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Theorem matunitlindflem2 34771
Description: One direction of matunitlindf 34772. (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 2821 . . . . . . 7 (𝐼 Mat 𝑅) = (𝐼 Mat 𝑅)
2 eqid 2821 . . . . . . 7 (Base‘(𝐼 Mat 𝑅)) = (Base‘(𝐼 Mat 𝑅))
31, 2matrcl 20951 . . . . . 6 (𝑀 ∈ (Base‘(𝐼 Mat 𝑅)) → (𝐼 ∈ Fin ∧ 𝑅 ∈ V))
43simpld 495 . . . . 5 (𝑀 ∈ (Base‘(𝐼 Mat 𝑅)) → 𝐼 ∈ Fin)
54ad3antlr 727 . . . 4 ((((𝑅 ∈ Field ∧ 𝑀 ∈ (Base‘(𝐼 Mat 𝑅))) ∧ 𝐼 ≠ ∅) ∧ curry 𝑀 LIndF (𝑅 freeLMod 𝐼)) → 𝐼 ∈ Fin)
6 isfld 19442 . . . . . . 7 (𝑅 ∈ Field ↔ (𝑅 ∈ DivRing ∧ 𝑅 ∈ CRing))
76simplbi 498 . . . . . 6 (𝑅 ∈ Field → 𝑅 ∈ DivRing)
87anim1i 614 . . . . 5 ((𝑅 ∈ Field ∧ 𝑀 ∈ (Base‘(𝐼 Mat 𝑅))) → (𝑅 ∈ DivRing ∧ 𝑀 ∈ (Base‘(𝐼 Mat 𝑅))))
94ad2antrl 724 . . . . . . . . . . . 12 ((𝑅 ∈ DivRing ∧ (𝑀 ∈ (Base‘(𝐼 Mat 𝑅)) ∧ 𝐼 ≠ ∅)) → 𝐼 ∈ Fin)
10 simpr 485 . . . . . . . . . . . . . . 15 ((𝑅 ∈ DivRing ∧ 𝑀 ∈ (Base‘(𝐼 Mat 𝑅))) → 𝑀 ∈ (Base‘(𝐼 Mat 𝑅)))
11 xpfi 8778 . . . . . . . . . . . . . . . . . . . . 21 ((𝐼 ∈ Fin ∧ 𝐼 ∈ Fin) → (𝐼 × 𝐼) ∈ Fin)
1211anidms 567 . . . . . . . . . . . . . . . . . . . 20 (𝐼 ∈ Fin → (𝐼 × 𝐼) ∈ Fin)
13 eqid 2821 . . . . . . . . . . . . . . . . . . . . 21 (𝑅 freeLMod (𝐼 × 𝐼)) = (𝑅 freeLMod (𝐼 × 𝐼))
14 eqid 2821 . . . . . . . . . . . . . . . . . . . . 21 (Base‘𝑅) = (Base‘𝑅)
1513, 14frlmfibas 20836 . . . . . . . . . . . . . . . . . . . 20 ((𝑅 ∈ DivRing ∧ (𝐼 × 𝐼) ∈ Fin) → ((Base‘𝑅) ↑m (𝐼 × 𝐼)) = (Base‘(𝑅 freeLMod (𝐼 × 𝐼))))
1612, 15sylan2 592 . . . . . . . . . . . . . . . . . . 19 ((𝑅 ∈ DivRing ∧ 𝐼 ∈ Fin) → ((Base‘𝑅) ↑m (𝐼 × 𝐼)) = (Base‘(𝑅 freeLMod (𝐼 × 𝐼))))
171, 13matbas 20952 . . . . . . . . . . . . . . . . . . . 20 ((𝐼 ∈ Fin ∧ 𝑅 ∈ DivRing) → (Base‘(𝑅 freeLMod (𝐼 × 𝐼))) = (Base‘(𝐼 Mat 𝑅)))
1817ancoms 459 . . . . . . . . . . . . . . . . . . 19 ((𝑅 ∈ DivRing ∧ 𝐼 ∈ Fin) → (Base‘(𝑅 freeLMod (𝐼 × 𝐼))) = (Base‘(𝐼 Mat 𝑅)))
1916, 18eqtrd 2856 . . . . . . . . . . . . . . . . . 18 ((𝑅 ∈ DivRing ∧ 𝐼 ∈ Fin) → ((Base‘𝑅) ↑m (𝐼 × 𝐼)) = (Base‘(𝐼 Mat 𝑅)))
2019eleq2d 2898 . . . . . . . . . . . . . . . . 17 ((𝑅 ∈ DivRing ∧ 𝐼 ∈ Fin) → (𝑀 ∈ ((Base‘𝑅) ↑m (𝐼 × 𝐼)) ↔ 𝑀 ∈ (Base‘(𝐼 Mat 𝑅))))
214, 20sylan2 592 . . . . . . . . . . . . . . . 16 ((𝑅 ∈ DivRing ∧ 𝑀 ∈ (Base‘(𝐼 Mat 𝑅))) → (𝑀 ∈ ((Base‘𝑅) ↑m (𝐼 × 𝐼)) ↔ 𝑀 ∈ (Base‘(𝐼 Mat 𝑅))))
22 fvex 6677 . . . . . . . . . . . . . . . . . 18 (Base‘𝑅) ∈ V
234, 4, 11syl2anc 584 . . . . . . . . . . . . . . . . . 18 (𝑀 ∈ (Base‘(𝐼 Mat 𝑅)) → (𝐼 × 𝐼) ∈ Fin)
24 elmapg 8409 . . . . . . . . . . . . . . . . . 18 (((Base‘𝑅) ∈ V ∧ (𝐼 × 𝐼) ∈ Fin) → (𝑀 ∈ ((Base‘𝑅) ↑m (𝐼 × 𝐼)) ↔ 𝑀:(𝐼 × 𝐼)⟶(Base‘𝑅)))
2522, 23, 24sylancr 587 . . . . . . . . . . . . . . . . 17 (𝑀 ∈ (Base‘(𝐼 Mat 𝑅)) → (𝑀 ∈ ((Base‘𝑅) ↑m (𝐼 × 𝐼)) ↔ 𝑀:(𝐼 × 𝐼)⟶(Base‘𝑅)))
2625adantl 482 . . . . . . . . . . . . . . . 16 ((𝑅 ∈ DivRing ∧ 𝑀 ∈ (Base‘(𝐼 Mat 𝑅))) → (𝑀 ∈ ((Base‘𝑅) ↑m (𝐼 × 𝐼)) ↔ 𝑀:(𝐼 × 𝐼)⟶(Base‘𝑅)))
2721, 26bitr3d 282 . . . . . . . . . . . . . . 15 ((𝑅 ∈ DivRing ∧ 𝑀 ∈ (Base‘(𝐼 Mat 𝑅))) → (𝑀 ∈ (Base‘(𝐼 Mat 𝑅)) ↔ 𝑀:(𝐼 × 𝐼)⟶(Base‘𝑅)))
2810, 27mpbid 233 . . . . . . . . . . . . . 14 ((𝑅 ∈ DivRing ∧ 𝑀 ∈ (Base‘(𝐼 Mat 𝑅))) → 𝑀:(𝐼 × 𝐼)⟶(Base‘𝑅))
2928adantrr 713 . . . . . . . . . . . . 13 ((𝑅 ∈ DivRing ∧ (𝑀 ∈ (Base‘(𝐼 Mat 𝑅)) ∧ 𝐼 ≠ ∅)) → 𝑀:(𝐼 × 𝐼)⟶(Base‘𝑅))
30 eldifsn 4713 . . . . . . . . . . . . . . . 16 (𝐼 ∈ (Fin ∖ {∅}) ↔ (𝐼 ∈ Fin ∧ 𝐼 ≠ ∅))
3130biimpri 229 . . . . . . . . . . . . . . 15 ((𝐼 ∈ Fin ∧ 𝐼 ≠ ∅) → 𝐼 ∈ (Fin ∖ {∅}))
324, 31sylan 580 . . . . . . . . . . . . . 14 ((𝑀 ∈ (Base‘(𝐼 Mat 𝑅)) ∧ 𝐼 ≠ ∅) → 𝐼 ∈ (Fin ∖ {∅}))
3332adantl 482 . . . . . . . . . . . . 13 ((𝑅 ∈ DivRing ∧ (𝑀 ∈ (Base‘(𝐼 Mat 𝑅)) ∧ 𝐼 ≠ ∅)) → 𝐼 ∈ (Fin ∖ {∅}))
34 curf 34752 . . . . . . . . . . . . . 14 ((𝑀:(𝐼 × 𝐼)⟶(Base‘𝑅) ∧ 𝐼 ∈ (Fin ∖ {∅}) ∧ (Base‘𝑅) ∈ V) → curry 𝑀:𝐼⟶((Base‘𝑅) ↑m 𝐼))
3522, 34mp3an3 1441 . . . . . . . . . . . . 13 ((𝑀:(𝐼 × 𝐼)⟶(Base‘𝑅) ∧ 𝐼 ∈ (Fin ∖ {∅})) → curry 𝑀:𝐼⟶((Base‘𝑅) ↑m 𝐼))
3629, 33, 35syl2anc 584 . . . . . . . . . . . 12 ((𝑅 ∈ DivRing ∧ (𝑀 ∈ (Base‘(𝐼 Mat 𝑅)) ∧ 𝐼 ≠ ∅)) → curry 𝑀:𝐼⟶((Base‘𝑅) ↑m 𝐼))
379, 36jca 512 . . . . . . . . . . 11 ((𝑅 ∈ DivRing ∧ (𝑀 ∈ (Base‘(𝐼 Mat 𝑅)) ∧ 𝐼 ≠ ∅)) → (𝐼 ∈ Fin ∧ curry 𝑀:𝐼⟶((Base‘𝑅) ↑m 𝐼)))
3837ex 413 . . . . . . . . . 10 (𝑅 ∈ DivRing → ((𝑀 ∈ (Base‘(𝐼 Mat 𝑅)) ∧ 𝐼 ≠ ∅) → (𝐼 ∈ Fin ∧ curry 𝑀:𝐼⟶((Base‘𝑅) ↑m 𝐼))))
3938imdistani 569 . . . . . . . . 9 ((𝑅 ∈ DivRing ∧ (𝑀 ∈ (Base‘(𝐼 Mat 𝑅)) ∧ 𝐼 ≠ ∅)) → (𝑅 ∈ DivRing ∧ (𝐼 ∈ Fin ∧ curry 𝑀:𝐼⟶((Base‘𝑅) ↑m 𝐼))))
4039anassrs 468 . . . . . . . 8 (((𝑅 ∈ DivRing ∧ 𝑀 ∈ (Base‘(𝐼 Mat 𝑅))) ∧ 𝐼 ≠ ∅) → (𝑅 ∈ DivRing ∧ (𝐼 ∈ Fin ∧ curry 𝑀:𝐼⟶((Base‘𝑅) ↑m 𝐼))))
41 anass 469 . . . . . . . 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 235 . . . . . . 7 (((𝑅 ∈ DivRing ∧ 𝑀 ∈ (Base‘(𝐼 Mat 𝑅))) ∧ 𝐼 ≠ ∅) → ((𝑅 ∈ DivRing ∧ 𝐼 ∈ Fin) ∧ curry 𝑀:𝐼⟶((Base‘𝑅) ↑m 𝐼)))
43 drngring 19440 . . . . . . . . . . . . 13 (𝑅 ∈ DivRing → 𝑅 ∈ Ring)
44 eqid 2821 . . . . . . . . . . . . . 14 (𝑅 unitVec 𝐼) = (𝑅 unitVec 𝐼)
45 eqid 2821 . . . . . . . . . . . . . 14 (𝑅 freeLMod 𝐼) = (𝑅 freeLMod 𝐼)
46 eqid 2821 . . . . . . . . . . . . . 14 (Base‘(𝑅 freeLMod 𝐼)) = (Base‘(𝑅 freeLMod 𝐼))
4744, 45, 46uvcff 20865 . . . . . . . . . . . . 13 ((𝑅 ∈ Ring ∧ 𝐼 ∈ Fin) → (𝑅 unitVec 𝐼):𝐼⟶(Base‘(𝑅 freeLMod 𝐼)))
4843, 47sylan 580 . . . . . . . . . . . 12 ((𝑅 ∈ DivRing ∧ 𝐼 ∈ Fin) → (𝑅 unitVec 𝐼):𝐼⟶(Base‘(𝑅 freeLMod 𝐼)))
4948ffvelrnda 6844 . . . . . . . . . . 11 (((𝑅 ∈ DivRing ∧ 𝐼 ∈ Fin) ∧ 𝑖𝐼) → ((𝑅 unitVec 𝐼)‘𝑖) ∈ (Base‘(𝑅 freeLMod 𝐼)))
