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Theorem mendassa 43178
Description: The module endomorphism algebra is an algebra. (Contributed by Mario Carneiro, 22-Sep-2015.)
Hypotheses
Ref Expression
mendassa.a 𝐴 = (MEndo‘𝑀)
mendassa.s 𝑆 = (Scalar‘𝑀)
Assertion
Ref Expression
mendassa ((𝑀 ∈ LMod ∧ 𝑆 ∈ CRing) → 𝐴 ∈ AssAlg)

Proof of Theorem mendassa
Dummy variables 𝑥 𝑦 𝑧 𝑣 𝑤 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 mendassa.a . . . 4 𝐴 = (MEndo‘𝑀)
21mendbas 43168 . . 3 (𝑀 LMHom 𝑀) = (Base‘𝐴)
32a1i 11 . 2 ((𝑀 ∈ LMod ∧ 𝑆 ∈ CRing) → (𝑀 LMHom 𝑀) = (Base‘𝐴))
4 mendassa.s . . . 4 𝑆 = (Scalar‘𝑀)
51, 4mendsca 43173 . . 3 𝑆 = (Scalar‘𝐴)
65a1i 11 . 2 ((𝑀 ∈ LMod ∧ 𝑆 ∈ CRing) → 𝑆 = (Scalar‘𝐴))
7 eqidd 2735 . 2 ((𝑀 ∈ LMod ∧ 𝑆 ∈ CRing) → (Base‘𝑆) = (Base‘𝑆))
8 eqidd 2735 . 2 ((𝑀 ∈ LMod ∧ 𝑆 ∈ CRing) → ( ·𝑠𝐴) = ( ·𝑠𝐴))
9 eqidd 2735 . 2 ((𝑀 ∈ LMod ∧ 𝑆 ∈ CRing) → (.r𝐴) = (.r𝐴))
101, 4mendlmod 43177 . 2 ((𝑀 ∈ LMod ∧ 𝑆 ∈ CRing) → 𝐴 ∈ LMod)
111mendring 43176 . . 3 (𝑀 ∈ LMod → 𝐴 ∈ Ring)
1211adantr 480 . 2 ((𝑀 ∈ LMod ∧ 𝑆 ∈ CRing) → 𝐴 ∈ Ring)
13 simpr3 1195 . . . . . . 7 (((𝑀 ∈ LMod ∧ 𝑆 ∈ CRing) ∧ (𝑥 ∈ (Base‘𝑆) ∧ 𝑦 ∈ (𝑀 LMHom 𝑀) ∧ 𝑧 ∈ (𝑀 LMHom 𝑀))) → 𝑧 ∈ (𝑀 LMHom 𝑀))
14 eqid 2734 . . . . . . . 8 (Base‘𝑀) = (Base‘𝑀)
1514, 14lmhmf 21050 . . . . . . 7 (𝑧 ∈ (𝑀 LMHom 𝑀) → 𝑧:(Base‘𝑀)⟶(Base‘𝑀))
1613, 15syl 17 . . . . . 6 (((𝑀 ∈ LMod ∧ 𝑆 ∈ CRing) ∧ (𝑥 ∈ (Base‘𝑆) ∧ 𝑦 ∈ (𝑀 LMHom 𝑀) ∧ 𝑧 ∈ (𝑀 LMHom 𝑀))) → 𝑧:(Base‘𝑀)⟶(Base‘𝑀))
1716ffvelcdmda 7103 . . . . 5 ((((𝑀 ∈ LMod ∧ 𝑆 ∈ CRing) ∧ (𝑥 ∈ (Base‘𝑆) ∧ 𝑦 ∈ (𝑀 LMHom 𝑀) ∧ 𝑧 ∈ (𝑀 LMHom 𝑀))) ∧ 𝑣 ∈ (Base‘𝑀)) → (𝑧𝑣) ∈ (Base‘𝑀))
