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Theorem mendlmod 41018
Description: The module endomorphism algebra is a left module. (Contributed by Mario Carneiro, 22-Sep-2015.)
Hypotheses
Ref Expression
mendassa.a 𝐴 = (MEndo‘𝑀)
mendassa.s 𝑆 = (Scalar‘𝑀)
Assertion
Ref Expression
mendlmod ((𝑀 ∈ LMod ∧ 𝑆 ∈ CRing) → 𝐴 ∈ LMod)

Proof of Theorem mendlmod
Dummy variables 𝑥 𝑦 𝑧 𝑢 𝑘 𝑣 𝑤 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 mendassa.a . . . 4 𝐴 = (MEndo‘𝑀)
21mendbas 41009 . . 3 (𝑀 LMHom 𝑀) = (Base‘𝐴)
32a1i 11 . 2 ((𝑀 ∈ LMod ∧ 𝑆 ∈ CRing) → (𝑀 LMHom 𝑀) = (Base‘𝐴))
4 eqidd 2739 . 2 ((𝑀 ∈ LMod ∧ 𝑆 ∈ CRing) → (+g𝐴) = (+g𝐴))
5 mendassa.s . . . 4 𝑆 = (Scalar‘𝑀)
61, 5mendsca 41014 . . 3 𝑆 = (Scalar‘𝐴)
76a1i 11 . 2 ((𝑀 ∈ LMod ∧ 𝑆 ∈ CRing) → 𝑆 = (Scalar‘𝐴))
8 eqidd 2739 . 2 ((𝑀 ∈ LMod ∧ 𝑆 ∈ CRing) → ( ·𝑠𝐴) = ( ·𝑠𝐴))
9 eqidd 2739 . 2 ((𝑀 ∈ LMod ∧ 𝑆 ∈ CRing) → (Base‘𝑆) = (Base‘𝑆))
10 eqidd 2739 . 2 ((𝑀 ∈ LMod ∧ 𝑆 ∈ CRing) → (+g𝑆) = (+g𝑆))
11 eqidd 2739 . 2 ((𝑀 ∈ LMod ∧ 𝑆 ∈ CRing) → (.r𝑆) = (.r𝑆))
12 eqidd 2739 . 2 ((𝑀 ∈ LMod ∧ 𝑆 ∈ CRing) → (1r𝑆) = (1r𝑆))
13 crngring 19795 . . 3 (𝑆 ∈ CRing → 𝑆 ∈ Ring)
1413adantl 482 . 2 ((𝑀 ∈ LMod ∧ 𝑆 ∈ CRing) → 𝑆 ∈ Ring)
151mendring 41017 . . . 4 (𝑀 ∈ LMod → 𝐴 ∈ Ring)
1615adantr 481 . . 3 ((𝑀 ∈ LMod ∧ 𝑆 ∈ CRing) → 𝐴 ∈ Ring)
17 ringgrp 19788 . . 3 (𝐴 ∈ Ring → 𝐴 ∈ Grp)
1816, 17syl 17 . 2 ((𝑀 ∈ LMod ∧ 𝑆 ∈ CRing) → 𝐴 ∈ Grp)
19 eqid 2738 . . . . 5 ( ·𝑠𝑀) = ( ·𝑠𝑀)
20 eqid 2738 . . . . 5 (Base‘𝑆) = (Base‘𝑆)
21 eqid 2738 . . . . 5 (Base‘𝑀) = (Base‘𝑀)
22 eqid 2738 . . . . 5 ( ·𝑠𝐴) = ( ·𝑠𝐴)
231, 19, 2, 5, 20, 21, 22mendvsca 41016 . . . 4 ((𝑥 ∈ (Base‘𝑆) ∧ 𝑦 ∈ (𝑀 LMHom 𝑀)) → (𝑥( ·𝑠𝐴)𝑦) = (((Base‘𝑀) × {𝑥}) ∘f ( ·𝑠𝑀)𝑦))
24233adant1 1129 . . 3 (((𝑀 ∈ LMod ∧ 𝑆 ∈ CRing) ∧ 𝑥 ∈ (Base‘𝑆) ∧ 𝑦 ∈ (𝑀 LMHom 𝑀)) → (𝑥( ·𝑠𝐴)𝑦) = (((Base‘𝑀) × {𝑥}) ∘f ( ·𝑠𝑀)𝑦))
2521, 19, 5, 20lmhmvsca 20307 . . . 4 ((𝑆 ∈ CRing ∧ 𝑥 ∈ (Base‘𝑆) ∧ 𝑦 ∈ (𝑀 LMHom 𝑀)) → (((Base‘𝑀) × {𝑥}) ∘f ( ·𝑠𝑀)𝑦) ∈ (𝑀 LMHom 𝑀))
26253adant1l 1175 . . 3 (((𝑀 ∈ LMod ∧ 𝑆 ∈ CRing) ∧ 𝑥 ∈ (Base‘𝑆) ∧ 𝑦 ∈ (𝑀 LMHom 𝑀)) → (((Base‘𝑀) × {𝑥}) ∘f ( ·𝑠𝑀)𝑦) ∈ (𝑀 LMHom 𝑀))
2724, 26eqeltrd 2839 . 2 (((𝑀 ∈ LMod ∧ 𝑆 ∈ CRing) ∧ 𝑥 ∈ (Base‘𝑆) ∧ 𝑦 ∈ (𝑀 LMHom 𝑀)) → (𝑥( ·𝑠𝐴)𝑦) ∈ (𝑀 LMHom 𝑀))
28 simpr2 1194 . . . . 5 (((𝑀 ∈ LMod ∧ 𝑆 ∈ CRing) ∧ (𝑥 ∈ (Base‘𝑆) ∧ 𝑦 ∈ (𝑀 LMHom 𝑀) ∧ 𝑧 ∈ (𝑀 LMHom 𝑀))) → 𝑦 ∈ (𝑀 LMHom 𝑀))
29 simpr3 1195 . . . . 5 (((𝑀 ∈ LMod ∧ 𝑆 ∈ CRing) ∧ (𝑥 ∈ (Base‘𝑆) ∧ 𝑦 ∈ (𝑀 LMHom 𝑀) ∧ 𝑧 ∈ (𝑀 LMHom 𝑀))) → 𝑧 ∈ (𝑀 LMHom 𝑀))
30 eqid 2738 . . . . . 6 (+g𝑀) = (+g𝑀)
31 eqid 2738 . . . . . 6 (+g𝐴) = (+g𝐴)
321, 2, 30, 31mendplusg 41011 . . . . 5 ((𝑦 ∈ (𝑀 LMHom 𝑀) ∧ 𝑧 ∈ (𝑀 LMHom 𝑀)) → (𝑦(+g𝐴)𝑧) = (𝑦f (+g𝑀)𝑧))
