Theorem mendring 43640

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.

Step	Hyp	Ref	Expression
1		mendassa.a	. . . 4 ⊢ 𝐴 = (MEndo‘𝑀)
2	1	mendbas 43632	. . 3 ⊢ (𝑀 LMHom 𝑀) = (Base‘𝐴)
3	2	a1i 11	. 2 ⊢ (𝑀 ∈ LMod → (𝑀 LMHom 𝑀) = (Base‘𝐴))
4		eqidd 2741	. 2 ⊢ (𝑀 ∈ LMod → (+g‘𝐴) = (+g‘𝐴))
5		eqidd 2741	. 2 ⊢ (𝑀 ∈ LMod → (.r‘𝐴) = (.r‘𝐴))
6		eqid 2740	. . . . . 6 ⊢ (+g‘𝑀) = (+g‘𝑀)
7		eqid 2740	. . . . . 6 ⊢ (+g‘𝐴) = (+g‘𝐴)
8	1, 2, 6, 7	mendplusg 43634	. . . . 5 ⊢ ((𝑥 ∈ (𝑀 LMHom 𝑀) ∧ 𝑦 ∈ (𝑀 LMHom 𝑀)) → (𝑥(+g‘𝐴)𝑦) = (𝑥 ∘f (+g‘𝑀)𝑦))
9	6	lmhmplusg 21041	. . . . 5 ⊢ ((𝑥 ∈ (𝑀 LMHom 𝑀) ∧ 𝑦 ∈ (𝑀 LMHom 𝑀)) → (𝑥 ∘f (+g‘𝑀)𝑦) ∈ (𝑀 LMHom 𝑀))
10	8, 9	eqeltrd 2840	. . . 4 ⊢ ((𝑥 ∈ (𝑀 LMHom 𝑀) ∧ 𝑦 ∈ (𝑀 LMHom 𝑀)) → (𝑥(+g‘𝐴)𝑦) ∈ (𝑀 LMHom 𝑀))
11	10	3adant1 1136	. . 3 ⊢ ((𝑀 ∈ LMod ∧ 𝑥 ∈ (𝑀 LMHom 𝑀) ∧ 𝑦 ∈ (𝑀 LMHom 𝑀)) → (𝑥(+g‘𝐴)𝑦) ∈ (𝑀 LMHom 𝑀))
12		simpr1 1201	. . . . . 6 ⊢ ((𝑀 ∈ LMod ∧ (𝑥 ∈ (𝑀 LMHom 𝑀) ∧ 𝑦 ∈ (𝑀 LMHom 𝑀) ∧ 𝑧 ∈ (𝑀 LMHom 𝑀))) → 𝑥 ∈ (𝑀 LMHom 𝑀))
13		simpr2 1202	. . . . . 6 ⊢ ((𝑀 ∈ LMod ∧ (𝑥 ∈ (𝑀 LMHom 𝑀) ∧ 𝑦 ∈ (𝑀 LMHom 𝑀) ∧ 𝑧 ∈ (𝑀 LMHom 𝑀))) → 𝑦 ∈ (𝑀 LMHom 𝑀))
14	12, 13, 9	syl2anc 590	. . . . 5 ⊢ ((𝑀 ∈ LMod ∧ (𝑥 ∈ (𝑀 LMHom 𝑀) ∧ 𝑦 ∈ (𝑀 LMHom 𝑀) ∧ 𝑧 ∈ (𝑀 LMHom 𝑀))) → (𝑥 ∘f (+g‘𝑀)𝑦) ∈ (𝑀 LMHom 𝑀))
15		simpr3 1203	. . . . 5 ⊢ ((𝑀 ∈ LMod ∧ (𝑥 ∈ (𝑀 LMHom 𝑀) ∧ 𝑦 ∈ (𝑀 LMHom 𝑀) ∧ 𝑧 ∈ (𝑀 LMHom 𝑀))) → 𝑧 ∈ (𝑀 LMHom 𝑀))
16	1, 2, 6, 7	mendplusg 43634	. . . . 5 ⊢ (((𝑥 ∘f (+g‘𝑀)𝑦) ∈ (𝑀 LMHom 𝑀) ∧ 𝑧 ∈ (𝑀 LMHom 𝑀)) → ((𝑥 ∘f (+g‘𝑀)𝑦)(+g‘𝐴)𝑧) = ((𝑥 ∘f (+g‘𝑀)𝑦) ∘f (+g‘𝑀)𝑧))
17	14, 15, 16	syl2anc 590	. . . 4 ⊢ ((𝑀 ∈ LMod ∧ (𝑥 ∈ (𝑀 LMHom 𝑀) ∧ 𝑦 ∈ (𝑀 LMHom 𝑀) ∧ 𝑧 ∈ (𝑀 LMHom 𝑀))) → ((𝑥 ∘f (+g‘𝑀)𝑦)(+g‘𝐴)𝑧) = ((𝑥 ∘f (+g‘𝑀)𝑦) ∘f (+g‘𝑀)𝑧))
18	12, 13, 8	syl2anc 590	. . . . 5 ⊢ ((𝑀 ∈ LMod ∧ (𝑥 ∈ (𝑀 LMHom 𝑀) ∧ 𝑦 ∈ (𝑀 LMHom 𝑀) ∧ 𝑧 ∈ (𝑀 LMHom 𝑀))) → (𝑥(+g‘𝐴)𝑦) = (𝑥 ∘f (+g‘𝑀)𝑦))
19	18	oveq1d 7378	. . . 4 ⊢ ((𝑀 ∈ LMod ∧ (𝑥 ∈ (𝑀 LMHom 𝑀) ∧ 𝑦 ∈ (𝑀 LMHom 𝑀) ∧ 𝑧 ∈ (𝑀 LMHom 𝑀))) → ((𝑥(+g‘𝐴)𝑦)(+g‘𝐴)𝑧) = ((𝑥 ∘f (+g‘𝑀)𝑦)(+g‘𝐴)𝑧))
20	6	lmhmplusg 21041	. . . . . . 7 ⊢ ((𝑦 ∈ (𝑀 LMHom 𝑀) ∧ 𝑧 ∈ (𝑀 LMHom 𝑀)) → (𝑦 ∘f (+g‘𝑀)𝑧) ∈ (𝑀 LMHom 𝑀))
21	13, 15, 20	syl2anc 590	. . . . . 6 ⊢ ((𝑀 ∈ LMod ∧ (𝑥 ∈ (𝑀 LMHom 𝑀) ∧ 𝑦 ∈ (𝑀 LMHom 𝑀) ∧ 𝑧 ∈ (𝑀 LMHom 𝑀))) → (𝑦 ∘f (+g‘𝑀)𝑧) ∈ (𝑀 LMHom 𝑀))
22	1, 2, 6, 7	mendplusg 43634	. . . . . 6 ⊢ ((𝑥 ∈ (𝑀 LMHom 𝑀) ∧ (𝑦 ∘f (+g‘𝑀)𝑧) ∈ (𝑀 LMHom 𝑀)) → (𝑥(+g‘𝐴)(𝑦 ∘f (+g‘𝑀)𝑧)) = (𝑥 ∘f (+g‘𝑀)(𝑦 ∘f (+g‘𝑀)𝑧)))
