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Theorem coe1termlem 23818
 Description: The coefficient function of a monomial. (Contributed by Mario Carneiro, 26-Jul-2014.) (Revised by Mario Carneiro, 23-Aug-2014.)
Hypothesis
Ref Expression
coe1term.1 𝐹 = (𝑧 ∈ ℂ ↦ (𝐴 · (𝑧𝑁)))
Assertion
Ref Expression
coe1termlem ((𝐴 ∈ ℂ ∧ 𝑁 ∈ ℕ0) → ((coeff‘𝐹) = (𝑛 ∈ ℕ0 ↦ if(𝑛 = 𝑁, 𝐴, 0)) ∧ (𝐴 ≠ 0 → (deg‘𝐹) = 𝑁)))
Distinct variable groups:   𝑧,𝑛,𝐴   𝑛,𝑁,𝑧
Allowed substitution hints:   𝐹(𝑧,𝑛)

Proof of Theorem coe1termlem
Dummy variable 𝑘 is distinct from all other variables.
StepHypRef Expression
1 ssid 3587 . . . 4 ℂ ⊆ ℂ
2 coe1term.1 . . . . 5 𝐹 = (𝑧 ∈ ℂ ↦ (𝐴 · (𝑧𝑁)))
32ply1term 23764 . . . 4 ((ℂ ⊆ ℂ ∧ 𝐴 ∈ ℂ ∧ 𝑁 ∈ ℕ0) → 𝐹 ∈ (Poly‘ℂ))
41, 3mp3an1 1403 . . 3 ((𝐴 ∈ ℂ ∧ 𝑁 ∈ ℕ0) → 𝐹 ∈ (Poly‘ℂ))
5 simpr 476 . . 3 ((𝐴 ∈ ℂ ∧ 𝑁 ∈ ℕ0) → 𝑁 ∈ ℕ0)
6 simpl 472 . . . . . 6 ((𝐴 ∈ ℂ ∧ 𝑁 ∈ ℕ0) → 𝐴 ∈ ℂ)
7 0cn 9911 . . . . . 6 0 ∈ ℂ
8 ifcl 4080 . . . . . 6 ((𝐴 ∈ ℂ ∧ 0 ∈ ℂ) → if(𝑛 = 𝑁, 𝐴, 0) ∈ ℂ)
96, 7, 8sylancl 693 . . . . 5 ((𝐴 ∈ ℂ ∧ 𝑁 ∈ ℕ0) → if(𝑛 = 𝑁, 𝐴, 0) ∈ ℂ)
109adantr 480 . . . 4 (((𝐴 ∈ ℂ ∧ 𝑁 ∈ ℕ0) ∧ 𝑛 ∈ ℕ0) → if(𝑛 = 𝑁, 𝐴, 0) ∈ ℂ)
11 eqid 2610 . . . 4 (𝑛 ∈ ℕ0 ↦ if(𝑛 = 𝑁, 𝐴, 0)) = (𝑛 ∈ ℕ0 ↦ if(𝑛 = 𝑁, 𝐴, 0))
1210, 11fmptd 6292 . . 3 ((𝐴 ∈ ℂ ∧ 𝑁 ∈ ℕ0) → (𝑛 ∈ ℕ0 ↦ if(𝑛 = 𝑁, 𝐴, 0)):ℕ0⟶ℂ)
13 simpr 476 . . . . . . . 8 (((𝐴 ∈ ℂ ∧ 𝑁 ∈ ℕ0) ∧ 𝑘 ∈ ℕ0) → 𝑘 ∈ ℕ0)
14 ifcl 4080 . . . . . . . . . 10 ((𝐴 ∈ ℂ ∧ 0 ∈ ℂ) → if(𝑘 = 𝑁, 𝐴, 0) ∈ ℂ)
156, 7, 14sylancl 693 . . . . . . . . 9 ((𝐴 ∈ ℂ ∧ 𝑁 ∈ ℕ0) → if(𝑘 = 𝑁, 𝐴, 0) ∈ ℂ)
1615adantr 480 . . . . . . . 8 (((𝐴 ∈ ℂ ∧ 𝑁 ∈ ℕ0) ∧ 𝑘 ∈ ℕ0) → if(𝑘 = 𝑁, 𝐴, 0) ∈ ℂ)
17 eqeq1 2614 . . . . . . . . . 10 (𝑛 = 𝑘 → (𝑛 = 𝑁𝑘 = 𝑁))
