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Theorem mendlmod 36782
 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 36773 . . 3 (𝑀 LMHom 𝑀) = (Base‘𝐴)
32a1i 11 . 2 ((𝑀 ∈ LMod ∧ 𝑆 ∈ CRing) → (𝑀 LMHom 𝑀) = (Base‘𝐴))
4 eqidd 2611 . 2 ((𝑀 ∈ LMod ∧ 𝑆 ∈ CRing) → (+g𝐴) = (+g𝐴))
5 mendassa.s . . . 4 𝑆 = (Scalar‘𝑀)
61, 5mendsca 36778 . . 3 𝑆 = (Scalar‘𝐴)
76a1i 11 . 2 ((𝑀 ∈ LMod ∧ 𝑆 ∈ CRing) → 𝑆 = (Scalar‘𝐴))
8 eqidd 2611 . 2 ((𝑀 ∈ LMod ∧ 𝑆 ∈ CRing) → ( ·𝑠𝐴) = ( ·𝑠𝐴))
9 eqidd 2611 . 2 ((𝑀 ∈ LMod ∧ 𝑆 ∈ CRing) → (Base‘𝑆) = (Base‘𝑆))
10 eqidd 2611 . 2 ((𝑀 ∈ LMod ∧ 𝑆 ∈ CRing) → (+g𝑆) = (+g𝑆))
11 eqidd 2611 . 2 ((𝑀 ∈ LMod ∧ 𝑆 ∈ CRing) → (.r𝑆) = (.r𝑆))
12 eqidd 2611 . 2 ((𝑀 ∈ LMod ∧ 𝑆 ∈ CRing) → (1r𝑆) = (1r𝑆))
13 crngring 18381 . . 3 (𝑆 ∈ CRing → 𝑆 ∈ Ring)
1413adantl 481 . 2 ((𝑀 ∈ LMod ∧ 𝑆 ∈ CRing) → 𝑆 ∈ Ring)
151mendring 36781 . . . 4 (𝑀 ∈ LMod → 𝐴 ∈ Ring)
1615adantr 480 . . 3 ((𝑀 ∈ LMod ∧ 𝑆 ∈ CRing) → 𝐴 ∈ Ring)
17 ringgrp 18375 . . 3 (𝐴 ∈ Ring → 𝐴 ∈ Grp)
1816, 17syl 17 . 2 ((𝑀 ∈ LMod ∧ 𝑆 ∈ CRing) → 𝐴 ∈ Grp)
19 eqid 2610 . . . . 5 ( ·𝑠𝑀) = ( ·𝑠𝑀)
20 eqid 2610 . . . . 5 (Base‘𝑆) = (Base‘𝑆)
21 eqid 2610 . . . . 5 (Base‘𝑀) = (Base‘𝑀)
22 eqid 2610 . . . . 5 ( ·𝑠𝐴) = ( ·𝑠𝐴)
231, 19, 2, 5, 20, 21, 22mendvsca 36780 . . . 4 ((𝑥 ∈ (Base‘𝑆) ∧ 𝑦 ∈ (𝑀 LMHom 𝑀)) → (𝑥( ·𝑠𝐴)𝑦) = (((Base‘𝑀) × {𝑥}) ∘𝑓 ( ·𝑠𝑀)𝑦))
24233adant1 1072 . . 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 18866 . . . 4 ((𝑆 ∈ CRing ∧ 𝑥 ∈ (Base‘𝑆) ∧ 𝑦 ∈ (𝑀 LMHom 𝑀)) → (((Base‘𝑀) × {𝑥}) ∘𝑓 ( ·𝑠𝑀)𝑦) ∈ (𝑀 LMHom 𝑀))
26253adant1l 1310 . . 3 (((𝑀 ∈ LMod ∧ 𝑆 ∈ CRing) ∧ 𝑥 ∈ (Base‘𝑆) ∧ 𝑦 ∈ (𝑀 LMHom 𝑀)) → (((Base‘𝑀) × {𝑥}) ∘𝑓 ( ·𝑠𝑀)𝑦) ∈ (𝑀 LMHom 𝑀))
2724, 26eqeltrd 2688 . 2 (((𝑀 ∈ LMod ∧ 𝑆 ∈ CRing) ∧ 𝑥 ∈ (Base‘𝑆) ∧ 𝑦 ∈ (𝑀 LMHom 𝑀)) → (𝑥( ·𝑠𝐴)𝑦) ∈ (𝑀 LMHom 𝑀))
28 simpr2 1061 . . . . 5 (((𝑀 ∈ LMod ∧ 𝑆 ∈ CRing) ∧ (𝑥 ∈ (Base‘𝑆) ∧ 𝑦 ∈ (𝑀 LMHom 𝑀) ∧ 𝑧 ∈ (𝑀 LMHom 𝑀))) → 𝑦 ∈ (𝑀 LMHom 𝑀))
29 simpr3 1062 . . . . 5 (((𝑀 ∈ LMod ∧ 𝑆 ∈ CRing) ∧ (𝑥 ∈ (Base‘𝑆) ∧ 𝑦 ∈ (𝑀 LMHom 𝑀) ∧ 𝑧 ∈ (𝑀 LMHom 𝑀))) → 𝑧 ∈ (𝑀 LMHom 𝑀))
30 eqid 2610 . . . . . 6 (+g𝑀) = (+g𝑀)
31 eqid 2610 . . . . . 6 (+g𝐴) = (+g𝐴)
321, 2, 30, 31mendplusg 36775 . . . . 5 ((𝑦 ∈ (𝑀 LMHom 𝑀) ∧ 𝑧 ∈ (𝑀 LMHom 𝑀)) → (𝑦(+g𝐴)𝑧) = (𝑦𝑓 (+g𝑀)𝑧))
