Theorem matunitlindflem2 34891

Description: One direction of matunitlindf 34892. (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 2823	. . . . . . 7 ⊢ (𝐼 Mat 𝑅) = (𝐼 Mat 𝑅)
2		eqid 2823	. . . . . . 7 ⊢ (Base‘(𝐼 Mat 𝑅)) = (Base‘(𝐼 Mat 𝑅))
3	1, 2	matrcl 21023	. . . . . 6 ⊢ (𝑀 ∈ (Base‘(𝐼 Mat 𝑅)) → (𝐼 ∈ Fin ∧ 𝑅 ∈ V))
4	3	simpld 497	. . . . 5 ⊢ (𝑀 ∈ (Base‘(𝐼 Mat 𝑅)) → 𝐼 ∈ Fin)
5	4	ad3antlr 729	. . . 4 ⊢ ((((𝑅 ∈ Field ∧ 𝑀 ∈ (Base‘(𝐼 Mat 𝑅))) ∧ 𝐼 ≠ ∅) ∧ curry 𝑀 LIndF (𝑅 freeLMod 𝐼)) → 𝐼 ∈ Fin)
6		isfld 19513	. . . . . . 7 ⊢ (𝑅 ∈ Field ↔ (𝑅 ∈ DivRing ∧ 𝑅 ∈ CRing))
7	6	simplbi 500	. . . . . 6 ⊢ (𝑅 ∈ Field → 𝑅 ∈ DivRing)
8	7	anim1i 616	. . . . 5 ⊢ ((𝑅 ∈ Field ∧ 𝑀 ∈ (Base‘(𝐼 Mat 𝑅))) → (𝑅 ∈ DivRing ∧ 𝑀 ∈ (Base‘(𝐼 Mat 𝑅))))
9	4	ad2antrl 726	. . . . . . . . . . . 12 ⊢ ((𝑅 ∈ DivRing ∧ (𝑀 ∈ (Base‘(𝐼 Mat 𝑅)) ∧ 𝐼 ≠ ∅)) → 𝐼 ∈ Fin)
10		simpr 487	. . . . . . . . . . . . . . 15 ⊢ ((𝑅 ∈ DivRing ∧ 𝑀 ∈ (Base‘(𝐼 Mat 𝑅))) → 𝑀 ∈ (Base‘(𝐼 Mat 𝑅)))
11		xpfi 8791	. . . . . . . . . . . . . . . . . . . . 21 ⊢ ((𝐼 ∈ Fin ∧ 𝐼 ∈ Fin) → (𝐼 × 𝐼) ∈ Fin)
12	11	anidms 569	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝐼 ∈ Fin → (𝐼 × 𝐼) ∈ Fin)
13		eqid 2823	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (𝑅 freeLMod (𝐼 × 𝐼)) = (𝑅 freeLMod (𝐼 × 𝐼))
14		eqid 2823	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (Base‘𝑅) = (Base‘𝑅)
15	13, 14	frlmfibas 20908	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((𝑅 ∈ DivRing ∧ (𝐼 × 𝐼) ∈ Fin) → ((Base‘𝑅) ↑m (𝐼 × 𝐼)) = (Base‘(𝑅 freeLMod (𝐼 × 𝐼))))
16	12, 15	sylan2 594	. . . . . . . . . . . . . . . . . . 19 ⊢ ((𝑅 ∈ DivRing ∧ 𝐼 ∈ Fin) → ((Base‘𝑅) ↑m (𝐼 × 𝐼)) = (Base‘(𝑅 freeLMod (𝐼 × 𝐼))))
17	1, 13	matbas 21024	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((𝐼 ∈ Fin ∧ 𝑅 ∈ DivRing) → (Base‘(𝑅 freeLMod (𝐼 × 𝐼))) = (Base‘(𝐼 Mat 𝑅)))
18	17	ancoms 461	. . . . . . . . . . . . . . . . . . 19 ⊢ ((𝑅 ∈ DivRing ∧ 𝐼 ∈ Fin) → (Base‘(𝑅 freeLMod (𝐼 × 𝐼))) = (Base‘(𝐼 Mat 𝑅)))
19	16, 18	eqtrd 2858	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝑅 ∈ DivRing ∧ 𝐼 ∈ Fin) → ((Base‘𝑅) ↑m (𝐼 × 𝐼)) = (Base‘(𝐼 Mat 𝑅)))
20	19	eleq2d 2900	. . . . . . . . . . . . . . . . 17 ⊢ ((𝑅 ∈ DivRing ∧ 𝐼 ∈ Fin) → (𝑀 ∈ ((Base‘𝑅) ↑m (𝐼 × 𝐼)) ↔ 𝑀 ∈ (Base‘(𝐼 Mat 𝑅))))
21	4, 20	sylan2 594	. . . . . . . . . . . . . . . 16 ⊢ ((𝑅 ∈ DivRing ∧ 𝑀 ∈ (Base‘(𝐼 Mat 𝑅))) → (𝑀 ∈ ((Base‘𝑅) ↑m (𝐼 × 𝐼)) ↔ 𝑀 ∈ (Base‘(𝐼 Mat 𝑅))))
22		fvex 6685	. . . . . . . . . . . . . . . . . 18 ⊢ (Base‘𝑅) ∈ V
23	4, 4, 11	syl2anc 586	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑀 ∈ (Base‘(𝐼 Mat 𝑅)) → (𝐼 × 𝐼) ∈ Fin)
24		elmapg 8421	. . . . . . . . . . . . . . . . . 18 ⊢ (((Base‘𝑅) ∈ V ∧ (𝐼 × 𝐼) ∈ Fin) → (𝑀 ∈ ((Base‘𝑅) ↑m (𝐼 × 𝐼)) ↔ 𝑀:(𝐼 × 𝐼)⟶(Base‘𝑅)))
25	22, 23, 24	sylancr 589	. . . . . . . . . . . . . . . . 17 ⊢ (𝑀 ∈ (Base‘(𝐼 Mat 𝑅)) → (𝑀 ∈ ((Base‘𝑅) ↑m (𝐼 × 𝐼)) ↔ 𝑀:(𝐼 × 𝐼)⟶(Base‘𝑅)))
26	25	adantl 484	. . . . . . . . . . . . . . . 16 ⊢ ((𝑅 ∈ DivRing ∧ 𝑀 ∈ (Base‘(𝐼 Mat 𝑅))) → (𝑀 ∈ ((Base‘𝑅) ↑m (𝐼 × 𝐼)) ↔ 𝑀:(𝐼 × 𝐼)⟶(Base‘𝑅)))
27	21, 26	bitr3d 283	. . . . . . . . . . . . . . 15 ⊢ ((𝑅 ∈ DivRing ∧ 𝑀 ∈ (Base‘(𝐼 Mat 𝑅))) → (𝑀 ∈ (Base‘(𝐼 Mat 𝑅)) ↔ 𝑀:(𝐼 × 𝐼)⟶(Base‘𝑅)))
28	10, 27	mpbid 234	. . . . . . . . . . . . . 14 ⊢ ((𝑅 ∈ DivRing ∧ 𝑀 ∈ (Base‘(𝐼 Mat 𝑅))) → 𝑀:(𝐼 × 𝐼)⟶(Base‘𝑅))
29	28	adantrr 715	. . . . . . . . . . . . 13 ⊢ ((𝑅 ∈ DivRing ∧ (𝑀 ∈ (Base‘(𝐼 Mat 𝑅)) ∧ 𝐼 ≠ ∅)) → 𝑀:(𝐼 × 𝐼)⟶(Base‘𝑅))
30		eldifsn 4721	. . . . . . . . . . . . . . . 16 ⊢ (𝐼 ∈ (Fin ∖ {∅}) ↔ (𝐼 ∈ Fin ∧ 𝐼 ≠ ∅))
31	30	biimpri 230	. . . . . . . . . . . . . . 15 ⊢ ((𝐼 ∈ Fin ∧ 𝐼 ≠ ∅) → 𝐼 ∈ (Fin ∖ {∅}))
32	4, 31	sylan 582	. . . . . . . . . . . . . 14 ⊢ ((𝑀 ∈ (Base‘(𝐼 Mat 𝑅)) ∧ 𝐼 ≠ ∅) → 𝐼 ∈ (Fin ∖ {∅}))
33	32	adantl 484	. . . . . . . . . . . . 13 ⊢ ((𝑅 ∈ DivRing ∧ (𝑀 ∈ (Base‘(𝐼 Mat 𝑅)) ∧ 𝐼 ≠ ∅)) → 𝐼 ∈ (Fin ∖ {∅}))
34		curf 34872	. . . . . . . . . . . . . 14 ⊢ ((𝑀:(𝐼 × 𝐼)⟶(Base‘𝑅) ∧ 𝐼 ∈ (Fin ∖ {∅}) ∧ (Base‘𝑅) ∈ V) → curry 𝑀:𝐼⟶((Base‘𝑅) ↑m 𝐼))
35	22, 34	mp3an3 1446	. . . . . . . . . . . . 13 ⊢ ((𝑀:(𝐼 × 𝐼)⟶(Base‘𝑅) ∧ 𝐼 ∈ (Fin ∖ {∅})) → curry 𝑀:𝐼⟶((Base‘𝑅) ↑m 𝐼))
36	29, 33, 35	syl2anc 586	. . . . . . . . . . . 12 ⊢ ((𝑅 ∈ DivRing ∧ (𝑀 ∈ (Base‘(𝐼 Mat 𝑅)) ∧ 𝐼 ≠ ∅)) → curry 𝑀:𝐼⟶((Base‘𝑅) ↑m 𝐼))
37	9, 36	jca 514	. . . . . . . . . . 11 ⊢ ((𝑅 ∈ DivRing ∧ (𝑀 ∈ (Base‘(𝐼 Mat 𝑅)) ∧ 𝐼 ≠ ∅)) → (𝐼 ∈ Fin ∧ curry 𝑀:𝐼⟶((Base‘𝑅) ↑m 𝐼)))
38	37	ex 415	. . . . . . . . . 10 ⊢ (𝑅 ∈ DivRing → ((𝑀 ∈ (Base‘(𝐼 Mat 𝑅)) ∧ 𝐼 ≠ ∅) → (𝐼 ∈ Fin ∧ curry 𝑀:𝐼⟶((Base‘𝑅) ↑m 𝐼))))
39	38	imdistani 571	. . . . . . . . 9 ⊢ ((𝑅 ∈ DivRing ∧ (𝑀 ∈ (Base‘(𝐼 Mat 𝑅)) ∧ 𝐼 ≠ ∅)) → (𝑅 ∈ DivRing ∧ (𝐼 ∈ Fin ∧ curry 𝑀:𝐼⟶((Base‘𝑅) ↑m 𝐼))))
40	39	anassrs 470	. . . . . . . 8 ⊢ (((𝑅 ∈ DivRing ∧ 𝑀 ∈ (Base‘(𝐼 Mat 𝑅))) ∧ 𝐼 ≠ ∅) → (𝑅 ∈ DivRing ∧ (𝐼 ∈ Fin ∧ curry 𝑀:𝐼⟶((Base‘𝑅) ↑m 𝐼))))
41		anass 471	. . . . . . . 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 236	. . . . . . 7 ⊢ (((𝑅 ∈ DivRing ∧ 𝑀 ∈ (Base‘(𝐼 Mat 𝑅))) ∧ 𝐼 ≠ ∅) → ((𝑅 ∈ DivRing ∧ 𝐼 ∈ Fin) ∧ curry 𝑀:𝐼⟶((Base‘𝑅) ↑m 𝐼)))
