Theorem coe11 24009

Description: The coefficient function is one-to-one, so if the coefficients are equal then the functions are equal and vice-versa. (Contributed by Mario Carneiro, 24-Jul-2014.) (Revised by Mario Carneiro, 23-Aug-2014.)

Hypotheses

Ref	Expression
coefv0.1	⊢ 𝐴 = (coeff‘𝐹)
coeadd.2	⊢ 𝐵 = (coeff‘𝐺)

Assertion

Ref	Expression
coe11	⊢ ((𝐹 ∈ (Poly‘𝑆) ∧ 𝐺 ∈ (Poly‘𝑆)) → (𝐹 = 𝐺 ↔ 𝐴 = 𝐵))

Proof of Theorem coe11

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

Step	Hyp	Ref	Expression
1		fveq2 6191	. . 3 ⊢ (𝐹 = 𝐺 → (coeff‘𝐹) = (coeff‘𝐺))
2		coefv0.1	. . 3 ⊢ 𝐴 = (coeff‘𝐹)
3		coeadd.2	. . 3 ⊢ 𝐵 = (coeff‘𝐺)
4	1, 2, 3	3eqtr4g 2681	. 2 ⊢ (𝐹 = 𝐺 → 𝐴 = 𝐵)
5		simp3 1063	. . . . . . . . . . 11 ⊢ ((𝐹 ∈ (Poly‘𝑆) ∧ 𝐺 ∈ (Poly‘𝑆) ∧ 𝐴 = 𝐵) → 𝐴 = 𝐵)
6	5	cnveqd 5298	. . . . . . . . . 10 ⊢ ((𝐹 ∈ (Poly‘𝑆) ∧ 𝐺 ∈ (Poly‘𝑆) ∧ 𝐴 = 𝐵) → ◡𝐴 = ◡𝐵)
7	6	imaeq1d 5465	. . . . . . . . 9 ⊢ ((𝐹 ∈ (Poly‘𝑆) ∧ 𝐺 ∈ (Poly‘𝑆) ∧ 𝐴 = 𝐵) → (◡𝐴 “ (ℂ ∖ {0})) = (◡𝐵 “ (ℂ ∖ {0})))
8	7	supeq1d 8352	. . . . . . . 8 ⊢ ((𝐹 ∈ (Poly‘𝑆) ∧ 𝐺 ∈ (Poly‘𝑆) ∧ 𝐴 = 𝐵) → sup((◡𝐴 “ (ℂ ∖ {0})), ℕ0, < ) = sup((◡𝐵 “ (ℂ ∖ {0})), ℕ0, < ))
9	2	dgrval 23984	. . . . . . . . 9 ⊢ (𝐹 ∈ (Poly‘𝑆) → (deg‘𝐹) = sup((◡𝐴 “ (ℂ ∖ {0})), ℕ0, < ))
10	9	3ad2ant1 1082	. . . . . . . 8 ⊢ ((𝐹 ∈ (Poly‘𝑆) ∧ 𝐺 ∈ (Poly‘𝑆) ∧ 𝐴 = 𝐵) → (deg‘𝐹) = sup((◡𝐴 “ (ℂ ∖ {0})), ℕ0, < ))
11	3	dgrval 23984	. . . . . . . . 9 ⊢ (𝐺 ∈ (Poly‘𝑆) → (deg‘𝐺) = sup((◡𝐵 “ (ℂ ∖ {0})), ℕ0, < ))
12	11	3ad2ant2 1083	. . . . . . . 8 ⊢ ((𝐹 ∈ (Poly‘𝑆) ∧ 𝐺 ∈ (Poly‘𝑆) ∧ 𝐴 = 𝐵) → (deg‘𝐺) = sup((◡𝐵 “ (ℂ ∖ {0})), ℕ0, < ))
