Theorem matunitlindflem1 23049

Description: One direction of matunitlindf 23051. (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 21044	. . . . 5 ⊢ (𝑅 ∈ Field ↔ (𝑅 ∈ DivRing ∧ 𝑅 ∈ CRing))
2	1	simplbi 502	. . . 4 ⊢ (𝑅 ∈ Field → 𝑅 ∈ DivRing)
3		drngring 21038	. . . 4 ⊢ (𝑅 ∈ DivRing → 𝑅 ∈ Ring)
4	2, 3	syl 18	. . 3 ⊢ (𝑅 ∈ Field → 𝑅 ∈ Ring)
5		eqid 2763	. . . . . . . . 9 ⊢ (𝑅 freeLMod 𝐼) = (𝑅 freeLMod 𝐼)
6	5	frlmlmod 22110	. . . . . . . 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 4070	. . . . . . . . . 10 ⊢ (𝐼 ∈ (Fin ∖ {∅}) → 𝐼 ∈ Fin)
10		eqid 2763	. . . . . . . . . . 11 ⊢ (Base‘𝑅) = (Base‘𝑅)
11	5, 10	frlmfibas 22123	. . . . . . . . . 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 6929	. . . . . . . . . 10 ⊢ (Base‘𝑅) ∈ V
14		curf 8928	. . . . . . . . . 10 ⊢ ((𝑀:(𝐼 × 𝐼)⟶(Base‘𝑅) ∧ 𝐼 ∈ (Fin ∖ {∅}) ∧ (Base‘𝑅) ∈ V) → curry 𝑀:𝐼⟶((Base‘𝑅) ↑m 𝐼))
15	13, 14	mp3an3 1480	. . . . . . . . 9 ⊢ ((𝑀:(𝐼 × 𝐼)⟶(Base‘𝑅) ∧ 𝐼 ∈ (Fin ∖ {∅})) → curry 𝑀:𝐼⟶((Base‘𝑅) ↑m 𝐼))
16		feq3 6719	. . . . . . . . . 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 2763	. . . . . . . 8 ⊢ (Base‘(𝑅 freeLMod 𝐼)) = (Base‘(𝑅 freeLMod 𝐼))
21		eqid 2763	. . . . . . . 8 ⊢ (Scalar‘(𝑅 freeLMod 𝐼)) = (Scalar‘(𝑅 freeLMod 𝐼))
22		eqid 2763	. . . . . . . 8 ⊢ ( ·𝑠 ‘(𝑅 freeLMod 𝐼)) = ( ·𝑠 ‘(𝑅 freeLMod 𝐼))
23		eqid 2763	. . . . . . . 8 ⊢ (0g‘(𝑅 freeLMod 𝐼)) = (0g‘(𝑅 freeLMod 𝐼))
24		eqid 2763	. . . . . . . 8 ⊢ (0g‘(Scalar‘(𝑅 freeLMod 𝐼))) = (0g‘(Scalar‘(𝑅 freeLMod 𝐼)))
25		eqid 2763	. . . . . . . 8 ⊢ (Base‘((Scalar‘(𝑅 freeLMod 𝐼)) freeLMod 𝐼)) = (Base‘((Scalar‘(𝑅 freeLMod 𝐼)) freeLMod 𝐼))
26	20, 21, 22, 23, 24, 25	islindf4 22199	. . . . . . 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 1399	. . . . . 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 22114	. . . . . . . . . 10 ⊢ ((𝑅 ∈ Ring ∧ 𝐼 ∈ (Fin ∖ {∅})) → 𝑅 = (Scalar‘(𝑅 freeLMod 𝐼)))
29	28	fvoveq1d 7473	. . . . . . . . 9 ⊢ ((𝑅 ∈ Ring ∧ 𝐼 ∈ (Fin ∖ {∅})) → (Base‘(𝑅 freeLMod 𝐼)) = (Base‘((Scalar‘(𝑅 freeLMod 𝐼)) freeLMod 𝐼)))
30	12, 29	eqtrd 2798	. . . . . . . 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 8907	. . . . . . . . . 10 ⊢ (𝑓 ∈ ((Base‘𝑅) ↑m 𝐼) → 𝑓:𝐼⟶(Base‘𝑅))
33		ffn 6739	. . . . . . . . . . . . . . 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 6740	. . . . . . . . . . . . . . 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 4126	. . . . . . . . . . . . . 14 ⊢ (𝐼 ∩ 𝐼) = 𝐼
39		eqidd 2764	. . . . . . . . . . . . . 14 ⊢ (((((𝑅 ∈ Ring ∧ 𝑀:(𝐼 × 𝐼)⟶(Base‘𝑅)) ∧ 𝐼 ∈ (Fin ∖ {∅})) ∧ 𝑓:𝐼⟶(Base‘𝑅)) ∧ 𝑛 ∈ 𝐼) → (𝑓‘𝑛) = (𝑓‘𝑛))
40		eqidd 2764	. . . . . . . . . . . . . 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 7734	. . . . . . . . . . . . 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 7113	. . . . . . . . . . . . . . . . 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 7116	. . . . . . . . . . . . . . . . 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 2763	. . . . . . . . . . . . . . . 16 ⊢ (.r‘𝑅) = (.r‘𝑅)
48	5, 20, 10, 42, 44, 46, 22, 47	frlmvscafval 22127	. . . . . . . . . . . . . . 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 6929	. . . . . . . . . . . . . . . . 17 ⊢ (𝑓‘𝑛) ∈ V
50		fnconstg 6800	. . . . . . . . . . . . . . . . 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 7116	. . . . . . . . . . . . . . . . . . 19 ⊢ (((𝑀:(𝐼 × 𝐼)⟶(Base‘𝑅) ∧ 𝐼 ∈ (Fin ∖ {∅})) ∧ 𝑛 ∈ 𝐼) → (curry 𝑀‘𝑛) ∈ ((Base‘𝑅) ↑m 𝐼))
