Theorem mendring 43956

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 43948	. . 3 ⊢ (𝑀 LMHom 𝑀) = (Base‘𝐴)
3	2	a1i 11	. 2 ⊢ (𝑀 ∈ LMod → (𝑀 LMHom 𝑀) = (Base‘𝐴))
4		eqidd 2767	. 2 ⊢ (𝑀 ∈ LMod → (+g‘𝐴) = (+g‘𝐴))
5		eqidd 2767	. 2 ⊢ (𝑀 ∈ LMod → (.r‘𝐴) = (.r‘𝐴))
6		eqid 2766	. . . . . 6 ⊢ (+g‘𝑀) = (+g‘𝑀)
7		eqid 2766	. . . . . 6 ⊢ (+g‘𝐴) = (+g‘𝐴)
8	1, 2, 6, 7	mendplusg 43950	. . . . 5 ⊢ ((𝑥 ∈ (𝑀 LMHom 𝑀) ∧ 𝑦 ∈ (𝑀 LMHom 𝑀)) → (𝑥(+g‘𝐴)𝑦) = (𝑥 ∘f (+g‘𝑀)𝑦))
9	6	lmhmplusg 21202	. . . . 5 ⊢ ((𝑥 ∈ (𝑀 LMHom 𝑀) ∧ 𝑦 ∈ (𝑀 LMHom 𝑀)) → (𝑥 ∘f (+g‘𝑀)𝑦) ∈ (𝑀 LMHom 𝑀))
10	8, 9	eqeltrd 2866	. . . 4 ⊢ ((𝑥 ∈ (𝑀 LMHom 𝑀) ∧ 𝑦 ∈ (𝑀 LMHom 𝑀)) → (𝑥(+g‘𝐴)𝑦) ∈ (𝑀 LMHom 𝑀))
11	10	3adant1 1148	. . 3 ⊢ ((𝑀 ∈ LMod ∧ 𝑥 ∈ (𝑀 LMHom 𝑀) ∧ 𝑦 ∈ (𝑀 LMHom 𝑀)) → (𝑥(+g‘𝐴)𝑦) ∈ (𝑀 LMHom 𝑀))
12		simpr1 1213	. . . . . 6 ⊢ ((𝑀 ∈ LMod ∧ (𝑥 ∈ (𝑀 LMHom 𝑀) ∧ 𝑦 ∈ (𝑀 LMHom 𝑀) ∧ 𝑧 ∈ (𝑀 LMHom 𝑀))) → 𝑥 ∈ (𝑀 LMHom 𝑀))
13		simpr2 1214	. . . . . 6 ⊢ ((𝑀 ∈ LMod ∧ (𝑥 ∈ (𝑀 LMHom 𝑀) ∧ 𝑦 ∈ (𝑀 LMHom 𝑀) ∧ 𝑧 ∈ (𝑀 LMHom 𝑀))) → 𝑦 ∈ (𝑀 LMHom 𝑀))
14	12, 13, 9	syl2anc 596	. . . . 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 1215	. . . . 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 43950	. . . . 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 596	. . . 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 596	. . . . 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 7438	. . . 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 21202	. . . . . . 7 ⊢ ((𝑦 ∈ (𝑀 LMHom 𝑀) ∧ 𝑧 ∈ (𝑀 LMHom 𝑀)) → (𝑦 ∘f (+g‘𝑀)𝑧) ∈ (𝑀 LMHom 𝑀))
21	13, 15, 20	syl2anc 596	. . . . . 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 43950	. . . . . 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 596	. . . . 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 43950	. . . . . . 7 ⊢ ((𝑦 ∈ (𝑀 LMHom 𝑀) ∧ 𝑧 ∈ (𝑀 LMHom 𝑀)) → (𝑦(+g‘𝐴)𝑧) = (𝑦 ∘f (+g‘𝑀)𝑧))
25	13, 15, 24	syl2anc 596	. . . . . 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 7439	. . . . 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 21025	. . . . . . . 8 ⊢ (𝑀 ∈ LMod → 𝑀 ∈ Grp)
28	27	grpmndd 19044	. . . . . . 7 ⊢ (𝑀 ∈ LMod → 𝑀 ∈ Mnd)
29	28	adantr 486	. . . . . 6 ⊢ ((𝑀 ∈ LMod ∧ (𝑥 ∈ (𝑀 LMHom 𝑀) ∧ 𝑦 ∈ (𝑀 LMHom 𝑀) ∧ 𝑧 ∈ (𝑀 LMHom 𝑀))) → 𝑀 ∈ Mnd)
30		eqid 2766	. . . . . . . . 9 ⊢ (Base‘𝑀) = (Base‘𝑀)
