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Theorem matunitlindflem2 33717
Description: One direction of matunitlindf 33718. (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 2806 . . . . . . 7 (𝐼 Mat 𝑅) = (𝐼 Mat 𝑅)
2 eqid 2806 . . . . . . 7 (Base‘(𝐼 Mat 𝑅)) = (Base‘(𝐼 Mat 𝑅))
31, 2matrcl 20425 . . . . . 6 (𝑀 ∈ (Base‘(𝐼 Mat 𝑅)) → (𝐼 ∈ Fin ∧ 𝑅 ∈ V))
43simpld 484 . . . . 5 (𝑀 ∈ (Base‘(𝐼 Mat 𝑅)) → 𝐼 ∈ Fin)
54ad3antlr 713 . . . 4 ((((𝑅 ∈ Field ∧ 𝑀 ∈ (Base‘(𝐼 Mat 𝑅))) ∧ 𝐼 ≠ ∅) ∧ curry 𝑀 LIndF (𝑅 freeLMod 𝐼)) → 𝐼 ∈ Fin)
6 isfld 18956 . . . . . . 7 (𝑅 ∈ Field ↔ (𝑅 ∈ DivRing ∧ 𝑅 ∈ CRing))
76simplbi 487 . . . . . 6 (𝑅 ∈ Field → 𝑅 ∈ DivRing)
87anim1i 604 . . . . 5 ((𝑅 ∈ Field ∧ 𝑀 ∈ (Base‘(𝐼 Mat 𝑅))) → (𝑅 ∈ DivRing ∧ 𝑀 ∈ (Base‘(𝐼 Mat 𝑅))))
94ad2antrl 710 . . . . . . . . . . . 12 ((𝑅 ∈ DivRing ∧ (𝑀 ∈ (Base‘(𝐼 Mat 𝑅)) ∧ 𝐼 ≠ ∅)) → 𝐼 ∈ Fin)
10 simpr 473 . . . . . . . . . . . . . . 15 ((𝑅 ∈ DivRing ∧ 𝑀 ∈ (Base‘(𝐼 Mat 𝑅))) → 𝑀 ∈ (Base‘(𝐼 Mat 𝑅)))
11 xpfi 8466 . . . . . . . . . . . . . . . . . . . . 21 ((𝐼 ∈ Fin ∧ 𝐼 ∈ Fin) → (𝐼 × 𝐼) ∈ Fin)
1211anidms 558 . . . . . . . . . . . . . . . . . . . 20 (𝐼 ∈ Fin → (𝐼 × 𝐼) ∈ Fin)
13 eqid 2806 . . . . . . . . . . . . . . . . . . . . 21 (𝑅 freeLMod (𝐼 × 𝐼)) = (𝑅 freeLMod (𝐼 × 𝐼))
14 eqid 2806 . . . . . . . . . . . . . . . . . . . . 21 (Base‘𝑅) = (Base‘𝑅)
1513, 14frlmfibas 20312 . . . . . . . . . . . . . . . . . . . 20 ((𝑅 ∈ DivRing ∧ (𝐼 × 𝐼) ∈ Fin) → ((Base‘𝑅) ↑𝑚 (𝐼 × 𝐼)) = (Base‘(𝑅 freeLMod (𝐼 × 𝐼))))
1612, 15sylan2 582 . . . . . . . . . . . . . . . . . . 19 ((𝑅 ∈ DivRing ∧ 𝐼 ∈ Fin) → ((Base‘𝑅) ↑𝑚 (𝐼 × 𝐼)) = (Base‘(𝑅 freeLMod (𝐼 × 𝐼))))
171, 13matbas 20426 . . . . . . . . . . . . . . . . . . . 20 ((𝐼 ∈ Fin ∧ 𝑅 ∈ DivRing) → (Base‘(𝑅 freeLMod (𝐼 × 𝐼))) = (Base‘(𝐼 Mat 𝑅)))
1817ancoms 448 . . . . . . . . . . . . . . . . . . 19 ((𝑅 ∈ DivRing ∧ 𝐼 ∈ Fin) → (Base‘(𝑅 freeLMod (𝐼 × 𝐼))) = (Base‘(𝐼 Mat 𝑅)))
1916, 18eqtrd 2840 . . . . . . . . . . . . . . . . . 18 ((𝑅 ∈ DivRing ∧ 𝐼 ∈ Fin) → ((Base‘𝑅) ↑𝑚 (𝐼 × 𝐼)) = (Base‘(𝐼 Mat 𝑅)))
2019eleq2d 2871 . . . . . . . . . . . . . . . . 17 ((𝑅 ∈ DivRing ∧ 𝐼 ∈ Fin) → (𝑀 ∈ ((Base‘𝑅) ↑𝑚 (𝐼 × 𝐼)) ↔ 𝑀 ∈ (Base‘(𝐼 Mat 𝑅))))
214, 20sylan2 582 . . . . . . . . . . . . . . . 16 ((𝑅 ∈ DivRing ∧ 𝑀 ∈ (Base‘(𝐼 Mat 𝑅))) → (𝑀 ∈ ((Base‘𝑅) ↑𝑚 (𝐼 × 𝐼)) ↔ 𝑀 ∈ (Base‘(𝐼 Mat 𝑅))))
22 fvex 6417 . . . . . . . . . . . . . . . . . 18 (Base‘𝑅) ∈ V
234, 4, 11syl2anc 575 . . . . . . . . . . . . . . . . . 18 (𝑀 ∈ (Base‘(𝐼 Mat 𝑅)) → (𝐼 × 𝐼) ∈ Fin)
24 elmapg 8101 . . . . . . . . . . . . . . . . . 18 (((Base‘𝑅) ∈ V ∧ (𝐼 × 𝐼) ∈ Fin) → (𝑀 ∈ ((Base‘𝑅) ↑𝑚 (𝐼 × 𝐼)) ↔ 𝑀:(𝐼 × 𝐼)⟶(Base‘𝑅)))
2522, 23, 24sylancr 577 . . . . . . . . . . . . . . . . 17 (𝑀 ∈ (Base‘(𝐼 Mat 𝑅)) → (𝑀 ∈ ((Base‘𝑅) ↑𝑚 (𝐼 × 𝐼)) ↔ 𝑀:(𝐼 × 𝐼)⟶(Base‘𝑅)))
2625adantl 469 . . . . . . . . . . . . . . . 16 ((𝑅 ∈ DivRing ∧ 𝑀 ∈ (Base‘(𝐼 Mat 𝑅))) → (𝑀 ∈ ((Base‘𝑅) ↑𝑚 (𝐼 × 𝐼)) ↔ 𝑀:(𝐼 × 𝐼)⟶(Base‘𝑅)))
2721, 26bitr3d 272 . . . . . . . . . . . . . . 15 ((𝑅 ∈ DivRing ∧ 𝑀 ∈ (Base‘(𝐼 Mat 𝑅))) → (𝑀 ∈ (Base‘(𝐼 Mat 𝑅)) ↔ 𝑀:(𝐼 × 𝐼)⟶(Base‘𝑅)))
2810, 27mpbid 223 . . . . . . . . . . . . . 14 ((𝑅 ∈ DivRing ∧ 𝑀 ∈ (Base‘(𝐼 Mat 𝑅))) → 𝑀:(𝐼 × 𝐼)⟶(Base‘𝑅))
2928adantrr 699 . . . . . . . . . . . . 13 ((𝑅 ∈ DivRing ∧ (𝑀 ∈ (Base‘(𝐼 Mat 𝑅)) ∧ 𝐼 ≠ ∅)) → 𝑀:(𝐼 × 𝐼)⟶(Base‘𝑅))
30 eldifsn 4508 . . . . . . . . . . . . . . . 16 (𝐼 ∈ (Fin ∖ {∅}) ↔ (𝐼 ∈ Fin ∧ 𝐼 ≠ ∅))
3130biimpri 219 . . . . . . . . . . . . . . 15 ((𝐼 ∈ Fin ∧ 𝐼 ≠ ∅) → 𝐼 ∈ (Fin ∖ {∅}))
324, 31sylan 571 . . . . . . . . . . . . . 14 ((𝑀 ∈ (Base‘(𝐼 Mat 𝑅)) ∧ 𝐼 ≠ ∅) → 𝐼 ∈ (Fin ∖ {∅}))
3332adantl 469 . . . . . . . . . . . . 13 ((𝑅 ∈ DivRing ∧ (𝑀 ∈ (Base‘(𝐼 Mat 𝑅)) ∧ 𝐼 ≠ ∅)) → 𝐼 ∈ (Fin ∖ {∅}))
34 curf 33698 . . . . . . . . . . . . . 14 ((𝑀:(𝐼 × 𝐼)⟶(Base‘𝑅) ∧ 𝐼 ∈ (Fin ∖ {∅}) ∧ (Base‘𝑅) ∈ V) → curry 𝑀:𝐼⟶((Base‘𝑅) ↑𝑚 𝐼))
3522, 34mp3an3 1567 . . . . . . . . . . . . 13 ((𝑀:(𝐼 × 𝐼)⟶(Base‘𝑅) ∧ 𝐼 ∈ (Fin ∖ {∅})) → curry 𝑀:𝐼⟶((Base‘𝑅) ↑𝑚 𝐼))
3629, 33, 35syl2anc 575 . . . . . . . . . . . 12 ((𝑅 ∈ DivRing ∧ (𝑀 ∈ (Base‘(𝐼 Mat 𝑅)) ∧ 𝐼 ≠ ∅)) → curry 𝑀:𝐼⟶((Base‘𝑅) ↑𝑚 𝐼))
379, 36jca 503 . . . . . . . . . . 11 ((𝑅 ∈ DivRing ∧ (𝑀 ∈ (Base‘(𝐼 Mat 𝑅)) ∧ 𝐼 ≠ ∅)) → (𝐼 ∈ Fin ∧ curry 𝑀:𝐼⟶((Base‘𝑅) ↑𝑚 𝐼)))
3837ex 399 . . . . . . . . . 10 (𝑅 ∈ DivRing → ((𝑀 ∈ (Base‘(𝐼 Mat 𝑅)) ∧ 𝐼 ≠ ∅) → (𝐼 ∈ Fin ∧ curry 𝑀:𝐼⟶((Base‘𝑅) ↑𝑚 𝐼))))
3938imdistani 560 . . . . . . . . 9 ((𝑅 ∈ DivRing ∧ (𝑀 ∈ (Base‘(𝐼 Mat 𝑅)) ∧ 𝐼 ≠ ∅)) → (𝑅 ∈ DivRing ∧ (𝐼 ∈ Fin ∧ curry 𝑀:𝐼⟶((Base‘𝑅) ↑𝑚 𝐼))))
4039anassrs 455 . . . . . . . 8 (((𝑅 ∈ DivRing ∧ 𝑀 ∈ (Base‘(𝐼 Mat 𝑅))) ∧ 𝐼 ≠ ∅) → (𝑅 ∈ DivRing ∧ (𝐼 ∈ Fin ∧ curry 𝑀:𝐼⟶((Base‘𝑅) ↑𝑚 𝐼))))
41 anass 456 . . . . . . . 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 225 . . . . . . 7 (((𝑅 ∈ DivRing ∧ 𝑀 ∈ (Base‘(𝐼 Mat 𝑅))) ∧ 𝐼 ≠ ∅) → ((𝑅 ∈ DivRing ∧ 𝐼 ∈ Fin) ∧ curry 𝑀:𝐼⟶((Base‘𝑅) ↑𝑚 𝐼)))
43 drngring 18954 . . . . . . . . . . . . 13 (𝑅 ∈ DivRing → 𝑅 ∈ Ring)
44 eqid 2806 . . . . . . . . . . . . . 14 (𝑅 unitVec 𝐼) = (𝑅 unitVec 𝐼)
45 eqid 2806 . . . . . . . . . . . . . 14 (𝑅 freeLMod 𝐼) = (𝑅 freeLMod 𝐼)
46 eqid 2806 . . . . . . . . . . . . . 14 (Base‘(𝑅 freeLMod 𝐼)) = (Base‘(𝑅 freeLMod 𝐼))
4744, 45, 46uvcff 20337 . . . . . . . . . . . . 13 ((𝑅 ∈ Ring ∧ 𝐼 ∈ Fin) → (𝑅 unitVec 𝐼):𝐼⟶(Base‘(𝑅 freeLMod 𝐼)))