1816feqmptd 6976 . . . . 5 (((𝑀 ∈ LMod ∧ 𝑆 ∈ CRing) ∧ (𝑥 ∈ (Base‘𝑆) ∧ 𝑦 ∈ (𝑀 LMHom 𝑀) ∧ 𝑧 ∈ (𝑀 LMHom 𝑀))) → 𝑧 = (𝑣 ∈ (Base‘𝑀) ↦ (𝑧𝑣)))
19 simpr1 1193 . . . . . . 7 (((𝑀 ∈ LMod ∧ 𝑆 ∈ CRing) ∧ (𝑥 ∈ (Base‘𝑆) ∧ 𝑦 ∈ (𝑀 LMHom 𝑀) ∧ 𝑧 ∈ (𝑀 LMHom 𝑀))) → 𝑥 ∈ (Base‘𝑆))
20 simpr2 1194 . . . . . . 7 (((𝑀 ∈ LMod ∧ 𝑆 ∈ CRing) ∧ (𝑥 ∈ (Base‘𝑆) ∧ 𝑦 ∈ (𝑀 LMHom 𝑀) ∧ 𝑧 ∈ (𝑀 LMHom 𝑀))) → 𝑦 ∈ (𝑀 LMHom 𝑀))
21 eqid 2734 . . . . . . . 8 ( ·𝑠𝑀) = ( ·𝑠𝑀)
22 eqid 2734 . . . . . . . 8 (Base‘𝑆) = (Base‘𝑆)
23 eqid 2734 . . . . . . . 8 ( ·𝑠𝐴) = ( ·𝑠𝐴)
241, 21, 2, 4, 22, 14, 23mendvsca 43175 . . . . . . 7 ((𝑥 ∈ (Base‘𝑆) ∧ 𝑦 ∈ (𝑀 LMHom 𝑀)) → (𝑥( ·𝑠𝐴)𝑦) = (((Base‘𝑀) × {𝑥}) ∘f ( ·𝑠𝑀)𝑦))
2519, 20, 24syl2anc 584 . . . . . 6 (((𝑀 ∈ LMod ∧ 𝑆 ∈ CRing) ∧ (𝑥 ∈ (Base‘𝑆) ∧ 𝑦 ∈ (𝑀 LMHom 𝑀) ∧ 𝑧 ∈ (𝑀 LMHom 𝑀))) → (𝑥( ·𝑠𝐴)𝑦) = (((Base‘𝑀) × {𝑥}) ∘f ( ·𝑠𝑀)𝑦))
26 fvexd 6921 . . . . . . 7 (((𝑀 ∈ LMod ∧ 𝑆 ∈ CRing) ∧ (𝑥 ∈ (Base‘𝑆) ∧ 𝑦 ∈ (𝑀 LMHom 𝑀) ∧ 𝑧 ∈ (𝑀 LMHom 𝑀))) → (Base‘𝑀) ∈ V)
27 simplr1 1214 . . . . . . 7 ((((𝑀 ∈ LMod ∧ 𝑆 ∈ CRing) ∧ (𝑥 ∈ (Base‘𝑆) ∧ 𝑦 ∈ (𝑀 LMHom 𝑀) ∧ 𝑧 ∈ (𝑀 LMHom 𝑀))) ∧ 𝑤 ∈ (Base‘𝑀)) → 𝑥 ∈ (Base‘𝑆))
28 fvexd 6921 . . . . . . 7 ((((𝑀 ∈ LMod ∧ 𝑆 ∈ CRing) ∧ (𝑥 ∈ (Base‘𝑆) ∧ 𝑦 ∈ (𝑀 LMHom 𝑀) ∧ 𝑧 ∈ (𝑀 LMHom 𝑀))) ∧ 𝑤 ∈ (Base‘𝑀)) → (𝑦𝑤) ∈ V)
29 fconstmpt 5750 . . . . . . . 8 ((Base‘𝑀) × {𝑥}) = (𝑤 ∈ (Base‘𝑀) ↦ 𝑥)
3029a1i 11 . . . . . . 7 (((𝑀 ∈ LMod ∧ 𝑆 ∈ CRing) ∧ (𝑥 ∈ (Base‘𝑆) ∧ 𝑦 ∈ (𝑀 LMHom 𝑀) ∧ 𝑧 ∈ (𝑀 LMHom 𝑀))) → ((Base‘𝑀) × {𝑥}) = (𝑤 ∈ (Base‘𝑀) ↦ 𝑥))
3114, 14lmhmf 21050 . . . . . . . . 9 (𝑦 ∈ (𝑀 LMHom 𝑀) → 𝑦:(Base‘𝑀)⟶(Base‘𝑀))
3220, 31syl 17 . . . . . . . 8 (((𝑀 ∈ LMod ∧ 𝑆 ∈ CRing) ∧ (𝑥 ∈ (Base‘𝑆) ∧ 𝑦 ∈ (𝑀 LMHom 𝑀) ∧ 𝑧 ∈ (𝑀 LMHom 𝑀))) → 𝑦:(Base‘𝑀)⟶(Base‘𝑀))
