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Theorem d1mat2pmat 22232
Description: The transformation of a matrix of dimenson 1. (Contributed by AV, 4-Aug-2019.)
Hypotheses
Ref Expression
d1mat2pmat.t 𝑇 = (𝑁 matToPolyMat 𝑅)
d1mat2pmat.b 𝐡 = (Baseβ€˜(𝑁 Mat 𝑅))
d1mat2pmat.p 𝑃 = (Poly1β€˜π‘…)
d1mat2pmat.s 𝑆 = (algScβ€˜π‘ƒ)
Assertion
Ref Expression
d1mat2pmat ((𝑅 ∈ 𝑉 ∧ (𝑁 = {𝐴} ∧ 𝐴 ∈ 𝑉) ∧ 𝑀 ∈ 𝐡) β†’ (π‘‡β€˜π‘€) = {⟨⟨𝐴, 𝐴⟩, (π‘†β€˜(𝐴𝑀𝐴))⟩})

Proof of Theorem d1mat2pmat
Dummy variables 𝑖 𝑗 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 snfi 9040 . . . . . 6 {𝐴} ∈ Fin
2 eleq1 2821 . . . . . 6 (𝑁 = {𝐴} β†’ (𝑁 ∈ Fin ↔ {𝐴} ∈ Fin))
31, 2mpbiri 257 . . . . 5 (𝑁 = {𝐴} β†’ 𝑁 ∈ Fin)
43adantr 481 . . . 4 ((𝑁 = {𝐴} ∧ 𝐴 ∈ 𝑉) β†’ 𝑁 ∈ Fin)
543ad2ant2 1134 . . 3 ((𝑅 ∈ 𝑉 ∧ (𝑁 = {𝐴} ∧ 𝐴 ∈ 𝑉) ∧ 𝑀 ∈ 𝐡) β†’ 𝑁 ∈ Fin)
6 simp1 1136 . . 3 ((𝑅 ∈ 𝑉 ∧ (𝑁 = {𝐴} ∧ 𝐴 ∈ 𝑉) ∧ 𝑀 ∈ 𝐡) β†’ 𝑅 ∈ 𝑉)
7 simp3 1138 . . 3 ((𝑅 ∈ 𝑉 ∧ (𝑁 = {𝐴} ∧ 𝐴 ∈ 𝑉) ∧ 𝑀 ∈ 𝐡) β†’ 𝑀 ∈ 𝐡)
8 d1mat2pmat.t . . . 4 𝑇 = (𝑁 matToPolyMat 𝑅)
9 eqid 2732 . . . 4 (𝑁 Mat 𝑅) = (𝑁 Mat 𝑅)
10 d1mat2pmat.b . . . 4 𝐡 = (Baseβ€˜(𝑁 Mat 𝑅))
11 d1mat2pmat.p . . . 4 𝑃 = (Poly1β€˜π‘…)
12 d1mat2pmat.s . . . 4 𝑆 = (algScβ€˜π‘ƒ)
138, 9, 10, 11, 12mat2pmatval 22217 . . 3 ((𝑁 ∈ Fin ∧ 𝑅 ∈ 𝑉 ∧ 𝑀 ∈ 𝐡) β†’ (π‘‡β€˜π‘€) = (𝑖 ∈ 𝑁, 𝑗 ∈ 𝑁 ↦ (π‘†β€˜(𝑖𝑀𝑗))))
145, 6, 7, 13syl3anc 1371 . 2 ((𝑅 ∈ 𝑉 ∧ (𝑁 = {𝐴} ∧ 𝐴 ∈ 𝑉) ∧ 𝑀 ∈ 𝐡) β†’ (π‘‡β€˜π‘€) = (𝑖 ∈ 𝑁, 𝑗 ∈ 𝑁 ↦ (π‘†β€˜(𝑖𝑀𝑗))))
15 id 22 . . . . . . 7 (𝐴 ∈ 𝑉 β†’ 𝐴 ∈ 𝑉)
16 fvexd 6903 . . . . . . 7 (𝐴 ∈ 𝑉 β†’ (π‘†β€˜(𝐴𝑀𝐴)) ∈ V)
1715, 15, 163jca 1128 . . . . . 6 (𝐴 ∈ 𝑉 β†’ (𝐴 ∈ 𝑉 ∧ 𝐴 ∈ 𝑉 ∧ (π‘†β€˜(𝐴𝑀𝐴)) ∈ V))
1817adantl 482 . . . . 5 ((𝑁 = {𝐴} ∧ 𝐴 ∈ 𝑉) β†’ (𝐴 ∈ 𝑉 ∧ 𝐴 ∈ 𝑉 ∧ (π‘†β€˜(𝐴𝑀𝐴)) ∈ V))
