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Theorem mendring 41920
Description: The module endomorphism algebra is a ring. (Contributed by Stefan O'Rear, 5-Sep-2015.)
Hypothesis
Ref Expression
mendassa.a 𝐴 = (MEndoβ€˜π‘€)
Assertion
Ref Expression
mendring (𝑀 ∈ LMod β†’ 𝐴 ∈ Ring)

Proof of Theorem mendring
Dummy variables π‘₯ 𝑦 𝑧 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 mendassa.a . . . 4 𝐴 = (MEndoβ€˜π‘€)
21mendbas 41912 . . 3 (𝑀 LMHom 𝑀) = (Baseβ€˜π΄)
32a1i 11 . 2 (𝑀 ∈ LMod β†’ (𝑀 LMHom 𝑀) = (Baseβ€˜π΄))
4 eqidd 2734 . 2 (𝑀 ∈ LMod β†’ (+gβ€˜π΄) = (+gβ€˜π΄))
5 eqidd 2734 . 2 (𝑀 ∈ LMod β†’ (.rβ€˜π΄) = (.rβ€˜π΄))
6 eqid 2733 . . . . . 6 (+gβ€˜π‘€) = (+gβ€˜π‘€)
7 eqid 2733 . . . . . 6 (+gβ€˜π΄) = (+gβ€˜π΄)
81, 2, 6, 7mendplusg 41914 . . . . 5 ((π‘₯ ∈ (𝑀 LMHom 𝑀) ∧ 𝑦 ∈ (𝑀 LMHom 𝑀)) β†’ (π‘₯(+gβ€˜π΄)𝑦) = (π‘₯ ∘f (+gβ€˜π‘€)𝑦))
96lmhmplusg 20648 . . . . 5 ((π‘₯ ∈ (𝑀 LMHom 𝑀) ∧ 𝑦 ∈ (𝑀 LMHom 𝑀)) β†’ (π‘₯ ∘f (+gβ€˜π‘€)𝑦) ∈ (𝑀 LMHom 𝑀))
108, 9eqeltrd 2834 . . . 4 ((π‘₯ ∈ (𝑀 LMHom 𝑀) ∧ 𝑦 ∈ (𝑀 LMHom 𝑀)) β†’ (π‘₯(+gβ€˜π΄)𝑦) ∈ (𝑀 LMHom 𝑀))
11103adant1 1131 . . 3 ((𝑀 ∈ LMod ∧ π‘₯ ∈ (𝑀 LMHom 𝑀) ∧ 𝑦 ∈ (𝑀 LMHom 𝑀)) β†’ (π‘₯(+gβ€˜π΄)𝑦) ∈ (𝑀 LMHom 𝑀))
12 simpr1 1195 . . . . . 6 ((𝑀 ∈ LMod ∧ (π‘₯ ∈ (𝑀 LMHom 𝑀) ∧ 𝑦 ∈ (𝑀 LMHom 𝑀) ∧ 𝑧 ∈ (𝑀 LMHom 𝑀))) β†’ π‘₯ ∈ (𝑀 LMHom 𝑀))
13 simpr2 1196 . . . . . 6 ((𝑀 ∈ LMod ∧ (π‘₯ ∈ (𝑀 LMHom 𝑀) ∧ 𝑦 ∈ (𝑀 LMHom 𝑀) ∧ 𝑧 ∈ (𝑀 LMHom 𝑀))) β†’ 𝑦 ∈ (𝑀 LMHom 𝑀))
1412, 13, 9syl2anc 585 . . . . 5 ((𝑀 ∈ LMod ∧ (π‘₯ ∈ (𝑀 LMHom 𝑀) ∧ 𝑦 ∈ (𝑀 LMHom 𝑀) ∧ 𝑧 ∈ (𝑀 LMHom 𝑀))) β†’ (π‘₯ ∘f (+gβ€˜π‘€)𝑦) ∈ (𝑀 LMHom 𝑀))
15 simpr3 1197 . . . . 5 ((𝑀 ∈ LMod ∧ (π‘₯ ∈ (𝑀 LMHom 𝑀) ∧ 𝑦 ∈ (𝑀 LMHom 𝑀) ∧ 𝑧 ∈ (𝑀 LMHom 𝑀))) β†’ 𝑧 ∈ (𝑀 LMHom 𝑀))
161, 2, 6, 7mendplusg 41914 . . . . 5 (((π‘₯ ∘f (+gβ€˜π‘€)𝑦) ∈ (𝑀 LMHom 𝑀) ∧ 𝑧 ∈ (𝑀 LMHom 𝑀)) β†’ ((π‘₯ ∘f (+gβ€˜π‘€)𝑦)(+gβ€˜π΄)𝑧) = ((π‘₯ ∘f (+gβ€˜π‘€)𝑦) ∘f (+gβ€˜π‘€)𝑧))
1714, 15, 16syl2anc 585 . . . 4 ((𝑀 ∈ LMod ∧ (π‘₯ ∈ (𝑀 LMHom 𝑀) ∧ 𝑦 ∈ (𝑀 LMHom 𝑀) ∧ 𝑧 ∈ (𝑀 LMHom 𝑀))) β†’ ((π‘₯ ∘f (+gβ€˜π‘€)𝑦)(+gβ€˜π΄)𝑧) = ((π‘₯ ∘f (+gβ€˜π‘€)𝑦) ∘f (+gβ€˜π‘€)𝑧))
1812, 13, 8syl2anc 585 . . . . 5 ((𝑀 ∈ LMod ∧ (π‘₯ ∈ (𝑀 LMHom 𝑀) ∧ 𝑦 ∈ (𝑀 LMHom 𝑀) ∧ 𝑧 ∈ (𝑀 LMHom 𝑀))) β†’ (π‘₯(+gβ€˜π΄)𝑦) = (π‘₯ ∘f (+gβ€˜π‘€)𝑦))
1918oveq1d 7421 . . . 4 ((𝑀 ∈ LMod ∧ (π‘₯ ∈ (𝑀 LMHom 𝑀) ∧ 𝑦 ∈ (𝑀 LMHom 𝑀) ∧ 𝑧 ∈ (𝑀 LMHom 𝑀))) β†’ ((π‘₯(+gβ€˜π΄)𝑦)(+gβ€˜π΄)𝑧) = ((π‘₯ ∘f (+gβ€˜π‘€)𝑦)(+gβ€˜π΄)𝑧))
206lmhmplusg 20648 . . . . . . 7 ((𝑦 ∈ (𝑀 LMHom 𝑀) ∧ 𝑧 ∈ (𝑀 LMHom 𝑀)) β†’ (𝑦 ∘f (+gβ€˜π‘€)𝑧) ∈ (𝑀 LMHom 𝑀))
