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Theorem d1mat2pmat 22745
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 𝐵 = (Base‘(𝑁 Mat 𝑅))
d1mat2pmat.p 𝑃 = (Poly1𝑅)
d1mat2pmat.s 𝑆 = (algSc‘𝑃)
Assertion
Ref Expression
d1mat2pmat ((𝑅𝑉 ∧ (𝑁 = {𝐴} ∧ 𝐴𝑉) ∧ 𝑀𝐵) → (𝑇𝑀) = {⟨⟨𝐴, 𝐴⟩, (𝑆‘(𝐴𝑀𝐴))⟩})

Proof of Theorem d1mat2pmat
Dummy variables 𝑖 𝑗 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 snfi 9083 . . . . . 6 {𝐴} ∈ Fin
2 eleq1 2829 . . . . . 6 (𝑁 = {𝐴} → (𝑁 ∈ Fin ↔ {𝐴} ∈ Fin))
31, 2mpbiri 258 . . . . 5 (𝑁 = {𝐴} → 𝑁 ∈ Fin)
43adantr 480 . . . 4 ((𝑁 = {𝐴} ∧ 𝐴𝑉) → 𝑁 ∈ Fin)
543ad2ant2 1135 . . 3 ((𝑅𝑉 ∧ (𝑁 = {𝐴} ∧ 𝐴𝑉) ∧ 𝑀𝐵) → 𝑁 ∈ Fin)
6 simp1 1137 . . 3 ((𝑅𝑉 ∧ (𝑁 = {𝐴} ∧ 𝐴𝑉) ∧ 𝑀𝐵) → 𝑅𝑉)
7 simp3 1139 . . 3 ((𝑅𝑉 ∧ (𝑁 = {𝐴} ∧ 𝐴𝑉) ∧ 𝑀𝐵) → 𝑀𝐵)
8 d1mat2pmat.t . . . 4 𝑇 = (𝑁 matToPolyMat 𝑅)
9 eqid 2737 . . . 4 (𝑁 Mat 𝑅) = (𝑁 Mat 𝑅)
10 d1mat2pmat.b . . . 4 𝐵 = (Base‘(𝑁 Mat 𝑅))
11 d1mat2pmat.p . . . 4 𝑃 = (Poly1𝑅)
12 d1mat2pmat.s . . . 4 𝑆 = (algSc‘𝑃)
138, 9, 10, 11, 12mat2pmatval 22730 . . 3 ((𝑁 ∈ Fin ∧ 𝑅𝑉𝑀𝐵) → (𝑇𝑀) = (𝑖𝑁, 𝑗𝑁 ↦ (𝑆‘(𝑖𝑀𝑗))))
145, 6, 7, 13syl3anc 1373 . 2 ((𝑅𝑉 ∧ (𝑁 = {𝐴} ∧ 𝐴𝑉) ∧ 𝑀𝐵) → (𝑇𝑀) = (𝑖𝑁, 𝑗𝑁 ↦ (𝑆‘(𝑖𝑀𝑗))))
15 id 22 . . . . . . 7 (𝐴𝑉𝐴𝑉)
16 fvexd 6921 . . . . . . 7 (𝐴𝑉 → (𝑆‘(𝐴𝑀𝐴)) ∈ V)
1715, 15, 163jca 1129 . . . . . 6 (𝐴𝑉 → (𝐴𝑉𝐴𝑉 ∧ (𝑆‘(𝐴𝑀𝐴)) ∈ V))
1817adantl 481 . . . . 5 ((𝑁 = {𝐴} ∧ 𝐴𝑉) → (𝐴𝑉𝐴𝑉 ∧ (𝑆‘(𝐴𝑀𝐴)) ∈ V))
19183ad2ant2 1135 . . . 4 ((𝑅𝑉 ∧ (𝑁 = {𝐴} ∧ 𝐴𝑉) ∧ 𝑀𝐵) → (𝐴𝑉𝐴𝑉 ∧ (𝑆‘(𝐴𝑀𝐴)) ∈ V))
20 eqid 2737 . . . . 5 (𝑖 ∈ {𝐴}, 𝑗 ∈ {𝐴} ↦ (𝑆‘(𝑖𝑀𝑗))) = (𝑖 ∈ {𝐴}, 𝑗 ∈ {𝐴} ↦ (𝑆‘(𝑖𝑀𝑗)))
21 fvoveq1 7454 . . . . 5 (𝑖 = 𝐴 → (𝑆‘(𝑖𝑀𝑗)) = (𝑆‘(𝐴𝑀𝑗)))
22 oveq2 7439 . . . . . 6 (𝑗 = 𝐴 → (𝐴𝑀𝑗) = (𝐴𝑀𝐴))
2322fveq2d 6910 . . . . 5 (𝑗 = 𝐴 → (𝑆‘(𝐴𝑀𝑗)) = (𝑆‘(𝐴𝑀𝐴)))
2420, 21, 23mposn 8128 . . . 4 ((𝐴𝑉𝐴𝑉 ∧ (𝑆‘(𝐴𝑀𝐴)) ∈ V) → (𝑖 ∈ {𝐴}, 𝑗 ∈ {𝐴} ↦ (𝑆‘(𝑖𝑀𝑗))) = {⟨⟨𝐴, 𝐴⟩, (𝑆‘(𝐴𝑀𝐴))⟩})
2519, 24syl 17 . . 3 ((𝑅𝑉 ∧ (𝑁 = {𝐴} ∧ 𝐴𝑉) ∧ 𝑀𝐵) → (𝑖 ∈ {𝐴}, 𝑗 ∈ {𝐴} ↦ (𝑆‘(𝑖𝑀𝑗))) = {⟨⟨𝐴, 𝐴⟩, (𝑆‘(𝐴𝑀𝐴))⟩})
26 mpoeq12 7506 . . . . . . 7 ((𝑁 = {𝐴} ∧ 𝑁 = {𝐴}) → (𝑖𝑁, 𝑗𝑁 ↦ (𝑆‘(𝑖𝑀𝑗))) = (𝑖 ∈ {𝐴}, 𝑗 ∈ {𝐴} ↦ (𝑆‘(𝑖𝑀𝑗))))
2726eqeq1d 2739 . . . . . 6 ((𝑁 = {𝐴} ∧ 𝑁 = {𝐴}) → ((𝑖𝑁, 𝑗𝑁 ↦ (𝑆‘(𝑖𝑀𝑗))) = {⟨⟨𝐴, 𝐴⟩, (𝑆‘(𝐴𝑀𝐴))⟩} ↔ (𝑖 ∈ {𝐴}, 𝑗 ∈ {𝐴} ↦ (𝑆‘(𝑖𝑀𝑗))) = {⟨⟨𝐴, 𝐴⟩, (𝑆‘(𝐴𝑀𝐴))⟩}))
