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Theorem d1mat2pmat 20363
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 7923 . . . . . 6 {𝐴} ∈ Fin
2 eleq1 2676 . . . . . 6 (𝑁 = {𝐴} → (𝑁 ∈ Fin ↔ {𝐴} ∈ Fin))
31, 2mpbiri 247 . . . . 5 (𝑁 = {𝐴} → 𝑁 ∈ Fin)
43adantr 480 . . . 4 ((𝑁 = {𝐴} ∧ 𝐴𝑉) → 𝑁 ∈ Fin)
543ad2ant2 1076 . . 3 ((𝑅𝑉 ∧ (𝑁 = {𝐴} ∧ 𝐴𝑉) ∧ 𝑀𝐵) → 𝑁 ∈ Fin)
6 simp1 1054 . . 3 ((𝑅𝑉 ∧ (𝑁 = {𝐴} ∧ 𝐴𝑉) ∧ 𝑀𝐵) → 𝑅𝑉)
7 simp3 1056 . . 3 ((𝑅𝑉 ∧ (𝑁 = {𝐴} ∧ 𝐴𝑉) ∧ 𝑀𝐵) → 𝑀𝐵)
8 d1mat2pmat.t . . . 4 𝑇 = (𝑁 matToPolyMat 𝑅)
9 eqid 2610 . . . 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 20348 . . 3 ((𝑁 ∈ Fin ∧ 𝑅𝑉𝑀𝐵) → (𝑇𝑀) = (𝑖𝑁, 𝑗𝑁 ↦ (𝑆‘(𝑖𝑀𝑗))))
145, 6, 7, 13syl3anc 1318 . 2 ((𝑅𝑉 ∧ (𝑁 = {𝐴} ∧ 𝐴𝑉) ∧ 𝑀𝐵) → (𝑇𝑀) = (𝑖𝑁, 𝑗𝑁 ↦ (𝑆‘(𝑖𝑀𝑗))))
15 id 22 . . . . . . 7 (𝐴𝑉𝐴𝑉)
16 fvex 6113 . . . . . . . 8 (𝑆‘(𝐴𝑀𝐴)) ∈ V
1716a1i 11 . . . . . . 7 (𝐴𝑉 → (𝑆‘(𝐴𝑀𝐴)) ∈ V)
1815, 15, 173jca 1235 . . . . . 6 (𝐴𝑉 → (𝐴𝑉𝐴𝑉 ∧ (𝑆‘(𝐴𝑀𝐴)) ∈ V))
1918adantl 481 . . . . 5 ((𝑁 = {𝐴} ∧ 𝐴𝑉) → (𝐴𝑉𝐴𝑉 ∧ (𝑆‘(𝐴𝑀𝐴)) ∈ V))
20193ad2ant2 1076 . . . 4 ((𝑅𝑉 ∧ (𝑁 = {𝐴} ∧ 𝐴𝑉) ∧ 𝑀𝐵) → (𝐴𝑉𝐴𝑉 ∧ (𝑆‘(𝐴𝑀𝐴)) ∈ V))
21 eqid 2610 . . . . 5 (𝑖 ∈ {𝐴}, 𝑗 ∈ {𝐴} ↦ (𝑆‘(𝑖𝑀𝑗))) = (𝑖 ∈ {𝐴}, 𝑗 ∈ {𝐴} ↦ (𝑆‘(𝑖𝑀𝑗)))
22 oveq1 6556 . . . . . 6 (𝑖 = 𝐴 → (𝑖𝑀𝑗) = (𝐴𝑀𝑗))
2322fveq2d 6107 . . . . 5 (𝑖 = 𝐴 → (𝑆‘(𝑖𝑀𝑗)) = (𝑆‘(𝐴𝑀𝑗)))
24 oveq2 6557 . . . . . 6 (𝑗 = 𝐴 → (𝐴𝑀𝑗) = (𝐴𝑀𝐴))
2524fveq2d 6107 . . . . 5 (𝑗 = 𝐴 → (𝑆‘(𝐴𝑀𝑗)) = (𝑆‘(𝐴𝑀𝐴)))
2621, 23, 25mpt2sn 7155 . . . 4 ((𝐴𝑉𝐴𝑉 ∧ (𝑆‘(𝐴𝑀𝐴)) ∈ V) → (𝑖 ∈ {𝐴}, 𝑗 ∈ {𝐴} ↦ (𝑆‘(𝑖𝑀𝑗))) = {⟨⟨𝐴, 𝐴⟩, (𝑆‘(𝐴𝑀𝐴))⟩})
2720, 26syl 17 . . 3 ((𝑅𝑉 ∧ (𝑁 = {𝐴} ∧ 𝐴𝑉) ∧ 𝑀𝐵) → (𝑖 ∈ {𝐴}, 𝑗 ∈ {𝐴} ↦ (𝑆‘(𝑖𝑀𝑗))) = {⟨⟨𝐴, 𝐴⟩, (𝑆‘(𝐴𝑀𝐴))⟩})
28 mpt2eq12 6613 . . . . . . 7 ((𝑁 = {𝐴} ∧ 𝑁 = {𝐴}) → (𝑖𝑁, 𝑗𝑁 ↦ (𝑆‘(𝑖𝑀𝑗))) = (𝑖 ∈ {𝐴}, 𝑗 ∈ {𝐴} ↦ (𝑆‘(𝑖𝑀𝑗))))
2928eqeq1d 2612 . . . . . 6 ((𝑁 = {𝐴} ∧ 𝑁 = {𝐴}) → ((𝑖𝑁, 𝑗𝑁 ↦ (𝑆‘(𝑖𝑀𝑗))) = {⟨⟨𝐴, 𝐴⟩, (𝑆‘(𝐴𝑀𝐴))⟩} ↔ (𝑖 ∈ {𝐴}, 𝑗 ∈ {𝐴} ↦ (𝑆‘(𝑖𝑀𝑗))) = {⟨⟨𝐴, 𝐴⟩, (𝑆‘(𝐴𝑀𝐴))⟩}))
3029anidms 675 . . . . 5 (𝑁 = {𝐴} → ((𝑖𝑁, 𝑗𝑁 ↦ (𝑆‘(𝑖𝑀𝑗))) = {⟨⟨𝐴, 𝐴⟩, (𝑆‘(𝐴𝑀𝐴))⟩} ↔ (𝑖 ∈ {𝐴}, 𝑗 ∈ {𝐴} ↦ (𝑆‘(𝑖𝑀𝑗))) = {⟨⟨𝐴, 𝐴⟩, (𝑆‘(𝐴𝑀𝐴))⟩}))
