Theorem matunitlindflem2 37625

Description: One direction of matunitlindf 37626. (Contributed by Brendan Leahy, 2-Jun-2021.)

Assertion

Ref	Expression
matunitlindflem2	⊢ ((((𝑅 ∈ Field ∧ 𝑀 ∈ (Base‘(𝐼 Mat 𝑅))) ∧ 𝐼 ≠ ∅) ∧ curry 𝑀 LIndF (𝑅 freeLMod 𝐼)) → ((𝐼 maDet 𝑅)‘𝑀) ∈ (Unit‘𝑅))

Proof of Theorem matunitlindflem2

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

Step	Hyp	Ref	Expression
1		eqid 2736	. . . . . . 7 ⊢ (𝐼 Mat 𝑅) = (𝐼 Mat 𝑅)
2		eqid 2736	. . . . . . 7 ⊢ (Base‘(𝐼 Mat 𝑅)) = (Base‘(𝐼 Mat 𝑅))
3	1, 2	matrcl 22417	. . . . . 6 ⊢ (𝑀 ∈ (Base‘(𝐼 Mat 𝑅)) → (𝐼 ∈ Fin ∧ 𝑅 ∈ V))
4	3	simpld 494	. . . . 5 ⊢ (𝑀 ∈ (Base‘(𝐼 Mat 𝑅)) → 𝐼 ∈ Fin)
5	4	ad3antlr 731	. . . 4 ⊢ ((((𝑅 ∈ Field ∧ 𝑀 ∈ (Base‘(𝐼 Mat 𝑅))) ∧ 𝐼 ≠ ∅) ∧ curry 𝑀 LIndF (𝑅 freeLMod 𝐼)) → 𝐼 ∈ Fin)
6		isfld 20741	. . . . . . 7 ⊢ (𝑅 ∈ Field ↔ (𝑅 ∈ DivRing ∧ 𝑅 ∈ CRing))
7	6	simplbi 497	. . . . . 6 ⊢ (𝑅 ∈ Field → 𝑅 ∈ DivRing)
8	7	anim1i 615	. . . . 5 ⊢ ((𝑅 ∈ Field ∧ 𝑀 ∈ (Base‘(𝐼 Mat 𝑅))) → (𝑅 ∈ DivRing ∧ 𝑀 ∈ (Base‘(𝐼 Mat 𝑅))))
9	4	ad2antrl 728	. . . . . . . . . . . 12 ⊢ ((𝑅 ∈ DivRing ∧ (𝑀 ∈ (Base‘(𝐼 Mat 𝑅)) ∧ 𝐼 ≠ ∅)) → 𝐼 ∈ Fin)
10		simpr 484	. . . . . . . . . . . . . . 15 ⊢ ((𝑅 ∈ DivRing ∧ 𝑀 ∈ (Base‘(𝐼 Mat 𝑅))) → 𝑀 ∈ (Base‘(𝐼 Mat 𝑅)))
11		xpfi 9359	. . . . . . . . . . . . . . . . . . . . 21 ⊢ ((𝐼 ∈ Fin ∧ 𝐼 ∈ Fin) → (𝐼 × 𝐼) ∈ Fin)
12	11	anidms 566	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝐼 ∈ Fin → (𝐼 × 𝐼) ∈ Fin)
13		eqid 2736	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (𝑅 freeLMod (𝐼 × 𝐼)) = (𝑅 freeLMod (𝐼 × 𝐼))
14		eqid 2736	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (Base‘𝑅) = (Base‘𝑅)
15	13, 14	frlmfibas 21783	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((𝑅 ∈ DivRing ∧ (𝐼 × 𝐼) ∈ Fin) → ((Base‘𝑅) ↑m (𝐼 × 𝐼)) = (Base‘(𝑅 freeLMod (𝐼 × 𝐼))))
16	12, 15	sylan2 593	. . . . . . . . . . . . . . . . . . 19 ⊢ ((𝑅 ∈ DivRing ∧ 𝐼 ∈ Fin) → ((Base‘𝑅) ↑m (𝐼 × 𝐼)) = (Base‘(𝑅 freeLMod (𝐼 × 𝐼))))
17	1, 13	matbas 22418	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((𝐼 ∈ Fin ∧ 𝑅 ∈ DivRing) → (Base‘(𝑅 freeLMod (𝐼 × 𝐼))) = (Base‘(𝐼 Mat 𝑅)))
18	17	ancoms 458	. . . . . . . . . . . . . . . . . . 19 ⊢ ((𝑅 ∈ DivRing ∧ 𝐼 ∈ Fin) → (Base‘(𝑅 freeLMod (𝐼 × 𝐼))) = (Base‘(𝐼 Mat 𝑅)))
19	16, 18	eqtrd 2776	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝑅 ∈ DivRing ∧ 𝐼 ∈ Fin) → ((Base‘𝑅) ↑m (𝐼 × 𝐼)) = (Base‘(𝐼 Mat 𝑅)))
20	19	eleq2d 2826	. . . . . . . . . . . . . . . . 17 ⊢ ((𝑅 ∈ DivRing ∧ 𝐼 ∈ Fin) → (𝑀 ∈ ((Base‘𝑅) ↑m (𝐼 × 𝐼)) ↔ 𝑀 ∈ (Base‘(𝐼 Mat 𝑅))))
21	4, 20	sylan2 593	. . . . . . . . . . . . . . . 16 ⊢ ((𝑅 ∈ DivRing ∧ 𝑀 ∈ (Base‘(𝐼 Mat 𝑅))) → (𝑀 ∈ ((Base‘𝑅) ↑m (𝐼 × 𝐼)) ↔ 𝑀 ∈ (Base‘(𝐼 Mat 𝑅))))
22		fvex 6918	. . . . . . . . . . . . . . . . . 18 ⊢ (Base‘𝑅) ∈ V
23	4, 4, 11	syl2anc 584	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑀 ∈ (Base‘(𝐼 Mat 𝑅)) → (𝐼 × 𝐼) ∈ Fin)
24		elmapg 8880	. . . . . . . . . . . . . . . . . 18 ⊢ (((Base‘𝑅) ∈ V ∧ (𝐼 × 𝐼) ∈ Fin) → (𝑀 ∈ ((Base‘𝑅) ↑m (𝐼 × 𝐼)) ↔ 𝑀:(𝐼 × 𝐼)⟶(Base‘𝑅)))
25	22, 23, 24	sylancr 587	. . . . . . . . . . . . . . . . 17 ⊢ (𝑀 ∈ (Base‘(𝐼 Mat 𝑅)) → (𝑀 ∈ ((Base‘𝑅) ↑m (𝐼 × 𝐼)) ↔ 𝑀:(𝐼 × 𝐼)⟶(Base‘𝑅)))
26	25	adantl 481	. . . . . . . . . . . . . . . 16 ⊢ ((𝑅 ∈ DivRing ∧ 𝑀 ∈ (Base‘(𝐼 Mat 𝑅))) → (𝑀 ∈ ((Base‘𝑅) ↑m (𝐼 × 𝐼)) ↔ 𝑀:(𝐼 × 𝐼)⟶(Base‘𝑅)))
27	21, 26	bitr3d 281	. . . . . . . . . . . . . . 15 ⊢ ((𝑅 ∈ DivRing ∧ 𝑀 ∈ (Base‘(𝐼 Mat 𝑅))) → (𝑀 ∈ (Base‘(𝐼 Mat 𝑅)) ↔ 𝑀:(𝐼 × 𝐼)⟶(Base‘𝑅)))
28	10, 27	mpbid 232	. . . . . . . . . . . . . 14 ⊢ ((𝑅 ∈ DivRing ∧ 𝑀 ∈ (Base‘(𝐼 Mat 𝑅))) → 𝑀:(𝐼 × 𝐼)⟶(Base‘𝑅))
29	28	adantrr 717	. . . . . . . . . . . . 13 ⊢ ((𝑅 ∈ DivRing ∧ (𝑀 ∈ (Base‘(𝐼 Mat 𝑅)) ∧ 𝐼 ≠ ∅)) → 𝑀:(𝐼 × 𝐼)⟶(Base‘𝑅))
30		eldifsn 4785	. . . . . . . . . . . . . . . 16 ⊢ (𝐼 ∈ (Fin ∖ {∅}) ↔ (𝐼 ∈ Fin ∧ 𝐼 ≠ ∅))
31	30	biimpri 228	. . . . . . . . . . . . . . 15 ⊢ ((𝐼 ∈ Fin ∧ 𝐼 ≠ ∅) → 𝐼 ∈ (Fin ∖ {∅}))
32	4, 31	sylan 580	. . . . . . . . . . . . . 14 ⊢ ((𝑀 ∈ (Base‘(𝐼 Mat 𝑅)) ∧ 𝐼 ≠ ∅) → 𝐼 ∈ (Fin ∖ {∅}))
33	32	adantl 481	. . . . . . . . . . . . 13 ⊢ ((𝑅 ∈ DivRing ∧ (𝑀 ∈ (Base‘(𝐼 Mat 𝑅)) ∧ 𝐼 ≠ ∅)) → 𝐼 ∈ (Fin ∖ {∅}))
34		curf 37606	. . . . . . . . . . . . . 14 ⊢ ((𝑀:(𝐼 × 𝐼)⟶(Base‘𝑅) ∧ 𝐼 ∈ (Fin ∖ {∅}) ∧ (Base‘𝑅) ∈ V) → curry 𝑀:𝐼⟶((Base‘𝑅) ↑m 𝐼))
35	22, 34	mp3an3 1451	. . . . . . . . . . . . 13 ⊢ ((𝑀:(𝐼 × 𝐼)⟶(Base‘𝑅) ∧ 𝐼 ∈ (Fin ∖ {∅})) → curry 𝑀:𝐼⟶((Base‘𝑅) ↑m 𝐼))
36	29, 33, 35	syl2anc 584	. . . . . . . . . . . 12 ⊢ ((𝑅 ∈ DivRing ∧ (𝑀 ∈ (Base‘(𝐼 Mat 𝑅)) ∧ 𝐼 ≠ ∅)) → curry 𝑀:𝐼⟶((Base‘𝑅) ↑m 𝐼))
37	9, 36	jca 511	. . . . . . . . . . 11 ⊢ ((𝑅 ∈ DivRing ∧ (𝑀 ∈ (Base‘(𝐼 Mat 𝑅)) ∧ 𝐼 ≠ ∅)) → (𝐼 ∈ Fin ∧ curry 𝑀:𝐼⟶((Base‘𝑅) ↑m 𝐼)))
38	37	ex 412	. . . . . . . . . 10 ⊢ (𝑅 ∈ DivRing → ((𝑀 ∈ (Base‘(𝐼 Mat 𝑅)) ∧ 𝐼 ≠ ∅) → (𝐼 ∈ Fin ∧ curry 𝑀:𝐼⟶((Base‘𝑅) ↑m 𝐼))))
39	38	imdistani 568	. . . . . . . . 9 ⊢ ((𝑅 ∈ DivRing ∧ (𝑀 ∈ (Base‘(𝐼 Mat 𝑅)) ∧ 𝐼 ≠ ∅)) → (𝑅 ∈ DivRing ∧ (𝐼 ∈ Fin ∧ curry 𝑀:𝐼⟶((Base‘𝑅) ↑m 𝐼))))
40	39	anassrs 467	. . . . . . . 8 ⊢ (((𝑅 ∈ DivRing ∧ 𝑀 ∈ (Base‘(𝐼 Mat 𝑅))) ∧ 𝐼 ≠ ∅) → (𝑅 ∈ DivRing ∧ (𝐼 ∈ Fin ∧ curry 𝑀:𝐼⟶((Base‘𝑅) ↑m 𝐼))))
41		anass 468	. . . . . . . 8 ⊢ (((𝑅 ∈ DivRing ∧ 𝐼 ∈ Fin) ∧ curry 𝑀:𝐼⟶((Base‘𝑅) ↑m 𝐼)) ↔ (𝑅 ∈ DivRing ∧ (𝐼 ∈ Fin ∧ curry 𝑀:𝐼⟶((Base‘𝑅) ↑m 𝐼))))
42	40, 41	sylibr 234	. . . . . . 7 ⊢ (((𝑅 ∈ DivRing ∧ 𝑀 ∈ (Base‘(𝐼 Mat 𝑅))) ∧ 𝐼 ≠ ∅) → ((𝑅 ∈ DivRing ∧ 𝐼 ∈ Fin) ∧ curry 𝑀:𝐼⟶((Base‘𝑅) ↑m 𝐼)))
43		drngring 20737	. . . . . . . . . . . . 13 ⊢ (𝑅 ∈ DivRing → 𝑅 ∈ Ring)
44		eqid 2736	. . . . . . . . . . . . . 14 ⊢ (𝑅 unitVec 𝐼) = (𝑅 unitVec 𝐼)
45		eqid 2736	. . . . . . . . . . . . . 14 ⊢ (𝑅 freeLMod 𝐼) = (𝑅 freeLMod 𝐼)
46		eqid 2736	. . . . . . . . . . . . . 14 ⊢ (Base‘(𝑅 freeLMod 𝐼)) = (Base‘(𝑅 freeLMod 𝐼))
47	44, 45, 46	uvcff 21812	. . . . . . . . . . . . 13 ⊢ ((𝑅 ∈ Ring ∧ 𝐼 ∈ Fin) → (𝑅 unitVec 𝐼):𝐼⟶(Base‘(𝑅 freeLMod 𝐼)))
48	43, 47	sylan 580	. . . . . . . . . . . 12 ⊢ ((𝑅 ∈ DivRing ∧ 𝐼 ∈ Fin) → (𝑅 unitVec 𝐼):𝐼⟶(Base‘(𝑅 freeLMod 𝐼)))
49	48	ffvelcdmda 7103	. . . . . . . . . . 11 ⊢ (((𝑅 ∈ DivRing ∧ 𝐼 ∈ Fin) ∧ 𝑖 ∈ 𝐼) → ((𝑅 unitVec 𝐼)‘𝑖) ∈ (Base‘(𝑅 freeLMod 𝐼)))
50	49	ad4ant14 752	. . . . . . . . . 10 ⊢ (((((𝑅 ∈ DivRing ∧ 𝐼 ∈ Fin) ∧ curry 𝑀:𝐼⟶((Base‘𝑅) ↑m 𝐼)) ∧ curry 𝑀 LIndF (𝑅 freeLMod 𝐼)) ∧ 𝑖 ∈ 𝐼) → ((𝑅 unitVec 𝐼)‘𝑖) ∈ (Base‘(𝑅 freeLMod 𝐼)))
51		ffn 6735	. . . . . . . . . . . . . . . 16 ⊢ (curry 𝑀:𝐼⟶((Base‘𝑅) ↑m 𝐼) → curry 𝑀 Fn 𝐼)
52		fnima 6697	. . . . . . . . . . . . . . . 16 ⊢ (curry 𝑀 Fn 𝐼 → (curry 𝑀 “ 𝐼) = ran curry 𝑀)
53	51, 52	syl 17	. . . . . . . . . . . . . . 15 ⊢ (curry 𝑀:𝐼⟶((Base‘𝑅) ↑m 𝐼) → (curry 𝑀 “ 𝐼) = ran curry 𝑀)
54	53	adantl 481	. . . . . . . . . . . . . 14 ⊢ (((𝑅 ∈ DivRing ∧ 𝐼 ∈ Fin) ∧ curry 𝑀:𝐼⟶((Base‘𝑅) ↑m 𝐼)) → (curry 𝑀 “ 𝐼) = ran curry 𝑀)
55	54	fveq2d 6909	. . . . . . . . . . . . 13 ⊢ (((𝑅 ∈ DivRing ∧ 𝐼 ∈ Fin) ∧ curry 𝑀:𝐼⟶((Base‘𝑅) ↑m 𝐼)) → ((LSpan‘(𝑅 freeLMod 𝐼))‘(curry 𝑀 “ 𝐼)) = ((LSpan‘(𝑅 freeLMod 𝐼))‘ran curry 𝑀))
56	55	adantr 480	. . . . . . . . . . . 12 ⊢ ((((𝑅 ∈ DivRing ∧ 𝐼 ∈ Fin) ∧ curry 𝑀:𝐼⟶((Base‘𝑅) ↑m 𝐼)) ∧ curry 𝑀 LIndF (𝑅 freeLMod 𝐼)) → ((LSpan‘(𝑅 freeLMod 𝐼))‘(curry 𝑀 “ 𝐼)) = ((LSpan‘(𝑅 freeLMod 𝐼))‘ran curry 𝑀))
57		simplll 774	. . . . . . . . . . . . . 14 ⊢ ((((𝑅 ∈ DivRing ∧ 𝐼 ∈ Fin) ∧ curry 𝑀:𝐼⟶((Base‘𝑅) ↑m 𝐼)) ∧ curry 𝑀 LIndF (𝑅 freeLMod 𝐼)) → 𝑅 ∈ DivRing)
58		simpllr 775	. . . . . . . . . . . . . 14 ⊢ ((((𝑅 ∈ DivRing ∧ 𝐼 ∈ Fin) ∧ curry 𝑀:𝐼⟶((Base‘𝑅) ↑m 𝐼)) ∧ curry 𝑀 LIndF (𝑅 freeLMod 𝐼)) → 𝐼 ∈ Fin)
59	45	frlmlmod 21770	. . . . . . . . . . . . . . . . 17 ⊢ ((𝑅 ∈ Ring ∧ 𝐼 ∈ Fin) → (𝑅 freeLMod 𝐼) ∈ LMod)
