Theorem coe1fval 22119

Description: Value of the univariate polynomial coefficient function. (Contributed by Stefan O'Rear, 21-Mar-2015.)

Hypothesis

Ref	Expression
coe1fval.a	⊢ 𝐴 = (coe1‘𝐹)

Assertion

Ref	Expression
coe1fval	⊢ (𝐹 ∈ 𝑉 → 𝐴 = (𝑛 ∈ ℕ0 ↦ (𝐹‘(1o × {𝑛}))))

Distinct variable group:   𝑛,𝐹

Allowed substitution hints:   𝐴(𝑛)   𝑉(𝑛)

Proof of Theorem coe1fval

Dummy variable 𝑓 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		elex 3458	. 2 ⊢ (𝐹 ∈ 𝑉 → 𝐹 ∈ V)
2		coe1fval.a	. . 3 ⊢ 𝐴 = (coe1‘𝐹)
3		fveq1 6827	. . . . 5 ⊢ (𝑓 = 𝐹 → (𝑓‘(1o × {𝑛})) = (𝐹‘(1o × {𝑛})))
4	3	mpteq2dv 5187	. . . 4 ⊢ (𝑓 = 𝐹 → (𝑛 ∈ ℕ0 ↦ (𝑓‘(1o × {𝑛}))) = (𝑛 ∈ ℕ0 ↦ (𝐹‘(1o × {𝑛}))))
5		df-coe1 22096	. . . 4 ⊢ coe1 = (𝑓 ∈ V ↦ (𝑛 ∈ ℕ0 ↦ (𝑓‘(1o × {𝑛}))))
6		nn0ex 12394	. . . . 5 ⊢ ℕ0 ∈ V
7	6	mptex 7163	. . . 4 ⊢ (𝑛 ∈ ℕ0 ↦ (𝐹‘(1o × {𝑛}))) ∈ V
8	4, 5, 7	fvmpt 6935	. . 3 ⊢ (𝐹 ∈ V → (coe1‘𝐹) = (𝑛 ∈ ℕ0 ↦ (𝐹‘(1o × {𝑛}))))
9	2, 8	eqtrid 2780	. 2 ⊢ (𝐹 ∈ V → 𝐴 = (𝑛 ∈ ℕ0 ↦ (𝐹‘(1o × {𝑛}))))
10	1, 9	syl 17	1 ⊢ (𝐹 ∈ 𝑉 → 𝐴 = (𝑛 ∈ ℕ0 ↦ (𝐹‘(1o × {𝑛}))))

Syntax hints:   → wi 4   = wceq 1541   ∈ wcel 2113  Vcvv 3437  {csn 4575   ↦ cmpt 5174   × cxp 5617  ‘cfv 6486  1_oc1o 8384  ℕ₀cn0 12388  coe₁cco1 22091

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 2115  ax-9 2123  ax-10 2146  ax-11 2162  ax-12 2182  ax-ext 2705  ax-rep 5219  ax-sep 5236  ax-nul 5246  ax-pr 5372  ax-un 7674  ax-cnex 11069  ax-1cn 11071  ax-addcl 11073

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 2537  df-eu 2566  df-clab 2712  df-cleq 2725  df-clel 2808  df-nfc 2882  df-ne 2930  df-ral 3049  df-rex 3058  df-reu 3348  df-rab 3397  df-v 3439  df-sbc 3738  df-csb 3847  df-dif 3901  df-un 3903  df-in 3905  df-ss 3915  df-pss 3918  df-nul 4283  df-if 4475  df-pw 4551  df-sn 4576  df-pr 4578  df-op 4582  df-uni 4859  df-iun 4943  df-br 5094  df-opab 5156  df-mpt 5175  df-tr 5201  df-id 5514  df-eprel 5519  df-po 5527  df-so 5528  df-fr 5572  df-we 5574  df-xp 5625  df-rel 5626  df-cnv 5627  df-co 5628  df-dm 5629  df-rn 5630  df-res 5631  df-ima 5632  df-pred 6253  df-ord 6314  df-on 6315  df-lim 6316  df-suc 6317  df-iota 6442  df-fun 6488  df-fn 6489  df-f 6490  df-f1 6491  df-fo 6492  df-f1o 6493  df-fv 6494  df-ov 7355  df-om 7803  df-2nd 7928  df-frecs 8217  df-wrecs 8248  df-recs 8297  df-rdg 8335  df-nn 12133  df-n0 12389  df-coe1 22096

This theorem is referenced by:  coe1fv  22120  coe1fval3  22122