5049ad4ant14 748 . . . . . . . . . 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 6508 . . . . . . . . . . . . . . . 16 (curry 𝑀:𝐼⟶((Base‘𝑅) ↑m 𝐼) → curry 𝑀 Fn 𝐼)
52 fnima 6472 . . . . . . . . . . . . . . . 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 482 . . . . . . . . . . . . . 14 (((𝑅 ∈ DivRing ∧ 𝐼 ∈ Fin) ∧ curry 𝑀:𝐼⟶((Base‘𝑅) ↑m 𝐼)) → (curry 𝑀𝐼) = ran curry 𝑀)
5554fveq2d 6668 . . . . . . . . . . . . 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 481 . . . . . . . . . . . 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 771 . . . . . . . . . . . . . 14 ((((𝑅 ∈ DivRing ∧ 𝐼 ∈ Fin) ∧ curry 𝑀:𝐼⟶((Base‘𝑅) ↑m 𝐼)) ∧ curry 𝑀 LIndF (𝑅 freeLMod 𝐼)) → 𝑅 ∈ DivRing)
58 simpllr 772 . . . . . . . . . . . . . 14 ((((𝑅 ∈ DivRing ∧ 𝐼 ∈ Fin) ∧ curry 𝑀:𝐼⟶((Base‘𝑅) ↑m 𝐼)) ∧ curry 𝑀 LIndF (𝑅 freeLMod 𝐼)) → 𝐼 ∈ Fin)
5945frlmlmod 20823 . . . . . . . . . . . . . . . . 17 ((𝑅 ∈ Ring ∧ 𝐼 ∈ Fin) → (𝑅 freeLMod 𝐼) ∈ LMod)
6043, 59sylan 580 . . . . . . . . . . . . . . . 16 ((𝑅 ∈ DivRing ∧ 𝐼 ∈ Fin) → (𝑅 freeLMod 𝐼) ∈ LMod)
6160adantr 481 . . . . . . . . . . . . . . 15 (((𝑅 ∈ DivRing ∧ 𝐼 ∈ Fin) ∧ curry 𝑀:𝐼⟶((Base‘𝑅) ↑m 𝐼)) → (𝑅 freeLMod 𝐼) ∈ LMod)
62 lindfrn 20895 . . . . . . . . . . . . . . 15 (((𝑅 freeLMod 𝐼) ∈ LMod ∧ curry 𝑀 LIndF (𝑅 freeLMod 𝐼)) → ran curry 𝑀 ∈ (LIndS‘(𝑅 freeLMod 𝐼)))
6361, 62sylan 580 . . . . . . . . . . . . . 14 ((((𝑅 ∈ DivRing ∧ 𝐼 ∈ Fin) ∧ curry 𝑀:𝐼⟶((Base‘𝑅) ↑m 𝐼)) ∧ curry 𝑀 LIndF (𝑅 freeLMod 𝐼)) → ran curry 𝑀 ∈ (LIndS‘(𝑅 freeLMod 𝐼)))
6445frlmsca 20827 . . . . . . . . . . . . . . . . . . . . . 22 ((𝑅 ∈ DivRing ∧ 𝐼 ∈ Fin) → 𝑅 = (Scalar‘(𝑅 freeLMod 𝐼)))
65 drngnzr 19965 . . . . . . . . . . . . . . . . . . . . . . 23 (𝑅 ∈ DivRing → 𝑅 ∈ NzRing)
6665adantr 481 . . . . . . . . . . . . . . . . . . . . . 22 ((𝑅 ∈ DivRing ∧ 𝐼 ∈ Fin) → 𝑅 ∈ NzRing)
6764, 66eqeltrrd 2914 . . . . . . . . . . . . . . . . . . . . 21 ((𝑅 ∈ DivRing ∧ 𝐼 ∈ Fin) → (Scalar‘(𝑅 freeLMod 𝐼)) ∈ NzRing)
6860, 67jca 512 . . . . . . . . . . . . . . . . . . . 20 ((𝑅 ∈ DivRing ∧ 𝐼 ∈ Fin) → ((𝑅 freeLMod 𝐼) ∈ LMod ∧ (Scalar‘(𝑅 freeLMod 𝐼)) ∈ NzRing))
69 eqid 2821 . . . . . . . . . . . . . . . . . . . . . 22 (Scalar‘(𝑅 freeLMod 𝐼)) = (Scalar‘(𝑅 freeLMod 𝐼))
7046, 69lindff1 20894 . . . . . . . . . . . . . . . . . . . . 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 1110 . . . . . . . . . . . . . . . . . . . 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 580 . . . . . . . . . . . . . . . . . . 19 (((𝑅 ∈ DivRing ∧ 𝐼 ∈ Fin) ∧ curry 𝑀 LIndF (𝑅 freeLMod 𝐼)) → curry 𝑀:dom curry 𝑀1-1→(Base‘(𝑅 freeLMod 𝐼)))
73 fdm 6516 . . . . . . . . . . . . . . . . . . 19 (curry 𝑀:𝐼⟶((Base‘𝑅) ↑m 𝐼) → dom curry 𝑀 = 𝐼)
74 f1eq2 6565 . . . . . . . . . . . . . . . . . . . 20 (dom curry 𝑀 = 𝐼 → (curry 𝑀:dom curry 𝑀1-1→(Base‘(𝑅 freeLMod 𝐼)) ↔ curry 𝑀:𝐼1-1→(Base‘(𝑅 freeLMod 𝐼))))
7574biimpac 479 . . . . . . . . . . . . . . . . . . 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 595 . . . . . . . . . . . . . . . . . 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 648 . . . . . . . . . . . . . . . . 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 6620 . . . . . . . . . . . . . . . . 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 8517 . . . . . . . . . . . . . . . 16 ((𝐼 ∈ Fin ∧ curry 𝑀:𝐼1-1-onto→ran curry 𝑀) → 𝐼 ≈ ran curry 𝑀)
8158, 79, 80syl2anc 584 . . . . . . . . . . . . . . 15 ((((𝑅 ∈ DivRing ∧ 𝐼 ∈ Fin) ∧ curry 𝑀:𝐼⟶((Base‘𝑅) ↑m 𝐼)) ∧ curry 𝑀 LIndF (𝑅 freeLMod 𝐼)) → 𝐼 ≈ ran curry 𝑀)
8281ensymd 8549 . . . . . . . . . . . . . 14 ((((𝑅 ∈ DivRing ∧ 𝐼 ∈ Fin) ∧ curry 𝑀:𝐼⟶((Base‘𝑅) ↑m 𝐼)) ∧ curry 𝑀 LIndF (𝑅 freeLMod 𝐼)) → ran curry 𝑀𝐼)
83 lindsenlbs 34769 . . . . . . . . . . . . . 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 1365 . . . . . . . . . . . . 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 2821 . . . . . . . . . . . . . 14 (LBasis‘(𝑅 freeLMod 𝐼)) = (LBasis‘(𝑅 freeLMod 𝐼))
86 eqid 2821 . . . . . . . . . . . . . 14 (LSpan‘(𝑅 freeLMod 𝐼)) = (LSpan‘(𝑅 freeLMod 𝐼))
8746, 85, 86lbssp 19782 . . . . . . . . . . . . 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 2856 . . . . . . . . . . 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 481 . . . . . . . . . 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 2916 . . . . . . . . 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 2821 . . . . . . . . . . . . 13 (Base‘(Scalar‘(𝑅 freeLMod 𝐼))) = (Base‘(Scalar‘(𝑅 freeLMod 𝐼)))
93 eqid 2821 . . . . . . . . . . . . 13 (0g‘(Scalar‘(𝑅 freeLMod 𝐼))) = (0g‘(Scalar‘(𝑅 freeLMod 𝐼)))
94 eqid 2821 . . . . . . . . . . . . 13 ( ·𝑠 ‘(𝑅 freeLMod 𝐼)) = ( ·𝑠 ‘(𝑅 freeLMod 𝐼))
9545, 14frlmfibas 20836 . . . . . . . . . . . . . . 15 ((𝑅 ∈ Ring ∧ 𝐼 ∈ Fin) → ((Base‘𝑅) ↑m 𝐼) = (Base‘(𝑅 freeLMod 𝐼)))
9695feq3d 6495 . . . . . . . . . . . . . 14 ((𝑅 ∈ Ring ∧ 𝐼 ∈ Fin) → (curry 𝑀:𝐼⟶((Base‘𝑅) ↑m 𝐼) ↔ curry 𝑀:𝐼⟶(Base‘(𝑅 freeLMod 𝐼))))
9796biimpa 477 . . . . . . . . . . . . 13 (((𝑅 ∈ Ring ∧ 𝐼 ∈ Fin) ∧ curry 𝑀:𝐼⟶((Base‘𝑅) ↑m 𝐼)) → curry 𝑀:𝐼⟶(Base‘(𝑅 freeLMod 𝐼)))
9859adantr 481 . . . . . . . . . . . . 13 (((𝑅 ∈ Ring ∧ 𝐼 ∈ Fin) ∧ curry 𝑀:𝐼⟶((Base‘𝑅) ↑m 𝐼)) → (𝑅 freeLMod 𝐼) ∈ LMod)
99 simplr 765 . . . . . . . . . . . . 13 (((𝑅 ∈ Ring ∧ 𝐼 ∈ Fin) ∧ curry 𝑀:𝐼⟶((Base‘𝑅) ↑m 𝐼)) → 𝐼 ∈ Fin)
