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Theorem mendlmod 40590
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 40581 . . 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 40586 . . 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 19428 . . 3 (𝑆 ∈ CRing → 𝑆 ∈ Ring)
1413adantl 485 . 2 ((𝑀 ∈ LMod ∧ 𝑆 ∈ CRing) → 𝑆 ∈ Ring)
151mendring 40589 . . . 4 (𝑀 ∈ LMod → 𝐴 ∈ Ring)
1615adantr 484 . . 3 ((𝑀 ∈ LMod ∧ 𝑆 ∈ CRing) → 𝐴 ∈ Ring)
17 ringgrp 19421 . . 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 40588 . . . 4 ((𝑥 ∈ (Base‘𝑆) ∧ 𝑦 ∈ (𝑀 LMHom 𝑀)) → (𝑥( ·𝑠𝐴)𝑦) = (((Base‘𝑀) × {𝑥}) ∘f ( ·𝑠𝑀)𝑦))
24233adant1 1131 . . 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 19936 . . . 4 ((𝑆 ∈ CRing ∧ 𝑥 ∈ (Base‘𝑆) ∧ 𝑦 ∈ (𝑀 LMHom 𝑀)) → (((Base‘𝑀) × {𝑥}) ∘f ( ·𝑠𝑀)𝑦) ∈ (𝑀 LMHom 𝑀))
26253adant1l 1177 . . 3 (((𝑀 ∈ LMod ∧ 𝑆 ∈ CRing) ∧ 𝑥 ∈ (Base‘𝑆) ∧ 𝑦 ∈ (𝑀 LMHom 𝑀)) → (((Base‘𝑀) × {𝑥}) ∘f ( ·𝑠𝑀)𝑦) ∈ (𝑀 LMHom 𝑀))
2724, 26eqeltrd 2833 . 2 (((𝑀 ∈ LMod ∧ 𝑆 ∈ CRing) ∧ 𝑥 ∈ (Base‘𝑆) ∧ 𝑦 ∈ (𝑀 LMHom 𝑀)) → (𝑥( ·𝑠𝐴)𝑦) ∈ (𝑀 LMHom 𝑀))
28 simpr2 1196 . . . . 5 (((𝑀 ∈ LMod ∧ 𝑆 ∈ CRing) ∧ (𝑥 ∈ (Base‘𝑆) ∧ 𝑦 ∈ (𝑀 LMHom 𝑀) ∧ 𝑧 ∈ (𝑀 LMHom 𝑀))) → 𝑦 ∈ (𝑀 LMHom 𝑀))
29 simpr3 1197 . . . . 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 40583 . . . . 5 ((𝑦 ∈ (𝑀 LMHom 𝑀) ∧ 𝑧 ∈ (𝑀 LMHom 𝑀)) → (𝑦(+g𝐴)𝑧) = (𝑦f (+g𝑀)𝑧))
3328, 29, 32syl2anc 587 . . . 4 (((𝑀 ∈ LMod ∧ 𝑆 ∈ CRing) ∧ (𝑥 ∈ (Base‘𝑆) ∧ 𝑦 ∈ (𝑀 LMHom 𝑀) ∧ 𝑧 ∈ (𝑀 LMHom 𝑀))) → (𝑦(+g𝐴)𝑧) = (𝑦f (+g𝑀)𝑧))
3433oveq2d 7186 . . 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 1195 . . . 4 (((𝑀 ∈ LMod ∧ 𝑆 ∈ CRing) ∧ (𝑥 ∈ (Base‘𝑆) ∧ 𝑦 ∈ (𝑀 LMHom 𝑀) ∧ 𝑧 ∈ (𝑀 LMHom 𝑀))) → 𝑥 ∈ (Base‘𝑆))
3618adantr 484 . . . . 5 (((𝑀 ∈ LMod ∧ 𝑆 ∈ CRing) ∧ (𝑥 ∈ (Base‘𝑆) ∧ 𝑦 ∈ (𝑀 LMHom 𝑀) ∧ 𝑧 ∈ (𝑀 LMHom 𝑀))) → 𝐴 ∈ Grp)
372, 31grpcl 18227 . . . . 5 ((𝐴 ∈ Grp ∧ 𝑦 ∈ (𝑀 LMHom 𝑀) ∧ 𝑧 ∈ (𝑀 LMHom 𝑀)) → (𝑦(+g𝐴)𝑧) ∈ (𝑀 LMHom 𝑀))
3836, 28, 29, 37syl3anc 1372 . . . 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 40588 . . . 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 587 . . 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 587 . . . . 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 40588 . . . . . 6 ((𝑥 ∈ (Base‘𝑆) ∧ 𝑧 ∈ (𝑀 LMHom 𝑀)) → (𝑥( ·𝑠𝐴)𝑧) = (((Base‘𝑀) × {𝑥}) ∘f ( ·𝑠𝑀)𝑧))
4335, 29, 42syl2anc 587 . . . . 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 7188 . . . 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 1185 . . . . 5 (((𝑀 ∈ LMod ∧ 𝑆 ∈ CRing) ∧ (𝑥 ∈ (Base‘𝑆) ∧ 𝑦 ∈ (𝑀 LMHom 𝑀) ∧ 𝑧 ∈ (𝑀 LMHom 𝑀))) → (𝑥( ·𝑠𝐴)𝑦) ∈ (𝑀 LMHom 𝑀))
