Theorem cpm2mfval 22980

Description: Value of the inverse matrix transformation. (Contributed by AV, 14-Dec-2019.)

Hypotheses

Ref	Expression
cpm2mfval.i	⊢ 𝐼 = (𝑁 cPolyMatToMat 𝑅)
cpm2mfval.s	⊢ 𝑆 = (𝑁 ConstPolyMat 𝑅)

Assertion

Ref	Expression
cpm2mfval	⊢ ((𝑁 ∈ Fin ∧ 𝑅 ∈ 𝑉) → 𝐼 = (𝑚 ∈ 𝑆 ↦ (𝑥 ∈ 𝑁, 𝑦 ∈ 𝑁 ↦ ((coe1‘(𝑥𝑚𝑦))‘0))))

Distinct variable groups:   𝑚,𝑁,𝑥,𝑦   𝑅,𝑚,𝑥,𝑦   𝑆,𝑚

Allowed substitution hints:   𝑆(𝑥, 𝑦)   𝐼(𝑥, 𝑦, 𝑚)   𝑉(𝑥, 𝑦, 𝑚)

Proof of Theorem cpm2mfval

Dummy variables 𝑛 𝑟 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		cpm2mfval.i	. 2 ⊢ 𝐼 = (𝑁 cPolyMatToMat 𝑅)
2		df-cpmat2mat 22939	. . . 4 ⊢ cPolyMatToMat = (𝑛 ∈ Fin, 𝑟 ∈ V ↦ (𝑚 ∈ (𝑛 ConstPolyMat 𝑟) ↦ (𝑥 ∈ 𝑛, 𝑦 ∈ 𝑛 ↦ ((coe1‘(𝑥𝑚𝑦))‘0))))
3	2	a1i 11	. . 3 ⊢ ((𝑁 ∈ Fin ∧ 𝑅 ∈ 𝑉) → cPolyMatToMat = (𝑛 ∈ Fin, 𝑟 ∈ V ↦ (𝑚 ∈ (𝑛 ConstPolyMat 𝑟) ↦ (𝑥 ∈ 𝑛, 𝑦 ∈ 𝑛 ↦ ((coe1‘(𝑥𝑚𝑦))‘0)))))
4		oveq12 7426	. . . . . 6 ⊢ ((𝑛 = 𝑁 ∧ 𝑟 = 𝑅) → (𝑛 ConstPolyMat 𝑟) = (𝑁 ConstPolyMat 𝑅))
5		cpm2mfval.s	. . . . . 6 ⊢ 𝑆 = (𝑁 ConstPolyMat 𝑅)
6	4, 5	eqtr4di 2815	. . . . 5 ⊢ ((𝑛 = 𝑁 ∧ 𝑟 = 𝑅) → (𝑛 ConstPolyMat 𝑟) = 𝑆)
7		simpl 488	. . . . . 6 ⊢ ((𝑛 = 𝑁 ∧ 𝑟 = 𝑅) → 𝑛 = 𝑁)
8		eqidd 2763	. . . . . 6 ⊢ ((𝑛 = 𝑁 ∧ 𝑟 = 𝑅) → ((coe1‘(𝑥𝑚𝑦))‘0) = ((coe1‘(𝑥𝑚𝑦))‘0))
9	7, 7, 8	mpoeq123dv 7492	. . . . 5 ⊢ ((𝑛 = 𝑁 ∧ 𝑟 = 𝑅) → (𝑥 ∈ 𝑛, 𝑦 ∈ 𝑛 ↦ ((coe1‘(𝑥𝑚𝑦))‘0)) = (𝑥 ∈ 𝑁, 𝑦 ∈ 𝑁 ↦ ((coe1‘(𝑥𝑚𝑦))‘0)))
10	6, 9	mpteq12dv 5196	. . . 4 ⊢ ((𝑛 = 𝑁 ∧ 𝑟 = 𝑅) → (𝑚 ∈ (𝑛 ConstPolyMat 𝑟) ↦ (𝑥 ∈ 𝑛, 𝑦 ∈ 𝑛 ↦ ((coe1‘(𝑥𝑚𝑦))‘0))) = (𝑚 ∈ 𝑆 ↦ (𝑥 ∈ 𝑁, 𝑦 ∈ 𝑁 ↦ ((coe1‘(𝑥𝑚𝑦))‘0))))
11	10	adantl 487	. . 3 ⊢ (((𝑁 ∈ Fin ∧ 𝑅 ∈ 𝑉) ∧ (𝑛 = 𝑁 ∧ 𝑟 = 𝑅)) → (𝑚 ∈ (𝑛 ConstPolyMat 𝑟) ↦ (𝑥 ∈ 𝑛, 𝑦 ∈ 𝑛 ↦ ((coe1‘(𝑥𝑚𝑦))‘0))) = (𝑚 ∈ 𝑆 ↦ (𝑥 ∈ 𝑁, 𝑦 ∈ 𝑁 ↦ ((coe1‘(𝑥𝑚𝑦))‘0))))
12		simpl 488	. . 3 ⊢ ((𝑁 ∈ Fin ∧ 𝑅 ∈ 𝑉) → 𝑁 ∈ Fin)
13		elex 3474	. . . 4 ⊢ (𝑅 ∈ 𝑉 → 𝑅 ∈ V)
14	13	adantl 487	. . 3 ⊢ ((𝑁 ∈ Fin ∧ 𝑅 ∈ 𝑉) → 𝑅 ∈ V)
15	5	ovexi 7451	. . . 4 ⊢ 𝑆 ∈ V
16		mptexg 7224	. . . 4 ⊢ (𝑆 ∈ V → (𝑚 ∈ 𝑆 ↦ (𝑥 ∈ 𝑁, 𝑦 ∈ 𝑁 ↦ ((coe1‘(𝑥𝑚𝑦))‘0))) ∈ V)
17	15, 16	mp1i 14	. . 3 ⊢ ((𝑁 ∈ Fin ∧ 𝑅 ∈ 𝑉) → (𝑚 ∈ 𝑆 ↦ (𝑥 ∈ 𝑁, 𝑦 ∈ 𝑁 ↦ ((coe1‘(𝑥𝑚𝑦))‘0))) ∈ V)
18	3, 11, 12, 14, 17	ovmpod 7569	. 2 ⊢ ((𝑁 ∈ Fin ∧ 𝑅 ∈ 𝑉) → (𝑁 cPolyMatToMat 𝑅) = (𝑚 ∈ 𝑆 ↦ (𝑥 ∈ 𝑁, 𝑦 ∈ 𝑁 ↦ ((coe1‘(𝑥𝑚𝑦))‘0))))
19	1, 18	eqtrid 2809	1 ⊢ ((𝑁 ∈ Fin ∧ 𝑅 ∈ 𝑉) → 𝐼 = (𝑚 ∈ 𝑆 ↦ (𝑥 ∈ 𝑁, 𝑦 ∈ 𝑁 ↦ ((coe1‘(𝑥𝑚𝑦))‘0))))