10086, 46, 92, 69, 93, 94, 97, 98, 99elfilspd 20877 . . . . . . . . . . . 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 20827 . . . . . . . . . . . . . . . 16 ((𝑅 ∈ Ring ∧ 𝐼 ∈ Fin) → 𝑅 = (Scalar‘(𝑅 freeLMod 𝐼)))
102101fveq2d 6668 . . . . . . . . . . . . . . 15 ((𝑅 ∈ Ring ∧ 𝐼 ∈ Fin) → (Base‘𝑅) = (Base‘(Scalar‘(𝑅 freeLMod 𝐼))))
103102oveq1d 7160 . . . . . . . . . . . . . 14 ((𝑅 ∈ Ring ∧ 𝐼 ∈ Fin) → ((Base‘𝑅) ↑m 𝐼) = ((Base‘(Scalar‘(𝑅 freeLMod 𝐼))) ↑m 𝐼))
104103adantr 481 . . . . . . . . . . . . 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 8418 . . . . . . . . . . . . . . 15 (𝑛 ∈ ((Base‘𝑅) ↑m 𝐼) → 𝑛:𝐼⟶(Base‘𝑅))
106 ffn 6508 . . . . . . . . . . . . . . . . . . . 20 (𝑛:𝐼⟶(Base‘𝑅) → 𝑛 Fn 𝐼)
107106adantl 482 . . . . . . . . . . . . . . . . . . 19 ((((𝑅 ∈ Ring ∧ 𝐼 ∈ Fin) ∧ curry 𝑀:𝐼⟶((Base‘𝑅) ↑m 𝐼)) ∧ 𝑛:𝐼⟶(Base‘𝑅)) → 𝑛 Fn 𝐼)
10851ad2antlr 723 . . . . . . . . . . . . . . . . . . 19 ((((𝑅 ∈ Ring ∧ 𝐼 ∈ Fin) ∧ curry 𝑀:𝐼⟶((Base‘𝑅) ↑m 𝐼)) ∧ 𝑛:𝐼⟶(Base‘𝑅)) → curry 𝑀 Fn 𝐼)
109 simpllr 772 . . . . . . . . . . . . . . . . . . 19 ((((𝑅 ∈ Ring ∧ 𝐼 ∈ Fin) ∧ curry 𝑀:𝐼⟶((Base‘𝑅) ↑m 𝐼)) ∧ 𝑛:𝐼⟶(Base‘𝑅)) → 𝐼 ∈ Fin)
110 inidm 4194 . . . . . . . . . . . . . . . . . . 19 (𝐼𝐼) = 𝐼
111 eqidd 2822 . . . . . . . . . . . . . . . . . . 19 (((((𝑅 ∈ Ring ∧ 𝐼 ∈ Fin) ∧ curry 𝑀:𝐼⟶((Base‘𝑅) ↑m 𝐼)) ∧ 𝑛:𝐼⟶(Base‘𝑅)) ∧ 𝑘𝐼) → (𝑛𝑘) = (𝑛𝑘))
112 eqidd 2822 . . . . . . . . . . . . . . . . . . 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 7405 . . . . . . . . . . . . . . . . . 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 780 . . . . . . . . . . . . . . . . . . . . 21 (((((𝑅 ∈ Ring ∧ 𝐼 ∈ Fin) ∧ curry 𝑀:𝐼⟶((Base‘𝑅) ↑m 𝐼)) ∧ 𝑛:𝐼⟶(Base‘𝑅)) ∧ 𝑘𝐼) → 𝐼 ∈ Fin)
115 ffvelrn 6842 . . . . . . . . . . . . . . . . . . . . . 22 ((𝑛:𝐼⟶(Base‘𝑅) ∧ 𝑘𝐼) → (𝑛𝑘) ∈ (Base‘𝑅))
116115adantll 710 . . . . . . . . . . . . . . . . . . . . 21 (((((𝑅 ∈ Ring ∧ 𝐼 ∈ Fin) ∧ curry 𝑀:𝐼⟶((Base‘𝑅) ↑m 𝐼)) ∧ 𝑛:𝐼⟶(Base‘𝑅)) ∧ 𝑘𝐼) → (𝑛𝑘) ∈ (Base‘𝑅))
117 ffvelrn 6842 . . . . . . . . . . . . . . . . . . . . . . 23 ((curry 𝑀:𝐼⟶((Base‘𝑅) ↑m 𝐼) ∧ 𝑘𝐼) → (curry 𝑀𝑘) ∈ ((Base‘𝑅) ↑m 𝐼))
118117ad4ant24 750 . . . . . . . . . . . . . . . . . . . . . 22 (((((𝑅 ∈ Ring ∧ 𝐼 ∈ Fin) ∧ curry 𝑀:𝐼⟶((Base‘𝑅) ↑m 𝐼)) ∧ 𝑛:𝐼⟶(Base‘𝑅)) ∧ 𝑘𝐼) → (curry 𝑀𝑘) ∈ ((Base‘𝑅) ↑m 𝐼))
11995ad3antrrr 726 . . . . . . . . . . . . . . . . . . . . . 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 2915 . . . . . . . . . . . . . . . . . . . . 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 2821 . . . . . . . . . . . . . . . . . . . . 21 (.r𝑅) = (.r𝑅)
12245, 46, 14, 114, 116, 120, 94, 121frlmvscafval 20840 . . . . . . . . . . . . . . . . . . . 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 6677 . . . . . . . . . . . . . . . . . . . . . 22 (𝑛𝑘) ∈ V
124 fnconstg 6561 . . . . . . . . . . . . . . . . . . . . . 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 8419 . . . . . . . . . . . . . . . . . . . . . . 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 750 . . . . . . . . . . . . . . . . . . . . 21 (((((𝑅 ∈ Ring ∧ 𝐼 ∈ Fin) ∧ curry 𝑀:𝐼⟶((Base‘𝑅) ↑m 𝐼)) ∧ 𝑛:𝐼⟶(Base‘𝑅)) ∧ 𝑘𝐼) → (curry 𝑀𝑘) Fn 𝐼)
129123fvconst2 6959 . . . . . . . . . . . . . . . . . . . . . 22 (𝑗𝐼 → ((𝐼 × {(𝑛𝑘)})‘𝑗) = (𝑛𝑘))
130129adantl 482 . . . . . . . . . . . . . . . . . . . . 21 ((((((𝑅 ∈ Ring ∧ 𝐼 ∈ Fin) ∧ curry 𝑀:𝐼⟶((Base‘𝑅) ↑m 𝐼)) ∧ 𝑛:𝐼⟶(Base‘𝑅)) ∧ 𝑘𝐼) ∧ 𝑗𝐼) → ((𝐼 × {(𝑛𝑘)})‘𝑗) = (𝑛𝑘))
131 eqidd 2822 . . . . . . . . . . . . . . . . . . . . 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 7405 . . . . . . . . . . . . . . . . . . . 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 2856 . . . . . . . . . . . . . . . . . . 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 5153 . . . . . . . . . . . . . . . . . 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 2856 . . . . . . . . . . . . . . . . 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 7161 . . . . . . . . . . . . . . . 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 2821 . . . . . . . . . . . . . . . . 17 (0g‘(𝑅 freeLMod 𝐼)) = (0g‘(𝑅 freeLMod 𝐼))
138 simplll 771 . . . . . . . . . . . . . . . . 17 ((((𝑅 ∈ Ring ∧ 𝐼 ∈ Fin) ∧ curry 𝑀:𝐼⟶((Base‘𝑅) ↑m 𝐼)) ∧ 𝑛:𝐼⟶(Base‘𝑅)) → 𝑅 ∈ Ring)
139 simp-5l 781 . . . . . . . . . . . . . . . . . . . 20 ((((((𝑅 ∈ Ring ∧ 𝐼 ∈ Fin) ∧ curry 𝑀:𝐼⟶((Base‘𝑅) ↑m 𝐼)) ∧ 𝑛:𝐼⟶(Base‘𝑅)) ∧ 𝑘𝐼) ∧ 𝑗𝐼) → 𝑅 ∈ Ring)
140115ad4ant23 749 . . . . . . . . . . . . . . . . . . . 20 ((((((𝑅 ∈ Ring ∧ 𝐼 ∈ Fin) ∧ curry 𝑀:𝐼⟶((Base‘𝑅) ↑m 𝐼)) ∧ 𝑛:𝐼⟶(Base‘𝑅)) ∧ 𝑘𝐼) ∧ 𝑗𝐼) → (𝑛𝑘) ∈ (Base‘𝑅))
141 simplr 765 . . . . . . . . . . . . . . . . . . . . 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 8418 . . . . . . . . . . . . . . . . . . . . . . 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 6844 . . . . . . . . . . . . . . . . . . . . 21 (((curry 𝑀:𝐼⟶((Base‘𝑅) ↑m 𝐼) ∧ 𝑘𝐼) ∧ 𝑗𝐼) → ((curry 𝑀𝑘)‘𝑗) ∈ (Base‘𝑅))
145141, 144sylanl1 676 . . . . . . . . . . . . . . . . . . . 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 19242 . . . . . . . . . . . . . . . . . . . 20 ((𝑅 ∈ Ring ∧ (𝑛𝑘) ∈ (Base‘𝑅) ∧ ((curry 𝑀𝑘)‘𝑗) ∈ (Base‘𝑅)) → ((𝑛𝑘)(.r𝑅)((curry 𝑀𝑘)‘𝑗)) ∈ (Base‘𝑅))
147139, 140, 145, 146syl3anc 1363 . . . . . . . . . . . . . . . . . . 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))
148147fmpttd 6872 . . . . . . . . . . . . . . . . . 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))
149 elmapg 8409 . . . . . . . . . . . . . . . . . . . . . 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)))