43		drngring 19511	. . . . . . . . . . . . 13 ⊢ (𝑅 ∈ DivRing → 𝑅 ∈ Ring)
44		eqid 2823	. . . . . . . . . . . . . 14 ⊢ (𝑅 unitVec 𝐼) = (𝑅 unitVec 𝐼)
45		eqid 2823	. . . . . . . . . . . . . 14 ⊢ (𝑅 freeLMod 𝐼) = (𝑅 freeLMod 𝐼)
46		eqid 2823	. . . . . . . . . . . . . 14 ⊢ (Base‘(𝑅 freeLMod 𝐼)) = (Base‘(𝑅 freeLMod 𝐼))
47	44, 45, 46	uvcff 20937	. . . . . . . . . . . . 13 ⊢ ((𝑅 ∈ Ring ∧ 𝐼 ∈ Fin) → (𝑅 unitVec 𝐼):𝐼⟶(Base‘(𝑅 freeLMod 𝐼)))
48	43, 47	sylan 582	. . . . . . . . . . . 12 ⊢ ((𝑅 ∈ DivRing ∧ 𝐼 ∈ Fin) → (𝑅 unitVec 𝐼):𝐼⟶(Base‘(𝑅 freeLMod 𝐼)))
49	48	ffvelrnda 6853	. . . . . . . . . . 11 ⊢ (((𝑅 ∈ DivRing ∧ 𝐼 ∈ Fin) ∧ 𝑖 ∈ 𝐼) → ((𝑅 unitVec 𝐼)‘𝑖) ∈ (Base‘(𝑅 freeLMod 𝐼)))
50	49	ad4ant14 750	. . . . . . . . . 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 6516	. . . . . . . . . . . . . . . 16 ⊢ (curry 𝑀:𝐼⟶((Base‘𝑅) ↑m 𝐼) → curry 𝑀 Fn 𝐼)
52		fnima 6480	. . . . . . . . . . . . . . . 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 484	. . . . . . . . . . . . . 14 ⊢ (((𝑅 ∈ DivRing ∧ 𝐼 ∈ Fin) ∧ curry 𝑀:𝐼⟶((Base‘𝑅) ↑m 𝐼)) → (curry 𝑀 “ 𝐼) = ran curry 𝑀)
55	54	fveq2d 6676	. . . . . . . . . . . . 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 483	. . . . . . . . . . . 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 773	. . . . . . . . . . . . . 14 ⊢ ((((𝑅 ∈ DivRing ∧ 𝐼 ∈ Fin) ∧ curry 𝑀:𝐼⟶((Base‘𝑅) ↑m 𝐼)) ∧ curry 𝑀 LIndF (𝑅 freeLMod 𝐼)) → 𝑅 ∈ DivRing)
58		simpllr 774	. . . . . . . . . . . . . 14 ⊢ ((((𝑅 ∈ DivRing ∧ 𝐼 ∈ Fin) ∧ curry 𝑀:𝐼⟶((Base‘𝑅) ↑m 𝐼)) ∧ curry 𝑀 LIndF (𝑅 freeLMod 𝐼)) → 𝐼 ∈ Fin)
59	45	frlmlmod 20895	. . . . . . . . . . . . . . . . 17 ⊢ ((𝑅 ∈ Ring ∧ 𝐼 ∈ Fin) → (𝑅 freeLMod 𝐼) ∈ LMod)
60	43, 59	sylan 582	. . . . . . . . . . . . . . . 16 ⊢ ((𝑅 ∈ DivRing ∧ 𝐼 ∈ Fin) → (𝑅 freeLMod 𝐼) ∈ LMod)
61	60	adantr 483	. . . . . . . . . . . . . . 15 ⊢ (((𝑅 ∈ DivRing ∧ 𝐼 ∈ Fin) ∧ curry 𝑀:𝐼⟶((Base‘𝑅) ↑m 𝐼)) → (𝑅 freeLMod 𝐼) ∈ LMod)
62		lindfrn 20967	. . . . . . . . . . . . . . 15 ⊢ (((𝑅 freeLMod 𝐼) ∈ LMod ∧ curry 𝑀 LIndF (𝑅 freeLMod 𝐼)) → ran curry 𝑀 ∈ (LIndS‘(𝑅 freeLMod 𝐼)))
63	61, 62	sylan 582	. . . . . . . . . . . . . 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 20899	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ ((𝑅 ∈ DivRing ∧ 𝐼 ∈ Fin) → 𝑅 = (Scalar‘(𝑅 freeLMod 𝐼)))
65		drngnzr 20037	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ (𝑅 ∈ DivRing → 𝑅 ∈ NzRing)
66	65	adantr 483	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ ((𝑅 ∈ DivRing ∧ 𝐼 ∈ Fin) → 𝑅 ∈ NzRing)
67	64, 66	eqeltrrd 2916	. . . . . . . . . . . . . . . . . . . . 21 ⊢ ((𝑅 ∈ DivRing ∧ 𝐼 ∈ Fin) → (Scalar‘(𝑅 freeLMod 𝐼)) ∈ NzRing)
68	60, 67	jca 514	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((𝑅 ∈ DivRing ∧ 𝐼 ∈ Fin) → ((𝑅 freeLMod 𝐼) ∈ LMod ∧ (Scalar‘(𝑅 freeLMod 𝐼)) ∈ NzRing))
69		eqid 2823	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (Scalar‘(𝑅 freeLMod 𝐼)) = (Scalar‘(𝑅 freeLMod 𝐼))
70	46, 69	lindff1 20966	. . . . . . . . . . . . . . . . . . . . 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 1114	. . . . . . . . . . . . . . . . . . . 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 582	. . . . . . . . . . . . . . . . . . 19 ⊢ (((𝑅 ∈ DivRing ∧ 𝐼 ∈ Fin) ∧ curry 𝑀 LIndF (𝑅 freeLMod 𝐼)) → curry 𝑀:dom curry 𝑀–1-1→(Base‘(𝑅 freeLMod 𝐼)))
73		fdm 6524	. . . . . . . . . . . . . . . . . . 19 ⊢ (curry 𝑀:𝐼⟶((Base‘𝑅) ↑m 𝐼) → dom curry 𝑀 = 𝐼)
74		f1eq2 6573	. . . . . . . . . . . . . . . . . . . 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 481	. . . . . . . . . . . . . . . . . . 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 597	. . . . . . . . . . . . . . . . . 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 650	. . . . . . . . . . . . . . . . 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 6628	. . . . . . . . . . . . . . . . 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 8530	. . . . . . . . . . . . . . . 16 ⊢ ((𝐼 ∈ Fin ∧ curry 𝑀:𝐼–1-1-onto→ran curry 𝑀) → 𝐼 ≈ ran curry 𝑀)
81	58, 79, 80	syl2anc 586	. . . . . . . . . . . . . . 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 8562	. . . . . . . . . . . . . 14 ⊢ ((((𝑅 ∈ DivRing ∧ 𝐼 ∈ Fin) ∧ curry 𝑀:𝐼⟶((Base‘𝑅) ↑m 𝐼)) ∧ curry 𝑀 LIndF (𝑅 freeLMod 𝐼)) → ran curry 𝑀 ≈ 𝐼)
83		lindsenlbs 34889	. . . . . . . . . . . . . 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 1369	. . . . . . . . . . . . 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 2823	. . . . . . . . . . . . . 14 ⊢ (LBasis‘(𝑅 freeLMod 𝐼)) = (LBasis‘(𝑅 freeLMod 𝐼))
86		eqid 2823	. . . . . . . . . . . . . 14 ⊢ (LSpan‘(𝑅 freeLMod 𝐼)) = (LSpan‘(𝑅 freeLMod 𝐼))
87	46, 85, 86	lbssp 19853	. . . . . . . . . . . . 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 2858	. . . . . . . . . . 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 483	. . . . . . . . . 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 2918	. . . . . . . . 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 2823	. . . . . . . . . . . . 13 ⊢ (Base‘(Scalar‘(𝑅 freeLMod 𝐼))) = (Base‘(Scalar‘(𝑅 freeLMod 𝐼)))