53		elmapfn 8925	. . . . . . . . . . . . . . . . . . 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 7241	. . . . . . . . . . . . . . . . 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 6739	. . . . . . . . . . . . . . . . . . . 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 8930	. . . . . . . . . . . . . . . . . . . 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 7734	. . . . . . . . . . . . . . 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 2798	. . . . . . . . . . . . . 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 5204	. . . . . . . . . . . . 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 2798	. . . . . . . . . . . 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 7467	. . . . . . . . . . 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 7623	. . . . . . . . . . . . . . . . 17 ⊢ ((𝑀:(𝐼 × 𝐼)⟶(Base‘𝑅) ∧ 𝑛 ∈ 𝐼 ∧ 𝑘 ∈ 𝐼) → (𝑛𝑀𝑘) ∈ (Base‘𝑅))
77	76	ad5ant245 1385	. . . . . . . . . . . . . . . 16 ⊢ (((((𝑅 ∈ Ring ∧ 𝑀:(𝐼 × 𝐼)⟶(Base‘𝑅)) ∧ 𝑓:𝐼⟶(Base‘𝑅)) ∧ 𝑛 ∈ 𝐼) ∧ 𝑘 ∈ 𝐼) → (𝑛𝑀𝑘) ∈ (Base‘𝑅))
78	10, 47	ringcl 20528	. . . . . . . . . . . . . . . 16 ⊢ ((𝑅 ∈ Ring ∧ (𝑓‘𝑛) ∈ (Base‘𝑅) ∧ (𝑛𝑀𝑘) ∈ (Base‘𝑅)) → ((𝑓‘𝑛)(.r‘𝑅)(𝑛𝑀𝑘)) ∈ (Base‘𝑅))
79	74, 75, 77, 78	syl3anc 1399	. . . . . . . . . . . . . . 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 7147	. . . . . . . . . . . . . 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 8897	. . . . . . . . . . . . . . . . 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 2850	. . . . . . . . . . . . . . 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 7258	. . . . . . . . . . . . . . . 16 ⊢ (𝐼 ∈ (Fin ∖ {∅}) → (𝑘 ∈ 𝐼 ↦ ((𝑓‘𝑛)(.r‘𝑅)(𝑛𝑀𝑘))) ∈ V)
90	89	ralrimivw 3162	. . . . . . . . . . . . . . 15 ⊢ (𝐼 ∈ (Fin ∖ {∅}) → ∀𝑛 ∈ 𝐼 (𝑘 ∈ 𝐼 ↦ ((𝑓‘𝑛)(.r‘𝑅)(𝑛𝑀𝑘))) ∈ V)
91		eqid 2763	. . . . . . . . . . . . . . . 16 ⊢ (𝑛 ∈ 𝐼 ↦ (𝑘 ∈ 𝐼 ↦ ((𝑓‘𝑛)(.r‘𝑅)(𝑛𝑀𝑘)))) = (𝑛 ∈ 𝐼 ↦ (𝑘 ∈ 𝐼 ↦ ((𝑓‘𝑛)(.r‘𝑅)(𝑛𝑀𝑘))))
92	91	fnmpt 6709	. . . . . . . . . . . . . . 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 6931	. . . . . . . . . . . . . 14 ⊢ (𝐼 ∈ (Fin ∖ {∅}) → (0g‘(𝑅 freeLMod 𝐼)) ∈ V)
95	93, 9, 94	fndmfifsupp 9407	. . . . . . . . . . . . 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 22133	. . . . . . . . . . 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 2799	. . . . . . . . . 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 2763	. . . . . . . . . . 11 ⊢ (0g‘𝑅) = (0g‘𝑅)
101	5, 100	frlm0 22115	. . . . . . . . . 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 2779	. . . . . . . 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 6920	. . . . . . . . . . . 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 5688	. . . . . . . . . 10 ⊢ ((𝑅 ∈ Ring ∧ 𝐼 ∈ (Fin ∖ {∅})) → (𝐼 × {(0g‘𝑅)}) = (𝐼 × {(0g‘(Scalar‘(𝑅 freeLMod 𝐼)))}))
107	106	eqeq2d 2774	. . . . . . . . 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 3329	. . . . . 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 3120	. . . 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 7249	. . . . . . . . . . . 12 ⊢ (𝑓:𝐼⟶{(0g‘𝑅)} ↔ (𝑓 Fn 𝐼 ∧ ∀𝑖 ∈ 𝐼 (𝑓‘𝑖) = (0g‘𝑅)))
117		fvex 6929	. . . . . . . . . . . . 13 ⊢ (0g‘𝑅) ∈ V
118	117	fconst2 7242	. . . . . . . . . . . 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 2960	. . . . . . . . . . 11 ⊢ ((𝑓‘𝑖) ≠ (0g‘𝑅) ↔ ¬ (𝑓‘𝑖) = (0g‘𝑅))
123	122	rexbii 3113	. . . . . . . . . 10 ⊢ (∃𝑖 ∈ 𝐼 (𝑓‘𝑖) ≠ (0g‘𝑅) ↔ ∃𝑖 ∈ 𝐼 ¬ (𝑓‘𝑖) = (0g‘𝑅))
124		rexnal 3118	. . . . . . . . . 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 4757	. . . . . . . . . . 11 ⊢ ¬ 𝑖 ∈ (𝐼 ∖ {𝑖})
130		difss 4075	. . . . . . . . . . 11 ⊢ (𝐼 ∖ {𝑖}) ⊆ 𝐼
131		diffi 9228	. . . . . . . . . . . . 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 2853	. . . . . . . . . . . . . . . . 17 ⊢ (𝑦 = ∅ → (𝑖 ∈ 𝑦 ↔ 𝑖 ∈ ∅))
134	133	notbid 321	. . . . . . . . . . . . . . . 16 ⊢ (𝑦 = ∅ → (¬ 𝑖 ∈ 𝑦 ↔ ¬ 𝑖 ∈ ∅))