This proof depends on syntax axioms:   → wi 4   ∧ wa 401   = wceq 1570   ∈ wcel 2145  Vcvv 3453   ↦ cmpt 5190  ‘cfv 6537  (class class class)co 7417   ∈ cmpo 7419  Fincfn 8956  0cc0 11128  coe₁cco1 22409   ConstPolyMat ccpmat 22934   cPolyMatToMat ccpmat2mat 22936

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1828  ax-4 1842  ax-5 1943  ax-6 2000  ax-7 2041  ax-8 2147  ax-9 2155  ax-10 2178  ax-11 2194  ax-12 2215  ax-ext 2734  ax-rep 5236  ax-sep 5255  ax-nul 5267  ax-pr 5402

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3an 1105  df-tru 1573  df-fal 1583  df-ex 1813  df-nf 1817  df-sb 2100  df-mo 2566  df-eu 2596  df-clab 2741  df-cleq 2754  df-clel 2837  df-nfc 2911  df-ne 2958  df-ral 3079  df-rex 3089  df-reu 3368  df-rab 3415  df-v 3455  df-sbc 3743  df-csb 3851  df-dif 3905  df-un 3907  df-in 3909  df-ss 3919  df-nul 4283  df-if 4486  df-sn 4588  df-pr 4590  df-op 4594  df-uni 4871  df-iun 4956  df-br 5108  df-opab 5172  df-mpt 5191  df-id 5554  df-xp 5665  df-rel 5666  df-cnv 5667  df-co 5668  df-dm 5669  df-rn 5670  df-res 5671  df-ima 5672  df-iota 6493  df-fun 6539  df-fn 6540  df-f 6541  df-f1 6542  df-fo 6543  df-f1o 6544  df-fv 6545  df-ov 7420  df-oprab 7421  df-mpo 7422  df-cpmat2mat 22939

This theorem is used by:  cpm2mval  22981  cpm2mf  22983  m2cpmfo  22987  cayleyhamiltonALT  23122