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Theorem pmatcollpw3lem 20999
Description: Lemma for pmatcollpw3 21000 and pmatcollpw3fi 21001: Write a polynomial matrix (over a commutative ring) as a sum of products of variable powers and constant matrices with scalar entries. (Contributed by AV, 8-Dec-2019.)
Hypotheses
Ref Expression
pmatcollpw.p 𝑃 = (Poly1𝑅)
pmatcollpw.c 𝐶 = (𝑁 Mat 𝑃)
pmatcollpw.b 𝐵 = (Base‘𝐶)
pmatcollpw.m = ( ·𝑠𝐶)
pmatcollpw.e = (.g‘(mulGrp‘𝑃))
pmatcollpw.x 𝑋 = (var1𝑅)
pmatcollpw.t 𝑇 = (𝑁 matToPolyMat 𝑅)
pmatcollpw3.a 𝐴 = (𝑁 Mat 𝑅)
pmatcollpw3.d 𝐷 = (Base‘𝐴)
Assertion
Ref Expression
pmatcollpw3lem (((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀𝐵) ∧ (𝐼 ⊆ ℕ0𝐼 ≠ ∅)) → (𝑀 = (𝐶 Σg (𝑛𝐼 ↦ ((𝑛 𝑋) (𝑇‘(𝑀 decompPMat 𝑛))))) → ∃𝑓 ∈ (𝐷𝑚 𝐼)𝑀 = (𝐶 Σg (𝑛𝐼 ↦ ((𝑛 𝑋) (𝑇‘(𝑓𝑛)))))))
Distinct variable groups:   𝐵,𝑛   𝑛,𝑀   𝑛,𝑁   𝑃,𝑛   𝑅,𝑛   𝑛,𝑋   ,𝑛   𝐶,𝑛   𝐵,𝑓   𝐶,𝑓,𝑛   𝐷,𝑓   𝑓,𝐼,𝑛   𝑓,𝑀   𝑓,𝑁   𝑅,𝑓   𝑇,𝑓   𝑓,𝑋   ,𝑓   ,𝑓
Allowed substitution hints:   𝐴(𝑓,𝑛)   𝐷(𝑛)   𝑃(𝑓)   𝑇(𝑛)   (𝑛)

Proof of Theorem pmatcollpw3lem
Dummy variables 𝑖 𝑗 𝑘 𝑙 𝑥 𝑦 𝑚 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 dmeq 5571 . . . . . . . . 9 (𝑥 = 𝑦 → dom 𝑥 = dom 𝑦)
21dmeqd 5573 . . . . . . . 8 (𝑥 = 𝑦 → dom dom 𝑥 = dom dom 𝑦)
3 oveq 6930 . . . . . . . . . 10 (𝑥 = 𝑦 → (𝑖𝑥𝑗) = (𝑖𝑦𝑗))
43fveq2d 6452 . . . . . . . . 9 (𝑥 = 𝑦 → (coe1‘(𝑖𝑥𝑗)) = (coe1‘(𝑖𝑦𝑗)))
54fveq1d 6450 . . . . . . . 8 (𝑥 = 𝑦 → ((coe1‘(𝑖𝑥𝑗))‘𝑘) = ((coe1‘(𝑖𝑦𝑗))‘𝑘))
62, 2, 5mpt2eq123dv 6996 . . . . . . 7 (𝑥 = 𝑦 → (𝑖 ∈ dom dom 𝑥, 𝑗 ∈ dom dom 𝑥 ↦ ((coe1‘(𝑖𝑥𝑗))‘𝑘)) = (𝑖 ∈ dom dom 𝑦, 𝑗 ∈ dom dom 𝑦 ↦ ((coe1‘(𝑖𝑦𝑗))‘𝑘)))
7 fveq2 6448 . . . . . . . 8 (𝑘 = 𝑙 → ((coe1‘(𝑖𝑦𝑗))‘𝑘) = ((coe1‘(𝑖𝑦𝑗))‘𝑙))
87mpt2eq3dv 7000 . . . . . . 7 (𝑘 = 𝑙 → (𝑖 ∈ dom dom 𝑦, 𝑗 ∈ dom dom 𝑦 ↦ ((coe1‘(𝑖𝑦𝑗))‘𝑘)) = (𝑖 ∈ dom dom 𝑦, 𝑗 ∈ dom dom 𝑦 ↦ ((coe1‘(𝑖𝑦𝑗))‘𝑙)))
96, 8cbvmpt2v 7014 . . . . . 6 (𝑥𝐵, 𝑘𝐼 ↦ (𝑖 ∈ dom dom 𝑥, 𝑗 ∈ dom dom 𝑥 ↦ ((coe1‘(𝑖𝑥𝑗))‘𝑘))) = (𝑦𝐵, 𝑙𝐼 ↦ (𝑖 ∈ dom dom 𝑦, 𝑗 ∈ dom dom 𝑦 ↦ ((coe1‘(𝑖𝑦𝑗))‘𝑙)))
10 dmexg 7377 . . . . . . . . . . 11 (𝑦𝐵 → dom 𝑦 ∈ V)
1110dmexd 7379 . . . . . . . . . 10 (𝑦𝐵 → dom dom 𝑦 ∈ V)
1211, 11jca 507 . . . . . . . . 9 (𝑦𝐵 → (dom dom 𝑦 ∈ V ∧ dom dom 𝑦 ∈ V))
1312ad2antrl 718 . . . . . . . 8 ((((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀𝐵) ∧ (𝐼 ⊆ ℕ0𝐼 ≠ ∅)) ∧ (𝑦𝐵𝑙𝐼)) → (dom dom 𝑦 ∈ V ∧ dom dom 𝑦 ∈ V))
14 mpt2exga 7528 . . . . . . . 8 ((dom dom 𝑦 ∈ V ∧ dom dom 𝑦 ∈ V) → (𝑖 ∈ dom dom 𝑦, 𝑗 ∈ dom dom 𝑦 ↦ ((coe1‘(𝑖𝑦𝑗))‘𝑙)) ∈ V)
1513, 14syl 17 . . . . . . 7 ((((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀𝐵) ∧ (𝐼 ⊆ ℕ0𝐼 ≠ ∅)) ∧ (𝑦𝐵𝑙𝐼)) → (𝑖 ∈ dom dom 𝑦, 𝑗 ∈ dom dom 𝑦 ↦ ((coe1‘(𝑖𝑦𝑗))‘𝑙)) ∈ V)
1615ralrimivva 3153 . . . . . 6 (((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀𝐵) ∧ (𝐼 ⊆ ℕ0𝐼 ≠ ∅)) → ∀𝑦𝐵𝑙𝐼 (𝑖 ∈ dom dom 𝑦, 𝑗 ∈ dom dom 𝑦 ↦ ((coe1‘(𝑖𝑦𝑗))‘𝑙)) ∈ V)
17 simprr 763 . . . . . 6 (((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀𝐵) ∧ (𝐼 ⊆ ℕ0𝐼 ≠ ∅)) → 𝐼 ≠ ∅)
18 nn0ex 11653 . . . . . . . 8 0 ∈ V
1918ssex 5041 . . . . . . 7 (𝐼 ⊆ ℕ0𝐼 ∈ V)
