Theorem matunitlindflem1 37220

Description: One direction of matunitlindf 37222. (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 20647	. . . . 5 ⊢ (𝑅 ∈ Field ↔ (𝑅 ∈ DivRing ∧ 𝑅 ∈ CRing))
2	1	simplbi 496	. . . 4 ⊢ (𝑅 ∈ Field → 𝑅 ∈ DivRing)
3		drngring 20643	. . . 4 ⊢ (𝑅 ∈ DivRing → 𝑅 ∈ Ring)
4	2, 3	syl 17	. . 3 ⊢ (𝑅 ∈ Field → 𝑅 ∈ Ring)
5		eqid 2725	. . . . . . . . 9 ⊢ (𝑅 freeLMod 𝐼) = (𝑅 freeLMod 𝐼)
6	5	frlmlmod 21700	. . . . . . . 8 ⊢ ((𝑅 ∈ Ring ∧ 𝐼 ∈ (Fin ∖ {∅})) → (𝑅 freeLMod 𝐼) ∈ LMod)
7	6	adantlr 713	. . . . . . 7 ⊢ (((𝑅 ∈ Ring ∧ 𝑀:(𝐼 × 𝐼)⟶(Base‘𝑅)) ∧ 𝐼 ∈ (Fin ∖ {∅})) → (𝑅 freeLMod 𝐼) ∈ LMod)
8		simpr 483	. . . . . . 7 ⊢ (((𝑅 ∈ Ring ∧ 𝑀:(𝐼 × 𝐼)⟶(Base‘𝑅)) ∧ 𝐼 ∈ (Fin ∖ {∅})) → 𝐼 ∈ (Fin ∖ {∅}))
9		eldifi 4123	. . . . . . . . . 10 ⊢ (𝐼 ∈ (Fin ∖ {∅}) → 𝐼 ∈ Fin)
10		eqid 2725	. . . . . . . . . . 11 ⊢ (Base‘𝑅) = (Base‘𝑅)
11	5, 10	frlmfibas 21713	. . . . . . . . . 10 ⊢ ((𝑅 ∈ Ring ∧ 𝐼 ∈ Fin) → ((Base‘𝑅) ↑m 𝐼) = (Base‘(𝑅 freeLMod 𝐼)))
12	9, 11	sylan2 591	. . . . . . . . 9 ⊢ ((𝑅 ∈ Ring ∧ 𝐼 ∈ (Fin ∖ {∅})) → ((Base‘𝑅) ↑m 𝐼) = (Base‘(𝑅 freeLMod 𝐼)))
13		fvex 6909	. . . . . . . . . 10 ⊢ (Base‘𝑅) ∈ V
14		curf 37202	. . . . . . . . . 10 ⊢ ((𝑀:(𝐼 × 𝐼)⟶(Base‘𝑅) ∧ 𝐼 ∈ (Fin ∖ {∅}) ∧ (Base‘𝑅) ∈ V) → curry 𝑀:𝐼⟶((Base‘𝑅) ↑m 𝐼))
15	13, 14	mp3an3 1446	. . . . . . . . 9 ⊢ ((𝑀:(𝐼 × 𝐼)⟶(Base‘𝑅) ∧ 𝐼 ∈ (Fin ∖ {∅})) → curry 𝑀:𝐼⟶((Base‘𝑅) ↑m 𝐼))
16		feq3 6706	. . . . . . . . . 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 475	. . . . . . . . 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 594	. . . . . . . 8 ⊢ (((𝑅 ∈ Ring ∧ 𝐼 ∈ (Fin ∖ {∅})) ∧ (𝑀:(𝐼 × 𝐼)⟶(Base‘𝑅) ∧ 𝐼 ∈ (Fin ∖ {∅}))) → curry 𝑀:𝐼⟶(Base‘(𝑅 freeLMod 𝐼)))
19	18	anandirs 677	. . . . . . 7 ⊢ (((𝑅 ∈ Ring ∧ 𝑀:(𝐼 × 𝐼)⟶(Base‘𝑅)) ∧ 𝐼 ∈ (Fin ∖ {∅})) → curry 𝑀:𝐼⟶(Base‘(𝑅 freeLMod 𝐼)))
20		eqid 2725	. . . . . . . 8 ⊢ (Base‘(𝑅 freeLMod 𝐼)) = (Base‘(𝑅 freeLMod 𝐼))
21		eqid 2725	. . . . . . . 8 ⊢ (Scalar‘(𝑅 freeLMod 𝐼)) = (Scalar‘(𝑅 freeLMod 𝐼))
22		eqid 2725	. . . . . . . 8 ⊢ ( ·𝑠 ‘(𝑅 freeLMod 𝐼)) = ( ·𝑠 ‘(𝑅 freeLMod 𝐼))
23		eqid 2725	. . . . . . . 8 ⊢ (0g‘(𝑅 freeLMod 𝐼)) = (0g‘(𝑅 freeLMod 𝐼))
24		eqid 2725	. . . . . . . 8 ⊢ (0g‘(Scalar‘(𝑅 freeLMod 𝐼))) = (0g‘(Scalar‘(𝑅 freeLMod 𝐼)))
25		eqid 2725	. . . . . . . 8 ⊢ (Base‘((Scalar‘(𝑅 freeLMod 𝐼)) freeLMod 𝐼)) = (Base‘((Scalar‘(𝑅 freeLMod 𝐼)) freeLMod 𝐼))
26	20, 21, 22, 23, 24, 25	islindf4 21789	. . . . . . 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 1368	. . . . . 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 21704	. . . . . . . . . 10 ⊢ ((𝑅 ∈ Ring ∧ 𝐼 ∈ (Fin ∖ {∅})) → 𝑅 = (Scalar‘(𝑅 freeLMod 𝐼)))
29	28	fvoveq1d 7441	. . . . . . . . 9 ⊢ ((𝑅 ∈ Ring ∧ 𝐼 ∈ (Fin ∖ {∅})) → (Base‘(𝑅 freeLMod 𝐼)) = (Base‘((Scalar‘(𝑅 freeLMod 𝐼)) freeLMod 𝐼)))
30	12, 29	eqtrd 2765	. . . . . . . 8 ⊢ ((𝑅 ∈ Ring ∧ 𝐼 ∈ (Fin ∖ {∅})) → ((Base‘𝑅) ↑m 𝐼) = (Base‘((Scalar‘(𝑅 freeLMod 𝐼)) freeLMod 𝐼)))
31	30	adantlr 713	. . . . . . 7 ⊢ (((𝑅 ∈ Ring ∧ 𝑀:(𝐼 × 𝐼)⟶(Base‘𝑅)) ∧ 𝐼 ∈ (Fin ∖ {∅})) → ((Base‘𝑅) ↑m 𝐼) = (Base‘((Scalar‘(𝑅 freeLMod 𝐼)) freeLMod 𝐼)))
32		elmapi 8868	. . . . . . . . . 10 ⊢ (𝑓 ∈ ((Base‘𝑅) ↑m 𝐼) → 𝑓:𝐼⟶(Base‘𝑅))
33		ffn 6723	. . . . . . . . . . . . . . 15 ⊢ (𝑓:𝐼⟶(Base‘𝑅) → 𝑓 Fn 𝐼)
34	33	adantl 480	. . . . . . . . . . . . . 14 ⊢ ((((𝑅 ∈ Ring ∧ 𝑀:(𝐼 × 𝐼)⟶(Base‘𝑅)) ∧ 𝐼 ∈ (Fin ∖ {∅})) ∧ 𝑓:𝐼⟶(Base‘𝑅)) → 𝑓 Fn 𝐼)
35	19	ffnd 6724	. . . . . . . . . . . . . . 15 ⊢ (((𝑅 ∈ Ring ∧ 𝑀:(𝐼 × 𝐼)⟶(Base‘𝑅)) ∧ 𝐼 ∈ (Fin ∖ {∅})) → curry 𝑀 Fn 𝐼)
36	35	adantr 479	. . . . . . . . . . . . . 14 ⊢ ((((𝑅 ∈ Ring ∧ 𝑀:(𝐼 × 𝐼)⟶(Base‘𝑅)) ∧ 𝐼 ∈ (Fin ∖ {∅})) ∧ 𝑓:𝐼⟶(Base‘𝑅)) → curry 𝑀 Fn 𝐼)
37		simplr 767	. . . . . . . . . . . . . 14 ⊢ ((((𝑅 ∈ Ring ∧ 𝑀:(𝐼 × 𝐼)⟶(Base‘𝑅)) ∧ 𝐼 ∈ (Fin ∖ {∅})) ∧ 𝑓:𝐼⟶(Base‘𝑅)) → 𝐼 ∈ (Fin ∖ {∅}))
38		inidm 4217	. . . . . . . . . . . . . 14 ⊢ (𝐼 ∩ 𝐼) = 𝐼
39		eqidd 2726	. . . . . . . . . . . . . 14 ⊢ (((((𝑅 ∈ Ring ∧ 𝑀:(𝐼 × 𝐼)⟶(Base‘𝑅)) ∧ 𝐼 ∈ (Fin ∖ {∅})) ∧ 𝑓:𝐼⟶(Base‘𝑅)) ∧ 𝑛 ∈ 𝐼) → (𝑓‘𝑛) = (𝑓‘𝑛))
40		eqidd 2726	. . . . . . . . . . . . . 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 7694	. . . . . . . . . . . . 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 774	. . . . . . . . . . . . . . . 16 ⊢ (((((𝑅 ∈ Ring ∧ 𝑀:(𝐼 × 𝐼)⟶(Base‘𝑅)) ∧ 𝐼 ∈ (Fin ∖ {∅})) ∧ 𝑓:𝐼⟶(Base‘𝑅)) ∧ 𝑛 ∈ 𝐼) → 𝐼 ∈ (Fin ∖ {∅}))
43		ffvelcdm 7090	. . . . . . . . . . . . . . . . 17 ⊢ ((𝑓:𝐼⟶(Base‘𝑅) ∧ 𝑛 ∈ 𝐼) → (𝑓‘𝑛) ∈ (Base‘𝑅))
44	43	adantll 712	. . . . . . . . . . . . . . . 16 ⊢ (((((𝑅 ∈ Ring ∧ 𝑀:(𝐼 × 𝐼)⟶(Base‘𝑅)) ∧ 𝐼 ∈ (Fin ∖ {∅})) ∧ 𝑓:𝐼⟶(Base‘𝑅)) ∧ 𝑛 ∈ 𝐼) → (𝑓‘𝑛) ∈ (Base‘𝑅))
45	19	ffvelcdmda 7093	. . . . . . . . . . . . . . . . 17 ⊢ ((((𝑅 ∈ Ring ∧ 𝑀:(𝐼 × 𝐼)⟶(Base‘𝑅)) ∧ 𝐼 ∈ (Fin ∖ {∅})) ∧ 𝑛 ∈ 𝐼) → (curry 𝑀‘𝑛) ∈ (Base‘(𝑅 freeLMod 𝐼)))
46	45	adantlr 713	. . . . . . . . . . . . . . . 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 2725	. . . . . . . . . . . . . . . 16 ⊢ (.r‘𝑅) = (.r‘𝑅)
48	5, 20, 10, 42, 44, 46, 22, 47	frlmvscafval 21717	. . . . . . . . . . . . . . 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 6909	. . . . . . . . . . . . . . . . 17 ⊢ (𝑓‘𝑛) ∈ V
50		fnconstg 6785	. . . . . . . . . . . . . . . . 17 ⊢ ((𝑓‘𝑛) ∈ V → (𝐼 × {(𝑓‘𝑛)}) Fn 𝐼)
51	49, 50	mp1i 13	. . . . . . . . . . . . . . . 16 ⊢ (((((𝑅 ∈ Ring ∧ 𝑀:(𝐼 × 𝐼)⟶(Base‘𝑅)) ∧ 𝐼 ∈ (Fin ∖ {∅})) ∧ 𝑓:𝐼⟶(Base‘𝑅)) ∧ 𝑛 ∈ 𝐼) → (𝐼 × {(𝑓‘𝑛)}) Fn 𝐼)
52	15	ffvelcdmda 7093	. . . . . . . . . . . . . . . . . . 19 ⊢ (((𝑀:(𝐼 × 𝐼)⟶(Base‘𝑅) ∧ 𝐼 ∈ (Fin ∖ {∅})) ∧ 𝑛 ∈ 𝐼) → (curry 𝑀‘𝑛) ∈ ((Base‘𝑅) ↑m 𝐼))
53		elmapfn 8884	. . . . . . . . . . . . . . . . . . 19 ⊢ ((curry 𝑀‘𝑛) ∈ ((Base‘𝑅) ↑m 𝐼) → (curry 𝑀‘𝑛) Fn 𝐼)
54	52, 53	syl 17	. . . . . . . . . . . . . . . . . 18 ⊢ (((𝑀:(𝐼 × 𝐼)⟶(Base‘𝑅) ∧ 𝐼 ∈ (Fin ∖ {∅})) ∧ 𝑛 ∈ 𝐼) → (curry 𝑀‘𝑛) Fn 𝐼)
55	54	adantlll 716	. . . . . . . . . . . . . . . . 17 ⊢ ((((𝑅 ∈ Ring ∧ 𝑀:(𝐼 × 𝐼)⟶(Base‘𝑅)) ∧ 𝐼 ∈ (Fin ∖ {∅})) ∧ 𝑛 ∈ 𝐼) → (curry 𝑀‘𝑛) Fn 𝐼)
56	55	adantlr 713	. . . . . . . . . . . . . . . 16 ⊢ (((((𝑅 ∈ Ring ∧ 𝑀:(𝐼 × 𝐼)⟶(Base‘𝑅)) ∧ 𝐼 ∈ (Fin ∖ {∅})) ∧ 𝑓:𝐼⟶(Base‘𝑅)) ∧ 𝑛 ∈ 𝐼) → (curry 𝑀‘𝑛) Fn 𝐼)
57	49	fvconst2 7216	. . . . . . . . . . . . . . . . 17 ⊢ (𝑘 ∈ 𝐼 → ((𝐼 × {(𝑓‘𝑛)})‘𝑘) = (𝑓‘𝑛))
58	57	adantl 480	. . . . . . . . . . . . . . . 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 6723	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝑀:(𝐼 × 𝐼)⟶(Base‘𝑅) → 𝑀 Fn (𝐼 × 𝐼))
60	59	anim2i 615	. . . . . . . . . . . . . . . . . . 19 ⊢ ((𝐼 ∈ (Fin ∖ {∅}) ∧ 𝑀:(𝐼 × 𝐼)⟶(Base‘𝑅)) → (𝐼 ∈ (Fin ∖ {∅}) ∧ 𝑀 Fn (𝐼 × 𝐼)))
61	60	ancoms 457	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝑀:(𝐼 × 𝐼)⟶(Base‘𝑅) ∧ 𝐼 ∈ (Fin ∖ {∅})) → (𝐼 ∈ (Fin ∖ {∅}) ∧ 𝑀 Fn (𝐼 × 𝐼)))
62	61	ad4ant23 751	. . . . . . . . . . . . . . . . 17 ⊢ ((((𝑅 ∈ Ring ∧ 𝑀:(𝐼 × 𝐼)⟶(Base‘𝑅)) ∧ 𝐼 ∈ (Fin ∖ {∅})) ∧ 𝑓:𝐼⟶(Base‘𝑅)) → (𝐼 ∈ (Fin ∖ {∅}) ∧ 𝑀 Fn (𝐼 × 𝐼)))
63		curfv 37204	. . . . . . . . . . . . . . . . . . . 20 ⊢ (((𝑀 Fn (𝐼 × 𝐼) ∧ 𝑛 ∈ 𝐼 ∧ 𝑘 ∈ 𝐼) ∧ 𝐼 ∈ (Fin ∖ {∅})) → ((curry 𝑀‘𝑛)‘𝑘) = (𝑛𝑀𝑘))
64	63	3exp1 1349	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝑀 Fn (𝐼 × 𝐼) → (𝑛 ∈ 𝐼 → (𝑘 ∈ 𝐼 → (𝐼 ∈ (Fin ∖ {∅}) → ((curry 𝑀‘𝑛)‘𝑘) = (𝑛𝑀𝑘)))))
65	64	com4r 94	. . . . . . . . . . . . . . . . . 18 ⊢ (𝐼 ∈ (Fin ∖ {∅}) → (𝑀 Fn (𝐼 × 𝐼) → (𝑛 ∈ 𝐼 → (𝑘 ∈ 𝐼 → ((curry 𝑀‘𝑛)‘𝑘) = (𝑛𝑀𝑘)))))
66	65	imp41 424	. . . . . . . . . . . . . . . . 17 ⊢ ((((𝐼 ∈ (Fin ∖ {∅}) ∧ 𝑀 Fn (𝐼 × 𝐼)) ∧ 𝑛 ∈ 𝐼) ∧ 𝑘 ∈ 𝐼) → ((curry 𝑀‘𝑛)‘𝑘) = (𝑛𝑀𝑘))
