Theorem matunitlindflem2 32576

Description: One direction of matunitlindf 32577. (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 2610	. . . . . . 7 ⊢ (𝐼 Mat 𝑅) = (𝐼 Mat 𝑅)
2		eqid 2610	. . . . . . 7 ⊢ (Base‘(𝐼 Mat 𝑅)) = (Base‘(𝐼 Mat 𝑅))
3	1, 2	matrcl 20037	. . . . . 6 ⊢ (𝑀 ∈ (Base‘(𝐼 Mat 𝑅)) → (𝐼 ∈ Fin ∧ 𝑅 ∈ V))
4	3	simpld 474	. . . . 5 ⊢ (𝑀 ∈ (Base‘(𝐼 Mat 𝑅)) → 𝐼 ∈ Fin)
5	4	ad3antlr 763	. . . 4 ⊢ ((((𝑅 ∈ Field ∧ 𝑀 ∈ (Base‘(𝐼 Mat 𝑅))) ∧ 𝐼 ≠ ∅) ∧ curry 𝑀 LIndF (𝑅 freeLMod 𝐼)) → 𝐼 ∈ Fin)
6		isfld 18579	. . . . . . 7 ⊢ (𝑅 ∈ Field ↔ (𝑅 ∈ DivRing ∧ 𝑅 ∈ CRing))
7	6	simplbi 475	. . . . . 6 ⊢ (𝑅 ∈ Field → 𝑅 ∈ DivRing)
8	7	anim1i 590	. . . . 5 ⊢ ((𝑅 ∈ Field ∧ 𝑀 ∈ (Base‘(𝐼 Mat 𝑅))) → (𝑅 ∈ DivRing ∧ 𝑀 ∈ (Base‘(𝐼 Mat 𝑅))))
9	4	ad2antrl 760	. . . . . . . . . . . 12 ⊢ ((𝑅 ∈ DivRing ∧ (𝑀 ∈ (Base‘(𝐼 Mat 𝑅)) ∧ 𝐼 ≠ ∅)) → 𝐼 ∈ Fin)
10		simpr 476	. . . . . . . . . . . . . . 15 ⊢ ((𝑅 ∈ DivRing ∧ 𝑀 ∈ (Base‘(𝐼 Mat 𝑅))) → 𝑀 ∈ (Base‘(𝐼 Mat 𝑅)))
11		xpfi 8116	. . . . . . . . . . . . . . . . . . . . 21 ⊢ ((𝐼 ∈ Fin ∧ 𝐼 ∈ Fin) → (𝐼 × 𝐼) ∈ Fin)
12	11	anidms 675	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝐼 ∈ Fin → (𝐼 × 𝐼) ∈ Fin)
13		eqid 2610	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (𝑅 freeLMod (𝐼 × 𝐼)) = (𝑅 freeLMod (𝐼 × 𝐼))
14		eqid 2610	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (Base‘𝑅) = (Base‘𝑅)
15	13, 14	frlmfibas 19924	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((𝑅 ∈ DivRing ∧ (𝐼 × 𝐼) ∈ Fin) → ((Base‘𝑅) ↑𝑚 (𝐼 × 𝐼)) = (Base‘(𝑅 freeLMod (𝐼 × 𝐼))))
16	12, 15	sylan2 490	. . . . . . . . . . . . . . . . . . 19 ⊢ ((𝑅 ∈ DivRing ∧ 𝐼 ∈ Fin) → ((Base‘𝑅) ↑𝑚 (𝐼 × 𝐼)) = (Base‘(𝑅 freeLMod (𝐼 × 𝐼))))
17	1, 13	matbas 20038	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((𝐼 ∈ Fin ∧ 𝑅 ∈ DivRing) → (Base‘(𝑅 freeLMod (𝐼 × 𝐼))) = (Base‘(𝐼 Mat 𝑅)))
18	17	ancoms 468	. . . . . . . . . . . . . . . . . . 19 ⊢ ((𝑅 ∈ DivRing ∧ 𝐼 ∈ Fin) → (Base‘(𝑅 freeLMod (𝐼 × 𝐼))) = (Base‘(𝐼 Mat 𝑅)))
19	16, 18	eqtrd 2644	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝑅 ∈ DivRing ∧ 𝐼 ∈ Fin) → ((Base‘𝑅) ↑𝑚 (𝐼 × 𝐼)) = (Base‘(𝐼 Mat 𝑅)))
20	19	eleq2d 2673	. . . . . . . . . . . . . . . . 17 ⊢ ((𝑅 ∈ DivRing ∧ 𝐼 ∈ Fin) → (𝑀 ∈ ((Base‘𝑅) ↑𝑚 (𝐼 × 𝐼)) ↔ 𝑀 ∈ (Base‘(𝐼 Mat 𝑅))))
21	4, 20	sylan2 490	. . . . . . . . . . . . . . . 16 ⊢ ((𝑅 ∈ DivRing ∧ 𝑀 ∈ (Base‘(𝐼 Mat 𝑅))) → (𝑀 ∈ ((Base‘𝑅) ↑𝑚 (𝐼 × 𝐼)) ↔ 𝑀 ∈ (Base‘(𝐼 Mat 𝑅))))
22		fvex 6113	. . . . . . . . . . . . . . . . . 18 ⊢ (Base‘𝑅) ∈ V
23	4, 4, 11	syl2anc 691	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑀 ∈ (Base‘(𝐼 Mat 𝑅)) → (𝐼 × 𝐼) ∈ Fin)
24		elmapg 7757	. . . . . . . . . . . . . . . . . 18 ⊢ (((Base‘𝑅) ∈ V ∧ (𝐼 × 𝐼) ∈ Fin) → (𝑀 ∈ ((Base‘𝑅) ↑𝑚 (𝐼 × 𝐼)) ↔ 𝑀:(𝐼 × 𝐼)⟶(Base‘𝑅)))
25	22, 23, 24	sylancr 694	. . . . . . . . . . . . . . . . 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 269	. . . . . . . . . . . . . . 15 ⊢ ((𝑅 ∈ DivRing ∧ 𝑀 ∈ (Base‘(𝐼 Mat 𝑅))) → (𝑀 ∈ (Base‘(𝐼 Mat 𝑅)) ↔ 𝑀:(𝐼 × 𝐼)⟶(Base‘𝑅)))
28	10, 27	mpbid 221	. . . . . . . . . . . . . 14 ⊢ ((𝑅 ∈ DivRing ∧ 𝑀 ∈ (Base‘(𝐼 Mat 𝑅))) → 𝑀:(𝐼 × 𝐼)⟶(Base‘𝑅))
29	28	adantrr 749	. . . . . . . . . . . . 13 ⊢ ((𝑅 ∈ DivRing ∧ (𝑀 ∈ (Base‘(𝐼 Mat 𝑅)) ∧ 𝐼 ≠ ∅)) → 𝑀:(𝐼 × 𝐼)⟶(Base‘𝑅))
30		eldifsn 4260	. . . . . . . . . . . . . . . 16 ⊢ (𝐼 ∈ (Fin ∖ {∅}) ↔ (𝐼 ∈ Fin ∧ 𝐼 ≠ ∅))
31	30	biimpri 217	. . . . . . . . . . . . . . 15 ⊢ ((𝐼 ∈ Fin ∧ 𝐼 ≠ ∅) → 𝐼 ∈ (Fin ∖ {∅}))
32	4, 31	sylan 487	. . . . . . . . . . . . . 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 32557	. . . . . . . . . . . . . 14 ⊢ ((𝑀:(𝐼 × 𝐼)⟶(Base‘𝑅) ∧ 𝐼 ∈ (Fin ∖ {∅}) ∧ (Base‘𝑅) ∈ V) → curry 𝑀:𝐼⟶((Base‘𝑅) ↑𝑚 𝐼))
35	22, 34	mp3an3 1405	. . . . . . . . . . . . 13 ⊢ ((𝑀:(𝐼 × 𝐼)⟶(Base‘𝑅) ∧ 𝐼 ∈ (Fin ∖ {∅})) → curry 𝑀:𝐼⟶((Base‘𝑅) ↑𝑚 𝐼))
36	29, 33, 35	syl2anc 691	. . . . . . . . . . . 12 ⊢ ((𝑅 ∈ DivRing ∧ (𝑀 ∈ (Base‘(𝐼 Mat 𝑅)) ∧ 𝐼 ≠ ∅)) → curry 𝑀:𝐼⟶((Base‘𝑅) ↑𝑚 𝐼))
37	9, 36	jca 553	. . . . . . . . . . 11 ⊢ ((𝑅 ∈ DivRing ∧ (𝑀 ∈ (Base‘(𝐼 Mat 𝑅)) ∧ 𝐼 ≠ ∅)) → (𝐼 ∈ Fin ∧ curry 𝑀:𝐼⟶((Base‘𝑅) ↑𝑚 𝐼)))
38	37	ex 449	. . . . . . . . . 10 ⊢ (𝑅 ∈ DivRing → ((𝑀 ∈ (Base‘(𝐼 Mat 𝑅)) ∧ 𝐼 ≠ ∅) → (𝐼 ∈ Fin ∧ curry 𝑀:𝐼⟶((Base‘𝑅) ↑𝑚 𝐼))))
39	38	imdistani 722	. . . . . . . . 9 ⊢ ((𝑅 ∈ DivRing ∧ (𝑀 ∈ (Base‘(𝐼 Mat 𝑅)) ∧ 𝐼 ≠ ∅)) → (𝑅 ∈ DivRing ∧ (𝐼 ∈ Fin ∧ curry 𝑀:𝐼⟶((Base‘𝑅) ↑𝑚 𝐼))))
40	39	anassrs 678	. . . . . . . 8 ⊢ (((𝑅 ∈ DivRing ∧ 𝑀 ∈ (Base‘(𝐼 Mat 𝑅))) ∧ 𝐼 ≠ ∅) → (𝑅 ∈ DivRing ∧ (𝐼 ∈ Fin ∧ curry 𝑀:𝐼⟶((Base‘𝑅) ↑𝑚 𝐼))))
41		anass 679	. . . . . . . 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 223	. . . . . . 7 ⊢ (((𝑅 ∈ DivRing ∧ 𝑀 ∈ (Base‘(𝐼 Mat 𝑅))) ∧ 𝐼 ≠ ∅) → ((𝑅 ∈ DivRing ∧ 𝐼 ∈ Fin) ∧ curry 𝑀:𝐼⟶((Base‘𝑅) ↑𝑚 𝐼)))
43		drngring 18577	. . . . . . . . . . . . 13 ⊢ (𝑅 ∈ DivRing → 𝑅 ∈ Ring)
44		eqid 2610	. . . . . . . . . . . . . 14 ⊢ (𝑅 unitVec 𝐼) = (𝑅 unitVec 𝐼)
45		eqid 2610	. . . . . . . . . . . . . 14 ⊢ (𝑅 freeLMod 𝐼) = (𝑅 freeLMod 𝐼)
46		eqid 2610	. . . . . . . . . . . . . 14 ⊢ (Base‘(𝑅 freeLMod 𝐼)) = (Base‘(𝑅 freeLMod 𝐼))
47	44, 45, 46	uvcff 19949	. . . . . . . . . . . . 13 ⊢ ((𝑅 ∈ Ring ∧ 𝐼 ∈ Fin) → (𝑅 unitVec 𝐼):𝐼⟶(Base‘(𝑅 freeLMod 𝐼)))
48	43, 47	sylan 487	. . . . . . . . . . . 12 ⊢ ((𝑅 ∈ DivRing ∧ 𝐼 ∈ Fin) → (𝑅 unitVec 𝐼):𝐼⟶(Base‘(𝑅 freeLMod 𝐼)))
