Theorem matunitlindflem1 22994

Description: One direction of matunitlindf 22996. (Contributed by Brendan Leahy, 2-Jun-2021.)

Assertion

Ref	Expression
matunitlindflem1	⊢ (((𝑅 ∈ Field ∧ 𝑀:(𝐼 × 𝐼)⟶(Base‘𝑅)) ∧ 𝐼 ∈ (Fin ∖ {∅})) → (¬ curry 𝑀 LIndF (𝑅 freeLMod 𝐼) → ((𝐼 maDet 𝑅)‘𝑀) = (0g‘𝑅)))

Proof of Theorem matunitlindflem1

Dummy variables 𝑥 𝑓 𝑦 𝑧 𝑖 𝑗 𝑘 𝑛 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		isfld 20993	. . . . 5 ⊢ (𝑅 ∈ Field ↔ (𝑅 ∈ DivRing ∧ 𝑅 ∈ CRing))
2	1	simplbi 502	. . . 4 ⊢ (𝑅 ∈ Field → 𝑅 ∈ DivRing)
3		drngring 20987	. . . 4 ⊢ (𝑅 ∈ DivRing → 𝑅 ∈ Ring)
4	2, 3	syl 18	. . 3 ⊢ (𝑅 ∈ Field → 𝑅 ∈ Ring)
5		eqid 2761	. . . . . . . . 9 ⊢ (𝑅 freeLMod 𝐼) = (𝑅 freeLMod 𝐼)
6	5	frlmlmod 22055	. . . . . . . 8 ⊢ ((𝑅 ∈ Ring ∧ 𝐼 ∈ (Fin ∖ {∅})) → (𝑅 freeLMod 𝐼) ∈ LMod)
7	6	adantlr 728	. . . . . . 7 ⊢ (((𝑅 ∈ Ring ∧ 𝑀:(𝐼 × 𝐼)⟶(Base‘𝑅)) ∧ 𝐼 ∈ (Fin ∖ {∅})) → (𝑅 freeLMod 𝐼) ∈ LMod)
8		simpr 490	. . . . . . 7 ⊢ (((𝑅 ∈ Ring ∧ 𝑀:(𝐼 × 𝐼)⟶(Base‘𝑅)) ∧ 𝐼 ∈ (Fin ∖ {∅})) → 𝐼 ∈ (Fin ∖ {∅}))
9		eldifi 4078	. . . . . . . . . 10 ⊢ (𝐼 ∈ (Fin ∖ {∅}) → 𝐼 ∈ Fin)
10		eqid 2761	. . . . . . . . . . 11 ⊢ (Base‘𝑅) = (Base‘𝑅)
11	5, 10	frlmfibas 22068	. . . . . . . . . 10 ⊢ ((𝑅 ∈ Ring ∧ 𝐼 ∈ Fin) → ((Base‘𝑅) ↑m 𝐼) = (Base‘(𝑅 freeLMod 𝐼)))
12	9, 11	sylan2 605	. . . . . . . . 9 ⊢ ((𝑅 ∈ Ring ∧ 𝐼 ∈ (Fin ∖ {∅})) → ((Base‘𝑅) ↑m 𝐼) = (Base‘(𝑅 freeLMod 𝐼)))
13		fvex 6898	. . . . . . . . . 10 ⊢ (Base‘𝑅) ∈ V
14		curf 8890	. . . . . . . . . 10 ⊢ ((𝑀:(𝐼 × 𝐼)⟶(Base‘𝑅) ∧ 𝐼 ∈ (Fin ∖ {∅}) ∧ (Base‘𝑅) ∈ V) → curry 𝑀:𝐼⟶((Base‘𝑅) ↑m 𝐼))
15	13, 14	mp3an3 1479	. . . . . . . . 9 ⊢ ((𝑀:(𝐼 × 𝐼)⟶(Base‘𝑅) ∧ 𝐼 ∈ (Fin ∖ {∅})) → curry 𝑀:𝐼⟶((Base‘𝑅) ↑m 𝐼))
16		feq3 6689	. . . . . . . . . 10 ⊢ (((Base‘𝑅) ↑m 𝐼) = (Base‘(𝑅 freeLMod 𝐼)) → (curry 𝑀:𝐼⟶((Base‘𝑅) ↑m 𝐼) ↔ curry 𝑀:𝐼⟶(Base‘(𝑅 freeLMod 𝐼))))
17	16	biimpa 482	. . . . . . . . 9 ⊢ ((((Base‘𝑅) ↑m 𝐼) = (Base‘(𝑅 freeLMod 𝐼)) ∧ curry 𝑀:𝐼⟶((Base‘𝑅) ↑m 𝐼)) → curry 𝑀:𝐼⟶(Base‘(𝑅 freeLMod 𝐼)))
18	12, 15, 17	syl2an 608	. . . . . . . 8 ⊢ (((𝑅 ∈ Ring ∧ 𝐼 ∈ (Fin ∖ {∅})) ∧ (𝑀:(𝐼 × 𝐼)⟶(Base‘𝑅) ∧ 𝐼 ∈ (Fin ∖ {∅}))) → curry 𝑀:𝐼⟶(Base‘(𝑅 freeLMod 𝐼)))
19	18	anandirs 692	. . . . . . 7 ⊢ (((𝑅 ∈ Ring ∧ 𝑀:(𝐼 × 𝐼)⟶(Base‘𝑅)) ∧ 𝐼 ∈ (Fin ∖ {∅})) → curry 𝑀:𝐼⟶(Base‘(𝑅 freeLMod 𝐼)))
20		eqid 2761	. . . . . . . 8 ⊢ (Base‘(𝑅 freeLMod 𝐼)) = (Base‘(𝑅 freeLMod 𝐼))
21		eqid 2761	. . . . . . . 8 ⊢ (Scalar‘(𝑅 freeLMod 𝐼)) = (Scalar‘(𝑅 freeLMod 𝐼))
22		eqid 2761	. . . . . . . 8 ⊢ ( ·𝑠 ‘(𝑅 freeLMod 𝐼)) = ( ·𝑠 ‘(𝑅 freeLMod 𝐼))
23		eqid 2761	. . . . . . . 8 ⊢ (0g‘(𝑅 freeLMod 𝐼)) = (0g‘(𝑅 freeLMod 𝐼))
24		eqid 2761	. . . . . . . 8 ⊢ (0g‘(Scalar‘(𝑅 freeLMod 𝐼))) = (0g‘(Scalar‘(𝑅 freeLMod 𝐼)))
25		eqid 2761	. . . . . . . 8 ⊢ (Base‘((Scalar‘(𝑅 freeLMod 𝐼)) freeLMod 𝐼)) = (Base‘((Scalar‘(𝑅 freeLMod 𝐼)) freeLMod 𝐼))
26	20, 21, 22, 23, 24, 25	islindf4 22144	. . . . . . 7 ⊢ (((𝑅 freeLMod 𝐼) ∈ LMod ∧ 𝐼 ∈ (Fin ∖ {∅}) ∧ curry 𝑀:𝐼⟶(Base‘(𝑅 freeLMod 𝐼))) → (curry 𝑀 LIndF (𝑅 freeLMod 𝐼) ↔ ∀𝑓 ∈ (Base‘((Scalar‘(𝑅 freeLMod 𝐼)) freeLMod 𝐼))(((𝑅 freeLMod 𝐼) Σg (𝑓 ∘f ( ·𝑠 ‘(𝑅 freeLMod 𝐼))curry 𝑀)) = (0g‘(𝑅 freeLMod 𝐼)) → 𝑓 = (𝐼 × {(0g‘(Scalar‘(𝑅 freeLMod 𝐼)))}))))
27	7, 8, 19, 26	syl3anc 1398	. . . . . 6 ⊢ (((𝑅 ∈ Ring ∧ 𝑀:(𝐼 × 𝐼)⟶(Base‘𝑅)) ∧ 𝐼 ∈ (Fin ∖ {∅})) → (curry 𝑀 LIndF (𝑅 freeLMod 𝐼) ↔ ∀𝑓 ∈ (Base‘((Scalar‘(𝑅 freeLMod 𝐼)) freeLMod 𝐼))(((𝑅 freeLMod 𝐼) Σg (𝑓 ∘f ( ·𝑠 ‘(𝑅 freeLMod 𝐼))curry 𝑀)) = (0g‘(𝑅 freeLMod 𝐼)) → 𝑓 = (𝐼 × {(0g‘(Scalar‘(𝑅 freeLMod 𝐼)))}))))
28	5	frlmsca 22059	. . . . . . . . . 10 ⊢ ((𝑅 ∈ Ring ∧ 𝐼 ∈ (Fin ∖ {∅})) → 𝑅 = (Scalar‘(𝑅 freeLMod 𝐼)))
29	28	fvoveq1d 7442	. . . . . . . . 9 ⊢ ((𝑅 ∈ Ring ∧ 𝐼 ∈ (Fin ∖ {∅})) → (Base‘(𝑅 freeLMod 𝐼)) = (Base‘((Scalar‘(𝑅 freeLMod 𝐼)) freeLMod 𝐼)))
30	12, 29	eqtrd 2796	. . . . . . . 8 ⊢ ((𝑅 ∈ Ring ∧ 𝐼 ∈ (Fin ∖ {∅})) → ((Base‘𝑅) ↑m 𝐼) = (Base‘((Scalar‘(𝑅 freeLMod 𝐼)) freeLMod 𝐼)))
31	30	adantlr 728	. . . . . . 7 ⊢ (((𝑅 ∈ Ring ∧ 𝑀:(𝐼 × 𝐼)⟶(Base‘𝑅)) ∧ 𝐼 ∈ (Fin ∖ {∅})) → ((Base‘𝑅) ↑m 𝐼) = (Base‘((Scalar‘(𝑅 freeLMod 𝐼)) freeLMod 𝐼)))
32		elmapi 8869	. . . . . . . . . 10 ⊢ (𝑓 ∈ ((Base‘𝑅) ↑m 𝐼) → 𝑓:𝐼⟶(Base‘𝑅))
33		ffn 6709	. . . . . . . . . . . . . . 15 ⊢ (𝑓:𝐼⟶(Base‘𝑅) → 𝑓 Fn 𝐼)
34	33	adantl 487	. . . . . . . . . . . . . 14 ⊢ ((((𝑅 ∈ Ring ∧ 𝑀:(𝐼 × 𝐼)⟶(Base‘𝑅)) ∧ 𝐼 ∈ (Fin ∖ {∅})) ∧ 𝑓:𝐼⟶(Base‘𝑅)) → 𝑓 Fn 𝐼)
35	19	ffnd 6710	. . . . . . . . . . . . . . 15 ⊢ (((𝑅 ∈ Ring ∧ 𝑀:(𝐼 × 𝐼)⟶(Base‘𝑅)) ∧ 𝐼 ∈ (Fin ∖ {∅})) → curry 𝑀 Fn 𝐼)
36	35	adantr 486	. . . . . . . . . . . . . 14 ⊢ ((((𝑅 ∈ Ring ∧ 𝑀:(𝐼 × 𝐼)⟶(Base‘𝑅)) ∧ 𝐼 ∈ (Fin ∖ {∅})) ∧ 𝑓:𝐼⟶(Base‘𝑅)) → curry 𝑀 Fn 𝐼)
37		simplr 781	. . . . . . . . . . . . . 14 ⊢ ((((𝑅 ∈ Ring ∧ 𝑀:(𝐼 × 𝐼)⟶(Base‘𝑅)) ∧ 𝐼 ∈ (Fin ∖ {∅})) ∧ 𝑓:𝐼⟶(Base‘𝑅)) → 𝐼 ∈ (Fin ∖ {∅}))
38		inidm 4172	. . . . . . . . . . . . . 14 ⊢ (𝐼 ∩ 𝐼) = 𝐼
39		eqidd 2762	. . . . . . . . . . . . . 14 ⊢ (((((𝑅 ∈ Ring ∧ 𝑀:(𝐼 × 𝐼)⟶(Base‘𝑅)) ∧ 𝐼 ∈ (Fin ∖ {∅})) ∧ 𝑓:𝐼⟶(Base‘𝑅)) ∧ 𝑛 ∈ 𝐼) → (𝑓‘𝑛) = (𝑓‘𝑛))
40		eqidd 2762	. . . . . . . . . . . . . 14 ⊢ (((((𝑅 ∈ Ring ∧ 𝑀:(𝐼 × 𝐼)⟶(Base‘𝑅)) ∧ 𝐼 ∈ (Fin ∖ {∅})) ∧ 𝑓:𝐼⟶(Base‘𝑅)) ∧ 𝑛 ∈ 𝐼) → (curry 𝑀‘𝑛) = (curry 𝑀‘𝑛))
41	34, 36, 37, 37, 38, 39, 40	offval 7702	. . . . . . . . . . . . 13 ⊢ ((((𝑅 ∈ Ring ∧ 𝑀:(𝐼 × 𝐼)⟶(Base‘𝑅)) ∧ 𝐼 ∈ (Fin ∖ {∅})) ∧ 𝑓:𝐼⟶(Base‘𝑅)) → (𝑓 ∘f ( ·𝑠 ‘(𝑅 freeLMod 𝐼))curry 𝑀) = (𝑛 ∈ 𝐼 ↦ ((𝑓‘𝑛)( ·𝑠 ‘(𝑅 freeLMod 𝐼))(curry 𝑀‘𝑛))))
42		simpllr 788	. . . . . . . . . . . . . . . 16 ⊢ (((((𝑅 ∈ Ring ∧ 𝑀:(𝐼 × 𝐼)⟶(Base‘𝑅)) ∧ 𝐼 ∈ (Fin ∖ {∅})) ∧ 𝑓:𝐼⟶(Base‘𝑅)) ∧ 𝑛 ∈ 𝐼) → 𝐼 ∈ (Fin ∖ {∅}))
43		ffvelcdm 7081	. . . . . . . . . . . . . . . . 17 ⊢ ((𝑓:𝐼⟶(Base‘𝑅) ∧ 𝑛 ∈ 𝐼) → (𝑓‘𝑛) ∈ (Base‘𝑅))
44	43	adantll 727	. . . . . . . . . . . . . . . 16 ⊢ (((((𝑅 ∈ Ring ∧ 𝑀:(𝐼 × 𝐼)⟶(Base‘𝑅)) ∧ 𝐼 ∈ (Fin ∖ {∅})) ∧ 𝑓:𝐼⟶(Base‘𝑅)) ∧ 𝑛 ∈ 𝐼) → (𝑓‘𝑛) ∈ (Base‘𝑅))
45	19	ffvelcdmda 7084	. . . . . . . . . . . . . . . . 17 ⊢ ((((𝑅 ∈ Ring ∧ 𝑀:(𝐼 × 𝐼)⟶(Base‘𝑅)) ∧ 𝐼 ∈ (Fin ∖ {∅})) ∧ 𝑛 ∈ 𝐼) → (curry 𝑀‘𝑛) ∈ (Base‘(𝑅 freeLMod 𝐼)))
46	45	adantlr 728	. . . . . . . . . . . . . . . 16 ⊢ (((((𝑅 ∈ Ring ∧ 𝑀:(𝐼 × 𝐼)⟶(Base‘𝑅)) ∧ 𝐼 ∈ (Fin ∖ {∅})) ∧ 𝑓:𝐼⟶(Base‘𝑅)) ∧ 𝑛 ∈ 𝐼) → (curry 𝑀‘𝑛) ∈ (Base‘(𝑅 freeLMod 𝐼)))
47		eqid 2761	. . . . . . . . . . . . . . . 16 ⊢ (.r‘𝑅) = (.r‘𝑅)
48	5, 20, 10, 42, 44, 46, 22, 47	frlmvscafval 22072	. . . . . . . . . . . . . . 15 ⊢ (((((𝑅 ∈ Ring ∧ 𝑀:(𝐼 × 𝐼)⟶(Base‘𝑅)) ∧ 𝐼 ∈ (Fin ∖ {∅})) ∧ 𝑓:𝐼⟶(Base‘𝑅)) ∧ 𝑛 ∈ 𝐼) → ((𝑓‘𝑛)( ·𝑠 ‘(𝑅 freeLMod 𝐼))(curry 𝑀‘𝑛)) = ((𝐼 × {(𝑓‘𝑛)}) ∘f (.r‘𝑅)(curry 𝑀‘𝑛)))
49		fvex 6898	. . . . . . . . . . . . . . . . 17 ⊢ (𝑓‘𝑛) ∈ V
50		fnconstg 6770	. . . . . . . . . . . . . . . . 17 ⊢ ((𝑓‘𝑛) ∈ V → (𝐼 × {(𝑓‘𝑛)}) Fn 𝐼)
51	49, 50	mp1i 14	. . . . . . . . . . . . . . . 16 ⊢ (((((𝑅 ∈ Ring ∧ 𝑀:(𝐼 × 𝐼)⟶(Base‘𝑅)) ∧ 𝐼 ∈ (Fin ∖ {∅})) ∧ 𝑓:𝐼⟶(Base‘𝑅)) ∧ 𝑛 ∈ 𝐼) → (𝐼 × {(𝑓‘𝑛)}) Fn 𝐼)
52	15	ffvelcdmda 7084	. . . . . . . . . . . . . . . . . . 19 ⊢ (((𝑀:(𝐼 × 𝐼)⟶(Base‘𝑅) ∧ 𝐼 ∈ (Fin ∖ {∅})) ∧ 𝑛 ∈ 𝐼) → (curry 𝑀‘𝑛) ∈ ((Base‘𝑅) ↑m 𝐼))
53		elmapfn 8887	. . . . . . . . . . . . . . . . . . 19 ⊢ ((curry 𝑀‘𝑛) ∈ ((Base‘𝑅) ↑m 𝐼) → (curry 𝑀‘𝑛) Fn 𝐼)
54	52, 53	syl 18	. . . . . . . . . . . . . . . . . 18 ⊢ (((𝑀:(𝐼 × 𝐼)⟶(Base‘𝑅) ∧ 𝐼 ∈ (Fin ∖ {∅})) ∧ 𝑛 ∈ 𝐼) → (curry 𝑀‘𝑛) Fn 𝐼)
55	54	adantlll 731	. . . . . . . . . . . . . . . . 17 ⊢ ((((𝑅 ∈ Ring ∧ 𝑀:(𝐼 × 𝐼)⟶(Base‘𝑅)) ∧ 𝐼 ∈ (Fin ∖ {∅})) ∧ 𝑛 ∈ 𝐼) → (curry 𝑀‘𝑛) Fn 𝐼)
56	55	adantlr 728	. . . . . . . . . . . . . . . 16 ⊢ (((((𝑅 ∈ Ring ∧ 𝑀:(𝐼 × 𝐼)⟶(Base‘𝑅)) ∧ 𝐼 ∈ (Fin ∖ {∅})) ∧ 𝑓:𝐼⟶(Base‘𝑅)) ∧ 𝑛 ∈ 𝐼) → (curry 𝑀‘𝑛) Fn 𝐼)
57	49	fvconst2 7210	. . . . . . . . . . . . . . . . 17 ⊢ (𝑘 ∈ 𝐼 → ((𝐼 × {(𝑓‘𝑛)})‘𝑘) = (𝑓‘𝑛))
58	57	adantl 487	. . . . . . . . . . . . . . . 16 ⊢ ((((((𝑅 ∈ Ring ∧ 𝑀:(𝐼 × 𝐼)⟶(Base‘𝑅)) ∧ 𝐼 ∈ (Fin ∖ {∅})) ∧ 𝑓:𝐼⟶(Base‘𝑅)) ∧ 𝑛 ∈ 𝐼) ∧ 𝑘 ∈ 𝐼) → ((𝐼 × {(𝑓‘𝑛)})‘𝑘) = (𝑓‘𝑛))
59		ffn 6709	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝑀:(𝐼 × 𝐼)⟶(Base‘𝑅) → 𝑀 Fn (𝐼 × 𝐼))
60	59	anim2i 629	. . . . . . . . . . . . . . . . . . 19 ⊢ ((𝐼 ∈ (Fin ∖ {∅}) ∧ 𝑀:(𝐼 × 𝐼)⟶(Base‘𝑅)) → (𝐼 ∈ (Fin ∖ {∅}) ∧ 𝑀 Fn (𝐼 × 𝐼)))
61	60	ancoms 464	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝑀:(𝐼 × 𝐼)⟶(Base‘𝑅) ∧ 𝐼 ∈ (Fin ∖ {∅})) → (𝐼 ∈ (Fin ∖ {∅}) ∧ 𝑀 Fn (𝐼 × 𝐼)))
62	61	ad4ant23 766	. . . . . . . . . . . . . . . . 17 ⊢ ((((𝑅 ∈ Ring ∧ 𝑀:(𝐼 × 𝐼)⟶(Base‘𝑅)) ∧ 𝐼 ∈ (Fin ∖ {∅})) ∧ 𝑓:𝐼⟶(Base‘𝑅)) → (𝐼 ∈ (Fin ∖ {∅}) ∧ 𝑀 Fn (𝐼 × 𝐼)))
63		curfv 8892	. . . . . . . . . . . . . . . . . . . 20 ⊢ (((𝑀 Fn (𝐼 × 𝐼) ∧ 𝑛 ∈ 𝐼 ∧ 𝑘 ∈ 𝐼) ∧ 𝐼 ∈ (Fin ∖ {∅})) → ((curry 𝑀‘𝑛)‘𝑘) = (𝑛𝑀𝑘))
64	63	3exp1 1371	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝑀 Fn (𝐼 × 𝐼) → (𝑛 ∈ 𝐼 → (𝑘 ∈ 𝐼 → (𝐼 ∈ (Fin ∖ {∅}) → ((curry 𝑀‘𝑛)‘𝑘) = (𝑛𝑀𝑘)))))
65	64	com4r 95	. . . . . . . . . . . . . . . . . 18 ⊢ (𝐼 ∈ (Fin ∖ {∅}) → (𝑀 Fn (𝐼 × 𝐼) → (𝑛 ∈ 𝐼 → (𝑘 ∈ 𝐼 → ((curry 𝑀‘𝑛)‘𝑘) = (𝑛𝑀𝑘)))))
66	65	imp41 431	. . . . . . . . . . . . . . . . 17 ⊢ ((((𝐼 ∈ (Fin ∖ {∅}) ∧ 𝑀 Fn (𝐼 × 𝐼)) ∧ 𝑛 ∈ 𝐼) ∧ 𝑘 ∈ 𝐼) → ((curry 𝑀‘𝑛)‘𝑘) = (𝑛𝑀𝑘))
67	62, 66	sylanl1 693	. . . . . . . . . . . . . . . 16 ⊢ ((((((𝑅 ∈ Ring ∧ 𝑀:(𝐼 × 𝐼)⟶(Base‘𝑅)) ∧ 𝐼 ∈ (Fin ∖ {∅})) ∧ 𝑓:𝐼⟶(Base‘𝑅)) ∧ 𝑛 ∈ 𝐼) ∧ 𝑘 ∈ 𝐼) → ((curry 𝑀‘𝑛)‘𝑘) = (𝑛𝑀𝑘))
68	51, 56, 42, 42, 38, 58, 67	offval 7702	. . . . . . . . . . . . . . 15 ⊢ (((((𝑅 ∈ Ring ∧ 𝑀:(𝐼 × 𝐼)⟶(Base‘𝑅)) ∧ 𝐼 ∈ (Fin ∖ {∅})) ∧ 𝑓:𝐼⟶(Base‘𝑅)) ∧ 𝑛 ∈ 𝐼) → ((𝐼 × {(𝑓‘𝑛)}) ∘f (.r‘𝑅)(curry 𝑀‘𝑛)) = (𝑘 ∈ 𝐼 ↦ ((𝑓‘𝑛)(.r‘𝑅)(𝑛𝑀𝑘))))
69	48, 68	eqtrd 2796	. . . . . . . . . . . . . 14 ⊢ (((((𝑅 ∈ Ring ∧ 𝑀:(𝐼 × 𝐼)⟶(Base‘𝑅)) ∧ 𝐼 ∈ (Fin ∖ {∅})) ∧ 𝑓:𝐼⟶(Base‘𝑅)) ∧ 𝑛 ∈ 𝐼) → ((𝑓‘𝑛)( ·𝑠 ‘(𝑅 freeLMod 𝐼))(curry 𝑀‘𝑛)) = (𝑘 ∈ 𝐼 ↦ ((𝑓‘𝑛)(.r‘𝑅)(𝑛𝑀𝑘))))
70	69	mpteq2dva 5198	. . . . . . . . . . . . 13 ⊢ ((((𝑅 ∈ Ring ∧ 𝑀:(𝐼 × 𝐼)⟶(Base‘𝑅)) ∧ 𝐼 ∈ (Fin ∖ {∅})) ∧ 𝑓:𝐼⟶(Base‘𝑅)) → (𝑛 ∈ 𝐼 ↦ ((𝑓‘𝑛)( ·𝑠 ‘(𝑅 freeLMod 𝐼))(curry 𝑀‘𝑛))) = (𝑛 ∈ 𝐼 ↦ (𝑘 ∈ 𝐼 ↦ ((𝑓‘𝑛)(.r‘𝑅)(𝑛𝑀𝑘)))))
71	41, 70	eqtrd 2796	. . . . . . . . . . . 12 ⊢ ((((𝑅 ∈ Ring ∧ 𝑀:(𝐼 × 𝐼)⟶(Base‘𝑅)) ∧ 𝐼 ∈ (Fin ∖ {∅})) ∧ 𝑓:𝐼⟶(Base‘𝑅)) → (𝑓 ∘f ( ·𝑠 ‘(𝑅 freeLMod 𝐼))curry 𝑀) = (𝑛 ∈ 𝐼 ↦ (𝑘 ∈ 𝐼 ↦ ((𝑓‘𝑛)(.r‘𝑅)(𝑛𝑀𝑘)))))
72	71	oveq2d 7436	. . . . . . . . . . 11 ⊢ ((((𝑅 ∈ Ring ∧ 𝑀:(𝐼 × 𝐼)⟶(Base‘𝑅)) ∧ 𝐼 ∈ (Fin ∖ {∅})) ∧ 𝑓:𝐼⟶(Base‘𝑅)) → ((𝑅 freeLMod 𝐼) Σg (𝑓 ∘f ( ·𝑠 ‘(𝑅 freeLMod 𝐼))curry 𝑀)) = ((𝑅 freeLMod 𝐼) Σg (𝑛 ∈ 𝐼 ↦ (𝑘 ∈ 𝐼 ↦ ((𝑓‘𝑛)(.r‘𝑅)(𝑛𝑀𝑘))))))
73		simplll 787	. . . . . . . . . . . 12 ⊢ ((((𝑅 ∈ Ring ∧ 𝑀:(𝐼 × 𝐼)⟶(Base‘𝑅)) ∧ 𝐼 ∈ (Fin ∖ {∅})) ∧ 𝑓:𝐼⟶(Base‘𝑅)) → 𝑅 ∈ Ring)
74		simp-4l 795	. . . . . . . . . . . . . . . 16 ⊢ (((((𝑅 ∈ Ring ∧ 𝑀:(𝐼 × 𝐼)⟶(Base‘𝑅)) ∧ 𝑓:𝐼⟶(Base‘𝑅)) ∧ 𝑛 ∈ 𝐼) ∧ 𝑘 ∈ 𝐼) → 𝑅 ∈ Ring)
75	43	ad4ant23 766	. . . . . . . . . . . . . . . 16 ⊢ (((((𝑅 ∈ Ring ∧ 𝑀:(𝐼 × 𝐼)⟶(Base‘𝑅)) ∧ 𝑓:𝐼⟶(Base‘𝑅)) ∧ 𝑛 ∈ 𝐼) ∧ 𝑘 ∈ 𝐼) → (𝑓‘𝑛) ∈ (Base‘𝑅))
76		fovcdm 7591	. . . . . . . . . . . . . . . . 17 ⊢ ((𝑀:(𝐼 × 𝐼)⟶(Base‘𝑅) ∧ 𝑛 ∈ 𝐼 ∧ 𝑘 ∈ 𝐼) → (𝑛𝑀𝑘) ∈ (Base‘𝑅))
77	76	ad5ant245 1384	. . . . . . . . . . . . . . . 16 ⊢ (((((𝑅 ∈ Ring ∧ 𝑀:(𝐼 × 𝐼)⟶(Base‘𝑅)) ∧ 𝑓:𝐼⟶(Base‘𝑅)) ∧ 𝑛 ∈ 𝐼) ∧ 𝑘 ∈ 𝐼) → (𝑛𝑀𝑘) ∈ (Base‘𝑅))
78	10, 47	ringcl 20477	. . . . . . . . . . . . . . . 16 ⊢ ((𝑅 ∈ Ring ∧ (𝑓‘𝑛) ∈ (Base‘𝑅) ∧ (𝑛𝑀𝑘) ∈ (Base‘𝑅)) → ((𝑓‘𝑛)(.r‘𝑅)(𝑛𝑀𝑘)) ∈ (Base‘𝑅))
79	74, 75, 77, 78	syl3anc 1398	. . . . . . . . . . . . . . 15 ⊢ (((((𝑅 ∈ Ring ∧ 𝑀:(𝐼 × 𝐼)⟶(Base‘𝑅)) ∧ 𝑓:𝐼⟶(Base‘𝑅)) ∧ 𝑛 ∈ 𝐼) ∧ 𝑘 ∈ 𝐼) → ((𝑓‘𝑛)(.r‘𝑅)(𝑛𝑀𝑘)) ∈ (Base‘𝑅))
80	79	fmpttd 7115	. . . . . . . . . . . . . 14 ⊢ ((((𝑅 ∈ Ring ∧ 𝑀:(𝐼 × 𝐼)⟶(Base‘𝑅)) ∧ 𝑓:𝐼⟶(Base‘𝑅)) ∧ 𝑛 ∈ 𝐼) → (𝑘 ∈ 𝐼 ↦ ((𝑓‘𝑛)(.r‘𝑅)(𝑛𝑀𝑘))):𝐼⟶(Base‘𝑅))
81	80	adantllr 732	. . . . . . . . . . . . 13 ⊢ (((((𝑅 ∈ Ring ∧ 𝑀:(𝐼 × 𝐼)⟶(Base‘𝑅)) ∧ 𝐼 ∈ (Fin ∖ {∅})) ∧ 𝑓:𝐼⟶(Base‘𝑅)) ∧ 𝑛 ∈ 𝐼) → (𝑘 ∈ 𝐼 ↦ ((𝑓‘𝑛)(.r‘𝑅)(𝑛𝑀𝑘))):𝐼⟶(Base‘𝑅))
82		elmapg 8859	. . . . . . . . . . . . . . . . 17 ⊢ (((Base‘𝑅) ∈ V ∧ 𝐼 ∈ (Fin ∖ {∅})) → ((𝑘 ∈ 𝐼 ↦ ((𝑓‘𝑛)(.r‘𝑅)(𝑛𝑀𝑘))) ∈ ((Base‘𝑅) ↑m 𝐼) ↔ (𝑘 ∈ 𝐼 ↦ ((𝑓‘𝑛)(.r‘𝑅)(𝑛𝑀𝑘))):𝐼⟶(Base‘𝑅)))
83	13, 82	mpan 703	. . . . . . . . . . . . . . . 16 ⊢ (𝐼 ∈ (Fin ∖ {∅}) → ((𝑘 ∈ 𝐼 ↦ ((𝑓‘𝑛)(.r‘𝑅)(𝑛𝑀𝑘))) ∈ ((Base‘𝑅) ↑m 𝐼) ↔ (𝑘 ∈ 𝐼 ↦ ((𝑓‘𝑛)(.r‘𝑅)(𝑛𝑀𝑘))):𝐼⟶(Base‘𝑅)))
84	83	adantl 487	. . . . . . . . . . . . . . 15 ⊢ ((𝑅 ∈ Ring ∧ 𝐼 ∈ (Fin ∖ {∅})) → ((𝑘 ∈ 𝐼 ↦ ((𝑓‘𝑛)(.r‘𝑅)(𝑛𝑀𝑘))) ∈ ((Base‘𝑅) ↑m 𝐼) ↔ (𝑘 ∈ 𝐼 ↦ ((𝑓‘𝑛)(.r‘𝑅)(𝑛𝑀𝑘))):𝐼⟶(Base‘𝑅)))
85	12	eleq2d 2847	. . . . . . . . . . . . . . 15 ⊢ ((𝑅 ∈ Ring ∧ 𝐼 ∈ (Fin ∖ {∅})) → ((𝑘 ∈ 𝐼 ↦ ((𝑓‘𝑛)(.r‘𝑅)(𝑛𝑀𝑘))) ∈ ((Base‘𝑅) ↑m 𝐼) ↔ (𝑘 ∈ 𝐼 ↦ ((𝑓‘𝑛)(.r‘𝑅)(𝑛𝑀𝑘))) ∈ (Base‘(𝑅 freeLMod 𝐼))))
86	84, 85	bitr3d 284	. . . . . . . . . . . . . 14 ⊢ ((𝑅 ∈ Ring ∧ 𝐼 ∈ (Fin ∖ {∅})) → ((𝑘 ∈ 𝐼 ↦ ((𝑓‘𝑛)(.r‘𝑅)(𝑛𝑀𝑘))):𝐼⟶(Base‘𝑅) ↔ (𝑘 ∈ 𝐼 ↦ ((𝑓‘𝑛)(.r‘𝑅)(𝑛𝑀𝑘))) ∈ (Base‘(𝑅 freeLMod 𝐼))))
87	86	ad5ant13 769	. . . . . . . . . . . . 13 ⊢ (((((𝑅 ∈ Ring ∧ 𝑀:(𝐼 × 𝐼)⟶(Base‘𝑅)) ∧ 𝐼 ∈ (Fin ∖ {∅})) ∧ 𝑓:𝐼⟶(Base‘𝑅)) ∧ 𝑛 ∈ 𝐼) → ((𝑘 ∈ 𝐼 ↦ ((𝑓‘𝑛)(.r‘𝑅)(𝑛𝑀𝑘))):𝐼⟶(Base‘𝑅) ↔ (𝑘 ∈ 𝐼 ↦ ((𝑓‘𝑛)(.r‘𝑅)(𝑛𝑀𝑘))) ∈ (Base‘(𝑅 freeLMod 𝐼))))