67	62, 66	sylanl1 678	. . . . . . . . . . . . . . . 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 7694	. . . . . . . . . . . . . . 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 2765	. . . . . . . . . . . . . 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 5249	. . . . . . . . . . . . 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 2765	. . . . . . . . . . . 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 7435	. . . . . . . . . . 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 773	. . . . . . . . . . . 12 ⊢ ((((𝑅 ∈ Ring ∧ 𝑀:(𝐼 × 𝐼)⟶(Base‘𝑅)) ∧ 𝐼 ∈ (Fin ∖ {∅})) ∧ 𝑓:𝐼⟶(Base‘𝑅)) → 𝑅 ∈ Ring)
74		simp-4l 781	. . . . . . . . . . . . . . . 16 ⊢ (((((𝑅 ∈ Ring ∧ 𝑀:(𝐼 × 𝐼)⟶(Base‘𝑅)) ∧ 𝑓:𝐼⟶(Base‘𝑅)) ∧ 𝑛 ∈ 𝐼) ∧ 𝑘 ∈ 𝐼) → 𝑅 ∈ Ring)
75	43	ad4ant23 751	. . . . . . . . . . . . . . . 16 ⊢ (((((𝑅 ∈ Ring ∧ 𝑀:(𝐼 × 𝐼)⟶(Base‘𝑅)) ∧ 𝑓:𝐼⟶(Base‘𝑅)) ∧ 𝑛 ∈ 𝐼) ∧ 𝑘 ∈ 𝐼) → (𝑓‘𝑛) ∈ (Base‘𝑅))
76		fovcdm 7591	. . . . . . . . . . . . . . . . 17 ⊢ ((𝑀:(𝐼 × 𝐼)⟶(Base‘𝑅) ∧ 𝑛 ∈ 𝐼 ∧ 𝑘 ∈ 𝐼) → (𝑛𝑀𝑘) ∈ (Base‘𝑅))
77	76	ad5ant245 1358	. . . . . . . . . . . . . . . 16 ⊢ (((((𝑅 ∈ Ring ∧ 𝑀:(𝐼 × 𝐼)⟶(Base‘𝑅)) ∧ 𝑓:𝐼⟶(Base‘𝑅)) ∧ 𝑛 ∈ 𝐼) ∧ 𝑘 ∈ 𝐼) → (𝑛𝑀𝑘) ∈ (Base‘𝑅))
78	10, 47	ringcl 20202	. . . . . . . . . . . . . . . 16 ⊢ ((𝑅 ∈ Ring ∧ (𝑓‘𝑛) ∈ (Base‘𝑅) ∧ (𝑛𝑀𝑘) ∈ (Base‘𝑅)) → ((𝑓‘𝑛)(.r‘𝑅)(𝑛𝑀𝑘)) ∈ (Base‘𝑅))
79	74, 75, 77, 78	syl3anc 1368	. . . . . . . . . . . . . . 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 7124	. . . . . . . . . . . . . 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 717	. . . . . . . . . . . . 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 8858	. . . . . . . . . . . . . . . . 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 688	. . . . . . . . . . . . . . . 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 480	. . . . . . . . . . . . . . 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 2811	. . . . . . . . . . . . . . 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 280	. . . . . . . . . . . . . 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 755	. . . . . . . . . . . . 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 231	. . . . . . . . . . . 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 7233	. . . . . . . . . . . . . . . 16 ⊢ (𝐼 ∈ (Fin ∖ {∅}) → (𝑘 ∈ 𝐼 ↦ ((𝑓‘𝑛)(.r‘𝑅)(𝑛𝑀𝑘))) ∈ V)
90	89	ralrimivw 3139	. . . . . . . . . . . . . . 15 ⊢ (𝐼 ∈ (Fin ∖ {∅}) → ∀𝑛 ∈ 𝐼 (𝑘 ∈ 𝐼 ↦ ((𝑓‘𝑛)(.r‘𝑅)(𝑛𝑀𝑘))) ∈ V)
91		eqid 2725	. . . . . . . . . . . . . . . 16 ⊢ (𝑛 ∈ 𝐼 ↦ (𝑘 ∈ 𝐼 ↦ ((𝑓‘𝑛)(.r‘𝑅)(𝑛𝑀𝑘)))) = (𝑛 ∈ 𝐼 ↦ (𝑘 ∈ 𝐼 ↦ ((𝑓‘𝑛)(.r‘𝑅)(𝑛𝑀𝑘))))
92	91	fnmpt 6696	. . . . . . . . . . . . . . 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 17	. . . . . . . . . . . . . 14 ⊢ (𝐼 ∈ (Fin ∖ {∅}) → (𝑛 ∈ 𝐼 ↦ (𝑘 ∈ 𝐼 ↦ ((𝑓‘𝑛)(.r‘𝑅)(𝑛𝑀𝑘)))) Fn 𝐼)
94		fvexd 6911	. . . . . . . . . . . . . 14 ⊢ (𝐼 ∈ (Fin ∖ {∅}) → (0g‘(𝑅 freeLMod 𝐼)) ∈ V)
95	93, 9, 94	fndmfifsupp 9403	. . . . . . . . . . . . 13 ⊢ (𝐼 ∈ (Fin ∖ {∅}) → (𝑛 ∈ 𝐼 ↦ (𝑘 ∈ 𝐼 ↦ ((𝑓‘𝑛)(.r‘𝑅)(𝑛𝑀𝑘)))) finSupp (0g‘(𝑅 freeLMod 𝐼)))
96	95	ad2antlr 725	. . . . . . . . . . . 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 21723	. . . . . . . . . . 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 2766	. . . . . . . . . 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 591	. . . . . . . . 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 2725	. . . . . . . . . . 11 ⊢ (0g‘𝑅) = (0g‘𝑅)
101	5, 100	frlm0 21705	. . . . . . . . . 10 ⊢ ((𝑅 ∈ Ring ∧ 𝐼 ∈ (Fin ∖ {∅})) → (𝐼 × {(0g‘𝑅)}) = (0g‘(𝑅 freeLMod 𝐼)))
102	101	ad4ant13 749	. . . . . . . . 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 2741	. . . . . . . 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 6900	. . . . . . . . . . . 12 ⊢ ((𝑅 ∈ Ring ∧ 𝐼 ∈ (Fin ∖ {∅})) → (0g‘𝑅) = (0g‘(Scalar‘(𝑅 freeLMod 𝐼))))
105	104	sneqd 4642	. . . . . . . . . . 11 ⊢ ((𝑅 ∈ Ring ∧ 𝐼 ∈ (Fin ∖ {∅})) → {(0g‘𝑅)} = {(0g‘(Scalar‘(𝑅 freeLMod 𝐼)))})
106	105	xpeq2d 5708	. . . . . . . . . 10 ⊢ ((𝑅 ∈ Ring ∧ 𝐼 ∈ (Fin ∖ {∅})) → (𝐼 × {(0g‘𝑅)}) = (𝐼 × {(0g‘(Scalar‘(𝑅 freeLMod 𝐼)))}))
107	106	eqeq2d 2736	. . . . . . . . 9 ⊢ ((𝑅 ∈ Ring ∧ 𝐼 ∈ (Fin ∖ {∅})) → (𝑓 = (𝐼 × {(0g‘𝑅)}) ↔ 𝑓 = (𝐼 × {(0g‘(Scalar‘(𝑅 freeLMod 𝐼)))})))
108	107	ad4ant13 749	. . . . . . . 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 343	. . . . . . 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 3316	. . . . . 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 281	. . . . 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 317	. . . 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 3091	. . . 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 288	. . 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 678	. 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 7224	. . . . . . . . . . . 12 ⊢ (𝑓:𝐼⟶{(0g‘𝑅)} ↔ (𝑓 Fn 𝐼 ∧ ∀𝑖 ∈ 𝐼 (𝑓‘𝑖) = (0g‘𝑅)))
117		fvex 6909	. . . . . . . . . . . . 13 ⊢ (0g‘𝑅) ∈ V
118	117	fconst2 7217	. . . . . . . . . . . 12 ⊢ (𝑓:𝐼⟶{(0g‘𝑅)} ↔ 𝑓 = (𝐼 × {(0g‘𝑅)}))
119	116, 118	sylbb1 236	. . . . . . . . . . 11 ⊢ ((𝑓 Fn 𝐼 ∧ ∀𝑖 ∈ 𝐼 (𝑓‘𝑖) = (0g‘𝑅)) → 𝑓 = (𝐼 × {(0g‘𝑅)}))
120	119	ex 411	. . . . . . . . . 10 ⊢ (𝑓 Fn 𝐼 → (∀𝑖 ∈ 𝐼 (𝑓‘𝑖) = (0g‘𝑅) → 𝑓 = (𝐼 × {(0g‘𝑅)})))
121	120	con3d 152	. . . . . . . . 9 ⊢ (𝑓 Fn 𝐼 → (¬ 𝑓 = (𝐼 × {(0g‘𝑅)}) → ¬ ∀𝑖 ∈ 𝐼 (𝑓‘𝑖) = (0g‘𝑅)))
122		df-ne 2930	. . . . . . . . . . 11 ⊢ ((𝑓‘𝑖) ≠ (0g‘𝑅) ↔ ¬ (𝑓‘𝑖) = (0g‘𝑅))
123	122	rexbii 3083	. . . . . . . . . 10 ⊢ (∃𝑖 ∈ 𝐼 (𝑓‘𝑖) ≠ (0g‘𝑅) ↔ ∃𝑖 ∈ 𝐼 ¬ (𝑓‘𝑖) = (0g‘𝑅))
124		rexnal 3089	. . . . . . . . . 10 ⊢ (∃𝑖 ∈ 𝐼 ¬ (𝑓‘𝑖) = (0g‘𝑅) ↔ ¬ ∀𝑖 ∈ 𝐼 (𝑓‘𝑖) = (0g‘𝑅))
125	123, 124	bitri 274	. . . . . . . . 9 ⊢ (∃𝑖 ∈ 𝐼 (𝑓‘𝑖) ≠ (0g‘𝑅) ↔ ¬ ∀𝑖 ∈ 𝐼 (𝑓‘𝑖) = (0g‘𝑅))
126	121, 125	imbitrrdi 251	. . . . . . . 8 ⊢ (𝑓 Fn 𝐼 → (¬ 𝑓 = (𝐼 × {(0g‘𝑅)}) → ∃𝑖 ∈ 𝐼 (𝑓‘𝑖) ≠ (0g‘𝑅)))
127	33, 126	syl 17	. . . . . . 7 ⊢ (𝑓:𝐼⟶(Base‘𝑅) → (¬ 𝑓 = (𝐼 × {(0g‘𝑅)}) → ∃𝑖 ∈ 𝐼 (𝑓‘𝑖) ≠ (0g‘𝑅)))
128	127	adantl 480	. . . . . 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 4797	. . . . . . . . . . 11 ⊢ ¬ 𝑖 ∈ (𝐼 ∖ {𝑖})
130		difss 4128	. . . . . . . . . . 11 ⊢ (𝐼 ∖ {𝑖}) ⊆ 𝐼
131		diffi 9204	. . . . . . . . . . . . 13 ⊢ (𝐼 ∈ Fin → (𝐼 ∖ {𝑖}) ∈ Fin)
132	131	ad4antlr 731	. . . . . . . . . . . 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 2814	. . . . . . . . . . . . . . . . 17 ⊢ (𝑦 = ∅ → (𝑖 ∈ 𝑦 ↔ 𝑖 ∈ ∅))
134	133	notbid 317	. . . . . . . . . . . . . . . 16 ⊢ (𝑦 = ∅ → (¬ 𝑖 ∈ 𝑦 ↔ ¬ 𝑖 ∈ ∅))
135		sseq1 4002	. . . . . . . . . . . . . . . 16 ⊢ (𝑦 = ∅ → (𝑦 ⊆ 𝐼 ↔ ∅ ⊆ 𝐼))
136	134, 135	anbi12d 630	. . . . . . . . . . . . . . 15 ⊢ (𝑦 = ∅ → ((¬ 𝑖 ∈ 𝑦 ∧ 𝑦 ⊆ 𝐼) ↔ (¬ 𝑖 ∈ ∅ ∧ ∅ ⊆ 𝐼)))
137	136	anbi2d 628	. . . . . . . . . . . . . 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 5242	. . . . . . . . . . . . . . . . . . . . . 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 6698	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (𝑛 ∈ ∅ ↦ (((invr‘𝑅)‘(𝑓‘𝑖))(.r‘𝑅)((𝑓‘𝑛)(.r‘𝑅)(𝑛𝑀𝑘)))) = ∅
140	138, 139	eqtrdi 2781	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (𝑦 = ∅ → (𝑛 ∈ 𝑦 ↦ (((invr‘𝑅)‘(𝑓‘𝑖))(.r‘𝑅)((𝑓‘𝑛)(.r‘𝑅)(𝑛𝑀𝑘)))) = ∅)
141	140	oveq2d 7435	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝑦 = ∅ → (𝑅 Σg (𝑛 ∈ 𝑦 ↦ (((invr‘𝑅)‘(𝑓‘𝑖))(.r‘𝑅)((𝑓‘𝑛)(.r‘𝑅)(𝑛𝑀𝑘))))) = (𝑅 Σg ∅))
142	100	gsum0 18647	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝑅 Σg ∅) = (0g‘𝑅)
