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

Theorem mendring 43200
Description: The module endomorphism algebra is a ring. (Contributed by Stefan O'Rear, 5-Sep-2015.)
Hypothesis
Ref Expression
mendassa.a 𝐴 = (MEndo‘𝑀)
Assertion
Ref Expression
mendring (𝑀 ∈ LMod → 𝐴 ∈ Ring)

Proof of Theorem mendring
Dummy variables 𝑥 𝑦 𝑧 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 → (𝑀 LMHom 𝑀) = (Base‘𝐴))
4 eqidd 2738 . 2 (𝑀 ∈ LMod → (+g𝐴) = (+g𝐴))
5 eqidd 2738 . 2 (𝑀 ∈ LMod → (.r𝐴) = (.r𝐴))
6 eqid 2737 . . . . . 6 (+g𝑀) = (+g𝑀)
7 eqid 2737 . . . . . 6 (+g𝐴) = (+g𝐴)
81, 2, 6, 7mendplusg 43194 . . . . 5 ((𝑥 ∈ (𝑀 LMHom 𝑀) ∧ 𝑦 ∈ (𝑀 LMHom 𝑀)) → (𝑥(+g𝐴)𝑦) = (𝑥f (+g𝑀)𝑦))
96lmhmplusg 21043 . . . . 5 ((𝑥 ∈ (𝑀 LMHom 𝑀) ∧ 𝑦 ∈ (𝑀 LMHom 𝑀)) → (𝑥f (+g𝑀)𝑦) ∈ (𝑀 LMHom 𝑀))
108, 9eqeltrd 2841 . . . 4 ((𝑥 ∈ (𝑀 LMHom 𝑀) ∧ 𝑦 ∈ (𝑀 LMHom 𝑀)) → (𝑥(+g𝐴)𝑦) ∈ (𝑀 LMHom 𝑀))
11103adant1 1131 . . 3 ((𝑀 ∈ LMod ∧ 𝑥 ∈ (𝑀 LMHom 𝑀) ∧ 𝑦 ∈ (𝑀 LMHom 𝑀)) → (𝑥(+g𝐴)𝑦) ∈ (𝑀 LMHom 𝑀))
12 simpr1 1195 . . . . . 6 ((𝑀 ∈ LMod ∧ (𝑥 ∈ (𝑀 LMHom 𝑀) ∧ 𝑦 ∈ (𝑀 LMHom 𝑀) ∧ 𝑧 ∈ (𝑀 LMHom 𝑀))) → 𝑥 ∈ (𝑀 LMHom 𝑀))
13 simpr2 1196 . . . . . 6 ((𝑀 ∈ LMod ∧ (𝑥 ∈ (𝑀 LMHom 𝑀) ∧ 𝑦 ∈ (𝑀 LMHom 𝑀) ∧ 𝑧 ∈ (𝑀 LMHom 𝑀))) → 𝑦 ∈ (𝑀 LMHom 𝑀))
1412, 13, 9syl2anc 584 . . . . 5 ((𝑀 ∈ LMod ∧ (𝑥 ∈ (𝑀 LMHom 𝑀) ∧ 𝑦 ∈ (𝑀 LMHom 𝑀) ∧ 𝑧 ∈ (𝑀 LMHom 𝑀))) → (𝑥f (+g𝑀)𝑦) ∈ (𝑀 LMHom 𝑀))
15 simpr3 1197 . . . . 5 ((𝑀 ∈ LMod ∧ (𝑥 ∈ (𝑀 LMHom 𝑀) ∧ 𝑦 ∈ (𝑀 LMHom 𝑀) ∧ 𝑧 ∈ (𝑀 LMHom 𝑀))) → 𝑧 ∈ (𝑀 LMHom 𝑀))
161, 2, 6, 7mendplusg 43194 . . . . 5 (((𝑥f (+g𝑀)𝑦) ∈ (𝑀 LMHom 𝑀) ∧ 𝑧 ∈ (𝑀 LMHom 𝑀)) → ((𝑥f (+g𝑀)𝑦)(+g𝐴)𝑧) = ((𝑥f (+g𝑀)𝑦) ∘f (+g𝑀)𝑧))
1714, 15, 16syl2anc 584 . . . 4 ((𝑀 ∈ LMod ∧ (𝑥 ∈ (𝑀 LMHom 𝑀) ∧ 𝑦 ∈ (𝑀 LMHom 𝑀) ∧ 𝑧 ∈ (𝑀 LMHom 𝑀))) → ((𝑥f (+g𝑀)𝑦)(+g𝐴)𝑧) = ((𝑥f (+g𝑀)𝑦) ∘f (+g𝑀)𝑧))
1812, 13, 8syl2anc 584 . . . . 5 ((𝑀 ∈ LMod ∧ (𝑥 ∈ (𝑀 LMHom 𝑀) ∧ 𝑦 ∈ (𝑀 LMHom 𝑀) ∧ 𝑧 ∈ (𝑀 LMHom 𝑀))) → (𝑥(+g𝐴)𝑦) = (𝑥f (+g𝑀)𝑦))