23	12, 21, 22	syl2anc 590	. . . . 5 ⊢ ((𝑀 ∈ LMod ∧ (𝑥 ∈ (𝑀 LMHom 𝑀) ∧ 𝑦 ∈ (𝑀 LMHom 𝑀) ∧ 𝑧 ∈ (𝑀 LMHom 𝑀))) → (𝑥(+g‘𝐴)(𝑦 ∘f (+g‘𝑀)𝑧)) = (𝑥 ∘f (+g‘𝑀)(𝑦 ∘f (+g‘𝑀)𝑧)))
24	1, 2, 6, 7	mendplusg 43634	. . . . . . 7 ⊢ ((𝑦 ∈ (𝑀 LMHom 𝑀) ∧ 𝑧 ∈ (𝑀 LMHom 𝑀)) → (𝑦(+g‘𝐴)𝑧) = (𝑦 ∘f (+g‘𝑀)𝑧))
25	13, 15, 24	syl2anc 590	. . . . . 6 ⊢ ((𝑀 ∈ LMod ∧ (𝑥 ∈ (𝑀 LMHom 𝑀) ∧ 𝑦 ∈ (𝑀 LMHom 𝑀) ∧ 𝑧 ∈ (𝑀 LMHom 𝑀))) → (𝑦(+g‘𝐴)𝑧) = (𝑦 ∘f (+g‘𝑀)𝑧))
26	25	oveq2d 7379	. . . . 5 ⊢ ((𝑀 ∈ LMod ∧ (𝑥 ∈ (𝑀 LMHom 𝑀) ∧ 𝑦 ∈ (𝑀 LMHom 𝑀) ∧ 𝑧 ∈ (𝑀 LMHom 𝑀))) → (𝑥(+g‘𝐴)(𝑦(+g‘𝐴)𝑧)) = (𝑥(+g‘𝐴)(𝑦 ∘f (+g‘𝑀)𝑧)))
27		lmodgrp 20864	. . . . . . . 8 ⊢ (𝑀 ∈ LMod → 𝑀 ∈ Grp)
28	27	grpmndd 18920	. . . . . . 7 ⊢ (𝑀 ∈ LMod → 𝑀 ∈ Mnd)
29	28	adantr 481	. . . . . 6 ⊢ ((𝑀 ∈ LMod ∧ (𝑥 ∈ (𝑀 LMHom 𝑀) ∧ 𝑦 ∈ (𝑀 LMHom 𝑀) ∧ 𝑧 ∈ (𝑀 LMHom 𝑀))) → 𝑀 ∈ Mnd)
30		eqid 2740	. . . . . . . . 9 ⊢ (Base‘𝑀) = (Base‘𝑀)
31	30, 30	lmhmf 21031	. . . . . . . 8 ⊢ (𝑥 ∈ (𝑀 LMHom 𝑀) → 𝑥:(Base‘𝑀)⟶(Base‘𝑀))
32	12, 31	syl 17	. . . . . . 7 ⊢ ((𝑀 ∈ LMod ∧ (𝑥 ∈ (𝑀 LMHom 𝑀) ∧ 𝑦 ∈ (𝑀 LMHom 𝑀) ∧ 𝑧 ∈ (𝑀 LMHom 𝑀))) → 𝑥:(Base‘𝑀)⟶(Base‘𝑀))
33		fvex 6847	. . . . . . . 8 ⊢ (Base‘𝑀) ∈ V
34	33, 33	elmap 8816	. . . . . . 7 ⊢ (𝑥 ∈ ((Base‘𝑀) ↑m (Base‘𝑀)) ↔ 𝑥:(Base‘𝑀)⟶(Base‘𝑀))
35	32, 34	sylibr 235	. . . . . 6 ⊢ ((𝑀 ∈ LMod ∧ (𝑥 ∈ (𝑀 LMHom 𝑀) ∧ 𝑦 ∈ (𝑀 LMHom 𝑀) ∧ 𝑧 ∈ (𝑀 LMHom 𝑀))) → 𝑥 ∈ ((Base‘𝑀) ↑m (Base‘𝑀)))
36	30, 30	lmhmf 21031	. . . . . . . 8 ⊢ (𝑦 ∈ (𝑀 LMHom 𝑀) → 𝑦:(Base‘𝑀)⟶(Base‘𝑀))
37	13, 36	syl 17	. . . . . . 7 ⊢ ((𝑀 ∈ LMod ∧ (𝑥 ∈ (𝑀 LMHom 𝑀) ∧ 𝑦 ∈ (𝑀 LMHom 𝑀) ∧ 𝑧 ∈ (𝑀 LMHom 𝑀))) → 𝑦:(Base‘𝑀)⟶(Base‘𝑀))
38	33, 33	elmap 8816	. . . . . . 7 ⊢ (𝑦 ∈ ((Base‘𝑀) ↑m (Base‘𝑀)) ↔ 𝑦:(Base‘𝑀)⟶(Base‘𝑀))
39	37, 38	sylibr 235	. . . . . 6 ⊢ ((𝑀 ∈ LMod ∧ (𝑥 ∈ (𝑀 LMHom 𝑀) ∧ 𝑦 ∈ (𝑀 LMHom 𝑀) ∧ 𝑧 ∈ (𝑀 LMHom 𝑀))) → 𝑦 ∈ ((Base‘𝑀) ↑m (Base‘𝑀)))
40	30, 30	lmhmf 21031	. . . . . . . 8 ⊢ (𝑧 ∈ (𝑀 LMHom 𝑀) → 𝑧:(Base‘𝑀)⟶(Base‘𝑀))
41	15, 40	syl 17	. . . . . . 7 ⊢ ((𝑀 ∈ LMod ∧ (𝑥 ∈ (𝑀 LMHom 𝑀) ∧ 𝑦 ∈ (𝑀 LMHom 𝑀) ∧ 𝑧 ∈ (𝑀 LMHom 𝑀))) → 𝑧:(Base‘𝑀)⟶(Base‘𝑀))
42	33, 33	elmap 8816	. . . . . . 7 ⊢ (𝑧 ∈ ((Base‘𝑀) ↑m (Base‘𝑀)) ↔ 𝑧:(Base‘𝑀)⟶(Base‘𝑀))
43	41, 42	sylibr 235	. . . . . 6 ⊢ ((𝑀 ∈ LMod ∧ (𝑥 ∈ (𝑀 LMHom 𝑀) ∧ 𝑦 ∈ (𝑀 LMHom 𝑀) ∧ 𝑧 ∈ (𝑀 LMHom 𝑀))) → 𝑧 ∈ ((Base‘𝑀) ↑m (Base‘𝑀)))