60	43, 59	sylan 580	. . . . . . . . . . . . . . . 16 ⊢ ((𝑅 ∈ DivRing ∧ 𝐼 ∈ Fin) → (𝑅 freeLMod 𝐼) ∈ LMod)
61	60	adantr 480	. . . . . . . . . . . . . . 15 ⊢ (((𝑅 ∈ DivRing ∧ 𝐼 ∈ Fin) ∧ curry 𝑀:𝐼⟶((Base‘𝑅) ↑m 𝐼)) → (𝑅 freeLMod 𝐼) ∈ LMod)
62		lindfrn 21842	. . . . . . . . . . . . . . 15 ⊢ (((𝑅 freeLMod 𝐼) ∈ LMod ∧ curry 𝑀 LIndF (𝑅 freeLMod 𝐼)) → ran curry 𝑀 ∈ (LIndS‘(𝑅 freeLMod 𝐼)))
63	61, 62	sylan 580	. . . . . . . . . . . . . 14 ⊢ ((((𝑅 ∈ DivRing ∧ 𝐼 ∈ Fin) ∧ curry 𝑀:𝐼⟶((Base‘𝑅) ↑m 𝐼)) ∧ curry 𝑀 LIndF (𝑅 freeLMod 𝐼)) → ran curry 𝑀 ∈ (LIndS‘(𝑅 freeLMod 𝐼)))
64	45	frlmsca 21774	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ ((𝑅 ∈ DivRing ∧ 𝐼 ∈ Fin) → 𝑅 = (Scalar‘(𝑅 freeLMod 𝐼)))
65		drngnzr 20749	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ (𝑅 ∈ DivRing → 𝑅 ∈ NzRing)
66	65	adantr 480	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ ((𝑅 ∈ DivRing ∧ 𝐼 ∈ Fin) → 𝑅 ∈ NzRing)
67	64, 66	eqeltrrd 2841	. . . . . . . . . . . . . . . . . . . . 21 ⊢ ((𝑅 ∈ DivRing ∧ 𝐼 ∈ Fin) → (Scalar‘(𝑅 freeLMod 𝐼)) ∈ NzRing)
68	60, 67	jca 511	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((𝑅 ∈ DivRing ∧ 𝐼 ∈ Fin) → ((𝑅 freeLMod 𝐼) ∈ LMod ∧ (Scalar‘(𝑅 freeLMod 𝐼)) ∈ NzRing))
69		eqid 2736	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (Scalar‘(𝑅 freeLMod 𝐼)) = (Scalar‘(𝑅 freeLMod 𝐼))
70	46, 69	lindff1 21841	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (((𝑅 freeLMod 𝐼) ∈ LMod ∧ (Scalar‘(𝑅 freeLMod 𝐼)) ∈ NzRing ∧ curry 𝑀 LIndF (𝑅 freeLMod 𝐼)) → curry 𝑀:dom curry 𝑀–1-1→(Base‘(𝑅 freeLMod 𝐼)))
71	70	3expa 1118	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((((𝑅 freeLMod 𝐼) ∈ LMod ∧ (Scalar‘(𝑅 freeLMod 𝐼)) ∈ NzRing) ∧ curry 𝑀 LIndF (𝑅 freeLMod 𝐼)) → curry 𝑀:dom curry 𝑀–1-1→(Base‘(𝑅 freeLMod 𝐼)))
72	68, 71	sylan 580	. . . . . . . . . . . . . . . . . . 19 ⊢ (((𝑅 ∈ DivRing ∧ 𝐼 ∈ Fin) ∧ curry 𝑀 LIndF (𝑅 freeLMod 𝐼)) → curry 𝑀:dom curry 𝑀–1-1→(Base‘(𝑅 freeLMod 𝐼)))
73		fdm 6744	. . . . . . . . . . . . . . . . . . 19 ⊢ (curry 𝑀:𝐼⟶((Base‘𝑅) ↑m 𝐼) → dom curry 𝑀 = 𝐼)
74		f1eq2 6799	. . . . . . . . . . . . . . . . . . . 20 ⊢ (dom curry 𝑀 = 𝐼 → (curry 𝑀:dom curry 𝑀–1-1→(Base‘(𝑅 freeLMod 𝐼)) ↔ curry 𝑀:𝐼–1-1→(Base‘(𝑅 freeLMod 𝐼))))
75	74	biimpac 478	. . . . . . . . . . . . . . . . . . 19 ⊢ ((curry 𝑀:dom curry 𝑀–1-1→(Base‘(𝑅 freeLMod 𝐼)) ∧ dom curry 𝑀 = 𝐼) → curry 𝑀:𝐼–1-1→(Base‘(𝑅 freeLMod 𝐼)))
76	72, 73, 75	syl2an 596	. . . . . . . . . . . . . . . . . 18 ⊢ ((((𝑅 ∈ DivRing ∧ 𝐼 ∈ Fin) ∧ curry 𝑀 LIndF (𝑅 freeLMod 𝐼)) ∧ curry 𝑀:𝐼⟶((Base‘𝑅) ↑m 𝐼)) → curry 𝑀:𝐼–1-1→(Base‘(𝑅 freeLMod 𝐼)))
77	76	an32s 652	. . . . . . . . . . . . . . . . 17 ⊢ ((((𝑅 ∈ DivRing ∧ 𝐼 ∈ Fin) ∧ curry 𝑀:𝐼⟶((Base‘𝑅) ↑m 𝐼)) ∧ curry 𝑀 LIndF (𝑅 freeLMod 𝐼)) → curry 𝑀:𝐼–1-1→(Base‘(𝑅 freeLMod 𝐼)))
78		f1f1orn 6858	. . . . . . . . . . . . . . . . 17 ⊢ (curry 𝑀:𝐼–1-1→(Base‘(𝑅 freeLMod 𝐼)) → curry 𝑀:𝐼–1-1-onto→ran curry 𝑀)
79	77, 78	syl 17	. . . . . . . . . . . . . . . 16 ⊢ ((((𝑅 ∈ DivRing ∧ 𝐼 ∈ Fin) ∧ curry 𝑀:𝐼⟶((Base‘𝑅) ↑m 𝐼)) ∧ curry 𝑀 LIndF (𝑅 freeLMod 𝐼)) → curry 𝑀:𝐼–1-1-onto→ran curry 𝑀)
80		f1oeng 9012	. . . . . . . . . . . . . . . 16 ⊢ ((𝐼 ∈ Fin ∧ curry 𝑀:𝐼–1-1-onto→ran curry 𝑀) → 𝐼 ≈ ran curry 𝑀)
81	58, 79, 80	syl2anc 584	. . . . . . . . . . . . . . 15 ⊢ ((((𝑅 ∈ DivRing ∧ 𝐼 ∈ Fin) ∧ curry 𝑀:𝐼⟶((Base‘𝑅) ↑m 𝐼)) ∧ curry 𝑀 LIndF (𝑅 freeLMod 𝐼)) → 𝐼 ≈ ran curry 𝑀)
82	81	ensymd 9046	. . . . . . . . . . . . . 14 ⊢ ((((𝑅 ∈ DivRing ∧ 𝐼 ∈ Fin) ∧ curry 𝑀:𝐼⟶((Base‘𝑅) ↑m 𝐼)) ∧ curry 𝑀 LIndF (𝑅 freeLMod 𝐼)) → ran curry 𝑀 ≈ 𝐼)
83		lindsenlbs 37623	. . . . . . . . . . . . . 14 ⊢ (((𝑅 ∈ DivRing ∧ 𝐼 ∈ Fin ∧ ran curry 𝑀 ∈ (LIndS‘(𝑅 freeLMod 𝐼))) ∧ ran curry 𝑀 ≈ 𝐼) → ran curry 𝑀 ∈ (LBasis‘(𝑅 freeLMod 𝐼)))
84	57, 58, 63, 82, 83	syl31anc 1374	. . . . . . . . . . . . 13 ⊢ ((((𝑅 ∈ DivRing ∧ 𝐼 ∈ Fin) ∧ curry 𝑀:𝐼⟶((Base‘𝑅) ↑m 𝐼)) ∧ curry 𝑀 LIndF (𝑅 freeLMod 𝐼)) → ran curry 𝑀 ∈ (LBasis‘(𝑅 freeLMod 𝐼)))
85		eqid 2736	. . . . . . . . . . . . . 14 ⊢ (LBasis‘(𝑅 freeLMod 𝐼)) = (LBasis‘(𝑅 freeLMod 𝐼))
86		eqid 2736	. . . . . . . . . . . . . 14 ⊢ (LSpan‘(𝑅 freeLMod 𝐼)) = (LSpan‘(𝑅 freeLMod 𝐼))
87	46, 85, 86	lbssp 21079	. . . . . . . . . . . . 13 ⊢ (ran curry 𝑀 ∈ (LBasis‘(𝑅 freeLMod 𝐼)) → ((LSpan‘(𝑅 freeLMod 𝐼))‘ran curry 𝑀) = (Base‘(𝑅 freeLMod 𝐼)))
88	84, 87	syl 17	. . . . . . . . . . . 12 ⊢ ((((𝑅 ∈ DivRing ∧ 𝐼 ∈ Fin) ∧ curry 𝑀:𝐼⟶((Base‘𝑅) ↑m 𝐼)) ∧ curry 𝑀 LIndF (𝑅 freeLMod 𝐼)) → ((LSpan‘(𝑅 freeLMod 𝐼))‘ran curry 𝑀) = (Base‘(𝑅 freeLMod 𝐼)))
89	56, 88	eqtrd 2776	. . . . . . . . . . 11 ⊢ ((((𝑅 ∈ DivRing ∧ 𝐼 ∈ Fin) ∧ curry 𝑀:𝐼⟶((Base‘𝑅) ↑m 𝐼)) ∧ curry 𝑀 LIndF (𝑅 freeLMod 𝐼)) → ((LSpan‘(𝑅 freeLMod 𝐼))‘(curry 𝑀 “ 𝐼)) = (Base‘(𝑅 freeLMod 𝐼)))
90	89	adantr 480	. . . . . . . . . 10 ⊢ (((((𝑅 ∈ DivRing ∧ 𝐼 ∈ Fin) ∧ curry 𝑀:𝐼⟶((Base‘𝑅) ↑m 𝐼)) ∧ curry 𝑀 LIndF (𝑅 freeLMod 𝐼)) ∧ 𝑖 ∈ 𝐼) → ((LSpan‘(𝑅 freeLMod 𝐼))‘(curry 𝑀 “ 𝐼)) = (Base‘(𝑅 freeLMod 𝐼)))
91	50, 90	eleqtrrd 2843	. . . . . . . . 9 ⊢ (((((𝑅 ∈ DivRing ∧ 𝐼 ∈ Fin) ∧ curry 𝑀:𝐼⟶((Base‘𝑅) ↑m 𝐼)) ∧ curry 𝑀 LIndF (𝑅 freeLMod 𝐼)) ∧ 𝑖 ∈ 𝐼) → ((𝑅 unitVec 𝐼)‘𝑖) ∈ ((LSpan‘(𝑅 freeLMod 𝐼))‘(curry 𝑀 “ 𝐼)))
92		eqid 2736	. . . . . . . . . . . . 13 ⊢ (Base‘(Scalar‘(𝑅 freeLMod 𝐼))) = (Base‘(Scalar‘(𝑅 freeLMod 𝐼)))
93		eqid 2736	. . . . . . . . . . . . 13 ⊢ (0g‘(Scalar‘(𝑅 freeLMod 𝐼))) = (0g‘(Scalar‘(𝑅 freeLMod 𝐼)))
94		eqid 2736	. . . . . . . . . . . . 13 ⊢ ( ·𝑠 ‘(𝑅 freeLMod 𝐼)) = ( ·𝑠 ‘(𝑅 freeLMod 𝐼))
95	45, 14	frlmfibas 21783	. . . . . . . . . . . . . . 15 ⊢ ((𝑅 ∈ Ring ∧ 𝐼 ∈ Fin) → ((Base‘𝑅) ↑m 𝐼) = (Base‘(𝑅 freeLMod 𝐼)))
96	95	feq3d 6722	. . . . . . . . . . . . . 14 ⊢ ((𝑅 ∈ Ring ∧ 𝐼 ∈ Fin) → (curry 𝑀:𝐼⟶((Base‘𝑅) ↑m 𝐼) ↔ curry 𝑀:𝐼⟶(Base‘(𝑅 freeLMod 𝐼))))
97	96	biimpa 476	. . . . . . . . . . . . 13 ⊢ (((𝑅 ∈ Ring ∧ 𝐼 ∈ Fin) ∧ curry 𝑀:𝐼⟶((Base‘𝑅) ↑m 𝐼)) → curry 𝑀:𝐼⟶(Base‘(𝑅 freeLMod 𝐼)))
98	59	adantr 480	. . . . . . . . . . . . 13 ⊢ (((𝑅 ∈ Ring ∧ 𝐼 ∈ Fin) ∧ curry 𝑀:𝐼⟶((Base‘𝑅) ↑m 𝐼)) → (𝑅 freeLMod 𝐼) ∈ LMod)
99		simplr 768	. . . . . . . . . . . . 13 ⊢ (((𝑅 ∈ Ring ∧ 𝐼 ∈ Fin) ∧ curry 𝑀:𝐼⟶((Base‘𝑅) ↑m 𝐼)) → 𝐼 ∈ Fin)
100	86, 46, 92, 69, 93, 94, 97, 98, 99	elfilspd 21824	. . . . . . . . . . . 12 ⊢ (((𝑅 ∈ Ring ∧ 𝐼 ∈ Fin) ∧ curry 𝑀:𝐼⟶((Base‘𝑅) ↑m 𝐼)) → (((𝑅 unitVec 𝐼)‘𝑖) ∈ ((LSpan‘(𝑅 freeLMod 𝐼))‘(curry 𝑀 “ 𝐼)) ↔ ∃𝑛 ∈ ((Base‘(Scalar‘(𝑅 freeLMod 𝐼))) ↑m 𝐼)((𝑅 unitVec 𝐼)‘𝑖) = ((𝑅 freeLMod 𝐼) Σg (𝑛 ∘f ( ·𝑠 ‘(𝑅 freeLMod 𝐼))curry 𝑀))))
101	45	frlmsca 21774	. . . . . . . . . . . . . . . 16 ⊢ ((𝑅 ∈ Ring ∧ 𝐼 ∈ Fin) → 𝑅 = (Scalar‘(𝑅 freeLMod 𝐼)))
102	101	fveq2d 6909	. . . . . . . . . . . . . . 15 ⊢ ((𝑅 ∈ Ring ∧ 𝐼 ∈ Fin) → (Base‘𝑅) = (Base‘(Scalar‘(𝑅 freeLMod 𝐼))))
103	102	oveq1d 7447	. . . . . . . . . . . . . 14 ⊢ ((𝑅 ∈ Ring ∧ 𝐼 ∈ Fin) → ((Base‘𝑅) ↑m 𝐼) = ((Base‘(Scalar‘(𝑅 freeLMod 𝐼))) ↑m 𝐼))
104	103	adantr 480	. . . . . . . . . . . . 13 ⊢ (((𝑅 ∈ Ring ∧ 𝐼 ∈ Fin) ∧ curry 𝑀:𝐼⟶((Base‘𝑅) ↑m 𝐼)) → ((Base‘𝑅) ↑m 𝐼) = ((Base‘(Scalar‘(𝑅 freeLMod 𝐼))) ↑m 𝐼))
105		elmapi 8890	. . . . . . . . . . . . . . 15 ⊢ (𝑛 ∈ ((Base‘𝑅) ↑m 𝐼) → 𝑛:𝐼⟶(Base‘𝑅))
106		ffn 6735	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝑛:𝐼⟶(Base‘𝑅) → 𝑛 Fn 𝐼)
107	106	adantl 481	. . . . . . . . . . . . . . . . . . 19 ⊢ ((((𝑅 ∈ Ring ∧ 𝐼 ∈ Fin) ∧ curry 𝑀:𝐼⟶((Base‘𝑅) ↑m 𝐼)) ∧ 𝑛:𝐼⟶(Base‘𝑅)) → 𝑛 Fn 𝐼)
108	51	ad2antlr 727	. . . . . . . . . . . . . . . . . . 19 ⊢ ((((𝑅 ∈ Ring ∧ 𝐼 ∈ Fin) ∧ curry 𝑀:𝐼⟶((Base‘𝑅) ↑m 𝐼)) ∧ 𝑛:𝐼⟶(Base‘𝑅)) → curry 𝑀 Fn 𝐼)
109		simpllr 775	. . . . . . . . . . . . . . . . . . 19 ⊢ ((((𝑅 ∈ Ring ∧ 𝐼 ∈ Fin) ∧ curry 𝑀:𝐼⟶((Base‘𝑅) ↑m 𝐼)) ∧ 𝑛:𝐼⟶(Base‘𝑅)) → 𝐼 ∈ Fin)
