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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.
StepHypRef Expression
1 eqid 2610 . . . . . . 7 (𝐼 Mat 𝑅) = (𝐼 Mat 𝑅)
2 eqid 2610 . . . . . . 7 (Base‘(𝐼 Mat 𝑅)) = (Base‘(𝐼 Mat 𝑅))
31, 2matrcl 20037 . . . . . 6 (𝑀 ∈ (Base‘(𝐼 Mat 𝑅)) → (𝐼 ∈ Fin ∧ 𝑅 ∈ V))
43simpld 474 . . . . 5 (𝑀 ∈ (Base‘(𝐼 Mat 𝑅)) → 𝐼 ∈ Fin)
54ad3antlr 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))
76simplbi 475 . . . . . 6 (𝑅 ∈ Field → 𝑅 ∈ DivRing)
87anim1i 590 . . . . 5 ((𝑅 ∈ Field ∧ 𝑀 ∈ (Base‘(𝐼 Mat 𝑅))) → (𝑅 ∈ DivRing ∧ 𝑀 ∈ (Base‘(𝐼 Mat 𝑅))))
94ad2antrl 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)
1211anidms 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)
1513, 14frlmfibas 19924 . . . . . . . . . . . . . . . . . . . 20 ((𝑅 ∈ DivRing ∧ (𝐼 × 𝐼) ∈ Fin) → ((Base‘𝑅) ↑𝑚 (𝐼 × 𝐼)) = (Base‘(𝑅 freeLMod (𝐼 × 𝐼))))
1612, 15sylan2 490 . . . . . . . . . . . . . . . . . . 19 ((𝑅 ∈ DivRing ∧ 𝐼 ∈ Fin) → ((Base‘𝑅) ↑𝑚 (𝐼 × 𝐼)) = (Base‘(𝑅 freeLMod (𝐼 × 𝐼))))
171, 13matbas 20038 . . . . . . . . . . . . . . . . . . . 20 ((𝐼 ∈ Fin ∧ 𝑅 ∈ DivRing) → (Base‘(𝑅 freeLMod (𝐼 × 𝐼))) = (Base‘(𝐼 Mat 𝑅)))
1817ancoms 468 . . . . . . . . . . . . . . . . . . 19 ((𝑅 ∈ DivRing ∧ 𝐼 ∈ Fin) → (Base‘(𝑅 freeLMod (𝐼 × 𝐼))) = (Base‘(𝐼 Mat 𝑅)))
1916, 18eqtrd 2644 . . . . . . . . . . . . . . . . . 18 ((𝑅 ∈ DivRing ∧ 𝐼 ∈ Fin) → ((Base‘𝑅) ↑𝑚 (𝐼 × 𝐼)) = (Base‘(𝐼 Mat 𝑅)))
2019eleq2d 2673 . . . . . . . . . . . . . . . . 17 ((𝑅 ∈ DivRing ∧ 𝐼 ∈ Fin) → (𝑀 ∈ ((Base‘𝑅) ↑𝑚 (𝐼 × 𝐼)) ↔ 𝑀 ∈ (Base‘(𝐼 Mat 𝑅))))
214, 20sylan2 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
234, 4, 11syl2anc 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)))
2522, 23, 24sylancr 694 . . . . . . . . . . . . . . . . 17 (𝑀 ∈ (Base‘(𝐼 Mat 𝑅)) → (𝑀 ∈ ((Base‘𝑅) ↑𝑚 (𝐼 × 𝐼)) ↔ 𝑀:(𝐼 × 𝐼)⟶(Base‘𝑅)))
2625adantl 481 . . . . . . . . . . . . . . . 16 ((𝑅 ∈ DivRing ∧ 𝑀 ∈ (Base‘(𝐼 Mat 𝑅))) → (𝑀 ∈ ((Base‘𝑅) ↑𝑚 (𝐼 × 𝐼)) ↔ 𝑀:(𝐼 × 𝐼)⟶(Base‘𝑅)))
2721, 26bitr3d 269 . . . . . . . . . . . . . . 15 ((𝑅 ∈ DivRing ∧ 𝑀 ∈ (Base‘(𝐼 Mat 𝑅))) → (𝑀 ∈ (Base‘(𝐼 Mat 𝑅)) ↔ 𝑀:(𝐼 × 𝐼)⟶(Base‘𝑅)))
2810, 27mpbid 221 . . . . . . . . . . . . . 14 ((𝑅 ∈ DivRing ∧ 𝑀 ∈ (Base‘(𝐼 Mat 𝑅))) → 𝑀:(𝐼 × 𝐼)⟶(Base‘𝑅))
2928adantrr 749 . . . . . . . . . . . . 13 ((𝑅 ∈ DivRing ∧ (𝑀 ∈ (Base‘(𝐼 Mat 𝑅)) ∧ 𝐼 ≠ ∅)) → 𝑀:(𝐼 × 𝐼)⟶(Base‘𝑅))
30 eldifsn 4260 . . . . . . . . . . . . . . . 16 (𝐼 ∈ (Fin ∖ {∅}) ↔ (𝐼 ∈ Fin ∧ 𝐼 ≠ ∅))
3130biimpri 217 . . . . . . . . . . . . . . 15 ((𝐼 ∈ Fin ∧ 𝐼 ≠ ∅) → 𝐼 ∈ (Fin ∖ {∅}))
324, 31sylan 487 . . . . . . . . . . . . . 14 ((𝑀 ∈ (Base‘(𝐼 Mat 𝑅)) ∧ 𝐼 ≠ ∅) → 𝐼 ∈ (Fin ∖ {∅}))
3332adantl 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))
3522, 34mp3an3 1405 . . . . . . . . . . . . 13 ((𝑀:(𝐼 × 𝐼)⟶(Base‘𝑅) ∧ 𝐼 ∈ (Fin ∖ {∅})) → curry 𝑀:𝐼⟶((Base‘𝑅) ↑𝑚 𝐼))
3629, 33, 35syl2anc 691 . . . . . . . . . . . 12 ((𝑅 ∈ DivRing ∧ (𝑀 ∈ (Base‘(𝐼 Mat 𝑅)) ∧ 𝐼 ≠ ∅)) → curry 𝑀:𝐼⟶((Base‘𝑅) ↑𝑚 𝐼))
379, 36jca 553 . . . . . . . . . . 11 ((𝑅 ∈ DivRing ∧ (𝑀 ∈ (Base‘(𝐼 Mat 𝑅)) ∧ 𝐼 ≠ ∅)) → (𝐼 ∈ Fin ∧ curry 𝑀:𝐼⟶((Base‘𝑅) ↑𝑚 𝐼)))
3837ex 449 . . . . . . . . . 10 (𝑅 ∈ DivRing → ((𝑀 ∈ (Base‘(𝐼 Mat 𝑅)) ∧ 𝐼 ≠ ∅) → (𝐼 ∈ Fin ∧ curry 𝑀:𝐼⟶((Base‘𝑅) ↑𝑚 𝐼))))
3938imdistani 722 . . . . . . . . 9 ((𝑅 ∈ DivRing ∧ (𝑀 ∈ (Base‘(𝐼 Mat 𝑅)) ∧ 𝐼 ≠ ∅)) → (𝑅 ∈ DivRing ∧ (𝐼 ∈ Fin ∧ curry 𝑀:𝐼⟶((Base‘𝑅) ↑𝑚 𝐼))))
4039anassrs 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))))
4240, 41sylibr 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))
4744, 45, 46uvcff 19949 . . . . . . . . . . . . 13 ((𝑅 ∈ Ring ∧ 𝐼 ∈ Fin) → (𝑅 unitVec 𝐼):𝐼⟶(Base‘(𝑅 freeLMod 𝐼)))
4843, 47sylan 487 . . . . . . . . . . . 12 ((𝑅 ∈ DivRing ∧ 𝐼 ∈ Fin) → (𝑅 unitVec 𝐼):𝐼⟶(Base‘(𝑅 freeLMod 𝐼)))
4948ffvelrnda 6267 . . . . . . . . . . 11 (((𝑅 ∈ DivRing ∧ 𝐼 ∈ Fin) ∧ 𝑖𝐼) → ((𝑅 unitVec 𝐼)‘𝑖) ∈ (Base‘(𝑅 freeLMod 𝐼)))
5049ad4ant14 1285 . . . . . . . . . 10 (((((𝑅 ∈ DivRing ∧ 𝐼 ∈ Fin) ∧ curry 𝑀:𝐼⟶((Base‘𝑅) ↑𝑚 𝐼)) ∧ curry 𝑀 LIndF (𝑅 freeLMod 𝐼)) ∧ 𝑖𝐼) → ((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 𝑀𝐼) = ran curry 𝑀)
5351, 52syl 17 . . . . . . . . . . . . . . 15 (curry 𝑀:𝐼⟶((Base‘𝑅) ↑𝑚 𝐼) → (curry 𝑀𝐼) = ran curry 𝑀)
5453adantl 481 . . . . . . . . . . . . . 14 (((𝑅 ∈ DivRing ∧ 𝐼 ∈ Fin) ∧ curry 𝑀:𝐼⟶((Base‘𝑅) ↑𝑚 𝐼)) → (curry 𝑀𝐼) = ran curry 𝑀)
5554fveq2d 6107 . . . . . . . . . . . . 13 (((𝑅 ∈ DivRing ∧ 𝐼 ∈ Fin) ∧ curry 𝑀:𝐼⟶((Base‘𝑅) ↑𝑚 𝐼)) → ((LSpan‘(𝑅 freeLMod 𝐼))‘(curry 𝑀𝐼)) = ((LSpan‘(𝑅 freeLMod 𝐼))‘ran curry 𝑀))
5655adantr 480 . . . . . . . . . . . 12 ((((𝑅 ∈ DivRing ∧ 𝐼 ∈ Fin) ∧ curry 𝑀:𝐼⟶((Base‘𝑅) ↑𝑚 𝐼)) ∧ curry 𝑀 LIndF (𝑅 freeLMod 𝐼)) → ((LSpan‘(𝑅 freeLMod 𝐼))‘(curry 𝑀𝐼)) = ((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)
5945frlmlmod 19912 . . . . . . . . . . . . . . . . 17 ((𝑅 ∈ Ring ∧ 𝐼 ∈ Fin) → (𝑅 freeLMod 𝐼) ∈ LMod)
6043, 59sylan 487 . . . . . . . . . . . . . . . 16 ((𝑅 ∈ DivRing ∧ 𝐼 ∈ Fin) → (𝑅 freeLMod 𝐼) ∈ LMod)
6160adantr 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)))
6361, 62sylan 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)))
6445frlmsca 19916 . . . . . . . . . . . . . . . . . . . . . 22 ((𝑅 ∈ DivRing ∧ 𝐼 ∈ Fin) → 𝑅 = (Scalar‘(𝑅 freeLMod 𝐼)))
65 drngnzr 19083 . . . . . . . . . . . . . . . . . . . . . . 23 (𝑅 ∈ DivRing → 𝑅 ∈ NzRing)
6665adantr 480 . . . . . . . . . . . . . . . . . . . . . 22 ((𝑅 ∈ DivRing ∧ 𝐼 ∈ Fin) → 𝑅 ∈ NzRing)
6764, 66eqeltrrd 2689 . . . . . . . . . . . . . . . . . . . . 21 ((𝑅 ∈ DivRing ∧ 𝐼 ∈ Fin) → (Scalar‘(𝑅 freeLMod 𝐼)) ∈ NzRing)
