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Theorem pmatcollpw3lem 22803
Description: Lemma for pmatcollpw3 22804 and pmatcollpw3fi 22805: 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 5927 . . . . . . . . 9 (𝑥 = 𝑦 → dom 𝑥 = dom 𝑦)
21dmeqd 5929 . . . . . . . 8 (𝑥 = 𝑦 → dom dom 𝑥 = dom dom 𝑦)
3 oveq 7451 . . . . . . . . . 10 (𝑥 = 𝑦 → (𝑖𝑥𝑗) = (𝑖𝑦𝑗))
43fveq2d 6923 . . . . . . . . 9 (𝑥 = 𝑦 → (coe1‘(𝑖𝑥𝑗)) = (coe1‘(𝑖𝑦𝑗)))
54fveq1d 6921 . . . . . . . 8 (𝑥 = 𝑦 → ((coe1‘(𝑖𝑥𝑗))‘𝑘) = ((coe1‘(𝑖𝑦𝑗))‘𝑘))
62, 2, 5mpoeq123dv 7521 . . . . . . 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 6919 . . . . . . . 8 (𝑘 = 𝑙 → ((coe1‘(𝑖𝑦𝑗))‘𝑘) = ((coe1‘(𝑖𝑦𝑗))‘𝑙))
87mpoeq3dv 7525 . . . . . . 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, 8cbvmpov 7541 . . . . . 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 7937 . . . . . . . . . . 11 (𝑦𝐵 → dom 𝑦 ∈ V)
1110dmexd 7939 . . . . . . . . . 10 (𝑦𝐵 → dom dom 𝑦 ∈ V)
1211, 11jca 511 . . . . . . . . 9 (𝑦𝐵 → (dom dom 𝑦 ∈ V ∧ dom dom 𝑦 ∈ V))
1312ad2antrl 727 . . . . . . . 8 ((((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀𝐵) ∧ (𝐼 ⊆ ℕ0𝐼 ≠ ∅)) ∧ (𝑦𝐵𝑙𝐼)) → (dom dom 𝑦 ∈ V ∧ dom dom 𝑦 ∈ V))
14 mpoexga 8114 . . . . . . . 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 3204 . . . . . 6 (((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀𝐵) ∧ (𝐼 ⊆ ℕ0𝐼 ≠ ∅)) → ∀𝑦𝐵𝑙𝐼 (𝑖 ∈ dom dom 𝑦, 𝑗 ∈ dom dom 𝑦 ↦ ((coe1‘(𝑖𝑦𝑗))‘𝑙)) ∈ V)
17 simprr 772 . . . . . 6 (((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀𝐵) ∧ (𝐼 ⊆ ℕ0𝐼 ≠ ∅)) → 𝐼 ≠ ∅)
18 nn0ex 12555 . . . . . . . 8 0 ∈ V
1918ssex 5342 . . . . . . 7 (𝐼 ⊆ ℕ0𝐼 ∈ V)
2019ad2antrl 727 . . . . . 6 (((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀𝐵) ∧ (𝐼 ⊆ ℕ0𝐼 ≠ ∅)) → 𝐼 ∈ V)
21 simp3 1138 . . . . . . 7 ((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀𝐵) → 𝑀𝐵)
2221adantr 480 . . . . . 6 (((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀𝐵) ∧ (𝐼 ⊆ ℕ0𝐼 ≠ ∅)) → 𝑀𝐵)
239, 16, 17, 20, 22mpocurryvald 8307 . . . . 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 6919 . . . . . . . . 9 (𝑙 = 𝑘 → ((coe1‘(𝑖𝑦𝑗))‘𝑙) = ((coe1‘(𝑖𝑦𝑗))‘𝑘))
2524mpoeq3dv 7525 . . . . . . . 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 3922 . . . . . . 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 2741 . . . . . . . . 9 (𝑥 = 𝑦𝑦 = 𝑥)
28 eqcom 2741 . . . . . . . . 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 291 . . . . . . . 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 3927 . . . . . . 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, 30eqtrdi 2790 . . . . . 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 5282 . . . . 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, 32eqtrdi 2790 . . . 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 5927 . . . . . . . . . . 11 (𝑥 = 𝑀 → dom 𝑥 = dom 𝑀)