110		inidm 4226	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝐼 ∩ 𝐼) = 𝐼
111		eqidd 2737	. . . . . . . . . . . . . . . . . . 19 ⊢ (((((𝑅 ∈ Ring ∧ 𝐼 ∈ Fin) ∧ curry 𝑀:𝐼⟶((Base‘𝑅) ↑m 𝐼)) ∧ 𝑛:𝐼⟶(Base‘𝑅)) ∧ 𝑘 ∈ 𝐼) → (𝑛‘𝑘) = (𝑛‘𝑘))
112		eqidd 2737	. . . . . . . . . . . . . . . . . . 19 ⊢ (((((𝑅 ∈ Ring ∧ 𝐼 ∈ Fin) ∧ curry 𝑀:𝐼⟶((Base‘𝑅) ↑m 𝐼)) ∧ 𝑛:𝐼⟶(Base‘𝑅)) ∧ 𝑘 ∈ 𝐼) → (curry 𝑀‘𝑘) = (curry 𝑀‘𝑘))
113	107, 108, 109, 109, 110, 111, 112	offval 7707	. . . . . . . . . . . . . . . . . 18 ⊢ ((((𝑅 ∈ Ring ∧ 𝐼 ∈ Fin) ∧ curry 𝑀:𝐼⟶((Base‘𝑅) ↑m 𝐼)) ∧ 𝑛:𝐼⟶(Base‘𝑅)) → (𝑛 ∘f ( ·𝑠 ‘(𝑅 freeLMod 𝐼))curry 𝑀) = (𝑘 ∈ 𝐼 ↦ ((𝑛‘𝑘)( ·𝑠 ‘(𝑅 freeLMod 𝐼))(curry 𝑀‘𝑘))))
114		simp-4r 783	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (((((𝑅 ∈ Ring ∧ 𝐼 ∈ Fin) ∧ curry 𝑀:𝐼⟶((Base‘𝑅) ↑m 𝐼)) ∧ 𝑛:𝐼⟶(Base‘𝑅)) ∧ 𝑘 ∈ 𝐼) → 𝐼 ∈ Fin)
115		ffvelcdm 7100	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ ((𝑛:𝐼⟶(Base‘𝑅) ∧ 𝑘 ∈ 𝐼) → (𝑛‘𝑘) ∈ (Base‘𝑅))
116	115	adantll 714	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (((((𝑅 ∈ Ring ∧ 𝐼 ∈ Fin) ∧ curry 𝑀:𝐼⟶((Base‘𝑅) ↑m 𝐼)) ∧ 𝑛:𝐼⟶(Base‘𝑅)) ∧ 𝑘 ∈ 𝐼) → (𝑛‘𝑘) ∈ (Base‘𝑅))
117		ffvelcdm 7100	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ ((curry 𝑀:𝐼⟶((Base‘𝑅) ↑m 𝐼) ∧ 𝑘 ∈ 𝐼) → (curry 𝑀‘𝑘) ∈ ((Base‘𝑅) ↑m 𝐼))
118	117	ad4ant24 754	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (((((𝑅 ∈ Ring ∧ 𝐼 ∈ Fin) ∧ curry 𝑀:𝐼⟶((Base‘𝑅) ↑m 𝐼)) ∧ 𝑛:𝐼⟶(Base‘𝑅)) ∧ 𝑘 ∈ 𝐼) → (curry 𝑀‘𝑘) ∈ ((Base‘𝑅) ↑m 𝐼))
119	95	ad3antrrr 730	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (((((𝑅 ∈ Ring ∧ 𝐼 ∈ Fin) ∧ curry 𝑀:𝐼⟶((Base‘𝑅) ↑m 𝐼)) ∧ 𝑛:𝐼⟶(Base‘𝑅)) ∧ 𝑘 ∈ 𝐼) → ((Base‘𝑅) ↑m 𝐼) = (Base‘(𝑅 freeLMod 𝐼)))
120	118, 119	eleqtrd 2842	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (((((𝑅 ∈ Ring ∧ 𝐼 ∈ Fin) ∧ curry 𝑀:𝐼⟶((Base‘𝑅) ↑m 𝐼)) ∧ 𝑛:𝐼⟶(Base‘𝑅)) ∧ 𝑘 ∈ 𝐼) → (curry 𝑀‘𝑘) ∈ (Base‘(𝑅 freeLMod 𝐼)))
121		eqid 2736	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (.r‘𝑅) = (.r‘𝑅)
122	45, 46, 14, 114, 116, 120, 94, 121	frlmvscafval 21787	. . . . . . . . . . . . . . . . . . . 20 ⊢ (((((𝑅 ∈ Ring ∧ 𝐼 ∈ Fin) ∧ curry 𝑀:𝐼⟶((Base‘𝑅) ↑m 𝐼)) ∧ 𝑛:𝐼⟶(Base‘𝑅)) ∧ 𝑘 ∈ 𝐼) → ((𝑛‘𝑘)( ·𝑠 ‘(𝑅 freeLMod 𝐼))(curry 𝑀‘𝑘)) = ((𝐼 × {(𝑛‘𝑘)}) ∘f (.r‘𝑅)(curry 𝑀‘𝑘)))
123		fvex 6918	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (𝑛‘𝑘) ∈ V
124		fnconstg 6795	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ ((𝑛‘𝑘) ∈ V → (𝐼 × {(𝑛‘𝑘)}) Fn 𝐼)
125	123, 124	mp1i 13	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (((((𝑅 ∈ Ring ∧ 𝐼 ∈ Fin) ∧ curry 𝑀:𝐼⟶((Base‘𝑅) ↑m 𝐼)) ∧ 𝑛:𝐼⟶(Base‘𝑅)) ∧ 𝑘 ∈ 𝐼) → (𝐼 × {(𝑛‘𝑘)}) Fn 𝐼)
126		elmapfn 8906	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ ((curry 𝑀‘𝑘) ∈ ((Base‘𝑅) ↑m 𝐼) → (curry 𝑀‘𝑘) Fn 𝐼)
127	117, 126	syl 17	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ ((curry 𝑀:𝐼⟶((Base‘𝑅) ↑m 𝐼) ∧ 𝑘 ∈ 𝐼) → (curry 𝑀‘𝑘) Fn 𝐼)
128	127	ad4ant24 754	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (((((𝑅 ∈ Ring ∧ 𝐼 ∈ Fin) ∧ curry 𝑀:𝐼⟶((Base‘𝑅) ↑m 𝐼)) ∧ 𝑛:𝐼⟶(Base‘𝑅)) ∧ 𝑘 ∈ 𝐼) → (curry 𝑀‘𝑘) Fn 𝐼)
129	123	fvconst2 7225	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (𝑗 ∈ 𝐼 → ((𝐼 × {(𝑛‘𝑘)})‘𝑗) = (𝑛‘𝑘))
130	129	adantl 481	. . . . . . . . . . . . . . . . . . . . 21 ⊢ ((((((𝑅 ∈ Ring ∧ 𝐼 ∈ Fin) ∧ curry 𝑀:𝐼⟶((Base‘𝑅) ↑m 𝐼)) ∧ 𝑛:𝐼⟶(Base‘𝑅)) ∧ 𝑘 ∈ 𝐼) ∧ 𝑗 ∈ 𝐼) → ((𝐼 × {(𝑛‘𝑘)})‘𝑗) = (𝑛‘𝑘))
131		eqidd 2737	. . . . . . . . . . . . . . . . . . . . 21 ⊢ ((((((𝑅 ∈ Ring ∧ 𝐼 ∈ Fin) ∧ curry 𝑀:𝐼⟶((Base‘𝑅) ↑m 𝐼)) ∧ 𝑛:𝐼⟶(Base‘𝑅)) ∧ 𝑘 ∈ 𝐼) ∧ 𝑗 ∈ 𝐼) → ((curry 𝑀‘𝑘)‘𝑗) = ((curry 𝑀‘𝑘)‘𝑗))
132	125, 128, 114, 114, 110, 130, 131	offval 7707	. . . . . . . . . . . . . . . . . . . 20 ⊢ (((((𝑅 ∈ Ring ∧ 𝐼 ∈ Fin) ∧ curry 𝑀:𝐼⟶((Base‘𝑅) ↑m 𝐼)) ∧ 𝑛:𝐼⟶(Base‘𝑅)) ∧ 𝑘 ∈ 𝐼) → ((𝐼 × {(𝑛‘𝑘)}) ∘f (.r‘𝑅)(curry 𝑀‘𝑘)) = (𝑗 ∈ 𝐼 ↦ ((𝑛‘𝑘)(.r‘𝑅)((curry 𝑀‘𝑘)‘𝑗))))
133	122, 132	eqtrd 2776	. . . . . . . . . . . . . . . . . . 19 ⊢ (((((𝑅 ∈ Ring ∧ 𝐼 ∈ Fin) ∧ curry 𝑀:𝐼⟶((Base‘𝑅) ↑m 𝐼)) ∧ 𝑛:𝐼⟶(Base‘𝑅)) ∧ 𝑘 ∈ 𝐼) → ((𝑛‘𝑘)( ·𝑠 ‘(𝑅 freeLMod 𝐼))(curry 𝑀‘𝑘)) = (𝑗 ∈ 𝐼 ↦ ((𝑛‘𝑘)(.r‘𝑅)((curry 𝑀‘𝑘)‘𝑗))))
134	133	mpteq2dva 5241	. . . . . . . . . . . . . . . . . 18 ⊢ ((((𝑅 ∈ Ring ∧ 𝐼 ∈ Fin) ∧ curry 𝑀:𝐼⟶((Base‘𝑅) ↑m 𝐼)) ∧ 𝑛:𝐼⟶(Base‘𝑅)) → (𝑘 ∈ 𝐼 ↦ ((𝑛‘𝑘)( ·𝑠 ‘(𝑅 freeLMod 𝐼))(curry 𝑀‘𝑘))) = (𝑘 ∈ 𝐼 ↦ (𝑗 ∈ 𝐼 ↦ ((𝑛‘𝑘)(.r‘𝑅)((curry 𝑀‘𝑘)‘𝑗)))))
135	113, 134	eqtrd 2776	. . . . . . . . . . . . . . . . 17 ⊢ ((((𝑅 ∈ Ring ∧ 𝐼 ∈ Fin) ∧ curry 𝑀:𝐼⟶((Base‘𝑅) ↑m 𝐼)) ∧ 𝑛:𝐼⟶(Base‘𝑅)) → (𝑛 ∘f ( ·𝑠 ‘(𝑅 freeLMod 𝐼))curry 𝑀) = (𝑘 ∈ 𝐼 ↦ (𝑗 ∈ 𝐼 ↦ ((𝑛‘𝑘)(.r‘𝑅)((curry 𝑀‘𝑘)‘𝑗)))))
136	135	oveq2d 7448	. . . . . . . . . . . . . . . 16 ⊢ ((((𝑅 ∈ Ring ∧ 𝐼 ∈ Fin) ∧ curry 𝑀:𝐼⟶((Base‘𝑅) ↑m 𝐼)) ∧ 𝑛:𝐼⟶(Base‘𝑅)) → ((𝑅 freeLMod 𝐼) Σg (𝑛 ∘f ( ·𝑠 ‘(𝑅 freeLMod 𝐼))curry 𝑀)) = ((𝑅 freeLMod 𝐼) Σg (𝑘 ∈ 𝐼 ↦ (𝑗 ∈ 𝐼 ↦ ((𝑛‘𝑘)(.r‘𝑅)((curry 𝑀‘𝑘)‘𝑗))))))
137		eqid 2736	. . . . . . . . . . . . . . . . 17 ⊢ (0g‘(𝑅 freeLMod 𝐼)) = (0g‘(𝑅 freeLMod 𝐼))
138		simplll 774	. . . . . . . . . . . . . . . . 17 ⊢ ((((𝑅 ∈ Ring ∧ 𝐼 ∈ Fin) ∧ curry 𝑀:𝐼⟶((Base‘𝑅) ↑m 𝐼)) ∧ 𝑛:𝐼⟶(Base‘𝑅)) → 𝑅 ∈ Ring)
139		simp-5l 784	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((((((𝑅 ∈ Ring ∧ 𝐼 ∈ Fin) ∧ curry 𝑀:𝐼⟶((Base‘𝑅) ↑m 𝐼)) ∧ 𝑛:𝐼⟶(Base‘𝑅)) ∧ 𝑘 ∈ 𝐼) ∧ 𝑗 ∈ 𝐼) → 𝑅 ∈ Ring)
140	115	ad4ant23 753	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((((((𝑅 ∈ Ring ∧ 𝐼 ∈ Fin) ∧ curry 𝑀:𝐼⟶((Base‘𝑅) ↑m 𝐼)) ∧ 𝑛:𝐼⟶(Base‘𝑅)) ∧ 𝑘 ∈ 𝐼) ∧ 𝑗 ∈ 𝐼) → (𝑛‘𝑘) ∈ (Base‘𝑅))
141		simplr 768	. . . . . . . . . . . . . . . . . . . . 21 ⊢ ((((𝑅 ∈ Ring ∧ 𝐼 ∈ Fin) ∧ curry 𝑀:𝐼⟶((Base‘𝑅) ↑m 𝐼)) ∧ 𝑛:𝐼⟶(Base‘𝑅)) → curry 𝑀:𝐼⟶((Base‘𝑅) ↑m 𝐼))
142		elmapi 8890	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ ((curry 𝑀‘𝑘) ∈ ((Base‘𝑅) ↑m 𝐼) → (curry 𝑀‘𝑘):𝐼⟶(Base‘𝑅))
143	117, 142	syl 17	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ ((curry 𝑀:𝐼⟶((Base‘𝑅) ↑m 𝐼) ∧ 𝑘 ∈ 𝐼) → (curry 𝑀‘𝑘):𝐼⟶(Base‘𝑅))
144	143	ffvelcdmda 7103	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (((curry 𝑀:𝐼⟶((Base‘𝑅) ↑m 𝐼) ∧ 𝑘 ∈ 𝐼) ∧ 𝑗 ∈ 𝐼) → ((curry 𝑀‘𝑘)‘𝑗) ∈ (Base‘𝑅))
145	141, 144	sylanl1 680	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((((((𝑅 ∈ Ring ∧ 𝐼 ∈ Fin) ∧ curry 𝑀:𝐼⟶((Base‘𝑅) ↑m 𝐼)) ∧ 𝑛:𝐼⟶(Base‘𝑅)) ∧ 𝑘 ∈ 𝐼) ∧ 𝑗 ∈ 𝐼) → ((curry 𝑀‘𝑘)‘𝑗) ∈ (Base‘𝑅))
146	14, 121	ringcl 20248	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((𝑅 ∈ Ring ∧ (𝑛‘𝑘) ∈ (Base‘𝑅) ∧ ((curry 𝑀‘𝑘)‘𝑗) ∈ (Base‘𝑅)) → ((𝑛‘𝑘)(.r‘𝑅)((curry 𝑀‘𝑘)‘𝑗)) ∈ (Base‘𝑅))
147	139, 140, 145, 146	syl3anc 1372	. . . . . . . . . . . . . . . . . . 19 ⊢ ((((((𝑅 ∈ Ring ∧ 𝐼 ∈ Fin) ∧ curry 𝑀:𝐼⟶((Base‘𝑅) ↑m 𝐼)) ∧ 𝑛:𝐼⟶(Base‘𝑅)) ∧ 𝑘 ∈ 𝐼) ∧ 𝑗 ∈ 𝐼) → ((𝑛‘𝑘)(.r‘𝑅)((curry 𝑀‘𝑘)‘𝑗)) ∈ (Base‘𝑅))
148	147	fmpttd 7134	. . . . . . . . . . . . . . . . . 18 ⊢ (((((𝑅 ∈ Ring ∧ 𝐼 ∈ Fin) ∧ curry 𝑀:𝐼⟶((Base‘𝑅) ↑m 𝐼)) ∧ 𝑛:𝐼⟶(Base‘𝑅)) ∧ 𝑘 ∈ 𝐼) → (𝑗 ∈ 𝐼 ↦ ((𝑛‘𝑘)(.r‘𝑅)((curry 𝑀‘𝑘)‘𝑗))):𝐼⟶(Base‘𝑅))
149		elmapg 8880	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (((Base‘𝑅) ∈ V ∧ 𝐼 ∈ Fin) → ((𝑗 ∈ 𝐼 ↦ ((𝑛‘𝑘)(.r‘𝑅)((curry 𝑀‘𝑘)‘𝑗))) ∈ ((Base‘𝑅) ↑m 𝐼) ↔ (𝑗 ∈ 𝐼 ↦ ((𝑛‘𝑘)(.r‘𝑅)((curry 𝑀‘𝑘)‘𝑗))):𝐼⟶(Base‘𝑅)))
150	22, 149	mpan 690	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (𝐼 ∈ Fin → ((𝑗 ∈ 𝐼 ↦ ((𝑛‘𝑘)(.r‘𝑅)((curry 𝑀‘𝑘)‘𝑗))) ∈ ((Base‘𝑅) ↑m 𝐼) ↔ (𝑗 ∈ 𝐼 ↦ ((𝑛‘𝑘)(.r‘𝑅)((curry 𝑀‘𝑘)‘𝑗))):𝐼⟶(Base‘𝑅)))
151	150	adantl 481	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((𝑅 ∈ Ring ∧ 𝐼 ∈ Fin) → ((𝑗 ∈ 𝐼 ↦ ((𝑛‘𝑘)(.r‘𝑅)((curry 𝑀‘𝑘)‘𝑗))) ∈ ((Base‘𝑅) ↑m 𝐼) ↔ (𝑗 ∈ 𝐼 ↦ ((𝑛‘𝑘)(.r‘𝑅)((curry 𝑀‘𝑘)‘𝑗))):𝐼⟶(Base‘𝑅)))
152	95	eleq2d 2826	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((𝑅 ∈ Ring ∧ 𝐼 ∈ Fin) → ((𝑗 ∈ 𝐼 ↦ ((𝑛‘𝑘)(.r‘𝑅)((curry 𝑀‘𝑘)‘𝑗))) ∈ ((Base‘𝑅) ↑m 𝐼) ↔ (𝑗 ∈ 𝐼 ↦ ((𝑛‘𝑘)(.r‘𝑅)((curry 𝑀‘𝑘)‘𝑗))) ∈ (Base‘(𝑅 freeLMod 𝐼))))