2019ad2antrl 718 . . . . . 6 (((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀𝐵) ∧ (𝐼 ⊆ ℕ0𝐼 ≠ ∅)) → 𝐼 ∈ V)
21 simp3 1129 . . . . . . 7 ((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀𝐵) → 𝑀𝐵)
2221adantr 474 . . . . . 6 (((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀𝐵) ∧ (𝐼 ⊆ ℕ0𝐼 ≠ ∅)) → 𝑀𝐵)
239, 16, 17, 20, 22mpt2curryvald 7680 . . . . 5 (((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀𝐵) ∧ (𝐼 ⊆ ℕ0𝐼 ≠ ∅)) → (curry (𝑥𝐵, 𝑘𝐼 ↦ (𝑖 ∈ dom dom 𝑥, 𝑗 ∈ dom dom 𝑥 ↦ ((coe1‘(𝑖𝑥𝑗))‘𝑘)))‘𝑀) = (𝑙𝐼𝑀 / 𝑦(𝑖 ∈ dom dom 𝑦, 𝑗 ∈ dom dom 𝑦 ↦ ((coe1‘(𝑖𝑦𝑗))‘𝑙))))
24 fveq2 6448 . . . . . . . . 9 (𝑙 = 𝑘 → ((coe1‘(𝑖𝑦𝑗))‘𝑙) = ((coe1‘(𝑖𝑦𝑗))‘𝑘))
2524mpt2eq3dv 7000 . . . . . . . 8 (𝑙 = 𝑘 → (𝑖 ∈ dom dom 𝑦, 𝑗 ∈ dom dom 𝑦 ↦ ((coe1‘(𝑖𝑦𝑗))‘𝑙)) = (𝑖 ∈ dom dom 𝑦, 𝑗 ∈ dom dom 𝑦 ↦ ((coe1‘(𝑖𝑦𝑗))‘𝑘)))
2625csbeq2dv 4217 . . . . . . 7 (𝑙 = 𝑘𝑀 / 𝑦(𝑖 ∈ dom dom 𝑦, 𝑗 ∈ dom dom 𝑦 ↦ ((coe1‘(𝑖𝑦𝑗))‘𝑙)) = 𝑀 / 𝑦(𝑖 ∈ dom dom 𝑦, 𝑗 ∈ dom dom 𝑦 ↦ ((coe1‘(𝑖𝑦𝑗))‘𝑘)))
27 eqcom 2785 . . . . . . . . 9 (𝑥 = 𝑦𝑦 = 𝑥)
28 eqcom 2785 . . . . . . . . 9 ((𝑖 ∈ dom dom 𝑥, 𝑗 ∈ dom dom 𝑥 ↦ ((coe1‘(𝑖𝑥𝑗))‘𝑘)) = (𝑖 ∈ dom dom 𝑦, 𝑗 ∈ dom dom 𝑦 ↦ ((coe1‘(𝑖𝑦𝑗))‘𝑘)) ↔ (𝑖 ∈ dom dom 𝑦, 𝑗 ∈ dom dom 𝑦 ↦ ((coe1‘(𝑖𝑦𝑗))‘𝑘)) = (𝑖 ∈ dom dom 𝑥, 𝑗 ∈ dom dom 𝑥 ↦ ((coe1‘(𝑖𝑥𝑗))‘𝑘)))
296, 27, 283imtr3i 283 . . . . . . . 8 (𝑦 = 𝑥 → (𝑖 ∈ dom dom 𝑦, 𝑗 ∈ dom dom 𝑦 ↦ ((coe1‘(𝑖𝑦𝑗))‘𝑘)) = (𝑖 ∈ dom dom 𝑥, 𝑗 ∈ dom dom 𝑥 ↦ ((coe1‘(𝑖𝑥𝑗))‘𝑘)))
3029cbvcsbv 3757 . . . . . . 7 𝑀 / 𝑦(𝑖 ∈ dom dom 𝑦, 𝑗 ∈ dom dom 𝑦 ↦ ((coe1‘(𝑖𝑦𝑗))‘𝑘)) = 𝑀 / 𝑥(𝑖 ∈ dom dom 𝑥, 𝑗 ∈ dom dom 𝑥 ↦ ((coe1‘(𝑖𝑥𝑗))‘𝑘))
3126, 30syl6eq 2830 . . . . . 6 (𝑙 = 𝑘𝑀 / 𝑦(𝑖 ∈ dom dom 𝑦, 𝑗 ∈ dom dom 𝑦 ↦ ((coe1‘(𝑖𝑦𝑗))‘𝑙)) = 𝑀 / 𝑥(𝑖 ∈ dom dom 𝑥, 𝑗 ∈ dom dom 𝑥 ↦ ((coe1‘(𝑖𝑥𝑗))‘𝑘)))
3231cbvmptv 4987 . . . . 5 (𝑙𝐼𝑀 / 𝑦(𝑖 ∈ dom dom 𝑦, 𝑗 ∈ dom dom 𝑦 ↦ ((coe1‘(𝑖𝑦𝑗))‘𝑙))) = (𝑘𝐼𝑀 / 𝑥(𝑖 ∈ dom dom 𝑥, 𝑗 ∈ dom dom 𝑥 ↦ ((coe1‘(𝑖𝑥𝑗))‘𝑘)))
3323, 32syl6eq 2830 . . . 4 (((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀𝐵) ∧ (𝐼 ⊆ ℕ0𝐼 ≠ ∅)) → (curry (𝑥𝐵, 𝑘𝐼 ↦ (𝑖 ∈ dom dom 𝑥, 𝑗 ∈ dom dom 𝑥 ↦ ((coe1‘(𝑖𝑥𝑗))‘𝑘)))‘𝑀) = (𝑘𝐼𝑀 / 𝑥(𝑖 ∈ dom dom 𝑥, 𝑗 ∈ dom dom 𝑥 ↦ ((coe1‘(𝑖𝑥𝑗))‘𝑘))))
34 dmeq 5571 . . . . . . . . . . 11 (𝑥 = 𝑀 → dom 𝑥 = dom 𝑀)
3534dmeqd 5573 . . . . . . . . . 10 (𝑥 = 𝑀 → dom dom 𝑥 = dom dom 𝑀)
36 oveq 6930 . . . . . . . . . . . 12 (𝑥 = 𝑀 → (𝑖𝑥𝑗) = (𝑖𝑀𝑗))
3736fveq2d 6452 . . . . . . . . . . 11 (𝑥 = 𝑀 → (coe1‘(𝑖𝑥𝑗)) = (coe1‘(𝑖𝑀𝑗)))
3837fveq1d 6450 . . . . . . . . . 10 (𝑥 = 𝑀 → ((coe1‘(𝑖𝑥𝑗))‘𝑘) = ((coe1‘(𝑖𝑀𝑗))‘𝑘))
3935, 35, 38mpt2eq123dv 6996 . . . . . . . . 9 (𝑥 = 𝑀 → (𝑖 ∈ dom dom 𝑥, 𝑗 ∈ dom dom 𝑥 ↦ ((coe1‘(𝑖𝑥𝑗))‘𝑘)) = (𝑖 ∈ dom dom 𝑀, 𝑗 ∈ dom dom 𝑀 ↦ ((coe1‘(𝑖𝑀𝑗))‘𝑘)))
4039adantl 475 . . . . . . . 8 (((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀𝐵) ∧ 𝑥 = 𝑀) → (𝑖 ∈ dom dom 𝑥, 𝑗 ∈ dom dom 𝑥 ↦ ((coe1‘(𝑖𝑥𝑗))‘𝑘)) = (𝑖 ∈ dom dom 𝑀, 𝑗 ∈ dom dom 𝑀 ↦ ((coe1‘(𝑖𝑀𝑗))‘𝑘)))
4121, 40csbied 3778 . . . . . . 7 ((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀𝐵) → 𝑀 / 𝑥(𝑖 ∈ dom dom 𝑥, 𝑗 ∈ dom dom 𝑥 ↦ ((coe1‘(𝑖𝑥𝑗))‘𝑘)) = (𝑖 ∈ dom dom 𝑀, 𝑗 ∈ dom dom 𝑀 ↦ ((coe1‘(𝑖𝑀𝑗))‘𝑘)))
