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Theorem mendring 43176
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 43168 . . 3 (𝑀 LMHom 𝑀) = (Base‘𝐴)
32a1i 11 . 2 (𝑀 ∈ LMod → (𝑀 LMHom 𝑀) = (Base‘𝐴))
4 eqidd 2735 . 2 (𝑀 ∈ LMod → (+g𝐴) = (+g𝐴))
5 eqidd 2735 . 2 (𝑀 ∈ LMod → (.r𝐴) = (.r𝐴))
6 eqid 2734 . . . . . 6 (+g𝑀) = (+g𝑀)
7 eqid 2734 . . . . . 6 (+g𝐴) = (+g𝐴)
81, 2, 6, 7mendplusg 43170 . . . . 5 ((𝑥 ∈ (𝑀 LMHom 𝑀) ∧ 𝑦 ∈ (𝑀 LMHom 𝑀)) → (𝑥(+g𝐴)𝑦) = (𝑥f (+g𝑀)𝑦))
96lmhmplusg 21060 . . . . 5 ((𝑥 ∈ (𝑀 LMHom 𝑀) ∧ 𝑦 ∈ (𝑀 LMHom 𝑀)) → (𝑥f (+g𝑀)𝑦) ∈ (𝑀 LMHom 𝑀))
108, 9eqeltrd 2838 . . . 4 ((𝑥 ∈ (𝑀 LMHom 𝑀) ∧ 𝑦 ∈ (𝑀 LMHom 𝑀)) → (𝑥(+g𝐴)𝑦) ∈ (𝑀 LMHom 𝑀))
11103adant1 1129 . . 3 ((𝑀 ∈ LMod ∧ 𝑥 ∈ (𝑀 LMHom 𝑀) ∧ 𝑦 ∈ (𝑀 LMHom 𝑀)) → (𝑥(+g𝐴)𝑦) ∈ (𝑀 LMHom 𝑀))
12 simpr1 1193 . . . . . 6 ((𝑀 ∈ LMod ∧ (𝑥 ∈ (𝑀 LMHom 𝑀) ∧ 𝑦 ∈ (𝑀 LMHom 𝑀) ∧ 𝑧 ∈ (𝑀 LMHom 𝑀))) → 𝑥 ∈ (𝑀 LMHom 𝑀))
13 simpr2 1194 . . . . . 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 1195 . . . . 5 ((𝑀 ∈ LMod ∧ (𝑥 ∈ (𝑀 LMHom 𝑀) ∧ 𝑦 ∈ (𝑀 LMHom 𝑀) ∧ 𝑧 ∈ (𝑀 LMHom 𝑀))) → 𝑧 ∈ (𝑀 LMHom 𝑀))
161, 2, 6, 7mendplusg 43170 . . . . 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 7445 . . . 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 21060 . . . . . . 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 43170 . . . . . 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 43170 . . . . . . 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 7446 . . . . 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 20881 . . . . . . . 8 (𝑀 ∈ LMod → 𝑀 ∈ Grp)
2827grpmndd 18976 . . . . . . 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 2734 . . . . . . . . 9 (Base‘𝑀) = (Base‘𝑀)
3130, 30lmhmf 21050 . . . . . . . 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 8909 . . . . . . 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 21050 . . . . . . . 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 8909 . . . . . . 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 21050 . . . . . . . 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 8909 . . . . . . 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 18823 . . . . . 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 1371 . . . . 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 2784 . . . 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 2784 . . 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 2735 . . . 4 (𝑀 ∈ LMod → (Scalar‘𝑀) = (Scalar‘𝑀))
50 eqid 2734 . . . . 5 (0g𝑀) = (0g𝑀)
51 eqid 2734 . . . . 5 (Scalar‘𝑀) = (Scalar‘𝑀)
5250, 30, 51, 510lmhm 21056 . . . 4 ((𝑀 ∈ LMod ∧ 𝑀 ∈ LMod ∧ (Scalar‘𝑀) = (Scalar‘𝑀)) → ((Base‘𝑀) × {(0g𝑀)}) ∈ (𝑀 LMHom 𝑀))
5348, 48, 49, 52syl3anc 1370 . . 3 (𝑀 ∈ LMod → ((Base‘𝑀) × {(0g𝑀)}) ∈ (𝑀 LMHom 𝑀))
541, 2, 6, 7mendplusg 43170 . . . . 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 18824 . . . . 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 2774 . . 3 ((𝑀 ∈ LMod ∧ 𝑥 ∈ (𝑀 LMHom 𝑀)) → (((Base‘𝑀) × {(0g𝑀)})(+g𝐴)𝑥) = 𝑥)
60 eqid 2734 . . . . 5 (invg𝑀) = (invg𝑀)
6160invlmhm 21058 . . . 4 (𝑀 ∈ LMod → (invg𝑀) ∈ (𝑀 LMHom 𝑀))
62 lmhmco 21059 . . . 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 43170 . . . . 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 22417 . . . . 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 2774 . . 3 ((𝑀 ∈ LMod ∧ 𝑥 ∈ (𝑀 LMHom 𝑀)) → (((invg𝑀) ∘ 𝑥)(+g𝐴)𝑥) = ((Base‘𝑀) × {(0g𝑀)}))
693, 4, 11, 47, 53, 59, 63, 68isgrpd 18988 . 2 (𝑀 ∈ LMod → 𝐴 ∈ Grp)
70 eqid 2734 . . . . 5 (.r𝐴) = (.r𝐴)