15022, 149mpan 686 . . . . . . . . . . . . . . . . . . . . 21 (𝐼 ∈ Fin → ((𝑗𝐼 ↦ ((𝑛𝑘)(.r𝑅)((curry 𝑀𝑘)‘𝑗))) ∈ ((Base‘𝑅) ↑m 𝐼) ↔ (𝑗𝐼 ↦ ((𝑛𝑘)(.r𝑅)((curry 𝑀𝑘)‘𝑗))):𝐼⟶(Base‘𝑅)))
151150adantl 482 . . . . . . . . . . . . . . . . . . . 20 ((𝑅 ∈ Ring ∧ 𝐼 ∈ Fin) → ((𝑗𝐼 ↦ ((𝑛𝑘)(.r𝑅)((curry 𝑀𝑘)‘𝑗))) ∈ ((Base‘𝑅) ↑m 𝐼) ↔ (𝑗𝐼 ↦ ((𝑛𝑘)(.r𝑅)((curry 𝑀𝑘)‘𝑗))):𝐼⟶(Base‘𝑅)))
15295eleq2d 2898 . . . . . . . . . . . . . . . . . . . 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))))
153151, 152bitr3d 282 . . . . . . . . . . . . . . . . . . 19 ((𝑅 ∈ Ring ∧ 𝐼 ∈ Fin) → ((𝑗𝐼 ↦ ((𝑛𝑘)(.r𝑅)((curry 𝑀𝑘)‘𝑗))):𝐼⟶(Base‘𝑅) ↔ (𝑗𝐼 ↦ ((𝑛𝑘)(.r𝑅)((curry 𝑀𝑘)‘𝑗))) ∈ (Base‘(𝑅 freeLMod 𝐼))))
154153ad3antrrr 726 . . . . . . . . . . . . . . . . . 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))))
155148, 154mpbid 233 . . . . . . . . . . . . . . . . 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)))
156 mptexg 6976 . . . . . . . . . . . . . . . . . . . . 21 (𝐼 ∈ Fin → (𝑗𝐼 ↦ ((𝑛𝑘)(.r𝑅)((curry 𝑀𝑘)‘𝑗))) ∈ V)
157156ralrimivw 3183 . . . . . . . . . . . . . . . . . . . 20 (𝐼 ∈ Fin → ∀𝑘𝐼 (𝑗𝐼 ↦ ((𝑛𝑘)(.r𝑅)((curry 𝑀𝑘)‘𝑗))) ∈ V)
158 eqid 2821 . . . . . . . . . . . . . . . . . . . . 21 (𝑘𝐼 ↦ (𝑗𝐼 ↦ ((𝑛𝑘)(.r𝑅)((curry 𝑀𝑘)‘𝑗)))) = (𝑘𝐼 ↦ (𝑗𝐼 ↦ ((𝑛𝑘)(.r𝑅)((curry 𝑀𝑘)‘𝑗))))
159158fnmpt 6482 . . . . . . . . . . . . . . . . . . . 20 (∀𝑘𝐼 (𝑗𝐼 ↦ ((𝑛𝑘)(.r𝑅)((curry 𝑀𝑘)‘𝑗))) ∈ V → (𝑘𝐼 ↦ (𝑗𝐼 ↦ ((𝑛𝑘)(.r𝑅)((curry 𝑀𝑘)‘𝑗)))) Fn 𝐼)
160157, 159syl 17 . . . . . . . . . . . . . . . . . . 19 (𝐼 ∈ Fin → (𝑘𝐼 ↦ (𝑗𝐼 ↦ ((𝑛𝑘)(.r𝑅)((curry 𝑀𝑘)‘𝑗)))) Fn 𝐼)
161 id 22 . . . . . . . . . . . . . . . . . . 19 (𝐼 ∈ Fin → 𝐼 ∈ Fin)
162 fvexd 6679 . . . . . . . . . . . . . . . . . . 19 (𝐼 ∈ Fin → (0g‘(𝑅 freeLMod 𝐼)) ∈ V)
163160, 161, 162fndmfifsupp 8835 . . . . . . . . . . . . . . . . . 18 (𝐼 ∈ Fin → (𝑘𝐼 ↦ (𝑗𝐼 ↦ ((𝑛𝑘)(.r𝑅)((curry 𝑀𝑘)‘𝑗)))) finSupp (0g‘(𝑅 freeLMod 𝐼)))
164163ad3antlr 727 . . . . . . . . . . . . . . . . 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)))
16545, 46, 137, 109, 109, 138, 155, 164frlmgsum 20846 . . . . . . . . . . . . . . . 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))))))
166136, 165eqtr2d 2857 . . . . . . . . . . . . . . 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)))
167105, 166sylan2 592 . . . . . . . . . . . . . 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)))
168167eqeq2d 2832 . . . . . . . . . . . . 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))))
169104, 168rexeqbidva 3427 . . . . . . . . . . . 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))))
170100, 169bitr4d 283 . . . . . . . . . . 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)))))))
17143, 170sylanl1 676 . . . . . . . . . 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)))))))
172171ad2antrr 722 . . . . . . . . 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)))))))
17391, 172mpbid 233 . . . . . . . 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))))))
174173ralrimiva 3182 . . . . . . 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))))))
17542, 174sylan 580 . . . . . 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))))))
17610, 21mpbird 258 . . . . . . . . 9 ((𝑅 ∈ DivRing ∧ 𝑀 ∈ (Base‘(𝐼 Mat 𝑅))) → 𝑀 ∈ ((Base‘𝑅) ↑m (𝐼 × 𝐼)))
177 elmapfn 8419 . . . . . . . . 9 (𝑀 ∈ ((Base‘𝑅) ↑m (𝐼 × 𝐼)) → 𝑀 Fn (𝐼 × 𝐼))
178176, 177syl 17 . . . . . . . 8 ((𝑅 ∈ DivRing ∧ 𝑀 ∈ (Base‘(𝐼 Mat 𝑅))) → 𝑀 Fn (𝐼 × 𝐼))
1794adantl 482 . . . . . . . 8 ((𝑅 ∈ DivRing ∧ 𝑀 ∈ (Base‘(𝐼 Mat 𝑅))) → 𝐼 ∈ Fin)
180 an32 642 . . . . . . . . . . . . . . . . . . 19 (((𝑀 Fn (𝐼 × 𝐼) ∧ 𝑗𝐼) ∧ 𝑘𝐼) ↔ ((𝑀 Fn (𝐼 × 𝐼) ∧ 𝑘𝐼) ∧ 𝑗𝐼))
181 df-3an 1081 . . . . . . . . . . . . . . . . . . 19 ((𝑀 Fn (𝐼 × 𝐼) ∧ 𝑘𝐼𝑗𝐼) ↔ ((𝑀 Fn (𝐼 × 𝐼) ∧ 𝑘𝐼) ∧ 𝑗𝐼))
182180, 181bitr4i 279 . . . . . . . . . . . . . . . . . 18 (((𝑀 Fn (𝐼 × 𝐼) ∧ 𝑗𝐼) ∧ 𝑘𝐼) ↔ (𝑀 Fn (𝐼 × 𝐼) ∧ 𝑘𝐼𝑗𝐼))
183 curfv 34754 . . . . . . . . . . . . . . . . . 18 (((𝑀 Fn (𝐼 × 𝐼) ∧ 𝑘𝐼𝑗𝐼) ∧ 𝐼 ∈ Fin) → ((curry 𝑀𝑘)‘𝑗) = (𝑘𝑀𝑗))
184182, 183sylanb 581 . . . . . . . . . . . . . . . . 17 ((((𝑀 Fn (𝐼 × 𝐼) ∧ 𝑗𝐼) ∧ 𝑘𝐼) ∧ 𝐼 ∈ Fin) → ((curry 𝑀𝑘)‘𝑗) = (𝑘𝑀𝑗))
185184an32s 648 . . . . . . . . . . . . . . . 16 ((((𝑀 Fn (𝐼 × 𝐼) ∧ 𝑗𝐼) ∧ 𝐼 ∈ Fin) ∧ 𝑘𝐼) → ((curry 𝑀𝑘)‘𝑗) = (𝑘𝑀𝑗))
186185oveq2d 7161 . . . . . . . . . . . . . . 15 ((((𝑀 Fn (𝐼 × 𝐼) ∧ 𝑗𝐼) ∧ 𝐼 ∈ Fin) ∧ 𝑘𝐼) → ((𝑛𝑘)(.r𝑅)((curry 𝑀𝑘)‘𝑗)) = ((𝑛𝑘)(.r𝑅)(𝑘𝑀𝑗)))
187186mpteq2dva 5153 . . . . . . . . . . . . . 14 (((𝑀 Fn (𝐼 × 𝐼) ∧ 𝑗𝐼) ∧ 𝐼 ∈ Fin) → (𝑘𝐼 ↦ ((𝑛𝑘)(.r𝑅)((curry 𝑀𝑘)‘𝑗))) = (𝑘𝐼 ↦ ((𝑛𝑘)(.r𝑅)(𝑘𝑀𝑗))))
188187an32s 648 . . . . . . . . . . . . 13 (((𝑀 Fn (𝐼 × 𝐼) ∧ 𝐼 ∈ Fin) ∧ 𝑗𝐼) → (𝑘𝐼 ↦ ((𝑛𝑘)(.r𝑅)((curry 𝑀𝑘)‘𝑗))) = (𝑘𝐼 ↦ ((𝑛𝑘)(.r𝑅)(𝑘𝑀𝑗))))
189188oveq2d 7161 . . . . . . . . . . . 12 (((𝑀 Fn (𝐼 × 𝐼) ∧ 𝐼 ∈ Fin) ∧ 𝑗𝐼) → (𝑅 Σg (𝑘𝐼 ↦ ((𝑛𝑘)(.r𝑅)((curry 𝑀𝑘)‘𝑗)))) = (𝑅 Σg (𝑘𝐼 ↦ ((𝑛𝑘)(.r𝑅)(𝑘𝑀𝑗)))))
190189mpteq2dva 5153 . . . . . . . . . . 11 ((𝑀 Fn (𝐼 × 𝐼) ∧ 𝐼 ∈ Fin) → (𝑗𝐼 ↦ (𝑅 Σg (𝑘𝐼 ↦ ((𝑛𝑘)(.r𝑅)((curry 𝑀𝑘)‘𝑗))))) = (𝑗𝐼 ↦ (𝑅 Σg (𝑘𝐼 ↦ ((𝑛𝑘)(.r𝑅)(𝑘𝑀𝑗))))))
191190eqeq2d 2832 . . . . . . . . . 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)))))))
192191rexbidv 3297 . . . . . . . . 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)))))))
193192ralbidv 3197 . . . . . . . 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)))))))
194178, 179, 193syl2anc 584 . . . . . . 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)))))))
195194ad2antrr 722 . . . . . 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)))))))
