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Theorem d1mat2pmat 22657
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 9065 . . . . . 6 {𝐴} ∈ Fin
2 eleq1 2813 . . . . . 6 (𝑁 = {𝐴} β†’ (𝑁 ∈ Fin ↔ {𝐴} ∈ Fin))
31, 2mpbiri 257 . . . . 5 (𝑁 = {𝐴} β†’ 𝑁 ∈ Fin)
43adantr 479 . . . 4 ((𝑁 = {𝐴} ∧ 𝐴 ∈ 𝑉) β†’ 𝑁 ∈ Fin)
543ad2ant2 1131 . . 3 ((𝑅 ∈ 𝑉 ∧ (𝑁 = {𝐴} ∧ 𝐴 ∈ 𝑉) ∧ 𝑀 ∈ 𝐡) β†’ 𝑁 ∈ Fin)
6 simp1 1133 . . 3 ((𝑅 ∈ 𝑉 ∧ (𝑁 = {𝐴} ∧ 𝐴 ∈ 𝑉) ∧ 𝑀 ∈ 𝐡) β†’ 𝑅 ∈ 𝑉)
7 simp3 1135 . . 3 ((𝑅 ∈ 𝑉 ∧ (𝑁 = {𝐴} ∧ 𝐴 ∈ 𝑉) ∧ 𝑀 ∈ 𝐡) β†’ 𝑀 ∈ 𝐡)
8 d1mat2pmat.t . . . 4 𝑇 = (𝑁 matToPolyMat 𝑅)
9 eqid 2725 . . . 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 22642 . . 3 ((𝑁 ∈ Fin ∧ 𝑅 ∈ 𝑉 ∧ 𝑀 ∈ 𝐡) β†’ (π‘‡β€˜π‘€) = (𝑖 ∈ 𝑁, 𝑗 ∈ 𝑁 ↦ (π‘†β€˜(𝑖𝑀𝑗))))
145, 6, 7, 13syl3anc 1368 . 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 6906 . . . . . . 7 (𝐴 ∈ 𝑉 β†’ (π‘†β€˜(𝐴𝑀𝐴)) ∈ V)
1715, 15, 163jca 1125 . . . . . 6 (𝐴 ∈ 𝑉 β†’ (𝐴 ∈ 𝑉 ∧ 𝐴 ∈ 𝑉 ∧ (π‘†β€˜(𝐴𝑀𝐴)) ∈ V))
1817adantl 480 . . . . 5 ((𝑁 = {𝐴} ∧ 𝐴 ∈ 𝑉) β†’ (𝐴 ∈ 𝑉 ∧ 𝐴 ∈ 𝑉 ∧ (π‘†β€˜(𝐴𝑀𝐴)) ∈ V))
19183ad2ant2 1131 . . . 4 ((𝑅 ∈ 𝑉 ∧ (𝑁 = {𝐴} ∧ 𝐴 ∈ 𝑉) ∧ 𝑀 ∈ 𝐡) β†’ (𝐴 ∈ 𝑉 ∧ 𝐴 ∈ 𝑉 ∧ (π‘†β€˜(𝐴𝑀𝐴)) ∈ V))
20 eqid 2725 . . . . 5 (𝑖 ∈ {𝐴}, 𝑗 ∈ {𝐴} ↦ (π‘†β€˜(𝑖𝑀𝑗))) = (𝑖 ∈ {𝐴}, 𝑗 ∈ {𝐴} ↦ (π‘†β€˜(𝑖𝑀𝑗)))
21 fvoveq1 7438 . . . . 5 (𝑖 = 𝐴 β†’ (π‘†β€˜(𝑖𝑀𝑗)) = (π‘†β€˜(𝐴𝑀𝑗)))
22 oveq2 7423 . . . . . 6 (𝑗 = 𝐴 β†’ (𝐴𝑀𝑗) = (𝐴𝑀𝐴))
2322fveq2d 6895 . . . . 5 (𝑗 = 𝐴 β†’ (π‘†β€˜(𝐴𝑀𝑗)) = (π‘†β€˜(𝐴𝑀𝐴)))
2420, 21, 23mposn 8104 . . . 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 7489 . . . . . . 7 ((𝑁 = {𝐴} ∧ 𝑁 = {𝐴}) β†’ (𝑖 ∈ 𝑁, 𝑗 ∈ 𝑁 ↦ (π‘†β€˜(𝑖𝑀𝑗))) = (𝑖 ∈ {𝐴}, 𝑗 ∈ {𝐴} ↦ (π‘†β€˜(𝑖𝑀𝑗))))
2726eqeq1d 2727 . . . . . 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 565 . . . . 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 479 . . . 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 1131 . . 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 2765 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 394   ∧ w3a 1084   = wceq 1533   ∈ wcel 2098  Vcvv 3463  {csn 4624  βŸ¨cop 4630  β€˜cfv 6542  (class class class)co 7415   ∈ cmpo 7417  Fincfn 8960  Basecbs 17177  algSccascl 21788  Poly1cpl1 22102   Mat cmat 22323   matToPolyMat cmat2pmat 22622
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 5280  ax-sep 5294  ax-nul 5301  ax-pow 5359  ax-pr 5423  ax-un 7737
This theorem depends on definitions:  df-bi 206  df-an 395  df-or 846  df-3or 1085  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 3770  df-csb 3886  df-dif 3943  df-un 3945  df-in 3947  df-ss 3957  df-pss 3960  df-nul 4319  df-if 4525  df-pw 4600  df-sn 4625  df-pr 4627  df-op 4631  df-uni 4904  df-iun 4993  df-br 5144  df-opab 5206  df-mpt 5227  df-tr 5261  df-id 5570  df-eprel 5576  df-po 5584  df-so 5585  df-fr 5627  df-we 5629  df-xp 5678  df-rel 5679  df-cnv 5680  df-co 5681  df-dm 5682  df-rn 5683  df-res 5684  df-ima 5685  df-ord 6367  df-on 6368  df-lim 6369  df-suc 6370  df-iota 6494  df-fun 6544  df-fn 6545  df-f 6546  df-f1 6547  df-fo 6548  df-f1o 6549  df-fv 6550  df-ov 7418  df-oprab 7419  df-mpo 7420  df-om 7868  df-1st 7989  df-2nd 7990  df-1o 8483  df-en 8961  df-fin 8964  df-mat2pmat 22625
This theorem is referenced by: (None)
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