19183ad2ant2 1134 . . . 4 ((𝑅 ∈ 𝑉 ∧ (𝑁 = {𝐴} ∧ 𝐴 ∈ 𝑉) ∧ 𝑀 ∈ 𝐡) β†’ (𝐴 ∈ 𝑉 ∧ 𝐴 ∈ 𝑉 ∧ (π‘†β€˜(𝐴𝑀𝐴)) ∈ V))
20 eqid 2732 . . . . 5 (𝑖 ∈ {𝐴}, 𝑗 ∈ {𝐴} ↦ (π‘†β€˜(𝑖𝑀𝑗))) = (𝑖 ∈ {𝐴}, 𝑗 ∈ {𝐴} ↦ (π‘†β€˜(𝑖𝑀𝑗)))
21 fvoveq1 7428 . . . . 5 (𝑖 = 𝐴 β†’ (π‘†β€˜(𝑖𝑀𝑗)) = (π‘†β€˜(𝐴𝑀𝑗)))
22 oveq2 7413 . . . . . 6 (𝑗 = 𝐴 β†’ (𝐴𝑀𝑗) = (𝐴𝑀𝐴))
2322fveq2d 6892 . . . . 5 (𝑗 = 𝐴 β†’ (π‘†β€˜(𝐴𝑀𝑗)) = (π‘†β€˜(𝐴𝑀𝐴)))
2420, 21, 23mposn 8085 . . . 4 ((𝐴 ∈ 𝑉 ∧ 𝐴 ∈ 𝑉 ∧ (π‘†β€˜(𝐴𝑀𝐴)) ∈ V) β†’ (𝑖 ∈ {𝐴}, 𝑗 ∈ {𝐴} ↦ (π‘†β€˜(𝑖𝑀𝑗))) = {⟨⟨𝐴, 𝐴⟩, (π‘†β€˜(𝐴𝑀𝐴))⟩})
2519, 24syl 17 . . 3 ((𝑅 ∈ 𝑉 ∧ (𝑁 = {𝐴} ∧ 𝐴 ∈ 𝑉) ∧ 𝑀 ∈ 𝐡) β†’ (𝑖 ∈ {𝐴}, 𝑗 ∈ {𝐴} ↦ (π‘†β€˜(𝑖𝑀𝑗))) = {⟨⟨𝐴, 𝐴⟩, (π‘†β€˜(𝐴𝑀𝐴))⟩})
26 mpoeq12 7478 . . . . . . 7 ((𝑁 = {𝐴} ∧ 𝑁 = {𝐴}) β†’ (𝑖 ∈ 𝑁, 𝑗 ∈ 𝑁 ↦ (π‘†β€˜(𝑖𝑀𝑗))) = (𝑖 ∈ {𝐴}, 𝑗 ∈ {𝐴} ↦ (π‘†β€˜(𝑖𝑀𝑗))))
2726eqeq1d 2734 . . . . . 6 ((𝑁 = {𝐴} ∧ 𝑁 = {𝐴}) β†’ ((𝑖 ∈ 𝑁, 𝑗 ∈ 𝑁 ↦ (π‘†β€˜(𝑖𝑀𝑗))) = {⟨⟨𝐴, 𝐴⟩, (π‘†β€˜(𝐴𝑀𝐴))⟩} ↔ (𝑖 ∈ {𝐴}, 𝑗 ∈ {𝐴} ↦ (π‘†β€˜(𝑖𝑀𝑗))) = {⟨⟨𝐴, 𝐴⟩, (π‘†β€˜(𝐴𝑀𝐴))⟩}))
2827anidms 567 . . . . 5 (𝑁 = {𝐴} β†’ ((𝑖 ∈ 𝑁, 𝑗 ∈ 𝑁 ↦ (π‘†β€˜(𝑖𝑀𝑗))) = {⟨⟨𝐴, 𝐴⟩, (π‘†β€˜(𝐴𝑀𝐴))⟩} ↔ (𝑖 ∈ {𝐴}, 𝑗 ∈ {𝐴} ↦ (π‘†β€˜(𝑖𝑀𝑗))) = {⟨⟨𝐴, 𝐴⟩, (π‘†β€˜(𝐴𝑀𝐴))⟩}))
2928adantr 481 . . . 4 ((𝑁 = {𝐴} ∧ 𝐴 ∈ 𝑉) β†’ ((𝑖 ∈ 𝑁, 𝑗 ∈ 𝑁 ↦ (π‘†β€˜(𝑖𝑀𝑗))) = {⟨⟨𝐴, 𝐴⟩, (π‘†β€˜(𝐴𝑀𝐴))⟩} ↔ (𝑖 ∈ {𝐴}, 𝑗 ∈ {𝐴} ↦ (π‘†β€˜(𝑖𝑀𝑗))) = {⟨⟨𝐴, 𝐴⟩, (π‘†β€˜(𝐴𝑀𝐴))⟩}))
30293ad2ant2 1134 . . 3 ((𝑅 ∈ 𝑉 ∧ (𝑁 = {𝐴} ∧ 𝐴 ∈ 𝑉) ∧ 𝑀 ∈ 𝐡) β†’ ((𝑖 ∈ 𝑁, 𝑗 ∈ 𝑁 ↦ (π‘†β€˜(𝑖𝑀𝑗))) = {⟨⟨𝐴, 𝐴⟩, (π‘†β€˜(𝐴𝑀𝐴))⟩} ↔ (𝑖 ∈ {𝐴}, 𝑗 ∈ {𝐴} ↦ (π‘†β€˜(𝑖𝑀𝑗))) = {⟨⟨𝐴, 𝐴⟩, (π‘†β€˜(𝐴𝑀𝐴))⟩}))
3125, 30mpbird 256 . 2 ((𝑅 ∈ 𝑉 ∧ (𝑁 = {𝐴} ∧ 𝐴 ∈ 𝑉) ∧ 𝑀 ∈ 𝐡) β†’ (𝑖 ∈ 𝑁, 𝑗 ∈ 𝑁 ↦ (π‘†β€˜(𝑖𝑀𝑗))) = {⟨⟨𝐴, 𝐴⟩, (π‘†β€˜(𝐴𝑀𝐴))⟩})
