Theorem 0dgrb 23806

Description: A function has degree zero iff it is a constant function. (Contributed by Mario Carneiro, 23-Jul-2014.)

Assertion

Ref	Expression
0dgrb	⊢ (𝐹 ∈ (Poly‘𝑆) → ((deg‘𝐹) = 0 ↔ 𝐹 = (ℂ × {(𝐹‘0)})))

Proof of Theorem 0dgrb

Dummy variables 𝑧 𝑘 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		eqid 2610	. . . . . . . 8 ⊢ (coeff‘𝐹) = (coeff‘𝐹)
2		eqid 2610	. . . . . . . 8 ⊢ (deg‘𝐹) = (deg‘𝐹)
3	1, 2	coeid 23798	. . . . . . 7 ⊢ (𝐹 ∈ (Poly‘𝑆) → 𝐹 = (𝑧 ∈ ℂ ↦ Σ𝑘 ∈ (0...(deg‘𝐹))(((coeff‘𝐹)‘𝑘) · (𝑧↑𝑘))))
4	3	adantr 480	. . . . . 6 ⊢ ((𝐹 ∈ (Poly‘𝑆) ∧ (deg‘𝐹) = 0) → 𝐹 = (𝑧 ∈ ℂ ↦ Σ𝑘 ∈ (0...(deg‘𝐹))(((coeff‘𝐹)‘𝑘) · (𝑧↑𝑘))))
5		simplr 788	. . . . . . . . . 10 ⊢ (((𝐹 ∈ (Poly‘𝑆) ∧ (deg‘𝐹) = 0) ∧ 𝑧 ∈ ℂ) → (deg‘𝐹) = 0)
6	5	oveq2d 6565	. . . . . . . . 9 ⊢ (((𝐹 ∈ (Poly‘𝑆) ∧ (deg‘𝐹) = 0) ∧ 𝑧 ∈ ℂ) → (0...(deg‘𝐹)) = (0...0))
7	6	sumeq1d 14279	. . . . . . . 8 ⊢ (((𝐹 ∈ (Poly‘𝑆) ∧ (deg‘𝐹) = 0) ∧ 𝑧 ∈ ℂ) → Σ𝑘 ∈ (0...(deg‘𝐹))(((coeff‘𝐹)‘𝑘) · (𝑧↑𝑘)) = Σ𝑘 ∈ (0...0)(((coeff‘𝐹)‘𝑘) · (𝑧↑𝑘)))
8		0z 11265	. . . . . . . . . 10 ⊢ 0 ∈ ℤ
9		exp0 12726	. . . . . . . . . . . . . 14 ⊢ (𝑧 ∈ ℂ → (𝑧↑0) = 1)
10	9	adantl 481	. . . . . . . . . . . . 13 ⊢ (((𝐹 ∈ (Poly‘𝑆) ∧ (deg‘𝐹) = 0) ∧ 𝑧 ∈ ℂ) → (𝑧↑0) = 1)
11	10	oveq2d 6565	. . . . . . . . . . . 12 ⊢ (((𝐹 ∈ (Poly‘𝑆) ∧ (deg‘𝐹) = 0) ∧ 𝑧 ∈ ℂ) → (((coeff‘𝐹)‘0) · (𝑧↑0)) = (((coeff‘𝐹)‘0) · 1))
12	1	coef3 23792	. . . . . . . . . . . . . . 15 ⊢ (𝐹 ∈ (Poly‘𝑆) → (coeff‘𝐹):ℕ0⟶ℂ)
13		0nn0 11184	. . . . . . . . . . . . . . 15 ⊢ 0 ∈ ℕ0
14		ffvelrn 6265	. . . . . . . . . . . . . . 15 ⊢ (((coeff‘𝐹):ℕ0⟶ℂ ∧ 0 ∈ ℕ0) → ((coeff‘𝐹)‘0) ∈ ℂ)
15	12, 13, 14	sylancl 693	. . . . . . . . . . . . . 14 ⊢ (𝐹 ∈ (Poly‘𝑆) → ((coeff‘𝐹)‘0) ∈ ℂ)
16	15	ad2antrr 758	. . . . . . . . . . . . 13 ⊢ (((𝐹 ∈ (Poly‘𝑆) ∧ (deg‘𝐹) = 0) ∧ 𝑧 ∈ ℂ) → ((coeff‘𝐹)‘0) ∈ ℂ)