88	81, 87	mpbid 235	. . . . . . . . . . . 12 ⊢ (((((𝑅 ∈ Ring ∧ 𝑀:(𝐼 × 𝐼)⟶(Base‘𝑅)) ∧ 𝐼 ∈ (Fin ∖ {∅})) ∧ 𝑓:𝐼⟶(Base‘𝑅)) ∧ 𝑛 ∈ 𝐼) → (𝑘 ∈ 𝐼 ↦ ((𝑓‘𝑛)(.r‘𝑅)(𝑛𝑀𝑘))) ∈ (Base‘(𝑅 freeLMod 𝐼)))
89		mptexg 7227	. . . . . . . . . . . . . . . 16 ⊢ (𝐼 ∈ (Fin ∖ {∅}) → (𝑘 ∈ 𝐼 ↦ ((𝑓‘𝑛)(.r‘𝑅)(𝑛𝑀𝑘))) ∈ V)
90	89	ralrimivw 3159	. . . . . . . . . . . . . . 15 ⊢ (𝐼 ∈ (Fin ∖ {∅}) → ∀𝑛 ∈ 𝐼 (𝑘 ∈ 𝐼 ↦ ((𝑓‘𝑛)(.r‘𝑅)(𝑛𝑀𝑘))) ∈ V)
91		eqid 2761	. . . . . . . . . . . . . . . 16 ⊢ (𝑛 ∈ 𝐼 ↦ (𝑘 ∈ 𝐼 ↦ ((𝑓‘𝑛)(.r‘𝑅)(𝑛𝑀𝑘)))) = (𝑛 ∈ 𝐼 ↦ (𝑘 ∈ 𝐼 ↦ ((𝑓‘𝑛)(.r‘𝑅)(𝑛𝑀𝑘))))
92	91	fnmpt 6679	. . . . . . . . . . . . . . 15 ⊢ (∀𝑛 ∈ 𝐼 (𝑘 ∈ 𝐼 ↦ ((𝑓‘𝑛)(.r‘𝑅)(𝑛𝑀𝑘))) ∈ V → (𝑛 ∈ 𝐼 ↦ (𝑘 ∈ 𝐼 ↦ ((𝑓‘𝑛)(.r‘𝑅)(𝑛𝑀𝑘)))) Fn 𝐼)
93	90, 92	syl 18	. . . . . . . . . . . . . 14 ⊢ (𝐼 ∈ (Fin ∖ {∅}) → (𝑛 ∈ 𝐼 ↦ (𝑘 ∈ 𝐼 ↦ ((𝑓‘𝑛)(.r‘𝑅)(𝑛𝑀𝑘)))) Fn 𝐼)
94		fvexd 6900	. . . . . . . . . . . . . 14 ⊢ (𝐼 ∈ (Fin ∖ {∅}) → (0g‘(𝑅 freeLMod 𝐼)) ∈ V)
95	93, 9, 94	fndmfifsupp 9370	. . . . . . . . . . . . 13 ⊢ (𝐼 ∈ (Fin ∖ {∅}) → (𝑛 ∈ 𝐼 ↦ (𝑘 ∈ 𝐼 ↦ ((𝑓‘𝑛)(.r‘𝑅)(𝑛𝑀𝑘)))) finSupp (0g‘(𝑅 freeLMod 𝐼)))
96	95	ad2antlr 740	. . . . . . . . . . . 12 ⊢ ((((𝑅 ∈ Ring ∧ 𝑀:(𝐼 × 𝐼)⟶(Base‘𝑅)) ∧ 𝐼 ∈ (Fin ∖ {∅})) ∧ 𝑓:𝐼⟶(Base‘𝑅)) → (𝑛 ∈ 𝐼 ↦ (𝑘 ∈ 𝐼 ↦ ((𝑓‘𝑛)(.r‘𝑅)(𝑛𝑀𝑘)))) finSupp (0g‘(𝑅 freeLMod 𝐼)))
97	5, 20, 23, 37, 37, 73, 88, 96	frlmgsum 22078	. . . . . . . . . . 11 ⊢ ((((𝑅 ∈ Ring ∧ 𝑀:(𝐼 × 𝐼)⟶(Base‘𝑅)) ∧ 𝐼 ∈ (Fin ∖ {∅})) ∧ 𝑓:𝐼⟶(Base‘𝑅)) → ((𝑅 freeLMod 𝐼) Σg (𝑛 ∈ 𝐼 ↦ (𝑘 ∈ 𝐼 ↦ ((𝑓‘𝑛)(.r‘𝑅)(𝑛𝑀𝑘))))) = (𝑘 ∈ 𝐼 ↦ (𝑅 Σg (𝑛 ∈ 𝐼 ↦ ((𝑓‘𝑛)(.r‘𝑅)(𝑛𝑀𝑘))))))
98	72, 97	eqtr2d 2797	. . . . . . . . . 10 ⊢ ((((𝑅 ∈ Ring ∧ 𝑀:(𝐼 × 𝐼)⟶(Base‘𝑅)) ∧ 𝐼 ∈ (Fin ∖ {∅})) ∧ 𝑓:𝐼⟶(Base‘𝑅)) → (𝑘 ∈ 𝐼 ↦ (𝑅 Σg (𝑛 ∈ 𝐼 ↦ ((𝑓‘𝑛)(.r‘𝑅)(𝑛𝑀𝑘))))) = ((𝑅 freeLMod 𝐼) Σg (𝑓 ∘f ( ·𝑠 ‘(𝑅 freeLMod 𝐼))curry 𝑀)))
99	32, 98	sylan2 605	. . . . . . . . 9 ⊢ ((((𝑅 ∈ Ring ∧ 𝑀:(𝐼 × 𝐼)⟶(Base‘𝑅)) ∧ 𝐼 ∈ (Fin ∖ {∅})) ∧ 𝑓 ∈ ((Base‘𝑅) ↑m 𝐼)) → (𝑘 ∈ 𝐼 ↦ (𝑅 Σg (𝑛 ∈ 𝐼 ↦ ((𝑓‘𝑛)(.r‘𝑅)(𝑛𝑀𝑘))))) = ((𝑅 freeLMod 𝐼) Σg (𝑓 ∘f ( ·𝑠 ‘(𝑅 freeLMod 𝐼))curry 𝑀)))
100		eqid 2761	. . . . . . . . . . 11 ⊢ (0g‘𝑅) = (0g‘𝑅)
101	5, 100	frlm0 22060	. . . . . . . . . 10 ⊢ ((𝑅 ∈ Ring ∧ 𝐼 ∈ (Fin ∖ {∅})) → (𝐼 × {(0g‘𝑅)}) = (0g‘(𝑅 freeLMod 𝐼)))
102	101	ad4ant13 764	. . . . . . . . 9 ⊢ ((((𝑅 ∈ Ring ∧ 𝑀:(𝐼 × 𝐼)⟶(Base‘𝑅)) ∧ 𝐼 ∈ (Fin ∖ {∅})) ∧ 𝑓 ∈ ((Base‘𝑅) ↑m 𝐼)) → (𝐼 × {(0g‘𝑅)}) = (0g‘(𝑅 freeLMod 𝐼)))
103	99, 102	eqeq12d 2777	. . . . . . . 8 ⊢ ((((𝑅 ∈ Ring ∧ 𝑀:(𝐼 × 𝐼)⟶(Base‘𝑅)) ∧ 𝐼 ∈ (Fin ∖ {∅})) ∧ 𝑓 ∈ ((Base‘𝑅) ↑m 𝐼)) → ((𝑘 ∈ 𝐼 ↦ (𝑅 Σg (𝑛 ∈ 𝐼 ↦ ((𝑓‘𝑛)(.r‘𝑅)(𝑛𝑀𝑘))))) = (𝐼 × {(0g‘𝑅)}) ↔ ((𝑅 freeLMod 𝐼) Σg (𝑓 ∘f ( ·𝑠 ‘(𝑅 freeLMod 𝐼))curry 𝑀)) = (0g‘(𝑅 freeLMod 𝐼))))
104	28	fveq2d 6889	. . . . . . . . . . . 12 ⊢ ((𝑅 ∈ Ring ∧ 𝐼 ∈ (Fin ∖ {∅})) → (0g‘𝑅) = (0g‘(Scalar‘(𝑅 freeLMod 𝐼))))
105	104	sneqd 4596	. . . . . . . . . . 11 ⊢ ((𝑅 ∈ Ring ∧ 𝐼 ∈ (Fin ∖ {∅})) → {(0g‘𝑅)} = {(0g‘(Scalar‘(𝑅 freeLMod 𝐼)))})
106	105	xpeq2d 5681	. . . . . . . . . 10 ⊢ ((𝑅 ∈ Ring ∧ 𝐼 ∈ (Fin ∖ {∅})) → (𝐼 × {(0g‘𝑅)}) = (𝐼 × {(0g‘(Scalar‘(𝑅 freeLMod 𝐼)))}))
107	106	eqeq2d 2772	. . . . . . . . 9 ⊢ ((𝑅 ∈ Ring ∧ 𝐼 ∈ (Fin ∖ {∅})) → (𝑓 = (𝐼 × {(0g‘𝑅)}) ↔ 𝑓 = (𝐼 × {(0g‘(Scalar‘(𝑅 freeLMod 𝐼)))})))
108	107	ad4ant13 764	. . . . . . . 8 ⊢ ((((𝑅 ∈ Ring ∧ 𝑀:(𝐼 × 𝐼)⟶(Base‘𝑅)) ∧ 𝐼 ∈ (Fin ∖ {∅})) ∧ 𝑓 ∈ ((Base‘𝑅) ↑m 𝐼)) → (𝑓 = (𝐼 × {(0g‘𝑅)}) ↔ 𝑓 = (𝐼 × {(0g‘(Scalar‘(𝑅 freeLMod 𝐼)))})))
109	103, 108	imbi12d 347	. . . . . . 7 ⊢ ((((𝑅 ∈ Ring ∧ 𝑀:(𝐼 × 𝐼)⟶(Base‘𝑅)) ∧ 𝐼 ∈ (Fin ∖ {∅})) ∧ 𝑓 ∈ ((Base‘𝑅) ↑m 𝐼)) → (((𝑘 ∈ 𝐼 ↦ (𝑅 Σg (𝑛 ∈ 𝐼 ↦ ((𝑓‘𝑛)(.r‘𝑅)(𝑛𝑀𝑘))))) = (𝐼 × {(0g‘𝑅)}) → 𝑓 = (𝐼 × {(0g‘𝑅)})) ↔ (((𝑅 freeLMod 𝐼) Σg (𝑓 ∘f ( ·𝑠 ‘(𝑅 freeLMod 𝐼))curry 𝑀)) = (0g‘(𝑅 freeLMod 𝐼)) → 𝑓 = (𝐼 × {(0g‘(Scalar‘(𝑅 freeLMod 𝐼)))}))))
110	31, 109	raleqbidva 3326	. . . . . 6 ⊢ (((𝑅 ∈ Ring ∧ 𝑀:(𝐼 × 𝐼)⟶(Base‘𝑅)) ∧ 𝐼 ∈ (Fin ∖ {∅})) → (∀𝑓 ∈ ((Base‘𝑅) ↑m 𝐼)((𝑘 ∈ 𝐼 ↦ (𝑅 Σg (𝑛 ∈ 𝐼 ↦ ((𝑓‘𝑛)(.r‘𝑅)(𝑛𝑀𝑘))))) = (𝐼 × {(0g‘𝑅)}) → 𝑓 = (𝐼 × {(0g‘𝑅)})) ↔ ∀𝑓 ∈ (Base‘((Scalar‘(𝑅 freeLMod 𝐼)) freeLMod 𝐼))(((𝑅 freeLMod 𝐼) Σg (𝑓 ∘f ( ·𝑠 ‘(𝑅 freeLMod 𝐼))curry 𝑀)) = (0g‘(𝑅 freeLMod 𝐼)) → 𝑓 = (𝐼 × {(0g‘(Scalar‘(𝑅 freeLMod 𝐼)))}))))
111	27, 110	bitr4d 285	. . . . 5 ⊢ (((𝑅 ∈ Ring ∧ 𝑀:(𝐼 × 𝐼)⟶(Base‘𝑅)) ∧ 𝐼 ∈ (Fin ∖ {∅})) → (curry 𝑀 LIndF (𝑅 freeLMod 𝐼) ↔ ∀𝑓 ∈ ((Base‘𝑅) ↑m 𝐼)((𝑘 ∈ 𝐼 ↦ (𝑅 Σg (𝑛 ∈ 𝐼 ↦ ((𝑓‘𝑛)(.r‘𝑅)(𝑛𝑀𝑘))))) = (𝐼 × {(0g‘𝑅)}) → 𝑓 = (𝐼 × {(0g‘𝑅)}))))
112	111	notbid 321	. . . 4 ⊢ (((𝑅 ∈ Ring ∧ 𝑀:(𝐼 × 𝐼)⟶(Base‘𝑅)) ∧ 𝐼 ∈ (Fin ∖ {∅})) → (¬ curry 𝑀 LIndF (𝑅 freeLMod 𝐼) ↔ ¬ ∀𝑓 ∈ ((Base‘𝑅) ↑m 𝐼)((𝑘 ∈ 𝐼 ↦ (𝑅 Σg (𝑛 ∈ 𝐼 ↦ ((𝑓‘𝑛)(.r‘𝑅)(𝑛𝑀𝑘))))) = (𝐼 × {(0g‘𝑅)}) → 𝑓 = (𝐼 × {(0g‘𝑅)}))))
113		rexanali 3117	. . . 4 ⊢ (∃𝑓 ∈ ((Base‘𝑅) ↑m 𝐼)((𝑘 ∈ 𝐼 ↦ (𝑅 Σg (𝑛 ∈ 𝐼 ↦ ((𝑓‘𝑛)(.r‘𝑅)(𝑛𝑀𝑘))))) = (𝐼 × {(0g‘𝑅)}) ∧ ¬ 𝑓 = (𝐼 × {(0g‘𝑅)})) ↔ ¬ ∀𝑓 ∈ ((Base‘𝑅) ↑m 𝐼)((𝑘 ∈ 𝐼 ↦ (𝑅 Σg (𝑛 ∈ 𝐼 ↦ ((𝑓‘𝑛)(.r‘𝑅)(𝑛𝑀𝑘))))) = (𝐼 × {(0g‘𝑅)}) → 𝑓 = (𝐼 × {(0g‘𝑅)})))
114	112, 113	bitr4di 292	. . 3 ⊢ (((𝑅 ∈ Ring ∧ 𝑀:(𝐼 × 𝐼)⟶(Base‘𝑅)) ∧ 𝐼 ∈ (Fin ∖ {∅})) → (¬ curry 𝑀 LIndF (𝑅 freeLMod 𝐼) ↔ ∃𝑓 ∈ ((Base‘𝑅) ↑m 𝐼)((𝑘 ∈ 𝐼 ↦ (𝑅 Σg (𝑛 ∈ 𝐼 ↦ ((𝑓‘𝑛)(.r‘𝑅)(𝑛𝑀𝑘))))) = (𝐼 × {(0g‘𝑅)}) ∧ ¬ 𝑓 = (𝐼 × {(0g‘𝑅)}))))
115	4, 114	sylanl1 693	. 2 ⊢ (((𝑅 ∈ Field ∧ 𝑀:(𝐼 × 𝐼)⟶(Base‘𝑅)) ∧ 𝐼 ∈ (Fin ∖ {∅})) → (¬ curry 𝑀 LIndF (𝑅 freeLMod 𝐼) ↔ ∃𝑓 ∈ ((Base‘𝑅) ↑m 𝐼)((𝑘 ∈ 𝐼 ↦ (𝑅 Σg (𝑛 ∈ 𝐼 ↦ ((𝑓‘𝑛)(.r‘𝑅)(𝑛𝑀𝑘))))) = (𝐼 × {(0g‘𝑅)}) ∧ ¬ 𝑓 = (𝐼 × {(0g‘𝑅)}))))
116		fconstfv 7218	. . . . . . . . . . . 12 ⊢ (𝑓:𝐼⟶{(0g‘𝑅)} ↔ (𝑓 Fn 𝐼 ∧ ∀𝑖 ∈ 𝐼 (𝑓‘𝑖) = (0g‘𝑅)))
117		fvex 6898	. . . . . . . . . . . . 13 ⊢ (0g‘𝑅) ∈ V
118	117	fconst2 7211	. . . . . . . . . . . 12 ⊢ (𝑓:𝐼⟶{(0g‘𝑅)} ↔ 𝑓 = (𝐼 × {(0g‘𝑅)}))
119	116, 118	sylbb1 240	. . . . . . . . . . 11 ⊢ ((𝑓 Fn 𝐼 ∧ ∀𝑖 ∈ 𝐼 (𝑓‘𝑖) = (0g‘𝑅)) → 𝑓 = (𝐼 × {(0g‘𝑅)}))
120	119	ex 418	. . . . . . . . . 10 ⊢ (𝑓 Fn 𝐼 → (∀𝑖 ∈ 𝐼 (𝑓‘𝑖) = (0g‘𝑅) → 𝑓 = (𝐼 × {(0g‘𝑅)})))
121	120	con3d 153	. . . . . . . . 9 ⊢ (𝑓 Fn 𝐼 → (¬ 𝑓 = (𝐼 × {(0g‘𝑅)}) → ¬ ∀𝑖 ∈ 𝐼 (𝑓‘𝑖) = (0g‘𝑅)))
122		df-ne 2957	. . . . . . . . . . 11 ⊢ ((𝑓‘𝑖) ≠ (0g‘𝑅) ↔ ¬ (𝑓‘𝑖) = (0g‘𝑅))
123	122	rexbii 3110	. . . . . . . . . 10 ⊢ (∃𝑖 ∈ 𝐼 (𝑓‘𝑖) ≠ (0g‘𝑅) ↔ ∃𝑖 ∈ 𝐼 ¬ (𝑓‘𝑖) = (0g‘𝑅))
124		rexnal 3115	. . . . . . . . . 10 ⊢ (∃𝑖 ∈ 𝐼 ¬ (𝑓‘𝑖) = (0g‘𝑅) ↔ ¬ ∀𝑖 ∈ 𝐼 (𝑓‘𝑖) = (0g‘𝑅))
125	123, 124	bitri 278	. . . . . . . . 9 ⊢ (∃𝑖 ∈ 𝐼 (𝑓‘𝑖) ≠ (0g‘𝑅) ↔ ¬ ∀𝑖 ∈ 𝐼 (𝑓‘𝑖) = (0g‘𝑅))
126	121, 125	imbitrrdi 255	. . . . . . . 8 ⊢ (𝑓 Fn 𝐼 → (¬ 𝑓 = (𝐼 × {(0g‘𝑅)}) → ∃𝑖 ∈ 𝐼 (𝑓‘𝑖) ≠ (0g‘𝑅)))
127	33, 126	syl 18	. . . . . . 7 ⊢ (𝑓:𝐼⟶(Base‘𝑅) → (¬ 𝑓 = (𝐼 × {(0g‘𝑅)}) → ∃𝑖 ∈ 𝐼 (𝑓‘𝑖) ≠ (0g‘𝑅)))
128	127	adantl 487	. . . . . 6 ⊢ ((((𝑅 ∈ Field ∧ 𝑀:(𝐼 × 𝐼)⟶(Base‘𝑅)) ∧ 𝐼 ∈ Fin) ∧ 𝑓:𝐼⟶(Base‘𝑅)) → (¬ 𝑓 = (𝐼 × {(0g‘𝑅)}) → ∃𝑖 ∈ 𝐼 (𝑓‘𝑖) ≠ (0g‘𝑅)))
129		neldifsn 4755	. . . . . . . . . . 11 ⊢ ¬ 𝑖 ∈ (𝐼 ∖ {𝑖})
130		difss 4083	. . . . . . . . . . 11 ⊢ (𝐼 ∖ {𝑖}) ⊆ 𝐼
131		diffi 9190	. . . . . . . . . . . . 13 ⊢ (𝐼 ∈ Fin → (𝐼 ∖ {𝑖}) ∈ Fin)
132	131	ad4antlr 746	. . . . . . . . . . . 12 ⊢ ((((((𝑅 ∈ Field ∧ 𝑀:(𝐼 × 𝐼)⟶(Base‘𝑅)) ∧ 𝐼 ∈ Fin) ∧ 𝑓:𝐼⟶(Base‘𝑅)) ∧ (𝑖 ∈ 𝐼 ∧ (𝑓‘𝑖) ≠ (0g‘𝑅))) ∧ (¬ 𝑖 ∈ (𝐼 ∖ {𝑖}) ∧ (𝐼 ∖ {𝑖}) ⊆ 𝐼)) → (𝐼 ∖ {𝑖}) ∈ Fin)
133		eleq2 2850	. . . . . . . . . . . . . . . . 17 ⊢ (𝑦 = ∅ → (𝑖 ∈ 𝑦 ↔ 𝑖 ∈ ∅))
134	133	notbid 321	. . . . . . . . . . . . . . . 16 ⊢ (𝑦 = ∅ → (¬ 𝑖 ∈ 𝑦 ↔ ¬ 𝑖 ∈ ∅))
135		sseq1 3956	. . . . . . . . . . . . . . . 16 ⊢ (𝑦 = ∅ → (𝑦 ⊆ 𝐼 ↔ ∅ ⊆ 𝐼))
136	134, 135	anbi12d 644	. . . . . . . . . . . . . . 15 ⊢ (𝑦 = ∅ → ((¬ 𝑖 ∈ 𝑦 ∧ 𝑦 ⊆ 𝐼) ↔ (¬ 𝑖 ∈ ∅ ∧ ∅ ⊆ 𝐼)))
137	136	anbi2d 642	. . . . . . . . . . . . . 14 ⊢ (𝑦 = ∅ → ((((((𝑅 ∈ Field ∧ 𝑀:(𝐼 × 𝐼)⟶(Base‘𝑅)) ∧ 𝐼 ∈ Fin) ∧ 𝑓:𝐼⟶(Base‘𝑅)) ∧ (𝑖 ∈ 𝐼 ∧ (𝑓‘𝑖) ≠ (0g‘𝑅))) ∧ (¬ 𝑖 ∈ 𝑦 ∧ 𝑦 ⊆ 𝐼)) ↔ (((((𝑅 ∈ Field ∧ 𝑀:(𝐼 × 𝐼)⟶(Base‘𝑅)) ∧ 𝐼 ∈ Fin) ∧ 𝑓:𝐼⟶(Base‘𝑅)) ∧ (𝑖 ∈ 𝐼 ∧ (𝑓‘𝑖) ≠ (0g‘𝑅))) ∧ (¬ 𝑖 ∈ ∅ ∧ ∅ ⊆ 𝐼))))
138		mpteq1 5194	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (𝑦 = ∅ → (𝑛 ∈ 𝑦 ↦ (((invr‘𝑅)‘(𝑓‘𝑖))(.r‘𝑅)((𝑓‘𝑛)(.r‘𝑅)(𝑛𝑀𝑘)))) = (𝑛 ∈ ∅ ↦ (((invr‘𝑅)‘(𝑓‘𝑖))(.r‘𝑅)((𝑓‘𝑛)(.r‘𝑅)(𝑛𝑀𝑘)))))
139		mpt0 6681	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (𝑛 ∈ ∅ ↦ (((invr‘𝑅)‘(𝑓‘𝑖))(.r‘𝑅)((𝑓‘𝑛)(.r‘𝑅)(𝑛𝑀𝑘)))) = ∅
140	138, 139	eqtrdi 2812	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (𝑦 = ∅ → (𝑛 ∈ 𝑦 ↦ (((invr‘𝑅)‘(𝑓‘𝑖))(.r‘𝑅)((𝑓‘𝑛)(.r‘𝑅)(𝑛𝑀𝑘)))) = ∅)
141	140	oveq2d 7436	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝑦 = ∅ → (𝑅 Σg (𝑛 ∈ 𝑦 ↦ (((invr‘𝑅)‘(𝑓‘𝑖))(.r‘𝑅)((𝑓‘𝑛)(.r‘𝑅)(𝑛𝑀𝑘))))) = (𝑅 Σg ∅))
142	100	gsum0 18873	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝑅 Σg ∅) = (0g‘𝑅)
143	141, 142	eqtrdi 2812	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝑦 = ∅ → (𝑅 Σg (𝑛 ∈ 𝑦 ↦ (((invr‘𝑅)‘(𝑓‘𝑖))(.r‘𝑅)((𝑓‘𝑛)(.r‘𝑅)(𝑛𝑀𝑘))))) = (0g‘𝑅))
144	143	oveq1d 7435	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑦 = ∅ → ((𝑅 Σg (𝑛 ∈ 𝑦 ↦ (((invr‘𝑅)‘(𝑓‘𝑖))(.r‘𝑅)((𝑓‘𝑛)(.r‘𝑅)(𝑛𝑀𝑘)))))(+g‘𝑅)(𝑖𝑀𝑘)) = ((0g‘𝑅)(+g‘𝑅)(𝑖𝑀𝑘)))
145	144	ifeq1d 4502	. . . . . . . . . . . . . . . . 17 ⊢ (𝑦 = ∅ → if(𝑗 = 𝑖, ((𝑅 Σg (𝑛 ∈ 𝑦 ↦ (((invr‘𝑅)‘(𝑓‘𝑖))(.r‘𝑅)((𝑓‘𝑛)(.r‘𝑅)(𝑛𝑀𝑘)))))(+g‘𝑅)(𝑖𝑀𝑘)), (𝑗𝑀𝑘)) = if(𝑗 = 𝑖, ((0g‘𝑅)(+g‘𝑅)(𝑖𝑀𝑘)), (𝑗𝑀𝑘)))
146	145	mpoeq3dv 7499	. . . . . . . . . . . . . . . 16 ⊢ (𝑦 = ∅ → (𝑗 ∈ 𝐼, 𝑘 ∈ 𝐼 ↦ if(𝑗 = 𝑖, ((𝑅 Σg (𝑛 ∈ 𝑦 ↦ (((invr‘𝑅)‘(𝑓‘𝑖))(.r‘𝑅)((𝑓‘𝑛)(.r‘𝑅)(𝑛𝑀𝑘)))))(+g‘𝑅)(𝑖𝑀𝑘)), (𝑗𝑀𝑘))) = (𝑗 ∈ 𝐼, 𝑘 ∈ 𝐼 ↦ if(𝑗 = 𝑖, ((0g‘𝑅)(+g‘𝑅)(𝑖𝑀𝑘)), (𝑗𝑀𝑘))))
147	146	fveq2d 6889	. . . . . . . . . . . . . . 15 ⊢ (𝑦 = ∅ → ((𝐼 maDet 𝑅)‘(𝑗 ∈ 𝐼, 𝑘 ∈ 𝐼 ↦ if(𝑗 = 𝑖, ((𝑅 Σg (𝑛 ∈ 𝑦 ↦ (((invr‘𝑅)‘(𝑓‘𝑖))(.r‘𝑅)((𝑓‘𝑛)(.r‘𝑅)(𝑛𝑀𝑘)))))(+g‘𝑅)(𝑖𝑀𝑘)), (𝑗𝑀𝑘)))) = ((𝐼 maDet 𝑅)‘(𝑗 ∈ 𝐼, 𝑘 ∈ 𝐼 ↦ if(𝑗 = 𝑖, ((0g‘𝑅)(+g‘𝑅)(𝑖𝑀𝑘)), (𝑗𝑀𝑘)))))
148	147	eqeq2d 2772	. . . . . . . . . . . . . 14 ⊢ (𝑦 = ∅ → (((𝐼 maDet 𝑅)‘𝑀) = ((𝐼 maDet 𝑅)‘(𝑗 ∈ 𝐼, 𝑘 ∈ 𝐼 ↦ if(𝑗 = 𝑖, ((𝑅 Σg (𝑛 ∈ 𝑦 ↦ (((invr‘𝑅)‘(𝑓‘𝑖))(.r‘𝑅)((𝑓‘𝑛)(.r‘𝑅)(𝑛𝑀𝑘)))))(+g‘𝑅)(𝑖𝑀𝑘)), (𝑗𝑀𝑘)))) ↔ ((𝐼 maDet 𝑅)‘𝑀) = ((𝐼 maDet 𝑅)‘(𝑗 ∈ 𝐼, 𝑘 ∈ 𝐼 ↦ if(𝑗 = 𝑖, ((0g‘𝑅)(+g‘𝑅)(𝑖𝑀𝑘)), (𝑗𝑀𝑘))))))
149	137, 148	imbi12d 347	. . . . . . . . . . . . 13 ⊢ (𝑦 = ∅ → (((((((𝑅 ∈ Field ∧ 𝑀:(𝐼 × 𝐼)⟶(Base‘𝑅)) ∧ 𝐼 ∈ Fin) ∧ 𝑓:𝐼⟶(Base‘𝑅)) ∧ (𝑖 ∈ 𝐼 ∧ (𝑓‘𝑖) ≠ (0g‘𝑅))) ∧ (¬ 𝑖 ∈ 𝑦 ∧ 𝑦 ⊆ 𝐼)) → ((𝐼 maDet 𝑅)‘𝑀) = ((𝐼 maDet 𝑅)‘(𝑗 ∈ 𝐼, 𝑘 ∈ 𝐼 ↦ if(𝑗 = 𝑖, ((𝑅 Σg (𝑛 ∈ 𝑦 ↦ (((invr‘𝑅)‘(𝑓‘𝑖))(.r‘𝑅)((𝑓‘𝑛)(.r‘𝑅)(𝑛𝑀𝑘)))))(+g‘𝑅)(𝑖𝑀𝑘)), (𝑗𝑀𝑘))))) ↔ ((((((𝑅 ∈ Field ∧ 𝑀:(𝐼 × 𝐼)⟶(Base‘𝑅)) ∧ 𝐼 ∈ Fin) ∧ 𝑓:𝐼⟶(Base‘𝑅)) ∧ (𝑖 ∈ 𝐼 ∧ (𝑓‘𝑖) ≠ (0g‘𝑅))) ∧ (¬ 𝑖 ∈ ∅ ∧ ∅ ⊆ 𝐼)) → ((𝐼 maDet 𝑅)‘𝑀) = ((𝐼 maDet 𝑅)‘(𝑗 ∈ 𝐼, 𝑘 ∈ 𝐼 ↦ if(𝑗 = 𝑖, ((0g‘𝑅)(+g‘𝑅)(𝑖𝑀𝑘)), (𝑗𝑀𝑘)))))))
150		elequ2 2160	. . . . . . . . . . . . . . . . 17 ⊢ (𝑦 = 𝑥 → (𝑖 ∈ 𝑦 ↔ 𝑖 ∈ 𝑥))
151	150	notbid 321	. . . . . . . . . . . . . . . 16 ⊢ (𝑦 = 𝑥 → (¬ 𝑖 ∈ 𝑦 ↔ ¬ 𝑖 ∈ 𝑥))
152		sseq1 3956	. . . . . . . . . . . . . . . 16 ⊢ (𝑦 = 𝑥 → (𝑦 ⊆ 𝐼 ↔ 𝑥 ⊆ 𝐼))
153	151, 152	anbi12d 644	. . . . . . . . . . . . . . 15 ⊢ (𝑦 = 𝑥 → ((¬ 𝑖 ∈ 𝑦 ∧ 𝑦 ⊆ 𝐼) ↔ (¬ 𝑖 ∈ 𝑥 ∧ 𝑥 ⊆ 𝐼)))
154	153	anbi2d 642	. . . . . . . . . . . . . 14 ⊢ (𝑦 = 𝑥 → ((((((𝑅 ∈ Field ∧ 𝑀:(𝐼 × 𝐼)⟶(Base‘𝑅)) ∧ 𝐼 ∈ Fin) ∧ 𝑓:𝐼⟶(Base‘𝑅)) ∧ (𝑖 ∈ 𝐼 ∧ (𝑓‘𝑖) ≠ (0g‘𝑅))) ∧ (¬ 𝑖 ∈ 𝑦 ∧ 𝑦 ⊆ 𝐼)) ↔ (((((𝑅 ∈ Field ∧ 𝑀:(𝐼 × 𝐼)⟶(Base‘𝑅)) ∧ 𝐼 ∈ Fin) ∧ 𝑓:𝐼⟶(Base‘𝑅)) ∧ (𝑖 ∈ 𝐼 ∧ (𝑓‘𝑖) ≠ (0g‘𝑅))) ∧ (¬ 𝑖 ∈ 𝑥 ∧ 𝑥 ⊆ 𝐼))))
155		mpteq1 5194	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝑦 = 𝑥 → (𝑛 ∈ 𝑦 ↦ (((invr‘𝑅)‘(𝑓‘𝑖))(.r‘𝑅)((𝑓‘𝑛)(.r‘𝑅)(𝑛𝑀𝑘)))) = (𝑛 ∈ 𝑥 ↦ (((invr‘𝑅)‘(𝑓‘𝑖))(.r‘𝑅)((𝑓‘𝑛)(.r‘𝑅)(𝑛𝑀𝑘)))))
156	155	oveq2d 7436	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝑦 = 𝑥 → (𝑅 Σg (𝑛 ∈ 𝑦 ↦ (((invr‘𝑅)‘(𝑓‘𝑖))(.r‘𝑅)((𝑓‘𝑛)(.r‘𝑅)(𝑛𝑀𝑘))))) = (𝑅 Σg (𝑛 ∈ 𝑥 ↦ (((invr‘𝑅)‘(𝑓‘𝑖))(.r‘𝑅)((𝑓‘𝑛)(.r‘𝑅)(𝑛𝑀𝑘))))))
157	156	oveq1d 7435	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑦 = 𝑥 → ((𝑅 Σg (𝑛 ∈ 𝑦 ↦ (((invr‘𝑅)‘(𝑓‘𝑖))(.r‘𝑅)((𝑓‘𝑛)(.r‘𝑅)(𝑛𝑀𝑘)))))(+g‘𝑅)(𝑖𝑀𝑘)) = ((𝑅 Σg (𝑛 ∈ 𝑥 ↦ (((invr‘𝑅)‘(𝑓‘𝑖))(.r‘𝑅)((𝑓‘𝑛)(.r‘𝑅)(𝑛𝑀𝑘)))))(+g‘𝑅)(𝑖𝑀𝑘)))
158	157	ifeq1d 4502	. . . . . . . . . . . . . . . . 17 ⊢ (𝑦 = 𝑥 → if(𝑗 = 𝑖, ((𝑅 Σg (𝑛 ∈ 𝑦 ↦ (((invr‘𝑅)‘(𝑓‘𝑖))(.r‘𝑅)((𝑓‘𝑛)(.r‘𝑅)(𝑛𝑀𝑘)))))(+g‘𝑅)(𝑖𝑀𝑘)), (𝑗𝑀𝑘)) = if(𝑗 = 𝑖, ((𝑅 Σg (𝑛 ∈ 𝑥 ↦ (((invr‘𝑅)‘(𝑓‘𝑖))(.r‘𝑅)((𝑓‘𝑛)(.r‘𝑅)(𝑛𝑀𝑘)))))(+g‘𝑅)(𝑖𝑀𝑘)), (𝑗𝑀𝑘)))
159	158	mpoeq3dv 7499	. . . . . . . . . . . . . . . 16 ⊢ (𝑦 = 𝑥 → (𝑗 ∈ 𝐼, 𝑘 ∈ 𝐼 ↦ if(𝑗 = 𝑖, ((𝑅 Σg (𝑛 ∈ 𝑦 ↦ (((invr‘𝑅)‘(𝑓‘𝑖))(.r‘𝑅)((𝑓‘𝑛)(.r‘𝑅)(𝑛𝑀𝑘)))))(+g‘𝑅)(𝑖𝑀𝑘)), (𝑗𝑀𝑘))) = (𝑗 ∈ 𝐼, 𝑘 ∈ 𝐼 ↦ if(𝑗 = 𝑖, ((𝑅 Σg (𝑛 ∈ 𝑥 ↦ (((invr‘𝑅)‘(𝑓‘𝑖))(.r‘𝑅)((𝑓‘𝑛)(.r‘𝑅)(𝑛𝑀𝑘)))))(+g‘𝑅)(𝑖𝑀𝑘)), (𝑗𝑀𝑘))))
160	159	fveq2d 6889	. . . . . . . . . . . . . . 15 ⊢ (𝑦 = 𝑥 → ((𝐼 maDet 𝑅)‘(𝑗 ∈ 𝐼, 𝑘 ∈ 𝐼 ↦ if(𝑗 = 𝑖, ((𝑅 Σg (𝑛 ∈ 𝑦 ↦ (((invr‘𝑅)‘(𝑓‘𝑖))(.r‘𝑅)((𝑓‘𝑛)(.r‘𝑅)(𝑛𝑀𝑘)))))(+g‘𝑅)(𝑖𝑀𝑘)), (𝑗𝑀𝑘)))) = ((𝐼 maDet 𝑅)‘(𝑗 ∈ 𝐼, 𝑘 ∈ 𝐼 ↦ if(𝑗 = 𝑖, ((𝑅 Σg (𝑛 ∈ 𝑥 ↦ (((invr‘𝑅)‘(𝑓‘𝑖))(.r‘𝑅)((𝑓‘𝑛)(.r‘𝑅)(𝑛𝑀𝑘)))))(+g‘𝑅)(𝑖𝑀𝑘)), (𝑗𝑀𝑘)))))
