Theorem coe1fval 22197

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 3453	. 2 ⊢ (𝐹 ∈ 𝑉 → 𝐹 ∈ V)
2		coe1fval.a	. . 3 ⊢ 𝐴 = (coe1‘𝐹)
3		fveq1 6833	. . . . 5 ⊢ (𝑓 = 𝐹 → (𝑓‘(1o × {𝑛})) = (𝐹‘(1o × {𝑛})))
4	3	mpteq2dv 5173	. . . 4 ⊢ (𝑓 = 𝐹 → (𝑛 ∈ ℕ0 ↦ (𝑓‘(1o × {𝑛}))) = (𝑛 ∈ ℕ0 ↦ (𝐹‘(1o × {𝑛}))))
5		df-coe1 22175	. . . 4 ⊢ coe1 = (𝑓 ∈ V ↦ (𝑛 ∈ ℕ0 ↦ (𝑓‘(1o × {𝑛}))))
6		nn0ex 12441	. . . . 5 ⊢ ℕ0 ∈ V
7	6	mptex 7174	. . . 4 ⊢ (𝑛 ∈ ℕ0 ↦ (𝐹‘(1o × {𝑛}))) ∈ V
8	4, 5, 7	fvmpt 6942	. . 3 ⊢ (𝐹 ∈ V → (coe1‘𝐹) = (𝑛 ∈ ℕ0 ↦ (𝐹‘(1o × {𝑛}))))
9	2, 8	eqtrid 2787	. 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 1547   ∈ wcel 2119  Vcvv 3432  {csn 4562   ↦ cmpt 5160   × cxp 5623  ‘cfv 6492  1_oc1o 8395  ℕ₀cn0 12435  coe₁cco1 22170

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1802  ax-4 1816  ax-5 1917  ax-6 1974  ax-7 2015  ax-8 2121  ax-9 2129  ax-10 2152  ax-11 2168  ax-12 2189  ax-ext 2712  ax-rep 5206  ax-sep 5225  ax-nul 5235  ax-pr 5369  ax-un 7685  ax-cnex 11092  ax-1cn 11094  ax-addcl 11096

This theorem depends on definitions:  df-bi 208  df-an 397  df-or 854  df-3or 1093  df-3an 1094  df-tru 1550  df-fal 1560  df-ex 1787  df-nf 1791  df-sb 2074  df-mo 2543  df-eu 2573  df-clab 2719  df-cleq 2732  df-clel 2815  df-nfc 2889  df-ne 2936  df-ral 3055  df-rex 3065  df-reu 3346  df-rab 3393  df-v 3434  df-sbc 3731  df-csb 3839  df-dif 3893  df-un 3895  df-in 3897  df-ss 3907  df-pss 3910  df-nul 4269  df-if 4462  df-pw 4538  df-sn 4563  df-pr 4565  df-op 4569  df-uni 4846  df-iun 4930  df-br 5080  df-opab 5142  df-mpt 5161  df-tr 5187  df-id 5520  df-eprel 5525  df-po 5533  df-so 5534  df-fr 5578  df-we 5580  df-xp 5631  df-rel 5632  df-cnv 5633  df-co 5634  df-dm 5635  df-rn 5636  df-res 5637  df-ima 5638  df-pred 6259  df-ord 6320  df-on 6321  df-lim 6322  df-suc 6323  df-iota 6448  df-fun 6494  df-fn 6495  df-f 6496  df-f1 6497  df-fo 6498  df-f1o 6499  df-fv 6500  df-ov 7366  df-om 7814  df-2nd 7939  df-frecs 8228  df-wrecs 8259  df-recs 8308  df-rdg 8346  df-nn 12173  df-n0 12436  df-coe1 22175

This theorem is referenced by:  coe1fv  22198  coe1fval3  22200