93		eqid 2823	. . . . . . . . . . . . 13 ⊢ (0g‘(Scalar‘(𝑅 freeLMod 𝐼))) = (0g‘(Scalar‘(𝑅 freeLMod 𝐼)))
94		eqid 2823	. . . . . . . . . . . . 13 ⊢ ( ·𝑠 ‘(𝑅 freeLMod 𝐼)) = ( ·𝑠 ‘(𝑅 freeLMod 𝐼))
95	45, 14	frlmfibas 20908	. . . . . . . . . . . . . . 15 ⊢ ((𝑅 ∈ Ring ∧ 𝐼 ∈ Fin) → ((Base‘𝑅) ↑m 𝐼) = (Base‘(𝑅 freeLMod 𝐼)))
96	95	feq3d 6503	. . . . . . . . . . . . . 14 ⊢ ((𝑅 ∈ Ring ∧ 𝐼 ∈ Fin) → (curry 𝑀:𝐼⟶((Base‘𝑅) ↑m 𝐼) ↔ curry 𝑀:𝐼⟶(Base‘(𝑅 freeLMod 𝐼))))
97	96	biimpa 479	. . . . . . . . . . . . 13 ⊢ (((𝑅 ∈ Ring ∧ 𝐼 ∈ Fin) ∧ curry 𝑀:𝐼⟶((Base‘𝑅) ↑m 𝐼)) → curry 𝑀:𝐼⟶(Base‘(𝑅 freeLMod 𝐼)))
98	59	adantr 483	. . . . . . . . . . . . 13 ⊢ (((𝑅 ∈ Ring ∧ 𝐼 ∈ Fin) ∧ curry 𝑀:𝐼⟶((Base‘𝑅) ↑m 𝐼)) → (𝑅 freeLMod 𝐼) ∈ LMod)
99		simplr 767	. . . . . . . . . . . . 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 20949	. . . . . . . . . . . 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 20899	. . . . . . . . . . . . . . . 16 ⊢ ((𝑅 ∈ Ring ∧ 𝐼 ∈ Fin) → 𝑅 = (Scalar‘(𝑅 freeLMod 𝐼)))
102	101	fveq2d 6676	. . . . . . . . . . . . . . 15 ⊢ ((𝑅 ∈ Ring ∧ 𝐼 ∈ Fin) → (Base‘𝑅) = (Base‘(Scalar‘(𝑅 freeLMod 𝐼))))
103	102	oveq1d 7173	. . . . . . . . . . . . . 14 ⊢ ((𝑅 ∈ Ring ∧ 𝐼 ∈ Fin) → ((Base‘𝑅) ↑m 𝐼) = ((Base‘(Scalar‘(𝑅 freeLMod 𝐼))) ↑m 𝐼))
104	103	adantr 483	. . . . . . . . . . . . 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 8430	. . . . . . . . . . . . . . 15 ⊢ (𝑛 ∈ ((Base‘𝑅) ↑m 𝐼) → 𝑛:𝐼⟶(Base‘𝑅))
106		ffn 6516	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝑛:𝐼⟶(Base‘𝑅) → 𝑛 Fn 𝐼)
107	106	adantl 484	. . . . . . . . . . . . . . . . . . 19 ⊢ ((((𝑅 ∈ Ring ∧ 𝐼 ∈ Fin) ∧ curry 𝑀:𝐼⟶((Base‘𝑅) ↑m 𝐼)) ∧ 𝑛:𝐼⟶(Base‘𝑅)) → 𝑛 Fn 𝐼)
108	51	ad2antlr 725	. . . . . . . . . . . . . . . . . . 19 ⊢ ((((𝑅 ∈ Ring ∧ 𝐼 ∈ Fin) ∧ curry 𝑀:𝐼⟶((Base‘𝑅) ↑m 𝐼)) ∧ 𝑛:𝐼⟶(Base‘𝑅)) → curry 𝑀 Fn 𝐼)
109		simpllr 774	. . . . . . . . . . . . . . . . . . 19 ⊢ ((((𝑅 ∈ Ring ∧ 𝐼 ∈ Fin) ∧ curry 𝑀:𝐼⟶((Base‘𝑅) ↑m 𝐼)) ∧ 𝑛:𝐼⟶(Base‘𝑅)) → 𝐼 ∈ Fin)
110		inidm 4197	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝐼 ∩ 𝐼) = 𝐼
111		eqidd 2824	. . . . . . . . . . . . . . . . . . 19 ⊢ (((((𝑅 ∈ Ring ∧ 𝐼 ∈ Fin) ∧ curry 𝑀:𝐼⟶((Base‘𝑅) ↑m 𝐼)) ∧ 𝑛:𝐼⟶(Base‘𝑅)) ∧ 𝑘 ∈ 𝐼) → (𝑛‘𝑘) = (𝑛‘𝑘))
112		eqidd 2824	. . . . . . . . . . . . . . . . . . 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 7418	. . . . . . . . . . . . . . . . . 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 782	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (((((𝑅 ∈ Ring ∧ 𝐼 ∈ Fin) ∧ curry 𝑀:𝐼⟶((Base‘𝑅) ↑m 𝐼)) ∧ 𝑛:𝐼⟶(Base‘𝑅)) ∧ 𝑘 ∈ 𝐼) → 𝐼 ∈ Fin)
115		ffvelrn 6851	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ ((𝑛:𝐼⟶(Base‘𝑅) ∧ 𝑘 ∈ 𝐼) → (𝑛‘𝑘) ∈ (Base‘𝑅))
116	115	adantll 712	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (((((𝑅 ∈ Ring ∧ 𝐼 ∈ Fin) ∧ curry 𝑀:𝐼⟶((Base‘𝑅) ↑m 𝐼)) ∧ 𝑛:𝐼⟶(Base‘𝑅)) ∧ 𝑘 ∈ 𝐼) → (𝑛‘𝑘) ∈ (Base‘𝑅))
117		ffvelrn 6851	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ ((curry 𝑀:𝐼⟶((Base‘𝑅) ↑m 𝐼) ∧ 𝑘 ∈ 𝐼) → (curry 𝑀‘𝑘) ∈ ((Base‘𝑅) ↑m 𝐼))
118	117	ad4ant24 752	. . . . . . . . . . . . . . . . . . . . . 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 728	. . . . . . . . . . . . . . . . . . . . . 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 2917	. . . . . . . . . . . . . . . . . . . . 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 2823	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (.r‘𝑅) = (.r‘𝑅)
122	45, 46, 14, 114, 116, 120, 94, 121	frlmvscafval 20912	. . . . . . . . . . . . . . . . . . . 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 6685	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (𝑛‘𝑘) ∈ V
124		fnconstg 6569	. . . . . . . . . . . . . . . . . . . . . 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 8431	. . . . . . . . . . . . . . . . . . . . . . 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 752	. . . . . . . . . . . . . . . . . . . . 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 6968	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (𝑗 ∈ 𝐼 → ((𝐼 × {(𝑛‘𝑘)})‘𝑗) = (𝑛‘𝑘))
130	129	adantl 484	. . . . . . . . . . . . . . . . . . . . 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 2824	. . . . . . . . . . . . . . . . . . . . 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 7418	. . . . . . . . . . . . . . . . . . . 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 2858	. . . . . . . . . . . . . . . . . . 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 5163	. . . . . . . . . . . . . . . . . 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 2858	. . . . . . . . . . . . . . . . 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 7174	. . . . . . . . . . . . . . . 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 2823	. . . . . . . . . . . . . . . . 17 ⊢ (0g‘(𝑅 freeLMod 𝐼)) = (0g‘(𝑅 freeLMod 𝐼))