42 pmatcollpw.c . . . . . . . . . . . . 13 𝐶 = (𝑁 Mat 𝑃)
43 eqid 2778 . . . . . . . . . . . . 13 (Base‘𝑃) = (Base‘𝑃)
44 pmatcollpw.b . . . . . . . . . . . . 13 𝐵 = (Base‘𝐶)
4542, 43, 44matbas2i 20636 . . . . . . . . . . . 12 (𝑀𝐵𝑀 ∈ ((Base‘𝑃) ↑𝑚 (𝑁 × 𝑁)))
46 elmapi 8164 . . . . . . . . . . . 12 (𝑀 ∈ ((Base‘𝑃) ↑𝑚 (𝑁 × 𝑁)) → 𝑀:(𝑁 × 𝑁)⟶(Base‘𝑃))
47 fdm 6301 . . . . . . . . . . . . . 14 (𝑀:(𝑁 × 𝑁)⟶(Base‘𝑃) → dom 𝑀 = (𝑁 × 𝑁))
4847dmeqd 5573 . . . . . . . . . . . . 13 (𝑀:(𝑁 × 𝑁)⟶(Base‘𝑃) → dom dom 𝑀 = dom (𝑁 × 𝑁))
49 dmxpid 5592 . . . . . . . . . . . . 13 dom (𝑁 × 𝑁) = 𝑁
5048, 49syl6req 2831 . . . . . . . . . . . 12 (𝑀:(𝑁 × 𝑁)⟶(Base‘𝑃) → 𝑁 = dom dom 𝑀)
5145, 46, 503syl 18 . . . . . . . . . . 11 (𝑀𝐵𝑁 = dom dom 𝑀)
52513ad2ant3 1126 . . . . . . . . . 10 ((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀𝐵) → 𝑁 = dom dom 𝑀)
5352adantr 474 . . . . . . . . 9 (((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀𝐵) ∧ 𝑚 = 𝑀) → 𝑁 = dom dom 𝑀)
54 simpr 479 . . . . . . . . . . . 12 (((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀𝐵) ∧ 𝑚 = 𝑀) → 𝑚 = 𝑀)
5554oveqd 6941 . . . . . . . . . . 11 (((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀𝐵) ∧ 𝑚 = 𝑀) → (𝑖𝑚𝑗) = (𝑖𝑀𝑗))
5655fveq2d 6452 . . . . . . . . . 10 (((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀𝐵) ∧ 𝑚 = 𝑀) → (coe1‘(𝑖𝑚𝑗)) = (coe1‘(𝑖𝑀𝑗)))
5756fveq1d 6450 . . . . . . . . 9 (((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀𝐵) ∧ 𝑚 = 𝑀) → ((coe1‘(𝑖𝑚𝑗))‘𝑘) = ((coe1‘(𝑖𝑀𝑗))‘𝑘))
5853, 53, 57mpt2eq123dv 6996 . . . . . . . 8 (((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀𝐵) ∧ 𝑚 = 𝑀) → (𝑖𝑁, 𝑗𝑁 ↦ ((coe1‘(𝑖𝑚𝑗))‘𝑘)) = (𝑖 ∈ dom dom 𝑀, 𝑗 ∈ dom dom 𝑀 ↦ ((coe1‘(𝑖𝑀𝑗))‘𝑘)))
5921, 58csbied 3778 . . . . . . 7 ((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀𝐵) → 𝑀 / 𝑚(𝑖𝑁, 𝑗𝑁 ↦ ((coe1‘(𝑖𝑚𝑗))‘𝑘)) = (𝑖 ∈ dom dom 𝑀, 𝑗 ∈ dom dom 𝑀 ↦ ((coe1‘(𝑖𝑀𝑗))‘𝑘)))
6041, 59eqtr4d 2817 . . . . . 6 ((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀𝐵) → 𝑀 / 𝑥(𝑖 ∈ dom dom 𝑥, 𝑗 ∈ dom dom 𝑥 ↦ ((coe1‘(𝑖𝑥𝑗))‘𝑘)) = 𝑀 / 𝑚(𝑖𝑁, 𝑗𝑁 ↦ ((coe1‘(𝑖𝑚𝑗))‘𝑘)))
6160adantr 474 . . . . 5 (((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀𝐵) ∧ (𝐼 ⊆ ℕ0𝐼 ≠ ∅)) → 𝑀 / 𝑥(𝑖 ∈ dom dom 𝑥, 𝑗 ∈ dom dom 𝑥 ↦ ((coe1‘(𝑖𝑥𝑗))‘𝑘)) = 𝑀 / 𝑚(𝑖𝑁, 𝑗𝑁 ↦ ((coe1‘(𝑖𝑚𝑗))‘𝑘)))
6261mpteq2dv 4982 . . . 4 (((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀𝐵) ∧ (𝐼 ⊆ ℕ0𝐼 ≠ ∅)) → (𝑘𝐼𝑀 / 𝑥(𝑖 ∈ dom dom 𝑥, 𝑗 ∈ dom dom 𝑥 ↦ ((coe1‘(𝑖𝑥𝑗))‘𝑘))) = (𝑘𝐼𝑀 / 𝑚(𝑖𝑁, 𝑗𝑁 ↦ ((coe1‘(𝑖𝑚𝑗))‘𝑘))))
6333, 62eqtrd 2814 . . 3 (((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀𝐵) ∧ (𝐼 ⊆ ℕ0𝐼 ≠ ∅)) → (curry (𝑥𝐵, 𝑘𝐼 ↦ (𝑖 ∈ dom dom 𝑥, 𝑗 ∈ dom dom 𝑥 ↦ ((coe1‘(𝑖𝑥𝑗))‘𝑘)))‘𝑀) = (𝑘𝐼𝑀 / 𝑚(𝑖𝑁, 𝑗𝑁 ↦ ((coe1‘(𝑖𝑚𝑗))‘𝑘))))
64 oveq 6930 . . . . . . . . . . . 12 (𝑚 = 𝑀 → (𝑖𝑚𝑗) = (𝑖𝑀𝑗))
6564adantl 475 . . . . . . . . . . 11 (((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀𝐵) ∧ 𝑚 = 𝑀) → (𝑖𝑚𝑗) = (𝑖𝑀𝑗))
6665fveq2d 6452 . . . . . . . . . 10 (((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀𝐵) ∧ 𝑚 = 𝑀) → (coe1‘(𝑖𝑚𝑗)) = (coe1‘(𝑖𝑀𝑗)))