4843, 47sylan 571 . . . . . . . . . . . 12 ((𝑅 ∈ DivRing ∧ 𝐼 ∈ Fin) → (𝑅 unitVec 𝐼):𝐼⟶(Base‘(𝑅 freeLMod 𝐼)))
4948ffvelrnda 6577 . . . . . . . . . . 11 (((𝑅 ∈ DivRing ∧ 𝐼 ∈ Fin) ∧ 𝑖𝐼) → ((𝑅 unitVec 𝐼)‘𝑖) ∈ (Base‘(𝑅 freeLMod 𝐼)))
5049ad4ant14 750 . . . . . . . . . 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 6252 . . . . . . . . . . . . . . . 16 (curry 𝑀:𝐼⟶((Base‘𝑅) ↑𝑚 𝐼) → curry 𝑀 Fn 𝐼)
52 fnima 6217 . . . . . . . . . . . . . . . 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 469 . . . . . . . . . . . . . 14 (((𝑅 ∈ DivRing ∧ 𝐼 ∈ Fin) ∧ curry 𝑀:𝐼⟶((Base‘𝑅) ↑𝑚 𝐼)) → (curry 𝑀𝐼) = ran curry 𝑀)
5554fveq2d 6408 . . . . . . . . . . . . 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 468 . . . . . . . . . . . 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 782 . . . . . . . . . . . . . 14 ((((𝑅 ∈ DivRing ∧ 𝐼 ∈ Fin) ∧ curry 𝑀:𝐼⟶((Base‘𝑅) ↑𝑚 𝐼)) ∧ curry 𝑀 LIndF (𝑅 freeLMod 𝐼)) → 𝑅 ∈ DivRing)
58 simpllr 784 . . . . . . . . . . . . . 14 ((((𝑅 ∈ DivRing ∧ 𝐼 ∈ Fin) ∧ curry 𝑀:𝐼⟶((Base‘𝑅) ↑𝑚 𝐼)) ∧ curry 𝑀 LIndF (𝑅 freeLMod 𝐼)) → 𝐼 ∈ Fin)
5945frlmlmod 20300 . . . . . . . . . . . . . . . . 17 ((𝑅 ∈ Ring ∧ 𝐼 ∈ Fin) → (𝑅 freeLMod 𝐼) ∈ LMod)
6043, 59sylan 571 . . . . . . . . . . . . . . . 16 ((𝑅 ∈ DivRing ∧ 𝐼 ∈ Fin) → (𝑅 freeLMod 𝐼) ∈ LMod)
6160adantr 468 . . . . . . . . . . . . . . 15 (((𝑅 ∈ DivRing ∧ 𝐼 ∈ Fin) ∧ curry 𝑀:𝐼⟶((Base‘𝑅) ↑𝑚 𝐼)) → (𝑅 freeLMod 𝐼) ∈ LMod)
62 lindfrn 20367 . . . . . . . . . . . . . . 15 (((𝑅 freeLMod 𝐼) ∈ LMod ∧ curry 𝑀 LIndF (𝑅 freeLMod 𝐼)) → ran curry 𝑀 ∈ (LIndS‘(𝑅 freeLMod 𝐼)))
6361, 62sylan 571 . . . . . . . . . . . . . 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 20304 . . . . . . . . . . . . . . . . . . . . . 22 ((𝑅 ∈ DivRing ∧ 𝐼 ∈ Fin) → 𝑅 = (Scalar‘(𝑅 freeLMod 𝐼)))
65 drngnzr 19467 . . . . . . . . . . . . . . . . . . . . . . 23 (𝑅 ∈ DivRing → 𝑅 ∈ NzRing)
6665adantr 468 . . . . . . . . . . . . . . . . . . . . . 22 ((𝑅 ∈ DivRing ∧ 𝐼 ∈ Fin) → 𝑅 ∈ NzRing)
6764, 66eqeltrrd 2886 . . . . . . . . . . . . . . . . . . . . 21 ((𝑅 ∈ DivRing ∧ 𝐼 ∈ Fin) → (Scalar‘(𝑅 freeLMod 𝐼)) ∈ NzRing)
6860, 67jca 503 . . . . . . . . . . . . . . . . . . . 20 ((𝑅 ∈ DivRing ∧ 𝐼 ∈ Fin) → ((𝑅 freeLMod 𝐼) ∈ LMod ∧ (Scalar‘(𝑅 freeLMod 𝐼)) ∈ NzRing))
69 eqid 2806 . . . . . . . . . . . . . . . . . . . . . 22 (Scalar‘(𝑅 freeLMod 𝐼)) = (Scalar‘(𝑅 freeLMod 𝐼))
7046, 69lindff1 20366 . . . . . . . . . . . . . . . . . . . . 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 1140 . . . . . . . . . . . . . . . . . . . 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 571 . . . . . . . . . . . . . . . . . . 19 (((𝑅 ∈ DivRing ∧ 𝐼 ∈ Fin) ∧ curry 𝑀 LIndF (𝑅 freeLMod 𝐼)) → curry 𝑀:dom curry 𝑀1-1→(Base‘(𝑅 freeLMod 𝐼)))
73 fdm 6260 . . . . . . . . . . . . . . . . . . 19 (curry 𝑀:𝐼⟶((Base‘𝑅) ↑𝑚 𝐼) → dom curry 𝑀 = 𝐼)
74 f1eq2 6308 . . . . . . . . . . . . . . . . . . . 20 (dom curry 𝑀 = 𝐼 → (curry 𝑀:dom curry 𝑀1-1→(Base‘(𝑅 freeLMod 𝐼)) ↔ curry 𝑀:𝐼1-1→(Base‘(𝑅 freeLMod 𝐼))))
7574biimpac 466 . . . . . . . . . . . . . . . . . . 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 585 . . . . . . . . . . . . . . . . . 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 634 . . . . . . . . . . . . . . . . 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 6360 . . . . . . . . . . . . . . . . 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 8207 . . . . . . . . . . . . . . . 16 ((𝐼 ∈ Fin ∧ curry 𝑀:𝐼1-1-onto→ran curry 𝑀) → 𝐼 ≈ ran curry 𝑀)
8158, 79, 80syl2anc 575 . . . . . . . . . . . . . . 15 ((((𝑅 ∈ DivRing ∧ 𝐼 ∈ Fin) ∧ curry 𝑀:𝐼⟶((Base‘𝑅) ↑𝑚 𝐼)) ∧ curry 𝑀 LIndF (𝑅 freeLMod 𝐼)) → 𝐼 ≈ ran curry 𝑀)
8281ensymd 8239 . . . . . . . . . . . . . 14 ((((𝑅 ∈ DivRing ∧ 𝐼 ∈ Fin) ∧ curry 𝑀:𝐼⟶((Base‘𝑅) ↑𝑚 𝐼)) ∧ curry 𝑀 LIndF (𝑅 freeLMod 𝐼)) → ran curry 𝑀𝐼)
83 lindsenlbs 33715 . . . . . . . . . . . . . 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 1485 . . . . . . . . . . . . 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 2806 . . . . . . . . . . . . . 14 (LBasis‘(𝑅 freeLMod 𝐼)) = (LBasis‘(𝑅 freeLMod 𝐼))
86 eqid 2806 . . . . . . . . . . . . . 14 (LSpan‘(𝑅 freeLMod 𝐼)) = (LSpan‘(𝑅 freeLMod 𝐼))
8746, 85, 86lbssp 19282 . . . . . . . . . . . . 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 2840 . . . . . . . . . . 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 468 . . . . . . . . . 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 2888 . . . . . . . . 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 2806 . . . . . . . . . . . . 13 (Base‘(Scalar‘(𝑅 freeLMod 𝐼))) = (Base‘(Scalar‘(𝑅 freeLMod 𝐼)))
93 eqid 2806 . . . . . . . . . . . . 13 (0g‘(Scalar‘(𝑅 freeLMod 𝐼))) = (0g‘(Scalar‘(𝑅 freeLMod 𝐼)))
94 eqid 2806 . . . . . . . . . . . . 13 ( ·𝑠 ‘(𝑅 freeLMod 𝐼)) = ( ·𝑠 ‘(𝑅 freeLMod 𝐼))
9545, 14frlmfibas 20312 . . . . . . . . . . . . . . 15 ((𝑅 ∈ Ring ∧ 𝐼 ∈ Fin) → ((Base‘𝑅) ↑𝑚 𝐼) = (Base‘(𝑅 freeLMod 𝐼)))
9695feq3d 6239 . . . . . . . . . . . . . 14 ((𝑅 ∈ Ring ∧ 𝐼 ∈ Fin) → (curry 𝑀:𝐼⟶((Base‘𝑅) ↑𝑚 𝐼) ↔ curry 𝑀:𝐼⟶(Base‘(𝑅 freeLMod 𝐼))))
9796biimpa 464 . . . . . . . . . . . . 13 (((𝑅 ∈ Ring ∧ 𝐼 ∈ Fin) ∧ curry 𝑀:𝐼⟶((Base‘𝑅) ↑𝑚 𝐼)) → curry 𝑀:𝐼⟶(Base‘(𝑅 freeLMod 𝐼)))
9859adantr 468 . . . . . . . . . . . . 13 (((𝑅 ∈ Ring ∧ 𝐼 ∈ Fin) ∧ curry 𝑀:𝐼⟶((Base‘𝑅) ↑𝑚 𝐼)) → (𝑅 freeLMod 𝐼) ∈ LMod)