3534dmeqd 5929 . . . . . . . . . 10 (𝑥 = 𝑀 → dom dom 𝑥 = dom dom 𝑀)
36 oveq 7451 . . . . . . . . . . . 12 (𝑥 = 𝑀 → (𝑖𝑥𝑗) = (𝑖𝑀𝑗))
3736fveq2d 6923 . . . . . . . . . . 11 (𝑥 = 𝑀 → (coe1‘(𝑖𝑥𝑗)) = (coe1‘(𝑖𝑀𝑗)))
3837fveq1d 6921 . . . . . . . . . 10 (𝑥 = 𝑀 → ((coe1‘(𝑖𝑥𝑗))‘𝑘) = ((coe1‘(𝑖𝑀𝑗))‘𝑘))
3935, 35, 38mpoeq123dv 7521 . . . . . . . . 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 481 . . . . . . . 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 3953 . . . . . . 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 2734 . . . . . . . . . . . . 13 (Base‘𝑃) = (Base‘𝑃)
44 pmatcollpw.b . . . . . . . . . . . . 13 𝐵 = (Base‘𝐶)
4542, 43, 44matbas2i 22442 . . . . . . . . . . . 12 (𝑀𝐵𝑀 ∈ ((Base‘𝑃) ↑m (𝑁 × 𝑁)))
46 elmapi 8903 . . . . . . . . . . . 12 (𝑀 ∈ ((Base‘𝑃) ↑m (𝑁 × 𝑁)) → 𝑀:(𝑁 × 𝑁)⟶(Base‘𝑃))
47 fdm 6755 . . . . . . . . . . . . . 14 (𝑀:(𝑁 × 𝑁)⟶(Base‘𝑃) → dom 𝑀 = (𝑁 × 𝑁))
4847dmeqd 5929 . . . . . . . . . . . . 13 (𝑀:(𝑁 × 𝑁)⟶(Base‘𝑃) → dom dom 𝑀 = dom (𝑁 × 𝑁))
49 dmxpid 5954 . . . . . . . . . . . . 13 dom (𝑁 × 𝑁) = 𝑁
5048, 49eqtr2di 2791 . . . . . . . . . . . 12 (𝑀:(𝑁 × 𝑁)⟶(Base‘𝑃) → 𝑁 = dom dom 𝑀)
5145, 46, 503syl 18 . . . . . . . . . . 11 (𝑀𝐵𝑁 = dom dom 𝑀)
52513ad2ant3 1135 . . . . . . . . . 10 ((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀𝐵) → 𝑁 = dom dom 𝑀)
5352adantr 480 . . . . . . . . 9 (((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀𝐵) ∧ 𝑚 = 𝑀) → 𝑁 = dom dom 𝑀)
54 simpr 484 . . . . . . . . . . . 12 (((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀𝐵) ∧ 𝑚 = 𝑀) → 𝑚 = 𝑀)
5554oveqd 7462 . . . . . . . . . . 11 (((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀𝐵) ∧ 𝑚 = 𝑀) → (𝑖𝑚𝑗) = (𝑖𝑀𝑗))
5655fveq2d 6923 . . . . . . . . . 10 (((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀𝐵) ∧ 𝑚 = 𝑀) → (coe1‘(𝑖𝑚𝑗)) = (coe1‘(𝑖𝑀𝑗)))
5756fveq1d 6921 . . . . . . . . 9 (((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀𝐵) ∧ 𝑚 = 𝑀) → ((coe1‘(𝑖𝑚𝑗))‘𝑘) = ((coe1‘(𝑖𝑀𝑗))‘𝑘))
5853, 53, 57mpoeq123dv 7521 . . . . . . . 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 3953 . . . . . . 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 2777 . . . . . 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 480 . . . . 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 5271 . . . 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 2774 . . 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 7451 . . . . . . . . . . . 12 (𝑚 = 𝑀 → (𝑖𝑚𝑗) = (𝑖𝑀𝑗))