196175, 195mpbid 233 . . . . 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))))))
1978, 196sylanl1 676 . . . 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))))))
198 fveq1 6663 . . . . . . . . . . 11 (𝑛 = (𝑓𝑖) → (𝑛𝑘) = ((𝑓𝑖)‘𝑘))
199 uncov 34755 . . . . . . . . . . . 12 ((𝑖 ∈ V ∧ 𝑘 ∈ V) → (𝑖uncurry 𝑓𝑘) = ((𝑓𝑖)‘𝑘))
200199el2v 3502 . . . . . . . . . . 11 (𝑖uncurry 𝑓𝑘) = ((𝑓𝑖)‘𝑘)
201198, 200syl6eqr 2874 . . . . . . . . . 10 (𝑛 = (𝑓𝑖) → (𝑛𝑘) = (𝑖uncurry 𝑓𝑘))
202201oveq1d 7160 . . . . . . . . 9 (𝑛 = (𝑓𝑖) → ((𝑛𝑘)(.r𝑅)(𝑘𝑀𝑗)) = ((𝑖uncurry 𝑓𝑘)(.r𝑅)(𝑘𝑀𝑗)))
203202mpteq2dv 5154 . . . . . . . 8 (𝑛 = (𝑓𝑖) → (𝑘𝐼 ↦ ((𝑛𝑘)(.r𝑅)(𝑘𝑀𝑗))) = (𝑘𝐼 ↦ ((𝑖uncurry 𝑓𝑘)(.r𝑅)(𝑘𝑀𝑗))))
204203oveq2d 7161 . . . . . . 7 (𝑛 = (𝑓𝑖) → (𝑅 Σg (𝑘𝐼 ↦ ((𝑛𝑘)(.r𝑅)(𝑘𝑀𝑗)))) = (𝑅 Σg (𝑘𝐼 ↦ ((𝑖uncurry 𝑓𝑘)(.r𝑅)(𝑘𝑀𝑗)))))
205204mpteq2dv 5154 . . . . . 6 (𝑛 = (𝑓𝑖) → (𝑗𝐼 ↦ (𝑅 Σg (𝑘𝐼 ↦ ((𝑛𝑘)(.r𝑅)(𝑘𝑀𝑗))))) = (𝑗𝐼 ↦ (𝑅 Σg (𝑘𝐼 ↦ ((𝑖uncurry 𝑓𝑘)(.r𝑅)(𝑘𝑀𝑗))))))
206205eqeq2d 2832 . . . . 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)))))))
207206ac6sfi 8751 . . . 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)))))))
2085, 197, 207syl2anc 584 . . 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)))))))
209 uncf 34753 . . . . . . 7 (𝑓:𝐼⟶((Base‘𝑅) ↑m 𝐼) → uncurry 𝑓:(𝐼 × 𝐼)⟶(Base‘𝑅))
21013, 14frlmfibas 20836 . . . . . . . . . . . . . . . 16 ((𝑅 ∈ Field ∧ (𝐼 × 𝐼) ∈ Fin) → ((Base‘𝑅) ↑m (𝐼 × 𝐼)) = (Base‘(𝑅 freeLMod (𝐼 × 𝐼))))
21112, 210sylan2 592 . . . . . . . . . . . . . . 15 ((𝑅 ∈ Field ∧ 𝐼 ∈ Fin) → ((Base‘𝑅) ↑m (𝐼 × 𝐼)) = (Base‘(𝑅 freeLMod (𝐼 × 𝐼))))
2121, 13matbas 20952 . . . . . . . . . . . . . . . 16 ((𝐼 ∈ Fin ∧ 𝑅 ∈ Field) → (Base‘(𝑅 freeLMod (𝐼 × 𝐼))) = (Base‘(𝐼 Mat 𝑅)))
213212ancoms 459 . . . . . . . . . . . . . . 15 ((𝑅 ∈ Field ∧ 𝐼 ∈ Fin) → (Base‘(𝑅 freeLMod (𝐼 × 𝐼))) = (Base‘(𝐼 Mat 𝑅)))
214211, 213eqtrd 2856 . . . . . . . . . . . . . 14 ((𝑅 ∈ Field ∧ 𝐼 ∈ Fin) → ((Base‘𝑅) ↑m (𝐼 × 𝐼)) = (Base‘(𝐼 Mat 𝑅)))
2154, 214sylan2 592 . . . . . . . . . . . . 13 ((𝑅 ∈ Field ∧ 𝑀 ∈ (Base‘(𝐼 Mat 𝑅))) → ((Base‘𝑅) ↑m (𝐼 × 𝐼)) = (Base‘(𝐼 Mat 𝑅)))
216215eleq2d 2898 . . . . . . . . . . . 12 ((𝑅 ∈ Field ∧ 𝑀 ∈ (Base‘(𝐼 Mat 𝑅))) → (uncurry 𝑓 ∈ ((Base‘𝑅) ↑m (𝐼 × 𝐼)) ↔ uncurry 𝑓 ∈ (Base‘(𝐼 Mat 𝑅))))
217 elmapg 8409 . . . . . . . . . . . . . 14 (((Base‘𝑅) ∈ V ∧ (𝐼 × 𝐼) ∈ Fin) → (uncurry 𝑓 ∈ ((Base‘𝑅) ↑m (𝐼 × 𝐼)) ↔ uncurry 𝑓:(𝐼 × 𝐼)⟶(Base‘𝑅)))
21822, 23, 217sylancr 587 . . . . . . . . . . . . 13 (𝑀 ∈ (Base‘(𝐼 Mat 𝑅)) → (uncurry 𝑓 ∈ ((Base‘𝑅) ↑m (𝐼 × 𝐼)) ↔ uncurry 𝑓:(𝐼 × 𝐼)⟶(Base‘𝑅)))
219218adantl 482 . . . . . . . . . . . 12 ((𝑅 ∈ Field ∧ 𝑀 ∈ (Base‘(𝐼 Mat 𝑅))) → (uncurry 𝑓 ∈ ((Base‘𝑅) ↑m (𝐼 × 𝐼)) ↔ uncurry 𝑓:(𝐼 × 𝐼)⟶(Base‘𝑅)))
220216, 219bitr3d 282 . . . . . . . . . . 11 ((𝑅 ∈ Field ∧ 𝑀 ∈ (Base‘(𝐼 Mat 𝑅))) → (uncurry 𝑓 ∈ (Base‘(𝐼 Mat 𝑅)) ↔ uncurry 𝑓:(𝐼 × 𝐼)⟶(Base‘𝑅)))
221220biimpar 478 . . . . . . . . . 10 (((𝑅 ∈ Field ∧ 𝑀 ∈ (Base‘(𝐼 Mat 𝑅))) ∧ uncurry 𝑓:(𝐼 × 𝐼)⟶(Base‘𝑅)) → uncurry 𝑓 ∈ (Base‘(𝐼 Mat 𝑅)))
222221adantr 481 . . . . . . . . 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)))
223 nfv 1906 . . . . . . . . . . . . . 14 𝑗(((𝑅 ∈ Field ∧ 𝑀 ∈ (Base‘(𝐼 Mat 𝑅))) ∧ uncurry 𝑓:(𝐼 × 𝐼)⟶(Base‘𝑅)) ∧ 𝑖𝐼)
224 nfmpt1 5156 . . . . . . . . . . . . . . 15 𝑗(𝑗𝐼 ↦ (𝑅 Σg (𝑘𝐼 ↦ ((𝑖uncurry 𝑓𝑘)(.r𝑅)(𝑘𝑀𝑗)))))
225224nfeq2 2995 . . . . . . . . . . . . . 14 𝑗((𝑅 unitVec 𝐼)‘𝑖) = (𝑗𝐼 ↦ (𝑅 Σg (𝑘𝐼 ↦ ((𝑖uncurry 𝑓𝑘)(.r𝑅)(𝑘𝑀𝑗)))))
226 fveq1 6663 . . . . . . . . . . . . . . . . 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))
2277, 43syl 17 . . . . . . . . . . . . . . . . . . . . 21 (𝑅 ∈ Field → 𝑅 ∈ Ring)
228227, 4anim12i 612 . . . . . . . . . . . . . . . . . . . 20 ((𝑅 ∈ Field ∧ 𝑀 ∈ (Base‘(𝐼 Mat 𝑅))) → (𝑅 ∈ Ring ∧ 𝐼 ∈ Fin))
229228adantr 481 . . . . . . . . . . . . . . . . . . 19 (((𝑅 ∈ Field ∧ 𝑀 ∈ (Base‘(𝐼 Mat 𝑅))) ∧ uncurry 𝑓:(𝐼 × 𝐼)⟶(Base‘𝑅)) → (𝑅 ∈ Ring ∧ 𝐼 ∈ Fin))
230 equcom 2016 . . . . . . . . . . . . . . . . . . . . 21 (𝑖 = 𝑗𝑗 = 𝑖)
231 ifbi 4486 . . . . . . . . . . . . . . . . . . . . 21 ((𝑖 = 𝑗𝑗 = 𝑖) → if(𝑖 = 𝑗, (1r𝑅), (0g𝑅)) = if(𝑗 = 𝑖, (1r𝑅), (0g𝑅)))
232230, 231ax-mp 5 . . . . . . . . . . . . . . . . . . . 20 if(𝑖 = 𝑗, (1r𝑅), (0g𝑅)) = if(𝑗 = 𝑖, (1r𝑅), (0g𝑅))
233 eqid 2821 . . . . . . . . . . . . . . . . . . . . 21 (1r𝑅) = (1r𝑅)
234 eqid 2821 . . . . . . . . . . . . . . . . . . . . 21 (0g𝑅) = (0g𝑅)
235 simpllr 772 . . . . . . . . . . . . . . . . . . . . 21 ((((𝑅 ∈ Ring ∧ 𝐼 ∈ Fin) ∧ 𝑖𝐼) ∧ 𝑗𝐼) → 𝐼 ∈ Fin)
236 simplll 771 . . . . . . . . . . . . . . . . . . . . 21 ((((𝑅 ∈ Ring ∧ 𝐼 ∈ Fin) ∧ 𝑖𝐼) ∧ 𝑗𝐼) → 𝑅 ∈ Ring)
237 simplr 765 . . . . . . . . . . . . . . . . . . . . 21 ((((𝑅 ∈ Ring ∧ 𝐼 ∈ Fin) ∧ 𝑖𝐼) ∧ 𝑗𝐼) → 𝑖𝐼)
238 simpr 485 . . . . . . . . . . . . . . . . . . . . 21 ((((𝑅 ∈ Ring ∧ 𝐼 ∈ Fin) ∧ 𝑖𝐼) ∧ 𝑗𝐼) → 𝑗𝐼)
239 eqid 2821 . . . . . . . . . . . . . . . . . . . . 21 (1r‘(𝐼 Mat 𝑅)) = (1r‘(𝐼 Mat 𝑅))
2401, 233, 234, 235, 236, 237, 238, 239mat1ov 20987 . . . . . . . . . . . . . . . . . . . 20 ((((𝑅 ∈ Ring ∧ 𝐼 ∈ Fin) ∧ 𝑖𝐼) ∧ 𝑗𝐼) → (𝑖(1r‘(𝐼 Mat 𝑅))𝑗) = if(𝑖 = 𝑗, (1r𝑅), (0g𝑅)))
241 df-3an 1081 . . . . . . . . . . . . . . . . . . . . 21 ((𝑅 ∈ Ring ∧ 𝐼 ∈ Fin ∧ 𝑖𝐼) ↔ ((𝑅 ∈ Ring ∧ 𝐼 ∈ Fin) ∧ 𝑖𝐼))
24244, 233, 234uvcvval 20860 . . . . . . . . . . . . . . . . . . . . 21 (((𝑅 ∈ Ring ∧ 𝐼 ∈ Fin ∧ 𝑖𝐼) ∧ 𝑗𝐼) → (((𝑅 unitVec 𝐼)‘𝑖)‘𝑗) = if(𝑗 = 𝑖, (1r𝑅), (0g𝑅)))
243241, 242sylanbr 582 . . . . . . . . . . . . . . . . . . . 20 ((((𝑅 ∈ Ring ∧ 𝐼 ∈ Fin) ∧ 𝑖𝐼) ∧ 𝑗𝐼) → (((𝑅 unitVec 𝐼)‘𝑖)‘𝑗) = if(𝑗 = 𝑖, (1r𝑅), (0g𝑅)))