153	151, 152	bitr3d 281	. . . . . . . . . . . . . . . . . . 19 ⊢ ((𝑅 ∈ Ring ∧ 𝐼 ∈ Fin) → ((𝑗 ∈ 𝐼 ↦ ((𝑛‘𝑘)(.r‘𝑅)((curry 𝑀‘𝑘)‘𝑗))):𝐼⟶(Base‘𝑅) ↔ (𝑗 ∈ 𝐼 ↦ ((𝑛‘𝑘)(.r‘𝑅)((curry 𝑀‘𝑘)‘𝑗))) ∈ (Base‘(𝑅 freeLMod 𝐼))))
154	153	ad3antrrr 730	. . . . . . . . . . . . . . . . . 18 ⊢ (((((𝑅 ∈ Ring ∧ 𝐼 ∈ Fin) ∧ curry 𝑀:𝐼⟶((Base‘𝑅) ↑m 𝐼)) ∧ 𝑛:𝐼⟶(Base‘𝑅)) ∧ 𝑘 ∈ 𝐼) → ((𝑗 ∈ 𝐼 ↦ ((𝑛‘𝑘)(.r‘𝑅)((curry 𝑀‘𝑘)‘𝑗))):𝐼⟶(Base‘𝑅) ↔ (𝑗 ∈ 𝐼 ↦ ((𝑛‘𝑘)(.r‘𝑅)((curry 𝑀‘𝑘)‘𝑗))) ∈ (Base‘(𝑅 freeLMod 𝐼))))
155	148, 154	mpbid 232	. . . . . . . . . . . . . . . . 17 ⊢ (((((𝑅 ∈ Ring ∧ 𝐼 ∈ Fin) ∧ curry 𝑀:𝐼⟶((Base‘𝑅) ↑m 𝐼)) ∧ 𝑛:𝐼⟶(Base‘𝑅)) ∧ 𝑘 ∈ 𝐼) → (𝑗 ∈ 𝐼 ↦ ((𝑛‘𝑘)(.r‘𝑅)((curry 𝑀‘𝑘)‘𝑗))) ∈ (Base‘(𝑅 freeLMod 𝐼)))
156		mptexg 7242	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (𝐼 ∈ Fin → (𝑗 ∈ 𝐼 ↦ ((𝑛‘𝑘)(.r‘𝑅)((curry 𝑀‘𝑘)‘𝑗))) ∈ V)
157	156	ralrimivw 3149	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝐼 ∈ Fin → ∀𝑘 ∈ 𝐼 (𝑗 ∈ 𝐼 ↦ ((𝑛‘𝑘)(.r‘𝑅)((curry 𝑀‘𝑘)‘𝑗))) ∈ V)
158		eqid 2736	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (𝑘 ∈ 𝐼 ↦ (𝑗 ∈ 𝐼 ↦ ((𝑛‘𝑘)(.r‘𝑅)((curry 𝑀‘𝑘)‘𝑗)))) = (𝑘 ∈ 𝐼 ↦ (𝑗 ∈ 𝐼 ↦ ((𝑛‘𝑘)(.r‘𝑅)((curry 𝑀‘𝑘)‘𝑗))))
159	158	fnmpt 6707	. . . . . . . . . . . . . . . . . . . 20 ⊢ (∀𝑘 ∈ 𝐼 (𝑗 ∈ 𝐼 ↦ ((𝑛‘𝑘)(.r‘𝑅)((curry 𝑀‘𝑘)‘𝑗))) ∈ V → (𝑘 ∈ 𝐼 ↦ (𝑗 ∈ 𝐼 ↦ ((𝑛‘𝑘)(.r‘𝑅)((curry 𝑀‘𝑘)‘𝑗)))) Fn 𝐼)
160	157, 159	syl 17	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝐼 ∈ Fin → (𝑘 ∈ 𝐼 ↦ (𝑗 ∈ 𝐼 ↦ ((𝑛‘𝑘)(.r‘𝑅)((curry 𝑀‘𝑘)‘𝑗)))) Fn 𝐼)
161		id 22	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝐼 ∈ Fin → 𝐼 ∈ Fin)
162		fvexd 6920	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝐼 ∈ Fin → (0g‘(𝑅 freeLMod 𝐼)) ∈ V)
163	160, 161, 162	fndmfifsupp 9419	. . . . . . . . . . . . . . . . . 18 ⊢ (𝐼 ∈ Fin → (𝑘 ∈ 𝐼 ↦ (𝑗 ∈ 𝐼 ↦ ((𝑛‘𝑘)(.r‘𝑅)((curry 𝑀‘𝑘)‘𝑗)))) finSupp (0g‘(𝑅 freeLMod 𝐼)))
164	163	ad3antlr 731	. . . . . . . . . . . . . . . . 17 ⊢ ((((𝑅 ∈ Ring ∧ 𝐼 ∈ Fin) ∧ curry 𝑀:𝐼⟶((Base‘𝑅) ↑m 𝐼)) ∧ 𝑛:𝐼⟶(Base‘𝑅)) → (𝑘 ∈ 𝐼 ↦ (𝑗 ∈ 𝐼 ↦ ((𝑛‘𝑘)(.r‘𝑅)((curry 𝑀‘𝑘)‘𝑗)))) finSupp (0g‘(𝑅 freeLMod 𝐼)))
165	45, 46, 137, 109, 109, 138, 155, 164	frlmgsum 21793	. . . . . . . . . . . . . . . 16 ⊢ ((((𝑅 ∈ Ring ∧ 𝐼 ∈ Fin) ∧ curry 𝑀:𝐼⟶((Base‘𝑅) ↑m 𝐼)) ∧ 𝑛:𝐼⟶(Base‘𝑅)) → ((𝑅 freeLMod 𝐼) Σg (𝑘 ∈ 𝐼 ↦ (𝑗 ∈ 𝐼 ↦ ((𝑛‘𝑘)(.r‘𝑅)((curry 𝑀‘𝑘)‘𝑗))))) = (𝑗 ∈ 𝐼 ↦ (𝑅 Σg (𝑘 ∈ 𝐼 ↦ ((𝑛‘𝑘)(.r‘𝑅)((curry 𝑀‘𝑘)‘𝑗))))))
166	136, 165	eqtr2d 2777	. . . . . . . . . . . . . . 15 ⊢ ((((𝑅 ∈ Ring ∧ 𝐼 ∈ Fin) ∧ curry 𝑀:𝐼⟶((Base‘𝑅) ↑m 𝐼)) ∧ 𝑛:𝐼⟶(Base‘𝑅)) → (𝑗 ∈ 𝐼 ↦ (𝑅 Σg (𝑘 ∈ 𝐼 ↦ ((𝑛‘𝑘)(.r‘𝑅)((curry 𝑀‘𝑘)‘𝑗))))) = ((𝑅 freeLMod 𝐼) Σg (𝑛 ∘f ( ·𝑠 ‘(𝑅 freeLMod 𝐼))curry 𝑀)))
167	105, 166	sylan2 593	. . . . . . . . . . . . . 14 ⊢ ((((𝑅 ∈ Ring ∧ 𝐼 ∈ Fin) ∧ curry 𝑀:𝐼⟶((Base‘𝑅) ↑m 𝐼)) ∧ 𝑛 ∈ ((Base‘𝑅) ↑m 𝐼)) → (𝑗 ∈ 𝐼 ↦ (𝑅 Σg (𝑘 ∈ 𝐼 ↦ ((𝑛‘𝑘)(.r‘𝑅)((curry 𝑀‘𝑘)‘𝑗))))) = ((𝑅 freeLMod 𝐼) Σg (𝑛 ∘f ( ·𝑠 ‘(𝑅 freeLMod 𝐼))curry 𝑀)))
168	167	eqeq2d 2747	. . . . . . . . . . . . 13 ⊢ ((((𝑅 ∈ Ring ∧ 𝐼 ∈ Fin) ∧ curry 𝑀:𝐼⟶((Base‘𝑅) ↑m 𝐼)) ∧ 𝑛 ∈ ((Base‘𝑅) ↑m 𝐼)) → (((𝑅 unitVec 𝐼)‘𝑖) = (𝑗 ∈ 𝐼 ↦ (𝑅 Σg (𝑘 ∈ 𝐼 ↦ ((𝑛‘𝑘)(.r‘𝑅)((curry 𝑀‘𝑘)‘𝑗))))) ↔ ((𝑅 unitVec 𝐼)‘𝑖) = ((𝑅 freeLMod 𝐼) Σg (𝑛 ∘f ( ·𝑠 ‘(𝑅 freeLMod 𝐼))curry 𝑀))))
169	104, 168	rexeqbidva 3332	. . . . . . . . . . . 12 ⊢ (((𝑅 ∈ Ring ∧ 𝐼 ∈ Fin) ∧ curry 𝑀:𝐼⟶((Base‘𝑅) ↑m 𝐼)) → (∃𝑛 ∈ ((Base‘𝑅) ↑m 𝐼)((𝑅 unitVec 𝐼)‘𝑖) = (𝑗 ∈ 𝐼 ↦ (𝑅 Σg (𝑘 ∈ 𝐼 ↦ ((𝑛‘𝑘)(.r‘𝑅)((curry 𝑀‘𝑘)‘𝑗))))) ↔ ∃𝑛 ∈ ((Base‘(Scalar‘(𝑅 freeLMod 𝐼))) ↑m 𝐼)((𝑅 unitVec 𝐼)‘𝑖) = ((𝑅 freeLMod 𝐼) Σg (𝑛 ∘f ( ·𝑠 ‘(𝑅 freeLMod 𝐼))curry 𝑀))))
170	100, 169	bitr4d 282	. . . . . . . . . . 11 ⊢ (((𝑅 ∈ Ring ∧ 𝐼 ∈ Fin) ∧ curry 𝑀:𝐼⟶((Base‘𝑅) ↑m 𝐼)) → (((𝑅 unitVec 𝐼)‘𝑖) ∈ ((LSpan‘(𝑅 freeLMod 𝐼))‘(curry 𝑀 “ 𝐼)) ↔ ∃𝑛 ∈ ((Base‘𝑅) ↑m 𝐼)((𝑅 unitVec 𝐼)‘𝑖) = (𝑗 ∈ 𝐼 ↦ (𝑅 Σg (𝑘 ∈ 𝐼 ↦ ((𝑛‘𝑘)(.r‘𝑅)((curry 𝑀‘𝑘)‘𝑗)))))))
171	43, 170	sylanl1 680	. . . . . . . . . 10 ⊢ (((𝑅 ∈ DivRing ∧ 𝐼 ∈ Fin) ∧ curry 𝑀:𝐼⟶((Base‘𝑅) ↑m 𝐼)) → (((𝑅 unitVec 𝐼)‘𝑖) ∈ ((LSpan‘(𝑅 freeLMod 𝐼))‘(curry 𝑀 “ 𝐼)) ↔ ∃𝑛 ∈ ((Base‘𝑅) ↑m 𝐼)((𝑅 unitVec 𝐼)‘𝑖) = (𝑗 ∈ 𝐼 ↦ (𝑅 Σg (𝑘 ∈ 𝐼 ↦ ((𝑛‘𝑘)(.r‘𝑅)((curry 𝑀‘𝑘)‘𝑗)))))))
172	171	ad2antrr 726	. . . . . . . . 9 ⊢ (((((𝑅 ∈ DivRing ∧ 𝐼 ∈ Fin) ∧ curry 𝑀:𝐼⟶((Base‘𝑅) ↑m 𝐼)) ∧ curry 𝑀 LIndF (𝑅 freeLMod 𝐼)) ∧ 𝑖 ∈ 𝐼) → (((𝑅 unitVec 𝐼)‘𝑖) ∈ ((LSpan‘(𝑅 freeLMod 𝐼))‘(curry 𝑀 “ 𝐼)) ↔ ∃𝑛 ∈ ((Base‘𝑅) ↑m 𝐼)((𝑅 unitVec 𝐼)‘𝑖) = (𝑗 ∈ 𝐼 ↦ (𝑅 Σg (𝑘 ∈ 𝐼 ↦ ((𝑛‘𝑘)(.r‘𝑅)((curry 𝑀‘𝑘)‘𝑗)))))))
173	91, 172	mpbid 232	. . . . . . . 8 ⊢ (((((𝑅 ∈ DivRing ∧ 𝐼 ∈ Fin) ∧ curry 𝑀:𝐼⟶((Base‘𝑅) ↑m 𝐼)) ∧ curry 𝑀 LIndF (𝑅 freeLMod 𝐼)) ∧ 𝑖 ∈ 𝐼) → ∃𝑛 ∈ ((Base‘𝑅) ↑m 𝐼)((𝑅 unitVec 𝐼)‘𝑖) = (𝑗 ∈ 𝐼 ↦ (𝑅 Σg (𝑘 ∈ 𝐼 ↦ ((𝑛‘𝑘)(.r‘𝑅)((curry 𝑀‘𝑘)‘𝑗))))))
174	173	ralrimiva 3145	. . . . . . 7 ⊢ ((((𝑅 ∈ DivRing ∧ 𝐼 ∈ Fin) ∧ curry 𝑀:𝐼⟶((Base‘𝑅) ↑m 𝐼)) ∧ curry 𝑀 LIndF (𝑅 freeLMod 𝐼)) → ∀𝑖 ∈ 𝐼 ∃𝑛 ∈ ((Base‘𝑅) ↑m 𝐼)((𝑅 unitVec 𝐼)‘𝑖) = (𝑗 ∈ 𝐼 ↦ (𝑅 Σg (𝑘 ∈ 𝐼 ↦ ((𝑛‘𝑘)(.r‘𝑅)((curry 𝑀‘𝑘)‘𝑗))))))
175	42, 174	sylan 580	. . . . . 6 ⊢ ((((𝑅 ∈ DivRing ∧ 𝑀 ∈ (Base‘(𝐼 Mat 𝑅))) ∧ 𝐼 ≠ ∅) ∧ curry 𝑀 LIndF (𝑅 freeLMod 𝐼)) → ∀𝑖 ∈ 𝐼 ∃𝑛 ∈ ((Base‘𝑅) ↑m 𝐼)((𝑅 unitVec 𝐼)‘𝑖) = (𝑗 ∈ 𝐼 ↦ (𝑅 Σg (𝑘 ∈ 𝐼 ↦ ((𝑛‘𝑘)(.r‘𝑅)((curry 𝑀‘𝑘)‘𝑗))))))
176	10, 21	mpbird 257	. . . . . . . . 9 ⊢ ((𝑅 ∈ DivRing ∧ 𝑀 ∈ (Base‘(𝐼 Mat 𝑅))) → 𝑀 ∈ ((Base‘𝑅) ↑m (𝐼 × 𝐼)))
177		elmapfn 8906	. . . . . . . . 9 ⊢ (𝑀 ∈ ((Base‘𝑅) ↑m (𝐼 × 𝐼)) → 𝑀 Fn (𝐼 × 𝐼))
178	176, 177	syl 17	. . . . . . . 8 ⊢ ((𝑅 ∈ DivRing ∧ 𝑀 ∈ (Base‘(𝐼 Mat 𝑅))) → 𝑀 Fn (𝐼 × 𝐼))
179	4	adantl 481	. . . . . . . 8 ⊢ ((𝑅 ∈ DivRing ∧ 𝑀 ∈ (Base‘(𝐼 Mat 𝑅))) → 𝐼 ∈ Fin)
180		an32 646	. . . . . . . . . . . . . . . . . . 19 ⊢ (((𝑀 Fn (𝐼 × 𝐼) ∧ 𝑗 ∈ 𝐼) ∧ 𝑘 ∈ 𝐼) ↔ ((𝑀 Fn (𝐼 × 𝐼) ∧ 𝑘 ∈ 𝐼) ∧ 𝑗 ∈ 𝐼))
181		df-3an 1088	. . . . . . . . . . . . . . . . . . 19 ⊢ ((𝑀 Fn (𝐼 × 𝐼) ∧ 𝑘 ∈ 𝐼 ∧ 𝑗 ∈ 𝐼) ↔ ((𝑀 Fn (𝐼 × 𝐼) ∧ 𝑘 ∈ 𝐼) ∧ 𝑗 ∈ 𝐼))
182	180, 181	bitr4i 278	. . . . . . . . . . . . . . . . . 18 ⊢ (((𝑀 Fn (𝐼 × 𝐼) ∧ 𝑗 ∈ 𝐼) ∧ 𝑘 ∈ 𝐼) ↔ (𝑀 Fn (𝐼 × 𝐼) ∧ 𝑘 ∈ 𝐼 ∧ 𝑗 ∈ 𝐼))
183		curfv 37608	. . . . . . . . . . . . . . . . . 18 ⊢ (((𝑀 Fn (𝐼 × 𝐼) ∧ 𝑘 ∈ 𝐼 ∧ 𝑗 ∈ 𝐼) ∧ 𝐼 ∈ Fin) → ((curry 𝑀‘𝑘)‘𝑗) = (𝑘𝑀𝑗))
184	182, 183	sylanb 581	. . . . . . . . . . . . . . . . 17 ⊢ ((((𝑀 Fn (𝐼 × 𝐼) ∧ 𝑗 ∈ 𝐼) ∧ 𝑘 ∈ 𝐼) ∧ 𝐼 ∈ Fin) → ((curry 𝑀‘𝑘)‘𝑗) = (𝑘𝑀𝑗))
185	184	an32s 652	. . . . . . . . . . . . . . . 16 ⊢ ((((𝑀 Fn (𝐼 × 𝐼) ∧ 𝑗 ∈ 𝐼) ∧ 𝐼 ∈ Fin) ∧ 𝑘 ∈ 𝐼) → ((curry 𝑀‘𝑘)‘𝑗) = (𝑘𝑀𝑗))
186	185	oveq2d 7448	. . . . . . . . . . . . . . 15 ⊢ ((((𝑀 Fn (𝐼 × 𝐼) ∧ 𝑗 ∈ 𝐼) ∧ 𝐼 ∈ Fin) ∧ 𝑘 ∈ 𝐼) → ((𝑛‘𝑘)(.r‘𝑅)((curry 𝑀‘𝑘)‘𝑗)) = ((𝑛‘𝑘)(.r‘𝑅)(𝑘𝑀𝑗)))
187	186	mpteq2dva 5241	. . . . . . . . . . . . . 14 ⊢ (((𝑀 Fn (𝐼 × 𝐼) ∧ 𝑗 ∈ 𝐼) ∧ 𝐼 ∈ Fin) → (𝑘 ∈ 𝐼 ↦ ((𝑛‘𝑘)(.r‘𝑅)((curry 𝑀‘𝑘)‘𝑗))) = (𝑘 ∈ 𝐼 ↦ ((𝑛‘𝑘)(.r‘𝑅)(𝑘𝑀𝑗))))
188	187	an32s 652	. . . . . . . . . . . . 13 ⊢ (((𝑀 Fn (𝐼 × 𝐼) ∧ 𝐼 ∈ Fin) ∧ 𝑗 ∈ 𝐼) → (𝑘 ∈ 𝐼 ↦ ((𝑛‘𝑘)(.r‘𝑅)((curry 𝑀‘𝑘)‘𝑗))) = (𝑘 ∈ 𝐼 ↦ ((𝑛‘𝑘)(.r‘𝑅)(𝑘𝑀𝑗))))
189	188	oveq2d 7448	. . . . . . . . . . . 12 ⊢ (((𝑀 Fn (𝐼 × 𝐼) ∧ 𝐼 ∈ Fin) ∧ 𝑗 ∈ 𝐼) → (𝑅 Σg (𝑘 ∈ 𝐼 ↦ ((𝑛‘𝑘)(.r‘𝑅)((curry 𝑀‘𝑘)‘𝑗)))) = (𝑅 Σg (𝑘 ∈ 𝐼 ↦ ((𝑛‘𝑘)(.r‘𝑅)(𝑘𝑀𝑗)))))
190	189	mpteq2dva 5241	. . . . . . . . . . 11 ⊢ ((𝑀 Fn (𝐼 × 𝐼) ∧ 𝐼 ∈ Fin) → (𝑗 ∈ 𝐼 ↦ (𝑅 Σg (𝑘 ∈ 𝐼 ↦ ((𝑛‘𝑘)(.r‘𝑅)((curry 𝑀‘𝑘)‘𝑗))))) = (𝑗 ∈ 𝐼 ↦ (𝑅 Σg (𝑘 ∈ 𝐼 ↦ ((𝑛‘𝑘)(.r‘𝑅)(𝑘𝑀𝑗))))))
191	190	eqeq2d 2747	. . . . . . . . . 10 ⊢ ((𝑀 Fn (𝐼 × 𝐼) ∧ 𝐼 ∈ Fin) → (((𝑅 unitVec 𝐼)‘𝑖) = (𝑗 ∈ 𝐼 ↦ (𝑅 Σg (𝑘 ∈ 𝐼 ↦ ((𝑛‘𝑘)(.r‘𝑅)((curry 𝑀‘𝑘)‘𝑗))))) ↔ ((𝑅 unitVec 𝐼)‘𝑖) = (𝑗 ∈ 𝐼 ↦ (𝑅 Σg (𝑘 ∈ 𝐼 ↦ ((𝑛‘𝑘)(.r‘𝑅)(𝑘𝑀𝑗)))))))
192	191	rexbidv 3178	. . . . . . . . 9 ⊢ ((𝑀 Fn (𝐼 × 𝐼) ∧ 𝐼 ∈ Fin) → (∃𝑛 ∈ ((Base‘𝑅) ↑m 𝐼)((𝑅 unitVec 𝐼)‘𝑖) = (𝑗 ∈ 𝐼 ↦ (𝑅 Σg (𝑘 ∈ 𝐼 ↦ ((𝑛‘𝑘)(.r‘𝑅)((curry 𝑀‘𝑘)‘𝑗))))) ↔ ∃𝑛 ∈ ((Base‘𝑅) ↑m 𝐼)((𝑅 unitVec 𝐼)‘𝑖) = (𝑗 ∈ 𝐼 ↦ (𝑅 Σg (𝑘 ∈ 𝐼 ↦ ((𝑛‘𝑘)(.r‘𝑅)(𝑘𝑀𝑗)))))))
