Theorem mat2pmatvalel 22619

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 22618	. . 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 7399	. . . 4 ⊢ ((𝑥 = 𝑋 ∧ 𝑦 = 𝑌) → (𝑥𝑀𝑦) = (𝑋𝑀𝑌))
9	8	fveq2d 6865	. . 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 6876	. 2 ⊢ (((𝑁 ∈ Fin ∧ 𝑅 ∈ 𝑉 ∧ 𝑀 ∈ 𝐵) ∧ (𝑋 ∈ 𝑁 ∧ 𝑌 ∈ 𝑁)) → (𝑆‘(𝑋𝑀𝑌)) ∈ V)
14	7, 10, 11, 12, 13	ovmpod 7544	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 1540   ∈ wcel 2109  Vcvv 3450  ‘cfv 6514  (class class class)co 7390   ∈ cmpo 7392  Fincfn 8921  Basecbs 17186  algSccascl 21768  Poly₁cpl1 22068   Mat cmat 22301   matToPolyMat cmat2pmat 22598

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 2008  ax-8 2111  ax-9 2119  ax-10 2142  ax-11 2158  ax-12 2178  ax-ext 2702  ax-rep 5237  ax-sep 5254  ax-nul 5264  ax-pow 5323  ax-pr 5390  ax-un 7714

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 2066  df-mo 2534  df-eu 2563  df-clab 2709  df-cleq 2722  df-clel 2804  df-nfc 2879  df-ne 2927  df-ral 3046  df-rex 3055  df-reu 3357  df-rab 3409  df-v 3452  df-sbc 3757  df-csb 3866  df-dif 3920  df-un 3922  df-in 3924  df-ss 3934  df-nul 4300  df-if 4492  df-pw 4568  df-sn 4593  df-pr 4595  df-op 4599  df-uni 4875  df-iun 4960  df-br 5111  df-opab 5173  df-mpt 5192  df-id 5536  df-xp 5647  df-rel 5648  df-cnv 5649  df-co 5650  df-dm 5651  df-rn 5652  df-res 5653  df-ima 5654  df-iota 6467  df-fun 6516  df-fn 6517  df-f 6518  df-f1 6519  df-fo 6520  df-f1o 6521  df-fv 6522  df-ov 7393  df-oprab 7394  df-mpo 7395  df-1st 7971  df-2nd 7972  df-mat2pmat 22601

This theorem is referenced by:  mat2pmatf1  22623  mat2pmat1  22626  mat2pmatlin  22629  m2cpm  22635  m2cpminvid  22647  monmatcollpw  22673  chpmat1dlem  22729  chpdmatlem2  22733  chpdmatlem3  22734