1918oveq1d 7446 . . . 4 ((𝑀 ∈ LMod ∧ (𝑥 ∈ (𝑀 LMHom 𝑀) ∧ 𝑦 ∈ (𝑀 LMHom 𝑀) ∧ 𝑧 ∈ (𝑀 LMHom 𝑀))) → ((𝑥(+g𝐴)𝑦)(+g𝐴)𝑧) = ((𝑥f (+g𝑀)𝑦)(+g𝐴)𝑧))
206lmhmplusg 21043 . . . . . . 7 ((𝑦 ∈ (𝑀 LMHom 𝑀) ∧ 𝑧 ∈ (𝑀 LMHom 𝑀)) → (𝑦f (+g𝑀)𝑧) ∈ (𝑀 LMHom 𝑀))
2113, 15, 20syl2anc 584 . . . . . 6 ((𝑀 ∈ LMod ∧ (𝑥 ∈ (𝑀 LMHom 𝑀) ∧ 𝑦 ∈ (𝑀 LMHom 𝑀) ∧ 𝑧 ∈ (𝑀 LMHom 𝑀))) → (𝑦f (+g𝑀)𝑧) ∈ (𝑀 LMHom 𝑀))
221, 2, 6, 7mendplusg 43194 . . . . . 6 ((𝑥 ∈ (𝑀 LMHom 𝑀) ∧ (𝑦f (+g𝑀)𝑧) ∈ (𝑀 LMHom 𝑀)) → (𝑥(+g𝐴)(𝑦f (+g𝑀)𝑧)) = (𝑥f (+g𝑀)(𝑦f (+g𝑀)𝑧)))
2312, 21, 22syl2anc 584 . . . . 5 ((𝑀 ∈ LMod ∧ (𝑥 ∈ (𝑀 LMHom 𝑀) ∧ 𝑦 ∈ (𝑀 LMHom 𝑀) ∧ 𝑧 ∈ (𝑀 LMHom 𝑀))) → (𝑥(+g𝐴)(𝑦f (+g𝑀)𝑧)) = (𝑥f (+g𝑀)(𝑦f (+g𝑀)𝑧)))
241, 2, 6, 7mendplusg 43194 . . . . . . 7 ((𝑦 ∈ (𝑀 LMHom 𝑀) ∧ 𝑧 ∈ (𝑀 LMHom 𝑀)) → (𝑦(+g𝐴)𝑧) = (𝑦f (+g𝑀)𝑧))
2513, 15, 24syl2anc 584 . . . . . 6 ((𝑀 ∈ LMod ∧ (𝑥 ∈ (𝑀 LMHom 𝑀) ∧ 𝑦 ∈ (𝑀 LMHom 𝑀) ∧ 𝑧 ∈ (𝑀 LMHom 𝑀))) → (𝑦(+g𝐴)𝑧) = (𝑦f (+g𝑀)𝑧))
2625oveq2d 7447 . . . . 5 ((𝑀 ∈ LMod ∧ (𝑥 ∈ (𝑀 LMHom 𝑀) ∧ 𝑦 ∈ (𝑀 LMHom 𝑀) ∧ 𝑧 ∈ (𝑀 LMHom 𝑀))) → (𝑥(+g𝐴)(𝑦(+g𝐴)𝑧)) = (𝑥(+g𝐴)(𝑦f (+g𝑀)𝑧)))
27 lmodgrp 20865 . . . . . . . 8 (𝑀 ∈ LMod → 𝑀 ∈ Grp)
2827grpmndd 18964 . . . . . . 7 (𝑀 ∈ LMod → 𝑀 ∈ Mnd)
2928adantr 480 . . . . . 6 ((𝑀 ∈ LMod ∧ (𝑥 ∈ (𝑀 LMHom 𝑀) ∧ 𝑦 ∈ (𝑀 LMHom 𝑀) ∧ 𝑧 ∈ (𝑀 LMHom 𝑀))) → 𝑀 ∈ Mnd)
30 eqid 2737 . . . . . . . . 9 (Base‘𝑀) = (Base‘𝑀)
3130, 30lmhmf 21033 . . . . . . . 8 (𝑥 ∈ (𝑀 LMHom 𝑀) → 𝑥:(Base‘𝑀)⟶(Base‘𝑀))
3212, 31syl 17 . . . . . . 7 ((𝑀 ∈ LMod ∧ (𝑥 ∈ (𝑀 LMHom 𝑀) ∧ 𝑦 ∈ (𝑀 LMHom 𝑀) ∧ 𝑧 ∈ (𝑀 LMHom 𝑀))) → 𝑥:(Base‘𝑀)⟶(Base‘𝑀))
33 fvex 6919 . . . . . . . 8 (Base‘𝑀) ∈ V
3433, 33elmap 8911 . . . . . . 7 (𝑥 ∈ ((Base‘𝑀) ↑m (Base‘𝑀)) ↔ 𝑥:(Base‘𝑀)⟶(Base‘𝑀))
3532, 34sylibr 234 . . . . . 6 ((𝑀 ∈ LMod ∧ (𝑥 ∈ (𝑀 LMHom 𝑀) ∧ 𝑦 ∈ (𝑀 LMHom 𝑀) ∧ 𝑧 ∈ (𝑀 LMHom 𝑀))) → 𝑥 ∈ ((Base‘𝑀) ↑m (Base‘𝑀)))
3630, 30lmhmf 21033 . . . . . . . 8 (𝑦 ∈ (𝑀 LMHom 𝑀) → 𝑦:(Base‘𝑀)⟶(Base‘𝑀))
3713, 36syl 17 . . . . . . 7 ((𝑀 ∈ LMod ∧ (𝑥 ∈ (𝑀 LMHom 𝑀) ∧ 𝑦 ∈ (𝑀 LMHom 𝑀) ∧ 𝑧 ∈ (𝑀 LMHom 𝑀))) → 𝑦:(Base‘𝑀)⟶(Base‘𝑀))
