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Theorem mendassa 43202
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 43192 . . 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 43197 . . 3 𝑆 = (Scalar‘𝐴)
65a1i 11 . 2 ((𝑀 ∈ LMod ∧ 𝑆 ∈ CRing) → 𝑆 = (Scalar‘𝐴))
7 eqidd 2738 . 2 ((𝑀 ∈ LMod ∧ 𝑆 ∈ CRing) → (Base‘𝑆) = (Base‘𝑆))
8 eqidd 2738 . 2 ((𝑀 ∈ LMod ∧ 𝑆 ∈ CRing) → ( ·𝑠𝐴) = ( ·𝑠𝐴))
9 eqidd 2738 . 2 ((𝑀 ∈ LMod ∧ 𝑆 ∈ CRing) → (.r𝐴) = (.r𝐴))
101, 4mendlmod 43201 . 2 ((𝑀 ∈ LMod ∧ 𝑆 ∈ CRing) → 𝐴 ∈ LMod)
111mendring 43200 . . 3 (𝑀 ∈ LMod → 𝐴 ∈ Ring)
1211adantr 480 . 2 ((𝑀 ∈ LMod ∧ 𝑆 ∈ CRing) → 𝐴 ∈ Ring)
13 simpr3 1197 . . . . . . 7 (((𝑀 ∈ LMod ∧ 𝑆 ∈ CRing) ∧ (𝑥 ∈ (Base‘𝑆) ∧ 𝑦 ∈ (𝑀 LMHom 𝑀) ∧ 𝑧 ∈ (𝑀 LMHom 𝑀))) → 𝑧 ∈ (𝑀 LMHom 𝑀))
14 eqid 2737 . . . . . . . 8 (Base‘𝑀) = (Base‘𝑀)
1514, 14lmhmf 21033 . . . . . . 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 7104 . . . . 5 ((((𝑀 ∈ LMod ∧ 𝑆 ∈ CRing) ∧ (𝑥 ∈ (Base‘𝑆) ∧ 𝑦 ∈ (𝑀 LMHom 𝑀) ∧ 𝑧 ∈ (𝑀 LMHom 𝑀))) ∧ 𝑣 ∈ (Base‘𝑀)) → (𝑧𝑣) ∈ (Base‘𝑀))
1816feqmptd 6977 . . . . 5 (((𝑀 ∈ LMod ∧ 𝑆 ∈ CRing) ∧ (𝑥 ∈ (Base‘𝑆) ∧ 𝑦 ∈ (𝑀 LMHom 𝑀) ∧ 𝑧 ∈ (𝑀 LMHom 𝑀))) → 𝑧 = (𝑣 ∈ (Base‘𝑀) ↦ (𝑧𝑣)))
19 simpr1 1195 . . . . . . 7 (((𝑀 ∈ LMod ∧ 𝑆 ∈ CRing) ∧ (𝑥 ∈ (Base‘𝑆) ∧ 𝑦 ∈ (𝑀 LMHom 𝑀) ∧ 𝑧 ∈ (𝑀 LMHom 𝑀))) → 𝑥 ∈ (Base‘𝑆))
20 simpr2 1196 . . . . . . 7 (((𝑀 ∈ LMod ∧ 𝑆 ∈ CRing) ∧ (𝑥 ∈ (Base‘𝑆) ∧ 𝑦 ∈ (𝑀 LMHom 𝑀) ∧ 𝑧 ∈ (𝑀 LMHom 𝑀))) → 𝑦 ∈ (𝑀 LMHom 𝑀))
21 eqid 2737 . . . . . . . 8 ( ·𝑠𝑀) = ( ·𝑠𝑀)
22 eqid 2737 . . . . . . . 8 (Base‘𝑆) = (Base‘𝑆)
23 eqid 2737 . . . . . . . 8 ( ·𝑠𝐴) = ( ·𝑠𝐴)
241, 21, 2, 4, 22, 14, 23mendvsca 43199 . . . . . . 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 1216 . . . . . . 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 5747 . . . . . . . 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 21033 . . . . . . . . 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 6977 . . . . . . 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 7717 . . . . . 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 2777 . . . . 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 7447 . . . . 5 (𝑤 = (𝑧𝑣) → (𝑥( ·𝑠𝑀)(𝑦𝑤)) = (𝑥( ·𝑠𝑀)(𝑦‘(𝑧𝑣))))
3817, 18, 35, 37fmptco 7149 . . . 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 1216 . . . . 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 5747 . . . . . 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 2737 . . . . . . . 8 (.r𝐴) = (.r𝐴)
441, 2, 43mendmulr 43196 . . . . . . 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 7153 . . . . . . 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 2777 . . . . 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 7717 . . . 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 2780 . . 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 20876 . . . . 5 ((𝐴 ∈ LMod ∧ 𝑥 ∈ (Base‘𝑆) ∧ 𝑦 ∈ (𝑀 LMHom 𝑀)) → (𝑥( ·𝑠𝐴)𝑦) ∈ (𝑀 LMHom 𝑀))
5351, 19, 20, 52syl3anc 1373 . . . 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 43196 . . . 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 20247 . . . . 5 ((𝐴 ∈ Ring ∧ 𝑦 ∈ (𝑀 LMHom 𝑀) ∧ 𝑧 ∈ (𝑀 LMHom 𝑀)) → (𝑦(.r𝐴)𝑧) ∈ (𝑀 LMHom 𝑀))
5856, 20, 13, 57syl3anc 1373 . . . 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 43199 . . . 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 2787 . 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 1217 . . . . . 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 21034 . . . . . 6 ((𝑦 ∈ (𝑀 LMHom 𝑀) ∧ 𝑥 ∈ (Base‘𝑆) ∧ (𝑧𝑣) ∈ (Base‘𝑀)) → (𝑦‘(𝑥( ·𝑠𝑀)(𝑧𝑣))) = (𝑥( ·𝑠𝑀)(𝑦‘(𝑧𝑣))))
6462, 39, 17, 63syl3anc 1373 . . . . 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 5242 . . . 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 20876 . . . . . 6 ((𝑀 ∈ LMod ∧ 𝑥 ∈ (Base‘𝑆) ∧ (𝑧𝑣) ∈ (Base‘𝑀)) → (𝑥( ·𝑠𝑀)(𝑧𝑣)) ∈ (Base‘𝑀))