49	48	ffvelrnda 6267	. . . . . . . . . . 11 ⊢ (((𝑅 ∈ DivRing ∧ 𝐼 ∈ Fin) ∧ 𝑖 ∈ 𝐼) → ((𝑅 unitVec 𝐼)‘𝑖) ∈ (Base‘(𝑅 freeLMod 𝐼)))
50	49	ad4ant14 1285	. . . . . . . . . 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 5958	. . . . . . . . . . . . . . . 16 ⊢ (curry 𝑀:𝐼⟶((Base‘𝑅) ↑𝑚 𝐼) → curry 𝑀 Fn 𝐼)
52		fnima 5923	. . . . . . . . . . . . . . . 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 6107	. . . . . . . . . . . . 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 794	. . . . . . . . . . . . . 14 ⊢ ((((𝑅 ∈ DivRing ∧ 𝐼 ∈ Fin) ∧ curry 𝑀:𝐼⟶((Base‘𝑅) ↑𝑚 𝐼)) ∧ curry 𝑀 LIndF (𝑅 freeLMod 𝐼)) → 𝑅 ∈ DivRing)
58		simpllr 795	. . . . . . . . . . . . . 14 ⊢ ((((𝑅 ∈ DivRing ∧ 𝐼 ∈ Fin) ∧ curry 𝑀:𝐼⟶((Base‘𝑅) ↑𝑚 𝐼)) ∧ curry 𝑀 LIndF (𝑅 freeLMod 𝐼)) → 𝐼 ∈ Fin)
59	45	frlmlmod 19912	. . . . . . . . . . . . . . . . 17 ⊢ ((𝑅 ∈ Ring ∧ 𝐼 ∈ Fin) → (𝑅 freeLMod 𝐼) ∈ LMod)
60	43, 59	sylan 487	. . . . . . . . . . . . . . . 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 19979	. . . . . . . . . . . . . . 15 ⊢ (((𝑅 freeLMod 𝐼) ∈ LMod ∧ curry 𝑀 LIndF (𝑅 freeLMod 𝐼)) → ran curry 𝑀 ∈ (LIndS‘(𝑅 freeLMod 𝐼)))
63	61, 62	sylan 487	. . . . . . . . . . . . . 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 19916	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ ((𝑅 ∈ DivRing ∧ 𝐼 ∈ Fin) → 𝑅 = (Scalar‘(𝑅 freeLMod 𝐼)))
65		drngnzr 19083	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ (𝑅 ∈ DivRing → 𝑅 ∈ NzRing)
66	65	adantr 480	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ ((𝑅 ∈ DivRing ∧ 𝐼 ∈ Fin) → 𝑅 ∈ NzRing)
67	64, 66	eqeltrrd 2689	. . . . . . . . . . . . . . . . . . . . 21 ⊢ ((𝑅 ∈ DivRing ∧ 𝐼 ∈ Fin) → (Scalar‘(𝑅 freeLMod 𝐼)) ∈ NzRing)
68	60, 67	jca 553	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((𝑅 ∈ DivRing ∧ 𝐼 ∈ Fin) → ((𝑅 freeLMod 𝐼) ∈ LMod ∧ (Scalar‘(𝑅 freeLMod 𝐼)) ∈ NzRing))
69		eqid 2610	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (Scalar‘(𝑅 freeLMod 𝐼)) = (Scalar‘(𝑅 freeLMod 𝐼))
70	46, 69	lindff1 19978	. . . . . . . . . . . . . . . . . . . . 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 1257	. . . . . . . . . . . . . . . . . . . 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 487	. . . . . . . . . . . . . . . . . . 19 ⊢ (((𝑅 ∈ DivRing ∧ 𝐼 ∈ Fin) ∧ curry 𝑀 LIndF (𝑅 freeLMod 𝐼)) → curry 𝑀:dom curry 𝑀–1-1→(Base‘(𝑅 freeLMod 𝐼)))
73		fdm 5964	. . . . . . . . . . . . . . . . . . 19 ⊢ (curry 𝑀:𝐼⟶((Base‘𝑅) ↑𝑚 𝐼) → dom curry 𝑀 = 𝐼)
74		f1eq2 6010	. . . . . . . . . . . . . . . . . . . 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 502	. . . . . . . . . . . . . . . . . . 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 493	. . . . . . . . . . . . . . . . . 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 842	. . . . . . . . . . . . . . . . 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 6061	. . . . . . . . . . . . . . . . 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 7860	. . . . . . . . . . . . . . . 16 ⊢ ((𝐼 ∈ Fin ∧ curry 𝑀:𝐼–1-1-onto→ran curry 𝑀) → 𝐼 ≈ ran curry 𝑀)
81	58, 79, 80	syl2anc 691	. . . . . . . . . . . . . . 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 7893	. . . . . . . . . . . . . 14 ⊢ ((((𝑅 ∈ DivRing ∧ 𝐼 ∈ Fin) ∧ curry 𝑀:𝐼⟶((Base‘𝑅) ↑𝑚 𝐼)) ∧ curry 𝑀 LIndF (𝑅 freeLMod 𝐼)) → ran curry 𝑀 ≈ 𝐼)
83		lindsenlbs 32574	. . . . . . . . . . . . . 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 1321	. . . . . . . . . . . . 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 2610	. . . . . . . . . . . . . 14 ⊢ (LBasis‘(𝑅 freeLMod 𝐼)) = (LBasis‘(𝑅 freeLMod 𝐼))
86		eqid 2610	. . . . . . . . . . . . . 14 ⊢ (LSpan‘(𝑅 freeLMod 𝐼)) = (LSpan‘(𝑅 freeLMod 𝐼))
87	46, 85, 86	lbssp 18900	. . . . . . . . . . . . 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 2644	. . . . . . . . . . 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 2691	. . . . . . . . 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 2610	. . . . . . . . . . . . 13 ⊢ (Base‘(Scalar‘(𝑅 freeLMod 𝐼))) = (Base‘(Scalar‘(𝑅 freeLMod 𝐼)))
93		eqid 2610	. . . . . . . . . . . . 13 ⊢ (0g‘(Scalar‘(𝑅 freeLMod 𝐼))) = (0g‘(Scalar‘(𝑅 freeLMod 𝐼)))
94		eqid 2610	. . . . . . . . . . . . 13 ⊢ ( ·𝑠 ‘(𝑅 freeLMod 𝐼)) = ( ·𝑠 ‘(𝑅 freeLMod 𝐼))
95	45, 14	frlmfibas 19924	. . . . . . . . . . . . . . 15 ⊢ ((𝑅 ∈ Ring ∧ 𝐼 ∈ Fin) → ((Base‘𝑅) ↑𝑚 𝐼) = (Base‘(𝑅 freeLMod 𝐼)))
96	95	feq3d 5945	. . . . . . . . . . . . . 14 ⊢ ((𝑅 ∈ Ring ∧ 𝐼 ∈ Fin) → (curry 𝑀:𝐼⟶((Base‘𝑅) ↑𝑚 𝐼) ↔ curry 𝑀:𝐼⟶(Base‘(𝑅 freeLMod 𝐼))))
97	96	biimpa 500	. . . . . . . . . . . . 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 788	. . . . . . . . . . . . 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 19961	. . . . . . . . . . . 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 19916	. . . . . . . . . . . . . . . 16 ⊢ ((𝑅 ∈ Ring ∧ 𝐼 ∈ Fin) → 𝑅 = (Scalar‘(𝑅 freeLMod 𝐼)))
102	101	fveq2d 6107	. . . . . . . . . . . . . . 15 ⊢ ((𝑅 ∈ Ring ∧ 𝐼 ∈ Fin) → (Base‘𝑅) = (Base‘(Scalar‘(𝑅 freeLMod 𝐼))))
103	102	oveq1d 6564	. . . . . . . . . . . . . 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 7765	. . . . . . . . . . . . . . 15 ⊢ (𝑛 ∈ ((Base‘𝑅) ↑𝑚 𝐼) → 𝑛:𝐼⟶(Base‘𝑅))