3332feqmptd 6976 . . . . . . 7 (((𝑀 ∈ LMod ∧ 𝑆 ∈ CRing) ∧ (𝑥 ∈ (Base‘𝑆) ∧ 𝑦 ∈ (𝑀 LMHom 𝑀) ∧ 𝑧 ∈ (𝑀 LMHom 𝑀))) → 𝑦 = (𝑤 ∈ (Base‘𝑀) ↦ (𝑦𝑤)))
3426, 27, 28, 30, 33offval2 7716 . . . . . 6 (((𝑀 ∈ LMod ∧ 𝑆 ∈ CRing) ∧ (𝑥 ∈ (Base‘𝑆) ∧ 𝑦 ∈ (𝑀 LMHom 𝑀) ∧ 𝑧 ∈ (𝑀 LMHom 𝑀))) → (((Base‘𝑀) × {𝑥}) ∘f ( ·𝑠𝑀)𝑦) = (𝑤 ∈ (Base‘𝑀) ↦ (𝑥( ·𝑠𝑀)(𝑦𝑤))))
3525, 34eqtrd 2774 . . . . 5 (((𝑀 ∈ LMod ∧ 𝑆 ∈ CRing) ∧ (𝑥 ∈ (Base‘𝑆) ∧ 𝑦 ∈ (𝑀 LMHom 𝑀) ∧ 𝑧 ∈ (𝑀 LMHom 𝑀))) → (𝑥( ·𝑠𝐴)𝑦) = (𝑤 ∈ (Base‘𝑀) ↦ (𝑥( ·𝑠𝑀)(𝑦𝑤))))
36 fveq2 6906 . . . . . 6 (𝑤 = (𝑧𝑣) → (𝑦𝑤) = (𝑦‘(𝑧𝑣)))
3736oveq2d 7446 . . . . 5 (𝑤 = (𝑧𝑣) → (𝑥( ·𝑠𝑀)(𝑦𝑤)) = (𝑥( ·𝑠𝑀)(𝑦‘(𝑧𝑣))))
3817, 18, 35, 37fmptco 7148 . . . 4 (((𝑀 ∈ LMod ∧ 𝑆 ∈ CRing) ∧ (𝑥 ∈ (Base‘𝑆) ∧ 𝑦 ∈ (𝑀 LMHom 𝑀) ∧ 𝑧 ∈ (𝑀 LMHom 𝑀))) → ((𝑥( ·𝑠𝐴)𝑦) ∘ 𝑧) = (𝑣 ∈ (Base‘𝑀) ↦ (𝑥( ·𝑠𝑀)(𝑦‘(𝑧𝑣)))))
39 simplr1 1214 . . . . 5 ((((𝑀 ∈ LMod ∧ 𝑆 ∈ CRing) ∧ (𝑥 ∈ (Base‘𝑆) ∧ 𝑦 ∈ (𝑀 LMHom 𝑀) ∧ 𝑧 ∈ (𝑀 LMHom 𝑀))) ∧ 𝑣 ∈ (Base‘𝑀)) → 𝑥 ∈ (Base‘𝑆))
40 fvexd 6921 . . . . 5 ((((𝑀 ∈ LMod ∧ 𝑆 ∈ CRing) ∧ (𝑥 ∈ (Base‘𝑆) ∧ 𝑦 ∈ (𝑀 LMHom 𝑀) ∧ 𝑧 ∈ (𝑀 LMHom 𝑀))) ∧ 𝑣 ∈ (Base‘𝑀)) → (𝑦‘(𝑧𝑣)) ∈ V)
41 fconstmpt 5750 . . . . . 6 ((Base‘𝑀) × {𝑥}) = (𝑣 ∈ (Base‘𝑀) ↦ 𝑥)
4241a1i 11 . . . . 5 (((𝑀 ∈ LMod ∧ 𝑆 ∈ CRing) ∧ (𝑥 ∈ (Base‘𝑆) ∧ 𝑦 ∈ (𝑀 LMHom 𝑀) ∧ 𝑧 ∈ (𝑀 LMHom 𝑀))) → ((Base‘𝑀) × {𝑥}) = (𝑣 ∈ (Base‘𝑀) ↦ 𝑥))
43 eqid 2734 . . . . . . . 8 (.r𝐴) = (.r𝐴)
441, 2, 43mendmulr 43172 . . . . . . 7 ((𝑦 ∈ (𝑀 LMHom 𝑀) ∧ 𝑧 ∈ (𝑀 LMHom 𝑀)) → (𝑦(.r𝐴)𝑧) = (𝑦𝑧))
4520, 13, 44syl2anc 584 . . . . . 6 (((𝑀 ∈ LMod ∧ 𝑆 ∈ CRing) ∧ (𝑥 ∈ (Base‘𝑆) ∧ 𝑦 ∈ (𝑀 LMHom 𝑀) ∧ 𝑧 ∈ (𝑀 LMHom 𝑀))) → (𝑦(.r𝐴)𝑧) = (𝑦𝑧))
46 fcompt 7152 . . . . . . 7 ((𝑦:(Base‘𝑀)⟶(Base‘𝑀) ∧ 𝑧:(Base‘𝑀)⟶(Base‘𝑀)) → (𝑦𝑧) = (𝑣 ∈ (Base‘𝑀) ↦ (𝑦‘(𝑧𝑣))))