6766fveq1d 6450 . . . . . . . . 9 (((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀𝐵) ∧ 𝑚 = 𝑀) → ((coe1‘(𝑖𝑚𝑗))‘𝑘) = ((coe1‘(𝑖𝑀𝑗))‘𝑘))
6867mpt2eq3dv 7000 . . . . . . . 8 (((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀𝐵) ∧ 𝑚 = 𝑀) → (𝑖𝑁, 𝑗𝑁 ↦ ((coe1‘(𝑖𝑚𝑗))‘𝑘)) = (𝑖𝑁, 𝑗𝑁 ↦ ((coe1‘(𝑖𝑀𝑗))‘𝑘)))
6921, 68csbied 3778 . . . . . . 7 ((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀𝐵) → 𝑀 / 𝑚(𝑖𝑁, 𝑗𝑁 ↦ ((coe1‘(𝑖𝑚𝑗))‘𝑘)) = (𝑖𝑁, 𝑗𝑁 ↦ ((coe1‘(𝑖𝑀𝑗))‘𝑘)))
7069ad2antrr 716 . . . . . 6 ((((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀𝐵) ∧ (𝐼 ⊆ ℕ0𝐼 ≠ ∅)) ∧ 𝑘𝐼) → 𝑀 / 𝑚(𝑖𝑁, 𝑗𝑁 ↦ ((coe1‘(𝑖𝑚𝑗))‘𝑘)) = (𝑖𝑁, 𝑗𝑁 ↦ ((coe1‘(𝑖𝑀𝑗))‘𝑘)))
71 pmatcollpw3.a . . . . . . 7 𝐴 = (𝑁 Mat 𝑅)
72 eqid 2778 . . . . . . 7 (Base‘𝑅) = (Base‘𝑅)
73 pmatcollpw3.d . . . . . . 7 𝐷 = (Base‘𝐴)
74 simpll1 1226 . . . . . . 7 ((((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀𝐵) ∧ (𝐼 ⊆ ℕ0𝐼 ≠ ∅)) ∧ 𝑘𝐼) → 𝑁 ∈ Fin)
75 simpll2 1228 . . . . . . 7 ((((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀𝐵) ∧ (𝐼 ⊆ ℕ0𝐼 ≠ ∅)) ∧ 𝑘𝐼) → 𝑅 ∈ CRing)
76 simp2 1128 . . . . . . . . 9 (((((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀𝐵) ∧ (𝐼 ⊆ ℕ0𝐼 ≠ ∅)) ∧ 𝑘𝐼) ∧ 𝑖𝑁𝑗𝑁) → 𝑖𝑁)
77 simp3 1129 . . . . . . . . 9 (((((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀𝐵) ∧ (𝐼 ⊆ ℕ0𝐼 ≠ ∅)) ∧ 𝑘𝐼) ∧ 𝑖𝑁𝑗𝑁) → 𝑗𝑁)
7822adantr 474 . . . . . . . . . 10 ((((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀𝐵) ∧ (𝐼 ⊆ ℕ0𝐼 ≠ ∅)) ∧ 𝑘𝐼) → 𝑀𝐵)
79783ad2ant1 1124 . . . . . . . . 9 (((((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀𝐵) ∧ (𝐼 ⊆ ℕ0𝐼 ≠ ∅)) ∧ 𝑘𝐼) ∧ 𝑖𝑁𝑗𝑁) → 𝑀𝐵)
8042, 43, 44, 76, 77, 79matecld 20640 . . . . . . . 8 (((((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀𝐵) ∧ (𝐼 ⊆ ℕ0𝐼 ≠ ∅)) ∧ 𝑘𝐼) ∧ 𝑖𝑁𝑗𝑁) → (𝑖𝑀𝑗) ∈ (Base‘𝑃))
81 ssel 3815 . . . . . . . . . . 11 (𝐼 ⊆ ℕ0 → (𝑘𝐼𝑘 ∈ ℕ0))
8281ad2antrl 718 . . . . . . . . . 10 (((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀𝐵) ∧ (𝐼 ⊆ ℕ0𝐼 ≠ ∅)) → (𝑘𝐼𝑘 ∈ ℕ0))
8382imp 397 . . . . . . . . 9 ((((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀𝐵) ∧ (𝐼 ⊆ ℕ0𝐼 ≠ ∅)) ∧ 𝑘𝐼) → 𝑘 ∈ ℕ0)
84833ad2ant1 1124 . . . . . . . 8 (((((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀𝐵) ∧ (𝐼 ⊆ ℕ0𝐼 ≠ ∅)) ∧ 𝑘𝐼) ∧ 𝑖𝑁𝑗𝑁) → 𝑘 ∈ ℕ0)
85 eqid 2778 . . . . . . . . 9 (coe1‘(𝑖𝑀𝑗)) = (coe1‘(𝑖𝑀𝑗))
86 pmatcollpw.p . . . . . . . . 9 𝑃 = (Poly1𝑅)
8785, 43, 86, 72coe1fvalcl 19982 . . . . . . . 8 (((𝑖𝑀𝑗) ∈ (Base‘𝑃) ∧ 𝑘 ∈ ℕ0) → ((coe1‘(𝑖𝑀𝑗))‘𝑘) ∈ (Base‘𝑅))
8880, 84, 87syl2anc 579 . . . . . . 7 (((((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀𝐵) ∧ (𝐼 ⊆ ℕ0𝐼 ≠ ∅)) ∧ 𝑘𝐼) ∧ 𝑖𝑁𝑗𝑁) → ((coe1‘(𝑖𝑀𝑗))‘𝑘) ∈ (Base‘𝑅))
8971, 72, 73, 74, 75, 88matbas2d 20637 . . . . . 6 ((((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀𝐵) ∧ (𝐼 ⊆ ℕ0𝐼 ≠ ∅)) ∧ 𝑘𝐼) → (𝑖𝑁, 𝑗𝑁 ↦ ((coe1‘(𝑖𝑀𝑗))‘𝑘)) ∈ 𝐷)
9070, 89eqeltrd 2859 . . . . 5 ((((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀𝐵) ∧ (𝐼 ⊆ ℕ0𝐼 ≠ ∅)) ∧ 𝑘𝐼) → 𝑀 / 𝑚(𝑖𝑁, 𝑗𝑁 ↦ ((coe1‘(𝑖𝑚𝑗))‘𝑘)) ∈ 𝐷)
