Theorem coe1fval 20375

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 3514	. 2 ⊢ (𝐹 ∈ 𝑉 → 𝐹 ∈ V)
2		coe1fval.a	. . 3 ⊢ 𝐴 = (coe1‘𝐹)
3		fveq1 6671	. . . . 5 ⊢ (𝑓 = 𝐹 → (𝑓‘(1o × {𝑛})) = (𝐹‘(1o × {𝑛})))
4	3	mpteq2dv 5164	. . . 4 ⊢ (𝑓 = 𝐹 → (𝑛 ∈ ℕ0 ↦ (𝑓‘(1o × {𝑛}))) = (𝑛 ∈ ℕ0 ↦ (𝐹‘(1o × {𝑛}))))
5		df-coe1 20353	. . . 4 ⊢ coe1 = (𝑓 ∈ V ↦ (𝑛 ∈ ℕ0 ↦ (𝑓‘(1o × {𝑛}))))
6		nn0ex 11906	. . . . 5 ⊢ ℕ0 ∈ V
7	6	mptex 6988	. . . 4 ⊢ (𝑛 ∈ ℕ0 ↦ (𝐹‘(1o × {𝑛}))) ∈ V
8	4, 5, 7	fvmpt 6770	. . 3 ⊢ (𝐹 ∈ V → (coe1‘𝐹) = (𝑛 ∈ ℕ0 ↦ (𝐹‘(1o × {𝑛}))))
9	2, 8	syl5eq 2870	. 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 1537   ∈ wcel 2114  Vcvv 3496  {csn 4569   ↦ cmpt 5148   × cxp 5555  ‘cfv 6357  1_oc1o 8097  ℕ₀cn0 11900  coe₁cco1 20348

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 1970  ax-7 2015  ax-8 2116  ax-9 2124  ax-10 2145  ax-11 2161  ax-12 2177  ax-ext 2795  ax-rep 5192  ax-sep 5205  ax-nul 5212  ax-pow 5268  ax-pr 5332  ax-un 7463  ax-cnex 10595  ax-1cn 10597  ax-addcl 10599

This theorem depends on definitions:  df-bi 209  df-an 399  df-or 844  df-3or 1084  df-3an 1085  df-tru 1540  df-ex 1781  df-nf 1785  df-sb 2070  df-mo 2622  df-eu 2654  df-clab 2802  df-cleq 2816  df-clel 2895  df-nfc 2965  df-ne 3019  df-ral 3145  df-rex 3146  df-reu 3147  df-rab 3149  df-v 3498  df-sbc 3775  df-csb 3886  df-dif 3941  df-un 3943  df-in 3945  df-ss 3954  df-pss 3956  df-nul 4294  df-if 4470  df-pw 4543  df-sn 4570  df-pr 4572  df-tp 4574  df-op 4576  df-uni 4841  df-iun 4923  df-br 5069  df-opab 5131  df-mpt 5149  df-tr 5175  df-id 5462  df-eprel 5467  df-po 5476  df-so 5477  df-fr 5516  df-we 5518  df-xp 5563  df-rel 5564  df-cnv 5565  df-co 5566  df-dm 5567  df-rn 5568  df-res 5569  df-ima 5570  df-pred 6150  df-ord 6196  df-on 6197  df-lim 6198  df-suc 6199  df-iota 6316  df-fun 6359  df-fn 6360  df-f 6361  df-f1 6362  df-fo 6363  df-f1o 6364  df-fv 6365  df-ov 7161  df-om 7583  df-wrecs 7949  df-recs 8010  df-rdg 8048  df-nn 11641  df-n0 11901  df-coe1 20353

This theorem is referenced by:  coe1fv  20376  coe1fval3  20378