138		simplll 773	. . . . . . . . . . . . . . . . 17 ⊢ ((((𝑅 ∈ Ring ∧ 𝐼 ∈ Fin) ∧ curry 𝑀:𝐼⟶((Base‘𝑅) ↑m 𝐼)) ∧ 𝑛:𝐼⟶(Base‘𝑅)) → 𝑅 ∈ Ring)
139		simp-5l 783	. . . . . . . . . . . . . . . . . . . 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 751	. . . . . . . . . . . . . . . . . . . 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 767	. . . . . . . . . . . . . . . . . . . . 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 8430	. . . . . . . . . . . . . . . . . . . . . . 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	ffvelrnda 6853	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (((curry 𝑀:𝐼⟶((Base‘𝑅) ↑m 𝐼) ∧ 𝑘 ∈ 𝐼) ∧ 𝑗 ∈ 𝐼) → ((curry 𝑀‘𝑘)‘𝑗) ∈ (Base‘𝑅))
145	141, 144	sylanl1 678	. . . . . . . . . . . . . . . . . . . 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 19313	. . . . . . . . . . . . . . . . . . . 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 1367	. . . . . . . . . . . . . . . . . . 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 6881	. . . . . . . . . . . . . . . . . 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 8421	. . . . . . . . . . . . . . . . . . . . . 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 688	. . . . . . . . . . . . . . . . . . . . 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 484	. . . . . . . . . . . . . . . . . . . 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 2900	. . . . . . . . . . . . . . . . . . . 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 283	. . . . . . . . . . . . . . . . . . 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 728	. . . . . . . . . . . . . . . . . 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 234	. . . . . . . . . . . . . . . . 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 6986	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (𝐼 ∈ Fin → (𝑗 ∈ 𝐼 ↦ ((𝑛‘𝑘)(.r‘𝑅)((curry 𝑀‘𝑘)‘𝑗))) ∈ V)
157	156	ralrimivw 3185	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝐼 ∈ Fin → ∀𝑘 ∈ 𝐼 (𝑗 ∈ 𝐼 ↦ ((𝑛‘𝑘)(.r‘𝑅)((curry 𝑀‘𝑘)‘𝑗))) ∈ V)
158		eqid 2823	. . . . . . . . . . . . . . . . . . . . 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 6490	. . . . . . . . . . . . . . . . . . . 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 6687	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝐼 ∈ Fin → (0g‘(𝑅 freeLMod 𝐼)) ∈ V)
163	160, 161, 162	fndmfifsupp 8848	. . . . . . . . . . . . . . . . . 18 ⊢ (𝐼 ∈ Fin → (𝑘 ∈ 𝐼 ↦ (𝑗 ∈ 𝐼 ↦ ((𝑛‘𝑘)(.r‘𝑅)((curry 𝑀‘𝑘)‘𝑗)))) finSupp (0g‘(𝑅 freeLMod 𝐼)))
164	163	ad3antlr 729	. . . . . . . . . . . . . . . . 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 20918	. . . . . . . . . . . . . . . 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 2859	. . . . . . . . . . . . . . 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 594	. . . . . . . . . . . . . 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 2834	. . . . . . . . . . . . 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 3428	. . . . . . . . . . . 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 284	. . . . . . . . . . 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 678	. . . . . . . . . 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 724	. . . . . . . . 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 234	. . . . . . . 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 3184	. . . . . . 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 582	. . . . . 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 259	. . . . . . . . 9 ⊢ ((𝑅 ∈ DivRing ∧ 𝑀 ∈ (Base‘(𝐼 Mat 𝑅))) → 𝑀 ∈ ((Base‘𝑅) ↑m (𝐼 × 𝐼)))
177		elmapfn 8431	. . . . . . . . 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 484	. . . . . . . 8 ⊢ ((𝑅 ∈ DivRing ∧ 𝑀 ∈ (Base‘(𝐼 Mat 𝑅))) → 𝐼 ∈ Fin)
180		an32 644	. . . . . . . . . . . . . . . . . . 19 ⊢ (((𝑀 Fn (𝐼 × 𝐼) ∧ 𝑗 ∈ 𝐼) ∧ 𝑘 ∈ 𝐼) ↔ ((𝑀 Fn (𝐼 × 𝐼) ∧ 𝑘 ∈ 𝐼) ∧ 𝑗 ∈ 𝐼))
181		df-3an 1085	. . . . . . . . . . . . . . . . . . 19 ⊢ ((𝑀 Fn (𝐼 × 𝐼) ∧ 𝑘 ∈ 𝐼 ∧ 𝑗 ∈ 𝐼) ↔ ((𝑀 Fn (𝐼 × 𝐼) ∧ 𝑘 ∈ 𝐼) ∧ 𝑗 ∈ 𝐼))
182	180, 181	bitr4i 280	. . . . . . . . . . . . . . . . . 18 ⊢ (((𝑀 Fn (𝐼 × 𝐼) ∧ 𝑗 ∈ 𝐼) ∧ 𝑘 ∈ 𝐼) ↔ (𝑀 Fn (𝐼 × 𝐼) ∧ 𝑘 ∈ 𝐼 ∧ 𝑗 ∈ 𝐼))
183		curfv 34874	. . . . . . . . . . . . . . . . . 18 ⊢ (((𝑀 Fn (𝐼 × 𝐼) ∧ 𝑘 ∈ 𝐼 ∧ 𝑗 ∈ 𝐼) ∧ 𝐼 ∈ Fin) → ((curry 𝑀‘𝑘)‘𝑗) = (𝑘𝑀𝑗))
184	182, 183	sylanb 583	. . . . . . . . . . . . . . . . 17 ⊢ ((((𝑀 Fn (𝐼 × 𝐼) ∧ 𝑗 ∈ 𝐼) ∧ 𝑘 ∈ 𝐼) ∧ 𝐼 ∈ Fin) → ((curry 𝑀‘𝑘)‘𝑗) = (𝑘𝑀𝑗))
185	184	an32s 650	. . . . . . . . . . . . . . . 16 ⊢ ((((𝑀 Fn (𝐼 × 𝐼) ∧ 𝑗 ∈ 𝐼) ∧ 𝐼 ∈ Fin) ∧ 𝑘 ∈ 𝐼) → ((curry 𝑀‘𝑘)‘𝑗) = (𝑘𝑀𝑗))