161	160	eqeq2d 2772	. . . . . . . . . . . . . 14 ⊢ (𝑦 = 𝑥 → (((𝐼 maDet 𝑅)‘𝑀) = ((𝐼 maDet 𝑅)‘(𝑗 ∈ 𝐼, 𝑘 ∈ 𝐼 ↦ if(𝑗 = 𝑖, ((𝑅 Σg (𝑛 ∈ 𝑦 ↦ (((invr‘𝑅)‘(𝑓‘𝑖))(.r‘𝑅)((𝑓‘𝑛)(.r‘𝑅)(𝑛𝑀𝑘)))))(+g‘𝑅)(𝑖𝑀𝑘)), (𝑗𝑀𝑘)))) ↔ ((𝐼 maDet 𝑅)‘𝑀) = ((𝐼 maDet 𝑅)‘(𝑗 ∈ 𝐼, 𝑘 ∈ 𝐼 ↦ if(𝑗 = 𝑖, ((𝑅 Σg (𝑛 ∈ 𝑥 ↦ (((invr‘𝑅)‘(𝑓‘𝑖))(.r‘𝑅)((𝑓‘𝑛)(.r‘𝑅)(𝑛𝑀𝑘)))))(+g‘𝑅)(𝑖𝑀𝑘)), (𝑗𝑀𝑘))))))
162	154, 161	imbi12d 347	. . . . . . . . . . . . 13 ⊢ (𝑦 = 𝑥 → (((((((𝑅 ∈ Field ∧ 𝑀:(𝐼 × 𝐼)⟶(Base‘𝑅)) ∧ 𝐼 ∈ Fin) ∧ 𝑓:𝐼⟶(Base‘𝑅)) ∧ (𝑖 ∈ 𝐼 ∧ (𝑓‘𝑖) ≠ (0g‘𝑅))) ∧ (¬ 𝑖 ∈ 𝑦 ∧ 𝑦 ⊆ 𝐼)) → ((𝐼 maDet 𝑅)‘𝑀) = ((𝐼 maDet 𝑅)‘(𝑗 ∈ 𝐼, 𝑘 ∈ 𝐼 ↦ if(𝑗 = 𝑖, ((𝑅 Σg (𝑛 ∈ 𝑦 ↦ (((invr‘𝑅)‘(𝑓‘𝑖))(.r‘𝑅)((𝑓‘𝑛)(.r‘𝑅)(𝑛𝑀𝑘)))))(+g‘𝑅)(𝑖𝑀𝑘)), (𝑗𝑀𝑘))))) ↔ ((((((𝑅 ∈ Field ∧ 𝑀:(𝐼 × 𝐼)⟶(Base‘𝑅)) ∧ 𝐼 ∈ Fin) ∧ 𝑓:𝐼⟶(Base‘𝑅)) ∧ (𝑖 ∈ 𝐼 ∧ (𝑓‘𝑖) ≠ (0g‘𝑅))) ∧ (¬ 𝑖 ∈ 𝑥 ∧ 𝑥 ⊆ 𝐼)) → ((𝐼 maDet 𝑅)‘𝑀) = ((𝐼 maDet 𝑅)‘(𝑗 ∈ 𝐼, 𝑘 ∈ 𝐼 ↦ if(𝑗 = 𝑖, ((𝑅 Σg (𝑛 ∈ 𝑥 ↦ (((invr‘𝑅)‘(𝑓‘𝑖))(.r‘𝑅)((𝑓‘𝑛)(.r‘𝑅)(𝑛𝑀𝑘)))))(+g‘𝑅)(𝑖𝑀𝑘)), (𝑗𝑀𝑘)))))))
163		eleq2 2850	. . . . . . . . . . . . . . . . 17 ⊢ (𝑦 = (𝑥 ∪ {𝑧}) → (𝑖 ∈ 𝑦 ↔ 𝑖 ∈ (𝑥 ∪ {𝑧})))
164	163	notbid 321	. . . . . . . . . . . . . . . 16 ⊢ (𝑦 = (𝑥 ∪ {𝑧}) → (¬ 𝑖 ∈ 𝑦 ↔ ¬ 𝑖 ∈ (𝑥 ∪ {𝑧})))
165		sseq1 3956	. . . . . . . . . . . . . . . 16 ⊢ (𝑦 = (𝑥 ∪ {𝑧}) → (𝑦 ⊆ 𝐼 ↔ (𝑥 ∪ {𝑧}) ⊆ 𝐼))
166	164, 165	anbi12d 644	. . . . . . . . . . . . . . 15 ⊢ (𝑦 = (𝑥 ∪ {𝑧}) → ((¬ 𝑖 ∈ 𝑦 ∧ 𝑦 ⊆ 𝐼) ↔ (¬ 𝑖 ∈ (𝑥 ∪ {𝑧}) ∧ (𝑥 ∪ {𝑧}) ⊆ 𝐼)))
167	166	anbi2d 642	. . . . . . . . . . . . . 14 ⊢ (𝑦 = (𝑥 ∪ {𝑧}) → ((((((𝑅 ∈ Field ∧ 𝑀:(𝐼 × 𝐼)⟶(Base‘𝑅)) ∧ 𝐼 ∈ Fin) ∧ 𝑓:𝐼⟶(Base‘𝑅)) ∧ (𝑖 ∈ 𝐼 ∧ (𝑓‘𝑖) ≠ (0g‘𝑅))) ∧ (¬ 𝑖 ∈ 𝑦 ∧ 𝑦 ⊆ 𝐼)) ↔ (((((𝑅 ∈ Field ∧ 𝑀:(𝐼 × 𝐼)⟶(Base‘𝑅)) ∧ 𝐼 ∈ Fin) ∧ 𝑓:𝐼⟶(Base‘𝑅)) ∧ (𝑖 ∈ 𝐼 ∧ (𝑓‘𝑖) ≠ (0g‘𝑅))) ∧ (¬ 𝑖 ∈ (𝑥 ∪ {𝑧}) ∧ (𝑥 ∪ {𝑧}) ⊆ 𝐼))))
168		mpteq1 5194	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝑦 = (𝑥 ∪ {𝑧}) → (𝑛 ∈ 𝑦 ↦ (((invr‘𝑅)‘(𝑓‘𝑖))(.r‘𝑅)((𝑓‘𝑛)(.r‘𝑅)(𝑛𝑀𝑘)))) = (𝑛 ∈ (𝑥 ∪ {𝑧}) ↦ (((invr‘𝑅)‘(𝑓‘𝑖))(.r‘𝑅)((𝑓‘𝑛)(.r‘𝑅)(𝑛𝑀𝑘)))))
169	168	oveq2d 7436	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝑦 = (𝑥 ∪ {𝑧}) → (𝑅 Σg (𝑛 ∈ 𝑦 ↦ (((invr‘𝑅)‘(𝑓‘𝑖))(.r‘𝑅)((𝑓‘𝑛)(.r‘𝑅)(𝑛𝑀𝑘))))) = (𝑅 Σg (𝑛 ∈ (𝑥 ∪ {𝑧}) ↦ (((invr‘𝑅)‘(𝑓‘𝑖))(.r‘𝑅)((𝑓‘𝑛)(.r‘𝑅)(𝑛𝑀𝑘))))))
170	169	oveq1d 7435	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑦 = (𝑥 ∪ {𝑧}) → ((𝑅 Σg (𝑛 ∈ 𝑦 ↦ (((invr‘𝑅)‘(𝑓‘𝑖))(.r‘𝑅)((𝑓‘𝑛)(.r‘𝑅)(𝑛𝑀𝑘)))))(+g‘𝑅)(𝑖𝑀𝑘)) = ((𝑅 Σg (𝑛 ∈ (𝑥 ∪ {𝑧}) ↦ (((invr‘𝑅)‘(𝑓‘𝑖))(.r‘𝑅)((𝑓‘𝑛)(.r‘𝑅)(𝑛𝑀𝑘)))))(+g‘𝑅)(𝑖𝑀𝑘)))
171	170	ifeq1d 4502	. . . . . . . . . . . . . . . . 17 ⊢ (𝑦 = (𝑥 ∪ {𝑧}) → if(𝑗 = 𝑖, ((𝑅 Σg (𝑛 ∈ 𝑦 ↦ (((invr‘𝑅)‘(𝑓‘𝑖))(.r‘𝑅)((𝑓‘𝑛)(.r‘𝑅)(𝑛𝑀𝑘)))))(+g‘𝑅)(𝑖𝑀𝑘)), (𝑗𝑀𝑘)) = if(𝑗 = 𝑖, ((𝑅 Σg (𝑛 ∈ (𝑥 ∪ {𝑧}) ↦ (((invr‘𝑅)‘(𝑓‘𝑖))(.r‘𝑅)((𝑓‘𝑛)(.r‘𝑅)(𝑛𝑀𝑘)))))(+g‘𝑅)(𝑖𝑀𝑘)), (𝑗𝑀𝑘)))
172	171	mpoeq3dv 7499	. . . . . . . . . . . . . . . 16 ⊢ (𝑦 = (𝑥 ∪ {𝑧}) → (𝑗 ∈ 𝐼, 𝑘 ∈ 𝐼 ↦ if(𝑗 = 𝑖, ((𝑅 Σg (𝑛 ∈ 𝑦 ↦ (((invr‘𝑅)‘(𝑓‘𝑖))(.r‘𝑅)((𝑓‘𝑛)(.r‘𝑅)(𝑛𝑀𝑘)))))(+g‘𝑅)(𝑖𝑀𝑘)), (𝑗𝑀𝑘))) = (𝑗 ∈ 𝐼, 𝑘 ∈ 𝐼 ↦ if(𝑗 = 𝑖, ((𝑅 Σg (𝑛 ∈ (𝑥 ∪ {𝑧}) ↦ (((invr‘𝑅)‘(𝑓‘𝑖))(.r‘𝑅)((𝑓‘𝑛)(.r‘𝑅)(𝑛𝑀𝑘)))))(+g‘𝑅)(𝑖𝑀𝑘)), (𝑗𝑀𝑘))))
173	172	fveq2d 6889	. . . . . . . . . . . . . . 15 ⊢ (𝑦 = (𝑥 ∪ {𝑧}) → ((𝐼 maDet 𝑅)‘(𝑗 ∈ 𝐼, 𝑘 ∈ 𝐼 ↦ if(𝑗 = 𝑖, ((𝑅 Σg (𝑛 ∈ 𝑦 ↦ (((invr‘𝑅)‘(𝑓‘𝑖))(.r‘𝑅)((𝑓‘𝑛)(.r‘𝑅)(𝑛𝑀𝑘)))))(+g‘𝑅)(𝑖𝑀𝑘)), (𝑗𝑀𝑘)))) = ((𝐼 maDet 𝑅)‘(𝑗 ∈ 𝐼, 𝑘 ∈ 𝐼 ↦ if(𝑗 = 𝑖, ((𝑅 Σg (𝑛 ∈ (𝑥 ∪ {𝑧}) ↦ (((invr‘𝑅)‘(𝑓‘𝑖))(.r‘𝑅)((𝑓‘𝑛)(.r‘𝑅)(𝑛𝑀𝑘)))))(+g‘𝑅)(𝑖𝑀𝑘)), (𝑗𝑀𝑘)))))
174	173	eqeq2d 2772	. . . . . . . . . . . . . 14 ⊢ (𝑦 = (𝑥 ∪ {𝑧}) → (((𝐼 maDet 𝑅)‘𝑀) = ((𝐼 maDet 𝑅)‘(𝑗 ∈ 𝐼, 𝑘 ∈ 𝐼 ↦ if(𝑗 = 𝑖, ((𝑅 Σg (𝑛 ∈ 𝑦 ↦ (((invr‘𝑅)‘(𝑓‘𝑖))(.r‘𝑅)((𝑓‘𝑛)(.r‘𝑅)(𝑛𝑀𝑘)))))(+g‘𝑅)(𝑖𝑀𝑘)), (𝑗𝑀𝑘)))) ↔ ((𝐼 maDet 𝑅)‘𝑀) = ((𝐼 maDet 𝑅)‘(𝑗 ∈ 𝐼, 𝑘 ∈ 𝐼 ↦ if(𝑗 = 𝑖, ((𝑅 Σg (𝑛 ∈ (𝑥 ∪ {𝑧}) ↦ (((invr‘𝑅)‘(𝑓‘𝑖))(.r‘𝑅)((𝑓‘𝑛)(.r‘𝑅)(𝑛𝑀𝑘)))))(+g‘𝑅)(𝑖𝑀𝑘)), (𝑗𝑀𝑘))))))
175	167, 174	imbi12d 347	. . . . . . . . . . . . 13 ⊢ (𝑦 = (𝑥 ∪ {𝑧}) → (((((((𝑅 ∈ Field ∧ 𝑀:(𝐼 × 𝐼)⟶(Base‘𝑅)) ∧ 𝐼 ∈ Fin) ∧ 𝑓:𝐼⟶(Base‘𝑅)) ∧ (𝑖 ∈ 𝐼 ∧ (𝑓‘𝑖) ≠ (0g‘𝑅))) ∧ (¬ 𝑖 ∈ 𝑦 ∧ 𝑦 ⊆ 𝐼)) → ((𝐼 maDet 𝑅)‘𝑀) = ((𝐼 maDet 𝑅)‘(𝑗 ∈ 𝐼, 𝑘 ∈ 𝐼 ↦ if(𝑗 = 𝑖, ((𝑅 Σg (𝑛 ∈ 𝑦 ↦ (((invr‘𝑅)‘(𝑓‘𝑖))(.r‘𝑅)((𝑓‘𝑛)(.r‘𝑅)(𝑛𝑀𝑘)))))(+g‘𝑅)(𝑖𝑀𝑘)), (𝑗𝑀𝑘))))) ↔ ((((((𝑅 ∈ Field ∧ 𝑀:(𝐼 × 𝐼)⟶(Base‘𝑅)) ∧ 𝐼 ∈ Fin) ∧ 𝑓:𝐼⟶(Base‘𝑅)) ∧ (𝑖 ∈ 𝐼 ∧ (𝑓‘𝑖) ≠ (0g‘𝑅))) ∧ (¬ 𝑖 ∈ (𝑥 ∪ {𝑧}) ∧ (𝑥 ∪ {𝑧}) ⊆ 𝐼)) → ((𝐼 maDet 𝑅)‘𝑀) = ((𝐼 maDet 𝑅)‘(𝑗 ∈ 𝐼, 𝑘 ∈ 𝐼 ↦ if(𝑗 = 𝑖, ((𝑅 Σg (𝑛 ∈ (𝑥 ∪ {𝑧}) ↦ (((invr‘𝑅)‘(𝑓‘𝑖))(.r‘𝑅)((𝑓‘𝑛)(.r‘𝑅)(𝑛𝑀𝑘)))))(+g‘𝑅)(𝑖𝑀𝑘)), (𝑗𝑀𝑘)))))))
176		eleq2 2850	. . . . . . . . . . . . . . . . 17 ⊢ (𝑦 = (𝐼 ∖ {𝑖}) → (𝑖 ∈ 𝑦 ↔ 𝑖 ∈ (𝐼 ∖ {𝑖})))
177	176	notbid 321	. . . . . . . . . . . . . . . 16 ⊢ (𝑦 = (𝐼 ∖ {𝑖}) → (¬ 𝑖 ∈ 𝑦 ↔ ¬ 𝑖 ∈ (𝐼 ∖ {𝑖})))
178		sseq1 3956	. . . . . . . . . . . . . . . 16 ⊢ (𝑦 = (𝐼 ∖ {𝑖}) → (𝑦 ⊆ 𝐼 ↔ (𝐼 ∖ {𝑖}) ⊆ 𝐼))
179	177, 178	anbi12d 644	. . . . . . . . . . . . . . 15 ⊢ (𝑦 = (𝐼 ∖ {𝑖}) → ((¬ 𝑖 ∈ 𝑦 ∧ 𝑦 ⊆ 𝐼) ↔ (¬ 𝑖 ∈ (𝐼 ∖ {𝑖}) ∧ (𝐼 ∖ {𝑖}) ⊆ 𝐼)))
180	179	anbi2d 642	. . . . . . . . . . . . . 14 ⊢ (𝑦 = (𝐼 ∖ {𝑖}) → ((((((𝑅 ∈ Field ∧ 𝑀:(𝐼 × 𝐼)⟶(Base‘𝑅)) ∧ 𝐼 ∈ Fin) ∧ 𝑓:𝐼⟶(Base‘𝑅)) ∧ (𝑖 ∈ 𝐼 ∧ (𝑓‘𝑖) ≠ (0g‘𝑅))) ∧ (¬ 𝑖 ∈ 𝑦 ∧ 𝑦 ⊆ 𝐼)) ↔ (((((𝑅 ∈ Field ∧ 𝑀:(𝐼 × 𝐼)⟶(Base‘𝑅)) ∧ 𝐼 ∈ Fin) ∧ 𝑓:𝐼⟶(Base‘𝑅)) ∧ (𝑖 ∈ 𝐼 ∧ (𝑓‘𝑖) ≠ (0g‘𝑅))) ∧ (¬ 𝑖 ∈ (𝐼 ∖ {𝑖}) ∧ (𝐼 ∖ {𝑖}) ⊆ 𝐼))))
181		mpteq1 5194	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝑦 = (𝐼 ∖ {𝑖}) → (𝑛 ∈ 𝑦 ↦ (((invr‘𝑅)‘(𝑓‘𝑖))(.r‘𝑅)((𝑓‘𝑛)(.r‘𝑅)(𝑛𝑀𝑘)))) = (𝑛 ∈ (𝐼 ∖ {𝑖}) ↦ (((invr‘𝑅)‘(𝑓‘𝑖))(.r‘𝑅)((𝑓‘𝑛)(.r‘𝑅)(𝑛𝑀𝑘)))))
182	181	oveq2d 7436	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝑦 = (𝐼 ∖ {𝑖}) → (𝑅 Σg (𝑛 ∈ 𝑦 ↦ (((invr‘𝑅)‘(𝑓‘𝑖))(.r‘𝑅)((𝑓‘𝑛)(.r‘𝑅)(𝑛𝑀𝑘))))) = (𝑅 Σg (𝑛 ∈ (𝐼 ∖ {𝑖}) ↦ (((invr‘𝑅)‘(𝑓‘𝑖))(.r‘𝑅)((𝑓‘𝑛)(.r‘𝑅)(𝑛𝑀𝑘))))))
183	182	oveq1d 7435	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑦 = (𝐼 ∖ {𝑖}) → ((𝑅 Σg (𝑛 ∈ 𝑦 ↦ (((invr‘𝑅)‘(𝑓‘𝑖))(.r‘𝑅)((𝑓‘𝑛)(.r‘𝑅)(𝑛𝑀𝑘)))))(+g‘𝑅)(𝑖𝑀𝑘)) = ((𝑅 Σg (𝑛 ∈ (𝐼 ∖ {𝑖}) ↦ (((invr‘𝑅)‘(𝑓‘𝑖))(.r‘𝑅)((𝑓‘𝑛)(.r‘𝑅)(𝑛𝑀𝑘)))))(+g‘𝑅)(𝑖𝑀𝑘)))
184	183	ifeq1d 4502	. . . . . . . . . . . . . . . . 17 ⊢ (𝑦 = (𝐼 ∖ {𝑖}) → if(𝑗 = 𝑖, ((𝑅 Σg (𝑛 ∈ 𝑦 ↦ (((invr‘𝑅)‘(𝑓‘𝑖))(.r‘𝑅)((𝑓‘𝑛)(.r‘𝑅)(𝑛𝑀𝑘)))))(+g‘𝑅)(𝑖𝑀𝑘)), (𝑗𝑀𝑘)) = if(𝑗 = 𝑖, ((𝑅 Σg (𝑛 ∈ (𝐼 ∖ {𝑖}) ↦ (((invr‘𝑅)‘(𝑓‘𝑖))(.r‘𝑅)((𝑓‘𝑛)(.r‘𝑅)(𝑛𝑀𝑘)))))(+g‘𝑅)(𝑖𝑀𝑘)), (𝑗𝑀𝑘)))
185	184	mpoeq3dv 7499	. . . . . . . . . . . . . . . 16 ⊢ (𝑦 = (𝐼 ∖ {𝑖}) → (𝑗 ∈ 𝐼, 𝑘 ∈ 𝐼 ↦ if(𝑗 = 𝑖, ((𝑅 Σg (𝑛 ∈ 𝑦 ↦ (((invr‘𝑅)‘(𝑓‘𝑖))(.r‘𝑅)((𝑓‘𝑛)(.r‘𝑅)(𝑛𝑀𝑘)))))(+g‘𝑅)(𝑖𝑀𝑘)), (𝑗𝑀𝑘))) = (𝑗 ∈ 𝐼, 𝑘 ∈ 𝐼 ↦ if(𝑗 = 𝑖, ((𝑅 Σg (𝑛 ∈ (𝐼 ∖ {𝑖}) ↦ (((invr‘𝑅)‘(𝑓‘𝑖))(.r‘𝑅)((𝑓‘𝑛)(.r‘𝑅)(𝑛𝑀𝑘)))))(+g‘𝑅)(𝑖𝑀𝑘)), (𝑗𝑀𝑘))))
186	185	fveq2d 6889	. . . . . . . . . . . . . . 15 ⊢ (𝑦 = (𝐼 ∖ {𝑖}) → ((𝐼 maDet 𝑅)‘(𝑗 ∈ 𝐼, 𝑘 ∈ 𝐼 ↦ if(𝑗 = 𝑖, ((𝑅 Σg (𝑛 ∈ 𝑦 ↦ (((invr‘𝑅)‘(𝑓‘𝑖))(.r‘𝑅)((𝑓‘𝑛)(.r‘𝑅)(𝑛𝑀𝑘)))))(+g‘𝑅)(𝑖𝑀𝑘)), (𝑗𝑀𝑘)))) = ((𝐼 maDet 𝑅)‘(𝑗 ∈ 𝐼, 𝑘 ∈ 𝐼 ↦ if(𝑗 = 𝑖, ((𝑅 Σg (𝑛 ∈ (𝐼 ∖ {𝑖}) ↦ (((invr‘𝑅)‘(𝑓‘𝑖))(.r‘𝑅)((𝑓‘𝑛)(.r‘𝑅)(𝑛𝑀𝑘)))))(+g‘𝑅)(𝑖𝑀𝑘)), (𝑗𝑀𝑘)))))
187	186	eqeq2d 2772	. . . . . . . . . . . . . 14 ⊢ (𝑦 = (𝐼 ∖ {𝑖}) → (((𝐼 maDet 𝑅)‘𝑀) = ((𝐼 maDet 𝑅)‘(𝑗 ∈ 𝐼, 𝑘 ∈ 𝐼 ↦ if(𝑗 = 𝑖, ((𝑅 Σg (𝑛 ∈ 𝑦 ↦ (((invr‘𝑅)‘(𝑓‘𝑖))(.r‘𝑅)((𝑓‘𝑛)(.r‘𝑅)(𝑛𝑀𝑘)))))(+g‘𝑅)(𝑖𝑀𝑘)), (𝑗𝑀𝑘)))) ↔ ((𝐼 maDet 𝑅)‘𝑀) = ((𝐼 maDet 𝑅)‘(𝑗 ∈ 𝐼, 𝑘 ∈ 𝐼 ↦ if(𝑗 = 𝑖, ((𝑅 Σg (𝑛 ∈ (𝐼 ∖ {𝑖}) ↦ (((invr‘𝑅)‘(𝑓‘𝑖))(.r‘𝑅)((𝑓‘𝑛)(.r‘𝑅)(𝑛𝑀𝑘)))))(+g‘𝑅)(𝑖𝑀𝑘)), (𝑗𝑀𝑘))))))
188	180, 187	imbi12d 347	. . . . . . . . . . . . 13 ⊢ (𝑦 = (𝐼 ∖ {𝑖}) → (((((((𝑅 ∈ Field ∧ 𝑀:(𝐼 × 𝐼)⟶(Base‘𝑅)) ∧ 𝐼 ∈ Fin) ∧ 𝑓:𝐼⟶(Base‘𝑅)) ∧ (𝑖 ∈ 𝐼 ∧ (𝑓‘𝑖) ≠ (0g‘𝑅))) ∧ (¬ 𝑖 ∈ 𝑦 ∧ 𝑦 ⊆ 𝐼)) → ((𝐼 maDet 𝑅)‘𝑀) = ((𝐼 maDet 𝑅)‘(𝑗 ∈ 𝐼, 𝑘 ∈ 𝐼 ↦ if(𝑗 = 𝑖, ((𝑅 Σg (𝑛 ∈ 𝑦 ↦ (((invr‘𝑅)‘(𝑓‘𝑖))(.r‘𝑅)((𝑓‘𝑛)(.r‘𝑅)(𝑛𝑀𝑘)))))(+g‘𝑅)(𝑖𝑀𝑘)), (𝑗𝑀𝑘))))) ↔ ((((((𝑅 ∈ Field ∧ 𝑀:(𝐼 × 𝐼)⟶(Base‘𝑅)) ∧ 𝐼 ∈ Fin) ∧ 𝑓:𝐼⟶(Base‘𝑅)) ∧ (𝑖 ∈ 𝐼 ∧ (𝑓‘𝑖) ≠ (0g‘𝑅))) ∧ (¬ 𝑖 ∈ (𝐼 ∖ {𝑖}) ∧ (𝐼 ∖ {𝑖}) ⊆ 𝐼)) → ((𝐼 maDet 𝑅)‘𝑀) = ((𝐼 maDet 𝑅)‘(𝑗 ∈ 𝐼, 𝑘 ∈ 𝐼 ↦ if(𝑗 = 𝑖, ((𝑅 Σg (𝑛 ∈ (𝐼 ∖ {𝑖}) ↦ (((invr‘𝑅)‘(𝑓‘𝑖))(.r‘𝑅)((𝑓‘𝑛)(.r‘𝑅)(𝑛𝑀𝑘)))))(+g‘𝑅)(𝑖𝑀𝑘)), (𝑗𝑀𝑘)))))))
189		fnov 7551	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑀 Fn (𝐼 × 𝐼) ↔ 𝑀 = (𝑗 ∈ 𝐼, 𝑘 ∈ 𝐼 ↦ (𝑗𝑀𝑘)))
190	59, 189	sylib 221	. . . . . . . . . . . . . . . . 17 ⊢ (𝑀:(𝐼 × 𝐼)⟶(Base‘𝑅) → 𝑀 = (𝑗 ∈ 𝐼, 𝑘 ∈ 𝐼 ↦ (𝑗𝑀𝑘)))
191	190	adantl 487	. . . . . . . . . . . . . . . 16 ⊢ ((𝑅 ∈ Field ∧ 𝑀:(𝐼 × 𝐼)⟶(Base‘𝑅)) → 𝑀 = (𝑗 ∈ 𝐼, 𝑘 ∈ 𝐼 ↦ (𝑗𝑀𝑘)))
192		ringgrp 20464	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑅 ∈ Ring → 𝑅 ∈ Grp)
193	4, 192	syl 18	. . . . . . . . . . . . . . . . 17 ⊢ (𝑅 ∈ Field → 𝑅 ∈ Grp)
194		oveq1 7427	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (𝑖 = 𝑗 → (𝑖𝑀𝑘) = (𝑗𝑀𝑘))
195	194	equcoms 2053	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (𝑗 = 𝑖 → (𝑖𝑀𝑘) = (𝑗𝑀𝑘))
196	195	oveq2d 7436	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝑗 = 𝑖 → ((0g‘𝑅)(+g‘𝑅)(𝑖𝑀𝑘)) = ((0g‘𝑅)(+g‘𝑅)(𝑗𝑀𝑘)))
197		simp1l 1216	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (((𝑅 ∈ Grp ∧ 𝑀:(𝐼 × 𝐼)⟶(Base‘𝑅)) ∧ 𝑗 ∈ 𝐼 ∧ 𝑘 ∈ 𝐼) → 𝑅 ∈ Grp)
198		fovcdm 7591	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ ((𝑀:(𝐼 × 𝐼)⟶(Base‘𝑅) ∧ 𝑗 ∈ 𝐼 ∧ 𝑘 ∈ 𝐼) → (𝑗𝑀𝑘) ∈ (Base‘𝑅))
199	198	3adant1l 1195	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (((𝑅 ∈ Grp ∧ 𝑀:(𝐼 × 𝐼)⟶(Base‘𝑅)) ∧ 𝑗 ∈ 𝐼 ∧ 𝑘 ∈ 𝐼) → (𝑗𝑀𝑘) ∈ (Base‘𝑅))
200		eqid 2761	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (+g‘𝑅) = (+g‘𝑅)
201	10, 200, 100	grplid 19178	. . . . . . . . . . . . . . . . . . . . 21 ⊢ ((𝑅 ∈ Grp ∧ (𝑗𝑀𝑘) ∈ (Base‘𝑅)) → ((0g‘𝑅)(+g‘𝑅)(𝑗𝑀𝑘)) = (𝑗𝑀𝑘))
202	197, 199, 201	syl2anc 596	. . . . . . . . . . . . . . . . . . . 20 ⊢ (((𝑅 ∈ Grp ∧ 𝑀:(𝐼 × 𝐼)⟶(Base‘𝑅)) ∧ 𝑗 ∈ 𝐼 ∧ 𝑘 ∈ 𝐼) → ((0g‘𝑅)(+g‘𝑅)(𝑗𝑀𝑘)) = (𝑗𝑀𝑘))
203	196, 202	sylan9eqr 2818	. . . . . . . . . . . . . . . . . . 19 ⊢ ((((𝑅 ∈ Grp ∧ 𝑀:(𝐼 × 𝐼)⟶(Base‘𝑅)) ∧ 𝑗 ∈ 𝐼 ∧ 𝑘 ∈ 𝐼) ∧ 𝑗 = 𝑖) → ((0g‘𝑅)(+g‘𝑅)(𝑖𝑀𝑘)) = (𝑗𝑀𝑘))
204		eqidd 2762	. . . . . . . . . . . . . . . . . . 19 ⊢ ((((𝑅 ∈ Grp ∧ 𝑀:(𝐼 × 𝐼)⟶(Base‘𝑅)) ∧ 𝑗 ∈ 𝐼 ∧ 𝑘 ∈ 𝐼) ∧ ¬ 𝑗 = 𝑖) → (𝑗𝑀𝑘) = (𝑗𝑀𝑘))
205	203, 204	ifeqda 4519	. . . . . . . . . . . . . . . . . 18 ⊢ (((𝑅 ∈ Grp ∧ 𝑀:(𝐼 × 𝐼)⟶(Base‘𝑅)) ∧ 𝑗 ∈ 𝐼 ∧ 𝑘 ∈ 𝐼) → if(𝑗 = 𝑖, ((0g‘𝑅)(+g‘𝑅)(𝑖𝑀𝑘)), (𝑗𝑀𝑘)) = (𝑗𝑀𝑘))
206	205	mpoeq3dva 7497	. . . . . . . . . . . . . . . . 17 ⊢ ((𝑅 ∈ Grp ∧ 𝑀:(𝐼 × 𝐼)⟶(Base‘𝑅)) → (𝑗 ∈ 𝐼, 𝑘 ∈ 𝐼 ↦ if(𝑗 = 𝑖, ((0g‘𝑅)(+g‘𝑅)(𝑖𝑀𝑘)), (𝑗𝑀𝑘))) = (𝑗 ∈ 𝐼, 𝑘 ∈ 𝐼 ↦ (𝑗𝑀𝑘)))
207	193, 206	sylan 592	. . . . . . . . . . . . . . . 16 ⊢ ((𝑅 ∈ Field ∧ 𝑀:(𝐼 × 𝐼)⟶(Base‘𝑅)) → (𝑗 ∈ 𝐼, 𝑘 ∈ 𝐼 ↦ if(𝑗 = 𝑖, ((0g‘𝑅)(+g‘𝑅)(𝑖𝑀𝑘)), (𝑗𝑀𝑘))) = (𝑗 ∈ 𝐼, 𝑘 ∈ 𝐼 ↦ (𝑗𝑀𝑘)))
208	191, 207	eqtr4d 2799	. . . . . . . . . . . . . . 15 ⊢ ((𝑅 ∈ Field ∧ 𝑀:(𝐼 × 𝐼)⟶(Base‘𝑅)) → 𝑀 = (𝑗 ∈ 𝐼, 𝑘 ∈ 𝐼 ↦ if(𝑗 = 𝑖, ((0g‘𝑅)(+g‘𝑅)(𝑖𝑀𝑘)), (𝑗𝑀𝑘))))
209	208	fveq2d 6889	. . . . . . . . . . . . . 14 ⊢ ((𝑅 ∈ Field ∧ 𝑀:(𝐼 × 𝐼)⟶(Base‘𝑅)) → ((𝐼 maDet 𝑅)‘𝑀) = ((𝐼 maDet 𝑅)‘(𝑗 ∈ 𝐼, 𝑘 ∈ 𝐼 ↦ if(𝑗 = 𝑖, ((0g‘𝑅)(+g‘𝑅)(𝑖𝑀𝑘)), (𝑗𝑀𝑘)))))
210	209	ad4antr 745	. . . . . . . . . . . . 13 ⊢ ((((((𝑅 ∈ Field ∧ 𝑀:(𝐼 × 𝐼)⟶(Base‘𝑅)) ∧ 𝐼 ∈ Fin) ∧ 𝑓:𝐼⟶(Base‘𝑅)) ∧ (𝑖 ∈ 𝐼 ∧ (𝑓‘𝑖) ≠ (0g‘𝑅))) ∧ (¬ 𝑖 ∈ ∅ ∧ ∅ ⊆ 𝐼)) → ((𝐼 maDet 𝑅)‘𝑀) = ((𝐼 maDet 𝑅)‘(𝑗 ∈ 𝐼, 𝑘 ∈ 𝐼 ↦ if(𝑗 = 𝑖, ((0g‘𝑅)(+g‘𝑅)(𝑖𝑀𝑘)), (𝑗𝑀𝑘)))))
211		elun1 4128	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (𝑖 ∈ 𝑥 → 𝑖 ∈ (𝑥 ∪ {𝑧}))
212	211	con3i 155	. . . . . . . . . . . . . . . . . . . 20 ⊢ (¬ 𝑖 ∈ (𝑥 ∪ {𝑧}) → ¬ 𝑖 ∈ 𝑥)
213		ssun1 4124	. . . . . . . . . . . . . . . . . . . . 21 ⊢ 𝑥 ⊆ (𝑥 ∪ {𝑧})
214		sstr 3939	. . . . . . . . . . . . . . . . . . . . 21 ⊢ ((𝑥 ⊆ (𝑥 ∪ {𝑧}) ∧ (𝑥 ∪ {𝑧}) ⊆ 𝐼) → 𝑥 ⊆ 𝐼)
215	213, 214	mpan 703	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((𝑥 ∪ {𝑧}) ⊆ 𝐼 → 𝑥 ⊆ 𝐼)
216	212, 215	anim12i 625	. . . . . . . . . . . . . . . . . . 19 ⊢ ((¬ 𝑖 ∈ (𝑥 ∪ {𝑧}) ∧ (𝑥 ∪ {𝑧}) ⊆ 𝐼) → (¬ 𝑖 ∈ 𝑥 ∧ 𝑥 ⊆ 𝐼))
217	216	anim2i 629	. . . . . . . . . . . . . . . . . 18 ⊢ ((((((𝑅 ∈ Field ∧ 𝑀:(𝐼 × 𝐼)⟶(Base‘𝑅)) ∧ 𝐼 ∈ Fin) ∧ 𝑓:𝐼⟶(Base‘𝑅)) ∧ (𝑖 ∈ 𝐼 ∧ (𝑓‘𝑖) ≠ (0g‘𝑅))) ∧ (¬ 𝑖 ∈ (𝑥 ∪ {𝑧}) ∧ (𝑥 ∪ {𝑧}) ⊆ 𝐼)) → (((((𝑅 ∈ Field ∧ 𝑀:(𝐼 × 𝐼)⟶(Base‘𝑅)) ∧ 𝐼 ∈ Fin) ∧ 𝑓:𝐼⟶(Base‘𝑅)) ∧ (𝑖 ∈ 𝐼 ∧ (𝑓‘𝑖) ≠ (0g‘𝑅))) ∧ (¬ 𝑖 ∈ 𝑥 ∧ 𝑥 ⊆ 𝐼)))
218	217	adantr 486	. . . . . . . . . . . . . . . . 17 ⊢ (((((((𝑅 ∈ Field ∧ 𝑀:(𝐼 × 𝐼)⟶(Base‘𝑅)) ∧ 𝐼 ∈ Fin) ∧ 𝑓:𝐼⟶(Base‘𝑅)) ∧ (𝑖 ∈ 𝐼 ∧ (𝑓‘𝑖) ≠ (0g‘𝑅))) ∧ (¬ 𝑖 ∈ (𝑥 ∪ {𝑧}) ∧ (𝑥 ∪ {𝑧}) ⊆ 𝐼)) ∧ ¬ 𝑧 ∈ 𝑥) → (((((𝑅 ∈ Field ∧ 𝑀:(𝐼 × 𝐼)⟶(Base‘𝑅)) ∧ 𝐼 ∈ Fin) ∧ 𝑓:𝐼⟶(Base‘𝑅)) ∧ (𝑖 ∈ 𝐼 ∧ (𝑓‘𝑖) ≠ (0g‘𝑅))) ∧ (¬ 𝑖 ∈ 𝑥 ∧ 𝑥 ⊆ 𝐼)))
219		velsn 4600	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ (𝑖 ∈ {𝑧} ↔ 𝑖 = 𝑧)
220		elun2 4129	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ (𝑖 ∈ {𝑧} → 𝑖 ∈ (𝑥 ∪ {𝑧}))
221	219, 220	sylbir 238	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (𝑖 = 𝑧 → 𝑖 ∈ (𝑥 ∪ {𝑧}))
222	221	necon3bi 2982	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (¬ 𝑖 ∈ (𝑥 ∪ {𝑧}) → 𝑖 ≠ 𝑧)
223	222	anim1i 627	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((¬ 𝑖 ∈ (𝑥 ∪ {𝑧}) ∧ (𝑥 ∪ {𝑧}) ⊆ 𝐼) → (𝑖 ≠ 𝑧 ∧ (𝑥 ∪ {𝑧}) ⊆ 𝐼))
224		ringcmn 20511	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31 ⊢ (𝑅 ∈ Ring → 𝑅 ∈ CMnd)
