Theorem mat2pmatfval 22661

Description: Value of the matrix transformation. (Contributed by AV, 31-Jul-2019.)

Hypotheses

Ref	Expression
mat2pmatfval.t	⊢ 𝑇 = (𝑁 matToPolyMat 𝑅)
mat2pmatfval.a	⊢ 𝐴 = (𝑁 Mat 𝑅)
mat2pmatfval.b	⊢ 𝐵 = (Base‘𝐴)
mat2pmatfval.p	⊢ 𝑃 = (Poly1‘𝑅)
mat2pmatfval.s	⊢ 𝑆 = (algSc‘𝑃)

Assertion

Ref	Expression
mat2pmatfval	⊢ ((𝑁 ∈ Fin ∧ 𝑅 ∈ 𝑉) → 𝑇 = (𝑚 ∈ 𝐵 ↦ (𝑥 ∈ 𝑁, 𝑦 ∈ 𝑁 ↦ (𝑆‘(𝑥𝑚𝑦)))))

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

Allowed substitution hints:   𝐴(𝑥,𝑦,𝑚)   𝐵(𝑥,𝑦)   𝑃(𝑥,𝑦,𝑚)   𝑆(𝑥,𝑦,𝑚)   𝑇(𝑥,𝑦,𝑚)   𝑉(𝑥,𝑦,𝑚)

Proof of Theorem mat2pmatfval

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

Step	Hyp	Ref	Expression
1		mat2pmatfval.t	. 2 ⊢ 𝑇 = (𝑁 matToPolyMat 𝑅)
2		df-mat2pmat 22645	. . . 4 ⊢ matToPolyMat = (𝑛 ∈ Fin, 𝑟 ∈ V ↦ (𝑚 ∈ (Base‘(𝑛 Mat 𝑟)) ↦ (𝑥 ∈ 𝑛, 𝑦 ∈ 𝑛 ↦ ((algSc‘(Poly1‘𝑟))‘(𝑥𝑚𝑦)))))
3	2	a1i 11	. . 3 ⊢ ((𝑁 ∈ Fin ∧ 𝑅 ∈ 𝑉) → matToPolyMat = (𝑛 ∈ Fin, 𝑟 ∈ V ↦ (𝑚 ∈ (Base‘(𝑛 Mat 𝑟)) ↦ (𝑥 ∈ 𝑛, 𝑦 ∈ 𝑛 ↦ ((algSc‘(Poly1‘𝑟))‘(𝑥𝑚𝑦))))))
4		oveq12 7414	. . . . . . 7 ⊢ ((𝑛 = 𝑁 ∧ 𝑟 = 𝑅) → (𝑛 Mat 𝑟) = (𝑁 Mat 𝑅))
5	4	fveq2d 6880	. . . . . 6 ⊢ ((𝑛 = 𝑁 ∧ 𝑟 = 𝑅) → (Base‘(𝑛 Mat 𝑟)) = (Base‘(𝑁 Mat 𝑅)))
6		mat2pmatfval.b	. . . . . . 7 ⊢ 𝐵 = (Base‘𝐴)
7		mat2pmatfval.a	. . . . . . . 8 ⊢ 𝐴 = (𝑁 Mat 𝑅)
8	7	fveq2i 6879	. . . . . . 7 ⊢ (Base‘𝐴) = (Base‘(𝑁 Mat 𝑅))
9	6, 8	eqtr2i 2759	. . . . . 6 ⊢ (Base‘(𝑁 Mat 𝑅)) = 𝐵
10	5, 9	eqtrdi 2786	. . . . 5 ⊢ ((𝑛 = 𝑁 ∧ 𝑟 = 𝑅) → (Base‘(𝑛 Mat 𝑟)) = 𝐵)
11		simpl 482	. . . . . 6 ⊢ ((𝑛 = 𝑁 ∧ 𝑟 = 𝑅) → 𝑛 = 𝑁)
12		2fveq3 6881	. . . . . . . . 9 ⊢ (𝑟 = 𝑅 → (algSc‘(Poly1‘𝑟)) = (algSc‘(Poly1‘𝑅)))
13	12	adantl 481	. . . . . . . 8 ⊢ ((𝑛 = 𝑁 ∧ 𝑟 = 𝑅) → (algSc‘(Poly1‘𝑟)) = (algSc‘(Poly1‘𝑅)))
14		mat2pmatfval.s	. . . . . . . . 9 ⊢ 𝑆 = (algSc‘𝑃)
15		mat2pmatfval.p	. . . . . . . . . 10 ⊢ 𝑃 = (Poly1‘𝑅)
16	15	fveq2i 6879	. . . . . . . . 9 ⊢ (algSc‘𝑃) = (algSc‘(Poly1‘𝑅))
17	14, 16	eqtr2i 2759	. . . . . . . 8 ⊢ (algSc‘(Poly1‘𝑅)) = 𝑆
18	13, 17	eqtrdi 2786	. . . . . . 7 ⊢ ((𝑛 = 𝑁 ∧ 𝑟 = 𝑅) → (algSc‘(Poly1‘𝑟)) = 𝑆)
19	18	fveq1d 6878	. . . . . 6 ⊢ ((𝑛 = 𝑁 ∧ 𝑟 = 𝑅) → ((algSc‘(Poly1‘𝑟))‘(𝑥𝑚𝑦)) = (𝑆‘(𝑥𝑚𝑦)))
20	11, 11, 19	mpoeq123dv 7482	. . . . 5 ⊢ ((𝑛 = 𝑁 ∧ 𝑟 = 𝑅) → (𝑥 ∈ 𝑛, 𝑦 ∈ 𝑛 ↦ ((algSc‘(Poly1‘𝑟))‘(𝑥𝑚𝑦))) = (𝑥 ∈ 𝑁, 𝑦 ∈ 𝑁 ↦ (𝑆‘(𝑥𝑚𝑦))))
21	10, 20	mpteq12dv 5207	. . . 4 ⊢ ((𝑛 = 𝑁 ∧ 𝑟 = 𝑅) → (𝑚 ∈ (Base‘(𝑛 Mat 𝑟)) ↦ (𝑥 ∈ 𝑛, 𝑦 ∈ 𝑛 ↦ ((algSc‘(Poly1‘𝑟))‘(𝑥𝑚𝑦)))) = (𝑚 ∈ 𝐵 ↦ (𝑥 ∈ 𝑁, 𝑦 ∈ 𝑁 ↦ (𝑆‘(𝑥𝑚𝑦)))))
22	21	adantl 481	. . 3 ⊢ (((𝑁 ∈ Fin ∧ 𝑅 ∈ 𝑉) ∧ (𝑛 = 𝑁 ∧ 𝑟 = 𝑅)) → (𝑚 ∈ (Base‘(𝑛 Mat 𝑟)) ↦ (𝑥 ∈ 𝑛, 𝑦 ∈ 𝑛 ↦ ((algSc‘(Poly1‘𝑟))‘(𝑥𝑚𝑦)))) = (𝑚 ∈ 𝐵 ↦ (𝑥 ∈ 𝑁, 𝑦 ∈ 𝑁 ↦ (𝑆‘(𝑥𝑚𝑦)))))
23		simpl 482	. . 3 ⊢ ((𝑁 ∈ Fin ∧ 𝑅 ∈ 𝑉) → 𝑁 ∈ Fin)
24		elex 3480	. . . 4 ⊢ (𝑅 ∈ 𝑉 → 𝑅 ∈ V)
25	24	adantl 481	. . 3 ⊢ ((𝑁 ∈ Fin ∧ 𝑅 ∈ 𝑉) → 𝑅 ∈ V)
26	6	fvexi 6890	. . . 4 ⊢ 𝐵 ∈ V
27		mptexg 7213	. . . 4 ⊢ (𝐵 ∈ V → (𝑚 ∈ 𝐵 ↦ (𝑥 ∈ 𝑁, 𝑦 ∈ 𝑁 ↦ (𝑆‘(𝑥𝑚𝑦)))) ∈ V)
28	26, 27	mp1i 13	. . 3 ⊢ ((𝑁 ∈ Fin ∧ 𝑅 ∈ 𝑉) → (𝑚 ∈ 𝐵 ↦ (𝑥 ∈ 𝑁, 𝑦 ∈ 𝑁 ↦ (𝑆‘(𝑥𝑚𝑦)))) ∈ V)
29	3, 22, 23, 25, 28	ovmpod 7559	. 2 ⊢ ((𝑁 ∈ Fin ∧ 𝑅 ∈ 𝑉) → (𝑁 matToPolyMat 𝑅) = (𝑚 ∈ 𝐵 ↦ (𝑥 ∈ 𝑁, 𝑦 ∈ 𝑁 ↦ (𝑆‘(𝑥𝑚𝑦)))))
30	1, 29	eqtrid 2782	1 ⊢ ((𝑁 ∈ Fin ∧ 𝑅 ∈ 𝑉) → 𝑇 = (𝑚 ∈ 𝐵 ↦ (𝑥 ∈ 𝑁, 𝑦 ∈ 𝑁 ↦ (𝑆‘(𝑥𝑚𝑦)))))