2113, 15, 20syl2anc 585 . . . . . 6 ((𝑀 ∈ LMod ∧ (π‘₯ ∈ (𝑀 LMHom 𝑀) ∧ 𝑦 ∈ (𝑀 LMHom 𝑀) ∧ 𝑧 ∈ (𝑀 LMHom 𝑀))) β†’ (𝑦 ∘f (+gβ€˜π‘€)𝑧) ∈ (𝑀 LMHom 𝑀))
221, 2, 6, 7mendplusg 41914 . . . . . 6 ((π‘₯ ∈ (𝑀 LMHom 𝑀) ∧ (𝑦 ∘f (+gβ€˜π‘€)𝑧) ∈ (𝑀 LMHom 𝑀)) β†’ (π‘₯(+gβ€˜π΄)(𝑦 ∘f (+gβ€˜π‘€)𝑧)) = (π‘₯ ∘f (+gβ€˜π‘€)(𝑦 ∘f (+gβ€˜π‘€)𝑧)))
2312, 21, 22syl2anc 585 . . . . 5 ((𝑀 ∈ LMod ∧ (π‘₯ ∈ (𝑀 LMHom 𝑀) ∧ 𝑦 ∈ (𝑀 LMHom 𝑀) ∧ 𝑧 ∈ (𝑀 LMHom 𝑀))) β†’ (π‘₯(+gβ€˜π΄)(𝑦 ∘f (+gβ€˜π‘€)𝑧)) = (π‘₯ ∘f (+gβ€˜π‘€)(𝑦 ∘f (+gβ€˜π‘€)𝑧)))
241, 2, 6, 7mendplusg 41914 . . . . . . 7 ((𝑦 ∈ (𝑀 LMHom 𝑀) ∧ 𝑧 ∈ (𝑀 LMHom 𝑀)) β†’ (𝑦(+gβ€˜π΄)𝑧) = (𝑦 ∘f (+gβ€˜π‘€)𝑧))
2513, 15, 24syl2anc 585 . . . . . 6 ((𝑀 ∈ LMod ∧ (π‘₯ ∈ (𝑀 LMHom 𝑀) ∧ 𝑦 ∈ (𝑀 LMHom 𝑀) ∧ 𝑧 ∈ (𝑀 LMHom 𝑀))) β†’ (𝑦(+gβ€˜π΄)𝑧) = (𝑦 ∘f (+gβ€˜π‘€)𝑧))
2625oveq2d 7422 . . . . 5 ((𝑀 ∈ LMod ∧ (π‘₯ ∈ (𝑀 LMHom 𝑀) ∧ 𝑦 ∈ (𝑀 LMHom 𝑀) ∧ 𝑧 ∈ (𝑀 LMHom 𝑀))) β†’ (π‘₯(+gβ€˜π΄)(𝑦(+gβ€˜π΄)𝑧)) = (π‘₯(+gβ€˜π΄)(𝑦 ∘f (+gβ€˜π‘€)𝑧)))
27 lmodgrp 20471 . . . . . . . 8 (𝑀 ∈ LMod β†’ 𝑀 ∈ Grp)
2827grpmndd 18829 . . . . . . 7 (𝑀 ∈ LMod β†’ 𝑀 ∈ Mnd)
2928adantr 482 . . . . . 6 ((𝑀 ∈ LMod ∧ (π‘₯ ∈ (𝑀 LMHom 𝑀) ∧ 𝑦 ∈ (𝑀 LMHom 𝑀) ∧ 𝑧 ∈ (𝑀 LMHom 𝑀))) β†’ 𝑀 ∈ Mnd)
30 eqid 2733 . . . . . . . . 9 (Baseβ€˜π‘€) = (Baseβ€˜π‘€)
3130, 30lmhmf 20638 . . . . . . . 8 (π‘₯ ∈ (𝑀 LMHom 𝑀) β†’ π‘₯:(Baseβ€˜π‘€)⟢(Baseβ€˜π‘€))
3212, 31syl 17 . . . . . . 7 ((𝑀 ∈ LMod ∧ (π‘₯ ∈ (𝑀 LMHom 𝑀) ∧ 𝑦 ∈ (𝑀 LMHom 𝑀) ∧ 𝑧 ∈ (𝑀 LMHom 𝑀))) β†’ π‘₯:(Baseβ€˜π‘€)⟢(Baseβ€˜π‘€))
33 fvex 6902 . . . . . . . 8 (Baseβ€˜π‘€) ∈ V
3433, 33elmap 8862 . . . . . . 7 (π‘₯ ∈ ((Baseβ€˜π‘€) ↑m (Baseβ€˜π‘€)) ↔ π‘₯:(Baseβ€˜π‘€)⟢(Baseβ€˜π‘€))
3532, 34sylibr 233 . . . . . 6 ((𝑀 ∈ LMod ∧ (π‘₯ ∈ (𝑀 LMHom 𝑀) ∧ 𝑦 ∈ (𝑀 LMHom 𝑀) ∧ 𝑧 ∈ (𝑀 LMHom 𝑀))) β†’ π‘₯ ∈ ((Baseβ€˜π‘€) ↑m (Baseβ€˜π‘€)))
3630, 30lmhmf 20638 . . . . . . . 8 (𝑦 ∈ (𝑀 LMHom 𝑀) β†’ 𝑦:(Baseβ€˜π‘€)⟢(Baseβ€˜π‘€))
3713, 36syl 17 . . . . . . 7 ((𝑀 ∈ LMod ∧ (π‘₯ ∈ (𝑀 LMHom 𝑀) ∧ 𝑦 ∈ (𝑀 LMHom 𝑀) ∧ 𝑧 ∈ (𝑀 LMHom 𝑀))) β†’ 𝑦:(Baseβ€˜π‘€)⟢(Baseβ€˜π‘€))
3833, 33elmap 8862 . . . . . . 7 (𝑦 ∈ ((Baseβ€˜π‘€) ↑m (Baseβ€˜π‘€)) ↔ 𝑦:(Baseβ€˜π‘€)⟢(Baseβ€˜π‘€))
3937, 38sylibr 233 . . . . . 6 ((𝑀 ∈ LMod ∧ (π‘₯ ∈ (𝑀 LMHom 𝑀) ∧ 𝑦 ∈ (𝑀 LMHom 𝑀) ∧ 𝑧 ∈ (𝑀 LMHom 𝑀))) β†’ 𝑦 ∈ ((Baseβ€˜π‘€) ↑m (Baseβ€˜π‘€)))
4030, 30lmhmf 20638 . . . . . . . 8 (𝑧 ∈ (𝑀 LMHom 𝑀) β†’ 𝑧:(Baseβ€˜π‘€)⟢(Baseβ€˜π‘€))
4115, 40syl 17 . . . . . . 7 ((𝑀 ∈ LMod ∧ (π‘₯ ∈ (𝑀 LMHom 𝑀) ∧ 𝑦 ∈ (𝑀 LMHom 𝑀) ∧ 𝑧 ∈ (𝑀 LMHom 𝑀))) β†’ 𝑧:(Baseβ€˜π‘€)⟢(Baseβ€˜π‘€))
4233, 33elmap 8862 . . . . . . 7 (𝑧 ∈ ((Baseβ€˜π‘€) ↑m (Baseβ€˜π‘€)) ↔ 𝑧:(Baseβ€˜π‘€)⟢(Baseβ€˜π‘€))