135		sseq1 3938	. . . . . . . . . . . . . . . 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 5200	. . . . . . . . . . . . . . . . . . . . . 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 6711	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (𝑛 ∈ ∅ ↦ (((invr‘𝑅)‘(𝑓‘𝑖))(.r‘𝑅)((𝑓‘𝑛)(.r‘𝑅)(𝑛𝑀𝑘)))) = ∅
140	138, 139	eqtrdi 2814	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (𝑦 = ∅ → (𝑛 ∈ 𝑦 ↦ (((invr‘𝑅)‘(𝑓‘𝑖))(.r‘𝑅)((𝑓‘𝑛)(.r‘𝑅)(𝑛𝑀𝑘)))) = ∅)
141	140	oveq2d 7467	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝑦 = ∅ → (𝑅 Σg (𝑛 ∈ 𝑦 ↦ (((invr‘𝑅)‘(𝑓‘𝑖))(.r‘𝑅)((𝑓‘𝑛)(.r‘𝑅)(𝑛𝑀𝑘))))) = (𝑅 Σg ∅))
142	100	gsum0 18923	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝑅 Σg ∅) = (0g‘𝑅)
143	141, 142	eqtrdi 2814	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝑦 = ∅ → (𝑅 Σg (𝑛 ∈ 𝑦 ↦ (((invr‘𝑅)‘(𝑓‘𝑖))(.r‘𝑅)((𝑓‘𝑛)(.r‘𝑅)(𝑛𝑀𝑘))))) = (0g‘𝑅))
144	143	oveq1d 7466	. . . . . . . . . . . . . . . . . 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 4505	. . . . . . . . . . . . . . . . 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 7530	. . . . . . . . . . . . . . . 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 6920	. . . . . . . . . . . . . . 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 2774	. . . . . . . . . . . . . 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 2162	. . . . . . . . . . . . . . . . 17 ⊢ (𝑦 = 𝑥 → (𝑖 ∈ 𝑦 ↔ 𝑖 ∈ 𝑥))
151	150	notbid 321	. . . . . . . . . . . . . . . 16 ⊢ (𝑦 = 𝑥 → (¬ 𝑖 ∈ 𝑦 ↔ ¬ 𝑖 ∈ 𝑥))
152		sseq1 3938	. . . . . . . . . . . . . . . 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 5200	. . . . . . . . . . . . . . . . . . . 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 7467	. . . . . . . . . . . . . . . . . . 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 7466	. . . . . . . . . . . . . . . . . 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 4505	. . . . . . . . . . . . . . . . 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 7530	. . . . . . . . . . . . . . . 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 6920	. . . . . . . . . . . . . . 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 2774	. . . . . . . . . . . . . 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 2853	. . . . . . . . . . . . . . . . 17 ⊢ (𝑦 = (𝑥 ∪ {𝑧}) → (𝑖 ∈ 𝑦 ↔ 𝑖 ∈ (𝑥 ∪ {𝑧})))
164	163	notbid 321	. . . . . . . . . . . . . . . 16 ⊢ (𝑦 = (𝑥 ∪ {𝑧}) → (¬ 𝑖 ∈ 𝑦 ↔ ¬ 𝑖 ∈ (𝑥 ∪ {𝑧})))
165		sseq1 3938	. . . . . . . . . . . . . . . 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 5200	. . . . . . . . . . . . . . . . . . . 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 7467	. . . . . . . . . . . . . . . . . . 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 7466	. . . . . . . . . . . . . . . . . 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 4505	. . . . . . . . . . . . . . . . 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 7530	. . . . . . . . . . . . . . . 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 6920	. . . . . . . . . . . . . . 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 2774	. . . . . . . . . . . . . 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 2853	. . . . . . . . . . . . . . . . 17 ⊢ (𝑦 = (𝐼 ∖ {𝑖}) → (𝑖 ∈ 𝑦 ↔ 𝑖 ∈ (𝐼 ∖ {𝑖})))
177	176	notbid 321	. . . . . . . . . . . . . . . 16 ⊢ (𝑦 = (𝐼 ∖ {𝑖}) → (¬ 𝑖 ∈ 𝑦 ↔ ¬ 𝑖 ∈ (𝐼 ∖ {𝑖})))