225	4, 224	syl 18	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30 ⊢ (𝑅 ∈ Field → 𝑅 ∈ CMnd)
226	225	ad7antr 751	. . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 ⊢ ((((((((𝑅 ∈ Field ∧ 𝑀:(𝐼 × 𝐼)⟶(Base‘𝑅)) ∧ 𝐼 ∈ Fin) ∧ 𝑓:𝐼⟶(Base‘𝑅)) ∧ (𝑖 ∈ 𝐼 ∧ (𝑓‘𝑖) ≠ (0g‘𝑅))) ∧ (𝑖 ≠ 𝑧 ∧ (𝑥 ∪ {𝑧}) ⊆ 𝐼)) ∧ ¬ 𝑧 ∈ 𝑥) ∧ 𝑘 ∈ 𝐼) → 𝑅 ∈ CMnd)
227		simplr 781	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31 ⊢ ((((𝑅 ∈ Field ∧ 𝑀:(𝐼 × 𝐼)⟶(Base‘𝑅)) ∧ 𝐼 ∈ Fin) ∧ 𝑓:𝐼⟶(Base‘𝑅)) → 𝐼 ∈ Fin)
228	215	adantl 487	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31 ⊢ ((𝑖 ≠ 𝑧 ∧ (𝑥 ∪ {𝑧}) ⊆ 𝐼) → 𝑥 ⊆ 𝐼)
229		ssfi 9188	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31 ⊢ ((𝐼 ∈ Fin ∧ 𝑥 ⊆ 𝐼) → 𝑥 ∈ Fin)
230	227, 228, 229	syl2an 608	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30 ⊢ (((((𝑅 ∈ Field ∧ 𝑀:(𝐼 × 𝐼)⟶(Base‘𝑅)) ∧ 𝐼 ∈ Fin) ∧ 𝑓:𝐼⟶(Base‘𝑅)) ∧ (𝑖 ≠ 𝑧 ∧ (𝑥 ∪ {𝑧}) ⊆ 𝐼)) → 𝑥 ∈ Fin)
231	230	ad5ant13 769	. . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 ⊢ ((((((((𝑅 ∈ Field ∧ 𝑀:(𝐼 × 𝐼)⟶(Base‘𝑅)) ∧ 𝐼 ∈ Fin) ∧ 𝑓:𝐼⟶(Base‘𝑅)) ∧ (𝑖 ∈ 𝐼 ∧ (𝑓‘𝑖) ≠ (0g‘𝑅))) ∧ (𝑖 ≠ 𝑧 ∧ (𝑥 ∪ {𝑧}) ⊆ 𝐼)) ∧ ¬ 𝑧 ∈ 𝑥) ∧ 𝑘 ∈ 𝐼) → 𝑥 ∈ Fin)
232	215	sselda 3931	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33 ⊢ (((𝑥 ∪ {𝑧}) ⊆ 𝐼 ∧ 𝑛 ∈ 𝑥) → 𝑛 ∈ 𝐼)
233	232	adantll 727	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32 ⊢ (((𝑖 ≠ 𝑧 ∧ (𝑥 ∪ {𝑧}) ⊆ 𝐼) ∧ 𝑛 ∈ 𝑥) → 𝑛 ∈ 𝐼)
234	233	ad4ant24 767	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31 ⊢ ((((((((𝑅 ∈ Field ∧ 𝑀:(𝐼 × 𝐼)⟶(Base‘𝑅)) ∧ 𝐼 ∈ Fin) ∧ 𝑓:𝐼⟶(Base‘𝑅)) ∧ (𝑖 ∈ 𝐼 ∧ (𝑓‘𝑖) ≠ (0g‘𝑅))) ∧ (𝑖 ≠ 𝑧 ∧ (𝑥 ∪ {𝑧}) ⊆ 𝐼)) ∧ 𝑘 ∈ 𝐼) ∧ 𝑛 ∈ 𝑥) → 𝑛 ∈ 𝐼)
235	4	ad6antr 749	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33 ⊢ (((((((𝑅 ∈ Field ∧ 𝑀:(𝐼 × 𝐼)⟶(Base‘𝑅)) ∧ 𝐼 ∈ Fin) ∧ 𝑓:𝐼⟶(Base‘𝑅)) ∧ (𝑖 ∈ 𝐼 ∧ (𝑓‘𝑖) ≠ (0g‘𝑅))) ∧ 𝑘 ∈ 𝐼) ∧ 𝑛 ∈ 𝐼) → 𝑅 ∈ Ring)
236	2	ad2antrr 739	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35 ⊢ (((𝑅 ∈ Field ∧ 𝑀:(𝐼 × 𝐼)⟶(Base‘𝑅)) ∧ 𝐼 ∈ Fin) → 𝑅 ∈ DivRing)
237		ffvelcdm 7081	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 39 ⊢ ((𝑓:𝐼⟶(Base‘𝑅) ∧ 𝑖 ∈ 𝐼) → (𝑓‘𝑖) ∈ (Base‘𝑅))
238	237	anim2i 629	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 38 ⊢ ((𝑅 ∈ DivRing ∧ (𝑓:𝐼⟶(Base‘𝑅) ∧ 𝑖 ∈ 𝐼)) → (𝑅 ∈ DivRing ∧ (𝑓‘𝑖) ∈ (Base‘𝑅)))
239	238	anassrs 473	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 37 ⊢ (((𝑅 ∈ DivRing ∧ 𝑓:𝐼⟶(Base‘𝑅)) ∧ 𝑖 ∈ 𝐼) → (𝑅 ∈ DivRing ∧ (𝑓‘𝑖) ∈ (Base‘𝑅)))
240		eqid 2761	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 39 ⊢ (invr‘𝑅) = (invr‘𝑅)
241	10, 100, 240	drnginvrcl 21011	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 38 ⊢ ((𝑅 ∈ DivRing ∧ (𝑓‘𝑖) ∈ (Base‘𝑅) ∧ (𝑓‘𝑖) ≠ (0g‘𝑅)) → ((invr‘𝑅)‘(𝑓‘𝑖)) ∈ (Base‘𝑅))
242	241	3expa 1136	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 37 ⊢ (((𝑅 ∈ DivRing ∧ (𝑓‘𝑖) ∈ (Base‘𝑅)) ∧ (𝑓‘𝑖) ≠ (0g‘𝑅)) → ((invr‘𝑅)‘(𝑓‘𝑖)) ∈ (Base‘𝑅))
243	239, 242	sylan 592	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 36 ⊢ ((((𝑅 ∈ DivRing ∧ 𝑓:𝐼⟶(Base‘𝑅)) ∧ 𝑖 ∈ 𝐼) ∧ (𝑓‘𝑖) ≠ (0g‘𝑅)) → ((invr‘𝑅)‘(𝑓‘𝑖)) ∈ (Base‘𝑅))
244	243	anasss 472	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35 ⊢ (((𝑅 ∈ DivRing ∧ 𝑓:𝐼⟶(Base‘𝑅)) ∧ (𝑖 ∈ 𝐼 ∧ (𝑓‘𝑖) ≠ (0g‘𝑅))) → ((invr‘𝑅)‘(𝑓‘𝑖)) ∈ (Base‘𝑅))
245	236, 244	sylanl1 693	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34 ⊢ (((((𝑅 ∈ Field ∧ 𝑀:(𝐼 × 𝐼)⟶(Base‘𝑅)) ∧ 𝐼 ∈ Fin) ∧ 𝑓:𝐼⟶(Base‘𝑅)) ∧ (𝑖 ∈ 𝐼 ∧ (𝑓‘𝑖) ≠ (0g‘𝑅))) → ((invr‘𝑅)‘(𝑓‘𝑖)) ∈ (Base‘𝑅))
246	245	ad2antrr 739	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33 ⊢ (((((((𝑅 ∈ Field ∧ 𝑀:(𝐼 × 𝐼)⟶(Base‘𝑅)) ∧ 𝐼 ∈ Fin) ∧ 𝑓:𝐼⟶(Base‘𝑅)) ∧ (𝑖 ∈ 𝐼 ∧ (𝑓‘𝑖) ≠ (0g‘𝑅))) ∧ 𝑘 ∈ 𝐼) ∧ 𝑛 ∈ 𝐼) → ((invr‘𝑅)‘(𝑓‘𝑖)) ∈ (Base‘𝑅))
247	43	ad5ant25 774	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34 ⊢ (((((((𝑅 ∈ Field ∧ 𝑀:(𝐼 × 𝐼)⟶(Base‘𝑅)) ∧ 𝐼 ∈ Fin) ∧ 𝑓:𝐼⟶(Base‘𝑅)) ∧ (𝑖 ∈ 𝐼 ∧ (𝑓‘𝑖) ≠ (0g‘𝑅))) ∧ 𝑘 ∈ 𝐼) ∧ 𝑛 ∈ 𝐼) → (𝑓‘𝑛) ∈ (Base‘𝑅))
248		simp-4r 796	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35 ⊢ (((((𝑅 ∈ Field ∧ 𝑀:(𝐼 × 𝐼)⟶(Base‘𝑅)) ∧ 𝐼 ∈ Fin) ∧ 𝑓:𝐼⟶(Base‘𝑅)) ∧ (𝑖 ∈ 𝐼 ∧ (𝑓‘𝑖) ≠ (0g‘𝑅))) → 𝑀:(𝐼 × 𝐼)⟶(Base‘𝑅))
249	76	3expa 1136	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 36 ⊢ (((𝑀:(𝐼 × 𝐼)⟶(Base‘𝑅) ∧ 𝑛 ∈ 𝐼) ∧ 𝑘 ∈ 𝐼) → (𝑛𝑀𝑘) ∈ (Base‘𝑅))
250	249	an32s 665	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35 ⊢ (((𝑀:(𝐼 × 𝐼)⟶(Base‘𝑅) ∧ 𝑘 ∈ 𝐼) ∧ 𝑛 ∈ 𝐼) → (𝑛𝑀𝑘) ∈ (Base‘𝑅))
251	248, 250	sylanl1 693	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34 ⊢ (((((((𝑅 ∈ Field ∧ 𝑀:(𝐼 × 𝐼)⟶(Base‘𝑅)) ∧ 𝐼 ∈ Fin) ∧ 𝑓:𝐼⟶(Base‘𝑅)) ∧ (𝑖 ∈ 𝐼 ∧ (𝑓‘𝑖) ≠ (0g‘𝑅))) ∧ 𝑘 ∈ 𝐼) ∧ 𝑛 ∈ 𝐼) → (𝑛𝑀𝑘) ∈ (Base‘𝑅))
252	235, 247, 251, 78	syl3anc 1398	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33 ⊢ (((((((𝑅 ∈ Field ∧ 𝑀:(𝐼 × 𝐼)⟶(Base‘𝑅)) ∧ 𝐼 ∈ Fin) ∧ 𝑓:𝐼⟶(Base‘𝑅)) ∧ (𝑖 ∈ 𝐼 ∧ (𝑓‘𝑖) ≠ (0g‘𝑅))) ∧ 𝑘 ∈ 𝐼) ∧ 𝑛 ∈ 𝐼) → ((𝑓‘𝑛)(.r‘𝑅)(𝑛𝑀𝑘)) ∈ (Base‘𝑅))
253	10, 47	ringcl 20477	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33 ⊢ ((𝑅 ∈ Ring ∧ ((invr‘𝑅)‘(𝑓‘𝑖)) ∈ (Base‘𝑅) ∧ ((𝑓‘𝑛)(.r‘𝑅)(𝑛𝑀𝑘)) ∈ (Base‘𝑅)) → (((invr‘𝑅)‘(𝑓‘𝑖))(.r‘𝑅)((𝑓‘𝑛)(.r‘𝑅)(𝑛𝑀𝑘))) ∈ (Base‘𝑅))
254	235, 246, 252, 253	syl3anc 1398	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32 ⊢ (((((((𝑅 ∈ Field ∧ 𝑀:(𝐼 × 𝐼)⟶(Base‘𝑅)) ∧ 𝐼 ∈ Fin) ∧ 𝑓:𝐼⟶(Base‘𝑅)) ∧ (𝑖 ∈ 𝐼 ∧ (𝑓‘𝑖) ≠ (0g‘𝑅))) ∧ 𝑘 ∈ 𝐼) ∧ 𝑛 ∈ 𝐼) → (((invr‘𝑅)‘(𝑓‘𝑖))(.r‘𝑅)((𝑓‘𝑛)(.r‘𝑅)(𝑛𝑀𝑘))) ∈ (Base‘𝑅))
255	254	adantllr 732	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31 ⊢ ((((((((𝑅 ∈ Field ∧ 𝑀:(𝐼 × 𝐼)⟶(Base‘𝑅)) ∧ 𝐼 ∈ Fin) ∧ 𝑓:𝐼⟶(Base‘𝑅)) ∧ (𝑖 ∈ 𝐼 ∧ (𝑓‘𝑖) ≠ (0g‘𝑅))) ∧ (𝑖 ≠ 𝑧 ∧ (𝑥 ∪ {𝑧}) ⊆ 𝐼)) ∧ 𝑘 ∈ 𝐼) ∧ 𝑛 ∈ 𝐼) → (((invr‘𝑅)‘(𝑓‘𝑖))(.r‘𝑅)((𝑓‘𝑛)(.r‘𝑅)(𝑛𝑀𝑘))) ∈ (Base‘𝑅))
256	234, 255	syldan 603	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30 ⊢ ((((((((𝑅 ∈ Field ∧ 𝑀:(𝐼 × 𝐼)⟶(Base‘𝑅)) ∧ 𝐼 ∈ Fin) ∧ 𝑓:𝐼⟶(Base‘𝑅)) ∧ (𝑖 ∈ 𝐼 ∧ (𝑓‘𝑖) ≠ (0g‘𝑅))) ∧ (𝑖 ≠ 𝑧 ∧ (𝑥 ∪ {𝑧}) ⊆ 𝐼)) ∧ 𝑘 ∈ 𝐼) ∧ 𝑛 ∈ 𝑥) → (((invr‘𝑅)‘(𝑓‘𝑖))(.r‘𝑅)((𝑓‘𝑛)(.r‘𝑅)(𝑛𝑀𝑘))) ∈ (Base‘𝑅))
257	256	adantllr 732	. . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 ⊢ (((((((((𝑅 ∈ Field ∧ 𝑀:(𝐼 × 𝐼)⟶(Base‘𝑅)) ∧ 𝐼 ∈ Fin) ∧ 𝑓:𝐼⟶(Base‘𝑅)) ∧ (𝑖 ∈ 𝐼 ∧ (𝑓‘𝑖) ≠ (0g‘𝑅))) ∧ (𝑖 ≠ 𝑧 ∧ (𝑥 ∪ {𝑧}) ⊆ 𝐼)) ∧ ¬ 𝑧 ∈ 𝑥) ∧ 𝑘 ∈ 𝐼) ∧ 𝑛 ∈ 𝑥) → (((invr‘𝑅)‘(𝑓‘𝑖))(.r‘𝑅)((𝑓‘𝑛)(.r‘𝑅)(𝑛𝑀𝑘))) ∈ (Base‘𝑅))
258		vex 3455	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30 ⊢ 𝑧 ∈ V
259	258	a1i 11	. . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 ⊢ ((((((((𝑅 ∈ Field ∧ 𝑀:(𝐼 × 𝐼)⟶(Base‘𝑅)) ∧ 𝐼 ∈ Fin) ∧ 𝑓:𝐼⟶(Base‘𝑅)) ∧ (𝑖 ∈ 𝐼 ∧ (𝑓‘𝑖) ≠ (0g‘𝑅))) ∧ (𝑖 ≠ 𝑧 ∧ (𝑥 ∪ {𝑧}) ⊆ 𝐼)) ∧ ¬ 𝑧 ∈ 𝑥) ∧ 𝑘 ∈ 𝐼) → 𝑧 ∈ V)
260		simplr 781	. . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 ⊢ ((((((((𝑅 ∈ Field ∧ 𝑀:(𝐼 × 𝐼)⟶(Base‘𝑅)) ∧ 𝐼 ∈ Fin) ∧ 𝑓:𝐼⟶(Base‘𝑅)) ∧ (𝑖 ∈ 𝐼 ∧ (𝑓‘𝑖) ≠ (0g‘𝑅))) ∧ (𝑖 ≠ 𝑧 ∧ (𝑥 ∪ {𝑧}) ⊆ 𝐼)) ∧ ¬ 𝑧 ∈ 𝑥) ∧ 𝑘 ∈ 𝐼) → ¬ 𝑧 ∈ 𝑥)
261		ssun2 4125	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34 ⊢ {𝑧} ⊆ (𝑥 ∪ {𝑧})
262		sstr 3939	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34 ⊢ (({𝑧} ⊆ (𝑥 ∪ {𝑧}) ∧ (𝑥 ∪ {𝑧}) ⊆ 𝐼) → {𝑧} ⊆ 𝐼)
263	261, 262	mpan 703	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33 ⊢ ((𝑥 ∪ {𝑧}) ⊆ 𝐼 → {𝑧} ⊆ 𝐼)
264	258	snss 4745	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33 ⊢ (𝑧 ∈ 𝐼 ↔ {𝑧} ⊆ 𝐼)
265	263, 264	sylibr 237	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32 ⊢ ((𝑥 ∪ {𝑧}) ⊆ 𝐼 → 𝑧 ∈ 𝐼)
266	265	adantl 487	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31 ⊢ ((𝑖 ≠ 𝑧 ∧ (𝑥 ∪ {𝑧}) ⊆ 𝐼) → 𝑧 ∈ 𝐼)
267	4	ad6antr 749	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32 ⊢ (((((((𝑅 ∈ Field ∧ 𝑀:(𝐼 × 𝐼)⟶(Base‘𝑅)) ∧ 𝐼 ∈ Fin) ∧ 𝑓:𝐼⟶(Base‘𝑅)) ∧ (𝑖 ∈ 𝐼 ∧ (𝑓‘𝑖) ≠ (0g‘𝑅))) ∧ 𝑧 ∈ 𝐼) ∧ 𝑘 ∈ 𝐼) → 𝑅 ∈ Ring)
268	4	ad5antr 747	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34 ⊢ ((((((𝑅 ∈ Field ∧ 𝑀:(𝐼 × 𝐼)⟶(Base‘𝑅)) ∧ 𝐼 ∈ Fin) ∧ 𝑓:𝐼⟶(Base‘𝑅)) ∧ (𝑖 ∈ 𝐼 ∧ (𝑓‘𝑖) ≠ (0g‘𝑅))) ∧ 𝑧 ∈ 𝐼) → 𝑅 ∈ Ring)
269	245	adantr 486	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34 ⊢ ((((((𝑅 ∈ Field ∧ 𝑀:(𝐼 × 𝐼)⟶(Base‘𝑅)) ∧ 𝐼 ∈ Fin) ∧ 𝑓:𝐼⟶(Base‘𝑅)) ∧ (𝑖 ∈ 𝐼 ∧ (𝑓‘𝑖) ≠ (0g‘𝑅))) ∧ 𝑧 ∈ 𝐼) → ((invr‘𝑅)‘(𝑓‘𝑖)) ∈ (Base‘𝑅))
270		ffvelcdm 7081	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35 ⊢ ((𝑓:𝐼⟶(Base‘𝑅) ∧ 𝑧 ∈ 𝐼) → (𝑓‘𝑧) ∈ (Base‘𝑅))
271	270	ad4ant24 767	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34 ⊢ ((((((𝑅 ∈ Field ∧ 𝑀:(𝐼 × 𝐼)⟶(Base‘𝑅)) ∧ 𝐼 ∈ Fin) ∧ 𝑓:𝐼⟶(Base‘𝑅)) ∧ (𝑖 ∈ 𝐼 ∧ (𝑓‘𝑖) ≠ (0g‘𝑅))) ∧ 𝑧 ∈ 𝐼) → (𝑓‘𝑧) ∈ (Base‘𝑅))
272	10, 47	ringcl 20477	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34 ⊢ ((𝑅 ∈ Ring ∧ ((invr‘𝑅)‘(𝑓‘𝑖)) ∈ (Base‘𝑅) ∧ (𝑓‘𝑧) ∈ (Base‘𝑅)) → (((invr‘𝑅)‘(𝑓‘𝑖))(.r‘𝑅)(𝑓‘𝑧)) ∈ (Base‘𝑅))
273	268, 269, 271, 272	syl3anc 1398	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33 ⊢ ((((((𝑅 ∈ Field ∧ 𝑀:(𝐼 × 𝐼)⟶(Base‘𝑅)) ∧ 𝐼 ∈ Fin) ∧ 𝑓:𝐼⟶(Base‘𝑅)) ∧ (𝑖 ∈ 𝐼 ∧ (𝑓‘𝑖) ≠ (0g‘𝑅))) ∧ 𝑧 ∈ 𝐼) → (((invr‘𝑅)‘(𝑓‘𝑖))(.r‘𝑅)(𝑓‘𝑧)) ∈ (Base‘𝑅))
274	273	adantr 486	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32 ⊢ (((((((𝑅 ∈ Field ∧ 𝑀:(𝐼 × 𝐼)⟶(Base‘𝑅)) ∧ 𝐼 ∈ Fin) ∧ 𝑓:𝐼⟶(Base‘𝑅)) ∧ (𝑖 ∈ 𝐼 ∧ (𝑓‘𝑖) ≠ (0g‘𝑅))) ∧ 𝑧 ∈ 𝐼) ∧ 𝑘 ∈ 𝐼) → (((invr‘𝑅)‘(𝑓‘𝑖))(.r‘𝑅)(𝑓‘𝑧)) ∈ (Base‘𝑅))
275		fovcdm 7591	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34 ⊢ ((𝑀:(𝐼 × 𝐼)⟶(Base‘𝑅) ∧ 𝑧 ∈ 𝐼 ∧ 𝑘 ∈ 𝐼) → (𝑧𝑀𝑘) ∈ (Base‘𝑅))
276	275	3expa 1136	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33 ⊢ (((𝑀:(𝐼 × 𝐼)⟶(Base‘𝑅) ∧ 𝑧 ∈ 𝐼) ∧ 𝑘 ∈ 𝐼) → (𝑧𝑀𝑘) ∈ (Base‘𝑅))
277	248, 276	sylanl1 693	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32 ⊢ (((((((𝑅 ∈ Field ∧ 𝑀:(𝐼 × 𝐼)⟶(Base‘𝑅)) ∧ 𝐼 ∈ Fin) ∧ 𝑓:𝐼⟶(Base‘𝑅)) ∧ (𝑖 ∈ 𝐼 ∧ (𝑓‘𝑖) ≠ (0g‘𝑅))) ∧ 𝑧 ∈ 𝐼) ∧ 𝑘 ∈ 𝐼) → (𝑧𝑀𝑘) ∈ (Base‘𝑅))
278	10, 47	ringcl 20477	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32 ⊢ ((𝑅 ∈ Ring ∧ (((invr‘𝑅)‘(𝑓‘𝑖))(.r‘𝑅)(𝑓‘𝑧)) ∈ (Base‘𝑅) ∧ (𝑧𝑀𝑘) ∈ (Base‘𝑅)) → ((((invr‘𝑅)‘(𝑓‘𝑖))(.r‘𝑅)(𝑓‘𝑧))(.r‘𝑅)(𝑧𝑀𝑘)) ∈ (Base‘𝑅))
279	267, 274, 277, 278	syl3anc 1398	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31 ⊢ (((((((𝑅 ∈ Field ∧ 𝑀:(𝐼 × 𝐼)⟶(Base‘𝑅)) ∧ 𝐼 ∈ Fin) ∧ 𝑓:𝐼⟶(Base‘𝑅)) ∧ (𝑖 ∈ 𝐼 ∧ (𝑓‘𝑖) ≠ (0g‘𝑅))) ∧ 𝑧 ∈ 𝐼) ∧ 𝑘 ∈ 𝐼) → ((((invr‘𝑅)‘(𝑓‘𝑖))(.r‘𝑅)(𝑓‘𝑧))(.r‘𝑅)(𝑧𝑀𝑘)) ∈ (Base‘𝑅))
280	266, 279	sylanl2 694	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30 ⊢ (((((((𝑅 ∈ Field ∧ 𝑀:(𝐼 × 𝐼)⟶(Base‘𝑅)) ∧ 𝐼 ∈ Fin) ∧ 𝑓:𝐼⟶(Base‘𝑅)) ∧ (𝑖 ∈ 𝐼 ∧ (𝑓‘𝑖) ≠ (0g‘𝑅))) ∧ (𝑖 ≠ 𝑧 ∧ (𝑥 ∪ {𝑧}) ⊆ 𝐼)) ∧ 𝑘 ∈ 𝐼) → ((((invr‘𝑅)‘(𝑓‘𝑖))(.r‘𝑅)(𝑓‘𝑧))(.r‘𝑅)(𝑧𝑀𝑘)) ∈ (Base‘𝑅))
281	280	adantlr 728	. . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 ⊢ ((((((((𝑅 ∈ Field ∧ 𝑀:(𝐼 × 𝐼)⟶(Base‘𝑅)) ∧ 𝐼 ∈ Fin) ∧ 𝑓:𝐼⟶(Base‘𝑅)) ∧ (𝑖 ∈ 𝐼 ∧ (𝑓‘𝑖) ≠ (0g‘𝑅))) ∧ (𝑖 ≠ 𝑧 ∧ (𝑥 ∪ {𝑧}) ⊆ 𝐼)) ∧ ¬ 𝑧 ∈ 𝑥) ∧ 𝑘 ∈ 𝐼) → ((((invr‘𝑅)‘(𝑓‘𝑖))(.r‘𝑅)(𝑓‘𝑧))(.r‘𝑅)(𝑧𝑀𝑘)) ∈ (Base‘𝑅))
282		fveq2 6885	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33 ⊢ (𝑛 = 𝑧 → (𝑓‘𝑛) = (𝑓‘𝑧))
283		oveq1 7427	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33 ⊢ (𝑛 = 𝑧 → (𝑛𝑀𝑘) = (𝑧𝑀𝑘))
284	282, 283	oveq12d 7438	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32 ⊢ (𝑛 = 𝑧 → ((𝑓‘𝑛)(.r‘𝑅)(𝑛𝑀𝑘)) = ((𝑓‘𝑧)(.r‘𝑅)(𝑧𝑀𝑘)))
285	284	oveq2d 7436	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31 ⊢ (𝑛 = 𝑧 → (((invr‘𝑅)‘(𝑓‘𝑖))(.r‘𝑅)((𝑓‘𝑛)(.r‘𝑅)(𝑛𝑀𝑘))) = (((invr‘𝑅)‘(𝑓‘𝑖))(.r‘𝑅)((𝑓‘𝑧)(.r‘𝑅)(𝑧𝑀𝑘))))
286	245	ad2antrr 739	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34 ⊢ (((((((𝑅 ∈ Field ∧ 𝑀:(𝐼 × 𝐼)⟶(Base‘𝑅)) ∧ 𝐼 ∈ Fin) ∧ 𝑓:𝐼⟶(Base‘𝑅)) ∧ (𝑖 ∈ 𝐼 ∧ (𝑓‘𝑖) ≠ (0g‘𝑅))) ∧ 𝑧 ∈ 𝐼) ∧ 𝑘 ∈ 𝐼) → ((invr‘𝑅)‘(𝑓‘𝑖)) ∈ (Base‘𝑅))
287	270	ad5ant24 773	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34 ⊢ (((((((𝑅 ∈ Field ∧ 𝑀:(𝐼 × 𝐼)⟶(Base‘𝑅)) ∧ 𝐼 ∈ Fin) ∧ 𝑓:𝐼⟶(Base‘𝑅)) ∧ (𝑖 ∈ 𝐼 ∧ (𝑓‘𝑖) ≠ (0g‘𝑅))) ∧ 𝑧 ∈ 𝐼) ∧ 𝑘 ∈ 𝐼) → (𝑓‘𝑧) ∈ (Base‘𝑅))
288	10, 47	ringass 20480	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34 ⊢ ((𝑅 ∈ Ring ∧ (((invr‘𝑅)‘(𝑓‘𝑖)) ∈ (Base‘𝑅) ∧ (𝑓‘𝑧) ∈ (Base‘𝑅) ∧ (𝑧𝑀𝑘) ∈ (Base‘𝑅))) → ((((invr‘𝑅)‘(𝑓‘𝑖))(.r‘𝑅)(𝑓‘𝑧))(.r‘𝑅)(𝑧𝑀𝑘)) = (((invr‘𝑅)‘(𝑓‘𝑖))(.r‘𝑅)((𝑓‘𝑧)(.r‘𝑅)(𝑧𝑀𝑘))))
289	267, 286, 287, 277, 288	syl13anc 1399	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33 ⊢ (((((((𝑅 ∈ Field ∧ 𝑀:(𝐼 × 𝐼)⟶(Base‘𝑅)) ∧ 𝐼 ∈ Fin) ∧ 𝑓:𝐼⟶(Base‘𝑅)) ∧ (𝑖 ∈ 𝐼 ∧ (𝑓‘𝑖) ≠ (0g‘𝑅))) ∧ 𝑧 ∈ 𝐼) ∧ 𝑘 ∈ 𝐼) → ((((invr‘𝑅)‘(𝑓‘𝑖))(.r‘𝑅)(𝑓‘𝑧))(.r‘𝑅)(𝑧𝑀𝑘)) = (((invr‘𝑅)‘(𝑓‘𝑖))(.r‘𝑅)((𝑓‘𝑧)(.r‘𝑅)(𝑧𝑀𝑘))))
290	289	eqcomd 2767	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32 ⊢ (((((((𝑅 ∈ Field ∧ 𝑀:(𝐼 × 𝐼)⟶(Base‘𝑅)) ∧ 𝐼 ∈ Fin) ∧ 𝑓:𝐼⟶(Base‘𝑅)) ∧ (𝑖 ∈ 𝐼 ∧ (𝑓‘𝑖) ≠ (0g‘𝑅))) ∧ 𝑧 ∈ 𝐼) ∧ 𝑘 ∈ 𝐼) → (((invr‘𝑅)‘(𝑓‘𝑖))(.r‘𝑅)((𝑓‘𝑧)(.r‘𝑅)(𝑧𝑀𝑘))) = ((((invr‘𝑅)‘(𝑓‘𝑖))(.r‘𝑅)(𝑓‘𝑧))(.r‘𝑅)(𝑧𝑀𝑘)))
291	266, 290	sylanl2 694	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31 ⊢ (((((((𝑅 ∈ Field ∧ 𝑀:(𝐼 × 𝐼)⟶(Base‘𝑅)) ∧ 𝐼 ∈ Fin) ∧ 𝑓:𝐼⟶(Base‘𝑅)) ∧ (𝑖 ∈ 𝐼 ∧ (𝑓‘𝑖) ≠ (0g‘𝑅))) ∧ (𝑖 ≠ 𝑧 ∧ (𝑥 ∪ {𝑧}) ⊆ 𝐼)) ∧ 𝑘 ∈ 𝐼) → (((invr‘𝑅)‘(𝑓‘𝑖))(.r‘𝑅)((𝑓‘𝑧)(.r‘𝑅)(𝑧𝑀𝑘))) = ((((invr‘𝑅)‘(𝑓‘𝑖))(.r‘𝑅)(𝑓‘𝑧))(.r‘𝑅)(𝑧𝑀𝑘)))