193	192	ralbidv 3177	. . . . . . . 8 ⊢ ((𝑀 Fn (𝐼 × 𝐼) ∧ 𝐼 ∈ Fin) → (∀𝑖 ∈ 𝐼 ∃𝑛 ∈ ((Base‘𝑅) ↑m 𝐼)((𝑅 unitVec 𝐼)‘𝑖) = (𝑗 ∈ 𝐼 ↦ (𝑅 Σg (𝑘 ∈ 𝐼 ↦ ((𝑛‘𝑘)(.r‘𝑅)((curry 𝑀‘𝑘)‘𝑗))))) ↔ ∀𝑖 ∈ 𝐼 ∃𝑛 ∈ ((Base‘𝑅) ↑m 𝐼)((𝑅 unitVec 𝐼)‘𝑖) = (𝑗 ∈ 𝐼 ↦ (𝑅 Σg (𝑘 ∈ 𝐼 ↦ ((𝑛‘𝑘)(.r‘𝑅)(𝑘𝑀𝑗)))))))
194	178, 179, 193	syl2anc 584	. . . . . . 7 ⊢ ((𝑅 ∈ DivRing ∧ 𝑀 ∈ (Base‘(𝐼 Mat 𝑅))) → (∀𝑖 ∈ 𝐼 ∃𝑛 ∈ ((Base‘𝑅) ↑m 𝐼)((𝑅 unitVec 𝐼)‘𝑖) = (𝑗 ∈ 𝐼 ↦ (𝑅 Σg (𝑘 ∈ 𝐼 ↦ ((𝑛‘𝑘)(.r‘𝑅)((curry 𝑀‘𝑘)‘𝑗))))) ↔ ∀𝑖 ∈ 𝐼 ∃𝑛 ∈ ((Base‘𝑅) ↑m 𝐼)((𝑅 unitVec 𝐼)‘𝑖) = (𝑗 ∈ 𝐼 ↦ (𝑅 Σg (𝑘 ∈ 𝐼 ↦ ((𝑛‘𝑘)(.r‘𝑅)(𝑘𝑀𝑗)))))))
195	194	ad2antrr 726	. . . . . 6 ⊢ ((((𝑅 ∈ DivRing ∧ 𝑀 ∈ (Base‘(𝐼 Mat 𝑅))) ∧ 𝐼 ≠ ∅) ∧ curry 𝑀 LIndF (𝑅 freeLMod 𝐼)) → (∀𝑖 ∈ 𝐼 ∃𝑛 ∈ ((Base‘𝑅) ↑m 𝐼)((𝑅 unitVec 𝐼)‘𝑖) = (𝑗 ∈ 𝐼 ↦ (𝑅 Σg (𝑘 ∈ 𝐼 ↦ ((𝑛‘𝑘)(.r‘𝑅)((curry 𝑀‘𝑘)‘𝑗))))) ↔ ∀𝑖 ∈ 𝐼 ∃𝑛 ∈ ((Base‘𝑅) ↑m 𝐼)((𝑅 unitVec 𝐼)‘𝑖) = (𝑗 ∈ 𝐼 ↦ (𝑅 Σg (𝑘 ∈ 𝐼 ↦ ((𝑛‘𝑘)(.r‘𝑅)(𝑘𝑀𝑗)))))))
196	175, 195	mpbid 232	. . . . 5 ⊢ ((((𝑅 ∈ DivRing ∧ 𝑀 ∈ (Base‘(𝐼 Mat 𝑅))) ∧ 𝐼 ≠ ∅) ∧ curry 𝑀 LIndF (𝑅 freeLMod 𝐼)) → ∀𝑖 ∈ 𝐼 ∃𝑛 ∈ ((Base‘𝑅) ↑m 𝐼)((𝑅 unitVec 𝐼)‘𝑖) = (𝑗 ∈ 𝐼 ↦ (𝑅 Σg (𝑘 ∈ 𝐼 ↦ ((𝑛‘𝑘)(.r‘𝑅)(𝑘𝑀𝑗))))))
197	8, 196	sylanl1 680	. . . 4 ⊢ ((((𝑅 ∈ Field ∧ 𝑀 ∈ (Base‘(𝐼 Mat 𝑅))) ∧ 𝐼 ≠ ∅) ∧ curry 𝑀 LIndF (𝑅 freeLMod 𝐼)) → ∀𝑖 ∈ 𝐼 ∃𝑛 ∈ ((Base‘𝑅) ↑m 𝐼)((𝑅 unitVec 𝐼)‘𝑖) = (𝑗 ∈ 𝐼 ↦ (𝑅 Σg (𝑘 ∈ 𝐼 ↦ ((𝑛‘𝑘)(.r‘𝑅)(𝑘𝑀𝑗))))))
198		fveq1 6904	. . . . . . . . . . 11 ⊢ (𝑛 = (𝑓‘𝑖) → (𝑛‘𝑘) = ((𝑓‘𝑖)‘𝑘))
199		uncov 37609	. . . . . . . . . . . 12 ⊢ ((𝑖 ∈ V ∧ 𝑘 ∈ V) → (𝑖uncurry 𝑓𝑘) = ((𝑓‘𝑖)‘𝑘))
200	199	el2v 3486	. . . . . . . . . . 11 ⊢ (𝑖uncurry 𝑓𝑘) = ((𝑓‘𝑖)‘𝑘)
201	198, 200	eqtr4di 2794	. . . . . . . . . 10 ⊢ (𝑛 = (𝑓‘𝑖) → (𝑛‘𝑘) = (𝑖uncurry 𝑓𝑘))
202	201	oveq1d 7447	. . . . . . . . 9 ⊢ (𝑛 = (𝑓‘𝑖) → ((𝑛‘𝑘)(.r‘𝑅)(𝑘𝑀𝑗)) = ((𝑖uncurry 𝑓𝑘)(.r‘𝑅)(𝑘𝑀𝑗)))
203	202	mpteq2dv 5243	. . . . . . . 8 ⊢ (𝑛 = (𝑓‘𝑖) → (𝑘 ∈ 𝐼 ↦ ((𝑛‘𝑘)(.r‘𝑅)(𝑘𝑀𝑗))) = (𝑘 ∈ 𝐼 ↦ ((𝑖uncurry 𝑓𝑘)(.r‘𝑅)(𝑘𝑀𝑗))))
204	203	oveq2d 7448	. . . . . . 7 ⊢ (𝑛 = (𝑓‘𝑖) → (𝑅 Σg (𝑘 ∈ 𝐼 ↦ ((𝑛‘𝑘)(.r‘𝑅)(𝑘𝑀𝑗)))) = (𝑅 Σg (𝑘 ∈ 𝐼 ↦ ((𝑖uncurry 𝑓𝑘)(.r‘𝑅)(𝑘𝑀𝑗)))))
205	204	mpteq2dv 5243	. . . . . 6 ⊢ (𝑛 = (𝑓‘𝑖) → (𝑗 ∈ 𝐼 ↦ (𝑅 Σg (𝑘 ∈ 𝐼 ↦ ((𝑛‘𝑘)(.r‘𝑅)(𝑘𝑀𝑗))))) = (𝑗 ∈ 𝐼 ↦ (𝑅 Σg (𝑘 ∈ 𝐼 ↦ ((𝑖uncurry 𝑓𝑘)(.r‘𝑅)(𝑘𝑀𝑗))))))
206	205	eqeq2d 2747	. . . . 5 ⊢ (𝑛 = (𝑓‘𝑖) → (((𝑅 unitVec 𝐼)‘𝑖) = (𝑗 ∈ 𝐼 ↦ (𝑅 Σg (𝑘 ∈ 𝐼 ↦ ((𝑛‘𝑘)(.r‘𝑅)(𝑘𝑀𝑗))))) ↔ ((𝑅 unitVec 𝐼)‘𝑖) = (𝑗 ∈ 𝐼 ↦ (𝑅 Σg (𝑘 ∈ 𝐼 ↦ ((𝑖uncurry 𝑓𝑘)(.r‘𝑅)(𝑘𝑀𝑗)))))))
207	206	ac6sfi 9321	. . . 4 ⊢ ((𝐼 ∈ Fin ∧ ∀𝑖 ∈ 𝐼 ∃𝑛 ∈ ((Base‘𝑅) ↑m 𝐼)((𝑅 unitVec 𝐼)‘𝑖) = (𝑗 ∈ 𝐼 ↦ (𝑅 Σg (𝑘 ∈ 𝐼 ↦ ((𝑛‘𝑘)(.r‘𝑅)(𝑘𝑀𝑗)))))) → ∃𝑓(𝑓:𝐼⟶((Base‘𝑅) ↑m 𝐼) ∧ ∀𝑖 ∈ 𝐼 ((𝑅 unitVec 𝐼)‘𝑖) = (𝑗 ∈ 𝐼 ↦ (𝑅 Σg (𝑘 ∈ 𝐼 ↦ ((𝑖uncurry 𝑓𝑘)(.r‘𝑅)(𝑘𝑀𝑗)))))))
208	5, 197, 207	syl2anc 584	. . 3 ⊢ ((((𝑅 ∈ Field ∧ 𝑀 ∈ (Base‘(𝐼 Mat 𝑅))) ∧ 𝐼 ≠ ∅) ∧ curry 𝑀 LIndF (𝑅 freeLMod 𝐼)) → ∃𝑓(𝑓:𝐼⟶((Base‘𝑅) ↑m 𝐼) ∧ ∀𝑖 ∈ 𝐼 ((𝑅 unitVec 𝐼)‘𝑖) = (𝑗 ∈ 𝐼 ↦ (𝑅 Σg (𝑘 ∈ 𝐼 ↦ ((𝑖uncurry 𝑓𝑘)(.r‘𝑅)(𝑘𝑀𝑗)))))))
209		uncf 37607	. . . . . . 7 ⊢ (𝑓:𝐼⟶((Base‘𝑅) ↑m 𝐼) → uncurry 𝑓:(𝐼 × 𝐼)⟶(Base‘𝑅))
210	13, 14	frlmfibas 21783	. . . . . . . . . . . . . . . 16 ⊢ ((𝑅 ∈ Field ∧ (𝐼 × 𝐼) ∈ Fin) → ((Base‘𝑅) ↑m (𝐼 × 𝐼)) = (Base‘(𝑅 freeLMod (𝐼 × 𝐼))))
211	12, 210	sylan2 593	. . . . . . . . . . . . . . 15 ⊢ ((𝑅 ∈ Field ∧ 𝐼 ∈ Fin) → ((Base‘𝑅) ↑m (𝐼 × 𝐼)) = (Base‘(𝑅 freeLMod (𝐼 × 𝐼))))
212	1, 13	matbas 22418	. . . . . . . . . . . . . . . 16 ⊢ ((𝐼 ∈ Fin ∧ 𝑅 ∈ Field) → (Base‘(𝑅 freeLMod (𝐼 × 𝐼))) = (Base‘(𝐼 Mat 𝑅)))
213	212	ancoms 458	. . . . . . . . . . . . . . 15 ⊢ ((𝑅 ∈ Field ∧ 𝐼 ∈ Fin) → (Base‘(𝑅 freeLMod (𝐼 × 𝐼))) = (Base‘(𝐼 Mat 𝑅)))
214	211, 213	eqtrd 2776	. . . . . . . . . . . . . 14 ⊢ ((𝑅 ∈ Field ∧ 𝐼 ∈ Fin) → ((Base‘𝑅) ↑m (𝐼 × 𝐼)) = (Base‘(𝐼 Mat 𝑅)))
215	4, 214	sylan2 593	. . . . . . . . . . . . 13 ⊢ ((𝑅 ∈ Field ∧ 𝑀 ∈ (Base‘(𝐼 Mat 𝑅))) → ((Base‘𝑅) ↑m (𝐼 × 𝐼)) = (Base‘(𝐼 Mat 𝑅)))
216	215	eleq2d 2826	. . . . . . . . . . . 12 ⊢ ((𝑅 ∈ Field ∧ 𝑀 ∈ (Base‘(𝐼 Mat 𝑅))) → (uncurry 𝑓 ∈ ((Base‘𝑅) ↑m (𝐼 × 𝐼)) ↔ uncurry 𝑓 ∈ (Base‘(𝐼 Mat 𝑅))))
217		elmapg 8880	. . . . . . . . . . . . . 14 ⊢ (((Base‘𝑅) ∈ V ∧ (𝐼 × 𝐼) ∈ Fin) → (uncurry 𝑓 ∈ ((Base‘𝑅) ↑m (𝐼 × 𝐼)) ↔ uncurry 𝑓:(𝐼 × 𝐼)⟶(Base‘𝑅)))
218	22, 23, 217	sylancr 587	. . . . . . . . . . . . 13 ⊢ (𝑀 ∈ (Base‘(𝐼 Mat 𝑅)) → (uncurry 𝑓 ∈ ((Base‘𝑅) ↑m (𝐼 × 𝐼)) ↔ uncurry 𝑓:(𝐼 × 𝐼)⟶(Base‘𝑅)))
219	218	adantl 481	. . . . . . . . . . . 12 ⊢ ((𝑅 ∈ Field ∧ 𝑀 ∈ (Base‘(𝐼 Mat 𝑅))) → (uncurry 𝑓 ∈ ((Base‘𝑅) ↑m (𝐼 × 𝐼)) ↔ uncurry 𝑓:(𝐼 × 𝐼)⟶(Base‘𝑅)))
220	216, 219	bitr3d 281	. . . . . . . . . . 11 ⊢ ((𝑅 ∈ Field ∧ 𝑀 ∈ (Base‘(𝐼 Mat 𝑅))) → (uncurry 𝑓 ∈ (Base‘(𝐼 Mat 𝑅)) ↔ uncurry 𝑓:(𝐼 × 𝐼)⟶(Base‘𝑅)))
221	220	biimpar 477	. . . . . . . . . 10 ⊢ (((𝑅 ∈ Field ∧ 𝑀 ∈ (Base‘(𝐼 Mat 𝑅))) ∧ uncurry 𝑓:(𝐼 × 𝐼)⟶(Base‘𝑅)) → uncurry 𝑓 ∈ (Base‘(𝐼 Mat 𝑅)))
222	221	adantr 480	. . . . . . . . 9 ⊢ ((((𝑅 ∈ Field ∧ 𝑀 ∈ (Base‘(𝐼 Mat 𝑅))) ∧ uncurry 𝑓:(𝐼 × 𝐼)⟶(Base‘𝑅)) ∧ ∀𝑖 ∈ 𝐼 ((𝑅 unitVec 𝐼)‘𝑖) = (𝑗 ∈ 𝐼 ↦ (𝑅 Σg (𝑘 ∈ 𝐼 ↦ ((𝑖uncurry 𝑓𝑘)(.r‘𝑅)(𝑘𝑀𝑗)))))) → uncurry 𝑓 ∈ (Base‘(𝐼 Mat 𝑅)))
223		nfv 1913	. . . . . . . . . . . . . 14 ⊢ Ⅎ𝑗(((𝑅 ∈ Field ∧ 𝑀 ∈ (Base‘(𝐼 Mat 𝑅))) ∧ uncurry 𝑓:(𝐼 × 𝐼)⟶(Base‘𝑅)) ∧ 𝑖 ∈ 𝐼)
224		nfmpt1 5249	. . . . . . . . . . . . . . 15 ⊢ Ⅎ𝑗(𝑗 ∈ 𝐼 ↦ (𝑅 Σg (𝑘 ∈ 𝐼 ↦ ((𝑖uncurry 𝑓𝑘)(.r‘𝑅)(𝑘𝑀𝑗)))))
225	224	nfeq2 2922	. . . . . . . . . . . . . 14 ⊢ Ⅎ𝑗((𝑅 unitVec 𝐼)‘𝑖) = (𝑗 ∈ 𝐼 ↦ (𝑅 Σg (𝑘 ∈ 𝐼 ↦ ((𝑖uncurry 𝑓𝑘)(.r‘𝑅)(𝑘𝑀𝑗)))))
226		fveq1 6904	. . . . . . . . . . . . . . . . 17 ⊢ (((𝑅 unitVec 𝐼)‘𝑖) = (𝑗 ∈ 𝐼 ↦ (𝑅 Σg (𝑘 ∈ 𝐼 ↦ ((𝑖uncurry 𝑓𝑘)(.r‘𝑅)(𝑘𝑀𝑗))))) → (((𝑅 unitVec 𝐼)‘𝑖)‘𝑗) = ((𝑗 ∈ 𝐼 ↦ (𝑅 Σg (𝑘 ∈ 𝐼 ↦ ((𝑖uncurry 𝑓𝑘)(.r‘𝑅)(𝑘𝑀𝑗)))))‘𝑗))
227	7, 43	syl 17	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (𝑅 ∈ Field → 𝑅 ∈ Ring)
228	227, 4	anim12i 613	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((𝑅 ∈ Field ∧ 𝑀 ∈ (Base‘(𝐼 Mat 𝑅))) → (𝑅 ∈ Ring ∧ 𝐼 ∈ Fin))
229	228	adantr 480	. . . . . . . . . . . . . . . . . . 19 ⊢ (((𝑅 ∈ Field ∧ 𝑀 ∈ (Base‘(𝐼 Mat 𝑅))) ∧ uncurry 𝑓:(𝐼 × 𝐼)⟶(Base‘𝑅)) → (𝑅 ∈ Ring ∧ 𝐼 ∈ Fin))
230		equcom 2016	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (𝑖 = 𝑗 ↔ 𝑗 = 𝑖)
231		ifbi 4547	. . . . . . . . . . . . . . . . . . . . 21 ⊢ ((𝑖 = 𝑗 ↔ 𝑗 = 𝑖) → if(𝑖 = 𝑗, (1r‘𝑅), (0g‘𝑅)) = if(𝑗 = 𝑖, (1r‘𝑅), (0g‘𝑅)))
232	230, 231	ax-mp 5	. . . . . . . . . . . . . . . . . . . 20 ⊢ if(𝑖 = 𝑗, (1r‘𝑅), (0g‘𝑅)) = if(𝑗 = 𝑖, (1r‘𝑅), (0g‘𝑅))
233		eqid 2736	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (1r‘𝑅) = (1r‘𝑅)
234		eqid 2736	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (0g‘𝑅) = (0g‘𝑅)
235		simpllr 775	. . . . . . . . . . . . . . . . . . . . 21 ⊢ ((((𝑅 ∈ Ring ∧ 𝐼 ∈ Fin) ∧ 𝑖 ∈ 𝐼) ∧ 𝑗 ∈ 𝐼) → 𝐼 ∈ Fin)
236		simplll 774	. . . . . . . . . . . . . . . . . . . . 21 ⊢ ((((𝑅 ∈ Ring ∧ 𝐼 ∈ Fin) ∧ 𝑖 ∈ 𝐼) ∧ 𝑗 ∈ 𝐼) → 𝑅 ∈ Ring)
237		simplr 768	. . . . . . . . . . . . . . . . . . . . 21 ⊢ ((((𝑅 ∈ Ring ∧ 𝐼 ∈ Fin) ∧ 𝑖 ∈ 𝐼) ∧ 𝑗 ∈ 𝐼) → 𝑖 ∈ 𝐼)
238		simpr 484	. . . . . . . . . . . . . . . . . . . . 21 ⊢ ((((𝑅 ∈ Ring ∧ 𝐼 ∈ Fin) ∧ 𝑖 ∈ 𝐼) ∧ 𝑗 ∈ 𝐼) → 𝑗 ∈ 𝐼)
239		eqid 2736	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (1r‘(𝐼 Mat 𝑅)) = (1r‘(𝐼 Mat 𝑅))
240	1, 233, 234, 235, 236, 237, 238, 239	mat1ov 22455	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((((𝑅 ∈ Ring ∧ 𝐼 ∈ Fin) ∧ 𝑖 ∈ 𝐼) ∧ 𝑗 ∈ 𝐼) → (𝑖(1r‘(𝐼 Mat 𝑅))𝑗) = if(𝑖 = 𝑗, (1r‘𝑅), (0g‘𝑅)))
241		df-3an 1088	. . . . . . . . . . . . . . . . . . . . 21 ⊢ ((𝑅 ∈ Ring ∧ 𝐼 ∈ Fin ∧ 𝑖 ∈ 𝐼) ↔ ((𝑅 ∈ Ring ∧ 𝐼 ∈ Fin) ∧ 𝑖 ∈ 𝐼))