46 eleq1w 2815 . . . . . . . . 9 (𝑦 = 𝑧 → (𝑦 ∈ (𝑀 LMHom 𝑀) ↔ 𝑧 ∈ (𝑀 LMHom 𝑀)))
47463anbi3d 1443 . . . . . . . 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 7178 . . . . . . . . 9 (𝑦 = 𝑧 → (𝑥( ·𝑠𝐴)𝑦) = (𝑥( ·𝑠𝐴)𝑧))
4948eleq1d 2817 . . . . . . . 8 (𝑦 = 𝑧 → ((𝑥( ·𝑠𝐴)𝑦) ∈ (𝑀 LMHom 𝑀) ↔ (𝑥( ·𝑠𝐴)𝑧) ∈ (𝑀 LMHom 𝑀)))
5047, 49imbi12d 348 . . . . . . 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 2010 . . . . . 6 (((𝑀 ∈ LMod ∧ 𝑆 ∈ CRing) ∧ 𝑥 ∈ (Base‘𝑆) ∧ 𝑧 ∈ (𝑀 LMHom 𝑀)) → (𝑥( ·𝑠𝐴)𝑧) ∈ (𝑀 LMHom 𝑀))
52513adant3r2 1184 . . . . 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 40583 . . . . 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 587 . . . 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 6689 . . . . 5 (((𝑀 ∈ LMod ∧ 𝑆 ∈ CRing) ∧ (𝑥 ∈ (Base‘𝑆) ∧ 𝑦 ∈ (𝑀 LMHom 𝑀) ∧ 𝑧 ∈ (𝑀 LMHom 𝑀))) → (Base‘𝑀) ∈ V)
56 fconst6g 6567 . . . . . 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 19925 . . . . . 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 19925 . . . . . 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 767 . . . . . 6 (((𝑀 ∈ LMod ∧ 𝑆 ∈ CRing) ∧ (𝑥 ∈ (Base‘𝑆) ∧ 𝑦 ∈ (𝑀 LMHom 𝑀) ∧ 𝑧 ∈ (𝑀 LMHom 𝑀))) → 𝑀 ∈ LMod)
6321, 30, 5, 19, 20lmodvsdi 19776 . . . . . 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 583 . . . . 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 7463 . . . 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 2783 . . 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 2783 . 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 6689 . . . 4 (((𝑀 ∈ LMod ∧ 𝑆 ∈ CRing) ∧ (𝑥 ∈ (Base‘𝑆) ∧ 𝑦 ∈ (Base‘𝑆) ∧ 𝑧 ∈ (𝑀 LMHom 𝑀))) → (Base‘𝑀) ∈ V)
69 simpr3 1197 . . . . 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 1195 . . . . 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 1196 . . . . 5 (((𝑀 ∈ LMod ∧ 𝑆 ∈ CRing) ∧ (𝑥 ∈ (Base‘𝑆) ∧ 𝑦 ∈ (Base‘𝑆) ∧ 𝑧 ∈ (𝑀 LMHom 𝑀))) → 𝑦 ∈ (Base‘𝑆))
74 fconst6g 6567 . . . . 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 767 . . . . 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 19777 . . . . 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 583 . . . 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 7464 . . 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 484 . . . . . 6 (((𝑀 ∈ LMod ∧ 𝑆 ∈ CRing) ∧ (𝑥 ∈ (Base‘𝑆) ∧ 𝑦 ∈ (Base‘𝑆) ∧ 𝑧 ∈ (𝑀 LMHom 𝑀))) → 𝑆 ∈ Ring)
8220, 77ringacl 19450 . . . . . 6 ((𝑆 ∈ Ring ∧ 𝑥 ∈ (Base‘𝑆) ∧ 𝑦 ∈ (Base‘𝑆)) → (𝑥(+g𝑆)𝑦) ∈ (Base‘𝑆))