4732, 16, 46syl2anc 584 . . . . . 6 (((𝑀 ∈ LMod ∧ 𝑆 ∈ CRing) ∧ (𝑥 ∈ (Base‘𝑆) ∧ 𝑦 ∈ (𝑀 LMHom 𝑀) ∧ 𝑧 ∈ (𝑀 LMHom 𝑀))) → (𝑦𝑧) = (𝑣 ∈ (Base‘𝑀) ↦ (𝑦‘(𝑧𝑣))))
4845, 47eqtrd 2774 . . . . 5 (((𝑀 ∈ LMod ∧ 𝑆 ∈ CRing) ∧ (𝑥 ∈ (Base‘𝑆) ∧ 𝑦 ∈ (𝑀 LMHom 𝑀) ∧ 𝑧 ∈ (𝑀 LMHom 𝑀))) → (𝑦(.r𝐴)𝑧) = (𝑣 ∈ (Base‘𝑀) ↦ (𝑦‘(𝑧𝑣))))
4926, 39, 40, 42, 48offval2 7716 . . . 4 (((𝑀 ∈ LMod ∧ 𝑆 ∈ CRing) ∧ (𝑥 ∈ (Base‘𝑆) ∧ 𝑦 ∈ (𝑀 LMHom 𝑀) ∧ 𝑧 ∈ (𝑀 LMHom 𝑀))) → (((Base‘𝑀) × {𝑥}) ∘f ( ·𝑠𝑀)(𝑦(.r𝐴)𝑧)) = (𝑣 ∈ (Base‘𝑀) ↦ (𝑥( ·𝑠𝑀)(𝑦‘(𝑧𝑣)))))
5038, 49eqtr4d 2777 . . 3 (((𝑀 ∈ LMod ∧ 𝑆 ∈ CRing) ∧ (𝑥 ∈ (Base‘𝑆) ∧ 𝑦 ∈ (𝑀 LMHom 𝑀) ∧ 𝑧 ∈ (𝑀 LMHom 𝑀))) → ((𝑥( ·𝑠𝐴)𝑦) ∘ 𝑧) = (((Base‘𝑀) × {𝑥}) ∘f ( ·𝑠𝑀)(𝑦(.r𝐴)𝑧)))
5110adantr 480 . . . . 5 (((𝑀 ∈ LMod ∧ 𝑆 ∈ CRing) ∧ (𝑥 ∈ (Base‘𝑆) ∧ 𝑦 ∈ (𝑀 LMHom 𝑀) ∧ 𝑧 ∈ (𝑀 LMHom 𝑀))) → 𝐴 ∈ LMod)
522, 5, 23, 22lmodvscl 20892 . . . . 5 ((𝐴 ∈ LMod ∧ 𝑥 ∈ (Base‘𝑆) ∧ 𝑦 ∈ (𝑀 LMHom 𝑀)) → (𝑥( ·𝑠𝐴)𝑦) ∈ (𝑀 LMHom 𝑀))
5351, 19, 20, 52syl3anc 1370 . . . 4 (((𝑀 ∈ LMod ∧ 𝑆 ∈ CRing) ∧ (𝑥 ∈ (Base‘𝑆) ∧ 𝑦 ∈ (𝑀 LMHom 𝑀) ∧ 𝑧 ∈ (𝑀 LMHom 𝑀))) → (𝑥( ·𝑠𝐴)𝑦) ∈ (𝑀 LMHom 𝑀))
541, 2, 43mendmulr 43172 . . . 4 (((𝑥( ·𝑠𝐴)𝑦) ∈ (𝑀 LMHom 𝑀) ∧ 𝑧 ∈ (𝑀 LMHom 𝑀)) → ((𝑥( ·𝑠𝐴)𝑦)(.r𝐴)𝑧) = ((𝑥( ·𝑠𝐴)𝑦) ∘ 𝑧))
5553, 13, 54syl2anc 584 . . 3 (((𝑀 ∈ LMod ∧ 𝑆 ∈ CRing) ∧ (𝑥 ∈ (Base‘𝑆) ∧ 𝑦 ∈ (𝑀 LMHom 𝑀) ∧ 𝑧 ∈ (𝑀 LMHom 𝑀))) → ((𝑥( ·𝑠𝐴)𝑦)(.r𝐴)𝑧) = ((𝑥( ·𝑠𝐴)𝑦) ∘ 𝑧))
5612adantr 480 . . . . 5 (((𝑀 ∈ LMod ∧ 𝑆 ∈ CRing) ∧ (𝑥 ∈ (Base‘𝑆) ∧ 𝑦 ∈ (𝑀 LMHom 𝑀) ∧ 𝑧 ∈ (𝑀 LMHom 𝑀))) → 𝐴 ∈ Ring)
572, 43ringcl 20267 . . . . 5 ((𝐴 ∈ Ring ∧ 𝑦 ∈ (𝑀 LMHom 𝑀) ∧ 𝑧 ∈ (𝑀 LMHom 𝑀)) → (𝑦(.r𝐴)𝑧) ∈ (𝑀 LMHom 𝑀))