99 simplr 776 . . . . . . . . . . . . 13 (((𝑅 ∈ Ring ∧ 𝐼 ∈ Fin) ∧ curry 𝑀:𝐼⟶((Base‘𝑅) ↑𝑚 𝐼)) → 𝐼 ∈ Fin)
10086, 46, 92, 69, 93, 94, 97, 98, 99elfilspd 20349 . . . . . . . . . . . 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 20304 . . . . . . . . . . . . . . . 16 ((𝑅 ∈ Ring ∧ 𝐼 ∈ Fin) → 𝑅 = (Scalar‘(𝑅 freeLMod 𝐼)))
102101fveq2d 6408 . . . . . . . . . . . . . . 15 ((𝑅 ∈ Ring ∧ 𝐼 ∈ Fin) → (Base‘𝑅) = (Base‘(Scalar‘(𝑅 freeLMod 𝐼))))
103102oveq1d 6885 . . . . . . . . . . . . . 14 ((𝑅 ∈ Ring ∧ 𝐼 ∈ Fin) → ((Base‘𝑅) ↑𝑚 𝐼) = ((Base‘(Scalar‘(𝑅 freeLMod 𝐼))) ↑𝑚 𝐼))
104103adantr 468 . . . . . . . . . . . . 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 8110 . . . . . . . . . . . . . . 15 (𝑛 ∈ ((Base‘𝑅) ↑𝑚 𝐼) → 𝑛:𝐼⟶(Base‘𝑅))
106 ffn 6252 . . . . . . . . . . . . . . . . . . . 20 (𝑛:𝐼⟶(Base‘𝑅) → 𝑛 Fn 𝐼)
107106adantl 469 . . . . . . . . . . . . . . . . . . 19 ((((𝑅 ∈ Ring ∧ 𝐼 ∈ Fin) ∧ curry 𝑀:𝐼⟶((Base‘𝑅) ↑𝑚 𝐼)) ∧ 𝑛:𝐼⟶(Base‘𝑅)) → 𝑛 Fn 𝐼)
10851ad2antlr 709 . . . . . . . . . . . . . . . . . . 19 ((((𝑅 ∈ Ring ∧ 𝐼 ∈ Fin) ∧ curry 𝑀:𝐼⟶((Base‘𝑅) ↑𝑚 𝐼)) ∧ 𝑛:𝐼⟶(Base‘𝑅)) → curry 𝑀 Fn 𝐼)
109 simpllr 784 . . . . . . . . . . . . . . . . . . 19 ((((𝑅 ∈ Ring ∧ 𝐼 ∈ Fin) ∧ curry 𝑀:𝐼⟶((Base‘𝑅) ↑𝑚 𝐼)) ∧ 𝑛:𝐼⟶(Base‘𝑅)) → 𝐼 ∈ Fin)
110 inidm 4019 . . . . . . . . . . . . . . . . . . 19 (𝐼𝐼) = 𝐼
111 eqidd 2807 . . . . . . . . . . . . . . . . . . 19 (((((𝑅 ∈ Ring ∧ 𝐼 ∈ Fin) ∧ curry 𝑀:𝐼⟶((Base‘𝑅) ↑𝑚 𝐼)) ∧ 𝑛:𝐼⟶(Base‘𝑅)) ∧ 𝑘𝐼) → (𝑛𝑘) = (𝑛𝑘))
112 eqidd 2807 . . . . . . . . . . . . . . . . . . 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 7130 . . . . . . . . . . . . . . . . . 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 794 . . . . . . . . . . . . . . . . . . . . 21 (((((𝑅 ∈ Ring ∧ 𝐼 ∈ Fin) ∧ curry 𝑀:𝐼⟶((Base‘𝑅) ↑𝑚 𝐼)) ∧ 𝑛:𝐼⟶(Base‘𝑅)) ∧ 𝑘𝐼) → 𝐼 ∈ Fin)
115 ffvelrn 6575 . . . . . . . . . . . . . . . . . . . . . 22 ((𝑛:𝐼⟶(Base‘𝑅) ∧ 𝑘𝐼) → (𝑛𝑘) ∈ (Base‘𝑅))
116115adantll 696 . . . . . . . . . . . . . . . . . . . . 21 (((((𝑅 ∈ Ring ∧ 𝐼 ∈ Fin) ∧ curry 𝑀:𝐼⟶((Base‘𝑅) ↑𝑚 𝐼)) ∧ 𝑛:𝐼⟶(Base‘𝑅)) ∧ 𝑘𝐼) → (𝑛𝑘) ∈ (Base‘𝑅))
117 ffvelrn 6575 . . . . . . . . . . . . . . . . . . . . . . 23 ((curry 𝑀:𝐼⟶((Base‘𝑅) ↑𝑚 𝐼) ∧ 𝑘𝐼) → (curry 𝑀𝑘) ∈ ((Base‘𝑅) ↑𝑚 𝐼))
118117ad4ant24 754 . . . . . . . . . . . . . . . . . . . . . 22 (((((𝑅 ∈ Ring ∧ 𝐼 ∈ Fin) ∧ curry 𝑀:𝐼⟶((Base‘𝑅) ↑𝑚 𝐼)) ∧ 𝑛:𝐼⟶(Base‘𝑅)) ∧ 𝑘𝐼) → (curry 𝑀𝑘) ∈ ((Base‘𝑅) ↑𝑚 𝐼))
11995ad3antrrr 712 . . . . . . . . . . . . . . . . . . . . . 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 2887 . . . . . . . . . . . . . . . . . . . . 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 2806 . . . . . . . . . . . . . . . . . . . . 21 (.r𝑅) = (.r𝑅)
12245, 46, 14, 114, 116, 120, 94, 121frlmvscafval 20316 . . . . . . . . . . . . . . . . . . . 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 6417 . . . . . . . . . . . . . . . . . . . . . 22 (𝑛𝑘) ∈ V
124 fnconstg 6304 . . . . . . . . . . . . . . . . . . . . . 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 8111 . . . . . . . . . . . . . . . . . . . . . . 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 754 . . . . . . . . . . . . . . . . . . . . 21 (((((𝑅 ∈ Ring ∧ 𝐼 ∈ Fin) ∧ curry 𝑀:𝐼⟶((Base‘𝑅) ↑𝑚 𝐼)) ∧ 𝑛:𝐼⟶(Base‘𝑅)) ∧ 𝑘𝐼) → (curry 𝑀𝑘) Fn 𝐼)
129123fvconst2 6690 . . . . . . . . . . . . . . . . . . . . . 22 (𝑗𝐼 → ((𝐼 × {(𝑛𝑘)})‘𝑗) = (𝑛𝑘))
130129adantl 469 . . . . . . . . . . . . . . . . . . . . 21 ((((((𝑅 ∈ Ring ∧ 𝐼 ∈ Fin) ∧ curry 𝑀:𝐼⟶((Base‘𝑅) ↑𝑚 𝐼)) ∧ 𝑛:𝐼⟶(Base‘𝑅)) ∧ 𝑘𝐼) ∧ 𝑗𝐼) → ((𝐼 × {(𝑛𝑘)})‘𝑗) = (𝑛𝑘))
131 eqidd 2807 . . . . . . . . . . . . . . . . . . . . 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 7130 . . . . . . . . . . . . . . . . . . . 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 2840 . . . . . . . . . . . . . . . . . . 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 4938 . . . . . . . . . . . . . . . . . 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 2840 . . . . . . . . . . . . . . . . 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 6886 . . . . . . . . . . . . . . . 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 2806 . . . . . . . . . . . . . . . . 17 (0g‘(𝑅 freeLMod 𝐼)) = (0g‘(𝑅 freeLMod 𝐼))
138 simplll 782 . . . . . . . . . . . . . . . . 17 ((((𝑅 ∈ Ring ∧ 𝐼 ∈ Fin) ∧ curry 𝑀:𝐼⟶((Base‘𝑅) ↑𝑚 𝐼)) ∧ 𝑛:𝐼⟶(Base‘𝑅)) → 𝑅 ∈ Ring)
139 simp-5l 796 . . . . . . . . . . . . . . . . . . . 20 ((((((𝑅 ∈ Ring ∧ 𝐼 ∈ Fin) ∧ curry 𝑀:𝐼⟶((Base‘𝑅) ↑𝑚 𝐼)) ∧ 𝑛:𝐼⟶(Base‘𝑅)) ∧ 𝑘𝐼) ∧ 𝑗𝐼) → 𝑅 ∈ Ring)
140115ad4ant23 752 . . . . . . . . . . . . . . . . . . . 20 ((((((𝑅 ∈ Ring ∧ 𝐼 ∈ Fin) ∧ curry 𝑀:𝐼⟶((Base‘𝑅) ↑𝑚 𝐼)) ∧ 𝑛:𝐼⟶(Base‘𝑅)) ∧ 𝑘𝐼) ∧ 𝑗𝐼) → (𝑛𝑘) ∈ (Base‘𝑅))
141 simplr 776 . . . . . . . . . . . . . . . . . . . . 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 8110 . . . . . . . . . . . . . . . . . . . . . . 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 6577 . . . . . . . . . . . . . . . . . . . . 21 (((curry 𝑀:𝐼⟶((Base‘𝑅) ↑𝑚 𝐼) ∧ 𝑘𝐼) ∧ 𝑗𝐼) → ((curry 𝑀𝑘)‘𝑗) ∈ (Base‘𝑅))
145141, 144sylanl1 662 . . . . . . . . . . . . . . . . . . . 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 18759 . . . . . . . . . . . . . . . . . . . 20 ((𝑅 ∈ Ring ∧ (𝑛𝑘) ∈ (Base‘𝑅) ∧ ((curry 𝑀𝑘)‘𝑗) ∈ (Base‘𝑅)) → ((𝑛𝑘)(.r𝑅)((curry 𝑀𝑘)‘𝑗)) ∈ (Base‘𝑅))