242	44, 233, 234	uvcvval 21807	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (((𝑅 ∈ Ring ∧ 𝐼 ∈ Fin ∧ 𝑖 ∈ 𝐼) ∧ 𝑗 ∈ 𝐼) → (((𝑅 unitVec 𝐼)‘𝑖)‘𝑗) = if(𝑗 = 𝑖, (1r‘𝑅), (0g‘𝑅)))
243	241, 242	sylanbr 582	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((((𝑅 ∈ Ring ∧ 𝐼 ∈ Fin) ∧ 𝑖 ∈ 𝐼) ∧ 𝑗 ∈ 𝐼) → (((𝑅 unitVec 𝐼)‘𝑖)‘𝑗) = if(𝑗 = 𝑖, (1r‘𝑅), (0g‘𝑅)))
244	232, 240, 243	3eqtr4a 2802	. . . . . . . . . . . . . . . . . . 19 ⊢ ((((𝑅 ∈ Ring ∧ 𝐼 ∈ Fin) ∧ 𝑖 ∈ 𝐼) ∧ 𝑗 ∈ 𝐼) → (𝑖(1r‘(𝐼 Mat 𝑅))𝑗) = (((𝑅 unitVec 𝐼)‘𝑖)‘𝑗))
245	229, 244	sylanl1 680	. . . . . . . . . . . . . . . . . 18 ⊢ (((((𝑅 ∈ Field ∧ 𝑀 ∈ (Base‘(𝐼 Mat 𝑅))) ∧ uncurry 𝑓:(𝐼 × 𝐼)⟶(Base‘𝑅)) ∧ 𝑖 ∈ 𝐼) ∧ 𝑗 ∈ 𝐼) → (𝑖(1r‘(𝐼 Mat 𝑅))𝑗) = (((𝑅 unitVec 𝐼)‘𝑖)‘𝑗))
246		ovex 7465	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (𝑅 Σg (𝑘 ∈ 𝐼 ↦ ((𝑖uncurry 𝑓𝑘)(.r‘𝑅)(𝑘𝑀𝑗)))) ∈ V
247		eqid 2736	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (𝑗 ∈ 𝐼 ↦ (𝑅 Σg (𝑘 ∈ 𝐼 ↦ ((𝑖uncurry 𝑓𝑘)(.r‘𝑅)(𝑘𝑀𝑗))))) = (𝑗 ∈ 𝐼 ↦ (𝑅 Σg (𝑘 ∈ 𝐼 ↦ ((𝑖uncurry 𝑓𝑘)(.r‘𝑅)(𝑘𝑀𝑗)))))
248	247	fvmpt2 7026	. . . . . . . . . . . . . . . . . . . . 21 ⊢ ((𝑗 ∈ 𝐼 ∧ (𝑅 Σg (𝑘 ∈ 𝐼 ↦ ((𝑖uncurry 𝑓𝑘)(.r‘𝑅)(𝑘𝑀𝑗)))) ∈ V) → ((𝑗 ∈ 𝐼 ↦ (𝑅 Σg (𝑘 ∈ 𝐼 ↦ ((𝑖uncurry 𝑓𝑘)(.r‘𝑅)(𝑘𝑀𝑗)))))‘𝑗) = (𝑅 Σg (𝑘 ∈ 𝐼 ↦ ((𝑖uncurry 𝑓𝑘)(.r‘𝑅)(𝑘𝑀𝑗)))))
249	246, 248	mpan2 691	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝑗 ∈ 𝐼 → ((𝑗 ∈ 𝐼 ↦ (𝑅 Σg (𝑘 ∈ 𝐼 ↦ ((𝑖uncurry 𝑓𝑘)(.r‘𝑅)(𝑘𝑀𝑗)))))‘𝑗) = (𝑅 Σg (𝑘 ∈ 𝐼 ↦ ((𝑖uncurry 𝑓𝑘)(.r‘𝑅)(𝑘𝑀𝑗)))))
250	249	adantl 481	. . . . . . . . . . . . . . . . . . 19 ⊢ (((((𝑅 ∈ Field ∧ 𝑀 ∈ (Base‘(𝐼 Mat 𝑅))) ∧ uncurry 𝑓:(𝐼 × 𝐼)⟶(Base‘𝑅)) ∧ 𝑖 ∈ 𝐼) ∧ 𝑗 ∈ 𝐼) → ((𝑗 ∈ 𝐼 ↦ (𝑅 Σg (𝑘 ∈ 𝐼 ↦ ((𝑖uncurry 𝑓𝑘)(.r‘𝑅)(𝑘𝑀𝑗)))))‘𝑗) = (𝑅 Σg (𝑘 ∈ 𝐼 ↦ ((𝑖uncurry 𝑓𝑘)(.r‘𝑅)(𝑘𝑀𝑗)))))
251		eqid 2736	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝑅 maMul ⟨𝐼, 𝐼, 𝐼⟩) = (𝑅 maMul ⟨𝐼, 𝐼, 𝐼⟩)
252		simp-4l 782	. . . . . . . . . . . . . . . . . . . 20 ⊢ (((((𝑅 ∈ Field ∧ 𝑀 ∈ (Base‘(𝐼 Mat 𝑅))) ∧ uncurry 𝑓:(𝐼 × 𝐼)⟶(Base‘𝑅)) ∧ 𝑖 ∈ 𝐼) ∧ 𝑗 ∈ 𝐼) → 𝑅 ∈ Field)
253	4	ad4antlr 733	. . . . . . . . . . . . . . . . . . . 20 ⊢ (((((𝑅 ∈ Field ∧ 𝑀 ∈ (Base‘(𝐼 Mat 𝑅))) ∧ uncurry 𝑓:(𝐼 × 𝐼)⟶(Base‘𝑅)) ∧ 𝑖 ∈ 𝐼) ∧ 𝑗 ∈ 𝐼) → 𝐼 ∈ Fin)
254	218	biimpar 477	. . . . . . . . . . . . . . . . . . . . 21 ⊢ ((𝑀 ∈ (Base‘(𝐼 Mat 𝑅)) ∧ uncurry 𝑓:(𝐼 × 𝐼)⟶(Base‘𝑅)) → uncurry 𝑓 ∈ ((Base‘𝑅) ↑m (𝐼 × 𝐼)))
255	254	ad5ant23 759	. . . . . . . . . . . . . . . . . . . 20 ⊢ (((((𝑅 ∈ Field ∧ 𝑀 ∈ (Base‘(𝐼 Mat 𝑅))) ∧ uncurry 𝑓:(𝐼 × 𝐼)⟶(Base‘𝑅)) ∧ 𝑖 ∈ 𝐼) ∧ 𝑗 ∈ 𝐼) → uncurry 𝑓 ∈ ((Base‘𝑅) ↑m (𝐼 × 𝐼)))
256		simpr 484	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ ((𝑅 ∈ Field ∧ 𝑀 ∈ (Base‘(𝐼 Mat 𝑅))) → 𝑀 ∈ (Base‘(𝐼 Mat 𝑅)))
257	256, 215	eleqtrrd 2843	. . . . . . . . . . . . . . . . . . . . 21 ⊢ ((𝑅 ∈ Field ∧ 𝑀 ∈ (Base‘(𝐼 Mat 𝑅))) → 𝑀 ∈ ((Base‘𝑅) ↑m (𝐼 × 𝐼)))
258	257	ad3antrrr 730	. . . . . . . . . . . . . . . . . . . 20 ⊢ (((((𝑅 ∈ Field ∧ 𝑀 ∈ (Base‘(𝐼 Mat 𝑅))) ∧ uncurry 𝑓:(𝐼 × 𝐼)⟶(Base‘𝑅)) ∧ 𝑖 ∈ 𝐼) ∧ 𝑗 ∈ 𝐼) → 𝑀 ∈ ((Base‘𝑅) ↑m (𝐼 × 𝐼)))
259		simplr 768	. . . . . . . . . . . . . . . . . . . 20 ⊢ (((((𝑅 ∈ Field ∧ 𝑀 ∈ (Base‘(𝐼 Mat 𝑅))) ∧ uncurry 𝑓:(𝐼 × 𝐼)⟶(Base‘𝑅)) ∧ 𝑖 ∈ 𝐼) ∧ 𝑗 ∈ 𝐼) → 𝑖 ∈ 𝐼)
260		simpr 484	. . . . . . . . . . . . . . . . . . . 20 ⊢ (((((𝑅 ∈ Field ∧ 𝑀 ∈ (Base‘(𝐼 Mat 𝑅))) ∧ uncurry 𝑓:(𝐼 × 𝐼)⟶(Base‘𝑅)) ∧ 𝑖 ∈ 𝐼) ∧ 𝑗 ∈ 𝐼) → 𝑗 ∈ 𝐼)
261	251, 14, 121, 252, 253, 253, 253, 255, 258, 259, 260	mamufv 22399	. . . . . . . . . . . . . . . . . . 19 ⊢ (((((𝑅 ∈ Field ∧ 𝑀 ∈ (Base‘(𝐼 Mat 𝑅))) ∧ uncurry 𝑓:(𝐼 × 𝐼)⟶(Base‘𝑅)) ∧ 𝑖 ∈ 𝐼) ∧ 𝑗 ∈ 𝐼) → (𝑖(uncurry 𝑓(𝑅 maMul ⟨𝐼, 𝐼, 𝐼⟩)𝑀)𝑗) = (𝑅 Σg (𝑘 ∈ 𝐼 ↦ ((𝑖uncurry 𝑓𝑘)(.r‘𝑅)(𝑘𝑀𝑗)))))
262	1, 251	matmulr 22445	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ ((𝐼 ∈ Fin ∧ 𝑅 ∈ Field) → (𝑅 maMul ⟨𝐼, 𝐼, 𝐼⟩) = (.r‘(𝐼 Mat 𝑅)))
263	262	ancoms 458	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ ((𝑅 ∈ Field ∧ 𝐼 ∈ Fin) → (𝑅 maMul ⟨𝐼, 𝐼, 𝐼⟩) = (.r‘(𝐼 Mat 𝑅)))
264	263	oveqd 7449	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ ((𝑅 ∈ Field ∧ 𝐼 ∈ Fin) → (uncurry 𝑓(𝑅 maMul ⟨𝐼, 𝐼, 𝐼⟩)𝑀) = (uncurry 𝑓(.r‘(𝐼 Mat 𝑅))𝑀))
265	264	oveqd 7449	. . . . . . . . . . . . . . . . . . . . 21 ⊢ ((𝑅 ∈ Field ∧ 𝐼 ∈ Fin) → (𝑖(uncurry 𝑓(𝑅 maMul ⟨𝐼, 𝐼, 𝐼⟩)𝑀)𝑗) = (𝑖(uncurry 𝑓(.r‘(𝐼 Mat 𝑅))𝑀)𝑗))
266	4, 265	sylan2 593	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((𝑅 ∈ Field ∧ 𝑀 ∈ (Base‘(𝐼 Mat 𝑅))) → (𝑖(uncurry 𝑓(𝑅 maMul ⟨𝐼, 𝐼, 𝐼⟩)𝑀)𝑗) = (𝑖(uncurry 𝑓(.r‘(𝐼 Mat 𝑅))𝑀)𝑗))
267	266	ad3antrrr 730	. . . . . . . . . . . . . . . . . . 19 ⊢ (((((𝑅 ∈ Field ∧ 𝑀 ∈ (Base‘(𝐼 Mat 𝑅))) ∧ uncurry 𝑓:(𝐼 × 𝐼)⟶(Base‘𝑅)) ∧ 𝑖 ∈ 𝐼) ∧ 𝑗 ∈ 𝐼) → (𝑖(uncurry 𝑓(𝑅 maMul ⟨𝐼, 𝐼, 𝐼⟩)𝑀)𝑗) = (𝑖(uncurry 𝑓(.r‘(𝐼 Mat 𝑅))𝑀)𝑗))
268	250, 261, 267	3eqtr2rd 2783	. . . . . . . . . . . . . . . . . 18 ⊢ (((((𝑅 ∈ Field ∧ 𝑀 ∈ (Base‘(𝐼 Mat 𝑅))) ∧ uncurry 𝑓:(𝐼 × 𝐼)⟶(Base‘𝑅)) ∧ 𝑖 ∈ 𝐼) ∧ 𝑗 ∈ 𝐼) → (𝑖(uncurry 𝑓(.r‘(𝐼 Mat 𝑅))𝑀)𝑗) = ((𝑗 ∈ 𝐼 ↦ (𝑅 Σg (𝑘 ∈ 𝐼 ↦ ((𝑖uncurry 𝑓𝑘)(.r‘𝑅)(𝑘𝑀𝑗)))))‘𝑗))
269	245, 268	eqeq12d 2752	. . . . . . . . . . . . . . . . 17 ⊢ (((((𝑅 ∈ Field ∧ 𝑀 ∈ (Base‘(𝐼 Mat 𝑅))) ∧ uncurry 𝑓:(𝐼 × 𝐼)⟶(Base‘𝑅)) ∧ 𝑖 ∈ 𝐼) ∧ 𝑗 ∈ 𝐼) → ((𝑖(1r‘(𝐼 Mat 𝑅))𝑗) = (𝑖(uncurry 𝑓(.r‘(𝐼 Mat 𝑅))𝑀)𝑗) ↔ (((𝑅 unitVec 𝐼)‘𝑖)‘𝑗) = ((𝑗 ∈ 𝐼 ↦ (𝑅 Σg (𝑘 ∈ 𝐼 ↦ ((𝑖uncurry 𝑓𝑘)(.r‘𝑅)(𝑘𝑀𝑗)))))‘𝑗)))
270	226, 269	imbitrrid 246	. . . . . . . . . . . . . . . 16 ⊢ (((((𝑅 ∈ Field ∧ 𝑀 ∈ (Base‘(𝐼 Mat 𝑅))) ∧ uncurry 𝑓:(𝐼 × 𝐼)⟶(Base‘𝑅)) ∧ 𝑖 ∈ 𝐼) ∧ 𝑗 ∈ 𝐼) → (((𝑅 unitVec 𝐼)‘𝑖) = (𝑗 ∈ 𝐼 ↦ (𝑅 Σg (𝑘 ∈ 𝐼 ↦ ((𝑖uncurry 𝑓𝑘)(.r‘𝑅)(𝑘𝑀𝑗))))) → (𝑖(1r‘(𝐼 Mat 𝑅))𝑗) = (𝑖(uncurry 𝑓(.r‘(𝐼 Mat 𝑅))𝑀)𝑗)))
271	270	ex 412	. . . . . . . . . . . . . . 15 ⊢ ((((𝑅 ∈ Field ∧ 𝑀 ∈ (Base‘(𝐼 Mat 𝑅))) ∧ uncurry 𝑓:(𝐼 × 𝐼)⟶(Base‘𝑅)) ∧ 𝑖 ∈ 𝐼) → (𝑗 ∈ 𝐼 → (((𝑅 unitVec 𝐼)‘𝑖) = (𝑗 ∈ 𝐼 ↦ (𝑅 Σg (𝑘 ∈ 𝐼 ↦ ((𝑖uncurry 𝑓𝑘)(.r‘𝑅)(𝑘𝑀𝑗))))) → (𝑖(1r‘(𝐼 Mat 𝑅))𝑗) = (𝑖(uncurry 𝑓(.r‘(𝐼 Mat 𝑅))𝑀)𝑗))))
272	271	com23 86	. . . . . . . . . . . . . 14 ⊢ ((((𝑅 ∈ Field ∧ 𝑀 ∈ (Base‘(𝐼 Mat 𝑅))) ∧ uncurry 𝑓:(𝐼 × 𝐼)⟶(Base‘𝑅)) ∧ 𝑖 ∈ 𝐼) → (((𝑅 unitVec 𝐼)‘𝑖) = (𝑗 ∈ 𝐼 ↦ (𝑅 Σg (𝑘 ∈ 𝐼 ↦ ((𝑖uncurry 𝑓𝑘)(.r‘𝑅)(𝑘𝑀𝑗))))) → (𝑗 ∈ 𝐼 → (𝑖(1r‘(𝐼 Mat 𝑅))𝑗) = (𝑖(uncurry 𝑓(.r‘(𝐼 Mat 𝑅))𝑀)𝑗))))
273	223, 225, 272	ralrimd 3263	. . . . . . . . . . . . 13 ⊢ ((((𝑅 ∈ Field ∧ 𝑀 ∈ (Base‘(𝐼 Mat 𝑅))) ∧ uncurry 𝑓:(𝐼 × 𝐼)⟶(Base‘𝑅)) ∧ 𝑖 ∈ 𝐼) → (((𝑅 unitVec 𝐼)‘𝑖) = (𝑗 ∈ 𝐼 ↦ (𝑅 Σg (𝑘 ∈ 𝐼 ↦ ((𝑖uncurry 𝑓𝑘)(.r‘𝑅)(𝑘𝑀𝑗))))) → ∀𝑗 ∈ 𝐼 (𝑖(1r‘(𝐼 Mat 𝑅))𝑗) = (𝑖(uncurry 𝑓(.r‘(𝐼 Mat 𝑅))𝑀)𝑗)))
274	273	ralimdva 3166	. . . . . . . . . . . 12 ⊢ (((𝑅 ∈ Field ∧ 𝑀 ∈ (Base‘(𝐼 Mat 𝑅))) ∧ uncurry 𝑓:(𝐼 × 𝐼)⟶(Base‘𝑅)) → (∀𝑖 ∈ 𝐼 ((𝑅 unitVec 𝐼)‘𝑖) = (𝑗 ∈ 𝐼 ↦ (𝑅 Σg (𝑘 ∈ 𝐼 ↦ ((𝑖uncurry 𝑓𝑘)(.r‘𝑅)(𝑘𝑀𝑗))))) → ∀𝑖 ∈ 𝐼 ∀𝑗 ∈ 𝐼 (𝑖(1r‘(𝐼 Mat 𝑅))𝑗) = (𝑖(uncurry 𝑓(.r‘(𝐼 Mat 𝑅))𝑀)𝑗)))
275	1, 2, 239	mat1bas 22456	. . . . . . . . . . . . . . . . 17 ⊢ ((𝑅 ∈ Ring ∧ 𝐼 ∈ Fin) → (1r‘(𝐼 Mat 𝑅)) ∈ (Base‘(𝐼 Mat 𝑅)))
276	13, 14	frlmfibas 21783	. . . . . . . . . . . . . . . . . . 19 ⊢ ((𝑅 ∈ Ring ∧ (𝐼 × 𝐼) ∈ Fin) → ((Base‘𝑅) ↑m (𝐼 × 𝐼)) = (Base‘(𝑅 freeLMod (𝐼 × 𝐼))))
277	12, 276	sylan2 593	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝑅 ∈ Ring ∧ 𝐼 ∈ Fin) → ((Base‘𝑅) ↑m (𝐼 × 𝐼)) = (Base‘(𝑅 freeLMod (𝐼 × 𝐼))))
278	1, 13	matbas 22418	. . . . . . . . . . . . . . . . . . 19 ⊢ ((𝐼 ∈ Fin ∧ 𝑅 ∈ Ring) → (Base‘(𝑅 freeLMod (𝐼 × 𝐼))) = (Base‘(𝐼 Mat 𝑅)))
279	278	ancoms 458	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝑅 ∈ Ring ∧ 𝐼 ∈ Fin) → (Base‘(𝑅 freeLMod (𝐼 × 𝐼))) = (Base‘(𝐼 Mat 𝑅)))
280	277, 279	eqtrd 2776	. . . . . . . . . . . . . . . . 17 ⊢ ((𝑅 ∈ Ring ∧ 𝐼 ∈ Fin) → ((Base‘𝑅) ↑m (𝐼 × 𝐼)) = (Base‘(𝐼 Mat 𝑅)))
281	275, 280	eleqtrrd 2843	. . . . . . . . . . . . . . . 16 ⊢ ((𝑅 ∈ Ring ∧ 𝐼 ∈ Fin) → (1r‘(𝐼 Mat 𝑅)) ∈ ((Base‘𝑅) ↑m (𝐼 × 𝐼)))