3833, 33elmap 8911 . . . . . . 7 (𝑦 ∈ ((Base‘𝑀) ↑m (Base‘𝑀)) ↔ 𝑦:(Base‘𝑀)⟶(Base‘𝑀))
3937, 38sylibr 234 . . . . . 6 ((𝑀 ∈ LMod ∧ (𝑥 ∈ (𝑀 LMHom 𝑀) ∧ 𝑦 ∈ (𝑀 LMHom 𝑀) ∧ 𝑧 ∈ (𝑀 LMHom 𝑀))) → 𝑦 ∈ ((Base‘𝑀) ↑m (Base‘𝑀)))
4030, 30lmhmf 21033 . . . . . . . 8 (𝑧 ∈ (𝑀 LMHom 𝑀) → 𝑧:(Base‘𝑀)⟶(Base‘𝑀))
4115, 40syl 17 . . . . . . 7 ((𝑀 ∈ LMod ∧ (𝑥 ∈ (𝑀 LMHom 𝑀) ∧ 𝑦 ∈ (𝑀 LMHom 𝑀) ∧ 𝑧 ∈ (𝑀 LMHom 𝑀))) → 𝑧:(Base‘𝑀)⟶(Base‘𝑀))
4233, 33elmap 8911 . . . . . . 7 (𝑧 ∈ ((Base‘𝑀) ↑m (Base‘𝑀)) ↔ 𝑧:(Base‘𝑀)⟶(Base‘𝑀))
4341, 42sylibr 234 . . . . . 6 ((𝑀 ∈ LMod ∧ (𝑥 ∈ (𝑀 LMHom 𝑀) ∧ 𝑦 ∈ (𝑀 LMHom 𝑀) ∧ 𝑧 ∈ (𝑀 LMHom 𝑀))) → 𝑧 ∈ ((Base‘𝑀) ↑m (Base‘𝑀)))
4430, 6mndvass 18811 . . . . . 6 ((𝑀 ∈ Mnd ∧ (𝑥 ∈ ((Base‘𝑀) ↑m (Base‘𝑀)) ∧ 𝑦 ∈ ((Base‘𝑀) ↑m (Base‘𝑀)) ∧ 𝑧 ∈ ((Base‘𝑀) ↑m (Base‘𝑀)))) → ((𝑥f (+g𝑀)𝑦) ∘f (+g𝑀)𝑧) = (𝑥f (+g𝑀)(𝑦f (+g𝑀)𝑧)))
4529, 35, 39, 43, 44syl13anc 1374 . . . . 5 ((𝑀 ∈ LMod ∧ (𝑥 ∈ (𝑀 LMHom 𝑀) ∧ 𝑦 ∈ (𝑀 LMHom 𝑀) ∧ 𝑧 ∈ (𝑀 LMHom 𝑀))) → ((𝑥f (+g𝑀)𝑦) ∘f (+g𝑀)𝑧) = (𝑥f (+g𝑀)(𝑦f (+g𝑀)𝑧)))
4623, 26, 453eqtr4d 2787 . . . 4 ((𝑀 ∈ LMod ∧ (𝑥 ∈ (𝑀 LMHom 𝑀) ∧ 𝑦 ∈ (𝑀 LMHom 𝑀) ∧ 𝑧 ∈ (𝑀 LMHom 𝑀))) → (𝑥(+g𝐴)(𝑦(+g𝐴)𝑧)) = ((𝑥f (+g𝑀)𝑦) ∘f (+g𝑀)𝑧))
4717, 19, 463eqtr4d 2787 . . 3 ((𝑀 ∈ LMod ∧ (𝑥 ∈ (𝑀 LMHom 𝑀) ∧ 𝑦 ∈ (𝑀 LMHom 𝑀) ∧ 𝑧 ∈ (𝑀 LMHom 𝑀))) → ((𝑥(+g𝐴)𝑦)(+g𝐴)𝑧) = (𝑥(+g𝐴)(𝑦(+g𝐴)𝑧)))
48 id 22 . . . 4 (𝑀 ∈ LMod → 𝑀 ∈ LMod)
49 eqidd 2738 . . . 4 (𝑀 ∈ LMod → (Scalar‘𝑀) = (Scalar‘𝑀))
50 eqid 2737 . . . . 5 (0g𝑀) = (0g𝑀)
51 eqid 2737 . . . . 5 (Scalar‘𝑀) = (Scalar‘𝑀)
5250, 30, 51, 510lmhm 21039 . . . 4 ((𝑀 ∈ LMod ∧ 𝑀 ∈ LMod ∧ (Scalar‘𝑀) = (Scalar‘𝑀)) → ((Base‘𝑀) × {(0g𝑀)}) ∈ (𝑀 LMHom 𝑀))
5348, 48, 49, 52syl3anc 1373 . . 3 (𝑀 ∈ LMod → ((Base‘𝑀) × {(0g𝑀)}) ∈ (𝑀 LMHom 𝑀))
541, 2, 6, 7mendplusg 43194 . . . . 5 ((((Base‘𝑀) × {(0g𝑀)}) ∈ (𝑀 LMHom 𝑀) ∧ 𝑥 ∈ (𝑀 LMHom 𝑀)) → (((Base‘𝑀) × {(0g𝑀)})(+g𝐴)𝑥) = (((Base‘𝑀) × {(0g𝑀)}) ∘f (+g𝑀)𝑥))
5553, 54sylan 580 . . . 4 ((𝑀 ∈ LMod ∧ 𝑥 ∈ (𝑀 LMHom 𝑀)) → (((Base‘𝑀) × {(0g𝑀)})(+g𝐴)𝑥) = (((Base‘𝑀) × {(0g𝑀)}) ∘f (+g𝑀)𝑥))