17	16	mulid1d 9936	. . . . . . . . . . . 12 ⊢ (((𝐹 ∈ (Poly‘𝑆) ∧ (deg‘𝐹) = 0) ∧ 𝑧 ∈ ℂ) → (((coeff‘𝐹)‘0) · 1) = ((coeff‘𝐹)‘0))
18	11, 17	eqtrd 2644	. . . . . . . . . . 11 ⊢ (((𝐹 ∈ (Poly‘𝑆) ∧ (deg‘𝐹) = 0) ∧ 𝑧 ∈ ℂ) → (((coeff‘𝐹)‘0) · (𝑧↑0)) = ((coeff‘𝐹)‘0))
19	18, 16	eqeltrd 2688	. . . . . . . . . 10 ⊢ (((𝐹 ∈ (Poly‘𝑆) ∧ (deg‘𝐹) = 0) ∧ 𝑧 ∈ ℂ) → (((coeff‘𝐹)‘0) · (𝑧↑0)) ∈ ℂ)
20		fveq2 6103	. . . . . . . . . . . 12 ⊢ (𝑘 = 0 → ((coeff‘𝐹)‘𝑘) = ((coeff‘𝐹)‘0))
21		oveq2 6557	. . . . . . . . . . . 12 ⊢ (𝑘 = 0 → (𝑧↑𝑘) = (𝑧↑0))
22	20, 21	oveq12d 6567	. . . . . . . . . . 11 ⊢ (𝑘 = 0 → (((coeff‘𝐹)‘𝑘) · (𝑧↑𝑘)) = (((coeff‘𝐹)‘0) · (𝑧↑0)))
23	22	fsum1 14320	. . . . . . . . . 10 ⊢ ((0 ∈ ℤ ∧ (((coeff‘𝐹)‘0) · (𝑧↑0)) ∈ ℂ) → Σ𝑘 ∈ (0...0)(((coeff‘𝐹)‘𝑘) · (𝑧↑𝑘)) = (((coeff‘𝐹)‘0) · (𝑧↑0)))
24	8, 19, 23	sylancr 694	. . . . . . . . 9 ⊢ (((𝐹 ∈ (Poly‘𝑆) ∧ (deg‘𝐹) = 0) ∧ 𝑧 ∈ ℂ) → Σ𝑘 ∈ (0...0)(((coeff‘𝐹)‘𝑘) · (𝑧↑𝑘)) = (((coeff‘𝐹)‘0) · (𝑧↑0)))
25	24, 18	eqtrd 2644	. . . . . . . 8 ⊢ (((𝐹 ∈ (Poly‘𝑆) ∧ (deg‘𝐹) = 0) ∧ 𝑧 ∈ ℂ) → Σ𝑘 ∈ (0...0)(((coeff‘𝐹)‘𝑘) · (𝑧↑𝑘)) = ((coeff‘𝐹)‘0))
26	7, 25	eqtrd 2644	. . . . . . 7 ⊢ (((𝐹 ∈ (Poly‘𝑆) ∧ (deg‘𝐹) = 0) ∧ 𝑧 ∈ ℂ) → Σ𝑘 ∈ (0...(deg‘𝐹))(((coeff‘𝐹)‘𝑘) · (𝑧↑𝑘)) = ((coeff‘𝐹)‘0))
27	26	mpteq2dva 4672	. . . . . 6 ⊢ ((𝐹 ∈ (Poly‘𝑆) ∧ (deg‘𝐹) = 0) → (𝑧 ∈ ℂ ↦ Σ𝑘 ∈ (0...(deg‘𝐹))(((coeff‘𝐹)‘𝑘) · (𝑧↑𝑘))) = (𝑧 ∈ ℂ ↦ ((coeff‘𝐹)‘0)))
28	4, 27	eqtrd 2644	. . . . 5 ⊢ ((𝐹 ∈ (Poly‘𝑆) ∧ (deg‘𝐹) = 0) → 𝐹 = (𝑧 ∈ ℂ ↦ ((coeff‘𝐹)‘0)))
29		fconstmpt 5085	. . . . 5 ⊢ (ℂ × {((coeff‘𝐹)‘0)}) = (𝑧 ∈ ℂ ↦ ((coeff‘𝐹)‘0))
30	28, 29	syl6eqr 2662	. . . 4 ⊢ ((𝐹 ∈ (Poly‘𝑆) ∧ (deg‘𝐹) = 0) → 𝐹 = (ℂ × {((coeff‘𝐹)‘0)}))
31	30	fveq1d 6105	. . . . . . 7 ⊢ ((𝐹 ∈ (Poly‘𝑆) ∧ (deg‘𝐹) = 0) → (𝐹‘0) = ((ℂ × {((coeff‘𝐹)‘0)})‘0))