31	30, 30	lmhmf 21192	. . . . . . . 8 ⊢ (𝑥 ∈ (𝑀 LMHom 𝑀) → 𝑥:(Base‘𝑀)⟶(Base‘𝑀))
32	12, 31	syl 18	. . . . . . 7 ⊢ ((𝑀 ∈ LMod ∧ (𝑥 ∈ (𝑀 LMHom 𝑀) ∧ 𝑦 ∈ (𝑀 LMHom 𝑀) ∧ 𝑧 ∈ (𝑀 LMHom 𝑀))) → 𝑥:(Base‘𝑀)⟶(Base‘𝑀))
33		fvex 6901	. . . . . . . 8 ⊢ (Base‘𝑀) ∈ V
34	33, 33	elmap 8878	. . . . . . 7 ⊢ (𝑥 ∈ ((Base‘𝑀) ↑m (Base‘𝑀)) ↔ 𝑥:(Base‘𝑀)⟶(Base‘𝑀))
35	32, 34	sylibr 237	. . . . . 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 21192	. . . . . . . 8 ⊢ (𝑦 ∈ (𝑀 LMHom 𝑀) → 𝑦:(Base‘𝑀)⟶(Base‘𝑀))
37	13, 36	syl 18	. . . . . . 7 ⊢ ((𝑀 ∈ LMod ∧ (𝑥 ∈ (𝑀 LMHom 𝑀) ∧ 𝑦 ∈ (𝑀 LMHom 𝑀) ∧ 𝑧 ∈ (𝑀 LMHom 𝑀))) → 𝑦:(Base‘𝑀)⟶(Base‘𝑀))
38	33, 33	elmap 8878	. . . . . . 7 ⊢ (𝑦 ∈ ((Base‘𝑀) ↑m (Base‘𝑀)) ↔ 𝑦:(Base‘𝑀)⟶(Base‘𝑀))
39	37, 38	sylibr 237	. . . . . 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 21192	. . . . . . . 8 ⊢ (𝑧 ∈ (𝑀 LMHom 𝑀) → 𝑧:(Base‘𝑀)⟶(Base‘𝑀))
41	15, 40	syl 18	. . . . . . 7 ⊢ ((𝑀 ∈ LMod ∧ (𝑥 ∈ (𝑀 LMHom 𝑀) ∧ 𝑦 ∈ (𝑀 LMHom 𝑀) ∧ 𝑧 ∈ (𝑀 LMHom 𝑀))) → 𝑧:(Base‘𝑀)⟶(Base‘𝑀))
42	33, 33	elmap 8878	. . . . . . 7 ⊢ (𝑧 ∈ ((Base‘𝑀) ↑m (Base‘𝑀)) ↔ 𝑧:(Base‘𝑀)⟶(Base‘𝑀))
43	41, 42	sylibr 237	. . . . . 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 18887	. . . . . 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 1399	. . . . 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 2811	. . . 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 2811	. . 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 23	. . . 4 ⊢ (𝑀 ∈ LMod → 𝑀 ∈ LMod)
49		eqidd 2767	. . . 4 ⊢ (𝑀 ∈ LMod → (Scalar‘𝑀) = (Scalar‘𝑀))
50		eqid 2766	. . . . 5 ⊢ (0g‘𝑀) = (0g‘𝑀)
51		eqid 2766	. . . . 5 ⊢ (Scalar‘𝑀) = (Scalar‘𝑀)
52	50, 30, 51, 51	0lmhm 21198	. . . 4 ⊢ ((𝑀 ∈ LMod ∧ 𝑀 ∈ LMod ∧ (Scalar‘𝑀) = (Scalar‘𝑀)) → ((Base‘𝑀) × {(0g‘𝑀)}) ∈ (𝑀 LMHom 𝑀))
53	48, 48, 49, 52	syl3anc 1398	. . 3 ⊢ (𝑀 ∈ LMod → ((Base‘𝑀) × {(0g‘𝑀)}) ∈ (𝑀 LMHom 𝑀))
54	1, 2, 6, 7	mendplusg 43950	. . . . 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 592	. . . 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 237	. . . . 5 ⊢ (𝑥 ∈ (𝑀 LMHom 𝑀) → 𝑥 ∈ ((Base‘𝑀) ↑m (Base‘𝑀)))
57	30, 6, 50	mndvlid 18888	. . . . 5 ⊢ ((𝑀 ∈ Mnd ∧ 𝑥 ∈ ((Base‘𝑀) ↑m (Base‘𝑀))) → (((Base‘𝑀) × {(0g‘𝑀)}) ∘f (+g‘𝑀)𝑥) = 𝑥)
58	28, 56, 57	syl2an 608	. . . 4 ⊢ ((𝑀 ∈ LMod ∧ 𝑥 ∈ (𝑀 LMHom 𝑀)) → (((Base‘𝑀) × {(0g‘𝑀)}) ∘f (+g‘𝑀)𝑥) = 𝑥)