292	285, 291	sylan9eqr 2818	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30 ⊢ ((((((((𝑅 ∈ Field ∧ 𝑀:(𝐼 × 𝐼)⟶(Base‘𝑅)) ∧ 𝐼 ∈ Fin) ∧ 𝑓:𝐼⟶(Base‘𝑅)) ∧ (𝑖 ∈ 𝐼 ∧ (𝑓‘𝑖) ≠ (0g‘𝑅))) ∧ (𝑖 ≠ 𝑧 ∧ (𝑥 ∪ {𝑧}) ⊆ 𝐼)) ∧ 𝑘 ∈ 𝐼) ∧ 𝑛 = 𝑧) → (((invr‘𝑅)‘(𝑓‘𝑖))(.r‘𝑅)((𝑓‘𝑛)(.r‘𝑅)(𝑛𝑀𝑘))) = ((((invr‘𝑅)‘(𝑓‘𝑖))(.r‘𝑅)(𝑓‘𝑧))(.r‘𝑅)(𝑧𝑀𝑘)))
293	292	adantllr 732	. . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 ⊢ (((((((((𝑅 ∈ Field ∧ 𝑀:(𝐼 × 𝐼)⟶(Base‘𝑅)) ∧ 𝐼 ∈ Fin) ∧ 𝑓:𝐼⟶(Base‘𝑅)) ∧ (𝑖 ∈ 𝐼 ∧ (𝑓‘𝑖) ≠ (0g‘𝑅))) ∧ (𝑖 ≠ 𝑧 ∧ (𝑥 ∪ {𝑧}) ⊆ 𝐼)) ∧ ¬ 𝑧 ∈ 𝑥) ∧ 𝑘 ∈ 𝐼) ∧ 𝑛 = 𝑧) → (((invr‘𝑅)‘(𝑓‘𝑖))(.r‘𝑅)((𝑓‘𝑛)(.r‘𝑅)(𝑛𝑀𝑘))) = ((((invr‘𝑅)‘(𝑓‘𝑖))(.r‘𝑅)(𝑓‘𝑧))(.r‘𝑅)(𝑧𝑀𝑘)))
294	10, 200, 226, 231, 257, 259, 260, 281, 293	gsumunsnd 20172	. . . . . . . . . . . . . . . . . . . . . . . . . . . 28 ⊢ ((((((((𝑅 ∈ Field ∧ 𝑀:(𝐼 × 𝐼)⟶(Base‘𝑅)) ∧ 𝐼 ∈ Fin) ∧ 𝑓:𝐼⟶(Base‘𝑅)) ∧ (𝑖 ∈ 𝐼 ∧ (𝑓‘𝑖) ≠ (0g‘𝑅))) ∧ (𝑖 ≠ 𝑧 ∧ (𝑥 ∪ {𝑧}) ⊆ 𝐼)) ∧ ¬ 𝑧 ∈ 𝑥) ∧ 𝑘 ∈ 𝐼) → (𝑅 Σg (𝑛 ∈ (𝑥 ∪ {𝑧}) ↦ (((invr‘𝑅)‘(𝑓‘𝑖))(.r‘𝑅)((𝑓‘𝑛)(.r‘𝑅)(𝑛𝑀𝑘))))) = ((𝑅 Σg (𝑛 ∈ 𝑥 ↦ (((invr‘𝑅)‘(𝑓‘𝑖))(.r‘𝑅)((𝑓‘𝑛)(.r‘𝑅)(𝑛𝑀𝑘)))))(+g‘𝑅)((((invr‘𝑅)‘(𝑓‘𝑖))(.r‘𝑅)(𝑓‘𝑧))(.r‘𝑅)(𝑧𝑀𝑘))))
295	294	oveq1d 7435	. . . . . . . . . . . . . . . . . . . . . . . . . . 27 ⊢ ((((((((𝑅 ∈ Field ∧ 𝑀:(𝐼 × 𝐼)⟶(Base‘𝑅)) ∧ 𝐼 ∈ Fin) ∧ 𝑓:𝐼⟶(Base‘𝑅)) ∧ (𝑖 ∈ 𝐼 ∧ (𝑓‘𝑖) ≠ (0g‘𝑅))) ∧ (𝑖 ≠ 𝑧 ∧ (𝑥 ∪ {𝑧}) ⊆ 𝐼)) ∧ ¬ 𝑧 ∈ 𝑥) ∧ 𝑘 ∈ 𝐼) → ((𝑅 Σg (𝑛 ∈ (𝑥 ∪ {𝑧}) ↦ (((invr‘𝑅)‘(𝑓‘𝑖))(.r‘𝑅)((𝑓‘𝑛)(.r‘𝑅)(𝑛𝑀𝑘)))))(+g‘𝑅)(𝑖𝑀𝑘)) = (((𝑅 Σg (𝑛 ∈ 𝑥 ↦ (((invr‘𝑅)‘(𝑓‘𝑖))(.r‘𝑅)((𝑓‘𝑛)(.r‘𝑅)(𝑛𝑀𝑘)))))(+g‘𝑅)((((invr‘𝑅)‘(𝑓‘𝑖))(.r‘𝑅)(𝑓‘𝑧))(.r‘𝑅)(𝑧𝑀𝑘)))(+g‘𝑅)(𝑖𝑀𝑘)))
296		ringabl 20510	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32 ⊢ (𝑅 ∈ Ring → 𝑅 ∈ Abel)
297	4, 296	syl 18	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31 ⊢ (𝑅 ∈ Field → 𝑅 ∈ Abel)
298	297	ad6antr 749	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30 ⊢ (((((((𝑅 ∈ Field ∧ 𝑀:(𝐼 × 𝐼)⟶(Base‘𝑅)) ∧ 𝐼 ∈ Fin) ∧ 𝑓:𝐼⟶(Base‘𝑅)) ∧ (𝑖 ∈ 𝐼 ∧ (𝑓‘𝑖) ≠ (0g‘𝑅))) ∧ (𝑥 ∪ {𝑧}) ⊆ 𝐼) ∧ 𝑘 ∈ 𝐼) → 𝑅 ∈ Abel)
299	225	ad6antr 749	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32 ⊢ (((((((𝑅 ∈ Field ∧ 𝑀:(𝐼 × 𝐼)⟶(Base‘𝑅)) ∧ 𝐼 ∈ Fin) ∧ 𝑓:𝐼⟶(Base‘𝑅)) ∧ (𝑖 ∈ 𝐼 ∧ (𝑓‘𝑖) ≠ (0g‘𝑅))) ∧ 𝑥 ⊆ 𝐼) ∧ 𝑘 ∈ 𝐼) → 𝑅 ∈ CMnd)
300		vex 3455	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33 ⊢ 𝑥 ∈ V
301	300	a1i 11	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32 ⊢ (((((((𝑅 ∈ Field ∧ 𝑀:(𝐼 × 𝐼)⟶(Base‘𝑅)) ∧ 𝐼 ∈ Fin) ∧ 𝑓:𝐼⟶(Base‘𝑅)) ∧ (𝑖 ∈ 𝐼 ∧ (𝑓‘𝑖) ≠ (0g‘𝑅))) ∧ 𝑥 ⊆ 𝐼) ∧ 𝑘 ∈ 𝐼) → 𝑥 ∈ V)
302		ssel2 3926	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 36 ⊢ ((𝑥 ⊆ 𝐼 ∧ 𝑛 ∈ 𝑥) → 𝑛 ∈ 𝐼)
303	302, 254	sylan2 605	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35 ⊢ (((((((𝑅 ∈ Field ∧ 𝑀:(𝐼 × 𝐼)⟶(Base‘𝑅)) ∧ 𝐼 ∈ Fin) ∧ 𝑓:𝐼⟶(Base‘𝑅)) ∧ (𝑖 ∈ 𝐼 ∧ (𝑓‘𝑖) ≠ (0g‘𝑅))) ∧ 𝑘 ∈ 𝐼) ∧ (𝑥 ⊆ 𝐼 ∧ 𝑛 ∈ 𝑥)) → (((invr‘𝑅)‘(𝑓‘𝑖))(.r‘𝑅)((𝑓‘𝑛)(.r‘𝑅)(𝑛𝑀𝑘))) ∈ (Base‘𝑅))
304	303	anassrs 473	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34 ⊢ ((((((((𝑅 ∈ Field ∧ 𝑀:(𝐼 × 𝐼)⟶(Base‘𝑅)) ∧ 𝐼 ∈ Fin) ∧ 𝑓:𝐼⟶(Base‘𝑅)) ∧ (𝑖 ∈ 𝐼 ∧ (𝑓‘𝑖) ≠ (0g‘𝑅))) ∧ 𝑘 ∈ 𝐼) ∧ 𝑥 ⊆ 𝐼) ∧ 𝑛 ∈ 𝑥) → (((invr‘𝑅)‘(𝑓‘𝑖))(.r‘𝑅)((𝑓‘𝑛)(.r‘𝑅)(𝑛𝑀𝑘))) ∈ (Base‘𝑅))
305	304	fmpttd 7115	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33 ⊢ (((((((𝑅 ∈ Field ∧ 𝑀:(𝐼 × 𝐼)⟶(Base‘𝑅)) ∧ 𝐼 ∈ Fin) ∧ 𝑓:𝐼⟶(Base‘𝑅)) ∧ (𝑖 ∈ 𝐼 ∧ (𝑓‘𝑖) ≠ (0g‘𝑅))) ∧ 𝑘 ∈ 𝐼) ∧ 𝑥 ⊆ 𝐼) → (𝑛 ∈ 𝑥 ↦ (((invr‘𝑅)‘(𝑓‘𝑖))(.r‘𝑅)((𝑓‘𝑛)(.r‘𝑅)(𝑛𝑀𝑘)))):𝑥⟶(Base‘𝑅))
306	305	an32s 665	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32 ⊢ (((((((𝑅 ∈ Field ∧ 𝑀:(𝐼 × 𝐼)⟶(Base‘𝑅)) ∧ 𝐼 ∈ Fin) ∧ 𝑓:𝐼⟶(Base‘𝑅)) ∧ (𝑖 ∈ 𝐼 ∧ (𝑓‘𝑖) ≠ (0g‘𝑅))) ∧ 𝑥 ⊆ 𝐼) ∧ 𝑘 ∈ 𝐼) → (𝑛 ∈ 𝑥 ↦ (((invr‘𝑅)‘(𝑓‘𝑖))(.r‘𝑅)((𝑓‘𝑛)(.r‘𝑅)(𝑛𝑀𝑘)))):𝑥⟶(Base‘𝑅))
307		ovex 7453	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 37 ⊢ (((invr‘𝑅)‘(𝑓‘𝑖))(.r‘𝑅)((𝑓‘𝑛)(.r‘𝑅)(𝑛𝑀𝑘))) ∈ V
308		eqid 2761	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 37 ⊢ (𝑛 ∈ 𝑥 ↦ (((invr‘𝑅)‘(𝑓‘𝑖))(.r‘𝑅)((𝑓‘𝑛)(.r‘𝑅)(𝑛𝑀𝑘)))) = (𝑛 ∈ 𝑥 ↦ (((invr‘𝑅)‘(𝑓‘𝑖))(.r‘𝑅)((𝑓‘𝑛)(.r‘𝑅)(𝑛𝑀𝑘))))
309	307, 308	fnmpti 6682	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 36 ⊢ (𝑛 ∈ 𝑥 ↦ (((invr‘𝑅)‘(𝑓‘𝑖))(.r‘𝑅)((𝑓‘𝑛)(.r‘𝑅)(𝑛𝑀𝑘)))) Fn 𝑥
310	309	a1i 11	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35 ⊢ ((𝐼 ∈ Fin ∧ 𝑥 ⊆ 𝐼) → (𝑛 ∈ 𝑥 ↦ (((invr‘𝑅)‘(𝑓‘𝑖))(.r‘𝑅)((𝑓‘𝑛)(.r‘𝑅)(𝑛𝑀𝑘)))) Fn 𝑥)
311		fvexd 6900	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35 ⊢ ((𝐼 ∈ Fin ∧ 𝑥 ⊆ 𝐼) → (0g‘𝑅) ∈ V)
312	310, 229, 311	fndmfifsupp 9370	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34 ⊢ ((𝐼 ∈ Fin ∧ 𝑥 ⊆ 𝐼) → (𝑛 ∈ 𝑥 ↦ (((invr‘𝑅)‘(𝑓‘𝑖))(.r‘𝑅)((𝑓‘𝑛)(.r‘𝑅)(𝑛𝑀𝑘)))) finSupp (0g‘𝑅))
313	312	adantll 727	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33 ⊢ ((((𝑅 ∈ Field ∧ 𝑀:(𝐼 × 𝐼)⟶(Base‘𝑅)) ∧ 𝐼 ∈ Fin) ∧ 𝑥 ⊆ 𝐼) → (𝑛 ∈ 𝑥 ↦ (((invr‘𝑅)‘(𝑓‘𝑖))(.r‘𝑅)((𝑓‘𝑛)(.r‘𝑅)(𝑛𝑀𝑘)))) finSupp (0g‘𝑅))
314	313	ad5ant14 770	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32 ⊢ (((((((𝑅 ∈ Field ∧ 𝑀:(𝐼 × 𝐼)⟶(Base‘𝑅)) ∧ 𝐼 ∈ Fin) ∧ 𝑓:𝐼⟶(Base‘𝑅)) ∧ (𝑖 ∈ 𝐼 ∧ (𝑓‘𝑖) ≠ (0g‘𝑅))) ∧ 𝑥 ⊆ 𝐼) ∧ 𝑘 ∈ 𝐼) → (𝑛 ∈ 𝑥 ↦ (((invr‘𝑅)‘(𝑓‘𝑖))(.r‘𝑅)((𝑓‘𝑛)(.r‘𝑅)(𝑛𝑀𝑘)))) finSupp (0g‘𝑅))
315	10, 100, 299, 301, 306, 314	gsumcl 20129	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31 ⊢ (((((((𝑅 ∈ Field ∧ 𝑀:(𝐼 × 𝐼)⟶(Base‘𝑅)) ∧ 𝐼 ∈ Fin) ∧ 𝑓:𝐼⟶(Base‘𝑅)) ∧ (𝑖 ∈ 𝐼 ∧ (𝑓‘𝑖) ≠ (0g‘𝑅))) ∧ 𝑥 ⊆ 𝐼) ∧ 𝑘 ∈ 𝐼) → (𝑅 Σg (𝑛 ∈ 𝑥 ↦ (((invr‘𝑅)‘(𝑓‘𝑖))(.r‘𝑅)((𝑓‘𝑛)(.r‘𝑅)(𝑛𝑀𝑘))))) ∈ (Base‘𝑅))
316	215, 315	sylanl2 694	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30 ⊢ (((((((𝑅 ∈ Field ∧ 𝑀:(𝐼 × 𝐼)⟶(Base‘𝑅)) ∧ 𝐼 ∈ Fin) ∧ 𝑓:𝐼⟶(Base‘𝑅)) ∧ (𝑖 ∈ 𝐼 ∧ (𝑓‘𝑖) ≠ (0g‘𝑅))) ∧ (𝑥 ∪ {𝑧}) ⊆ 𝐼) ∧ 𝑘 ∈ 𝐼) → (𝑅 Σg (𝑛 ∈ 𝑥 ↦ (((invr‘𝑅)‘(𝑓‘𝑖))(.r‘𝑅)((𝑓‘𝑛)(.r‘𝑅)(𝑛𝑀𝑘))))) ∈ (Base‘𝑅))
317	265, 279	sylanl2 694	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30 ⊢ (((((((𝑅 ∈ Field ∧ 𝑀:(𝐼 × 𝐼)⟶(Base‘𝑅)) ∧ 𝐼 ∈ Fin) ∧ 𝑓:𝐼⟶(Base‘𝑅)) ∧ (𝑖 ∈ 𝐼 ∧ (𝑓‘𝑖) ≠ (0g‘𝑅))) ∧ (𝑥 ∪ {𝑧}) ⊆ 𝐼) ∧ 𝑘 ∈ 𝐼) → ((((invr‘𝑅)‘(𝑓‘𝑖))(.r‘𝑅)(𝑓‘𝑧))(.r‘𝑅)(𝑧𝑀𝑘)) ∈ (Base‘𝑅))
318		simpllr 788	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33 ⊢ ((((𝑅 ∈ Field ∧ 𝑀:(𝐼 × 𝐼)⟶(Base‘𝑅)) ∧ 𝐼 ∈ Fin) ∧ 𝑓:𝐼⟶(Base‘𝑅)) → 𝑀:(𝐼 × 𝐼)⟶(Base‘𝑅))
319		simpl 488	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33 ⊢ ((𝑖 ∈ 𝐼 ∧ (𝑓‘𝑖) ≠ (0g‘𝑅)) → 𝑖 ∈ 𝐼)
320	318, 319	anim12i 625	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32 ⊢ (((((𝑅 ∈ Field ∧ 𝑀:(𝐼 × 𝐼)⟶(Base‘𝑅)) ∧ 𝐼 ∈ Fin) ∧ 𝑓:𝐼⟶(Base‘𝑅)) ∧ (𝑖 ∈ 𝐼 ∧ (𝑓‘𝑖) ≠ (0g‘𝑅))) → (𝑀:(𝐼 × 𝐼)⟶(Base‘𝑅) ∧ 𝑖 ∈ 𝐼))
321	320	adantr 486	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31 ⊢ ((((((𝑅 ∈ Field ∧ 𝑀:(𝐼 × 𝐼)⟶(Base‘𝑅)) ∧ 𝐼 ∈ Fin) ∧ 𝑓:𝐼⟶(Base‘𝑅)) ∧ (𝑖 ∈ 𝐼 ∧ (𝑓‘𝑖) ≠ (0g‘𝑅))) ∧ (𝑥 ∪ {𝑧}) ⊆ 𝐼) → (𝑀:(𝐼 × 𝐼)⟶(Base‘𝑅) ∧ 𝑖 ∈ 𝐼))
322		fovcdm 7591	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32 ⊢ ((𝑀:(𝐼 × 𝐼)⟶(Base‘𝑅) ∧ 𝑖 ∈ 𝐼 ∧ 𝑘 ∈ 𝐼) → (𝑖𝑀𝑘) ∈ (Base‘𝑅))
323	322	3expa 1136	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31 ⊢ (((𝑀:(𝐼 × 𝐼)⟶(Base‘𝑅) ∧ 𝑖 ∈ 𝐼) ∧ 𝑘 ∈ 𝐼) → (𝑖𝑀𝑘) ∈ (Base‘𝑅))
324	321, 323	sylan 592	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30 ⊢ (((((((𝑅 ∈ Field ∧ 𝑀:(𝐼 × 𝐼)⟶(Base‘𝑅)) ∧ 𝐼 ∈ Fin) ∧ 𝑓:𝐼⟶(Base‘𝑅)) ∧ (𝑖 ∈ 𝐼 ∧ (𝑓‘𝑖) ≠ (0g‘𝑅))) ∧ (𝑥 ∪ {𝑧}) ⊆ 𝐼) ∧ 𝑘 ∈ 𝐼) → (𝑖𝑀𝑘) ∈ (Base‘𝑅))
325	10, 200, 298, 316, 317, 324	abl32 20017	. . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 ⊢ (((((((𝑅 ∈ Field ∧ 𝑀:(𝐼 × 𝐼)⟶(Base‘𝑅)) ∧ 𝐼 ∈ Fin) ∧ 𝑓:𝐼⟶(Base‘𝑅)) ∧ (𝑖 ∈ 𝐼 ∧ (𝑓‘𝑖) ≠ (0g‘𝑅))) ∧ (𝑥 ∪ {𝑧}) ⊆ 𝐼) ∧ 𝑘 ∈ 𝐼) → (((𝑅 Σg (𝑛 ∈ 𝑥 ↦ (((invr‘𝑅)‘(𝑓‘𝑖))(.r‘𝑅)((𝑓‘𝑛)(.r‘𝑅)(𝑛𝑀𝑘)))))(+g‘𝑅)((((invr‘𝑅)‘(𝑓‘𝑖))(.r‘𝑅)(𝑓‘𝑧))(.r‘𝑅)(𝑧𝑀𝑘)))(+g‘𝑅)(𝑖𝑀𝑘)) = (((𝑅 Σg (𝑛 ∈ 𝑥 ↦ (((invr‘𝑅)‘(𝑓‘𝑖))(.r‘𝑅)((𝑓‘𝑛)(.r‘𝑅)(𝑛𝑀𝑘)))))(+g‘𝑅)(𝑖𝑀𝑘))(+g‘𝑅)((((invr‘𝑅)‘(𝑓‘𝑖))(.r‘𝑅)(𝑓‘𝑧))(.r‘𝑅)(𝑧𝑀𝑘))))
326	325	adantlrl 733	. . . . . . . . . . . . . . . . . . . . . . . . . . . 28 ⊢ (((((((𝑅 ∈ Field ∧ 𝑀:(𝐼 × 𝐼)⟶(Base‘𝑅)) ∧ 𝐼 ∈ Fin) ∧ 𝑓:𝐼⟶(Base‘𝑅)) ∧ (𝑖 ∈ 𝐼 ∧ (𝑓‘𝑖) ≠ (0g‘𝑅))) ∧ (𝑖 ≠ 𝑧 ∧ (𝑥 ∪ {𝑧}) ⊆ 𝐼)) ∧ 𝑘 ∈ 𝐼) → (((𝑅 Σg (𝑛 ∈ 𝑥 ↦ (((invr‘𝑅)‘(𝑓‘𝑖))(.r‘𝑅)((𝑓‘𝑛)(.r‘𝑅)(𝑛𝑀𝑘)))))(+g‘𝑅)((((invr‘𝑅)‘(𝑓‘𝑖))(.r‘𝑅)(𝑓‘𝑧))(.r‘𝑅)(𝑧𝑀𝑘)))(+g‘𝑅)(𝑖𝑀𝑘)) = (((𝑅 Σg (𝑛 ∈ 𝑥 ↦ (((invr‘𝑅)‘(𝑓‘𝑖))(.r‘𝑅)((𝑓‘𝑛)(.r‘𝑅)(𝑛𝑀𝑘)))))(+g‘𝑅)(𝑖𝑀𝑘))(+g‘𝑅)((((invr‘𝑅)‘(𝑓‘𝑖))(.r‘𝑅)(𝑓‘𝑧))(.r‘𝑅)(𝑧𝑀𝑘))))
327	326	adantlr 728	. . . . . . . . . . . . . . . . . . . . . . . . . . 27 ⊢ ((((((((𝑅 ∈ Field ∧ 𝑀:(𝐼 × 𝐼)⟶(Base‘𝑅)) ∧ 𝐼 ∈ Fin) ∧ 𝑓:𝐼⟶(Base‘𝑅)) ∧ (𝑖 ∈ 𝐼 ∧ (𝑓‘𝑖) ≠ (0g‘𝑅))) ∧ (𝑖 ≠ 𝑧 ∧ (𝑥 ∪ {𝑧}) ⊆ 𝐼)) ∧ ¬ 𝑧 ∈ 𝑥) ∧ 𝑘 ∈ 𝐼) → (((𝑅 Σg (𝑛 ∈ 𝑥 ↦ (((invr‘𝑅)‘(𝑓‘𝑖))(.r‘𝑅)((𝑓‘𝑛)(.r‘𝑅)(𝑛𝑀𝑘)))))(+g‘𝑅)((((invr‘𝑅)‘(𝑓‘𝑖))(.r‘𝑅)(𝑓‘𝑧))(.r‘𝑅)(𝑧𝑀𝑘)))(+g‘𝑅)(𝑖𝑀𝑘)) = (((𝑅 Σg (𝑛 ∈ 𝑥 ↦ (((invr‘𝑅)‘(𝑓‘𝑖))(.r‘𝑅)((𝑓‘𝑛)(.r‘𝑅)(𝑛𝑀𝑘)))))(+g‘𝑅)(𝑖𝑀𝑘))(+g‘𝑅)((((invr‘𝑅)‘(𝑓‘𝑖))(.r‘𝑅)(𝑓‘𝑧))(.r‘𝑅)(𝑧𝑀𝑘))))
328	295, 327	eqtrd 2796	. . . . . . . . . . . . . . . . . . . . . . . . . 26 ⊢ ((((((((𝑅 ∈ Field ∧ 𝑀:(𝐼 × 𝐼)⟶(Base‘𝑅)) ∧ 𝐼 ∈ Fin) ∧ 𝑓:𝐼⟶(Base‘𝑅)) ∧ (𝑖 ∈ 𝐼 ∧ (𝑓‘𝑖) ≠ (0g‘𝑅))) ∧ (𝑖 ≠ 𝑧 ∧ (𝑥 ∪ {𝑧}) ⊆ 𝐼)) ∧ ¬ 𝑧 ∈ 𝑥) ∧ 𝑘 ∈ 𝐼) → ((𝑅 Σg (𝑛 ∈ (𝑥 ∪ {𝑧}) ↦ (((invr‘𝑅)‘(𝑓‘𝑖))(.r‘𝑅)((𝑓‘𝑛)(.r‘𝑅)(𝑛𝑀𝑘)))))(+g‘𝑅)(𝑖𝑀𝑘)) = (((𝑅 Σg (𝑛 ∈ 𝑥 ↦ (((invr‘𝑅)‘(𝑓‘𝑖))(.r‘𝑅)((𝑓‘𝑛)(.r‘𝑅)(𝑛𝑀𝑘)))))(+g‘𝑅)(𝑖𝑀𝑘))(+g‘𝑅)((((invr‘𝑅)‘(𝑓‘𝑖))(.r‘𝑅)(𝑓‘𝑧))(.r‘𝑅)(𝑧𝑀𝑘))))
329	328	ifeq1d 4502	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ ((((((((𝑅 ∈ Field ∧ 𝑀:(𝐼 × 𝐼)⟶(Base‘𝑅)) ∧ 𝐼 ∈ Fin) ∧ 𝑓:𝐼⟶(Base‘𝑅)) ∧ (𝑖 ∈ 𝐼 ∧ (𝑓‘𝑖) ≠ (0g‘𝑅))) ∧ (𝑖 ≠ 𝑧 ∧ (𝑥 ∪ {𝑧}) ⊆ 𝐼)) ∧ ¬ 𝑧 ∈ 𝑥) ∧ 𝑘 ∈ 𝐼) → if(𝑗 = 𝑖, ((𝑅 Σg (𝑛 ∈ (𝑥 ∪ {𝑧}) ↦ (((invr‘𝑅)‘(𝑓‘𝑖))(.r‘𝑅)((𝑓‘𝑛)(.r‘𝑅)(𝑛𝑀𝑘)))))(+g‘𝑅)(𝑖𝑀𝑘)), if(𝑗 = 𝑧, (𝑧𝑀𝑘), (𝑗𝑀𝑘))) = if(𝑗 = 𝑖, (((𝑅 Σg (𝑛 ∈ 𝑥 ↦ (((invr‘𝑅)‘(𝑓‘𝑖))(.r‘𝑅)((𝑓‘𝑛)(.r‘𝑅)(𝑛𝑀𝑘)))))(+g‘𝑅)(𝑖𝑀𝑘))(+g‘𝑅)((((invr‘𝑅)‘(𝑓‘𝑖))(.r‘𝑅)(𝑓‘𝑧))(.r‘𝑅)(𝑧𝑀𝑘))), if(𝑗 = 𝑧, (𝑧𝑀𝑘), (𝑗𝑀𝑘))))
330	329	3adant2 1149	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ ((((((((𝑅 ∈ Field ∧ 𝑀:(𝐼 × 𝐼)⟶(Base‘𝑅)) ∧ 𝐼 ∈ Fin) ∧ 𝑓:𝐼⟶(Base‘𝑅)) ∧ (𝑖 ∈ 𝐼 ∧ (𝑓‘𝑖) ≠ (0g‘𝑅))) ∧ (𝑖 ≠ 𝑧 ∧ (𝑥 ∪ {𝑧}) ⊆ 𝐼)) ∧ ¬ 𝑧 ∈ 𝑥) ∧ 𝑗 ∈ 𝐼 ∧ 𝑘 ∈ 𝐼) → if(𝑗 = 𝑖, ((𝑅 Σg (𝑛 ∈ (𝑥 ∪ {𝑧}) ↦ (((invr‘𝑅)‘(𝑓‘𝑖))(.r‘𝑅)((𝑓‘𝑛)(.r‘𝑅)(𝑛𝑀𝑘)))))(+g‘𝑅)(𝑖𝑀𝑘)), if(𝑗 = 𝑧, (𝑧𝑀𝑘), (𝑗𝑀𝑘))) = if(𝑗 = 𝑖, (((𝑅 Σg (𝑛 ∈ 𝑥 ↦ (((invr‘𝑅)‘(𝑓‘𝑖))(.r‘𝑅)((𝑓‘𝑛)(.r‘𝑅)(𝑛𝑀𝑘)))))(+g‘𝑅)(𝑖𝑀𝑘))(+g‘𝑅)((((invr‘𝑅)‘(𝑓‘𝑖))(.r‘𝑅)(𝑓‘𝑧))(.r‘𝑅)(𝑧𝑀𝑘))), if(𝑗 = 𝑧, (𝑧𝑀𝑘), (𝑗𝑀𝑘))))
331	330	mpoeq3dva 7497	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ (((((((𝑅 ∈ Field ∧ 𝑀:(𝐼 × 𝐼)⟶(Base‘𝑅)) ∧ 𝐼 ∈ Fin) ∧ 𝑓:𝐼⟶(Base‘𝑅)) ∧ (𝑖 ∈ 𝐼 ∧ (𝑓‘𝑖) ≠ (0g‘𝑅))) ∧ (𝑖 ≠ 𝑧 ∧ (𝑥 ∪ {𝑧}) ⊆ 𝐼)) ∧ ¬ 𝑧 ∈ 𝑥) → (𝑗 ∈ 𝐼, 𝑘 ∈ 𝐼 ↦ if(𝑗 = 𝑖, ((𝑅 Σg (𝑛 ∈ (𝑥 ∪ {𝑧}) ↦ (((invr‘𝑅)‘(𝑓‘𝑖))(.r‘𝑅)((𝑓‘𝑛)(.r‘𝑅)(𝑛𝑀𝑘)))))(+g‘𝑅)(𝑖𝑀𝑘)), if(𝑗 = 𝑧, (𝑧𝑀𝑘), (𝑗𝑀𝑘)))) = (𝑗 ∈ 𝐼, 𝑘 ∈ 𝐼 ↦ if(𝑗 = 𝑖, (((𝑅 Σg (𝑛 ∈ 𝑥 ↦ (((invr‘𝑅)‘(𝑓‘𝑖))(.r‘𝑅)((𝑓‘𝑛)(.r‘𝑅)(𝑛𝑀𝑘)))))(+g‘𝑅)(𝑖𝑀𝑘))(+g‘𝑅)((((invr‘𝑅)‘(𝑓‘𝑖))(.r‘𝑅)(𝑓‘𝑧))(.r‘𝑅)(𝑧𝑀𝑘))), if(𝑗 = 𝑧, (𝑧𝑀𝑘), (𝑗𝑀𝑘)))))
332	331	fveq2d 6889	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (((((((𝑅 ∈ Field ∧ 𝑀:(𝐼 × 𝐼)⟶(Base‘𝑅)) ∧ 𝐼 ∈ Fin) ∧ 𝑓:𝐼⟶(Base‘𝑅)) ∧ (𝑖 ∈ 𝐼 ∧ (𝑓‘𝑖) ≠ (0g‘𝑅))) ∧ (𝑖 ≠ 𝑧 ∧ (𝑥 ∪ {𝑧}) ⊆ 𝐼)) ∧ ¬ 𝑧 ∈ 𝑥) → ((𝐼 maDet 𝑅)‘(𝑗 ∈ 𝐼, 𝑘 ∈ 𝐼 ↦ if(𝑗 = 𝑖, ((𝑅 Σg (𝑛 ∈ (𝑥 ∪ {𝑧}) ↦ (((invr‘𝑅)‘(𝑓‘𝑖))(.r‘𝑅)((𝑓‘𝑛)(.r‘𝑅)(𝑛𝑀𝑘)))))(+g‘𝑅)(𝑖𝑀𝑘)), if(𝑗 = 𝑧, (𝑧𝑀𝑘), (𝑗𝑀𝑘))))) = ((𝐼 maDet 𝑅)‘(𝑗 ∈ 𝐼, 𝑘 ∈ 𝐼 ↦ if(𝑗 = 𝑖, (((𝑅 Σg (𝑛 ∈ 𝑥 ↦ (((invr‘𝑅)‘(𝑓‘𝑖))(.r‘𝑅)((𝑓‘𝑛)(.r‘𝑅)(𝑛𝑀𝑘)))))(+g‘𝑅)(𝑖𝑀𝑘))(+g‘𝑅)((((invr‘𝑅)‘(𝑓‘𝑖))(.r‘𝑅)(𝑓‘𝑧))(.r‘𝑅)(𝑧𝑀𝑘))), if(𝑗 = 𝑧, (𝑧𝑀𝑘), (𝑗𝑀𝑘))))))
333		eqid 2761	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ (𝐼 maDet 𝑅) = (𝐼 maDet 𝑅)
334	1	simprbi 503	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ (𝑅 ∈ Field → 𝑅 ∈ CRing)
335	334	ad5antr 747	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ ((((((𝑅 ∈ Field ∧ 𝑀:(𝐼 × 𝐼)⟶(Base‘𝑅)) ∧ 𝐼 ∈ Fin) ∧ 𝑓:𝐼⟶(Base‘𝑅)) ∧ (𝑖 ∈ 𝐼 ∧ (𝑓‘𝑖) ≠ (0g‘𝑅))) ∧ (𝑖 ≠ 𝑧 ∧ (𝑥 ∪ {𝑧}) ⊆ 𝐼)) → 𝑅 ∈ CRing)
336		simp-4r 796	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ ((((((𝑅 ∈ Field ∧ 𝑀:(𝐼 × 𝐼)⟶(Base‘𝑅)) ∧ 𝐼 ∈ Fin) ∧ 𝑓:𝐼⟶(Base‘𝑅)) ∧ (𝑖 ∈ 𝐼 ∧ (𝑓‘𝑖) ≠ (0g‘𝑅))) ∧ (𝑖 ≠ 𝑧 ∧ (𝑥 ∪ {𝑧}) ⊆ 𝐼)) → 𝐼 ∈ Fin)
337	193	ad6antr 749	. . . . . . . . . . . . . . . . . . . . . . . . . 26 ⊢ (((((((𝑅 ∈ Field ∧ 𝑀:(𝐼 × 𝐼)⟶(Base‘𝑅)) ∧ 𝐼 ∈ Fin) ∧ 𝑓:𝐼⟶(Base‘𝑅)) ∧ (𝑖 ∈ 𝐼 ∧ (𝑓‘𝑖) ≠ (0g‘𝑅))) ∧ 𝑥 ⊆ 𝐼) ∧ 𝑘 ∈ 𝐼) → 𝑅 ∈ Grp)
338	320	adantr 486	. . . . . . . . . . . . . . . . . . . . . . . . . . 27 ⊢ ((((((𝑅 ∈ Field ∧ 𝑀:(𝐼 × 𝐼)⟶(Base‘𝑅)) ∧ 𝐼 ∈ Fin) ∧ 𝑓:𝐼⟶(Base‘𝑅)) ∧ (𝑖 ∈ 𝐼 ∧ (𝑓‘𝑖) ≠ (0g‘𝑅))) ∧ 𝑥 ⊆ 𝐼) → (𝑀:(𝐼 × 𝐼)⟶(Base‘𝑅) ∧ 𝑖 ∈ 𝐼))
339	338, 323	sylan 592	. . . . . . . . . . . . . . . . . . . . . . . . . 26 ⊢ (((((((𝑅 ∈ Field ∧ 𝑀:(𝐼 × 𝐼)⟶(Base‘𝑅)) ∧ 𝐼 ∈ Fin) ∧ 𝑓:𝐼⟶(Base‘𝑅)) ∧ (𝑖 ∈ 𝐼 ∧ (𝑓‘𝑖) ≠ (0g‘𝑅))) ∧ 𝑥 ⊆ 𝐼) ∧ 𝑘 ∈ 𝐼) → (𝑖𝑀𝑘) ∈ (Base‘𝑅))
