Users' Mathboxes Mathbox for Stefan O'Rear < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >   Mathboxes  >  mendlmod Structured version   Visualization version   GIF version

Theorem mendlmod 37241
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 37232 . . 3 (𝑀 LMHom 𝑀) = (Base‘𝐴)
32a1i 11 . 2 ((𝑀 ∈ LMod ∧ 𝑆 ∈ CRing) → (𝑀 LMHom 𝑀) = (Base‘𝐴))
4 eqidd 2622 . 2 ((𝑀 ∈ LMod ∧ 𝑆 ∈ CRing) → (+g𝐴) = (+g𝐴))
5 mendassa.s . . . 4 𝑆 = (Scalar‘𝑀)
61, 5mendsca 37237 . . 3 𝑆 = (Scalar‘𝐴)
76a1i 11 . 2 ((𝑀 ∈ LMod ∧ 𝑆 ∈ CRing) → 𝑆 = (Scalar‘𝐴))
8 eqidd 2622 . 2 ((𝑀 ∈ LMod ∧ 𝑆 ∈ CRing) → ( ·𝑠𝐴) = ( ·𝑠𝐴))
9 eqidd 2622 . 2 ((𝑀 ∈ LMod ∧ 𝑆 ∈ CRing) → (Base‘𝑆) = (Base‘𝑆))
10 eqidd 2622 . 2 ((𝑀 ∈ LMod ∧ 𝑆 ∈ CRing) → (+g𝑆) = (+g𝑆))
11 eqidd 2622 . 2 ((𝑀 ∈ LMod ∧ 𝑆 ∈ CRing) → (.r𝑆) = (.r𝑆))
12 eqidd 2622 . 2 ((𝑀 ∈ LMod ∧ 𝑆 ∈ CRing) → (1r𝑆) = (1r𝑆))
13 crngring 18479 . . 3 (𝑆 ∈ CRing → 𝑆 ∈ Ring)
1413adantl 482 . 2 ((𝑀 ∈ LMod ∧ 𝑆 ∈ CRing) → 𝑆 ∈ Ring)
151mendring 37240 . . . 4 (𝑀 ∈ LMod → 𝐴 ∈ Ring)
1615adantr 481 . . 3 ((𝑀 ∈ LMod ∧ 𝑆 ∈ CRing) → 𝐴 ∈ Ring)
17 ringgrp 18473 . . 3 (𝐴 ∈ Ring → 𝐴 ∈ Grp)
1816, 17syl 17 . 2 ((𝑀 ∈ LMod ∧ 𝑆 ∈ CRing) → 𝐴 ∈ Grp)
19 eqid 2621 . . . . 5 ( ·𝑠𝑀) = ( ·𝑠𝑀)
20 eqid 2621 . . . . 5 (Base‘𝑆) = (Base‘𝑆)
21 eqid 2621 . . . . 5 (Base‘𝑀) = (Base‘𝑀)
22 eqid 2621 . . . . 5 ( ·𝑠𝐴) = ( ·𝑠𝐴)
231, 19, 2, 5, 20, 21, 22mendvsca 37239 . . . 4 ((𝑥 ∈ (Base‘𝑆) ∧ 𝑦 ∈ (𝑀 LMHom 𝑀)) → (𝑥( ·𝑠𝐴)𝑦) = (((Base‘𝑀) × {𝑥}) ∘𝑓 ( ·𝑠𝑀)𝑦))
24233adant1 1077 . . 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 18964 . . . 4 ((𝑆 ∈ CRing ∧ 𝑥 ∈ (Base‘𝑆) ∧ 𝑦 ∈ (𝑀 LMHom 𝑀)) → (((Base‘𝑀) × {𝑥}) ∘𝑓 ( ·𝑠𝑀)𝑦) ∈ (𝑀 LMHom 𝑀))
26253adant1l 1315 . . 3 (((𝑀 ∈ LMod ∧ 𝑆 ∈ CRing) ∧ 𝑥 ∈ (Base‘𝑆) ∧ 𝑦 ∈ (𝑀 LMHom 𝑀)) → (((Base‘𝑀) × {𝑥}) ∘𝑓 ( ·𝑠𝑀)𝑦) ∈ (𝑀 LMHom 𝑀))
2724, 26eqeltrd 2698 . 2 (((𝑀 ∈ LMod ∧ 𝑆 ∈ CRing) ∧ 𝑥 ∈ (Base‘𝑆) ∧ 𝑦 ∈ (𝑀 LMHom 𝑀)) → (𝑥( ·𝑠𝐴)𝑦) ∈ (𝑀 LMHom 𝑀))
28 simpr2 1066 . . . . 5 (((𝑀 ∈ LMod ∧ 𝑆 ∈ CRing) ∧ (𝑥 ∈ (Base‘𝑆) ∧ 𝑦 ∈ (𝑀 LMHom 𝑀) ∧ 𝑧 ∈ (𝑀 LMHom 𝑀))) → 𝑦 ∈ (𝑀 LMHom 𝑀))
29 simpr3 1067 . . . . 5 (((𝑀 ∈ LMod ∧ 𝑆 ∈ CRing) ∧ (𝑥 ∈ (Base‘𝑆) ∧ 𝑦 ∈ (𝑀 LMHom 𝑀) ∧ 𝑧 ∈ (𝑀 LMHom 𝑀))) → 𝑧 ∈ (𝑀 LMHom 𝑀))
30 eqid 2621 . . . . . 6 (+g𝑀) = (+g𝑀)
31 eqid 2621 . . . . . 6 (+g𝐴) = (+g𝐴)
321, 2, 30, 31mendplusg 37234 . . . . 5 ((𝑦 ∈ (𝑀 LMHom 𝑀) ∧ 𝑧 ∈ (𝑀 LMHom 𝑀)) → (𝑦(+g𝐴)𝑧) = (𝑦𝑓 (+g𝑀)𝑧))
3328, 29, 32syl2anc 692 . . . 4 (((𝑀 ∈ LMod ∧ 𝑆 ∈ CRing) ∧ (𝑥 ∈ (Base‘𝑆) ∧ 𝑦 ∈ (𝑀 LMHom 𝑀) ∧ 𝑧 ∈ (𝑀 LMHom 𝑀))) → (𝑦(+g𝐴)𝑧) = (𝑦𝑓 (+g𝑀)𝑧))
3433oveq2d 6620 . . 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 1065 . . . 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 17351 . . . . 5 ((𝐴 ∈ Grp ∧ 𝑦 ∈ (𝑀 LMHom 𝑀) ∧ 𝑧 ∈ (𝑀 LMHom 𝑀)) → (𝑦(+g𝐴)𝑧) ∈ (𝑀 LMHom 𝑀))
