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 38261
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 38252 . . 3 (𝑀 LMHom 𝑀) = (Base‘𝐴)
32a1i 11 . 2 ((𝑀 ∈ LMod ∧ 𝑆 ∈ CRing) → (𝑀 LMHom 𝑀) = (Base‘𝐴))
4 eqidd 2757 . 2 ((𝑀 ∈ LMod ∧ 𝑆 ∈ CRing) → (+g𝐴) = (+g𝐴))
5 mendassa.s . . . 4 𝑆 = (Scalar‘𝑀)
61, 5mendsca 38257 . . 3 𝑆 = (Scalar‘𝐴)
76a1i 11 . 2 ((𝑀 ∈ LMod ∧ 𝑆 ∈ CRing) → 𝑆 = (Scalar‘𝐴))
8 eqidd 2757 . 2 ((𝑀 ∈ LMod ∧ 𝑆 ∈ CRing) → ( ·𝑠𝐴) = ( ·𝑠𝐴))
9 eqidd 2757 . 2 ((𝑀 ∈ LMod ∧ 𝑆 ∈ CRing) → (Base‘𝑆) = (Base‘𝑆))
10 eqidd 2757 . 2 ((𝑀 ∈ LMod ∧ 𝑆 ∈ CRing) → (+g𝑆) = (+g𝑆))
11 eqidd 2757 . 2 ((𝑀 ∈ LMod ∧ 𝑆 ∈ CRing) → (.r𝑆) = (.r𝑆))
12 eqidd 2757 . 2 ((𝑀 ∈ LMod ∧ 𝑆 ∈ CRing) → (1r𝑆) = (1r𝑆))
13 crngring 18754 . . 3 (𝑆 ∈ CRing → 𝑆 ∈ Ring)
1413adantl 473 . 2 ((𝑀 ∈ LMod ∧ 𝑆 ∈ CRing) → 𝑆 ∈ Ring)
151mendring 38260 . . . 4 (𝑀 ∈ LMod → 𝐴 ∈ Ring)
1615adantr 472 . . 3 ((𝑀 ∈ LMod ∧ 𝑆 ∈ CRing) → 𝐴 ∈ Ring)
17 ringgrp 18748 . . 3 (𝐴 ∈ Ring → 𝐴 ∈ Grp)
1816, 17syl 17 . 2 ((𝑀 ∈ LMod ∧ 𝑆 ∈ CRing) → 𝐴 ∈ Grp)
19 eqid 2756 . . . . 5 ( ·𝑠𝑀) = ( ·𝑠𝑀)
20 eqid 2756 . . . . 5 (Base‘𝑆) = (Base‘𝑆)
21 eqid 2756 . . . . 5 (Base‘𝑀) = (Base‘𝑀)
22 eqid 2756 . . . . 5 ( ·𝑠𝐴) = ( ·𝑠𝐴)
231, 19, 2, 5, 20, 21, 22mendvsca 38259 . . . 4 ((𝑥 ∈ (Base‘𝑆) ∧ 𝑦 ∈ (𝑀 LMHom 𝑀)) → (𝑥( ·𝑠𝐴)𝑦) = (((Base‘𝑀) × {𝑥}) ∘𝑓 ( ·𝑠𝑀)𝑦))
24233adant1 1125 . . 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 19243 . . . 4 ((𝑆 ∈ CRing ∧ 𝑥 ∈ (Base‘𝑆) ∧ 𝑦 ∈ (𝑀 LMHom 𝑀)) → (((Base‘𝑀) × {𝑥}) ∘𝑓 ( ·𝑠𝑀)𝑦) ∈ (𝑀 LMHom 𝑀))
26253adant1l 1186 . . 3 (((𝑀 ∈ LMod ∧ 𝑆 ∈ CRing) ∧ 𝑥 ∈ (Base‘𝑆) ∧ 𝑦 ∈ (𝑀 LMHom 𝑀)) → (((Base‘𝑀) × {𝑥}) ∘𝑓 ( ·𝑠𝑀)𝑦) ∈ (𝑀 LMHom 𝑀))
2724, 26eqeltrd 2835 . 2 (((𝑀 ∈ LMod ∧ 𝑆 ∈ CRing) ∧ 𝑥 ∈ (Base‘𝑆) ∧ 𝑦 ∈ (𝑀 LMHom 𝑀)) → (𝑥( ·𝑠𝐴)𝑦) ∈ (𝑀 LMHom 𝑀))
28 simpr2 1236 . . . . 5 (((𝑀 ∈ LMod ∧ 𝑆 ∈ CRing) ∧ (𝑥 ∈ (Base‘𝑆) ∧ 𝑦 ∈ (𝑀 LMHom 𝑀) ∧ 𝑧 ∈ (𝑀 LMHom 𝑀))) → 𝑦 ∈ (𝑀 LMHom 𝑀))
29 simpr3 1238 . . . . 5 (((𝑀 ∈ LMod ∧ 𝑆 ∈ CRing) ∧ (𝑥 ∈ (Base‘𝑆) ∧ 𝑦 ∈ (𝑀 LMHom 𝑀) ∧ 𝑧 ∈ (𝑀 LMHom 𝑀))) → 𝑧 ∈ (𝑀 LMHom 𝑀))
30 eqid 2756 . . . . . 6 (+g𝑀) = (+g𝑀)
31 eqid 2756 . . . . . 6 (+g𝐴) = (+g𝐴)
321, 2, 30, 31mendplusg 38254 . . . . 5 ((𝑦 ∈ (𝑀 LMHom 𝑀) ∧ 𝑧 ∈ (𝑀 LMHom 𝑀)) → (𝑦(+g𝐴)𝑧) = (𝑦𝑓 (+g𝑀)𝑧))
3328, 29, 32syl2anc 696 . . . 4 (((𝑀 ∈ LMod ∧ 𝑆 ∈ CRing) ∧ (𝑥 ∈ (Base‘𝑆) ∧ 𝑦 ∈ (𝑀 LMHom 𝑀) ∧ 𝑧 ∈ (𝑀 LMHom 𝑀))) → (𝑦(+g𝐴)𝑧) = (𝑦𝑓 (+g𝑀)𝑧))