244232, 240, 2433eqtr4a 2882 . . . . . . . . . . . . . . . . . . 19 ((((𝑅 ∈ Ring ∧ 𝐼 ∈ Fin) ∧ 𝑖𝐼) ∧ 𝑗𝐼) → (𝑖(1r‘(𝐼 Mat 𝑅))𝑗) = (((𝑅 unitVec 𝐼)‘𝑖)‘𝑗))
245229, 244sylanl1 676 . . . . . . . . . . . . . . . . . 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))
246 ovex 7178 . . . . . . . . . . . . . . . . . . . . 21 (𝑅 Σg (𝑘𝐼 ↦ ((𝑖uncurry 𝑓𝑘)(.r𝑅)(𝑘𝑀𝑗)))) ∈ V
247 eqid 2821 . . . . . . . . . . . . . . . . . . . . . 22 (𝑗𝐼 ↦ (𝑅 Σg (𝑘𝐼 ↦ ((𝑖uncurry 𝑓𝑘)(.r𝑅)(𝑘𝑀𝑗))))) = (𝑗𝐼 ↦ (𝑅 Σg (𝑘𝐼 ↦ ((𝑖uncurry 𝑓𝑘)(.r𝑅)(𝑘𝑀𝑗)))))
248247fvmpt2 6772 . . . . . . . . . . . . . . . . . . . . 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)))))
249246, 248mpan2 687 . . . . . . . . . . . . . . . . . . . 20 (𝑗𝐼 → ((𝑗𝐼 ↦ (𝑅 Σg (𝑘𝐼 ↦ ((𝑖uncurry 𝑓𝑘)(.r𝑅)(𝑘𝑀𝑗)))))‘𝑗) = (𝑅 Σg (𝑘𝐼 ↦ ((𝑖uncurry 𝑓𝑘)(.r𝑅)(𝑘𝑀𝑗)))))
250249adantl 482 . . . . . . . . . . . . . . . . . . 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)))))
251 eqid 2821 . . . . . . . . . . . . . . . . . . . 20 (𝑅 maMul ⟨𝐼, 𝐼, 𝐼⟩) = (𝑅 maMul ⟨𝐼, 𝐼, 𝐼⟩)
252 simp-4l 779 . . . . . . . . . . . . . . . . . . . 20 (((((𝑅 ∈ Field ∧ 𝑀 ∈ (Base‘(𝐼 Mat 𝑅))) ∧ uncurry 𝑓:(𝐼 × 𝐼)⟶(Base‘𝑅)) ∧ 𝑖𝐼) ∧ 𝑗𝐼) → 𝑅 ∈ Field)
2534ad4antlr 729 . . . . . . . . . . . . . . . . . . . 20 (((((𝑅 ∈ Field ∧ 𝑀 ∈ (Base‘(𝐼 Mat 𝑅))) ∧ uncurry 𝑓:(𝐼 × 𝐼)⟶(Base‘𝑅)) ∧ 𝑖𝐼) ∧ 𝑗𝐼) → 𝐼 ∈ Fin)
254218biimpar 478 . . . . . . . . . . . . . . . . . . . . 21 ((𝑀 ∈ (Base‘(𝐼 Mat 𝑅)) ∧ uncurry 𝑓:(𝐼 × 𝐼)⟶(Base‘𝑅)) → uncurry 𝑓 ∈ ((Base‘𝑅) ↑m (𝐼 × 𝐼)))
255254ad5ant23 756 . . . . . . . . . . . . . . . . . . . 20 (((((𝑅 ∈ Field ∧ 𝑀 ∈ (Base‘(𝐼 Mat 𝑅))) ∧ uncurry 𝑓:(𝐼 × 𝐼)⟶(Base‘𝑅)) ∧ 𝑖𝐼) ∧ 𝑗𝐼) → uncurry 𝑓 ∈ ((Base‘𝑅) ↑m (𝐼 × 𝐼)))
256 simpr 485 . . . . . . . . . . . . . . . . . . . . . 22 ((𝑅 ∈ Field ∧ 𝑀 ∈ (Base‘(𝐼 Mat 𝑅))) → 𝑀 ∈ (Base‘(𝐼 Mat 𝑅)))
257256, 215eleqtrrd 2916 . . . . . . . . . . . . . . . . . . . . 21 ((𝑅 ∈ Field ∧ 𝑀 ∈ (Base‘(𝐼 Mat 𝑅))) → 𝑀 ∈ ((Base‘𝑅) ↑m (𝐼 × 𝐼)))
258257ad3antrrr 726 . . . . . . . . . . . . . . . . . . . 20 (((((𝑅 ∈ Field ∧ 𝑀 ∈ (Base‘(𝐼 Mat 𝑅))) ∧ uncurry 𝑓:(𝐼 × 𝐼)⟶(Base‘𝑅)) ∧ 𝑖𝐼) ∧ 𝑗𝐼) → 𝑀 ∈ ((Base‘𝑅) ↑m (𝐼 × 𝐼)))
259 simplr 765 . . . . . . . . . . . . . . . . . . . 20 (((((𝑅 ∈ Field ∧ 𝑀 ∈ (Base‘(𝐼 Mat 𝑅))) ∧ uncurry 𝑓:(𝐼 × 𝐼)⟶(Base‘𝑅)) ∧ 𝑖𝐼) ∧ 𝑗𝐼) → 𝑖𝐼)
260 simpr 485 . . . . . . . . . . . . . . . . . . . 20 (((((𝑅 ∈ Field ∧ 𝑀 ∈ (Base‘(𝐼 Mat 𝑅))) ∧ uncurry 𝑓:(𝐼 × 𝐼)⟶(Base‘𝑅)) ∧ 𝑖𝐼) ∧ 𝑗𝐼) → 𝑗𝐼)
261251, 14, 121, 252, 253, 253, 253, 255, 258, 259, 260mamufv 20928 . . . . . . . . . . . . . . . . . . 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)))))
2621, 251matmulr 20977 . . . . . . . . . . . . . . . . . . . . . . . 24 ((𝐼 ∈ Fin ∧ 𝑅 ∈ Field) → (𝑅 maMul ⟨𝐼, 𝐼, 𝐼⟩) = (.r‘(𝐼 Mat 𝑅)))
263262ancoms 459 . . . . . . . . . . . . . . . . . . . . . . 23 ((𝑅 ∈ Field ∧ 𝐼 ∈ Fin) → (𝑅 maMul ⟨𝐼, 𝐼, 𝐼⟩) = (.r‘(𝐼 Mat 𝑅)))
264263oveqd 7162 . . . . . . . . . . . . . . . . . . . . . 22 ((𝑅 ∈ Field ∧ 𝐼 ∈ Fin) → (uncurry 𝑓(𝑅 maMul ⟨𝐼, 𝐼, 𝐼⟩)𝑀) = (uncurry 𝑓(.r‘(𝐼 Mat 𝑅))𝑀))
265264oveqd 7162 . . . . . . . . . . . . . . . . . . . . 21 ((𝑅 ∈ Field ∧ 𝐼 ∈ Fin) → (𝑖(uncurry 𝑓(𝑅 maMul ⟨𝐼, 𝐼, 𝐼⟩)𝑀)𝑗) = (𝑖(uncurry 𝑓(.r‘(𝐼 Mat 𝑅))𝑀)𝑗))
2664, 265sylan2 592 . . . . . . . . . . . . . . . . . . . 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))
267266ad3antrrr 726 . . . . . . . . . . . . . . . . . . 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))
268250, 261, 2673eqtr2rd 2863 . . . . . . . . . . . . . . . . . 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))
269245, 268eqeq12d 2837 . . . . . . . . . . . . . . . . 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)))
270226, 269syl5ibr 247 . . . . . . . . . . . . . . . 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)))
271270ex 413 . . . . . . . . . . . . . . 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))))
272271com23 86 . . . . . . . . . . . . . 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))))
273223, 225, 272ralrimd 3218 . . . . . . . . . . . . 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)))
274273ralimdva 3177 . . . . . . . . . . . 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)))
2751, 2, 239mat1bas 20988 . . . . . . . . . . . . . . . . 17 ((𝑅 ∈ Ring ∧ 𝐼 ∈ Fin) → (1r‘(𝐼 Mat 𝑅)) ∈ (Base‘(𝐼 Mat 𝑅)))
27613, 14frlmfibas 20836 . . . . . . . . . . . . . . . . . . 19 ((𝑅 ∈ Ring ∧ (𝐼 × 𝐼) ∈ Fin) → ((Base‘𝑅) ↑m (𝐼 × 𝐼)) = (Base‘(𝑅 freeLMod (𝐼 × 𝐼))))
27712, 276sylan2 592 . . . . . . . . . . . . . . . . . 18 ((𝑅 ∈ Ring ∧ 𝐼 ∈ Fin) → ((Base‘𝑅) ↑m (𝐼 × 𝐼)) = (Base‘(𝑅 freeLMod (𝐼 × 𝐼))))
2781, 13matbas 20952 . . . . . . . . . . . . . . . . . . 19 ((𝐼 ∈ Fin ∧ 𝑅 ∈ Ring) → (Base‘(𝑅 freeLMod (𝐼 × 𝐼))) = (Base‘(𝐼 Mat 𝑅)))
279278ancoms 459 . . . . . . . . . . . . . . . . . 18 ((𝑅 ∈ Ring ∧ 𝐼 ∈ Fin) → (Base‘(𝑅 freeLMod (𝐼 × 𝐼))) = (Base‘(𝐼 Mat 𝑅)))
280277, 279eqtrd 2856 . . . . . . . . . . . . . . . . 17 ((𝑅 ∈ Ring ∧ 𝐼 ∈ Fin) → ((Base‘𝑅) ↑m (𝐼 × 𝐼)) = (Base‘(𝐼 Mat 𝑅)))
281275, 280eleqtrrd 2916 . . . . . . . . . . . . . . . 16 ((𝑅 ∈ Ring ∧ 𝐼 ∈ Fin) → (1r‘(𝐼 Mat 𝑅)) ∈ ((Base‘𝑅) ↑m (𝐼 × 𝐼)))
282 elmapfn 8419 . . . . . . . . . . . . . . . 16 ((1r‘(𝐼 Mat 𝑅)) ∈ ((Base‘𝑅) ↑m (𝐼 × 𝐼)) → (1r‘(𝐼 Mat 𝑅)) Fn (𝐼 × 𝐼))
283281, 282syl 17 . . . . . . . . . . . . . . 15 ((𝑅 ∈ Ring ∧ 𝐼 ∈ Fin) → (1r‘(𝐼 Mat 𝑅)) Fn (𝐼 × 𝐼))