3836, 28, 29, 37syl3anc 1323 . . . 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 37239 . . . 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 692 . . 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 692 . . . . 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 37239 . . . . . 6 ((𝑥 ∈ (Base‘𝑆) ∧ 𝑧 ∈ (𝑀 LMHom 𝑀)) → (𝑥( ·𝑠𝐴)𝑧) = (((Base‘𝑀) × {𝑥}) ∘𝑓 ( ·𝑠𝑀)𝑧))
4335, 29, 42syl2anc 692 . . . . 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 6622 . . . 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 1273 . . . . 5 (((𝑀 ∈ LMod ∧ 𝑆 ∈ CRing) ∧ (𝑥 ∈ (Base‘𝑆) ∧ 𝑦 ∈ (𝑀 LMHom 𝑀) ∧ 𝑧 ∈ (𝑀 LMHom 𝑀))) → (𝑥( ·𝑠𝐴)𝑦) ∈ (𝑀 LMHom 𝑀))
46 eleq1 2686 . . . . . . . . 9 (𝑦 = 𝑧 → (𝑦 ∈ (𝑀 LMHom 𝑀) ↔ 𝑧 ∈ (𝑀 LMHom 𝑀)))
47463anbi3d 1402 . . . . . . . 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 6612 . . . . . . . . 9 (𝑦 = 𝑧 → (𝑥( ·𝑠𝐴)𝑦) = (𝑥( ·𝑠𝐴)𝑧))
4948eleq1d 2683 . . . . . . . 8 (𝑦 = 𝑧 → ((𝑥( ·𝑠𝐴)𝑦) ∈ (𝑀 LMHom 𝑀) ↔ (𝑥( ·𝑠𝐴)𝑧) ∈ (𝑀 LMHom 𝑀)))
5047, 49imbi12d 334 . . . . . . 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 2262 . . . . . 6 (((𝑀 ∈ LMod ∧ 𝑆 ∈ CRing) ∧ 𝑥 ∈ (Base‘𝑆) ∧ 𝑧 ∈ (𝑀 LMHom 𝑀)) → (𝑥( ·𝑠𝐴)𝑧) ∈ (𝑀 LMHom 𝑀))
52513adant3r2 1272 . . . . 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 37234 . . . . 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 692 . . . 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 6158 . . . . . 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 6051 . . . . . 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 18953 . . . . . 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 18953 . . . . . 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 789 . . . . . 6 (((𝑀 ∈ LMod ∧ 𝑆 ∈ CRing) ∧ (𝑥 ∈ (Base‘𝑆) ∧ 𝑦 ∈ (𝑀 LMHom 𝑀) ∧ 𝑧 ∈ (𝑀 LMHom 𝑀))) → 𝑀 ∈ LMod)
6421, 30, 5, 19, 20lmodvsdi 18807 . . . . . 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 488 . . . . 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 6886 . . . 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 2665 . . 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 2665 . 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 1067 . . . . 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 1065 . . . . 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 1066 . . . . 5 (((𝑀 ∈ LMod ∧ 𝑆 ∈ CRing) ∧ (𝑥 ∈ (Base‘𝑆) ∧ 𝑦 ∈ (Base‘𝑆) ∧ 𝑧 ∈ (𝑀 LMHom 𝑀))) → 𝑦 ∈ (Base‘𝑆))
75 fconst6g 6051 . . . . 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 789 . . . . 5 (((𝑀 ∈ LMod ∧ 𝑆 ∈ CRing) ∧ (𝑥 ∈ (Base‘𝑆) ∧ 𝑦 ∈ (Base‘𝑆) ∧ 𝑧 ∈ (𝑀 LMHom 𝑀))) → 𝑀 ∈ LMod)
78 eqid 2621 . . . . . 6 (+g𝑆) = (+g𝑆)