6860, 67jca 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))
7046, 69lindff1 19978 . . . . . . . . . . . . . . . . . . . . 21 (((𝑅 freeLMod 𝐼) ∈ LMod ∧ (Scalar‘(𝑅 freeLMod 𝐼)) ∈ NzRing ∧ curry 𝑀 LIndF (𝑅 freeLMod 𝐼)) → curry 𝑀:dom curry 𝑀1-1→(Base‘(𝑅 freeLMod 𝐼)))
71703expa 1257 . . . . . . . . . . . . . . . . . . . 20 ((((𝑅 freeLMod 𝐼) ∈ LMod ∧ (Scalar‘(𝑅 freeLMod 𝐼)) ∈ NzRing) ∧ curry 𝑀 LIndF (𝑅 freeLMod 𝐼)) → curry 𝑀:dom curry 𝑀1-1→(Base‘(𝑅 freeLMod 𝐼)))
7268, 71sylan 487 . . . . . . . . . . . . . . . . . . 19 (((𝑅 ∈ DivRing ∧ 𝐼 ∈ Fin) ∧ curry 𝑀 LIndF (𝑅 freeLMod 𝐼)) → curry 𝑀:dom curry 𝑀1-1→(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 𝑀1-1→(Base‘(𝑅 freeLMod 𝐼)) ↔ curry 𝑀:𝐼1-1→(Base‘(𝑅 freeLMod 𝐼))))
7574biimpac 502 . . . . . . . . . . . . . . . . . . 19 ((curry 𝑀:dom curry 𝑀1-1→(Base‘(𝑅 freeLMod 𝐼)) ∧ dom curry 𝑀 = 𝐼) → curry 𝑀:𝐼1-1→(Base‘(𝑅 freeLMod 𝐼)))
7672, 73, 75syl2an 493 . . . . . . . . . . . . . . . . . 18 ((((𝑅 ∈ DivRing ∧ 𝐼 ∈ Fin) ∧ curry 𝑀 LIndF (𝑅 freeLMod 𝐼)) ∧ curry 𝑀:𝐼⟶((Base‘𝑅) ↑𝑚 𝐼)) → curry 𝑀:𝐼1-1→(Base‘(𝑅 freeLMod 𝐼)))
7776an32s 842 . . . . . . . . . . . . . . . . 17 ((((𝑅 ∈ DivRing ∧ 𝐼 ∈ Fin) ∧ curry 𝑀:𝐼⟶((Base‘𝑅) ↑𝑚 𝐼)) ∧ curry 𝑀 LIndF (𝑅 freeLMod 𝐼)) → curry 𝑀:𝐼1-1→(Base‘(𝑅 freeLMod 𝐼)))
78 f1f1orn 6061 . . . . . . . . . . . . . . . . 17 (curry 𝑀:𝐼1-1→(Base‘(𝑅 freeLMod 𝐼)) → curry 𝑀:𝐼1-1-onto→ran curry 𝑀)
7977, 78syl 17 . . . . . . . . . . . . . . . 16 ((((𝑅 ∈ DivRing ∧ 𝐼 ∈ Fin) ∧ curry 𝑀:𝐼⟶((Base‘𝑅) ↑𝑚 𝐼)) ∧ curry 𝑀 LIndF (𝑅 freeLMod 𝐼)) → curry 𝑀:𝐼1-1-onto→ran curry 𝑀)
80 f1oeng 7860 . . . . . . . . . . . . . . . 16 ((𝐼 ∈ Fin ∧ curry 𝑀:𝐼1-1-onto→ran curry 𝑀) → 𝐼 ≈ ran curry 𝑀)
8158, 79, 80syl2anc 691 . . . . . . . . . . . . . . 15 ((((𝑅 ∈ DivRing ∧ 𝐼 ∈ Fin) ∧ curry 𝑀:𝐼⟶((Base‘𝑅) ↑𝑚 𝐼)) ∧ curry 𝑀 LIndF (𝑅 freeLMod 𝐼)) → 𝐼 ≈ ran curry 𝑀)
8281ensymd 7893 . . . . . . . . . . . . . 14 ((((𝑅 ∈ DivRing ∧ 𝐼 ∈ Fin) ∧ curry 𝑀:𝐼⟶((Base‘𝑅) ↑𝑚 𝐼)) ∧ curry 𝑀 LIndF (𝑅 freeLMod 𝐼)) → ran curry 𝑀𝐼)
83 lindsenlbs 32574 . . . . . . . . . . . . . 14 (((𝑅 ∈ DivRing ∧ 𝐼 ∈ Fin ∧ ran curry 𝑀 ∈ (LIndS‘(𝑅 freeLMod 𝐼))) ∧ ran curry 𝑀𝐼) → ran curry 𝑀 ∈ (LBasis‘(𝑅 freeLMod 𝐼)))
8457, 58, 63, 82, 83syl31anc 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))
8746, 85, 86lbssp 18900 . . . . . . . . . . . . 13 (ran curry 𝑀 ∈ (LBasis‘(𝑅 freeLMod 𝐼)) → ((LSpan‘(𝑅 freeLMod 𝐼))‘ran curry 𝑀) = (Base‘(𝑅 freeLMod 𝐼)))
8884, 87syl 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)))
8956, 88eqtrd 2644 . . . . . . . . . . 11 ((((𝑅 ∈ DivRing ∧ 𝐼 ∈ Fin) ∧ curry 𝑀:𝐼⟶((Base‘𝑅) ↑𝑚 𝐼)) ∧ curry 𝑀 LIndF (𝑅 freeLMod 𝐼)) → ((LSpan‘(𝑅 freeLMod 𝐼))‘(curry 𝑀𝐼)) = (Base‘(𝑅 freeLMod 𝐼)))
9089adantr 480 . . . . . . . . . 10 (((((𝑅 ∈ DivRing ∧ 𝐼 ∈ Fin) ∧ curry 𝑀:𝐼⟶((Base‘𝑅) ↑𝑚 𝐼)) ∧ curry 𝑀 LIndF (𝑅 freeLMod 𝐼)) ∧ 𝑖𝐼) → ((LSpan‘(𝑅 freeLMod 𝐼))‘(curry 𝑀𝐼)) = (Base‘(𝑅 freeLMod 𝐼)))
9150, 90eleqtrrd 2691 . . . . . . . . 9 (((((𝑅 ∈ DivRing ∧ 𝐼 ∈ Fin) ∧ curry 𝑀:𝐼⟶((Base‘𝑅) ↑𝑚 𝐼)) ∧ curry 𝑀 LIndF (𝑅 freeLMod 𝐼)) ∧ 𝑖𝐼) → ((𝑅 unitVec 𝐼)‘𝑖) ∈ ((LSpan‘(𝑅 freeLMod 𝐼))‘(curry 𝑀𝐼)))
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))
9545, 14frlmfibas 19924 . . . . . . . . . . . . . . 15 ((𝑅 ∈ Ring ∧ 𝐼 ∈ Fin) → ((Base‘𝑅) ↑𝑚 𝐼) = (Base‘(𝑅 freeLMod 𝐼)))
9695feq3d 5945 . . . . . . . . . . . . . 14 ((𝑅 ∈ Ring ∧ 𝐼 ∈ Fin) → (curry 𝑀:𝐼⟶((Base‘𝑅) ↑𝑚 𝐼) ↔ curry 𝑀:𝐼⟶(Base‘(𝑅 freeLMod 𝐼))))
9796biimpa 500 . . . . . . . . . . . . 13 (((𝑅 ∈ Ring ∧ 𝐼 ∈ Fin) ∧ curry 𝑀:𝐼⟶((Base‘𝑅) ↑𝑚 𝐼)) → curry 𝑀:𝐼⟶(Base‘(𝑅 freeLMod 𝐼)))
9859adantr 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)
10086, 46, 92, 69, 93, 94, 97, 98, 99elfilspd 19961 . . . . . . . . . . . 12 (((𝑅 ∈ Ring ∧ 𝐼 ∈ Fin) ∧ curry 𝑀:𝐼⟶((Base‘𝑅) ↑𝑚 𝐼)) → (((𝑅 unitVec 𝐼)‘𝑖) ∈ ((LSpan‘(𝑅 freeLMod 𝐼))‘(curry 𝑀𝐼)) ↔ ∃𝑛 ∈ ((Base‘(Scalar‘(𝑅 freeLMod 𝐼))) ↑𝑚 𝐼)((𝑅 unitVec 𝐼)‘𝑖) = ((𝑅 freeLMod 𝐼) Σg (𝑛𝑓 ( ·𝑠 ‘(𝑅 freeLMod 𝐼))curry 𝑀))))
10145frlmsca 19916 . . . . . . . . . . . . . . . 16 ((𝑅 ∈ Ring ∧ 𝐼 ∈ Fin) → 𝑅 = (Scalar‘(𝑅 freeLMod 𝐼)))
102101fveq2d 6107 . . . . . . . . . . . . . . 15 ((𝑅 ∈ Ring ∧ 𝐼 ∈ Fin) → (Base‘𝑅) = (Base‘(Scalar‘(𝑅 freeLMod 𝐼))))
103102oveq1d 6564 . . . . . . . . . . . . . 14 ((𝑅 ∈ Ring ∧ 𝐼 ∈ Fin) → ((Base‘𝑅) ↑𝑚 𝐼) = ((Base‘(Scalar‘(𝑅 freeLMod 𝐼))) ↑𝑚 𝐼))
104103adantr 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)
107106adantl 481 . . . . . . . . . . . . . . . . . . 19 ((((𝑅 ∈ Ring ∧ 𝐼 ∈ Fin) ∧ curry 𝑀:𝐼⟶((Base‘𝑅) ↑𝑚 𝐼)) ∧ 𝑛:𝐼⟶(Base‘𝑅)) → 𝑛 Fn 𝐼)
10851ad2antlr 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 (𝐼𝐼) = 𝐼
111 eqidd 2611 . . . . . . . . . . . . . . . . . . 19 (((((𝑅 ∈ Ring ∧ 𝐼 ∈ Fin) ∧ curry 𝑀:𝐼⟶((Base‘𝑅) ↑𝑚 𝐼)) ∧ 𝑛:𝐼⟶(Base‘𝑅)) ∧ 𝑘𝐼) → (𝑛𝑘) = (𝑛𝑘))
112 eqidd 2611 . . . . . . . . . . . . . . . . . . 19 (((((𝑅 ∈ Ring ∧ 𝐼 ∈ Fin) ∧ curry 𝑀:𝐼⟶((Base‘𝑅) ↑𝑚 𝐼)) ∧ 𝑛:𝐼⟶(Base‘𝑅)) ∧ 𝑘𝐼) → (curry 𝑀𝑘) = (curry 𝑀𝑘))
113107, 108, 109, 109, 110, 111, 112offval 6802 . . . . . . . . . . . . . . . . . 18 ((((𝑅 ∈ Ring ∧ 𝐼 ∈ Fin) ∧ curry 𝑀:𝐼⟶((Base‘𝑅) ↑𝑚 𝐼)) ∧ 𝑛:𝐼⟶(Base‘𝑅)) → (𝑛𝑓 ( ·𝑠 ‘(𝑅 freeLMod 𝐼))curry 𝑀) = (𝑘𝐼 ↦ ((𝑛𝑘)( ·𝑠 ‘(𝑅 freeLMod 𝐼))(curry 𝑀𝑘))))
114 simp-4r 803 . . . . . . . . . . . . . . . . . . . . 21 (((((𝑅 ∈ Ring ∧ 𝐼 ∈ Fin) ∧ curry 𝑀:𝐼⟶((Base‘𝑅) ↑𝑚 𝐼)) ∧ 𝑛:𝐼⟶(Base‘𝑅)) ∧ 𝑘𝐼) → 𝐼 ∈ Fin)
115 ffvelrn 6265 . . . . . . . . . . . . . . . . . . . . . 22 ((𝑛:𝐼⟶(Base‘𝑅) ∧ 𝑘𝐼) → (𝑛𝑘) ∈ (Base‘𝑅))