59	55, 58	eqtrd 2801	. . 3 ⊢ ((𝑀 ∈ LMod ∧ 𝑥 ∈ (𝑀 LMHom 𝑀)) → (((Base‘𝑀) × {(0g‘𝑀)})(+g‘𝐴)𝑥) = 𝑥)
60		eqid 2766	. . . . 5 ⊢ (invg‘𝑀) = (invg‘𝑀)
61	60	invlmhm 21200	. . . 4 ⊢ (𝑀 ∈ LMod → (invg‘𝑀) ∈ (𝑀 LMHom 𝑀))
62		lmhmco 21201	. . . 4 ⊢ (((invg‘𝑀) ∈ (𝑀 LMHom 𝑀) ∧ 𝑥 ∈ (𝑀 LMHom 𝑀)) → ((invg‘𝑀) ∘ 𝑥) ∈ (𝑀 LMHom 𝑀))
63	61, 62	sylan 592	. . 3 ⊢ ((𝑀 ∈ LMod ∧ 𝑥 ∈ (𝑀 LMHom 𝑀)) → ((invg‘𝑀) ∘ 𝑥) ∈ (𝑀 LMHom 𝑀))
64	1, 2, 6, 7	mendplusg 43950	. . . . 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 600	. . . 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 22592	. . . . 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 608	. . . 4 ⊢ ((𝑀 ∈ LMod ∧ 𝑥 ∈ (𝑀 LMHom 𝑀)) → (((invg‘𝑀) ∘ 𝑥) ∘f (+g‘𝑀)𝑥) = ((Base‘𝑀) × {(0g‘𝑀)}))
68	65, 67	eqtrd 2801	. . 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 19056	. 2 ⊢ (𝑀 ∈ LMod → 𝐴 ∈ Grp)
70		eqid 2766	. . . . 5 ⊢ (.r‘𝐴) = (.r‘𝐴)
71	1, 2, 70	mendmulr 43952	. . . 4 ⊢ ((𝑥 ∈ (𝑀 LMHom 𝑀) ∧ 𝑦 ∈ (𝑀 LMHom 𝑀)) → (𝑥(.r‘𝐴)𝑦) = (𝑥 ∘ 𝑦))
72		lmhmco 21201	. . . 4 ⊢ ((𝑥 ∈ (𝑀 LMHom 𝑀) ∧ 𝑦 ∈ (𝑀 LMHom 𝑀)) → (𝑥 ∘ 𝑦) ∈ (𝑀 LMHom 𝑀))
73	71, 72	eqeltrd 2866	. . 3 ⊢ ((𝑥 ∈ (𝑀 LMHom 𝑀) ∧ 𝑦 ∈ (𝑀 LMHom 𝑀)) → (𝑥(.r‘𝐴)𝑦) ∈ (𝑀 LMHom 𝑀))
74	73	3adant1 1148	. 2 ⊢ ((𝑀 ∈ LMod ∧ 𝑥 ∈ (𝑀 LMHom 𝑀) ∧ 𝑦 ∈ (𝑀 LMHom 𝑀)) → (𝑥(.r‘𝐴)𝑦) ∈ (𝑀 LMHom 𝑀))
75		coass 6272	. . 3 ⊢ ((𝑥 ∘ 𝑦) ∘ 𝑧) = (𝑥 ∘ (𝑦 ∘ 𝑧))
76	12, 13, 71	syl2anc 596	. . . . 5 ⊢ ((𝑀 ∈ LMod ∧ (𝑥 ∈ (𝑀 LMHom 𝑀) ∧ 𝑦 ∈ (𝑀 LMHom 𝑀) ∧ 𝑧 ∈ (𝑀 LMHom 𝑀))) → (𝑥(.r‘𝐴)𝑦) = (𝑥 ∘ 𝑦))
77	76	oveq1d 7438	. . . 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 596	. . . . 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 43952	. . . . 5 ⊢ (((𝑥 ∘ 𝑦) ∈ (𝑀 LMHom 𝑀) ∧ 𝑧 ∈ (𝑀 LMHom 𝑀)) → ((𝑥 ∘ 𝑦)(.r‘𝐴)𝑧) = ((𝑥 ∘ 𝑦) ∘ 𝑧))
80	78, 15, 79	syl2anc 596	. . . 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 2801	. . 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 43952	. . . . . 6 ⊢ ((𝑦 ∈ (𝑀 LMHom 𝑀) ∧ 𝑧 ∈ (𝑀 LMHom 𝑀)) → (𝑦(.r‘𝐴)𝑧) = (𝑦 ∘ 𝑧))
83	13, 15, 82	syl2anc 596	. . . . 5 ⊢ ((𝑀 ∈ LMod ∧ (𝑥 ∈ (𝑀 LMHom 𝑀) ∧ 𝑦 ∈ (𝑀 LMHom 𝑀) ∧ 𝑧 ∈ (𝑀 LMHom 𝑀))) → (𝑦(.r‘𝐴)𝑧) = (𝑦 ∘ 𝑧))
84	83	oveq2d 7439	. . . 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 21201	. . . . . 6 ⊢ ((𝑦 ∈ (𝑀 LMHom 𝑀) ∧ 𝑧 ∈ (𝑀 LMHom 𝑀)) → (𝑦 ∘ 𝑧) ∈ (𝑀 LMHom 𝑀))