3130adantr 480 . . . 4 ((𝑁 = {𝐴} ∧ 𝐴𝑉) → ((𝑖𝑁, 𝑗𝑁 ↦ (𝑆‘(𝑖𝑀𝑗))) = {⟨⟨𝐴, 𝐴⟩, (𝑆‘(𝐴𝑀𝐴))⟩} ↔ (𝑖 ∈ {𝐴}, 𝑗 ∈ {𝐴} ↦ (𝑆‘(𝑖𝑀𝑗))) = {⟨⟨𝐴, 𝐴⟩, (𝑆‘(𝐴𝑀𝐴))⟩}))
32313ad2ant2 1076 . . 3 ((𝑅𝑉 ∧ (𝑁 = {𝐴} ∧ 𝐴𝑉) ∧ 𝑀𝐵) → ((𝑖𝑁, 𝑗𝑁 ↦ (𝑆‘(𝑖𝑀𝑗))) = {⟨⟨𝐴, 𝐴⟩, (𝑆‘(𝐴𝑀𝐴))⟩} ↔ (𝑖 ∈ {𝐴}, 𝑗 ∈ {𝐴} ↦ (𝑆‘(𝑖𝑀𝑗))) = {⟨⟨𝐴, 𝐴⟩, (𝑆‘(𝐴𝑀𝐴))⟩}))
3327, 32mpbird 246 . 2 ((𝑅𝑉 ∧ (𝑁 = {𝐴} ∧ 𝐴𝑉) ∧ 𝑀𝐵) → (𝑖𝑁, 𝑗𝑁 ↦ (𝑆‘(𝑖𝑀𝑗))) = {⟨⟨𝐴, 𝐴⟩, (𝑆‘(𝐴𝑀𝐴))⟩})
3414, 33eqtrd 2644 1 ((𝑅𝑉 ∧ (𝑁 = {𝐴} ∧ 𝐴𝑉) ∧ 𝑀𝐵) → (𝑇𝑀) = {⟨⟨𝐴, 𝐴⟩, (𝑆‘(𝐴𝑀𝐴))⟩})
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 195  wa 383  w3a 1031   = wceq 1475  wcel 1977  Vcvv 3173  {csn 4125  cop 4131  cfv 5804  (class class class)co 6549  cmpt2 6551  Fincfn 7841  Basecbs 15695  algSccascl 19132  Poly1cpl1 19368   Mat cmat 20032   matToPolyMat cmat2pmat 20328
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1713  ax-4 1728  ax-5 1827  ax-6 1875  ax-7 1922  ax-8 1979  ax-9 1986  ax-10 2006  ax-11 2021  ax-12 2034  ax-13 2234  ax-ext 2590  ax-rep 4699  ax-sep 4709  ax-nul 4717  ax-pow 4769  ax-pr 4833  ax-un 6847
This theorem depends on definitions:  df-bi 196  df-or 384  df-an 385  df-3or 1032  df-3an 1033  df-tru 1478  df-ex 1696  df-nf 1701  df-sb 1868  df-eu 2462  df-mo 2463  df-clab 2597  df-cleq 2603  df-clel 2606  df-nfc 2740  df-ne 2782  df-ral 2901  df-rex 2902  df-reu 2903  df-rab 2905  df-v 3175  df-sbc 3403  df-csb 3500  df-dif 3543  df-un 3545  df-in 3547  df-ss 3554  df-pss 3556  df-nul 3875  df-if 4037  df-pw 4110  df-sn 4126  df-pr 4128  df-tp 4130  df-op 4132  df-uni 4373  df-iun 4457  df-br 4584  df-opab 4644  df-mpt 4645  df-tr 4681  df-eprel 4949  df-id 4953  df-po 4959  df-so 4960  df-fr 4997  df-we 4999  df-xp 5044  df-rel 5045  df-cnv 5046  df-co 5047  df-dm 5048  df-rn 5049  df-res 5050  df-ima 5051  df-ord 5643  df-on 5644  df-lim 5645  df-suc 5646  df-iota 5768  df-fun 5806  df-fn 5807  df-f 5808  df-f1 5809  df-fo 5810  df-f1o 5811  df-fv 5812  df-ov 6552  df-oprab 6553  df-mpt2 6554  df-om 6958  df-1st 7059  df-2nd 7060  df-1o 7447  df-en 7842  df-fin 7845  df-mat2pmat 20331
This theorem is referenced by: (None)
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