186	185	oveq2d 7174	. . . . . . . . . . . . . . 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 5163	. . . . . . . . . . . . . 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 650	. . . . . . . . . . . . 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 7174	. . . . . . . . . . . 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 5163	. . . . . . . . . . 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 2834	. . . . . . . . . 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 3299	. . . . . . . . 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 3199	. . . . . . . 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 586	. . . . . . 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 724	. . . . . 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 234	. . . . 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 678	. . . 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 6671	. . . . . . . . . . 11 ⊢ (𝑛 = (𝑓‘𝑖) → (𝑛‘𝑘) = ((𝑓‘𝑖)‘𝑘))
199		uncov 34875	. . . . . . . . . . . 12 ⊢ ((𝑖 ∈ V ∧ 𝑘 ∈ V) → (𝑖uncurry 𝑓𝑘) = ((𝑓‘𝑖)‘𝑘))
200	199	el2v 3503	. . . . . . . . . . 11 ⊢ (𝑖uncurry 𝑓𝑘) = ((𝑓‘𝑖)‘𝑘)
201	198, 200	syl6eqr 2876	. . . . . . . . . 10 ⊢ (𝑛 = (𝑓‘𝑖) → (𝑛‘𝑘) = (𝑖uncurry 𝑓𝑘))
202	201	oveq1d 7173	. . . . . . . . 9 ⊢ (𝑛 = (𝑓‘𝑖) → ((𝑛‘𝑘)(.r‘𝑅)(𝑘𝑀𝑗)) = ((𝑖uncurry 𝑓𝑘)(.r‘𝑅)(𝑘𝑀𝑗)))
203	202	mpteq2dv 5164	. . . . . . . 8 ⊢ (𝑛 = (𝑓‘𝑖) → (𝑘 ∈ 𝐼 ↦ ((𝑛‘𝑘)(.r‘𝑅)(𝑘𝑀𝑗))) = (𝑘 ∈ 𝐼 ↦ ((𝑖uncurry 𝑓𝑘)(.r‘𝑅)(𝑘𝑀𝑗))))
204	203	oveq2d 7174	. . . . . . 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 5164	. . . . . 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 2834	. . . . 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 8764	. . . 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 586	. . 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 34873	. . . . . . 7 ⊢ (𝑓:𝐼⟶((Base‘𝑅) ↑m 𝐼) → uncurry 𝑓:(𝐼 × 𝐼)⟶(Base‘𝑅))
210	13, 14	frlmfibas 20908	. . . . . . . . . . . . . . . 16 ⊢ ((𝑅 ∈ Field ∧ (𝐼 × 𝐼) ∈ Fin) → ((Base‘𝑅) ↑m (𝐼 × 𝐼)) = (Base‘(𝑅 freeLMod (𝐼 × 𝐼))))
211	12, 210	sylan2 594	. . . . . . . . . . . . . . 15 ⊢ ((𝑅 ∈ Field ∧ 𝐼 ∈ Fin) → ((Base‘𝑅) ↑m (𝐼 × 𝐼)) = (Base‘(𝑅 freeLMod (𝐼 × 𝐼))))
212	1, 13	matbas 21024	. . . . . . . . . . . . . . . 16 ⊢ ((𝐼 ∈ Fin ∧ 𝑅 ∈ Field) → (Base‘(𝑅 freeLMod (𝐼 × 𝐼))) = (Base‘(𝐼 Mat 𝑅)))
213	212	ancoms 461	. . . . . . . . . . . . . . 15 ⊢ ((𝑅 ∈ Field ∧ 𝐼 ∈ Fin) → (Base‘(𝑅 freeLMod (𝐼 × 𝐼))) = (Base‘(𝐼 Mat 𝑅)))
214	211, 213	eqtrd 2858	. . . . . . . . . . . . . 14 ⊢ ((𝑅 ∈ Field ∧ 𝐼 ∈ Fin) → ((Base‘𝑅) ↑m (𝐼 × 𝐼)) = (Base‘(𝐼 Mat 𝑅)))
215	4, 214	sylan2 594	. . . . . . . . . . . . 13 ⊢ ((𝑅 ∈ Field ∧ 𝑀 ∈ (Base‘(𝐼 Mat 𝑅))) → ((Base‘𝑅) ↑m (𝐼 × 𝐼)) = (Base‘(𝐼 Mat 𝑅)))
216	215	eleq2d 2900	. . . . . . . . . . . 12 ⊢ ((𝑅 ∈ Field ∧ 𝑀 ∈ (Base‘(𝐼 Mat 𝑅))) → (uncurry 𝑓 ∈ ((Base‘𝑅) ↑m (𝐼 × 𝐼)) ↔ uncurry 𝑓 ∈ (Base‘(𝐼 Mat 𝑅))))
217		elmapg 8421	. . . . . . . . . . . . . 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 589	. . . . . . . . . . . . 13 ⊢ (𝑀 ∈ (Base‘(𝐼 Mat 𝑅)) → (uncurry 𝑓 ∈ ((Base‘𝑅) ↑m (𝐼 × 𝐼)) ↔ uncurry 𝑓:(𝐼 × 𝐼)⟶(Base‘𝑅)))
219	218	adantl 484	. . . . . . . . . . . 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 283	. . . . . . . . . . 11 ⊢ ((𝑅 ∈ Field ∧ 𝑀 ∈ (Base‘(𝐼 Mat 𝑅))) → (uncurry 𝑓 ∈ (Base‘(𝐼 Mat 𝑅)) ↔ uncurry 𝑓:(𝐼 × 𝐼)⟶(Base‘𝑅)))
221	220	biimpar 480	. . . . . . . . . 10 ⊢ (((𝑅 ∈ Field ∧ 𝑀 ∈ (Base‘(𝐼 Mat 𝑅))) ∧ uncurry 𝑓:(𝐼 × 𝐼)⟶(Base‘𝑅)) → uncurry 𝑓 ∈ (Base‘(𝐼 Mat 𝑅)))
222	221	adantr 483	. . . . . . . . 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 1915	. . . . . . . . . . . . . 14 ⊢ Ⅎ𝑗(((𝑅 ∈ Field ∧ 𝑀 ∈ (Base‘(𝐼 Mat 𝑅))) ∧ uncurry 𝑓:(𝐼 × 𝐼)⟶(Base‘𝑅)) ∧ 𝑖 ∈ 𝐼)
224		nfmpt1 5166	. . . . . . . . . . . . . . 15 ⊢ Ⅎ𝑗(𝑗 ∈ 𝐼 ↦ (𝑅 Σg (𝑘 ∈ 𝐼 ↦ ((𝑖uncurry 𝑓𝑘)(.r‘𝑅)(𝑘𝑀𝑗)))))
225	224	nfeq2 2997	. . . . . . . . . . . . . 14 ⊢ Ⅎ𝑗((𝑅 unitVec 𝐼)‘𝑖) = (𝑗 ∈ 𝐼 ↦ (𝑅 Σg (𝑘 ∈ 𝐼 ↦ ((𝑖uncurry 𝑓𝑘)(.r‘𝑅)(𝑘𝑀𝑗)))))
226		fveq1 6671	. . . . . . . . . . . . . . . . 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 614	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((𝑅 ∈ Field ∧ 𝑀 ∈ (Base‘(𝐼 Mat 𝑅))) → (𝑅 ∈ Ring ∧ 𝐼 ∈ Fin))
229	228	adantr 483	. . . . . . . . . . . . . . . . . . 19 ⊢ (((𝑅 ∈ Field ∧ 𝑀 ∈ (Base‘(𝐼 Mat 𝑅))) ∧ uncurry 𝑓:(𝐼 × 𝐼)⟶(Base‘𝑅)) → (𝑅 ∈ Ring ∧ 𝐼 ∈ Fin))
230		equcom 2025	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (𝑖 = 𝑗 ↔ 𝑗 = 𝑖)
231		ifbi 4490	. . . . . . . . . . . . . . . . . . . . 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 2823	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (1r‘𝑅) = (1r‘𝑅)
234		eqid 2823	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (0g‘𝑅) = (0g‘𝑅)
235		simpllr 774	. . . . . . . . . . . . . . . . . . . . 21 ⊢ ((((𝑅 ∈ Ring ∧ 𝐼 ∈ Fin) ∧ 𝑖 ∈ 𝐼) ∧ 𝑗 ∈ 𝐼) → 𝐼 ∈ Fin)
236		simplll 773	. . . . . . . . . . . . . . . . . . . . 21 ⊢ ((((𝑅 ∈ Ring ∧ 𝐼 ∈ Fin) ∧ 𝑖 ∈ 𝐼) ∧ 𝑗 ∈ 𝐼) → 𝑅 ∈ Ring)
237		simplr 767	. . . . . . . . . . . . . . . . . . . . 21 ⊢ ((((𝑅 ∈ Ring ∧ 𝐼 ∈ Fin) ∧ 𝑖 ∈ 𝐼) ∧ 𝑗 ∈ 𝐼) → 𝑖 ∈ 𝐼)
238		simpr 487	. . . . . . . . . . . . . . . . . . . . 21 ⊢ ((((𝑅 ∈ Ring ∧ 𝐼 ∈ Fin) ∧ 𝑖 ∈ 𝐼) ∧ 𝑗 ∈ 𝐼) → 𝑗 ∈ 𝐼)