3433oveq2d 6825 . . 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 1234 . . . 4 (((𝑀 ∈ LMod ∧ 𝑆 ∈ CRing) ∧ (𝑥 ∈ (Base‘𝑆) ∧ 𝑦 ∈ (𝑀 LMHom 𝑀) ∧ 𝑧 ∈ (𝑀 LMHom 𝑀))) → 𝑥 ∈ (Base‘𝑆))
3618adantr 472 . . . . 5 (((𝑀 ∈ LMod ∧ 𝑆 ∈ CRing) ∧ (𝑥 ∈ (Base‘𝑆) ∧ 𝑦 ∈ (𝑀 LMHom 𝑀) ∧ 𝑧 ∈ (𝑀 LMHom 𝑀))) → 𝐴 ∈ Grp)
372, 31grpcl 17627 . . . . 5 ((𝐴 ∈ Grp ∧ 𝑦 ∈ (𝑀 LMHom 𝑀) ∧ 𝑧 ∈ (𝑀 LMHom 𝑀)) → (𝑦(+g𝐴)𝑧) ∈ (𝑀 LMHom 𝑀))
3836, 28, 29, 37syl3anc 1477 . . . 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 38259 . . . 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 696 . . 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 696 . . . . 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 38259 . . . . . 6 ((𝑥 ∈ (Base‘𝑆) ∧ 𝑧 ∈ (𝑀 LMHom 𝑀)) → (𝑥( ·𝑠𝐴)𝑧) = (((Base‘𝑀) × {𝑥}) ∘𝑓 ( ·𝑠𝑀)𝑧))
4335, 29, 42syl2anc 696 . . . . 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 6827 . . . 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 1200 . . . . 5 (((𝑀 ∈ LMod ∧ 𝑆 ∈ CRing) ∧ (𝑥 ∈ (Base‘𝑆) ∧ 𝑦 ∈ (𝑀 LMHom 𝑀) ∧ 𝑧 ∈ (𝑀 LMHom 𝑀))) → (𝑥( ·𝑠𝐴)𝑦) ∈ (𝑀 LMHom 𝑀))
46 eleq1w 2818 . . . . . . . . 9 (𝑦 = 𝑧 → (𝑦 ∈ (𝑀 LMHom 𝑀) ↔ 𝑧 ∈ (𝑀 LMHom 𝑀)))
47463anbi3d 1550 . . . . . . . 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 6817 . . . . . . . . 9 (𝑦 = 𝑧 → (𝑥( ·𝑠𝐴)𝑦) = (𝑥( ·𝑠𝐴)𝑧))
4948eleq1d 2820 . . . . . . . 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 2404 . . . . . 6 (((𝑀 ∈ LMod ∧ 𝑆 ∈ CRing) ∧ 𝑥 ∈ (Base‘𝑆) ∧ 𝑧 ∈ (𝑀 LMHom 𝑀)) → (𝑥( ·𝑠𝐴)𝑧) ∈ (𝑀 LMHom 𝑀))
52513adant3r2 1199 . . . . 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 38254 . . . . 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 696 . . . 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 6360 . . . . 5 (((𝑀 ∈ LMod ∧ 𝑆 ∈ CRing) ∧ (𝑥 ∈ (Base‘𝑆) ∧ 𝑦 ∈ (𝑀 LMHom 𝑀) ∧ 𝑧 ∈ (𝑀 LMHom 𝑀))) → (Base‘𝑀) ∈ V)
56 fconst6g 6251 . . . . . 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 19232 . . . . . 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 19232 . . . . . 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 807 . . . . . 6 (((𝑀 ∈ LMod ∧ 𝑆 ∈ CRing) ∧ (𝑥 ∈ (Base‘𝑆) ∧ 𝑦 ∈ (𝑀 LMHom 𝑀) ∧ 𝑧 ∈ (𝑀 LMHom 𝑀))) → 𝑀 ∈ LMod)