147139, 140, 145, 146syl3anc 1483 . . . . . . . . . . . . . . . . . . 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 6603 . . . . . . . . . . . . . . . . . 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 8101 . . . . . . . . . . . . . . . . . . . . . 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 673 . . . . . . . . . . . . . . . . . . . . 21 (𝐼 ∈ Fin → ((𝑗𝐼 ↦ ((𝑛𝑘)(.r𝑅)((curry 𝑀𝑘)‘𝑗))) ∈ ((Base‘𝑅) ↑𝑚 𝐼) ↔ (𝑗𝐼 ↦ ((𝑛𝑘)(.r𝑅)((curry 𝑀𝑘)‘𝑗))):𝐼⟶(Base‘𝑅)))
151150adantl 469 . . . . . . . . . . . . . . . . . . . 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 2871 . . . . . . . . . . . . . . . . . . . 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 272 . . . . . . . . . . . . . . . . . . 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 712 . . . . . . . . . . . . . . . . . 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 223 . . . . . . . . . . . . . . . . 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 6705 . . . . . . . . . . . . . . . . . . . . 21 (𝐼 ∈ Fin → (𝑗𝐼 ↦ ((𝑛𝑘)(.r𝑅)((curry 𝑀𝑘)‘𝑗))) ∈ V)
157156ralrimivw 3155 . . . . . . . . . . . . . . . . . . . 20 (𝐼 ∈ Fin → ∀𝑘𝐼 (𝑗𝐼 ↦ ((𝑛𝑘)(.r𝑅)((curry 𝑀𝑘)‘𝑗))) ∈ V)
158 eqid 2806 . . . . . . . . . . . . . . . . . . . . 21 (𝑘𝐼 ↦ (𝑗𝐼 ↦ ((𝑛𝑘)(.r𝑅)((curry 𝑀𝑘)‘𝑗)))) = (𝑘𝐼 ↦ (𝑗𝐼 ↦ ((𝑛𝑘)(.r𝑅)((curry 𝑀𝑘)‘𝑗))))
159158fnmpt 6227 . . . . . . . . . . . . . . . . . . . 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 6419 . . . . . . . . . . . . . . . . . . 19 (𝐼 ∈ Fin → (0g‘(𝑅 freeLMod 𝐼)) ∈ V)
163160, 161, 162fndmfifsupp 8523 . . . . . . . . . . . . . . . . . 18 (𝐼 ∈ Fin → (𝑘𝐼 ↦ (𝑗𝐼 ↦ ((𝑛𝑘)(.r𝑅)((curry 𝑀𝑘)‘𝑗)))) finSupp (0g‘(𝑅 freeLMod 𝐼)))
164163ad3antlr 713 . . . . . . . . . . . . . . . . 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 20318 . . . . . . . . . . . . . . . 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 2841 . . . . . . . . . . . . . . 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 582 . . . . . . . . . . . . . 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 2816 . . . . . . . . . . . . 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 3344 . . . . . . . . . . . 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 273 . . . . . . . . . . 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 662 . . . . . . . . . 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 708 . . . . . . . . 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 223 . . . . . . . 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 3154 . . . . . . 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 571 . . . . . 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 248 . . . . . . . . 9 ((𝑅 ∈ DivRing ∧ 𝑀 ∈ (Base‘(𝐼 Mat 𝑅))) → 𝑀 ∈ ((Base‘𝑅) ↑𝑚 (𝐼 × 𝐼)))
177 elmapfn 8111 . . . . . . . . 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 469 . . . . . . . 8 ((𝑅 ∈ DivRing ∧ 𝑀 ∈ (Base‘(𝐼 Mat 𝑅))) → 𝐼 ∈ Fin)
180 an32 628 . . . . . . . . . . . . . . . . . . 19 (((𝑀 Fn (𝐼 × 𝐼) ∧ 𝑗𝐼) ∧ 𝑘𝐼) ↔ ((𝑀 Fn (𝐼 × 𝐼) ∧ 𝑘𝐼) ∧ 𝑗𝐼))
181 df-3an 1102 . . . . . . . . . . . . . . . . . . 19 ((𝑀 Fn (𝐼 × 𝐼) ∧ 𝑘𝐼𝑗𝐼) ↔ ((𝑀 Fn (𝐼 × 𝐼) ∧ 𝑘𝐼) ∧ 𝑗𝐼))
182180, 181bitr4i 269 . . . . . . . . . . . . . . . . . 18 (((𝑀 Fn (𝐼 × 𝐼) ∧ 𝑗𝐼) ∧ 𝑘𝐼) ↔ (𝑀 Fn (𝐼 × 𝐼) ∧ 𝑘𝐼𝑗𝐼))
183 curfv 33700 . . . . . . . . . . . . . . . . . 18 (((𝑀 Fn (𝐼 × 𝐼) ∧ 𝑘𝐼𝑗𝐼) ∧ 𝐼 ∈ Fin) → ((curry 𝑀𝑘)‘𝑗) = (𝑘𝑀𝑗))
184182, 183sylanb 572 . . . . . . . . . . . . . . . . 17 ((((𝑀 Fn (𝐼 × 𝐼) ∧ 𝑗𝐼) ∧ 𝑘𝐼) ∧ 𝐼 ∈ Fin) → ((curry 𝑀𝑘)‘𝑗) = (𝑘𝑀𝑗))
185184an32s 634 . . . . . . . . . . . . . . . 16 ((((𝑀 Fn (𝐼 × 𝐼) ∧ 𝑗𝐼) ∧ 𝐼 ∈ Fin) ∧ 𝑘𝐼) → ((curry 𝑀𝑘)‘𝑗) = (𝑘𝑀𝑗))
186185oveq2d 6886 . . . . . . . . . . . . . . 15 ((((𝑀 Fn (𝐼 × 𝐼) ∧ 𝑗𝐼) ∧ 𝐼 ∈ Fin) ∧ 𝑘𝐼) → ((𝑛𝑘)(.r𝑅)((curry 𝑀𝑘)‘𝑗)) = ((𝑛𝑘)(.r𝑅)(𝑘𝑀𝑗)))
187186mpteq2dva 4938 . . . . . . . . . . . . . 14 (((𝑀 Fn (𝐼 × 𝐼) ∧ 𝑗𝐼) ∧ 𝐼 ∈ Fin) → (𝑘𝐼 ↦ ((𝑛𝑘)(.r𝑅)((curry 𝑀𝑘)‘𝑗))) = (𝑘𝐼 ↦ ((𝑛𝑘)(.r𝑅)(𝑘𝑀𝑗))))
188187an32s 634 . . . . . . . . . . . . 13 (((𝑀 Fn (𝐼 × 𝐼) ∧ 𝐼 ∈ Fin) ∧ 𝑗𝐼) → (𝑘𝐼 ↦ ((𝑛𝑘)(.r𝑅)((curry 𝑀𝑘)‘𝑗))) = (𝑘𝐼 ↦ ((𝑛𝑘)(.r𝑅)(𝑘𝑀𝑗))))
189188oveq2d 6886 . . . . . . . . . . . 12 (((𝑀 Fn (𝐼 × 𝐼) ∧ 𝐼 ∈ Fin) ∧ 𝑗𝐼) → (𝑅 Σg (𝑘𝐼 ↦ ((𝑛𝑘)(.r𝑅)((curry 𝑀𝑘)‘𝑗)))) = (𝑅 Σg (𝑘𝐼 ↦ ((𝑛𝑘)(.r𝑅)(𝑘𝑀𝑗)))))
190189mpteq2dva 4938 . . . . . . . . . . 11 ((𝑀 Fn (𝐼 × 𝐼) ∧ 𝐼 ∈ Fin) → (𝑗𝐼 ↦ (𝑅 Σg (𝑘𝐼 ↦ ((𝑛𝑘)(.r𝑅)((curry 𝑀𝑘)‘𝑗))))) = (𝑗𝐼 ↦ (𝑅 Σg (𝑘𝐼 ↦ ((𝑛𝑘)(.r𝑅)(𝑘𝑀𝑗))))))
191190eqeq2d 2816 . . . . . . . . . 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 3240 . . . . . . . . 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 3174 . . . . . . . 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 575 . . . . . . 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 708 . . . . . 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 223 . . . . 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 662 . . . 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 6403 . . . . . . . . . . 11 (𝑛 = (𝑓𝑖) → (𝑛𝑘) = ((𝑓𝑖)‘𝑘))
199 vex 3394 . . . . . . . . . . . 12 𝑖 ∈ V
200 vex 3394 . . . . . . . . . . . 12 𝑘 ∈ V
201 uncov 33701 . . . . . . . . . . . 12 ((𝑖 ∈ V ∧ 𝑘 ∈ V) → (𝑖uncurry 𝑓𝑘) = ((𝑓𝑖)‘𝑘))
202199, 200, 201mp2an 675 . . . . . . . . . . 11 (𝑖uncurry 𝑓𝑘) = ((𝑓𝑖)‘𝑘)
