Theorem mendring 41017

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 41009	. . 3 ⊢ (𝑀 LMHom 𝑀) = (Base‘𝐴)
3	2	a1i 11	. 2 ⊢ (𝑀 ∈ LMod → (𝑀 LMHom 𝑀) = (Base‘𝐴))
4		eqidd 2739	. 2 ⊢ (𝑀 ∈ LMod → (+g‘𝐴) = (+g‘𝐴))
5		eqidd 2739	. 2 ⊢ (𝑀 ∈ LMod → (.r‘𝐴) = (.r‘𝐴))
6		eqid 2738	. . . . . 6 ⊢ (+g‘𝑀) = (+g‘𝑀)
7		eqid 2738	. . . . . 6 ⊢ (+g‘𝐴) = (+g‘𝐴)
8	1, 2, 6, 7	mendplusg 41011	. . . . 5 ⊢ ((𝑥 ∈ (𝑀 LMHom 𝑀) ∧ 𝑦 ∈ (𝑀 LMHom 𝑀)) → (𝑥(+g‘𝐴)𝑦) = (𝑥 ∘f (+g‘𝑀)𝑦))
9	6	lmhmplusg 20306	. . . . 5 ⊢ ((𝑥 ∈ (𝑀 LMHom 𝑀) ∧ 𝑦 ∈ (𝑀 LMHom 𝑀)) → (𝑥 ∘f (+g‘𝑀)𝑦) ∈ (𝑀 LMHom 𝑀))
10	8, 9	eqeltrd 2839	. . . 4 ⊢ ((𝑥 ∈ (𝑀 LMHom 𝑀) ∧ 𝑦 ∈ (𝑀 LMHom 𝑀)) → (𝑥(+g‘𝐴)𝑦) ∈ (𝑀 LMHom 𝑀))
11	10	3adant1 1129	. . 3 ⊢ ((𝑀 ∈ LMod ∧ 𝑥 ∈ (𝑀 LMHom 𝑀) ∧ 𝑦 ∈ (𝑀 LMHom 𝑀)) → (𝑥(+g‘𝐴)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))
14	12, 13, 9	syl2anc 584	. . . . 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 1195	. . . . 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 41011	. . . . 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 584	. . . 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 584	. . . . 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 7290	. . . 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 20306	. . . . . . 7 ⊢ ((𝑦 ∈ (𝑀 LMHom 𝑀) ∧ 𝑧 ∈ (𝑀 LMHom 𝑀)) → (𝑦 ∘f (+g‘𝑀)𝑧) ∈ (𝑀 LMHom 𝑀))
21	13, 15, 20	syl2anc 584	. . . . . 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 41011	. . . . . 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 584	. . . . 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 41011	. . . . . . 7 ⊢ ((𝑦 ∈ (𝑀 LMHom 𝑀) ∧ 𝑧 ∈ (𝑀 LMHom 𝑀)) → (𝑦(+g‘𝐴)𝑧) = (𝑦 ∘f (+g‘𝑀)𝑧))
25	13, 15, 24	syl2anc 584	. . . . . 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 7291	. . . . 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 20130	. . . . . . . 8 ⊢ (𝑀 ∈ LMod → 𝑀 ∈ Grp)
28	27	grpmndd 18589	. . . . . . 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 2738	. . . . . . . . 9 ⊢ (Base‘𝑀) = (Base‘𝑀)
31	30, 30	lmhmf 20296	. . . . . . . 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 6787	. . . . . . . 8 ⊢ (Base‘𝑀) ∈ V
34	33, 33	elmap 8659	. . . . . . 7 ⊢ (𝑥 ∈ ((Base‘𝑀) ↑m (Base‘𝑀)) ↔ 𝑥:(Base‘𝑀)⟶(Base‘𝑀))
35	32, 34	sylibr 233	. . . . . 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 20296	. . . . . . . 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 8659	. . . . . . 7 ⊢ (𝑦 ∈ ((Base‘𝑀) ↑m (Base‘𝑀)) ↔ 𝑦:(Base‘𝑀)⟶(Base‘𝑀))
39	37, 38	sylibr 233	. . . . . 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 20296	. . . . . . . 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 8659	. . . . . . 7 ⊢ (𝑧 ∈ ((Base‘𝑀) ↑m (Base‘𝑀)) ↔ 𝑧:(Base‘𝑀)⟶(Base‘𝑀))
43	41, 42	sylibr 233	. . . . . 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 21541	. . . . . 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 1371	. . . . 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 2788	. . . 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 2788	. . 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 2739	. . . 4 ⊢ (𝑀 ∈ LMod → (Scalar‘𝑀) = (Scalar‘𝑀))