5631, 34sylibr 234 . . . . 5 (𝑥 ∈ (𝑀 LMHom 𝑀) → 𝑥 ∈ ((Base‘𝑀) ↑m (Base‘𝑀)))
5730, 6, 50mndvlid 18812 . . . . 5 ((𝑀 ∈ Mnd ∧ 𝑥 ∈ ((Base‘𝑀) ↑m (Base‘𝑀))) → (((Base‘𝑀) × {(0g𝑀)}) ∘f (+g𝑀)𝑥) = 𝑥)
5828, 56, 57syl2an 596 . . . 4 ((𝑀 ∈ LMod ∧ 𝑥 ∈ (𝑀 LMHom 𝑀)) → (((Base‘𝑀) × {(0g𝑀)}) ∘f (+g𝑀)𝑥) = 𝑥)
5955, 58eqtrd 2777 . . 3 ((𝑀 ∈ LMod ∧ 𝑥 ∈ (𝑀 LMHom 𝑀)) → (((Base‘𝑀) × {(0g𝑀)})(+g𝐴)𝑥) = 𝑥)
60 eqid 2737 . . . . 5 (invg𝑀) = (invg𝑀)
6160invlmhm 21041 . . . 4 (𝑀 ∈ LMod → (invg𝑀) ∈ (𝑀 LMHom 𝑀))
62 lmhmco 21042 . . . 4 (((invg𝑀) ∈ (𝑀 LMHom 𝑀) ∧ 𝑥 ∈ (𝑀 LMHom 𝑀)) → ((invg𝑀) ∘ 𝑥) ∈ (𝑀 LMHom 𝑀))
6361, 62sylan 580 . . 3 ((𝑀 ∈ LMod ∧ 𝑥 ∈ (𝑀 LMHom 𝑀)) → ((invg𝑀) ∘ 𝑥) ∈ (𝑀 LMHom 𝑀))
641, 2, 6, 7mendplusg 43194 . . . . 5 ((((invg𝑀) ∘ 𝑥) ∈ (𝑀 LMHom 𝑀) ∧ 𝑥 ∈ (𝑀 LMHom 𝑀)) → (((invg𝑀) ∘ 𝑥)(+g𝐴)𝑥) = (((invg𝑀) ∘ 𝑥) ∘f (+g𝑀)𝑥))
6563, 64sylancom 588 . . . 4 ((𝑀 ∈ LMod ∧ 𝑥 ∈ (𝑀 LMHom 𝑀)) → (((invg𝑀) ∘ 𝑥)(+g𝐴)𝑥) = (((invg𝑀) ∘ 𝑥) ∘f (+g𝑀)𝑥))
6630, 6, 60, 50grpvlinv 22402 . . . . 5 ((𝑀 ∈ Grp ∧ 𝑥 ∈ ((Base‘𝑀) ↑m (Base‘𝑀))) → (((invg𝑀) ∘ 𝑥) ∘f (+g𝑀)𝑥) = ((Base‘𝑀) × {(0g𝑀)}))
6727, 56, 66syl2an 596 . . . 4 ((𝑀 ∈ LMod ∧ 𝑥 ∈ (𝑀 LMHom 𝑀)) → (((invg𝑀) ∘ 𝑥) ∘f (+g𝑀)𝑥) = ((Base‘𝑀) × {(0g𝑀)}))
6865, 67eqtrd 2777 . . 3 ((𝑀 ∈ LMod ∧ 𝑥 ∈ (𝑀 LMHom 𝑀)) → (((invg𝑀) ∘ 𝑥)(+g𝐴)𝑥) = ((Base‘𝑀) × {(0g𝑀)}))
693, 4, 11, 47, 53, 59, 63, 68isgrpd 18976 . 2 (𝑀 ∈ LMod → 𝐴 ∈ Grp)
70 eqid 2737 . . . . 5 (.r𝐴) = (.r𝐴)
711, 2, 70mendmulr 43196 . . . 4 ((𝑥 ∈ (𝑀 LMHom 𝑀) ∧ 𝑦 ∈ (𝑀 LMHom 𝑀)) → (𝑥(.r𝐴)𝑦) = (𝑥𝑦))
72 lmhmco 21042 . . . 4 ((𝑥 ∈ (𝑀 LMHom 𝑀) ∧ 𝑦 ∈ (𝑀 LMHom 𝑀)) → (𝑥𝑦) ∈ (𝑀 LMHom 𝑀))
7371, 72eqeltrd 2841 . . 3 ((𝑥 ∈ (𝑀 LMHom 𝑀) ∧ 𝑦 ∈ (𝑀 LMHom 𝑀)) → (𝑥(.r𝐴)𝑦) ∈ (𝑀 LMHom 𝑀))
74733adant1 1131 . 2 ((𝑀 ∈ LMod ∧ 𝑥 ∈ (𝑀 LMHom 𝑀) ∧ 𝑦 ∈ (𝑀 LMHom 𝑀)) → (𝑥(.r𝐴)𝑦) ∈ (𝑀 LMHom 𝑀))
75 coass 6285 . . 3 ((𝑥𝑦) ∘ 𝑧) = (𝑥 ∘ (𝑦𝑧))
7612, 13, 71syl2anc 584 . . . . 5 ((𝑀 ∈ LMod ∧ (𝑥 ∈ (𝑀 LMHom 𝑀) ∧ 𝑦 ∈ (𝑀 LMHom 𝑀) ∧ 𝑧 ∈ (𝑀 LMHom 𝑀))) → (𝑥(.r𝐴)𝑦) = (𝑥𝑦))