4341, 42sylibr 233 . . . . . 6 ((𝑀 ∈ LMod ∧ (π‘₯ ∈ (𝑀 LMHom 𝑀) ∧ 𝑦 ∈ (𝑀 LMHom 𝑀) ∧ 𝑧 ∈ (𝑀 LMHom 𝑀))) β†’ 𝑧 ∈ ((Baseβ€˜π‘€) ↑m (Baseβ€˜π‘€)))
4430, 6mndvass 21886 . . . . . 6 ((𝑀 ∈ Mnd ∧ (π‘₯ ∈ ((Baseβ€˜π‘€) ↑m (Baseβ€˜π‘€)) ∧ 𝑦 ∈ ((Baseβ€˜π‘€) ↑m (Baseβ€˜π‘€)) ∧ 𝑧 ∈ ((Baseβ€˜π‘€) ↑m (Baseβ€˜π‘€)))) β†’ ((π‘₯ ∘f (+gβ€˜π‘€)𝑦) ∘f (+gβ€˜π‘€)𝑧) = (π‘₯ ∘f (+gβ€˜π‘€)(𝑦 ∘f (+gβ€˜π‘€)𝑧)))
4529, 35, 39, 43, 44syl13anc 1373 . . . . 5 ((𝑀 ∈ LMod ∧ (π‘₯ ∈ (𝑀 LMHom 𝑀) ∧ 𝑦 ∈ (𝑀 LMHom 𝑀) ∧ 𝑧 ∈ (𝑀 LMHom 𝑀))) β†’ ((π‘₯ ∘f (+gβ€˜π‘€)𝑦) ∘f (+gβ€˜π‘€)𝑧) = (π‘₯ ∘f (+gβ€˜π‘€)(𝑦 ∘f (+gβ€˜π‘€)𝑧)))
4623, 26, 453eqtr4d 2783 . . . 4 ((𝑀 ∈ LMod ∧ (π‘₯ ∈ (𝑀 LMHom 𝑀) ∧ 𝑦 ∈ (𝑀 LMHom 𝑀) ∧ 𝑧 ∈ (𝑀 LMHom 𝑀))) β†’ (π‘₯(+gβ€˜π΄)(𝑦(+gβ€˜π΄)𝑧)) = ((π‘₯ ∘f (+gβ€˜π‘€)𝑦) ∘f (+gβ€˜π‘€)𝑧))
4717, 19, 463eqtr4d 2783 . . 3 ((𝑀 ∈ LMod ∧ (π‘₯ ∈ (𝑀 LMHom 𝑀) ∧ 𝑦 ∈ (𝑀 LMHom 𝑀) ∧ 𝑧 ∈ (𝑀 LMHom 𝑀))) β†’ ((π‘₯(+gβ€˜π΄)𝑦)(+gβ€˜π΄)𝑧) = (π‘₯(+gβ€˜π΄)(𝑦(+gβ€˜π΄)𝑧)))
48 id 22 . . . 4 (𝑀 ∈ LMod β†’ 𝑀 ∈ LMod)
49 eqidd 2734 . . . 4 (𝑀 ∈ LMod β†’ (Scalarβ€˜π‘€) = (Scalarβ€˜π‘€))
50 eqid 2733 . . . . 5 (0gβ€˜π‘€) = (0gβ€˜π‘€)
51 eqid 2733 . . . . 5 (Scalarβ€˜π‘€) = (Scalarβ€˜π‘€)
5250, 30, 51, 510lmhm 20644 . . . 4 ((𝑀 ∈ LMod ∧ 𝑀 ∈ LMod ∧ (Scalarβ€˜π‘€) = (Scalarβ€˜π‘€)) β†’ ((Baseβ€˜π‘€) Γ— {(0gβ€˜π‘€)}) ∈ (𝑀 LMHom 𝑀))
5348, 48, 49, 52syl3anc 1372 . . 3 (𝑀 ∈ LMod β†’ ((Baseβ€˜π‘€) Γ— {(0gβ€˜π‘€)}) ∈ (𝑀 LMHom 𝑀))
541, 2, 6, 7mendplusg 41914 . . . . 5 ((((Baseβ€˜π‘€) Γ— {(0gβ€˜π‘€)}) ∈ (𝑀 LMHom 𝑀) ∧ π‘₯ ∈ (𝑀 LMHom 𝑀)) β†’ (((Baseβ€˜π‘€) Γ— {(0gβ€˜π‘€)})(+gβ€˜π΄)π‘₯) = (((Baseβ€˜π‘€) Γ— {(0gβ€˜π‘€)}) ∘f (+gβ€˜π‘€)π‘₯))
5553, 54sylan 581 . . . 4 ((𝑀 ∈ LMod ∧ π‘₯ ∈ (𝑀 LMHom 𝑀)) β†’ (((Baseβ€˜π‘€) Γ— {(0gβ€˜π‘€)})(+gβ€˜π΄)π‘₯) = (((Baseβ€˜π‘€) Γ— {(0gβ€˜π‘€)}) ∘f (+gβ€˜π‘€)π‘₯))
5631, 34sylibr 233 . . . . 5 (π‘₯ ∈ (𝑀 LMHom 𝑀) β†’ π‘₯ ∈ ((Baseβ€˜π‘€) ↑m (Baseβ€˜π‘€)))
5730, 6, 50mndvlid 21887 . . . . 5 ((𝑀 ∈ Mnd ∧ π‘₯ ∈ ((Baseβ€˜π‘€) ↑m (Baseβ€˜π‘€))) β†’ (((Baseβ€˜π‘€) Γ— {(0gβ€˜π‘€)}) ∘f (+gβ€˜π‘€)π‘₯) = π‘₯)
5828, 56, 57syl2an 597 . . . 4 ((𝑀 ∈ LMod ∧ π‘₯ ∈ (𝑀 LMHom 𝑀)) β†’ (((Baseβ€˜π‘€) Γ— {(0gβ€˜π‘€)}) ∘f (+gβ€˜π‘€)π‘₯) = π‘₯)
5955, 58eqtrd 2773 . . 3 ((𝑀 ∈ LMod ∧ π‘₯ ∈ (𝑀 LMHom 𝑀)) β†’ (((Baseβ€˜π‘€) Γ— {(0gβ€˜π‘€)})(+gβ€˜π΄)π‘₯) = π‘₯)
60 eqid 2733 . . . . 5 (invgβ€˜π‘€) = (invgβ€˜π‘€)
6160invlmhm 20646 . . . 4 (𝑀 ∈ LMod β†’ (invgβ€˜π‘€) ∈ (𝑀 LMHom 𝑀))
62 lmhmco 20647 . . . 4 (((invgβ€˜π‘€) ∈ (𝑀 LMHom 𝑀) ∧ π‘₯ ∈ (𝑀 LMHom 𝑀)) β†’ ((invgβ€˜π‘€) ∘ π‘₯) ∈ (𝑀 LMHom 𝑀))
6361, 62sylan 581 . . 3 ((𝑀 ∈ LMod ∧ π‘₯ ∈ (𝑀 LMHom 𝑀)) β†’ ((invgβ€˜π‘€) ∘ π‘₯) ∈ (𝑀 LMHom 𝑀))
