Theorem mat2pmatvalel 22640

Description: A (matrix) element of the result of a 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
mat2pmatvalel	⊢ (((𝑁 ∈ Fin ∧ 𝑅 ∈ 𝑉 ∧ 𝑀 ∈ 𝐵) ∧ (𝑋 ∈ 𝑁 ∧ 𝑌 ∈ 𝑁)) → (𝑋(𝑇‘𝑀)𝑌) = (𝑆‘(𝑋𝑀𝑌)))

Proof of Theorem mat2pmatvalel

Dummy variables 𝑥 𝑦 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		mat2pmatfval.t	. . . 4 ⊢ 𝑇 = (𝑁 matToPolyMat 𝑅)
2		mat2pmatfval.a	. . . 4 ⊢ 𝐴 = (𝑁 Mat 𝑅)
3		mat2pmatfval.b	. . . 4 ⊢ 𝐵 = (Base‘𝐴)
4		mat2pmatfval.p	. . . 4 ⊢ 𝑃 = (Poly1‘𝑅)
5		mat2pmatfval.s	. . . 4 ⊢ 𝑆 = (algSc‘𝑃)
6	1, 2, 3, 4, 5	mat2pmatval 22639	. . 3 ⊢ ((𝑁 ∈ Fin ∧ 𝑅 ∈ 𝑉 ∧ 𝑀 ∈ 𝐵) → (𝑇‘𝑀) = (𝑥 ∈ 𝑁, 𝑦 ∈ 𝑁 ↦ (𝑆‘(𝑥𝑀𝑦))))
7	6	adantr 480	. 2 ⊢ (((𝑁 ∈ Fin ∧ 𝑅 ∈ 𝑉 ∧ 𝑀 ∈ 𝐵) ∧ (𝑋 ∈ 𝑁 ∧ 𝑌 ∈ 𝑁)) → (𝑇‘𝑀) = (𝑥 ∈ 𝑁, 𝑦 ∈ 𝑁 ↦ (𝑆‘(𝑥𝑀𝑦))))
8		oveq12 7355	. . . 4 ⊢ ((𝑥 = 𝑋 ∧ 𝑦 = 𝑌) → (𝑥𝑀𝑦) = (𝑋𝑀𝑌))
9	8	fveq2d 6826	. . 3 ⊢ ((𝑥 = 𝑋 ∧ 𝑦 = 𝑌) → (𝑆‘(𝑥𝑀𝑦)) = (𝑆‘(𝑋𝑀𝑌)))
10	9	adantl 481	. 2 ⊢ ((((𝑁 ∈ Fin ∧ 𝑅 ∈ 𝑉 ∧ 𝑀 ∈ 𝐵) ∧ (𝑋 ∈ 𝑁 ∧ 𝑌 ∈ 𝑁)) ∧ (𝑥 = 𝑋 ∧ 𝑦 = 𝑌)) → (𝑆‘(𝑥𝑀𝑦)) = (𝑆‘(𝑋𝑀𝑌)))
11		simprl 770	. 2 ⊢ (((𝑁 ∈ Fin ∧ 𝑅 ∈ 𝑉 ∧ 𝑀 ∈ 𝐵) ∧ (𝑋 ∈ 𝑁 ∧ 𝑌 ∈ 𝑁)) → 𝑋 ∈ 𝑁)
12		simprr 772	. 2 ⊢ (((𝑁 ∈ Fin ∧ 𝑅 ∈ 𝑉 ∧ 𝑀 ∈ 𝐵) ∧ (𝑋 ∈ 𝑁 ∧ 𝑌 ∈ 𝑁)) → 𝑌 ∈ 𝑁)
13		fvexd 6837	. 2 ⊢ (((𝑁 ∈ Fin ∧ 𝑅 ∈ 𝑉 ∧ 𝑀 ∈ 𝐵) ∧ (𝑋 ∈ 𝑁 ∧ 𝑌 ∈ 𝑁)) → (𝑆‘(𝑋𝑀𝑌)) ∈ V)
14	7, 10, 11, 12, 13	ovmpod 7498	1 ⊢ (((𝑁 ∈ Fin ∧ 𝑅 ∈ 𝑉 ∧ 𝑀 ∈ 𝐵) ∧ (𝑋 ∈ 𝑁 ∧ 𝑌 ∈ 𝑁)) → (𝑋(𝑇‘𝑀)𝑌) = (𝑆‘(𝑋𝑀𝑌)))

Syntax hints:   → wi 4   ∧ wa 395   ∧ w3a 1086   = wceq 1541   ∈ wcel 2111  Vcvv 3436  ‘cfv 6481  (class class class)co 7346   ∈ cmpo 7348  Fincfn 8869  Basecbs 17120  algSccascl 21789  Poly₁cpl1 22089   Mat cmat 22322   matToPolyMat cmat2pmat 22619

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1911  ax-6 1968  ax-7 2009  ax-8 2113  ax-9 2121  ax-10 2144  ax-11 2160  ax-12 2180  ax-ext 2703  ax-rep 5215  ax-sep 5232  ax-nul 5242  ax-pow 5301  ax-pr 5368  ax-un 7668

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1544  df-fal 1554  df-ex 1781  df-nf 1785  df-sb 2068  df-mo 2535  df-eu 2564  df-clab 2710  df-cleq 2723  df-clel 2806  df-nfc 2881  df-ne 2929  df-ral 3048  df-rex 3057  df-reu 3347  df-rab 3396  df-v 3438  df-sbc 3737  df-csb 3846  df-dif 3900  df-un 3902  df-in 3904  df-ss 3914  df-nul 4281  df-if 4473  df-pw 4549  df-sn 4574  df-pr 4576  df-op 4580  df-uni 4857  df-iun 4941  df-br 5090  df-opab 5152  df-mpt 5171  df-id 5509  df-xp 5620  df-rel 5621  df-cnv 5622  df-co 5623  df-dm 5624  df-rn 5625  df-res 5626  df-ima 5627  df-iota 6437  df-fun 6483  df-fn 6484  df-f 6485  df-f1 6486  df-fo 6487  df-f1o 6488  df-fv 6489  df-ov 7349  df-oprab 7350  df-mpo 7351  df-1st 7921  df-2nd 7922  df-mat2pmat 22622

This theorem is referenced by:  mat2pmatf1  22644  mat2pmat1  22647  mat2pmatlin  22650  m2cpm  22656  m2cpminvid  22668  monmatcollpw  22694  chpmat1dlem  22750  chpdmatlem2  22754  chpdmatlem3  22755