6321, 30, 5, 19, 20lmodvsdi 19084 . . . . . 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 489 . . . . 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 7094 . . . 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 2800 . . 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 2800 . 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 6360 . . . 4 (((𝑀 ∈ LMod ∧ 𝑆 ∈ CRing) ∧ (𝑥 ∈ (Base‘𝑆) ∧ 𝑦 ∈ (Base‘𝑆) ∧ 𝑧 ∈ (𝑀 LMHom 𝑀))) → (Base‘𝑀) ∈ V)
69 simpr3 1238 . . . . 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 1234 . . . . 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 1236 . . . . 5 (((𝑀 ∈ LMod ∧ 𝑆 ∈ CRing) ∧ (𝑥 ∈ (Base‘𝑆) ∧ 𝑦 ∈ (Base‘𝑆) ∧ 𝑧 ∈ (𝑀 LMHom 𝑀))) → 𝑦 ∈ (Base‘𝑆))
74 fconst6g 6251 . . . . 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 807 . . . . 5 (((𝑀 ∈ LMod ∧ 𝑆 ∈ CRing) ∧ (𝑥 ∈ (Base‘𝑆) ∧ 𝑦 ∈ (Base‘𝑆) ∧ 𝑧 ∈ (𝑀 LMHom 𝑀))) → 𝑀 ∈ LMod)
77 eqid 2756 . . . . . 6 (+g𝑆) = (+g𝑆)
7821, 30, 5, 19, 20, 77lmodvsdir 19085 . . . . 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 489 . . . 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 7095 . . 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 472 . . . . . 6 (((𝑀 ∈ LMod ∧ 𝑆 ∈ CRing) ∧ (𝑥 ∈ (Base‘𝑆) ∧ 𝑦 ∈ (Base‘𝑆) ∧ 𝑧 ∈ (𝑀 LMHom 𝑀))) → 𝑆 ∈ Ring)
8220, 77ringacl 18774 . . . . . 6 ((𝑆 ∈ Ring ∧ 𝑥 ∈ (Base‘𝑆) ∧ 𝑦 ∈ (Base‘𝑆)) → (𝑥(+g𝑆)𝑦) ∈ (Base‘𝑆))
8381, 71, 73, 82syl3anc 1477 . . . . 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 38259 . . . . 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 696 . . . 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 7083 . . . . 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 6824 . . . 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 2793 . . 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 1199 . . . . 5 (((𝑀 ∈ LMod ∧ 𝑆 ∈ CRing) ∧ (𝑥 ∈ (Base‘𝑆) ∧ 𝑦 ∈ (Base‘𝑆) ∧ 𝑧 ∈ (𝑀 LMHom 𝑀))) → (𝑥( ·𝑠𝐴)𝑧) ∈ (𝑀 LMHom 𝑀))
90 eleq1w 2818 . . . . . . . . 9 (𝑥 = 𝑦 → (𝑥 ∈ (Base‘𝑆) ↔ 𝑦 ∈ (Base‘𝑆)))
91903anbi2d 1549 . . . . . . . 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 6816 . . . . . . . . 9 (𝑥 = 𝑦 → (𝑥( ·𝑠𝐴)𝑧) = (𝑦( ·𝑠𝐴)𝑧))
9392eleq1d 2820 . . . . . . . 8 (𝑥 = 𝑦 → ((𝑥( ·𝑠𝐴)𝑧) ∈ (𝑀 LMHom 𝑀) ↔ (𝑦( ·𝑠𝐴)𝑧) ∈ (𝑀 LMHom 𝑀)))
9491, 93imbi12d 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))))
9594, 51chvarv 2404 . . . . . 6 (((𝑀 ∈ LMod ∧ 𝑆 ∈ CRing) ∧ 𝑦 ∈ (Base‘𝑆) ∧ 𝑧 ∈ (𝑀 LMHom 𝑀)) → (𝑦( ·𝑠𝐴)𝑧) ∈ (𝑀 LMHom 𝑀))