3328, 29, 32syl2anc 584 . . . 4 (((𝑀 ∈ LMod ∧ 𝑆 ∈ CRing) ∧ (𝑥 ∈ (Base‘𝑆) ∧ 𝑦 ∈ (𝑀 LMHom 𝑀) ∧ 𝑧 ∈ (𝑀 LMHom 𝑀))) → (𝑦(+g𝐴)𝑧) = (𝑦f (+g𝑀)𝑧))
3433oveq2d 7291 . . 3 (((𝑀 ∈ LMod ∧ 𝑆 ∈ CRing) ∧ (𝑥 ∈ (Base‘𝑆) ∧ 𝑦 ∈ (𝑀 LMHom 𝑀) ∧ 𝑧 ∈ (𝑀 LMHom 𝑀))) → (((Base‘𝑀) × {𝑥}) ∘f ( ·𝑠𝑀)(𝑦(+g𝐴)𝑧)) = (((Base‘𝑀) × {𝑥}) ∘f ( ·𝑠𝑀)(𝑦f (+g𝑀)𝑧)))
35 simpr1 1193 . . . 4 (((𝑀 ∈ LMod ∧ 𝑆 ∈ CRing) ∧ (𝑥 ∈ (Base‘𝑆) ∧ 𝑦 ∈ (𝑀 LMHom 𝑀) ∧ 𝑧 ∈ (𝑀 LMHom 𝑀))) → 𝑥 ∈ (Base‘𝑆))
3618adantr 481 . . . . 5 (((𝑀 ∈ LMod ∧ 𝑆 ∈ CRing) ∧ (𝑥 ∈ (Base‘𝑆) ∧ 𝑦 ∈ (𝑀 LMHom 𝑀) ∧ 𝑧 ∈ (𝑀 LMHom 𝑀))) → 𝐴 ∈ Grp)
372, 31grpcl 18585 . . . . 5 ((𝐴 ∈ Grp ∧ 𝑦 ∈ (𝑀 LMHom 𝑀) ∧ 𝑧 ∈ (𝑀 LMHom 𝑀)) → (𝑦(+g𝐴)𝑧) ∈ (𝑀 LMHom 𝑀))
3836, 28, 29, 37syl3anc 1370 . . . 4 (((𝑀 ∈ LMod ∧ 𝑆 ∈ CRing) ∧ (𝑥 ∈ (Base‘𝑆) ∧ 𝑦 ∈ (𝑀 LMHom 𝑀) ∧ 𝑧 ∈ (𝑀 LMHom 𝑀))) → (𝑦(+g𝐴)𝑧) ∈ (𝑀 LMHom 𝑀))
391, 19, 2, 5, 20, 21, 22mendvsca 41016 . . . 4 ((𝑥 ∈ (Base‘𝑆) ∧ (𝑦(+g𝐴)𝑧) ∈ (𝑀 LMHom 𝑀)) → (𝑥( ·𝑠𝐴)(𝑦(+g𝐴)𝑧)) = (((Base‘𝑀) × {𝑥}) ∘f ( ·𝑠𝑀)(𝑦(+g𝐴)𝑧)))
4035, 38, 39syl2anc 584 . . 3 (((𝑀 ∈ LMod ∧ 𝑆 ∈ CRing) ∧ (𝑥 ∈ (Base‘𝑆) ∧ 𝑦 ∈ (𝑀 LMHom 𝑀) ∧ 𝑧 ∈ (𝑀 LMHom 𝑀))) → (𝑥( ·𝑠𝐴)(𝑦(+g𝐴)𝑧)) = (((Base‘𝑀) × {𝑥}) ∘f ( ·𝑠𝑀)(𝑦(+g𝐴)𝑧)))
4135, 28, 23syl2anc 584 . . . . 5 (((𝑀 ∈ LMod ∧ 𝑆 ∈ CRing) ∧ (𝑥 ∈ (Base‘𝑆) ∧ 𝑦 ∈ (𝑀 LMHom 𝑀) ∧ 𝑧 ∈ (𝑀 LMHom 𝑀))) → (𝑥( ·𝑠𝐴)𝑦) = (((Base‘𝑀) × {𝑥}) ∘f ( ·𝑠𝑀)𝑦))
421, 19, 2, 5, 20, 21, 22mendvsca 41016 . . . . . 6 ((𝑥 ∈ (Base‘𝑆) ∧ 𝑧 ∈ (𝑀 LMHom 𝑀)) → (𝑥( ·𝑠𝐴)𝑧) = (((Base‘𝑀) × {𝑥}) ∘f ( ·𝑠𝑀)𝑧))
4335, 29, 42syl2anc 584 . . . . 5 (((𝑀 ∈ LMod ∧ 𝑆 ∈ CRing) ∧ (𝑥 ∈ (Base‘𝑆) ∧ 𝑦 ∈ (𝑀 LMHom 𝑀) ∧ 𝑧 ∈ (𝑀 LMHom 𝑀))) → (𝑥( ·𝑠𝐴)𝑧) = (((Base‘𝑀) × {𝑥}) ∘f ( ·𝑠𝑀)𝑧))
4441, 43oveq12d 7293 . . . 4 (((𝑀 ∈ LMod ∧ 𝑆 ∈ CRing) ∧ (𝑥 ∈ (Base‘𝑆) ∧ 𝑦 ∈ (𝑀 LMHom 𝑀) ∧ 𝑧 ∈ (𝑀 LMHom 𝑀))) → ((𝑥( ·𝑠𝐴)𝑦) ∘f (+g𝑀)(𝑥( ·𝑠𝐴)𝑧)) = ((((Base‘𝑀) × {𝑥}) ∘f ( ·𝑠𝑀)𝑦) ∘f (+g𝑀)(((Base‘𝑀) × {𝑥}) ∘f ( ·𝑠𝑀)𝑧)))
45273adant3r3 1183 . . . . 5 (((𝑀 ∈ LMod ∧ 𝑆 ∈ CRing) ∧ (𝑥 ∈ (Base‘𝑆) ∧ 𝑦 ∈ (𝑀 LMHom 𝑀) ∧ 𝑧 ∈ (𝑀 LMHom 𝑀))) → (𝑥( ·𝑠𝐴)𝑦) ∈ (𝑀 LMHom 𝑀))
46 eleq1w 2821 . . . . . . . . 9 (𝑦 = 𝑧 → (𝑦 ∈ (𝑀 LMHom 𝑀) ↔ 𝑧 ∈ (𝑀 LMHom 𝑀)))
47463anbi3d 1441 . . . . . . . 8 (𝑦 = 𝑧 → (((𝑀 ∈ LMod ∧ 𝑆 ∈ CRing) ∧ 𝑥 ∈ (Base‘𝑆) ∧ 𝑦 ∈ (𝑀 LMHom 𝑀)) ↔ ((𝑀 ∈ LMod ∧ 𝑆 ∈ CRing) ∧ 𝑥 ∈ (Base‘𝑆) ∧ 𝑧 ∈ (𝑀 LMHom 𝑀))))
48 oveq2 7283 . . . . . . . . 9 (𝑦 = 𝑧 → (𝑥( ·𝑠𝐴)𝑦) = (𝑥( ·𝑠𝐴)𝑧))
4948eleq1d 2823 . . . . . . . 8 (𝑦 = 𝑧 → ((𝑥( ·𝑠𝐴)𝑦) ∈ (𝑀 LMHom 𝑀) ↔ (𝑥( ·𝑠𝐴)𝑧) ∈ (𝑀 LMHom 𝑀)))
5047, 49imbi12d 345 . . . . . . 7 (𝑦 = 𝑧 → ((((𝑀 ∈ LMod ∧ 𝑆 ∈ CRing) ∧ 𝑥 ∈ (Base‘𝑆) ∧ 𝑦 ∈ (𝑀 LMHom 𝑀)) → (𝑥( ·𝑠𝐴)𝑦) ∈ (𝑀 LMHom 𝑀)) ↔ (((𝑀 ∈ LMod ∧ 𝑆 ∈ CRing) ∧ 𝑥 ∈ (Base‘𝑆) ∧ 𝑧 ∈ (𝑀 LMHom 𝑀)) → (𝑥( ·𝑠𝐴)𝑧) ∈ (𝑀 LMHom 𝑀))))
5150, 27chvarvv 2002 . . . . . 6 (((𝑀 ∈ LMod ∧ 𝑆 ∈ CRing) ∧ 𝑥 ∈ (Base‘𝑆) ∧ 𝑧 ∈ (𝑀 LMHom 𝑀)) → (𝑥( ·𝑠𝐴)𝑧) ∈ (𝑀 LMHom 𝑀))