143	141, 142	eqtrdi 2781	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝑦 = ∅ → (𝑅 Σg (𝑛 ∈ 𝑦 ↦ (((invr‘𝑅)‘(𝑓‘𝑖))(.r‘𝑅)((𝑓‘𝑛)(.r‘𝑅)(𝑛𝑀𝑘))))) = (0g‘𝑅))
144	143	oveq1d 7434	. . . . . . . . . . . . . . . . . 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 4549	. . . . . . . . . . . . . . . . 17 ⊢ (𝑦 = ∅ → if(𝑗 = 𝑖, ((𝑅 Σg (𝑛 ∈ 𝑦 ↦ (((invr‘𝑅)‘(𝑓‘𝑖))(.r‘𝑅)((𝑓‘𝑛)(.r‘𝑅)(𝑛𝑀𝑘)))))(+g‘𝑅)(𝑖𝑀𝑘)), (𝑗𝑀𝑘)) = if(𝑗 = 𝑖, ((0g‘𝑅)(+g‘𝑅)(𝑖𝑀𝑘)), (𝑗𝑀𝑘)))
146	145	mpoeq3dv 7499	. . . . . . . . . . . . . . . 16 ⊢ (𝑦 = ∅ → (𝑗 ∈ 𝐼, 𝑘 ∈ 𝐼 ↦ if(𝑗 = 𝑖, ((𝑅 Σg (𝑛 ∈ 𝑦 ↦ (((invr‘𝑅)‘(𝑓‘𝑖))(.r‘𝑅)((𝑓‘𝑛)(.r‘𝑅)(𝑛𝑀𝑘)))))(+g‘𝑅)(𝑖𝑀𝑘)), (𝑗𝑀𝑘))) = (𝑗 ∈ 𝐼, 𝑘 ∈ 𝐼 ↦ if(𝑗 = 𝑖, ((0g‘𝑅)(+g‘𝑅)(𝑖𝑀𝑘)), (𝑗𝑀𝑘))))
147	146	fveq2d 6900	. . . . . . . . . . . . . . 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 2736	. . . . . . . . . . . . . 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 343	. . . . . . . . . . . . 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 2113	. . . . . . . . . . . . . . . . 17 ⊢ (𝑦 = 𝑥 → (𝑖 ∈ 𝑦 ↔ 𝑖 ∈ 𝑥))
151	150	notbid 317	. . . . . . . . . . . . . . . 16 ⊢ (𝑦 = 𝑥 → (¬ 𝑖 ∈ 𝑦 ↔ ¬ 𝑖 ∈ 𝑥))
152		sseq1 4002	. . . . . . . . . . . . . . . 16 ⊢ (𝑦 = 𝑥 → (𝑦 ⊆ 𝐼 ↔ 𝑥 ⊆ 𝐼))
153	151, 152	anbi12d 630	. . . . . . . . . . . . . . 15 ⊢ (𝑦 = 𝑥 → ((¬ 𝑖 ∈ 𝑦 ∧ 𝑦 ⊆ 𝐼) ↔ (¬ 𝑖 ∈ 𝑥 ∧ 𝑥 ⊆ 𝐼)))
154	153	anbi2d 628	. . . . . . . . . . . . . 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 5242	. . . . . . . . . . . . . . . . . . . 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 7435	. . . . . . . . . . . . . . . . . . 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 7434	. . . . . . . . . . . . . . . . . 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 4549	. . . . . . . . . . . . . . . . 17 ⊢ (𝑦 = 𝑥 → if(𝑗 = 𝑖, ((𝑅 Σg (𝑛 ∈ 𝑦 ↦ (((invr‘𝑅)‘(𝑓‘𝑖))(.r‘𝑅)((𝑓‘𝑛)(.r‘𝑅)(𝑛𝑀𝑘)))))(+g‘𝑅)(𝑖𝑀𝑘)), (𝑗𝑀𝑘)) = if(𝑗 = 𝑖, ((𝑅 Σg (𝑛 ∈ 𝑥 ↦ (((invr‘𝑅)‘(𝑓‘𝑖))(.r‘𝑅)((𝑓‘𝑛)(.r‘𝑅)(𝑛𝑀𝑘)))))(+g‘𝑅)(𝑖𝑀𝑘)), (𝑗𝑀𝑘)))
159	158	mpoeq3dv 7499	. . . . . . . . . . . . . . . 16 ⊢ (𝑦 = 𝑥 → (𝑗 ∈ 𝐼, 𝑘 ∈ 𝐼 ↦ if(𝑗 = 𝑖, ((𝑅 Σg (𝑛 ∈ 𝑦 ↦ (((invr‘𝑅)‘(𝑓‘𝑖))(.r‘𝑅)((𝑓‘𝑛)(.r‘𝑅)(𝑛𝑀𝑘)))))(+g‘𝑅)(𝑖𝑀𝑘)), (𝑗𝑀𝑘))) = (𝑗 ∈ 𝐼, 𝑘 ∈ 𝐼 ↦ if(𝑗 = 𝑖, ((𝑅 Σg (𝑛 ∈ 𝑥 ↦ (((invr‘𝑅)‘(𝑓‘𝑖))(.r‘𝑅)((𝑓‘𝑛)(.r‘𝑅)(𝑛𝑀𝑘)))))(+g‘𝑅)(𝑖𝑀𝑘)), (𝑗𝑀𝑘))))
160	159	fveq2d 6900	. . . . . . . . . . . . . . 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 2736	. . . . . . . . . . . . . 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 343	. . . . . . . . . . . . 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 2814	. . . . . . . . . . . . . . . . 17 ⊢ (𝑦 = (𝑥 ∪ {𝑧}) → (𝑖 ∈ 𝑦 ↔ 𝑖 ∈ (𝑥 ∪ {𝑧})))
164	163	notbid 317	. . . . . . . . . . . . . . . 16 ⊢ (𝑦 = (𝑥 ∪ {𝑧}) → (¬ 𝑖 ∈ 𝑦 ↔ ¬ 𝑖 ∈ (𝑥 ∪ {𝑧})))
165		sseq1 4002	. . . . . . . . . . . . . . . 16 ⊢ (𝑦 = (𝑥 ∪ {𝑧}) → (𝑦 ⊆ 𝐼 ↔ (𝑥 ∪ {𝑧}) ⊆ 𝐼))
166	164, 165	anbi12d 630	. . . . . . . . . . . . . . 15 ⊢ (𝑦 = (𝑥 ∪ {𝑧}) → ((¬ 𝑖 ∈ 𝑦 ∧ 𝑦 ⊆ 𝐼) ↔ (¬ 𝑖 ∈ (𝑥 ∪ {𝑧}) ∧ (𝑥 ∪ {𝑧}) ⊆ 𝐼)))
167	166	anbi2d 628	. . . . . . . . . . . . . 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 5242	. . . . . . . . . . . . . . . . . . . 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 7435	. . . . . . . . . . . . . . . . . . 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 7434	. . . . . . . . . . . . . . . . . 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 4549	. . . . . . . . . . . . . . . . 17 ⊢ (𝑦 = (𝑥 ∪ {𝑧}) → if(𝑗 = 𝑖, ((𝑅 Σg (𝑛 ∈ 𝑦 ↦ (((invr‘𝑅)‘(𝑓‘𝑖))(.r‘𝑅)((𝑓‘𝑛)(.r‘𝑅)(𝑛𝑀𝑘)))))(+g‘𝑅)(𝑖𝑀𝑘)), (𝑗𝑀𝑘)) = if(𝑗 = 𝑖, ((𝑅 Σg (𝑛 ∈ (𝑥 ∪ {𝑧}) ↦ (((invr‘𝑅)‘(𝑓‘𝑖))(.r‘𝑅)((𝑓‘𝑛)(.r‘𝑅)(𝑛𝑀𝑘)))))(+g‘𝑅)(𝑖𝑀𝑘)), (𝑗𝑀𝑘)))
172	171	mpoeq3dv 7499	. . . . . . . . . . . . . . . 16 ⊢ (𝑦 = (𝑥 ∪ {𝑧}) → (𝑗 ∈ 𝐼, 𝑘 ∈ 𝐼 ↦ if(𝑗 = 𝑖, ((𝑅 Σg (𝑛 ∈ 𝑦 ↦ (((invr‘𝑅)‘(𝑓‘𝑖))(.r‘𝑅)((𝑓‘𝑛)(.r‘𝑅)(𝑛𝑀𝑘)))))(+g‘𝑅)(𝑖𝑀𝑘)), (𝑗𝑀𝑘))) = (𝑗 ∈ 𝐼, 𝑘 ∈ 𝐼 ↦ if(𝑗 = 𝑖, ((𝑅 Σg (𝑛 ∈ (𝑥 ∪ {𝑧}) ↦ (((invr‘𝑅)‘(𝑓‘𝑖))(.r‘𝑅)((𝑓‘𝑛)(.r‘𝑅)(𝑛𝑀𝑘)))))(+g‘𝑅)(𝑖𝑀𝑘)), (𝑗𝑀𝑘))))
173	172	fveq2d 6900	. . . . . . . . . . . . . . 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 2736	. . . . . . . . . . . . . 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 343	. . . . . . . . . . . . 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 2814	. . . . . . . . . . . . . . . . 17 ⊢ (𝑦 = (𝐼 ∖ {𝑖}) → (𝑖 ∈ 𝑦 ↔ 𝑖 ∈ (𝐼 ∖ {𝑖})))
177	176	notbid 317	. . . . . . . . . . . . . . . 16 ⊢ (𝑦 = (𝐼 ∖ {𝑖}) → (¬ 𝑖 ∈ 𝑦 ↔ ¬ 𝑖 ∈ (𝐼 ∖ {𝑖})))
178		sseq1 4002	. . . . . . . . . . . . . . . 16 ⊢ (𝑦 = (𝐼 ∖ {𝑖}) → (𝑦 ⊆ 𝐼 ↔ (𝐼 ∖ {𝑖}) ⊆ 𝐼))
179	177, 178	anbi12d 630	. . . . . . . . . . . . . . 15 ⊢ (𝑦 = (𝐼 ∖ {𝑖}) → ((¬ 𝑖 ∈ 𝑦 ∧ 𝑦 ⊆ 𝐼) ↔ (¬ 𝑖 ∈ (𝐼 ∖ {𝑖}) ∧ (𝐼 ∖ {𝑖}) ⊆ 𝐼)))
180	179	anbi2d 628	. . . . . . . . . . . . . 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 5242	. . . . . . . . . . . . . . . . . . . 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 7435	. . . . . . . . . . . . . . . . . . 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 7434	. . . . . . . . . . . . . . . . . 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 4549	. . . . . . . . . . . . . . . . 17 ⊢ (𝑦 = (𝐼 ∖ {𝑖}) → if(𝑗 = 𝑖, ((𝑅 Σg (𝑛 ∈ 𝑦 ↦ (((invr‘𝑅)‘(𝑓‘𝑖))(.r‘𝑅)((𝑓‘𝑛)(.r‘𝑅)(𝑛𝑀𝑘)))))(+g‘𝑅)(𝑖𝑀𝑘)), (𝑗𝑀𝑘)) = if(𝑗 = 𝑖, ((𝑅 Σg (𝑛 ∈ (𝐼 ∖ {𝑖}) ↦ (((invr‘𝑅)‘(𝑓‘𝑖))(.r‘𝑅)((𝑓‘𝑛)(.r‘𝑅)(𝑛𝑀𝑘)))))(+g‘𝑅)(𝑖𝑀𝑘)), (𝑗𝑀𝑘)))
185	184	mpoeq3dv 7499	. . . . . . . . . . . . . . . 16 ⊢ (𝑦 = (𝐼 ∖ {𝑖}) → (𝑗 ∈ 𝐼, 𝑘 ∈ 𝐼 ↦ if(𝑗 = 𝑖, ((𝑅 Σg (𝑛 ∈ 𝑦 ↦ (((invr‘𝑅)‘(𝑓‘𝑖))(.r‘𝑅)((𝑓‘𝑛)(.r‘𝑅)(𝑛𝑀𝑘)))))(+g‘𝑅)(𝑖𝑀𝑘)), (𝑗𝑀𝑘))) = (𝑗 ∈ 𝐼, 𝑘 ∈ 𝐼 ↦ if(𝑗 = 𝑖, ((𝑅 Σg (𝑛 ∈ (𝐼 ∖ {𝑖}) ↦ (((invr‘𝑅)‘(𝑓‘𝑖))(.r‘𝑅)((𝑓‘𝑛)(.r‘𝑅)(𝑛𝑀𝑘)))))(+g‘𝑅)(𝑖𝑀𝑘)), (𝑗𝑀𝑘))))
186	185	fveq2d 6900	. . . . . . . . . . . . . . 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 2736	. . . . . . . . . . . . . 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 343	. . . . . . . . . . . . 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 7552	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑀 Fn (𝐼 × 𝐼) ↔ 𝑀 = (𝑗 ∈ 𝐼, 𝑘 ∈ 𝐼 ↦ (𝑗𝑀𝑘)))
190	59, 189	sylib 217	. . . . . . . . . . . . . . . . 17 ⊢ (𝑀:(𝐼 × 𝐼)⟶(Base‘𝑅) → 𝑀 = (𝑗 ∈ 𝐼, 𝑘 ∈ 𝐼 ↦ (𝑗𝑀𝑘)))
191	190	adantl 480	. . . . . . . . . . . . . . . 16 ⊢ ((𝑅 ∈ Field ∧ 𝑀:(𝐼 × 𝐼)⟶(Base‘𝑅)) → 𝑀 = (𝑗 ∈ 𝐼, 𝑘 ∈ 𝐼 ↦ (𝑗𝑀𝑘)))
192		ringgrp 20190	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑅 ∈ Ring → 𝑅 ∈ Grp)
193	4, 192	syl 17	. . . . . . . . . . . . . . . . 17 ⊢ (𝑅 ∈ Field → 𝑅 ∈ Grp)
194		oveq1 7426	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (𝑖 = 𝑗 → (𝑖𝑀𝑘) = (𝑗𝑀𝑘))
195	194	equcoms 2015	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (𝑗 = 𝑖 → (𝑖𝑀𝑘) = (𝑗𝑀𝑘))
196	195	oveq2d 7435	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝑗 = 𝑖 → ((0g‘𝑅)(+g‘𝑅)(𝑖𝑀𝑘)) = ((0g‘𝑅)(+g‘𝑅)(𝑗𝑀𝑘)))
197		simp1l 1194	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (((𝑅 ∈ Grp ∧ 𝑀:(𝐼 × 𝐼)⟶(Base‘𝑅)) ∧ 𝑗 ∈ 𝐼 ∧ 𝑘 ∈ 𝐼) → 𝑅 ∈ Grp)
198		fovcdm 7591	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ ((𝑀:(𝐼 × 𝐼)⟶(Base‘𝑅) ∧ 𝑗 ∈ 𝐼 ∧ 𝑘 ∈ 𝐼) → (𝑗𝑀𝑘) ∈ (Base‘𝑅))
199	198	3adant1l 1173	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (((𝑅 ∈ Grp ∧ 𝑀:(𝐼 × 𝐼)⟶(Base‘𝑅)) ∧ 𝑗 ∈ 𝐼 ∧ 𝑘 ∈ 𝐼) → (𝑗𝑀𝑘) ∈ (Base‘𝑅))
200		eqid 2725	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (+g‘𝑅) = (+g‘𝑅)
201	10, 200, 100	grplid 18932	. . . . . . . . . . . . . . . . . . . . 21 ⊢ ((𝑅 ∈ Grp ∧ (𝑗𝑀𝑘) ∈ (Base‘𝑅)) → ((0g‘𝑅)(+g‘𝑅)(𝑗𝑀𝑘)) = (𝑗𝑀𝑘))