7776oveq1d 7446 . . . 4 ((𝑀 ∈ LMod ∧ (𝑥 ∈ (𝑀 LMHom 𝑀) ∧ 𝑦 ∈ (𝑀 LMHom 𝑀) ∧ 𝑧 ∈ (𝑀 LMHom 𝑀))) → ((𝑥(.r𝐴)𝑦)(.r𝐴)𝑧) = ((𝑥𝑦)(.r𝐴)𝑧))
7812, 13, 72syl2anc 584 . . . . 5 ((𝑀 ∈ LMod ∧ (𝑥 ∈ (𝑀 LMHom 𝑀) ∧ 𝑦 ∈ (𝑀 LMHom 𝑀) ∧ 𝑧 ∈ (𝑀 LMHom 𝑀))) → (𝑥𝑦) ∈ (𝑀 LMHom 𝑀))
791, 2, 70mendmulr 43196 . . . . 5 (((𝑥𝑦) ∈ (𝑀 LMHom 𝑀) ∧ 𝑧 ∈ (𝑀 LMHom 𝑀)) → ((𝑥𝑦)(.r𝐴)𝑧) = ((𝑥𝑦) ∘ 𝑧))
8078, 15, 79syl2anc 584 . . . 4 ((𝑀 ∈ LMod ∧ (𝑥 ∈ (𝑀 LMHom 𝑀) ∧ 𝑦 ∈ (𝑀 LMHom 𝑀) ∧ 𝑧 ∈ (𝑀 LMHom 𝑀))) → ((𝑥𝑦)(.r𝐴)𝑧) = ((𝑥𝑦) ∘ 𝑧))
8177, 80eqtrd 2777 . . 3 ((𝑀 ∈ LMod ∧ (𝑥 ∈ (𝑀 LMHom 𝑀) ∧ 𝑦 ∈ (𝑀 LMHom 𝑀) ∧ 𝑧 ∈ (𝑀 LMHom 𝑀))) → ((𝑥(.r𝐴)𝑦)(.r𝐴)𝑧) = ((𝑥𝑦) ∘ 𝑧))
821, 2, 70mendmulr 43196 . . . . . 6 ((𝑦 ∈ (𝑀 LMHom 𝑀) ∧ 𝑧 ∈ (𝑀 LMHom 𝑀)) → (𝑦(.r𝐴)𝑧) = (𝑦𝑧))
8313, 15, 82syl2anc 584 . . . . 5 ((𝑀 ∈ LMod ∧ (𝑥 ∈ (𝑀 LMHom 𝑀) ∧ 𝑦 ∈ (𝑀 LMHom 𝑀) ∧ 𝑧 ∈ (𝑀 LMHom 𝑀))) → (𝑦(.r𝐴)𝑧) = (𝑦𝑧))
8483oveq2d 7447 . . . 4 ((𝑀 ∈ LMod ∧ (𝑥 ∈ (𝑀 LMHom 𝑀) ∧ 𝑦 ∈ (𝑀 LMHom 𝑀) ∧ 𝑧 ∈ (𝑀 LMHom 𝑀))) → (𝑥(.r𝐴)(𝑦(.r𝐴)𝑧)) = (𝑥(.r𝐴)(𝑦𝑧)))
85 lmhmco 21042 . . . . . 6 ((𝑦 ∈ (𝑀 LMHom 𝑀) ∧ 𝑧 ∈ (𝑀 LMHom 𝑀)) → (𝑦𝑧) ∈ (𝑀 LMHom 𝑀))
8613, 15, 85syl2anc 584 . . . . 5 ((𝑀 ∈ LMod ∧ (𝑥 ∈ (𝑀 LMHom 𝑀) ∧ 𝑦 ∈ (𝑀 LMHom 𝑀) ∧ 𝑧 ∈ (𝑀 LMHom 𝑀))) → (𝑦𝑧) ∈ (𝑀 LMHom 𝑀))
871, 2, 70mendmulr 43196 . . . . 5 ((𝑥 ∈ (𝑀 LMHom 𝑀) ∧ (𝑦𝑧) ∈ (𝑀 LMHom 𝑀)) → (𝑥(.r𝐴)(𝑦𝑧)) = (𝑥 ∘ (𝑦𝑧)))
8812, 86, 87syl2anc 584 . . . 4 ((𝑀 ∈ LMod ∧ (𝑥 ∈ (𝑀 LMHom 𝑀) ∧ 𝑦 ∈ (𝑀 LMHom 𝑀) ∧ 𝑧 ∈ (𝑀 LMHom 𝑀))) → (𝑥(.r𝐴)(𝑦𝑧)) = (𝑥 ∘ (𝑦𝑧)))
8984, 88eqtrd 2777 . . 3 ((𝑀 ∈ LMod ∧ (𝑥 ∈ (𝑀 LMHom 𝑀) ∧ 𝑦 ∈ (𝑀 LMHom 𝑀) ∧ 𝑧 ∈ (𝑀 LMHom 𝑀))) → (𝑥(.r𝐴)(𝑦(.r𝐴)𝑧)) = (𝑥 ∘ (𝑦𝑧)))
9075, 81, 893eqtr4a 2803 . 2 ((𝑀 ∈ LMod ∧ (𝑥 ∈ (𝑀 LMHom 𝑀) ∧ 𝑦 ∈ (𝑀 LMHom 𝑀) ∧ 𝑧 ∈ (𝑀 LMHom 𝑀))) → ((𝑥(.r𝐴)𝑦)(.r𝐴)𝑧) = (𝑥(.r𝐴)(𝑦(.r𝐴)𝑧)))
911, 2, 70mendmulr 43196 . . . 4 ((𝑥 ∈ (𝑀 LMHom 𝑀) ∧ (𝑦f (+g𝑀)𝑧) ∈ (𝑀 LMHom 𝑀)) → (𝑥(.r𝐴)(𝑦f (+g𝑀)𝑧)) = (𝑥 ∘ (𝑦f (+g𝑀)𝑧)))
