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Theorem coe1fv 19397
 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 19396 . . 3 (𝐹𝑉𝐴 = (𝑛 ∈ ℕ0 ↦ (𝐹‘(1𝑜 × {𝑛}))))
32fveq1d 6105 . 2 (𝐹𝑉 → (𝐴𝑁) = ((𝑛 ∈ ℕ0 ↦ (𝐹‘(1𝑜 × {𝑛})))‘𝑁))
4 sneq 4135 . . . . 5 (𝑛 = 𝑁 → {𝑛} = {𝑁})
54xpeq2d 5063 . . . 4 (𝑛 = 𝑁 → (1𝑜 × {𝑛}) = (1𝑜 × {𝑁}))
65fveq2d 6107 . . 3 (𝑛 = 𝑁 → (𝐹‘(1𝑜 × {𝑛})) = (𝐹‘(1𝑜 × {𝑁})))
7 eqid 2610 . . 3 (𝑛 ∈ ℕ0 ↦ (𝐹‘(1𝑜 × {𝑛}))) = (𝑛 ∈ ℕ0 ↦ (𝐹‘(1𝑜 × {𝑛})))
8 fvex 6113 . . 3 (𝐹‘(1𝑜 × {𝑁})) ∈ V
96, 7, 8fvmpt 6191 . 2 (𝑁 ∈ ℕ0 → ((𝑛 ∈ ℕ0 ↦ (𝐹‘(1𝑜 × {𝑛})))‘𝑁) = (𝐹‘(1𝑜 × {𝑁})))
103, 9sylan9eq 2664 1 ((𝐹𝑉𝑁 ∈ ℕ0) → (𝐴𝑁) = (𝐹‘(1𝑜 × {𝑁})))
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ∧ wa 383   = wceq 1475   ∈ wcel 1977  {csn 4125   ↦ cmpt 4643   × cxp 5036  ‘cfv 5804  1𝑜c1o 7440  ℕ0cn0 11169  coe1cco1 19369 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  ax-cnex 9871  ax-resscn 9872  ax-1cn 9873  ax-icn 9874  ax-addcl 9875  ax-addrcl 9876  ax-mulcl 9877  ax-mulrcl 9878  ax-i2m1 9883  ax-1ne0 9884  ax-rrecex 9887  ax-cnre 9888 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-pred 5597  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-om 6958  df-wrecs 7294  df-recs 7355  df-rdg 7393  df-nn 10898  df-n0 11170  df-coe1 19374 This theorem is referenced by:  fvcoe1  19398  coe1mul2  19460  deg1le0  23675
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