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Theorem coe1fv 21722
Description: Value of an evaluated coefficient in a polynomial coefficient vector. (Contributed by Stefan O'Rear, 21-Mar-2015.)
Hypothesis
Ref Expression
coe1fval.a 𝐴 = (coe1𝐹)
Assertion
Ref Expression
coe1fv ((𝐹𝑉𝑁 ∈ ℕ0) → (𝐴𝑁) = (𝐹‘(1o × {𝑁})))

Proof of Theorem coe1fv
Dummy variable 𝑛 is distinct from all other variables.
StepHypRef Expression
1 coe1fval.a . . . 4 𝐴 = (coe1𝐹)
21coe1fval 21721 . . 3 (𝐹𝑉𝐴 = (𝑛 ∈ ℕ0 ↦ (𝐹‘(1o × {𝑛}))))
32fveq1d 6891 . 2 (𝐹𝑉 → (𝐴𝑁) = ((𝑛 ∈ ℕ0 ↦ (𝐹‘(1o × {𝑛})))‘𝑁))
4 sneq 4638 . . . . 5 (𝑛 = 𝑁 → {𝑛} = {𝑁})
54xpeq2d 5706 . . . 4 (𝑛 = 𝑁 → (1o × {𝑛}) = (1o × {𝑁}))
65fveq2d 6893 . . 3 (𝑛 = 𝑁 → (𝐹‘(1o × {𝑛})) = (𝐹‘(1o × {𝑁})))
7 eqid 2733 . . 3 (𝑛 ∈ ℕ0 ↦ (𝐹‘(1o × {𝑛}))) = (𝑛 ∈ ℕ0 ↦ (𝐹‘(1o × {𝑛})))
8 fvex 6902 . . 3 (𝐹‘(1o × {𝑁})) ∈ V
96, 7, 8fvmpt 6996 . 2 (𝑁 ∈ ℕ0 → ((𝑛 ∈ ℕ0 ↦ (𝐹‘(1o × {𝑛})))‘𝑁) = (𝐹‘(1o × {𝑁})))
103, 9sylan9eq 2793 1 ((𝐹𝑉𝑁 ∈ ℕ0) → (𝐴𝑁) = (𝐹‘(1o × {𝑁})))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 397   = wceq 1542  wcel 2107  {csn 4628  cmpt 5231   × cxp 5674  cfv 6541  1oc1o 8456  0cn0 12469  coe1cco1 21694
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2109  ax-9 2117  ax-10 2138  ax-11 2155  ax-12 2172  ax-ext 2704  ax-rep 5285  ax-sep 5299  ax-nul 5306  ax-pr 5427  ax-un 7722  ax-cnex 11163  ax-1cn 11165  ax-addcl 11167
This theorem depends on definitions:  df-bi 206  df-an 398  df-or 847  df-3or 1089  df-3an 1090  df-tru 1545  df-fal 1555  df-ex 1783  df-nf 1787  df-sb 2069  df-mo 2535  df-eu 2564  df-clab 2711  df-cleq 2725  df-clel 2811  df-nfc 2886  df-ne 2942  df-ral 3063  df-rex 3072  df-reu 3378  df-rab 3434  df-v 3477  df-sbc 3778  df-csb 3894  df-dif 3951  df-un 3953  df-in 3955  df-ss 3965  df-pss 3967  df-nul 4323  df-if 4529  df-pw 4604  df-sn 4629  df-pr 4631  df-op 4635  df-uni 4909  df-iun 4999  df-br 5149  df-opab 5211  df-mpt 5232  df-tr 5266  df-id 5574  df-eprel 5580  df-po 5588  df-so 5589  df-fr 5631  df-we 5633  df-xp 5682  df-rel 5683  df-cnv 5684  df-co 5685  df-dm 5686  df-rn 5687  df-res 5688  df-ima 5689  df-pred 6298  df-ord 6365  df-on 6366  df-lim 6367  df-suc 6368  df-iota 6493  df-fun 6543  df-fn 6544  df-f 6545  df-f1 6546  df-fo 6547  df-f1o 6548  df-fv 6549  df-ov 7409  df-om 7853  df-2nd 7973  df-frecs 8263  df-wrecs 8294  df-recs 8368  df-rdg 8407  df-nn 12210  df-n0 12470  df-coe1 21699
This theorem is referenced by:  fvcoe1  21723  coe1mul2  21783  deg1le0  25621
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