50		eqid 2738	. . . . 5 ⊢ (0g‘𝑀) = (0g‘𝑀)
51		eqid 2738	. . . . 5 ⊢ (Scalar‘𝑀) = (Scalar‘𝑀)
52	50, 30, 51, 51	0lmhm 20302	. . . 4 ⊢ ((𝑀 ∈ LMod ∧ 𝑀 ∈ LMod ∧ (Scalar‘𝑀) = (Scalar‘𝑀)) → ((Base‘𝑀) × {(0g‘𝑀)}) ∈ (𝑀 LMHom 𝑀))
53	48, 48, 49, 52	syl3anc 1370	. . 3 ⊢ (𝑀 ∈ LMod → ((Base‘𝑀) × {(0g‘𝑀)}) ∈ (𝑀 LMHom 𝑀))
54	1, 2, 6, 7	mendplusg 41011	. . . . 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 580	. . . 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 233	. . . . 5 ⊢ (𝑥 ∈ (𝑀 LMHom 𝑀) → 𝑥 ∈ ((Base‘𝑀) ↑m (Base‘𝑀)))
57	30, 6, 50	mndvlid 21542	. . . . 5 ⊢ ((𝑀 ∈ Mnd ∧ 𝑥 ∈ ((Base‘𝑀) ↑m (Base‘𝑀))) → (((Base‘𝑀) × {(0g‘𝑀)}) ∘f (+g‘𝑀)𝑥) = 𝑥)
58	28, 56, 57	syl2an 596	. . . 4 ⊢ ((𝑀 ∈ LMod ∧ 𝑥 ∈ (𝑀 LMHom 𝑀)) → (((Base‘𝑀) × {(0g‘𝑀)}) ∘f (+g‘𝑀)𝑥) = 𝑥)
59	55, 58	eqtrd 2778	. . 3 ⊢ ((𝑀 ∈ LMod ∧ 𝑥 ∈ (𝑀 LMHom 𝑀)) → (((Base‘𝑀) × {(0g‘𝑀)})(+g‘𝐴)𝑥) = 𝑥)
60		eqid 2738	. . . . 5 ⊢ (invg‘𝑀) = (invg‘𝑀)
61	60	invlmhm 20304	. . . 4 ⊢ (𝑀 ∈ LMod → (invg‘𝑀) ∈ (𝑀 LMHom 𝑀))
62		lmhmco 20305	. . . 4 ⊢ (((invg‘𝑀) ∈ (𝑀 LMHom 𝑀) ∧ 𝑥 ∈ (𝑀 LMHom 𝑀)) → ((invg‘𝑀) ∘ 𝑥) ∈ (𝑀 LMHom 𝑀))
63	61, 62	sylan 580	. . 3 ⊢ ((𝑀 ∈ LMod ∧ 𝑥 ∈ (𝑀 LMHom 𝑀)) → ((invg‘𝑀) ∘ 𝑥) ∈ (𝑀 LMHom 𝑀))
64	1, 2, 6, 7	mendplusg 41011	. . . . 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 588	. . . 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 21544	. . . . 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 596	. . . 4 ⊢ ((𝑀 ∈ LMod ∧ 𝑥 ∈ (𝑀 LMHom 𝑀)) → (((invg‘𝑀) ∘ 𝑥) ∘f (+g‘𝑀)𝑥) = ((Base‘𝑀) × {(0g‘𝑀)}))
68	65, 67	eqtrd 2778	. . 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 18601	. 2 ⊢ (𝑀 ∈ LMod → 𝐴 ∈ Grp)
70		eqid 2738	. . . . 5 ⊢ (.r‘𝐴) = (.r‘𝐴)
71	1, 2, 70	mendmulr 41013	. . . 4 ⊢ ((𝑥 ∈ (𝑀 LMHom 𝑀) ∧ 𝑦 ∈ (𝑀 LMHom 𝑀)) → (𝑥(.r‘𝐴)𝑦) = (𝑥 ∘ 𝑦))
72		lmhmco 20305	. . . 4 ⊢ ((𝑥 ∈ (𝑀 LMHom 𝑀) ∧ 𝑦 ∈ (𝑀 LMHom 𝑀)) → (𝑥 ∘ 𝑦) ∈ (𝑀 LMHom 𝑀))
73	71, 72	eqeltrd 2839	. . 3 ⊢ ((𝑥 ∈ (𝑀 LMHom 𝑀) ∧ 𝑦 ∈ (𝑀 LMHom 𝑀)) → (𝑥(.r‘𝐴)𝑦) ∈ (𝑀 LMHom 𝑀))
74	73	3adant1 1129	. 2 ⊢ ((𝑀 ∈ LMod ∧ 𝑥 ∈ (𝑀 LMHom 𝑀) ∧ 𝑦 ∈ (𝑀 LMHom 𝑀)) → (𝑥(.r‘𝐴)𝑦) ∈ (𝑀 LMHom 𝑀))
75		coass 6169	. . 3 ⊢ ((𝑥 ∘ 𝑦) ∘ 𝑧) = (𝑥 ∘ (𝑦 ∘ 𝑧))
76	12, 13, 71	syl2anc 584	. . . . 5 ⊢ ((𝑀 ∈ LMod ∧ (𝑥 ∈ (𝑀 LMHom 𝑀) ∧ 𝑦 ∈ (𝑀 LMHom 𝑀) ∧ 𝑧 ∈ (𝑀 LMHom 𝑀))) → (𝑥(.r‘𝐴)𝑦) = (𝑥 ∘ 𝑦))
77	76	oveq1d 7290	. . . 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 584	. . . . 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 41013	. . . . 5 ⊢ (((𝑥 ∘ 𝑦) ∈ (𝑀 LMHom 𝑀) ∧ 𝑧 ∈ (𝑀 LMHom 𝑀)) → ((𝑥 ∘ 𝑦)(.r‘𝐴)𝑧) = ((𝑥 ∘ 𝑦) ∘ 𝑧))
80	78, 15, 79	syl2anc 584	. . . 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 2778	. . 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 41013	. . . . . 6 ⊢ ((𝑦 ∈ (𝑀 LMHom 𝑀) ∧ 𝑧 ∈ (𝑀 LMHom 𝑀)) → (𝑦(.r‘𝐴)𝑧) = (𝑦 ∘ 𝑧))
83	13, 15, 82	syl2anc 584	. . . . 5 ⊢ ((𝑀 ∈ LMod ∧ (𝑥 ∈ (𝑀 LMHom 𝑀) ∧ 𝑦 ∈ (𝑀 LMHom 𝑀) ∧ 𝑧 ∈ (𝑀 LMHom 𝑀))) → (𝑦(.r‘𝐴)𝑧) = (𝑦 ∘ 𝑧))
84	83	oveq2d 7291	. . . 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 20305	. . . . . 6 ⊢ ((𝑦 ∈ (𝑀 LMHom 𝑀) ∧ 𝑧 ∈ (𝑀 LMHom 𝑀)) → (𝑦 ∘ 𝑧) ∈ (𝑀 LMHom 𝑀))