3328, 29, 32syl2anc 691 . . . 4 (((𝑀 ∈ LMod ∧ 𝑆 ∈ CRing) ∧ (𝑥 ∈ (Base‘𝑆) ∧ 𝑦 ∈ (𝑀 LMHom 𝑀) ∧ 𝑧 ∈ (𝑀 LMHom 𝑀))) → (𝑦(+g𝐴)𝑧) = (𝑦𝑓 (+g𝑀)𝑧))
3433oveq2d 6565 . . 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 1060 . . . 4 (((𝑀 ∈ LMod ∧ 𝑆 ∈ CRing) ∧ (𝑥 ∈ (Base‘𝑆) ∧ 𝑦 ∈ (𝑀 LMHom 𝑀) ∧ 𝑧 ∈ (𝑀 LMHom 𝑀))) → 𝑥 ∈ (Base‘𝑆))
3618adantr 480 . . . . 5 (((𝑀 ∈ LMod ∧ 𝑆 ∈ CRing) ∧ (𝑥 ∈ (Base‘𝑆) ∧ 𝑦 ∈ (𝑀 LMHom 𝑀) ∧ 𝑧 ∈ (𝑀 LMHom 𝑀))) → 𝐴 ∈ Grp)
372, 31grpcl 17253 . . . . 5 ((𝐴 ∈ Grp ∧ 𝑦 ∈ (𝑀 LMHom 𝑀) ∧ 𝑧 ∈ (𝑀 LMHom 𝑀)) → (𝑦(+g𝐴)𝑧) ∈ (𝑀 LMHom 𝑀))
3836, 28, 29, 37syl3anc 1318 . . . 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 36780 . . . 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 691 . . 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 691 . . . . 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 36780 . . . . . 6 ((𝑥 ∈ (Base‘𝑆) ∧ 𝑧 ∈ (𝑀 LMHom 𝑀)) → (𝑥( ·𝑠𝐴)𝑧) = (((Base‘𝑀) × {𝑥}) ∘𝑓 ( ·𝑠𝑀)𝑧))
4335, 29, 42syl2anc 691 . . . . 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 6567 . . . 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 1268 . . . . 5 (((𝑀 ∈ LMod ∧ 𝑆 ∈ CRing) ∧ (𝑥 ∈ (Base‘𝑆) ∧ 𝑦 ∈ (𝑀 LMHom 𝑀) ∧ 𝑧 ∈ (𝑀 LMHom 𝑀))) → (𝑥( ·𝑠𝐴)𝑦) ∈ (𝑀 LMHom 𝑀))
46 eleq1 2676 . . . . . . . . 9 (𝑦 = 𝑧 → (𝑦 ∈ (𝑀 LMHom 𝑀) ↔ 𝑧 ∈ (𝑀 LMHom 𝑀)))
47463anbi3d 1397 . . . . . . . 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 6557 . . . . . . . . 9 (𝑦 = 𝑧 → (𝑥( ·𝑠𝐴)𝑦) = (𝑥( ·𝑠𝐴)𝑧))
4948eleq1d 2672 . . . . . . . 8 (𝑦 = 𝑧 → ((𝑥( ·𝑠𝐴)𝑦) ∈ (𝑀 LMHom 𝑀) ↔ (𝑥( ·𝑠𝐴)𝑧) ∈ (𝑀 LMHom 𝑀)))
5047, 49imbi12d 333 . . . . . . 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, 27chvarv 2251 . . . . . 6 (((𝑀 ∈ LMod ∧ 𝑆 ∈ CRing) ∧ 𝑥 ∈ (Base‘𝑆) ∧ 𝑧 ∈ (𝑀 LMHom 𝑀)) → (𝑥( ·𝑠𝐴)𝑧) ∈ (𝑀 LMHom 𝑀))
52513adant3r2 1267 . . . . 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 36775 . . . . 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 691 . . . 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 fvex 6113 . . . . . 6 (Base‘𝑀) ∈ V
5655a1i 11 . . . . 5 (((𝑀 ∈ LMod ∧ 𝑆 ∈ CRing) ∧ (𝑥 ∈ (Base‘𝑆) ∧ 𝑦 ∈ (𝑀 LMHom 𝑀) ∧ 𝑧 ∈ (𝑀 LMHom 𝑀))) → (Base‘𝑀) ∈ V)