116115adantll 746 . . . . . . . . . . . . . . . . . . . . 21 (((((𝑅 ∈ Ring ∧ 𝐼 ∈ Fin) ∧ curry 𝑀:𝐼⟶((Base‘𝑅) ↑𝑚 𝐼)) ∧ 𝑛:𝐼⟶(Base‘𝑅)) ∧ 𝑘𝐼) → (𝑛𝑘) ∈ (Base‘𝑅))
117 ffvelrn 6265 . . . . . . . . . . . . . . . . . . . . . . 23 ((curry 𝑀:𝐼⟶((Base‘𝑅) ↑𝑚 𝐼) ∧ 𝑘𝐼) → (curry 𝑀𝑘) ∈ ((Base‘𝑅) ↑𝑚 𝐼))
118117ad4ant24 1290 . . . . . . . . . . . . . . . . . . . . . 22 (((((𝑅 ∈ Ring ∧ 𝐼 ∈ Fin) ∧ curry 𝑀:𝐼⟶((Base‘𝑅) ↑𝑚 𝐼)) ∧ 𝑛:𝐼⟶(Base‘𝑅)) ∧ 𝑘𝐼) → (curry 𝑀𝑘) ∈ ((Base‘𝑅) ↑𝑚 𝐼))
11995ad3antrrr 762 . . . . . . . . . . . . . . . . . . . . . 22 (((((𝑅 ∈ Ring ∧ 𝐼 ∈ Fin) ∧ curry 𝑀:𝐼⟶((Base‘𝑅) ↑𝑚 𝐼)) ∧ 𝑛:𝐼⟶(Base‘𝑅)) ∧ 𝑘𝐼) → ((Base‘𝑅) ↑𝑚 𝐼) = (Base‘(𝑅 freeLMod 𝐼)))
120118, 119eleqtrd 2690 . . . . . . . . . . . . . . . . . . . . 21 (((((𝑅 ∈ Ring ∧ 𝐼 ∈ Fin) ∧ curry 𝑀:𝐼⟶((Base‘𝑅) ↑𝑚 𝐼)) ∧ 𝑛:𝐼⟶(Base‘𝑅)) ∧ 𝑘𝐼) → (curry 𝑀𝑘) ∈ (Base‘(𝑅 freeLMod 𝐼)))
121 eqid 2610 . . . . . . . . . . . . . . . . . . . . 21 (.r𝑅) = (.r𝑅)
12245, 46, 14, 114, 116, 120, 94, 121frlmvscafval 19928 . . . . . . . . . . . . . . . . . . . 20 (((((𝑅 ∈ Ring ∧ 𝐼 ∈ Fin) ∧ curry 𝑀:𝐼⟶((Base‘𝑅) ↑𝑚 𝐼)) ∧ 𝑛:𝐼⟶(Base‘𝑅)) ∧ 𝑘𝐼) → ((𝑛𝑘)( ·𝑠 ‘(𝑅 freeLMod 𝐼))(curry 𝑀𝑘)) = ((𝐼 × {(𝑛𝑘)}) ∘𝑓 (.r𝑅)(curry 𝑀𝑘)))
123 fvex 6113 . . . . . . . . . . . . . . . . . . . . . 22 (𝑛𝑘) ∈ V
124 fnconstg 6006 . . . . . . . . . . . . . . . . . . . . . 22 ((𝑛𝑘) ∈ V → (𝐼 × {(𝑛𝑘)}) Fn 𝐼)
125123, 124mp1i 13 . . . . . . . . . . . . . . . . . . . . 21 (((((𝑅 ∈ Ring ∧ 𝐼 ∈ Fin) ∧ curry 𝑀:𝐼⟶((Base‘𝑅) ↑𝑚 𝐼)) ∧ 𝑛:𝐼⟶(Base‘𝑅)) ∧ 𝑘𝐼) → (𝐼 × {(𝑛𝑘)}) Fn 𝐼)
126 elmapfn 7766 . . . . . . . . . . . . . . . . . . . . . . 23 ((curry 𝑀𝑘) ∈ ((Base‘𝑅) ↑𝑚 𝐼) → (curry 𝑀𝑘) Fn 𝐼)
127117, 126syl 17 . . . . . . . . . . . . . . . . . . . . . 22 ((curry 𝑀:𝐼⟶((Base‘𝑅) ↑𝑚 𝐼) ∧ 𝑘𝐼) → (curry 𝑀𝑘) Fn 𝐼)
128127ad4ant24 1290 . . . . . . . . . . . . . . . . . . . . 21 (((((𝑅 ∈ Ring ∧ 𝐼 ∈ Fin) ∧ curry 𝑀:𝐼⟶((Base‘𝑅) ↑𝑚 𝐼)) ∧ 𝑛:𝐼⟶(Base‘𝑅)) ∧ 𝑘𝐼) → (curry 𝑀𝑘) Fn 𝐼)
129123fvconst2 6374 . . . . . . . . . . . . . . . . . . . . . 22 (𝑗𝐼 → ((𝐼 × {(𝑛𝑘)})‘𝑗) = (𝑛𝑘))
130129adantl 481 . . . . . . . . . . . . . . . . . . . . 21 ((((((𝑅 ∈ Ring ∧ 𝐼 ∈ Fin) ∧ curry 𝑀:𝐼⟶((Base‘𝑅) ↑𝑚 𝐼)) ∧ 𝑛:𝐼⟶(Base‘𝑅)) ∧ 𝑘𝐼) ∧ 𝑗𝐼) → ((𝐼 × {(𝑛𝑘)})‘𝑗) = (𝑛𝑘))
131 eqidd 2611 . . . . . . . . . . . . . . . . . . . . 21 ((((((𝑅 ∈ Ring ∧ 𝐼 ∈ Fin) ∧ curry 𝑀:𝐼⟶((Base‘𝑅) ↑𝑚 𝐼)) ∧ 𝑛:𝐼⟶(Base‘𝑅)) ∧ 𝑘𝐼) ∧ 𝑗𝐼) → ((curry 𝑀𝑘)‘𝑗) = ((curry 𝑀𝑘)‘𝑗))
132125, 128, 114, 114, 110, 130, 131offval 6802 . . . . . . . . . . . . . . . . . . . 20 (((((𝑅 ∈ Ring ∧ 𝐼 ∈ Fin) ∧ curry 𝑀:𝐼⟶((Base‘𝑅) ↑𝑚 𝐼)) ∧ 𝑛:𝐼⟶(Base‘𝑅)) ∧ 𝑘𝐼) → ((𝐼 × {(𝑛𝑘)}) ∘𝑓 (.r𝑅)(curry 𝑀𝑘)) = (𝑗𝐼 ↦ ((𝑛𝑘)(.r𝑅)((curry 𝑀𝑘)‘𝑗))))
133122, 132eqtrd 2644 . . . . . . . . . . . . . . . . . . 19 (((((𝑅 ∈ Ring ∧ 𝐼 ∈ Fin) ∧ curry 𝑀:𝐼⟶((Base‘𝑅) ↑𝑚 𝐼)) ∧ 𝑛:𝐼⟶(Base‘𝑅)) ∧ 𝑘𝐼) → ((𝑛𝑘)( ·𝑠 ‘(𝑅 freeLMod 𝐼))(curry 𝑀𝑘)) = (𝑗𝐼 ↦ ((𝑛𝑘)(.r𝑅)((curry 𝑀𝑘)‘𝑗))))
134133mpteq2dva 4672 . . . . . . . . . . . . . . . . . 18 ((((𝑅 ∈ Ring ∧ 𝐼 ∈ Fin) ∧ curry 𝑀:𝐼⟶((Base‘𝑅) ↑𝑚 𝐼)) ∧ 𝑛:𝐼⟶(Base‘𝑅)) → (𝑘𝐼 ↦ ((𝑛𝑘)( ·𝑠 ‘(𝑅 freeLMod 𝐼))(curry 𝑀𝑘))) = (𝑘𝐼 ↦ (𝑗𝐼 ↦ ((𝑛𝑘)(.r𝑅)((curry 𝑀𝑘)‘𝑗)))))
135113, 134eqtrd 2644 . . . . . . . . . . . . . . . . 17 ((((𝑅 ∈ Ring ∧ 𝐼 ∈ Fin) ∧ curry 𝑀:𝐼⟶((Base‘𝑅) ↑𝑚 𝐼)) ∧ 𝑛:𝐼⟶(Base‘𝑅)) → (𝑛𝑓 ( ·𝑠 ‘(𝑅 freeLMod 𝐼))curry 𝑀) = (𝑘𝐼 ↦ (𝑗𝐼 ↦ ((𝑛𝑘)(.r𝑅)((curry 𝑀𝑘)‘𝑗)))))
136135oveq2d 6565 . . . . . . . . . . . . . . . 16 ((((𝑅 ∈ Ring ∧ 𝐼 ∈ Fin) ∧ curry 𝑀:𝐼⟶((Base‘𝑅) ↑𝑚 𝐼)) ∧ 𝑛:𝐼⟶(Base‘𝑅)) → ((𝑅 freeLMod 𝐼) Σg (𝑛𝑓 ( ·𝑠 ‘(𝑅 freeLMod 𝐼))curry 𝑀)) = ((𝑅 freeLMod 𝐼) Σg (𝑘𝐼 ↦ (𝑗𝐼 ↦ ((𝑛𝑘)(.r𝑅)((curry 𝑀𝑘)‘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)) ∧ 𝑘𝐼) ∧ 𝑗𝐼) → 𝑅 ∈ Ring)
140115ad4ant23 1289 . . . . . . . . . . . . . . . . . . . 20 ((((((𝑅 ∈ Ring ∧ 𝐼 ∈ Fin) ∧ curry 𝑀:𝐼⟶((Base‘𝑅) ↑𝑚 𝐼)) ∧ 𝑛:𝐼⟶(Base‘𝑅)) ∧ 𝑘𝐼) ∧ 𝑗𝐼) → (𝑛𝑘) ∈ (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 𝑀𝑘) ∈ ((Base‘𝑅) ↑𝑚 𝐼) → (curry 𝑀𝑘):𝐼⟶(Base‘𝑅))
143117, 142syl 17 . . . . . . . . . . . . . . . . . . . . . 22 ((curry 𝑀:𝐼⟶((Base‘𝑅) ↑𝑚 𝐼) ∧ 𝑘𝐼) → (curry 𝑀𝑘):𝐼⟶(Base‘𝑅))
144143ffvelrnda 6267 . . . . . . . . . . . . . . . . . . . . 21 (((curry 𝑀:𝐼⟶((Base‘𝑅) ↑𝑚 𝐼) ∧ 𝑘𝐼) ∧ 𝑗𝐼) → ((curry 𝑀𝑘)‘𝑗) ∈ (Base‘𝑅))
145141, 144sylanl1 680 . . . . . . . . . . . . . . . . . . . 20 ((((((𝑅 ∈ Ring ∧ 𝐼 ∈ Fin) ∧ curry 𝑀:𝐼⟶((Base‘𝑅) ↑𝑚 𝐼)) ∧ 𝑛:𝐼⟶(Base‘𝑅)) ∧ 𝑘𝐼) ∧ 𝑗𝐼) → ((curry 𝑀𝑘)‘𝑗) ∈ (Base‘𝑅))
14614, 121ringcl 18384 . . . . . . . . . . . . . . . . . . . 20 ((𝑅 ∈ Ring ∧ (𝑛𝑘) ∈ (Base‘𝑅) ∧ ((curry 𝑀𝑘)‘𝑗) ∈ (Base‘𝑅)) → ((𝑛𝑘)(.r𝑅)((curry 𝑀𝑘)‘𝑗)) ∈ (Base‘𝑅))
147139, 140, 145, 146syl3anc 1318 . . . . . . . . . . . . . . . . . . 19 ((((((𝑅 ∈ Ring ∧ 𝐼 ∈ Fin) ∧ curry 𝑀:𝐼⟶((Base‘𝑅) ↑𝑚 𝐼)) ∧ 𝑛:𝐼⟶(Base‘𝑅)) ∧ 𝑘𝐼) ∧ 𝑗𝐼) → ((𝑛𝑘)(.r𝑅)((curry 𝑀𝑘)‘𝑗)) ∈ (Base‘𝑅))
148 eqid 2610 . . . . . . . . . . . . . . . . . . 19 (𝑗𝐼 ↦ ((𝑛𝑘)(.r𝑅)((curry 𝑀𝑘)‘𝑗))) = (𝑗𝐼 ↦ ((𝑛𝑘)(.r𝑅)((curry 𝑀𝑘)‘𝑗)))
149147, 148fmptd 6292 . . . . . . . . . . . . . . . . . 18 (((((𝑅 ∈ Ring ∧ 𝐼 ∈ Fin) ∧ curry 𝑀:𝐼⟶((Base‘𝑅) ↑𝑚 𝐼)) ∧ 𝑛:𝐼⟶(Base‘𝑅)) ∧ 𝑘𝐼) → (𝑗𝐼 ↦ ((𝑛𝑘)(.r𝑅)((curry 𝑀𝑘)‘𝑗))):𝐼⟶(Base‘𝑅))
150 elmapg 7757 . . . . . . . . . . . . . . . . . . . . . 22 (((Base‘𝑅) ∈ V ∧ 𝐼 ∈ Fin) → ((𝑗𝐼 ↦ ((𝑛𝑘)(.r𝑅)((curry 𝑀𝑘)‘𝑗))) ∈ ((Base‘𝑅) ↑𝑚 𝐼) ↔ (𝑗𝐼 ↦ ((𝑛𝑘)(.r𝑅)((curry 𝑀𝑘)‘𝑗))):𝐼⟶(Base‘𝑅)))
15122, 150mpan 702 . . . . . . . . . . . . . . . . . . . . 21 (𝐼 ∈ Fin → ((𝑗𝐼 ↦ ((𝑛𝑘)(.r𝑅)((curry 𝑀𝑘)‘𝑗))) ∈ ((Base‘𝑅) ↑𝑚 𝐼) ↔ (𝑗𝐼 ↦ ((𝑛𝑘)(.r𝑅)((curry 𝑀𝑘)‘𝑗))):𝐼⟶(Base‘𝑅)))
152151adantl 481 . . . . . . . . . . . . . . . . . . . 20 ((𝑅 ∈ Ring ∧ 𝐼 ∈ Fin) → ((𝑗𝐼 ↦ ((𝑛𝑘)(.r𝑅)((curry 𝑀𝑘)‘𝑗))) ∈ ((Base‘𝑅) ↑𝑚 𝐼) ↔ (𝑗𝐼 ↦ ((𝑛𝑘)(.r𝑅)((curry 𝑀𝑘)‘𝑗))):𝐼⟶(Base‘𝑅)))