202	197, 199, 201	syl2anc 582	. . . . . . . . . . . . . . . . . . . 20 ⊢ (((𝑅 ∈ Grp ∧ 𝑀:(𝐼 × 𝐼)⟶(Base‘𝑅)) ∧ 𝑗 ∈ 𝐼 ∧ 𝑘 ∈ 𝐼) → ((0g‘𝑅)(+g‘𝑅)(𝑗𝑀𝑘)) = (𝑗𝑀𝑘))
203	196, 202	sylan9eqr 2787	. . . . . . . . . . . . . . . . . . 19 ⊢ ((((𝑅 ∈ Grp ∧ 𝑀:(𝐼 × 𝐼)⟶(Base‘𝑅)) ∧ 𝑗 ∈ 𝐼 ∧ 𝑘 ∈ 𝐼) ∧ 𝑗 = 𝑖) → ((0g‘𝑅)(+g‘𝑅)(𝑖𝑀𝑘)) = (𝑗𝑀𝑘))
204		eqidd 2726	. . . . . . . . . . . . . . . . . . 19 ⊢ ((((𝑅 ∈ Grp ∧ 𝑀:(𝐼 × 𝐼)⟶(Base‘𝑅)) ∧ 𝑗 ∈ 𝐼 ∧ 𝑘 ∈ 𝐼) ∧ ¬ 𝑗 = 𝑖) → (𝑗𝑀𝑘) = (𝑗𝑀𝑘))
205	203, 204	ifeqda 4566	. . . . . . . . . . . . . . . . . 18 ⊢ (((𝑅 ∈ Grp ∧ 𝑀:(𝐼 × 𝐼)⟶(Base‘𝑅)) ∧ 𝑗 ∈ 𝐼 ∧ 𝑘 ∈ 𝐼) → if(𝑗 = 𝑖, ((0g‘𝑅)(+g‘𝑅)(𝑖𝑀𝑘)), (𝑗𝑀𝑘)) = (𝑗𝑀𝑘))
206	205	mpoeq3dva 7497	. . . . . . . . . . . . . . . . 17 ⊢ ((𝑅 ∈ Grp ∧ 𝑀:(𝐼 × 𝐼)⟶(Base‘𝑅)) → (𝑗 ∈ 𝐼, 𝑘 ∈ 𝐼 ↦ if(𝑗 = 𝑖, ((0g‘𝑅)(+g‘𝑅)(𝑖𝑀𝑘)), (𝑗𝑀𝑘))) = (𝑗 ∈ 𝐼, 𝑘 ∈ 𝐼 ↦ (𝑗𝑀𝑘)))
207	193, 206	sylan 578	. . . . . . . . . . . . . . . 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 2768	. . . . . . . . . . . . . . 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 6900	. . . . . . . . . . . . . 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 730	. . . . . . . . . . . . 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 4174	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (𝑖 ∈ 𝑥 → 𝑖 ∈ (𝑥 ∪ {𝑧}))
212	211	con3i 154	. . . . . . . . . . . . . . . . . . . 20 ⊢ (¬ 𝑖 ∈ (𝑥 ∪ {𝑧}) → ¬ 𝑖 ∈ 𝑥)
213		ssun1 4170	. . . . . . . . . . . . . . . . . . . . 21 ⊢ 𝑥 ⊆ (𝑥 ∪ {𝑧})
214		sstr 3985	. . . . . . . . . . . . . . . . . . . . 21 ⊢ ((𝑥 ⊆ (𝑥 ∪ {𝑧}) ∧ (𝑥 ∪ {𝑧}) ⊆ 𝐼) → 𝑥 ⊆ 𝐼)
215	213, 214	mpan 688	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((𝑥 ∪ {𝑧}) ⊆ 𝐼 → 𝑥 ⊆ 𝐼)
216	212, 215	anim12i 611	. . . . . . . . . . . . . . . . . . 19 ⊢ ((¬ 𝑖 ∈ (𝑥 ∪ {𝑧}) ∧ (𝑥 ∪ {𝑧}) ⊆ 𝐼) → (¬ 𝑖 ∈ 𝑥 ∧ 𝑥 ⊆ 𝐼))
217	216	anim2i 615	. . . . . . . . . . . . . . . . . 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 479	. . . . . . . . . . . . . . . . 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 4646	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ (𝑖 ∈ {𝑧} ↔ 𝑖 = 𝑧)
220		elun2 4175	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ (𝑖 ∈ {𝑧} → 𝑖 ∈ (𝑥 ∪ {𝑧}))
221	219, 220	sylbir 234	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (𝑖 = 𝑧 → 𝑖 ∈ (𝑥 ∪ {𝑧}))
222	221	necon3bi 2956	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (¬ 𝑖 ∈ (𝑥 ∪ {𝑧}) → 𝑖 ≠ 𝑧)
223	222	anim1i 613	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((¬ 𝑖 ∈ (𝑥 ∪ {𝑧}) ∧ (𝑥 ∪ {𝑧}) ⊆ 𝐼) → (𝑖 ≠ 𝑧 ∧ (𝑥 ∪ {𝑧}) ⊆ 𝐼))
224		ringcmn 20230	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31 ⊢ (𝑅 ∈ Ring → 𝑅 ∈ CMnd)
225	4, 224	syl 17	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30 ⊢ (𝑅 ∈ Field → 𝑅 ∈ CMnd)
226	225	ad7antr 736	. . . . . . . . . . . . . . . . . . . . . . . . . . . . 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 767	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31 ⊢ ((((𝑅 ∈ Field ∧ 𝑀:(𝐼 × 𝐼)⟶(Base‘𝑅)) ∧ 𝐼 ∈ Fin) ∧ 𝑓:𝐼⟶(Base‘𝑅)) → 𝐼 ∈ Fin)
228	215	adantl 480	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31 ⊢ ((𝑖 ≠ 𝑧 ∧ (𝑥 ∪ {𝑧}) ⊆ 𝐼) → 𝑥 ⊆ 𝐼)
229		ssfi 9198	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31 ⊢ ((𝐼 ∈ Fin ∧ 𝑥 ⊆ 𝐼) → 𝑥 ∈ Fin)
230	227, 228, 229	syl2an 594	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30 ⊢ (((((𝑅 ∈ Field ∧ 𝑀:(𝐼 × 𝐼)⟶(Base‘𝑅)) ∧ 𝐼 ∈ Fin) ∧ 𝑓:𝐼⟶(Base‘𝑅)) ∧ (𝑖 ≠ 𝑧 ∧ (𝑥 ∪ {𝑧}) ⊆ 𝐼)) → 𝑥 ∈ Fin)
231	230	ad5ant13 755	. . . . . . . . . . . . . . . . . . . . . . . . . . . . 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 3976	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33 ⊢ (((𝑥 ∪ {𝑧}) ⊆ 𝐼 ∧ 𝑛 ∈ 𝑥) → 𝑛 ∈ 𝐼)
233	232	adantll 712	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32 ⊢ (((𝑖 ≠ 𝑧 ∧ (𝑥 ∪ {𝑧}) ⊆ 𝐼) ∧ 𝑛 ∈ 𝑥) → 𝑛 ∈ 𝐼)
234	233	ad4ant24 752	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 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 734	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 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 724	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35 ⊢ (((𝑅 ∈ Field ∧ 𝑀:(𝐼 × 𝐼)⟶(Base‘𝑅)) ∧ 𝐼 ∈ Fin) → 𝑅 ∈ DivRing)
237		ffvelcdm 7090	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 39 ⊢ ((𝑓:𝐼⟶(Base‘𝑅) ∧ 𝑖 ∈ 𝐼) → (𝑓‘𝑖) ∈ (Base‘𝑅))
238	237	anim2i 615	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 38 ⊢ ((𝑅 ∈ DivRing ∧ (𝑓:𝐼⟶(Base‘𝑅) ∧ 𝑖 ∈ 𝐼)) → (𝑅 ∈ DivRing ∧ (𝑓‘𝑖) ∈ (Base‘𝑅)))
239	238	anassrs 466	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 37 ⊢ (((𝑅 ∈ DivRing ∧ 𝑓:𝐼⟶(Base‘𝑅)) ∧ 𝑖 ∈ 𝐼) → (𝑅 ∈ DivRing ∧ (𝑓‘𝑖) ∈ (Base‘𝑅)))
240		eqid 2725	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 39 ⊢ (invr‘𝑅) = (invr‘𝑅)
241	10, 100, 240	drnginvrcl 20658	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 38 ⊢ ((𝑅 ∈ DivRing ∧ (𝑓‘𝑖) ∈ (Base‘𝑅) ∧ (𝑓‘𝑖) ≠ (0g‘𝑅)) → ((invr‘𝑅)‘(𝑓‘𝑖)) ∈ (Base‘𝑅))
242	241	3expa 1115	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 37 ⊢ (((𝑅 ∈ DivRing ∧ (𝑓‘𝑖) ∈ (Base‘𝑅)) ∧ (𝑓‘𝑖) ≠ (0g‘𝑅)) → ((invr‘𝑅)‘(𝑓‘𝑖)) ∈ (Base‘𝑅))
243	239, 242	sylan 578	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 36 ⊢ ((((𝑅 ∈ DivRing ∧ 𝑓:𝐼⟶(Base‘𝑅)) ∧ 𝑖 ∈ 𝐼) ∧ (𝑓‘𝑖) ≠ (0g‘𝑅)) → ((invr‘𝑅)‘(𝑓‘𝑖)) ∈ (Base‘𝑅))
244	243	anasss 465	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35 ⊢ (((𝑅 ∈ DivRing ∧ 𝑓:𝐼⟶(Base‘𝑅)) ∧ (𝑖 ∈ 𝐼 ∧ (𝑓‘𝑖) ≠ (0g‘𝑅))) → ((invr‘𝑅)‘(𝑓‘𝑖)) ∈ (Base‘𝑅))
245	236, 244	sylanl1 678	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 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 724	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 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 760	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 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 782	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 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 1115	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 36 ⊢ (((𝑀:(𝐼 × 𝐼)⟶(Base‘𝑅) ∧ 𝑛 ∈ 𝐼) ∧ 𝑘 ∈ 𝐼) → (𝑛𝑀𝑘) ∈ (Base‘𝑅))
250	249	an32s 650	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35 ⊢ (((𝑀:(𝐼 × 𝐼)⟶(Base‘𝑅) ∧ 𝑘 ∈ 𝐼) ∧ 𝑛 ∈ 𝐼) → (𝑛𝑀𝑘) ∈ (Base‘𝑅))
251	248, 250	sylanl1 678	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 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 1368	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 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 20202	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 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 1368	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 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 717	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 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 589	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 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 717	. . . . . . . . . . . . . . . . . . . . . . . . . . . . 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 3465	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 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 767	. . . . . . . . . . . . . . . . . . . . . . . . . . . . 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 4171	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34 ⊢ {𝑧} ⊆ (𝑥 ∪ {𝑧})
262		sstr 3985	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34 ⊢ (({𝑧} ⊆ (𝑥 ∪ {𝑧}) ∧ (𝑥 ∪ {𝑧}) ⊆ 𝐼) → {𝑧} ⊆ 𝐼)