44	30, 6	mndvass 18764	. . . . . 6 ⊢ ((𝑀 ∈ Mnd ∧ (𝑥 ∈ ((Base‘𝑀) ↑m (Base‘𝑀)) ∧ 𝑦 ∈ ((Base‘𝑀) ↑m (Base‘𝑀)) ∧ 𝑧 ∈ ((Base‘𝑀) ↑m (Base‘𝑀)))) → ((𝑥 ∘f (+g‘𝑀)𝑦) ∘f (+g‘𝑀)𝑧) = (𝑥 ∘f (+g‘𝑀)(𝑦 ∘f (+g‘𝑀)𝑧)))
45	29, 35, 39, 43, 44	syl13anc 1380	. . . . 5 ⊢ ((𝑀 ∈ LMod ∧ (𝑥 ∈ (𝑀 LMHom 𝑀) ∧ 𝑦 ∈ (𝑀 LMHom 𝑀) ∧ 𝑧 ∈ (𝑀 LMHom 𝑀))) → ((𝑥 ∘f (+g‘𝑀)𝑦) ∘f (+g‘𝑀)𝑧) = (𝑥 ∘f (+g‘𝑀)(𝑦 ∘f (+g‘𝑀)𝑧)))
46	23, 26, 45	3eqtr4d 2785	. . . 4 ⊢ ((𝑀 ∈ LMod ∧ (𝑥 ∈ (𝑀 LMHom 𝑀) ∧ 𝑦 ∈ (𝑀 LMHom 𝑀) ∧ 𝑧 ∈ (𝑀 LMHom 𝑀))) → (𝑥(+g‘𝐴)(𝑦(+g‘𝐴)𝑧)) = ((𝑥 ∘f (+g‘𝑀)𝑦) ∘f (+g‘𝑀)𝑧))
47	17, 19, 46	3eqtr4d 2785	. . 3 ⊢ ((𝑀 ∈ LMod ∧ (𝑥 ∈ (𝑀 LMHom 𝑀) ∧ 𝑦 ∈ (𝑀 LMHom 𝑀) ∧ 𝑧 ∈ (𝑀 LMHom 𝑀))) → ((𝑥(+g‘𝐴)𝑦)(+g‘𝐴)𝑧) = (𝑥(+g‘𝐴)(𝑦(+g‘𝐴)𝑧)))
48		id 22	. . . 4 ⊢ (𝑀 ∈ LMod → 𝑀 ∈ LMod)
49		eqidd 2741	. . . 4 ⊢ (𝑀 ∈ LMod → (Scalar‘𝑀) = (Scalar‘𝑀))
50		eqid 2740	. . . . 5 ⊢ (0g‘𝑀) = (0g‘𝑀)
51		eqid 2740	. . . . 5 ⊢ (Scalar‘𝑀) = (Scalar‘𝑀)
52	50, 30, 51, 51	0lmhm 21037	. . . 4 ⊢ ((𝑀 ∈ LMod ∧ 𝑀 ∈ LMod ∧ (Scalar‘𝑀) = (Scalar‘𝑀)) → ((Base‘𝑀) × {(0g‘𝑀)}) ∈ (𝑀 LMHom 𝑀))
53	48, 48, 49, 52	syl3anc 1379	. . 3 ⊢ (𝑀 ∈ LMod → ((Base‘𝑀) × {(0g‘𝑀)}) ∈ (𝑀 LMHom 𝑀))
54	1, 2, 6, 7	mendplusg 43634	. . . . 5 ⊢ ((((Base‘𝑀) × {(0g‘𝑀)}) ∈ (𝑀 LMHom 𝑀) ∧ 𝑥 ∈ (𝑀 LMHom 𝑀)) → (((Base‘𝑀) × {(0g‘𝑀)})(+g‘𝐴)𝑥) = (((Base‘𝑀) × {(0g‘𝑀)}) ∘f (+g‘𝑀)𝑥))
55	53, 54	sylan 586	. . . 4 ⊢ ((𝑀 ∈ LMod ∧ 𝑥 ∈ (𝑀 LMHom 𝑀)) → (((Base‘𝑀) × {(0g‘𝑀)})(+g‘𝐴)𝑥) = (((Base‘𝑀) × {(0g‘𝑀)}) ∘f (+g‘𝑀)𝑥))
56	31, 34	sylibr 235	. . . . 5 ⊢ (𝑥 ∈ (𝑀 LMHom 𝑀) → 𝑥 ∈ ((Base‘𝑀) ↑m (Base‘𝑀)))
57	30, 6, 50	mndvlid 18765	. . . . 5 ⊢ ((𝑀 ∈ Mnd ∧ 𝑥 ∈ ((Base‘𝑀) ↑m (Base‘𝑀))) → (((Base‘𝑀) × {(0g‘𝑀)}) ∘f (+g‘𝑀)𝑥) = 𝑥)
58	28, 56, 57	syl2an 602	. . . 4 ⊢ ((𝑀 ∈ LMod ∧ 𝑥 ∈ (𝑀 LMHom 𝑀)) → (((Base‘𝑀) × {(0g‘𝑀)}) ∘f (+g‘𝑀)𝑥) = 𝑥)
59	55, 58	eqtrd 2775	. . 3 ⊢ ((𝑀 ∈ LMod ∧ 𝑥 ∈ (𝑀 LMHom 𝑀)) → (((Base‘𝑀) × {(0g‘𝑀)})(+g‘𝐴)𝑥) = 𝑥)
60		eqid 2740	. . . . 5 ⊢ (invg‘𝑀) = (invg‘𝑀)
61	60	invlmhm 21039	. . . 4 ⊢ (𝑀 ∈ LMod → (invg‘𝑀) ∈ (𝑀 LMHom 𝑀))
62		lmhmco 21040	. . . 4 ⊢ (((invg‘𝑀) ∈ (𝑀 LMHom 𝑀) ∧ 𝑥 ∈ (𝑀 LMHom 𝑀)) → ((invg‘𝑀) ∘ 𝑥) ∈ (𝑀 LMHom 𝑀))
63	61, 62	sylan 586	. . 3 ⊢ ((𝑀 ∈ LMod ∧ 𝑥 ∈ (𝑀 LMHom 𝑀)) → ((invg‘𝑀) ∘ 𝑥) ∈ (𝑀 LMHom 𝑀))
64	1, 2, 6, 7	mendplusg 43634	. . . . 5 ⊢ ((((invg‘𝑀) ∘ 𝑥) ∈ (𝑀 LMHom 𝑀) ∧ 𝑥 ∈ (𝑀 LMHom 𝑀)) → (((invg‘𝑀) ∘ 𝑥)(+g‘𝐴)𝑥) = (((invg‘𝑀) ∘ 𝑥) ∘f (+g‘𝑀)𝑥))
65	63, 64	sylancom 594	. . . 4 ⊢ ((𝑀 ∈ LMod ∧ 𝑥 ∈ (𝑀 LMHom 𝑀)) → (((invg‘𝑀) ∘ 𝑥)(+g‘𝐴)𝑥) = (((invg‘𝑀) ∘ 𝑥) ∘f (+g‘𝑀)𝑥))
