ (Baseβ€˜π‘€) ∈ V)
56 fconst6g 6791 . . . . . 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 20926 . . . . . 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 20926 . . . . . 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 765 . . . . . 6 (((𝑀 ∈ LMod ∧ 𝑆 ∈ CRing) ∧ (π‘₯ ∈ (Baseβ€˜π‘†) ∧ 𝑦 ∈ (𝑀 LMHom 𝑀) ∧ 𝑧 ∈ (𝑀 LMHom 𝑀))) β†’ 𝑀 ∈ LMod)
6321, 30, 5, 19, 20lmodvsdi 20775 . . . . . 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 578 . . . . 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 7730 . . . 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 2778 . . 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 2778 . 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 6917 . . . 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 6791 . . . . 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 765 . . . . 5 (((𝑀 ∈ LMod ∧ 𝑆 ∈ CRing) ∧ (π‘₯ ∈ (Baseβ€˜π‘†) ∧ 𝑦 ∈ (Baseβ€˜π‘†) ∧ 𝑧 ∈ (𝑀 LMHom 𝑀))) β†’ 𝑀 ∈ LMod)
77 eqid 2728 . . . . . 6 (+gβ€˜π‘†) = (+gβ€˜π‘†)
7821, 30, 5, 19, 20, 77lmodvsdir 20776 . . . . 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 578 . . . 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 7731 . . 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 479 . . . . . 6 (((𝑀 ∈ LMod ∧ 𝑆 ∈ CRing) ∧ (π‘₯ ∈ (Baseβ€˜π‘†) ∧ 𝑦 ∈ (Baseβ€˜π‘†) ∧ 𝑧 ∈ (𝑀 LMHom 𝑀))) β†’ 𝑆 ∈ Ring)
8220, 77ringacl 20221 . . . . . 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 42646 . . . . 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 582 . . . 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 7719 . . . . 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 7441 . . . 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 2771 . . 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 2812 . . . . . . . . 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 7433 . . . . . . . . 9 (π‘₯ = 𝑦 β†’ (π‘₯( ·𝑠 β€˜π΄)𝑧) = (𝑦( ·𝑠 β€˜π΄)𝑧))
9392eleq1d 2814 . . . . . . . 8 (π‘₯ = 𝑦 β†’ ((π‘₯( ·𝑠 β€˜π΄)𝑧) ∈ (𝑀 LMHom 𝑀) ↔ (𝑦( ·𝑠 β€˜π΄)𝑧) ∈ (𝑀 LMHom 𝑀)))
9491, 93imbi12d 343 . . . . . . 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 42641 . . . . 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 582 . . . 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 582 . . . . 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 42646 . . . . . 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 582 . . . . 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 7444 . . . 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 2768 . . 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 2778 . 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 7461 . . . 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 7099 . . . 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 5744 . . . . 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 6972 . . . 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 7711 . . 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 2728 . . . . . 6 (.rβ€˜π‘†) = (.rβ€˜π‘†)
11220, 111ringcl 20197 . . . . 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 42646 . . . 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 582 . . 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 479 . . . . 5 ((((𝑀 ∈ LMod ∧ 𝑆 ∈ CRing) ∧ (π‘₯ ∈ (Baseβ€˜π‘†) ∧ 𝑦 ∈ (Baseβ€˜π‘†) ∧ 𝑧 ∈ (𝑀 LMHom 𝑀))) ∧ π‘˜ ∈ (Baseβ€˜π‘€)) β†’ π‘₯ ∈ (Baseβ€˜π‘†))
117 ovexd 7461 . . . . 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 5744 . . . . . 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 5744 . . . . . . . 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 7711 . . . . . 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 2768 . . . . 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 7711 . . . 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 42646 . . . . 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 582 . . . 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 479 . . . . . 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 20777 . . . . . 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 5252 . . . 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 2778 . . 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 2778 . 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 479 . . . . 5 (((𝑀 ∈ LMod ∧ 𝑆 ∈ CRing) ∧ π‘₯ ∈ (𝑀 LMHom 𝑀)) β†’ 𝑆 ∈ Ring)