263	261, 262	mpan 688	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33 ⊢ ((𝑥 ∪ {𝑧}) ⊆ 𝐼 → {𝑧} ⊆ 𝐼)
264	258	snss 4791	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33 ⊢ (𝑧 ∈ 𝐼 ↔ {𝑧} ⊆ 𝐼)
265	263, 264	sylibr 233	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32 ⊢ ((𝑥 ∪ {𝑧}) ⊆ 𝐼 → 𝑧 ∈ 𝐼)
266	265	adantl 480	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31 ⊢ ((𝑖 ≠ 𝑧 ∧ (𝑥 ∪ {𝑧}) ⊆ 𝐼) → 𝑧 ∈ 𝐼)
267	4	ad6antr 734	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 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 732	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 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 479	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 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 7090	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35 ⊢ ((𝑓:𝐼⟶(Base‘𝑅) ∧ 𝑧 ∈ 𝐼) → (𝑓‘𝑧) ∈ (Base‘𝑅))
271	270	ad4ant24 752	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 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 20202	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 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 1368	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 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 479	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32 ⊢ (((((((𝑅 ∈ Field ∧ 𝑀:(𝐼 × 𝐼)⟶(Base‘𝑅)) ∧ 𝐼 ∈ Fin) ∧ 𝑓:𝐼⟶(Base‘𝑅)) ∧ (𝑖 ∈ 𝐼 ∧ (𝑓‘𝑖) ≠ (0g‘𝑅))) ∧ 𝑧 ∈ 𝐼) ∧ 𝑘 ∈ 𝐼) → (((invr‘𝑅)‘(𝑓‘𝑖))(.r‘𝑅)(𝑓‘𝑧)) ∈ (Base‘𝑅))
275		fovcdm 7591	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34 ⊢ ((𝑀:(𝐼 × 𝐼)⟶(Base‘𝑅) ∧ 𝑧 ∈ 𝐼 ∧ 𝑘 ∈ 𝐼) → (𝑧𝑀𝑘) ∈ (Base‘𝑅))
276	275	3expa 1115	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33 ⊢ (((𝑀:(𝐼 × 𝐼)⟶(Base‘𝑅) ∧ 𝑧 ∈ 𝐼) ∧ 𝑘 ∈ 𝐼) → (𝑧𝑀𝑘) ∈ (Base‘𝑅))
277	248, 276	sylanl1 678	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 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 20202	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 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 1368	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 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 679	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 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 713	. . . . . . . . . . . . . . . . . . . . . . . . . . . . 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 6896	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33 ⊢ (𝑛 = 𝑧 → (𝑓‘𝑛) = (𝑓‘𝑧))
283		oveq1 7426	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33 ⊢ (𝑛 = 𝑧 → (𝑛𝑀𝑘) = (𝑧𝑀𝑘))
284	282, 283	oveq12d 7437	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32 ⊢ (𝑛 = 𝑧 → ((𝑓‘𝑛)(.r‘𝑅)(𝑛𝑀𝑘)) = ((𝑓‘𝑧)(.r‘𝑅)(𝑧𝑀𝑘)))
285	284	oveq2d 7435	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 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 724	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 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 759	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 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 20205	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 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 1369	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 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 2731	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 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 679	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 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 2787	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 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 717	. . . . . . . . . . . . . . . . . . . . . . . . . . . . 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 19925	. . . . . . . . . . . . . . . . . . . . . . . . . . . 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 7434	. . . . . . . . . . . . . . . . . . . . . . . . . . 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 20229	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32 ⊢ (𝑅 ∈ Ring → 𝑅 ∈ Abel)
297	4, 296	syl 17	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31 ⊢ (𝑅 ∈ Field → 𝑅 ∈ Abel)
298	297	ad6antr 734	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 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 734	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 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 3465	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 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 3971	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 36 ⊢ ((𝑥 ⊆ 𝐼 ∧ 𝑛 ∈ 𝑥) → 𝑛 ∈ 𝐼)
303	302, 254	sylan2 591	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 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 466	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 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 7124	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 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 650	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 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 7452	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 37 ⊢ (((invr‘𝑅)‘(𝑓‘𝑖))(.r‘𝑅)((𝑓‘𝑛)(.r‘𝑅)(𝑛𝑀𝑘))) ∈ V
308		eqid 2725	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 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 6699	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 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 6911	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35 ⊢ ((𝐼 ∈ Fin ∧ 𝑥 ⊆ 𝐼) → (0g‘𝑅) ∈ V)
312	310, 229, 311	fndmfifsupp 9403	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34 ⊢ ((𝐼 ∈ Fin ∧ 𝑥 ⊆ 𝐼) → (𝑛 ∈ 𝑥 ↦ (((invr‘𝑅)‘(𝑓‘𝑖))(.r‘𝑅)((𝑓‘𝑛)(.r‘𝑅)(𝑛𝑀𝑘)))) finSupp (0g‘𝑅))
313	312	adantll 712	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 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 756	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 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 19882	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 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 679	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 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 679	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 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 774	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33 ⊢ ((((𝑅 ∈ Field ∧ 𝑀:(𝐼 × 𝐼)⟶(Base‘𝑅)) ∧ 𝐼 ∈ Fin) ∧ 𝑓:𝐼⟶(Base‘𝑅)) → 𝑀:(𝐼 × 𝐼)⟶(Base‘𝑅))
319		simpl 481	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33 ⊢ ((𝑖 ∈ 𝐼 ∧ (𝑓‘𝑖) ≠ (0g‘𝑅)) → 𝑖 ∈ 𝐼)
320	318, 319	anim12i 611	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 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 479	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31 ⊢ ((((((𝑅 ∈ Field ∧ 𝑀:(𝐼 × 𝐼)⟶(Base‘𝑅)) ∧ 𝐼 ∈ Fin) ∧ 𝑓:𝐼⟶(Base‘𝑅)) ∧ (𝑖 ∈ 𝐼 ∧ (𝑓‘𝑖) ≠ (0g‘𝑅))) ∧ (𝑥 ∪ {𝑧}) ⊆ 𝐼) → (𝑀:(𝐼 × 𝐼)⟶(Base‘𝑅) ∧ 𝑖 ∈ 𝐼))
322		fovcdm 7591	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32 ⊢ ((𝑀:(𝐼 × 𝐼)⟶(Base‘𝑅) ∧ 𝑖 ∈ 𝐼 ∧ 𝑘 ∈ 𝐼) → (𝑖𝑀𝑘) ∈ (Base‘𝑅))
323	322	3expa 1115	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31 ⊢ (((𝑀:(𝐼 × 𝐼)⟶(Base‘𝑅) ∧ 𝑖 ∈ 𝐼) ∧ 𝑘 ∈ 𝐼) → (𝑖𝑀𝑘) ∈ (Base‘𝑅))