57 fconst6g 6007 . . . . . 6 (𝑥 ∈ (Base‘𝑆) → ((Base‘𝑀) × {𝑥}):(Base‘𝑀)⟶(Base‘𝑆))
5835, 57syl 17 . . . . 5 (((𝑀 ∈ LMod ∧ 𝑆 ∈ CRing) ∧ (𝑥 ∈ (Base‘𝑆) ∧ 𝑦 ∈ (𝑀 LMHom 𝑀) ∧ 𝑧 ∈ (𝑀 LMHom 𝑀))) → ((Base‘𝑀) × {𝑥}):(Base‘𝑀)⟶(Base‘𝑆))
5921, 21lmhmf 18855 . . . . . 6 (𝑦 ∈ (𝑀 LMHom 𝑀) → 𝑦:(Base‘𝑀)⟶(Base‘𝑀))
6028, 59syl 17 . . . . 5 (((𝑀 ∈ LMod ∧ 𝑆 ∈ CRing) ∧ (𝑥 ∈ (Base‘𝑆) ∧ 𝑦 ∈ (𝑀 LMHom 𝑀) ∧ 𝑧 ∈ (𝑀 LMHom 𝑀))) → 𝑦:(Base‘𝑀)⟶(Base‘𝑀))
6121, 21lmhmf 18855 . . . . . 6 (𝑧 ∈ (𝑀 LMHom 𝑀) → 𝑧:(Base‘𝑀)⟶(Base‘𝑀))
6229, 61syl 17 . . . . 5 (((𝑀 ∈ LMod ∧ 𝑆 ∈ CRing) ∧ (𝑥 ∈ (Base‘𝑆) ∧ 𝑦 ∈ (𝑀 LMHom 𝑀) ∧ 𝑧 ∈ (𝑀 LMHom 𝑀))) → 𝑧:(Base‘𝑀)⟶(Base‘𝑀))
63 simpll 786 . . . . . 6 (((𝑀 ∈ LMod ∧ 𝑆 ∈ CRing) ∧ (𝑥 ∈ (Base‘𝑆) ∧ 𝑦 ∈ (𝑀 LMHom 𝑀) ∧ 𝑧 ∈ (𝑀 LMHom 𝑀))) → 𝑀 ∈ LMod)
6421, 30, 5, 19, 20lmodvsdi 18709 . . . . . 6 ((𝑀 ∈ LMod ∧ (𝑤 ∈ (Base‘𝑆) ∧ 𝑣 ∈ (Base‘𝑀) ∧ 𝑢 ∈ (Base‘𝑀))) → (𝑤( ·𝑠𝑀)(𝑣(+g𝑀)𝑢)) = ((𝑤( ·𝑠𝑀)𝑣)(+g𝑀)(𝑤( ·𝑠𝑀)𝑢)))
6563, 64sylan 487 . . . . 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)))
6656, 58, 60, 62, 65caofdi 6831 . . . 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)))
6744, 54, 663eqtr4d 2654 . . 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)))
6834, 40, 673eqtr4d 2654 . 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)))
6955a1i 11 . . . 4 (((𝑀 ∈ LMod ∧ 𝑆 ∈ CRing) ∧ (𝑥 ∈ (Base‘𝑆) ∧ 𝑦 ∈ (Base‘𝑆) ∧ 𝑧 ∈ (𝑀 LMHom 𝑀))) → (Base‘𝑀) ∈ V)
70 simpr3 1062 . . . . 5 (((𝑀 ∈ LMod ∧ 𝑆 ∈ CRing) ∧ (𝑥 ∈ (Base‘𝑆) ∧ 𝑦 ∈ (Base‘𝑆) ∧ 𝑧 ∈ (𝑀 LMHom 𝑀))) → 𝑧 ∈ (𝑀 LMHom 𝑀))
7170, 61syl 17 . . . 4 (((𝑀 ∈ LMod ∧ 𝑆 ∈ CRing) ∧ (𝑥 ∈ (Base‘𝑆) ∧ 𝑦 ∈ (Base‘𝑆) ∧ 𝑧 ∈ (𝑀 LMHom 𝑀))) → 𝑧:(Base‘𝑀)⟶(Base‘𝑀))
72 simpr1 1060 . . . . 5 (((𝑀 ∈ LMod ∧ 𝑆 ∈ CRing) ∧ (𝑥 ∈ (Base‘𝑆) ∧ 𝑦 ∈ (Base‘𝑆) ∧ 𝑧 ∈ (𝑀 LMHom 𝑀))) → 𝑥 ∈ (Base‘𝑆))
7372, 57syl 17 . . . 4 (((𝑀 ∈ LMod ∧ 𝑆 ∈ CRing) ∧ (𝑥 ∈ (Base‘𝑆) ∧ 𝑦 ∈ (Base‘𝑆) ∧ 𝑧 ∈ (𝑀 LMHom 𝑀))) → ((Base‘𝑀) × {𝑥}):(Base‘𝑀)⟶(Base‘𝑆))
74 simpr2 1061 . . . . 5 (((𝑀 ∈ LMod ∧ 𝑆 ∈ CRing) ∧ (𝑥 ∈ (Base‘𝑆) ∧ 𝑦 ∈ (Base‘𝑆) ∧ 𝑧 ∈ (𝑀 LMHom 𝑀))) → 𝑦 ∈ (Base‘𝑆))
75 fconst6g 6007 . . . . 5 (𝑦 ∈ (Base‘𝑆) → ((Base‘𝑀) × {𝑦}):(Base‘𝑀)⟶(Base‘𝑆))
7674, 75syl 17 . . . 4 (((𝑀 ∈ LMod ∧ 𝑆 ∈ CRing) ∧ (𝑥 ∈ (Base‘𝑆) ∧ 𝑦 ∈ (Base‘𝑆) ∧ 𝑧 ∈ (𝑀 LMHom 𝑀))) → ((Base‘𝑀) × {𝑦}):(Base‘𝑀)⟶(Base‘𝑆))
