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Theorem coe1fval 19503
Description: Value of the univariate polynomial coefficient function. (Contributed by Stefan O'Rear, 21-Mar-2015.)
Hypothesis
Ref Expression
coe1fval.a 𝐴 = (coe1𝐹)
Assertion
Ref Expression
coe1fval (𝐹𝑉𝐴 = (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.
StepHypRef Expression
1 elex 3201 . 2 (𝐹𝑉𝐹 ∈ V)
2 coe1fval.a . . 3 𝐴 = (coe1𝐹)
3 fveq1 6152 . . . . 5 (𝑓 = 𝐹 → (𝑓‘(1𝑜 × {𝑛})) = (𝐹‘(1𝑜 × {𝑛})))
43mpteq2dv 4710 . . . 4 (𝑓 = 𝐹 → (𝑛 ∈ ℕ0 ↦ (𝑓‘(1𝑜 × {𝑛}))) = (𝑛 ∈ ℕ0 ↦ (𝐹‘(1𝑜 × {𝑛}))))
5 df-coe1 19481 . . . 4 coe1 = (𝑓 ∈ V ↦ (𝑛 ∈ ℕ0 ↦ (𝑓‘(1𝑜 × {𝑛}))))
6 nn0ex 11249 . . . . 5 0 ∈ V
76mptex 6446 . . . 4 (𝑛 ∈ ℕ0 ↦ (𝐹‘(1𝑜 × {𝑛}))) ∈ V
84, 5, 7fvmpt 6244 . . 3 (𝐹 ∈ V → (coe1𝐹) = (𝑛 ∈ ℕ0 ↦ (𝐹‘(1𝑜 × {𝑛}))))
92, 8syl5eq 2667 . 2 (𝐹 ∈ V → 𝐴 = (𝑛 ∈ ℕ0 ↦ (𝐹‘(1𝑜 × {𝑛}))))
101, 9syl 17 1 (𝐹𝑉𝐴 = (𝑛 ∈ ℕ0 ↦ (𝐹‘(1𝑜 × {𝑛}))))
Colors of variables: wff setvar class
Syntax hints:  wi 4   = wceq 1480  wcel 1987  Vcvv 3189  {csn 4153  cmpt 4678   × cxp 5077  cfv 5852  1𝑜c1o 7505  0cn0 11243  coe1cco1 19476
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1719  ax-4 1734  ax-5 1836  ax-6 1885  ax-7 1932  ax-8 1989  ax-9 1996  ax-10 2016  ax-11 2031  ax-12 2044  ax-13 2245  ax-ext 2601  ax-rep 4736  ax-sep 4746  ax-nul 4754  ax-pow 4808  ax-pr 4872  ax-un 6909  ax-cnex 9943  ax-resscn 9944  ax-1cn 9945  ax-icn 9946  ax-addcl 9947  ax-addrcl 9948  ax-mulcl 9949  ax-mulrcl 9950  ax-i2m1 9955  ax-1ne0 9956  ax-rrecex 9959  ax-cnre 9960
This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3or 1037  df-3an 1038  df-tru 1483  df-ex 1702  df-nf 1707  df-sb 1878  df-eu 2473  df-mo 2474  df-clab 2608  df-cleq 2614  df-clel 2617  df-nfc 2750  df-ne 2791  df-ral 2912  df-rex 2913  df-reu 2914  df-rab 2916  df-v 3191  df-sbc 3422  df-csb 3519  df-dif 3562  df-un 3564  df-in 3566  df-ss 3573  df-pss 3575  df-nul 3897  df-if 4064  df-pw 4137  df-sn 4154  df-pr 4156  df-tp 4158  df-op 4160  df-uni 4408  df-iun 4492  df-br 4619  df-opab 4679  df-mpt 4680  df-tr 4718  df-eprel 4990  df-id 4994  df-po 5000  df-so 5001  df-fr 5038  df-we 5040  df-xp 5085  df-rel 5086  df-cnv 5087  df-co 5088  df-dm 5089  df-rn 5090  df-res 5091  df-ima 5092  df-pred 5644  df-ord 5690  df-on 5691  df-lim 5692  df-suc 5693  df-iota 5815  df-fun 5854  df-fn 5855  df-f 5856  df-f1 5857  df-fo 5858  df-f1o 5859  df-fv 5860  df-ov 6613  df-om 7020  df-wrecs 7359  df-recs 7420  df-rdg 7458  df-nn 10972  df-n0 11244  df-coe1 19481
This theorem is referenced by:  coe1fv  19504  coe1fval3  19506
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