9212, 21, 91syl2anc 584 . . 3 ((𝑀 ∈ LMod ∧ (𝑥 ∈ (𝑀 LMHom 𝑀) ∧ 𝑦 ∈ (𝑀 LMHom 𝑀) ∧ 𝑧 ∈ (𝑀 LMHom 𝑀))) → (𝑥(.r𝐴)(𝑦f (+g𝑀)𝑧)) = (𝑥 ∘ (𝑦f (+g𝑀)𝑧)))
9325oveq2d 7447 . . 3 ((𝑀 ∈ LMod ∧ (𝑥 ∈ (𝑀 LMHom 𝑀) ∧ 𝑦 ∈ (𝑀 LMHom 𝑀) ∧ 𝑧 ∈ (𝑀 LMHom 𝑀))) → (𝑥(.r𝐴)(𝑦(+g𝐴)𝑧)) = (𝑥(.r𝐴)(𝑦f (+g𝑀)𝑧)))
94 lmhmco 21042 . . . . . 6 ((𝑥 ∈ (𝑀 LMHom 𝑀) ∧ 𝑧 ∈ (𝑀 LMHom 𝑀)) → (𝑥𝑧) ∈ (𝑀 LMHom 𝑀))
9512, 15, 94syl2anc 584 . . . . 5 ((𝑀 ∈ LMod ∧ (𝑥 ∈ (𝑀 LMHom 𝑀) ∧ 𝑦 ∈ (𝑀 LMHom 𝑀) ∧ 𝑧 ∈ (𝑀 LMHom 𝑀))) → (𝑥𝑧) ∈ (𝑀 LMHom 𝑀))
961, 2, 6, 7mendplusg 43194 . . . . 5 (((𝑥𝑦) ∈ (𝑀 LMHom 𝑀) ∧ (𝑥𝑧) ∈ (𝑀 LMHom 𝑀)) → ((𝑥𝑦)(+g𝐴)(𝑥𝑧)) = ((𝑥𝑦) ∘f (+g𝑀)(𝑥𝑧)))
9778, 95, 96syl2anc 584 . . . 4 ((𝑀 ∈ LMod ∧ (𝑥 ∈ (𝑀 LMHom 𝑀) ∧ 𝑦 ∈ (𝑀 LMHom 𝑀) ∧ 𝑧 ∈ (𝑀 LMHom 𝑀))) → ((𝑥𝑦)(+g𝐴)(𝑥𝑧)) = ((𝑥𝑦) ∘f (+g𝑀)(𝑥𝑧)))
981, 2, 70mendmulr 43196 . . . . . 6 ((𝑥 ∈ (𝑀 LMHom 𝑀) ∧ 𝑧 ∈ (𝑀 LMHom 𝑀)) → (𝑥(.r𝐴)𝑧) = (𝑥𝑧))
9912, 15, 98syl2anc 584 . . . . 5 ((𝑀 ∈ LMod ∧ (𝑥 ∈ (𝑀 LMHom 𝑀) ∧ 𝑦 ∈ (𝑀 LMHom 𝑀) ∧ 𝑧 ∈ (𝑀 LMHom 𝑀))) → (𝑥(.r𝐴)𝑧) = (𝑥𝑧))
10076, 99oveq12d 7449 . . . 4 ((𝑀 ∈ LMod ∧ (𝑥 ∈ (𝑀 LMHom 𝑀) ∧ 𝑦 ∈ (𝑀 LMHom 𝑀) ∧ 𝑧 ∈ (𝑀 LMHom 𝑀))) → ((𝑥(.r𝐴)𝑦)(+g𝐴)(𝑥(.r𝐴)𝑧)) = ((𝑥𝑦)(+g𝐴)(𝑥𝑧)))
101 lmghm 21030 . . . . . 6 (𝑥 ∈ (𝑀 LMHom 𝑀) → 𝑥 ∈ (𝑀 GrpHom 𝑀))
102 ghmmhm 19244 . . . . . 6 (𝑥 ∈ (𝑀 GrpHom 𝑀) → 𝑥 ∈ (𝑀 MndHom 𝑀))
10312, 101, 1023syl 18 . . . . 5 ((𝑀 ∈ LMod ∧ (𝑥 ∈ (𝑀 LMHom 𝑀) ∧ 𝑦 ∈ (𝑀 LMHom 𝑀) ∧ 𝑧 ∈ (𝑀 LMHom 𝑀))) → 𝑥 ∈ (𝑀 MndHom 𝑀))
10430, 6, 6mhmvlin 18814 . . . . 5 ((𝑥 ∈ (𝑀 MndHom 𝑀) ∧ 𝑦 ∈ ((Base‘𝑀) ↑m (Base‘𝑀)) ∧ 𝑧 ∈ ((Base‘𝑀) ↑m (Base‘𝑀))) → (𝑥 ∘ (𝑦f (+g𝑀)𝑧)) = ((𝑥𝑦) ∘f (+g𝑀)(𝑥𝑧)))
105103, 39, 43, 104syl3anc 1373 . . . 4 ((𝑀 ∈ LMod ∧ (𝑥 ∈ (𝑀 LMHom 𝑀) ∧ 𝑦 ∈ (𝑀 LMHom 𝑀) ∧ 𝑧 ∈ (𝑀 LMHom 𝑀))) → (𝑥 ∘ (𝑦f (+g𝑀)𝑧)) = ((𝑥𝑦) ∘f (+g𝑀)(𝑥𝑧)))
10697, 100, 1053eqtr4d 2787 . . 3 ((𝑀 ∈ LMod ∧ (𝑥 ∈ (𝑀 LMHom 𝑀) ∧ 𝑦 ∈ (𝑀 LMHom 𝑀) ∧ 𝑧 ∈ (𝑀 LMHom 𝑀))) → ((𝑥(.r𝐴)𝑦)(+g𝐴)(𝑥(.r𝐴)𝑧)) = (𝑥 ∘ (𝑦f (+g𝑀)𝑧)))
10792, 93, 1063eqtr4d 2787 . 2 ((𝑀 ∈ LMod ∧ (𝑥 ∈ (𝑀 LMHom 𝑀) ∧ 𝑦 ∈ (𝑀 LMHom 𝑀) ∧ 𝑧 ∈ (𝑀 LMHom 𝑀))) → (𝑥(.r𝐴)(𝑦(+g𝐴)𝑧)) = ((𝑥(.r𝐴)𝑦)(+g𝐴)(𝑥(.r𝐴)𝑧)))