203198, 202syl6eqr 2858 . . . . . . . . . 10 (𝑛 = (𝑓𝑖) → (𝑛𝑘) = (𝑖uncurry 𝑓𝑘))
204203oveq1d 6885 . . . . . . . . 9 (𝑛 = (𝑓𝑖) → ((𝑛𝑘)(.r𝑅)(𝑘𝑀𝑗)) = ((𝑖uncurry 𝑓𝑘)(.r𝑅)(𝑘𝑀𝑗)))
205204mpteq2dv 4939 . . . . . . . 8 (𝑛 = (𝑓𝑖) → (𝑘𝐼 ↦ ((𝑛𝑘)(.r𝑅)(𝑘𝑀𝑗))) = (𝑘𝐼 ↦ ((𝑖uncurry 𝑓𝑘)(.r𝑅)(𝑘𝑀𝑗))))
206205oveq2d 6886 . . . . . . 7 (𝑛 = (𝑓𝑖) → (𝑅 Σg (𝑘𝐼 ↦ ((𝑛𝑘)(.r𝑅)(𝑘𝑀𝑗)))) = (𝑅 Σg (𝑘𝐼 ↦ ((𝑖uncurry 𝑓𝑘)(.r𝑅)(𝑘𝑀𝑗)))))
207206mpteq2dv 4939 . . . . . 6 (𝑛 = (𝑓𝑖) → (𝑗𝐼 ↦ (𝑅 Σg (𝑘𝐼 ↦ ((𝑛𝑘)(.r𝑅)(𝑘𝑀𝑗))))) = (𝑗𝐼 ↦ (𝑅 Σg (𝑘𝐼 ↦ ((𝑖uncurry 𝑓𝑘)(.r𝑅)(𝑘𝑀𝑗))))))
208207eqeq2d 2816 . . . . 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)))))))
209208ac6sfi 8439 . . . 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)))))))
2105, 197, 209syl2anc 575 . . 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)))))))
211 uncf 33699 . . . . . . 7 (𝑓:𝐼⟶((Base‘𝑅) ↑𝑚 𝐼) → uncurry 𝑓:(𝐼 × 𝐼)⟶(Base‘𝑅))
21213, 14frlmfibas 20312 . . . . . . . . . . . . . . . 16 ((𝑅 ∈ Field ∧ (𝐼 × 𝐼) ∈ Fin) → ((Base‘𝑅) ↑𝑚 (𝐼 × 𝐼)) = (Base‘(𝑅 freeLMod (𝐼 × 𝐼))))
21312, 212sylan2 582 . . . . . . . . . . . . . . 15 ((𝑅 ∈ Field ∧ 𝐼 ∈ Fin) → ((Base‘𝑅) ↑𝑚 (𝐼 × 𝐼)) = (Base‘(𝑅 freeLMod (𝐼 × 𝐼))))
2141, 13matbas 20426 . . . . . . . . . . . . . . . 16 ((𝐼 ∈ Fin ∧ 𝑅 ∈ Field) → (Base‘(𝑅 freeLMod (𝐼 × 𝐼))) = (Base‘(𝐼 Mat 𝑅)))
215214ancoms 448 . . . . . . . . . . . . . . 15 ((𝑅 ∈ Field ∧ 𝐼 ∈ Fin) → (Base‘(𝑅 freeLMod (𝐼 × 𝐼))) = (Base‘(𝐼 Mat 𝑅)))
216213, 215eqtrd 2840 . . . . . . . . . . . . . 14 ((𝑅 ∈ Field ∧ 𝐼 ∈ Fin) → ((Base‘𝑅) ↑𝑚 (𝐼 × 𝐼)) = (Base‘(𝐼 Mat 𝑅)))
2174, 216sylan2 582 . . . . . . . . . . . . 13 ((𝑅 ∈ Field ∧ 𝑀 ∈ (Base‘(𝐼 Mat 𝑅))) → ((Base‘𝑅) ↑𝑚 (𝐼 × 𝐼)) = (Base‘(𝐼 Mat 𝑅)))
218217eleq2d 2871 . . . . . . . . . . . 12 ((𝑅 ∈ Field ∧ 𝑀 ∈ (Base‘(𝐼 Mat 𝑅))) → (uncurry 𝑓 ∈ ((Base‘𝑅) ↑𝑚 (𝐼 × 𝐼)) ↔ uncurry 𝑓 ∈ (Base‘(𝐼 Mat 𝑅))))
219 elmapg 8101 . . . . . . . . . . . . . 14 (((Base‘𝑅) ∈ V ∧ (𝐼 × 𝐼) ∈ Fin) → (uncurry 𝑓 ∈ ((Base‘𝑅) ↑𝑚 (𝐼 × 𝐼)) ↔ uncurry 𝑓:(𝐼 × 𝐼)⟶(Base‘𝑅)))
22022, 23, 219sylancr 577 . . . . . . . . . . . . 13 (𝑀 ∈ (Base‘(𝐼 Mat 𝑅)) → (uncurry 𝑓 ∈ ((Base‘𝑅) ↑𝑚 (𝐼 × 𝐼)) ↔ uncurry 𝑓:(𝐼 × 𝐼)⟶(Base‘𝑅)))
221220adantl 469 . . . . . . . . . . . 12 ((𝑅 ∈ Field ∧ 𝑀 ∈ (Base‘(𝐼 Mat 𝑅))) → (uncurry 𝑓 ∈ ((Base‘𝑅) ↑𝑚 (𝐼 × 𝐼)) ↔ uncurry 𝑓:(𝐼 × 𝐼)⟶(Base‘𝑅)))
222218, 221bitr3d 272 . . . . . . . . . . 11 ((𝑅 ∈ Field ∧ 𝑀 ∈ (Base‘(𝐼 Mat 𝑅))) → (uncurry 𝑓 ∈ (Base‘(𝐼 Mat 𝑅)) ↔ uncurry 𝑓:(𝐼 × 𝐼)⟶(Base‘𝑅)))
223222biimpar 465 . . . . . . . . . 10 (((𝑅 ∈ Field ∧ 𝑀 ∈ (Base‘(𝐼 Mat 𝑅))) ∧ uncurry 𝑓:(𝐼 × 𝐼)⟶(Base‘𝑅)) → uncurry 𝑓 ∈ (Base‘(𝐼 Mat 𝑅)))
224223adantr 468 . . . . . . . . 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)))
225 nfv 2005 . . . . . . . . . . . . . 14 𝑗(((𝑅 ∈ Field ∧ 𝑀 ∈ (Base‘(𝐼 Mat 𝑅))) ∧ uncurry 𝑓:(𝐼 × 𝐼)⟶(Base‘𝑅)) ∧ 𝑖𝐼)
226 nfmpt1 4941 . . . . . . . . . . . . . . 15 𝑗(𝑗𝐼 ↦ (𝑅 Σg (𝑘𝐼 ↦ ((𝑖uncurry 𝑓𝑘)(.r𝑅)(𝑘𝑀𝑗)))))
227226nfeq2 2964 . . . . . . . . . . . . . 14 𝑗((𝑅 unitVec 𝐼)‘𝑖) = (𝑗𝐼 ↦ (𝑅 Σg (𝑘𝐼 ↦ ((𝑖uncurry 𝑓𝑘)(.r𝑅)(𝑘𝑀𝑗)))))
228 fveq1 6403 . . . . . . . . . . . . . . . . 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))
2297, 43syl 17 . . . . . . . . . . . . . . . . . . . . 21 (𝑅 ∈ Field → 𝑅 ∈ Ring)
230229, 4anim12i 602 . . . . . . . . . . . . . . . . . . . 20 ((𝑅 ∈ Field ∧ 𝑀 ∈ (Base‘(𝐼 Mat 𝑅))) → (𝑅 ∈ Ring ∧ 𝐼 ∈ Fin))
231230adantr 468 . . . . . . . . . . . . . . . . . . 19 (((𝑅 ∈ Field ∧ 𝑀 ∈ (Base‘(𝐼 Mat 𝑅))) ∧ uncurry 𝑓:(𝐼 × 𝐼)⟶(Base‘𝑅)) → (𝑅 ∈ Ring ∧ 𝐼 ∈ Fin))
232 equcom 2114 . . . . . . . . . . . . . . . . . . . . 21 (𝑖 = 𝑗𝑗 = 𝑖)
233 ifbi 4300 . . . . . . . . . . . . . . . . . . . . 21 ((𝑖 = 𝑗𝑗 = 𝑖) → if(𝑖 = 𝑗, (1r𝑅), (0g𝑅)) = if(𝑗 = 𝑖, (1r𝑅), (0g𝑅)))
234232, 233ax-mp 5 . . . . . . . . . . . . . . . . . . . 20 if(𝑖 = 𝑗, (1r𝑅), (0g𝑅)) = if(𝑗 = 𝑖, (1r𝑅), (0g𝑅))
235 eqid 2806 . . . . . . . . . . . . . . . . . . . . 21 (1r𝑅) = (1r𝑅)
236 eqid 2806 . . . . . . . . . . . . . . . . . . . . 21 (0g𝑅) = (0g𝑅)
237 simpllr 784 . . . . . . . . . . . . . . . . . . . . 21 ((((𝑅 ∈ Ring ∧ 𝐼 ∈ Fin) ∧ 𝑖𝐼) ∧ 𝑗𝐼) → 𝐼 ∈ Fin)
238 simplll 782 . . . . . . . . . . . . . . . . . . . . 21 ((((𝑅 ∈ Ring ∧ 𝐼 ∈ Fin) ∧ 𝑖𝐼) ∧ 𝑗𝐼) → 𝑅 ∈ Ring)
239 simplr 776 . . . . . . . . . . . . . . . . . . . . 21 ((((𝑅 ∈ Ring ∧ 𝐼 ∈ Fin) ∧ 𝑖𝐼) ∧ 𝑗𝐼) → 𝑖𝐼)
240 simpr 473 . . . . . . . . . . . . . . . . . . . . 21 ((((𝑅 ∈ Ring ∧ 𝐼 ∈ Fin) ∧ 𝑖𝐼) ∧ 𝑗𝐼) → 𝑗𝐼)
241 eqid 2806 . . . . . . . . . . . . . . . . . . . . 21 (1r‘(𝐼 Mat 𝑅)) = (1r‘(𝐼 Mat 𝑅))
2421, 235, 236, 237, 238, 239, 240, 241mat1ov 20462 . . . . . . . . . . . . . . . . . . . 20 ((((𝑅 ∈ Ring ∧ 𝐼 ∈ Fin) ∧ 𝑖𝐼) ∧ 𝑗𝐼) → (𝑖(1r‘(𝐼 Mat 𝑅))𝑗) = if(𝑖 = 𝑗, (1r𝑅), (0g𝑅)))
243 df-3an 1102 . . . . . . . . . . . . . . . . . . . . 21 ((𝑅 ∈ Ring ∧ 𝐼 ∈ Fin ∧ 𝑖𝐼) ↔ ((𝑅 ∈ Ring ∧ 𝐼 ∈ Fin) ∧ 𝑖𝐼))
24444, 235, 236uvcvval 20332 . . . . . . . . . . . . . . . . . . . . 21 (((𝑅 ∈ Ring ∧ 𝐼 ∈ Fin ∧ 𝑖𝐼) ∧ 𝑗𝐼) → (((𝑅 unitVec 𝐼)‘𝑖)‘𝑗) = if(𝑗 = 𝑖, (1r𝑅), (0g𝑅)))
245243, 244sylanbr 573 . . . . . . . . . . . . . . . . . . . 20 ((((𝑅 ∈ Ring ∧ 𝐼 ∈ Fin) ∧ 𝑖𝐼) ∧ 𝑗𝐼) → (((𝑅 unitVec 𝐼)‘𝑖)‘𝑗) = if(𝑗 = 𝑖, (1r𝑅), (0g𝑅)))
246234, 242, 2453eqtr4a 2866 . . . . . . . . . . . . . . . . . . 19 ((((𝑅 ∈ Ring ∧ 𝐼 ∈ Fin) ∧ 𝑖𝐼) ∧ 𝑗𝐼) → (𝑖(1r‘(𝐼 Mat 𝑅))𝑗) = (((𝑅 unitVec 𝐼)‘𝑖)‘𝑗))