641, 2, 6, 7mendplusg 41914 . . . . 5 ((((invgβ€˜π‘€) ∘ π‘₯) ∈ (𝑀 LMHom 𝑀) ∧ π‘₯ ∈ (𝑀 LMHom 𝑀)) β†’ (((invgβ€˜π‘€) ∘ π‘₯)(+gβ€˜π΄)π‘₯) = (((invgβ€˜π‘€) ∘ π‘₯) ∘f (+gβ€˜π‘€)π‘₯))
6563, 64sylancom 589 . . . 4 ((𝑀 ∈ LMod ∧ π‘₯ ∈ (𝑀 LMHom 𝑀)) β†’ (((invgβ€˜π‘€) ∘ π‘₯)(+gβ€˜π΄)π‘₯) = (((invgβ€˜π‘€) ∘ π‘₯) ∘f (+gβ€˜π‘€)π‘₯))
6630, 6, 60, 50grpvlinv 21889 . . . . 5 ((𝑀 ∈ Grp ∧ π‘₯ ∈ ((Baseβ€˜π‘€) ↑m (Baseβ€˜π‘€))) β†’ (((invgβ€˜π‘€) ∘ π‘₯) ∘f (+gβ€˜π‘€)π‘₯) = ((Baseβ€˜π‘€) Γ— {(0gβ€˜π‘€)}))
6727, 56, 66syl2an 597 . . . 4 ((𝑀 ∈ LMod ∧ π‘₯ ∈ (𝑀 LMHom 𝑀)) β†’ (((invgβ€˜π‘€) ∘ π‘₯) ∘f (+gβ€˜π‘€)π‘₯) = ((Baseβ€˜π‘€) Γ— {(0gβ€˜π‘€)}))
6865, 67eqtrd 2773 . . 3 ((𝑀 ∈ LMod ∧ π‘₯ ∈ (𝑀 LMHom 𝑀)) β†’ (((invgβ€˜π‘€) ∘ π‘₯)(+gβ€˜π΄)π‘₯) = ((Baseβ€˜π‘€) Γ— {(0gβ€˜π‘€)}))
693, 4, 11, 47, 53, 59, 63, 68isgrpd 18841 . 2 (𝑀 ∈ LMod β†’ 𝐴 ∈ Grp)
70 eqid 2733 . . . . 5 (.rβ€˜π΄) = (.rβ€˜π΄)
711, 2, 70mendmulr 41916 . . . 4 ((π‘₯ ∈ (𝑀 LMHom 𝑀) ∧ 𝑦 ∈ (𝑀 LMHom 𝑀)) β†’ (π‘₯(.rβ€˜π΄)𝑦) = (π‘₯ ∘ 𝑦))
72 lmhmco 20647 . . . 4 ((π‘₯ ∈ (𝑀 LMHom 𝑀) ∧ 𝑦 ∈ (𝑀 LMHom 𝑀)) β†’ (π‘₯ ∘ 𝑦) ∈ (𝑀 LMHom 𝑀))
7371, 72eqeltrd 2834 . . 3 ((π‘₯ ∈ (𝑀 LMHom 𝑀) ∧ 𝑦 ∈ (𝑀 LMHom 𝑀)) β†’ (π‘₯(.rβ€˜π΄)𝑦) ∈ (𝑀 LMHom 𝑀))
74733adant1 1131 . 2 ((𝑀 ∈ LMod ∧ π‘₯ ∈ (𝑀 LMHom 𝑀) ∧ 𝑦 ∈ (𝑀 LMHom 𝑀)) β†’ (π‘₯(.rβ€˜π΄)𝑦) ∈ (𝑀 LMHom 𝑀))
75 coass 6262 . . 3 ((π‘₯ ∘ 𝑦) ∘ 𝑧) = (π‘₯ ∘ (𝑦 ∘ 𝑧))
7612, 13, 71syl2anc 585 . . . . 5 ((𝑀 ∈ LMod ∧ (π‘₯ ∈ (𝑀 LMHom 𝑀) ∧ 𝑦 ∈ (𝑀 LMHom 𝑀) ∧ 𝑧 ∈ (𝑀 LMHom 𝑀))) β†’ (π‘₯(.rβ€˜π΄)𝑦) = (π‘₯ ∘ 𝑦))
7776oveq1d 7421 . . . 4 ((𝑀 ∈ LMod ∧ (π‘₯ ∈ (𝑀 LMHom 𝑀) ∧ 𝑦 ∈ (𝑀 LMHom 𝑀) ∧ 𝑧 ∈ (𝑀 LMHom 𝑀))) β†’ ((π‘₯(.rβ€˜π΄)𝑦)(.rβ€˜π΄)𝑧) = ((π‘₯ ∘ 𝑦)(.rβ€˜π΄)𝑧))
7812, 13, 72syl2anc 585 . . . . 5 ((𝑀 ∈ LMod ∧ (π‘₯ ∈ (𝑀 LMHom 𝑀) ∧ 𝑦 ∈ (𝑀 LMHom 𝑀) ∧ 𝑧 ∈ (𝑀 LMHom 𝑀))) β†’ (π‘₯ ∘ 𝑦) ∈ (𝑀 LMHom 𝑀))
791, 2, 70mendmulr 41916 . . . . 5 (((π‘₯ ∘ 𝑦) ∈ (𝑀 LMHom 𝑀) ∧ 𝑧 ∈ (𝑀 LMHom 𝑀)) β†’ ((π‘₯ ∘ 𝑦)(.rβ€˜π΄)𝑧) = ((π‘₯ ∘ 𝑦) ∘ 𝑧))
8078, 15, 79syl2anc 585 . . . 4 ((𝑀 ∈ LMod ∧ (π‘₯ ∈ (𝑀 LMHom 𝑀) ∧ 𝑦 ∈ (𝑀 LMHom 𝑀) ∧ 𝑧 ∈ (𝑀 LMHom 𝑀))) β†’ ((π‘₯ ∘ 𝑦)(.rβ€˜π΄)𝑧) = ((π‘₯ ∘ 𝑦) ∘ 𝑧))
8177, 80eqtrd 2773 . . 3 ((𝑀 ∈ LMod ∧ (π‘₯ ∈ (𝑀 LMHom 𝑀) ∧ 𝑦 ∈ (𝑀 LMHom 𝑀) ∧ 𝑧 ∈ (𝑀 LMHom 𝑀))) β†’ ((π‘₯(.rβ€˜π΄)𝑦)(.rβ€˜π΄)𝑧) = ((π‘₯ ∘ 𝑦) ∘ 𝑧))
821, 2, 70mendmulr 41916 . . . . . 6 ((𝑦 ∈ (𝑀 LMHom 𝑀) ∧ 𝑧 ∈ (𝑀 LMHom 𝑀)) β†’ (𝑦(.rβ€˜π΄)𝑧) = (𝑦 ∘ 𝑧))
8313, 15, 82syl2anc 585 . . . . 5 ((𝑀 ∈ LMod ∧ (π‘₯ ∈ (𝑀 LMHom 𝑀) ∧ 𝑦 ∈ (𝑀 LMHom 𝑀) ∧ 𝑧 ∈ (𝑀 LMHom 𝑀))) β†’ (𝑦(.rβ€˜π΄)𝑧) = (𝑦 ∘ 𝑧))
8483oveq2d 7422 . . . 4 ((𝑀 ∈ LMod ∧ (π‘₯ ∈ (𝑀 LMHom 𝑀) ∧ 𝑦 ∈ (𝑀 LMHom 𝑀) ∧ 𝑧 ∈ (𝑀 LMHom 𝑀))) β†’ (π‘₯(.rβ€˜π΄)(𝑦(.rβ€˜π΄)𝑧)) = (π‘₯(.rβ€˜π΄)(𝑦 ∘ 𝑧)))