106		ffn 5958	. . . . . . . . . . . . . . . . . . . 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 759	. . . . . . . . . . . . . . . . . . 19 ⊢ ((((𝑅 ∈ Ring ∧ 𝐼 ∈ Fin) ∧ curry 𝑀:𝐼⟶((Base‘𝑅) ↑𝑚 𝐼)) ∧ 𝑛:𝐼⟶(Base‘𝑅)) → curry 𝑀 Fn 𝐼)
109		simpllr 795	. . . . . . . . . . . . . . . . . . 19 ⊢ ((((𝑅 ∈ Ring ∧ 𝐼 ∈ Fin) ∧ curry 𝑀:𝐼⟶((Base‘𝑅) ↑𝑚 𝐼)) ∧ 𝑛:𝐼⟶(Base‘𝑅)) → 𝐼 ∈ Fin)
110		inidm 3784	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝐼 ∩ 𝐼) = 𝐼
111		eqidd 2611	. . . . . . . . . . . . . . . . . . 19 ⊢ (((((𝑅 ∈ Ring ∧ 𝐼 ∈ Fin) ∧ curry 𝑀:𝐼⟶((Base‘𝑅) ↑𝑚 𝐼)) ∧ 𝑛:𝐼⟶(Base‘𝑅)) ∧ 𝑘 ∈ 𝐼) → (𝑛‘𝑘) = (𝑛‘𝑘))
112		eqidd 2611	. . . . . . . . . . . . . . . . . . 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 6802	. . . . . . . . . . . . . . . . . 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 803	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (((((𝑅 ∈ Ring ∧ 𝐼 ∈ Fin) ∧ curry 𝑀:𝐼⟶((Base‘𝑅) ↑𝑚 𝐼)) ∧ 𝑛:𝐼⟶(Base‘𝑅)) ∧ 𝑘 ∈ 𝐼) → 𝐼 ∈ Fin)
115		ffvelrn 6265	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ ((𝑛:𝐼⟶(Base‘𝑅) ∧ 𝑘 ∈ 𝐼) → (𝑛‘𝑘) ∈ (Base‘𝑅))
116	115	adantll 746	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (((((𝑅 ∈ Ring ∧ 𝐼 ∈ Fin) ∧ curry 𝑀:𝐼⟶((Base‘𝑅) ↑𝑚 𝐼)) ∧ 𝑛:𝐼⟶(Base‘𝑅)) ∧ 𝑘 ∈ 𝐼) → (𝑛‘𝑘) ∈ (Base‘𝑅))
117		ffvelrn 6265	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ ((curry 𝑀:𝐼⟶((Base‘𝑅) ↑𝑚 𝐼) ∧ 𝑘 ∈ 𝐼) → (curry 𝑀‘𝑘) ∈ ((Base‘𝑅) ↑𝑚 𝐼))
118	117	ad4ant24 1290	. . . . . . . . . . . . . . . . . . . . . 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 762	. . . . . . . . . . . . . . . . . . . . . 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 2690	. . . . . . . . . . . . . . . . . . . . 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 2610	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (.r‘𝑅) = (.r‘𝑅)
122	45, 46, 14, 114, 116, 120, 94, 121	frlmvscafval 19928	. . . . . . . . . . . . . . . . . . . 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 6113	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (𝑛‘𝑘) ∈ V
124		fnconstg 6006	. . . . . . . . . . . . . . . . . . . . . 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 7766	. . . . . . . . . . . . . . . . . . . . . . 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 1290	. . . . . . . . . . . . . . . . . . . . 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 6374	. . . . . . . . . . . . . . . . . . . . . 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 2611	. . . . . . . . . . . . . . . . . . . . 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 6802	. . . . . . . . . . . . . . . . . . . 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 2644	. . . . . . . . . . . . . . . . . . 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 4672	. . . . . . . . . . . . . . . . . 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 2644	. . . . . . . . . . . . . . . . 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 6565	. . . . . . . . . . . . . . . 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 2610	. . . . . . . . . . . . . . . . 17 ⊢ (0g‘(𝑅 freeLMod 𝐼)) = (0g‘(𝑅 freeLMod 𝐼))
138		simplll 794	. . . . . . . . . . . . . . . . 17 ⊢ ((((𝑅 ∈ Ring ∧ 𝐼 ∈ Fin) ∧ curry 𝑀:𝐼⟶((Base‘𝑅) ↑𝑚 𝐼)) ∧ 𝑛:𝐼⟶(Base‘𝑅)) → 𝑅 ∈ Ring)
139		simp-5l 804	. . . . . . . . . . . . . . . . . . . 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 1289	. . . . . . . . . . . . . . . . . . . 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 788	. . . . . . . . . . . . . . . . . . . . 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 7765	. . . . . . . . . . . . . . . . . . . . . . 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 6267	. . . . . . . . . . . . . . . . . . . . 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 18384	. . . . . . . . . . . . . . . . . . . 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 1318	. . . . . . . . . . . . . . . . . . 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		eqid 2610	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝑗 ∈ 𝐼 ↦ ((𝑛‘𝑘)(.r‘𝑅)((curry 𝑀‘𝑘)‘𝑗))) = (𝑗 ∈ 𝐼 ↦ ((𝑛‘𝑘)(.r‘𝑅)((curry 𝑀‘𝑘)‘𝑗)))
149	147, 148	fmptd 6292	. . . . . . . . . . . . . . . . . 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))
150		elmapg 7757	. . . . . . . . . . . . . . . . . . . . . 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)))
151	22, 150	mpan 702	. . . . . . . . . . . . . . . . . . . . 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)))
152	151	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)))
153	95	eleq2d 2673	. . . . . . . . . . . . . . . . . . . 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))))
154	152, 153	bitr3d 269	. . . . . . . . . . . . . . . . . . 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))))
155	154	ad3antrrr 762	. . . . . . . . . . . . . . . . . 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))))
156	149, 155	mpbid 221	. . . . . . . . . . . . . . . . 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)))
157		mptexg 6389	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (𝐼 ∈ Fin → (𝑗 ∈ 𝐼 ↦ ((𝑛‘𝑘)(.r‘𝑅)((curry 𝑀‘𝑘)‘𝑗))) ∈ V)
158	157	ralrimivw 2950	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝐼 ∈ Fin → ∀𝑘 ∈ 𝐼 (𝑗 ∈ 𝐼 ↦ ((𝑛‘𝑘)(.r‘𝑅)((curry 𝑀‘𝑘)‘𝑗))) ∈ V)
