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 479	. 2 ⊢ (((𝑁 ∈ Fin ∧ 𝑅 ∈ 𝑉 ∧ 𝑀 ∈ 𝐵) ∧ (𝑋 ∈ 𝑁 ∧ 𝑌 ∈ 𝑁)) → (𝑇‘𝑀) = (𝑥 ∈ 𝑁, 𝑦 ∈ 𝑁 ↦ (𝑆‘(𝑥𝑀𝑦))))
8		oveq12 7422	. . . 4 ⊢ ((𝑥 = 𝑋 ∧ 𝑦 = 𝑌) → (𝑥𝑀𝑦) = (𝑋𝑀𝑌))
9	8	fveq2d 6894	. . 3 ⊢ ((𝑥 = 𝑋 ∧ 𝑦 = 𝑌) → (𝑆‘(𝑥𝑀𝑦)) = (𝑆‘(𝑋𝑀𝑌)))
10	9	adantl 480	. 2 ⊢ ((((𝑁 ∈ Fin ∧ 𝑅 ∈ 𝑉 ∧ 𝑀 ∈ 𝐵) ∧ (𝑋 ∈ 𝑁 ∧ 𝑌 ∈ 𝑁)) ∧ (𝑥 = 𝑋 ∧ 𝑦 = 𝑌)) → (𝑆‘(𝑥𝑀𝑦)) = (𝑆‘(𝑋𝑀𝑌)))
11		simprl 769	. 2 ⊢ (((𝑁 ∈ Fin ∧ 𝑅 ∈ 𝑉 ∧ 𝑀 ∈ 𝐵) ∧ (𝑋 ∈ 𝑁 ∧ 𝑌 ∈ 𝑁)) → 𝑋 ∈ 𝑁)
12		simprr 771	. 2 ⊢ (((𝑁 ∈ Fin ∧ 𝑅 ∈ 𝑉 ∧ 𝑀 ∈ 𝐵) ∧ (𝑋 ∈ 𝑁 ∧ 𝑌 ∈ 𝑁)) → 𝑌 ∈ 𝑁)
13		fvexd 6905	. 2 ⊢ (((𝑁 ∈ Fin ∧ 𝑅 ∈ 𝑉 ∧ 𝑀 ∈ 𝐵) ∧ (𝑋 ∈ 𝑁 ∧ 𝑌 ∈ 𝑁)) → (𝑆‘(𝑋𝑀𝑌)) ∈ V)
14	7, 10, 11, 12, 13	ovmpod 7567	1 ⊢ (((𝑁 ∈ Fin ∧ 𝑅 ∈ 𝑉 ∧ 𝑀 ∈ 𝐵) ∧ (𝑋 ∈ 𝑁 ∧ 𝑌 ∈ 𝑁)) → (𝑋(𝑇‘𝑀)𝑌) = (𝑆‘(𝑋𝑀𝑌)))

Syntax hints:   → wi 4   ∧ wa 394   ∧ w3a 1084   = wceq 1533   ∈ wcel 2098  Vcvv 3463  ‘cfv 6543  (class class class)co 7413   ∈ cmpo 7415  Fincfn 8957  Basecbs 17174  algSccascl 21785  Poly₁cpl1 22099   Mat cmat 22320   matToPolyMat cmat2pmat 22619

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1905  ax-6 1963  ax-7 2003  ax-8 2100  ax-9 2108  ax-10 2129  ax-11 2146  ax-12 2166  ax-ext 2696  ax-rep 5281  ax-sep 5295  ax-nul 5302  ax-pow 5360  ax-pr 5424  ax-un 7735

This theorem depends on definitions:  df-bi 206  df-an 395  df-or 846  df-3an 1086  df-tru 1536  df-fal 1546  df-ex 1774  df-nf 1778  df-sb 2060  df-mo 2528  df-eu 2557  df-clab 2703  df-cleq 2717  df-clel 2802  df-nfc 2877  df-ne 2931  df-ral 3052  df-rex 3061  df-reu 3365  df-rab 3420  df-v 3465  df-sbc 3771  df-csb 3887  df-dif 3944  df-un 3946  df-in 3948  df-ss 3958  df-nul 4320  df-if 4526  df-pw 4601  df-sn 4626  df-pr 4628  df-op 4632  df-uni 4905  df-iun 4994  df-br 5145  df-opab 5207  df-mpt 5228  df-id 5571  df-xp 5679  df-rel 5680  df-cnv 5681  df-co 5682  df-dm 5683  df-rn 5684  df-res 5685  df-ima 5686  df-iota 6495  df-fun 6545  df-fn 6546  df-f 6547  df-f1 6548  df-fo 6549  df-f1o 6550  df-fv 6551  df-ov 7416  df-oprab 7417  df-mpo 7418  df-1st 7987  df-2nd 7988  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