247231, 246sylanl1 662 . . . . . . . . . . . . . . . . . 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))
248 ovex 6902 . . . . . . . . . . . . . . . . . . . . 21 (𝑅 Σg (𝑘𝐼 ↦ ((𝑖uncurry 𝑓𝑘)(.r𝑅)(𝑘𝑀𝑗)))) ∈ V
249 eqid 2806 . . . . . . . . . . . . . . . . . . . . . 22 (𝑗𝐼 ↦ (𝑅 Σg (𝑘𝐼 ↦ ((𝑖uncurry 𝑓𝑘)(.r𝑅)(𝑘𝑀𝑗))))) = (𝑗𝐼 ↦ (𝑅 Σg (𝑘𝐼 ↦ ((𝑖uncurry 𝑓𝑘)(.r𝑅)(𝑘𝑀𝑗)))))
250249fvmpt2 6508 . . . . . . . . . . . . . . . . . . . . 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)))))
251248, 250mpan2 674 . . . . . . . . . . . . . . . . . . . 20 (𝑗𝐼 → ((𝑗𝐼 ↦ (𝑅 Σg (𝑘𝐼 ↦ ((𝑖uncurry 𝑓𝑘)(.r𝑅)(𝑘𝑀𝑗)))))‘𝑗) = (𝑅 Σg (𝑘𝐼 ↦ ((𝑖uncurry 𝑓𝑘)(.r𝑅)(𝑘𝑀𝑗)))))
252251adantl 469 . . . . . . . . . . . . . . . . . . 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)))))
253 eqid 2806 . . . . . . . . . . . . . . . . . . . 20 (𝑅 maMul ⟨𝐼, 𝐼, 𝐼⟩) = (𝑅 maMul ⟨𝐼, 𝐼, 𝐼⟩)
254 simp-4l 792 . . . . . . . . . . . . . . . . . . . 20 (((((𝑅 ∈ Field ∧ 𝑀 ∈ (Base‘(𝐼 Mat 𝑅))) ∧ uncurry 𝑓:(𝐼 × 𝐼)⟶(Base‘𝑅)) ∧ 𝑖𝐼) ∧ 𝑗𝐼) → 𝑅 ∈ Field)
2554ad4antlr 717 . . . . . . . . . . . . . . . . . . . 20 (((((𝑅 ∈ Field ∧ 𝑀 ∈ (Base‘(𝐼 Mat 𝑅))) ∧ uncurry 𝑓:(𝐼 × 𝐼)⟶(Base‘𝑅)) ∧ 𝑖𝐼) ∧ 𝑗𝐼) → 𝐼 ∈ Fin)
256220biimpar 465 . . . . . . . . . . . . . . . . . . . . 21 ((𝑀 ∈ (Base‘(𝐼 Mat 𝑅)) ∧ uncurry 𝑓:(𝐼 × 𝐼)⟶(Base‘𝑅)) → uncurry 𝑓 ∈ ((Base‘𝑅) ↑𝑚 (𝐼 × 𝐼)))
257256ad5ant23 764 . . . . . . . . . . . . . . . . . . . 20 (((((𝑅 ∈ Field ∧ 𝑀 ∈ (Base‘(𝐼 Mat 𝑅))) ∧ uncurry 𝑓:(𝐼 × 𝐼)⟶(Base‘𝑅)) ∧ 𝑖𝐼) ∧ 𝑗𝐼) → uncurry 𝑓 ∈ ((Base‘𝑅) ↑𝑚 (𝐼 × 𝐼)))
258 simpr 473 . . . . . . . . . . . . . . . . . . . . . 22 ((𝑅 ∈ Field ∧ 𝑀 ∈ (Base‘(𝐼 Mat 𝑅))) → 𝑀 ∈ (Base‘(𝐼 Mat 𝑅)))
259258, 217eleqtrrd 2888 . . . . . . . . . . . . . . . . . . . . 21 ((𝑅 ∈ Field ∧ 𝑀 ∈ (Base‘(𝐼 Mat 𝑅))) → 𝑀 ∈ ((Base‘𝑅) ↑𝑚 (𝐼 × 𝐼)))
260259ad3antrrr 712 . . . . . . . . . . . . . . . . . . . 20 (((((𝑅 ∈ Field ∧ 𝑀 ∈ (Base‘(𝐼 Mat 𝑅))) ∧ uncurry 𝑓:(𝐼 × 𝐼)⟶(Base‘𝑅)) ∧ 𝑖𝐼) ∧ 𝑗𝐼) → 𝑀 ∈ ((Base‘𝑅) ↑𝑚 (𝐼 × 𝐼)))
261 simplr 776 . . . . . . . . . . . . . . . . . . . 20 (((((𝑅 ∈ Field ∧ 𝑀 ∈ (Base‘(𝐼 Mat 𝑅))) ∧ uncurry 𝑓:(𝐼 × 𝐼)⟶(Base‘𝑅)) ∧ 𝑖𝐼) ∧ 𝑗𝐼) → 𝑖𝐼)
262 simpr 473 . . . . . . . . . . . . . . . . . . . 20 (((((𝑅 ∈ Field ∧ 𝑀 ∈ (Base‘(𝐼 Mat 𝑅))) ∧ uncurry 𝑓:(𝐼 × 𝐼)⟶(Base‘𝑅)) ∧ 𝑖𝐼) ∧ 𝑗𝐼) → 𝑗𝐼)
263253, 14, 121, 254, 255, 255, 255, 257, 260, 261, 262mamufv 20400 . . . . . . . . . . . . . . . . . . 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)))))
2641, 253matmulr 20451 . . . . . . . . . . . . . . . . . . . . . . . 24 ((𝐼 ∈ Fin ∧ 𝑅 ∈ Field) → (𝑅 maMul ⟨𝐼, 𝐼, 𝐼⟩) = (.r‘(𝐼 Mat 𝑅)))
265264ancoms 448 . . . . . . . . . . . . . . . . . . . . . . 23 ((𝑅 ∈ Field ∧ 𝐼 ∈ Fin) → (𝑅 maMul ⟨𝐼, 𝐼, 𝐼⟩) = (.r‘(𝐼 Mat 𝑅)))
266265oveqd 6887 . . . . . . . . . . . . . . . . . . . . . 22 ((𝑅 ∈ Field ∧ 𝐼 ∈ Fin) → (uncurry 𝑓(𝑅 maMul ⟨𝐼, 𝐼, 𝐼⟩)𝑀) = (uncurry 𝑓(.r‘(𝐼 Mat 𝑅))𝑀))
267266oveqd 6887 . . . . . . . . . . . . . . . . . . . . 21 ((𝑅 ∈ Field ∧ 𝐼 ∈ Fin) → (𝑖(uncurry 𝑓(𝑅 maMul ⟨𝐼, 𝐼, 𝐼⟩)𝑀)𝑗) = (𝑖(uncurry 𝑓(.r‘(𝐼 Mat 𝑅))𝑀)𝑗))
2684, 267sylan2 582 . . . . . . . . . . . . . . . . . . . 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))
269268ad3antrrr 712 . . . . . . . . . . . . . . . . . . 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))
270252, 263, 2693eqtr2rd 2847 . . . . . . . . . . . . . . . . . 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))
271247, 270eqeq12d 2821 . . . . . . . . . . . . . . . . 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)))
272228, 271syl5ibr 237 . . . . . . . . . . . . . . . 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)))
273272ex 399 . . . . . . . . . . . . . . 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))))
274273com23 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))))
275225, 227, 274ralrimd 3147 . . . . . . . . . . . . 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)))
276275ralimdva 3150 . . . . . . . . . . . 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)))
2771, 2, 241mat1bas 20463 . . . . . . . . . . . . . . . . 17 ((𝑅 ∈ Ring ∧ 𝐼 ∈ Fin) → (1r‘(𝐼 Mat 𝑅)) ∈ (Base‘(𝐼 Mat 𝑅)))
27813, 14frlmfibas 20312 . . . . . . . . . . . . . . . . . . 19 ((𝑅 ∈ Ring ∧ (𝐼 × 𝐼) ∈ Fin) → ((Base‘𝑅) ↑𝑚 (𝐼 × 𝐼)) = (Base‘(𝑅 freeLMod (𝐼 × 𝐼))))
27912, 278sylan2 582 . . . . . . . . . . . . . . . . . 18 ((𝑅 ∈ Ring ∧ 𝐼 ∈ Fin) → ((Base‘𝑅) ↑𝑚 (𝐼 × 𝐼)) = (Base‘(𝑅 freeLMod (𝐼 × 𝐼))))
2801, 13matbas 20426 . . . . . . . . . . . . . . . . . . 19 ((𝐼 ∈ Fin ∧ 𝑅 ∈ Ring) → (Base‘(𝑅 freeLMod (𝐼 × 𝐼))) = (Base‘(𝐼 Mat 𝑅)))
281280ancoms 448 . . . . . . . . . . . . . . . . . 18 ((𝑅 ∈ Ring ∧ 𝐼 ∈ Fin) → (Base‘(𝑅 freeLMod (𝐼 × 𝐼))) = (Base‘(𝐼 Mat 𝑅)))
282279, 281eqtrd 2840 . . . . . . . . . . . . . . . . 17 ((𝑅 ∈ Ring ∧ 𝐼 ∈ Fin) → ((Base‘𝑅) ↑𝑚 (𝐼 × 𝐼)) = (Base‘(𝐼 Mat 𝑅)))
283277, 282eleqtrrd 2888 . . . . . . . . . . . . . . . 16 ((𝑅 ∈ Ring ∧ 𝐼 ∈ Fin) → (1r‘(𝐼 Mat 𝑅)) ∈ ((Base‘𝑅) ↑𝑚 (𝐼 × 𝐼)))
284 elmapfn 8111 . . . . . . . . . . . . . . . 16 ((1r‘(𝐼 Mat 𝑅)) ∈ ((Base‘𝑅) ↑𝑚 (𝐼 × 𝐼)) → (1r‘(𝐼 Mat 𝑅)) Fn (𝐼 × 𝐼))
285283, 284syl 17 . . . . . . . . . . . . . . 15 ((𝑅 ∈ Ring ∧ 𝐼 ∈ Fin) → (1r‘(𝐼 Mat 𝑅)) Fn (𝐼 × 𝐼))
286229, 4, 285syl2an 585 . . . . . . . . . . . . . 14 ((𝑅 ∈ Field ∧ 𝑀 ∈ (Base‘(𝐼 Mat 𝑅))) → (1r‘(𝐼 Mat 𝑅)) Fn (𝐼 × 𝐼))
