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Theorem d1mat2pmat 22088
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 8988 . . . . . 6 {𝐴} ∈ Fin
2 eleq1 2825 . . . . . 6 (𝑁 = {𝐴} → (𝑁 ∈ Fin ↔ {𝐴} ∈ Fin))
31, 2mpbiri 257 . . . . 5 (𝑁 = {𝐴} → 𝑁 ∈ Fin)
43adantr 481 . . . 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 2736 . . . 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 22073 . . 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 6857 . . . . . . 7 (𝐴𝑉 → (𝑆‘(𝐴𝑀𝐴)) ∈ V)
1715, 15, 163jca 1128 . . . . . 6 (𝐴𝑉 → (𝐴𝑉𝐴𝑉 ∧ (𝑆‘(𝐴𝑀𝐴)) ∈ V))
1817adantl 482 . . . . 5 ((𝑁 = {𝐴} ∧ 𝐴𝑉) → (𝐴𝑉𝐴𝑉 ∧ (𝑆‘(𝐴𝑀𝐴)) ∈ V))
19183ad2ant2 1134 . . . 4 ((𝑅𝑉 ∧ (𝑁 = {𝐴} ∧ 𝐴𝑉) ∧ 𝑀𝐵) → (𝐴𝑉𝐴𝑉 ∧ (𝑆‘(𝐴𝑀𝐴)) ∈ V))
20 eqid 2736 . . . . 5 (𝑖 ∈ {𝐴}, 𝑗 ∈ {𝐴} ↦ (𝑆‘(𝑖𝑀𝑗))) = (𝑖 ∈ {𝐴}, 𝑗 ∈ {𝐴} ↦ (𝑆‘(𝑖𝑀𝑗)))
21 fvoveq1 7380 . . . . 5 (𝑖 = 𝐴 → (𝑆‘(𝑖𝑀𝑗)) = (𝑆‘(𝐴𝑀𝑗)))
22 oveq2 7365 . . . . . 6 (𝑗 = 𝐴 → (𝐴𝑀𝑗) = (𝐴𝑀𝐴))
2322fveq2d 6846 . . . . 5 (𝑗 = 𝐴 → (𝑆‘(𝐴𝑀𝑗)) = (𝑆‘(𝐴𝑀𝐴)))
2420, 21, 23mposn 8035 . . . 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 7430 . . . . . . 7 ((𝑁 = {𝐴} ∧ 𝑁 = {𝐴}) → (𝑖𝑁, 𝑗𝑁 ↦ (𝑆‘(𝑖𝑀𝑗))) = (𝑖 ∈ {𝐴}, 𝑗 ∈ {𝐴} ↦ (𝑆‘(𝑖𝑀𝑗))))
2726eqeq1d 2738 . . . . . 6 ((𝑁 = {𝐴} ∧ 𝑁 = {𝐴}) → ((𝑖𝑁, 𝑗𝑁 ↦ (𝑆‘(𝑖𝑀𝑗))) = {⟨⟨𝐴, 𝐴⟩, (𝑆‘(𝐴𝑀𝐴))⟩} ↔ (𝑖 ∈ {𝐴}, 𝑗 ∈ {𝐴} ↦ (𝑆‘(𝑖𝑀𝑗))) = {⟨⟨𝐴, 𝐴⟩, (𝑆‘(𝐴𝑀𝐴))⟩}))
2827anidms 567 . . . . 5 (𝑁 = {𝐴} → ((𝑖𝑁, 𝑗𝑁 ↦ (𝑆‘(𝑖𝑀𝑗))) = {⟨⟨𝐴, 𝐴⟩, (𝑆‘(𝐴𝑀𝐴))⟩} ↔ (𝑖 ∈ {𝐴}, 𝑗 ∈ {𝐴} ↦ (𝑆‘(𝑖𝑀𝑗))) = {⟨⟨𝐴, 𝐴⟩, (𝑆‘(𝐴𝑀𝐴))⟩}))
2928adantr 481 . . . 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 256 . 2 ((𝑅𝑉 ∧ (𝑁 = {𝐴} ∧ 𝐴𝑉) ∧ 𝑀𝐵) → (𝑖𝑁, 𝑗𝑁 ↦ (𝑆‘(𝑖𝑀𝑗))) = {⟨⟨𝐴, 𝐴⟩, (𝑆‘(𝐴𝑀𝐴))⟩})
3214, 31eqtrd 2776 1 ((𝑅𝑉 ∧ (𝑁 = {𝐴} ∧ 𝐴𝑉) ∧ 𝑀𝐵) → (𝑇𝑀) = {⟨⟨𝐴, 𝐴⟩, (𝑆‘(𝐴𝑀𝐴))⟩})
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 205  wa 396  w3a 1087   = wceq 1541  wcel 2106  Vcvv 3445  {csn 4586  cop 4592  cfv 6496  (class class class)co 7357  cmpo 7359  Fincfn 8883  Basecbs 17083  algSccascl 21258  Poly1cpl1 21548   Mat cmat 21754   matToPolyMat cmat2pmat 22053
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 2707  ax-rep 5242  ax-sep 5256  ax-nul 5263  ax-pow 5320  ax-pr 5384  ax-un 7672
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 2538  df-eu 2567  df-clab 2714  df-cleq 2728  df-clel 2814  df-nfc 2889  df-ne 2944  df-ral 3065  df-rex 3074  df-reu 3354  df-rab 3408  df-v 3447  df-sbc 3740  df-csb 3856  df-dif 3913  df-un 3915  df-in 3917  df-ss 3927  df-pss 3929  df-nul 4283  df-if 4487  df-pw 4562  df-sn 4587  df-pr 4589  df-op 4593  df-uni 4866  df-iun 4956  df-br 5106  df-opab 5168  df-mpt 5189  df-tr 5223  df-id 5531  df-eprel 5537  df-po 5545  df-so 5546  df-fr 5588  df-we 5590  df-xp 5639  df-rel 5640  df-cnv 5641  df-co 5642  df-dm 5643  df-rn 5644  df-res 5645  df-ima 5646  df-ord 6320  df-on 6321  df-lim 6322  df-suc 6323  df-iota 6448  df-fun 6498  df-fn 6499  df-f 6500  df-f1 6501  df-fo 6502  df-f1o 6503  df-fv 6504  df-ov 7360  df-oprab 7361  df-mpo 7362  df-om 7803  df-1st 7921  df-2nd 7922  df-1o 8412  df-en 8884  df-fin 8887  df-mat2pmat 22056
This theorem is referenced by: (None)
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