77 simpll 786 . . . . 5 (((𝑀 ∈ LMod ∧ 𝑆 ∈ CRing) ∧ (𝑥 ∈ (Base‘𝑆) ∧ 𝑦 ∈ (Base‘𝑆) ∧ 𝑧 ∈ (𝑀 LMHom 𝑀))) → 𝑀 ∈ LMod)
78 eqid 2610 . . . . . 6 (+g𝑆) = (+g𝑆)
7921, 30, 5, 19, 20, 78lmodvsdir 18710 . . . . 5 ((𝑀 ∈ LMod ∧ (𝑤 ∈ (Base‘𝑆) ∧ 𝑣 ∈ (Base‘𝑆) ∧ 𝑢 ∈ (Base‘𝑀))) → ((𝑤(+g𝑆)𝑣)( ·𝑠𝑀)𝑢) = ((𝑤( ·𝑠𝑀)𝑢)(+g𝑀)(𝑣( ·𝑠𝑀)𝑢)))
8077, 79sylan 487 . . . 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)))
8169, 71, 73, 76, 80caofdir 6832 . . 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)))
8214adantr 480 . . . . . 6 (((𝑀 ∈ LMod ∧ 𝑆 ∈ CRing) ∧ (𝑥 ∈ (Base‘𝑆) ∧ 𝑦 ∈ (Base‘𝑆) ∧ 𝑧 ∈ (𝑀 LMHom 𝑀))) → 𝑆 ∈ Ring)
8320, 78ringacl 18401 . . . . . 6 ((𝑆 ∈ Ring ∧ 𝑥 ∈ (Base‘𝑆) ∧ 𝑦 ∈ (Base‘𝑆)) → (𝑥(+g𝑆)𝑦) ∈ (Base‘𝑆))
8482, 72, 74, 83syl3anc 1318 . . . . 5 (((𝑀 ∈ LMod ∧ 𝑆 ∈ CRing) ∧ (𝑥 ∈ (Base‘𝑆) ∧ 𝑦 ∈ (Base‘𝑆) ∧ 𝑧 ∈ (𝑀 LMHom 𝑀))) → (𝑥(+g𝑆)𝑦) ∈ (Base‘𝑆))
851, 19, 2, 5, 20, 21, 22mendvsca 36780 . . . . 5 (((𝑥(+g𝑆)𝑦) ∈ (Base‘𝑆) ∧ 𝑧 ∈ (𝑀 LMHom 𝑀)) → ((𝑥(+g𝑆)𝑦)( ·𝑠𝐴)𝑧) = (((Base‘𝑀) × {(𝑥(+g𝑆)𝑦)}) ∘𝑓 ( ·𝑠𝑀)𝑧))
8684, 70, 85syl2anc 691 . . . 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))
8769, 72, 74ofc12 6820 . . . . 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)}))
8887oveq1d 6564 . . . 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))
8986, 88eqtr4d 2647 . . 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))
90513adant3r2 1267 . . . . 5 (((𝑀 ∈ LMod ∧ 𝑆 ∈ CRing) ∧ (𝑥 ∈ (Base‘𝑆) ∧ 𝑦 ∈ (Base‘𝑆) ∧ 𝑧 ∈ (𝑀 LMHom 𝑀))) → (𝑥( ·𝑠𝐴)𝑧) ∈ (𝑀 LMHom 𝑀))
91 eleq1 2676 . . . . . . . . 9 (𝑥 = 𝑦 → (𝑥 ∈ (Base‘𝑆) ↔ 𝑦 ∈ (Base‘𝑆)))
92913anbi2d 1396 . . . . . . . 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))))
93 oveq1 6556 . . . . . . . . 9 (𝑥 = 𝑦 → (𝑥( ·𝑠𝐴)𝑧) = (𝑦( ·𝑠𝐴)𝑧))
9493eleq1d 2672 . . . . . . . 8 (𝑥 = 𝑦 → ((𝑥( ·𝑠𝐴)𝑧) ∈ (𝑀 LMHom 𝑀) ↔ (𝑦( ·𝑠𝐴)𝑧) ∈ (𝑀 LMHom 𝑀)))
9592, 94imbi12d 333 . . . . . . 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))))
9695, 51chvarv 2251 . . . . . 6 (((𝑀 ∈ LMod ∧ 𝑆 ∈ CRing) ∧ 𝑦 ∈ (Base‘𝑆) ∧ 𝑧 ∈ (𝑀 LMHom 𝑀)) → (𝑦( ·𝑠𝐴)𝑧) ∈ (𝑀 LMHom 𝑀))
97963adant3r1 1266 . . . . 5 (((𝑀 ∈ LMod ∧ 𝑆 ∈ CRing) ∧ (𝑥 ∈ (Base‘𝑆) ∧ 𝑦 ∈ (Base‘𝑆) ∧ 𝑧 ∈ (𝑀 LMHom 𝑀))) → (𝑦( ·𝑠𝐴)𝑧) ∈ (𝑀 LMHom 𝑀))