8381, 71, 73, 82syl3anc 1372 . . . . 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 40588 . . . . 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 587 . . . 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 7452 . . . . 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 7185 . . . 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 2776 . . 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 1184 . . . . 5 (((𝑀 ∈ LMod ∧ 𝑆 ∈ CRing) ∧ (𝑥 ∈ (Base‘𝑆) ∧ 𝑦 ∈ (Base‘𝑆) ∧ 𝑧 ∈ (𝑀 LMHom 𝑀))) → (𝑥( ·𝑠𝐴)𝑧) ∈ (𝑀 LMHom 𝑀))
90 eleq1w 2815 . . . . . . . . 9 (𝑥 = 𝑦 → (𝑥 ∈ (Base‘𝑆) ↔ 𝑦 ∈ (Base‘𝑆)))
91903anbi2d 1442 . . . . . . . 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 7177 . . . . . . . . 9 (𝑥 = 𝑦 → (𝑥( ·𝑠𝐴)𝑧) = (𝑦( ·𝑠𝐴)𝑧))
9392eleq1d 2817 . . . . . . . 8 (𝑥 = 𝑦 → ((𝑥( ·𝑠𝐴)𝑧) ∈ (𝑀 LMHom 𝑀) ↔ (𝑦( ·𝑠𝐴)𝑧) ∈ (𝑀 LMHom 𝑀)))
9491, 93imbi12d 348 . . . . . . 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 2010 . . . . . 6 (((𝑀 ∈ LMod ∧ 𝑆 ∈ CRing) ∧ 𝑦 ∈ (Base‘𝑆) ∧ 𝑧 ∈ (𝑀 LMHom 𝑀)) → (𝑦( ·𝑠𝐴)𝑧) ∈ (𝑀 LMHom 𝑀))
96953adant3r1 1183 . . . . 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 40583 . . . . 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 587 . . . 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 587 . . . . 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 40588 . . . . . 6 ((𝑦 ∈ (Base‘𝑆) ∧ 𝑧 ∈ (𝑀 LMHom 𝑀)) → (𝑦( ·𝑠𝐴)𝑧) = (((Base‘𝑀) × {𝑦}) ∘f ( ·𝑠𝑀)𝑧))
10173, 69, 100syl2anc 587 . . . . 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 7188 . . . 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 2773 . . 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 2783 . 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 7205 . . . 4 ((((𝑀 ∈ LMod ∧ 𝑆 ∈ CRing) ∧ (𝑥 ∈ (Base‘𝑆) ∧ 𝑦 ∈ (Base‘𝑆) ∧ 𝑧 ∈ (𝑀 LMHom 𝑀))) ∧ 𝑘 ∈ (Base‘𝑀)) → (𝑥(.r𝑆)𝑦) ∈ V)
10670ffvelrnda 6861 . . . 4 ((((𝑀 ∈ LMod ∧ 𝑆 ∈ CRing) ∧ (𝑥 ∈ (Base‘𝑆) ∧ 𝑦 ∈ (Base‘𝑆) ∧ 𝑧 ∈ (𝑀 LMHom 𝑀))) ∧ 𝑘 ∈ (Base‘𝑀)) → (𝑧𝑘) ∈ (Base‘𝑀))
107 fconstmpt 5585 . . . . 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 6737 . . . 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 7444 . . 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 19433 . . . . 5 ((𝑆 ∈ Ring ∧ 𝑥 ∈ (Base‘𝑆) ∧ 𝑦 ∈ (Base‘𝑆)) → (𝑥(.r𝑆)𝑦) ∈ (Base‘𝑆))
11381, 71, 73, 112syl3anc 1372 . . . 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 40588 . . . 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 587 . . 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 484 . . . . 5 ((((𝑀 ∈ LMod ∧ 𝑆 ∈ CRing) ∧ (𝑥 ∈ (Base‘𝑆) ∧ 𝑦 ∈ (Base‘𝑆) ∧ 𝑧 ∈ (𝑀 LMHom 𝑀))) ∧ 𝑘 ∈ (Base‘𝑀)) → 𝑥 ∈ (Base‘𝑆))
117 ovexd 7205 . . . . 5 ((((𝑀 ∈ LMod ∧ 𝑆 ∈ CRing) ∧ (𝑥 ∈ (Base‘𝑆) ∧ 𝑦 ∈ (Base‘𝑆) ∧ 𝑧 ∈ (𝑀 LMHom 𝑀))) ∧ 𝑘 ∈ (Base‘𝑀)) → (𝑦( ·𝑠𝑀)(𝑧𝑘)) ∈ V)