711, 2, 70mendmulr 43172 . . . 4 ((𝑥 ∈ (𝑀 LMHom 𝑀) ∧ 𝑦 ∈ (𝑀 LMHom 𝑀)) → (𝑥(.r𝐴)𝑦) = (𝑥𝑦))
72 lmhmco 21059 . . . 4 ((𝑥 ∈ (𝑀 LMHom 𝑀) ∧ 𝑦 ∈ (𝑀 LMHom 𝑀)) → (𝑥𝑦) ∈ (𝑀 LMHom 𝑀))
7371, 72eqeltrd 2838 . . 3 ((𝑥 ∈ (𝑀 LMHom 𝑀) ∧ 𝑦 ∈ (𝑀 LMHom 𝑀)) → (𝑥(.r𝐴)𝑦) ∈ (𝑀 LMHom 𝑀))
74733adant1 1129 . 2 ((𝑀 ∈ LMod ∧ 𝑥 ∈ (𝑀 LMHom 𝑀) ∧ 𝑦 ∈ (𝑀 LMHom 𝑀)) → (𝑥(.r𝐴)𝑦) ∈ (𝑀 LMHom 𝑀))
75 coass 6286 . . 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 7445 . . . 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 43172 . . . . 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 2774 . . 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 43172 . . . . . 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 7446 . . . 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 21059 . . . . . 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 43172 . . . . 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 2774 . . 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 2800 . 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 43172 . . . 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 7446 . . 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 21059 . . . . . 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 43170 . . . . 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 43172 . . . . . 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 7448 . . . 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 21047 . . . . . 6 (𝑥 ∈ (𝑀 LMHom 𝑀) → 𝑥 ∈ (𝑀 GrpHom 𝑀))
102 ghmmhm 19256 . . . . . 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 18826 . . . . 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 1370 . . . 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 2784 . . 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 2784 . 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 43172 . . . 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 7445 . . 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 43170 . . . . 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 7448 . . . 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 4234 . . . . 5 ((Base‘𝑀) ∩ (Base‘𝑀)) = (Base‘𝑀)
120115, 117, 41, 118, 118, 118, 119ofco 7721 . . . 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 2784 . . 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 2784 . 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 21057 . 2 (𝑀 ∈ LMod → ( I ↾ (Base‘𝑀)) ∈ (𝑀 LMHom 𝑀))
1241, 2, 70mendmulr 43172 . . . 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 2774 . 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 43172 . . . 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 2774 . 2 ((𝑀 ∈ LMod ∧ 𝑥 ∈ (𝑀 LMHom 𝑀)) → (𝑥(.r𝐴)( I ↾ (Base‘𝑀))) = 𝑥)
1363, 4, 5, 69, 74, 90, 107, 122, 123, 129, 135isringd 20304 1 (𝑀 ∈ LMod → 𝐴 ∈ Ring)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 395  w3a 1086   = wceq 1536  wcel 2105  Vcvv 3477  {csn 4630   I cid 5581   × cxp 5686  cres 5690  ccom 5692   Fn wfn 6557  wf 6558  cfv 6562  (class class class)co 7430  f cof 7694  m cmap 8864  Basecbs 17244  +gcplusg 17297  .rcmulr 17298  Scalarcsca 17300  0gc0g 17485  Mndcmnd 18759   MndHom cmhm 18806  Grpcgrp 18963  invgcminusg 18964   GrpHom cghm 19242  Ringcrg 20250  LModclmod 20874   LMHom clmhm 21035  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-lmod 20876  df-lmhm 21038  df-mend 43160
This theorem is referenced by:  mendlmod  43177  mendassa  43178
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