340	10, 200	grpcl 19152	. . . . . . . . . . . . . . . . . . . . . . . . . 26 ⊢ ((𝑅 ∈ Grp ∧ (𝑅 Σg (𝑛 ∈ 𝑥 ↦ (((invr‘𝑅)‘(𝑓‘𝑖))(.r‘𝑅)((𝑓‘𝑛)(.r‘𝑅)(𝑛𝑀𝑘))))) ∈ (Base‘𝑅) ∧ (𝑖𝑀𝑘) ∈ (Base‘𝑅)) → ((𝑅 Σg (𝑛 ∈ 𝑥 ↦ (((invr‘𝑅)‘(𝑓‘𝑖))(.r‘𝑅)((𝑓‘𝑛)(.r‘𝑅)(𝑛𝑀𝑘)))))(+g‘𝑅)(𝑖𝑀𝑘)) ∈ (Base‘𝑅))
341	337, 315, 339, 340	syl3anc 1398	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ (((((((𝑅 ∈ Field ∧ 𝑀:(𝐼 × 𝐼)⟶(Base‘𝑅)) ∧ 𝐼 ∈ Fin) ∧ 𝑓:𝐼⟶(Base‘𝑅)) ∧ (𝑖 ∈ 𝐼 ∧ (𝑓‘𝑖) ≠ (0g‘𝑅))) ∧ 𝑥 ⊆ 𝐼) ∧ 𝑘 ∈ 𝐼) → ((𝑅 Σg (𝑛 ∈ 𝑥 ↦ (((invr‘𝑅)‘(𝑓‘𝑖))(.r‘𝑅)((𝑓‘𝑛)(.r‘𝑅)(𝑛𝑀𝑘)))))(+g‘𝑅)(𝑖𝑀𝑘)) ∈ (Base‘𝑅))
342	228, 341	sylanl2 694	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ (((((((𝑅 ∈ Field ∧ 𝑀:(𝐼 × 𝐼)⟶(Base‘𝑅)) ∧ 𝐼 ∈ Fin) ∧ 𝑓:𝐼⟶(Base‘𝑅)) ∧ (𝑖 ∈ 𝐼 ∧ (𝑓‘𝑖) ≠ (0g‘𝑅))) ∧ (𝑖 ≠ 𝑧 ∧ (𝑥 ∪ {𝑧}) ⊆ 𝐼)) ∧ 𝑘 ∈ 𝐼) → ((𝑅 Σg (𝑛 ∈ 𝑥 ↦ (((invr‘𝑅)‘(𝑓‘𝑖))(.r‘𝑅)((𝑓‘𝑛)(.r‘𝑅)(𝑛𝑀𝑘)))))(+g‘𝑅)(𝑖𝑀𝑘)) ∈ (Base‘𝑅))
343	248, 266	anim12i 625	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ ((((((𝑅 ∈ Field ∧ 𝑀:(𝐼 × 𝐼)⟶(Base‘𝑅)) ∧ 𝐼 ∈ Fin) ∧ 𝑓:𝐼⟶(Base‘𝑅)) ∧ (𝑖 ∈ 𝐼 ∧ (𝑓‘𝑖) ≠ (0g‘𝑅))) ∧ (𝑖 ≠ 𝑧 ∧ (𝑥 ∪ {𝑧}) ⊆ 𝐼)) → (𝑀:(𝐼 × 𝐼)⟶(Base‘𝑅) ∧ 𝑧 ∈ 𝐼))
344	343, 276	sylan 592	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ (((((((𝑅 ∈ Field ∧ 𝑀:(𝐼 × 𝐼)⟶(Base‘𝑅)) ∧ 𝐼 ∈ Fin) ∧ 𝑓:𝐼⟶(Base‘𝑅)) ∧ (𝑖 ∈ 𝐼 ∧ (𝑓‘𝑖) ≠ (0g‘𝑅))) ∧ (𝑖 ≠ 𝑧 ∧ (𝑥 ∪ {𝑧}) ⊆ 𝐼)) ∧ 𝑘 ∈ 𝐼) → (𝑧𝑀𝑘) ∈ (Base‘𝑅))
345		simp-5r 798	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ ((((((𝑅 ∈ Field ∧ 𝑀:(𝐼 × 𝐼)⟶(Base‘𝑅)) ∧ 𝐼 ∈ Fin) ∧ 𝑓:𝐼⟶(Base‘𝑅)) ∧ (𝑖 ∈ 𝐼 ∧ (𝑓‘𝑖) ≠ (0g‘𝑅))) ∧ (𝑖 ≠ 𝑧 ∧ (𝑥 ∪ {𝑧}) ⊆ 𝐼)) → 𝑀:(𝐼 × 𝐼)⟶(Base‘𝑅))
346	345, 198	syl3an1 1181	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ (((((((𝑅 ∈ Field ∧ 𝑀:(𝐼 × 𝐼)⟶(Base‘𝑅)) ∧ 𝐼 ∈ Fin) ∧ 𝑓:𝐼⟶(Base‘𝑅)) ∧ (𝑖 ∈ 𝐼 ∧ (𝑓‘𝑖) ≠ (0g‘𝑅))) ∧ (𝑖 ≠ 𝑧 ∧ (𝑥 ∪ {𝑧}) ⊆ 𝐼)) ∧ 𝑗 ∈ 𝐼 ∧ 𝑘 ∈ 𝐼) → (𝑗𝑀𝑘) ∈ (Base‘𝑅))
347	266, 273	sylan2 605	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ ((((((𝑅 ∈ Field ∧ 𝑀:(𝐼 × 𝐼)⟶(Base‘𝑅)) ∧ 𝐼 ∈ Fin) ∧ 𝑓:𝐼⟶(Base‘𝑅)) ∧ (𝑖 ∈ 𝐼 ∧ (𝑓‘𝑖) ≠ (0g‘𝑅))) ∧ (𝑖 ≠ 𝑧 ∧ (𝑥 ∪ {𝑧}) ⊆ 𝐼)) → (((invr‘𝑅)‘(𝑓‘𝑖))(.r‘𝑅)(𝑓‘𝑧)) ∈ (Base‘𝑅))
348		simplrl 789	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ ((((((𝑅 ∈ Field ∧ 𝑀:(𝐼 × 𝐼)⟶(Base‘𝑅)) ∧ 𝐼 ∈ Fin) ∧ 𝑓:𝐼⟶(Base‘𝑅)) ∧ (𝑖 ∈ 𝐼 ∧ (𝑓‘𝑖) ≠ (0g‘𝑅))) ∧ (𝑖 ≠ 𝑧 ∧ (𝑥 ∪ {𝑧}) ⊆ 𝐼)) → 𝑖 ∈ 𝐼)
349	265	ad2antll 742	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ ((((((𝑅 ∈ Field ∧ 𝑀:(𝐼 × 𝐼)⟶(Base‘𝑅)) ∧ 𝐼 ∈ Fin) ∧ 𝑓:𝐼⟶(Base‘𝑅)) ∧ (𝑖 ∈ 𝐼 ∧ (𝑓‘𝑖) ≠ (0g‘𝑅))) ∧ (𝑖 ≠ 𝑧 ∧ (𝑥 ∪ {𝑧}) ⊆ 𝐼)) → 𝑧 ∈ 𝐼)
350		simprl 783	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ ((((((𝑅 ∈ Field ∧ 𝑀:(𝐼 × 𝐼)⟶(Base‘𝑅)) ∧ 𝐼 ∈ Fin) ∧ 𝑓:𝐼⟶(Base‘𝑅)) ∧ (𝑖 ∈ 𝐼 ∧ (𝑓‘𝑖) ≠ (0g‘𝑅))) ∧ (𝑖 ≠ 𝑧 ∧ (𝑥 ∪ {𝑧}) ⊆ 𝐼)) → 𝑖 ≠ 𝑧)
351	333, 10, 200, 47, 335, 336, 342, 344, 346, 347, 348, 349, 350	mdetero 22925	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ ((((((𝑅 ∈ Field ∧ 𝑀:(𝐼 × 𝐼)⟶(Base‘𝑅)) ∧ 𝐼 ∈ Fin) ∧ 𝑓:𝐼⟶(Base‘𝑅)) ∧ (𝑖 ∈ 𝐼 ∧ (𝑓‘𝑖) ≠ (0g‘𝑅))) ∧ (𝑖 ≠ 𝑧 ∧ (𝑥 ∪ {𝑧}) ⊆ 𝐼)) → ((𝐼 maDet 𝑅)‘(𝑗 ∈ 𝐼, 𝑘 ∈ 𝐼 ↦ if(𝑗 = 𝑖, (((𝑅 Σg (𝑛 ∈ 𝑥 ↦ (((invr‘𝑅)‘(𝑓‘𝑖))(.r‘𝑅)((𝑓‘𝑛)(.r‘𝑅)(𝑛𝑀𝑘)))))(+g‘𝑅)(𝑖𝑀𝑘))(+g‘𝑅)((((invr‘𝑅)‘(𝑓‘𝑖))(.r‘𝑅)(𝑓‘𝑧))(.r‘𝑅)(𝑧𝑀𝑘))), if(𝑗 = 𝑧, (𝑧𝑀𝑘), (𝑗𝑀𝑘))))) = ((𝐼 maDet 𝑅)‘(𝑗 ∈ 𝐼, 𝑘 ∈ 𝐼 ↦ if(𝑗 = 𝑖, ((𝑅 Σg (𝑛 ∈ 𝑥 ↦ (((invr‘𝑅)‘(𝑓‘𝑖))(.r‘𝑅)((𝑓‘𝑛)(.r‘𝑅)(𝑛𝑀𝑘)))))(+g‘𝑅)(𝑖𝑀𝑘)), if(𝑗 = 𝑧, (𝑧𝑀𝑘), (𝑗𝑀𝑘))))))
352	351	adantr 486	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (((((((𝑅 ∈ Field ∧ 𝑀:(𝐼 × 𝐼)⟶(Base‘𝑅)) ∧ 𝐼 ∈ Fin) ∧ 𝑓:𝐼⟶(Base‘𝑅)) ∧ (𝑖 ∈ 𝐼 ∧ (𝑓‘𝑖) ≠ (0g‘𝑅))) ∧ (𝑖 ≠ 𝑧 ∧ (𝑥 ∪ {𝑧}) ⊆ 𝐼)) ∧ ¬ 𝑧 ∈ 𝑥) → ((𝐼 maDet 𝑅)‘(𝑗 ∈ 𝐼, 𝑘 ∈ 𝐼 ↦ if(𝑗 = 𝑖, (((𝑅 Σg (𝑛 ∈ 𝑥 ↦ (((invr‘𝑅)‘(𝑓‘𝑖))(.r‘𝑅)((𝑓‘𝑛)(.r‘𝑅)(𝑛𝑀𝑘)))))(+g‘𝑅)(𝑖𝑀𝑘))(+g‘𝑅)((((invr‘𝑅)‘(𝑓‘𝑖))(.r‘𝑅)(𝑓‘𝑧))(.r‘𝑅)(𝑧𝑀𝑘))), if(𝑗 = 𝑧, (𝑧𝑀𝑘), (𝑗𝑀𝑘))))) = ((𝐼 maDet 𝑅)‘(𝑗 ∈ 𝐼, 𝑘 ∈ 𝐼 ↦ if(𝑗 = 𝑖, ((𝑅 Σg (𝑛 ∈ 𝑥 ↦ (((invr‘𝑅)‘(𝑓‘𝑖))(.r‘𝑅)((𝑓‘𝑛)(.r‘𝑅)(𝑛𝑀𝑘)))))(+g‘𝑅)(𝑖𝑀𝑘)), if(𝑗 = 𝑧, (𝑧𝑀𝑘), (𝑗𝑀𝑘))))))
353	332, 352	eqtrd 2796	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (((((((𝑅 ∈ Field ∧ 𝑀:(𝐼 × 𝐼)⟶(Base‘𝑅)) ∧ 𝐼 ∈ Fin) ∧ 𝑓:𝐼⟶(Base‘𝑅)) ∧ (𝑖 ∈ 𝐼 ∧ (𝑓‘𝑖) ≠ (0g‘𝑅))) ∧ (𝑖 ≠ 𝑧 ∧ (𝑥 ∪ {𝑧}) ⊆ 𝐼)) ∧ ¬ 𝑧 ∈ 𝑥) → ((𝐼 maDet 𝑅)‘(𝑗 ∈ 𝐼, 𝑘 ∈ 𝐼 ↦ if(𝑗 = 𝑖, ((𝑅 Σg (𝑛 ∈ (𝑥 ∪ {𝑧}) ↦ (((invr‘𝑅)‘(𝑓‘𝑖))(.r‘𝑅)((𝑓‘𝑛)(.r‘𝑅)(𝑛𝑀𝑘)))))(+g‘𝑅)(𝑖𝑀𝑘)), if(𝑗 = 𝑧, (𝑧𝑀𝑘), (𝑗𝑀𝑘))))) = ((𝐼 maDet 𝑅)‘(𝑗 ∈ 𝐼, 𝑘 ∈ 𝐼 ↦ if(𝑗 = 𝑖, ((𝑅 Σg (𝑛 ∈ 𝑥 ↦ (((invr‘𝑅)‘(𝑓‘𝑖))(.r‘𝑅)((𝑓‘𝑛)(.r‘𝑅)(𝑛𝑀𝑘)))))(+g‘𝑅)(𝑖𝑀𝑘)), if(𝑗 = 𝑧, (𝑧𝑀𝑘), (𝑗𝑀𝑘))))))
354		iftrue 4488	. . . . . . . . . . . . . . . . . . . . . . . . . 26 ⊢ (𝑗 = 𝑧 → if(𝑗 = 𝑧, (𝑧𝑀𝑘), (𝑗𝑀𝑘)) = (𝑧𝑀𝑘))
355		oveq1 7427	. . . . . . . . . . . . . . . . . . . . . . . . . 26 ⊢ (𝑗 = 𝑧 → (𝑗𝑀𝑘) = (𝑧𝑀𝑘))
356	354, 355	eqtr4d 2799	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ (𝑗 = 𝑧 → if(𝑗 = 𝑧, (𝑧𝑀𝑘), (𝑗𝑀𝑘)) = (𝑗𝑀𝑘))
357		iffalse 4491	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ (¬ 𝑗 = 𝑧 → if(𝑗 = 𝑧, (𝑧𝑀𝑘), (𝑗𝑀𝑘)) = (𝑗𝑀𝑘))
358	356, 357	pm2.61i 184	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ if(𝑗 = 𝑧, (𝑧𝑀𝑘), (𝑗𝑀𝑘)) = (𝑗𝑀𝑘)
359		ifeq2 4487	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ (if(𝑗 = 𝑧, (𝑧𝑀𝑘), (𝑗𝑀𝑘)) = (𝑗𝑀𝑘) → if(𝑗 = 𝑖, ((𝑅 Σg (𝑛 ∈ (𝑥 ∪ {𝑧}) ↦ (((invr‘𝑅)‘(𝑓‘𝑖))(.r‘𝑅)((𝑓‘𝑛)(.r‘𝑅)(𝑛𝑀𝑘)))))(+g‘𝑅)(𝑖𝑀𝑘)), if(𝑗 = 𝑧, (𝑧𝑀𝑘), (𝑗𝑀𝑘))) = if(𝑗 = 𝑖, ((𝑅 Σg (𝑛 ∈ (𝑥 ∪ {𝑧}) ↦ (((invr‘𝑅)‘(𝑓‘𝑖))(.r‘𝑅)((𝑓‘𝑛)(.r‘𝑅)(𝑛𝑀𝑘)))))(+g‘𝑅)(𝑖𝑀𝑘)), (𝑗𝑀𝑘)))
360	358, 359	mp1i 14	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ ((𝑗 ∈ 𝐼 ∧ 𝑘 ∈ 𝐼) → if(𝑗 = 𝑖, ((𝑅 Σg (𝑛 ∈ (𝑥 ∪ {𝑧}) ↦ (((invr‘𝑅)‘(𝑓‘𝑖))(.r‘𝑅)((𝑓‘𝑛)(.r‘𝑅)(𝑛𝑀𝑘)))))(+g‘𝑅)(𝑖𝑀𝑘)), if(𝑗 = 𝑧, (𝑧𝑀𝑘), (𝑗𝑀𝑘))) = if(𝑗 = 𝑖, ((𝑅 Σg (𝑛 ∈ (𝑥 ∪ {𝑧}) ↦ (((invr‘𝑅)‘(𝑓‘𝑖))(.r‘𝑅)((𝑓‘𝑛)(.r‘𝑅)(𝑛𝑀𝑘)))))(+g‘𝑅)(𝑖𝑀𝑘)), (𝑗𝑀𝑘)))
361	360	mpoeq3ia 7498	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (𝑗 ∈ 𝐼, 𝑘 ∈ 𝐼 ↦ if(𝑗 = 𝑖, ((𝑅 Σg (𝑛 ∈ (𝑥 ∪ {𝑧}) ↦ (((invr‘𝑅)‘(𝑓‘𝑖))(.r‘𝑅)((𝑓‘𝑛)(.r‘𝑅)(𝑛𝑀𝑘)))))(+g‘𝑅)(𝑖𝑀𝑘)), if(𝑗 = 𝑧, (𝑧𝑀𝑘), (𝑗𝑀𝑘)))) = (𝑗 ∈ 𝐼, 𝑘 ∈ 𝐼 ↦ if(𝑗 = 𝑖, ((𝑅 Σg (𝑛 ∈ (𝑥 ∪ {𝑧}) ↦ (((invr‘𝑅)‘(𝑓‘𝑖))(.r‘𝑅)((𝑓‘𝑛)(.r‘𝑅)(𝑛𝑀𝑘)))))(+g‘𝑅)(𝑖𝑀𝑘)), (𝑗𝑀𝑘)))
362	361	fveq2i 6888	. . . . . . . . . . . . . . . . . . . . 21 ⊢ ((𝐼 maDet 𝑅)‘(𝑗 ∈ 𝐼, 𝑘 ∈ 𝐼 ↦ if(𝑗 = 𝑖, ((𝑅 Σg (𝑛 ∈ (𝑥 ∪ {𝑧}) ↦ (((invr‘𝑅)‘(𝑓‘𝑖))(.r‘𝑅)((𝑓‘𝑛)(.r‘𝑅)(𝑛𝑀𝑘)))))(+g‘𝑅)(𝑖𝑀𝑘)), if(𝑗 = 𝑧, (𝑧𝑀𝑘), (𝑗𝑀𝑘))))) = ((𝐼 maDet 𝑅)‘(𝑗 ∈ 𝐼, 𝑘 ∈ 𝐼 ↦ if(𝑗 = 𝑖, ((𝑅 Σg (𝑛 ∈ (𝑥 ∪ {𝑧}) ↦ (((invr‘𝑅)‘(𝑓‘𝑖))(.r‘𝑅)((𝑓‘𝑛)(.r‘𝑅)(𝑛𝑀𝑘)))))(+g‘𝑅)(𝑖𝑀𝑘)), (𝑗𝑀𝑘))))
363		ifeq2 4487	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ (if(𝑗 = 𝑧, (𝑧𝑀𝑘), (𝑗𝑀𝑘)) = (𝑗𝑀𝑘) → if(𝑗 = 𝑖, ((𝑅 Σg (𝑛 ∈ 𝑥 ↦ (((invr‘𝑅)‘(𝑓‘𝑖))(.r‘𝑅)((𝑓‘𝑛)(.r‘𝑅)(𝑛𝑀𝑘)))))(+g‘𝑅)(𝑖𝑀𝑘)), if(𝑗 = 𝑧, (𝑧𝑀𝑘), (𝑗𝑀𝑘))) = if(𝑗 = 𝑖, ((𝑅 Σg (𝑛 ∈ 𝑥 ↦ (((invr‘𝑅)‘(𝑓‘𝑖))(.r‘𝑅)((𝑓‘𝑛)(.r‘𝑅)(𝑛𝑀𝑘)))))(+g‘𝑅)(𝑖𝑀𝑘)), (𝑗𝑀𝑘)))
364	358, 363	mp1i 14	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ ((𝑗 ∈ 𝐼 ∧ 𝑘 ∈ 𝐼) → if(𝑗 = 𝑖, ((𝑅 Σg (𝑛 ∈ 𝑥 ↦ (((invr‘𝑅)‘(𝑓‘𝑖))(.r‘𝑅)((𝑓‘𝑛)(.r‘𝑅)(𝑛𝑀𝑘)))))(+g‘𝑅)(𝑖𝑀𝑘)), if(𝑗 = 𝑧, (𝑧𝑀𝑘), (𝑗𝑀𝑘))) = if(𝑗 = 𝑖, ((𝑅 Σg (𝑛 ∈ 𝑥 ↦ (((invr‘𝑅)‘(𝑓‘𝑖))(.r‘𝑅)((𝑓‘𝑛)(.r‘𝑅)(𝑛𝑀𝑘)))))(+g‘𝑅)(𝑖𝑀𝑘)), (𝑗𝑀𝑘)))
365	364	mpoeq3ia 7498	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (𝑗 ∈ 𝐼, 𝑘 ∈ 𝐼 ↦ if(𝑗 = 𝑖, ((𝑅 Σg (𝑛 ∈ 𝑥 ↦ (((invr‘𝑅)‘(𝑓‘𝑖))(.r‘𝑅)((𝑓‘𝑛)(.r‘𝑅)(𝑛𝑀𝑘)))))(+g‘𝑅)(𝑖𝑀𝑘)), if(𝑗 = 𝑧, (𝑧𝑀𝑘), (𝑗𝑀𝑘)))) = (𝑗 ∈ 𝐼, 𝑘 ∈ 𝐼 ↦ if(𝑗 = 𝑖, ((𝑅 Σg (𝑛 ∈ 𝑥 ↦ (((invr‘𝑅)‘(𝑓‘𝑖))(.r‘𝑅)((𝑓‘𝑛)(.r‘𝑅)(𝑛𝑀𝑘)))))(+g‘𝑅)(𝑖𝑀𝑘)), (𝑗𝑀𝑘)))
366	365	fveq2i 6888	. . . . . . . . . . . . . . . . . . . . 21 ⊢ ((𝐼 maDet 𝑅)‘(𝑗 ∈ 𝐼, 𝑘 ∈ 𝐼 ↦ if(𝑗 = 𝑖, ((𝑅 Σg (𝑛 ∈ 𝑥 ↦ (((invr‘𝑅)‘(𝑓‘𝑖))(.r‘𝑅)((𝑓‘𝑛)(.r‘𝑅)(𝑛𝑀𝑘)))))(+g‘𝑅)(𝑖𝑀𝑘)), if(𝑗 = 𝑧, (𝑧𝑀𝑘), (𝑗𝑀𝑘))))) = ((𝐼 maDet 𝑅)‘(𝑗 ∈ 𝐼, 𝑘 ∈ 𝐼 ↦ if(𝑗 = 𝑖, ((𝑅 Σg (𝑛 ∈ 𝑥 ↦ (((invr‘𝑅)‘(𝑓‘𝑖))(.r‘𝑅)((𝑓‘𝑛)(.r‘𝑅)(𝑛𝑀𝑘)))))(+g‘𝑅)(𝑖𝑀𝑘)), (𝑗𝑀𝑘))))
367	353, 362, 366	3eqtr3g 2819	. . . . . . . . . . . . . . . . . . . 20 ⊢ (((((((𝑅 ∈ Field ∧ 𝑀:(𝐼 × 𝐼)⟶(Base‘𝑅)) ∧ 𝐼 ∈ Fin) ∧ 𝑓:𝐼⟶(Base‘𝑅)) ∧ (𝑖 ∈ 𝐼 ∧ (𝑓‘𝑖) ≠ (0g‘𝑅))) ∧ (𝑖 ≠ 𝑧 ∧ (𝑥 ∪ {𝑧}) ⊆ 𝐼)) ∧ ¬ 𝑧 ∈ 𝑥) → ((𝐼 maDet 𝑅)‘(𝑗 ∈ 𝐼, 𝑘 ∈ 𝐼 ↦ if(𝑗 = 𝑖, ((𝑅 Σg (𝑛 ∈ (𝑥 ∪ {𝑧}) ↦ (((invr‘𝑅)‘(𝑓‘𝑖))(.r‘𝑅)((𝑓‘𝑛)(.r‘𝑅)(𝑛𝑀𝑘)))))(+g‘𝑅)(𝑖𝑀𝑘)), (𝑗𝑀𝑘)))) = ((𝐼 maDet 𝑅)‘(𝑗 ∈ 𝐼, 𝑘 ∈ 𝐼 ↦ if(𝑗 = 𝑖, ((𝑅 Σg (𝑛 ∈ 𝑥 ↦ (((invr‘𝑅)‘(𝑓‘𝑖))(.r‘𝑅)((𝑓‘𝑛)(.r‘𝑅)(𝑛𝑀𝑘)))))(+g‘𝑅)(𝑖𝑀𝑘)), (𝑗𝑀𝑘)))))
368	223, 367	sylanl2 694	. . . . . . . . . . . . . . . . . . 19 ⊢ (((((((𝑅 ∈ Field ∧ 𝑀:(𝐼 × 𝐼)⟶(Base‘𝑅)) ∧ 𝐼 ∈ Fin) ∧ 𝑓:𝐼⟶(Base‘𝑅)) ∧ (𝑖 ∈ 𝐼 ∧ (𝑓‘𝑖) ≠ (0g‘𝑅))) ∧ (¬ 𝑖 ∈ (𝑥 ∪ {𝑧}) ∧ (𝑥 ∪ {𝑧}) ⊆ 𝐼)) ∧ ¬ 𝑧 ∈ 𝑥) → ((𝐼 maDet 𝑅)‘(𝑗 ∈ 𝐼, 𝑘 ∈ 𝐼 ↦ if(𝑗 = 𝑖, ((𝑅 Σg (𝑛 ∈ (𝑥 ∪ {𝑧}) ↦ (((invr‘𝑅)‘(𝑓‘𝑖))(.r‘𝑅)((𝑓‘𝑛)(.r‘𝑅)(𝑛𝑀𝑘)))))(+g‘𝑅)(𝑖𝑀𝑘)), (𝑗𝑀𝑘)))) = ((𝐼 maDet 𝑅)‘(𝑗 ∈ 𝐼, 𝑘 ∈ 𝐼 ↦ if(𝑗 = 𝑖, ((𝑅 Σg (𝑛 ∈ 𝑥 ↦ (((invr‘𝑅)‘(𝑓‘𝑖))(.r‘𝑅)((𝑓‘𝑛)(.r‘𝑅)(𝑛𝑀𝑘)))))(+g‘𝑅)(𝑖𝑀𝑘)), (𝑗𝑀𝑘)))))
369	368	eqeq2d 2772	. . . . . . . . . . . . . . . . . 18 ⊢ (((((((𝑅 ∈ Field ∧ 𝑀:(𝐼 × 𝐼)⟶(Base‘𝑅)) ∧ 𝐼 ∈ Fin) ∧ 𝑓:𝐼⟶(Base‘𝑅)) ∧ (𝑖 ∈ 𝐼 ∧ (𝑓‘𝑖) ≠ (0g‘𝑅))) ∧ (¬ 𝑖 ∈ (𝑥 ∪ {𝑧}) ∧ (𝑥 ∪ {𝑧}) ⊆ 𝐼)) ∧ ¬ 𝑧 ∈ 𝑥) → (((𝐼 maDet 𝑅)‘𝑀) = ((𝐼 maDet 𝑅)‘(𝑗 ∈ 𝐼, 𝑘 ∈ 𝐼 ↦ if(𝑗 = 𝑖, ((𝑅 Σg (𝑛 ∈ (𝑥 ∪ {𝑧}) ↦ (((invr‘𝑅)‘(𝑓‘𝑖))(.r‘𝑅)((𝑓‘𝑛)(.r‘𝑅)(𝑛𝑀𝑘)))))(+g‘𝑅)(𝑖𝑀𝑘)), (𝑗𝑀𝑘)))) ↔ ((𝐼 maDet 𝑅)‘𝑀) = ((𝐼 maDet 𝑅)‘(𝑗 ∈ 𝐼, 𝑘 ∈ 𝐼 ↦ if(𝑗 = 𝑖, ((𝑅 Σg (𝑛 ∈ 𝑥 ↦ (((invr‘𝑅)‘(𝑓‘𝑖))(.r‘𝑅)((𝑓‘𝑛)(.r‘𝑅)(𝑛𝑀𝑘)))))(+g‘𝑅)(𝑖𝑀𝑘)), (𝑗𝑀𝑘))))))
370	369	biimprd 251	. . . . . . . . . . . . . . . . 17 ⊢ (((((((𝑅 ∈ Field ∧ 𝑀:(𝐼 × 𝐼)⟶(Base‘𝑅)) ∧ 𝐼 ∈ Fin) ∧ 𝑓:𝐼⟶(Base‘𝑅)) ∧ (𝑖 ∈ 𝐼 ∧ (𝑓‘𝑖) ≠ (0g‘𝑅))) ∧ (¬ 𝑖 ∈ (𝑥 ∪ {𝑧}) ∧ (𝑥 ∪ {𝑧}) ⊆ 𝐼)) ∧ ¬ 𝑧 ∈ 𝑥) → (((𝐼 maDet 𝑅)‘𝑀) = ((𝐼 maDet 𝑅)‘(𝑗 ∈ 𝐼, 𝑘 ∈ 𝐼 ↦ if(𝑗 = 𝑖, ((𝑅 Σg (𝑛 ∈ 𝑥 ↦ (((invr‘𝑅)‘(𝑓‘𝑖))(.r‘𝑅)((𝑓‘𝑛)(.r‘𝑅)(𝑛𝑀𝑘)))))(+g‘𝑅)(𝑖𝑀𝑘)), (𝑗𝑀𝑘)))) → ((𝐼 maDet 𝑅)‘𝑀) = ((𝐼 maDet 𝑅)‘(𝑗 ∈ 𝐼, 𝑘 ∈ 𝐼 ↦ if(𝑗 = 𝑖, ((𝑅 Σg (𝑛 ∈ (𝑥 ∪ {𝑧}) ↦ (((invr‘𝑅)‘(𝑓‘𝑖))(.r‘𝑅)((𝑓‘𝑛)(.r‘𝑅)(𝑛𝑀𝑘)))))(+g‘𝑅)(𝑖𝑀𝑘)), (𝑗𝑀𝑘))))))
371	218, 370	embantd 60	. . . . . . . . . . . . . . . 16 ⊢ (((((((𝑅 ∈ Field ∧ 𝑀:(𝐼 × 𝐼)⟶(Base‘𝑅)) ∧ 𝐼 ∈ Fin) ∧ 𝑓:𝐼⟶(Base‘𝑅)) ∧ (𝑖 ∈ 𝐼 ∧ (𝑓‘𝑖) ≠ (0g‘𝑅))) ∧ (¬ 𝑖 ∈ (𝑥 ∪ {𝑧}) ∧ (𝑥 ∪ {𝑧}) ⊆ 𝐼)) ∧ ¬ 𝑧 ∈ 𝑥) → (((((((𝑅 ∈ Field ∧ 𝑀:(𝐼 × 𝐼)⟶(Base‘𝑅)) ∧ 𝐼 ∈ Fin) ∧ 𝑓:𝐼⟶(Base‘𝑅)) ∧ (𝑖 ∈ 𝐼 ∧ (𝑓‘𝑖) ≠ (0g‘𝑅))) ∧ (¬ 𝑖 ∈ 𝑥 ∧ 𝑥 ⊆ 𝐼)) → ((𝐼 maDet 𝑅)‘𝑀) = ((𝐼 maDet 𝑅)‘(𝑗 ∈ 𝐼, 𝑘 ∈ 𝐼 ↦ if(𝑗 = 𝑖, ((𝑅 Σg (𝑛 ∈ 𝑥 ↦ (((invr‘𝑅)‘(𝑓‘𝑖))(.r‘𝑅)((𝑓‘𝑛)(.r‘𝑅)(𝑛𝑀𝑘)))))(+g‘𝑅)(𝑖𝑀𝑘)), (𝑗𝑀𝑘))))) → ((𝐼 maDet 𝑅)‘𝑀) = ((𝐼 maDet 𝑅)‘(𝑗 ∈ 𝐼, 𝑘 ∈ 𝐼 ↦ if(𝑗 = 𝑖, ((𝑅 Σg (𝑛 ∈ (𝑥 ∪ {𝑧}) ↦ (((invr‘𝑅)‘(𝑓‘𝑖))(.r‘𝑅)((𝑓‘𝑛)(.r‘𝑅)(𝑛𝑀𝑘)))))(+g‘𝑅)(𝑖𝑀𝑘)), (𝑗𝑀𝑘))))))
372	371	expcom 419	. . . . . . . . . . . . . . 15 ⊢ (¬ 𝑧 ∈ 𝑥 → ((((((𝑅 ∈ Field ∧ 𝑀:(𝐼 × 𝐼)⟶(Base‘𝑅)) ∧ 𝐼 ∈ Fin) ∧ 𝑓:𝐼⟶(Base‘𝑅)) ∧ (𝑖 ∈ 𝐼 ∧ (𝑓‘𝑖) ≠ (0g‘𝑅))) ∧ (¬ 𝑖 ∈ (𝑥 ∪ {𝑧}) ∧ (𝑥 ∪ {𝑧}) ⊆ 𝐼)) → (((((((𝑅 ∈ Field ∧ 𝑀:(𝐼 × 𝐼)⟶(Base‘𝑅)) ∧ 𝐼 ∈ Fin) ∧ 𝑓:𝐼⟶(Base‘𝑅)) ∧ (𝑖 ∈ 𝐼 ∧ (𝑓‘𝑖) ≠ (0g‘𝑅))) ∧ (¬ 𝑖 ∈ 𝑥 ∧ 𝑥 ⊆ 𝐼)) → ((𝐼 maDet 𝑅)‘𝑀) = ((𝐼 maDet 𝑅)‘(𝑗 ∈ 𝐼, 𝑘 ∈ 𝐼 ↦ if(𝑗 = 𝑖, ((𝑅 Σg (𝑛 ∈ 𝑥 ↦ (((invr‘𝑅)‘(𝑓‘𝑖))(.r‘𝑅)((𝑓‘𝑛)(.r‘𝑅)(𝑛𝑀𝑘)))))(+g‘𝑅)(𝑖𝑀𝑘)), (𝑗𝑀𝑘))))) → ((𝐼 maDet 𝑅)‘𝑀) = ((𝐼 maDet 𝑅)‘(𝑗 ∈ 𝐼, 𝑘 ∈ 𝐼 ↦ if(𝑗 = 𝑖, ((𝑅 Σg (𝑛 ∈ (𝑥 ∪ {𝑧}) ↦ (((invr‘𝑅)‘(𝑓‘𝑖))(.r‘𝑅)((𝑓‘𝑛)(.r‘𝑅)(𝑛𝑀𝑘)))))(+g‘𝑅)(𝑖𝑀𝑘)), (𝑗𝑀𝑘)))))))
373	372	com23 87	. . . . . . . . . . . . . 14 ⊢ (¬ 𝑧 ∈ 𝑥 → (((((((𝑅 ∈ Field ∧ 𝑀:(𝐼 × 𝐼)⟶(Base‘𝑅)) ∧ 𝐼 ∈ Fin) ∧ 𝑓:𝐼⟶(Base‘𝑅)) ∧ (𝑖 ∈ 𝐼 ∧ (𝑓‘𝑖) ≠ (0g‘𝑅))) ∧ (¬ 𝑖 ∈ 𝑥 ∧ 𝑥 ⊆ 𝐼)) → ((𝐼 maDet 𝑅)‘𝑀) = ((𝐼 maDet 𝑅)‘(𝑗 ∈ 𝐼, 𝑘 ∈ 𝐼 ↦ if(𝑗 = 𝑖, ((𝑅 Σg (𝑛 ∈ 𝑥 ↦ (((invr‘𝑅)‘(𝑓‘𝑖))(.r‘𝑅)((𝑓‘𝑛)(.r‘𝑅)(𝑛𝑀𝑘)))))(+g‘𝑅)(𝑖𝑀𝑘)), (𝑗𝑀𝑘))))) → ((((((𝑅 ∈ Field ∧ 𝑀:(𝐼 × 𝐼)⟶(Base‘𝑅)) ∧ 𝐼 ∈ Fin) ∧ 𝑓:𝐼⟶(Base‘𝑅)) ∧ (𝑖 ∈ 𝐼 ∧ (𝑓‘𝑖) ≠ (0g‘𝑅))) ∧ (¬ 𝑖 ∈ (𝑥 ∪ {𝑧}) ∧ (𝑥 ∪ {𝑧}) ⊆ 𝐼)) → ((𝐼 maDet 𝑅)‘𝑀) = ((𝐼 maDet 𝑅)‘(𝑗 ∈ 𝐼, 𝑘 ∈ 𝐼 ↦ if(𝑗 = 𝑖, ((𝑅 Σg (𝑛 ∈ (𝑥 ∪ {𝑧}) ↦ (((invr‘𝑅)‘(𝑓‘𝑖))(.r‘𝑅)((𝑓‘𝑛)(.r‘𝑅)(𝑛𝑀𝑘)))))(+g‘𝑅)(𝑖𝑀𝑘)), (𝑗𝑀𝑘)))))))
374	373	adantl 487	. . . . . . . . . . . . 13 ⊢ ((𝑥 ∈ Fin ∧ ¬ 𝑧 ∈ 𝑥) → (((((((𝑅 ∈ Field ∧ 𝑀:(𝐼 × 𝐼)⟶(Base‘𝑅)) ∧ 𝐼 ∈ Fin) ∧ 𝑓:𝐼⟶(Base‘𝑅)) ∧ (𝑖 ∈ 𝐼 ∧ (𝑓‘𝑖) ≠ (0g‘𝑅))) ∧ (¬ 𝑖 ∈ 𝑥 ∧ 𝑥 ⊆ 𝐼)) → ((𝐼 maDet 𝑅)‘𝑀) = ((𝐼 maDet 𝑅)‘(𝑗 ∈ 𝐼, 𝑘 ∈ 𝐼 ↦ if(𝑗 = 𝑖, ((𝑅 Σg (𝑛 ∈ 𝑥 ↦ (((invr‘𝑅)‘(𝑓‘𝑖))(.r‘𝑅)((𝑓‘𝑛)(.r‘𝑅)(𝑛𝑀𝑘)))))(+g‘𝑅)(𝑖𝑀𝑘)), (𝑗𝑀𝑘))))) → ((((((𝑅 ∈ Field ∧ 𝑀:(𝐼 × 𝐼)⟶(Base‘𝑅)) ∧ 𝐼 ∈ Fin) ∧ 𝑓:𝐼⟶(Base‘𝑅)) ∧ (𝑖 ∈ 𝐼 ∧ (𝑓‘𝑖) ≠ (0g‘𝑅))) ∧ (¬ 𝑖 ∈ (𝑥 ∪ {𝑧}) ∧ (𝑥 ∪ {𝑧}) ⊆ 𝐼)) → ((𝐼 maDet 𝑅)‘𝑀) = ((𝐼 maDet 𝑅)‘(𝑗 ∈ 𝐼, 𝑘 ∈ 𝐼 ↦ if(𝑗 = 𝑖, ((𝑅 Σg (𝑛 ∈ (𝑥 ∪ {𝑧}) ↦ (((invr‘𝑅)‘(𝑓‘𝑖))(.r‘𝑅)((𝑓‘𝑛)(.r‘𝑅)(𝑛𝑀𝑘)))))(+g‘𝑅)(𝑖𝑀𝑘)), (𝑗𝑀𝑘)))))))
375	149, 162, 175, 188, 210, 374	findcard2s 9181	. . . . . . . . . . . 12 ⊢ ((𝐼 ∖ {𝑖}) ∈ Fin → ((((((𝑅 ∈ Field ∧ 𝑀:(𝐼 × 𝐼)⟶(Base‘𝑅)) ∧ 𝐼 ∈ Fin) ∧ 𝑓:𝐼⟶(Base‘𝑅)) ∧ (𝑖 ∈ 𝐼 ∧ (𝑓‘𝑖) ≠ (0g‘𝑅))) ∧ (¬ 𝑖 ∈ (𝐼 ∖ {𝑖}) ∧ (𝐼 ∖ {𝑖}) ⊆ 𝐼)) → ((𝐼 maDet 𝑅)‘𝑀) = ((𝐼 maDet 𝑅)‘(𝑗 ∈ 𝐼, 𝑘 ∈ 𝐼 ↦ if(𝑗 = 𝑖, ((𝑅 Σg (𝑛 ∈ (𝐼 ∖ {𝑖}) ↦ (((invr‘𝑅)‘(𝑓‘𝑖))(.r‘𝑅)((𝑓‘𝑛)(.r‘𝑅)(𝑛𝑀𝑘)))))(+g‘𝑅)(𝑖𝑀𝑘)), (𝑗𝑀𝑘))))))