86	13, 15, 85	syl2anc 584	. . . . 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 41013	. . . . 5 ⊢ ((𝑥 ∈ (𝑀 LMHom 𝑀) ∧ (𝑦 ∘ 𝑧) ∈ (𝑀 LMHom 𝑀)) → (𝑥(.r‘𝐴)(𝑦 ∘ 𝑧)) = (𝑥 ∘ (𝑦 ∘ 𝑧)))
88	12, 86, 87	syl2anc 584	. . . 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 2778	. . 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 2804	. 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 41013	. . . 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 584	. . 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 7291	. . 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 20305	. . . . . 6 ⊢ ((𝑥 ∈ (𝑀 LMHom 𝑀) ∧ 𝑧 ∈ (𝑀 LMHom 𝑀)) → (𝑥 ∘ 𝑧) ∈ (𝑀 LMHom 𝑀))
95	12, 15, 94	syl2anc 584	. . . . 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 41011	. . . . 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 584	. . . 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 41013	. . . . . 6 ⊢ ((𝑥 ∈ (𝑀 LMHom 𝑀) ∧ 𝑧 ∈ (𝑀 LMHom 𝑀)) → (𝑥(.r‘𝐴)𝑧) = (𝑥 ∘ 𝑧))
99	12, 15, 98	syl2anc 584	. . . . 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 7293	. . . 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 20293	. . . . . 6 ⊢ (𝑥 ∈ (𝑀 LMHom 𝑀) → 𝑥 ∈ (𝑀 GrpHom 𝑀))
102		ghmmhm 18844	. . . . . 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 21546	. . . . 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 1370	. . . 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 2788	. . 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 2788	. 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 41013	. . . 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 584	. . 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 7290	. . 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 41011	. . . . 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 584	. . . 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 7293	. . . 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 6600	. . . . . 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 6600	. . . . . 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 4152	. . . . 5 ⊢ ((Base‘𝑀) ∩ (Base‘𝑀)) = (Base‘𝑀)
120	115, 117, 41, 118, 118, 118, 119	ofco 7556	. . . 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 2788	. . 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 2788	. 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 20303	. 2 ⊢ (𝑀 ∈ LMod → ( I ↾ (Base‘𝑀)) ∈ (𝑀 LMHom 𝑀))
124	1, 2, 70	mendmulr 41013	. . . 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 580	. . 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 6649	. . . 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 2778	. 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 41013	. . . 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 597	. . 3 ⊢ ((𝑀 ∈ LMod ∧ 𝑥 ∈ (𝑀 LMHom 𝑀)) → (𝑥(.r‘𝐴)( I ↾ (Base‘𝑀))) = (𝑥 ∘ ( I ↾ (Base‘𝑀))))
133		fcoi1 6648	. . . 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 2778	. 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 19824	1 ⊢ (𝑀 ∈ LMod → 𝐴 ∈ Ring)