9190fmpttd 6651 . . . 4 (((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀𝐵) ∧ (𝐼 ⊆ ℕ0𝐼 ≠ ∅)) → (𝑘𝐼𝑀 / 𝑚(𝑖𝑁, 𝑗𝑁 ↦ ((coe1‘(𝑖𝑚𝑗))‘𝑘))):𝐼𝐷)
9273fvexi 6462 . . . . . 6 𝐷 ∈ V
9392a1i 11 . . . . 5 ((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀𝐵) → 𝐷 ∈ V)
9419adantr 474 . . . . 5 ((𝐼 ⊆ ℕ0𝐼 ≠ ∅) → 𝐼 ∈ V)
95 elmapg 8155 . . . . 5 ((𝐷 ∈ V ∧ 𝐼 ∈ V) → ((𝑘𝐼𝑀 / 𝑚(𝑖𝑁, 𝑗𝑁 ↦ ((coe1‘(𝑖𝑚𝑗))‘𝑘))) ∈ (𝐷𝑚 𝐼) ↔ (𝑘𝐼𝑀 / 𝑚(𝑖𝑁, 𝑗𝑁 ↦ ((coe1‘(𝑖𝑚𝑗))‘𝑘))):𝐼𝐷))
9693, 94, 95syl2an 589 . . . 4 (((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀𝐵) ∧ (𝐼 ⊆ ℕ0𝐼 ≠ ∅)) → ((𝑘𝐼𝑀 / 𝑚(𝑖𝑁, 𝑗𝑁 ↦ ((coe1‘(𝑖𝑚𝑗))‘𝑘))) ∈ (𝐷𝑚 𝐼) ↔ (𝑘𝐼𝑀 / 𝑚(𝑖𝑁, 𝑗𝑁 ↦ ((coe1‘(𝑖𝑚𝑗))‘𝑘))):𝐼𝐷))
9791, 96mpbird 249 . . 3 (((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀𝐵) ∧ (𝐼 ⊆ ℕ0𝐼 ≠ ∅)) → (𝑘𝐼𝑀 / 𝑚(𝑖𝑁, 𝑗𝑁 ↦ ((coe1‘(𝑖𝑚𝑗))‘𝑘))) ∈ (𝐷𝑚 𝐼))
9863, 97eqeltrd 2859 . 2 (((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀𝐵) ∧ (𝐼 ⊆ ℕ0𝐼 ≠ ∅)) → (curry (𝑥𝐵, 𝑘𝐼 ↦ (𝑖 ∈ dom dom 𝑥, 𝑗 ∈ dom dom 𝑥 ↦ ((coe1‘(𝑖𝑥𝑗))‘𝑘)))‘𝑀) ∈ (𝐷𝑚 𝐼))
99 fveq1 6447 . . . . . . . . . . 11 (𝑓 = (curry (𝑥𝐵, 𝑘𝐼 ↦ (𝑖 ∈ dom dom 𝑥, 𝑗 ∈ dom dom 𝑥 ↦ ((coe1‘(𝑖𝑥𝑗))‘𝑘)))‘𝑀) → (𝑓𝑛) = ((curry (𝑥𝐵, 𝑘𝐼 ↦ (𝑖 ∈ dom dom 𝑥, 𝑗 ∈ dom dom 𝑥 ↦ ((coe1‘(𝑖𝑥𝑗))‘𝑘)))‘𝑀)‘𝑛))
10099adantl 475 . . . . . . . . . 10 ((((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀𝐵) ∧ (𝐼 ⊆ ℕ0𝐼 ≠ ∅)) ∧ 𝑓 = (curry (𝑥𝐵, 𝑘𝐼 ↦ (𝑖 ∈ dom dom 𝑥, 𝑗 ∈ dom dom 𝑥 ↦ ((coe1‘(𝑖𝑥𝑗))‘𝑘)))‘𝑀)) → (𝑓𝑛) = ((curry (𝑥𝐵, 𝑘𝐼 ↦ (𝑖 ∈ dom dom 𝑥, 𝑗 ∈ dom dom 𝑥 ↦ ((coe1‘(𝑖𝑥𝑗))‘𝑘)))‘𝑀)‘𝑛))
101100adantr 474 . . . . . . . . 9 (((((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀𝐵) ∧ (𝐼 ⊆ ℕ0𝐼 ≠ ∅)) ∧ 𝑓 = (curry (𝑥𝐵, 𝑘𝐼 ↦ (𝑖 ∈ dom dom 𝑥, 𝑗 ∈ dom dom 𝑥 ↦ ((coe1‘(𝑖𝑥𝑗))‘𝑘)))‘𝑀)) ∧ 𝑛𝐼) → (𝑓𝑛) = ((curry (𝑥𝐵, 𝑘𝐼 ↦ (𝑖 ∈ dom dom 𝑥, 𝑗 ∈ dom dom 𝑥 ↦ ((coe1‘(𝑖𝑥𝑗))‘𝑘)))‘𝑀)‘𝑛))
102 eqid 2778 . . . . . . . . . . . 12 (𝑥𝐵, 𝑘𝐼 ↦ (𝑖 ∈ dom dom 𝑥, 𝑗 ∈ dom dom 𝑥 ↦ ((coe1‘(𝑖𝑥𝑗))‘𝑘))) = (𝑥𝐵, 𝑘𝐼 ↦ (𝑖 ∈ dom dom 𝑥, 𝑗 ∈ dom dom 𝑥 ↦ ((coe1‘(𝑖𝑥𝑗))‘𝑘)))
103 dmexg 7377 . . . . . . . . . . . . . . . . 17 (𝑥𝐵 → dom 𝑥 ∈ V)
104103dmexd 7379 . . . . . . . . . . . . . . . 16 (𝑥𝐵 → dom dom 𝑥 ∈ V)
105104, 104jca 507 . . . . . . . . . . . . . . 15 (𝑥𝐵 → (dom dom 𝑥 ∈ V ∧ dom dom 𝑥 ∈ V))
106105ad2antrl 718 . . . . . . . . . . . . . 14 (((((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀𝐵) ∧ (𝐼 ⊆ ℕ0𝐼 ≠ ∅)) ∧ 𝑛𝐼) ∧ (𝑥𝐵𝑘𝐼)) → (dom dom 𝑥 ∈ V ∧ dom dom 𝑥 ∈ V))
107 mpt2exga 7528 . . . . . . . . . . . . . 14 ((dom dom 𝑥 ∈ V ∧ dom dom 𝑥 ∈ V) → (𝑖 ∈ dom dom 𝑥, 𝑗 ∈ dom dom 𝑥 ↦ ((coe1‘(𝑖𝑥𝑗))‘𝑘)) ∈ V)
108106, 107syl 17 . . . . . . . . . . . . 13 (((((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀𝐵) ∧ (𝐼 ⊆ ℕ0𝐼 ≠ ∅)) ∧ 𝑛𝐼) ∧ (𝑥𝐵𝑘𝐼)) → (𝑖 ∈ dom dom 𝑥, 𝑗 ∈ dom dom 𝑥 ↦ ((coe1‘(𝑖𝑥𝑗))‘𝑘)) ∈ V)
109108ralrimivva 3153 . . . . . . . . . . . 12 ((((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀𝐵) ∧ (𝐼 ⊆ ℕ0𝐼 ≠ ∅)) ∧ 𝑛𝐼) → ∀𝑥𝐵𝑘𝐼 (𝑖 ∈ dom dom 𝑥, 𝑗 ∈ dom dom 𝑥 ↦ ((coe1‘(𝑖𝑥𝑗))‘𝑘)) ∈ V)
11020adantr 474 . . . . . . . . . . . 12 ((((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀𝐵) ∧ (𝐼 ⊆ ℕ0𝐼 ≠ ∅)) ∧ 𝑛𝐼) → 𝐼 ∈ V)