6564adantl 481 . . . . . . . . . . 11 (((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀𝐵) ∧ 𝑚 = 𝑀) → (𝑖𝑚𝑗) = (𝑖𝑀𝑗))
6665fveq2d 6923 . . . . . . . . . 10 (((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀𝐵) ∧ 𝑚 = 𝑀) → (coe1‘(𝑖𝑚𝑗)) = (coe1‘(𝑖𝑀𝑗)))
6766fveq1d 6921 . . . . . . . . 9 (((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀𝐵) ∧ 𝑚 = 𝑀) → ((coe1‘(𝑖𝑚𝑗))‘𝑘) = ((coe1‘(𝑖𝑀𝑗))‘𝑘))
6867mpoeq3dv 7525 . . . . . . . 8 (((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀𝐵) ∧ 𝑚 = 𝑀) → (𝑖𝑁, 𝑗𝑁 ↦ ((coe1‘(𝑖𝑚𝑗))‘𝑘)) = (𝑖𝑁, 𝑗𝑁 ↦ ((coe1‘(𝑖𝑀𝑗))‘𝑘)))
6921, 68csbied 3953 . . . . . . 7 ((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀𝐵) → 𝑀 / 𝑚(𝑖𝑁, 𝑗𝑁 ↦ ((coe1‘(𝑖𝑚𝑗))‘𝑘)) = (𝑖𝑁, 𝑗𝑁 ↦ ((coe1‘(𝑖𝑀𝑗))‘𝑘)))
7069ad2antrr 725 . . . . . 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 2734 . . . . . . 7 (Base‘𝑅) = (Base‘𝑅)
73 pmatcollpw3.d . . . . . . 7 𝐷 = (Base‘𝐴)
74 simpll1 1212 . . . . . . 7 ((((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀𝐵) ∧ (𝐼 ⊆ ℕ0𝐼 ≠ ∅)) ∧ 𝑘𝐼) → 𝑁 ∈ Fin)
75 simpll2 1213 . . . . . . 7 ((((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀𝐵) ∧ (𝐼 ⊆ ℕ0𝐼 ≠ ∅)) ∧ 𝑘𝐼) → 𝑅 ∈ CRing)
76 simp2 1137 . . . . . . . . 9 (((((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀𝐵) ∧ (𝐼 ⊆ ℕ0𝐼 ≠ ∅)) ∧ 𝑘𝐼) ∧ 𝑖𝑁𝑗𝑁) → 𝑖𝑁)
77 simp3 1138 . . . . . . . . 9 (((((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀𝐵) ∧ (𝐼 ⊆ ℕ0𝐼 ≠ ∅)) ∧ 𝑘𝐼) ∧ 𝑖𝑁𝑗𝑁) → 𝑗𝑁)
7822adantr 480 . . . . . . . . . 10 ((((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀𝐵) ∧ (𝐼 ⊆ ℕ0𝐼 ≠ ∅)) ∧ 𝑘𝐼) → 𝑀𝐵)
79783ad2ant1 1133 . . . . . . . . 9 (((((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀𝐵) ∧ (𝐼 ⊆ ℕ0𝐼 ≠ ∅)) ∧ 𝑘𝐼) ∧ 𝑖𝑁𝑗𝑁) → 𝑀𝐵)
8042, 43, 44, 76, 77, 79matecld 22446 . . . . . . . 8 (((((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀𝐵) ∧ (𝐼 ⊆ ℕ0𝐼 ≠ ∅)) ∧ 𝑘𝐼) ∧ 𝑖𝑁𝑗𝑁) → (𝑖𝑀𝑗) ∈ (Base‘𝑃))
81 ssel 3996 . . . . . . . . . . 11 (𝐼 ⊆ ℕ0 → (𝑘𝐼𝑘 ∈ ℕ0))
8281ad2antrl 727 . . . . . . . . . 10 (((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀𝐵) ∧ (𝐼 ⊆ ℕ0𝐼 ≠ ∅)) → (𝑘𝐼𝑘 ∈ ℕ0))
8382imp 406 . . . . . . . . 9 ((((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀𝐵) ∧ (𝐼 ⊆ ℕ0𝐼 ≠ ∅)) ∧ 𝑘𝐼) → 𝑘 ∈ ℕ0)
84833ad2ant1 1133 . . . . . . . 8 (((((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀𝐵) ∧ (𝐼 ⊆ ℕ0𝐼 ≠ ∅)) ∧ 𝑘𝐼) ∧ 𝑖𝑁𝑗𝑁) → 𝑘 ∈ ℕ0)
85 eqid 2734 . . . . . . . . 9 (coe1‘(𝑖𝑀𝑗)) = (coe1‘(𝑖𝑀𝑗))
86 pmatcollpw.p . . . . . . . . 9 𝑃 = (Poly1𝑅)
8785, 43, 86, 72coe1fvalcl 22228 . . . . . . . 8 (((𝑖𝑀𝑗) ∈ (Base‘𝑃) ∧ 𝑘 ∈ ℕ0) → ((coe1‘(𝑖𝑀𝑗))‘𝑘) ∈ (Base‘𝑅))
8880, 84, 87syl2anc 583 . . . . . . 7 (((((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀𝐵) ∧ (𝐼 ⊆ ℕ0𝐼 ≠ ∅)) ∧ 𝑘𝐼) ∧ 𝑖𝑁𝑗𝑁) → ((coe1‘(𝑖𝑀𝑗))‘𝑘) ∈ (Base‘𝑅))