981, 2, 30, 31mendplusg 36775 . . . . 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)))
9990, 97, 98syl2anc 691 . . . 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)))
10072, 70, 42syl2anc 691 . . . . 5 (((𝑀 ∈ LMod ∧ 𝑆 ∈ CRing) ∧ (𝑥 ∈ (Base‘𝑆) ∧ 𝑦 ∈ (Base‘𝑆) ∧ 𝑧 ∈ (𝑀 LMHom 𝑀))) → (𝑥( ·𝑠𝐴)𝑧) = (((Base‘𝑀) × {𝑥}) ∘𝑓 ( ·𝑠𝑀)𝑧))
1011, 19, 2, 5, 20, 21, 22mendvsca 36780 . . . . . 6 ((𝑦 ∈ (Base‘𝑆) ∧ 𝑧 ∈ (𝑀 LMHom 𝑀)) → (𝑦( ·𝑠𝐴)𝑧) = (((Base‘𝑀) × {𝑦}) ∘𝑓 ( ·𝑠𝑀)𝑧))
10274, 70, 101syl2anc 691 . . . . 5 (((𝑀 ∈ LMod ∧ 𝑆 ∈ CRing) ∧ (𝑥 ∈ (Base‘𝑆) ∧ 𝑦 ∈ (Base‘𝑆) ∧ 𝑧 ∈ (𝑀 LMHom 𝑀))) → (𝑦( ·𝑠𝐴)𝑧) = (((Base‘𝑀) × {𝑦}) ∘𝑓 ( ·𝑠𝑀)𝑧))
103100, 102oveq12d 6567 . . . 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)))
10499, 103eqtrd 2644 . . 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)))
10581, 89, 1043eqtr4d 2654 . 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)))
106 ovex 6577 . . . . 5 (𝑥(.r𝑆)𝑦) ∈ V
107106a1i 11 . . . 4 ((((𝑀 ∈ LMod ∧ 𝑆 ∈ CRing) ∧ (𝑥 ∈ (Base‘𝑆) ∧ 𝑦 ∈ (Base‘𝑆) ∧ 𝑧 ∈ (𝑀 LMHom 𝑀))) ∧ 𝑘 ∈ (Base‘𝑀)) → (𝑥(.r𝑆)𝑦) ∈ V)
10871ffvelrnda 6267 . . . 4 ((((𝑀 ∈ LMod ∧ 𝑆 ∈ CRing) ∧ (𝑥 ∈ (Base‘𝑆) ∧ 𝑦 ∈ (Base‘𝑆) ∧ 𝑧 ∈ (𝑀 LMHom 𝑀))) ∧ 𝑘 ∈ (Base‘𝑀)) → (𝑧𝑘) ∈ (Base‘𝑀))
109 fconstmpt 5085 . . . . 5 ((Base‘𝑀) × {(𝑥(.r𝑆)𝑦)}) = (𝑘 ∈ (Base‘𝑀) ↦ (𝑥(.r𝑆)𝑦))
110109a1i 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)))
11171feqmptd 6159 . . . 4 (((𝑀 ∈ LMod ∧ 𝑆 ∈ CRing) ∧ (𝑥 ∈ (Base‘𝑆) ∧ 𝑦 ∈ (Base‘𝑆) ∧ 𝑧 ∈ (𝑀 LMHom 𝑀))) → 𝑧 = (𝑘 ∈ (Base‘𝑀) ↦ (𝑧𝑘)))
11269, 107, 108, 110, 111offval2 6812 . . 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))))
113 eqid 2610 . . . . . 6 (.r𝑆) = (.r𝑆)
11420, 113ringcl 18384 . . . . 5 ((𝑆 ∈ Ring ∧ 𝑥 ∈ (Base‘𝑆) ∧ 𝑦 ∈ (Base‘𝑆)) → (𝑥(.r𝑆)𝑦) ∈ (Base‘𝑆))
11582, 72, 74, 114syl3anc 1318 . . . 4 (((𝑀 ∈ LMod ∧ 𝑆 ∈ CRing) ∧ (𝑥 ∈ (Base‘𝑆) ∧ 𝑦 ∈ (Base‘𝑆) ∧ 𝑧 ∈ (𝑀 LMHom 𝑀))) → (𝑥(.r𝑆)𝑦) ∈ (Base‘𝑆))
1161, 19, 2, 5, 20, 21, 22mendvsca 36780 . . . 4 (((𝑥(.r𝑆)𝑦) ∈ (Base‘𝑆) ∧ 𝑧 ∈ (𝑀 LMHom 𝑀)) → ((𝑥(.r𝑆)𝑦)( ·𝑠𝐴)𝑧) = (((Base‘𝑀) × {(𝑥(.r𝑆)𝑦)}) ∘𝑓 ( ·𝑠𝑀)𝑧))
117115, 70, 116syl2anc 691 . . 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))