287286adantr 468 . . . . . . . . . . . . 13 (((𝑅 ∈ Field ∧ 𝑀 ∈ (Base‘(𝐼 Mat 𝑅))) ∧ uncurry 𝑓:(𝐼 × 𝐼)⟶(Base‘𝑅)) → (1r‘(𝐼 Mat 𝑅)) Fn (𝐼 × 𝐼))
2881matring 20456 . . . . . . . . . . . . . . . . . 18 ((𝐼 ∈ Fin ∧ 𝑅 ∈ Ring) → (𝐼 Mat 𝑅) ∈ Ring)
2894, 229, 288syl2anr 586 . . . . . . . . . . . . . . . . 17 ((𝑅 ∈ Field ∧ 𝑀 ∈ (Base‘(𝐼 Mat 𝑅))) → (𝐼 Mat 𝑅) ∈ Ring)
290289adantr 468 . . . . . . . . . . . . . . . 16 (((𝑅 ∈ Field ∧ 𝑀 ∈ (Base‘(𝐼 Mat 𝑅))) ∧ uncurry 𝑓:(𝐼 × 𝐼)⟶(Base‘𝑅)) → (𝐼 Mat 𝑅) ∈ Ring)
291 simplr 776 . . . . . . . . . . . . . . . 16 (((𝑅 ∈ Field ∧ 𝑀 ∈ (Base‘(𝐼 Mat 𝑅))) ∧ uncurry 𝑓:(𝐼 × 𝐼)⟶(Base‘𝑅)) → 𝑀 ∈ (Base‘(𝐼 Mat 𝑅)))
292 eqid 2806 . . . . . . . . . . . . . . . . 17 (.r‘(𝐼 Mat 𝑅)) = (.r‘(𝐼 Mat 𝑅))
2932, 292ringcl 18759 . . . . . . . . . . . . . . . 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)))
294290, 223, 291, 293syl3anc 1483 . . . . . . . . . . . . . . 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)))
295217adantr 468 . . . . . . . . . . . . . . 15 (((𝑅 ∈ Field ∧ 𝑀 ∈ (Base‘(𝐼 Mat 𝑅))) ∧ uncurry 𝑓:(𝐼 × 𝐼)⟶(Base‘𝑅)) → ((Base‘𝑅) ↑𝑚 (𝐼 × 𝐼)) = (Base‘(𝐼 Mat 𝑅)))
296294, 295eleqtrrd 2888 . . . . . . . . . . . . . 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)))
297 elmapfn 8111 . . . . . . . . . . . . . 14 ((uncurry 𝑓(.r‘(𝐼 Mat 𝑅))𝑀) ∈ ((Base‘𝑅) ↑𝑚 (𝐼 × 𝐼)) → (uncurry 𝑓(.r‘(𝐼 Mat 𝑅))𝑀) Fn (𝐼 × 𝐼))
298296, 297syl 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))
299 eqfnov2 6993 . . . . . . . . . . . . 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)))
300287, 298, 299syl2anc 575 . . . . . . . . . . . 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)))
301276, 300sylibrd 250 . . . . . . . . . . 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)))
302301imp 395 . . . . . . . . . 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))
303302eqcomd 2812 . . . . . . . . 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)))
304 oveq1 6877 . . . . . . . . . . 11 (𝑛 = uncurry 𝑓 → (𝑛(.r‘(𝐼 Mat 𝑅))𝑀) = (uncurry 𝑓(.r‘(𝐼 Mat 𝑅))𝑀))
305304eqeq1d 2808 . . . . . . . . . 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))))
306305rspcev 3502 . . . . . . . . 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)))
307224, 303, 306syl2anc 575 . . . . . . . 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)))
308307expl 447 . . . . . . 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))))
309211, 308sylani 593 . . . . . 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))))
310309exlimdv 2024 . . . . 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))))
311310imp 395 . . . 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)))
312311adantlr 697 . . 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)))
313210, 312syldan 581 . 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)))
3146simprbi 486 . . . 4 (𝑅 ∈ Field → 𝑅 ∈ CRing)
315 eqid 2806 . . . . . . . . . 10 (𝐼 maDet 𝑅) = (𝐼 maDet 𝑅)
316315, 1, 2, 14mdetcl 20610 . . . . . . . . 9 ((𝑅 ∈ CRing ∧ 𝑀 ∈ (Base‘(𝐼 Mat 𝑅))) → ((𝐼 maDet 𝑅)‘𝑀) ∈ (Base‘𝑅))
317315, 1, 2, 14mdetcl 20610 . . . . . . . . 9 ((𝑅 ∈ CRing ∧ 𝑛 ∈ (Base‘(𝐼 Mat 𝑅))) → ((𝐼 maDet 𝑅)‘𝑛) ∈ (Base‘𝑅))
318 eqid 2806 . . . . . . . . . 10 (∥r𝑅) = (∥r𝑅)
31914, 318, 121dvdsrmul 18846 . . . . . . . . 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)))
320316, 317, 319syl2an 585 . . . . . . . 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)))
321320anandis 660 . . . . . . 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)))
322321anassrs 455 . . . . . 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)))
323322adantrr 699 . . . . 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)))
324 fveq2 6404 . . . . . . . . 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))))
3251, 2, 315, 121, 292mdetmul 20637 . . . . . . . . . . . 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)))
3263253expa 1140 . . . . . . . . . . 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)))
327326an32s 634 . . . . . . . . . 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)))
328315, 1, 241, 235mdet1 20615 . . . . . . . . . . . 12 ((𝑅 ∈ CRing ∧ 𝐼 ∈ Fin) → ((𝐼 maDet 𝑅)‘(1r‘(𝐼 Mat 𝑅))) = (1r𝑅))
3294, 328sylan2 582 . . . . . . . . . . 11 ((𝑅 ∈ CRing ∧ 𝑀 ∈ (Base‘(𝐼 Mat 𝑅))) → ((𝐼 maDet 𝑅)‘(1r‘(𝐼 Mat 𝑅))) = (1r𝑅))
330329adantr 468 . . . . . . . . . 10 (((𝑅 ∈ CRing ∧ 𝑀 ∈ (Base‘(𝐼 Mat 𝑅))) ∧ 𝑛 ∈ (Base‘(𝐼 Mat 𝑅))) → ((𝐼 maDet 𝑅)‘(1r‘(𝐼 Mat 𝑅))) = (1r𝑅))
331327, 330eqeq12d 2821 . . . . . . . . 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)))
332324, 331syl5ib 235 . . . . . . . 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)))
333332impr 444 . . . . . . 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))
334333breq2d 4856 . . . . . 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)))
335 eqid 2806 . . . . . . . 8 (Unit‘𝑅) = (Unit‘𝑅)
336335, 235, 318crngunit 18860 . . . . . . 7 (𝑅 ∈ CRing → (((𝐼 maDet 𝑅)‘𝑀) ∈ (Unit‘𝑅) ↔ ((𝐼 maDet 𝑅)‘𝑀)(∥r𝑅)(1r𝑅)))
337336ad2antrr 708 . . . . . 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)))
338334, 337bitr4d 273 . . . . 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)))
339323, 338mpbid 223 . . . 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))