52513adant3r2 1182 . . . . 5 (((𝑀 ∈ LMod ∧ 𝑆 ∈ CRing) ∧ (𝑥 ∈ (Base‘𝑆) ∧ 𝑦 ∈ (𝑀 LMHom 𝑀) ∧ 𝑧 ∈ (𝑀 LMHom 𝑀))) → (𝑥( ·𝑠𝐴)𝑧) ∈ (𝑀 LMHom 𝑀))
531, 2, 30, 31mendplusg 41011 . . . . 5 (((𝑥( ·𝑠𝐴)𝑦) ∈ (𝑀 LMHom 𝑀) ∧ (𝑥( ·𝑠𝐴)𝑧) ∈ (𝑀 LMHom 𝑀)) → ((𝑥( ·𝑠𝐴)𝑦)(+g𝐴)(𝑥( ·𝑠𝐴)𝑧)) = ((𝑥( ·𝑠𝐴)𝑦) ∘f (+g𝑀)(𝑥( ·𝑠𝐴)𝑧)))
5445, 52, 53syl2anc 584 . . . 4 (((𝑀 ∈ LMod ∧ 𝑆 ∈ CRing) ∧ (𝑥 ∈ (Base‘𝑆) ∧ 𝑦 ∈ (𝑀 LMHom 𝑀) ∧ 𝑧 ∈ (𝑀 LMHom 𝑀))) → ((𝑥( ·𝑠𝐴)𝑦)(+g𝐴)(𝑥( ·𝑠𝐴)𝑧)) = ((𝑥( ·𝑠𝐴)𝑦) ∘f (+g𝑀)(𝑥( ·𝑠𝐴)𝑧)))
55 fvexd 6789 . . . . 5 (((𝑀 ∈ LMod ∧ 𝑆 ∈ CRing) ∧ (𝑥 ∈ (Base‘𝑆) ∧ 𝑦 ∈ (𝑀 LMHom 𝑀) ∧ 𝑧 ∈ (𝑀 LMHom 𝑀))) → (Base‘𝑀) ∈ V)
56 fconst6g 6663 . . . . . 6 (𝑥 ∈ (Base‘𝑆) → ((Base‘𝑀) × {𝑥}):(Base‘𝑀)⟶(Base‘𝑆))
5735, 56syl 17 . . . . 5 (((𝑀 ∈ LMod ∧ 𝑆 ∈ CRing) ∧ (𝑥 ∈ (Base‘𝑆) ∧ 𝑦 ∈ (𝑀 LMHom 𝑀) ∧ 𝑧 ∈ (𝑀 LMHom 𝑀))) → ((Base‘𝑀) × {𝑥}):(Base‘𝑀)⟶(Base‘𝑆))
5821, 21lmhmf 20296 . . . . . 6 (𝑦 ∈ (𝑀 LMHom 𝑀) → 𝑦:(Base‘𝑀)⟶(Base‘𝑀))
5928, 58syl 17 . . . . 5 (((𝑀 ∈ LMod ∧ 𝑆 ∈ CRing) ∧ (𝑥 ∈ (Base‘𝑆) ∧ 𝑦 ∈ (𝑀 LMHom 𝑀) ∧ 𝑧 ∈ (𝑀 LMHom 𝑀))) → 𝑦:(Base‘𝑀)⟶(Base‘𝑀))
6021, 21lmhmf 20296 . . . . . 6 (𝑧 ∈ (𝑀 LMHom 𝑀) → 𝑧:(Base‘𝑀)⟶(Base‘𝑀))
6129, 60syl 17 . . . . 5 (((𝑀 ∈ LMod ∧ 𝑆 ∈ CRing) ∧ (𝑥 ∈ (Base‘𝑆) ∧ 𝑦 ∈ (𝑀 LMHom 𝑀) ∧ 𝑧 ∈ (𝑀 LMHom 𝑀))) → 𝑧:(Base‘𝑀)⟶(Base‘𝑀))
62 simpll 764 . . . . . 6 (((𝑀 ∈ LMod ∧ 𝑆 ∈ CRing) ∧ (𝑥 ∈ (Base‘𝑆) ∧ 𝑦 ∈ (𝑀 LMHom 𝑀) ∧ 𝑧 ∈ (𝑀 LMHom 𝑀))) → 𝑀 ∈ LMod)
6321, 30, 5, 19, 20lmodvsdi 20146 . . . . . 6 ((𝑀 ∈ LMod ∧ (𝑤 ∈ (Base‘𝑆) ∧ 𝑣 ∈ (Base‘𝑀) ∧ 𝑢 ∈ (Base‘𝑀))) → (𝑤( ·𝑠𝑀)(𝑣(+g𝑀)𝑢)) = ((𝑤( ·𝑠𝑀)𝑣)(+g𝑀)(𝑤( ·𝑠𝑀)𝑢)))
6462, 63sylan 580 . . . . 5 ((((𝑀 ∈ LMod ∧ 𝑆 ∈ CRing) ∧ (𝑥 ∈ (Base‘𝑆) ∧ 𝑦 ∈ (𝑀 LMHom 𝑀) ∧ 𝑧 ∈ (𝑀 LMHom 𝑀))) ∧ (𝑤 ∈ (Base‘𝑆) ∧ 𝑣 ∈ (Base‘𝑀) ∧ 𝑢 ∈ (Base‘𝑀))) → (𝑤( ·𝑠𝑀)(𝑣(+g𝑀)𝑢)) = ((𝑤( ·𝑠𝑀)𝑣)(+g𝑀)(𝑤( ·𝑠𝑀)𝑢)))
6555, 57, 59, 61, 64caofdi 7572 . . . 4 (((𝑀 ∈ LMod ∧ 𝑆 ∈ CRing) ∧ (𝑥 ∈ (Base‘𝑆) ∧ 𝑦 ∈ (𝑀 LMHom 𝑀) ∧ 𝑧 ∈ (𝑀 LMHom 𝑀))) → (((Base‘𝑀) × {𝑥}) ∘f ( ·𝑠𝑀)(𝑦f (+g𝑀)𝑧)) = ((((Base‘𝑀) × {𝑥}) ∘f ( ·𝑠𝑀)𝑦) ∘f (+g𝑀)(((Base‘𝑀) × {𝑥}) ∘f ( ·𝑠𝑀)𝑧)))
6644, 54, 653eqtr4d 2788 . . 3 (((𝑀 ∈ LMod ∧ 𝑆 ∈ CRing) ∧ (𝑥 ∈ (Base‘𝑆) ∧ 𝑦 ∈ (𝑀 LMHom 𝑀) ∧ 𝑧 ∈ (𝑀 LMHom 𝑀))) → ((𝑥( ·𝑠𝐴)𝑦)(+g𝐴)(𝑥( ·𝑠𝐴)𝑧)) = (((Base‘𝑀) × {𝑥}) ∘f ( ·𝑠𝑀)(𝑦f (+g𝑀)𝑧)))
6734, 40, 663eqtr4d 2788 . 2 (((𝑀 ∈ LMod ∧ 𝑆 ∈ CRing) ∧ (𝑥 ∈ (Base‘𝑆) ∧ 𝑦 ∈ (𝑀 LMHom 𝑀) ∧ 𝑧 ∈ (𝑀 LMHom 𝑀))) → (𝑥( ·𝑠𝐴)(𝑦(+g𝐴)𝑧)) = ((𝑥( ·𝑠𝐴)𝑦)(+g𝐴)(𝑥( ·𝑠𝐴)𝑧)))
68 fvexd 6789 . . . 4 (((𝑀 ∈ LMod ∧ 𝑆 ∈ CRing) ∧ (𝑥 ∈ (Base‘𝑆) ∧ 𝑦 ∈ (Base‘𝑆) ∧ 𝑧 ∈ (𝑀 LMHom 𝑀))) → (Base‘𝑀) ∈ V)
69 simpr3 1195 . . . . 5 (((𝑀 ∈ LMod ∧ 𝑆 ∈ CRing) ∧ (𝑥 ∈ (Base‘𝑆) ∧ 𝑦 ∈ (Base‘𝑆) ∧ 𝑧 ∈ (𝑀 LMHom 𝑀))) → 𝑧 ∈ (𝑀 LMHom 𝑀))
7069, 60syl 17 . . . 4 (((𝑀 ∈ LMod ∧ 𝑆 ∈ CRing) ∧ (𝑥 ∈ (Base‘𝑆) ∧ 𝑦 ∈ (Base‘𝑆) ∧ 𝑧 ∈ (𝑀 LMHom 𝑀))) → 𝑧:(Base‘𝑀)⟶(Base‘𝑀))
71 simpr1 1193 . . . . 5 (((𝑀 ∈ LMod ∧ 𝑆 ∈ CRing) ∧ (𝑥 ∈ (Base‘𝑆) ∧ 𝑦 ∈ (Base‘𝑆) ∧ 𝑧 ∈ (𝑀 LMHom 𝑀))) → 𝑥 ∈ (Base‘𝑆))