86	13, 15, 85	syl2anc 596	. . . . 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 43952	. . . . 5 ⊢ ((𝑥 ∈ (𝑀 LMHom 𝑀) ∧ (𝑦 ∘ 𝑧) ∈ (𝑀 LMHom 𝑀)) → (𝑥(.r‘𝐴)(𝑦 ∘ 𝑧)) = (𝑥 ∘ (𝑦 ∘ 𝑧)))
88	12, 86, 87	syl2anc 596	. . . 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 2801	. . 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 2827	. 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 43952	. . . 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 596	. . 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 7439	. . 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 21201	. . . . . 6 ⊢ ((𝑥 ∈ (𝑀 LMHom 𝑀) ∧ 𝑧 ∈ (𝑀 LMHom 𝑀)) → (𝑥 ∘ 𝑧) ∈ (𝑀 LMHom 𝑀))
95	12, 15, 94	syl2anc 596	. . . . 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 43950	. . . . 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 596	. . . 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 43952	. . . . . 6 ⊢ ((𝑥 ∈ (𝑀 LMHom 𝑀) ∧ 𝑧 ∈ (𝑀 LMHom 𝑀)) → (𝑥(.r‘𝐴)𝑧) = (𝑥 ∘ 𝑧))
99	12, 15, 98	syl2anc 596	. . . . 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 7441	. . . 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 21189	. . . . . 6 ⊢ (𝑥 ∈ (𝑀 LMHom 𝑀) → 𝑥 ∈ (𝑀 GrpHom 𝑀))
102		ghmmhm 19327	. . . . . 6 ⊢ (𝑥 ∈ (𝑀 GrpHom 𝑀) → 𝑥 ∈ (𝑀 MndHom 𝑀))
103	12, 101, 102	3syl 19	. . . . 5 ⊢ ((𝑀 ∈ LMod ∧ (𝑥 ∈ (𝑀 LMHom 𝑀) ∧ 𝑦 ∈ (𝑀 LMHom 𝑀) ∧ 𝑧 ∈ (𝑀 LMHom 𝑀))) → 𝑥 ∈ (𝑀 MndHom 𝑀))
104	30, 6, 6	mhmvlin 18890	. . . . 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 1398	. . . 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 2811	. . 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 2811	. 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 43952	. . . 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 596	. . 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 7438	. . 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 43950	. . . . 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 596	. . . 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 7441	. . . 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 6712	. . . . . 6 ⊢ (𝑥:(Base‘𝑀)⟶(Base‘𝑀) → 𝑥 Fn (Base‘𝑀))
115	12, 31, 114	3syl 19	. . . . 5 ⊢ ((𝑀 ∈ LMod ∧ (𝑥 ∈ (𝑀 LMHom 𝑀) ∧ 𝑦 ∈ (𝑀 LMHom 𝑀) ∧ 𝑧 ∈ (𝑀 LMHom 𝑀))) → 𝑥 Fn (Base‘𝑀))