8971, 72, 73, 74, 75, 88matbas2d 22443 . . . . . 6 ((((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀𝐵) ∧ (𝐼 ⊆ ℕ0𝐼 ≠ ∅)) ∧ 𝑘𝐼) → (𝑖𝑁, 𝑗𝑁 ↦ ((coe1‘(𝑖𝑀𝑗))‘𝑘)) ∈ 𝐷)
9070, 89eqeltrd 2838 . . . . 5 ((((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀𝐵) ∧ (𝐼 ⊆ ℕ0𝐼 ≠ ∅)) ∧ 𝑘𝐼) → 𝑀 / 𝑚(𝑖𝑁, 𝑗𝑁 ↦ ((coe1‘(𝑖𝑚𝑗))‘𝑘)) ∈ 𝐷)
9190fmpttd 7147 . . . 4 (((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀𝐵) ∧ (𝐼 ⊆ ℕ0𝐼 ≠ ∅)) → (𝑘𝐼𝑀 / 𝑚(𝑖𝑁, 𝑗𝑁 ↦ ((coe1‘(𝑖𝑚𝑗))‘𝑘))):𝐼𝐷)
9273fvexi 6933 . . . . . 6 𝐷 ∈ V
9392a1i 11 . . . . 5 ((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀𝐵) → 𝐷 ∈ V)
9419adantr 480 . . . . 5 ((𝐼 ⊆ ℕ0𝐼 ≠ ∅) → 𝐼 ∈ V)
95 elmapg 8893 . . . . 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 595 . . . 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 257 . . 3 (((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀𝐵) ∧ (𝐼 ⊆ ℕ0𝐼 ≠ ∅)) → (𝑘𝐼𝑀 / 𝑚(𝑖𝑁, 𝑗𝑁 ↦ ((coe1‘(𝑖𝑚𝑗))‘𝑘))) ∈ (𝐷m 𝐼))
9863, 97eqeltrd 2838 . 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 6918 . . . . . . . . . . 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 481 . . . . . . . . . 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 480 . . . . . . . . 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 2734 . . . . . . . . . . . 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 7937 . . . . . . . . . . . . . . . . 17 (𝑥𝐵 → dom 𝑥 ∈ V)
104103dmexd 7939 . . . . . . . . . . . . . . . 16 (𝑥𝐵 → dom dom 𝑥 ∈ V)
105104, 104jca 511 . . . . . . . . . . . . . . 15 (𝑥𝐵 → (dom dom 𝑥 ∈ V ∧ dom dom 𝑥 ∈ V))
106105ad2antrl 727 . . . . . . . . . . . . . 14 (((((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀𝐵) ∧ (𝐼 ⊆ ℕ0𝐼 ≠ ∅)) ∧ 𝑛𝐼) ∧ (𝑥𝐵𝑘𝐼)) → (dom dom 𝑥 ∈ V ∧ dom dom 𝑥 ∈ V))
107 mpoexga 8114 . . . . . . . . . . . . . 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 3204 . . . . . . . . . . . 12 ((((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀𝐵) ∧ (𝐼 ⊆ ℕ0𝐼 ≠ ∅)) ∧ 𝑛𝐼) → ∀𝑥𝐵𝑘𝐼 (𝑖 ∈ dom dom 𝑥, 𝑗 ∈ dom dom 𝑥 ↦ ((coe1‘(𝑖𝑥𝑗))‘𝑘)) ∈ V)
11020adantr 480 . . . . . . . . . . . 12 ((((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀𝐵) ∧ (𝐼 ⊆ ℕ0𝐼 ≠ ∅)) ∧ 𝑛𝐼) → 𝐼 ∈ V)
11122adantr 480 . . . . . . . . . . . 12 ((((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀𝐵) ∧ (𝐼 ⊆ ℕ0𝐼 ≠ ∅)) ∧ 𝑛𝐼) → 𝑀𝐵)
112 simpr 484 . . . . . . . . . . . 12 ((((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀𝐵) ∧ (𝐼 ⊆ ℕ0𝐼 ≠ ∅)) ∧ 𝑛𝐼) → 𝑛𝐼)
113102, 109, 110, 111, 112fvmpocurryd 8308 . . . . . . . . . . 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 22783 . . . . . . . . . . . . . 14 decompPMat = (𝑥 ∈ V, 𝑘 ∈ ℕ0 ↦ (𝑖 ∈ dom dom 𝑥, 𝑗 ∈ dom dom 𝑥 ↦ ((coe1‘(𝑖𝑥𝑗))‘𝑘)))