376	132, 375	mpcom 39	. . . . . . . . . . 11 ⊢ ((((((𝑅 ∈ Field ∧ 𝑀:(𝐼 × 𝐼)⟶(Base‘𝑅)) ∧ 𝐼 ∈ Fin) ∧ 𝑓:𝐼⟶(Base‘𝑅)) ∧ (𝑖 ∈ 𝐼 ∧ (𝑓‘𝑖) ≠ (0g‘𝑅))) ∧ (¬ 𝑖 ∈ (𝐼 ∖ {𝑖}) ∧ (𝐼 ∖ {𝑖}) ⊆ 𝐼)) → ((𝐼 maDet 𝑅)‘𝑀) = ((𝐼 maDet 𝑅)‘(𝑗 ∈ 𝐼, 𝑘 ∈ 𝐼 ↦ if(𝑗 = 𝑖, ((𝑅 Σg (𝑛 ∈ (𝐼 ∖ {𝑖}) ↦ (((invr‘𝑅)‘(𝑓‘𝑖))(.r‘𝑅)((𝑓‘𝑛)(.r‘𝑅)(𝑛𝑀𝑘)))))(+g‘𝑅)(𝑖𝑀𝑘)), (𝑗𝑀𝑘)))))
377	129, 130, 376	mpanr12 718	. . . . . . . . . 10 ⊢ (((((𝑅 ∈ Field ∧ 𝑀:(𝐼 × 𝐼)⟶(Base‘𝑅)) ∧ 𝐼 ∈ Fin) ∧ 𝑓:𝐼⟶(Base‘𝑅)) ∧ (𝑖 ∈ 𝐼 ∧ (𝑓‘𝑖) ≠ (0g‘𝑅))) → ((𝐼 maDet 𝑅)‘𝑀) = ((𝐼 maDet 𝑅)‘(𝑗 ∈ 𝐼, 𝑘 ∈ 𝐼 ↦ if(𝑗 = 𝑖, ((𝑅 Σg (𝑛 ∈ (𝐼 ∖ {𝑖}) ↦ (((invr‘𝑅)‘(𝑓‘𝑖))(.r‘𝑅)((𝑓‘𝑛)(.r‘𝑅)(𝑛𝑀𝑘)))))(+g‘𝑅)(𝑖𝑀𝑘)), (𝑗𝑀𝑘)))))
378	377	adantlr 728	. . . . . . . . 9 ⊢ ((((((𝑅 ∈ Field ∧ 𝑀:(𝐼 × 𝐼)⟶(Base‘𝑅)) ∧ 𝐼 ∈ Fin) ∧ 𝑓:𝐼⟶(Base‘𝑅)) ∧ (𝑘 ∈ 𝐼 ↦ (𝑅 Σg (𝑛 ∈ 𝐼 ↦ ((𝑓‘𝑛)(.r‘𝑅)(𝑛𝑀𝑘))))) = (𝐼 × {(0g‘𝑅)})) ∧ (𝑖 ∈ 𝐼 ∧ (𝑓‘𝑖) ≠ (0g‘𝑅))) → ((𝐼 maDet 𝑅)‘𝑀) = ((𝐼 maDet 𝑅)‘(𝑗 ∈ 𝐼, 𝑘 ∈ 𝐼 ↦ if(𝑗 = 𝑖, ((𝑅 Σg (𝑛 ∈ (𝐼 ∖ {𝑖}) ↦ (((invr‘𝑅)‘(𝑓‘𝑖))(.r‘𝑅)((𝑓‘𝑛)(.r‘𝑅)(𝑛𝑀𝑘)))))(+g‘𝑅)(𝑖𝑀𝑘)), (𝑗𝑀𝑘)))))
379		eqid 2761	. . . . . . . . . . . 12 ⊢ 𝐼 = 𝐼
380		fconstmpt 5713	. . . . . . . . . . . . . . . . 17 ⊢ (𝐼 × {(0g‘𝑅)}) = (𝑘 ∈ 𝐼 ↦ (0g‘𝑅))
381	380	eqeq2i 2774	. . . . . . . . . . . . . . . 16 ⊢ ((𝑘 ∈ 𝐼 ↦ (𝑅 Σg (𝑛 ∈ 𝐼 ↦ ((𝑓‘𝑛)(.r‘𝑅)(𝑛𝑀𝑘))))) = (𝐼 × {(0g‘𝑅)}) ↔ (𝑘 ∈ 𝐼 ↦ (𝑅 Σg (𝑛 ∈ 𝐼 ↦ ((𝑓‘𝑛)(.r‘𝑅)(𝑛𝑀𝑘))))) = (𝑘 ∈ 𝐼 ↦ (0g‘𝑅)))
382		ovex 7453	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑅 Σg (𝑛 ∈ 𝐼 ↦ ((𝑓‘𝑛)(.r‘𝑅)(𝑛𝑀𝑘)))) ∈ V
383	382	rgenw 3081	. . . . . . . . . . . . . . . . 17 ⊢ ∀𝑘 ∈ 𝐼 (𝑅 Σg (𝑛 ∈ 𝐼 ↦ ((𝑓‘𝑛)(.r‘𝑅)(𝑛𝑀𝑘)))) ∈ V
384		mpteqb 7013	. . . . . . . . . . . . . . . . 17 ⊢ (∀𝑘 ∈ 𝐼 (𝑅 Σg (𝑛 ∈ 𝐼 ↦ ((𝑓‘𝑛)(.r‘𝑅)(𝑛𝑀𝑘)))) ∈ V → ((𝑘 ∈ 𝐼 ↦ (𝑅 Σg (𝑛 ∈ 𝐼 ↦ ((𝑓‘𝑛)(.r‘𝑅)(𝑛𝑀𝑘))))) = (𝑘 ∈ 𝐼 ↦ (0g‘𝑅)) ↔ ∀𝑘 ∈ 𝐼 (𝑅 Σg (𝑛 ∈ 𝐼 ↦ ((𝑓‘𝑛)(.r‘𝑅)(𝑛𝑀𝑘)))) = (0g‘𝑅)))
385	383, 384	ax-mp 5	. . . . . . . . . . . . . . . 16 ⊢ ((𝑘 ∈ 𝐼 ↦ (𝑅 Σg (𝑛 ∈ 𝐼 ↦ ((𝑓‘𝑛)(.r‘𝑅)(𝑛𝑀𝑘))))) = (𝑘 ∈ 𝐼 ↦ (0g‘𝑅)) ↔ ∀𝑘 ∈ 𝐼 (𝑅 Σg (𝑛 ∈ 𝐼 ↦ ((𝑓‘𝑛)(.r‘𝑅)(𝑛𝑀𝑘)))) = (0g‘𝑅))
386	381, 385	bitri 278	. . . . . . . . . . . . . . 15 ⊢ ((𝑘 ∈ 𝐼 ↦ (𝑅 Σg (𝑛 ∈ 𝐼 ↦ ((𝑓‘𝑛)(.r‘𝑅)(𝑛𝑀𝑘))))) = (𝐼 × {(0g‘𝑅)}) ↔ ∀𝑘 ∈ 𝐼 (𝑅 Σg (𝑛 ∈ 𝐼 ↦ ((𝑓‘𝑛)(.r‘𝑅)(𝑛𝑀𝑘)))) = (0g‘𝑅))
387	225	ad5antr 747	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ ((((((𝑅 ∈ Field ∧ 𝑀:(𝐼 × 𝐼)⟶(Base‘𝑅)) ∧ 𝐼 ∈ Fin) ∧ 𝑓:𝐼⟶(Base‘𝑅)) ∧ (𝑖 ∈ 𝐼 ∧ (𝑓‘𝑖) ≠ (0g‘𝑅))) ∧ 𝑘 ∈ 𝐼) → 𝑅 ∈ CMnd)
388		simp-4r 796	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ ((((((𝑅 ∈ Field ∧ 𝑀:(𝐼 × 𝐼)⟶(Base‘𝑅)) ∧ 𝐼 ∈ Fin) ∧ 𝑓:𝐼⟶(Base‘𝑅)) ∧ (𝑖 ∈ 𝐼 ∧ (𝑓‘𝑖) ≠ (0g‘𝑅))) ∧ 𝑘 ∈ 𝐼) → 𝐼 ∈ Fin)
389		eqid 2761	. . . . . . . . . . . . . . . . . . . . . . . . . . 27 ⊢ (𝑛 ∈ 𝐼 ↦ (((invr‘𝑅)‘(𝑓‘𝑖))(.r‘𝑅)((𝑓‘𝑛)(.r‘𝑅)(𝑛𝑀𝑘)))) = (𝑛 ∈ 𝐼 ↦ (((invr‘𝑅)‘(𝑓‘𝑖))(.r‘𝑅)((𝑓‘𝑛)(.r‘𝑅)(𝑛𝑀𝑘))))
390	307, 389	fnmpti 6682	. . . . . . . . . . . . . . . . . . . . . . . . . 26 ⊢ (𝑛 ∈ 𝐼 ↦ (((invr‘𝑅)‘(𝑓‘𝑖))(.r‘𝑅)((𝑓‘𝑛)(.r‘𝑅)(𝑛𝑀𝑘)))) Fn 𝐼
391	390	a1i 11	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ (𝐼 ∈ Fin → (𝑛 ∈ 𝐼 ↦ (((invr‘𝑅)‘(𝑓‘𝑖))(.r‘𝑅)((𝑓‘𝑛)(.r‘𝑅)(𝑛𝑀𝑘)))) Fn 𝐼)
392		id 23	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ (𝐼 ∈ Fin → 𝐼 ∈ Fin)
393		fvexd 6900	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ (𝐼 ∈ Fin → (0g‘𝑅) ∈ V)
394	391, 392, 393	fndmfifsupp 9370	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ (𝐼 ∈ Fin → (𝑛 ∈ 𝐼 ↦ (((invr‘𝑅)‘(𝑓‘𝑖))(.r‘𝑅)((𝑓‘𝑛)(.r‘𝑅)(𝑛𝑀𝑘)))) finSupp (0g‘𝑅))
395	394	ad4antlr 746	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ ((((((𝑅 ∈ Field ∧ 𝑀:(𝐼 × 𝐼)⟶(Base‘𝑅)) ∧ 𝐼 ∈ Fin) ∧ 𝑓:𝐼⟶(Base‘𝑅)) ∧ (𝑖 ∈ 𝐼 ∧ (𝑓‘𝑖) ≠ (0g‘𝑅))) ∧ 𝑘 ∈ 𝐼) → (𝑛 ∈ 𝐼 ↦ (((invr‘𝑅)‘(𝑓‘𝑖))(.r‘𝑅)((𝑓‘𝑛)(.r‘𝑅)(𝑛𝑀𝑘)))) finSupp (0g‘𝑅))
396		simplrl 789	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ ((((((𝑅 ∈ Field ∧ 𝑀:(𝐼 × 𝐼)⟶(Base‘𝑅)) ∧ 𝐼 ∈ Fin) ∧ 𝑓:𝐼⟶(Base‘𝑅)) ∧ (𝑖 ∈ 𝐼 ∧ (𝑓‘𝑖) ≠ (0g‘𝑅))) ∧ 𝑘 ∈ 𝐼) → 𝑖 ∈ 𝐼)
397	320, 323	sylan 592	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ ((((((𝑅 ∈ Field ∧ 𝑀:(𝐼 × 𝐼)⟶(Base‘𝑅)) ∧ 𝐼 ∈ Fin) ∧ 𝑓:𝐼⟶(Base‘𝑅)) ∧ (𝑖 ∈ 𝐼 ∧ (𝑓‘𝑖) ≠ (0g‘𝑅))) ∧ 𝑘 ∈ 𝐼) → (𝑖𝑀𝑘) ∈ (Base‘𝑅))
398		fveq2 6885	. . . . . . . . . . . . . . . . . . . . . . . . . 26 ⊢ (𝑛 = 𝑖 → (𝑓‘𝑛) = (𝑓‘𝑖))
399		oveq1 7427	. . . . . . . . . . . . . . . . . . . . . . . . . 26 ⊢ (𝑛 = 𝑖 → (𝑛𝑀𝑘) = (𝑖𝑀𝑘))
400	398, 399	oveq12d 7438	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ (𝑛 = 𝑖 → ((𝑓‘𝑛)(.r‘𝑅)(𝑛𝑀𝑘)) = ((𝑓‘𝑖)(.r‘𝑅)(𝑖𝑀𝑘)))
401	400	oveq2d 7436	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ (𝑛 = 𝑖 → (((invr‘𝑅)‘(𝑓‘𝑖))(.r‘𝑅)((𝑓‘𝑛)(.r‘𝑅)(𝑛𝑀𝑘))) = (((invr‘𝑅)‘(𝑓‘𝑖))(.r‘𝑅)((𝑓‘𝑖)(.r‘𝑅)(𝑖𝑀𝑘))))
402		simpll 779	. . . . . . . . . . . . . . . . . . . . . . . . . . 27 ⊢ (((𝑅 ∈ Field ∧ 𝑀:(𝐼 × 𝐼)⟶(Base‘𝑅)) ∧ 𝐼 ∈ Fin) → 𝑅 ∈ Field)
403	2, 237	anim12i 625	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31 ⊢ ((𝑅 ∈ Field ∧ (𝑓:𝐼⟶(Base‘𝑅) ∧ 𝑖 ∈ 𝐼)) → (𝑅 ∈ DivRing ∧ (𝑓‘𝑖) ∈ (Base‘𝑅)))
404	403	anassrs 473	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30 ⊢ (((𝑅 ∈ Field ∧ 𝑓:𝐼⟶(Base‘𝑅)) ∧ 𝑖 ∈ 𝐼) → (𝑅 ∈ DivRing ∧ (𝑓‘𝑖) ∈ (Base‘𝑅)))
405		eqid 2761	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32 ⊢ (1r‘𝑅) = (1r‘𝑅)
406	10, 100, 47, 405, 240	drnginvrl 21014	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31 ⊢ ((𝑅 ∈ DivRing ∧ (𝑓‘𝑖) ∈ (Base‘𝑅) ∧ (𝑓‘𝑖) ≠ (0g‘𝑅)) → (((invr‘𝑅)‘(𝑓‘𝑖))(.r‘𝑅)(𝑓‘𝑖)) = (1r‘𝑅))
407	406	3expa 1136	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30 ⊢ (((𝑅 ∈ DivRing ∧ (𝑓‘𝑖) ∈ (Base‘𝑅)) ∧ (𝑓‘𝑖) ≠ (0g‘𝑅)) → (((invr‘𝑅)‘(𝑓‘𝑖))(.r‘𝑅)(𝑓‘𝑖)) = (1r‘𝑅))
408	404, 407	sylan 592	. . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 ⊢ ((((𝑅 ∈ Field ∧ 𝑓:𝐼⟶(Base‘𝑅)) ∧ 𝑖 ∈ 𝐼) ∧ (𝑓‘𝑖) ≠ (0g‘𝑅)) → (((invr‘𝑅)‘(𝑓‘𝑖))(.r‘𝑅)(𝑓‘𝑖)) = (1r‘𝑅))
409	408	anasss 472	. . . . . . . . . . . . . . . . . . . . . . . . . . . 28 ⊢ (((𝑅 ∈ Field ∧ 𝑓:𝐼⟶(Base‘𝑅)) ∧ (𝑖 ∈ 𝐼 ∧ (𝑓‘𝑖) ≠ (0g‘𝑅))) → (((invr‘𝑅)‘(𝑓‘𝑖))(.r‘𝑅)(𝑓‘𝑖)) = (1r‘𝑅))
410	409	oveq1d 7435	. . . . . . . . . . . . . . . . . . . . . . . . . . 27 ⊢ (((𝑅 ∈ Field ∧ 𝑓:𝐼⟶(Base‘𝑅)) ∧ (𝑖 ∈ 𝐼 ∧ (𝑓‘𝑖) ≠ (0g‘𝑅))) → ((((invr‘𝑅)‘(𝑓‘𝑖))(.r‘𝑅)(𝑓‘𝑖))(.r‘𝑅)(𝑖𝑀𝑘)) = ((1r‘𝑅)(.r‘𝑅)(𝑖𝑀𝑘)))
411	402, 410	sylanl1 693	. . . . . . . . . . . . . . . . . . . . . . . . . 26 ⊢ (((((𝑅 ∈ Field ∧ 𝑀:(𝐼 × 𝐼)⟶(Base‘𝑅)) ∧ 𝐼 ∈ Fin) ∧ 𝑓:𝐼⟶(Base‘𝑅)) ∧ (𝑖 ∈ 𝐼 ∧ (𝑓‘𝑖) ≠ (0g‘𝑅))) → ((((invr‘𝑅)‘(𝑓‘𝑖))(.r‘𝑅)(𝑓‘𝑖))(.r‘𝑅)(𝑖𝑀𝑘)) = ((1r‘𝑅)(.r‘𝑅)(𝑖𝑀𝑘)))
412	411	adantr 486	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ ((((((𝑅 ∈ Field ∧ 𝑀:(𝐼 × 𝐼)⟶(Base‘𝑅)) ∧ 𝐼 ∈ Fin) ∧ 𝑓:𝐼⟶(Base‘𝑅)) ∧ (𝑖 ∈ 𝐼 ∧ (𝑓‘𝑖) ≠ (0g‘𝑅))) ∧ 𝑘 ∈ 𝐼) → ((((invr‘𝑅)‘(𝑓‘𝑖))(.r‘𝑅)(𝑓‘𝑖))(.r‘𝑅)(𝑖𝑀𝑘)) = ((1r‘𝑅)(.r‘𝑅)(𝑖𝑀𝑘)))
413	4	ad5antr 747	. . . . . . . . . . . . . . . . . . . . . . . . . 26 ⊢ ((((((𝑅 ∈ Field ∧ 𝑀:(𝐼 × 𝐼)⟶(Base‘𝑅)) ∧ 𝐼 ∈ Fin) ∧ 𝑓:𝐼⟶(Base‘𝑅)) ∧ (𝑖 ∈ 𝐼 ∧ (𝑓‘𝑖) ≠ (0g‘𝑅))) ∧ 𝑘 ∈ 𝐼) → 𝑅 ∈ Ring)
414	245	adantr 486	. . . . . . . . . . . . . . . . . . . . . . . . . 26 ⊢ ((((((𝑅 ∈ Field ∧ 𝑀:(𝐼 × 𝐼)⟶(Base‘𝑅)) ∧ 𝐼 ∈ Fin) ∧ 𝑓:𝐼⟶(Base‘𝑅)) ∧ (𝑖 ∈ 𝐼 ∧ (𝑓‘𝑖) ≠ (0g‘𝑅))) ∧ 𝑘 ∈ 𝐼) → ((invr‘𝑅)‘(𝑓‘𝑖)) ∈ (Base‘𝑅))
415	237	ad2ant2lr 761	. . . . . . . . . . . . . . . . . . . . . . . . . . 27 ⊢ (((((𝑅 ∈ Field ∧ 𝑀:(𝐼 × 𝐼)⟶(Base‘𝑅)) ∧ 𝐼 ∈ Fin) ∧ 𝑓:𝐼⟶(Base‘𝑅)) ∧ (𝑖 ∈ 𝐼 ∧ (𝑓‘𝑖) ≠ (0g‘𝑅))) → (𝑓‘𝑖) ∈ (Base‘𝑅))
416	415	adantr 486	. . . . . . . . . . . . . . . . . . . . . . . . . 26 ⊢ ((((((𝑅 ∈ Field ∧ 𝑀:(𝐼 × 𝐼)⟶(Base‘𝑅)) ∧ 𝐼 ∈ Fin) ∧ 𝑓:𝐼⟶(Base‘𝑅)) ∧ (𝑖 ∈ 𝐼 ∧ (𝑓‘𝑖) ≠ (0g‘𝑅))) ∧ 𝑘 ∈ 𝐼) → (𝑓‘𝑖) ∈ (Base‘𝑅))
417	10, 47	ringass 20480	. . . . . . . . . . . . . . . . . . . . . . . . . 26 ⊢ ((𝑅 ∈ Ring ∧ (((invr‘𝑅)‘(𝑓‘𝑖)) ∈ (Base‘𝑅) ∧ (𝑓‘𝑖) ∈ (Base‘𝑅) ∧ (𝑖𝑀𝑘) ∈ (Base‘𝑅))) → ((((invr‘𝑅)‘(𝑓‘𝑖))(.r‘𝑅)(𝑓‘𝑖))(.r‘𝑅)(𝑖𝑀𝑘)) = (((invr‘𝑅)‘(𝑓‘𝑖))(.r‘𝑅)((𝑓‘𝑖)(.r‘𝑅)(𝑖𝑀𝑘))))
418	413, 414, 416, 397, 417	syl13anc 1399	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ ((((((𝑅 ∈ Field ∧ 𝑀:(𝐼 × 𝐼)⟶(Base‘𝑅)) ∧ 𝐼 ∈ Fin) ∧ 𝑓:𝐼⟶(Base‘𝑅)) ∧ (𝑖 ∈ 𝐼 ∧ (𝑓‘𝑖) ≠ (0g‘𝑅))) ∧ 𝑘 ∈ 𝐼) → ((((invr‘𝑅)‘(𝑓‘𝑖))(.r‘𝑅)(𝑓‘𝑖))(.r‘𝑅)(𝑖𝑀𝑘)) = (((invr‘𝑅)‘(𝑓‘𝑖))(.r‘𝑅)((𝑓‘𝑖)(.r‘𝑅)(𝑖𝑀𝑘))))
419	4	adantr 486	. . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 ⊢ ((𝑅 ∈ Field ∧ 𝑀:(𝐼 × 𝐼)⟶(Base‘𝑅)) → 𝑅 ∈ Ring)
420	419	3ad2ant1 1151	. . . . . . . . . . . . . . . . . . . . . . . . . . . 28 ⊢ (((𝑅 ∈ Field ∧ 𝑀:(𝐼 × 𝐼)⟶(Base‘𝑅)) ∧ 𝑖 ∈ 𝐼 ∧ 𝑘 ∈ 𝐼) → 𝑅 ∈ Ring)
421	322	3adant1l 1195	. . . . . . . . . . . . . . . . . . . . . . . . . . . 28 ⊢ (((𝑅 ∈ Field ∧ 𝑀:(𝐼 × 𝐼)⟶(Base‘𝑅)) ∧ 𝑖 ∈ 𝐼 ∧ 𝑘 ∈ 𝐼) → (𝑖𝑀𝑘) ∈ (Base‘𝑅))
422	10, 47, 405	ringlidm 20498	. . . . . . . . . . . . . . . . . . . . . . . . . . . 28 ⊢ ((𝑅 ∈ Ring ∧ (𝑖𝑀𝑘) ∈ (Base‘𝑅)) → ((1r‘𝑅)(.r‘𝑅)(𝑖𝑀𝑘)) = (𝑖𝑀𝑘))
423	420, 421, 422	syl2anc 596	. . . . . . . . . . . . . . . . . . . . . . . . . . 27 ⊢ (((𝑅 ∈ Field ∧ 𝑀:(𝐼 × 𝐼)⟶(Base‘𝑅)) ∧ 𝑖 ∈ 𝐼 ∧ 𝑘 ∈ 𝐼) → ((1r‘𝑅)(.r‘𝑅)(𝑖𝑀𝑘)) = (𝑖𝑀𝑘))
424	423	ad5ant145 1396	. . . . . . . . . . . . . . . . . . . . . . . . . 26 ⊢ ((((((𝑅 ∈ Field ∧ 𝑀:(𝐼 × 𝐼)⟶(Base‘𝑅)) ∧ 𝐼 ∈ Fin) ∧ 𝑓:𝐼⟶(Base‘𝑅)) ∧ 𝑖 ∈ 𝐼) ∧ 𝑘 ∈ 𝐼) → ((1r‘𝑅)(.r‘𝑅)(𝑖𝑀𝑘)) = (𝑖𝑀𝑘))
425	424	adantlrr 734	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ ((((((𝑅 ∈ Field ∧ 𝑀:(𝐼 × 𝐼)⟶(Base‘𝑅)) ∧ 𝐼 ∈ Fin) ∧ 𝑓:𝐼⟶(Base‘𝑅)) ∧ (𝑖 ∈ 𝐼 ∧ (𝑓‘𝑖) ≠ (0g‘𝑅))) ∧ 𝑘 ∈ 𝐼) → ((1r‘𝑅)(.r‘𝑅)(𝑖𝑀𝑘)) = (𝑖𝑀𝑘))
426	412, 418, 425	3eqtr3d 2804	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ ((((((𝑅 ∈ Field ∧ 𝑀:(𝐼 × 𝐼)⟶(Base‘𝑅)) ∧ 𝐼 ∈ Fin) ∧ 𝑓:𝐼⟶(Base‘𝑅)) ∧ (𝑖 ∈ 𝐼 ∧ (𝑓‘𝑖) ≠ (0g‘𝑅))) ∧ 𝑘 ∈ 𝐼) → (((invr‘𝑅)‘(𝑓‘𝑖))(.r‘𝑅)((𝑓‘𝑖)(.r‘𝑅)(𝑖𝑀𝑘))) = (𝑖𝑀𝑘))
427	401, 426	sylan9eqr 2818	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ (((((((𝑅 ∈ Field ∧ 𝑀:(𝐼 × 𝐼)⟶(Base‘𝑅)) ∧ 𝐼 ∈ Fin) ∧ 𝑓:𝐼⟶(Base‘𝑅)) ∧ (𝑖 ∈ 𝐼 ∧ (𝑓‘𝑖) ≠ (0g‘𝑅))) ∧ 𝑘 ∈ 𝐼) ∧ 𝑛 = 𝑖) → (((invr‘𝑅)‘(𝑓‘𝑖))(.r‘𝑅)((𝑓‘𝑛)(.r‘𝑅)(𝑛𝑀𝑘))) = (𝑖𝑀𝑘))
428	10, 200, 387, 388, 395, 254, 396, 397, 427	gsumdifsnd 20175	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ ((((((𝑅 ∈ Field ∧ 𝑀:(𝐼 × 𝐼)⟶(Base‘𝑅)) ∧ 𝐼 ∈ Fin) ∧ 𝑓:𝐼⟶(Base‘𝑅)) ∧ (𝑖 ∈ 𝐼 ∧ (𝑓‘𝑖) ≠ (0g‘𝑅))) ∧ 𝑘 ∈ 𝐼) → (𝑅 Σg (𝑛 ∈ 𝐼 ↦ (((invr‘𝑅)‘(𝑓‘𝑖))(.r‘𝑅)((𝑓‘𝑛)(.r‘𝑅)(𝑛𝑀𝑘))))) = ((𝑅 Σg (𝑛 ∈ (𝐼 ∖ {𝑖}) ↦ (((invr‘𝑅)‘(𝑓‘𝑖))(.r‘𝑅)((𝑓‘𝑛)(.r‘𝑅)(𝑛𝑀𝑘)))))(+g‘𝑅)(𝑖𝑀𝑘)))
429		ovex 7453	. . . . . . . . . . . . . . . . . . . . . . . . . . 27 ⊢ ((𝑓‘𝑛)(.r‘𝑅)(𝑛𝑀𝑘)) ∈ V
430		eqid 2761	. . . . . . . . . . . . . . . . . . . . . . . . . . 27 ⊢ (𝑛 ∈ 𝐼 ↦ ((𝑓‘𝑛)(.r‘𝑅)(𝑛𝑀𝑘))) = (𝑛 ∈ 𝐼 ↦ ((𝑓‘𝑛)(.r‘𝑅)(𝑛𝑀𝑘)))
431	429, 430	fnmpti 6682	. . . . . . . . . . . . . . . . . . . . . . . . . 26 ⊢ (𝑛 ∈ 𝐼 ↦ ((𝑓‘𝑛)(.r‘𝑅)(𝑛𝑀𝑘))) Fn 𝐼
432	431	a1i 11	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ (𝐼 ∈ Fin → (𝑛 ∈ 𝐼 ↦ ((𝑓‘𝑛)(.r‘𝑅)(𝑛𝑀𝑘))) Fn 𝐼)
433	432, 392, 393	fndmfifsupp 9370	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ (𝐼 ∈ Fin → (𝑛 ∈ 𝐼 ↦ ((𝑓‘𝑛)(.r‘𝑅)(𝑛𝑀𝑘))) finSupp (0g‘𝑅))
434	433	ad4antlr 746	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ ((((((𝑅 ∈ Field ∧ 𝑀:(𝐼 × 𝐼)⟶(Base‘𝑅)) ∧ 𝐼 ∈ Fin) ∧ 𝑓:𝐼⟶(Base‘𝑅)) ∧ (𝑖 ∈ 𝐼 ∧ (𝑓‘𝑖) ≠ (0g‘𝑅))) ∧ 𝑘 ∈ 𝐼) → (𝑛 ∈ 𝐼 ↦ ((𝑓‘𝑛)(.r‘𝑅)(𝑛𝑀𝑘))) finSupp (0g‘𝑅))
435	10, 100, 47, 413, 388, 414, 252, 434	gsummulc2 20546	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ ((((((𝑅 ∈ Field ∧ 𝑀:(𝐼 × 𝐼)⟶(Base‘𝑅)) ∧ 𝐼 ∈ Fin) ∧ 𝑓:𝐼⟶(Base‘𝑅)) ∧ (𝑖 ∈ 𝐼 ∧ (𝑓‘𝑖) ≠ (0g‘𝑅))) ∧ 𝑘 ∈ 𝐼) → (𝑅 Σg (𝑛 ∈ 𝐼 ↦ (((invr‘𝑅)‘(𝑓‘𝑖))(.r‘𝑅)((𝑓‘𝑛)(.r‘𝑅)(𝑛𝑀𝑘))))) = (((invr‘𝑅)‘(𝑓‘𝑖))(.r‘𝑅)(𝑅 Σg (𝑛 ∈ 𝐼 ↦ ((𝑓‘𝑛)(.r‘𝑅)(𝑛𝑀𝑘))))))
436	428, 435	eqtr3d 2798	. . . . . . . . . . . . . . . . . . . . 21 ⊢ ((((((𝑅 ∈ Field ∧ 𝑀:(𝐼 × 𝐼)⟶(Base‘𝑅)) ∧ 𝐼 ∈ Fin) ∧ 𝑓:𝐼⟶(Base‘𝑅)) ∧ (𝑖 ∈ 𝐼 ∧ (𝑓‘𝑖) ≠ (0g‘𝑅))) ∧ 𝑘 ∈ 𝐼) → ((𝑅 Σg (𝑛 ∈ (𝐼 ∖ {𝑖}) ↦ (((invr‘𝑅)‘(𝑓‘𝑖))(.r‘𝑅)((𝑓‘𝑛)(.r‘𝑅)(𝑛𝑀𝑘)))))(+g‘𝑅)(𝑖𝑀𝑘)) = (((invr‘𝑅)‘(𝑓‘𝑖))(.r‘𝑅)(𝑅 Σg (𝑛 ∈ 𝐼 ↦ ((𝑓‘𝑛)(.r‘𝑅)(𝑛𝑀𝑘))))))
437	436	adantr 486	. . . . . . . . . . . . . . . . . . . 20 ⊢ (((((((𝑅 ∈ Field ∧ 𝑀:(𝐼 × 𝐼)⟶(Base‘𝑅)) ∧ 𝐼 ∈ Fin) ∧ 𝑓:𝐼⟶(Base‘𝑅)) ∧ (𝑖 ∈ 𝐼 ∧ (𝑓‘𝑖) ≠ (0g‘𝑅))) ∧ 𝑘 ∈ 𝐼) ∧ (𝑅 Σg (𝑛 ∈ 𝐼 ↦ ((𝑓‘𝑛)(.r‘𝑅)(𝑛𝑀𝑘)))) = (0g‘𝑅)) → ((𝑅 Σg (𝑛 ∈ (𝐼 ∖ {𝑖}) ↦ (((invr‘𝑅)‘(𝑓‘𝑖))(.r‘𝑅)((𝑓‘𝑛)(.r‘𝑅)(𝑛𝑀𝑘)))))(+g‘𝑅)(𝑖𝑀𝑘)) = (((invr‘𝑅)‘(𝑓‘𝑖))(.r‘𝑅)(𝑅 Σg (𝑛 ∈ 𝐼 ↦ ((𝑓‘𝑛)(.r‘𝑅)(𝑛𝑀𝑘))))))
438		oveq2 7428	. . . . . . . . . . . . . . . . . . . . 21 ⊢ ((𝑅 Σg (𝑛 ∈ 𝐼 ↦ ((𝑓‘𝑛)(.r‘𝑅)(𝑛𝑀𝑘)))) = (0g‘𝑅) → (((invr‘𝑅)‘(𝑓‘𝑖))(.r‘𝑅)(𝑅 Σg (𝑛 ∈ 𝐼 ↦ ((𝑓‘𝑛)(.r‘𝑅)(𝑛𝑀𝑘))))) = (((invr‘𝑅)‘(𝑓‘𝑖))(.r‘𝑅)(0g‘𝑅)))
439	438	adantl 487	. . . . . . . . . . . . . . . . . . . 20 ⊢ (((((((𝑅 ∈ Field ∧ 𝑀:(𝐼 × 𝐼)⟶(Base‘𝑅)) ∧ 𝐼 ∈ Fin) ∧ 𝑓:𝐼⟶(Base‘𝑅)) ∧ (𝑖 ∈ 𝐼 ∧ (𝑓‘𝑖) ≠ (0g‘𝑅))) ∧ 𝑘 ∈ 𝐼) ∧ (𝑅 Σg (𝑛 ∈ 𝐼 ↦ ((𝑓‘𝑛)(.r‘𝑅)(𝑛𝑀𝑘)))) = (0g‘𝑅)) → (((invr‘𝑅)‘(𝑓‘𝑖))(.r‘𝑅)(𝑅 Σg (𝑛 ∈ 𝐼 ↦ ((𝑓‘𝑛)(.r‘𝑅)(𝑛𝑀𝑘))))) = (((invr‘𝑅)‘(𝑓‘𝑖))(.r‘𝑅)(0g‘𝑅)))
440	4	ad4antr 745	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (((((𝑅 ∈ Field ∧ 𝑀:(𝐼 × 𝐼)⟶(Base‘𝑅)) ∧ 𝐼 ∈ Fin) ∧ 𝑓:𝐼⟶(Base‘𝑅)) ∧ (𝑖 ∈ 𝐼 ∧ (𝑓‘𝑖) ≠ (0g‘𝑅))) → 𝑅 ∈ Ring)