239		eqid 2823	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (1r‘(𝐼 Mat 𝑅)) = (1r‘(𝐼 Mat 𝑅))
240	1, 233, 234, 235, 236, 237, 238, 239	mat1ov 21059	. . . . . . . . . . . . . . . . . . . 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 1085	. . . . . . . . . . . . . . . . . . . . 21 ⊢ ((𝑅 ∈ Ring ∧ 𝐼 ∈ Fin ∧ 𝑖 ∈ 𝐼) ↔ ((𝑅 ∈ Ring ∧ 𝐼 ∈ Fin) ∧ 𝑖 ∈ 𝐼))
242	44, 233, 234	uvcvval 20932	. . . . . . . . . . . . . . . . . . . . 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 584	. . . . . . . . . . . . . . . . . . . 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 2884	. . . . . . . . . . . . . . . . . . 19 ⊢ ((((𝑅 ∈ Ring ∧ 𝐼 ∈ Fin) ∧ 𝑖 ∈ 𝐼) ∧ 𝑗 ∈ 𝐼) → (𝑖(1r‘(𝐼 Mat 𝑅))𝑗) = (((𝑅 unitVec 𝐼)‘𝑖)‘𝑗))
245	229, 244	sylanl1 678	. . . . . . . . . . . . . . . . . 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 7191	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (𝑅 Σg (𝑘 ∈ 𝐼 ↦ ((𝑖uncurry 𝑓𝑘)(.r‘𝑅)(𝑘𝑀𝑗)))) ∈ V
247		eqid 2823	. . . . . . . . . . . . . . . . . . . . . 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 6781	. . . . . . . . . . . . . . . . . . . . 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 689	. . . . . . . . . . . . . . . . . . . 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 484	. . . . . . . . . . . . . . . . . . 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 2823	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝑅 maMul ⟨𝐼, 𝐼, 𝐼⟩) = (𝑅 maMul ⟨𝐼, 𝐼, 𝐼⟩)
252		simp-4l 781	. . . . . . . . . . . . . . . . . . . 20 ⊢ (((((𝑅 ∈ Field ∧ 𝑀 ∈ (Base‘(𝐼 Mat 𝑅))) ∧ uncurry 𝑓:(𝐼 × 𝐼)⟶(Base‘𝑅)) ∧ 𝑖 ∈ 𝐼) ∧ 𝑗 ∈ 𝐼) → 𝑅 ∈ Field)
253	4	ad4antlr 731	. . . . . . . . . . . . . . . . . . . 20 ⊢ (((((𝑅 ∈ Field ∧ 𝑀 ∈ (Base‘(𝐼 Mat 𝑅))) ∧ uncurry 𝑓:(𝐼 × 𝐼)⟶(Base‘𝑅)) ∧ 𝑖 ∈ 𝐼) ∧ 𝑗 ∈ 𝐼) → 𝐼 ∈ Fin)
254	218	biimpar 480	. . . . . . . . . . . . . . . . . . . . 21 ⊢ ((𝑀 ∈ (Base‘(𝐼 Mat 𝑅)) ∧ uncurry 𝑓:(𝐼 × 𝐼)⟶(Base‘𝑅)) → uncurry 𝑓 ∈ ((Base‘𝑅) ↑m (𝐼 × 𝐼)))
255	254	ad5ant23 758	. . . . . . . . . . . . . . . . . . . 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 487	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ ((𝑅 ∈ Field ∧ 𝑀 ∈ (Base‘(𝐼 Mat 𝑅))) → 𝑀 ∈ (Base‘(𝐼 Mat 𝑅)))
257	256, 215	eleqtrrd 2918	. . . . . . . . . . . . . . . . . . . . 21 ⊢ ((𝑅 ∈ Field ∧ 𝑀 ∈ (Base‘(𝐼 Mat 𝑅))) → 𝑀 ∈ ((Base‘𝑅) ↑m (𝐼 × 𝐼)))
258	257	ad3antrrr 728	. . . . . . . . . . . . . . . . . . . 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 767	. . . . . . . . . . . . . . . . . . . 20 ⊢ (((((𝑅 ∈ Field ∧ 𝑀 ∈ (Base‘(𝐼 Mat 𝑅))) ∧ uncurry 𝑓:(𝐼 × 𝐼)⟶(Base‘𝑅)) ∧ 𝑖 ∈ 𝐼) ∧ 𝑗 ∈ 𝐼) → 𝑖 ∈ 𝐼)
260		simpr 487	. . . . . . . . . . . . . . . . . . . 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 21000	. . . . . . . . . . . . . . . . . . 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 21049	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ ((𝐼 ∈ Fin ∧ 𝑅 ∈ Field) → (𝑅 maMul ⟨𝐼, 𝐼, 𝐼⟩) = (.r‘(𝐼 Mat 𝑅)))
263	262	ancoms 461	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ ((𝑅 ∈ Field ∧ 𝐼 ∈ Fin) → (𝑅 maMul ⟨𝐼, 𝐼, 𝐼⟩) = (.r‘(𝐼 Mat 𝑅)))
264	263	oveqd 7175	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ ((𝑅 ∈ Field ∧ 𝐼 ∈ Fin) → (uncurry 𝑓(𝑅 maMul ⟨𝐼, 𝐼, 𝐼⟩)𝑀) = (uncurry 𝑓(.r‘(𝐼 Mat 𝑅))𝑀))
265	264	oveqd 7175	. . . . . . . . . . . . . . . . . . . . 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 594	. . . . . . . . . . . . . . . . . . . 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 728	. . . . . . . . . . . . . . . . . . 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 2865	. . . . . . . . . . . . . . . . . 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 2839	. . . . . . . . . . . . . . . . 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	syl5ibr 248	. . . . . . . . . . . . . . . 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 415	. . . . . . . . . . . . . . 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 3220	. . . . . . . . . . . . 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 3179	. . . . . . . . . . . 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 21060	. . . . . . . . . . . . . . . . 17 ⊢ ((𝑅 ∈ Ring ∧ 𝐼 ∈ Fin) → (1r‘(𝐼 Mat 𝑅)) ∈ (Base‘(𝐼 Mat 𝑅)))
276	13, 14	frlmfibas 20908	. . . . . . . . . . . . . . . . . . 19 ⊢ ((𝑅 ∈ Ring ∧ (𝐼 × 𝐼) ∈ Fin) → ((Base‘𝑅) ↑m (𝐼 × 𝐼)) = (Base‘(𝑅 freeLMod (𝐼 × 𝐼))))
277	12, 276	sylan2 594	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝑅 ∈ Ring ∧ 𝐼 ∈ Fin) → ((Base‘𝑅) ↑m (𝐼 × 𝐼)) = (Base‘(𝑅 freeLMod (𝐼 × 𝐼))))
278	1, 13	matbas 21024	. . . . . . . . . . . . . . . . . . 19 ⊢ ((𝐼 ∈ Fin ∧ 𝑅 ∈ Ring) → (Base‘(𝑅 freeLMod (𝐼 × 𝐼))) = (Base‘(𝐼 Mat 𝑅)))
279	278	ancoms 461	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝑅 ∈ Ring ∧ 𝐼 ∈ Fin) → (Base‘(𝑅 freeLMod (𝐼 × 𝐼))) = (Base‘(𝐼 Mat 𝑅)))
280	277, 279	eqtrd 2858	. . . . . . . . . . . . . . . . 17 ⊢ ((𝑅 ∈ Ring ∧ 𝐼 ∈ Fin) → ((Base‘𝑅) ↑m (𝐼 × 𝐼)) = (Base‘(𝐼 Mat 𝑅)))