115114reseq1i 6004 . . . . . . . . . . . . 13 ( decompPMat ↾ (𝐵 × 𝐼)) = ((𝑥 ∈ V, 𝑘 ∈ ℕ0 ↦ (𝑖 ∈ dom dom 𝑥, 𝑗 ∈ dom dom 𝑥 ↦ ((coe1‘(𝑖𝑥𝑗))‘𝑘))) ↾ (𝐵 × 𝐼))
116 ssv 4027 . . . . . . . . . . . . . . . . 17 𝐵 ⊆ V
117116a1i 11 . . . . . . . . . . . . . . . 16 ((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀𝐵) → 𝐵 ⊆ V)
118 simpl 482 . . . . . . . . . . . . . . . 16 ((𝐼 ⊆ ℕ0𝐼 ≠ ∅) → 𝐼 ⊆ ℕ0)
119117, 118anim12i 612 . . . . . . . . . . . . . . 15 (((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀𝐵) ∧ (𝐼 ⊆ ℕ0𝐼 ≠ ∅)) → (𝐵 ⊆ V ∧ 𝐼 ⊆ ℕ0))
120119adantr 480 . . . . . . . . . . . . . 14 ((((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀𝐵) ∧ (𝐼 ⊆ ℕ0𝐼 ≠ ∅)) ∧ 𝑛𝐼) → (𝐵 ⊆ V ∧ 𝐼 ⊆ ℕ0))
121 resmpo 7566 . . . . . . . . . . . . . 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, 122eqtr2id 2787 . . . . . . . . . . . 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 7462 . . . . . . . . . . 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 2774 . . . . . . . . . 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 714 . . . . . . . . 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 2774 . . . . . . . 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 6923 . . . . . . 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 725 . . . . . . . . 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 7612 . . . . . . . . 9 ((𝑀𝐵𝑛𝐼) → (𝑀( decompPMat ↾ (𝐵 × 𝐼))𝑛) = (𝑀 decompPMat 𝑛))
131129, 130sylan 579 . . . . . . . 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 6923 . . . . . . 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 2774 . . . . . 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 7461 . . . . 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 5269 . . . 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 7461 . . 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 2745 . 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 3624 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 206  wa 395  w3a 1087   = wceq 1537  wcel 2103  wne 2942  wrex 3072  Vcvv 3482  csb 3915  wss 3970  c0 4347  cmpt 5252   × cxp 5697  dom cdm 5699  cres 5701  wf 6568  cfv 6572  (class class class)co 7445  cmpo 7447  curry ccur 8302  m cmap 8880  Fincfn 8999  0cn0 12549  Basecbs 17253   ·𝑠 cvsca 17310   Σg cgsu 17495  .gcmg 19102  mulGrpcmgp 20156  CRingccrg 20256  var1cv1 22191  Poly1cpl1 22192  coe1cco1 22193   Mat cmat 22425   matToPolyMat cmat2pmat 22724   decompPMat cdecpmat 22782