13	8, 10, 12	3eqtr4d 2666	. . . . . . 7 ⊢ ((𝐹 ∈ (Poly‘𝑆) ∧ 𝐺 ∈ (Poly‘𝑆) ∧ 𝐴 = 𝐵) → (deg‘𝐹) = (deg‘𝐺))
14	13	oveq2d 6666	. . . . . 6 ⊢ ((𝐹 ∈ (Poly‘𝑆) ∧ 𝐺 ∈ (Poly‘𝑆) ∧ 𝐴 = 𝐵) → (0...(deg‘𝐹)) = (0...(deg‘𝐺)))
15		simpl3 1066	. . . . . . . 8 ⊢ (((𝐹 ∈ (Poly‘𝑆) ∧ 𝐺 ∈ (Poly‘𝑆) ∧ 𝐴 = 𝐵) ∧ 𝑘 ∈ (0...(deg‘𝐹))) → 𝐴 = 𝐵)
16	15	fveq1d 6193	. . . . . . 7 ⊢ (((𝐹 ∈ (Poly‘𝑆) ∧ 𝐺 ∈ (Poly‘𝑆) ∧ 𝐴 = 𝐵) ∧ 𝑘 ∈ (0...(deg‘𝐹))) → (𝐴‘𝑘) = (𝐵‘𝑘))
17	16	oveq1d 6665	. . . . . 6 ⊢ (((𝐹 ∈ (Poly‘𝑆) ∧ 𝐺 ∈ (Poly‘𝑆) ∧ 𝐴 = 𝐵) ∧ 𝑘 ∈ (0...(deg‘𝐹))) → ((𝐴‘𝑘) · (𝑧↑𝑘)) = ((𝐵‘𝑘) · (𝑧↑𝑘)))
18	14, 17	sumeq12dv 14437	. . . . 5 ⊢ ((𝐹 ∈ (Poly‘𝑆) ∧ 𝐺 ∈ (Poly‘𝑆) ∧ 𝐴 = 𝐵) → Σ𝑘 ∈ (0...(deg‘𝐹))((𝐴‘𝑘) · (𝑧↑𝑘)) = Σ𝑘 ∈ (0...(deg‘𝐺))((𝐵‘𝑘) · (𝑧↑𝑘)))
19	18	mpteq2dv 4745	. . . 4 ⊢ ((𝐹 ∈ (Poly‘𝑆) ∧ 𝐺 ∈ (Poly‘𝑆) ∧ 𝐴 = 𝐵) → (𝑧 ∈ ℂ ↦ Σ𝑘 ∈ (0...(deg‘𝐹))((𝐴‘𝑘) · (𝑧↑𝑘))) = (𝑧 ∈ ℂ ↦ Σ𝑘 ∈ (0...(deg‘𝐺))((𝐵‘𝑘) · (𝑧↑𝑘))))
20		eqid 2622	. . . . . 6 ⊢ (deg‘𝐹) = (deg‘𝐹)
21	2, 20	coeid 23994	. . . . 5 ⊢ (𝐹 ∈ (Poly‘𝑆) → 𝐹 = (𝑧 ∈ ℂ ↦ Σ𝑘 ∈ (0...(deg‘𝐹))((𝐴‘𝑘) · (𝑧↑𝑘))))
22	21	3ad2ant1 1082	. . . 4 ⊢ ((𝐹 ∈ (Poly‘𝑆) ∧ 𝐺 ∈ (Poly‘𝑆) ∧ 𝐴 = 𝐵) → 𝐹 = (𝑧 ∈ ℂ ↦ Σ𝑘 ∈ (0...(deg‘𝐹))((𝐴‘𝑘) · (𝑧↑𝑘))))
23		eqid 2622	. . . . . 6 ⊢ (deg‘𝐺) = (deg‘𝐺)
24	3, 23	coeid 23994	. . . . 5 ⊢ (𝐺 ∈ (Poly‘𝑆) → 𝐺 = (𝑧 ∈ ℂ ↦ Σ𝑘 ∈ (0...(deg‘𝐺))((𝐵‘𝑘) · (𝑧↑𝑘))))
25	24	3ad2ant2 1083	. . . 4 ⊢ ((𝐹 ∈ (Poly‘𝑆) ∧ 𝐺 ∈ (Poly‘𝑆) ∧ 𝐴 = 𝐵) → 𝐺 = (𝑧 ∈ ℂ ↦ Σ𝑘 ∈ (0...(deg‘𝐺))((𝐵‘𝑘) · (𝑧↑𝑘))))
26	19, 22, 25	3eqtr4d 2666	. . 3 ⊢ ((𝐹 ∈ (Poly‘𝑆) ∧ 𝐺 ∈ (Poly‘𝑆) ∧ 𝐴 = 𝐵) → 𝐹 = 𝐺)
27	26	3expia 1267	. 2 ⊢ ((𝐹 ∈ (Poly‘𝑆) ∧ 𝐺 ∈ (Poly‘𝑆)) → (𝐴 = 𝐵 → 𝐹 = 𝐺))
28	4, 27	impbid2 216	1 ⊢ ((𝐹 ∈ (Poly‘𝑆) ∧ 𝐺 ∈ (Poly‘𝑆)) → (𝐹 = 𝐺 ↔ 𝐴 = 𝐵))