96953adant3r1 1198 . . . . 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 38254 . . . . 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 696 . . . 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 696 . . . . 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 38259 . . . . . 6 ((𝑦 ∈ (Base‘𝑆) ∧ 𝑧 ∈ (𝑀 LMHom 𝑀)) → (𝑦( ·𝑠𝐴)𝑧) = (((Base‘𝑀) × {𝑦}) ∘𝑓 ( ·𝑠𝑀)𝑧))
10173, 69, 100syl2anc 696 . . . . 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 6827 . . . 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 2790 . . 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 2800 . 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 6839 . . . 4 ((((𝑀 ∈ LMod ∧ 𝑆 ∈ CRing) ∧ (𝑥 ∈ (Base‘𝑆) ∧ 𝑦 ∈ (Base‘𝑆) ∧ 𝑧 ∈ (𝑀 LMHom 𝑀))) ∧ 𝑘 ∈ (Base‘𝑀)) → (𝑥(.r𝑆)𝑦) ∈ V)
10670ffvelrnda 6518 . . . 4 ((((𝑀 ∈ LMod ∧ 𝑆 ∈ CRing) ∧ (𝑥 ∈ (Base‘𝑆) ∧ 𝑦 ∈ (Base‘𝑆) ∧ 𝑧 ∈ (𝑀 LMHom 𝑀))) ∧ 𝑘 ∈ (Base‘𝑀)) → (𝑧𝑘) ∈ (Base‘𝑀))
107 fconstmpt 5316 . . . . 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 6407 . . . 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 7075 . . 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 2756 . . . . . 6 (.r𝑆) = (.r𝑆)
11220, 111ringcl 18757 . . . . 5 ((𝑆 ∈ Ring ∧ 𝑥 ∈ (Base‘𝑆) ∧ 𝑦 ∈ (Base‘𝑆)) → (𝑥(.r𝑆)𝑦) ∈ (Base‘𝑆))
11381, 71, 73, 112syl3anc 1477 . . . 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 38259 . . . 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 696 . . 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 472 . . . . 5 ((((𝑀 ∈ LMod ∧ 𝑆 ∈ CRing) ∧ (𝑥 ∈ (Base‘𝑆) ∧ 𝑦 ∈ (Base‘𝑆) ∧ 𝑧 ∈ (𝑀 LMHom 𝑀))) ∧ 𝑘 ∈ (Base‘𝑀)) → 𝑥 ∈ (Base‘𝑆))
117 ovexd 6839 . . . . 5 ((((𝑀 ∈ LMod ∧ 𝑆 ∈ CRing) ∧ (𝑥 ∈ (Base‘𝑆) ∧ 𝑦 ∈ (Base‘𝑆) ∧ 𝑧 ∈ (𝑀 LMHom 𝑀))) ∧ 𝑘 ∈ (Base‘𝑀)) → (𝑦( ·𝑠𝑀)(𝑧𝑘)) ∈ V)
118 fconstmpt 5316 . . . . . 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 1263 . . . . . . 7 ((((𝑀 ∈ LMod ∧ 𝑆 ∈ CRing) ∧ (𝑥 ∈ (Base‘𝑆) ∧ 𝑦 ∈ (Base‘𝑆) ∧ 𝑧 ∈ (𝑀 LMHom 𝑀))) ∧ 𝑘 ∈ (Base‘𝑀)) → 𝑦 ∈ (Base‘𝑆))
121 fconstmpt 5316 . . . . . . . 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 7075 . . . . . 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 2790 . . . . 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 7075 . . . 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 38259 . . . . 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 696 . . . 