7271, 56syl 17 . . . 4 (((𝑀 ∈ LMod ∧ 𝑆 ∈ CRing) ∧ (𝑥 ∈ (Base‘𝑆) ∧ 𝑦 ∈ (Base‘𝑆) ∧ 𝑧 ∈ (𝑀 LMHom 𝑀))) → ((Base‘𝑀) × {𝑥}):(Base‘𝑀)⟶(Base‘𝑆))
73 simpr2 1194 . . . . 5 (((𝑀 ∈ LMod ∧ 𝑆 ∈ CRing) ∧ (𝑥 ∈ (Base‘𝑆) ∧ 𝑦 ∈ (Base‘𝑆) ∧ 𝑧 ∈ (𝑀 LMHom 𝑀))) → 𝑦 ∈ (Base‘𝑆))
74 fconst6g 6663 . . . . 5 (𝑦 ∈ (Base‘𝑆) → ((Base‘𝑀) × {𝑦}):(Base‘𝑀)⟶(Base‘𝑆))
7573, 74syl 17 . . . 4 (((𝑀 ∈ LMod ∧ 𝑆 ∈ CRing) ∧ (𝑥 ∈ (Base‘𝑆) ∧ 𝑦 ∈ (Base‘𝑆) ∧ 𝑧 ∈ (𝑀 LMHom 𝑀))) → ((Base‘𝑀) × {𝑦}):(Base‘𝑀)⟶(Base‘𝑆))
76 simpll 764 . . . . 5 (((𝑀 ∈ LMod ∧ 𝑆 ∈ CRing) ∧ (𝑥 ∈ (Base‘𝑆) ∧ 𝑦 ∈ (Base‘𝑆) ∧ 𝑧 ∈ (𝑀 LMHom 𝑀))) → 𝑀 ∈ LMod)
77 eqid 2738 . . . . . 6 (+g𝑆) = (+g𝑆)
7821, 30, 5, 19, 20, 77lmodvsdir 20147 . . . . 5 ((𝑀 ∈ LMod ∧ (𝑤 ∈ (Base‘𝑆) ∧ 𝑣 ∈ (Base‘𝑆) ∧ 𝑢 ∈ (Base‘𝑀))) → ((𝑤(+g𝑆)𝑣)( ·𝑠𝑀)𝑢) = ((𝑤( ·𝑠𝑀)𝑢)(+g𝑀)(𝑣( ·𝑠𝑀)𝑢)))
7976, 78sylan 580 . . . 4 ((((𝑀 ∈ LMod ∧ 𝑆 ∈ CRing) ∧ (𝑥 ∈ (Base‘𝑆) ∧ 𝑦 ∈ (Base‘𝑆) ∧ 𝑧 ∈ (𝑀 LMHom 𝑀))) ∧ (𝑤 ∈ (Base‘𝑆) ∧ 𝑣 ∈ (Base‘𝑆) ∧ 𝑢 ∈ (Base‘𝑀))) → ((𝑤(+g𝑆)𝑣)( ·𝑠𝑀)𝑢) = ((𝑤( ·𝑠𝑀)𝑢)(+g𝑀)(𝑣( ·𝑠𝑀)𝑢)))
8068, 70, 72, 75, 79caofdir 7573 . . 3 (((𝑀 ∈ LMod ∧ 𝑆 ∈ CRing) ∧ (𝑥 ∈ (Base‘𝑆) ∧ 𝑦 ∈ (Base‘𝑆) ∧ 𝑧 ∈ (𝑀 LMHom 𝑀))) → ((((Base‘𝑀) × {𝑥}) ∘f (+g𝑆)((Base‘𝑀) × {𝑦})) ∘f ( ·𝑠𝑀)𝑧) = ((((Base‘𝑀) × {𝑥}) ∘f ( ·𝑠𝑀)𝑧) ∘f (+g𝑀)(((Base‘𝑀) × {𝑦}) ∘f ( ·𝑠𝑀)𝑧)))
8114adantr 481 . . . . . 6 (((𝑀 ∈ LMod ∧ 𝑆 ∈ CRing) ∧ (𝑥 ∈ (Base‘𝑆) ∧ 𝑦 ∈ (Base‘𝑆) ∧ 𝑧 ∈ (𝑀 LMHom 𝑀))) → 𝑆 ∈ Ring)
8220, 77ringacl 19817 . . . . . 6 ((𝑆 ∈ Ring ∧ 𝑥 ∈ (Base‘𝑆) ∧ 𝑦 ∈ (Base‘𝑆)) → (𝑥(+g𝑆)𝑦) ∈ (Base‘𝑆))
8381, 71, 73, 82syl3anc 1370 . . . . 5 (((𝑀 ∈ LMod ∧ 𝑆 ∈ CRing) ∧ (𝑥 ∈ (Base‘𝑆) ∧ 𝑦 ∈ (Base‘𝑆) ∧ 𝑧 ∈ (𝑀 LMHom 𝑀))) → (𝑥(+g𝑆)𝑦) ∈ (Base‘𝑆))
841, 19, 2, 5, 20, 21, 22mendvsca 41016 . . . . 5 (((𝑥(+g𝑆)𝑦) ∈ (Base‘𝑆) ∧ 𝑧 ∈ (𝑀 LMHom 𝑀)) → ((𝑥(+g𝑆)𝑦)( ·𝑠𝐴)𝑧) = (((Base‘𝑀) × {(𝑥(+g𝑆)𝑦)}) ∘f ( ·𝑠𝑀)𝑧))
8583, 69, 84syl2anc 584 . . . 4 (((𝑀 ∈ LMod ∧ 𝑆 ∈ CRing) ∧ (𝑥 ∈ (Base‘𝑆) ∧ 𝑦 ∈ (Base‘𝑆) ∧ 𝑧 ∈ (𝑀 LMHom 𝑀))) → ((𝑥(+g𝑆)𝑦)( ·𝑠𝐴)𝑧) = (((Base‘𝑀) × {(𝑥(+g𝑆)𝑦)}) ∘f ( ·𝑠𝑀)𝑧))
8668, 71, 73ofc12 7561 . . . . 5 (((𝑀 ∈ LMod ∧ 𝑆 ∈ CRing) ∧ (𝑥 ∈ (Base‘𝑆) ∧ 𝑦 ∈ (Base‘𝑆) ∧ 𝑧 ∈ (𝑀 LMHom 𝑀))) → (((Base‘𝑀) × {𝑥}) ∘f (+g𝑆)((Base‘𝑀) × {𝑦})) = ((Base‘𝑀) × {(𝑥(+g𝑆)𝑦)}))
8786oveq1d 7290 . . . 4 (((𝑀 ∈ LMod ∧ 𝑆 ∈ CRing) ∧ (𝑥 ∈ (Base‘𝑆) ∧ 𝑦 ∈ (Base‘𝑆) ∧ 𝑧 ∈ (𝑀 LMHom 𝑀))) → ((((Base‘𝑀) × {𝑥}) ∘f (+g𝑆)((Base‘𝑀) × {𝑦})) ∘f ( ·𝑠𝑀)𝑧) = (((Base‘𝑀) × {(𝑥(+g𝑆)𝑦)}) ∘f ( ·𝑠𝑀)𝑧))
8885, 87eqtr4d 2781 . . 3 (((𝑀 ∈ LMod ∧ 𝑆 ∈ CRing) ∧ (𝑥 ∈ (Base‘𝑆) ∧ 𝑦 ∈ (Base‘𝑆) ∧ 𝑧 ∈ (𝑀 LMHom 𝑀))) → ((𝑥(+g𝑆)𝑦)( ·𝑠𝐴)𝑧) = ((((Base‘𝑀) × {𝑥}) ∘f (+g𝑆)((Base‘𝑀) × {𝑦})) ∘f ( ·𝑠𝑀)𝑧))
89513adant3r2 1182 . . . . 5 (((𝑀 ∈ LMod ∧ 𝑆 ∈ CRing) ∧ (𝑥 ∈ (Base‘𝑆) ∧ 𝑦 ∈ (Base‘𝑆) ∧ 𝑧 ∈ (𝑀 LMHom 𝑀))) → (𝑥( ·𝑠𝐴)𝑧) ∈ (𝑀 LMHom 𝑀))