284227, 4, 283syl2an 595 . . . . . . . . . . . . . 14 ((𝑅 ∈ Field ∧ 𝑀 ∈ (Base‘(𝐼 Mat 𝑅))) → (1r‘(𝐼 Mat 𝑅)) Fn (𝐼 × 𝐼))
285284adantr 481 . . . . . . . . . . . . 13 (((𝑅 ∈ Field ∧ 𝑀 ∈ (Base‘(𝐼 Mat 𝑅))) ∧ uncurry 𝑓:(𝐼 × 𝐼)⟶(Base‘𝑅)) → (1r‘(𝐼 Mat 𝑅)) Fn (𝐼 × 𝐼))
2861matring 20982 . . . . . . . . . . . . . . . . . 18 ((𝐼 ∈ Fin ∧ 𝑅 ∈ Ring) → (𝐼 Mat 𝑅) ∈ Ring)
2874, 227, 286syl2anr 596 . . . . . . . . . . . . . . . . 17 ((𝑅 ∈ Field ∧ 𝑀 ∈ (Base‘(𝐼 Mat 𝑅))) → (𝐼 Mat 𝑅) ∈ Ring)
288287adantr 481 . . . . . . . . . . . . . . . 16 (((𝑅 ∈ Field ∧ 𝑀 ∈ (Base‘(𝐼 Mat 𝑅))) ∧ uncurry 𝑓:(𝐼 × 𝐼)⟶(Base‘𝑅)) → (𝐼 Mat 𝑅) ∈ Ring)
289 simplr 765 . . . . . . . . . . . . . . . 16 (((𝑅 ∈ Field ∧ 𝑀 ∈ (Base‘(𝐼 Mat 𝑅))) ∧ uncurry 𝑓:(𝐼 × 𝐼)⟶(Base‘𝑅)) → 𝑀 ∈ (Base‘(𝐼 Mat 𝑅)))
290 eqid 2821 . . . . . . . . . . . . . . . . 17 (.r‘(𝐼 Mat 𝑅)) = (.r‘(𝐼 Mat 𝑅))
2912, 290ringcl 19242 . . . . . . . . . . . . . . . 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)))
292288, 221, 289, 291syl3anc 1363 . . . . . . . . . . . . . . 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)))
293215adantr 481 . . . . . . . . . . . . . . 15 (((𝑅 ∈ Field ∧ 𝑀 ∈ (Base‘(𝐼 Mat 𝑅))) ∧ uncurry 𝑓:(𝐼 × 𝐼)⟶(Base‘𝑅)) → ((Base‘𝑅) ↑m (𝐼 × 𝐼)) = (Base‘(𝐼 Mat 𝑅)))
294292, 293eleqtrrd 2916 . . . . . . . . . . . . . 14 (((𝑅 ∈ Field ∧ 𝑀 ∈ (Base‘(𝐼 Mat 𝑅))) ∧ uncurry 𝑓:(𝐼 × 𝐼)⟶(Base‘𝑅)) → (uncurry 𝑓(.r‘(𝐼 Mat 𝑅))𝑀) ∈ ((Base‘𝑅) ↑m (𝐼 × 𝐼)))
295 elmapfn 8419 . . . . . . . . . . . . . 14 ((uncurry 𝑓(.r‘(𝐼 Mat 𝑅))𝑀) ∈ ((Base‘𝑅) ↑m (𝐼 × 𝐼)) → (uncurry 𝑓(.r‘(𝐼 Mat 𝑅))𝑀) Fn (𝐼 × 𝐼))
296294, 295syl 17 . . . . . . . . . . . . 13 (((𝑅 ∈ Field ∧ 𝑀 ∈ (Base‘(𝐼 Mat 𝑅))) ∧ uncurry 𝑓:(𝐼 × 𝐼)⟶(Base‘𝑅)) → (uncurry 𝑓(.r‘(𝐼 Mat 𝑅))𝑀) Fn (𝐼 × 𝐼))
297 eqfnov2 7270 . . . . . . . . . . . . 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)))
298285, 296, 297syl2anc 584 . . . . . . . . . . . 12 (((𝑅 ∈ Field ∧ 𝑀 ∈ (Base‘(𝐼 Mat 𝑅))) ∧ uncurry 𝑓:(𝐼 × 𝐼)⟶(Base‘𝑅)) → ((1r‘(𝐼 Mat 𝑅)) = (uncurry 𝑓(.r‘(𝐼 Mat 𝑅))𝑀) ↔ ∀𝑖𝐼𝑗𝐼 (𝑖(1r‘(𝐼 Mat 𝑅))𝑗) = (𝑖(uncurry 𝑓(.r‘(𝐼 Mat 𝑅))𝑀)𝑗)))
299274, 298sylibrd 260 . . . . . . . . . . 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)))
300299imp 407 . . . . . . . . . 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))
301300eqcomd 2827 . . . . . . . . 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)))
302 oveq1 7152 . . . . . . . . . . 11 (𝑛 = uncurry 𝑓 → (𝑛(.r‘(𝐼 Mat 𝑅))𝑀) = (uncurry 𝑓(.r‘(𝐼 Mat 𝑅))𝑀))
303302eqeq1d 2823 . . . . . . . . . 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))))
304303rspcev 3622 . . . . . . . . 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)))
305222, 301, 304syl2anc 584 . . . . . . . 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)))
306305expl 458 . . . . . . 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))))
307209, 306sylani 603 . . . . . 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))))
308307exlimdv 1925 . . . . 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))))
309308imp 407 . . . 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)))
310309adantlr 711 . . 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)))
311208, 310syldan 591 . 2 ((((𝑅 ∈ Field ∧ 𝑀 ∈ (Base‘(𝐼 Mat 𝑅))) ∧ 𝐼 ≠ ∅) ∧ curry 𝑀 LIndF (𝑅 freeLMod 𝐼)) → ∃𝑛 ∈ (Base‘(𝐼 Mat 𝑅))(𝑛(.r‘(𝐼 Mat 𝑅))𝑀) = (1r‘(𝐼 Mat 𝑅)))
3126simprbi 497 . . . 4 (𝑅 ∈ Field → 𝑅 ∈ CRing)
313 eqid 2821 . . . . . . . . . 10 (𝐼 maDet 𝑅) = (𝐼 maDet 𝑅)
314313, 1, 2, 14mdetcl 21135 . . . . . . . . 9 ((𝑅 ∈ CRing ∧ 𝑀 ∈ (Base‘(𝐼 Mat 𝑅))) → ((𝐼 maDet 𝑅)‘𝑀) ∈ (Base‘𝑅))
315313, 1, 2, 14mdetcl 21135 . . . . . . . . 9 ((𝑅 ∈ CRing ∧ 𝑛 ∈ (Base‘(𝐼 Mat 𝑅))) → ((𝐼 maDet 𝑅)‘𝑛) ∈ (Base‘𝑅))
316 eqid 2821 . . . . . . . . . 10 (∥r𝑅) = (∥r𝑅)
31714, 316, 121dvdsrmul 19329 . . . . . . . . 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)))
318314, 315, 317syl2an 595 . . . . . . . 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)))
319318anandis 674 . . . . . . 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)))
320319anassrs 468 . . . . . 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)))
321320adantrr 713 . . . . 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)))
322 fveq2 6664 . . . . . . . . 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))))
3231, 2, 313, 121, 290mdetmul 21162 . . . . . . . . . . . 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)))
3243233expa 1110 . . . . . . . . . . 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)))
325324an32s 648 . . . . . . . . . 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)))
326313, 1, 239, 233mdet1 21140 . . . . . . . . . . . 12 ((𝑅 ∈ CRing ∧ 𝐼 ∈ Fin) → ((𝐼 maDet 𝑅)‘(1r‘(𝐼 Mat 𝑅))) = (1r𝑅))
3274, 326sylan2 592 . . . . . . . . . . 11 ((𝑅 ∈ CRing ∧ 𝑀 ∈ (Base‘(𝐼 Mat 𝑅))) → ((𝐼 maDet 𝑅)‘(1r‘(𝐼 Mat 𝑅))) = (1r𝑅))
328327adantr 481 . . . . . . . . . 10 (((𝑅 ∈ CRing ∧ 𝑀 ∈ (Base‘(𝐼 Mat 𝑅))) ∧ 𝑛 ∈ (Base‘(𝐼 Mat 𝑅))) → ((𝐼 maDet 𝑅)‘(1r‘(𝐼 Mat 𝑅))) = (1r𝑅))
329325, 328eqeq12d 2837 . . . . . . . . 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)))
330322, 329syl5ib 245 . . . . . . . 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)))
331330impr 455 . . . . . . 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))
332331breq2d 5070 . . . . . 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)))
333 eqid 2821 . . . . . . . 8 (Unit‘𝑅) = (Unit‘𝑅)