118 fconstmpt 5585 . . . . . 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 1217 . . . . . . 7 ((((𝑀 ∈ LMod ∧ 𝑆 ∈ CRing) ∧ (𝑥 ∈ (Base‘𝑆) ∧ 𝑦 ∈ (Base‘𝑆) ∧ 𝑧 ∈ (𝑀 LMHom 𝑀))) ∧ 𝑘 ∈ (Base‘𝑀)) → 𝑦 ∈ (Base‘𝑆))
121 fconstmpt 5585 . . . . . . . 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 7444 . . . . . 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 2773 . . . . 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 7444 . . . 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 40588 . . . . 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 587 . . . 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 484 . . . . . 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 19778 . . . . . 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 1373 . . . . 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 5125 . . . 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 2783 . . 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 2783 . 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 484 . . . . 5 (((𝑀 ∈ LMod ∧ 𝑆 ∈ CRing) ∧ 𝑥 ∈ (𝑀 LMHom 𝑀)) → 𝑆 ∈ Ring)
135 eqid 2738 . . . . . 6 (1r𝑆) = (1r𝑆)
13620, 135ringidcl 19440 . . . . 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 40588 . . . 4 (((1r𝑆) ∈ (Base‘𝑆) ∧ 𝑥 ∈ (𝑀 LMHom 𝑀)) → ((1r𝑆)( ·𝑠𝐴)𝑥) = (((Base‘𝑀) × {(1r𝑆)}) ∘f ( ·𝑠𝑀)𝑥))
139137, 138sylancom 591 . . 3 (((𝑀 ∈ LMod ∧ 𝑆 ∈ CRing) ∧ 𝑥 ∈ (𝑀 LMHom 𝑀)) → ((1r𝑆)( ·𝑠𝐴)𝑥) = (((Base‘𝑀) × {(1r𝑆)}) ∘f ( ·𝑠𝑀)𝑥))
140 fvexd 6689 . . . 4 (((𝑀 ∈ LMod ∧ 𝑆 ∈ CRing) ∧ 𝑥 ∈ (𝑀 LMHom 𝑀)) → (Base‘𝑀) ∈ V)
14121, 21lmhmf 19925 . . . . 5 (𝑥 ∈ (𝑀 LMHom 𝑀) → 𝑥:(Base‘𝑀)⟶(Base‘𝑀))
142141adantl 485 . . . 4 (((𝑀 ∈ LMod ∧ 𝑆 ∈ CRing) ∧ 𝑥 ∈ (𝑀 LMHom 𝑀)) → 𝑥:(Base‘𝑀)⟶(Base‘𝑀))
143 simpll 767 . . . . 5 (((𝑀 ∈ LMod ∧ 𝑆 ∈ CRing) ∧ 𝑥 ∈ (𝑀 LMHom 𝑀)) → 𝑀 ∈ LMod)
14421, 5, 19, 135lmodvs1 19781 . . . . 5 ((𝑀 ∈ LMod ∧ 𝑦 ∈ (Base‘𝑀)) → ((1r𝑆)( ·𝑠𝑀)𝑦) = 𝑦)
145143, 144sylan 583 . . . 4 ((((𝑀 ∈ LMod ∧ 𝑆 ∈ CRing) ∧ 𝑥 ∈ (𝑀 LMHom 𝑀)) ∧ 𝑦 ∈ (Base‘𝑀)) → ((1r𝑆)( ·𝑠𝑀)𝑦) = 𝑦)
146140, 142, 137, 145caofid0l 7455 . . 3 (((𝑀 ∈ LMod ∧ 𝑆 ∈ CRing) ∧ 𝑥 ∈ (𝑀 LMHom 𝑀)) → (((Base‘𝑀) × {(1r𝑆)}) ∘f ( ·𝑠𝑀)𝑥) = 𝑥)
147139, 146eqtrd 2773 . 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 19759 1 ((𝑀 ∈ LMod ∧ 𝑆 ∈ CRing) → 𝐴 ∈ LMod)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 399  w3a 1088   = wceq 1542  wcel 2114  Vcvv 3398  {csn 4516  cmpt 5110   × cxp 5523  wf 6335  cfv 6339  (class class class)co 7170  f cof 7423  Basecbs 16586  +gcplusg 16668  .rcmulr 16669  Scalarcsca 16671   ·𝑠 cvsca 16672  Grpcgrp 18219  1rcur 19370  Ringcrg 19416  CRingccrg 19417  LModclmod 19753   LMHom clmhm 19910  MEndocmend 40572