66	30, 6, 60, 50	grpvlinv 22388	. . . . 5 ⊢ ((𝑀 ∈ Grp ∧ 𝑥 ∈ ((Base‘𝑀) ↑m (Base‘𝑀))) → (((invg‘𝑀) ∘ 𝑥) ∘f (+g‘𝑀)𝑥) = ((Base‘𝑀) × {(0g‘𝑀)}))
67	27, 56, 66	syl2an 602	. . . 4 ⊢ ((𝑀 ∈ LMod ∧ 𝑥 ∈ (𝑀 LMHom 𝑀)) → (((invg‘𝑀) ∘ 𝑥) ∘f (+g‘𝑀)𝑥) = ((Base‘𝑀) × {(0g‘𝑀)}))
68	65, 67	eqtrd 2775	. . 3 ⊢ ((𝑀 ∈ LMod ∧ 𝑥 ∈ (𝑀 LMHom 𝑀)) → (((invg‘𝑀) ∘ 𝑥)(+g‘𝐴)𝑥) = ((Base‘𝑀) × {(0g‘𝑀)}))
69	3, 4, 11, 47, 53, 59, 63, 68	isgrpd 18932	. 2 ⊢ (𝑀 ∈ LMod → 𝐴 ∈ Grp)
70		eqid 2740	. . . . 5 ⊢ (.r‘𝐴) = (.r‘𝐴)
71	1, 2, 70	mendmulr 43636	. . . 4 ⊢ ((𝑥 ∈ (𝑀 LMHom 𝑀) ∧ 𝑦 ∈ (𝑀 LMHom 𝑀)) → (𝑥(.r‘𝐴)𝑦) = (𝑥 ∘ 𝑦))
72		lmhmco 21040	. . . 4 ⊢ ((𝑥 ∈ (𝑀 LMHom 𝑀) ∧ 𝑦 ∈ (𝑀 LMHom 𝑀)) → (𝑥 ∘ 𝑦) ∈ (𝑀 LMHom 𝑀))
73	71, 72	eqeltrd 2840	. . 3 ⊢ ((𝑥 ∈ (𝑀 LMHom 𝑀) ∧ 𝑦 ∈ (𝑀 LMHom 𝑀)) → (𝑥(.r‘𝐴)𝑦) ∈ (𝑀 LMHom 𝑀))
74	73	3adant1 1136	. 2 ⊢ ((𝑀 ∈ LMod ∧ 𝑥 ∈ (𝑀 LMHom 𝑀) ∧ 𝑦 ∈ (𝑀 LMHom 𝑀)) → (𝑥(.r‘𝐴)𝑦) ∈ (𝑀 LMHom 𝑀))
75		coass 6224	. . 3 ⊢ ((𝑥 ∘ 𝑦) ∘ 𝑧) = (𝑥 ∘ (𝑦 ∘ 𝑧))
76	12, 13, 71	syl2anc 590	. . . . 5 ⊢ ((𝑀 ∈ LMod ∧ (𝑥 ∈ (𝑀 LMHom 𝑀) ∧ 𝑦 ∈ (𝑀 LMHom 𝑀) ∧ 𝑧 ∈ (𝑀 LMHom 𝑀))) → (𝑥(.r‘𝐴)𝑦) = (𝑥 ∘ 𝑦))
77	76	oveq1d 7378	. . . 4 ⊢ ((𝑀 ∈ LMod ∧ (𝑥 ∈ (𝑀 LMHom 𝑀) ∧ 𝑦 ∈ (𝑀 LMHom 𝑀) ∧ 𝑧 ∈ (𝑀 LMHom 𝑀))) → ((𝑥(.r‘𝐴)𝑦)(.r‘𝐴)𝑧) = ((𝑥 ∘ 𝑦)(.r‘𝐴)𝑧))
78	12, 13, 72	syl2anc 590	. . . . 5 ⊢ ((𝑀 ∈ LMod ∧ (𝑥 ∈ (𝑀 LMHom 𝑀) ∧ 𝑦 ∈ (𝑀 LMHom 𝑀) ∧ 𝑧 ∈ (𝑀 LMHom 𝑀))) → (𝑥 ∘ 𝑦) ∈ (𝑀 LMHom 𝑀))
79	1, 2, 70	mendmulr 43636	. . . . 5 ⊢ (((𝑥 ∘ 𝑦) ∈ (𝑀 LMHom 𝑀) ∧ 𝑧 ∈ (𝑀 LMHom 𝑀)) → ((𝑥 ∘ 𝑦)(.r‘𝐴)𝑧) = ((𝑥 ∘ 𝑦) ∘ 𝑧))
80	78, 15, 79	syl2anc 590	. . . 4 ⊢ ((𝑀 ∈ LMod ∧ (𝑥 ∈ (𝑀 LMHom 𝑀) ∧ 𝑦 ∈ (𝑀 LMHom 𝑀) ∧ 𝑧 ∈ (𝑀 LMHom 𝑀))) → ((𝑥 ∘ 𝑦)(.r‘𝐴)𝑧) = ((𝑥 ∘ 𝑦) ∘ 𝑧))
81	77, 80	eqtrd 2775	. . 3 ⊢ ((𝑀 ∈ LMod ∧ (𝑥 ∈ (𝑀 LMHom 𝑀) ∧ 𝑦 ∈ (𝑀 LMHom 𝑀) ∧ 𝑧 ∈ (𝑀 LMHom 𝑀))) → ((𝑥(.r‘𝐴)𝑦)(.r‘𝐴)𝑧) = ((𝑥 ∘ 𝑦) ∘ 𝑧))
82	1, 2, 70	mendmulr 43636	. . . . . 6 ⊢ ((𝑦 ∈ (𝑀 LMHom 𝑀) ∧ 𝑧 ∈ (𝑀 LMHom 𝑀)) → (𝑦(.r‘𝐴)𝑧) = (𝑦 ∘ 𝑧))
83	13, 15, 82	syl2anc 590	. . . . 5 ⊢ ((𝑀 ∈ LMod ∧ (𝑥 ∈ (𝑀 LMHom 𝑀) ∧ 𝑦 ∈ (𝑀 LMHom 𝑀) ∧ 𝑧 ∈ (𝑀 LMHom 𝑀))) → (𝑦(.r‘𝐴)𝑧) = (𝑦 ∘ 𝑧))
84	83	oveq2d 7379	. . . 4 ⊢ ((𝑀 ∈ LMod ∧ (𝑥 ∈ (𝑀 LMHom 𝑀) ∧ 𝑦 ∈ (𝑀 LMHom 𝑀) ∧ 𝑧 ∈ (𝑀 LMHom 𝑀))) → (𝑥(.r‘𝐴)(𝑦(.r‘𝐴)𝑧)) = (𝑥(.r‘𝐴)(𝑦 ∘ 𝑧)))
85		lmhmco 21040	. . . . . 6 ⊢ ((𝑦 ∈ (𝑀 LMHom 𝑀) ∧ 𝑧 ∈ (𝑀 LMHom 𝑀)) → (𝑦 ∘ 𝑧) ∈ (𝑀 LMHom 𝑀))
86	13, 15, 85	syl2anc 590	. . . . 5 ⊢ ((𝑀 ∈ LMod ∧ (𝑥 ∈ (𝑀 LMHom 𝑀) ∧ 𝑦 ∈ (𝑀 LMHom 𝑀) ∧ 𝑧 ∈ (𝑀 LMHom 𝑀))) → (𝑦 ∘ 𝑧) ∈ (𝑀 LMHom 𝑀))
