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Theorem pmatcollpw3lem 20790
Description: Lemma for pmatcollpw3 20791 and pmatcollpw3fi 20792: 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 5479 . . . . . . . . 9 (𝑥 = 𝑦 → dom 𝑥 = dom 𝑦)
21dmeqd 5481 . . . . . . . 8 (𝑥 = 𝑦 → dom dom 𝑥 = dom dom 𝑦)
3 oveq 6819 . . . . . . . . . 10 (𝑥 = 𝑦 → (𝑖𝑥𝑗) = (𝑖𝑦𝑗))
43fveq2d 6356 . . . . . . . . 9 (𝑥 = 𝑦 → (coe1‘(𝑖𝑥𝑗)) = (coe1‘(𝑖𝑦𝑗)))
54fveq1d 6354 . . . . . . . 8 (𝑥 = 𝑦 → ((coe1‘(𝑖𝑥𝑗))‘𝑘) = ((coe1‘(𝑖𝑦𝑗))‘𝑘))
62, 2, 5mpt2eq123dv 6882 . . . . . . 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 6352 . . . . . . . 8 (𝑘 = 𝑙 → ((coe1‘(𝑖𝑦𝑗))‘𝑘) = ((coe1‘(𝑖𝑦𝑗))‘𝑙))
87mpt2eq3dv 6886 . . . . . . 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 6900 . . . . . 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 7262 . . . . . . . . . . 11 (𝑦𝐵 → dom 𝑦 ∈ V)
11 dmexg 7262 . . . . . . . . . . 11 (dom 𝑦 ∈ V → dom dom 𝑦 ∈ V)
1210, 11syl 17 . . . . . . . . . 10 (𝑦𝐵 → dom dom 𝑦 ∈ V)
1312, 12jca 555 . . . . . . . . 9 (𝑦𝐵 → (dom dom 𝑦 ∈ V ∧ dom dom 𝑦 ∈ V))
1413ad2antrl 766 . . . . . . . 8 ((((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀𝐵) ∧ (𝐼 ⊆ ℕ0𝐼 ≠ ∅)) ∧ (𝑦𝐵𝑙𝐼)) → (dom dom 𝑦 ∈ V ∧ dom dom 𝑦 ∈ V))
15 mpt2exga 7414 . . . . . . . 8 ((dom dom 𝑦 ∈ V ∧ dom dom 𝑦 ∈ V) → (𝑖 ∈ dom dom 𝑦, 𝑗 ∈ dom dom 𝑦 ↦ ((coe1‘(𝑖𝑦𝑗))‘𝑙)) ∈ V)
1614, 15syl 17 . . . . . . 7 ((((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀𝐵) ∧ (𝐼 ⊆ ℕ0𝐼 ≠ ∅)) ∧ (𝑦𝐵𝑙𝐼)) → (𝑖 ∈ dom dom 𝑦, 𝑗 ∈ dom dom 𝑦 ↦ ((coe1‘(𝑖𝑦𝑗))‘𝑙)) ∈ V)
1716ralrimivva 3109 . . . . . 6 (((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀𝐵) ∧ (𝐼 ⊆ ℕ0𝐼 ≠ ∅)) → ∀𝑦𝐵𝑙𝐼 (𝑖 ∈ dom dom 𝑦, 𝑗 ∈ dom dom 𝑦 ↦ ((coe1‘(𝑖𝑦𝑗))‘𝑙)) ∈ V)
18 simprr 813 . . . . . 6 (((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀𝐵) ∧ (𝐼 ⊆ ℕ0𝐼 ≠ ∅)) → 𝐼 ≠ ∅)
19 nn0ex 11490 . . . . . . . 8 0 ∈ V
2019ssex 4954 . . . . . . 7 (𝐼 ⊆ ℕ0𝐼 ∈ V)
2120ad2antrl 766 . . . . . 6 (((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀𝐵) ∧ (𝐼 ⊆ ℕ0𝐼 ≠ ∅)) → 𝐼 ∈ V)
22 simp3 1133 . . . . . . 7 ((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀𝐵) → 𝑀𝐵)
2322adantr 472 . . . . . 6 (((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀𝐵) ∧ (𝐼 ⊆ ℕ0𝐼 ≠ ∅)) → 𝑀𝐵)
249, 17, 18, 21, 23mpt2curryvald 7565 . . . . 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))))
25 fveq2 6352 . . . . . . . . 9 (𝑙 = 𝑘 → ((coe1‘(𝑖𝑦𝑗))‘𝑙) = ((coe1‘(𝑖𝑦𝑗))‘𝑘))
2625mpt2eq3dv 6886 . . . . . . . 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)))