15395eleq2d 2673 . . . . . . . . . . . . . . . . . . . 20 ((𝑅 ∈ Ring ∧ 𝐼 ∈ Fin) → ((𝑗𝐼 ↦ ((𝑛𝑘)(.r𝑅)((curry 𝑀𝑘)‘𝑗))) ∈ ((Base‘𝑅) ↑𝑚 𝐼) ↔ (𝑗𝐼 ↦ ((𝑛𝑘)(.r𝑅)((curry 𝑀𝑘)‘𝑗))) ∈ (Base‘(𝑅 freeLMod 𝐼))))
154152, 153bitr3d 269 . . . . . . . . . . . . . . . . . . 19 ((𝑅 ∈ Ring ∧ 𝐼 ∈ Fin) → ((𝑗𝐼 ↦ ((𝑛𝑘)(.r𝑅)((curry 𝑀𝑘)‘𝑗))):𝐼⟶(Base‘𝑅) ↔ (𝑗𝐼 ↦ ((𝑛𝑘)(.r𝑅)((curry 𝑀𝑘)‘𝑗))) ∈ (Base‘(𝑅 freeLMod 𝐼))))
155154ad3antrrr 762 . . . . . . . . . . . . . . . . . 18 (((((𝑅 ∈ Ring ∧ 𝐼 ∈ Fin) ∧ curry 𝑀:𝐼⟶((Base‘𝑅) ↑𝑚 𝐼)) ∧ 𝑛:𝐼⟶(Base‘𝑅)) ∧ 𝑘𝐼) → ((𝑗𝐼 ↦ ((𝑛𝑘)(.r𝑅)((curry 𝑀𝑘)‘𝑗))):𝐼⟶(Base‘𝑅) ↔ (𝑗𝐼 ↦ ((𝑛𝑘)(.r𝑅)((curry 𝑀𝑘)‘𝑗))) ∈ (Base‘(𝑅 freeLMod 𝐼))))
156149, 155mpbid 221 . . . . . . . . . . . . . . . . 17 (((((𝑅 ∈ Ring ∧ 𝐼 ∈ Fin) ∧ curry 𝑀:𝐼⟶((Base‘𝑅) ↑𝑚 𝐼)) ∧ 𝑛:𝐼⟶(Base‘𝑅)) ∧ 𝑘𝐼) → (𝑗𝐼 ↦ ((𝑛𝑘)(.r𝑅)((curry 𝑀𝑘)‘𝑗))) ∈ (Base‘(𝑅 freeLMod 𝐼)))
157 mptexg 6389 . . . . . . . . . . . . . . . . . . . . 21 (𝐼 ∈ Fin → (𝑗𝐼 ↦ ((𝑛𝑘)(.r𝑅)((curry 𝑀𝑘)‘𝑗))) ∈ V)
158157ralrimivw 2950 . . . . . . . . . . . . . . . . . . . 20 (𝐼 ∈ Fin → ∀𝑘𝐼 (𝑗𝐼 ↦ ((𝑛𝑘)(.r𝑅)((curry 𝑀𝑘)‘𝑗))) ∈ V)
159 eqid 2610 . . . . . . . . . . . . . . . . . . . . 21 (𝑘𝐼 ↦ (𝑗𝐼 ↦ ((𝑛𝑘)(.r𝑅)((curry 𝑀𝑘)‘𝑗)))) = (𝑘𝐼 ↦ (𝑗𝐼 ↦ ((𝑛𝑘)(.r𝑅)((curry 𝑀𝑘)‘𝑗))))
160159fnmpt 5933 . . . . . . . . . . . . . . . . . . . 20 (∀𝑘𝐼 (𝑗𝐼 ↦ ((𝑛𝑘)(.r𝑅)((curry 𝑀𝑘)‘𝑗))) ∈ V → (𝑘𝐼 ↦ (𝑗𝐼 ↦ ((𝑛𝑘)(.r𝑅)((curry 𝑀𝑘)‘𝑗)))) Fn 𝐼)
161158, 160syl 17 . . . . . . . . . . . . . . . . . . 19 (𝐼 ∈ Fin → (𝑘𝐼 ↦ (𝑗𝐼 ↦ ((𝑛𝑘)(.r𝑅)((curry 𝑀𝑘)‘𝑗)))) Fn 𝐼)
162 id 22 . . . . . . . . . . . . . . . . . . 19 (𝐼 ∈ Fin → 𝐼 ∈ Fin)
163 fvex 6113 . . . . . . . . . . . . . . . . . . . 20 (0g‘(𝑅 freeLMod 𝐼)) ∈ V
164163a1i 11 . . . . . . . . . . . . . . . . . . 19 (𝐼 ∈ Fin → (0g‘(𝑅 freeLMod 𝐼)) ∈ V)
165161, 162, 164fndmfifsupp 8171 . . . . . . . . . . . . . . . . . 18 (𝐼 ∈ Fin → (𝑘𝐼 ↦ (𝑗𝐼 ↦ ((𝑛𝑘)(.r𝑅)((curry 𝑀𝑘)‘𝑗)))) finSupp (0g‘(𝑅 freeLMod 𝐼)))
166165ad3antlr 763 . . . . . . . . . . . . . . . . 17 ((((𝑅 ∈ Ring ∧ 𝐼 ∈ Fin) ∧ curry 𝑀:𝐼⟶((Base‘𝑅) ↑𝑚 𝐼)) ∧ 𝑛:𝐼⟶(Base‘𝑅)) → (𝑘𝐼 ↦ (𝑗𝐼 ↦ ((𝑛𝑘)(.r𝑅)((curry 𝑀𝑘)‘𝑗)))) finSupp (0g‘(𝑅 freeLMod 𝐼)))
16745, 46, 137, 109, 109, 138, 156, 166frlmgsum 19930 . . . . . . . . . . . . . . . 16 ((((𝑅 ∈ Ring ∧ 𝐼 ∈ Fin) ∧ curry 𝑀:𝐼⟶((Base‘𝑅) ↑𝑚 𝐼)) ∧ 𝑛:𝐼⟶(Base‘𝑅)) → ((𝑅 freeLMod 𝐼) Σg (𝑘𝐼 ↦ (𝑗𝐼 ↦ ((𝑛𝑘)(.r𝑅)((curry 𝑀𝑘)‘𝑗))))) = (𝑗𝐼 ↦ (𝑅 Σg (𝑘𝐼 ↦ ((𝑛𝑘)(.r𝑅)((curry 𝑀𝑘)‘𝑗))))))
168136, 167eqtr2d 2645 . . . . . . . . . . . . . . 15 ((((𝑅 ∈ Ring ∧ 𝐼 ∈ Fin) ∧ curry 𝑀:𝐼⟶((Base‘𝑅) ↑𝑚 𝐼)) ∧ 𝑛:𝐼⟶(Base‘𝑅)) → (𝑗𝐼 ↦ (𝑅 Σg (𝑘𝐼 ↦ ((𝑛𝑘)(.r𝑅)((curry 𝑀𝑘)‘𝑗))))) = ((𝑅 freeLMod 𝐼) Σg (𝑛𝑓 ( ·𝑠 ‘(𝑅 freeLMod 𝐼))curry 𝑀)))
169105, 168sylan2 490 . . . . . . . . . . . . . 14 ((((𝑅 ∈ Ring ∧ 𝐼 ∈ Fin) ∧ curry 𝑀:𝐼⟶((Base‘𝑅) ↑𝑚 𝐼)) ∧ 𝑛 ∈ ((Base‘𝑅) ↑𝑚 𝐼)) → (𝑗𝐼 ↦ (𝑅 Σg (𝑘𝐼 ↦ ((𝑛𝑘)(.r𝑅)((curry 𝑀𝑘)‘𝑗))))) = ((𝑅 freeLMod 𝐼) Σg (𝑛𝑓 ( ·𝑠 ‘(𝑅 freeLMod 𝐼))curry 𝑀)))
170169eqeq2d 2620 . . . . . . . . . . . . 13 ((((𝑅 ∈ Ring ∧ 𝐼 ∈ Fin) ∧ curry 𝑀:𝐼⟶((Base‘𝑅) ↑𝑚 𝐼)) ∧ 𝑛 ∈ ((Base‘𝑅) ↑𝑚 𝐼)) → (((𝑅 unitVec 𝐼)‘𝑖) = (𝑗𝐼 ↦ (𝑅 Σg (𝑘𝐼 ↦ ((𝑛𝑘)(.r𝑅)((curry 𝑀𝑘)‘𝑗))))) ↔ ((𝑅 unitVec 𝐼)‘𝑖) = ((𝑅 freeLMod 𝐼) Σg (𝑛𝑓 ( ·𝑠 ‘(𝑅 freeLMod 𝐼))curry 𝑀))))
171104, 170rexeqbidva 3132 . . . . . . . . . . . 12 (((𝑅 ∈ Ring ∧ 𝐼 ∈ Fin) ∧ curry 𝑀:𝐼⟶((Base‘𝑅) ↑𝑚 𝐼)) → (∃𝑛 ∈ ((Base‘𝑅) ↑𝑚 𝐼)((𝑅 unitVec 𝐼)‘𝑖) = (𝑗𝐼 ↦ (𝑅 Σg (𝑘𝐼 ↦ ((𝑛𝑘)(.r𝑅)((curry 𝑀𝑘)‘𝑗))))) ↔ ∃𝑛 ∈ ((Base‘(Scalar‘(𝑅 freeLMod 𝐼))) ↑𝑚 𝐼)((𝑅 unitVec 𝐼)‘𝑖) = ((𝑅 freeLMod 𝐼) Σg (𝑛𝑓 ( ·𝑠 ‘(𝑅 freeLMod 𝐼))curry 𝑀))))
172100, 171bitr4d 270 . . . . . . . . . . 11 (((𝑅 ∈ Ring ∧ 𝐼 ∈ Fin) ∧ curry 𝑀:𝐼⟶((Base‘𝑅) ↑𝑚 𝐼)) → (((𝑅 unitVec 𝐼)‘𝑖) ∈ ((LSpan‘(𝑅 freeLMod 𝐼))‘(curry 𝑀𝐼)) ↔ ∃𝑛 ∈ ((Base‘𝑅) ↑𝑚 𝐼)((𝑅 unitVec 𝐼)‘𝑖) = (𝑗𝐼 ↦ (𝑅 Σg (𝑘𝐼 ↦ ((𝑛𝑘)(.r𝑅)((curry 𝑀𝑘)‘𝑗)))))))
17343, 172sylanl1 680 . . . . . . . . . 10 (((𝑅 ∈ DivRing ∧ 𝐼 ∈ Fin) ∧ curry 𝑀:𝐼⟶((Base‘𝑅) ↑𝑚 𝐼)) → (((𝑅 unitVec 𝐼)‘𝑖) ∈ ((LSpan‘(𝑅 freeLMod 𝐼))‘(curry 𝑀𝐼)) ↔ ∃𝑛 ∈ ((Base‘𝑅) ↑𝑚 𝐼)((𝑅 unitVec 𝐼)‘𝑖) = (𝑗𝐼 ↦ (𝑅 Σg (𝑘𝐼 ↦ ((𝑛𝑘)(.r𝑅)((curry 𝑀𝑘)‘𝑗)))))))
174173ad2antrr 758 . . . . . . . . 9 (((((𝑅 ∈ DivRing ∧ 𝐼 ∈ Fin) ∧ curry 𝑀:𝐼⟶((Base‘𝑅) ↑𝑚 𝐼)) ∧ curry 𝑀 LIndF (𝑅 freeLMod 𝐼)) ∧ 𝑖𝐼) → (((𝑅 unitVec 𝐼)‘𝑖) ∈ ((LSpan‘(𝑅 freeLMod 𝐼))‘(curry 𝑀𝐼)) ↔ ∃𝑛 ∈ ((Base‘𝑅) ↑𝑚 𝐼)((𝑅 unitVec 𝐼)‘𝑖) = (𝑗𝐼 ↦ (𝑅 Σg (𝑘𝐼 ↦ ((𝑛𝑘)(.r𝑅)((curry 𝑀𝑘)‘𝑗)))))))
17591, 174mpbid 221 . . . . . . . 8 (((((𝑅 ∈ DivRing ∧ 𝐼 ∈ Fin) ∧ curry 𝑀:𝐼⟶((Base‘𝑅) ↑𝑚 𝐼)) ∧ curry 𝑀 LIndF (𝑅 freeLMod 𝐼)) ∧ 𝑖𝐼) → ∃𝑛 ∈ ((Base‘𝑅) ↑𝑚 𝐼)((𝑅 unitVec 𝐼)‘𝑖) = (𝑗𝐼 ↦ (𝑅 Σg (𝑘𝐼 ↦ ((𝑛𝑘)(.r𝑅)((curry 𝑀𝑘)‘𝑗))))))
176175ralrimiva 2949 . . . . . . 7 ((((𝑅 ∈ DivRing ∧ 𝐼 ∈ Fin) ∧ curry 𝑀:𝐼⟶((Base‘𝑅) ↑𝑚 𝐼)) ∧ curry 𝑀 LIndF (𝑅 freeLMod 𝐼)) → ∀𝑖𝐼𝑛 ∈ ((Base‘𝑅) ↑𝑚 𝐼)((𝑅 unitVec 𝐼)‘𝑖) = (𝑗𝐼 ↦ (𝑅 Σg (𝑘𝐼 ↦ ((𝑛𝑘)(.r𝑅)((curry 𝑀𝑘)‘𝑗))))))
17742, 176sylan 487 . . . . . 6 ((((𝑅 ∈ DivRing ∧ 𝑀 ∈ (Base‘(𝐼 Mat 𝑅))) ∧ 𝐼 ≠ ∅) ∧ curry 𝑀 LIndF (𝑅 freeLMod 𝐼)) → ∀𝑖𝐼𝑛 ∈ ((Base‘𝑅) ↑𝑚 𝐼)((𝑅 unitVec 𝐼)‘𝑖) = (𝑗𝐼 ↦ (𝑅 Σg (𝑘𝐼 ↦ ((𝑛𝑘)(.r𝑅)((curry 𝑀𝑘)‘𝑗))))))
17810, 21mpbird 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))
180178, 179syl 17 . . . . . . . 8 ((𝑅 ∈ DivRing ∧ 𝑀 ∈ (Base‘(𝐼 Mat 𝑅))) → 𝑀 Fn (𝐼 × 𝐼))
1814adantl 481 . . . . . . . 8 ((𝑅 ∈ DivRing ∧ 𝑀 ∈ (Base‘(𝐼 Mat 𝑅))) → 𝐼 ∈ Fin)
182 an32 835 . . . . . . . . . . . . . . . . . . 19 (((𝑀 Fn (𝐼 × 𝐼) ∧ 𝑗𝐼) ∧ 𝑘𝐼) ↔ ((𝑀 Fn (𝐼 × 𝐼) ∧ 𝑘𝐼) ∧ 𝑗𝐼))
183 df-3an 1033 . . . . . . . . . . . . . . . . . . 19 ((𝑀 Fn (𝐼 × 𝐼) ∧ 𝑘𝐼𝑗𝐼) ↔ ((𝑀 Fn (𝐼 × 𝐼) ∧ 𝑘𝐼) ∧ 𝑗𝐼))
184182, 183bitr4i 266 . . . . . . . . . . . . . . . . . 18 (((𝑀 Fn (𝐼 × 𝐼) ∧ 𝑗𝐼) ∧ 𝑘𝐼) ↔ (𝑀 Fn (𝐼 × 𝐼) ∧ 𝑘𝐼𝑗𝐼))