2827anidms 566 . . . . 5 (𝑁 = {𝐴} → ((𝑖𝑁, 𝑗𝑁 ↦ (𝑆‘(𝑖𝑀𝑗))) = {⟨⟨𝐴, 𝐴⟩, (𝑆‘(𝐴𝑀𝐴))⟩} ↔ (𝑖 ∈ {𝐴}, 𝑗 ∈ {𝐴} ↦ (𝑆‘(𝑖𝑀𝑗))) = {⟨⟨𝐴, 𝐴⟩, (𝑆‘(𝐴𝑀𝐴))⟩}))
2928adantr 480 . . . 4 ((𝑁 = {𝐴} ∧ 𝐴𝑉) → ((𝑖𝑁, 𝑗𝑁 ↦ (𝑆‘(𝑖𝑀𝑗))) = {⟨⟨𝐴, 𝐴⟩, (𝑆‘(𝐴𝑀𝐴))⟩} ↔ (𝑖 ∈ {𝐴}, 𝑗 ∈ {𝐴} ↦ (𝑆‘(𝑖𝑀𝑗))) = {⟨⟨𝐴, 𝐴⟩, (𝑆‘(𝐴𝑀𝐴))⟩}))
30293ad2ant2 1135 . . 3 ((𝑅𝑉 ∧ (𝑁 = {𝐴} ∧ 𝐴𝑉) ∧ 𝑀𝐵) → ((𝑖𝑁, 𝑗𝑁 ↦ (𝑆‘(𝑖𝑀𝑗))) = {⟨⟨𝐴, 𝐴⟩, (𝑆‘(𝐴𝑀𝐴))⟩} ↔ (𝑖 ∈ {𝐴}, 𝑗 ∈ {𝐴} ↦ (𝑆‘(𝑖𝑀𝑗))) = {⟨⟨𝐴, 𝐴⟩, (𝑆‘(𝐴𝑀𝐴))⟩}))
3125, 30mpbird 257 . 2 ((𝑅𝑉 ∧ (𝑁 = {𝐴} ∧ 𝐴𝑉) ∧ 𝑀𝐵) → (𝑖𝑁, 𝑗𝑁 ↦ (𝑆‘(𝑖𝑀𝑗))) = {⟨⟨𝐴, 𝐴⟩, (𝑆‘(𝐴𝑀𝐴))⟩})
3214, 31eqtrd 2777 1 ((𝑅𝑉 ∧ (𝑁 = {𝐴} ∧ 𝐴𝑉) ∧ 𝑀𝐵) → (𝑇𝑀) = {⟨⟨𝐴, 𝐴⟩, (𝑆‘(𝐴𝑀𝐴))⟩})
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 206  wa 395  w3a 1087   = wceq 1540  wcel 2108  Vcvv 3480  {csn 4626  cop 4632  cfv 6561  (class class class)co 7431  cmpo 7433  Fincfn 8985  Basecbs 17247  algSccascl 21872  Poly1cpl1 22178   Mat cmat 22411   matToPolyMat cmat2pmat 22710
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 2007  ax-8 2110  ax-9 2118  ax-10 2141  ax-11 2157  ax-12 2177  ax-ext 2708  ax-rep 5279  ax-sep 5296  ax-nul 5306  ax-pow 5365  ax-pr 5432  ax-un 7755
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3or 1088  df-3an 1089  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2065  df-mo 2540  df-eu 2569  df-clab 2715  df-cleq 2729  df-clel 2816  df-nfc 2892  df-ne 2941  df-ral 3062  df-rex 3071  df-reu 3381  df-rab 3437  df-v 3482  df-sbc 3789  df-csb 3900  df-dif 3954  df-un 3956  df-in 3958  df-ss 3968  df-pss 3971  df-nul 4334  df-if 4526  df-pw 4602  df-sn 4627  df-pr 4629  df-op 4633  df-uni 4908  df-iun 4993  df-br 5144  df-opab 5206  df-mpt 5226  df-tr 5260  df-id 5578  df-eprel 5584  df-po 5592  df-so 5593  df-fr 5637  df-we 5639  df-xp 5691  df-rel 5692  df-cnv 5693  df-co 5694  df-dm 5695  df-rn 5696  df-res 5697  df-ima 5698  df-ord 6387  df-on 6388  df-lim 6389  df-suc 6390  df-iota 6514  df-fun 6563  df-fn 6564  df-f 6565  df-f1 6566  df-fo 6567  df-f1o 6568  df-fv 6569  df-ov 7434  df-oprab 7435  df-mpo 7436  df-om 7888  df-1st 8014  df-2nd 8015  df-1o 8506  df-en 8986  df-fin 8989  df-mat2pmat 22713
This theorem is referenced by: (None)
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