2726csbeq2dv 4135 . . . . . . 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)))
28 eqcom 2767 . . . . . . . . 9 (𝑥 = 𝑦𝑦 = 𝑥)
29 eqcom 2767 . . . . . . . . 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)))
306, 28, 293imtr3i 280 . . . . . . . 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)))
3130cbvcsbv 3680 . . . . . . 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))
3227, 31syl6eq 2810 . . . . . 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)))
3332cbvmptv 4902 . . . . 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)))
3424, 33syl6eq 2810 . . . 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))))
35 dmeq 5479 . . . . . . . . . . 11 (𝑥 = 𝑀 → dom 𝑥 = dom 𝑀)
3635dmeqd 5481 . . . . . . . . . 10 (𝑥 = 𝑀 → dom dom 𝑥 = dom dom 𝑀)
37 oveq 6819 . . . . . . . . . . . 12 (𝑥 = 𝑀 → (𝑖𝑥𝑗) = (𝑖𝑀𝑗))
3837fveq2d 6356 . . . . . . . . . . 11 (𝑥 = 𝑀 → (coe1‘(𝑖𝑥𝑗)) = (coe1‘(𝑖𝑀𝑗)))
3938fveq1d 6354 . . . . . . . . . 10 (𝑥 = 𝑀 → ((coe1‘(𝑖𝑥𝑗))‘𝑘) = ((coe1‘(𝑖𝑀𝑗))‘𝑘))
4036, 36, 39mpt2eq123dv 6882 . . . . . . . . 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)))
4140adantl 473 . . . . . . . 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)))
4222, 41csbied 3701 . . . . . . 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)))
43 pmatcollpw.c . . . . . . . . . . . . 13 𝐶 = (𝑁 Mat 𝑃)
44 eqid 2760 . . . . . . . . . . . . 13 (Base‘𝑃) = (Base‘𝑃)
45 pmatcollpw.b . . . . . . . . . . . . 13 𝐵 = (Base‘𝐶)
4643, 44, 45matbas2i 20430 . . . . . . . . . . . 12 (𝑀𝐵𝑀 ∈ ((Base‘𝑃) ↑𝑚 (𝑁 × 𝑁)))
47 elmapi 8045 . . . . . . . . . . . 12 (𝑀 ∈ ((Base‘𝑃) ↑𝑚 (𝑁 × 𝑁)) → 𝑀:(𝑁 × 𝑁)⟶(Base‘𝑃))
48 fdm 6212 . . . . . . . . . . . . . 14 (𝑀:(𝑁 × 𝑁)⟶(Base‘𝑃) → dom 𝑀 = (𝑁 × 𝑁))
4948dmeqd 5481 . . . . . . . . . . . . 13 (𝑀:(𝑁 × 𝑁)⟶(Base‘𝑃) → dom dom 𝑀 = dom (𝑁 × 𝑁))
50 dmxpid 5500 . . . . . . . . . . . . 13 dom (𝑁 × 𝑁) = 𝑁
5149, 50syl6req 2811 . . . . . . . . . . . 12 (𝑀:(𝑁 × 𝑁)⟶(Base‘𝑃) → 𝑁 = dom dom 𝑀)
5246, 47, 513syl 18 . . . . . . . . . . 11 (𝑀𝐵𝑁 = dom dom 𝑀)
53523ad2ant3 1130 . . . . . . . . . 10 ((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀𝐵) → 𝑁 = dom dom 𝑀)
5453adantr 472 . . . . . . . . 9 (((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀𝐵) ∧ 𝑚 = 𝑀) → 𝑁 = dom dom 𝑀)
55 simpr 479 . . . . . . . . . . . 12 (((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀𝐵) ∧ 𝑚 = 𝑀) → 𝑚 = 𝑀)
5655oveqd 6830 . . . . . . . . . . 11 (((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀𝐵) ∧ 𝑚 = 𝑀) → (𝑖𝑚𝑗) = (𝑖𝑀𝑗))
5756fveq2d 6356 . . . . . . . . . 10 (((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀𝐵) ∧ 𝑚 = 𝑀) → (coe1‘(𝑖𝑚𝑗)) = (coe1‘(𝑖𝑀𝑗)))