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 472 . . . . . 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 19086 . . . . . 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 1479 . . . . 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 4892 . . . 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 2800 . . 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 2800 . 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 472 . . . . 5 (((𝑀 ∈ LMod ∧ 𝑆 ∈ CRing) ∧ 𝑥 ∈ (𝑀 LMHom 𝑀)) → 𝑆 ∈ Ring)
135 eqid 2756 . . . . . 6 (1r𝑆) = (1r𝑆)
13620, 135ringidcl 18764 . . . . 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 38259 . . . 4 (((1r𝑆) ∈ (Base‘𝑆) ∧ 𝑥 ∈ (𝑀 LMHom 𝑀)) → ((1r𝑆)( ·𝑠𝐴)𝑥) = (((Base‘𝑀) × {(1r𝑆)}) ∘𝑓 ( ·𝑠𝑀)𝑥))
139137, 138sylancom 704 . . 3 (((𝑀 ∈ LMod ∧ 𝑆 ∈ CRing) ∧ 𝑥 ∈ (𝑀 LMHom 𝑀)) → ((1r𝑆)( ·𝑠𝐴)𝑥) = (((Base‘𝑀) × {(1r𝑆)}) ∘𝑓 ( ·𝑠𝑀)𝑥))
140 fvexd 6360 . . . 4 (((𝑀 ∈ LMod ∧ 𝑆 ∈ CRing) ∧ 𝑥 ∈ (𝑀 LMHom 𝑀)) → (Base‘𝑀) ∈ V)
14121, 21lmhmf 19232 . . . . 5 (𝑥 ∈ (𝑀 LMHom 𝑀) → 𝑥:(Base‘𝑀)⟶(Base‘𝑀))
142141adantl 473 . . . 4 (((𝑀 ∈ LMod ∧ 𝑆 ∈ CRing) ∧ 𝑥 ∈ (𝑀 LMHom 𝑀)) → 𝑥:(Base‘𝑀)⟶(Base‘𝑀))
143 simpll 807 . . . . 5 (((𝑀 ∈ LMod ∧ 𝑆 ∈ CRing) ∧ 𝑥 ∈ (𝑀 LMHom 𝑀)) → 𝑀 ∈ LMod)
14421, 5, 19, 135lmodvs1 19089 . . . . 5 ((𝑀 ∈ LMod ∧ 𝑦 ∈ (Base‘𝑀)) → ((1r𝑆)( ·𝑠𝑀)𝑦) = 𝑦)
145143, 144sylan 489 . . . 4 ((((𝑀 ∈ LMod ∧ 𝑆 ∈ CRing) ∧ 𝑥 ∈ (𝑀 LMHom 𝑀)) ∧ 𝑦 ∈ (Base‘𝑀)) → ((1r𝑆)( ·𝑠𝑀)𝑦) = 𝑦)
146140, 142, 137, 145caofid0l 7086 . . 3 (((𝑀 ∈ LMod ∧ 𝑆 ∈ CRing) ∧ 𝑥 ∈ (𝑀 LMHom 𝑀)) → (((Base‘𝑀) × {(1r𝑆)}) ∘𝑓 ( ·𝑠𝑀)𝑥) = 𝑥)
147139, 146eqtrd 2790 . 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 19067 1 ((𝑀 ∈ LMod ∧ 𝑆 ∈ CRing) → 𝐴 ∈ LMod)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 383  w3a 1072   = wceq 1628  wcel 2135  Vcvv 3336  {csn 4317  cmpt 4877   × cxp 5260  wf 6041  cfv 6045  (class class class)co 6809  𝑓 cof 7056  Basecbs 16055  +gcplusg 16139  .rcmulr 16140  Scalarcsca 16142   ·𝑠 cvsca 16143  Grpcgrp 17619  1rcur 18697  Ringcrg 18743  CRingccrg 18744  LModclmod 19061   LMHom clmhm 19217  MEndocmend 38243