87	1, 2, 70	mendmulr 43636	. . . . 5 ⊢ ((𝑥 ∈ (𝑀 LMHom 𝑀) ∧ (𝑦 ∘ 𝑧) ∈ (𝑀 LMHom 𝑀)) → (𝑥(.r‘𝐴)(𝑦 ∘ 𝑧)) = (𝑥 ∘ (𝑦 ∘ 𝑧)))
88	12, 86, 87	syl2anc 590	. . . 4 ⊢ ((𝑀 ∈ LMod ∧ (𝑥 ∈ (𝑀 LMHom 𝑀) ∧ 𝑦 ∈ (𝑀 LMHom 𝑀) ∧ 𝑧 ∈ (𝑀 LMHom 𝑀))) → (𝑥(.r‘𝐴)(𝑦 ∘ 𝑧)) = (𝑥 ∘ (𝑦 ∘ 𝑧)))
89	84, 88	eqtrd 2775	. . 3 ⊢ ((𝑀 ∈ LMod ∧ (𝑥 ∈ (𝑀 LMHom 𝑀) ∧ 𝑦 ∈ (𝑀 LMHom 𝑀) ∧ 𝑧 ∈ (𝑀 LMHom 𝑀))) → (𝑥(.r‘𝐴)(𝑦(.r‘𝐴)𝑧)) = (𝑥 ∘ (𝑦 ∘ 𝑧)))
90	75, 81, 89	3eqtr4a 2801	. 2 ⊢ ((𝑀 ∈ LMod ∧ (𝑥 ∈ (𝑀 LMHom 𝑀) ∧ 𝑦 ∈ (𝑀 LMHom 𝑀) ∧ 𝑧 ∈ (𝑀 LMHom 𝑀))) → ((𝑥(.r‘𝐴)𝑦)(.r‘𝐴)𝑧) = (𝑥(.r‘𝐴)(𝑦(.r‘𝐴)𝑧)))
91	1, 2, 70	mendmulr 43636	. . . 4 ⊢ ((𝑥 ∈ (𝑀 LMHom 𝑀) ∧ (𝑦 ∘f (+g‘𝑀)𝑧) ∈ (𝑀 LMHom 𝑀)) → (𝑥(.r‘𝐴)(𝑦 ∘f (+g‘𝑀)𝑧)) = (𝑥 ∘ (𝑦 ∘f (+g‘𝑀)𝑧)))
92	12, 21, 91	syl2anc 590	. . 3 ⊢ ((𝑀 ∈ LMod ∧ (𝑥 ∈ (𝑀 LMHom 𝑀) ∧ 𝑦 ∈ (𝑀 LMHom 𝑀) ∧ 𝑧 ∈ (𝑀 LMHom 𝑀))) → (𝑥(.r‘𝐴)(𝑦 ∘f (+g‘𝑀)𝑧)) = (𝑥 ∘ (𝑦 ∘f (+g‘𝑀)𝑧)))
93	25	oveq2d 7379	. . 3 ⊢ ((𝑀 ∈ LMod ∧ (𝑥 ∈ (𝑀 LMHom 𝑀) ∧ 𝑦 ∈ (𝑀 LMHom 𝑀) ∧ 𝑧 ∈ (𝑀 LMHom 𝑀))) → (𝑥(.r‘𝐴)(𝑦(+g‘𝐴)𝑧)) = (𝑥(.r‘𝐴)(𝑦 ∘f (+g‘𝑀)𝑧)))
94		lmhmco 21040	. . . . . 6 ⊢ ((𝑥 ∈ (𝑀 LMHom 𝑀) ∧ 𝑧 ∈ (𝑀 LMHom 𝑀)) → (𝑥 ∘ 𝑧) ∈ (𝑀 LMHom 𝑀))
95	12, 15, 94	syl2anc 590	. . . . 5 ⊢ ((𝑀 ∈ LMod ∧ (𝑥 ∈ (𝑀 LMHom 𝑀) ∧ 𝑦 ∈ (𝑀 LMHom 𝑀) ∧ 𝑧 ∈ (𝑀 LMHom 𝑀))) → (𝑥 ∘ 𝑧) ∈ (𝑀 LMHom 𝑀))
96	1, 2, 6, 7	mendplusg 43634	. . . . 5 ⊢ (((𝑥 ∘ 𝑦) ∈ (𝑀 LMHom 𝑀) ∧ (𝑥 ∘ 𝑧) ∈ (𝑀 LMHom 𝑀)) → ((𝑥 ∘ 𝑦)(+g‘𝐴)(𝑥 ∘ 𝑧)) = ((𝑥 ∘ 𝑦) ∘f (+g‘𝑀)(𝑥 ∘ 𝑧)))
97	78, 95, 96	syl2anc 590	. . . 4 ⊢ ((𝑀 ∈ LMod ∧ (𝑥 ∈ (𝑀 LMHom 𝑀) ∧ 𝑦 ∈ (𝑀 LMHom 𝑀) ∧ 𝑧 ∈ (𝑀 LMHom 𝑀))) → ((𝑥 ∘ 𝑦)(+g‘𝐴)(𝑥 ∘ 𝑧)) = ((𝑥 ∘ 𝑦) ∘f (+g‘𝑀)(𝑥 ∘ 𝑧)))
98	1, 2, 70	mendmulr 43636	. . . . . 6 ⊢ ((𝑥 ∈ (𝑀 LMHom 𝑀) ∧ 𝑧 ∈ (𝑀 LMHom 𝑀)) → (𝑥(.r‘𝐴)𝑧) = (𝑥 ∘ 𝑧))
99	12, 15, 98	syl2anc 590	. . . . 5 ⊢ ((𝑀 ∈ LMod ∧ (𝑥 ∈ (𝑀 LMHom 𝑀) ∧ 𝑦 ∈ (𝑀 LMHom 𝑀) ∧ 𝑧 ∈ (𝑀 LMHom 𝑀))) → (𝑥(.r‘𝐴)𝑧) = (𝑥 ∘ 𝑧))
100	76, 99	oveq12d 7381	. . . 4 ⊢ ((𝑀 ∈ LMod ∧ (𝑥 ∈ (𝑀 LMHom 𝑀) ∧ 𝑦 ∈ (𝑀 LMHom 𝑀) ∧ 𝑧 ∈ (𝑀 LMHom 𝑀))) → ((𝑥(.r‘𝐴)𝑦)(+g‘𝐴)(𝑥(.r‘𝐴)𝑧)) = ((𝑥 ∘ 𝑦)(+g‘𝐴)(𝑥 ∘ 𝑧)))
101		lmghm 21028	. . . . . 6 ⊢ (𝑥 ∈ (𝑀 LMHom 𝑀) → 𝑥 ∈ (𝑀 GrpHom 𝑀))
102		ghmmhm 19199	. . . . . 6 ⊢ (𝑥 ∈ (𝑀 GrpHom 𝑀) → 𝑥 ∈ (𝑀 MndHom 𝑀))
103	12, 101, 102	3syl 18	. . . . 5 ⊢ ((𝑀 ∈ LMod ∧ (𝑥 ∈ (𝑀 LMHom 𝑀) ∧ 𝑦 ∈ (𝑀 LMHom 𝑀) ∧ 𝑧 ∈ (𝑀 LMHom 𝑀))) → 𝑥 ∈ (𝑀 MndHom 𝑀))