Syntax hints:   → wi 4   ∧ wa 396   ∧ w3a 1086   = wceq 1539   ∈ wcel 2106  Vcvv 3432  {csn 4561   I cid 5488   × cxp 5587   ↾ cres 5591   ∘ ccom 5593   Fn wfn 6428  ⟶wf 6429  ‘cfv 6433  (class class class)co 7275   ∘_f cof 7531   ↑_m cmap 8615  Basecbs 16912  +_gcplusg 16962  ._rcmulr 16963  Scalarcsca 16965  0_gc0g 17150  Mndcmnd 18385   MndHom cmhm 18428  Grpcgrp 18577  inv_gcminusg 18578   GrpHom cghm 18831  Ringcrg 19783  LModclmod 20123   LMHom clmhm 20281  MEndocmend 41000

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 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-10 2137  ax-11 2154  ax-12 2171  ax-ext 2709  ax-rep 5209  ax-sep 5223  ax-nul 5230  ax-pow 5288  ax-pr 5352  ax-un 7588  ax-cnex 10927  ax-resscn 10928  ax-1cn 10929  ax-icn 10930  ax-addcl 10931  ax-addrcl 10932  ax-mulcl 10933  ax-mulrcl 10934  ax-mulcom 10935  ax-addass 10936  ax-mulass 10937  ax-distr 10938  ax-i2m1 10939  ax-1ne0 10940  ax-1rid 10941  ax-rnegex 10942  ax-rrecex 10943  ax-cnre 10944  ax-pre-lttri 10945  ax-pre-lttrn 10946  ax-pre-ltadd 10947  ax-pre-mulgt0 10948

This theorem depends on definitions:  df-bi 206  df-an 397  df-or 845  df-3or 1087  df-3an 1088  df-tru 1542  df-fal 1552  df-ex 1783  df-nf 1787  df-sb 2068  df-mo 2540  df-eu 2569  df-clab 2716  df-cleq 2730  df-clel 2816  df-nfc 2889  df-ne 2944  df-nel 3050  df-ral 3069  df-rex 3070  df-rmo 3071  df-reu 3072  df-rab 3073  df-v 3434  df-sbc 3717  df-csb 3833  df-dif 3890  df-un 3892  df-in 3894  df-ss 3904  df-pss 3906  df-nul 4257  df-if 4460  df-pw 4535  df-sn 4562  df-pr 4564  df-tp 4566  df-op 4568  df-uni 4840  df-iun 4926  df-br 5075  df-opab 5137  df-mpt 5158  df-tr 5192  df-id 5489  df-eprel 5495  df-po 5503  df-so 5504  df-fr 5544  df-we 5546  df-xp 5595  df-rel 5596  df-cnv 5597  df-co 5598  df-dm 5599  df-rn 5600  df-res 5601  df-ima 5602  df-pred 6202  df-ord 6269  df-on 6270  df-lim 6271  df-suc 6272  df-iota 6391  df-fun 6435  df-fn 6436  df-f 6437  df-f1 6438  df-fo 6439  df-f1o 6440  df-fv 6441  df-riota 7232  df-ov 7278  df-oprab 7279  df-mpo 7280  df-of 7533  df-om 7713  df-1st 7831  df-2nd 7832  df-frecs 8097  df-wrecs 8128  df-recs 8202  df-rdg 8241  df-1o 8297  df-er 8498  df-map 8617  df-en 8734  df-dom 8735  df-sdom 8736  df-fin 8737  df-pnf 11011  df-mnf 11012  df-xr 11013  df-ltxr 11014  df-le 11015  df-sub 11207  df-neg 11208  df-nn 11974  df-2 12036  df-3 12037  df-4 12038  df-5 12039  df-6 12040  df-n0 12234  df-z 12320  df-uz 12583  df-fz 13240  df-struct 16848  df-sets 16865  df-slot 16883  df-ndx 16895  df-base 16913  df-plusg 16975  df-mulr 16976  df-sca 16978  df-vsca 16979  df-0g 17152  df-mgm 18326  df-sgrp 18375  df-mnd 18386  df-mhm 18430  df-grp 18580  df-minusg 18581  df-ghm 18832  df-cmn 19388  df-abl 19389  df-mgp 19721  df-ur 19738  df-ring 19785  df-lmod 20125  df-lmhm 20284  df-mend 41001

This theorem is referenced by:  mendlmod  41018  mendassa  41019