Theorem coe1fval 22158

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 3463	. 2 ⊢ (𝐹 ∈ 𝑉 → 𝐹 ∈ V)
2		coe1fval.a	. . 3 ⊢ 𝐴 = (coe1‘𝐹)
3		fveq1 6841	. . . . 5 ⊢ (𝑓 = 𝐹 → (𝑓‘(1o × {𝑛})) = (𝐹‘(1o × {𝑛})))
4	3	mpteq2dv 5194	. . . 4 ⊢ (𝑓 = 𝐹 → (𝑛 ∈ ℕ0 ↦ (𝑓‘(1o × {𝑛}))) = (𝑛 ∈ ℕ0 ↦ (𝐹‘(1o × {𝑛}))))
5		df-coe1 22135	. . . 4 ⊢ coe1 = (𝑓 ∈ V ↦ (𝑛 ∈ ℕ0 ↦ (𝑓‘(1o × {𝑛}))))
6		nn0ex 12419	. . . . 5 ⊢ ℕ0 ∈ V
7	6	mptex 7179	. . . 4 ⊢ (𝑛 ∈ ℕ0 ↦ (𝐹‘(1o × {𝑛}))) ∈ V
8	4, 5, 7	fvmpt 6949	. . 3 ⊢ (𝐹 ∈ V → (coe1‘𝐹) = (𝑛 ∈ ℕ0 ↦ (𝐹‘(1o × {𝑛}))))
9	2, 8	eqtrid 2784	. 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 1542   ∈ wcel 2114  Vcvv 3442  {csn 4582   ↦ cmpt 5181   × cxp 5630  ‘cfv 6500  1_oc1o 8400  ℕ₀cn0 12413  coe₁cco1 22130

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1912  ax-6 1969  ax-7 2010  ax-8 2116  ax-9 2124  ax-10 2147  ax-11 2163  ax-12 2185  ax-ext 2709  ax-rep 5226  ax-sep 5243  ax-nul 5253  ax-pr 5379  ax-un 7690  ax-cnex 11094  ax-1cn 11096  ax-addcl 11098

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3or 1088  df-3an 1089  df-tru 1545  df-fal 1555  df-ex 1782  df-nf 1786  df-sb 2069  df-mo 2540  df-eu 2570  df-clab 2716  df-cleq 2729  df-clel 2812  df-nfc 2886  df-ne 2934  df-ral 3053  df-rex 3063  df-reu 3353  df-rab 3402  df-v 3444  df-sbc 3743  df-csb 3852  df-dif 3906  df-un 3908  df-in 3910  df-ss 3920  df-pss 3923  df-nul 4288  df-if 4482  df-pw 4558  df-sn 4583  df-pr 4585  df-op 4589  df-uni 4866  df-iun 4950  df-br 5101  df-opab 5163  df-mpt 5182  df-tr 5208  df-id 5527  df-eprel 5532  df-po 5540  df-so 5541  df-fr 5585  df-we 5587  df-xp 5638  df-rel 5639  df-cnv 5640  df-co 5641  df-dm 5642  df-rn 5643  df-res 5644  df-ima 5645  df-pred 6267  df-ord 6328  df-on 6329  df-lim 6330  df-suc 6331  df-iota 6456  df-fun 6502  df-fn 6503  df-f 6504  df-f1 6505  df-fo 6506  df-f1o 6507  df-fv 6508  df-ov 7371  df-om 7819  df-2nd 7944  df-frecs 8233  df-wrecs 8264  df-recs 8313  df-rdg 8351  df-nn 12158  df-n0 12414  df-coe1 22135

This theorem is referenced by:  coe1fv  22159  coe1fval3  22161