104	30, 6, 6	mhmvlin 18767	. . . . 5 ⊢ ((𝑥 ∈ (𝑀 MndHom 𝑀) ∧ 𝑦 ∈ ((Base‘𝑀) ↑m (Base‘𝑀)) ∧ 𝑧 ∈ ((Base‘𝑀) ↑m (Base‘𝑀))) → (𝑥 ∘ (𝑦 ∘f (+g‘𝑀)𝑧)) = ((𝑥 ∘ 𝑦) ∘f (+g‘𝑀)(𝑥 ∘ 𝑧)))
105	103, 39, 43, 104	syl3anc 1379	. . . 4 ⊢ ((𝑀 ∈ LMod ∧ (𝑥 ∈ (𝑀 LMHom 𝑀) ∧ 𝑦 ∈ (𝑀 LMHom 𝑀) ∧ 𝑧 ∈ (𝑀 LMHom 𝑀))) → (𝑥 ∘ (𝑦 ∘f (+g‘𝑀)𝑧)) = ((𝑥 ∘ 𝑦) ∘f (+g‘𝑀)(𝑥 ∘ 𝑧)))
106	97, 100, 105	3eqtr4d 2785	. . 3 ⊢ ((𝑀 ∈ LMod ∧ (𝑥 ∈ (𝑀 LMHom 𝑀) ∧ 𝑦 ∈ (𝑀 LMHom 𝑀) ∧ 𝑧 ∈ (𝑀 LMHom 𝑀))) → ((𝑥(.r‘𝐴)𝑦)(+g‘𝐴)(𝑥(.r‘𝐴)𝑧)) = (𝑥 ∘ (𝑦 ∘f (+g‘𝑀)𝑧)))
107	92, 93, 106	3eqtr4d 2785	. 2 ⊢ ((𝑀 ∈ LMod ∧ (𝑥 ∈ (𝑀 LMHom 𝑀) ∧ 𝑦 ∈ (𝑀 LMHom 𝑀) ∧ 𝑧 ∈ (𝑀 LMHom 𝑀))) → (𝑥(.r‘𝐴)(𝑦(+g‘𝐴)𝑧)) = ((𝑥(.r‘𝐴)𝑦)(+g‘𝐴)(𝑥(.r‘𝐴)𝑧)))
108	1, 2, 70	mendmulr 43636	. . . 4 ⊢ (((𝑥 ∘f (+g‘𝑀)𝑦) ∈ (𝑀 LMHom 𝑀) ∧ 𝑧 ∈ (𝑀 LMHom 𝑀)) → ((𝑥 ∘f (+g‘𝑀)𝑦)(.r‘𝐴)𝑧) = ((𝑥 ∘f (+g‘𝑀)𝑦) ∘ 𝑧))
109	14, 15, 108	syl2anc 590	. . 3 ⊢ ((𝑀 ∈ LMod ∧ (𝑥 ∈ (𝑀 LMHom 𝑀) ∧ 𝑦 ∈ (𝑀 LMHom 𝑀) ∧ 𝑧 ∈ (𝑀 LMHom 𝑀))) → ((𝑥 ∘f (+g‘𝑀)𝑦)(.r‘𝐴)𝑧) = ((𝑥 ∘f (+g‘𝑀)𝑦) ∘ 𝑧))
110	18	oveq1d 7378	. . 3 ⊢ ((𝑀 ∈ LMod ∧ (𝑥 ∈ (𝑀 LMHom 𝑀) ∧ 𝑦 ∈ (𝑀 LMHom 𝑀) ∧ 𝑧 ∈ (𝑀 LMHom 𝑀))) → ((𝑥(+g‘𝐴)𝑦)(.r‘𝐴)𝑧) = ((𝑥 ∘f (+g‘𝑀)𝑦)(.r‘𝐴)𝑧))
111	1, 2, 6, 7	mendplusg 43634	. . . . 5 ⊢ (((𝑥 ∘ 𝑧) ∈ (𝑀 LMHom 𝑀) ∧ (𝑦 ∘ 𝑧) ∈ (𝑀 LMHom 𝑀)) → ((𝑥 ∘ 𝑧)(+g‘𝐴)(𝑦 ∘ 𝑧)) = ((𝑥 ∘ 𝑧) ∘f (+g‘𝑀)(𝑦 ∘ 𝑧)))
112	95, 86, 111	syl2anc 590	. . . 4 ⊢ ((𝑀 ∈ LMod ∧ (𝑥 ∈ (𝑀 LMHom 𝑀) ∧ 𝑦 ∈ (𝑀 LMHom 𝑀) ∧ 𝑧 ∈ (𝑀 LMHom 𝑀))) → ((𝑥 ∘ 𝑧)(+g‘𝐴)(𝑦 ∘ 𝑧)) = ((𝑥 ∘ 𝑧) ∘f (+g‘𝑀)(𝑦 ∘ 𝑧)))
113	99, 83	oveq12d 7381	. . . 4 ⊢ ((𝑀 ∈ LMod ∧ (𝑥 ∈ (𝑀 LMHom 𝑀) ∧ 𝑦 ∈ (𝑀 LMHom 𝑀) ∧ 𝑧 ∈ (𝑀 LMHom 𝑀))) → ((𝑥(.r‘𝐴)𝑧)(+g‘𝐴)(𝑦(.r‘𝐴)𝑧)) = ((𝑥 ∘ 𝑧)(+g‘𝐴)(𝑦 ∘ 𝑧)))
114		ffn 6662	. . . . . 6 ⊢ (𝑥:(Base‘𝑀)⟶(Base‘𝑀) → 𝑥 Fn (Base‘𝑀))
115	12, 31, 114	3syl 18	. . . . 5 ⊢ ((𝑀 ∈ LMod ∧ (𝑥 ∈ (𝑀 LMHom 𝑀) ∧ 𝑦 ∈ (𝑀 LMHom 𝑀) ∧ 𝑧 ∈ (𝑀 LMHom 𝑀))) → 𝑥 Fn (Base‘𝑀))
116		ffn 6662	. . . . . 6 ⊢ (𝑦:(Base‘𝑀)⟶(Base‘𝑀) → 𝑦 Fn (Base‘𝑀))
117	13, 36, 116	3syl 18	. . . . 5 ⊢ ((𝑀 ∈ LMod ∧ (𝑥 ∈ (𝑀 LMHom 𝑀) ∧ 𝑦 ∈ (𝑀 LMHom 𝑀) ∧ 𝑧 ∈ (𝑀 LMHom 𝑀))) → 𝑦 Fn (Base‘𝑀))
118	33	a1i 11	. . . . 5 ⊢ ((𝑀 ∈ LMod ∧ (𝑥 ∈ (𝑀 LMHom 𝑀) ∧ 𝑦 ∈ (𝑀 LMHom 𝑀) ∧ 𝑧 ∈ (𝑀 LMHom 𝑀))) → (Base‘𝑀) ∈ V)
119		inidm 4162	. . . . 5 ⊢ ((Base‘𝑀) ∩ (Base‘𝑀)) = (Base‘𝑀)
120	115, 117, 41, 118, 118, 118, 119	ofco 7652	. . . 4 ⊢ ((𝑀 ∈ LMod ∧ (𝑥 ∈ (𝑀 LMHom 𝑀) ∧ 𝑦 ∈ (𝑀 LMHom 𝑀) ∧ 𝑧 ∈ (𝑀 LMHom 𝑀))) → ((𝑥 ∘f (+g‘𝑀)𝑦) ∘ 𝑧) = ((𝑥 ∘ 𝑧) ∘f (+g‘𝑀)(𝑦 ∘ 𝑧)))