90 eleq1w 2821 . . . . . . . . 9 (𝑥 = 𝑦 → (𝑥 ∈ (Base‘𝑆) ↔ 𝑦 ∈ (Base‘𝑆)))
91903anbi2d 1440 . . . . . . . 8 (𝑥 = 𝑦 → (((𝑀 ∈ LMod ∧ 𝑆 ∈ CRing) ∧ 𝑥 ∈ (Base‘𝑆) ∧ 𝑧 ∈ (𝑀 LMHom 𝑀)) ↔ ((𝑀 ∈ LMod ∧ 𝑆 ∈ CRing) ∧ 𝑦 ∈ (Base‘𝑆) ∧ 𝑧 ∈ (𝑀 LMHom 𝑀))))
92 oveq1 7282 . . . . . . . . 9 (𝑥 = 𝑦 → (𝑥( ·𝑠𝐴)𝑧) = (𝑦( ·𝑠𝐴)𝑧))
9392eleq1d 2823 . . . . . . . 8 (𝑥 = 𝑦 → ((𝑥( ·𝑠𝐴)𝑧) ∈ (𝑀 LMHom 𝑀) ↔ (𝑦( ·𝑠𝐴)𝑧) ∈ (𝑀 LMHom 𝑀)))
9491, 93imbi12d 345 . . . . . . 7 (𝑥 = 𝑦 → ((((𝑀 ∈ LMod ∧ 𝑆 ∈ CRing) ∧ 𝑥 ∈ (Base‘𝑆) ∧ 𝑧 ∈ (𝑀 LMHom 𝑀)) → (𝑥( ·𝑠𝐴)𝑧) ∈ (𝑀 LMHom 𝑀)) ↔ (((𝑀 ∈ LMod ∧ 𝑆 ∈ CRing) ∧ 𝑦 ∈ (Base‘𝑆) ∧ 𝑧 ∈ (𝑀 LMHom 𝑀)) → (𝑦( ·𝑠𝐴)𝑧) ∈ (𝑀 LMHom 𝑀))))
9594, 51chvarvv 2002 . . . . . 6 (((𝑀 ∈ LMod ∧ 𝑆 ∈ CRing) ∧ 𝑦 ∈ (Base‘𝑆) ∧ 𝑧 ∈ (𝑀 LMHom 𝑀)) → (𝑦( ·𝑠𝐴)𝑧) ∈ (𝑀 LMHom 𝑀))
96953adant3r1 1181 . . . . 5 (((𝑀 ∈ LMod ∧ 𝑆 ∈ CRing) ∧ (𝑥 ∈ (Base‘𝑆) ∧ 𝑦 ∈ (Base‘𝑆) ∧ 𝑧 ∈ (𝑀 LMHom 𝑀))) → (𝑦( ·𝑠𝐴)𝑧) ∈ (𝑀 LMHom 𝑀))
971, 2, 30, 31mendplusg 41011 . . . . 5 (((𝑥( ·𝑠𝐴)𝑧) ∈ (𝑀 LMHom 𝑀) ∧ (𝑦( ·𝑠𝐴)𝑧) ∈ (𝑀 LMHom 𝑀)) → ((𝑥( ·𝑠𝐴)𝑧)(+g𝐴)(𝑦( ·𝑠𝐴)𝑧)) = ((𝑥( ·𝑠𝐴)𝑧) ∘f (+g𝑀)(𝑦( ·𝑠𝐴)𝑧)))
9889, 96, 97syl2anc 584 . . . 4 (((𝑀 ∈ LMod ∧ 𝑆 ∈ CRing) ∧ (𝑥 ∈ (Base‘𝑆) ∧ 𝑦 ∈ (Base‘𝑆) ∧ 𝑧 ∈ (𝑀 LMHom 𝑀))) → ((𝑥( ·𝑠𝐴)𝑧)(+g𝐴)(𝑦( ·𝑠𝐴)𝑧)) = ((𝑥( ·𝑠𝐴)𝑧) ∘f (+g𝑀)(𝑦( ·𝑠𝐴)𝑧)))
9971, 69, 42syl2anc 584 . . . . 5 (((𝑀 ∈ LMod ∧ 𝑆 ∈ CRing) ∧ (𝑥 ∈ (Base‘𝑆) ∧ 𝑦 ∈ (Base‘𝑆) ∧ 𝑧 ∈ (𝑀 LMHom 𝑀))) → (𝑥( ·𝑠𝐴)𝑧) = (((Base‘𝑀) × {𝑥}) ∘f ( ·𝑠𝑀)𝑧))
1001, 19, 2, 5, 20, 21, 22mendvsca 41016 . . . . . 6 ((𝑦 ∈ (Base‘𝑆) ∧ 𝑧 ∈ (𝑀 LMHom 𝑀)) → (𝑦( ·𝑠𝐴)𝑧) = (((Base‘𝑀) × {𝑦}) ∘f ( ·𝑠𝑀)𝑧))
10173, 69, 100syl2anc 584 . . . . 5 (((𝑀 ∈ LMod ∧ 𝑆 ∈ CRing) ∧ (𝑥 ∈ (Base‘𝑆) ∧ 𝑦 ∈ (Base‘𝑆) ∧ 𝑧 ∈ (𝑀 LMHom 𝑀))) → (𝑦( ·𝑠𝐴)𝑧) = (((Base‘𝑀) × {𝑦}) ∘f ( ·𝑠𝑀)𝑧))
10299, 101oveq12d 7293 . . . 4 (((𝑀 ∈ LMod ∧ 𝑆 ∈ CRing) ∧ (𝑥 ∈ (Base‘𝑆) ∧ 𝑦 ∈ (Base‘𝑆) ∧ 𝑧 ∈ (𝑀 LMHom 𝑀))) → ((𝑥( ·𝑠𝐴)𝑧) ∘f (+g𝑀)(𝑦( ·𝑠𝐴)𝑧)) = ((((Base‘𝑀) × {𝑥}) ∘f ( ·𝑠𝑀)𝑧) ∘f (+g𝑀)(((Base‘𝑀) × {𝑦}) ∘f ( ·𝑠𝑀)𝑧)))
10398, 102eqtrd 2778 . . 3 (((𝑀 ∈ LMod ∧ 𝑆 ∈ CRing) ∧ (𝑥 ∈ (Base‘𝑆) ∧ 𝑦 ∈ (Base‘𝑆) ∧ 𝑧 ∈ (𝑀 LMHom 𝑀))) → ((𝑥( ·𝑠𝐴)𝑧)(+g𝐴)(𝑦( ·𝑠𝐴)𝑧)) = ((((Base‘𝑀) × {𝑥}) ∘f ( ·𝑠𝑀)𝑧) ∘f (+g𝑀)(((Base‘𝑀) × {𝑦}) ∘f ( ·𝑠𝑀)𝑧)))
10480, 88, 1033eqtr4d 2788 . 2 (((𝑀 ∈ LMod ∧ 𝑆 ∈ CRing) ∧ (𝑥 ∈ (Base‘𝑆) ∧ 𝑦 ∈ (Base‘𝑆) ∧ 𝑧 ∈ (𝑀 LMHom 𝑀))) → ((𝑥(+g𝑆)𝑦)( ·𝑠𝐴)𝑧) = ((𝑥( ·𝑠𝐴)𝑧)(+g𝐴)(𝑦( ·𝑠𝐴)𝑧)))
105 ovexd 7310 . . . 4 ((((𝑀 ∈ LMod ∧ 𝑆 ∈ CRing) ∧ (𝑥 ∈ (Base‘𝑆) ∧ 𝑦 ∈ (Base‘𝑆) ∧ 𝑧 ∈ (𝑀 LMHom 𝑀))) ∧ 𝑘 ∈ (Base‘𝑀)) → (𝑥(.r𝑆)𝑦) ∈ V)
10670ffvelrnda 6961 . . . 4 ((((𝑀 ∈ LMod ∧ 𝑆 ∈ CRing) ∧ (𝑥 ∈ (Base‘𝑆) ∧ 𝑦 ∈ (Base‘𝑆) ∧ 𝑧 ∈ (𝑀 LMHom 𝑀))) ∧ 𝑘 ∈ (Base‘𝑀)) → (𝑧𝑘) ∈ (Base‘𝑀))
107 fconstmpt 5649 . . . . 5 ((Base‘𝑀) × {(𝑥(.r𝑆)𝑦)}) = (𝑘 ∈ (Base‘𝑀) ↦ (𝑥(.r𝑆)𝑦))
108107a1i 11 . . . 4 (((𝑀 ∈ LMod ∧ 𝑆 ∈ CRing) ∧ (𝑥 ∈ (Base‘𝑆) ∧ 𝑦 ∈ (Base‘𝑆) ∧ 𝑧 ∈ (𝑀 LMHom 𝑀))) → ((Base‘𝑀) × {(𝑥(.r𝑆)𝑦)}) = (𝑘 ∈ (Base‘𝑀) ↦ (𝑥(.r𝑆)𝑦)))