135 eqid 2728 . . . . . 6 (1rβ€˜π‘†) = (1rβ€˜π‘†)
13620, 135ringidcl 20209 . . . . 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 42646 . . . 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 586 . . 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 6917 . . . 4 (((𝑀 ∈ LMod ∧ 𝑆 ∈ CRing) ∧ π‘₯ ∈ (𝑀 LMHom 𝑀)) β†’ (Baseβ€˜π‘€) ∈ V)
14121, 21lmhmf 20926 . . . . 5 (π‘₯ ∈ (𝑀 LMHom 𝑀) β†’ π‘₯:(Baseβ€˜π‘€)⟢(Baseβ€˜π‘€))
142141adantl 480 . . . 4 (((𝑀 ∈ LMod ∧ 𝑆 ∈ CRing) ∧ π‘₯ ∈ (𝑀 LMHom 𝑀)) β†’ π‘₯:(Baseβ€˜π‘€)⟢(Baseβ€˜π‘€))
143 simpll 765 . . . . 5 (((𝑀 ∈ LMod ∧ 𝑆 ∈ CRing) ∧ π‘₯ ∈ (𝑀 LMHom 𝑀)) β†’ 𝑀 ∈ LMod)
14421, 5, 19, 135lmodvs1 20780 . . . . 5 ((𝑀 ∈ LMod ∧ 𝑦 ∈ (Baseβ€˜π‘€)) β†’ ((1rβ€˜π‘†)( ·𝑠 β€˜π‘€)𝑦) = 𝑦)
145143, 144sylan 578 . . . 4 ((((𝑀 ∈ LMod ∧ 𝑆 ∈ CRing) ∧ π‘₯ ∈ (𝑀 LMHom 𝑀)) ∧ 𝑦 ∈ (Baseβ€˜π‘€)) β†’ ((1rβ€˜π‘†)( ·𝑠 β€˜π‘€)𝑦) = 𝑦)
146140, 142, 137, 145caofid0l 7722 . . 3 (((𝑀 ∈ LMod ∧ 𝑆 ∈ CRing) ∧ π‘₯ ∈ (𝑀 LMHom 𝑀)) β†’ (((Baseβ€˜π‘€) Γ— {(1rβ€˜π‘†)}) ∘f ( ·𝑠 β€˜π‘€)π‘₯) = π‘₯)
147139, 146eqtrd 2768 . 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 20756 1 ((𝑀 ∈ LMod ∧ 𝑆 ∈ CRing) β†’ 𝐴 ∈ LMod)
Colors of variables: wff setvar class
Syntax hints:   β†’ wi 4   ∧ wa 394   ∧ w3a 1084   = wceq 1533   ∈ wcel 2098  Vcvv 3473  {csn 4632   ↦ cmpt 5235   Γ— cxp 5680  βŸΆwf 6549  β€˜cfv 6553  (class class class)co 7426   ∘f cof 7689  Basecbs 17187  +gcplusg 17240  .rcmulr 17241  Scalarcsca 17243   ·𝑠 cvsca 17244  Grpcgrp 18897  1rcur 20128  Ringcrg 20180  CRingccrg 20181  LModclmod 20750   LMHom clmhm 20911  MEndocmend 42630
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 2166  ax-ext 2699  ax-rep 5289  ax-sep 5303  ax-nul 5310  ax-pow 5369  ax-pr 5433  ax-un 7746  ax-cnex 11202  ax-resscn 11203  ax-1cn 11204  ax-icn 11205  ax-addcl 11206  ax-addrcl 11207  ax-mulcl 11208  ax-mulrcl 11209  ax-mulcom 11210  ax-addass 11211  ax-mulass 11212  ax-distr 11213  ax-i2m1 11214  ax-1ne0 11215  ax-1rid 11216  ax-rnegex 11217  ax-rrecex 11218  ax-cnre 11219  ax-pre-lttri 11220  ax-pre-lttrn 11221  ax-pre-ltadd 11222  ax-pre-mulgt0 11223
This theorem depends on definitions:  df-bi 206  df-an 395  df-or 846  df-3or 1085  df-3an 1086  df-tru 1536  df-fal 1546  df-ex 1774  df-nf 1778  df-sb 2060  df-mo 2529  df-eu 2558  df-clab 2706  df-cleq 2720  df-clel 2806  df-nfc 2881  df-ne 2938  df-nel 3044  df-ral 3059  df-rex 3068  df-rmo 3374  df-reu 3375  df-rab 3431  df-v 3475  df-sbc 3779  df-csb 3895  df-dif 3952  df-un 3954  df-in 3956  df-ss 3966  df-pss 3968  df-nul 4327  df-if 4533  df-pw 4608  df-sn 4633  df-pr 4635  df-tp 4637  df-op 4639  df-uni 4913  df-iun 5002  df-br 5153  df-opab 5215  df-mpt 5236  df-tr 5270  df-id 5580  df-eprel 5586  df-po 5594  df-so 5595  df-fr 5637  df-we 5639  df-xp 5688  df-rel 5689  df-cnv 5690  df-co 5691  df-dm 5692  df-rn 5693  df-res 5694  df-ima 5695  df-pred 6310  df-ord 6377  df-on 6378  df-lim 6379  df-suc 6380  df-iota 6505  df-fun 6555  df-fn 6556  df-f 6557  df-f1 6558  df-fo 6559  df-f1o 6560  df-fv 6561  df-riota 7382  df-ov 7429  df-oprab 7430  df-mpo 7431  df-of 7691  df-om 7877  df-1st 7999  df-2nd 8000  df-frecs 8293  df-wrecs 8324  df-recs 8398  df-rdg 8437  df-1o 8493  df-er 8731  df-map 8853  df-en 8971  df-dom 8972  df-sdom 8973  df-fin 8974  df-pnf 11288  df-mnf 11289  df-xr 11290  df-ltxr 11291  df-le 11292  df-sub 11484  df-neg 11485  df-nn 12251  df-2 12313  df-3 12314  df-4 12315  df-5 12316  df-6 12317  df-n0 12511  df-z 12597  df-uz 12861  df-fz 13525  df-struct 17123  df-sets 17140  df-slot 17158  df-ndx 17170  df-base 17188  df-plusg 17253  df-mulr 17254  df-sca 17256  df-vsca 17257  df-0g 17430  df-mgm 18607  df-sgrp 18686  df-mnd 18702  df-mhm 18747  df-grp 18900  df-minusg 18901  df-ghm 19175  df-cmn 19744  df-abl 19745  df-mgp 20082  df-rng 20100  df-ur 20129  df-ring 20182  df-cring 20183  df-lmod 20752  df-lmhm 20914  df-mend 42631
This theorem is referenced by:  mendassa  42649
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