Syntax hints:   → wi 4   ∧ wa 395   = wceq 1540   ∈ wcel 2108  Vcvv 3459   ↦ cmpt 5201  ‘cfv 6531  (class class class)co 7405   ∈ cmpo 7407  Fincfn 8959  Basecbs 17228  algSccascl 21812  Poly₁cpl1 22112   Mat cmat 22345   matToPolyMat cmat2pmat 22642

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-10 2141  ax-11 2157  ax-12 2177  ax-ext 2707  ax-rep 5249  ax-sep 5266  ax-nul 5276  ax-pr 5402

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2065  df-mo 2539  df-eu 2568  df-clab 2714  df-cleq 2727  df-clel 2809  df-nfc 2885  df-ne 2933  df-ral 3052  df-rex 3061  df-reu 3360  df-rab 3416  df-v 3461  df-sbc 3766  df-csb 3875  df-dif 3929  df-un 3931  df-in 3933  df-ss 3943  df-nul 4309  df-if 4501  df-sn 4602  df-pr 4604  df-op 4608  df-uni 4884  df-iun 4969  df-br 5120  df-opab 5182  df-mpt 5202  df-id 5548  df-xp 5660  df-rel 5661  df-cnv 5662  df-co 5663  df-dm 5664  df-rn 5665  df-res 5666  df-ima 5667  df-iota 6484  df-fun 6533  df-fn 6534  df-f 6535  df-f1 6536  df-fo 6537  df-f1o 6538  df-fv 6539  df-ov 7408  df-oprab 7409  df-mpo 7410  df-mat2pmat 22645

This theorem is referenced by:  mat2pmatval  22662  mat2pmatf  22666  m2cpmf  22680