178		sseq1 3938	. . . . . . . . . . . . . . . 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 5200	. . . . . . . . . . . . . . . . . . . 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 7467	. . . . . . . . . . . . . . . . . . 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 7466	. . . . . . . . . . . . . . . . . 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 4505	. . . . . . . . . . . . . . . . 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 7530	. . . . . . . . . . . . . . . 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 6920	. . . . . . . . . . . . . . 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 2774	. . . . . . . . . . . . . 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 7582	. . . . . . . . . . . . . . . . . 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 20515	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑅 ∈ Ring → 𝑅 ∈ Grp)
193	4, 192	syl 18	. . . . . . . . . . . . . . . . 17 ⊢ (𝑅 ∈ Field → 𝑅 ∈ Grp)
194		oveq1 7458	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (𝑖 = 𝑗 → (𝑖𝑀𝑘) = (𝑗𝑀𝑘))
195	194	equcoms 2055	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (𝑗 = 𝑖 → (𝑖𝑀𝑘) = (𝑗𝑀𝑘))
196	195	oveq2d 7467	. . . . . . . . . . . . . . . . . . . 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 7623	. . . . . . . . . . . . . . . . . . . . . 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 2763	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (+g‘𝑅) = (+g‘𝑅)
201	10, 200, 100	grplid 19228	. . . . . . . . . . . . . . . . . . . . 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 2820	. . . . . . . . . . . . . . . . . . 19 ⊢ ((((𝑅 ∈ Grp ∧ 𝑀:(𝐼 × 𝐼)⟶(Base‘𝑅)) ∧ 𝑗 ∈ 𝐼 ∧ 𝑘 ∈ 𝐼) ∧ 𝑗 = 𝑖) → ((0g‘𝑅)(+g‘𝑅)(𝑖𝑀𝑘)) = (𝑗𝑀𝑘))
204		eqidd 2764	. . . . . . . . . . . . . . . . . . 19 ⊢ ((((𝑅 ∈ Grp ∧ 𝑀:(𝐼 × 𝐼)⟶(Base‘𝑅)) ∧ 𝑗 ∈ 𝐼 ∧ 𝑘 ∈ 𝐼) ∧ ¬ 𝑗 = 𝑖) → (𝑗𝑀𝑘) = (𝑗𝑀𝑘))
205	203, 204	ifeqda 4522	. . . . . . . . . . . . . . . . . 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 7528	. . . . . . . . . . . . . . . . 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 2801	. . . . . . . . . . . . . . 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 6920	. . . . . . . . . . . . . 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 4183	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (𝑖 ∈ 𝑥 → 𝑖 ∈ (𝑥 ∪ {𝑧}))
212	211	con3i 155	. . . . . . . . . . . . . . . . . . . 20 ⊢ (¬ 𝑖 ∈ (𝑥 ∪ {𝑧}) → ¬ 𝑖 ∈ 𝑥)
213		ssun1 4179	. . . . . . . . . . . . . . . . . . . . 21 ⊢ 𝑥 ⊆ (𝑥 ∪ {𝑧})
214		sstr 3921	. . . . . . . . . . . . . . . . . . . . 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 4184	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ (𝑖 ∈ {𝑧} → 𝑖 ∈ (𝑥 ∪ {𝑧}))
221	219, 220	sylbir 238	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (𝑖 = 𝑧 → 𝑖 ∈ (𝑥 ∪ {𝑧}))
222	221	necon3bi 2985	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (¬ 𝑖 ∈ (𝑥 ∪ {𝑧}) → 𝑖 ≠ 𝑧)
223	222	anim1i 627	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((¬ 𝑖 ∈ (𝑥 ∪ {𝑧}) ∧ (𝑥 ∪ {𝑧}) ⊆ 𝐼) → (𝑖 ≠ 𝑧 ∧ (𝑥 ∪ {𝑧}) ⊆ 𝐼))
224		ringcmn 20562	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 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 9226	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 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 3913	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 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 7113	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 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 2763	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 39 ⊢ (invr‘𝑅) = (invr‘𝑅)
241	10, 100, 240	drnginvrcl 21062	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 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 1399	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 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 20528	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 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 1399	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 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 3458	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 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 4180	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34 ⊢ {𝑧} ⊆ (𝑥 ∪ {𝑧})