5857fveq1d 6354 . . . . . . . . 9 (((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀𝐵) ∧ 𝑚 = 𝑀) → ((coe1‘(𝑖𝑚𝑗))‘𝑘) = ((coe1‘(𝑖𝑀𝑗))‘𝑘))
5954, 54, 58mpt2eq123dv 6882 . . . . . . . 8 (((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀𝐵) ∧ 𝑚 = 𝑀) → (𝑖𝑁, 𝑗𝑁 ↦ ((coe1‘(𝑖𝑚𝑗))‘𝑘)) = (𝑖 ∈ dom dom 𝑀, 𝑗 ∈ dom dom 𝑀 ↦ ((coe1‘(𝑖𝑀𝑗))‘𝑘)))
6022, 59csbied 3701 . . . . . . 7 ((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀𝐵) → 𝑀 / 𝑚(𝑖𝑁, 𝑗𝑁 ↦ ((coe1‘(𝑖𝑚𝑗))‘𝑘)) = (𝑖 ∈ dom dom 𝑀, 𝑗 ∈ dom dom 𝑀 ↦ ((coe1‘(𝑖𝑀𝑗))‘𝑘)))
6142, 60eqtr4d 2797 . . . . . 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)))
6261adantr 472 . . . . 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)))
6362mpteq2dv 4897 . . . 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))))
6434, 63eqtrd 2794 . . 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))))
65 oveq 6819 . . . . . . . . . . . 12 (𝑚 = 𝑀 → (𝑖𝑚𝑗) = (𝑖𝑀𝑗))
6665adantl 473 . . . . . . . . . . 11 (((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀𝐵) ∧ 𝑚 = 𝑀) → (𝑖𝑚𝑗) = (𝑖𝑀𝑗))
6766fveq2d 6356 . . . . . . . . . 10 (((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀𝐵) ∧ 𝑚 = 𝑀) → (coe1‘(𝑖𝑚𝑗)) = (coe1‘(𝑖𝑀𝑗)))
6867fveq1d 6354 . . . . . . . . 9 (((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀𝐵) ∧ 𝑚 = 𝑀) → ((coe1‘(𝑖𝑚𝑗))‘𝑘) = ((coe1‘(𝑖𝑀𝑗))‘𝑘))
6968mpt2eq3dv 6886 . . . . . . . 8 (((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀𝐵) ∧ 𝑚 = 𝑀) → (𝑖𝑁, 𝑗𝑁 ↦ ((coe1‘(𝑖𝑚𝑗))‘𝑘)) = (𝑖𝑁, 𝑗𝑁 ↦ ((coe1‘(𝑖𝑀𝑗))‘𝑘)))
7022, 69csbied 3701 . . . . . . 7 ((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀𝐵) → 𝑀 / 𝑚(𝑖𝑁, 𝑗𝑁 ↦ ((coe1‘(𝑖𝑚𝑗))‘𝑘)) = (𝑖𝑁, 𝑗𝑁 ↦ ((coe1‘(𝑖𝑀𝑗))‘𝑘)))
7170ad2antrr 764 . . . . . 6 ((((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀𝐵) ∧ (𝐼 ⊆ ℕ0𝐼 ≠ ∅)) ∧ 𝑘𝐼) → 𝑀 / 𝑚(𝑖𝑁, 𝑗𝑁 ↦ ((coe1‘(𝑖𝑚𝑗))‘𝑘)) = (𝑖𝑁, 𝑗𝑁 ↦ ((coe1‘(𝑖𝑀𝑗))‘𝑘)))
72 pmatcollpw3.a . . . . . . 7 𝐴 = (𝑁 Mat 𝑅)
73 eqid 2760 . . . . . . 7 (Base‘𝑅) = (Base‘𝑅)
74 pmatcollpw3.d . . . . . . 7 𝐷 = (Base‘𝐴)
75 simpll1 1255 . . . . . . 7 ((((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀𝐵) ∧ (𝐼 ⊆ ℕ0𝐼 ≠ ∅)) ∧ 𝑘𝐼) → 𝑁 ∈ Fin)
76 simpll2 1257 . . . . . . 7 ((((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀𝐵) ∧ (𝐼 ⊆ ℕ0𝐼 ≠ ∅)) ∧ 𝑘𝐼) → 𝑅 ∈ CRing)
77 simp2 1132 . . . . . . . . 9 (((((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀𝐵) ∧ (𝐼 ⊆ ℕ0𝐼 ≠ ∅)) ∧ 𝑘𝐼) ∧ 𝑖𝑁𝑗𝑁) → 𝑖𝑁)
78 simp3 1133 . . . . . . . . 9 (((((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀𝐵) ∧ (𝐼 ⊆ ℕ0𝐼 ≠ ∅)) ∧ 𝑘𝐼) ∧ 𝑖𝑁𝑗𝑁) → 𝑗𝑁)