Syntax hints:   → wi 4   ↔ wb 196   ∧ wa 384   ∧ w3a 1037   = wceq 1483   ∈ wcel 1990   ∖ cdif 3571  {csn 4177   ↦ cmpt 4729  ^◡ccnv 5113   “ cima 5117  ‘cfv 5888  (class class class)co 6650  supcsup 8346  ℂcc 9934  0cc0 9936   · cmul 9941   < clt 10074  ℕ₀cn0 11292  ...cfz 12326  ↑cexp 12860  Σcsu 14416  Polycply 23940  coeffccoe 23942  degcdgr 23943

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-8 1992  ax-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246  ax-ext 2602  ax-rep 4771  ax-sep 4781  ax-nul 4789  ax-pow 4843  ax-pr 4906  ax-un 6949  ax-inf2 8538  ax-cnex 9992  ax-resscn 9993  ax-1cn 9994  ax-icn 9995  ax-addcl 9996  ax-addrcl 9997  ax-mulcl 9998  ax-mulrcl 9999  ax-mulcom 10000  ax-addass 10001  ax-mulass 10002  ax-distr 10003  ax-i2m1 10004  ax-1ne0 10005  ax-1rid 10006  ax-rnegex 10007  ax-rrecex 10008  ax-cnre 10009  ax-pre-lttri 10010  ax-pre-lttrn 10011  ax-pre-ltadd 10012  ax-pre-mulgt0 10013  ax-pre-sup 10014  ax-addf 10015

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3or 1038  df-3an 1039  df-tru 1486  df-fal 1489  df-ex 1705  df-nf 1710  df-sb 1881  df-eu 2474  df-mo 2475  df-clab 2609  df-cleq 2615  df-clel 2618  df-nfc 2753  df-ne 2795  df-nel 2898  df-ral 2917  df-rex 2918  df-reu 2919  df-rmo 2920  df-rab 2921  df-v 3202  df-sbc 3436  df-csb 3534  df-dif 3577  df-un 3579  df-in 3581  df-ss 3588  df-pss 3590  df-nul 3916  df-if 4087  df-pw 4160  df-sn 4178  df-pr 4180  df-tp 4182  df-op 4184  df-uni 4437  df-int 4476  df-iun 4522  df-br 4654  df-opab 4713  df-mpt 4730  df-tr 4753  df-id 5024  df-eprel 5029  df-po 5035  df-so 5036  df-fr 5073  df-se 5074  df-we 5075  df-xp 5120  df-rel 5121  df-cnv 5122  df-co 5123  df-dm 5124  df-rn 5125  df-res 5126  df-ima 5127  df-pred 5680  df-ord 5726  df-on 5727  df-lim 5728  df-suc 5729  df-iota 5851  df-fun 5890  df-fn 5891  df-f 5892  df-f1 5893  df-fo 5894  df-f1o 5895  df-fv 5896  df-isom 5897  df-riota 6611  df-ov 6653  df-oprab 6654  df-mpt2 6655  df-of 6897  df-om 7066  df-1st 7168  df-2nd 7169  df-wrecs 7407  df-recs 7468  df-rdg 7506  df-1o 7560  df-oadd 7564  df-er 7742  df-map 7859  df-pm 7860  df-en 7956  df-dom 7957  df-sdom 7958  df-fin 7959  df-sup 8348  df-inf 8349  df-oi 8415  df-card 8765  df-pnf 10076  df-mnf 10077  df-xr 10078  df-ltxr 10079  df-le 10080  df-sub 10268  df-neg 10269  df-div 10685  df-nn 11021  df-2 11079  df-3 11080  df-n0 11293  df-z 11378  df-uz 11688  df-rp 11833  df-fz 12327  df-fzo 12466  df-fl 12593  df-seq 12802  df-exp 12861  df-hash 13118  df-cj 13839  df-re 13840  df-im 13841  df-sqrt 13975  df-abs 13976  df-clim 14219  df-rlim 14220  df-sum 14417  df-0p 23437  df-ply 23944  df-coe 23946  df-dgr 23947

This theorem is referenced by: (None)