3214, 31eqtrd 2772 1 ((𝑅 ∈ 𝑉 ∧ (𝑁 = {𝐴} ∧ 𝐴 ∈ 𝑉) ∧ 𝑀 ∈ 𝐡) β†’ (π‘‡β€˜π‘€) = {⟨⟨𝐴, 𝐴⟩, (π‘†β€˜(𝐴𝑀𝐴))⟩})
Colors of variables: wff setvar class
Syntax hints:   β†’ wi 4   ↔ wb 205   ∧ wa 396   ∧ w3a 1087   = wceq 1541   ∈ wcel 2106  Vcvv 3474  {csn 4627  βŸ¨cop 4633  β€˜cfv 6540  (class class class)co 7405   ∈ cmpo 7407  Fincfn 8935  Basecbs 17140  algSccascl 21398  Poly1cpl1 21692   Mat cmat 21898   matToPolyMat cmat2pmat 22197
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-10 2137  ax-11 2154  ax-12 2171  ax-ext 2703  ax-rep 5284  ax-sep 5298  ax-nul 5305  ax-pow 5362  ax-pr 5426  ax-un 7721
This theorem depends on definitions:  df-bi 206  df-an 397  df-or 846  df-3or 1088  df-3an 1089  df-tru 1544  df-fal 1554  df-ex 1782  df-nf 1786  df-sb 2068  df-mo 2534  df-eu 2563  df-clab 2710  df-cleq 2724  df-clel 2810  df-nfc 2885  df-ne 2941  df-ral 3062  df-rex 3071  df-reu 3377  df-rab 3433  df-v 3476  df-sbc 3777  df-csb 3893  df-dif 3950  df-un 3952  df-in 3954  df-ss 3964  df-pss 3966  df-nul 4322  df-if 4528  df-pw 4603  df-sn 4628  df-pr 4630  df-op 4634  df-uni 4908  df-iun 4998  df-br 5148  df-opab 5210  df-mpt 5231  df-tr 5265  df-id 5573  df-eprel 5579  df-po 5587  df-so 5588  df-fr 5630  df-we 5632  df-xp 5681  df-rel 5682  df-cnv 5683  df-co 5684  df-dm 5685  df-rn 5686  df-res 5687  df-ima 5688  df-ord 6364  df-on 6365  df-lim 6366  df-suc 6367  df-iota 6492  df-fun 6542  df-fn 6543  df-f 6544  df-f1 6545  df-fo 6546  df-f1o 6547  df-fv 6548  df-ov 7408  df-oprab 7409  df-mpo 7410  df-om 7852  df-1st 7971  df-2nd 7972  df-1o 8462  df-en 8936  df-fin 8939  df-mat2pmat 22200
This theorem is referenced by: (None)
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