441	10, 47, 100	ringrz 20525	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ ((𝑅 ∈ Ring ∧ ((invr‘𝑅)‘(𝑓‘𝑖)) ∈ (Base‘𝑅)) → (((invr‘𝑅)‘(𝑓‘𝑖))(.r‘𝑅)(0g‘𝑅)) = (0g‘𝑅))
442	440, 245, 441	syl2anc 596	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (((((𝑅 ∈ Field ∧ 𝑀:(𝐼 × 𝐼)⟶(Base‘𝑅)) ∧ 𝐼 ∈ Fin) ∧ 𝑓:𝐼⟶(Base‘𝑅)) ∧ (𝑖 ∈ 𝐼 ∧ (𝑓‘𝑖) ≠ (0g‘𝑅))) → (((invr‘𝑅)‘(𝑓‘𝑖))(.r‘𝑅)(0g‘𝑅)) = (0g‘𝑅))
443	442	ad2antrr 739	. . . . . . . . . . . . . . . . . . . 20 ⊢ (((((((𝑅 ∈ Field ∧ 𝑀:(𝐼 × 𝐼)⟶(Base‘𝑅)) ∧ 𝐼 ∈ Fin) ∧ 𝑓:𝐼⟶(Base‘𝑅)) ∧ (𝑖 ∈ 𝐼 ∧ (𝑓‘𝑖) ≠ (0g‘𝑅))) ∧ 𝑘 ∈ 𝐼) ∧ (𝑅 Σg (𝑛 ∈ 𝐼 ↦ ((𝑓‘𝑛)(.r‘𝑅)(𝑛𝑀𝑘)))) = (0g‘𝑅)) → (((invr‘𝑅)‘(𝑓‘𝑖))(.r‘𝑅)(0g‘𝑅)) = (0g‘𝑅))
444	437, 439, 443	3eqtrd 2800	. . . . . . . . . . . . . . . . . . 19 ⊢ (((((((𝑅 ∈ Field ∧ 𝑀:(𝐼 × 𝐼)⟶(Base‘𝑅)) ∧ 𝐼 ∈ Fin) ∧ 𝑓:𝐼⟶(Base‘𝑅)) ∧ (𝑖 ∈ 𝐼 ∧ (𝑓‘𝑖) ≠ (0g‘𝑅))) ∧ 𝑘 ∈ 𝐼) ∧ (𝑅 Σg (𝑛 ∈ 𝐼 ↦ ((𝑓‘𝑛)(.r‘𝑅)(𝑛𝑀𝑘)))) = (0g‘𝑅)) → ((𝑅 Σg (𝑛 ∈ (𝐼 ∖ {𝑖}) ↦ (((invr‘𝑅)‘(𝑓‘𝑖))(.r‘𝑅)((𝑓‘𝑛)(.r‘𝑅)(𝑛𝑀𝑘)))))(+g‘𝑅)(𝑖𝑀𝑘)) = (0g‘𝑅))
445	444	ifeq1d 4502	. . . . . . . . . . . . . . . . . 18 ⊢ (((((((𝑅 ∈ Field ∧ 𝑀:(𝐼 × 𝐼)⟶(Base‘𝑅)) ∧ 𝐼 ∈ Fin) ∧ 𝑓:𝐼⟶(Base‘𝑅)) ∧ (𝑖 ∈ 𝐼 ∧ (𝑓‘𝑖) ≠ (0g‘𝑅))) ∧ 𝑘 ∈ 𝐼) ∧ (𝑅 Σg (𝑛 ∈ 𝐼 ↦ ((𝑓‘𝑛)(.r‘𝑅)(𝑛𝑀𝑘)))) = (0g‘𝑅)) → if(𝑗 = 𝑖, ((𝑅 Σg (𝑛 ∈ (𝐼 ∖ {𝑖}) ↦ (((invr‘𝑅)‘(𝑓‘𝑖))(.r‘𝑅)((𝑓‘𝑛)(.r‘𝑅)(𝑛𝑀𝑘)))))(+g‘𝑅)(𝑖𝑀𝑘)), (𝑗𝑀𝑘)) = if(𝑗 = 𝑖, (0g‘𝑅), (𝑗𝑀𝑘)))
446	445	ex 418	. . . . . . . . . . . . . . . . 17 ⊢ ((((((𝑅 ∈ Field ∧ 𝑀:(𝐼 × 𝐼)⟶(Base‘𝑅)) ∧ 𝐼 ∈ Fin) ∧ 𝑓:𝐼⟶(Base‘𝑅)) ∧ (𝑖 ∈ 𝐼 ∧ (𝑓‘𝑖) ≠ (0g‘𝑅))) ∧ 𝑘 ∈ 𝐼) → ((𝑅 Σg (𝑛 ∈ 𝐼 ↦ ((𝑓‘𝑛)(.r‘𝑅)(𝑛𝑀𝑘)))) = (0g‘𝑅) → if(𝑗 = 𝑖, ((𝑅 Σg (𝑛 ∈ (𝐼 ∖ {𝑖}) ↦ (((invr‘𝑅)‘(𝑓‘𝑖))(.r‘𝑅)((𝑓‘𝑛)(.r‘𝑅)(𝑛𝑀𝑘)))))(+g‘𝑅)(𝑖𝑀𝑘)), (𝑗𝑀𝑘)) = if(𝑗 = 𝑖, (0g‘𝑅), (𝑗𝑀𝑘))))
447	446	ralimdva 3175	. . . . . . . . . . . . . . . 16 ⊢ (((((𝑅 ∈ Field ∧ 𝑀:(𝐼 × 𝐼)⟶(Base‘𝑅)) ∧ 𝐼 ∈ Fin) ∧ 𝑓:𝐼⟶(Base‘𝑅)) ∧ (𝑖 ∈ 𝐼 ∧ (𝑓‘𝑖) ≠ (0g‘𝑅))) → (∀𝑘 ∈ 𝐼 (𝑅 Σg (𝑛 ∈ 𝐼 ↦ ((𝑓‘𝑛)(.r‘𝑅)(𝑛𝑀𝑘)))) = (0g‘𝑅) → ∀𝑘 ∈ 𝐼 if(𝑗 = 𝑖, ((𝑅 Σg (𝑛 ∈ (𝐼 ∖ {𝑖}) ↦ (((invr‘𝑅)‘(𝑓‘𝑖))(.r‘𝑅)((𝑓‘𝑛)(.r‘𝑅)(𝑛𝑀𝑘)))))(+g‘𝑅)(𝑖𝑀𝑘)), (𝑗𝑀𝑘)) = if(𝑗 = 𝑖, (0g‘𝑅), (𝑗𝑀𝑘))))
448	447	imp 412	. . . . . . . . . . . . . . 15 ⊢ ((((((𝑅 ∈ Field ∧ 𝑀:(𝐼 × 𝐼)⟶(Base‘𝑅)) ∧ 𝐼 ∈ Fin) ∧ 𝑓:𝐼⟶(Base‘𝑅)) ∧ (𝑖 ∈ 𝐼 ∧ (𝑓‘𝑖) ≠ (0g‘𝑅))) ∧ ∀𝑘 ∈ 𝐼 (𝑅 Σg (𝑛 ∈ 𝐼 ↦ ((𝑓‘𝑛)(.r‘𝑅)(𝑛𝑀𝑘)))) = (0g‘𝑅)) → ∀𝑘 ∈ 𝐼 if(𝑗 = 𝑖, ((𝑅 Σg (𝑛 ∈ (𝐼 ∖ {𝑖}) ↦ (((invr‘𝑅)‘(𝑓‘𝑖))(.r‘𝑅)((𝑓‘𝑛)(.r‘𝑅)(𝑛𝑀𝑘)))))(+g‘𝑅)(𝑖𝑀𝑘)), (𝑗𝑀𝑘)) = if(𝑗 = 𝑖, (0g‘𝑅), (𝑗𝑀𝑘)))
449	386, 448	sylan2b 606	. . . . . . . . . . . . . 14 ⊢ ((((((𝑅 ∈ Field ∧ 𝑀:(𝐼 × 𝐼)⟶(Base‘𝑅)) ∧ 𝐼 ∈ Fin) ∧ 𝑓:𝐼⟶(Base‘𝑅)) ∧ (𝑖 ∈ 𝐼 ∧ (𝑓‘𝑖) ≠ (0g‘𝑅))) ∧ (𝑘 ∈ 𝐼 ↦ (𝑅 Σg (𝑛 ∈ 𝐼 ↦ ((𝑓‘𝑛)(.r‘𝑅)(𝑛𝑀𝑘))))) = (𝐼 × {(0g‘𝑅)})) → ∀𝑘 ∈ 𝐼 if(𝑗 = 𝑖, ((𝑅 Σg (𝑛 ∈ (𝐼 ∖ {𝑖}) ↦ (((invr‘𝑅)‘(𝑓‘𝑖))(.r‘𝑅)((𝑓‘𝑛)(.r‘𝑅)(𝑛𝑀𝑘)))))(+g‘𝑅)(𝑖𝑀𝑘)), (𝑗𝑀𝑘)) = if(𝑗 = 𝑖, (0g‘𝑅), (𝑗𝑀𝑘)))
450	449, 379	jctil 529	. . . . . . . . . . . . 13 ⊢ ((((((𝑅 ∈ Field ∧ 𝑀:(𝐼 × 𝐼)⟶(Base‘𝑅)) ∧ 𝐼 ∈ Fin) ∧ 𝑓:𝐼⟶(Base‘𝑅)) ∧ (𝑖 ∈ 𝐼 ∧ (𝑓‘𝑖) ≠ (0g‘𝑅))) ∧ (𝑘 ∈ 𝐼 ↦ (𝑅 Σg (𝑛 ∈ 𝐼 ↦ ((𝑓‘𝑛)(.r‘𝑅)(𝑛𝑀𝑘))))) = (𝐼 × {(0g‘𝑅)})) → (𝐼 = 𝐼 ∧ ∀𝑘 ∈ 𝐼 if(𝑗 = 𝑖, ((𝑅 Σg (𝑛 ∈ (𝐼 ∖ {𝑖}) ↦ (((invr‘𝑅)‘(𝑓‘𝑖))(.r‘𝑅)((𝑓‘𝑛)(.r‘𝑅)(𝑛𝑀𝑘)))))(+g‘𝑅)(𝑖𝑀𝑘)), (𝑗𝑀𝑘)) = if(𝑗 = 𝑖, (0g‘𝑅), (𝑗𝑀𝑘))))
451	450	ralrimivw 3159	. . . . . . . . . . . 12 ⊢ ((((((𝑅 ∈ Field ∧ 𝑀:(𝐼 × 𝐼)⟶(Base‘𝑅)) ∧ 𝐼 ∈ Fin) ∧ 𝑓:𝐼⟶(Base‘𝑅)) ∧ (𝑖 ∈ 𝐼 ∧ (𝑓‘𝑖) ≠ (0g‘𝑅))) ∧ (𝑘 ∈ 𝐼 ↦ (𝑅 Σg (𝑛 ∈ 𝐼 ↦ ((𝑓‘𝑛)(.r‘𝑅)(𝑛𝑀𝑘))))) = (𝐼 × {(0g‘𝑅)})) → ∀𝑗 ∈ 𝐼 (𝐼 = 𝐼 ∧ ∀𝑘 ∈ 𝐼 if(𝑗 = 𝑖, ((𝑅 Σg (𝑛 ∈ (𝐼 ∖ {𝑖}) ↦ (((invr‘𝑅)‘(𝑓‘𝑖))(.r‘𝑅)((𝑓‘𝑛)(.r‘𝑅)(𝑛𝑀𝑘)))))(+g‘𝑅)(𝑖𝑀𝑘)), (𝑗𝑀𝑘)) = if(𝑗 = 𝑖, (0g‘𝑅), (𝑗𝑀𝑘))))
452		mpoeq123 7492	. . . . . . . . . . . 12 ⊢ ((𝐼 = 𝐼 ∧ ∀𝑗 ∈ 𝐼 (𝐼 = 𝐼 ∧ ∀𝑘 ∈ 𝐼 if(𝑗 = 𝑖, ((𝑅 Σg (𝑛 ∈ (𝐼 ∖ {𝑖}) ↦ (((invr‘𝑅)‘(𝑓‘𝑖))(.r‘𝑅)((𝑓‘𝑛)(.r‘𝑅)(𝑛𝑀𝑘)))))(+g‘𝑅)(𝑖𝑀𝑘)), (𝑗𝑀𝑘)) = if(𝑗 = 𝑖, (0g‘𝑅), (𝑗𝑀𝑘)))) → (𝑗 ∈ 𝐼, 𝑘 ∈ 𝐼 ↦ if(𝑗 = 𝑖, ((𝑅 Σg (𝑛 ∈ (𝐼 ∖ {𝑖}) ↦ (((invr‘𝑅)‘(𝑓‘𝑖))(.r‘𝑅)((𝑓‘𝑛)(.r‘𝑅)(𝑛𝑀𝑘)))))(+g‘𝑅)(𝑖𝑀𝑘)), (𝑗𝑀𝑘))) = (𝑗 ∈ 𝐼, 𝑘 ∈ 𝐼 ↦ if(𝑗 = 𝑖, (0g‘𝑅), (𝑗𝑀𝑘))))
453	379, 451, 452	sylancr 599	. . . . . . . . . . 11 ⊢ ((((((𝑅 ∈ Field ∧ 𝑀:(𝐼 × 𝐼)⟶(Base‘𝑅)) ∧ 𝐼 ∈ Fin) ∧ 𝑓:𝐼⟶(Base‘𝑅)) ∧ (𝑖 ∈ 𝐼 ∧ (𝑓‘𝑖) ≠ (0g‘𝑅))) ∧ (𝑘 ∈ 𝐼 ↦ (𝑅 Σg (𝑛 ∈ 𝐼 ↦ ((𝑓‘𝑛)(.r‘𝑅)(𝑛𝑀𝑘))))) = (𝐼 × {(0g‘𝑅)})) → (𝑗 ∈ 𝐼, 𝑘 ∈ 𝐼 ↦ if(𝑗 = 𝑖, ((𝑅 Σg (𝑛 ∈ (𝐼 ∖ {𝑖}) ↦ (((invr‘𝑅)‘(𝑓‘𝑖))(.r‘𝑅)((𝑓‘𝑛)(.r‘𝑅)(𝑛𝑀𝑘)))))(+g‘𝑅)(𝑖𝑀𝑘)), (𝑗𝑀𝑘))) = (𝑗 ∈ 𝐼, 𝑘 ∈ 𝐼 ↦ if(𝑗 = 𝑖, (0g‘𝑅), (𝑗𝑀𝑘))))
454	453	an32s 665	. . . . . . . . . 10 ⊢ ((((((𝑅 ∈ Field ∧ 𝑀:(𝐼 × 𝐼)⟶(Base‘𝑅)) ∧ 𝐼 ∈ Fin) ∧ 𝑓:𝐼⟶(Base‘𝑅)) ∧ (𝑘 ∈ 𝐼 ↦ (𝑅 Σg (𝑛 ∈ 𝐼 ↦ ((𝑓‘𝑛)(.r‘𝑅)(𝑛𝑀𝑘))))) = (𝐼 × {(0g‘𝑅)})) ∧ (𝑖 ∈ 𝐼 ∧ (𝑓‘𝑖) ≠ (0g‘𝑅))) → (𝑗 ∈ 𝐼, 𝑘 ∈ 𝐼 ↦ if(𝑗 = 𝑖, ((𝑅 Σg (𝑛 ∈ (𝐼 ∖ {𝑖}) ↦ (((invr‘𝑅)‘(𝑓‘𝑖))(.r‘𝑅)((𝑓‘𝑛)(.r‘𝑅)(𝑛𝑀𝑘)))))(+g‘𝑅)(𝑖𝑀𝑘)), (𝑗𝑀𝑘))) = (𝑗 ∈ 𝐼, 𝑘 ∈ 𝐼 ↦ if(𝑗 = 𝑖, (0g‘𝑅), (𝑗𝑀𝑘))))
455	454	fveq2d 6889	. . . . . . . . 9 ⊢ ((((((𝑅 ∈ Field ∧ 𝑀:(𝐼 × 𝐼)⟶(Base‘𝑅)) ∧ 𝐼 ∈ Fin) ∧ 𝑓:𝐼⟶(Base‘𝑅)) ∧ (𝑘 ∈ 𝐼 ↦ (𝑅 Σg (𝑛 ∈ 𝐼 ↦ ((𝑓‘𝑛)(.r‘𝑅)(𝑛𝑀𝑘))))) = (𝐼 × {(0g‘𝑅)})) ∧ (𝑖 ∈ 𝐼 ∧ (𝑓‘𝑖) ≠ (0g‘𝑅))) → ((𝐼 maDet 𝑅)‘(𝑗 ∈ 𝐼, 𝑘 ∈ 𝐼 ↦ if(𝑗 = 𝑖, ((𝑅 Σg (𝑛 ∈ (𝐼 ∖ {𝑖}) ↦ (((invr‘𝑅)‘(𝑓‘𝑖))(.r‘𝑅)((𝑓‘𝑛)(.r‘𝑅)(𝑛𝑀𝑘)))))(+g‘𝑅)(𝑖𝑀𝑘)), (𝑗𝑀𝑘)))) = ((𝐼 maDet 𝑅)‘(𝑗 ∈ 𝐼, 𝑘 ∈ 𝐼 ↦ if(𝑗 = 𝑖, (0g‘𝑅), (𝑗𝑀𝑘)))))
456	334	ad3antrrr 743	. . . . . . . . . . 11 ⊢ ((((𝑅 ∈ Field ∧ 𝑀:(𝐼 × 𝐼)⟶(Base‘𝑅)) ∧ 𝐼 ∈ Fin) ∧ (𝑖 ∈ 𝐼 ∧ (𝑓‘𝑖) ≠ (0g‘𝑅))) → 𝑅 ∈ CRing)
457		simplr 781	. . . . . . . . . . 11 ⊢ ((((𝑅 ∈ Field ∧ 𝑀:(𝐼 × 𝐼)⟶(Base‘𝑅)) ∧ 𝐼 ∈ Fin) ∧ (𝑖 ∈ 𝐼 ∧ (𝑓‘𝑖) ≠ (0g‘𝑅))) → 𝐼 ∈ Fin)
458		simpllr 788	. . . . . . . . . . . 12 ⊢ ((((𝑅 ∈ Field ∧ 𝑀:(𝐼 × 𝐼)⟶(Base‘𝑅)) ∧ 𝐼 ∈ Fin) ∧ (𝑖 ∈ 𝐼 ∧ (𝑓‘𝑖) ≠ (0g‘𝑅))) → 𝑀:(𝐼 × 𝐼)⟶(Base‘𝑅))
459	458, 198	syl3an1 1181	. . . . . . . . . . 11 ⊢ (((((𝑅 ∈ Field ∧ 𝑀:(𝐼 × 𝐼)⟶(Base‘𝑅)) ∧ 𝐼 ∈ Fin) ∧ (𝑖 ∈ 𝐼 ∧ (𝑓‘𝑖) ≠ (0g‘𝑅))) ∧ 𝑗 ∈ 𝐼 ∧ 𝑘 ∈ 𝐼) → (𝑗𝑀𝑘) ∈ (Base‘𝑅))
460		simprl 783	. . . . . . . . . . 11 ⊢ ((((𝑅 ∈ Field ∧ 𝑀:(𝐼 × 𝐼)⟶(Base‘𝑅)) ∧ 𝐼 ∈ Fin) ∧ (𝑖 ∈ 𝐼 ∧ (𝑓‘𝑖) ≠ (0g‘𝑅))) → 𝑖 ∈ 𝐼)
461	333, 10, 100, 456, 457, 459, 460	mdetr0 22920	. . . . . . . . . 10 ⊢ ((((𝑅 ∈ Field ∧ 𝑀:(𝐼 × 𝐼)⟶(Base‘𝑅)) ∧ 𝐼 ∈ Fin) ∧ (𝑖 ∈ 𝐼 ∧ (𝑓‘𝑖) ≠ (0g‘𝑅))) → ((𝐼 maDet 𝑅)‘(𝑗 ∈ 𝐼, 𝑘 ∈ 𝐼 ↦ if(𝑗 = 𝑖, (0g‘𝑅), (𝑗𝑀𝑘)))) = (0g‘𝑅))
462	461	ad4ant14 765	. . . . . . . . 9 ⊢ ((((((𝑅 ∈ Field ∧ 𝑀:(𝐼 × 𝐼)⟶(Base‘𝑅)) ∧ 𝐼 ∈ Fin) ∧ 𝑓:𝐼⟶(Base‘𝑅)) ∧ (𝑘 ∈ 𝐼 ↦ (𝑅 Σg (𝑛 ∈ 𝐼 ↦ ((𝑓‘𝑛)(.r‘𝑅)(𝑛𝑀𝑘))))) = (𝐼 × {(0g‘𝑅)})) ∧ (𝑖 ∈ 𝐼 ∧ (𝑓‘𝑖) ≠ (0g‘𝑅))) → ((𝐼 maDet 𝑅)‘(𝑗 ∈ 𝐼, 𝑘 ∈ 𝐼 ↦ if(𝑗 = 𝑖, (0g‘𝑅), (𝑗𝑀𝑘)))) = (0g‘𝑅))
463	378, 455, 462	3eqtrd 2800	. . . . . . . 8 ⊢ ((((((𝑅 ∈ Field ∧ 𝑀:(𝐼 × 𝐼)⟶(Base‘𝑅)) ∧ 𝐼 ∈ Fin) ∧ 𝑓:𝐼⟶(Base‘𝑅)) ∧ (𝑘 ∈ 𝐼 ↦ (𝑅 Σg (𝑛 ∈ 𝐼 ↦ ((𝑓‘𝑛)(.r‘𝑅)(𝑛𝑀𝑘))))) = (𝐼 × {(0g‘𝑅)})) ∧ (𝑖 ∈ 𝐼 ∧ (𝑓‘𝑖) ≠ (0g‘𝑅))) → ((𝐼 maDet 𝑅)‘𝑀) = (0g‘𝑅))
464	463	rexlimdvaa 3165	. . . . . . 7 ⊢ (((((𝑅 ∈ Field ∧ 𝑀:(𝐼 × 𝐼)⟶(Base‘𝑅)) ∧ 𝐼 ∈ Fin) ∧ 𝑓:𝐼⟶(Base‘𝑅)) ∧ (𝑘 ∈ 𝐼 ↦ (𝑅 Σg (𝑛 ∈ 𝐼 ↦ ((𝑓‘𝑛)(.r‘𝑅)(𝑛𝑀𝑘))))) = (𝐼 × {(0g‘𝑅)})) → (∃𝑖 ∈ 𝐼 (𝑓‘𝑖) ≠ (0g‘𝑅) → ((𝐼 maDet 𝑅)‘𝑀) = (0g‘𝑅)))
465	464	expimpd 459	. . . . . 6 ⊢ ((((𝑅 ∈ Field ∧ 𝑀:(𝐼 × 𝐼)⟶(Base‘𝑅)) ∧ 𝐼 ∈ Fin) ∧ 𝑓:𝐼⟶(Base‘𝑅)) → (((𝑘 ∈ 𝐼 ↦ (𝑅 Σg (𝑛 ∈ 𝐼 ↦ ((𝑓‘𝑛)(.r‘𝑅)(𝑛𝑀𝑘))))) = (𝐼 × {(0g‘𝑅)}) ∧ ∃𝑖 ∈ 𝐼 (𝑓‘𝑖) ≠ (0g‘𝑅)) → ((𝐼 maDet 𝑅)‘𝑀) = (0g‘𝑅)))
466	128, 465	sylan2d 617	. . . . 5 ⊢ ((((𝑅 ∈ Field ∧ 𝑀:(𝐼 × 𝐼)⟶(Base‘𝑅)) ∧ 𝐼 ∈ Fin) ∧ 𝑓:𝐼⟶(Base‘𝑅)) → (((𝑘 ∈ 𝐼 ↦ (𝑅 Σg (𝑛 ∈ 𝐼 ↦ ((𝑓‘𝑛)(.r‘𝑅)(𝑛𝑀𝑘))))) = (𝐼 × {(0g‘𝑅)}) ∧ ¬ 𝑓 = (𝐼 × {(0g‘𝑅)})) → ((𝐼 maDet 𝑅)‘𝑀) = (0g‘𝑅)))
467	32, 466	sylan2 605	. . . 4 ⊢ ((((𝑅 ∈ Field ∧ 𝑀:(𝐼 × 𝐼)⟶(Base‘𝑅)) ∧ 𝐼 ∈ Fin) ∧ 𝑓 ∈ ((Base‘𝑅) ↑m 𝐼)) → (((𝑘 ∈ 𝐼 ↦ (𝑅 Σg (𝑛 ∈ 𝐼 ↦ ((𝑓‘𝑛)(.r‘𝑅)(𝑛𝑀𝑘))))) = (𝐼 × {(0g‘𝑅)}) ∧ ¬ 𝑓 = (𝐼 × {(0g‘𝑅)})) → ((𝐼 maDet 𝑅)‘𝑀) = (0g‘𝑅)))
468	467	rexlimdva 3164	. . 3 ⊢ (((𝑅 ∈ Field ∧ 𝑀:(𝐼 × 𝐼)⟶(Base‘𝑅)) ∧ 𝐼 ∈ Fin) → (∃𝑓 ∈ ((Base‘𝑅) ↑m 𝐼)((𝑘 ∈ 𝐼 ↦ (𝑅 Σg (𝑛 ∈ 𝐼 ↦ ((𝑓‘𝑛)(.r‘𝑅)(𝑛𝑀𝑘))))) = (𝐼 × {(0g‘𝑅)}) ∧ ¬ 𝑓 = (𝐼 × {(0g‘𝑅)})) → ((𝐼 maDet 𝑅)‘𝑀) = (0g‘𝑅)))
469	9, 468	sylan2 605	. 2 ⊢ (((𝑅 ∈ Field ∧ 𝑀:(𝐼 × 𝐼)⟶(Base‘𝑅)) ∧ 𝐼 ∈ (Fin ∖ {∅})) → (∃𝑓 ∈ ((Base‘𝑅) ↑m 𝐼)((𝑘 ∈ 𝐼 ↦ (𝑅 Σg (𝑛 ∈ 𝐼 ↦ ((𝑓‘𝑛)(.r‘𝑅)(𝑛𝑀𝑘))))) = (𝐼 × {(0g‘𝑅)}) ∧ ¬ 𝑓 = (𝐼 × {(0g‘𝑅)})) → ((𝐼 maDet 𝑅)‘𝑀) = (0g‘𝑅)))
470	115, 469	sylbid 243	1 ⊢ (((𝑅 ∈ Field ∧ 𝑀:(𝐼 × 𝐼)⟶(Base‘𝑅)) ∧ 𝐼 ∈ (Fin ∖ {∅})) → (¬ curry 𝑀 LIndF (𝑅 freeLMod 𝐼) → ((𝐼 maDet 𝑅)‘𝑀) = (0g‘𝑅)))

This proof depends on syntax axioms:  ¬ wn 3   → wi 4   ↔ wb 209   ∧ wa 401   ∧ w3a 1103   = wceq 1570   ∈ wcel 2145   ≠ wne 2956  ∀wral 3077  ∃wrex 3087  Vcvv 3451   ∖ cdif 3896   ∪ cun 3897   ⊆ wss 3899  ∅c0 4279  ifcif 4482  {csn 4584   class class class wbr 5103   ↦ cmpt 5186   × cxp 5649   Fn wfn 6533  ⟶wf 6534  ‘cfv 6538  (class class class)co 7420   ∈ cmpo 7422   ∘_f cof 7691  curry ccur 8282   ↑_m cmap 8847  Fincfn 8973   finSupp cfsupp 9353  Basecbs 17387  +_gcplusg 17428  ._rcmulr 17429  Scalarcsca 17431   ·_𝑠 cvsca 17432  0_gc0g 17610   Σ_g cgsu 17611  Grpcgrp 19144  CMndccmn 19994  Abelcabl 19995  1_rcur 20407  Ringcrg 20459  CRingccrg 20460  inv_rcinvr 20617  DivRingcdr 20980  Fieldcfield 20981  LModclmod 21135   freeLMod cfrlm 22052   LIndF clindf 22110   maDet cmdat 22899

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 2147  ax-9 2155  ax-10 2178  ax-11 2194  ax-12 2213  ax-ext 2733  ax-rep 5232  ax-sep 5249  ax-nul 5260  ax-pow 5327  ax-pr 5391  ax-un 7751  ax-cnex 11256  ax-resscn 11257  ax-1cn 11258  ax-icn 11259  ax-addcl 11260  ax-addrcl 11261  ax-mulcl 11262  ax-mulrcl 11263  ax-mulcom 11264  ax-addass 11265  ax-mulass 11266  ax-distr 11267  ax-i2m1 11268  ax-1ne0 11269  ax-1rid 11270  ax-rnegex 11271  ax-rrecex 11272  ax-cnre 11273  ax-pre-lttri 11274  ax-pre-lttrn 11275  ax-pre-ltadd 11276  ax-pre-mulgt0 11277  ax-addf 11279  ax-mulf 11280

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3or 1104  df-3an 1105  df-xor 1542  df-tru 1573  df-fal 1583  df-ex 1813  df-nf 1817  df-sb 2100  df-mo 2565  df-eu 2595  df-clab 2740  df-cleq 2753  df-clel 2836  df-nfc 2910  df-ne 2957  df-nel 3063  df-ral 3078  df-rex 3088  df-rmo 3366  df-reu 3367  df-rab 3414  df-v 3453  df-sbc 3740  df-csb 3848  df-dif 3902  df-un 3904  df-in 3906  df-ss 3916  df-pss 3919  df-nul 4280  df-if 4483  df-pw 4559  df-sn 4585  df-pr 4587  df-tp 4589  df-op 4591  df-ot 4593  df-uni 4868  df-int 4908  df-iun 4953  df-iin 4954  df-br 5104  df-opab 5168  df-mpt 5187  df-tr 5213  df-id 5546  df-eprel 5551  df-po 5559  df-so 5560  df-fr 5604  df-se 5605  df-we 5606  df-xp 5657  df-rel 5658  df-cnv 5659  df-co 5660  df-dm 5661  df-rn 5662  df-res 5663  df-ima 5664  df-pred 6304  df-ord 6365  df-on 6366  df-lim 6367  df-suc 6368  df-iota 6494  df-fun 6540  df-fn 6541  df-f 6542  df-f1 6543  df-fo 6544  df-f1o 6545  df-fv 6546  df-isom 6547  df-riota 7377  df-ov 7423  df-oprab 7424  df-mpo 7425  df-of 7693  df-om 7878  df-1st 8001  df-2nd 8002  df-supp 8178  df-tpos 8243  df-cur 8284  df-frecs 8299  df-wrecs 8330  df-recs 8379  df-rdg 8418  df-1o 8476  df-2o 8477  df-er 8717  df-map 8849  df-pm 8850  df-ixp 8926  df-en 8974  df-dom 8975  df-sdom 8976  df-fin 8977  df-fsupp 9354  df-sup 9434  df-oi 9504  df-card 10020  df-pnf 11345  df-mnf 11346  df-xr 11347  df-ltxr 11348  df-le 11349  df-sub 11543  df-neg 11544  df-div 11974  df-nn 12336  df-2 12405  df-3 12406  df-4 12407  df-5 12408  df-6 12409  df-7 12410  df-8 12411  df-9 12412  df-n0 12607  df-xnn0 12680  df-z 12694  df-dec 12815  df-uz 12966  df-rp 13121  df-fz 13640  df-fzo 13789  df-seq 14145  df-exp 14205  df-hash 14475  df-word 14659  df-lsw 14708  df-concat 14716  df-s1 14743  df-substr 14789  df-pfx 14821  df-splice 14899  df-reverse 14908  df-s2 14999  df-struct 17325  df-sets 17342  df-slot 17360  df-ndx 17372  df-base 17388  df-ress 17409  df-plusg 17441  df-mulr 17442  df-starv 17443  df-sca 17444  df-vsca 17445  df-ip 17446  df-tset 17447  df-ple 17448  df-ds 17450  df-unif 17451  df-hom 17452  df-cco 17453  df-0g 17612  df-gsum 17613  df-prds 17618  df-pws 17620  df-mre 17756  df-mrc 17757  df-acs 17759  df-mgm 18816  df-sgrp 18908  df-mnd 18924  df-mhm 18978  df-submnd 18979  df-efmnd 19065  df-grp 19147  df-minusg 19148  df-sbg 19149  df-mulg 19278  df-subg 19333  df-ghm 19428  df-gim 19473  df-cntz 19531  df-oppg 19560  df-symg 19584  df-pmtr 19656  df-psgn 19705  df-evpm 19706  df-cmn 19996  df-abl 19997  df-mgp 20361  df-rng 20375  df-ur 20408  df-ring 20461  df-cring 20462  df-oppr 20567  df-dvdsr 20587  df-unit 20588  df-invr 20618  df-dvr 20631  df-rhm 20702  df-nzr 20763  df-subrng 20798  df-subrg 20822  df-drng 20982  df-field 20983  df-lmod 21137  df-lss 21207  df-lsp 21247  df-lmhm 21297  df-lbs 21350  df-sra 21448  df-rgmod 21449  df-cnfld 21679  df-zring 21753  df-zrh 21809  df-dsmm 22038  df-frlm 22053  df-uvc 22089  df-lindf 22112  df-mat 22723  df-mdet 22900

This theorem is used by:  matunitlindf  22996