11122adantr 474 . . . . . . . . . . . 12 ((((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀𝐵) ∧ (𝐼 ⊆ ℕ0𝐼 ≠ ∅)) ∧ 𝑛𝐼) → 𝑀𝐵)
112 simpr 479 . . . . . . . . . . . 12 ((((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀𝐵) ∧ (𝐼 ⊆ ℕ0𝐼 ≠ ∅)) ∧ 𝑛𝐼) → 𝑛𝐼)
113102, 109, 110, 111, 112fvmpt2curryd 7681 . . . . . . . . . . 11 ((((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀𝐵) ∧ (𝐼 ⊆ ℕ0𝐼 ≠ ∅)) ∧ 𝑛𝐼) → ((curry (𝑥𝐵, 𝑘𝐼 ↦ (𝑖 ∈ dom dom 𝑥, 𝑗 ∈ dom dom 𝑥 ↦ ((coe1‘(𝑖𝑥𝑗))‘𝑘)))‘𝑀)‘𝑛) = (𝑀(𝑥𝐵, 𝑘𝐼 ↦ (𝑖 ∈ dom dom 𝑥, 𝑗 ∈ dom dom 𝑥 ↦ ((coe1‘(𝑖𝑥𝑗))‘𝑘)))𝑛))
114 df-decpmat 20979 . . . . . . . . . . . . . 14 decompPMat = (𝑥 ∈ V, 𝑘 ∈ ℕ0 ↦ (𝑖 ∈ dom dom 𝑥, 𝑗 ∈ dom dom 𝑥 ↦ ((coe1‘(𝑖𝑥𝑗))‘𝑘)))
115114reseq1i 5640 . . . . . . . . . . . . 13 ( decompPMat ↾ (𝐵 × 𝐼)) = ((𝑥 ∈ V, 𝑘 ∈ ℕ0 ↦ (𝑖 ∈ dom dom 𝑥, 𝑗 ∈ dom dom 𝑥 ↦ ((coe1‘(𝑖𝑥𝑗))‘𝑘))) ↾ (𝐵 × 𝐼))
116 ssv 3844 . . . . . . . . . . . . . . . . 17 𝐵 ⊆ V
117116a1i 11 . . . . . . . . . . . . . . . 16 ((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀𝐵) → 𝐵 ⊆ V)
118 simpl 476 . . . . . . . . . . . . . . . 16 ((𝐼 ⊆ ℕ0𝐼 ≠ ∅) → 𝐼 ⊆ ℕ0)
119117, 118anim12i 606 . . . . . . . . . . . . . . 15 (((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀𝐵) ∧ (𝐼 ⊆ ℕ0𝐼 ≠ ∅)) → (𝐵 ⊆ V ∧ 𝐼 ⊆ ℕ0))
120119adantr 474 . . . . . . . . . . . . . 14 ((((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀𝐵) ∧ (𝐼 ⊆ ℕ0𝐼 ≠ ∅)) ∧ 𝑛𝐼) → (𝐵 ⊆ V ∧ 𝐼 ⊆ ℕ0))
121 resmpt2 7037 . . . . . . . . . . . . . 14 ((𝐵 ⊆ V ∧ 𝐼 ⊆ ℕ0) → ((𝑥 ∈ V, 𝑘 ∈ ℕ0 ↦ (𝑖 ∈ dom dom 𝑥, 𝑗 ∈ dom dom 𝑥 ↦ ((coe1‘(𝑖𝑥𝑗))‘𝑘))) ↾ (𝐵 × 𝐼)) = (𝑥𝐵, 𝑘𝐼 ↦ (𝑖 ∈ dom dom 𝑥, 𝑗 ∈ dom dom 𝑥 ↦ ((coe1‘(𝑖𝑥𝑗))‘𝑘))))
122120, 121syl 17 . . . . . . . . . . . . 13 ((((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀𝐵) ∧ (𝐼 ⊆ ℕ0𝐼 ≠ ∅)) ∧ 𝑛𝐼) → ((𝑥 ∈ V, 𝑘 ∈ ℕ0 ↦ (𝑖 ∈ dom dom 𝑥, 𝑗 ∈ dom dom 𝑥 ↦ ((coe1‘(𝑖𝑥𝑗))‘𝑘))) ↾ (𝐵 × 𝐼)) = (𝑥𝐵, 𝑘𝐼 ↦ (𝑖 ∈ dom dom 𝑥, 𝑗 ∈ dom dom 𝑥 ↦ ((coe1‘(𝑖𝑥𝑗))‘𝑘))))
123115, 122syl5req 2827 . . . . . . . . . . . 12 ((((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀𝐵) ∧ (𝐼 ⊆ ℕ0𝐼 ≠ ∅)) ∧ 𝑛𝐼) → (𝑥𝐵, 𝑘𝐼 ↦ (𝑖 ∈ dom dom 𝑥, 𝑗 ∈ dom dom 𝑥 ↦ ((coe1‘(𝑖𝑥𝑗))‘𝑘))) = ( decompPMat ↾ (𝐵 × 𝐼)))
124123oveqd 6941 . . . . . . . . . . 11 ((((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀𝐵) ∧ (𝐼 ⊆ ℕ0𝐼 ≠ ∅)) ∧ 𝑛𝐼) → (𝑀(𝑥𝐵, 𝑘𝐼 ↦ (𝑖 ∈ dom dom 𝑥, 𝑗 ∈ dom dom 𝑥 ↦ ((coe1‘(𝑖𝑥𝑗))‘𝑘)))𝑛) = (𝑀( decompPMat ↾ (𝐵 × 𝐼))𝑛))
125113, 124eqtrd 2814 . . . . . . . . . 10 ((((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀𝐵) ∧ (𝐼 ⊆ ℕ0𝐼 ≠ ∅)) ∧ 𝑛𝐼) → ((curry (𝑥𝐵, 𝑘𝐼 ↦ (𝑖 ∈ dom dom 𝑥, 𝑗 ∈ dom dom 𝑥 ↦ ((coe1‘(𝑖𝑥𝑗))‘𝑘)))‘𝑀)‘𝑛) = (𝑀( decompPMat ↾ (𝐵 × 𝐼))𝑛))
126125adantlr 705 . . . . . . . . 9 (((((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀𝐵) ∧ (𝐼 ⊆ ℕ0𝐼 ≠ ∅)) ∧ 𝑓 = (curry (𝑥𝐵, 𝑘𝐼 ↦ (𝑖 ∈ dom dom 𝑥, 𝑗 ∈ dom dom 𝑥 ↦ ((coe1‘(𝑖𝑥𝑗))‘𝑘)))‘𝑀)) ∧ 𝑛𝐼) → ((curry (𝑥𝐵, 𝑘𝐼 ↦ (𝑖 ∈ dom dom 𝑥, 𝑗 ∈ dom dom 𝑥 ↦ ((coe1‘(𝑖𝑥𝑗))‘𝑘)))‘𝑀)‘𝑛) = (𝑀( decompPMat ↾ (𝐵 × 𝐼))𝑛))