85 lmhmco 20647 . . . . . 6 ((𝑦 ∈ (𝑀 LMHom 𝑀) ∧ 𝑧 ∈ (𝑀 LMHom 𝑀)) β†’ (𝑦 ∘ 𝑧) ∈ (𝑀 LMHom 𝑀))
8613, 15, 85syl2anc 585 . . . . 5 ((𝑀 ∈ LMod ∧ (π‘₯ ∈ (𝑀 LMHom 𝑀) ∧ 𝑦 ∈ (𝑀 LMHom 𝑀) ∧ 𝑧 ∈ (𝑀 LMHom 𝑀))) β†’ (𝑦 ∘ 𝑧) ∈ (𝑀 LMHom 𝑀))
871, 2, 70mendmulr 41916 . . . . 5 ((π‘₯ ∈ (𝑀 LMHom 𝑀) ∧ (𝑦 ∘ 𝑧) ∈ (𝑀 LMHom 𝑀)) β†’ (π‘₯(.rβ€˜π΄)(𝑦 ∘ 𝑧)) = (π‘₯ ∘ (𝑦 ∘ 𝑧)))
8812, 86, 87syl2anc 585 . . . 4 ((𝑀 ∈ LMod ∧ (π‘₯ ∈ (𝑀 LMHom 𝑀) ∧ 𝑦 ∈ (𝑀 LMHom 𝑀) ∧ 𝑧 ∈ (𝑀 LMHom 𝑀))) β†’ (π‘₯(.rβ€˜π΄)(𝑦 ∘ 𝑧)) = (π‘₯ ∘ (𝑦 ∘ 𝑧)))
8984, 88eqtrd 2773 . . 3 ((𝑀 ∈ LMod ∧ (π‘₯ ∈ (𝑀 LMHom 𝑀) ∧ 𝑦 ∈ (𝑀 LMHom 𝑀) ∧ 𝑧 ∈ (𝑀 LMHom 𝑀))) β†’ (π‘₯(.rβ€˜π΄)(𝑦(.rβ€˜π΄)𝑧)) = (π‘₯ ∘ (𝑦 ∘ 𝑧)))
9075, 81, 893eqtr4a 2799 . 2 ((𝑀 ∈ LMod ∧ (π‘₯ ∈ (𝑀 LMHom 𝑀) ∧ 𝑦 ∈ (𝑀 LMHom 𝑀) ∧ 𝑧 ∈ (𝑀 LMHom 𝑀))) β†’ ((π‘₯(.rβ€˜π΄)𝑦)(.rβ€˜π΄)𝑧) = (π‘₯(.rβ€˜π΄)(𝑦(.rβ€˜π΄)𝑧)))
911, 2, 70mendmulr 41916 . . . 4 ((π‘₯ ∈ (𝑀 LMHom 𝑀) ∧ (𝑦 ∘f (+gβ€˜π‘€)𝑧) ∈ (𝑀 LMHom 𝑀)) β†’ (π‘₯(.rβ€˜π΄)(𝑦 ∘f (+gβ€˜π‘€)𝑧)) = (π‘₯ ∘ (𝑦 ∘f (+gβ€˜π‘€)𝑧)))
9212, 21, 91syl2anc 585 . . 3 ((𝑀 ∈ LMod ∧ (π‘₯ ∈ (𝑀 LMHom 𝑀) ∧ 𝑦 ∈ (𝑀 LMHom 𝑀) ∧ 𝑧 ∈ (𝑀 LMHom 𝑀))) β†’ (π‘₯(.rβ€˜π΄)(𝑦 ∘f (+gβ€˜π‘€)𝑧)) = (π‘₯ ∘ (𝑦 ∘f (+gβ€˜π‘€)𝑧)))
9325oveq2d 7422 . . 3 ((𝑀 ∈ LMod ∧ (π‘₯ ∈ (𝑀 LMHom 𝑀) ∧ 𝑦 ∈ (𝑀 LMHom 𝑀) ∧ 𝑧 ∈ (𝑀 LMHom 𝑀))) β†’ (π‘₯(.rβ€˜π΄)(𝑦(+gβ€˜π΄)𝑧)) = (π‘₯(.rβ€˜π΄)(𝑦 ∘f (+gβ€˜π‘€)𝑧)))
94 lmhmco 20647 . . . . . 6 ((π‘₯ ∈ (𝑀 LMHom 𝑀) ∧ 𝑧 ∈ (𝑀 LMHom 𝑀)) β†’ (π‘₯ ∘ 𝑧) ∈ (𝑀 LMHom 𝑀))
9512, 15, 94syl2anc 585 . . . . 5 ((𝑀 ∈ LMod ∧ (π‘₯ ∈ (𝑀 LMHom 𝑀) ∧ 𝑦 ∈ (𝑀 LMHom 𝑀) ∧ 𝑧 ∈ (𝑀 LMHom 𝑀))) β†’ (π‘₯ ∘ 𝑧) ∈ (𝑀 LMHom 𝑀))
961, 2, 6, 7mendplusg 41914 . . . . 5 (((π‘₯ ∘ 𝑦) ∈ (𝑀 LMHom 𝑀) ∧ (π‘₯ ∘ 𝑧) ∈ (𝑀 LMHom 𝑀)) β†’ ((π‘₯ ∘ 𝑦)(+gβ€˜π΄)(π‘₯ ∘ 𝑧)) = ((π‘₯ ∘ 𝑦) ∘f (+gβ€˜π‘€)(π‘₯ ∘ 𝑧)))
9778, 95, 96syl2anc 585 . . . 4 ((𝑀 ∈ LMod ∧ (π‘₯ ∈ (𝑀 LMHom 𝑀) ∧ 𝑦 ∈ (𝑀 LMHom 𝑀) ∧ 𝑧 ∈ (𝑀 LMHom 𝑀))) β†’ ((π‘₯ ∘ 𝑦)(+gβ€˜π΄)(π‘₯ ∘ 𝑧)) = ((π‘₯ ∘ 𝑦) ∘f (+gβ€˜π‘€)(π‘₯ ∘ 𝑧)))
981, 2, 70mendmulr 41916 . . . . . 6 ((π‘₯ ∈ (𝑀 LMHom 𝑀) ∧ 𝑧 ∈ (𝑀 LMHom 𝑀)) β†’ (π‘₯(.rβ€˜π΄)𝑧) = (π‘₯ ∘ 𝑧))
9912, 15, 98syl2anc 585 . . . . 5 ((𝑀 ∈ LMod ∧ (π‘₯ ∈ (𝑀 LMHom 𝑀) ∧ 𝑦 ∈ (𝑀 LMHom 𝑀) ∧ 𝑧 ∈ (𝑀 LMHom 𝑀))) β†’ (π‘₯(.rβ€˜π΄)𝑧) = (π‘₯ ∘ 𝑧))
10076, 99oveq12d 7424 . . . 4 ((𝑀 ∈ LMod ∧ (π‘₯ ∈ (𝑀 LMHom 𝑀) ∧ 𝑦 ∈ (𝑀 LMHom 𝑀) ∧ 𝑧 ∈ (𝑀 LMHom 𝑀))) β†’ ((π‘₯(.rβ€˜π΄)𝑦)(+gβ€˜π΄)(π‘₯(.rβ€˜π΄)𝑧)) = ((π‘₯ ∘ 𝑦)(+gβ€˜π΄)(π‘₯ ∘ 𝑧)))
101 lmghm 20635 . . . . . 6 (π‘₯ ∈ (𝑀 LMHom 𝑀) β†’ π‘₯ ∈ (𝑀 GrpHom 𝑀))
102 ghmmhm 19097 . . . . . 6 (π‘₯ ∈ (𝑀 GrpHom 𝑀) β†’ π‘₯ ∈ (𝑀 MndHom 𝑀))