159		eqid 2610	. . . . . . . . . . . . . . . . . . . . 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))))
160	159	fnmpt 5933	. . . . . . . . . . . . . . . . . . . 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)
161	158, 160	syl 17	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝐼 ∈ Fin → (𝑘 ∈ 𝐼 ↦ (𝑗 ∈ 𝐼 ↦ ((𝑛‘𝑘)(.r‘𝑅)((curry 𝑀‘𝑘)‘𝑗)))) Fn 𝐼)
162		id 22	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝐼 ∈ Fin → 𝐼 ∈ Fin)
163		fvex 6113	. . . . . . . . . . . . . . . . . . . 20 ⊢ (0g‘(𝑅 freeLMod 𝐼)) ∈ V
164	163	a1i 11	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝐼 ∈ Fin → (0g‘(𝑅 freeLMod 𝐼)) ∈ V)
165	161, 162, 164	fndmfifsupp 8171	. . . . . . . . . . . . . . . . . 18 ⊢ (𝐼 ∈ Fin → (𝑘 ∈ 𝐼 ↦ (𝑗 ∈ 𝐼 ↦ ((𝑛‘𝑘)(.r‘𝑅)((curry 𝑀‘𝑘)‘𝑗)))) finSupp (0g‘(𝑅 freeLMod 𝐼)))
166	165	ad3antlr 763	. . . . . . . . . . . . . . . . 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)))
167	45, 46, 137, 109, 109, 138, 156, 166	frlmgsum 19930	. . . . . . . . . . . . . . . 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))))))
168	136, 167	eqtr2d 2645	. . . . . . . . . . . . . . 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)))
169	105, 168	sylan2 490	. . . . . . . . . . . . . 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)))
170	169	eqeq2d 2620	. . . . . . . . . . . . 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))))
171	104, 170	rexeqbidva 3132	. . . . . . . . . . . 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))))
172	100, 171	bitr4d 270	. . . . . . . . . . 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)))))))
173	43, 172	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)))))))
174	173	ad2antrr 758	. . . . . . . . 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)))))))
175	91, 174	mpbid 221	. . . . . . . 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))))))
176	175	ralrimiva 2949	. . . . . . 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))))))
177	42, 176	sylan 487	. . . . . 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))))))
178	10, 21	mpbird 246	. . . . . . . . 9 ⊢ ((𝑅 ∈ DivRing ∧ 𝑀 ∈ (Base‘(𝐼 Mat 𝑅))) → 𝑀 ∈ ((Base‘𝑅) ↑𝑚 (𝐼 × 𝐼)))
179		elmapfn 7766	. . . . . . . . 9 ⊢ (𝑀 ∈ ((Base‘𝑅) ↑𝑚 (𝐼 × 𝐼)) → 𝑀 Fn (𝐼 × 𝐼))
180	178, 179	syl 17	. . . . . . . 8 ⊢ ((𝑅 ∈ DivRing ∧ 𝑀 ∈ (Base‘(𝐼 Mat 𝑅))) → 𝑀 Fn (𝐼 × 𝐼))
181	4	adantl 481	. . . . . . . 8 ⊢ ((𝑅 ∈ DivRing ∧ 𝑀 ∈ (Base‘(𝐼 Mat 𝑅))) → 𝐼 ∈ Fin)
182		an32 835	. . . . . . . . . . . . . . . . . . 19 ⊢ (((𝑀 Fn (𝐼 × 𝐼) ∧ 𝑗 ∈ 𝐼) ∧ 𝑘 ∈ 𝐼) ↔ ((𝑀 Fn (𝐼 × 𝐼) ∧ 𝑘 ∈ 𝐼) ∧ 𝑗 ∈ 𝐼))
183		df-3an 1033	. . . . . . . . . . . . . . . . . . 19 ⊢ ((𝑀 Fn (𝐼 × 𝐼) ∧ 𝑘 ∈ 𝐼 ∧ 𝑗 ∈ 𝐼) ↔ ((𝑀 Fn (𝐼 × 𝐼) ∧ 𝑘 ∈ 𝐼) ∧ 𝑗 ∈ 𝐼))
184	182, 183	bitr4i 266	. . . . . . . . . . . . . . . . . 18 ⊢ (((𝑀 Fn (𝐼 × 𝐼) ∧ 𝑗 ∈ 𝐼) ∧ 𝑘 ∈ 𝐼) ↔ (𝑀 Fn (𝐼 × 𝐼) ∧ 𝑘 ∈ 𝐼 ∧ 𝑗 ∈ 𝐼))
185		curfv 32559	. . . . . . . . . . . . . . . . . 18 ⊢ (((𝑀 Fn (𝐼 × 𝐼) ∧ 𝑘 ∈ 𝐼 ∧ 𝑗 ∈ 𝐼) ∧ 𝐼 ∈ Fin) → ((curry 𝑀‘𝑘)‘𝑗) = (𝑘𝑀𝑗))
186	184, 185	sylanb 488	. . . . . . . . . . . . . . . . 17 ⊢ ((((𝑀 Fn (𝐼 × 𝐼) ∧ 𝑗 ∈ 𝐼) ∧ 𝑘 ∈ 𝐼) ∧ 𝐼 ∈ Fin) → ((curry 𝑀‘𝑘)‘𝑗) = (𝑘𝑀𝑗))
187	186	an32s 842	. . . . . . . . . . . . . . . 16 ⊢ ((((𝑀 Fn (𝐼 × 𝐼) ∧ 𝑗 ∈ 𝐼) ∧ 𝐼 ∈ Fin) ∧ 𝑘 ∈ 𝐼) → ((curry 𝑀‘𝑘)‘𝑗) = (𝑘𝑀𝑗))
188	187	oveq2d 6565	. . . . . . . . . . . . . . 15 ⊢ ((((𝑀 Fn (𝐼 × 𝐼) ∧ 𝑗 ∈ 𝐼) ∧ 𝐼 ∈ Fin) ∧ 𝑘 ∈ 𝐼) → ((𝑛‘𝑘)(.r‘𝑅)((curry 𝑀‘𝑘)‘𝑗)) = ((𝑛‘𝑘)(.r‘𝑅)(𝑘𝑀𝑗)))
189	188	mpteq2dva 4672	. . . . . . . . . . . . . 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))))
190	189	an32s 842	. . . . . . . . . . . . 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))))
191	190	oveq2d 6565	. . . . . . . . . . . 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)))))
192	191	mpteq2dva 4672	. . . . . . . . . . 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))))))
193	192	eqeq2d 2620	. . . . . . . . . 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)))))))
194	193	rexbidv 3034	. . . . . . . . 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)))))))
195	194	ralbidv 2969	. . . . . . . 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)))))))
196	180, 181, 195	syl2anc 691	. . . . . . 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)))))))
197	196	ad2antrr 758	. . . . . 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)))))))
198	177, 197	mpbid 221	. . . . 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))))))
199	8, 198	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))))))
200		fveq1 6102	. . . . . . . . . . 11 ⊢ (𝑛 = (𝑓‘𝑖) → (𝑛‘𝑘) = ((𝑓‘𝑖)‘𝑘))
201		vex 3176	. . . . . . . . . . . 12 ⊢ 𝑖 ∈ V
202		vex 3176	. . . . . . . . . . . 12 ⊢ 𝑘 ∈ V
203		uncov 32560	. . . . . . . . . . . 12 ⊢ ((𝑖 ∈ V ∧ 𝑘 ∈ V) → (𝑖uncurry 𝑓𝑘) = ((𝑓‘𝑖)‘𝑘))
204	201, 202, 203	mp2an 704	. . . . . . . . . . 11 ⊢ (𝑖uncurry 𝑓𝑘) = ((𝑓‘𝑖)‘𝑘)
205	200, 204	syl6eqr 2662	. . . . . . . . . 10 ⊢ (𝑛 = (𝑓‘𝑖) → (𝑛‘𝑘) = (𝑖uncurry 𝑓𝑘))
206	205	oveq1d 6564	. . . . . . . . 9 ⊢ (𝑛 = (𝑓‘𝑖) → ((𝑛‘𝑘)(.r‘𝑅)(𝑘𝑀𝑗)) = ((𝑖uncurry 𝑓𝑘)(.r‘𝑅)(𝑘𝑀𝑗)))
207	206	mpteq2dv 4673	. . . . . . . 8 ⊢ (𝑛 = (𝑓‘𝑖) → (𝑘 ∈ 𝐼 ↦ ((𝑛‘𝑘)(.r‘𝑅)(𝑘𝑀𝑗))) = (𝑘 ∈ 𝐼 ↦ ((𝑖uncurry 𝑓𝑘)(.r‘𝑅)(𝑘𝑀𝑗))))