7923adantr 472 . . . . . . . . . 10 ((((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀𝐵) ∧ (𝐼 ⊆ ℕ0𝐼 ≠ ∅)) ∧ 𝑘𝐼) → 𝑀𝐵)
80793ad2ant1 1128 . . . . . . . . 9 (((((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀𝐵) ∧ (𝐼 ⊆ ℕ0𝐼 ≠ ∅)) ∧ 𝑘𝐼) ∧ 𝑖𝑁𝑗𝑁) → 𝑀𝐵)
8143, 44, 45, 77, 78, 80matecld 20434 . . . . . . . 8 (((((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀𝐵) ∧ (𝐼 ⊆ ℕ0𝐼 ≠ ∅)) ∧ 𝑘𝐼) ∧ 𝑖𝑁𝑗𝑁) → (𝑖𝑀𝑗) ∈ (Base‘𝑃))
82 ssel 3738 . . . . . . . . . . 11 (𝐼 ⊆ ℕ0 → (𝑘𝐼𝑘 ∈ ℕ0))
8382ad2antrl 766 . . . . . . . . . 10 (((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀𝐵) ∧ (𝐼 ⊆ ℕ0𝐼 ≠ ∅)) → (𝑘𝐼𝑘 ∈ ℕ0))
8483imp 444 . . . . . . . . 9 ((((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀𝐵) ∧ (𝐼 ⊆ ℕ0𝐼 ≠ ∅)) ∧ 𝑘𝐼) → 𝑘 ∈ ℕ0)
85843ad2ant1 1128 . . . . . . . 8 (((((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀𝐵) ∧ (𝐼 ⊆ ℕ0𝐼 ≠ ∅)) ∧ 𝑘𝐼) ∧ 𝑖𝑁𝑗𝑁) → 𝑘 ∈ ℕ0)
86 eqid 2760 . . . . . . . . 9 (coe1‘(𝑖𝑀𝑗)) = (coe1‘(𝑖𝑀𝑗))
87 pmatcollpw.p . . . . . . . . 9 𝑃 = (Poly1𝑅)
8886, 44, 87, 73coe1fvalcl 19784 . . . . . . . 8 (((𝑖𝑀𝑗) ∈ (Base‘𝑃) ∧ 𝑘 ∈ ℕ0) → ((coe1‘(𝑖𝑀𝑗))‘𝑘) ∈ (Base‘𝑅))
8981, 85, 88syl2anc 696 . . . . . . 7 (((((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀𝐵) ∧ (𝐼 ⊆ ℕ0𝐼 ≠ ∅)) ∧ 𝑘𝐼) ∧ 𝑖𝑁𝑗𝑁) → ((coe1‘(𝑖𝑀𝑗))‘𝑘) ∈ (Base‘𝑅))
9072, 73, 74, 75, 76, 89matbas2d 20431 . . . . . 6 ((((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀𝐵) ∧ (𝐼 ⊆ ℕ0𝐼 ≠ ∅)) ∧ 𝑘𝐼) → (𝑖𝑁, 𝑗𝑁 ↦ ((coe1‘(𝑖𝑀𝑗))‘𝑘)) ∈ 𝐷)
9171, 90eqeltrd 2839 . . . . 5 ((((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀𝐵) ∧ (𝐼 ⊆ ℕ0𝐼 ≠ ∅)) ∧ 𝑘𝐼) → 𝑀 / 𝑚(𝑖𝑁, 𝑗𝑁 ↦ ((coe1‘(𝑖𝑚𝑗))‘𝑘)) ∈ 𝐷)
92 eqid 2760 . . . . 5 (𝑘𝐼𝑀 / 𝑚(𝑖𝑁, 𝑗𝑁 ↦ ((coe1‘(𝑖𝑚𝑗))‘𝑘))) = (𝑘𝐼𝑀 / 𝑚(𝑖𝑁, 𝑗𝑁 ↦ ((coe1‘(𝑖𝑚𝑗))‘𝑘)))
9391, 92fmptd 6548 . . . 4 (((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀𝐵) ∧ (𝐼 ⊆ ℕ0𝐼 ≠ ∅)) → (𝑘𝐼𝑀 / 𝑚(𝑖𝑁, 𝑗𝑁 ↦ ((coe1‘(𝑖𝑚𝑗))‘𝑘))):𝐼𝐷)
94 fvex 6362 . . . . . . 7 (Base‘𝐴) ∈ V
9574, 94eqeltri 2835 . . . . . 6 𝐷 ∈ V
9695a1i 11 . . . . 5 ((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀𝐵) → 𝐷 ∈ V)
9720adantr 472 . . . . 5 ((𝐼 ⊆ ℕ0𝐼 ≠ ∅) → 𝐼 ∈ V)
98 elmapg 8036 . . . . 5 ((𝐷 ∈ V ∧ 𝐼 ∈ V) → ((𝑘𝐼𝑀 / 𝑚(𝑖𝑁, 𝑗𝑁 ↦ ((coe1‘(𝑖𝑚𝑗))‘𝑘))) ∈ (𝐷𝑚 𝐼) ↔ (𝑘𝐼𝑀 / 𝑚(𝑖𝑁, 𝑗𝑁 ↦ ((coe1‘(𝑖𝑚𝑗))‘𝑘))):𝐼𝐷))
9996, 97, 98syl2an 495 . . . 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))
10093, 99mpbird 247 . . 3 (((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀𝐵) ∧ (𝐼 ⊆ ℕ0𝐼 ≠ ∅)) → (𝑘𝐼𝑀 / 𝑚(𝑖𝑁, 𝑗𝑁 ↦ ((coe1‘(𝑖𝑚𝑗))‘𝑘))) ∈ (𝐷𝑚 𝐼))
10164, 100eqeltrd 2839 . 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))