5856, 20, 13, 57syl3anc 1370 . . . 4 (((𝑀 ∈ LMod ∧ 𝑆 ∈ CRing) ∧ (𝑥 ∈ (Base‘𝑆) ∧ 𝑦 ∈ (𝑀 LMHom 𝑀) ∧ 𝑧 ∈ (𝑀 LMHom 𝑀))) → (𝑦(.r𝐴)𝑧) ∈ (𝑀 LMHom 𝑀))
591, 21, 2, 4, 22, 14, 23mendvsca 43175 . . . 4 ((𝑥 ∈ (Base‘𝑆) ∧ (𝑦(.r𝐴)𝑧) ∈ (𝑀 LMHom 𝑀)) → (𝑥( ·𝑠𝐴)(𝑦(.r𝐴)𝑧)) = (((Base‘𝑀) × {𝑥}) ∘f ( ·𝑠𝑀)(𝑦(.r𝐴)𝑧)))
6019, 58, 59syl2anc 584 . . 3 (((𝑀 ∈ LMod ∧ 𝑆 ∈ CRing) ∧ (𝑥 ∈ (Base‘𝑆) ∧ 𝑦 ∈ (𝑀 LMHom 𝑀) ∧ 𝑧 ∈ (𝑀 LMHom 𝑀))) → (𝑥( ·𝑠𝐴)(𝑦(.r𝐴)𝑧)) = (((Base‘𝑀) × {𝑥}) ∘f ( ·𝑠𝑀)(𝑦(.r𝐴)𝑧)))
6150, 55, 603eqtr4d 2784 . 2 (((𝑀 ∈ LMod ∧ 𝑆 ∈ CRing) ∧ (𝑥 ∈ (Base‘𝑆) ∧ 𝑦 ∈ (𝑀 LMHom 𝑀) ∧ 𝑧 ∈ (𝑀 LMHom 𝑀))) → ((𝑥( ·𝑠𝐴)𝑦)(.r𝐴)𝑧) = (𝑥( ·𝑠𝐴)(𝑦(.r𝐴)𝑧)))
62 simplr2 1215 . . . . . 6 ((((𝑀 ∈ LMod ∧ 𝑆 ∈ CRing) ∧ (𝑥 ∈ (Base‘𝑆) ∧ 𝑦 ∈ (𝑀 LMHom 𝑀) ∧ 𝑧 ∈ (𝑀 LMHom 𝑀))) ∧ 𝑣 ∈ (Base‘𝑀)) → 𝑦 ∈ (𝑀 LMHom 𝑀))
634, 22, 14, 21, 21lmhmlin 21051 . . . . . 6 ((𝑦 ∈ (𝑀 LMHom 𝑀) ∧ 𝑥 ∈ (Base‘𝑆) ∧ (𝑧𝑣) ∈ (Base‘𝑀)) → (𝑦‘(𝑥( ·𝑠𝑀)(𝑧𝑣))) = (𝑥( ·𝑠𝑀)(𝑦‘(𝑧𝑣))))
6462, 39, 17, 63syl3anc 1370 . . . . 5 ((((𝑀 ∈ LMod ∧ 𝑆 ∈ CRing) ∧ (𝑥 ∈ (Base‘𝑆) ∧ 𝑦 ∈ (𝑀 LMHom 𝑀) ∧ 𝑧 ∈ (𝑀 LMHom 𝑀))) ∧ 𝑣 ∈ (Base‘𝑀)) → (𝑦‘(𝑥( ·𝑠𝑀)(𝑧𝑣))) = (𝑥( ·𝑠𝑀)(𝑦‘(𝑧𝑣))))
6564mpteq2dva 5247 . . . 4 (((𝑀 ∈ LMod ∧ 𝑆 ∈ CRing) ∧ (𝑥 ∈ (Base‘𝑆) ∧ 𝑦 ∈ (𝑀 LMHom 𝑀) ∧ 𝑧 ∈ (𝑀 LMHom 𝑀))) → (𝑣 ∈ (Base‘𝑀) ↦ (𝑦‘(𝑥( ·𝑠𝑀)(𝑧𝑣)))) = (𝑣 ∈ (Base‘𝑀) ↦ (𝑥( ·𝑠𝑀)(𝑦‘(𝑧𝑣)))))
66 simplll 775 . . . . . 6 ((((𝑀 ∈ LMod ∧ 𝑆 ∈ CRing) ∧ (𝑥 ∈ (Base‘𝑆) ∧ 𝑦 ∈ (𝑀 LMHom 𝑀) ∧ 𝑧 ∈ (𝑀 LMHom 𝑀))) ∧ 𝑣 ∈ (Base‘𝑀)) → 𝑀 ∈ LMod)
6714, 4, 21, 22lmodvscl 20892 . . . . . 6 ((𝑀 ∈ LMod ∧ 𝑥 ∈ (Base‘𝑆) ∧ (𝑧𝑣) ∈ (Base‘𝑀)) → (𝑥( ·𝑠𝑀)(𝑧𝑣)) ∈ (Base‘𝑀))
6866, 39, 17, 67syl3anc 1370 . . . . 5 ((((𝑀 ∈ LMod ∧ 𝑆 ∈ CRing) ∧ (𝑥 ∈ (Base‘𝑆) ∧ 𝑦 ∈ (𝑀 LMHom 𝑀) ∧ 𝑧 ∈ (𝑀 LMHom 𝑀))) ∧ 𝑣 ∈ (Base‘𝑀)) → (𝑥( ·𝑠𝑀)(𝑧𝑣)) ∈ (Base‘𝑀))