185 curfv 32559 . . . . . . . . . . . . . . . . . 18 (((𝑀 Fn (𝐼 × 𝐼) ∧ 𝑘𝐼𝑗𝐼) ∧ 𝐼 ∈ Fin) → ((curry 𝑀𝑘)‘𝑗) = (𝑘𝑀𝑗))
186184, 185sylanb 488 . . . . . . . . . . . . . . . . 17 ((((𝑀 Fn (𝐼 × 𝐼) ∧ 𝑗𝐼) ∧ 𝑘𝐼) ∧ 𝐼 ∈ Fin) → ((curry 𝑀𝑘)‘𝑗) = (𝑘𝑀𝑗))
187186an32s 842 . . . . . . . . . . . . . . . 16 ((((𝑀 Fn (𝐼 × 𝐼) ∧ 𝑗𝐼) ∧ 𝐼 ∈ Fin) ∧ 𝑘𝐼) → ((curry 𝑀𝑘)‘𝑗) = (𝑘𝑀𝑗))
188187oveq2d 6565 . . . . . . . . . . . . . . 15 ((((𝑀 Fn (𝐼 × 𝐼) ∧ 𝑗𝐼) ∧ 𝐼 ∈ Fin) ∧ 𝑘𝐼) → ((𝑛𝑘)(.r𝑅)((curry 𝑀𝑘)‘𝑗)) = ((𝑛𝑘)(.r𝑅)(𝑘𝑀𝑗)))
189188mpteq2dva 4672 . . . . . . . . . . . . . 14 (((𝑀 Fn (𝐼 × 𝐼) ∧ 𝑗𝐼) ∧ 𝐼 ∈ Fin) → (𝑘𝐼 ↦ ((𝑛𝑘)(.r𝑅)((curry 𝑀𝑘)‘𝑗))) = (𝑘𝐼 ↦ ((𝑛𝑘)(.r𝑅)(𝑘𝑀𝑗))))
190189an32s 842 . . . . . . . . . . . . 13 (((𝑀 Fn (𝐼 × 𝐼) ∧ 𝐼 ∈ Fin) ∧ 𝑗𝐼) → (𝑘𝐼 ↦ ((𝑛𝑘)(.r𝑅)((curry 𝑀𝑘)‘𝑗))) = (𝑘𝐼 ↦ ((𝑛𝑘)(.r𝑅)(𝑘𝑀𝑗))))
191190oveq2d 6565 . . . . . . . . . . . 12 (((𝑀 Fn (𝐼 × 𝐼) ∧ 𝐼 ∈ Fin) ∧ 𝑗𝐼) → (𝑅 Σg (𝑘𝐼 ↦ ((𝑛𝑘)(.r𝑅)((curry 𝑀𝑘)‘𝑗)))) = (𝑅 Σg (𝑘𝐼 ↦ ((𝑛𝑘)(.r𝑅)(𝑘𝑀𝑗)))))
192191mpteq2dva 4672 . . . . . . . . . . 11 ((𝑀 Fn (𝐼 × 𝐼) ∧ 𝐼 ∈ Fin) → (𝑗𝐼 ↦ (𝑅 Σg (𝑘𝐼 ↦ ((𝑛𝑘)(.r𝑅)((curry 𝑀𝑘)‘𝑗))))) = (𝑗𝐼 ↦ (𝑅 Σg (𝑘𝐼 ↦ ((𝑛𝑘)(.r𝑅)(𝑘𝑀𝑗))))))
193192eqeq2d 2620 . . . . . . . . . 10 ((𝑀 Fn (𝐼 × 𝐼) ∧ 𝐼 ∈ Fin) → (((𝑅 unitVec 𝐼)‘𝑖) = (𝑗𝐼 ↦ (𝑅 Σg (𝑘𝐼 ↦ ((𝑛𝑘)(.r𝑅)((curry 𝑀𝑘)‘𝑗))))) ↔ ((𝑅 unitVec 𝐼)‘𝑖) = (𝑗𝐼 ↦ (𝑅 Σg (𝑘𝐼 ↦ ((𝑛𝑘)(.r𝑅)(𝑘𝑀𝑗)))))))
194193rexbidv 3034 . . . . . . . . 9 ((𝑀 Fn (𝐼 × 𝐼) ∧ 𝐼 ∈ Fin) → (∃𝑛 ∈ ((Base‘𝑅) ↑𝑚 𝐼)((𝑅 unitVec 𝐼)‘𝑖) = (𝑗𝐼 ↦ (𝑅 Σg (𝑘𝐼 ↦ ((𝑛𝑘)(.r𝑅)((curry 𝑀𝑘)‘𝑗))))) ↔ ∃𝑛 ∈ ((Base‘𝑅) ↑𝑚 𝐼)((𝑅 unitVec 𝐼)‘𝑖) = (𝑗𝐼 ↦ (𝑅 Σg (𝑘𝐼 ↦ ((𝑛𝑘)(.r𝑅)(𝑘𝑀𝑗)))))))
195194ralbidv 2969 . . . . . . . 8 ((𝑀 Fn (𝐼 × 𝐼) ∧ 𝐼 ∈ Fin) → (∀𝑖𝐼𝑛 ∈ ((Base‘𝑅) ↑𝑚 𝐼)((𝑅 unitVec 𝐼)‘𝑖) = (𝑗𝐼 ↦ (𝑅 Σg (𝑘𝐼 ↦ ((𝑛𝑘)(.r𝑅)((curry 𝑀𝑘)‘𝑗))))) ↔ ∀𝑖𝐼𝑛 ∈ ((Base‘𝑅) ↑𝑚 𝐼)((𝑅 unitVec 𝐼)‘𝑖) = (𝑗𝐼 ↦ (𝑅 Σg (𝑘𝐼 ↦ ((𝑛𝑘)(.r𝑅)(𝑘𝑀𝑗)))))))
196180, 181, 195syl2anc 691 . . . . . . 7 ((𝑅 ∈ DivRing ∧ 𝑀 ∈ (Base‘(𝐼 Mat 𝑅))) → (∀𝑖𝐼𝑛 ∈ ((Base‘𝑅) ↑𝑚 𝐼)((𝑅 unitVec 𝐼)‘𝑖) = (𝑗𝐼 ↦ (𝑅 Σg (𝑘𝐼 ↦ ((𝑛𝑘)(.r𝑅)((curry 𝑀𝑘)‘𝑗))))) ↔ ∀𝑖𝐼𝑛 ∈ ((Base‘𝑅) ↑𝑚 𝐼)((𝑅 unitVec 𝐼)‘𝑖) = (𝑗𝐼 ↦ (𝑅 Σg (𝑘𝐼 ↦ ((𝑛𝑘)(.r𝑅)(𝑘𝑀𝑗)))))))
197196ad2antrr 758 . . . . . 6 ((((𝑅 ∈ DivRing ∧ 𝑀 ∈ (Base‘(𝐼 Mat 𝑅))) ∧ 𝐼 ≠ ∅) ∧ curry 𝑀 LIndF (𝑅 freeLMod 𝐼)) → (∀𝑖𝐼𝑛 ∈ ((Base‘𝑅) ↑𝑚 𝐼)((𝑅 unitVec 𝐼)‘𝑖) = (𝑗𝐼 ↦ (𝑅 Σg (𝑘𝐼 ↦ ((𝑛𝑘)(.r𝑅)((curry 𝑀𝑘)‘𝑗))))) ↔ ∀𝑖𝐼𝑛 ∈ ((Base‘𝑅) ↑𝑚 𝐼)((𝑅 unitVec 𝐼)‘𝑖) = (𝑗𝐼 ↦ (𝑅 Σg (𝑘𝐼 ↦ ((𝑛𝑘)(.r𝑅)(𝑘𝑀𝑗)))))))
198177, 197mpbid 221 . . . . 5 ((((𝑅 ∈ DivRing ∧ 𝑀 ∈ (Base‘(𝐼 Mat 𝑅))) ∧ 𝐼 ≠ ∅) ∧ curry 𝑀 LIndF (𝑅 freeLMod 𝐼)) → ∀𝑖𝐼𝑛 ∈ ((Base‘𝑅) ↑𝑚 𝐼)((𝑅 unitVec 𝐼)‘𝑖) = (𝑗𝐼 ↦ (𝑅 Σg (𝑘𝐼 ↦ ((𝑛𝑘)(.r𝑅)(𝑘𝑀𝑗))))))
1998, 198sylanl1 680 . . . 4 ((((𝑅 ∈ Field ∧ 𝑀 ∈ (Base‘(𝐼 Mat 𝑅))) ∧ 𝐼 ≠ ∅) ∧ curry 𝑀 LIndF (𝑅 freeLMod 𝐼)) → ∀𝑖𝐼𝑛 ∈ ((Base‘𝑅) ↑𝑚 𝐼)((𝑅 unitVec 𝐼)‘𝑖) = (𝑗𝐼 ↦ (𝑅 Σg (𝑘𝐼 ↦ ((𝑛𝑘)(.r𝑅)(𝑘𝑀𝑗))))))
200 fveq1 6102 . . . . . . . . . . 11 (𝑛 = (𝑓𝑖) → (𝑛𝑘) = ((𝑓𝑖)‘𝑘))
201 vex 3176 . . . . . . . . . . . 12 𝑖 ∈ V
202 vex 3176 . . . . . . . . . . . 12 𝑘 ∈ V
203 uncov 32560 . . . . . . . . . . . 12 ((𝑖 ∈ V ∧ 𝑘 ∈ V) → (𝑖uncurry 𝑓𝑘) = ((𝑓𝑖)‘𝑘))
204201, 202, 203mp2an 704 . . . . . . . . . . 11 (𝑖uncurry 𝑓𝑘) = ((𝑓𝑖)‘𝑘)
205200, 204syl6eqr 2662 . . . . . . . . . 10 (𝑛 = (𝑓𝑖) → (𝑛𝑘) = (𝑖uncurry 𝑓𝑘))
206205oveq1d 6564 . . . . . . . . 9 (𝑛 = (𝑓𝑖) → ((𝑛𝑘)(.r𝑅)(𝑘𝑀𝑗)) = ((𝑖uncurry 𝑓𝑘)(.r𝑅)(𝑘𝑀𝑗)))
207206mpteq2dv 4673 . . . . . . . 8 (𝑛 = (𝑓𝑖) → (𝑘𝐼 ↦ ((𝑛𝑘)(.r𝑅)(𝑘𝑀𝑗))) = (𝑘𝐼 ↦ ((𝑖uncurry 𝑓𝑘)(.r𝑅)(𝑘𝑀𝑗))))
208207oveq2d 6565 . . . . . . 7 (𝑛 = (𝑓𝑖) → (𝑅 Σg (𝑘𝐼 ↦ ((𝑛𝑘)(.r𝑅)(𝑘𝑀𝑗)))) = (𝑅 Σg (𝑘𝐼 ↦ ((𝑖uncurry 𝑓𝑘)(.r𝑅)(𝑘𝑀𝑗)))))
209208mpteq2dv 4673 . . . . . 6 (𝑛 = (𝑓𝑖) → (𝑗𝐼 ↦ (𝑅 Σg (𝑘𝐼 ↦ ((𝑛𝑘)(.r𝑅)(𝑘𝑀𝑗))))) = (𝑗𝐼 ↦ (𝑅 Σg (𝑘𝐼 ↦ ((𝑖uncurry 𝑓𝑘)(.r𝑅)(𝑘𝑀𝑗))))))
210209eqeq2d 2620 . . . . 5 (𝑛 = (𝑓𝑖) → (((𝑅 unitVec 𝐼)‘𝑖) = (𝑗𝐼 ↦ (𝑅 Σg (𝑘𝐼 ↦ ((𝑛𝑘)(.r𝑅)(𝑘𝑀𝑗))))) ↔ ((𝑅 unitVec 𝐼)‘𝑖) = (𝑗𝐼 ↦ (𝑅 Σg (𝑘𝐼 ↦ ((𝑖uncurry 𝑓𝑘)(.r𝑅)(𝑘𝑀𝑗)))))))
211210ac6sfi 8089 . . . 4 ((𝐼 ∈ Fin ∧ ∀𝑖𝐼𝑛 ∈ ((Base‘𝑅) ↑𝑚 𝐼)((𝑅 unitVec 𝐼)‘𝑖) = (𝑗𝐼 ↦ (𝑅 Σg (𝑘𝐼 ↦ ((𝑛𝑘)(.r𝑅)(𝑘𝑀𝑗)))))) → ∃𝑓(𝑓:𝐼⟶((Base‘𝑅) ↑𝑚 𝐼) ∧ ∀𝑖𝐼 ((𝑅 unitVec 𝐼)‘𝑖) = (𝑗𝐼 ↦ (𝑅 Σg (𝑘𝐼 ↦ ((𝑖uncurry 𝑓𝑘)(.r𝑅)(𝑘𝑀𝑗)))))))
2125, 199, 211syl2anc 691 . . 3 ((((𝑅 ∈ Field ∧ 𝑀 ∈ (Base‘(𝐼 Mat 𝑅))) ∧ 𝐼 ≠ ∅) ∧ curry 𝑀 LIndF (𝑅 freeLMod 𝐼)) → ∃𝑓(𝑓:𝐼⟶((Base‘𝑅) ↑𝑚 𝐼) ∧ ∀𝑖𝐼 ((𝑅 unitVec 𝐼)‘𝑖) = (𝑗𝐼 ↦ (𝑅 Σg (𝑘𝐼 ↦ ((𝑖uncurry 𝑓𝑘)(.r𝑅)(𝑘𝑀𝑗)))))))
213 uncf 32558 . . . . . . 7 (𝑓:𝐼⟶((Base‘𝑅) ↑𝑚 𝐼) → uncurry 𝑓:(𝐼 × 𝐼)⟶(Base‘𝑅))
21413, 14frlmfibas 19924 . . . . . . . . . . . . . . . 16 ((𝑅 ∈ Field ∧ (𝐼 × 𝐼) ∈ Fin) → ((Base‘𝑅) ↑𝑚 (𝐼 × 𝐼)) = (Base‘(𝑅 freeLMod (𝐼 × 𝐼))))
21512, 214sylan2 490 . . . . . . . . . . . . . . 15 ((𝑅 ∈ Field ∧ 𝐼 ∈ Fin) → ((Base‘𝑅) ↑𝑚 (𝐼 × 𝐼)) = (Base‘(𝑅 freeLMod (𝐼 × 𝐼))))
2161, 13matbas 20038 . . . . . . . . . . . . . . . 16 ((𝐼 ∈ Fin ∧ 𝑅 ∈ Field) → (Base‘(𝑅 freeLMod (𝐼 × 𝐼))) = (Base‘(𝐼 Mat 𝑅)))
217216ancoms 468 . . . . . . . . . . . . . . 15 ((𝑅 ∈ Field ∧ 𝐼 ∈ Fin) → (Base‘(𝑅 freeLMod (𝐼 × 𝐼))) = (Base‘(𝐼 Mat 𝑅)))
218215, 217eqtrd 2644 . . . . . . . . . . . . . 14 ((𝑅 ∈ Field ∧ 𝐼 ∈ Fin) → ((Base‘𝑅) ↑𝑚 (𝐼 × 𝐼)) = (Base‘(𝐼 Mat 𝑅)))
2194, 218sylan2 490 . . . . . . . . . . . . 13 ((𝑅 ∈ Field ∧ 𝑀 ∈ (Base‘(𝐼 Mat 𝑅))) → ((Base‘𝑅) ↑𝑚 (𝐼 × 𝐼)) = (Base‘(𝐼 Mat 𝑅)))
220219eleq2d 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)))
22222, 23, 221sylancr 694 . . . . . . . . . . . . 13 (𝑀 ∈ (Base‘(𝐼 Mat 𝑅)) → (uncurry 𝑓 ∈ ((Base‘𝑅) ↑𝑚 (𝐼 × 𝐼)) ↔ uncurry 𝑓:(𝐼 × 𝐼)⟶(Base‘𝑅)))
223222adantl 481 . . . . . . . . . . . 12 ((𝑅 ∈ Field ∧ 𝑀 ∈ (Base‘(𝐼 Mat 𝑅))) → (uncurry 𝑓 ∈ ((Base‘𝑅) ↑𝑚 (𝐼 × 𝐼)) ↔ uncurry 𝑓:(𝐼 × 𝐼)⟶(Base‘𝑅)))