121	112, 113, 120	3eqtr4d 2785	. . 3 ⊢ ((𝑀 ∈ LMod ∧ (𝑥 ∈ (𝑀 LMHom 𝑀) ∧ 𝑦 ∈ (𝑀 LMHom 𝑀) ∧ 𝑧 ∈ (𝑀 LMHom 𝑀))) → ((𝑥(.r‘𝐴)𝑧)(+g‘𝐴)(𝑦(.r‘𝐴)𝑧)) = ((𝑥 ∘f (+g‘𝑀)𝑦) ∘ 𝑧))
122	109, 110, 121	3eqtr4d 2785	. 2 ⊢ ((𝑀 ∈ LMod ∧ (𝑥 ∈ (𝑀 LMHom 𝑀) ∧ 𝑦 ∈ (𝑀 LMHom 𝑀) ∧ 𝑧 ∈ (𝑀 LMHom 𝑀))) → ((𝑥(+g‘𝐴)𝑦)(.r‘𝐴)𝑧) = ((𝑥(.r‘𝐴)𝑧)(+g‘𝐴)(𝑦(.r‘𝐴)𝑧)))
123	30	idlmhm 21038	. 2 ⊢ (𝑀 ∈ LMod → ( I ↾ (Base‘𝑀)) ∈ (𝑀 LMHom 𝑀))
124	1, 2, 70	mendmulr 43636	. . . 4 ⊢ ((( I ↾ (Base‘𝑀)) ∈ (𝑀 LMHom 𝑀) ∧ 𝑥 ∈ (𝑀 LMHom 𝑀)) → (( I ↾ (Base‘𝑀))(.r‘𝐴)𝑥) = (( I ↾ (Base‘𝑀)) ∘ 𝑥))
125	123, 124	sylan 586	. . 3 ⊢ ((𝑀 ∈ LMod ∧ 𝑥 ∈ (𝑀 LMHom 𝑀)) → (( I ↾ (Base‘𝑀))(.r‘𝐴)𝑥) = (( I ↾ (Base‘𝑀)) ∘ 𝑥))
126	31	adantl 482	. . . 4 ⊢ ((𝑀 ∈ LMod ∧ 𝑥 ∈ (𝑀 LMHom 𝑀)) → 𝑥:(Base‘𝑀)⟶(Base‘𝑀))
127		fcoi2 6709	. . . 4 ⊢ (𝑥:(Base‘𝑀)⟶(Base‘𝑀) → (( I ↾ (Base‘𝑀)) ∘ 𝑥) = 𝑥)
128	126, 127	syl 17	. . 3 ⊢ ((𝑀 ∈ LMod ∧ 𝑥 ∈ (𝑀 LMHom 𝑀)) → (( I ↾ (Base‘𝑀)) ∘ 𝑥) = 𝑥)
129	125, 128	eqtrd 2775	. 2 ⊢ ((𝑀 ∈ LMod ∧ 𝑥 ∈ (𝑀 LMHom 𝑀)) → (( I ↾ (Base‘𝑀))(.r‘𝐴)𝑥) = 𝑥)
130		id 22	. . . 4 ⊢ (𝑥 ∈ (𝑀 LMHom 𝑀) → 𝑥 ∈ (𝑀 LMHom 𝑀))
131	1, 2, 70	mendmulr 43636	. . . 4 ⊢ ((𝑥 ∈ (𝑀 LMHom 𝑀) ∧ ( I ↾ (Base‘𝑀)) ∈ (𝑀 LMHom 𝑀)) → (𝑥(.r‘𝐴)( I ↾ (Base‘𝑀))) = (𝑥 ∘ ( I ↾ (Base‘𝑀))))
132	130, 123, 131	syl2anr 603	. . 3 ⊢ ((𝑀 ∈ LMod ∧ 𝑥 ∈ (𝑀 LMHom 𝑀)) → (𝑥(.r‘𝐴)( I ↾ (Base‘𝑀))) = (𝑥 ∘ ( I ↾ (Base‘𝑀))))
133		fcoi1 6708	. . . 4 ⊢ (𝑥:(Base‘𝑀)⟶(Base‘𝑀) → (𝑥 ∘ ( I ↾ (Base‘𝑀))) = 𝑥)
134	126, 133	syl 17	. . 3 ⊢ ((𝑀 ∈ LMod ∧ 𝑥 ∈ (𝑀 LMHom 𝑀)) → (𝑥 ∘ ( I ↾ (Base‘𝑀))) = 𝑥)
135	132, 134	eqtrd 2775	. 2 ⊢ ((𝑀 ∈ LMod ∧ 𝑥 ∈ (𝑀 LMHom 𝑀)) → (𝑥(.r‘𝐴)( I ↾ (Base‘𝑀))) = 𝑥)
136	3, 4, 5, 69, 74, 90, 107, 122, 123, 129, 135	isringd 20270	1 ⊢ (𝑀 ∈ LMod → 𝐴 ∈ Ring)

Syntax hints:   → wi 4   ∧ wa 396   ∧ w3a 1092   = wceq 1547   ∈ wcel 2119  Vcvv 3432  {csn 4562   I cid 5519   × cxp 5623   ↾ cres 5627   ∘ ccom 5629   Fn wfn 6487  ⟶wf 6488  ‘cfv 6492  (class class class)co 7363   ∘_f cof 7625   ↑_m cmap 8770  Basecbs 17177  +_gcplusg 17218  ._rcmulr 17219  Scalarcsca 17221  0_gc0g 17400  Mndcmnd 18700   MndHom cmhm 18747  Grpcgrp 18907  inv_gcminusg 18908   GrpHom cghm 19185  Ringcrg 20212  LModclmod 20857   LMHom clmhm 21016  MEndocmend 43623