116		ffn 6712	. . . . . 6 ⊢ (𝑦:(Base‘𝑀)⟶(Base‘𝑀) → 𝑦 Fn (Base‘𝑀))
117	13, 36, 116	3syl 19	. . . . 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 4182	. . . . 5 ⊢ ((Base‘𝑀) ∩ (Base‘𝑀)) = (Base‘𝑀)
120	115, 117, 41, 118, 118, 118, 119	ofco 7712	. . . 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 2811	. . 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 2811	. 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 21199	. 2 ⊢ (𝑀 ∈ LMod → ( I ↾ (Base‘𝑀)) ∈ (𝑀 LMHom 𝑀))
124	1, 2, 70	mendmulr 43952	. . . 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 592	. . 3 ⊢ ((𝑀 ∈ LMod ∧ 𝑥 ∈ (𝑀 LMHom 𝑀)) → (( I ↾ (Base‘𝑀))(.r‘𝐴)𝑥) = (( I ↾ (Base‘𝑀)) ∘ 𝑥))
126	31	adantl 487	. . . 4 ⊢ ((𝑀 ∈ LMod ∧ 𝑥 ∈ (𝑀 LMHom 𝑀)) → 𝑥:(Base‘𝑀)⟶(Base‘𝑀))
127		fcoi2 6760	. . . 4 ⊢ (𝑥:(Base‘𝑀)⟶(Base‘𝑀) → (( I ↾ (Base‘𝑀)) ∘ 𝑥) = 𝑥)
128	126, 127	syl 18	. . 3 ⊢ ((𝑀 ∈ LMod ∧ 𝑥 ∈ (𝑀 LMHom 𝑀)) → (( I ↾ (Base‘𝑀)) ∘ 𝑥) = 𝑥)
129	125, 128	eqtrd 2801	. 2 ⊢ ((𝑀 ∈ LMod ∧ 𝑥 ∈ (𝑀 LMHom 𝑀)) → (( I ↾ (Base‘𝑀))(.r‘𝐴)𝑥) = 𝑥)
130		id 23	. . . 4 ⊢ (𝑥 ∈ (𝑀 LMHom 𝑀) → 𝑥 ∈ (𝑀 LMHom 𝑀))
131	1, 2, 70	mendmulr 43952	. . . 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 609	. . 3 ⊢ ((𝑀 ∈ LMod ∧ 𝑥 ∈ (𝑀 LMHom 𝑀)) → (𝑥(.r‘𝐴)( I ↾ (Base‘𝑀))) = (𝑥 ∘ ( I ↾ (Base‘𝑀))))
133		fcoi1 6759	. . . 4 ⊢ (𝑥:(Base‘𝑀)⟶(Base‘𝑀) → (𝑥 ∘ ( I ↾ (Base‘𝑀))) = 𝑥)
134	126, 133	syl 18	. . 3 ⊢ ((𝑀 ∈ LMod ∧ 𝑥 ∈ (𝑀 LMHom 𝑀)) → (𝑥 ∘ ( I ↾ (Base‘𝑀))) = 𝑥)
135	132, 134	eqtrd 2801	. 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 20407	1 ⊢ (𝑀 ∈ LMod → 𝐴 ∈ Ring)

This proof depends on syntax axioms:   → wi 4   ∧ wa 401   ∧ w3a 1103   = wceq 1570   ∈ wcel 2146  Vcvv 3458  {csn 4594   I cid 5560   × cxp 5664   ↾ cres 5668   ∘ ccom 5670   Fn wfn 6538  ⟶wf 6539  ‘cfv 6543  (class class class)co 7423   ∘_f cof 7685   ↑_m cmap 8833  Basecbs 17294  +_gcplusg 17335  ._rcmulr 17336  Scalarcsca 17338  0_gc0g 17517  Mndcmnd 18821   MndHom cmhm 18870  Grpcgrp 19031  inv_gcminusg 19032   GrpHom cghm 19314  Ringcrg 20346  LModclmod 21018   LMHom clmhm 21177  MEndocmend 43939