340314, 339sylanl1 662 . . 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))
341340ad4ant14 750 . 2 (((((𝑅 ∈ Field ∧ 𝑀 ∈ (Base‘(𝐼 Mat 𝑅))) ∧ 𝐼 ≠ ∅) ∧ curry 𝑀 LIndF (𝑅 freeLMod 𝐼)) ∧ (𝑛 ∈ (Base‘(𝐼 Mat 𝑅)) ∧ (𝑛(.r‘(𝐼 Mat 𝑅))𝑀) = (1r‘(𝐼 Mat 𝑅)))) → ((𝐼 maDet 𝑅)‘𝑀) ∈ (Unit‘𝑅))
342313, 341rexlimddv 3223 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 197  wa 384  w3a 1100   = wceq 1637  wex 1859  wcel 2156  wne 2978  wral 3096  wrex 3097  Vcvv 3391  cdif 3766  c0 4116  ifcif 4279  {csn 4370  cotp 4378   class class class wbr 4844  cmpt 4923   × cxp 5309  dom cdm 5311  ran crn 5312  cima 5314   Fn wfn 6092  wf 6093  1-1wf1 6094  1-1-ontowf1o 6096  cfv 6097  (class class class)co 6870  𝑓 cof 7121  curry ccur 7622  uncurry cunc 7623  𝑚 cmap 8088  cen 8185  Fincfn 8188   finSupp cfsupp 8510  Basecbs 16064  .rcmulr 16150  Scalarcsca 16152   ·𝑠 cvsca 16153  0gc0g 16301   Σg cgsu 16302  1rcur 18699  Ringcrg 18745  CRingccrg 18746  rcdsr 18836  Unitcui 18837  DivRingcdr 18947  Fieldcfield 18948  LModclmod 19063  LSpanclspn 19174  LBasisclbs 19277  NzRingcnzr 19462   freeLMod cfrlm 20297   unitVec cuvc 20328   LIndF clindf 20350  LIndSclinds 20351   maMul cmmul 20396   Mat cmat 20420   maDet cmdat 20598
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1877  ax-4 1894  ax-5 2001  ax-6 2068  ax-7 2104  ax-8 2158  ax-9 2165  ax-10 2185  ax-11 2201  ax-12 2214  ax-13 2420  ax-ext 2784  ax-rep 4964  ax-sep 4975  ax-nul 4983  ax-pow 5035  ax-pr 5096  ax-un 7175  ax-inf2 8781  ax-cnex 10273  ax-resscn 10274  ax-1cn 10275  ax-icn 10276  ax-addcl 10277  ax-addrcl 10278  ax-mulcl 10279  ax-mulrcl 10280  ax-mulcom 10281  ax-addass 10282  ax-mulass 10283  ax-distr 10284  ax-i2m1 10285  ax-1ne0 10286  ax-1rid 10287  ax-rnegex 10288  ax-rrecex 10289  ax-cnre 10290  ax-pre-lttri 10291  ax-pre-lttrn 10292  ax-pre-ltadd 10293  ax-pre-mulgt0 10294  ax-addf 10296  ax-mulf 10297
This theorem depends on definitions:  df-bi 198  df-an 385  df-or 866  df-3or 1101  df-3an 1102  df-xor 1619  df-tru 1641  df-fal 1651  df-ex 1860  df-nf 1864  df-sb 2061  df-eu 2634  df-mo 2635  df-clab 2793  df-cleq 2799  df-clel 2802  df-nfc 2937  df-ne 2979  df-nel 3082  df-ral 3101  df-rex 3102  df-reu 3103  df-rmo 3104  df-rab 3105  df-v 3393  df-sbc 3634  df-csb 3729  df-dif 3772  df-un 3774  df-in 3776  df-ss 3783  df-pss 3785  df-nul 4117  df-if 4280  df-pw 4353  df-sn 4371  df-pr 4373  df-tp 4375  df-op 4377  df-ot 4379  df-uni 4631  df-int 4670  df-iun 4714  df-iin 4715  df-br 4845  df-opab 4907  df-mpt 4924  df-tr 4947  df-id 5219  df-eprel 5224  df-po 5232  df-so 5233  df-fr 5270  df-se 5271  df-we 5272  df-xp 5317  df-rel 5318  df-cnv 5319  df-co 5320  df-dm 5321  df-rn 5322  df-res 5323  df-ima 5324  df-pred 5893  df-ord 5939  df-on 5940  df-lim 5941  df-suc 5942  df-iota 6060  df-fun 6099  df-fn 6100  df-f 6101  df-f1 6102  df-fo 6103  df-f1o 6104  df-fv 6105  df-isom 6106  df-riota 6831  df-ov 6873  df-oprab 6874  df-mpt2 6875  df-of 7123  df-om 7292  df-1st 7394  df-2nd 7395  df-supp 7526  df-tpos 7583  df-cur 7624  df-unc 7625  df-wrecs 7638  df-recs 7700  df-rdg 7738  df-1o 7792  df-2o 7793  df-oadd 7796  df-er 7975  df-map 8090  df-pm 8091  df-ixp 8142  df-en 8189  df-dom 8190  df-sdom 8191  df-fin 8192  df-fsupp 8511  df-sup 8583  df-oi 8650  df-card 9044  df-pnf 10357  df-mnf 10358  df-xr 10359  df-ltxr 10360  df-le 10361  df-sub 10549  df-neg 10550  df-div 10966  df-nn 11302  df-2 11360  df-3 11361  df-4 11362  df-5 11363  df-6 11364  df-7 11365  df-8 11366  df-9 11367  df-n0 11556  df-xnn0 11626  df-z 11640  df-dec 11756  df-uz 11901  df-rp 12043  df-fz 12546  df-fzo 12686  df-seq 13021  df-exp 13080  df-hash 13334  df-word 13506  df-lsw 13507  df-concat 13508  df-s1 13509  df-substr 13510  df-splice 13511  df-reverse 13512  df-s2 13813  df-struct 16066  df-ndx 16067  df-slot 16068  df-base 16070  df-sets 16071  df-ress 16072  df-plusg 16162  df-mulr 16163  df-starv 16164  df-sca 16165  df-vsca 16166  df-ip 16167  df-tset 16168  df-ple 16169  df-ds 16171  df-unif 16172  df-hom 16173  df-cco 16174  df-0g 16303  df-gsum 16304  df-prds 16309  df-pws 16311  df-mre 16447  df-mrc 16448  df-mri 16449  df-acs 16450  df-mgm 17443  df-sgrp 17485  df-mnd 17496  df-mhm 17536  df-submnd 17537  df-grp 17626  df-minusg 17627  df-sbg 17628  df-mulg 17742  df-subg 17789  df-ghm 17856  df-gim 17899  df-cntz 17947  df-oppg 17973  df-symg 17995  df-pmtr 18059  df-psgn 18108  df-evpm 18109  df-cmn 18392  df-abl 18393  df-mgp 18688  df-ur 18700  df-srg 18704  df-ring 18747  df-cring 18748  df-oppr 18821  df-dvdsr 18839  df-unit 18840  df-invr 18870  df-dvr 18881  df-rnghom 18915  df-drng 18949  df-field 18950  df-subrg 18978  df-lmod 19065  df-lss 19133  df-lsp 19175  df-lmhm 19225  df-lbs 19278  df-lvec 19306  df-sra 19377  df-rgmod 19378  df-nzr 19463  df-cnfld 19951  df-zring 20023  df-zrh 20056  df-dsmm 20283  df-frlm 20298  df-uvc 20329  df-lindf 20352  df-linds 20353  df-mamu 20397  df-mat 20421  df-mdet 20599
This theorem is referenced by:  matunitlindf  33718
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