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1802  ax-4 1816  ax-5 1917  ax-6 1974  ax-7 2015  ax-8 2121  ax-9 2129  ax-10 2152  ax-11 2168  ax-12 2189  ax-ext 2712  ax-rep 5206  ax-sep 5225  ax-nul 5235  ax-pow 5301  ax-pr 5369  ax-un 7685  ax-cnex 11092  ax-resscn 11093  ax-1cn 11094  ax-icn 11095  ax-addcl 11096  ax-addrcl 11097  ax-mulcl 11098  ax-mulrcl 11099  ax-mulcom 11100  ax-addass 11101  ax-mulass 11102  ax-distr 11103  ax-i2m1 11104  ax-1ne0 11105  ax-1rid 11106  ax-rnegex 11107  ax-rrecex 11108  ax-cnre 11109  ax-pre-lttri 11110  ax-pre-lttrn 11111  ax-pre-ltadd 11112  ax-pre-mulgt0 11113

This theorem depends on definitions:  df-bi 208  df-an 397  df-or 854  df-3or 1093  df-3an 1094  df-tru 1550  df-fal 1560  df-ex 1787  df-nf 1791  df-sb 2074  df-mo 2543  df-eu 2573  df-clab 2719  df-cleq 2732  df-clel 2815  df-nfc 2889  df-ne 2936  df-nel 3040  df-ral 3055  df-rex 3065  df-rmo 3345  df-reu 3346  df-rab 3393  df-v 3434  df-sbc 3731  df-csb 3839  df-dif 3893  df-un 3895  df-in 3897  df-ss 3907  df-pss 3910  df-nul 4269  df-if 4462  df-pw 4538  df-sn 4563  df-pr 4565  df-tp 4567  df-op 4569  df-uni 4846  df-iun 4930  df-br 5080  df-opab 5142  df-mpt 5161  df-tr 5187  df-id 5520  df-eprel 5525  df-po 5533  df-so 5534  df-fr 5578  df-we 5580  df-xp 5631  df-rel 5632  df-cnv 5633  df-co 5634  df-dm 5635  df-rn 5636  df-res 5637  df-ima 5638  df-pred 6259  df-ord 6320  df-on 6321  df-lim 6322  df-suc 6323  df-iota 6448  df-fun 6494  df-fn 6495  df-f 6496  df-f1 6497  df-fo 6498  df-f1o 6499  df-fv 6500  df-riota 7320  df-ov 7366  df-oprab 7367  df-mpo 7368  df-of 7627  df-om 7814  df-1st 7938  df-2nd 7939  df-frecs 8228  df-wrecs 8259  df-recs 8308  df-rdg 8346  df-1o 8402  df-er 8640  df-map 8772  df-en 8891  df-dom 8892  df-sdom 8893  df-fin 8894  df-pnf 11179  df-mnf 11180  df-xr 11181  df-ltxr 11182  df-le 11183  df-sub 11377  df-neg 11378  df-nn 12173  df-2 12242  df-3 12243  df-4 12244  df-5 12245  df-6 12246  df-n0 12436  df-z 12523  df-uz 12787  df-fz 13460  df-struct 17115  df-sets 17132  df-slot 17150  df-ndx 17162  df-base 17178  df-plusg 17231  df-mulr 17232  df-sca 17234  df-vsca 17235  df-0g 17402  df-mgm 18606  df-sgrp 18685  df-mnd 18701  df-mhm 18749  df-grp 18910  df-minusg 18911  df-ghm 19186  df-cmn 19755  df-abl 19756  df-mgp 20120  df-rng 20132  df-ur 20161  df-ring 20214  df-lmod 20859  df-lmhm 21019  df-mend 43624

This theorem is referenced by:  mendlmod  43641  mendassa  43642