324	321, 323	sylan 578	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 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 19770	. . . . . . . . . . . . . . . . . . . . . . . . . . . . 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 718	. . . . . . . . . . . . . . . . . . . . . . . . . . . 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 713	. . . . . . . . . . . . . . . . . . . . . . . . . . 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 2765	. . . . . . . . . . . . . . . . . . . . . . . . . 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 4549	. . . . . . . . . . . . . . . . . . . . . . . . 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 1128	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ ((((((((𝑅 ∈ Field ∧ 𝑀:(𝐼 × 𝐼)⟶(Base‘𝑅)) ∧ 𝐼 ∈ Fin) ∧ 𝑓:𝐼⟶(Base‘𝑅)) ∧ (𝑖 ∈ 𝐼 ∧ (𝑓‘𝑖) ≠ (0g‘𝑅))) ∧ (𝑖 ≠ 𝑧 ∧ (𝑥 ∪ {𝑧}) ⊆ 𝐼)) ∧ ¬ 𝑧 ∈ 𝑥) ∧ 𝑗 ∈ 𝐼 ∧ 𝑘 ∈ 𝐼) → if(𝑗 = 𝑖, ((𝑅 Σg (𝑛 ∈ (𝑥 ∪ {𝑧}) ↦ (((invr‘𝑅)‘(𝑓‘𝑖))(.r‘𝑅)((𝑓‘𝑛)(.r‘𝑅)(𝑛𝑀𝑘)))))(+g‘𝑅)(𝑖𝑀𝑘)), if(𝑗 = 𝑧, (𝑧𝑀𝑘), (𝑗𝑀𝑘))) = if(𝑗 = 𝑖, (((𝑅 Σg (𝑛 ∈ 𝑥 ↦ (((invr‘𝑅)‘(𝑓‘𝑖))(.r‘𝑅)((𝑓‘𝑛)(.r‘𝑅)(𝑛𝑀𝑘)))))(+g‘𝑅)(𝑖𝑀𝑘))(+g‘𝑅)((((invr‘𝑅)‘(𝑓‘𝑖))(.r‘𝑅)(𝑓‘𝑧))(.r‘𝑅)(𝑧𝑀𝑘))), if(𝑗 = 𝑧, (𝑧𝑀𝑘), (𝑗𝑀𝑘))))
331	330	mpoeq3dva 7497	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ (((((((𝑅 ∈ Field ∧ 𝑀:(𝐼 × 𝐼)⟶(Base‘𝑅)) ∧ 𝐼 ∈ Fin) ∧ 𝑓:𝐼⟶(Base‘𝑅)) ∧ (𝑖 ∈ 𝐼 ∧ (𝑓‘𝑖) ≠ (0g‘𝑅))) ∧ (𝑖 ≠ 𝑧 ∧ (𝑥 ∪ {𝑧}) ⊆ 𝐼)) ∧ ¬ 𝑧 ∈ 𝑥) → (𝑗 ∈ 𝐼, 𝑘 ∈ 𝐼 ↦ if(𝑗 = 𝑖, ((𝑅 Σg (𝑛 ∈ (𝑥 ∪ {𝑧}) ↦ (((invr‘𝑅)‘(𝑓‘𝑖))(.r‘𝑅)((𝑓‘𝑛)(.r‘𝑅)(𝑛𝑀𝑘)))))(+g‘𝑅)(𝑖𝑀𝑘)), if(𝑗 = 𝑧, (𝑧𝑀𝑘), (𝑗𝑀𝑘)))) = (𝑗 ∈ 𝐼, 𝑘 ∈ 𝐼 ↦ if(𝑗 = 𝑖, (((𝑅 Σg (𝑛 ∈ 𝑥 ↦ (((invr‘𝑅)‘(𝑓‘𝑖))(.r‘𝑅)((𝑓‘𝑛)(.r‘𝑅)(𝑛𝑀𝑘)))))(+g‘𝑅)(𝑖𝑀𝑘))(+g‘𝑅)((((invr‘𝑅)‘(𝑓‘𝑖))(.r‘𝑅)(𝑓‘𝑧))(.r‘𝑅)(𝑧𝑀𝑘))), if(𝑗 = 𝑧, (𝑧𝑀𝑘), (𝑗𝑀𝑘)))))
332	331	fveq2d 6900	. . . . . . . . . . . . . . . . . . . . . 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 2725	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ (𝐼 maDet 𝑅) = (𝐼 maDet 𝑅)
334	1	simprbi 495	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ (𝑅 ∈ Field → 𝑅 ∈ CRing)
335	334	ad5antr 732	. . . . . . . . . . . . . . . . . . . . . . . 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 782	. . . . . . . . . . . . . . . . . . . . . . . 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 734	. . . . . . . . . . . . . . . . . . . . . . . . . 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 479	. . . . . . . . . . . . . . . . . . . . . . . . . . 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 578	. . . . . . . . . . . . . . . . . . . . . . . . . 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 18906	. . . . . . . . . . . . . . . . . . . . . . . . . 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 1368	. . . . . . . . . . . . . . . . . . . . . . . . 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 679	. . . . . . . . . . . . . . . . . . . . . . . 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 611	. . . . . . . . . . . . . . . . . . . . . . . . 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 578	. . . . . . . . . . . . . . . . . . . . . . . 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 784	. . . . . . . . . . . . . . . . . . . . . . . . 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 1160	. . . . . . . . . . . . . . . . . . . . . . . 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 591	. . . . . . . . . . . . . . . . . . . . . . . 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 775	. . . . . . . . . . . . . . . . . . . . . . . 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 727	. . . . . . . . . . . . . . . . . . . . . . . 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 769	. . . . . . . . . . . . . . . . . . . . . . . 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 22556	. . . . . . . . . . . . . . . . . . . . . . 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 479	. . . . . . . . . . . . . . . . . . . . . 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 2765	. . . . . . . . . . . . . . . . . . . . 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 4536	. . . . . . . . . . . . . . . . . . . . . . . . . 26 ⊢ (𝑗 = 𝑧 → if(𝑗 = 𝑧, (𝑧𝑀𝑘), (𝑗𝑀𝑘)) = (𝑧𝑀𝑘))
355		oveq1 7426	. . . . . . . . . . . . . . . . . . . . . . . . . 26 ⊢ (𝑗 = 𝑧 → (𝑗𝑀𝑘) = (𝑧𝑀𝑘))
356	354, 355	eqtr4d 2768	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ (𝑗 = 𝑧 → if(𝑗 = 𝑧, (𝑧𝑀𝑘), (𝑗𝑀𝑘)) = (𝑗𝑀𝑘))
357		iffalse 4539	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ (¬ 𝑗 = 𝑧 → if(𝑗 = 𝑧, (𝑧𝑀𝑘), (𝑗𝑀𝑘)) = (𝑗𝑀𝑘))
358	356, 357	pm2.61i 182	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ if(𝑗 = 𝑧, (𝑧𝑀𝑘), (𝑗𝑀𝑘)) = (𝑗𝑀𝑘)
359		ifeq2 4535	. . . . . . . . . . . . . . . . . . . . . . . 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 13	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ ((𝑗 ∈ 𝐼 ∧ 𝑘 ∈ 𝐼) → if(𝑗 = 𝑖, ((𝑅 Σg (𝑛 ∈ (𝑥 ∪ {𝑧}) ↦ (((invr‘𝑅)‘(𝑓‘𝑖))(.r‘𝑅)((𝑓‘𝑛)(.r‘𝑅)(𝑛𝑀𝑘)))))(+g‘𝑅)(𝑖𝑀𝑘)), if(𝑗 = 𝑧, (𝑧𝑀𝑘), (𝑗𝑀𝑘))) = if(𝑗 = 𝑖, ((𝑅 Σg (𝑛 ∈ (𝑥 ∪ {𝑧}) ↦ (((invr‘𝑅)‘(𝑓‘𝑖))(.r‘𝑅)((𝑓‘𝑛)(.r‘𝑅)(𝑛𝑀𝑘)))))(+g‘𝑅)(𝑖𝑀𝑘)), (𝑗𝑀𝑘)))
361	360	mpoeq3ia 7498	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (𝑗 ∈ 𝐼, 𝑘 ∈ 𝐼 ↦ if(𝑗 = 𝑖, ((𝑅 Σg (𝑛 ∈ (𝑥 ∪ {𝑧}) ↦ (((invr‘𝑅)‘(𝑓‘𝑖))(.r‘𝑅)((𝑓‘𝑛)(.r‘𝑅)(𝑛𝑀𝑘)))))(+g‘𝑅)(𝑖𝑀𝑘)), if(𝑗 = 𝑧, (𝑧𝑀𝑘), (𝑗𝑀𝑘)))) = (𝑗 ∈ 𝐼, 𝑘 ∈ 𝐼 ↦ if(𝑗 = 𝑖, ((𝑅 Σg (𝑛 ∈ (𝑥 ∪ {𝑧}) ↦ (((invr‘𝑅)‘(𝑓‘𝑖))(.r‘𝑅)((𝑓‘𝑛)(.r‘𝑅)(𝑛𝑀𝑘)))))(+g‘𝑅)(𝑖𝑀𝑘)), (𝑗𝑀𝑘)))
362	361	fveq2i 6899	. . . . . . . . . . . . . . . . . . . . 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 4535	. . . . . . . . . . . . . . . . . . . . . . . 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 13	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ ((𝑗 ∈ 𝐼 ∧ 𝑘 ∈ 𝐼) → if(𝑗 = 𝑖, ((𝑅 Σg (𝑛 ∈ 𝑥 ↦ (((invr‘𝑅)‘(𝑓‘𝑖))(.r‘𝑅)((𝑓‘𝑛)(.r‘𝑅)(𝑛𝑀𝑘)))))(+g‘𝑅)(𝑖𝑀𝑘)), if(𝑗 = 𝑧, (𝑧𝑀𝑘), (𝑗𝑀𝑘))) = if(𝑗 = 𝑖, ((𝑅 Σg (𝑛 ∈ 𝑥 ↦ (((invr‘𝑅)‘(𝑓‘𝑖))(.r‘𝑅)((𝑓‘𝑛)(.r‘𝑅)(𝑛𝑀𝑘)))))(+g‘𝑅)(𝑖𝑀𝑘)), (𝑗𝑀𝑘)))
365	364	mpoeq3ia 7498	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (𝑗 ∈ 𝐼, 𝑘 ∈ 𝐼 ↦ if(𝑗 = 𝑖, ((𝑅 Σg (𝑛 ∈ 𝑥 ↦ (((invr‘𝑅)‘(𝑓‘𝑖))(.r‘𝑅)((𝑓‘𝑛)(.r‘𝑅)(𝑛𝑀𝑘)))))(+g‘𝑅)(𝑖𝑀𝑘)), if(𝑗 = 𝑧, (𝑧𝑀𝑘), (𝑗𝑀𝑘)))) = (𝑗 ∈ 𝐼, 𝑘 ∈ 𝐼 ↦ if(𝑗 = 𝑖, ((𝑅 Σg (𝑛 ∈ 𝑥 ↦ (((invr‘𝑅)‘(𝑓‘𝑖))(.r‘𝑅)((𝑓‘𝑛)(.r‘𝑅)(𝑛𝑀𝑘)))))(+g‘𝑅)(𝑖𝑀𝑘)), (𝑗𝑀𝑘)))
366	365	fveq2i 6899	. . . . . . . . . . . . . . . . . . . . 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 2788	. . . . . . . . . . . . . . . . . . . 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 679	. . . . . . . . . . . . . . . . . . 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 2736	. . . . . . . . . . . . . . . . . 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 247	. . . . . . . . . . . . . . . . 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 59	. . . . . . . . . . . . . . . 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 412	. . . . . . . . . . . . . . 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 86	. . . . . . . . . . . . . 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 480	. . . . . . . . . . . . 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 9190	. . . . . . . . . . . 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 38	. . . . . . . . . . 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 703	. . . . . . . . . 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 713	. . . . . . . . 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 2725	. . . . . . . . . . . 12 ⊢ 𝐼 = 𝐼