102 fveq1 6351 . . . . . . . . . . 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))
103102adantl 473 . . . . . . . . . 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))
104103adantr 472 . . . . . . . . 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))
105 eqid 2760 . . . . . . . . . . . 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)))
106 dmexg 7262 . . . . . . . . . . . . . . . . 17 (𝑥𝐵 → dom 𝑥 ∈ V)
107 dmexg 7262 . . . . . . . . . . . . . . . . 17 (dom 𝑥 ∈ V → dom dom 𝑥 ∈ V)
108106, 107syl 17 . . . . . . . . . . . . . . . 16 (𝑥𝐵 → dom dom 𝑥 ∈ V)
109108, 108jca 555 . . . . . . . . . . . . . . 15 (𝑥𝐵 → (dom dom 𝑥 ∈ V ∧ dom dom 𝑥 ∈ V))
110109ad2antrl 766 . . . . . . . . . . . . . 14 (((((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀𝐵) ∧ (𝐼 ⊆ ℕ0𝐼 ≠ ∅)) ∧ 𝑛𝐼) ∧ (𝑥𝐵𝑘𝐼)) → (dom dom 𝑥 ∈ V ∧ dom dom 𝑥 ∈ V))
111 mpt2exga 7414 . . . . . . . . . . . . . 14 ((dom dom 𝑥 ∈ V ∧ dom dom 𝑥 ∈ V) → (𝑖 ∈ dom dom 𝑥, 𝑗 ∈ dom dom 𝑥 ↦ ((coe1‘(𝑖𝑥𝑗))‘𝑘)) ∈ V)
112110, 111syl 17 . . . . . . . . . . . . 13 (((((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀𝐵) ∧ (𝐼 ⊆ ℕ0𝐼 ≠ ∅)) ∧ 𝑛𝐼) ∧ (𝑥𝐵𝑘𝐼)) → (𝑖 ∈ dom dom 𝑥, 𝑗 ∈ dom dom 𝑥 ↦ ((coe1‘(𝑖𝑥𝑗))‘𝑘)) ∈ V)
113112ralrimivva 3109 . . . . . . . . . . . 12 ((((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀𝐵) ∧ (𝐼 ⊆ ℕ0𝐼 ≠ ∅)) ∧ 𝑛𝐼) → ∀𝑥𝐵𝑘𝐼 (𝑖 ∈ dom dom 𝑥, 𝑗 ∈ dom dom 𝑥 ↦ ((coe1‘(𝑖𝑥𝑗))‘𝑘)) ∈ V)
11421adantr 472 . . . . . . . . . . . 12 ((((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀𝐵) ∧ (𝐼 ⊆ ℕ0𝐼 ≠ ∅)) ∧ 𝑛𝐼) → 𝐼 ∈ V)
11523adantr 472 . . . . . . . . . . . 12 ((((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀𝐵) ∧ (𝐼 ⊆ ℕ0𝐼 ≠ ∅)) ∧ 𝑛𝐼) → 𝑀𝐵)
116 simpr 479 . . . . . . . . . . . 12 ((((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀𝐵) ∧ (𝐼 ⊆ ℕ0𝐼 ≠ ∅)) ∧ 𝑛𝐼) → 𝑛𝐼)
117105, 113, 114, 115, 116fvmpt2curryd 7566 . . . . . . . . . . 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))
118 df-decpmat 20770 . . . . . . . . . . . . . 14 decompPMat = (𝑥 ∈ V, 𝑘 ∈ ℕ0 ↦ (𝑖 ∈ dom dom 𝑥, 𝑗 ∈ dom dom 𝑥 ↦ ((coe1‘(𝑖𝑥𝑗))‘𝑘)))
119118reseq1i 5547 . . . . . . . . . . . . 13 ( decompPMat ↾ (𝐵 × 𝐼)) = ((𝑥 ∈ V, 𝑘 ∈ ℕ0 ↦ (𝑖 ∈ dom dom 𝑥, 𝑗 ∈ dom dom 𝑥 ↦ ((coe1‘(𝑖𝑥𝑗))‘𝑘))) ↾ (𝐵 × 𝐼))
120 ssv 3766 . . . . . . . . . . . . . . . . 17 𝐵 ⊆ V
121120a1i 11 . . . . . . . . . . . . . . . 16 ((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀𝐵) → 𝐵 ⊆ V)
122 simpl 474 . . . . . . . . . . . . . . . 16 ((𝐼 ⊆ ℕ0𝐼 ≠ ∅) → 𝐼 ⊆ ℕ0)
123121, 122anim12i 591 . . . . . . . . . . . . . . 15 (((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀𝐵) ∧ (𝐼 ⊆ ℕ0𝐼 ≠ ∅)) → (𝐵 ⊆ V ∧ 𝐼 ⊆ ℕ0))