11872adantr 480 . . . . 5 ((((𝑀 ∈ LMod ∧ 𝑆 ∈ CRing) ∧ (𝑥 ∈ (Base‘𝑆) ∧ 𝑦 ∈ (Base‘𝑆) ∧ 𝑧 ∈ (𝑀 LMHom 𝑀))) ∧ 𝑘 ∈ (Base‘𝑀)) → 𝑥 ∈ (Base‘𝑆))
119 ovex 6577 . . . . . 6 (𝑦( ·𝑠𝑀)(𝑧𝑘)) ∈ V
120119a1i 11 . . . . 5 ((((𝑀 ∈ LMod ∧ 𝑆 ∈ CRing) ∧ (𝑥 ∈ (Base‘𝑆) ∧ 𝑦 ∈ (Base‘𝑆) ∧ 𝑧 ∈ (𝑀 LMHom 𝑀))) ∧ 𝑘 ∈ (Base‘𝑀)) → (𝑦( ·𝑠𝑀)(𝑧𝑘)) ∈ V)
121 fconstmpt 5085 . . . . . 6 ((Base‘𝑀) × {𝑥}) = (𝑘 ∈ (Base‘𝑀) ↦ 𝑥)
122121a1i 11 . . . . 5 (((𝑀 ∈ LMod ∧ 𝑆 ∈ CRing) ∧ (𝑥 ∈ (Base‘𝑆) ∧ 𝑦 ∈ (Base‘𝑆) ∧ 𝑧 ∈ (𝑀 LMHom 𝑀))) → ((Base‘𝑀) × {𝑥}) = (𝑘 ∈ (Base‘𝑀) ↦ 𝑥))
123 simplr2 1097 . . . . . . 7 ((((𝑀 ∈ LMod ∧ 𝑆 ∈ CRing) ∧ (𝑥 ∈ (Base‘𝑆) ∧ 𝑦 ∈ (Base‘𝑆) ∧ 𝑧 ∈ (𝑀 LMHom 𝑀))) ∧ 𝑘 ∈ (Base‘𝑀)) → 𝑦 ∈ (Base‘𝑆))
124 fconstmpt 5085 . . . . . . . 8 ((Base‘𝑀) × {𝑦}) = (𝑘 ∈ (Base‘𝑀) ↦ 𝑦)
125124a1i 11 . . . . . . 7 (((𝑀 ∈ LMod ∧ 𝑆 ∈ CRing) ∧ (𝑥 ∈ (Base‘𝑆) ∧ 𝑦 ∈ (Base‘𝑆) ∧ 𝑧 ∈ (𝑀 LMHom 𝑀))) → ((Base‘𝑀) × {𝑦}) = (𝑘 ∈ (Base‘𝑀) ↦ 𝑦))
12669, 123, 108, 125, 111offval2 6812 . . . . . 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))))
127102, 126eqtrd 2644 . . . . 5 (((𝑀 ∈ LMod ∧ 𝑆 ∈ CRing) ∧ (𝑥 ∈ (Base‘𝑆) ∧ 𝑦 ∈ (Base‘𝑆) ∧ 𝑧 ∈ (𝑀 LMHom 𝑀))) → (𝑦( ·𝑠𝐴)𝑧) = (𝑘 ∈ (Base‘𝑀) ↦ (𝑦( ·𝑠𝑀)(𝑧𝑘))))
12869, 118, 120, 122, 127offval2 6812 . . . 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)))))
1291, 19, 2, 5, 20, 21, 22mendvsca 36780 . . . . 5 ((𝑥 ∈ (Base‘𝑆) ∧ (𝑦( ·𝑠𝐴)𝑧) ∈ (𝑀 LMHom 𝑀)) → (𝑥( ·𝑠𝐴)(𝑦( ·𝑠𝐴)𝑧)) = (((Base‘𝑀) × {𝑥}) ∘𝑓 ( ·𝑠𝑀)(𝑦( ·𝑠𝐴)𝑧)))
13072, 97, 129syl2anc 691 . . . 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)))
13177adantr 480 . . . . . 6 ((((𝑀 ∈ LMod ∧ 𝑆 ∈ CRing) ∧ (𝑥 ∈ (Base‘𝑆) ∧ 𝑦 ∈ (Base‘𝑆) ∧ 𝑧 ∈ (𝑀 LMHom 𝑀))) ∧ 𝑘 ∈ (Base‘𝑀)) → 𝑀 ∈ LMod)
13221, 5, 19, 20, 113lmodvsass 18711 . . . . . 6 ((𝑀 ∈ LMod ∧ (𝑥 ∈ (Base‘𝑆) ∧ 𝑦 ∈ (Base‘𝑆) ∧ (𝑧𝑘) ∈ (Base‘𝑀))) → ((𝑥(.r𝑆)𝑦)( ·𝑠𝑀)(𝑧𝑘)) = (𝑥( ·𝑠𝑀)(𝑦( ·𝑠𝑀)(𝑧𝑘))))
133131, 118, 123, 108, 132syl13anc 1320 . . . . 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))))
134133mpteq2dva 4672 . . . 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)))))
135128, 130, 1343eqtr4d 2654 . . 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))))
136112, 117, 1353eqtr4d 2654 . 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)))