380		fconstmpt 5740	. . . . . . . . . . . . . . . . 17 ⊢ (𝐼 × {(0g‘𝑅)}) = (𝑘 ∈ 𝐼 ↦ (0g‘𝑅))
381	380	eqeq2i 2738	. . . . . . . . . . . . . . . 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 7452	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑅 Σg (𝑛 ∈ 𝐼 ↦ ((𝑓‘𝑛)(.r‘𝑅)(𝑛𝑀𝑘)))) ∈ V
383	382	rgenw 3054	. . . . . . . . . . . . . . . . 17 ⊢ ∀𝑘 ∈ 𝐼 (𝑅 Σg (𝑛 ∈ 𝐼 ↦ ((𝑓‘𝑛)(.r‘𝑅)(𝑛𝑀𝑘)))) ∈ V
384		mpteqb 7023	. . . . . . . . . . . . . . . . 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 274	. . . . . . . . . . . . . . 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 732	. . . . . . . . . . . . . . . . . . . . . . 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 782	. . . . . . . . . . . . . . . . . . . . . . 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 2725	. . . . . . . . . . . . . . . . . . . . . . . . . . 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 6699	. . . . . . . . . . . . . . . . . . . . . . . . . 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 22	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ (𝐼 ∈ Fin → 𝐼 ∈ Fin)
393		fvexd 6911	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ (𝐼 ∈ Fin → (0g‘𝑅) ∈ V)
394	391, 392, 393	fndmfifsupp 9403	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ (𝐼 ∈ Fin → (𝑛 ∈ 𝐼 ↦ (((invr‘𝑅)‘(𝑓‘𝑖))(.r‘𝑅)((𝑓‘𝑛)(.r‘𝑅)(𝑛𝑀𝑘)))) finSupp (0g‘𝑅))
395	394	ad4antlr 731	. . . . . . . . . . . . . . . . . . . . . . 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 775	. . . . . . . . . . . . . . . . . . . . . . 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 578	. . . . . . . . . . . . . . . . . . . . . . 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 6896	. . . . . . . . . . . . . . . . . . . . . . . . . 26 ⊢ (𝑛 = 𝑖 → (𝑓‘𝑛) = (𝑓‘𝑖))
399		oveq1 7426	. . . . . . . . . . . . . . . . . . . . . . . . . 26 ⊢ (𝑛 = 𝑖 → (𝑛𝑀𝑘) = (𝑖𝑀𝑘))
400	398, 399	oveq12d 7437	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ (𝑛 = 𝑖 → ((𝑓‘𝑛)(.r‘𝑅)(𝑛𝑀𝑘)) = ((𝑓‘𝑖)(.r‘𝑅)(𝑖𝑀𝑘)))
401	400	oveq2d 7435	. . . . . . . . . . . . . . . . . . . . . . . 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 765	. . . . . . . . . . . . . . . . . . . . . . . . . . 27 ⊢ (((𝑅 ∈ Field ∧ 𝑀:(𝐼 × 𝐼)⟶(Base‘𝑅)) ∧ 𝐼 ∈ Fin) → 𝑅 ∈ Field)
403	2, 237	anim12i 611	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31 ⊢ ((𝑅 ∈ Field ∧ (𝑓:𝐼⟶(Base‘𝑅) ∧ 𝑖 ∈ 𝐼)) → (𝑅 ∈ DivRing ∧ (𝑓‘𝑖) ∈ (Base‘𝑅)))
404	403	anassrs 466	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30 ⊢ (((𝑅 ∈ Field ∧ 𝑓:𝐼⟶(Base‘𝑅)) ∧ 𝑖 ∈ 𝐼) → (𝑅 ∈ DivRing ∧ (𝑓‘𝑖) ∈ (Base‘𝑅)))
405		eqid 2725	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32 ⊢ (1r‘𝑅) = (1r‘𝑅)
406	10, 100, 47, 405, 240	drnginvrl 20661	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31 ⊢ ((𝑅 ∈ DivRing ∧ (𝑓‘𝑖) ∈ (Base‘𝑅) ∧ (𝑓‘𝑖) ≠ (0g‘𝑅)) → (((invr‘𝑅)‘(𝑓‘𝑖))(.r‘𝑅)(𝑓‘𝑖)) = (1r‘𝑅))
407	406	3expa 1115	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 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 578	. . . . . . . . . . . . . . . . . . . . . . . . . . . . 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 465	. . . . . . . . . . . . . . . . . . . . . . . . . . . 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 7434	. . . . . . . . . . . . . . . . . . . . . . . . . . 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 678	. . . . . . . . . . . . . . . . . . . . . . . . . 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 479	. . . . . . . . . . . . . . . . . . . . . . . . 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 732	. . . . . . . . . . . . . . . . . . . . . . . . . 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 479	. . . . . . . . . . . . . . . . . . . . . . . . . 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 746	. . . . . . . . . . . . . . . . . . . . . . . . . . 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 479	. . . . . . . . . . . . . . . . . . . . . . . . . 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 20205	. . . . . . . . . . . . . . . . . . . . . . . . . 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 1369	. . . . . . . . . . . . . . . . . . . . . . . . 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 479	. . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 ⊢ ((𝑅 ∈ Field ∧ 𝑀:(𝐼 × 𝐼)⟶(Base‘𝑅)) → 𝑅 ∈ Ring)
420	419	3ad2ant1 1130	. . . . . . . . . . . . . . . . . . . . . . . . . . . 28 ⊢ (((𝑅 ∈ Field ∧ 𝑀:(𝐼 × 𝐼)⟶(Base‘𝑅)) ∧ 𝑖 ∈ 𝐼 ∧ 𝑘 ∈ 𝐼) → 𝑅 ∈ Ring)
421	322	3adant1l 1173	. . . . . . . . . . . . . . . . . . . . . . . . . . . 28 ⊢ (((𝑅 ∈ Field ∧ 𝑀:(𝐼 × 𝐼)⟶(Base‘𝑅)) ∧ 𝑖 ∈ 𝐼 ∧ 𝑘 ∈ 𝐼) → (𝑖𝑀𝑘) ∈ (Base‘𝑅))
422	10, 47, 405	ringlidm 20217	. . . . . . . . . . . . . . . . . . . . . . . . . . . 28 ⊢ ((𝑅 ∈ Ring ∧ (𝑖𝑀𝑘) ∈ (Base‘𝑅)) → ((1r‘𝑅)(.r‘𝑅)(𝑖𝑀𝑘)) = (𝑖𝑀𝑘))
423	420, 421, 422	syl2anc 582	. . . . . . . . . . . . . . . . . . . . . . . . . . 27 ⊢ (((𝑅 ∈ Field ∧ 𝑀:(𝐼 × 𝐼)⟶(Base‘𝑅)) ∧ 𝑖 ∈ 𝐼 ∧ 𝑘 ∈ 𝐼) → ((1r‘𝑅)(.r‘𝑅)(𝑖𝑀𝑘)) = (𝑖𝑀𝑘))
424	423	ad5ant145 1366	. . . . . . . . . . . . . . . . . . . . . . . . . 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 719	. . . . . . . . . . . . . . . . . . . . . . . . 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 2773	. . . . . . . . . . . . . . . . . . . . . . . 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 2787	. . . . . . . . . . . . . . . . . . . . . . 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 19928	. . . . . . . . . . . . . . . . . . . . . 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 7452	. . . . . . . . . . . . . . . . . . . . . . . . . . 27 ⊢ ((𝑓‘𝑛)(.r‘𝑅)(𝑛𝑀𝑘)) ∈ V
430		eqid 2725	. . . . . . . . . . . . . . . . . . . . . . . . . . 27 ⊢ (𝑛 ∈ 𝐼 ↦ ((𝑓‘𝑛)(.r‘𝑅)(𝑛𝑀𝑘))) = (𝑛 ∈ 𝐼 ↦ ((𝑓‘𝑛)(.r‘𝑅)(𝑛𝑀𝑘)))
431	429, 430	fnmpti 6699	. . . . . . . . . . . . . . . . . . . . . . . . . 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 9403	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ (𝐼 ∈ Fin → (𝑛 ∈ 𝐼 ↦ ((𝑓‘𝑛)(.r‘𝑅)(𝑛𝑀𝑘))) finSupp (0g‘𝑅))
434	433	ad4antlr 731	. . . . . . . . . . . . . . . . . . . . . . 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 20265	. . . . . . . . . . . . . . . . . . . . . 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 2767	. . . . . . . . . . . . . . . . . . . . 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 479	. . . . . . . . . . . . . . . . . . . 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 7427	. . . . . . . . . . . . . . . . . . . . 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 480	. . . . . . . . . . . . . . . . . . . 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 730	. . . . . . . . . . . . . . . . . . . . . 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 20242	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ ((𝑅 ∈ Ring ∧ ((invr‘𝑅)‘(𝑓‘𝑖)) ∈ (Base‘𝑅)) → (((invr‘𝑅)‘(𝑓‘𝑖))(.r‘𝑅)(0g‘𝑅)) = (0g‘𝑅))