224220, 223bitr3d 269 . . . . . . . . . . 11 ((𝑅 ∈ Field ∧ 𝑀 ∈ (Base‘(𝐼 Mat 𝑅))) → (uncurry 𝑓 ∈ (Base‘(𝐼 Mat 𝑅)) ↔ uncurry 𝑓:(𝐼 × 𝐼)⟶(Base‘𝑅)))
225224biimpar 501 . . . . . . . . . 10 (((𝑅 ∈ Field ∧ 𝑀 ∈ (Base‘(𝐼 Mat 𝑅))) ∧ uncurry 𝑓:(𝐼 × 𝐼)⟶(Base‘𝑅)) → uncurry 𝑓 ∈ (Base‘(𝐼 Mat 𝑅)))
226225adantr 480 . . . . . . . . 9 ((((𝑅 ∈ Field ∧ 𝑀 ∈ (Base‘(𝐼 Mat 𝑅))) ∧ uncurry 𝑓:(𝐼 × 𝐼)⟶(Base‘𝑅)) ∧ ∀𝑖𝐼 ((𝑅 unitVec 𝐼)‘𝑖) = (𝑗𝐼 ↦ (𝑅 Σg (𝑘𝐼 ↦ ((𝑖uncurry 𝑓𝑘)(.r𝑅)(𝑘𝑀𝑗)))))) → uncurry 𝑓 ∈ (Base‘(𝐼 Mat 𝑅)))
227 nfv 1830 . . . . . . . . . . . . . 14 𝑗(((𝑅 ∈ Field ∧ 𝑀 ∈ (Base‘(𝐼 Mat 𝑅))) ∧ uncurry 𝑓:(𝐼 × 𝐼)⟶(Base‘𝑅)) ∧ 𝑖𝐼)
228 nfmpt1 4675 . . . . . . . . . . . . . . 15 𝑗(𝑗𝐼 ↦ (𝑅 Σg (𝑘𝐼 ↦ ((𝑖uncurry 𝑓𝑘)(.r𝑅)(𝑘𝑀𝑗)))))
229228nfeq2 2766 . . . . . . . . . . . . . 14 𝑗((𝑅 unitVec 𝐼)‘𝑖) = (𝑗𝐼 ↦ (𝑅 Σg (𝑘𝐼 ↦ ((𝑖uncurry 𝑓𝑘)(.r𝑅)(𝑘𝑀𝑗)))))
230 fveq1 6102 . . . . . . . . . . . . . . . . 17 (((𝑅 unitVec 𝐼)‘𝑖) = (𝑗𝐼 ↦ (𝑅 Σg (𝑘𝐼 ↦ ((𝑖uncurry 𝑓𝑘)(.r𝑅)(𝑘𝑀𝑗))))) → (((𝑅 unitVec 𝐼)‘𝑖)‘𝑗) = ((𝑗𝐼 ↦ (𝑅 Σg (𝑘𝐼 ↦ ((𝑖uncurry 𝑓𝑘)(.r𝑅)(𝑘𝑀𝑗)))))‘𝑗))
2317, 43syl 17 . . . . . . . . . . . . . . . . . . . . 21 (𝑅 ∈ Field → 𝑅 ∈ Ring)
232231, 4anim12i 588 . . . . . . . . . . . . . . . . . . . 20 ((𝑅 ∈ Field ∧ 𝑀 ∈ (Base‘(𝐼 Mat 𝑅))) → (𝑅 ∈ Ring ∧ 𝐼 ∈ Fin))
233232adantr 480 . . . . . . . . . . . . . . . . . . 19 (((𝑅 ∈ Field ∧ 𝑀 ∈ (Base‘(𝐼 Mat 𝑅))) ∧ uncurry 𝑓:(𝐼 × 𝐼)⟶(Base‘𝑅)) → (𝑅 ∈ Ring ∧ 𝐼 ∈ Fin))
234 equcom 1932 . . . . . . . . . . . . . . . . . . . . 21 (𝑖 = 𝑗𝑗 = 𝑖)
235 ifbi 4057 . . . . . . . . . . . . . . . . . . . . 21 ((𝑖 = 𝑗𝑗 = 𝑖) → if(𝑖 = 𝑗, (1r𝑅), (0g𝑅)) = if(𝑗 = 𝑖, (1r𝑅), (0g𝑅)))
236234, 235ax-mp 5 . . . . . . . . . . . . . . . . . . . 20 if(𝑖 = 𝑗, (1r𝑅), (0g𝑅)) = if(𝑗 = 𝑖, (1r𝑅), (0g𝑅))
237 eqid 2610 . . . . . . . . . . . . . . . . . . . . 21 (1r𝑅) = (1r𝑅)
238 eqid 2610 . . . . . . . . . . . . . . . . . . . . 21 (0g𝑅) = (0g𝑅)
239 simpllr 795 . . . . . . . . . . . . . . . . . . . . 21 ((((𝑅 ∈ Ring ∧ 𝐼 ∈ Fin) ∧ 𝑖𝐼) ∧ 𝑗𝐼) → 𝐼 ∈ Fin)
240 simplll 794 . . . . . . . . . . . . . . . . . . . . 21 ((((𝑅 ∈ Ring ∧ 𝐼 ∈ Fin) ∧ 𝑖𝐼) ∧ 𝑗𝐼) → 𝑅 ∈ Ring)
241 simplr 788 . . . . . . . . . . . . . . . . . . . . 21 ((((𝑅 ∈ Ring ∧ 𝐼 ∈ Fin) ∧ 𝑖𝐼) ∧ 𝑗𝐼) → 𝑖𝐼)
242 simpr 476 . . . . . . . . . . . . . . . . . . . . 21 ((((𝑅 ∈ Ring ∧ 𝐼 ∈ Fin) ∧ 𝑖𝐼) ∧ 𝑗𝐼) → 𝑗𝐼)
243 eqid 2610 . . . . . . . . . . . . . . . . . . . . 21 (1r‘(𝐼 Mat 𝑅)) = (1r‘(𝐼 Mat 𝑅))
2441, 237, 238, 239, 240, 241, 242, 243mat1ov 20073 . . . . . . . . . . . . . . . . . . . 20 ((((𝑅 ∈ Ring ∧ 𝐼 ∈ Fin) ∧ 𝑖𝐼) ∧ 𝑗𝐼) → (𝑖(1r‘(𝐼 Mat 𝑅))𝑗) = if(𝑖 = 𝑗, (1r𝑅), (0g𝑅)))
245 df-3an 1033 . . . . . . . . . . . . . . . . . . . . 21 ((𝑅 ∈ Ring ∧ 𝐼 ∈ Fin ∧ 𝑖𝐼) ↔ ((𝑅 ∈ Ring ∧ 𝐼 ∈ Fin) ∧ 𝑖𝐼))
24644, 237, 238uvcvval 19944 . . . . . . . . . . . . . . . . . . . . 21 (((𝑅 ∈ Ring ∧ 𝐼 ∈ Fin ∧ 𝑖𝐼) ∧ 𝑗𝐼) → (((𝑅 unitVec 𝐼)‘𝑖)‘𝑗) = if(𝑗 = 𝑖, (1r𝑅), (0g𝑅)))
247245, 246sylanbr 489 . . . . . . . . . . . . . . . . . . . 20 ((((𝑅 ∈ Ring ∧ 𝐼 ∈ Fin) ∧ 𝑖𝐼) ∧ 𝑗𝐼) → (((𝑅 unitVec 𝐼)‘𝑖)‘𝑗) = if(𝑗 = 𝑖, (1r𝑅), (0g𝑅)))
248236, 244, 2473eqtr4a 2670 . . . . . . . . . . . . . . . . . . 19 ((((𝑅 ∈ Ring ∧ 𝐼 ∈ Fin) ∧ 𝑖𝐼) ∧ 𝑗𝐼) → (𝑖(1r‘(𝐼 Mat 𝑅))𝑗) = (((𝑅 unitVec 𝐼)‘𝑖)‘𝑗))
249233, 248sylanl1 680 . . . . . . . . . . . . . . . . . 18 (((((𝑅 ∈ Field ∧ 𝑀 ∈ (Base‘(𝐼 Mat 𝑅))) ∧ uncurry 𝑓:(𝐼 × 𝐼)⟶(Base‘𝑅)) ∧ 𝑖𝐼) ∧ 𝑗𝐼) → (𝑖(1r‘(𝐼 Mat 𝑅))𝑗) = (((𝑅 unitVec 𝐼)‘𝑖)‘𝑗))
250 ovex 6577 . . . . . . . . . . . . . . . . . . . . 21 (𝑅 Σg (𝑘𝐼 ↦ ((𝑖uncurry 𝑓𝑘)(.r𝑅)(𝑘𝑀𝑗)))) ∈ V
251 eqid 2610 . . . . . . . . . . . . . . . . . . . . . 22 (𝑗𝐼 ↦ (𝑅 Σg (𝑘𝐼 ↦ ((𝑖uncurry 𝑓𝑘)(.r𝑅)(𝑘𝑀𝑗))))) = (𝑗𝐼 ↦ (𝑅 Σg (𝑘𝐼 ↦ ((𝑖uncurry 𝑓𝑘)(.r𝑅)(𝑘𝑀𝑗)))))
252251fvmpt2 6200 . . . . . . . . . . . . . . . . . . . . 21 ((𝑗𝐼 ∧ (𝑅 Σg (𝑘𝐼 ↦ ((𝑖uncurry 𝑓𝑘)(.r𝑅)(𝑘𝑀𝑗)))) ∈ V) → ((𝑗𝐼 ↦ (𝑅 Σg (𝑘𝐼 ↦ ((𝑖uncurry 𝑓𝑘)(.r𝑅)(𝑘𝑀𝑗)))))‘𝑗) = (𝑅 Σg (𝑘𝐼 ↦ ((𝑖uncurry 𝑓𝑘)(.r𝑅)(𝑘𝑀𝑗)))))
253250, 252mpan2 703 . . . . . . . . . . . . . . . . . . . 20 (𝑗𝐼 → ((𝑗𝐼 ↦ (𝑅 Σg (𝑘𝐼 ↦ ((𝑖uncurry 𝑓𝑘)(.r𝑅)(𝑘𝑀𝑗)))))‘𝑗) = (𝑅 Σg (𝑘𝐼 ↦ ((𝑖uncurry 𝑓𝑘)(.r𝑅)(𝑘𝑀𝑗)))))
254253adantl 481 . . . . . . . . . . . . . . . . . . 19 (((((𝑅 ∈ Field ∧ 𝑀 ∈ (Base‘(𝐼 Mat 𝑅))) ∧ uncurry 𝑓:(𝐼 × 𝐼)⟶(Base‘𝑅)) ∧ 𝑖𝐼) ∧ 𝑗𝐼) → ((𝑗𝐼 ↦ (𝑅 Σg (𝑘𝐼 ↦ ((𝑖uncurry 𝑓𝑘)(.r𝑅)(𝑘𝑀𝑗)))))‘𝑗) = (𝑅 Σg (𝑘𝐼 ↦ ((𝑖uncurry 𝑓𝑘)(.r𝑅)(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)) ∧ 𝑖𝐼) ∧ 𝑗𝐼) → 𝑅 ∈ Field)
2574ad4antlr 765 . . . . . . . . . . . . . . . . . . . 20 (((((𝑅 ∈ Field ∧ 𝑀 ∈ (Base‘(𝐼 Mat 𝑅))) ∧ uncurry 𝑓:(𝐼 × 𝐼)⟶(Base‘𝑅)) ∧ 𝑖𝐼) ∧ 𝑗𝐼) → 𝐼 ∈ Fin)
258222biimpar 501 . . . . . . . . . . . . . . . . . . . . 21 ((𝑀 ∈ (Base‘(𝐼 Mat 𝑅)) ∧ uncurry 𝑓:(𝐼 × 𝐼)⟶(Base‘𝑅)) → uncurry 𝑓 ∈ ((Base‘𝑅) ↑𝑚 (𝐼 × 𝐼)))
259258ad5ant23 1296 . . . . . . . . . . . . . . . . . . . 20 (((((𝑅 ∈ Field ∧ 𝑀 ∈ (Base‘(𝐼 Mat 𝑅))) ∧ uncurry 𝑓:(𝐼 × 𝐼)⟶(Base‘𝑅)) ∧ 𝑖𝐼) ∧ 𝑗𝐼) → uncurry 𝑓 ∈ ((Base‘𝑅) ↑𝑚 (𝐼 × 𝐼)))
260 simpr 476 . . . . . . . . . . . . . . . . . . . . . 22 ((𝑅 ∈ Field ∧ 𝑀 ∈ (Base‘(𝐼 Mat 𝑅))) → 𝑀 ∈ (Base‘(𝐼 Mat 𝑅)))
261260, 219eleqtrrd 2691 . . . . . . . . . . . . . . . . . . . . 21 ((𝑅 ∈ Field ∧ 𝑀 ∈ (Base‘(𝐼 Mat 𝑅))) → 𝑀 ∈ ((Base‘𝑅) ↑𝑚 (𝐼 × 𝐼)))
262261ad3antrrr 762 . . . . . . . . . . . . . . . . . . . 20 (((((𝑅 ∈ Field ∧ 𝑀 ∈ (Base‘(𝐼 Mat 𝑅))) ∧ uncurry 𝑓:(𝐼 × 𝐼)⟶(Base‘𝑅)) ∧ 𝑖𝐼) ∧ 𝑗𝐼) → 𝑀 ∈ ((Base‘𝑅) ↑𝑚 (𝐼 × 𝐼)))
263 simplr 788 . . . . . . . . . . . . . . . . . . . 20 (((((𝑅 ∈ Field ∧ 𝑀 ∈ (Base‘(𝐼 Mat 𝑅))) ∧ uncurry 𝑓:(𝐼 × 𝐼)⟶(Base‘𝑅)) ∧ 𝑖𝐼) ∧ 𝑗𝐼) → 𝑖𝐼)
