Theorem mat2pmatvalel 22712

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 22711	. . 3 ⊢ ((𝑁 ∈ Fin ∧ 𝑅 ∈ 𝑉 ∧ 𝑀 ∈ 𝐵) → (𝑇‘𝑀) = (𝑥 ∈ 𝑁, 𝑦 ∈ 𝑁 ↦ (𝑆‘(𝑥𝑀𝑦))))
7	6	adantr 482	. 2 ⊢ (((𝑁 ∈ Fin ∧ 𝑅 ∈ 𝑉 ∧ 𝑀 ∈ 𝐵) ∧ (𝑋 ∈ 𝑁 ∧ 𝑌 ∈ 𝑁)) → (𝑇‘𝑀) = (𝑥 ∈ 𝑁, 𝑦 ∈ 𝑁 ↦ (𝑆‘(𝑥𝑀𝑦))))
8		oveq12 7369	. . . 4 ⊢ ((𝑥 = 𝑋 ∧ 𝑦 = 𝑌) → (𝑥𝑀𝑦) = (𝑋𝑀𝑌))
9	8	fveq2d 6835	. . 3 ⊢ ((𝑥 = 𝑋 ∧ 𝑦 = 𝑌) → (𝑆‘(𝑥𝑀𝑦)) = (𝑆‘(𝑋𝑀𝑌)))
10	9	adantl 483	. 2 ⊢ ((((𝑁 ∈ Fin ∧ 𝑅 ∈ 𝑉 ∧ 𝑀 ∈ 𝐵) ∧ (𝑋 ∈ 𝑁 ∧ 𝑌 ∈ 𝑁)) ∧ (𝑥 = 𝑋 ∧ 𝑦 = 𝑌)) → (𝑆‘(𝑥𝑀𝑦)) = (𝑆‘(𝑋𝑀𝑌)))
11		simprl 777	. 2 ⊢ (((𝑁 ∈ Fin ∧ 𝑅 ∈ 𝑉 ∧ 𝑀 ∈ 𝐵) ∧ (𝑋 ∈ 𝑁 ∧ 𝑌 ∈ 𝑁)) → 𝑋 ∈ 𝑁)
12		simprr 779	. 2 ⊢ (((𝑁 ∈ Fin ∧ 𝑅 ∈ 𝑉 ∧ 𝑀 ∈ 𝐵) ∧ (𝑋 ∈ 𝑁 ∧ 𝑌 ∈ 𝑁)) → 𝑌 ∈ 𝑁)
13		fvexd 6846	. 2 ⊢ (((𝑁 ∈ Fin ∧ 𝑅 ∈ 𝑉 ∧ 𝑀 ∈ 𝐵) ∧ (𝑋 ∈ 𝑁 ∧ 𝑌 ∈ 𝑁)) → (𝑆‘(𝑋𝑀𝑌)) ∈ V)
14	7, 10, 11, 12, 13	ovmpod 7512	1 ⊢ (((𝑁 ∈ Fin ∧ 𝑅 ∈ 𝑉 ∧ 𝑀 ∈ 𝐵) ∧ (𝑋 ∈ 𝑁 ∧ 𝑌 ∈ 𝑁)) → (𝑋(𝑇‘𝑀)𝑌) = (𝑆‘(𝑋𝑀𝑌)))

Syntax hints:   → wi 4   ∧ wa 397   ∧ w3a 1093   = wceq 1548   ∈ wcel 2121  Vcvv 3433  ‘cfv 6489  (class class class)co 7360   ∈ cmpo 7362  Fincfn 8887  Basecbs 17174  algSccascl 21831  Poly₁cpl1 22166   Mat cmat 22394   matToPolyMat cmat2pmat 22691

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1803  ax-4 1817  ax-5 1918  ax-6 1975  ax-7 2016  ax-8 2123  ax-9 2131  ax-10 2154  ax-11 2170  ax-12 2191  ax-ext 2713  ax-rep 5202  ax-sep 5221  ax-nul 5231  ax-pow 5297  ax-pr 5365  ax-un 7682

This theorem depends on definitions:  df-bi 209  df-an 398  df-or 855  df-3an 1095  df-tru 1551  df-fal 1561  df-ex 1788  df-nf 1792  df-sb 2075  df-mo 2545  df-eu 2575  df-clab 2720  df-cleq 2733  df-clel 2816  df-nfc 2890  df-ne 2937  df-ral 3056  df-rex 3066  df-reu 3347  df-rab 3394  df-v 3435  df-sbc 3726  df-csb 3834  df-dif 3888  df-un 3890  df-in 3892  df-ss 3902  df-nul 4265  df-if 4458  df-pw 4534  df-sn 4559  df-pr 4561  df-op 4565  df-uni 4842  df-iun 4926  df-br 5076  df-opab 5138  df-mpt 5157  df-id 5516  df-xp 5627  df-rel 5628  df-cnv 5629  df-co 5630  df-dm 5631  df-rn 5632  df-res 5633  df-ima 5634  df-iota 6445  df-fun 6491  df-fn 6492  df-f 6493  df-f1 6494  df-fo 6495  df-f1o 6496  df-fv 6497  df-ov 7363  df-oprab 7364  df-mpo 7365  df-1st 7935  df-2nd 7936  df-mat2pmat 22694

This theorem is referenced by:  mat2pmatf1  22716  mat2pmat1  22719  mat2pmatlin  22722  m2cpm  22728  m2cpminvid  22740  monmatcollpw  22766  chpmat1dlem  22822  chpdmatlem2  22826  chpdmatlem3  22827