13714adantr 480 . . . . 5 (((𝑀 ∈ LMod ∧ 𝑆 ∈ CRing) ∧ 𝑥 ∈ (𝑀 LMHom 𝑀)) → 𝑆 ∈ Ring)
138 eqid 2610 . . . . . 6 (1r𝑆) = (1r𝑆)
13920, 138ringidcl 18391 . . . . 5 (𝑆 ∈ Ring → (1r𝑆) ∈ (Base‘𝑆))
140137, 139syl 17 . . . 4 (((𝑀 ∈ LMod ∧ 𝑆 ∈ CRing) ∧ 𝑥 ∈ (𝑀 LMHom 𝑀)) → (1r𝑆) ∈ (Base‘𝑆))
1411, 19, 2, 5, 20, 21, 22mendvsca 36780 . . . 4 (((1r𝑆) ∈ (Base‘𝑆) ∧ 𝑥 ∈ (𝑀 LMHom 𝑀)) → ((1r𝑆)( ·𝑠𝐴)𝑥) = (((Base‘𝑀) × {(1r𝑆)}) ∘𝑓 ( ·𝑠𝑀)𝑥))
142140, 141sylancom 698 . . 3 (((𝑀 ∈ LMod ∧ 𝑆 ∈ CRing) ∧ 𝑥 ∈ (𝑀 LMHom 𝑀)) → ((1r𝑆)( ·𝑠𝐴)𝑥) = (((Base‘𝑀) × {(1r𝑆)}) ∘𝑓 ( ·𝑠𝑀)𝑥))
14355a1i 11 . . . 4 (((𝑀 ∈ LMod ∧ 𝑆 ∈ CRing) ∧ 𝑥 ∈ (𝑀 LMHom 𝑀)) → (Base‘𝑀) ∈ V)
14421, 21lmhmf 18855 . . . . 5 (𝑥 ∈ (𝑀 LMHom 𝑀) → 𝑥:(Base‘𝑀)⟶(Base‘𝑀))
145144adantl 481 . . . 4 (((𝑀 ∈ LMod ∧ 𝑆 ∈ CRing) ∧ 𝑥 ∈ (𝑀 LMHom 𝑀)) → 𝑥:(Base‘𝑀)⟶(Base‘𝑀))
146 simpll 786 . . . . 5 (((𝑀 ∈ LMod ∧ 𝑆 ∈ CRing) ∧ 𝑥 ∈ (𝑀 LMHom 𝑀)) → 𝑀 ∈ LMod)
14721, 5, 19, 138lmodvs1 18714 . . . . 5 ((𝑀 ∈ LMod ∧ 𝑦 ∈ (Base‘𝑀)) → ((1r𝑆)( ·𝑠𝑀)𝑦) = 𝑦)
148146, 147sylan 487 . . . 4 ((((𝑀 ∈ LMod ∧ 𝑆 ∈ CRing) ∧ 𝑥 ∈ (𝑀 LMHom 𝑀)) ∧ 𝑦 ∈ (Base‘𝑀)) → ((1r𝑆)( ·𝑠𝑀)𝑦) = 𝑦)
149143, 145, 140, 148caofid0l 6823 . . 3 (((𝑀 ∈ LMod ∧ 𝑆 ∈ CRing) ∧ 𝑥 ∈ (𝑀 LMHom 𝑀)) → (((Base‘𝑀) × {(1r𝑆)}) ∘𝑓 ( ·𝑠𝑀)𝑥) = 𝑥)
150142, 149eqtrd 2644 . 2 (((𝑀 ∈ LMod ∧ 𝑆 ∈ CRing) ∧ 𝑥 ∈ (𝑀 LMHom 𝑀)) → ((1r𝑆)( ·𝑠𝐴)𝑥) = 𝑥)
1513, 4, 7, 8, 9, 10, 11, 12, 14, 18, 27, 68, 105, 136, 150islmodd 18692 1 ((𝑀 ∈ LMod ∧ 𝑆 ∈ CRing) → 𝐴 ∈ LMod)
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ∧ wa 383   ∧ w3a 1031   = wceq 1475   ∈ wcel 1977  Vcvv 3173  {csn 4125   ↦ cmpt 4643   × cxp 5036  ⟶wf 5800  ‘cfv 5804  (class class class)co 6549   ∘𝑓 cof 6793  Basecbs 15695  +gcplusg 15768  .rcmulr 15769  Scalarcsca 15771   ·𝑠 cvsca 15772  Grpcgrp 17245  1rcur 18324  Ringcrg 18370  CRingccrg 18371  LModclmod 18686   LMHom clmhm 18840  MEndocmend 36764 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1713  ax-4 1728  ax-5 1827  ax-6 1875  ax-7 1922  ax-8 1979  ax-9 1986  ax-10 2006  ax-11 2021  ax-12 2034  ax-13 2234  ax-ext 2590  ax-rep 4699  ax-sep 4709  ax-nul 4717  ax-pow 4769  ax-pr 4833  ax-un 6847  ax-cnex 9871  ax-resscn 9872  ax-1cn 9873  ax-icn 9874  ax-addcl 9875  ax-addrcl 9876  