124123adantr 472 . . . . . . . . . . . . . 14 ((((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀𝐵) ∧ (𝐼 ⊆ ℕ0𝐼 ≠ ∅)) ∧ 𝑛𝐼) → (𝐵 ⊆ V ∧ 𝐼 ⊆ ℕ0))
125 resmpt2 6923 . . . . . . . . . . . . . 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))))
126124, 125syl 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))))
127119, 126syl5req 2807 . . . . . . . . . . . 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)))
128127oveqd 6830 . . . . . . . . . . 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))
129117, 128eqtrd 2794 . . . . . . . . . 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))
130129adantlr 753 . . . . . . . . 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))
131104, 130eqtrd 2794 . . . . . . . 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))
132131fveq2d 6356 . . . . . . 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)))
13322ad2antrr 764 . . . . . . . . 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)
134 ovres 6965 . . . . . . . . 9 ((𝑀𝐵𝑛𝐼) → (𝑀( decompPMat ↾ (𝐵 × 𝐼))𝑛) = (𝑀 decompPMat 𝑛))
135133, 134sylan 489 . . . . . . . 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))
136135fveq2d 6356 . . . . . . 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)))
137132, 136eqtrd 2794 . . . . . 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)))
138137oveq2d 6829 . . . . 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))))
139138mpteq2dva 4896 . . . 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)))))
140139oveq2d 6829 . . 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))))))
141140eqeq2d 2770 . 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)))))))
142101, 141rspcedv 3453 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 196  wa 383  w3a 1072   = wceq 1632  wcel 2139  wne 2932  wrex 3051  Vcvv 3340  csb 3674  wss 3715  c0 4058  cmpt 4881   × cxp 5264  dom cdm 5266  cres 5268  wf 6045  cfv 6049  (class class class)co 6813  cmpt2 6815  curry ccur 7560  𝑚 cmap 8023  Fincfn 8121  0cn0 11484  Basecbs 16059   ·𝑠 cvsca 16147   Σg cgsu 16303  .gcmg 17741  mulGrpcmgp 18689  CRingccrg 18748  var1cv1 19748  Poly1cpl1 19749  coe1cco1 19750   Mat cmat 20415   matToPolyMat cmat2pmat 20711   decompPMat cdecpmat 20769