264 simpr 476 . . . . . . . . . . . . . . . . . . . 20 (((((𝑅 ∈ Field ∧ 𝑀 ∈ (Base‘(𝐼 Mat 𝑅))) ∧ uncurry 𝑓:(𝐼 × 𝐼)⟶(Base‘𝑅)) ∧ 𝑖𝐼) ∧ 𝑗𝐼) → 𝑗𝐼)
265255, 14, 121, 256, 257, 257, 257, 259, 262, 263, 264mamufv 20012 . . . . . . . . . . . . . . . . . . 19 (((((𝑅 ∈ Field ∧ 𝑀 ∈ (Base‘(𝐼 Mat 𝑅))) ∧ uncurry 𝑓:(𝐼 × 𝐼)⟶(Base‘𝑅)) ∧ 𝑖𝐼) ∧ 𝑗𝐼) → (𝑖(uncurry 𝑓(𝑅 maMul ⟨𝐼, 𝐼, 𝐼⟩)𝑀)𝑗) = (𝑅 Σg (𝑘𝐼 ↦ ((𝑖uncurry 𝑓𝑘)(.r𝑅)(𝑘𝑀𝑗)))))
2661, 255matmulr 20063 . . . . . . . . . . . . . . . . . . . . . . . 24 ((𝐼 ∈ Fin ∧ 𝑅 ∈ Field) → (𝑅 maMul ⟨𝐼, 𝐼, 𝐼⟩) = (.r‘(𝐼 Mat 𝑅)))
267266ancoms 468 . . . . . . . . . . . . . . . . . . . . . . 23 ((𝑅 ∈ Field ∧ 𝐼 ∈ Fin) → (𝑅 maMul ⟨𝐼, 𝐼, 𝐼⟩) = (.r‘(𝐼 Mat 𝑅)))
268267oveqd 6566 . . . . . . . . . . . . . . . . . . . . . 22 ((𝑅 ∈ Field ∧ 𝐼 ∈ Fin) → (uncurry 𝑓(𝑅 maMul ⟨𝐼, 𝐼, 𝐼⟩)𝑀) = (uncurry 𝑓(.r‘(𝐼 Mat 𝑅))𝑀))
269268oveqd 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))
2704, 269sylan2 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))
271270ad3antrrr 762 . . . . . . . . . . . . . . . . . . 19 (((((𝑅 ∈ Field ∧ 𝑀 ∈ (Base‘(𝐼 Mat 𝑅))) ∧ uncurry 𝑓:(𝐼 × 𝐼)⟶(Base‘𝑅)) ∧ 𝑖𝐼) ∧ 𝑗𝐼) → (𝑖(uncurry 𝑓(𝑅 maMul ⟨𝐼, 𝐼, 𝐼⟩)𝑀)𝑗) = (𝑖(uncurry 𝑓(.r‘(𝐼 Mat 𝑅))𝑀)𝑗))
272254, 265, 2713eqtr2rd 2651 . . . . . . . . . . . . . . . . . 18 (((((𝑅 ∈ Field ∧ 𝑀 ∈ (Base‘(𝐼 Mat 𝑅))) ∧ uncurry 𝑓:(𝐼 × 𝐼)⟶(Base‘𝑅)) ∧ 𝑖𝐼) ∧ 𝑗𝐼) → (𝑖(uncurry 𝑓(.r‘(𝐼 Mat 𝑅))𝑀)𝑗) = ((𝑗𝐼 ↦ (𝑅 Σg (𝑘𝐼 ↦ ((𝑖uncurry 𝑓𝑘)(.r𝑅)(𝑘𝑀𝑗)))))‘𝑗))
273249, 272eqeq12d 2625 . . . . . . . . . . . . . . . . 17 (((((𝑅 ∈ Field ∧ 𝑀 ∈ (Base‘(𝐼 Mat 𝑅))) ∧ uncurry 𝑓:(𝐼 × 𝐼)⟶(Base‘𝑅)) ∧ 𝑖𝐼) ∧ 𝑗𝐼) → ((𝑖(1r‘(𝐼 Mat 𝑅))𝑗) = (𝑖(uncurry 𝑓(.r‘(𝐼 Mat 𝑅))𝑀)𝑗) ↔ (((𝑅 unitVec 𝐼)‘𝑖)‘𝑗) = ((𝑗𝐼 ↦ (𝑅 Σg (𝑘𝐼 ↦ ((𝑖uncurry 𝑓𝑘)(.r𝑅)(𝑘𝑀𝑗)))))‘𝑗)))
274230, 273syl5ibr 235 . . . . . . . . . . . . . . . 16 (((((𝑅 ∈ Field ∧ 𝑀 ∈ (Base‘(𝐼 Mat 𝑅))) ∧ uncurry 𝑓:(𝐼 × 𝐼)⟶(Base‘𝑅)) ∧ 𝑖𝐼) ∧ 𝑗𝐼) → (((𝑅 unitVec 𝐼)‘𝑖) = (𝑗𝐼 ↦ (𝑅 Σg (𝑘𝐼 ↦ ((𝑖uncurry 𝑓𝑘)(.r𝑅)(𝑘𝑀𝑗))))) → (𝑖(1r‘(𝐼 Mat 𝑅))𝑗) = (𝑖(uncurry 𝑓(.r‘(𝐼 Mat 𝑅))𝑀)𝑗)))
275274ex 449 . . . . . . . . . . . . . . 15 ((((𝑅 ∈ Field ∧ 𝑀 ∈ (Base‘(𝐼 Mat 𝑅))) ∧ uncurry 𝑓:(𝐼 × 𝐼)⟶(Base‘𝑅)) ∧ 𝑖𝐼) → (𝑗𝐼 → (((𝑅 unitVec 𝐼)‘𝑖) = (𝑗𝐼 ↦ (𝑅 Σg (𝑘𝐼 ↦ ((𝑖uncurry 𝑓𝑘)(.r𝑅)(𝑘𝑀𝑗))))) → (𝑖(1r‘(𝐼 Mat 𝑅))𝑗) = (𝑖(uncurry 𝑓(.r‘(𝐼 Mat 𝑅))𝑀)𝑗))))
276275com23 84 . . . . . . . . . . . . . 14 ((((𝑅 ∈ Field ∧ 𝑀 ∈ (Base‘(𝐼 Mat 𝑅))) ∧ uncurry 𝑓:(𝐼 × 𝐼)⟶(Base‘𝑅)) ∧ 𝑖𝐼) → (((𝑅 unitVec 𝐼)‘𝑖) = (𝑗𝐼 ↦ (𝑅 Σg (𝑘𝐼 ↦ ((𝑖uncurry 𝑓𝑘)(.r𝑅)(𝑘𝑀𝑗))))) → (𝑗𝐼 → (𝑖(1r‘(𝐼 Mat 𝑅))𝑗) = (𝑖(uncurry 𝑓(.r‘(𝐼 Mat 𝑅))𝑀)𝑗))))
277227, 229, 276ralrimd 2942 . . . . . . . . . . . . 13 ((((𝑅 ∈ Field ∧ 𝑀 ∈ (Base‘(𝐼 Mat 𝑅))) ∧ uncurry 𝑓:(𝐼 × 𝐼)⟶(Base‘𝑅)) ∧ 𝑖𝐼) → (((𝑅 unitVec 𝐼)‘𝑖) = (𝑗𝐼 ↦ (𝑅 Σg (𝑘𝐼 ↦ ((𝑖uncurry 𝑓𝑘)(.r𝑅)(𝑘𝑀𝑗))))) → ∀𝑗𝐼 (𝑖(1r‘(𝐼 Mat 𝑅))𝑗) = (𝑖(uncurry 𝑓(.r‘(𝐼 Mat 𝑅))𝑀)𝑗)))
278277ralimdva 2945 . . . . . . . . . . . 12 (((𝑅 ∈ Field ∧ 𝑀 ∈ (Base‘(𝐼 Mat 𝑅))) ∧ uncurry 𝑓:(𝐼 × 𝐼)⟶(Base‘𝑅)) → (∀𝑖𝐼 ((𝑅 unitVec 𝐼)‘𝑖) = (𝑗𝐼 ↦ (𝑅 Σg (𝑘𝐼 ↦ ((𝑖uncurry 𝑓𝑘)(.r𝑅)(𝑘𝑀𝑗))))) → ∀𝑖𝐼𝑗𝐼 (𝑖(1r‘(𝐼 Mat 𝑅))𝑗) = (𝑖(uncurry 𝑓(.r‘(𝐼 Mat 𝑅))𝑀)𝑗)))
2791, 2, 243mat1bas 20074 . . . . . . . . . . . . . . . . 17 ((𝑅 ∈ Ring ∧ 𝐼 ∈ Fin) → (1r‘(𝐼 Mat 𝑅)) ∈ (Base‘(𝐼 Mat 𝑅)))
28013, 14frlmfibas 19924 . . . . . . . . . . . . . . . . . . 19 ((𝑅 ∈ Ring ∧ (𝐼 × 𝐼) ∈ Fin) → ((Base‘𝑅) ↑𝑚 (𝐼 × 𝐼)) = (Base‘(𝑅 freeLMod (𝐼 × 𝐼))))
28112, 280sylan2 490 . . . . . . . . . . . . . . . . . 18 ((𝑅 ∈ Ring ∧ 𝐼 ∈ Fin) → ((Base‘𝑅) ↑𝑚 (𝐼 × 𝐼)) = (Base‘(𝑅 freeLMod (𝐼 × 𝐼))))
2821, 13matbas 20038 . . . . . . . . . . . . . . . . . . 19 ((𝐼 ∈ Fin ∧ 𝑅 ∈ Ring) → (Base‘(𝑅 freeLMod (𝐼 × 𝐼))) = (Base‘(𝐼 Mat 𝑅)))
283282ancoms 468 . . . . . . . . . . . . . . . . . 18 ((𝑅 ∈ Ring ∧ 𝐼 ∈ Fin) → (Base‘(𝑅 freeLMod (𝐼 × 𝐼))) = (Base‘(𝐼 Mat 𝑅)))
284281, 283eqtrd 2644 . . . . . . . . . . . . . . . . 17 ((𝑅 ∈ Ring ∧ 𝐼 ∈ Fin) → ((Base‘𝑅) ↑𝑚 (𝐼 × 𝐼)) = (Base‘(𝐼 Mat 𝑅)))
285279, 284eleqtrrd 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))
287285, 286syl 17 . . . . . . . . . . . . . . 15 ((𝑅 ∈ Ring ∧ 𝐼 ∈ Fin) → (1r‘(𝐼 Mat 𝑅)) Fn (𝐼 × 𝐼))
288231, 4, 287syl2an 493 . . . . . . . . . . . . . 14 ((𝑅 ∈ Field ∧ 𝑀 ∈ (Base‘(𝐼 Mat 𝑅))) → (1r‘(𝐼 Mat 𝑅)) Fn (𝐼 × 𝐼))
289288adantr 480 . . . . . . . . . . . . 13 (((𝑅 ∈ Field ∧ 𝑀 ∈ (Base‘(𝐼 Mat 𝑅))) ∧ uncurry 𝑓:(𝐼 × 𝐼)⟶(Base‘𝑅)) → (1r‘(𝐼 Mat 𝑅)) Fn (𝐼 × 𝐼))
2901matring 20068 . . . . . . . . . . . . . . . . . 18 ((𝐼 ∈ Fin ∧ 𝑅 ∈ Ring) → (𝐼 Mat 𝑅) ∈ Ring)
2914, 231, 290syl2anr 494 . . . . . . . . . . . . . . . . 17 ((𝑅 ∈ Field ∧ 𝑀 ∈ (Base‘(𝐼 Mat 𝑅))) → (𝐼 Mat 𝑅) ∈ Ring)
292291adantr 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))
2952, 294ringcl 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)))
296292, 225, 293, 295syl3anc 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)))
297219adantr 480 . . . . . . . . . . . . . . 15 (((𝑅 ∈ Field ∧ 𝑀 ∈ (Base‘(𝐼 Mat 𝑅))) ∧ uncurry 𝑓:(𝐼 × 𝐼)⟶(Base‘𝑅)) → ((Base‘𝑅) ↑𝑚 (𝐼 × 𝐼)) = (Base‘(𝐼 Mat 𝑅)))
298296, 297eleqtrrd 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))
300298, 299syl 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) ↔ ∀𝑖𝐼𝑗𝐼 (𝑖(1r‘(𝐼 Mat 𝑅))𝑗) = (𝑖(uncurry 𝑓(.r‘(𝐼 Mat 𝑅))𝑀)𝑗)))
302289, 300, 301syl2anc 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) ↔ ∀𝑖𝐼𝑗𝐼 (𝑖(1r‘(𝐼 Mat 𝑅))𝑗) = (𝑖(uncurry 𝑓(.r‘(𝐼 Mat 𝑅))𝑀)𝑗)))
303278, 302sylibrd 248 . . . . . . . . . . 11 (((𝑅 ∈ Field ∧ 𝑀 ∈ (Base‘(𝐼 Mat 𝑅))) ∧ uncurry 𝑓:(𝐼 × 𝐼)⟶(Base‘𝑅)) → (∀𝑖𝐼 ((𝑅 unitVec 𝐼)‘𝑖) = (𝑗𝐼 ↦ (𝑅 Σg (𝑘𝐼 ↦ ((𝑖uncurry 𝑓𝑘)(.r𝑅)(𝑘𝑀𝑗))))) → (1r‘(𝐼 Mat 𝑅)) = (uncurry 𝑓(.r‘(𝐼 Mat 𝑅))𝑀)))