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1802  ax-4 1816  ax-5 1917  ax-6 1975  ax-7 2020  ax-8 2116  ax-9 2124  ax-10 2145  ax-11 2162  ax-12 2179  ax-ext 2710  ax-rep 5154  ax-sep 5167  ax-nul 5174  ax-pow 5232  ax-pr 5296  ax-un 7479  ax-cnex 10671  ax-resscn 10672  ax-1cn 10673  ax-icn 10674  ax-addcl 10675  ax-addrcl 10676  ax-mulcl 10677  ax-mulrcl 10678  ax-mulcom 10679  ax-addass 10680  ax-mulass 10681  ax-distr 10682  ax-i2m1 10683  ax-1ne0 10684  ax-1rid 10685  ax-rnegex 10686  ax-rrecex 10687  ax-cnre 10688  ax-pre-lttri 10689  ax-pre-lttrn 10690  ax-pre-ltadd 10691  ax-pre-mulgt0 10692
This theorem depends on definitions:  df-bi 210  df-an 400  df-or 847  df-3or 1089  df-3an 1090  df-tru 1545  df-fal 1555  df-ex 1787  df-nf 1791  df-sb 2075  df-mo 2540  df-eu 2570  df-clab 2717  df-cleq 2730  df-clel 2811  df-nfc 2881  df-ne 2935  df-nel 3039  df-ral 3058  df-rex 3059  df-reu 3060  df-rmo 3061  df-rab 3062  df-v 3400  df-sbc 3681  df-csb 3791  df-dif 3846  df-un 3848  df-in 3850  df-ss 3860  df-pss 3862  df-nul 4212  df-if 4415  df-pw 4490  df-sn 4517  df-pr 4519  df-tp 4521  df-op 4523  df-uni 4797  df-iun 4883  df-br 5031  df-opab 5093  df-mpt 5111  df-tr 5137  df-id 5429  df-eprel 5434  df-po 5442  df-so 5443  df-fr 5483  df-we 5485  df-xp 5531  df-rel 5532  df-cnv 5533  df-co 5534  df-dm 5535  df-rn 5536  df-res 5537  df-ima 5538  df-pred 6129  df-ord 6175  df-on 6176  df-lim 6177  df-suc 6178  df-iota 6297  df-fun 6341  df-fn 6342  df-f 6343  df-f1 6344  df-fo 6345  df-f1o 6346  df-fv 6347  df-riota 7127  df-ov 7173  df-oprab 7174  df-mpo 7175  df-of 7425  df-om 7600  df-1st 7714  df-2nd 7715  df-wrecs 7976  df-recs 8037  df-rdg 8075  df-1o 8131  df-er 8320  df-map 8439  df-en 8556  df-dom 8557  df-sdom 8558  df-fin 8559  df-pnf 10755  df-mnf 10756  df-xr 10757  df-ltxr 10758  df-le 10759  df-sub 10950  df-neg 10951  df-nn 11717  df-2 11779  df-3 11780  df-4 11781  df-5 11782  df-6 11783  df-n0 11977  df-z 12063  df-uz 12325  df-fz 12982  df-struct 16588  df-ndx 16589  df-slot 16590  df-base 16592  df-sets 16593  df-plusg 16681  df-mulr 16682  df-sca 16684  df-vsca 16685  df-0g 16818  df-mgm 17968  df-sgrp 18017  df-mnd 18028  df-mhm 18072  df-grp 18222  df-minusg 18223  df-ghm 18474  df-cmn 19026  df-abl 19027  df-mgp 19359  df-ur 19371  df-ring 19418  df-cring 19419  df-lmod 19755  df-lmhm 19913  df-mend 40573
This theorem is referenced by:  mendassa  40591
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