1817ifbid 4058 . . . . . . . . 9 (𝑛 = 𝑘 → if(𝑛 = 𝑁, 𝐴, 0) = if(𝑘 = 𝑁, 𝐴, 0))
1918, 11fvmptg 6189 . . . . . . . 8 ((𝑘 ∈ ℕ0 ∧ if(𝑘 = 𝑁, 𝐴, 0) ∈ ℂ) → ((𝑛 ∈ ℕ0 ↦ if(𝑛 = 𝑁, 𝐴, 0))‘𝑘) = if(𝑘 = 𝑁, 𝐴, 0))
2013, 16, 19syl2anc 691 . . . . . . 7 (((𝐴 ∈ ℂ ∧ 𝑁 ∈ ℕ0) ∧ 𝑘 ∈ ℕ0) → ((𝑛 ∈ ℕ0 ↦ if(𝑛 = 𝑁, 𝐴, 0))‘𝑘) = if(𝑘 = 𝑁, 𝐴, 0))
2120neeq1d 2841 . . . . . 6 (((𝐴 ∈ ℂ ∧ 𝑁 ∈ ℕ0) ∧ 𝑘 ∈ ℕ0) → (((𝑛 ∈ ℕ0 ↦ if(𝑛 = 𝑁, 𝐴, 0))‘𝑘) ≠ 0 ↔ if(𝑘 = 𝑁, 𝐴, 0) ≠ 0))
22 nn0re 11178 . . . . . . . . 9 (𝑁 ∈ ℕ0𝑁 ∈ ℝ)
2322leidd 10473 . . . . . . . 8 (𝑁 ∈ ℕ0𝑁𝑁)
2423ad2antlr 759 . . . . . . 7 (((𝐴 ∈ ℂ ∧ 𝑁 ∈ ℕ0) ∧ 𝑘 ∈ ℕ0) → 𝑁𝑁)
25 iffalse 4045 . . . . . . . . 9 𝑘 = 𝑁 → if(𝑘 = 𝑁, 𝐴, 0) = 0)
2625necon1ai 2809 . . . . . . . 8 (if(𝑘 = 𝑁, 𝐴, 0) ≠ 0 → 𝑘 = 𝑁)
2726breq1d 4593 . . . . . . 7 (if(𝑘 = 𝑁, 𝐴, 0) ≠ 0 → (𝑘𝑁𝑁𝑁))
2824, 27syl5ibrcom 236 . . . . . 6 (((𝐴 ∈ ℂ ∧ 𝑁 ∈ ℕ0) ∧ 𝑘 ∈ ℕ0) → (if(𝑘 = 𝑁, 𝐴, 0) ≠ 0 → 𝑘𝑁))
2921, 28sylbid 229 . . . . 5 (((𝐴 ∈ ℂ ∧ 𝑁 ∈ ℕ0) ∧ 𝑘 ∈ ℕ0) → (((𝑛 ∈ ℕ0 ↦ if(𝑛 = 𝑁, 𝐴, 0))‘𝑘) ≠ 0 → 𝑘𝑁))
3029ralrimiva 2949 . . . 4 ((𝐴 ∈ ℂ ∧ 𝑁 ∈ ℕ0) → ∀𝑘 ∈ ℕ0 (((𝑛 ∈ ℕ0 ↦ if(𝑛 = 𝑁, 𝐴, 0))‘𝑘) ≠ 0 → 𝑘𝑁))
31 plyco0 23752 . . . . 5 ((𝑁 ∈ ℕ0 ∧ (𝑛 ∈ ℕ0 ↦ if(𝑛 = 𝑁, 𝐴, 0)):ℕ0⟶ℂ) → (((𝑛 ∈ ℕ0 ↦ if(𝑛 = 𝑁, 𝐴, 0)) “ (ℤ‘(𝑁 + 1))) = {0} ↔ ∀𝑘 ∈ ℕ0 (((𝑛 ∈ ℕ0 ↦ if(𝑛 = 𝑁, 𝐴, 0))‘𝑘) ≠ 0 → 𝑘𝑁)))
325, 12, 31syl2anc 691 . . . 4 ((𝐴 ∈ ℂ ∧ 𝑁 ∈ ℕ0) → (((𝑛 ∈ ℕ0 ↦ if(𝑛 = 𝑁, 𝐴, 0)) “ (ℤ‘(𝑁 + 1))) = {0} ↔ ∀𝑘 ∈ ℕ0 (((𝑛 ∈ ℕ0 ↦ if(𝑛 = 𝑁, 𝐴, 0))‘𝑘) ≠ 0 → 𝑘𝑁)))
3330, 32mpbird 246 . . 3 ((𝐴 ∈ ℂ ∧ 𝑁 ∈ ℕ0) → ((𝑛 ∈ ℕ0 ↦ if(𝑛 = 𝑁, 𝐴, 0)) “ (ℤ‘(𝑁 + 1))) = {0})
342ply1termlem 23763 . . . 4 ((𝐴 ∈ ℂ ∧ 𝑁 ∈ ℕ0) → 𝐹 = (𝑧 ∈ ℂ ↦ Σ𝑘 ∈ (0...𝑁)(if(𝑘 = 𝑁, 𝐴, 0) · (𝑧𝑘))))