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1871  ax-4 1886  ax-5 1988  ax-6 2054  ax-7 2090  ax-8 2141  ax-9 2148  ax-10 2168  ax-11 2183  ax-12 2196  ax-13 2391  ax-ext 2740  ax-rep 4923  ax-sep 4933  ax-nul 4941  ax-pow 4992  ax-pr 5055  ax-un 7114  ax-cnex 10184  ax-resscn 10185  ax-1cn 10186  ax-icn 10187  ax-addcl 10188  ax-addrcl 10189  ax-mulcl 10190  ax-mulrcl 10191  ax-mulcom 10192  ax-addass 10193  ax-mulass 10194  ax-distr 10195  ax-i2m1 10196  ax-1ne0 10197  ax-1rid 10198  ax-rnegex 10199  ax-rrecex 10200  ax-cnre 10201  ax-pre-lttri 10202  ax-pre-lttrn 10203  ax-pre-ltadd 10204  ax-pre-mulgt0 10205
This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-3or 1073  df-3an 1074  df-tru 1635  df-fal 1638  df-ex 1854  df-nf 1859  df-sb 2047  df-eu 2611  df-mo 2612  df-clab 2747  df-cleq 2753  df-clel 2756  df-nfc 2891  df-ne 2933  df-nel 3036  df-ral 3055  df-rex 3056  df-reu 3057  df-rab 3059  df-v 3342  df-sbc 3577  df-csb 3675  df-dif 3718  df-un 3720  df-in 3722  df-ss 3729  df-pss 3731  df-nul 4059  df-if 4231  df-pw 4304  df-sn 4322  df-pr 4324  df-tp 4326  df-op 4328  df-ot 4330  df-uni 4589  df-int 4628  df-iun 4674  df-br 4805  df-opab 4865  df-mpt 4882  df-tr 4905  df-id 5174  df-eprel 5179  df-po 5187  df-so 5188  df-fr 5225  df-we 5227  df-xp 5272  df-rel 5273  df-cnv 5274  df-co 5275  df-dm 5276  df-rn 5277  df-res 5278  df-ima 5279  df-pred 5841  df-ord 5887  df-on 5888  df-lim 5889  df-suc 5890  df-iota 6012  df-fun 6051  df-fn 6052  df-f 6053  df-f1 6054  df-fo 6055  df-f1o 6056  df-fv 6057  df-riota 6774  df-ov 6816  df-oprab 6817  df-mpt2 6818  df-of 7062  df-om 7231  df-1st 7333  df-2nd 7334  df-supp 7464  df-cur 7562  df-wrecs 7576  df-recs 7637  df-rdg 7675  df-1o 7729  df-oadd 7733  df-er 7911  df-map 8025  df-ixp 8075  df-en 8122  df-dom 8123  df-sdom 8124  df-fin 8125  df-fsupp 8441  df-sup 8513  df-pnf 10268  df-mnf 10269  df-xr 10270  df-ltxr 10271  df-le 10272  df-sub 10460  df-neg 10461  df-nn 11213  df-2 11271  df-3 11272  df-4 11273  df-5 11274  df-6 11275  df-7 11276  df-8 11277  df-9 11278  df-n0 11485  df-z 11570  df-dec 11686  df-uz 11880  df-fz 12520  df-struct 16061  df-ndx 16062  df-slot 16063  df-base 16065  df-sets 16066  df-ress 16067  df-plusg 16156  df-mulr 16157  df-sca 16159  df-vsca 16160  df-ip 16161  df-tset 16162  df-ple 16163  df-ds 16166  df-hom 16168  df-cco 16169  df-0g 16304  df-prds 16310  df-pws 16312  df-sra 19374  df-rgmod 19375  df-psr 19558  df-opsr 19562  df-psr1 19752  df-ply1 19754  df-coe1 19755  df-dsmm 20278  df-frlm 20293  df-mat 20416  df-decpmat 20770
This theorem is referenced by:  pmatcollpw3  20791  pmatcollpw3fi  20792
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