7921, 30, 5, 19, 20, 78lmodvsdir 18808 . . . . 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 488 . . . 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 6887 . . 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 481 . . . . . 6 (((𝑀 ∈ LMod ∧ 𝑆 ∈ CRing) ∧ (𝑥 ∈ (Base‘𝑆) ∧ 𝑦 ∈ (Base‘𝑆) ∧ 𝑧 ∈ (𝑀 LMHom 𝑀))) → 𝑆 ∈ Ring)
8320, 78ringacl 18499 . . . . . 6 ((𝑆 ∈ Ring ∧ 𝑥 ∈ (Base‘𝑆) ∧ 𝑦 ∈ (Base‘𝑆)) → (𝑥(+g𝑆)𝑦) ∈ (Base‘𝑆))
8482, 72, 74, 83syl3anc 1323 . . . . 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 37239 . . . . 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 692 . . . 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 6875 . . . . 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 6619 . . . 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 2658 . . 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 1272 . . . . 5 (((𝑀 ∈ LMod ∧ 𝑆 ∈ CRing) ∧ (𝑥 ∈ (Base‘𝑆) ∧ 𝑦 ∈ (Base‘𝑆) ∧ 𝑧 ∈ (𝑀 LMHom 𝑀))) → (𝑥( ·𝑠𝐴)𝑧) ∈ (𝑀 LMHom 𝑀))
91 eleq1 2686 . . . . . . . . 9 (𝑥 = 𝑦 → (𝑥 ∈ (Base‘𝑆) ↔ 𝑦 ∈ (Base‘𝑆)))
92913anbi2d 1401 . . . . . . . 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 6611 . . . . . . . . 9 (𝑥 = 𝑦 → (𝑥( ·𝑠𝐴)𝑧) = (𝑦( ·𝑠𝐴)𝑧))
9493eleq1d 2683 . . . . . . . 8 (𝑥 = 𝑦 → ((𝑥( ·𝑠𝐴)𝑧) ∈ (𝑀 LMHom 𝑀) ↔ (𝑦( ·𝑠𝐴)𝑧) ∈ (𝑀 LMHom 𝑀)))
9592, 94imbi12d 334 . . . . . . 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 2262 . . . . . 6 (((𝑀 ∈ LMod ∧ 𝑆 ∈ CRing) ∧ 𝑦 ∈ (Base‘𝑆) ∧ 𝑧 ∈ (𝑀 LMHom 𝑀)) → (𝑦( ·𝑠𝐴)𝑧) ∈ (𝑀 LMHom 𝑀))
97963adant3r1 1271 . . . . 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 37234 . . . . 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 692 . . . 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 692 . . . . 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 37239 . . . . . 6 ((𝑦 ∈ (Base‘𝑆) ∧ 𝑧 ∈ (𝑀 LMHom 𝑀)) → (𝑦( ·𝑠𝐴)𝑧) = (((Base‘𝑀) × {𝑦}) ∘𝑓 ( ·𝑠𝑀)𝑧))
10274, 70, 101syl2anc 692 . . . . 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 6622 . . . 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 2655 . . 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 2665 . 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 6632 . . . . 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 6315 . . . 4 ((((𝑀 ∈ LMod ∧ 𝑆 ∈ CRing) ∧ (𝑥 ∈ (Base‘𝑆) ∧ 𝑦 ∈ (Base‘𝑆) ∧ 𝑧 ∈ (𝑀 LMHom 𝑀))) ∧ 𝑘 ∈ (Base‘𝑀)) → (𝑧𝑘) ∈ (Base‘𝑀))
109 fconstmpt 5123 . . . . 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 6206 . . . 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 6867 . . 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 2621 . . . . . 6 (.r𝑆) = (.r𝑆)
11420, 113ringcl 18482 . . . . 5 ((𝑆 ∈ Ring ∧ 𝑥 ∈ (Base‘𝑆) ∧ 𝑦 ∈ (Base‘𝑆)) → (𝑥(.r𝑆)𝑦) ∈ (Base‘𝑆))