442	440, 245, 441	syl2anc 582	. . . . . . . . . . . . . . . . . . . . 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 724	. . . . . . . . . . . . . . . . . . . 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 2769	. . . . . . . . . . . . . . . . . . 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 4549	. . . . . . . . . . . . . . . . . 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 411	. . . . . . . . . . . . . . . . 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 3156	. . . . . . . . . . . . . . . 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 405	. . . . . . . . . . . . . . 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 592	. . . . . . . . . . . . . 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 518	. . . . . . . . . . . . 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 3139	. . . . . . . . . . . 12 ⊢ ((((((𝑅 ∈ Field ∧ 𝑀:(𝐼 × 𝐼)⟶(Base‘𝑅)) ∧ 𝐼 ∈ Fin) ∧ 𝑓:𝐼⟶(Base‘𝑅)) ∧ (𝑖 ∈ 𝐼 ∧ (𝑓‘𝑖) ≠ (0g‘𝑅))) ∧ (𝑘 ∈ 𝐼 ↦ (𝑅 Σg (𝑛 ∈ 𝐼 ↦ ((𝑓‘𝑛)(.r‘𝑅)(𝑛𝑀𝑘))))) = (𝐼 × {(0g‘𝑅)})) → ∀𝑗 ∈ 𝐼 (𝐼 = 𝐼 ∧ ∀𝑘 ∈ 𝐼 if(𝑗 = 𝑖, ((𝑅 Σg (𝑛 ∈ (𝐼 ∖ {𝑖}) ↦ (((invr‘𝑅)‘(𝑓‘𝑖))(.r‘𝑅)((𝑓‘𝑛)(.r‘𝑅)(𝑛𝑀𝑘)))))(+g‘𝑅)(𝑖𝑀𝑘)), (𝑗𝑀𝑘)) = if(𝑗 = 𝑖, (0g‘𝑅), (𝑗𝑀𝑘))))
452		mpoeq123 7492	. . . . . . . . . . . 12 ⊢ ((𝐼 = 𝐼 ∧ ∀𝑗 ∈ 𝐼 (𝐼 = 𝐼 ∧ ∀𝑘 ∈ 𝐼 if(𝑗 = 𝑖, ((𝑅 Σg (𝑛 ∈ (𝐼 ∖ {𝑖}) ↦ (((invr‘𝑅)‘(𝑓‘𝑖))(.r‘𝑅)((𝑓‘𝑛)(.r‘𝑅)(𝑛𝑀𝑘)))))(+g‘𝑅)(𝑖𝑀𝑘)), (𝑗𝑀𝑘)) = if(𝑗 = 𝑖, (0g‘𝑅), (𝑗𝑀𝑘)))) → (𝑗 ∈ 𝐼, 𝑘 ∈ 𝐼 ↦ if(𝑗 = 𝑖, ((𝑅 Σg (𝑛 ∈ (𝐼 ∖ {𝑖}) ↦ (((invr‘𝑅)‘(𝑓‘𝑖))(.r‘𝑅)((𝑓‘𝑛)(.r‘𝑅)(𝑛𝑀𝑘)))))(+g‘𝑅)(𝑖𝑀𝑘)), (𝑗𝑀𝑘))) = (𝑗 ∈ 𝐼, 𝑘 ∈ 𝐼 ↦ if(𝑗 = 𝑖, (0g‘𝑅), (𝑗𝑀𝑘))))
453	379, 451, 452	sylancr 585	. . . . . . . . . . 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 650	. . . . . . . . . 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 6900	. . . . . . . . 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 728	. . . . . . . . . . 11 ⊢ ((((𝑅 ∈ Field ∧ 𝑀:(𝐼 × 𝐼)⟶(Base‘𝑅)) ∧ 𝐼 ∈ Fin) ∧ (𝑖 ∈ 𝐼 ∧ (𝑓‘𝑖) ≠ (0g‘𝑅))) → 𝑅 ∈ CRing)
457		simplr 767	. . . . . . . . . . 11 ⊢ ((((𝑅 ∈ Field ∧ 𝑀:(𝐼 × 𝐼)⟶(Base‘𝑅)) ∧ 𝐼 ∈ Fin) ∧ (𝑖 ∈ 𝐼 ∧ (𝑓‘𝑖) ≠ (0g‘𝑅))) → 𝐼 ∈ Fin)
458		simpllr 774	. . . . . . . . . . . 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 1160	. . . . . . . . . . 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 769	. . . . . . . . . . 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 22551	. . . . . . . . . 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 750	. . . . . . . . 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 2769	. . . . . . . 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 3145	. . . . . . 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 452	. . . . . 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 603	. . . . 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 591	. . . 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 3144	. . 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 591	. 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 239	1 ⊢ (((𝑅 ∈ Field ∧ 𝑀:(𝐼 × 𝐼)⟶(Base‘𝑅)) ∧ 𝐼 ∈ (Fin ∖ {∅})) → (¬ curry 𝑀 LIndF (𝑅 freeLMod 𝐼) → ((𝐼 maDet 𝑅)‘𝑀) = (0g‘𝑅)))

Syntax hints:  ¬ wn 3   → wi 4   ↔ wb 205   ∧ wa 394   ∧ w3a 1084   = wceq 1533   ∈ wcel 2098   ≠ wne 2929  ∀wral 3050  ∃wrex 3059  Vcvv 3461   ∖ cdif 3941   ∪ cun 3942   ⊆ wss 3944  ∅c0 4322  ifcif 4530  {csn 4630   class class class wbr 5149   ↦ cmpt 5232   × cxp 5676   Fn wfn 6544  ⟶wf 6545  ‘cfv 6549  (class class class)co 7419   ∈ cmpo 7421   ∘_f cof 7683  curry ccur 8271   ↑_m cmap 8845  Fincfn 8964   finSupp cfsupp 9387  Basecbs 17183  +_gcplusg 17236  ._rcmulr 17237  Scalarcsca 17239   ·_𝑠 cvsca 17240  0_gc0g 17424   Σ_g cgsu 17425  Grpcgrp 18898  CMndccmn 19747  Abelcabl 19748  1_rcur 20133  Ringcrg 20185  CRingccrg 20186  inv_rcinvr 20338  DivRingcdr 20636  Fieldcfield 20637  LModclmod 20755   freeLMod cfrlm 21697   LIndF clindf 21755   maDet cmdat 22530

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1905  ax-6 1963  ax-7 2003  ax-8 2100  ax-9 2108  ax-10 2129  ax-11 2146  ax-12 2166  ax-ext 2696  ax-rep 5286  ax-sep 5300  ax-nul 5307  ax-pow 5365  ax-pr 5429  ax-un 7741  ax-cnex 11196  ax-resscn 11197  ax-1cn 11198  ax-icn 11199  ax-addcl 11200  ax-addrcl 11201  ax-mulcl 11202  ax-mulrcl 11203  ax-mulcom 11204  ax-addass 11205  ax-mulass 11206  ax-distr 11207  ax-i2m1 11208  ax-1ne0 11209  ax-1rid 11210  ax-rnegex 11211  ax-rrecex 11212  ax-cnre 11213  ax-pre-lttri 11214  ax-pre-lttrn 11215  ax-pre-ltadd 11216  ax-pre-mulgt0 11217  ax-addf 11219  ax-mulf 11220

This theorem depends on definitions:  df-bi 206  df-an 395  df-or 846  df-3or 1085  df-3an 1086  df-xor 1505  df-tru 1536  df-fal 1546  df-ex 1774  df-nf 1778  df-sb 2060  df-mo 2528  df-eu 2557  df-clab 2703  df-cleq 2717  df-clel 2802  df-nfc 2877  df-ne 2930  df-nel 3036  df-ral 3051  df-rex 3060  df-rmo 3363  df-reu 3364  df-rab 3419  df-v 3463  df-sbc 3774  df-csb 3890  df-dif 3947  df-un 3949  df-in 3951  df-ss 3961  df-pss 3964  df-nul 4323  df-if 4531  df-pw 4606  df-sn 4631  df-pr 4633  df-tp 4635  df-op 4637  df-ot 4639  df-uni 4910  df-int 4951  df-iun 4999  df-iin 5000  df-br 5150  df-opab 5212  df-mpt 5233  df-tr 5267  df-id 5576  df-eprel 5582  df-po 5590  df-so 5591  df-fr 5633  df-se 5634  df-we 5635  df-xp 5684  df-rel 5685  df-cnv 5686  df-co 5687  df-dm 5688  df-rn 5689  df-res 5690  df-ima 5691  df-pred 6307  df-ord 6374  df-on 6375  df-lim 6376  df-suc 6377  df-iota 6501  df-fun 6551  df-fn 6552  df-f 6553  df-f1 6554  df-fo 6555  df-f1o 6556  df-fv 6557  df-isom 6558  df-riota 7375  df-ov 7422  df-oprab 7423  df-mpo 7424  df-of 7685  df-om 7872  df-1st 7994  df-2nd 7995  df-supp 8166  df-tpos 8232  df-cur 8273  df-frecs 8287  df-wrecs 8318  df-recs 8392  df-rdg 8431  df-1o 8487  df-2o 8488  df-er 8725  df-map 8847  df-pm 8848  df-ixp 8917  df-en 8965  df-dom 8966  df-sdom 8967  df-fin 8968  df-fsupp 9388  df-sup 9467  df-oi 9535  df-card 9964  df-pnf 11282  df-mnf 11283  df-xr 11284  df-ltxr 11285  df-le 11286  df-sub 11478  df-neg 11479  df-div 11904  df-nn 12246  df-2 12308  df-3 12309  df-4 12310  df-5 12311  df-6 12312  df-7 12313  df-8 12314  df-9 12315  df-n0 12506  df-xnn0 12578  df-z 12592  df-dec 12711  df-uz 12856  df-rp 13010  df-fz 13520  df-fzo 13663  df-seq 14003  df-exp 14063  df-hash 14326  df-word 14501  df-lsw 14549  df-concat 14557  df-s1 14582  df-substr 14627  df-pfx 14657  df-splice 14736  df-reverse 14745  df-s2 14835  df-struct 17119  df-sets 17136  df-slot 17154  df-ndx 17166  df-base 17184  df-ress 17213  df-plusg 17249  df-mulr 17250  df-starv 17251  df-sca 17252  df-vsca 17253  df-ip 17254  df-tset 17255  df-ple 17256  df-ds 17258  df-unif 17259  df-hom 17260  df-cco 17261  df-0g 17426  df-gsum 17427  df-prds 17432  df-pws 17434  df-mre 17569  df-mrc 17570  df-acs 17572  df-mgm 18603  df-sgrp 18682  df-mnd 18698  df-mhm 18743  df-submnd 18744  df-efmnd 18829  df-grp 18901  df-minusg 18902  df-sbg 18903  df-mulg 19032  df-subg 19086  df-ghm 19176  df-gim 19222  df-cntz 19280  df-oppg 19309  df-symg 19334  df-pmtr 19409  df-psgn 19458  df-evpm 19459  df-cmn 19749  df-abl 19750  df-mgp 20087  df-rng 20105  df-ur 20134  df-ring 20187  df-cring 20188  df-oppr 20285  df-dvdsr 20308  df-unit 20309  df-invr 20339  df-dvr 20352  df-rhm 20423  df-nzr 20464  df-subrng 20495  df-subrg 20520  df-drng 20638  df-field 20639  df-lmod 20757  df-lss 20828  df-lsp 20868  df-lmhm 20919  df-lbs 20972  df-sra 21070  df-rgmod 21071  df-cnfld 21297  df-zring 21390  df-zrh 21446  df-dsmm 21683  df-frlm 21698  df-uvc 21734  df-lindf 21757  df-mat 22352  df-mdet 22531

This theorem is referenced by:  matunitlindf  37222