262		sstr 3921	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34 ⊢ (({𝑧} ⊆ (𝑥 ∪ {𝑧}) ∧ (𝑥 ∪ {𝑧}) ⊆ 𝐼) → {𝑧} ⊆ 𝐼)
263	261, 262	mpan 703	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33 ⊢ ((𝑥 ∪ {𝑧}) ⊆ 𝐼 → {𝑧} ⊆ 𝐼)
264	258	snss 4747	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 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 7113	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 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 20528	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 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 1399	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 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 7623	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 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 20528	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 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 1399	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 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 6916	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33 ⊢ (𝑛 = 𝑧 → (𝑓‘𝑛) = (𝑓‘𝑧))
283		oveq1 7458	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33 ⊢ (𝑛 = 𝑧 → (𝑛𝑀𝑘) = (𝑧𝑀𝑘))
284	282, 283	oveq12d 7469	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32 ⊢ (𝑛 = 𝑧 → ((𝑓‘𝑛)(.r‘𝑅)(𝑛𝑀𝑘)) = ((𝑓‘𝑧)(.r‘𝑅)(𝑧𝑀𝑘)))
285	284	oveq2d 7467	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 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 20531	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 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 1400	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 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 2769	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 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 2820	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 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 20223	. . . . . . . . . . . . . . . . . . . . . . . . . . . 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 7466	. . . . . . . . . . . . . . . . . . . . . . . . . . 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 20561	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 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 3458	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 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 3908	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 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 7147	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 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 7484	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 37 ⊢ (((invr‘𝑅)‘(𝑓‘𝑖))(.r‘𝑅)((𝑓‘𝑛)(.r‘𝑅)(𝑛𝑀𝑘))) ∈ V
308		eqid 2763	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 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 6712	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 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 6931	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35 ⊢ ((𝐼 ∈ Fin ∧ 𝑥 ⊆ 𝐼) → (0g‘𝑅) ∈ V)