11582, 72, 74, 114syl3anc 1323 . . . 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 37239 . . . 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 692 . . 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 481 . . . . 5 ((((𝑀 ∈ LMod ∧ 𝑆 ∈ CRing) ∧ (𝑥 ∈ (Base‘𝑆) ∧ 𝑦 ∈ (Base‘𝑆) ∧ 𝑧 ∈ (𝑀 LMHom 𝑀))) ∧ 𝑘 ∈ (Base‘𝑀)) → 𝑥 ∈ (Base‘𝑆))
119 ovex 6632 . . . . . 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 5123 . . . . . 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 1102 . . . . . . 7 ((((𝑀 ∈ LMod ∧ 𝑆 ∈ CRing) ∧ (𝑥 ∈ (Base‘𝑆) ∧ 𝑦 ∈ (Base‘𝑆) ∧ 𝑧 ∈ (𝑀 LMHom 𝑀))) ∧ 𝑘 ∈ (Base‘𝑀)) → 𝑦 ∈ (Base‘𝑆))
124 fconstmpt 5123 . . . . . . . 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 6867 . . . . . 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 2655 . . . . 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 6867 . . . 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 37239 . . . . 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 692 . . . 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 481 . . . . . 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 18809 . . . . . 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 1325 . . . . 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 4704 . . . 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 2665 . . 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 2665 . 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 481 . . . . 5 (((𝑀 ∈ LMod ∧ 𝑆 ∈ CRing) ∧ 𝑥 ∈ (𝑀 LMHom 𝑀)) → 𝑆 ∈ Ring)
138 eqid 2621 . . . . . 6 (1r𝑆) = (1r𝑆)
13920, 138ringidcl 18489 . . . . 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 37239 . . . 4 (((1r𝑆) ∈ (Base‘𝑆) ∧ 𝑥 ∈ (𝑀 LMHom 𝑀)) → ((1r𝑆)( ·𝑠𝐴)𝑥) = (((Base‘𝑀) × {(1r𝑆)}) ∘𝑓 ( ·𝑠𝑀)𝑥))
142140, 141sylancom 700 . . 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 18953 . . . . 5 (𝑥 ∈ (𝑀 LMHom 𝑀) → 𝑥:(Base‘𝑀)⟶(Base‘𝑀))
145144adantl 482 . . . 4 (((𝑀 ∈ LMod ∧ 𝑆 ∈ CRing) ∧ 𝑥 ∈ (𝑀 LMHom 𝑀)) → 𝑥:(Base‘𝑀)⟶(Base‘𝑀))
146 simpll 789 . . . . 5 (((𝑀 ∈ LMod ∧ 𝑆 ∈ CRing) ∧ 𝑥 ∈ (𝑀 LMHom 𝑀)) → 𝑀 ∈ LMod)
14721, 5, 19, 138lmodvs1 18812 . . . . 5 ((𝑀 ∈ LMod ∧ 𝑦 ∈ (Base‘𝑀)) → ((1r𝑆)( ·𝑠𝑀)𝑦) = 𝑦)
148146, 147sylan 488 . . . 4 ((((𝑀 ∈ LMod ∧ 𝑆 ∈ CRing) ∧ 𝑥 ∈ (𝑀 LMHom 𝑀)) ∧ 𝑦 ∈ (Base‘𝑀)) → ((1r𝑆)( ·𝑠𝑀)𝑦) = 𝑦)
149143, 145, 140, 148caofid0l 6878 . . 3 (((𝑀 ∈ LMod ∧ 𝑆 ∈ CRing) ∧ 𝑥 ∈ (𝑀 LMHom 𝑀)) → (((Base‘𝑀) × {(1r𝑆)}) ∘𝑓 ( ·𝑠𝑀)𝑥) = 𝑥)
150142, 149eqtrd 2655 . 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 18790 1 ((𝑀 ∈ LMod ∧ 𝑆 ∈ CRing) → 𝐴 ∈ LMod)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 384  w3a 1036   = wceq 1480  wcel 1987  Vcvv 3186  {csn 4148  cmpt 4673   × cxp 5072  wf 5843  cfv 5847  (class class class)co 6604  𝑓 cof 6848  Basecbs 15781  +gcplusg 15862  .rcmulr 15863  Scalarcsca 15865   ·𝑠 cvsca 15866  Grpcgrp 17343  1rcur 18422  Ringcrg 18468  CRingccrg 18469  LModclmod 18784   LMHom clmhm 18938  MEndocmend 37223