208	207	oveq2d 6565	. . . . . . 7 ⊢ (𝑛 = (𝑓‘𝑖) → (𝑅 Σg (𝑘 ∈ 𝐼 ↦ ((𝑛‘𝑘)(.r‘𝑅)(𝑘𝑀𝑗)))) = (𝑅 Σg (𝑘 ∈ 𝐼 ↦ ((𝑖uncurry 𝑓𝑘)(.r‘𝑅)(𝑘𝑀𝑗)))))
209	208	mpteq2dv 4673	. . . . . 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))))))
210	209	eqeq2d 2620	. . . . 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)))))))
211	210	ac6sfi 8089	. . . 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)))))))
212	5, 199, 211	syl2anc 691	. . 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)))))))
213		uncf 32558	. . . . . . 7 ⊢ (𝑓:𝐼⟶((Base‘𝑅) ↑𝑚 𝐼) → uncurry 𝑓:(𝐼 × 𝐼)⟶(Base‘𝑅))
214	13, 14	frlmfibas 19924	. . . . . . . . . . . . . . . 16 ⊢ ((𝑅 ∈ Field ∧ (𝐼 × 𝐼) ∈ Fin) → ((Base‘𝑅) ↑𝑚 (𝐼 × 𝐼)) = (Base‘(𝑅 freeLMod (𝐼 × 𝐼))))
215	12, 214	sylan2 490	. . . . . . . . . . . . . . 15 ⊢ ((𝑅 ∈ Field ∧ 𝐼 ∈ Fin) → ((Base‘𝑅) ↑𝑚 (𝐼 × 𝐼)) = (Base‘(𝑅 freeLMod (𝐼 × 𝐼))))
216	1, 13	matbas 20038	. . . . . . . . . . . . . . . 16 ⊢ ((𝐼 ∈ Fin ∧ 𝑅 ∈ Field) → (Base‘(𝑅 freeLMod (𝐼 × 𝐼))) = (Base‘(𝐼 Mat 𝑅)))
217	216	ancoms 468	. . . . . . . . . . . . . . 15 ⊢ ((𝑅 ∈ Field ∧ 𝐼 ∈ Fin) → (Base‘(𝑅 freeLMod (𝐼 × 𝐼))) = (Base‘(𝐼 Mat 𝑅)))
218	215, 217	eqtrd 2644	. . . . . . . . . . . . . 14 ⊢ ((𝑅 ∈ Field ∧ 𝐼 ∈ Fin) → ((Base‘𝑅) ↑𝑚 (𝐼 × 𝐼)) = (Base‘(𝐼 Mat 𝑅)))
219	4, 218	sylan2 490	. . . . . . . . . . . . 13 ⊢ ((𝑅 ∈ Field ∧ 𝑀 ∈ (Base‘(𝐼 Mat 𝑅))) → ((Base‘𝑅) ↑𝑚 (𝐼 × 𝐼)) = (Base‘(𝐼 Mat 𝑅)))
220	219	eleq2d 2673	. . . . . . . . . . . 12 ⊢ ((𝑅 ∈ Field ∧ 𝑀 ∈ (Base‘(𝐼 Mat 𝑅))) → (uncurry 𝑓 ∈ ((Base‘𝑅) ↑𝑚 (𝐼 × 𝐼)) ↔ uncurry 𝑓 ∈ (Base‘(𝐼 Mat 𝑅))))
221		elmapg 7757	. . . . . . . . . . . . . 14 ⊢ (((Base‘𝑅) ∈ V ∧ (𝐼 × 𝐼) ∈ Fin) → (uncurry 𝑓 ∈ ((Base‘𝑅) ↑𝑚 (𝐼 × 𝐼)) ↔ uncurry 𝑓:(𝐼 × 𝐼)⟶(Base‘𝑅)))
222	22, 23, 221	sylancr 694	. . . . . . . . . . . . 13 ⊢ (𝑀 ∈ (Base‘(𝐼 Mat 𝑅)) → (uncurry 𝑓 ∈ ((Base‘𝑅) ↑𝑚 (𝐼 × 𝐼)) ↔ uncurry 𝑓:(𝐼 × 𝐼)⟶(Base‘𝑅)))
223	222	adantl 481	. . . . . . . . . . . 12 ⊢ ((𝑅 ∈ Field ∧ 𝑀 ∈ (Base‘(𝐼 Mat 𝑅))) → (uncurry 𝑓 ∈ ((Base‘𝑅) ↑𝑚 (𝐼 × 𝐼)) ↔ uncurry 𝑓:(𝐼 × 𝐼)⟶(Base‘𝑅)))
224	220, 223	bitr3d 269	. . . . . . . . . . 11 ⊢ ((𝑅 ∈ Field ∧ 𝑀 ∈ (Base‘(𝐼 Mat 𝑅))) → (uncurry 𝑓 ∈ (Base‘(𝐼 Mat 𝑅)) ↔ uncurry 𝑓:(𝐼 × 𝐼)⟶(Base‘𝑅)))
225	224	biimpar 501	. . . . . . . . . 10 ⊢ (((𝑅 ∈ Field ∧ 𝑀 ∈ (Base‘(𝐼 Mat 𝑅))) ∧ uncurry 𝑓:(𝐼 × 𝐼)⟶(Base‘𝑅)) → uncurry 𝑓 ∈ (Base‘(𝐼 Mat 𝑅)))
226	225	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)))
227		nfv 1830	. . . . . . . . . . . . . 14 ⊢ Ⅎ𝑗(((𝑅 ∈ Field ∧ 𝑀 ∈ (Base‘(𝐼 Mat 𝑅))) ∧ uncurry 𝑓:(𝐼 × 𝐼)⟶(Base‘𝑅)) ∧ 𝑖 ∈ 𝐼)
228		nfmpt1 4675	. . . . . . . . . . . . . . 15 ⊢ Ⅎ𝑗(𝑗 ∈ 𝐼 ↦ (𝑅 Σg (𝑘 ∈ 𝐼 ↦ ((𝑖uncurry 𝑓𝑘)(.r‘𝑅)(𝑘𝑀𝑗)))))
229	228	nfeq2 2766	. . . . . . . . . . . . . 14 ⊢ Ⅎ𝑗((𝑅 unitVec 𝐼)‘𝑖) = (𝑗 ∈ 𝐼 ↦ (𝑅 Σg (𝑘 ∈ 𝐼 ↦ ((𝑖uncurry 𝑓𝑘)(.r‘𝑅)(𝑘𝑀𝑗)))))
230		fveq1 6102	. . . . . . . . . . . . . . . . 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))
231	7, 43	syl 17	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (𝑅 ∈ Field → 𝑅 ∈ Ring)
232	231, 4	anim12i 588	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((𝑅 ∈ Field ∧ 𝑀 ∈ (Base‘(𝐼 Mat 𝑅))) → (𝑅 ∈ Ring ∧ 𝐼 ∈ Fin))
233	232	adantr 480	. . . . . . . . . . . . . . . . . . 19 ⊢ (((𝑅 ∈ Field ∧ 𝑀 ∈ (Base‘(𝐼 Mat 𝑅))) ∧ uncurry 𝑓:(𝐼 × 𝐼)⟶(Base‘𝑅)) → (𝑅 ∈ Ring ∧ 𝐼 ∈ Fin))
234		equcom 1932	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (𝑖 = 𝑗 ↔ 𝑗 = 𝑖)
235		ifbi 4057	. . . . . . . . . . . . . . . . . . . . 21 ⊢ ((𝑖 = 𝑗 ↔ 𝑗 = 𝑖) → if(𝑖 = 𝑗, (1r‘𝑅), (0g‘𝑅)) = if(𝑗 = 𝑖, (1r‘𝑅), (0g‘𝑅)))
236	234, 235	ax-mp 5	. . . . . . . . . . . . . . . . . . . 20 ⊢ if(𝑖 = 𝑗, (1r‘𝑅), (0g‘𝑅)) = if(𝑗 = 𝑖, (1r‘𝑅), (0g‘𝑅))
237		eqid 2610	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (1r‘𝑅) = (1r‘𝑅)
238		eqid 2610	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (0g‘𝑅) = (0g‘𝑅)
239		simpllr 795	. . . . . . . . . . . . . . . . . . . . 21 ⊢ ((((𝑅 ∈ Ring ∧ 𝐼 ∈ Fin) ∧ 𝑖 ∈ 𝐼) ∧ 𝑗 ∈ 𝐼) → 𝐼 ∈ Fin)
240		simplll 794	. . . . . . . . . . . . . . . . . . . . 21 ⊢ ((((𝑅 ∈ Ring ∧ 𝐼 ∈ Fin) ∧ 𝑖 ∈ 𝐼) ∧ 𝑗 ∈ 𝐼) → 𝑅 ∈ Ring)
241		simplr 788	. . . . . . . . . . . . . . . . . . . . 21 ⊢ ((((𝑅 ∈ Ring ∧ 𝐼 ∈ Fin) ∧ 𝑖 ∈ 𝐼) ∧ 𝑗 ∈ 𝐼) → 𝑖 ∈ 𝐼)
242		simpr 476	. . . . . . . . . . . . . . . . . . . . 21 ⊢ ((((𝑅 ∈ Ring ∧ 𝐼 ∈ Fin) ∧ 𝑖 ∈ 𝐼) ∧ 𝑗 ∈ 𝐼) → 𝑗 ∈ 𝐼)
243		eqid 2610	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (1r‘(𝐼 Mat 𝑅)) = (1r‘(𝐼 Mat 𝑅))
244	1, 237, 238, 239, 240, 241, 242, 243	mat1ov 20073	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((((𝑅 ∈ Ring ∧ 𝐼 ∈ Fin) ∧ 𝑖 ∈ 𝐼) ∧ 𝑗 ∈ 𝐼) → (𝑖(1r‘(𝐼 Mat 𝑅))𝑗) = if(𝑖 = 𝑗, (1r‘𝑅), (0g‘𝑅)))
245		df-3an 1033	. . . . . . . . . . . . . . . . . . . . 21 ⊢ ((𝑅 ∈ Ring ∧ 𝐼 ∈ Fin ∧ 𝑖 ∈ 𝐼) ↔ ((𝑅 ∈ Ring ∧ 𝐼 ∈ Fin) ∧ 𝑖 ∈ 𝐼))
246	44, 237, 238	uvcvval 19944	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (((𝑅 ∈ Ring ∧ 𝐼 ∈ Fin ∧ 𝑖 ∈ 𝐼) ∧ 𝑗 ∈ 𝐼) → (((𝑅 unitVec 𝐼)‘𝑖)‘𝑗) = if(𝑗 = 𝑖, (1r‘𝑅), (0g‘𝑅)))
247	245, 246	sylanbr 489	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((((𝑅 ∈ Ring ∧ 𝐼 ∈ Fin) ∧ 𝑖 ∈ 𝐼) ∧ 𝑗 ∈ 𝐼) → (((𝑅 unitVec 𝐼)‘𝑖)‘𝑗) = if(𝑗 = 𝑖, (1r‘𝑅), (0g‘𝑅)))
248	236, 244, 247	3eqtr4a 2670	. . . . . . . . . . . . . . . . . . 19 ⊢ ((((𝑅 ∈ Ring ∧ 𝐼 ∈ Fin) ∧ 𝑖 ∈ 𝐼) ∧ 𝑗 ∈ 𝐼) → (𝑖(1r‘(𝐼 Mat 𝑅))𝑗) = (((𝑅 unitVec 𝐼)‘𝑖)‘𝑗))
249	233, 248	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))