127101, 126eqtrd 2814 . . . . . . . 8 (((((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀𝐵) ∧ (𝐼 ⊆ ℕ0𝐼 ≠ ∅)) ∧ 𝑓 = (curry (𝑥𝐵, 𝑘𝐼 ↦ (𝑖 ∈ dom dom 𝑥, 𝑗 ∈ dom dom 𝑥 ↦ ((coe1‘(𝑖𝑥𝑗))‘𝑘)))‘𝑀)) ∧ 𝑛𝐼) → (𝑓𝑛) = (𝑀( decompPMat ↾ (𝐵 × 𝐼))𝑛))
128127fveq2d 6452 . . . . . . 7 (((((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀𝐵) ∧ (𝐼 ⊆ ℕ0𝐼 ≠ ∅)) ∧ 𝑓 = (curry (𝑥𝐵, 𝑘𝐼 ↦ (𝑖 ∈ dom dom 𝑥, 𝑗 ∈ dom dom 𝑥 ↦ ((coe1‘(𝑖𝑥𝑗))‘𝑘)))‘𝑀)) ∧ 𝑛𝐼) → (𝑇‘(𝑓𝑛)) = (𝑇‘(𝑀( decompPMat ↾ (𝐵 × 𝐼))𝑛)))
12921ad2antrr 716 . . . . . . . . 9 ((((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀𝐵) ∧ (𝐼 ⊆ ℕ0𝐼 ≠ ∅)) ∧ 𝑓 = (curry (𝑥𝐵, 𝑘𝐼 ↦ (𝑖 ∈ dom dom 𝑥, 𝑗 ∈ dom dom 𝑥 ↦ ((coe1‘(𝑖𝑥𝑗))‘𝑘)))‘𝑀)) → 𝑀𝐵)
130 ovres 7079 . . . . . . . . 9 ((𝑀𝐵𝑛𝐼) → (𝑀( decompPMat ↾ (𝐵 × 𝐼))𝑛) = (𝑀 decompPMat 𝑛))
131129, 130sylan 575 . . . . . . . 8 (((((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀𝐵) ∧ (𝐼 ⊆ ℕ0𝐼 ≠ ∅)) ∧ 𝑓 = (curry (𝑥𝐵, 𝑘𝐼 ↦ (𝑖 ∈ dom dom 𝑥, 𝑗 ∈ dom dom 𝑥 ↦ ((coe1‘(𝑖𝑥𝑗))‘𝑘)))‘𝑀)) ∧ 𝑛𝐼) → (𝑀( decompPMat ↾ (𝐵 × 𝐼))𝑛) = (𝑀 decompPMat 𝑛))
132131fveq2d 6452 . . . . . . 7 (((((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀𝐵) ∧ (𝐼 ⊆ ℕ0𝐼 ≠ ∅)) ∧ 𝑓 = (curry (𝑥𝐵, 𝑘𝐼 ↦ (𝑖 ∈ dom dom 𝑥, 𝑗 ∈ dom dom 𝑥 ↦ ((coe1‘(𝑖𝑥𝑗))‘𝑘)))‘𝑀)) ∧ 𝑛𝐼) → (𝑇‘(𝑀( decompPMat ↾ (𝐵 × 𝐼))𝑛)) = (𝑇‘(𝑀 decompPMat 𝑛)))
133128, 132eqtrd 2814 . . . . . 6 (((((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀𝐵) ∧ (𝐼 ⊆ ℕ0𝐼 ≠ ∅)) ∧ 𝑓 = (curry (𝑥𝐵, 𝑘𝐼 ↦ (𝑖 ∈ dom dom 𝑥, 𝑗 ∈ dom dom 𝑥 ↦ ((coe1‘(𝑖𝑥𝑗))‘𝑘)))‘𝑀)) ∧ 𝑛𝐼) → (𝑇‘(𝑓𝑛)) = (𝑇‘(𝑀 decompPMat 𝑛)))
134133oveq2d 6940 . . . . 5 (((((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀𝐵) ∧ (𝐼 ⊆ ℕ0𝐼 ≠ ∅)) ∧ 𝑓 = (curry (𝑥𝐵, 𝑘𝐼 ↦ (𝑖 ∈ dom dom 𝑥, 𝑗 ∈ dom dom 𝑥 ↦ ((coe1‘(𝑖𝑥𝑗))‘𝑘)))‘𝑀)) ∧ 𝑛𝐼) → ((𝑛 𝑋) (𝑇‘(𝑓𝑛))) = ((𝑛 𝑋) (𝑇‘(𝑀 decompPMat 𝑛))))
135134mpteq2dva 4981 . . . 4 ((((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀𝐵) ∧ (𝐼 ⊆ ℕ0𝐼 ≠ ∅)) ∧ 𝑓 = (curry (𝑥𝐵, 𝑘𝐼 ↦ (𝑖 ∈ dom dom 𝑥, 𝑗 ∈ dom dom 𝑥 ↦ ((coe1‘(𝑖𝑥𝑗))‘𝑘)))‘𝑀)) → (𝑛𝐼 ↦ ((𝑛 𝑋) (𝑇‘(𝑓𝑛)))) = (𝑛𝐼 ↦ ((𝑛 𝑋) (𝑇‘(𝑀 decompPMat 𝑛)))))
136135oveq2d 6940 . . 3 ((((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀𝐵) ∧ (𝐼 ⊆ ℕ0𝐼 ≠ ∅)) ∧ 𝑓 = (curry (𝑥𝐵, 𝑘𝐼 ↦ (𝑖 ∈ dom dom 𝑥, 𝑗 ∈ dom dom 𝑥 ↦ ((coe1‘(𝑖𝑥𝑗))‘𝑘)))‘𝑀)) → (𝐶 Σg (𝑛𝐼 ↦ ((𝑛 𝑋) (𝑇‘(𝑓𝑛))))) = (𝐶 Σg (𝑛𝐼 ↦ ((𝑛 𝑋) (𝑇‘(𝑀 decompPMat 𝑛))))))
137136eqeq2d 2788 . 2 ((((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀𝐵) ∧ (𝐼 ⊆ ℕ0𝐼 ≠ ∅)) ∧ 𝑓 = (curry (𝑥𝐵, 𝑘𝐼 ↦ (𝑖 ∈ dom dom 𝑥, 𝑗 ∈ dom dom 𝑥 ↦ ((coe1‘(𝑖𝑥𝑗))‘𝑘)))‘𝑀)) → (𝑀 = (𝐶 Σg (𝑛𝐼 ↦ ((𝑛 𝑋) (𝑇‘(𝑓𝑛))))) ↔ 𝑀 = (𝐶 Σg (𝑛𝐼 ↦ ((𝑛 𝑋) (𝑇‘(𝑀 decompPMat 𝑛)))))))
13898, 137rspcedv 3515 1 (((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀𝐵) ∧ (𝐼 ⊆ ℕ0𝐼 ≠ ∅)) → (𝑀 = (𝐶 Σg (𝑛𝐼 ↦ ((𝑛 𝑋) (𝑇‘(𝑀 decompPMat 𝑛))))) → ∃𝑓 ∈ (𝐷𝑚 𝐼)𝑀 = (𝐶 Σg (𝑛𝐼 ↦ ((𝑛 𝑋) (𝑇‘(𝑓𝑛)))))))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 198  wa 386  w3a 1071   = wceq 1601  wcel 2107  wne 2969  wrex 3091  Vcvv 3398  csb 3751  wss 3792  c0 4141  cmpt 4967   × cxp 5355  dom cdm 5357  cres 5359  wf 6133  cfv 6137  (class class class)co 6924  cmpt2 6926  curry ccur 7675  𝑚 cmap 8142  Fincfn 8243  0cn0 11646  Basecbs 16259   ·𝑠 cvsca 16346   Σg cgsu 16491  .gcmg 17931  mulGrpcmgp 18880  CRingccrg 18939  var1cv1 19946  Poly1cpl1 19947  coe1cco1 19948   Mat cmat 20621   matToPolyMat cmat2pmat 20920   decompPMat cdecpmat 20978