1081, 2, 70mendmulr 43196 . . . 4 (((𝑥f (+g𝑀)𝑦) ∈ (𝑀 LMHom 𝑀) ∧ 𝑧 ∈ (𝑀 LMHom 𝑀)) → ((𝑥f (+g𝑀)𝑦)(.r𝐴)𝑧) = ((𝑥f (+g𝑀)𝑦) ∘ 𝑧))
10914, 15, 108syl2anc 584 . . 3 ((𝑀 ∈ LMod ∧ (𝑥 ∈ (𝑀 LMHom 𝑀) ∧ 𝑦 ∈ (𝑀 LMHom 𝑀) ∧ 𝑧 ∈ (𝑀 LMHom 𝑀))) → ((𝑥f (+g𝑀)𝑦)(.r𝐴)𝑧) = ((𝑥f (+g𝑀)𝑦) ∘ 𝑧))
11018oveq1d 7446 . . 3 ((𝑀 ∈ LMod ∧ (𝑥 ∈ (𝑀 LMHom 𝑀) ∧ 𝑦 ∈ (𝑀 LMHom 𝑀) ∧ 𝑧 ∈ (𝑀 LMHom 𝑀))) → ((𝑥(+g𝐴)𝑦)(.r𝐴)𝑧) = ((𝑥f (+g𝑀)𝑦)(.r𝐴)𝑧))
1111, 2, 6, 7mendplusg 43194 . . . . 5 (((𝑥𝑧) ∈ (𝑀 LMHom 𝑀) ∧ (𝑦𝑧) ∈ (𝑀 LMHom 𝑀)) → ((𝑥𝑧)(+g𝐴)(𝑦𝑧)) = ((𝑥𝑧) ∘f (+g𝑀)(𝑦𝑧)))
11295, 86, 111syl2anc 584 . . . 4 ((𝑀 ∈ LMod ∧ (𝑥 ∈ (𝑀 LMHom 𝑀) ∧ 𝑦 ∈ (𝑀 LMHom 𝑀) ∧ 𝑧 ∈ (𝑀 LMHom 𝑀))) → ((𝑥𝑧)(+g𝐴)(𝑦𝑧)) = ((𝑥𝑧) ∘f (+g𝑀)(𝑦𝑧)))
11399, 83oveq12d 7449 . . . 4 ((𝑀 ∈ LMod ∧ (𝑥 ∈ (𝑀 LMHom 𝑀) ∧ 𝑦 ∈ (𝑀 LMHom 𝑀) ∧ 𝑧 ∈ (𝑀 LMHom 𝑀))) → ((𝑥(.r𝐴)𝑧)(+g𝐴)(𝑦(.r𝐴)𝑧)) = ((𝑥𝑧)(+g𝐴)(𝑦𝑧)))
114 ffn 6736 . . . . . 6 (𝑥:(Base‘𝑀)⟶(Base‘𝑀) → 𝑥 Fn (Base‘𝑀))
11512, 31, 1143syl 18 . . . . 5 ((𝑀 ∈ LMod ∧ (𝑥 ∈ (𝑀 LMHom 𝑀) ∧ 𝑦 ∈ (𝑀 LMHom 𝑀) ∧ 𝑧 ∈ (𝑀 LMHom 𝑀))) → 𝑥 Fn (Base‘𝑀))
116 ffn 6736 . . . . . 6 (𝑦:(Base‘𝑀)⟶(Base‘𝑀) → 𝑦 Fn (Base‘𝑀))
11713, 36, 1163syl 18 . . . . 5 ((𝑀 ∈ LMod ∧ (𝑥 ∈ (𝑀 LMHom 𝑀) ∧ 𝑦 ∈ (𝑀 LMHom 𝑀) ∧ 𝑧 ∈ (𝑀 LMHom 𝑀))) → 𝑦 Fn (Base‘𝑀))
11833a1i 11 . . . . 5 ((𝑀 ∈ LMod ∧ (𝑥 ∈ (𝑀 LMHom 𝑀) ∧ 𝑦 ∈ (𝑀 LMHom 𝑀) ∧ 𝑧 ∈ (𝑀 LMHom 𝑀))) → (Base‘𝑀) ∈ V)
119 inidm 4227 . . . . 5 ((Base‘𝑀) ∩ (Base‘𝑀)) = (Base‘𝑀)
120115, 117, 41, 118, 118, 118, 119ofco 7722 . . . 4 ((𝑀 ∈ LMod ∧ (𝑥 ∈ (𝑀 LMHom 𝑀) ∧ 𝑦 ∈ (𝑀 LMHom 𝑀) ∧ 𝑧 ∈ (𝑀 LMHom 𝑀))) → ((𝑥f (+g𝑀)𝑦) ∘ 𝑧) = ((𝑥𝑧) ∘f (+g𝑀)(𝑦𝑧)))
121112, 113, 1203eqtr4d 2787 . . 3 ((𝑀 ∈ LMod ∧ (𝑥 ∈ (𝑀 LMHom 𝑀) ∧ 𝑦 ∈ (𝑀 LMHom 𝑀) ∧ 𝑧 ∈ (𝑀 LMHom 𝑀))) → ((𝑥(.r𝐴)𝑧)(+g𝐴)(𝑦(.r𝐴)𝑧)) = ((𝑥f (+g𝑀)𝑦) ∘ 𝑧))
122109, 110, 1213eqtr4d 2787 . 2 ((𝑀 ∈ LMod ∧ (𝑥 ∈ (𝑀 LMHom 𝑀) ∧ 𝑦 ∈ (𝑀 LMHom 𝑀) ∧ 𝑧 ∈ (𝑀 LMHom 𝑀))) → ((𝑥(+g𝐴)𝑦)(.r𝐴)𝑧) = ((𝑥(.r𝐴)𝑧)(+g𝐴)(𝑦(.r𝐴)𝑧)))
12330idlmhm 21040 . 2 (𝑀 ∈ LMod → ( I ↾ (Base‘𝑀)) ∈ (𝑀 LMHom 𝑀))