250		ovex 6577	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (𝑅 Σg (𝑘 ∈ 𝐼 ↦ ((𝑖uncurry 𝑓𝑘)(.r‘𝑅)(𝑘𝑀𝑗)))) ∈ V
251		eqid 2610	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (𝑗 ∈ 𝐼 ↦ (𝑅 Σg (𝑘 ∈ 𝐼 ↦ ((𝑖uncurry 𝑓𝑘)(.r‘𝑅)(𝑘𝑀𝑗))))) = (𝑗 ∈ 𝐼 ↦ (𝑅 Σg (𝑘 ∈ 𝐼 ↦ ((𝑖uncurry 𝑓𝑘)(.r‘𝑅)(𝑘𝑀𝑗)))))
252	251	fvmpt2 6200	. . . . . . . . . . . . . . . . . . . . 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)))))
253	250, 252	mpan2 703	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝑗 ∈ 𝐼 → ((𝑗 ∈ 𝐼 ↦ (𝑅 Σg (𝑘 ∈ 𝐼 ↦ ((𝑖uncurry 𝑓𝑘)(.r‘𝑅)(𝑘𝑀𝑗)))))‘𝑗) = (𝑅 Σg (𝑘 ∈ 𝐼 ↦ ((𝑖uncurry 𝑓𝑘)(.r‘𝑅)(𝑘𝑀𝑗)))))
254	253	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)))))
255		eqid 2610	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝑅 maMul ⟨𝐼, 𝐼, 𝐼⟩) = (𝑅 maMul ⟨𝐼, 𝐼, 𝐼⟩)
256		simp-4l 802	. . . . . . . . . . . . . . . . . . . 20 ⊢ (((((𝑅 ∈ Field ∧ 𝑀 ∈ (Base‘(𝐼 Mat 𝑅))) ∧ uncurry 𝑓:(𝐼 × 𝐼)⟶(Base‘𝑅)) ∧ 𝑖 ∈ 𝐼) ∧ 𝑗 ∈ 𝐼) → 𝑅 ∈ Field)
257	4	ad4antlr 765	. . . . . . . . . . . . . . . . . . . 20 ⊢ (((((𝑅 ∈ Field ∧ 𝑀 ∈ (Base‘(𝐼 Mat 𝑅))) ∧ uncurry 𝑓:(𝐼 × 𝐼)⟶(Base‘𝑅)) ∧ 𝑖 ∈ 𝐼) ∧ 𝑗 ∈ 𝐼) → 𝐼 ∈ Fin)
258	222	biimpar 501	. . . . . . . . . . . . . . . . . . . . 21 ⊢ ((𝑀 ∈ (Base‘(𝐼 Mat 𝑅)) ∧ uncurry 𝑓:(𝐼 × 𝐼)⟶(Base‘𝑅)) → uncurry 𝑓 ∈ ((Base‘𝑅) ↑𝑚 (𝐼 × 𝐼)))
259	258	ad5ant23 1296	. . . . . . . . . . . . . . . . . . . 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)))
260		simpr 476	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ ((𝑅 ∈ Field ∧ 𝑀 ∈ (Base‘(𝐼 Mat 𝑅))) → 𝑀 ∈ (Base‘(𝐼 Mat 𝑅)))
261	260, 219	eleqtrrd 2691	. . . . . . . . . . . . . . . . . . . . 21 ⊢ ((𝑅 ∈ Field ∧ 𝑀 ∈ (Base‘(𝐼 Mat 𝑅))) → 𝑀 ∈ ((Base‘𝑅) ↑𝑚 (𝐼 × 𝐼)))
262	261	ad3antrrr 762	. . . . . . . . . . . . . . . . . . . 20 ⊢ (((((𝑅 ∈ Field ∧ 𝑀 ∈ (Base‘(𝐼 Mat 𝑅))) ∧ uncurry 𝑓:(𝐼 × 𝐼)⟶(Base‘𝑅)) ∧ 𝑖 ∈ 𝐼) ∧ 𝑗 ∈ 𝐼) → 𝑀 ∈ ((Base‘𝑅) ↑𝑚 (𝐼 × 𝐼)))
263		simplr 788	. . . . . . . . . . . . . . . . . . . 20 ⊢ (((((𝑅 ∈ Field ∧ 𝑀 ∈ (Base‘(𝐼 Mat 𝑅))) ∧ uncurry 𝑓:(𝐼 × 𝐼)⟶(Base‘𝑅)) ∧ 𝑖 ∈ 𝐼) ∧ 𝑗 ∈ 𝐼) → 𝑖 ∈ 𝐼)
264		simpr 476	. . . . . . . . . . . . . . . . . . . 20 ⊢ (((((𝑅 ∈ Field ∧ 𝑀 ∈ (Base‘(𝐼 Mat 𝑅))) ∧ uncurry 𝑓:(𝐼 × 𝐼)⟶(Base‘𝑅)) ∧ 𝑖 ∈ 𝐼) ∧ 𝑗 ∈ 𝐼) → 𝑗 ∈ 𝐼)
265	255, 14, 121, 256, 257, 257, 257, 259, 262, 263, 264	mamufv 20012	. . . . . . . . . . . . . . . . . . 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)))))
266	1, 255	matmulr 20063	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ ((𝐼 ∈ Fin ∧ 𝑅 ∈ Field) → (𝑅 maMul ⟨𝐼, 𝐼, 𝐼⟩) = (.r‘(𝐼 Mat 𝑅)))
267	266	ancoms 468	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ ((𝑅 ∈ Field ∧ 𝐼 ∈ Fin) → (𝑅 maMul ⟨𝐼, 𝐼, 𝐼⟩) = (.r‘(𝐼 Mat 𝑅)))
268	267	oveqd 6566	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ ((𝑅 ∈ Field ∧ 𝐼 ∈ Fin) → (uncurry 𝑓(𝑅 maMul ⟨𝐼, 𝐼, 𝐼⟩)𝑀) = (uncurry 𝑓(.r‘(𝐼 Mat 𝑅))𝑀))
269	268	oveqd 6566	. . . . . . . . . . . . . . . . . . . . 21 ⊢ ((𝑅 ∈ Field ∧ 𝐼 ∈ Fin) → (𝑖(uncurry 𝑓(𝑅 maMul ⟨𝐼, 𝐼, 𝐼⟩)𝑀)𝑗) = (𝑖(uncurry 𝑓(.r‘(𝐼 Mat 𝑅))𝑀)𝑗))
270	4, 269	sylan2 490	. . . . . . . . . . . . . . . . . . . 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))
271	270	ad3antrrr 762	. . . . . . . . . . . . . . . . . . 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))
272	254, 265, 271	3eqtr2rd 2651	. . . . . . . . . . . . . . . . . 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))
273	249, 272	eqeq12d 2625	. . . . . . . . . . . . . . . . 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)))
274	230, 273	syl5ibr 235	. . . . . . . . . . . . . . . 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)))
275	274	ex 449	. . . . . . . . . . . . . . 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))))
276	275	com23 84	. . . . . . . . . . . . . 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))))
277	227, 229, 276	ralrimd 2942	. . . . . . . . . . . . 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)))
278	277	ralimdva 2945	. . . . . . . . . . . 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)))
279	1, 2, 243	mat1bas 20074	. . . . . . . . . . . . . . . . 17 ⊢ ((𝑅 ∈ Ring ∧ 𝐼 ∈ Fin) → (1r‘(𝐼 Mat 𝑅)) ∈ (Base‘(𝐼 Mat 𝑅)))
280	13, 14	frlmfibas 19924	. . . . . . . . . . . . . . . . . . 19 ⊢ ((𝑅 ∈ Ring ∧ (𝐼 × 𝐼) ∈ Fin) → ((Base‘𝑅) ↑𝑚 (𝐼 × 𝐼)) = (Base‘(𝑅 freeLMod (𝐼 × 𝐼))))
281	12, 280	sylan2 490	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝑅 ∈ Ring ∧ 𝐼 ∈ Fin) → ((Base‘𝑅) ↑𝑚 (𝐼 × 𝐼)) = (Base‘(𝑅 freeLMod (𝐼 × 𝐼))))
282	1, 13	matbas 20038	. . . . . . . . . . . . . . . . . . 19 ⊢ ((𝐼 ∈ Fin ∧ 𝑅 ∈ Ring) → (Base‘(𝑅 freeLMod (𝐼 × 𝐼))) = (Base‘(𝐼 Mat 𝑅)))
283	282	ancoms 468	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝑅 ∈ Ring ∧ 𝐼 ∈ Fin) → (Base‘(𝑅 freeLMod (𝐼 × 𝐼))) = (Base‘(𝐼 Mat 𝑅)))
284	281, 283	eqtrd 2644	. . . . . . . . . . . . . . . . 17 ⊢ ((𝑅 ∈ Ring ∧ 𝐼 ∈ Fin) → ((Base‘𝑅) ↑𝑚 (𝐼 × 𝐼)) = (Base‘(𝐼 Mat 𝑅)))
285	279, 284	eleqtrrd 2691	. . . . . . . . . . . . . . . 16 ⊢ ((𝑅 ∈ Ring ∧ 𝐼 ∈ Fin) → (1r‘(𝐼 Mat 𝑅)) ∈ ((Base‘𝑅) ↑𝑚 (𝐼 × 𝐼)))
286		elmapfn 7766	. . . . . . . . . . . . . . . 16 ⊢ ((1r‘(𝐼 Mat 𝑅)) ∈ ((Base‘𝑅) ↑𝑚 (𝐼 × 𝐼)) → (1r‘(𝐼 Mat 𝑅)) Fn (𝐼 × 𝐼))
287	285, 286	syl 17	. . . . . . . . . . . . . . 15 ⊢ ((𝑅 ∈ Ring ∧ 𝐼 ∈ Fin) → (1r‘(𝐼 Mat 𝑅)) Fn (𝐼 × 𝐼))
288	231, 4, 287	syl2an 493	. . . . . . . . . . . . . 14 ⊢ ((𝑅 ∈ Field ∧ 𝑀 ∈ (Base‘(𝐼 Mat 𝑅))) → (1r‘(𝐼 Mat 𝑅)) Fn (𝐼 × 𝐼))