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1793  ax-4 1807  ax-5 1909  ax-6 1967  ax-7 2007  ax-8 2105  ax-9 2113  ax-10 2136  ax-11 2153  ax-12 2173  ax-ext 2705  ax-rep 5306  ax-sep 5320  ax-nul 5327  ax-pow 5386  ax-pr 5450  ax-un 7766  ax-cnex 11236  ax-resscn 11237  ax-1cn 11238  ax-icn 11239  ax-addcl 11240  ax-addrcl 11241  ax-mulcl 11242  ax-mulrcl 11243  ax-mulcom 11244  ax-addass 11245  ax-mulass 11246  ax-distr 11247  ax-i2m1 11248  ax-1ne0 11249  ax-1rid 11250  ax-rnegex 11251  ax-rrecex 11252  ax-cnre 11253  ax-pre-lttri 11254  ax-pre-lttrn 11255  ax-pre-ltadd 11256  ax-pre-mulgt0 11257
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 847  df-3or 1088  df-3an 1089  df-tru 1540  df-fal 1550  df-ex 1778  df-nf 1782  df-sb 2065  df-mo 2537  df-eu 2566  df-clab 2712  df-cleq 2726  df-clel 2813  df-nfc 2890  df-ne 2943  df-nel 3049  df-ral 3064  df-rex 3073  df-reu 3384  df-rab 3439  df-v 3484  df-sbc 3799  df-csb 3916  df-dif 3973  df-un 3975  df-in 3977  df-ss 3987  df-pss 3990  df-nul 4348  df-if 4549  df-pw 4624  df-sn 4649  df-pr 4651  df-tp 4653  df-op 4655  df-ot 4657  df-uni 4932  df-iun 5021  df-br 5170  df-opab 5232  df-mpt 5253  df-tr 5287  df-id 5597  df-eprel 5603  df-po 5611  df-so 5612  df-fr 5654  df-we 5656  df-xp 5705  df-rel 5706  df-cnv 5707  df-co 5708  df-dm 5709  df-rn 5710  df-res 5711  df-ima 5712  df-pred 6331  df-ord 6397  df-on 6398  df-lim 6399  df-suc 6400  df-iota 6524  df-fun 6574  df-fn 6575  df-f 6576  df-f1 6577  df-fo 6578  df-f1o 6579  df-fv 6580  df-riota 7401  df-ov 7448  df-oprab 7449  df-mpo 7450  df-of 7710  df-om 7900  df-1st 8026  df-2nd 8027  df-supp 8198  df-cur 8304  df-frecs 8318  df-wrecs 8349  df-recs 8423  df-rdg 8462  df-1o 8518  df-er 8759  df-map 8882  df-ixp 8952  df-en 9000  df-dom 9001  df-sdom 9002  df-fin 9003  df-fsupp 9428  df-sup 9507  df-pnf 11322  df-mnf 11323  df-xr 11324  df-ltxr 11325  df-le 11326  df-sub 11518  df-neg 11519  df-nn 12290  df-2 12352  df-3 12353  df-4 12354  df-5 12355  df-6 12356  df-7 12357  df-8 12358  df-9 12359  df-n0 12550  df-z 12636  df-dec 12755  df-uz 12900  df-fz 13564  df-struct 17189  df-sets 17206  df-slot 17224  df-ndx 17236  df-base 17254  df-ress 17283  df-plusg 17319  df-mulr 17320  df-sca 17322  df-vsca 17323  df-ip 17324  df-tset 17325  df-ple 17326  df-ds 17328  df-hom 17330  df-cco 17331  df-0g 17496  df-prds 17502  df-pws 17504  df-sra 21190  df-rgmod 21191  df-dsmm 21770  df-frlm 21785  df-psr 21946  df-opsr 21950  df-psr1 22195  df-ply1 22197  df-coe1 22198  df-mat 22426  df-decpmat 22783
This theorem is referenced by:  pmatcollpw3  22804  pmatcollpw3fi  22805
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