1241, 2, 70mendmulr 43196 . . . 4 ((( I ↾ (Base‘𝑀)) ∈ (𝑀 LMHom 𝑀) ∧ 𝑥 ∈ (𝑀 LMHom 𝑀)) → (( I ↾ (Base‘𝑀))(.r𝐴)𝑥) = (( I ↾ (Base‘𝑀)) ∘ 𝑥))
125123, 124sylan 580 . . 3 ((𝑀 ∈ LMod ∧ 𝑥 ∈ (𝑀 LMHom 𝑀)) → (( I ↾ (Base‘𝑀))(.r𝐴)𝑥) = (( I ↾ (Base‘𝑀)) ∘ 𝑥))
12631adantl 481 . . . 4 ((𝑀 ∈ LMod ∧ 𝑥 ∈ (𝑀 LMHom 𝑀)) → 𝑥:(Base‘𝑀)⟶(Base‘𝑀))
127 fcoi2 6783 . . . 4 (𝑥:(Base‘𝑀)⟶(Base‘𝑀) → (( I ↾ (Base‘𝑀)) ∘ 𝑥) = 𝑥)
128126, 127syl 17 . . 3 ((𝑀 ∈ LMod ∧ 𝑥 ∈ (𝑀 LMHom 𝑀)) → (( I ↾ (Base‘𝑀)) ∘ 𝑥) = 𝑥)
129125, 128eqtrd 2777 . 2 ((𝑀 ∈ LMod ∧ 𝑥 ∈ (𝑀 LMHom 𝑀)) → (( I ↾ (Base‘𝑀))(.r𝐴)𝑥) = 𝑥)
130 id 22 . . . 4 (𝑥 ∈ (𝑀 LMHom 𝑀) → 𝑥 ∈ (𝑀 LMHom 𝑀))
1311, 2, 70mendmulr 43196 . . . 4 ((𝑥 ∈ (𝑀 LMHom 𝑀) ∧ ( I ↾ (Base‘𝑀)) ∈ (𝑀 LMHom 𝑀)) → (𝑥(.r𝐴)( I ↾ (Base‘𝑀))) = (𝑥 ∘ ( I ↾ (Base‘𝑀))))
132130, 123, 131syl2anr 597 . . 3 ((𝑀 ∈ LMod ∧ 𝑥 ∈ (𝑀 LMHom 𝑀)) → (𝑥(.r𝐴)( I ↾ (Base‘𝑀))) = (𝑥 ∘ ( I ↾ (Base‘𝑀))))
133 fcoi1 6782 . . . 4 (𝑥:(Base‘𝑀)⟶(Base‘𝑀) → (𝑥 ∘ ( I ↾ (Base‘𝑀))) = 𝑥)
134126, 133syl 17 . . 3 ((𝑀 ∈ LMod ∧ 𝑥 ∈ (𝑀 LMHom 𝑀)) → (𝑥 ∘ ( I ↾ (Base‘𝑀))) = 𝑥)
135132, 134eqtrd 2777 . 2 ((𝑀 ∈ LMod ∧ 𝑥 ∈ (𝑀 LMHom 𝑀)) → (𝑥(.r𝐴)( I ↾ (Base‘𝑀))) = 𝑥)
1363, 4, 5, 69, 74, 90, 107, 122, 123, 129, 135isringd 20288 1 (𝑀 ∈ LMod → 𝐴 ∈ Ring)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 395  w3a 1087   = wceq 1540  wcel 2108  Vcvv 3480  {csn 4626   I cid 5577   × cxp 5683  cres 5687  ccom 5689   Fn wfn 6556  wf 6557  cfv 6561  (class class class)co 7431  f cof 7695  m cmap 8866  Basecbs 17247  +gcplusg 17297  .rcmulr 17298  Scalarcsca 17300  0gc0g 17484  Mndcmnd 18747   MndHom cmhm 18794  Grpcgrp 18951  invgcminusg 18952   GrpHom cghm 19230  Ringcrg 20230  LModclmod 20858   LMHom clmhm 21018  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-lmod 20860  df-lmhm 21021  df-mend 43184
This theorem is referenced by:  mendlmod  43201  mendassa  43202
  Copyright terms: Public domain W3C validator