289	288	adantr 480	. . . . . . . . . . . . 13 ⊢ (((𝑅 ∈ Field ∧ 𝑀 ∈ (Base‘(𝐼 Mat 𝑅))) ∧ uncurry 𝑓:(𝐼 × 𝐼)⟶(Base‘𝑅)) → (1r‘(𝐼 Mat 𝑅)) Fn (𝐼 × 𝐼))
290	1	matring 20068	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝐼 ∈ Fin ∧ 𝑅 ∈ Ring) → (𝐼 Mat 𝑅) ∈ Ring)
291	4, 231, 290	syl2anr 494	. . . . . . . . . . . . . . . . 17 ⊢ ((𝑅 ∈ Field ∧ 𝑀 ∈ (Base‘(𝐼 Mat 𝑅))) → (𝐼 Mat 𝑅) ∈ Ring)
292	291	adantr 480	. . . . . . . . . . . . . . . 16 ⊢ (((𝑅 ∈ Field ∧ 𝑀 ∈ (Base‘(𝐼 Mat 𝑅))) ∧ uncurry 𝑓:(𝐼 × 𝐼)⟶(Base‘𝑅)) → (𝐼 Mat 𝑅) ∈ Ring)
293		simplr 788	. . . . . . . . . . . . . . . 16 ⊢ (((𝑅 ∈ Field ∧ 𝑀 ∈ (Base‘(𝐼 Mat 𝑅))) ∧ uncurry 𝑓:(𝐼 × 𝐼)⟶(Base‘𝑅)) → 𝑀 ∈ (Base‘(𝐼 Mat 𝑅)))
294		eqid 2610	. . . . . . . . . . . . . . . . 17 ⊢ (.r‘(𝐼 Mat 𝑅)) = (.r‘(𝐼 Mat 𝑅))
295	2, 294	ringcl 18384	. . . . . . . . . . . . . . . 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)))
296	292, 225, 293, 295	syl3anc 1318	. . . . . . . . . . . . . . 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)))
297	219	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)))
298	296, 297	eleqtrrd 2691	. . . . . . . . . . . . . 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)))
299		elmapfn 7766	. . . . . . . . . . . . . 14 ⊢ ((uncurry 𝑓(.r‘(𝐼 Mat 𝑅))𝑀) ∈ ((Base‘𝑅) ↑𝑚 (𝐼 × 𝐼)) → (uncurry 𝑓(.r‘(𝐼 Mat 𝑅))𝑀) Fn (𝐼 × 𝐼))
300	298, 299	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))
301		eqfnov2 6665	. . . . . . . . . . . . 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)))
302	289, 300, 301	syl2anc 691	. . . . . . . . . . . 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)))
303	278, 302	sylibrd 248	. . . . . . . . . . 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)))
304	303	imp 444	. . . . . . . . . 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))
305	304	eqcomd 2616	. . . . . . . . 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)))
306		oveq1 6556	. . . . . . . . . . 11 ⊢ (𝑛 = uncurry 𝑓 → (𝑛(.r‘(𝐼 Mat 𝑅))𝑀) = (uncurry 𝑓(.r‘(𝐼 Mat 𝑅))𝑀))
307	306	eqeq1d 2612	. . . . . . . . . 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))))
308	307	rspcev 3282	. . . . . . . . 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)))
309	226, 305, 308	syl2anc 691	. . . . . . . 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)))
310	309	expl 646	. . . . . . 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))))
311	213, 310	sylani 684	. . . . . 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))))
312	311	exlimdv 1848	. . . . 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))))
313	312	imp 444	. . . 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)))
314	313	adantlr 747	. . 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)))
315	212, 314	syldan 486	. 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)))
316	6	simprbi 479	. . . 4 ⊢ (𝑅 ∈ Field → 𝑅 ∈ CRing)
317		eqid 2610	. . . . . . . . . 10 ⊢ (𝐼 maDet 𝑅) = (𝐼 maDet 𝑅)
318	317, 1, 2, 14	mdetcl 20221	. . . . . . . . 9 ⊢ ((𝑅 ∈ CRing ∧ 𝑀 ∈ (Base‘(𝐼 Mat 𝑅))) → ((𝐼 maDet 𝑅)‘𝑀) ∈ (Base‘𝑅))
319	317, 1, 2, 14	mdetcl 20221	. . . . . . . . 9 ⊢ ((𝑅 ∈ CRing ∧ 𝑛 ∈ (Base‘(𝐼 Mat 𝑅))) → ((𝐼 maDet 𝑅)‘𝑛) ∈ (Base‘𝑅))
320		eqid 2610	. . . . . . . . . 10 ⊢ (∥r‘𝑅) = (∥r‘𝑅)
321	14, 320, 121	dvdsrmul 18471	. . . . . . . . 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)))
322	318, 319, 321	syl2an 493	. . . . . . . 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)))
323	322	anandis 869	. . . . . . 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)))
324	323	anassrs 678	. . . . . 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)))
325	324	adantrr 749	. . . . 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)))
326		fveq2 6103	. . . . . . . . 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))))
327	1, 2, 317, 121, 294	mdetmul 20248	. . . . . . . . . . . 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)))
328	327	3expa 1257	. . . . . . . . . . 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)))
329	328	an32s 842	. . . . . . . . . 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)))
330	317, 1, 243, 237	mdet1 20226	. . . . . . . . . . . 12 ⊢ ((𝑅 ∈ CRing ∧ 𝐼 ∈ Fin) → ((𝐼 maDet 𝑅)‘(1r‘(𝐼 Mat 𝑅))) = (1r‘𝑅))
331	4, 330	sylan2 490	. . . . . . . . . . 11 ⊢ ((𝑅 ∈ CRing ∧ 𝑀 ∈ (Base‘(𝐼 Mat 𝑅))) → ((𝐼 maDet 𝑅)‘(1r‘(𝐼 Mat 𝑅))) = (1r‘𝑅))
332	331	adantr 480	. . . . . . . . . 10 ⊢ (((𝑅 ∈ CRing ∧ 𝑀 ∈ (Base‘(𝐼 Mat 𝑅))) ∧ 𝑛 ∈ (Base‘(𝐼 Mat 𝑅))) → ((𝐼 maDet 𝑅)‘(1r‘(𝐼 Mat 𝑅))) = (1r‘𝑅))
333	329, 332	eqeq12d 2625	. . . . . . . . 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)))
334	326, 333	syl5ib 233	. . . . . . . 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)))
335	334	impr 647	. . . . . . 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))
336	335	breq2d 4595	. . . . . 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)))
337		eqid 2610	. . . . . . . 8 ⊢ (Unit‘𝑅) = (Unit‘𝑅)
338	337, 237, 320	crngunit 18485	. . . . . . 7 ⊢ (𝑅 ∈ CRing → (((𝐼 maDet 𝑅)‘𝑀) ∈ (Unit‘𝑅) ↔ ((𝐼 maDet 𝑅)‘𝑀)(∥r‘𝑅)(1r‘𝑅)))
339	338	ad2antrr 758	. . . . . 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)))