304303imp 444 . . . . . . . . . 10 ((((𝑅 ∈ Field ∧ 𝑀 ∈ (Base‘(𝐼 Mat 𝑅))) ∧ uncurry 𝑓:(𝐼 × 𝐼)⟶(Base‘𝑅)) ∧ ∀𝑖𝐼 ((𝑅 unitVec 𝐼)‘𝑖) = (𝑗𝐼 ↦ (𝑅 Σg (𝑘𝐼 ↦ ((𝑖uncurry 𝑓𝑘)(.r𝑅)(𝑘𝑀𝑗)))))) → (1r‘(𝐼 Mat 𝑅)) = (uncurry 𝑓(.r‘(𝐼 Mat 𝑅))𝑀))
305304eqcomd 2616 . . . . . . . . 9 ((((𝑅 ∈ Field ∧ 𝑀 ∈ (Base‘(𝐼 Mat 𝑅))) ∧ uncurry 𝑓:(𝐼 × 𝐼)⟶(Base‘𝑅)) ∧ ∀𝑖𝐼 ((𝑅 unitVec 𝐼)‘𝑖) = (𝑗𝐼 ↦ (𝑅 Σg (𝑘𝐼 ↦ ((𝑖uncurry 𝑓𝑘)(.r𝑅)(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))
307306eqeq1d 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))))
308307rspcev 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)))
309226, 305, 308syl2anc 691 . . . . . . . 8 ((((𝑅 ∈ Field ∧ 𝑀 ∈ (Base‘(𝐼 Mat 𝑅))) ∧ uncurry 𝑓:(𝐼 × 𝐼)⟶(Base‘𝑅)) ∧ ∀𝑖𝐼 ((𝑅 unitVec 𝐼)‘𝑖) = (𝑗𝐼 ↦ (𝑅 Σg (𝑘𝐼 ↦ ((𝑖uncurry 𝑓𝑘)(.r𝑅)(𝑘𝑀𝑗)))))) → ∃𝑛 ∈ (Base‘(𝐼 Mat 𝑅))(𝑛(.r‘(𝐼 Mat 𝑅))𝑀) = (1r‘(𝐼 Mat 𝑅)))
310309expl 646 . . . . . . 7 ((𝑅 ∈ Field ∧ 𝑀 ∈ (Base‘(𝐼 Mat 𝑅))) → ((uncurry 𝑓:(𝐼 × 𝐼)⟶(Base‘𝑅) ∧ ∀𝑖𝐼 ((𝑅 unitVec 𝐼)‘𝑖) = (𝑗𝐼 ↦ (𝑅 Σg (𝑘𝐼 ↦ ((𝑖uncurry 𝑓𝑘)(.r𝑅)(𝑘𝑀𝑗)))))) → ∃𝑛 ∈ (Base‘(𝐼 Mat 𝑅))(𝑛(.r‘(𝐼 Mat 𝑅))𝑀) = (1r‘(𝐼 Mat 𝑅))))
311213, 310sylani 684 . . . . . 6 ((𝑅 ∈ Field ∧ 𝑀 ∈ (Base‘(𝐼 Mat 𝑅))) → ((𝑓:𝐼⟶((Base‘𝑅) ↑𝑚 𝐼) ∧ ∀𝑖𝐼 ((𝑅 unitVec 𝐼)‘𝑖) = (𝑗𝐼 ↦ (𝑅 Σg (𝑘𝐼 ↦ ((𝑖uncurry 𝑓𝑘)(.r𝑅)(𝑘𝑀𝑗)))))) → ∃𝑛 ∈ (Base‘(𝐼 Mat 𝑅))(𝑛(.r‘(𝐼 Mat 𝑅))𝑀) = (1r‘(𝐼 Mat 𝑅))))
312311exlimdv 1848 . . . . 5 ((𝑅 ∈ Field ∧ 𝑀 ∈ (Base‘(𝐼 Mat 𝑅))) → (∃𝑓(𝑓:𝐼⟶((Base‘𝑅) ↑𝑚 𝐼) ∧ ∀𝑖𝐼 ((𝑅 unitVec 𝐼)‘𝑖) = (𝑗𝐼 ↦ (𝑅 Σg (𝑘𝐼 ↦ ((𝑖uncurry 𝑓𝑘)(.r𝑅)(𝑘𝑀𝑗)))))) → ∃𝑛 ∈ (Base‘(𝐼 Mat 𝑅))(𝑛(.r‘(𝐼 Mat 𝑅))𝑀) = (1r‘(𝐼 Mat 𝑅))))
313312imp 444 . . . 4 (((𝑅 ∈ Field ∧ 𝑀 ∈ (Base‘(𝐼 Mat 𝑅))) ∧ ∃𝑓(𝑓:𝐼⟶((Base‘𝑅) ↑𝑚 𝐼) ∧ ∀𝑖𝐼 ((𝑅 unitVec 𝐼)‘𝑖) = (𝑗𝐼 ↦ (𝑅 Σg (𝑘𝐼 ↦ ((𝑖uncurry 𝑓𝑘)(.r𝑅)(𝑘𝑀𝑗))))))) → ∃𝑛 ∈ (Base‘(𝐼 Mat 𝑅))(𝑛(.r‘(𝐼 Mat 𝑅))𝑀) = (1r‘(𝐼 Mat 𝑅)))
314313adantlr 747 . . 3 ((((𝑅 ∈ Field ∧ 𝑀 ∈ (Base‘(𝐼 Mat 𝑅))) ∧ 𝐼 ≠ ∅) ∧ ∃𝑓(𝑓:𝐼⟶((Base‘𝑅) ↑𝑚 𝐼) ∧ ∀𝑖𝐼 ((𝑅 unitVec 𝐼)‘𝑖) = (𝑗𝐼 ↦ (𝑅 Σg (𝑘𝐼 ↦ ((𝑖uncurry 𝑓𝑘)(.r𝑅)(𝑘𝑀𝑗))))))) → ∃𝑛 ∈ (Base‘(𝐼 Mat 𝑅))(𝑛(.r‘(𝐼 Mat 𝑅))𝑀) = (1r‘(𝐼 Mat 𝑅)))
315212, 314syldan 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)))
3166simprbi 479 . . . 4 (𝑅 ∈ Field → 𝑅 ∈ CRing)
317 eqid 2610 . . . . . . . . . 10 (𝐼 maDet 𝑅) = (𝐼 maDet 𝑅)
318317, 1, 2, 14mdetcl 20221 . . . . . . . . 9 ((𝑅 ∈ CRing ∧ 𝑀 ∈ (Base‘(𝐼 Mat 𝑅))) → ((𝐼 maDet 𝑅)‘𝑀) ∈ (Base‘𝑅))
319317, 1, 2, 14mdetcl 20221 . . . . . . . . 9 ((𝑅 ∈ CRing ∧ 𝑛 ∈ (Base‘(𝐼 Mat 𝑅))) → ((𝐼 maDet 𝑅)‘𝑛) ∈ (Base‘𝑅))
320 eqid 2610 . . . . . . . . . 10 (∥r𝑅) = (∥r𝑅)
32114, 320, 121dvdsrmul 18471 . . . . . . . . 9 ((((𝐼 maDet 𝑅)‘𝑀) ∈ (Base‘𝑅) ∧ ((𝐼 maDet 𝑅)‘𝑛) ∈ (Base‘𝑅)) → ((𝐼 maDet 𝑅)‘𝑀)(∥r𝑅)(((𝐼 maDet 𝑅)‘𝑛)(.r𝑅)((𝐼 maDet 𝑅)‘𝑀)))
322318, 319, 321syl2an 493 . . . . . . . 8 (((𝑅 ∈ CRing ∧ 𝑀 ∈ (Base‘(𝐼 Mat 𝑅))) ∧ (𝑅 ∈ CRing ∧ 𝑛 ∈ (Base‘(𝐼 Mat 𝑅)))) → ((𝐼 maDet 𝑅)‘𝑀)(∥r𝑅)(((𝐼 maDet 𝑅)‘𝑛)(.r𝑅)((𝐼 maDet 𝑅)‘𝑀)))
323322anandis 869 . . . . . . 7 ((𝑅 ∈ CRing ∧ (𝑀 ∈ (Base‘(𝐼 Mat 𝑅)) ∧ 𝑛 ∈ (Base‘(𝐼 Mat 𝑅)))) → ((𝐼 maDet 𝑅)‘𝑀)(∥r𝑅)(((𝐼 maDet 𝑅)‘𝑛)(.r𝑅)((𝐼 maDet 𝑅)‘𝑀)))
324323anassrs 678 . . . . . 6 (((𝑅 ∈ CRing ∧ 𝑀 ∈ (Base‘(𝐼 Mat 𝑅))) ∧ 𝑛 ∈ (Base‘(𝐼 Mat 𝑅))) → ((𝐼 maDet 𝑅)‘𝑀)(∥r𝑅)(((𝐼 maDet 𝑅)‘𝑛)(.r𝑅)((𝐼 maDet 𝑅)‘𝑀)))
325324adantrr 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)(∥r𝑅)(((𝐼 maDet 𝑅)‘𝑛)(.r𝑅)((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))))
3271, 2, 317, 121, 294mdetmul 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)(.r𝑅)((𝐼 maDet 𝑅)‘𝑀)))
3283273expa 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)(.r𝑅)((𝐼 maDet 𝑅)‘𝑀)))
329328an32s 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)(.r𝑅)((𝐼 maDet 𝑅)‘𝑀)))
330317, 1, 243, 237mdet1 20226 . . . . . . . . . . . 12 ((𝑅 ∈ CRing ∧ 𝐼 ∈ Fin) → ((𝐼 maDet 𝑅)‘(1r‘(𝐼 Mat 𝑅))) = (1r𝑅))
3314, 330sylan2 490 . . . . . . . . . . 11 ((𝑅 ∈ CRing ∧ 𝑀 ∈ (Base‘(𝐼 Mat 𝑅))) → ((𝐼 maDet 𝑅)‘(1r‘(𝐼 Mat 𝑅))) = (1r𝑅))
332331adantr 480 . . . . . . . . . 10 (((𝑅 ∈ CRing ∧ 𝑀 ∈ (Base‘(𝐼 Mat 𝑅))) ∧ 𝑛 ∈ (Base‘(𝐼 Mat 𝑅))) → ((𝐼 maDet 𝑅)‘(1r‘(𝐼 Mat 𝑅))) = (1r𝑅))
333329, 332eqeq12d 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)(.r𝑅)((𝐼 maDet 𝑅)‘𝑀)) = (1r𝑅)))
334326, 333syl5ib 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)(.r𝑅)((𝐼 maDet 𝑅)‘𝑀)) = (1r𝑅)))
335334impr 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)(.r𝑅)((𝐼 maDet 𝑅)‘𝑀)) = (1r𝑅))
336335breq2d 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)(∥r𝑅)(((𝐼 maDet 𝑅)‘𝑛)(.r𝑅)((𝐼 maDet 𝑅)‘𝑀)) ↔ ((𝐼 maDet 𝑅)‘𝑀)(∥r𝑅)(1r𝑅)))
337 eqid 2610 . . . . . . . 8 (Unit‘𝑅) = (Unit‘𝑅)
338337, 237, 320crngunit 18485 . . . . . . 7 (𝑅 ∈ CRing → (((𝐼 maDet 𝑅)‘𝑀) ∈ (Unit‘𝑅) ↔ ((𝐼 maDet 𝑅)‘𝑀)(∥r𝑅)(1r𝑅)))
339338ad2antrr 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)(∥r𝑅)(1r𝑅)))
340336, 339bitr4d 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)(∥r𝑅)(((𝐼 maDet 𝑅)‘𝑛)(.r𝑅)((𝐼 maDet 𝑅)‘𝑀)) ↔ ((𝐼 maDet 𝑅)‘𝑀) ∈ (Unit‘𝑅)))
341325, 340mpbid 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))
342316, 341sylanl1 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))
343342ad4ant14 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))
344315, 343rexlimddv 3017 1 ((((𝑅 ∈ Field ∧ 𝑀 ∈ (Base‘(𝐼 Mat 𝑅))) ∧ 𝐼 ≠ ∅) ∧ curry 𝑀 LIndF (𝑅 freeLMod 𝐼)) → ((𝐼 maDet 𝑅)‘𝑀) ∈ (Unit‘𝑅))
 Colors of variables: wff setvar class 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  0gc0g 15923   Σg cgsu 15924  1rcur 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
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