Theorem coe1fv 22114

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 22113	. . 3 ⊢ (𝐹 ∈ 𝑉 → 𝐴 = (𝑛 ∈ ℕ0 ↦ (𝐹‘(1o × {𝑛}))))
3	2	fveq1d 6819	. 2 ⊢ (𝐹 ∈ 𝑉 → (𝐴‘𝑁) = ((𝑛 ∈ ℕ0 ↦ (𝐹‘(1o × {𝑛})))‘𝑁))
4		sneq 4581	. . . . 5 ⊢ (𝑛 = 𝑁 → {𝑛} = {𝑁})
5	4	xpeq2d 5641	. . . 4 ⊢ (𝑛 = 𝑁 → (1o × {𝑛}) = (1o × {𝑁}))
6	5	fveq2d 6821	. . 3 ⊢ (𝑛 = 𝑁 → (𝐹‘(1o × {𝑛})) = (𝐹‘(1o × {𝑁})))
7		eqid 2731	. . 3 ⊢ (𝑛 ∈ ℕ0 ↦ (𝐹‘(1o × {𝑛}))) = (𝑛 ∈ ℕ0 ↦ (𝐹‘(1o × {𝑛})))
8		fvex 6830	. . 3 ⊢ (𝐹‘(1o × {𝑁})) ∈ V
9	6, 7, 8	fvmpt 6924	. 2 ⊢ (𝑁 ∈ ℕ0 → ((𝑛 ∈ ℕ0 ↦ (𝐹‘(1o × {𝑛})))‘𝑁) = (𝐹‘(1o × {𝑁})))
10	3, 9	sylan9eq 2786	1 ⊢ ((𝐹 ∈ 𝑉 ∧ 𝑁 ∈ ℕ0) → (𝐴‘𝑁) = (𝐹‘(1o × {𝑁})))

Syntax hints:   → wi 4   ∧ wa 395   = wceq 1541   ∈ wcel 2111  {csn 4571   ↦ cmpt 5167   × cxp 5609  ‘cfv 6476  1_oc1o 8373  ℕ₀cn0 12376  coe₁cco1 22085

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1911  ax-6 1968  ax-7 2009  ax-8 2113  ax-9 2121  ax-10 2144  ax-11 2160  ax-12 2180  ax-ext 2703  ax-rep 5212  ax-sep 5229  ax-nul 5239  ax-pr 5365  ax-un 7663  ax-cnex 11057  ax-1cn 11059  ax-addcl 11061

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3or 1087  df-3an 1088  df-tru 1544  df-fal 1554  df-ex 1781  df-nf 1785  df-sb 2068  df-mo 2535  df-eu 2564  df-clab 2710  df-cleq 2723  df-clel 2806  df-nfc 2881  df-ne 2929  df-ral 3048  df-rex 3057  df-reu 3347  df-rab 3396  df-v 3438  df-sbc 3737  df-csb 3846  df-dif 3900  df-un 3902  df-in 3904  df-ss 3914  df-pss 3917  df-nul 4279  df-if 4471  df-pw 4547  df-sn 4572  df-pr 4574  df-op 4578  df-uni 4855  df-iun 4938  df-br 5087  df-opab 5149  df-mpt 5168  df-tr 5194  df-id 5506  df-eprel 5511  df-po 5519  df-so 5520  df-fr 5564  df-we 5566  df-xp 5617  df-rel 5618  df-cnv 5619  df-co 5620  df-dm 5621  df-rn 5622  df-res 5623  df-ima 5624  df-pred 6243  df-ord 6304  df-on 6305  df-lim 6306  df-suc 6307  df-iota 6432  df-fun 6478  df-fn 6479  df-f 6480  df-f1 6481  df-fo 6482  df-f1o 6483  df-fv 6484  df-ov 7344  df-om 7792  df-2nd 7917  df-frecs 8206  df-wrecs 8237  df-recs 8286  df-rdg 8324  df-nn 12121  df-n0 12377  df-coe1 22090

This theorem is referenced by:  fvcoe1  22115  coe1mul2  22178  deg1le0  26038