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1839  ax-4 1853  ax-5 1953  ax-6 2021  ax-7 2055  ax-8 2109  ax-9 2116  ax-10 2135  ax-11 2150  ax-12 2163  ax-13 2334  ax-ext 2754  ax-rep 5008  ax-sep 5019  ax-nul 5027  ax-pow 5079  ax-pr 5140  ax-un 7228  ax-cnex 10330  ax-resscn 10331  ax-1cn 10332  ax-icn 10333  ax-addcl 10334  ax-addrcl 10335  ax-mulcl 10336  ax-mulrcl 10337  ax-mulcom 10338  ax-addass 10339  ax-mulass 10340  ax-distr 10341  ax-i2m1 10342  ax-1ne0 10343  ax-1rid 10344  ax-rnegex 10345  ax-rrecex 10346  ax-cnre 10347  ax-pre-lttri 10348  ax-pre-lttrn 10349  ax-pre-ltadd 10350  ax-pre-mulgt0 10351
This theorem depends on definitions:  df-bi 199  df-an 387  df-or 837  df-3or 1072  df-3an 1073  df-tru 1605  df-fal 1615  df-ex 1824  df-nf 1828  df-sb 2012  df-mo 2551  df-eu 2587  df-clab 2764  df-cleq 2770  df-clel 2774  df-nfc 2921  df-ne 2970  df-nel 3076  df-ral 3095  df-rex 3096  df-reu 3097  df-rab 3099  df-v 3400  df-sbc 3653  df-csb 3752  df-dif 3795  df-un 3797  df-in 3799  df-ss 3806  df-pss 3808  df-nul 4142  df-if 4308  df-pw 4381  df-sn 4399  df-pr 4401  df-tp 4403  df-op 4405  df-ot 4407  df-uni 4674  df-int 4713  df-iun 4757  df-br 4889  df-opab 4951  df-mpt 4968  df-tr 4990  df-id 5263  df-eprel 5268  df-po 5276  df-so 5277  df-fr 5316  df-we 5318  df-xp 5363  df-rel 5364  df-cnv 5365  df-co 5366  df-dm 5367  df-rn 5368  df-res 5369  df-ima 5370  df-pred 5935  df-ord 5981  df-on 5982  df-lim 5983  df-suc 5984  df-iota 6101  df-fun 6139  df-fn 6140  df-f 6141  df-f1 6142  df-fo 6143  df-f1o 6144  df-fv 6145  df-riota 6885  df-ov 6927  df-oprab 6928  df-mpt2 6929  df-of 7176  df-om 7346  df-1st 7447  df-2nd 7448  df-supp 7579  df-cur 7677  df-wrecs 7691  df-recs 7753  df-rdg 7791  df-1o 7845  df-oadd 7849  df-er 8028  df-map 8144  df-ixp 8197  df-en 8244  df-dom 8245  df-sdom 8246  df-fin 8247  df-fsupp 8566  df-sup 8638  df-pnf 10415  df-mnf 10416  df-xr 10417  df-ltxr 10418  df-le 10419  df-sub 10610  df-neg 10611  df-nn 11379  df-2 11442  df-3 11443  df-4 11444  df-5 11445  df-6 11446  df-7 11447  df-8 11448  df-9 11449  df-n0 11647  df-z 11733  df-dec 11850  df-uz 11997  df-fz 12648  df-struct 16261  df-ndx 16262  df-slot 16263  df-base 16265  df-sets 16266  df-ress 16267  df-plusg 16355  df-mulr 16356  df-sca 16358  df-vsca 16359  df-ip 16360  df-tset 16361  df-ple 16362  df-ds 16364  df-hom 16366  df-cco 16367  df-0g 16492  df-prds 16498  df-pws 16500  df-sra 19573  df-rgmod 19574  df-psr 19757  df-opsr 19761  df-psr1 19950  df-ply1 19952  df-coe1 19953  df-dsmm 20479  df-frlm 20494  df-mat 20622  df-decpmat 20979
This theorem is referenced by:  pmatcollpw3  21000  pmatcollpw3fi  21001
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