ax-mulcl 9877  ax-mulrcl 9878  ax-mulcom 9879  ax-addass 9880  ax-mulass 9881  ax-distr 9882  ax-i2m1 9883  ax-1ne0 9884  ax-1rid 9885  ax-rnegex 9886  ax-rrecex 9887  ax-cnre 9888  ax-pre-lttri 9889  ax-pre-lttrn 9890  ax-pre-ltadd 9891  ax-pre-mulgt0 9892 This theorem depends on definitions:  df-bi 196  df-or 384  df-an 385  df-3or 1032  df-3an 1033  df-tru 1478  df-ex 1696  df-nf 1701  df-sb 1868  df-eu 2462  df-mo 2463  df-clab 2597  df-cleq 2603  df-clel 2606  df-nfc 2740  df-ne 2782  df-nel 2783  df-ral 2901  df-rex 2902  df-reu 2903  df-rmo 2904  df-rab 2905  df-v 3175  df-sbc 3403  df-csb 3500  df-dif 3543  df-un 3545  df-in 3547  df-ss 3554  df-pss 3556  df-nul 3875  df-if 4037  df-pw 4110  df-sn 4126  df-pr 4128  df-tp 4130  df-op 4132  df-uni 4373  df-int 4411  df-iun 4457  df-br 4584  df-opab 4644  df-mpt 4645  df-tr 4681  df-eprel 4949  df-id 4953  df-po 4959  df-so 4960  df-fr 4997  df-we 4999  df-xp 5044  df-rel 5045  df-cnv 5046  df-co 5047  df-dm 5048  df-rn 5049  df-res 5050  df-ima 5051  df-pred 5597  df-ord 5643  df-on 5644  df-lim 5645  df-suc 5646  df-iota 5768  df-fun 5806  df-fn 5807  df-f 5808  df-f1 5809  df-fo 5810  df-f1o 5811  df-fv 5812  df-riota 6511  df-ov 6552  df-oprab 6553  df-mpt2 6554  df-of 6795  df-om 6958  df-1st 7059  df-2nd 7060  df-wrecs 7294  df-recs 7355  df-rdg 7393  df-1o 7447  df-oadd 7451  df-er 7629  df-map 7746  df-en 7842  df-dom 7843  df-sdom 7844  df-fin 7845  df-pnf 9955  df-mnf 9956  df-xr 9957  df-ltxr 9958  df-le 9959  df-sub 10147  df-neg 10148  df-nn 10898  df-2 10956  df-3 10957  df-4 10958  df-5 10959  df-6 10960  df-n0 11170  df-z 11255  df-uz 11564  df-fz 12198  df-struct 15697  df-ndx 15698  df-slot 15699  df-base 15700  df-sets 15701  df-plusg 15781  df-mulr 15782  df-sca 15784  df-vsca 15785  df-0g 15925  df-mgm 17065  df-sgrp 17107  df-mnd 17118  df-mhm 17158  df-grp 17248  df-minusg 17249  df-ghm 17481  df-cmn 18018  df-abl 18019  df-mgp 18313  df-ur 18325  df-ring 18372  df-cring 18373  df-lmod 18688  df-lmhm 18843  df-mend 36765 This theorem is referenced by:  mendassa  36783
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