10312, 101, 1023syl 18 . . . . 5 ((𝑀 ∈ LMod ∧ (π‘₯ ∈ (𝑀 LMHom 𝑀) ∧ 𝑦 ∈ (𝑀 LMHom 𝑀) ∧ 𝑧 ∈ (𝑀 LMHom 𝑀))) β†’ π‘₯ ∈ (𝑀 MndHom 𝑀))
10430, 6, 6mhmvlin 21891 . . . . 5 ((π‘₯ ∈ (𝑀 MndHom 𝑀) ∧ 𝑦 ∈ ((Baseβ€˜π‘€) ↑m (Baseβ€˜π‘€)) ∧ 𝑧 ∈ ((Baseβ€˜π‘€) ↑m (Baseβ€˜π‘€))) β†’ (π‘₯ ∘ (𝑦 ∘f (+gβ€˜π‘€)𝑧)) = ((π‘₯ ∘ 𝑦) ∘f (+gβ€˜π‘€)(π‘₯ ∘ 𝑧)))
105103, 39, 43, 104syl3anc 1372 . . . 4 ((𝑀 ∈ LMod ∧ (π‘₯ ∈ (𝑀 LMHom 𝑀) ∧ 𝑦 ∈ (𝑀 LMHom 𝑀) ∧ 𝑧 ∈ (𝑀 LMHom 𝑀))) β†’ (π‘₯ ∘ (𝑦 ∘f (+gβ€˜π‘€)𝑧)) = ((π‘₯ ∘ 𝑦) ∘f (+gβ€˜π‘€)(π‘₯ ∘ 𝑧)))
10697, 100, 1053eqtr4d 2783 . . 3 ((𝑀 ∈ LMod ∧ (π‘₯ ∈ (𝑀 LMHom 𝑀) ∧ 𝑦 ∈ (𝑀 LMHom 𝑀) ∧ 𝑧 ∈ (𝑀 LMHom 𝑀))) β†’ ((π‘₯(.rβ€˜π΄)𝑦)(+gβ€˜π΄)(π‘₯(.rβ€˜π΄)𝑧)) = (π‘₯ ∘ (𝑦 ∘f (+gβ€˜π‘€)𝑧)))
10792, 93, 1063eqtr4d 2783 . 2 ((𝑀 ∈ LMod ∧ (π‘₯ ∈ (𝑀 LMHom 𝑀) ∧ 𝑦 ∈ (𝑀 LMHom 𝑀) ∧ 𝑧 ∈ (𝑀 LMHom 𝑀))) β†’ (π‘₯(.rβ€˜π΄)(𝑦(+gβ€˜π΄)𝑧)) = ((π‘₯(.rβ€˜π΄)𝑦)(+gβ€˜π΄)(π‘₯(.rβ€˜π΄)𝑧)))
1081, 2, 70mendmulr 41916 . . . 4 (((π‘₯ ∘f (+gβ€˜π‘€)𝑦) ∈ (𝑀 LMHom 𝑀) ∧ 𝑧 ∈ (𝑀 LMHom 𝑀)) β†’ ((π‘₯ ∘f (+gβ€˜π‘€)𝑦)(.rβ€˜π΄)𝑧) = ((π‘₯ ∘f (+gβ€˜π‘€)𝑦) ∘ 𝑧))
10914, 15, 108syl2anc 585 . . 3 ((𝑀 ∈ LMod ∧ (π‘₯ ∈ (𝑀 LMHom 𝑀) ∧ 𝑦 ∈ (𝑀 LMHom 𝑀) ∧ 𝑧 ∈ (𝑀 LMHom 𝑀))) β†’ ((π‘₯ ∘f (+gβ€˜π‘€)𝑦)(.rβ€˜π΄)𝑧) = ((π‘₯ ∘f (+gβ€˜π‘€)𝑦) ∘ 𝑧))
11018oveq1d 7421 . . 3 ((𝑀 ∈ LMod ∧ (π‘₯ ∈ (𝑀 LMHom 𝑀) ∧ 𝑦 ∈ (𝑀 LMHom 𝑀) ∧ 𝑧 ∈ (𝑀 LMHom 𝑀))) β†’ ((π‘₯(+gβ€˜π΄)𝑦)(.rβ€˜π΄)𝑧) = ((π‘₯ ∘f (+gβ€˜π‘€)𝑦)(.rβ€˜π΄)𝑧))
1111, 2, 6, 7mendplusg 41914 . . . . 5 (((π‘₯ ∘ 𝑧) ∈ (𝑀 LMHom 𝑀) ∧ (𝑦 ∘ 𝑧) ∈ (𝑀 LMHom 𝑀)) β†’ ((π‘₯ ∘ 𝑧)(+gβ€˜π΄)(𝑦 ∘ 𝑧)) = ((π‘₯ ∘ 𝑧) ∘f (+gβ€˜π‘€)(𝑦 ∘ 𝑧)))
11295, 86, 111syl2anc 585 . . . 4 ((𝑀 ∈ LMod ∧ (π‘₯ ∈ (𝑀 LMHom 𝑀) ∧ 𝑦 ∈ (𝑀 LMHom 𝑀) ∧ 𝑧 ∈ (𝑀 LMHom 𝑀))) β†’ ((π‘₯ ∘ 𝑧)(+gβ€˜π΄)(𝑦 ∘ 𝑧)) = ((π‘₯ ∘ 𝑧) ∘f (+gβ€˜π‘€)(𝑦 ∘ 𝑧)))
11399, 83oveq12d 7424 . . . 4 ((𝑀 ∈ LMod ∧ (π‘₯ ∈ (𝑀 LMHom 𝑀) ∧ 𝑦 ∈ (𝑀 LMHom 𝑀) ∧ 𝑧 ∈ (𝑀 LMHom 𝑀))) β†’ ((π‘₯(.rβ€˜π΄)𝑧)(+gβ€˜π΄)(𝑦(.rβ€˜π΄)𝑧)) = ((π‘₯ ∘ 𝑧)(+gβ€˜π΄)(𝑦 ∘ 𝑧)))
114 ffn 6715 . . . . . 6 (π‘₯:(Baseβ€˜π‘€)⟢(Baseβ€˜π‘€) β†’ π‘₯ Fn (Baseβ€˜π‘€))
11512, 31, 1143syl 18 . . . . 5 ((𝑀 ∈ LMod ∧ (π‘₯ ∈ (𝑀 LMHom 𝑀) ∧ 𝑦 ∈ (𝑀 LMHom 𝑀) ∧ 𝑧 ∈ (𝑀 LMHom 𝑀))) β†’ π‘₯ Fn (Baseβ€˜π‘€))
116 ffn 6715 . . . . . 6 (𝑦:(Baseβ€˜π‘€)⟢(Baseβ€˜π‘€) β†’ 𝑦 Fn (Baseβ€˜π‘€))
11713, 36, 1163syl 18 . . . . 5 ((𝑀 ∈ LMod ∧ (π‘₯ ∈ (𝑀 LMHom 𝑀) ∧ 𝑦 ∈ (𝑀 LMHom 𝑀) ∧ 𝑧 ∈ (𝑀 LMHom 𝑀))) β†’ 𝑦 Fn (Baseβ€˜π‘€))
11833a1i 11 . . . . 5 ((𝑀 ∈ LMod ∧ (π‘₯ ∈ (𝑀 LMHom 𝑀) ∧ 𝑦 ∈ (𝑀 LMHom 𝑀) ∧ 𝑧 ∈ (𝑀 LMHom 𝑀))) β†’ (Baseβ€˜π‘€) ∈ V)
119 inidm 4218 . . . . 5 ((Baseβ€˜π‘€) ∩ (Baseβ€˜π‘€)) = (Baseβ€˜π‘€)