691, 21, 2, 4, 22, 14, 23mendvsca 43175 . . . . . . 7 ((𝑥 ∈ (Base‘𝑆) ∧ 𝑧 ∈ (𝑀 LMHom 𝑀)) → (𝑥( ·𝑠𝐴)𝑧) = (((Base‘𝑀) × {𝑥}) ∘f ( ·𝑠𝑀)𝑧))
7019, 13, 69syl2anc 584 . . . . . 6 (((𝑀 ∈ LMod ∧ 𝑆 ∈ CRing) ∧ (𝑥 ∈ (Base‘𝑆) ∧ 𝑦 ∈ (𝑀 LMHom 𝑀) ∧ 𝑧 ∈ (𝑀 LMHom 𝑀))) → (𝑥( ·𝑠𝐴)𝑧) = (((Base‘𝑀) × {𝑥}) ∘f ( ·𝑠𝑀)𝑧))
71 fvexd 6921 . . . . . . 7 ((((𝑀 ∈ LMod ∧ 𝑆 ∈ CRing) ∧ (𝑥 ∈ (Base‘𝑆) ∧ 𝑦 ∈ (𝑀 LMHom 𝑀) ∧ 𝑧 ∈ (𝑀 LMHom 𝑀))) ∧ 𝑣 ∈ (Base‘𝑀)) → (𝑧𝑣) ∈ V)
7226, 39, 71, 42, 18offval2 7716 . . . . . 6 (((𝑀 ∈ LMod ∧ 𝑆 ∈ CRing) ∧ (𝑥 ∈ (Base‘𝑆) ∧ 𝑦 ∈ (𝑀 LMHom 𝑀) ∧ 𝑧 ∈ (𝑀 LMHom 𝑀))) → (((Base‘𝑀) × {𝑥}) ∘f ( ·𝑠𝑀)𝑧) = (𝑣 ∈ (Base‘𝑀) ↦ (𝑥( ·𝑠𝑀)(𝑧𝑣))))
7370, 72eqtrd 2774 . . . . 5 (((𝑀 ∈ LMod ∧ 𝑆 ∈ CRing) ∧ (𝑥 ∈ (Base‘𝑆) ∧ 𝑦 ∈ (𝑀 LMHom 𝑀) ∧ 𝑧 ∈ (𝑀 LMHom 𝑀))) → (𝑥( ·𝑠𝐴)𝑧) = (𝑣 ∈ (Base‘𝑀) ↦ (𝑥( ·𝑠𝑀)(𝑧𝑣))))
74 fveq2 6906 . . . . 5 (𝑤 = (𝑥( ·𝑠𝑀)(𝑧𝑣)) → (𝑦𝑤) = (𝑦‘(𝑥( ·𝑠𝑀)(𝑧𝑣))))
7568, 73, 33, 74fmptco 7148 . . . 4 (((𝑀 ∈ LMod ∧ 𝑆 ∈ CRing) ∧ (𝑥 ∈ (Base‘𝑆) ∧ 𝑦 ∈ (𝑀 LMHom 𝑀) ∧ 𝑧 ∈ (𝑀 LMHom 𝑀))) → (𝑦 ∘ (𝑥( ·𝑠𝐴)𝑧)) = (𝑣 ∈ (Base‘𝑀) ↦ (𝑦‘(𝑥( ·𝑠𝑀)(𝑧𝑣)))))
7665, 75, 493eqtr4d 2784 . . 3 (((𝑀 ∈ LMod ∧ 𝑆 ∈ CRing) ∧ (𝑥 ∈ (Base‘𝑆) ∧ 𝑦 ∈ (𝑀 LMHom 𝑀) ∧ 𝑧 ∈ (𝑀 LMHom 𝑀))) → (𝑦 ∘ (𝑥( ·𝑠𝐴)𝑧)) = (((Base‘𝑀) × {𝑥}) ∘f ( ·𝑠𝑀)(𝑦(.r𝐴)𝑧)))
772, 5, 23, 22lmodvscl 20892 . . . . 5 ((𝐴 ∈ LMod ∧ 𝑥 ∈ (Base‘𝑆) ∧ 𝑧 ∈ (𝑀 LMHom 𝑀)) → (𝑥( ·𝑠𝐴)𝑧) ∈ (𝑀 LMHom 𝑀))
7851, 19, 13, 77syl3anc 1370 . . . 4 (((𝑀 ∈ LMod ∧ 𝑆 ∈ CRing) ∧ (𝑥 ∈ (Base‘𝑆) ∧ 𝑦 ∈ (𝑀 LMHom 𝑀) ∧ 𝑧 ∈ (𝑀 LMHom 𝑀))) → (𝑥( ·𝑠𝐴)𝑧) ∈ (𝑀 LMHom 𝑀))
791, 2, 43mendmulr 43172 . . . 4 ((𝑦 ∈ (𝑀 LMHom 𝑀) ∧ (𝑥( ·𝑠𝐴)𝑧) ∈ (𝑀 LMHom 𝑀)) → (𝑦(.r𝐴)(𝑥( ·𝑠𝐴)𝑧)) = (𝑦 ∘ (𝑥( ·𝑠𝐴)𝑧)))