10970feqmptd 6837 . . . 4 (((𝑀 ∈ LMod ∧ 𝑆 ∈ CRing) ∧ (𝑥 ∈ (Base‘𝑆) ∧ 𝑦 ∈ (Base‘𝑆) ∧ 𝑧 ∈ (𝑀 LMHom 𝑀))) → 𝑧 = (𝑘 ∈ (Base‘𝑀) ↦ (𝑧𝑘)))
11068, 105, 106, 108, 109offval2 7553 . . 3 (((𝑀 ∈ LMod ∧ 𝑆 ∈ CRing) ∧ (𝑥 ∈ (Base‘𝑆) ∧ 𝑦 ∈ (Base‘𝑆) ∧ 𝑧 ∈ (𝑀 LMHom 𝑀))) → (((Base‘𝑀) × {(𝑥(.r𝑆)𝑦)}) ∘f ( ·𝑠𝑀)𝑧) = (𝑘 ∈ (Base‘𝑀) ↦ ((𝑥(.r𝑆)𝑦)( ·𝑠𝑀)(𝑧𝑘))))
111 eqid 2738 . . . . . 6 (.r𝑆) = (.r𝑆)
11220, 111ringcl 19800 . . . . 5 ((𝑆 ∈ Ring ∧ 𝑥 ∈ (Base‘𝑆) ∧ 𝑦 ∈ (Base‘𝑆)) → (𝑥(.r𝑆)𝑦) ∈ (Base‘𝑆))
11381, 71, 73, 112syl3anc 1370 . . . 4 (((𝑀 ∈ LMod ∧ 𝑆 ∈ CRing) ∧ (𝑥 ∈ (Base‘𝑆) ∧ 𝑦 ∈ (Base‘𝑆) ∧ 𝑧 ∈ (𝑀 LMHom 𝑀))) → (𝑥(.r𝑆)𝑦) ∈ (Base‘𝑆))
1141, 19, 2, 5, 20, 21, 22mendvsca 41016 . . . 4 (((𝑥(.r𝑆)𝑦) ∈ (Base‘𝑆) ∧ 𝑧 ∈ (𝑀 LMHom 𝑀)) → ((𝑥(.r𝑆)𝑦)( ·𝑠𝐴)𝑧) = (((Base‘𝑀) × {(𝑥(.r𝑆)𝑦)}) ∘f ( ·𝑠𝑀)𝑧))
115113, 69, 114syl2anc 584 . . 3 (((𝑀 ∈ LMod ∧ 𝑆 ∈ CRing) ∧ (𝑥 ∈ (Base‘𝑆) ∧ 𝑦 ∈ (Base‘𝑆) ∧ 𝑧 ∈ (𝑀 LMHom 𝑀))) → ((𝑥(.r𝑆)𝑦)( ·𝑠𝐴)𝑧) = (((Base‘𝑀) × {(𝑥(.r𝑆)𝑦)}) ∘f ( ·𝑠𝑀)𝑧))
11671adantr 481 . . . . 5 ((((𝑀 ∈ LMod ∧ 𝑆 ∈ CRing) ∧ (𝑥 ∈ (Base‘𝑆) ∧ 𝑦 ∈ (Base‘𝑆) ∧ 𝑧 ∈ (𝑀 LMHom 𝑀))) ∧ 𝑘 ∈ (Base‘𝑀)) → 𝑥 ∈ (Base‘𝑆))
117 ovexd 7310 . . . . 5 ((((𝑀 ∈ LMod ∧ 𝑆 ∈ CRing) ∧ (𝑥 ∈ (Base‘𝑆) ∧ 𝑦 ∈ (Base‘𝑆) ∧ 𝑧 ∈ (𝑀 LMHom 𝑀))) ∧ 𝑘 ∈ (Base‘𝑀)) → (𝑦( ·𝑠𝑀)(𝑧𝑘)) ∈ V)
118 fconstmpt 5649 . . . . . 6 ((Base‘𝑀) × {𝑥}) = (𝑘 ∈ (Base‘𝑀) ↦ 𝑥)
119118a1i 11 . . . . 5 (((𝑀 ∈ LMod ∧ 𝑆 ∈ CRing) ∧ (𝑥 ∈ (Base‘𝑆) ∧ 𝑦 ∈ (Base‘𝑆) ∧ 𝑧 ∈ (𝑀 LMHom 𝑀))) → ((Base‘𝑀) × {𝑥}) = (𝑘 ∈ (Base‘𝑀) ↦ 𝑥))
120 simplr2 1215 . . . . . . 7 ((((𝑀 ∈ LMod ∧ 𝑆 ∈ CRing) ∧ (𝑥 ∈ (Base‘𝑆) ∧ 𝑦 ∈ (Base‘𝑆) ∧ 𝑧 ∈ (𝑀 LMHom 𝑀))) ∧ 𝑘 ∈ (Base‘𝑀)) → 𝑦 ∈ (Base‘𝑆))
121 fconstmpt 5649 . . . . . . . 8 ((Base‘𝑀) × {𝑦}) = (𝑘 ∈ (Base‘𝑀) ↦ 𝑦)
122121a1i 11 . . . . . . 7 (((𝑀 ∈ LMod ∧ 𝑆 ∈ CRing) ∧ (𝑥 ∈ (Base‘𝑆) ∧ 𝑦 ∈ (Base‘𝑆) ∧ 𝑧 ∈ (𝑀 LMHom 𝑀))) → ((Base‘𝑀) × {𝑦}) = (𝑘 ∈ (Base‘𝑀) ↦ 𝑦))
12368, 120, 106, 122, 109offval2 7553 . . . . . 6 (((𝑀 ∈ LMod ∧ 𝑆 ∈ CRing) ∧ (𝑥 ∈ (Base‘𝑆) ∧ 𝑦 ∈ (Base‘𝑆) ∧ 𝑧 ∈ (𝑀 LMHom 𝑀))) → (((Base‘𝑀) × {𝑦}) ∘f ( ·𝑠𝑀)𝑧) = (𝑘 ∈ (Base‘𝑀) ↦ (𝑦( ·𝑠𝑀)(𝑧𝑘))))
124101, 123eqtrd 2778 . . . . 5 (((𝑀 ∈ LMod ∧ 𝑆 ∈ CRing) ∧ (𝑥 ∈ (Base‘𝑆) ∧ 𝑦 ∈ (Base‘𝑆) ∧ 𝑧 ∈ (𝑀 LMHom 𝑀))) → (𝑦( ·𝑠𝐴)𝑧) = (𝑘 ∈ (Base‘𝑀) ↦ (𝑦( ·𝑠𝑀)(𝑧𝑘))))
12568, 116, 117, 119, 124offval2 7553 . . . 4 (((𝑀 ∈ LMod ∧ 𝑆 ∈ CRing) ∧ (𝑥 ∈ (Base‘𝑆) ∧ 𝑦 ∈ (Base‘𝑆) ∧ 𝑧 ∈ (𝑀 LMHom 𝑀))) → (((Base‘𝑀) × {𝑥}) ∘f ( ·𝑠𝑀)(𝑦( ·𝑠𝐴)𝑧)) = (𝑘 ∈ (Base‘𝑀) ↦ (𝑥( ·𝑠𝑀)(𝑦( ·𝑠𝑀)(𝑧𝑘)))))
1261, 19, 2, 5, 20, 21, 22mendvsca 41016 . . . . 5 ((𝑥 ∈ (Base‘𝑆) ∧ (𝑦( ·𝑠𝐴)𝑧) ∈ (𝑀 LMHom 𝑀)) → (𝑥( ·𝑠𝐴)(𝑦( ·𝑠𝐴)𝑧)) = (((Base‘𝑀) × {𝑥}) ∘f ( ·𝑠𝑀)(𝑦( ·𝑠𝐴)𝑧)))
12771, 96, 126syl2anc 584 . . . 4 (((𝑀 ∈ LMod ∧ 𝑆 ∈ CRing) ∧ (𝑥 ∈ (Base‘𝑆) ∧ 𝑦 ∈ (Base‘𝑆) ∧ 𝑧 ∈ (𝑀 LMHom 𝑀))) → (𝑥( ·𝑠𝐴)(𝑦( ·𝑠𝐴)𝑧)) = (((Base‘𝑀) × {𝑥}) ∘f ( ·𝑠𝑀)(𝑦( ·𝑠𝐴)𝑧)))
