Theorem coe1fv 21287

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 = (coe1‘𝐹)

Assertion

Ref	Expression
coe1fv	⊢ ((𝐹 ∈ 𝑉 ∧ 𝑁 ∈ ℕ0) → (𝐴‘𝑁) = (𝐹‘(1o × {𝑁})))

Proof of Theorem coe1fv

Dummy variable 𝑛 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		coe1fval.a	. . . 4 ⊢ 𝐴 = (coe1‘𝐹)
2	1	coe1fval 21286	. . 3 ⊢ (𝐹 ∈ 𝑉 → 𝐴 = (𝑛 ∈ ℕ0 ↦ (𝐹‘(1o × {𝑛}))))
3	2	fveq1d 6758	. 2 ⊢ (𝐹 ∈ 𝑉 → (𝐴‘𝑁) = ((𝑛 ∈ ℕ0 ↦ (𝐹‘(1o × {𝑛})))‘𝑁))
4		sneq 4568	. . . . 5 ⊢ (𝑛 = 𝑁 → {𝑛} = {𝑁})
5	4	xpeq2d 5610	. . . 4 ⊢ (𝑛 = 𝑁 → (1o × {𝑛}) = (1o × {𝑁}))
6	5	fveq2d 6760	. . 3 ⊢ (𝑛 = 𝑁 → (𝐹‘(1o × {𝑛})) = (𝐹‘(1o × {𝑁})))
7		eqid 2738	. . 3 ⊢ (𝑛 ∈ ℕ0 ↦ (𝐹‘(1o × {𝑛}))) = (𝑛 ∈ ℕ0 ↦ (𝐹‘(1o × {𝑛})))
8		fvex 6769	. . 3 ⊢ (𝐹‘(1o × {𝑁})) ∈ V
9	6, 7, 8	fvmpt 6857	. 2 ⊢ (𝑁 ∈ ℕ0 → ((𝑛 ∈ ℕ0 ↦ (𝐹‘(1o × {𝑛})))‘𝑁) = (𝐹‘(1o × {𝑁})))
10	3, 9	sylan9eq 2799	1 ⊢ ((𝐹 ∈ 𝑉 ∧ 𝑁 ∈ ℕ0) → (𝐴‘𝑁) = (𝐹‘(1o × {𝑁})))

Syntax hints:   → wi 4   ∧ wa 395   = wceq 1539   ∈ wcel 2108  {csn 4558   ↦ cmpt 5153   × cxp 5578  ‘cfv 6418  1_oc1o 8260  ℕ₀cn0 12163  coe₁cco1 21259

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1799  ax-4 1813  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2110  ax-9 2118  ax-10 2139  ax-11 2156  ax-12 2173  ax-ext 2709  ax-rep 5205  ax-sep 5218  ax-nul 5225  ax-pr 5347  ax-un 7566  ax-cnex 10858  ax-1cn 10860  ax-addcl 10862

This theorem depends on definitions:  df-bi 206  df-an 396  df-or 844  df-3or 1086  df-3an 1087  df-tru 1542  df-fal 1552  df-ex 1784  df-nf 1788  df-sb 2069  df-mo 2540  df-eu 2569  df-clab 2716  df-cleq 2730  df-clel 2817  df-nfc 2888  df-ne 2943  df-ral 3068  df-rex 3069  df-reu 3070  df-rab 3072  df-v 3424  df-sbc 3712  df-csb 3829  df-dif 3886  df-un 3888  df-in 3890  df-ss 3900  df-pss 3902  df-nul 4254  df-if 4457  df-pw 4532  df-sn 4559  df-pr 4561  df-tp 4563  df-op 4565  df-uni 4837  df-iun 4923  df-br 5071  df-opab 5133  df-mpt 5154  df-tr 5188  df-id 5480  df-eprel 5486  df-po 5494  df-so 5495  df-fr 5535  df-we 5537  df-xp 5586  df-rel 5587  df-cnv 5588  df-co 5589  df-dm 5590  df-rn 5591  df-res 5592  df-ima 5593  df-pred 6191  df-ord 6254  df-on 6255  df-lim 6256  df-suc 6257  df-iota 6376  df-fun 6420  df-fn 6421  df-f 6422  df-f1 6423  df-fo 6424  df-f1o 6425  df-fv 6426  df-ov 7258  df-om 7688  df-2nd 7805  df-frecs 8068  df-wrecs 8099  df-recs 8173  df-rdg 8212  df-nn 11904  df-n0 12164  df-coe1 21264

This theorem is referenced by:  fvcoe1  21288  coe1mul2  21350  deg1le0  25181