312	310, 229, 311	fndmfifsupp 9407	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 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 20180	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 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 7623	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 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 20068	. . . . . . . . . . . . . . . . . . . . . . . . . . . . 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 2798	. . . . . . . . . . . . . . . . . . . . . . . . . 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 4505	. . . . . . . . . . . . . . . . . . . . . . . . 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 7528	. . . . . . . . . . . . . . . . . . . . . . 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 6920	. . . . . . . . . . . . . . . . . . . . . 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 2763	. . . . . . . . . . . . . . . . . . . . . . . 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 19202	. . . . . . . . . . . . . . . . . . . . . . . . . 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 1399	. . . . . . . . . . . . . . . . . . . . . . . . 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 22980	. . . . . . . . . . . . . . . . . . . . . . 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 2798	. . . . . . . . . . . . . . . . . . . . 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 4491	. . . . . . . . . . . . . . . . . . . . . . . . . 26 ⊢ (𝑗 = 𝑧 → if(𝑗 = 𝑧, (𝑧𝑀𝑘), (𝑗𝑀𝑘)) = (𝑧𝑀𝑘))
355		oveq1 7458	. . . . . . . . . . . . . . . . . . . . . . . . . 26 ⊢ (𝑗 = 𝑧 → (𝑗𝑀𝑘) = (𝑧𝑀𝑘))
356	354, 355	eqtr4d 2801	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ (𝑗 = 𝑧 → if(𝑗 = 𝑧, (𝑧𝑀𝑘), (𝑗𝑀𝑘)) = (𝑗𝑀𝑘))
357		iffalse 4494	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ (¬ 𝑗 = 𝑧 → if(𝑗 = 𝑧, (𝑧𝑀𝑘), (𝑗𝑀𝑘)) = (𝑗𝑀𝑘))
358	356, 357	pm2.61i 184	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ if(𝑗 = 𝑧, (𝑧𝑀𝑘), (𝑗𝑀𝑘)) = (𝑗𝑀𝑘)
359		ifeq2 4490	. . . . . . . . . . . . . . . . . . . . . . . 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 7529	. . . . . . . . . . . . . . . . . . . . . 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 6919	. . . . . . . . . . . . . . . . . . . . 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 4490	. . . . . . . . . . . . . . . . . . . . . . . 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 7529	. . . . . . . . . . . . . . . . . . . . . 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 6919	. . . . . . . . . . . . . . . . . . . . 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 2821	. . . . . . . . . . . . . . . . . . . 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 2774	. . . . . . . . . . . . . . . . . 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 9219	. . . . . . . . . . . 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 2763	. . . . . . . . . . . 12 ⊢ 𝐼 = 𝐼
380		fconstmpt 5720	. . . . . . . . . . . . . . . . 17 ⊢ (𝐼 × {(0g‘𝑅)}) = (𝑘 ∈ 𝐼 ↦ (0g‘𝑅))
381	380	eqeq2i 2776	. . . . . . . . . . . . . . . 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 7484	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑅 Σg (𝑛 ∈ 𝐼 ↦ ((𝑓‘𝑛)(.r‘𝑅)(𝑛𝑀𝑘)))) ∈ V
383	382	rgenw 3084	. . . . . . . . . . . . . . . . 17 ⊢ ∀𝑘 ∈ 𝐼 (𝑅 Σg (𝑛 ∈ 𝐼 ↦ ((𝑓‘𝑛)(.r‘𝑅)(𝑛𝑀𝑘)))) ∈ V
384		mpteqb 7044	. . . . . . . . . . . . . . . . 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 2763	. . . . . . . . . . . . . . . . . . . . . . . . . . 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 6712	. . . . . . . . . . . . . . . . . . . . . . . . . 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 6931	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ (𝐼 ∈ Fin → (0g‘𝑅) ∈ V)
394	391, 392, 393	fndmfifsupp 9407	. . . . . . . . . . . . . . . . . . . . . . . 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 6916	. . . . . . . . . . . . . . . . . . . . . . . . . 26 ⊢ (𝑛 = 𝑖 → (𝑓‘𝑛) = (𝑓‘𝑖))
399		oveq1 7458	. . . . . . . . . . . . . . . . . . . . . . . . . 26 ⊢ (𝑛 = 𝑖 → (𝑛𝑀𝑘) = (𝑖𝑀𝑘))