12876adantr 481 . . . . . 6 ((((𝑀 ∈ LMod ∧ 𝑆 ∈ CRing) ∧ (𝑥 ∈ (Base‘𝑆) ∧ 𝑦 ∈ (Base‘𝑆) ∧ 𝑧 ∈ (𝑀 LMHom 𝑀))) ∧ 𝑘 ∈ (Base‘𝑀)) → 𝑀 ∈ LMod)
12921, 5, 19, 20, 111lmodvsass 20148 . . . . . 6 ((𝑀 ∈ LMod ∧ (𝑥 ∈ (Base‘𝑆) ∧ 𝑦 ∈ (Base‘𝑆) ∧ (𝑧𝑘) ∈ (Base‘𝑀))) → ((𝑥(.r𝑆)𝑦)( ·𝑠𝑀)(𝑧𝑘)) = (𝑥( ·𝑠𝑀)(𝑦( ·𝑠𝑀)(𝑧𝑘))))
130128, 116, 120, 106, 129syl13anc 1371 . . . . 5 ((((𝑀 ∈ LMod ∧ 𝑆 ∈ CRing) ∧ (𝑥 ∈ (Base‘𝑆) ∧ 𝑦 ∈ (Base‘𝑆) ∧ 𝑧 ∈ (𝑀 LMHom 𝑀))) ∧ 𝑘 ∈ (Base‘𝑀)) → ((𝑥(.r𝑆)𝑦)( ·𝑠𝑀)(𝑧𝑘)) = (𝑥( ·𝑠𝑀)(𝑦( ·𝑠𝑀)(𝑧𝑘))))
131130mpteq2dva 5174 . . . 4 (((𝑀 ∈ LMod ∧ 𝑆 ∈ CRing) ∧ (𝑥 ∈ (Base‘𝑆) ∧ 𝑦 ∈ (Base‘𝑆) ∧ 𝑧 ∈ (𝑀 LMHom 𝑀))) → (𝑘 ∈ (Base‘𝑀) ↦ ((𝑥(.r𝑆)𝑦)( ·𝑠𝑀)(𝑧𝑘))) = (𝑘 ∈ (Base‘𝑀) ↦ (𝑥( ·𝑠𝑀)(𝑦( ·𝑠𝑀)(𝑧𝑘)))))
132125, 127, 1313eqtr4d 2788 . . 3 (((𝑀 ∈ LMod ∧ 𝑆 ∈ CRing) ∧ (𝑥 ∈ (Base‘𝑆) ∧ 𝑦 ∈ (Base‘𝑆) ∧ 𝑧 ∈ (𝑀 LMHom 𝑀))) → (𝑥( ·𝑠𝐴)(𝑦( ·𝑠𝐴)𝑧)) = (𝑘 ∈ (Base‘𝑀) ↦ ((𝑥(.r𝑆)𝑦)( ·𝑠𝑀)(𝑧𝑘))))
133110, 115, 1323eqtr4d 2788 . 2 (((𝑀 ∈ LMod ∧ 𝑆 ∈ CRing) ∧ (𝑥 ∈ (Base‘𝑆) ∧ 𝑦 ∈ (Base‘𝑆) ∧ 𝑧 ∈ (𝑀 LMHom 𝑀))) → ((𝑥(.r𝑆)𝑦)( ·𝑠𝐴)𝑧) = (𝑥( ·𝑠𝐴)(𝑦( ·𝑠𝐴)𝑧)))
13414adantr 481 . . . . 5 (((𝑀 ∈ LMod ∧ 𝑆 ∈ CRing) ∧ 𝑥 ∈ (𝑀 LMHom 𝑀)) → 𝑆 ∈ Ring)
135 eqid 2738 . . . . . 6 (1r𝑆) = (1r𝑆)
13620, 135ringidcl 19807 . . . . 5 (𝑆 ∈ Ring → (1r𝑆) ∈ (Base‘𝑆))
137134, 136syl 17 . . . 4 (((𝑀 ∈ LMod ∧ 𝑆 ∈ CRing) ∧ 𝑥 ∈ (𝑀 LMHom 𝑀)) → (1r𝑆) ∈ (Base‘𝑆))
1381, 19, 2, 5, 20, 21, 22mendvsca 41016 . . . 4 (((1r𝑆) ∈ (Base‘𝑆) ∧ 𝑥 ∈ (𝑀 LMHom 𝑀)) → ((1r𝑆)( ·𝑠𝐴)𝑥) = (((Base‘𝑀) × {(1r𝑆)}) ∘f ( ·𝑠𝑀)𝑥))
139137, 138sylancom 588 . . 3 (((𝑀 ∈ LMod ∧ 𝑆 ∈ CRing) ∧ 𝑥 ∈ (𝑀 LMHom 𝑀)) → ((1r𝑆)( ·𝑠𝐴)𝑥) = (((Base‘𝑀) × {(1r𝑆)}) ∘f ( ·𝑠𝑀)𝑥))
140 fvexd 6789 . . . 4 (((𝑀 ∈ LMod ∧ 𝑆 ∈ CRing) ∧ 𝑥 ∈ (𝑀 LMHom 𝑀)) → (Base‘𝑀) ∈ V)
14121, 21lmhmf 20296 . . . . 5 (𝑥 ∈ (𝑀 LMHom 𝑀) → 𝑥:(Base‘𝑀)⟶(Base‘𝑀))
142141adantl 482 . . . 4 (((𝑀 ∈ LMod ∧ 𝑆 ∈ CRing) ∧ 𝑥 ∈ (𝑀 LMHom 𝑀)) → 𝑥:(Base‘𝑀)⟶(Base‘𝑀))
143 simpll 764 . . . . 5 (((𝑀 ∈ LMod ∧ 𝑆 ∈ CRing) ∧ 𝑥 ∈ (𝑀 LMHom 𝑀)) → 𝑀 ∈ LMod)
14421, 5, 19, 135lmodvs1 20151 . . . . 5 ((𝑀 ∈ LMod ∧ 𝑦 ∈ (Base‘𝑀)) → ((1r𝑆)( ·𝑠𝑀)𝑦) = 𝑦)
145143, 144sylan 580 . . . 4 ((((𝑀 ∈ LMod ∧ 𝑆 ∈ CRing) ∧ 𝑥 ∈ (𝑀 LMHom 𝑀)) ∧ 𝑦 ∈ (Base‘𝑀)) → ((1r𝑆)( ·𝑠𝑀)𝑦) = 𝑦)
146140, 142, 137, 145caofid0l 7564 . . 3 (((𝑀 ∈ LMod ∧ 𝑆 ∈ CRing) ∧ 𝑥 ∈ (𝑀 LMHom 𝑀)) → (((Base‘𝑀) × {(1r𝑆)}) ∘f ( ·𝑠𝑀)𝑥) = 𝑥)
147139, 146eqtrd 2778 . 2 (((𝑀 ∈ LMod ∧ 𝑆 ∈ CRing) ∧ 𝑥 ∈ (𝑀 LMHom 𝑀)) → ((1r𝑆)( ·𝑠𝐴)𝑥) = 𝑥)
1483, 4, 7, 8, 9, 10, 11, 12, 14, 18, 27, 67, 104, 133, 147islmodd 20129 1 ((𝑀 ∈ LMod ∧ 𝑆 ∈ CRing) → 𝐴 ∈ LMod)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 396  w3a 1086   = wceq 1539  wcel 2106  Vcvv 3432  {csn 4561  cmpt 5157   × cxp 5587  wf 6429  cfv 6433  (class class class)co 7275  f cof 7531  Basecbs 16912  +gcplusg 16962  .rcmulr 16963  Scalarcsca 16965   ·𝑠 cvsca 16966  Grpcgrp 18577  1rcur 19737  Ringcrg 19783  CRingccrg 19784  LModclmod 20123   LMHom clmhm 20281  MEndocmend 41000