282		elmapfn 8906	. . . . . . . . . . . . . . . 16 ⊢ ((1r‘(𝐼 Mat 𝑅)) ∈ ((Base‘𝑅) ↑m (𝐼 × 𝐼)) → (1r‘(𝐼 Mat 𝑅)) Fn (𝐼 × 𝐼))
283	281, 282	syl 17	. . . . . . . . . . . . . . 15 ⊢ ((𝑅 ∈ Ring ∧ 𝐼 ∈ Fin) → (1r‘(𝐼 Mat 𝑅)) Fn (𝐼 × 𝐼))
284	227, 4, 283	syl2an 596	. . . . . . . . . . . . . 14 ⊢ ((𝑅 ∈ Field ∧ 𝑀 ∈ (Base‘(𝐼 Mat 𝑅))) → (1r‘(𝐼 Mat 𝑅)) Fn (𝐼 × 𝐼))
285	284	adantr 480	. . . . . . . . . . . . 13 ⊢ (((𝑅 ∈ Field ∧ 𝑀 ∈ (Base‘(𝐼 Mat 𝑅))) ∧ uncurry 𝑓:(𝐼 × 𝐼)⟶(Base‘𝑅)) → (1r‘(𝐼 Mat 𝑅)) Fn (𝐼 × 𝐼))
286	1	matring 22450	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝐼 ∈ Fin ∧ 𝑅 ∈ Ring) → (𝐼 Mat 𝑅) ∈ Ring)
287	4, 227, 286	syl2anr 597	. . . . . . . . . . . . . . . . 17 ⊢ ((𝑅 ∈ Field ∧ 𝑀 ∈ (Base‘(𝐼 Mat 𝑅))) → (𝐼 Mat 𝑅) ∈ Ring)
288	287	adantr 480	. . . . . . . . . . . . . . . 16 ⊢ (((𝑅 ∈ Field ∧ 𝑀 ∈ (Base‘(𝐼 Mat 𝑅))) ∧ uncurry 𝑓:(𝐼 × 𝐼)⟶(Base‘𝑅)) → (𝐼 Mat 𝑅) ∈ Ring)
289		simplr 768	. . . . . . . . . . . . . . . 16 ⊢ (((𝑅 ∈ Field ∧ 𝑀 ∈ (Base‘(𝐼 Mat 𝑅))) ∧ uncurry 𝑓:(𝐼 × 𝐼)⟶(Base‘𝑅)) → 𝑀 ∈ (Base‘(𝐼 Mat 𝑅)))
290		eqid 2736	. . . . . . . . . . . . . . . . 17 ⊢ (.r‘(𝐼 Mat 𝑅)) = (.r‘(𝐼 Mat 𝑅))
291	2, 290	ringcl 20248	. . . . . . . . . . . . . . . 16 ⊢ (((𝐼 Mat 𝑅) ∈ Ring ∧ uncurry 𝑓 ∈ (Base‘(𝐼 Mat 𝑅)) ∧ 𝑀 ∈ (Base‘(𝐼 Mat 𝑅))) → (uncurry 𝑓(.r‘(𝐼 Mat 𝑅))𝑀) ∈ (Base‘(𝐼 Mat 𝑅)))
292	288, 221, 289, 291	syl3anc 1372	. . . . . . . . . . . . . . 15 ⊢ (((𝑅 ∈ Field ∧ 𝑀 ∈ (Base‘(𝐼 Mat 𝑅))) ∧ uncurry 𝑓:(𝐼 × 𝐼)⟶(Base‘𝑅)) → (uncurry 𝑓(.r‘(𝐼 Mat 𝑅))𝑀) ∈ (Base‘(𝐼 Mat 𝑅)))
293	215	adantr 480	. . . . . . . . . . . . . . 15 ⊢ (((𝑅 ∈ Field ∧ 𝑀 ∈ (Base‘(𝐼 Mat 𝑅))) ∧ uncurry 𝑓:(𝐼 × 𝐼)⟶(Base‘𝑅)) → ((Base‘𝑅) ↑m (𝐼 × 𝐼)) = (Base‘(𝐼 Mat 𝑅)))
294	292, 293	eleqtrrd 2843	. . . . . . . . . . . . . 14 ⊢ (((𝑅 ∈ Field ∧ 𝑀 ∈ (Base‘(𝐼 Mat 𝑅))) ∧ uncurry 𝑓:(𝐼 × 𝐼)⟶(Base‘𝑅)) → (uncurry 𝑓(.r‘(𝐼 Mat 𝑅))𝑀) ∈ ((Base‘𝑅) ↑m (𝐼 × 𝐼)))
295		elmapfn 8906	. . . . . . . . . . . . . 14 ⊢ ((uncurry 𝑓(.r‘(𝐼 Mat 𝑅))𝑀) ∈ ((Base‘𝑅) ↑m (𝐼 × 𝐼)) → (uncurry 𝑓(.r‘(𝐼 Mat 𝑅))𝑀) Fn (𝐼 × 𝐼))
296	294, 295	syl 17	. . . . . . . . . . . . 13 ⊢ (((𝑅 ∈ Field ∧ 𝑀 ∈ (Base‘(𝐼 Mat 𝑅))) ∧ uncurry 𝑓:(𝐼 × 𝐼)⟶(Base‘𝑅)) → (uncurry 𝑓(.r‘(𝐼 Mat 𝑅))𝑀) Fn (𝐼 × 𝐼))
297		eqfnov2 7564	. . . . . . . . . . . . 13 ⊢ (((1r‘(𝐼 Mat 𝑅)) Fn (𝐼 × 𝐼) ∧ (uncurry 𝑓(.r‘(𝐼 Mat 𝑅))𝑀) Fn (𝐼 × 𝐼)) → ((1r‘(𝐼 Mat 𝑅)) = (uncurry 𝑓(.r‘(𝐼 Mat 𝑅))𝑀) ↔ ∀𝑖 ∈ 𝐼 ∀𝑗 ∈ 𝐼 (𝑖(1r‘(𝐼 Mat 𝑅))𝑗) = (𝑖(uncurry 𝑓(.r‘(𝐼 Mat 𝑅))𝑀)𝑗)))
298	285, 296, 297	syl2anc 584	. . . . . . . . . . . 12 ⊢ (((𝑅 ∈ Field ∧ 𝑀 ∈ (Base‘(𝐼 Mat 𝑅))) ∧ uncurry 𝑓:(𝐼 × 𝐼)⟶(Base‘𝑅)) → ((1r‘(𝐼 Mat 𝑅)) = (uncurry 𝑓(.r‘(𝐼 Mat 𝑅))𝑀) ↔ ∀𝑖 ∈ 𝐼 ∀𝑗 ∈ 𝐼 (𝑖(1r‘(𝐼 Mat 𝑅))𝑗) = (𝑖(uncurry 𝑓(.r‘(𝐼 Mat 𝑅))𝑀)𝑗)))
299	274, 298	sylibrd 259	. . . . . . . . . . 11 ⊢ (((𝑅 ∈ Field ∧ 𝑀 ∈ (Base‘(𝐼 Mat 𝑅))) ∧ uncurry 𝑓:(𝐼 × 𝐼)⟶(Base‘𝑅)) → (∀𝑖 ∈ 𝐼 ((𝑅 unitVec 𝐼)‘𝑖) = (𝑗 ∈ 𝐼 ↦ (𝑅 Σg (𝑘 ∈ 𝐼 ↦ ((𝑖uncurry 𝑓𝑘)(.r‘𝑅)(𝑘𝑀𝑗))))) → (1r‘(𝐼 Mat 𝑅)) = (uncurry 𝑓(.r‘(𝐼 Mat 𝑅))𝑀)))
300	299	imp 406	. . . . . . . . . 10 ⊢ ((((𝑅 ∈ Field ∧ 𝑀 ∈ (Base‘(𝐼 Mat 𝑅))) ∧ uncurry 𝑓:(𝐼 × 𝐼)⟶(Base‘𝑅)) ∧ ∀𝑖 ∈ 𝐼 ((𝑅 unitVec 𝐼)‘𝑖) = (𝑗 ∈ 𝐼 ↦ (𝑅 Σg (𝑘 ∈ 𝐼 ↦ ((𝑖uncurry 𝑓𝑘)(.r‘𝑅)(𝑘𝑀𝑗)))))) → (1r‘(𝐼 Mat 𝑅)) = (uncurry 𝑓(.r‘(𝐼 Mat 𝑅))𝑀))
301	300	eqcomd 2742	. . . . . . . . 9 ⊢ ((((𝑅 ∈ Field ∧ 𝑀 ∈ (Base‘(𝐼 Mat 𝑅))) ∧ uncurry 𝑓:(𝐼 × 𝐼)⟶(Base‘𝑅)) ∧ ∀𝑖 ∈ 𝐼 ((𝑅 unitVec 𝐼)‘𝑖) = (𝑗 ∈ 𝐼 ↦ (𝑅 Σg (𝑘 ∈ 𝐼 ↦ ((𝑖uncurry 𝑓𝑘)(.r‘𝑅)(𝑘𝑀𝑗)))))) → (uncurry 𝑓(.r‘(𝐼 Mat 𝑅))𝑀) = (1r‘(𝐼 Mat 𝑅)))
302		oveq1 7439	. . . . . . . . . . 11 ⊢ (𝑛 = uncurry 𝑓 → (𝑛(.r‘(𝐼 Mat 𝑅))𝑀) = (uncurry 𝑓(.r‘(𝐼 Mat 𝑅))𝑀))
303	302	eqeq1d 2738	. . . . . . . . . 10 ⊢ (𝑛 = uncurry 𝑓 → ((𝑛(.r‘(𝐼 Mat 𝑅))𝑀) = (1r‘(𝐼 Mat 𝑅)) ↔ (uncurry 𝑓(.r‘(𝐼 Mat 𝑅))𝑀) = (1r‘(𝐼 Mat 𝑅))))
304	303	rspcev 3621	. . . . . . . . 9 ⊢ ((uncurry 𝑓 ∈ (Base‘(𝐼 Mat 𝑅)) ∧ (uncurry 𝑓(.r‘(𝐼 Mat 𝑅))𝑀) = (1r‘(𝐼 Mat 𝑅))) → ∃𝑛 ∈ (Base‘(𝐼 Mat 𝑅))(𝑛(.r‘(𝐼 Mat 𝑅))𝑀) = (1r‘(𝐼 Mat 𝑅)))
305	222, 301, 304	syl2anc 584	. . . . . . . 8 ⊢ ((((𝑅 ∈ Field ∧ 𝑀 ∈ (Base‘(𝐼 Mat 𝑅))) ∧ uncurry 𝑓:(𝐼 × 𝐼)⟶(Base‘𝑅)) ∧ ∀𝑖 ∈ 𝐼 ((𝑅 unitVec 𝐼)‘𝑖) = (𝑗 ∈ 𝐼 ↦ (𝑅 Σg (𝑘 ∈ 𝐼 ↦ ((𝑖uncurry 𝑓𝑘)(.r‘𝑅)(𝑘𝑀𝑗)))))) → ∃𝑛 ∈ (Base‘(𝐼 Mat 𝑅))(𝑛(.r‘(𝐼 Mat 𝑅))𝑀) = (1r‘(𝐼 Mat 𝑅)))
306	305	expl 457	. . . . . . 7 ⊢ ((𝑅 ∈ Field ∧ 𝑀 ∈ (Base‘(𝐼 Mat 𝑅))) → ((uncurry 𝑓:(𝐼 × 𝐼)⟶(Base‘𝑅) ∧ ∀𝑖 ∈ 𝐼 ((𝑅 unitVec 𝐼)‘𝑖) = (𝑗 ∈ 𝐼 ↦ (𝑅 Σg (𝑘 ∈ 𝐼 ↦ ((𝑖uncurry 𝑓𝑘)(.r‘𝑅)(𝑘𝑀𝑗)))))) → ∃𝑛 ∈ (Base‘(𝐼 Mat 𝑅))(𝑛(.r‘(𝐼 Mat 𝑅))𝑀) = (1r‘(𝐼 Mat 𝑅))))
307	209, 306	sylani 604	. . . . . 6 ⊢ ((𝑅 ∈ Field ∧ 𝑀 ∈ (Base‘(𝐼 Mat 𝑅))) → ((𝑓:𝐼⟶((Base‘𝑅) ↑m 𝐼) ∧ ∀𝑖 ∈ 𝐼 ((𝑅 unitVec 𝐼)‘𝑖) = (𝑗 ∈ 𝐼 ↦ (𝑅 Σg (𝑘 ∈ 𝐼 ↦ ((𝑖uncurry 𝑓𝑘)(.r‘𝑅)(𝑘𝑀𝑗)))))) → ∃𝑛 ∈ (Base‘(𝐼 Mat 𝑅))(𝑛(.r‘(𝐼 Mat 𝑅))𝑀) = (1r‘(𝐼 Mat 𝑅))))
308	307	exlimdv 1932	. . . . 5 ⊢ ((𝑅 ∈ Field ∧ 𝑀 ∈ (Base‘(𝐼 Mat 𝑅))) → (∃𝑓(𝑓:𝐼⟶((Base‘𝑅) ↑m 𝐼) ∧ ∀𝑖 ∈ 𝐼 ((𝑅 unitVec 𝐼)‘𝑖) = (𝑗 ∈ 𝐼 ↦ (𝑅 Σg (𝑘 ∈ 𝐼 ↦ ((𝑖uncurry 𝑓𝑘)(.r‘𝑅)(𝑘𝑀𝑗)))))) → ∃𝑛 ∈ (Base‘(𝐼 Mat 𝑅))(𝑛(.r‘(𝐼 Mat 𝑅))𝑀) = (1r‘(𝐼 Mat 𝑅))))
309	308	imp 406	. . . 4 ⊢ (((𝑅 ∈ Field ∧ 𝑀 ∈ (Base‘(𝐼 Mat 𝑅))) ∧ ∃𝑓(𝑓:𝐼⟶((Base‘𝑅) ↑m 𝐼) ∧ ∀𝑖 ∈ 𝐼 ((𝑅 unitVec 𝐼)‘𝑖) = (𝑗 ∈ 𝐼 ↦ (𝑅 Σg (𝑘 ∈ 𝐼 ↦ ((𝑖uncurry 𝑓𝑘)(.r‘𝑅)(𝑘𝑀𝑗))))))) → ∃𝑛 ∈ (Base‘(𝐼 Mat 𝑅))(𝑛(.r‘(𝐼 Mat 𝑅))𝑀) = (1r‘(𝐼 Mat 𝑅)))
310	309	adantlr 715	. . 3 ⊢ ((((𝑅 ∈ Field ∧ 𝑀 ∈ (Base‘(𝐼 Mat 𝑅))) ∧ 𝐼 ≠ ∅) ∧ ∃𝑓(𝑓:𝐼⟶((Base‘𝑅) ↑m 𝐼) ∧ ∀𝑖 ∈ 𝐼 ((𝑅 unitVec 𝐼)‘𝑖) = (𝑗 ∈ 𝐼 ↦ (𝑅 Σg (𝑘 ∈ 𝐼 ↦ ((𝑖uncurry 𝑓𝑘)(.r‘𝑅)(𝑘𝑀𝑗))))))) → ∃𝑛 ∈ (Base‘(𝐼 Mat 𝑅))(𝑛(.r‘(𝐼 Mat 𝑅))𝑀) = (1r‘(𝐼 Mat 𝑅)))
311	208, 310	syldan 591	. 2 ⊢ ((((𝑅 ∈ Field ∧ 𝑀 ∈ (Base‘(𝐼 Mat 𝑅))) ∧ 𝐼 ≠ ∅) ∧ curry 𝑀 LIndF (𝑅 freeLMod 𝐼)) → ∃𝑛 ∈ (Base‘(𝐼 Mat 𝑅))(𝑛(.r‘(𝐼 Mat 𝑅))𝑀) = (1r‘(𝐼 Mat 𝑅)))
312	6	simprbi 496	. . . 4 ⊢ (𝑅 ∈ Field → 𝑅 ∈ CRing)
313		eqid 2736	. . . . . . . . . 10 ⊢ (𝐼 maDet 𝑅) = (𝐼 maDet 𝑅)
314	313, 1, 2, 14	mdetcl 22603	. . . . . . . . 9 ⊢ ((𝑅 ∈ CRing ∧ 𝑀 ∈ (Base‘(𝐼 Mat 𝑅))) → ((𝐼 maDet 𝑅)‘𝑀) ∈ (Base‘𝑅))
315	313, 1, 2, 14	mdetcl 22603	. . . . . . . . 9 ⊢ ((𝑅 ∈ CRing ∧ 𝑛 ∈ (Base‘(𝐼 Mat 𝑅))) → ((𝐼 maDet 𝑅)‘𝑛) ∈ (Base‘𝑅))
316		eqid 2736	. . . . . . . . . 10 ⊢ (∥r‘𝑅) = (∥r‘𝑅)
317	14, 316, 121	dvdsrmul 20365	. . . . . . . . 9 ⊢ ((((𝐼 maDet 𝑅)‘𝑀) ∈ (Base‘𝑅) ∧ ((𝐼 maDet 𝑅)‘𝑛) ∈ (Base‘𝑅)) → ((𝐼 maDet 𝑅)‘𝑀)(∥r‘𝑅)(((𝐼 maDet 𝑅)‘𝑛)(.r‘𝑅)((𝐼 maDet 𝑅)‘𝑀)))
318	314, 315, 317	syl2an 596	. . . . . . . 8 ⊢ (((𝑅 ∈ CRing ∧ 𝑀 ∈ (Base‘(𝐼 Mat 𝑅))) ∧ (𝑅 ∈ CRing ∧ 𝑛 ∈ (Base‘(𝐼 Mat 𝑅)))) → ((𝐼 maDet 𝑅)‘𝑀)(∥r‘𝑅)(((𝐼 maDet 𝑅)‘𝑛)(.r‘𝑅)((𝐼 maDet 𝑅)‘𝑀)))
319	318	anandis 678	. . . . . . 7 ⊢ ((𝑅 ∈ CRing ∧ (𝑀 ∈ (Base‘(𝐼 Mat 𝑅)) ∧ 𝑛 ∈ (Base‘(𝐼 Mat 𝑅)))) → ((𝐼 maDet 𝑅)‘𝑀)(∥r‘𝑅)(((𝐼 maDet 𝑅)‘𝑛)(.r‘𝑅)((𝐼 maDet 𝑅)‘𝑀)))
320	319	anassrs 467	. . . . . 6 ⊢ (((𝑅 ∈ CRing ∧ 𝑀 ∈ (Base‘(𝐼 Mat 𝑅))) ∧ 𝑛 ∈ (Base‘(𝐼 Mat 𝑅))) → ((𝐼 maDet 𝑅)‘𝑀)(∥r‘𝑅)(((𝐼 maDet 𝑅)‘𝑛)(.r‘𝑅)((𝐼 maDet 𝑅)‘𝑀)))
321	320	adantrr 717	. . . . 5 ⊢ (((𝑅 ∈ CRing ∧ 𝑀 ∈ (Base‘(𝐼 Mat 𝑅))) ∧ (𝑛 ∈ (Base‘(𝐼 Mat 𝑅)) ∧ (𝑛(.r‘(𝐼 Mat 𝑅))𝑀) = (1r‘(𝐼 Mat 𝑅)))) → ((𝐼 maDet 𝑅)‘𝑀)(∥r‘𝑅)(((𝐼 maDet 𝑅)‘𝑛)(.r‘𝑅)((𝐼 maDet 𝑅)‘𝑀)))
322		fveq2 6905	. . . . . . . . 9 ⊢ ((𝑛(.r‘(𝐼 Mat 𝑅))𝑀) = (1r‘(𝐼 Mat 𝑅)) → ((𝐼 maDet 𝑅)‘(𝑛(.r‘(𝐼 Mat 𝑅))𝑀)) = ((𝐼 maDet 𝑅)‘(1r‘(𝐼 Mat 𝑅))))
323	1, 2, 313, 121, 290	mdetmul 22630	. . . . . . . . . . . 12 ⊢ ((𝑅 ∈ CRing ∧ 𝑛 ∈ (Base‘(𝐼 Mat 𝑅)) ∧ 𝑀 ∈ (Base‘(𝐼 Mat 𝑅))) → ((𝐼 maDet 𝑅)‘(𝑛(.r‘(𝐼 Mat 𝑅))𝑀)) = (((𝐼 maDet 𝑅)‘𝑛)(.r‘𝑅)((𝐼 maDet 𝑅)‘𝑀)))
324	323	3expa 1118	. . . . . . . . . . 11 ⊢ (((𝑅 ∈ CRing ∧ 𝑛 ∈ (Base‘(𝐼 Mat 𝑅))) ∧ 𝑀 ∈ (Base‘(𝐼 Mat 𝑅))) → ((𝐼 maDet 𝑅)‘(𝑛(.r‘(𝐼 Mat 𝑅))𝑀)) = (((𝐼 maDet 𝑅)‘𝑛)(.r‘𝑅)((𝐼 maDet 𝑅)‘𝑀)))
325	324	an32s 652	. . . . . . . . . 10 ⊢ (((𝑅 ∈ CRing ∧ 𝑀 ∈ (Base‘(𝐼 Mat 𝑅))) ∧ 𝑛 ∈ (Base‘(𝐼 Mat 𝑅))) → ((𝐼 maDet 𝑅)‘(𝑛(.r‘(𝐼 Mat 𝑅))𝑀)) = (((𝐼 maDet 𝑅)‘𝑛)(.r‘𝑅)((𝐼 maDet 𝑅)‘𝑀)))
