Theorem cpm2mfval 22789

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 22748	. . . 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 7401	. . . . . 6 ⊢ ((𝑛 = 𝑁 ∧ 𝑟 = 𝑅) → (𝑛 ConstPolyMat 𝑟) = (𝑁 ConstPolyMat 𝑅))
5		cpm2mfval.s	. . . . . 6 ⊢ 𝑆 = (𝑁 ConstPolyMat 𝑅)
6	4, 5	eqtr4di 2814	. . . . 5 ⊢ ((𝑛 = 𝑁 ∧ 𝑟 = 𝑅) → (𝑛 ConstPolyMat 𝑟) = 𝑆)
7		simpl 486	. . . . . 6 ⊢ ((𝑛 = 𝑁 ∧ 𝑟 = 𝑅) → 𝑛 = 𝑁)
8		eqidd 2762	. . . . . 6 ⊢ ((𝑛 = 𝑁 ∧ 𝑟 = 𝑅) → ((coe1‘(𝑥𝑚𝑦))‘0) = ((coe1‘(𝑥𝑚𝑦))‘0))
9	7, 7, 8	mpoeq123dv 7467	. . . . 5 ⊢ ((𝑛 = 𝑁 ∧ 𝑟 = 𝑅) → (𝑥 ∈ 𝑛, 𝑦 ∈ 𝑛 ↦ ((coe1‘(𝑥𝑚𝑦))‘0)) = (𝑥 ∈ 𝑁, 𝑦 ∈ 𝑁 ↦ ((coe1‘(𝑥𝑚𝑦))‘0)))
10	6, 9	mpteq12dv 5186	. . . 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 485	. . 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 486	. . 3 ⊢ ((𝑁 ∈ Fin ∧ 𝑅 ∈ 𝑉) → 𝑁 ∈ Fin)
13		elex 3474	. . . 4 ⊢ (𝑅 ∈ 𝑉 → 𝑅 ∈ V)
14	13	adantl 485	. . 3 ⊢ ((𝑁 ∈ Fin ∧ 𝑅 ∈ 𝑉) → 𝑅 ∈ V)
15	5	ovexi 7426	. . . 4 ⊢ 𝑆 ∈ V
16		mptexg 7201	. . . 4 ⊢ (𝑆 ∈ V → (𝑚 ∈ 𝑆 ↦ (𝑥 ∈ 𝑁, 𝑦 ∈ 𝑁 ↦ ((coe1‘(𝑥𝑚𝑦))‘0))) ∈ V)
17	15, 16	mp1i 13	. . 3 ⊢ ((𝑁 ∈ Fin ∧ 𝑅 ∈ 𝑉) → (𝑚 ∈ 𝑆 ↦ (𝑥 ∈ 𝑁, 𝑦 ∈ 𝑁 ↦ ((coe1‘(𝑥𝑚𝑦))‘0))) ∈ V)
18	3, 11, 12, 14, 17	ovmpod 7544	. 2 ⊢ ((𝑁 ∈ Fin ∧ 𝑅 ∈ 𝑉) → (𝑁 cPolyMatToMat 𝑅) = (𝑚 ∈ 𝑆 ↦ (𝑥 ∈ 𝑁, 𝑦 ∈ 𝑁 ↦ ((coe1‘(𝑥𝑚𝑦))‘0))))
19	1, 18	eqtrid 2808	1 ⊢ ((𝑁 ∈ Fin ∧ 𝑅 ∈ 𝑉) → 𝐼 = (𝑚 ∈ 𝑆 ↦ (𝑥 ∈ 𝑁, 𝑦 ∈ 𝑁 ↦ ((coe1‘(𝑥𝑚𝑦))‘0))))

Syntax hints:   → wi 4   ∧ wa 399   = wceq 1559   ∈ wcel 2141  Vcvv 3453   ↦ cmpt 5180  ‘cfv 6517  (class class class)co 7392   ∈ cmpo 7394  Fincfn 8923  0cc0 11070  coe₁cco1 22220   ConstPolyMat ccpmat 22743   cPolyMatToMat ccpmat2mat 22745

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1814  ax-4 1828  ax-5 1929  ax-6 1986  ax-7 2027  ax-8 2143  ax-9 2151  ax-10 2174  ax-11 2190  ax-12 2211  ax-ext 2733  ax-rep 5226  ax-sep 5245  ax-nul 5255  ax-pr 5389

This theorem depends on definitions:  df-bi 209  df-an 400  df-or 859  df-3an 1099  df-tru 1562  df-fal 1572  df-ex 1799  df-nf 1803  df-sb 2090  df-mo 2565  df-eu 2595  df-clab 2740  df-cleq 2753  df-clel 2836  df-nfc 2910  df-ne 2957  df-ral 3076  df-rex 3086  df-reu 3367  df-rab 3414  df-v 3455  df-sbc 3745  df-csb 3853  df-dif 3907  df-un 3909  df-in 3911  df-ss 3921  df-nul 4286  df-if 4480  df-sn 4582  df-pr 4584  df-op 4588  df-uni 4865  df-iun 4950  df-br 5100  df-opab 5162  df-mpt 5181  df-id 5540  df-xp 5651  df-rel 5652  df-cnv 5653  df-co 5654  df-dm 5655  df-rn 5656  df-res 5657  df-ima 5658  df-iota 6473  df-fun 6519  df-fn 6520  df-f 6521  df-f1 6522  df-fo 6523  df-f1o 6524  df-fv 6525  df-ov 7395  df-oprab 7396  df-mpo 7397  df-cpmat2mat 22748

This theorem is referenced by:  cpm2mval  22790  cpm2mf  22792  m2cpmfo  22796  cayleyhamiltonALT  22931