32		0cn 9911	. . . . . . . 8 ⊢ 0 ∈ ℂ
33		fvex 6113	. . . . . . . . 9 ⊢ ((coeff‘𝐹)‘0) ∈ V
34	33	fvconst2 6374	. . . . . . . 8 ⊢ (0 ∈ ℂ → ((ℂ × {((coeff‘𝐹)‘0)})‘0) = ((coeff‘𝐹)‘0))
35	32, 34	ax-mp 5	. . . . . . 7 ⊢ ((ℂ × {((coeff‘𝐹)‘0)})‘0) = ((coeff‘𝐹)‘0)
36	31, 35	syl6eq 2660	. . . . . 6 ⊢ ((𝐹 ∈ (Poly‘𝑆) ∧ (deg‘𝐹) = 0) → (𝐹‘0) = ((coeff‘𝐹)‘0))
37	36	sneqd 4137	. . . . 5 ⊢ ((𝐹 ∈ (Poly‘𝑆) ∧ (deg‘𝐹) = 0) → {(𝐹‘0)} = {((coeff‘𝐹)‘0)})
38	37	xpeq2d 5063	. . . 4 ⊢ ((𝐹 ∈ (Poly‘𝑆) ∧ (deg‘𝐹) = 0) → (ℂ × {(𝐹‘0)}) = (ℂ × {((coeff‘𝐹)‘0)}))
39	30, 38	eqtr4d 2647	. . 3 ⊢ ((𝐹 ∈ (Poly‘𝑆) ∧ (deg‘𝐹) = 0) → 𝐹 = (ℂ × {(𝐹‘0)}))
40	39	ex 449	. 2 ⊢ (𝐹 ∈ (Poly‘𝑆) → ((deg‘𝐹) = 0 → 𝐹 = (ℂ × {(𝐹‘0)})))
41		plyf 23758	. . . . 5 ⊢ (𝐹 ∈ (Poly‘𝑆) → 𝐹:ℂ⟶ℂ)
42		ffvelrn 6265	. . . . 5 ⊢ ((𝐹:ℂ⟶ℂ ∧ 0 ∈ ℂ) → (𝐹‘0) ∈ ℂ)
43	41, 32, 42	sylancl 693	. . . 4 ⊢ (𝐹 ∈ (Poly‘𝑆) → (𝐹‘0) ∈ ℂ)
44		0dgr 23805	. . . 4 ⊢ ((𝐹‘0) ∈ ℂ → (deg‘(ℂ × {(𝐹‘0)})) = 0)
45	43, 44	syl 17	. . 3 ⊢ (𝐹 ∈ (Poly‘𝑆) → (deg‘(ℂ × {(𝐹‘0)})) = 0)
46		fveq2 6103	. . . 4 ⊢ (𝐹 = (ℂ × {(𝐹‘0)}) → (deg‘𝐹) = (deg‘(ℂ × {(𝐹‘0)})))
47	46	eqeq1d 2612	. . 3 ⊢ (𝐹 = (ℂ × {(𝐹‘0)}) → ((deg‘𝐹) = 0 ↔ (deg‘(ℂ × {(𝐹‘0)})) = 0))
48	45, 47	syl5ibrcom 236	. 2 ⊢ (𝐹 ∈ (Poly‘𝑆) → (𝐹 = (ℂ × {(𝐹‘0)}) → (deg‘𝐹) = 0))
49	40, 48	impbid 201	1 ⊢ (𝐹 ∈ (Poly‘𝑆) → ((deg‘𝐹) = 0 ↔ 𝐹 = (ℂ × {(𝐹‘0)})))

Syntax hints:   → wi 4   ↔ wb 195   ∧ wa 383   = wceq 1475   ∈ wcel 1977  {csn 4125   ↦ cmpt 4643   × cxp 5036  ⟶wf 5800  ‘cfv 5804  (class class class)co 6549  ℂcc 9813  0cc0 9815  1c1 9816   · cmul 9820  ℕ₀cn0 11169  ℤcz 11254  ...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:  dgrnznn  23807  dgreq0  23825  dgrcolem2  23834  dgrco  23835  plyrem  23864  fta1  23867  aaliou2  23899