6866, 39, 17, 67syl3anc 1373 . . . . 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 43199 . . . . . . 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 7717 . . . . . 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 2777 . . . . 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 7149 . . . 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 2787 . . 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 20876 . . . . 5 ((𝐴 ∈ LMod ∧ 𝑥 ∈ (Base‘𝑆) ∧ 𝑧 ∈ (𝑀 LMHom 𝑀)) → (𝑥( ·𝑠𝐴)𝑧) ∈ (𝑀 LMHom 𝑀))
7851, 19, 13, 77syl3anc 1373 . . . 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 43196 . . . 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 2787 . 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 21885 1 ((𝑀 ∈ LMod ∧ 𝑆 ∈ CRing) → 𝐴 ∈ AssAlg)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 395  w3a 1087   = wceq 1540  wcel 2108  Vcvv 3480  {csn 4626  cmpt 5225   × cxp 5683  ccom 5689  wf 6557  cfv 6561  (class class class)co 7431  f cof 7695  Basecbs 17247  .rcmulr 17298  Scalarcsca 17300   ·𝑠 cvsca 17301  Ringcrg 20230  CRingccrg 20231  LModclmod 20858   LMHom clmhm 21018  AssAlgcasa 21870  MEndocmend 43183
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-10 2141  ax-11 2157  ax-12 2177  ax-ext 2708  ax-rep 5279  ax-sep 5296  ax-nul 5306  ax-pow 5365  ax-pr 5432  ax-un 7755  ax-cnex 11211  ax-resscn 11212  ax-1cn 11213  ax-icn 11214  ax-addcl 11215  ax-addrcl 11216  ax-mulcl 11217  ax-mulrcl 11218  ax-mulcom 11219  ax-addass 11220  ax-mulass 11221  ax-distr 11222  ax-i2m1 11223  ax-1ne0 11224  ax-1rid 11225  ax-rnegex 11226  ax-rrecex 11227  ax-cnre 11228  ax-pre-lttri 11229  ax-pre-lttrn 11230  ax-pre-ltadd 11231  ax-pre-mulgt0 11232
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3or 1088  df-3an 1089  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2065  df-mo 2540  df-eu 2569  df-clab 2715  df-cleq 2729  df-clel 2816  df-nfc 2892  df-ne 2941  df-nel 3047  df-ral 3062  df-rex 3071  df-rmo 3380  df-reu 3381  df-rab 3437  df-v 3482  df-sbc 3789  df-csb 3900  df-dif 3954  df-un 3956  df-in 3958  df-ss 3968  df-pss 3971  df-nul 4334  df-if 4526  df-pw 4602  df-sn 4627  df-pr 4629  df-tp 4631  df-op 4633  df-uni 4908  df-iun 4993  df-br 5144  df-opab 5206  df-mpt 5226  df-tr 5260  df-id 5578  df-eprel 5584  df-po 5592  df-so 5593  df-fr 5637  df-we 5639  df-xp 5691  df-rel 5692  df-cnv 5693  df-co 5694  df-dm 5695  df-rn 5696  df-res 5697  df-ima 5698  df-pred 6321  df-ord 6387  df-on 6388  df-lim 6389  df-suc 6390  df-iota 6514  df-fun 6563  df-fn 6564  df-f 6565  df-f1 6566  df-fo 6567  df-f1o 6568  df-fv 6569  df-riota 7388  df-ov 7434  df-oprab 7435  df-mpo 7436  df-of 7697  df-om 7888  df-1st 8014  df-2nd 8015  df-frecs 8306  df-wrecs 8337  df-recs 8411  df-rdg 8450  df-1o 8506  df-er 8745  df-map 8868  df-en 8986  df-dom 8987  df-sdom 8988  df-fin 8989  df-pnf 11297  df-mnf 11298  df-xr 11299  df-ltxr 11300  df-le 11301  df-sub 11494  df-neg 11495  df-nn 12267  df-2 12329  df-3 12330  df-4 12331  df-5 12332  df-6 12333  df-n0 12527  df-z 12614  df-uz 12879  df-fz 13548  df-struct 17184  df-sets 17201  df-slot 17219  df-ndx 17231  df-base 17248  df-plusg 17310  df-mulr 17311  df-sca 17313  df-vsca 17314  df-0g 17486  df-mgm 18653  df-sgrp 18732  df-mnd 18748  df-mhm 18796  df-grp 18954  df-minusg 18955  df-ghm 19231  df-cmn 19800  df-abl 19801  df-mgp 20138  df-rng 20150  df-ur 20179  df-ring 20232  df-cring 20233  df-lmod 20860  df-lmhm 21021  df-assa 21873  df-mend 43184
This theorem is referenced by: (None)
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