120115, 117, 41, 118, 118, 118, 119ofco 7690 . . . 4 ((𝑀 ∈ LMod ∧ (π‘₯ ∈ (𝑀 LMHom 𝑀) ∧ 𝑦 ∈ (𝑀 LMHom 𝑀) ∧ 𝑧 ∈ (𝑀 LMHom 𝑀))) β†’ ((π‘₯ ∘f (+gβ€˜π‘€)𝑦) ∘ 𝑧) = ((π‘₯ ∘ 𝑧) ∘f (+gβ€˜π‘€)(𝑦 ∘ 𝑧)))
121112, 113, 1203eqtr4d 2783 . . 3 ((𝑀 ∈ LMod ∧ (π‘₯ ∈ (𝑀 LMHom 𝑀) ∧ 𝑦 ∈ (𝑀 LMHom 𝑀) ∧ 𝑧 ∈ (𝑀 LMHom 𝑀))) β†’ ((π‘₯(.rβ€˜π΄)𝑧)(+gβ€˜π΄)(𝑦(.rβ€˜π΄)𝑧)) = ((π‘₯ ∘f (+gβ€˜π‘€)𝑦) ∘ 𝑧))
122109, 110, 1213eqtr4d 2783 . 2 ((𝑀 ∈ LMod ∧ (π‘₯ ∈ (𝑀 LMHom 𝑀) ∧ 𝑦 ∈ (𝑀 LMHom 𝑀) ∧ 𝑧 ∈ (𝑀 LMHom 𝑀))) β†’ ((π‘₯(+gβ€˜π΄)𝑦)(.rβ€˜π΄)𝑧) = ((π‘₯(.rβ€˜π΄)𝑧)(+gβ€˜π΄)(𝑦(.rβ€˜π΄)𝑧)))
12330idlmhm 20645 . 2 (𝑀 ∈ LMod β†’ ( I β†Ύ (Baseβ€˜π‘€)) ∈ (𝑀 LMHom 𝑀))
1241, 2, 70mendmulr 41916 . . . 4 ((( I β†Ύ (Baseβ€˜π‘€)) ∈ (𝑀 LMHom 𝑀) ∧ π‘₯ ∈ (𝑀 LMHom 𝑀)) β†’ (( I β†Ύ (Baseβ€˜π‘€))(.rβ€˜π΄)π‘₯) = (( I β†Ύ (Baseβ€˜π‘€)) ∘ π‘₯))
125123, 124sylan 581 . . 3 ((𝑀 ∈ LMod ∧ π‘₯ ∈ (𝑀 LMHom 𝑀)) β†’ (( I β†Ύ (Baseβ€˜π‘€))(.rβ€˜π΄)π‘₯) = (( I β†Ύ (Baseβ€˜π‘€)) ∘ π‘₯))
12631adantl 483 . . . 4 ((𝑀 ∈ LMod ∧ π‘₯ ∈ (𝑀 LMHom 𝑀)) β†’ π‘₯:(Baseβ€˜π‘€)⟢(Baseβ€˜π‘€))
127 fcoi2 6764 . . . 4 (π‘₯:(Baseβ€˜π‘€)⟢(Baseβ€˜π‘€) β†’ (( I β†Ύ (Baseβ€˜π‘€)) ∘ π‘₯) = π‘₯)
128126, 127syl 17 . . 3 ((𝑀 ∈ LMod ∧ π‘₯ ∈ (𝑀 LMHom 𝑀)) β†’ (( I β†Ύ (Baseβ€˜π‘€)) ∘ π‘₯) = π‘₯)
129125, 128eqtrd 2773 . 2 ((𝑀 ∈ LMod ∧ π‘₯ ∈ (𝑀 LMHom 𝑀)) β†’ (( I β†Ύ (Baseβ€˜π‘€))(.rβ€˜π΄)π‘₯) = π‘₯)
130 id 22 . . . 4 (π‘₯ ∈ (𝑀 LMHom 𝑀) β†’ π‘₯ ∈ (𝑀 LMHom 𝑀))
1311, 2, 70mendmulr 41916 . . . 4 ((π‘₯ ∈ (𝑀 LMHom 𝑀) ∧ ( I β†Ύ (Baseβ€˜π‘€)) ∈ (𝑀 LMHom 𝑀)) β†’ (π‘₯(.rβ€˜π΄)( I β†Ύ (Baseβ€˜π‘€))) = (π‘₯ ∘ ( I β†Ύ (Baseβ€˜π‘€))))
132130, 123, 131syl2anr 598 . . 3 ((𝑀 ∈ LMod ∧ π‘₯ ∈ (𝑀 LMHom 𝑀)) β†’ (π‘₯(.rβ€˜π΄)( I β†Ύ (Baseβ€˜π‘€))) = (π‘₯ ∘ ( I β†Ύ (Baseβ€˜π‘€))))
133 fcoi1 6763 . . . 4 (π‘₯:(Baseβ€˜π‘€)⟢(Baseβ€˜π‘€) β†’ (π‘₯ ∘ ( I β†Ύ (Baseβ€˜π‘€))) = π‘₯)
134126, 133syl 17 . . 3 ((𝑀 ∈ LMod ∧ π‘₯ ∈ (𝑀 LMHom 𝑀)) β†’ (π‘₯ ∘ ( I β†Ύ (Baseβ€˜π‘€))) = π‘₯)
135132, 134eqtrd 2773 . 2 ((𝑀 ∈ LMod ∧ π‘₯ ∈ (𝑀 LMHom 𝑀)) β†’ (π‘₯(.rβ€˜π΄)( I β†Ύ (Baseβ€˜π‘€))) = π‘₯)
1363, 4, 5, 69, 74, 90, 107, 122, 123, 129, 135isringd 20099 1 (𝑀 ∈ LMod β†’ 𝐴 ∈ Ring)
Colors of variables: wff setvar class
Syntax hints:   β†’ wi 4   ∧ wa 397   ∧ w3a 1088   = wceq 1542   ∈ wcel 2107  Vcvv 3475  {csn 4628   I cid 5573   Γ— cxp 5674   β†Ύ cres 5678   ∘ ccom 5680   Fn wfn 6536  βŸΆwf 6537  β€˜cfv 6541  (class class class)co 7406   ∘f cof 7665   ↑m cmap 8817  Basecbs 17141  +gcplusg 17194  .rcmulr 17195  Scalarcsca 17197  0gc0g 17382  Mndcmnd 18622   MndHom cmhm 18666  Grpcgrp 18816  invgcminusg 18817   GrpHom cghm 19084  Ringcrg 20050  LModclmod 20464   LMHom clmhm 20623  MEndocmend 41903