400	398, 399	oveq12d 7469	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ (𝑛 = 𝑖 → ((𝑓‘𝑛)(.r‘𝑅)(𝑛𝑀𝑘)) = ((𝑓‘𝑖)(.r‘𝑅)(𝑖𝑀𝑘)))
401	400	oveq2d 7467	. . . . . . . . . . . . . . . . . . . . . . . 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 2763	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32 ⊢ (1r‘𝑅) = (1r‘𝑅)
406	10, 100, 47, 405, 240	drnginvrl 21065	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 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 7466	. . . . . . . . . . . . . . . . . . . . . . . . . . 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 20531	. . . . . . . . . . . . . . . . . . . . . . . . . 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 1400	. . . . . . . . . . . . . . . . . . . . . . . . 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 20549	. . . . . . . . . . . . . . . . . . . . . . . . . . . 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 1397	. . . . . . . . . . . . . . . . . . . . . . . . . 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 2806	. . . . . . . . . . . . . . . . . . . . . . . 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 2820	. . . . . . . . . . . . . . . . . . . . . . 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 20226	. . . . . . . . . . . . . . . . . . . . . 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 7484	. . . . . . . . . . . . . . . . . . . . . . . . . . 27 ⊢ ((𝑓‘𝑛)(.r‘𝑅)(𝑛𝑀𝑘)) ∈ V
430		eqid 2763	. . . . . . . . . . . . . . . . . . . . . . . . . . 27 ⊢ (𝑛 ∈ 𝐼 ↦ ((𝑓‘𝑛)(.r‘𝑅)(𝑛𝑀𝑘))) = (𝑛 ∈ 𝐼 ↦ ((𝑓‘𝑛)(.r‘𝑅)(𝑛𝑀𝑘)))
431	429, 430	fnmpti 6712	. . . . . . . . . . . . . . . . . . . . . . . . . 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 9407	. . . . . . . . . . . . . . . . . . . . . . . 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 20597	. . . . . . . . . . . . . . . . . . . . . 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 2800	. . . . . . . . . . . . . . . . . . . . 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 7459	. . . . . . . . . . . . . . . . . . . . 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 20576	. . . . . . . . . . . . . . . . . . . . . 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 2802	. . . . . . . . . . . . . . . . . . 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 4505	. . . . . . . . . . . . . . . . . 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 3178	. . . . . . . . . . . . . . . 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 3162	. . . . . . . . . . . 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 7523	. . . . . . . . . . . 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 6920	. . . . . . . . 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 22975	. . . . . . . . . 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 2802	. . . . . . . 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 3168	. . . . . . 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 3167	. . 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 2147   ≠ wne 2959  ∀wral 3080  ∃wrex 3090  Vcvv 3454   ⊆ wss 3900   ∖ cdif 4048   ∪ cun 4152  ∅c0 4281  ifcif 4485  {csn 4588   class class class wbr 5108   ↦ cmpt 5192   × cxp 5656   Fn wfn 6561  ⟶wf 6562  ‘cfv 6566  (class class class)co 7451   ∈ cmpo 7453   ∘_f cof 7723  curry ccur 8320   ↑_m cmap 8885  Fincfn 9011   finSupp cfsupp 9391  Basecbs 17431  +_gcplusg 17472  ._rcmulr 17473  Scalarcsca 17475   ·_𝑠 cvsca 17476  0_gc0g 17660   Σ_g cgsu 17661  Grpcgrp 19194  CMndccmn 20045  Abelcabl 20046  1_rcur 20458  Ringcrg 20510  CRingccrg 20511  inv_rcinvr 20668  DivRingcdr 21031  Fieldcfield 21032  LModclmod 21186   freeLMod cfrlm 22107   LIndF clindf 22165   maDet cmdat 22954