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 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-10 2137  ax-11 2154  ax-12 2171  ax-ext 2709  ax-rep 5209  ax-sep 5223  ax-nul 5230  ax-pow 5288  ax-pr 5352  ax-un 7588  ax-cnex 10927  ax-resscn 10928  ax-1cn 10929  ax-icn 10930  ax-addcl 10931  ax-addrcl 10932  ax-mulcl 10933  ax-mulrcl 10934  ax-mulcom 10935  ax-addass 10936  ax-mulass 10937  ax-distr 10938  ax-i2m1 10939  ax-1ne0 10940  ax-1rid 10941  ax-rnegex 10942  ax-rrecex 10943  ax-cnre 10944  ax-pre-lttri 10945  ax-pre-lttrn 10946  ax-pre-ltadd 10947  ax-pre-mulgt0 10948
This theorem depends on definitions:  df-bi 206  df-an 397  df-or 845  df-3or 1087  df-3an 1088  df-tru 1542  df-fal 1552  df-ex 1783  df-nf 1787  df-sb 2068  df-mo 2540  df-eu 2569  df-clab 2716  df-cleq 2730  df-clel 2816  df-nfc 2889  df-ne 2944  df-nel 3050  df-ral 3069  df-rex 3070  df-rmo 3071  df-reu 3072  df-rab 3073  df-v 3434  df-sbc 3717  df-csb 3833  df-dif 3890  df-un 3892  df-in 3894  df-ss 3904  df-pss 3906  df-nul 4257  df-if 4460  df-pw 4535  df-sn 4562  df-pr 4564  df-tp 4566  df-op 4568  df-uni 4840  df-iun 4926  df-br 5075  df-opab 5137  df-mpt 5158  df-tr 5192  df-id 5489  df-eprel 5495  df-po 5503  df-so 5504  df-fr 5544  df-we 5546  df-xp 5595  df-rel 5596  df-cnv 5597  df-co 5598  df-dm 5599  df-rn 5600  df-res 5601  df-ima 5602  df-pred 6202  df-ord 6269  df-on 6270  df-lim 6271  df-suc 6272  df-iota 6391  df-fun 6435  df-fn 6436  df-f 6437  df-f1 6438  df-fo 6439  df-f1o 6440  df-fv 6441  df-riota 7232  df-ov 7278  df-oprab 7279  df-mpo 7280  df-of 7533  df-om 7713  df-1st 7831  df-2nd 7832  df-frecs 8097  df-wrecs 8128  df-recs 8202  df-rdg 8241  df-1o 8297  df-er 8498  df-map 8617  df-en 8734  df-dom 8735  df-sdom 8736  df-fin 8737  df-pnf 11011  df-mnf 11012  df-xr 11013  df-ltxr 11014  df-le 11015  df-sub 11207  df-neg 11208  df-nn 11974  df-2 12036  df-3 12037  df-4 12038  df-5 12039  df-6 12040  df-n0 12234  df-z 12320  df-uz 12583  df-fz 13240  df-struct 16848  df-sets 16865  df-slot 16883  df-ndx 16895  df-base 16913  df-plusg 16975  df-mulr 16976  df-sca 16978  df-vsca 16979  df-0g 17152  df-mgm 18326  df-sgrp 18375  df-mnd 18386  df-mhm 18430  df-grp 18580  df-minusg 18581  df-ghm 18832  df-cmn 19388  df-abl 19389  df-mgp 19721  df-ur 19738  df-ring 19785  df-cring 19786  df-lmod 20125  df-lmhm 20284  df-mend 41001
This theorem is referenced by:  mendassa  41019
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