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2109  ax-9 2117  ax-10 2138  ax-11 2155  ax-12 2172  ax-ext 2704  ax-rep 5285  ax-sep 5299  ax-nul 5306  ax-pow 5363  ax-pr 5427  ax-un 7722  ax-cnex 11163  ax-resscn 11164  ax-1cn 11165  ax-icn 11166  ax-addcl 11167  ax-addrcl 11168  ax-mulcl 11169  ax-mulrcl 11170  ax-mulcom 11171  ax-addass 11172  ax-mulass 11173  ax-distr 11174  ax-i2m1 11175  ax-1ne0 11176  ax-1rid 11177  ax-rnegex 11178  ax-rrecex 11179  ax-cnre 11180  ax-pre-lttri 11181  ax-pre-lttrn 11182  ax-pre-ltadd 11183  ax-pre-mulgt0 11184
This theorem depends on definitions:  df-bi 206  df-an 398  df-or 847  df-3or 1089  df-3an 1090  df-tru 1545  df-fal 1555  df-ex 1783  df-nf 1787  df-sb 2069  df-mo 2535  df-eu 2564  df-clab 2711  df-cleq 2725  df-clel 2811  df-nfc 2886  df-ne 2942  df-nel 3048  df-ral 3063  df-rex 3072  df-rmo 3377  df-reu 3378  df-rab 3434  df-v 3477  df-sbc 3778  df-csb 3894  df-dif 3951  df-un 3953  df-in 3955  df-ss 3965  df-pss 3967  df-nul 4323  df-if 4529  df-pw 4604  df-sn 4629  df-pr 4631  df-tp 4633  df-op 4635  df-uni 4909  df-iun 4999  df-br 5149  df-opab 5211  df-mpt 5232  df-tr 5266  df-id 5574  df-eprel 5580  df-po 5588  df-so 5589  df-fr 5631  df-we 5633  df-xp 5682  df-rel 5683  df-cnv 5684  df-co 5685  df-dm 5686  df-rn 5687  df-res 5688  df-ima 5689  df-pred 6298  df-ord 6365  df-on 6366  df-lim 6367  df-suc 6368  df-iota 6493  df-fun 6543  df-fn 6544  df-f 6545  df-f1 6546  df-fo 6547  df-f1o 6548  df-fv 6549  df-riota 7362  df-ov 7409  df-oprab 7410  df-mpo 7411  df-of 7667  df-om 7853  df-1st 7972  df-2nd 7973  df-frecs 8263  df-wrecs 8294  df-recs 8368  df-rdg 8407  df-1o 8463  df-er 8700  df-map 8819  df-en 8937  df-dom 8938  df-sdom 8939  df-fin 8940  df-pnf 11247  df-mnf 11248  df-xr 11249  df-ltxr 11250  df-le 11251  df-sub 11443  df-neg 11444  df-nn 12210  df-2 12272  df-3 12273  df-4 12274  df-5 12275  df-6 12276  df-n0 12470  df-z 12556  df-uz 12820  df-fz 13482  df-struct 17077  df-sets 17094  df-slot 17112  df-ndx 17124  df-base 17142  df-plusg 17207  df-mulr 17208  df-sca 17210  df-vsca 17211  df-0g 17384  df-mgm 18558  df-sgrp 18607  df-mnd 18623  df-mhm 18668  df-grp 18819  df-minusg 18820  df-ghm 19085  df-cmn 19645  df-abl 19646  df-mgp 19983  df-ur 20000  df-ring 20052  df-lmod 20466  df-lmhm 20626  df-mend 41904
This theorem is referenced by:  mendlmod  41921  mendassa  41922
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