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 1945  ax-6 2002  ax-7 2043  ax-8 2149  ax-9 2157  ax-10 2180  ax-11 2197  ax-12 2216  ax-ext 2735  ax-rep 5238  ax-sep 5255  ax-nul 5266  ax-pow 5334  ax-pr 5398  ax-un 7783  ax-cnex 11295  ax-resscn 11296  ax-1cn 11297  ax-icn 11298  ax-addcl 11299  ax-addrcl 11300  ax-mulcl 11301  ax-mulrcl 11302  ax-mulcom 11303  ax-addass 11304  ax-mulass 11305  ax-distr 11306  ax-i2m1 11307  ax-1ne0 11308  ax-1rid 11309  ax-rnegex 11310  ax-rrecex 11311  ax-cnre 11312  ax-pre-lttri 11313  ax-pre-lttrn 11314  ax-pre-ltadd 11315  ax-pre-mulgt0 11316  ax-addf 11318  ax-mulf 11319

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 2102  df-mo 2567  df-eu 2597  df-clab 2742  df-cleq 2755  df-clel 2839  df-nfc 2913  df-ne 2960  df-nel 3066  df-ral 3081  df-rex 3091  df-rmo 3369  df-reu 3370  df-rab 3417  df-v 3456  df-sbc 3743  df-csb 3852  df-ss 3901  df-pss 4017  df-dif 4050  df-in 4094  df-un 4154  df-nul 4282  df-if 4486  df-pw 4562  df-sn 4589  df-pr 4591  df-tp 4593  df-op 4841  df-ot 4864  df-uni 4873  df-int 4913  df-iun 4958  df-iin 4959  df-br 5109  df-opab 5174  df-mpt 5193  df-tr 5219  df-id 5553  df-eprel 5558  df-po 5566  df-so 5567  df-fr 5611  df-se 5612  df-we 5613  df-xp 5664  df-rel 5665  df-cnv 5666  df-co 5667  df-dm 5668  df-rn 5669  df-res 5670  df-ima 5671  df-pred 6330  df-ord 6391  df-on 6392  df-lim 6393  df-suc 6394  df-iota 6520  df-fun 6568  df-fn 6569  df-f 6570  df-f1 6571  df-fo 6572  df-f1o 6573  df-fv 6574  df-isom 6575  df-riota 7408  df-ov 7454  df-oprab 7455  df-mpo 7456  df-of 7725  df-om 7912  df-1st 8035  df-2nd 8036  df-supp 8215  df-tpos 8281  df-cur 8322  df-frecs 8337  df-wrecs 8368  df-recs 8417  df-rdg 8456  df-1o 8514  df-2o 8515  df-er 8755  df-map 8887  df-pm 8888  df-ixp 8964  df-en 9012  df-dom 9013  df-sdom 9014  df-fin 9015  df-fsupp 9392  df-sup 9471  df-oi 9541  df-card 10057  df-pnf 11384  df-mnf 11385  df-xr 11386  df-ltxr 11387  df-le 11388  df-sub 11582  df-neg 11583  df-div 12014  df-nn 12376  df-2 12445  df-3 12446  df-4 12447  df-5 12448  df-6 12449  df-7 12450  df-8 12451  df-9 12452  df-n0 12647  df-xnn0 12720  df-z 12734  df-dec 12855  df-uz 13006  df-rp 13161  df-fz 13680  df-fzo 13829  df-seq 14185  df-exp 14245  df-hash 14516  df-word 14700  df-lsw 14749  df-concat 14757  df-s1 14784  df-substr 14831  df-pfx 14863  df-splice 14941  df-reverse 14950  df-s2 15041  df-struct 17369  df-sets 17386  df-slot 17404  df-ndx 17416  df-base 17432  df-ress 17453  df-plusg 17486  df-mulr 17487  df-starv 17488  df-sca 17489  df-vsca 17490  df-ip 17491  df-tset 17492  df-ple 17493  df-ds 17495  df-unif 17496  df-hom 17497  df-cco 17498  df-euf 17499  df-0g 17662  df-gsum 17663  df-prds 17668  df-pws 17670  df-mre 17806  df-mrc 17807  df-acs 17809  df-mgm 18866  df-sgrp 18958  df-mnd 18974  df-mhm 19028  df-submnd 19029  df-efmnd 19115  df-grp 19197  df-minusg 19198  df-sbg 19199  df-mulg 19328  df-subg 19383  df-ghm 19478  df-gim 19523  df-cntz 19582  df-oppg 19611  df-symg 19635  df-pmtr 19707  df-psgn 19756  df-evpm 19757  df-cmn 20047  df-abl 20048  df-mgp 20412  df-rng 20426  df-ur 20459  df-ring 20512  df-cring 20513  df-oppr 20618  df-dvdsr 20638  df-unit 20639  df-invr 20669  df-dvr 20682  df-rhm 20753  df-nzr 20814  df-subrng 20849  df-subrg 20873  df-drng 21033  df-field 21034  df-lmod 21188  df-lss 21258  df-lsp 21298  df-lmhm 21348  df-lbs 21401  df-sra 21499  df-rgmod 21500  df-cnfld 21730  df-zring 21807  df-zrh 21864  df-dsmm 22093  df-frlm 22108  df-uvc 22144  df-lindf 22167  df-mat 22778  df-mdet 22955

This theorem is used by:  matunitlindf  23051