334333, 233, 316crngunit 19343 . . . . . . 7 (𝑅 ∈ CRing → (((𝐼 maDet 𝑅)‘𝑀) ∈ (Unit‘𝑅) ↔ ((𝐼 maDet 𝑅)‘𝑀)(∥r𝑅)(1r𝑅)))
335334ad2antrr 722 . . . . . 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)))
336332, 335bitr4d 283 . . . . 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)))
337321, 336mpbid 233 . . . 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))
338312, 337sylanl1 676 . . 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))
339338ad4ant14 748 . 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))
340311, 339rexlimddv 3291 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 207  wa 396  w3a 1079   = wceq 1528  wex 1771  wcel 2105  wne 3016  wral 3138  wrex 3139  Vcvv 3495  cdif 3932  c0 4290  ifcif 4465  {csn 4559  cotp 4567   class class class wbr 5058  cmpt 5138   × cxp 5547  dom cdm 5549  ran crn 5550  cima 5552   Fn wfn 6344  wf 6345  1-1wf1 6346  1-1-ontowf1o 6348  cfv 6349  (class class class)co 7145  f cof 7396  curry ccur 7922  uncurry cunc 7923  m cmap 8396  cen 8495  Fincfn 8498   finSupp cfsupp 8822  Basecbs 16473  .rcmulr 16556  Scalarcsca 16558   ·𝑠 cvsca 16559  0gc0g 16703   Σg cgsu 16704  1rcur 19182  Ringcrg 19228  CRingccrg 19229  rcdsr 19319  Unitcui 19320  DivRingcdr 19433  Fieldcfield 19434  LModclmod 19565  LSpanclspn 19674  LBasisclbs 19777  NzRingcnzr 19960   freeLMod cfrlm 20820   unitVec cuvc 20856   LIndF clindf 20878  LIndSclinds 20879   maMul cmmul 20924   Mat cmat 20946   maDet cmdat 21123
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1787  ax-4 1801  ax-5 1902  ax-6 1961  ax-7 2006  ax-8 2107  ax-9 2115  ax-10 2136  ax-11 2151  ax-12 2167  ax-ext 2793  ax-rep 5182  ax-sep 5195  ax-nul 5202  ax-pow 5258  ax-pr 5321  ax-un 7450  ax-cnex 10582  ax-resscn 10583  ax-1cn 10584  ax-icn 10585  ax-addcl 10586  ax-addrcl 10587  ax-mulcl 10588  ax-mulrcl 10589  ax-mulcom 10590  ax-addass 10591  ax-mulass 10592  ax-distr 10593  ax-i2m1 10594  ax-1ne0 10595  ax-1rid 10596  ax-rnegex 10597  ax-rrecex 10598  ax-cnre 10599  ax-pre-lttri 10600  ax-pre-lttrn 10601  ax-pre-ltadd 10602  ax-pre-mulgt0 10603  ax-addf 10605  ax-mulf 10606
This theorem depends on definitions:  df-bi 208  df-an 397  df-or 842  df-3or 1080  df-3an 1081  df-xor 1496  df-tru 1531  df-fal 1541  df-ex 1772  df-nf 1776  df-sb 2061  df-mo 2618  df-eu 2650  df-clab 2800  df-cleq 2814  df-clel 2893  df-nfc 2963  df-ne 3017  df-nel 3124  df-ral 3143  df-rex 3144  df-reu 3145  df-rmo 3146  df-rab 3147  df-v 3497  df-sbc 3772  df-csb 3883  df-dif 3938  df-un 3940  df-in 3942  df-ss 3951  df-pss 3953  df-nul 4291  df-if 4466  df-pw 4539  df-sn 4560  df-pr 4562  df-tp 4564  df-op 4566  df-ot 4568  df-uni 4833  df-int 4870  df-iun 4914  df-iin 4915  df-br 5059  df-opab 5121  df-mpt 5139  df-tr 5165  df-id 5454  df-eprel 5459  df-po 5468  df-so 5469  df-fr 5508  df-se 5509  df-we 5510  df-xp 5555  df-rel 5556  df-cnv 5557  df-co 5558  df-dm 5559  df-rn 5560  df-res 5561  df-ima 5562  df-pred 6142  df-ord 6188  df-on 6189  df-lim 6190  df-suc 6191  df-iota 6308  df-fun 6351  df-fn 6352  df-f 6353  df-f1 6354  df-fo 6355  df-f1o 6356  df-fv 6357  df-isom 6358  df-riota 7103  df-ov 7148  df-oprab 7149  df-mpo 7150  df-of 7398  df-om 7569  df-1st 7680  df-2nd 7681  df-supp 7822  df-tpos 7883  df-cur 7924  df-unc 7925  df-wrecs 7938  df-recs 7999  df-rdg 8037  df-1o 8093  df-2o 8094  df-oadd 8097  df-er 8279  df-map 8398  df-pm 8399  df-ixp 8451  df-en 8499  df-dom 8500  df-sdom 8501  df-fin 8502  df-fsupp 8823  df-sup 8895  df-oi 8963  df-card 9357  df-pnf 10666  df-mnf 10667  df-xr 10668  df-ltxr 10669  df-le 10670  df-sub 10861  df-neg 10862  df-div 11287  df-nn 11628  df-2 11689  df-3 11690  df-4 11691  df-5 11692  df-6 11693  df-7 11694  df-8 11695  df-9 11696  df-n0 11887  df-xnn0 11957  df-z 11971  df-dec 12088  df-uz 12233  df-rp 12380  df-fz 12883  df-fzo 13024  df-seq 13360  df-exp 13420  df-hash 13681  df-word 13852  df-lsw 13905  df-concat 13913  df-s1 13940  df-substr 13993  df-pfx 14023  df-splice 14102  df-reverse 14111  df-s2 14200  df-struct 16475  df-ndx 16476  df-slot 16477  df-base 16479  df-sets 16480  df-ress 16481  df-plusg 16568  df-mulr 16569  df-starv 16570  df-sca 16571  df-vsca 16572  df-ip 16573  df-tset 16574  df-ple 16575  df-ds 16577  df-unif 16578  df-hom 16579  df-cco 16580  df-0g 16705  df-gsum 16706  df-prds 16711  df-pws 16713  df-mre 16847  df-mrc 16848  df-mri 16849  df-acs 16850  df-mgm 17842  df-sgrp 17891  df-mnd 17902  df-mhm 17946  df-submnd 17947  df-grp 18046  df-minusg 18047  df-sbg 18048  df-mulg 18165  df-subg 18216  df-ghm 18296  df-gim 18339  df-cntz 18387  df-oppg 18414  df-symg 18436  df-pmtr 18501  df-psgn 18550  df-evpm 18551  df-cmn 18839  df-abl 18840  df-mgp 19171  df-ur 19183  df-srg 19187  df-ring 19230  df-cring 19231  df-oppr 19304  df-dvdsr 19322  df-unit 19323  df-invr 19353  df-dvr 19364  df-rnghom 19398  df-drng 19435  df-field 19436  df-subrg 19464  df-lmod 19567  df-lss 19635  df-lsp 19675  df-lmhm 19725  df-lbs 19778  df-lvec 19806  df-sra 19875  df-rgmod 19876  df-nzr 19961  df-cnfld 20476  df-zring 20548  df-zrh 20581  df-dsmm 20806  df-frlm 20821  df-uvc 20857  df-lindf 20880  df-linds 20881  df-mamu 20925  df-mat 20947  df-mdet 21124
This theorem is referenced by:  matunitlindf  34772
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