35 elfznn0 12302 . . . . . . 7 (𝑘 ∈ (0...𝑁) → 𝑘 ∈ ℕ0)
3620oveq1d 6564 . . . . . . 7 (((𝐴 ∈ ℂ ∧ 𝑁 ∈ ℕ0) ∧ 𝑘 ∈ ℕ0) → (((𝑛 ∈ ℕ0 ↦ if(𝑛 = 𝑁, 𝐴, 0))‘𝑘) · (𝑧𝑘)) = (if(𝑘 = 𝑁, 𝐴, 0) · (𝑧𝑘)))
3735, 36sylan2 490 . . . . . 6 (((𝐴 ∈ ℂ ∧ 𝑁 ∈ ℕ0) ∧ 𝑘 ∈ (0...𝑁)) → (((𝑛 ∈ ℕ0 ↦ if(𝑛 = 𝑁, 𝐴, 0))‘𝑘) · (𝑧𝑘)) = (if(𝑘 = 𝑁, 𝐴, 0) · (𝑧𝑘)))
3837sumeq2dv 14281 . . . . 5 ((𝐴 ∈ ℂ ∧ 𝑁 ∈ ℕ0) → Σ𝑘 ∈ (0...𝑁)(((𝑛 ∈ ℕ0 ↦ if(𝑛 = 𝑁, 𝐴, 0))‘𝑘) · (𝑧𝑘)) = Σ𝑘 ∈ (0...𝑁)(if(𝑘 = 𝑁, 𝐴, 0) · (𝑧𝑘)))
3938mpteq2dv 4673 . . . 4 ((𝐴 ∈ ℂ ∧ 𝑁 ∈ ℕ0) → (𝑧 ∈ ℂ ↦ Σ𝑘 ∈ (0...𝑁)(((𝑛 ∈ ℕ0 ↦ if(𝑛 = 𝑁, 𝐴, 0))‘𝑘) · (𝑧𝑘))) = (𝑧 ∈ ℂ ↦ Σ𝑘 ∈ (0...𝑁)(if(𝑘 = 𝑁, 𝐴, 0) · (𝑧𝑘))))
4034, 39eqtr4d 2647 . . 3 ((𝐴 ∈ ℂ ∧ 𝑁 ∈ ℕ0) → 𝐹 = (𝑧 ∈ ℂ ↦ Σ𝑘 ∈ (0...𝑁)(((𝑛 ∈ ℕ0 ↦ if(𝑛 = 𝑁, 𝐴, 0))‘𝑘) · (𝑧𝑘))))
414, 5, 12, 33, 40coeeq 23787 . 2 ((𝐴 ∈ ℂ ∧ 𝑁 ∈ ℕ0) → (coeff‘𝐹) = (𝑛 ∈ ℕ0 ↦ if(𝑛 = 𝑁, 𝐴, 0)))
424adantr 480 . . . 4 (((𝐴 ∈ ℂ ∧ 𝑁 ∈ ℕ0) ∧ 𝐴 ≠ 0) → 𝐹 ∈ (Poly‘ℂ))
435adantr 480 . . . 4 (((𝐴 ∈ ℂ ∧ 𝑁 ∈ ℕ0) ∧ 𝐴 ≠ 0) → 𝑁 ∈ ℕ0)
4412adantr 480 . . . 4 (((𝐴 ∈ ℂ ∧ 𝑁 ∈ ℕ0) ∧ 𝐴 ≠ 0) → (𝑛 ∈ ℕ0 ↦ if(𝑛 = 𝑁, 𝐴, 0)):ℕ0⟶ℂ)
4533adantr 480 . . . 4 (((𝐴 ∈ ℂ ∧ 𝑁 ∈ ℕ0) ∧ 𝐴 ≠ 0) → ((𝑛 ∈ ℕ0 ↦ if(𝑛 = 𝑁, 𝐴, 0)) “ (ℤ‘(𝑁 + 1))) = {0})
4640adantr 480 . . . 4 (((𝐴 ∈ ℂ ∧ 𝑁 ∈ ℕ0) ∧ 𝐴 ≠ 0) → 𝐹 = (𝑧 ∈ ℂ ↦ Σ𝑘 ∈ (0...𝑁)(((𝑛 ∈ ℕ0 ↦ if(𝑛 = 𝑁, 𝐴, 0))‘𝑘) · (𝑧𝑘))))
47 iftrue 4042 . . . . . . . 8 (𝑛 = 𝑁 → if(𝑛 = 𝑁, 𝐴, 0) = 𝐴)
4847, 11fvmptg 6189 . . . . . . 7 ((𝑁 ∈ ℕ0𝐴 ∈ ℂ) → ((𝑛 ∈ ℕ0 ↦ if(𝑛 = 𝑁, 𝐴, 0))‘𝑁) = 𝐴)
4948ancoms 468 . . . . . 6 ((𝐴 ∈ ℂ ∧ 𝑁 ∈ ℕ0) → ((𝑛 ∈ ℕ0 ↦ if(𝑛 = 𝑁, 𝐴, 0))‘𝑁) = 𝐴)
5049neeq1d 2841 . . . . 5 ((𝐴 ∈ ℂ ∧ 𝑁 ∈ ℕ0) → (((𝑛 ∈ ℕ0 ↦ if(𝑛 = 𝑁, 𝐴, 0))‘𝑁) ≠ 0 ↔ 𝐴 ≠ 0))