340	336, 339	bitr4d 270	. . . . 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)))
341	325, 340	mpbid 221	. . . 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))
342	316, 341	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))
343	342	ad4ant14 1285	. 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))
344	315, 343	rexlimddv 3017	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 195   ∧ wa 383   ∧ w3a 1031   = wceq 1475  ∃wex 1695   ∈ wcel 1977   ≠ wne 2780  ∀wral 2896  ∃wrex 2897  Vcvv 3173   ∖ cdif 3537  ∅c0 3874  ifcif 4036  {csn 4125  ⟨cotp 4133   class class class wbr 4583   ↦ cmpt 4643   × cxp 5036  dom cdm 5038  ran crn 5039   “ cima 5041   Fn wfn 5799  ⟶wf 5800  –1-1→wf1 5801  –1-1-onto→wf1o 5803  ‘cfv 5804  (class class class)co 6549   ∘_𝑓 cof 6793  curry ccur 7278  uncurry cunc 7279   ↑_𝑚 cmap 7744   ≈ cen 7838  Fincfn 7841   finSupp cfsupp 8158  Basecbs 15695  ._rcmulr 15769  Scalarcsca 15771   ·_𝑠 cvsca 15772  0_gc0g 15923   Σ_g cgsu 15924  1_rcur 18324  Ringcrg 18370  CRingccrg 18371  ∥_rcdsr 18461  Unitcui 18462  DivRingcdr 18570  Fieldcfield 18571  LModclmod 18686  LSpanclspn 18792  LBasisclbs 18895  NzRingcnzr 19078   freeLMod cfrlm 19909   unitVec cuvc 19940   LIndF clindf 19962  LIndSclinds 19963   maMul cmmul 20008   Mat cmat 20032   maDet cmdat 20209

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1713  ax-4 1728  ax-5 1827  ax-6 1875  ax-7 1922  ax-8 1979  ax-9 1986  ax-10 2006  ax-11 2021  ax-12 2034  ax-13 2234  ax-ext 2590  ax-rep 4699  ax-sep 4709  ax-nul 4717  ax-pow 4769  ax-pr 4833  ax-un 6847  ax-inf2 8421  ax-cnex 9871  ax-resscn 9872  ax-1cn 9873  ax-icn 9874  ax-addcl 9875  ax-addrcl 9876  ax-mulcl 9877  ax-mulrcl 9878  ax-mulcom 9879  ax-addass 9880  ax-mulass 9881  ax-distr 9882  ax-i2m1 9883  ax-1ne0 9884  ax-1rid 9885  ax-rnegex 9886  ax-rrecex 9887  ax-cnre 9888  ax-pre-lttri 9889  ax-pre-lttrn 9890  ax-pre-ltadd 9891  ax-pre-mulgt0 9892  ax-addf 9894  ax-mulf 9895

This theorem depends on definitions:  df-bi 196  df-or 384  df-an 385  df-3or 1032  df-3an 1033  df-xor 1457  df-tru 1478  df-fal 1481  df-ex 1696  df-nf 1701  df-sb 1868  df-eu 2462  df-mo 2463  df-clab 2597  df-cleq 2603  df-clel 2606  df-nfc 2740  df-ne 2782  df-nel 2783  df-ral 2901  df-rex 2902  df-reu 2903  df-rmo 2904  df-rab 2905  df-v 3175  df-sbc 3403  df-csb 3500  df-dif 3543  df-un 3545  df-in 3547  df-ss 3554  df-pss 3556  df-nul 3875  df-if 4037  df-pw 4110  df-sn 4126  df-pr 4128  df-tp 4130  df-op 4132  df-ot 4134  df-uni 4373  df-int 4411  df-iun 4457  df-iin 4458  df-br 4584  df-opab 4644  df-mpt 4645  df-tr 4681  df-eprel 4949  df-id 4953  df-po 4959  df-so 4960  df-fr 4997  df-se 4998  df-we 4999  df-xp 5044  df-rel 5045  df-cnv 5046  df-co 5047  df-dm 5048  df-rn 5049  df-res 5050  df-ima 5051  df-pred 5597  df-ord 5643  df-on 5644  df-lim 5645  df-suc 5646  df-iota 5768  df-fun 5806  df-fn 5807  df-f 5808  df-f1 5809  df-fo 5810  df-f1o 5811  df-fv 5812  df-isom 5813  df-riota 6511  df-ov 6552  df-oprab 6553  df-mpt2 6554  df-of 6795  df-om 6958  df-1st 7059  df-2nd 7060  df-supp 7183  df-tpos 7239  df-cur 7280  df-unc 7281  df-wrecs 7294  df-recs 7355  df-rdg 7393  df-1o 7447  df-2o 7448  df-oadd 7451  df-er 7629  df-map 7746  df-pm 7747  df-ixp 7795  df-en 7842  df-dom 7843  df-sdom 7844  df-fin 7845  df-fsupp 8159  df-sup 8231  df-oi 8298  df-card 8648  df-pnf 9955  df-mnf 9956  df-xr 9957  df-ltxr 9958  df-le 9959  df-sub 10147  df-neg 10148  df-div 10564  df-nn 10898  df-2 10956  df-3 10957  df-4 10958  df-5 10959  df-6 10960  df-7 10961  df-8 10962  df-9 10963  df-n0 11170  df-xnn0 11241  df-z 11255  df-dec 11370  df-uz 11564  df-rp 11709  df-fz 12198  df-fzo 12335  df-seq 12664  df-exp 12723  df-hash 12980  df-word 13154  df-lsw 13155  df-concat 13156  df-s1 13157  df-substr 13158  df-splice 13159  df-reverse 13160  df-s2 13444  df-struct 15697  df-ndx 15698  df-slot 15699  df-base 15700  df-sets 15701  df-ress 15702  df-plusg 15781  df-mulr 15782  df-starv 15783  df-sca 15784  df-vsca 15785  df-ip 15786  df-tset 15787  df-ple 15788  df-ds 15791  df-unif 15792  df-hom 15793  df-cco 15794  df-0g 15925  df-gsum 15926  df-prds 15931  df-pws 15933  df-mre 16069  df-mrc 16070  df-mri 16071  df-acs 16072  df-mgm 17065  df-sgrp 17107  df-mnd 17118  df-mhm 17158  df-submnd 17159  df-grp 17248  df-minusg 17249  df-sbg 17250  df-mulg 17364  df-subg 17414  df-ghm 17481  df-gim 17524  df-cntz 17573  df-oppg 17599  df-symg 17621  df-pmtr 17685  df-psgn 17734  df-evpm 17735  df-cmn 18018  df-abl 18019  df-mgp 18313  df-ur 18325  df-srg 18329  df-ring 18372  df-cring 18373  df-oppr 18446  df-dvdsr 18464  df-unit 18465  df-invr 18495  df-dvr 18506  df-rnghom 18538  df-drng 18572  df-field 18573  df-subrg 18601  df-lmod 18688  df-lss 18754  df-lsp 18793  df-lmhm 18843  df-lbs 18896  df-lvec 18924  df-sra 18993  df-rgmod 18994  df-nzr 19079  df-cnfld 19568  df-zring 19638  df-zrh 19671  df-dsmm 19895  df-frlm 19910  df-uvc 19941  df-lindf 19964  df-linds 19965  df-mamu 20009  df-mat 20033  df-mdet 20210

This theorem is referenced by:  matunitlindf  32577