281	275, 280	eleqtrrd 2918	. . . . . . . . . . . . . . . 16 ⊢ ((𝑅 ∈ Ring ∧ 𝐼 ∈ Fin) → (1r‘(𝐼 Mat 𝑅)) ∈ ((Base‘𝑅) ↑m (𝐼 × 𝐼)))
282		elmapfn 8431	. . . . . . . . . . . . . . . 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 597	. . . . . . . . . . . . . 14 ⊢ ((𝑅 ∈ Field ∧ 𝑀 ∈ (Base‘(𝐼 Mat 𝑅))) → (1r‘(𝐼 Mat 𝑅)) Fn (𝐼 × 𝐼))
285	284	adantr 483	. . . . . . . . . . . . 13 ⊢ (((𝑅 ∈ Field ∧ 𝑀 ∈ (Base‘(𝐼 Mat 𝑅))) ∧ uncurry 𝑓:(𝐼 × 𝐼)⟶(Base‘𝑅)) → (1r‘(𝐼 Mat 𝑅)) Fn (𝐼 × 𝐼))
286	1	matring 21054	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝐼 ∈ Fin ∧ 𝑅 ∈ Ring) → (𝐼 Mat 𝑅) ∈ Ring)
287	4, 227, 286	syl2anr 598	. . . . . . . . . . . . . . . . 17 ⊢ ((𝑅 ∈ Field ∧ 𝑀 ∈ (Base‘(𝐼 Mat 𝑅))) → (𝐼 Mat 𝑅) ∈ Ring)
288	287	adantr 483	. . . . . . . . . . . . . . . 16 ⊢ (((𝑅 ∈ Field ∧ 𝑀 ∈ (Base‘(𝐼 Mat 𝑅))) ∧ uncurry 𝑓:(𝐼 × 𝐼)⟶(Base‘𝑅)) → (𝐼 Mat 𝑅) ∈ Ring)
289		simplr 767	. . . . . . . . . . . . . . . 16 ⊢ (((𝑅 ∈ Field ∧ 𝑀 ∈ (Base‘(𝐼 Mat 𝑅))) ∧ uncurry 𝑓:(𝐼 × 𝐼)⟶(Base‘𝑅)) → 𝑀 ∈ (Base‘(𝐼 Mat 𝑅)))
290		eqid 2823	. . . . . . . . . . . . . . . . 17 ⊢ (.r‘(𝐼 Mat 𝑅)) = (.r‘(𝐼 Mat 𝑅))
291	2, 290	ringcl 19313	. . . . . . . . . . . . . . . 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 1367	. . . . . . . . . . . . . . 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 483	. . . . . . . . . . . . . . 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 2918	. . . . . . . . . . . . . 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 8431	. . . . . . . . . . . . . 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 7283	. . . . . . . . . . . . 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 586	. . . . . . . . . . . 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 261	. . . . . . . . . . 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 409	. . . . . . . . . 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 2829	. . . . . . . . 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 7165	. . . . . . . . . . 11 ⊢ (𝑛 = uncurry 𝑓 → (𝑛(.r‘(𝐼 Mat 𝑅))𝑀) = (uncurry 𝑓(.r‘(𝐼 Mat 𝑅))𝑀))
303	302	eqeq1d 2825	. . . . . . . . . 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 3625	. . . . . . . . 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 586	. . . . . . . 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 460	. . . . . . 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 605	. . . . . 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 1934	. . . . 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 409	. . . 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 713	. . 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 593	. 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 499	. . . 4 ⊢ (𝑅 ∈ Field → 𝑅 ∈ CRing)
313		eqid 2823	. . . . . . . . . 10 ⊢ (𝐼 maDet 𝑅) = (𝐼 maDet 𝑅)
314	313, 1, 2, 14	mdetcl 21207	. . . . . . . . 9 ⊢ ((𝑅 ∈ CRing ∧ 𝑀 ∈ (Base‘(𝐼 Mat 𝑅))) → ((𝐼 maDet 𝑅)‘𝑀) ∈ (Base‘𝑅))
315	313, 1, 2, 14	mdetcl 21207	. . . . . . . . 9 ⊢ ((𝑅 ∈ CRing ∧ 𝑛 ∈ (Base‘(𝐼 Mat 𝑅))) → ((𝐼 maDet 𝑅)‘𝑛) ∈ (Base‘𝑅))
316		eqid 2823	. . . . . . . . . 10 ⊢ (∥r‘𝑅) = (∥r‘𝑅)
317	14, 316, 121	dvdsrmul 19400	. . . . . . . . 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 597	. . . . . . . 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 676	. . . . . . 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 470	. . . . . 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 715	. . . . 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 6672	. . . . . . . . 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 21234	. . . . . . . . . . . 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 1114	. . . . . . . . . . 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 650	. . . . . . . . . 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 21212	. . . . . . . . . . . 12 ⊢ ((𝑅 ∈ CRing ∧ 𝐼 ∈ Fin) → ((𝐼 maDet 𝑅)‘(1r‘(𝐼 Mat 𝑅))) = (1r‘𝑅))
327	4, 326	sylan2 594	. . . . . . . . . . 11 ⊢ ((𝑅 ∈ CRing ∧ 𝑀 ∈ (Base‘(𝐼 Mat 𝑅))) → ((𝐼 maDet 𝑅)‘(1r‘(𝐼 Mat 𝑅))) = (1r‘𝑅))
328	327	adantr 483	. . . . . . . . . 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 2839	. . . . . . . . 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	syl5ib 246	. . . . . . . 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 457	. . . . . . 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 5080	. . . . . 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 2823	. . . . . . . 8 ⊢ (Unit‘𝑅) = (Unit‘𝑅)
334	333, 233, 316	crngunit 19414	. . . . . . 7 ⊢ (𝑅 ∈ CRing → (((𝐼 maDet 𝑅)‘𝑀) ∈ (Unit‘𝑅) ↔ ((𝐼 maDet 𝑅)‘𝑀)(∥r‘𝑅)(1r‘𝑅)))