5150biimpar 501 . . . 4 (((𝐴 ∈ ℂ ∧ 𝑁 ∈ ℕ0) ∧ 𝐴 ≠ 0) → ((𝑛 ∈ ℕ0 ↦ if(𝑛 = 𝑁, 𝐴, 0))‘𝑁) ≠ 0)
5242, 43, 44, 45, 46, 51dgreq 23804 . . 3 (((𝐴 ∈ ℂ ∧ 𝑁 ∈ ℕ0) ∧ 𝐴 ≠ 0) → (deg‘𝐹) = 𝑁)
5352ex 449 . 2 ((𝐴 ∈ ℂ ∧ 𝑁 ∈ ℕ0) → (𝐴 ≠ 0 → (deg‘𝐹) = 𝑁))
5441, 53jca 553 1 ((𝐴 ∈ ℂ ∧ 𝑁 ∈ ℕ0) → ((coeff‘𝐹) = (𝑛 ∈ ℕ0 ↦ if(𝑛 = 𝑁, 𝐴, 0)) ∧ (𝐴 ≠ 0 → (deg‘𝐹) = 𝑁)))
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ↔ wb 195   ∧ wa 383   = wceq 1475   ∈ wcel 1977   ≠ wne 2780  ∀wral 2896   ⊆ wss 3540  ifcif 4036  {csn 4125   class class class wbr 4583   ↦ cmpt 4643   “ cima 5041  ⟶wf 5800  ‘cfv 5804  (class class class)co 6549  ℂcc 9813  0cc0 9815  1c1 9816   + caddc 9818   · cmul 9820   ≤ cle 9954  ℕ0cn0 11169  ℤ≥cuz 11563  ...cfz 12197  ↑cexp 12722  Σcsu 14264  Polycply 23744  coeffccoe 23746  degcdgr 23747 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1713  ax-4 1728  ax-5 1827  ax-6 1875  ax-7 1922  ax-8 1979  ax-9 1986  ax-10 2006  ax-11 2021  ax-12 2034  ax-13 2234  ax-ext 2590  ax-rep 4699  ax-sep 4709  ax-nul 4717  ax-pow 4769  ax-pr 4833  ax-un 6847  ax-inf2 8421  ax-cnex 9871  ax-resscn 9872  ax-1cn 9873  ax-icn 9874  ax-addcl 9875  ax-addrcl 9876  ax-mulcl 9877  ax-mulrcl 9878  ax-mulcom 9879  ax-addass 9880  ax-mulass 9881  ax-distr 9882  ax-i2m1 9883  ax-1ne0 9884  ax-1rid 9885  ax-rnegex 9886  ax-rrecex 9887  ax-cnre 9888  ax-pre-lttri 9889  ax-pre-lttrn 9890  ax-pre-ltadd 9891  ax-pre-mulgt0 9892  ax-pre-sup 9893  ax-addf 9894 This theorem depends on definitions:  df-bi 196  df-or 384  df-an 385  df-3or 1032  df-3an 1033  df-tru 1478  df-fal 1481  df-ex 1696  df-nf 1701  df-sb 1868  df-eu 2462  df-mo 2463  df-clab 2597  df-cleq 2603  df-clel 