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1867  ax-4 1882  ax-5 1984  ax-6 2050  ax-7 2086  ax-8 2137  ax-9 2144  ax-10 2164  ax-11 2179  ax-12 2192  ax-13 2387  ax-ext 2736  ax-rep 4919  ax-sep 4929  ax-nul 4937  ax-pow 4988  ax-pr 5051  ax-un 7110  ax-cnex 10180  ax-resscn 10181  ax-1cn 10182  ax-icn 10183  ax-addcl 10184  ax-addrcl 10185  ax-mulcl 10186  ax-mulrcl 10187  ax-mulcom 10188  ax-addass 10189  ax-mulass 10190  ax-distr 10191  ax-i2m1 10192  ax-1ne0 10193  ax-1rid 10194  ax-rnegex 10195  ax-rrecex 10196  ax-cnre 10197  ax-pre-lttri 10198  ax-pre-lttrn 10199  ax-pre-ltadd 10200  ax-pre-mulgt0 10201
This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-3or 1073  df-3an 1074  df-tru 1631  df-ex 1850  df-nf 1855  df-sb 2043  df-eu 2607  df-mo 2608  df-clab 2743  df-cleq 2749  df-clel 2752  df-nfc 2887  df-ne 2929  df-nel 3032  df-ral 3051  df-rex 3052  df-reu 3053  df-rmo 3054  df-rab 3055  df-v 3338  df-sbc 3573  df-csb 3671  df-dif 3714  df-un 3716  df-in 3718  df-ss 3725  df-pss 3727  df-nul 4055  df-if 4227  df-pw 4300  df-sn 4318  df-pr 4320  df-tp 4322  df-op 4324  df-uni 4585  df-int 4624  df-iun 4670  df-br 4801  df-opab 4861  df-mpt 4878  df-tr 4901  df-id 5170  df-eprel 5175  df-po 5183  df-so 5184  df-fr 5221  df-we 5223  df-xp 5268  df-rel 5269  df-cnv 5270  df-co 5271  df-dm 5272  df-rn 5273  df-res 5274  df-ima 5275  df-pred 5837  df-ord 5883  df-on 5884  df-lim 5885  df-suc 5886  df-iota 6008  df-fun 6047  df-fn 6048  df-f 6049  df-f1 6050  df-fo 6051  df-f1o 6052  df-fv 6053  df-riota 6770  df-ov 6812  df-oprab 6813  df-mpt2 6814  df-of 7058  df-om 7227  df-1st 7329  df-2nd 7330  df-wrecs 7572  df-recs 7633  df-rdg 7671  df-1o 7725  df-oadd 7729  df-er 7907  df-map 8021  df-en 8118  df-dom 8119  df-sdom 8120  df-fin 8121  df-pnf 10264  df-mnf 10265  df-xr 10266  df-ltxr 10267  df-le 10268  df-sub 10456  df-neg 10457  df-nn 11209  df-2 11267  df-3 11268  df-4 11269  df-5 11270  df-6 11271  df-n0 11481  df-z 11566  df-uz 11876  df-fz 12516  df-struct 16057  df-ndx 16058  df-slot 16059  df-base 16061  df-sets 16062  df-plusg 16152  df-mulr 16153  df-sca 16155  df-vsca 16156  df-0g 16300  df-mgm 17439  df-sgrp 17481  df-mnd 17492  df-mhm 17532  df-grp 17622  df-minusg 17623  df-ghm 17855  df-cmn 18391  df-abl 18392  df-mgp 18686  df-ur 18698  df-ring 18745  df-cring 18746  df-lmod 19063  df-lmhm 19220  df-mend 38244
This theorem is referenced by:  mendassa  38262
  Copyright terms: Public domain W3C validator