8020, 78, 79syl2anc 584 . . 3 (((𝑀 ∈ LMod ∧ 𝑆 ∈ CRing) ∧ (𝑥 ∈ (Base‘𝑆) ∧ 𝑦 ∈ (𝑀 LMHom 𝑀) ∧ 𝑧 ∈ (𝑀 LMHom 𝑀))) → (𝑦(.r𝐴)(𝑥( ·𝑠𝐴)𝑧)) = (𝑦 ∘ (𝑥( ·𝑠𝐴)𝑧)))
8176, 80, 603eqtr4d 2784 . 2 (((𝑀 ∈ LMod ∧ 𝑆 ∈ CRing) ∧ (𝑥 ∈ (Base‘𝑆) ∧ 𝑦 ∈ (𝑀 LMHom 𝑀) ∧ 𝑧 ∈ (𝑀 LMHom 𝑀))) → (𝑦(.r𝐴)(𝑥( ·𝑠𝐴)𝑧)) = (𝑥( ·𝑠𝐴)(𝑦(.r𝐴)𝑧)))
823, 6, 7, 8, 9, 10, 12, 61, 81isassad 21902 1 ((𝑀 ∈ LMod ∧ 𝑆 ∈ CRing) → 𝐴 ∈ AssAlg)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 395  w3a 1086   = wceq 1536  wcel 2105  Vcvv 3477  {csn 4630  cmpt 5230   × cxp 5686  ccom 5692  wf 6558  cfv 6562  (class class class)co 7430  f cof 7694  Basecbs 17244  .rcmulr 17298  Scalarcsca 17300   ·𝑠 cvsca 17301  Ringcrg 20250  CRingccrg 20251  LModclmod 20874   LMHom clmhm 21035  AssAlgcasa 21887  MEndocmend 43159
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1791  ax-4 1805  ax-5 1907  ax-6 1964  ax-7 2004  ax-8 2107  ax-9 2115  ax-10 2138  ax-11 2154  ax-12 2174  ax-ext 2705  ax-rep 5284  ax-sep 5301  ax-nul 5311  ax-pow 5370  ax-pr 5437  ax-un 7753  ax-cnex 11208  ax-resscn 11209  ax-1cn 11210  ax-icn 11211  ax-addcl 11212  ax-addrcl 11213  ax-mulcl 11214  ax-mulrcl 11215  ax-mulcom 11216  ax-addass 11217  ax-mulass 11218  ax-distr 11219  ax-i2m1 11220  ax-1ne0 11221  ax-1rid 11222  ax-rnegex 11223  ax-rrecex 11224  ax-cnre 11225  ax-pre-lttri 11226  ax-pre-lttrn 11227  ax-pre-ltadd 11228  ax-pre-mulgt0 11229
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3or 1087  df-3an 1088  df-tru 1539  df-fal 1549  df-ex 1776  df-nf 1780  df-sb 2062  df-mo 2537  df-eu 2566  df-clab 2712  df-cleq 2726  df-clel 2813  df-nfc 2889  df-ne 2938  df-nel 3044  df-ral 3059  df-rex 3068  