326	313, 1, 239, 233	mdet1 22608	. . . . . . . . . . . 12 ⊢ ((𝑅 ∈ CRing ∧ 𝐼 ∈ Fin) → ((𝐼 maDet 𝑅)‘(1r‘(𝐼 Mat 𝑅))) = (1r‘𝑅))
327	4, 326	sylan2 593	. . . . . . . . . . 11 ⊢ ((𝑅 ∈ CRing ∧ 𝑀 ∈ (Base‘(𝐼 Mat 𝑅))) → ((𝐼 maDet 𝑅)‘(1r‘(𝐼 Mat 𝑅))) = (1r‘𝑅))
328	327	adantr 480	. . . . . . . . . 10 ⊢ (((𝑅 ∈ CRing ∧ 𝑀 ∈ (Base‘(𝐼 Mat 𝑅))) ∧ 𝑛 ∈ (Base‘(𝐼 Mat 𝑅))) → ((𝐼 maDet 𝑅)‘(1r‘(𝐼 Mat 𝑅))) = (1r‘𝑅))
329	325, 328	eqeq12d 2752	. . . . . . . . 9 ⊢ (((𝑅 ∈ CRing ∧ 𝑀 ∈ (Base‘(𝐼 Mat 𝑅))) ∧ 𝑛 ∈ (Base‘(𝐼 Mat 𝑅))) → (((𝐼 maDet 𝑅)‘(𝑛(.r‘(𝐼 Mat 𝑅))𝑀)) = ((𝐼 maDet 𝑅)‘(1r‘(𝐼 Mat 𝑅))) ↔ (((𝐼 maDet 𝑅)‘𝑛)(.r‘𝑅)((𝐼 maDet 𝑅)‘𝑀)) = (1r‘𝑅)))
330	322, 329	imbitrid 244	. . . . . . . 8 ⊢ (((𝑅 ∈ CRing ∧ 𝑀 ∈ (Base‘(𝐼 Mat 𝑅))) ∧ 𝑛 ∈ (Base‘(𝐼 Mat 𝑅))) → ((𝑛(.r‘(𝐼 Mat 𝑅))𝑀) = (1r‘(𝐼 Mat 𝑅)) → (((𝐼 maDet 𝑅)‘𝑛)(.r‘𝑅)((𝐼 maDet 𝑅)‘𝑀)) = (1r‘𝑅)))
331	330	impr 454	. . . . . . 7 ⊢ (((𝑅 ∈ CRing ∧ 𝑀 ∈ (Base‘(𝐼 Mat 𝑅))) ∧ (𝑛 ∈ (Base‘(𝐼 Mat 𝑅)) ∧ (𝑛(.r‘(𝐼 Mat 𝑅))𝑀) = (1r‘(𝐼 Mat 𝑅)))) → (((𝐼 maDet 𝑅)‘𝑛)(.r‘𝑅)((𝐼 maDet 𝑅)‘𝑀)) = (1r‘𝑅))
332	331	breq2d 5154	. . . . . 6 ⊢ (((𝑅 ∈ CRing ∧ 𝑀 ∈ (Base‘(𝐼 Mat 𝑅))) ∧ (𝑛 ∈ (Base‘(𝐼 Mat 𝑅)) ∧ (𝑛(.r‘(𝐼 Mat 𝑅))𝑀) = (1r‘(𝐼 Mat 𝑅)))) → (((𝐼 maDet 𝑅)‘𝑀)(∥r‘𝑅)(((𝐼 maDet 𝑅)‘𝑛)(.r‘𝑅)((𝐼 maDet 𝑅)‘𝑀)) ↔ ((𝐼 maDet 𝑅)‘𝑀)(∥r‘𝑅)(1r‘𝑅)))
333		eqid 2736	. . . . . . . 8 ⊢ (Unit‘𝑅) = (Unit‘𝑅)
334	333, 233, 316	crngunit 20379	. . . . . . 7 ⊢ (𝑅 ∈ CRing → (((𝐼 maDet 𝑅)‘𝑀) ∈ (Unit‘𝑅) ↔ ((𝐼 maDet 𝑅)‘𝑀)(∥r‘𝑅)(1r‘𝑅)))
335	334	ad2antrr 726	. . . . . 6 ⊢ (((𝑅 ∈ CRing ∧ 𝑀 ∈ (Base‘(𝐼 Mat 𝑅))) ∧ (𝑛 ∈ (Base‘(𝐼 Mat 𝑅)) ∧ (𝑛(.r‘(𝐼 Mat 𝑅))𝑀) = (1r‘(𝐼 Mat 𝑅)))) → (((𝐼 maDet 𝑅)‘𝑀) ∈ (Unit‘𝑅) ↔ ((𝐼 maDet 𝑅)‘𝑀)(∥r‘𝑅)(1r‘𝑅)))
336	332, 335	bitr4d 282	. . . . 5 ⊢ (((𝑅 ∈ CRing ∧ 𝑀 ∈ (Base‘(𝐼 Mat 𝑅))) ∧ (𝑛 ∈ (Base‘(𝐼 Mat 𝑅)) ∧ (𝑛(.r‘(𝐼 Mat 𝑅))𝑀) = (1r‘(𝐼 Mat 𝑅)))) → (((𝐼 maDet 𝑅)‘𝑀)(∥r‘𝑅)(((𝐼 maDet 𝑅)‘𝑛)(.r‘𝑅)((𝐼 maDet 𝑅)‘𝑀)) ↔ ((𝐼 maDet 𝑅)‘𝑀) ∈ (Unit‘𝑅)))
337	321, 336	mpbid 232	. . . 4 ⊢ (((𝑅 ∈ CRing ∧ 𝑀 ∈ (Base‘(𝐼 Mat 𝑅))) ∧ (𝑛 ∈ (Base‘(𝐼 Mat 𝑅)) ∧ (𝑛(.r‘(𝐼 Mat 𝑅))𝑀) = (1r‘(𝐼 Mat 𝑅)))) → ((𝐼 maDet 𝑅)‘𝑀) ∈ (Unit‘𝑅))
338	312, 337	sylanl1 680	. . 3 ⊢ (((𝑅 ∈ Field ∧ 𝑀 ∈ (Base‘(𝐼 Mat 𝑅))) ∧ (𝑛 ∈ (Base‘(𝐼 Mat 𝑅)) ∧ (𝑛(.r‘(𝐼 Mat 𝑅))𝑀) = (1r‘(𝐼 Mat 𝑅)))) → ((𝐼 maDet 𝑅)‘𝑀) ∈ (Unit‘𝑅))
339	338	ad4ant14 752	. 2 ⊢ (((((𝑅 ∈ Field ∧ 𝑀 ∈ (Base‘(𝐼 Mat 𝑅))) ∧ 𝐼 ≠ ∅) ∧ curry 𝑀 LIndF (𝑅 freeLMod 𝐼)) ∧ (𝑛 ∈ (Base‘(𝐼 Mat 𝑅)) ∧ (𝑛(.r‘(𝐼 Mat 𝑅))𝑀) = (1r‘(𝐼 Mat 𝑅)))) → ((𝐼 maDet 𝑅)‘𝑀) ∈ (Unit‘𝑅))
340	311, 339	rexlimddv 3160	1 ⊢ ((((𝑅 ∈ Field ∧ 𝑀 ∈ (Base‘(𝐼 Mat 𝑅))) ∧ 𝐼 ≠ ∅) ∧ curry 𝑀 LIndF (𝑅 freeLMod 𝐼)) → ((𝐼 maDet 𝑅)‘𝑀) ∈ (Unit‘𝑅))

Syntax hints:   → wi 4   ↔ wb 206   ∧ wa 395   ∧ w3a 1086   = wceq 1539  ∃wex 1778   ∈ wcel 2107   ≠ wne 2939  ∀wral 3060  ∃wrex 3069  Vcvv 3479   ∖ cdif 3947  ∅c0 4332  ifcif 4524  {csn 4625  ⟨cotp 4633   class class class wbr 5142   ↦ cmpt 5224   × cxp 5682  dom cdm 5684  ran crn 5685   “ cima 5687   Fn wfn 6555  ⟶wf 6556  –1-1→wf1 6557  –1-1-onto→wf1o 6559  ‘cfv 6560  (class class class)co 7432   ∘_f cof 7696  curry ccur 8291  uncurry cunc 8292   ↑_m cmap 8867   ≈ cen 8983  Fincfn 8986   finSupp cfsupp 9402  Basecbs 17248  ._rcmulr 17299  Scalarcsca 17301   ·_𝑠 cvsca 17302  0_gc0g 17485   Σ_g cgsu 17486  1_rcur 20179  Ringcrg 20231  CRingccrg 20232  ∥_rcdsr 20355  Unitcui 20356  NzRingcnzr 20513  DivRingcdr 20730  Fieldcfield 20731  LModclmod 20859  LSpanclspn 20970  LBasisclbs 21074   freeLMod cfrlm 21767   unitVec cuvc 21803   LIndF clindf 21825  LIndSclinds 21826   maMul cmmul 22395   Mat cmat 22412   maDet cmdat 22591

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1794  ax-4 1808  ax-5 1909  ax-6 1966  ax-7 2006  ax-8 2109  ax-9 2117  ax-10 2140  ax-11 2156  ax-12 2176  ax-ext 2707  ax-rep 5278  ax-sep 5295  ax-nul 5305  ax-pow 5364  ax-pr 5431  ax-un 7756  ax-cnex 11212  ax-resscn 11213  ax-1cn 11214  ax-icn 11215  ax-addcl 11216  ax-addrcl 11217  ax-mulcl 11218  ax-mulrcl 11219  ax-mulcom 11220  ax-addass 11221  ax-mulass 11222  ax-distr 11223  ax-i2m1 11224  ax-1ne0 11225  ax-1rid 11226  ax-rnegex 11227  ax-rrecex 11228  ax-cnre 11229  ax-pre-lttri 11230  ax-pre-lttrn 11231  ax-pre-ltadd 11232  ax-pre-mulgt0 11233  ax-addf 11235  ax-mulf 11236

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3or 1087  df-3an 1088  df-xor 1511  df-tru 1542  df-fal 1552  df-ex 1779  df-nf 1783  df-sb 2064  df-mo 2539  df-eu 2568  df-clab 2714  df-cleq 2728  df-clel 2815  df-nfc 2891  df-ne 2940  df-nel 3046  df-ral 3061  df-rex 3070  df-rmo 3379  df-reu 3380  df-rab 3436  df-v 3481  df-sbc 3788  df-csb 3899  df-dif 3953  df-un 3955  df-in 3957  df-ss 3967  df-pss 3970  df-nul 4333  df-if 4525  df-pw 4601  df-sn 4626  df-pr 4628  df-tp 4630  df-op 4632  df-ot 4634  df-uni 4907  df-int 4946  df-iun 4992  df-iin 4993  df-br 5143  df-opab 5205  df-mpt 5225  df-tr 5259  df-id 5577  df-eprel 5583  df-po 5591  df-so 5592  df-fr 5636  df-se 5637  df-we 5638  df-xp 5690  df-rel 5691  df-cnv 5692  df-co 5693  df-dm 5694  df-rn 5695  df-res 5696  df-ima 5697  df-pred 6320  df-ord 6386  df-on 6387  df-lim 6388  df-suc 6389  df-iota 6513  df-fun 6562  df-fn 6563  df-f 6564  df-f1 6565  df-fo 6566  df-f1o 6567  df-fv 6568  df-isom 6569  df-riota 7389  df-ov 7435  df-oprab 7436  df-mpo 7437  df-of 7698  df-om 7889  df-1st 8015  df-2nd 8016  df-supp 8187  df-tpos 8252  df-cur 8293  df-unc 8294  df-frecs 8307  df-wrecs 8338  df-recs 8412  df-rdg 8451  df-1o 8507  df-2o 8508  df-er 8746  df-map 8869  df-pm 8870  df-ixp 8939  df-en 8987  df-dom 8988  df-sdom 8989  df-fin 8990  df-fsupp 9403  df-sup 9483  df-oi 9551  df-card 9980  df-pnf 11298  df-mnf 11299  df-xr 11300  df-ltxr 11301  df-le 11302  df-sub 11495  df-neg 11496  df-div 11922  df-nn 12268  df-2 12330  df-3 12331  df-4 12332  df-5 12333  df-6 12334  df-7 12335  df-8 12336  df-9 12337  df-n0 12529  df-xnn0 12602  df-z 12616  df-dec 12736  df-uz 12880  df-rp 13036  df-fz 13549  df-fzo 13696  df-seq 14044  df-exp 14104  df-hash 14371  df-word 14554  df-lsw 14602  df-concat 14610  df-s1 14635  df-substr 14680  df-pfx 14710  df-splice 14789  df-reverse 14798  df-s2 14888  df-struct 17185  df-sets 17202  df-slot 17220  df-ndx 17232  df-base 17249  df-ress 17276  df-plusg 17311  df-mulr 17312  df-starv 17313  df-sca 17314  df-vsca 17315  df-ip 17316  df-tset 17317  df-ple 17318  df-ds 17320  df-unif 17321  df-hom 17322  df-cco 17323  df-0g 17487  df-gsum 17488  df-prds 17493  df-pws 17495  df-mre 17630  df-mrc 17631  df-mri 17632  df-acs 17633  df-mgm 18654  df-sgrp 18733  df-mnd 18749  df-mhm 18797  df-submnd 18798  df-efmnd 18883  df-grp 18955  df-minusg 18956  df-sbg 18957  df-mulg 19087  df-subg 19142  df-ghm 19232  df-gim 19278  df-cntz 19336  df-oppg 19365  df-symg 19388  df-pmtr 19461  df-psgn 19510  df-evpm 19511  df-cmn 19801  df-abl 19802  df-mgp 20139  df-rng 20151  df-ur 20180  df-srg 20185  df-ring 20233  df-cring 20234  df-oppr 20335  df-dvdsr 20358  df-unit 20359  df-invr 20389  df-dvr 20402  df-rhm 20473  df-nzr 20514  df-subrng 20547  df-subrg 20571  df-drng 20732  df-field 20733  df-lmod 20861  df-lss 20931  df-lsp 20971  df-lmhm 21022  df-lbs 21075  df-lvec 21103  df-sra 21173  df-rgmod 21174  df-cnfld 21366  df-zring 21459  df-zrh 21515  df-dsmm 21753  df-frlm 21768  df-uvc 21804  df-lindf 21827  df-linds 21828  df-mamu 22396  df-mat 22413  df-mdet 22592

This theorem is referenced by:  matunitlindf  37626