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Theorem d1mat2pmat 22766
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 9109 . . . . . 6 {𝐴} ∈ Fin
2 eleq1 2832 . . . . . 6 (𝑁 = {𝐴} → (𝑁 ∈ Fin ↔ {𝐴} ∈ Fin))
31, 2mpbiri 258 . . . . 5 (𝑁 = {𝐴} → 𝑁 ∈ Fin)
43adantr 480 . . . 4 ((𝑁 = {𝐴} ∧ 𝐴𝑉) → 𝑁 ∈ Fin)
543ad2ant2 1134 . . 3 ((𝑅𝑉 ∧ (𝑁 = {𝐴} ∧ 𝐴𝑉) ∧ 𝑀𝐵) → 𝑁 ∈ Fin)
6 simp1 1136 . . 3 ((𝑅𝑉 ∧ (𝑁 = {𝐴} ∧ 𝐴𝑉) ∧ 𝑀𝐵) → 𝑅𝑉)
7 simp3 1138 . . 3 ((𝑅𝑉 ∧ (𝑁 = {𝐴} ∧ 𝐴𝑉) ∧ 𝑀𝐵) → 𝑀𝐵)
8 d1mat2pmat.t . . . 4 𝑇 = (𝑁 matToPolyMat 𝑅)
9 eqid 2740 . . . 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 22751 . . 3 ((𝑁 ∈ Fin ∧ 𝑅𝑉𝑀𝐵) → (𝑇𝑀) = (𝑖𝑁, 𝑗𝑁 ↦ (𝑆‘(𝑖𝑀𝑗))))
145, 6, 7, 13syl3anc 1371 . 2 ((𝑅𝑉 ∧ (𝑁 = {𝐴} ∧ 𝐴𝑉) ∧ 𝑀𝐵) → (𝑇𝑀) = (𝑖𝑁, 𝑗𝑁 ↦ (𝑆‘(𝑖𝑀𝑗))))
15 id 22 . . . . . . 7 (𝐴𝑉𝐴𝑉)
16 fvexd 6935 . . . . . . 7 (𝐴𝑉 → (𝑆‘(𝐴𝑀𝐴)) ∈ V)
1715, 15, 163jca 1128 . . . . . 6 (𝐴𝑉 → (𝐴𝑉𝐴𝑉 ∧ (𝑆‘(𝐴𝑀𝐴)) ∈ V))
1817adantl 481 . . . . 5 ((𝑁 = {𝐴} ∧ 𝐴𝑉) → (𝐴𝑉𝐴𝑉 ∧ (𝑆‘(𝐴𝑀𝐴)) ∈ V))
19183ad2ant2 1134 . . . 4 ((𝑅𝑉 ∧ (𝑁 = {𝐴} ∧ 𝐴𝑉) ∧ 𝑀𝐵) → (𝐴𝑉𝐴𝑉 ∧ (𝑆‘(𝐴𝑀𝐴)) ∈ V))
20 eqid 2740 . . . . 5 (𝑖 ∈ {𝐴}, 𝑗 ∈ {𝐴} ↦ (𝑆‘(𝑖𝑀𝑗))) = (𝑖 ∈ {𝐴}, 𝑗 ∈ {𝐴} ↦ (𝑆‘(𝑖𝑀𝑗)))
21 fvoveq1 7471 . . . . 5 (𝑖 = 𝐴 → (𝑆‘(𝑖𝑀𝑗)) = (𝑆‘(𝐴𝑀𝑗)))
22 oveq2 7456 . . . . . 6 (𝑗 = 𝐴 → (𝐴𝑀𝑗) = (𝐴𝑀𝐴))
2322fveq2d 6924 . . . . 5 (𝑗 = 𝐴 → (𝑆‘(𝐴𝑀𝑗)) = (𝑆‘(𝐴𝑀𝐴)))
2420, 21, 23mposn 8144 . . . 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 7523 . . . . . . 7 ((𝑁 = {𝐴} ∧ 𝑁 = {𝐴}) → (𝑖𝑁, 𝑗𝑁 ↦ (𝑆‘(𝑖𝑀𝑗))) = (𝑖 ∈ {𝐴}, 𝑗 ∈ {𝐴} ↦ (𝑆‘(𝑖𝑀𝑗))))
2726eqeq1d 2742 . . . . . 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 1134 . . 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 2780 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 1537  wcel 2108  Vcvv 3488  {csn 4648  cop 4654  cfv 6573  (class class class)co 7448  cmpo 7450  Fincfn 9003  Basecbs 17258  algSccascl 21895  Poly1cpl1 22199   Mat cmat 22432   matToPolyMat cmat2pmat 22731
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1793  ax-4 1807  ax-5 1909  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-10 2141  ax-11 2158  ax-12 2178  ax-ext 2711  ax-rep 5303  ax-sep 5317  ax-nul 5324  ax-pow 5383  ax-pr 5447  ax-un 7770
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 847  df-3or 1088  df-3an 1089  df-tru 1540  df-fal 1550  df-ex 1778  df-nf 1782  df-sb 2065  df-mo 2543  df-eu 2572  df-clab 2718  df-cleq 2732  df-clel 2819  df-nfc 2895  df-ne 2947  df-ral 3068  df-rex 3077  df-reu 3389  df-rab 3444  df-v 3490  df-sbc 3805  df-csb 3922  df-dif 3979  df-un 3981  df-in 3983  df-ss 3993  df-pss 3996  df-nul 4353  df-if 4549  df-pw 4624  df-sn 4649  df-pr 4651  df-op 4655  df-uni 4932  df-iun 5017  df-br 5167  df-opab 5229  df-mpt 5250  df-tr 5284  df-id 5593  df-eprel 5599  df-po 5607  df-so 5608  df-fr 5652  df-we 5654  df-xp 5706  df-rel 5707  df-cnv 5708  df-co 5709  df-dm 5710  df-rn 5711  df-res 5712  df-ima 5713  df-ord 6398  df-on 6399  df-lim 6400  df-suc 6401  df-iota 6525  df-fun 6575  df-fn 6576  df-f 6577  df-f1 6578  df-fo 6579  df-f1o 6580  df-fv 6581  df-ov 7451  df-oprab 7452  df-mpo 7453  df-om 7904  df-1st 8030  df-2nd 8031  df-1o 8522  df-en 9004  df-fin 9007  df-mat2pmat 22734
This theorem is referenced by: (None)
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