2606  df-nfc 2740  df-ne 2782  df-nel 2783  df-ral 2901  df-rex 2902  df-reu 2903  df-rmo 2904  df-rab 2905  df-v 3175  df-sbc 3403  df-csb 3500  df-dif 3543  df-un 3545  df-in 3547  df-ss 3554  df-pss 3556  df-nul 3875  df-if 4037  df-pw 4110  df-sn 4126  df-pr 4128  df-tp 4130  df-op 4132  df-uni 4373  df-int 4411  df-iun 4457  df-br 4584  df-opab 4644  df-mpt 4645  df-tr 4681  df-eprel 4949  df-id 4953  df-po 4959  df-so 4960  df-fr 4997  df-se 4998  df-we 4999  df-xp 5044  df-rel 5045  df-cnv 5046  df-co 5047  df-dm 5048  df-rn 5049  df-res 5050  df-ima 5051  df-pred 5597  df-ord 5643  df-on 5644  df-lim 5645  df-suc 5646  df-iota 5768  df-fun 5806  df-fn 5807  df-f 5808  df-f1 5809  df-fo 5810  df-f1o 5811  df-fv 5812  df-isom 5813  df-riota 6511  df-ov 6552  df-oprab 6553  df-mpt2 6554  df-of 6795  df-om 6958  df-1st 7059  df-2nd 7060  df-wrecs 7294  df-recs 7355  df-rdg 7393  df-1o 7447  df-oadd 7451  df-er 7629  df-map 7746  df-pm 7747  df-en 7842  df-dom 7843  df-sdom 7844  df-fin 7845  df-sup 8231  df-inf 8232  df-oi 8298  df-card 8648  df-pnf 9955  df-mnf 9956  df-xr 9957  df-ltxr 9958  df-le 9959  df-sub 10147  df-neg 10148  df-div 10564  df-nn 10898  df-2 10956  df-3 10957  df-n0 11170  df-z 11255  df-uz 11564  df-rp 11709  df-fz 12198  df-fzo 12335  df-fl 12455  df-seq 12664  df-exp 12723  df-hash 12980  df-cj 13687  df-re 13688  df-im 13689  df-sqrt 13823  df-abs 13824  df-clim 14067  df-rlim 14068  df-sum 14265  df-0p 23243  df-ply 23748  df-coe 23750  df-dgr 23751 This theorem is referenced by:  coe1term  23819  dgr1term  23820
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