335	334	ad2antrr 724	. . . . . 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 284	. . . . 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 234	. . . 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 678	. . 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 750	. 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 3293	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 208   ∧ wa 398   ∧ w3a 1083   = wceq 1537  ∃wex 1780   ∈ wcel 2114   ≠ wne 3018  ∀wral 3140  ∃wrex 3141  Vcvv 3496   ∖ cdif 3935  ∅c0 4293  ifcif 4469  {csn 4569  ⟨cotp 4577   class class class wbr 5068   ↦ cmpt 5148   × cxp 5555  dom cdm 5557  ran crn 5558   “ cima 5560   Fn wfn 6352  ⟶wf 6353  –1-1→wf1 6354  –1-1-onto→wf1o 6356  ‘cfv 6357  (class class class)co 7158   ∘_f cof 7409  curry ccur 7933  uncurry cunc 7934   ↑_m cmap 8408   ≈ cen 8508  Fincfn 8511   finSupp cfsupp 8835  Basecbs 16485  ._rcmulr 16568  Scalarcsca 16570   ·_𝑠 cvsca 16571  0_gc0g 16715   Σ_g cgsu 16716  1_rcur 19253  Ringcrg 19299  CRingccrg 19300  ∥_rcdsr 19390  Unitcui 19391  DivRingcdr 19504  Fieldcfield 19505  LModclmod 19636  LSpanclspn 19745  LBasisclbs 19848  NzRingcnzr 20032   freeLMod cfrlm 20892   unitVec cuvc 20928   LIndF clindf 20950  LIndSclinds 20951   maMul cmmul 20996   Mat cmat 21018   maDet cmdat 21195

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1911  ax-6 1970  ax-7 2015  ax-8 2116  ax-9 2124  ax-10 2145  ax-11 2161  ax-12 2177  ax-ext 2795  ax-rep 5192  ax-sep 5205  ax-nul 5212  ax-pow 5268  ax-pr 5332  ax-un 7463  ax-cnex 10595  ax-resscn 10596  ax-1cn 10597  ax-icn 10598  ax-addcl 10599  ax-addrcl 10600  ax-mulcl 10601  ax-mulrcl 10602  ax-mulcom 10603  ax-addass 10604  ax-mulass 10605  ax-distr 10606  ax-i2m1 10607  ax-1ne0 10608  ax-1rid 10609  ax-rnegex 10610  ax-rrecex 10611  ax-cnre 10612  ax-pre-lttri 10613  ax-pre-lttrn 10614  ax-pre-ltadd 10615  ax-pre-mulgt0 10616  ax-addf 10618  ax-mulf 10619

This theorem depends on definitions:  df-bi 209  df-an 399  df-or 844  df-3or 1084  df-3an 1085  df-xor 1502  df-tru 1540  df-fal 1550  df-ex 1781  df-nf 1785  df-sb 2070  df-mo 2622  df-eu 2654  df-clab 2802  df-cleq 2816  df-clel 2895  df-nfc 2965  df-ne 3019  df-nel 3126  df-ral 3145  df-rex 3146  df-reu 3147  df-rmo 3148  df-rab 3149  df-v 3498  df-sbc 3775  df-csb 3886  df-dif 3941  df-un 3943  df-in 3945  df-ss 3954  df-pss 3956  df-nul 4294  df-if 4470  df-pw 4543  df-sn 4570  df-pr 4572  df-tp 4574  df-op 4576  df-ot 4578  df-uni 4841  df-int 4879  df-iun 4923  df-iin 4924  df-br 5069  df-opab 5131  df-mpt 5149  df-tr 5175  df-id 5462  df-eprel 5467  df-po 5476  df-so 5477  df-fr 5516  df-se 5517  df-we 5518  df-xp 5563  df-rel 5564  df-cnv 5565  df-co 5566  df-dm 5567  df-rn 5568  df-res 5569  df-ima 5570  df-pred 6150  df-ord 6196  df-on 6197  df-lim 6198  df-suc 6199  df-iota 6316  df-fun 6359  df-fn 6360  df-f 6361  df-f1 6362  df-fo 6363  df-f1o 6364  df-fv 6365  df-isom 6366  df-riota 7116  df-ov 7161  df-oprab 7162  df-mpo 7163  df-of 7411  df-om 7583  df-1st 7691  df-2nd 7692  df-supp 7833  df-tpos 7894  df-cur 7935  df-unc 7936  df-wrecs 7949  df-recs 8010  df-rdg 8048  df-1o 8104  df-2o 8105  df-oadd 8108  df-er 8291  df-map 8410  df-pm 8411  df-ixp 8464  df-en 8512  df-dom 8513  df-sdom 8514  df-fin 8515  df-fsupp 8836  df-sup 8908  df-oi 8976  df-card 9370  df-pnf 10679  df-mnf 10680  df-xr 10681  df-ltxr 10682  df-le 10683  df-sub 10874  df-neg 10875  df-div 11300  df-nn 11641  df-2 11703  df-3 11704  df-4 11705  df-5 11706  df-6 11707  df-7 11708  df-8 11709  df-9 11710  df-n0 11901  df-xnn0 11971  df-z 11985  df-dec 12102  df-uz 12247  df-rp 12393  df-fz 12896  df-fzo 13037  df-seq 13373  df-exp 13433  df-hash 13694  df-word 13865  df-lsw 13917  df-concat 13925  df-s1 13952  df-substr 14005  df-pfx 14035  df-splice 14114  df-reverse 14123  df-s2 14212  df-struct 16487  df-ndx 16488  df-slot 16489  df-base 16491  df-sets 16492  df-ress 16493  df-plusg 16580  df-mulr 16581  df-starv 16582  df-sca 16583  df-vsca 16584  df-ip 16585  df-tset 16586  df-ple 16587  df-ds 16589  df-unif 16590  df-hom 16591  df-cco 16592  df-0g 16717  df-gsum 16718  df-prds 16723  df-pws 16725  df-mre 16859  df-mrc 16860  df-mri 16861  df-acs 16862  df-mgm 17854  df-sgrp 17903  df-mnd 17914  df-mhm 17958  df-submnd 17959  df-efmnd 18036  df-grp 18108  df-minusg 18109  df-sbg 18110  df-mulg 18227  df-subg 18278  df-ghm 18358  df-gim 18401  df-cntz 18449  df-oppg 18476  df-symg 18498  df-pmtr 18572  df-psgn 18621  df-evpm 18622  df-cmn 18910  df-abl 18911  df-mgp 19242  df-ur 19254  df-srg 19258  df-ring 19301  df-cring 19302  df-oppr 19375  df-dvdsr 19393  df-unit 19394  df-invr 19424  df-dvr 19435  df-rnghom 19469  df-drng 19506  df-field 19507  df-subrg 19535  df-lmod 19638  df-lss 19706  df-lsp 19746  df-lmhm 19796  df-lbs 19849  df-lvec 19877  df-sra 19946  df-rgmod 19947  df-nzr 20033  df-cnfld 20548  df-zring 20620  df-zrh 20653  df-dsmm 20878  df-frlm 20893  df-uvc 20929  df-lindf 20952  df-linds 20953  df-mamu 20997  df-mat 21019  df-mdet 21196

This theorem is referenced by:  matunitlindf  34892