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1828  ax-4 1842  ax-5 1943  ax-6 2000  ax-7 2041  ax-8 2148  ax-9 2156  ax-10 2179  ax-11 2195  ax-12 2216  ax-ext 2738  ax-rep 5243  ax-sep 5262  ax-nul 5274  ax-pow 5341  ax-pr 5409  ax-un 7745  ax-cnex 11174  ax-resscn 11175  ax-1cn 11176  ax-icn 11177  ax-addcl 11178  ax-addrcl 11179  ax-mulcl 11180  ax-mulrcl 11181  ax-mulcom 11182  ax-addass 11183  ax-mulass 11184  ax-distr 11185  ax-i2m1 11186  ax-1ne0 11187  ax-1rid 11188  ax-rnegex 11189  ax-rrecex 11190  ax-cnre 11191  ax-pre-lttri 11192  ax-pre-lttrn 11193  ax-pre-ltadd 11194  ax-pre-mulgt0 11195

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3or 1104  df-3an 1105  df-tru 1573  df-fal 1583  df-ex 1813  df-nf 1817  df-sb 2100  df-mo 2570  df-eu 2600  df-clab 2745  df-cleq 2758  df-clel 2841  df-nfc 2915  df-ne 2962  df-nel 3068  df-ral 3083  df-rex 3093  df-rmo 3372  df-reu 3373  df-rab 3420  df-v 3460  df-sbc 3748  df-csb 3857  df-dif 3911  df-un 3913  df-in 3915  df-ss 3925  df-pss 3928  df-nul 4290  df-if 4493  df-pw 4569  df-sn 4595  df-pr 4597  df-tp 4599  df-op 4601  df-uni 4878  df-iun 4963  df-br 5115  df-opab 5179  df-mpt 5198  df-tr 5224  df-id 5561  df-eprel 5566  df-po 5574  df-so 5575  df-fr 5619  df-we 5621  df-xp 5672  df-rel 5673  df-cnv 5674  df-co 5675  df-dm 5676  df-rn 5677  df-res 5678  df-ima 5679  df-pred 6309  df-ord 6370  df-on 6371  df-lim 6372  df-suc 6373  df-iota 6499  df-fun 6545  df-fn 6546  df-f 6547  df-f1 6548  df-fo 6549  df-f1o 6550  df-fv 6551  df-riota 7380  df-ov 7426  df-oprab 7427  df-mpo 7428  df-of 7687  df-om 7872  df-1st 7995  df-2nd 7996  df-frecs 8287  df-wrecs 8318  df-recs 8367  df-rdg 8406  df-1o 8462  df-er 8703  df-map 8835  df-en 8953  df-dom 8954  df-sdom 8955  df-fin 8956  df-pnf 11263  df-mnf 11264  df-xr 11265  df-ltxr 11266  df-le 11267  df-sub 11461  df-neg 11462  df-nn 12252  df-2 12321  df-3 12322  df-4 12323  df-5 12324  df-6 12325  df-n0 12523  df-z 12610  df-uz 12881  df-fz 13554  df-struct 17232  df-sets 17249  df-slot 17267  df-ndx 17279  df-base 17295  df-plusg 17348  df-mulr 17349  df-sca 17351  df-vsca 17352  df-0g 17519  df-mgm 18723  df-sgrp 18806  df-mnd 18822  df-mhm 18872  df-grp 19034  df-minusg 19035  df-ghm 19315  df-cmn 19883  df-abl 19884  df-mgp 20248  df-rng 20262  df-ur 20295  df-ring 20348  df-lmod 21020  df-lmhm 21180  df-mend 43940

This theorem is used by:  mendlmod  43957  mendassa  43958