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1719  ax-4 1734  ax-5 1836  ax-6 1885  ax-7 1932  ax-8 1989  ax-9 1996  ax-10 2016  ax-11 2031  ax-12 2044  ax-13 2245  ax-ext 2601  ax-rep 4731  ax-sep 4741  ax-nul 4749  ax-pow 4803  ax-pr 4867  ax-un 6902  ax-cnex 9936  ax-resscn 9937  ax-1cn 9938  ax-icn 9939  ax-addcl 9940  ax-addrcl 9941  ax-mulcl 9942  ax-mulrcl 9943  ax-mulcom 9944  ax-addass 9945  ax-mulass 9946  ax-distr 9947  ax-i2m1 9948  ax-1ne0 9949  ax-1rid 9950  ax-rnegex 9951  ax-rrecex 9952  ax-cnre 9953  ax-pre-lttri 9954  ax-pre-lttrn 9955  ax-pre-ltadd 9956  ax-pre-mulgt0 9957
This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3or 1037  df-3an 1038  df-tru 1483  df-ex 1702  df-nf 1707  df-sb 1878  df-eu 2473  df-mo 2474  df-clab 2608  df-cleq 2614  df-clel 2617  df-nfc 2750  df-ne 2791  df-nel 2894  df-ral 2912  df-rex 2913  df-reu 2914  df-rmo 2915  df-rab 2916  df-v 3188  df-sbc 3418  df-csb 3515  df-dif 3558  df-un 3560  df-in 3562  df-ss 3569  df-pss 3571  df-nul 3892  df-if 4059  df-pw 4132  df-sn 4149  df-pr 4151  df-tp 4153  df-op 4155  df-uni 4403  df-int 4441  df-iun 4487  df-br 4614  df-opab 4674  df-mpt 4675  df-tr 4713  df-eprel 4985  df-id 4989  df-po 4995  df-so 4996  df-fr 5033  df-we 5035  df-xp 5080  df-rel 5081  df-cnv 5082  df-co 5083  df-dm 5084  df-rn 5085  df-res 5086  df-ima 5087  df-pred 5639  df-ord 5685  df-on 5686  df-lim 5687  df-suc 5688  df-iota 5810  df-fun 5849  df-fn 5850  df-f 5851  df-f1 5852  df-fo 5853  df-f1o 5854  df-fv 5855  df-riota 6565  df-ov 6607  df-oprab 6608  df-mpt2 6609  df-of 6850  df-om 7013  df-1st 7113  df-2nd 7114  df-wrecs 7352  df-recs 7413  df-rdg 7451  df-1o 7505  df-oadd 7509  df-er 7687  df-map 7804  df-en 7900  df-dom 7901  df-sdom 7902  df-fin 7903  df-pnf 10020  df-mnf 10021  df-xr 10022  df-ltxr 10023  df-le 10024  df-sub 10212  df-neg 10213  df-nn 10965  df-2 11023  df-3 11024  df-4 11025  df-5 11026  df-6 11027  df-n0 11237  df-z 11322  df-uz 11632  df-fz 12269  df-struct 15783  df-ndx 15784  df-slot 15785  df-base 15786  df-sets 15787  df-plusg 15875  df-mulr 15876  df-sca 15878  df-vsca 15879  df-0g 16023  df-mgm 17163  df-sgrp 17205  df-mnd 17216  df-mhm 17256  df-grp 17346  df-minusg 17347  df-ghm 17579  df-cmn 18116  df-abl 18117  df-mgp 18411  df-ur 18423  df-ring 18470  df-cring 18471  df-lmod 18786  df-lmhm 18941  df-mend 37224
This theorem is referenced by:  mendassa  37242
  Copyright terms: Public domain W3C validator