df-rmo 3377  df-reu 3378  df-rab 3433  df-v 3479  df-sbc 3791  df-csb 3908  df-dif 3965  df-un 3967  df-in 3969  df-ss 3979  df-pss 3982  df-nul 4339  df-if 4531  df-pw 4606  df-sn 4631  df-pr 4633  df-tp 4635  df-op 4637  df-uni 4912  df-iun 4997  df-br 5148  df-opab 5210  df-mpt 5231  df-tr 5265  df-id 5582  df-eprel 5588  df-po 5596  df-so 5597  df-fr 5640  df-we 5642  df-xp 5694  df-rel 5695  df-cnv 5696  df-co 5697  df-dm 5698  df-rn 5699  df-res 5700  df-ima 5701  df-pred 6322  df-ord 6388  df-on 6389  df-lim 6390  df-suc 6391  df-iota 6515  df-fun 6564  df-fn 6565  df-f 6566  df-f1 6567  df-fo 6568  df-f1o 6569  df-fv 6570  df-riota 7387  df-ov 7433  df-oprab 7434  df-mpo 7435  df-of 7696  df-om 7887  df-1st 8012  df-2nd 8013  df-frecs 8304  df-wrecs 8335  df-recs 8409  df-rdg 8448  df-1o 8504  df-er 8743  df-map 8866  df-en 8984  df-dom 8985  df-sdom 8986  df-fin 8987  df-pnf 11294  df-mnf 11295  df-xr 11296  df-ltxr 11297  df-le 11298  df-sub 11491  df-neg 11492  df-nn 12264  df-2 12326  df-3 12327  df-4 12328  df-5 12329  df-6 12330  df-n0 12524  df-z 12611  df-uz 12876  df-fz 13544  df-struct 17180  df-sets 17197  df-slot 17215  df-ndx 17227  df-base 17245  df-plusg 17310  df-mulr 17311  df-sca 17313  df-vsca 17314  df-0g 17487  df-mgm 18665  df-sgrp 18744  df-mnd 18760  df-mhm 18808  df-grp 18966  df-minusg 18967  df-ghm 19243  df-cmn 19814  df-abl 19815  df-mgp 20152  df-rng 20170  df-ur 20199  df-ring 20252  df-cring 20253  df-lmod 20876  df-lmhm 21038  df-assa 21890  df-mend 43160
This theorem is referenced by: (None)
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